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=22.5cm -1.5cm -0.3cm i BONN-TH-99-17 [Target-space Duality\ in Heterotic and Type I Effective Lagrangians\ ]{} 0.5cm **Zygmunt Lalak${}^{1,2}$ Stéphane Lavignac${}^{1,3}$ and Hans Peter Nilles${}^{1}$** *${}^{1}$Physikalisches Institut, Universität Bonn* *Nussallee 12, D-53115 Bonn, Germany* 0.3cm *${}^{2}$Institute of Theoretical Physics* *Warsaw University, Poland* 0.3cm *${}^{3}$Service de Physique Théorique, CEA-Saclay* *F-91191 Gif-sur-Yvette Cédex, France* **Abstract** .3cm We study the implications of target-space duality symmetries for low-energy effective actions of various four-dimensional string theories. In the heterotic case such symmetries can be incorporated in simple orbifold examples. At present a similar statement cannot be made about the simplest type IIB orientifolds due to an obstruction at the level of gravitational anomalies. This fact confirms previous doubts concerning a conjectured heterotic-type IIB orientifold duality and shows that target-space symmetries can be a powerful tool in studying relations between various string theories at the level of the effective low-energy action. Contraints on effective Lagrangians from these symmetries are discussed in detail. In particular, we consider ways of extending $T$-duality to include additional corrections to the Kähler potential in heterotic string models with $N=2$ subsectors. Introduction ============ The idea of the string-driven unification of fundamental interactions has been given a new perspective with the discovery of the web of links, usually referred to as dualities, possibly connecting different string theories. Evidence in favour of the existence of these links has been accumulated through the comparison of various compactifications of different ten-dimensional models and of eleven-dimensional supergravity. However, for the application of these new developments to phenomenologically relevant models in four dimensions, progress has been made in two specific areas. First, the low-energy limit of the strongly coupled heterotic $E_8 \times E_8$ superstring and properties of its four-dimensional effective Lagrangian have been worked out in some detail [@witten]–[@ls_anom]. Second, the novel class of four dimensional chiral type I models has been constructed through the compactification of the type IIB string theory on six-dimensional orientifolds [@orientifolds]–[@lln]. The type I–heterotic duality in ten dimensions of Polchinski and Witten [@Polchinski], which exchanges the strongly coupled sector of one theory with the weakly coupled sector of the other, suggests then the existence of links between heterotic and type I models in lower dimensions. In addition, it has been realized that the volume of the compact six-dimensional space appears as a new parameter relevant for duality symmetries. This allows the possibility that heterotic–type I duality can link weakly coupled models on one side to weakly coupled models on the other side in four dimensions. This has even lead to the concept of a generalized duality between heterotic orbifolds and type IIB orientifolds [@Sagnotti_Z_3], relating models with different numbers of antisymmetric tensor fields. And indeed, candidate dual models in the compactifications of the heterotic $SO(32)$ string have been found. However, since the underlying string theories look quite different, a detailed comparison between type IIB orientifolds and heterotic models is needed either to establish the conjectured duality firmly, or to diagnose points where it breaks down. One obvious test of the 4d heterotic-type IIB orientifold duality is related to the presence of anomalous local $U(1)$ symmetries. In orientifold models several independent anomalous $U(1)$ factors may be present, whereas in the explicitly known models on the heterotic side there is always only a single anomalous $U(1)$. Attempts to understand the physics of the presumably dual models with anomalous abelian factors have shed new light on the above duality conjecture, and in fact have lead us to identify certain doubts on the validity of 4d heterotic-type IIB duality [@lln]. A nontrivial test of duality at the level of the 4d effective Lagrangians is associated with isometries of the 4d moduli spaces of prospective dual partners. The experience from the heterotic superstring models tells us that very often such isometries can be extended to symmetries of the full classical effective Lagrangian. In the heterotic string models the best known example of these symmetries are target-space dualities, which are reflections of the underlying symmetry of string theory. As such, target-space dualities are expected to be good quantum symmetries of the effective Lagrangian, which means that currents associated with them should be free of triangle gauge and gravitational anomalies. The requirement of exact quantum T-duality turns out to be a powerful tool in studies of the four-dimensional string models. This symmetry restricts tree-level couplings in the effective action, and in addition allows us to determine the structure of one-loop corrections to that action. This can be shown to be the case of threshold corrections to the gauge couplings in orbifold models. In the present paper we use target-space duality in this spirit, to generalize the form of one-loop corrections to the Kähler potential in the heterotic Lagrangian. To decide whether the 4d heterotic–type IIB orientifold duality holds, one has to understand what exactly happens to target-space duality in the type IIB orientifold models. In addition, if target-space duality is there in the type IIB orientifold models, this might help to reconstruct the form of their effective Lagrangians, which is crucial from the point of view of phenomenological applications. The problem of realization of target-space duality as a quantum symmetry in effective Lagrangians which might describe type IIB orientifold models in four dimensions is discussed in detail in section 3 of the present paper. We show that even in the simplest models with only D9-branes, it is impossible to enforce cancellation of both gauge and gravitational target-space duality anomalies by a Green-Schwarz mechanism. Furthermore, even if one disregards the gravitational duality anomalies, the structure of the recently computed threshold corrections [@ABD] is not compatible with the Green-Schwarz cancellation of gauge duality anomalies and seems to indicate that target-space duality does not hold at the one-loop level in orientifold models. However, there is more to say about the one-loop effective Lagrangian. In section 4 we examine in detail specific heterotic models with respect to T-duality invariance. We explain that further corrections to the effective one-loop Lagrangian or to target-space duality transformations must arise in models with a plane fixed under some of the orbifold twists. This necessity is due to those one-loop corrections to the Kähler potential which come together with the well known holomorphic threshold corrections in models with $N=2$ subsectors. The corrections that we discuss were not taken into account in the earlier analysis of modular anomalies. In the version of the effective Lagrangian, where only holomorphic thresholds are corrected to become covariant with respect to T-duality, the nonholomorphic corrections argued for in section 4 violate T-duality severely. They do it in a way that cannot be easily repaired without introducing additional kinetic mixing between $S$ and $T$ moduli. We propose such a modification, which generalizes the nonholomorphic corrections to the form which is invariant over the full range of the values of the $T$ modulus. The proper inclusion of the nonholomorphic corrections discussed in section 4 may modify somewhat phenomenological implications of the well-known heterotic models. Effective Lagrangians in heterotic models ========================================= Classical symmetries of moduli space ------------------------------------ Dimensional reduction of the ten-dimensional supergravity gives a nontrivial kinetic Lagrangian for the universal dilaton and geometric moduli superfields, which is invariant under the action of various symmetry transformations. Geometry of the moduli space is reflected by the Kähler potential K= -(S + |[S]{}) - \_[i=1]{}\^[3]{} (T\_i + |[T]{}\_i) \[e1\] There are two obvious symmetries[^1] of the kinetic part of the Lagrangian obtained by using this $K$ (for simplicity we put $T_1=T_2=T_3=T$ in the rest of this section): \[e2\] T 0.7cm [and]{} 0.7cm S . The first symmetry, target-space duality, is believed to be, in its discrete form, the target-space version of a symmetry of the underlying string theory. The second is broken down at the level of the perturbative Lagrangian to an axionic shift through the coupling to the gauge fields. This target-space $S$-duality may be restored by nonperturbative effects, but this issue lies beyond the scope of the present paper. To make the target-space T-duality (\[e2\]) the classical symmetry of the supergravity Lagrangian [@flat; @lmn; @flt; @lidl], one needs to transform the superpotential, W, as well K K + 3 ( i c T + d ) + h.c. , W (i c T + d)\^[-3]{} W \[eq:spp\] If one switches off all the matter and nonuniversal moduli superfields, the suitable form of the superpotential is $W (T) = \frac{const}{\eta^6 (T)}$ where $\eta^2$ is the squared Dedekind’s modular form of weight one. Further, it is possible to extend this clasical symmetry of the tree-level Lagrangian to include also matter and nonuniversal moduli fields. These fields transform as tensors, i.e. linearly, and their entries in the Kähler potential are modular invariant on their own (see the Appendix). For instance, the Kähler potential for untwisted matter is $K_A = A \bar{A} / (T + \bar{T})$, for twisted moduli C it turns out to be $K_C = C \bar{C} / (T + \bar{T})^3$, and for twisted matter $A_C$ $K_{A_C} = A_C \bar{A}_C / (T + \bar{T} )^2$ (like in the $Z_3$ orbifold example). Cancellation of one-loop anomalies ---------------------------------- Target-space duality transformations involve sigma-model transformations, and we have to check whether they are anomalous at the one-loop order [@Derendinger_sigma], [@cardoso1], [@cardoso2]. By the term sigma model we understand the supersymmetric sigma model defined through the kinetic terms of the form $g_{i \bar{j}} (T, \bar{T}) \partial_\mu \phi^i \partial^\mu \bar{\phi}^{\bar{j}}$ where $g_{i \bar{j}} = \partial K / \partial \phi^i \partial \bar{\phi}^{\bar{j}}$ with the Kähler potential[^2] $K$. The form of the sigma model metric $g_{i \bar{j}}$ for indices corresponding to various fields was given at the end of the previous subsection. We can see, that upon $T$-duality transformations the form of $g_{i \bar{j}}$ changes, and to compensate for that change one needs to ‘rotate’ the $\phi$’s. These rotations, due to supersymmetry, act also on all fermions present in the model, and transform them in the way of chiral rotations, with charges related to the modular weights of the fields (see the Appendix). This chiral transformation results in an anomaly in the divergence of the current associated with $T$-duality. In addition, under $T$-duality the Kähler potential is not invariant, but suffers a shift of the form $F(T) + \bar{F} (\bar{T})$ where $F$ is holomorphic. This shift is absorbed in the redefinition of the superpotential $W$, which in turn cancels against the transformation (\[eq:spp\]) of $W$. Such Kähler transformation also results in the rotation of chiral fermions, this time with the same charge for all chiral multiplets and just the opposite one for gauginos. This rotation is also anomalous and the anomaly must be taken into account in addition to the sigma-model anomaly. The anomalous diagrams can be visualized as triangle diagrams with two gauge bosons or two gravitons and one composite connection which plays the role of the gauge field of $T$-duality. In fact, the part of the composite connection whose $T$-duality variation leads to nonvanishing variation of the diagrams is $V_\mu \sim \frac{\partial_\mu ( T - \bar{T} )}{ T + \bar{T}}$, since one typically assumes that all other non-inert fields fluctuate around vanishing vacuum values, hence the anomalous graphs are simply these with $\partial_\mu Im (T)$ at one vertex and gauge bosons or gravitons at remaining vertices. The sigma model anomalies can be computed in field theory limit of string theories, and should be cancelled for the sake of the quantum exactness of the target-space dualities. The form of the one-loop terms needed in the effective Lagrangian to cancel anomalies can be worked out in field theory limit, but these terms can also be directly computed in string theory in various orbifold models, and there they have simply the status of one-loop corrections to the effective action, with their coefficients given from string theory. In the rest of this subsection we recapitulate briefly the status of these corrections in relevant classes of orbifold models. Let us start with the the simplest cases when all the orbifold planes are rotated by the orbifold twists (e.g. the $Z_3$ and $Z_7$ orbifolds). Then there are neither holomorphic threshold corrections, nor the associated corrections to the Kähler potential that will be described in section 4, and all T-duality anomalies, gauge and gravitational, are cancelled through the universal four dimensional Green-Schwarz mechanism. More precisely, the 1-loop anomalies get cancelled by the shift of the dilaton superfield S: $S \rightarrow S - \frac{3 \delta_{GS}}{8 \pi^2} \log (i c T + d) $. Then the dilaton Kähler potential is modified to $K = - \log ( S + \bar{S} - \frac{3 \delta_{GS}}{8 \pi^2} \log (T + \bar{T}))$ which is T-duality invariant. The details of that cancellation are given in the Appendix. This four-dimensional Green-Schwarz mechanism [@GS], [@dsw], [@agnt] is sufficient to cancel all anomalies in models as e.g. the $Z_3$ and $Z_7$ orbifolds. In orbifold models with invariant planes, which contain N=2 subsectors, anomalies are no longer universal and one needs additional counterterms to cancel them completely [@Derendinger_sigma]. These counterterms depend on moduli in a holomorphic way, and they are interpreted as holomorphic one-loop corrections to the tree-level gauge kinetic functions. In the most general case, using eq. (\[eq:L\_nl\]), (\[eq:threshold\]) and (\[eq:b’\_i\_a\]) from the Appendix, one can rewrite the one-loop effective lagrangian for the gauge fields as[^3]: $$\begin{aligned} {\cal L}_{GK} & = & \frac{1}{4}\: \sum_a \int \! \mbox{d}^2 \theta\ W^a W^a\ P_C \left\{\, \left[\, S + \bar S\ + \sigma + \bar \sigma\ -\ \frac{1}{8 \pi^2}\ \sum_i \, \delta^i_{GS} \ln (T_i + \bar T_i)\, \right] \right. \nonumber \\ & & \left. -\ \frac{1}{8 \pi^2}\ \sum_i\, (b^{\prime i}_a - \delta^i_{GS})\, \ln \left[\, |\eta ( T_i)|^4 (T_i + \bar T_i)\, \right]\, \right\}\ +\ \mbox{h.c.}\end{aligned}$$ where $\sigma (T)$ is the holomorphic part of the universal one-loop threshold correction invariant under $SL(2,Z)$ $T$-duality transformations and approaching $-\frac{1}{4 \pi} T$ when $T \rightarrow \infty$. The two terms in the bracket are separately modular invariant. From this expression one can extract the evolution of gauge couplings, upon adding the field-independent terms due to loops of massless charged states: $$\begin{aligned} \frac{1}{g^2_a}\: \left|_{\mbox{1-loop}} \right. & = & \mbox{Re} S\ + \mbox{Re} \sigma\ -\ \frac{1}{16 \pi^2}\ \sum_i \, \delta^i_{GS} \ln (T_i + \bar T_i)\ -\ \frac{b_a}{16 \pi^2}\: \ln \frac{\mu^2}{M^2_H} \nonumber \\ & & -\ \frac{1}{16 \pi^2}\ \sum_i\, b^{N=2}_{a, i}\, \ln \left[\, |\eta ( T_i)|^4 (T_i + \bar T_i)\, \right]\end{aligned}$$ where $M_H$ is the (heterotic) string scale. This expression reflects already the knowledge of the explicit string computations of the nonuniversal threshold corrections to gauge couplings, as the coefficients $b^{N=2}_{a, i}$ are one-loop beta functions coming from $N=2$ subsectors associated with invariant planes. The holomorphic part of the universal threshold correction – $\sigma (T)$ – remains to be determined for arbitrary $T$ via string calculations. String results are naturally formulated in the linear multiplet formalism (see the Appendix), since the string degrees of freedom are antisymmetric tensor fields themselves and not the pseudoscalars dual to them. Later in this paper we shall try to find relations between these string degrees of freedom and the chiral multiplets suitable for low-energy supergravity description of the type IIB orientifold models. Here we merely point out that the expression of the one-loop string coupling $1/l$ in terms of the effective supergravity moduli fields $\mbox{Re} S$ and $\mbox{Re} T_i$ (the one-loop chiral-linear duality relation (\[eq:L\_S\_duality\_loop\])) is determined by the cancellation of sigma-model anomalies (see the Appendix). We can now move on to the discussion of the $Z_3$ and $Z_7$ type IIB orientifold compactifications and their relation to heterotic compactifications. Target-space duality in $D=4$, $N=1$ type IIB $Z_N$ orientifolds ================================================================ Let us consider orientifolds [@orientifolds; @gp] of type IIB string theory compactified on a $T^6 / Z_N$ orbifold [@Sagnotti_Z_3]–[@Lykken]. Consistency of these models (absence of RR tadpoles that would spoil the UV finiteness of the theory) requires the introduction of D-branes on which open strings can end. If the orientifold projector is chosen to be the world-sheet parity $\Omega$, one can have either only D9-branes or both D9-branes and D5-branes. In addition, not all twists leading to $N=1$ supersymmetry are allowed by tadpole conditions (for a classification see ref. [@Ibanez_orientifolds]); the consistent $Z_N$ orientifolds that contain only 9-branes are the odd $N$ ($Z_3$ and $Z_7$) orientifolds. It has been noticed [@Ibanez_sigma] that the classical Lagrangian of orientifolds with only D9-branes is invariant under $SL(2,R)_{T_i}$ transformations. Indeed, the effective Lagrangian describing the dynamics of the open string and untwisted closed string sectors of these models, which can be obtained by reduction and truncation from the $D=10$ type I supergravity action, has the same structure as the Lagrangian of the untwisted sector of heterotic orbifolds: $$\begin{aligned} f_9 & = & S\ , \hskip 2cm W\ \sim\ C^9_1\: C^9_2\: C^9_3\ , \\ K & = & -\ \ln (S + \bar S)\ -\, \sum_i\ \ln \left( T_i + \bar T_i + |C^9_i|^2 \right)\ ,\end{aligned}$$ where $f_9$ denotes the gauge kinetic function for gauge fields of the D9-brane sector, and $C^9_i$ is a generic (99) matter field associated with the $i^{\rm th}$ complex plane. The addition of the twisted closed string states (which are gauge singlets) does not spoil this invariance. In models containing $5_i$-branes (5-branes wrapping on the $i^{\rm th}$ compact plane), $SL(2,R)_{T_i}$ is explicitly broken by the gauge kinetic function of the ($5_i 5_i$) gauge bosons, $f_{5_i} = T_i$, but modular invariance with respect to the $j^{\rm th}$ compact plane still holds as soon as no $5_j$-branes are present [@Ibanez_sigma]. One may then ask whether target-space modular invariance is a good quantum symmetry of $D=4$, $N=1$ type IIB orientifolds. Although it is not clear what would be the origin of this symmetry in the underlying string theory[^4], it is expected to hold at the one-loop level if heterotic - type IIB duality is valid, because then sigma-model anomalies are compensated for in the dual heterotic orbifold models (strictly speaking, this argument does not apply to orientifolds containing 5-branes, since these models do not seem to have any perturbative heterotic dual). Thus investigating the possibility of cancelling sigma-model anomalies in orientifolds amounts to testing duality. Also, the question of whether target-space modular invariance is a good quantum symmetry in orientifolds is interesting on its own, even if this string duality does not hold, because it could give us information on the effective Lagrangian of those models, which are not so well known. The purpose of this section is to investigate this question at the level of the effective field theory, using the information given by recent string computations. The authors of ref. [@Ibanez_sigma] have proposed a mechanism for the cancellation of sigma-model anomalies that is reminiscent of the way Abelian gauge anomalies are compensated for in orientifolds [@Ibanez_anomalous]. The gauge group of orientifold models often contains several anomalous Abelian factors $U(1)_i$. Their anomalies are cancelled by a generalized [@Sagnotti_generalized] Green-Schwarz mechanism involving $RR$ twisted antisymmetric tensors $B^k_{\mu \nu}$ with appropriate couplings to the gauge fields. In a more familiar chiral superfield language, those antisymmetric tensors are described by their pseudoscalar duals $a_k$, which lie in the same chiral multiplets $M_k$ as the scalars corresponding to the blowing-up modes of the orientifold, $a_k = \mbox{Im} M_k \! \mid_{\theta = \bar \theta = 0}$. The basic ingredients of the generalized Green-Schwarz mechanism are a coupling of the $M_k$ to the gauge fields, $$f_a\ =\ f_p\ +\ \sum_{k=1}^{[\frac{N-1}{2}]}\, s_{ak}\, M_k\ , \label{eq:f_a_tree}$$ with $f_p=S$ for gauge group coming from 9-branes, and a shift of the twisted axions $a_k$ under a $U(1)_i$ gauge transformation: $$M_k\ \rightarrow\ M_k\, +\, i\, \delta^k_i \Lambda_i\ . \label{eq:M_k_shift}$$ In eq. (\[eq:f\_a\_tree\]), the sum goes over independent twisted sectors, and for a twist $\theta^k$ with no fixed plane, the $M_k$ fields are defined by $M_k = \frac{1}{\sqrt{N_k}} \sum_{f=1}^{N_k} M^f_k$, where $\{ M^f_k \}_{f=1 \ldots N_k}$ are the states from the $k^{\rm th}$ twisted sector, each of them living at one of the $N_k$ points that are fixed under $\theta^k$ (for a twist leaving some plane unrotated, not all twisted states fit into linear multiplets, see ref. [@Klein]). It is interesting to note that the shift (\[eq:M\_k\_shift\]) is a one-loop effect in the low-energy effective field theory, although its string origin is a tree-level Green-Schwarz coupling at the level of the orientifold. From eq. (\[eq:f\_a\_tree\]) and (\[eq:M\_k\_shift\]) one obtains that mixed $U(1)G_aG_a$ anomalies are cancelled if the following conditions are fulfilled: $$C^i_a\ =\ 8\, \pi^2 \sum_k\: c^k_a\, \delta^k_i\ .$$ This Green-Schwarz cancellation of $U(1)$-gauge anomalies has been confirmed by a string computation of the couplings $s_{ak}$ and $\delta^k_i$ [@ABD]. Since the absence of $U(1)$-gravitational anomalies is also required for the model to be consistent, it is natural to assume [@Ibanez_sigma] that they are compensated for by the same mechanism (this proposal has been confirmed in the recent paper [@scr_ser]), i.e. that the twisted axions $a_k$ couple to the $R \widetilde R$ term: $$-\, \frac{1}{4}\: \left( a + t_k\, a_k \right)\, R \widetilde R\ . \label{eq:a_k_R2}$$ $U(1)$-gravitational anomalies must then satisfy the conditions: $$\frac{\mbox{Tr} X_i}{12}\ =\ 8\, \pi^2 \sum_k\: t_k\, \delta^k_i\ . \label{eq:U1_grav}$$ Contrary to the $s_{ak}$, the couplings $t_k$ have not been computed yet; however, they can be extracted from eq. (\[eq:U1\_grav\]), since the $\delta^k_i$ are known from ref. [@ABD]. The authors of ref. [@Ibanez_sigma] have proposed that sigma-model anomalies associated with complex planes that are rotated by all twists are cancelled by a similar Green-Schwarz mechanism. The most general possibility is that both $S$ and the $M_k$ shift under $SL(2,R)_{T_i}$ transformations: $S \rightarrow S - \frac{1}{8 \pi^2}\, \delta^{i,S}_{GS} \ln (i c_i T_i + d_i)$, $M_k \rightarrow M_k - \frac{1}{8 \pi^2}\, \delta^{i,k}_{GS} \ln (i c_i T_i + d_i)$. The sigma-gauge anomalies must then satisfy: $$b^{\prime i}_a\ =\ \delta^{i,S}_{GS}\ +\ \sum_k\, s_{ak}\, \delta^{i,k}_{GS}\ . \label{eq:sigma_gauge_GS}$$ Since there are generically as many Green-Schwarz parameters $\delta^{i,S}_{GS}$ and $\delta^{i,k}_{GS}$ as anomalies, it is always possible to cancel the sigma-gauge anomalies by means of a Green-Schwarz mechanism. Given the couplings $s_{ak}$, which are known from ref. [@ABD], and the anomaly coefficients $b^{\prime i}_a$, which are computed from the massless spectrum, eq. (\[eq:sigma\_gauge\_GS\]) determines uniquely the Green-Schwarz parameters that are needed in order to ensure anomaly cancellation. One finds [@Ibanez_sigma] $\delta^{i,S}_{GS} = 0$ for all $Z_N$ orientifolds of ref. [@Ibanez_orientifolds], i.e. the dilaton does not play any role in the cancellation of sigma-gauge anomalies. Although this last feature, which is reminiscent of the mechanism of Abelian gauge anomaly cancellation, may appear to be suggestive, it should be stressed that it is not possible to check the conjecture of a Green-Schwarz mechanism on the basis of an analysis of the mixed gauge anomalies, since there are as many potential counterterms as anomalies. This situation is to be contrasted with the case of heterotic orbifolds with no fixed planes, in which three parameters $\delta^i_{GS}$, $i=1,2,3$ must cancel all anomalies, thus implying several consistency relations that can be checked in explicit models (namely the anomalies $b^{\prime i}_a$ must be gauge-group independent). Sigma-gravitational anomalies ----------------------------- However, one can test this conjecture by considering sigma-gravitational anomalies. Indeed, any shift of the universal and/or twisted axions under target-space modular transformations induces, through the couplings (\[eq:a\_k\_R2\]), a variation of the Lagrangian $\delta {\cal L} = \frac{\theta_i}{768 \pi^2}\, \bar b^{\prime i}_{grav} R \widetilde R$ that cancels part of the triangle anomaly $b^{\prime i}_{grav}$. Assuming that the sigma-gauge anomalies are compensated for by a Green-Schwarz mechanism, the shifts of the axions $a$ and $a_k$ under $SL(2,R)_{T_i}$ are determined in an unambiguous way by eq. (\[eq:sigma\_gauge\_GS\]), therefore the corresponding coefficient $\bar b^{\prime i}_{grav}$ read, given the fact that $\delta^{i,S}_{GS} = 0$ in all cases: $$\bar b^{\prime i}_{grav}\ =\ 24\, \sum_k\, t_k\, \delta^{i,k}_{GS}$$ If a Green-Schwarz mechanism ensures the validity of target-space duality at the one-loop level, then one should have $b^{\prime i}_{grav} = \bar b^{\prime i}_{grav}$. It is straightforward to check this relation in explicit models. For definiteness we restrict ourselves to the odd order $Z_N$ orientifolds, which contain only 9-branes and do not have any fixed plane. The values of the triangle anomaly $b^{\prime i}_{grav}$ and of the Green-Schwarz contribution $\bar b^{\prime i}_{grav}$ are displayed in Table 1, where $b^{\prime i}_M$ denotes the contribution of the $M_k$ fields to $b^{\prime i}_{closed}$; since they transform non-linearly under $SL(2,R)_{T_i}$, they have zero modular weights and $b^{\prime i}_M$ is just the number of $M_k$ fields ($b^{\prime i}_M = 27$ and $21$ for $Z_3$ and $Z_7$ respectively). Following ref. [@Ibanez_sigma], we also give, for later discussion, the separate contributions of the open and closed string states to the triangle anomaly: $$\begin{aligned} b^{\prime i}_{open} & = & -\ \dim G\ +\ \sum_{\alpha}\, (1 + 2 n^i_{\alpha})\ , \\ b^{\prime i}_{closed} & = & 21\ +\ 1\ +\ b^{\prime i}_{mod}\ ,\end{aligned}$$ where $b^{\prime i}_{mod}$ denotes the contribution of the other modulinos than the dilatino, and by definition $b^{\prime i}_{grav} = b^{\prime i}_{open} + b^{\prime i}_{closed}$. .8cm ---------------------------------------- ------------------------------------------------ ------------------------- ----------------------- ------------------------------------------------------------------- $\hskip .3cm \mbox{model} \hskip .3cm$ $\hskip .3cm b^{\prime i}_{open} \hskip .3cm$ $b^{\prime i}_{closed}$ $b^{\prime i}_{grav}$ $\bar b^{\prime i}_{grav} \equiv 24 \sum_k t_k \delta^{i,k}_{GS}$ $Z_3$ $-10$ $19 + b^{\prime i}_M$ $9 + b^{\prime i}_M$ $-18$ $Z_7$ $-6$ $21 + b^{\prime i}_M$ $15 + b^{\prime i}_M$ $-6$ ---------------------------------------- ------------------------------------------------ ------------------------- ----------------------- ------------------------------------------------------------------- .5cm Table 1 .3cm From Table 1 it is obvious that $\bar b^{\prime i}_{grav} \neq b^{\prime i}_{grav}$, i.e. the Green-Schwarz mechanism alone cannot compensate for both sigma-gauge and sigma-gravitational anomalies. However, there may exist some other effect that would cancel the remaining anomaly $b^{\prime i}_{grav} - \bar b^{\prime i}_{grav}$. This is actually suggested by the string diagramatics relevant for sigma-gravitational anomalies in open string models, as explained in ref. [@Ibanez_sigma]. Indeed, only diagrams with an open string loop (the annulus and Moebius amplitudes) can contribute to sigma-gauge anomalies (as well as $U(1)$ anomalies), but in the case of sigma-gravitational anomalies, there are additional contributions coming from diagrams with a closed string loop (the torus and Klein bottle amplitudes). Such diagrams are not present for sigma-gauge anomalies, because they would be of higher order in the string coupling. The statement that sigma-gravitational anomalies are cancelled is equivalent to the statement that all four diagram topologies should sum up to zero, and that the field theory limit of these string diagrams should contain both the triangle anomaly (which one can split into two separate contributions, $b^{\prime i}_{open}$ and $b^{\prime i}_{closed}$) and the corresponding counterterms. The open string diagrams are expected to provide the field theory anomalous tree graphs that are characteristic of a Green-Schwarz mechanism, since these diagrams, when considered in the closed string (tree) channel, are suggestive of a factorized form with closed string states propagating in the cylinder (from the cancellation of sigma-gauge anomalies, we know that those states must be twisted RR antisymmetric tensors). The interpretation of the closed string diagrams is less obvious. The authors of ref. [@Ibanez_sigma] assume that factorization is possible and that these diagrams generate a one-loop mixing between the dilaton and the $T$ moduli that make up the sigma-model connection. However, this proposal appears to be incompatible with the above field theory analysis, since it would imply a one-loop[^5] shift of the universal axion under $SL(2,R)_{T_i}$ transformations, which has already been excluded on the basis of sigma-gauge anomaly cancellation (the same conclusion would hold for an additional mixing between the twisted $M_k$ fields and the $T$ moduli). This suggests that the possible counterterms originating from closed string loops are not Green-Schwarz terms. The only alternative possibility we can think of would be a $T_i$-dependent correction to the CP-odd $R^2$ terms with the appropriate behaviour under $SL(2,R)_{T_i}$, i.e. $${\cal L}_{CT}\ =\ \frac{1}{32 \pi^2}\ \mbox{Im}\, \Delta (T)\, R \widetilde R\ , \hskip .7cm \Delta (T)\ \stackrel{SL(2,R)_{T_i}}{\longrightarrow}\ \Delta (T)\ +\ \frac{b^{\prime i}_{grav} - \bar b^{\prime i}_{grav}}{24}\ \ln (i c_i T_i + d_i)\ .$$ However, such corrections are not expected in models which do not possess any $N=2$ sectors. Moreover, they cannot appear at the perturbative level because of the Peccei-Quinn symmetries associated to the $T_i$ axions. Still one cannot exclude the possibility that they are generated nonperturbatively. Before concluding this subsection, let us comment more precisely on the concrete proposal of ref. [@Ibanez_sigma]. There it was assumed that the contribution of the open string sector to sigma-gravitational anomalies, $b^{\prime i}_{open}$, was exactly compensated for by the $SL(2,R)_{T_i}$ shift of the $M_k$ moduli, whereas $b^{\prime i}_{closed}$ was taken care of by closed string loop diagrams (interpreted as a mixing between the dilaton and the $T$ moduli, which as explained above seems to be strongly disfavoured by the field theory analysis). If this picture were correct, one would find $b^{\prime i}_{open} = \bar b^{\prime i}_{grav}$ in all explicit orientifold models. However, Table 1 shows that it is not verified in the $Z_3$ case, although it accidentally holds in the $Z_7$ case. We are therefore led to the following conclusion: either modular invariance is not a good quantum symmetry in orientifold models, which casts a new doubt on the validity of heterotic - type IIB duality; or it is a good quantum symmetry and the anomalies cannot be cancelled by a pure Green-Schwarz mechanism, even in models that do not possess any fixed planes. One would then require $T_i$-dependent, holomorphic corrections to the $R^2$ terms, which however cannot arise perturbatively. One-loop corrections to the gauge couplings ------------------------------------------- As stressed above, the consideration of field theory anomalies is not enough to decide whether target-space duality is a good quantum symmetry of type IIB orientifolds, mainly because of the large number of possible counterterms. Although the cancellation of sigma-gravitational anomalies already appears to be problematic, further tests of this symmetry are needed. The one-loop corrections to gauge couplings that have been recently computed in type IIB orientifolds [@ABD] give us the opportunity to perform such a test. Indeed, the string result should agree with the one-loop corrections computed in the effective supergravity theory. While the former depends on the linear multiplets of the theory, the latter is expressed in terms of chiral multiplets only; a duality transformation relates the two formulations. The relations between the linear multiplets and their dual chiral multiplets are defined order by order in perturbation theory. As is well known from the heterotic case[^6], the presence of Green-Schwarz counterterms modifies these relations at the one-loop order (see eq. (\[eq:L\_S\_duality\_loop\]) in Appendix B). In the following, we shall then try to relate the string and supergravity expressions for the one-loop gauge couplings through an explicit linear-chiral duality transformation. The duality relations obtained in this way should tell us whether the couplings needed for the cancellation of sigma-model anomalies are indeed generated in orientifold models. Let us first establish the tree-level duality relations. The linear multiplets we have to deal with are the universal linear multiplet $L \sim (l, B_{\mu \nu}, \chi)$ and a model-dependent number of twisted linear multiplets $L_k \sim (m_k, B_{k \mu \nu}, \chi_k)$, which describe the dilaton and the blowing-up modes, respectively, as well as their antisymmetric tensor partner. The tree-level Lagrangian for $L$ and $L_k$ has to reproduce the tree-level gauge couplings (here we consider only gauge groups coming from the $9$-brane sector) [@ABD] $$\frac{1}{g^2_a}\ =\ \frac{1}{l}\ +\ \sum_k\, s_{ak}\, m_k\ . \label{eq:g_I_9_ABD_tree}$$ Assuming no kinetic mixing between the different linear multiplets (this is indeed the case in the basis in which formulae (\[eq:g\_I\_9\_ABD\_tree\]) is written), one finds: $${\cal L}\ =\ \int \! \mbox{d}^4 \theta\ \Phi\, (\widehat L, \widehat L_k)\ , \hskip 1cm \Phi\, (\widehat L, \widehat L_k)\ =\ \ln \widehat L\ -\ \sum_k\, \widehat L^2_k\ , \label{eq:L_k_lagr}$$ where $\widehat L = L - 2\, \Omega$ and $\widehat L_k = L_k + \sum_a s_{ak} \Omega_a$ are the modified, gauge invariant multiplets. Following the standard procedure (see Appendix B), one then obtains the tree-level duality relations: $$\frac{1}{\widehat L}\ =\ \frac{S + \bar S}{2}\ , \hskip 1cm \widehat L_k\ =\ \frac{M_k + \bar M_k}{2}\ , \label{eq:L_M_k_duality_tree}$$ as well as the Kähler potential and gauge kinetic function that define the Lagrangian describing the dual chiral superfields $S$ and $M_k$: $$f_a\ =\ S\ +\ \sum_k\, s_{ak}\, M_k\ , \hskip 1cm K (S, \bar S, M_k, \bar M_k)\ =\ -\, \ln (S + \bar S)\ +\ \frac{1}{4}\: (M_k + \bar M_k)^2\ . \label{eq:M_k_lagr}$$ One recovers, as expected, the gauge kinetic function (\[eq:f\_a\_tree\]). Note that, since we started from eq. (\[eq:g\_I\_9\_ABD\_tree\]), which is the first order in a perturbative expansion around the orientifold point $m_k=0$, eq. (\[eq:L\_k\_lagr\]), (\[eq:L\_M\_k\_duality\_tree\]) and the Kähler potential in eq. (\[eq:M\_k\_lagr\]) are valid at leading order in $M_k$ only. At the one-loop level, the Lagrangian (\[eq:L\_k\_lagr\]) receives corrections which modify the duality relations (\[eq:L\_M\_k\_duality\_tree\]) and the Lagrangian in the chiral basis (\[eq:M\_k\_lagr\]). The loop-corrected Lagrangian in the linear basis (respectively the chiral basis) should reproduce the one-loop gauge couplings computed in ref. [@ABD] (respectively the one-loop gauge couplings computed in the effective supergravity theory [@KL]). For the sake of simplicity, we restrict our discussion to the case of odd $N$ $Z_N$ orientifolds. These models do not possess any $N=2$ sectors, so that the only way to compensate for sigma-model anomalies is the Green-Schwarz mechanism, as discussed in the previous subsection. The one-loop gauge couplings obtained from the string computation of ref. [@ABD] read: $$\left. \frac{1}{g^2_a\, (\mu^2)}\ \right|_{\mbox{1-loop}}\ =\ \frac{1}{l}\ +\ \sum_k\, s_{ak}\, m_k\ -\ \frac{b_a}{16 \pi^2}\: \ln \frac{\mu^2}{M^2_I}\ . \label{eq:g_I_9_ABD}$$ The generic expression for one-loop gauge couplings in effective supergravity theories read [@KL]: $$\begin{aligned} \frac{1}{g^2_a (\mu^2)}\: \left|_{\mbox{1-loop}} \right. & = & \mbox{Re} f_a \mid_{\mbox{1-loop}}\ -\ \frac{b_a}{16 \pi^2}\: \ln \frac{\mu^2}{M^2_{pl}}\ \nonumber \\ & + & \frac{1}{16 \pi^2}\: \left[\, c_a\, K\ +\ 2\, C_2(G_a)\, \ln g^{-2}_a\ -\ 2\, \sum_\alpha\, T(R_\alpha)\, \ln \det Z_{R_\alpha}\, \right]\ , \label{eq:g_I_KL}\end{aligned}$$ where $Z_{R_\alpha}$ is the effective wave function normalization matrix for the representation $R_\alpha$, and $c_a = \sum_\alpha T(R_\alpha) - C_2(G_a)$. In the right hand side of eq. (\[eq:g\_I\_KL\]), the functions $K$, $g^{-2}_a$ and $Z_{R_\alpha}$ are truncated at tree-level, while $f_a \mid_{\mbox{1-loop}}$ includes the one-loop string corrections. Since formula (\[eq:g\_I\_KL\]) is valid in any supergravity effective theory, independently of its string origin, we can apply it to the orienfold models we are considering. Then $f_a$ is given by its tree-level expression (\[eq:M\_k\_lagr\]) (the string threshold corrections vanish in the absence of $N=2$ sectors), $K = K (S, \bar S; M_k, \bar M_k) - \sum_i \ln (T_i + \bar T_i)$ where $K (S, \bar S; M_k, \bar M_k)$ is given by (\[eq:M\_k\_lagr\]), and $Z_{R_\alpha} = \Pi_i (T_i + \bar T_i)^{n^i_{R_\alpha}}$. Eq. (\[eq:g\_I\_KL\]) then becomes: $$\frac{1}{g^2_a (\mu^2)}\ =\ \mbox{Re} S\ +\ \sum_k\, s_{ak}\, \mbox{Re} M_k\ -\ \sum_i \frac{b^{\prime i}_a}{16 \pi^2}\: \ln (T_i + \bar T_i)\ -\ \frac{b_a}{16 \pi^2}\: \ln \frac{\mbox{Re} S\, \mu^2}{M^2_{pl}}\ , \label{eq:g_I_KL_2}$$ up to terms that are suppressed both by a one-loop factor and a dependence on $\mbox{Re} M_k$, which we can neglect. Note that the $T$-dependent non-harmonic corrections (third piece in the right hand side of eq. (\[eq:g\_I\_KL\_2\])) are related, through supersymmetry, to the anomalous triangle diagrams associated with the sigma-model connection. Assuming that it is possible to perform a linear-chiral duality transformation at the one-loop level in a consistent way, one can obtain the one-loop linear-chiral duality relations for $\widehat L$ and $\widehat L_k$, as was done in ref. [@ABD], by comparing the string expression (\[eq:g\_I\_9\_ABD\]) with the effective supergravity expression (\[eq:g\_I\_KL\_2\]). This gives: $$\begin{aligned} \frac{1}{l} & = & \mbox{Re} S\ , \label{eq:l_S_duality}\\ m_k & = & \mbox{Re} M_k\ -\ \frac{1}{16 \pi^2}\ \sum_i\, \delta^{i,k}_{GS} \ln (T_i + \bar T_i)\ -\ \frac{b_k}{16 \pi^2}\: \ln \left( \frac{\mbox{Re} S}{\Pi_i\, \mbox{Re} T_i} \right)^{\! 1/2}\ , \label{eq:m_M_duality_1}\end{aligned}$$ where the coefficients $\delta^{i,k}_{GS}$ are defined by eq. (\[eq:sigma\_gauge\_GS\]) (with $\delta^{i,S}_{GS} = 0$ in all models), and the coefficients $b_k$ are defined by $b^{N=1}_a\ =\ \sum_k\, s_{ak}\, b_k$. The second piece in the right hand side of the duality relation (\[eq:m\_M\_duality\_1\]) then comes precisely with the coefficients needed for the Green-Schwarz cancellation of the sigma-gauge anomalies; however, the third piece transforms under $SL(2,R)_{T_i}$ in the same way as Green-Schwarz counterterms and seems to spoil anomaly cancellation. The presence of this troublesome term can be traced back to the difficulty of relating the string expression (\[eq:g\_I\_9\_ABD\]) to the supergravity expression (\[eq:g\_I\_KL\_2\]) through a linear-chiral duality transformation, as we discuss below. The expression for the one-loop gauge couplings in the linear basis, eq. (\[eq:g\_I\_9\_ABD\]), does not contain any term proportional to $\ln (T_i + \bar T_i)$. This means that the one-loop Lagrangian for the linear multiplets must contains a term[^7] $$\Delta {\cal L}_{GS}\ =\ -\, \frac{1}{8 \pi^2}\: \sum_{i,k}\, \delta^{i,k}_{GS}\, \widehat L_k \ln (T_i + \bar T_i) \label{eq:GS_counterterms}$$ whose contribution to the gauge couplings exactly cancels the non-harmonic corrections coming from the supersymmetric partners of the anomaly diagrams, $- \sum_i \frac{b^{\prime i}_a}{16 \pi^2}\, \ln (T_i + \bar T_i)$. Note that this Green-Schwarz mechanism is not enough to ensure $SL(2,R)_{T_i}$ invariance of the gauge couplings, since the upper scale of logarithmic running, the string scale $M_I$, is not invariant on its own. Indeed, when expressed in units of the Planck mass, $M_I$ depends on the $T_i$ moduli: $M^2_I = \frac{\lambda_I M^2_{Pl}}{2\, {\rm Re} S} = (\frac{{\rm Re} S}{\Pi_i {\rm Re} T_i})^{1/2} \frac{M^2_{Pl}}{2\, {\rm Re} S}$ (we have used the identities $\mbox{Re} S = V_I M^6_I / \lambda_I$ and $\mbox{Re} T_i = V^i_I M^2_I / \lambda_I$ [@Ibanez_orientifolds], where $\lambda_I$ is the ten-dimensional string coupling and $V^i_I$ is the volume of the $i^{\rm th}$ compact torus in the string metric). Now the addition of the Green-Schwarz counterterms (\[eq:GS\_counterterms\]) to the Lagrangian (\[eq:L\_k\_lagr\]) leads to the modified duality relations $$\begin{aligned} \frac{1}{\widehat L} & = & \frac{S + \bar S}{2}\ , \\ \widehat L_k & = & \frac{1}{2}\ \left[\, M_k + \bar M_k\ -\ \frac{1}{8 \pi^2}\ \sum_i\, \delta^i_{GS} \ln (T_i + \bar T_i)\, \right]\ , \label{eq:m_M_duality_2}\end{aligned}$$ which do not agree with eq. (\[eq:m\_M\_duality\_1\]) and therefore yield an expression for the gauge couplings in the chiral basis that does not fit the supergravity expression (\[eq:g\_I\_KL\_2\]). One can try to solve this problem by adding to the Lagrangian of the linear multiplets, beyond the terms (\[eq:GS\_counterterms\]), the piece that reproduces the fitted duality relation (\[eq:m\_M\_duality\_1\]), i.e. $$\Delta {\cal L}_{2}\ =\ \frac{1}{16 \pi^2}\: \sum_{k}\, b_k \widehat L_k \ln \left[ \widehat L\, \prod_i (T_i + \bar T_i) \right] \label{eq:fitted_counterterms}$$ ($\Delta {\cal L}_{2}$ also modifies the duality relation between $\widehat L$ and $S$, but the corrections can be neglected, because it is proportional both to a one-loop factor and to $m_k$). This reproduces the supergravity expression (\[eq:g\_I\_KL\_2\]), but looks somewhat ad hoc; in particular, this amounts to shift the upper scale of running to the invariant scale $M^2_{Pl} / \mbox{Re} S$ and to absorb the residual, non-invariant logarithmic term into the effective one-loop Lagrangian. Note that the non-invariant part of $\Delta {\cal L}_{2}$ has the same form as the Green-Schwarz counterterms $\Delta {\cal L}_{GS}$; as a result, this new term modifies the shift of the $M_k$ fields under $SL(2,R)_{T_i}$ and only a part of the sigma-model anomalies is cancelled. Indeed, the modified Kähler potential obtained from the duality transformation is a function of the combinations of chiral superfields corresponding to the linear multiplets $\widehat L_k$; then requiring invariance of $K$ amounts to modify the shift of the $M_k$ to: $$M_k\ \rightarrow\ M_k\ -\ \frac{1}{8 \pi^2}\: \left( \delta^{i,k}_{GS} - \frac{b_k}{2} \right)\, \ln (i c_i T_i + d_i)\ .$$ The uncancelled anomaly is $b^{\prime i}_a - \sum_k s_{ak} (\delta^{i,k}_{GS} - \frac{b_k}{2}) = b_a / 2$. Alternatively, one may absorb the terms (\[eq:fitted\_counterterms\]) into a redefinition of the twisted linear multiplets at the one-loop level, $\widetilde L_k = L_k - \frac{b_k}{32 \pi^2}\, \ln\, [\widehat L\, \Pi_i (T_i + \bar T_i)]$. The one-loop duality relations are then given by eq. (\[eq:m\_M\_duality\_2\]), and the Green-Schwarz couplings ensure exact cancellation of sigma-gauge anomalies. However, target-space duality is violated at the level of one-loop corrections to the Kähler potential, due to the fact that the constraints $D^2 L_k = D^2 (\widetilde L_k - \frac{b_k}{32 \pi^2}\, \ln\, [\widehat L\, \Pi_i (T_i + \bar T_i)]) = 0$ and $\bar D^2 L_k = 0$ are no longer invariant. We conclude that the structure of the one-loop corrections to gauge couplings in type IIB orientifolds does not seem to be compatible with the cancellation of sigma-gauge anomalies by a pure Green-Schwarz mechanism, even in the simplest models with no $N=2$ subsectors, contrary to what the mere analysis of anomalies would suggest. This, together with the difficulty observed at the level of mixed gravitational anomalies, suggests that target-space duality is merely an accidental, tree-level symmetry of the effective supergravity Lagrangian of orientifold models, and strengthen previous doubts [@lln] on the validity of the heterotic - type IIB duality. Note that it is not completely clear how the agreement between the string and effective supergravity expressions for the one-loop gauge couplings is obtained; in this respect, an explicit string computation of corrections to the twisted moduli Kähler potential would give very useful information. Threshold corrections and heterotic - type IIB duality ------------------------------------------------------ We have seen that target-space duality can be used as a tool to test heterotic - type IIB duality at the one-loop level. For completeness, we would like to mention another nontrivial check based on the comparison of the threshold corrections in candidate dual orbifold/orientifold models, which also raises doubts on the validity of this duality (the following discussion is taken from ref. [@ABD]). Specifically, we shall consider the $Z_3$ models, in which a perfect matching of the massless spectra and gauge groups of both low-energy effective field theories is obtained after decoupling of the anomalous $U(1)$. The one-loop gauge couplings of the $Z_3$ orbifold/orientifold computed in string theory read [@ABD]: $$\begin{aligned} \frac{1}{g^2_a (\mu^2) \mid_H} & = & \frac{1}{l_H}\ -\ \frac{b^H_a}{16 \pi^2}\: \ln \frac{\mu^2}{M^2_H}\ , \\ \frac{1}{g^2_a (\mu^2) \mid_I} & = & \frac{1}{l_I}\ +\ s_a\, m\ -\ \frac{b^I_a}{16 \pi^2}\: \ln \frac{\mu^2}{M^2_I}\ , \label{eq:g_Z_ABD}\end{aligned}$$ where $M_H$ (respectively $M_I$) is the heterotic (respectively type I) string scale, and $m$ is the symmetric combination of the 27 twisted scalars. At tree level[^8], the $D=10$ heterotic - type I duality implies $l_H = l_I$, but this relation does no longer hold at the one-loop level. The correct duality relation is found by expressing the linear multiplets in terms of the chiral fields $S$ and $T_i$, and using the duality dictionary derived by dimensional reduction of the $D=10$ duality relations [@Polchinski] $\lambda_H = \lambda^{-1}_I$ and $M^2_H = \lambda^{-1}_I M^2_I$ (where $\lambda_H$, respectively $\lambda_I$, is the $D=10$ heterotic, respectively type I dilaton). One finds $V_H = V_I$, where $V_{H(I)} = \int d x^6 \scriptstyle{\sqrt{g^{(6)}_{H(I)}}}$ is the compact volume in the string metric, as well as[^9] $(\mbox{Re} S)_H = (\mbox{Re} S)_I$ and $(\mbox{Re} T_i)_H = (\mbox{Re} T_i)_I$. On the heterotic side, the expression of $l_H$ in terms of $S$ and $T_i$ is given by the linear-chiral relation (\[eq:L\_S\_duality\_loop\]), which in the particular case of the orbifold considered can be rewritten as $\frac{1}{l} = \mbox{Re} S - \frac{b^H_{SU(12)}}{16 \pi^2}\, \ln (V^{1/3}_H M^2_H)$, thus yielding: $$\frac{1}{g^2_a (\mu^2) \mid_H}\ =\ \mbox{Re} S\ -\ \frac{b^H_{SU(12)}}{16 \pi^2}\: \ln\, (V^{1/3}_H M^2_H)\ -\ \frac{b^H_a}{16 \pi^2}\: \ln \frac{\mu^2}{M^2_H}\ . \label{eq:g_H_Z_ABD_2}$$ This is however not yet the relevant expression for the low-energy gauge couplings, since it holds in the trivial vacuum with a nonzero anomalous $U(1)$ $D$-term, $D_X = \xi^2_H$. Considering the physical flat direction with maximal gauge symmetry $G = SU(12) \times SO(8)$, along which $\xi^2_H$ is compensated for by vevs of the twisted moduli $M_k$, one finally finds: $$\begin{aligned} \frac{1}{g^2_a (\mu^2) \mid_H}\ =\ \mbox{Re} S\ -\ \frac{b^I_a}{16 \pi^2}\: \ln\, (V^{1/3}_H \mu^2)\ . \label{eq:g_H_Z_ABD_3}\end{aligned}$$ The change from eq. (\[eq:g\_H\_Z\_ABD\_2\]) to eq. (\[eq:g\_H\_Z\_ABD\_3\]) is due to the fact that along the flat direction considered, some twisted charged states become massive and decouple, yielding $b^H_{SU(12)} \rightarrow b^I_a$ and $b^H_a \rightarrow b^I_a$. This results in a shift of the unification scale from the string scale to the compactification scale $M_c = V^{1/3}_H$. On the orientifold side, using the dilaton linear-chiral duality relation (\[eq:l\_S\_duality\]), we obtain: $$\frac{1}{g^2_a (\mu^2) \mid_I}\ =\ \mbox{Re} S\ +\ s_a\, m\ -\ \frac{b^I_a}{16 \pi^2}\: \ln \frac{\mu^2}{M^2_I}\ . \label{eq:g_I_Z_ABD_2}$$ Unlike in the heterotic case, the Fayet-Iliopoulos term depends now on the twisted moduli [@Ibanez_anomalous; @Poppitz; @lln] and the point of maximal gauge symmetry corresponds to $\xi^2_I =0$; as can be seen from an explicit linear-chiral duality transformation, $\xi^2_I$ is proportional to the twisted modulus $m$ in the linear basis. Therefore, at the point of maximal symmetry, one has $m=0$ and: $$\frac{1}{g^2_a (\mu^2) \mid_I}\ =\ \mbox{Re} S\ -\ \frac{b^I_a}{16 \pi^2}\: \ln \frac{\mu^2}{M^2_I}\ , \label{eq:g_I_Z_ABD_3}$$ which is obviously not dual to eq. (\[eq:g\_H\_Z\_ABD\_3\]). Note that, as was noticed in [@ABD], duality would be restored if some (presumably nonperturbative) mechanism ensured that $\xi^2_I =0$ for $\mbox{Re} M = 0$. More terms in the one-loop heterotic Lagrangian =============================================== One-loop corrections to the Kähler potential -------------------------------------------- We now consider specific one-loop corrections to the Kähler potential that are related to the well known holomorphic threshold corrections. In the remaining part of this paper we call these one-loop terms $\kappa$–corrections. Both types of corrections are uniquely correlated through the anomaly cancellation ten-dimensional Green-Schwarz terms [@GS]. Let us start with the perturbative heterotic string. As pointed out by Green and Schwarz for the gauge groups $E_8 \times E_8$ and $SO(32)$ the anomaly twelve form $I_{12}$ factorizes as $I_4 X_8$ and all gauge, gravitational and mixed gauge-gravitational anomalies are cancelled by the variation of well defined local counterterms. The crucial role in this cancellation is played by the antisymmetric field $B_{MN}$, which must transform nontrivially under gauge transformations. To recover classical gauge invariance at tree level the strength of the antisymmetric tensor field is modified so that it is gauge invariant and obeys $dH = I_4$. In the Horava-Witten model [@wh] representing the low-energy effective theory of the strongly coupled heterotic $E_8 \times E_8$ string, where the two $E_8$ gauge sectors live on opposite boundaries of the eleven-diemnsional bulk, the role of the effective antisymmetric tensor fields participating in the Green-Schwarz mechanism is played by the components $C_{AB11}$ of the three-form $C$. In general, there can be as many factorizable components of the anomaly form $I_{D+2}$ in $D$ dimensions, as many antisymmetric tensor fields are available in the model under considarations. A generalized Green-Schwarz mechanism involving more than one antisymmetric tensor field is at work in Type I/Type IIB orientifold models [@Sagnotti_Z_3], [@Ibanez_anomalous]. The local Green-Schwarz counterterms must be added to the ten-dimensional action of the heterotic superstring, and must be taken into account when one performs the dimensional reduction/compactification down to four dimensions. Partial reduction of the GS terms has been performed in [@hpn1], [@hpn2], [@itoyama], and more recently in [@bd], [@lpt] in the context of the strongly coupled heterotic string. In particular reference [@itoyama] contains essentially the complete result. Among the terms coming from this reduction the best known ones are the axionic parts of the holomorphic threshold corrections, $\pm \epsilon \, \theta F \tilde{F}$, where $\theta=Im(T)$, which have exactly the same form in the strongly coupled and in the weakly coupled heterotic $E_8 \times E_8$ string cases (they can be directly computed as the large $T$ limit of the threshold corrections in weakly coupled orbifolds [@hpnss], [@ss]). However, those are not the only relevant terms that contribute one-loop terms to the four dimensional Lagrangian. The other ones, ignored so far, are terms involving derivatives of the matter fields. To see these terms arising from the compactification we need the explicit form of the GS counterterms in the case of the $E_8 \times E_8$ heterotic string. If we denote the anomaly associated with the twelve-form $I_{12}$ by $G$, the new part in the 10d action satisfying $\delta_{gauge} S_{GS} + G = 0 $ can be written as S\_[GS]{} = ( 4 ( \_[3L]{} - \_[3Y]{} ) X\_7 - 6 B X\_8 ) \[eq5\] where $d X_7 = X_8$ and $X_8 = \frac{1}{24} Tr F^4 - \frac{1}{7200} (Tr F^2 )^2 - \frac{1}{240} Tr F^2 tr R^2 + \frac{1}{8} tr R^4 + \frac{1}{32} (tr R^2)^2$. To be specific, let us take the case of the standard embedding where $F^{(1)}_{MN}=R_{MN}$ and the Bianchi identity is fullfilled pointwise. The index on $F$ indicates that we use the gauge connection of only one of the two $E_8$’s, say the first one, to solve the Bianchi identity. As is well known, in that case the first $E_8$ is broken down to its $E_6$ subgroup, and the components of its connection with compact indices give rise to scalars $A$ in $h_{1,1}$ ${\bf 27}$ and $h_{1,2}$ ${\bf \bar{27}}$ representations of $E_6$. There are no matter fields associated with the unbroken $E_8$. The numbers $h_{a,b}$ are the Hodge numbers of the Calabi-Yau manifold which forms the compact 6d space. The usual axionic thresholds come from the terms which couple compact components of B, the $B_{MN}$, to the $F^{(i)} \tilde F^{(i)}$ term composed of 4d gauge field strengths. These come from the integral of $-6 \, B \, X_8$ over the Calabi-Yau space. Noting that the expansion of the compact components of $B$ in harmonics on the CY space is $B_{MN} = \sum_{1}^{h_{1,1}} \theta^Z (x) \Omega^{Z}_{MN} (y)$ where $\Omega^Z $ are the harmonic $(1,1)$ forms, the resulting 4d coupling is L\_= \^Z ( F\^[(1)]{} \^[(1)]{} \_K \^Z tr ( F\^[(1)]{} F\^[(1)]{}) - F\^[(2)]{} \^[(2)]{} \_K \^Z tr ( F\^[(2)]{} F\^[(2)]{}) ) \[eq6\] i.e. the couplings have exactly the same magnitude and opposite sign for $E_6$ and $E_8$ factors. The second coupling between 4d zero modes comes from the terms in $S_{GS}$ which contain space-time components of $B$ i.e. those proportional to $B_{\mu \nu}$. The physical degree of freedom associated with these components is the pseudoscalar dual to $H=d B + \omega_{3L} - \frac{1}{30} \omega_{3Y}$. After integration by parts one obtains the relevant part of the GS terms S\_[GS, H]{} = - 6 \_K H X\^[1]{}\_[7]{} \[sgs\] where the standard embedding is assumed and $X^{1}_{7} = \frac{1}{120} \omega^{1}_{3Y} ( tr F^{2}_{1} - \frac{1}{2} tr R^2 ) $. Let us note that if we were looking at corresponding terms with $\omega^{(2)}_{3Y}$ instead of $\omega^{(1)}_{3Y}$ then we would obtain the same expression with the opposite sign. The couplings of interest come from the terms \^[M N PQRS]{} H\_ Tr ([A]{}\_M \_\_N) ( tr F\^[2]{}\_[1]{} - tr R\^2 )\_[PQRS]{} \[haa\] Assuming the Calabi-Yau space with $h_{1,2} = 0$, going over to complex coordinates ${\cal B}_1 = 1/ \sqrt{2} ({\cal A}_4 + i {\cal A}_5),..., {\cal B}_3=1 / \sqrt{2} ({\cal A}_8 + i {\cal A}_9)$ and using the expansion [@redw] ${\cal B}_1 = T_{ax} A^{Kx} g^{a \bar{n} } \Omega^{K}_{1 \bar{n}}, \hfill \bar{{\cal B}}_{\bar{1}} = \bar{T}_{\bar{a} \bar{x}} \bar{A}^{K\bar{x}} g^{\bar{a} n } \bar{\Omega}^{K}_{\bar{1} n}, \, etc.$ with orthogonality relation for the gauge group generators $Tr ( T_{a x} \bar{T}_{ \bar{b} \bar{y}} ) = g_{a \bar{b} } \delta_{x \bar{y}}$ one readily obtains the 4d coupling of the form \[eqmix\] L = \^ H\_ ( A\^[Z x]{} |[A]{}\^[Y |[y]{}]{} ) \_[x |[y]{}]{} \_K \^[MNPQRS]{} g\^[TU]{} \^[Z]{}\_[MT]{} \^[Y]{}\_[NU]{} ( tr F\^[2]{}\_[1]{} - tr R\^2 )\_[PQRS]{} In the next step one needs to use the duality relation between $H$ and the universal axion $D$: $H_{\mu \nu \rho} = g^{-1/2}_{4} e^{-6 \sigma} \phi^{3/2} \epsilon_{\mu \nu \rho \delta} \partial^{\delta} D $ where $g^{(10)}_{AB} = ( e^{-3 \sigma} g^{(4)}_{\mu \nu}; \, e^\sigma g^{(0)}_{MN} )$ and $\phi $ is related to the string dilaton[^10]. Then one obtains the terms $\partial^\mu D A \stackrel{\leftrightarrow}{\partial_\mu} \bar{A}$ which give rise to kinetic mixing of the dilaton superfield $S$ and untwisted matter superfields $A$. To find the relation between the integrals which are coefficients in the terms (\[eqmix\]) and (\[eq6\]) we go now to the case $h_{1,1}=1$. Then the cohomology group $H^{1,1}$ consists only of the Kähler class generated by the Kähler form $k= i g_{i \bar{j}} dz^i \wedge d \bar{z}^{\bar{j}}$ where $g_{i \bar{j}}$ is the CY metric. Going over to the holomorphic coordinates one can see that the integrand in (\[eqmix\]) contains the factor g\^[m |[n]{}]{} k\_[|[q]{} m]{} k\_[p |[n]{}]{} = - g\_[p |[q]{}]{} = i k\_[p |[q]{}]{} which finally allows one to write the coefficient of the 4d operator in (\[eqmix\]) in the form of the topological integral $- \frac{1}{2} \int_K k \wedge tr (F^{(1)} \wedge F^{(1)})$ which is the same as the integral in the axionic threshold formula (\[eq6\]). The simple derivation we have given here is equivalent to Witten’s truncation on the six-torus when one replaces $g_{m \bar{n}} \rightarrow \delta_{m \bar{n}}$ and $k_{m \bar{n}} \rightarrow i \delta_{m \bar{n}} = \epsilon_{m \bar{n}}$. One should notice that the antisymmetric symbol $\epsilon$ introduced above reduces to the usual 2d antisymmetric symbol for each two-plane of the torus in the real coordinates. The above field-theoretical model calculation establishes the intimate relation between the axionic threshold correction (which is imaginary part of the holomorphic threshold correction) and the nonholomorphic correction (\[eqmix\]): either both of them vanish or both are present in the 4d effective Lagrangian. In terms of the four dimensional chiral superfields the nonholomorphic correction leads to the modified Kähler potential K\_S= - ( S + |[S]{} - A |[A]{}) This mixing with the 4d dilaton superfield is expected to hold for all untwisted matter multiplets, including the case of $h_{1,2} \neq 0$, in orbifold compactifications and is also valid in the case of nonstandard embeddings of the gauge group. The corrections to the Kähler potential arise obviously at one-loop order in the string coupling, since they come from the ten-dimensional Green-Schwarz terms. The sign of the mixing terms depends on the gauge group under consideration: it would be different for matter coming from the breaking of the second $E_8$. The same effect is present in the context of the Horava-Witten model which describes low-energy behaviour of the strongly coupled heterotic $E_8 \times E_8$ superstring. Let us complete this argument by summarizing the relevant calculation in the Horava-Witten model. In that case the decomposition of the eleven-dimensional metric which leads to canonical Einstein-Hilbert terms in the four-dimensional action and exhibits relevant four-dimensional degrees of freedom is $g^{(11)}_{AB} = ( e^{- 2 \beta (x) - \gamma (x)} g^{(4)} ; e^{- 2 \beta (x) + 2 \gamma (x)}; e^{\beta (x)} g^{(0)}_{IJ} )$. In this expression $e^{ 3 \beta}$ is the fluctuating volume $V$ of the 6d Calabi-Yau space in units of the reference volume $V_0 = \int_K \sqrt{g^{(0)}} $. The real parts of the chiral moduli superfields in the effective action are $Re(S) = e^{3 \beta}$ and $Re(T) = e^\gamma $. One should note, that the 10d source of kinetic terms for the 4d gauge fields and for 4d scalars is the operator $ \frac{1}{ 8 \pi (4 \pi k_{11}^{2})^{2/3}} \int d^{10} x \sqrt{g} Tr F_{AB} F^{AB}$. The relevant parts of it are d\^[10]{} x Tr F\_ F\^ + d\^[10]{} x Tr F\_[M]{} F\^[M]{} \[ttm\] where $\mu, \nu$ are noncompact and $M$ the compact indices, and $k_{11}^{2}$ is the eleven-dimensional gravitational constant. Using the decomposition of the gauge fields ${\cal A}_M$ given below formula (\[sgs\]) and the decomposition of the 11d metric in terms of $\beta, \, \gamma$ one obtains L\_[kin]{} = d\^[4]{} x e\^[ 3 ]{} Tr F\_ F\^ + d\^[4]{} x |\_A|\^2 . As pointed out in [@wh] computing corrections to the effective Lagrangian in the linear order in perturbations of the metric of the compact six-dimensional space corresponds to substitution $V = e^{3 \beta} V_0 \, \rightarrow V_{v,h} = ( e^{3 \beta} \pm \xi_0 e^\gamma ) V_0$ at the visible and hidden walls respectively. The parameter $\xi_0$ is given by the topological integral \[e:xpl\] \_0 &=& - \_X k ( trF\^[(1)]{} F\^[(1)]{} - trR R ). The normalization that gives direct correspondence with the weakly coupled case is established through $V_0 = 2 \pi (4 \pi k_{11}^{2})^{2/3}$ and $\alpha' = \frac{1}{(4 \pi )^{2/3} \pi^2 } \frac{k_{11}^{2/3}}{\rho_0} $ where one typically puts $\alpha' = 1/2$. In the kinetic terms of the gauge fields this leads to the well known difference between gauge couplings (gauge kinetic functions) on different walls, and upon substitution into the second term in (\[ttm\]) this gives immediately the corrections to the kinetic terms for scalars[^11] L\_[kin]{} = \_0 . These corrections have opposite sign for matter on opposite walls, and are interpreted as corrections to the $S$-dependent part of the 4d Kähler potential K\_S = - ( S + |[S]{}) \_0 -(S + |[S]{} \_0 A\_[v,h]{} |[A]{}\_[v,h]{} ) which is exactly the same expression that we have obtained in the weakly coupled heterotic string for matter living on the visible wall (one should observe that with the standard normalization of the integrals $\xi_0 = \kappa$). As we can see clearly in the context of the Horava-Witten model these corrections are indeed uniquely related to the corrections to the holomorphic gauge kinetic functions $f_{v,h} = S \pm \xi_0 T$. They both have here the geometric interpretation of the correction to the volume of the 6d compact space induced by different vacuum gauge fluxes on different walls. In the weakly coupled regime the same effects are seen as one-loop quantum effects, and from the point of view of the four-dimensional effective theory the matching is exact. The reason for this matching is anomaly cancellation. The crucial observation is that the massless spectrum of weakly and strongly coupled theories is the same, and, hence, that ten-dimensional and eleven-dimensional Green-Schwarz terms participating in anomaly cancellation must be strictly related to each other. The most relevant term in the compactification down to four dimensions is the topological term $C \wedge G \wedge G$ of eleven dimensional Horava-Witten Lagrangian, where $C$ is the eleven-dimensional three-form field and $G$ is its modified strength. It turns out that compactification[^12] of the components with the index structure $\epsilon^{\mu \nu \rho \delta \, IJKL \, 11 MN} G_{\mu \nu \rho \delta} G_{IJKL} C_{11 MN} $ produces in four dimensions axionic parts of the threshold corrections \[e:23\] &L\^[(4)]{} = ( )\^[4/3]{} tr(F\^[(1)]{} \^[(1)]{}) &\ &+ ( )\^[4/3]{} tr(F\^[(2)]{} \^[(2)]{}) .& These expressions, after substitution of $\rho_0$, coincide with the axionic threshold corrections given in (\[eq6\]). In the same way the results of the compactification of the part of the $ C \wedge G \wedge G$ with the index structure $\epsilon^{M \rho N \delta \, IJKL \, 11 \mu \nu } G_{M \rho N \delta} G_{IJKL} C_{11 \mu \nu} $ coincide precisely with the $\partial^\mu D A \stackrel{\leftrightarrow}{\partial_\mu} \bar{A}$ terms given in (\[eqmix\]) for matter in the visible sector (the sign would be the opposite one if we had matter in the hidden $E_8$ sector). Zero modes of the $C_{11 MN}$ and $C_{11 \mu \nu}$ coincide with the axions which are zero modes of $B_{MN}$ and $B_{\mu \nu}$. Thus we can see, that the relation between holomorphic threshold corrections and the $\kappa$-corrections to the Kähler potential receives even stronger support when viewed from the perspective of the strongly coupled heterotic models. There both types of 4d terms come from the topological term of the eleven dimensional supergravity, which is uniquely constrained by supersymmetry and by anomaly cancellation. Either both types of terms are present, or both should be absent in any heterotic model. With the present observation taken into account the 4d effective Lagrangians from the weakly and strongly coupled theories look exactly equivalent at the - respectively - one loop and linear in CY deformation orders[^13]. Equally important from the point of view of the rest of the present paper is the observation, that it is natural to expect the corrections to $K_S$ to have the same nature also in the heterotic $SO(32)$ models. The structure of the Green-Schwarz terms is in this case very much the same as described above and to be specific one can think of the decomposition $SO(32) \rightarrow SU(3) \times U(1) \times SO(26)$ where $SU(3)$ is identified with the holonomy group of the Calabi-Yau space in analogy with the calculation presented earlier in this section. Modular transformations in the presence of $\kappa$-corrections to the Kähler function -------------------------------------------------------------------------------------- As argued in the previous section in the (2,2) compactifications with holomorphic threshold corrections to the gauge couplings we expect the presence of nonholomorphic corrections which have a natural interpretation of one-loop contributions to the Kähler function. In what follows we restrict ourselves to the stndard embedding, with matter in the visible $E_6$ sector only. Then in the most symmetric case, with just the single T-modulus, the relevant parts of the Kähler function and the kinetic function are K = - ( S + |[S]{} + (T + |[T]{}) - A |[A]{}) - 3 ( T + |[T]{}) + f=S - (+ ) \^2 (T) + (T) where $ \kappa$ is a computable numerical parameter and $\sigma (T)$ is the holomorhic part of universal one-loop threshold correction which is invariant under the $SL(2,Z)$ T-duality transformations and approaches $-\frac{1}{4 \pi} T$ as $T \rightarrow \infty$. With the transformations $S \rightarrow S + \frac{3 \delta_{GS}}{4 \pi^2} \log (i c T + d), \; T \rightarrow \frac{a T -i b}{i c T + d} $ the model given by the above $K$ and $f$ is no longer invariant at one loop. One can cancel the variation of the $\kappa$-term in $K_S$ with a combination of the two modified transformations[^14]: - $S \rightarrow S + \frac{3 \delta_{GS}}{4 \pi^2} \log (i c T + d) + \gamma_S \, \kappa A \bar{A} ( \frac{1}{(i c T + d)(- i c \bar{T} + d) } -1 )/2 $, - $T \rightarrow \tilde{\Gamma} T = \frac{ a T - i b}{i c T + d} - \gamma_T \, \frac{ 2 \pi^2 \kappa}{3 \delta_{GS}} \frac{ (i c T + d)(- i c \bar{T} + d) -1}{(i c T + d )^2 (- i c \bar{T} + d )^2} (T + \bar{T}) A \bar{A} $. where $\gamma_S + \gamma_T = 1 $. Any combination of $\gamma$’s is troublesome. First, let us note that the new terms should not be counted in $P_C K $ which enters the nonlocal term representing the triangle graphs with a Kähler or sigma-model connection, as that would be a higher order effect. Second, the new terms turn a chiral superfield into a general superfield. Let us look more closely at various possibilities. The new terms in the transformation of S do not spoil the anomaly cancellation, but they vary at one-loop order the physical, 1PI, gauge coupling constant, which is not consistent with the symmetry. These terms in the transformation of T are proportional to the ratio of two parameters $\kappa$ and $\delta_{GS}$. One can assume that this ratio is small and treat these new terms as one-loop contributions. Then one should drop it in the T-dependent threshold corrections to gauge couplings, but the Kähler function $K_T = - \log ( T + \bar{T} )$ becomes non-covariant, and there is no suitable term to restore covariance. One should observe for instance, that although the $T + \bar{T}$ and $|\eta (T) |^{-4}$ both transform identically with respect to the tree-level modular transformations, their derivatives are completly different, hence the higher order terms in the expansion in powers of $|A|^2$ of their images under the full transformations cannot be matched to restore the covariance.\ To see this more explicitly lets us assume the Kähler potential for the superfield T in the form K\_T = - (T + |[T]{} + |(T)|\^[-4]{} ) The requirement that the covariance is restored at one-loop gives a solution for the coefficient $\beta$ (we put here $\gamma_T =1$ for convenience) = - where $\Gamma T = (a T - i b)/(i c T + d)$. This solution obviously does not make sense, as the coefficient $\beta $ turns out to be a function depending on the parameters of the modular transformation. Let us note, that the form of the above solution suggests that one could try another counterterm, K\_T = - (T + |[T]{} + ) where $\hat{G}_2$ is the covariant version of the Eisenstein form $G_2 (T)$, transforming with modular weight 2. It is a straightforward calculation to find out that this form of the counterterm does not work, however, as under $\tilde{\Gamma}$ &\_2 (T) \_2 (T) = \_2 (T) ( 1 + A |[A]{} ( (icT + d)\^2 + . . &\ & . . - ( (i c T + d)\^2 - - 4 i c (i c T + d) + ) ) ) & with $\alpha = - \frac{ 2 \pi^2 \kappa}{3 \delta_{GS}} \frac{ (i c T + d)(- i c \bar{T} + d) -1}{(i c T + d )^2 (- i c \bar{T} + d )^2} (T + \bar{T})$. Hence, once again, one obtains a coefficient $\beta$ that depends explicitly on the transformation parameters.\ Finally, one could ask whether, when we take $\gamma_S \neq 0$, in the gauge kinetic function $f$ the new term due to transformation of $S$, $\delta_{A S}$, would not cancel against the term $\delta_{A T}$ from the transformation of $T$. The point is that $\delta_{A S} \propto \delta_{GS}$ and $\delta_{A T} \propto \kappa^2 / \delta_{GS}$, hence the terms are of different orders in loop expansion parameters. In other words, such cancellation would require certain conspiracy between $\kappa$ and $\delta_{GS}$ – quantities which have very different microscopic origin and in principle do not need to be related. In addition, even a successful cancellation in $f$ would not solve the problem of the covariance of $K_T$. The problem which we have described in this section does not imply neccesarily that target-space duality does not hold in the heterotic string models. On the contrary, there are very good reasons to believe that it is an exact symmetry of many heterotic compactifications and as such should be representable in the effective Lagrangian. We would rather argue that there should exist further nonperturbative contributions to the Kähler function for moduli and matter fields. To illustrate these statements, let us recall that we have identified the $\kappa$–corrections to the kinetic terms in field theoretical limit, i.e. in the limiting domain of large $Re(T)$. In this limit the gauge kinetic function is $ f = S \pm \kappa T$ (for $E_6$ and $E_8$ sectors) which is at odds with the usual form of $T$-duality. To establish $T$-duality one needs to promote $T$ in the $f$ to $\log \eta^2 (T)$ plus $SL(2,Z)$-invariant universal terms. This is the necessary extension of $f$ to the case of arbitrary, both large and small, values of $Re (T)$. Similar nontrivial extension of $K_S$ is likely to be necessary to restore one-loop duality in the full effective Lagrangian, valid over the whole moduli space. The possible form of the generalized Kähler function could be K(S,|[S]{}; A, |[A]{}) = - ( S + |[S]{} - A |[A]{} ) \[res\] where $j(T)$ is the $SL(2,Z)$-invariant form. With the expression (\[res\]) substituted into the Kähler function the Lagrangian becomes modular invariant at the one-loop level without the need to modify the standard T-duality transformation for moduli and matter fields. Also, in the large $T$ limit it reduces to the expressions which we have computed in the section $4.1 \;$. The expression (\[res\]) is not the only one which fulfills these conditions. There exist other possible choices[^15], with a different behaviour at small $T$. However, it still makes sense to discuss the effective Lagrangians with the perturbative form of $K_S$ (including $\kappa$-terms) and to compare them to perturbative Lagrangians coming from other compactifications, having in mind that one restricts oneself to a part of moduli space where $Re(T) \, >> \, 1$ in natural units and compare terms that might violate $T$-duality. The kinetic mixing between the dilaton modulus $S$ and matter fields transforming like tensors under $SL(2,Z)$ leaves some questions concerning heterotic–type IIB orientifold duality. On one hand it spoils the one-loop duality invariance of the naive effective Lagrangian, thus making the situation more symmetric between the two classes of models - T-duality could be violated at one-loop level in a region of moduli space in both of the models. However, the way it would actually be violated seems to be completely different on both sides. Moreover, we recall, that we believe that in orbifold models $\kappa$–corrections are due to the existence of $N=2$ subsectors. Hence, in this respect, making relations with orbifold models is justified in the analysis of the orientifolds like the $Z_{6}'$ one (whose dual partner is not known). In addition, we do not expect similar matter-dilaton kinetic mixing in type IIB orientifold models. The reason is that there exists no one-loop coupling between the untwisted antisymmetric tensor field, corresponding to the imaginary part of the dilaton, and gauge fields. If the intuition gained during the analysis of the anomaly cancelling counterterms in heterotic models is correct, the kinetic mixing terms would rather be partners of the $B^{(k)} \wedge F^{(k)}$ couplings. Speculating further, by the same token we would expect modifications of the form $\frac{1}{4} (M_k + \bar{M}_k + \kappa_k A_k \bar{A}_k)^2$ to the Kähler functions of twisted moduli dual to twisted antisymmetric tensor fields, which have not been seen in explicit calculations. Conclusions =========== Many of the properties of string theory can be understood through a study of symmetries. In view of possible phenomenological applications it is important to incorporate such symmetries (if possible) into the low-energy effective field theory actions. We have seen that target-space duality is a very useful symmetry, able to constrain severely those effective actions. In the framework of simple compactification schemes of the heterotic theories, target-space duality can be incorporated successfully. This even goes beyond the classical level and constrains the one-loop effective action, including a mechanism for cancellation of all target-space anomalies. In the first part of the paper we recall the discussion of the simplest cases, $Z_3$ and $Z_7$ heterotic orbifolds, in detail. We then try to extend this picture to certain type IIB orientifold models (again $Z_3$ and $Z_7$), that have received some attention recently. One of the results of this paper is the observation that target-space duality now suffers from anomalies that cannot be cancelled by a mechanism similar to that in the heterotic case. The failure at the level of sigma-gravitational anomalies could in principle be repaired by $T$-dependent corrections to the CP-odd $R^2$ terms, but those would have to arise nonperturbatively. Our results (independently of the question of interpretation) strengthen the earlier arguments against a conjectured duality of a certain class of heterotic-type IIB orientifold models. Target-space dualities can be cancelled on the heterotic side, while no such satisfactory field theoretical cancellation mechanism seems to be at work in the considered $Z_3$ and $Z_7$ type IIB orientifolds. Another face of the problem with T-duality, and consequently with heterotic–type IIB orientifold duality, shows up when one tries to interpret the threshold corrections that have been recently computed in $Z_N$ orientifold models in the light of the cancellation mechanism suggested by the field-theoretical analysis of anomalies. In heterotic orbifolds, the structure of threshold corrections is intimately linked to the mechanism of mixed sigma-model-gauge anomaly cancellation, and one can actually infer from the explicit form of the one-loop corrections to gauge couplings how these anomalies are precisely compensated for. The interpretation of threshold corrections in orientifolds is less obvious due to the fact that the upper scale of logarithmic running is not modular invariant. Upon performing the linear-chiral duality transformation that relates the string result to the one-loop gauge couplings computed in the effective supergravity theory, one finds that this results in the impossibility to ensure target-space duality at the one-loop level: either sigma-gauge anomalies are cancelled, but the twisted moduli Kähler potential is not invariant; or the shift of the twisted moduli leaves an uncancelled anomaly. Of course, these results may simply point out that some nonperturbative terms in the orientifold Lagrangian are still missing, and in any case further string calculations in these models are truly needed. However, as far as one can trust the field theory approach, we have seen that target-space symmetries are a powerful tool to study and test conjectured relations between various string theories and the structure of their effective Lagrangians. It is well established that on the heterotic side, a symmetry that corresponds to $T$-duality does exist at the level of string theory. Therefore, a suitable description of this symmetry should be possible at the level of the field theoretic low-energy effective Lagrangian. As we have seen earlier, this works without problems in the simplest models. In section 4 we report on an attempt to generalize this to more general cases (e.g. to those models which contain $N=2$ subsectors). We point out that a manifestly symmetric incorporation of T-duality becomes problematic due to the appearance of one-loop corrections to the Kähler potential. Several possibilities to make these terms consistent with T-duality are proposed. Here the question arises, whether string theory can be approximated by a single unique low-energy effective action. After all, the traditional approach treated this action as relevant for the large T limit. Maybe different low-energy effective actions might be necessary to describe other patches of $T$ moduli space. Nevertheless, we demonstrate that such a unique description might be possible for the heterotic models under consideration. [**Acknowledgments**]{} .3cm The authors would like to thank I. Antoniadis, C. Bachas, E. Dudas, M. Klein and N. Obers for useful conversations. This work has been supported by TMR programs ERBFMRX–CT96–0045 and CT96–0090. Z.L. is additionaly supported by the Polish Committee for Scientific Research grant 2 P03B 037 15 and by M. Curie-Sklodowska Foundation. [**Appendix A: Sigma-model anomalies in heterotic orbifolds**]{} In this appendix, we recall some basic facts about sigma-model anomalies in $D=4$, $N=1$ heterotic orbifolds. At tree level, the Lagrangian is invariant under $SL(2,{R})$ reparametrizations of the geometric moduli[^16] [@flat; @lmn; @flt], $$T_i\ \rightarrow\ \frac{a_i T_i - i b_i}{i c_i T_i + d_i}\ , \hskip 2cm a_i d_i - b_i c_i = 1\ , \label{eq:SL_2_R_T}$$ together with linear transformations of the matter fields: $$\Phi_{\alpha}\ \rightarrow\ \prod_{i=1}^3\, (i c_i T_i + d_i)^{n^i_{\alpha}}\, \Phi_{\alpha}\ , \label{eq:SL_2_R_Phi}$$ where the $n^i_{\alpha}$ are the modular weights of the chiral superfield $\Phi_{\alpha}$. At the one-loop level, this symmetry known as target-space duality is generally anomalous; indeed, it acts as a chiral rotation on fermions and can have mixed anomalies with gauge symmetries and gravity[^17]. Under an $SL(2,{R})_{T_i}$ transformation, the one-loop Lagrangian undergoes an anomalous variation $$\delta {\cal L}_{\rm anomaly}\ =\ \frac{\theta_i}{32 \pi^2}\ \sum_a\, b^{\prime i}_a\, F^a \widetilde{F}^a\ -\ \frac{\theta_i}{768 \pi^2}\ b^{\prime i}_{grav}\, R \widetilde{R}\ ,$$ where $\theta_i = \arg (i c_i T_i + d_i)$ is the angle of the chiral rotation, and the mixed sigma-gauge and sigma-gravitational anomaly coefficients $b^{\prime i}_a$ and $b^{\prime i}_{grav}$ are given by [@Derendinger_sigma; @lidl]: $$\begin{aligned} b^{\prime i}_a & = & -\, C_2 (G_a)\, +\, \sum_{\alpha}\, (1 + 2 n^i_{\alpha})\, T(R_{\alpha})\ , \label{eq:sigma_gauge} \\ b^{\prime i}_{grav} & = & 21\ +\ 1\ +\ b^{\prime i}_{mod}\ -\ \dim G\ +\ \sum_{\alpha}\, (1 + 2 n^i_{\alpha})\ . \label{eq:sigma_grav}\end{aligned}$$ In Eq. (\[eq:sigma\_gauge\]), $C_2 (G_a)$ is the quadratic Casimir of the gauge group $G_a$ and $T(R_{\alpha})$ is the index of the representation $R_{\alpha}$ of $G_a$; in Eq. (\[eq:sigma\_grav\]), $\dim G$ is the dimension of the total gauge group, $21$ and $1$ stand for the contribution of the gravitino and the dilatino respectively, and $b^{\prime i}_{mod}$ denotes the contribution of the other (gauge singlet) modulinos, which is model-dependent. The mixed sigma-gauge anomaly is reproduced by the variation of the following nonlocal term [@cardoso1; @Derendinger_sigma]: $${\cal L}_{\rm n.l.}\ =\ -\, \frac{1}{32 \pi^2}\: \sum_a \int \! \mbox{d}^2 \theta\ W^a W^a\, \sum_i\, b^{\prime i}_a\ P_C \left[\, \ln (T_i + \bar T_i)\, \right]\ +\ \mbox{h.c.}\ , \label{eq:L_nl}$$ where $P_C = -\, \frac{1}{16}\, {\Box}^{-1} \bar D^2 D^2$ is the chiral projector, defined such that $\bar D\, (P_C H) = 0$ for any superfield $H$, and $P_C H = H$ if $H$ is a chiral superfield. In terms of components, ${\cal L}_{\mbox{n.l.}}$ also contains a non-harmonic contribution to gauge couplings: $$-\, \frac{1}{4}\, \sum_a\, F^a F^a\, \left(\, -\, \sum_i\, \frac{b^{\prime i}_a}{16 \pi^2}\: \ln (T_i + \bar T_i)\, \right)\ .$$ This shows that non-harmonic one-loop corrections to the gauge couplings are related, through supersymmetry, to the presence of sigma-model anomalies. Since the discrete version of the above symmetries is related to $T$-duality, which is an exact symmetry of the heterotic string, these anomalies must be compensated in some way. Two mechanisms can be at work [@Derendinger_sigma]: (i) a Green-Schwarz mechanism [@GS] similar to the one responsible for the cancellation of abelian gauge anomalies [@dsw], which involves a non-linear transformation of the dilaton superfield at the one-loop level: $$S\ \rightarrow\ S\ -\ \frac{1}{8 \pi^2}\ \delta^i_{GS} \ln (i c_i T_i + d_i)\ ; \label{eq:S_shift}$$ (ii) non-invariant, $T_i$-dependent holomorphic corrections to the gauge kinetic function. Such corrections arise from loops of massive string states and are associated with complex planes that are left invariant by some of the orbifold twists (corresponding to $N=2$ subsectors of the orbifold). In a large class of models, they take the form [@DKL] $$\Delta f^{\rm 1-loop}_a\ =\ -\, \frac{1}{4 \pi^2}\: \sum_i\, c_{a, i}\, \ln \left[ \eta (T_i) \right]\ , \label{eq:threshold}$$ where the coefficient $c_{a, i}$, to be determined by an explicit string computation, vanishes when the $i^{\rm th}$ complex plane is rotated by all twists. Note that $T_i$-dependent threshold corrections explicitly break the continuous $SL(2,{R})_{T_i}$ symmetry to its discrete version $SL(2,{Z})_{T_i}$, while the Green-Schwarz mechanism preserves it. Under an $SL(2,{Z})_{T_i}$ transformation, one has $\eta^2 (T_i) \rightarrow (i c_i T_i + d_i)\, \eta^2 (T_i)$ and $$f^{\rm 1-loop}_a\ \rightarrow\ f^{\rm 1-loop}_a\ -\ \frac{1}{8 \pi^2}\: \left(\, \delta^i_{GS} + c_{a, i}\, \right)\, \ln (i c_i T_i + d_i)\ .$$ Anomaly cancellation occurs provided the following relations are satisfied: $$b^{\prime i}_a\ =\ \delta^i_{GS}\, +\, c_{a, i}\ . \label{eq:b'_i_a}$$ Since $c_{a, i} = 0$ when the $i^{\rm th}$ complex plane is rotated by all twists, Eq. (\[eq:b’\_i\_a\]) imposes a strong constraint on the corresponding sigma-model anomalies, which must then be gauge-group independent (exactly as happens for abelian gauge anomalies compensated by a Green-Schwarz mechanism). Cancellation of mixed sigma-gravitational anomalies is realized in a similar manner. Collecting all contributions to the one-loop effective Lagrangian for the gauge fields, Eq. (\[eq:L\_nl\]) and (\[eq:threshold\]), and using Eq. (\[eq:b’\_i\_a\]), one obtains [@Derendinger_sigma]: $$\begin{aligned} {\cal L}_{\rm gauge} & = & \frac{1}{4}\: \sum_a \int \! \mbox{d}^2 \theta\ W^a W^a\ P_C \left\{\, \left[\, S + \bar S\ -\ \frac{1}{8 \pi^2}\ \delta^i_{GS} \ln (T_i + \bar T_i)\, \right] \right. \nonumber \\ & & \left. -\ \frac{1}{8 \pi^2}\ \sum_i\, (b^{\prime i}_a - \delta^i_{GS})\, \ln \left[\, |\eta (T_i)|^4 (T_i + \bar T_i)\, \right]\, \right\}\ +\ \mbox{h.c.}\ . \label{eq:L_GK_loop}\end{aligned}$$ Eq. (\[eq:L\_GK\_loop\]), together with Eq. (\[eq:K\_S\_loop\]), shows that the structure of one-loop corrections to the effective string action is strongly constrained by target-space duality anomaly cancellation. Note that anomaly considerations do not tell us anything about possible holomorphic, modular invariant corrections [@hpnss] that are not included in Eq. (\[eq:L\_GK\_loop\]). [**Appendix B: Green-Schwarz mechanism in the linear multiplet formalism**]{} The Green-Schwarz mechanism that cancels part of the sigma-model anomalies in heterotic orbifolds can be naturally described in the linear multiplet formalism[^18]. Indeed, in terms of the string massless states, the axion-dilaton-dilatino system fits into a linear multiplet $L = (l, B_{\mu \nu}, \chi)$, where the antisymmetric two-tensor $B_{\mu \nu}$ is dual to the model-independent axion $a = \mbox{Im} S$, $\partial_\mu a \sim \epsilon_{\mu \nu \rho \sigma}\, \partial^{\nu} B^{\rho \sigma}$. $L$ couples to the gauge fields in such a way that the combination $\widehat L = L - 2 \Omega$, where $\Omega$ is the Chern-Simons superfield defined by $\bar D^2 \Omega = \sum_a W^a W^a$ and $D^2 \Omega = \sum_a \bar W^a \bar W^a$, is gauge invariant. A gauge-invariant Lagrangian for $L$ then takes the simple form ${\cal L}_L = \int \! \mbox{d}^4 \theta\ \Phi (\widehat L)$. The transformation to the dual formulation in terms of the dilaton chiral superfield is accomplished by treating $\widehat L$ as an unconstrained superfield and adding the constraint $\bar D^2 (\widehat L + 2 \Omega) = D^2 (\widehat L + 2 \Omega) = 0$ to the Lagrangian: $${\cal L}\ =\ \int \! \mbox{d}^4 \theta\ \left[\, \Phi (\widehat L)\ -\ \frac{1}{2}\: (S + \bar S)\, (\widehat L + 2 \Omega)\, \right]\ , \label{eq:L_S_lagr}$$ where $S$ is the dilaton chiral superfield, and an unconstrained superfield $\Sigma$ defined by $S = \bar D^2 \Sigma$ plays the role of the Lagrange multiplier. The equation of motion for $\Sigma$ is nothing else but the constraint for $\widehat L$, while the equation of motion for $\widetilde L$ gives the duality relation: $$\frac{\partial \Phi}{\partial \widehat L}\ =\ \frac{1}{2}\: (S + \bar S) \hskip 1cm \Rightarrow \hskip 1cm \widehat L\ =\ \widehat L\, (S, \bar S)\ . \label{eq:L_S_duality}$$ Putting (\[eq:L\_S\_duality\]) into (\[eq:L\_S\_lagr\]), one obtains the Lagrangian for $S$, ${\cal L}_S = \int \! \mbox{d}^4 \theta\: K (S, \bar S) + \frac{1}{4}\: \int \! \mbox{d}^2 \theta\: f(S)$, with $f(S) = S$ and the Kähler potential given by: $$K\, (S, \bar S)\ =\ \left[\, \Phi (\widehat L)\ -\ \frac{1}{2}\: (S + \bar S)\, \widehat L\, \right]_{\ \widehat L\ =\ \widehat L\, (S, \bar S)}\ .$$ At tree level, one has $\Phi (\widehat L) = \ln \widehat L$, which leads to the duality relation $1 / \widehat L = (S + \bar S) / 2$ and the Kähler potential $K (S, \bar S) = - \ln (S + \bar S)$. The Green-Schwarz terms needed for the cancellation of sigma-model anomalies come at one loop and take the form $\Delta {\cal L}_{GS} = \sum_i \frac{\delta^i_{GS}}{16 \pi^2}\: \widehat L\, \ln (T_i + \bar T_i)$. In components, this contains a coupling between the antisymmetric tensor $B_{\mu \nu}$ and the sigma-model connection. Through a linear-chiral duality transformation, $\Delta {\cal L}_{GS}$ translates into a one-loop mixing between the dilaton and the geometric moduli. Indeed, the one-loop duality relation reads $$\frac{1}{\widehat L}\ =\ \frac{1}{2}\ \left[\, S + \bar S\ -\ \frac{1}{8 \pi^2}\ \sum_i\, \delta^i_{GS} \ln (T_i + \bar T_i)\, \right]\ , \label{eq:L_S_duality_loop}$$ implying a Kähler potential of the form (the dots refer to further corrections that are not related to sigma-model anomaly cancellation): $$K^{\rm 1-loop}\, (S, \bar S)\ =\ -\, \ln \left[\, S + \bar S\ -\ \frac{1}{8 \pi^2}\ \sum_i\, \delta^i_{GS} \ln (T_i + \bar T_i)\ +\ \ldots\, \right]\ . \label{eq:K_S_loop}$$ Target-space modular invariance then requires that $S$ transforms at one loop according to (\[eq:S\_shift\]). In type IIB orientifold compactifications we need to add more linear multiplets $m_k$, $k=1,...,n_f$ where $f$ is the number of twisted sectors. Such an extension of the linear multiplet formalism becomes somewhat subtle in the context of supergravity. To illustrate the trouble, let us take the simple case of just a single additional linear multiplet. If one neglects the superpotential couplings, the Lagrangian is given by ${\cal L}= S_0 \bar{S}_0 \Phi (\hat{L}, \hat{m})$ where $S_0 = (z_0, \psi_0, f_0)$ is the conformal compensator. Let us take = e\^[K/3]{} ( - X\^[-1/2]{} + X\^[1/2]{} Y\^2 ) , \[eqphi\] with $X= e^{K/3} \frac{\hat{L}}{S_0 \bar{S}_0 }, \; Y= e^{K/3} \frac{\hat{m}}{S_0 \bar{S}_0 } $. Using this form of $\Phi$ one obtains the graviton kinetic term in the form - R ( + ) . It is clear that choosing the value of $z_0$ which corresponds to the Einstein frame, $|z_0|^3 = (1 - \frac{s}{4} m^2)(2 e^K l)^{1/2}$ for small $m$, leads in general to a mixing between $m$ and $l$. The effective Lagrangian becomes simple and similar in its form to the heterotic effective Lagrangian only in the vicinity of the point $<m>=0$. 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[^4]: In heterotic compactifications, discrete modular transformations of the $T_i$ moduli correspond to a fundamental quantum string symmetry, $T$-duality. On the contrary, there is no such connection between the target-space modular invariance observed at the level of the effective Lagrangian of orientifolds and $T$-duality, since the latter exchanges Dirichlet and Neumann boundary conditions, and therefore does not leave a given D-brane configuration invariant. [^5]: Although it has been argued [@Ibanez_sigma] on the basis of string diagramatics that a one-loop mixing between the dilaton and the $T$ moduli should not affect sigma-gauge anomalies, because it would give a higher order contribution, this is not the case in the effective field theory, where such a mixing would yield terms proportional to $F^a \widetilde F^a$ in the variation of the Lagrangian, together with the $R \widetilde R$ piece. [^6]: Green-Schwarz cancellation of sigma-model anomalies in the presence of several linear multiplets $L$ and $L_k$ has been described in detail in ref. [@Klein]. [^7]: As stressed before in the context of Abelian gauge anomalies, the couplings $\delta^{i,k}_{GS}$ arise at the one-loop level in the effective field theory, although they are presumably present at tree level in the orientifold sense. [^8]: Needless to say, there is a perfect matching of the gauge couplings of both models at tree-level. Indeed, at the point of maximal gauge symmetry, one has $m=0$ (corresponding to $\xi^2_I = 0$) on the orientifold side, so that $\frac{1}{g^2_a \mid_I} = \frac{1}{l_I}$ is indeed dual to $\frac{1}{g^2_a \mid_H} = \frac{1}{l_H}$. [^9]: Recall that $(\mbox{Re} S)_H = V_H M^6_H / \lambda^2_H$ and $(\mbox{Re} T_i)_H = V^i_H M^2_H$ (where $V^i$ denotes the volume of the compact 2-torus $T^i$) on the heterotic side, and $(\mbox{Re} S)_I = V_I M^6_I / \lambda_I$ and $(\mbox{Re} T_i)_I = V^i_I M^2_I / \lambda_I$ on the orientifold side [@Ibanez_orientifolds]. [^10]: We recall that definitions of the universal moduli complex scalars in the weakly coupled string convention are: $S=e^{3 \sigma} \phi^{-3/4} + 3 i \sqrt{2} D$ and $T = e^{\sigma} \phi^{3/4} + i \sqrt{2} \theta$. [^11]: We remove the factor of $12$ through rescaling of matter fields $A$. [^12]: The compactification includes here substitution of the lowest order nontrivial solutions for $G$ along the eleventh dimension. [^13]: This concerns here models without five-branes in the 5d bulk on the strongly coupled side. [^14]: This form of modified $T$-duality transformation is reminiscent of the situation in the (2,2) orbifold models when the moduli C, charged under the subgroup H from the decomposition $H \times E_6 \times E_8$ of the orbifold gauge group are to obtain an expectation value. Then the transformation of T is no longer an $SL(2,Z)$ transformation, but becomes extended in a holomorphic way by terms which can be represented in the form of the power-law expansion in the blowing-up moduli C: $ T \rightarrow \frac{a T - i b}{i c T + d} + g_n (T) C^n, \; n \geq 3 $ [@flt]. One should notice, that when the duality transformation gets modified at tree level due to blowing up of the orbifold, either the transformation of S must get further modified at one loop, or the T-dependent counterterm under the logarithm must change to keep the $K(S,\bar{S})$ invariant. [^15]: For instance: &K(S,|[S]{}; A, |[A]{}) = - ( S + |[S]{} - A |[A]{} |(T) |\^4 |j(T) -744|\^[1/6]{} ) ,&\ &K(S,|[S]{}; A, |[A]{}) = - ( S + |[S]{} - A |[A]{} | \_2 (T) | ( - )\^[-1]{} / \^2 ) .& to name just two examples. Also, these are lowest order terms in the expansion in $A \bar{A}$. The exact form has to be determined by string calculations. [^16]: We consider only the diagonal moduli $T_i$, $i=1,2,3$, which are common to all orbifolds. In the presence of off-diagonal moduli $T_{i \bar j}$ ($i \neq j$), the target-space duality group is actually larger than $\prod_{i=1}^{3} SL(2,{R})_{T_i}$. [^17]: Strictly speaking, as explained in subsection 2.2, target-space duality transformations combine a sigma-model reparametrization and a Kähler transformation, both of which are anomalous [@Derendinger_sigma]. Following the literature, we shall call the resulting anomalies (\[eq:sigma\_gauge\]) and (\[eq:sigma\_grav\]) indifferently “target-space duality anomalies” or “sigma-model anomalies”. [^18]: For a useful reference about the linear multiplet in effective heterotic string theories, see [@Derendinger_linear] and references therein.

--- abstract: 'The notion of $2$-almost Gorenstein local ring ($2$-$\AGL$ ring for short) is a generalization of the notion of almost Gorenstein local ring from the point of view of Sally modules of canonical ideals. In this paper, for further developments of the theory, we discuss three different topics on $2$-$\AGL$ rings. The first one is to clarify the structure of minimal presentations of canonical ideals, and the second one is the study of the question of when certain fiber products, so called amalgamated duplications are $2$-$\AGL$ rings. We also explore Ulrich ideals in $2$-$\AGL$ rings, mainly two-generated ones.' address: - 'Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan' - 'Department of Mathematics and Informatics, Graduate School of Science and Technology, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522, Japan' - 'Global Education Center, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo 169-8050, Japan' author: - Shiro Goto - Ryotaro Isobe - Naoki Taniguchi title: 'Ulrich ideals and $2$-$\AGL$ rings' --- Introduction ============ The series [@CGKM; @GGHV; @GMP; @GMTY1; @GMTY2; @GMTY3; @GMTY4; @GRTT; @GTT; @GTT2] of researches are motivated and supported by the strong desire to stratify Cohen-Macaulay rings, finding new and interesting classes which naturally include that of Gorenstein rings. As is already pointed out by these works, the class of [*almost Gorenstein local rings*]{} (AGL rings for short) could be a very nice candidate for such classes. The prototype of AGL rings is found in the work [@BF] of V. Barucci and R. Fröberg in 1997, where they introduced the notion of AGL ring for one-dimensional analytically unramified local rings, developing a beautiful theory on numerical semigroups. In 2013, the first author, N. Matsuoka, and T. T. Phuong [@GMP] extended the notion of AGL ring given by [@BF] to arbitrary one-dimensional Cohen-Macaulay local rings, by means of the first Hilbert coefficients of canonical ideals. They broadly opened up the theory in dimension one, which prepared for the higher dimensional notion of AGL ring provided in 2015 by [@GTT]. Subsequently in 2017, T. D. M. Chau, the first author, S. Kumashiro, and N. Matsuoka [@CGKM] defined the notion of $2$-${\rm AGL}$ ring as a possible successor of AGL rings of dimension one. To explain the motivations for the present researches, we need to remind the reader of $2$-AGL rings more precisely. Throughout, let $(R, \m)$ be a Cohen-Macaulay local ring with $\dim R=1$, possessing the canonical module ${\mathrm{K}}_R$. We say that an ideal $I$ in $R$ is a [*canonical ideal*]{} of $R$, if $I \neq R$, and $I \cong {\mathrm{K}}_R$ as an $R$-module. In what follows, we assume that the ring $R$ possesses a canonical ideal, which contains a parameter ideal $Q=(a)$ of $R$ as a reduction. This assumption is automatically satisfied if $R$ has an infinite residue class field. Let ${\mathcal{T}}= {\mathcal{R}}(Q)=R[Qt]$ and ${\mathcal{R}}= {\mathcal{R}}(I)=R[It]$ be the Rees algebras of $Q$ and $I$ respectively, where $t$ denotes an indeterminate. We set ${\mathcal{S}}_Q(I) = I{\mathcal{R}}/I{\mathcal{T}}$ and call it the [*Sally module*]{} of $I$ with respect to $Q$ ([@V1]). Let ${\mathrm{e}}_i(I)~(i=0, 1)$ be the $i$-th Hilbert coefficients of $R$ with respect to $I$, that is, the integers satisfy the equality $$\ell_R(R/I^{n+1}) = {\mathrm{e}}_0(I) \binom{n+1}{1} - {\mathrm{e}}_1(I) \ \ \mbox{for all}\ \ n \gg 0$$ where $\ell_R(M)$ denotes, for each $R$-module $M$, the length of $M$. We set $\rank~{\mathcal{S}}_Q(I) = \ell_{{\mathcal{T}}_{\mathfrak{p}}}([{\mathcal{S}}_Q(I)]_{{\mathfrak{p}}})$ which is called the [*rank*]{} of ${\mathcal{S}}_Q(I)$, where ${\mathfrak{p}}= \m {\mathcal{T}}$. We then have $$\rank~{\mathcal{S}}_Q(I) = {\mathrm{e}}_1(I) - \left[{\mathrm{e}}_0(I) - \ell_R(R/I)\right]$$ ([@GNO Proposition 2.2 (3)]). Note that $\rank~{\mathcal{S}}_Q(I)$ is an invariant of $R$, independent of the choice of canonical ideals $I$ and the reductions $Q$ of $I$ (see [@CGKM Theorem 2.5]). With this notation we have the following. ([@CGKM Definition 1.3])\[1.1\] We say that $R$ is a [*$2$-almost Gorenstein local ring*]{} ($2$-${\rm AGL}$ ring for short), if $\rank~{\mathcal{S}}_Q(I) =2$, that is, ${\mathrm{e}}_1(I) = {\mathrm{e}}_0(I) - \ell_R(R/I) + 2$. Because $R$ is a non-Gorenstein AGL ring if and only if $\rank~{\mathcal{S}}_Q(I) =1$ ([@GMP Theorem 3.16]), $2$-AGL rings could be considered to be one of the successors of AGL rings. We set $K = a^{-1}I$ in the total ring ${\mathrm{Q}}(R)$ of fractions of $R$. Therefore, $K$ is a fractional ideal of $R$ such that $R \subseteq K \subseteq \overline{R}$ (here $\overline{R}$ stands for the integral closure of $R$ in ${\mathrm{Q}}(R)$) and $K \cong {\mathrm{K}}_R$, which we call a [*canonical fractional ideal*]{} of $R$. We set $S =R[K]$. Hence, $S$ is a module-finite birational extension of $R$, and it is independent of the choice of $K$ ([@CGKM Theorem 2.5 (3)]). Let ${\mathfrak{c}}= R:S$. We are now able to state the characterization of $2$-AGL rings given by [@CGKM], which we shall often refer to, in the present paper. \[mainref\] The following conditions are equivalent. 1. $R$ is a $2$-$\AGL$ ring. 2. There is an exact sequence $0 \to {\mathcal{B}}(-1) \to {\mathcal{S}}_Q(I) \to {\mathcal{B}}(-1) \to 0$ of graded ${\mathcal{T}}$-modules, where ${\mathcal{B}}={\mathcal{T}}/\m {\mathcal{T}}\ (\cong (R/\m)[t])$. 3. $K^2 = K^3$ and $\ell_R(K^2/K) = 2$. 4. $I^3=QI^2$ and $\ell_R(I^2/QI) = 2$. 5. $R$ is not a Gorenstein ring but $\ell_R(S/[K:{\mathfrak{m}}])=1$. 6. $\ell_R(S/K)=2$. 7. $\ell_R(R/{\mathfrak{c}}) = 2$. When this is the case, ${\mathfrak{m}}{\cdot}{\mathcal{S}}_Q(I) \ne (0)$, whence the exact sequence given by condition $(2)$ is not split, and we have $$\ell_R(R/I^{n+1}) = {\mathrm{e}}_0(I)\binom{n+1}{1}- \left({\mathrm{e}}_0(I) - \ell_R(R/I) +2\right)$$ for all $n \ge 1$. As is noted above, the notion of $2$-AGL ring could be considered to be one of the successors of the notion of AGL ring. However, if $2$-AGL rings claim that they are orthodox successors of AGL rings, it must be proved, showing that they really inherit several distinctive properties which AGL rings usually keep. In the present article, to certify the orthodoxy of $2$-AGL rings for the further studies, we investigate three topics on $2$-AGL rings, which are closely studied already for the case of AGL rings. The first topic concerns minimal presentations of canonical ideals. In Section 2, we will give a necessary and sufficient condition for a given one-dimensional Cohen-Macaulay local ring $R$ to be a $2$-AGL ring, in terms of minimal presentations of canonical fractional ideals. Our results Theorems \[3.2\] and \[3.4a\] exactly correspond to those about AGL rings given by [@GTT Theorem 7.8]. In Section 3, we investigate a generalization of so called [*amalgamated duplications*]{} of $R$ ([@marco]), including certain fiber products, and prove that $R$ is a $2$-$\AGL$ ring if and only if so is the fiber product $R \times_{R/{\mathfrak{c}}} R$. By [@CGKM Theorem 4.2] $R$ is a $2$-$\AGL$ ring if and only if so is the trivial extension $R \ltimes {\mathfrak{c}}$ of ${\mathfrak{c}}$ over $R$, which corresponds to [@GMP Theotem 6.5] for the case of AGL rings. In Sections 4 and 5, we are interested in Ulrich ideals in $2$-AGL rings. The existence of two-generated Ulrich ideals is basically a substantially strong condition for $R$, which we closely discuss in Section 4, especially in the case where $R$ is a $2$-AGL ring. Here, we should not rush, but should explain about what are Ulrich ideals. The notion of Ulrich ideal/module dates back to the work [@GOTWY] in 2014, where the authors introduced the notion, generalizing that of MGMCM modules (maximally generated maximal Cohen-Macaulay modules) ([@BHU]), and started the basic theory. The maximal ideal of a Cohen-Macaulay local ring with minimal multiplicity is a typical example of Ulrich ideals, and the higher syzygy modules of Ulrich ideals are Ulrich modules. In [@GOTWY; @GOTWY2], all the Ulrich ideals of Gorenstein local rings of finite CM-representation type and of dimension at most $2$ are determined, by means of the classification in the representation theory. On the other hand, in [@GTT2], the first author, R. Takahashi, and the third author studied the structure of the complex $\RHom_R(R/I, R)$ for Ulrich ideals $I$ in a Cohen-Macaulay local ring $R$ of arbitrary dimension, and proved that in a one-dimensional non-Gorenstein AGL ring $(R,\m)$, the only possible Ulrich ideal is the maximal ideal $\m$ ([@GTT2 Theorem 2.14 (1)]). In Section 5, we study the natural question of how and what happens about $2$-AGL rings. To state our conclusion, let ${\mathcal{X}}_R$ denote the set of Ulrich ideals in $R$. We then have the following, which we will prove in Section 5. The assertion exactly corresponds to [@GTT2 Theorem 2.14 (1)], the result of the case where $R$ is an AGL ring of dimension one. Suppose that $(R,\m)$ is a $2$-${\rm AGL}$ ring with minimal multiplicity, possessing a canonical fractional ideal $K$. Then $${\mathcal{X}}_R= \begin{cases} \{{\mathfrak{c}}, \m\}, & \ \text{if} \ K/R~\text{is}~R/{\mathfrak{c}}\text{-free},\\ \{\m\}, & \ \text{otherwise}. \end{cases}$$ For one-dimensional Gorenstein local rings $R$ of finite CM-representation type, the list of Ulrich ideals is known by [@GOTWY]. The proof given by [@GOTWY] is based on the techniques in the representation theory of maximal Cohen-Macaulay modules. It might have some interests to give a straightforward proof, making use of the results in [@GTT Section 12] from a different point of view. In Section 6 we shall perform it as an appendix. In what follows, unless otherwise specified, let $R$ be a one-dimensional Cohen-Macaulay local ring with maximal ideal ${\mathfrak{m}}$. For each finitely generated $R$-module $M$, let $\mu_R(M)$ (resp. $\ell_R(M)$) denote the number of elements in a minimal system of generators of $M$ (resp. the length of $M$). We denote by $\mathrm{K}_R$ the canonical module of $R$. Minimal presentations of canonical ideals in $2$-$\AGL$ rings ============================================================= In this section, we explore the structure of minimal presentations of canonical ideals of $2$-${\rm AGL}$ rings. Before going ahead, we summarize some known results on $2$-AGL rings, which we shall often refer to throughout this paper. Let $(R,\m)$ be a Cohen-Macaulay local ring with $\dim R = 1$, admitting the canonical module ${\mathrm{K}}_R$. We assume that $R$ possesses a canonical fractional ideal $K$, that is an $R$-submodule of ${\mathrm{Q}}(R)$ such that $R \subseteq K \subseteq \overline{R}$, where $\overline{R}$ denotes the integral closure of $R$ in ${\mathrm{Q}}(R)$, and $K \cong {\mathrm{K}}_R$ as an $R$-module. Let $S =R[K]$ and set ${\mathfrak{c}}= R : S$. We denote by ${\mathrm{r}}(R) =\ell_R(\Ext_R^1(R/\m, R))$ the Cohen-Macaulay type of $R$. \[2.3a\] Suppose that $R$ is a $2$-${\rm AGL}$ ring with $r = {\mathrm{r}}(R)$. Then the following assertions hold true. 1. ${\mathfrak{c}}= K:S = R:K$. 2. There is a minimal system $x_1, x_2, \ldots, x_n$ of generators of $\m$ such that ${\mathfrak{c}}=(x_1^2) + (x_2, x_3, \ldots, x_n)$. 3. $S/K \cong R/{\mathfrak{c}}$ and $S/R \cong K/R \oplus R/{\mathfrak{c}}$ as $R/{\mathfrak{c}}$-modules. 4. $K/R\cong (R/{\mathfrak{c}})^{\oplus\ell} \oplus (R/\m)^{\oplus m}$ as an $R/{\mathfrak{c}}$-module for some $\ell>0$, $m \ge 0$ such that $\ell + m = r-1$. 5. $\mu_R(S) = r+1$. Therefore, if $R$ is a $2$-AGL ring, then $\ell_R(K/R)=2\ell + m$. Hence, $K/R$ is a free $R/{\mathfrak{c}}$-module if and only if $\ell_R(K/R) = 2(r-1)$. Let us now fix the setting of this section. In what follows, we assume that $R=T/{\mathfrak{a}}$, $\m = \n/{\mathfrak{a}}$, for some regular local ring $(T, \n)$ with $\dim T= n \ge 3$ and an ideal ${\mathfrak{a}}$ of $T$ such that ${\mathfrak{a}}\subseteq \n^2$. Suppose that $R$ is not a Gorenstein ring. For each $a \in T$, let $\overline{a}$ denote the image of $a$ in $R$. Firstly, suppose that $R$ is a $2$-AGL ring, and write ${\mathfrak{c}}= (x_1^2) + (x_2, x_3, \ldots, x_n)$ with a minimal system $x_1, x_2, \ldots, x_n$ of generators of $\m$ (see Proposition \[2.3a\] (2)). We choose $X_i \in \n$ so that $x_i=\overline{X_i}$ in $R$, whence $\n = (X_1, X_2, \ldots, X_n)$. Let $J = (X_1^2)+(X_2, X_3, \ldots, X_n)$. We then have $ T/J \cong R/{\mathfrak{c}}$, since $\ell_T(T/J) = \ell_R(R/{\mathfrak{c}})=2$, so that ${\mathfrak{a}}\subseteq J$ and ${\mathfrak{c}}= J/{\mathfrak{a}}$. On the other hand, by Proposition \[2.3a\] (4) we have $$K/R \cong (R/{\mathfrak{c}})^{\oplus \ell} \oplus (R/\m)^{\oplus m}$$ with $\ell >0, m \ge 0$ such that $\ell + m = {\mathrm{r}}(R)-1$. Hence, letting $K= R + \sum_{i=1}^\ell Rf_i + \sum_{j=1}^m R g_j$ with $f_i, g_j \in K$, we may assume that $$\sum_{i=1}^{\ell}(R/{\mathfrak{c}}){\cdot} \overline{f_i} \cong (R/{\mathfrak{c}})^{\oplus \ell}\ \ \text{and} \ \ \sum_{j=1}^m (R/{\mathfrak{c}}){\cdot}\overline{g_j} \cong (R/\m)^{\oplus m},$$ where $\overline{f_i}, \overline{g_j}$ denote the images of $f_i, g_j$ in $K/R$. With this notation, we have the following, which corresponds to [@GTT Theorem 7.8] for AGL rings. \[3.2\] The $T$-module $K$ has a minimal free presentation of the form $$F_1 \overset{\Bbb M}{\longrightarrow} F_0 \overset{\Bbb N}{\longrightarrow} K \to 0,$$ where the matrices $\Bbb N$ and $\Bbb M$ are given by $$\Bbb N= [ \begin{smallmatrix} -1 & f_1 f_2 \cdots f_{\ell} & g_1 g_2 \cdots g_m \end{smallmatrix}]$$ and $$\Bbb M =\left[ \begin{smallmatrix} a_{11} a_{12} \cdots a_{1n} & \cdots & a_{\ell1} a_{\ell2} \cdots a_{\ell n} & b_{11} b_{12} \cdots b_{1n} & \cdots & b_{m1} b_{m2} \cdots b_{mn} & c_1 c_2 \cdots c_q \\ X^2_1 X_2 \cdots X_n & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \ddots & 0 & 0 & 0 & 0 & 0 \\ \vdots & \vdots & X^2_1 X_2 \cdots X_n & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & X_1 X_2 \cdots X_n & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & \ddots & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & X_1 X_2 \cdots X_n & 0 \end{smallmatrix}\right]$$ with $a_{ij} \in J$ $($$1 \le i \le \ell$, $1 \le j \le n$$)$, $b_{ik} \in J$ $($$1 \le i \le m$, $2 \le k \le n$$)$, and $q \ge 0$. The matrix $\Bbb M$ involves the information on a system of generators of ${\mathfrak{a}}$, and we have $${\mathfrak{a}}= \sum_{i=1}^{\ell} {\rm I}_2 \left(\begin{smallmatrix} a_{i1} & a_{i2} & \cdots & a_{in} \\ X^2_1 & X_2 & \cdots & X_n \end{smallmatrix}\right) + \sum_{i=1}^m {\rm I}_2 \left(\begin{smallmatrix} b_{i1} & b_{i2} & \cdots & b_{in} \\ X_1 & X_2 & \cdots & X_n \end{smallmatrix}\right) + (c_1, c_2, \ldots, c_q),$$ where ${\mathrm{I}}_2(\Bbb L)$ denotes, for a $2 \times n$ matrix $\Bbb L$ with entries in $T$, the ideal of $T$ generated by $2 \times 2$ minors of $\Bbb L$. Let $$F_1 \overset{\Bbb A}{\longrightarrow} F_0 \overset{[ \begin{smallmatrix} -1 & f_1 f_2 \cdots f_{\ell} & g_1 g_2 \cdots g_m \end{smallmatrix}]}{\longrightarrow} K \longrightarrow 0$$ be a part of a minimal $T$-free resolution of $K$ with $F_0 = T \oplus T^{\oplus \ell} \oplus T^{\oplus m}$, which gives rise to a presentation $$F_1 \overset{\Bbb A'}{\longrightarrow} G_0 \overset{\Bbb N'}{\longrightarrow} K/R \longrightarrow 0$$ of $K/R$, where $\Bbb N'=[ \begin{smallmatrix} \bar{f_1} \bar{f_2} \cdots \bar{f_{\ell}} & \bar{g_1} \bar{g_2} \cdots \bar{g_m} \end{smallmatrix}]$, and $\Bbb A'$ is the $(\ell + m) \times s$ matrix obtained from $\Bbb A$ by deleting the first row. On the other hand, since $K/R \cong (T/J)^{\oplus \ell}\oplus (T/\n)^{\oplus m}$, the $T$-module $K/R$ has a minimal presentation of the form $$G_1 = [T^{\oplus \ell} \oplus T^{\oplus m}]^{\oplus n}= T^{\oplus \ell n} \oplus T^{\oplus mn} \overset{\Bbb B}{\longrightarrow} G_0 = T^{\oplus \ell} \oplus T^{\oplus m} \overset{\Bbb N'}{\longrightarrow} K/R \longrightarrow 0,$$ where the matrix $\Bbb B$ is given by $$\Bbb B =\left[ \begin{smallmatrix} X^2_1 X_2 \cdots X_n & 0 & 0 & 0 & 0 & 0 \\ 0 & \ddots & 0 & 0 & 0 & 0 \\ \vdots & \vdots & X^2_1 X_2 \cdots X_n & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & X_1 X_2 \cdots X_n & 0 & 0 \\ 0 & 0 & 0 & 0 & \ddots & 0 \\ 0 & 0 & 0 & 0 & 0 & X_1 X_2 \cdots X_n \end{smallmatrix}\right].\vspace{1em}$$ Therefore, comparing with two presentations of $K/R$, we get a commutative diagram of $T$-modules, where $\eta \circ \xi$ is an isomorphism. Hence, $ \Bbb A'\ Q= \left( \Bbb B \mid O \right) $ for some $s \times s$ invertible matrix $Q$ with entries in $T$ (here $O$ denotes the null matrix). Setting $\Bbb M = \Bbb A Q$, we get $ \text{\large $\Bbb M$} = \arraycolsep5pt \left( \begin{array}{@{\,}ccc@{\,}} ~ & * &~\\ \hline &\multicolumn{1}{c}{\raisebox{-10pt}[0pt][0pt]{\large $\Bbb A'$}}\\ &&\\ \end{array} \right){\text{\large $Q$}} \ = \arraycolsep5pt \left( \begin{array}{@{\,}c|c@{\,}} ~ * ~ & ~ * ~\\ \hline \raisebox{-10pt}[0pt][0pt]{\large $\Bbb B$}&\raisebox{-10pt}[0pt][0pt]{\large $O$}\\ &\\ \end{array} \right) $, whence a required minimal presentation $$F_1 \overset{\Bbb M}{\longrightarrow} F_0 \overset{\Bbb N}{\longrightarrow} K \longrightarrow 0$$ of $K$ follows. Let us prove that $a_{ij}, b_{ij} \in J$. We set $Z_1 =X_1^2$, and $Z_i = X_i$ for each $2 \le i \le n$. Then, $a_{ij}\cdot(-1) + Z_j\cdot f_i = 0$ for every $1 \le i \le \ell$ and $1 \le j \le n$, whence $a_{ij} \in J$, because ${\mathfrak{c}}K \subseteq {\mathfrak{c}}S = {\mathfrak{c}}$ and ${\mathfrak{c}}=J/{\mathfrak{a}}$. Since $b_{ij}\cdot(-1) + Z_j\cdot g_i = 0$ for $j \ge 2$, we have $b_{ij} \in J$. The last assertion about the generating system of the defining ideal ${\mathfrak{a}}$ of $R$ follows from the fact that $Z_1, Z_2, \ldots, Z_n$ forms a regular sequence on $T$. We refer to [@GTT Proof of Theorem 7.8] for details. As a consequence of Theorem \[3.2\], we have the following. It exactly corresponds to [@GTT Corollary 7.10] for AGL rings. With the same notation as in Theorem $\ref{3.2}$, the following assertions hold true. 1. Suppose that $n = 3$. Then, ${\mathrm{r}}(R) = 2$, $q = 0$, $\ell =1$, and $m=0$, so that $\Bbb M = \left[\begin{smallmatrix} a_{11} & a_{12} & a_{13} \\ X^2_1 & X_2 & X_3 \end{smallmatrix}\right]$. 2. If $R$ has minimal multiplicity, then $q = 0$. \(1) Consider the minimal $T$-free resolution $$0 \longrightarrow F_2 \overset{{}^t \Bbb M}{\longrightarrow} F_1 \longrightarrow F_0 \longrightarrow R \longrightarrow 0,$$ where the matrix $\Bbb M$ has the form stated in Theorem \[3.2\]. We then have $${\mathrm{r}}(R)+1 = \rank_TF_1 = \ell n + mn + q = 3{\cdot}[{\mathrm{r}}(R)-1]+q,$$ so that $4-2{\cdot}{\mathrm{r}}(R)=q \ge 0$. Therefore, ${\mathrm{r}}(R)=2$, and $q=0$, since $R$ is not a Gorenstein ring. Thus, $\ell=1$, $m=0$, because $\ell + m = {\mathrm{r}}(R) -1$. \(2) Since $R$ has multiplicity $n$, we have ${\mathrm{r}}(R)=n-1$, while by [@S2 [Theorem]{} 1 (iii)], $ n(n-2)=\ell n + mn + q. $ Hence, $q=0$, because $\ell + m + 1 =n$. In this paper we will refer so often to examples arising from numerical semigroup rings, that let us explain here about a canonical form of generators for their canonical modules. Let $0 < a_1, a_2, \ldots, a_\ell \in \Bbb Z~(\ell >0)$ be positive integers such that $\mathrm{GCD}~(a_1, a_2, \ldots, a_\ell)=1$. We set $$H = \left<a_1, a_2, \ldots, a_\ell\right>=\left\{\sum_{i=1}^\ell c_ia_i \mid 0 \le c_i \in \Bbb Z~\text{for~all}~1 \le i \le \ell \right\}$$ and call it the numerical semigroup generated by the numbers $\{a_i\}_{1 \le i \le \ell}$. Let $V = k[[t]]$ be the formal power series ring over a field $k$. We set $R = k[[H]] = k[[t^{a_1}, t^{a_2}, \ldots, t^{a_\ell}]]$ in $V$ and call it the semigroup ring of $H$ over $k$. The ring $R$ is a one-dimensional Cohen-Macaulay local domain with $\overline{R} = V$ and $\m = (t^{a_1},t^{a_2}, \ldots, t^{a_\ell} )$, the maximal ideal. Let $${\mathrm{c}}(H) = \min \{n \in \Bbb Z \mid m \in H~\text{for~all}~m \in \Bbb Z~\text{such~that~}m \ge n\}$$ be the conductor of $H$ and set ${\mathrm{f}}(H) = {\mathrm{c}}(H) -1$. Hence, ${\mathrm{f}}(H) = \max ~(\Bbb Z \setminus H)$, which is called the Frobenius number of $H$. Let $$\mathrm{PF}(H) = \{n \in \Bbb Z \setminus H \mid n + a_i \in H~\text{for~all}~1 \le i \le \ell\}$$ denote the set of pseudo-Frobenius numbers of $H$. Therefore, ${\mathrm{f}}(H)$ coincides with the ${\mathrm{a}}$-invariant of the graded $k$-algebra $k[t^{a_1}, t^{a_2}, \ldots, t^{a_\ell}]$ and $\sharp \mathrm{PF}(H) = {\mathrm{r}}(R)$ ([@GW Example (2.1.9), Definition (3.1.4)]). We set $f = {\mathrm{f}}(H)$ and $$K = \sum_{c \in \mathrm{PF}(H)}Rt^{f-c}$$ in $V$. Then $K$ is a fractional ideal of $R$ such that $R \subseteq K \subseteq \overline{R}$ and $$K \cong {\mathrm{K}}_R = \sum_{c \in \mathrm{PF}(H)}Rt^{-c}$$ as an $R$-module ([@GW Example (2.1.9)]). Therefore, $K$ is a canonical fractional ideal of $R$. Notice that $t^f \not\in K$ but $\m t^f \subseteq R$, whence $K:\m = K + Rt^f$. Before stating the concrete example, let us explore the properties of $2$-$\AGL$ numerical semigroup rings. \[2.4\] Suppose that $R$ is a $2$-$\AGL$ ring. Then $$\displaystyle K/R = \bigoplus_{c \in {\rm PF}(H)\setminus \{f\}} R {\cdot} \overline{t^{f-c}}$$ where $\overline{(*)}$ denotes the image in $K/R$. We set $r ={\mathrm{r}}(R)$, $f=c_r$ and write ${\rm PF}(H) =\{c_1, c_2, \ldots, c_r\}$. Let us consider $$I = \{i \in \Lambda \mid \operatorname{Ann}_{R/{\mathfrak{c}}} \overline{t^{f-c_i}} =(0)\}, \ \ J = \{i \in \Lambda \mid \operatorname{Ann}_{R/{\mathfrak{c}}} \overline{t^{f-c_i}} \ne (0)\}$$ where ${\mathfrak{c}}= R:R[K]$ and $\Lambda =\{1, 2, \cdots, r-1\}$. Notice that $I \cup J = \Lambda$ and $I \ne \emptyset$. Since $R$ is a $2$-$\AGL$ ring, there exists $b \in H$ such that $(0):_{R/{\mathfrak{c}}}\m = {\mathfrak{m}}/{\mathfrak{c}}= [(t^b) + {\mathfrak{c}}]/{\mathfrak{c}}$. Then, for each $i \in I$, we have $t^b {\cdot} \overline{t^{f-c_i}} \ne 0$ and $\m {\cdot}\overline{t^{f-c_i+b}} =(0)$ in $K/R$. Hence $f+b-c_i \in {\rm PF}(H)$, and the elements $\{t^b{\cdot} \overline{t^{f-c_i}}\}_{i \in I}$ in $K/R$ are linearly independent over $k$. Therefore $$K/R = \sum_{i \in I} R{\cdot}\overline{t^{f-c_i}} \bigoplus \sum_{j \in J} R{\cdot}\overline{t^{f-c_j}} = \bigoplus_{i \in \Lambda} R {\cdot} \overline{t^{f-c_i}}$$ as desired. For the moment, suppose that $R$ is a $2$-$\AGL$ ring and we maintain the notation as in the proof of Proposition \[2.4\]. Choose $b = a_j$ for some $1 \le j \le \ell$. We then have $f+b-c_i \in {\rm PF}(H)$ for each $i \in I$, while $f-c_j \in {\rm PF}(H)$ for each $j \in J$ if $J \ne \emptyset$. By writing $I = \{c_1 < c_2 < \cdots < c_p\}~(p>0)$ and $J = \{d_1 < d_2 < \cdots <d_p\}~(q \ge 0)$, we have the following. The following assertions hold true. 1. $f+b=c_i + c_{p+1 -i}$ for every $1 \le i \le p$. 2. If $J \ne \emptyset$, then $f=d_j +d_{q+1-j}$ for every $1 \le j \le q$. The assertions follow from the fact that the maps $$\{c_i \mid i \in I\} \to \{c_i \mid i \in I\}, x \mapsto f+b-x, \quad \{c_j \mid j \in J\} \to \{c_j \mid j \in J\}, x \mapsto f-x$$ are well-defined and bijective. As a consequence, we get the following, which corresponds to the case where $J = \emptyset$. Suppose that $R$ is a $2$-$\AGL$ ring. Then the following conditions are equivalent. 1. $K/R \cong (R/{\mathfrak{c}})^{\oplus (r-1)}$ as an $R$-module. 2. There is an integer $1 \le j \le \ell$ such that $f + a_j = c_i + c_{r-i}$ for every $1 \le i \le r-1$. Let us now go back to state the example of Theorem \[3.2\]. With the notation of Theorem \[3.2\], we cannot expect $q=0$ in general, as we show in the following. \[3.5\] Let $V = k[[t]]$ be the formal power series ring over a field $k$, and set $R = k[[t^5,t^7,t^9,t^{13}]]$. Hence, $R=k[[H]]$, the semigroup ring of the numerical semigroup $H=\left<5, 7, 9, 13\right>$. We then have ${\rm f}(H) = 11$ and $\mathrm{PF}(H) =\{8, 11\}$, whence $K=R + Rt^3$ and $R[K] = k[[t^3, t^5, t^7]] = R + Rt^3+Rt^6$. Therefore, ${\mathfrak{c}}= (t^{10},t^7,t^9,t^{13})$ and $\ell_R(R/{\mathfrak{c}}) =2$, so that by Theorem \[mainref\], $R$ is a $2$-AGL ring with ${\mathrm{r}}(R)= 2$. We are interested in the defining ideal ${\mathfrak{a}}$ of $R$. Let $T=k[[X,Y,Z,W]]$ be the formal power series ring, and let $\varphi : T \to R$ be the $k$-algebra map defined by $\varphi (X) = t^5, \varphi(Y)=t^7, \varphi(Z) = t^9$, and $\varphi(W) = t^{13}$. Then, $R$ has a minimal $T$-free resolution of the form $$\Bbb F : \ 0 \to T^2 \overset{\Bbb M}{\to} T^6 \overset{\Bbb N}{\to} T^5 \overset{\Bbb L}{\to} T \to R \to 0,$$ where the matrices $\Bbb M, \Bbb N,$ and $\Bbb L$ are given by $${}^t\Bbb M =\left[ \begin{smallmatrix} W & X^2 & XY & YZ & Y^2 - XZ & Z^2 - XW \\ X^2 & Y & Z & W & 0 & 0\\ \end{smallmatrix}\right],$$ $$\Bbb N =\left[ \begin{smallmatrix} -Z^2 + XW & 0 & X^2Z & -X^3 & 0 & W \\ Y^2 - XZ & -X^2Y & X^3 & 0 & -W & 0 \\ 0 & 0 & W & -Z & 0 & Y \\ 0 & -W & 0 & Y & -Z & -X \\ 0 & Z & -Y & 0 & X & 0\\ \end{smallmatrix}\right], $$ $$\Bbb L =\left[ \begin{smallmatrix} Y^2-XZ & Z^2-XW & X^4-YW & X^3Y-ZW & X^2YZ-W^2\\ \end{smallmatrix}\right].$$ The $T$-dual of $\Bbb F$ gives rise to the presentation $$T^6 \overset{{}^t\Bbb M}{\to} T^2 \to K \to 0$$ of the canonical fractional ideal $K=R + Rt^3$, so that $$K/R \cong T/(X^2, Y, Z, W) \cong R/{{\mathfrak{c}}}.$$ We have ${\mathfrak{a}}=\Ker \varphi = {\mathrm{I}}_2 \left(\begin{smallmatrix} W & X^2 & XY & YZ \\ X^2 & Y & Z & W \\ \end{smallmatrix}\right) + (Y^2 - XZ, Z^2 - XW) $. We note one example of $2$-${\rm AGL}$ rings of minimal multiplicity, whence $q=0$. \[3.6\] Let $V = k[[t]]$ be the formal power series ring over a field $k$, and set $R = k[[H]]$, where $H=\left<4, 9, 11, 14\right>$. Then, ${\rm f}(H) = 10$ and ${\mathrm PF}(H) = \{5, 7, 10\}$, whence $K=R + Rt^3+ Rt^5$ and $R[K] = k[[t^3, t^4, t^5]] = R + Rt^3 + Rt^5 + Rt^6$. Therefore, ${\mathfrak{c}}= (t^8,t^9,t^{11},t^{14})$ and $\ell_R(R/{\mathfrak{c}}) =2$, so that by Theorem \[mainref\], $R$ is a $2$-AGL ring possessing minimal multiplicity $4$ and ${\mathrm{r}}(R)=3$. We consider the $k$-algebra map $\varphi : T \to R$ defined by $\varphi (X) = t^4, \varphi(Y)=t^9, \varphi(Z) = t^{11}$, and $\varphi(W) = t^{14}$, where $T=k[[X,Y,Z,W]]$ denotes the formal power series ring. Then, $R$ has a minimal $T$-free resolution $$\Bbb F : \ 0 \to T^3 \overset{\Bbb M}{\to} T^8 \overset{\Bbb N}{\to} T^6 \overset{\Bbb L}{\to} T \to R \to 0$$ where the matrices $\Bbb M, \Bbb N,$ and $\Bbb L$ are given by $${}^t\Bbb M = \left[ \begin{smallmatrix} -Z & -X^3 & -W & -X^2Y & Y & W & X^4 & X^2Z \\ X^2 & Y & Z & W & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & X & Y & Z & W \end{smallmatrix}\right],$$ $$\Bbb N =\left[ \begin{smallmatrix} -X^2Z & 0 & X^4 & 0 & 0 & 0 & W & -Z \\ 0 & 0 & W & -Z & 0 & W & 0 & -Y \\ 0 & W & 0 & -Y & -X^2Y & X^3 & 0 & 0 \\ -W & 0 & 0 & X^2 & 0 & -Z & Y & 0 \\ 0 & Z & -Y & 0 & -W & 0 & 0 & X \\ Y & -X^2 & 0 & 0 & Z & 0 & -X & 0 \\ \end{smallmatrix}\right], $$ $$\Bbb L =\left[ \begin{smallmatrix} Y^2-XW & X^5-YZ & Z^2-X^2W & X^3Z-YW & X^4Y-ZW & X^2YZ - W^2\\ \end{smallmatrix}\right].$$ Taking $T$-dual of $\Bbb F$, we have the presentation $$T^8 \overset{{}^t\Bbb M}{\to} T^3 \to K \to 0$$ of $K = R + Rt^3 + Rt^5$, so that $$K/R \cong T/(X^2, Y, Z, W) \oplus T/\n \cong R/{\mathfrak{c}}\oplus R/{\mathfrak{m}}.$$ Hence, $K/R$ is not $R/{\mathfrak{c}}$-free. We have $ \Ker \varphi = {\mathrm{I}}_2 \left(\begin{smallmatrix} -Z & -X^3 & -W & -X^2Y \\ X^2 & Y & Z & W \\ \end{smallmatrix}\right) + {\mathrm{I}}_2 \left(\begin{smallmatrix} Y & W & X^4 & X^2Z \\ X & Y & Z & W \\ \end{smallmatrix}\right). $ We are now asking for a sufficient condition for $R = T/{\mathfrak{a}}$ to be a $2$-AGL ring in terms of the presentation of the canonical ideal. Let us maintain the setting in the preamble of this section, assuming $R$ possesses a canonical fractional ideal $K$ of the form $$K=R+ \sum_{i=1}^\ell Rf_i + \sum_{j=1}^m Rg_j$$ where $f_i, g_j \in K$, and $\ell >0$, $m \ge 0$ with $\ell + m + 1={\mathrm{r}}(R)$. We then have the following. We should compare it with [@GTT Theorem 7.8]. \[3.4a\] Let $X_1, X_2, \ldots, X_n$ be a regular system of parameters of $T$ and assume that $K$ has a presentation of the form $$F_1 \overset{\Bbb M}{\longrightarrow} F_0 \overset{\Bbb N}{\longrightarrow} K \to 0 \quad \quad \quad (\sharp)$$ where $\Bbb N$ and $\Bbb M$ are matrices of the form stated in Theorem $\ref{3.2}$, satisfying the condition that $a_{ij}, b_{pk} \in (X_1^2) + (X_2, X_3, \ldots, X_n)$ for every $1 \le i \le \ell$, $1 \le j \le n$, $1 \le p \le m$, and $2 \le k \le n$. Then $R$ is a $2$-${\rm AGL}$ ring. The presentation $(\sharp)$ gives rise to a presentation $$F_1 \overset{\Bbb B}{\longrightarrow} G_0 \overset{\Bbb L}{\longrightarrow} K/R \longrightarrow 0$$ of $K/R$, where $\Bbb L=\left[\begin{smallmatrix} \bar{f_1} \bar{f_2} \cdots \bar{f_{\ell}} & \bar{g_1} \bar{g_2} \cdots \bar{g_m}\end{smallmatrix}\right]$ (here $\overline{*}$ denotes the image in $K/R$), and the matrix $\Bbb B$ has the form stated in the proof of Theorem \[3.2\]. Hence$$K/R \cong (T/J)^{\oplus \ell} \oplus (T/\n)^{\oplus m},$$ so that $\n{\cdot}(K/R) \ne (0)$, since $\ell > 0$. Therefore, ${\mathfrak{c}}\subsetneq \m$. We set $J = (X_1^2)+(X_2, X_3, \ldots, X_n)$ and let $I = JR$. Then, since $a_{ik} \in J$, inside of $K/R$ we get $$X_1^2{\cdot}f_i = \overline{a_{i1}} \ \ \text{and} \ \ X_k{\cdot}f_i = \overline{a_{ik}}$$ for every $1 \le i \le \ell$ and $2 \le k \le n$. Hence, $X_1^2{\cdot}f_i, X_k{\cdot}f_i \in I$. We similarly have $X_k{\cdot}g_j \in I$ for all $1 \le j \le m$ and $2 \le k \le n$, because $b_{jk} \in J$. Moreover, $X_1^2{\cdot}g_j \in J$ for every $1 \le j \le m$. Thus, $IK \subseteq I$, whence $IS \subseteq I$, because $S = R[K]=K^q$ for $q \gg 0$. Therefore, $I \subseteq {\mathfrak{c}}\subsetneq \m$, so that $I = {\mathfrak{c}}$, since $\ell_R(R/I) \le 2$. Thus, $\ell_R(R/{\mathfrak{c}}) = 2$, and $R$ is a $2$-AGL ring by Theorem \[mainref\]. As a consequence of Theorem \[3.4a\], we have the following. Let $(T,\n)$ be a regular local ring with $\dim T=n \ge 3$ and $\n = (X_1, X_2, \ldots, X_n)$. Choose positive integers $\ell_1, \ell_2, \ldots, \ell_n >0$ so that $\ell_1\ge 2$ and set ${\mathfrak{a}}= {\mathrm{I}}_2 \left(\begin{smallmatrix} X_1^2 & X_2 & \cdots & X_{n-1} & X_n \\ X_2^{\ell_2} & X_3^{\ell_3} & \cdots & X_n^{\ell_n} & X_1^{\ell_1}\\ \end{smallmatrix}\right)$. Then $R=T/{\mathfrak{a}}$ is a $2$-$\AGL$ ring, for which $K/R$ is a free $R/{\mathfrak{c}}$-module of rank $n-2$. Since $\sqrt{{\mathfrak{a}}+ (X_1)} = \n$, $\grade_T{\mathfrak{a}}= n-1$, so that ${\mathfrak{a}}$ is a perfect ideal of $T$, whence $R=T/{\mathfrak{a}}$ is a Cohen-Macaulay local ring with $\dim R=1$, and a minimal $T$-free resolution of $R$ is given by the Eagon-Northcott complex associated to the matrix $\left(\begin{smallmatrix} X_1^2 & X_2 & \cdots & X_{n-1} & X_n \\ X_2^{\ell_2} & X_3^{\ell_3} & \cdots & X_n^{\ell_n} & X_1^{\ell_1}\\ \end{smallmatrix}\right)$ ([@EN]). We take the $T$-dual of the resolution and get the following presentation $$T^{\oplus n(n-2)} \overset{{\Bbb M}'}{\to} T^{\oplus (n-1)} \overset{\varepsilon}{\to} K_R \to 0$$ of the canonical module $K_R$ of $R$, where the matrix ${\Bbb M}'$ is given by [$${\Bbb M}'=\left[ \begin{smallmatrix} X_2^{\ell_2} -X_3^{\ell_3} \cdots (-1)^nX_n^{\ell_n}(-1)^{n+1}X_1^{\ell_1} & 0 & & & & \\ X_1^2 -X_2 \cdots (-1)^{n+1}X_n & X_2^{\ell_2} -X_3^{\ell_3} \cdots (-1)^nX_n^{\ell_n}(-1)^{n+1}X_1^{\ell_1} & & & & \\ & & \ddots & \\ & & & & X_1^2 -X_2 \cdots (-1)^{n+1}X_n & X_2^{\ell_2} -X_3^{\ell_3} \cdots (-1)^nX_n^{\ell_n}(-1)^{n+1}X_1^{\ell_1} \\ & & & & 0 & X_1^2 -X_2 \cdots (-1)^{n+1}X_n \end{smallmatrix}\right].$$ ]{} Let $x_i$ denote, for each $1 \le i\le n$, the image of $X_i$ in $R$. Since $x_1^2x_1^{\ell_1}= x_2^{\ell_2}x_n$, $x_i x_1^{\ell_1} = x_{i+1}^{\ell_{i+1}}x_n$ for every $2 \le i \le n-1$ and $x_1$ is a parameter of $R$, we have that every $x_i$ is a non-zerodivisor in $R$. We set $y = \frac{x_2^{\ell_2}}{x_1^{2}}$, and $$f_i = \begin{cases} x_{i+1}^{\ell_{i+1}} & \text{if} \ \ 1 \le i \le n-1\\ x_1^{\ell_1} & \text{if}\ \ i = n \end{cases} \ \ \ \ g_i = \begin{cases} x_1^{2} & \text{if} \ \ i=1\\ x_i & \text{if}\ \ 2 \le i \le n \end{cases} .$$ Then $f_i = g_iy$ for all $1 \le i \le n$, so that $y^{n} = \frac{\prod_{i=1}^nf_i}{\prod_{i=1}^ng_i}= x_1^{\ell_1-2}x_2^{\ell_2 - 1} \cdots x_n^{\ell_n -1} \in R$. Hence, $y \in \overline{R}$. Let $K = \sum_{i=0}^{n-2}Ry^i$. Therefore, $R \subseteq K \subseteq \overline{R}$. We will show that $K$ is a canonical fractional ideal of $R$. Indeed, because $ [\begin{smallmatrix} -1& y& -y^2& \cdots (-1)^{n-1}y^{n-2} \end{smallmatrix}]{\cdot}{\Bbb M}' = {\mathbf 0}$, the $T$-linear map $\psi : T^{\oplus (n-1)} \to K$ defined by $\psi ({\mathbf e}_i)= (-1)^{i}y^{i-1}$ for $1 \le i \le n-1$ (here $\{{\mathbf e}_i\}_{1 \le i \le n-1}$ denotes the standard basis of $T^{\oplus n-1}$) factors through $K_R$. Let $\sigma : K_R \to K$ be the $R$-linear map such that $\psi = \sigma \varepsilon$. Then, $K = \Im \sigma$, and $\sigma$ is a monomorphism. Indeed, assume that $X = \Ker \sigma \ne (0)$, and choose ${\mathfrak{p}}\in \Ass_RX$. Then, $({\mathrm{K}}_R)_{\mathfrak{p}}\cong {\mathrm{K}}_{R_{\mathfrak{p}}}$, since ${\mathfrak{p}}\in \Ass_R {\mathrm{K}}_R$, while $K_{\mathfrak{p}}\cong R_{\mathfrak{p}}$, since $K$ is isomprphic to some $\m$-primary ideal of $R$ (here $\m$ denotes the maximal ideal of $R$). Consequently, we get the exact sequence $$0 \to X_{\mathfrak{p}}\to {\mathrm{K}}_{R_{\mathfrak{p}}} \to R_{\mathfrak{p}}\to 0$$ of $R_{\mathfrak{p}}$-modules, which forces $X_{\mathfrak{p}}=(0)$, because $\ell_{R_{\mathfrak{p}}}({\mathrm{K}}_{R_{\mathfrak{p}}})= \ell_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})$. This is a contradiction. Thus, ${\mathrm{K}}_R \cong K$. We identify ${\mathrm{K}}_R = K$ and $\varepsilon = \psi$. Then, because $(X_1^{\ell_1}, X_2^{\ell_2}, \ldots, X_n^{\ell_n}) \subseteq (X_1^{2}, X_2, \ldots, X_n)$, the matrix ${\Bbb M}'$ is transformed with elementary column operations into the following matrix $$\Bbb M =\left[ \begin{smallmatrix} a_{11} a_{12} \cdots a_{1n} & \cdots & \cdots & a_{n-2, 1} a_{n-2, 2} \cdots a_{n-2, n} \\ X^2_1 X_2 \cdots X_n & 0 & \cdots & 0 \\ 0 & \ddots & & 0 \\ \vdots & & \ \ddots & \vdots \\ 0 & 0 & \cdots & X_1^2 X_2 \cdots X_n \\ \end{smallmatrix}\right]$$ with $a_{ij} \in (X_1^{\ell_1}, X_2^{\ell_2}, \ldots, X_n^{\ell_n})$, so that Theorem \[3.4a\] shows $R$ is a $2$-$\AGL$ ring. Since $K/R \cong (T/(X_1^2, X_2, \ldots, X_n))^{\oplus n-2}$, $K/R$ is a free $R/{\mathfrak{c}}$-module of rank $n-2$. $2$-${\rm AGL}$ rings obtained by fiber products ================================================ In this section we study the problem of when certain fiber products, or more generally, quasi-trivial extensions of one-dimensional Cohen-Macaulay local rings are $2$-${\rm AGL}$ rings. Let $R$ be a commutative ring and $I$ an ideal of $R$. For an element $\alpha \in R$, we set $A(\alpha)=R\oplus I$ as an additive group and define the multiplication on $A(\alpha)$ by $$(a, x) \cdot (b, y) = (ab, ay + bx + \alpha (xy))$$ for $(a, x), (b, y) \in A(\alpha)$. Then, $A(\alpha)$ forms a commutative ring which we denote by $A(\alpha) = R \overset{\alpha}{\ltimes} I$, and call it [*the quasi-trivial extension of $R$ by $I$ with respect to $\alpha$*]{}. We consider $A(\alpha)$ to be an $R$-algebra via the homomorphism $\xi : R \to A(\alpha), ~a \mapsto (a, 0)$. Therefore, $A(\alpha)$ is a ring extension of $R$, and $A(\alpha)$ is a finitely generated $R$-module, when $I$ is a finitely generated ideal of $R$. Notice that if $\alpha = 0$, then $A(0) = R \ltimes I$ is the ordinary idealization $I$ over $R$, introduced by M. Nagata [@N Page 2], and $[(0) \times I]^2=(0)$ in $A(0)$. If $\alpha = 1$, then $A(1)$ is called in [@marco] the amalgamated duplication of $R$ along $I$, and $$A(1) \cong R \times_{R/I} R, \ (a,i) \mapsto (a, a+i),$$ the fiber product of the two copies of the natural homomorphism $R \to R/I$. Hence, if $R$ is a reduced ring, then so is $A(1)$. Let us note the following. \[lemma 3.1\] Let $(R,\m)$ be a $($not necessarily Noetherian$)$ local ring. Assume that $I \ne R$ or $\alpha \in \m$. Then $A(\alpha)$ is a local ring with maximal ideal $\m \times I$. Let $(a,x) \in A(\alpha) \setminus (\m \times I)$. Then, $a+\alpha x \not\in \m$, since $a \not\in \m$ but $\alpha x \in \m$. Therefore, setting $b = a^{-1}$ and $y = -(a+\alpha x)^{-1}{\cdot}xb$, we get $(a,x)(b,y)=1$ in $A(\alpha)$. Hence, $A(\alpha)$ is a local ring, because $\m \times I$ is an ideal of $A(\alpha)$. When $I=R$, $A(-1)$ is not a local ring, even if $(R,\m)$ is a local ring. In fact, assume that $A(-1)$ is a local ring. Then, because $\m \times R$ is a maximal ideal of $A(-1)$ and $(1,1) \not\in \m \times R$, we have $(1,1)(b,y)= (1,0)$ for some $(b,y) \in A(-1)$, so that $b=1$ and $y+b + (-1){\cdot}1{\cdot}y=0$. This is absurd. In what follows, let $(R,\m)$ be a one-dimensional Cohen-Macaulay local ring with a canonical fractional ideal $K$. We set $S=R[K]$ and ${\mathfrak{c}}= R:S$. Let $T$ be a birational module-finite extension of $R$ (hence $R \subseteq T \subseteq \overline{R}$), and assume that $K \subseteq T$ but $R \ne T$. We set $I = R:T$. Hence, $I = K:T$ by [@GMP Lemma 3.5 (1)], so that $K:I=T$. \[lemma 3.5\] $T/K \cong {\mathrm{K}}_{R/I}$. Hence, $\ell_R(T/K)= \ell_R(R/I)$. Take the $K$-dual of the exact sequence $0 \to I \to R \to R/I \to 0$, and consider the resulting exact sequence $0 \to K \to K:I \to \Ext_R^1(R/I, K) \to 0$. We then have $T/K \cong \Ext_R^1(R/I, K) = {\mathrm{K}}_{R/I}$, since $K :I=T$. Therefore, $\ell_R(T/K) = \ell_R({\mathrm{K}}_{R/I})= \ell_R(R/I)$. Let $\alpha \in R$ and set $A = R \overset{\alpha}{\ltimes} I$. Then, since $I \ne R$, $A$ is a Cohen-Macaulay local ring with $\dim A=1$ and $\n = \m \times I$, the unique maximal ideal (Lemma \[lemma 3.1\]). We are now interested in the question of when $A$ is a $2$-$\AGL$ ring. Notice that we have the extensions $$A \subseteq T \overset{\alpha}{\ltimes} T \subseteq {\mathrm{Q}}(R) \overset{\alpha}{\ltimes} {\mathrm{Q}}(R)= {\mathrm{Q}}(A)$$ of rings. We set $L = T \times K$ in $T \overset{\alpha}{\ltimes} T$. Hence, $L$ is an $A$-submodule of $T \overset{\alpha}{\ltimes} T$, and $A \subseteq L \subseteq \overline{A}$. We begin with the following, which plays a key role in this section. \[3.1\] $L \cong {\mathrm{K}}_A$ and $A[L] = T \overset{\alpha}{\ltimes} T$. Since $I = K:T$, $I^\vee \cong T$ where $(-)^{\vee} = \Hom_R(-, K)$, and we have the natural isomorphism $$\sigma: A^{\vee} = \Hom_R(R \oplus I, K) \overset{\cong}{\to} I^{\vee} \oplus R^\vee \overset{\cong}{\to} T \oplus K=L$$ of $R$-modules. Let $Z = T \oplus T$. Then, the $R$-module $Z$ becomes a $T \overset{\alpha}{\ltimes} T$-module by the following action $$(a, x) \rightharpoonup (b, y) = \left((a+\alpha x)b, ay + bx\right)$$ for each $(a, x) \in T \overset{\alpha}{\ltimes} T$ and $(b, y) \in Z$. It is routine to check that $L$ which is considered inside of $Z$ is an $A$-submodule of $Z$, and that the above $R$-isomorphism $\sigma : A^\vee \to L$ is actually an $A$-isomorphism. We now consider the homomorphism $\psi: T \overset{\alpha}{\ltimes} T \to Z$ of $T \overset{\alpha}{\ltimes} T$-modules defined by $\psi (1) = (1,0)$. Then, $\psi$ is an isomorphism, since $\psi(a,x) = (a+\alpha x, x) $ for each $(a,x) \in T \overset{\alpha}{\ltimes} T$. Notice that for each $(a,x) \in T \overset{\alpha}{\ltimes} T$, $(a,x) \in T \times K$ if and only if $\psi (a,x) \in T \times K$. Therefore, $L$ which is considered inside of $T \overset{\alpha}{\ltimes} T$ is isomorphic to ${\mathrm{K}}_A$, because $L$ which is considered inside of $Z$ is isomorphic to $A^\vee = {\mathrm{K}}_A$. Since $KT=T$, we have $L^n = T \overset{\alpha}{\ltimes} T$ for every $n \ge 2$. Thus, $A[L] = T \overset{\alpha}{\ltimes} T$, since $A[L]= \bigcup_{\ell \ge 1}L^\ell = L^n$ for $n \gg 0$. Let ${\mathrm{r}}_R(I) = \ell_R(\Ext_R^1(R/\m,I))$ denote the Cohen-Macaulay type of the $R$-module $I$. \[type\] ${\mathrm{r}}(A(\alpha))= \mu_R(T)+{\mathrm{r}}(R)= {\mathrm{r}}_R(I) + \mu_R(K/I)$. Hence, the Cohen-Macaulay type of $A(\alpha)$ is independent of the choice of $\alpha \in R$. With the same notation as in Proposition \[3.1\], because $\n L = (\m \times I)(T \times K) = \m T \times \m K$, we have an $R$-isomorphism $L/\n L \cong T/\m T \oplus K/\m K$. Therefore, since $R/\m = A/\n$, ${\mathrm{r}}(A) = \mu_R(T) + {\mathrm{r}}(R)$, which is independent of $\alpha$. Consequently, because $\sum_{f \in \Hom_R(I,K)}f(I)= (K:I)I = TI =I$ where the second equality follows from the fact that $I = K:T$, by [@GKL Theorem 3.3] we get ${\mathrm{r}}(A) = {\mathrm{r}}_R(I) + \mu_R(K/I)$. We now come to the main result of this section. \[3.2b\] With the same notation as above, the following conditions are equivalent. 1. The fiber product $R \times_{R/I}R$ is a $2$-$\AGL$ ring. 2. The idealization $R \ltimes I$ is a $2$-$\AGL$ ring. 3. $A(\alpha)=R \overset{\alpha}{\ltimes} I$ is a $2$-$\AGL$ ring for every $\alpha \in R$. 4. $A(\alpha)=R \overset{\alpha}{\ltimes} I$ is a $2$-$\AGL$ ring for some $\alpha \in R$. 5. $\ell_R(T/K)=2$. 6. $\ell_R(R/I)=2$. We maintain the same notation as in Proposition \[3.1\]. Since $A[L] = T \overset{\alpha}{\ltimes} T$, $A[L]/L\cong T/K$ as an $R$-module, so that $\ell_A(A[L]/L)= \ell_R(T/K)$, because $R/\m = A/\n$. Thus, the assertion readily follows from Proposition \[lemma 3.5\], Theorem \[mainref\], and Proposition \[2.3a\]. \[3.3\] Suppose that $R$ is a $2$-$\AGL$ ring. If $A(\alpha)=R \overset{\alpha}{\ltimes} I$ is a $2$-$\AGL$ ring for some $\alpha \in R$, then $T=S$ and $I={\mathfrak{c}}$. We have $S = R[K] \subseteq T$, since $K \subseteq T$. Therefore, $S=T$, because $\ell_R(T/K)=\ell_R(S/K)=2$ by Theorems \[mainref\] and \[3.2b\]. Choosing $T=S$, we have the following. The equivalence of assertions (2) and (3) covers [@CGKM Theorem 4.2]. We should compare the result with [@GMP Theorem 6.5] for the assertion on AGL rings. \[3.4\] Let $R$ be a one-dimensional Cohen-Macaulay local ring with a canonical fractional ideal $K$ and assume that $R$ is not a Gorenstein ring. We set $S=R[K]$ and ${\mathfrak{c}}= R:S$. Then the following conditions are equivalent. 1. $R \times_{R/{\mathfrak{c}}}R$ is a $2$-$\AGL$ ring. 2. $R \ltimes {\mathfrak{c}}$ is a $2$-$\AGL$ ring. 3. $R$ is a $2$-$\AGL$ ring. We note one example. \[3.5\] Let $k$ be a field and set $R=k[[t^4, t^7, t^9]]$. Then $K=R+Rt^5$, so that $R$ is an AGL ring with ${\mathrm{r}}(R)=2$, because $\m K \subseteq R$ ([@GMP Theorem 3.11]). Hence ${\mathfrak{c}}= \m$. Let $T = k[[t^4, t^5, t^6, t^7]]$. Then, $T = R + Rt^5+Rt^6$, and $I=R:T = (t^7, t^8, t^9)$. Therefore, because $\ell_R(R/I) = 2$, by Theorem \[3.2b\] and Corollary \[type\] $A(\alpha) = R \overset{\alpha}{\ltimes} I$ is a $2$-$\AGL$ ring with ${\mathrm{r}}(A(\alpha))= \mu_R(T) + {\mathrm{r}}(R)=3+2=5$ for every $\alpha \in R$. In particular, $R \times_{R/I} R$ and $R \ltimes I$ are $2$-$\AGL$ rings. Two-generated Ulrich ideals in $2$-${\rm AGL}$ rings ==================================================== In this section, we explore Ulrich ideals in $2$-${\rm AGL} $ rings, mainly two-generated ones. One can find in [@GIK], for arbitrary Cohen-Macaulay local rings of dimension one, a beautiful and complete theory of Ulrich ideals which are not two-generated. First of all, let us briefly recall the definition of Ulrich ideals. The notion of Ulrich ideal was given by [@GOTWY] in arbitrary dimension. Although we will soon restrict our attention on the one-dimensional case, let us give it for arbitrary dimension. So, let $(R, \m) $ be a Cohen-Macaulay local ring with $\dim R=d \ge 0$, and $I$ an $\m$-primary ideal of $R$. We assume that $I$ contains a parameter ideal $Q$ of $R$ as a reduction. ([@GOTWY Definition 1.1])\[2.1\] We say that $I$ is an [*Ulrich ideal*]{} in $R$, if the following conditions are satisfied. 1. $I \ne Q$, but $I^2=QI$. 2. $I/I^2$ is a free $R/I$-module. In Definition \[2.1\], condition $(1)$ is equivalent to saying that the associated graded ring $\gr_I(R) = \bigoplus_{n\ge 0} I^n/I^{n+1}$ is a Cohen-Macaulay ring with ${\mathrm{a}}(\gr_I(R))=1-d$, where ${\mathrm{a}}(\gr_I(R))$ denotes the ${\mathrm{a}}$-invariant of $\gr_I(R)$. Therefore, condition $(1)$ of Definition \[2.1\] is independent of the choice of reductions $Q$ of $I$. When $I=\m$, condition $(2)$ is automatically satisfied, and condition $(1)$ is equivalent to saying that $R$ has minimal multiplicity greater than one. Here let us summarize a few basic result on Ulrich ideals, which we later use in this section. To state them, we need the notion of G-dimension. For the moment, let $R$ be a Noetherian ring. A [*totally reflexive*]{} $R$-module is by definition a finitely generated reflexive $R$-module $G$ such that $\Ext_R^{p}(G,R) = (0)$ and $\Ext^{p}_R(\Hom_R(G,R),R)=(0)$ for all $p >0$. Note that every finitely generated free $R$-module is totally reflexive. The [*Gorenstein dimension*]{} (G-dimension for short) of a finitely generated $R$-module $M$, denoted by ${\rm G}$-${\rm dim}_RM$, is defined as the infimum of integers $n \ge 0$ such that there exists an exact sequence $$0 \to G_n \to G_{n-1} \to \cdots \to G_0 \to M \to 0$$ of $R$-modules with each $G_i$ totally reflexive. A Noetherian local ring $R$ is called G[*-regular*]{}, if every totally reflexive $R$-module is free. This is equivalent to saying that the equality ${\rm G}$-${\rm dim}_RM = \pd_RM$ holds true for every finitely generated $R$-modules $M$ ([@greg]). \[Ulrich\] Let $I$ be an Ulrich ideal in $R$ and set $n = \mu_R(I)$. Then the following assertions hold true. 1. $(n-d){\cdot}{\mathrm{r}}(R/I)= {\mathrm{r}}(R)$. 2. Suppose that there exists an exact sequence $0 \to R \to {\mathrm{K}}_R \to C \to 0$ of $R$-modules where ${\mathrm{K}}_R$ denotes the canonical module of $R$. If $n \ge d+2$, then ${\mathrm{Ann}}_RC \subseteq I$. 3. $n= d+1$ if and only if ${\rm G}$-${\rm dim}_RR/I < \infty$. Let $I$ be an $\m$-primary ideal of $R$, containing a parameter ideal $Q$ of $R$ as a reduction. Assume that $I^2=QI$ and consider the exact sequence $$0 \to Q/QI \to I/I^2 \to I/Q \to 0$$ of $R$-modules. We then have that $I/I^2$ is a free $R/I$-module if and only if so is $I/Q$. If $I^2=QI$ and $\mu_R(I) =d+1$, the latter condition is equivalent to saying that $Q:_RI =I$, that is $I$ is exactly a [*good ideal*]{} in the sense of [@GIW]. It is known by [@GOTWY] that when $R$ is a Gorenstein ring, every Ulrich ideal $I$ in $R$ is $(d+1)$-generated (if it exists), and $I$ is a good ideal of $R$ (see [@GOTWY Lemma 2.3, Corollary 2.6]). Similarly as good ideals, Ulrich ideals are characteristic ideals, but behave very well in their nature ([@GOTWY; @GOTWY2]). The existence of $(d+1)$-generated Ulrich ideals gives a strong influence to the structure of $R$, which we shall confirm in this section. We now be back to the following setting. Let $(R,\m)$ be a Cohen-Macaulay local ring with $\dim R = 1$, and let ${\mathcal{X}}_{R}$ be the set of Ulrich ideals in $R$. In general, it is quite difficult to list up the members of ${\mathcal{X}}_R$ (see, e.g., [@GOTWY]). Here, to grasp what kind of sets ${\mathcal{X}}_R$ is, first of all we explore one example. To do this, we need the following. \[lem3.1\] Let $R$ be a Gorenstein local ring with $\dim R=1$. We denote by ${\mathcal{A}}_R$ the set of birational module-finite extensions $A$ of $R$ such that $A$ is a Gorenstein ring, and set ${\mathcal{A}}_R^0 = \{A \in {\mathcal{A}}_R \mid \mu_R(A)=2\}$. Then, there exist bijective correspondences $${\mathcal{A}}_R^0 \to {\mathcal{X}}_R, \ A \mapsto R:A, \ \ \text{and}\ \ {\mathcal{X}}_R \to {\mathcal{A}}_R^0, \ I \mapsto I:I.$$ Let ${\mathcal{G}}_R$ be the set of ideals $I$ in $R$ such that $I^2=aI$ and $I = (a):_RI$ for some non-zerodivisor $a \in I$. We then have by [@GIK2 Proposition 3.1] a bijective correspondence ${\mathcal{G}}_R \to {\mathcal{A}}_R, ~I \mapsto I:I$. Because ${\mathcal{X}}_R = \{I \in {\mathcal{G}}_R \mid \mu_R(I) = 2\}$ and $I:I = a^{-1}I$ for every $I \in {\mathcal{G}}_R$ and every reduction $(a)$ of $I$, we get $\mu_R(I:I) = \mu_R(I)$, so that $I:I \in {\mathcal{A}}_R^0$ for each $I \in {\mathcal{X}}_R$. Conversely, let $A \in {\mathcal{A}}_R^0$ and write $A = I:I$ with $I \in {\mathcal{G}}_R$. Let $(a)$ be a reduction of $I$. Then, $A = I:I = a^{-1}I$, so that $\mu_R(I) = \mu_R(A) = 2$, while $R:A=I$, because $A= R:I$ by [@GIK2 Proposition 2.5] and $I = R:(R:I)$ (remember that $R$ is a Gorenstein ring). Hence, $I \in {\mathcal{X}}_R$, and the correspondences follow. \[3.2a\] Let $k$ be a field and set $R = k[[t^n, t^{n+1}, \ldots, t^{2n-2}]]~(n \ge 3)$, where $t$ is an indeterminate. Then, $R$ is a Gorenstein ring, and $${\mathcal{X}}_{R} = \begin{cases} \{(t^4, t^6)\} & (n=3), \\ \{(t^4-\alpha t^5, t^6) \mid \alpha \in k \} & (n=4), \\ \ \ \ \emptyset & (n \ge 5). \end{cases}$$ When $n =4$, we have $(t^4-\alpha t^5, t^6)=(t^4-\beta t^5, t^6)$, only if $\alpha =\beta$. Our ring $R$ is a Gorenstein ring, since the numerical semigroup $H =\left<n, n+1, \ldots, 2n-2\right>$ is symmetric ([@HK2]). Therefore, in order to determine the members of ${\mathcal{X}}_R$, by Lemma \[lem3.1\] it suffices to list the members of ${\mathcal{A}}_R^0$, taking $R:A$ for each $A\in {\mathcal{A}}_R^0$. We set $V = k[[t]]$. [(1)]{} [*$($The case where $n=3$$)$*]{} Let $A \in {\mathcal{A}}_R^0$. Then $R \subsetneq A \subsetneq V$, whence $t^5 \in R:\m \subseteq A$, which follows from the fact that the image of $t^5$ in ${\mathrm{Q}}(R)/R$ is a unique socle of ${\mathrm{Q}}(R)/R$ and $(0) \ne A/R \subseteq {\mathrm{Q}}(R)/R$. Therefore $$k[[t^3, t^4, t^5]] \subseteq A \subseteq k[[t^2, t^3]].$$ Hence $A=k[[t^2, t^3]]$, because $k[[t^3, t^4, t^5]]$ is not a Gorenstein ring and $\ell_{k}( k[[t^2, t^3]]/k[[t^3, t^4, t^5]])=1$. It is direct to show $R:A = R: t^2 =(t^4, t^6)$. Hence ${\mathcal{X}}_R=\{(t^4,t^6)\}$. [(2)]{} [*$($The case where $n=4$$)$*]{} Let $A \in {\mathcal{A}}_R^0$. Then, $t^7 \in R:\m \subseteq A$, and $ k[[t^4,t^5, t^6, t^7]] \subseteq A \subseteq k[[t^2, t^3]]$. We have $A \not\subseteq k[[t^3,t^4, t^5]]$. Indeed, if $A \subseteq k[[t^3,t^4, t^5]]$, then $A=k[[t^3, t^4, t^5]]$ or $A= k[[t^4,t^5, t^6, t^7]]$, because $\ell_k(k[[t^3,t^4, t^5]]/k[[t^4,t^5, t^6, t^7]])=1$. This is, however, impossible, since both $k[[t^3, t^4, t^5]]$ and $k[[t^4,t^5, t^6, t^7]]$ are not a Gorenstein rings. Hence $$k[[t^4,t^5, t^6, t^7]] \subsetneq A \subseteq k[[t^2, t^3]], \ \ A \not\subseteq k[[t^3,t^4,t^5]].$$ We choose $\xi \in A$ so that $\xi \not\in k[[t^3,t^4,t^5]]$. Then, since $k[[t^4,t^5, t^6, t^7]] = k + t^4V \subseteq A$, we may assume that $\xi = t^2 + \alpha t^3$ with $\alpha \in k$. Therefore, because $$k[[t^4, t^5, t^6, t^7]] \subsetneq R[\xi]=k[[t^2+\alpha t^3, t^4, t^5, t^6]] \subseteq A \subseteq k[[t^2,t^3]]$$ and $\ell_k(k[[t^2,t^3]]/k[[t^4, t^5, t^6, t^7]]) = 2$, we have $ \ell_k(k[[t^2, t^3]]/R[\xi]) \le 1. $ Hence, $A=R[\xi]$ or $A=k[[t^2, t^3]]$, where $k[[t^2,t^3]] \not\in {\mathcal{A}}_R^0$ since $\mu_R(k[[t^2,t^3]])= 3$. Thus, $A= R[\xi]$, and we have $R:A = R:R[\xi] = R: \xi = (t^4-\alpha t^5, t^6)$. Hence, ${\mathcal{X}}_R=\{(t^4-\alpha t^5, t^6) \mid \alpha \in k\}$, because ${\mathcal{A}}_R^0=\{R[t^2+\alpha t^3] \mid \alpha \in k\}$. [(3)]{} [*$($The case where $n= 2q + 1$ with $q \ge 2$$)$*]{} Assume that ${\mathcal{X}}_R \ne \emptyset$ and choose $I \in {\mathcal{X}}_R$. We set $A = I:I$. Then $A \in {\mathcal{A}}_R^0$. We have $ t^nV \subseteq k[[t^n, t^{n+1}, \ldots, t^{2n-1}]] \subseteq A, $ since the image of $t^{2n -1}$ in ${\mathrm{Q}}(R)/R$ is a unique socle of ${\mathrm{Q}}(R)/R$. We set $ {\mathcal{C}}= A : V = t^cV~(c \ge 0), $ the conductor of $A$. Hence, $c \le n =2q+1$, because $t^nV \subseteq A$. Let $\ell = \ell_k(V/A)$. We then have $2\ell = c$, since $A$ is a Gorenstein ring ([@HK Korollar 3.5]), so that $\ell \le q$. Let $\m_A$ denote the maximal ideal of $A$. Then, $(\m_A/\m A)^2=(0)$, since $\ell_k(A/\m A)=\ell_R(A/\m A) = \mu_R(A)=2$. We look at the chain $$A/{\m A} \supsetneq \m_A/\m A \supsetneq (\m_A/\m A)^2 = (0)$$ of ideals in the ring $\overline{A}=A/\m A$, and take $\xi \in \m_A$, so that $\m_A/\m A = (\overline{\xi})$ (here $\overline{\xi}$ denotes the image of $\xi$ in $\overline{A}$). Then, $\overline{\xi} \neq 0$, but ${\overline{\xi}}^2 = 0$ in $\overline{A}$. Consequently, $ \xi^2 \in \m A \subseteq t^{n}V$ and $A = R + R\xi$, since $A/{\mathfrak{m}}A = k + k \overline{\xi}$. Therefore, $ 2{\rm \nu}(\xi) \ge n = 2q+1, $ so that ${\rm \nu}(\xi) \ge q+1$ (here $\nu(*)$ denotes the valuation of $V$). Thus, $ A = R + R\xi \subseteq k[[t^{q+1}, t^{q+2}, \ldots, t^{2q+1}]], $ whence $A = k[[t^{q+1}, t^{q+2}, \ldots, t^{2q+1}]]$, because $\ell_k(V/k[[t^{q+1}, t^{q+2}, \ldots, t^{2q+1}]]) = q$ and $\ell_k(V/A) =\ell\le q$. This is, however, impossible, since $A$ is a Gorenstein ring but $k[[t^{q+1}, t^{q+2}, \ldots, t^{2q+1}]]$ is not. Thus ${\mathcal{X}}_R = \emptyset$. [(4)]{} [*$($The case where $n = 2q$ with $q \ge 3$$)$*]{} Assume that ${\mathcal{X}}_R \ne \emptyset$ and choose $I \in {\mathcal{X}}_R$. We set $A = I:I$. We then have $t^{2n-1} \in A$, considering the image of $t^{2n-1}$ in ${\mathrm{Q}}(R)/R$. We set $\ell = \ell_k(V/A)$ and ${\mathcal{C}}= A : V$. Then ${\mathcal{C}}= t^{2\ell}V$, since $A$ is a Gorenstein ring. Therefore, $\ell \le q$, because $t^n V \subseteq A$ and $n = 2q$. On the other hand, considering the chain $$A/{\m A} \supsetneq \m_A/\m A \supsetneq (\m_A/\m A)^2 = (0)$$ of ideals in the ring $\overline{A}=A/{\m A}$ and taking $\xi \in \m_A$ so that $\m_A/\m A= (\overline{\xi})$, we get $\overline{\xi} \neq 0$ and ${\overline{\xi}}^2 = 0$ in $\overline{A}$. Therefore, $ \xi^2 \in \m A \subseteq t^{n}V \ \ \text{and} \ \ A = R + R\xi$, because $A/{\mathfrak{m}}A = k + k \overline{\xi}$. Consequently, $2{\rm \nu}(\xi) \ge n = 2q$. Hence, ${\rm \nu}(\xi) \ge q$, so that $$A= R + R\xi\subsetneq k[[t^q, t^{q+1}, \ldots, t^{2q-1}]] \subseteq V,$$ where the strictness of the first inclusion follows from the fact that $k[[t^q, t^{q+1}, \ldots, t^{2q-1}]]$ is not a Gorenstein ring. Therefore, because $\ell_k(V/A)= \ell$ and $\ell_k(V/k[[t^q, t^{q+1}, \ldots, t^{2q-1}]]) = q-1$, we get $q-1 < \ell$, whence $\ell = q$. We set $T = k[[t^{2q}, t^{q+1}, \ldots, t^{4q-1}]]$. Then $\ell_k(A/T) = q -1$, since $\ell_k(V/T)= 2q-1$. Because $$A/T \subseteq V/T = k\overline{t} + k\overline{t^2} + \cdots + k\overline{t^{2q-1}},$$ where $\overline{*}$ denotes the image in $V/T$, we obtain elements $\xi_1, \xi_2, \ldots, \xi_{q-1} \in \sum_{i=1}^{2q-1}kt^i$ so that $A = T + \sum_{i=1}^{q-1} k\xi_i$. Therefore $$A = T[\xi_1, \xi_2, \ldots, \xi_{q-1}] = k[[\xi_1, \xi_2, \ldots, \xi_{q-1}, t^{2q}, \ldots, t^{4q-1}]],$$ whence $\xi_1, \xi_2, \ldots, \xi_{q-1} \in \m_A$ and $(\overline{\xi_1}, \overline{\xi_2}, \ldots, \overline{\xi}_{q-1}) \subseteq \m_A/\m A$. We now notice that if $\sum_{i=1}^{q-1} a_i\xi_i \in \m A$ with $a_i \in k$, then $\sum_{i=1}^{q-1} a_i\xi_i \in t^{2q}V$, whence $a_i=0$ for all $1\le i \le q-1 $. Therefore, $1 = \ell_k(\m_A/\m A) \ge q-1 \ge 2$. This is a required contradiction. Let us make sure of the last assertion. Suppose that $n=4$ and $(t^4-\alpha t^5, t^6) = (t^4-\beta t^5, t^6)$ where $\alpha, \beta \in k$. We write $t^4- \alpha t^5 = f (t^4- \alpha t^5) + g t^6$ for some $f, g \in R$. By setting $f = c_0 + c_1t^4+c_2t^5+c_3t^6 + \eta$, $g = d_0+ d_1t^4 + d_2t^5+d_3t^6 + \xi$ for some $c_i, d_j \in k$ and $\eta, \xi \in t^8V$, we then have $t^4-\alpha t^5 = c_0t^4 -\beta c_0t^5 + d_0 t^6 + \text{(higher terms)}$. Hence, $c_0=1$ and $\alpha = \beta c_0 = \beta$, as desired. Let us give here simple examples of $2$-AGL rings, which contain numerous two-generated Ulrich ideals. \[2.7\] Let $(R, \m)$ be an ${\rm AGL}$ ring with $\dim R=1$ and suppose that $R$ is not a Gorenstein ring, say $R= k[[t^3,t^4,t^5]]$, the semigroup ring of $H = \left<3,4,5\right>$ over a field $k$. Let $\alpha \in \m$ and consider the quasi-trivial extension $A = R \overset{\alpha}{\ltimes} R$ of $R$ with respect to $\alpha$ (see Section 3) Then, $A$ is a $2$-AGL ring by [@CGKM Theorem 3.10], because $A$ is a free $R$-module with $\ell_R(A/\m A) = 2$. Let ${\mathfrak{q}}$ be a parameter ideal of $R$ and assume that $\alpha \in {\mathfrak{q}}$. We set $I = {\mathfrak{q}}\times R$. Then, $I$ is an Ulrich ideal of $A$ with $\mu_A(I)=2$. Therefore, if $\alpha = 0$, then ${\mathfrak{q}}\times R$ is an Ulrich ideal of $A$ for every parameter ideal ${\mathfrak{q}}$ of $R$ ([@GOTWY Example 2.2]). Let ${\mathfrak{q}}= (a)$ and set $f = (a,0) \in I$. Then, $I^2 = fI$, since $I^2 = (a^2) \times (aR + \alpha R) = (a^2) \times aR=fI$. Note that $I/fA = [(a) \oplus R]/[(a) \oplus (a)] \cong R/(a)$ and $A/I = [R \oplus R]/[(a) \oplus R] \cong R/(a)$ as $R$-modules. We then have $\ell_A(A/I) =\ell_A(I/fA) = \ell_R(R/(a))$. Hence, $A/I \cong I/fA$ as an $A$-module, because $I/fA$ is a cyclic $A$-module. Thus, $I \in {\mathcal{X}}_A$ with $\mu_A(I)=2$. Two-generated Ulrich ideals are totally reflexive $R$-modules (Proposition \[Ulrich\] (3)), possessing minimal free resolutions of a very restricted form. Let us note the following, which we need to prove Theorem \[2.3\]. We include a brief proof for the sake of completeness. \[2.2\] Suppose that $I$ is an Ulrich ideal of $R$ and assume that $\mu_R(I)=2$. We write $I=(a, b)$ with $(a)$ a reduction of $I$. Therefore, $b^2 = ac$ for some $c \in I$. With this notation, the minimal free resolution of $I$ is given by $$\Bbb F : \ \ \cdots \to R^2 \overset{ \begin{pmatrix} -b & -c\\ a & b \end{pmatrix}}{\longrightarrow} R^2 \overset{ \begin{pmatrix} -b & -c\\ a & b \end{pmatrix}}{\longrightarrow}R^2 \overset{\begin{pmatrix} a & b \end{pmatrix}}{\longrightarrow} I \to 0,$$ Hence $\pd_RI= \infty$. The ideal $I$ is so called a totally reflexive $R$-module, because $I$ is reflexive, $\Ext_R^p(I, R) =(0)$, and $\Ext_R^p(\Hom_R(I,R), R) = (0)$ for all $p >0$. Here we don’t assume that $R$ is a Gorenstein ring, but the proof given in [[@GOTWY Example 7.3]]{} still works to get the minimal free resolution $\Bbb F$ of $I$. Since $$I= (a):_RI = (a) : I = a(R:I),$$ we have $I \cong R:I \cong I^*$, where $I^* = \Hom_R(I,R)$. Note that the $R$-dual $\Bbb F^*$ of $\Bbb F$ remains exact. In fact, assume that $\left(\begin{smallmatrix} -b&a\\ -c&b \end{smallmatrix}\right)\left(\begin{smallmatrix} x\\ y \end{smallmatrix}\right)=0$. Then, since $-bx + ay = 0$, we have $\left(\begin{smallmatrix} -y\\ x \end{smallmatrix}\right)=\left(\begin{smallmatrix} -b&-c\\ a&b \end{smallmatrix}\right)\left(\begin{smallmatrix} f\\ g \end{smallmatrix}\right)$ for some $f,g \in R$. Therefore, $\left(\begin{smallmatrix} x\\ y \end{smallmatrix}\right)=\left(\begin{smallmatrix} -b&a\\ -c&b \end{smallmatrix}\right)\left(\begin{smallmatrix} -g\\ f \end{smallmatrix}\right)$, which shows that $\Bbb F^*$ is exact, because $\left(\begin{smallmatrix} -b&a\\ -c&b \end{smallmatrix}\right)^2=0$. Consequently, $\Ext_R^p(I,R)=(0)$ for all $p > 0$, whence $\Ext_R^p(I^*,R)=(0)$ for all $p >0$, because $I \cong I^*$. On the other hand, by the above argument we have an exact sequence $$0 \to I \to R^{\oplus 2} \to I \to 0$$ whose $R$-dual $0 \to I^* \to R^{\oplus 2} \to I^* \to 0$ remains exact. Therefore, $I$ is a reflexive $R$-module. Thus, $I$ is a totally reflexive $R$-module. We now start the analysis of the question of how many two-generated Ulrich ideals are contained in a given $2$-AGL ring. Let $K$ be a canonical fractional ideal of $R$. Let $S =R[K]$ and set ${\mathfrak{c}}= R : S$. We then have the following, which shows the existence of two-generated Ulrich ideals is a substantially strong restriction. \[2.3\] Suppose that $R$ is a $2$-${\rm AGL}$ ring and let $K$ be a canonical fractional ideal of $R$. Let ${\mathfrak{c}}= (x_1^2) + (x_2, x_3, \ldots, x_n)$ with a minimal system $x_1, x_2, \ldots, x_n$ of generators of $\m$. Assume that $R$ contains an Ulrich ideal $I$ with $\mu_R(I)=2$. Then the following assertions hold true. 1. $K/R$ is a free $R/{\mathfrak{c}}$-module. 2. $I + {\mathfrak{c}}= \m$. 3. ${\mathfrak{c}}= (x_2, x_3, \ldots, x_n)$. Consequently, $\mu_R({\mathfrak{c}}) =n-1$, and $x_1^2 \in (x_2, x_3, \ldots, x_n)$. \(1) We have $K/R \cong (R/{\mathfrak{c}})^{\oplus \ell} \oplus (R/\m)^{\oplus m}$ with $\ell > 0, m \ge 0$ such that $\ell +m = {\mathrm{r}}(R)+1$ (Proposition \[2.3a\] (4)). To show assertion $(1)$, let us assume that $m>0$. Then, since $I$ is totally reflexive (Proposition \[2.2\]) and $\Ext_R^p(I,K)= (0)$ for every $p > 0$, we get $\Ext_R^p(I, K/R)=(0)$, so that $$\Ext_R^p(I, R/\m)=(0) \ \ \text{for all} \ \ p>0,$$ because $R/\m$ is a direct summand of $K/R$. This is impossible, since $\pd_RI=\infty$. Hence, $m=0$, and $K/R$ is $R/{\mathfrak{c}}$-free. \(2) Let us use the same notation as in Proposition \[2.2\]. Hence, $I$ has a minimal free resolution of the form $$\Bbb F : \ \ \cdots \to R^2 \overset{ \begin{pmatrix} -b & -c\\ a & b \end{pmatrix}}{\longrightarrow} R^2 \overset{ \begin{pmatrix} -b & -c\\ a & b \end{pmatrix}}{\longrightarrow}R^2 \overset{\begin{pmatrix} a & b \end{pmatrix}}{\longrightarrow} I \to 0.$$ Remember that $\Ext_R^p(I, R/{\mathfrak{c}}) = (0)$ for all $p >0$, because $\Ext_R^p(I,K)= \Ext^p_R(I,R)= (0)$ for all $p >0$ and $R/{\mathfrak{c}}$ is a direct summand of $K/R$. Let $\overline{x}$ denote, for each $x \in R$, the image of $x$ in $R/{\mathfrak{c}}$. Then, taking the $R/{\mathfrak{c}}$-dual of the resolution $\Bbb F$, we get the exact sequence $$(E)\ \ \ 0 \to \Hom_R(I, R/{\mathfrak{c}}) \to (R/{\mathfrak{c}})^{\oplus 2} \overset{ \begin{pmatrix} -\overline{b} & \overline{a}\\ -\overline{c} & \overline{b} \end{pmatrix}}{\longrightarrow} (R/{\mathfrak{c}})^{\oplus 2} \overset{ \begin{pmatrix} -\overline{b} & \overline{a}\\ -\overline{c} & \overline{b} \end{pmatrix}}{\longrightarrow} (R/{\mathfrak{c}})^{\oplus 2} \to \cdots,$$ which shows that $I \nsubseteq {\mathfrak{c}}$. Therefore, $I+{\mathfrak{c}}=\m$, since $\ell_R(R/{\mathfrak{c}}) = 2$. \(3) To show assertion $(3)$, we notice that $\m/{\mathfrak{c}}= (\overline{x_1}) = (\overline{a}, \overline{b})$, and $\ell_R(\m/{\mathfrak{c}})=1$. Hence $\m^2 \subseteq {\mathfrak{c}}$. We set $J = (x_2, x_3, \ldots, x_n)$, and consider the following two cases. [Case 1 ($\overline{a} \ne 0$).]{} Let us write $a = \alpha x_1 + \xi$ for some $\alpha \in R$ and $\xi \in J$. Then, since $\overline{a} \ne 0$, $\alpha \notin \m$ and $\overline{b} \in {\mathfrak{m}}/{\mathfrak{c}}= (\overline{a})$. Let $b = \beta a + \gamma$ with $\beta \in R$ and $\gamma \in {\mathfrak{c}}$. Then, $I=(a,b) = (a, \gamma)$, whence replacing $b$ with $\gamma$, we may assume that $\alpha=1$ and $b \in {\mathfrak{c}}$. Therefore, $ \left(\begin{smallmatrix} -\overline{b} & \overline{a}\\ -\overline{c} & \overline{b} \end{smallmatrix}\right)= \left(\begin{smallmatrix} 0 & \overline{x_1}\\ -\overline{c} & 0 \end{smallmatrix}\right), $ so that $\overline{c} \ne 0$ by the exactness of the sequence ($E$). Consequently, writing $c= \delta x_1 + \rho$ with $\delta \notin \m$ and $\rho \in J$, we have $ \delta x_1^2 \equiv ac = b^2 \equiv 0 \ \text{mod} \ J. $ Hence $x_1^2 \in J$, so that ${\mathfrak{c}}=J$, as claimed. [Case 2 ($\overline{a} = 0$).]{} Let $a = \alpha x_1^2 + \beta$ with $\alpha \in R$ and $\beta \in J$. Let us write $b = \gamma x_1 + \delta$ with $\gamma \in R$ and $\delta \in J$. Then, since $\m/{\mathfrak{c}}= (\overline{b}) \ne (0)$, we get $\gamma \not\in \m$. Let $c= \rho x_1 + \eta$ with $\rho \in R$ and $\eta \in J$. Then, since $b^2 = ac$, we have $ \gamma^2 x_1^2 \equiv \alpha \rho x_1^3 \ \text{mod} \ J. $ Hence, $x_1^2 \in J$, and ${\mathfrak{c}}= J$. Suppose that $(R,\m)$ is a $2$-${\rm AGL}$ ring with infinite residue class field. Let $I$ be an Ulrich ideal $I$ in $R$ with $\mu_R(I)=2$. Then, there exists a minimal system $x_1, x_2, \ldots, x_n$ of generators of $\m$ and $b \in {\mathfrak{c}}$ such that ${\mathfrak{c}}= (x_2, x_3, \ldots, x_n)$ and $I = (x_1, b)$ with $I^2 = x_1I$. Choose a minimal system $x_1, x_2, \ldots, x_n$ of generators of $\m$ such that ${\mathfrak{c}}= (x_1^2) + (x_2, x_3, \ldots, x_n)$. Then, ${\mathfrak{c}}= (x_2, x_3, \ldots, x_n)$ by Theorem \[2.3\]. We write $I = (a,b)$ where both the ideals $(a), (b)$ are reductions of $I$. If $a \not\in {\mathfrak{c}}$, then since $\m/{\mathfrak{c}}= (\overline{a})$ where $\overline{*}$ denotes the image in $\m/{\mathfrak{c}}$, we get $b = \alpha a + \beta$ with $\alpha \in R$ and $\beta \in {\mathfrak{c}}$. Hence $I = (a,b) = (a, \beta)$ and $\m = (a) + {\mathfrak{c}}$. If $a \in {\mathfrak{c}}$, then $\m/{\mathfrak{c}}= (\overline{b})$, so that $a = \alpha b + \beta$, whence $I = (\beta, b)$. The following result is involved in [@GTT2 Theorem 2.8], since Cohen-Macaulay local rings of minimal multiplicity are G-regular ([@greg]). Let us give a brief proof in our context. \[1.7\] Let $R$ be a $2$-${\rm AGL}$ ring and let $K$ be a canonical fractional ideal of $R$. Assume that $S=R[K]$ is a Gorenstein ring. If $R$ has minimal multiplicity, then $R$ contains no two-generated Ulrich ideals. Because $R=K:K$ ([@HK Bemerkung 2.5]) and $KS = S$, $${\mathfrak{c}}= R : S= K :S \cong \Hom_R(S,K).$$ Therefore, since $S$ is a Gorenstein ring, we have ${\mathfrak{c}}\cong S$, so that $n-1 = \mu_R({\mathfrak{c}}) = \mu_R(S) = {\mathrm{r}}(R) +1, $ where the first (resp. third) equality follows from Theorem \[2.3\] (resp. Proposition \[2.3a\] (4)). Thus, $R$ doesn’t have minimal multiplicity, because ${\mathrm{r}}(R) = n-1$ otherwise. The condition that ${\mathfrak{c}}\in {\mathcal{X}}_R$ is a strong restriction on $2$-${\rm AGL}$ rings $R$. We need the following, in order to see that $2$-AGL rings might contain Ulrich ideals, which are not two-generated. \[1.6\] Suppose that $R$ is a $2$-${\rm AGL}$ ring, possessing a canonical fractional ideal $K$. Let $S = R[K]$ and set ${\mathfrak{c}}= R:S$. Then the following conditions are equivalent. 1. ${\mathfrak{c}}\in {\mathcal{X}}_R$. 2. $S$ is a Gorenstein ring and $K/R$ is a free $R/{\mathfrak{c}}$-module. \(2) $\Rightarrow$ (1) Since ${\mathfrak{c}}= K:S \cong \Hom_R(S,K)$, we have ${\mathfrak{c}}= fS$ for some $f \in S$, whence ${\mathfrak{c}}^2 = f {\mathfrak{c}}$. Therefore, ${\mathfrak{c}}/{\mathfrak{c}}^2$ is a free $R/{\mathfrak{c}}$-module if and only if so is $S/R$, because ${\mathfrak{c}}/fR \cong S/R$. The latter condition is equivalent to saying that $K/R$ is a free $R/{\mathfrak{c}}$-module, which follows from the exact sequence $$0 \to R/{\mathfrak{c}}\to S/{\mathfrak{c}}\to S/R \to 0$$ and the fact that $S/R \cong K/R \oplus R/{\mathfrak{c}}$ (Proposition \[2.3a\] (3)). \(1) $\Rightarrow$ (2) By [@GMP Corollary 3.8], $S$ is a Gorenstein ring, since ${\mathfrak{c}}^2 = f {\mathfrak{c}}$ for some $f \in {\mathfrak{c}}$. Therefore, ${\mathfrak{c}}= fS$ for some $f \in {\mathfrak{c}}$, since ${\mathfrak{c}}= K:S$. Thus, ${\mathfrak{c}}/{\mathfrak{c}}^2 \cong S/fS = S/{\mathfrak{c}}$, whence $S/{\mathfrak{c}}$ is a free $R/{\mathfrak{c}}$-module. Consequently, $K/R$ is a free $R/{\mathfrak{c}}$-module, since $S/R \cong K/R \oplus R/{\mathfrak{c}}$. Let us explore an example, which shows the set ${\mathcal{X}}_R$ depends on the characteristic of the base fields. For the ring stated in Example \[difficult\], we have the complete list of Ulrich ideals in it. \[difficult\] Let $V = k[[t]]$ be the formal power series ring over a field $k$ and set $R = k[[t^6,t^8,t^{10},t^{11}]]$. Then the following assertions hold true. 1. $R$ is a $2$-AGL ring with ${\mathrm{r}}(R) = 2$ and $S= k[[t^2, t^{11}]]$ is a Gorenstein ring with ${\mathfrak{c}}= (t^6, t^8, t^{10}) \in {\mathcal{X}}_R$. We have ${\mathfrak{c}}= (x_2,x_3,x_4)$ and $x_1^2 \in {\mathfrak{c}}$, where $x_1 =t^{11}, x_2=t^6, x_3=t^8$, and $x_4=t^{10}$. 2. Let $I \in {\mathcal{X}}_R$ and set $n = \mu_R(I)$. Then, $n =2, 3$, and $n=3$ if and only if $I = {\mathfrak{c}}$. 3. If $\ch k \ne 2$, then the set of two-generated Ulrich ideals is $$\left\{(t^6 + c_1 t^8 + c_2 t^{10}, t^{11}) \mid c_1, c_2 \in k\right\} \cup \left\{(t^8+c_1t^{10} + c_2 t^{12}, t^{11})\mid c_1, c_2 \in k\right\}$$ and we have the following. 1. $(t^6 + c_1 t^8 + c_2 t^{10}, t^{11})=(t^6 + d_1 t^8 + d_2 t^{10}, t^{11})$, only if $c_1 =d_1$ and $c_2 =d_2$. 2. $(t^8+c_1t^{10} + c_2 t^{12}, t^{11}) = (t^8+d_1t^{10} + d_2 t^{12}, t^{11})$, only if $c_1 =d_1$ and $c_2 =d_2$. 4. If $\ch k = 2$, then the set of two-generated Ulrich ideals is $$\begin{aligned} &&\left\{(t^6 + c_1 t^8 + c_2 t^{10}, t^{11}) \mid c_1, c_2 \in k\right\} \cup \left\{(t^8+c_1t^{10} + c_2 t^{12}, t^{11} +d t^{12})\mid c_1, c_2, d \in k\right\} \\ && \cup \left\{(t^6 + c_1t^8+c_2t^{11}, t^{10} + d t^{11})\mid c_1, c_2, d \in k, d \ne 0\right\}\end{aligned}$$ and we have the following. 1. $(t^6 + c_1 t^8 + c_2 t^{10}, t^{11})=(t^6 + d_1 t^8 + d_2 t^{10}, t^{11})$, only if $c_1 =d_1$ and $c_2 =d_2$. 2. $(t^8+c_1t^{10} + c_2 t^{12}, t^{11} +d t^{12}) = (t^8+d_1t^{10} + d_2 t^{12}, t^{11} +e t^{12})$, only if $c_1 =d_1$, $c_2 =d_2$, and $d=e$. 3. $(t^6 + c_1t^8+c_2t^{11}, t^{10} + d t^{11})= (t^6 + d_1t^8+d_2t^{11}, t^{10} + e t^{11})$, only if $c_1 =d_1$, $c_2 =d_2$, and $d=e$. 5. The Ulrich ideals in $R$ generated by monomials in $t$ are $\left\{(t^6, t^{11}), (t^8, t^{11}), {\mathfrak{c}}= (t^6,t^{8},t^{10})\right\}$. \(1) Because $K = R + Rt^2$, we have ${\mathrm{r}}(R)=2$ and $S= k[[t^2, t^{11}]]$, so that $S$ is a Gorenstein ring, and ${\mathfrak{c}}= (t^6,t^{8},t^{10})$, since $S = R + Rt^2 + Rt^4$. We have $\ell_R(S/K) = 2$, since $S/K = k{\cdot}\overline{t^4} + k{\cdot}\overline{t^{15}}$, where $\overline{t^4}$ and $\overline{t^{15}}$ denote the images of $t^4$ and $t^{15}$ in $S/K$, respectively. Therefore, $R$ is a $2$-AGL ring by Theorem \[mainref\]. Because $K/R$ is a cyclic $R$-module, $K/R \cong R/{\mathfrak{c}}$, whence ${\mathfrak{c}}\in {\mathcal{X}}_R$ by Proposition \[1.6\]. \(2) Since $(n-1){\cdot}{\mathrm{r}}(R/I) = {\mathrm{r}}(R)=2$ by Proposition \[Ulrich\] (1), we get $n = 2,3$. Suppose $n = 3$. Then, ${\mathfrak{c}}\subseteq I$ by Proposition \[Ulrich\] (2), since $K/R \cong R/{\mathfrak{c}}$. On the other hand, if ${\mathfrak{c}}\subsetneq I$, we then have by [@GIK Theorem 3.1] ${\mathfrak{c}}= bcS$ for some $b, c \in \m$. This is, however, impossible, because ${\mathfrak{c}}= t^6S$ and $b,c \in \m \subseteq t^6V$. Therefore, $I= {\mathfrak{c}}$, if $n=3$. (3), (4) We denote by $\nu(*)$ the valuation of $V$. Let $I \in {\mathcal{X}}_R$ and suppose that $\mu_R(I) =2$. Let us write $I=(a, b)$ where $a, b \in R$. First we may assume $I^2 = a I$ and $\nu(a) < \nu(b)$. We then have $\nu(a) < 11$. Indeed, if $\nu(a) \ge 12$, then $a, b \in {\mathfrak{c}}=(t^6, t^8, t^{10})$, so that $I \subseteq {\mathfrak{c}}$, which is absurd (remember that $I + {\mathfrak{c}}= \m$). Besides, we notice that $\nu(a)$ is even, because $I/(a) \cong R/I$ as an $R$-module. Therefore, $\nu(a) = 6, 8, 10$. In addition, we have the following. \[4.12\] The following assertions hold true. 1. If $\nu(a) = 6$, then $\nu(b) = 10, 11$. 2. If $\nu(a) = 8$, then $\nu(b) = 11$. 3. One has $\nu(a) \ne 10$. 4. If $\ch k \ne 2$, then $(\nu(a), \nu(b)) \ne (6, 10)$. $(i)$ We first consider the case where $\nu(a) = 6$. Then we get $\nu(b) <12$. In fact, if $\nu(b) \ge 12$, then the images of $1, t^8, t^{10}, t^{11}$ in $R/I$ are linearly independent over the field $k$, so that $\ell_R(R/I) \ge 4$. This makes a contradiction, because $I/(a) \cong R/I$. Hence $\nu(b) \le 11$. We are now assuming that $\nu(b) = 8$. Since $b^2 = ac$ for some $c \in I$, we notice that $\nu(c) = 10$. Let us write $c = a \rho + b \eta$ where $\rho, \xi \in \m$. We then have $c \in t^{12}V$, which is impossible. Consequently, $\nu(b) = 10, 11$ as claimed. $(ii)$ Suppose that $\nu(a) = 8$. By setting $b^2 = a^2 \varphi + ab \psi$ for some $\varphi, \psi \in \m$, we have $\nu(b) \ne 10$. Let us write $a = t^8 + \alpha t^{10} + \beta t^{11} + \xi$, where $\alpha, \beta \in k$ and $\xi \in R$ with $\nu(\xi) \ge 12$. If $\nu(b) \ge 12$, then $b \in {\mathfrak{c}}$, so that $a \notin {\mathfrak{c}}$, because $I+{\mathfrak{c}}= {\mathfrak{m}}$. Hence $\beta \ne 0$ (remember that ${\mathfrak{c}}=(t^6, t^8, t^{10})$). Therefore, if $\nu(b) \ge 14$ (resp. $\nu(b) = 12$), then the images of $1, t^6, t^8, t^{10}, t^{12}$ (resp. $1, t^6, t^8, t^{10}, t^{14}$) in $R/I$ are linearly independent over $k$, so that $\ell_R(R/I) \ge 5$, which makes a contradiction, because $R/I \cong I/(a)$. Hence $\nu(b) = 11$. $(iii)$ Let us assume that $\nu(a) = 10$. Since $b^2 \in (a^2, ab)$, we have $\nu(b) \ne 11, 12$, whence $\nu(b) \ge 14$. Thus $b \in {\mathfrak{c}}$ and $a \notin {\mathfrak{c}}$. Then the images of $1, t^6, t^8, t^{10}, t^{14}, t^{16}$ in $R/I$ are linearly independent over $k$, which is absurd. $(iv)$ Suppose that $\nu(a) = 6$ and $\nu(a) = 10$. We may assume $a = t^6 + c_1t^8 + c_2t^{11} + c_3 t^{19}$ and $b= t^{10} + d_1t^{11} + d_2 t^{19}$, where $c_i, d_ j \in k$. Look at the isomorphism $R/I \cong k[Y, W]/{\mathfrak{a}}$, where ${\mathfrak{a}}$ is the ideal of $k[Y, W]$ generated by [$$(-c_1Y -c_2 W-c_3YW)^3 - Y(-d_1 W -d_2YW), Y^2 -(-c_1 Y -c_2 W -c_3 YW)(-d_1W -d_2YW),$$ $$(-d_1W - d_2 YW)^2 - (-c_1Y -c_2 W-c_3YW)^2Y, \ \ \text{and} \ \ W^2 -(-c_1Y -c_2 W-c_3YW)^2(-d_1W - d_2 YW).$$]{} Hence $(Y, W)^3 + {\mathfrak{a}}= (Y, W)^3 + (Y^2, d_1YW, W^2)$. If $d_1 = 0$, then $\ell_R(R/I) \ge 4$, which is impossible. Therefore $d_1 \ne 0$. Since $I^2 = a I$, we can write $b^2 = a^2 \varphi + ab \psi$ for some $\varphi, \psi \in \m$. By comparing the coefficients of $t^{21}$, we have $2d_1 = 0$, so that $\ch k =2$. Consequently, if $\ch k \ne 2$, then $(\nu(a), \nu(b)) \ne (6, 10)$, as desired. Notice that, for each $0 \ne f \in R$, we have $t^{n+16}V \subseteq (f)$, where $n =\nu(f)$. It follows from the equalities $t^{n+16}V = fV {\cdot} t^{16}V = f{\cdot} (R:V)$ and the fact that $(R:V)$ is an ideal of $R$. \(3) Suppose that $\ch k \ne 2$. First we consider the case where $\nu(a) = 6$ and $\nu(b) = 11$. Then $t^{33}V \subseteq (ab)$, so that $I = (t^6 + c_1 t^8 + c_2 t^{10}, t^{11})$ for some $c_1, c_2 \in k$. On the other hand, if we set $J = (t^6 + c_1 t^8 + c_2 t^{10}, t^{11})$ with $c_1, c_2 \in k$, then $J$ is an Ulrich ideal of $R$. Let $a = t^6 + c_1 t^8 + c_2 t^{10}$. Notice that $t^n \in a J$ for each even integer $n \ge 18$, because $t^n = t^{n-12}{\cdot}a^2 - t^{n-12}{\cdot} (c_1^2t^{16} + \cdots + c_2^2 t^{20})$. Therefore, $J^2 = a J + (t^{22}) = a J$. Moreover, we have the isomorphism $R/J \cong k[Y, Z]/{\mathfrak{a}}$, where $${\mathfrak{a}}= \left((-c_1 Y-c_2Z)^3 -YZ, Y^2 - (-c_1 Y-c_2Z)Z, Z^2 -(-c_1 Y-c_2Z)^2Y, (-c_1 Y-c_2Z)Z\right)$$ which yields $\ell_R(R/J) = 3$, because ${\mathfrak{a}}+ (Y, Z)^3 = (Y, Z)^2$. Hence $R/J \cong J/(a)$, so that $J \in {\mathcal{X}}_R$. Let us assume $\nu(a) = 8$ and $\nu(b) = 11$. We may assume $a = t^8 + c_1t^{10} + c_2t^{12}$ and $b = t^{11} + dt^{12}$ where $c_1, c_2, d \in k$. The equality $I^2 = aI$ yields that $2d=0$ by comparing the coefficients of $t^{23}$. Hence $d=0$. Conversely, let $J = (t^8 + c_1t^{10} + c_2t^{12}, t^{11})$ for some $c_1, c_2 \in k$. Then $t^n \in a J$ for each even integer $n \ge 22$, where $a = t^8 + c_1t^{10} + c_2t^{12}$. We have the isomorphism $R/J \cong k[X, Z]/{\mathfrak{a}}$, where $${\mathfrak{a}}= \left(X^3 - (-c_1 Z-c_2X^2)Z, (-c_1 Z-c_2X^2)^2 - XZ, Z^2 -X^2(-c_1 Z-c_2X^2), -X^2Z\right)$$ while ${\mathfrak{a}}= (X^3, Z^2, X^2Z, XZ) = (X^3, XZ, Z^2)$. Therefore, $\ell_R(R/J)=4$ and $J \in {\mathcal{X}}_R$. The last assertions follow from the same technique as in the proof of Example \[3.2a\]. \(4) Suppose that $\ch k = 2$. Thanks to the proof of (3), if $\nu(a) = 6, \nu(b) = 11$ (resp. $\nu(a) = 8, \nu(b) = 11$), then we have $I = (t^6 + c_1 t^8 + c_2 t^{10}, t^{11})$ (resp. $I = (t^8 + c_1 t^{10} + c_2 t^{12}, t^{11} + dt^{12})$) where $c_1, c_2 \in k$ (resp. $c_1, c_2, d \in k$). Let us assume $\nu(a) = 6$ and $\nu(b) = 10$. We then have $I = (t^6 + c_1 t^8 + c_2 t^{11} + c_3 t^{19}, t^{10} + d_1t ^{11} + d_2t^{19})$ for some $c_1, c_2, c_3, d_1, d_2 \in k$. Consider the same isomorphism $R/I \cong k[Y, W]/{\mathfrak{a}}$ as in the proof of Claim \[4.12\] $(iv)$. Then $(Y, W)^3 + {\mathfrak{a}}= (Y, W)^3 + (Y^2, d_1YW, W^2)$. Since $I \in {\mathcal{X}}_R$, we have $\ell_R(R/I) = 3$, whence $d_1 \ne 0$ and ${\mathfrak{a}}= (Y, W)^2$. Therefore, $YW \in {\mathfrak{a}}$ and $t^{19} \in I$. Consequently, $I = (t^6 + c_1 t^8 + c_2 t^{11}, t^{10} + d_1t ^{11})$. For the converse, let $J = (t^6 + c_1 t^8 + c_2 t^{11}, t^{10} + d_1t ^{11})$ and set $a=t^6 + c_1 t^8 + c_2 t^{11}$, where $c_1, c_2, d_1 \in k$ and $d_1 \ne 0$. Since $d_1 \ne 0$, we see that $\ell_R(R/J) = 3$ by the above isomorphism $R/J \cong k[Y, W]/{\mathfrak{a}}$. The fact that $t^n \in (a^2)$ for each even integer $n \ge 20$ implies $(t^{10} + d_1t ^{11})^2 \in (a^2)$. Hence $J^2 = aJ$, so that $J \in {\mathcal{X}}_R$. Similarly for the proof of Example \[3.2a\], we have the last assertions. \(5) Follows from the assertions (2), (3), and (4). Closing this section, since the ring as in Example \[difficult\] is obtained from the gluing of the numerical semigroup $\left<3, 4, 5\right>$, let us explore the $2$-$\AGL$ rings arising as gluing of numerical semigroup rings. In what follows, let $0 < a_1, a_2, \ldots, a_{\ell} \in \Bbb Z~(\ell >0)$ be positive integers such that $\GCD(a_1, a_2, \ldots, a_{\ell}) =1$. We set $H_1 =\left<a_1, a_2, \ldots, a_{\ell}\right>$ and assume that $a_1, a_2, \ldots, a_{\ell}$ forms a minimal system of generators of $H_1$. Let $0<\alpha \in H_1$ be an odd integer such that $\alpha \ne a_i$ for every $1 \le i \le \ell$. We consider $H = \left< 2a_1, 2a_2, \ldots, 2a_{\ell}, \alpha \right>$ the gluing of $H_1$ and the set of non-negative integers $\Bbb N$. The reader is referred to [@RG Chapter 8] for basic properties of gluing of numerical semigroups. Let $V=k[[t]]$ be the formal power series ring over a field $k$ and set $R_1 = k[[H_1]]$, $R=k[[H]]$ the semigroup rings of $H_1$ and $H$, respectively. We denote by $\m_1$ (resp. $\m$) the maximal ideal of $R_1$ (resp. $R$). Notice that $\mu_R(\m) = \ell + 1$ and $R$ is a free $R_1$-module of rank $2$. By letting ${\rm PF}(H_1) =\{p_1, p_2, \ldots, p_r\}$, the canonical fractional ideal $K_1$ of $R_1$ has the form $K_1 = \sum_{i=1}^r R_1{\cdot}t^{p_r-p_i}$, while $K = \sum_{i=1}^r R{\cdot}t^{2(p_r-p_i)}$ is the canonical fractional ideal of $R$, where $r = {\mathrm{r}}(R_1)$ and $p_r = f(H_1)$. We then have $R \otimes_{R_1} K_1 \cong K$ and hence $K/R \cong R \otimes_{R_1}(K_1/R_1)$ as an $R$-module. We set ${\mathfrak{c}}= R:R[K]$. With this notation we have the following. \[4.13\] Suppose that $R_1$ is an $\AGL$ ring, but not a Gorenstein ring. Then the following assertions hold true. 1. $R$ is a $2$-$\AGL$ ring, ${\mathfrak{c}}= \m_1 R$, and $\mu_R({\mathfrak{c}}) = \ell \ge 3$. 2. ${\mathfrak{c}}\in {\mathcal{X}}_R$ if and only if $R_1$ has minimal multiplicity. 3. $R$ doesn’t have minimal multiplicity. Therefore, $\m \notin {\mathcal{X}}_R$. \(1) Since $R$ is a free $R_1$-module of rank $2$ and $\ell_R(R/\m_1 R) =2$, we conclude that $R$ is a $2$-$\AGL$ ring ([@CGKM Theorem 3.10]). Besides, we have ${\mathfrak{c}}= \operatorname{Ann}_RK/R = (\operatorname{Ann}_{R_1}{K_1}/{R_1})R = \m_1 R$, whence $\mu_R({\mathfrak{c}}) = \ell \ge 3$. \(2) The isomorphisms ${\mathfrak{c}}/{\mathfrak{c}}^2 \cong R \otimes_{R_1}(\m_1/{{\m_1}^2}) \cong R\otimes_{R_1}(R_1/\m_1)^{\oplus \ell} \cong (R/{\mathfrak{c}})^{\oplus \ell}$ show that ${\mathfrak{c}}/{\mathfrak{c}}^2$ is a free $R/{\mathfrak{c}}$-module. Hence, ${\mathfrak{c}}\in {\mathcal{X}}_R$ if and only if ${\mathfrak{c}}^2 = f {\mathfrak{c}}$ for some $f \in {\mathfrak{c}}$. The latter condition is equivalent to saying that ${\mathfrak{c}}^2 = t^{2a_i}{\mathfrak{c}}$ for some $1 \le i \le \ell$, that is ${\m_1}^2 = t^{2a_i}\m_1$, as desired. \(3) We notice that $\mu_R(\m) = \ell + 1$ and ${\mathrm{e}}(R) =\min\{2a_1, 2a_2, \ldots, 2a_{\ell}, \alpha\}$. Suppose that ${\mathrm{e}}(R) = 2a_i$ for some $1 \le i \le \ell$. Since $\ell = \mu_{R_1}(\m_1) \le {\mathrm{e}}(R_1) \le a_1$, we get $${\mathrm{e}}(R) - \mu_R(\m) = 2a_i - (\ell + 1) \ge 2 \ell - (\ell + 1) = \ell -1 \ge 2$$ which implies that $R$ doesn’t have minimal multiplicity. Thereafter, we consider the case where ${\mathrm{e}}(R) = \alpha$. Suppose that $R$ has minimal multiplicity, that is ${\mathrm{e}}(R) = \mu_R(\m)$, in order to seek a contradiction. Since $\alpha$ is an odd integer, we notice that $\ell$ is even, because $\alpha = {\mathrm{e}}(R) = \mu_R(\m) = \ell + 1$. Besides, $\alpha < 2 a_i$ for each $1 \le i \le \ell$. Let us write $\alpha = \alpha_1 a_1 + \alpha_2 a_2 + \cdots + \alpha_{\ell}a_{\ell}$ where $\alpha_i \ge 0$. Then one of the $\{\alpha_i\}_{1 \le i \le \ell}$ is positive. Therefore, $\alpha = \alpha_ia_i$ for some $1 \le i \le \ell$, so that $\alpha_i = 1$ and $\alpha = a_i$. This makes a contradiction. Hence $R$ doesn’t have minimal multiplicity. Consequently, we have the following. \[4.14\] Suppose that $R_1$ is an $\AGL$ ring, but not a Gorenstein ring. Then the following assertions hold true. 1. Let $I \in {\mathcal{X}}_R$. Then either $\mu_R(I) = 2$ or $I = {\mathfrak{c}}$. 2. The set of two-generated Ulrich ideals which are generated by monomials in $t$ is $$\left\{ (t^{2m}, t^{\alpha}) \mid 0<m \in H_1,\ \alpha - m \in H_1,\ 2(\alpha - 2m) \in H\right\}.$$ \(1) Thanks to Proposition \[Ulrich\] (2), if $\mu_R(I) \ge 3$, then ${\mathfrak{c}}\subseteq I$. Since $R$ is a $2$-$\AGL$ ring and $\m \notin {\mathcal{X}}_R$, we conclude that $I = {\mathfrak{c}}$. \(2) Let $I \in {\mathcal{X}}_R$ such that $\mu_R(I) =2$ and $I$ is generated by monomials in $t$. We write $I = (t^p, t^q)$ where $0 < p < q$ and $p, q \in H$. Notice that, for each $0 < h \in H$ with $h \ne \alpha$, we have that $t^h \in {\mathfrak{c}}$. Since $I + {\mathfrak{c}}= {\mathfrak{m}}$ by Theorem \[2.3\] (2), we get $I \not\subseteq {\mathfrak{c}}$, which yields that $p=\alpha$ or $q = \alpha$. The isomorphism $R/I \cong I/(t^p)$ ensures that $p$ is even, so that $\alpha = q$. Therefore $0 < p < \alpha$. Let us write $p = \sum_{i=1}^{\ell}\left(2a_i\right)c_i + c\alpha = 2\left(\sum_{i=1}^{\ell}a_ic_i\right) + c\alpha$ where $c_i, c \ge 0$. As $p < \alpha$, we have $c = 0$. Therefore, $p= 2m$ for some $0 < m \in H_1$. Moreover, because $I^2 = t^{2m}I$, we have $2(\alpha-2m) \in H$, but $\alpha-2m \notin H$. Since $R/I = R/(t^{2m}, t^{\alpha}) \cong R_1/(t^m, t^{\alpha})$ and $\ell_R(R/I) = m$, we obtain that $t^{\alpha} \in t^m R_1$. Hence $\alpha - m \in H_1$. Conversely, let $I = (t^{2m}, t^{\alpha})$ where $0< m_1 \in H_1$, $\alpha - m \in H_1$, and $2(\alpha - 2m) \in H$. We then have $I^2 = t^{2m}I + (t^{2\alpha}) = t^{2m}I$, while $R/I \cong R_1/(t^m, t^{\alpha}) = R_1/t^{m}R_1$, so that $\ell_R(R/I) = m$. Therefore $I \in {\mathcal{X}}_R$, as desired. \[4.16\] Let $H_1 = \left<4, 7, 9\right>$ and $\alpha \ge 11$ an odd integer. We set $R_1=k[[t^4, t^7, t^9]]$ the numerical semigroup ring of $H_1$ over a field $k$. By Example \[3.5\], $R_1$ is an $\AGL$ ring with ${\mathrm{r}}(R_1) =2$. Let $H = \left<8, 14, 18, \alpha\right>$ and set $R=k[[H]]$. Then $\mu_R(I) = 2$ for each $I \in {\mathcal{X}}_R$. Moreover, we have the following. 1. If $\alpha = 11, 13$, then ${\mathcal{X}}_R = \emptyset$. 2. If $\alpha \ge 15$, then $(t^8, t^{\alpha}) \in {\mathcal{X}}_R$. 3. If $\alpha =15$ and $\ch k = 2$, then $(t^8 + ct^{14}, t^{\alpha}) \in {\mathcal{X}}_R$ for every $c \in k$, and we have $(t^8 + c_1t^{14}, t^{\alpha}) = (t^8 + c_1t^{14}, t^{\alpha})$, only if $c_1 =c_2$. 4. If $\alpha \ge 17$, then $(t^8 + ct^{14}, t^{\alpha}) \in {\mathcal{X}}_R$ for every $c \in k$, and we have $(t^8 + c_1t^{14}, t^{\alpha}) = (t^8 + c_1t^{14}, t^{\alpha})$, only if $c_1 =c_2$. G-regularity in $2$-${\rm AGL}$ rings ===================================== The condition that $K/R$ is a free $R/{\mathfrak{c}}$-module gives an agreeable restriction on the behavior of $2$-AGL rings, as we have shown in Proposition \[1.6\] (see also [@CGKM Section 5]). However, even though $K/R$ is not $R/{\mathfrak{c}}$-free, $2$-AGL rings also enjoy nice properties. We will show in the following that every $2$-AGL ring $R$ is G[*-regular*]{} in the sense of [@greg], namely, totally reflexive $R$-modules are all free, provided $K/R$ is not $R/{\mathfrak{c}}$-free. \[1.5\] Suppose that $R$ is a $2$-${\rm AGL}$ ring, possessing a canonical fractional ideal $K$. We set ${\mathfrak{c}}= R : R[K]$, and assume that $K/R$ is not a free $R/{\mathfrak{c}}$-module. Let $M$ be a finitely generated $R$-module. If $\Ext_R^p(M, R) =(0)$ for all $p \gg 0$, then $\pd_RM < \infty$. Hence $R$ is $G$-regular in the sense of [@greg]. Let $L=\Omega_R^1(M)$ be the first syzygy module of $M$. For every $p \ge 2$ we have $ \Ext_R^{p-1}(L, R) \cong \Ext_R^p(M, R), $ which shows $ \Ext_R^p(L, K/R) = (0)$ for all $p \gg 0$, because $\Ext_R^p(L,K) = (0)$. Therefore, since $R/\m$ is a direct summand of $K/R$ (Proposition \[2.3a\] (4)), $\Ext_R^p(L, R/\m) =(0)$ for $p \gg 0$, so that $\pd_RL < \infty$. Hence $\pd_R M < \infty$. We should compare the following result with [@GTT2 Theorem 2.14 (1)], where a corresponding result for one-dimensional AGL rings is given. \[4.3\] Suppose that $(R,\m)$ is a $2$-${\rm AGL}$ ring with minimal multiplicity, possessing a canonical fractional ideal $K$ and ${\mathfrak{c}}= R: R[K]$. Then $${\mathcal{X}}_R= \begin{cases} \{{\mathfrak{c}}, \m\}, & \ \text{if} \ K/R~\text{is}~R/{\mathfrak{c}}\text{-free},\\ \{\m\}, & \ \text{otherwise}. \end{cases}$$ Since $R$ has minimal multiplicity, $\m \in {\mathcal{X}}_R$, so that ${\mathcal{X}}_R \ne \emptyset$. $(1)$ Suppose that $K/R$ is $R/{\mathfrak{c}}$-free. Then, by [@CGKM Proposition 5.7 (1)], $\m:\m$ is a local ring, while $S=R[K]$ is a Gorenstein ring, since $R$ is a $2$-AGL ring with minimal multiplicity ([@CGKM Corollary 5.3]). Therefore, thanks to Proposition \[1.6\], ${\mathfrak{c}}=R:S \in {\mathcal{X}}_R$, so that $\{{\mathfrak{c}}, \m\} \subseteq {\mathcal{X}}_R$. Let $I \in {\mathcal{X}}_R$. Then, because $R$ has minimal multiplicity, $\mu_R(I) \ge 3$ by Corollary \[1.7\]. Therefore, since $K/R$ is $R/{\mathfrak{c}}$-free, we get ${\mathfrak{c}}= (0):_RK/R \subseteq I$ ([@GTT2 Corollary 2.13]). Thus, $I = {\mathfrak{c}}$ or $I = \m$, because $\ell_R(R/{\mathfrak{c}})=2$. $(2)$ Suppose that $K/R$ is not $R/{\mathfrak{c}}$-free and let $I$ be an Ulrich ideal of $R$. Then, $\mu_R(I) \ge 3$ by Theorem \[2.3\]. Therefore, thanks to the proof of case (1), $I={\mathfrak{c}}$ or $I=\m$. Thus, $I = {\mathfrak{m}}$, because ${\mathfrak{c}}\not\in {\mathcal{X}}_R$ by Proposition \[1.6\]. We close this paper with the following, where two kinds of $2$-${\rm AGL}$ rings of minimal multiplicity are given, one is $R/{\mathfrak{c}}$-free and the other one is not. Let $V=k[[t]]$ denote the formal power series ring over a field $k$ and set $R_1 =k[[t^3, t^7, t^8]]$, $R_2=k[[t^4, t^9, t^{11}, t^{14}]]$. Let $K_i$ be a canonical fractional ideal of $R_i$. Then, both $R_1$ and $R_2$ are $2$-AGL rings. We have $K_1/R_1$ is a free $R/{\mathfrak{c}}_1$-module, but $K_2/R_2$ is not $R/{\mathfrak{c}}_2$-free, where ${\mathfrak{c}}_i = R_i : R_i[K_i]$. Therefore, ${\mathcal{X}}_{R_1}=\{(t^6, t^7, t^8), (t^3, t^7, t^8)\}$, and ${\mathcal{X}}_{R_2}=\{(t^4, t^9, t^{11}, t^{14})\}$. We have $K_1 = R + Rt$ and $K_2 = R + Rt + Rt^5$. Hence, $R_1[K_1] = R[t] = V$, and $R_2[K_2] = R[t^3, t^5] = k[[t^3, t^4, t^5]]$, so that $ \ell_{R_1}({R_1[K_1]}/{K_1}) = \ell_{R_2}({R_2[K_2]}/{K_2}) = 2. $ Therefore, by Theorem \[mainref\], both $R_1$ and $R_2$ are $2$-AGL rings. Because $ \ell_{R_1}(K_1/{R_1}) = 2 \ \ \text{and} \ \ \ell_{R_2}(K_2]/{R_2}) = 3, $ $K_1/{R_1}$ is a free $R/{\mathfrak{c}}_1$-module, but $K_2/R_2$ is not $R/{\mathfrak{c}}_2$-free (use Proposition \[2.3a\] (4)). Notice that $R_1$ and $R_2$ have minimal multiplicity $3$ and $4$, respectively. Hence, the results readily follow from Corollary \[4.3\], since ${\mathfrak{c}}_1 =R_1:V= t^6 V = (t^6, t^7, t^8)$. Appendix: Ulrich ideals in one-dimensional Gorenstein local rings of finite Cohen-Macaulay representation type ============================================================================================================== In [@GOTWY], the authors determined all the Ulrich ideals in one-dimensional Gorenstein local rings $R$ of finite CM-representation type, while in [@GTT Section 12] most birational module-finite extensions of these rings have been searched. Since the proof given by [@GOTWY] depends on the techniques in the representation theory of maximal Cohen-Macaulay modules, it might have some interests to give a straightforward proof, making use of the results of [@GTT Section 12] and determining the members of ${\mathcal{A}}_R^0$ by Lemma \[lem3.1\], as well. We note it as an appendix. In this appendix, let $(R, \m)$ be a Gorenstein complete local ring of dimension one with algebraically closed residue class field $k$ of characteristic $0$. Suppose that $R$ has finite CM-representation type. Then, by [@Y (8.5), (8.10), and (8.15)] we get $$R \cong k[[X, Y]]/(f),$$ where $k[[X, Y]]$ is the formal power series ring over $k$, and $f$ is one of the following polynomials. - $X^2-Y^{n+1}$ $(n \ge 1)$ - $X^2Y-Y^{n-1}$ $(n \ge 4)$ - $X^3-Y^4$ - $X^3-XY^3$ - $X^3-Y^5$ With this notation we have the following. \[5.1\] The set ${\mathcal{X}}_R$ is given by the following. 1. ${\mathcal{X}}_R = \begin{cases} \left\{(x, y^q) \mid 0 < q \le \ell \right\} & \text{if} \ n=2 \ell -1 \ \text{with} \ \ell \ge 1,\\ \left\{(x, y^q) \mid 0 < q \le \ell \right\} & \text{if} \ n=2\ell \ \text{with} \ \ell \ge 1. \end{cases}$ 2. ${\mathcal{X}}_R = \begin{cases} \left\{(x^2, y), (x, y^{\ell+1}) \right\} & \text{if} \ n=2 \ell +3 \ \text{with} \ \ell \ge 1,\\ \left\{(x^2, y), (x-y^{\ell}, y(x+y^{\ell})), (x+y^{\ell}, y(x-y^{\ell})) \right\} & \text{if} \ n=2(\ell + 1) \ \text{with} \ \ell \ge 1. \end{cases}$ 3. ${\mathcal{X}}_R= \left\{(x, y^2)\right\}$ 4. ${\mathcal{X}}_R = \left\{(x,y^3)\right\}$ 5. ${\mathcal{X}}_R = \emptyset$ where $x$ and $y$ denote the images of $X$ and $Y$ in the corresponding rings, respectively. For a ring $A$, let $J(A)$ denote its Jacobson radical. We denote by $\overline{R}$ the integral closure of $R$ in ${\mathrm{Q}}(R)$, and by ${\mathcal{B}}_R$ the set of birational module-finite extensions of $R$. [**(1)**]{} ($E_6$) See Example \[3.2a\]. [**(2)**]{} ($E_8$) Let $R=k[[t^3,t^5]]$ and $V=k[[t]]$. By [@GTT Proposition 12.7 (3)], ${\mathcal{B}}_R= \{R, k[[t^3,t^5, t^7]], k[[t^3,t^4,t^5]], k[[t^2,t^3]], V\}$, among which $k[[t^3,t^5, t^7]], k[[t^3,t^4,t^5]]$ are not Gorenstein rings, and $\mu_R(V)=\mu_R(k[[t^2,t^3]])=3$. Hence, ${\mathcal{A}}_R^0=\emptyset$, so that ${\mathcal{X}}_R= \emptyset$ by Lemma \[lem3.1\]. [**(3)**]{} ($E_7$) Let $R=k[[X,Y]]/(X^3-XY^3)$. We set $S=k[[X, Y]]$, $V=k[[t]]$, and $f=X^3-XY^3$. Then, since $(f)=(X)\cap (X^2-Y^3)$, we get the tower $$R=S/(f) \subseteq S/(X)\oplus S/(X^2-Y^3) = k[[Y]] \oplus k[[t^2, t^3]] \subseteq k[[Y]] \oplus V = \overline{R}$$ of rings, where we identify $S/(X) = k[[Y]]$ and $S/(X^2-Y^3) = k[[t^2, t^3]] \subseteq V$. \[5.2\] $ {\mathcal{A}}_R = \{ R, k[[Y]] \oplus k[[t^2, t^3]], k[[Y]] \oplus V, k + J(\overline{R}) \}. $ Let $A \in {\mathcal{B}}_R$ such that $R \ne A$ and let $p_2 : \overline{R} \to V$ denote the projection. We set $B=p_2(A)$. Since $k[[t^2, t^3]] \subseteq B \subseteq V$, $B= k[[t^2, t^3]]$ or $B=V$. Suppose that $A$ is not a local ring. Then, $A$ decomposes into a direct product of local rings, since $A$ is a module-finite extension of the complete local ring $R$, so that we may choose a non-trivial idempotent $e \in A$. Then, since $\overline{R} = k[[X]] \oplus V$, we get $e= (1,0)$, or $(0,1)$, whence $(1,0), (0,1) \in A$, so that $A=A(1,0) + A(0,1) = k[[Y]] \oplus B$. Suppose that $A$ is a local ring. In this case, the argument in [@GTT Pages 2708–2710] shows that if $B=V$, then $A \cong k[[Y, Z]]/(Z(Y-Z^2)) = k[[(Y, t^2), (0, t)]] =k + J(\overline{R})$, and that if $B=k[[t^2, t^3]]$, then $A$ is an AGL but not a Gorenstein ring. Thus we have the assertion. Since $J(\overline{R})=R(Y,t^2)+R(0,t)+R(0,t^2)$, we have $k+J(\overline{R})= R + R(0,t)+R(0,t^2)$, whence $\mu_R(k+J(\overline{R})) = 3$. Therefore, ${\mathcal{A}}_R^0 = \left\{k[[Y]] \oplus k[[t^2, t^3]]\right\}$, so that by Lemma \[lem3.1\] ${\mathcal{X}}_R=\{(x,y^3)\}$, since $R:(k[[Y]] \oplus k[[t^2, t^3]])=(x,y^3)$. [**(4)**]{} ($D_n$) [*$(\rm i)$ $($The case where $n=2 \ell + 3$ with $\ell \ge 1$$)$.*]{} Let $R=k[[X,Y]]/(X^2Y-Y^{2\ell +2})$. We set $S=k[[X, Y]]$, $V=k[[t]]$, and $f= Y(X^2-Y^{2\ell + 1})$. We consider the tower $$R=S/(f) \subseteq S/(Y)\oplus S/(X^2-Y^{2\ell + 1}) = k[[X]] \oplus k[[t^2, t^{2\ell + 1}]] \subseteq k[[X]] \oplus V = \overline{R}$$ of rings, where we identify $S/(Y) = k[[X]]$ and $S/(X^2-Y^{2\ell + 1}) = k[[t^2, t^{2\ell + 1}]]$. We then have the following. \[5.3\] ${\mathcal{A}}_R = \left\{R, k+J(\overline{R})\right\} \cup \left\{k[[X]] \oplus k[[t^2, t^{2q+1}]] \mid 0 \le q \le \ell \right\}$ Let $A \in {\mathcal{B}}_R$ such that $R \ne A$ and let $p_2 : \overline{R} \to V$ denote the projection. We set $B=p_2(A)$. Then, by [@GTT Corollary 12.5 (1)] $B= k[[t^2, t^{2q+1}]]$ for some $0 \le q \le \ell$, since $k[[t^2, t^{2\ell +1}]] \subseteq B \subseteq V$. If $A$ is not a local ring, then the same proof as in Claim \[5.2\] works, to get $A= k[[X]] \oplus B$. If $A$ is a local ring, then by the argument in [@GTT Pages 2710–2711] we have $ A \cong k[[X, Z]]/[(Z)\cap(X-Z^{2\ell + 1})] = k + J(\overline{R})$. Consequently, ${\mathcal{A}}_R^0=\left\{k[[X]]\oplus k[[t^2,t^{2\ell +1}]], k+J(\overline{R})\right\}$. We have $$\left(k[[X]]\oplus k[[t^2,t^{2\ell +1}]]\right)/R \cong S/(X^2, Y)$$ and $k+J(\overline{R})=R+R(0,t)$. Therefore, Lemma \[lem3.1\] shows the assertion, because $$R:\left(k[[X]]\oplus k[[t^2,t^{2\ell +1}]]\right) = (x^2, y) \ \ \text{and}\ \ R:\left(k+J(\overline{R})\right)=(x, y^{\ell+1}).$$ [**(4)**]{} ($D_n$) [*$(\rm ii)$ $($The case where $n=2 \ell + 2$ with $\ell \ge 1$$)$.*]{} Let $R=k[[X,Y]]/(X^2Y-Y^{2\ell +1})$. We set $S=k[[X, Y]]$, $V=k[[t]]$, and $f=Y(X^2-Y^{2\ell})$. Consider the tower $$R=S/(f) \subseteq S/(Y)\oplus T = k[[X]] \oplus \overline{T} = \overline{R}$$ of rings, where $T=S/(X^2-Y^{2\ell})$. By [@GTT Page 2711] an intermediate ring $R \subsetneq A \subseteq \overline{R}$ is an AGL ring but not a Gorenstein ring, if $A$ is a local ring. Therefore, every $A \in {\mathcal{A}}_R$ is not local, if $R\ne A$. \[5.5\] ${\mathcal{A}}_R = \left\{R, S/(X-Y^{\ell})\oplus S/(Y(X+Y^{\ell})), S/(X+Y^{\ell})\oplus S/(Y(X-Y^{\ell})\right)\} \cup \left\{k[[X]] \oplus T[\frac{x}{y^q}] \mid 0 \le q \le \ell \right\}$ Let $A \in {\mathcal{A}}_R$ such that $R \ne A$. Note that $\overline{R}=k[[X]] \oplus S/(X-Y^{\ell}) \oplus S/(X+Y^{\ell})$. Let $\{e_i\}_{i=1,2,3}$ be the orthogonal primitive idempotents of $\overline{R}$. Then, $e_i \in A$ for some $1 \le i \le 3$, since $A$ is not a local ring. If $A \ne \overline{R}$, such $e_i$ is unique for $A$. [*$(\rm i)$ $($The case where $e_1 \in A$$)$.*]{} Let $p:\overline{R} \to S/(X-Y^{\ell}) \oplus S/(X+Y^{\ell})$ denote the projection. Then $$T:=S/{(X-Y^{\ell}) \cap (X+Y^{\ell})} \subseteq p(A) \subseteq \overline{T} = S/(X-Y^{\ell}) \oplus S/(X+Y^{\ell})$$ so that, by [@GTT Corollary 12.5 (2)] $p(A) = T[\frac{x}{y^q}]$ for some $0 \le q < \ell$. Hence, $A = k[[X]] \oplus T[\frac{x}{y^q}]$. [*$(\rm ii)$ $($The case where $e_2 \in A$$)$.*]{} Let $p:\overline{R} \to k[[X]] \oplus S/(X+Y^{\ell})$ denote the projection. Because $A \ne \overline{R}$, we have $$S/{(Y) \cap (X+Y^{\ell})} \subseteq p(A) \subsetneq k[[X]] \oplus S/(X+Y^{\ell}),$$ which shows $p(A) = S/{(Y) \cap (X+Y^{\ell})} = S/(Y(X+Y^{\ell}))$. Thus, $A = S/(X-Y^{\ell}) \oplus S/(Y(X+Y^{\ell}))$. Similarly, $A=S/(X+Y^{\ell})\oplus S/(Y(X-Y^{\ell}))$ if $e_3 \in A$, which proves Claim \[5.5\]. Therefore, $${\mathcal{A}}_R^0 =\left\{k[[X]] \oplus T, S/(X-Y^{\ell})\oplus S/(Y(X+Y^{\ell})), S/(X+Y^{\ell})\oplus S/(Y(X-Y^{\ell}))\right\},$$ so that ${\mathcal{X}}_R= \left\{(x^2, y), (x-y^{\ell}, y(x+y^{\ell})), (x+y^{\ell}, y(x-y^{\ell})) \right\}$. [**(5)**]{} ($A_n$) [*$(\rm i)$ $($The case where $n=2 \ell$ with $\ell \ge 1$$)$.*]{} Let $R=k[[t^2, t^{2\ell +1}]]$. Then, ${\mathcal{A}}_R^0 = \{k[[t^2, t^{2q+1} \mid 0 \le q \le \ell -1]]\}$ by [@GTT Corollary 12.5 (1)], whence ${\mathcal{X}}_R =\{ (x, y^q) \mid 0 < q \le \ell \}$. [**(5)**]{} ($A_n$) [*$(\rm ii)$ $($The case where $n=2 \ell - 1$ with $\ell \ge 1$$)$.*]{} Let $R=k[[X,Y]]/(X^2-Y^{2\ell})$. We set $S=k[[X, Y]]$ and $f=X^2-Y^{2\ell}=(X-Y^{\ell})(X+Y^{\ell})$. We then have $\ell_R(\overline{R}/R)=\ell$ by the exact sequence $$0 \to R=S/(f) \longrightarrow \overline{R}=S/(X-Y^\ell)\oplus S/(X+Y^\ell) \longrightarrow S/(X, Y^\ell) \to 0$$ of $R$-modules. Let $A \in {\mathcal{A}}_R$ such that $R \ne A$. Then, by [@GTT Corollary 12.5 (2)] $ A = R\left[\frac{x}{y^q}\right] \ \ \text{for some} \ \ 0 < q \le \ell $ in ${\mathrm{Q}}(R)$. If $n=\ell$, then $A = \overline{R}$ is a Gorenstein ring with $\mu_R(\overline{R})=2$, so that $(x, y^\ell)= R:\overline{R} \in {\mathcal{X}}_R$. Let us now assume that $0 < q < \ell$. Since $(\frac{x}{y^q})^2 =x^2y^{-2q} = y^{2\ell}y^{-2q}=y^{2(\ell - q)}\in R$, we have $ A=R+R{\cdot}\frac{x}{y^q}. $ We will show that $A$ is a Gorenstein local ring with $\mu_R(A) =2$. Indeed, set let $\n = \m A+ \frac{x}{y^q}A$ of $A$, and let $M$ be an arbitrary maximal ideal of $A$. We choose a maximal ideal $N$ of $\overline{R}$ so that $M = N \cap A$. We then have $ N \supseteq J(\overline{R}) \supseteq y\overline{R} + \frac{x}{y^q}\overline{R}, $ whence $M=N\cap A \supseteq \n$, so that $M=\n$ because $\n$ is a maximal ideal of $A$. Hence, $(A,\n)$ is a local ring. Consequently, $2 \le \mu_R(A) = \ell_R(A/\m A) \le \e(A) \le \e(R)=2$. Thus $A \in {\mathcal{X}}_R^0$. Note that $R:A = R:_R\frac{x}{y^q}$, because $A=R+R\frac{x}{y^q}$. We now take $a \in R:\frac{x}{y^n}$. Then, setting $b=a{\cdot}\frac{x}{y^q} \in R$, we have $ax=by^q$, so that $AX-BY^q=C(X^2-Y^{2\ell})$ for some $C \in S$. Here $a, b$ are the images of $A, B$ respectively. 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--- abstract: 'Phase curves and secondary eclipses of gaseous exoplanets are diagnostic of atmospheric composition and meteorology, and the long observational baseline and high photometric precision from the [[*Kepler*]{}]{} Mission make its dataset well-suited for exploring phase curve variability, which provides additional insights into atmospheric dynamics. Observations of the hot Jupiter Kepler-76b span more than 1,000 days, providing an ideal dataset to search for atmospheric variability. In this study, we find that Kepler-76b’s secondary eclipse, with a depth of $87 \pm 6$ parts-per-million (ppm), corresponds to an effective temperature of 2,830$^{+50}_{-30}$ K. Our results also show clear indications of variability in Kepler-76b’s atmospheric emission and reflectivity, with the phase curve amplitude typically $50.5 \pm 1.3$ ppm but varying between 35 and 70 ppm over tens of days. As is common for hot Jupiters, Kepler-76b’s phase curve shows a discernible offset of $\left( 9 \pm 1.3 \right)^\circ$ eastward of the sub-stellar point and varying in concert with the amplitude. These variations may arise from the advance and retreat of thermal structures and cloud formations in Kepler-76b’s atmosphere; the resulting thermal perturbations may couple with the super-rotation expected to transport aerosols, giving rise to a feedback loop. Looking forward, the [[*TESS*]{}]{} Mission can provide new insight into planetary atmospheres, with good prospects to observe both secondary eclipses and phase curves among targets from the mission. [[*TESS*]{}]{}’s increased sensitivity in red wavelengths as compared to [[*Kepler*]{}]{} means that it will probably probe different aspects of planetary atmospheres.' author: - Brian Jackson - Elisabeth Adams - Wesley Sandidge - Steven Kreyche - Jennifer Briggs bibliography: - 'bibliography.bib' title: 'Variability in the Atmosphere of the Hot Jupiter Kepler-76b' --- Introduction {#sec:Introduction} ============ The age of exoplanet meteorology has arrived – from the first detection of atmospheric thermal emission [@2005Natur.434..740D; @2005ApJ...626..523C] to the recent detection of the secondary eclipses of an Earth-sized planet [@2016Natur.532..207D], observation, characterization, and modeling of exoplanet atmospheres has blossomed into a rich field approaching that of solar system meteorology in scope and complexity. Indeed, the much wider range of dynamical, compositional, and radiative environments across which exoplanets have been discovered means the study of these systems will undoubtedly provide insight into the planetary atmospheres in our own solar system. Among the first key results in the study of exoplanet meteorology was the mapping of atmospheric emission from the transiting hot Jupiter HD189733b [@2007Natur.447..183K]. In that study, observations in the *Spitzer* Space Telescope’s 8 $\mu$m band revealed a deep secondary eclipse (0.3381% in differential flux) corresponding to a dayside brightness temperature of more than 1,200 K. Perhaps of more interest to meteorological studies, the data also showed signs of a planetary emission during the entire orbit, peaking as the tidally locked planet’s dayside rotated into view. However, instead of being centered around the eclipse, the planet’s phase curve peak occurred about 2 hours before, indicating the brightest and hottest region in the atmosphere was shifted about 16$^\circ$ eastward from the sub-stellar point. @2007Natur.447..183K attributed this offset to advective transport within the atmosphere, a result anticipated by earlier studies of hot Jupiter atmospheres . The mountain of data made available by the [[*Kepler*]{}]{} Mission has allowed analysis of phase curves for numerous planets [e.g., @2015ApJ...804..150E]. However, since the [[*Kepler*]{}]{} data involve visible wavelengths (with a bandpass spanning 400 to 900 nm), these phase curves likely involve a combination of thermal emission and reflected light from high-altitude aerosols [@2000ApJ...538..885S]. The aerosols obscure the lower atmosphere, complicating the determination of atmospheric composition [@2016Natur.529...59S]. Although the composition, size distribution, and vertical mixing of aerosols are difficult to determine *a priori*, their influence on the planetary phase curve, in particular the resulting phase curve offset, may be diagnostic. @2016ApJ...828...22P summarize the distribution of observed phase curve offsets and highlight a clear correlation with planetary effective temperature. Cooler planets show offsets toward the west, consistent with the expectation that clouds dominate these planets’ phase curves. By contrast, some hotter planets, such as HD189733 b, show eastward offsets, symptomatic of either homogeneous or little dayside cloud cover and downwind transport of thermally emitting regions, although there are important exceptions, such as CoRoT-2b [@2018NatAs...2..220D]. @2016ApJ...828...22P also suggest that the same super-rotating winds that drive hot regions eastward can draw aerosols that form on the nightside across a hot planet’s western (night-to-day) terminator, forming a dayside cloud front whose eastern-most boundary depends on the aerosol composition – the more refractory the aerosols, the farther east the boundary. These clouds can contribute to the planetary phase curve and, since they may be confined to the western hemisphere, they may compete with eastern-hemisphere emission and drive the observed phase curve offset toward the west. All these effects take place in a highly dynamic atmosphere, in which winds approach sonic speeds [@2018ApJ...853..133K], and the interplay between transport of aerosols, which act to cool the atmosphere, and potent stellar insolation, which drives the winds, produce variable atmospheric structures. This variability likely shows up in the reflected and emitted light from the planets’ atmospheres. Studying aerosol transport in the atmosphere of HD209458b, predicted variations in the secondary eclipse depths as large as 50% in IR wavelengths over tens of days. Analyzing hundreds of days of [[*Kepler*]{}]{} observations of the hot Jupiter HAT-P-7 b, @2016NatAs...1E...4A, indeeed, found significant variations in the phase curve offset, which wandered back and forth across the sub-stellar point over tens of days, likely driven, at least in part, by variations in aerosol transport. Although @2016NatAs...1E...4A were not able to robustly detect them, variations in the phase curve amplitude probably accompanied the offset fluctuations since variations in aerosol transport would also be expected to modify the planet’s total geometric albedo. @2019ApJ...872L..27H explored the effects of magnetic fields on hot Jupiter phase curves and found that planetary-scale equatorial magneto-Kelvin waves could drive westward tilting eddies and produce the observed periodic westward offset in HAT-P-7b’s phase curve reported in @2016NatAs...1E...4A. In principle, it should be possible to connect variations in a planet’s atmospheric reflection and emission to phase curve variability, but this requires a large enough population of planets that spans a range of conditions to piece together a cogent story. Following this thread, we analyzed [[*Kepler*]{}]{} observations of the hot Jupiter Kepler-76b, a two Jupiter-mass (${\rm M_{Jup}}$) planet with an orbital period of 1.5 days around a hot (6,300 K) star [@2013ApJ...771...26F]. In addition to having a clear atmospheric signal, Kepler-76b also induces observable ellipsoidal variations on its host star (the photometric signature of tidal bulges on the star rotating in and out of view – [@2010ApJ...713L.145W]) and a beaming signal due to the radial velocity of the star [@2003ApJ...588L.117L]. With nearly 1,400 days of observation, we find significant variability in the planet’s phase curve amplitude, $A_{\rm planet}$, and offset, $\delta$, qualitatively consistent with the variations seen for HAT-P-7 b. We have also updated the transit parameters for Kepler-76b. This article is organized as follows: in Section \[sec:Data\_Analysis\], we describe our process for conditioning the [[*Kepler*]{}]{} dataset and then our methodology for modeling Kepler-76b’s transit, eclipse, and other photometric signals. In Section \[sec:Results\], we discuss our results, and in Section \[sec:Discussion\], we discuss their implications and future prospects for using [[*TESS*]{}]{} data to conduct similar analyses. Data Analysis {#sec:Data_Analysis} ============= Data Conditioning ----------------- We applied the following steps to condition all fourteen available quarters (Quarters 1-5, 7-9, 11-13, 15-17) of [[*Kepler*]{}]{} long-cadence (i.e., 30-min integration times) data: 1. To reduce variations in each quarter’s data to the same scale, we subtracted the median value from each quarter’s PDCSAP\_FLUX time-series before dividing through by that same median value. 2. We next applied a median boxcar filter with a window size equal to four orbital periods. We experimented with window sizes from 1-15 orbital periods (holding fixed the transit parameters to those reported in @2013ApJ...771...26F), then fitting the out-of-transit photometric signals described below using the Levenberg-Marquardt algorithm (LM) [@newville_2014_11813], including the planet’s eclipse and phase curve; four orbital periods maximized the eclipse depth and minimized the distortion to the phase curve using all available data. To mitigate edge-effect distortions from our boxcar filter, we extended the time-series out a full window length beyond both ends by reflecting the original time-series across its boundary. 3. Finally, we stitched each quarter’s conditioned data into one long time-series. Figure \[fig:raw-conditioned-data\_Analysis\_of\_Kepler76b\] illustrates a portion of the raw and conditioned time-series for Kepler-76b. ![The quarter-by-quarter (a) raw and (b) conditioned PDCSAP\_FLUX. Time along the x-axis is shown in the [[*Kepler*]{}]{} mission’s barycentric Julian date minus 2454833 (midnight on 2009 January 1), and the y-axes show variations in flux.\[fig:raw-conditioned-data\_Analysis\_of\_Kepler76b\]](f1.png){width="\textwidth"} Analyzing All Data Folded Together ---------------------------------- [lcr]{} Transit & &\ $T_0$ (BJD) & $2454965.00396 \pm 5 \times 10^{-5}$ & Time of Primary Transit\ Period (days) & $1.54492871 \pm 9 \times 10^{-8} $ & Orbital Period\ $R_{\rm p}/R{\star}$ & $0.085 \pm 0.001$ & Planetary Radius/Stellar Radius\ $a/R_{\star}$ & $5.103^{+0.058}_{-0.060}$ & Orbital Semi-Major Axis/Stellar Radius\ $b$ & $0.908 \pm 0.003$ & Impact Parameter\ $i$ (degrees) & $79.7 \pm 0.1$ & Orbital Inclination\ $(\gamma_1, \gamma_2)$ & (0.313, 0.304) & Limb-Darkening Coefficients [@2013ApJ...771...26F]\ Eclipse/Photometric Variations & &\ $D$ (ppm) & $87 \pm 6$ & Secondary Eclipse Depth\ $A_{\rm planet}$ (ppm) & $50.5 \pm 1.3$ & Phase Curve Amplitude\ $\delta$ (degrees) & $-9.0 \pm 1.3$ & Phase Curve Offset\ $A_{\rm ellip}$ (ppm) & $13 \pm 1$ & Ellipsoidal Variation (EV) Amplitude\ $A_{\rm beam}$ (ppm) & $3.8 \pm 0.3$ & Beaming Effect Amplitude\ Other Parameters & &\ $M_{\star}\ ({\rm M_{\odot}})$ & $1.2 \pm 0.2$ & Stellar Mass [@2013ApJ...771...26F]\ $g$ & 0.07 & Gravity-Darkening Parameter [@2011AA...529A..75C]\ $\alpha_{\rm beam}$ & 0.92 & Beaming Coefficient [@2013ApJ...771...26F]\ $\alpha_{\rm ellip}$ & 1.02 & EV Coefficient\ $K_{\rm z}$ (km/s) & $0.306 \pm 0.020$ & Radial Velocity [@2013ApJ...771...26F]\ $q_{\rm ellip}$ & $\left( 1.9 \pm 0.2 \right) \times 10^{-3}$ & Planet-Star EV Mass Ratio\ $M_{\rm p, ellip}\ ({\rm M_{\rm Jup}})$ & $2.4 \pm 0.3$ & Planetary EV Mass\ $q_{\rm beam}$ & $\left( 1.7\pm 0.1 \right) \times 10^{-3}$ & Planet-Star Beaming Mass Ratio\ $M_{\rm p, beam}\ ({\rm M_{\rm Jup}})$ & $2.1 \pm 0.2$ & Planetary Beaming Mass After applying the above conditioning procedure, we fit a series of photometric models to all the available data. We first analyzed the out-of-transit portion of Kepler-76b’s light curve since those signals (the ellipsoidal variations, the planet’s phase curve, and the Doppler beaming signal) are independent of the transit and can, in principle, contribute to the transit portion (although we found their effects are negligible). We folded all data on the best-fit period reported by @2013ApJ...771...26F and masked out the transit and eclipse portions of the light curve and fit the following model: $$ \Delta F = F_0 - A_{\rm ellip} \cos \left(2\times2\pi\phi \right) + A_{\rm sin} \sin \left(2 \pi\phi \right) - A_{\rm cos} \cos \left(2\pi \phi \right), \label{eqn:BEER_curve}$$ where $F_0$ represents a constant baseline and $\phi$ the orbital phase ($\phi = 0$ at mid-transit). The second term represents the ellipsoidal variations induced by the planet’s tidal gravity [@2010ApJ...713L.145W], with $A_{\rm ellip}$ its amplitude. The third and fourth terms are a combination of the Doppler beaming signal [@2003ApJ...588L.117L] and the planet’s reflected/thermally emitted phase curve, with allowance for an offset in the curve’s maximum from superior conjunction at $\phi = 0.5$ [@2013ApJ...771...26F]. We also allowed the per-point uncertainty (noise) to be a free parameter and estimated the model likelihood $L$ as $$\ln L = -\frac{1}{2} \sum_i \bigg( \frac{ d(t_{\rm i}) - m(t_{\rm i}) }{\sigma} \bigg)^2 + \ln \left( 2 \pi \sigma^2 \right), \label{eqn:likelihood}$$ where $d(t_{\rm i})$ is the datum at time $t_{\rm i}$, $m(t_{\rm i})$ the model for that same time, and $\sigma$ the per-point uncertainty. With this likelihood, we used a Markov Chain Monte Carlo (MCMC) analysis [@2013PASP..125..306F] with 100 walker chains, each with 5,000 links and a burn-in phase of 2,500 links. We aimed for convergence by requiring small auto-correlation times [e.g., @geyer1992] for the mean chain of each fit parameter (none exceeded 80 links) and all Geweke Z-scores [@Geweke92evaluatingthe] of about 3 or less. Traditionally, the beaming and planetary phase curve signals are treated separately through the combination $A_{\rm beam} \sin \left(2 \pi\phi \right) - A_{\rm planet} \cos \big(2\pi \left( \phi - \delta \right)\big)$. This expression can be re-cast using $A_{\rm sin} = A_{\rm beam} - A_{\rm planet} \sin \left( 2 \pi \delta \right)$ and $A_{\rm cos} = A_{\rm planet} \cos \left( 2 \pi \delta \right)$. Since the beaming and planetary phase curve signals have the same frequency (once per orbit) and the phase offset $\delta$ allows them to phase up (at least in principle), they are degenerate; completely independent fits for $A_{\rm beam}$, $A_{\rm planet}$, and $\delta$ are not possible. Figure \[fig:BEER-curve-fit\_Analysis\_of\_Kepler76b\] shows the fit and residuals for Equation \[eqn:BEER\_curve\] to the dataset. Figure \[fig:BEER-curve-fit-params\_Analysis-of-Kepler76b\] shows the resulting best-fit parameters, while Figure \[fig:Aplanet-delta-fit-params\_Analysis-of-Kepler76b\] shows the corresponding range of values for $A_{\rm planet}$ and $\delta$. To generate the latter figure, we calculated $A_{\rm beam}$ from the radial-velocity semi-amplitude $K_{\rm z} = 0.306 \pm 0.020$ km/s reported in @2013ApJ...771...26F using $A_{\rm beam} = 4\ \alpha_{\rm beam}$ and $K_{\rm z}/c = 3.8 \pm 0.3$ ppm, where $c$ is the speed of light [@2003ApJ...588L.117L] – we did not use the $A_{\rm beam}$-value reported in @2013ApJ...771...26F. Then, we Monte-Carlo-sampled 250,000 times from a Gaussian distribution for $A_{\rm beam}$ (with the mean and width given above), as well as the MCMC chains for $A_{\rm sin}$ and $A_{\rm cos}$. Figure \[fig:Aplanet-delta-fit-params\_Analysis-of-Kepler76b\] shows the clear correlation between $A_{\rm beam}$ and $\delta$. As a final check, we fit the out-of-transit/out-of-eclipse data with an LM approach, holding $A_{\rm beam}$ fixed at 13.5 ppm, as reported in @2013ApJ...771...26F, and were able to recover the reported $A_{\rm planet}$ but not the $\delta$-value. ![The blue points in the top panel show photometric measurements for the Kepler-76 system, outside of the planet’s eclipse and transit, folded on the ephemeris reported as in @2013ApJ...771...26F. The white points show these same data binned in 1-min-wide bins, with error bars showing the standard deviation in each bin. The solid orange line shows our best-fit model (Equation \[eqn:BEER\_curve\]), while the dashed black line shows that from @2013ApJ...771...26F. The bottom panel shows the residuals between our model and the data.\[fig:BEER-curve-fit\_Analysis\_of\_Kepler76b\]](f2.png){width="\textwidth"} ![The posterior distributions for the best-fit parameters for the model described by Equation \[eqn:BEER\_curve\] and corresponding to the orange curve in Figure \[fig:BEER-curve-fit\_Analysis\_of\_Kepler76b\]. The histograms along the top-right of each row/column shows the distribution for a parameter marginalized over all the other parameters, while distributions in the shaded contour plots are marginalized over the parameters not labeled to illustrate the correlations between parameters, which appear to be negligible. \[fig:BEER-curve-fit-params\_Analysis-of-Kepler76b\]](f3.png){width="\textwidth"} ![The posterior distributions for the amplitude of the planet’s phase curve and its phase offset, assuming the beaming signal amplitude implied by the radial velocity semi-amplitude reported in @2013ApJ...771...26F. \[fig:Aplanet-delta-fit-params\_Analysis-of-Kepler76b\]](f4.png){width="\textwidth"} Next, we analyzed the in-transit portion of the light curve. Importantly, we accounted for [[*Kepler*]{}]{}’s finite integration time of 30-min-per-point by super-sampling each point, i.e. we modeled the light curve at a cadence of 3-min-per-point and then downsampled to 30-min-per-point [cf. @2010MNRAS.408.1758K] (the finite integration had no significant effect on the Equation \[eqn:BEER\_curve\] model). By first subtracting the Equation \[eqn:BEER\_curve\] model from the in-transit portion of the light curve, we removed its (very small) influence on the transit. Next, we used LM to fit a standard transit curve with quadratic limb-darkening [@2002ApJ...580L.171M][^1] to that portion of the time-series within two transit durations of the reported mid-transit time, allowing the out-of-transit baseline to vary. This analysis netted initial best-fit transit parameters: $a/R_\star$, the ratio of the planet’s semi-major axis to the stellar radius; $b$, the impact parameter; and $R_{\rm p}/R_\star$, the planet-to-star radius ratio. We assumed the orbital eccentricity is equal to zero based on the photometric and radial velocity analyses in @2013ApJ...771...26F and held fixed the limb-darkening coefficients $\gamma_1 = 0.313$ and $\gamma_2 = 0.304$. We explored two other sets of quadratic limb-darkening coefficients, estimated assuming the stellar parameters from @2013ApJ...771...26F and using the model from @2015MNRAS.450.1879E but found different values had negligible impact on our transit analysis. To check for transit-timing variations, we fit each individual transit (with the full orbital light curve this time and not just the in-transit portion) using LM again and held fixed all transit parameters except the mid-transit times and out-of-transit baseline. For uncertainties, we took the square root of the diagonal elements of the resulting covariance matrix [@Press:2007:NRE:1403886 p. 790] and confirmed the accuracy of the mid-transit times and uncertainties by fitting transit curves with an MCMC analysis [@2013PASP..125..306F] to several individual transits – we achieved convergence with 100 walkers, each a chain of 1,000 links and a burn-in phase of 300 links. Although most single transits are detectable, they usually comprise one or two data points and so provide no useful constraints on parameters other than the mid-transit times. Finally, we fit the collection of mid-transit times with both linear and quadratic ephmerides using an MCMC analysis (100 walker chains, 5,000 links in each with a burn-in of 1,500 links) and found no significant departure from the latter. We also found no significant periodicities using a Lomb-Scargle periodogram . Figure \[fig:TTVs\_Analysis\_of\_Kepler76b\] shows the resulting transit ephemeris, which is consistent to within uncertainties with that reported in @2013ApJ...771...26F. ![The observed mid-transit times ($O$) compared to those calculated ($C$) from a linear ephmeris. The orbital period $P$ and initial mid-transit time $T_0$ are shown. The gaps in the data near orbit numbers 300, 550, and 800 represent gaps in the available [[*Kepler*]{}]{} data.\[fig:TTVs\_Analysis\_of\_Kepler76b\]](f5.png){width="\textwidth"} With this updated ephemeris, we folded together all available data and fit the resulting transit portion of the light curve, with free parameters $a/R_\star$, $b$, $R_{\rm p}/R_\star$, and an estimate of the per-point noise (again using Equation \[eqn:likelihood\] for the likelihood). We used the same MCMC analysis as above, with 100 walker chains, each with 32,000 links and a burn-in phase of 16,000 links. We aimed for convergence by requiring small auto-correlation times [e.g., @geyer1992] for the mean chain of each fit parameter (none exceeded 250 links) and all Geweke Z-scores [@Geweke92evaluatingthe] of about 3 or less. Figure \[fig:folded-transit-corner-plot\_Analysis-of-Kepler76b\] shows the resulting parameter distributions, with the best-fit values and uncertainties given in Table \[tbl:Model\_Parameters\], and Figure \[fig:final\_best\_fit\_transit\_Analysis\_of\_Kepler76b\] shows the resulting transit curve and residuals. ![The distributions of best-fit parameters for all the data phase-folded on the ephemeris shown in Figure \[fig:TTVs\_Analysis\_of\_Kepler76b\]: the planet-star radius ratio $R_{\rm p}/R_\star$, the ratio of the planet’s orbital semi-major axis to the stellar radius $a/R_\star$, the impact parameter $b$, and an estimate of the per-point noise in parts-per-million.[]{data-label="fig:folded-transit-corner-plot_Analysis-of-Kepler76b"}](f6.png){width="\textwidth"} ![The top panel shows best-fit transit curve for all the in-transit data phase-folded on the ephemeris shown in Figure \[fig:TTVs\_Analysis\_of\_Kepler76b\], with residuals between the model (orange line) and data (blue points) in the bottom panel.\[fig:final\_best\_fit\_transit\_Analysis\_of\_Kepler76b\]](f7.png){width="\textwidth"} Searching for Eclipse and Planetary Phase Curve Variability {#sec:Searching} ----------------------------------------------------------- We turned to a search for orbit-to-orbit variability in the planet’s phase curve and secondary eclipse. First, we folded together all 944 orbits-worth of data on our best-fit ephemeris and used the LM algorithm to fit the out-of-transit and eclipse portions (accounting for [[*Kepler*]{}]{}’s finite integration time) to establish the average fit-values – the resulting eclipse depth $D = 87 \pm 6$ ppm, a 14.5-$\sigma$ detection. In our search for variability, ideally, we’d analyze the data for each orbit individually to maximize our time resolution. However, with a 14.5-$\sigma$ detection over 944 orbits, we expect that a 3-$\sigma$ detection of the eclipse in the presence of Gaussian noise requires folding together data from about 40 ($=\left( \frac{3}{14.5} \right)^2 \times 944 $) orbits, giving a time resolution of about 60 days. We expect estimates of the parameters $A_{\rm sin}$ and $A_{\rm cos}$ to be more robust, however, since those signals are present throughout most of the orbit (the eclipse only occupies one or two points each orbit). For this analysis, we marched a window spanning 10, 20, or 40 orbits from the beginning to the end of the dataset, moving one orbital period at a time. We folded all the out-of-transit points in the window together and fit both Equation \[eqn:BEER\_curve\] and an eclipse, which is represented by the same transit light curve model as above but with a variable depth $D$ and limb-darkening parameters set to zero (i.e., a uniform disk). We used the LM algorithm and took the square root of the diagonal of the covariance matrix scaled by the square root of the resulting reduced $\chi^2$ as the parameter uncertainties – we compared these uncertainties to those derived from an MCMC analysis for several stacked orbits and found good agreement (to within 1 ppm). To avoid having our window span large gaps in the data, we required a window to include a large enough number of data points that at least half the desired number of orbits were represented. For example, for the window spanning 10 orbits, we required at least 5 orbital periods worth of data (i.e., about $360 = 5 \times 1.5\ {\rm days}/30\ {\rm min}$ points). We took the mid-eclipse time to be the mid-transit time plus half an orbital period, again assuming zero orbital eccentricity. Figure \[fig:planet-phase-curve-var\_Analysis\_of\_Kepler76b\] shows the resulting collection of best-fit parameters for the different windows, where the x-coordinate of each point represents the median observational time for all the points in the window. Since the windows span several orbits and the points are only spaced by one orbital period, the parameter fits are not all statistically independent. ![The leftmost three panels show variations over time in the eclipse depth $D$ and amplitudes $A_{\rm sin}$ and $A_{\rm cos}$ after stacking and folding data points from consecutive orbits in a window 10 orbits wide (blue points), 20 orbits wide (orange points), and 40 orbits wide (green points). The blue horizontal lines show the best-fit average values for each parameter. The rightmost panels show histograms of how much $D$, $A_{\rm sin}$, and $A_{\rm cos}$ deviate from their respective average values, normalized to their respective uncertainties, and the dashed grey lines illustrate $\pm$3-$\sigma$. \[fig:planet-phase-curve-var\_Analysis\_of\_Kepler76b\]](f8.png){width="\textwidth"} The plots seem to show considerable variability, but comparing the variations to the best-fit average values (top right panel of Figure \[fig:planet-phase-curve-var\_Analysis\_of\_Kepler76b\]) shows that the eclipse depth departs from its average value, 87 ppm, by less than 3-$\sigma$, where $\sigma$ is calculated by adding uncertainties in quadrature [@1997ieas.book.....T p. 58]. By contrast, $A_{\rm sin}$ and $A_{\rm cos}$ frequently depart by more than 3- and sometimes 5-$\sigma$ from their average values. As expected, though, the more orbits we stacked together, the more the variations are averaged out, but even the points with 40 orbits stacked together (in green) show statistically significant variation. These results, of course, rely on the accuracy of our uncertainty estimates, so we conducted several numerical tests to check them. First, we generated and fit several synthetic datasets incorporating Gaussian noise and with the same observational times as the real dataset but holding the phase curve parameters and eclipse depth constant. These fits showed variations visually similar to those in Figure \[fig:planet-phase-curve-var\_Analysis\_of\_Kepler76b\], but always the variations in fit parameters were within 3-$\sigma$ of the assumed constant values. We next gathered and phased up about 7,000 datasets involving 10, 20, or 40 orbits worth of out-of-transit data (10 times the number of points in the left panels in Figure \[fig:planet-phase-curve-var\_Analysis\_of\_Kepler76b\]) but randomly scattered throughout the whole [[*Kepler*]{}]{} time-series, i.e., orbits that were not necessarily adjacent in time. We expect that parameter fits to these scattered datasets will show only random scatter, with orbit-to-orbit coherent variations averaged out. We used the Kolmogorov-Smirnov (KS) test [@Press:2007:NRE:1403886 p. 730] to compare the distributions of best-fit parameters (scaled by their respective uncertainties) for these scattered data to the same scaled distributions for the unscattered data (right panels in Figure \[fig:planet-phase-curve-var\_Analysis\_of\_Kepler76b\]). To judge the range of KS scores we should expect, we randomly sub-sampled the distribution of best-fit parameters for the scattered data 10,000 times and compared these sub-samples to one another. The KS test comparisons indicated that the distribution of eclipse depths for the unscattered data shown in Figure \[fig:planet-phase-curve-var\_Analysis\_of\_Kepler76b\] resembled closely the distribution for the scattered data, indicating that the variations in the top left panel of Figure \[fig:planet-phase-curve-var\_Analysis\_of\_Kepler76b\] are consistent with statistical noise. The KS test comparisons for $A_{\rm sin}$ suggests the distribution for the unscattered data is probably inconsistent with random scatter (with a KS probability of about 0.1%), and the KS test comparisons for $A_{\rm cos}$ are totally inconsistent with random scatter (KS probabilities $<10^{-11}$ when 40 orbits are stacked together). This trend is not surprising. We typically detected the $A_{\rm cos}$ signal at somewhat greater significance than the $A_{\rm sin}$, and both were detected at much greater significance than the eclipse. We also employed the same orbit stacking and phasing and fit transit radii (holding other transit parameters fixed) to look for correlations with the phase curve parameters. If the variations in the phase curve arise from some stellar or instrumental effect, we might expect the transit depth to vary in concert; however we found no robust linear correlation. As a final check, we performed an injection-recovery test for which we generated a synthetic dataset designed to closely mimic the original dataset and included the variations in phase curve parameters shown in Figure \[fig:planet-phase-curve-var\_Analysis\_of\_Kepler76b\]. For this test, we divided up the full time-series into individual orbits, calculating the mid-transit time for each orbit. We linearly interpolated to that mid-transit time from among the phase curve parameters for 10 orbits stacked together (blue points in Figure \[fig:planet-phase-curve-var\_Analysis\_of\_Kepler76b\]) to determine the phase curve parameters for the orbit. We generated a phase curve model for that orbit and subtracted it from the original, unconditioned time-series (i.e., from the data shown in the top panel of Figure \[fig:raw-conditioned-data\_Analysis\_of\_Kepler76b\]). Then, we shifted all the mid-transit times forward in time by about 800 days, i.e., half the maximum observational time in the [[*Kepler*]{}]{} dataset – mid-transit times that moved out beyond this maximum time, we wrapped back around to the beginning of the dataset. We next injected synthetic phase curve signals back into these shifted data but this time using the phase curve parameters from the original mid-transit times. This approach has the effect of retaining the time-ordered structure of the phase curve variations while planting the phase curves in a different noise environment. Presumably, if we can recover the phase curve parameters, they are robust against noise. Indeed, after fully re-conditioning these shifted, synthetic data, we were able to recover the vast majority of parameters to within uncertainties. Not surprisingly, for the points not recovered successfully, most were associated with the eclipse depth, which was detected with the least signal-to-noise originally. Results {#sec:Results} ======= Our analysis provides an updated set of transit parameters for the Kepler-76 planetary system that was originally announced in @2013ApJ...771...26F; our ephemeris agrees well with that previous study, while our transit parameters are different. For instance, @2013ApJ...771...26F estimated $R_{\rm p}/R_\star = 0.0968 \pm 0.0003$, while we estimate $R_{\rm p}/R_\star = 0.085 \pm 0.001$, an 11-$\sigma$ discrepancy. We conducted a variety of tests (ignoring the finite-time [[*Kepler*]{}]{} integration, applying an alternative data conditioning approach, etc.) and were unable to reproduce the previously reported value. @2019arXiv190101730H provided equations for calculating the transit depth for a limb-darkened star. Using these equations and the transit parameters given in @2013ApJ...771...26F, we find that the depth expected (0.66%) does not seem to agree with the depth of the transit light curve from our study or that shown in Figure 6 of @2013ApJ...771...26F (about 0.56%). In any case, we corroborate detection of planet-induced ellipsoidal variations and the beaming signal, albeit with a different approach and results. Our amplitude for the ellipsoidal variations $A_{\rm ellip}$ is 4-$\sigma$ smaller than that of @2013ApJ...771...26F. For the beaming signal, we retrieve an amplitude $A_{\rm beam}$ almost 5-$\sigma$ discrepant, although this disagreement may simply arise from our different approach (fitting the planetary phase curve and beaming signals in a non-degenerate way). Using our new transit parameters, we can estimate a new planet-star mass ratio $q$ through the relation $A_{\rm ellip} = \alpha_{\rm ellip} \left( q \sin^2 i \right) \left( R_\star / a \right)^{3}$. In this equation, $\alpha_{\rm ellip}$ depends on the gravity-darkening parameter (0.07 – [@2011AA...529A..75C]) and linear limb-darkening parameter (0.552 – [@2015MNRAS.450.1879E]) as described in @1985ApJ...295..143M. With these values, $\alpha_{\rm ellip} = 1.02$. By re-casting the $A_{\rm ellip}$ equation above in terms of mass ratio $q$ and sampling the distributions of fit parameters from our analysis, we can convert the ellipsoidal signal into $q_{\rm ellip} = \left( 1.9 \pm 0.2 \right) \times 10^{-3}$. In a similar way, we can also estimate a mass ratio for the beaming signal, whose amplitude $A_{\rm beam} = \alpha_{\rm beam} 4 \left( K_{\rm Z} / c \right)$. Taking $\alpha_{\rm beam} = 0.92$, as in @2013ApJ...771...26F, we find $q_{\rm beam} = \left( 1.7\pm 0.1 \right) \times 10^{-3}$, which is in good agreement with our estimate for $q_{\rm ellip}$. With the stellar mass $M_\star = 1.2 \pm 0.2$ solar masses $M_{\odot}$ from @2013ApJ...771...26F, these ratios give planetary masses $M_{\rm p, ellip} = 2.4 \pm 0.3$ Jupiter masses ${\rm M_{Jup}}$ and $M_{\rm p, beam} = 2.1 \pm 0.2\ {\rm M_{Jup}}$, respectively. These values agree with the mass estimates in @2013ApJ...771...26F. Turning to the results from our phase curve analysis, *a priori*, we expect that the dayside temperature of Kepler-76b, in radiative equilibrium, lies between 2,000 and 2,300 K (between the extremes of complete atmospheric redistribution and a uniform temperature dayside with no redistribution to the nightside), either of which puts a significant fraction of its blackbody curve within the [[*Kepler*]{}]{} bandpass. If the planet’s secondary eclipse represented only thermally emitted light, these temperatures correspond to a depth between 4 and 20 ppm. However, Kepler-76b’s average eclipse depth $D = 87 \pm 6$ ppm (within 1.3-$\sigma$ of the result in @2013ApJ...771...26F). With a nightside flux equal to zero, $D$ should be equal to $2\times A_{\rm planet} \cos \left( \phi - \delta \right)$, the planet’s phase curve at mid-eclipse. Converting our average $A_{\rm cos}$ and $A_{\rm sin}$-values, we find that these values agree to within 2.3-$\sigma$, consistent with no nightside flux. This eclipse depth corresponds to a brightness temperature of 2,830$^{+50}_{-30}$ K and suggests there is a significant reflected component, $\geq 67$ ppm. Following @2011MNRAS.415.3921F, we estimate a [[*Kepler*]{}]{} bandpass-integrated geometric albedo $p_{\rm geo} \geq 0.25$, also consistent with @2013ApJ...771...26F. Finally, we turn to the apparent variations in the planet’s phase curve. From Figure \[fig:planet-phase-curve-var\_Analysis\_of\_Kepler76b\], variations in $A_{\rm cos}$ and $A_{\rm sin}$ amount to variations of about 40% in $A_{\rm planet}$ around a median value of $50.5 \pm 1.3$ ppm. Meanwhile, $\delta$ has a median value of $-9^\circ \pm 1.3^\circ$ (an offset east of the sub-stellar point) and swings between about $-40^\circ$ (an eastward offset) and $+20^\circ$ (a westward offset). Figure \[fig:Aplanet-delta-var\_Analysis\_of\_Kepler76b\] shows the range of $A_{\rm planet}$- and $\delta$-values, along with a linear fit which incorporated the per-point uncertainties in $A_{\rm planet}$ and $\delta$ [@boggs1990orthogonal]. The Kepler-76 system closely resembles the HAT-P-7 system – both have hot ($>$ 6,000 K) stars hosting large ($>$ 1 ${\rm M_{Jup}}$) short-period planets with observed [[*Kepler*]{}]{} secondary eclipses indicating dayside brightness temperatures $>$ 2,700 K. As discussed in Section \[sec:Introduction\], @2016NatAs...1E...4A applied a model similar to ours, searching for changes in the planet’s phase curve, finding statistically significant fluctuations in $\delta$ between about $6^\circ$ eastward and $9^\circ$ westward of the substellar point over timescales of tens to hundreds of days. That study then applied a model involving atmospheric transport of aerosols and found good qualitative agreement between the model and their observational results. Our results are qualitatively similar, although with much larger apparent swings in $\delta$. In addition, we were able to detect statistically significant variations in the planet’s phase curve amplitude, while @2016NatAs...1E...4A did not. Discussion and Conclusions {#sec:Discussion} ========================== As discussed in @2016ApJ...828...22P, the phase curve offset for an eclipsing planet arises, in part, from the competition between thermal emission from the hottest regions and reflection from highly refractive cloud regions on the dayside. Although the daysides of hot Jupiters like Kepler-76b are thought to be too hot for aerosol condensation, super-rotation in their atmospheres can transport nightside aersols across the western terminator onto the dayside before they evaporate. The same super-rotation is thought to shift the hottest region on the dayside eastward of the sub-stellar point. Thus, either with homogeneous clouds or without clouds at all, a hot Jupiter’s phase curve would exhibit an eastward (toward $\delta <$ 0) shift. On the other hand, reflective clouds swept over the western terminator would drive the phase curve peak west (toward $\delta > 0$). According to @2016ApJ...828...22P, the degree of phase shift depends, at least in part, on the cloud composition, with some of the more refractory aerosols able to travel farther east than the less refractory ones. Variations in the phase curve may help clear up this cloudy story. For planets where reflection from clouds contributes significantly to the phase curve, we might expect a correlation between the phase curve amplitude and offset – more clouds would produce more signal with a larger westward shift. Figure \[fig:Aplanet-delta-var\_Analysis\_of\_Kepler76b\] shows a robust (17-$\sigma$) positive correlation between $A_{\rm planet}$ and $\delta$. Since the contributions from clouds and from the dayside hotspot are thought to drive the phase curve peak in opposite directions, their relative influences are encoded in this correlation, and future work with more advanced models for the planetary disk [e.g., @2017ascl.soft11019L] could help to tease them out. ![The green points show variations in $A_{\rm planet}$ and $\delta$ for 40 orbits stacked together (i.e., based on the green points in Figure \[fig:final\_best\_fit\_transit\_Analysis\_of\_Kepler76b\]). The black crosses show typical uncertainties, while the dashed blue line shows a linear fit to the green points, with a slope, $m = \left( 1.7 \pm 0.1 \right) ^\circ/{\rm ppm}$. \[fig:Aplanet-delta-var\_Analysis\_of\_Kepler76b\]](f9.png){width="\textwidth"} By making some simplifying assumptions about the phase curve, however, we can draw a few tentative conclusions regarding Kepler-76b’s atmosphere. As discussed above, a plausible maximum equilibrium temperature expected for Kepler-76b is about 2,300 K, and therefore the maximum contribution to the eclipse is probably about 20 ppm. The other 67 ppm must arise from reflected and/or scattered light. Including Rayleigh scattering and thermal emission, the cloudless atmosphere models from @2016ApJ...828...22P suggest an effective geometric albedo in the [[*Kepler*]{}]{} bandpass for Kepler-76b $p_{\rm geo} \approx 0.15$, considerably less than we infer. Thus, some additional component seems to be required. @2016ApJ...828...22P also indicate that aerosols swept from the nightside and evaporated on the dayside are likely confined to near the western (night-to-day) terminator. If reflection from these aerosols made up the remaining 67 ppm signal for the eclipse, however, we might expect the phase curve to be offset to the west (i.e., $\delta > 0$). This line of reasoning suggests Kepler-76b’s dayside is enshrouded by a reflective and homogeneous cloud deck, and the increase in phase curve amplitude and westward shift in offset in Figure \[fig:planet-phase-curve-var\_Analysis\_of\_Kepler76b\] may arise from periodic arrival of additional aerosols from the nightside. As discussed in @2016NatAs...1E...4A, the transport of reflective clouds to the dayside would likely perturb the dayside temperature profiles, potentially reducing the day-night temperature contrast. Since the strength of the super-rotation depends, in part, on this contrast [@2011ApJ...738...71S], a reduction in contrast might reduce both windspeeds and the degree of dayside cloud cover, setting up a feedback. Such a feedback could align with the scenario above, and detailed atmospheric models that include this feedback may help determine the timescales for this feedback for Kepler-76b. In addition to the planet’s day-night temperature contrast and rotation rates (which are relatively well constrained), the timescales presumably depend on the optical properties and distribution of the aerosols, which determine their influence on the temperature structure. Thus, comparing the expected to the observed fluctuation timescales could constrain the aerosol properties and mixing in the planet’s atmosphere. Although the full promise of [[*Kepler*]{}]{} observations for studying phase curve variations remains unfulfilled, the advent of the [[*TESS*]{}]{} Mission portends even more insight into planetary atmospheres. Already, [[*TESS*]{}]{}  is detecting phase curves and secondary eclipses from hot Jupiters [@2018arXiv181106020S], and more are expected. For example, Figure \[fig:eclipse\_estimates\] shows estimates of signal-to-noise ratios (SNR) expected for secondary eclipses observed by [[*TESS*]{}]{} based on the synthetic population of planets in the [[*TESS*]{}]{} yield calculations from @2018ApJS..239....2B. For each synthetic planet from that study, we estimated the eclipse depth $D$ for (1) the case in which the planet reflects all of the light it receives from the star (shown in blue) and (2) the case in which the planet emits (and absorbs) light as a perfect blackbody (shown in orange). For the former case, the eclipse depth is given as $\frac{1}{4} \left( \frac{R_{\rm p}}{a} \right)^2$. We then estimated the corresponding SNR by multiplying the transit SNRs given for each planet by planet-star radius ratio squared and then by $D_{\rm reflected}$. For the reflected light eclipse depths, we used the stellar insolation given for each planet and assumed dayside thermal equilibrium with zero albedo for each planet to estimate a brightness temperature and convolved the resulting blackbody curve against the [[*TESS*]{}]{} spectral response function[^2]. We combined the insolation and planet’s orbital distance to estimate the stellar effective temperature and, likewise, convolved the corresponding blackbody curve against the response function. Finally, we divided the planet’s convolved brightness by the star’s and used the re-scaled transit SNR to calculate the SNR for the thermally emitted eclipse. Figure \[fig:eclipse\_estimates\] only shows those SNR-values $> 3$, among which the vast majority (54 of 62 total) have $R_{\rm p} \geq 10\ R_{\rm Earth}$. The synthetic population from @2018ApJS..239....2B included 4,553 planets, and Figure \[fig:eclipse\_estimates\] suggests only about 1% of these may have secondary eclipses detectable by [[*TESS*]{}]{}. This result differs from a similar analysis in @2015ApJ...809...77S, which suggested only 0.01% of [[*TESS*]{}]{} planets (about 2) will exhibit observable eclipses, with the difference due to the larger population of short-period, giant planets included in the population we used [@2018ApJS..239....2B]. Yet, among those with observable eclipses, our calculation suggests that the eclipse signals will be considerably more sensitive to emitted rather than reflected light, in contrast to the eclipses observed by [[*Kepler*]{}]{}; this is largely due to the fact that the [[*TESS*]{}]{} spectral response function is more sensitive at longer wavelengths than [[*Kepler*]{}]{}. Presumably, this different sensitivity means secondary eclipses detectable by [[*TESS*]{}]{} will reveal atmospheric thermal structures in new ways. However, [[*TESS*]{}]{} will observe many targets for fewer than 30 days, meaning the kind of variability study presented here would be difficult for all but the most pronounced planetary phase curves or those near the continuous viewing area. The planets with the more easily detectable phase curves, hot Jupiters, also produce the deepest transit signals and are most easily studied from the ground. @2018ApJS..239....2B estimated that a significant fraction of [[*TESS*]{}]{} discoveries will be these largest planets, many of which will lie within [[*TESS*]{}]{}’s continuous viewing zone, providing nearly a year of monitoring. For a planet like Kepler-76b, with an orbital period of 1.5 days, these observations would span hundreds of orbits. Thus, in addition to an exciting menagerie of Earths and super-Earths, [[*TESS*]{}]{} is poised to reveal a large population of hot Jupiters, potentially ripe for atmospheric characterization. ![The $\log$ of the signal-to-noise ratio $SNR$ for planetary secondary eclipses for a population of synthetic planets from the [[*TESS*]{}]{} mission yield study of @2018ApJS..239....2B. The blue dots in (a) show eclipses for perfectly reflecting planets, while the orange dots show eclipses for perfect blackbody planets, with black lines connecting the two eclipses for each planet. Panel (b) shows a histogram of SNR-values. \[fig:eclipse\_estimates\]](f10.png){width="\textwidth"} This study is based on work supported by NASA under Grant no. NNX15AB78G issued through the Astrophysical Data Analysis Program by the Science Mission Directorate. The authors gratefully acknowledge assistance from Loretta Cannon in crafting this manuscript and helpful input from Nick Cowan, Lisa Dang, Tamara Rogers, Alexander Hindle, and an anonymous referee. . [^1]: As implemented in PyAstronomy – <https://github.com/sczesla/PyAstronomy>. [^2]: https://tessgi.github.io/TessGiWebsite/the-tess-space-telescope.html

--- abstract: 'Let $T$ be a closed linear relation from a Hilbert space $\sH$ to a Hilbert space $\sK$ and let $B \in \mathbf{B}(\sK)$ be selfadjoint. It will be shown that the relation $T^{*}(I+iB)T$ is maximal sectorial via a matrix decomposition of $B$ with respect to the orthogonal decomposition $\sH=\cdom T^* \oplus \mul T$. This leads to an explicit expression of the corresponding closed sectorial form. These results include the case where $\mul T$ is invariant under $B$. The more general description makes it possible to give an expression for the extremal maximal sectorial extensions of the sum of sectorial relations. In particular, one can characterize when the form sum extension is extremal.' address: - | Department of Mathematics and Statistics\ University of Vaasa\ P.O. Box 700, 65101 Vaasa\ Finland - | Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence\ University of Groningen\ P.O. Box 407, 9700 AK Groningen\ Nederland author: - 'S. Hassi' - 'H.S.V. de Snoo' title: A class of sectorial relations and the associated closed forms --- [^1] Introduction ============ A linear relation $H$ in a Hilbert space $\sH$ is said to be *accretive* if $\RE (h',h) \geq 0$, $\{h,h'\} \in H$. Note that the closure of an accretive relation is also accretive. An accretive relation $H$ in $\sH$ is said to be *maximal accretive* if the existence of an accretive relation $H'$ in $\sH$ with $H \subset H'$ implies $H' = H$. A maximal accretive relation is automatically closed. In a similar way, a linear relation $H$ in a Hilbert space $\sH$ is said to be *sectorial* with vertex at the origin and semi-angle $\alpha$, $\alpha \in [0,\pi /2)$, if $$\label{sect0-} | \IM (h',h) | \le (\tan \alpha) \,\RE (h',h), \quad \{ h, h'\} \in H.$$ The closure of a sectorial relation is also sectorial. A sectorial relation $H$ in a Hilbert space $\sH$ is said to be *maximal sectorial* if the existence of a sectorial relation $H'$ in $\sH$ with $H \subset H'$ implies $H' = H$. A maximal sectorial relation is automatically closed. Note that a sectorial relation is maximal sectorial if and only if it is maximal as an accretive relation; see [@HSnSz09]. A sesquilinear form $\st=\st[\cdot, \cdot]$ in a Hilbert space $\sH$ is a mapping from $\dom \st \subset \sH$ to $\dC$ which is linear in its first entry and antilinear in its second entry. The adjoint $\st^{*}$ is defined by $\st^{*}[h,k]=\overline{\st[k,h]}$, $h,k \in \dom \st$; for the diagonal of $\st$ the notation $\st[\cdot]$ will be used. A (sesquilinear) form is said to be sectorial with vertex at the origin and semi-angle $\alpha$, $\alpha \in [0,\pi/2)$, if $$\label{sec} |\st_{\I} [h]| \le (\tan \alpha) \, \st_{\r} [h], \quad h \in \dom \st,$$ where the real part $\st_{\rm r}$ and the imaginary part $\st_{\I}$ are defined by $$\label{secc} \st_{\rm r}=\frac{\st+\st^{*}}{2}, \quad \st_{\rm i}=\frac{\st-\st^{*}}{2i}, \quad \dom \st_{\rm r}=\dom \st_{\I}=\dom \st.$$ A sesquilinear form will be called a form in the rest of this note. Observe that the form $\st_{\rm r}$ is nonnegative and that the form $\st_{\rm i}$ is symmetric, while $\st=\st_{\rm r}+i\,\st_{\rm i}$. A sectorial form $\st$ is said to be *closed* if $$h_n \to h, \quad \st[h_n-h_m] \to 0 \quad \Rightarrow \quad h \in \dom \st \quad \mbox{and} \quad \st[h_n-h] \to 0.$$ A sectorial form $\st$ is closed if and only if its real part $\st_r$ is closed; see [@Kato]. The connection between maximal sectorial relations and closed sectorial forms is given in the so-called first representation theorem; cf. [@Ar96], [@HSSW17], [@Kato], [@RB90]. \[s-first\] Let $\st$ be a closed sectorial form in a Hilbert space $\sH$ with vertex at the origin and semi-angle $\alpha$, $\alpha \in [0,\pi/2)$. Then there exists a unique maximal sectorial relation $H$ in $\sH$ with vertex at the origin and semi-angle $\alpha$ in $\sH$ such that $$\label{glij} \dom H \subset \dom \st,$$ and for every $\{ h,h' \} \in H$ and $k \in \dom \st$ one has $$\label{eqn:firstrepresentation} \st[h,k] = ( h', k ).$$ Conversely, for every maximal sectorial relation $H$ with vertex at the origin and semi-angle $\alpha$, $\alpha \in [0,\pi/2)$, there exists a unique closed sectorial form $\st$ such that and are satisfied. This result contains as a special case the connection between nonnegative selfadjoint relations and closed nonnegative forms. The nonnegative selfadjoint relation $H_r$ corresponding to the real part $\st_{\r}$ of the form $\st$ is called the real part of $H$; this notion should not to be confused with the real part introduced in [@HSnSz09].\ In the theory of sectorial operators one encounters expressions $T^{*}(I+iB)T$ where $T$ is a linear operator from a Hilbert space $\sH$ to a Hilbert space $\sK$ and $B \in \mathbf{B}(\sK)$ is a selfadjoint operator. In the context of sectorial relations the operator $T$ may be replaced by a linear relation $T$. A frequently used observation is that when $T$ is a closed linear relation and the multivalued part $\mul T$ is invariant under $B$, then the product is a maximal sectorial relation; cf. [@HSSW17]. However, in fact, the relation $$\label{tbt} T^{*}(I+iB)T$$ is maximal sectorial for any closed linear relation $T$. This will be shown in this note via a matrix decomposition of $B$ with respect to the orthogonal decomposition $\sH=\cdom T^* \oplus \mul T$. In addition the closed sectorial form corresponding to $T^{*}(I+iB)T$ will be determined. The main argument consists of a reduction to the case where $T$ is an operator. For the convenience of the reader the arguments in the operator case are included. Note that if $T$ is not closed, then $T^{*}(I+iB)T$ is a sectorial relation which may have maximal sectorial extensions, such as $T^{*}(I+iB)T^{**}$ and some of these extensions have been determined in [@HSS19]; cf. [@ST12]. It is clear that the sum of two sectorial relations is a sectorial relation and there will be maximal sectorial extensions. In [@HSS19] the Friedrichs extension has been determined in general, while the Kreĭn extension has been determined only under additional conditions. As an application of the above results for the relation in the Kreĭn extension and, in fact, all extremal maximal sectorial extensions of the sum of two sectorial relations will be characterized in general. With this characterization one can determine when the form sum extension is extremal. A preliminary result ==================== The first case to be considered is the linear relation $T^{*}(I+iB)T$, where $T$ a closed linear operator, which is not necessarily densely defined, and $B \in \mathbf{B}(\sK)$ is selfadjoint. In this case one can write down a natural closed sectorial form and verify that $T^{*}(I+iB)T$ is the maximal sectorial relation corresponding to the form via Theorem \[s-first\]. \[s-repr0o\] Let $T$ be a closed linear operator from a Hilbert space $\sH$ to a Hilbert space $\sK$ and let the operator $B \in \mathbf{B}(\sK)$ be selfadjoint. Then the form $\st$ in $\sH$ defined by $$\label{henil} \st[h,k]=((I+iB)T h ,T k), \quad h,k \in \dom \st=\dom T,$$ is closed and sectorial with vertex at the origin and semi-angle $\alpha \leq \arctan \|B\|$ and the maximal sectorial relation $H$ corresponding to the form $\st$ is given by $$\label{henill} H= T^*(I+iB)T,$$ with $\mul H=\mul T^*=(\dom T)^\perp$. A subset of $\dom \st = \dom T$ is a core of the form $\st$ if and only if it is a core of the operator $T$. Moreover, the nonnegative selfadjoint relation $H_r$ corresponding to the real part $(\st_H)_{\r}$ of the form $\st$ is given by $$\label{henilll} H_\r=T^*T.$$ It is straightforward to check that $\st$ in is a closed sectorial form as indicated, since $$\st_{r}[h,k]=(Th,Tk), \quad \st_{\I}[h,k]=(B Th,Tk).$$ Therefore, $|\st_{i}[h]|=|(B Th,Th)| \leq \|B\| \|Th\|^{2}=\|B\| \st_{r}[h]$, so that $\st$ is closed and sectorial with vertex at the origin and semi-angle $\alpha \leq \arctan \|B\|$. Moreover, since $T$ is closed, it is clear that $\st_{r}$ and hence $\st$ is closed. Now let $\{h,h'\} \in T^*(I+iB)T$, then there exists $\varphi \in \sK$ such that $$\{h,\varphi\} \in T, \quad \{(I+iB)\varphi, h'\} \in T^*,$$ from which it follows that $$(h',h)=(\varphi, \varphi)+i(B \varphi, \varphi).$$ Consequently, one sees that $$| \IM (h'h) |=|(B\varphi, \varphi)| \leq \|B\|\,\|\varphi\|^2= \|B\| \,\RE (h',h),$$ which implies that $T^*(I+iB)T$ is a sectorial relation with vertex at the origin and semi-angle $\alpha \leq \arctan \|B\|$. Furthermore, observe that the above calculation also shows that $\mul T^*(I+iB)T=\mul T^*$. To see that $T^*(I+iB)T$ is closed, let $\{h_n, h_n'\} \in T^*(I+iB)T$ converge to $\{h,h'\}$. Then there exist $\varphi_n \in \sK$ such that $$\{h_n,\varphi_n\} \in T, \quad \{(I+iB)\varphi_n, h_n'\} \in T^*,$$ and the identity $\RE(h_n',h_n)=\|\varphi_n\|^2$ shows that $(\varphi_n)$ is a Cauchy sequence in $\sK$, so that $\varphi_n \to \varphi$ with $\varphi \in \sK$. Thus $$\{h_n,\varphi_n\} \to \{h,\varphi\}, \quad \{(I+iB)\varphi_n, h_n'\} \to \{(I+iB)\varphi, h'\}.$$ Since $T$ and $T^*$ are closed, one concludes that $ \{h,\varphi\} \in T$ and $\{(I+iB)\varphi, h'\} \in T^*$, which implies that $\{h,h'\} \in T^*(I+iB)T$. Hence $T^*(I+iB)T$ is closed. Now let $H$ be the maximal sectorial relation corresponding to $\st$ in . Assume that $\{h,h'\} \in H$, then for all $k \in \dom \st=\dom T$ $$\st[h,k]=(h',k) \quad \mbox{or} \quad ( (I+iB)Th,Tk)= (h',k),$$ which implies that $$\{(I+iB)Th, h'\} \in T^* \quad \mbox{or} \quad \{h,h'\} \in T^*(I+iB)T.$$ Consequently, it follows that $H \subset T^*(I+iB)T$. Since $T^*(I+iB)T$ is sectorial and $H$ is maximal sectorial, it follows that $H=T^*(I+iB)T$. In particular, one sees that the closed relation $T^*(I+iB)T$ is maximal sectorial. With the closed linear operator $T$ from $\sH$ to $\sK$ and the selfadjoint operator $B \in \bB(\sK)$, consider the following matrix decomposition of $B$ $$\label{Bdec1+} B=\begin{pmatrix} B_{aa}& B_{ab}\\ B_{ba}^*& B_{bb}\end{pmatrix}: \, \begin{pmatrix} \ker T^* \\ \cran T \end{pmatrix} \to \begin{pmatrix} \ker T^* \\ \cran T \end{pmatrix}.$$ Then it is clear that $$\label{Bbb} \st[h,k]=((I+iB)T h ,T k)=((I+iB_{bb})T h ,T k), \quad h,h \in \dom \st=\dom T,$$ which shows that only the compression of $B$ to $\cran T$ plays a role in . In applications involving Theorem \[s-repr0o\], it is therefore useful to recall the following corollary. \[newnew\] Let $T'$ be a closed linear operator from the Hilbert space $\sH$ to a Hilbert space $\sK'$ and let the operator $B' \in \mathbf{B}(\sK')$ be selfadjoint. Assume that the form $\st$ in Theorem \[s-repr0o\] is also given by $$ \st[h,k]=((I+iB')T' h ,T' k), \quad h,k \in \dom \st=\dom T'.$$Then there is a unitary mapping $U$ from $\cran T$ onto $\cran T'$, such that $$T'=UT, \quad B_{bb}'=UB_{bb}U^*,$$ where $B_{bb}$ and $B_{bb}'$ stand for the compressions of $B$ and $B'$ to $\cran T$ and $\cran T'$, respectively. By assumption $((I+iB')T' h ,T' k)= ((I+iB)T h ,T k)$ for all $h,k \in \dom \st$. This leads to $$(T' h ,T' k)= (T h ,T k) \quad \mbox{and} \quad (B' T' h ,T' k)= (B T h ,T k)$$ for all $h,k \in \dom \st$. Hence the mapping $Th \mapsto T'h$ is unitary, and denote it by $U$. Then $T'=UT$ and it follows that $(B' T' h ,T' k)= (B U^*T' h , U^*T' k)$. A matrix decomposition for $T^*(I+iB)T$ {#sect3} ======================================= Let $T$ be a linear relation from $\sH$ to $\sK$ which is closed; observe that then the subspace $\mul T$ is closed. The adjoint $T^{*}$ of $T$ is the set of all $\{h,h'\} \in \sK \times \sH$ for which $$(h',f)=(h,f') \quad \mbox{for all} \quad \{f,f'\} \in T.$$ Hence, the definition of $T^{*}$ depends on the Hilbert spaces $\sH$ and $\sK$ in which $T$ is assumed to act. Let $\sK$ have the orthogonal decomposition $$\label{hilbert} \sK=\cdom T^* \oplus \mul T,$$ and let $P$ be the orthogonal projection onto $\cdom T^*$. Observe that $PT \subset T$, since $\{0\} \times \mul T \subset T$. Therefore $T^{*} \subset (PT)^{*}=T^*P$, where the last equality holds since $P \in \mathbf{B}(\sK)$. Then one has $$\label{hilbert1} (PT)^*=T^* \operatorname{\, \widehat \oplus \,}\, (\mul T \times \{0\}).$$ The *orthogonal operator part* $T_{\rm s}$ of $T$ is defined as $T_{\rm s}=PT$. Hence $T_\op$ is an operator from the Hilbert space $\sH$ to the Hilbert space $\sK$ and $T_\op \subset T$. Note that $\ran T_\op \subset \cdom T^*=\sK \ominus \mul T$. Thus one may interpret $T_\op$ as an operator from the Hilbert space $\sH$ to the Hilbert space $\cdom T^*$ and one may also consider the adjoint $(T_\op)^\times$ of $T_{\s}$ with respect to these spaces. It is not difficult to see the connection between these adjoints: if $\{h,h'\} \in \sK \times \sH$, then $$\label{hilbert17} \{h,h'\} \in T^{*} \quad \Leftrightarrow \quad \{h,h'\} \in (T_{\s})^{\times}.$$ The identity shows the difference between $(T_{\s})^{*}$ and $(T_\op)^\times$. Let $T$ be a closed linear relation from a Hilbert space $\sH$ to a Hilbert space $\sK$ and let $B \in \mathbf{B}(\sK)$ be selfadjoint. In order to study the linear relation $$T^{*}(I+iB)T,$$ decompose the Hilbert space $\sK$ as in and decompose the selfadjoint operator $B \in \mathbf{B}(\sK)$ accordingly: $$\label{Bdec1} B=\begin{pmatrix} B_{11}& B_{12}\\ B_{12}^*& B_{22}\end{pmatrix}: \, \begin{pmatrix} \cdom T^* \\ \mul T \end{pmatrix} \to \begin{pmatrix} \cdom T^* \\ \mul T \end{pmatrix}.$$ Here the operators $B_{11} \in \mathbf{B}(\cdom T^*)$ and $B_{22} \in \mathbf{B}(\mul T)$ are selfadjoint, while $B_{12} \in \mathbf{B}(\mul T, \cdom T^*)$ and $B_{12}^* \in \mathbf{B}(\cdom T^*,\mul T)$. By means of the decomposition the following auxiliary operators will be introduced. First, define the operator $C_0 \in \mathbf{B}(\cdom T^*)$ by $$\label{c0} C_0=I+B_{12}(I+B_{22}^2)^{-1}B_{12}^*.$$ Observe that $C_0 \geq I$ and that $(C_0)^{-1}$ belongs to $\mathbf{B}(\cdom T^*)$ and is a nonnegative operator. Next, define the operator $C \in \mathbf{B}(\cdom T^*)$ by $$\label{c} C=C_0^{-\half}\left[B_{11}-B_{12}(I+B_{22}^2)^{-\half}B_{22} (I+B_{22}^2)^{-\half}B_{12}^*\right]C_0^{-\half},$$ which is clearly selfadjoint. \[redux\] Let $T$ be a closed linear relation from a Hilbert space $\sH$ to a Hilbert space $\sK$, let $T_\op$ be the orthogonal operator part of $T$, and let the selfadjoint operator $B \in \mathbf{B}(\sK)$ be decomposed as in . Let the operators $C_0$ and $C$ be defined by and . Then $$\label{redu} T^*(I + i B)T = (T_{\s})^{\times}C_0^{1/2}(I+i C) C_0^{1/2}T_{\s},$$ and, consequently, $T^*(I + i B)T$ is maximal sectorial and $$\label{redu1} \mul T^*(I + i B)T=\mul T^{*}=\mul (T_{\s})^{\times}.$$ In order to prove the equality in , assume that $\{h,h'\}\in T^* (I + i B) T$. This means that $$\label{eka-} \{h,\varphi\}\in T \quad \mbox{and} \quad \{(I + i B)\varphi,h'\}\in T^*$$ for some $\varphi\in\sK$. Decompose the element $\varphi$ as $$\label{eka--} \varphi=\varphi_1+\varphi_2, \quad \varphi_{1} \in \cdom T^{*}, \,\,\varphi_2\in\mul T.$$ Since $\{0, \varphi_{2}\} \in T$, it is clear that $$\label{eka---} \{h,\varphi\}\in T \quad \Leftrightarrow \quad \{h,\varphi_{1}\}\in T_{\s}.$$ Using and the above decomposition of $B$, one observes that $$\{(I + i B)\varphi,h'\} =\left\{ \begin{pmatrix} (I+i B_{11})\varphi_1+iB_{12}\varphi_2 \\ iB_{12}^*\varphi_1+(I+i B_{22})\varphi_2 \end{pmatrix}, h' \right\},$$ which implies that the condition $\{(I + i B)\varphi,h'\}\in T^*$ is equivalent to $$\left\{ \begin{array}{l} \{ (I+i B_{11})\varphi_1+iB_{12}\varphi_2, h' \} \in T^*, \\ iB_{12}^*\varphi_1+(I+i B_{22})\varphi_2=0, \end{array} \right.$$ or, what is the same thing, $$\label{eka} \left\{ \begin{array}{l} \{ [I+i B_{11} + B_{12}(I+i B_{22})^{-1}B_{12}^*]\varphi_1, h'\} \in T^*, \\ \varphi_2=-i(I+i B_{22})^{-1}B_{12}^*\varphi_1. \end{array} \right.$$ Due to the definitions and and the identity $$(I+i B_{22})^{-1} = (I+B_{22}^2)^{-\half}(I-iB_{22})(I+B_{22}^2)^{-\half},$$ observe that $$\begin{split} &I+i B_{11} + B_{12}(I+i B_{22})^{-1}B_{12}^* \\ &\hspace{1cm} =C_0 + i[B_{11}-B_{12}(I+B_{22}^2)^{-\half}B_{22}(I+B_{22}^2)^{-\half}B_{12}^*]\\ &\hspace{1cm} = C_0^{1/2}(I+i C) C_0^{1/2}. \end{split}$$Therefore, it follows from , via the equivalence in , that $$\label{eka+} \{(I + i B)\varphi, h'\} \in T^* \quad \Leftrightarrow \quad \left\{ \begin{array}{l} \{C_0^{1/2}(I+i C) C_0^{1/2} \varphi_1,h'\} \in (T_{s})^{\times}, \\ \varphi_2=-i(I+i B_{22})^{-1}B_{12}^*\varphi_1. \end{array} \right.$$ Combining and , one sees that $$\{h,h'\} \in (T_{\s})^{\times} C_0^{1/2}(I+i C) C_0^{1/2} T_{\s}.$$ Conversely, if this inclusion holds, then there exists $\varphi_{1} \in \cdom T^{*}$, such that $$\{h, \varphi_{1}\} \in T_{\s} \quad \mbox{and} \quad \{C_0^{1/2}(I+i C) C_0^{1/2} \varphi_1,h'\} \in (T_{\s})^{\times}.$$ Then define $\varphi_2=-i(I+i B_{22})^{-1}B_{12}^*\varphi_1$, so that $\varphi_{2} \in \mul T$. Furthermore, define $\varphi=\varphi_{1}+\varphi_{2}$. Hence $\{h, \varphi\} \in T$, and it follows from that $$\{h,h'\}\in T^* (I + i B) T.$$ Therefore one can rewrite $T^* (I + i B) T$ in the form . Observe that $C_{0}^{\half}T_{\s}$ is a closed linear operator from the Hilbert space $\sH$ to the Hilbert space $\cdom T^{*}$ whose adjoint is given by $$\label{C0Tadjoint} ( C_0^{1/2}T_{\rm s})^{\times}=(T_{\rm s})^{\times} \, C_0^{1/2}.$$ Hence, by Theorem \[s-repr0o\] $(T_{\s})^{\times} C_0^{1/2}(I+i C) C_0^{1/2} T_{\s}$ is a maximal sectorial relation in $\sH$ and by the identity the same is true for $T^*(I+iB)T$. The statement in follows by tracing the above equivalences for an element $\{0,h'\}$. \[rem3.2\] Let $\varphi=\varphi_1+\varphi_2\in \sK$ be decomposed as in . Then one has the following equivalence: $$(I+iB)\varphi\in\cdom T^* \quad \Leftrightarrow\quad (I+iB)\varphi=C_0^{1/2}(I+iC)C_0^{1/2}\varphi_1.$$ To see this, let $\eta =(I+iB)\varphi$. Then $\eta \in \cdom T^*$ if and only if $$\begin{pmatrix} I+i B_{11} & iB_{12} \\ iB_{12}^* & I+i B_{22} \end{pmatrix} \begin{pmatrix} \varphi_1 \\ \varphi_2 \end{pmatrix} = \begin{pmatrix} \eta \\ 0 \end{pmatrix},$$ where $\cdom T^*$ is interpreted as the subspace $\cdom T^*\times \{0\}$ of $\sK$. Now apply . A class of maximal sectorial relations and associated forms {#sect4} =========================================================== The linear relation $T^*(I+iB)T$ is maximal sectorial for any selfadjoint $B \in \mathbf{B}(\sK)$ and any closed linear relation $T$ from $\sH$ to $\sK$. Now the corresponding closed sectorial form will be determined. This gives the appropriate version of Theorem \[s-repr0o\] in terms of relations. In fact, the general result is based on a reduction via Lemma \[redux\] to Theorem \[s-repr0o\]. \[s-repr\] Let $T$ be a closed linear relation from a Hilbert space $\sH$ to a Hilbert space $\sK$ and let the selfadjoint operator $B \in \mathbf{B}(\sK)$ be decomposed as in . Let the operators $C_0$ and $C$ be defined by and . Then the form $\st$ defined by $$\label{henilB} \st[h,k]=((I+iC) \,C_0^{\half}T_{\s} \,h , C_0^{\half}T_{\s}\, k), \quad h,k \in \dom \st= \dom T,$$ is closed and sectorial with vertex at the origin and semi-angle $\gamma \leq \arctan \|C\|$. Moreover, the maximal sectorial relation $H$ corresponding to the form $\st$ is given by $$\label{HwithC} H = (T_{\s})^{\times}\, C_0^{1/2}(I+i C) C_0^{1/2}\,T_{\s}=T^*(I + i B)T.$$ A subset of $\dom \st = \dom T$ is a core of the form $\st$ if and only if it is a core of the operator $T_\op$. Moreover, the nonnegative selfadjoint relation $H_r$ corresponding to the real part $(\st_H)_r$ of the form $\st$ is given by $$H_r=(T_{\s})^{\times}C_0T_{\s}.$$ Since $C_{0}^{\half}T_{\s}$ is a closed linear operator from the Hilbert space $\sH$ to the Hilbert space $\cdom T^{*}$, Theorem \[s-repr0o\] (with $\sK$ replaced by $\cdom T^{*}$, $B$ by $C$, and $T$ by $C_0^{1/2}T_{\rm s}$) shows that the form $\st$ in is closed and sectorial with vertex at the origin and semi-angle $\gamma \leq \arctan \|C\|$. Moreover, the maximal sectorial relation associated with the form $\st$ is given by $$( C_0^{1/2}T_{\rm s})^{\times}(I+i C) C_0^{1/2}T_{\rm s} = (T_{\rm s})^{\times}C_0^{1/2}(I+i C) C_0^{1/2}T_{\rm s},$$ cf. , , and . The identities in are clear from Lemma \[redux\]. The assertion concerning the core holds, since the factor $C_0$ is bounded with bounded inverse. The formula shows that $$(\st_H)_r[h,k]=(C_0^{\half}T_\op h ,C_0^{\half}T_\op k), \quad h,k \in \dom \st = \dom T,$$ and hence $H_{\rm r}=(C_0^{1/2}T_\op)^{\times}C_0^{1/2}T_\op =(T_\op)^{\times}C_0 T_\op$ (cf.the discussion above). Recall that if $\{h,h'\}\in T^* (I + i B) T$, then $\{h,\varphi\}\in T$ and $\{(I + i B)\varphi,h'\}\in T^*$. The last inclusion implies the condition $(I+iB)\varphi \in \dom T^{*}\subset \cdom T^{*}$, giving rise to $\varphi_2=-i(I+i B_{22})^{-1}B_{12}^*\varphi_1$. Thus, for instance, when $B=\diag (B_{11}, B_{22})$, it follows that $\varphi_{2}=0$, so that it is immediately clear that $\gamma \leq \arctan \|B_{11}\|$, independent of $B_{22}$. Note that the following assertions are equivalent: 1. $B=\diag (B_{11}, B_{22})$; 2. $B_{12}=0$; 3. $C_0=I$; 4. $\mul T$ is invariant under $B$, in which case $C=B_{11}$. Hence, if $\mul T$ is invariant under $B$, i.e., if any of the assertions (i)–(iv) hold, then Theorem \[s-repr\] gives the following corollary, which coincides with [@HSSW17 Theorem 5.1]. In the case where $\mul T=\{0\}$ the corollary reduces to Theorem \[s-repr0o\]. \[old\] Let $T$ be a closed linear relation from a Hilbert space $\sH$ to a Hilbert space $\sK$, let $T_\op$ be the orthogonal operator part of $T$, and let $\mul T$ be invariant under the selfadjoint operator $B \in \mathbf{B}(\sK)$, so that $B=\diag (B_{11}, B_{22})$. Then the form $\st$ defined by $$\st[h,k]=((I+iB_{11})T_{\s}h, T_{\s}k), \quad h,k \in \dom \st=\dom T,$$ is closed and sectorial with vertex at the origin and semi-angle $\gamma \leq \arctan \|B_{11}\|$. Moreover, the maximal sectorial relation $H$ corresponding to the form $\st$ is given by $$H=(T_{\s})^{\times} (I+iB_{11}) T_{\s}=T^{*}(I+iB)T.$$ In the case that $\mul T$ is not invariant under $B$, one has $C_0\neq I$, and the formulas are different: for instance, the real part $(\st_H)_r$ in Theorem \[s-repr\] is of the form $$(\st_H)_r[h,k]=(C_0^{\half}T_\op h ,C_0^{\half}T_\op k), \quad h,k \in \dom \st=\dom T_\op = \dom T.$$ \[examp\] Assume that $B_{11} \neq 0$ and $$B_{11}=B_{12}(I+B_{22}^2)^{-\half}B_{22}(I+B_{22}^2)^{-\half}B_{12}^*,$$ so that $C=0$. In this case the maximal sectorial relation $H=T^*(I+iB)T$ in Theorem \[s-repr\] is selfadjoint, i.e., $H=H_r$ and the associated form $\st$ is nonnegative. On the other hand, with such a choice of $B$ the operator part of $T$ determines the maximal sectorial relation $(T_{\rm s})^*(I + iB)T_{\rm s}$ with semi-angle $\arctan \|B_{11}\|>0$, while $T^*(I + i B)T$ has semi-angle $\gamma=0$. Maximal sectorial relations and their representations ===================================================== Let $H$ be a maximal sectorial relation in $\sH$ and let the closed sectorial form $\st_H$ correspond to $H$; cf. Theorem \[s-first\]. Since the closed form $\st_H$ is sectorial, one has the inequality $$\label{henillla0} |(\st_{H})_{\rm i}[h] | \leq (\tan \alpha) (\st_{H})_{\rm r}[h], \quad h \in \dom \st,$$ and in this situation the real part $(\st_{H})_{\rm r}$ is a closed nonnegative form. Hence by the first representation theorem there exists a nonnegative selfadjoint relation $H_{\rm r}$, the so-called *real part* of $H$, such that $\dom H_{\rm r} \subset \dom (\st_{H})_{\rm r}=\dom \st_{H}$ and $$(\st_{H})_{\rm r}[h,k] = ( h', k ), \quad \{ h,h' \} \in H_{\rm r}, \quad k \in \dom (\st_{H})_{\rm r}=\dom \st_{H}.$$This real part $H_{\rm r}$, not to be confused with the real part introduced in [@HSnSz09], will play an important role in formulating the second representation theorem below. First the case where $H$ is a maximal sectorial operator will be considered, in which case $H$ is automatically densely defined; see [@Kato]. \[s-thirdLemma\] Let $H$ be an maximal sectorial operator in $\sH$, let the closed sectorial form $\st_H$ correspond to $H$ via Theorem \[s-first\], and let $H_{\rm r}$ be the real part of $H$. Then there exists a unique selfadjoint operator $G \in \mathbf{B}(\sH)$ with $\| G \| = \tan \alpha$, of the form $$\label{Bdec1++} G=\begin{pmatrix} 0& 0\\ 0& G_{bb}\end{pmatrix}: \, \begin{pmatrix} \ker H_{\rm r} \\ \cran H_{\rm r} \end{pmatrix} \to \begin{pmatrix} \ker H_{\rm r} \\ \cran H_{\rm r} \end{pmatrix},$$ such that $$\label{sss--} \st_H[h,k]=((I +iG) (H_{r})^{\half} h, (H_{r})^{\half} k), \quad h,k \in \dom \st_H =\dom H_{r}^{\half}.$$ Moreover, the corresponding maximal sectorial operator $H$ is given by $$H=(H_{\rm r})^{\half}(I+iG)(H_{\rm r})^{\half},$$ with $\mul H=\mul H_{\rm r}$. The inequality $$| (\st_{H})_{\rm i}[h,k] |^{2} \leq C \st_{\rm r}[h] \st_{\rm r}[k] =C\|H_{\rm r}^{\half} h\| \|H_{\rm r}^{\half} k\|, \quad h,k \in \dom,$$ shows the existence of a selfadjoint operator $G$ in $\sH \ominus \ker H$ such that $$\label{henilllaa} (\st_{H})_{\rm i}[h,k]=(G (H_{\rm r})^{\half} h, (H_{\rm r})^{\half} k), \quad h, k \in \dom (H_{\rm r})^{\half}.$$ Extend $G$ to all of $\sH$ in a trivial way, so that the same formula remains valid; see Corollary \[newnew\]. It follows from the decomposition $\st=\st_{\r} +i \st_{\I}$, cf. , and the identities and , that $$\st_{H}= (\st_{H})_{\rm r}+i (\st_{H})_{\rm i},$$ so that $$\st_{H}=[h,k]= ( (H_{r})_{\s}^{\half}h, (H_{r})_{\s}^{\half} k) +i ( G(H_{r})_{\s}^{\half}h, (H_{r})_{\s}^{\half} k).$$ This last identity immediately gives . The rest follows from Corollary \[old\]. Now let $H$ be a maximal sectorial relation, let $H_{\r}$ be its real part, and let $(H_{r})_{\s}$ be its orthogonal operator part. Then one obtains the representation $$\label{henillla} (\st_{H})_{\rm r}[h,k]=( ((H_{r}))_{\s}^{\half}h, ((H_{r})_{\s})^{\half} k), \quad h,k \in \dom (\st_{H})_{\rm r}=\dom ((H_{r})_{\s})^{\half},$$ cf. Theorem \[s-repr0o\]. Now apply Corollary \[old\] and therefore one may formulate the second representation theorem as follows. \[s-third\] Let $H$ be a maximal sectorial relation in $\sH$, let the closed sectorial form $\st_H$ correspond to $H$ via Theorem \[s-first\], and let $H_{\rm r}$ be the real part of $H$. Then there exists a selfadjoint operator $G \in \mathbf{B}(\sH)$ with $\| G \| = \tan \alpha$, such that $G$ is trivial on $\ker H_{\rm r} \oplus \mul H_{\rm r}$, and $$\label{sss-} \st_H[h,k]=((I +iG) ((H_{\rm r})_{\rm s})^{\half} h, ((H_{\rm r})_{\rm s})^{\half} k), \quad h,k \in \dom \st_H =\dom H_{\rm r}^{\half}.$$ Moreover, the maximal sectorial relation $H$ is given by $$\label{sss-+} H=(((H_{\rm r})_{s})^{\half})^{\times}(I+iG)((H_{\rm r})_{s})^{\half},$$ with $\mul H=\mul H_{\rm r}$. Next, it is assumed that $H$ is a maximal sectorial relation of the form $H=T^{*}(I+iB)T$, where $T$ is a closed linear relation from a Hilbert space $\sH$ to a Hilbert space $\sK$ and the operator $B \in \mathbf{B}(\sK)$ is selfadjoint. Let the operators $C_0$ and $C$ be defined by and , then $$H= (T_{\s})^{\times}\, C_0^{1/2}(I+i C) C_0^{1/2}\,T_{\s},$$ while the corresponding closed sectorial form is given $$\st[h,k]=((I+iC) \,C_0^{\half}T_{\s} \,h , C_0^{\half}T_{\s}\, k), \quad h,k \in \dom \st= \dom T.$$ To compare these expressions with and , observe that $$(\,C_0^{\half}T_{\s} \,h , C_0^{\half}T_{\s}\, k) =((H_{r})_{\rm s})^{\half} h, ((H_{r})_{\rm s})^{\half} k)$$ and $$(C \,C_0^{\half}T_{\s} \,h , C_0^{\half}T_{\s}\, k) =(G ((H_{r})_{\rm s})^{\half} h, ((H_{r})_{\rm s})^{\half} k).$$ It is clear from that only the (selfadjoint) compression of $C$ to $\cran C_0^{1/2}T_{\rm s}$ contributes to the form , so that it is straightforward to set up a unitary mapping; cf. Corollary \[newnew\]. Extremal maximal sectorial extensions of sums of maximal sectorial relations ============================================================================ Let $H_{1}$ and $H_{2}$ be maximal sectorial relations in a Hilbert space $\sH$. Then the sum $H_{1}+H_{2}$ is a sectorial relation in $\sH$ with $$\dom (H_{1}+H_{2})=\dom H_{1} \cap \dom H_{2},$$ so that the sum is not necessarily densely defined. In particular, $H_1+H_2$ and its closure need not be operators, since $$\label{s-sum-mulll} \mul (H_{1}+H_{2})=\mul H_{1} + \mul H_{2}.$$ To describe the class of extremal maximal sectorial extensions of $H_1+H_2$ some basic notations are recalled from [@HSS19], together with the description of the Friedrichs and Kreĭn extensions $$(H_1+H_2)_F \quad \mbox{and} \quad (H_1+H_2)_K$$ of $H_1+H_2$, respectively. In order to describe the whole class of extremal extensions of $H_1+H_2$ and the corresponding closed forms a proper description of the closed sectorial form $\st_K$ is essential. The results in Sections \[sect3\] and \[sect4\] allow a general treatment that will relax the additional conditions in [@HSS19]. Basic notions ------------- Let $H_{1}$ and $H_{2}$ be maximal sectorial relations and decompose them as follows $$\label{H12} H_{j} = A_{j}^{\half} (I+iB_{j}) A_{j}^{\half}, \quad 1 \le j \le 2,$$ where $A_{j}$ (the real part of $H_{j}$), $1 \le j \le 2,$ are nonnegative selfadjoint relations in $\sH$ and $B_{j}$, $1 \le j \le 2,$ are bounded selfadjoint operators in $\sH$ which are trivial on $\ker A_j \oplus \mul A_j$; cf. Theorem \[s-third\]. Furthermore, if $A_{1}$ and $A_{2}$ are decomposed as $$A_{j}=A_{j\s} \oplus A_{j \infty}, \quad 1 \le j \le 2,$$ where $A_{j \infty} =\{0\} \times \mul A_{j}$, $1 \le j \le 2$, and $A_{j\s}$, $1 \le j \le 2$, are densely defined nonnegative selfadjoint operators (defined as orthogonal complements in the graph sense), then the uniquely determined square roots of $A_{j}$, $1 \le j \le 2$ are given by $$A_{j}^\half=A_{j\s}^\half \oplus A_{j \infty}, \quad 1 \le j \le 2.$$ Associated with $H_{1}$ and $H_{2}$ is the relation $\Phi$ from $\sH \times \sH$ to $\sH $, defined by $$\label{s-s-Einz} \Phi= \left\{\, \left\{ \{f_{1},f_{2}\}, f_{1}' + f_{2}' \right\} :\, \{f_{j},f_{j}' \} \in A_{j}^{\half} , \, 1 \le j \le 2\,\right\}.$$ Clearly, $\Phi$ is a relation whose domain and multivalued part are given by $$\dom \Phi = \dom A_{1}^{\half} \times \dom A_{2}^{\half}, \quad \mul \Phi=\mul H_{1} +\mul H_{2}.$$ The relation $\Phi$ is not necessarily densely defined in $\sH \times \sH$, so that in general $\Phi^*$ is a relation as $\mul \Phi^*=(\dom \Phi)^\perp$. Furthermore, the adjoint $\Phi^*$ of $\Phi$ is the relation from $\sH$ to $\sH \times \sH$, given by $$\label{s-s-Zwei} \Phi^* = \left\{ \left\{ h, \{ h_{1}', h_{2}'\} \right\} \, : \{h, h_{j}'\} \in A_{j}^{\half}, \, 1 \le j \le 2 \right\}.$$ The identity shows that the (orthogonal) operator part $(\Phi^*)_\s$ of $\Phi^*$ is given by: $$\begin{aligned} \label{s-s-qus} (\Phi^*)_\s & = &\left\{ \left\{ h, \{ h_{1}', h_{2}'\} \right\} \, : \{h, h_{j}'\} \in A_{j\s}^{\half}, \, 1 \le j \le 2 \right\} \\ & = & \left\{ \left\{ h, \{ A_{1\s}^{\half} h, A_{2\s}^{\half} h\} \right\} \, : h \in \dom A_{1}^{\half} \cap \dom A_{2}^{\half} \right\}. \nonumber\end{aligned}$$ The identities and show that $$\dom \Phi^{*} = \dom A_{1}^{\half} \cap \dom A_{2}^{\half}, \, \, \mul \Phi^*=\mul H_{1} \times \mul H_{2}, \, \, \ran (\Phi^{*})_\s = \sF_{0},$$ where the subspace $\sF_{0} \subset \sH \times \sH$ is defined by $$\label{F0} \sF_{0} = \left\{ \left\{ A_{1\s}^{\half} h , A_{2\s}^{\half} h \right\} \, : \, h \in \dom A_{1}^{\half} \cap \dom A_{2}^{\half} \right\}.$$ The closure of $\sF_{0}$ in $\sH \times \sH$ will be denoted by $\sF$. Define the relation $\Psi$ from $\sH$ to $\sH \times \sH$ by $$\label{s-s-qu} \Psi=\left\{\, \left\{h, \left\{A_{1s}^{\half} h, A_{2s}^{\half} h\right\}\right\} :\, h \in \dom H_{1} \cap \dom H_{2} \,\right\} \subset \sH \times (\sH \times \sH).$$ It follows from this definition that $$\dom \Psi=\dom H_{1} \cap \dom H_{2}, \quad \mul \Psi=\{0\}, \quad \ran \Psi=\sE_0,$$ where the space $\sE_0 \subset \sH \times \sH$ is defined by $$\label{E0} \sE_0 = \left\{\, \left\{A_{1\s}^{\half}f , A_{2\s}^{\half}f\right\}:\, f\in \dom H_{1} \cap \dom H_{2} \,\right\}.$$ Observe that $\sE_0 \subset \sF_0$. The closure of $\sE_{0}$ in $\sH \times \sH$ will be denoted by $\sE$. Hence, $$\label{s-s-ef} \sE \subset \sF.$$ Comparison of and shows $$\label{Psiclos} \Psi \subset (\Phi^*)_\s,$$ and thus the operator $\Psi$ is closable and $\Psi^{**} \subset (\Phi^*)_\s$. It follows from $\cdom \Psi^*=(\mul \Psi^{**})^\perp$ and $\mul \Psi^*=(\dom \Psi)^\perp$, that $$\cdom \Psi^*=\sH , \quad \mul \Psi^* =(\dom H_{1} \cap \dom H_{2})^\perp.$$ Next, define the relation $K $ from $\sH \times \sH$ to $\sH$ by $$\begin{aligned} \label{s-s-opK} K & = & \big\{ \{\{ (I+iB_{1})A_{1\s}^{\half} f, (I+iB_{2})A_{2\s}^{\half}f \}, f_{1}' + f_{2}' \}\, : \\ && \quad \quad \quad \{ (I+iB_{1})A_{1\s}^{\half} f, f_{1}' \} \in A_{1}^{\half}, \{ (I+iB_{2})A_{2\s}^{\half} f, f_{2}' \} \in A_{2}^{\half} \big\} \nonumber \\ && \hspace{-0.6cm} \subset \hspace{0.2cm} (\sH \times \sH) \times \sH. \nonumber\end{aligned}$$ Clearly, the domain and multivalued part of $K$ are given by $$\dom K= \sD_0, \quad \mul K=\mul (H_{1} + H_{2}),$$ where $$\label{domK} \sD_{0}= \left\{\,\{(I+iB_{1}) A_{1\s}^{1/2}f, (I+iB_{2}) A_{2\s}^{1/2}f\} : \, f\in \dom H_{1} \cap \dom H_{2} \,\right\} .$$ The closure of $\sD_{0}$ in $\sH \times \sH$ will be denoted by $\sD$. \[lem3.1\] The relations $K$, $\Phi$, and $\Psi$ satisfy the following inclusions: $$\label{s-s-trits} K \subset \Phi \subset \Psi^*, \quad \Psi \subset \Phi^* \subset K^*.$$ To see this note that $K \subset \Phi$ follows from and , and that $\Psi \subset \Phi^*$ follows from and . Therefore, also $\Phi^* \subset K^*$ and $\Phi \subset \Phi^{**} \subset \Psi^*$. The Friedrichs and the Kreĭn extensions of $H_1+H_2$ {#sec3.2} ---------------------------------------------------- The descriptions of the Friedrichs extension and the Kreĭn extension $(H_{1}+H_{2})_F$ and $(H_{1}+H_{2})_K$ of $H_1+H_2$ are now recalled from [@HSS19]. For this, define the orthogonal sum of the operators $B_{1}$ and $B_{2}$ in $\sH \times \sH$ by $$B_\oplus:=B_1 \oplus B_2 = \begin{pmatrix} B_{1} & 0 \\ 0 & B_{2} \end{pmatrix}.$$ The descriptions of $(H_{1}+H_{2})_F$ and $(H_{1}+H_{2})_K$ incorporate the initial data on the factorizations of $H_1$ and $H_2$ via the mappings $\Phi$, $\Psi$, and $K$ in Subsection \[s-sum-mulll\]. The construction of the Friedrichs extension was given in [@HSS19 Theorem 3.2], where some further details and a proof of the following result can be found. The new additions in the next theorem are the second representations for $(H_{1}+H_{2})_F$ and $\st_{F}$ that will be needed in the rest of this paper. \[ss-twee\] Let $H_{1}$ and $H_{2}$ be maximal sectorial and let $\Psi$ be defined by . Then the Friedrichs extension of $H_{1}+H_{2}$ has the expression $$\label{HF2} (H_{1}+H_{2})_F=\Psi^* (I + i B_\oplus) \Psi^{**}=\Psi^* C_0^{1/2}(I+iC)C_0^{1/2} P_\sD (\Psi^{**})_{\s}.$$ The closed sectorial form $\st_{F}$ associated with $(H_{1}+H_{2})_F$ is given by $$\label{tF2} \st_{F} [f,g]= ((I + iB_\oplus ) \Psi^{**} f, \Psi^{**} g) = (C_0^{1/2}(I+iC)C_0^{1/2} P_\sD (\Psi^{**})_{\s} f,P_\sD (\Psi^{**})_{\s} g),$$ for all $f, \, g \in \dom \st_{F} = \dom \Psi^{**}$. As indicated the first expressions for $(H_{1}+H_{2})_F$ in and $\st_F$ in have been proved in [@HSS19 Theorem 3.2] and, hence, it suffices to derive the second expressions in and . By definition, one has $\ran \Psi=\sE_0$ (see , ), and by Lemma \[lem3.1\] one has $\Psi\subset \Psi^{**}\subset K^*$, which after projection onto $\sD=\cdom K$ yields $$P_\sD \Psi^{**}\subset P_\sD K^*=(K^*)_\s.$$ Notice that $\sD_0=\dom K=(I+i B_\oplus)\sE_0$ (see , ). Since the operator $I+iB_\oplus$ is bounded with bounded inverse, one has the equality $$\label{sDsE} \sD=(I+i B_\oplus)\sE.$$ It follows that the range of $(I+i B_\oplus)\Psi^{**}$ belongs to $\sD=\cdom K$. Now by Remark \[rem3.2\] this implies that for all $f\in \dom \Psi^{**}$ one has the equality $$\label{neweq00} (I+B_\oplus)(\Psi^{**})_{\s}f = C_0^{1/2}(I+iC)C_0^{1/2} P_\sD (\Psi^{**})_{\s}f.$$ This leads to $$\Psi^* (I + i B_\oplus) \Psi^{**} = \Psi^* C_0^{1/2}(I+iC)C_0^{1/2} P_\sD (\Psi^{**})_{\s},$$ which proves . Similarly by substituting into the first formula for $\st_F$ and noting that $P_\sD C_0^{1/2}=P_\sD$, one obtains the second formula in . Also the construction of the Kreĭn extension for the sum $H_1+H_2$ can be found in [@HSS19 Theorem 3.2]. However, the corresponding form $\st_K$ was described only under additional conditions to prevent the difficulty that appears by the fact that the multivalued part of $(H_1+H_2)_K$ is in general not invariant under the mapping $B_\oplus$. Theorem \[s-repr\] allows a removal of these additional conditions and leads to a description of the form $\st_K$ in the general situation. For this purpose, decompose the Hilbert space $\sH\times \sH$ as follows $$\label{hilbertn} \sH\times\sH=\cdom K \oplus \mul K^*,$$ and let $P$ be the orthogonal projection onto $\cdom K$. Moreover, decompose the selfadjoint operator $B_\oplus \in \mathbf{B}(\sH\times\sH)$ accordingly: $$\label{Bplusdec1} B_\oplus=\begin{pmatrix} B_{11}& B_{12}\\ B_{12}^*& B_{22}\end{pmatrix}: \, \begin{pmatrix} \cdom K \\ \mul K^* \end{pmatrix} \to \begin{pmatrix} \cdom K \\ \mul K^* \end{pmatrix}.$$ Next define the operator $C_0 \in \mathbf{B}(\cdom K^*)$ by $$\label{c0plus} C_0=I+B_{12}(I+B_{22}^2)^{-1}B_{12}^*,$$ and the operator $C \in \mathbf{B}(\cdom K^*)$ by $$\label{cplus} C=C_0^{-\half}\left[B_{11}-B_{12}(I+B_{22}^2)^{-\half}B_{22} (I+B_{22}^2)^{-\half}B_{12}^*\right]C_0^{-\half},$$ which is clearly selfadjoint. \[KVNext\] Let $H_{1}$ and $H_{2}$ be maximal sectorial relations in a Hilbert space $\sH$, let $K$ be defined by , and let $C_0$ and $C$ be given by and , respectively. Then the Kreĭn extension of $H_{1} + H_{2}$ has the expression $$(H_{1} + H_{2})_K=K^{**}(I + i B_\oplus)K^{*} = ((K^*)_{\s})^{\times}\, C_0^{1/2}(I+i C) C_0^{1/2}\,(K^*)_{\s}.$$ The closed sectorial form $\st_K$ associated with $(H_{1} + H_{2})_K$ is given by $$\st_{K} [f,g] = ((I+i C) C_0^{1/2} (K^{*})_{\s} f,C_0^{1/2} (K^{*})_{\s} g), \quad f, \, g \in \dom \st_{K} = \dom K^{*}.$$ The first equality in the first statement is proved in [@HSS19 Theorem 3.2]. The second equality is obtained by applying Theorem \[s-repr\] to the sectorial relation $K^{**}(I + i B_\oplus)K^{*}$. The statement concerning the form $\st_{K}$ is a consequence of this second representation of $(H_{1} + H_{2})_K$, since $C_0^{1/2} (K^{*})_{s}$ is a closed operator and hence one can apply Theorem \[s-repr0o\] to get the desired expression for the corresponding form $\st_K$. The form $\st_K$ described in Theorem \[KVNext\] can be used to give a complete description of all *extremal maximal sectorial extensions* of the sum $H_1+H_2$. Namely, a maximal sectorial extension $\wt H$ of a sectorial relation $S$ is extremal precisely when the corresponding closed sectorial form $\st_{\wt H}$ is a restriction of the closed sectorial form $\st_K$ generated by the Kreĭn extension $S_K$ of $S$; see e.g. [@HSSW17 Definition 7.7, Theorems 8.2, 8.4, 8.5]. Therefore, Theorem \[KVNext\] implies the following description of all extremal maximal sectorial extensions of $H_1+H_2$. \[s-sum-caracter\] Let $H_{1}$ and $H_{2}$ be maximal sectorial relations in $\sH$, let $\Psi$ and $K$ be defined by and , respectively, and let $P_\sD$ be the orthogonal projection from $\sH\times\sH$ onto $\sD=\cdom K$. Then the following statements are equivalent: 1. $\widetilde{H}$ is an extremal maximal sectorial extension of $H_{1}+H_{2}$; 2. $\widetilde{H} = R^{\ast} (I+i C ) R$, where $R$ is a closed linear operator satisfying $$C_0^{1/2} P_\sD \Psi^{**} \subset R \subset C_0^{1/2}(K^*)_\s.$$ For comparison with the abstract results this statement will be proved by means of the constructions used in [@HSSW17]. Let $S=H_1+H_2$ then the sectorial relation $S$ gives rise to a Hilbert space $\sH_S$ and a selfadjoint operator $B_S \in \mathbf{B}(\sH_S)$ such that the Friedrichs extension $S_F$ and the Kreĭn extension $S_K$ of $S$ are given by $$S_F=Q^*(I+iB_S)Q^{**}, \quad t_F=J^{**}(I+iB_S)J^*,$$ with corresponding forms $$\st_F[f,g]=((I+iB_S)Q^{**} f, Q^{**}g), \quad f,g \in \dom Q^{**},$$ and $$\st_K[f,g]=((I+iB_S)J^{*} f, J^{*}g), \quad f,g \in \dom J^{*};$$ see [@HSSW17 Theorem 8.3]. Here $Q: \sH \to \sH_S$ is an operator and $J: \sH_S \to \sH$ is a densely defined linear relation such that $$J \subset Q^*, \quad Q \subset J^*;$$ in particular, the adjoint $J^*$ is an operator. Recall from Theorem \[KVNext\] that $$\st_{K} [f,g] = ((I+i C) C_0^{1/2} (K^{*})_{\s} f,C_0^{1/2} (K^{*})_{\s} g),$$ while Theorem \[ss-twee\] gives $$\st_{F} [f,g]= ((I + iB_\oplus ) \Psi^{**} f, \Psi^{**} g) = (C_0^{1/2}(I+iC)C_0^{1/2} P_\sD (\Psi^{**})_{s} f,P_\sD (\Psi^{**})_{s} g).$$ Now apply [@HSSW17 Theorem 8.4] and Corollary \[newnew\]. The form sum construction {#sec3.3} ------------------------- The maximal sectorial relations $H_{1}$ and $H_{2}$ generate the following closed sectorial form $$\label{s-s-fs} ((I+iB_{1})A_{1\s}^{\half} h, A_{1\s}^{\half} k) +((I+iB_{2})A_{2\s}^{\half} h, A_{2\s}^{\half} k), \quad h,k \in \dom A_{1}^{\half} \cap \dom A_{2}^{\half}.$$ Observe that the restriction of this form to $\dom \Psi^{**}$ is equal to $$\label{s-sum-form} (\Psi^{**}h, \Psi^{**}k) = ((I+iB_{1})A_{1\s}^{\half} h, A_{1\s}^{\half} k) +((I+iB_{2})A_{2\s}^{\half} h, A_{2\s}^{\half} k), \quad h,k \in \dom \Psi^{**},$$ since $\Psi^{**} \subset (\Phi^*)_\s$, cf. . Thus, the form in has a natural domain which is in general larger than $\dom \Psi^{**}$. \[s-sum-een\] Let $H_{1}$ and $H_{2}$ be maximal sectorial relations in $\sH$, let $\Phi$ be given by , and let $\sE=\clos \sE_0$ and $\sF=\clos \sF_0$ be defined by amd . Then the maximal sectorial relation $$\Phi^{**} (I+iB_\oplus ) \Phi^*$$ is an extension of the relation $H_{1}+H_{2}$, which corresponds to the closed sectorial form in . Moreover, the following statements are equivalent: 1. $\Phi^{**} (I+iB_\oplus ) \Phi^*$ is extremal; 2. $\sE=\sF$. The first statement is proved in [@HSS19 Theorem 3.5]. For the proof of the equivalence of (i) and (ii) appropriate modifications are needed in the arguments used in the proof of [@HSS19 Theorem 3.5]. The special case treated there was based on the additional assumption that $\sD=\sE$, where $\sD=\cdom K$; a condition which implies the invariance of $\mul K^*$ under the operator $B_\oplus$. In the present general case such an invariance property cannot be assumed. Now for simplicity denote the form sum extension of $H_1+H_2$ briefly by $\wh H=\Phi^{**} (I+iB_\oplus ) \Phi^*$. \(i) $\Rightarrow$ (ii) Assume that $\wh H$ is extremal. Since $\sE \subset \sF$ by , it is enough to prove the inclusion $\sF \subset \sE$. By Theorem \[s-sum-caracter\] and $\mul \Phi^*=\mul H_{1} \times \mul H_{2}$ one sees $$\label{NICE1} \wh H=((\Phi^{*})_{\s})^{*} (I+iB_\oplus ) (\Phi^{*})_{\s} = R^{\ast} (I+i C ) R,$$ for some closed operator $R$ satisfying $$\label{Rincl} C_0^{1/2} P_\sD \Psi^{**} \subset R \subset C_0^{1/2}(K^*)_\s,$$ where $P_\sD$ is the orthogonal projection of $\sH \times \sH$ onto $\sD=\cdom K$. Recall that $(\Phi^{*})_{\s} \subset \Phi^{*} \subset K^{*}$ and hence $P_\sD(\Phi^{*})_{\s}\subset P_\sD K^*=(K^*)_{\s}$. Moreover, one has $\dom P_\sD(\Phi^{*})_{\s} =\dom (\Phi^{*})_{\s} = \dom R$, since by assumption these two domains coincide with the corresponding joint form domain. Denoting $\wh R= C_0^{-1/2}R$, one has $\dom \wh R=\dom P_\sD(\Phi^{*})_{\s}$ and can be rewritten as $$\label{NICE2} \wh H=((\Phi^{*})_{\s})^{*} (I+iB_\oplus ) (\Phi^{*})_{\s} = \wh R^{\ast}C_0^{1/2}(I+iC)C_0^{1/2}\wh R,$$ where $\wh R$ satisfies $P_\sD \Psi^{**} \subset \wh R \subset (K^*)_\s$. One concludes that $P_\sD(\Phi^{*})_{\s}=\wh R$, since both operators are restrictions of $(K^*)_\s$, and thus $$\label{Rhat*} ((\Phi^{*})_{\s})^* P_\sD = \wh R^*.$$ Now one obtains from the equalities $$\begin{aligned} ((\Phi^{*})_{\s})^{*} (I+iB_\oplus ) (\Phi^{*})_{\s} & = & \wh R^{\ast}C_0^{1/2}(I+iC)C_0^{1/2}\wh R \nonumber \\ & = & ((\Phi^{*})_{\s})^{*}P_{\sD} C_0^{1/2}(I+iC)C_0^{1/2}\wh R \nonumber \\ & = & ((\Phi^{*})_{\s})^{*} C_0^{1/2}(I+iC)C_0^{1/2}\wh R . \nonumber\end{aligned}$$ Hence, for every $f\in\dom \wh H$ one has $$(I+iB_\oplus )(\Phi^{*})_{\s}f-C_0^{1/2}(I+iC)C_0^{1/2}\wh R f \in \ker ((\Phi^{*})_{\s})^{*}.$$ Here $C_0^{1/2}(I+iC)C_0^{1/2}\wh R f \in \sD=\cdom K$ and $\sD=\cdom K=(I+iB_\oplus) \sE$; see . Therefore, there exists $\varphi\in \sE$ such that $C_0^{1/2}(I+iC)C_0^{1/2}\wh R f=(I+iB_\oplus)\varphi$. On the other hand, $(\Phi^{*})_{\s}f \in \sF=\cran(\Phi^{*})_{\s}=(\ker ((\Phi^{*})_{\s})^{*})^\perp$, see , . Since $\varphi\in \sE\subset \sF$, this yields $$\left((I+iB_\oplus )((\Phi^{*})_{\s}f-\varphi ), (\Phi^{*})_{\s}f-\varphi \right)=0,$$ and thus $(\Phi^{*})_{\s}f-\varphi=0$. Consequently, for all $f\in \dom \wh H$ one has $$(\Phi^{*})_{\s}f\in \sE.$$ Since $\dom \wh H$ is a core for the corresponding closed form, or equivalently, the closure of $(\Phi^{*})_{\s} \uphar\dom \wh H$ is equal to $(\Phi^{*})_{\s}$, the claim follows: $\sF = \cran (\Phi^{*})_{\s} \subset \sE.$ \(ii) $\Rightarrow$ (i) Assume that $\sE = \sF$. Then $\sF_{0}=\ran (\Phi^{*})_{\s}\subset \sE$ and hence for all $f\in \dom (\Phi^{*})_{\s}$ one has $(I+B_\oplus)(\Phi^{*})_{\s}f \in \cdom K$. By Remark \[rem3.2\] this implies that $$\label{neweq0} (I+B_\oplus)(\Phi^{*})_{\s}f = C_0^{1/2}(I+iC)C_0^{1/2} P_\sD (\Phi^{*})_{\s}f.$$ On the other hand, as shown above $P_\sD(\Phi^{*})_{\s}\subset P_\sD K^*=(K^*)_{\s}$. Let $\wh R$ be the closure of $(K^*)_{\s}\uphar\dom (\Phi^{*})_{\s}$. Then $\wh R^*$ satisfies the identity . Since $\Psi^{**}\subset (\Phi^*)_\s$ (see ) one obtains $P_\sD\Psi^{**}\subset \wh R$. The identities and imply that for all $f\in \dom \wh H$ the equalities $$\begin{split} ((\Phi^{*})_{\s})^{*} (I+B_\oplus)(\Phi^{*})_{\s}f & = ((\Phi^{*})_{\s})^{*} P_\sD C_0^{1/2}(I+iC)C_0^{1/2} P_\sD (\Phi^{*})_{\s} f \\ & = \wh R^* C_0^{1/2}(I+iC)C_0^{1/2} \wh R f \end{split}$$ hold. Then the closed operator $R=C_0^{1/2}\wh R$ satisfies the inclusions as well as the desired identity $((\Phi^{*})_{\s})^{*} (I+B_\oplus)(\Phi^{*})_{s}=R^*(I+iC)R$, and thus $\wh H$ is extremal, cf. Theorem \[s-sum-caracter\]. Theorem \[s-sum-een\] is a generalization of [@HSS19 Theorem 3.5], where an additional invariance of $\mul K^*$ under the operator $B_\oplus$ was used. Moreover, Theorem \[s-sum-een\] generalizes a corresponding result for the form sum of two closed nonnegative forms established earlier in [@HSSW2007 Theorem 4.1]. The present result relies on Theorem \[s-repr\], where the description of the closed sectorial form generated by a general maximal sectorial relation of the form $H=T^*(I+iB)T$ where $T$ is a closed relation. This generality implies that with special choices of $B$ the relation $H$ can be taken to be nonnegative and selfadjoint, i.e., the corresponding closed form $\st$ becomes nonnegative; see Example \[examp\]. [33]{} Yu.M. Arlinskiĭ, “Maximal sectorial extensions and closed forms associated with them”, Ukrainian Math. J., 48 (1996), 723–739. M. Haase, *The functional calculus for sectorial operators*, Oper. Theory Adv. Appl., 169, Birkhaüser Verlag, 2006. S. Hassi, A. Sandovici, H.S.V. de Snoo, and H. Winkler, “Extremal extensions for the sum of nonnegative selfadjoint relations”, Proc. Amer. Math. Soc., 135 (2007), 3193–3204. S. Hassi, A. Sandovici, H.S.V. de Snoo, and H. Winkler, “Extremal maximal sectorial extensions of sectorial relations”, Indag. Math., 28 (2017), 1019–1055. S. Hassi, A. Sandovici, and H.S.V. de Snoo, “Factorized sectorial relations, their maximal sectorial extensions, and form sums”, Banach J. Math. Anal., 13 (2019), 538–564. S. Hassi, H.S.V. de Snoo, and F.H. Szafraniec, “Componentwise and Cartesian decompositions of linear relations”, Dissertationes Mathematicae, 465 (2009), 59 pages. T. Kato, *Perturbation theory for linear operators*, Springer-Verlag, Berlin, 1980. F.S. Rofe-Beketov, “Numerical range of a linear relation and maximal relations”, Teor. Funktsii Funktsional. Anal. i Prilozhen, 44 (1985), 103–112 (Russian) \[English translation: J. Math. Sciences, 48 (1990), 329–336\]. A. Sandovici, *Contributions to the extension theory in Hilbert spaces*, Doctoral dissertation, University of Groningen, 2006. Z. Sebestyén and Zs. Tarcsay, “$T^{*}T$ always has a positive selfadjoint extension”, Acta Math. Hungar., 135 (2012), 116–129. [^1]: The second author thanks the University of Vaasa for its hospitality during the preparation of this work.

--- abstract: 'Real-world complex systems such as ecological communities and neuron networks are essential parts of our everyday lives. These systems are composed of units which interact through intricate networks. The ability to predict sudden changes in network behaviour, known as critical transitions, from data is important to avert disastrous consequences of major disruptions. Predicting such changes is a major challenge as it requires forecasting the behaviour for parameter ranges for which no data on the system is available. In this paper, we address this issue for networks with weak individual interactions and chaotic local dynamics. We do this by building a model network, termed an [*effective network*]{}, consisting of the underlying local dynamics at each node and a statistical description of their interactions. We illustrate this approach by reconstructing the dynamics and structure of realistic neuronal interaction networks of the cat cerebral cortex. We reconstruct the community structure by analysing the stochastic fluctuations generated by the network and predict critical transitions for coupling parameters outside the observed range.' author: - 'Deniz Eroglu,$^{1,2,3}$ Matteo Tanzi,$^{2,4}$ Sebastian van Strien,$^2$ Tiago Pereira$^{1,2}$' title: 'Effective networks: a model to predict network structure and critical transitions from datasets ' --- Instituto de Ciências Matemáticas e Computação, Universidade de São Paulo, São Carlos, Brazil Department of Mathematics, Imperial College London, London SW7 2AZ, UK Department of Bioinformatics and Genetics, Kadir Has University, 34083 Istanbul, Turkey Department of Mathematics and Statistics, University of Victoria, Victoria BC, CA We are surrounded by a range of complex networks consisting of a large number of intricately coupled nodes. Neuron networks form an important class examples where the interaction structure is heterogeneous [@kandel2000]. Because changes in the interaction can have massive ramifications on the system as a whole, it is desirable to predict such disturbances and thus enact precautionary measures to avert potential disasters. For instance, neurological disorders such as Parkinson’s disease, schizophrenia, and epilepsy, are thought to be associated with anomalous interaction structure among neurons [@bohland2009]. Often, as in the case of neuron networks, it is impossible to directly determine the interaction structure. Therefore, a major scientific challenge is to develop techniques which use measurements of the time evolution of the nodes state to indirectly recover the network structure and predict the network’s behaviour when essential characteristics of the interactions change. The literature on data-based network reconstruction is vast. Reconstruction methods can be classified into [*model-free*]{} methods and *model-based* methods. The former identify the presence and strength of a connection between two nodes by measuring the [*dependence*]{} between their time-series in terms of: correlations [@de2004discovery; @reverter2008combining], mutual information [@butte1999mutual], maximum entropy distributions [@braunstein2008inference; @cocco2009neuronal], Granger causality, and causation entropy [@bressler2011wiener; @ladroue2009beyond]. Such methods alone do not provide information on the dynamics of the network, which is necessary to predict critical transitions. Model-based methods instead provide estimates (or assume a priori knowledge) of the dynamics and interactions in the system, and use this knowledge to reconstruct the network topology. When the interactions are sufficiently strong, the network structure can be recovered [@casadiego2017; @han2015; @wang2016]. For a more extensive account of reconstruction (model-free and -based) methods see the reviews [@wang2016; @nitzan2017revealing; @stankovski2017] and references therein. In many real-world applications, the behaviour of isolated nodes is erratic (*chaotic*) and the interaction is weak [@schneidman2006; @haas2015; @kandel2000]. Moreover, the network structure typically has community structures and hierarchical organisations such as the rich-club networks found in the brain [@heuvel2011]. As the interaction strength per connection is weak and the statistical behaviour of the nodes robust, the influence of a single link on the overall dynamics is negligible and only the cumulative contribution of many links matter. Furthermore, because of the chaotic dynamics and weak coupling, the influence of a single node on the rest of the network corresponds essentially to a random signal that is superposed to the randomness generated by the chaos in the local dynamics. The existing reconstruction techniques fail to reconstruct a model from the data, as they require the interaction to be of the same magnitude as the isolated dynamics. In this paper, we introduce the notion of an [*effective network*]{}. Its aim is to make a model of a complex system from observations of the nodes evolution when the network has heterogeneous structure, the strength of interaction is small and local dynamics are highly erratic. This approach starts by reconstructing the local dynamics from observations of nodes with relatively few connections, and then recovers the interaction function from observations of the highly connected nodes whose dynamics are the most affected by the interactions as a result of the multitude of connections they receive from the rest of the networks[@park2013; @pereira2017]. A key achievement is that we are able to identify community structures even in the presence of weak coupling. An outcome of the effective network is that it recovers enough information to forecast and anticipate the network behaviour, even in situations where the parameters of the system change into ranges that have not been previously encountered. [**Complex networks of nonlinear systems.**]{} We consider networks with $N$ nodes with chaotic isolated dynamics and pairwise interactions satisfying some additional assumptions described below. The network is described by its adjacency matrix ${\boldsymbol}A$, whose entry $A_{ij}$ equals $1$ if node $i$ receives a connection from $j$ and equals $0$ otherwise. We assume that the time evolution of the state ${\boldsymbol}x_i(t)$ of node $i$ at time $t$ is expressed as $$\begin{aligned} \bm{x}_i(t+1)&=&\bm{F}_i (\bm{x}_i(t))+\alpha \sum_{j=1}^{N} A_{ij} \bm{H}(\bm{x}_i(t),\bm{x}_j(t)). \label{eq1}\end{aligned}$$ When performing reconstruction from data, the isolated local dynamics $\bm{F}_i\colon M\to M$, the coupling function $\bm{H}$, the coupling parameter $\alpha$ (that is small), the adjacency matrix ${\boldsymbol}A$, and even the dimension $k$ of the space $M$, are assumed to be [*unknown*]{}. This class of equations can be applied to modelling many important complex systems found in neuroscience (e.g. coupled equations in neuron networks) [@izhikevich2007], engineering (e.g. smart grids) [@yadav2017; @dorfler2013], material sciences (e.g. superconductors) [@watanabe1994], and biology (e.g. cardiac pacemaker cells) [@winfree2001], among other systems. Our three assumptions are: $(a)$ the local dynamics are close to some unknown [*ergodic*]{} and [*chaotic*]{} map $\bm{F}$ (that is, $\| \bm{F} - \bm{F}_i \| \le \delta$, which is often the case in applications [@pinto2000; @eroglu2017]). $(b)$ The network connectivity is [*heterogeneous*]{}, which means that the number of incoming connections at a node $i$ (given by its degree $k_i = \sum_j A_{ij}$) varies widely across the network. More precisely, $k_i$ is larger for a few nodes called [*hubs*]{}. $(c)$ $\alpha$ is such that, denoting by $\Delta=\max_i k_i$ the maximum number of connections $\alpha \Delta$ is of the same magnitude of $\bm{DF}$. Assumptions $(a)$ and $(c)$ imply that the the total effect of the network on a node $i$, which is of the order $\alpha k_i$, ranges from small (for nodes $i$ with few incoming connections) to of order 1 for the hub nodes. A prime example is the cat cerebral cortex which possesses inter-connected regions split into communities with a hierarchical organization. This network has heterogeneous connectivity, chaotic motion and weak coupling [@scannell1993; @scannell1995; @zamora-lopez2010]. Other examples, includes the drosophila optic lobe network [@takemura2013; @garcia-perez2018]. For a given dataset, our effective network first tests whether the underlying system satisfies assumptions $(a)-(c)$, and, if so, reconstructs the model (see the Methods for more detail). We assume the availability of a time series of observations $y_i(t)=\phi( {{\boldsymbol}x}_i(t))$, where $\phi$ is a projection to a variable on which unit interactions depend. This situation occurs frequently in applications; for example, with measurements of membrane potentials in neurons or voltages in electronic circuits. Here, we demonstrate how to obtain a model for the system by constructing an effective network from the time series $y_i(t)$. Effective networks recover both structure and dynamics {#effective-networks-recover-both-structure-and-dynamics .unnumbered} ====================================================== To obtain an [*effective network*]{} from observations of a complex system, we combine statistical analysis, machine-learning techniques, and dynamical systems theory for networks. An effective network provides (i) local evolution laws and averaged interactions for each unit that, in combination, closely approximate the unit dynamics, and (ii) a network with the same degree distribution and communities as the original system. We use the term “effective" because, even if it does not carry information on each link and interaction in the original system, the network gathers sufficient data to reproduce the behaviour of the original network and predict its critical transitions. [1.0]{} ![[**Reconstruction scheme with [the effective network.]{}**]{} From the time series, we build a model for the local evolution $f_i$ at each node. Under the assumption that such rules change from node to node depending on their connectivity, we estimate the coupling function. Using the fluctuations of the time series with respect to the low-dimensional rules, we recover the community structures. Gathering all this information, we obtain an effective network which is a model for the original system that can be used to predict critical transitions](fig3_alt4_Scheme_reduced.pdf "fig:"){width="\linewidth"} . \[RC\] Using our assumptions for the network and local dynamics, we can show that the evolution at each node will have low-dimensional excursions over finite time scales. In particular, the approximated evolution rule at node $i$ has a low dimensional description over large time scales given by . More precisely, the evolution rule at node $i$ is given by $$\begin{aligned} {\boldsymbol}G_i({\boldsymbol}x_i) = {\boldsymbol}F_i({\boldsymbol}x_i) + \beta_i {\boldsymbol}V({\boldsymbol}x_i(t)) \nonumber\end{aligned}$$ where ${\boldsymbol}F_i \approx {\boldsymbol}F$ is the isolated dynamics, $\beta_i = \alpha k_i$ is the rescaled degree, and ${\boldsymbol}V$ is the averaged mean-field effect of the interaction function (effective coupling function) that takes into account the cumulative effect of interactions on node $i$. The true dynamics ${\boldsymbol}x_i(t)$ slightly deviates from this rule and is given by $$\begin{aligned} {\boldsymbol}x_i(t+1) = {\boldsymbol}G_i({\boldsymbol}x_i(t)) + {\boldsymbol}\xi_i(t) \nonumber\end{aligned}$$ where ${\boldsymbol}\xi_i(t)$ is a fluctuation term that is small for an interval of time which is exponentially large in terms of the size of the network. This low-dimensional reduction has been rigorously established in important test cases over long time scales (see [@pereira2017]). This approximation holds, roughly speaking, for two reasons. Firstly, a low degree node $i$ has a small number of connections compared to the maximum degree, and the interactions slightly perturb its statistical behaviour. Secondly, hub nodes receive a huge number of connections so that the sum of their interactions with the other nodes in the network, $\alpha \sum_j A_{ij} \bm{H}(\bm{x}_i,\bm{x}_j)$, can be approximated as the integral $\alpha k_i \int \bm{H}(\bm{x}_i,\bm{y})\rho (\bm{y}) \, d\bm{y}$ where $\rho(\bm{y})$ is the stationary distribution of typical orbits of the low degree nodes. This approximation holds up to a small fluctuation, ${\boldsymbol}\xi(t)$, which depends on the state of neighbours of the $i$th node. This fact will be key to detecting communities. The approximation described above also applies to the measured state variable $y_i(t)$. Depending on the system, we need to preprocess the data (see Methods). The processed variable is still referred to as $y_i(t)$. The Takens reconstruction technique tells us that $y_i(t+1)$ is a nonlinear function of $k+1$ past points $y_i(t), \dots y_i(t-k)$, for a given number $k$ provided by the approach. Here, we focus on the case when $k=1$, which occurs in many real-world examples, and discuss cases with $k\ge2$ in the Supplementary Materials. This means that $$\begin{aligned} y_i(t+1) = g_i(y(t))+ \xi_i(t) \label{eq:meanfield}\end{aligned}$$ where $$g_i=f_i(y_i(t))+\beta_i v(y_i(t))$$ is a low-dimensional law that approximates the evolution of $y_i(t)$. We first employ Takens reconstruction to estimate $g_i$ and coarsely classify the nodes by their degrees. For instance, in rich-club motifs, the network has low-degree nodes organised in communities and high-degree in the rich-club. Since $\beta_i=\alpha k_i$ is small for low-degree nodes $i$, the evolution rules at such nodes will be similar to the isolated dynamics. Thus we identify which nodes are low-degree nodes and we can recover the local dynamics $f$. Next, we use Eq. (\[eq:meanfield\]) and a classification of nodes by their time-series, to obtain the coupling function and estimate the degree distribution $k_i$ and the coupling parameter $\alpha$. More precisely, an effective network is obtained in three main steps (see Methods for additional details): - [**Reduced dynamics.**]{} Since the fluctuation term $\xi_i(t)$ is small, the rule $g_i$ can be estimated fitting the points $(y_i(t), y_i(t+1))$ using machine-learning techniques. Here, we estimate the function $g_i$ by decomposing it as a linear combination of basis functions, which is chosen according to the application. The parameters of the basis functions are obtained by performing a $10$-fold cross-validation with $90\%$ training and $10\%$ test [@shandilya2011; @james2013] (see Methods for details). The particular technique used in this step is not essential as the dynamics will be low-dimensional. Other techniques such as compressive sensing based techniques [@brunton2015; @mangan2016; @wang2011] could be also employed. - [**Isolated dynamics and effective coupling.**]{} Since the network is heterogeneous, it has many low degree nodes for which $g_i$ is close to the uncoupled dynamics $f_i\approx f$. We first run a model-free estimation to coarsely classify nodes into low-degree nodes and hubs and estimate $f$ (see Supplementary Material). We then use the expression of $g_i$ recovered at low degree nodes to obtain an approximation for $f$, while $g_i$ recovered at hub nodes allows to estimate $\beta_i v$. We estimate the parameter $\beta_i$ by using Bayesian inference (see Methods). - [**Network structure and communities.**]{} Since $\beta_i=\alpha k_i$, we can recover the network’s degree distribution from knowledge of $\beta_i$ at every node. To reconstruct the community structures, we use the fact that the term $\xi_i(t)$ at a node $i$ depends on the state of all nodes interacting with $i$. Thus the correlation between $\xi_i$ and $\xi_j$ is proportional to the matching index (number of common neighbors) of the nodes $i$ and $j$. Thus, we can create a network that has the same statistical properties of the actual one. Further details are provided in the next section. [**Benchmark model for the isolated dynamics.**]{} We consider heterogeneous networks of neurons described by the Rulkov model (see Methods for details). The model has two variables, $u$ and $w$, evolving at different time scales. The fast variable $u$ describes the membrane potential and $w$ the slow currents. In the following discussion, we choose the parameters such that the isolated dynamics presents a tonic spiking behaviour. In the Supplementary Materials, we also show the results for bursting dynamics, i.e. when the parameters are such that the membrane potential oscillates between two phases: a bursting phase of a rapid spiking, and a quiescent phase. We also validated our methods for other types of isolate chaotic dynamics (also in the Supplementary materials). Revealing community structure: the rich-club motif {#revealing-community-structure-the-rich-club-motif .unnumbered} ================================================== We apply the effective network in a wide range of setups as discussed in the supplementary material. Here we focus on the network structure of the cat cerebral cortex for which the mesoscopic connectivity has been probed and detailed [@zamora-lopez2010]. The regions and their connections were discovered by using datasets from tract-tracing experiments analyzed via nonmetric multidimensional scaling, an optimization approach that minimizes the distance between connected structures and maximizes between the unconnected ones [@scannell1993; @scannell1995]. The network contains 53 meso-regions arranged in four communities that closely follow functional subdivisions; namely, visual (16 nodes), auditory (7 nodes), somatomotor (16 nodes) and frontolimbic (14 nodes), as shown in Fig \[RC2\] (a). In addition to these cortical regions, some cortical areas (hubs) form a hidden layer called a *rich-club* and are densely connected to each other and the communities. A set of nodes form a rich-club if their level of connectivity exceeds what would be expected by chance alone. The maximum number of connections in this network is $\Delta = 37$. The network obtained is weighted [@scannell1993; @scannell1995]. For simplicity and to improve the performance in detecting communities, we turn the network into an undirected simple graph [@zamora-lopez2010]. We simulate each mesoregion as a neuron interacting via electrical synapses (see Methods for details) and obtain a multivariate data $\{ y_1(t), y_2(t), \dots, y_N(t) \}$ for a time $T=5000$. For simplicity, we will denote $y_i = \{ y_i(t)\}_{t=0}^T$, whenever there is no risk of confusion. For comparison, we first test the recovery of the network using the functional network framework [@bettinardi2017; @eguiluz2005; @bullmore2009]. The intuition behind this approach is that nodes with similar time series perform analogous functions in the network and are assumed to have similar characteristics. The functional network can thus be constructed by the matrix of the similarities between nodes via statistical analysis [@zhang2006; @greicius2003]. Here we employ a covariance analysis between the time series. The functional network alone does not detect communities from the data because the bursts are erratic (Fig. \[RC2\] (b)). Other similarity measures such as mutual information give no significative improvement. We also compared our method with a reconstruction by the sparse recovery technique [@brunton2015; @wang2016]. There, the dynamics is written as linear combination of chosen basis functions and the unknowns are the coefficients. Sparse recovery works well when many coefficients are exactly zero. The presence of a link is determined when any coefficient of the corresponding interaction is nonzero. In our setting, each link provides a small contribution to the individual dynamics and only their cumulative effect can influence the dynamics. Therefore, the coefficients to be recovered are close to zero, and cannot be distinguished from zero terms. In this setup, sparse recovery can also uncover a model for the isolated dynamics, however, it misidentifies the network structure. We implemented the sparse recovery method to our benchmark method and could assign only $20\%$ of the nodes to the right cluster. These results are discussed in the Methods. Remarkably, however, the effective network is able to recover the community structures (Fig. \[RC2\] (c)). To every pair of $y_i$ and $y_j$ we assign a *Pearson distance* $s_{ij}\ge0$ which measures the difference between the observed dynamics at nodes $i$ and $j$ so that $s_{ij} \approx 0$ if the attractors of $i$ and $j$ are similar and $s_{ij} \approx 1$ if they are reasonably distinguishable (see Methods for details). The higher the number of nodes with behaviour different from $i$, the larger the intensity $S_{i} = \sum_{j}s_{ij}$. The analysis of intensities $S_i$ allows us to distinguish nodes in the rich-club. Next, we find the local low-dimensional rules $g_i$ for each time series $y_i$ by applying Step (i). These rules are similar and display a chaotic evolution, thus, confirming that the similarity analysis by the functional networks would provide little insight into the reconstruction. Having the local rules $g_i$, we can decompose the time series in terms of a low-dimensional deterministic part and a fluctuation term $\xi_i$, which depends on the neighbours of the $i$th node. One of our main key points is that these fluctuations allow us to reveal the community structure. Indeed, if nodes $i$ and $j$ interact with the same nodes, they are subject to the same inputs and the covariance Cov$(\xi_j,\xi_i)$ is high. If not, Cov$(\xi_i,\xi_j)$ is nearly zero due to the decay of correlations in the deterministic part. Therefore, Cov$(\xi_j,\xi_i)$ is high when nodes $i$ and $j$ have high [*matching index*]{} (high fraction of common connections). It is crucial that the correlation analysis is restricted to the dynamical fluctuations $\xi_i$. Indeed, since the variance of the deterministic part of $y_{i}$ is larger than that of the small fluctuations $\xi_i$, performing a direct correlation analysis between $y_{i}$ and $y_{j}$ hides all the contributions to the correlations coming from the covariance between $\xi_i$ and $\xi_j$. Consequently, the correlation of the deterministic part is close to zero due to the chaotic nature of the dynamics (see Supplementary Materials). So, remarkably, the noise terms $\xi_i$ are needed to access the community structure. Using steps (i)-(iii) we obtained a model for the isolated dynamics, coupling function, and distribution of degrees. To apply current methods of community detection, we obtain an adjacency matrix with entries equal to zero or one from the matrix of correlations Fig. \[RC\] (c). This is done by thresholding the entries and considering a link only when the correlations were greater than $0.5$. We tested threshold values ranging from $0.3$ to $0.6$ and obtained the same results. Indeed, the distribution of the entries of the matrix of correlations is unimodal and has a peak near 0.5. We computed rich-club coefficients for each node [@colizza2006] (see details in Methods), and nodes with the highest coefficients formed the rich-club displayed at the centre of the network. By applying the community detection method [@blondel2008] to the thresholded effective matrix, we recovered the communities according to their function as shown in Figure \[RC2\] (d). There is no reason to expect that the links in the effective network correspond to links in the actual network. However, since every node makes most of its interactions within a cluster, two nodes with highly correlated fluctuations $\xi(t)$ are likely to belong to the same community, and this can be enforced in the effective network by adding a connection between them. Further results for a variety of rich-club motifs are shown in the Supplementary Materials. [1]{} ![[**Effective network of the cat cerebral cortex.**]{} We use the local dynamics as a spiking neuron coupled via electric synapses. (a) The cat cerebral cortex network with nodes colour coded according to the four functional modules. Rich-club members are indicated by red encircled nodes. (b) *The covariance matrix of the data* cannot detect communities. (c) *The covariance matrix of the fluctuations* can distinguish clusters of interconnected nodes. This matrix has entries color coded according to the key on the right with red entries corresponding to couple of nodes sharing a large numbers of nearest neighbours in the network, while blue nodes correspond to couple of nodes that share a small number of common neighbours. (d) A model in the cat cortex constructed via the effective network approach. From the matrix in (c) we can recover a representative effective network. The reconstructed network represents the real network in (a) with good accuracy. See Methods for the details of the detection of communities and rich-club members.[]{data-label="RC2"}](cat_map_rc_reduced.pdf "fig:"){width="\linewidth"} #### Performance of the communities reconstruction. To quantify the effectiveness of community reconstruction, we compute $PE= \frac{m}{N}$, where $N$ is the total number of nodes and $m$ is the number of nodes assigned to the wrong community. We compute $PE$ for different values for the coupling strength $\Delta \alpha$ between $0.05$ and $0.4$. For each value of $\alpha$, we considered 50 different simulations of the dynamics obtained by choosing different initial conditions. The figure shows the plot of the mean of $PE$ and a shaded region corresponding to the standard deviation. For $\Delta \alpha$ values larger than $0.4$, the reconstructing procedure cannot identify the communities correctly. This is due to the synchronization rich club which appears around this value. [0.8]{} ![[**Prediction error for misidentification of communities in the reconstructed cat cerebral cortex from synthetic data.**]{} For each realisation, the chosen parameters are the same as in Figure \[RC2\] and only the overall coupling is changed. Mean and standard deviation of prediction error (PE) computed for the network over 50 realizations for each value of $\alpha$. If $\Delta \alpha > 0.42$, the system synchronizes and the procedure cannot reconstruct the community structures.[]{data-label="fig:pe"}](construction_err.pdf "fig:"){width="\linewidth"} Predicting critical transitions in rich-clubs {#predicting-critical-transitions-in-rich-clubs .unnumbered} ============================================= The ability to reconstruct the network and dynamics from data can be exploited to predict critical transitions that may occur when the coupling strength. In the cat brain, for example, a transition to collective dynamics in the rich-club has drastic repercussions for the functionality of the network [@zamora-lopez2010; @lopes2017]. Our goal now is to obtain data when the network is far from a collective dynamics and predict the onset of collective motion in the rich-club. The effective network can predict the onset of such collective dynamics based on a single multivariate time series for fixed coupling strength in a regime far from the synchronized state. We analyze time-series obtained simulating the dynamics at a value of the coupling strength $\Delta \alpha = 0.3$, and apply our reconstruction procedure to obtain the network structure and a model of the isolated dynamics. [*Estimating the transition to synchronization in the rich-club.*]{} Transitions to synchronization between the bursts is possible while the fast spikes remain out of synchronization [@rulkov2001]. To estimate the transition to burst synchronization, we obtain the slow variable as a filter over the membrane potential (fast variable). Indeed, since we measure the membrane potential $y_i(t) = u_i(t)$, the slow variable is given as $ z_i(t) = \mu \sum_{k=1}^t (y_i(k) - \sigma) $ and for a good choice of parameters $\mu$ and $\sigma$ this can be identified with the slow variable of the model $w$. In the methods section, we obtain an equation for the slow variable and analyze the effect of the network of the dynamics of the slow variable. We can use the data on the network and the dynamics recovered from the time-series recorded at $\Delta \alpha = 0.3$ to predict that at the value $\Delta \alpha \approx 0.42$ the rich-club will develop a burst synchronization (details in the Methods). To capture such a transition in the bursts of the rich club, we introduce a phase $\theta_j(t)$ for the slow variable, the definition of $\theta_j(t)$ can be found in the Methods section. Once we have the phase we compute the order parameter $$r(t) e^{i \psi(t)} = \frac{1}{N_c} \sum_{j=1}^{N_c} e^{i \theta_j (t)}$$ Loosely speaking, a small value of the order parameter, $r \approx 0$, means that no collective state is present in the system, whereas $r(t)\approx 1$ means that the bursts are synchronized. Figure \[fig:criticality\] shows that behaviour of the order parameter as a function of the coupling. The rich-club undergoes a transition to burst synchronization at $\Delta \alpha \approx 0.4$ that corresponds to an increase of roughly $40\%$ of the coupling strength and is close to the predicted value $\Delta \alpha \approx 0.42$. In the Supplementary Materials, we present other examples where the local dynamics is fully chaotic. [0.9]{} ![[**Prediction of critical transitions in the rich-club of the cat cerebral cortex.**]{} The level of average synchronization $r$ of the rich-club is shown for different values of the coupling strength. Insets show time series of neuronal dynamics of four rich-club members and color of time series matches with the color of nodes in Fig. \[RC2\]. For values in the grey shaded region, $r$ is increasing towards close to one and the rich-club exhibits collective behavior. We can predict the critical coupling $\alpha_c$ with some standard deviation (grey shaded region) by studying the effective network obtained from a time series measured when $\Delta \alpha=0.3$.[]{data-label="fig:criticality"}](alpha_vs_r_v2-01.png "fig:"){width="0.9\linewidth"} Obtaining a statistical description of the network {#obtaining-a-statistical-description-of-the-network .unnumbered} ================================================== The effective network can also provide detailed information on the structure of each community and a statistical description of the network. To illustrate this, we reconstruct the statistical properties of scale-free networks. ![ [**Example of an effective network for a scale-free network where the interaction dynamics and structure are unknown.**]{} An effective network is constructed starting from a time series and provides models for the network connectivity and unit dynamics and interactions. The actual network and its degree distribution are shown in panel (a). The time series of some representative nodes are shown in panel (b). Correlation analysis of the time series yields the functional network represented in panel (c). Here, the $(i,j)$ entry of the matrix is the Pearson distance $s_{ij}$ color coded according to the key on the right, i.e. blue entries correspond to pairs of nodes having different time-series while yellow entries correspond to pairs with similar time-series. Thus one can coarsely distinguish between high- and low-degree nodes. This analysis on its own leads to a gross overestimate of the structural parameter, see panel (d). However, using this coarse classification and mathematical results on coupled systems (\[eq1\]) satisfying assumptions (i), (ii) and (iii) makes it possible to find an approximation of the evolution rule at each node, see panel (e). Using this additional information, one can obtain a description of the network connectivity structure, see panel (f). The network used in this figure has 1000 nodes, but only 50 are shown for clarity.[]{data-label="fig1"}](main_fig_v5_low_3_reduced.pdf){width="\linewidth"} [**Scale-free networks of coupled [ bursting]{} neurons**]{}. We consider coupled bursting neurons with excitatory synapses [@rulkov2001] in scale-free networks (see Methods for details on simulation of scale-free networks and formulation of the model). Our techniques work equally well for spiking dynamics. We generate a scale-free network with $N=10^4$ nodes such that the probability of having a node of degree $k$ is proportional to $k^{-\gamma}$, where $\gamma=2.53$. For this reconstruction we only need 2000 data points for each node. Again, to every pair of time series $y_i$ and $y_j$ we assign a *Pearson distance* $s_{ij}\ge0$ and the node intensity $S_{i} = \sum_{j}s_{ij}$. The empirical distribution of the intensities $S_i$ approximates the degree distribution of the network, see Fig \[fig1\](d). In the example here, the estimated structural exponent from the distribution of $S_i$ is $\gamma_{\rm est} = 3.1$, which yields a relative error of nearly 25% with respect to the true value of $\gamma$ (see also the plots in Figure \[fig2\] a)). The functional network therefore overestimates $\gamma$, which has drastic consequences for the predicted character of the network. For example, the number of connections of a hub for a scale-free network is concentrated at $k_{\rm max} \sim N^{1/(\gamma - 1)}$, so the relative inaccuracy for the estimate $ k_{\rm est}$ of the maximal degree is $ k_{\rm max}/k_{\rm est} = N^{1/\gamma - 1/ \gamma_{\rm est}}$, which is about 500%. Such inaccuracy has important repercussions for the ability to predict the emergence of collective behaviour [@pereira2010; @pereira2017]. The statistical measure used for the construction of functional network typically depend in a nonlinear way on the degrees, thus causing a distortion in the statistics. We will discuss the case of Pearson distance. Suppose that the signals $\{(y_i(t),y_i(t+1))\}$ are purely deterministic, $y_i(t+1)=g_i(y_i(t))$, and thus perfectly fall on the graphs of the functions $g_i$, that are determined by $\beta_i$ and so by the degree $k_i$ of the node. The Pearson distance $s_{ij}$ between the signal at $i$ and $j$ is going to be a number between 0 and 1, depending on how close these graphs are. Unfortunately this distance depends nonlinearly on the degrees $k_i$ and $k_j$. Devising another distance $s'_{ij}$ without the knowledge of the interaction, in general, would still carry the nonlinear dependence on the degrees. Moreover, once fluctuations from the network are included as $y_i(t+1)=g_i(y_i(t))+\xi_i(t)$ the differences between time-series $\{(y_i(t),y_i(t+1))\}$ for different $i$s can be due to fluctuations rather than differences in the degrees. So the decomposition of the rules in terms of interactions and fluctuations is essential to recover precise details of the real degree distribution. ![ [**Reconstruction of structural power-law exponents $\gamma$ of scale-free networks from data.**]{} We estimated $\gamma$ from the multivariate time series obtained by simulating chaotic coupled dynamics on random scale-free networks with degree distribution $P(k) \propto k^{-\gamma}$. The three plots in panel (a) compare the functional and effective network approach in recovering the degree distribution. The first plot shows the real distribution of degrees, the second shows the distribution obtained with the functional network, and the last figure shows the distribution recovered with the effective network. The functional approach is able to recover that the degree distribution follows some power law and can distinguish low from high degree nodes. However, it does not recover the parameter of the power law accurately at all, see the second plot in panel (a). Much better estimates are obtained using the effective network, see the third plot in panel (a). Panel (b) shows the degree distribution of the original system (in blue) and that estimated from an effective model (in red) for the neural network in the optical lobe of Drosophila melanogaster. We obtained an accuracy of $3\%$ in the structural exponent of the tail. Panel (c) shows the true exponent $\gamma$ versus $\gamma_{\rm {est}}$ obtained with an effective network from data for spiking neuron maps coupled with chemical synapses on networks with different values of $\gamma$. We generated 1000 networks, from which the $\gamma_{\rm est}$ estimate is within $2\%$ accuracy. []{data-label="fig2"}](Fig2newnewnew.pdf){width="0.52\linewidth"} The effective network provides a better statistical description of the network structure. To compare with the functional network approach described above, we constructed an effective network of the same system tested for the functional network approach. The estimate for $\gamma$ from the effective network is $\gamma_{\rm est}=2.55$, which has an error of only $1\%$ ( Fig. \[fig1\] (d)). We repeat the analysis on a different network with different parameters $\gamma$ in the degree distribution. The estimated $ \gamma_{\rm est}$ values are shown in Fig. \[fig2\] (c) as a function of the true parameter $\gamma$. The relative error on the estimated exponent is within $2\%$. [*Real-world scale-free networks: the optic lobe of Drosophila melanogaster.*]{} Effective networks can be used to provide estimates for real-world systems. As one example, we applied our method to data simulated from the neuronal network in the D. melanogaster optic lobe, which constitutes $>$50% of the total brain volume and contains 1781 nodes [@takemura2013]. The degree distribution has a power-law tail [@garcia-perez2018]. We used spiking neurons with chemical coupling to simulate the multivariate time series, from which we constructed an effective model capable of estimating the degree distribution with great accuracy (Fig. \[fig2\] (b) and Methods). Additional applications of our procedure for different types of chaotic systems (doubling map, logistic maps, spiking neurons with electrical synapses, Rössler systems, and Hénon maps) can be found in the Supplementary Materials, together with results demonstrating that effective networks are robust against noise. Conclusions {#conclusions .unnumbered} =========== We have introduced an [*effective network*]{} obtained from time-series of a complex network (by observing the dynamics at each node). Our method complements the existing ones in two ways. Firstly, it encompasses the case of isolated chaotic dynamics. Secondly it deals with weak coupling among the nodes. Both cases are commonly found in applications. Key to the success of the reconstruction is the heterogeneity of the network which allows us to separate the contribution from the local dynamics and the interactions and thus perform a multi-level reduction. We have compared our procedure with methodologies most relevant for the systems we considered. We have thus excluded results tailored to specific experimental setups or dynamics (binary dynamics coupled on networks [@li2017universal], and see [@wang2016] for a review). We also did not consider methods that rely on measurements obtained intervening on the system with controlled inputs (see e.g. [@nitzan2017revealing]) and restrict our attention to the case where the time-series is recorded under constant conditions. When the coupling is strong, sparse recovery can be applied [@brunton2015]. On the other hand, when the coupling is weak sparse recovery cannot distinguish small parameters from those that are identically zero thus misidentifying connections between nodes as it can be observed in Figure \[SR\]. Also model-free methods are ill-suited to analyse weakly coupled chaotic dynamical systems studied here. Indeed, the influence of a single pairwise interaction on the time-series is so weak that its effect on the correlation of the measurements can hardly be detected by direct measurements. Our reconstruction works when no collective dynamics is induced by the weak coupling. When the rich-club synchronizes. A synchronous rich-club induces correlation between the fluctuations in the communities. This happens because two nodes in different communities but coupled to the rich-club would receive a similar forcing. Since we use the fluctuation to identify the community structure, the method would identify these nodes as belonging to the same community. While our method works well for various isolated dynamical systems, in two scenarios it didn’t provide a good reconstruction. First, when the local dynamics is the Lorenz with classical parameters. In this case, the passage near a fixed points smears the fluctuations. Second, in the bursting dynamics, when the quiescent state is too long. This failure seems to be related as well to the long passage near a fixed point. Our method works with large networks, up to $~10^5-10^6$ nodes after which the computation time can be long. In fact, the number of operations to fit the the dynamics at all nodes, as in point (i), is proportional to Number of nodes $\times$ (Lengthofthetimeseries) $^\beta$, where the power $\beta$ depends on the optimization algorithm adopted. The number of operations to determine the correlations between the fluctuations, point (iii) above, is of the order of (Numberofnodes) $^2$ $\times$ Lengthofthetimeseries. The effective network exploits the network heterogeneous structure. This allows to single out different dynamical behaviours across the network and use this information to reconstruct the local dynamics and the effective interaction function. To obtain the community structures we use that certain noise terms associated with the time series at two nodes in the same community are correlated. By collecting data when the network is far from critical transitions, our method can reconstruct the effective network and enables predictions regarding critical transitions occurring in a network. [**Methods**]{} Random scale-free networks {#random-scale-free-networks .unnumbered} -------------------------- A scale-free network has degree distribution $P(k) = C k^{-\gamma}$, where $\gamma>0$ is the characteristic exponent and $C$ is a normalising constant. We use a random network model ${\mathscr}{G}(\bm{w})$ which is an extension of the Erdös-Rényi model for random graphs with a general degree distribution [@chung2002]. Given ${\bm w} = (w_1 , w_2 , \cdots, w_n )$, each potential edge between $i$ and $j$ is chosen with probability $ p_{ij} = w_i w_j \rho, $ and where $\rho = (\sum_{i=1}^n w_i)^{-1}.$ To ensure that $p_{ij} \le 1$, we consider $\max_i w_i \rho \le 1.$ Then $w_i$ is the expected degree of the $i$th node. Taking $w_i = ci^{-1/(\gamma-1)}$, where $\gamma \ge 2$ and $c = \frac{\gamma - 2}{\gamma - 1}dn^{1/\gamma - 1}$ with $d$ denoting the mean degree, the distribution of expected degrees follows a power law, and so $P(w) \propto w^{-\gamma}$ is the expected exponent of power-law distribution [@chung2002; @chung2003]. If the random graph generated is not connected, we use only the largest connected component (the giant component). [*Finding power-law distribution parameters from empirical data.*]{} For estimation of the power-law distribution parameters, we use the maximum likelihood estimator first introduced by Muniruzzaman [@muniruzzaman1957], which is equivalent to the well-known Hill estimator [@hill1975]. After that, we test the reliability between the data and the power law by using the goodness-of-fit method. If the resulting $p$-value is larger than 0.1, the power-law estimation is an appropriate hypothesis for the data. A complete procedure for the analysis of power-law data can be found in Ref. [@clauset2009]. Functional networks {#sec:functional_networks .unnumbered} ------------------- For networks composed of chaotic oscillators, building the functional network from the standard Pearson correlation between time series gives no meaningful results because of the decay of correlation intrinsic to the chaotic dynamics. Hence, functional networks are built using a Pearson distance $s_{ij}\ge 0$, which describes the proximity of the dynamics (i.e. the attractor) at two nodes $i$ and $j$. To do this, we consider the time series ${z}_i(t):=({y}_i(t),{y}_i(t+1))$, $t=0,\dots,T-1$ reordered in $z_i^{\rm lex} (t)$ according to the lexicon order; that is, according to the magnitude of the first component of $z_i(t)$. Then, let $r_{ij}$ be the Pearson correlation, $r_{ij}=$ Cor$(z_i^{\rm lex}, z_j^{\rm lex})$, so that $r_{ij}=1$ indicates that the attractors at nodes $i$ and $j$ fully agree. Finally, define the Pearson distance $s_{ij}=1-|r_{ij}|$ so that $s_{ij}=0$ indicates full agreement of the dynamics and $s_{ij}>0$ measures the difference between the attractors. The intensity $S_i = \sum_{j}s_{ij}$ approximates how many nodes have a dynamical rule different from $i$ and helps to distinguish between poorly connected nodes and hubs. Since most of the network is composed of poorly connected nodes with similar evolution rules, they exhibit a smaller $S_i$ than high-degree nodes, which are scarcer and have different dynamics from the low-degree nodes. In the case of a scale-free network where the degree distribution is monotonic, the empirical distribution of $S_i$ is also expected to have the same trend. Effective network {#effective-network .unnumbered} ----------------- [*Dimensional reduction over finite time scales*]{}. To construct an effective network, we combine statistical estimation with mathematical results for high-dimensional dynamical systems that are based on theorems describing the evolution of dynamical systems coupled on a heterogeneous networks. The assumptions we make are: (a) the dynamics determined by $\bm F$ is ergodic, (b) the network is heterogeneous in the sense that most nodes are low degree and a few hubs make many connections, and (c) the coupling strength $\alpha$ is small so that $\alpha \max_i k_i = O(1)$. These assumptions play a role in the reduction process. Indeed, (c) means that the contribution of the low-degree nodes to the dynamics of the hub nodes is an average, (b) means that the ergodic behaviour of low-degree nodes is similar to that of ${\boldsymbol}F$, and finally, (a) means that we obtain a hierarchy of reductions with different dynamics (depending on connectivity) from which the structure of the network can be inferred. In heterogeneous networks, and up to a small error (relative to the network size), the dynamics at each node evolves according to a low-dimensional dynamical system for a certain time scale where the effect of the interaction of one node on the entire network is averaged to a give net interaction. This is rigorously proven in [@pereira2017] under some additional technical assumptions. One can write the interaction term in equation (\[eq1\]) at node $i$ as $$\begin{aligned} \label{dimred2} \alpha\sum_{j=1}^{N} A_{ij} \bm{H}(\bm{x}_j(t),\bm{x}_i(t)) = \alpha k_i \bm{V}(\bm{x}_i(t))+ \alpha k_i\bm{\xi}_i(t)\end{aligned}$$ where $$\begin{aligned} \bm V(\bm x)=\int \bm H(\bm x, \bm y)\, d\mu (\bm y) \nonumber\end{aligned}$$ is the averaged interaction in terms of the invariant measure $\mu$ for the isolated dynamics and $\beta_i = \alpha k_i$ where $k_i$ is the degree of node $i$. The term $\bm{\xi}_i(t)$ is determined by the states of the $i$th node neighbours. The low-dimensional system, called the [*reduced system*]{}, plus a fluctuation term thus reads as $$\begin{aligned} \label{Eq:Red} \bm{x}_i(t+1) = {{\boldsymbol}G_i}({\boldsymbol}x_i(t)) +\alpha k_i \bm{\xi}_i(t)\end{aligned}$$ where ${{\boldsymbol}G_i}(\bm{x})=\bm{F}_i (\bm{x}) + \alpha k_i \bm{V}(\bm{x}), $ for $i=1,\dots,N$. Using the chaotic and ergodic properties of the dynamics and concentration inequalities, we can show that $|\alpha k_i\bm\xi_i(t)|$ is small over exponentially large time scales, in terms of the system size, and for most initial conditions. Not all structures of the graph will follow this reduced equation. If a fully connected graph is weakly coupled to a scale-free network, then it can develop dynamics different from those of the reduced model. In this case, we cannot use the physical measure $\mu$ in the expectation; however, the reduction is still valid whenever $\mu$ is substituted with a measure that satisfies self-consistency relationships (see below). k-Fold Cross-Validation {#k-fold-cross-validation .unnumbered} ======================= Cross-validation is a resampling method used to evaluate dynamical models on a limited data sample. The parameter $k$ refers to the number of groups that a given data is to be divided into. We use $k=10$, since it is known that this value produces the lowest bias in several tests, so that the method can be called 10-fold cross-validation. The procedure is the following: First, randomly shuffle the data and divide it into k groups. Hold few groups to test the system, use the remaining ones as training data. In our case, we used 1 group to test the system and 9 groups to train our model. We fit a model on the training set and evaluate it on the test one. According to the evaluation scores, we summarize the dynamical properties of the model. Further details on how this standard procedure works can be found in [@james2013]. Reconstruction of the effective network representation from data {#reconstruction-of-the-effective-network-representation-from-data .unnumbered} ================================================================ [**Step 1 : Reduced dynamics.**]{} Given the multivariate time series $\{y_1(t), \dots, y_N(t) \}$, we consider each time series separately and then perform a Takens reconstruction of the attractor for each time series $y_i(t)$. The reduction guarantees with high probability that the dynamics is low dimensional. If the time series is high dimensional, we discard it, and otherwise, once we are in the appropriate dimension, we continue to the next step. Many cases can be captured by a one-dimensional map and Takens reconstruction yields the return map $$\begin{aligned} y_i(t+1) = g_i (y_i(t)) \nonumber\end{aligned}$$ where $g_i : \mathbb{R} \rightarrow \mathbb{R}$. In fact, even when the local dynamics is three dimensional, by introducing a Poincarè section, we can sometimes reduce the analysis to one dimension (see Supplementary Materials for details). [**Step 2: Isolated dynamics and effective coupling function**]{}. We first obtain a model for the isolated dynamics. Here we describe the procedure when the reduced dynamics can be modelled by a one-dimensional map, since the method works similarly in the higher-dimensional setting as shown in the Supplementary Materials. To estimate the deterministic rule $g_i$ for each time series, we use a $10$-fold cross-validation with $90\%$ of the time series for training and $10\%$ for testing (see k-Fold Cross-Validation [above]{}). To obtain a model for the isolated dynamics we first build a functional network (see Functional Networks section in Methods). For the low-degree nodes, $\alpha \kapppa_i v$ is negligible and the dynamics at the low-degree nodes are close to $f$. We identify these nodes by analysing the distribution of $S_i$. We use the top $N_{\rm top}$ nodes of the highest intensity nodes to obtain a proxy for the isolated dynamics. We then average the rules obtained at those nodes to get $\langle g \rangle \approx f$. The choice of $N_{\rm top}$ is not fixed and depends upon the number of nodes and the fluctuation $\sigma_g^2 = \langle (g_i - \langle g \rangle)^2 \rangle$, which is averaged from the $N_{top}$ highest intensity nodes. A good heuristic to choose $N_{\rm top}$ is $\sigma_g^2 / N^{1/2}_{\rm top} \ll 1$, as we have used here. [*The effective coupling function*]{}. Since $\alpha \kapppa_i v= g_i - \langle g \rangle$, analysing the family $\{ g_i - \langle g \rangle \}_{i=1}^{N}$ can yield the shape of $v$ up to a multiplicative constant via a nonlinear regression by imposing that $g_i - \langle g \rangle$ and $g_j - \langle g \rangle$ are linearly dependent. The choice of the base function for the fitting is supervised and depends on the particular application (see Supplementary Material for additional examples). [**Step 3. Network Structural Statistics**]{}. After selecting a $v$ that satisfactorily approximates $g_i- \langle g \rangle$ up to a multiplicative constant over all indices $i$, we estimate the parameter $\beta_i$ using a dynamic Bayesian inference. Because the fluctuations $\xi_i(t)$ are close to Gaussian we use a Gaussian likelihood function and a Gaussian prior for the distribution of the values of $\beta_i$, and hence obtain equations for the mean and variance. We split the data into epochs of $200$ points and update the mean and variance iteratively. [**Matching index:**]{} Recall that the degree of node $i$ is $ k_{i} = \sum_{j}^N A_{ij}$ and count the number of neighbours it has. We now define the *matching index* of a graph [@zamora-lopez2010]. Consider the neighbourhood of node $i$ as $\Gamma (i) = \{j \in \{ 1, \dots, N\} \, | \, A_{ij} = 1\}$. This is the set of nodes that shares an edge with the node $i$. The matching index of nodes $i$ and $\ell$ is the cardinality of the overlap of their neighbourhoods $\mu_{i \ell} = |\Gamma(i) \cap \Gamma(\ell)|$. We are interested in the normalised matching index: $$\begin{aligned} \widehat{\mu}_{i\ell} = \frac{|\Gamma(i) \cap \Gamma(\ell)|}{|\Gamma(i) \cup \Gamma(\ell)|} \end{aligned}$$ or equivalently in terms of the adjacency matrix $$\begin{aligned} \widehat{\mu}_{i \ell} = \frac{(A + A^2)_{i \ell}}{k_{i} + k_{\ell} - (A + A^2)_{i\ell}}.\end{aligned}$$ Clearly $\widehat{\mu}_{i\ell} = 1$ if and only if $i$ and $l$ are connected to exactly the same nodes, i.e., $\Gamma(i) = \Gamma(\ell)$; whereas $\widehat{\mu}_{i \ell} = 0$ if nodes $i$ and $\ell$ have no common neighbours. [**Community structures.**]{} Once we have obtained the rules $g_i$, we filter the deterministic part of the time series $y_i$ and access the fluctuations $\xi_i$. We decompose the fluctuations $\xi_i = \xi_i^{\rm c} + \xi_i^{\rm o} $ where $\xi_i^{\rm c}$ is the fluctuation of the local mean field from nodes in the cluster containing $i$, and $\xi_i^{\rm o}$ is the contribution from outside the cluster. Since a node makes most of its connections within its cluster, $\xi_i^c\gg\xi_i^{o}$ with high probability, and thus if $i$ and $j$ belong to the same cluster $ \mbox{Corr}(\xi_i , \xi_j)=\mbox{Corr}(\xi^c_i , \xi^c_j)$. Next, we notice that the common noise is generated by the common connections between nodes $i$ and $j$. In fact, for fixed isolated dynamics and coupling function $$\begin{aligned} \mbox{Corr}(\xi^c_i , \xi^c_j) \propto \widehat{\mu}_{ij}.\end{aligned}$$ It is well known that in the cat cerebral cortex nodes in the same community has a high matching index while nodes are distinct communities has a low matching index. In fact, this tends to be typically in modular networks [@zamora-lopez2010]. Therefore, for nodes in distinct clusters the component $\xi_i^{\rm c} \approx 0$, so $ \mbox{Corr}(\xi_i , \xi_j) \approx 0.$ Hence, we can recover the network structure by performing a [*covariance analysis of the noise*]{}. [**Filtering out the deterministic part**]{}. Filtering out the deterministic part plays a major role in recovering community structures. Suppose we have two signals of the form $y_i(t) = {Y}_i(t) + \zeta(t), \quad i=1,2 $, where ${Y}_i$ is independent of $i$ and $\zeta(t)$ is a common noise term. $Y_i$ represents the superposition of the deterministic chaos and the independent fluctuations of the network setting. For the correlation, we have $$\begin{aligned} \mbox{Corr}(y_i , y_j) \approx \frac{\mbox{Cov}(\zeta,\zeta)}{\sigma^2_{Y_1} \sigma_{Y_2}^2}\end{aligned}$$ Hence, the large values of the variance of the time series ($\sigma_{y_i}\approx \sigma_{Y_i}\gg \sigma_{\zeta}$) suppress the contribution of the common noise, and an analysis solely based on the the original time series $y_i$ will overlook the common noise contribution. In fact, in the real system, we cannot assume ${Y}_i$ to be independently distributed. However, if the system is sufficiently chaotic, it will exhibit fast decay of correlations. Generating the connectivity structure {#generating-the-connectivity-structure .unnumbered} ===================================== To generate the adjacency matrix from a given degree distribution, we use the [*configuration model*]{} [@newman2003]. We know the number of stubs (half-links) for each node and the model randomly connects these stubs (half-links) to each other and generates an adjacency matrix that respects the given distribution. Dynamical Models {#dynamical-models .unnumbered} ================ We tested the effective network on a wide range of dynamics and networks satisfying assumptions (a)-(c). We focused on scale free networks and on rich club networks to evaluate the performance of the reconstruction. As a benchmark model, we restricted our attention to a neuron dynamics. Further models can be found in Supplementary Material. Neuron Model {#neuron-model .unnumbered} ============ We use Rulkov maps to model spiking and bursting neurons [@rulkov2001]. In this setting, the variable $\bm{x}$ at each node is two dimensional and we will denote $\bm{x} = (u, w)$. The local uncoupled dynamics is $\bm{F}(\bm{x}) = (F_1(u,w),F_2(u,w))$ with $$\begin{aligned} F_1(u,w) = \frac{\beta}{1+u^2} + w \quad\mbox{ and }\quad F_2(u,w) = w - \nu u - \sigma.\end{aligned}$$ The variable $u$ represents the membrane potential of a neuron, and, in this case, $u$ is the state variable measured by the observed time series $y_i(t)$; that is, $\phi({\boldsymbol}x)=u$. Different combinations of parameters $\sigma$ and $\beta$ give rise to different dynamical states of the neuron, such as resting, tonic spiking, and chaotic bursts. To test our procedure we considered two cases where $\sigma=\nu=0.001$ and $\beta=5.9$, which correspond to tonic spiking and $\beta=4.4$ for bursting. [*Chemical synapsis.*]{} Synaptic coupling occurs only through the membrane potential $u$. That is, $ \bm{H}(\bm{x}_i, \bm{x}_j) = (h(u_i,u_j),0)$ with $h(u_i,u_j) = (u_i - V_s) \Gamma (u_j)$, where $$\begin{aligned} \Gamma(u_j) = 1/ (1+\exp\{ \lambda( u_j - \Theta_s )\} )\end{aligned}$$ and $V_s$ is a parameter called reverse potential. Choosing $V_s > u_i(t)$, the synaptic connection is excitatory. We take $V_s = 20$, $\Theta_s = - 0.25$, and $\lambda=10$. Applying the dimensional reduction in , we obtain a map with averaged value of the interactions $ \bar F_1(u) = F_1(u) + \alpha k_i \langle \Gamma \rangle (u - V_s). $ One can *a posteriori* verify that $ \langle \Gamma \rangle = \int \Gamma \mu(dx) $, where $\mu$ is the physical measure for the map $F_1$ and can be estimated from the dynamics of the low-degree nodes. [*Electrical synapsis.*]{} Again, the synaptic coupling occurs only through the membrane potential $u$. That is, $ \bm{H}(\bm{x}_i, \bm{x}_j) = (h(u_i,u_j),0)$ with $$\begin{aligned} h(u_i,u_j) = u_j - u_i\end{aligned}$$ so the coupling depends only on the difference of states. The cat cerebral cortex {#the-cat-cerebral-cortex .unnumbered} ----------------------- The dataset for connections in the cat cerebral cortex consists of 53 geographic cortical areas investigated in detail with data-mining methods and was taken from [@scannell1993; @scannell1995]. The network is organised into four functional areas: visual, auditory, somatosensory-motor, and frontolimbic. [**Functional network.**]{} Several methods can be used to create a functional network to recover the clusters. The approach is to compute pairwise correlations of all nodes and to consider the similarities as interactions between nodes. This method is not appropriate for weakly coupled chaotic systems since they exhibit exponential decay of correlations. [**Effective network representation.**]{} The connectivity matrix can be estimated using the correlation matrix $ \rho_{ij} = \mbox{Corr}(\xi_i , \xi_j)$. The adjacency matrix $\bm{A}$ can be computed by thresholding the correlation matrix as $A_{ij}= \Theta(\rho_{ij} > \tau)$, where $\Theta$ is the Heaviside step function and $\tau=0.5$ is an arbitrary threshold. Note that we tested different thresholds with values from 0.3 to 0.6 and obtained similar results. [**Finding communities.**]{} Given the reconstructed connectivity matrix $\bm{A}$, we used the modularity-based Louvain method [@blondel2008] to detect communities. [**Finding rich-club members.**]{} Colizza [*et al.*]{} developed an algorithm to detect members of the rich-club [@colizza2006]. The algorithm provides a rich-club coefficient $\phi(k) \in [0,1]$ for each degree value in the network. We considered nodes with $\phi(k) > 0.8$ to be the members of the rich-club. Predicting critical transitions {#predicting-critical-transitions .unnumbered} =============================== Once we reconstruct the relevant information, we can perform numerical analysis of the recovered equations to explore the dynamics of the systems outside the observed range of parameters. Here we explain how to gather the information for a theoretical prediction of the critical transition. [**Burst synchronization.**]{} We introduce the slow variable as discussed in the main body of the manuscript. To introduce a phase variable $\theta$ for the dynamics $y$ we first smooth the time series [@footnotesmoothing]. Then, we find the time $t_n$ of local maxima as the $n$th maximum point of the slow variable. We do this after performing a Gaussian filter to denoise slightly the data. Finally, we introduce the phase variable $\theta$ as $$\begin{aligned} \theta(t) = 2\pi \left(\frac{t - t_n}{t_{n+1} - t_n} +t_n \right), \quad t_n < t < t_{n+1}\end{aligned}$$ as discussed in Refs. [@pereira2007] [**Reduction in the rich-club**]{}. Nodes in the rich-club have degrees of approximately $\Delta$ and make $\kappa \Delta$ connections inside the rich-club and $(1-\kappa)\Delta$ connections to the rest of the network. Following our reduction scheme, the interactions within and outside the rich-club can be described by the expected value of the interactions with respect to the invariant measure associated with each of them. Let $C$ denote the set of nodes in the rich-club, then the coupling term for the $i$th node in the rich-club is $$\begin{aligned} \sum_{j} A_{ij} \bm{H}(\bm{x}_i,\bm{x}_j) = \sum_{j \in C} A_{ij} \bm{H}(\bm{x}_i,\bm{x}_j) + \sum_{j \not \in C} A_{ij} \bm{H}(\bm{x}_i,\bm{x}_j)\end{aligned}$$ However, $$\begin{aligned} \sum_{j \not \in C} A_{ij} \bm{H}(\bm{x}_i,\bm{x}_j) = (1 - \kappa)\Delta \int \bm{h}(\bm{x}_i, \bm{y}) d\mu(\bm{y}) + \bm{\xi}^o_i(t)\end{aligned}$$ where $\mu$ is the invariant measure for the nodes outside the rich-club. Hence, for the rich-club we obtain $$\begin{aligned} \label{RCeq} \bm{x}_i(t+1) = \bm{q}_i(x_i(t)) + \sum_{j \in C} A_{ij} \bm{H}(\bm{x}_i,\bm{x}_j) + \bm{\xi}^o_i(t),\end{aligned}$$ where $\bm{q}_i(\bm{x}_i(t)) = \bm{F}_i(\bm{x}_i(t)) + (1 - \kappa)\Delta\alpha \int \bm{H}(\bm{x}_i, \bm{y}) d\mu(\bm{y})$. [**Predicting the transition to collective behaviour** ]{} Let us recall that when isolated $ u_i(t+1) = F_{1,i}(u_i(t))+ w_i(t)$ where $F_{1,i} \approx F_1$ and $ w_i (t+1) = w_i (t) + \mu (w_i(t) - 1)$. After some algebra we obtain $$\begin{aligned} \label{yeq} w(t+1) = w_0 + \mu \sum_{n=0}^t (u(n) - 1)\end{aligned}$$ Using the reduction Eq. (\[RCeq\]), in the network we obtain $$\begin{aligned} u_i(t+1) = F_{1,i}(u_i(t)) + u_i(t) + \alpha \Delta [ \langle u \rangle - u_i(t)] + \xi_i(t)\end{aligned}$$ where $i$ denotes the $i$th nodes in the rich-club, $\langle u \rangle$ is the mean membrane potential in the rich-club and $\xi_i$ are fluctuations. We fix two nodes $y_i = u_i$ and $y_j = u_j$ in the rich-club. Let us introduce the disturbance $$\begin{aligned} \zeta(t) = u_i(t) - u_j(t)\end{aligned}$$ Applying the mean value theorem and using that $F_{1,i} \approx F_1$ we obtain $$\begin{aligned} \zeta(t+1) = DF_1 (x_i(t)) \zeta(t) + \mu \sum_{n=0}^t \zeta(n) - \alpha \Delta \zeta(t)\end{aligned}$$ and introducing a proxy for the dynamics of the slow variables $$\begin{aligned} \label{ueq} \eta(t) = \sum_{n=0}^t z(n)\end{aligned}$$ and considering $$\begin{aligned} \label{Approx} \sum_{n=0}^t DF_1(x_i(n)) \zeta(n) \approx \lambda \sum_{n=0}^t \zeta(n)\end{aligned}$$ where we used that $\sum_{n=0}^t \zeta(n)$ is a slow variable. We obtain $$\begin{aligned} \label{udyn} \eta(t+1) = (\lambda - \alpha \Delta ) \eta(t) + \mu \sum_{n=0}^t \eta(n) \end{aligned}$$ Recall that for the cat cerebral cortex $\Delta = 37$. We given the time series $\{y_i \}$ for $\alpha \Delta = 0.3$, we estimate $F_1$ using our method $i$ as the slow variables are constants over short time scales, and the obtain slow variables as a filter over the fast variables. Then, the parameters of the rich-club such as the mean number of connections among the rich-club in terms of the reduction and external connections in terms of strength of fluctuations (We will provide a throughly discussion about the estimation of these parameters for a fully chaotic system in the Methods). From the data and using Eq. (\[Approx\]) we estimate $ \lambda = 1.42 $ and thus we obtain $ \alpha \Delta = 0.42. $ At this critical value the slow variables tend the stay together due to the contraction in the dynamics. This is related to the onset of synchronization in the bursts, which is captured via a phase variable through the order parameter. [**Sparse Recovery**]{}. The method gives a way of recovering the equations from data by considering the evolution map as a linear combination of a well chosen basis of functions, called library. The main assumption is that many coefficients will be zero. That is, the vector of coefficients will be sparse [@brunton2015]. Since we have $N$ nodes in our network, we consider the library $ {\mathscr}{L} = [ \phi_1(x_1),\phi_1(x_2),\cdots,\phi_1(x_N),\phi_2(x_1,x_N), \cdots , \phi_k(x_N,x_N) ] $ as the set of basis functions. And denote the $ X = \left(x_1({t_{2}}), x_2({t_{2}}), \cdots, x_N(t_1), x_1(t_2),\cdots,x_N(t_n)\right)^* $ where $^*$ denotes the transpose and we introduce $$\begin{aligned} \Theta = \left( \begin{array}{cccc} \phi_{1}(x_1(t_{1})) & \phi_1(x_2({t_{1}})) &\cdots & \phi_{k}(x_N({t_{1}}), x_N({t_{1}})) \\ \phi_{1}(x_1(t_{2})) & \phi_1(x_2({t_{2}})) &\cdots & \phi_{k}(x_N({t_{2}}), x_N({t_{2}})) \\ \vdots & \vdots & \vdots & \vdots \\ \phi_{1}(x_1(t_{n-1})) & \phi_1(x_2({t_{n-1}})) &\cdots & \phi_{k}(x_N({t_{n-1}}), x_N({t_{n-1}})) \\ \end{array} \right)\end{aligned}$$ We then look for a solution of the system $ X = \Theta \Xi $ where $$\Xi = \left( \zeta_{1,1} , \zeta_{1,2} , \cdots, \zeta_{1,N}, \zeta_{2,1} , \zeta_{2,2} , \cdots \zeta_{kN} \right)^*$$is the vector of coefficients. The sparse recovery technique then solves the linear equation for $\Xi$ iteratively enforcing the sparsity of $\Xi$ by introducing $\sigma$ such that if $|\zeta_{ij}| \le \sigma$ we set such entry to zero [@brunton2015]. =-9 In our case, we will consider the scenario where the synapsis between neurons are electric and we generate the multivariate data for the cat cerebral cortex. We will assume we have knowledge of the coupling function so we can easily read the network structure from the sparse recovery. We choose a library of polynomials as $\phi_j(x_i) = x_i^j$ for $j=\{1,\cdots,5\}$ and the pairwise $\phi_j(x_i,x_k)$ to be homogeneous polynomials of degree two in the variables $x_i$ and $x_k$ for $i,k\in\{1,\cdots,N\}$. We solve the minimization problem and group dependencies of the network through the coefficient vector as the network structure [@brunton2015]. The strength of each connection is of order $\alpha \approx 0.015$. Hence we have chosen values of $\sigma$ close to this value. The reconstructed network does not identity the clusters correctly as can be seen by comparing the blue and red markers in Figure \[SR\]. Moreover, employing the technique of community detection shows that the probability of identifying a node in the correct cluster is around 45%. This happens as each single connection is not strong enough. In fact, since the dynamics of the neurons are chaotic each connection plays a role of noise which leads to the misidentification. Thus we have shown that the sparse recovery method is not appropriate in our setting. ![Sparse recovery method is applied to the data generated by bursting neurons electrically coupled on the cat cerebral cortex. Selecting the threshold parameter $\sigma$ in the method changes the reconstructed network. Here we show the results of sparse recovery method for different enforced sparsity $\sigma$. The nonzero entries of the original network’s adjacency matrix are in blue. The red filled circles represent the nonzero entries in the adjacency matrix of the network reconstructed with the sparse recovery method. As each connection is small in comparison with the isolated dynamics, the sparse recovery tends to neglect them. []{data-label="SR"}](SparseRecovery_v2_reduced.pdf){width="1.0\linewidth"} **Acknowledgements** We would like to thank Tomislav Stankovski, Chiranjit Mitra, Mauro Copelli, Dmitry Turaev and Jeroen Lamb for enlightening discussions. This work was supported in part by the Center for Research in Mathematics Applied to Industry (FAPESP Cemeai grant 2013/07375-0), the European Research Council (ERC AdG grant number 339523 RGDD), and the Serrapilheira Institute. **Data Availability.** The connection matrices of cat cortex can be found at <https://sites.google.com/site/bctnet/datasets>. 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For a precise statement and details of the general setting see [@Tanzi]. The theorem has three main assumptions concerning the uncoupled dynamics ${\boldsymbol}F$, the network encoded in its adjacency matrix $A_{ij}$, and the reduced dynamics ${\boldsymbol}G_j$. We discuss a particular case and use it to illustrate the reconstruction step-by-step. - [**The local dynamics must increase the distance between points**]{} by a constant factor. A paradigmatic one-dimensional example is $ {\boldsymbol}F( x)=2 x$ mod $1$ that presents some of the most interesting characteristics of chaotic maps. - [**The networks are heterogeneous**]{}, having nodes with disparate degrees. This assumption is made precise in [@Tanzi] by a set of conditions involving the size of the network $N$, the number of hub nodes $M$, the maximum degree of a hub node $\Delta$, and the maximum degree of a low degree node $\delta$. One instance in which these conditions are satisfied and that includes many situations of interest is the following. Suppose $M$ is constant or slowly increasing with the size of the network, $M\sim \log N$, $\Delta$ grows faster than the square root of the system size, $\Delta\sim N^{\frac{1}{2}+\epsilon}$, and $\delta$ increases slowly, but polynomially in the system size, $\delta\sim N^{\frac{\epsilon}{2}}$. Then for a sufficiently large $N$, the network satisfies the conditions. - [**The reduced dynamics must be hyperbolic**]{}. This means that we assume the maps ${\boldsymbol}G_j$ to be either expanding (possibly by a non-constant factor), or to have a finite number of attracting periodic orbits. Every map can be perturbed by an arbitrarily small amount to obtain such an hyperbolic map [@Strien]. Under these assumptions, we can prove the following approximation result For every hub node $j$, the dynamics at the hub is given by $${\boldsymbol}x_j(t+1)={\boldsymbol}G_j({\boldsymbol}x_j(t))+{\boldsymbol}\xi_j(t)$$ where $|{\boldsymbol}\xi_j(t)|<\xi$ for time $T$ with $1\leq T\leq \exp[C\xi^2\Delta]$, and a set of initial condition of measure $1-T/\exp[C\xi^2\Delta].$, where $C$ is constant in $\Delta$, and $\xi$. Notice that one can pick the time scale $T$ exponentially large, but such that $T/\exp[C\xi^2\Delta]$ is very small so that, for large $\Delta$, the approximation result holds for very long time and for a large set of initial conditions. The reduced dynamics given by ${\boldsymbol}G_j$ depends on the hub’s state only. The term ${\boldsymbol}\xi(t)$ gives the fluctuations of the sum of all interactions from their expectation with respect to the physical measure of the unperturbed dynamics. Reconstruction of degree distributions ====================================== [**Scale-free networks.**]{} We create ensembles of scale-free network of $N=6000$ nodes with distinct structural exponents $\gamma$ following the same technique as in the main body of the manuscript. As in the main body, we only consider the largest connected component of the network (giant component). For each network realisation we compute the largest degree $\Delta$ and, for simplicity, we fix the coupling strength $\alpha / \Delta = 0.5$ throughout the rest of the section. Doubling map ------------ Let us now apply our approach when $\bm{F}_i$ is a perturbed version of the doubling map. Since the dynamics is one dimensional, we denote $\bm{x} = x$ and $\bm{F}_i(\bm{x}) = f_i(x)$ with $$f_i(x) = 2x + \varepsilon_i \sin 2\pi x \mod 1$$ and where we take $\varepsilon_i$ to be i.i.d. random variables (i.e. independent over $i$) uniformly distributed on $[0,10^{-3}]$. Likewise we write $\bm{H} = h$ with $$\label{Eq:DiffCoup} h(x_j,x_i) = \sin2\pi x_j-\sin2\pi x_i.$$ We fixed $\alpha = 10^{-2}.$ Since the unique absolutely continuous invariant measure for the doubling map is the Lebesgue measure $m$, we have $${\boldsymbol}V({\boldsymbol}x)=v(x) = \int h(y,x)d m(y)$$ yielding $v(x) = -\sin 2\pi x$ and the reduced dynamics takes the form $$\begin{aligned} x_i(t+1) = f_i(x(t)) - \alpha k_i v(x(t)) + \alpha \xi_i(t). \label{eq:hubs2}\end{aligned}$$ We aim to recover the reduced dynamics and in particular $f$ and $v$ from the data of a multivariate time series with $T=2000$ time steps. [**From data to Model.**]{} We assume not to have access to the network structure and only measure the time-series $\{x_i(t)\}$ at each node, as illustrated in Figure \[fig:Approach\] Data. By performing the steps described in the main body of the manuscript, we can recover an effective network. We now illustrate in detail our procedure in the case above of the doubling map. [**(i) Reduced dynamics.**]{} From the time series observed at each node, we construct the attractor for that particular node. Computing the embedding dimension we notice that the dynamic at each site (node) is well described by a one dimensional map, $g_i(x)= f_i(x) - \alpha k_i v(x)$, up to a fluctuation $\xi_i$. Hence, to reconstruct the attractor it suffices to obtain the return map. Return maps at different nodes are shown in Figure \[fig:Approach\] (i). [**(ii) Isolated dynamics and effective coupling.**]{} We start by introducing a similarity measure. [*Similarity of the time series.*]{} Two time series are considered almost the same, if whenever $y_i(t)$ and $y_j(t')$ are close then $y_i(t+1)$ and $y_j(t'+1)$ are also close. This means that one considers the new times series ${z}_i(t):=({y}_i(t),{y}_i(t+1))$, $z_{j}(t):=({y}_j(t),{y}_j(t+1))$, $t=0,\dots,T-1$, reshuffle them according to the lexicographical ordering (i.e. according to the magnitude of the first coordinate), and then take $s_{ij}$ to be the Pearson correlation distance between these reshuffled sequences. So if the time series at $i$ and $j$ are just out of phase, then, thanks to the reordering, their distance $s_{ij}=0$. We then calculate the intensities $S_i = \sum_{j} s_{ij}$ and use this to classify the nodes. The similarity matrix is shown in Figure \[fig:Approach\](ii). Using the correlation distance described above, we determine for which nodes $S_i$ is minimal and in this way we identify the low degree nodes. For the low degree nodes, the sequence $\{(y_i(t),y_i(t+1))\}_{t=0,\dots,T-1}$ lies close to the graph of the return map for $f$, and thus we obtain a good estimate for the isolated dynamics $f$, see Figure \[fig:Approach\](ii). Next, we apply the reduction to estimate the coupling function. To obtain the effective coupling $v$ from data, take a hub $j$ and consider the sequence $\{(y_j(t),y_j(t+1))\}_{t=1}^{2000}$. The resulting time series approximates the graph of $g_i$ and subtracting $f$ gives an approximation of the function $v$ up to a multiplicative constant (shown in Figure \[fig:Approach\](ii)). [**(iii) Network Structural Statistics.**]{} There are two ways to obtain information about the statistics of the degrees $k_i$. The first one uses the noise variance. In fact, the size of the fluctuations $\xi_i$ depends on the number of connections the node $i$ makes, and with good approximation Var$(\xi_i) \propto k_i$. The second one uses $f$ and $v$ recovered at the previous step. For every node $i$, choose $\beta_i$ to fit the time series $\{(y_i(t),y_i(t+1))\}_{t=0,\dots,T-1}$ with the map $$g_i(y) = f(y) - \beta_i v(y).$$ The value of $\beta_i$ is obtained by Bayesian inference on the return maps is shown in the first panel of Figure \[fig:Approach\](i) as $f$ and $v$. The distribution of $\beta_i$ will be linear proportional to the degree distribution and so we can use it to obtain the the structural parameter. At this point we can construct an effective network with degree distribution close to that of the real network can be constructed with the configuration model as discussed in the main body of the manuscript. We can also check whether there are communities in the network by analyzing the covariance of the fluctuations $\xi_i$. In Figure \[gamma\_est\] a) we reconstruct from the distribution of $\beta_i$ the structural exponent $\gamma$ for 1000 scale-free network with exponents ranging from $2.4$ to $3.6$ ![\[Color Online\] Step-by-step construction of an effective network for the doubling map. Our approach recovers the local dynamics of the doubling map, interaction function and statistical structure from the time series. Starting from datasets we uncover the approximate evolution rule for the time series using machine learning techniques. Such rules will be different for different nodes (depending on their degree) as shown in panel (i). Analysing the differences between these rules by means of a similarity analysis, we are able to obtain model of the isolated dynamics and an estimate the coupling function, see panel (ii). Finally, by using the theory of dynamical systems and dimensional reduction, we are able to estimate the number of input each node receives and the community structures. We then create a random presentation of the network structure by using a network model such as the configuration model.[]{data-label="fig:Approach"}](reconstruction_alt6_reduced.pdf){width="0.37\linewidth"} ### Robustness of the reconstruction under noise Adding some small independent noise to the dynamics does not influence much the reconstruction procedure for the doubling map. This is a consequence of stochastic stability of the local dynamics together with the persistence of the reduction [@Tanzi]. On the other hand, when the fluctuations become too large the reconstruction will underestimate the network structure. To illustrate these effects, we consider the randomly perturbed doubling maps $$x_i(t+1) = f_i(x(t)) + \eta_i(t)$$ where the random variables $\eta_i(t)$ are independent over $i$ and $t$, and identically distributed uniformly in the interval $[-\eta_0,\eta_0]$. Intuitively, as long as $\eta_0 < \alpha \min k_i$ the reconstruction will go through as the noise fluctuation will not compete with the coupling term. Notice that we normalize $\| v \| =1$. This is illustrated in Figure \[Doubling\_noise\]. In inset a) the noise has a large support $\varepsilon = 0.1$, and in particular larger than the coupling $\alpha \min k_i = 10^{-2}$. As a consequence, the reconstruction understimate the number of low degree nodes. In inset b) the noise has a small support $\varepsilon = 10^{-3}$. ![\[Color Online\] Effective networks are robust under random perturbations. When stochastic perturbations are moderate effective networks provide sharp estimates on the network structure. If the noise is large, the difference between the time-series at low and high degree nodes becomes blurred. In this case even the effective network underestimates the structural parameter $\gamma$. Inset a) shows the reconstruction of the degree distribution for $\epsilon_0=0.1$. Inset b) shows that the reconstruction is unaffected by the noise for $\varepsilon_0 = 10^{-3}$.[]{data-label="Doubling_noise"}](sm_noise.pdf){width="0.8\linewidth"} Logistic Maps on Scale-Free Networks ------------------------------------ Consider $M=[0,1]$, $\bm{x} = x$ and $\bm{F}_i(\bm{x}) = f_i(x)$ where $$f(x) := 4 x(1-x),$$ It is known that for such parameter, the map $f$ has an absolutely continuous invariant probability measure. In contrast to the doubling map, a small perturbation of the logistic map might result in a map that has a non-smooth physical measure, namely the measure is a distribution given by the sum of Dirac delta masses at a finite number of points that constitute the periodic orbits of the system. This compromises the validity of the rigorous result analysis that we used in the case of the doubling map, for which small perturbations produced small changes in the invariant measure [@Strien]. Nonetheless, as we will show, the reconstruction analysis gives good results in this case as well. We believe that this happens because our reconstruction only needs a finite number of points and gives a representation of the dynamics over finite time-scales. As a coupling function, we consider $$h(x_j,x_i) = \sin(2\pi x_j-2\pi x_i)$$ and the reduction is given by $ x_i(t+1) = f(x_i(t)) + \alpha k_i v(x_i(t)),$ where $v$ is defined as above. Again, we consider the reduced model $g_i(x)=f(x) + \beta_i v(x)$ in terms of the free parameter $\beta_i = \alpha k_i$. Analysing the distribution of $\beta_i$ we access the distribution of the degrees. Results are presented in Figure\[gamma\_est\]b). ![Recovering the structural parameter of scale free networks using effective networks. The power-law exponent $\gamma$ versus the estimated $\gamma_{\text{est}}$ from data for 1000 distinct networks. Inset a) shows the reconstruction for coupled doubling maps with diffusive coupling. Inset b) shows the reconstruction for coupled logistic maps with Kuramoto interactions. Inset c) Spiking neurons with electrical coupling, and finally, inset d) shows the reconstruction for the Hénon maps with the $y-$component diffusive coupled at the $x-$component. []{data-label="gamma_est"}](sm_gamma.pdf){width="0.6\linewidth"} Spiking Neurons with Electrical Synapses ---------------------------------------- We use the same models for the local dynamics of the spiking neurons which are described in the section Methods. We consider here a different coupling function. Denoting $\bm{x} = (u,w)$, the coupling is $$\bm{H}(\bm{x_i},\bm{x_j})=\bm{E} ( \bm{x}_j - \bm{x}_i ) = (u_j - u_i,0)$$ Results of the reconstruction are presented in Figure \[gamma\_est\]c). Henon Maps on Scale-Free Networks --------------------------------- Using the same notation, $\bm{x} = (u,w)$, the coupled Hénon maps are given by $$\bm{F}(\bm{x})=\left\{ \begin{array}{ll} 1-1.4u^2 + w\\ 0.3 w \end{array} \right. \mbox{~ ~ and ~ ~ } \bm{H}(\bm{x_i},\bm{x_j}) = \left\{ \begin{array}{ll} w_j-w_i \\ 0 \end{array} \right.$$ We observe the dynamics of the first component of the Hénon map, that is, $ y = \phi(\bm{x}) = u$. In this case, the reconstruction starts by determining the dimension of the reduced system. Takens embedding reveals that the dimension is two for large time excursions. Hence, we aim at learning a function of the kind $$y_i(t+1) = g_i(y_i(t),y_i(t-1)) + \xi_i(t).$$ We use polynomial functions for the fitting and perform a 10-fold cross-validation. Just our theory implies that $$g_i(y_i(t),y_i(t-1)) = f (y_i(t),y_i(t-1)) + \alpha k_i v(y_i(t),y_i(t-1))$$ where $f$ models the isolated dynamics and $h$ the coupling. Again we obtain $f$ from the low-degree nodes via a similarity analysis. We learn the function $h$ by $$\alpha k_i v(y_i(t),y_i(t-1)) = g_i(y_i(t),y_i(t-1)) - f (y_i(t),y_i(t-1))$$ In our case, $v(y_i(t),y_i(t-1)) \propto y_i(t-1)$ and we can obtain the degree distributions, Fig. \[gamma\_est\]d). Coupled Roessler Oscillators ---------------------------- Assume that the local dynamics is modelled by a Rössler oscillator [@Roessler]. The dynamics is now in continuous time and our method can also be applied by using a suitable Poincaré section. This gives an induced map that describes the dynamics of the system at specific instants of time (when the system hits a selected subset of phase space). Denoting $\bm{x} = (x, y, z)^*$ the vector field is given by $ \bm{F}(\bm{x}) = (y - z, x + 0.2 y , 0.2 + z(x - 9) ) $ and the coupling function, assumed to be diffusive, is given by $\bm{H}(\bm{x}_i,\bm{x}_j) = \bm{E}(\bm{x}_j-\bm{x}_i)$, where $\bm{E}$ projects to the first component, i.e., $\bm{E}(x,y,z) = (x,0,0)$. So, our main equation reads as $$\dot{\bm{x}} = \bm{F}(\bm{x}) + \alpha \sum_{j=1}^N A_{ij} \bm{E}(\bm{x}_j - \bm{x}_i)$$ We perform a numerical integration of the equations on the Rich-Club network using a 4th order Runge-Kutta with integration step $10^{-4}$ and get the data $\{ \bm{x}_i(t) \}_{t\ge 0}$. Using a statistical analysis of the time-series of the state variables, we are not able to reveal the connectivity structure. The data is phase coherent, that is, taking a Hilbert Transform we can decompose the time series in terms of amplitude and phase we conclude that the spread in the phase variable is small and thus the return time to a given section is nearly constant. So, we consider the Poincaré section defined by the maxima $w_i$ of the time series $x_i (t)$. This gives us a time series $\{ w_i(n)\} $ indexing all maxima. We then apply all the steps of the reconstruction procedure to this time series. Because of the coherent dynamics of the phase, the coupling form is preserved in the Poincaré section. The results of the network structure estimation are presented in Figure \[fig:Approach2\]. ![Main estimations of the reconstruction for the Rössler systems. In inset a) we show the return maps obtained from the time-series of a hub and a low-degree node. Inset b) shows a return plot for the coupling function which is used to estimate the reduced dynamics and the degrees. Inset (c) shows the power-law distribution of the degrees estimated from data and for the original network.[]{data-label="fig:Approach2"}](sm_roessler.pdf){width="1\linewidth"} Reconstruction of Rich-Club Mofits ================================== We report the performance of the method in the setting of a network of $100$ nodes having five clusters of $20$ nodes each. Four of these clusters are modelled as Erdös-Renyi random graph with connection probability $p=0.3$. The remaining cluster is the integrating clusters with connection probability $p=0.8$. The network resembles Fig. \[RC\]. ![\[Color Online\] Illustration of the rich-club motif. The network is composed of communities and certain nodes from each community are highly connected among themselves forming an integrating cluster.[]{data-label="RC"}](Illustration_Fig4_reduced.pdf){width="6cm"} [**Filtering out the deterministic chaos.**]{} We need to filter the contributions of the deterministic parts to reconstruct the community structure. Indeed, for two nodes $i$ and $j$ in the same cluster, the signals have the form $ x_i(t) = \tilde{X}_i(t) + \zeta(t), $ and $x_j(t)=\tilde{X}_j(t)+\zeta(t)$ where $\tilde{X}_i$ and $\tilde{X}_j$ is a superposition of the deterministic chaos depending on the variable at the node and independent fluctuations coming from the rest of the network, while $\zeta$ is common noise. $\tilde{X}_i$ and $\tilde{X}_j$ have fast decay of correlations, depends on different sets of variables, and for the sake of the following argument, can be assumed to be independent between each other and with $\zeta$. Under these assumptions, $ \mbox{Corr}( \tilde X_i(t) + \zeta(t) , \tilde X_j(t) + \zeta(t)) = \mbox{Var}(\zeta)/ \sigma_{x_i} \sigma_{x_j}$. Hence, the large values of the variance of the time series leads to strong suppression of the correlation coming from the small common noise $\zeta$. Condition for recovering the community -------------------------------------- In general, the coupling function is a sum of terms $ h(x,y) = u(x) v(y). $ This leads to noise terms $$\xi_i (t) = u(x_i) \left( \frac{1}{\Delta}\sum_{j} A_{ij} v(y_j) - k_i \int v(y) d\mu(y) \right)$$ where $\mu$ is the physical measure of the local dynamics. Given $i$ and $j$ the sum can be split into connections commons the $i$ and $j$ and to the independent connections. So, for such $i$ and $j$ we can write $$\xi_i = u(x_i)[\zeta_i(t) + w(t)] \mbox{~ and ~} \xi_j = u(x_j)[\zeta_j(t) + w(t)]$$ where $w$ is the noise due to the common connections. Notice that $w$ has zero mean. Let’s estimate the covariance of the component $\xi_i$ and $\xi_j$. By abuse of notation we will omit the time index $t$ and write the covariance and by the previous computations $ \mbox{Cov}(\xi_i, \xi_i) = \mathbb{E} [(u(x_i) w)( u(x_j) w) ]. $ After some manipulations, we obtain $$\begin{aligned} \mbox{Cov}(\xi_i, \xi_i) &= & \langle u \rangle^2 \mbox{Var} (w) \end{aligned}$$ so, if $ \int u (x) d\mu(x) = 0, $ the correlation between the noise will vanish even though they have a common term. Thus, the generic condition in order for the above scheme to be able to recover communities is that $ \langle v \rangle \not=0$. If this condition is not met, the network reconstruction via the $g_i$’s is also not possible. We recall that $\langle v \rangle=0$ is not a generic condition and this is why the effective network approach works in most cases. Doubling Map on Rich Club Network --------------------------------- Using the doubling map with the diffusive coupling described in the above section on the reconstruction for scale-free networks, we test the community detection from time-series using the correlation between time series and correlations between the noise $\xi_i$. Then we apply our technique to recover the network structure. We show the results in Fig. \[doubling\] ![\[Color Online\] [Reconstruction of a rich-club motif for coupled doubling maps with diffusive coupling $h(x,y) = \sin 2\pi y - \sin 2\pi x$.]{} The left panel is a color map of the pairwise Pearson-correlation between the time-series $x_i$ and $x_j$. Due to the strong chaotic behaviour this analysis does not give any information on the community structures. In the right panel, we show the colormap of the noise correlation. The clusters are manifested in the correlation structure of the noise, namely, the four communities and the integrating cluster. []{data-label="doubling"}](dm_rc.pdf){width="0.8\linewidth"} Logistic Maps on a Rich Club Network ------------------------------------ The dynamic of the logistic map and the coupling are described in the above section on the reconstruction for scale-free networks. We test the community detection approach on time-series using the correlation between time series and correlations between the noise. To this end, we fix the coupling strength $\alpha=10^{-4}$ and simulate the network dynamics. Then we apply our technique to recover the network structure. We show the results in Fig. \[logistic\]. ![\[Color Online\] Reconstruction of rich club for logistic maps with Kuramoto coupling $h(x,y) = \sin 2\pi( y - x)$. The left panel shows a color map of the pairwise Pearson-correlation between the time-series $x_i$ and $x_j$. In the right panel, we show the colormap of the noise correlation that reveals the community structure and, in particular, the integrating cluster.[]{data-label="logistic"}](lm_rc.pdf){width="0.8\linewidth"} Spiking neurons coupled with Electric Synapsis on a Rich-Club ------------------------------------------------------------- Again, we consider the spiking neurons and the electrical coupling described above on the same rich-club motif as before. The reconstruction and analysis of the noise correlation is able to detect the rich-club clusters. To this end we fix the coupling strength $\alpha=5 \times 10^{-4}$ and simulate the network dynamics. Then we apply our technique to recover the network structure. We show the results in Fig. \[spiking\]. ![\[Color Online\] Reconstruction of rich club for spiking neurons with electrical coupling. The left panel shows a color map of the pairwise Pearson-correlation between the time-series of the membrane potential. The right panel, shows the colormap of the noise correlation revealing the community structure and in particular, the integrating cluster. []{data-label="spiking"}](rmec_rc.pdf){width="0.8\linewidth"} Bursting neurons coupled with Electric Synapsis on a Rich-Club -------------------------------------------------------------- Our technique applies equally well then the local dynamics has multiple time-scales such as a bursting neuron. Our extensive numerical investigation reveals that when the resting time is not much larger then the total bursting time the reduced dynamics is capable of extracting the relevant information of the time series. Thus, we fixed the neuron parameter $\beta = 4.4$ to obtain a bursting dynamics. The reconstruction and analysis of the noise correlation is able to detect the rich-club clusters. To this end we fix the coupling strength $\alpha = 10^{-3}$ and simulate the network dynamics. Then we apply our technique to recover the network structure. We show the results in Fig. \[bursting\] ![\[Color Online\] Reconstruction of rich club for bursting neurons with electrical coupling. The left panel shows a color map of the pairwise Pearson-correlation between the time-series of the membrane potential. The right panel, shows the colormap of the noise correlation revealing the community structure and in particular, the integrating cluster.[]{data-label="bursting"}](dm_rc.pdf){width="0.8\linewidth"} Predicting Critical Transitions in Rich-Club Networks ===================================================== We present another example of how to use the effective network methodology to predict critical transitions. In this case we choose the doubling map for the local dynamics coupled on a rich-club network with clusters sampled as Erdös-Renyi graphs. Consider a rich club network of 2200 elements with 5 clusters. Four of the clusters are made of $N_\ell=500$ nodes with small degrees, and one cluster (called integrating cluster or rich club) has $N_{I}=200$ nodes which are connected with most of the network. The edges within a cluster of low degree nodes are assigned as described above. As a model for the isolated dynamics, we use the doubling map $f(x)=2x$ mod 1 with the diffusive coupling $ {\boldsymbol}H({\boldsymbol}x_i,{\boldsymbol}x_j)=h(x_j,x_i) = \varphi(x_j)-\varphi(x_i)$ where we picked $\varphi(x)=\sin(2\pi x)$. We use the function $$E(t) = \frac{1}{N_I(N_I-1)}\sum_{i, j\in IC} \|x_i(t)-x_j(t)\|$$ as an empirical measure of the synchronization level at time $t$. From a single multivariate time-series, when the coupling is fixed at $\alpha_0 \Delta = 0.2$ (red marker Figure \[New\]), the analysis of $ \langle E \rangle$ gives no sign of critical transitions. A statistical analysis shows that the variance of the averaged synchronization error is not amplified, and the extreme value statistics fails to reveal a transition as it also does an analysis of dynamical correlations. As described above, we can use the effective network methodology to recover the local dynamics $f$ and the effective coupling $v$. In particular we recover the isolated dynamics $f(x)=2x$ within $4\%$ accuracy for the poorly connected nodes. The unique physical equilibrium measure for this map is the uniform distribution $m$ on $[0,1)$. Furthermore, we recover the effective coupling $$\begin{aligned} v(x) = \int h(y,x)dm(y)=-\sin 2\pi x.\end{aligned}$$ At this point, we know that the approximate evolution of any node $i$ is given by $g_i=f_i+\alpha k_i v_i$, and we can study numerically or analitically this rule to find those values of $\alpha$ for which the integrating cluster exhibits synchrony. The analysis shows that excursions towards synchronization will start once the coupling is increased by $15\%$ (see Figure \[New\]). Below we provide the details on how to recover this critical value of $\alpha$ for which a transition arises.. Nodes in the integrating cluster have roughly degree $\Delta$ and make $\kappa \Delta$ connections inside the integrating cluster and $(1-\kappa)\Delta$ to the rest of the network.The interactions felt by a node in the rich-club can be split in those coming from nodes in other clusters and those coming from nodes within rich-club itself: $$\begin{aligned} \sum_{j} A_{ij} h(x_i,x_j) = \sum_{j \in RC} A_{ij} h(x_i,x_j) + \sum_{j \not \in RC} A_{ij} h(x_i,x_j)\end{aligned}$$ But, $$\begin{aligned} \sum_{j \not \in RC} A_{ij} h(x_i,x_j) = (1 - \kappa)\Delta \int h(x_i, y) d\mu(y) + \xi^o_i(t)\end{aligned}$$ where $\mu$ is the invariant measure for the nodes outside the integrating cluster [^1]. Hence, the equation of the integrating cluster can be written as $$\begin{aligned} x_i(t+1) = q_i(x_i(t)) + \sum_{j \in RC} A_{ij} h(x_i,x_j) + \xi^o_i(t),\end{aligned}$$ where $q_i(x_i(t)) = f_i(x_i(t)) + (1 - \kappa)\Delta\alpha \int h(x_i, y) d\mu(y)$. We can estimate $\mu$ empirically analyzing each cluster. Assuming that $h(x,y)=\varphi(y) -\varphi(x)$, we can recover $\varphi$ from the analysis of $q_i$ using the reconstruction techniques. In fact, for this particular choice of $h$, $\int h(x,y) d\mu(y)$ is equal to $-\varphi(x)$ plus a constant. Now if $\nu$ is the measure that describes the behaviour of a node in the integrating cluster, then the interaction within the rich-club can be written as $$\sum_{j \in RC} A_{ij} h(x_i,x_j)=\kappa\Delta\int h(x_i, y) d\nu(y)+\xi^c(t)=\kappa \Delta\left(-\varphi(x)+\int \varphi(s)d\nu(s)\right)+\xi^c(t).$$ Putting the two equations together one has that $$\label{Eq:nuself} x_i(t+1) = g_i(x_i(t)) + \alpha \Delta c(\mu,\nu) + \zeta_i(t)$$ where $g_i = f_i-\alpha \Delta \varphi $ models the reduced dynamics and the $c(\mu,\nu) = (1 - \kappa)\Delta\alpha\int\varphi (s)d\mu(s)+\kappa\Delta\alpha\int \varphi(s)d\nu(s)$ is the mean contribution from all interactions (inside and outside the integrating cluster). Finally $\zeta_i = \xi^o_i(t)+\xi^c(t)$ combines the effect of the fluctuations. From a single multivariate time series at a given coupling parameter, shown in Figure \[New\] as a red dot, we reconstruct the model. First, we obtain the the rule $g_i$ which we uncover to be $g_i(x) = 2 x- \beta_i \sin 2\pi x $ mod $1$, hence $\beta_i = 0.169$ and this number is nearly independent of the node in the integrating cluster. Hence, we obtain an estimate $(\alpha \Delta)_{\rm est} = 0.169$. We also obtain that the dynamics of nodes in the communities is well approximated by $f(x) = 2x $ mod $1$. Next, we need to estimate $\kappa$ to construct a model for the connectivity of the integrating cluster. From the recovered local dynamics $f$, one knows that $\mu$ is the Lebesgue measure and, since $\int \varphi d\mu = 0$, we obtain $ c(\mu,\nu) = \kappa \int \varphi d\nu$. In the regime of parameters where the measurements have been made, $\nu$ can be obtained empirically and this allows to recover $\kappa$ since $$\frac{1}{\int \varphi d\nu} \frac{1}{\alpha \Delta} \langle x_i(t+1) - g_i(x_i(t)) \rangle \rightarrow \kappa,$$ where we evaluate $\int\phi d\nu$ with respect to the empirically retrieved $\nu$, we substitute $\alpha\Delta$ with the estimated value above, and $\langle \cdot \rangle$ denotes the time average. We obtain that $\kappa = 0.47$. The data analysis reveals that such $\kappa$ value is nearly independent of the node in the integrating cluster. Moreover, the analysis of the covariance of the noise in the integrating clusters shows a lack of communities and the functional analysis network indicates that the integrating cluster of is a random network 200 nodes with $p=0.83$. From this analysis, we obtain an estimate for $\Delta \approx p \times 200 / \kappa = 353=\Delta_{est}$. We are now able to estimate $\alpha$ by $\alpha_{est} = (\alpha\Delta)_{est}/\Delta_{est}= 5\times 10^{-4}$. With the reconstructed data $(f,v,A)$, we obtain that for a node in the integrating cluster, reads as $$x_i(t+1) = 2x_i(t)-\alpha \Delta\sin(2\pi x_i(t)) +\kappa\Delta\alpha \int \sin(2\pi s)d\nu(s)+ \xi(t).$$ When $1/2\pi<\alpha\Delta<3/2\pi$, the map $ 2x_i(t)-\alpha \Delta \sin(2\pi x_i(t))$ mod $1$ has an attracting fixed point at 0. In this new regime, $\nu=\delta_0$ is a self-consistent measure for the integrating cluster, meaning that the measure $\nu$ gives rise to an approximated dynamical rule $g_i$ that has $\nu$ as equilibrium measure. This can be easily verified since $\int \sin(2\pi s)d\delta_0(s)=0$. We can then conclude that by selecting a value of the coupling strength such that $\alpha \Delta$ is in the range above, one expects all the states at the nodes in the integrating cluster to evolve towards the point $0$ and fluctuate around this point by $\xi(t)$. To obtain the range of $\alpha \Delta$ such that the fluctuations of $E$ are around twice $\max\| \xi \|$, we notice that since $\nu=\delta_0$ the stability properties are given by the linear stability around $x=0$. Let $u$ be a small displacement around $0$ and let us denote at $J(\alpha) = Dg_i(0)$ the Jacobian of the map $g_i$ at $0$. Then we obtain that $u(t+1) = \sum_{i=0}^t J(\alpha)^{i} \xi_i$ and so $\| u \| \le \max \| \xi \| / (1 - J(\alpha))$, which yields that $1/(1-J(\alpha)) < 2$. Hence, we obtain $3/4\pi<\alpha\Delta<5/4\pi$. Because we measured $\alpha \Delta$ as $0.169$ in the data given we predict that a high quality coherent state in the rich-clubwill appear then $\alpha \Delta$ is increased by $40\%$. This is a agreement with the experiments. ![[**Prediction of critical transition in a network with a rich-club motif.**]{} The level of average synchronisation, $\langle E \rangle$ of the integrating cluster is shown for different values of the coupling strength $\alpha$. Insets show $E(t)$ plotted as a function of time for five points indicated by arrows. For $\alpha$ values in the grey shaded region, $\langle E \rangle$ is close to zero and the integrating cluster exhibits collective behaviour. We can predict the extrema of the shaded region by studying the effective network obtained from a time series without any knowledge of parameters, including $\alpha_0 \Delta$. For this prediction, we used the time series when $\alpha_0 \Delta =0.2$ (red point).[]{data-label="New"}](f_fig4_1.pdf){width="\linewidth"} -5mm Cat cerebral cortex =================== Bursting neurons with Chemical Synapesis ---------------------------------------- We simulated each mesoregion of the cat cerebral cortex network with bursting Rulkov oscillators coupled through chemical synapsis. For such dynamics the parameters are given as $\beta=4.4$ and $\Delta \alpha=0.05$ where there is no synchrony between oscillators Fig. \[catmap\_bursting\]. We use the simulated data to reconstruct the network structure as shown in Figure \[catmap\_bursting\]. ![Effective network of the cat cerebral cortex. We use the local dynamics as a spiking neuron coupled via electric synapses with the parameters $\Delta \alpha=0.05$ and $\beta=4.4$. (a) The cat cerebral cortex network with nodes colour coded according to the four functional modules. Rich-club members are indicated by red encircled nodes. (b) The covariance matrix of the data cannot detect communities. (c) The covariance matrix of the fluctuations can distinguish clusters of interconnected nodes. (d) A model in the cat cortex constructed via the effective network approach. From the matrix in (c) we can recover a representative effective network. The reconstructed network represents the real network in (a) with good accuracy. See Methods for the details of the detection of communities and rich-club members.[]{data-label="catmap_bursting"}](cat_map_rc_beta44_alpha00001_reduced.pdf){width="1.0\linewidth"} [99]{} Pereira, T., van Strien, S., & Tanzi, M. [*Heterogeneously coupled maps: hub dynamics and emergence across connectivity layers*]{}. J. Eur. Math. Soc., preprint: arXiv [**1704.06163**]{}, 1–63 (2017) de Melo, W., and Van Strien, S., [*One-dimensional dynamics*]{}. Springer (2012). Rössler, O. E. [*An equation for continuous chaos*]{}. Physics Letters A 57, no. 5 (1976): 397-398. [^1]: In the example above $\mu$ is the Lebesgue measure $m$.

**Realization of the probability laws in the quantum central limit theorems by a quantum walk** Takuya Machida *Meiji Institute for Advanced Study of Mathematical Sciences,* *Meiji University, 1-1-1 Higashimita, Tamaku, Kawasaki 214-8571, Japan* Introduction ============ Quantum walks (QWs), which are quantum versions of random walks, are expected to be one of the simple dynamics to understand quantum systems, and they are permeating more and more in science. The behavior of the QWs is different from that of random walks. With a proper rescaling, probability distributions of the QWs on the line $\mathbb{Z}=\left\{0,\pm 1,\pm2,\ldots\right\}$ after many steps are approximately expressed by probability density functions with a compact support. Even as time evolution of the QWs is determined by a space-homogeneous dynamics, limit density functions of the QWs have singularity in space. To interpret the interesting property of the QWs, a lot of long-time limit theorems have been investigated, because getting such a limit theorem is equivalent to understanding asymptotic behavior of the QWs after long time. To forecast the behavior of the QWs, it is worth studying the long-time limit theorems. On the other hand, quantum probability theory is algebraic generalization of the Kolmogorov probability theory. One of the major ideas in probability is the notion of independence. In quantum probability theory, there are four notions of independence (free, monotone, commuting, and boolean independence) and quantum central limit theorems associated to each independence have been proved [@Voiculescu1985; @Lu1997; @Muraki1997; @CushenHudson1971; @GiriWaldenfels1978; @SpeicherWoroudi1997]. As we well know, a long-time limit distribution of simple random walks, that is the Gaussian distribution, follows from the central limit theorem for independence in the Kolmogorov probability theory. The relation between the random walks and the central limit theorem has supported fruitful applications of the random walks. Meanwhile, there are only a few current applications of the QWs, which are mainly construction of quantum algorithms in quantum computation (see, for instance, [@Ambainis2003; @ShenviKempeWhaley2003; @ChildsCleveJordanYonge-Mallo2009; @FeldmanHilleryLeeReitznerZhengBuvzek2010; @ReitznerHilleryFeldmanBuvzek2009; @Venegas-Andraca2008]). If the QWs connect to the quantum central limit theorems, they would be more useful via quantum probability theory. To discuss the potential of the QWs in quantum probability theory, we focus on the limit distribution of a discrete-time QW on the line in this paper. The first derivation of a limit theorem for discrete-time 2-state QWs on the line was done by a path counting method in 2002 [@Konno2002] (see also Konno [@Konno2005]). In probability distributions of the QWs on time- or space-inhomogeneous environments, localization can occur [@Konno2010; @KonnoLuczakSegawa2013; @Machida2011]. Inui et al. [@InuiKonnoSegawa2005] obtained a limit distribution with localization for a 3-state QW. In addition, localization of multi-state walks was also discussed in Inui and Konno [@InuiKonno2005] and Segawa and Konno [@SegawaKonno2008]. Konno and Machida [@KonnoMachida2010] rewrote a 2-state QW with memory introduced by McGettrick [@McGettrick2010] as a 4-state QW and got two limit theorems. Konno [@Konno2009] calculated the QW on random environments whose probability distribution after long time isn’t localized at all. Via a weak convergence theorem, Chisaki et al. [@ChisakiKonnoSegawaShikano2011] investigated crossover from QWs to random walks. Recently, Konno et al. [@KonnoMachidaWakasainpress] found relations between long-time limit density functions of discrete- or continuous-time QWs on the line and well-known Fucksian linear differential equations of the second order (the Heun equation, the Gauss hypergeometric equation). These limit theorems are results for the QWs with a localized initial condition. Since we have not obtained the probability density functions related with the central limit theorems in quantum probability theory from the QWs with a localized initial state, we will move our focus to the QWs with a non-localized initial state in this paper. There are a few results for the QWs distributed widely in an initial state [@AbalDonangeloRomanelliSiri2006; @AbalSiriRomanelliDonangelo2006; @ValcarcelRoldanRomanelli2010; @ChandrashekarBusch2012]. Abal et. al. [@AbalDonangeloRomanelliSiri2006; @AbalSiriRomanelliDonangelo2006] analyzed a QW starting from two positions. Chandrashekar and Busch [@ChandrashekarBusch2012] reported numerical results for a QW with an initial state ranging over an area. Valc[á]{}rcel et al. [@ValcarcelRoldanRomanelli2010] treated the QW initialized by a Gaussian-like distribution in a continuum limit. A uniform stationary measure of the Hadamard walk with a non-localized initial state was discussed in Konno et al. [@KonnoLuczakSegawa2013]. In the rest of this paper, we deal with the following topics. We introduce the definition of a discrete-time 2-state QW on the line in Sec. \[sec:definition\]. Section \[sec:limit\_th\] is devoted to show our limit theorem. In addition, the limit theorem derives a corollary. After introducing a non-localized initial state which is defined in Sec. \[sec:initial\], we report that the QW generates the probability laws associated to the independence in quantum probability theory. In the final section, we conclude our results and discuss a future problem. Definition of a discrete-time QW on the line {#sec:definition} ============================================ Total system of discrete-time 2-state QWs on the line is defined in a tensor space $\mathcal{H}_p\otimes\mathcal{H}_c$, where $\mathcal{H}_p$ is called a position Hilbert space which is spanned by a basis $\left\{\ket{x}:\,x\in\mathbb{Z}\right\}$ and $\mathcal{H}_c$ is called a coin Hilbert space which is spanned by a basis $\left\{\ket{0},\ket{1}\right\}$ with the vectors $\bra{0}=[1,0],\,\bra{1}=[0,1]$. Let $\ket{\psi_{t}(x)} \in \mathcal{H}_c$ be the probability amplitudes of the walker at position $x$ at time $t \in\left\{0,1,2,\ldots\right\}$. The state of the 2-state QWs on the line at time $t$ is expressed by $\ket{\Psi_t}=\sum_{x\in\mathbb{Z}}\ket{x}\otimes\ket{\psi_{t}(x)}$. Time evolution of the QWs is described by a unitary matrix $$U=\cos\theta\ket{0}\bra{0}+\sin\theta\ket{0}\bra{1}+\sin\theta\ket{1}\bra{0}-\cos\theta\ket{1}\bra{1}$$ with $\theta\in [0,2\pi)$. The behavior of the QW with $\theta= 0, \pi/2, \pi, 3\pi/2$ is trivial. So, in the present paper we don’t treat such a case. The amplitudes evolve according to $$\ket{\psi_{t+1}(x)}=\ket{0}\bra{0}U\ket{\psi_t(x+1)}+\ket{1}\bra{1}U\ket{\psi_t(x-1)}.$$ The probability that the quantum walker $X_t$ can be found at position $x$ at time $t$ is defined by $$\mathbb{P}(X_t=x)=\braket{\psi_t(x)|\psi_t(x)}.$$ By giving the initial states $\ket{\psi_0(x)}$, we can determine the probability distribution $\mathbb{P}(X_t=x)$ for any time $t$. In this paper, we focus on a special non-localized initial state which will be introduced in Sec. \[sec:initial\]. Limit theorem {#sec:limit_th} ============= In this section, we present a limit theorem of the QW as $t\to\infty$ by using the Fourier analysis which is one of the standard methods to derive limit theorems of the QWs [@Konno2010; @Machida2011; @InuiKonnoSegawa2005; @InuiKonno2005; @SegawaKonno2008; @KonnoMachida2010; @ChisakiKonnoSegawaShikano2011; @GrimmettJansonScudo2004]. The long-time limit theorem plays an essential role to draw asymptotic behavior of the QW after many steps. The time evolution of the QW provides the Fourier transform $\ket{\hat{\Psi}_{t}(k)}=\sum_{x\in\mathbb{Z}} e^{-ikx}\ket{\psi_t(x)}\,(k\in\left[-\pi,\pi\right))$, the relation $\ket{\hat{\Psi}_{t+1}(k)}= \hat U(k)\ket{\hat{\Psi}_{t}(k)}=\hat U(k)^t \ket{\hat\Psi_{0}(k)}$ , where $\hat U(k)=(e^{ik}\ket{0}\bra{0}+e^{-ik}\ket{1}\bra{1})U$. After straightforward calculation by the Fourier analysis, for $r=0,1,2,\ldots$, we get $$\begin{aligned} \lim_{t\to\infty}\mathbb{E}\biggl[\biggl(\frac{X_t}{t}\biggr)^r\biggr]=&\int_{0}^{\pi} \frac{dk}{2\pi} h(k)^r\,\biggl[\biggl\{\left|\braket{v(k)|\hat\Psi_0(k)}\right|^2+\left|\braket{v(-k)|\hat\Psi_0(-k)}\right|^2\biggr\}\nonumber\\ &+(-1)^r\biggl\{\left|\braket{\overline{v(\pi-k)}|\hat\Psi_0(k)}\right|^2+\left|\braket{\overline{v(\pi+k)}|\hat\Psi_0(-k)}\right|^2\biggr\}\biggr],\end{aligned}$$ where $$\begin{aligned} h(k)=&\frac{c\cos k}{\sqrt{1-c^2\sin^2 k}},\\ \ket{v(k)}=&\frac{e^{ik}s}{\sqrt{N(k)}}\ket{0}-\frac{c\cos k+\sqrt{1-c^2\sin^2 k}}{\sqrt{N(k)}}\ket{1},\\ N(k)=&1+s^2+c^2\cos 2k+2c\cos k\sqrt{1-c^2\sin^2 k},\end{aligned}$$ and $c=\cos\theta, s=\sin\theta$. The detail can be found in Grimmett et al. [@GrimmettJansonScudo2004]. By putting $h(k)=x$, we obtain the following lemma. For $r=0,1,2,\ldots$, we have $$\lim_{t\to\infty}\mathbb{E}\left[\left(\frac{X_t}{t}\right)^r\right]=\int_{-\infty}^\infty x^r \frac{|s|}{2\pi(1-x^2)\sqrt{c^2-x^2}}\eta(x)I_{(-|c|,|c|)}(x)\,dx,$$ where $I_A(x)=1$ if $x\in A$, $I_A(x)=0$ if $x\notin A$ and $$\begin{aligned} \eta(x)=&\left|\braket{v(\kappa(x))|\hat\Psi_0(\kappa(x))}\right|^2+\left|\braket{v(\kappa(x))|\hat\Psi_0(\kappa(x)-\pi)}\right|^2\nonumber\\ &+\left|\braket{v(-\kappa(x))|\hat\Psi_0(-\kappa(x))}\right|^2+\left|\braket{v(-\kappa(x))|\hat\Psi_0(\pi-\kappa(x))}\right|^2,\\ \kappa(x)=&\arccos\left(\frac{|s|x}{c\sqrt{1-x^2}}\right)\,\in\left[0,\pi\right]. \end{aligned}$$ \[lem:limit\] If we give a special initial state to the Fourier transform $\ket{\hat\Psi_0(k)}$, the following limit theorem, which can be immediately proved from Lemma \[lem:limit\], is obtained. Let $F:\mathbb{R}\longrightarrow\mathbb{R}$ be a function that satisfies $F(k+2\pi)=F(k)$, $\int_{-\pi}^{\pi}F(k)^2 dk=2\pi$ and $F(k)\in C^{\infty}[-\pi,\pi]$ almost everywhere, where $\mathbb{R}$ means the set of real numbers. If we assume $\ket{\hat\Psi_0(k)}=F(k)(\alpha\ket{0}+\beta\ket{1})$ with $\alpha,\beta\in\mathbb{C}$ and $|\alpha|^2+|\beta|^2=1$, $$\lim_{t\to\infty}\mathbb{E}\biggl[\left(\frac{X_t}{t}\right)^r\biggr]=\int_{-\infty}^\infty x^r\, \Bigl\{f_1(x;\alpha,\beta)\eta_1(x)+f_2(x;\alpha,\beta)\eta_2(x)\Bigr\}I_{(-|c|,|c|)}(x)\,dx,$$ where $\mathbb{C}$ is the set of complex numbers, $$\begin{aligned} f_1(x;\alpha,\beta)=&\frac{|s|}{\pi(1-x^2)\sqrt{c^2-x^2}}\biggl[1-\biggl\{|\alpha|^2-|\beta|^2+\frac{2s\Re(\alpha\overline{\beta})}{c}\biggr\}x\biggr],\\ f_2(x;\alpha,\beta)=&-\frac{s\Im(\alpha\overline{\beta})}{|c|\pi(1-x^2)},\\ \eta_1(x)=&\frac{1}{4}\Bigl\{F(\kappa(x))^2+F(-\kappa(x))^2+F(\kappa(x)-\pi)^2+F(\pi-\kappa(x))^2\Bigr\},\\ \eta_2(x)=&\frac{1}{2}\Bigl\{F(\kappa(x))^2-F(-\kappa(x))^2+F(\kappa(x)-\pi)^2-F(\pi-\kappa(x))^2\Bigr\},\end{aligned}$$ \[th:limit\] and $\Re(z)$ (resp. $\Im(z)$) denotes the real (resp. imaginary) part of $z\in\mathbb{C}$. Theorem \[th:limit\], moreover, leads us to the following corollary. If the function $F(k)$ satisfies $|F(k-\pi)|=|F(-k)|=|F(k)|$, $$\lim_{t\to\infty}\mathbb{E}\bigg[\bigg(\frac{X_t}{t}\biggr)^r\biggr]=\int_{-\infty}^\infty x^r f_1(x;\alpha,\beta)F(\kappa(x))^2 I_{(-|c|,|c|)}(x)\,dx.$$ \[cor:1\] In Sec. \[sec:example\], we will show four examples of the QW as $t\to\infty$ by using our limit theorem. Non-localized initial state {#sec:initial} =========================== In this section, by using the Fourier series expansion, we construct a non-localized initial state. Let $w:\mathbb{R}\longrightarrow\mathbb{R}$ be the function that satisfies $w(k+2\pi)=w(k)$ and $w\in L^2[-\pi,\pi]$. Assuming $W(w)=\int_{-\pi}^{\pi} w(k)^2\,dk > 0$, we take the initial state of 2-state QWs on the line as $$\ket{\psi_0(x)}=\frac{1}{\sqrt{2\pi W(w)}}\left(\int_{-\pi}^{\pi} w(k)e^{ikx}dk\right)\left(\alpha\ket{0}+\beta\ket{1}\right),\label{eq:upis}$$ with $\alpha,\beta\in\mathbb{C}$ and $|\alpha|^2+|\beta|^2=1$. We should note that $\sum_{x\in\mathbb{Z}}\mathbb{P}(X_0=x)=\sum_{x\in\mathbb{Z}}\braket{\psi_0(x)|\psi_0(x)}=1$ by Parseval’s theorem. The Carleson-Hunt theorem changes the initial state of the Fourier transform, $$\begin{aligned} \ket{\hat\Psi_0(k)}=\sqrt{\frac{2\pi}{W(w)}}\,w(k)\left(\alpha\ket{0}+\beta\ket{1}\right)\quad \mbox{a.e. on }[-\pi,\pi],\label{eq:fourierexpand}\end{aligned}$$ because we have $$\sum_{x\in\mathbb{Z}}\left(\frac{1}{2\pi}\int_{-\pi}^{\pi}w(\tilde{k})e^{i\tilde{k}x}d\tilde{k}\right)e^{-ikx}=w(k)$$ almost everywhere on $[-\pi,\pi]$ [@JorsboeMejlbro1982; @Reyna2002]. Equation (\[eq:fourierexpand\]) is obtained by the Fourier series expansion. The function $\sqrt{2\pi/W(w)}\,w(k)$ corresponds to the function $F(k)$ in Sec. \[sec:limit\_th\]. Note that the function $w(k)=1$ gives the QW starting from the origin with $\ket{\psi_0(0)}=\alpha\ket{0}+\beta\ket{1}$. In the next section, we will extract the effects of this non-localized initial state in limit distributions. Limit distributions {#sec:example} =================== Limit distribution of the QW starting from the origin with $\ket{\psi_0(0)}=\alpha\ket{0}+\beta\ket{1}$ was derived in Konno [@Konno2002; @Konno2005]. If we pick $w(k)=1$, his result is obtained, $$\lim_{t\to\infty}\mathbb{E}\bigg[\bigg(\frac{X_t}{t}\biggr)^r\biggr]=\int_{-\infty}^\infty x^r \frac{|s|}{\pi(1-x^2)\sqrt{c^2-x^2}}\biggl[1-\biggl\{|\alpha|^2-|\beta|^2+\frac{2s\Re(\alpha\overline{\beta})}{c}\biggr\}x\biggr] I_{(-|c|,|c|)}(x)\,dx.$$ In this section, given the function $w(k)\in L^2[-\pi,\pi]$, we realize limit distributions in quantum central limit theorems from the discrete-time 2-state QW with a non-localized initial state. Wigner semicircle law --------------------- If we set $$w(k)=\frac{\sin k}{1-c^2\sin^2 k},$$ a limit theorem follows from Corollary \[cor:1\], $$\lim_{t\to\infty}\mathbb{E}\left[\left(\frac{X_t}{t}\right)^r\right]=\int_{-\infty}^{\infty}x^r\frac{2}{\pi c^2}\sqrt{c^2-x^2}\biggl[1-\left\{|\alpha|^2-|\beta|^2+\frac{2s\Re(\alpha\overline{\beta})}{c}\right\}x\biggr]I_{(-|c|,|c|)}(x)\,dx.\label{eq:result_case1}$$ Note that $W(w)=\pi/|s|^3$. When we choose $\alpha,\beta$ such that $|\alpha|^2-|\beta|^2+\frac{2s\Re(\alpha\overline{\beta})}{c}=0$ (e.g. $\alpha=1/\sqrt{2},\beta=i/\sqrt{2}$), the limit density function becomes the Wigner semicircle distribution, which is also gotten in the free central limit theorem associated to the free independence [@Voiculescu1985]. Arcsine law ----------- The non-localized initial state determined by the function $$w(k)=\frac{1}{\sqrt{1-c^2\sin^2 k}}$$ yields a convergence theorem, $$\lim_{t\to\infty}\mathbb{E}\left[\left(\frac{X_t}{t}\right)^r\right]=\int_{-\infty}^{\infty}x^r\frac{1}{\pi\sqrt{c^2-x^2}}\biggl[1-\left\{|\alpha|^2-|\beta|^2+\frac{2s\Re(\alpha\overline{\beta})}{c}\right\}x\biggr]I_{(-|c|,|c|)}(x)\,dx.\label{eq:result_case2}$$ We should note $W(w)=2\pi/|s|$. The density function in Eq. (\[eq:result\_case2\]) includes the arcsine law which also appears in the monotone central limit theorem [@Lu1997; @Muraki1997]. Gaussian distribution --------------------- Assuming $$w(k)=\sqrt{\frac{|\sin k|}{\left(1-c^2\sin^2 k\right)^{\frac{3}{2}}}}\exp\left\{-\frac{c^2\cos^2 k}{4\sigma^2\left(1-c^2\sin^2 k\right)}\right\},\label{eq:w_case3}$$ then we have $W(w)=\frac{2\sqrt{2\pi}\sigma}{|c|s^2}\erf\left(\frac{|c|}{\sqrt{2}\sigma}\right)$ and get a limit theorem, $$\lim_{t\to\infty}\mathbb{E}\left[\left(\frac{X_t}{t}\right)^r\right]=\int_{-\infty}^{\infty}x^r\frac{\exp\left(-\frac{x^2}{2\sigma^2}\right)}{\sqrt{2\pi}\sigma\erf\left(\frac{|c|}{\sqrt{2}\sigma}\right)}\biggl[1-\left\{|\alpha|^2-|\beta|^2+\frac{2s\Re(\alpha\overline{\beta})}{c}\right\}x\biggr]I_{(-|c|,|c|)}(x)\,dx,\label{eq:result_case3}$$ where $\sigma$ is a positive constant and the function $\erf(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt$ denotes the Gauss error function. This result implies Eq. (\[eq:w\_case3\]) can lead the QW to the Gaussian distribution as $t\to\infty$. The commuting central limit theorem associated to the commuting independence is represented by the Gaussian distribution [@CushenHudson1971; @GiriWaldenfels1978]. Uniform distribution -------------------- In this subsection, we show that the uniform distribution can be obtained by the QW. The uniform distribution isn’t related with the central limit theorems associated to the independence in quantum probability theory. However, from the view point of quantum computation, it is interesting for the QW to generate the uniform distribution. Given the function $$w(k)=\sqrt{\frac{|\sin k|}{\left(1-c^2\sin^2 k\right)^\frac{3}{2}}},$$ we get the following limit theorem, $$\lim_{t\to\infty}\mathbb{E}\left[\left(\frac{X_t}{t}\right)^r\right]=\int_{-\infty}^{\infty}x^r\frac{1}{2|c|}\biggl[1-\left\{|\alpha|^2-|\beta|^2+\frac{2s\Re(\alpha\overline{\beta})}{c}\right\}x\biggr]I_{(-|c|,|c|)}(x)\,dx.\label{eq:result_case4}$$ Remark that $W(w)=4/s^2$. Equation (\[eq:result\_case4\]) proves that the QW produces the uniform distribution. The relation between a QW with a non-localized initial state and the uniform distribution was also discussed numerically in Valcárcel et al. [@ValcarcelRoldanRomanelli2010]. Summary {#sec:summary} ======= In this section, we argue about the conclusion and discussion for our results. To discover a connection between QWs and quantum probability theory, we focused on a QW with a non-localized initial state. In quantum probability theory, there are four notions of independence currently, and the quantum central limit theorems associated to them have been obtained. Probability density functions in the central limit theorems don’t agree with limit distributions of the QWs with a localized initial state. So, we turned to limit distributions of a discrete-time 2-state QW with a non-localized initial state. As a result, we showed that the QW can create the probability laws in the quantum central theorems. If we take $\alpha=1/\sqrt{2}, \beta=i/\sqrt{2}$, then the density functions in Eqs. (\[eq:result\_case1\]), (\[eq:result\_case2\]) and (\[eq:result\_case3\]) consist with those in the central limit theorems induced by the free, monotone and commuting independence, respectively. We remark that when the QW operated by $U=\ket{0}\bra{0}-\ket{1}\bra{1}$ (i.e. $\theta=0$ case) starts from the origin, $\lim_{t\to\infty}\mathbb{P}(X_t/t\leq x)=\int_{-\infty}^x\,\frac{1}{2}(\delta_{-1}(y)+\delta_1(y))\,dy$, where $\delta_x(y)$ means the Dirac delta function. Also, the boolean central limit theorem exhibits this probability law [@SpeicherWoroudi1997]. So, we have found that four probability laws in quantum central limit theorems can be created by the QW. It, however, is not understood what the non-localized initial states given in this paper mean to the definition of independence in quantum probability theory. The relation between the initial states and independence should be clear in the future. Moreover our limit theorem supports that we can realize some desired probability density functions with a compact support in quantum computation by controlling the function $F(k)$. As an example of the application, we generated the uniform random valuable by the QW (see Eq. (\[eq:result\_case4\])). Figure \[fig:comparison\] illustrates comparisons between the limit density function and the probability distribution at time $t=5000$ in the case of $\alpha=1/\sqrt{2},\,\beta=i/\sqrt{2}$. \ [(a) semicircle]{} \ [(b) arcsine]{} \ [(c) Gaussian]{} \ [(d) uniform]{} \[fig:comparison\] One of the interesting future problems is to discuss convergence rate in the quantum central limit theorems and the limit theorem of the QW. The rate of our limit theorem is $t$, while that of the quantum central limit theorems is $\sqrt{t}$. 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--- abstract: 'Matching the number counts of high-$z$ sub-millimetre-selected galaxies (SMGs) has been a long standing problem for galaxy formation models. In this paper, we use 3D dust radiative transfer to model the sub-mm emission from galaxies in the  cosmological hydrodynamic simulations, and compare predictions to the latest single-dish observational constraints on the abundance of -selected sources. We find unprecedented agreement with the integrated  luminosity function, along with good agreement in the redshift distribution of bright SMGs. The excellent agreement is driven primarily by ’s good match to infrared measures of the star formation rate (SFR) function between $z = 2-4$ at high SFRs. Also important is the self-consistent on-the-fly dust model in , which predicts, on average, higher dust masses (by up to a factor of 7) compared to using a fixed dust-to-metals ratio of 0.3. We construct a lightcone to investigate the effect of far-field blending, and find minimal contribution to the shape and normalisation of the luminosity function. We provide new fits to the  luminosity as a function of SFR and dust mass. Our results demonstrate that exotic solutions to the discrepancy between sub-mm counts in simulations and observations, such as a top-heavy IMF, are unnecessary, and that sub-millimetre-bright phases are a natural consequence of massive galaxy evolution.' author: - | Christopher C. Lovell,$^{1}$[^1] James E. Geach,$^{1}$ Romeel Davé,$^{2,3,4}$ Desika Narayanan$^{5,6,7}$ & Qi Li$^{5}$\ $^{1}$Centre for Astrophysics Research, School of Physics, Astronomy & Mathematics, University of Hertfordshire, Hatfield AL10 9AB\ $^{2}$Institute for Astronomy, Royal Observatory, University of Edinburgh, Edinburgh EH9 3HJ\ $^{3}$University of the Western Cape, Bellville, Cape Town 7535, South Africa\ $^{4}$South African Astronomical Observatories, Observatory, Cape Town 7925, South Africa\ $^{5}$Department of Astronomy, University of Florida, 211 Bryant Space Sciences Center, Gainesville, FL, USA\ $^{6}$University of Florida Informatics Institute, 432 Newell Drive, CISE Bldg E251, Gainesville, FL, USA\ $^{7}$Cosmic Dawn Center, Niels Bohr Institute, University of Copenhagen and DTU-Space, Technical University of Denmark bibliography: - 'smg\_paper.bib' - 'custom.bib' - 'pd\_refs.bib' date: 'Accepted XXX. Received YYY; in original form ZZZ' title: 'Reproducing sub-millimetre galaxy number counts with cosmological hydrodynamic simulations' --- \[firstpage\] galaxies: active – galaxies: evolution – galaxies: formation – galaxies: high-redshift – galaxies: abundances Acknowledgements {#acknowledgements .unnumbered} ================ The authors wish to thank James Trayford and Maarten Baes for helpful comments and suggestions. C.C.L. and J.E.G. acknowledge financial support from the Royal Society by way of grants RGF\\EA\\181016 and URF\\R\\180014.  was run at the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC capital grants ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. DiRAC is part of the National e-Infrastructure. Partial support for D.N. and Q.L. were provided from the US National Science Foundation via NSF AST-1715206 and AST-1909153. We used the following open source software packages in the analysis, unless already mentioned: <span style="font-variant:small-caps;">Astropy</span> [@robitaille_astropy:_2013], <span style="font-variant:small-caps;">Scipy</span> [@2020SciPy-NMeth] and Matplotlib [@Hunter:2007]. Data will be made available upon request to the corresponding author \[lastpage\] [^1]: E-mail: [email protected] (CCL)

--- abstract: 'The electromagnetic calorimeters of the various magnetic spectrometers in Hall C at Jefferson Lab are presented. For the existing HMS and SOS spectrometers design considerations, relevant construction information, and comparisons of simulated and experimental results are included. The energy resolution of the HMS and SOS calorimeters is better than $\sigma/E \sim 6\%/\sqrt E $, and pion/electron ($\pi/e$) separation of about 100:1 has been achieved in energy range 1 – 5 GeV. Good agreement has been observed between the experimental and simulated energy resolutions, but simulations systematically exceed experimentally determined $\pi^-$ suppression factors by close to a factor of two. For the SHMS spectrometer presently under construction details on the design and accompanying GEANT4 simulation efforts are given. The anticipated performance of the new calorimeter is predicted over the full momentum range of the SHMS. Good electron/hadron separation is anticipated by combining the energy deposited in an initial (preshower) calorimeter layer with the total energy deposited in the calorimeter.' author: - 'H. Mkrtchyan' - 'R. Carlini' - 'V. Tadevosyan' - 'J. Arrington' - 'A. Asaturyan' - 'M. E. Christy' - 'D. Dutta' - 'R. Ent' - 'H. C. Fenker' - 'D. Gaskell' - 'T. Horn' - 'M. K. Jones' - 'C. E. Keppel' - 'D. J. Mack' - 'S. P. Malace' - 'A. Mkrtchyan' - 'M. I. Niculescu' - 'J. Seely' - 'V. Tvaskis' - 'S. A. Wood' - 'S. Zhamkochyan' title: 'The lead-glass electromagnetic calorimeters for the magnetic spectrometers in Hall C at Jefferson Lab' --- Introduction {#intro} ============ The experimental program at Jefferson Lab focuses on the studies of the electromagnetic structure of nucleons and nuclei, in particular in a region where a transition is expected from a nucleon-meson description into a quark-gluon description of matter. In experimental Hall C the emphasis has been on inclusive (e,e$^\prime$) electron scattering and proton knockout (e,e$^\prime$p) experiments at the highest four-momentum transfer ($Q^2$) accessible, deuteron photodisintegration experiments, and both exclusive and semi-inclusive pion electroproduction reactions. In particular, the Hall C experimental program has studied the onset of the quark-parton model description of such reactions. To accomplish such a diverse program, a highly flexible set of instruments capable of accurate measurements of final momenta and angles is required, including both efficient background rejection and good particle identification properties. This remains very much in place after the 12-GeV Upgrade of Jefferson Lab (JLab) has been completed, with Hall C emphasizing precision measurements at high luminosities, with detection of high-energy reaction products approaching the beam energy at very forward angles. The initial base equipment of Hall C was well suited to the JLab scientific program that required high luminosity, intermediate detector acceptances and resolution [@CDR1990]. With the high luminosities needed to access neutrino-like scattering probabilities comes a high-background suppression requirement. The magnetic spectrometer pair that constituted the base equipment pointed to a common pivot with scattering chamber. The Short Orbit Spectrometer (SOS), with a QDD configuration, accessed a momentum range of 0.3 - 1.7 GeV/c, and an angular range of $13.3^{\circ}$ - $168.4^{\circ}$. It was explicitly designed to measure pions and kaons with short life times. The High Momentum Spectrometer (HMS), with a QQQD magnetic configuration, covered a momentum range 0.5 - 7.3 GeV/c, but was to date not used above 5.7 GeV/c. The HMS accessed an angular range between $10.5^{\circ}$ - $80^{\circ}$. After the JLab 12-GeV Upgrade [@CDR-12], the Hall C scientific program is again focused on high luminosity measurements with detection of high energy reaction products at small forward angles. Such a physics program can be accessed only by a spectrometer system providing high acceptance for, given the larger boosts associated with the energy upgrade, very forward-going particles, and analyzing power for particle momenta approaching that of the incoming beam. To accomplish this, and maintain a spectrometer pair rotating around a common pivot for precision coincidence measurements, the SOS will be superseded by the newly built Super High Momentum Spectrometer (SHMS). The SHMS will achieve a minimum (maximum) scattering angle of 5.5$^\circ$ (40$^\circ$) with acceptable solid angle and do so at high luminosity. The maximum momentum will be 11 GeV/c, well matched to the maximum energy available in Hall C. The basic parameters of the HMS, SOS and SHMS are listed in Table \[hms-shms-param\]. Parameter HMS SOS SHMS ---------------------------- --------------- ---------------- -------------- Momentum Range (GeV/c) 0.5-7.3 0.3-1.7 1.5-11.0 Momentum Acceptance (%) $\pm 10$ $\pm 20$ -10 - +22 Momentum resolution (%) 0.10-0.15 $<$0.1 0.03-0.08 Horiz. Angl. Accept.(mrad) $\pm$32 $\pm$40 $\pm$18 Vert. Angl. Accept. (mrad) $\pm$85 $\pm$70 $\pm$50 Solid angle (msr) 8.1 9.0 $>$ 4.5 Maximum scattering angle $\leq 80^o$ $\leq 168.4^o$ $\leq 40^o$ Minimum scattering angle $\geq 10.5^o$ $\geq 13.3^o$ $\geq 5.5^o$ Horiz. Angl. res. (mrad) 0.8 0.5 0.5-1.2 Vertical Angl. res. (mrad) 1.0 1.0 0.3-1.1 Vertex Reconstr. res. (cm) 0.3 2-3 0.1-0.3 : \[hms-shms-param\] The basic parameters of the HMS, SOS and SHMS spectrometers. The standard detector packages in the HMS and SOS were designed from inception to be very similar [@arrington]. The detector stacks, shown for the HMS in Fig. \[hms\_hut\], are located inside the respective concrete spectrometer shield houses. A pair of six-plane drift chambers (DC1 and DC2) is situated immediately after the dipole magnet, in the forefront of shield house to allow for particle tracking. They are followed by two pairs of x-y scintillator hodoscopes sandwiching a gas Čerenkov. In some experiments, an aerogel Čerenkov detector was added either before (HMS) or after (SOS) the pairs of scintillators. The last detector in the detection stack is the lead-glass electromagnetic calorimeter, positioned at the very back of the shield house. Its support structure is in fact mounted on the concrete wall of the shield house. The two sets of drift chambers are used for track reconstruction, the four scintillating hodoscope arrays for triggering and time-of-flight measurements, and the threshold gas (and aerogel) Čerenkov detectors and lead-glass calorimeters for electron/hadron separation. Hall C experiments typically demand well-understood detection efficiencies of better than 99%, and background particle suppression of 1,000:1 in $e/\pi$ separation, typically. This can be achieved by combining 100:1 suppression in the electromagnetic calorimeter, with the remaining suppression in a gas Čerenkov counter. Several experiments used signals from the calorimeter and gas Čerenkov counters already in a hardware trigger to reject pions or electrons by a factor of 25:1 in the online data acquisition system. The Particle Identification (PID) systems of both spectrometers performed remarkably stable over more than a decade of use. The combination of gas Čerenkov and lead-glass electromagnetic calorimeter ensured pion suppressions of typically a few 1,000:1, for electron detection efficiency of better than 98%. =3.40in =1.70in The detector package of the SHMS will be a near-clone of the HMS. It will again include a pair of multiwire drift chambers for tracking, and scintillator and quartz hodoscopes for timing. As the SHMS will both detect a variety of hadrons ($\pi$,$K$,p) in a number of coincidence experiments with HMS, and electrons in single-arm (e,e$^\prime$) experiments, special attention is again paid to the PID system. It must provide similar particle identification as mentioned above, even at the higher energies. In its basic configuration the SHMS detection stack includes a heavy gas Čerenkov for hadron selection, and a noble-gas Čerenkov and lead-glass electromagnetic calorimeter for electron/hadron separation. It is again envisaged to augment the detector stack with aerogel Čerenkov detectors, primarily for kaon identification. The approved experiments demand a suppression of pion background for electron/hadron separation of 1,000:1, with suppression in the electromagnetic calorimeter alone on the level of 100:1. An experiment to measure the pion form factor at the highest $Q^2$ accessible at JLab with 11 GeV beam [@Fpi-12] requires a strong suppression of electrons against negative pions of a few 1,000:1, with a requirement on the electromagnetic calorimeter of a 200:1 suppression. This paper describes the electromagnetic calorimeters in the various magnetic spectrometers, be it existing or under construction, in Hall C at Jefferson Lab. Section \[hcal\_scal\] describes in detail the pre-assembly studies, the component selection, construction and assembly of the HMS and SOS calorimeters. Section \[early\_mc\] explains the Monte Carlo simulation package used, and highlights the structure and some details of the simulation software. Sections \[electronic\_calibr\] and  \[hcal\_scal\_perform\] cover the electronics and calibration of the calorimeters. We present resolution, efficiency and hadron rejection capability of the calorimeters in both HMS and SOS, and compare experimental data with simulation results. In Section \[shms\_calo\] we describe details of the newly designed calorimeter for the SHMS, including information on the component selection and construction. We also present results of pre-assembly component checkout, and the anticipated performance of the SHMS calorimeter from simulation studies. HMS and SOS Calorimeters {#hcal_scal} ======================== Particle detection using electromagnetic calorimeters is based on the production of electromagnetic showers in a material. The total amount of the light radiated in this case is proportional to the energy of the primary particle. Electrons (as well as positrons and photons), will deposit their entire energy in the calorimeter giving a detected energy fraction of one. The energy fraction is the ratio of energy detected in the calorimeter to particle energy. Charged hadrons entering a calorimeter have a low probability to interact and produce a shower, and may pass through without interaction. In this case they will deposit a constant amount of energy in the calorimeter. However, they may undergo nuclear interactions in the lead-glass and produce particle showers similar to the electron and positron induced particle showers. Hadrons that interact inelastically near the front surface of the calorimeter and transfer a sufficiently large fraction of their energy to neutral pions will mimic electrons. The maximum attainable electron/hadron rejection factor is limited mainly by the cross section of such interactions. Construction {#hcal_construct} ------------ R&D, design and construction of the calorimeters for the HMS and SOS magnetic spectrometers started in 1991-1992. In 1994 both calorimeters were assembled and installed as part of the instrumentation of Hall C spectrometers, becoming the first operational detectors at JLab. Since the first commissioning experiment, the calorimeters have been successfully used in nearly all experiments carried out in Hall C. In 2008, the SOS spectrometer was retired and its calorimeter blocks removed to be used for the preshower of the newly designed SHMS spectrometer. The HMS calorimeter will remain in place for use after the Continuous Electron Beam Accelerator Facility’s (CEBAF) 12–GeV upgrade. The HMS/SOS calorimeters are of identical design and construction except for their total size. Blocks in each calorimeter are arranged in four planes and stacked 13 and 11 blocks high in the HMS (see Fig. \[hms\_lg\]) and SOS respectively. The planes are shifted relative to each other in the vertical direction by $\sim$5 mm. In addition, the entire detector is tilted by $5^o$ relative to the central ray of the spectrometer. These shifts make it impossible for particles to pass through the calorimeter without interaction. The total thickness of the material along the particle direction of $\sim$14.6 radiation lengths is enough to absorb the major part of energy of electrons within the HMS momentum range. All blocks were produced in early 1990’s by a Russian factory in Lytkarino [@lytkarino], whose products of good optical quality were well known. The blocks are $10~{\rm cm}\times 10~{\rm cm}\times 70~{\rm cm}$ in size and machined with a precision of 0.05 mm. They may contain bubbles or stones with a diameter less than 300 $\mu$m with an impurity frequency of less than 5-10 per kg of glass. The optics and acceptances of the spectrometers (see Table \[hms-shms-param\]) required the calorimeters to have frontal dimensions about $60\times 120~{\rm cm}^2$ for the HMS and $60\times 100~{\rm cm}^2$ for the SOS. To avoid any shower leakage from the calorimeter volume, we chose to extend the physical dimensions of the calorimeters at least 5 cm beyond the sizes required by spectrometer acceptance. This gave calorimeter physical areas of $70\times 130~{\rm cm}^2$ for the HMS and $70\times 110~{\rm cm}^2$ for the SOS. Looking from the side, the HMS calorimeter consists of 52 modules stacked in 4 columns (each layer 3.65 rad. length thick) (see Fig. \[hms\_lg\]). In addition to total energy deposition of the particle, a modular calorimeter gives information on the longitudinal development of the shower (which is different for electromagnetic and hadronic showers). This additional information can be used for more effective electron/hadron selection. =3.40in =3.40in Since the modules are oriented transversely to the incident particles, to detect photons from Čerenkov radiation one needs to attach photomultipliers (PMTs) from the side of the block and cover the area $10\times 10~{\rm cm}^2$ of the blocks as effectively as possible. The energy resolution of a lead-glass shower counter depends strongly on the ratio of the photocathode area to the output area of the radiator [@davydov]. Photomultipliers with a photocathode diameter of 3.0“-3.5” were considered to be the optimal choice for the HMS/SOS calorimeters since they could provide a relatively high value of $\sim$0.44 - 0.50 for this ratio. The single module assembly {#hcal_module} -------------------------- The requirement that the lead glass blocks must be optically isolated and optically coupled to PMTs was the primary guidance for the construction. The individual module design is shown in Fig. \[calo\_block\]. To ensure light-tightness, each block is wrapped in 25 $\mu$m thick aluminized Mylar and 40 $\mu$m thick Tedlar type film. There is a thin layer of air between the block and Mylar, for optical insulation was not completely tight wrapped. Each block is also equipped with ST type optical fiber adapter for light monitoring system. The blocks are slightly different in sizes, but on average the spread in length is less than $\pm$0.250 mm and less than $\pm$0.100 mm (100$\pm$0.10 mm) in transverse size. The gaps between the modules in final assembly are less than 250 $\mu$m. The calorimeter signals from the blocks are read out by 8-stage Philips XP3462B photomultiplier tubes. The PMTs are shielded by six turns of 100 $\mu$m thick $\mu$-metal foil. Since the PMTs operate at negative high voltage and the photocathodes are near the magnetic shields and other mechanical parts at ground potential, special protection is required to avoid current leakage between the photocathodes and ground. For this reason, the PMT bulbs were wrapped in several layers of thin Teflon and black electrical tape. After full assembly, the current leak for each block was measured with a high voltage about 200 V above nominal operating setting. =3.40in =1.70in Silicone grease ND-703 with high viscosity is used for the PMT – block optical contact (index of refraction $\sim$1.46). Originally PMTs were attached to only right side of the blocks (looking along the central ray of the HMS). The PMTs on the left side in the first two layers were added in the late 1998 in order to enhance signal output, especially at low energies. In addition, this weakens dependence of the aggregate signal from a module on the particle’s point of impact. Photomultiplier tube selection and studies {#hcal-pmt} ------------------------------------------ The choice of the photomultiplier tube depends on the intensity of light to be measured and the regime of its operation. One of the most important requirements for the PMTs used in the HMS/SOS calorimeters was high efficiency for the electrons above $\sim$100 MeV, and good linearity up to the energies of several GeV. At low energy (or low light intensity) the PMT must have relatively high gain in order to keep electron trigger efficiency high. But, in all cases its operation regime must be optimized for best signal-to-noise ratio. Ideally, the gain of a PMT with $n$ dynode stages and an average secondary emission ratio $\delta$ per stage is $G\sim{\delta}^n$. While the secondary emission ratio is given by ${\delta=A\cdot\triangle V^\alpha}$, where $A$ is a constant, $\triangle V\approx V/(n+1)$ is the interstage voltage, and $\alpha$ is a coefficient which depends on the dynode material and geometric structure (typically $\alpha\approx$0.7-0.8). For a voltage $V$ applied between the cathode and the anode, the gain is roughly $G\approx k\cdot V^{\alpha n}$, where $k$ is a constant. So the gain (or the PMT output signal amplitude) is proportional to the applied voltage $V$ and will increase as $V^{\alpha n}$ (in the linearity range of the PMT). But with the applied high voltage the anode dark current will also increase (current in the PMT even when it is not illuminated). Major sources of dark current are thermoelectric emission of electrons from the materials, ionization of residual gases, glass scintillation, leakage current from imperfect insulation. The resulting noise from the dark current is a critical factor in determining the low limit of light detection, in the optimization of the PMT gain ( via high voltage), especially when the rate of dark current change varies. The choice of XP3462B PMT was made after studies of several other 3 inch and 3.5 inch photomultiplier tubes on the matter of having good linearity, photocathode uniformity, high quantum efficiency, and good timing properties. Gain variations with HV and dark currents also were measured. In order to understand the limits imposed by the PMTs on the performance of the detector, several tests were performed on a set of candidate PMTs  [@Amatuni96]. All showed excellent linearity over a 3500:1 dynamic range of a reasonably chosen high voltage, as well as good time and amplitude resolutions. For pulses corresponding to photo-electron (pe) yield of ${N_{pe}=10^3}$, which is the expected signal from 1 GeV electron, the amplitude and time resolutions were ${\sigma_A/A \approx 4\%}$, and ${\sigma_t\approx}$100-150 ps (measured with a pulsed variable intensity UV laser). These tests served as a guide for specifying requirements for the procurement of the PMTs. Following these tests, as a time and cost effective solution, Photonis XP3462B PMTs were chosen for the equipment of the HMS and SOS calorimeters. These 8-stage PMTs have a 3” diameter ($\approx$68 mm) semitransparent bi-alkaline photocathode, and a linear focused cube dynode structure with a peak quantum efficiency (QE) of $\sim$29% at 400nm (Fig. \[xp3462b-qe\]). =3.40in =2.40in Using the criteria of high quantum efficiency, low dark current and high gain at relatively low HV, the best PMTs (150 out of 180 available) were selected. The negative operating voltages were set in the range $\sim$1.4-1.8 kV to match the gain $\sim 10^6$. The outputs were gain matched to within $\sim20\%$, and the remaining differences were corrected in software. The PMT output signal may vary with respect to the incident photon’s hit position on the photocathode. In general, this is caused by the photocathode and the multiplier (dynode section) non-uniformities. Although the focusing electrodes of a phototube are designed so that electrons emitted from the photocathode are collected effectively by the first dynode, some electrons may deviate from their desired trajectories causing lower collection efficiency. The collection efficiency varies with position on the photocathode from which the photoelectrons are emitted and influences the spatial uniformity of a PMT. The spatial uniformity is also determined by the photocathode surface uniformity itself. If the cathode-to-first dynode voltage is low, the number of photoelectrons that enter the effective area of the first dynode becomes low, resulting in a slight decrease in the collection efficiency. For samples of PMTs the photocathode uniformity and effective diameter have been studied with a laser scanner. A $\sim$1 mm diameter fiber was positioned on the front of the PMT at a small distance from the photocathode. The light generated by the laser was split into two parts: one for the PMT scan, and another to monitor incident light intensity by a photo-diode. The PMT was mounted on a special stand, which could be moved remotely in 2-5 mm steps. At each position of the PMT, the coordinate information ($x_i$) from the scanner, PMT signal amplitude ($A^i_{pmt}$), and reference photo-diode signal ($A_0$) were readout and written to a data file. The PMT photocathode uniformity and effective diameter were found from the analysis of the $A^i_{pmt}/A_0$ distribution versus $x_i$. Nearly all the tested XP3462B PMTs had a photocathode of good uniformity and effective diameter of no less than $\sim$2.8 inch. The measured effective diameter only weakly depends on the PMT high voltage. This is likely an indication that the effective diameter is largely determined by the collection efficiency between the photocathode and the first dynode. Gain variation has been studied for the phototubes, under experimental conditions typical for CEBAF beam, as a function of the mean anode current, light pulse intensity and the high voltage distribution applied to the dynode system [@Amatuni96]. These studies suggest that at mean anode current $\sim 20~\mu$A the PMT gain may change up to $\sim 15\%$. Examples of gain variation with mean anode current measured at the different light pulse heights are shown in Fig. \[gain\_var\]. =3.40in =2.40in Samples of the assembled modules were tested in a magnetic field to evaluate the quality of the PMT shields. At a fixed high voltage the blocks were illuminated through the ST connectors with a constant light intensity. Signal amplitude from the PMT was measured at gradually increasing values of the magnetic field. Measurements were performed at two different orientations of the PMT relative to the magnetic field: axial and transverse. As expected, the effect of the magnetic field was much stronger for the axial orientation. For both axial and transverse magnetic fields up to 2 Gauss, no effect was detected. Even at field values of about 4 Gauss, no effect was observed when the field was oriented transversely relative to PMT axis, while an axial field of the same strength reduced the PMT signal by 20–30%. We concluded that the PMT magnetic shields were sufficient, since in HMS and SOS detector huts the calorimeters are located far from the magnets where fringe fields are less that 0.5 Gauss. Studies on optical properties of TF-1 type lead glass blocks {#tf1-block} ------------------------------------------------------------ With its index of refraction $\sim$1.65, radiation length 2.74 cm and density of $3.86~ {\rm g}/{\rm cm}^3$ TF-1 type lead glass is well suited for serving as Čerenkov radiator in electromagnetic calorimeters. Note, the TF-1 radiative length found in different sources varies from 2.5 to 2.8cm. We cite the value obtained by means of PEGS4 (preprocessor for EGS4 [@EGS4]) and GEANT4 [@geant4] packages. The fractional composition consists primarily of ${\rm PbO}$ (51.2%), ${\rm SiO}_2$ (41.3%), ${\rm K}_2{\rm O}$ (3.5%) and ${\rm Na}_2{\rm O}$ (3.5%). Before assembly, the light transmittance of all the blocks was measured using a spectrophotometer from the JLab Detector Group [@Zorn]. The wave-length was scanned from 200 nm to 700 nm in steps of 10 nm. The blocks were oriented transversely, and the light intensity passing through the 10 cm thickness was measured. Two measurements were carried out: with and without blocks (to subtract dark current of the light detector and light loss in air). Those measurements were repeated in 2008 on a set of blocks taken from the decommissioned SOS calorimeter for re-use in the SHMS preshower counter. The blocks had been in use under the beam conditions for 15 years, and thus checks for possible degradation of the lead glass from radiation were necessary. Reliability of the measurements was checked by measuring spared, unused blocks and comparing with 1992 data. Results from 1992 and 2008 measurements are compared in Fig. \[tf1\_transp\]. =3.40in =2.40in Signs of marginal degradation can be noticed. =3.40in =2.40in =3.40in =2.40in In 1992 the transmittance of some of the blocks had been measured in the longitudinal direction also. From pairs of the transverse and longitudinal measurements both refractive index (shown in Fig.\[tf1\_ri\]) and attenuation length of the glass were extracted. From single measurements of the blocks in transverse orientation, only attenuation lengths are extracted by assuming the nominal refractive index of the glass of 1.65. As shown in Fig.\[tf1\_attl\], the light attenuation length varies significantly in the range of sensitivity of the XP3462B photocathode, and is $\sim$100 cm at the peak of sensitivity $\sim$400 nm. The slight shift between 1992 and 2008 year measurements is partly due to different absolute calibrations of the setup. The block to block variations in light transmission were compensated by pairing high quantum efficiency PMTs with low transparency blocks and vice versa in the module assemblies. Thus when all the PMTs were operated at a gain of $\sim 10^6$, the responses of modules to cosmic muons were equalized to within $\sim 20\%$. For straight through muons, signal of 60-70 photo-electrons on average from a block, and a pulse height resolution of $\sim 10-15\%$ were observed. The response of a module to cosmic rays passing at different distances to the PMT was studied. Two small ($5~{\rm cm}\times 5~{\rm cm}$) scintillator counters, placed on top and below the module and aligned vertically, were used to localize particles and to trigger signal readout. For single PMT modules, the signal variation at the edges was $\sim\pm15-20\%$ relative to the center (shown in Fig. \[ydep\]). For the two PMT modules, variation of the summed signal was on the level of $\sim\pm7\%$. and the light output was about 1.5 times higher than for the single tube case [@Gasp92]. =3.40in =2.40in The relative light transmittance of all the assembled modules was measured by use of green and blue Light Emitting Diodes (LEDs). The ratio of light transmission efficiency for blue and green LEDs, $\kappa = A_{\rm Blue}/A_{\rm Green}$, (see Fig. \[tf1\_fom\]) depends on optical properties of the blocks and is a measure of block quality. As the SOS spectrometer typically detected lower momentum particles than the HMS, the blocks with higher $\kappa$, and thus a higher transmission efficiency for Čerenkov light, were used in the SOS calorimeter. This also had the benefit of ensuring to some extent uniformity in the calorimeters. =3.40in =2.40in Final equalization of the PMT output signals, determination of the function parameters for amplitude–distance corrections and overall calibration of the calorimeters were performed with electron beam, by using “clean electron” data after their installation in the spectrometer detector huts. Choice of high voltage divider {#xp3462-hvdiv} ------------------------------ Special studies were performed to optimize the PMT high voltage base design for the requirements of good linearity (better than 1%), high rate capability and a weak variation of PMT gain with anode current [@Amatuni96]. Two manufacturers [@philips] recommended high voltage divider designs, optimized for high gain and linearity respectively. The bases had different relative fractions of the applied HV between the successive dynodes (from cathode to anode, including the focusing electrodes). We selected a design, which is a compromise between the two, but has also high anode current capability. This third design is a purely resistive, high current (2.3 mA at 1.5 kV), surface mounted divider ($\sim 0.640~M\Omega$), operating at negative HV (see Fig. \[hv\_divider\]). The relative fractions of the applied HV between the dynodes (from cathode to anode) are: 3.12/1.50/1.25/1.25/1.50/1.75/2.00/2.75/2.75. The supply voltage for a gain of $10^6$ is approximately 1750 V. =3.40in =1.70in The PMT resistive base assembly is linear to within $\sim 2\%$ up to the peak anode current of 120 $\mu$A ($\sim 5\times 10^4$ pe). The dark current is typically less than 3 nA. The base has anode and dynode output signals. Channel-to-channel adjustable high voltages are provided by a system of CAEN SY-403 high voltage power supplies (64 channel, $V_{max}$ = 3.0 kV, $I_{max}= 3.0~mA$). Monte Carlo simulation codes {#early_mc} ============================ The first versions of simulation codes for the HMS/SOS calorimeters were based on the ELSS [@ELSS] and EGS4 [@EGS4] packages for simulations of electromagnetic showers. Dedicated code was added for Čerenkov light generation, optical photon tracing and photoelectron knockout from PMT photocathodes. The optics took into account light absorption in the lead glass, reflections from the block sides, and passage through the optical coupling to the PMT photocathode. However, the software did not take into account block to block variations of lead glass absorption length and electronic effects. The first simulations revealed sufficient signal ($\sim$900 photoelectrons from a 1 GeV incident electron), good linearity and reasonable resolution in the GeV range for the calorimeter designs. Subsequent simulations of HMS calorimeter are based on the GEANT4 package, version 9.1. The QGSP\_BERT physics list [@qgsp-bert] was chosen to model hadron interactions, which is recommended by the GEANT4 developers for high energy physics calorimetry [@geant-www]. This list includes the parton string model [@parton-string] at energies above 12 GeV, intra-nuclear Bertini cascade [@Bertini] below 9.9 GeV, and a nuclear evaporation model [@nucl-evap] at low energies. The GHEISHA model [@gheisha] is used at energies 9.5 – 25 GeV. Electromagnetic processes are modeled to good accuracy within the framework of the GEANT4 standard electromagnetic package. The code closely emulates the geometry and the composition of the detector. Particularly, the optical characteristics of the setup were thoroughly implemented in the light tracing part of the code summarized below. The light attenuation length is randomly varied from block to block within the observed experimental limits (see Fig. \[tf1\_attl\]). The optical insulation of the module has multi-layer composition: air gap between aluminized Mylar and lead glass block, and Mylar support layer facing the block. The reflective and absorptive properties of aluminum reflector are expressed by means of real and imaginary parts of refractive index [@sopra] (see Fig. \[almylar\]). =3.40in =2.40in Instead of GEANT4 optical photon handling, the generated light is traced by means of a dedicated fast Fortran code which takes care of the modular construction and is suited to the particular geometry of the module. Few compromises and simplifications took place in the code: a strict rectangular geometry of the glass blocks is assumed; all the boundaries are flat and perfectly smooth, diffuse reflections from the walls are neglected; Rayleigh scattering in the glass is neglected as well; nor the polarization of Čerenkov light in the reflections/transmissions is taken into account. Light reflectance from the block walls and passage from block to PMT photocathode is treated as reflection/transmission from/through a plane-parallel plate sandwiched between two optical media of different refractive indices. In the first case it is a layer of air in between the lead glass and the reflector aluminum, in the second case it is a layer of optical grease between the lead glass and PMT window glass. The layers are assumed thin enough to neglect light absorption, and thick enough to neglect light interference effects. With these assumptions, expressions for reflectivity and transmittivity of the boundaries were derived in the limit of infinite series of Fresnel reflections/transitions from the surfaces of the plate (similar to [@BornWolf], p. 360). This model was checked against GEANT4 calculations, and good agreement was found between the two. In terms of the detector signal, the difference was less than a few percent. A typical quantum efficiency of XP3462B photocathode (Fig. \[xp3462b-qe\]) is assigned to all the PMTs. Electronic effects are taken into account by assigning a random multiplicative “gain” factor to each channel in order to transform the number of photoelectrons into ADC channels. This factor is varied from channel to channel by 50% around a mean value of 2. The electronic noise is modeled by adding a random pedestal of normal distribution with $\sigma=10$ ADC channels. Both, the “gain” factor and the pedestal width roughly correspond to experimental conditions. The projectiles are sampled at the focal plane of the spectrometer using the coordinate, angular and momentum distributions observed in the Meson Duality experiment [@mduality]. The momenta are scaled to the settings of the studies. Material traversed by particles before reaching the calorimeter smears the energy and coordinates of the particles. Therefore, all the material between the focal plane and calorimeter is also modeled (see Table \[hms-mat\]). ----------------- -------------- ---------- ------------------ ------------------------ Component Material position thickness density (cm) (cm) $({\rm g}/{\rm cm}^3)$ DC2 gas Ethane/Ar 29.3 15 0.00143 DC2 foils Mylar 2$\times$0.00254 1.4 S1X hodoscope BC408 scint. 77.8 1.067 1.032 S1Y hodoscope BC408 scint. 97.5 1.067 1.032 Aero. entrance Al 40 0.15 2.6989 Aero. radiator Aerogel 9 0.152 Aerogel air gap air 25.5 0.0012 Aerogel exit Al 0.1 2.6989 Gas Č gas $C_4F_1O$ 198 150 0.0047 Gas Č wind. Al 2$\times$0.1 2.6989 Gas Č mir.sup. Rohacell 230 1.8 0.050 S2X hodoscope BC408 scint. 298.8 1.067 1.032 S2Y hodoscope BC408 scint. 318.5 1.067 1.032 Calo. support Al 350 0.55 2.6989 ----------------- -------------- ---------- ------------------ ------------------------ : \[hms-mat\] Materials between HMS focal plane and calorimeter that are taken into account in the simulation. The listed positions are at the fronts of components Electronics and Calibration {#electronic_calibr} =========================== Electronics {#calo-electronic} ----------- The readout electronics were identical for both calorimeters. The raw anode signals from the phototubes were taken from the detector hut to the electronics room through $\sim$30 feet RG58, then $\sim$450 feet RG8 coaxial cables. The signals were then split 50/50, with one output sent through 400 ns RG58 delay cable to a 64-channel LeCroy 1881M Fastbus ADC module, and the other to a Philips 740 linear fan-in modules to be summed. A schematic diagram of the electronics for the calorimeters is shown in Fig. \[hcal\_electronic\]. =3.40in =2.80in Data from the Fastbus modules were acquired in the “sparsified” mode, in which only significant data were read from each ADC channel. The ADCs have programmable thresholds which were set ab initio fifteen channels above zero. The zero (or “pedestal”) of an ADC channel was determined at the beginning of each run by creating 1000 artificial triggers. These thousand events show up as a narrow peak in a histogram of an ADC output. Typical pedestal widths were about 5-7 channels for the ADC gate width $\sim$100 ns. Then the new threshold for each ADC channel was calculated as three times the width above the pedestal. The automatically determined thresholds then can be used as input to the data acquisition code such that it only reads out above the threshold, hence minimizing data flow. Because of the high pion to electron ratio for some of the experiments, events are required to pass loose particle identification cuts before generating a trigger. In order to have a high efficiency for electrons, a trigger was accepted as an electron if either the gas Čerenkov detector fired or if the electromagnetic calorimeter had a large enough signal. The threshold on the gas Čerenkov counter signal was typically set near the 1 pe level, and the threshold on the calorimeter signal was set just above the pion peak, which is independent of the spectrometer momentum setting. This allowed for extremely high electron efficiency even if one of the two detectors had a low efficiency. On the other hand, the pion rejection was conditioned by the low, in this case, threshold on the calorimeter signal. Raw signals from the whole calorimeter and from the front layer alone are summed for use as an option in the first level electronic trigger for $e/\pi$ discrimination [@arrington]. The fourth layer of the SOS calorimeter is not summed, since due to the 1.74 GeV maximum electron energy in the SOS, most of the electromagnetic shower is contained in the first 3 layers, and removing the last layer has almost no impact on the electron signal, but reduces the pion signal (for straight through pions by 25%). The sum in the first layer (PRSUM) and the sum in the entire calorimeter (SHSUM) are discriminated to give three logic signals for the trigger: PRHI and PRLO are high and low threshold signals from the first layer, and SHLO from the entire calorimeter. Also, groups of four modules are summed and sent through discriminators to scalers in order to call attention to dead or noisy tubes. The electron trigger (ELREAL) had two components: Electron High (ELHI) and Electron Low (ELLO). ELLO was designed to trigger for all electrons, even those depositing low shower energy. Thus, it provided increased efficiency at the low electron momenta. ELLO required a Čerenkov detector signal, a hodoscope signal (SCIN), and a shower signal (PRLO). ELHI required a high calorimeter signal, but no Čerenkov detector signal, and it was composed of preradiator high signal (PRHI), a three-out-of-four coincidence scintillator signal (SCIN) and the shower counter signal (SHLO). Calorimeter Calibration {#calo_calibr} ----------------------- The ability of particle identification of a calorimeter is based on differences in the energy deposition from different types of projectiles. The deposited energy is obtained by converting the recorded ADC channel value of each module into equivalent energy. To obtain an accurate measurement of it, two main issues must be overcome: the light attenuation in the lead-glass block, and block to block PMT gain variation. To correct the attenuation, the signal from each block is multiplied by a correction factor that depends on track position. This correction factor was different for the blocks with one and two PMT readouts. The correction was checked by looking at the distributions of corrected energy as a function of distance from the PMTs. The PMT gains had been matched in the hardware in order to make the calorimeter trigger uniform within acceptances of the calorimeters as much as possible. At first, using scattered electrons in each spectrometer the operating high voltages for the PMTs were adjusted so that the ADC signals were nearly identical (to $\sim 10\%$) for blocks in the same layer. Electrons with larger momenta are bent less in the spectrometer, and populate the bottom blocks in the calorimeter. Because the bottom blocks detect higher energy electrons, their gain must be kept lower than for the top blocks so that the output signals are of the same size. Therefore, setting the gain such that the output signal is constant as a function of vertical position in the calorimeter means having a gain variation between the blocks roughly equal to the momentum acceptance of the spectrometers ($\sim 20\%$ in the HMS, $\sim 40\%$ in the SOS). The output signals were made equal (rather than gains) in order to make the calorimeter trigger efficiency as uniform as possible over the entire calorimeter. The data analysis procedure corrects for the gain differences in the process of calorimeter calibration. Good electron events are selected by means of gas Čerenkov detector. The standard calibration algorithm [@amatuni] is based on minimization of the variance of the estimated energy with respect to the calibration constants, subject to the constraint that the estimate is unbiased (relative to the primary energy). The momentum of the primary electron is obtained from the tracking in the magnetic field of the spectrometer. The deposited energy per channel is estimated by $$\label{eq:calo-module-e} { e_i = c_i \times (A_i - ped_i) \times f(y) },$$ where $i$ is the channel number, $c_i$ is the calibration constant, $A_i$ is the raw ADC signal, $ped_i$ is the pedestal position, $f(y)$ is correction for the light attenuation for the horizontal hit coordinate $y$. Due to the segmentation in the vertical direction, the calorimeters have a coarse tracking capability which is helpful when separating multiple tracks (see, for instance, [@pruning]), [@tvaskis]). In the calorimeter analysis code hits on adjacent blocks are grouped into clusters for which the deposited energy and center of gravity are calculated. These clusters are matched with tracks from the upstream detectors if the distance from the track to cluster in the vertical direction is less than a predefined “slop” parameter (usually 7.5 cm). The calorimeter energy corresponding to a track is divided by the track momentum and used for particle identification. In the few GeV/c range pions and electrons are well separated (see Fig. \[fpi2\_edep\]), a cut at 0.7 ensures an electron detection efficiency better than 99% and  30:1 pion suppression (see [@malace] and Section \[hcal\_scal\_perform\]). =3.40in =2.40in Performance of HMS/SOS calorimeters {#hcal_scal_perform} =================================== Selection of calorimeter experimental data {#exp-data} ------------------------------------------ For these studies, HMS calorimeter data from the E01-004 (Fpi-2) [@fpi2] and E00-108 (Meson Duality) [@mduality] experiments have been collected for comparison with simulations. Fpi-2 measured the charged pion form factor at $Q^2$=1.6 and 2.45 (GeV/c)$^2$ via exclusive pion production, while Meson Duality looked for signatures of quark-hadron duality in semi-inclusive pion production. The experiments ran back to back in summer of 2003, and both detected pions in the HMS in coincidence with electrons in the SOS. Fpi-2 also detected electrons at elastic kinematics for HMS acceptance studies. Some other JLab experiments also used HMS or SOS calorimeters for good pion rejection and studied the devices, such as E89-008 [@arrington], E02-019 [@fomin], E03-103 ([@seely], [@daniel]). In order to obtain high purity samples of electrons and pions, tight cuts were applied to spectrometer events. Only events with single tracks in HMS and SOS passing through the collimators were used. Spectrometer acceptances were restricted to ensure good tracking accuracies, and, on the HMS side, efficient particle identification with gas Čerenkov counter. Electrons in HMS were identified by applying a high cut on the gas Čerenkov signal greater than 4 photoelectrons, while pions were identified with null signal. Pion samples were selected in the HMS from $(e'\pi)$ coincidence events, by posing tight electron PID cuts on the SOS gas Čerenkov detector signal greater than 3 photoelectrons, and normalized energy deposition in the SOS calorimeter $E_{Dep}/P_{SOS}$ greater than 0.9. Furthermore, a coincidence timing cut $\mid$cointime$\mid < $1 ns was also applied. Accidental events were selected and subtracted from energy deposition histograms by off coincidence timing cut 3 $< \mid$cointime$\mid < $ 13 ns. In addition, a kinematic cut of exclusive pion production on the missing mass was applied for Fpi-2. Finally, for these studies HMS calorimeter was calibrated on a run by run basis. Examples of the resultant distributions of the energy depositions in the calorimeter from incident electrons and pions are shown in Fig. \[fpi2\_edep\]. Resolution of HMS/SOS calorimeters {#hcal-scal-res} ---------------------------------- =3.40in =2.40in We define calorimeter resolution as the width of a Gaussian fit to the electron peak (Fig. \[fpi2\_edep\]) in the distribution of energy deposition. The resolution of HMS calorimeter from a number of Hall C experiments is compared with simulation in Fig. \[hcal\_sigma\]. Experiments before the modification of the detector in 1998 (see subsection \[hcal\_module\]), like E89-008 shown in the figure, report resolution $\sim6\%/\sqrt{E}$ ($E$ in GeV) ([@arrington], [@niculescu]). Experiments carried out afterward found improved energy resolution. Exception is the E99-118 experiment [@tvaskis] with resolution 8%/$\sqrt{E}$. E00-116, the first experiment to actually analyze data with the modified calorimeter, states resolution $5.4\%/\sqrt{E}$ [@malace]. E03-103, despite of a gain shift problem in the calorimeter electronics, obtained somewhat scattered data close to the simulation [@seely]. E04-001 got very good resolution in the wide range of HMS momenta, in agreement with simulation, presumably due to relatively low rate, good tracking conditions, and run by run calibration [@mamyan]. A somewhat worse resolution is obtained from online analysis of the E01-006 [@RSS] experiment at high energies up to $\sim$4.7 GeV/c. As for the SOS calorimeter, an on-line data analysis during the Hall C Spring03 experiments (E00-002, E01-002, E00-116) gave a resolution of $6\%/\sqrt{E}+1\%$, within the range of SOS momentum setting 0.5 – 1.74 GeV/c. The E00-108 experiment reported a resolution consistent with $\sim5\%/\sqrt{E}$ for SOS momentum range 1.2 – 1.7 GeV/c [@navasardyan]. In general, resolution from an experiment depends on multiple factors related both to hardware and software. Some of them, like trigger rate, background rate, performance of tracking detectors, tracking algorithm itself affect performance of the calorimeter indirectly, through the tracking conditions. Other factors, like electronic noise, stability of high voltage supply, low energy background, calibration affect the performance directly. The conventional 3-parameter fit [@pdg] to the simulated HMS resolution (in %) gives a dependence on energy in the form $3.75/\sqrt(E) \oplus 1.64 \oplus 1.96/E$. The first term is purely of stochastic origin, the second term reflects systematics from non-uniformity of the detector and calibration uncertainty, the third term, poorly constrained here by limited statistics, comes from electronic noise. In the simulated data stochastic and systematic terms dominate, electronic noise is tangible only at low energies $\lesssim$1.5 GeV. Overall, the resolution of the HMS/SOS calorimeters is close to resolutions of the lead-glass calorimeters of similar thicknesses (see [@Avakian98] and references therein, also [@e705_93; @wa91_95]). Electron detection efficiency and pion rejection {#hcal-eff-rej} ------------------------------------------------ The experimental efficiency of electron detection, which is defined as the fraction of events with the normalized energy deposition above threshold, at momenta within the range 2.8–4.1 GeV/c, for different cuts is in reasonable agreement with the simulation (see Fig. \[hms\_eff\_vs\_p\]). =3.40in =3.40in The simulation predicts a steady rise of $e^-$ detection efficiency with energy due to the improvement in resolution. However, as shown in Fig. \[hms\_cut\_eff\_07\] there is a growing disagreement with experiment for energies below 2 GeV. =3.40in =2.40in The $\pi^-$ suppression factor, the ratio of total number of pionic events and misidentified as pions, at different momenta and cut values is shown in Fig. \[hms\_cut\_sup\_vs\_p\]. Experimental data for comparison are mostly from the Meson Duality experiment. At 3 GeV/c there are data from Fpi-2 as well. The Fpi-2 data are presumably of better quality due to favorable background conditions and exclusive kinematics for pion production. Good agreement between the two experiments at 3 GeV ensure the quality of the pion suppression data found in Meson Duality. =3.40in =3.40in Both experiment and simulation show a momentum dependence of the suppression factor peaking at several GeV/c. While in the experiment the peak value is reached at $\sim$2.5 GeV/c independent of the cut, in the Monte Carlo it shifts to higher momenta as the cut is raised. Overall, agreement between experiment and simulation is satisfactory for the rejection studies. Few Hall C experiments report on the rejection capabilities of HMS or SOS calorimeters. E89-008, the first Hall C experiment [@arrington] states pion suppression by 25:1 for $E_{Dep}/P> 0.7$ at 1 GeV/c HMS momentum, which agrees with experimental data in this study, and fast improvement with energy due to moving the threshold to higher positions. E94-014 reports a pion rejection 95% at SOS momenta 1.4-1.5 GeV/c for the cut value of 0.7 [@armstrong]. Note that these two experiments ran before the calorimeters had been modified. The same rejection is reported in E00-108 for HMS at 1.7 GeV/c [@navasardyan], again in rough agreement with this study. The higher suppression factor obtained in this study comes from the cleaner selection of the particle samples (see subsection \[exp-data\]). =3.40in =2.40in Segmentation of HMS calorimeter allows for using the difference in longitudinal development of electromagnetic and hadronic showers for PID. In particular, energy deposition in the forward layer is most indicative. This is elaborated in subsection \[shms-calo-perf\], with regard to the SHMS calorimeter. As it is seen in Fig. \[hms\_sup\_vs\_ef\], one can gain substantially in PID capability of the HMS counter, by combining energy depositions in the first layer and in the whole calorimeter. Relative to the ordinary rejection, at the 3 GeV/c spectrometer setting, improvement in pion suppression is more than twice at low electron detection efficiencies 90 – 95%, and $\sim$1.5 times at high efficiencies above 99.7%. Alternatively, one can keep the suppression factor constant and gain in detection efficiency. For instance, at 250:1 $\pi^-$ suppression factor one can boost $e^-$ detection efficiency from $\sim$93% to $\sim$98.5%. Long-term stability of calorimeters {#hcal-scal-stab} ----------------------------------- The HMS/SOS calorimeters’ resolution shows only slight changes during the years of usage (see Fig.\[hcal\_sigma\]). These changes include variations in electronics, calibration technique and possible degradation of the calorimeter components. Stability of both calorimeters also has been evaluated by tracking changes in the ADC pedestal and PMT gain values. These values have been found to be stable within accuracy of the measurements during the entire time of operation. The long-term stability of the calorimeters’ responses have been monitored by tracking the variations in the width of normalized energy deposition (E/p) distribution, and variations in the PMT gain calibration constants from run to run, and from experiment to experiment. No significant degradation of HMS/SOS calorimeters’ performances after 15 years of operation have been noticed. SHMS Calorimeter {#shms_calo} ================ Design construction {#shms-calo-constr} ------------------- As a full absorption detector, the SHMS calorimeter is situated at the very end of detector stack of the spectrometer [@CDR-12]. The relatively large beam envelope of the SHMS dictated a different calorimeter design from HMS/SOS, with a wider acceptance coverage. In order to exclude possible energy leaks at higher energies, it was necessary to consider a shower counter for SHMS thicker than in HMS. The deeper calorimeter, the less energy leak of the electromagnetic shower from the radiator, but more light loss due to absorption in the glass and reflections is expected. Therefore, there should be an optimum in the detector dimension along the particle trajectory. For an energy range of a few tens of GeV it was found that the optimum is at the radiator length of $\sim$40 cm [@Avakian96; @Binon]. The general requirements for the SHMS calorimeter are:\ - Effective area: $120 \times 140~{\rm cm}^2$;\ - Total thickness: $\sim$20 rad. length;\ - Dynamic range: 1.0 - 11.0 GeV/c;\ - Energy resolution: $\sim 6\%/\sqrt E $, $E$ in GeV;\ - Pion rejection: $\sim$100:1 at $P\gtrsim$1.5-2.0 GeV/c;\ - Electron detection efficiency: $>98\%$. Studies of different versions and choice of assembling {#choice-assembl} ------------------------------------------------------ A few different versions of calorimeter assembly for the SHMS spectrometer have been considered ([@A-Mkrt06; @A-Mkrt07; @H-Mkrt10]) before it was optimized for cost/performance. A possible choice is a construction similar to the HMS and SOS calorimeters. An alternative is a calorimeter similar to HERMES [@Avakian98] and Hall A [@Alcorn] shower counters. The goal of these studies was to explore a few proposed versions of the SHMS calorimeter based on commercially produced lead glass. The configurations considered are a total absorption part (called Shower in the following), or a combination preshower and shower parts (“Preshower+Shower” in the following). The Preshower is a slab of a few radiative length thick lead-glass before the Shower part. For each version the energy resolution, electron detection efficiency and pion/electron separation capabilities were determined by simulations. The Shower and Preshower were made from modules, which consist of an optically isolated rectangular lead-glass block and optically coupled to it a PMT. For all versions we assumed only modular construction of the calorimeters since this gives more flexibility in assembling and allows for localizing the position of energy deposition clusters. Different types and sizes of the lead-glass blocks were also considered. We found the energy resolution for all versions with and without Preshower to be nearly similar, but different versions required different number of modules (channels) to cover the acceptance of the SHMS. Adding a Preshower dramatically improves the $\pi/e$ rejection factor. Our studies allowed selection of the optimum calorimeter geometry while maintaining the good energy resolution and pion rejection capabilities. The newly designed SHMS calorimeter consists of two parts (see Fig. \[shms\_calo\_sk\]): the main part at the rear (Shower), and Preshower before the Shower to augment PID capability of the detector. An optimal and cost-effective choice was found by using available modules from HERMES calorimeter for Shower part, and modules from SOS calorimeter for Preshower. With this choice the Shower becomes 18.2 radiative length deep and almost entirely absorbs showers from $\sim$10 GeV electromagnetic projectiles, and Preshower becomes 3.6 radiation length thick. =3.40in =3.40in Description of constructive elements {#shms-calo-elements} ------------------------------------ The SHMS Preshower radiator consists of a layer of 28 TF-1 type lead glass blocks from the calorimeter of the retired SOS spectrometer in Hall C, stacked in two columns in an aluminum enclosure (not shown in Fig. \[shms\_calo\_sk\]). 28 PMT assemblies, one per block, are attached to the left and right sides of the enclosure. The Shower part consists of 224 modules from the decommissioned HERMES detector [@Avakian98] stacked in a “fly eye” configuration of 14 columns and 16 rows. $\sim 120\times130~{\rm cm}^2$ of effective area of detector covers the beam envelope at the calorimeter. The Preshower enclosure adds little to the material on the pass of particles. On the front and back are 2” Honeycomb plate and a 1 $mm$ sheet of aluminum respectively, which add up to 1.7% of radiation length only. The optical insulation of the $10~{\rm cm}\times 10~{\rm cm}\times 70~{\rm cm}$ TF-1 blocks (see Section \[hcal\_scal\] for details) in the Preshower is optimized to minimize the dead material between them, without compromising the light tightness. First, the blocks are loosely wrapped in a single layer of 50 $\mu$m thick reflective aluminized Mylar film, with Mylar layer facing the block surface. Then, every other block is wrapped with a 10 $cm$ wide strip of 50 $\mu$m thick black Tedlar film, to cover its top, bottom, left and right sides but the circular openings for the PMT attachments. Looking at the face of detector, the wrapped and unwrapped blocks are arranged in a chess pattern. Insulation of the remaining front and back sides of the blocks are provided by facing inner surfaces of the front and rear plates of the enclosure, covered also with Tedlar. In addition, a layer of Tedlar separates the left and the right columns. The PMT assembly tubings are screwed in 90 $mm$ circular openings on both sides of the enclosure. The spacing of the openings matches the height of the blocks, so that a PMT faces to each of the blocks. The 3” XP3462B PMTs are optically coupled to the blocks using ND-703 type Bycron grease of refractive index 1.46. The HERMES modules to be used in the Shower part are similar in construction to the HMS/SOS modules but differ in details. The radiator is an optically isolated $8.9\times8.9\times50~{\rm cm}^3$ block of F-101 lead-glass, which is similar to TF-1 in physical parameters. The typical density of F-101 type lead-glass is 3.86 ${\rm g}/{\rm cm}^3$, radiation length 2.78 cm, and refraction index  1.65. The chemical composition of F-101 is: $Pb_2O_4$ (51.23%), $SiO_2$ (41.53%), $K_2O$ (7%) and $CeO$ (0.2%) by weight [@Avakian96]. The small amount of Cerium, added for the sake of radiation hardness ([@Kobayashi], [@Adams]), absorbs light at small wavelengths, and thus restricts the band of optical transparency to higher wavelengths (see Fig. \[f101\_tf1\_attl\]). Results of F-101 type lead-glass block transmittance measurements are shown in Fig. \[f101\_norad\_trans\]. =3.40in =2.40in For unused blocks, a $\sim 10\%$ shift in transmittance has been found between the 1994 year measurements by the HERMES collaboration [@Avakian96] and our measurements at JLab in 2008. We believe that the shift between the two sets of measurements is due to different calibration techniques of the setups. Each F-101 block is coupled to a 3” XP3461 PMT from Photonis, with green extended bialkali photocathode, of the same sizes and internal structure as the XP3462B in the HMS/SOS calorimeters and in the Preshower. Typical quantum efficiency of the photocathode is $\sim30\%$ for $\lambda\sim$400 $nm$ light (see Fig. \[xp3461-qe\]), and the gain is $\sim10^6$ at $\sim$1500 V. Silgard-184 silicone glue of refractive index 1.41 is used for optical coupling of the PMTs to lead-glass blocks. =3.40in =2.40in A $\mu$-metal sheet of 1.5 mm thickness and two layers of Teflon foil are used for magnetic shielding and electrical insulation of the PMTs. The blocks are wrapped with 50 $\mu$m aluminized Mylar and 125 $\mu$m black Tedlar paper for optical insulation. A surrounding aluminum tube which houses the $\mu$-metal, is fixed to a flange, which is glued to the surface of the lead-glass. The flange is made of titanium, which matches the thermal expansion coefficient of F-101 lead-glass [@Avakian96]. Beyond simple repairs, no adjustment has been made to the original HERMES construction of the modules for re-use in the SHMS calorimeter. Pre-assembling checks and tests {#rad-test} ------------------------------- As both the TF-1 and F-101 lead-glass blocks have been in use for more than 14 years under conditions of high luminosity, there was concern about possible radiation degradation of the blocks and the PMTs. Changes in transparency of TF-1 and F-101 lead-glasses, irradiated with 70 GeV protons and 30 GeV $\pi^-$ mesons have been reported in [@Inyakin]. It was found that the resistance of TF-1 lead-glass against irradiation is 50 times less than that of F-101. An accumulated dose of 2 krad produces a degradation of transmittance of F-101 glass of less than 1%. It was also found that the darkening of lead-glass radiators due to irradiation can be considerably reversed by intensive light illumination. Ref. [@Goldberg] reports that exposure of radiation-damaged glass to UV irradiation or to high temperature can bring about recovery of the glass. The changes in transparency of TF-1 and F-101 type lead-glass radiators have been studied in [@H-Mkrt10; @A-Mkrt11]. The estimated radiation dose for the used blocks was about 2 krad. For several samples of F-101 and TF-1 type blocks the light transmittance has been measured before and after 5 days of curing with UV light (of wavelength $\lambda$=200-400 $nm$). The transmission for F-101 type blocks from HERMES before and after the UV curing is shown in Fig. \[f101\_rad\_and\_uv\_trans\]. We do not find significant changes in transmittance. =3.40in =3.40in Note that for the TF-1 type blocks taken from the SOS calorimeter, our measurements again show negligible degradation over more than 15 years of operation (see Fig. \[tf1\_transp\] in Section \[tf1-block\]). This is due to efficient shielding of the SOS (HMS) spectrometer detector huts. To summarize the results of our studies on the radiation effects, there is no evidence for noticeable radiation damage of TF-1 and F-101 lead-glass blocks to be used in the construction of the calorimeter for SHMS spectrometer. As a cross check, we performed similar studies for the TF-1 type lead-glass blocks taken from the BigCal calorimeter, which had been used in Hall C experiment Gep-III [@gep3]. This calorimeter was operated in open geometry, and accumulated a dose of $\sim$2-6 krad. The results presented in Fig. \[tf1\_rad\_and\_uv\_trans\] show the effect of UV curing, indicating strong radiation degradation. =3.40in =3.40in The gain and relative quantum efficiencies for randomly selected PMTs from the SOS calorimeter (XP3462B) and from the HERMES detector (XP3461) have been measured to check possible degradation effects in the PMTs. A simple setup with a LED light source was used to localize the Single Electron Peak (SEP) at a given HV and define the gain for each PMT. Examples of gain variation versus high voltage for the Photonis XP3462B PMT are shown in Fig. \[xp3462b\_gain\]. =3.40in =3.40in While 1992 and 2010 data sets agree within the errors, a systematic offset of $\sim10-15\%$ can be seen between the two, which is related to different setups used in the measurements. For a set of PMTs dismounted from HERMES modules we have compared relative quantum efficiencies with new XP3461 PMTs. The HV for each PMT was adjusted to the gain $\approx1.5\times10^6$. The light intensity was adjusted to get about 100 photoelectrons from the unused new PMTs, and this intensity was monitored by a reference PMT. Since we kept the LED at fixed intensity and operated all the PMTs at a fixed gain, the difference between the detected number of photoelectrons may only come from the difference in the PMT QEs. The number of detected photoelectrons will in this case be directly related with the quantum efficiencies of PMTs. The comparison is shown in Fig. \[xp3461\_qe\_hermes\_new\]. A hint of aging, a 15% systematic decrease in quantum efficiency can be noticed. However, this is not taken into account in the simulations, for the decrease is marginal when compared to the accuracy of the measurements. =3.40in =2.40in Simulation code for SHMS calorimeter {#geant4-code} ------------------------------------ The code is based on GEANT4 simulation package [@geant4], release 9.2. As in the simulations of the HMS calorimeter (see section \[early\_mc\]), the QGSP\_BERT physics list was chosen to model hadron interactions. The code closely follows the parameters of the detector components mentioned in the previous sections. Other features are added into the model in order to bring it closer to reality as described below. As optical measurements of both TF-1 and F-101 glasses revealed block to block variation in transparency, the attenuation lengths were randomly varied from block to block accordingly, around their mean values shown in Fig. \[f101\_tf1\_attl\]. =3.40in =2.40in The quantum efficiencies of XP3462B and XP3461 PMT photocathodes are taken from the graphs provided by Photonis (see Fig. \[xp3462b-qe\] and Fig. \[xp3461-qe\]). The electronic effects in data acquisition system are taken into account assuming same performance as for the HMS calorimeter (see section \[early\_mc\]). As in the HMS case, particles originate at the focal plane and traverse detector material and support structures in front of the calorimeter (see Table \[shms-mat\]). Note, the two SHMS aerogel detectors for kaon identification [@E12-09-011] are not considered here, since their design was not finalized by the time of the calculations. Focal plane coordinates, directions and deviations of momentum from spectrometer setting were sampled by means of a Monte Carlo code of SHMS magnetic optics. ---------------- -------------- ---------- ------------------ ------------------------ Component Material position thickness density (cm) (cm) $({\rm g}/{\rm cm}^3)$ DC2 gas Ethane/Ar 40 3.81 0.00143 DC2 foils Mylar 7$\times$0.00254 1.4 S1X hodoscope BC408 scint. 50 0.5 1.032 S1Y hodoscope BC408 scint. 60 0.5 1.032 Gas Č gas $C_4F_8O$ 80 109.5 0.0089 Gas Č wind. Al 2$\times$0.1 2.6989 Gas Č mir. glass 0.3 2.4 Gas Č mir.sup. carbon fiber 0.1 1.8 S2X hodoscope BC408 scint. 260 0.5 1.032 S2Y hodoscope Quartz 265 2.5 2.634 Preshower sup. Al 269 0.05 2.6989 Preshower cov. Al 280 0.1 2.6989 Shower sup. Al 282 0.05 2.6989 ---------------- -------------- ---------- ------------------ ------------------------ : \[shms-mat\] Materials between SHMS focal plane and calorimeter that are taken into account in the simulation. The listed positions are at the fronts of components Light tracing is done within the frame of GEANT4 optics model. All the components related to the tracking of optical photons — like lead glass blocks, reflective foil wrapper, air layer between the reflector and the block, PMT glass windows, optical couplings of the windows and the blocks — were coded in terms of their sizes and optical parameters. The calibration algorithm used in these studies is the same as for the HMS calorimeter (see subsection \[calo\_calibr\]): the variation of total energy deposition in Preshower and Shower relative to the energy of the primary electron is minimized with respect to the calibration constants for each signal channel. The signals from Preshower are corrected for the horizontal coordinate of impact point. The signals from Shower are not corrected for impact point coordinates. Performance of SHMS calorimeter {#shms-calo-perf} ------------------------------- Resolution of the modeled SHMS calorimeter (Fig. \[shms\_calo\_res\]) is analogous to what has been reported for other lead-glass shower counters (references 21 through 31 in [@Tadevos2010]), though it is somewhat lower compared to the HMS calorimeter (compare Fig. \[shms\_calo\_res\] with Fig. \[hcal\_sigma\]). =3.40in =2.40in Examination of the functional forms of energy dependencies of the two resolutions shows that the difference comes mainly from the stochastic term: compare 5.04$\%\sqrt{E}$ for the SHMS with 3.75$\%\sqrt{E}$ for the HMS. The stochastic term is sensitive to dead material before detector and to photoelectron statistics [@pdg], which is in turn sensitive to the quality of radiator and light detectors. Both of these conditions are less favorable for the SHMS counter: there is more material between the focal plane and the calorimeter in the SHMS than in the HMS $-\sim$0.38 versus $\sim$0.16 radiation lengths respectively; and the lead-glass in SHMS calorimeter is less transparent than in the HMS calorimeter. The latter, combined with larger sizes, noticeably reduces photoelectron statistics in the SHMS calorimeter. Despite that, decent electron/hadron separation can be achieved by using the signal from the Preshower in addition to the total energy deposition in the calorimeter. As an illustration of hadron/electron rejection capability, example histograms of energy depositions from $e^-$ and $\pi^-$ in the calorimeter and in the Preshower are presented in Fig. \[shms\_calo\_endep\]. As it is seen in the bottom panel, the minimum ionizing pions and the showering electrons are separable to some extent in Preshower. =3.40in =3.40in Electron detection efficiency and pion suppression factor for different cuts on the normalized total deposited energy are shown in Fig. \[shms\_calo\_eff\] (compare with Fig. \[hms\_eff\_vs\_p\] and Fig. \[hms\_cut\_sup\_vs\_p\] for HMS calorimeter). For a constant cut, $e^-$ detection improves with momentum, which is consistent with better resolutions at higher energies. Meanwhile, $\pi^-$ rejection tends to worsen because of the increase in electromagnetic component of hadron induced cascades. The cut $E_{Dep}/P>0.7$ ensures $e^-$ detection better than 99.8% but modest $\pi^-$ suppression of $\sim$10. By imposing higher cuts one can trade off $e^-$ detection efficiency for a higher $\pi^-$ suppression. =3.40in =3.40in When compared to the HMS calorimeter, at the same cuts on total deposited energy the SHMS calorimeter ensures somewhat better $e^-$ detection efficiency due to lower fraction of events of low visible energy deposition. Meanwhile, the $\pi^-$ suppression is noticeably decreased (compare bottom panel in Fig. \[shms\_calo\_eff\] with Fig. \[hms\_cut\_sup\_vs\_p\]). Calorimeter segmentation allows one to take advantage of the differences in the space development of electromagnetic and hadronic showers for PID. Electromagnetic showers develop earlier and deposit more energy at the start than hadronic cascades. Thus measuring energy deposited in the front layer of a detector along with total energy deposition improves the electron/hadron separation. Pion suppression with the two PID methods $-$ by using total energy deposition alone, and energy deposition in the Preshower together with total energy deposition $-$ are compared in Fig. \[shms\_calo\_sup\_p\]. =3.40in =3.40in Suppression factors on the top panel are obtained by imposing cuts on the total deposited energy. The cuts (shown in the top panel of Fig. \[shms\_calo\_cuts\]) are chosen to ensure the electron detection efficiencies listed in the figures. =3.40in =3.40in The suppression factors on the bottom panel are obtained by separation of pion and electron events of concurrent energy depositions in the Preshower and in the whole calorimeter (exemplified in the Fig. \[shms\_calo\_cuts\], bottom panel). The separation boundaries are tuned to the same electron detection efficiencies as in the first case, and are optimized for minimum error rate by means of SVM$^{light}$ neural network [@svm_light]. Details can be found in a similar case with HMS calorimeter  [@Tadevos2010], where the forward layer of the counter was used as preshower. There, for the PID with combined energy depositions, from comparison with experimental data it was found that the simulation overestimates pion suppression, by $\sim$70% at low electron detection efficiencies $\gtrsim$90%, and $\sim$40% at high efficiencies $\sim$99.7%. As it is seen in Fig. \[shms\_calo\_sup\_p\], in both cases there is a trend that suggests improvement of the $\pi^-$ rejection with increase of momentum. Combining the total energy deposition $E_{tot}$ with deposition in the Preshower $E_{pre}$ significantly improves pion rejection. Gain in suppression by a factor of 2 - 10 times is achievable, dependent on momentum and the chosen e- efficiency. Generally the gain is bigger at higher momenta and for lower $e^-$ detection efficiencies (see Fig. \[shms\_calo\_sup\_gain\]). Even for very high electron efficiencies, the combined cut yields a factor of two or more improvement in the pion rejection over the simple E$_{TOT}$ cut. By using the Preshower the PID capabilities of the SHMS calorimeter become as good as that of HMS calorimeter where the first layer serves as Preshower. =3.40in =2.40in To summarize results on the SHMS calorimeter, the GEANT4 simulations were conducted with realistic parameters of the detector. The simulations predict a resolution similar to other lead-glass counters, though somewhat worse than for the existing HMS calorimeter. Good electron/hadron separation can be achieved by using energy deposition in the Preshower along with total energy deposition in the calorimeter. In this case the PID capability is similar to the one attainable with the HMS calorimeter. A pion suppression factor of a few hundred is predicted at 99% electron efficiency. SUMMARY AND CONCLUSIONS ======================= In summary, we have developed and constructed electromagnetic calorimeters from TF-1 type lead-glass blocks for the HMS and SOS magnetic spectrometers at JLab Hall C. The energy resolution better than $\sigma/E \sim 6\% /\sqrt E$ and the pion suppression $\sim$100:1 for $\sim$99% $e^-$ detection efficiency have been achieved in the 1 – 5 GeV energy range. Performance of the HMS calorimeter within full momentum range of the spectrometer, attainable after CEBAF 12 GeV upgrade, is modeled by GEANT4 simulation. Within the limited momentum range the calculated resolution and $\pi^-$ suppression factor are in good agreement with experimental data. The simulated pion suppression systematically exceeds experiment, by less than a factor of two, which is acceptable for rejection studies. The HMS/SOS calorimeters have been used in nearly all the Hall C experiments, providing good energy resolution and high pion suppression factor. No significant deterioration in the performance is observed in the course of operation since 1994. Design construction of the electromagnetic calorimeter for the newly built SHMS spectrometer in Hall C has been finalized, based on extensive exploratory studies. From a few considered versions, the Preshower+Shower configuration was selected as most cost-effective. The Preshower will consist of a layer of 28 modules with TF-1 type lead glass radiators, stacked back to back in two columns. The Shower part will consist of 224 modules with F-101 type lead glass radiators, stacked in a “fly eye” configuration of 14 columns and 16 rows. $120\times130~{\rm cm}^2$ of active area will cover beam envelope at the calorimeter. A Monte Carlo program for the newly designed SHMS shower counter was developed, based on the GEANT4 simulation package, and simulations have been conducted with realistic parameters of the detector. The predicted resolution yields somewhat to the HMS calorimeter. Good electron/hadron separation can be achieved by using energy deposition in the Preshower along with total energy deposition in the calorimeter. In this case the PID capability is similar to or better than those attainable with HMS calorimeter. A pion suppression factor of a few hundreds is predicted for 99% electron detection efficiency. ACKNOWLEDGMENTS The authors wish to thank Tsolak Amatuni for the work on hardware and software of HMS/SOS calorimeters in the development and construction stages, and for the idea of using support vector machines for particle identification with segmented calorimeters. We thank Ashot Gasparyan for the work and fruitful ideas in the early stages of the hardware development. We thank Carl Zorn from Detector Group of the Physics Division for outstanding support during optical studies of the lead-glass blocks and PMTs. 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--- address: | Department of Mathematics\ The Royal Institute of Technology, Stockholm, Sweden\ author: - - - bibliography: - 'allbib.bib' title: Bayesian learning of weakly structural Markov graph laws using sequential Monte Carlo methods --- , Introduction {#sec:intro} ============ Preliminaries {#sec:prel} ============= Temporal embedding of weakly structural Markov laws {#sec:non-temp:FK:flows} =================================================== Particle approximation of temporalised weakly structural Markov laws {#sec:particle:methods:temp:models} ==================================================================== Particle Gibbs sampling {#sec:particle:Gibbs:sampling} ----------------------- Application to decomposable graphical models {#sec:application} ============================================ Numerical study {#sec:numerics} =============== Acknowledgements {#sec:acknowledgements .unnumbered} ================

--- abstract: 'Detailed physisorption data from experiment for the H$_2$ molecule on low-index Cu surfaces challenge theory. Recently, density-functional theory (DFT) has been developed to account for nonlocal correlation effects, including van der Waals (dispersion) forces. We show that the functional vdW-DF2 gives a potential-energy curve, potential-well energy levels, and difference in lateral corrugation promisingly close to the results obtained by resonant elastic backscattering-diffraction experiments. The backscattering barrier is found selective for choice of exchange-functional approximation. Further, the DFT-D3 and TS-vdW corrections to traditional DFT formulations are also benchmarked, and deviations are analyzed.' author: - Kyuho Lee - Kristian Berland - Mina Yoon - Stig Andersson - 'Elsebeth Schr[ö]{}der' - Per Hyldgaard - 'Bengt I. Lundqvist' date: 'May 16, 2012' title: | Benchmarking van der Waals Density Functionals with Experimental Data:\ Potential Energy Curves for H$_2$ Molecules on Cu(111), (100), and (110) Surfaces --- [^1] Introduction ============ The van der Waals (vdW) or dispersion interactions play important roles in defining structure, stability, and function for molecules and materials. Our understanding of chemistry, biology, solid state physics, and materials science benefits greatly from density-functional theory (DFT). This in principle exact theory for stability and structure of electron systems [@Kohn] is computationally feasible also for complex and extended systems. However, in practice, approximations have to be made to describe exchange and correlation (XC) of the participating electrons [@KohnSham]. This work aims at benchmarking some XC descriptions of nonlocal correlations that describe vdW interactions [@LeeEtAl10; @Grimme; @TS]. To boil down the intricate electronic dynamics behind the vdW interaction into a density functional $F[n]$ is a formidable task. $F[n]$ should depend only on the electron density $n(\mathbf{r})$ and do that in the right and generally applicable way. It should obey fundamental physical laws, like charge conservation and time invariance, and have a physically sound account of system and interactions. The vdW-DF method [@dion2004; @LeeEtAl10] has such ambitions. There are more pragmatic methods, including those correcting traditional DFT calculations with pairwise vdW potentials, like the DFT-D [@Grimme] and TS-vdW [@TS] methods. Further, first-principles electron-structure calculations are made efficient but still carry much higher computational costs than DFT. An example is the random-phase approximation (RPA) to the correlation energy used as a suitable complement to the exact exchange energy [@HarlKresse2009]. The results obtained from particular XC functionals and other vdW descriptions can be assessed by comparing with other accurate electron-structure theories like those presented in Refs. , or with experiments. Typically one or two measurable quantities are available, like in Ref. . In Ref.  some of us stressed the importance of exploiting extended and accurate experimental data sets when these are available. Here, we extend this comparison by considering several facets of the Cu surface. Surface physics has a long and successful tradition of detailed and informative experiment-theory comparisons and offers possibilities also here. Extensive data sets are available for systems and conditions where the weak vdW forces can be reached and accurately mapped. A full physisorption potential and a detailed characterization thereof have been derived from versatile, accurate, and clearly interpretable measurements. In the physisorption regime, resonant elastic backscattering-diffraction experiments from low-index crystal faces provide a detailed quantitative knowledge. The actual data bank is rich and covers results, for instance for the whole shape of the physisorption potential, for the differences in corrugation across several facets, and for the energy levels in the potential well. This Paper compares state-of-the-art vdW descriptions of physisorption of H$_2$ and D$_2$ molecules on the low-indexed Cu surfaces with physisorption potentials constructed from selective-adsorption bound-state measurements. These data were analyzed in the early 90s in model systems [@andersson1993; @perandersson1993; @persson2008]. In general terms, the measurements, calculations, and analysis underline the importance of building in the essential surface-physics into vdW functionals and other vdW accounts. An earlier study of H$_2$ on the close-packed Cu(111) surface shows some spread in the results from different vdW accounts and that one of the tested XC functionals (vdW-DF2) compares promisingly with the experimental physisorption potential [@h2cu]. This motivates an extension of the study to the hydrogen molecule on other, more corrugated Cu surfaces. This paper is a significant extension of Ref. , addressing questions that were not resolved back in 1993 [@andersson1993; @perandersson1993; @persson2008], for instance, trends with crystal face. The outline, beyond this introduction, is as follows: First a brief review of physisorption, in particular on metal surfaces, and a review of the traditional description. This is followed by a presentation of some DFTs with accounts of vdW forces, a presentation of the systems studied, and a review of the experimental benchmark sets. Next, calculated results for Potential-Energy Curves (PECs) and other physical quantities are presented, and the Paper is concluded with comparisons, analysis, and outlook for future functionals. Physisorption and Weak Adsorption\[sec:2\] ========================================== Chemically inert atoms and molecules adsorb physically on cold metal surfaces [@persson2008]. Characteristic desorption temperatures range from only a few K to tens of K, while adsorption energies range from a few meV to around 100 meV. These values may be determined from measurements of thermal desorption and isosteric heat of adsorption. For light adsorbates, like He and H$_2$, gas-surface-scattering experiments provide a more direct and elegant method which involves the elastic backscattering with resonance structure. The bound-level sequences in the potential well can be measured with accuracy and in detail. Isotopes with widely different masses ($^3$He, $^4$He, H$_2$, D$_2$) are available. This permits a unique assignment of the levels and a determination of the well depth and ultimately a qualified test of model potentials [@Roy]. The potential well is formed by the vdW attraction which arises from adsorbate-substrate electron correlation. At large distances from the surface the vdW attraction goes like $V_{{\mbox{\scriptsize vdW}}}(z) = -C_{{\mbox{\scriptsize vdW}}}/z^3$. Here $z$ is the distance normal to the surface, measured from the center-of-mass of the particle to a surface reference plane close to the outermost layer of ion-cores in the solid, the so-called vdW plane. Near the surface the short-range repulsion, the “corrugated wall", acts. Specifically, we consider molecules that physisorb on metal surfaces where no significant change in the electronic configuration takes place upon adsorption. The weak coupling to electronic excitations [@SchGunnars1980] makes the adsorption largely electronically adiabatic. The energy transfer occurs through the phonon system of the solid lattice [@Brenig1987]. These conditions are expected to hold for hydrogen molecules on simple or noble metals. In early days, atom- and molecule-diffraction studies of metallic single-crystal surfaces were lagging behind those of ion-crystal surfaces. On metals, diffraction spots appear much weaker, which reflects the much weaker corrugation of close-packed metal surfaces [@Boato], than on an ionic crystal, like LiF(100) [@Garcia]. In the traditional picture of physisorption, the interaction between an inert adparticle and a metal surface is approximated as a superposition of the long-ranged $V_{{\mbox{\scriptsize vdW}}}$ and a short-range Pauli repulsion, $V_R$. The latter is due to the overlap between wavefunction tails of the metal conduction electrons and the closed-shell electrons of the adparticle [@zaremba1977; @HarrisNordlander; @persson2008], $$V_0(z) = V_R (z) + V_{{\mbox{\scriptsize vdW}}}(z). \label{eq:1}$$ Here approximately $$V_R (z) = V^\prime_R \exp(-\alpha z), \label{eq:2}$$ and $$V_{{\mbox{\scriptsize vdW}}}(z) = - \frac{C_{{\mbox{\scriptsize vdW}}}}{(z - z_{{\mbox{\scriptsize vdW}}})^3} f(2k_c(z - z_{{\mbox{\scriptsize vdW}}})), \label{eq:3}$$ now with $z$ measured from the “jellium" edge [@jelliumedge]. $V_0(z)$ is an effective potential. It arises as a lateral and adsorption-angle average of an underlying adsorption potential. We use $V_1(z)$ to express the amplitude of the modulation around the average $V_0(z)$. The repulsive potential $V_R(z)$ has a prefactor $V^\prime_R$ that can be determined from the shifts of the metal one-electron energies caused by the adparticle. It can also be calculated by, for example perturbation theory in a pseudo-potential description of the adparticle and a jellium-model representation of the metal surface [@andersson1993]. The strength of the asymptotic vdW attraction, $C_{{\mbox{\scriptsize vdW}}}$, and the reference-plane position, $z_{{\mbox{\scriptsize vdW}}}$, depend on the dielectric properties of the metal substrate and the adsorbate [@ZarembaKohn1976; @Liebsch1986]. The prefactor $f(z)$ in the potential $V_{{\mbox{\scriptsize vdW}}}(z)$ of Eq. (\[eq:3\]) introduces a saturation of the attraction at atomic-scale separations. The function $f(x)$ \[$f(x)=1-(1+x+x^2/2)\exp(-x)$ in some accounts\] lacks a rigorous prescription and thus includes some level of arbitrariness for $V_{{\mbox{\scriptsize vdW}}}(z)$. Experimental data provides a possible empirical solution to this dilemma. The physisorption potential $V_0(z)$ in Eq. (\[eq:1\]) depends on the details of the surface electron structure both via the electron spill out ($V_R$) and the spatial decay of polarization properties in the surface region ($V_{{\mbox{\scriptsize vdW}}}$). Accordingly, there is a crystal-face dependence of $V_0(z)$ for a given adparticle [@persson2008]. Figure \[fig:3\] shows the electron density profiles calculated for the Cu(111), (100), and (110) surfaces with the method described below. They illustrate that the corrugations on these facets are small but differ, growing in order (111) $<$ (100) $<$ (110). For the scattering experiment, the density far out in the tails is particularly important. That He-atom diffraction from dense metal surfaces is weak (compared to for instance ionic crystals) was early observed [@Boato] and subsequently explained in terms of a simple tie between the scattering potential and the electron-density profile: The He-surface interaction energy $E_{{\mbox{\scriptsize He}}}(r)$ can reasonably well be expressed as [@Esbjerg] $$E_{{\mbox{\scriptsize He}}}(\mathbf{r}) \simeq E_{{\mbox{\scriptsize He}}}^{{\mbox{\scriptsize hom}}}(n_o(\mathbf{r})), \label{eq:4}$$ where $E_{{\mbox{\scriptsize He}}}^{{\mbox{\scriptsize hom}}}(n)$ is the energy change on embedding a free He atom in a homogeneous electron gas of density $n$, and $n_o(\mathbf{r})$ is the host electron density at point $\mathbf{r}$. On close-packed metal surfaces the electron distribution $n_o(\mathbf{r})$ is smeared out almost uniformly along the surface [@Smoluchowski1941], thus giving weak corrugation. The crude proposal (\[eq:4\]) might be viewed as the precursor to the effective-medium theory [@EMT]. {width="85.00000%"} The form (\[eq:4\]) provides an interpretation of the mechanism by which the increasing density corrugation for (111) $<$ (100) $<$ (110) causes increasing amplitudes of modulation $V_1(z)$ in the physisorption potentials. The min-to-max variation of the rotationally averaged, lateral periodic corrugation $V_1(z)$ is modeled with an amplitude function [@HarrisAndLiebsch1982b], like in Eq. (\[eq:2\]), $V_1(z) = V_1^\prime \exp(-\beta z)$. Here the exponent $\beta$ is related to the exponent $\alpha$ of $V_0(z)$ via $\beta = \alpha/2 + \sqrt{(\alpha/2)^2 + G_{10}^2}$. The strength prefactor $V_1^\prime$ is adjusted so that the calculated intensities of the first-order $\mathbf{G}_{10}$ diffraction beams agree with measured values. The simple message of the experimental characterization in Figure \[fig:1\](a) is that, at the optimal separation and out, the $V_1$ corrugation terms are rather weak compared to the $V_0$ averages. This observation confirms that the basic particle-surface interaction is predominantly one dimensional. ![\[fig:1\] (a) Physisorption interaction potentials, $V$, for H$_2$ (D$_2$) on Cu(111) (circles), Cu(100) (squares), and Cu(110) (triangles), in the form of lateral average, $V_0(z)$ and corrugation $V_1(z)$. The potential functions $V_0$ and $V_1$ are defined in the text. The position $z$ of the molecular center of mass is here given with respect to the classical turning point, i.e. the position $z_t$ where a classical particle at energy $\epsilon_i = 0$ would be reflected in the potential. Adapted from [@andersson1993]. (b) The corresponding PECs calculated with vdW-DF2, averaged according to Eq. (\[eq:7\]).](Fig2_v2.png){width="40.00000%"} ![\[fig:2\] Energy levels in the $V_0$ potentials of Fig. \[fig:1\](b) are plotted versus the mass-reduced level number $\eta= (n + 1/2)/\sqrt{m}$, where $n$ is the quantum number and $m$ the mass. Experimentally determined values are given by solid black and open black circles (the open circles being deuterium results). The theory results for the levels are identified by solid red and open red circles (the open circles being the deuterium results). Experimental curves adapted from [@andersson1993]. ](fig3EnergyLevels4.png){width="40.00000%"} Figure \[fig:1\](a) shows experiment-based PECs $V_0(z)$ for physisorption of H$_2$ on the Cu(111), Cu(100), and Cu(110) surfaces. The H$_2$ molecules are trapped in states $\epsilon_i$ that are quantized in the perpendicular direction but have an essentially free in-surface dynamics. The panel details the laterally (and rotationally) averaged potential $V_0(z)$ that reflects the perpendicular quantization, i.e., the physisorption levels $\epsilon_n$. The experiment-based forms of $V_0(z)$ \[Fig. \[fig:1\](a)\] are obtained by adjusting the parameters[^2] in the modeling framework [@andersson1993; @perandersson1993; @persson2008], Eqs. (\[eq:1\])-(\[eq:3\]), to accurately reproduce the set of $\epsilon_n$ values. The experiment-based PECs of Figure \[fig:1\](a) are characterized by minima position (separations from the last atom plane), and depths given as follows: 3.52 [Å]{} and 29.0 meV for Cu(111), 3.26 [Å]{} and 31.3 meV for Cu(100), 2.97 [Å]{} and 32.1 meV for Cu(110). In Figure \[fig:1\](a), however, the curves are shown with minima positions slightly translated so that the set of $V_0(z)$ curves coincide at the classical turning point and thus facilitate an easy comparison. The diffraction analysis of resonant back-scattering follows the reasoning: For light adsorbates, like He and H$_2$, in gas-surface-scattering experiments, the elastic backscattering has a resonance structure. This provides a direct and elegant method to characterize the PEC, as they give accurate and detailed measurements of bound-level energies $\epsilon_n$ in the potential well. Isotopes with widely different masses, like ${}^3$He, ${}^4$He, H$_2$, D$_2$, permit a unique assignment of the levels and a determination of the well depth and ultimately a qualified test of model potentials [@Roy]. For a resonance associated with a diffraction that involves a surface reciprocal lattice vector $\mathbf{G}$ there is a kinematical condition, $$\epsilon_i=\epsilon_n +\frac{\hbar^2}{2m_p}(\mathbf{K}_i-\mathbf{G})^2, \label{eq:5}$$ where $m_p$ is the particle mass and where $\epsilon_i$ and $\mathbf{K}_i$ are the energy and wavevector component parallel to the surface of the incident beam, respectively. At resonance, weak periodic lateral corrugations of the basic interaction induce large changes in the diffracted beam intensities. The narrow resonance is observed as features in the diffracted beam intensities upon variations in the experimental incidence conditions. The intrinsically sharp resonances in angular and energy space have line widths that depend on intermediate bound-state life-time. They are limited by elastic and phonon inelastic processes. Lifetime broadening is only a fraction of a meV, substantially smaller than separations between the lower-lying levels (a few meV), allowing a number of physisorption levels $\epsilon_n$ with a unique assignment to be sharply determined from Eq. (\[eq:5\]). H$_2$ is the only molecule for which a detailed mapping of the bound-level spectrum and the gas-surface interaction potential has been performed with resonance scattering measurements [@PerrauAndLapujoulade1982; @YuEtAl1985; @ChiesaEtAl1985; @HartenEtAl1986; @AnderssonEtAl1988; @andersson1993; @perandersson1993; @persson2008]. The sequences here were obtained using nozzle beams of para-H$_2$ and normal-D$_2$, that is, the beams are predominantly composed of $j = 0$ molecules. Two isotopes H$_2$ and D$_2$ of widely different masses and with the different rotational populations of para-H$_2$ (p-H$_2$) and ortho-D$_2$ (o-D$_2$) and the normal species (n-H$_2$, n-D$_2$) are thus available; this richness in data means that the data analysis is greatly simplified and the interpretation is clear. For instance, the rotational anisotropy of the interaction has been determined via analysis of resonance structure resulting from the rotational $(j, m)$ sub-level splittings observed for n-H$_2$ and p-H$_2$ beams [@ChiesaEtAl1985; @WilzenEtAl1991]. Such knowledge permits a firm conclusion that the here-discussed measured bound-state energies, $\epsilon_n$ (Fig. \[fig:2\]), refer to an isotropic distribution of the molecular orientation. The level assignment is compatible with a single gas-surface potential for the two hydrogen isotopes [@perandersson1993]. Figure \[fig:2\] illustrates the analysis that leads to a single accurate gas-surface potential curve for each of the facets from the experimentally observed energies $\epsilon_n$ [@andersson1993; @perandersson1993; @persson2008]. The black open and filled circles represent measured $\epsilon_n$ values. The procedure is an adaptation to surface physics of the Rydberg-Klein-Rees method of molecular physics [@Roy]. The ordering is experimentally known and in this ordering, all $\epsilon_n$ values fall accurately on a common curve \[when plotted versus the mass-reduced level number $\eta = (n + 1/2)/\sqrt{m}$\]. The variation in the quantization levels reflects the asymptotic behavior of the potential curve and thus determines the value of $C_{{\mbox{\scriptsize vdW}}}$ to a high accuracy and gives a good direct estimate of the well depth [@perandersson1993; @persson2008]. A third-order polynomial fit to the data yields for $\eta = 0$ a potential-well depth $D=29.5$, 31.4, and 32.3 meV for the (111), (100), and (110) surfaces, respectively. This direct construction of an effective physisorption potential supplements the above-described experiment-based procedure \[that instead uses the measured energies $\epsilon_n$ to fit $V_0(z)$ curves and obtain an even higher accuracy[^3]\]. Figure \[fig:1\](a) also shows experiment-based determinations of the (rotationally averaged) amplitudes $V_1(z)$ of the lateral corrugation. The measured intensities of the first-order diffraction beams provide (as described above) an estimate of the resulting lateral variation in the H$_2$-Cu potential. The corrugation is very small, $\sim$0.5 meV at the potential well minimum. However, the existence of finite amplitudes $V_1(z)$ is essential: The larger corrugation closer to the substrate contributes most importantly to the diffraction and resonance phenomena. In fact, it is a finite magnitude of $V_1(z)$ that ensures a coupling to the in-plane crystal momentum and allows an elastic scattering event to satisfy the kinematical condition (\[eq:5\]). van der Waals Accounts Used in DFT calculations =============================================== Noncovalent forces, such as hydrogen bonding and vdW interactions, are crucial for the formation, stability, and function of molecules and materials. In sparse matter the vdW forces are particularly relevant in regions with low electron density. For a long time, it has been possible to account for vdW interactions only by high-level quantum-chemical wave-function or Quantum Monte Carlo methods. The correct long-range interaction tail for separated molecules is absent from all popular local-density or gradient corrected XC functionals of density-functional theory, as well as from the Hartree-Fock approximation. Development of approximate DFT approaches that accurately model the London dispersion interactions [@Stone1997; @Kaplan2006] is a very active field of research (reviewed in, for example Refs. ). To account for vdW interactions in computational physics traditional DFT codes are natural starting points. The vdW energy emanates from the correlated motion of electrons and there are proposals to account for it, like (i) DFT extended atom-pair potentials, (ii) explicit density functionals, and (iii) RPA in perturbation theory. Chemical accuracy is aimed for. Extensive physical and chemical systems are of great interest, including bio- and nanosystems, where the vdW interactions are indispensable. Describing bonds in a variety of systems with “chemical accuracy" requires that both strong and weak bonds are calculated. Strong covalent bonds are well described by traditional approximations, like the generalized gradient approximations (GGAs) [@PBE; @LangrMehl; @PerdewEtAl], which are typically built in into approaches (i) and (ii). vdW-relevant systems range from small organic molecules to large and complex systems, like sparse materials and protein-DNA complexes. They all have noncovalent bonds of significance. Quite a number of naturally and technologically relevant materials have already been successfully treated [@langreth2009]. The methods (i) and (ii) are essentially cost free, speed is given by that of traditional DFT (for example GGA based). Such DFT calculations are competitive in terms of efficiency and broad applicability. Computational power is an important factor that is still relevant and an argument for choosing methods of types (i) and (ii) in many applications ahead of type (iii). DFT extended by atom-pair potentials ------------------------------------ A common remedy for the missing vdW interaction in GGA-based DFT consists of adding a pairwise interatomic $C_6/R^6$ term ($E_{{\mbox{\scriptsize vdW}}}$) to the DFT energy. Examples are DFT-D [@Grimme2004], TS-vdW [@TS], and alike [@ElstnerHFSK2001; @JureckaCHS2007]. Refs. [@SunKLZ2008; @ZhaoTruhlar2008] describe various other approaches also currently in use. The DFT-D method is a popular way to add on dispersion corrections to traditional Kohn-Sham (KS) density functional theory. It is implemented into several code packages. Successively it has been refined to obtain higher accuracy, a broader range of applicability, and less empiricism. In the recent DFT-D3 version [@Grimme], the main new ingredients are atom-pairwise specific dispersion coefficients and cutoff radii that are both computed from first principles. The coefficients for new eighth-order dispersion terms are computed using established recursion relations. Geometry-dependent information is here included by employing the new concept of fractional coordination numbers. They are used to interpolate between dispersion coefficients of atoms in different chemical environments. The method only requires adjustment of two global parameters for each density functional, is asymptotically exact for a gas of weakly interacting neutral atoms, and easily allows the computation of atomic forces. As recommended [@Grimme], three-body nonadditivity terms are not considered. Another almost parameter-free[^4] method for accounting for long-range vdW interactions from mean-field electronic structure calculations relies on the summation of interatomic $C_6/R^6$ terms, derived from the electron density of a molecule or solid and reference data for the free atoms [@TS]. The mean absolute error in the $C_6$ coefficients is 5.5%, when compared to accurate experimental values for 1225 intermolecular pairs, irrespective of the employed exchange-correlation functional. The effective atomic $C_6$ coefficients have been shown to depend strongly on the bonding environment of an atom in a molecule [@TS]. Explicit density functionals ---------------------------- Ground-state properties can be described by functionals of the electron density $n(\mathbf{r})$ [@Kohn]. The functional $E_{xc}[n(\mathbf{r})]$ for the XC energy is a central ingredient. The local-density approximation (LDA) [@KohnSham; @HedinLu] and GGAs [@LangrMehl; @PerdewEtAl; @PBE] do not describe the nonlocal correlations behind the vdW interactions. This subsection discusses explicit XC density functionals $E_{xc}[n(\mathbf{r})]$, focusing on the non-local correlation functional, $E_c^{{\mbox{\scriptsize nl}}}[n(\mathbf{r})]$. In the vdW-DF, the vdW interactions and correlations are expressed in terms of the density $n(\mathbf{r})$ as a truly nonlocal six-dimensional integral [@dion2004; @langreth2005; @thonhauser2007]. It originates in the adiabatic connection formula [@PerdLangrI; @GunnLund; @PerdLangrII], and uses an approximate coupling-constant integration and an approximate dielectric function with a single-pole form. The dielectric function is fully nonlocal and satisfies known limits, sum rules, and invariances, has a pole strength determined by a sum rule and is scaled to locally give the approximate gradient-corrected electron-gas ground-state energy. There are no empirical or fitted parameters, just references to general theoretical criteria. Account for inhomogeneity is approximately achieved by a gradient correction, which is obtained from a relevant reference system. In the original vdW-DF version [@dion2004; @thonhauser2007; @rydberg2000; @langreth2005], the slowly varying electron gas is used for this. The gradient correction is then taken from Ref. . Although promising results have been obtained for a variety of systems, including adsorption [@langreth2009; @Mats], there is room for improvements. Recently another reference system has been proposed, with the argument that adsorption systems have electrons in separate molecule-like regions, with exponentially decaying tails in between. The vdW-DF2 functional uses the gradient coefficient of the B88 exchange functional [@B88] for the determination of the internal functional \[Eq. (12) of Ref. \] within the nonlocal correlation functional. This is based on application of the large-$N$ asymptote [@Schwinger1980; @Schwinger1981] on appropriate molecular systems. Using this method, Elliott and Burke [@Elliott2009] have shown, from first principles, that the correct exchange gradient coefficient $\beta$ for an isolated atom (monomer) is essentially identical to the B88 value, which had been previously determined empirically [@B88]. Thus in the internal functional, vdW-DF2 [@LeeEtAl10] replaces $Z_{ab}$ in that equation with the value implied by the $\beta$ of B88. This procedure defines the relationship between the kernels of vdW-DF and vdW-DF2 for the nonlocal correlation energy. Like vdW-DF, vdW-DF2 is a transferable functional based on physical principles and approximations. It has no empirical input. The choices of exchange functional also differ. The original vdW-DF uses the revPBE [@revPBE] exchange functional, which is good at separations in typical vdW complexes [@dion2004; @langreth2005; @thonhauser2007]. At smaller separations [@puzder2006; @kannemann-becke2009; @murray2009; @klimes2010; @cooper2010], recent studies suggest that the PW86 exchange functional [@PW86] most closely duplicates Hartree-Fock interaction energies both for atoms [@kannemann-becke2009] and molecules [@murray2009]. The vdW-DF2 functional [@LeeEtAl10] employs the PW86R functional [@murray2009], which more closely reproduces the PW86 integral form at lower densities than those considered by the original PW86 authors. RPA --- For first-principles electron-structure calculations, the random-phase approximation (RPA) to the correlation energy is presumably a suitable complement to the exact exchange energy [@HarlKresse2009]. The RPA to the correlation energy [@NozieresPines1958] incorporates a screened nonlocal exchange term and long-range dynamic correlation effects that underpin vdW bonding [@HarlKresse2009]. Hubbard, Pines and Nozieres [@pinesbook] pointed out that RPA does have, at least, formal limitations in the description of local correlations (large momentum transfer). This is because RPA treats same- and opposite-spin scattering on the same footing, thereby neglecting effects of Pauli exclusion in the description of the RPA correlation term. There exist several suggestions for RPA corrections [@mahan]. A recent study [@Ren2011] suggests a single-excitation extension for RPA calculations in inhomogeneous systems, thus lowering the mean average error for noncovalent systems [@Ren2011]. Efficient RPA implementations have become increasingly available for solids [@MiyakeGR2002; @HarlKresse2009; @MariniGR2006] and molecular systems [@ScuseraHS; @Furche2001; @Furche2008; @RenRS2009]. One implementation [@HarlKresse2009] gives the XC functional as $$E_{xc} = E_{{{\mbox{\scriptsize EXX}}}} + E_c, \label{eq:6}$$ where the exact exchange energy $E_{{{\mbox{\scriptsize EXX}}}}$ (Hartree-Fock energy) and the correlation energy $E_c$, given as the independent-particle response function, are all evaluated from KS orbitals by using for example plane-wave code and suitably optimized projector augmented wave (PAW) potentials that describe high energy scattering properties very accurately up to 100 eV above the vacuum level [@ShishkinKresse2006]. The operations scale like $N^{6-7}$ and a high parallel efficiency can be reached [@HarlKresse2009]. Implementation aspects\[sec:3D\] -------------------------------- The vdW-DF2 calculations are performed by using the <span style="font-variant:small-caps;">abinit</span> [@abinit1; @abinit2] code with a plane-wave basis set and Troullier-Martins-type [@Troullier1992] norm-conserving pseudopotentials. The scalar-relativistic correction is included in the pseudopotentials for transition-metals. A kinetic energy cut-off of 70 Ry is used. For *k*-space integrations, a $4\times{}4\times{}1$ Monkhorst-Pack mesh is used. For the partial occupation of metallic bands, we use the Fermi-Dirac smearing scheme with a 0.1 eV broadening width. With this setup the adsorption energies are converged within 1 meV. The vdW-DF total energy is calculated in a fully self-consistent way [@thonhauser2007]. We adapted an implementation of the efficient vdW-DF algorithm [@soler] from <span style="font-variant:small-caps;">siesta</span> [@siesta] for use within a modified version of <span style="font-variant:small-caps;">abinit</span>. ![\[fig:4\] High-symmetry positions on the low-index Cu surfaces. In our calculations the H$_2$ molecule lies flat in one of these positions.](fig4HighSymmSites-v2.png){width="30.00000%"} The surfaces are modeled by a slab of four atomic layers with a vacuum region of 20 Å in a periodic supercell. For the calculations on the (111), (100), and (110) surfaces we use the surface unit cells of $3\times2\sqrt{3}$, $3\times3$, and $2\sqrt{2}\times3$, respectively. In the electron-structure calculations, the molecule is kept in a flat orientation above the high-symmetry positions or sites[^5] on the Cu surfaces, as indicated in Figure \[fig:4\]. Some test points indicate that the total energy depends very little on orientation. For instance, in a “worst-case" situation, the energy change by a 90-degree in-plane rotation of H$_2$ at the long bridge site of Cu(110) is 0.92 meV, in a fixed-height in-plane rotation. If H$_2$ is moved to the equilibrium adsorption height, which is 0.04 [Å]{} lower, the energy difference increases to 0.97 meV. This variation is much smaller than the lateral corrugation ($\sim 4$ meV in this facet) and out-of-plane rotation ($\sim 5$ meV). To estimate the magnitude of the error introduced by neglecting the angular average (that automatically is included in the experimental data) we perform a separate calculation of H$_2$ in an up-right position on the Cu(111) surface using the optimal H$_2$-to-surface separation. This calculation results in a downward energy shift of 4.8 meV, which is the amplitude of the angular variation at the optimal separation. However, for an isotropic H$_2$ wave function this shift corresponds roughly to lowering of the ground state energy by merely $1/3 \times 4.8$ meV = 1.6 meV, assuming a simple sinusoidal energy variation in the angular space. The energy of higher order states would also be lowered, but to a lesser extent. This estimate thus indicates that the effect of the angular energy dependence is merely a minor quantitative correction to the eigenvalues. The PECs are calculated with vdW-DF2 for the high-symmetry sites, and from these the laterally averaged potential $V_0$ is approximately obtained. In turn, the bound quantum states in the $V_0$ potential well are calculated by solving the corresponding Schrödinger equation. The theoretical values are generated for a number of discrete points, surface positions (atop, hollow, long bridge, and short bridge), and separations $z$, while experimental data (Fig. \[fig:1\]) are presented as lateral averages and functions of separation $z$. To connect the two, some approximation has to be made to extract the laterally averaged result out of the discrete one. To capture the effects of the surface topology and to use the surface-lattice points used in the DFT calculations, we find the following approximate average reasonable: $$V_0 = \frac{1}{4} ( V_{{\mbox{\scriptsize hollow}}}+V_{{\mbox{\scriptsize atop}}}+V_{{\mbox{\scriptsize long-bridge}}}+V_{{\mbox{\scriptsize short-bridge}}}) \label{eq:7}$$ where $V_{{\mbox{\scriptsize long-bridge}}}=V_{{\mbox{\scriptsize short-bridge}}}$ for the (100) and (111) surfaces. One argument in favor of this approximation is the fact that the (100) surface has twice as many bridge sites as there are atop sites. Approximation (\[eq:7\]) is used in our comparisons between our vdW-DF2-determined potential $V_0$ and the experimentally determined one (Fig. \[fig:1\]) and in our generation of eigenvalues used to relate to the experimental ones \[which are defined in terms of a laterally averaged and H$_2$ rotational-angle averaged potential $V_0(z)$\] (Fig. \[fig:2\]). Reference uses the atop PEC on Cu(111), which is an adequate choice due to the small corrugation of the Cu(111) surface. For the discussion of the relation between corrugation and $V_1$, the classical turning point is the relevant separation. The corrugation of the PEC minimum is smaller, but still representative of the expected variation in the probability for the H$_2$ trapping. The original report of experimental results also covers a potential $V_2(z)$. It represents the min-to-max variation of the lateral average of the rotational anisotropy. A full appreciation of the accuracy of the comparison between the experimentally determined $V_0(z)$ and $V_0^{{{\mbox{\scriptsize vdW-DF2}}}}(z)$, based on existing vdW-DF2 calculations, would benefit from an understanding of $V_2(z)$. To indicate that this is unlikely to have any large consequence for the three Cu surfaces, a simple estimate of the rotation angle effect is made above. This should be considered as a stimulus for further refinement of the testing. Results and Benchmarking ======================== We present a new benchmark, taken from surface physics, with extraordinary virtues. Data are provided for (i) energy eigenvalues, $\epsilon_n$, for H$_2$ and D$_2$ in the PEC well, which have direct ties to measured reflection intensity, (ii) the laterally averaged physisorption potential, $V_0$, which is derived from measured data, the extracted PEC, and (iii) the corrugation, $V_1$, also derived from measured data. Benchmarking strategies ----------------------- Evaluation of XC functionals is often made by comparing with other theoretical results in a systematic way, for instance, the common comparison with S22 data set [@Jurecka2006; @Sherrill2009; @Molnar2009; @Takatani2010; @Szalewicz2010]. These sets have twenty-two prototypical small molecular duplexes for non-covalent interactions (hydrogen-bonded, dispersion-dominated, and mixed) in biological molecules. They provide PECs at a very accurate level of wave-function methods, in particular the CCSD(T) method. However, by necessity, the electron systems in such sets are finite in size. The original vdW-DF performs well on the S22 data set, except for hydrogen-bonded duplexes (underbinding by about 15% [@langreth2009; @LeeEtAl10]). Use of the vdW-DF2 functional reduces the mean absolute deviations of binding energy and equilibrium separation significantly [@LeeEtAl10]. Shapes of PECs away from the equilibrium separation are greatly improved. The long-range part of the vdW interaction, particularly crucial for extended systems, has a weaker attraction in the vdW-DF2, thus reducing the error to 8 meV at separations 1 [Å]{} away from equilibrium [@LeeEtAl10]. Recently, other numbers for the S22 benchmark on vdW-DF2 have been published [@bligard]. The two calculations differ in the treatment of the intermolecular separation, being relaxed [@LeeEtAl10] and unrelaxed [@bligard], respectively. Of course, absence of relaxations does lead to an appearance of worse performance. Experimental information provides the ultimate basis for assessing functionals. The vdW-DF functional has been promising in applications to a variety of systems [@langreth2009], but primarily vdW-bonded ones. Typically, the calculated results are tested on measured binding-energy and/or bond-length values that happen to be available. The vdW-DF2 functional has also been successfully applied to some extended systems, like graphene and graphite [@LeeEtAl10], metal-organic-frameworks systems [@LeeEtAl10B], molecular crystal systems [@MolcrysDF2], physisorption systems [@LeeEtAl10C; @selfassembly], liquid water [@mogelhoj] and layered oxides [@londero]. However, the studies are of the common kind that focus on comparison against just a few accessible observations. Accurate experimental values for the eigenenergies of H$_2$ and D$_2$ molecules bound to Cu surfaces [@andersson1993; @perandersson1993] motivate theoretical account and assessment. This knowledge base covers results for the whole shape of the physisorption potentials. Here calculations on several Cu facets allow studies of trends and a deeper analysis. The extensive report of vdW-DF2 results in Figure \[fig:5\] serves as a starting point. PECs from vdW-DF2 ----------------- {width="95.00000%"} PECs are calculated for H$_2$ in atop, bridge, and hollow sites on the Cu(111), (100), and (110) surfaces. The resulting benchmark for vdW-DF2 is also compared (below) with those of two other vdW approximations, the DFT-D3 [@Grimme] and TS-vdW [@TS] methods. To emphasize various aspects of the PECs and make valuable use of the numerical accuracy, the next few sections (and figures) highlight various aspects of the vdW-DF2 results. For each position $z$ of the molecular center of mass, we compare the averaged potential functions $V_0(z)$ and $V_1(z)$, in Fig. \[fig:1\], and find strong qualitative agreement. For instance, at the potential minimum of $V_0(z)$, Figure \[fig:1\] gives for $V_1(z)$ the approximate values 1, 3, and 4 meV for (111), (100), and (110), respectively, in quantitative agreement with the illustration of theoretical results analysis. The insensitivity to facets and the corrugation is discussed in greater detail below. Isotropy of lateral averaged potentials --------------------------------------- An important feature of the experimental $V_0(z)$ in Figure \[fig:1\] is the similar sizes of the well depths (range 29–32 meV) and separations (around 3.5 [Å]{}) on the (111), (100), and (110) surfaces. This isotropy, i.e., similarity of physisorption potential among different facets, is interesting and perhaps surprising because Cu(111) contains a metallic surface state, whereas Cu(100) and Cu(110) do not. The most striking feature of the vdW-DF2 results for the H$_2$-Cu PEC is probably that it is able to reproduce this isotropy. From Figure \[fig:5\] we find physisorption depths in the interval 35–39 meV and separations in the range 3.3–3.6 [Å]{}. From the experimentally more relevant laterally averaged $V^{{\mbox{\scriptsize vdW-DF2}}}(z)$, Fig. \[fig:1\](b), we find physisorption depths 31–36 meV. ![\[fig:6\] PECs of H$_2$ on atop sites of the Cu(111), (100), and (110) surfaces calculated with the vdW-DF2 functional.](fig6H2-Cu-notitle.pdf){width="40.00000%"} The isotropy is emphasized in Figure \[fig:6\] by plotting the vdW-DF2 PECs on the atop sites of each surface. The curves lie very close to each other, both in the Pauli-repulsion region at short separations, dominated by $V_R$ in the traditional theory \[Eqs. (\[eq:1\])-(\[eq:3\])\], and in the vdW-attraction region at large separations. They differ only discernibly in the $V_R$-region, which can be understood in terms of the higher electron density, $n_o(\mathbf{r})$ on the atop site on the dense (111) surface. Both agreements and differences are found in the trends (with facets) of the experiment-based \[$V_0(z)$, $V_1(z)$\] and vdW-DF2 based \[$V^{{\mbox{\scriptsize vdW-DF2}}}(z)$\] characterizations, Figure \[fig:1\](a) and (b). vdW-DF2 reproduces the trend in ordering and roughly the magnitudes in the modulation amplitudes \[$V_1(z)$\] at the classical turning points (identified as position ‘0’ in Figure \[fig:1\]). As shown in Fig. \[fig:6\], vdW-DF2 also reproduces the ordering of separations corresponding to physisorption minima, correctly decreasing as (111) $>$ (100) $>$ (110). On the other hand, for $V_0(z)$ the physisorption depth varies as (111) $>$ (100) $>$ (110) whereas in $V^{{\mbox{\scriptsize vdW-DF2}}}(z)$ the depth varies (110) $>$ (100) $>$ (111). The largest relative difference, 25%, is found for Cu(111). The set of physisorption depths on the three facets are reproduced with an average confidence of 15%. Corrugation\[ssec:D\] --------------------- Another striking feature of the PECs is the variation with the density corrugation of each surface. In Figure \[fig:3\], the density profiles of the clean Cu(111), (100), and (110) surfaces indicate how the corrugation may vary. For fcc metals the (111) surface is the most dense, while the (100) and (110) surfaces are successively more open and thus corrugated. The trend is reflected in the PECs. These clear effects on the PEC are illustrated by the calculated PECs for H$_2$ in atop, bridge, and hollow sites on the Cu(111), (100), and (110) surfaces (Fig. \[fig:5\]). From being small on the flat and dense (111) surface \[Fig. \[fig:5\](a)\], the corrugation grows from (111) to (100) and from (100) to (110), just as expected from the above reasoning. Figure \[fig:7\] shows the variations in the adsorption energies relative to the value of the atop configuration on the various facets. We choose to report the corrugation at the PEC minimum. The corrugation at the classical turning point is likely a stronger indicator of the strength of the elastic scattering that traps the incoming H$_2$ molecules, Sec. \[sec:2\]. ![\[fig:7\] Corrugation for H$_2$ on Cu(111), Cu(100), and Cu(110) (a) illustrated by the lateral variation of the calculated adsorption energy of H$_2$ at each high-symmetry position on these surfaces; (b) illustrated by calculated adsorption-energy values of H$_2$ on these surfaces. The black horisontal lines indicate approximate site averages, analogous to Eq. (\[eq:7\]). Subtracting the atop value from each average gives 1.3, 1.6, and 2.5 meV as lateral variations on the (111), (100) and (110) surfaces, these numbers can be interpreted as measures of the corrugation.](fig7acorrA.png "fig:"){width="40.00000%"} ![\[fig:7\] Corrugation for H$_2$ on Cu(111), Cu(100), and Cu(110) (a) illustrated by the lateral variation of the calculated adsorption energy of H$_2$ at each high-symmetry position on these surfaces; (b) illustrated by calculated adsorption-energy values of H$_2$ on these surfaces. The black horisontal lines indicate approximate site averages, analogous to Eq. (\[eq:7\]). Subtracting the atop value from each average gives 1.3, 1.6, and 2.5 meV as lateral variations on the (111), (100) and (110) surfaces, these numbers can be interpreted as measures of the corrugation.](fig7bcorrB.png "fig:"){width="40.00000%"} On all three facets, the calculated stable site is atop (Figs. \[fig:5\] and \[fig:7\]). This result can be understood with a simple argument based on the traditional model and a tight-binding (TB) description of the electrons. Equations (\[eq:1\])–(\[eq:3\]) separate the potential energy into repulsive and attractive parts, $V_R$ and $V_{{\mbox{\scriptsize vdW}}}$, respectively. Close to the minimum point, the vdW attraction, $V_{{\mbox{\scriptsize vdW}}}$ (Eq. (\[eq:3\])) gets stronger in the direction towards the surface. The repulsion terms $V_R$ prevent the admolecule from benefiting from this by going even closer. Equation (\[eq:4\]) reflects that higher density gives higher repulsion, and the density profiles in Figure \[fig:3\] show the proper order. For a more general discussion, see Ref. [@Chen]. While the electron density (Fig. \[fig:3\]) is characterized by only one kind of corrugation, the corrugation of the PECs depends on where the probe hits the PEC. From Figure \[fig:5\] we can envisage different corrugation values for different $z$ values. For the reflection-diffraction experiment one can argue that H$_2$ molecules coming in to the surface with a positive kinetic energy, i.e., at or above the energy of the classical turning point, are particularly relevant. Corrugation and exchange functional ----------------------------------- ![\[fig:8\] Interaction-energy values for benzene at various positions on the Cu(111) surface, calculated with different XC functionals. It shows that corrugation energies are sensitive to choice of functional, according to calculations with vdW-DF [@dion2004], vdW-DF(C09) [@cooper2010], vdW-DF(optPBE) [@klimes2010], vdW-DF2 [@LeeEtAl10], and vdW-DF2(C09).](fig8BzCu111_corr2.png){width="40.00000%"} To clarify the underlying cause of corrugation, a different adsorption system is first studied. Benzene on the Cu(111) surface is known to be a true vdW system [@slidingrings]. Interaction-energy values for the benzene molecule at various positions on the Cu(111) are calculated with five different density functionals, all accounting for vdW forces, and shown in Figure \[fig:8\]. The functionals differ by the differing strengths of vdW attraction in vdW-DF and vdW-DF2, but in particular by different exchange approximations, revPBE [@dion2004; @revPBE], C09 [@cooper2010], optPBE [@klimes2010], and PW86R [@PW86; @murray2009; @LeeEtAl10]. While the binding-energy value is not so sensitive to choice of functional, corrugation-energies values are. Therefore, the accurate results of a reflection-diffraction experiment are valuable in several respects. On one hand, they support that the traditional model is right in its separation in Eqs. (\[eq:1\])-(\[eq:3\]) and there attaching the repulsive wall, $V_R$, to Pauli repulsion [@persson2008] and thus to exchange. On another hand, they are able to discriminate between different approximations for the exchange functional. Similar results have also been calculated for benzene on graphene [@berland12]. Comparison with experiment-related quantities --------------------------------------------- ![\[fig:9\] (a) Experimentally determined effective physisorption potential for H$_2$ at atop site on the Cu(111) surface [@andersson1993], compared with potential-energy curves for H$_2$ on the Cu(111) surface, calculated for the atop site in GGA-revPBE, GGA-PBE, vdW-DF2, vdW-DF [@h2cu], TS-vdW [@TS], and DFT-D3(PBE) [@Grimme]. Partly adapted from Ref. . (b) Comparison of PECs for atop and hollow sites on Cu(111) calculated with vdW-DF2 and DFT-D3. ](fig9.png){width="40.00000%"} Comparison of the calculated results on the H$_2$-Cu systems with the model used for the analysis of the experimental data [@andersson1993; @perandersson1993] is next done for the laterally averaged potential $V_0(z)$, the potential derived from experiment. The experiment-derived results in Figure \[fig:1\] are redrawn in Figure \[fig:9\] as the experimental physisorption potential for H$_2$ on Cu(111). Figure \[fig:9\](a) shows our comparison of PECs calculated using vdW-DF and vdW-DF2, respectively, drawn against the experimental physisorption potential for H$_2$, originally published in Ref. . The Cu(111) surface is chosen for its flatness that gives clarity in the analysis and eliminates several side-issues that could have made interpretations fuzzier. Several qualitative similarities are found for both vdW-DF and vdW-DF2 functionals. The vdW-DF2 functional gives PECs in a useful qualitative and quantitative agreement with the experimental physisorption curve, for instance with respect to well depth, equilibrium separation, and curvature of PEC near the well bottom, and thus zero-point vibration frequency. Comparisons of full PECs are also parts of the benchmarking. Other methods ------------- Well depths and equilibrium separations for H$_2$ on Cu(111) are seen as PEC minimum points in Figure \[fig:9\](a). As the corrugation is so small, it should suffice to use only the atop result. More precise value pairs [@h2cu] are ($-28.9$ meV; 3.5 [Å]{}) for $V_0(z)$ extracted from experiment [@andersson1993], ($-53$ meV; 3.8 [Å]{}) as calculated with the vdW-DF functional, ($-39$ meV; 3.6 [Å]{}) with the vdW-DF2 functional, ($-98$ meV; 2.8 [Å]{}) with the DFT-D3(PBE) method [@Grimme], and ($-66$ meV; 3.2 [Å]{}) with the TS-vdW method [@TS]. The striking discrepancy between the three major types of accounts for vdW in extended media is discussed below. As reported for example in Ref. , the LDA and GGA functionals do not describe the nonlocal correlation effects that give vdW forces. They also misrepresent the PECs. The minima are too shallow and the equilibrium separations are too large [@rydberg2000]. In Figure \[fig:9\], the DFT-D3(PBE) curves are the results of calculations with the DFT-D3 corrections [@Grimme] added on top of the PBE PECs. Figure \[fig:9\](b) compares a DFT-D3 PEC at the hollow site of the (111) surface with that of an atop site \[same as the one in Fig. \[fig:9\](a)\] [@h2cu]. The energy difference between atop and hollow adsorption-energy values is found to be 11 meV. If this corrugation at $z_{{\mbox{\scriptsize min}}}$ were a measure of the corrugation, the corrugation by DFT-D3 on Cu(111) would be 11 meV, or 11 times larger than that given by vdW-DF2 (1 meV). It is interesting to note that the TS-vdW method delivers a PEC \[Fig. \[fig:9\](a)\] similar to that from DFT-D3 \[Fig. \[fig:9\](a) and (b)\], although not quite as deep. Energy levels ------------- The quantum-mechanical motion of the H$_2$ molecule in the various (laterally averaged) potentials (Fig. \[fig:1\]) can be calculated and for the motion perpendicular to the surface be described by, e.g., the bound-state eigenenergy values. Figure \[fig:2\] presents and compares results from experiment and theory. The experimental curve, identified by filled (H$_2$) and empty (D$_2$) black circles may be analyzed [@andersson1993; @perandersson1993; @persson2008] within the traditional theoretical picture [@zaremba1977; @HarrisNordlander] of the interaction between inert adsorbates and metal surfaces, Section \[sec:2\]. The experimental level sequence in Figure \[fig:2\] can be accurately reproduced ($<0.3$ meV) by such a physisorption potential [@perandersson1993; @persson2008] (Fig. \[fig:1\]), having a well depth of 28.9 meV and a potential minimum located 3.5 [Å]{} outside the topmost layer of copper ion cores. From the measured intensities of the first-order diffraction beams, a very small lateral variation of the H$_2$-Cu(111) potential can be deduced, $\sim 0.5$ meV at the potential-well minimum. The vdW-DF2 theory results for the levels are identified by filled (H$_2$) and empty (D$_2$) red circles. The theoretical results are constructed from the calculated vdW-DF2 PECs in Figure \[fig:5\] by first providing an estimate for the laterally averaged potential $V^{{{\mbox{\scriptsize vdW-DF2}}}}_0(z)$ for each facet, according to Eq. (\[eq:7\]). We note that unlike the experimental results \[which define $V_0(z)$\], the variation in $V^{{{\mbox{\scriptsize vdW-DF2}}}}_0(z)$ does not reflect an average over the angles of the H$_2$. We also note that this is a small effect, Section \[sec:3D\]. Figure \[fig:2\] documents good agreement in results from vdW-DF2 and experiment for the energy levels on each of the Cu facets. The eigenvalues have the same order as in the experimental results (Fig. \[fig:2\]), indicating good agreement between the calculated and measured average potentials. There are some discrepancies between the eigenvalues for H$_2$. This signals that the vdW-DF2 functional might not give the right shape for the PEC of H$_2$ on Cu(111). The vdW-DF2 PEC is judged to lie close to the experimental physisorption potential, both at the equilibrium position and at separations further away from the surface, and is thus described as “promising" [@h2cu]. The same applies for H$_2$ on Cu(100) and Cu(110). Summary ------- Having access to the full PEC, including shape of potential and asymptotic behavior, allows a more stringent assessment of the theoretical results. This is in addition to the many other conclusions that Figure \[fig:9\] gives. *PECs from vdW-DF2:* The picture for H$_2$ on Cu(111) of Ref.  at large applies also to the Cu(100) and (110) surfaces. Going from the most dense surface, Cu(111), to the more open surfaces, the changes are small. The vdW-DF2 description is good to 25% in calculating the potential depth in the worst case, Cu(111). vdW-DF2 describes the mean physisorption well depths (averaged over all three facets) to within 15% of the experiments. *Lateral average:* Approximate lateral averages of the PECs, $V_0$, have a fair agreement with those derived from experiment (Fig. \[fig:1\]). *Isotropy:* On the fcc metal Cu, the PECs of H$_2$ are almost isotropic (Figs. \[fig:5\] and \[fig:6\]). *Corrugation:* On each surface, the PECs vary (Figs. \[fig:5\]–\[fig:8\]) with the density corrugation (Fig. \[fig:3\]). For fcc metals the (111) surface is most dense, and (100) and (110) are successively more open and corrugated. For the calculated PECs for H$_2$ in atop, bridge, and hollow sites on the Cu, trends and magnitudes of $V_1$ agree with experimental findings (Figs. \[fig:1\] and \[fig:7\]). *Corrugation and exchange functional:* By the example of the benzene molecule on the Cu(111) surface, calculations with several different density functionals that account for vdW forces show that corrugation-energies values are sensitive to functionals that differ by different exchange approximations. Therefore, the accurate results of a reflection-diffraction experiment are valuable for discriminating between exchange functionals (Fig. \[fig:8\]). The vdW-DF2 functional uses a good exchange approximation. *Comparison with experiment-related quantities:* The experiment-derived results are shown in Figures \[fig:1\] and \[fig:9\] as the experimental physisorption potential for H$_2$ on Cu(111). Comparison of PECs calculated with vdW-DF and vdW-DF2 show that the vdW-DF2 functional gives PECs in a useful qualitative and quantitative agreement with the experimental physisorption curve. *Other functionals:* PECs calculated with several different methods for H$_2$ on Cu(111) in atop and hollow positions show a striking discrepancy between the results from the DFT-D3 [@Grimme] and TS-vdW [@TS] methods on the one hand, and those of vdW-DF2 and experiment on the other hand. This discrepancy is traced back to the fact that pair potentials center the interactions on the nuclei and do not fully reflect that important binding contributions arise in the wave function tails outside the surface. *Energy levels:* The energy levels in the H$_2$-Cu PEC wells (Fig. \[fig:2\]) are calculated for all facets and compared with the experimental ones (Fig. \[fig:2\]). Agreement with experimental results is gratifying. We judge the performance of vdW-DF2 as very promising. In making this assessment we observe that (i) vdW-DF2 is a first-principles method, where characteristic electron energies are typically in the eV range, and (ii) the test system and results are very demanding. The second point is made evident by the fact that other popular methods deviate significantly more from the experimental curve. For instance, application of the DFT-D3(PBE) method [@Grimme] (with atom-pairwise specific dispersion coefficients and cutoff radii computed from first principles) gives ($-98$ meV; 2.8 [Å]{}) for the PEC minimum point. The good agreement of the minima of the vdW-DF2 and experimental curves are encouraging, and so is the relative closeness of experimental and calculated eigenenergy values in Figure \[fig:2\]. The discrepancies between the eigenvalues signal that the vdW-DF2 PEC might not express the exact shape of the physisorption potential for H$_2$ on Cu(111). Comparisons and Analysis ======================== Figures \[fig:9\](a) and (b) show that the vdW-DF2 functional and DFT-D3 and TS-vdW methods give very different results. We could have made this point even stronger by showing also results for corrugation and energy values. However, we believe that comparisons of the PECs at atop and hollow sites on the Cu(111) surface suffice. No doubt, H$_2$ on Cu is a demanding case for all methods. The local probe H$_2$ on Cu avoids smearing-out effects, unlike for example graphene and PAHs, and incipient covalency, unlike H$_2$O and CO, and is thus a pure vdW system and a lateral-sensitive one. We trace the differing of the results with the vdW-DF2 and DFT-D3 methods to the differences in the descriptions of the vdW forces. The vdW-DF2 and similar functionals describe interactions between all electrons, while DFT-D3 and TS-vdW methods rely on atom-pair interactions. Even if large efforts are put into mimicking the real electron-charge distribution by electron-charge clouds around each atom nucleus, this has to be a misrepresentation of surface-induced redistributions of electronic charge in a general-geometry correction of a traditional DFT calculation. There is no mechanism for Zaremba-Kohn effects. The Zaremba-Kohn formulation of physisorption is not build in explicitly into the vdW-DF2 functional. However, the interactions of the electrons are build in into the supporting formalism in the similar way as in the derivation of the Zaremba-Kohn formula. Conclusions =========== Accurate and extensive experimental data are used to benchmark calculational schemes for sparse matter, that is, methods that account for vdW forces. Reflection-diffraction experiments on light particles on well-characterized surfaces provide accurate data banks of experimental physisorption information, which challenge any such scheme to produce relevant physisorption PECs. PECs of H$_2$ on the Cu(111), (100), and (110) surfaces are here studied. Accuracy is high even by the surface-physics standards and is here provided thanks to diffraction kinematical conditions giving sharp resonances in diffraction beam intensities. We propose that such surface-related PEC benchmarking should find a broader usage. The vdW-DF, vdW-DF2, DFT-D3, and TS-vdW schemes are used, and results are compared. The first two are expressions of the vdW-DF method, that is, nonempirical nonlocal functionals in which the electrodynamical couplings of the plasmon response produces fully distributed contributions to vdW interactions; Like in the Zaremba-Kohn picture [@zaremba1977], they permit the extended conduction electrons to respond also in the density tails outside a surface. The latter two are examples of DFT extended with vdW pair potentials and represent the dispersive interaction through an effective response and pair potentials located on the nuclei positions. Several qualitative similarities are found between the vdW-DF and vdW-DF2 functionals. The vdW-DF2 functional gives PECs in a useful qualitative and quantitative agreement with the experimental PECs. This is looked at for well depths, equilibrium separations, and curvatures of PEC near the well bottom, and thus molecular zero-point vibration frequency. The DFT-D3 and TS-vdW schemes give PEC results that deviate significantly more from experimental PECs. The benchmark with the experimental H$_2$/Cu scattering data is thus able to discriminate between the results of pair-potential-extended DFT methods and vdW-DF2. The differences suggest that it is important to reflect the actual, distributed location of the fluctuations (plasmons) that give rise to vdW forces. The vdW-DF2 density functional benchmarks very well against the S22 data sets [@LeeEtAl10]. It is also the functional being more extensively compared between experiment and theory here. Certain very well-pronounced features, like isotropy of the H$_2$-Cu PEC, the (111), (100), and (110) PECs being close to identical, and the clear trend in its corrugation that grows in order (111) $<$ (100) $<$ (110), are well described. The calculated $V_1$ results are found to be close to the experimental ones, thereby being almost decisive on exchange functionals. The energy levels for the quantum-mechanical motion in the H$_2$-Cu PEC agree in a gratifying way. The accuracy of this experiment is also shown to be valuable for discriminating between exchange functionals (Fig. \[fig:6\]). The vdW-DF2 functional is found to apply a good exchange approximation. The vdW-DF2 is found promising for applications at short and intermediate distances, as is relevant for adsorption. However, the accuracy of experimental data is high enough to stimulate a more detailed analysis of all aspects of the theoretical description. This should be valuable for the further XC-functional development. For instance, some discrepancies are found for the eigenvalues. They signal that the vdW-DF2 PEC for H$_2$ on Cu(111) might not have a perfect shape. Additional physical effects could be searched for. The metallic surface state on Cu(111) might be one source; It is possible that the metallic nature of the H$_2$/Cu(111), H$_2$/Cu(100), and H$_2$/Cu(110) systems motivates modifications in the description of the electrodynamical response inside the nonlocal functional. For a well established conclusion, a more accurate theory is called for. In any case, H$_2$/Cu physisorption constitutes possibilities for benchmarking theory descriptions and represents a very strong challenge for the density functional development. The Swedish National Infrastructure for Computing (SNIC) at C3SE is acknowledged for providing computer allocation and the Swedish Research Council (VR) for providing support to KB, ES, and PH. 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It also takes off from an approximate Zaremba-Kohn type [@zaremba1977; @HarrisNordlander] specification of the repulsive wall. The location of the jellium edge [@jelliumedge] (relative to the position of the last atomic plane) is set at 1.97 $a_0$, at 1.71 $a_0$, and at 1.21 $a_0$ for the Cu(111), Cu(100) and for Cu(110) surfaces. The remaining set of parameters are split into two groups, those that are constrained to be identical for all facets and those that are assumed to be facet specific. Fitting against the set of measured quantization levels $\epsilon_n$ yields values $C_{{\mbox{\scriptsize vdW}}}=4740 a_0^3$ meV, $z_{{\mbox{\scriptsize vdW}}}= 0.563 a_0$, and $k_c=0.46 a_0^{-1}$ for parameters in the first group, as well as the facet specific determinations, $V_R'= 7480$ meV, $V_R'= 5610$ meV, and $V_R'= 5210$ meV for Cu(111), Cu(100), and Cu(110), respectively. [^3]: We have tested the accuracy and consistency among the two experimental determinations of the effective physisorption potential for Cu(111). Specifically, we constructed an alternative $V_0(z)$ form in which we directly inserted the Rydberg-Klein-Rees/Le Roy value of $C_{{\mbox{\scriptsize vdW}}}$ and used the directly extracted well depth 31.4 meV [@perandersson1993] to specify an effective value of $V_R'$. We found that the lowest 4 physisorption eigenvalues of this potential then coincided within 3 percent of the measured $\epsilon_n$ energies. In contrast, the experiment-based potentials $V_0(z)$ described in Ref. [@andersson1993] above and in potentials Fig. \[fig:1\](a) reproduce the full set of measured energies (for all facets) to within 0.3 meV. [^4]: Both DFT-D and TS-vdW have a need to fix a cross-over function that is designed to minimize double counting of the semilocal correlation in regular DFT and the vdW contribution. The parameter of this cross over is often fitted to a reference system, for example S22. [^5]: In the description of the DFT calculations we refer to these positions as “sites" but note that there is a large zero-point motion perpendicular to the surface and that the experimentally relevant molecules move with a large in-surface kinetic energy while trapped in the physisorption well.

--- abstract: 'At intermediate and high degree $l$, solar p- and f modes can be considered as surface waves. Using variational principle, we derive an integral expression for the group velocities of the surface waves in terms of adiabatic eigenfunctions of normal modes, and address the benefits of using group-velocity measurements as a supplementary diagnostic tool in solar seismology. The principal advantage of using group velocities, when compared with direct analysis of the oscillation frequencies, comes from their smaller sensitivity to the uncertainties in the near-photospheric layers. We address some numerical examples where group velocities are used to reveal inconsistencies between the solar models and the seismic data. Further, we implement the group-velocity measurements to the calibration of the specific entropy, helium abundance Y and heavy-element abundance Z in the adiabatically-stratified part of the solar convective envelope, using different recent versions of the equation of state. The results are in close agreement with our earlier measurements based on more sophisticated analysis of the solar oscillation frequencies (Vorontsov et al. 2013, MNRAS 430, 1636). These results bring further support to the downward revision of the solar heavy-element abundances in recent spectroscopic measurements.' date: 'Accepted 2012 December 00. Received 2012 December 00; in original form 2012 December 00' title: Helioseismic Measurements in the Solar Envelope Using Group Velocities of Surface Waves --- \[firstpage\] waves – equation of state – Sun: oscillations – Sun: helioseismology – Sun: abundances. Introduction ============ The major difficulty in the seismic measurements of the solar internal structure comes from the uncertain effects of the outermost solar layers (the photosphere and layers immediately below), where trapped acoustic waves are reflected to the solar interior. The difficulty originates from both the uncertainties in the theoretical modeling of these layers (e. g. effects of the penetrative convection), and from poor understanding of the physics of wave propagation there (non-adiabatic effects). This difficulty is behind the dominant source of mismatch between the observational and theoretical p-mode frequencies. The discrepancy increases with frequency (as the upper turning points of the p modes move upwards), reaching values of the order of one percent at frequencies of maximum oscillation power (about 3 mHz). In standards of solar seismology, one percent is a huge quantity: the frequencies of solar oscillations are measured with precision better than one part in $10^4$, and it is this high precision which enables the seismic data with its unique diagnostic capability. To allow an accurate diagnostic of the deep interior, the near-surface uncertainties are suppressed, in one way or another, by a proper design of helioseismic inversion technique. Principally, the separation of the uncertain effects is made possible by the relatively small values of the sound speed in the subsurface layers, which makes the acoustic ray paths nearly vertical there, when the degree $l$ of the oscillations is not too high. The possibility of separating the uncertain effects is most transparent when high-frequency asymptotic analysis is implemented to describe the solar p modes: the subsurface effects bring a frequency-dependent phase shift $\alpha(\omega)$ of the standing acoustic wave, which does not depend on the degree $l$ when $l$ is small. Separation of the near-surface uncertainties comes for a price of loosing valuable diagnostic information. An example is He II ionization region, the domain which is particularly important for measuring the solar abundances and for the calibration of the equation of state. For p modes of low degree $l$, the signal of He ionization is also seen as a frequency-dependent “surface phase shift”; this signal is suppressed together with near-surface uncertainties when an arbitrary function of frequency is allowed for $\alpha(\omega)$. At higher degree $l$, we meet another difficulty. In high-precision measurements, the acoustic waves in the subsurface layers can no longer be considered as purely vertical, and at least a first-order correction shall be added to $\alpha(\omega)$ to account for the resulted dependence of the surface phase shift on the degree $l$, as another function of frequency multiplied by $l(l+1)$ [@Brodsky93]. But according to the asymptotic description, allowing the degree dependence to the surface term widens the family of possible solutions in the deep interior [@Gough95]. It is desirable, therefore, to extend the set of diagnostic tools, implemented in solar seismology for analyzing the oscillation frequencies, by adding new tools which respond differently to the near-surface uncertainties, and suppress these uncertainties in a different way. This will allow more options for the cross-validation of the results, to make them more reliable. An issue of particular importance is possible effects of systematic errors in frequency measurements; these errors may be significantly bigger than the reported observational uncertainties [see e. g. @Vorontsov13a which we refer below as Paper I]. In general, systematic errors propagate differently to the results when different techniques of data analysis are implemented, and using different tools brings better chances to detect these errors. In this paper, we consider solar oscillations of intermediate and high degree $l$ as surface waves, and address the diagnostic properties of group velocities of these waves. The concept of group velocity is known to be a valuable tool in terrestrial seismology, where it is applied to study the propagation of Love’s and Rayleigh’s waves [see e. g. @Dahlen98]. Section 2 contains a general discussion, based on the integral representation of the group velocity in terms of adiabatic eigenfunctions of normal modes, developed in the Appendix. In section 3, we test the diagnostic potential of group velocities by addressing the agreement of several solar models with observational data. In section 4, we implement the group-velocity analysis to the calibration of the main parameters of the solar convective envelope: specific entropy in the adiabatically-stratified layers, helium abundance $Y$ and heavy-element abundance $Z$. Section 5 contains a short discussion. The group velocity ================== With temporal dependence separated as $\exp(-i\omega t)$, the displacement field of the oscillations specified by a particular spherical harmonic is $${\bf u}=\hat{\bf r}U(r)Y_{lm}(\theta,\phi)+V(r)\nabla_1Y_{lm}(\theta,\phi),$$ where $\nabla_1$ is horizontal component of the gradient operator, $\nabla_1=\hat{\bf\theta}\partial/\partial\theta+\hat{\bf\phi}\sin^{-1}\theta\,\partial/\partial\phi$, and unit vectors are designated by hats. The horizontal wavenumber at the photospheric level is $k_H=L/R_\odot$, with $L^2=l(l+1)$. The horizontal phase velocity $v_p$ and group velocity $v_g$ are $$v_p={\omega\over L}R_\odot,\quad\quad v_g=\left({\partial\omega\over\partial L}\right)_nR_\odot,$$ where the derivative is taken at constant radial order $n$. Using self-adjoint properties of the equations of linear adiabatic oscillations, it is shown in the Appendix that $${v_g\over v_p}\equiv \left({\partial\ln\omega\over\partial\ln L}\right)_n= {\int\limits_0^R\rho_0r^2L^2V^2dr +{L^2\over 4\pi G\omega^2}\int\limits_0^\infty P^2dr \over \int\limits_0^R\rho_0r^2\left(U^2+L^2V^2\right)\,dr}\,,$$ where $P=P(r)$ describes the Eulirean perturbation $\psi'$ to the gravitational potential as $$\psi'=-P(r)Y_{lm}(\theta,\phi).$$ At high degree $l$, the effects of gravity perturbation are small, and the second term in the nominator of the equation (3) can be neglected. The first term in the nominator is proportional to the mean kinetic energy of the horizontal motions; the denominator is proportional to the total kinetic energy. When the effects of gravity perturbations are small, the ratio $v_g/v_p$ is thus the ratio of the horizontal kinetic energy to the total kinetic energy. An important property of the group velocity (when compared with phase velocity) is its enhanced sensitivity to the stratification of the inner part of the acoustic cavity, where the horizontal kinetic energy is localized (Fig. 1). Closer to the surface, the acoustic ray paths become nearly vertical, and the kinetic energy is dominated by the vertical motion. ![Horizontal and vertical kinetic energy densities for $p_4$ mode of $l=300$. The energy per unit depth in the integrals in equation (3) was multiplied with adiabatic sound speed $c(r)$ to account for the rescaling of the independent variable from geometrical to acoustic radius.[]{data-label="f1"}](fig1.eps){width="1.0\linewidth"} The concept of group velocity is not restricted to high-degree modes; it extends formally to all the non-radial modes when we consider the degree $l$ as a continuous parameter. The ratio $v_g/v_p$ in the degree range $0\le l\le 300$, calculated for the reference solar model S of @Christensen96, is shown in Fig.2. ![Ratio of group and phase velocities calculated for the solar model in the degree range $0\le l\le 300$ and frequency range 1mHz$\le\omega/2\pi\le$5mHz. Red circles show modes with frequencies below 2mHz, green—between 2 and 3mHz, and blue—above 3mHz. Upper scale is the position of the inner turning point in radius.[]{data-label="f2"}](fig2.eps){width="1.0\linewidth"} For modes penetrating deep into the solar interior, the ratio $v_g/v_p$ tends to collapse to a single function of the penetration depth (Fig. 2), the behaviour which reflects the high-frequency asymptotic properties of solar p modes. The rapid variation seen at $r_1\approx 0.7R_\odot$ comes from the rapid change in the sound-speed gradient at the base of the convection zone. The prominent fluctuations in $c_g/c_p$ exhibited by modes with turning points closer to the surface are produced by the rapid variation of the adiabatic exponent in the He ionization region. For f modes, the ratio $v_g/v_p$ is close to 1/2, which reflects equipartition of the kinetic energy between horizontal and vertical motions. Indeed, using an approximate dispersion relation of high-degree solar f modes as $\omega^2\simeq k_Hg_0(R_\odot)$, where $g_0(R_\odot)$ is surface gravity, we have $(\partial\ln\omega/\partial\ln L)_{n=0}\simeq 1/2$. Introducing the derivative $(\partial\omega/\partial L)_n$, we extend the family of solutions to the oscillation equations from integer to continuous values of degree $l$. This extension can be achieved simply by allowing $l$ to take arbitrary values in the ordinary differential equations resulted from variable separation; it can be viewed as a result of relaxing the periodic boundary conditions in angular coordinates. We now proceed with similar generalization but allowing non-integer values to the radial order $n$. This is equivalent to allowing a continuous variation to the radial phase integral, which takes values of $\pi(n+1)$ at resonant frequencies. The only limitation here is that the phase has a well-defined measure only for functions with nearly-harmonic behaviour. Such a behaviour is exhibited by the solutions to the oscillation equations in the adiabatically-stratified part of the solar convective envelope, owing to their high-frequency asymptotic properties (the wave propagation here is close to that of purely acoustic waves). Specifically, the near-harmonic behavior is exhibited with best accuracy by eigenfunction $\psi_p(\tau)$ defined as [see @Vorontsov91b and Paper I] $$\psi_p=\rho_0^{-1/2}r\left({1\over c^2}-{\tilde w^2\over r^2}\right)^{1/4} \left(1-{N^2\over\omega^2}\right)^{-1/2}p_1,$$ where $p_1=p_1(r)$ describes the Eulerian pressure perturbations $p'$ as $$p'=p_1(r)Y_{lm}(\theta,\phi),$$ $$\tilde w^2={L^2\over\omega^2},$$ (this parameter specifies the radial position $r_1$ of the inner turning points), and independent variable $\tau$ satisfies $${d\tau\over dr}={1\over c}\left(1-\tilde w^2{c^2\over r^2}\right)^{1/2}.$$ Two linearly-independent solutions to the oscillation equations in Cowling approximation (which is applicable in the low-density envelope) are $\psi\simeq\sin(\omega\tau)$ and $\psi\simeq\cos(\omega\tau)$. Using variational principle for evaluating the variation with frequency of phases of the inner and of the outer solutions (which satisfy inner and outer boundary conditions, respectively) at a boundary taken in the domain where asymptotic description is applicable, we obtain (see Appendix) $$\left({\partial\omega\over\partial n}\right)_L ={\pi\over 2\omega^2}{\psi_p^2+{1\over\omega^2}\left({d\psi_p\over d\tau}\right)^2 \over\int\limits_0^R\rho_0r^2\left(U^2+L^2V^2\right)dr}.$$ In analogy with $(\partial\omega/\partial L)_n$, considered as an angular component of the group velocity, $(\partial\omega/\partial n)_L$ can be considered as a mean group velocity in radial direction. The dependence of both the $(\partial\omega/\partial L)_n$ and $(\partial\omega/\partial n)_L$ on the near-surface uncertainties comes principally from the dependence on these uncertainties of the denominator in the expressions (3) and (9), which is the same (mode’s kinetic energy). This observation suggests using the ratio $$\gamma(L,n)=\left({\partial\omega\over\partial L}\right)_n\Big/\left({\partial\omega\over\partial n}\right)_L$$ as diagnostic quantity, which will allow to suppress the effects of the near-surface uncertainties. In the simplest way, this quantity can be evaluated from the mode frequencies using central differences as $$\gamma_{ln}={\omega_{l+1,n}-\omega_{l-1,n}\over\omega_{l,n+1}-\omega_{l,n-1}}.$$ According to the equation (9), the inverse of $(\partial\omega/\partial n)_L$ can be considered as “mode mass”. The difference with traditional definition of the mode mass (mode energy at unit surface amplitude) is that the surface amplitude is replaced by the amplitude in the propagation domain. The principal advantage is that with this definition, the “mode mass” becomes an observable quantity. The diagnostic properties of $\gamma(L,n)$ defined by the equation (10) can be seen better if we extend the comparison with high-frequency asymptotic analysis somewhat further. In the leading-order approximation, the asymptotic eigenfrequency equation is $$\omega F(\tilde w)=\pi\left[n+\alpha(\omega)\right],$$ where $$F(\tilde w)=\int\limits_{r_1}^R\left(1-\tilde w^2{c^2\over r^2}\right)^{1/2}{dr\over c}.$$ The left-hand side of the equation (12) is the radial phase integral $\int_{r_1}^Rk_rdr$ of a purely acoustic wave, $\alpha(\omega)$ is the frequency-dependent “surface phase shift”. The sound-speed profile $c(r)$ can be recovered from $dF(\tilde w)/d\tilde w$ using Abel’s integral transform applied to the equation (13). Differentiating both sides of the equation (12) in frequency $\omega$, first at $L$=const, then at $n$=const, and subtracting the results, we have $${dF\over d\tilde w}=-\pi\left({\partial\omega\over\partial L}\right)_n\Big/\left({\partial\omega\over\partial n}\right)_L,$$ and we see that the influence of the unknown behaviour of $\alpha(\omega)$ on the results of the sound-speed inversion is successfully eliminated . Comparing the equations (10) and (14), we see that $\gamma(L,n)$ is expected to be largely insensitive to the near-surface uncertainties. Testing solar models with seismic data ====================================== In this section, we test the ability of group-velocity measurements to reveal inconsistencies between the solar models and the observational data. In these tests, we compare the $\gamma_{ln}$-values measured from the solar oscillation frequencies with those obtained from the eigenfrequencies of solar models. Fig. 3(a) shows the difference in $\gamma_{ln}$ between the Sun and the reference model S of @Christensen96. The observational frequencies were obtained by averaging the results of 15 years of SOHO MDI measurements (this observational data set is discussed in more detail in Paper I, where it is designated as data set 1). A prominent mismatch is seen in the group-velocity data for waves with turning points just below the convective envelope, which indicates an inadequate description of the seismic stratification in the solar tachocline. ![(a) difference between observational values of $\gamma_{ln}$ and those of the reference model S. (b) same residuals but obtained with interpolated values of $\gamma_{ln}$ (see text) (c) as (b), but after a slowly-varying function of frequency was subtracted from the residuals. Red circles show the results obtained at frequencies below 2 mHz, green circles—between 2 and 3 mHz, and blue circles—with data above 3 mHz.[]{data-label="f3"}](fig3.eps){width="1.0\linewidth"} For waves confined in the convective envelope, the mismatch is moderately small at low frequencies (below 2 mHz), but grows significantly when frequency increases. This behaviour is induced by the systematic difference between observational and theoretical frequencies, which grows when frequency increases. As a result, the $\gamma_{ln}$-values evaluated with using the equation (11) from observational and theoretical frequencies correspond to surface waves with different penetration depth. A simple way to reduce this effect is to replace the theoretical values of $\gamma_{ln}$ with values obtained by the interpolation along the p-mode ridge to proper values of $\tilde w$ (defined by equation 6). The result is shown in Fig. 3(b); as expected, the residuals now fall much closer to a single function of the penetration depth. Small, but systematic fluctuations around the common trend remain in the residuals even after the interpolation. These are due to the fact that that the observational and theoretical values of $\gamma$, reduced to the same penetration depth, are still measured at different frequencies (as a result, the phase of the wave function in the He II ionization region is distorted). We can make the signal which brings information about inconsistencies in deep interior cleaner still by subtracting a common function of frequency: the result is shown in Fig. 3(c) (in this computation, $f(\omega)$ was obtained by approximating the residuals in the domain $r_1>0.85 R_\odot$ by polynomial of degree 10 in frequency $\omega$). The model was than corrected by helioseismic inversion to bring it into agreement with seismic data. The resulted correction to the sound-speed profile is shown in Fig. 4(a). The inversion technique is described in Paper I; it results in another hydrostatic model, which allows a new set of eigenfrequencies to be calculated. We processed these frequencies in the same way as discussed above, to test the new model against the solar data. This is an important test: the solution in the deep interior may not be unique because two arbitrary functions of frequency were allowed by the inversion to account for near-surface effects. The result is shown in Fig. 4(b): there is no signal in the residuals. As the solar data were processed by the inversion in a very different manner, we now have better confidence in the results. ![(a) difference in the sound speed between the Sun and the reference model S, obtained in helioseismic structural inversion with observational frequencies. (b) As Fig. 3(c), but obtained with eigenfrequencies of the model resulted from the inversion.[]{data-label="f4"}](fig4.eps){width="1.0\linewidth"} Fig. 5 shows the results of similar tests but performed with two solar models having nearly-optimal parameters of the adiabatically-stratified part of the convective envelope (specific entropy and chemical composition) which were measured by the seismic calibration described in Paper I. Note that the vertical scale differs by an order of magnitude from that used in Fig. 3. A small but significant mismatch with observations is seen in both the two models. Comparing with Fig. 3(c) and with corresponding sound-speed difference (Fig. 4a), we can say that the solar sound speed is slightly bigger than in the first model (Fig. 5a) in the domain $0.85 R_\odot<r<0.9 R_\odot$ and slightly smaller in the domain $0.8 R_\odot<r<0.85 R_\odot$ (we can say nothing about deeper layers because, as with model S, the residuals become distorted by much bigger inaccuracy in the tachocline). The second model (Fig. 5b) shows better agreement in the interval $0.8 R_\odot<r<0.85 R_\odot$, but disagreement in the interval $0.85 R_\odot<r<0.9 R_\odot$ is made bigger. These two models were used as a reference in the structural inversions for the adiabatic exponent $\Gamma_1$ in Paper I. The results are shown in Paper I by Figs 12(a) and 12(b) for the first and the second model, respectively; they are in agreement with the conclusions drawn from the analysis of group velocities. The magnitude of the mismatch between the models and the data is quite small: it calls for corrections of about $1\cdot 10^{-4}$ in the profile of the adiabatic exponent. ![As Fig. 3(c), but obtained with eigenfrequencies of two solar models with nearly-optimal parameters of the adiabatic part of the convective envelope, constructed with SAHA-S3 equation of state and discussed in Paper I. (a): model with $Y=0.250,\,\,Z=0.008$; (b) model with $Y=0.245,\,\,Z=0.010$.[]{data-label="f5"}](fig5.eps){width="1.0\linewidth"} Calibration of the envelope model ================================= We now implement the group-velocity measurements to the calibration of the the global parameters of the solar convective envelope—specific entropy in the adiabatically-stratified layers and two parameters of the chemical composition. As in Paper I, we compare with seismic data the 3-dimensional grids of envelope models calculated with four different versions of the equation of state. The models are described in detail in Paper I. Fig. 6 illustrates the potential possibility of simultaneous measurement of the parameters of the convective envelope by showing the response of the mismatch in $\gamma_{ln}$ between the Sun and the model to the variation of the specific entropy, helium abundance, and heavy-element abundance in the model. Fig. 6(a) shows the residuals obtained with the best-fit model in the grid of models calculated with SAHA-S3 equation of state. Figs 6(b) and 6(c) show the response of the residuals to the variation of the specific entropy (controlled by a mixing-length parameter $\alpha$) and to the variation of the helium abundance $Y$. Each of the two variations produces a quasi-periodic signal in the residuals by changing the profile of the adiabatic exponent in the He II ionization region. The two signals look similar, but closer inspection reveals that they have different phase: smaller entropy (bigger $\alpha$) shifts the He II ionization to greater depths (it also shifts the location of the upper turning points, but these two effects do not compensate for each other). ![Difference between solar values of $\gamma_{ln}$ and model predictions, for the “best-fit” model in the 3-D grid of envelope models (a), for a model of the same chemical composition but with different specific entropy in the adiabatically-stratified layers (b), model which differs from the best-fit model in He abundance Y (c), and model which differs in the heavy-element abundance Z (c). Red circles show the results in the $\tilde w$-range between 4000 s and 7000 s (lower turning points between $0.85 R_\odot$ and $0.933 R_\odot$), blue circles—for $\tilde w>7000$ s.[]{data-label="f6"}](fig6.eps){width="1.0\linewidth"} Principally, it is the availability of both the amplitude and phase of the He II ionization signal in the solar oscillation frequencies which allows separate measurement of helium abundance and entropy; this property was used in the first seismic measurements of the solar He abundance [e. g. @Vorontsov91] [for extended discussion, see @Vorontsov92]. The signal produced by the variation of the heavy-element abundance (Fig. 6d) is more complicated; a distinctive feature of this signal is that bigger $Z$ shifts $\Delta\gamma_{ln}$ to negative values, which signals that the sound speed in the model is too small (smaller $Z$ brings smaller values to the adiabatic exponent $\Gamma_1$). The results of the calibration are illustrated by Fig. 7. In these computations, the frequency range of the input data was limited by lower frequencies (below 2 mHz), where the difference in absolute values of the observational and theoretical frequencies is relatively small, and observational and theoretical values of $\gamma_{ln}$ were compared directly, without any interpolation to common penetration depths (common values of $\tilde w$). The goodness of fit was measured by the merit function ($\chi^2$ per degree of freedom) $$M^2={1\over N}\sum\limits_{l,n} \left[{\gamma_{ln}^{\rm(obs)}-\gamma_{ln}^{\rm(model)}\over\delta\gamma_{ln}^{\rm obs}}\right]^2,$$ where $\delta\gamma_{ln}^{\rm obs}$ is the $1\sigma$ uncertainty in the observational values of $\gamma_{ln}$ (defined by equation 11) induced by the expected random errors in the oscillation frequencies. The solar p-mode frequencies were measured from the 1-yr SOHO MDI power spectra by the technique described in [@Vorontsov09; @Vorontsov13b] (data set 4 of Paper I). As in Paper I, the input date were limited by $\tilde w>4000$ s (inner turning points $r_1>0.85 R_\odot$) to eliminate modes with theoretical frequencies distorted by the inadequate description of the solar tachocline. (The same data were used in the results shown in Fig. 6). ![Goodness of fit (merit function $M$) of envelope models calculated with (a) OPAL-1996, (b) OPAL-5005, (c) SAHA-S2 and (d) SAHA-S3 equations of state. $M_{\rm opt}$ is the best value of the merit function, $\Delta M$ is the interval between contour lines (solid curves). For each pair of $Y$ and $Z$, the specific entropy was choosen to optimize $M$. Dashed level lines show the mass coordinate $m_{0.75}$ (equation 16) in the models optimized with respect to the specific entropy. The thick dashed line is for $m_{0.75}=0.9822$.[]{data-label="f7"}](fig7.eps){width="0.95\linewidth"} Dashed lines in Fig. 7 show the values of the dimensionless mass coordinate taken at $r=0.75 R_\odot$, $$m_{0.75}=m(0.75 R_\odot)/M_\odot,$$ for models which have optimal specific entropy. When helioseismic structural inversion is performed into the radiative interior, this parameter determines the density profile obtained in the solar core. The ability of the inversion to fit low-degree measurements depends on a proper value of this parameter (corresponding effects in the oscillation frequencies come from the effects of gravity perturbation in the high-density core). Successful inversion into the deep interior requires $m_{0.75}\simeq 0.9822$ (this finding is discussed in more detail in Paper I). The maximum-likelihood values of $Y$ and $Z$, resulted from the calibration, are in agreement with those reported in Paper I; they are in the range of $Y$=0.245–0.260 and $Z$=0.006–0.011. On average, the optimal values of $Y$ are now slightly bigger (by about 0.005), and optimal values of $Z$ are slightly smaller (by about 0.001). The major difference with the results reported in Paper I is that different versions of the equation of state allow to achieve nearly the same optimal values for the merit function. In our vision, the failure of our group-velocity analysis to distinguish between the performance of different equations of state comes principally from the more limited amount of the input data. The entire (and most valuable) p$_1$-ridge, for example, is only used for evaluating the group velocities of $n=2$ waves (see equation 11). Also, evaluating $(\partial\omega/\partial n)_L$ using central differences over large frequency intervals (up to 1 mHz) brings an excessive averaging and loss of spatial resolution. The attractive feature of the calibration with $\gamma_{ln}$ is its ultimate simplicity, which brings better confidence to the results. Two functions of frequency were allowed in the calibration of Paper I to account for the uncertain near-surface effects; in general, this strategy makes the results more ambiguous. No allowance for any uncertainties was given in the calibration which is described above. Another convenient feature is that $\gamma_{ln}$, as dimensionless quantity, is invariant to the homology rescaling of the hydrostatic model; the calibration reported in Paper I had to implement the rescaling as an extra (fourth) parameter, to allow small corrections to the solar radius. Since the near-surface uncertainties are not eliminated completely in the analysis, we performed several numerical experiments to address the stability of the calibration. Fig. 8(a) shows shows the result obtained with using interpolated model values of $\gamma_{ln}$ (see section 3). Fig. 8(b) shows the result obtained when the grid of models was recalculated using @Canuto91 convection theory as an alternative to standard prescription. This modification brings the absolute values of the theoretical frequencies to somewhat better agreement with observations. In Fig. 8(c) we address the result of an artificial experiment where outer boundary conditions in the eigenfrequency computations (here, we implemented “zero” boundary conditions, with setting Lagrangian pressure perturbation to zero) were shifted from the temperature minimum to the photospheric level. This modification makes the discrepancy between observational and theoretical frequencies bigger. Fig. 7(d) shows the result obtained when the observational frequency set was replaced with earlier measurements [@Schou99 data set 3 of Paper I]. We conclude from these experiments that calibration is moderately stable, but the accuracy of measuring the chemical-composition parameters of the solar envelope is not better than 0.005 in $Y$ and 0.002 in $Z$. ![Stability of the calibration of envelope models calculated with SAHA-S3 equation of state (each panel has to be compared with Fig. 7d). (a) the result obtained when observational and theoretical values of $\gamma_{ln}$ were compared with using interpolation to common values of $\tilde w$; (b) the result obtained with models calculated using an alternative prescription of the convection theory; (c) effect of changing the outer boundary conditions in eigenfrequency computations; (d) effect of changing the observational data set (see text).[]{data-label="f8"}](fig8.eps){width="0.95\linewidth"} Discussion ========== The analysis of the solar p-mode data in terms of group velocities of surface waves represents an alternative tool of helioseismic measurements, which can be used productively for validating the results obtained with more traditional methods of solar seismology. The distinctive feature of the group velocities is their enhanced sensitivity to the solar stratification in the bottom part of the acoustic cavity, and smaller sensitivity to the uncertain effects of the near-photospheric layers, when compared with oscillation frequencies. This property allows to make the technique of seismic analysis more simple and transparent. Calibration of the chemical-composition parameters $Y$ and $Z$ in the solar convective envelope confirms our previous results (Paper I), obtained with using a much more sophisticated analysis of the solar oscillation frequencies. All our results support strongly the downward revision of heavy-element abundances reported in recent spectroscopic measurements [@Asplund09]. In further work, the development of inversion techniques implementing the concept of group velocity may be a significant step forward. Another issue is related with raw data analysis at high degree $l$, where accurate frequency measurements represent a very difficult task [for an account of the current efforts, see @korz13]. An interesting approach may consist in measuring the group velocity directly, as a slope of the p-mode ridge in the $l-\nu$ power spectra. Acknowledgments {#acknowledgments .unnumbered} =============== In this work, V.A.B. and S.V.A. were supported by the RBRF grant 12-02-00135-a. [99]{} Asplund M., Grevesse N., Sauval A.J., Scott, P., 2009, Ann. Rev. A&A, 47, 481 Brodsky M., Vorontsov S. V., 1993, ApJ, 409, 455 Canuto V. M., Mazzitelli L., 1991, ApJ, 370, 295 Christensen-Dalsgaard J. et al., 1996, Sci, 272, 1286 Dahlen F. A., Tromp J., 1998, Theoretical Global Seismology, Princeton Univ. Press, Princeton Gough D. O., Vorontsov S. V., 1995, MNRAS, 273, 573 Korzennik S. G., Rabello-Soares M. C., Schou J., Larson T. P. 2013, ApJ, 772:87 Schou J., 1999, ApJ, 523, L181 Vorontsov S. V., 1991, Sov. Astron., 35, 400 Vorontsov S. V., 2006, in Fletcher K., Thompson M., eds, Beyond the Spherical Sun. ESA SP-624, Published on CDROM, p.46.1 Vorontsov S. V., Zharkov V. N., 1989, Soviet Sci. Rev. E. Astrophys. Space Phys., 7, 1 Vorontsov S. V., Jefferies S. M., 2013, ApJ, 778:75 Vorontsov S. V., Baturin V. A., Pamyatnikh A. A., 1991, Nature, 349, 49 Vorontsov S. V., Baturin V. A., Pamyatnikh A. A., 1992, MNRAS, 257, 32 Vorontsov S. V., Jefferies S. M., Giebink C., Schou J., 2009, Astron.Soc.Pacific Conf.Series, 416, 301 Vorontsov S. V., Baturin V. A., Ayukov S. V., Gryaznov V. K., 2013, MNRAS, 430, 1636 (Paper I) The variational principle ========================= In operator form, the equations of linear adiabatic oscillations of a spherically-symmetric star can be written as $$\rho_0\omega^2{\bf u}=H_0{\bf u},$$ where ${\bf u}$ is the displacement field, and linear integro-differential operator $H_0$ is defined as $$H_0{\bf u}=\nabla p'+\rho'\nabla\psi_0+\rho_0\nabla\psi',$$ $$p'=-\rho_0c^2\nabla\cdot{\bf u}+\rho_0{\bf u}\cdot\nabla\psi_0,$$ $$\rho'=-\nabla\cdot\left(\rho_0{\bf u}\right),$$ $$\nabla^2\psi'=4\pi G\rho',$$ where $\psi$ is gravitational potential, subscript zero designates equilibrium values of corresponding physical quantities, and their Eulerian perturbations are designated by prime. We take the scalar product of the both sides of equation (A1) with ${\bf u}^*$, were star stands for complex conjugate, and integrate over volume $V_R$ occupied by the star. Using Gauss theorem, it is straightforward to show that $$\begin{aligned} &&\omega^2\int\limits_{V_R}\rho_0{\bf u}^*\cdot{\bf u}\,dv =\int\limits_{V_R}\Bigg[{1\over\rho_0c^2}p'^*p'+\rho_0N^2u_r^*u_r\nonumber\\ &+&\rho_0\left({\bf u}^*\cdot\nabla\psi'+{\bf u}\cdot\nabla\psi'^*\right) +{1\over 4\pi G}\nabla\psi'^*\cdot\nabla\psi'\Bigg]dv\nonumber\\ &+&\int\limits_{S_R}\Bigg[u_r^*p'-{1\over 4\pi G}\psi'^*\left({\partial\psi'\over\partial r}+4\pi G\rho_0u_r\right)\Bigg]ds,\nonumber\\\end{aligned}$$ where $S_R$ is unperturbed spherical outer boundary, $u_r$ is radial component of ${\bf u}$, and $N$ is Brunt-Väisälä frequency, $$N^2=-g_0\left({d\ln\rho_0\over dr}+{g_0\over c^2}\right),$$ where $g_0$ is unperturbed gravitational acceleration. We now define a quadratic functional $$\Phi=\int\limits_{V_R}\!\left({\bf u}^*\cdot H_0{\bf u}-\omega^2\rho_0{\bf u}^*\cdot{\bf u}\right)dv,$$ and reduce the right-hand of this expression to the integral in radial coordinate using separation of spatial variables specified by equations (1, 4, 6). We transform the surface term using standard boundary condition for gravity perturbations $$\left[{dP\over dr}-4\pi G\rho_0U+{l+1\over r}P\right]_{r=R}=0$$ which physical meaning is the continuity of the gravitational potential and its gradient on the deformed solar surface. The result is $$\Phi=\int\limits_0^R{\cal L}\,dr+R^2\left[Up_1+{l+1\over 4\pi Gr}P^2\right]_{r=R}$$ with $$\begin{aligned} &&{\cal L}=r^2\Bigg\{{1\over\rho_0c^2}p_1^2+\rho_0N^2U^2 -2\rho_0\left(U{dP\over dr}+{L\over r}WP\right)\nonumber\\ &+&{1\over 4\pi G}\left[\left({dP\over dr}\right)^2+{L^2\over r^2}P^2\right] -\rho_0\omega^2\left(U^2+W^2\right)\Bigg\},\end{aligned}$$ where $$W=LV$$ and $$p_1=-\rho_0c^2{dU\over dr}+\left(\rho_0g_0-{2\rho_0c^2\over r}\right)U+{\rho_0c^2\over r}LW.$$ We now consider $\Phi$ as a homogeneous quadratic function of “fields” $U, W, P$ and their derivatives, which depend on $\omega,\,l$ and structural variables of the equilibrium model as “parameters” (note that the Eulerian pressure perturbation $p_1$ is not a “field” but an auxiliary variable). It can be verified directly that ${\cal L}$, defined by the equation (A11), satisfies the Euler-Lagrange equations $${d\over dr}{\partial{\cal L}\over\partial\dot{U}}-{\partial{\cal L}\over\partial U}=0,$$ $${d\over dr}{\partial{\cal L}\over\partial\dot{W}}-{\partial{\cal L}\over\partial W}=0,$$ $${d\over dr}{\partial{\cal L}\over\partial\dot{P}}-{\partial{\cal L}\over\partial P}=0,$$ where dot designates the radial derivative (equation A14 is equivalent to the radial component of the momentum equation, equation A15—to the horizontal component of the momentum equation, and equation A16—to the Poisson’s equation for gravity perturbations). We now designate as $\delta_U$ the first variation of a corresponding quantity induced by a small variation of $U$ with keeping the two other “fields” $W$ and $P$ and all the “parameters” unchanged. In a similar way, we introduce variations $\delta_W,\,\delta_P,\,\delta_L$, and $\delta_\omega$. Using integration by parts and equations (A14–A16),we have $$\delta_U\int\limits_0^R{\cal L}\,dr =\int\limits_0^R\left[{\partial{\cal L}\over\partial U}\delta U+{\partial{\cal L}\over\partial\dot{U}}{d\over dr}(\delta U)\right]dr =\left[{\partial{\cal L}\over\partial\dot{U}}\delta U\right]_0^R$$ and similar expressions for $\delta_W\int_0^R{\cal L}\,dr$ and $\delta_P\int_0^R{\cal L}\,dr$. For variations of $\Phi$ we obtain, using equations (A10, A11), $$\delta_U\Phi=R^2\left[U\delta_Up_1-p_1\delta U\right]_{r=R},$$ $$\delta_W\Phi=R^2\left[U\delta_Wp_1\right]_{r=R},$$ $$\delta_P\Phi=0,$$ $$\begin{aligned} \delta_L\Phi&=&\delta(L^2)\int\limits_0^R\left(\omega^2\rho_0r^2V^2+{1\over 4\pi G}P^2\right)dr\nonumber\\ &+&R^2\left[U\delta_Lp_1+{\delta l\over 4\pi Gr}P^2\right]_{r=R},\end{aligned}$$ $$\delta_\omega\Phi=-\delta(\omega^2)\int\limits_0^R\rho_0r^2\left(U^2+W^2\right)dr,$$ where boundary condition for gravity perturbations (equation A9) was used in deriving equation (A20). Due to the definition of $\Phi$ (equation A8), its variations sum to zero; since we do not change the parameters of the equilibrium model, we have $$\delta\Phi=\left(\delta_U+\delta_W+\delta_P+\delta_L+\delta_\omega\right)\Phi=0.$$ Using equations (A18-A22), we get $$\begin{aligned} &&\delta(L^2)\int\limits_0^R\left(\omega^2\rho_0r^2V^2+{1\over 4\pi G}P^2\right)dr\nonumber\\ &-&\delta(\omega^2)\int\limits_0^R\rho_0r^2\left(U^2+W^2\right)dr\nonumber\\ &+&R^2\left[U\delta p_1-p_1\delta U\right]_{r=R}+{\delta l\over 4\pi G}RP^2(R)=0,\end{aligned}$$ where $\delta p_1$ is the net variation of $p_1$. We will now assume that the (homogeneous and conservative) mechanical outer boundary condition can be written in a form $$AU+Bp_1=0,$$ where $A$ and $B$ do not depend on $l$ and $\omega$ (an example is the so-called “zero” boundary condition, $\nabla\cdot{\bf u}=0$). The variations $\delta U$ and $\delta p_1$ are then related as $U\delta p_1=p_1\delta U$, the third term in the equation (A24) vanishes, and we arrive to $$\begin{aligned} &&\delta(\omega^2)\int\limits_0^R\rho_0r^2\left(U^2+L^2V^2\right)dr\\ &=&\delta(L^2)\int\limits_0^R\left(\omega^2\rho_0r^2V^2+{1\over 4\pi G}P^2\right)dr +{\delta l\over 4\pi G}RP^2(R),\nonumber\end{aligned}$$ the equation which relates small variations of frequency $\omega$ and degree $l$. In the outer space ($r>R$), $P(r)$ is a regular solution to the Laplace equation, $P(r)\propto r^{-l-1}$, and we have $$\begin{aligned} \int\limits_R^\infty P^2dr&=&-\int\limits_R^\infty{r\over l+1}P{dP\over dr}dr\nonumber\\ &=&{1\over 2}{R\over l+1}P^2(R)+{1\over 2(l+1)}\int\limits_R^\infty P^2 dr,\end{aligned}$$ using integration by parts, which gives $$RP^2(R)=(2l+1)\int\limits_R^\infty P^2dr.$$ Using $(2l+1)\delta l=\delta(L^2)$, an alternative form of the equation (A26) is thus $$\begin{aligned} &&\delta(\omega^2)\int\limits_0^R\rho_0r^2\left(U^2+L^2V^2\right)dr\nonumber\\ &=&\delta(L^2)\left[\omega^2\int\limits_0^R\rho_0r^2V^2dr+{1\over 4\pi G}\int\limits_0^\infty P^2dr\right],\end{aligned}$$ which gives equation (3) of section 2. We note that similar derivation was addressed, using somewhat different approach, in [@Vorontsov06]. Due to inaccuracy in the treatment of the effects of gravity perturbations, term with gravity perturbations (second term in the right-hand side of the equation A29) has been lost in the final result of [@Vorontsov06]. We now address the derivation of the expression (9) for $(\partial\omega/\partial n)_L$. In the analysis which is described above, we replace volume $V_R$ occupied by the star by a smaller volume $V_b$ bounded by a spherical surface $S_b$ of radius $r=r_b$. We choose $r_b$ somewhere in the domain where the wave propagation is close to that of purely acoustic waves, in the low-density envelope where gravity perturbation $\psi'$ is described by a solution to the Laplace equation, which is regular at $r=\infty$. The boundary condition on gravity perturbations (equation A9) which was implemented at $r=R$ is also satisfied at $r=r_b$, as well as everywhere in between (term with $\rho_0$ in equation A9 is small, and was retained in the derivation of $(\partial\omega/\partial L)_n$ only to make it more general. When working with solar oscillations, gravity perturbations in the outer envelope may be discarded at any degree $l$ due to low density). Keeping $L$ constant, the equation (A24) is replaced with $$\delta(\omega^2)\int\limits_0^{r_b}\rho_0r^2\left(U_i^2+W_i^2\right)dr =r_b^2\left[U_i\delta p_{1,i}-p_{1,i}\delta U_i\right]_{r=r_b},$$ where we use subscript $i$ to designate solutions in the domain $r<r_b$ (solutions which satisfy central boundary conditions). We now consider solutions in the external domain $r_b\le r\le R$, which satisfy surface boundary condition specified by the equation (A25), with frequency-independent $A$ and $B$. In the similar way, we get $$\delta(\omega^2)\int\limits_{r_b}^R\rho_0r^2\left(U_e^2+W_e^2\right)dr =-r_b^2\left[U_e\delta p_{1,e}-p_{1,e}\delta U_e\right]_{r=r_b},$$ where subscript $e$ designates the external solutions. At resonant frequencies, the internal and external solutions match each other. Adding the equations (A31, A32), we have $$r_b^2\left[U_e p_{1,i}-p_{1,e}U_i\right]_{r=r_b} =\delta(\omega^2)\int\limits_0^R\rho_0r^2\left(U^2+W^2\right)dr.$$ The radial-displacement function $U(r)$ and Eulerian pressure perturbation $p_1(r)$ are related by the differential equation $${dp_1\over dr}=\left(\omega^2-N^2\right)\rho_0U-{g_0\over c^2}p_1,$$ which comes from the radial component of the momentum equation (A2) in Cowling approximation. Using equations (5) and (8), which specify the wave function $\psi_p(\tau)$, the Wronskian of the internal and external solutions in the left-hand side of the equation (A32) can be represented in terms of $\psi_p(\tau)$, and we have $${1\over\omega^2}\left(\psi_{p,i}{d\psi_{p,e}\over d\tau}-\psi_{p,e}{d\psi_{p,i}\over d\tau}\right) =\delta(\omega^2)\int\limits_0^R\rho_0r^2\left(U^2+W^2\right)dr.$$ Let $\theta$ is the radial phase integral, which takes values of $\pi(n+1)$ at resonant frequencies, and $\delta\theta=\pi\delta n$ is variation of this phase integral induced by a small deviation of frequency $\omega$ from a resonant frequency. Representing $\psi_{p,i}$ and $\psi_{p,e}$ in the vicinity of the matching point by harmonic functions of $\omega\tau$ with a small phase difference $\delta\theta$, we get $$\pi\delta n={\omega\over\psi_p^2+{1\over\omega^2}\left({d\psi_p\over d\tau}\right)^2} \delta(\omega^2)\int\limits_0^R\rho_0r^2\left(U^2+L^2V^2\right)dr,$$ where $\psi_p$ is the wave function at the resonant frequency, and we arrive to the equation (9) of section 2. Note that the derivation only assumes the applicability of the high-frequency approximation (harmonicity of $\psi_p(\tau)$) in the vicinity of the matching point, not in the entire acoustic cavity. \[lastpage\]

--- abstract: 'The model of a side information “vending machine (VM) accounts for scenarios in which the measurement of side information sequences can be controlled via the selection of cost-constrained actions. In this paper, the three-node cascade source coding problem is studied under the assumption that a side information VM is available and the intermediate and/or at the end node of the cascade. A single-letter characterization of the achievable trade-off among the transmission rates, the distortions in the reconstructions at the intermediate and at the end node, and the cost for acquiring the side information is derived for a number of relevant special cases. It is shown that a joint design of the description of the source and of the control signals used to guide the selection of the actions at downstream nodes is generally necessary for an efficient use of the available communication links. In particular, for all the considered models, layered coding strategies prove to be optimal, whereby the base layer fulfills two network objectives: determining the actions of downstream nodes and simultaneously providing a coarse description of the source. Design of the optimal coding strategy is shown via examples to depend on both the network topology and the action costs. Examples also illustrate the involved performance trade-offs across the network.' author: - 'Behzad Ahmadi, Chiranjib Choudhuri, Osvaldo Simeone and Urbashi Mitra [^1][^2]' title: 'Cascade Source Coding with a Side Information “Vending Machine”' --- Rate-distortion theory, cascade source coding, side information, vending machine, common reconstruction constraint. Introduction ============ The concept of a side information “vending machine (VM) was introduced in [@Permuter] for a point-to-point model, in order to account for source coding scenarios in which acquiring the side information at the receiver entails some cost and thus should be done efficiently. In this class of models, the quality of the side information $Y$ can be controlled at the decoder by selecting an action $A$ that affects the effective channel between the source $X$ and the side information $Y$ through a conditional distribution $p_{Y|X,A}(y|x,a)$. Each action $A$ is associated with a cost, and the problem is that of characterizing the available trade-offs between rate, distortion and action cost. Extending the point-to-point set-up, cascade models provide baseline scenarios in which to study fundamental aspects of communication in multi-hop networks, which are central to the operation of, e.g., sensor or computer networks (see Fig. \[fig:fig1\]). Standard information-theoretic models for cascade scenarios assume the availability of given side information sequences at the nodes (see e.g., [@Ravi]-[@Chia]). In this paper, instead, we account for the cost of acquiring the side information by introducing side information VMs at an intermediate node and/ or at the final destination of a cascade model. As an example of the applications of interest, consider the computer network of Fig. \[fig:fig1\], where the intermediate and end nodes can obtain side information from remote data bases, but only at the cost of investing system resources such as time or bandwidth. Another example is a sensor network in which acquiring measurements entails an energy cost. ![A multi-hop computer network in which intermediate and end nodes can access side information by interrogating remote data bases via cost-constrained actions.[]{data-label="fig:fig1"}](db2_mod) As shown in [@Permuter] for a point-to-point system, the optimal operation of a VM at the decoder requires taking actions that are guided by the message received from the encoder. This implies the exchange of an explicit control signal embedded in the message communicated to the decoder that instructs the latter on how to operate the VM. Generalizing to the cascade models under study, a key issue ** to be tackled in this work is the design of communication strategies that strike the right balance between control signaling and source compression across the two hops. Related Work ------------ As mentioned, the original paper [@Permuter] considered a point-to-point system with a single encoder and a single decoder. Various works have extended the results in [@Permuter] to multi-terminal models. Specifically, [@Weissman_multi; @Ahmadi_ISIT'11] considered a set-up analogous to the Heegard-Berger problem [@HB; @Kaspi], in which the side information may or may not be available at the decoder. The more general case in which both decoders have access to the same vending machine, and either the side information produced by the vending machine at the two decoders satisfy a degradedness condition, or lossless source reconstructions are required at the decoders is solved in [@Weissman_multi]. In [@Ahmadi_DSC], a distributed source coding setting that extends [@Berger-Yeung] to the case of a decoder with a side information VM is investigated, along with a cascade source coding model to be discussed below. Finally, in [@Compression; @with; @actions], a related problem is considered in which the sequence to be compressed is dependent on the actions taken by a separate encoder. The problem of characterizing the rate-distortion region for cascade source coding models, even with conventional side information sequences (i.e., without VMs as in Fig. \[fig:intro\]) at Node 2 and Node 3, is generally open. We refer to [@Ravi] and references therein for a review of the state of the art on the cascade problem and to [@Vasudevan] for the cascade-broadcast problem. In this work, we focus on the cascade source coding problem with side information VMs. The basic cascade source coding model consists of three nodes arranged so that Node 1 communicates with Node 2 and Node 2 to Node 3 over finite-rate links, as illustrated for a computer network scenario in Fig. \[fig:fig1\] and schematically in Fig. \[fig:intro\]-(a). Both Node 2 and Node 3 wish to reconstruct a, generally lossy, version of source $X$ and have access to different side information sequences. An extension of the cascade model is the cascade-broadcast model of Fig. \[fig:intro\]-(b), in which an additional broadcast link of rate $R_{b}$ exists that is received by both Node 2 and Node 3. ![($a$) Cascade source coding problem and ($b$) cascade-broadcast source coding problem.[]{data-label="fig:intro"}](intro) Two specific instances of the models in Fig. \[fig:intro\] for which a characterization of the rate-distortion performance has been found are the settings considered in [@Chia] and that in [@Ahmadi_CR], which we briefly review here for their relevance to the present work. In [@Chia], the cascade model in Fig. \[fig:intro\](a) was considered for the special case in which the side information $Y$ measured at Node 2 is also available at Node 1 (i.e., $X=(X,Y)$) and we have the Markov chain $X-Y-Z$ so that the side information at Node 3 is degraded with respect to that of Node 2. Instead, in [@Ahmadi_CR], the cascade-broadcast model in Fig. \[fig:intro\](b) was considered for the special case in which either rate $R_{b}$ or $R_{1}$ is zero, and the reconstructions at Node 1 and Node 2 are constrained to be retrievable also at the encoder in the sense of the Common Reconstruction (CR) introduced in [@Steinberg] (see below for a rigorous definition). Contributions ------------- In this paper, we investigate the source coding models of Fig. \[fig:intro\] by assuming that some of the side information sequences can be affected by the actions taken by the corresponding nodes via VMs. The main contributions are as follows. - *Cascade source coding problem with VM at Node 3* (Fig. \[fig:fig2\]): In Sec. \[sub:RD\_cascade\], we derive the achievable rate-distortion-cost trade-offs for the set-up in Fig. \[fig:fig2\], in which a side information VM exists at Node 3, while the side information $Y$ is known at both Node 1 and Node 2 and satisfies the Markov chain $X\textrm{---}Y\textrm{---}Z$. This characterization extends the result of [@Chia] discussed above to a model with a VM at Node 3. We remark that in [@Ahmadi_DSC], the rate-distortion-cost characterization for the model in Fig. \[fig:fig2\] was obtained, but under the assumption that the side information at Node 3 be available in a causal fashion in the sense of [@Weiss-Elgam]; - *Cascade-broadcast source coding problem with VM at Node 2 and Node 3, lossless compression* (Fig. \[fig:fig3\]): In Sec. \[sub:RD\_BC\_lossless\], we study the cascade-broadcast model in Fig. \[fig:fig3\] in which a VM exists at both Node 2 and Node 3. In order to enable the action to be taken by both Node 2 and Node 3, we assume that the information about which action should be taken by Node 2 and Node 3 is sent by Node 1 on the broadcast link of rate $R_{b}$. Under the constraint of lossless reconstruction at Node 2 and Node 3, we obtain a characterization of the rate-cost performance. This conclusion generalizes the result in [@Weissman_multi] discussed above to the case in which the rate $R_{1}$ and/or $R_{2}$ are non-zero; - *Cascade-broadcast source coding problem with VM at Node 2 and Node 3, lossy compression with CR constraint* (Fig. \[fig:fig3\]): In Sec. \[sub:CR\], we tackle the problem in Fig. \[fig:fig3\] but under the more general requirement of lossy reconstruction. Conclusive results are obtained under the additional constraints that the side information at Node 3 is degraded and that the source reconstructions at Node 2 and Node 3 can be recovered with arbitrarily small error probability at Node 1. This is referred to as the CR constraint following [@Steinberg], and is of relevance in applications in which the data being sent is of sensitive nature and unknown distortions in the receivers’ reconstructions are not acceptable (see [@Steinberg] for further discussion). This characterization extends the result of [@Ahmadi_CR] mentioned above to the set-up with a side information VM, and also in that both rates $R_{1}$ and $R_{b}$ are allowed to be non-zero; - *Adaptive actions*: Finally, we revisit the results above by allowing the decoders to select their actions in an adaptive way, based not only on the received messages but also on the previous samples of the side information extending [@Chiru]. Note that the effect of adaptive actions on rate–distortion–cost region was open even for simple point-to-point communication channel with decoder side non-causal side information VM until recently, when [@Chiru] has shown that adaptive action does not decrease the rate–distortion–cost region of point-to-point system. In this paper we have extended this result to the multi-terminal framework and we conclude that, in all of the considered examples, where applicable, adaptive ![Cascade source coding problem with a side information “vending machine” at Node 3.[]{data-label="fig:fig2"}](fig2) selection of the actions does not improve the achievable rate-distortion-cost trade-offs. Our results extends to multi-hop scenarios the conclusion in [@Permuter] that a joint representation of data and control messages enables an efficient use of the available communication links. In particular, layered coding strategies prove to be optimal for all the considered models, in which, the base layer fulfills two objectives: determining the actions of downstream nodes and simultaneously providing a coarse description of the source. Moreover, the examples provided in the paper demonstrate the dependence of the optimal coding design on network topology action costs. Throughout the paper, we closely follow the notation in [@Ahmadi_CR]. In particular, a random variable is denoted by an upper case letter (e.g., $X,Y,Z$) and its realization is denoted by a lower case letter (e.g., $x,y,z$). The shorthand notation $X^{n}$ is used to denote the tuple (or the column vector) of random variables $(X_{1},\ldots,X_{n})$, and $x^{n}$ is used to denote a realization. The notation $X^{n}\sim p(x^{n})$ indicates that $p(x^{n})$ is the probability mass function (pmf) of the random vector $X^{n}$. Similarly, $Y^{n}|\{X^{n}=x^{n}\}\sim p(y^{n}|x^{n})$ indicates that $p(y^{n}|x^{n})$ is the conditional pmf of $Y^{n}$ given $\{X^{n}=x^{n}\}$. We say that $X\textrm{---}Y\textrm{---}Z$ form a Markov chain if $p(x,y,z)=p(x)p(y|x)p(z|y)$, that is, $X$ and $Z$ are conditionally independent of each other given $Y$. Cascade Source Coding with A Side information Vending Machine ============================================================= In this section, we first describe the system model for the cascade source coding problem with a side information vending machine of Fig. \[fig:fig2\]. We then present the characterization of the corresponding rate-distortion-cost performance in Sec. \[sub:RD\_cascade\]. ![Cascade source coding problem with a side information “vending machine” at Node 2 and Node 3.[]{data-label="fig:fig3"}](fig3) System Model\[sub:System-Model\_cascade\] ----------------------------------------- The problem of cascade source coding of Fig. \[fig:fig2\], is defined by the probability mass functions (pmfs) $p_{XY}(x,y)$ and $p_{Z|AY}(z|a,y)$ and discrete alphabets $\mathcal{X},\mathcal{Y},\mathcal{Z},\mathcal{A},\mathcal{\hat{X}}_{1},\mathcal{\hat{X}}_{2},$ as follows. The source sequences $X^{n}$ and $Y^{n}$ with $X^{n}\in\mathcal{X}^{n}$ and $Y^{n}\in\mathcal{Y}^{n}$, respectively$,$ are such that the pairs $(X_{i},Y_{i})$ for $i\in[1,n]$ are independent and identically distributed (i.i.d.) with joint pmf $p_{XY}(x,y)$. Node 1 measures sequences $X^{n}$ and $Y^{n}$ and encodes them in a message $M_{1}$ of $nR_{1}$ bits, which is delivered to Node 2. Node 2 estimates a sequence $\hat{X}_{1}^{n}\in\mathcal{\hat{X}}_{1}^{n}$ within given distortion requirements to be discussed below. Moreover, Node 2 maps the message $M_{1}$ received from Node 1 and the locally available sequence $Y^{n}$ in a message $M_{2}$ of $nR_{2}$ bits, which is delivered to Node 3. Node 3 wishes to estimate a sequence $\hat{X}_{2}^{n}\in\mathcal{\hat{X}}_{2}^{n}$ within given distortion requirements. To this end, Node 3 receives message $M_{2}$ and based on this, it selects an action sequence $A^{n},$ where $A^{n}\in\mathcal{A}^{n}.$ The action sequence affects the quality of the measurement $Z^{n}$ of sequence $Y^{n}$ obtained at the Node 3. Specifically, given $A^{n}$ and $Y^{n}$, the sequence $Z^{n}$ is distributed as $p(z^{n}|a^{n},y^{n})=\prod_{i=1}^{n}p_{Z|A,Y}(z_{i}|y_{i},a_{i})$. The cost of the action sequence is defined by a cost function $\Lambda$: $\mathcal{A\rightarrow}[0,\Lambda_{\max}]$ with $0\leq\Lambda_{\max}<\infty,$ as $\Lambda(a^{n})=\sum_{i=1}^{n}\Lambda(a_{i})$. The estimated sequence $\hat{X}_{2}^{n}$ with $\hat{X}_{2}^{n}\in\mathcal{\hat{X}}_{2}^{n}$ is then obtained as a function of $M_{2}$ and $Z^{n}$. The estimated sequences $\hat{X}_{j}^{n}$ for $j=1,2$ must satisfy distortion constraints defined by functions $d_{j}(x,\hat{x}_{j})$: $\mathcal{X}\times\mathcal{\hat{X}}_{j}\rightarrow[0,D_{\max}]$ with $0\leq D_{\max}<\infty$ for $j=1,2,$ respectively. A formal description of the operations at the encoder and the decoder follows. ![Cascade-broadcast source coding problem with a side information “vending machine” at Node 2.[]{data-label="fig:fig4"}](fig4_modified) \[def\_cascade\]An $(n,R_{1},R_{2},D_{1},D_{2},\Gamma,\epsilon)$ code for the set-up of Fig. \[fig:fig2\] consists of two source encoders, namely $$\mathrm{g}_{1}\text{: }\mathcal{X}^{n}\times\mathcal{Y}^{n}\rightarrow[1,2^{nR_{1}}],\label{encoder1}$$ which maps the sequences $X^{n}$ and $Y^{n}$ into a message $M_{1};$ $$\mathrm{g}_{2}\text{:}\text{ }\mathcal{Y}^{n}\times[1,2^{nR_{1}}]\rightarrow[1,2^{nR_{2}}],\label{encoder2}$$ which maps the sequence $Y^{n}$ and message $M_{1}$ into a message $M_{2};$ an “action function $$\mathrm{\ell}\text{: }[1,2^{nR_{2}}]\rightarrow\mathcal{A}^{n},\label{action_fun}$$ which maps the message $M_{2}$ into an action sequence $A^{n};$ two decoders, namely $$\mathrm{h}_{1}\text{: }[1,2^{nR_{1}}]\times\mathcal{Y}^{n}\rightarrow\mathcal{\hat{X}}_{1}^{n},\label{decoder1}$$ which maps the message $M_{1}$ and the measured sequence $Y^{n}$ into the estimated sequence $\hat{X}_{1}^{n};$ $$\mathrm{h}_{2}\text{: }[1,2^{nR_{2}}]\times\mathcal{Z}^{n}\rightarrow\mathcal{\hat{X}}_{2}^{n},\label{decoder2}$$ which maps the message $M_{2}$ and the measured sequence $Z^{n}$ into the the estimated sequence $\hat{X}_{2}^{n};$ such that the action cost constraint $\Gamma$ and distortion constraints $D_{j}$ for $j=1,2$ are satisfied, i.e., $$\begin{aligned} \frac{1}{n}\underset{i=1}{\overset{n}{\sum}}\mathrm{E}\left[\Lambda(A_{i})\right] & \leq\Gamma\label{action cost}\\ \text{ and }\frac{1}{n}\underset{i=1}{\overset{n}{\sum}}\mathrm{E}\left[d_{j}(X_{ji},\textrm{h}_{ji})\right] & \leq D_{j}\text{ for }j=1,2,\label{dist const}\end{aligned}$$ where we have defined as $\textrm{h}_{1i}$ and $\textrm{h}_{2i}$ the $i$th symbol of the function $\textrm{h}_{1}(M_{1},Y^{n})$ and $\textrm{h}_{2}(M_{2},Z^{n})$, respectively. \[def\_ach\]Given a distortion-cost tuple $(D_{1},D_{2},\Gamma)$, a rate tuple $(R_{1},R_{2})$ is said to be achievable if, for any $\epsilon>0$, and sufficiently large $n$, there exists a $(n,R_{1},R_{2},D_{1}+\epsilon,D_{2}+\epsilon,\Gamma+\epsilon)$ code. \[def\_reg\]The *rate-distortion-cost region* $\mathcal{R}(D_{1},D_{2},\Gamma)$ is defined as the closure of all rate tuples $(R_{1},R_{2})$ that are achievable given the distortion-cost tuple $(D_{1},D_{2},\Gamma)$. For side information $Z$ available causally at Node 3, i.e., with decoding function (\[decoder2\]) at Node 3 modified so that $\hat{X}_{i}$ is a function of $M_{2}$ and $Z^{i}$ only, the rate-distortion region $\mathcal{R}(D_{1},D_{2},\Gamma)$ has been derived in [@Ahmadi_DSC]. In the rest of this section, for simplicity of notation, we drop the subscripts from the definition of the pmfs, thus identifying a pmf by its argument. Rate-Distortion-Cost Region \[sub:RD\_cascade\] ----------------------------------------------- In this section, a single-letter characterization of the rate-distortion-cost region is derived. \[prop:RD\_action\_cascade\]The rate-distortion-cost region $\mathcal{R\mbox{\ensuremath{(D_{1},D_{2},\Gamma)}}}$ for the cascade source coding problem illustrated in Fig. \[fig:fig2\] is given by the union of all rate pairs $(R_{1},R_{2})$ that satisfy the conditions \[eqn: RD\_action\_cascade\] $$\begin{aligned} R_{1} & \geq & I(X;\hat{X}_{1},A,U|Y)\label{eq:R1}\\ \ce{and}\mbox{ }R_{2} & \geq & I(X,Y;A)+I(X,Y;U|A,Z),\label{eq:R2}\end{aligned}$$ where the mutual information terms are evaluated with respect to the joint pmf $$\begin{aligned} p(x,y,z,a,\hat{x}_{1},u)=p(x,y)p(\hat{x}_{1},a,u|x,y)p(z|y,a) & ,\label{eq:joint}\end{aligned}$$ for some pmf $p(\hat{x}_{1},a,u|x,y)$ such that the inequalities \[eqn: action\_cascade\_const\] $$\begin{aligned} \ce{E}[d_{1}(X,\hat{X}_{1})] & \leq & D_{1},\label{eq:dist1}\\ \ce{E}[d_{2}(X,\ce{f}(U,Z))] & \leq & D_{2},\label{eq:dist2}\\ \ce{and}\textrm{ }\ce{E}[\Lambda(A)] & \leq & \Gamma,\label{eq:action_bound}\end{aligned}$$ are satisfied for some function $\ce{f}\textrm{: }\mathcal{U}\times\mathcal{Z}\rightarrow\hat{\mathcal{X}}_{2}$. Finally, $U$ is an auxiliary random variable whose alphabet cardinality can be constrained as $|\mathcal{U}|\leq|\mathcal{X}||\mathcal{Y}||\mathcal{A}|+3$, without loss of optimality. For side information $Z$ independent of the action $A$ given $Y$$,$ i.e., for $p(z|a,y)=p(z|y),$ the rate-distortion region $\mathcal{R}(D_{1},D_{2},\Gamma)$ in Proposition \[prop:RD\_action\_cascade\] reduces to that derived in [@Chia]. The proof of the converse is provided in Appendix A for a more general case of adaptive action to be defined in Sec \[sec:Adaptive-Actions\]. The achievability follows as a combination of the techniques proposed in [@Permuter] and [@Chia Theorem 1]. Here we briefly outline the main ideas, since the technical details follow from standard arguments. For the scheme at hand, Node 1 first maps sequences $X^{n}$ and $Y^{n}$ into the action sequence $A^{n}$ using the standard joint typicality criterion. This mapping requires a codebook of rate $I(X,Y;A)$ (see, e.g., [@Elgammal pp. 62-63])$.$ Given the sequence $A^{n}$, the sequences $X^{n}$ and $Y^{n}$ are further mapped into a sequence $U^{n}$. This requires a codebook of size $I(X,Y;U|A)$ for each action sequence $A^{n}$ from standard rate-distortion considerations [@Elgammal pp. 62-63]. Similarly, given the sequences $A^{n}$ and $U^{n},$ the sequences $X^{n}$ and $Y^{n}$ are further mapped into the estimate $\hat{X}_{1}^{n}$ for Node 2 using a codebook of rate $I(X,Y;\hat{X}_{1}|U,A)$ for each codeword pair $(U^{n},A^{n})$. The thus obtained codewords are then communicated to Node 2 and Node 3 as follows. By leveraging the side information $Y^{n}$ available at Node 2, conveying the codewords $A^{n},$ $U^{n}$ and $\hat{X}_{1}^{n}$ to Node 2 requires rate $I(X,Y;U,A)+I(X,Y;\hat{X}_{1}|U,A)-I(U,A,\hat{X}_{1};Y)$ by the Wyner-Ziv theorem [@Elgammal p. 280]$,$ which equals the right-hand side of (\[eq:R1\]). Then, sequences $A^{n}$ and $U^{n}$ are sent by Node 2 to Node 3, which requires a rate equal to the right-hand side of (\[eq:R2\]). This follows from the rates of the used codebooks and from the Wyner-Ziv theorem, due to the side information $Z^{n}$ available at Node 3 upon application of the action sequence $A^{n}$. Finally, Node 3 produces $\hat{X}_{2}^{n}$ that leverages through a symbol-by-symbol function as $\hat{X}_{2i}=\textrm{f}(U_{i},Z_{i})$ for $i\in[1,n].$ Lossless Compression -------------------- Suppose that the source sequence $X^{n}$ needs to be communicated [*losslessly*]{} at both Node 2 and Node 3, in the sense that $d_{j}(x,\hat{x}_{j})$ is the Hamming distortion measure for $j=1,2$ ($d_{j}(x,\hat{x}_{j})=0$ if $x=\hat{x}_{j}$ and $d_{j}(x,\hat{x}_{j})=1$ if $x\neq\hat{x}_{j}$) and $D_{1}=D_{2}=0$. We can establish the following immediate consequence of Proposition \[prop:RD\_action\_cascade\]. \[coro:cascade\_lossless\] The rate-distortion-cost region $\mathcal{R\mbox{\ensuremath{(0,0,\Gamma)}}}$ for the cascade source coding problem illustrated in Fig. \[fig:fig2\] with Hamming distortion metrics is given by the union of all rate pairs $(R_{1},R_{2})$ that satisfy the conditions \[eqn: R\_action\_cascade\_lossless\] $$\begin{aligned} R_{1} & \geq & I(X;A|Y)+H(X|A,Y)\label{eq:R1_lossless}\\ \ce{and}\mbox{ }R_{2} & \geq & I(X,Y;A)+H(X|A,Z),\label{eq:R2_lossless}\end{aligned}$$ where the mutual information terms are evaluated with respect to the joint pmf $$\begin{aligned} p(x,y,z,a)=p(x,y)p(a|x,y)p(z|y,a) & ,\label{eq:joint_lossless}\end{aligned}$$ for some pmf $p(a|x,y)$ such that $\ce{E}[\Lambda(A)]\leq\Gamma$. Cascade-Broadcast Source Coding with A Side Information Vending Machine ======================================================================= In this section, the cascade-broadcast source coding problem with a side information vending machine illustrated in Fig. \[fig:fig3\] is studied. At first, the rate-cost performance is characterized for the special case in which the reproductions at Node 2 and Node 3 are constrained to be lossless. Then, the lossy version of the problem is considered in Sec. \[sub:CR\], with an additional common reconstruction requirement in the sense of [@Steinberg] and assuming degradedness of the side information sequences. System Model\[sub:lossless\] ---------------------------- In this section, we describe the general system model for the cascade-broadcast source coding problem with a side information vending machine. We emphasize that, unlike the setup of Fig. \[fig:fig2\], here, the vending machine is at both Node 2 and Node 3. Moreover, we assume that an additional broadcast link of rate $R_{b}$ is available that is received by Node 2 and 3 to enable both Node 2 and Node 3 so as to take concerted actions in order to affect the side information sequences. We assume the action sequence taken by Node 2 and Node 3 to be a function of only the broadcast message $M_{b}$ sent over the broadcast link of rate $R_{b}$. The problem is defined by the pmfs $p_{X}(x)$, $p_{YZ|AX}(y,z|a,x)$ and discrete alphabets $\mathcal{X},\mathcal{Y},{\cal Z},\mathcal{A},$ $\mathcal{\hat{X}}_{1},\mathcal{\hat{X}}_{2},$ as follows. The source sequence $X^{n}$ with $X^{n}\in\mathcal{X}^{n}$ is i.i.d. with pmf $p_{X}(x)$. Node 1 measures sequence $X^{n}$ and encodes it into messages $M_{1}$ and $M_{b}$ of $nR_{1}$ and $nR_{b}$ bits, respectively, which are delivered to Node 2. Moreover, message $M_{b}$ is broadcast also to Node 3. Node 2 estimates a sequence $\hat{X}_{1}^{n}\in\mathcal{\hat{X}}_{1}^{n}$ and Node 3 estimates a sequence $\hat{X}_{2}^{n}\in\mathcal{\hat{X}}_{2}^{n}$. To this end, Node 2 receives messages $M_{1}$ and $M_{b}$ and, based only on the latter message, it selects an action sequence $A^{n},$ where $A^{n}\in\mathcal{A}^{n}.$ Node 2 maps messages $M_{1}$ and $M_{b}$, received from Node 1, and the locally available sequence $Y^{n}$ in a message $M_{2}$ of $nR_{2}$ bits, which is delivered to Node 3. Node 3 receives messages $M_{2}$ and $M_{b}$ and based only on the latter message, it selects an action sequence $A^{n},$ where $A^{n}\in\mathcal{A}^{n}.$ Given $A^{n}$ and $X^{n}$, the sequences $Y^{n}$ and $Z^{n}$ are distributed as $p(y^{n},z^{n}|a^{n},x^{n})=\prod_{i=1}^{n}p_{YZ|A,X}(y_{i},z_{i}|a_{i},x_{i})$. The cost of the action sequence is defined as in previous section. A formal description of the operations at encoder and decoder follows. \[def\_BC\_lossless\]An $(n,R_{1},R_{2},R_{b},D_{1},D_{2},\Gamma,\epsilon)$ code for the set-up of Fig. \[fig:fig4\] consists of two source encoders, namely $$\begin{aligned} \mathrm{g}_{1}\text{: \ensuremath{\mathcal{X}^{n}\rightarrow[1,2^{nR_{1}}]\times[1,2^{nR_{b}}],}} & & \text{ }\label{eq:encoder1}\end{aligned}$$ which maps the sequence $X^{n}$ into messages $M_{1}$ and $M_{b}$, respectively; $$\mathrm{g}_{2}\text{:}\text{ }[1,2^{nR_{1}}]\times[1,2^{nR_{b}}]\times\mathcal{Y}^{n}\rightarrow[1,2^{nR_{2}}]\label{encoder2_BC}$$ which maps the sequence $Y^{n}$ and messages $(M_{1},M_{b})$ into a message $M_{2};$ an “action function $$\mathrm{\ell}\text{: }[1,2^{nR_{b}}]\rightarrow\mathcal{A}^{n},\label{action_fun_BC}$$ which maps the message $M_{b}$ into an action sequence $A^{n};$ two decoders, namely $$\mathrm{h}_{1}\text{: }\text{ }[1,2^{nR_{1}}]\times[1,2^{nR_{b}}]\times\mathcal{Y}^{n}\rightarrow\mathcal{\hat{X}}_{1}^{n},\label{decoder1_BC}$$ which maps messages $M_{1}$ and $M_{b}$ and the measured sequence $Y^{n}$ into the estimated sequence $\hat{X}_{1}^{n};$ and $$\mathrm{h}_{2}\text{: }[1,2^{nR_{2}}]\times[1,2^{nR_{b}}]\times\mathcal{Z}^{n}\rightarrow\mathcal{\hat{X}}_{2}^{n},\label{decoder2_BC}$$ which maps the messages $M_{2}$ and $M_{b}$ into the the estimated sequence $\hat{X}_{2}^{n};$ such that the action cost constraint (\[action cost\]) and distortion constraint (\[dist const\]) are satisfied. Achievable rates $(R_{1},R_{2},R_{b})$ and rate-distortion-cost region are defined analogously to Definitions \[def\_ach\] and \[def\_reg\]. The rate–distortion–cost region for the system model described above is open even for the case without VM at Node 2 and Node 3 (see [@Vasudevan]). Hence, in the following subsections, we characterize the rate region for a few special cases. As in the previous section, subscripts are dropped from the pmf for simplicity of notation. Lossless Compression\[sub:RD\_BC\_lossless\] -------------------------------------------- In this section, a single-letter characterization of the rate-cost region $\mathcal{R\mbox{\ensuremath{(0,0,\Gamma)}}}$ is derived for the special case in which the distortion metrics are assumed to be Hamming and the distortion constraints are $D_{1}=0$ and $D_{2}=0$. \[prop:RD\_BC\_lossless\]The rate-cost region $\mathcal{R\mbox{\ensuremath{(0,0,\Gamma)}}}$ for the cascade-broadcast source coding problem illustrated in Fig. \[fig:fig3\] with Hamming distortion metrics is given by the union of all rate triples $(R_{1},R_{2},R_{b})$ that satisfy the conditions \[eqn: RD\_BC\_lossless\] $$\begin{aligned} R_{b} & \geq & I(X;A)\label{eq:Ra_lossless}\\ R_{1}+R_{b} & \geq & I(X;A)+H(X|A,Y)\label{eq:R1+Rb_lossless}\\ \ce{and}\mbox{ }R_{2}+R_{b} & \geq & I(X;A)+H(X|A,Z)\label{eq:R2+Rb_lossless}\end{aligned}$$ where the mutual information terms are evaluated with respect to the joint pmf $$\begin{aligned} p(x,y,z,a)=p(x,a)p(y,z|a,x),\label{eq:joint-BC_lossless}\end{aligned}$$ for some pmf $p(a|x)$ such that $\textrm{E}[\Lambda(A)]\leq\Gamma$. If $R_{1}=0$ and $R_{2}=0,$ the rate-cost region $\mathcal{R}(\Gamma)$ of Proposition \[prop:RD\_BC\_lossless\] reduces to the one derived in [@Weissman_multi Theorem 1]. The rate region (\[eqn: RD\_BC\_lossless\]) also describes the rate-distortion region under the more restrictive requirement of lossless reconstruction in the sense of the probabilities of error $\textrm{Pr}[X^{n}\neq\hat{X}_{j}^{n}]\leq\epsilon$ for $j=1,2$, as it follows from standard arguments (see [@Elgammal Sec. 3.6.4]). A similar conclusion applies for Corollary \[coro:cascade\_lossless\]. The converse proof for bound (\[eq:Ra\_lossless\]) follows immediately since $A^{n}$ is selected only as a function of message $M_{b}$. As for the other two bounds, namely (\[eq:R1+Rb\_lossless\])-(\[eq:R2+Rb\_lossless\]), the proof of the converse can be established following cut-set arguments and using the point-to-point result of [@Permuter]. For achievability, we use the code structure proposed in [@Permuter] along with rate splitting. Specifically, Node 1 first maps sequence $X^{n}$ into the action sequence $A^{n}$. This mapping requires a codebook of rate $I(X;A)$. This rate has to be conveyed over link $R_{b}$ by the definition of the problem and is thus received by both Node 2 and Node 3. Given the so obtained sequence $A^{n}$, communicating $X$ losslessly to Node 2 requires rate $H(X|A,Y)$. We split this rate into two rates $r_{1b}$ and $r_{1d}$, such that the message corresponding to the first rate is carried over the broadcast link of rate $R_{b}$ and the second on the direct link of rate $R_{1}$. Note that Node 2 can thus recover sequence $X$ losslessly. The rate $H(X|A,Z)$ which is required to send $X$ losslessly to Node 3, is then split into two parts, of rates $r_{2b}$ and $r_{2d}$. The message corresponding to the rate $r_{2b}$ is sent to Node 3 on the broadcast link of the rate $R_{b}$ by Node 1, while the message of rate $r_{2d}$ is sent by Node 2 to Node 3. This way, Node 1 and Node 2 cooperate to transmit $X$ to Node 3. As per the discussion above, the following inequalities have to be satisfied $$\begin{aligned} r_{2b}+r_{2d}+r_{1b} & \geq & H(X|A,Z),\\ r_{1b}+r_{1d} & \geq & H(X|A,Y),\\ R_{1} & \geq & r_{1d},\\ R_{2} & \geq & r_{2d},\\ \textrm{and }R_{b} & \geq & r_{1b}+r_{2b}+I(X;A),\end{aligned}$$ Applying Fourier-Motzkin elimination [@Elgammal Appendix C] to the inequalities above, the inequalities in (\[eqn: RD\_BC\_lossless\]) are obtained. Example: Switching-Dependent Side Information\[sub:lossless\_ex\] ----------------------------------------------------------------- We now consider the special case of the model in Fig. \[fig:fig3\] in which the actions $A\in{\cal A}=\{0,1,2,3\}$ acts a switch that decides whether Node 2, Node 3 or either node gets to observe a side information $W$. The side information $W$ is jointly distributed with source $X$ according to the joint pmf $p(x,w)$. Moreover, defining as $\textrm{e}$ an erasure symbol, the conditional pmf $p(y,z|x,a)$ is as follows: $Y=Z=\textrm{e}$ for $A=0$ (neither Node 2 nor Node 3 observes the side information $W$); $Y=W$ and $Z=\textrm{e}$ for $A=1$ (only Node 2 observes the side information $W$); $Y=\textrm{e}$ and $Z=W$ for $A=2$ (only Node 3 observes the side information $W$); and $Y=Z=W$ for $A=3$ (both nodes observe the side information $W$)[^3]. We also select the cost function such that $\Lambda(j)=\lambda_{j}$ for $j\in{\cal A}$. When $R_{1}=R_{2}=0$, this model reduces to the ones studied in [@Weissman_multi Sec. III]. The following is a consequence of Proposition 2. \[cor:ex\]For the setting of switching-dependent side information described above, the rate-cost region (\[eqn: RD\_BC\_lossless\]) is given by \[eqn: RD\_BC\_lossless\_ex\] $$\begin{aligned} R_{b} & \geq & I(X;A)\label{eq:Ra_lossless_ex}\\ R_{1}+R_{b} & \geq & H(X)-p_{1}I(X;W|A=1)-p_{3}I(X;W|A=3)\label{eq:R1+Rb_lossless_ex}\\ \ce{and}\mbox{ }R_{2}+R_{b} & \geq & H(X)-p_{2}I(X;W|A=2)-p_{3}I(X;W|A=3)\label{eq:R2+Rb_lossless_ex}\end{aligned}$$ where the mutual information terms are evaluated with respect to the joint pmf $$\begin{aligned} p(x,y,z,a)=p(x,a)p(y,z|a,x),\label{eq:joint-BC_lossless-1}\end{aligned}$$ for some pmf $p(a|x)$ such that $\sum_{j=0}^{3}p_{j}\lambda_{j}\leq\Gamma$, where we have denoted $p_{j}=\ce{Pr}[A=j]$ for $j\in{\cal A}$. The region (\[eqn: RD\_BC\_lossless\_ex\]) is obtained from the rate-cost region (\[eqn: RD\_BC\_lossless\]) by noting that in (\[eq:R1+Rb\_lossless\]) we have $I(X;A)+H(X|A,Y)=H(X)-I(X;Y|A)$ and similarly for (\[eq:R2+Rb\_lossless\]). In the following, we will elaborate upon two specific instances of the switching-dependent side information example. *Binary Symmetric Channel (BSC) between $X$ and $W$*: Let $(X,W)$ be binary and symmetric so that $p(x)=p(w)=1/2$ for $x,w\in\{0,1\}$ and $\Pr[X\neq W]=\delta$ for $\delta\in[0,1]$. Moreover, let $\lambda_{j}=\infty$ for $j=0,3$ and $\lambda_{j}=1$ otherwise. We set the action cost constraint to $\Gamma=1$. Note that, given this definition of $\Lambda(a)$, at each time, Node 1 can choose whether to provide the side information $W$ to Node 2 *or* to Node 3 with no further constraints. By symmetry, it can be seen that we can set the pmf $p(a|x)$ with $x\in\{0,1\}$ and $a\in\{1,2\}$ to be a BSC with transition probability $q$. This implies that $p_{1}=\textrm{Pr}[A=1]=q$ and $p_{2}=\textrm{Pr}[A=2]=1-q$. We now evaluate the inequality (\[eq:Ra\_lossless\_ex\]) as $R_{b}\geq0$; inequality (\[eq:R1+Rb\_lossless\_ex\]) as $R_{1}+R_{b}\geq1-p_{1}I(X;W|A=1)=1-qH(\delta)$; and similarly inequality (\[eq:R2+Rb\_lossless\]) as $R_{2}+R_{b}\geq1-(1-q)H(\delta).$ From these inequalities, it can be seen that, in order to trace the boundary of the rate-cost region, in general, one needs to consider all values of $q$ in the interval $[0,1]$. This ![The side information S-channel $p(w|x)$ used in the example of Sec. \[sub:lossless\_ex\].[]{data-label="fig:S-Channel"}](S-Channel) corresponds to appropriate time-sharing between providing side information to Node 2 (for a fraction of time $q$) and Node 3 (for the remaining fraction of time). Note that, as shown in [@Weissman_multi Sec. III], if $R_{1}=R_{2}=0$, it is optimal to set $q=\frac{1}{2}$, and thus equally share the side information between Node 2 and Node 3, in order to minimize the rate $R_{b}$. This difference is due to the fact that in the cascade model at hand, it can be advantageous to provide more side information to one of the two encoders depending on the desired trade-off between the rates $R_{1}$ and $R_{2}$ in the achievable rate-cost region. **S*-Channel between $X$ and $W$*: \[S-Channel\]We now consider the special case of Corollary \[cor:ex\] in which $(X,W)$ are jointly distributed so that $p(x)=1/2$ and $p(w|x)$ is the S-channel characterized by $p(0|0)=1-\delta$ and $p(1|1)=1$ (see Fig. \[fig:S-Channel\]). Moreover, we let $\lambda_{1}=1$, $\lambda_{2}=0$, $\lambda_{0}=\lambda_{3}=\infty$ as above, while the cost constraint is set to $\Gamma\leq1$. As discussed in [@Weissman_multi Sec. III] for this example with $R_{1}=R_{2}=0$, providing side information to Node 2 is more costly and thus should be done efficiently. In particular, given Fig. \[fig:S-Channel\], it is expected that biasing the choice $A=2$ when $X=1$ (i.e., providing side information to Node 2) may lead to some gain (see [@Weissman_multi]). Here we show that in the cascade model, this gain depends on the relative importance of rates $R_{1}$ and $R_{2}$. To this end, we set $p(a|x)$ as $p(1|0)=\alpha$ and $p(1|1)=\beta$ for $\alpha,\beta\in[0,1]$. We now evaluate the inequality (\[eq:Ra\_lossless\_ex\]) as $R_{b}\geq0$; inequality (\[eq:R1+Rb\_lossless\_ex\]) as $$\begin{aligned} R_{1}+R_{b} & \geq & 1-\Bigl(\frac{\alpha+\beta}{2}\Bigr)\Bigl(H\Bigl(\frac{(1-\delta)\alpha}{\alpha+\beta}\Bigr)-H(1-\delta)\frac{\alpha}{\alpha+\beta}\Bigr);\label{eq:ex_R1+Rb_ex_S}\end{aligned}$$ and inequality (\[eq:R2+Rb\_lossless\_ex\]) as $$\begin{aligned} R_{2}+R_{b} & \geq & 1-\Bigl(\frac{2-\alpha-\beta}{2}\Bigr)\Bigl(H\Bigl(\frac{(1-\delta)(1-\alpha)}{2-\alpha-\beta}\Bigr)-H(1-\delta)\frac{1-\alpha}{2-\alpha-\beta}\Bigr),\label{eq:ex_R2+Rb_ex_S}\end{aligned}$$ ![Difference between the weighted sum-rate $R_{1}+\eta R_{2}$ obtained with the greedy and with the optimal strategy as per Corollary \[cor:ex\] ($R_{b}=0.4$, $\delta=0.6$). []{data-label="fig:greedy-nongreedy"}](plot_greedyvsnongreedy) We now evaluate the minimum weighted sum-rate $R_{1}+\eta R_{2}$ obtained from (\[eq:ex\_R1+Rb\_ex\_S\])-(\[eq:ex\_R2+Rb\_ex\_S\]) for $R_{b}=0.4$, $\delta=0.6$ and both $\Gamma=0.1$ and $\Gamma=0.9$. Parameter $\eta\geq0$ rules on the relative importance of the two rates. For comparison, we also compute the performance attainable by imposing that the action $A$ be selected independent of $X$, which we refer to as the greedy approach [@Permuter]. Fig. \[fig:greedy-nongreedy\] plots the difference between the two weighted sum-rates $R_{1}+\eta R_{2}$ . It can be seen that, as $\eta$ decreases and thus minimizing rate $R_{1}$ to Node 2 becomes more important, one can achieve larger gains by choosing the action $A$ to be dependent on $X$. Moreover, this gain is more significant when the action cost budget $\Gamma$ allows Node 2 to collect a larger fraction of the side information samples. Lossy Compression with Common Reconstruction Constraint\[sub:CR\] ----------------------------------------------------------------- In this section, we turn to the problem of characterizing the rate-distortion-cost region ${\cal R}(D_{1},D_{2}$ $,\Gamma)$ for $D_{1},D_{2}>0$. In order to make the problem tractable [^4], we impose the degradedness condition $X-(A,Y)-Z$ (as in [@Weissman_multi]), which implies the factorization $$\begin{aligned} p(y,z|a,x) & = & p(y|a,x)p(z|y,a);\label{eq:degradedness}\end{aligned}$$ and that the reconstructions at Nodes 2 and 3 be reproducible by Node 1. As discussed, this latter condition is referred to as the CR constraint [@Steinberg]. Note that this constraint is automatically satisfied in the lossless case. To be more specific, an $(n,R_{1},R_{2},R_{b},D_{1},D_{2},\Gamma,\epsilon)$ code is defined per Definition \[def\_BC\_lossless\] with the difference that there are two additional functions for the encoder, namely \[eqn: en\_recons-1\] $$\begin{aligned} & \psi_{1}\textrm{: }\mathcal{X}^{n}\rightarrow\mathcal{\hat{X}}_{1}^{n}\\ \textrm{and } & \psi_{2}\textrm{: }\mathcal{X}^{n}\rightarrow\mathcal{\hat{X}}_{2}^{n},\end{aligned}$$ which map the source sequence into the estimated sequences at the encoder, namely $\psi_{1}(X^{n})$ and $\psi_{2}(X^{n})$, respectively; and the CR requirements are imposed, **i.e.**, \[eqn: CR\_req\] $$\begin{aligned} \textrm{Pr}\left[\psi_{1}(X^{n})\neq\textrm{h}_{1}(M_{1},M_{b},Y^{n})\right] & \leq & \epsilon\\ \textrm{and Pr}\left[\psi_{2}(X^{n})\neq\textrm{h}_{2}(M_{2},M_{b},Z^{n})\right] & \leq & \epsilon,\end{aligned}$$ so that the encoder’s estimates $\psi_{1}(\cdot$) and $\psi_{2}(\cdot)$ are equal to the decoders’ estimates (cf. (\[decoder1\_BC\])-(\[decoder2\_BC\])) with high probability. \[prop:RD\_action\_BC\]The rate-distortion region $\mathcal{R\mbox{\ensuremath{(D_{1},D_{2},\Gamma)}}}$ for the cascade-broadcast source coding problem illustrated in Fig. \[fig:fig3\] under the CR constraint and the degradedness condition (\[eq:degradedness\]) is given by the union of all rate triples $(R_{1},R_{2},R_{b})$ that satisfy the conditions \[eqn: RD\_action\_BC\] $$\begin{aligned} R_{b} & \geq & I(X;A)\label{eq:Ra}\\ R_{1}+R_{b} & \geq & I(X;A)+I(X;\hat{X}_{1},\hat{X}_{2}|A,Y)\\ R_{2}+R_{b} & \geq & I(X;A)+I(X;\hat{X}_{2}|A,Z)\\ \mbox{and}\mbox{ }R_{1}+R_{2}+R_{b} & \geq & I(X;A)+I(X;\hat{X}_{2}|A,Z)+I(X;\hat{X}_{1}|A,Y,\hat{X}_{2}),\label{eq:R1+R2+Ra}\end{aligned}$$ where the mutual information terms are evaluated with respect to the joint pmf $$\begin{aligned} p(x,y,z,a,\hat{x}_{1},\hat{x}_{2})=p(x)p(a|x)p(y|x,a)p(z|a,y)p(\hat{x}_{1},\hat{x}_{2}|x,a) & ,\label{eq:joint_BC}\end{aligned}$$ such that the inequalities \[eqn: action\_cascade\_BC\] $$\begin{aligned} \ce{E}[d_{j}(X,\hat{X}_{j})] & \leq & D_{j}\textrm{, }\;\mbox{for }j=1,2,\label{eq:-1}\\ \mbox{and}\textrm{ }\ce{E}[\Lambda(A)] & \leq & \Gamma,\end{aligned}$$ are satisfied. If either $R_{1}=0$ or $R_{b}=0$ and the side information $Y$ is independent of the action $A$ given $X$$,$ i.e., $\ p(y|a,x)=p(y|x),$ the rate-distortion region $\mathcal{R}(D_{1},D_{2},\Gamma)$ of Proposition \[prop:RD\_action\_BC\] reduces to the one derived in [@Ahmadi_CR Proposition 10]. The proof of the converse is provided in Appendix B. The achievability follows similar to Proposition \[prop:RD\_BC\_lossless\]. Specifically, Node 1 first maps sequence $X^{n}$ into the action sequence $A^{n}$. This mapping requires a codebook of rate $I(X;A)$. This rate has to be conveyed over link $R_{b}$ by the definition of the problem and is thus received by both Node 2 and Node 3. The source sequence $X^{n}$ is mapped into the estimate $\hat{X}_{2}^{n}$ for Node 3 using a codebook of rate $I(X;\hat{X}_{2}|A)$ for each sequence $A^{n}$. Communicating $\hat{X}_{2}^{n}$ to Node 2 requires rate $I(X;\hat{X}_{2}|A,Y)$ by the Wyner-Ziv theorem. We split this rate into two rates $r_{2b}$ and $r_{2d}$, such that the message corresponding to the first rate is carried over the broadcast link of rate $R_{b}$ and the second on the direct link of rate $R_{1}$. Note that Node 2 can thus recover sequence $\hat{X}_{2}^{n}$. Communicating $\hat{X}_{2}^{n}$ to Node 3 requires rate $I(X;\hat{X}_{2}|A,Z)$ by the Wyner-Ziv theorem. We split this rate into two rates $r_{0b}$ and $r_{0d}$. The message corresponding to the rate $r_{0b}$ is send to Node 3 on the broadcast link of the rate $R_{b}$ by Node 1, while the message of rate $r_{0d}$ is sent by Node 2 to Node 3. This way, Node 1 and Node 2 cooperate to transmit $\hat{X}_{2}$ to Node 3. Finally, the source sequence $X^{n}$ is mapped by Node 1 into the estimate $\hat{X}_{1}^{n}$ for Node 2 using a codebook of rate $I(X;\hat{X}_{1}|A,\hat{X}_{2})$ for each pair of sequences $(A^{n},\hat{X}_{2}^{n})$. Using the Wyner-Ziv coding, this rate is reduced to $I(X;\hat{X}_{1}|A,Y,\hat{X}_{2})$ and split into two rates $r_{1b}$ and $r_{1d}$, which are sent through links $R_{b}$ and $R_{1}$, respectively. As per discussion above, the following inequalities have to be satisfied $$\begin{aligned} r_{0b}+r_{0d}+r_{2b} & \geq & I(X;\hat{X}_{2}|A,Z),\\ r_{2b}+r_{2d} & \geq & I(X;\hat{X}_{2}|A,Y),\end{aligned}$$ $$\begin{aligned} r_{1b}+r_{1d} & \geq & I(X;\hat{X}_{1}|A,Y,\hat{X}_{2}),\\ R_{1} & \geq & r_{1d}+r_{2d},\\ R_{2} & \geq & r_{0d},\\ \textrm{and }R_{b} & \geq & r_{1b}+r_{2b}+r_{0b}+I(X;A),\end{aligned}$$ Applying Fourier-Motzkin elimination [@Elgammal Appendix C] to the inequalities above, the inequalities in (\[eqn: RD\_action\_BC\]) are obtained. Adaptive Actions\[sec:Adaptive-Actions\] ======================================== In this section, we assume that actions taken by the nodes are not only a function of the message $M_{2}$ for the model of Fig. \[fig:fig2\] or $M_{b}$ for the models of Fig. \[fig:fig3\] and Fig. \[fig:fig4\], respectively, but also a function of the past observed side information samples. Following [@Chiru], we refer to this case as the one with *adaptive actions*. Note that for the cascade-broadcast problem, we consider the model in Fig. \[fig:fig4\], which differs from the one in Fig. \[fig:fig3\] considered thus far in that the side information $Z$ is not available at Node 3. At this time, it appears to be problematic to define adaptive actions in the presence of two nodes that observe different side information sequences. For the cascade model in Fig. \[fig:fig2\], a $(n,R_{1},R_{2},D_{1},D_{2},\Gamma)$ code is defined per Definition \[def\_cascade\] with the difference that the action encoder (\[action\_fun\]) is modified to be $$\mathrm{\ell}\text{: }[1,2^{nR_{2}}]\times{\cal Z}^{i-1}\rightarrow\mathcal{A},\label{action_fun_adaptive}$$ which maps the message $M_{2}$ and the past observed decoder side information sequence $Z^{i-1}$ into the $i$th symbol of the action sequence $A_{i}$. Moreover, for the cascade-broadcast model of Fig. \[fig:fig4\], the “action function (\[action\_fun\_BC\]) in Definition \[def\_BC\_lossless\] is modified as $$\mathrm{\ell}\text{: }[1,2^{nR_{b}}]\times{\cal Y}^{i-1}\rightarrow\mathcal{A},\label{action_fun_adaptive-1}$$ which maps the message $M_{b}$ and the past observed decoder side information sequence $Y^{i-1}$ into the $i$th symbol of the action sequence $A_{i}$. \[prop:RD\_action\_cascade\_adaptive\]The rate-distortion-cost region $\mathcal{R\mbox{\ensuremath{(D_{1},D_{2},\Gamma)}}}$ for the cascade source coding problem illustrated in Fig. \[fig:fig2\] with adaptive action-dependent side information is given by the rate region described in Proposition \[prop:RD\_action\_cascade\]. \[prop:RD\_action\_BC\_adaptive\]The rate-distortion-cost region $\mathcal{R\mbox{\ensuremath{(D_{1},D_{2},\Gamma)}}}$ for the cascade-broadcast source coding problem under the CR illustrated in Fig. \[fig:fig4\] with adaptive action-dependent side information is given by the region described in Proposition \[prop:RD\_action\_BC\] by setting $Z=\emptyset$. The results above show that enabling adaptive actions does not increase the achievable rate-distortion-cost region. These results generalize the observations in [@Chiru] for the point-to-point setting, wherein a similar conclusion is drawn. To establish the propositions above, we only need to prove the converse. The proofs for Proposition \[prop:RD\_action\_cascade\_adaptive\] and Proposition \[prop:RD\_action\_BC\_adaptive\] are given in Appendix A and B, respectively. Concluding Remarks ================== In an increasing number of applications, communication networks are expected to be able to convey not only data, but also information about control for actuation over multiple hops. In this work, we have tackled the analysis of a baseline communication model with three nodes connected in a cascade with the possible presence of an additional broadcast link. We have characterized the optimal trade-off between rate, distortion and cost for actuation in a number of relevant cases of interest. In general, the results point to the advantages of leveraging a joint representation of data and control information in order to utilize in the most efficient way the available communication links. Specifically, in all the considered models, a layered coding strategy, possibly coupled with rate splitting, has been proved to be optimal. This strategy is such that the base layer has the double role of guiding the actions of the downstream nodes and of providing a coarse description of the source, similar to [@Permuter]. Moreover, it is shown that this base compression layer should be designed in a way that depends on the network topology and on the relative cost of activating the different links. ACKNOWLEDGMENTS =============== The work of O. Simeone is supported by the U.S. National Science Foundation under grant CCF-0914899, and the work of U. Mitra by ONR N00014-09-1-0700, NSF CCF-0917343 and DOT CA-26-7084-00. Appendix A: Converse Proof for Proposition \[prop:RD\_action\_cascade\] and \[prop:RD\_action\_cascade\_adaptive\] {#appendix-a-converse-proof-for-proposition-proprd_action_cascade-and-proprd_action_cascade_adaptive .unnumbered} ================================================================================================================== Here, we prove the converse part of Proposition \[prop:RD\_action\_cascade\_adaptive\]. Since the setting of Proposition \[prop:RD\_action\_cascade\] is more restrictive, as it does not allow for adaptive actions, the converse proof for Proposition \[prop:RD\_action\_cascade\] follows immediately. For any $(n,R_{1},R_{2},D_{1}+\epsilon,D_{2}+\epsilon,\Gamma+\epsilon)$ code, we have $$\begin{aligned} nR_{1} & \geq H(M_{1})\nonumber \\ & \geq H(M_{1}|Y^{n})\nonumber \\ & \stackrel{(a)}{=}I(M_{1};X^{n},Z^{n}|Y^{n})\nonumber \\ & =H(X^{n},Z^{n}|Y^{n})-H(X^{n},Z^{n}|M_{1},Y^{n})\nonumber \\ & =H(X^{n}|Y^{n})+H(Z^{n}|X^{n},Y^{n})-H(Z^{n}|Y^{n},M_{1})-H(X^{n}|Z^{n},Y^{n},M_{1})\nonumber \\ & \stackrel{(a,b)}{=}H(X^{n}|Y^{n})+H(Z^{n}|X^{n},Y^{n},M_{1},M_{2})-H(Z^{n}|Y^{n},M_{1},M_{2})-H(X^{n}|Z^{n},Y^{n},M_{1},M_{2})\nonumber \\ & \stackrel{(c)}{=}H(X^{n}|Y^{n})-H(X^{n}|Z^{n},Y^{n},M_{1},M_{2},A^{n},\hat{X}_{1}^{n})\nonumber \\ & +\sum_{i=1}^{n}H(Z_{i}|Z^{i-1},X^{n},Y^{n},M_{1},M_{2})-H(Z_{i}|Z^{i-1},Y^{n},M_{1},M_{2})\nonumber \\ & \stackrel{(c,d)}{\geq}\sum_{i=1}^{n}(H(X_{i}|Y_{i})-H(X_{i}|X^{i-1},Y^{i},M_{2},A^{i},Z^{n},\hat{X}_{1i}))\nonumber \\ & +\sum_{i=1}^{n}H(Z_{i}|Z^{i-1},X^{n},Y^{n},M_{1},M_{2},A_{i})-H(Z_{i}|Z^{i-1},Y^{n},M_{1},M_{2},A_{i})\nonumber \\ & \stackrel{(e)}{=}\sum_{i=1}^{n}I(X_{i};\hat{X}_{1i},A_{i},U_{i}|Y_{i})+H(Z_{i}|Y_{i},A_{i})-H(Z_{i}|Y_{i},A_{i})\nonumber \\ & =\sum_{i=1}^{n}I(X_{i};\hat{X}_{1i},A_{i},U_{i}|Y_{i}),\label{eq:conv1_end}\end{aligned}$$ where $(a)$ follows since $M_{1}$ is a function of $(X^{n},Y^{n})$; $(b)$ follows since $M_{2}$ is a function of $(M_{1},Y^{n})$; $(c)$ follows since $A_{i}$ is a function of $(M_{2},Z^{i-1})$ and since $\hat{X}_{1}^{n}$ is a function of $(M_{1},Y^{n})$; $(d)$ follows since conditioning decreases entropy and since $X^{n}$ and $Y^{n}$ are i.i.d.; and $(e)$ follows by defining $U_{i}=(M_{2},X^{i-1},Y^{i-1},A^{i-1},Z^{n\backslash i})$ and since $(Z^{i-1},X^{n},Y^{n\backslash i},M_{1},M_{2})\textrm{---}$ $(A_{i},Y_{i})\textrm{---}Z_{i}$ form a Markov chain by construction. We also have $$\begin{aligned} nR_{2} & \geq & H(M_{2})\\ & = & I(M_{2};X^{n},Y^{n},Z^{n})\end{aligned}$$ $$\begin{aligned} & = & H(X^{n},Y^{n},Z^{n})-H(X^{n},Y^{n},Z^{n}|M_{2})\nonumber \\ & = & H(X^{n},Y^{n})+H(Z^{n}|X^{n},Y^{n})-H(Z^{n}|M_{2})-H(X^{n},Y^{n}|M_{2},Z^{n})\nonumber \\ & = & \overset{n}{\underset{i=1}{\sum}}H(X_{i},Y_{i})+H(Z_{i}|Z^{i-1},X^{n},Y^{n})-H(Z_{i}|Z^{i-1},M_{2})\nonumber \\ & & -H(X_{i},Y_{i}|X^{i-1},Y^{i-1},M_{2},Z^{n})\nonumber \\ & \overset{(a)}{=} & \overset{n}{\underset{i=1}{\sum}}H(X_{i},Y_{i})\negmedspace+\negmedspace H(Z_{i}|Z^{i-1},X^{n},Y^{n},M_{2},A_{i})\negmedspace-\negmedspace H(Z_{i}|Z^{i-1},M_{2},A_{i})\negmedspace\nonumber \\ & & -\negmedspace H(X_{i},Y_{i}|X^{i-1},Y^{i-1},M_{2},Z^{n},A^{i})\nonumber \\ & \overset{(b)}{\geq} & \overset{n}{\underset{i=1}{\sum}}H(X_{i},Y_{i})+H(Z_{i}|X_{i},Y_{i},A_{i})-H(Z_{i}|A_{i})-H(X_{i},Y_{i}|U_{i},A_{i},Z_{i}),\label{eq:conv2_end}\end{aligned}$$ where (*a*) follows because $M_{2}$ is a function of $(M_{1},Y^{n})$ and thus of $(X^{n},Y^{n})$ and because $A^{i}$ is a function of $(M_{2},Z^{i-1})$ and ($b$) follows since conditioning decreases entropy, since the Markov chain relationship $Z_{i}\textrm{---}(X_{i},Y_{i},A_{i})\textrm{---}$ $(X^{n\backslash i},Y^{n\backslash i},M_{2})$ holds and by using the definition of $U_{i}$. Defining $Q$ to be a random variable uniformly distributed over $[1,n]$ and independent of all the other random variables and with $X\overset{\triangle}{=}X_{Q}$, $Y\overset{\triangle}{=}Y_{Q}$, $Z\overset{\triangle}{=}Z_{Q}$, $A\overset{\triangle}{=}A_{Q}$, $\hat{X}_{1}\overset{\triangle}{=}\hat{X}_{1Q}$, $\hat{X}_{2}\overset{\triangle}{=}\hat{X}_{2Q}$ and $U\overset{\triangle}{=}(U_{Q},Q),$ from (\[eq:conv1\_end\]) we have $$\begin{aligned} nR_{1} & \geq & I(X;\hat{X}_{1},A,U|Y,Q)\overset{(a)}{\geq}H(X|Y)-H(X|\hat{X}_{1},A,U,Y)=I(X;\hat{X}_{1},A,U|Y),\end{aligned}$$ where in ($a$) we have used the fact that $(X^{n},Y^{n})$ are i.i.d and conditioning reduces entropy. Moreover, from (\[eq:conv2\_end\]) we have $$\begin{aligned} nR_{2} & \geq & H(X,Y|Q)+H(Z|X,Y,A,Q)-H(Z|A,Q)-H(X,Y|U,A,Z,Q)\\ & \overset{(a)}{\geq} & H(XY)+H(Z|X,Y,A)-H(Z|A)-H(X,Y|U,A,Z)\\ & = & I(XY;U,A,Z)-I(Z;X,Y|A)\\ & = & I(XY;A)+I(X,Y;U|A,Z),\end{aligned}$$ where ($a$) follows since $(X^{n},Y^{n})$ are i.i.d, since conditioning decreases entropy, by the definition of $U$ and by the problem definition. We note that the defined random variables factorize as (\[eq:joint\]) since we have the Markov chain relationship $X$—$(A,Y)$—$Z$ by the problem definition and that $\hat{X}_{2}$ is a function $\textrm{f}(U,Z)$ of $U$ and $Z$ by the definition of $U$. Moreover, from the cost and distortion constraints (\[action cost\])-(\[dist const\]), we have \[eqn: action\_cascade\_const-1\] $$\begin{aligned} D_{j}+\epsilon & \geq\textrm{\ensuremath{\frac{1}{n}\sum_{i=1}^{n}}E}[d_{j}(X_{i},\hat{X}_{ji})]=\textrm{E}[d_{j}(X,\hat{X}_{j})],\textrm{ for }j=1,2,\label{eq:dist1-1}\\ \textrm{and }\Gamma+\epsilon & \geq\frac{1}{n}\sum_{i=1}^{n}\textrm{E}\left[\Lambda(A_{i})\right]=\textrm{E}\left[\Lambda(A)\right].\end{aligned}$$ To bound the cardinality of auxiliary random variable $U$, we fix $p(z|y,a)$ and factorize the joint pmf $p(x,y,z,a,u,\hat{x}_{1})$ as $$\begin{aligned} p(x,y,z,a,u,\hat{x}_{1}) & = & p(u)p(\hat{x}_{1},a,x,y|u)p(z|y,a).\end{aligned}$$ Therefore, for fixed $p(z|y,a)$, the quantities (\[eq:R1\])-(\[eq:action\_bound\]) can be expressed in terms of integrals given by $\int g_{j}(p(\hat{x}_{1},a,x,y|u))dF(u)$, for $j=1,...,|\mathcal{X}||\mathcal{Y}||\mathcal{A}|+3$, of functions $g_{j}(\cdot)$ that are continuous on the space of probabilities over alphabet $|\mathcal{X}|\text{\texttimes}|\mathcal{Y}|\text{\texttimes}|\mathcal{A}|\text{\texttimes}|\hat{\mathcal{X}}_{1}|$. Specifically, we have $g_{j}$ for $j=1,...,|\mathcal{X}||\mathcal{Y}||\mathcal{A}|-1$, given by the pmf $p(a,x,y)$ for all values of $x\in\mathcal{X}$, $y\in\mathcal{Y}$ and $a\in\mathcal{A}$, (except one), $g_{|\mathcal{X}||\mathcal{Y}||\mathcal{A}|}=H(X|A,Y,\hat{X}_{1},U=u)$, $g_{|\mathcal{X}||\mathcal{Y}||\mathcal{A}|+1}=H(X,Y|A,Z,U=u)$, and $g_{|\mathcal{X}||\mathcal{Y}||\mathcal{A}|+1+j}=\textrm{E}[d_{j}(X,\hat{X}_{j})|U=u],$ for $j=1,2$. The proof in concluded by invoking the Fenchel–Eggleston–Caratheodory theorem [@Elgammal Appendix C]. Appendix B: Proof of Proposition \[prop:RD\_action\_BC\] {#appendix-b-proof-of-proposition-proprd_action_bc .unnumbered} ======================================================== Here, we prove the converse parts of Proposition \[prop:RD\_action\_BC\] and Proposition \[prop:RD\_action\_BC\_adaptive\]. We start by proving Proposition \[prop:RD\_action\_BC\]. The proof of Proposition \[prop:RD\_action\_BC\_adaptive\] will follow by setting $Z=\emptyset$, and noting that in the proof below the action $A_{i}$ can be made to be a function of $Y^{i-1}$, in addition to being a function of $M_{b}$, without modifying any steps of the proof. By the CR requirements (\[eqn: CR\_req\]), we first observe that for any $(n,R_{1},R_{2},R_{b},D_{1}+\epsilon,D_{2}+\epsilon,\Gamma+\epsilon)$ code, we have the Fano inequalities \[eqn: Fano\] $$\begin{aligned} H(\psi_{1}(X^{n})|\textrm{h}_{1}(M_{1},M_{b},Y^{n})) & \leq & n\delta(\epsilon),\label{eq:Fano1}\\ \textrm{and }H(\psi_{2}(X^{n})|\textrm{h}_{2}(M_{2},M_{b},Z^{n})) & \leq & n\delta(\epsilon),\label{eq:Fano2}\end{aligned}$$ where $\delta(\epsilon)$ denotes any function such that $\delta(\epsilon)\rightarrow0$ if $\epsilon\rightarrow0$. Next, we have $$\begin{aligned} & nR_{b} & \geq H(M_{b})\\ & & \stackrel{(a)}{=}I(M_{b};X^{n},Y^{n})\end{aligned}$$ $$\begin{aligned} & =H(X^{n},Y^{n})-H(X^{n},Y^{n}|M_{b})\nonumber \\ & \stackrel{(a)}{=}H(X^{n})+H(Y^{n}|X^{n},M_{b})-H(X^{n},Y^{n}|M_{b})\nonumber \\ & \stackrel{(b)}{=}\sum_{i=1}^{n}H(X_{i})+H(Y_{i}|Y^{i-1},X^{n},M_{b},A_{i})-H(X_{i},Y_{i}|X^{i-1},Y^{i-1},M_{b},A_{i})\nonumber \\ & =\sum_{i=1}^{n}H(X_{i})+H(Y_{i}|Y^{i-1},X^{n},M_{b},A_{i})-H(X_{i}|X^{i-1},Y^{i-1},M_{b},A_{i})\nonumber \\ & -H(Y_{i}|X^{i},Y^{i-1},M_{b},A_{i})\nonumber \\ & \stackrel{(c)}{=}\sum_{i=1}^{n}H(X_{i})+H(Y_{i}|X_{i},A_{i})-H(X_{i}|X^{i-1},Y^{i-1},M_{b},A_{i})-H(Y_{i}|X_{i},A_{i})\nonumber \\ & \stackrel{(d)}{\geq}\sum_{i=1}^{n}H(X_{i})-H(X_{i}|A_{i}),\label{eq:conv_Rb}\end{aligned}$$ where $(a)$ follows since $M_{b}$ is a function of $X^{n}$; $(b)$ follows since $A_{i}$ is a function of $M_{b}$ and since $X^{n}$ is i.i.d.; $(c)$ follows since $(Y^{i-1},X^{n\backslash i},M_{b})\textrm{---}(A_{i},X_{i})\textrm{---}Y_{i}$ forms a Markov chain by problem definition; and $(d)$ follows conditioning reduces entropy. In the following, for simplicity of notation, we write $\textrm{h}_{1},\textrm{h}_{2},\psi_{1},\psi_{2}$ for the values of corresponding functions in Sec. \[sub:CR\]. Next, We can also write $$\begin{aligned} n(R_{1}+R_{b}) & \geq & H(M_{1},M_{b})\\ & \overset{(a)}{=} & I(M_{1},M_{b};X^{n},Y^{n},Z^{n})\\ & = & H(X^{n},Y^{n},Z^{n})-H(X^{n},Y^{n},Z^{n}|M_{1},M_{b})\\ & = & H(X^{n})+H(Y^{n},Z^{n}|X^{n})-H(Y^{n},Z^{n}|M_{1},M_{b})-H(X^{n}|Y^{n},Z^{n},M_{1},M_{b})\\ & \overset{(b)}{=} & H(X^{n})+H(Y^{n},Z^{n}|X^{n},M_{b})-H(Y^{n}|M_{1},M_{b})\\ & & -H(Z^{n}|M_{1},M_{b},Y^{n},A^{n})-H(X^{n}|Y^{n},Z^{n},M_{1},M_{b},M_{2},A^{n})\\ & \overset{(b,c)}{=} & \overset{n}{\underset{i=1}{\sum}}H(X_{i})+H(Y_{i},Z_{i}|X_{i},A_{i})-H(Y_{i}|Y^{i-1},M_{1},M_{b},A_{i})\\ & & -H(Z_{i}|Z^{i-1},M_{1},M_{b},Y^{n},A^{n})-H(X_{i}|X^{i-1},Y^{n},Z^{n},M_{1},M_{b},A^{n},M_{2},\textrm{h}_{1},\textrm{h}_{2})\\ & \overset{(d)}{\geq} & \overset{n}{\underset{i=1}{\sum}}H(X_{i})\negmedspace+\negmedspace H(Y_{i}|X_{i},A_{i})+\negmedspace H(Z_{i}|Y_{i},A_{i})-H(Y_{i}|A_{i})-H(Z_{i}|Y_{i},A_{i})\\ & & -H(X_{i}|Y_{i},A_{i},\textrm{h}_{1},\textrm{h}_{2})\negmedspace\end{aligned}$$ $$\begin{aligned} & = & \overset{n}{\underset{i=1}{\sum}}I(X_{i};Y_{i},A_{i},\textrm{h}_{1},\textrm{h}_{2})-I(Y_{i};X_{i}|A_{i})\nonumber \\ & = & \overset{n}{\underset{i=1}{\sum}}I(X_{i};Y_{i},A_{i},\textrm{h}_{1},\textrm{h}_{2},\psi_{1},\psi_{2})-I(X_{i};\psi_{1},\psi_{2}|Y_{i},A_{i},\textrm{h}_{1},\textrm{h}_{2})-I(Y_{i};X_{i}|A_{i})\nonumber \\ & \overset{(e)}{\geq} & \overset{n}{\underset{i=1}{\sum}}I(X_{i};Y_{i},A_{i},\psi_{1},\psi_{2})-H(\psi_{1},\psi_{2}|Y_{i},A_{i},\textrm{h}_{1},\textrm{h}_{2})\nonumber \\ & & +H(\psi_{1},\psi_{2}|Y_{i},A_{i},\textrm{h}_{1},\textrm{h}_{2},X_{i})-I(Y_{i};X_{i}|A_{i})\nonumber \\ & \overset{(f)}{\geq} & \overset{n}{\underset{i=1}{\sum}}I(X_{i};Y_{i},A_{i},\psi_{1},\psi_{2})-I(Y_{i};X_{i}|A_{i})+n\delta(\epsilon)\nonumber \\ & = & \overset{n}{\underset{i=1}{\sum}}I(X_{i};A_{i})+I(X_{i};\psi_{1},\psi_{2}|Y_{i},A_{i})+n\delta(\epsilon),\label{eq:conv_R1+Rb}\end{aligned}$$ where ($a$) follows because $(M_{1},M_{b})$ is a function of $X^{n}$; ($b$) follows because $M_{b}$ is a function of $X^{n}$, $A^{n}$ is a function of $M_{b}$ and $M_{2}$ is a function of $(M_{1},M_{b},Y^{n})$; ($c$) follows since $H(Y^{n},Z^{n}|X^{n},M_{b})=\sum_{i=1}^{n}H(Y_{i},Z_{i}|$ $Y^{i-1},Z^{i-1},X^{n},M_{b},A_{i})=\sum_{i=1}^{n}H(Y_{i},Z_{i}|X_{i},A_{i})$ and since $\textrm{h}_{1}$ and $\textrm{h}_{2}$ are functions of $(M_{1},M_{b},Y^{n})$ and $(M_{2},M_{b},Z^{n})$, respectively and because $(Y_{i},Z_{i})\textrm{---}(X_{i},A_{i})\textrm{---}$ $(X^{n\backslash i},Y^{i-1},Z^{i-1},M_{b})$ forms a Markov chain; ($d$) follows since conditioning reduces entropy, since side information VM follows $p(y^{n},z^{n}|a^{n},x^{n})$$=\prod_{i=1}^{n}p_{Y|A,X}(y_{i}|a_{i},x_{i})$ $p_{Z|A,Y}(z_{i}|a_{i},y_{i})$ from (\[eq:degradedness\]) and because $Z_{i}\textrm{---}(Y_{i},A_{i})\textrm{---}$ $(Y^{n\backslash i},Z^{i-1},M_{1},M_{b})$ forms a Markov chain; ($e$) follows by the chain rule for mutual information and the fact that mutual information is non-negative; and ($f$) follows by the Fano inequality (\[eqn: Fano\]) and because entropy is non-negative. We can also write $$\begin{aligned} n(R_{2}+R_{b}) & \geq & H(M_{2},M_{b})\\ & \stackrel{(a)}{=} & I(M_{2},M_{b};X^{n},Y^{n},Z^{n})\\ & = & H(X^{n},Y^{n},Z^{n})-H(X^{n},Y^{n},Z^{n}|M_{2},M_{b})\\ & \stackrel{(a)}{=} & H(X^{n})+H(Y^{n},Z^{n}|X^{n},M_{b})-H(Z^{n}|M_{2},M_{b})-H(X^{n},Y^{n}|Z^{n},M_{2},M_{b})\\ & \stackrel{(b)}{=} & \sum_{i=1}^{n}H(X_{i})+H(Y_{i},Z_{i}|Y^{i-1},Z^{i-1},X^{n},M_{b},A_{i})-H(Z_{i}|Z^{i-1},M_{2},M_{b},A_{i})\\ & - & H(X_{i},Y_{i}|X^{i-1},Y^{i-1},M_{2},M_{b},Z^{n},A_{i})\\ & = & \sum_{i=1}^{n}H(X_{i},Y_{i})-H(Y_{i}|X_{i})+H(Y_{i},Z_{i}|Y^{i-1},Z^{i-1},X^{n},M_{b},A_{i})\\ & - & H(Z_{i}|Z^{i-1},M_{2},M_{b},A_{i})-H(X_{i},Y_{i}|X^{i-1},Y^{i-1},M_{2},M_{b},Z^{n},A_{i})\end{aligned}$$ $$\begin{aligned} & \stackrel{(c)}{=}\sum_{i=1}^{n}H(X_{i},Y_{i})-H(Y_{i}|X_{i})+H(Y_{i}|X_{i},A_{i})+H(Z_{i}|A_{i},Y_{i},X_{i})\nonumber \\ & -H(Z_{i}|Z^{i-1},M_{2},M_{b},A_{i})-H(X_{i},Y_{i}|X^{i-1},Y^{i-1},M_{2},M_{b},Z^{n},A_{i})\nonumber \\ & \stackrel{(d)}{=}\sum_{i=1}^{n}H(X_{i},Y_{i})-I(Y_{i};A_{i}|X_{i})+H(Z_{i}|A_{i},Y_{i},X_{i})-H(Z_{i}|Z^{i-1},M_{2},M_{b},A_{i})\nonumber \\ & -H(X_{i},Y_{i}|X^{i-1},Y^{i-1},M_{2},M_{b},\textrm{h}_{2},Z^{n},A_{i})\nonumber \\ & \stackrel{(e)}{\geq}\sum_{i=1}^{n}H(X_{i},Y_{i})+I(X_{i};A_{i})-I(Y_{i},X_{i};A_{i})+H(Z_{i}|A_{i},Y_{i},X_{i})\nonumber \\ & -H(Z_{i}|A_{i})-H(X_{i},Y_{i}|\textrm{h}_{2},A_{i},Z_{i})\nonumber \\ & =\sum_{i=1}^{n}I(X_{i},Y_{i};\textrm{h}_{2},A_{i},Z_{i},\psi_{2i})-I(X_{i},Y_{i};\psi_{2i}|\textrm{h}_{2},A_{i},Z_{i})+I(X_{i};A_{i})\nonumber \\ & -I(Y_{i},X_{i};A_{i})-I(X_{i},Y_{i};Z_{i}|A_{i})\nonumber \\ & \geq\sum_{i=1}^{n}I(X_{i},Y_{i};A_{i},Z_{i},\psi_{2i})-H(\psi_{2i}|\textrm{h}_{2},A_{i},Z_{i})+H(\psi_{2i}|\textrm{h}_{2},A_{i},X_{i},Y_{i},Z_{i})\nonumber \\ & +I(X_{i};A_{i})-I(X_{i},Y_{i};Z_{i},A_{i})\nonumber \\ & \stackrel{(f)}{\geq}\sum_{i=1}^{n}I(X_{i};A_{i})+I(X_{i},Y_{i};\psi_{2i}|A_{i},Z_{i})+n\delta(\epsilon),\label{eq:conv_R2+Rb}\end{aligned}$$ where $(a)$ follows since $M_{b}$ is a function of $X^{n}$ and because $M_{2}$ is a function of $(M_{1},M_{b},Y^{n})$ and thus of $(X^{n},Y^{n})$; $(b)$ follows since $A_{i}$ is a function of $M_{b}$ and since $X^{n}$ is i.i.d.; $(c)$ follows since $(Y_{i},Z_{i})\textrm{---}(X_{i},A_{i})\textrm{---}$ $(X^{n\backslash i},Y^{i-1},Z^{i-1},M_{b})$ forms a Markov chain and since $p(y^{n},z^{n}|a^{n},x^{n})$ $=\prod_{i=1}^{n}p_{Y|A,X}(y_{i}|a_{i},x_{i})$$p_{Z|A,Y}(z_{i}|a_{i},y_{i})$; $(d)$ follows since $\textrm{h}_{2}$ is a function of $(M_{2},M_{b},Z^{n})$; $(e)$ follows since conditioning reduces entropy; and $(f)$ follows since entropy is non-negative and using the Fanos inequality. Moreover, with the definition $M=(M_{1},M_{2},M_{b})$, we have the chain of inequalities $$\begin{aligned} n(R_{1}+R_{2}+R_{b}) & \geq H(M)\\ & \stackrel{(a)}{=}I(M;X^{n},Y^{n},Z^{n})\\ & =H(X^{n},Y^{n},Z^{n})-H(X^{n},Y^{n},Z^{n}|M)\\ & \stackrel{(a)}{=}H(X^{n})+H(Y^{n},Z^{n}|X^{n},M_{b})-H(X^{n},Y^{n},Z^{n}|M)\end{aligned}$$ $$\begin{aligned} & =I(X^{n};A^{n})+H(Y^{n},Z^{n}|X^{n},M_{b})-H(Y^{n},Z^{n}|M)\nonumber \\ & -H(X^{n}|Y^{n},Z^{n},M)+H(X^{n}|A^{n})\nonumber \\ & =I(X^{n};A^{n})+H(Y^{n},Z^{n}|X^{n},M_{b})-H(Y^{n},Z^{n}|M)+I(X^{n};Y^{n},Z^{n},M|A^{n})\nonumber \\ & =I(X^{n};A^{n})+I(M;X^{n}|Y^{n},A^{n},Z^{n})+H(Y^{n},Z^{n}|X^{n},M_{b})\nonumber \\ & -H(Y^{n},Z^{n}|M)+I(X^{n};Y^{n},Z^{n}|A^{n})\nonumber \\ & \stackrel{(b)}{=}H(X^{n})-H(X^{n}|A^{n})+H(X^{n}|Y^{n},A^{n},Z^{n})-H(X^{n}|Y^{n},A^{n},Z^{n},M)\nonumber \\ & -H(Y^{n},Z^{n}|M)+H(Y^{n},Z^{n}|A^{n})\nonumber \\ & =H(X^{n})-H(X^{n}|A^{n})+H(X^{n},Y^{n},Z^{n}|A^{n})-H(X^{n}|Y^{n},A^{n},Z^{n},M)\nonumber \\ & -H(Y^{n},Z^{n}|M)\nonumber \\ & =H(X^{n})+H(Y^{n},Z^{n}|A^{n},X^{n})-H(X^{n}|Y^{n},A^{n},Z^{n},M)-H(Y^{n},Z^{n}|M)\nonumber \\ & \stackrel{(c)}{=}\sum_{i=1}^{n}H(X_{i})+H(Y_{i}|A_{i},X_{i})+H(Z_{i}|A_{i},Y_{i})-H(X_{i}|X^{i-1},Y^{n},A^{n},Z^{n},M)\nonumber \\ & -H(Z_{i}|Z^{i-1},M,A_{i})-H(Y_{i}|Y^{i-1},Z^{n},M,A_{i})\nonumber \\ & \stackrel{(d)}{=}\sum_{i=1}^{n}H(X_{i})\negmedspace+\negmedspace H(Y_{i}|A_{i},X_{i})\negmedspace+\negmedspace H(Z_{i}|A_{i},Y_{i})\negmedspace-\negmedspace H(X_{i}|X^{i-1},\negmedspace Y^{n},A^{n},Z^{n},M,\textrm{h}_{1},\textrm{h}_{2})\nonumber \\ & -H(Z_{i}|Z^{i-1},M,A_{i})-H(Y_{i}|Y^{i-1},Z^{n},M,A_{i},\textrm{h}_{2})\nonumber \\ & \geq\sum_{i=1}^{n}H(X_{i})+H(Y_{i}|A_{i},X_{i})+H(Z_{i}|A_{i},Y_{i})-H(X_{i}|Y_{i},A_{i},\textrm{h}_{1},\textrm{h}_{2})\nonumber \\ & -H(Z_{i}|A_{i})-H(Y_{i}|Z_{i},A_{i},\textrm{h}_{2})\nonumber \\ & \stackrel{(e)}{\geq}I(X_{i};A_{i},Y_{i},\psi_{1},\psi_{2})+H(Y_{i}|A_{i},X_{i})+H(Z_{i}|A_{i},Y_{i})\nonumber \\ & -H(Z_{i}|A_{i})-H(Y_{i}|Z_{i},A_{i},\psi_{2})-n\delta(\epsilon),\label{eq:conv_end-1}\end{aligned}$$ where $(a)$ follows since $(M_{1},M_{b})$ is a function of $X^{n}$ and $M_{2}$ is a function of $(M_{1},M_{b},Y^{n})$; $(b)$ follows since $H(Y^{n},Z^{n}|X^{n},M_{b})=\sum_{i=1}^{n}H(Y_{i},Z_{i}|$ $Y^{i-1},Z^{i-1},X^{n},M_{b},A_{i})=\sum_{i=1}^{n}H(Y_{i},Z_{i}|$ $X_{i},A_{i})=H(Y^{n},Z^{n}|X^{n},A^{n})$; $(c)$ follows since $A_{i}$ is a function of $M_{b}$; $(d)$ follows since $\textrm{h}_{1},\textrm{h}_{2}$ are functions of $(M,Y^{n})$ and $(M,Z^{n})$, respectively; and $(e)$ follows since entropy is non-negative and by Fano’s inequality. Next, from (\[eq:conv\_end-1\]) we have **$$\begin{aligned} n(R_{1}+R_{2}+R_{b}) & \geq I(X_{i};A_{i},Y_{i},\psi_{1},\psi_{2})+H(Y_{i}|A_{i},X_{i})+H(Z_{i}|A_{i},Y_{i})-H(Z_{i}|A_{i})\nonumber \\ & -H(Y_{i},Z_{i}|A_{i},\psi_{2})+H(Z_{i}|A_{i},\psi_{2})-n\delta(\epsilon)\nonumber \\ & =I(X_{i};A_{i},Y_{i},\psi_{1},\psi_{2})+H(Y_{i}|A_{i},X_{i})-H(Z_{i}|A_{i})-H(Y_{i}|A_{i},\psi_{2})\nonumber \\ & +H(Z_{i}|A_{i},\psi_{2})-n\delta(\epsilon)\nonumber \\ & =I(X_{i};A_{i},Y_{i},\psi_{1},\psi_{2})-I(X_{i};Y_{i}|A_{i},\psi_{2})-I(Z_{i};\psi_{2}|A_{i})-n\delta(\epsilon)\nonumber \\ & \stackrel{(a)}{=}I(X_{i};A_{i},Y_{i},\psi_{1},\psi_{2})-I(X_{i};Y_{i}|A_{i},\psi_{2})-I(Y_{i};A_{i}|X_{i})-I(Z_{i};Y_{i}|A_{i})\nonumber \\ & +I(Y_{i};A_{i},\psi_{2}|X_{i})+I(Z_{i};Y_{i}|\psi_{2},A_{i})-n\delta(\epsilon)\nonumber \\ & \stackrel{(b)}{=}I(X_{i};A_{i},Y_{i},\psi_{1},\psi_{2})-I(X_{i};Y_{i}|A_{i},\psi_{2})+I(X_{i};A_{i})-I(Y_{i},X_{i};A_{i})\nonumber \\ & -I(Z_{i};X_{i},Y_{i}|A_{i})\negmedspace+\negmedspace I(X_{i},Y_{i};A_{i},\psi_{2})\negmedspace+\negmedspace I(Z_{i};X_{i},Y_{i}|\psi_{2},A_{i})-\negmedspace I(X_{i};A_{i},\psi_{2})\negmedspace-\negmedspace n\delta(\epsilon)\nonumber \\ & =I(X_{i};A_{i})+I(X_{i};A_{i},Y_{i},\psi_{1},\psi_{2})+I(X_{i},Y_{i};A_{i},\psi_{2},Z_{i})-I(A_{i},Z_{i};X_{i},Y_{i})\nonumber \\ & -I(X_{i};Y_{i},A_{i},\psi_{2})-n\delta(\epsilon)\nonumber \\ & =I(X_{i};A_{i})\negmedspace+\negmedspace I(X_{i};A_{i},Y_{i},\psi_{1},\psi_{2})\negmedspace+\negmedspace I(X_{i},Y_{i};\psi_{2}|A_{i},Z_{i})\negmedspace-\negmedspace I(X_{i};Y_{i},A_{i},\psi_{2})\negmedspace-\negmedspace n\delta(\epsilon)\nonumber \\ & =I(X_{i};A_{i})+I(X_{i},Y_{i};\psi_{2}|A_{i},Z_{i})+I(X_{i};\psi_{1}|A_{i},Y_{i},\psi_{2})-n\delta(\epsilon),\label{eq:conv_end}\end{aligned}$$** where $(a)$ is true since $$\begin{aligned} & & I(Y_{i};A_{i}|X_{i})+I(Z_{i};Y_{i}|A_{i})-I(Y_{i};A_{i},\psi_{2}|X_{i})-I(Z_{i};Y_{i}|\psi_{2},A_{i})\\ & & =H(Y_{i}|X_{i})-H(Y_{i}|X_{i},A_{i})+H(Z_{i}|A_{i})-H(Z_{i}|A_{i},Y_{i})-H(Y_{i}|X_{i})+H(Y_{i}|X_{i},A_{i})\\ & & -H(Z_{i}|\psi_{2},A_{i})+H(Z_{i}|A_{i},Y_{i})\\ & & =H(Z_{i}|A_{i})-H(Z_{i}|\psi_{2},A_{i});\end{aligned}$$ $(b)$ follows because $I(Z_{i};X_{i},Y_{i}|A_{i})=I(Z_{i};Y_{i}|A_{i})$ and $I(Z_{i};X_{i},Y_{i}|A_{i},\psi_{2})=I(Z_{i};Y_{i}|A_{i},\psi_{2})$. Next, define $\hat{X}_{ji}=\psi_{ji}(X^{n})$ for $j=1,2$ and $i=1,2,...,n$ and let $Q$ be a random variable uniformly distributed over $[1,n]$ and independent of all the other random variables and with $X\overset{\triangle}{=}X_{Q}$, $Y\overset{\triangle}{=}Y_{Q}$, $A\overset{\triangle}{=}A_{Q}$, from (\[eq:conv\_Rb\]), we have $$\begin{aligned} nR_{b} & \overset{}{\geq}H(X|Q)-H(X|A,Q)\overset{(a)}{\geq}H(X)-H(X|A)=I(X;A),\end{aligned}$$ where ($a$) follows since $X^{n}$ is i.i.d. and since conditioning decreases entropy. Next, from (\[eq:conv\_R1+Rb\]), we have$ $ $$\begin{aligned} n(R_{1}+R_{b}) & \overset{}{\geq} & I(X;A|Q)+I(X;\hat{X}_{1},\hat{X}_{2}|Y,A,Q)\\ & \overset{(a)}{\geq} & I(X;A)+I(X;\hat{X}_{1},\hat{X}_{2}|Y,A),\end{aligned}$$ where ($a$) follows since $X^{n}$ is i.i.d., since conditioning decreases entropy and by the problem definition. From (\[eq:conv\_R2+Rb\]), we also have $$\begin{aligned} n(R_{2}+R_{b}) & \overset{}{\geq} & I(X;A|Q)+I(X,Y;\hat{X}_{2}|A,Z,Q)\\ & \overset{(a)}{\geq} & I(X;A)+H(X,Y|A,Z,Q)-H(X,Y|A,Z,\hat{X}_{2})\\ & \overset{(b)}{=} & I(X;A)+H(Y|A,Z)+H(X|A,Y,Z)-H(X,Y|A,Z,\hat{X}_{2})\\ & = & I(X;A)+I(X,Y;\hat{X}_{2}|A,Z)\\ & \geq & I(X;A)+I(X;\hat{X}_{2}|A,Z)\end{aligned}$$ where ($a$) follows since $X^{n}$ is i.i.d. and by conditioning reduces entropy; and ($b$) follows by the problem definition. Finally, from (\[eq:conv\_end\]), we have $$\begin{aligned} n(R_{1}+R_{2}+R_{b}) & \overset{}{\geq}I(X,A|Q)+I(X,Y;\hat{X}_{2}|A,Z,Q)+I(X;\hat{X}_{1}|A,Y,\hat{X}_{2},Q)\nonumber \\ & \overset{(a)}{\geq}I(X,A)+H(X,Y|A,Z,Q)-H(X,Y|A,Z,\hat{X}_{2})+I(X;\hat{X}_{1}|A,Y,\hat{X}_{2})\nonumber \\ & \overset{(b)}{=}I(X;A)\negmedspace+\negmedspace H(Y|A,Z)\negmedspace+\negmedspace H(X|A,Y,Z)\negmedspace-\negmedspace H(X,Y|A,Z,\hat{X}_{2})\negmedspace+\negmedspace I(X;\hat{X}_{1}|A,Y,\hat{X}_{2})\nonumber \\ & =I(X;A)+I(X,Y;\hat{X}_{2}|A,Z)+I(X;\hat{X}_{1}|A,Y,\hat{X}_{2})\nonumber \\ & \geq I(X;A)+I(X;\hat{X}_{2}|A,Z)+I(X;\hat{X}_{1}|A,Y,\hat{X}_{2})\end{aligned}$$ where ($a$) follows since $X^{n}$ is i.i.d, since conditioning decreases entropy, and by the problem definition; and ($b$) follows by the problem definition. From cost constraint (\[action cost\]), we have $$\begin{aligned} \Gamma+\epsilon & \geq\frac{1}{n}\sum_{i=1}^{n}\textrm{E}\left[\Lambda(A_{i})\right]=\textrm{E}\left[\Lambda(A)\right].\end{aligned}$$ Moreover, let $\mathcal{B}$ be the event $\mathcal{B}=\{\left(\psi_{1}\textrm{(}X^{n})\neq h_{1}(M_{1},M_{b},Y^{n})\right)\wedge\left(\psi_{2}\textrm{(}X^{n})\neq h_{2}(M_{2},M_{b})\right)\}$. Using the CR requirement (\[eqn: CR\_req\]), we have $\textrm{Pr}(\mathcal{B})\leq\epsilon$. For $j=1,2$, we have $$\begin{aligned} \textrm{E}\left[d(X_{j},\hat{X}_{j})\right] & = & \frac{1}{n}\sum_{i=1}^{n}\textrm{E}\left[d(X_{ji},\hat{X}_{ji})\right]\nonumber \\ & = & \frac{1}{n}\sum_{i=1}^{n}\textrm{E}\negmedspace\left[d(X_{ji},\hat{X}_{ji})\Big|\mathcal{B}\right]\negmedspace\textrm{Pr}(\mathcal{B})\negmedspace+\negmedspace\frac{1}{n}\sum_{i=1}^{n}\textrm{E}\negmedspace\left[d(X_{ji},\hat{X}_{ji})\Big|\mathcal{B}^{c}\right]\negmedspace\textrm{Pr}(\mathcal{B}^{c})\nonumber \\ & \negmedspace\negmedspace\overset{(a)}{\leq}\negmedspace\negmedspace & \frac{1}{n}\sum_{i=1}^{n}\textrm{E}\left[d(X_{ji},\hat{X}_{ji})\Big|\mathcal{B}^{c}\right]\textrm{Pr}(\mathcal{B}^{c})+\epsilon D_{max}\nonumber \\ & \negmedspace\negmedspace\overset{(b)}{\leq}\negmedspace\negmedspace & \frac{1}{n}\sum_{i=1}^{n}\textrm{E}\left[d(X_{ji},h_{ji})\right]+\epsilon D_{max}\nonumber \\ & \negmedspace\negmedspace\overset{(c)}{\leq}\negmedspace\negmedspace & D_{j}+\epsilon D_{max},\end{aligned}$$ where ($a$) follows using the fact that $\textrm{Pr}(\mathcal{B})\leq\epsilon$ and that the distortion is upper bounded by $D_{max}$; ($b$) follows by the definition of $\hat{X}_{ji}$ and $\mathcal{B}$; and ($c$) follows by (\[dist const\]). 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V. Poor, “Heegard-Berger and cascade source coding problems with common reconstruction constraints, arXiv:1112.1762v3, 2011. Y. Steinberg, “Coding and common reconstruction, *IEEE Trans. Inform. Theory*, vol. 55, no. 11, 2009. T. Weissman and A. El Gamal, “Source coding with limited-look-ahead side information at the decoder, *IEEE Trans. Inf. Theory*, vol. 52, no. 12, pp. 5218–5239, Dec. 2006. C. Choudhuri and U. Mitra, “How useful is adaptive action?, Submitted to Globecom 2012. A. El Gamal and Y. Kim, *Network Information Theory*, Cambridge University Press, Dec 2011. [^1]: B. Ahmadi and O. Simeone are with the CWCSPR, New Jersey Institute of Technology, Newark, NJ 07102 USA (e-mail: {behzad.ahmadi,osvaldo.simeone}@njit.edu). [^2]: C. Choudhuri and U. Mitra are with Ming Hsieh Dept. of Electrical Engineering, University of Southern California, Los Angeles, CA, 90089 USA (e-mail: {cchoudhu,ubli}@usc.edu). [^3]: This implies that $p(y,z|x,a)=\underset{w}{\sum}p(w|x)\delta(y-w)\delta(z-\ce{e})$ for $a=1$ and similarly for other values of $a$. [^4]: As noted earlier, the problem is open even in the case with no VM [@Vasudevan].

--- abstract: 'In this paper we show that the entropy of a cosmological horizon in topological Reissner-Nordström- de Sitter and Kerr-Newman-de Sitter spaces can be described by the Cardy-Verlinde formula, which is supposed to be an entropy formula of conformal field theory in any number of dimension. Furthermore, we find that the entropy of a black hole horizon can also be rewritten in terms of the Cardy-Verlinde formula for these black holes in de Sitter spaces, if we use the definition due to Abbott and Deser for conserved charges in asymptotically de Sitter spaces. Such result presume a well-defined dS/CFT correspondence, which has not yet attained the credibility of its AdS analogue.' author: - | M.R. Setare [^1]\ [Institute for Theoretical Physics and Mathematics, Tehran, Iran]{} title: '**[ The Cardy-Verlinde formula and entropy of black holes in de Sitter spaces ]{}**' --- Introduction ============ The holographic duality which connects $n+1$-dimensional gravity in Anti-de Sitter (AdS) background with $n$-dimensional conformal field theory (CFT) has been discussed vigorously for some years[@AdS]. But it seems that we live in a universe with a positive cosmological constant which will look like de Sitter space–time in the far future. Therefore, we should try to understand quantum gravity or string theory in de Sitter space preferably in a holographic way. Of course, physics in de Sitter space is interesting even without its connection to the real world; de Sitter entropy and temperature have always been mysterious aspects of quantum gravity[@GH].\ While string theory successfully has addressed the problem of entropy for black holes, dS entropy remains a mystery. One reason is that the finite entropy seems to suggest that the Hilbert space of quantum gravity for asymptotically de Sitter space is finite dimensional, . Another, related, reason is that the horizon and entropy in de Sitter space have an obvious observer dependence. For a black hole in flat space (or even in AdS) we can take the point of view of an outside observer who can assign a unique entropy to the black hole. The problem of  what an observer venturing inside the black hole experiences, is much more tricky and has not been given a satisfactory answer within string theory. While the idea of black hole complementarity provides useful clues, [@Susskind], rigorous calculations are still limited to the perspective of the outside observer. In de Sitter space there is no way to escape the problem of the observer dependent entropy. This contributes to the difficulty of de Sitter space.\ More recently, it has been proposed that defined in a manner analogous to the AdS/CFT correspondence, quantum gravity in a de Sitter (dS) space is dual to a certain Euclidean CFT living on a spacelike boundary of the dS space [@Strom] (see also earlier works [@Hull]-[@Bala]). Following this proposal, some investigations on the dS space have been carried out recently [@Mazu]-[@Ogus]. According to the dS/CFT correspondence, it might be expected that as in the case of AdS black holes [@Witten2], the thermodynamics of cosmological horizon in asymptotically dS spaces can be identified with that of a certain Euclidean CFT residing on a spacelike boundary of the asymptotically dS spaces.\ One of the remarkable outcomes of the AdS/CFT and dS/CFT correspondence has been the generalization of Cardy’s formula (Cardy-Verlinde formula) for arbitrary dimensionality, as well as a variety of AdS and dS backgrounds. In this paper, we will show that the entropy of a cosmological horizon in the topological Reissner-Nordström -de Sitter (TRNdS) and topological Kerr-Newman-de Sitter spaces (TKNdS) can also be rewritten in the form of the Cardy-Verlinde formula. We then show that if one uses the Abbott and Deser (AD) prescription [@AD], the entropy of black hole horizons in dS spaces can also be expressed by the Cardy-Verlinde formula. Topological Reissner-Nordström-de Sitter Black Holes ==================================================== We start with an $(n+2)$-dimensional TRNdS black hole solution, whose metric is $$\begin{aligned} && ds^2 = -f(r) dt^2 +f(r)^{-1}dr^2 +r^2 \gamma_{ij}dx^{i}dx^{j}, \nonumber \\ &&~~~~~~ f(r)=k -\frac{\omega_n M}{r^{n-1}} +\frac{n \omega_n^2 Q^2}{8(n-1) r^{2n-2}} -\frac{r^2}{l^2},\end{aligned}$$ where $$\omega_n=\frac{16\pi G_{n+2}}{n\mbox {Vol}(\Sigma)},$$ where $\gamma_{ij}$ denotes the line element of an $n-$dimensional hypersurface $\Sigma$ with constant curvature $n(n-1)k$ and volume $Vol(\Sigma)$ , $G_{n+2}$ is the $(n+2)-$dimensional Newtonian gravity constant, $M$ is an integration constant, $Q$ is the electric/magnetic charge of Maxwell field. When $k=1$, the metric Eq.(1) is just the Reissner-Nordström-de Sitter solution. For general $M$ and $Q$, the equation $f(r)=0$ may have four real roots. Three of them are real, the largest on is the cosmological horizon $r_{c}$, the smallest is the inner (Cauchy) horizon of black hole, the one in between is the outer horizon $r_{+}$ of the black hole. And the fourth is negative and has no physical meaning. The case $M=Q=0$ reduces to the de Sitter space with a cosmological horizon $r_{c}=l$.\ When $k=0$ or $k<0$, there is only one positive real root of $f(r)$, and this locates the position of cosmological horizon $r_{c}$.\ In the case of $k=0$, $\gamma_{ij}dx^{i}dx^{j}$ is an $n-$dimensional Ricci flat hypersurface, when $M=Q=0$ the solution Eq.(1) goes to pure de Sitter space $$ds^{2}=\frac{r^{2}}{l^{2}}dt^{2}-\frac{l^{2}}{r^{2}}dr^{2}+r^{2}dx_{n}^{2}$$ in which $r$ becomes a timelike coordinate.\ When $Q=0$, and $M\rightarrow -M$ the metric Eq.(1)is the TdS (topological de Sitter) solution , which have a cosmological horizon and a naked singularity, for this type of solution, the Cardy-Verlinde formula also work well.\ Here we review the BBM prescription [@BBM] for computing the conserved quantities of asymptotically de Sitter spacetimes briefly. In a theory of gravity, mass is a measure of how much a metric deviates near infinity from its natural vacuum behavior; i.e, mass measures the warping of space. Inspired by the analogous reasoning in AdS space one can construct a divergence-free Euclidean quasilocal stress tensor in de Sitter space by the response of the action to variation of the boundary metric we have $$\begin{aligned} T^{\mu \nu} &=& {2 \over \sqrt{h}} { \delta I \over \delta h_{\mu \nu}} = \ \ {1 \over 8\pi G} \left[ K^{\mu\nu} - K \, h^{\mu\nu} + {n \over l} \, h^{\mu\nu} +\frac{l}{n} \,G^{\mu\nu} \right] , \label{stressminus}\end{aligned}$$ where $h^{\mu\nu}$ is the metric induced on surfaces of fixed time, $K_{\mu\nu}$, $K$ are respectively extrinsic curvature and its trace, $G^{\mu\nu}$ is the Einstein tensor of the boundary geometry. To compute the mass and other conserved quantities, one can write the metric $h^{\mu\nu}$ in the following form $$h_{\mu\nu} \, dx^{\mu} \, dx^{\nu } = N_{\rho}^{2} \, d\rho^{2} + \sigma_{ab}\, (d\phi^a + N_\Sigma^a \, d\rho) \, (d\phi^b + N_\Sigma^b \, d\rho) \label{boundmet}$$ where the $\phi^{a}$ are angular variables parametrizing closed surfaces around the origin. When there is a Killing vector field $\xi^{\mu}$ on the boundary, then the conserved charge associated to $\xi^{\mu}$ can be written as $$Q = \oint_{\Sigma} d^{n}\phi \,\sqrt{\sigma } \, n^{\mu}\xi^{\mu} \,T_{\mu\nu} \label{chargedef}$$ where $n^{\mu}$ is the unit normal vector on the boundary, $\sigma$ is the determinant of the metric $\sigma_{ab}$. Therefore the mass of an asymptotically de Sitter space is as $$M = \oint_{\Sigma} d^{n}\phi \,\sqrt{ \sigma } \, N_{\rho} \, \epsilon ~~~~~;~~~~~ \epsilon \equiv n^{\mu}n^{\nu} \, T_{\mu\nu} \, , \label{massdef}$$ where Killing vector is normalized as $\xi^{\mu} = N_{\rho} n^{\mu}$. Using this prescription [@BBM], the gravitational mass, having subtracted the anomalous Casimir energy, of the TRNdS solution is $$\label{3eq3} E=-M =-\frac{r_c^{n-1}}{\omega_n} \left (k -\frac{r_c^2}{l^2} + \frac{n\omega_n^2 Q^2}{8(n-1)r_c^{2n-2}}\right).$$ Some thermodynamic quantities associated with the cosmological horizon are $$\begin{aligned} && T_{c}= \frac{1}{4\pi r_c} \left(-(n-1)k +(n+1)\frac{r_c^2}{l^2} +\frac{n\omega_n^2 Q^2}{8 r_c^{2n-2}}\right), \nonumber \\ && S_{c} =\frac{r_c^n\mbox{Vol}(\sigma)}{4G}, \nonumber \\ && \phi_{c} =-\frac{n}{4(n-1)}\frac{\omega_n Q}{r_c^{n-1}},\end{aligned}$$ where $\phi_{c}$ is the chemical potential conjugate to the charge $Q$.\ The Casimir energy $E_C$, defined as $E_C =(n+1) E-nTS-n\phi Q$ in this case, is found to be $$E_C=-\frac{2nkr_c^{n-1}\mbox{Vol}(\sigma)}{16\pi G},$$ when $k=0$, the Casimir energy vanishes, as the case of asymptotically AdS space. When $k=\pm 1$, we see from Eq.(10) that the sign of energy is just contrast to the case of TRNAdS space [@youm].\ Thus we can see that the entropy Eq.(9)of the cosmological horizon can be rewritten as $$S=\frac{2\pi l}{n}\sqrt{|\frac{E_{C}}{k}|(2(E-E_q)-E_C)},$$ where $$E_q = \frac{1}{2}\phi_{c} Q =-\frac{n}{8(n-1)}\frac{\omega_n Q^2}{r_c^{n-1}}.$$ We note that the entropy expression (11) has a similar form as the case of TRNAdS black holes [@youm].\ For the black hole horizon, which is only for the case $k=1$, associated thermodynamic quantities are $$\begin{aligned} \label{3eq8} && T_{b}=\frac{1}{4\pi r_b}\left( (n-1) -(n+1)\frac{r_{b}^2}{l^2} -\frac{n\omega_n^2 Q^2} {8r_{b}^{2n-2}}\right), \nonumber \\ && S_{b}=\frac{r_{b}^n \mbox{Vol}(\sigma)}{4G}, \nonumber \\ && \phi_{b} =\frac{n}{4(n-1)}\frac{\omega_n Q}{r_{b}^{n-1}}.\end{aligned}$$ Now if we uses the BBM mass Eq.(\[3eq3\]) the black hole horizon entropy cannot be expressed in a form like Cardy-Verlinde formula [@cai1]. The other way for computing conserved quantities of asymptotically de Sitter space is the Abbott and Deser (AD) prescription [@AD]. According to this prescription, the gravitational mass of asymptotically de Sitter space coincides with the ADM mass in asymptotically flat space, when the cosmological constant goes to zero. Using the AD prescription for calculating conserved quantities the black hole horizon entropy of TKNdS space can be expressed in term of the Cardy-Verlind formula [@cai1]. The AD mass of TRNdS solution can be expressed in terms of black hole horizon radius $r_b$ and charge $Q$, $$E' =M =\frac{r_{b}^{n-1}}{\omega_n} \left (1-\frac{r_{b}^2}{l^2} + \frac{n\omega_n^2 Q^2}{8(n-1)r_{b}^{2n-2}}\right).$$ In this case, the Casimir energy, defined as $ E'_C =(n+1) E' -n T_{b} S_{b}-n \phi_{b} Q$, is $$E'_C =\frac{2n r_{b}^{n-1}\mbox{Vol}(\sigma)}{16\pi G},$$ and the black hole entropy $ S_{b}$ can be rewritten as $$\label{3eq11} S_{b} =\frac{2\pi l}{n}\sqrt{ E'_C |2( E'-E'_q)-E'_C|},$$ where $$E'_q =\frac{1}{2} \phi_{b} Q=\frac{n\omega_n Q^2}{8(n-1)r_{b}^{n-1}},$$ which is the energy of electromagnetic field outside the black hole horizon. Thus we demonstrate that the black hole horizon entropy of TRNdS solution can be expressed in a form as the Cardy-Verlinde formula. However, if one uses the BBM mass Eq.(8), the black hole horizon entropy $S_b$ cannot be expressed by a form like the Cardy-Verlinde formula. Topological Kerr-Newman-de Sitter Black Holes ============================================= The line element of TKNdS black holes in 4-dimension case is given by $$\begin{aligned} ds^{2} &=&-\frac{\Delta _{r}}{\rho ^{2}}\left(dt-\frac{a}{\Xi }\sin ^{2}\theta d\phi \right)^{2}+\frac{\rho ^{2}}{\Delta _{r}}dr^{2}+\frac{\rho ^{2}}{\Delta _{\theta }}d\theta ^{2} \nonumber \\ &&+\frac{\Delta _{\theta }\sin ^{2}\theta }{\rho ^{2}}\left[a dt-\frac{ (r^{2}+a^{2})}{\Xi }d\phi \right]^{2}, \label{kdsmet}\end{aligned}$$ where $$\begin{aligned} \Delta _{r} &=&(r^{2}+a^{2})\left(k-\frac{r^{2}}{l^{2}}\right) -2Mr+q^2, \nonumber \\ \Delta _{\theta } &=&1+\frac{a^{2}\cos ^{2}\theta}{ l^{2}}, \nonumber \\ \Xi &=&1+\frac{a^{2}}{l^{2}}, \nonumber \\ \rho ^{2} &=&r^{2}+a^{2}\cos ^{2}\theta . \label{kdelt}\end{aligned}$$ Here the parameters $M$, $a$, and $q$ are associated with the mass, angular momentum, and electric charge parameters of the space-time, respectively. The topological metric Eq.(18) will only solve the Einstein equations if k=1, which is the spherical topology. In fact when $k=1$, the metric Eq.(18) is just the Kerr-Newman -de Sitter solution. Three real roots of the equation $\Delta _{r}=0$, are the locations of three horizons, the largest being the cosmological horizon $r_{c}$, the smallest is the inner horizon of black hole, the one in between is the outer horizon $r_{b}$ of the black hole.\ If we want in the $k=0,-1$ cases to solve the Einstein equations, then we must set $sin\theta \rightarrow \theta$, and $sin\theta \rightarrow sinh\theta$ respectively [@man1]-[@man4]. When $k=0$ or $k=-1$, there is only one positive real root of $\Delta _{r}$, and this locates the position of cosmological horizon $r_{c}$.\ In the BBM prescription[@BBM], the gravitational mass, subtracted the anomalous Casimir energy, of the 4-dimensional TKNdS solution is $$E=\frac{-M}{\Xi}. \label{bbmass}$$ Where the parameter $M$ can be obtained from the equation $\Delta_{r}=0$. On this basis, the following relation for the gravitational mass can be obtained $$E=\frac{-M}{\Xi} =\frac{(r_c^2+a^2)(r_c^2-k l^2)-q^2 l^2}{2\Xi r_c l^2}. \label{kmass}$$ The Hawking temperature of the cosmological horizon is given by $$T_{c}=\frac{-1}{4\pi}\frac{\Delta '_{r}(r_{c})}{(r_c^2+a^2)}=\frac{3r_c^4+r_c^2(a^2-k l^2)+(ka^2+q^2)l^2}{4\pi r_c l^2(r_c^2+a^2)}. \label{tem}$$ The entropy associated with the cosmological horizon can be calculated as $$S_{c}=\frac{\pi(r_{c}^{2}+a^{2})}{\Xi}. \label{ent}$$ The angular velocity of the cosmological horizon is given by $$\Omega_{c}=\frac{-a\Xi}{(r_{c}^{2}+a^{2})}. \label{ang}$$ The angular momentum $J_{c}$, the electric charge $Q$, and the electric potentials $\phi_{qc}$ and $\phi_{qc0}$ are given by $$\begin{aligned} & & {\mathcal{J}}_c=\frac{M a}{\Xi^2}, \nonumber \\ & & Q =\frac{q}{\Xi}, \nonumber \\ & & \Phi_{qc}=-\frac{q r_c}{r_c^2+a^2}, \nonumber \\ & & \Phi_{qc0}=-\frac{q}{r_c}, \label{ctherm}\end{aligned}$$ The obtained above quantities of the cosmological horizon satisfy the first law of thermodynamics: $$dE=T_cdS_c+\Omega_c d{\mathcal{J}}_c+(\Phi_{qc}+\Phi_{qc0}) dQ . \label{Flth}$$ Using the Eqs.(\[ent\],\[ctherm\]) for the cosmological horizon entropy, angular momentum and charge, and also the equation $\Delta_{r}(r_{c})=0$, we can obtain the mtric parameters $M$, $a$, $q$ as a function of $S_{c}$, ${\mathcal{J}}_c$ and $Q$, and after that we can write $E$ as a function of these thermodynamical quantities: $E(S_{c},{\mathcal{J}}_c,Q)$ (see [@cal]). Then one can define the quantities conjugate to $S_{c}$, ${\mathcal{J}}_c$ and $Q$, as $$T_c=\left( \frac{\partial E}{\partial S_c}\right) _{J_c,_Q},\ \ \Omega_c =\left( \frac{\partial E}{\partial {J}_c}\right) _{S_c,Q},\ \ \Phi_{qc} = \left( \frac{\partial E}{\partial Q}\right) _{S_c,J_c}\\\Phi_{qc0} =lim_{a\rightarrow 0} \left( \frac{\partial E}{\partial Q}\right) _{S_c,J_c}\\ \label{Dsmar}$$ Making use of the fact that the metric for the boundary CFT can be determined only up to a conformal factor, we rescale the boundary metric for the CFT to the following form: $$ds_{CFT}^2=\lim_{r \rightarrow \infty}\frac{R^2}{r^2}ds^2 , \label{euniverse}$$ Then the thermodynamic relations between the boundary CFT and the bulk TKNdS are given by $$E_{CFT}=\frac{l}{R}E,\hspace{0.07 cm}T_{CFT}=\frac{l}{R}T ,\hspace{0.07 cm}J_{CFT}=\frac{l}{R}J,\hspace{0.07 cm}\phi_{CFT}=\frac{l}{R}\phi,\hspace{0.07 cm}\phi_{0CFT}=\frac{l}{R}\phi_{0}, \label{CFT}$$ The Casimir energy $E_C$, defined as $E_C =(n+1) E-n(T_cS_c+J_c\Omega_c+Q/2\phi_{qc}+Q/2\phi_{qc0})$ , and $n=2$ in this case, is found to be $$E_C=-\frac{k(r_c^2+a^2)l }{R \Xi r_c}, \label{ckecas01}$$ in KNdS space case [@jing] the Casimir energy $E_c$ is always negative, but in TKNdS space case the Casimir energy can be positive, negative or vanishing depending on the choice of $k$. Thus we can see that the entropy Eq.(\[ent\])of the cosmological horizon can be rewritten as $$S=\frac{2\pi R}{n}\sqrt{|\frac{E_{c}}{k}|(2(E-E_q)-E_c)},\label{careq}$$ where $$E_q = \frac{1}{2}\phi_{c0} Q.\label{qeq}$$ We note that the entropy expression (\[careq\]) has a similar form as in the case of TRNdS black holes Eq.(11).\ For the black hole horizon, which only exists for the case $k=1$ the associated thermodynamic quantities are $$T_{b}=\frac{1}{4\pi}\frac{\Delta '_{r}(r_{b})}{(r_b^2+a^2)}=-\frac{3r_b^4+r_b^2(a^2-l^2)+(a^2+q^2)l^2}{4\pi r_b l^2(r_b^2+a^2)}. \label{tem2}$$ $$S_{b}=\frac{\pi(r_{b}^{2}+a^{2})}{\Xi}. \label{ent2}$$ $$\Omega_{b}=\frac{a\Xi}{(r_{b}^{2}+a^{2})}. \label{ang2}$$ $${\mathcal{J}}_b=\frac{M a}{\Xi^2},$$ $$Q =\frac{q}{\Xi},$$ $$\Phi_{qb}=\frac{q r_b}{r_b^2+a^2},$$ $$\Phi_{qb0}=\frac{q}{r_b}.$$ The AD mass of TKNdS solution can be expressed in terms of the black hole horizon radius $r_b$, $a$ and charge $q$: $$E'=\frac{M}{\Xi} =\frac{(r_b^2+a^2)(r_b^2-l^2)-q^2 l^2}{2\Xi r_b l^2}. \label{kmass2}$$ The quantities obtained above of the black hole horizon also satisfy the first law of thermodynamics: $$dE'=T_bdS_b+\Omega_b d{\mathcal{J}}_b+(\Phi_{qb}+\Phi_{qb0}) dQ . \label{Flth1}$$ The thermodynamics quantities of the CFT must be rescaled by a factor $\frac{l}{R}$ similar to the pervious case. In this case, the Casimir energy, defined by $ E'_C =(n+1) E' -n(T_bS_b+J_b\Omega_b+Q/2\phi_{qb}+Q/2\phi_{qb0})$, is $$E'_C =\frac{(r_b^2+a^2)l }{R \Xi r_b}, \label{ckecas02}$$ and the black hole entropy $ S_{b}$ can be rewritten as $$S_{b}=\frac{2\pi R}{n}\sqrt{E'_{C}|(2(E'-E'_q)-E'_C)|},\label{careq2}$$ where $$E'_q =\frac{1}{2}\phi_{qb0} Q.\label{qeq2}$$ This is the energy of an electromagnetic field outside the black hole horizon. Thus we demonstrate that the black hole horizon entropy of the TKNdS solution can be expressed in the form of the Cardy-Verlinde formula. However, if one uses the BBM mass Eq.(\[kmass\]) the black hole horizon entropy $S_{b}$ cannot be expressed in a form like the Cardy-Verlinde formula. Our result is in favour of the dS/CFT correspondence. Conclusion ========== The Cardy-Verlinde formula recently proposed by Verlinde [@Verl], relates the entropy of a certain CFT to its total energy and Casimir energy in arbitrary dimensions. In the spirit of dS/CFT correspondence, this formula has been shown to hold exactly for the cases of dS Schwarzschild, dS topological, dS Reissner-Nordström , dS Kerr, and dS Kerr-Newman black holes. In this paper we have further checked the Cardy-Verlinde formula with topological Reissner-Nordström- de Sitter and topological Kerr-Newman de Sitter black holes.\ It is well-known that there is no black hole solution whose event horizon is not a sphere, in a de Sitter background, although there are such solutions in an anti-de Sitter background; then in TRNdS, TKNdS spaces for the case k=0,-1 the black hole does not have an event horizon, however the cosmological horizon geometry is spherical, flat and hyperbolic for k=1,0,-1, respectively. As we have shown there exist two different temperatures and entropies associated with the cosmological horizon and black hole horizon, in TRNdS, TKNdS spacetimes. If the temperatures of the black hole and cosmological horizon are equal, then the entropy of the system is the sum of the entropies of cosmological and black hole horizons. The geometric features of the black hole temperature and entropy seem to imply that the black hole thermodynamics is closely related to nontrivial topological structure of spacetime. In [@cai2] Cai, et al in order to relate the entropy with the Euler characteristic $\chi$ of the corresponding Euclidean manifolds have presented the following relation: $$S=\frac{\chi_{1}A_{BH}}{8}+\frac{\chi_{2}A_{CH}}{8},$$ in which the Euler number of the manifolds is divided into two parts; the first part comes from the black hole horizon and the second part come from the cosmological horizon (see also ). If one uses the BBM mass of the asymptotically dS spaces, the black hole horizon entropy cannot be expressed in a form like the Cardy-Verlinde formula[@cai1]. 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--- author: - 'T.P.G. Wijnen, O.R. Pols, F.I. Pelupessy, S. Portegies Zwart' bibliography: - 'phdbib.bib' date: 'Received ..../ Accepted ....' title: 'Face-on accretion onto a protoplanetary disc' --- [Using assumptions that represent the (dynamical) conditions in a typical GC, we investigate whether a low-mass star of $0.4\,{\mbox{$\mathrm{M}_{\odot}$}}$ surrounded by a protoplanetary disc can accrete a sufficient amount of enriched material to account for the observed abundances in ‘second generation’ GC stars. In particular, we focus on the gas loading rate onto the disc and star as well as on the lifetime and stability of the disc.]{} [We perform simulations at multiple resolutions with two different smoothed particle hydrodynamics codes and compare the results. Each code uses a different implementation of the artificial viscosity.]{} [We find that the gas loading rate is about a factor of two smaller than the rate based on geometric arguments, because the effective cross section of the disc is smaller than its surface area. Furthermore, the loading rate is consistent for both codes, irrespective of resolution. Although the disc gains mass in the high resolution runs, it loses angular momentum on a time scale of $10^4$ years. Two effects determine the loss of (specific) angular momentum in our simulations: 1) continuous ram pressure stripping and 2) accretion of material with no azimuthal angular momentum. Our study, as well as previous work, suggests that the former, dominant process is mainly caused by numerical rather than physical effects, while the latter is not. The latter process, as expected theoretically, causes the disc to become more compact and increases the surface density profile considerably at smaller radii.]{} [[The disc size is determined in the first place by the ram pressure exerted by the flow when it first hits the disc. Further evolution is governed by the decrease in the specific angular momentum of the disc as it accretes material with no azimuthal angular momentum. Even taking into account the uncertainties in our simulations and the result that the loading rate is within a factor two of a simple geometric estimate]{.nodecor}, the size and lifetime of the disc are probably not sufficient to accrete the amount of mass required in the Early disc accretion scenario.]{} Introduction ============ Stars generally form in clusters [@lada03]. These dense environments affect the formation and evolution of the stars they host. For example, globular clusters (GCs) were once considered the archetype of coeval, simple stellar populations, but during the last two decades they have been shown to harbour multiple stellar populations. Observations imply that a considerable fraction (up to 70%) of the stars currently in GCs has a very different chemical composition from the initial population [see e.g. @gratton12]. They indicate that a second (and in some cases even higher order, e.g. @piotto07) population[^1] of stars has formed from material enriched by ejecta from first generation stars. To explain the formation of these enriched stellar populations, several scenarios have been proposed [see e.g. @decressin07-1; @dercole08; @de_mink09; @bastian13-2]. One of the recently proposed scenarios applies in particular to star formation in GCs, but could, in theory, also leave its signature on stellar systems that are less dense. @bastian13-2, hereafter B13, have suggested a scenario in which the enriched population is not formed by a second star formation event, but rather by the accretion of enriched material, that was expelled by the high-mass stars of the initial population, onto the low-mass stars of the same (initial) population. Because Bondi-Hoyle accretion, i.e. gravitational focusing of material onto the star, is unlikely to be efficient in a GC environment with a high velocity dispersion, they suggest that the protoplanetary discs of low-mass stars sweep up the enriched matter. In order to account for the observed abundances in the enriched population, the low-mass stars have to accrete of the order of their own mass, i.e. a 0.25 ${\mbox{$\mathrm{M}_{\odot}$}}$ star has to accrete about 0.25 ${\mbox{$\mathrm{M}_{\odot}$}}$ of enriched material in the most extreme case (as inferred from, e.g., the main sequence of NGC2808 [@piotto07]). The time scale of this scenario is limited by the lifetimes and sizes of protoplanetary discs. B13 assume that the protoplanetary discs can accrete material for up to 20 Myr. Current disc lifetimes are observed to be 5-15 Myr, but B13 argue that their lifetimes may have been considerably longer in GCs. The accretion rate averaged over 20 Myr therefore has to be about $10^{-8} {\mbox{$\mathrm{M}_{\odot}$}}/ yr$. In their scenario, they assume that the accretion rate is proportional to the size of the disc, $\pi R_{\rm disc}^2$, density of the ISM, $\rho_{\rm ISM}$, and the velocity, $v_{\rm ISM}$, of the disc with respect to the ISM, i.e. $\dot{M} \propto \rho_{\rm ISM} v_{\rm ISM} \pi R_{\rm disc}^2$. Furthermore, they assume an average and constant disc radius of 100 AU. In this work, we test several of these assumptions of the early disc accretion scenario. A similar scenario has been studied before by @moeckel09, M09 hereafter. They performed smoothed particle hydrodynamics simulations of a protoplanetary disc that is embedded in a flow of interstellar medium (ISM) with a velocity of $3\,\mathrm{km\, s^{-1}}$. They found that the mean accretion rate onto the star equals the rate expected from Bondi-Hoyle theory, whether a disc is present or not. We note that for the parameters they assumed, the theoretical Bondi-Hoyle radius is almost twice the radius of the disc. Here we follow up on the work by M09 by simulating the accretion process onto a protoplanetary disc for the typical conditions expected in a GC environment. We directly compare the outcome of two different smoothed particle hydrodynamics codes for the same set of initial conditions and different particle resolutions. We first discuss both the physical and numerical effects in our reference model and subsequently we compare the results of the different codes and particle numbers. Expected physical effects ========================= In dense stellar systems, where both the velocity dispersion and the possibility of close stellar encounters are high, we expect the following physical effects to play an important role in the process of accretion onto protoplanetary discs: ram pressure, angular momentum transport, dynamical encounters and external photo-evaporation. Ram pressure stripping {#sec:rampressure_eda1} ---------------------- As the protoplanetary disc moves through the ISM, it experiences ram pressure, $P_{\rm ram} = \rho_{\rm ISM} v_{\rm ISM}^2$. This drag force can truncate the disc, depending on the gravitational force of the disc that keeps the latter bound to the star. By equating the gravitational force per unit area, or ‘pressure’, $P_{\rm grav}=GM_*\Sigma(r)r^{-2}$, of the disc to $P_{\rm ram}$, we determine beyond which radius the ram pressure dominates and the disc is expected to be truncated. This truncation radius is given by: $$\label{eq:ramradius_eda1} R_{\rm trunc} = \left(\frac{GM_*\Sigma_0 \mathbf{r_0^n}}{\rho_{\rm ISM} v_{\rm ISM}^2}\right)^{\frac{1}{n+2}}$$ with $M_*$ the mass of the star and we have assumed that the surface density profile of the disc can be written as $\Sigma(r)=\Sigma_0 (r/r_0)^{-n}$, where $r_0$ is an arbitrary but constant radius to which $\Sigma_0$ is scaled. After the material at radii larger than $R_{\rm trunc}$ has been stripped from the disc, we expect further evolution of the disc to be determined by the redistribution of angular momentum due to accretion and viscous evolution. The pressure in the mid-plane of the disc is at least one order of magnitude smaller than the gravitational ‘pressure’ and therefore does not play a significant role in resisting the ram pressure. Redistribution of angular momentum {#sec:am_redistribution_eda1} ---------------------------------- The redistribution of angular momentum in the disc is governed by two processes:(1) the accretion of material with no angular momentum material with respect to the disc and (2) the viscous evolution of the disc and consequent redistribution of its mass and angular momentum. ### Accretion of ISM The accretion of material with no azimuthal angular momentum lowers the specific angular momentum of the disc. Since the total angular momentum of the disc has to be conserved, mass and angular momentum will be redistributed within the disc. We can estimate this redistribution to first order, if we consider the disc to consist of concentric rings with thickness $dr$ and mass $m_{\rm ring}(r) = 2 \pi r \Sigma(r) dr$. In a time interval $dt$, the ring will accrete an amount of mass from the ISM equal to $m_{\rm accr}(r) = 2 \pi r \rho_{\rm ISM} v_{\rm ISM} dt dr$. The specific angular momentum, $h$, in the ring will decrease by a factor $\Sigma(r)/(\Sigma(r) + \rho_{\rm ISM} v_{\rm ISM} dt)$ and the ring will migrate to a smaller radius that corresponds to its new specific angular momentum. This process causes inward migration of material that belongs to the disc and leads to contraction of the disc. This derivation does not take into account interaction between adjacent rings, which is determined by viscous processes. ### Viscous redistribution Although the nature of the viscous processes that occur in accretion discs are still debated [see e.g. @armitage11], we do know that they are responsible for transporting material inwards through the disc. When this happens, some material has to move outward to conserve the total angular momentum of the disc, $J_{\rm disc}$. Both viscous evolution and accretion of ISM cause material to drift inwards where it is eventually accreted onto the star and lost from the disc together with the angular momentum it carried. The outward spreading of material at the outer edge of the disc could make it more vulnerable to ram pressure stripping, which in turn also robs the disc of its angular momentum. Dynamical encounters -------------------- In dense stellar systems, disc radii have been shown to be truncated due to close stellar encounters [@breslau14; @rosotti14; @vincke15]. The last study shows that in dense clusters ($\bar{\rho}_{\rm cluster} \approx 500\,\mathrm{pc^{-3}}$), almost 40% of the discs is smaller than 100 AU and the median disc radius can be as small as 20 AU in the core. Close stellar encounters thus affect the surface area of the disc and thereby also the rate at which the disc can sweep up ISM. In this work we only focus on the hydrodynamic aspects of accretion onto protoplanetary discs, in particular the accretion rate. The disc radius we find in our simulations is of similar size as the 20 AU found by @vincke15. The question whether the process of ram pressure or dynamical encounters dominates the truncation of the disc in embedded dense stellar systems is beyond the scope of this paper. External photo-evaporation -------------------------- Globular clusters host a large number of massive stars in the early phases of their evolution and the large UV flux they produce may also strip material from the disc. Studies have shown that the fraction of stars that have discs can decrease by a factor of two close to O stars [see e.g. @balog07; @guarcello07; @guarcello09; @fang12], but @richert15 argue that these results could be partly affected by sample incompleteness. Recently, @facchini16 estimated that the mass loss rate from the outer edge of a protoplanetary disc due to photo-evaporative winds can be of the order of $10^{-8} - 10^{-7}\, {\mbox{$\mathrm{M}_{\odot}$}}/ \mathrm{yr}$, see their Fig. 12. We do not take radiative processes into account in this work but we note that any mass loss from the disc, additional to that found in this work, will shorten its lifetime with respect to our findings. Numerical set-up {#sec:set-up_eda1} ================ ![A schematic overview of the numerical set-up.\[fig:schematics\_eda1\]](Schematics.pdf){width="49.00000%"} Our simulations are performed using the AMUSE environment [@portegies_zwart13; @pelupessy13][^2]. AMUSE is a Python framework in which one can use and combine a variety of astrophysical codes that have been published and well tested by the community, i.e. ‘community codes’. The simulation is set up in Python and AMUSE takes care of the communication between Python and the code(s) that one wishes to use. This way, it is very easy to use the same set-up with different community codes and compare their outcome. Currently, two different smoothed particle hydrodynamics (SPH) codes are included in AMUSE, i.e. Fi [@pelupessy04] and Gadget2 [@springel05; @springel01]. These two codes solve the dynamics of a self-gravitating hydrodynamic fluid using a Tree gravity solver [@hernquist89]. In this work, we use Gadget2 for our reference model and we verify the consistency and robustness of our results by comparing with Fi. Both SPH codes include self-gravity. Because we want to be able to perform future simulations for a wide range of parameters, we need to find a balance between computational effort and convergence, i.e. the most efficient set-up. In order to test whether our particle resolution is sufficient, we perform simulations with different numbers of particles in the disc. This way, we can test whether the results converge and the numerical noise diminishes. The verification and validation will be discussed in Sec. \[sec:convergence\_eda1\]. For our reference model, we use the vanilla version of Gadget2 as it is freely available online[^3] and included in AMUSE, with the only exception that we have implemented the @morris97 viscosity formalism as is done in M09, see Sec. \[sec:visc\_eda1\]. The set-up of our simulations is as follows (see Fig. \[fig:schematics\_eda1\]): a stationary protoplanetary disc is positioned coaxially in a cylindrical, laminar flow of gas, representing the ISM. The protoplanetary disc is positioned in the centre of the cylinder. The inflow and outflow boundaries are each located at a distance of 500 AU from the centre of the protoplanetary disc. The radius of the cylinder is also set to 500 AU. Particles that flow outside the computational domain are removed from the simulation. The inflow consists of new particles. The parameters we have adopted are listed in Table \[tb:parameters\_eda1\]. **Parameter** **Value** **Description** ------------------------- ---------------------------------------- -------------------------------- $n$ $5 \times 10^6 \, \mathrm{cm^{-3}}$ Number density of ISM $v_{\rm ISM}$ $20\,\mathrm{km\, s^{-1}}$ $\mu$ 2.3 Mean molecular weight $M_*$ $0.4\,{\mbox{$\mathrm{M}_{\odot}$}}$ $M_{\rm disc}$ $0.004\,{\mbox{$\mathrm{M}_{\odot}$}}$ $R_{\rm disc, inner}$ 10AU $R_{\rm disc, outer}$ 100AU EoS Isothermal Equation of state $T$ 25K Temperature of gas particles $c_s$ $0.3\,\mathrm{km\, s^{-1}}$ Sound speed $R_{\rm sink}$ 0.09AU Sink particle radius $N_{\rm neighbours}$ $64\pm2$ $\epsilon_{\rm grav}$ $10^{-2}\,\mathrm{AU}$ Gravitational softening length $\alpha_{\rm SPH}$ 0.1 Artificial viscosity parameter $\mathrm{N_{\rm disc}}$ $4000-128000$ Number of disc particles : Initial values of the parameters in our simulations. The parameters are the same for every simulation, except for the number of disc particles which may vary from simulation to simulation.[]{data-label="tb:parameters_eda1"} Initial conditions and parameters --------------------------------- We adopt equal mass particles for the flow and the disc in order to prevent spurious forces on the interface between the ISM and the disc. We set the number of neighbours in the SPH codes to $64 \pm 2$ and use an isothermal equation of state. The sound speed, $c_{\rm s}$, equals $0.3\,\mathrm{km\, s^{-1}}$, which corresponds to a temperature of approximately 25 K for all particles. The isothermal equation of state can be justified by considering the cooling time scale: $$\label{eq:cooling_tau_eda1} \tau_{\rm cool} = \frac{\frac{3}{2}k_{B}T}{n \Lambda(n,T)}$$ where $n$ is the number density, $k_{B}$ the Boltzmann constant, $T$ the temperature and $\Lambda(n,T)$ the cooling rate. For $T$ = 25K and $n \geq 10^3 \mathrm{cm}^{-3}$, the cooling time scale $\tau_{\rm cool} \lesssim 16$ years, assuming $\Lambda \approx 10^{-26} \mathrm{erg \ cm^3 \ s^{-1}}$ [@neufeld95]. Any departure from the equilibrium temperature of 25K will therefore be restored quickly with respect to our simulation time scale. We discuss the parameters and assumptions for the ISM and the disc separately below in Sec. \[sec:ism\_eda1\] and \[sec:disc\_eda1\], respectively. ### Interstellar Medium {#sec:ism_eda1} By adopting a temperature of 25 K, we assume that cooling of the ISM is much more efficient than the radiative transfer of heat from nearby stars. It has been suggested that the Lyman-Werner flux plays an important role in the formation of multiple populations [@conroy11], but a survey on 130 young massive clusters have shown little to no ionized gas [@bastian13]. Young massive clusters are considered to be the modern-day counterpart of proto-GCs and this study therefore implies that heating does not play an important role in the environment where the accretion process is believed to take place. We can also estimate the cooling time scale for stellar ejecta from Eq. \[eq:cooling\_tau\_eda1\], to justify that the expelled gas cools fast to low temperatures. As a lower limit for the density of stellar ejecta we assume $n = 10^2 \mathrm{cm}^{-3}$ and $T = 10^4$ K [@smith07 see also B13]. Combined with an appropriate cooling rate of $\Lambda \approx 10^{-25} \mathrm{erg\,cm^3 / s}$ [@richings14] this results in a cooling time scale of roughly 6500 years. We consider this an upper limit, because assuming a higher density for the stellar ejecta would result in an even shorter cooling time scale. Consequently, the cooling time scale is at least three to four orders of magnitude smaller than the $10^{7}$ years time scale on which the early disc accretion scenario is expected to take place. Our assumption that the ISM has cooled sufficiently and can be approximated with a temperature of 25 K is therefore justified. To estimate the density of the ISM in the early disc accretion scenario we use the values given in B13 for the available processed material and the core radius of a typical GC, respectively $1.3\times10^5\,{\mbox{$\mathrm{M}_{\odot}$}}$ and 1 parsec. The average density would then be around $2 \times 10^{-18} \,\mathrm{g\,cm^{-3}}$. However, the density will vary and can locally be higher. To provide a better comparison with M09, we adopt their assumed number density of $n = 5 \times 10^{6}\,\mathrm{cm^{-3}}$ and mean molecular weight $\mu=2.3$, which translates to a mass density, $\rho_{\rm ISM}$, of $1.92 \times 10^{-17} \,\mathrm{g\,cm^{-3}}$. In the case of GCs with multiple stellar populations, $\mu$ may be somewhat larger because the enriched populations exhibit a helium enhancement compared to primordial molecular clouds. We assume an inflow velocity, $v_{\rm ISM}$, of 20 $\mathrm{km \ s}^{-1}$ in order to approximate the typical velocity dispersion in GCs. This means that our set-up is in the supersonic regime and the treatment of the artificial viscosity is important. We will discuss the artificial viscosity in Sec. \[sec:visc\_eda1\]. The high velocity gives a very small Bondi-Hoyle accretion radius, as we will discuss in Sec. \[sec:sink\_eda1\]. The inflow is modelled by adding a slice of ISM with thickness $v_{\rm ISM}dt$ at the inflow boundary and random uniformly distributing the SPH-particles within this slice. The ISM flow reaches the disc after $\approx$ 100 years and the computational domain contains $\approx 6N_{\rm disc}$ ISM-particles when it is completely filled. ### Disc {#sec:disc_eda1} In the case of a disc in a steady state, the mid-plane temperature profile follows a simple power-law, $T_c \propto r^{-p}$, and the surface density profile, $\Sigma$, is proportional to $r^{p-\frac{3}{2}}$, assuming a constant viscosity parameter $\alpha_{\nu}$ in the @shakura73 formalism [see e.g. @armitage11]. Since we assume a constant temperature in the whole disc, $p = 0$ and $\Sigma \propto r^{-\frac{3}{2}}$, which corresponds to a minimum mass solar nebula [@Weidenschilling77; @hayashi81] and is also assumed in e.g. M09 and @rosotti14. The typical temperature of a protoplanetary disc at radii $> 10$ AU is roughly consistent with a temperature of 25 K [$\approx$ 20K, see e.g. @armitage11]. Although heating of the outer layers of the disc may cause photoevaporation, the mid-plane of the disc is shielded. At approximately 10 AU, the disc has to be heated to temperatures $> 10^3$ K to effectively lose mass due to photoevaporation. An $0.4\,{\mbox{$\mathrm{M}_{\odot}$}}$ star has an effective temperature of $3\times 10^3$ K, which is not high enough to unbind gas from the surface layer of the disc at radii $> 10$ AU. At radii $< 10$ AU extreme-ultraviolet radiation from the star may cause mass loss from the disc in the order of $10^{-11}-10^{-10}\,{\mbox{$\mathrm{M}_{\odot}$}}/$yr [@armitage11; @font04]. This is 2 to 3 orders of magnitude less than the mass loss rates found in this work and we conclude that taking heating by the star into account would not affect our results. The stability of differentially rotating gaseous discs against self-gravity can be expressed in terms of the Toomre parameter $Q$ [@toomre64], which is defined as: $$Q = \frac{c_s \Omega}{\pi G \Sigma}$$ where $\Omega$ is the angular frequency. A disc becomes unstable if $Q$ is less than unity at the outer edge of the disc. In our set-up, $Q$ has a value of $\approx 44$. Self-gravity is therefore not expected to lead to instabilities. As in M09, we assume that the initial mass of the disc is 1% of the mass of the star and we set the outer radius of the disc to 100 AU and the inner radius to 10 AU. Realistically, the disc should probably extend inwards towards the radius of the star, but it is computationally very expensive to simulate the disc on the orbital time scales at these small radii. During the simulation, particles in the disc will migrate inwards due to viscous evolution of the disc and are accreted onto the star, resulting in a disc that extends further inwards. This set-up allows us to postpone the expensive calculations towards later times in our simulations, thereby significantly decreasing the duration of our simulations. We position the disc perpendicular to the flow (see Fig. \[fig:schematics\_eda1\]). This perfect alignment may not occur frequently in nature, but allows to discern the relevant physical processes more clearly. In future work we will investigate the influence of giving the disc an inclination with respect to the flow direction. ### Star {#sec:sink_eda1} We assume that the mass of the star is concentrated in a single point, which has a mass of $0.4\,{\mbox{$\mathrm{M}_{\odot}$}}$. This corresponds to the typical mass expected in the early disc accretion scenario. The point mass is added as a collisionless particle to the SPH code and we treat it as a sink particle, i.e. it can accrete gas particles that fall within a certain (fixed) radius. The sink particle absorbs the mass and momentum carried by the gas particles it accretes. We assume the radius of the sink particle is 5% of the Bondi-Hoyle accretion radius, $R_{\rm A}$, defined as [see also @Bondi52; @shima85]: $$\label{eq:ra_eda1} R_{\rm A} = \frac{2GM_{*}}{c_s^2 +v_{\rm ISM}^2}\\$$ in which $M_{*}$ is the mass of the star, $c_s$ is the sound speed and $v_{\rm ISM}$ the velocity of the ISM (with respect to the star). Using the values mentioned above, gives us an accretion radius of 1.8 AU, which is significantly smaller than in M09, where $R_{\rm A} \approx 2 R_{\rm disc}$. Artificial viscosity {#sec:visc_eda1} -------------------- Since the typical velocity dispersion in a GC is highly supersonic, it is important to resolve shocks. In order to do so, SPH codes introduce a numerical viscosity which is characterised by the parameters $\alpha_{\rm SPH}$ and $\beta_{\rm SPH}$, where $\beta_{\rm SPH}$ has been introduced to prevent particle interpenetration in shocks with high Mach numbers [@springel10]. Typical values for the parameters are $\alpha_{\rm SPH} \simeq 0.5-1$ and $\beta_{\rm SPH}=2\alpha_{\rm SPH}$. The numerical (shear) viscosity can also be used to model the physical viscous transport of matter in an accretion disc [@artymowicz94], but this implies lower values of $\alpha_{\rm SPH}$. This value can be derived using @artymowicz94 to relate the artificial viscosity parameter $\alpha_{\rm SPH}$ to the standard viscosity parameter $\alpha_{\nu}$ proposed by @shakura73. Assuming $\alpha_{\nu} \approx 0.01$ [@armitage11], would correspond to an $\alpha_{\rm SPH}$ of roughly 0.02 in our reference model ($\alpha_{\rm SPH} \propto \sqrt[3]{N_{\rm disc}}$), which is too low for numerical reliability. We therefore set $\alpha_{\rm SPH}$ initially to 0.1, as was also done in e.g. M09 and @rosotti14. We have implemented the viscosity switch proposed by @morris97 in Gadget2[^4]. In this treatment of the viscosity every particle has its individual viscous parameter $\alpha_{\rm SPH}$ (and $\beta_{\rm SPH}=2\alpha_{\rm SPH}$), which is important because we simultaneously need a low value of $\alpha_{\rm SPH}$ in the disc and a high value of $\alpha_{\rm SPH}$, up to 1, to resolve the shock in front of the disc. In the SPH code Fi, the viscosity remains constant throughout the simulation at a value of $\alpha_{\rm SPH} = 0.1$. We have adopted $\beta_{\rm SPH}=1$ for Fi. With these values we minimize the viscous stresses in the disc, which due to the low relative velocities are dominated by the first order $\alpha_{\rm SPH}$ term, while preventing particle interpenetration in the strong accretion shock. We have tested lowering the $\beta_{\rm SPH}$ value below the adopted value and found that it can cause numerical artefacts. Following @artymowicz96, we calculate the time scale on which the disc undergoes significant viscous evolution, $\tau_{\nu}=Re\,\Omega^{-1}$, assuming the Reynolds number, $Re$, and angular frequency, $\Omega$, are given by: $$\label{eq:reynolds_eda1} Re = \frac{(r/\langle h \rangle)^2}{\alpha_{\nu}}\\ \Omega = \sqrt{\frac{GM_{*}}{r^3}}\\$$ with $r$ the radial distance to the star and $\langle h \rangle$ the (resolution-dependent) average smoothing length in the disc at that radius. For a simulation of 16.000 disc particles, and assuming the corresponding $\alpha_{\nu}$, this gives us a viscous time scale, $\tau_{\nu}$, of roughly 10.000 years at 10 AU. We can also calculate the physical viscous time scale at 10 AU by replacing $\langle h \rangle$ with the scale height of the disc, $H=\frac{c_s}{\Omega}$, at that radius. This leads to a time scale of $3\times10^5$ years. In order to remain safely in the regime where the simulation is not dominated by viscous evolution of the disc, we evolve the whole set-up for 2500 years and start the inflow at $t=0$. The numerical viscosity in the disc changes during the simulations with Gadget2 and we will discuss the numerical effects in Sec. \[sec:convergence\_eda1\] and \[sec:consistency\_eda1\]. Distinguishing the disc from the ISM {#sec:disc_recognition_eda1} ------------------------------------ \ \ To determine how much ISM has been swept-up by the disc at any moment in time, we need to differentiate between the disc and the ISM flow. This is not straightforward since there is no clear separation between the outer edge of the disc and the flow. In order to distinguish between the disc and the ISM flow, we use a clump finding algorithm to identify different groups in the parameter space of the angular speed ($v_{\theta}$), the density ($\rho$) and the speed along the axis of the cylinder($v_x$) of each particle. In this parameter space, we expect the ISM particles to cluster around values of, respectively, 0 km/s, $\rho_{\rm ISM}$ and $v_{\rm ISM}$, while the disc particles will not. For the clump finding algorithm we use Hop [@eisenstein98]. Fig. \[fig:disc\_recognition\_eda1a\] shows a comparison of how well this algorithm (solid black line) performs for our reference model compared to drawing a coaxial cylinder around the disc, with a length of 100 AU and a radius of 120 AU, and adding the mass of all the SPH-particles in that volume (dotted black line). The difference between these two lines can be attributed almost completely to the ISM that is in the volume, but not part of the disc. To demonstrate this, Fig. \[fig:disc\_recognition\_eda1c\] shows the difference between these two methods (dotted black line), the mass of ISM in the volume as determined with the Hop algorithm (green dashed line) and assuming the volume is completely filled with ISM (solid green horizontal line). In the beginning of the simulation, when the disc is stripped of its outer parts by the ram pressure exerted by the flow, the algorithm has some difficulties differentiating the disc from the ISM flow, but as soon as a stable situation is reached and the stripped outer edges of the disc have left the computational domain, it performs very well. The few spikes visible in Fig. \[fig:disc\_recognition\_eda1\] are artefacts: some particles are marked as part of the disc, while they are either being stripped from the outer edge (and already carry some angular momentum) or flow along the outer edge of the disc (and pick up some angular momentum from the disc). These artefacts do not influence our results as they are smoothed out when we linearly fit the data from the moment a steady state is reached. Once we have determined which particles belong to the disc, we can also determine the surface density profile, $\Sigma(r)$, and radius, $R_{\rm disc}$, of the disc. In order to do so, we calculate the column density of the disc, viewing the disc face-on. We then bin the obtained 2-D surface density in concentric annuli, which gives us the surface density profile as a function of the radius. At $t=0$, we determine $\Sigma(100 {\rm AU})$, i.e. at the initial outer radius of the disc. At consecutive times, we recalculate the surface density profile and consider the radius at which $\Sigma(r)\big|_{t} =\Sigma(100 {\rm AU})\big|_{t=0}$ to be the radius of the disc at that moment in time. The snapshots of our reference model, shown in Fig. \[fig:snapshots\_eda1\], illustrate that this method performs very well if the simulation has reached a steady state. Furthermore, the radius determined in this way also agrees with the estimate of the truncation radius due to ram pressure stripping, Eq. \[eq:ramradius\_eda1\], as can be seen in Fig. \[fig:disc\_recognition\_eda1b\] where we have plotted both radii as a function of time. Much in the same way as we did for the disc radius, the truncation radius in Fig. \[fig:disc\_recognition\_eda1b\] is derived from the surface density profile at each moment in time. Angular momentum conservation {#sec:am_conservation_eda1} ----------------------------- For TreeSPH codes like Fi and Gadget2 angular momentum is not conserved exactly. The reasons for this are that 1) the gravity forces for a Barnes-Hut tree code are not exactly symmetric for particles in different parts of the domain and 2) interactions between particles in different levels of the time step hierarchy are not exactly symmetric. To make sure that the loss of angular moment that we measure is not due to (the lack of) angular momentum conservation, we determined how well angular momentum is conserved in our reference model. In order to do so, we look at the change of total angular momentum, i.e. of all particles that have been in the computational domain over the course of the simulation, between 1500 and 2500 years. The total angular momentum fluctuates around a mean value, without any increasing or decreasing trend with time. To establish a measure of angular momentum conservation, we determined the mean total angular momentum during the last 1000 years of the simulation, which is $1.5\times10^{44}$kgm$^2$s$^{-1}$, while the standard deviation is $1\times10^{40}$kgm$^2$s$^{-1}$. Thus angular momentum is conserved up to about 1 part in $10^4$. We compare this to the angular momentum loss of the disc in Sec. \[sec:reference\_eda1\]. Results ======= ------------------------- ------------------------------------- ------------------------ -------- ------------------------ $\mathbf{N_{\rm disc}}$ ($\mathbf{10^3}$) **Gadget2** $\mathbf{N_{\rm sim}}$ **Fi** $\mathbf{N_{\rm sim}}$ 4 G4 4 F4 1 8 G8 2 F8 1 16 G16 1 F16 1 G16NF (no ISM inflow) 1 - G16CV (constant $\alpha_{\rm SPH}$) 1 - 32 G32 1 - 64 G64 1 - 128 G128 1 - ------------------------- ------------------------------------- ------------------------ -------- ------------------------ : Different models for the convergence and consistency study[]{data-label="tb:convergence_eda1"} \ [*Columns 1 to 5*]{}: The number of disc particles in a simulation, the label of that simulation and the number of runs of that simulation, for Gadget2 and Fi respectively. Non-standard assumptions are mentioned between brackets. For every run we use the basic set-up as discussed in Sec. \[sec:set-up\_eda1\]. Note that the total number of particles in each simulation is a factor of 7 higher. We have performed a number of simulations with both Gadget2 and Fi at different resolutions, as listed in Table \[tb:convergence\_eda1\]. The main difference between both codes is the treatment of the artificial viscosity (see Sec. \[sec:visc\_eda1\]). Our reference model is a simulation with Gadget2 and 16.000 disc particles, labelled G16. We discuss our reference model in Sec. \[sec:reference\_eda1\] and we use the other models for the convergence and consistency study which we discuss in Sec. \[sec:convandcons\_eda1\]. Whenever we halve the number of disc particles, we double the number of simulations. We compare these results with runs from the SPH code Fi, which we have performed once for each resolution[^5]. We finish the comparison between both SPH codes by discussing a run of Gadget2 with the same artificial viscosity implementation as used in Fi. We choose simulation G16 as our reference model, so that we can compare our results with those of M09, who used the same code. We have not chosen one of the Gadget2 simulations with a higher resolution as our reference model, because we can not compare this model to a simulation with the same number of disc particles with Fi. As a benchmark for the accretion rate onto the star and the mass loss from the outer edge of the disc, we have performed a simulation of our reference model without inflow of ISM, labelled G16NF. We discuss this simulation first. Reference model without inflow of ISM {#sec:noflow_eda1} ------------------------------------- We have performed a simulation, labelled G16NF, of a protoplanetary disc without inflow of ISM to gauge the mass loss rates at the inner and outer edge of the disc. We compare the mass and angular momentum loss of subsequent simulations to the quantities derived for this simulation. The results of this simulation are summarized in Table \[tb:gadget\_eda1\], which contains the summary of the simulations with Gadget2. The mass and angular momentum change rates are determined by plotting the concerning quantity as a function of time, e.g. the mass of the disc, and fitting a linear function to it from the moment the simulation reaches a steady state, i.e. from 1500 years onward. Every rate has been defined in this way, i.e. by a linear fit between 1500 and 2500 years. We find an accretion rate onto the star, $\dot{M}_{\rm star}$, of $4.5\times10^{-8}\,{\mbox{$\mathrm{M}_{\odot}$}}/$yr. This is similar to what is observed for stars of $0.4\,{\mbox{$\mathrm{M}_{\odot}$}}$ [@muzerolle05] and is theoretically expected for a disc in a steady state and constant $\alpha_{\nu}$, corresponding to $\alpha_{\rm SPH}=0.1$ [@shakura73; @armitage11]. We note that M09 found an accretion rate onto the star of $1.5\times10^{-7}\,{\mbox{$\mathrm{M}_{\odot}$}}/$yr in their isolated disc simulation. Roughly half of the mass loss of the disc is due to accretion onto the star, because the inner edge of the disc migrates inwards. This can be seen in Fig. \[fig:surface\_density\_eda1c\], which shows the azimuthally averaged surface density at three different moments in time, i.e. at $t=0$ (solid black), 1500 (dashed purple) and 2500 years (dotted green). The other half is ‘lost’ due to outward spreading of the outer edge as it leaves our computational domain, i.e. when the radial distance of the SPH-particle to the star is more than 500 AU. The outward spreading can be inferred from the decreasing surface density profile at $R>60\,$AU in Fig. \[fig:surface\_density\_eda1c\]. The rates and time scales that are quoted for model G16NF in Table \[tb:gadget\_eda1\] under stripping and angular momentum loss are therefore actually associated with viscous spreading of the disc. We have determined the time scales for viscous spreading by taking the total disc mass and angular momentum at the beginning of the interval and dividing it by the $\dot{M}_{\rm disc}$ and $\dot{J}_{\rm disc}$ respectively. According to this extrapolation, the disc would viscously dissipate on a time scale of $7\times10^4$ years. This is almost half of the numerical viscous time scale at 100 AU, see Sec. \[sec:visc\_eda1\], but should be considered to be more representative for the disc as a whole. Reference model {#sec:reference_eda1} --------------- {width="\textwidth"}\[fig:snapshots\_eda1a\] Fig. \[fig:snapshots\_eda1\] shows snapshots from the simulations with our reference model. At $t=250$ yr, we can see the ram pressure at play as the flow drags along disc material from radii larger than $R_{\rm trunc}$, which is 55 AU in this case. The third column shows that the simulation has reached a steady state at $t=1500$ yr in which the disc is continuously accreting. The last snapshot illustrates that the disc is shrinking in size during the steady state. In Fig. \[fig:disc\_recognition\_eda1a\] we have plotted the mass of the disc in simulation G16 (solid black line) and the total amount of swept-up ISM (green dashed line) as a function of time. The swept-up ISM is defined as the sum of the ISM that is in the disc at each moment in time and the ISM that has been accreted by the star up to that moment. The total amount of swept-up ISM increases and we can determine the ISM loading rate from the slope of this line, as described in Sec. \[sec:noflow\_eda1\]. This gives a value of $\dot{M}_{\rm ISM} = 1.03\times10^{-7}\,{\mbox{$\mathrm{M}_{\odot}$}}/$yr. We can compare this value to the rate that we expect based on a simple geometric estimate: $$\label{eq:mdot_eda1} \dot{M}_{\rm ISM}=\rho_{\rm ISM} v_{\rm ISM} \pi R_{\rm disc}^2$$ where $\rho_{\rm ISM}$ and $v_{\rm ISM}$ are the density and velocity of the ISM respectively and $R_{\rm disc}$ the radius of the disc. The ISM loading rate in the simulation is a factor of about two lower than the geometric rate. We will show in Sec. \[sec:convergence\_eda1\] that the difference is independent of the resolution of our simulation or the code we used. The result that the ISM loading rate is consistently a factor of two lower than given by Eq. \[eq:mdot\_eda1\], can be understood because at the outer edge of the disc ISM is not entrained by the disc. When the ISM flow first hits the disc, at $t \approx 100$ years, the steep increase of swept-up ISM does agree with the theoretical rate. This is because initially almost all ISM colliding with the surface of the disc is considered part of the disc, seen as an increase in the mass of the disc in Fig. \[fig:disc\_recognition\_eda1a\] at $t\approx 100$ years. As the outer edges of the disc are dragged along with the flow, these regions are no longer associated with the disc and there is a large jump in the disc mass and a corresponding smaller one in the swept-up ISM. From that moment on, the effective cross section of the disc, i.e. the area with which it sweeps up ISM, is smaller than the actual surface area of the disc. This is illustrated in Fig. \[fig:surface\_density\_eda1a\], which shows the evolution of the surface density profile for the G16 simulation. The contribution of ISM to the surface density profile at each moment is indicated with filled regions in the corresponding colour. This illustrates that ISM is only entrained by the inner regions and not by the outer regions of the disc. \ \ Although the disc continuously sweeps up ISM, it actually loses mass, see Fig. \[fig:disc\_recognition\_eda1a\], at a rate of $1.68\times10^{-7}\,{\mbox{$\mathrm{M}_{\odot}$}}/$yr, which is faster than the rate at which the disc sweeps up ISM. The disc loses mass at the inner edge due to accretion onto the star and at the outer edge due to continuous stripping. The accretion rate onto the star is $1.27\times10^{-7}\,{\mbox{$\mathrm{M}_{\odot}$}}/$yr, roughly equal to the ISM loading rate and almost three times the accretion rate onto the star for an isolated disc. Sweeping up ISM thus enhances the accretion rate onto the star. The remaining mass is lost from the outer edges at a rate of $1.44\times10^{-7}\,{\mbox{$\mathrm{M}_{\odot}$}}/$yr. Not only is the ISM not entrained by the outer edge, it is actually removing mass from those regions. This result can be explained in the following way. Due to viscous torques within the disc, as discussed in Sec. \[sec:am\_redistribution\_eda1\], disc material migrates inwards. In order to conserve the total angular momentum of the disc, some disc material moves outwards. This outward diffusion lowers the surface density profile at the outer edge of the disc and material is therefore continuously stripped from the disc by the ram pressure of the ISM flow. The rate at which material is lost from the outer edge of the disc is four times higher than in the isolated case. The rate of angular momentum transport in the disc depends on the (artificial) viscosity and will be discussed in more detail in section \[sec:consistency\_eda1\], where we compare our two different methods of modelling the viscosity. The ablation at the outer edge of the disc causes both mass and angular momentum to be lost from the disc. During the last 1000 years of the simulation, the angular momentum lost by the disc equals $5.43\times10^{42}$kgm$^2$s$^{-1}$, which is more than 460 $\sigma$, as determined in Sec. \[sec:am\_conservation\_eda1\], and thus highly significant in comparison with errors in the angular momentum conservation in the code. Therefore, the loss of angular momentum of the disc can not be ascribed to errors in time integration, but must be due to (the modelling of) physical processes in our simulation. The angular momentum lost due to accretion onto the star in the same time interval, $1.4\times10^{40}$kgm$^2$s$^{-1}$, is insignificant compared to the loss of angular momentum from the outer edge of the disc. In Sec. \[sec:convergence\_eda1\] we discuss that at least half of the angular momentum lost from the disc is caused by interaction with the ISM at the outer edge of the disc in which angular momentum is transferred to the ISM and carried away in the flow. The remaining angular momentum loss is due to continuous stripping of original disc material from the outer edge of the disc. The angular momentum loss suggests a disc lifetime of $6\times10^3$ years, which is an order of magnitude shorter than the time scale derived in the simulation without flow. ### Evolution of the surface density profile {#sec:sdp_eda1} The slope of the surface density profile at a given radius in Fig. \[fig:surface\_density\_eda1a\] increases with time as more ISM is entrained and disc material is transported inwards due to viscous evolution. From the start of the simulation, the artificial viscosity in the mid-plane of the disc increases and is higher than the initial value of 0.1. During the simulation, the typical value of $\alpha_{\rm SPH}$ in the mid-plane of the disc is 0.6. This means that the transport of mass and angular momentum through the disc in the simulation is faster than estimated in Sec. \[sec:visc\_eda1\]. In Sec. \[sec:consistency\_eda1\] we discuss how this compares to a simulation in which $\alpha_{\rm SPH}$ remains equal to 0.1 in the disc. The purple and green filled regions show the contribution of ISM in the disc to the surface density profile at $t=1500$ and 2500 years. At each snapshot, the relative contribution of the ISM to the surface density profile is highest at small radii and decreases towards larger radii. The ISM in the disc migrates inwards due to viscous torques and thus follows the same trend as the total surface density profile of the disc. Furthermore, new ISM with no angular momentum is continuously entrained by the disc, which also contributes to the inward migration of disc material. Both processes, continuous accretion and viscous evolution, steepen the exterior slope of both the total and ISM surface density profile. The total fraction of ISM in the disc increases with time as more material is swept-up. Convergence and consistency {#sec:convandcons_eda1} --------------------------- \ \ In this section we compare the outcome of simulations with different numbers of disc particles and SPH codes. The runs for the convergence study are listed in Table \[tb:convergence\_eda1\]. Fig. \[fig:convergence\_rates\_eda1a\] shows the average mass gain and loss rates that are relevant for our study: the accretion rate onto the star, the net rate at which ISM material is swept-up, the rate at which initial disc material is stripped and the total rate of change in the disc mass. The rates are determined by considering the various mass quantities, e.g. the mass of the disc and the star, as a function of time during the last 1000 years of the simulation and applying a linear fit to their slope. We summarize all the quantities plotted in Fig. \[fig:convergence\_rates\_eda1\] in Tables \[tb:gadget\_eda1\] and \[tb:fi\_eda1\]. We first discuss the convergence of the simulations, i.e. the effect of changing the number of disc particles, and subsequently the consistency between Gadget2 and Fi. ### Convergence {#sec:convergence_eda1} Fig. \[fig:convergence\_rates\_eda1a\] shows that, except for the ISM loading rate, all rates have a decreasing trend with increasing resolution, with the exception of the accretion rate onto the star in the low resolution F4 simulation. The decreasing trends of the accretion rate onto the star, the stripping rate and the mass loss of the disc can be understood as follows. When the number of disc particles increases, the average smoothing length of the SPH particles decreases. Therefore, at lower resolution, the viscous time scale is shorter (according to Eq. \[eq:reynolds\_eda1\]) and the transport of mass and angular momentum through the disc is faster than at higher resolutions. Moreover, as we will discuss in Sec. \[sec:consistency\_eda1\], in our simulations with Gadget2 the artificial viscosity in the disc is higher for lower resolutions. This further increases the dependence of mass and angular momentum transport on the number of disc particles. As discussed in Sec. \[sec:reference\_eda1\], there is a physical relation between $\dot{M}$ at the inner edge of the disc, i.e. accretion onto the star, and $\dot{M}$ at the outer edge of the disc, i.e. the ablation rate, which is the trend we observe in Fig. \[fig:convergence\_rates\_eda1a\]. This implies that the transport of angular momentum in the simulations is dominated by numerical effects, as these rates are sensitive to the number of disc particles and have not converged yet. The rate of mass change of the disc follows the same trend as the accretion and ablation rate, since the disc loses less mass at both the inner and outer edge with increasing resolution. The ISM loading rate, however, appears to be independent of the number of disc particles and is almost constant at all resolutions. This implies that the effective cross section with which the disc entrains ISM is roughly equal for all resolutions. Fig. \[fig:convergence\_rates\_eda1c\] shows that the radius, i.e. total surface area, of the disc is larger for lower resolution, but the outskirts of the disc are also more diffuse in that case. As discussed above, if the outer edge of the disc is too diffuse, the ISM particles are not entrained by the disc. At higher resolution, the disc is more compact, as can be seen from both the decreasing radius and increasing mass in Fig. \[fig:convergence\_rates\_eda1c\]. This can again be understood from the concept of mass and angular momentum transport: at high resolution less mass is ablated and more mass remains in the disc, which in turn migrates inwards making the disc more compact. The surface density at the outer edge of the disc is therefore higher for a larger number of disc particles. The net effect across different resolutions is that the effective cross section of the disc remains approximately the same, i.e. the effective cross section depends on the height of the surface density profile in the outskirts of the disc. However, the ratio of the effective cross section over the actual surface area of the disc, $\sigma_{\rm disc}/A_{\rm disc}$, increases with resolution. ![The angular momentum lost from the disc during the final 1000 years in terms of the initial angular momentum of the disc for both Fi (dotted) and Gadget2 (dashed). The angular momentum loss associated with processes in the disc, e.g. accretion of ISM, angular momentum exchange between accreted ISM and original disc material, is plotted in green and angular momentum loss associated with ablation is shown in brown. We have omitted the angular momentum loss due to accretion onto the star, because it is insignificant (as discussed in \[sec:reference\_eda1\]).\[fig:AMlosscomponents\_eda1\]](relative_AM_lost_ref.pdf){width="49.00000%"} Fig. \[fig:convergence\_rates\_eda1b\] shows the amount of angular momentum lost from the disc during the last 1000 years of the simulation as a fraction of its angular momentum at the start of that interval. It shows that more angular momentum is lost at lower resolution. The time scales that can be derived from this angular momentum loss are shown in Tables \[tb:gadget\_eda1\] and \[tb:fi\_eda1\]. These time scales are of the same order as, although mostly consistently smaller than, the time scales derived from the mass loss of the disc. To better understand how the angular momentum is lost, we have split the angular momentum loss during the final 1000 years into two components in Fig. \[fig:AMlosscomponents\_eda1\]: (1) the angular momentum loss associated with ablation and (2) the angular momentum lost due to processes in the disc, i.e. the accretion of ISM and the exchange of angular momentum between the swept-up ISM and original disc material. The angular momentum loss is normalized to the initial angular momentum of the disc so we can compare the components on an absolute scale. We do not show the angular momentum lost due to accretion onto the star, because it is two orders of magnitude smaller than the angular momentum lost due to ablation (see Sec. \[sec:reference\_eda1\]). Ideally, the second component attributed to processes in the disc should be very small. The swept-up ISM should have no net azimuthal angular momentum and the amount of angular momentum gained by swept-up ISM in the disc should equal the amount that is lost by material that remains in the disc. It is not expected to be exactly zero, as the angular momentum that is lost from the outer edge of the disc has been gained at the expense of material that is still in the disc. The component related to stripped material can be separated into two constituents: angular momentum carried away by original disc material and angular momentum carried away by ISM that briefly interacts with the disc, gains angular momentum and then moves along with the flow[^6]. In our simulations with Gadget2 both constituents contribute equally to the angular momentum loss associated with stripping. However, in our simulations with Fi, 93% of the angular momentum loss associated with stripping is actually due to angular momentum taken away by the ISM. This explains why the angular momentum loss associated with stripping differs by roughly a factor two between both codes. Since stripping dominates the total angular momentum loss of the disc, the same factor of two difference between both codes is also seen in $\dot{J}_{\rm disc}$ in Tables \[tb:gadget\_eda1\] and \[tb:fi\_eda1\] when comparing simulations with the same resolution. We therefore argue that the angular momentum loss of the disc is dominated by frictional interactions between the ISM flow and the outer edge of the disc. The lifetime of the disc in our simulations is thus predominantly limited by angular momentum loss associated with this process. Both codes show a decreasing trend of angular momentum loss with increasing resolution and we will discuss in Sec. \[sec:dis\_amloss\_eda1\] to what extent this angular momentum loss is physical. ### Consistency {#sec:consistency_eda1} Although the trend of all quantities in Fig. \[fig:convergence\_rates\_eda1\] is the same for both codes, those that are determined from the simulations with Gadget2 are consistently lower than, or at most equal to, the same quantity determined from the simulations with Fi, with the exception of the disc radius at the lowest resolution. One of the fundamental differences between Gadget2 and Fi is the treatment of the artificial viscosity: in Fi it is constant, i.e. $\alpha_{\rm SPH}=0.1$, while in Gadget2 it can vary by an order of magnitude depending on the local velocity gradient. As mentioned in Sec. \[sec:sdp\_eda1\], during the simulations with Gadget2, $\alpha_{\rm SPH}$ is not equal to 0.1 in the disc, as derived from accretion disc theory and observations (see Sec. \[sec:visc\_eda1\]), but approximately 0.6. In fact, $\alpha_{\rm SPH}$ in the mid-plane of the disc increases steeply inwards as a function of the radius between about $\frac{1}{2}R_{\rm disc}$ and the star. At the outer edge of the disc $\alpha_{\rm SPH}$ is also higher, due to the high velocity gradient caused by the shock. The value of $\alpha_{\rm SPH}$ as a function of radius is reflected by the surface density profiles of the G16 simulation in Fig. \[fig:surface\_density\_eda1a\]. The radius at which $\alpha_{\rm SPH}$ has a minimum in the G16 simulation corresponds to the peak in the surface density profile. Thus matter accumulates at a radius in the disc where the transport of mass and angular momentum is slowest. In contrast, $\alpha_{\rm SPH}$ is constant in the F16 simulation and the surface density profile is therefore broader and less steep than in the G16 simulation (see Fig. \[fig:surface\_density\_eda1b\]). Furthermore, $\alpha_{\rm SPH}$ depends on the number of disc particles: the higher the resolution, the lower $\alpha_{\rm SPH}$ ($\alpha_{\rm SPH} \approx 0.4$ at the highest resolution). Therefore, in a given simulation with Gadget2, more mass is transported inwards, and angular momentum is transported outwards accordingly, than in the same simulation with Fi. Fig. \[fig:AMlosscomponents\_eda1\] shows, as discussed in the previous section, that in the simulations with Fi the angular momentum loss associated with ablation is consistently a factor two lower than in the simulations with Gadget2. This implies that the angular momentum loss not only depends on resolution, but also on the modelling of physical processes. In principle, the smoothing length should be the same in simulations with the same number of particles for both codes. However, since the artificial viscosity is higher at smaller radii in Gadget2, numerical effects start dominating at an earlier stage in the simulation and at these small radii the smoothing length increases as a result of the faster inward movement of particles. At radii smaller than 2 AU, the numerical viscous time scale in the disc becomes smaller than the orbital period. Despite these differences, the ISM loading rate in both codes and at all resolutions agree within a factor 1.5, because we are looking at the whole star and disc system. In the case of Fi, accreted ISM resides in the disc for a longer time before it is accreted onto the star. So to determine the ISM loading rate it is less relevant how rapidly material is transported through the disc, as long as it has been swept up by the disc. Both SPH codes agree that the disc will always lose angular momentum, even if the disc gains mass in some simulations (i.e. the F16 and G128 simulations), and that the angular momentum is predominantly lost from the outer edge of the disc. However, the time scales derived from the angular momentum loss of the disc in the simulations with Fi differ by a factor of three from those derived with Gadget2 (see Table \[tb:gadget\_eda1\] and \[tb:fi\_eda1\]). ### Gadget2 and Fi with the same viscosity implementation We have performed a simulation with 16.000 disc particles using Gadget2 with the same (constant) viscosity implementation as in Fi, i.e. $\alpha_{\rm SPH}=0.1$ and $\beta_{\rm SPH}=1$, to address the differences between the results of Gadget2 and Fi. We label this simulation G16CV and summarize the results in Table \[tb:gadget\_eda1\]. The amount of swept-up ISM and the ISM loading rate are almost the same as in the F16 simulation. The main difference between this simulation and the F16 simulation is that both disc and ISM material are accreted faster onto the star in the G16CV simulation than they are in the F16 simulation. As a result the disc is losing mass in the G16CV simulation, even though no original disc material is stripped from the outer edge of the disc, i.e. $\dot{M}_{\rm strip} = 0\,{\mbox{$\mathrm{M}_{\odot}$}}$/yr. The ISM loading rate in the G16CV simulation is dominated by the accretion rate of ISM onto the star; the amount of ISM in the disc remains roughly constant as a function of time when the system has reached a steady state. This also partly explains the difference with ${\mbox{$\dot{M}_{\mathrm{star}}$}}$ in the G16 simulation, where the accretion rate onto the star is dominated by the accretion of disc material and therefore a factor two lower. As in the F16 simulation, the disc loses its angular momentum predominantly through interaction with the ISM flow at the outer edge of the disc. However, in the G16CV more angular momentum is transferred from the outer edge of the disc to the ISM flow than in the F16 simulation. Even though the viscosity prescription is the same in both simulations, the inward transport of mass occurs faster in the G16CV simulation. We have not been able to pinpoint the exact cause of the difference between the two codes. It could be that the discrepancies are attributable to other differences in e.g. the implementation of the time-stepping or the limiters on the acceleration, that are more difficult to discern. Discussion {#sec:dis_eda1} ========== [l|ccccc]{} **Model**&${\mbox{$\dot{M}_{\mathrm{disc}}$}}$ (${\mbox{$\mathrm{M}_{\odot}$}}/$yr)&${\mbox{$\dot{M}_{\mathrm{star}}$}}$ (${\mbox{$\mathrm{M}_{\odot}$}}/$yr)&${\mbox{$\dot{M}_{\mathrm{ISM}}$}}$ (${\mbox{$\mathrm{M}_{\odot}$}}/$yr)&${\mbox{$\dot{M}_{\mathrm{strip}}$}}$ (${\mbox{$\mathrm{M}_{\odot}$}}/$yr)&$\tau_{\dot{M}}$ (yr)\ G4&$(-4.6 \pm 0.1)\times10^{-7}$&$(-3.20 \pm 0.1)\times10^{-7}$&$(0.79 \pm 0.06)\times10^{-7}$&$(-2.2 \pm 0.2)\times10^{-7}$&$(1.7 \pm 0.1)\times10^{3}$\ G8&$(-2.76 \pm 0.07)\times10^{-7}$&$(-1.93 \pm 0.03)\times10^{-7}$&$(0.94\pm 0.04)\times10^{-7}$&$(-1.78 \pm 0.06)\times10^{-7}$&$(4.5 \pm 0.01)\times10^{3}$\ G16&$-1.68\times10^{-7}$&$-1.27\times10^{-7}$&$ 1.03\times10^{-7}$&$-1.44\times10^{-7}$&$8.5\times10^{3}$\ G32&$-0.66\times10^{-7}$&$-0.86\times10^{-7}$&$ 1.10\times10^{-7}$&$-0.90\times10^{-7}$&$24.0\times10^{3}$\ G64&$-0.15\times10^{-7}$&$-0.54\times10^{-7}$&$ 1.09\times10^{-7}$&$-0.70\times10^{-7}$&$114.0\times10^{3}$\ G128&$0.11\times10^{-7}$&$-0.38\times10^{-7}$&$ 1.13\times10^{-7}$&$-0.64\times10^{-7}$&$\emph{167.5}\times10^{3}$\ G16NF[^7]&$-0.81\times10^{-7}$&$-0.45\times10^{-7}$&$ -$&$-0.35\times10^{-7}$&$48.0\times10^{3}$\ G16CV&$-1.10\times10^{-7}$&$-2.44\times10^{-7}$&$ 1.35\times10^{-7}$&$0\times10^{-7}$&$16.1\times10^{3}$\ \ &${\mbox{$M_\mathrm{disc}$}}$ (${\mbox{$\mathrm{M}_{\odot}$}}$)&${\mbox{$R_\mathrm{disc}$}}$ (AU)&$\dot{J}_{\mathrm{disc}}$ (kgm$^2$s$^{-2}$)&$\tau_{\dot{J}}$ (yr)\ G4&$(7.7 \pm 0.6)\times10^{-4}$&$28.5 \pm 0.6$&$(-2.86 \pm 0.09)\times10^{32}$&$(2.5 \pm 0.1)\times10^{3}$\ G8&$(12.33 \pm 0.02)\times10^{-4}$&$24.8$&$(-2.13 \pm 0.02)\times10^{32}$&$(4.2 \pm 0.1)\times10^{3}$\ G16&$14.31\times10^{-4}$&$21.0$&$-1.72\times10^{32}$&$5.6\times10^{3}$\ G32&$15.76\times10^{-4}$&$20.3$&$-1.22\times10^{32}$&$8.0\times10^{3}$\ G64&$16.76\times10^{-4}$&$20.3$&$-0.98\times10^{32}$&$10.3\times10^{3}$\ G128&$17.67\times10^{-4}$&$18.8$&$-0.90\times10^{32}$&$11.8\times10^{3}$\ G16NF[@xdefthefnmark[\[note1\]]{}footnotemark]{}&$38.61\times10^{-4}$&$98.3$&$-0.67\times10^{32}$&$70.9\times10^{3}$\ G16CV&$17.69\times10^{-4}$&$28.75$&$-1.37\times10^{32}$&$9.7\times10^{3}$\ \[tb:gadget\_eda1\] [l|ccccc]{} **Model**&${\mbox{$\dot{M}_{\mathrm{disc}}$}}$ (${\mbox{$\mathrm{M}_{\odot}$}}/$yr)&${\mbox{$\dot{M}_{\mathrm{star}}$}}$ (${\mbox{$\mathrm{M}_{\odot}$}}/$yr)&${\mbox{$\dot{M}_{\mathrm{ISM}}$}}$ (${\mbox{$\mathrm{M}_{\odot}$}}/$yr)&${\mbox{$\dot{M}_{\mathrm{strip}}$}}$ (${\mbox{$\mathrm{M}_{\odot}$}}/$yr)&$\tau_{\dot{M}}$ (yr)\ F4&$-1.52\times10^{-7}$&$-1.57\times10^{-7}$&$1.10\times10^{-7}$&$-1.06\times10^{-7}$&$11.6\times10^{3}$\ F8&$-0.50 \times10^{-7}$&$-1.70 \times10^{-7}$&$ 1.43\times10^{-7}$&$-0.23\times10^{-7}$&$37.8\times10^{3}$\ F16&$0.11\times10^{-7}$&$-1.29 \times10^{-7}$&$1.50\times10^{-7}$&$-0.10\times10^{-7}$&$\emph{187.4}\times10^{3}$\ \ &${\mbox{$M_\mathrm{disc}$}}$ (${\mbox{$\mathrm{M}_{\odot}$}}$)&${\mbox{$R_\mathrm{disc}$}}$ (AU)&$\dot{J}_{\mathrm{disc}}$ (kgm$^2$s$^{-2}$)&$\tau_{\dot{J}}$ (yr)\ F4&$17.6 \times10^{-4}$&$28.0$&$-1.60\times10^{32}$&$7.9\times10^{3}$\ F8&$18.7 \times10^{-4}$&$26.0$&$-1.04\times10^{32}$&$12.2\times10^{3}$\ F16&$20.0 \times10^{-4}$&$24.5$&$-0.73\times10^{32}$&$18.1\times10^{3}$\ \[tb:fi\_eda1\] We discuss the uncertainties in the modelling of the physical processes and other caveats of our simulations below. After discussing the limitations in our simulations, we compare to other work and finally we discuss the implications of our results. Angular momentum loss {#sec:dis_amloss_eda1} --------------------- As discussed in Sec. \[sec:consistency\_eda1\], the angular momentum loss due to ablation of the disc is strongly dependent on the artificial viscosity parameter $\alpha_{\rm SPH}$ in the disc and the ablation rate of the disc also shows a decreasing trend as a function of the resolution (see Sec. \[sec:convergence\_eda1\]). Furthermore, in our simulations with Fi angular momentum is predominantly lost to the ISM through angular momentum exchange at the outer edge of the disc, instead of being carried away by stripped disc material. The angular momentum loss associated with these processes shows a decreasing trend with increasing resolution. This suggests that the ablation in our simulation, and the associated angular momentum loss, is dominated by numerical effects and that physically it may not be the dominant process for the loss of angular momentum from the disc. In the simulation of M09 with the same version of Gadget2, but with a higher resolution and smaller flow velocity (i.e. lower ram pressure), the stripping of the outer edge does not play a significant role (see Sec. \[sec:dis\_m09\_eda1\]). If the ablation of the disc is predominantly numerical, then the time scale on which the disc loses its mass can not be considered a reliable indicator for the lifetime of the disc. Although the time scale of angular momentum loss is generally shorter than the time scale derived from the mass loss of the disc, we consider the former to be a more reliable estimator for the disc life time. In particular, we consider the time scale of angular momentum loss determined with Fi to be more indicative, because in those simulations the viscosity in the disc agrees better with accretion disc theory and the disc actually gains mass. The angular momentum loss time scale in the F16 simulation is similar to that in the G128 simulation, i.e. highest resolution with Gadget2, but it is still an order of magnitude smaller than the physical estimate in Sec. \[sec:visc\_eda1\]. We can not directly interpret the time scale of angular momentum loss as the lifetime of the disc, since it is dominated by numerical effects in our simulations. Physically, the decrease in specific angular momentum of the disc due to accretion of ISM with zero azimuthal angular momentum may contribute equally to, or even dominate, the decreasing disc size. Both processes are also non-linear and we consider our estimate of the angular momentum loss time scale as a lower limit to the actual lifetime of the disc. Viscous evolution ----------------- We have used two SPH codes that model the viscosity in a different way. It is clear that the transport of mass and angular momentum in the disc behaves differently in both codes even if the same viscosity prescription is implemented in both codes. To model the (viscous) evolution of the disc correctly, additional relevant physical processes, e.g. magnetohydrodynamics, radiative transfer, radial mixing (which are not all physically well understood and beyond the scope of this paper) should be incorporated. In this work, we are interested in how much mass a protoplanetary disc can sweep up and how this process would affect its lifetime. The result for the ISM loading appears to be independent of the viscous modelling of the disc and we therefore conclude that this result is robust. Furthermore, the simulations indicate that the process of face-on accretion onto a protoplanetary disc will likely shorten rather than prolong its lifetime. Resolution ---------- As mentioned in Sec. \[sec:set-up\_eda1\], we try to find a balance between computing time and convergence. Our reference model, G16, took 3.5 days to run on 32 cores, while the comparison F16 simulation took a little more than two months on 22 CPUs. When looking at the ablation rate as a function of increasing resolution, the number of particles that are stripped increases, but the mass that they carry away decreases more rapidly. We therefore argue that the ablation rate is not caused by numerical noise but rather, as discussed above, by the treatment of viscosity in the disc. This is supported by the result that the Poisson noise for the low-resolution models that we have performed multiple times is less than 5%, see table \[tb:gadget\_eda1\]. Furthermore, the ISM loading rate is robust for all our simulations and does not seem to be affected by numerical noise. Modelling of the ISM -------------------- In order to be able to discern the relevant physical processes from the simulations, we have modelled the ISM in a very idealized way: as a homogeneous gas with no clumps and no turbulence. To a certain extent we have modelled the reaction of the disc to a density gradient when the flow first hits the disc. In that case the size of the disc is determined by the ram pressure. If the disc were to encounter a region of gas with lower density, the surface area of the disc can extend to larger radii without being stripped and that would probably increase its effective cross section. Despite the lower density, the ISM loading rate could still be of the same order, because a lower ISM density allows a larger surface area of the disc to collect ISM. In that sense, accretion onto protoplanetary discs may be a self-regulating process. In a follow-up work we will perform simulations with different ISM densities and velocities to investigate if and how the ISM loading rate depends on these quantities. In principle it would be possible to add angular momentum to the disc, such that it maintains or prolongs its lifetime, if the ISM has turbulence or substructures on scales at or slightly smaller than the disc scale. However, even in that case the net effect is unlikely to increase the overall lifetime, since the disc would quickly encounter a different part of the ISM where an adverse, disruptive configuration of substructure might exist. Comparison to other work {#sec:dis_m09_eda1} ------------------------ A similar simulation has been performed by M09, but for a higher disc mass and lower velocity of the ISM with respect to the disc. In their work both the Bondi-Hoyle radius and the radius beyond which ram pressure would dominate are much larger than the radius of the disc. They used a higher resolution for the disc, initially $\sim 2.5 \times 10^5$ particles, which had 8 times the mass of their ISM particles. However, in their work the ISM also does not reside in the outer regions of the disc (see their Fig. 3), meaning that even at much higher resolution the effective surface area of the disc is smaller than its actual surface area. We interpret this as a physical effect: at the outer edge of the disc, the ISM is forced to flow around it and is not entrained. Furthermore, the disc in the simulation of M09 also shows a decreasing radius and steepening surface density profile as a function of time. They attribute this to the redistribution, i.e. inward movement, of disc material due to the accretion of ISM with no angular momentum. The loss of disc material to the ISM in their simulation is negligible, the disc mass only decreases due to accretion onto the star. This agrees with the trend suggested in our convergence study, i.e. that the continuous stripping of the disc at the outer edge is a numerical artefact. However, it could also be due to the much lower velocity in their work. A future parameter study at other, e.g. lower, densities and velocities of the ISM could provide a more decisive answer. Implications for the early disc accretion scenario -------------------------------------------------- Although the ISM loading rate we find corresponds relatively well to the geometric rate used in B13 for their early disc accretion scenario, they assume a disc radius that is significantly larger, i.e. 100 AU, than the radius found in our simulation. Moreover, the size of the disc decreases continuously during the process of accretion and we have assumed the idealized case in which the disc is positioned exactly perpendicular to the flow. The ISM loading rate derived in this work is therefore most probably an upper limit for the average rate one would expect for a population of discs that all have different inclinations between their rotational axis and velocity vector. Furthermore, B13 assume a disc lifetime of $10^7$ years from observations, while we find that angular momentum transport plays a significant role in shortening the lifetime of the disc in dense ISM environments. It therefore seems unlikely that a low-mass star can accrete of the order of its own mass via its protoplanetary disc, as required in the early disc accretion scenario. Implications for planet formation --------------------------------- The process of accretion onto protoplanetary discs may also affect planet formation. It could play a role in the formation of ‘hot Jupiters’, i.e. massive planets that have formed further out in the protoplanetary disc and migrated inwards. The mass loading and consequent redistribution may also increase the probability of forming a planet in the habitable zone. @ronco14 found that a steeper surface density profile, i.e. more mass at smaller radii, increases the probability of forming a planet with a significant water content in the habitable zone. The constraint for an initial steep surface density profile could perhaps be eased, when the entraining of ISM onto the protoplanetary disc causes the disc material to migrate inward. These suggestions could be tested by incorporating our findings in detailed planet formation models. Conclusions =========== We have performed simulations of accretion of interstellar material (ISM) onto a protoplanetary disc with two different smoothed particle hydrodynamics codes. We find that, as theoretically expected, when the flow of ISM first hits the disc, all disc material beyond the radius where ram pressure dominates is stripped. As ISM is being accreted and disc material migrates inwards, the disc becomes more compact and the surface density profile increases at smaller radii. We find that the ISM loading rate, i.e. the rate at which ISM is entrained by the disc and star, is approximately constant across all our simulations with both codes and is a factor of two lower than the rate expected from geometric arguments (see Eq. \[eq:mdot\_eda1\]). This difference arises because the outskirts of the disc do not entrain ISM and therefore the effective cross section of the disc is smaller than its physical surface area. We find that, despite the accretion of ISM, the net effect is that the disc loses mass, except in the highest-resolution simulations with both codes where the disc gains mass. This decreasing trend with resolution implies that the net mass loss from the disc in our low-resolution simulations is numerical. Considering the time scale on which the disc loses all of its angular momentum rather than its mass, provides an estimate of about $10^4$ years. The angular momentum loss from the disc in our simulations is dominated by continuous stripping of disc material and by transfer of angular momentum to the ISM as it flows past the outer edge of the disc. Our convergence and consistency study as well as previous work indicate that this these effects are predominantly numerical. The time scale estimated from the simulations with the highest resolution therefore provide a lower limit to the lifetime of the disc. The loss of angular momentum due to accretion of disc material onto the star, which is governed by the (modelling of) viscous processes in the disc, is two orders of magnitude smaller than the loss associated with stripping. Even if the disc grows in mass, the (specific) angular momentum of the disc will always decrease in this scenario, if not for the aforementioned angular momentum loss processes then by accretion of ISM with no azimuthal angular momentum. Either way, the disc will shrink in size, thereby decreasing its effective cross section. Although our ISM loading rate agrees within a factor of two with the geometrically estimated rate, the lifetime and size of the disc are probably not sufficient to accrete the amount of mass required in the early disc accretion scenario. In future work we will extend our simulations to explore the parameter-space and conditions that correspond to a broader range of stellar environments in order to find a parametrization for the mass loading rate onto a protoplanetary disc system in terms of the density and velocity of the ambient medium and the size of the disc. We thank Selma de Mink, Nate Bastian and Nickolas Moeckel for valuable discussions and input. We are also grateful to the referee whose valuable comments have improved this work. This research is funded by the Netherlands Organisation for Scientific Research (NWO) under grant 614.001.202. NWO also granted computational resources on Cartesius under grant SH-295-14. [^1]: Most scenarios proposed to date imply subsequent epochs of star formation and hence refer to multiple *generations* of stars in a GC. Since it is still not clear whether GCs can facilitate an extended star formation history or if the enriched stars actually belong to the initial population, we will refer to multiple *populations* of stars. [^2]: <http://www.amusecode.org> [^3]: <http://www.mpa-garching.mpg.de/gadget> [^4]: This implementation is done in the source code of Gadget2, which is not according to the philosophy of AMUSE, but always possible if required. [^5]: We have done only one simulation for each resolution with Fi, because it is computationally more expensive. 16000 disc particles is the highest resolution we could simulate with Fi in a reasonable amount of time. [^6]: The Hop algorithm considers this ISM to have been associated with the disc during a few time steps. This is an artefact of the Hop algorithm, but these interactions do carry away angular momentum from the outer edge of the disc nonetheless. In general it is difficult to truly disentangle all components, also considering that angular momentum stripped from the outer edge of the disc has been gained from material that still resides in the disc. We therefore only provide the total angular momentum loss rates and time scales. [^7]: \[note1\] The stripping rates and time scales derived for this simulation correspond to viscous spreading and not to actual stripping.

--- abstract: 'Multiwavelength data on star-forming galaxies provide strong evidence for large-scale galactic winds in both nearby and distant objects. The results from recent ground-based and space-borne programs are reviewed. The impact of these winds on the host galaxies and the surrounding environment is discussed in the context of galaxy evolution.' author: - Sylvain Veilleux title: 'Galactic Winds: Near and Far' --- \#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} = \#1 1.25in .125in .25in Introduction ============ Galactic winds that extend on a scale comparable to the host galaxies are now known to be a common feature both locally and at high redshifts. These winds are significant carriers of mass, momentum, and energies that may impact the formation and evolution of the host galaxies and the surrounding intergalactic medium. Given the scope of this conference, the present review focusses exclusively on starburst-driven winds. AGN-driven galactic winds, perhaps a very important phenomenon in the lives of galaxies with spheroids (Kormendy & Gebhardt 2001), are not discussed here (see, e.g., Veilleux et al. 2002a for a recent review of this topic). Due to space limitations, the emphasis of this review is on the recent ($\ga$ 1998) literature. Readers interested in results from earlier studies may refer to the reviews by Strickland (2002) and Heckman (2002). First, the basic physics of starburst-driven winds is described briefly in §2. An observational summary of the properties of local winds is given in the preamble to §3. The remainder of §3 describes detailed data on three well-studied cases of local starburst-driven winds, and summarizes the evidence for winds in luminous and ultraluminous infrared galaxies and distant Lyman break galaxies. This section often emphasizes the importance of using multiwavelength data to draw a complete picture of this complex multi-phase phenomenon. The impact of starburst-driven winds on the host galaxies and their environment is discussed briefly in §4. Here the focus of the discussion is on the existence and properties of the wind fluid and on the size of the “zone of influence” of these winds. A summary is given in §5. Basic Physics ============= The driving force behind starburst-driven winds is the mechanical energy from stellar winds and supernova events (e.g., Chevalier & Clegg 1985). This mechanical energy is quickly thermalized to produce a hot cavity with a temperature $\sim 10^8\;\Lambda^{-1}$ K, where $\Lambda = M_{\mathrm total}/M_{\mathrm ejecta} \ge 1$ is the mass-loading term. This over-pressured cavity expands through the ambient medium, sweeping this material up in the process to produce a bubble-like structure. The complex interaction between the wind and the ISM of the host galaxy has been the subject of several numerical simulations (e.g., MacLow & McCray 1988; Suchkov et al. 1994, 1996; MacLow & Ferrara 1999; D’Ercole & Brighenti 1999; Strickland & Stevens 2000; Silich & Tenorio-Tagle 2001). If radiative energy losses are negligible (probably a good assumption in some objects; e.g., Heckman et al. 2001), the bubble expands adiabatically through the galaxy ISM with a velocity $\sim 100~n_0^{-0.2}\;\dot{E}_{42}^{0.2}\;t_7^{-0.4}$ km s$^{-1}$, where $n_0$ is the ambient nucleon density in cm$^{-3}$, $\dot{E}_{42}$ is the rate of deposition of mechanical energy in 10$^{42}$ erg s$^{-1}$, and $t_7$ is the age of the bubble in 10$^7$ years (e.g., Weaver et al. 1977). A powerful starburst may inject enough energy to produce a cavity of hot gas that can burst out of the disk ISM, at which point the dense walls of the bubble start accelerating outward, become Rayleigh-Taylor unstable, and break up into cloudlets and filaments. If halo drag is negligible (probably [*not*]{} a good assumption in general), the wind fluid may reach terminal velocities as high as $\sim$ 3000 $\Lambda^{-1}$ km s$^{-1}$, well in excess of the escape velocity of the host galaxy. In contrast, the terminal velocities of clouds accelerated by the wind are more modest, of order $\sim 600~\dot{p}_{34}^{0.5}\;\Omega_w^{-0.5}\;r_{\rm 0,kpc}\;N_{\rm cloud,21}^{-0.5}$, where $\dot{p}_{34}$ is the wind momentum flux in 10$^{34}$ dynes, $\Omega_W$ is the solid angle of the wind in steradians, $r_{\rm 0,kpc}$ is the initial position of the cloud in kpc, and $N_{\rm cloud,21}$ is the column density of the cloud in 10$^{21}$ cm$^{-2}$ (Strel’nitskii & Sunyaev 1973; Heckman et al. 2000). A critical quantity in all of these calculations is the thermalization efficiency, or the percentage of the mechanical energy from the starburst that goes into heating the gas. Unfortunately, this quantity is poorly constrained observationally. Most simulations assume a thermalization efficiency of 100%, i.e. none of the energy injected by the starburst is radiated away. In reality, this efficiency depends critically on the environment, and is likely to be significantly less than 100% in the high-density environment of powerful nuclear starbursts (e.g., Thornton et al. 1998; Strickland & Stevens 2000; Silich, Tenorio-Tagle, & Muñoz-Tuñón 2003). Galactic magnetic fields may also “cushion” the effects of the starburst on the ISM, and reduce the impact of the galactic wind on the host galaxy and its environment (e.g., Tomisaka 1990; Ferrière et al. 1991; Slavin & Cox 1992; Mineshinge et al. 1993; Ferrière 1998). Observed Properties of Galactic Winds ===================================== A great number of surveys have provided important statistical information on galactic winds in the local universe (e.g., Heckman, Armus, & Miley 1990; Veilleux et al. 1995; Lehnert & Heckman 1995, 1996; Gonzalez Delgado et al. 1998; Heckman et al. 2000, Rupke, Veilleux, & Sanders 2002, 2003, in prep.). Galaxy-scale winds are common among galaxies with global star formation rates per unit area $\Sigma_* \equiv SFR / \pi R_{\rm opt}^2 \ga 0.1$ M$_\odot$ yr$^{-1}$ kpc$^{-2}$, where $R_{\rm opt}$ is the optical radius. This general rule-of-thumb also appears to apply to ultra/luminous infrared galaxies (see §3.3) and distant Lyman break galaxies (see §3.4). “Quiescent” galaxies with global star formation rates per unit area below this threshold often show signs of galactic fountaining in the forms of warm, ionized extraplanar material a few kpc above or below the galactic disks (e.g., Miller & Veilleux 2003a, 2003b and references therein). The energy input from stellar winds and supernovae in these objects elevates some of the ISM above the disk plane, but is not sufficient to produce large-scale winds. This rule-of-thumb is conservative since a number of known wind galaxies, including our own Galaxy (§3.1) and several dwarf galaxies, have $\Sigma_* <<$ 0.1 M$_\odot$ yr$^{-1}$ kpc$^{-2}$ (e.g., Hunter & Gallagher 1990, 1997; Meurer et al. 1992; Marlowe et al. 1995; Kunth et al. 1998; Martin 1998, 1999; see Kunth’s and Martin’s contributions at this conference). The production of detectable winds probably depends not only on the characteristics of the starburst (global [*and*]{} local $\Sigma_*$, starburst age), but also on the detailed properties of the ISM in the host galaxies (e.g., see the theoretical blowout criterion of MacLow & McCray 1988). The winds in actively star-forming galaxies in the local universe show a very broad range of properties, with opening angles of $\sim$ 0.1 – 0.5 $\times$ (4$\pi$ sr), radii ranging from $<$ 1 kpc to several 10s of kpc, outflow velocities of a few 10s of km s$^{-1}$ to more than 1000 km s$^{-1}$ (with clear evidence for a positive correlation with the temperature of the gas phase), total (kinetic and thermal) outflow energies of $\sim$ 10$^{53}$ – 10$^{57}$ ergs or 5 – 20% of the total mechanical energy returned to the ISM by the starburst, and mass outflow rates ranging from $<$ 1 M$_\odot$ yr$^{-1}$ to $>$ 100 M$_\odot$ and scaling roughly with the star formation rates (see §3.3 below). In the remainder of this section, we discuss a few well-studied cases of galactic winds in the local universe and summarize the evidence for winds in luminous and ultraluminous infrared galaxies at low and moderate redshifts as well as in distant Lyman break galaxies. The Milky Way ------------- By far the closest case of a large-scale outflow is the wind in our own Galaxy. Evidence for a dusty bipolar wind extending $\sim$ 350 pc ($\sim$ 1$^\circ$) above and below the disk of our Galaxy has recently been reported by Bland-Hawthorn & Cohen (2003) based on data from the Midcourse Space Experiment (MSX). The position of the warm dust structure coincides closely with the well-known Galactic Center Lobe detected at radio wavelengths (e.g., Sofue 2000 and references therein). Simple arguments suggest that the energy requirement for this structure is of order $\sim$ 10$^{55}$ ergs with a dynamical time scale of $\sim$ 1 Myr. Bland-Hawthorn & Cohen (2003) also argue that the North Polar Spur, a thermal X-ray/radio loop that extends from the Galactic plane to $b = +80^\circ$ (e.g., Sofue 2000), can naturally be explained as an open-ended bipolar wind, when viewed in projection in the near field. This structure extends on a scale of 10 – 20 kpc and implies an energy requirement of $\sim$ (1 – 30) $\times$ 10$^{55}$ ergs and a dynamical timescale of $\sim$ 15 Myr, i.e. considerably longer than that of the smaller structure seen in the MSX maps. If confirmed, this may indicate that the Milky Way Galaxy has gone through multiple galactic wind episodes. Bland-Hawthorn & Cohen (2003) point out that the North Polar Spur would escape detection in external galaxies; it is therefore possible that the number of galaxies with large-scale winds has been (severely?) underestimated. Nearby Starburst Galaxies ------------------------- Two classic examples of starburst-driven outflows are described in this section to illustrate the wide variety of processes taking place in these objects. 0.1in [**M 82.**]{} This archetype starburst galaxy hosts arguably the best studied galactic wind. Some of the strongest evidence for the wind is found at optical wavelengths, where long-slit and Fabry-Perot spectroscopy of the warm ionized filaments above and below the disk shows line splittings of up to $\sim$ 250 km s$^{-1}$, corresponding to deprojected velocities of order 525 – 655 km s$^{-1}$ (e.g., McKeith et al. 1995; Shopbell & Bland-Hawthorn 1998). Combining these velocities with estimates for the ionized masses of the outflowing filamentary complex, the kinetic energy involved in the warm ionized outflow is of order $\sim$ 2 $\times$ 10$^{55}$ ergs or $\sim$ 1% of the total mechanical energy input from the starburst. The ionized filaments are found to lie on the surface of cones with relatively narrow opening angles ($\sim$ 5 – 25$^\circ$) slightly tilted ($\sim$ 5 – 15$^\circ$) with respect to the spin axis of the galaxy. Deep narrow-band images of M82 have shown that the outflow extends out to at least 12 kpc on one side (e.g., Devine & Bally 1999), coincident with X-ray emitting material seen by $ROSAT$ (Lehnert, Heckman, & Weaver 1999) and $XMM$-Newton (Stevens, Read, & Bravo-Guerrero 2003). The wind fluid in this object has apparently been detected by both $CXO$ (Griffiths et al. 2000) and $XMM$-Newton (Stevens et al. 2003). The well-known H I complex around this system (e.g., Yun et al. 1994) may be taking part, and perhaps even focussing, the outflow on scales of a few kpc (Stevens et al. 2003). Recently published high-quality CO maps of this object now indicate that some of the molecular material in this system is also involved in the large-scale outflow (Walter, Weiss, & Scoville 2002; see also Garcia-Burillo et al. 2001). The outflow velocities derived from the CO data ($\sim$ 100 km s$^{-1}$ on average) are considerably lower than the velocities of the warm ionized gas, but the mass involved in the molecular outflow is substantially larger ($\sim$ 3 x 10$^8$ M$_\odot$), implying a kinetic energy ($\sim$ 3 $\times$ 10$^{55}$ ergs) that is comparable if not larger than that involved in the warm ionized filaments. The molecular gas is clearly a very important dynamical component of this outflow. 0.1in [**NGC 3079.** ]{} An outstanding example of starburst-driven superbubble is present in the edge-on disk galaxy, NGC 3079. High-resolution $HST$ H$\alpha$ maps of this object show that the bubble is made of four separate bundles of ionized filaments (Cecil et al. 2001). The two-dimensional velocity field of the ionized bubble material derived from Fabry-Perot data (Veilleux et al. 1994) indicates that the ionized bubble material is entrained in a mushroom vortex above the disk with velocities of up to $\sim$ 1500 km s$^{-1}$ (Cecil et al. 2001). A recently published X-ray map obtained with the $CXO$ (Cecil, Bland-Hawthorn, & Veilleux 2002) reveals excellent spatial correlation between the hot X-ray emitting gas and the warm optical line-emitting material of the bubble, suggesting that the X-rays are being emitted either as upstream, standoff bow shocks or by cooling at cloud/wind conductive interfaces. This good spatial correlation between the hot and warm gas phases appears to be common in galactic winds (Strickland et al. 2000, 2002; Veilleux et al. 2003, and references therein). The total energy involved in the outflow of NGC 3079 appears to be slightly smaller than that in M 82, although it is a lower limit since the total extent of the X-ray emitting material beyond the nuclear bubble of NGC 3079 is not well constrained (Cecil et al. 2002). Contrary to M 82, the hot wind fluid that drives the outflow in NGC 3079 has not yet been detected, and evidence for entrained molecular gas is sparse and controversial (e.g., Irwin & Sofue 1992; Baan & Irwin 1995; Israel et al. 1998; but see Koda et al. 2002). Luminous and Ultraluminous Infrared Galaxies. --------------------------------------------- Given that the far-infrared energy output of a (dusty) galaxy is a direct measure of its star formation rate, it is not surprising [*a posteriori*]{} to find evidence for large-scale galactic winds in several luminous and ultraluminous infrared galaxies (LIRGs and ULIRGs; e.g., Heckman et al. 1990; Veilleux et al. 1995). Systematic searches for winds have been carried out in recent years in these objects to look for the unambiguous wind signature of blueshifted absorbing material in front of the continuum source (Heckman et al. 2000; Rupke et al. 2002). The feature of choice to search for outflowing neutral material in galaxies of moderate redshifts ($z \la$ 0.6) is the Na ID interstellar absorption doublet at 5890, 5896 Å. The wind detection frequency derived from a set of 44 starburst-dominated LIRGs and ULIRGs is high, of order $\sim$ 70 – 80% (Rupke et al. 2002, 2003 in prep.; also see Rupke’s and Martin’s contributions at this conference). The outflow velocities reach values in excess of 1700 km s$^{-1}$ (even more extreme velocities are found in some AGN-dominated ULIRGs). A simple model of a mass-conserving free wind (details of the model are given in Rupke et al. 2002) is used to infer mass outflow rates in the range $\dot{M}_{\mathrm{tot}}$(H)$\;= {\mathrm few} - 120\;$ for galaxies hosting a wind. These values of $\dot{M}_{\mathrm{tot}}$, normalized to the corresponding global star formation rates inferred from infrared luminosities, are in the range $\eta \equiv \dot{M}_{\mathrm{tot}} / \mathrm{SFR} = 0.01 - 1$. The parameter $\eta$, often called the “mass entrainment efficiency” or “reheating efficiency” shows no dependence on the mass of the host (parameterized by host galaxy kinematics and absolute $R$- and $K^{\prime}$-band magnitudes), but there is a possible tendency for $\eta$ to decrease with increasing infrared luminosities (i.e. star formation rates). The large molecular gas content in ULIRGs may impede the formation of large-scale winds and reduce $\eta$ in these objects. A lower thermalization efficiency (i.e. higher radiative efficiency) in these dense gas-rich systems may also help explain the lower $\eta$ (Rupke et al. 2003, in prep.; see Rupke’s contribution at this conference). Lyman Break Galaxies -------------------- Evidence for galactic winds has now been found in a number of z $\sim$ 3 – 5 galaxies, including an important fraction of Lyman break galaxies (LBGs; e.g., Franx et al. 1997; Pettini et al. 2000, 2002; Frye, Broadhurst, & Benitez 2002; Dawson et al. 2002; Ajiki et al. 2002; Adelberger et al. 2003; Shapley et al. 2003). The best studied wind at high redshift is that of the gravitationally lensed LBG MS 1512-cB58 (Pettini et al. 2000, 2002). An outflow velocity of $\sim$ 255 km s$^{-1}$ is derived in this object, based on the positions of the low-ionization absorption lines relative to the rest-frame optical emission lines (Ly$\alpha$ is to be avoided for this purpose since resonant scattering and selective dust absorption of the Ly$\alpha$ photons may severely distort the profile of this line; e.g., Tenorio-Tagle et al. 1999). The mass-conserving free wind model of Rupke et al. (2002) applied to MS 1512-cB58 (for consistency) results in a mass outflow rate of $\sim$ 20 $M_\odot$ yr$^{-1}$, equivalent to about 50% the star formation rate of this galaxy based on the dust-corrected UV continuum level. Similar outflow velocities are derived in other LBGs (Pettini et al. 2001). The possibly strong impact of these LBG winds on the environment at high $z$ is discussed in the next section (§4.2). Implications ============ Starburst-driven winds may have a strong influence on structure formation at high redshifts, on the “porosity” of star-forming galaxies (i.e. the probability for ionizing photons to escape their host galaxies), hence the nature of the extragalactic UV and infrared background, and on the chemical and thermal evolution of galaxies and their environment. Due to space limitations, this review only addresses the last issue. Heating and Enrichment of the ISM and IGM ----------------------------------------- [**Hot Metal-Enriched Gas.**]{} Nuclear starbursts inject both mechanical energy and metals in the centers of galaxies. This hot, chemically-enriched material, the driving engine of galactic winds, is eventually deposited on the outskirts of the host galaxies, and contributes to the heating and metal enrichment of galaxy halos and the IGM. Surprisingly little evidence exists for the presence of this enriched wind fluid. This is due to the fact that the wind fluid is tenuous and hot and therefore very hard to detect in the X-rays. The current best evidence for the existence of the wind fluid is found in M 82 (Griffiths et al. 2000; Stevens et al. 2003), NGC 1569 (Martin, Kobulnicky, & Heckman 2002), and possibly the Milky Way (e.g., Koyama et al. 1989; Yamauchi et al. 1990). The ratio of alpha elements to iron appears to be slightly super-solar in the winds of both NGC 1569 and M 82, as expected if the stellar ejecta from SNe II are providing some, but not all of the wind fluid. 0.1in [**Selective Loss of Metals.** ]{} The outflow velocities in LIRGs and ULIRGs do not appear to be correlated with the rotation velocity (or equivalently, the escape velocity) of the host galaxy, implying selective loss of metal-enriched gas from shallower potentials (Heckman et al. 2000; Rupke et al. 2002). If confirmed over a broader range of galaxy masses (e.g., Martin 1999; but see the contribution by Martin at this conference for a word of warning), this result may help explain the mass-metallicity relation and radial metallicity gradients in elliptical galaxies and galaxy bulges and disks (e.g., Bender, Burstein, & Faber 1993; Franx & Illingworth 1990; Carollo & Danziger 1994; Zaritsky et al. 1994; Trager et al. 1998). The ejected gas may also contribute to the heating and chemical enrichment of the ICM in galaxy clusters (e.g., Dupke & Arnaud 2001; Finoguenov et al. 2002, and references therein). 0.1in [**Dust Outflows.**]{} Galactic winds also act as conveyor belts for the dust in the hosts. The evidence for a large-scale dusty outflow in our own Galaxy has already been mentioned in §3.1 (Bland-Hawthorn & Cohen 2003). Far-infrared maps of external galaxies with known galactic winds show extended dust emission along the galaxy minor axis, suggestive of dust entrainment in the outflow (e.g., Hughes, Gear, & Robson 1994; Alton et al. 1998, 1999; Radovich, Kahanpää, & Lemke 2001). Direct evidence is also found at optical wavelengths in the form of elevated dust filaments in a few galaxies (e.g., NGC 1808, Phillips 1993; NGC 3079, Cecil et al. 2001). A strong correlation between color excesses, $E(B - V)$, and the equivalent widths of the blueshifted low-ionization lines in star-forming galaxies at low (e.g., Armus, Heckman, & Miley 1989; Veilleux et al. 1995; Heckman et al. 2000; Rupke et al. 2003) and moderate-to-high redshifts (e.g., Rupke et al. 2003; Shapley et al. 2003) provides additional support for the prevalence of dust outflows. Assuming a Galactic dust-to-gas ratio, Heckman et al. (2000) estimate that the dust outflow rate is about 1% of the total mass outflow rate in LIRGs. Dust ejected from galaxies may help feed the reservoir of intergalactic dust (e.g., Coma cluster; Stickel et al. 1998). Zone of Influence of Winds -------------------------- The impact of galactic winds on the host galaxies and the environment depends sensitively on the size of the “zone of influence” of these winds, i.e. the region affected either directly (e.g., heating, metals) or indirectly (e.g., ionizing radiation) by these winds. This section summarizes the methods used to estimate this quantity. 0.1in [**Indirect Measurements based on Estimates of the Escape Velocity.**]{} The true extent of galactic winds is difficult to determine in practice due to the steeply declining density profile of both the wind material and the host ISM. The zone of influence of galactic winds is therefore often estimated using indirect means which rely on a number of assumptions. A popular method is to use the measured velocity of the outflow and compare it with the local escape velocity derived from some model for the gravitational potential of the host galaxy. If the measured outflow velocity exceeds the predicted escape velocity [*and*]{} if the halo drag is negligible, then the outflowing material is presumed to escape the host galaxy and be deposited in the IGM on scales $\ga$ 50 – 100 kpc. This method was used for instance by Rupke et al. (2002) to estimate the average escape fraction $\langle f_{\mathrm{esc}} \rangle \equiv \sum \dot{M}_{\mathrm{esc}}^i / \sum \dot{M}_{\mathrm{tot}}^i$ and “ejection efficiency” $\langle\delta\rangle \equiv \sum \dot{M}_{\mathrm{esc}}^i / \sum \mathrm{SFR}^i$ for 12 ULIRGs, which were found to be $\sim 0.4-0.5$ and $\sim 0.1$, respectively. These calculations assumed that the host galaxy could be modeled as a singular isothermal sphere truncated at some radius $r_{\rm max}$. Neither the escape fraction nor the ejection efficiency were found to be sensitive to the exact value of $r_{\rm max}$. Other strong cases for escaping material include the “H$\alpha$ cap” of M 82 and the $\sim$ 1500 km s$^{-1}$ line-emitting material in the superbubble of NGC 3079; both objects were discussed in §3.2. Note that the outflow velocities measured by Rupke et al. (2002) refer to the neutral component of the outflow, not the hot enriched wind fluid. Unfortunately, direct measurements of the wind velocity are not yet technically possible so one generally relies on the expected terminal velocity of an adiabatic wind at the measured X-ray temperature $T_X$ \[$v_X \sim (5KT_X/\mu)^{0.5}$, where $\mu$ is the mean mass per particle\] to provide a lower limit to the velocity of the wind fluid (this is a lower limit because it only takes into account the thermal energy of this gas and neglects any bulk motion; e.g., Chevalier & Clegg 1985; Martin 1999; Heckman et al. 2000). Arguably the single most important assumption made to determine the fate of the outflowing gas is that halo drag is negligible. Silich & Tenorio-Tagle (2001) have argued that halo drag may severely limit the extent of the wind and the escape fraction. Drag by a dense halo or a complex of tidal debris may be particularly important in ULIRGs if they are created by galaxy interactions (e.g., Veilleux, Kim, & Sanders 2002b). 0.1in [**Deep X-ray and Optical Maps of Local Starbursts.**]{} The fundamental limitation in directly measuring the zone of influence of winds is the sensitivity of the instruments. Fortunately, $CXO$ and $XMM$-Newton now provide powerful tools to better constrain the extent of the hot medium (e.g., M 82, Stevens et al. 2003; NGC 3079, Cecil et al. 2002; NGC 6240, Komossa et al. 2003; Veilleux et al. 2003; NGC 1511, Dahlem et 2003). The reader should refer to the contribution of M. Ehle at this conference for a summary of recent X-ray results (see also Strickland et al. 2003 and references therein). The present discussion focusses on optical constraints derived from the detection of warm ionized gas on the outskirts of wind hosts. Progress in this area of research has been possible thanks to advances in the fabrication of low-order Fabry-Perot etalons which are used as tunable filters to provide monochromatic images over a large fraction of the field of view of the imager. The central wavelength (3500 Å – 1.0 $\mu$m) is tuned to the emission-line feature of interest and the bandwidth (10 – 100 Å) is chosen to minimize the sky background. Continuum and emission-line images are produced nearly simultaneously thanks to a “charge shuffling/frequency switching” mode, where the charges are moved up and down within the detector at the same time as switching between two discrete frequencies with the tunable filter, therefore averaging out temporal variations associated with atmospheric lines and transparency, seeing, instrument and detector instabilities. The narrow-band images are obtained in a straddle mode, where the off-band image is made up of a pair of images that “straddle” the on-band image in wavelength (e.g., $\lambda_1$ = 6500 Å and $\lambda_2$ = 6625 Å for rest-frame H$\alpha$); this greatly improves the accuracy of the continuum removal since it corrects for slopes in the continuum and underlying absorption features. These techniques have been used with the Taurus Tunable Filter (TTF; Bland-Hawthorn & Jones 1998; Bland-Hawthorn & Kedziora-Chudczer 2003) on the AAT and WHT to produce emission-line images of several “quiescent” disk galaxies (Miller & Veilleux 2003a) and a few starburst galaxies (Veilleux et al. 2003) down to unprecedented emission-line fluxes. Gaseous complexes or filaments larger than $\sim$ 20 kpc have been discovered or confirmed in a number of wind hosts (e.g., NGC 1482 and NGC 6240; the presence of warm ionized gas at $\sim$ 12 kpc from the center of M 82 was discussed in §3.2). Multi-line imaging and long-slit spectroscopy of the gas found on large scale reveal line ratios which are generally not H II region-like. Shocks often contribute significantly to the ionization of the outflowing gas on the outskirts of starburst galaxies. As expected from shock models (e.g., Dopita & Sutherland 1995), the importance of shocks over photoionization by OB stars appears to scale with the velocity of the outflowing gas (e.g., NGC 1482, NGC 6240, or ESO484-G036 versus NGC 1705; NGC 3079 is an extreme example of a shock-excited wind nebula; Veilleux et al. 1994), although other factors like the starburst age, star formation rate, and the dynamical state of the outflowing structure (e.g., pre- or post-blowout) must also be important in determining the excitation properties of the gas at these large radii (e.g., Shopbell & Bland-Hawthorn 1998 and Veilleux & Rupke 2002). 0.1in [**Influence of the Wind on Companion Galaxies.**]{} Companion galaxies located within the zone of influence of the wind will be affected by the wind ram pressure. Irwin et al. (1987) noticed that the dwarf S0 galaxy NGC 3073 exhibits an elongated H I tail that is remarkably aligned with the nucleus of NGC 3079. Irwin et al. have argued that ram pressure due to the outflowing gas of NGC 3079 is responsible for this tail. If that is the case, the wind of NGC 3079 must extent to at least $\sim$ 50 kpc. This is the only system known so far where this phenomenon is suspected to take place. 0.1in [**Absorption-Line Studies.**]{} Absorption-line spectroscopy of bright background galaxies (e.g., high-$z$ quasars, Lyman break galaxies) can provide direct constraints on the zone of influence of galactic winds. Norman et al. (1996) have used this method to estimate the extent of the wind in NGC 520. A strong and possibly complex Mg II, Mg I, and Fe II absorption-line system was found near the systemic velocity of NGC 520 at a distance from the galactic nucleus of 24 $h^{-1}$ kpc. A weaker system at a distance of 52 $h^{-1}$ kpc is also possibly present. Unfortunately, NGC 520 is undergoing a tidal interaction so the absorption may arise from tidally disrupted gas rather than material in the purported wind. Norman et al. also looked for absorption-line systems associated with the wind of NGC 253, but the proximity of this system to our own Galaxy and to the line of sight to the Magellanic Stream makes the identification of the absorption-line systems ambiguous. No other local wind galaxy has been studied using this technique. Large absorption-line data sets collected on high-$z$ galaxies provide new constraints on the zone of influence of winds in the early universe. Adelberger et al. (2003) have recently presented tantalizing evidence for a deficit of neutral hydrogen clouds within a comoving radius of $\sim$ 0.5 $h^{-1}$ Mpc from $z \sim 3$ LBGs. The uncertainties are large and the results are significant at less than the $\sim$ 2$\sigma$ level. Adelberger et al. (2003) argue that this deficit, if real, is unlikely to be due solely to the ionizing radiation from LBGs (e.g., Steidel et al. 2001; Giallongo et al. 2002). They favor a scenario in which the winds in LBGs directly influence the surrounding IGM. They also argue that the excess of absorption-line systems with large CIV column densities near LBGs is evidence for chemical enrichment of the IGM by the LBG winds. Summary ======= Tremendous progress has been made over the past five years in understanding the physics and impact of starburst-driven winds in the local and distant universe. We now know that starburst-driven winds are common in galaxies with high star formation rates per unit area, both locally (nearby starbursts, luminous and ultraluminous infrared galaxies) and at high redshifts (e.g., Lyman break galaxies). There is [*direct*]{} evidence that starburst-driven winds have had a strong influence on the chemical evolution of the host ISM and possibly also that of the IGM: (1) Enriched wind fluid has been detected in a few nearby galaxies. (2) Approximately half of the outflowing material in powerful starburst galaxies (ULIRGs) have velocities in excess of the escape velocities. (3) Deep emission-line maps of local wind galaxies indicate that the zone of influence of the wind often extends beyond $\sim$ 10 kpc. (4) Recent results on Lyman break galaxies suggest tentatively that the zone of influence of LBG winds may extent out to $\sim$ 500 $h^{-1}$ kpc. Nevertheless, much work remains to be done in this area of research; the next five years promise to be equally exciting as the last five! It is a pleasure to thank S. Aalto for organizing a stimulating conference. Some of the results presented in this paper are part of a long-term effort involving many collaborators, including J. 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--- abstract: 'A [*test space*]{} is the set of outcome-sets associated with a collection of experiments. This notion provides a simple mathematical framework for the study of probabilistic theories – notably, quantum mechanics – in which one is faced with incommensurable random quantities. In the case of quantum mechanics, the relevant test space, the set of orthonormal bases of a Hilbert space, carries significant topological structure. This paper inaugurates a general study of topological test spaces. Among other things, we show that any topological test space with a compact space of outcomes is of finite rank. We also generalize results of Meyer and Clifton-Kent by showing that, under very weak assumptions, any second-countable topological test space contains a dense semi-classical test space.' --- [Topological Test Spaces]{}[^1]\ Alexander Wilce\ Department of Mathematical Sciences, Susquehanna University\ Selinsgrove, Pa 17870  email: [email protected] [0. Introduction]{} A [*test space*]{} in the sense of Foulis and Randall \[3, 4, 5\], is a pair $(X,{\frak A})$ where $X$ is a non-empty set and ${\frak A}$ is a covering of $X$ by non-empty subsets. [^2] The intended interpretation is that each set $E \in {\frak A}$ represents an exhaustive set of mutually exclusive possible [*outcomes*]{}, as of some experiment, decision, physical process, or [*test*]{}. A [*state*]{}, or [*probability weight*]{}, on $(X,{\frak A})$ is a mapping $\omega : X \rightarrow [0,1]$ summing to $1$ over each test. Obviously, this framework subsumes discrete classical probability theory, which deals with test spaces $(E,\{E\})$ having only a single test. It also accommodates quantum probability theory, as follows. Let ${{\bf H}}$ be a Hilbert space, let $S = S({{\bf H}})$ be the unit sphere of ${{\bf H}}$, and let ${\frak F} = {\frak F}({{\bf H}})$ denote the collection of all [*frames*]{}, i.e., maximal pairwise orthogonal subsets of $S$. The test space $(S,{\frak F})$ is a model for the set of maximally informative, discrete quantum-mechanical experiments. As long as $\dim({{\bf H}}) > 2$, Gleason’s theorem \[6\] tells us that every state $\omega$ on $(S,{\frak F})$ arises from a density operator $W$ on ${{\bf H}}$ via the rule $\omega(x) = \langle Wx,x\rangle$ for all $x \in S$. In this last example, the test space has a natural topological structure: $S$ is a metric space, and ${\frak F}$ can be topologized as well (in several ways). The purpose of this paper is to provide a framework for the study of topological test spaces generally. Section 1 develops basic properties of the Vietoris topology, which we use heavily in the sequel. Section 2 considers topological test spaces in general, and locally finite topological test spaces in particular. Section 3 addresses the problem of topologizing the logic of an algebraic topological test space. In section 4, we generalize results of Meyer \[8\] and Clifton and Kent \[2\] by showing that any second-countable topological test space satisfying a rather natural condition contains a dense semi-classical subspace. The balance of this section collects some essential background information concerning test spaces (see \[11\] for a detailed survey). Readers familiar with this material can proceed directly to section 1. [**0.1 Events**]{} Let $(X,{\frak A})$ be a test space. Two outcomes $x, y \in X$ are said to be [*orthogonal*]{}, or [*mutually exclusive*]{}, if they are distinct and belong to a common test. In this case, we write $x \perp y$. More generally, a set $A \subseteq X$ is called an [*event*]{} for $X$ if there exists a test $E \supseteq A$. The set of events is denoted by ${\cal E}(X,{\frak A})$. There is a natural orthogonality relation on ${\cal E}(X,{\frak A})$ extending that on $X$, namely, $A \perp B$ iff $A \cap B = \emptyset$ and $A \cup B \in {\cal E}(X,{\frak A})$. Every state $\omega$ on $(X,{\frak A})$ extends to a mapping $\omega : {\cal E}(X,{\frak A}) \rightarrow [0,1]$ given by $\omega(A) = \sum_{x \in A} \omega(x)$. If $A \perp B$, then $\omega(A \cup B) = \omega(A) + \omega(B)$ for every probability weight $\omega$. Two events $A$ and $C$ are [*complementary*]{} – abbreviated $A {{\sf{oc}}}C$ – if they partition a test, and [*perspective*]{} if they are complementary to a common third event $C$. In this case, we write $A \sim B$. Note that if $A$ and $B$ are perspective, then for every state $\omega$ on $(X,{\frak A})$, $\omega(A) = 1 - \omega(C) = \omega(B)$. [**0.2 Algebraic Test Spaces**]{} We say that $X$ is [*algebraic*]{} iff for all events $A, B, C \in {\cal E}(X,{\frak A})$, $A \sim B \ {\mbox}{and} \ B {{\sf{oc}}}C \ \Rightarrow \ A {{\sf{oc}}}C$. In this case, $\sim$ is an equivalence relation on ${\cal E}(X)$. Moreover, if $A \perp B$ and $B \sim C$, then $A \perp C$ as well, and $A \cup B \sim A \cup C$. Let $\Pi(X,{\frak A}) = {\cal E}(X,{\frak A})/\sim$, and write $p(A)$ for the $\sim$-equivalence class of an event $A \in {\cal E}(X)$. Then $\Pi$ carries a well-defined orthogonality relation, namely $p(A) \perp p(B) \Leftrightarrow A \perp B$, and also a partial binary operation $p(A) \oplus p(B) = p(A \cup B)$, defined for orthogonal pairs. We may also define $0 := p(\emptyset)$, $1 := p(E)$, $E \in {\frak A}$, and $p(A)' = p(C)$ where $C$ is any event complementary to $A$. The structure $(\Pi,\oplus, ', 0, 1)$, called the [**logic**]{} of $(X,{\frak A})$, satisfies the following conditions: $p \oplus q = q \oplus p$ and $p \oplus (q \oplus r) = (p \oplus q) \oplus r$[^3]; $p \oplus p$ is defined only if $p = 0$; $p \oplus 0 = 0 \oplus p = p$; For every $p \in \Pi$, there exists a unique element — namely, $p'$ —satisfying $p \oplus p' = 1$. For the test space $(S,{\frak F})$ of frames of a Hilbert space ${{\bf H}}$, events are simply orthonormal set of vectors in ${{\bf H}}$, and two events are perspective iff they have the same closed span. Hence, we can identify $\Pi(S,{\frak F})$ with the set of closed subspaces of ${{\bf H}}$, with $\oplus$ coinciding with the usual orthogonal sum operation. [**0.3 Orthoalgebras**]{} Abstractly, a structure satisfying (1) through (4) above is called an [*orthoalgebra*]{}. It can be shown that every orthoalgebra arises canonically (though not uniquely) as $\Pi(X,{\frak A})$ for an algebraic test space $(X,{\frak A})$. Indeed, if $L$ is an orthoalgebra, let $X_{L} = L \setminus \{0\}$ and let ${\frak A}_{L}$ denote the set of finite subsets $E = \{e_{1},...,e_{n}\}$ of $L \setminus {0}$ for which $e_{1} \oplus \cdots \oplus e_{n}$ exists and equals $1$. Then $(X_{L}, {\frak A}_{L})$ is an algebraic test space with logic canonically isomorphic to $L$. Any orthoalgebra $L$ carries a natural partial order, defined by setting $p \leq q$ iff there exists some $r \in L$ with $p \perp r$ and $p \oplus r = q$. With respect to this ordering, the mapping $p \mapsto p'$ is an orthocomplementation. [**0.4 Proposition \[3\]**]{}: *If $L$ is an orthoalgebra, the following are equivalent:* $L$ is [*orthocoherent*]{}, i.e., for all pairwise orthogonal elements $p, q, r \in L$, $p \oplus q \oplus r$ exists. $p \oplus q = p \vee q$ for all $p \perp q$ in $L$ $(L,\leq,')$ is an orthomodular poset Note also that if $(L, \leq, ')$ is any orthoposet, the partial binary operation of orthogonal join — that is, $p \oplus q = p \vee q$ for $p \leq q'$ – is associative iff $L$ is orthomodular, in which case, $(L, \oplus)$ is an orthoalgebra, the natural order on which coincides with the given order on $L$ \[11\]. Thus, orthomodular posets and orthomodular lattices can be regarded as essentially the same things as orthocoherent orthoalgebras and lattice-ordered orthoalgebras, respectively.\ [1. Background on the Vietoris Topology]{} General references for this section are \[7\] and \[9\]. If $X$ is any topological space, let $2^X$ denote the set of all closed subsets of $X$. If $A \subseteq X$, let $$[A] := \{ F \in 2^{X} | F \cap A \not = \emptyset\}.$$ Clearly, $[A \cap B] \subseteq [A] \cap [B]$ and $\bigcup_{i} [A_{i}] = [\bigcup_{i} A_{i}]$. The [*Vietoris topology*]{} on $2^{X}$ is the coarsest topology in which $[U]$ is open if $U \subseteq X$ is open and $[F]$ is closed if $F \subseteq X$ is closed. [^4] Thus, if $U$ is open, so is $(U) := [U^{c}]^{c} = \{ F \in 2^{X} | F \subseteq U\}$. Let ${\cal B}$ be any basis for the topology on $X$: then the collection of sets of the form $$\langle U_{1},...,U_{n} \rangle \ := \ [U_{1}] \cap \cdots \cap [U_{n}] \cap \left ( \bigcup_{i=1}^{n} U_{i} \right )$$ with $U_{1},...,U_{n}$ in ${\cal B}$, is a basis for the Vietoris topology on $2^{X}$. Note that $\langle U_{1},...,U_{n} \rangle$ consists of all closed sets contained in $\bigcup_{i=1}^{n} U_{i}$ and meeting each set $U_{i}$ at least once. If $X$ is a compact metric space, then the Vietoris topology on $2^{X}$ is just that induced by the Hausdorff metric. Two classical results concerning the Vietoris topology are [*Vietoris’ Theorem*]{}: $2^{X}$ is compact iff $X$ is compact, and [*Michael’s Theorem*]{}: a (Vietoris) compact union of compact sets is compact.[^5] The operation $\cup : 2^{X} \times 2^{X} \rightarrow 2^{X}$ is also Vietoris continuous, since $$\cup^{-1}([U]) = \{ (A,B) | A \cup B \in [U]\} = ([U] \times 2^{X}) \cup (2^{X} \times [U]),$$ which is open if $U$ is open and closed if $U$ is closed. In particular, for any fixed closed set $A$, the mapping $f_{A} : 2^{X} \rightarrow 2^{X}$ given by $f_{A} : B \mapsto A \cup B$ is continuous. Notice also that the mapping $\pi : 2^{X} \times 2^{X} \rightarrow 2^{X \times X}$ given by $\pi(A,B) = A \times B$ is continuous, as $\pi^{-1}([U \times V]) = [U] \times [V]$ and $\pi^{-1}((U \times V)) = (U) \times (V)$. Henceforth, we regard any collection $\frak A$ of closed subsets of a topological space $X$ as a subspace of $2^{X}$. In the special case in which ${\frak A}$ is a collection of finite sets of uniformly bounded cardinality, say $|E| < n$ for every $E \in {\frak A}$, there is a more direct approach to topologizing ${\frak A}$ that bears discussion. Let ${\frak A}^{o} \subseteq X^{n}$ denote the space of [*ordered*]{} versions $(x_{1},...,x_{n})$ of sets $\{x_{1},...,x_{n}\} \in {\frak A}$, with the relative product topology. We can give ${\frak A}$ the quotient topology induced by the natural surjection $\pi : {\frak A}^{o} \rightarrow {\frak A}$ that “forgets" the order. The following is doubtless well-known, but I include the short proof for completeness. [**1.1 Proposition:**]{} [*Let $X$ be Hausdorff and ${\frak A}$, a collection of non-empty finite subsets of $X$ of cardinality $\leq n$ (with the Vietoris topology). Then the canonical surjection $\pi : {\frak A}^{o} \rightarrow {\frak A}$ is an open continuous map. Hence, the Vietoris topology on ${\frak A}$ coincides with the quotient topology induced by $\pi$.* ]{} Proof: Let $U_{1},...,U_{n}$ be open subsets of $X$. Then $\pi((U_{1} \times \cdots \times U_{n})\cap {\frak A}^{o}) = \langle U_{1},...,U_{n}\rangle \cap {\frak A}$, so $\pi$ is open. Also $$\pi^{-1}(\langle U_{1},...,U_{n}\rangle \cap {\frak A}) \ = \ \bigcup_{\sigma} (U_{\sigma(1)} \times \cdots \times U_{\sigma(n)}) \cap {\frak A}^{o},$$ where $\sigma$ runs over all permutations of $\{1,2,...,n\}$, so $\pi$ is continuous. It follows immediately that the quotient and Vietoris topologies on ${\frak A}$ coincide. $\Box$\ [2. Topological Test Spaces]{} We come now to the subject of this paper. [**2.1 Definition:**]{} A [*topological test space*]{} is a test space $(X, {\frak A})$ where $X$ is a Hausdorff space and the relation $\perp$ is closed in $X \times X$. [**2.2 Examples**]{}\ (a) Let ${{\bf H}}$ be a Hilbert space. Let $S$ be the unit sphere of ${{\bf H}}$, in any topology making the inner product continuous. Then the test space $(S,{\frak F})$ defined above is a topological test space, since the orthogonality relation is closed in $S^{2}$.\ (b) Suppose that $X$ is Hausdorff, that every $E \in {\frak A}$ is finite, and that $(X,{\frak A})$ supports a set $\Gamma$ of [*continuous*]{} probability weights that are $\perp$-separating in the sense that $p \not \perp q$ iff $\exists \omega \in \Gamma$ with $\omega(p) + \omega(q) > 1$. Then $\perp$ is closed in $X^{2}$, so again $(X,{\frak A})$ is a topological test space.\ (c) Let $L$ be any topological orthomodular lattice \[1\]. The mapping $\phi: L^{2} \rightarrow L^{2}$ given by $\phi(p,q) = (p,p \wedge q')$ is continuous, and $\perp = \phi^{-1}(\Delta)$ where $\Delta$ is the diagonal of $L^{2}$. Since $L$ is Hausdorff, $\Delta$ is closed, whence, so is $\perp$. Hence, the test space $(L\setminus \{0\},{\frak A}_{L})$ (as described in 0.3 above) is topological. The following Lemma collects some basic facts about topological test spaces that will be used freely in the sequel. [**2.3 Lemma:**]{} *Let $(X, {\frak A})$ be a topological test space. Then* Each point $x \in X$ has an open neighborhood containing no two orthogonal outcomes. (We shall call such a neighborhood [*totally non- orthogonal*]{}.) For every set $A \subseteq X$, $A^{\perp}$ is closed. Each pairwise orthogonal subset of $X$ is discrete Each pairwise orthogonal subset of $X$ is closed. Proof: (a) Let $x \in X$. Since $(x,x) \not \in \perp$ and $\perp$ is closed, we can find open sets $V$ and $W$ about $x$ with $(V \times W) \cap \perp = \emptyset$. Taking $U = V \cap W$ gives the advertised result. \(b) Let $y \in X \setminus x^{\perp}$. Then $(x,y) \not \in \perp$. Since the latter is closed, there exist open sets $U, V \subseteq X$ with $(x,y) \in U \times V$ and $(U \times V) \cap \perp = \emptyset$. Thus, no element of $V$ lies orthogonal to any element of $U$; in particular, we have $y \in V \subseteq X \setminus x^{\perp}$. Thus, $X \setminus x^{\perp}$ is open, i.e., $x^{\perp}$ is closed. It now follows that for any set $A \subseteq X$, the set $A^{\perp} = \bigcap_{x \in A} x^{\perp}$ is closed. \(c) Let $D$ be pairwise orthogonal. Let $x \in D$: by part (b), $X \setminus x^{\perp}$ is open, whence, $\{x\} = D \cap (X \setminus x^{\perp})$ is relatively open in $D$. Thus, $D$ is discrete. \(d) Now suppose $D$ is pairwise orthogonal, and let $z \in \overline{D}$: if $z \not \in D$, then for every open neighborhood $U$ of $z$, $U \cap D$ is infinite; hence, we can find distinct elements $x, y \in D \cap U$. Since $D$ is pairwise orthogonal, this tells us that $(U \times U) \cap \perp \not = \emptyset$. But then $(x,x)$ is a limit point of $\perp$. Since $\perp$ is closed, $(x,x) \in \perp$, which is a contradiction. Thus, $z \in D$, i.e., $D$ is closed. $\Box$ It follows in particular that every test $E \in {\frak A}$ and every event $A \in {\cal E}(X,{\frak A})$ is a closed, discrete subset of $X$. Hence, we may construe $\frak A$ and ${\cal E}(X,{\frak A})$ of as subspaces of $2^{X}$ in the Vietoris topology. A test space $(X,{\frak A})$ is [*locally finite*]{} iff each test $E \in {\frak A}$ is a finite set. We shall say that a test space $(X,{\frak A})$ is of [*rank $n$*]{} if $n$ is the maximum cardinality of a test in ${\frak A}$. If all tests have cardinality [*equal*]{} to $n$, then $(X,{\frak A})$ is [*$n$-uniform*]{}. [**2.4 Theorem:**]{} [*Let $(X,{\frak A})$ be a topological test space with $X$ compact. Then all pairwise orthogonal subsets of $X$ are finite, and of uniformly bounded size. In particular, $\frak A$ is of finite rank.*]{} Proof: By Part (a) of Lemma 2.3, every point $x \in X$ is contained in some totally non- orthogonal open set. Since $X$ is compact, a finite number of these, say $U_{1},...,U_{n}$, cover $X$. A pairwise orthogonal set $D \subseteq X$ can meet each $U_{i}$ at most once; hence, $|D| \leq n$. $\Box$. For locally finite topological test spaces, the Vietoris topology on the space of events has a particularly nice description. Suppose $A$ is a finite event: By Part (a) of Lemma 2.3, we can find for each $x \in A$ a totally non-orthogonal open neighborhood $U_{x}$. Since $X$ is Hausdorff and $A$ is finite, we can arrange for these to be disjoint from one another. Consider now the Vietoris-open neighborhood ${\cal V} = \langle U_{x}, x \in A \rangle \cap {\cal E}$ of $A$ in ${\cal E}$: an event $B$ belonging to $\cal V$ is contained in $\bigcup_{x \in A} U_{x}$ and meets each $U_{x}$ in at least one point; however, being pairwise orthogonal, $B$ can meet each $U_{x}$ [*at most*]{} once. Thus, $B$ selects [*exactly one*]{} point from each of the disjoint sets $U_{x}$ (and hence, in particular, $|B| = |A|$). Note that, since the totally non-orthogonal sets form a basis for the topology on $X$, open sets of the form just described form a basis for the Vietoris topology on ${\cal E}$. As an immediate consequence of these remarks, we have the following: [**2.5 Proposition:**]{} [*Let $(X,{\frak A})$ be locally finite. Then the set ${\cal E}_{n}$ of all events of a given cardinality $n$ is clopen in ${\cal E}(X,{\frak A})$.*]{} A test space $(X,{\frak A})$ is [*UDF*]{} ([*unital, dispersion-free*]{}) iff for ever $x \in X$ there exists a $\{0,1\}$-valued state $\omega$ on $(X,{\frak A})$ with $\omega(x) = 1$. Let $U_{1},...,U_{n}$ be pairwise disjoint totally non-orthogonal open sets, and and let ${\cal U} = \langle U_{1},...,U_{n} \rangle$: then ${\cal U}$ can be regarded as a UDF test space (each $U_{i}$ selecting one outcome from each test in ${\cal V}$). The foregoing considerations thus have the further interesting consequence that any locally finite topological test space is [*locally UDF*]{}. In particular, for such test spaces, the existence or non-existence of dispersion-free states will depend entirely on the [*global*]{} topological structure of the space. If $(X,{\frak A})$ is a topological test space, let $\overline{\frak A}$ denote the (Vietoris) closure of ${\frak A}$ in $2^{X}$. We are going to show that $(X,\overline{{\frak A}})$ is again a topological test space, having in fact the same orthogonality relation as $(X,{\frak A})$. If $(X,{\frak A})$ is of finite rank, moreover, $(X,\overline{\frak A})$ has the same states as $(X,{\frak A})$. [**2.6 Lemma:**]{} [*Let $(X,{\frak A})$ be any topological test space, and let $E \in \overline{\frak A}$. Then $E$ is pairwise orthogonal (with respect to the orthogonality induced by $\frak A$).*]{} Proof: Let $x$ and $y$ be two distinct points of $E$. Let $U$ and $V$ be disjoint neighborhoods of $x$ and $y$ respectively, and let $(E_{\lambda})_{\lambda \in \Lambda}$ be a net of closed sets in ${\frak A}$ converging to $E$ in the Vietoris topology. Since $E \in [U] \cap [V]$, we can find $\lambda_{U,V} \in \Lambda$ such that $E_{\lambda} \in [U] \cap [V]$ for all $\lambda \geq \lambda_{U,V}$. In particular, we can find $x_{\lambda_{U,V}} \in E_{\lambda_{U,V}} \cap U$ and $y_{\lambda_{U,V}}\in E_{\lambda_{U,V}} \cap V$. Since $U$ and $V$ are disjoint, $x_{\lambda_{U,V}}$ and $y_{\lambda_{U,V}}$ are distinct, and hence, – since they belong to a common test $E_{\lambda}$ – orthogonal. This gives us a net $(x_{\lambda_{U,V}},y_{\lambda_{U,V}})$ in $X \times X$ converging to $(x,y)$ and with $(x_{\lambda_{U,V}},y_{\lambda_{U,V}}) \in \perp$. Since $\perp$ is closed, $(x,y) \in \perp$, i.e., $x \perp y$. $\Box$ It follows that the orthogonality relation on $X$ induced by $\overline{\frak A}$ is the same as that induced by ${\frak A}$. In particular, $(X,\overline{\frak A})$ is again a topological test space. Let ${\cal F}_{n}$ denote the set of finite subsets of $X$ having $n$ or fewer elements. [**2.7 Lemma:**]{} *Let $X$ be Hausdorff. Then for every $n$,* ${\cal F}_{n}$ is closed in $2^{X}$. If $f : X \rightarrow {\Bbb R}$ is continuous, then so is the mapping $\hat{f} : {\frak F}_{n} \rightarrow {\Bbb R}$ given by $\hat{f}(A) := \sum_{x \in A} f(x)$. Proof: (a) Let $F$ be a closed set (finite or infinite) of cardinality greater than $n$. Let $x_{1},...,x_{n+1}$ be distinct elements of $F$, and let $U_{1},....,U_{n}$ be pairwise disjoint open sets with $x_{i} \in U_{i}$ for each $i = 1,...,n$. Then no closed set in ${\cal U} := [U_{1}] \cap \cdots \cap [U_{n}]$ has fewer than $n+1$ points – i.e, ${\cal U}$ is an open neighborhood of $F$ disjoint from ${\cal F}_{n}$. This shows that $2^{X} \setminus {\cal F}_{n}$ is open, i.e., ${\cal F}_{n}$ is closed. \(b) By proposition 1.1, ${\frak F}_{n}$ is the quotient space of $X^{n}$ induced by the surjection surjection $q: (x_{1},...,x_{n}) \mapsto \{x_{1},...,x_{n}\}$. The mapping $\overline{f} : X^{n} \rightarrow {\Bbb R}$ given by $(x_{1},...,x_{n}) \mapsto \sum_{i=1}^{n} f(x_{i})$ is plainly continuous; hence, so is $\hat{f}$. $\Box$ [**2.8 Proposition:**]{} [*Let $(X,{\frak A})$ be a rank-$n$ (respectively, $n$-uniform) test space. Then $(X,\overline{\frak A})$ is also a rank-$n$ (respectively, $n$-uniform) test space having the same continuous states as $(X,{\frak A})$.* ]{} Proof: If ${\frak A}$ is rank-$n$, then ${\frak A} \subseteq {\frak F}_{n}$. Since the latter is closed, $\overline{\frak A} \subseteq {\frak F}_{n}$ also. Note that if ${\frak A}$ is $n$-uniform and $E \in \overline{\frak A}$, then any net $E_{\lambda} \rightarrow E$ is eventually in bijective correspondence with $E$, by Proposition 2.5. Hence, $(X,\overline{\frak A})$ is also $n$-uniform. Finally, every continuous state on $(X,{\frak A})$ lifts to a continuous state on $(X,\overline{\frak A})$ by Lemma 2.7 (b). $\Box$\ [3. The Logic of a Topological Test Space]{} In this section, we consider the logic $\Pi = \Pi(X,{\frak A})$ of an algebraic test space $(X,{\frak A})$. We endow this with the quotient topology induced by the canonical surjection $p : {\cal E} \rightarrow \Pi$ (where ${\cal E} = {\cal E}(X,{\frak A})$ has, as usual, its Vietoris topology). Our aim is to find conditions on $(X,{\frak A})$ that will guarantee reasonable continuity properties for the orthogonal sum operation and the orthocomplement. In this connection, we advance the following [**3.1 Definition:**]{} A [*topological orthoalgebra*]{} is an orthoalgbra $(L,\perp,\oplus,0,1)$ in which $L$ is a topological space, the relation $\perp \subseteq L^{2}$ is closed, and the mappings $\oplus : \perp \rightarrow L$ and $' : L \rightarrow L$ are continuous. A detailed study of topological orthoalgebras must wait for another paper. However, it is worth mentioning here that, while every topological orthomodular lattice is a topological orthoalgebra, there exist lattice-ordered topological orthoalgebras in which the meet and join are discontinuous – e.g., the orthoalgebra $L({{\bf H}})$ of closed subspaces of a Hilbert space, in its operator-norm topology. [**3.2 Lemma:**]{} *Let $(L,\perp,\oplus,0,1)$ be a topological orthoalgebra. Then* The order relation $\leq$ is closed in $L^{2}$ $L$ is a Hausdorff space. Proof: For (a), note that $a \leq b$ iff $a \perp b'$. Thus, $\leq \ = \ f^{- 1}(\perp)$ where $f : L \times L \rightarrow L \times L$ is the continuous mapping $f(a,b) = (a,b')$. Since $\perp$ is closed, so is $\leq$. The second statement now follows by standard arguments (cf. Nachbin \[10\]). $\Box$ We now return to the question: when is the logic of a topological test space, in the quotient topology, a topological orthoalgebra? [**3.3 Lemma:**]{} *Suppose ${\cal E}$ is closed in $2^{X}$. Then* The orthogonality relation $\perp_{\cal E}$ on ${\cal E}$ is closed in ${\cal E}^{2}$. The mapping $\cup : \perp_{\cal E} \rightarrow {\cal E}$ is continuous Proof: The mapping ${\cal E}^2 \rightarrow 2^{X}$ given by $(A,B) \mapsto A \cup B$ is continuous; hence, if ${\cal E}$ is closed in $2^{X}$, then so is the set ${{\bf C}}:= \{(A,B) \in {\cal E}^{2} | A \cup B \in {\cal E}\}$ of [*compatible*]{} pairs of events. It will suffice to show that the set ${{\bf O}}:= \{ (A,B) \in {\cal E} | A \subseteq B^{\perp}\}$ is also closed, since $\perp = {{\bf C}}\cap {{\bf O}}$. But $(A,B) \in {{\bf O}}$ iff $A \times B \subseteq \perp$, i.e., ${{\bf O}}= \pi^{-1}((\perp)) \cap {\cal E}$ where $\pi : 2^{X} \times 2^{X} \rightarrow 2^{X \times X}$ is the product mapping $(A,B) \mapsto A \times B$. As observed in section 1, this mapping is continuous, and since $\perp$ is closed in $2^{X \times X}$, so is $(\perp)$ in $2^{X \times X}$. Statement (b) follows immediately from the Vietoris continuity of $\cup$. $\Box$ [*Remarks:*]{} The hypothesis that ${\cal E}$ be closed in $2^{X}$ is not used in showing that the relation ${{\bf O}}$ is closed. If $(X,{\frak A})$ is [ *coherent*]{} \[10\], then ${{\bf O}}= \perp$, so in this case, the hypothesis can be avoided altogether. On the other hand, if $X$ is compact and ${\frak A}$ is closed, then ${\cal E}$ will also be compact and hence, closed. (To see this, note that if $X$ is compact then by Vietoris’ Theorem, $2^{X}$ is compact. Hence, so is the closed set $(E) = \{ A \in 2^{X} | A \subseteq E\}$ for each $E \in {\frak A}$. The mapping $2^{X} \rightarrow 2^{2^{X}}$ given by $E \mapsto (E)$ is easily seen to be continuous. Since $\frak A$ is closed, hence compact, in $2^{X}$, it follows that $\{(E) | E \in {\frak A}\}$ is a compact subset of $2^{2^{X}}$. By Michael’s theorem, ${\cal E} = \bigcup_{E \in {\frak A}} (E)$ is compact, hence closed, in $2^X$.) In order to apply Lemma 3.3 to show that $\perp \subseteq \Pi^{2}$ is closed and $\oplus : \perp \rightarrow \Pi$ is continuous, we would like to have the canonical surjection $p : {\cal E} \rightarrow \Pi$ open. The following condition is sufficient to secure this, plus the continuity of the orthocomplementation $' : \Pi \rightarrow \Pi$. [**3.3 Definition:**]{} Call a topological test space $(X,{\frak A})$ is [*stably complemented*]{} iff for any open set ${\cal U}$ in ${\cal E}$, the set ${\cal U}^{{{\sf{oc}}}}$ of events complementary to events in ${\cal U}$ is again open. [*Remark:*]{} If ${{\bf H}}$ is a finite-dimensional Hilbert space, it can be shown that the corresponding test space $(S,{\frak F})$ of frames is stably complemented \[12\]. [**3.5 Lemma:**]{} *Let $(X, {\frak A})$ be a topological test space, and let $p : {\cal E} \rightarrow \Pi$ be the canonical quotient mapping (with $\Pi$ having the quotient topology). Then the following are equivalent:* $(X,{\frak A})$ is stably complemented The mapping $p: {\cal E} \rightarrow \Pi$ is open and the mapping $' : \Pi \rightarrow \Pi$ is continuous. Proof: Suppose first that $(X,{\frak A})$ is stably complemented, and let ${\cal U}$ be an open set in ${\cal E}$. Then $$\begin{aligned} p^{-1}(p({\cal U}) ) & = & \{ A \in {\cal E} | \exists B \in {\cal U} A \sim B\}\\ & = & \{ A \in {\cal E} | \exists C \in {\cal U}^{{{\sf{oc}}}} A {{\sf{oc}}}C\}\\ & = & \left ( {\cal U}^{{{\sf{oc}}}} \right )^{{{\sf{oc}}}}\end{aligned}$$ which is open. Thus, $p({\cal U})$ is open. Now note that $' : \Pi \rightarrow \Pi$ is continuous iff, for every open set $V \subseteq \Pi$, the set $V' = \{p' | p \in V\}$ is also open. But $p^{-1}(V') = (p^{-1}(V))^{{{\sf{oc}}}}$: since $p$ is continuous and $(X,{\frak A})$ is stably complemented, this last is open. Hence, $V'$ is open. For the converse, note first that if $'$ is continuous, it is also open (since $a'' = a$ for all $a \in \Pi$). Now for any open set ${\cal U} \subseteq {\cal E}$, ${\cal U}^{{{\sf{oc}}}} = p^{-1}(p({\cal U})')$: Since $p$ and $'$ are continuous open mappings, this last is open as well. $\Box$ [**3.6 Proposition:**]{} [*Let $(X,{\frak A})$ be a stably complemented algebraic test space with ${\cal E}$ closed. Then $\Pi$ is a topological orthoalgebra.*]{} Proof: Continuity of $'$ has already been established. We show first that $\perp \subseteq \Pi^{2}$ is closed. If $(a,b) \not \in \perp$, then for all $A \in p^{-1}(a)$ and $B \in p^{-1}(b)$, $(A,B) \not \in \perp_{\cal E}$. The latter is closed, by Lemma 3.3 (a); hence, we can find Vietoris-open neighborhoods ${\cal U}$ and ${\cal V}$ of $A$ and $B$, respectively, with $({\cal U} \times {\cal V}) \cap \perp_{\cal E} = \emptyset$. Since $p$ is open, $U := p({\cal U})$ and $V := p({\cal V})$ are open neighborhoods of $a$ and $b$ with $(U \times V) \cap \perp = \emptyset$. To establish the continuity of $\oplus : \perp \rightarrow \Pi$, let $a \oplus b = c$ and let $A \in p^{-1}(a), B \in p^{-1}(B)$ and $C \in p^{-1}(c)$ be representative events. Note that $A \perp B$ and $A \cup B = C$. Let $W$ be an open set containing $c$: then ${\cal W} := p^{-1}(W)$ is an open set containing $C$. By Lemma 3.3 (b), $\cup : \perp_{\cal E} \rightarrow {\cal E}$ is continuous; hence, we can find open sets ${\cal U}$ about $A$ and ${\cal V}$ about $B$ with $A_{1} \cup B_{1} \in {\cal W}$ for every $(A_{1},B_{1}) \in ({\cal U} \times {\cal V}) \cap \perp_{\cal E}$. Now let $U = p({\cal U})$ and $V = p({\cal V})$: these are open neighborhoods of $a$ and $b$, and for every $a_{1} \in U$ and $b_{1} \in V$ with $a_{1} \perp b_{1}$, $a_{1} \oplus b_{1} \in p(p^{-1}(W)) = W$ (recalling here that $p$ is surjective). Thus, $(U \times V) \cap \perp \subseteq \oplus^{-1}(W)$, so $\oplus$ is continuous. $\Box$\ [5. Semi-classical Test Spaces]{} From a purely combinatorial point of view, the simplest test spaces are those in which distinct tests do not overlap. Such test spaces are said to be [ *semi-classical*]{}. In such a test space, the relation of perspectivity is the identity relation on events; consequently, the logic of a semi-classical test space $(X,{\frak A})$ is simply the horizontal sum of the boolean algebras $2^{E}$, $E$ ranging over ${\frak A}$. A state on a semi-classical test space $(X,{\frak A})$ is simply an assignment to each $E \in {\frak A}$ of a probability weight on $E$. (In particular, there is no obstruction to constructing “hidden variables" models for states on such test spaces.) Recent work of D. Meyer \[8\] and of R. Clifton and A. Kent \[2\] has shown that the test space $(S({{\bf H}}),{\frak F}({{\bf H}}))$ associated with a finite-dimensional Hilbert space contains (in our language) a dense semi-classical sub-test space. To conclude this paper, I’ll show that the this result in fact holds for a large and rather natural class of topological test spaces. [**4.1 Lemma:**]{} [*Let $X$ be any Hausdorff (indeed, $T_{1}$) space, and let $U \subseteq X$ be a dense open set. Then $(U) = \{ F \in 2^{X} | F \subseteq U\}$ is a dense open set in $2^{X}$.*]{} Proof: Since sets of the form $\langle U_{1},....,U_{n}\rangle$, $U_{1},...,U_{n}$ open in $X$, form a basis for the Vietoris topology on $2^{X}$, it will suffice to show that $(U) \cap \langle U_{1},...,U_{n} \rangle \not = 0$ for all choices of non-empty opens $U_{1},...,U_{n}$. Since $U$ is dense, we can select for each $i = 1,...,n$ a point $x_{i} \in U \cap U_{i}$. The finite set $F := \{x_{1},...,x_{n}\}$ is closed (since $X$ is $T_{1}$), and by construction lies in $(U) \cap \langle U_{1},...,U_{n} \rangle$. $\Box$ [**4.2 Corollary:**]{} [*Let $(X,{\frak A})$ be any topological test space with $X$ having no isolated points, and let $E$ be any test in ${\frak A}$. Then open set $(E^{c}) = [E]^{c}$ of tests disjoint from $E$ is dense in ${\frak A}$.*]{} Proof: Since $E$ is a closed set, its complement $E^{c}$ is an open set; since $E$ is discrete and includes no isolated point, $E^{c}$ is dense. The result follows from the preceding lemma. $\Box$ [**4.3 Theorem:**]{} [*Let $(X,{\frak A})$ be a topological test space with $X$ (and hence, ${\frak A}$) second countable, and without isolated points. Then there exists a countable, pairwise-disjoint sequence $E_{n} \in {\frak A}$ such that (i) $\{E_{n}\}$ is dense in ${\frak A}$, and (ii) $\bigcup_{n} E_{n}$ is dense in $X$.*]{} Proof: Since it is second countable, ${\frak A}$ has a countable basis of open sets ${\cal W}_{k}$, $k \in {\Bbb N}$. Selecting an element $F_{k} \in {\cal W}_{k}$ for each $k \in {\Bbb N}$, we obtain a countable dense subset of ${\frak A}$. We shall construct a countable dense pairwise-disjoint subsequence $\{E_{j}\}$ of $\{F_{k}\}$. Let $E_{1} = F_{1}$. By Corollary 4.2, $[E_{1}]^{c}$ is a dense open set; hence, it has a non-empty intersection with ${\cal W}_{2}$. As $\{F_{k}\}$ is dense, there exists an index $k(2)$ with $E_{2} := F_{k(2)} \in {\cal W}_{2} \cap [E_{1}]^{c}$. We now have $E_{1} \in W_{1}$, $E_{2} \in {\cal W}_{2}$, and $E_{1} \cap E_{2} = \emptyset$. Now proceed recursively: Since $[E_{1}]^{c} \cap [E_{2}]^{c} \cap \cdots \cap [E_{j}]^{c}$ is a dense open and ${\cal W}_{j+1}$ is a non-empty open, they have a non-empty intersection; hence, we can select $E_{j+1} = F_{k(j+1)}$ belonging to this intersection. This will give us a test belonging to ${\cal W}_{j+1}$ but disjoint from each of the pairwise disjoint sets $E_{1},...,E_{j}$. Thus, we obtain a sequence $E_{j} := F_{k(j)}$ of pairwise disjoint tests, one of which lies in each non-empty basic open set ${\cal W}_{j}$ – and which are, therefore, dense. For the second assertion, it now suffices to notice that for each open set $U \subseteq X$, $[U]$ is a non-empty open in ${\frak A}$, and hence contains some $E_{j}$. But then $E_{j} \cap U \not = \emptyset$, whence, $\bigcup_{j} E_{j}$ is dense in $X$. $\Box$\ [References]{} \[1\] Choe, T. H., Greechie, R. J., and Chae, Y., [*Representations of locally compact orthomodular lattices*]{}, Topology and its Applications [**56**]{} (1994) 165-173 \[2\] Clifton, R., and Kent, [*Simulating quantum mechanics with non-contextual hidden variables*]{}, The Proceedings of the Royal Society of London A [**456**]{} (2000) 2101-2114 \[3\] Foulis, D. J., Greechie, R. J., and Ruttimann, G. T., [*Filters and supports on orthoalgebras*]{} International Journal of Theoretical Physics [**31**]{} (1992) 789-807 \[4\] Foulis, D. J., Greechie, R. J., and Ruttimann, G. T., [*Logico-algebraic structures II: supports on test spaces*]{}, International Journal of Theoretical Physics [**32**]{} (1993) 1675-1690 \[5\] Foulis, D. J., and Randall, C.H., [*A mathematical language for quantum physics*]{}, in H. Gruber et al. (eds.), [*Les fondements de la méchanique quantique*]{}, AVCP: Lausanne (1983) \[6\] Gleason, A., [*Measures on the closed subspaces of Hilbert space*]{}, Journal of Mathematics and Mechanics [**6**]{} (1957) 885-894 \[7\] Illanes, A., and Nadler, S. B., [**Hyperspaces**]{}, Dekker: New York (1999) \[8\] Meyer, D., [*Finite precision measurement nullifies the Kochen-Specker theorem*]{}, Physical Review Letters [**83**]{} (1999) 3751-3754 \[9\] Michael, E., [*Topologies on spaces of subsets*]{}, Transactions of the American Mathematical Society [**7**]{} (1951) 152-182 \[10\] Nachbin, L., [**Topology and Order**]{}, van Nostrand: Princeton 1965 \[11\] Wilce, A., [*Test Spaces and Orthoalgebras*]{}, in Coecke et al (eds.) [**Current Research in Operational Quantum Logic**]{}, Kluwer: Dordrecht (2000) \[12\] Wilce, A., [*Symmetry and Compactness in Quantum Logic*]{}, in preparation. [^1]: I wish to dedicate this paper to the memory of Frank J. Hague III [^2]: It is also usual to assume that ${\frak A}$ is [*irredundant*]{}, i.e., that no set in ${\frak A}$ properly contain another. For convenience, we relax this assumption. [^3]: With one side defined iff the other is. [^4]: In particular, $\emptyset$ is an isolated point of $2^{X}$. Many authors omit $\emptyset$ from $2^{X}$. [^5]: More precisely, if ${\cal C}$ is a compact subset of $2^{X}$ with each $C \in {\cal C}$ compact, then $\bigcup_{C \in {\cal C}} C$ is again compact.

--- abstract: 'We present a simple model which allows to investigate equilibrium aspects of molecular recognition between rigid biomolecules on a generic level. Using a two-stage approach, which consists of a design and a testing step, the role of cooperativity and of varying bond strength in molecular recognition is investigated. Cooperativity is found to enhance selectivity. In complexes which require a high binding flexibility a small number of strong bonds seems to be favored compared to a situation with many but weak bonds.' author: - 'Hans Behringer, Andreas Degenhard, Friederike Schmid' title: 'A Coarse-Grained Lattice Model for Molecular Recognition' --- Living organisms could not function without the ability of biomolecules to specifically recognize each other [@Alberts_1994; @Kleanthous_2000]. Molecular recognition can be viewed as the ability of a biomolecule to interact preferentially with a particular target molecule among a vast variety of different but structurally similar rival molecules. Recognition processes are governed by an interplay of non-covalent interactions, in particular, hydrophobic interactions and hydrogen bonds. Such non-covalent bonds have typical energies of 1-2 kcal/mole (the relatively strong hydrogen bonds may contribute up to 8-10 kcal/mole) and are therefore only slightly stronger than the thermal energy $k_{{\scriptsize}{\mbox}{B}}T_{{\scriptsize}{\mbox}{Room}} \simeq 0.62$ kcal/mole at physiological conditions. Biomolecular recognition is thus only achieved if a large number of functional groups on the two partner molecules match precisely. This observation has lead to a “key-lock” picture: Two biomolecules recognize each other if their shapes at the recognition site and/or the interactions between the residues in contact are largely complementary [@pauling_1940]. In the present Letter, we introduce a coarse-grained approach which allows to investigate this “principle of complementarity” on a very general level, and use it to study the role of different factors for the selectivity of interactions between biomolecule surfaces. Specifically, we analyze two elements that have been discussed in the literature: the cooperativity, and the interplay of interaction strengths. We will show that our model can help to understand some of the features of real protein-protein interfaces. Previous theoretical studies have mostly dealt with the adsorption of heteropolymers on random and structured surfaces [@Chakraborty_2001; @Polotsky_2004a; @Bogner_2004]. Some works have adapted the random energy model from the theory of disordered systems to the problem of biomolecular binding [@Janin_1986; @Wang_2003]. In contrast, in the present approach, we consider explicitly systems of two interacting, rigid, heterogeneous surfaces. This is motivated by some basic findings about the biochemical structure of the recognition site, [[*i.e.*]{}]{}, the contact interface between recognizing proteins. In recent years the structural properties of proteins at the recognition site has been clarified [@Jones_1996; @LoConte_1999; @Kleanthous_2000]. Although different protein-protein complexes may differ considerably, a general picture of a standard recognition site containing approximately 30 residues, with a total size of 1200-2000 Å${}^2$ has emerged. Apart from notable exceptions, the association of the proteins is basically rigid, although minor rearrangements of amino acid side-chains do occur [@Jones_1996; @LoConte_1999]. We describe the structure of the proteins at the contact interfaces by two sets of classical spin variables $\sigma = (\sigma_1,\ldots, \sigma_N)$ and $\theta= (\theta_1, \ldots,\theta_N)$, whose values specify the various types of residues. The set $\sigma$ characterizes the structure of the recognition site on the target molecule, and $\theta$ that on the probe molecule, [[*i.e.*]{}]{}, the molecule that is supposed to recognize the target. The position of site $i$ on the surfaces can be specified arbitrarily. For simplicity, we assume that the positions $i$ on both surfaces match, and that the total number of contact residues is equal $N$ for both molecules. However, we take into account the possibility that the quality of the contact of two residues at position $i$ may vary, [[*e.g.*]{}]{}, due to steric hindrances or varying relative alignment of polar moments, caused by minor rearrangements of the amino acid side-chains. This is modeled by an additional variable $S_i$, $i=1, \ldots, N$. The total interaction is thus described by a Hamiltonian $\mathcal{H}(\sigma, \theta; S)$, which incorporates in a coarse-grained way both the structural properties of the recognition site and the interaction between residues. To study the recognition process between two biomolecules, we adopt a two-stage approach. We take the structure of the target recognition site, $\sigma^{(0)} = (\sigma_1^{(0)}, \ldots, \sigma_N^{(0)})$, to be given. In the first step, the probe “learns” the target structure at a given “design temperature” $1/\beta_{{\mbox}{\tiny D}}$. One obtains an ensemble of probe molecules with structures $\theta$ distributed according to a probability $P(\theta | \sigma^{(0)}) = \frac{1}{Z_{{\scriptsize}{\mbox}{\tiny D}}} \sum_{S} \exp\left(-\beta_{{\scriptsize}{\mbox}{\tiny D}} \mathcal{H}(\sigma^{(0)}, \theta;S) \right)$, which depends on the target structure. This first design step is introduced to mimic the design in biotechnological applications or the evolution process in nature. The parameter $\beta_{{\scriptsize}{\mbox}{\tiny D}}$ characterizes the conditions under which the design has been carried out, [[*i.e.*]{}]{}, it is a Lagrange parameter which fixes the achieved average interaction energy. A similar design procedure has been introduced in studies of protein folding [@Pande_2000] and the adsorption of polymers on structured surfaces [@Jayaraman_2005]. In the second step, the recognition ability of the designed probe ensemble is tested. To this end the probe molecules are exposed to both the original target structure $\sigma^{(0)}$ and a competing (different) rival structure $\sigma^{(1)}$ at some temperature $1/{\beta}$, which in general differs from the design temperature $1/\beta_{{\mbox}{\tiny D}}$. The thermal free energy $F(\theta|\sigma^{(\alpha)})$ for the interaction between $\sigma^{(\alpha)}$ ($\alpha = 0,1$) and a probe $\theta$ is given by $ F(\theta|\sigma^{(\alpha)}) = -\frac{1}{\beta} \ln \sum_{S} \exp\left(- \beta \mathcal{H}(\sigma^{(\alpha)}, \theta;S) \right) $. Averaged over all probe molecules, we obtain $\langle F^{(\alpha)} \rangle = \sum_\theta F(\theta|\sigma^{(\alpha)})P(\theta|\sigma^{(0)})$. The target is recognized if the average free energy difference $\Delta F = \langle F^{(0)} \rangle - \langle F^{(1)} \rangle$ is negative, [[*i.e.*]{}]{}, probe molecules exposed to equal amounts of target and rival molecules preferentially bind to the target. Note that our treatment does not account for kinetic effects, only equilibrium aspects are considered. The association of the proteins is accompanied by a reduction of the translational and rotational entropy. However, these additional entropic contributions to the free energy of association depend only weakly on the mass and shape of the rigid molecules, and can be considered, in a first approximation, to be of the same order for the association with the target and the rival molecule. Thus, these contributions cancel in the free energy difference. Similarly, contributions from the interaction with solvent molecules are also assumed to be of comparable size. A modified HP-model can serve as a first example to illustrate this general description. In the HP-model, which was introduced originally to study protein folding [@Dill_1985], residues are distinguished by their hydrophobicity only. Hydrophobic residues are represented by $\sigma_i,\theta_i = +1$, and polar residues by $\sigma_i,\theta_i = -1$. In addition, the variable $S_i$ describing the (geometric) quality of the contact can take on the values $\pm 1$ where $S_i = +1$ models a good contact and $S_i=-1$ a bad one. Only for good contacts does one get a contribution to the binding energy. The Hamiltonian is then given by $$\label{eq:HP-hamiltonian} \mathcal{H}(\sigma,\theta;S) =- \varepsilon \sum_{i}\frac{1+S_i}{2} \sigma_i\theta_i$$ where the sum extends over the $N$ positions of the residues of the recognition site and $\varepsilon$ being the interaction constant [@footnote]. Note that a “good” contact can nevertheless lead to an unfavorable energy contribution. For this simple model, the different steps of the two-stage approach described above can be worked out analytically. First, we analyze the efficiency of the design step by inspecting the achieved complementarities (of interactions) of the designed probe molecules with the target molecule. To this end, we define a complementarity parameter $K = \sum_i \sigma^{(0)}_i\theta_i$ which ranges from $-N$ to $+N$, with $K$ close to $+N$ signaling a large complementarity of the recognition sites. The probability distribution $P(\theta|\sigma^{(0)})$ can be converted to a distribution $P(K)$ for the probability of having a complementarity $K$. Up to a normalization factor, it is given by $$P(K) \propto {N \choose \frac{1}{2}(N+K)} \exp \left( \frac{\varepsilon\beta_{{\mbox}{\tiny D}}}{2} K\right).$$ Its first moment $\left<K\right> = \sum_K K P(K) = N \tanh\left( {\varepsilon\beta_{{\mbox}{\tiny D}}}/{2} \right)$ quantifies the quality of the design. For decreasing design temperatures $ 1/\beta_{{\mbox}{\tiny D}}$ the average complementarity per site $\left< K \right>/N$ approaches one, and thus the designed probe molecules are well optimized with respect to the target. In the second step the association of the probe molecules with the target and with a different rival molecule is compared. Introducing the quantity $Q = \sum_i \sigma^{(0)}_i\sigma^{(1)}_i$ as a measure for the similarity between the recognition sites of the target and the rival molecules, the free energy difference per site can be expressed in the form $\Delta F(Q)/N = -\frac{1}{2} \varepsilon \tanh\left(\frac{\varepsilon \beta_{{\mbox}{\tiny D}}}{2} \right) (1-Q/N).$ $\Delta F/N$ is negative, if the rival and the target are different and $Q$ is thus smaller than $N$. The probe molecule therefore binds preferentially to the target molecule, and thus the target is specifically recognized. The free energy difference increases with decreasing similarity parameter $Q$. After this introductory analysis of a simple system, we turn to consider more complex models which allow to investigate the influence of different factors on the specific recognition between surfaces. We begin with studying the role of cooperativity. Systematic mutagenesis experiments have revealed that cooperativity plays an important role in molecular recognition processes [@Cera_1998]. Cooperativity in biological processes basically means that the interaction strength of two residues depends on the interactions in their neighborhood. Physically, this can be caused by a physical rearrangement of amino acid side-chains or a readjustment of polar moments as a function of the local environment. In the simplified language of our model, cooperativity thus means that the quality of a contact depends on the quality of the neighbor contacts. This can be incorporated in the HP-model by the following extension: $$\label{eq:HP-coop} \mathcal{H}(\sigma, \theta;S) = - \varepsilon\sum_{i=1}^{N} \frac{1+S_i}{2} \sigma_i \theta_i - J \sum_{\left<ij\right>} S_i S_j.$$ The second sum accounts for the cooperative interaction and runs over neighbor residue positions $i$ and $j$. The interaction coefficient $J$ is positive for cooperative interactions and negative for anti-cooperativity. For $J > 0$, the cooperative term rewards additional contacts in the vicinity of a good contact between two residues. This leads to a better optimization of the side-chains and thus the complementarity between the probe and the target molecule is improved. Cooperativity is therefore expected to enhance the quality of the design step compared to an interaction without cooperativity. Similarly, one expects an improved recognition specificity. For non-zero, but finite values of $J$, the model can no longer be solved analytically. Therefore, we calculated numerically the density of states for the interaction between two proteins as a function of the energy and the complementarity parameter using efficient modern Monte Carlo algorithms [@Hueller_2002]. The density of states $\Omega_J(K, E)$ for a fixed target structure $\sigma^{(0)}$ is the number of configurations $(\theta,S)$ that have energy $E=\mathcal{H}$ when interacting with the target, and a complementarity $K$ with the target recognition site. The probability distribution of the complementarity $K$ is then (up to a normalization constant) given by $ P_{\beta_{{\mbox}{\tiny D}}}(K; J) \sim \sum_E \Omega_J(K, E) \exp(-\beta_{{\mbox}{\tiny D}}E)$. ![\[bild:coop\_comp\] Average complementarity per site of the designed probe ensemble for different values of $J$. For the lower dashed curve $J=0$, the upper dashed line represents the limit $J \to \infty$, which can be tackled analytically [@Behringer_etal_vor]. The curves in between from bottom up belong to values 0.1, 0.25, 0.5, 0.75 of $J$ in units of $\varepsilon$. The inset shows $\left<K\right>/N$ for $N=256 $ (full curve) and $N=36$ (dashed line) with $J=\varepsilon/2$. Only minor finite-size effects are visible.](figure1.eps) For simplicity, we consider asymptotically large interfaces on a square lattice. (The actual calculations shown here were carried out with $N=256$, and we checked that the results do not change any more for larger $N$). Fig. \[bild:coop\_comp\] shows the average complementarity $\left< K \right>/N $ for different cooperativities $J$. Cooperativity is found to increase the average complementarity of the designed probe molecules for large enough values of the parameter $\varepsilon \beta_{{\mbox}{\tiny D}}$. For $\varepsilon \beta_{{\mbox}{\tiny D}} \sim 1$, a small change in the cooperativity $J$ leads to a large difference in the average complementarity, [[*i.e.*]{}]{}, small changes in $J$ can have a large impact on the recognition process. As $\varepsilon$ is typically of the order of 1 kcal/mole, this regime indeed corresponds to physiological conditions for reasonable design temperatures, $1/\beta_{{\mbox}{\tiny D}} \lesssim 1/\beta_{{\mbox}{\tiny Room}}$. Fig. \[bild:coop\_freiedifferenz\] shows the free energy difference per site $\Delta F(Q)/N$ of the association of probe molecules with the target structure and a rival structure, for different values of the cooperativity constant $J$. Increasing the cooperativity increases the free energy difference. Relatively small cooperativities are sufficient to obtain an effect, and the maximum effect of cooperativity is already reached for a value $J \simeq \varepsilon$. Thus, we find that cooperativity indeed improves the recognition ability as expected for cooperativity constants $J \simeq \varepsilon$. The above findings were obtained for large interfaces. Although minor finite-size effects are visible for interfaces of realistic size (with $N \sim \mathcal{O}(30)$) the general findings discussed above still hold qualitatively (compare inset of Fig. \[bild:coop\_comp\]). ![\[bild:coop\_freiedifferenz\] Free energy difference per site (in arbitrary units) of the association of the probe ensemble with the two competing molecules as a function of their similarity for different cooperativities $J$ (with $\beta_{{\mbox}{\tiny D}}=\beta=0.5$). For the upper dashed line $J=0$, the lower dashed line describes the limiting case $J\to \infty$ for $Q/N$ close to one [@Behringer_etal_vor]. The full curves from top to bottom correspond to the same values of $J$ as in Fig. \[bild:coop\_comp\].](figure2.eps) [ In situations where one molecule is flexible conformational changes occur. However, cooperativity works on the level of residue interactions and thus we expect that the favorable effect of cooperativity to molecular recognition is not spoilt by the entropic contributions due to refolding. This, however, needs further investigation. Note that flexible binding has been addressed recently [@Wang_2006].]{} Next, we investigate the role of the interplay of interactions for molecular recognition. This study is motivated by the observation that antibody-antigen interfaces have very specific properties. Mutagenesis studies have revealed that the structural interface in these complexes is different from the functional recognition site made up of those residues that contribute to the binding energy. Only approximately one quarter of the residues at the interface contribute considerably to the binding energy [@Cunningham_1993; @LoConte_1999]. These contributing residues are sometimes called “hot spots”. In addition it has been shown that antigen-antibody interfaces are less hydrophobic, compared to other protein-protein interfaces, so that the relatively strong hydrogen bonds are more important [@LoConte_1999]. In the immune system molecular recognition must satisfy very specific requirements. The immune system has to recognize substances that have never been encountered before. Thus antigen-antibody recognition has to exhibit a large flexibility [@Jones_1996], and has to be able to adapt very rapidly by evolution. These peculiarities of antibody-antigen interfaces suggest that selective molecular interactions are obtained most efficiently with only a few strong interactions across the interface, so that a complementarity with the whole recognition site is not necessary. Within our two-stage approach, we can address the question whether few but strong bonds or many but weak bonds are more favorable. To this end we consider a model which distinguishes between active and inactive residues only. Only active residues contribute to a bond. The variables $\sigma$ and $\theta$ now take on the values $\sigma_i,\theta_i = +1$ for active and $\sigma_i,\theta_i = 0$ for inactive residues, and the Hamiltonian is given by $$\label{eq:AI-hamiltonian} \mathcal{H}(\sigma,\theta, S) =- \varepsilon_{{\scriptsize}{\mbox}{H}} \sum_{i=1}^N\frac{1+S_i}{2} \sigma_i\theta_i$$ with $S_i$ specifying again the quality of the contact of residues, and $\varepsilon_{{\mbox}{\tiny H}}$ giving the interaction strength. Moreover, we extend the design step by fixing the average number of active residues $A = \langle \sum_i \theta_i \rangle$ on the probe molecules with a Lagrange parameter. The total interaction energy $E$ is also subject to restrictions: It has to exceed the thermal energy to stabilize the complex, but on the other hand it has to be small enough to ensure the high flexibility of the target-probe complex that is crucial for the immune system. When increasing the average number of active residues $A$, one must therefore reduce the interaction energy $\varepsilon_{{\scriptsize}{\mbox}{H}}$ accordingly, [[*e.g.*]{}]{}, by keeping the product $E \approx A \varepsilon_{{\scriptsize}{\mbox}{H}}$ constant. Figure \[bild:lern\_NBenergie\] shows as a function of $A/N$ the average free energy difference per site $\Delta F/N$ of the association with the target molecule and a rival molecule, averaged over all possible target and rival structures $\sigma$. We find that $\langle \Delta F \rangle$ exhibits a minimum at a small fraction $A/N$ of active residues. The position of the minimum at small fractions of $A/N$ is fairly insensitive to a variation of the interaction parameters. Hence this simple coarse-grained model already predicts that molecular recognition is most efficient if the functional recognition site consists only of a small fraction of the structural recognition site, as is indeed observed in antibody-antigen complexes. ![\[bild:lern\_NBenergie\] Averaged free energy difference per site (in arbitrary units) as a function of the fraction $A/N$ of active residues for $\varepsilon_{{\scriptsize}{\mbox}{H}} A/N = 0.1$. The full curve corresponds to a ratio $\beta/\beta_{{\mbox}{\tiny D}} = 1$, the dashed curve to $\beta/\beta_{{\mbox}{\tiny D}} = 1/2$. ](figure3.eps) In conclusion, we have presented coarse-grained models which allow to study generic features of biomolecular recognition. A two-stage approach which distinguishes between the design of probe molecules and the test of their recognition abilities has been adopted. We have applied the approach to investigate the role of cooperativity and of hydrogen bonding for molecular recognition. It turned out that cooperativity can substantially influence the efficiency of both design and recognition ability of recognition sites. Our model also reproduces the observation that the structural recognition site has to be distinguished from a functional recognition site in highly flexible complexes such as antigen-antibody complexes. The approach can readily be generalized to study other aspects of molecular recognition. For example, it will be interesting to investigate the influence of the heterogeneity of the mixture of target and rival molecules in physiological situations. This can be incorporated by considering ensembles of targets and rivals differing in certain properties as for example correlations and length scales. 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--- abstract: | Given a linear equation of the form $a_1x_1 + a_2x_2 + a_3x_3 = 0$ with integer coefficients $a_i$, we are interested in maximising the number of solutions to this equation in a set $S \subseteq \mathbb{Z}$, for sets $S$ of a given size. We prove that, for any choice of constants $a_1, a_2$ and $a_3$, the maximum number of solutions is at least $\left(\frac{1}{12} + o(1)\right)|S|^2$. Furthermore, we show that this is optimal, in the following sense. For any $\varepsilon > 0,$ there are choices of $a_1, a_2$ and $a_3,$ for which any large set $S$ of integers has at most $\left(\frac{1}{12} + \varepsilon\right)|S|^2$ solutions. For equations in $k {\geqslant}3$ variables, we also show an analogous result. Set $\sigma_k = \int_{-\infty}^{\infty} (\frac{\sin \pi x}{\pi x})^k dx.$ Then, for any choice of constants $a_1, \dots, a_k$, there are sets $S$ with at least $\left(\frac{\sigma_k}{k^{k-1}} + o(1)\right)|S|^{k-1}$ solutions to $a_1x_1 + \dots + a_kx_k = 0$. Moreover, there are choices of coefficients $a_1, \dots, a_k$ for which any large set $S$ must have no more than $\left(\frac{\sigma_k}{k^{k-1}} + \varepsilon\right)|S|^{k-1}$ solutions, for any $\varepsilon > 0$. author: - James Aaronson nocite: '[@HLP]' title: Maximising the number of solutions to a linear equation in a set of integers --- Introduction {#sec:intro} ============ Let $a_1, a_2 {\text{ and }}a_3$ be fixed coprime integers, none of which is zero. We will consider the linear equation $$\label{eqn:to_solve} a_1x_1 + a_2x_2 + a_3x_3 = 0.$$ In this paper, we are interested in the problem of finding sets with as many solutions to (\[eqn:to\_solve\]) as possible. This leads to the following definition. Given a finite set $S {\subseteq}{\mathbb{Z}}$, define $T(S) = T_{a_1, a_2, a_3}(S)$ to be the number of triples $x_1, x_2, x_3 \in S$ satisfying (\[eqn:to\_solve\]). The trivial upper bound on $T(S)$ is $T(S) {\leqslant}|S|^2$. This is because, for any choice of $x_1 {\text{ and }}x_2$, there is at most one choice of $x_3$ such that $a_1x_1 + a_2x_2 + a_3x_3 = 0$, namely $x_3 = \frac{-a_1x_1-a_2x_2}{a_3}$. We are interested in making $T(S)$ as large as possible, for a fixed size $|S|$. For some choices of coefficients $a_1, a_2 {\text{ and }}a_3$, the exact maximal value of $T(S)$ is known. For example, consider the case $a_1 = a_2 = a_3 = 1$. Then, work of Hardy and Littlewood [@HLIneqNotes] and Gabriel [@Gabriel] shows that, when $|S|$ is odd, $T(S)$ is maximised when $S$ is an interval centred about 0. This was extended to even $|S|$ by Lev in [@lev_max]. In fact, their arguments show that if $S \subseteq {\mathbb{Z}}$ is a set, and $S'$ is an interval centred about 0 of the same size, then $T_{a_1, a_2, a_3}(S) {\leqslant}T_{1,1,1}(S')$. The ideas behind their approaches involve rearrangement inequalites, which are discussed in detail in [@HLP Chapter 10], and which inspire some of the arguments in this paper. Similarly, it is shown by Green and Sisask in [@GreenSisask Theorem 1.2] and by Lev and Pinchasi in [@LevPinchasi Theorem 2] respectively that, if $(a_1, a_2, a_3) = (1, -2, \pm 1)$, then $T(S)$ is again maximised when $S$ is an interval centred at 0. The set of solutions to $x_1 - 2x_2 + x_3 = 0$ is precisely the set of three-term arithmetic progressions; that is, the set of affine shifts of the set $\{0, 1, 2\}$. By analogy with this, Bhattacharya, Ganguly, Shao and Zhao considered longer arithmetic progressions; in [@uppertails Theorem 2.4], they proved that the number of $k$ term arithmetic progressions in a set $S$ of $n$ integers is maximised when $S$ is an interval. Ganguly asked [@pc_ganguly] about other affine patterns; in particular, finding sets $S$ with as many affine copies of $\{0, 1, 3\}$, or solutions to $x + 2y = 3z$, as possible. In this case, such a result would necessarily be less clean; for instance, there are more solutions to $x + 2y = 3z$ in $\{0,1,3\}$ than in $\{0,1,2\}$. Indeed, in general, much less is known. For a lower bound on the maximal value of $T(S)$, a fairly good bound is given by the following example. \[prop:twelfth\] Regardless of the values of $a_1, a_2 {\text{ and }}a_3$, there are choices of $S$ with $|S|$ arbitrarily large, for which $T(S) {\geqslant}\frac{1}{12}|S|^2 + O(|S|)$. The idea behind the construction is to split $S$ into three pieces $S_1, S_2 {\text{ and }}S_3$, of roughly equal size, for which there are many solutions to $a_1x_1 + a_2x_2 + a_3x_3 = 0$ with each $x_i$ taken from $S_i$. Let $M$ be a large integer, which we assume to be divisible by 6. We will define $$\begin{aligned} S_1 &= a_2a_3[-M/6, M/6] \\ S_2 &= a_1a_3[-M/6, M/6] \\ S_3 &= a_1a_2[-M/6, M/6], \end{aligned}$$ where $[-M/6, M/6]$ refers to the set of integers with absolute value no greater than $M/6$, and set $S = S_1 \cup S_2 \cup S_3$. Then, $|S|$ is certainly no more than $M$. However, we may find a large collection of triples $(x_1, x_2, x_3)$ by choosing $x_1 \in S_1 {\text{ and }}x_2 \in S_2$ arbitrarily, and selecting those for which $x_3 = \frac{-a_1x_1-a_2x_2}{a_3}$ is in $S_3$. If $x_1 = a_2a_3x_1' {\text{ and }}x_2 = a_1a_3x_2'$, then we have $x_3 = -a_1a_2(x_1' + x_2')$. Therefore, a pair $(x_1', x_2')$ will give rise to a solution precisely when $|x_1' + x_2'| {\leqslant}M/6$. We may compute the number of such pairs $(x_1', x_2')$ as the sum $$\sum_{x_1' = -M/6}^{M/6} M/3 + 1 - |x_1'| = \frac{1}{12}M^2 + O(M).$$ Thus, the number of triples is at least $\frac{1}{12}|S|^2 + O(|S|)$. Given this, it is natural to define the following quantity: Define $\gamma_{a_1,a_2,a_3}$ by $$\gamma_{a_1,a_2,a_3} = \limsup_{|S| \rightarrow \infty} \dfrac{T(S)}{|S|^2}$$ where $S$ runs over subsets of ${\mathbb{Z}}$. Thus, the assertion that $$\label{eqn:gamma_uniform_bound} \frac{1}{12} {\leqslant}\gamma_{a_1, a_2, a_3} {\leqslant}\frac{3}{4}$$ holds for all $a_1, a_2$ and $a_3$ follows from Proposition \[prop:twelfth\] and the work of Hardy and Littlewood in [@HLIneqNotes]. As far as the author is aware, exact values for $\gamma_{a_1, a_2, a_3}$ are only known in cases for which $|a_1a_2a_3| {\leqslant}2$ (this includes the cases previously discussed). In particular, we have $$\begin{aligned} \gamma_{1,1,\pm 1} &= \frac{3}{4} \label{eqn:trivial_case}\\ \gamma_{1,-2,\pm 1} &= \frac{1}{2} \label{eqn:abc_is_2}\end{aligned}$$ $\gamma_{1,-2,1}$ is [@GreenSisask Theorem 1.2], and $\gamma_{1,-2,-1} = \frac{1}{2}$ is [@LevPinchasi Theorem 2]. The same holds in the third non-equivalent case with $|a_1a_2a_3| = 2$, namely $\gamma_{1, 2, 1} = \frac{1}{2}$. Even the value of $\gamma_{1,2,-3}$ is not known, although the author conjectures that it is $\frac{1}{3}$, which is the value calculated for $S = [-M/2, M/2]$. The main theorem of this paper is a converse, of sorts, to Proposition \[prop:twelfth\]. In particular, we will prove the following. \[thm:main\_theorem\] The constant $\frac{1}{12}$ in the statement of Proposition \[prop:twelfth\] is optimal, in the following sense. For any ${\varepsilon}> 0$, there exists a choice of $a_1, a_2 {\text{ and }}a_3$ for which $\gamma_{a_1, a_2, a_3} {\leqslant}\frac{1}{12} + {\varepsilon}$. In view of this theorem, (\[eqn:gamma\_uniform\_bound\]) gives the best possible bounds on $\gamma_{a_1, a_2, a_3}$ that are independent of the coefficients $a_i$. The plan for this paper is as follows. In Section \[sec:import\_lemmas\], we will record some additive combinatorial lemmas that we will need in order to establish Theorem \[thm:main\_theorem\]. In Section \[sec:proof\_of\_thm\], we will use these lemmas to prove Theorem \[thm:main\_theorem\]. One might also ask about generalising Theorem \[thm:main\_theorem\] to other settings. For instance, given a system of $m$ linear equations in $k$ variables (where we assume that $m {\leqslant}k - 2$), can we prove an analogue of Theorem \[thm:main\_theorem\]? If $m = 1$, then an analogue of Proposition \[prop:twelfth\] holds for any value of $k {\geqslant}3$. Set $$\label{eqn:definition_of_sigma} \sigma_k = \int_{-\infty}^{\infty} \left(\frac{\sin \pi x}{\pi x} \right)^k dx.$$ Then, for any choice of coefficients $a_1, \dots, a_k$, there are sets $S$ with at least $\frac{\sigma_k}{k^{k-1}}|S|^{k-1} + O(|S|^{k-2})$ solutions to $a_1x_1 + \dots + a_kx_k = 0$. We will discuss (\[eqn:definition\_of\_sigma\]) further in Section \[sec:generalisation\_to\_k\_var\]. Furthermore, the corresponding analogue of Theorem \[thm:main\_theorem\] holds. For any ${\varepsilon}> 0$, there are choices of coefficients $a_1, \dots, a_k$ for which any large set $S$ must have no more than $\left(\frac{\sigma_k}{k^{k-1}} + \varepsilon\right)|S|^{k-1}$ solutions. For instance, for any small positive ${\varepsilon}$ we can find coefficients $a_1, a_2, a_3 {\text{ and }}a_4$ with the property that $ T(S) {\leqslant}\left( \frac{1}{96} + {\varepsilon}\right) |S|^3, $ where $T(S)$ counts the number of solutions to $a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4 = 0.$ We will discuss this in Section \[sec:generalisation\_to\_k\_var\]. On the other hand, the opposite is true in the case that $m > 1$. Indeed, it is possible to show that there is *no* constant $c > 0$, such that for any system of 2 equations in 4 variables, there are large sets $S$ with at least $c |S|^2$ solutions to the system. We will prove this fact in Section \[sec:generalisation\_to\_systems\]. Notation {#notation .unnumbered} -------- As we have already noted, $T(S)$ will be the number of solutions to $a_1x_1 + a_2x_2 + a_3x_3$ in S. We can extend this by defining $T(S_1, S_2, S_3)$ to be the number of solutions to $a_1x_1 + a_2x_2 + a_3x_3$, where $x_i \in S_i$. We will use the notation $a \cdot S$ to denote the set $\{ax, x \in S\}$. We will also make frequent use of the Vinogradov notation $f \ll g$ to mean that $f = O(g)$. When the $\ll$ is subscripted, we allow the implicit constant to depend on the subscripts. The author is supported by an EPSRC grant EP/N509711/1. The author would like to thank his supervisor, Ben Green, for his continued support and encouragement, and the anonymous referee for a thorough reading of a previous version of this paper. This version of the paper replaces a previous version [@me]. The argument used to prove Theorem \[thm:main\_theorem\] is replaced with a new argument which avoids appealing to the arithmetic regularity lemma (and can handle a wider class of equations), and the results of Section \[sec:generalisation\_to\_systems\] are new to this version. Additive Combinatorial Lemmas {#sec:import_lemmas} ============================= In this section, we will collect some lemmas that will be necessary for the proof of Theorem \[thm:main\_theorem\]. For any set $A \subseteq {\mathbb{Z}}$, let $\delta[A]$ be its growth under the differencing operator, $\frac{|A-A|}{|A|}$. If $A {\text{ and }}B$ are two sets of integers, let the *additive energy* between $A {\text{ and }}B, E(A, B)$, be defined by $$E(A,B) = \# \{(a_1,b_1,a_2,b_2) \in A \times B \times A \times B: a_1 + b_1 = a_2 + b_2\}.$$ It is easy to see that this satisfies the following inequalities: \[eqn:bound\_energy\] E(A,B) &|A|\^2|B|\ E(A,B) &|A||B|\^2\ E(A,B) &|A|\^[3/2]{}|B|\^[3/2]{}, the third of which follows immediately from the first two. We will require the following lemma, which states that, when two sets $A {\text{ and }}B$ have $\delta[A] {\text{ and }}\delta[B]$ small, and if $E(A, B)$ is large, then $|A - B|$ is also small. \[lem:gs\_doubling\_lem\] Suppose that $A, B \subseteq {\mathbb{Z}}$ are sets with $E(A, B) {\geqslant}\eta |A|^{3/2}|B|^{3/2}$. Then, $|A - B| {\leqslant}\frac{\delta[A]\delta[B]}{\eta} |A|^{1/2}|B|^{1/2}$. We will also require a weak form of a structure theorem due to Green and Sisask. \[thm:green\_sisask\_structure\] Let ${\varepsilon}_1 \in (0,1/2)$ be a parameter. Then there are choices of (large) integers $K_1 = K_1({\varepsilon}_1) {\text{ and }}K_2 = K_2({\varepsilon}_1)$ with the following property. For any set $S \subseteq {\mathbb{Z}}$, there is a decomposition of $S$ as a disjoint union $S_1 \amalg \dots \amalg S_n \amalg S_0$ such that 1. $|S_i| {\geqslant}|S|/K_1$ for $i=1,\dots,n$; 2. $\delta[S_i] {\leqslant}K_2$ for $i = 1,\dots,n$; 3. $E(S_0,S) {\leqslant}{\varepsilon}_1 |S|^3$. Observe that property (1) guarantees that $n {\leqslant}K_1$. The quantity $T(S_1, S_2, S_3)$ is related to the additive energy via the following lemma. \[lem:easy\_TNRG\_lemma\] Suppose that $S_1, S_2 {\text{ and }}S_3 \subseteq {\mathbb{Z}}$ are finite sets. Then $$T(S_1, S_2, S_3)^2 {\leqslant}E(a_1\cdot S_1, a_2\cdot S_2)|S_3|.$$ For any $t \in {\mathbb{Z}}$, let $\mu(t)$ denote the number of ways of writing $t = a_1x_1 + a_2x_2$, for $x_i \in S_i$. Thus, by definition, $$E(a_1\cdot S_1, a_2\cdot S_2) = \sum_{t} \mu(t)^2.$$ Now, we see that $$T(S_1, S_2, S_3)^2 = \left(\sum_{t \in -a_3 \cdot S_3} \mu(t)\right)^2 {\leqslant}\left(\sum_t \mu(t)^2\right)|S_3| = E(a_1\cdot S_1, a_2\cdot S_2) |S_3|,$$ the inequality following from Cauchy-Schwarz. This completes the proof of Lemma \[lem:easy\_TNRG\_lemma\]. The following two facts are standard results in additive combinatorics. \[lem:ruzsa\_triangle\] For sets $A, B, C \subseteq {\mathbb{Z}}$, $$|A - C| |B| {\leqslant}|A - B||B - C|.$$ \[lem:energy\_cs\] For sets $A, B \subseteq {\mathbb{Z}}$, $$E(A, B) {\leqslant}E(A,A)^{1/2} E(B,B)^{1/2}.$$ We will require the following lemma bounding $T(S_1, S_2, S_3)$. \[lem:uniform\_bound\_symmetric\] Suppose that $S_1, S_2, S_3 \subseteq {\mathbb{Z}}$ are sets with sizes $s_1, s_2 {\text{ and }}s_3$ respectively. Then, we have the bound $$\label{eqn:uniform_bound} T(S_1, S_2, S_3) {\leqslant}\frac{1}{4}\left(s_1s_2 + s_2s_3 + s_1s_3 + 1\right).$$ We will first prove Lemma \[lem:uniform\_bound\_symmetric\] in the case that $a_1, a_2 {\text{ and }}a_3$ are all 1. Without loss of generality, assume that $s_1 {\leqslant}s_2 {\leqslant}s_3$. Suppose first that $s_3 {\geqslant}s_1 + s_2$. In that case, we have $$\begin{aligned} T(S_1, S_2, S_3) &{\leqslant}s_1s_2 \\ &{\leqslant}\frac{1}{4} (s_1 + s_2)^2 \\ &{\leqslant}\frac{1}{4} (s_1s_3 + s_2s_3).\end{aligned}$$ The first line follows from the trivial observation that for each pair of $x \in S_1 {\text{ and }}y \in S_2$, there can be at most one solution to $x + y + z = 0$ with $z \in S_3$. The third line follows from our assumption on $s_3$. Thus, (\[eqn:uniform\_bound\]) follows in this case. Now, suppose that $s_3 < s_1 + s_2$. In this case, we may apply [@LevPinchasi Lemma 2], which states that $$T(S_1, S_2, S_3) {\leqslant}\frac{2(s_1s_2 + s_2s_3 + s_3s_1) - (s_1^2 + s_2^2 + s_3^2) + 1}{4}.$$ (\[eqn:uniform\_bound\]) follows in this case via an easy application of the Cauchy-Schwarz inequality. Finally, for arbitrary coefficients $a_1, a_2 {\text{ and }}a_3$, observe that $$\begin{aligned} T_{a_1, a_2, a_3}(S_1, S_2, S_3) &= T_{1,1,1}(a_1 \cdot S_1, a_2 \cdot S_2, a_3 \cdot S_3) \\ &{\leqslant}\frac{1}{4}\left(s_1s_2 + s_2s_3 + s_1s_3 + 1\right).\end{aligned}$$ This completes the proof of Lemma \[lem:uniform\_bound\_symmetric\]. Finally, we will require the following theorem of Bukh: \[thm:bukh\_sod\_thm\] Given two coprime integers $\lambda_1 {\text{ and }}\lambda_2$, we have that for any $S \subseteq {\mathbb{Z}}$, $$|\lambda_1 \cdot S + \lambda_2 \cdot S| {\geqslant}(|\lambda_1| + |\lambda_2|)|S| - o_{\lambda_1, \lambda_2}(|S|)$$ Proof of Theorem \[thm:main\_theorem\] {#sec:proof_of_thm} ====================================== In this section, we will use the lemmas of Section \[sec:import\_lemmas\] to prove Theorem \[thm:main\_theorem\]. We must prove that, given a suitable choice of $a_1, a_2 {\text{ and }}a_3$, all sufficiently large sets $S$ have $T(S) {\leqslant}\left(\frac{1}{12} + {\varepsilon}\right)|S|^2$. Let ${\varepsilon}> 0$. Given our choice of ${\varepsilon}$, we must choose the values of the coefficients $a_1, a_2 {\text{ and }}a_3$; we will do so later. Suppose that $S$ is a sufficiently large set. We will immediately apply the structure theorem, Theorem \[thm:green\_sisask\_structure\], to $S$, with ${\varepsilon}_1 = \left(\frac{{\varepsilon}}{6}\right)^4$. This gives us a decomposition $S = S_1 \amalg \dots \amalg S_n \amalg S_0$. We will start by showing that the contribution to $T(S)$ from solutions $a_1x_1 + a_2x_2 + a_3x_3 = 0$, with at least one of the $x_i$ taken from $S_0$, is small. \[lem:ignore\_S0\] The number of solutions to $a_1x_1 + a_2x_2 + a_3x_3 = 0$ in $S$, where some $x_i$ is taken from $S_0$, is no greater than $\frac{{\varepsilon}}{2}|S|^2$. The number of such solutions may be upper bounded by $$T(S_0, S, S) + T(S, S_0, S) + T(S, S, S_0),$$ and so it suffices to show that each term is no greater than $\frac{{\varepsilon}}{6}|S|^2$. Applying Lemmas \[lem:easy\_TNRG\_lemma\] and \[lem:energy\_cs\], we have $$\begin{aligned} T(S_0, S, S)^2 &{\leqslant}E(a_1 \cdot S_0, a_2 \cdot S) |S| \\ &{\leqslant}E(S_0, S_0)^{1/2} E(S, S)^{1/2} |S| \\ &{\leqslant}{\varepsilon}_1^{1/2} |S|^4,\end{aligned}$$ from which it follows that $T(S_0, S, S) {\leqslant}{\varepsilon}_1^{1/4}|S|^2$. By our choice of ${\varepsilon}_1$, this gives exactly what we claimed. At this point, we must bound the number of solutions to $a_1x_1 + a_2x_2 + a_3x_3 = 0$ where each of $x_1, x_2 {\text{ and }}x_3$ is taken from an $S_i$ with $i {\geqslant}1$. To do this, we will start by restricting which triples $(i, j, k)$ can have the property that there are many solutions with $x_1 \in S_i, x_2 \in S_j {\text{ and }}x_3 \in S_k$. For instance, the fact that $\delta[S_1]$ is small, together with an assumption that $a_1$ and $a_2$ are coprime and $|a_1 + a_2|$ is large, will imply that there cannot be too many solutions with $x_1, x_2 {\text{ and }}x_3$ all in $S_1$. In particular, this will give us a fairly rigid structure on the collection of triples $S_i, S_j, S_k$ such that $T(S_i, S_j, S_k)$ can give a non-trivial contribution to $T(S, S, S)$. In order to quantify this structure, we will draw a labelled digraph $G$ whose vertices correspond to the $S_i$ with $i {\geqslant}1$. We will draw an edge from $S_i$ to $S_j$ with label $\frac{a_1}{a_2}$ if and only if $T(S_i, S_j, S) {\geqslant}\frac{{\varepsilon}}{24 K_1^2} |S|^2$, where $K_1$ is as in the statement of Theorem \[thm:green\_sisask\_structure\]. Similarly, we will draw an edge with label $\frac{a_3}{a_2}$ if $T(S, S_j, S_i) {\geqslant}\frac{{\varepsilon}}{24 K_1^2} |S|^2$, and similarly for the other four possible labels. In particular, observe that if there is an edge from $S_i$ to $S_j$ with label $x$, then there will be an edge from $S_j$ to $S_i$ with label $x^{-1}$. Our definition of $G$ does not necessarily preclude the existence of multiple edges between $S_i$ and $S_j$ (with different labels), or edges from $S_i$ to $S_i$. However, as part of the proof, we will show that this cannot happen, provided that we assume a suitable hypothesis on $a_1, a_2 {\text{ and }}a_3$. First, we will show that $G$ captures almost all of the solutions to $a_1x_1 + a_2x_2 + a_3x_3 = 0$. \[lem:ignore\_bad\_triples\] Say that a triple $S_i, S_j {\text{ and }}S_k$ is *good* if and only if the six relevant edges are present. For example, $S_i \rightarrow S_j$ has label $\frac{a_1}{a_2}$, $S_k \rightarrow S_j$ has label $\frac{a_3}{a_2}$, and so on. Say that a triple is *bad* otherwise. Then, the total number of solutions to $a_1x_1 + a_2x_2 + a_3x_3 = 0$ among all of the bad triples is at most $\frac{{\varepsilon}}{4} |S|^2$. There are six ways a triple $(S_i, S_j, S_k)$ can be bad. One such way is if there is no edge from $S_i$ to $S_j$ with label $\frac{a_1}{a_2}$. Let us count the total number of solutions among triples for which the $\frac{a_1}{a_2}$ edge is missing. That is $$\begin{aligned} \sum_{\text{such } S_i, S_j} T(S_i, S_j, S) &{\leqslant}K_1^2 \frac{{\varepsilon}}{24 K_1^2} |S|^2 \\ &= \frac{{\varepsilon}}{24} |S|^2,\end{aligned}$$ since the number of pairs $S_i, S_j$ is bounded by $K_1^2$. Summing this over the six possible ways for a triple to be bad completes the proof of Lemma \[lem:ignore\_bad\_triples\]. In view of Lemmas \[lem:ignore\_S0\] and \[lem:ignore\_bad\_triples\], it remains to show that the number of solutions among the good triples is at most $\left(\frac{1}{12} + \frac{{\varepsilon}}{4}\right)|S|^2$, for a suitable choice of the coefficients $a_1, a_2 {\text{ and }}a_3$. The values we will choose are $a_1 = 1, a_2 = M {\text{ and }}a_3 = M+1$, where $$\label{eqn:condition_on_M} M > \left(\frac{1000 K_1^4K_2^2}{{\varepsilon}^2}\right)^{K_1}.$$ We can now prove the following lemma: \[lem:no\_bad\_cycles\] With the values of $a_1, a_2 {\text{ and }}a_3$ that we have chosen, the product of the labels along any cycle in $G$ must be 1. This immediately tells us that $G$ has no loops (edges from a vertex to itself). In view of the fact that an edge from $S_i$ to $S_j$ with label $x$ is accompanied by an edge from $S_j$ to $S_i$ with label $x^{-1}$, this also tells us that there can be at most one edge from $S_i$ to $S_j$. We have chosen particular values of the $a_i$ for simplicity; indeed, we only need a single choice of coefficients to work in order to establish Theorem \[thm:main\_theorem\]. However, the same argument is able to establish Lemma \[lem:no\_bad\_cycles\], and thus also Theorem \[thm:main\_theorem\], for a much wider class of equations. For example, whenever $a_1, a_2 {\text{ and }}a_3$ are coprime, and at least two of the three coefficients are large enough, then the analogue of Lemma \[lem:no\_bad\_cycles\] holds, and thus $\gamma_{a_1, a_2, a_3} < 1/12 + {\varepsilon}$. Conversely, it does not suffice for just one of the $a_i$ to be large. For example, if $a_1 = a_2 = 1$, then it can be shown that, for $S$ a slightly modified version of the set in Proposition \[prop:twelfth\], $T_{1, 1, a_3}(S) > \frac{1}{5} |S|^2$ for any $a_3$. Suppose there is a cycle whose label product is not 1; consider a shortest such cycle. By minimality, such a cycle may have no repeated vertices, and thus must have at most $K_1$ vertices. Thus, without loss of generality the cycle is $S_1, S_2, \dots, S_k, S_1$, where $S_i \rightarrow S_{i+1}$ has label $t_i$ (with $S_{k+1} = S_1$), and $k {\leqslant}K_1$. By Lemma \[lem:easy\_TNRG\_lemma\], we deduce that for each $i$, $$E(t_i \cdot S_i, S_{i+1}) {\geqslant}\frac{{\varepsilon}^2}{576 K_1^4}|S|^3.$$ Now, let us apply Lemma \[lem:gs\_doubling\_lem\] to $S_i {\text{ and }}S_{i+1}$. We have that $$E(t_i \cdot S_i, S_{i+1}) {\geqslant}\frac{{\varepsilon}^2}{576 K_1^4} |S_i|^{3/2}|S_{i+1}|^{3/2},$$ and so we deduce that $$\begin{aligned} |t_i \cdot S_i - S_{i+1}| &{\leqslant}\delta[S_i] \delta[S_{i+1}] |S_i|^{1/2}|S_{i+1}|^{1/2} \frac{576K_1^4}{{\varepsilon}^2} \nonumber\\ &{\leqslant}\frac{576 K_1^4K_2^2}{{\varepsilon}^2}|S_i|^{1/2}|S_{i+1}|^{1/2}.\end{aligned}$$ Now, we can prove, by inductively applying Lemma \[lem:ruzsa\_triangle\], that $$|t_1t_2 \dots t_i \cdot S_1 - S_{i+1}| {\leqslant}\left(\frac{576 K_1^4K_2^2}{{\varepsilon}^2}\right)^i |S_1|^{1/2}|S_{i+1}|^{1/2}.$$ Thus, setting $i=k$, we learn that $$|t_1t_2 \dots t_k \cdot S_1 - S_1| {\leqslant}\left(\frac{576 K_1^4K_2^2}{{\varepsilon}^2}\right)^{K_1} |S_1|,$$ since $k {\leqslant}K_1$. By hypothesis, $t_1t_2 \dots t_k \neq 1$. However, we know that $t_1t_2\dots t_k$ can be written in the form $M^{e_1}(M+1)^{e_2}$ for some integers $e_i$ not both zero. Suppose that $e_1$ is nonzero; the argument is similar if $e_2$ is nonzero. Write $t_1t_2 \dots t_k = \frac{r}{s}$ for coprime integers $r {\text{ and }}s$; our hypothesis tells us that $M$ must divide $r$ or $s$. Therefore, $$|r| + |s| > \left(\frac{1000 K_1^4K_2^2}{{\varepsilon}^2}\right)^{K_1},$$ as a consequence of (\[eqn:condition\_on\_M\]). Thus, we have shown that $|r \cdot S_1 - s \cdot S_1| {\leqslant}\left(\frac{576 K_1^4K_2^2}{{\varepsilon}^2}\right)^{K_1} |S_1|$. But, if $S_1$ is sufficiently large, this contradicts Theorem \[thm:bukh\_sod\_thm\], which states that $$|r \cdot S_1 - s \cdot S_1| > \left(\frac{1000 K_1^4K_2^2}{{\varepsilon}^2}\right)^{K_1} |S_1|,$$ whenever $|S_1|$ is sufficiently large. This contradiction completes the proof of Lemma \[lem:no\_bad\_cycles\]. To complete the proof of Theorem \[thm:main\_theorem\], we just need to bound the number of solutions to $a_1x_1 + a_2x_2 + a_3x_3 = 0$, with $x_1, x_2, x_3$ taken from a good triple. The following lemma will achieve this. \[lem:bounding\_good\_solutions\] Suppose we choose $a_1 = 1, a_2 = M {\text{ and }}a_3 = M+1$, as in the statement of Lemma \[lem:no\_bad\_cycles\]. Then the number of solutions to $a_1x_1 + a_2x_2 + a_3x_3 = 0$ taken from good triples is bounded above by $\left(\frac{1}{12} + \frac{{\varepsilon}}{4}\right)|S|^2$, whenever $|S|$ is large enough. We will start by defining a function $$\label{eqn:defn_of_depth} d : [n] \rightarrow {\mathbb{Q}},$$ with the property that if $S_i \rightarrow S_j$ has label $t$, then $d(j) = td(i)$. One way we can do this is as follows. For each connected component $G'$ of $G$, choose the smallest value of $i$ such that $S_i$ is in $G'$, and set $d(i) = 1$. Then, for any other $j$ with $S_j$ in $G'$, $d(j)$ is determined by the product of the labels on any path from $S_i$ to $S_j$. Lemma \[lem:no\_bad\_cycles\] guarantees that this value does not depend on the path chosen. Now, for each $d$, let $R_d = \cup_{i : d(i) = d} S_i$. Suppose that $S_i, S_j, S_k$ is a good triple, in that order (so, for example, the label on $S_i \rightarrow S_j$ is $a_2/a_1$). Then, setting $d$ = $a_1d(i)$, we have that $d(i) = d/a_1, d(j) = d/a_2 {\text{ and }}d(k) = d/a_3$. Therefore, all of the solutions coming from the good triple $S_i, S_j, S_k$ will be counted in $T(R_{d/a_1}, R_{d/a_2}, R_{d/a_3})$, and so an upper bound for the total number of solutions coming from good triples is $$\sum_{d} T(R_{d/a_1}, R_{d/a_2}, R_{d/a_3}),$$ where the sum is taken over all $d$ such that all three of the $R_i$ exist (in particular, there can be no more than $n$ terms in the sum). We may apply Lemma \[lem:uniform\_bound\_symmetric\] to give an upper bound for this. $$\begin{aligned} \sum_{d} T(R_{d/a_1}, R_{d/a_2}, R_{d/a_3}) &{\leqslant}\frac{1}{4}\sum_d |R_{d/a_1}| |R_{d/a_2}| + |R_{d/a_1}| |R_{d/a_3}| + |R_{d/a_2}| |R_{d/a_3}| + 1 \nonumber\\ &{\leqslant}\frac{1}{4}\sum_{d_1 \sim d_2} |R_{d_1}||R_{d_2}| + K_1, \label{eqn:sum_of_RR}\end{aligned}$$ where the sum on the second line is over *unordered* pairs $d_1, d_2$ such that $d_1/d_2$ is equal to the ratio between two of the $a_i$. The second inequality follows because if $d_1 \sim d_2$, then there is exactly one ratio $a_i/a_j$ such that $d_1/d_2 = a_i/a_j$. Thus, the term $|R_{d_1}||R_{d_2}|$ appears in at most one of the sums on the right hand side of the first line. Finally, for $i = 0, 1 {\text{ and }}2$, define the quantity $X_i$ by $$X_i = \sum_{\substack{d = M^{e_1}(M+1)^{e_2} \\ e_1 - e_2 = i \mod 3}} |R_d|.$$ By our construction of $d$, each $|R_d|$ appears as a term in exactly one of the $X_i$. Furthermore, $d_1 \sim d_2$ only if $R_{d_1}$ and $R_{d_2}$ are in different sums $X_i$, and any term $|R_{d_1}||R_{d_2}|$ appears at most once in (\[eqn:sum\_of\_RR\]). Consequently, we have the upper bound $$\begin{aligned} \frac{1}{4}\sum_{d_1 \sim d_2} |R_{d_1}||R_{d_2}| &{\leqslant}\frac{1}{4}(X_0X_1 + X_0X_2 + X_1X_2) \\ &{\leqslant}\left(\frac{1}{12} + \frac{{\varepsilon}}{4}\right)|S|^2,\end{aligned}$$ the latter inequality following from an easy application of Cauchy-Schwarz, since $X_0 + X_1 + X_2 {\leqslant}|S|$. This completes the proof of Lemma \[lem:bounding\_good\_solutions\]. We have now essentially proven Theorem \[thm:main\_theorem\]. Indeed, any solution to $a_1x_1 + a_2x_2 + a_3x_3 = 0$ must either have some $x_i$ in $S_0$, or must come from a bad triple, or must come from a good triple. Combining Lemmas \[lem:ignore\_S0\], \[lem:ignore\_bad\_triples\] and \[lem:bounding\_good\_solutions\] gives the result if $|S|$ is large enough. Equations in more than 3 variables {#sec:generalisation_to_k_var} ================================== A fairly natural extension of Theorem \[thm:main\_theorem\] is to ask if a similar result holds for $k$-variable equations $$\label{eqn:to_solve_k_var} a_1x_1 + \dots + a_kx_k = 0.$$ As before, let $T(S)$ be the number of solutions to (\[eqn:to\_solve\_k\_var\]) in $S$. Similarly, let $T(S_1, \dots, S_k)$ denote the number of solutions with $x_i$ taken from $S_i$. We have a trivial upper bound for $T(S)$, namely that $T(S) {\leqslant}|S|^{k-1}$. Before presenting our analogous example to Proposition \[prop:twelfth\], we require some notation and definitions. Let $I_x : {\mathbb{R}}\rightarrow {\mathbb{R}}$ denote the indicator function of a (real) interval of length $x$ centred at the origin, so $I_x(y) = 1$ if and only if $|y| {\leqslant}\frac{x}{2}$, and $I_x(y) = 0$ otherwise. \[def:defn\_of\_sigma\] For an integer $k {\geqslant}3$, define $$\label{eqn:interval_formula_for_sigma} \sigma_k = (\underbrace{I_1 \ast \dots \ast I_1}_k)(0).$$ In the introduction, we gave the following formula for $\sigma_k$: $$\tag{\ref{eqn:definition_of_sigma}} \sigma_k = \int_{-\infty}^{\infty} \left(\frac{\sin \pi x}{\pi x} \right)^k dx.$$ The equivalence of these forms follows from taking a Fourier transform and applying the convolution identity; the details can be seen in [@BorweinIntegral]. See also [@nathanson], where it can be shown that $\sigma_{2h}$ is the leading coefficient of the polynomial $\Psi_{h}(n)$. $\sigma_k$ obeys a simple asymptotic (see for example [@Polya4Asymp], or [@GoddardAsymptotic] for more terms): $$\sigma_k = \sqrt{\frac{6}{k \pi}} \left(1 + O(1/k)\right)$$ as $k \rightarrow \infty$. We may interpret $\sigma_k$ combinatorially. If $f_k$ is the probability density function of a sum of $k$ independent random variables distributed uniformly on $[-1/2, 1/2]$, then $\sigma_k = f_k(0)$. Thus, the form of the asymptotic for $\sigma_k$ is not surprising, in view of the Central Limit theorem. \[def:definition\_of\_phi\] For $t_1, \dots, t_k$ positive real numbers, define the function $\Phi = \Phi_k$ by $$\label{eqn:definition_of_phi} \Phi(t_1, \dots, t_k) = (I_{t_1} \ast \dots \ast I_{t_k})(0).$$ In particular, $\sigma_k = \Phi_k(1, \dots, 1)$. There is an explicit formula for $\Phi$. In general, we have $$\label{eqn:phi_closed_form_k_var} \Phi_k(t_1, \dots, t_k) = \frac{1}{(k-1)!2^k}\sum_{{\varepsilon}\in \{\pm 1\}^{k}} \omega ({\varepsilon}) ({\varepsilon}\cdot \mathbf{t})^{k-1} \operatorname{sgn}({\varepsilon}\cdot \mathbf{t}),$$ where $\omega({\varepsilon}) = \prod_i {\varepsilon}_i$ and ${\varepsilon}\cdot \mathbf{t} = \sum_i {\varepsilon}_it_i$ and $\operatorname{sgn}$ denotes the sign function. This is established in [@BorweinIntegral]. For $k = 3$, we can write (for $t_1 {\leqslant}t_2 {\leqslant}t_3$) $$\label{eqn:phi_closed_form_3_var} \Phi_3(t_1,t_2,t_3) = \begin{cases} t_1t_2 & t_3{\geqslant}t_1+t_2\\ \dfrac{2(t_1t_2+t_2t_3+t_1t_3) - (t_1^2+t_2^2+t_3^2)}{4} & \text{ otherwise}. \end{cases}$$ In analogy with Proposition \[prop:twelfth\], we have the following. \[prop:twelfth\_k\_var\] Let $k {\geqslant}3$ be an integer. For any equation $a_1x_1 + \dots + a_kx_k = 0,$ there are large sets $S$ for which $$\label{eqn:size_of_S_k_var} T(S) {\geqslant}\frac{\sigma_k}{k^{k-1}} |S|^{k-1} + O(|S|^{k-2}).$$ The proof of Proposition \[prop:twelfth\_k\_var\] will rely on the following fact, which states that, when the coefficients $a_i$ are all 1, long progressions behave somewhat like real intervals. \[prop:intervals\_like\_progressions\] Suppose $S_1, \dots, S_k$ are arithmetic progressions centred at the origin, with the same common difference. Let $s_i$ be the number of terms in $S_i$. Then, the number of solutions to $x_1 + \dots + x_k = 0$ where each $x_i \in S_i$ is $$\Phi(s_1, \dots, s_k) + O_k((s_1 + \dots + s_k)^{k-2}).$$ We may assume without loss of generality that the progressions $S_i$ have common difference 1. To prove Proposition \[prop:intervals\_like\_progressions\], it suffices to use the following observation. Suppose that $y_1, \dots, y_{k-1}$ are elements of the real intervals $I_{s_1}, \dots, I_{s_{k-1}}$. Then, we have the following two implications for $k-1$-tuples of real numbers $y_1, \dots, y_{k-1}$. - If $|y_1 + \dots + y_{k-1}| {\leqslant}\frac{s_k}{2},$ then $\left|\lfloor y_1\rfloor + \dots + \lfloor y_{k-1}\rfloor \right| {\leqslant}\frac{s_k}{2} + k;$ - If $\left|\lfloor y_1\rfloor + \dots + \lfloor y_{k-1}\rfloor \right| {\leqslant}\frac{s_k}{2} - k,$ then $|y_1 + \dots + y_{k-1}| {\leqslant}\frac{s_k}{2}.$ Now, $T(S_1, \dots, S_k)$ counts the number of $k-1$-tuples of integers $(x_i)_{i=1}^{k-1}$ with $x_i \in S_i$, such that $- \sum_i x_i \in S_k$. Up to an error which is at most $O_k((s_1 + \dots + s_k)^{k-2})$, this can be written as an integral $$\int_{-s_1/2}^{s_1/2} \dots \int_{-s_{k-1}/2}^{s_{k-1}/2} {\mathds{1}}_{|\lfloor y_1\rfloor + \dots + \lfloor y_{k-1}\rfloor | {\leqslant}s_k/2}dy_1 \dots dy_{k-1}.$$ The two implications above allow us to show that, up to acceptable error, this is equal to $$\int_{-s_1/2}^{s_1/2} \dots \int_{-s_{k-1}/2}^{s_{k-1}/2} {\mathds{1}}_{|y_1+ \dots + y_{k-1} | {\leqslant}s_k/2}dy_1 \dots dy_{k-1},$$ which is equal to $\Phi(s_1, \dots, s_k)$; we omit the details. We are now ready to prove Proposition \[prop:twelfth\_k\_var\]. As in Proposition \[prop:twelfth\], we will consider $S$ as the union of $k$ sets $S_1, \dots, S_k$, with the property that $T(S_1, \dots, S_k)$ is large. The way we will do this is as follows. Let $M$ be a large integer, which we assume to be divisible by $2k$. Define $$S_i = \frac{1}{a_i}[-M/2k, M/2k]$$ for each $i$ with $1 {\leqslant}i {\leqslant}k$, where we may normalise the sets to consist of integers by multiplying by $\prod_i a_i$. Then, let $S = \cup_i S_i$, so that $|S| {\leqslant}M$. It remains to show that $T(S_1, \dots, S_k) {\geqslant}\frac{\sigma_k}{k^{k-1}} M^{k-1} + O(M^{k-2}).$ But this follows as an easy consequence of Proposition \[prop:intervals\_like\_progressions\]. Indeed, $$\begin{aligned} T(S_1, \dots, S_k) &= \Phi_k(M/k, \dots, M/k) + O(M^{k-2})\\ &= \left(\frac{M}{k}\right)^{k-1}\Phi_k(1, \dots, 1) + O(M^{k-2}) \\ &= \frac{\sigma_k}{k^{k-1}} M^{k-1} + O(M^{k-2}).\end{aligned}$$ Perhaps unsurprisingly, Theorem \[thm:main\_theorem\] also generalises to this setting. \[thm:main\_theorem\_k\_var\] Let ${\varepsilon}> 0$. Then, there exist coefficients $a_1, \dots, a_k$ with the property that, for any suitably large set $S$, $$T(S) {\leqslant}\left( \frac{\sigma_k}{k^{k-1}} + {\varepsilon}\right)|S|^{k-1} + o(|S|^{k-1}).$$ The proof of Theorem \[thm:main\_theorem\_k\_var\] is broadly similar to the proof of Theorem \[thm:main\_theorem\]. There are two main places in which the argument slightly differs. Firstly, we must generalise Lemma \[lem:easy\_TNRG\_lemma\] to give a bound for $T(S_1, \dots, S_k)$ in terms of $E(S_1, S_2)$: \[lem:easy\_TNRG\_lemma\_k\_var\] Suppose that $S_1, \dots, S_k \subseteq {\mathbb{Z}}$ are finite sets. Then $$T(S_1, \dots, S_k)^2 {\leqslant}E(a_1 \cdot S_1, a_2 \cdot S_2) \left(|S_3||S_4|\dots |S_k|\right)^{2 - 1/(k-2)}.$$ For any $t \in {\mathbb{Z}}$, let $\mu(t)$ denote the number of ways of writing $t = a_1x_1 + a_2x_2$, for $x_i \in S_i$. Thus, by definition, $$E(a_1\cdot S_1, a_2\cdot S_2) = \sum_t \mu(t)^2.$$ Define $\nu(t)$ to be the number of ways of writing $t = -a_3x_3 - \dots - a_kx_k$, for $x_i \in S_i$. Thus, we see that $$\begin{aligned} T(S_1, \dots, S_k)^2 &= \left(\sum_t \mu(t) \nu(t)\right)^2 \\ &{\leqslant}\left(\sum_t \mu(t)^2\right)\left(\sum_t \nu(t)^2\right) \\ &= E(a_1\cdot S_1, a_2\cdot S_2)\left(\sum_t \nu(t)^2\right).\end{aligned}$$ Finally, we observe that $\sum_t \nu(t)^2$ represents the number of solutions to the equation $$a_3x_3 + \dots + a_kx_k = a_3x_3' + \dots + a_kx_k',$$ and so we can bound it by $(|S_3| \dots |S_k|)^{2 - \frac{1}{k-2}},$ by the same argument used in (\[eqn:bound\_energy\]) to bound the energy. Secondly, we will have to apply a $k$ variable analogue of Lemma \[lem:uniform\_bound\_symmetric\]. The analogue of this is the following: \[lem:uniform\_bound\_symmetric\_k\_var\] Suppose that $S_1, \dots, S_k \subseteq {\mathbb{Z}}$ are sets with $|S_i| = s_i$. Then $$T(S_1, \dots, S_k) {\leqslant}\frac{\sigma_k}{k} \left( \sum_i \hat{s_i} \right) + O\left(\sum_i s_i \right)^{k-2},$$ where $\hat{s_i} = \prod_{j \neq i} s_j$. This lemma is actually weaker than Lemma \[lem:uniform\_bound\_symmetric\], where the error term was $O(1)$. The weaker error term here comes from our reduction to the real case using Proposition \[prop:intervals\_like\_progressions\]; an inductive proof would likely give an $O(\sum s_i)^{k-3}$ error term. However, the $O(\sum s_i)^{k-2}$ error term is sufficient for our purpose. If $k$ is even, we can actually deduce a stronger version of (\[eqn:to\_prove\_about\_convolution\]) by using Hölder’s inequality. We have $$\begin{aligned} \Phi_k(s_1, \dots, s_k) &= \frac{1}{2\pi} \int_{-\infty}^{\infty} \widehat{I_{s_1}}(r) \dots \widehat{I_{s_k}}(r) dr \\ &{\leqslant}\frac{1}{2\pi}\left( \prod_i \int_{-\infty}^{\infty} \widehat{I_{s_i}}(r) dr \right)^{1/k} \\ &= \prod_i \Phi_k(s_i, \dots, s_i)^{1/k} \\ &= \sigma_k (s_1 \dots s_k)^{1 - 1/k},\end{aligned}$$ where the second line used Hölder’s inequality along with the fact that $k$ is even. This is stronger than (\[eqn:to\_prove\_about\_convolution\]) via an application of the AM-GM inequality. It is unclear whether the stronger version holds in the case that $k$ is odd; indeed, it is not too hard to establish for $k = 3$ by using (\[eqn:phi\_closed\_form\_3\_var\]). However, this stronger form is not necessary, so we only prove the version we need. First, observe that the statement of the lemma is unchanged if we assume without loss of generality that each $a_i$ is 1, since we may replace $S_i$ with $a_i \cdot S_i$. The first step in the proof is to apply [@lev_max Theorem 1], which says that we may take each $S_i$ to be an interval of length $s_i$, roughly centred at the origin (depending on the parity of $s_i$), in order to maximise $T(S_1, \dots, S_k)$. We may immediately apply Proposition \[prop:intervals\_like\_progressions\], which says that $$T(S_1, \dots, S_k) = \Phi_k(s_1, \dots, s_k) + O((s_1 + \dots + s_k)^{k-2}).$$ Thus, it suffices to prove that $$\label{eqn:to_prove_about_convolution} \Phi(s_1, \dots, s_k) {\leqslant}\frac{\sigma_k}{k} \left( \sum_i \hat{s_i} \right).$$ This will follow if we can prove that, for positive real numbers $t_1, \dots, t_k$, $$\label{eqn:to_prove_about_phi_flipped} t_1\dots t_k \Phi(t^{-1}_1, \dots, t^{-1}_k) {\leqslant}\frac{\sigma_k}{k}(t_1 + \dots + t_k).$$ To prove (\[eqn:to\_prove\_about\_phi\_flipped\]), first observe that equality holds in the case that all of the $t_i$ are equal. Indeed, when $t_i = 1$ the relation follows from the definition of $\sigma$, and for other constant values of $t_i$ the equality follows by homogeneity. Set $\Theta(t_1, \dots, t_k) = t_1 \dots t_k \Phi(t^{-1}_1, \dots, t^{-1}_k)$. To prove that $\Theta(t_1, \dots, t_k)$ achieves its maximum value (with $t_1 + \dots + t_k$ fixed) when all of the $t_i$ are equal, observe that it will suffice to prove the following claim. \[clm:make\_si\_equal\] If $t_1 + t_2$ is fixed (as well as each of $t_3, \dots, t_k$), then $\Theta(t_1, \dots, t_k)$ achieves its maximum when $t_1 = t_2$. To see that this claim is sufficient, observe that we may repeatedly replace the largest and smallest of the $t_i$ with their average. In doing so, $\max t_i - \min t_i$ will tend to 0, and we can use the continuity of $\Theta$ to obtain the result. To prove Claim \[clm:make\_si\_equal\], recall the expression for $\Theta(t_1, \dots, t_k)$: $$\begin{aligned} \Theta(t_1, \dots, t_k) &= t_1t_2 (t_3\dots t_k) (I_{t^{-1}_1} \ast I_{t^{-1}_2}) \ast (I_{t^{-1}_3} \ast \dots \ast I_{t^{-1}_k})(0) \\ &= (t_1 t_2 (I_{t^{-1}_1} \ast I_{t^{-1}_2}) \ast g)(0),\end{aligned}$$ where $g(x) = t_3 \dots t_k (I_{t^{-1}_3} \ast \dots \ast I_{t^{-1}_k})(x)$. Now, observe that $g$ may be written as a combination of intervals, in the following sense: $$g(x) = \int_0^{\infty} h(r) I_r(x) dr,$$ for some function $h : {\mathbb{R}}_{>0} \rightarrow {\mathbb{R}}_{>0}$ with bounded support. (The exception is when $k = 3$, in which case $g$ is just a single interval. But that will not affect the remainder of the proof of Claim \[clm:make\_si\_equal\].) To see why this is the case, we may use induction. If $k = 4$, then suppose without loss of generality that $t_3 {\leqslant}t_4$. Then, we take $h(r) = t_3t_4$ if $\frac{t_3^{-1} - t_4^{-1}}{2} {\leqslant}r {\leqslant}\frac{t_3^{-1} + t_4^{-1}}{2}$, and 0 otherwise. For $k > 4$, it is easiest to apply the induction hypothesis to $I_{t^{-1}_3} \ast \dots \ast I_{t^{-1}_{k-1}}$, and then use a similar decomposition to the one we used for the $k=4$ case. We omit the details. In view of this decomposition, proving Claim \[clm:make\_si\_equal\] may be reduced to the following claim: \[clm:make\_si\_equal\_against\_indicator\] Fix $t_1 + t_2$. Then, for any choice of $t$, we have that $t_1 t_2 (I_{t^{-1}_1} \ast I_{t^{-1}_2} \ast I_t)(0)$ is maximised when $t_1 = t_2$. In fact, the easiest way to prove Claim \[clm:make\_si\_equal\_against\_indicator\] is via the following explicit formula for $(I_a \ast I_b \ast I_c)(0)$: $$\label{eqn:formula_for_phi_3} (I_a \ast I_b \ast I_c)(0) = \begin{cases} ab & c{\geqslant}a+b\\ \dfrac{2(ab+bc+ca) - (a^2+b^2+c^2)}{4} & \text{ otherwise}, \end{cases}$$ assuming that $c {\geqslant}a,b$ without loss of generality. Given (\[eqn:formula\_for\_phi\_3\]), we can prove that $\Theta(a, b, c)$ is a concave function. If, for instance, $c^{-1} > a^{-1} + b^{-1}$, then $\Theta(a, b, c) = c$ which is clearly concave. When $a^{-1}, b^{-1} {\text{ and }}c^{-1}$ satisfy the triangle inequality, then $$\Theta(a, b, c) = \dfrac{2(a + b + c) - (abc^{-1} + ab^{-1}c + a^{-1}bc)}{4}.$$ We may prove that this is concave by computing the Hessian matrix and showing that it is nonpositive-definite everywhere; for instance, by using Sylvester’s Rule. We omit the details. In particular, $t \Theta(t_1, t_2, t^{-1}) = t_1 t_2 (I_{t^{-1}_1} \ast I_{t^{-1}_2} \ast I_t)(0)$ is concave as a function of $t_1 {\text{ and }}t_2$. Therefore, $$\frac{1}{2}\left(t \Theta(t_1, t_2, t^{-1}) + t \Theta(t_2, t_1, t^{-1})\right) {\leqslant}t \Theta\left(\frac{t_1+t_2}{2}, \frac{t_1+t_2}{2}, t^{-1}\right),$$ which is exactly the statement of Claim \[clm:make\_si\_equal\_against\_indicator\]. This completes the proof of Claim \[clm:make\_si\_equal\], and thus Lemma \[lem:uniform\_bound\_symmetric\_k\_var\]. Armed with our more general Lemmas \[lem:easy\_TNRG\_lemma\_k\_var\] and \[lem:uniform\_bound\_symmetric\_k\_var\], we may use an argument similar to the proof of Theorem \[thm:main\_theorem\] in section 3 in order to prove Theorem \[thm:main\_theorem\_k\_var\]. Select $a_1, \dots, a_k$ to be coprime integers so that $a_1 = 1$, and $|a_i|$ is sufficiently large for $i \neq 1$. Let $S$ be a large set of integers. With Lemma \[lem:easy\_TNRG\_lemma\_k\_var\] replacing Lemma \[lem:easy\_TNRG\_lemma\], much of the argument is the same as the proof of Theorem \[thm:main\_theorem\]: - We start by using Theorem \[thm:green\_sisask\_structure\] to split $S = S_1 \amalg \dots \amalg S_n \amalg S_0$, and show that $S_0$ can be ignored. - We can define the labelled digraph $G$ which captures almost all of the solutions to $a_1x_1 + \dots + a_kx_k = 0$. - We can prove that the product of the labels along a cycle must be 1, allowing us to define the function $d : [n] \rightarrow {\mathbb{Q}}$ as in (\[eqn:defn\_of\_depth\]). - This allows us to show that an upper bound for the number of solutions coming from good $k$-tuples is $$\sum_d T(R_{d/a_1}, \dots, R_{d/a_k}),$$ where $R_{d}$ is the union of the $S_i$ with $d(i) = d$ (as in the case $k=3$, this sum can have no more than $n$ terms). Lemma \[lem:uniform\_bound\_symmetric\_k\_var\] allows us to bound this: $$\begin{aligned} \sum_d T(R_{d/a_1}, \dots, R_{d/a_k}) &{\leqslant}\sum_d \frac{\sigma_k}{k} \sum_i \widehat{R_{d/a_i}} + O(K_1 |S|^{k-2}) \nonumber\\ &{\leqslant}\frac{\sigma_k}{k} \sum_{(d_1, \dots, d_{k-1})} |R_{d_1}|\dots|R_{d_{k-1}}| + O(K_1 |S|^{k-2}).\label{eqn:sum_as_Rdi_k}\end{aligned}$$ On the first line, $\widehat{R_{a/d_i}}$ denotes the product of the other $|R_{d/a_j}|$, and the error term comes from the fact that there are at most $n < K_1$ terms in the sum on the left hand side. On the second line, the sum is over unordered $k-1$-tuples $(d_1, \dots, d_{k-1})$ for which, for some ordering of the $a_i$, we have that $a_{i_1}d_1 = \dots = a_{i_{k-1}}d_{k-1}$; there can only be one such ordering by coprimality. Now, for $i = 0, 1, \dots, k-1$, define the quantity $X_i$ by $$X_i = \sum_{\substack{d = a_2^{e_2} \dots a_k^{e_k} \\ e_2 + 2e_3 + \dots + (k-1)e_k \equiv d \mod k}} |R_d|.$$ Note that if $d$ is such that $R_d$ is nonempty, then the representation of $d$ as a product $d = a_2^{e_2} \dots a_k^{e_k}$ exists due to how we constructed the labels, and is unique due to the coprimality of the $a_i$. Now, suppose that $R_{d_1}$ and $R_{d_2}$ appear together in at least one term on the right hand side of (\[eqn:sum\_as\_Rdi\_k\]). Then, $d_1/d_2 = a_i/a_j$ for some $i \neq j$, and so $R_{d_1} {\text{ and }}R_{d_2}$ contribute to different $X_i$. Thus, we may upper bound the sum in the right hand side of (\[eqn:sum\_as\_Rdi\_k\]): $$\begin{aligned} \sum_{(d_1, \dots, d_{k-1})} |R_{d_1}|\dots|R_{d_{k-1}}| {\leqslant}\sum_i \widehat{X_i},\end{aligned}$$ where $\widehat{X_i} = \prod_{j \neq i}X_j$. This bound follows from the fact that each unordered $k-1$-tuple $|R_{d_1}|\dots |R_{d_{k-1}}|$ on the left hand side contributes to exactly one of the terms on the right hand side. Finally, observe that $$\sum_i \widehat{X_i} {\leqslant}\frac{1}{k^{k-2}} \left(\sum_i X_i\right)^{k-1}.$$ To see why, observe that if $X_i + X_j$ is kept fixed, moving $X_i$ and $X_j$ closer together increases the value of the left hand side without changing the right hand side. Thus the left hand side is maximised when the $X_i$ are all the same, at which point equality occurs. Putting all of this together, we learn that $$\sum_d T(R_{d/a_1}, \dots, R_{d/a_k}) {\leqslant}\frac{\sigma_k}{k^{k-1}} |S|^{k-1} + O(|S|^{k-2}),$$ which gives the bound in the statement of Theorem \[thm:main\_theorem\_k\_var\] when $|S|$ is large enough. Systems of more than one equation {#sec:generalisation_to_systems} ================================= Another way in which one might wish to extend Theorem \[thm:main\_theorem\] is to ask if a similar result holds for systems of $m$ equations in $k$ variables. One might imagine that a result of the following form ought to hold. Suppose that $k {\geqslant}m+2$ and $m {\geqslant}1.$ Does there exist an explicit positive constant $\sigma_{m, k}$ with the following properties: - For any system $\mathcal{A}$ of $m$ equations in $k$ variables, there are be large sets $S$, for which there are at least $(\sigma_{m, k} - o(1)) |S|^{k-m}$ $k$-tuples in $S$ satisfying $\mathcal{A}$. - For any ${\varepsilon}> 0$, there are systems such that the number of $k$-tuples satisfying $\mathcal{A}$ in *any* large $S \subseteq {\mathbb{Z}}$ is no more than $(\sigma_{m, k} + {\varepsilon})|S|^{k - m}$. Thus, Theorems \[thm:main\_theorem\] and \[thm:main\_theorem\_k\_var\] tell us that $\sigma_{m, k}$ exists whenever $m = 1$, and that $\sigma_{1, k} = \sigma_k$. However, it turns out that when $m > 1$, not even the first of these has a positive answer, in the following sense. \[thm:no\_system\_thm\] Let ${\varepsilon}> 0$. Then, there exists a non-degenerate system of two equations in four variables with the property that for any large enough $S$, there are no more than ${\varepsilon}|S|^2$ solutions to the system in $S$. It is easy to see that Theorem \[thm:no\_system\_thm\] implies the analogous result for any choice of $k, m$ with $k {\geqslant}m+2$ and $m > 1$. The goal of this section is to prove Theorem \[thm:no\_system\_thm\]. We will prove Theorem \[thm:no\_system\_thm\] for the following system: \[eqn:the\_system\] x + y &= z\ x + My &= w, where $M$ is a sufficiently large constant (in terms of ${\varepsilon}$) to be chosen later. We will start by borrowing the following lemma, which appears as part of the proof of the Balog-Szemerédi-Gowers theorem. \[lem:BSGLemma\] Let $G$ be a bipartite graph with vertex sets $A$ and $B$ and edge set $E \subseteq A \times B$. Suppose that $|E| {\geqslant}{\varepsilon}|A||B|$, for some ${\varepsilon}> 0$. Then we can find subsets $A' \subseteq A$ and $B' \subseteq B$, with $|A'| \gg_{\varepsilon}|A|$ and $|B'| \gg_{\varepsilon}|B|,$ such that, whenever $a \in A'$ and $b \in B'$, there are $ \gg_{\varepsilon}|A||B|$ paths of length three from $a$ to $b$ in $G$. Let $S$ be a sufficiently large set (in terms of $M {\text{ and }}{\varepsilon}$), and suppose that there are more than ${\varepsilon}|S|^2$ solutions to (\[eqn:the\_system\]) in $S$. Consider the bipartite graph on vertex set $A \amalg B$, where $A = B = S$; that is, both parts of $G$ are $S$. Draw an edge from $a$ to $b$ if and only if there is a solution to (\[eqn:the\_system\]) with $x = a {\text{ and }}y = b$; in other words, if $a + b {\text{ and }}a + Mb$ are both in $S$. In particular, $G$ has at least ${\varepsilon}|S|^2$ edges. We may immediately apply Lemma \[lem:BSGLemma\] to $G$. This gives us sets $A' \subseteq A {\text{ and }}B' \subseteq B$ such that, for any $a \in A' {\text{ and }}b \in B'$, there are $\gg_{\varepsilon}|S|^2$ paths of length 3 in $G$ from $a$ to $b$. These sets $A'$ and $B'$ satisfy $|A' + B'| \ll_{\varepsilon}|S|$ and $|A' + M \cdot B'| \ll_{\varepsilon}|S|$. To prove this claim, we can use an argument similar to that used in the proof of the Balog-Szemerédi-Gowers theorem. Showing that $|A' + B'| \ll_{\varepsilon}|S|$ and $|A' + M \cdot B'| \ll_{\varepsilon}|S|$ are similar, so we will only do the former. Let $X$ denote the set of triples $(x, y, z)$ of elements of $(A + B) \cap S$, for which $x - y + z \in A' + B'$. We may trivially upper bound $|X|$; indeed, $|(A + B) \cap S| {\leqslant}|S|$, so $|X| {\leqslant}|S|^3$. For a lower bound on $|X|$, consider an element $a + b$ of $A' + B'$. By definition, there are $\gg_{\varepsilon}|S|^2$ paths of length 3 from $a$ to $b$ in $G$. Each such path may be written $a \sim b', a' \sim b', a' \sim b$ for some $a' \in A, b' \in B$. In other words, $a + b', a' + b' {\text{ and }}a + b'$ are all in $S$. Now, $(a+b') - (a'+b') + (a'+b) = (a+b),$ so we have located a triple $x, y, z \in (A+B)\cap S$ with $x - y + z = a+b$. These triples will be different for different paths, and so there must be $\gg_{\varepsilon}|S|^2$ such triples. There are $|A' + B'|$ elements of $A' + B'$, each of which gives $\gg_{\varepsilon}|S|^2$ triples $x, y, z$. Thus, we have that $|A' + B'| |S|^2 \ll_{\varepsilon}|S|^3$, and thus $|A' + B'| \ll_{\varepsilon}|S|$, as required. Let us now see how we may use this claim to complete the proof of Theorem \[thm:no\_system\_thm\]. 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--- abstract: 'There has recently been considerable interest in the stability of different fair bandwidth sharing policies for models that arise in the context of Internet congestion control. Here, we consider a connection level model, introduced by Massoulié and Roberts \[*Telecommunication Systems* **15** (2000) 185–201\], that represents the randomly varying number of flows present in a network. The weighted $\alpha$-fair and weighted max–min fair bandwidth sharing policies are among important policies that have been studied for this model. Stability results are known in both cases when the interarrival times and service times are exponentially distributed. Partial results for general service times are known for weighted $\alpha$-fair policies; no such results are known for weighted max–min fair policies. Here, we show that weighted max–min fair policies are stable for subcritical networks with general interarrival and service distributions, provided the latter have $2+\delta_{1}$ moments for some $\delta_{1}>0$. Our argument employs an appropriate Lyapunov function for the weighted max–min fair policy.' address: | School of Mathematics\ University of Minnesota\ Twin Cities Campus\ Institute of Technology\ 127 Vincent Hall\ 206 Church Street S.E.\ Minneapolis, Minnesota 55455\ USA\ author: - title: 'Network stability under max–min fair bandwidth sharing' --- . . Introduction {#intro} ============ We consider a connection level model for Internet congestion control that was first studied by Massoulié and Roberts [@r9]. This stochastic model represents the randomly varying number of flows in a network for which bandwidth is dynamically shared among flows that correspond to the transfer of documents along specified routes. Standard bandwidth sharing policies are the weighted $\alpha$-fair, $\alpha\in(0, \infty)$, and the weighted max–min fair policies. An important example of the former is the proportionally fair policy, which corresponds to $\alpha= 1$. The weighted max–min fair policy corresponds to $\alpha= \infty$. These policies allocate service uniformly to documents along a given route, and allocate service amongst different routes in a “fair” manner. A question of considerable interest is when such policies are stable. De Veciana, Lee and Konstantopoulos [@r5] studied the stability of weighted max–min fair and proportionally fair policies; Bonald and Massoulié [@r2] studied the stability of weighted $\alpha$-fair policies. Both papers assumed exponentially distributed interarrival and service times for documents. The first condition is equivalent to Poisson arrivals, and the second condition corresponds to exponentially distributed document sizes with documents processed at a constant rate. Both papers constructed Lyapunov functions which imply the stability of such models when the models are subcritical, that is, the underlying Markov process is positive Harris recurrent when the average load at each link is less than its capacity. Relatively little is currently known regarding the stability of subcritical networks with general interarrival and service times. Massoulié [@r8] showed stability for the proportionally fair policy for exponentially distributed interarrival times and general service times that are of phase type. A suitable Lyapunov function was employed to show stability. The stability problem for bandwidth sharing policies is in certain aspects similar to the analogous problem for multiclass queueing networks. A significant complication that arises in the context of bandwidth sharing policies is the requirement of simultaneous service of documents at all links along a route. This can reduce the efficiency of service, and complicates analysis when the interarrival and service times are not exponentially distributed. When the interarrival and service times are exponentially distributed, finer results are possible. In Kang et al. [@r7], a diffusion approximation is established under weighted proportionally fair policies. There and in Gromoll and Williams [@r6], summaries and a more detailed bibliography are provided for different bandwidth sharing policies, for both exponentially distributed and more general interarrival and service times. Here, we investigate the behavior of weighted max–min fair policies for subcritical networks whose interarrival and service times have general distributions. We show that such networks are stable, provided that the service distributions have $2+\delta_{1}$ moments for some $\delta_{1}>0$. No conclusion is reached when fewer moments exist. As in previous papers on stability, we construct a suitable Lyapunov function. Because of the more general framework here, the Markov process underlying the model will now have a general state space, and will require the machinery associated with positive Harris recurrence. We next give a more detailed description of the model we consider, after which we state our main results. We then provide some basic motivation behind their proof together with a summary of the remainder of the paper. Description of the model {#description-of-the-model .unnumbered} ------------------------ In the model we consider, *documents* are assumed to arrive at one of a finite number of *routes* $r \in\mathcal{R}$ according to independent renewal processes, with interarrival times denoted by $\xi_{r}(1), \xi_{r}(2), \ldots.$ Here, $\xi_{r}(1)$ are the initial residual interarrival times, and are considered part of the initial state. The remaining variables $\xi_{r}(2), \xi_{r}(3), \ldots$ are assumed to be i.i.d. with mean $1/\nu_{r}$, $\nu_{r}>0$, for each $r$, with the sequences being independent of one another; $\xi_{r}$ will denote a random variable with the corresponding distribution. The service times of documents are assumed to be independent of one another and of the interarrival times, and have distribution functions $H_{r}(\cdot)$ with means $m_{r}<\infty$. The initial state will include the residual service times of documents initially in the network. On each route $r$, there are a finite number of links $l$, where service is allocated to the documents on the route. For the models considered in [@r2; @r5; @r8] and [@r9], documents on a route $r$ receive service simultaneously at all links $l$ on the route, with all such documents being allocated the same rate of service $\lambda_{r}$ at all such links at a given time. Associated with such a network is an *incidence matrix* $A=(A_{l,r})$, $l \in\mathcal{L}$, $r \in\mathcal{R}$, with $A_{l,r} = 1$ if link $l$ lies on route $r$ and $A_{l,r} = 0$ otherwise. When $A_{l,r} = A_{l,r'} = 1$, with $r \ne r'$, the routes $r$ and $r'$ share a common link. Setting $z_{r}$ equal to the number of documents on route $r$, $\Lambda _{r} = \lambda_{r}z_{r}$ denotes the rate of service allocated to the totality of all documents on the route. Each link $l$ is assumed to have a given *bandwidth capacity* $c_{l}>0$. A *feasible policy* requires that this capacity not be exceeded, namely $$\label{eq1.12.1} \sum_{r\in\mathcal{R}}A_{l,r} \Lambda_{r} \le c_{l}\qquad \mbox{for all } l \in\mathcal{L}.$$ Denoting by $\Lambda= (\Lambda_{r})$ and $c=(c_{l})$ the corresponding column vectors, this is equivalent to $A\Lambda\le c$, with the inequality being interpreted coordinatewise. None of the results in this paper relies on the restriction that either $A_{l,r} = 1$ or $A_{l,r}=0$. Here, we continue to assume that (\[eq1.12.1\]) is satisfied, for given $A$, but with the weaker assumption $A_{l,r}\ge 0$. Under this new setup, each link may be interpreted as belonging to every route. A given link $l$ now allocates the same rate of service $\Lambda_{r}$ to each route $r$, which utilizes this service at rate $A_{l,r}$. For $A_{l,r}\in[0,1]$, $A_{l,r}$ may be interpreted as the proportion of this potential service that is actually utilized at link $l$ by route $r$. The *traffic intensity* $\rho_{r} = \nu_{r}m_{r}$ measures the average rate over time at which work enters a route $r$. We say a network is *subcritical* if $$\label{eq5.3.8} \sum_{r\in\mathcal{R}}A_{l,r}\rho_{r}<c_{l} \qquad\mbox{for all }l\in \mathcal{L},$$ or, in matrix form, $A\rho<c$, where $\rho=(\rho_{r})$ is the corresponding column vector. This corresponds to the definition of subcriticality that is employed in the context of queueing networks, where the load at each station (here, load at each link) is strictly less than its capacity. Condition (\[eq5.3.8\]) is needed for stability. It is assumed in, for example, [@r2; @r5] and [@r8]. The $\alpha$-fair and max–min fair policies are examples of feasible policies for which the allocation of service to documents at a given time is determined by the vector $z=(z_{r})$; the weighted $\alpha$-fair and max–min fair policies are defined analogously, but with a weight $w_{r}>0$ assigned to route $r$. We do not define $\alpha$-fair here, or, in particular, proportionally fair, referring the reader to the previous references. *Weighted max–min fair* (WMMF) is defined as a feasible policy that, at each time, allocates service so that $$\label{eq1.15.1} \min_{r \in\mathcal{R}'} \{\lambda_{r}/w_{r}\} \mbox{ is maximized},$$ among nonempty routes $\mathcal{R}'$. That is, the minimum amount of weighted service each document receives is maximized, on $r$ with $z_{r}>0$, subject to the constraint (\[eq1.12.1\]). As defined above, a WMMF policy always exists, although it need not be unique, since there may be some flexibility in allocating service among those routes where documents are receiving more than the minimal amount of service. Since our results apply to all such policies, we will not bother to select a “best” member that, for instance, maximizes service on the routes that are already receiving more than the minimum service. Such a “best” policy can be obtained by solving a hierarchy of optimization problems, as mentioned above display (2) in [@r5]. \[By employing the convexity that is inherent in the constraint (\[eq1.12.1\]), it is routine to verify the existence of such policies.\] Since the vector $z$ of documents changes as time evolves, so will the allocation of service. From this point on, we reserve the notation $\lambda_{r}(t)$ and $\Lambda_{r}(t)$ for the allocation of service for a WMMF policy at time $t$. We find it useful to also introduce $$\label{eq1.17.1} \lambda^{w}(t) = \min_{r \in\mathcal{R}'} \{\lambda_{r}(t)/w_{r}\}$$ with (\[eq1.15.1\]) in mind. Between arrivals and departures of documents, $\lambda(\cdot) = (\lambda_{r}(\cdot))$ and $\lambda^{w}(\cdot)$ will be constant; we specify that they be right continuous with left limits. The state of the network evolves over time as documents arrive in the network, are served, and then depart. For networks with exponentially distributed interarrival and service times and an assigned policy, $z=(z_{r})$ suffices to describe its state. As with queueing networks, one needs to specify the residual interarrival and service times in general. With this in mind, we employ the notation $z_{r}(B_{r})$ to denote the number of documents on route $r$ that have residual service times in $B_{r}\subseteq\mathbb{R}^{+}$, and $u_{r}$ to denote the residual interarrival time for $r$, with $z(B)=(z_{r}(B_{r}))$, $B=(B_{r})$, and $u=(u_{r})$ denoting the corresponding vectors. Setting $$\label{eq1.18.1} x = (z(\cdot), u),$$ the state $x$ contains this information. We will employ $X(t), Z(t,\cdot)$ and $U(t)$ for the corresponding random states at time $t$. The natural metric space $S$ that corresponds to the states $x$ is no longer discrete. We will describe $S$ in more detail in Section \[sec2\]. One can specify a Markov process $X(\cdot)$ on $S$ that corresponds to the network with the assigned WMMF policy. The process $X(\cdot)$ is constructed in the same manner as is its analog for a queueing network. More detail is again given in Section \[sec2\]. We note here that since $S$ is not discrete, the notion of positive recurrence needs to be replaced by that of positive Harris recurrence. When $X(\cdot)$ is positive Harris recurrent, we will say that the network is *stable*. In order to demonstrate positive Harris recurrence for $X(\cdot)$, we will define, in Section \[sec3\], an appropriate nonnegative function, or *norm*, $\|x\|$, for $x \in S$. It is defined in terms of the norms $|x|_{L}, |x|_{R}$ and $|x|_{A}$, by $$\label{eq1.20.1} \|x\| = |x|_{L} + |x|_{R} + |x|_{A}.$$ Without going into detail here, we note that $|x|_{L}$ and $|x|_{R}$ are defined from $z(\cdot)$, where $|x|_{L}$, in essence, measures residual service times smaller than $N$, for a given large $N$, $|x|_{R}$ measures residual service times greater than $N$, and $|x|_{A}$ is a function of the largest residual interarrival time. (When a distribution function $H_{r}$ has a thin enough tail, we actually replace $N$ by a smaller value $N_{H_{r}}$ that depends on $H_{r}$.) As one should expect, as either the total number of documents $\sum_{r}z_{r} \to\infty$ or $|u| \to\infty$, then $\|x\| \to \infty$. Main results {#main-results .unnumbered} ------------ We now state our two main results. \[thm5.7.1\] Suppose that a subcritical network with a weighted max–min fair policy has interarrival times with finite means and service times with $2+\delta_{1}$ moments, $\delta_{1}>0$. For the norm in (\[eq1.20.1\]), and appropriate $N,L$ and $\varepsilon_{1}> 0$, $$\label{eq5.3.10} E_{x}[\|X(N^{3})\|] \le(\|x\| \vee L) - \varepsilon_{1}N^{2}\qquad \mbox{for all } x \in S.$$ Inequality (\[eq5.3.10\]) states that, for large $\|x\|$, $X(\cdot )$ has an average negative drift over $[0,N^{3}]$ that is at least of order $1/N$. This rate will be a consequence of the application of $N$ in the construction of the norm $|x|_{L}$ that appears in (\[eq1.20.1\]). The reader will recognize (\[eq5.3.10\]) as a version of Foster’s criterion. It will imply the positive Harris recurrence of $X(\cdot)$, provided that the states in $S$ communicate with one another in an appropriate sense. Petite sets are typically employed for this purpose; they will be defined in Section \[sec2\]. A petite set $A$ has the property that each measurable set $B$ is “equally accessible” from all points in $A$ with respect to a given measure. \[thm1.24.1\] Suppose that a subcritical network with a weighted max–min fair policy has interarrival times with finite means and service times with $2+\delta_{1}$ moments, $\delta_{1}>0$. Also, suppose that $A_{L}=\{x\dvtx\|x\|\le L\}$ is petite for each $L>0$, for the norm in (\[eq1.20.1\]). Then, $X(\cdot)$ is positive Harris recurrent. Theorem \[thm1.24.1\] will follow from Theorem \[thm5.7.1\] by standard reasoning. More detail is given in Section \[sec2\]. A standard criterion that ensures the above sets $A_{L}$ are petite is given by the following two conditions on the interarrival times. The first condition is that the distribution of $\xi_{r}(2)$ is unbounded for all $r$, that is, $$\label{eq1.25.1} P\bigl(\xi_{r}(2) \ge s\bigr) > 0 \qquad\mbox{for all } s.$$ The second condition is that, for some $l_{r}\in\mathbb{Z}^{+}$, the $(l_{r}-1)$-fold convolution of $\xi_{r}(2)$ and Lebesque measure are not mutually singular. That is, for some nonnegative $q_{r}(\cdot)$ with $\int^{\infty }_{0}q_{r}(s)\,ds>0$, $$\label{eq1.25.2} P\bigl(\xi_{r}(2) + \cdots+ \xi_{r}(l_{r}) \in[c,d]\bigr) \ge\int^{d}_{c} q_{k}(s)\,ds$$ for all $c<d$. When the interarrival times are exponentially distributed, both (\[eq1.25.1\]) and (\[eq1.25.2\]) are immediate. More detail is given in Section \[sec2\]. We therefore have the following corollary of Theorem \[thm1.24.1\]. \[col1.26.1\] Suppose that a subcritical network with a weighted max–min fair policy has interarrival times with finite means that satisfy (\[eq1.25.1\]) and (\[eq1.25.2\]), and service times with $2+\delta_{1}$ moments, $\delta_{1}>0$. Then, $X(\cdot)$ is positive Harris recurrent. Outline of the paper and main ideas {#outline-of-the-paper-and-main-ideas .unnumbered} ----------------------------------- In Section \[sec2\], we will provide a brief background of Markov processes and will summarize the construction of the space $S$ and Markov process $X(\cdot )$ described above. We will also provide background that will be employed to derive Theorem \[thm1.24.1\] from Theorem \[thm5.7.1\] and to obtain Corollary \[col1.26.1\]. The machinery for this is standard in the context of queueing networks; we explain there the needed modifications. The remainder of the paper is devoted to the demonstration of Theorem \[thm5.7.1\]. (One minor result, Proposition \[prop3.13.1\], is needed for Theorem \[thm1.24.1\].) In Section \[sec3\], we will specify the norms , and that define in (\[eq1.20.1\]). Employing bounds on these three norms that will be derived in Sections \[sec4\], \[sec5\] and \[sec10\], we obtain the conclusion (\[eq5.3.10\]) of Theorem \[thm5.7.1\]. For large $\|x\|$, at least one of the norms $|x|_{V}$, with $V$ equal to $L, R$ or $A$, must also be large. When $|x|_{V}$ is large for given $V$, it will follow that $E_{x}[|X(N^{3})|_{V}]-|x|_{V}$ is sufficiently negative so that (\[eq5.3.10\]) will hold. The analysis for $|\cdot|_{A}$ is straightforward and is done in Section \[sec4\]. The behavior of $E_{x}[|X(N^{3})|_{R}]-|x|_{R}$ is analyzed in Section \[sec5\]. The remaining five sections are devoted to analyzing $E_{x}[|X(N^{3})|_{L}] - |x|_{L}$. In the last two cases, one needs to reason that, in an appropriate sense, the decrease in residual service times of existing documents more than compensates for the increase due to arriving documents, thus producing a net negative drift. For such an analysis, it makes sense to decompose the process $X(\cdot )$ into processes $\tilde{X}(\cdot)$ and $X^{A}(\cdot)$, with $$X(t) = \tilde{X}(t) + X^{A}(t) \qquad\mbox{for all } t.$$ The process $\tilde{X}(t)$ is obtained from $X(t)$ by retaining only those documents, the *original documents*, that were initially in the network, and $X^{A}(t)$ consists of the remaining documents. Neither $\tilde {X}(\cdot)$ nor $X^{A}(t)$ is Markov. One defines $\tilde{Z}(t,B)$ and $Z^{A}(t,B)$ analogously to $Z(t,B)$. Because of the WMMF policy, all documents that remain on a route $r$, over the time interval $[0,t]$, receive the same service $\Delta_{r}(t)$, with $\Delta_{r}(t) = \int^{t}_{0}\lambda_{r}(t')\,dt'$. Consequently, $$\label{eq1.30.1} \tilde{Z}_{r}(t,B) = z_{r}\bigl(B+\Delta_{r}(t)\bigr) \qquad\mbox{for } t \ge0, r \in \mathcal{R}, B \subseteq\mathbb{R}^{+}.$$ The norms $|\cdot|_{L}$ and $|\cdot|_{R}$ will be defined so that documents with greater residual service times contribute more heavily to the norms. On account of (\[eq1.30.1\]), $|\tilde{X}(t)|_{L}$ and $|\tilde{X}(t)|_{R}$ will therefore decrease over time; one can also obtain upper bounds on $|X^{A}(t)|_{L}$ and $|X^{A}(t)|_{R}$. One can use this to obtain a negative net drift on $E_{x}[|X(N^{3})|_{L}]-|x|_{L}$ and $E_{x}[|X(N^{3})|_{R}]-|x|_{R}$, as mentioned earlier. Only limited use of inequalities arising from (\[eq1.30.1\]) is needed in Section \[sec5\] for $|\cdot|_{R}$. More detailed versions are needed for $|\cdot|_{L}$, which are presented in the first part of Section \[sec6\]. In Section \[sec6\], we also introduce the sets $\mathcal{A}(t)$, along which we will be able to obtain good pathwise upper bounds on $|X^{A}(t)|_{L}$. We show in Section \[sec6\], by using elementary large deviation estimates, that the probabilities of the complements $\mathcal{A}(t)^{c}$ are small enough so that $$E_{x} [|X(N^{3})|_{L} - |x|_{L}; \mathcal{A}(N^{3})^{c} ]$$ is negligible with respect to $E_{x}[|X(N^{3})|_{L}]-|x|_{L}$. Sections \[sec7\]–\[sec10\] analyze the behavior of $|X(N^{3})|_{L}$ on $\mathcal {A}(N^{3})$. Section \[sec7\] considers the contribution to $|X(N^{3})|_{L}$ of residual times $s > N_{H_{r}}$; $N_{H_{r}}$ was mentioned parenthetically after (\[eq1.20.1\]) and satisfies $N_{H_{r}} \le N$. Sections \[sec8\] and \[sec9\] consider the contribution to $|X(N^{3})|_{L}$ of residual times $s \le N_{H_{r}}$. In Section \[sec8\], this is done for $\Delta_{r}(N^{3}) > 1/b^{3}$, for given $r$, with the constant $b$ introduced in (\[eq5.1.4\]). Here, service of individual documents is intense enough to provide straightforward upper bounds for $|X(N^{3})|_{L}-|x|_{L}$. Section \[sec9\] considers the case with $\Delta_{r}(N^{3}) \le1/b^{3}$. This is the only place in the paper where the subcriticality of the network is employed; estimation for $|X(N^{3})|_{L}$ must therefore be more precise. The short Section \[sec10\] combines the results of Sections \[sec6\]–\[sec9\] to give the desired bounds on $E_{x}[|X(N^{3})|_{L}]-|x|_{L}$. Notation {#notation .unnumbered} -------- For the reader’s convenience, we list here some of the notation in the paper, part of which has already been employed. We set $\bar{H}_{r}(s) = 1 - H_{r}(s)$; quantities such as $\bar{H}^{*}_{r}(s)$ and $\bar{\Phi}^{*}_{r}(s)$, are defined analogously in terms of $H^{*}_{r}(s)$ and $\Phi^{*}_{r}(s)$, which will be introduced later on. The term $x$ indicates a state in $S$ and the corresponding term $X(t)$ indicates a random state at time $t$; $z(\cdot)$ and $Z(t,\cdot)$, and $u$ and $U(t)$ play analogous roles. We will abbreviate $\Delta _{r} = \Delta_{r}(N^{3})$ and set $i_{r}(s) = s + \Delta_{r}$; $i_{r}(s)$ is the initial residual service time of an original document that has residual service time $s$ at time $N^{3}$. We employ $C_{1}, C_{2}, \ldots$ and $\varepsilon_{1}, \varepsilon_{2}, \ldots$ for different positive constants that appear in our bounds, whose precise values are unimportant. The symbols $\mathbb{Z}^{+}$ and $\mathbb{R}^{+}$ denote the positive integers and positive real numbers, and $\mathbb{Z}^{+,0} = \mathbb{Z}^{+} \cup\{0\}$; $\lfloor y \rfloor$ and $\lceil y \rceil$ denote the integer part of $y\in\mathbb{R}^{+}$ and the smallest integer $n$ with $n \ge y$; and $c\vee d$ and $c\wedge d$ denote the greater and smaller value of $c,d \in\mathbb{R}$. The acronyms LHS and RHS will stand for “left-hand side” and “right-hand side” when referring to equations or inequalities. Since the paper is devoted to demonstrating Theorems \[thm5.7.1\] and \[thm1.24.1\], we will implicitly assume that the network under consideration has a WMMF policy, except when stated otherwise, and that the moment conditions on the interarrival and service times given in Theorem \[thm5.7.1\] hold. We assume the network is subcritical only when explicitly stated. Markov process background {#sec2} ========================= In this section, we provide a more detailed description of the construction of the Markov process $X(\cdot)$ that underlies a WMMF network. We then show how Theorem \[thm1.24.1\] and its corollary follow from Theorem \[thm5.7.1\]. Analogs of this material for queueing networks are given in Bramson [@r1]. Because of the similarity of the two settings, we present a summary here and refer the reader to [@r1] for additional detail. Construction of the Markov process {#construction-of-the-markov-process .unnumbered} ---------------------------------- As in (\[eq1.18.1\]), we define the state space $S$ to be the set of pairs $x=(z(\cdot),u)$, where $z(\cdot) = (z_{r}(\cdot))$ and $z_{r}(\cdot )$ is a counting measure that maps $B \subseteq\mathbb{R}^{+}$ to $\mathbb{Z}^{+,0}$, and $u=(u_{r})$, $r \in \mathcal{R}$, has positive components. Here, $z(\cdot)$ corresponds to the residual service times of documents and $u$ to the residual interarrival times. (One could, as in (4.1) of [@r1], distinguish documents based on their “age,” which is not needed here.) For the purpose of constructing a metric $d(\cdot, \cdot)$ on $S$, we assign to each document the pair $(r_{i}, s_{i})$, $i=1,2,\ldots,$ where $r_{i} \in \{1, \ldots, |\mathcal{R}|\}$ denotes its route and $s_{i}>0$ its residual service time. Documents are ordered so that $s_{1} \le s_{2} \le\cdots ,$ with the decision for ties being made based on a given ordering of the routes. When $i$ exceeds the number of documents belonging to $x$, we assign the value $(r_{i}, s_{i}) = (0,0)$. For $x, x' \in S$, with the coordinates labeled correspondingly, we set $$\label{eq2.3.1} d(x,x') = \sum^{\infty}_{i=1} \bigl((|r_{i}-r'_{i}|+|s_{i}-s'_{i}|) \wedge 1 \bigr) + \sum_{r}|u_{r}-u'_{r}|.$$ One can check that $d(\cdot, \cdot)$ is separable and locally compact. (See page 82 of [@r1] for details.) We equip $S$ with the standard Borel $\sigma$-algebra inherited from $d(\cdot, \cdot)$, which we denote by $\mathscr{S}$. In Proposition \[prop3.13.1\], we will show $|\cdot|_{L}$, $|\cdot|_{R}$ and $|\cdot|_{A}$ are continuous in $d(\cdot, \cdot)$. The Markov process $X(t) = (Z(t,\cdot), U(t))$ underlying the network, with $Z(t, \cdot)$ and $U(t)$ taking values $z(\cdot)$ and $u$ as above, is defined to be the right continuous process whose evolution is determined by the assigned WMMF policy. Documents are allocated service according to the rates $\lambda_{r}(\cdot)$, which are constant in between arrivals and departures of documents on routes. Upon an arrival or departure, rates are re-assigned according to the policy. We note that this procedure is not policy specific, and also applies to $\alpha$-fair policies. By modifying the state space descriptor to contain more information, one could also include more general networks. Along the lines of page 85 of [@r1], a filtration $(\mathcal {F}_{t})$, $t\in [0,\infty]$, can be assigned to $X(\cdot)$ so that $X(\cdot)$ is Borel right and, in particular, is strong Markov. The processes $X(\cdot)$ fall into the class of piecewise-deterministic Markov processes, for which the reader is referred to Davis [@r4] for more detail. Recurrence {#recurrence .unnumbered} ---------- The Markov process $X(\cdot)$ is said to be *Harris recurrent* if, for some nontrivial $\sigma$-finite measure $\varphi$, $$\varphi(B) > 0 \mbox{ implies } P_{x}(\eta_{B} = \infty) = 1 \qquad\mbox{for all } x \in S,$$ where $\eta_{B} = \int^{\infty}_{0}1\{{X(t)\in B}\}\,dt$. If $X(\cdot )$ is Harris recurrent, it possesses a stationary measure $\pi$ that is unique up to a constant multiple. When $\pi$ is finite, $X(\cdot)$ is said to be *positive Harris recurrent*. A practical condition for determining positive Harris recurrence can be given by using petite sets. A nonempty set $A \in\mathscr{S}$ is said to be *petite* if for some fixed probability measure $a$ on $(0, \infty)$ and some nontrivial measure $\nu$ on $(S, \mathscr{S})$, $$\nu(B) \le\int^{\infty}_{0}P^{t}(x,B)a(dt)$$ for all $x \in A$ and $B \in\mathscr{S}$. Here, $P^{t}(\cdot, \cdot )$, $t\ge 0$, is the semigroup associated with $X(\cdot)$. As mentioned in the , a petite set $A$ has the property that each set $B$ is “equally accessible” from all points $x \in A$ with respect to the measure $\nu $. Note that any nonempty measurable subset of a petite set is also petite. For given $\delta> 0$, set $$\tau_{B}(\delta) = \inf\{t \ge\delta\dvtx X(t) \in B\}$$ and $\tau_{B} = \tau_{B}(0)$. Then, $\tau_{B}(\delta)$ is a stopping time. Employing petite sets and $\tau_{B}(\delta)$, one has the following characterization of Harris recurrence and positive Harris recurrence. (The Markov process and state space need to satisfy minimal regularity conditions, as on page 86 of [@r1].) The criteria are from Meyn and Tweedie [@r10]; discrete time analogs of the different parts of the proposition have long been known; see, for instance, Nummelin [@r11] and Orey [@r12]. \[thm2.8.1\] A Markov process $X(\cdot)$ is Harris recurrent if and only if there exists a closed petite set $A$ with $$\label{eq2.8.2} P_{x}(\tau_{A} < \infty) = 1 \qquad\mbox{for all } x \in S.$$ Suppose the Markov process $X(\cdot)$ is Harris recurrent. Then, $X(\cdot)$ is positive Harris recurrent if and only if there exists a closed petite set $A$ such that for some $\delta> 0$, $$\label{eq2.8.3} \sup_{x \in A}E_{x}[\tau_{A}(\delta)] < \infty.$$ One can apply Theorem \[thm2.8.1\], together with a stopping time argument, to show the following version of Foster’s criterion. It is contained in Proposition 4.5 in [@r1]. \[prop2.9.1\] Suppose that $X(\cdot)$ is a Markov process, with norm $\|\cdot\|$, such that for some $\varepsilon> 0$, $L > 0$ and $M>0$, $$\label{eq2.9.2} E_{x}[\|X(M)\|] \le(\|x\| \vee L) - \varepsilon \qquad\mbox{for all } x.$$ Then, for $0 < \delta\le M$, $$\label{eq2.9.3} E_{x}[\tau_{A_{L}}(\delta)] \le\frac{M}{\varepsilon}(\|x\| \vee L) \qquad\mbox{for all } x,$$ where $A_{L}=\{x\dvtx\|x\| \le L\}$. In particular, if $A_{L}$ is closed petite, then $X(\cdot)$ is positive Harris recurrent. Theorem \[thm1.24.1\] and its corollary {#theorem-thm1.24.1-and-its-corollary .unnumbered} --------------------------------------- Proposition \[prop2.9.1\] and Theorem \[thm5.7.1\] provide the main tools for demonstrating Theorem \[thm1.24.1\]. We also require Proposition \[prop3.13.1\], which states that the norm $\|\cdot\|$ in (\[eq1.20.1\]) is continuous in the metric $d(\cdot,\cdot)$, and hence that $A_{L}=\{ x\dvtx \|x\| \le L\}$ is closed for each $L$. Together, they give a quick proof of the theorem. [Proof of Theorem \[thm1.24.1\]]{} From the conclusion (\[eq5.3.10\]) in Theorem \[thm5.7.1\], we know that the assumption (\[eq2.9.2\]) in Proposition \[prop2.9.1\] is satisfied for some $L$, with $M=N^{3}$ and $\varepsilon= \varepsilon_{1}N^{2}$. In Theorem \[thm1.24.1\], it is assumed that $A_{L}$ is petite; by Proposition \[prop3.13.1\], we also know it is closed. So, all of the assumptions in Proposition \[prop2.9.1\] are satisfied, and hence $X(\cdot)$ is positive Harris recurrent. Corollary \[col1.26.1\] follows immediately from Theorem \[thm1.24.1\] and the assertion, before the statement of the corollary, that the sets $A_{L}$ are petite under the assumptions (\[eq1.25.1\]) and (\[eq1.25.2\]). A somewhat stronger version of the analogous assertion for queueing networks is demonstrated in Proposition 4.7 of [@r1]. (The proposition states that the sets $A$ are uniformly small.) The reasoning is the same in both cases and does not involve the policy of the network. The argument, in essence, requires that one wait long enough for the network to have at least a given positive probability of being empty; this time $t$ does not depend on $x$ for $\| x\|\le L$. One uses (\[eq1.25.1\]) for this. By using (\[eq1.25.2\]), one can also show that the joint distribution function of the residual interarrival times has an absolutely continuous component at this time, whose density is bounded away from 0. It will follow that the set $A_{L}$ is petite with respect to $\nu$, with $a$ chosen as the point mass at $t$, if $\nu$ is concentrated on the empty states, where it is a small enough multiple of $|\mathcal {R}|$-dimensional Lebesque measure restricted to a small cube. Summary of the proof of Theorem \[thm5.7.1\] {#sec3} ============================================ As mentioned in Section \[intro\], the norm $\|\cdot\|$ in Theorem \[thm5.7.1\] consists of three components, with $$\label{eq5.3.6} \|x\| = |x|_{L} + |x|_{R} + |x|_{A}$$ for each $x$. After introducing these components, we will state the bounds associated with each of them that we will need, leaving their proofs to the remaining sections. We then show how Theorem \[thm5.7.1\] follows from these bounds. Definition of norms {#definition-of-norms .unnumbered} ------------------- We first define $|x|_{L}$. This requires a fair amount of notation, which we will introduce shortly. We begin by expressing $|x|_{L}$ in terms of this notation; when the notation is then specified, we motivate it by referring back to $|x|_{L}$. We set $|x|_{L} = \sup_{r,s} |x|_{r,s}$ for $r \in\mathcal{R}$ and $s>0$, where $$\label{eq5.3.1} |x|_{r,s} = \frac{w_{r}(1+as_{N})z^{*}_{r}(s)}{\nu_{r}\Gamma (\bar{H}^{*}_{r}(s_{N}))} .%\label{eq5.3.1}$$ We need to define the terms ${H}^{*}_{r}(\cdot)$, $z^{*}_{r}(\cdot)$, $\Gamma(\cdot)$, $a$ and $s_{N}$. Starting with ${H}^{*}_{r}(\cdot)$ and $z^{*}_{r}(\cdot)$, we recall the distribution functions $H_{r}(\cdot)$ and counting measure $z_{r}(\cdot)$ from Section \[intro\]. In (\[eq5.3.1\]), we will require their analogs ${H}^{*}_{r}(\cdot) $ and $z^{*}_{r}(\cdot)$ to have densities with bounded first derivatives and to be “close” to $H_{r}(\cdot)$ and $z_{r}(\cdot)$. For this, we define $H^{*}_{r} (\cdot)$ and $z^{*}_{r}(\cdot)$ as the convolutions of $H_{r}(\cdot )$ and of $z_{r}(\cdot)$ by an appropriate distribution function $\Phi(\cdot)$ with density $\phi(\cdot)$. Setting $$\label{eq5.1.4} \phi(s) = \cases{ \frac{2}{3}ebe^{-bs}, &\quad for $s > 1/b$, \vspace*{2pt}\cr % \frac{2}{3}b^{2}s, &\quad for $s \in(0, 1/b]$, \vspace*{2pt}\cr % 0, &\quad for $s \le0$,}$$ for $b \in\mathbb{Z}^{+}$ with $b \ge2$, $\phi(\cdot)$ is the density of $\Phi(s) = \int^{s}_{-\infty}\phi(s')\,ds'$. We note that $\Phi(\cdot)$ has mean at most $2/b$ and that $\phi(\cdot)$ satisfies $$\label{eq5.1.10} \phi'(s) \le b^{2} \quad\mbox{and}\quad \phi(s+s')/\phi(s) \ge e^{-bs'}$$ for $s,s'>0$. The above properties and the exponential tail of $\phi (\cdot)$ will be useful later when analyzing $|\cdot|_{L}$ and $|\cdot|_{R}$ \[as in (\[eq5.2.1\]), (\[eq10.5.3\]), (\[eq6.88.1\]), (\[eq6.88.2\]) and (\[eq46.5.4\])\]. Convoluting by $\Phi(\cdot)$, we set $$\begin{aligned} \label{eq5.1.6} H^{*}_{r}(s) &=& (H_{r}*\Phi)(s) = \int^{\infty}_{0} \Phi (s-s')\,dH(s'),\nonumber\\[-8pt]\\[-8pt] % z^{*}_{r}((0,s]) &=& (z_{r}*\Phi)((0,s]) = \int^{\infty}_{0} \Phi(s-s')\, dz_{r}((0,s'])\nonumber\end{aligned}$$ with $z^{*}_{r}(B)$ being defined analogously for $B \subseteq\mathbb{R}^{+}$. Differentiating both quantities in (\[eq5.1.6\]), we also set $$\begin{aligned} \label{eq5.1.8} %h^{*}_{r}(s) \stackrel{\mathrm{def}}{=} (H_{r}*\Phi)'(s) = h^{*}_{r}(s) &=& (H_{r}*\Phi)'(s) = \int^{\infty}_{0} \phi(s-s') \,dH(s'), \nonumber\\[-8pt]\\[-8pt] % %z^{*}_{r}(s) \stackrel{\mathrm{def}}{=} (z_{r}*\Phi)'((0,s]) = z^{*}_{r}(s) &=& (z_{r}*\Phi)'((0,s]) = \int^{\infty}_{0} \phi(s-s') \,dz_{r}((0,s']).\nonumber\end{aligned}$$ Convolution by $\Phi(\cdot)$, as in (\[eq5.1.6\]), produces a measure $z_{r}^{*}(\cdot)$ that approximates $z_{r}(\cdot)$ and possesses a density. Since $H_{r}(\cdot)$ is assumed to have a finite $(2+\delta_{1})$th moment for all $r$, the same is true for $H^{*}_{r}(\cdot)$. This implies that for appropriate $C_{1}\ge1$, $$\label{eq5.2.1} \bar{H}^{*}_{r}(s) \le\frac{C_{1}}{(1+s)^{2+\delta_{1}}} \qquad\mbox{for all } s > 0 \mbox{ and } r \in\mathcal{R},$$ for $\delta_{1}$ chosen as in Theorem \[thm5.7.1\]. We assume wlog that $\delta_{1} \le1$. Since the difference of the means of $H^{*}_{r}(\cdot)$ and $H_{r}(\cdot)$ is at most $2/b$ for each $r$ and $H(\cdot)$ is subcritical, $H^{*}(\cdot)$ will also be subcritical for large enough $b$. We set $$\label{eq5.1.3} \Gamma(\sigma) = \sigma+ C_{2}a \sigma^{\gamma} \qquad\mbox{for } \sigma\in[0,1].$$ We choose $\gamma\in(0, \delta_{1}/24]$, $C_{2}\ge2C_{1}/\gamma$ and $a$ small enough so that $a \le(1/C_{2}) \wedge1$ and (\[eq5.2.5\]) is satisfied. One can think of $\Gamma(\cdot)$ as being almost linear for values of $\sigma$ that are not too small; the power $\gamma$ needs to be small in order to be able to bound $|x|_{r,s}$ later on for $H^{*}_{r}(s_{N})$ small; $\gamma>0$ is needed so that the integral in (\[eq5.2.4\]) is finite. We set $s_{N} = s \wedge(N_{H_{r}}+1)$ for $N \in\mathbb{Z}^{+}$, where $$\label{eq90.1.1} N_{H_{r}} = (\bar{H}^{*}_{r})^{-1}(1/N^{4})\wedge N.$$ It follows that $$\label{eq90.3.1} 1/N^{4} \le\bar{H}^{*}_{r}(N_{H_{r}}) \le C_{1}/ N^{2+\delta_{1}}.$$ If $H^{*}_{r}(\cdot)$ has a relatively fat tail, say $\bar {H}^{*}_{r}(s) \sim s^{-3}$, (\[eq90.1.1\]) implies that $N_{H_{r}} = N$; otherwise, $N_{H_{r}} < N$ and $\bar{H}^{*}_{r}(N_{H_{r}}) = 1/N^{4}$. In either case, it will follow from (\[eq90.3.1\]) that $\Gamma(\bar{H}^{*}_{r}(N_{H_{r}}))$ is “large enough” for us to adequately bound $|x|_{r,s}$. We will assume that $N\in \mathbb{Z}^{+}$ is chosen large enough so $N \ge1/a$ and $N_{H_{r}} \ge1$ for all $r$. The norm $|\cdot|_{L}$ has been defined with the following motivation. As the process $X(\cdot)$ evolves, documents arrive at each route, are served, and eventually depart. In Proposition \[prop46.3.2\], we will show that, under certain assumptions for $X(t)$ on $t \in[0, N^{3}]$, for large enough $b$, $$\label{eq3.6.1} \lambda^{w}(t) \ge(1 + \varepsilon_{2})/|x|_{L} \qquad\mbox{on } t\in[0, N^{3}],$$ for some $\varepsilon_{2}> 0$, because of the subcriticality of $H^{*}(\cdot)$. Reasoning as below (\[eq1.30.1\]), this will imply that individual documents receive enough service so that $|X(t)|_{L}$ decreases on average over $[0, N^{3}]$. More specifically, the increase in the term $\Gamma(\bar{H}^{*}_{r} (s_{N}))$ in (\[eq5.3.1\]), after translating $s_{N}$ according to the service of documents, will compensate for the arrival of new documents. For documents with residual service $s \le N_{H_{r}} \le N+1$ at $t=0$, the term $1+as_{N}$ in (\[eq5.3.1\]), after translating $s_{N}$ according to the service of documents, will decrease sufficiently over $[0, N^{3}]$ to produce the term $-\varepsilon_{1}N^{2}$ in (\[eq5.3.10\]). For documents with residual service $s > N_{H_{r}}$, we will instead need to employ the norm , which we introduce next. (On $(N_{H_{r}}, N_{H_{r}}+1]$, the intervals overlap.) The norm $|\cdot|_{R}$ in (\[eq5.3.6\]) is given by $$\label{eq5.3.2} |x|_{R} = M_{1}\sum_{r}\kappa_{N,r}\int^{\infty}_{N_{H_{r}}} N_{r}(s)z^{*}_{r}(s)\,ds.$$ We need to identify the terms $\kappa_{N,r}$, $N_{r}(\cdot)$ and $M_{1}$. We set $$\label{eq3.8.1} \kappa_{N,r} = 1/\Gamma(\bar{H}^{*}_{r}(N_{H_{r}}))$$ and $$\label{eq90.1.2} N_{r}(s) = \cases{ s^{2}/N, &\quad for $s > N$, \cr % s, &\quad for $s \le N$.}$$ Later on, we will also employ $\kappa_{N} \stackrel{\mathrm{def}}{=} \max_{r}\kappa_{N,r}$. For the term $M_{1}$, we will require that $$\label{eq3.8.2} M_{1} \ge8 C_{3}\Bigl(\max_{r,r'} w_{r}/w_{r'}\Bigr),$$ where $C_{3}$ is chosen as in (\[eq5.4.2\]). Since is given by a weighted sum of the residual service times of the different documents, it will be easier to work with than $|\cdot|_{L}$, which is a supremum. For smaller values of $s$, we required $|\cdot|_{L}$ because of the nature of the WMMF policy. Because of the bound on $\bar{H}^{*}_{r}(\cdot)$ in (\[eq5.2.1\]), the impact of large residual service times on the evolution of $X(\cdot)$ will typically be small, and so one can employ the “more generous” definition over $(N_{H_{r}}, \infty)$ given in (\[eq5.3.2\]). As we will see in Section \[sec5\], we will require the presence of the term $N_{r}(s)$ in the integrand in (\[eq5.3.2\]) to ensure that the integral decreases sufficiently rapidly from the service of documents when the integral is large. This will rely on $N'_{r}(s) \ge1$ on $(N_{H_{r}}, \infty )$. For $s > N$, the denominator $N$ in $s^{2}/N$ is needed so that the expected increase due to arrivals does not dominate the term $-\varepsilon_{1}N^{2}$ in (\[eq5.3.10\]), which was mentioned in the motivation for the definition of . This denominator is not needed for $s \in[N_{H_{r}},N)$ because (\[eq90.1.1\]) will guarantee that the integrand is already sufficiently small there. The terms $\kappa_{N,r}$ are needed when we combine the norms $|\cdot|_{L}$ and $|\cdot|_{R}$ in $\|\cdot\|$, because of the denominator $\Gamma (\cdot)$ in $|\cdot|_{r,s}$. The norm $|\cdot|_{A}$ in (\[eq5.3.6\]) is needed for the residual interarrival times. It is given by $$\label{eq5.3.5} |x|_{A} = \frac{1}{N}\max_{r}\theta(u_{r}),$$ where $\theta(y)$, $y>0$, satisfies the following properties. We assume that $\theta(y)>0$ for all $y$ and that $\theta(\cdot)$ and $\theta '(\cdot)$ are strictly increasing, with $$\label{eq5.8.1} \theta'(y) \to\infty\qquad\mbox{as } y \to\infty.$$ We also assume that $$\label{eq5.8.3} \theta(y) \le y^{2} \qquad\mbox{for all } y,$$ and that $\theta(\cdot)$ grows sufficiently slowly so that $$\label{eq5.8.2} E[\theta(\xi_{r})] < \infty\qquad\mbox{for all } r.$$ Since $E[\xi_{r}]<\infty$, it is possible to specify such $\theta (\cdot)$ that also satisfy the previous two displays. The above properties for $\theta(\cdot)$ will enable us to show that the expected value of $|X(t)|_{A}$ will decrease over time when $|X(t)|_{A}$ is large. In particular, because of (\[eq5.8.1\]) and (\[eq5.8.2\]), the decrease in $|\cdot|_{A}$ due to decreasing residual interarrival times will, on the average, dominate the increase in $|\cdot|_{A}$ due to new interarrival times that occur when a document joins a route. The argument for this is given in Section \[sec4\] and is fairly quick. We note that when $\xi_{r}$ are all exponentially distributed, the term $|\cdot|_{A}$ may be omitted in the definition of . The reader attempting to understand the norm should first concentrate on $|\cdot|_{L}$, which was chosen to accommodate the WMMF policy. When the service distributions $H_{r}(\cdot)$ all have compact support and the interarrival times are exponentially distributed, one may, in fact, set $\|x\|=|x|_{L}$ for a large enough choice of $N$. We note that the norm is not appropriate for weighted $\alpha$-fair policies. In particular, the supremum and the function $\Gamma(\cdot)$ in its definition are not appropriate factors in this context. On the other hand, $|\cdot|_{R}$, with suitable $M_1$, and $|\cdot|_{A}$ should still be applicable to $\alpha$-fair policies, provided a suitable replacement of can be found. In order to apply Proposition \[prop2.9.1\] in the proof of Theorem \[thm1.24.1\] in Section \[sec2\], we needed to know that the sets $A_{L} = \{x\dvtx\|x\|\le L\}$ are closed. For this, it suffices to show the norm is continuous in the metric $d(\cdot,\cdot)$ that is given in (\[eq2.3.1\]). \[prop3.13.1\] The norm $\|\cdot\|$ in (\[eq5.3.6\]) is continuous in the metric $d(\cdot,\cdot)$ given by (\[eq2.3.1\]). It suffices to show $|\cdot|_{L}$, $|\cdot|_{R}$ and $|\cdot|_{A}$ are each continuous in $d(\cdot,\cdot)$. For $|\cdot|_{L}$, note that the coefficients of $z^{*}_{r}(s)$ in (\[eq5.3.1\]) are bounded. On the other hand, if $d(x,x') \le\varepsilon< 1$, then one can show, by using the first part of (\[eq5.1.10\]), that $$\label{eq13.3.2} |z^{*}_{r}(s) - z{}^{\prime,*}_{r}(s)| \le b^{2} \varepsilon\qquad \mbox{for all } s \mbox{ and } r,$$ where $z^{\prime,*}_{r} = (z')^{*}_{r}(s)$. It follows from this and (\[eq5.3.1\]) that $|\cdot|_{L}$ is in fact Lipschitz in $d(\cdot,\cdot)$. For $|\cdot|_{R}$, one can apply both parts of (\[eq5.1.10\]) to show with a bit of work that, if $d(x,x') \le\varepsilon< 1$ and $x$ has no residual service times greater than $M$, for given $M$, then $$\label{eq13.3.3} \int^{\infty}_{N_{H_{r}}}N_{r}(s) |z^{*}_{r}(s) - z^{\prime,*}_{r}(s)|\,ds \le(M+1)^{2}b^{2}\varepsilon+ (1-e^{-b\varepsilon})|x|_{R}$$ for all $r$. Since the coefficients of $\int^{\infty}_{N_{H_{r}}}N_{r}(s) z^{*}_{r}(s)\,ds$ in $|x|_{R}$ are bounded and the RHS of (\[eq13.3.3\]) goes to 0 as $\varepsilon\to0$, the continuity of $|\cdot|_{R}$ follows. Since $\theta'(u_{r})$ is bounded for bounded values of $u_{r}$, $|\cdot|_{A}$ is also continuous. In addition to the norms in (\[eq5.3.6\]), we will employ the following norms in showing Theorem \[thm5.7.1\]: $$\label{eq5.3.7} |x| = \sum_{r}z_{r}(\mathbb{R}^{+}) = \sum_{r}z^{*}_{r}(\mathbb{R}^{+})$$ and $$\label{eq5.3.3} |x|_{K} = \sum_{r}\kappa_{N,r}z^{*}_{r}((N_{H_{r}},\infty)).$$ Although we will not employ them in this section, we also introduce the norms $$\label{eq5.3.11} |x|_{1} = \sum_{r}z^{*}_{r}((0,N_{H_{r}}]),\qquad % |x|_{2} = \sum_{r}z^{*}_{r}((N_{H_{r}}, \infty))$$ and $$\label{eq5.3.12} |x|_{S} = |x|_{L} + \max_{r}\frac{w_{r}}{\rho_{r}}z^{*}_{r}((N_{H_{r}},\infty)).$$ It obviously follows from (\[eq5.3.7\]) and (\[eq5.3.11\]) that $|x| = |x|_{1} + |x|_{2}$. The norm $|\cdot|_{S}$ will be employed in Proposition \[prop46.3.2\] to derive the bound given in (\[eq3.6.1\]). Bounds on $|\cdot|_{L}, |\cdot|_{R}$ and $|\cdot|_{A}$ {#bounds-on-cdot_l-cdot_r-and-cdot_a .unnumbered} ------------------------------------------------------ In order to derive (\[eq5.3.10\]), we need bounds on $|\cdot|_{L}$, $|\cdot|_{R}$ and $|\cdot|_{A}$ as the process $X(t)$ evolves from $t=0$ to $t=N^{3}$. We first need to specify the term $L$ appearing in (\[eq5.3.10\]). We choose $l_{1}$ large enough so that $$\label{eq5.8.6} \theta'(l_{1}/2) \ge M_{1}N$$ and, for all $r$, $$\label{eq5.8.7} E[\theta(\xi_{r}); \xi_{r}>l_{1}/2] \le(1 /|\mathcal{R}|) P(\xi_{r}>N^{3}).$$ We set $$\label{eq5.8.5} L_{1} = \frac{1}{N}\theta(l_{1})$$ and $$\label{eq5.8.8} L = 6(\kappa^{2}_{N}N^{17} \vee L_{1}).$$ For $|\cdot|_{L}$, we employ the bound from Proposition \[prop50.1.1\] that, for large enough $N$ and $b$, small enough $a$, and appropriate $C_{3}$ and $\varepsilon_{3}> 0$, $$\begin{aligned} \qquad \label{eq5.4.2} &&E_{x}[|X(N^{3})|_{L}] - |x|_{L} \nonumber\\[-8pt]\\[-8pt] &&\qquad\le C_{3}N^{3} \cdot1\{|x| \le N^{6}\} + [C_{3}(|x|_{K}/|x|)N^{3} - \varepsilon_{3}N^{2}] \cdot1\{|x|>N^{6}\}\nonumber\end{aligned}$$ for all $x$. The precise value of $\varepsilon_{3}$ is not important; in Proposition \[prop50.1.1\], it is given by $\frac{1}{4} \min_{r}w_{r}$. We assume wlog that $\varepsilon_{3}\le C_{3}$. For $|\cdot|_{R}$, we employ the bound from Proposition \[prop10.7.1\] that, for given $\varepsilon_{4}> 0$, large enough $N$, and $M_{2}=\frac{1}{8} (1\wedge\min_{l}c_{l})(\min_{r,r'}(w_{r}/w_{r'}))M_{1} \ge C_{3}$, $$\begin{aligned} \label{eq5.4.1} && E_{x}[|X(N^{3})|_{R}] - |x|_{R} \nonumber\\ % &&\qquad\le\varepsilon_{4}N^{2} - M_{2}(|x|_{K}/|x|) N^{3} \cdot1\{|x|>N^{6}\} \\ % &&\qquad\quad{} - \kappa_{N}N^{4} \cdot1 \{|x|_{R}>\kappa^{2}_{N}N^{17}, |x| \le N^{6}\}\nonumber\end{aligned}$$ for all $x$. We will later choose $\varepsilon_{4}$ small with respect to $\varepsilon_{3}$; the constant $C_{3}$ is chosen as in (\[eq3.8.2\]) and (\[eq5.4.2\]). For $|\cdot|_{A}$, we will show in Proposition \[prop6.1.4\] and Proposition \[prop6.2.5\] that, for this choice of $\varepsilon_{4}$ and large enough $N$, $$\label{eq5.5.1} E_{x}[|X(N^{3})|_{A}] - |x|_{A} \le\varepsilon_{4}N^{2} - M_{1} N^{3}\cdot1\{|x|_{A}>L/6\}$$ for all $x$. Derivation of (\[eq5.3.10\]) from (\[eq5.4.2\]), (\[eq5.4.1\]) and (\[eq5.5.1\]) {#derivation-of-eq5.3.10-from-eq5.4.2-eq5.4.1-and-eq5.5.1 .unnumbered} -------------------------------------------------------------------------------- We now derive (\[eq5.3.10\]) from these three bounds. Adding the RHS of (\[eq5.4.2\]), (\[eq5.4.1\]) and (\[eq5.5.1\]), one obtains, for large enough $N$ and $b$, and small enough $a$, $$\label{eq5.5.2} E_{x}[\|X(N^{3})\|] - \|x\| \le2 C_{3}N^{3}$$ for all $x$. We next consider the behavior of the LHS of (\[eq5.5.2\]) for $\|x\| > L/2$, where $L$ is given by (\[eq5.8.8\]). This condition implies that either $|x|_{L}>\kappa^{2}_{N} N^{17}$, $|x|_{R}>\kappa^{2}_{N}N^{17}$ or $|x|_{A}\ge L/6$. Suppose first that $|x|_{L}>\kappa^{2}_{N}N^{17}$. We note that if $|x| \le N^{6}$, then $$|x|_{L} \le C_{4}N^{8}$$ for some constant $C_{4}$. This bound follows from the definition of $|x|_{L}$ in (\[eq5.3.1\]), together with the bounds $z^{*}_{r}(s)\le 12b^{2}|x|$ for all $s$, $s_{N} \le N$, and $\Gamma(\bar{H}^{*}_{r}(s_{N})) \ge C_{5} /N$, for some $C_{5}> 0$ \[which follows from (\[eq90.3.1\]) and $\gamma\le1/4$\]. Therefore, if $|x|_{L} > \kappa^{2}_{N}N^{17}$ and $N$ is large enough so that $\kappa_{N} \ge1$, one must have $|x|>N^{6}$. On the other hand, it follows from (\[eq5.4.1\]) that, on $|x|>N^{6}$, $$\label{eq5.5.3} E_{x}[|X(N^{3})|_{R}] - |x|_{R} \le\varepsilon_{4}N^{2} - M_{2}(|x|_{K}/|x|) N^{3}.$$ Adding the terms corresponding to $|x|>N^{6}$ in (\[eq5.4.2\]) and (\[eq5.5.1\]) to this implies that, for $|x|>N^{6}$, and hence for $|x|_{L} > \kappa^{2}_{N} N^{17}$, $$\begin{aligned} \quad \label{eq5.5.4} E_{x}[\|X(N^{3})\|] - \|x\| &\le& (2\varepsilon_{4}- \varepsilon_{3})N^{2} + (C_{3}- M_{2}) (|x|_{K}/|x|)N^{3}\nonumber\\[-8pt]\\[-8pt] % &\le& -\varepsilon_{1}N^{2},\nonumber\end{aligned}$$ where the latter inequality follows for $\varepsilon_{4}\le\varepsilon _{3}/3$ and $\varepsilon_{1} \stackrel{\mathrm{def}}{=} \varepsilon_{3}/3$, since $M_{2} \ge C_{3}$. Suppose next that $|x|_{R} > \kappa^{2}_{N}N^{17}$ and $|x| \le N^{6}$. Adding up the corresponding terms from (\[eq5.4.2\]), (\[eq5.4.1\]) and (\[eq5.5.1\]) implies that the LHS of (\[eq5.5.4\]) is at most $$\label{eq5.11.1} 2\varepsilon_{4}N^{2} + C_{3}N^{3} - \kappa_{N} N^{4} \le- \varepsilon_{1}N^{3}$$ for large $N$, which is better than the bound in (\[eq5.5.4\]). Suppose finally that $|x|_{A} \ge L/6$. We need to consider only the case $|x| \le N^{6}$, since $|x| > N^{6}$ is covered by (\[eq5.5.4\]). In this case, it follows from (\[eq5.4.2\]), (\[eq5.4.1\]) and (\[eq5.5.1\]) that the LHS of (\[eq5.5.4\]) is at most $$\label{eq5.5.5} (3C_{3}- M_{1})N^{3} \le-\varepsilon_{1}N^{3},$$ since $M_{1} \ge4 C_{3}$. Together, (\[eq5.5.4\]), (\[eq5.11.1\]) and (\[eq5.5.5\]) imply that, for large enough $N$ and $b$, and small enough $a$, $$\label{eq3.22.1} E_{x}[\|X(N^{3})\|] - \|x\| \le- \varepsilon_{1}N^{2}$$ for all $\|x\| > L/2$. Since for large $N$, $$L - L/2 \ge2C_{3}N^{3} + \varepsilon_{1}N^{2},$$ (\[eq5.3.10\]) follows easily form (\[eq5.5.2\]) and (\[eq3.22.1\]). Upper bounds on $E_{x}[|X(N^{3})|_{A}]$ {#sec4} ======================================= In this section, we will demonstrate the inequality (\[eq5.5.1\]) for the upper bounds on $E_{x}[|X(N^{3})|_{A}] - |x|_{A}$. In Proposition \[prop6.1.4\], we obtain the first term on the RHS of (\[eq5.5.1\]); this holds for all $x$. We then obtain a better bound in Proposition \[prop6.2.5\], which is valid on $|x| \ge L/6$. Both parts require just standard techniques. The first bound employs the following elementary inequality on the residual interarrival times at time $N^{3}$: $$\label{eq6.1.1} |X(N^{3})|_{A} \le|x|_{A} \vee\frac{1}{N}\max \{\theta(\xi_{r}(k))\dvtx r \in\mathcal{R}, k \in[2, A_{r} (N^{3})+1] \}. %\{\theta(\xi_{r}(k))$$ Here and in later sections, $A_{r}(t)$ denotes the cumulative number of arrivals at the route $r$ by time $t$; $A(t)$ will denote the corresponding vector. The inequality $k \le A_{r}(t)+1$ implies that the interarrival epoch associated with $\xi_{r}(k)$ has already begun by time $t$. Recall that $\xi_{r}(1)$ is the initial residual time at route $r$ and $\xi_{r}(2), \xi_{r}(3), \ldots$ are i.i.d. random variables, and $\theta(\cdot)$ satisfies (\[eq5.3.5\])–(\[eq5.8.2\]). \[prop6.1.4\] For any $\varepsilon> 0$ and large enough $N$, not depending on $x$, $$\label{eq6.1.3} E_{x}[|X(N^{3})|_{A}] - |x|_{A} \le\varepsilon N^{2}.$$ By (\[eq5.8.2\]), $E[\theta(\xi_{r})] < \infty$ for all $r$. One can therefore show with some estimation that, for each $r$, $$\label{eq6.1.2} \frac{1}{t}E_{x} \Bigl[\max_{k\in[2,A_{r}(t)+1]} \theta (\xi_{r}(k)) \Bigr] \to0,$$ uniformly in $x$ as $t \to\infty$. For fixed $x$, (\[eq6.1.2\]) follows immediately from (4.83) of [@r1]; since $A_{r}(t)$ decreases when $\xi_{r}(1)$ increases, this limit is uniform in $x$. Inequality (\[eq6.1.3\]) follows immediately from (\[eq6.1.1\]) and (\[eq6.1.2\]), with $t = N^{3}$. We proceed to Proposition \[prop6.2.5\]. For the proposition, it will be useful to decompose $|X(t)|_{A} - |x|_{A}$ as $$\label{eq6.2.6} |X(t)|_{A} - |x|_{A} = I_{A}(t) - D_{A}(t),$$ where $I_{A}(t)$ and $D_{A}(t)$ are the nondecreasing functions corresponding to the cumulative increase and decrease of $|X(\cdot)|_{A}$ up to time $t$. That is, , with $I_{A}(t)$ being the jump process, with $$I_{A}(t) - I_{A}(t-) = |X(t)|_{A} - |X(t-)|_{A}$$ and $D'_{A}(t)$ being the rate of decrease of $|X(t)|_{A}$ at other times. We note that $D_{A}(t)$ is locally Lipschitz, with $D'_{A}(t)$ defined except at arrivals. In particular, since $U'_{r}(t)=-1$ except at arrivals, $$\label{eq6.2.7} D'_{A}(t) = \frac{1}{N}\max_{r}\theta'(U_{r}(t))\qquad \mbox{almost everywhere.}$$ We recall the definitions for $l_{1}, L_{1}$ and $M_{1}$ in (\[eq5.8.6\])–(\[eq5.8.5\]) and (\[eq3.8.2\]). \[prop6.2.5\] Suppose that $|x|_{A} \ge L/6$. Then, for large enough $N$ not depending on $x$, $$\label{eq6.2.8} E_{x}[|X(N^{3})|_{A}] - |x|_{A} \le1 - M_{1}N^{3} \le- M_{1}N^{3}/2.$$ We first show that $$\label{eq6.2.9} D_{A}(N^{3}) \ge M_{1}N^{3}.$$ Since $|x|_{A} \ge L/6 = \kappa^{2}_{N}N^{17} \vee L_{1}$ and $\theta (y) \le y^{2}$ for all $y$, one has, for $N \ge2$, that $\max_{r}u_{r} \ge N^{8} \vee l_{1}$. So, for all $t \in[0, N^{3}]$, $$\label{eq6.2.10} \max_{r}u_{r} - \max_{r}U_{r}(t) \le N^{3} \le\frac{1}{2} \max_{r}u_{r}.$$ Consequently, for all $t \in[0, N^{3}]$, $$\label{eq6.2.11} \max_{r}U_{r}(t) \ge\frac{1}{2} \max_{r}u_{r} \ge N^{3} \vee\frac{1}{2} l_{1}.$$ Moreover, by (\[eq5.8.6\]) and (\[eq6.2.7\]), for $\max_{r}U_{r}(t) \ge \frac{1}{2} l_{1}$, $D'_{A}(t) \ge M_{1}$ almost everywhere. Together with (\[eq6.2.11\]), this implies $D'_{A}(t) \ge M_{1}$ almost everywhere on $[0,N^{3}]$, and hence (\[eq6.2.9\]) holds. On account of (\[eq6.2.9\]), in order to show (\[eq6.2.8\]), it suffices to show $$\label{eq6.2.12} E_{x}[I_{A}(N^{3})] \le1 %\varepsilon_{d}N^{2}$$ for large $N$. To obtain (\[eq6.2.12\]), we first note that, for each $r$, there cannot be more than one interarrival time occurring over $(0,N^{3}]$ with value greater than $N^{3}$. Moreover, because of (\[eq6.2.11\]), only interarrival times with value at least $N^{3} \vee(l_{1}/2)$ can contribute to $I_{A}(N^{3})$. The expectation of $\theta(\xi_{r})$, for $\xi_{r}$ conditioned on being greater than $N^{3}$ and restricted to being greater than $l_{1}/2$, is $$\label{eq6.2.13} E [\theta(\xi_{r}); \xi_{r} > l_{1}/2 ]/P(\xi_{r} > N^{3}).$$ (If $\xi_{r}$ is bounded above by $N^{3}$, set the ratio equal to 0.) It follows that, for any $x$, $$\label{eq6.2.14} E_{x} [I_{A}(N^{3})] \le\frac{1}{N} \sum_{r} E[\theta(\xi_{r}); \xi_{r} > l_{1}/2]/P(\xi_{r} > N^{3}).$$ By (\[eq5.8.7\]), the RHS of (\[eq6.2.14\]) is at most $1/N$, which implies (\[eq6.2.12\]). Upper bounds on $E_{x}[|X(N^{3})|_{R}]$ {#sec5} ======================================= In this section, we will demonstrate the following proposition for the upper bounds on $E_{x}[|X(N^{3})|_{R}] - |x|_{R}$, where $|\cdot|_{R}$ is the norm introduced in (\[eq5.3.2\]). \[prop10.7.1\] For given $\varepsilon> 0$, large enough $N$ and all $x$, $$\begin{aligned} \label{eq10.7.2} E_{x}[|X(N^{3})|_{R}] - |x|_{R} &\le& \varepsilon N^{2} - M_{2}(|x|_{K}/|x|)N^{3} \cdot 1\{|x|>N^{6}\} \nonumber\\[-8pt]\\[-8pt] % &&{} - \kappa_{N}N^{4} \cdot1\{|x|_{R}>\kappa^{2}_{N}N^{17}, |x|\le N^{6}\},\nonumber\end{aligned}$$ where $M_{2}$ is specified before (\[eq5.4.1\]). The bound (\[eq10.7.2\]) implies (\[eq5.4.1\]), which was employed in Section \[sec3\], together with bounds on $E_{x}[|X(N^{3})|_{L}]$ and $E_{x} [|X(N^{3})|_{A}]$, to obtain (\[eq5.3.10\]) of Theorem \[thm5.7.1\]. The bound on $E_{x}[|X(N^{3})|_{A}]$ was derived relatively quickly, whereas the bound on $E_{x}[|X(N^{3})|_{L}]$ will require substantial estimation and will be derived in Sections \[sec6\]–\[sec10\]. The bound on $E_{x}[|X(N^{3})|_{R}]$ that is given here will require a moderate amount of work. In order to show Proposition \[prop10.7.1\], it will be useful to rewrite $|X(t)|_{R} - |x|_{R}$ as $$\label{eq10.7.3} |X(t)|_{R} - |x|_{R} = I_{R}(t) - D_{R}(t),$$ where $I_{A}(t)$ and $D_{A}(t)$ are the nondecreasing functions corresponding to the cumulative increase and decrease of $|X(\cdot)|_{R}$ up to time $t$. A similar decomposition was used in Section \[sec4\] for $|X(t)|_{A}$. Here, $I_{R}(0) = D_{R}(0) = 0$, with $I_{R}(t)$ being the jump process with $$I_{R}(t) - I_{R}(t-) = |X(t)|_{R} - |X(t-)|_{R}.$$ One can check that $D_{R}(\cdot)$ is continuous except when a document departs from a route. Its derivative is defined almost everywhere, being defined except at the arrival or departure of a document. Since $D_{R}(\cdot)$ is nondecreasing, $$D_{R}(t_{2}) - D_{R}(t_{1}) \ge\int^{t_{2}}_{t_{1}}D'_{R}(t)\,dt \qquad\mbox{for } t_{1} \le t_{2}.$$ It is easy to obtain a suitable upper bound on $E_{x}[I_{R}(N^{3})]$; a suitable lower bound on $E_{x}[D_{R}(N^{3})]$ requires more effort. We therefore first demonstrate Proposition \[prop10.7.4\], which analyzes $E_{x}[I_{R}(N^{3})]$. As in Section \[sec4\], $A_{r}(t)$ denotes the cumulative number of arrivals at route $r$ by time $t$. It follows from elementary renewal theory that, for appropriate $C_{6}$ and $t \ge1$, $$\label{eq10.1.2} E_{x}[A_{r}(t)] \le C_{6}t \qquad\mbox{for each } r$$ (see, e.g., [@r3], page 136). Since large residual interarrival times can only delay arrivals, the bound is uniform in $x$. \[prop10.7.4\] For given $\varepsilon>0$ and large enough $N$, $$\label{eq10.7.5} E_{x}[I_{R}(N^{3})] \le\varepsilon N^{2} \qquad\mbox{for all } x.$$ It follows from (\[eq5.3.2\]) that the expected increase in $I_{R}(\cdot)$, due to a document that arrives at route $r$, is $$M_{1}\kappa_{N,r}\int^{\infty}_{N_{H_{r}}}N_{r}(s)h^{*}_{r}(s)\,ds.$$ Since the number of arriving documents by time $N^{3}$ and their initial service times are independent, it follows that $$\label{eq10.7.6} E_{x}[I_{R}(N^{3})] = \biggl( M_{1}\kappa_{N,r}\int^{\infty}_{N_{H_{r}}} N_{r}(s)h^{*}_{r}(s)\,ds \biggr)E_{x}[A_{r}(N^{3})].$$ In order to bound the first term on the RHS of (\[eq10.7.6\]), we decompose the integral there into $\int^{N}_{N_{H_{r}}} + \int^{\infty}_{N}$. When $N \ge N_{H_{r}}$, one has, by (\[eq90.1.1\]) and (\[eq3.8.1\]), $$\begin{aligned} \label{eq10.7.7} \kappa_{N,r}\int^{N}_{N_{H_{r}}}N_{r}(s)h^{*}_{r}(s)\,ds &=& \frac {1}{\Gamma (1/N^{4})}\int^{N}_{N_{H_{r}}}s h^{*}_{r}(s)\,ds \nonumber\\[-8pt]\\[-8pt] % &\le& \frac{N}{\Gamma(1/N^{4})}\bar{H}^{*}_{r}(N_{H_{r}}) \le(N^{3} \Gamma (1/N^{4}) )^{-1}.\nonumber\end{aligned}$$ This is, for large enough $N$, at most $1/N^{2}$, because of the small power $\gamma$ in the definition of $\Gamma(\cdot)$. Also, $$\begin{aligned} \label{eq10.7.8} \kappa_{N,r}\int^{\infty}_{N}N_{r}(s)h^{*}_{r}(s)\,ds &\le& \frac {1}{N\Gamma (1/N^{4})}\int^{\infty}_{N}s^{2}h^{*}_{r}(s)\,ds \nonumber\\ % &\le& \frac{1}{N^{1+\delta_{1}/2}\Gamma(1/N^{4})}\int^{\infty}_{N} s^{2+\delta_{1}/2}h_{r}^{*}(s)\,ds \\ &\le&\frac{C_{7}}{N^{1+\delta _{1}/2}\Gamma (1/N^{4})}\nonumber\end{aligned}$$ for appropriate $C_{7}$, with the last inequality holding because of (\[eq5.2.1\]). Since $\gamma\le\delta_{1}/24$, this is, for large $N$, at most $1/N^{1+\delta_{1}/4}$. Together, the bounds for the two integrals imply that, for large enough $N$, $$\label{eq10.7.9} M_{1}\kappa_{N,r}\int^{\infty}_{N_{H_{r}}}N_{r}(s) h^{*}_{r}(s)\,ds \le 2/N^{1+\delta_{1}/4}.$$ Application of (\[eq10.7.9\]) and (\[eq10.1.2\]) to (\[eq10.7.6\]), with $t=N^{3}$ in (\[eq10.1.2\]), implies (\[eq10.7.5\]). We now derive a lower bound on $E_{x}[D_{R}(N^{3})]$. As in (\[eq10.7.2\]), we need to consider two separate cases, depending on whether $|x|>N^{6}$ or both $|x|_{R} > \kappa^{2}_{N}N^{17}$ and $|x|\le N^{6}$ hold. In both cases, we will employ the following lemma. Recall that $M_2 = \frac{1}{8}C_{8}M_1$, with $C_{8}=(1\wedge\min_{l}c_{l})(\min_{r,r'}(w_{r}/w_{r'}))$. \[lem10.8.1\] For all $t$, $$\label{eq10.8.2} D_{R}(t) \ge M_{1}\bigl(|x|_{K} - |X(t)|_{K}\bigr).$$ For almost all $t$, $$\begin{aligned} \label{eq10.8.3} D'_{R}(t) &\ge& \frac{C_{8}M_{1}}{|X (t)|}\sum_{r} \kappa_{N,r}\int^{\infty}_{N_{H_{r}}} \biggl(\frac{s}{N} \vee1 \biggr)Z^{*}_{r}(t,s)\,ds \nonumber\\[-8pt]\\[-8pt] % &\ge& 8M_{2}|X(t)|_{K}/ |X(t)|.\nonumber\end{aligned}$$ We first show (a). Recall that $\tilde{X}(\cdot)$ is the stochastic process constructed from $X(\cdot)$ in Section \[intro\], where service of documents is pathwise identical to $X(\cdot)$, but where the arrival of documents is suppressed. One can check that, for all $t$ and $\omega$, $$\label{eq10.10.1} |\tilde{X}(t)|_{K} \le|X(t)|_{K}$$ and $$\label{eq10.10.2} D_{R}(t) \ge|x|_{R} - |\tilde{X}(t)|_{R}.$$ Inequality (\[eq10.10.1\]) follows immediately from $\tilde {Z}^{*}(t,B)\le Z^{*}(t,B)$ for all $B \subseteq\mathbb{R}^{+}$. For (\[eq10.10.2\]), note that the LHS gives the cumulative decrease of $|X(\cdot)|_{R}$ over $[0,t]$ due to the service of all documents, whereas the RHS gives the decrease due to service of only the original documents while ignoring the decrease due to service of new documents. On account of (\[eq10.10.1\]) and (\[eq10.10.2\]), to show (\[eq10.8.2\]) it suffices to show $$\label{eq10.10.3} |x|_{R} - |\tilde{X}(t)|_{R} \ge M_{1}\bigl(|x|_{K} - |\tilde{X}(t)|_{K}\bigr).$$ Substituting in the definition of $|\cdot|_{R}$ given by (\[eq5.3.2\]) and integrating by parts on the LHS of (\[eq10.10.3\]) gives $$\begin{aligned} \label{eq10.10.4} && M_{1}\sum_{r}\kappa_{N,r}N_{r}(N_{H_{r}})\bigl(z^{*}_{r}((N_{H_{r}}, \infty)) - \tilde{Z}^{*}_{r}(t, (N_{H_{r}}, \infty))\bigr) \nonumber\\[-8pt]\\[-8pt] % &&\qquad{} + M_{1}\sum_{r}\kappa_{N,r}\int^{\infty}_{N_{H_{r}}}N'_{r}(s) \bigl(z^{*}_{r}((s, \infty)) - \tilde{Z}^{*}_{r}(t, (s, \infty))\bigr)\,ds.\nonumber\end{aligned}$$ It follows from (\[eq90.1.2\]) and $N_{H_{r}}\ge1$ that $N_{r}(N_{H_{r}}) \ge 1$ and that $N'_{r}(s) \ge1$ for all $s$. Consequently, (\[eq10.10.4\]) is at least $$\begin{aligned} && M_{1}\sum_{r}\kappa_{N,r}\bigl(z^{*}_{r}((N_{H_{r}}, \infty)) - \tilde{Z}^{*}_{r} (t, (N_{H_{r}}, \infty))\bigr) \\ % &&\qquad= M_{1}\bigl(|x|_{K} - |\tilde{X}(t)|_{K}\bigr),\end{aligned}$$ which implies (\[eq10.10.3\]). For (b), we first note that because of the weighted max–min fair protocol and (\[eq1.12.1\]), the rate at which each document is served is at least $$\label{eq10.8.4} \Bigl(\min_{l}c_{l}\Bigr)\Bigl(\min_{r,r'}(w_{r}/w'_{r})\Bigr)\big/|X(t)|.$$ Moreover, the rate of decrease of $|X(t)|_{R}$ per unit service of each document on route $r$ is at least $$\begin{aligned} \label{eq10.8.5}\quad M_{1}\kappa_{N,r}\int^{\infty }_{N_{H_{r}}}N'_{r}(s)Z^{*}_{r}(t,s)\,ds &\ge& M_{1}\kappa_{N,r}\int^{\infty}_{N_{H_{r}}} \biggl(\frac{s}{N}\vee1 \biggr) Z^{*}_{r}(t,s)\,ds \nonumber\\[-8pt]\\[-8pt] % &\ge& M_{1}\kappa_{N,r}Z^{*}_{r}(t, (N_{H_{r}}, \infty)).\nonumber\end{aligned}$$ Summing (\[eq10.8.5\]) over $r$ and multiplying by (\[eq10.8.4\]) gives each of the bounds in (\[eq10.8.3\]). We first derive a lower bound on $E_{x}[D_{R}(N^{3})]$ in the case where $|x| > N^{6}$. \[prop10.7.10\] For large enough $N$ and all $|x| > N^{6}$, $$\label{eq10.7.11} E_{x}[D_{R}(N^{3})] \ge M_{2}(|x|_{K}/|x|)N^{3}.$$ We restrict our attention to the set $$B_{1} = \{\omega\dvtx|X(t)| \le|x| + N^{6} \mbox{ for all } t \in [0,N^{3}]\}.$$ By applying Markov’s inequality to inequality (\[eq10.1.2\]) with $t=N^{3}$, one has that, for large $N$, $$\label{eq10.2.2} P_{x} \biggl(\sum_{r}A_{r}(N^{3})>N^{6} \biggr) \le\frac{C_{6}}{N^{3}}|\mathcal{R}| \le\frac{1}{2}$$ for all $x$. Consequently, $$\label{eq10.2.1} P(B_{1}) \ge1/2.$$ This bound does not depend on $|x|$. We now consider two cases, depending on whether the set $$B_{2} = \bigl\{\omega\dvtx|X(t)|_{K} > \tfrac{1}{2}|x|_{K} \mbox{ for all } t \in [0,N^{3}] \bigr\}$$ occurs. Since $|x|>N^{6}$, it follows from the second half of (\[eq10.8.3\]) that, for all $t \in[0, N^{3}]$, $$D'_{R}(t) \ge2M_{2}|x|_{K}/|x|$$ on $B_{1} \cap B_{2}$. Consequently, on $B_{1} \cap B_{2}$, $$\label{eq10.2.6} D_{R}(N^{3}) \ge2M_{2} (|x|_{K}/|x| ) N^{3}.$$ On the other hand, on $B_{1} \cap B_{2}^{c}$, $$\label{eq10.7.12} |x|_{K} - |X(\tau)|_{K} \ge\tfrac{1}{2}|x|_{K}$$ for some (random) $\tau\in[0, N^{3}]$. By (\[eq10.8.2\]), $$D_{R}(t) \ge M_{1}\bigl(|x|_{K} - |X(t)|_{K}\bigr)$$ for all $t$. Together with (\[eq10.7.12\]), this implies that $$\label{eq10.3.4} D_{R}(N^{3}) \ge D_{R}(\tau) \ge\tfrac{1}{2}M_{1}|x|_{K} \ge2M_{2} (|x|_{K}/|x| )N^{6},$$ where $|x|>N^{6}$ was used in the last inequality. Together, (\[eq10.2.6\]) and (\[eq10.3.4\]) imply that, on $B_{1}$, $$D_{R}(N^{3}) \ge2M_{2} (|x|_{K}/|x| ) N^{3}.$$ Inequality (\[eq10.7.11\]) follows from this and (\[eq10.2.1\]). We now derive a lower bound on $E_{x}[D_{R}(N^{3})]$ in the case where $|x|_{R} > \kappa^{2}_{N}N^{17}$ and $|x| \le N^{6}$ both hold. We note that, starting from (\[eq10.5.1\]), the argument relies on the discreteness of documents. If one wishes to employ a fluid limit based argument rather than the discrete setting employed in this paper, different reasoning will be required at this point; it is not obvious how one would proceed. \[prop10.7.14\] For large enough $N$, $$\label{eq10.7.15} E_{x}[D_{R}(N^{3})] \ge\kappa_{N}N^{4}$$ for all $|x|_{R} > \kappa^{2}_{N}N^{17}$ and $|x| \le N^{6}$. As in the proof of Proposition \[prop10.7.10\], we restrict attention to the set $B_{1}$ defined there. The bound $P(B_{1}) \ge1/2$ in (\[eq10.2.1\]) continues to hold here. In our present setting, since $|x| \le N^{6}$, $\omega \in B_{1}$ implies that $$|X(t)| \le2N^{6} \qquad\mbox{for all } t \in[0, N^{3}].$$ We also consider two cases, depending on whether $$B_{3} = \bigl\{\omega\dvtx|X(t)|_{R} > \tfrac{1}{2}\kappa^{2}_{N}N^{17} \mbox{ for all } t \in[0, N^{3}] \bigr\}$$ occurs. The case $B^{c}_{3}$ is almost immediate. It follows from (\[eq10.7.3\]) that, for large enough $N$ and for some $\tau \in(0, N^{3}]$, $$\label{eq10.7.16} D_{R}(N^{3}) \ge D_{R}(\tau) \ge|x|_{R} - |X(\tau)|_{R} \ge\tfrac{1}{2} \kappa^{2}_{N}N^{17} > 2\kappa_{N}N^{4}$$ for $\omega\in B^{c}_{3}$. The case $B_{3}$ requires some work. We first note that, by the first part of (\[eq10.8.3\]), $$\begin{aligned} \label{eq10.7.17} D'_{R}(t) &\ge& \frac{C_{8}M_{1}}{|X(t)|} \sum_{r}\kappa_{N,r} \biggl(\int^{\infty}_{N_{H_{r}}} \biggl(\frac{s}{N} \vee1 \biggr)Z^{*}_{r}(t,s)\,ds \biggr) \nonumber\\[-8pt]\\[-8pt] % &\ge& \frac{C_{8}M_{1}}{2N^{6}} \sum_{r}\kappa_{N,r} \biggl(\int^{\infty}_{N_{H_{r}}} \biggl(\frac{s}{N} \vee1 \biggr)Z^{*}_{r}(t,s)\,ds \biggr),\nonumber\end{aligned}$$ when $\omega\in B_{1}$. We will truncate the second integral in (\[eq10.7.17\]) in order to be able to introduce an additional factor $s$ into the integrand. We first note that, since $\Phi(0)=0$, if a document with residual service time at least $s$ is present at time $t$ on some route $r$, then, for large $N$, $$\label{eq10.5.1} |X(t)|_{R} \ge M_{1}\kappa_{N,r}s^{2}/N \ge M_{1}s^{2}/N.$$ Hence, there are no documents with residual service time $$\label{eq10.5.2} s > s_{1} \stackrel{\mathrm{def}}{=} \bigl((N/M_{1})|X(t)|_{R} \bigr)^{1/2}.$$ It follows that, for appropriate $C_{9}> 0$, (\[eq10.7.17\]) is at least $$\begin{aligned} \label{eq10.5.3} &&\frac{C_{8}M_{1}}{2N^{6}}\sum_{r} \kappa_{N,r} \biggl(\int^{s_{1}+1}_{N_{H_{r}}} \biggl(\frac{s}{N}\vee1 \biggr) Z^{*}_{r}(t,s)\,ds \biggr) \nonumber\\ % &&\qquad\ge\frac{C_{8}M_{1}^{3/2}}{4N^{13/2} |X(t)|^{1/2}_{R}} \sum_{r}\kappa_{N,r} \biggl(\int^{s_{1}+1}_{N_{H_{r}}} N_{r}(s)Z^{*}_{r}(t,s)\,ds \biggr) \nonumber\\[-8pt]\\[-8pt] % &&\qquad\ge\frac{2C_{9}M_{1}^{3/2}}{N^{13/2}|X(t)|^{1/2}_{R}}\sum _{r}\kappa_{N,r} \biggl(\int^{\infty}_{{N_{H_{r}}}}N_{r}(s)Z^{*}_{r}(t,s)\,ds \biggr) \nonumber\\ % &&\qquad= \frac{2C_{9}M_{1}^{1/2}}{N^{13/2}}|X(t)|^{1/2}_{R} \ge C_{9}M_{1}^{1/2} \kappa_{N}N^{2}\nonumber\end{aligned}$$ for all $t \in[0,N^{3}]$. The exponential tail of $\Phi(\cdot)$ is used in the last inequality; the equality relies on $\omega\in B_{3}$. Employing the bound on $D'_{R}(t)$ obtained from (\[eq10.7.17\]) and (\[eq10.5.3\]), and integrating over $t \in[0,N^{3}]$, it follows that, for large $N$, $$D_{R}(N^{3}) \ge C_{9}M_{1}^{1/2}\kappa_{N}N^{5} > 2\kappa_{N}N^{4}$$ on $B_{1} \cap B_{3}$. Together with (\[eq10.7.16\]), this implies that $D_{R}(N^{3}) > 2\kappa_{N}N^{4}$ on $B_{1}$. Inequality (\[eq10.7.15\]) follows from this and $P(B_{1}) \ge1/2$. Proposition \[prop10.7.1\] follows immediately from (\[eq10.7.3\]) and Propositions \[prop10.7.4\], \[prop10.7.10\] and \[prop10.7.14\]. Upper bounds on $E_{x}[|X(N^{3})|_{L}]$: Basic layout and bounds on exceptional sets {#sec6} ==================================================================================== In this section, we begin our investigation of upper bounds on $E_{x}[|X( N^{3})|_{L}] - |x|_{L}$. Since these bounds will require us to examine a number of subcases in Sections \[sec6\]–\[sec9\], we will only arrive at the desired bounds in Section \[sec10\]. In the current section, we first state certain elementary inequalities, mostly involving $|\cdot|_{r,s}$, that will be useful later on. We then define the “good” sets $\mathcal{A}(\cdot)$ of realizations of $X(\cdot)$ to which our bounds in Sections \[sec7\]–\[sec9\] will apply. The remainder of the section is spent demonstrating Proposition \[prop21.1.1\], which gives an upper bound on $E_{x}[|X(t)|_{L}-|x|_{L}$; $\mathcal{A}(t)^{c}]$, where $\mathcal{A}(t)^{c}$ is the small exceptional set. Elementary inequalities {#elementary-inequalities .unnumbered} ----------------------- Here we state a number of elementary inequalities that will be useful later on. Let $z_{i}(\cdot)$, $i=1,2,3$, denote configurations of particles on $\mathbb{R}^{+}$, with $z_{i}(B)$ denoting the number of particles (or documents) in $B \subseteq\mathbb{R}^{+}$. If one assumes $$\label{eq39.4.1} z_{3}(B) = z_{1}(B) + z_{2}(B) \qquad\mbox{for all } B \subseteq\mathbb{R}^{+},$$ it follows that $$\label{eq39.4.2} z^{*}_{3}(B) = z^{*}_{1}(B) + z^{*}_{2}(B) \qquad\mbox{for all } B \subseteq \mathbb{R}^{+},$$ where $z_{i}^{*}(B)$ is defined analogously to $z^{*}_{r}(B)$ below (\[eq5.1.6\]), with convolution being with respect to $\phi(\cdot )$. Several elementary equalities follow from (\[eq39.4.2\]), including $$\label{eq39.4.15} |x_{3}|_{r,s} = |x_{1}|_{r,s} + |x_{2}|_{r,s} \qquad\mbox{for all } r \in \mathcal{R} \mbox{ and } s > 0,$$ where $x_{i}$ are states in the metric space $S$ introduced in Section \[sec2\] for which the analog of (\[eq39.4.1\]) is satisfied for each $r$ and $|\cdot|_{r,s}$ is given by (\[eq5.3.1\]). Recall that $\tilde{X}(\cdot)$ and $X^{A}(\cdot)$ are the processes constructed from $X(\cdot)$ that were introduced in Section \[intro\], where service of each document is pathwise identical to $X(\cdot)$, but where, for $\tilde{X}(\cdot)$, the arrival of documents is suppressed and, for $X^{A}(\cdot)$, only new documents are included. One has $$Z(t,B) = \tilde{Z}(t,B) + Z^{A}(t,B) \qquad\mbox{for } t \ge0 \mbox{ and } B \subseteq\mathbb{R}^{+},$$ where the processes $Z(\cdot), \tilde{Z}(\cdot)$, and $Z^{A}(\cdot )$ correspond to $X(\cdot), \tilde{X}(\cdot)$ and $X^{A}(\cdot)$. From (\[eq39.4.2\]), $$\label{eq39.4.16} Z^{*}(t,B) = \tilde{Z}^{*}(t,B) + Z^{A,*}(t,B) \qquad\mbox{for } t \ge0 \mbox{ and } B \subseteq\mathbb{R}^{+},$$ and from (\[eq39.4.15\]), $$\label{eq39.1.3} |X(t)|_{r,s} = |\tilde{X}(t)|_{r,s} + |X^{A}(t)|_{r,s}\qquad \mbox{for } t \ge0, r \in\mathcal{R}, s > 0.$$ Another elementary equality involving $X(\cdot)$ is given by $$\label{eq39.1.4} \tilde{Z}_{r}(t,B) = z_{r}\bigl(B + \Delta_{r}(t)\bigr) \qquad\mbox{for } t \ge0, r \in \mathcal{R}, B \subseteq\mathbb{R}^{+},$$ where, we recall, $\Delta_{r}(t)$ is the translation that gives the amount of service an original document that has not yet completed service has received by time $t$. The equality relies on all documents on a given route $r$ receiving equal service at each time. \[If $\tilde{Z}_{r}(t,\mathbb{R}^{+})=0$, set $\Delta_{r}(t)=\infty$ and $z_{r}(\mathbb{R}^{+} + \infty) = 0$.\] From (\[eq39.1.4\]), one obtains $$\label{eq39.1.5} \tilde{Z}^{*}_{r}(t,B) \le z^{*}_{r}\bigl(B + \Delta_{r}(t)\bigr) \qquad\mbox{for } t \ge 0, r \in\mathcal{R}, B \subseteq\mathbb{R}^{+};$$ the inequality arises from the possibility that original documents have completed service by time $t$. A consequence of (\[eq5.3.1\]) and (\[eq39.1.5\]) is that $$\label{eq39.1.6} |\tilde{X}(t)|_{r,s} \le|x|_{r,s+\Delta_{r}(t)} \qquad \mbox{for }t\ge0, r \in \mathcal{R}, s > 0.$$ Combining (\[eq39.1.3\]) and (\[eq39.1.6\]) produces $$\label{eq39.1.7} |X(t)|_{r,s} \le|x|_{r,s+\Delta_{r}(t)} + |X^{A}(t)|_{r,s} \qquad\mbox{for } t \ge0, r \in\mathcal{R}, s > 0;$$ taking the supremum over all $r$ and $s$ therefore gives $$\label{eq39.2.1} |X(t)|_{L} \le|x|_{L} + |X^{A}(t)|_{L} \qquad\mbox{for all } t \ge0.$$ Application of (\[eq39.1.5\]) also implies $$\label{eq39.1.8} \tilde{Z}^{*}_{r}(t,s) \le z^{*}_{r}\bigl(s+\Delta_{r}(t)\bigr)\qquad \mbox{for } t \ge0, r \in\mathcal{R}, B \subseteq\mathbb{R}^{+},$$ and application of (\[eq39.1.5\]), together with (\[eq39.4.16\]), implies that $$\label{eq39.1.9} |X(t)|_{2} \le|x|_{2} + |X^{A}(t)|_{2} \qquad\mbox{for } t \ge0,$$ where $|\cdot|_{2}$ is given in (\[eq5.3.11\]). The term on the LHS of (\[eq10.8.5\]) can also be derived using (\[eq39.1.8\]). The sets $\mathcal{A}(t)$ {#the-sets-mathcalat .unnumbered} ------------------------- In this subsection, we define the random set $\mathcal{A}(t)$, which is a function of $X(t')$, for $t' \in[0,t]$. In Sections \[sec7\]–\[sec10\], we will establish upper bounds on $|X(N^{3})|_{r,s}$ for $\omega\in\mathcal {A}(N^{3})$; the exceptional small set $\mathcal{A}(N^{3})^{c}$ will be treated in the next subsection. The set $\mathcal{A}(t)$ will be a “good” set in the sense that the number of arrivals over $[0,t]$, for given $t$, is restricted by upper bounds, which will enable us to show that $|X(\cdot)|_{L}$ decreases in an appropriate manner. The set $\mathcal{A}(t)$ is given by $\mathcal{A}(t) = \mathcal {A}_{1}(t) \cap \mathcal{A}_{2}(t)$, with $$\label{eq20.1.5} \mathcal{A}_{i}(t) = \bigcap_{r,j} \mathcal{A}_{i,r,j}(t)\qquad \mbox{for } i=1,2,$$ where $\mathcal{A}_{i,r,j}(t)$ specify upper bounds on the numbers of weighted arrivals of documents with different service times. To define $\mathcal{A}_{i,r,j}(t)$, we denote by $v_{0}, v_{1}, \ldots, v_{J}$ the increasing sequence with $$\label{eq20.1.6} v_{j+1} = v_{j} + 1/b^{3} \qquad\mbox{for } j = 0, \ldots, J-1,$$ with $v_{0}=0$ and $v_{J}=N+1$, and where $b$ is as in (\[eq5.1.4\]). Note that it follows from the second half of (\[eq5.1.10\]) that, for $b \ge2$, $$\label{eq20.1.7} \bar{H}^{*}_{r}(v_{j+1})/\bar{H}^{*}_{r}(v_{j}) \ge1/2 \qquad\mbox{for all } r \mbox{ and } j.$$ We also denote by $S^{1}_{r}(k)$, $k = 1, \ldots, A_{r}(t)$, the service time of the $k$th arrival at route $r$, where $A_{r}(t)$ is the cumulative number of arrivals at $r$ by time $t$. We set, for $r \in\mathcal{R}$ and $j = 0, \ldots, J$, $$\label{eq20.1.2} \mathcal{A}_{1,r,j}(t) = \Biggl\{ \omega\dvtx\sum^{A_{r}(t)}_{k=1} \bar{\Phi }\bigl(v_{j} - S^{1}_{r}(k)\bigr) \le2\nu_{r}\bigl(\bar{H}^{*}_{r}(v_{j})t \vee t^{\eta}\bigr) \Biggr\}.$$ Here, we assume $\eta\in(0, 1/12]$, and, as elsewhere, we set $\bar{H}_{r} (\cdot) = 1 - H_{r}(\cdot)$ and $\bar{\Phi}(\cdot) = 1 - \Phi (\cdot)$. One has, as a special case of (\[eq20.1.2\]), that $$\label{eq20.2.1} A_{r}(t) \le2\nu_{r}t \qquad\mbox{on } \mathcal{A}_{1,r,0}(t).$$ Since $$\label{eq20.1.3} E\bigl[\bar{\Phi}\bigl(v_{j} - S^{1}_{r}(k)\bigr)\bigr] = \int^{\infty}_{0} \bar{\Phi}(v_{j}-s)\,dH_{r}(s) = \bar{H}^{*}_{r}(v_{j})$$ and $A_{r}(t) \sim\nu_{r}t$ for large $t$, the probability of the complement $\mathcal{A}_{1,r,j}(t)^{c}$ can be bounded above by using standard large derivation estimates. The term $t^{\eta}$ is included on the RHS of (\[eq20.1.2\]) so that, when $\bar{H}^{*}_{r}(v_{j})$ is small, the probability of the event remains small. We also set, for $r \in\mathcal{R}$ and $j = 0, \ldots, J$, $$\label{eq20.2.2}\quad \mathcal{A}_{2,r,j}(t) = \Biggl\{ \omega\dvtx\sum^{A_{r}(t)}_{k=1} \phi\bigl( v_{j} - S^{1}_{r}(k)\bigr) \le(1+\varepsilon_{5})\nu_{r}\bigl(h^{*}_{r}(v_{j})t\vee t^{\eta}\bigr) \Biggr\},$$ where $\varepsilon_{5}> 0$. Analogous to (\[eq20.1.3\]), one has $$\label{eq20.2.4} E\bigl[\phi\bigl(v_{j} - S^{1}_{j}(k)\bigr)\bigr] = \int^{\infty}_{0}\phi(v_{j} - s) \, dH_{r}(s) = h^{*}_{r}(v_{j}).$$ The probabilities $P_{x}(\mathcal{A}_{2,r,j}(t)^{c})$ will satisfy large deviation bounds as well. The constant $\varepsilon_{5}$ here will later be required to satisfy $\varepsilon_{5}\le\varepsilon_{7}/4$, where $\varepsilon_{7}$ is specified in (\[eq17.1.5\]) and measures how subcritical the network is. In (\[eq20.1.2\]), we only need to employ the constant 2, rather than $1+\varepsilon_{5}$ as in (\[eq20.2.2\]), because (\[eq20.1.2\]) will be applied to the right tail of $\bar {H}^{*}_{r}(\cdot)$, rather than the “main body” of $H^{*}_{r}(\cdot)$, as will (\[eq20.2.2\]). Upper bounds on $\mathcal{A}(t)^{c}$ {#upper-bounds-on-mathcalatc .unnumbered} ------------------------------------ The main result in this last subsection is the following proposition. \[prop21.1.1\] For large enough $t$, $$\label{eq21.1.2} E_{x}[|X(t)|_{L} - |x|_{L}; \mathcal{A}(t)^{c}] \le N^{3} e^{-C_{10}t^{\eta}}$$ for all $N, x$ and appropriate $C_{10}> 0$. Proposition \[prop21.1.1\] gives strong bounds on the growth of $|X(t)|_{L}$ on $\mathcal{A}(t)^{c}$. This behavior is primarily due to the small probability $P_{x}(\mathcal{A}(t)^{c})$, which is given in the next proposition. \[prop21.1.3\] For large enough $t$, $$\label{eq21.1.4} P_{x}(\mathcal{A}(t)^{c}) \le N e^{-C_{11}t^{\eta}}$$ for all $N, x$ and appropriate $C_{11}> 0$. The interarrival times are assumed to be independent, and large initial residual interarrival times only delay future arrivals. The initial state $x$ will therefore not affect the bounds in (\[eq21.1.2\]) and (\[eq21.1.4\]). Note that only the arrival process $A(\cdot)$ is relevant for the bounds in (\[eq21.1.4\]). Proposition \[prop21.1.3\] will serve as the main step in demonstrating Proposition \[prop21.1.1\]; it will also be used along with Proposition \[prop21.1.1\] in Section \[sec10\]. When we apply (\[eq21.1.2\]) and (\[eq21.1.4\]) there, we will set $t = N^{3}$ and so the factors $N^{3}$ and $N$ can be absorbed into the corresponding exponentials. We note that $C_{10}$ and $C_{11}$ in (\[eq21.1.2\]) and (\[eq21.1.4\]), and the bound on $t$ depend on our choices of $\varepsilon_{5}$ and $b$, and on $\nu_{r}$ and $w_{r}$. In order to show Proposition \[prop21.1.3\], we will employ elementary large deviation estimates, which are given in the following two lemmas. \[lem21.1.5\] Let $W(1), W(2), \ldots$ denote nonnegative i.i.d. random variables with mean $\beta< \infty$. Then, for each $\varepsilon> 0$, there exists $C_{12}> 0$, so that $$\label{eq21.2.1} P \Biggl( \sum^{n}_{k=1} W(k) \le(1 - \varepsilon) \beta n \Biggr) \le e^{-C_{12}n}.$$ When the support of $W(1)$ is contained in $[0,1]$ and $\varepsilon\in(0,1]$, $$\label{eq21.2.6} P \Biggl( \sum^{n}_{k=1} W(k) \ge(1 + \varepsilon) \beta n \Biggr) \le e^{{-C_{13}}{\varepsilon^{2} \beta n}},$$ where $C_{13}> 0$ does not depend on the distribution of $W(1)$ or on $\varepsilon$. Both (\[eq21.2.1\]) and (\[eq21.2.6\]) are elementary large deviation bounds. We summarize the argument for (\[eq21.2.6\]); (\[eq21.2.1\]) can be shown directly or by applying (\[eq21.2.6\]) after truncating $W(k)$. As usual, one employs the moment generating function $$\label{eq21.2.2} \psi_{\theta}(n) = E \bigl[e^{\theta\sum^{n}_{k=1}(W(k)-\beta)} \bigr]\qquad \mbox{for } \theta> 0.$$ By expanding the exponential for $n=1$, it follows that for appropriate $C_{14} \ge1$ and for $\theta\in(0,1]$, $$\label{eq21.2.3} \psi_{\theta}(1) \le1 + C_{14}\beta\theta^{2},$$ and hence $$\label{eq21.2.4} \psi_{\theta}(n) \le(1 + C_{14}\beta\theta^{2})^{n} \le e^{C_{14} \beta\theta^{2} n}.$$ By applying Markov’s inequality and setting $\theta= \varepsilon/2C_{14} $, it follows that the LHS of (\[eq21.2.6\]) is at most $$\label{eq21.2.5} e^{-\varepsilon\beta\theta n}\psi_{\theta}(n) \le e^{-\varepsilon ^{2}\beta n/4C_{14}} \le e^{-C_{13}\varepsilon^{2}\beta n}$$ for $C_{13}= 1/4 C_{14}$, as desired. Let $W(1), W(2), \ldots$ denote the successive interarrival times for a renewal process (with delay), with $A(t)= \max\{n\dvtx\sum^{n}_{k=1} W(k)\le t\}$ denoting the number of renewals by time $t$. Here, $W(2), W(3), \ldots $ are i.i.d., with $W(1)$ being the residual interarrival time. We also introduce i.i.d. random variables $Y(1), Y(2), \ldots,$ with $Y(1) \in[0,1]$ that are defined on the same space as $W(k)$. Set $E[W(2)]=\beta> 0$ and $E[Y(1)] = m$. \[lem21.3.1\] Let $W(1), W(2), \ldots$ and $Y(1), Y(2), \ldots$ be as above. Then, for given $\varepsilon\in(0,1]$ and large $t$, $$\label{eq21.3.2} P \Biggl(\sum^{A(t)}_{k=1}Y(k) > (1+\varepsilon)\beta^{-1}mt \Biggr) \le e^{-C_{15}mt},$$ where $C_{15}> 0$ does not depend on the distribution of $Y(1)$. $\{A(t) \ge n \}$ is contained in the event $ \{\sum^{n}_{k=1}W(k)\le t \}$. Consequently, by (\[eq21.2.1\]) of Lemma \[lem21.1.5\], substitution of $\varepsilon/3$ for $\varepsilon$ there implies that, for $n(t) = \lceil (1 - \varepsilon/3)^{-1}\beta^{-1}t\rceil$, $$\label{eq21.3.3} P\bigl(A(t) > n(t)\bigr) \le P \Biggl(\sum^{n(t)+1}_{k=2}W(k) \le t \Biggr) \le e^{-C_{16}t}$$ for appropriate $C_{16}> 0$ and large $t$ (which may depend on $\varepsilon $ and the distribution of $W$). We next consider the set where $A(t) \le n(t)$. It follows from (\[eq21.2.6\]) of Lemma \[lem21.1.5\] that $$\begin{aligned} \label{eq21.3.4} && P \Biggl(\sum^{A(t)}_{k=1}Y(k) > (1+\varepsilon)\beta^{-1}mt; A(t)\le n(t) \Biggr) \nonumber\\[-8pt]\\[-8pt] % &&\qquad \le P \Biggl(\sum^{n(t)}_{k=1}Y(k) > (1+\varepsilon)\beta^{-1}mt \Biggr) \le e^{-C_{13}\varepsilon^{2}\beta^{-1}mt/9}.\nonumber\end{aligned}$$ Inequality (\[eq21.3.2\]) follows from (\[eq21.3.3\]) and (\[eq21.3.4\]). We now employ Lemma \[lem21.3.1\] to prove Proposition \[prop21.1.3\]. [Proof of Proposition \[prop21.1.3\]]{} We first note that since $\mathcal{A}(t) = \mathcal{A}_{1}(t) \cap \mathcal{A}_{2}(t)$, with $\mathcal{A}_{i}(t) = \bigcap_{r \in \mathcal{R}} \bigcap_{j=0}^{J} \mathcal{A}_{i,r,j}(t)$, where $J = b^{3}(N+1)+1 \le2b^{3}N$, it suffices to show that for each $\mathcal{A}_{i,r,j}(t)$, $$\label{eq21.4.1} P_{x}(\mathcal{A}_{i,r,j}(t)^{c}) \le e^{-C_{17}t^{\eta}}$$ for $t \ge t_{0}$, for some fixed $t_{0}$ and appropriate $C_{17}> 0$. We consider the case where $i=1$. Denote by $W(1), W(2), \ldots$ the interarrival times of documents on route $r$ and set $Y(k) = \bar{\Phi}(v_{j} - S^{1}_{r}(k))$. Then, $Y(k)$ are i.i.d. random variables and, except for $W(1)$, so are $W(k)$. One has $$\label{eq21.4.2} \beta\stackrel{\mathrm{def}}{=} E[W(2)]=\nu^{-1}_{r} \quad\mbox{and}\quad m \stackrel{\mathrm{def}}{=}E[Y(1)]=\bar{H}^{*}_{r}(v_{j})$$ with the last equality following from (\[eq20.1.3\]). We break the problem into two cases, depending on whether or not $\bar{H}^{*}_{r}(v_{j}) \ge t^{\eta- 1}$, in each case applying Lemma \[lem21.3.1\], with $\varepsilon= 1$. Under $\bar{H}^{*}_{r} (v_{j}) \ge t^{\eta- 1}$, one has $$\label{eq21.4.3}\qquad P_{x}(\mathcal{A}_{1,r,j}(t)^{c}) = P_{x} \Biggl(\sum^{A_{r}(t)}_{k=1}\bar {\Phi} \bigl(v_{j}-S^{1}_{r}(k)\bigr) > 2\nu_{r}\bar{H}^{*}_{r} (v_{j})t \Biggr) \le e^{-C_{15}t^{\eta}}$$ for large $t$ and $C_{15}> 0$ as in the lemma, where neither depends on the particular value of $\bar{H}^{*}_{r} (v_{j})$. For $\bar{H}^{*}_{r} (v_{j}) < t^{\eta-1}$, we replace the random variables defined above (\[eq21.4.2\]) by i.i.d. random variables $Y'(k) \in(0,1]$, with $Y'(k)\ge Y(k)$ and $E[Y'(k)] = t^{\eta- 1}$. Then, again applying Lemma \[lem21.3.1\], but this time to $Y'(k)$, $k=1,2,\ldots,$ $$\label{eq21.5.5} P_{x}(\mathcal{A}_{1,r,j}(t)^{c}) \le P_{x} \Biggl(\sum ^{A_{r}(t)}_{k=1}Y'(k) > 2 \nu_{r} t^{\eta} \Biggr) \le e^{-C_{15}t^{\eta}}$$ as before. Together with (\[eq21.4.3\]), this implies (\[eq21.4.1\]) for $i=1$, with $C_{17}= C_{15}$. The reasoning for (\[eq21.4.1\]) when $i=2$ is the same, except that one now sets $Y(k) = \phi(v_{j} - S^{1}_{r}(k))$, from which one obtains $$\label{eq21.5.3} m\stackrel{\mathrm{def}}{=}E[Y(1)] = h^{*}_{r}(v_{j}).$$ Also, the coefficient 2 on the RHS of (\[eq20.1.2\]) is replaced by the coefficient $1+\varepsilon_{5}$ in (\[eq20.2.2\]). Setting $\varepsilon= \varepsilon_{5} $ in Lemma \[lem21.3.1\], one obtains $$\label{eq21.5.4} P_{x}\Biggl(\sum^{A_{r}(t)}_{k=1}\phi\bigl(v_{j}-S^{1}_{r}(k)\bigr) > (1+\varepsilon _{5})\nu_{r} \bigl(h^{*}_{r}(v_{j})\vee t^{\eta}\bigr) \Biggr) \le e^{-C_{15}t^{\eta}}$$ for large $t$ and appropriate $C_{15}> 0$, chosen as in the lemma. Setting $C_{17}= C_{15}$, one obtains (\[eq21.4.1\]) for $i=2$ as well. Setting $|A(t)| = \sum_{r}A_{r}(t)$, where $A_{r}(t)$ is the number of arrivals at each route by time $t$, it follows from elementary renewal theory that for appropriate $C_{18}$ and $t \ge1$, $$\label{eq21.6.2} E [|A(t)|^{2} ] \le C_{18}t^{2}$$ (see, e.g., [@r3], page 136). Inequality (\[eq21.6.2\]) is not difficult to show by applying a standard truncation argument. Here and later on, we will also use the two inequalities $$\label{eq6.88.1} z^{*}_{r}(s) \le bz^{*}_{r}((s,\infty)) \qquad\mbox{for all } s > 0,$$ and $$\label{eq6.88.2} \Gamma\bigl(\bar{H}^{*}_{r}(N_{H_{r}}+1)\bigr)/\Gamma(\bar {H}^{*}_{r}(N_{H_{r}})) \ge e^{-b},$$ which follow from the definition of $\phi(\cdot)$ and the second inequality in (\[eq5.1.10\]). Employing Proposition \[prop21.1.3\] and these inequalities, we now demonstrate Proposition \[prop21.1.1\]. [Proof of Proposition \[prop21.1.1\]]{} By Hölder’s inequality, $$\label{eq21.7.1}\qquad E_{x}[|X(t)|_{L} - |x|_{L}; \mathcal{A}(t)^{c}] \le\sqrt{P_{x} (\mathcal{A}(t)^{c})} \sqrt{E_{x} \bigl[\bigl(|X(t)|_{L} - |x|_{L}\bigr)^{2} \bigr]}.$$ Also, by Proposition \[prop21.1.3\], one has $$\label{eq21.7.2} \sqrt{P_{x}(\mathcal{A}(t)^{c})} \le\sqrt{N} e^{-C_{11}t^{\eta/2}}$$ for all $N,x$ and appropriate $C_{11}> 0$. So it remains to bound the expectation on the RHS of (\[eq21.7.1\]). It follows from the definitions of $|\cdot|_{L}$, $\phi(\cdot)$ and $\Gamma(\cdot)$, and from (\[eq90.3.1\]), (\[eq6.88.1\]) and (\[eq6.88.2\]), that $$\label{eq21.7.3} |x'|_{L} \le\biggl(\sup_{r}\frac{w_{r}}{\nu_{r}} \biggr) \frac{2be^{b}(1+aN)|x'|}{\Gamma(1/N^{4})} \le C_{19}N^{2}|x'|$$ for all $x' \in S$ and appropriate $C_{19}$. So application of (\[eq39.2.1\]), together with (\[eq21.7.3\]) for $x'=X^{A}(t)$, implies that $$\begin{aligned} E_{x}\bigl[\bigl(|X(t)|_{L}-|x|_{L}\bigr)^{2}\bigr] &\le& E_{x}[|X^{A}(t)|^{2}_{L}] \le C_{19}^{2} N^{4}E_{x}[|X^{A}(t)|^{2}] \\ &\le& C_{19}^{2} N^{4}E_{x}[|A(t)|^{2}].\end{aligned}$$ Together with (\[eq21.6.2\]), this implies $$\label{eq21.7.5} \sqrt{E_{x} \bigl[\bigl(|X(t)|_{L} - |x|_{L}\bigr)^{2} \bigr]} \le C_{20}N^{2}t$$ for appropriate $C_{20}$ and large $t$. Substitution of (\[eq21.7.2\]) and (\[eq21.7.5\]) into (\[eq21.7.1\]) implies that for large enough $t$, (\[eq21.1.2\]) holds, as desired. Upper bounds on $|X(N^{3})|_{r,s}$ for $s > N_{H_{r}}$ {#sec7} ====================================================== In Section \[sec6\], we obtained upper bounds on $E_{x}[|X(N^{3})|_{L} - |x|_{L}; \mathcal{A}(N^{3})^{c}]$; we still need to analyze the behavior of $|X(N^{3})|_{L}-|x|_{L}$ on $\mathcal{A}(N^{3})$. For this, we analyze $|X(N^{3})|_{r,s}$ for several cases that depend on whether or not $|x|>N^{6}$ and $s>N_{H_{r}}$. In this section, we consider the case where $|x|>N^{6}$ and $s>N_{H_{r}}$, which is the simplest case. The main result here is the following proposition. Recall that $|x|_2$ is defined in (\[eq5.3.11\]). \[prop40.2.3\] For given $\varepsilon_{3}>0$, large enough $N$, and $|x|>N^{6}$ and $|x|_{2}/|x|\le1/2$, $$\begin{aligned} \label{eq40.2.4} &&E_{x} \Bigl[{\sup_{r, s> N_{H_{r}}}}{|X(N^{3})|}_{r, s}-|x|_{L}; G \Bigr] \nonumber\\[-8pt]\\[-8pt] % &&\qquad\le C_{3}(|x|_{K}/|x|)N^{3} + \varepsilon_{3} N^{2} \biggl(\frac{1}{2}-P(G) \biggr)\nonumber\end{aligned}$$ for all measurable sets $G$, with $C_{3}> 0$ not depending on $N$, $G$ or $x$. In the proof of Proposition \[prop48.3.2\], we will employ Proposition \[prop40.2.3\] by setting $$\label{eq7.2.1} G =\mathcal{A}(N^{3})\cap\Bigl\{\omega\dvtx|X(N^{3})|_{L} = {\sup_{r, s > N_{H_{r}}}} |X(N^{3})|_{r, s} \Bigr\}.$$ Much of the work needed to demonstrate Proposition \[prop40.2.3\] is done in the following proposition. We recall that $i_{r}(s) = s + \Delta_{r}$, where $\Delta_{r} = \Delta_{r}(N^{3})$. \[prop39.3.2\] For given $\varepsilon> 0$, large enough $N$ and all $x$, $$\label{eq39.3.3} E_{x} \Bigl[\sup_{r, s > N_{H_{r}}} \bigl\{|X(N^{3})|_{r,s} - |x|_{r,i_{r}(s)} \bigr\}; G \Bigr] \le C_{21}\varepsilon N^{2}$$ for all measurable sets $G$, with $C_{21}$ not depending on $\varepsilon$, $N$, $G$ or $x$. We will instead show that $$\label{eq39.3.1} E_{x} \Bigl[{\sup_{r, s > N_{H_{r}}}}|X^{A}(N^{3})|_{r,s} \Bigr] \le C_{21}\varepsilon N^{2}.$$ Inequality (\[eq39.3.3\]) follows immediately from this and inequality (\[eq39.1.7\]) since$|X^{A}(N^{3})|_{r,s} \ge0$. To show (\[eq39.3.1\]), we first note that for all $r$ and $s$, $$\label{eq39.2.3} |X^{A}(N^{3})|_{r,s} \le C_{22}\kappa_{N,r} N_{H_{r}}Z^{A,*}_{r}(N^{3}, s)$$ for appropriate $C_{22}$, where $Z^{A,*}_{r}(N^{3}, s)\stackrel {\mathrm{def}}{=} (Z^{A})^{*}_{r}(N^{3}, s)$. The inequality uses (\[eq5.3.1\]) and (\[eq6.88.2\]). On $s>N_{H_{r}}$, the RHS of (\[eq39.2.3\]) is at most $$\begin{aligned} \label{eq39.2.5}\qquad\quad C_{22}b \kappa_{N,r} N_{H_{r}} Z^{A,*}_{r}(N^{3}, (s,\infty)) \le C_{22}b \kappa_{N,r} \int^{\infty }_{N_{H_{r}}}N_{r}(s')Z^{A,*}_{r}(N^{3}, s')\,ds'\end{aligned}$$ on account of (\[eq6.88.1\]) and $N_{r}(s)\ge s$. On the other hand, by (\[eq5.3.2\]), the RHS of (\[eq39.2.5\]) is at most $$\label{eq39.2.7} (C_{22}b/M_{1})|X^{A}(N^{3})|_{R} \le C_{21}I_{R}(N^{3}),$$ where $C_{21}\stackrel{\mathrm{def}}{=}C_{22}b/M_{1}$ and $I_{R}(\cdot )$ is as in Section \[sec5\]. Putting (\[eq39.2.3\])–(\[eq39.2.7\]) together, it follows that, for large $N$, $$\label{eq39.2.8} {\sup_{r, s > N_{H_{r}}}}|X^{A}(N^{3})|_{r,s} \le C_{21} I_{R}(N^{3}).$$ Also, by Proposition \[prop10.7.4\], for given $\varepsilon$, one has that for large enough $N$, $$\label{eq39.2.9} E_{x} [I_{R}(N^{3}) ] \le\varepsilon N^{2} \qquad\mbox{for all } x.$$ Taking expectations in (\[eq39.2.8\]) and applying (\[eq39.2.9\]) implies (\[eq39.3.1\]). In order to demonstrate Proposition \[prop40.2.3\], we need Lemma \[lem40.1.1\], which bounds $|x|_{K}$ from below in terms of $|x|$ when $({\sup_{r, s \ge N_{H_{r}}}}|x|_{r,s})/|x|_{L}$ is not small. For the lemma, we require the inequality $$\label{eq40.1.5} z^{*}_{r}((0, N_{H_{r}}]) \le C_{23}|x|_{L} \qquad\mbox{for } r \in \mathcal{R},$$ for appropriate $C_{23}$. This is a weaker version of (\[eq17.1.3\]), which we prove in Lemma \[lem17.1.6\]. \[Equation (\[eq40.1.5\]) does not require any additional assumptions on $a$ or $b$, unlike (\[eq17.1.3\]).\] If one supposes that $|x|_{2} \le|x|/2$, it then follows easily by summing (\[eq40.1.5\]) over $r$ that $$\label{eq7.5.1} |x|_{L} \ge C_{24}|x|$$ for $C_{24}= 1/2 C_{23}|\mathcal{R}|$. This inequality will be used in Proposition \[prop40.2.3\] and will also be used in Sections \[sec8\] and \[sec9\]. \[lem40.1.1\] Suppose that, for some $r_{0}$ and $s_{0} \ge N_{H_{r_{0}}}$, $$\label{eq40.1.2} |x|_{r_{0},s_{0}} \ge|x|_{L}/2.$$ Then, for appropriate $\varepsilon_{6}> 0$ not depending on $N$, $$\label{eq40.1.3} |x|_{K} \ge\varepsilon_{6}|x|/N.$$ Applying (\[eq40.1.5\]), and then substituting (\[eq40.1.2\]) into (\[eq5.3.1\]), one obtains for given $r$ that $$\begin{aligned} \label{eq40.1.4} z^{*}_{r}((0, N_{H_{r}}]) &\le& C_{25}Nz^{*}_{r_{0}}(s_{0})/\Gamma \bigl(\bar{H}^{*}_{r_{0}}(N_{H_{r_{0}}}+1)\bigr) \nonumber\\[-8pt]\\[-8pt] % &\le& C_{25}be^{b}N\kappa_{N,r_{0}}z^{*}_{r_{0}}((N_{H_{r_{0}}},\infty))\nonumber\end{aligned}$$ for appropriate $C_{25}> 0$, where the second inequality employs the assumption $s_{0}\ge N_{H_{r_{0}}}$, together with (\[eq6.88.1\]) and (\[eq6.88.2\]). Addition of $z^{*}_{r}((N_{H_{r}},\infty))$ to both sides of (\[eq40.1.4\]) gives $$\begin{aligned} z^{*}_{r}(\mathbb{R}^{+}) &\le& z^{*}_{r}((N_{H_{r}},\infty)) + C_{25} be^{b} N \kappa_{N,r_{0}} z^{*}_{r_{0}}((N_{H_{r_{0}}},\infty)) \\ % &\le& (1+C_{25}be^{b})N \sum_{r'} \kappa_{N,r'}z^{*}_{r'} ((N_{H_{r'}},\infty)).\end{aligned}$$ Summing over $r$ then implies $$\label{eq40.2.2} |x| \le\varepsilon_{6}^{-1}N|x|_{K}$$ with $\varepsilon_{6}= [|\mathcal{R}|(1+C_{25}be^{b}) ]^{-1}$. We now apply Proposition \[prop39.3.2\], together with Lemma \[lem40.1.1\] and (\[eq7.5.1\]), to demonstrate Proposition \[prop40.2.3\]. [Proof of Proposition \[prop40.2.3\]]{} Suppose first that $|x|_{r_{0},s_{0}}>|x|_{L}/2$ for some $r_{0}$ and $s_{0} > N_{H_{r_{0}}}$. Choosing $\varepsilon> 0$ and $C_{21}$ as in Proposition \[prop39.3.2\], with $\varepsilon$ small enough so $\varepsilon< \varepsilon_{3} /C_{21}$ for given $\varepsilon_{3}> 0$, it follows from the proposition and Lemma \[lem40.1.1\] that for large $N$ and any $G$, the LHS of (\[eq40.2.4\]) is at most $$\begin{aligned} \label{eq40.3.1} C_{21}\varepsilon N^{2} &\le& \varepsilon_{3}N^{2} \le2\varepsilon _{3}\varepsilon_{6} ^{-1}(|x|_{K}/|x|)N^{3} - \varepsilon_{3}N^{2} \nonumber\\[-8pt]\\[-8pt] % &\le& C_{3}(|x|_{K}/|x|)N^{3} - \varepsilon_{3}N^{2},\nonumber\end{aligned}$$ if $C_{3}$ is chosen to be at least $2\varepsilon_{3}\varepsilon_{6}^{-1}$, where $\varepsilon_{6} $ is as in the lemma. This is at most the RHS of (\[eq40.2.4\]). Suppose, on the other hand, that $|x|_{r,s} \le|x|_{L}/2$ for all $s>N_{H_{r}}$ and $r$. Under $|x|>N^{6}$ and $|x|_{2} \le|x|/2$, it follows from (\[eq7.5.1\]) that $|x|_{L} \ge C_{24}N^{6}$. Hence, $$\label{eq40.3.2} {\sup_{r, s>N_{H_{r}}}}|x|_{r,s} - |x|_{L} \le-\frac{1}{2} C_{24}N^{6}.$$ Since $i_{r}(s) \ge s > N_{H_{r}}$, it follows from Proposition \[prop39.3.2\] and (\[eq40.3.2\]) that the LHS of (\[eq40.2.4\]) is at most $$\begin{aligned} \label{eq40.3.3} C_{21}\varepsilon N^{2} + {\sup_{r, s>N_{H_{r}}}}|x|_{r,s} - |x|_{L} &\le& C_{21}\varepsilon N^{2}-\frac{1}{2}C_{24}N^{6}P(G)\nonumber\\[-8pt]\\[-8pt] %%\le\epsb N^{2}- % &\le& \varepsilon_{3}N^{2} \biggl(\frac{1}{2} - P(G) \biggr)\nonumber\end{aligned}$$ for large $N$, if we choose $\varepsilon\le\varepsilon_{3}/2C_{21}$. This is at most the RHS of (\[eq40.2.4\]), which completes the proof. Pathwise upper bounds on $|X(N^{3})|_{r,s}$ for $s \le N_{H_{r}}$ and $\Delta_{r}>1/b^{3}$ {#sec8} ====================================================== In the previous section, we analyzed the behavior of $|X(N^{3})|_{r,s} - |x|_{L}$ for $s>N_{H_{r}}$. When $s \le N_{H_{r}}$, we analyze the cases where $\Delta_{r} \le1/b^{3}$ and $\Delta_{r}>1/b^{3}$ separately. The latter case is quicker and we do it in this section, postponing the case $\Delta_{r} \le1/b^{3}$ until Section \[sec9\]. For both cases, we will require certain pathwise upper bounds on $|X^{A}(N^{3})|_{r,s}$ that hold on $\mathcal{A}_{1}(N^{3})$, which are given in Proposition \[prop24.4.1\]. We begin the section with these bounds. Upper bounds on $|X^{A}(N^{3})|_{r,s}$ on $\mathcal{A}_{1} (N^{3})$ {#upper-bounds-on-xan3_rs-on-mathcala_1n3 .unnumbered} ---------------------------------------------------------- In order to derive bounds on $|X^{A}(N^{3})|_{r,s}$, we first require bounds on $Z^{A,*}_{r}(\cdot,\cdot)$ that measure how quickly documents with the corresponding service times enter a route $r$ up to a given time. In Lemma \[lem24.2.5\], we provide uniform bounds on $Z^{A,*}_{r}(t,s)$ for $t \in [0,N^{3}]$ and $\omega\in\mathcal{A}_{1}(N^{3})$. As in previous sections, $S^{1}_{r}(k)$, $k=1, \ldots, A_{r}(t)$, denotes the positions of the arrivals of documents up to time $t$. We also denote here by $S^{2}_{r}(t,k)$ the amount of service such a document has received by time $t$; $S^{1}_{r}(k) - S^{2}_{r}(t,k)$ is therefore the residual service time of the $k$th document at time $t$. \[lem24.2.5\] Suppose $\omega\in\mathcal{A}_{1}(N^{3})$ for some $N$. Then, for all $r$, $s \in[0, N+1]$ and $t \in[0, N^{3}]$, $$\label{eq24.2.6} Z^{A,*}_{r}(t,s) \le4b\nu_{r}\bigl(\bar{H}^{*}_{r}(s)N^{3} \vee N^{3\eta}\bigr).$$ If instead $s>N+1$, then $$\label{eq24.3.1} Z^{A,*}_{r}(t,s) \le2b\nu_{r}\bigl(\bar{H}^{*}_{r}(N+1)N^{3} \vee N^{3\eta}\bigr).$$ For all $r$, $s \in[0,N+1]$ and $t \in[0,N^{3}]$, $$\begin{aligned} \label{eq24.3.2}\qquad Z^{A,*}_{r}(t,s) &=& \sum^{A_{r}(t)}_{k=1} \phi\bigl(s - S^{1}_{r}(k) + S^{2}_{r}(t,k)\bigr)\nonumber\\[-8pt]\\[-8pt] % &\le& \sum^{A_{r}(N^{3})}_{k=1} \sup_{s^{\prime}\in[0,\infty)}\phi \bigl(s - S^{1}_{r}(k)+s'\bigr)\le b\sum^{A_{r}(N^{3})}_{k=1} \bar{\Phi}\bigl(s - S^{1}_{r}(k)\bigr)\nonumber\end{aligned}$$ with the latter inequality employing $\phi(s) \le b\bar{\Phi}(s)$ and the monotonicity of $\bar{\Phi}(\cdot)$. Letting $j_{0}$ denote the largest $j$ with $v_{j} \le s$, the last term in (\[eq24.3.2\]) is at most $$\label{eq24.3.3} b\sum^{A_{r}(N^{3})}_{k=1} \bar{\Phi}\bigl(v_{j_{0}} - S^{1}_{r}(k)\bigr) \le2b\nu_{r} \bigl(\bar{H}^{*}_{r}(v_{j_{0}})N^{3} \vee N^{3\eta}\bigr)$$ on $\mathcal{A}_{1}(N^{3})$. The inequality in (\[eq24.3.1\]) follows from this, with $j_{0}=J$. The inequality in (\[eq24.2.6\]) follows by applying (\[eq20.1.7\]) to the RHS of (\[eq24.3.3\]). We now derive uniform upper bounds on $|X^{A}(t)|_{r,s}$ for $t \in[0,N^{3}]$ and $\omega\in\mathcal{A}_{1}(N^{3})$. In applications, we will be primarily interested in the behavior at $t = N^{3}$. \[prop24.4.1\] Suppose $\omega\in\mathcal{A}_{1}(N^{3})$ for some $N$. Then, for all $r$ and $s$, $$\label{eq24.4.7} |X^{A}(t)|_{r,s} \le C_{26}N^{3} \qquad\mbox{for } t \in[0, N^{3}] \mbox{ and all } x,$$ for appropriate $C_{26}$ not depending on $x, N, \omega, r$ or $s$. In particular, $$\label{eq24.4.2} |X(t)|_{L} - |x|_{L} \le C_{26}N^{3} \qquad\mbox{for } t \in[0, N^{3}] \mbox{ and all } x.$$ By (\[eq39.2.1\]), $$\label{eq24.4.4} |X(t)|_{L} - |x|_{L} \le|X^{A}(t)|_{L} \qquad\mbox{for all } t,$$ and so (\[eq24.4.2\]) follows immediately from (\[eq24.4.7\]). We now investigate $|X^{A}(t)|_{r,s}$. From (\[eq5.3.1\]) and Lemma \[lem24.2.5\], it follows that, for $t \in[0, N^{3}]$, $$\begin{aligned} \label{eq24.4.5} |X^{A}(t)|_{r,s} &=& \frac{w_{r}(1+as_{N})Z^{A,*}_{r}(t,s)}{\nu _{r}\Gamma (\bar{H}^{*}_{r}(s_{N}))} \nonumber\\ % &\le& 4bw_{r}N^{3}(1+as_{N})\bar{H}^{*}_{r}(s_{N})/\Gamma (\bar{H}^{*}_{r}(s_{N})) \\ % &&{} + 4bw_{r}N^{3\eta}(1+as_{N})/\Gamma(\bar{H}^{*}_{r}(s_{N})). \nonumber\end{aligned}$$ We proceed to analyze the two terms on the RHS of (\[eq24.4.5\]). It follows from the definition of $\Gamma(\cdot)$ in (\[eq5.1.3\]) that, for all $s$, $$\label{eq24.4.6} (1+as_{N})\bar{H}^{*}_{r}(s_{N})/\Gamma(\bar{H}^{*}_{r}(s_{N})) \le (1+as_{N}) (\bar{H}^{*}_{r}(s_{N}))^{1-\gamma}/a C_{2}.$$ Since by assumption, $\bar{H}^{*}_{r}(\cdot)$ has more than two moments and $\gamma\le1/2$, the RHS of (\[eq24.4.6\]) goes to 0 as $s_{N}\to \infty$. Hence, it is bounded for all $s_{N}$, which implies that the first term on the RHS of (\[eq24.4.5\]) is bounded above by $C_{27}N^{3}$, for some $C_{27}$ not depending on $t, r$ or $s$. On the other hand, for all $s$, $$\begin{aligned} \label{eq24.5.1} (1+as_{N})/\Gamma(\bar{H}^{*}_{r}(s_{N})) &\le& \bigl(1+a(N+1)\bigr)\bar {H}^{*}_{r}(N_{H_{r}} +1)^{-\gamma}/a C_{2}\nonumber\\[-8pt]\\[-8pt] % &\le& \bigl(1+a(N+1)\bigr)(e^{b}N^{4})^{\gamma}/C_{2}a.\nonumber\end{aligned}$$ Since $\gamma\le1/4$, $\eta\le1/3$ and $aN \ge1$, the latter term on the RHS of (\[eq24.4.5\]) is bounded above by $C_{28}N^{2}$, for some $C_{28}$ not depending on $t$, $r$ or $s$. The above bounds for the two terms on the RHS of (\[eq24.4.5\]) sum to $(C_{27} + C_{28})N^{3}$. Setting $C_{26}= C_{27}+ C_{28}$, this implies (\[eq24.4.7\]). Upper bounds on $|X(N^{3})|_{r,s}$ for $s \le N_{H_{r}}$ and $\Delta_{r}> 1/b^{3}$ {#upper-bounds-on-xn3_rs-for-s-le-n_h_r-and-delta_r-1b3 .unnumbered} ---------------------------------------------------------------------------------- Proposition \[prop41.2.1\] gives an upper bound on $|X(N^{3})|_{r,s}-|x|_{L}$ when $s \le N_{H_{r}}$ and $\Delta_{r}>1/b^{3}$. The proof, which employs Proposition \[prop24.4.1\], is quick. \[prop41.2.1\] Suppose that $|x|>N^{6}$, with $|x|_{2}/|x| \le1/2$. Then, for large enough $N$, $$\label{eq41.2.2}\quad {\sup_{\Delta_{r}>1/b^{3}}\, \sup_{s\le N_{H_{r}}}} |X(N^{3})|_{r,s} - |x|_{L} \le-N^{4} \qquad\mbox{for all }\omega\in \mathcal{A}_{1}(N^{3}),$$ where $N$ does not depend on $x$ or $\omega$. For each $r$ and $s$, $$\begin{aligned} \label{eq41.2.4} |X(N^{3})|_{r,s}-|x|_{L} &=& |X^{A}(N^{3})|_{r,s}- \bigl(|x|_{L}-|\tilde{X}(N^{3}) |_{r,s} \bigr) \nonumber\\[-8pt]\\[-8pt] % &\le& C_{26}N^{3} - \bigl(|x|_{L} - |\tilde{X}(N^{3})|_{r,s} \bigr)\nonumber\end{aligned}$$ with the last line following from Proposition \[prop24.4.1\]. We consider two cases, depending on whether $|x|_{r,i_{r}(s)}>|x|_{L}/2$ for given $r$ and $s$. Suppose first that $|x|_{r,i_{r}(s)}>|x|_{L}/2$, with $s \le N_{H_{r}}$ and $|x|_{2}\le|x|/2$. One has $$\label{eq41.2.6} |x|_{r,i_{r}(s)} - |\tilde{X}(N^{3})|_{r,s} \ge\frac{w_{r}}{\nu _{r}} \cdot \frac{a}{b^{3}}\cdot\frac{z^{*}_{r}(i_{r}(s))}{\Gamma(\bar {H}^{*}_{r}(i_{r} (s)_{N}) )}.$$ To see this, one applies (\[eq39.1.5\]) to the definition of $|x|_{r,s}$ in (\[eq5.3.1\]), noting that since $s \le N_{H_{r}}$, $$\label{eq8.9.1} i_{r}(s)_{N} - s_{N} = i_{r}(s) \wedge(N_{H_{r}}+1) - s \ge\Delta _{r} \wedge1 > 1/b^{3},$$ and that $\Gamma(\bar{H}^{*}_{r}(i_{r}(s)_{N})) \le\Gamma(\bar {H}^{*}_{r}(s))$. On account of (\[eq5.3.1\]) and $|x|_{r,i_{r}(s)}>|x|_{L}/2$, one obtains, from the RHS of (\[eq41.2.6\]), $$\begin{aligned} && \frac{a}{b^{3}} \cdot\biggl(\frac{w_{r}z^{*}_{r}(i_{r}(s))}{\nu _{r}\Gamma (\bar{H}^{*}_{r}(i_{r}(s)_{N}))} \Big/|x|_{r,i_{r}(s)} \biggr) \cdot \frac{|x|_{r,i_{r}(s)}}{|x|_{L}} \cdot|x|_{L} \\ % &&\qquad \ge\frac{a}{b^{3}} \cdot\bigl(1+ai_{r}(s)_{N}\bigr)^{-1} \cdot\frac {1}{2} \cdot |x|_{L}. %\ge\xxx N^{5},\end{aligned}$$ Because of $|x|_{2}\le|x|/2$, (\[eq7.5.1\]), $i_{r}(s)_{N}\le N$, $|x|>N^{6}$ and $aN \ge1$, this is at most $C_{29}N^{5}$, where $C_{29}> 0$ does not depend on $N$, $x$ or $\omega$. It follows from (\[eq41.2.6\]) and the succeeding inequalities that $$\label{eq41.3.2} |x|_{L}-|\tilde{X}(N^{3})|_{r, s} \ge|x|_{r,i_{r}(s)} - |\tilde {X}(N^{3})|_{r,s} \ge C_{29}N^{5}.$$ Together with (\[eq41.2.4\]), this gives the RHS of (\[eq41.2.2\]). Suppose, on the other hand, that $|x|_{r,i_{r}(s)} \le|x|_{L}/2$, with $|x|_{2} \le|x|/2$. Then, by (\[eq7.5.1\]) and (\[eq39.1.6\]), the RHS of (\[eq41.2.4\]) is at most $$\begin{aligned} \label{eq41.2.5} && C_{26}N^{3} - \tfrac{1}{2} |x|_{L} - \bigl(|x|_{r,i_{r}(s)} - |\tilde{X} (N^{3})|_{r,s} \bigr) \nonumber\\[-8pt]\\[-8pt] % &&\qquad \le C_{26}N^{3} - \tfrac{1}{2} C_{24}N^{6} \le- N^{5}\nonumber\end{aligned}$$ for large $N$. This implies (\[eq41.2.2\]) for $|x|_{r,i_{r}(s)} \le |x|_{L}/2$, and hence completes the proof. Pathwise upper bounds on $|X(N^{3})|_{r,s}$ for $s \le N_{H_{r}}$ and $\Delta_{r} \le1/b^{3}$ {#sec9} ====================================================== In Sections \[sec7\] and \[sec8\], we analyzed the behavior of $|X(N^{3})|_{r,s} - |x|_{L}$ for $s > N_{H_{r}}$, and for $s \le N_{H_{r}}$ with $\Delta_{r} > 1/b^{3}$. There remains the case $s \le N_{H_{r}}$ with $\Delta_{r} \le1/b^{3}$, which is the subject of this section. This is, in essence, the “main case” one needs to show in order to establish the stability of the network since the other cases dealt with less sensitive behavior and did not employ the subcriticality of the system that was given in (\[eq5.3.8\]). The same was also true for the computations of the $|\cdot|_{A}$ and $|\cdot|_{R}$ norms in Sections \[sec4\] and \[sec5\]. Section \[sec9\] consists of three subsections. First, in Proposition \[prop46.3.2\], we give lower bounds on the minimal service rates $\lambda^{w} (\cdot)$ of documents in terms of the norm $|\cdot|_{L}$. In the next subsection, we begin our analysis of $|X(N^{3})|_{r,s}$ for $s\le N_{H_{r}}$ and $\Delta_{r}\le1/b^{3}$. We decompose $|X(N^{3})|_{r,s} - |x|_{r,i_{r}(s)}$ into several parts that are easier to analyze. In Proposition \[prop44.1.2\], we then obtain upper bounds on the factor $Z^{*}_{r}(N^{3},s) - z^{*}_{r}(i_{r}(s))$ of one of the parts. In the third subsection, we do a detailed analysis of the decomposition from the previous subsection, which also employs the bounds on $\lambda^{w}(\cdot)$ from the first subsection. From this, we obtain in Proposition \[prop48.1.1\] the desired bound on $|X(N^{3})|_{r,s} - |x|_{L}$. We note that, whereas in Section \[sec8\], our results pertained to $\omega \in \mathcal{A}_{1} (N^{3})$, starting from the second subsection here, we require $\omega\in\mathcal{A}_{2}(t)$. Our final results on $|X(N^{3})|_{r,s}-|x|_{L}$, for $s \le N_{H_{r}}$, will therefore be valid on $\mathcal{A}(N^{3}) = \mathcal{A}_{1}(N^{3}) \cap\mathcal{A}_{2}(N^{3})$. Lower bounds on $\lambda^{w}(\cdot)$ {#lower-bounds-on-lambdawcdot .unnumbered} ------------------------------------ In order to demonstrate the stability of the network, its subcriticality needs to be employed at some point. With this in mind, we choose $\varepsilon _{7}\in(0,1]$ small enough so that $$\label{eq17.1.5} (1+\varepsilon_{7})^{2} \sum_{r \in\mathcal{R}}A_{l,r}\rho_{r} \le c_{l} \qquad\mbox{for all } l,$$ which is possible because of (\[eq5.3.8\]). We henceforth assume $\varepsilon_{5}\le \varepsilon_{7}/4$, where $\varepsilon_{5}$ was employed in (\[eq20.2.2\]) in the definition of $\mathcal{A}_{2}(\cdot)$. The main results in this subsection are Propositions \[prop17.2.1\] and \[prop46.3.2\]. Proposition \[prop17.2.1\] gives a lower bound on $\lambda^{w}(t)$ in terms of $|X(t)|_{S}$; Proposition \[prop46.3.2\], under additional assumptions, gives the bound in terms of $|x|_{L}$. \[prop17.2.1\] Assume (\[eq17.1.5\]) holds for some $\varepsilon_{7}> 0$. Then, for large enough $b$ and small enough $a$, $$\label{eq17.2.1} \lambda^{w}(t) \ge(1+\varepsilon_{7})/|X(t)|_{S}$$ for almost all $t$. In this and the previous subsection, we need to employ certain properties of $\Gamma(\bar{H}^{*}_{r}(\cdot))$, which appears in the denominator in (\[eq5.3.1\]). In Lemma \[lem5.2.6\], we state two such properties; the first is employed for Lemma \[lem46.8.2\] and the second is employed for Lemma \[lem17.1.6\]. Recall that $m_r$ is the mean of $H_{r}(\cdot)$. \[lem5.2.6\] For $\Gamma(\cdot)$ as defined in (\[eq5.1.3\]), $$\label{eq5.2.2} \Gamma'(\bar{H}^{*}_{r}(s)) \ge1 + as \qquad\mbox{for all $r$ and $s$}.$$ Moreover, for large enough $b$ and small enough $a$, $$\label{eq5.2.5} \int^{\infty}_{0}\frac{\Gamma(\bar{H}^{*}_{r}(s))}{1+as}\,ds \le (1+\varepsilon_{7}) m_{r}$$ for $\varepsilon_{7}> 0$ satisfying (\[eq17.1.5\]). By (\[eq5.1.3\]) and then (\[eq5.2.1\]), one has, for all $r$ and $s$, $$\begin{aligned} \label{eq5.2.7} \Gamma'(\bar{H}^{*}_{r}(s)) &=& 1+C_{2}\gamma a(\bar {H}^{*}_{r}(s))^{\gamma- 1} \nonumber\\[-8pt]\\[-8pt] % &\ge& 1+C_{2}C_{1}^{\gamma- 1}\gamma a(1+s)^{(1-\gamma)(2+\delta_{1})} \ge1 + as,\nonumber\end{aligned}$$ where the last inequality uses $\gamma\le1/2$ and $C_{2}\ge {C_{1}}^{(1-\gamma)/\gamma}$. This implies (\[eq5.2.2\]). For (\[eq5.2.5\]), we note from (\[eq5.1.3\]) and (\[eq5.2.1\]) that $$\label{eq5.2.3}\hspace*{28pt} \int^{\infty}_{0}\frac{\Gamma(\bar{H}^{*}_{r}(s))}{1+as}\,ds \le \int^{\infty}_{0}\bar{H}^{*}_{r}(s)\,ds + C_{1}^{2} a \int^{\infty}_{0} (1+s)^{-2\gamma}(1+as)^{-1}\,ds.$$ The constant $b$ can be chosen large enough so the first term on the RHS of (\[eq5.2.3\]) is at most $(1+\varepsilon_{7}/2)m_{r}$. Also, by choosing $a>0$ small enough, since the second term can be chosen as close to 0 as desired, by monotone convergence, $$\label{eq5.2.4}\hspace*{28pt} a \int^{\infty}_{0}(1+s)^{-2\gamma}(1+as)^{-1}\,ds = \int^{\infty}_{0} (1+s)^{-(1+2\gamma)}\frac{1+s}{1/a+s}\,ds \to0$$ as $a \searrow0$. So, for large enough $b$ and small enough $a$, (\[eq5.2.5\]) holds. By employing (\[eq5.2.5\]), we obtain upper bounds for $z^{*}_{r} ((0,N_{H_{r}}])$ and $z^{*}_{r}(\mathbb{R}^{+})$ in terms of $|x|_{L}$ and $|x|_{S}$. Inequality (\[eq17.1.4\]) will be crucial for Proposition \[prop17.2.1\]. \[lem17.1.6\] For large enough $b$ and small enough $a$, $$\label{eq17.1.3} z^{*}_{r}((0, N_{H_{r}}]) \le(1 + \varepsilon_{7})w^{-1}_{r}\rho_{r}|x|_{L}$$ and $$\label{eq17.1.4} z^{*}_{r}(\mathbb{R}^{+}) \le(1 + \varepsilon_{7})w^{-1}_{r}\rho_{r}|x|_{S}$$ for all $N$ and $r$, where $\varepsilon_{7}> 0$ is as in (\[eq17.1.5\]). We note that by (\[eq5.3.1\]), $$\label{eq17.1.1}\quad z^{*}_{r}((0,N_{H_{r}}]) = \int^{N_{H_{r}}}_{0}z^{*}_{r}(s)\,ds \le w^{-1}_{r} \nu_{r}|x|_{L} \int^{N_{H_{r}}}_{0} \frac{\Gamma(\bar{H}^{*}_{r} (s))}{1+as}\,ds.$$ By (\[eq5.2.5\]), for large enough $b$ and small enough $a$, the last term in (\[eq17.1.1\]) is at most $$\label{eq17.1.2} (1+\varepsilon_{7})w^{-1}_{r}\nu_{r}m_{r}|x|_{L} = (1+\varepsilon _{7})w^{-1}_{r}\rho_{r}|x|_{L}$$ for all $N$ and $r$, which implies (\[eq17.1.3\]). It follows from (\[eq17.1.3\]) and the definition of $|\cdot|_{S}$ in (\[eq5.3.12\]) that $$\begin{aligned} z^{*}_{r}(\mathbb{R}^{+}) &\le& (1+\varepsilon_{7})w^{-1}_{r}\rho_{r} \biggl[ |x|_{L} + \frac{w_{r}}{\rho_{r}}z^{*}_{r}((N_{H_{r}}, \infty)) \biggr] \\ % &\le& (1+\varepsilon_{7})w^{-1}_{r}\rho_{r}|x|_{S},\end{aligned}$$ which implies (\[eq17.1.4\]). A weaker version of the bound (\[eq17.1.3\]) was used in (\[eq40.1.5\]), where the RHS of (\[eq17.1.3\]) was replaced by $C_{23}|x|_L$, and no additional assumptions on $b$ and $a$ were required. This follows by noting that the second term on the RHS of (\[eq5.2.3\]) does not depend on $a$ (since $a\le1$). We now demonstrate Proposition \[prop17.2.1\]. [Proof of Proposition \[prop17.2.1\]]{} On account of (\[eq17.1.5\]), a feasible protocol is given by assigning service to each nonempty route $r$ at rate $\Lambda_{r,F}\stackrel {\mathrm{def}} {=}(1+\varepsilon_{7})^{2}\rho_{r}$. By (\[eq17.1.4\]), the rate at which each document is served is $$\label{eq17.2.3} \lambda_{r,F} = \frac{(1+\varepsilon_{7})^{2}\rho _{r}}{Z_{r}(t,\mathbb {R}^{+})} = \frac{ (1+\varepsilon_{7})^{2}\rho_{r}}{Z^{*}_{r}(t,\mathbb{R}^{+})} \ge \frac {(1+\varepsilon_{7})w_{r}}{ |X(t)|_{S}}$$ at almost all times $t$. It follows from this and the definition of the weighted max–min fair protocol that $$\lambda^{w}(t) = \min_{r \in\mathcal{R}'} \frac{\lambda_{r}(t)}{w_{r}} \ge\min_{r \in\mathcal{R}'} \frac{\lambda_{r,F}}{w_{r}} \ge\frac{(1+\varepsilon_{7})}{|X(t)|_{S}} \qquad\mbox{for almost all } t,$$ which implies (\[eq17.2.1\]). We apply Proposition \[prop17.2.1\] to derive the following lower bound of $\lambda_{r}(t)$ on $[0,N^{3}]$. We note that, by (\[eq24.4.2\]) of Proposition \[prop24.4.1\] and (\[eq7.5.1\]), for $\omega\in \mathcal{A}_{1}(N^{3})$, $|x|>N^{6}$ and $|x|_{2}\le|x|/2$, $$\label{eq46.3.1} |X(t)|_{L} \le|x|_{L} + C_{26}N^{3} \le(1+\varepsilon)|x|_{L}$$ holds for given $\varepsilon> 0$ and large enough $N$. In the proposition, we will use $$\varepsilon_{8}\stackrel{\mathrm{def}}{=} \biggl[\frac{C_{24}}{8} \biggl(\max_{r}\frac{w_{r}}{\rho_{r}} \biggr)^{-1}\varepsilon_{7}\biggr] \wedge\frac{1}{2}.$$ \[prop46.3.2\] Suppose that (\[eq17.1.5\]) holds for some $\varepsilon_{7}\in(0,1]$, and that $|x| > N^{6}$, with $|x|_{2} \le\varepsilon_{8}|x|$. Then, for large enough $N$ and $b$, and small enough $a$, $$\label{eq46.3.8} \lambda^{w}(t) \ge(1+\varepsilon_{7}/2)/|x|_{L}$$ for almost all $t \in[0,N^{3}]$ on $\omega\in\mathcal{A}_{1}(N^{3})$. It follows from Proposition \[prop17.2.1\] that $$\label{eq46.4.1} \lambda^{w}(t) \ge(1+\varepsilon_{7})/|X(t)|_{S} \qquad\mbox{almost everywhere},$$ for large enough $b$ and small enough $a$. On the other hand, it follows from (\[eq5.3.12\]), (\[eq46.3.1\]), (\[eq39.1.9\]) and (\[eq20.2.1\]) that, since $|x|>N^{6}$ and $|x|_{2} \le\varepsilon_{8}|x|$, $$\begin{aligned} \label{eq46.4.2} |X(t)|_{S} &\le& |X(t)|_{L} + \biggl(\max_{r} \frac{w_{r}} {\rho_{r}} \biggr)Z^{*}_{r}(t,(N_{H_{r}}, \infty)) \nonumber\\[-8pt]\\[-8pt] % &\le& (1+\varepsilon)|x|_{L} + \biggl(\max_{r} \frac{w_{r}} {\rho_{r}} \biggr) \Bigl[|x|_{2} + 2 \Bigl(\max_{r} \nu_{r} \Bigr)N^{3} \Bigr]\nonumber\end{aligned}$$ holds for given $\varepsilon> 0$ and large enough $N$, for all $\omega \in \mathcal{A}_{1}(N^{3})$ and $t \in[0,N^{3}]$. Applying $|x|>N^{6}$, $|x|_2 \le\varepsilon_{8}|x|$ and (\[eq7.5.1\]) to the RHS of (\[eq46.4.2\]) implies that it is at most $$\biggl(1+\varepsilon+\frac{\varepsilon_{7}}{8} \biggr)|x|_{L} + 2 \biggl(\max_{r} \frac{w_{r}}{\rho_{r}} \biggr) \Bigl(\max_{r} \nu_{r} \Bigr)|x|_{L}/C_{24}N^{2}.$$ Consequently, for small enough $\varepsilon> 0$, $$\label{eq46.4.4} |X(t)|_{S} \le(1+\varepsilon_{7}/4)|x|_{L} \qquad\mbox{for all } t \in[0,N^{3}].$$ Together with (\[eq46.4.1\]), this implies (\[eq46.3.8\]). Decomposition of $|X(N^{3})|_{r,s}-|x|_{r,i_{r}(s)}$ {#decomposition-of-xn3_rs-x_ri_rs .unnumbered} ---------------------------------------------------- In this short subsection, we decompose $|X(N^{3})|_{r,s}-|x|_{r,i_{r}(s)}$ into several parts, one of which contains the factor $Z^{*}_{r}(N^{3},s) - z^{*}_{r}(i_{r}(s))$. In Proposition \[prop44.1.2\], we then obtain upper bounds on this factor. In this and the remaining subsection, the estimates need to be more precise than in previous sections in order to make use of the subcriticality of $X(\cdot)$. The decomposition that was referred to above is given by $$\begin{aligned} \label{eq42.1.2} &&|X(N^{3})|_{r,s} - |x|_{r,i_{r}(s)} \nonumber\\ &&\qquad= \frac {w_{r}(1+as)(Z^{*}_{r}(N^{3},s) - z^{*}_{r}(i_{r}(s)))}{\nu_{r}\Gamma(\sigma_{r})} \\ % &&\qquad\quad{} - |x|_{r,i_{r}(s)}\frac{1+as}{1+ai_{r}(s)} \frac{\Gamma(\sigma _{r}) - \Gamma ({\sigma_{r}'})}{\Gamma(\sigma_{r})} - \frac{aw_{r}\Delta_{r}z^{*}_{r} (i_{r}(s))}{\nu_{r}\Gamma({\sigma_{r}'})}, \nonumber\end{aligned}$$ and holds for $s \le N_{H_{r}}$ and $\Delta_{r} \le1/b^{3}$. It will be employed in Corollary \[col46.2.1\]. Here and later on, we abbreviate, setting $\sigma_{r} = \bar{H}^{*}_{r}(s)$ and ${\sigma _{r}'} = \bar{H}^{*}_{r}(i_{r}(s))$. \[One can check that (\[eq42.1.2\]) holds as given, without employing either $s_{N}$ or $i_{r}(s)_{N}$, as in (\[eq5.3.1\]), since $i_{r}(s) = s + \Delta_{r} \le N_{H_{r}}+1$, and hence $s_{N} = s$ and $i_{r}(s)_{N} = i_{r}(s)$.\] To apply the bound (\[eq20.2.2\]) on $\omega\in\mathcal {A}_{2}(N^{3})$ and derive an upper bound on $Z^{*}_{r}(N^{3},s) - z^{*}_{r}(i_{r}(s))$, we need to select a $v_{j}$ from among $v_{0}, \ldots, v_{J}$, as given by (\[eq20.1.6\]). For this, we denote by $v(s)$ the value $v_{j}$ with $$\label{eq44.2.1} v_{j} \in\bigl[i_{r}(s), i_{r}(s) + 1/b^{3}\bigr).$$ Under $s \le N_{H_{r}}$ and $\Delta_{r} \le1/b^{3}$, such a $v(s)$ exists. \[prop44.1.2\] Suppose $\omega\in\mathcal{A}_{2}(N^{3})$, for some $N$ and $b$, with $b$ as in (\[eq5.1.4\]). Then, $$\label{eq44.1.3}\qquad Z^{*}_{r}(N^{3},s) - z^{*}_{r}(i_{r}(s)) \le(1 + \varepsilon_{5})(1 + 4/b^{2}) \nu_{r} [ h^{*}_{r}(v(s))N^{3} \vee N^{3\eta} ]$$ for all $r$ and $s$ with $\Delta_{r} \le1/b^{3}$ and $s \le N_{H_{r}}$, where $\varepsilon_{5}>0$ is as in (\[eq20.2.2\]) and $v(s)$ is given by (\[eq44.2.1\]). By (\[eq39.1.5\]), the LHS of (\[eq44.1.3\]) is at most $Z^{A,*}_{r} (N^{3},s)$. For $s \le N_{H_{r}}$, this equals $$\begin{aligned} \label{eq44.1.5}\qquad\quad \sum^{A_{r}(N^{3})}_{k=1}\phi\bigl(s - S^{1}_{r}(k) + S^{2}_{r}(N^{3},k)\bigr) &\le& e^{2/b^{2}} \sum^{A_{r}(N^{3})}_{k=1}\phi\bigl(v(s) - S^{1}_{r}(k)\bigr) \nonumber\\[-8pt]\\[-8pt] &\le&(1 + 4/b^{2}) \sum^{A_{r}(N^{3})}_{k=1}\phi\bigl(v(s) - S^{1}_{r}(k)\bigr).\nonumber\end{aligned}$$ To see (\[eq44.1.5\]), we note that since $S^{2}_{r}(N^{3}, k) \le \Delta_{r} \le1/b^{3}$, $$\label{eq44.2.5}\hspace*{28pt} v_{j} - S^{1}_{r}(k) \in[s - S^{1}_{r}(k) + S^{2}_{r}(N^{3},k), s - S^{1}_{r}(k) + S^{2}_{r}(N^{3},k) + 2/b^{3}].$$ Together with the second half of (\[eq5.1.10\]), this implies the first inequality. The second inequality follows by expanding $e^{2/b^{2}}$. Since $\omega\in\mathcal{A}_{2}(N^{3})$, the RHS of (\[eq44.1.3\]) then follows by applying (\[eq20.2.2\]). In the next subsection, we will also employ the following bound on $h^{*}_{r} (s_{2}) - h^{*}_{r}(s_{1})$ for $s_{1} \le s_{2}$. \[prop46.5.1\] For any $r$, $s_{1} \le s_{2}$ and $b$, $$\label{eq46.5.2} h^{*}_{r}(s_{2}) - h^{*}_{r}(s_{1}) \le eb^{2}(s_{2} - s_{1})\bar {H}^{*}_{r}(s_{1}).$$ Since $h^{*}_{r}(s) = \int^{\infty}_{0}\phi(s-s')\,dH_{r}(s')$ for each $s$, the LHS of (\[eq46.5.2\]) equals $$\label{eq46.5.3} \int^{\infty}_{0}\bigl(\phi(s_{2}-s') - \phi(s_{1}-s')\bigr)\,dH_{r}(s').$$ By the first part of (\[eq5.1.10\]) and the definition of $\phi (\cdot)$, $\phi'(s) \le b^{2}$ for all $s$ and $\phi(\cdot)$ is decreasing on $[1/b, \infty)$. So, (\[eq46.5.3\]) is at most $$\begin{aligned} \label{eq46.5.4}\hspace*{28pt} \int^{\infty}_{0}b^{2}(s_{2}-s_{1})1\{s'>s_{1}-1/b\}\,dH(s') &\le& b^{2} (s_{2}-s_{1})\bar{H}_{r}(s_{1}-1/b) \nonumber\\ % &\le& b^{2}(s_{2}-s_{1})\bar{H}^{*}_{r}(s_{1}-1/b)\\ &\le& eb^{2}(s_{2}-s_{1}) \bar{H}^{*}_{r}(s_{1}).\nonumber\end{aligned}$$ Upper bounds on $|X(N^{3})|_{r,s}$ {#upper-bounds-on-xn3_rs .unnumbered} ---------------------------------- In this subsection, we employ the previous two subsections to obtain upper bounds on $|X(N^{3})|_{r,s}-|x|_{L}$ for $\omega\in\mathcal {A}(N^{3})$, when $s \le N_{H_{r}}$ and $\Delta_{r} \le1/b^{3}$. Our main result is the following proposition. As elsewhere in this paper, we are assuming that $aN\ge1$. \[prop48.1.1\] Suppose that (\[eq17.1.5\]) holds for some $\varepsilon_{7}\in[0,1]$ and that $|x| > N^{6}$, with $|x|_{2} \le\varepsilon_{8}|x|$, where $\varepsilon_{8}$ is specified below (\[eq46.3.1\]). Then, for large enough $N$ and $b$, and small enough $a$, $$\label{eq48.1.2} |X(N^{3})|_{r,s} - |x|_{L} \le-\tfrac{1}{2}w_{r}N^{2}$$ for $\omega\in\mathcal{A}(N^{3})$, and all $r$ and $s$ with $\Delta _{r}\le 1/b^{3}$ and $s \le N_{H_{r}}$. Our main step in demonstrating Proposition \[prop48.1.1\] will be to demonstrate the following proposition. \[prop46.1.1\] Under the same assumptions as in Proposition \[prop48.1.1\], $$\begin{aligned} \label{eq46.1.2}\hspace*{32pt} \frac{w_{r}(1+as)(Z^{*}_{r}(N^{3},s) - z^{*}_{r}(i_{r}(s)))}{\nu_{r} \Gamma (\sigma_{r})} &\le& |x|_{L} \cdot\frac{1+as}{1+ai_{r}(s)} \cdot \frac{\Gamma (\sigma_{r}) - \Gamma(\sigma_{r}')}{\Gamma(\sigma_{r})}\nonumber\\[-8pt]\\[-8pt] % &&{} + \frac{C_{30}w_{r}N^{3}}{ab(1+as)} + C_{31}w_{r}N^{3/2}\nonumber\end{aligned}$$ for appropriate $C_{30}$ and $C_{31}$ not depending on $w, N, a, b, r$ or $s$. In order to demonstrate Proposition \[prop46.1.1\], we note that, on account of Proposition \[prop44.1.2\], the LHS of (\[eq46.1.2\]) is, under the assumptions for the latter proposition, at most $$\begin{aligned} \label{eq46.2.3} &&d_{r}(s) \bigl(h^{*}_{r}(v(s))N^{3}\vee N^{3\eta} \bigr) \nonumber\\ &&\qquad\le d_{r}(s) \Bigl(\inf_{s^{\prime} \in[s,i_{r}(s)]}h^{*}_{r}(s') \Bigr)N^{3} \\ % &&\qquad\quad{} + d_{r}(s) \Bigl(h^{*}_{r}(v(s)) - \inf_{s^{\prime} \in[s,i_{r}(s)]} h^{*}_{r}(s') \Bigr)N^{3} + d_{r}(s)N^{3\eta},\nonumber\end{aligned}$$ where $$d_{r}(s) = (1+\varepsilon_{5})(1+4/b^{2})w_{r}(1+as)/\Gamma(\sigma_{r}).$$ We will show in Lemmas \[lem46.8.2\], \[lem46.5.5\] and \[lem46.6.3\] that each of the three terms on the RHS of (\[eq46.2.3\]) is bounded above by the corresponding term on the RHS of (\[eq46.1.2\]). Proposition \[prop46.1.1\] then follows. We first show Lemma \[lem46.8.2\], which applies to the first term on the RHS of (\[eq46.2.3\]), and should be thought of as the “main term” there. \[lem46.8.2\] Under the same assumptions as in Proposition \[prop48.1.1\], $$\label{eq46.8.3} d_{r}(s) \Bigl(\inf_{s^{\prime} \in[s,i_{r}(s)]}h^{*}_{r}(s') \Bigr) N^{3} \le|x|_{L}\frac{1+as}{1+ai_{r}(s)} \cdot\frac{\Gamma (\sigma_{r})-\Gamma (\sigma_{r}')}{\Gamma(\sigma_{r})}.$$ It follows from Proposition \[prop46.3.2\] that $$\label{eq46.8.4} \lambda^{w}(t) \ge(1+\varepsilon_{7}/2)/|x|_{L} \qquad\mbox{for almost all } t \in [0,N^{3}],$$ for large enough $N$ and $b$, and small enough $a$, and therefore $$\label{eq46.8.5} \Delta_{r} \ge(1+\varepsilon_{7}/2)w_{r}N^{3}/|x|_{L} \qquad\mbox{for all } r.$$ Consequently, the LHS of (\[eq46.8.3\]) is at most $$\begin{aligned} \label{eq46.8.6} &&d_{r}(s)|x|_{L} \Bigl(\inf_{s^{\prime} \in[s,i_{r}(s)]}h^{*}_{r}(s') \Bigr) \Delta_{r}/w_{r}(1+\varepsilon_{7}/2) \nonumber\\[-8pt]\\[-8pt] % &&\qquad\le d_{r}(s)|x|_{L} \bigl(\bar{H}^{*}_{r}(s) - \bar{H}^{*}_{r}(i_{r}(s)) \bigr)/ w_{r}(1+\varepsilon_{7}/2).\nonumber\end{aligned}$$ This last quantity can be rewritten as $$\begin{aligned} \label{eq46.8.7} && \frac{(1+\varepsilon_{5})(1+4/b^{2})}{1+\varepsilon_{7}/2} \cdot|x|_{L} \cdot\frac{1+as} {1+ai_{r}(s)} \cdot\frac{\Gamma(\sigma_{r})-\Gamma(\sigma_{r}')} {\Gamma(\sigma_{r})} \nonumber\\[-8pt]\\[-8pt] &&\qquad{}\times \frac{1+ai_{r}(s)}{ ({\Gamma (\sigma_{r}) - \Gamma(\sigma_{r}')})/({\sigma_{r} - \sigma_{r}'} )}.\nonumber\end{aligned}$$ We proceed to bound the components of (\[eq46.8.7\]). Since $\varepsilon_{5}\le\varepsilon_{7}/4$, one has for large enough $b$, depending on $\varepsilon_{7}$, that $$\label{eq46.8.8} \frac{(1+\varepsilon_{5})(1+4/b^{2})}{1+\varepsilon_{7}/2} \le(1+1/b^{2})^{-1}.$$ Since $\Gamma(\cdot)$ is concave and $\sigma_{r}>\sigma_{r}'$, $$\label{eq46.8.9} \frac{\Gamma(\sigma_{r}) - \Gamma(\sigma_{r}')}{\sigma_{r} - \sigma_{r}'} \ge \Gamma'(\sigma_{r}) \ge1 + as,$$ with the second inequality holding on account of (\[eq5.2.2\]). So the last term in (\[eq46.8.7\]) is at most $$\label{eq46.8.10} \frac{1+ai_{r}(s)}{1+as} = 1 + \frac{a\Delta_{r}}{1+as} \le1 + 1/b^{3},$$ where the inequality uses $\Delta_{r} \le1/b^{3}$. Consequently, (\[eq46.8.7\]) is, for large $b$, at most $$(1+1/b^{2})^{-1}(1 + 1/b^{3})|x|_{L}\frac{1+as}{1+ai_{r}(s)} \cdot \frac{\Gamma(\sigma_{r}) - \Gamma(\sigma_{r}')}{\Gamma(\sigma_{r})},$$ which is at most as large as the RHS of (\[eq46.8.3\]). This implies the lemma. We next demonstrate Lemma \[lem46.5.5\], which applies to the second term on the RHS of (\[eq46.2.3\]). \[lem46.5.5\] For all $r$ and $s$ with $\Delta_{r} \le1/b^{3}$ and $s \le N_{H_{r}}$, $$\label{eq46.5.6} d_{r}(s) \Bigl(h^{*}_{r}(v(s)) - \inf_{s^{\prime} \in[s,i_{r}(s)]} h^{*}_{r}(s') \Bigr)N^{3} \le\frac{C_{30}w_{r}N^{3}}{ab(1+as)}$$ for appropriate $C_{30}$ not depending on $w, N, a, b, r$ or $s$. Since $v(s)-s \le2/b^{3}$, it follows from Proposition \[prop46.5.1\] that the LHS of (\[eq46.5.6\]) is at most $$\label{eq46.6.1} (1+\varepsilon_{5})(1+4/b^{2})\frac{2e b^{2}}{b^{3}}w_{r}N^{3}\bar {H}^{*}_{r}(s) \frac{(1+as)}{\Gamma(\sigma_{r})}.$$ On account of (\[eq5.1.3\]), since $\gamma\le\delta_{1}/4$, $b \ge2$ and $\varepsilon_{5}\le1$, this is at most $$\begin{aligned} \label{eq46.6.2}\qquad \frac{24w_{r}}{C_{2}ab}N^{3}(\bar{H}^{*}_{r}(s))^{1-\gamma}(1+as) &\le& \frac{24C_{1}w_{r}}{C_{2}ab}N^{3}(1+as)^{1-(1-\gamma)(2+\delta_{1})} \nonumber\\[-8pt]\\[-8pt] &\le&\frac{24 C_{1}w_{r}N^{3}}{C_{2}ab(1+as)} .\nonumber\end{aligned}$$ Recall that $C_{1}$ and $C_{2}$ do not depend on $w, N, a, b, r$ or $s$. The RHS of (\[eq46.5.6\]) follows from this last term by setting $C_{30}= 24 C_{1}/ C_{2}$. We now demonstrate Lemma \[lem46.6.3\], which applies to the third term on the RHS of (\[eq46.2.3\]). \[lem46.6.3\] For all $s \le N_{H_{r}}$, $$\label{eq46.6.4} d_{r}(s)N^{3\eta} \le C_{31}w_{r}N^{3/2}$$ for appropriate $C_{31}$ not depending on $w, N, a, b, r$ or $s$. Since $s \le N_{H_{r}} \le N$, $\gamma\le1/24$, $\eta\le1/12$, $b \ge2$ and $\varepsilon_{5}\le1$, it follows from (\[eq5.1.3\]) and (\[eq90.3.1\]) that the LHS of (\[eq46.6.4\]) is at most $$\label{eq46.7.1} \frac{4w_{r}N^{3\eta}(1+aN)}{C_{2}a \bar {H}^{*}_{r}(N_{H_{r}})^{\gamma}} \le \frac{4}{C_{2}a}w_{r}N^{1/2}(1+aN).$$ Since $aN \ge1$, this is at most $8w_{r}N^{3/2}/C_{2}$, which gives the RHS of (\[eq46.6.4\]) for $C_{31}= 8/C_{2}$. Proposition \[prop46.1.1\] follows by applying Lemmas \[lem46.8.2\], \[lem46.5.5\] and \[lem46.6.3\] to (\[eq46.2.3\]). We will apply the following corollary of the proposition to Proposition \[prop48.1.1\]. The corollary combines the inequality (\[eq46.1.2\]) with (\[eq42.1.2\]). \[col46.2.1\] Under the same assumptions as in Propositions \[prop48.1.1\] and \[prop46.1.1\], $$\label{eq46.2.2}\qquad |X(N^{3})|_{r,s}-|x|_{L} \le\frac{C_{30}w_{r}N^{3}}{ab(1+as)} + C_{31} w_{r}N^{3/2} - \frac{aw_{r}\Delta_{r}z^{*}_{r}(i_{r}(s))}{\nu_{r}\Gamma(\sigma_{r}')}$$ for appropriate $C_{30}$ and $C_{31}$ not depending on $w$, $N$, $a$, $b$, $r$ or $s$. The first term on the RHS of (\[eq46.1.2\]) of Proposition \[prop46.1.1\] is at most $$\label{eq46.1.3} |x|_{r,i_{r}(s)}\frac{1+as}{1+ai_{r}(s)} \cdot\frac{\Gamma(\sigma _{r})-\Gamma (\sigma_{r}')}{\Gamma(\sigma_{r})} + |x|_{L} - |x|_{r,i_{r}(s)}$$ since the coefficients of $|x|_{r,i_{r}(s)}$ in the first term in (\[eq46.1.3\]) are at most 1. Substituting (\[eq46.1.3\]) into (\[eq46.1.2\]) and then applying the resulting inequality to the RHS of (\[eq42.1.2\]), we note that the term on the LHS of (\[eq46.1.2\]) is the first term on the RHS of (\[eq42.1.2\]) and the first term in (\[eq46.1.3\]) is the negative of the second term on the RHS of (\[eq42.1.2\]). After the resulting cancellation, the last two terms on the RHS of (\[eq46.1.2\]), together with the last term on the RHS of (\[eq42.1.2\]), give the RHS of (\[eq46.2.2\]). In order to show Proposition \[prop48.1.1\], we will need a lower bound on the last term on the RHS of (\[eq46.2.2\]) and an upper bound on each of the first two terms. In the following lemma, we obtain the former. Note that the assumptions in the lemma are those of Proposition \[prop46.3.2\], with the additional assumption that $$\label{eq48.1.8} |x|_{r,i_{r}(s)} \ge|x|_{L}/(1+\varepsilon_{7}/2) \qquad\mbox{for some } s \le N_{H_{r}}.$$ \[lem48.1.3\] Suppose that (\[eq17.1.5\]) holds for some $\varepsilon_{7}\in(0,1]$, that $|x|>N^{6}$ with $|x|_{2} \le\varepsilon_{8}|x|$, and that (\[eq48.1.8\]) is satisfied for a given $s$. Then, for large enough $N$ and $b$, and small enough $a$, $$\label{eq48.1.4} \frac{\Delta_{r}z^{*}_{r}(i_{r}(s))}{\nu_{r}\Gamma(\sigma_{r}')} \ge\frac{N^{3}} {1+ai_{r}(s)} \qquad\mbox{on } \omega\in\mathcal{A}_{1}(N^{3}).$$ By Proposition \[prop46.3.2\], $$\lambda^{w}(t) \ge(1+\varepsilon_{7}/2)/|x|_{L} \qquad\mbox{for almost all } t \in [0,N^{3}].$$ Consequently, $$\label{eq48.2.1} \Delta_{r} \ge(1+\varepsilon_{7}/2)w_{r}N^{3}/|x|_{L} \qquad\mbox{for all } r.$$ It follows from (\[eq48.2.1\]), (\[eq5.3.1\]) and (\[eq48.1.8\]) that the LHS of (\[eq48.1.4\]) is at least $$\label{eq48.2.2}\hspace*{28pt} \frac{(1+\varepsilon_{7}/2)w_{r}N^{3}z^{*}_{r}(i_{r}(s))}{\nu _{r}\Gamma (\sigma_{r}')|x|_{L}} = \frac{(1+\varepsilon _{7}/2)N^{3}|x|_{r,i_{r}(s)}}{(1+ai_{r}(s))|x|_{L}} \ge \frac{N^{3}}{1+ai_{r}(s)}.$$ We now apply Corollary \[col46.2.1\] and Lemma \[lem48.1.3\] to demonstrate Proposition \[prop48.1.1\]. [Proof of Proposition \[prop48.1.1\]]{} We will consider two cases for a given $s \le N_{H_{r}}$, depending on whether (\[eq48.1.8\]) holds. Suppose it does. Then, by Lemma \[lem48.1.3\], $$\label{eq48.2.3} \frac{aw_{r}\Delta_{r}z^{*}_{r}(i_{r}(s))}{\nu_{r}\Gamma(\sigma _{r}')} \ge \frac{aw_{r}N^{3}}{1+ai_{r}(s)},$$ which is a lower bound for the third term on the RHS of (\[eq46.2.2\]). On the other hand, if one chooses $b \ge8C_{30}/a^{2}$, then, since $a \le1$ and $\Delta_{r}\le1/b^{3} \le1$, the first term on the RHS of (\[eq46.2.2\]) satisfies $$\label{eq48.2.6} \frac{C_{30}w_{r}N^{3}}{ab(1+as)} \le\frac{aw_{r}N^{3}}{4(1+ai_{r}(s))},$$ which is 1/4 of the RHS of (\[eq48.2.3\]). Since $s \le N_{H_{r}}$, $i_{r}(s) \le N+1$. So, the sum of the first and third terms on the RHS of (\[eq46.2.2\]) is, for large $N$, at most $$\label{eq48.2.7} -\frac{3aw_{r}N^{3}}{4(1+ai_{r}(s))} \le-\frac{5}{8}w_{r}N^{2}.$$ The second term on the RHS of (\[eq46.2.2\]) satisfies $$C_{31}w_{r}N^{3/2} \le\tfrac{1}{8}w_{r}N^{2}$$ for large $N$. Combining this with (\[eq48.2.7\]), one obtains from Corollary \[col46.2.1\] that $$|X(N^{3})|_{r,s} - |x|_{L} \le- \tfrac{1}{2}w_{r}N^{2},$$ which implies (\[eq48.1.2\]) under (\[eq48.1.8\]). When (\[eq48.1.8\]) fails for $s$, one has, for large $N$, $$\begin{aligned} \label{eq48.3.1} |X(N^{3})|_{r,s} - |x|_{L} &=& \bigl(|X(N^{3})|_{r,s} - |x|_{r,i_{r}(s)}\bigr) - \bigl(|x|_{L} - |x|_{r,i_{r}(s)}\bigr) \nonumber\\ % &\le& |X(N^{3})|_{r,s} - |x|_{r,i_{r}(s)} - \tfrac{1}{4}\varepsilon_{7}|x|_{L} \nonumber\\[-8pt]\\[-8pt] &\le& |X^{A}(N^{3})|_{r,s} - \tfrac{1}{4} C_{24}\varepsilon_{7}|x| \nonumber\\ % &\le& C_{26}N^{3} - \tfrac{1}{4} C_{24}\varepsilon_{7}N^{6} \le-N^{5}, \nonumber\end{aligned}$$ where, in the second inequality, we applied (\[eq39.1.7\]) and (\[eq7.5.1\]), and in the third inequality, we applied (\[eq24.4.7\]) of Proposition \[prop24.4.1\] and $|x|>N^{6}$. This implies (\[eq48.1.2\]) when (\[eq48.1.8\]) fails. Conclusion: Upper bounds on $E_{x}[|X(N^{3})|_{L}]$ {#sec10} =================================================== In the preceding four sections, we obtained upper bounds on $$|X(N^{3})|_{r,s}-|x|_{L}\quad\mbox{and}\quad E_{x} [|X(N^{3})|_{L} - |x|_{L}; \mathcal{A}(N^{3})^{c} ]$$ under various assumptions. In Propositions \[prop21.1.1\] and \[prop21.1.3\] we showed that$P_{x}(\mathcal {A}(N^{3})^{c})$ and the corresponding expectation $E_{x}[|X(N^{3})|_{L}$; $\mathcal{A}(N^{3})^{c}]$ are small. In Proposition \[prop40.2.3\], we showed that the expected value of $|X(N^{3})|_{r,s}-|x|_{L}$ is small for $s>N_{H_{r}}$. In Sections \[sec8\] and \[sec9\], we obtained pathwise estimates on $\mathcal{A}(N^{3})$ when $s \le N_{H_{r}}$, depending on whether $\Delta_{r}> 1/b^{3}$ or $\Delta_{r} \le1/b^{3}$. Proposition \[prop41.2.1\] gives an upper bound in the former subcase and Proposition \[prop48.1.1\] gives an upper bound in the latter subcase. Except for Propositions \[prop21.1.1\] and \[prop21.1.3\], we assumed that $|x|>N^{6}$; for the different results, we also required various side conditions. We tie these results together in Proposition \[prop50.1.1\] to obtain inequality (\[eq5.4.2\]) that was cited earlier. We do this in several steps, first combining the results for $s \le N_{H_{r}}$, then combining these with Proposition \[prop40.2.3\] for $s>N_{H_{r}}$, and lastly including the bound from Proposition \[prop21.1.1\] on $\mathcal{A}(N^{3})^{c}$. The first two steps are done in Proposition \[prop48.3.2\]. As elsewhere in the paper, $aN \ge1$ is assumed. \[prop48.3.2\] Suppose that (\[eq17.1.5\]) holds for some $\varepsilon_{7}\in(0,1]$ and that $|x| > N^{6}$, with $|x|_{2} \le\varepsilon_{8}|x|$, where $\varepsilon_{8}$ is specified below (\[eq46.3.1\]). Then, for large enough $N$ and $b$, and small enough $a$, $$\label{eq48.3.3} |X(N^{3})|_{r,s} - |x|_{L} \le-\tfrac{1}{2}w_{r}N^{2}$$ for $\omega\in\mathcal{A}(N^{3})$, all $r$, and $s$ with $s \le N_{H_{r}}$. Moreover, for large enough $N$ and $b$, and small enough $a$, $$\begin{aligned} \label{eq48.4.2} &&E_{x} [|X(N^{3})|_{L} - |x|_{L}; \mathcal{A}(N^{3}) ] \nonumber\\[-8pt]\\[-8pt] % &&\qquad\le C_{3}(|x|_{K}/|x|)N^{3}- \biggl(\frac{1}{4}\min_{r} w_{r} \biggr)N^{2} \bigl(2P_{x}(\mathcal{A}(N^{3})) - 1\bigr).\nonumber\end{aligned}$$ Note that the assumptions for the first half of Proposition \[prop48.3.2\] are the same as for Proposition \[prop48.1.1\], except that the restriction that $\Delta_{r} \le1/b^{3}$ has been removed. [Proof of Proposition \[prop48.3.2\]]{} Inequality (\[eq48.1.2\]) in Proposition \[prop48.1.1\] covers the case where $\Delta_{r} \le1/b^{3}$; (\[eq41.2.2\]) of Proposition \[prop41.2.1\] covers the case where $\Delta_{r}>1/b^{3}$. Together, they imply (\[eq48.3.3\]). In order to demonstrate (\[eq48.4.2\]), we partition $\mathcal {A}(N^{3})$ into $G \cup H$, with $$G = \Bigl\{ \omega\dvtx|X(N^{3})|_{L} = {\sup_{r, s > N_{H_{r}}}} |X(N^{3})|_{r,s} \Bigr\}.$$ Applying Proposition \[prop40.2.3\] to this $G$, with $\varepsilon_{3}= (\min_{r} w_{r})/2$, and applying (\[eq48.3.3\]) on $H$, it follows that the LHS of (\[eq48.4.2\]) equals $$\begin{aligned} \label{eq48.4.3}\qquad && E_{x} \Bigl[{\sup_{r, s > N_{H_{r}}}}|X(N^{3})|_{r,s} - |x|_{L}; G \Bigr] + E_{x} \Bigl[{\sup_{r, s \le N_{H_{r}}}} |X(N^{3})|_{r,s} - |x|_{L}; H \Bigr] \nonumber\\ % &&\qquad \le C_{3}(|x|_{K}/|x|)N^{3}- \biggl(\frac{1}{4}\min_{r} w_{r} \biggr)N^{2} \bigl(2P_{x}(G) + 2P_{x}(H) - 1\bigr) \\ % &&\qquad = C_{3}(|x|_{K}/|x|)N^{3}- \biggl(\frac{1}{4}\min_{r} w_{r} \biggr)N^{2} \bigl(2P_{x}(\mathcal{A}(N^{3})) - 1\bigr).\nonumber\end{aligned}$$ This implies (\[eq48.4.2\]). We now obtain our desired result, Proposition \[prop50.1.1\], which gives upper bounds on $E_{x}[|X(N^{3})|_{L}] - |x|_{L}$. The first part of the proposition applies to all $x$; the second part requires that $|x|>N^{6}$. \[prop50.1.1\] Suppose that (\[eq17.1.5\]) holds for some $\varepsilon_{7}\in(0,1]$. (a) For large enough $N$, $$\label{eq50.6.1} E_{x}[|X(N^{3})|_{L}] - |x|_{L} \le C_{3}N^{3} \qquad\mbox{for all $x$.}$$ (b) For $|x|>N^{6}$, large enough $N$ and $b$, and small enough $a$, $$\label{eq50.1.2} E_{x}[|X(N^{3})|_{L}] - |x|_{L} \le C_{3}(|x|_{K}/|x|)N^{3} - \biggl(\frac{1}{4} \min_{r} w_{r} \biggr)N^{2}.$$ In both parts, $C_{3}$ is an appropriate constant that does not depend on $x$ or $N$. We first show (a). By (\[eq39.2.1\]) and (\[eq24.4.2\]) of Proposition \[prop24.4.1\], $$\label{eq50.2.1} |X(N^{3})|_{L} - |x|_{L} \le C_{26}N^{3}$$ for all $\omega\in\mathcal{A}_{1}(N^{3})$ and appropriate $C_{26}> 0$ not depending on $x$, $N$, or $\omega$. Together with Proposition \[prop21.1.1\], this implies $$\begin{aligned} \label{eq50.6.2} &&E_{x}[|X(N^{3})|_{L}] - |x|_{L} \nonumber\\ % &&\qquad= E_{x}[|X(N^{3})|_{L}; \mathcal{A}(N^{3})] + E_{x} [|X(N^{3})|_{L}; \mathcal{A}(N^{3})^{c}] - |x|_{L} \\ % &&\qquad\le C_{26}N^{3} + N^{3}e^{-C_{10}N^{3\eta}} \le2C_{26}N^{3}\nonumber\end{aligned}$$ for large enough $N$. For $C_{3}\ge2C_{26}$, this implies (\[eq50.6.1\]). For (b), we suppose first that $|x|_{2} \le\varepsilon_{8}|x|$, where $\varepsilon_{8}$ is given below (\[eq46.3.1\]). Then, (\[eq48.4.2\]) of Proposition \[prop48.3.2\], together with Propositions \[prop21.1.1\] and \[prop21.1.3\], implies that the LHS of (\[eq50.1.2\]) is equal to $$\begin{aligned} \label{eq50.1.4} && E_{x}[|X(N^{3})|_{L}; \mathcal{A}(N^{3})]+E_{x}[|X(N^{3})|_{L}; \mathcal{A} (N^{3})^{c}] - |x|_{L} \nonumber\\ % &&\qquad \le C_{3}(|x|_{K}/|x|)N^{3} - \biggl(\frac{1}{4}\min_{r} w_{r} \biggr)N^{2}\bigl(2P_{x}(\mathcal{A}(N^{3}))-1\bigr) \nonumber\\[-8pt]\\[-8pt] % &&\qquad\quad{} + N^{3}e^{-C_{10}N^{3\eta}} \nonumber\\ % &&\qquad \le C_{3}(|x|_{K}/|x|)N^{3} - \biggl(\frac{1}{4}\min_{r} w_{r} \biggr)N^{2} \nonumber\end{aligned}$$ for large $N$ and $b$, and small $a$. This implies (\[eq50.1.2\]) for $|x|_{2} \le\varepsilon_{8}|x|$. Assume now that $|x|_{2}>\varepsilon_{8}|x|$. Choosing $C_{3}\ge (2C_{26}+\frac{1}{4} \min_{r}w_{r} ) /\varepsilon_{8}$, it follows from (\[eq50.6.2\]) that, for large $N$, $$\begin{aligned} E_{x}[|X(N^{3})|_{L}] - |x|_{L} &\le& \biggl(C_{3}\varepsilon_{8}- \frac{1}{4} \min_{r}w_{r} \biggr)N^{3} \\ % &\le& C_{3}(|x|_{2}/|x|)N^{3} - \biggl(\frac{1}{4}\min_{r} w_{r} \biggr)N^{3} \\ % &\le& C_{3}(|x|_{K}/|x|)N^{3} - \biggl(\frac{1}{4}\min_{r} w_{r} \biggr)N^{3}.\end{aligned}$$ This implies (\[eq50.1.2\]) for $|x|_{2} > \varepsilon_{8}|x|$. Acknowledgments {#acknowledgments .unnumbered} =============== The author thanks the referees for a detailed reading of the paper and for helpful comments. [99]{} (). . . , . (). . In . , . (). , ed. , . (). . . , . , (). . . (). . . , , (). . . (). . . (). . . (). . In . . , . (). . . , . (). . , .

--- abstract: 'Crystalline topological phases have recently attracted a lot of experimental and theoretical attention. Key advances include the complete elementary band representation analyses of crystalline matter by symmetry indicators and the discovery of higher-order hinge and corner states. However, current classification schemes of such phases are either implicit or limited in scope. We present a new scheme for the explicit classification of crystalline topological insulators and superconductors. These phases are protected by crystallographic point group symmetries and are characterized by bulk topological invariants. The classification paradigm generalizes the Clifford algebra extension process of each Altland-Zirnbauer symmetry class and utilizes algebras which incorporate the point group symmetry. Explicit results for all point group symmetries of three-dimensional crystals are presented as well as for all symmorphic layer groups of two-dimensional crystals. We discuss future extensions for treatment of magnetic crystals and defected or higher-dimensional systems as well as weak and fragile invariants.' author: - Eyal Cornfeld - Adam Chapman title: '[Classification of Crystalline Topological Insulators and Superconductors with Point Group Symmetries]{}' --- Introduction {#sec:intro} ============ Over the past decades, the discovery of topological phases such as topological insulators, superconductors, and semimetals, have transformed our understanding of condensed matter physics [@Hasan2010Colloquium; @Qi2011Topological; @Chiu2016Classification; @Armitage2018Weyl]. Study of such phenomena is being extensively used as a new tool for classifying phases of matter which can not be distinguished by broken symmetries. Many topological insulators arise in systems of weakly interacting fermions which feature a bulk gap, and topological superconductors are similarly described by the fermionic quasiparticle excitations of a BCS superconductor. Key aspects of these topological phases are the symmetries possessed by the material in question. If a gapped material possesses only a charge conservation symmetry, it may only realize an integer quantum Hall effect [@Thouless1982Quantized; @Avron1983Homotopy] characterized by a ${\mathbb{Z}}$ topological index. However, in the presence of other non-spatial symmetries, such as time-reversal or particle-hole symmetry, many other topological phases are possible [@Kane2005Topological; @Qi2009Time; @Schnyder2008Classification]. Examples of such phases include, the ${\mathbb{Z}}_2$ two-dimensional and three-dimensional topological insulators [@Hasan2010Colloquium; @Qi2011Topological; @Kane2005Quantum; @Kane2005Topological; @Konig2007Quantum; @FuKaneMele2007Topological; @Moore2007Topological; @Roy2009Topological], and one-dimensional and two-dimensional topological $p$-wave superconductors [@kitaev2001unpaired; @Read2000Paired]. Many materials in nature are however also characterized by spatial crystalline symmetries which emanate from their crystallographic structure. Over the past several years, these space group and point group symmetries of crystalline matter have been theoretically predicted to host a large and diverse variety of topological phases [@Mong2010Antiferromagnetic; @Fu2011Topological; @slager2013space; @Chiu2013Classification; @Benalcazar2014Classification; @Varjas2015Bulk; @Shiozaki2015Z2; @Cho2015Topological; @Yang2015Topological; @wang2016hourglass; @Ezawa2016Hourglass; @Varjas2017Space; @Yang2017Topological; @Wieder2018Wallpaper; @Bouhon2017Global; @bouhon2017bulk; @Kruthoff2017Topological; @kruthoff2017topology]. Many of these proposed topological phases have been measured in various experiments [@hsieh2012topological; @dziawa2012topological; @tanaka2012experimental; @xu2012observation] in materials such as $\mathrm{PbTe}$, $\mathrm{Pb_{1-x}Sn_xTe}$, and $\mathrm{Pb_{1-x}Sn_xSe}$. Other crystalline topological phases host exotic surface behaviour such as hinge and corner states [@parameswaran2017topological; @Benalcazar2017Quantized; @Benalcazar2017Electric; @Song2017d; @Langbehn2017Reflection; @Schindler2018Higher; @schindler2018bismuth; @xu2017topological; @Shapourian2018Topological; @lin2017topological; @Ezawa2018Higher; @Khalaf2018Higher; @Geier2018Second; @trifunovic2018higher; @fang2017rotation; @okuma2018topological]. Such higher-order topological insulators and superconductors have been also recently experimentally observed in bismuth by Schindler *et al.* \[\]. A recent survey by Vergniory *et al.* \[\] had found that a staggering 24% of materials in nature have some nontrivial band structure topology. In this paper we engage the vital challenge of achieving a complete classification of all possible topological phases in presence of all possible material symmetries. A major milestone in our understanding of topological phases came with the discovery of the periodic table of topological insulators and superconductors [@kitaev2009periodic; @schnyder2009classification; @ryu2010topological; @Hasan2010Colloquium; @moore2010birth; @stone2010symmetries; @Teo2010Topological; @franz2013topological; @witten2015three; @Chiu2016Classification]; see Table \[tab:per\]. The table provides a complete classification of all topological phases of non-interacting fermions in presence of non-spatial symmetries. The presence and absence of these symmetries are categorized into the 10 Altland-Zirnbauer (AZ) symmetry classes [@Altland1997Nonstandard; @heinzner2005symmetry; @zirnbauer2010symmetry] and form the “tenfold way". Each AZ symmetry class corresponds to a topological “classifying space" of possible free-fermion Hamiltonians respecting the symmetry, and these are the topological invariants of these classifying spaces which are the ${\mathbb{Z}}$ and ${\mathbb{Z}}_2$ topological indices. There are many different view-points for this profound classification [@Schnyder2008Classification; @schnyder2009classification; @ryu2010topological; @kitaev2009periodic; @stone2010symmetries]. The approach of Refs. \[\], which was pioneered by Kitaev \[\], takes an algebraic perspective. In this paradigm, the set of non-spatial symmetry transformations forms a Clifford algebra with a specific action over the Hilbert space. The enumeration of all possible topologically distinct Hamiltonians is equivalent to asking the following: How many actions can a Hamiltonian have on the Hilbert space which are compatible with the algebra of non-spatial symmetries? The answer to this question is exactly the “classifying space" for the AZ symmetry class; see Fig. \[fig:diag1\]. The twofold and eightfold periodic structures within the table then naturally follow from the Bott periodicity of Clifford modules [@atiyah1964clifford]. [ccccccc]{} & ------------- Non-spatial symmetries ------------- & & ------------- Symmetries + Hamiltonian ------------- & & & ------------- Classifying space ------------- \ & $\Downarrow$ & & $\Downarrow$ & & & $\Uparrow$\ $\Bigg($ & ---------- Clifford algebra ---------- & $\hookrightarrow$ & ------------------ Extended Clifford algebra ------------------ & $\Bigg)$ & $\mapsto$ & --------------- All Hilbert- space actions --------------- \ \ & ${{\mathrm{Cl}}}_{p,q}$ & & ${{\mathrm{Cl}}}_{p+1,q}$ & & & ${\mathcal{R}}_{p-q+2}$ In this paper we generalize this classification scheme as to include the effects of all point group symmetries of crystalline matter. One highly successful approach towards solving the classification problem is the use of symmetry indicators of elementary band representations [@Dong2016Classification; @po2017symmetry; @bradlyn2017topological; @Watanabe2018Structure; @Bradlyn2018Band; @song2018quantitative; @Cano2018Building; @Vergniory2017Graph; @Ono2018Unified; @vergniory2018high; @Khalaf2018Symmetry; @Khalaf2018Higher]. In this approach, the “symmetry indicators", which are a generalization of the Fu-Kane [@Fu2007Topological; @Fu2011Topological] formula ${\mathbb{Z}}_2$ invariant, are used to analyse the band structure into elementary band representation of the corresponding crystallographic group. It is shown that bands which form one subpart of a disconnected elementary band representation must always be topological [@Cano2018Building]. This is a very prolific classification scheme which is used to categorize numerous topological crystalline phases. However, this technique requires one to perform Berry-phase analyses of individual electronic Bloch wavefunctions along various Wilson loops in order to detect different topological phases which correspond to the same band representation, e.g., the integer quantum Hall effect. An alternate theoretically complete and rigorous approach is the formulation of the crystalline classification problem using twisted equivariant K-theory [@freed2013twisted; @Morimoto2013Topological; @Shiozaki2014Topology; @Hsieh2014CPT; @Shiozaki2016Topology; @Shiozaki2017Topological; @Trifunovic2017Bott; @shiozaki2018atiyah; @Geier2018Second; @trifunovic2018higher; @okuma2018topological]. Within twisted equivariant K-theory the crystallographic group action on the Hamiltonian is encoded by a twist on the Brillouin zone (BZ) which serves as the base space of the K-group [@Shiozaki2017Topological]. This approach was successful in obtaining a complete classification of all topological invariants of order-two magnetic space group crystals [@Morimoto2013Topological; @Shiozaki2014Topology] as well as of all wallpaper group crystals [@Shiozaki2017Topological; @graph]. However, due to the mathematically challenging nature of the paradigm, ongoing works [@shiozaki2018atiyah] are still being carried in an attempt to calculate all topological invariants of all crystallographic symmetry groups. Herein, we follow the approach of Freed and Moor \[\], Morimoto and Furusaki \[\], and Shiozaki, Sato, and Gomi \[\]. We focus on bulk topological invariants which are akin to the strong topological invariants of topological insulators and superconductors with non-spatial symmetries [@kitaev2009periodic; @schnyder2009classification; @ryu2010topological; @franz2013topological; @witten2015three; @abramovici2012clifford]. We generalize the Clifford algebra classification scheme (see Fig. \[fig:diag1\]) by incorporating the point group symmetry actions into the algebra of symmetries generated by the non-spatial symmetries, and a Hamiltonian compatible with this symmetry creates a natural ${\mathbb{Z}}_2$-graded algebraic structure. Since any operator, such as the Hamiltonian, can be thought of as a transformation acting on the Hilbert space, the study of all compatible Hamiltonians is thus equivalent to the study of all possible actions of the extended ${\mathbb{Z}}_2$-graded algebra on the Hilbert space; see Fig. \[fig:diag2\]. For each of the possible 32 crystallographic point group symmetries, we find the appropriate ${\mathbb{Z}}_2$-graded algebra and its corresponding “classifying space" of possible actions. The topological indices characterizing possible topological insulators and superconductors in any of the 10 AZ symmetry classes are the topological invariants of this classifying space, and these invariants are herein presented. The majority of phases found by our classification paradigm correspond to novel crystalline topological insulators and superconductors. Nevertheless, a tremendous amount of work on various crystalline topological phases has recently been carried out by numerous research groups around the globe, we thus also compare our results to many phases that had already been analysed (see Sec. \[sec:discussion\]). The rest of the paper is divided as follows: In Sec. \[sec:results\] we summarize our results which are presented in the tables throughout the paper. In Sec. \[sec:meth\] we present the physical paradigm of our classification and then provide the mathematical background needed for the analysis. In Sec. \[sec:exam\] we give some pedagogical examples for calculating the topological classification of all symmetries. In Sec. \[sec:discussion\] we discuss our results in comparison with previous classification schemes and suggest possible future extensions. [ccccccc]{} & --------------- Crystalline & non-spatial symmetries --------------- & & ------------- Symmetries + Hamiltonian ------------- & & & ------------- Classifying space ------------- \ & $\Downarrow$ & & $\Downarrow$ & & & $\Uparrow$\ $\Bigg($ & ------------ Algebra of symmetries ------------ & $\hookrightarrow$ & ------------------------- ${\mathbb{Z}}_2$-graded real algebra ------------------------- & $\Bigg)$ & $\mapsto$ & --------------- All Hilbert- space actions --------------- \ \ & $B^0$ & & $B$ & & & Summary of Results {#sec:results} ================== $$\begin{array}{l||c||l||l||c|c|c|c} q & \mathrm{Extension} & \mathrm{Classifying~space} & \mathrm{AZ~class} & d=0 & d=1 & d=2 & d=3 \\ \hline\hline 0 & {\mathbb{C}}\hookrightarrow{\mathbb{C}}\oplus{\mathbb{C}}& {\mathcal{C}}_0=\prod_{k+m=n} U(n)/(U(k)\times U(m)) & \mathrm{A} & {\mathbb{Z}}& 0 & {\mathbb{Z}}& 0 \\ \hline 1 & {\mathbb{C}}\oplus{\mathbb{C}}\hookrightarrow M_2({\mathbb{C}}) & {\mathcal{C}}_1=U(n) & \mathrm{AIII} & 0 & {\mathbb{Z}}& 0 & {\mathbb{Z}}\\ \hline\hline 0 & {\mathbb{R}}\hookrightarrow{\mathbb{R}}\oplus{\mathbb{R}}& {\mathcal{R}}_0=\prod_{k+m=n} O(n)/(O(k)\times O(m)) & \mathrm{AI} & {\mathbb{Z}}& 0 & 0 & 0 \\ \hline 1 & {\mathbb{R}}\oplus{\mathbb{R}}\hookrightarrow M_2({\mathbb{R}}) & {\mathcal{R}}_1=O(n) & \mathrm{BDI} & {\mathbb{Z}}_2 & {\mathbb{Z}}& 0 & 0 \\ \hline 2 & {\mathbb{R}}\hookrightarrow{\mathbb{C}}& {\mathcal{R}}_2=O(2n)/U(n) & \mathrm{D} & {\mathbb{Z}}_2 & {\mathbb{Z}}_2 & {\mathbb{Z}}& 0 \\ \hline 3 & {\mathbb{C}}\hookrightarrow{\mathbb{H}}& {\mathcal{R}}_3=U(2n)/Sp(n) & \mathrm{DIII} & 0 & {\mathbb{Z}}_2 & {\mathbb{Z}}_2 & {\mathbb{Z}}\\ \hline 4 & {\mathbb{H}}\hookrightarrow{\mathbb{H}}\oplus{\mathbb{H}}& {\mathcal{R}}_4=\prod_{k+m=n} Sp(n)/(Sp(k)\times Sp(m)) & \mathrm{AII} & {\mathbb{Z}}& 0 & {\mathbb{Z}}_2 & {\mathbb{Z}}_2 \\ \hline 5 & {\mathbb{H}}\oplus{\mathbb{H}}\hookrightarrow M_2({\mathbb{H}}) & {\mathcal{R}}_5=Sp(n) & \mathrm{CII} & 0 & {\mathbb{Z}}& 0 & {\mathbb{Z}}_2 \\ \hline 6 & {\mathbb{H}}\hookrightarrow M_2({\mathbb{C}}) & {\mathcal{R}}_6=Sp(n)/U(n) & \mathrm{C} & 0 & 0 & {\mathbb{Z}}& 0 \\ \hline 7 & {\mathbb{C}}\hookrightarrow M_2({\mathbb{R}}) & {\mathcal{R}}_7=U(n)/O(n) & \mathrm{CI} & 0 & 0 & 0 & {\mathbb{Z}}\end{array}$$ The results are brought for all point group symmetries in all crystal systems, triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. All groups are brought both in Schönflies (Schön.) notation and in Hermann–Mauguin (HM) notation, they are accompanied by a solid displaying the crystalline symmetry group, taken with permission from Ashcroft and Mermin [@ashcroft1976solid]. Bulk topological invariants for all AZ symmetry classes are presented in Tables \[tab:p1\]-\[tab:m1\]. All classifying spaces for all three-dimensional (3D) point group symmetries in all crystal systems are compactly presented in Table \[tab:class\]. In Appendix \[app:layer\] we use our techniques to calculate all classifying spaces for all two-dimensional (2D) symmorphic layer group symmetries [@graph]; these results are also compactly presented in Table \[tab:class\]. Possible extensions of our work in treatment magnetic crystals as well as defected and higher-dimensional systems are discussed in Sec. \[sec:discussion\]. Methodology of Classification {#sec:meth} ============================= Physical Paradigm {#sec:paradigm} ----------------- The many-body Hilbert space is a complex linear vector space. A symmetry action on the states within the Hilbert space is either a unitary or anti-unitary action. These imply that the Hilbert space forms a real module over the algebra of symmetries, denoted $B^0$. When no crystalline symmetry is present, $B^0$ is generated by some selection of the following non-spatial symmetries [@schnyder2009classification; @ryu2010topological; @kennedy2016bott]: charge conservation $Q$, time reversal $T$, particle-hole symmetry $C$, and spin rotations $S_1,S_2,S_3$. Such a selection corresponds to an AZ symmetry class [@Altland1997Nonstandard; @heinzner2005symmetry; @zirnbauer2010symmetry]; see Table \[tab:per\]. Any translation invariant non-interacting quantum dynamics is described by a Hamiltonian, $\mathcal{H}(\mathbf{k})$, which is quadratic in creation-annihilation(/Majorana) operators of the many-body Fock(/Nambu) space representation. When using this representation, the Hamiltonian may be linearized around the $\Gamma$-point, which is the high-symmetry time-reversal-invariant point, such that [@kitaev2009periodic; @schnyder2009classification; @ryu2010topological; @franz2013topological; @witten2015three; @abramovici2012clifford] $$\label{gammapt} \mathcal{H}(\mathbf{k})=iM+\bm{\gamma}\cdot\mathbf{k},{{\;\;\:}}\{\gamma_i,\gamma_j\}=2\delta_{ij}.$$ Here, the Dirac gamma matrices $\bm{\gamma}=\gamma_1,\gamma_2,\ldots,\gamma_d$ (in $d$ spatial dimensions) may be naturally incorporated into the algebra of symmetries, $B^0$. When studying either insulators or superconductors which correspond to gapped Hamiltonians, $\mathcal{H}(\mathbf{k})$, the mass matrix, $M$, which gaps the spectrum, may be spectrally flattened (i.e. ${M^2=-1}$), and, together with the algebra of symmetries, $B^0$, form an extended algebra, $B$. Classifying all gapped Dirac Hamiltonians is thus equivalent to classifying all Hilbert space actions [@abramovici2012clifford] of the extended algebra $B$ which are compatible with the action of $B^0$; see Figs. \[fig:diag1\] and \[fig:diag2\]. One finds a Morita equivalent Clifford algebra structure [@kitaev2009periodic; @atiyah1964clifford; @Vela2002Central; @cliff] for both algebras, ${B^0={{\mathrm{Cl}}}_{p,q}}$ and ${B={{\mathrm{Cl}}}_{p+1,q}}$. Moreover, one may also identify the extended algebra as a ${\mathbb{Z}}_2$-graded algebra (superalgebra) $B=B^0\oplus B^1$ whose even part is the algebra of symmetries $B^0$; see Sec. \[sec:graded\]. The space of Dirac Hamiltonians corresponding to the extension ${{{\mathrm{Cl}}}_{q+6,d}\hookrightarrow{{\mathrm{Cl}}}_{q+7,d}}$ is stably homotopic to a Cartan symmetric space ${\mathcal{R}}_{q-d}$ which is the classifying space [@kitaev2009periodic; @schnyder2009classification; @ryu2010topological; @Hasan2010Colloquium; @moore2010birth; @franz2013topological; @witten2015three] of the AZ symmetry class $q$ in $d$ dimensions. Every stable topological phase is therefore characterized by an invariant in the group $\pi_0({\mathcal{R}}_{q-d})$; see Sec. \[sec:ext\]. This structure is summarized in the periodic table [@kitaev2009periodic; @schnyder2009classification; @ryu2010topological; @Hasan2010Colloquium; @moore2010birth; @franz2013topological; @witten2015three] of topological insulators and superconductors, which follows from the Atiyah-Bott-Shapiro construction [@kitaev2009periodic; @atiyah1964clifford]; see Table \[tab:per\]. This classification of Dirac Hamiltonians captures the bulk topological invariants corresponding to strong topological phases [@kitaev2009periodic; @schnyder2009classification; @ryu2010topological; @franz2013topological; @witten2015three; @abramovici2012clifford], and it is these bulk strong topological invariants we herein generalize to incorporate spatial crystalline symmetry. Possible future analyses of weak [@FuKaneMele2007Topological; @Moore2007Topological; @Varjas2017Space] and fragile [@Po2018Fragile; @bouhon2018wilson; @bradlyn2018disconnected] crystalline topological phases are discussed in Sec. \[sec:discussion\]. $$\begin{array}{l||c|c||c|c|c||c|c|c||} & \parbox{\fgw}{\includegraphics[width=\fgw]{C1_3D.png}} & \parbox{\fgw}{\includegraphics[width=\fgw]{Ci_3D.png}} & \parbox{\fgw}{\includegraphics[width=\fgw]{C2_3D.png}} & \parbox{\fgw}{\includegraphics[width=\fgw]{Cs_3D.png}} & \parbox{\fgw}{\includegraphics[width=\fgw]{C2h_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{D2_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{C2v_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{D2h_3D.png}} \\ \hline \hline \mathrm{Sch\ddot{o}n.} & C_1 & C_i,S_2 & C_2,D_1 & C_s,C_{1h},C_{1v} & C_{2h},D_{1d} & D_2 & C_{2v},D_{1h} & D_{2h} \\ \hline \mathrm{HM} & 1 & \bar 1 & 2 & \mathrm{m} & 2/\mathrm{m} & 222 & \mathrm{mm}2 & \mathrm{mmm} \\ \hline\hline & {\mathcal{C}}_{q+1} & {\mathcal{C}}_{q} & {\mathcal{C}}_{q+1}^2 & {\mathcal{C}}_{q} & {\mathcal{C}}_{q}^2 & {\mathcal{C}}_{q+1}^4 & {\mathcal{C}}_{q}^2 & {\mathcal{C}}_{q}^4 \\ \hline\hline \mathrm{A} & 0 & {\mathbb{Z}}& 0 & {\mathbb{Z}}& {\mathbb{Z}}^2 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^4 \\ \hline \mathrm{AIII} & {\mathbb{Z}}& 0 & {\mathbb{Z}}^2 & 0 & 0 & {\mathbb{Z}}^4 & 0 & 0 \\ \hline\hline & {\mathcal{R}}_{q-3} & {\mathcal{R}}_{q-4} & {\mathcal{R}}_{q-3}^2 & {\mathcal{R}}_{q-4} & {\mathcal{R}}_{q-4}^2 & {\mathcal{R}}_{q-3}^4 & {\mathcal{R}}_{q-4}^2 & {\mathcal{R}}_{q-4}^4 \\ \hline\hline \mathrm{AI} & 0 & {\mathbb{Z}}& 0 & {\mathbb{Z}}& {\mathbb{Z}}^2 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^4 \\ \hline \mathrm{BDI} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \mathrm{D} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \mathrm{DIII} & {\mathbb{Z}}& 0 & {\mathbb{Z}}^2 & 0 & 0 & {\mathbb{Z}}^4 & 0 & 0 \\ \hline \mathrm{AII} & {\mathbb{Z}}_2 & {\mathbb{Z}}& {\mathbb{Z}}_2^2 & {\mathbb{Z}}& {\mathbb{Z}}^2 & {\mathbb{Z}}_2^4 & {\mathbb{Z}}^2 & {\mathbb{Z}}^4 \\ \hline \mathrm{CII} & {\mathbb{Z}}_2 & {\mathbb{Z}}_2 & {\mathbb{Z}}_2^2 & {\mathbb{Z}}_2 & {\mathbb{Z}}_2^2 & {\mathbb{Z}}_2^4 & {\mathbb{Z}}_2^2 & {\mathbb{Z}}_2^4 \\ \hline \mathrm{C} & 0 & {\mathbb{Z}}_2 & 0 & {\mathbb{Z}}_2 & {\mathbb{Z}}_2^2 & 0 & {\mathbb{Z}}_2^2 & {\mathbb{Z}}_2^4 \\ \hline \mathrm{CI} & {\mathbb{Z}}& 0 & {\mathbb{Z}}^2 & 0 & 0 & {\mathbb{Z}}^4 & 0 & 0 \end{array}$$ \[tab:o2\] $$\begin{array}{l||c||c|c|c|c|c|c|c||} \multicolumn{2}{l||}{} & \parbox{\fgw}{\includegraphics[height=\fgb]{C4_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{S4_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{C4h_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{D4_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{C4v_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{D2d_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{D4h_3D.png}} \\ \hline\hline \mathrm{Sch\ddot{o}n.} & C_1 & C_4 & S_4 & C_{4h} & D_4 & C_{4v} & D_{2d} & D_{4h} \\ \hline \mathrm{HM} & 1 & 4 & \bar 4 & 4/\mathrm{m} & 422 & 4\mathrm{mm} & \bar{4}2\mathrm{m} & 4/\mathrm{mmm} \\ \hline\hline & {\mathcal{C}}_{q+1} & {\mathcal{C}}_{q+1}^4 & {\mathcal{C}}_{q}^2 & {\mathcal{C}}_q^4 & {\mathcal{C}}_{q+1}^5 & {\mathcal{C}}_{q}^2\times{\mathcal{C}}_{q+1} & {\mathcal{C}}_{q}^2\times{\mathcal{C}}_{q+1} & {\mathcal{C}}_{q}^5 \\ \hline\hline \mathrm{A} & 0 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^4 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^2 & {\mathbb{Z}}^5 \\ \hline \mathrm{AIII} & {\mathbb{Z}}& {\mathbb{Z}}^4 & 0 & 0 & {\mathbb{Z}}^5 & {\mathbb{Z}}& {\mathbb{Z}}& 0 \\ \hline\hline & {\mathcal{R}}_{q-3} & {\mathcal{R}}_{q-3}^2\times{\mathcal{C}}_{q+1} & {\mathcal{R}}_{q-4}\times{\mathcal{R}}_{q-2} & {\mathcal{R}}_{q-4}^2\times{\mathcal{C}}_q & {\mathcal{R}}_{q-3}^5 & {\mathcal{R}}_{q-4}^2\times{\mathcal{R}}_{q-5} & {\mathcal{R}}_{q-4}^2\times{\mathcal{R}}_{q-3} & {\mathcal{R}}_{q-4}^5 \\ \hline\hline \mathrm{AI} & 0 & 0 & {\mathbb{Z}}& {\mathbb{Z}}^3 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^2 & {\mathbb{Z}}^5 \\ \hline \mathrm{BDI} & 0 & {\mathbb{Z}}& 0 & 0 & 0 & {\mathbb{Z}}& 0 & 0 \\ \hline \mathrm{D} & 0 & 0 & {\mathbb{Z}}& {\mathbb{Z}}& 0 & 0 & 0 & 0 \\ \hline \mathrm{DIII} & {\mathbb{Z}}& {\mathbb{Z}}^3 & {\mathbb{Z}}_2 & 0 & {\mathbb{Z}}^5 & 0 & {\mathbb{Z}}& 0 \\ \hline \mathrm{AII} & {\mathbb{Z}}_2 & {\mathbb{Z}}_2^2 & {\mathbb{Z}}\times{\mathbb{Z}}_2 & {\mathbb{Z}}^3 & {\mathbb{Z}}_2^5 & {\mathbb{Z}}^2 & {\mathbb{Z}}^2\times{\mathbb{Z}}_2 & {\mathbb{Z}}^5 \\ \hline \mathrm{CII} & {\mathbb{Z}}_2 & {\mathbb{Z}}_2^2\times{\mathbb{Z}}& {\mathbb{Z}}_2 & {\mathbb{Z}}_2^2 & {\mathbb{Z}}_2^5 & {\mathbb{Z}}_2^2\times{\mathbb{Z}}& {\mathbb{Z}}_2^3 & {\mathbb{Z}}_2^5 \\ \hline \mathrm{C} & 0 & 0 & {\mathbb{Z}}_2\times{\mathbb{Z}}& {\mathbb{Z}}_2^2\times{\mathbb{Z}}& 0 & {\mathbb{Z}}_2^3 & {\mathbb{Z}}_2^2 & {\mathbb{Z}}_2^5 \\ \hline \mathrm{CI} & {\mathbb{Z}}& {\mathbb{Z}}^3 & 0 & 0 & {\mathbb{Z}}^5 & {\mathbb{Z}}_2 & {\mathbb{Z}}& 0 \end{array}$$ $$\begin{array}{l||c||c|c|c|c|c||} \multicolumn{2}{l||}{} & \parbox{\fgw}{\includegraphics[height=\fgb]{C3_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{S6_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{D3_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{C3v_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{D3d_3D.png}} \\ \hline\hline \mathrm{Sch\ddot{o}n.} & C_1 & C_3 & C_{3i},S_6 & D_3 & C_{3v} & D_{3d} \\ \hline \mathrm{HM} & 1 & 3 & \bar 3 & 32 & 3\mathrm{m} & \bar{3}\mathrm{m} \\ \hline\hline & {\mathcal{C}}_{q+1} & {\mathcal{C}}_{q+1}^3 & {\mathcal{C}}_{q}^3 & {\mathcal{C}}_{q+1}^3 & {\mathcal{C}}_{q}\times{\mathcal{C}}_{q+1} & {\mathcal{C}}_{q}^3 \\ \hline\hline \mathrm{A} & 0 & 0 & {\mathbb{Z}}^3 & 0 & {\mathbb{Z}}& {\mathbb{Z}}^3 \\ \hline \mathrm{AIII} & {\mathbb{Z}}& {\mathbb{Z}}^3 & 0 & {\mathbb{Z}}^3 & {\mathbb{Z}}& 0 \\ \hline\hline & {\mathcal{R}}_{q-3} & {\mathcal{R}}_{q-3}\times{\mathcal{C}}_{q+1} & {\mathcal{R}}_{q-4}\times{\mathcal{C}}_{q} & {\mathcal{R}}_{q-3}^3 & {\mathcal{R}}_{q-4}\times{\mathcal{R}}_{q-5} & {\mathcal{R}}_{q-4}^3 \\ \hline\hline \mathrm{AI} & 0 & 0 & {\mathbb{Z}}^2 & 0 & {\mathbb{Z}}& {\mathbb{Z}}^3 \\ \hline \mathrm{BDI} & 0 & {\mathbb{Z}}& 0 & 0 & {\mathbb{Z}}& 0 \\ \hline \mathrm{D} & 0 & 0 & {\mathbb{Z}}& 0 & 0 & 0 \\ \hline \mathrm{DIII} & {\mathbb{Z}}& {\mathbb{Z}}^2 & 0 & {\mathbb{Z}}^3 & 0 & 0 \\ \hline \mathrm{AII} & {\mathbb{Z}}_2 & {\mathbb{Z}}_2 & {\mathbb{Z}}^2 & {\mathbb{Z}}_2^3 & {\mathbb{Z}}& {\mathbb{Z}}^3 \\ \hline \mathrm{CII} & {\mathbb{Z}}_2 & {\mathbb{Z}}_2\times{\mathbb{Z}}& {\mathbb{Z}}_2 & {\mathbb{Z}}_2^3 & {\mathbb{Z}}_2\times{\mathbb{Z}}& {\mathbb{Z}}_2^3 \\ \hline \mathrm{C} & 0 & 0 & {\mathbb{Z}}_2\times{\mathbb{Z}}& 0 & {\mathbb{Z}}_2^2 & {\mathbb{Z}}_2^3 \\ \hline \mathrm{CI} & {\mathbb{Z}}& {\mathbb{Z}}^2 & 0 & {\mathbb{Z}}^3 & {\mathbb{Z}}_2 & 0 \end{array}$$ $$\begin{array}{l||c||c|c|c|c|c|c|c||} \multicolumn{2}{l||}{} & \parbox{\fgw}{\includegraphics[height=\fgb]{C6_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{C3h_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{C6h_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{D6_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{C6v_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{D3h_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{D6h_3D.png}} \\ \hline\hline \mathrm{Sch\ddot{o}n.} & C_1 & C_6 & C_{3h} & C_{6h} & D_6 & C_{6v} & D_{3h} & D_{6h} \\ \hline \mathrm{HM} & 1 & 6 & 3/\mathrm{m} & 6/\mathrm{m} & 622 & 6\mathrm{mm} & \bar{6}\mathrm{m}2 & 6/\mathrm{mmm} \\ \hline\hline & {\mathcal{C}}_{q+1} & {\mathcal{C}}_{q+1}^6 & {\mathcal{C}}_{q}^3 & {\mathcal{C}}_q^6 & {\mathcal{C}}_{q+1}^6 & {\mathcal{C}}_{q}^2\times{\mathcal{C}}_{q+1}^2 & {\mathcal{C}}_{q}^3 & {\mathcal{C}}_{q}^6 \\ \hline\hline \mathrm{A} & 0 & 0 & {\mathbb{Z}}^3 & {\mathbb{Z}}^6 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^3 & {\mathbb{Z}}^6 \\ \hline \mathrm{AIII} & {\mathbb{Z}}& {\mathbb{Z}}^6 & 0 & 0 & {\mathbb{Z}}^6 & {\mathbb{Z}}^2 & 0 & 0 \\ \hline\hline & {\mathcal{R}}_{q-3} & {\mathcal{R}}_{q-3}^2\times{\mathcal{C}}_{q+1}^2 & {\mathcal{R}}_{q-4}\times{\mathcal{C}}_{q} & {\mathcal{R}}_{q-4}^2\times{\mathcal{C}}_q^2 & {\mathcal{R}}_{q-3}^6 & {\mathcal{R}}_{q-4}^2\times{\mathcal{R}}_{q-5}^2 & {\mathcal{R}}_{q-4}^3 & {\mathcal{R}}_{q-4}^6 \\ \hline\hline \mathrm{AI} & 0 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^4 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^3 & {\mathbb{Z}}^6 \\ \hline \mathrm{BDI} & 0 & {\mathbb{Z}}^2 & 0 & 0 & 0 & {\mathbb{Z}}^2 & 0 & 0 \\ \hline \mathrm{D} & 0 & 0 & {\mathbb{Z}}& {\mathbb{Z}}^2 & 0 & 0 & 0 & 0 \\ \hline \mathrm{DIII} & {\mathbb{Z}}& {\mathbb{Z}}^4 & 0 & 0 & {\mathbb{Z}}^6 & 0 & 0 & 0 \\ \hline \mathrm{AII} & {\mathbb{Z}}_2 & {\mathbb{Z}}_2^2 & {\mathbb{Z}}^2 & {\mathbb{Z}}^4 & {\mathbb{Z}}_2^6 & {\mathbb{Z}}^2 & {\mathbb{Z}}^3 & {\mathbb{Z}}^6 \\ \hline \mathrm{CII} & {\mathbb{Z}}_2 &{\mathbb{Z}}_2^2\times{\mathbb{Z}}^2 & {\mathbb{Z}}_2 & {\mathbb{Z}}_2^2 & {\mathbb{Z}}_2^6 &{\mathbb{Z}}_2^2\times{\mathbb{Z}}^2 & {\mathbb{Z}}_2^3 & {\mathbb{Z}}_2^6 \\ \hline \mathrm{C} & 0 & 0 & {\mathbb{Z}}_2\times{\mathbb{Z}}&{\mathbb{Z}}_2^2\times{\mathbb{Z}}^2 & 0 & {\mathbb{Z}}_2^4 & {\mathbb{Z}}_2^3 & {\mathbb{Z}}_2^6 \\ \hline \mathrm{CI} & {\mathbb{Z}}& {\mathbb{Z}}^4 & 0 & 0 & {\mathbb{Z}}^6 & {\mathbb{Z}}_2^2 & 0 & 0 \end{array}$$ $$\begin{array}{l||c||c|c|c|c|c||} \multicolumn{2}{l||}{} & \parbox{\fgw}{\includegraphics[width=\fgw]{T_3D.png}} & \parbox{\fgw}{\includegraphics[width=\fgw]{Th_3D.png}} & \parbox{\fgw}{\includegraphics[width=\fgw]{O_3D.png}} & \parbox{\fgw}{\includegraphics[width=\fgw]{Td_3D.png}} & \parbox{\fgw}{\includegraphics[width=\fgw]{Oh_3D.png}} \\ \hline\hline \mathrm{Sch\ddot{o}n.} & C_1 & T & T_h & O & T_d & O_h \\ \hline \mathrm{HM} & 1 & 23 & \mathrm{m}\bar{3} & 432 & \bar{4}3\mathrm{m} & \mathrm{m}\bar{3}\mathrm{m} \\ \hline\hline & {\mathcal{C}}_{q+1} & {\mathcal{C}}_{q+1}^4 & {\mathcal{C}}_{q}^4 & {\mathcal{C}}_{q+1}^5 & {\mathcal{C}}_{q}^2\times{\mathcal{C}}_{q+1} & {\mathcal{C}}_{q}^5 \\ \hline\hline \mathrm{A} & 0 & 0 & {\mathbb{Z}}^4 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^5 \\ \hline \mathrm{AIII} & {\mathbb{Z}}& {\mathbb{Z}}^4 & 0 & {\mathbb{Z}}^5 & {\mathbb{Z}}& 0 \\ \hline\hline & {\mathcal{R}}_{q-3} & {\mathcal{R}}_{q-3}^2\times{\mathcal{C}}_{q+1} & {\mathcal{R}}_{q-4}^2\times{\mathcal{C}}_{q} & {\mathcal{R}}_{q-3}^5 & {\mathcal{R}}_{q-4}^2\times{\mathcal{R}}_{q-5} & {\mathcal{R}}_{q-4}^5 \\ \hline\hline \mathrm{AI} & 0 & 0 & {\mathbb{Z}}^3 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^5 \\ \hline \mathrm{BDI} & 0 & {\mathbb{Z}}& 0 & 0 & {\mathbb{Z}}& 0 \\ \hline \mathrm{D} & 0 & 0 & {\mathbb{Z}}& 0 & 0 & 0 \\ \hline \mathrm{DIII} & {\mathbb{Z}}& {\mathbb{Z}}^3 & 0 & {\mathbb{Z}}^5 & 0 & 0 \\ \hline \mathrm{AII} & {\mathbb{Z}}_2 & {\mathbb{Z}}_2^2 & {\mathbb{Z}}^3 & {\mathbb{Z}}_2^5 & {\mathbb{Z}}^2 & {\mathbb{Z}}^5 \\ \hline \mathrm{CII} & {\mathbb{Z}}_2 & {\mathbb{Z}}_2^2\times{\mathbb{Z}}& {\mathbb{Z}}_2^2 & {\mathbb{Z}}_2^5 & {\mathbb{Z}}_2^2\times{\mathbb{Z}}& {\mathbb{Z}}_2^5 \\ \hline \mathrm{C} & 0 & 0 & {\mathbb{Z}}_2^2\times{\mathbb{Z}}& 0 & {\mathbb{Z}}_2^3 & {\mathbb{Z}}_2^5 \\ \hline \mathrm{CI} & {\mathbb{Z}}& {\mathbb{Z}}^3 & 0 & {\mathbb{Z}}^5 & {\mathbb{Z}}_2 & 0 \end{array}$$ \[tab:exTd\] In this paper, we study the structure of the ${\mathbb{Z}}_2$-graded structure algebra extension ${B^0\hookrightarrow B}$ in presence of additional symmetries of the point group, ${g\in G}$, which act on the momenta by $$\label{Og} U_g^{\phantom{|}}\mathcal{H}(\mathbf{k})U_g^{-1}=\mathcal{H}(O_g\mathbf{k}),$$ where ${O_g\in\mathrm{O}(d)}$ is an orthogonal transformation. This yields an action on the Clifford algebras such that $G$ acts only on the spatial generators $$U_g^{\phantom{|}}\bm{\gamma}U_g^{-1}=O_g^{\mathrm{T}}\bm{\gamma}.$$ However, due to the spin of the electrons, the unitary actions, $U_g$, form a projective representation, $\Pi(G)$, satisfying $$\label{spinor} U_{g}U_{g'}=\pm U_{g\cdot g'}$$ (see Sec. \[sec:pgs\]). These algebras of spatial and extended non-spatial symmetry actions on the Hilbert space are given by ${B=\langle{{\mathrm{Cl}}}_{p+1,q},\Pi(G)\rangle}$, which denotes the algebra generated by the elements of the extended non-spatial symmetry actions, ${{\mathrm{Cl}}}_{p+1,q}$, as well as by the projective spatial symmetries, $\Pi(G)$. We focus on the 32 possible point group symmetries of 3D crystals [@graph]. These are determined by an abstract group structure, $G$, and an action on 3D space, ${G\subset\mathrm{O}(3)}$. We identify the algebra extension structure of each point group symmetry within each AZ symmetry class and construct the appropriate classifying space and its bulk topological invariants which are manifested by model Hamiltonians. In the absence of crystalline symmetry, such bulk topological invariants always manifest in edge modes due to the bulk-boundary correspondence [@franz2013topological; @Kane2005Topological; @Kane2005Quantum; @FuKaneMele2007Topological; @Moore2007Topological; @Hsieh2009Observation; @Roy2009Topological; @Fu2007Topological; @Konig2007Quantum; @xia2009observation; @Chen2009Experimental; @Khalaf2018Symmetry]. However, in previously treated crystalline symmetries, it was noted that such crystalline bulk invariants may also either host no edge modes [@Hughes2011Inversion; @Shiozaki2014Topology], or in fact host higher-order [@parameswaran2017topological; @Benalcazar2017Quantized; @Benalcazar2017Electric; @Song2017d; @Langbehn2017Reflection; @Schindler2018Higher; @schindler2018bismuth; @xu2017topological; @Shapourian2018Topological; @lin2017topological; @Ezawa2018Higher; @Khalaf2018Higher; @Geier2018Second; @trifunovic2018higher; @fang2017rotation] hinge or corner protected topological modes. Possible treatment of the higher-order topological insulators and superconductors which correspond to the bulk invariants presented in this paper are discussed in Sec. \[sec:discussion\]. Detailed Derivation {#sec:analysis} ------------------- In this section we provide the mathematical background required for the complete understanding of our technique. The educated reader may skip to the examples in Sec. \[sec:exam\]. ### $\mathbb{Z}_2$-Graded Algebras and their Tensor Products {#sec:graded} An algebra $A$ (over $\mathbb{R}$) is called $\mathbb{Z}_2$-graded [@Vela2002Central] if ${A=A^0 \oplus A^1}$ where ${a \cdot a' \in A^{i+j \pmod{2}}}$ for every ${a \in A^i}$ and ${a'\in A^j}$. In this case, $A^0$ is called “the even part" of $A$, and $A^1$ “the odd part". The elements of $A^0 \cup A^1$ are called “homogeneous", the elements of $A^0$ “even" and the elements of $A^1$ “odd". The even part forms a subalgebra, ${A^0\hookrightarrow A}$, while the odd part is not an algebra as it is not closed under multiplication. For example, the Clifford algebra [@kitaev2009periodic; @cliff] ${A={{\mathrm{Cl}}}_{p,q}}$, which is the algebra generated over the reals by generators $x_1,\dots,x_p$ and $\gamma_1,\dots,\gamma_q$ which are anticommuting in pairs and satisfy ${x_j^2=-1}$ for every ${j \in \{1,\dots,p\}}$ and ${\gamma_j^2=1}$ for every ${j \in \{1,\dots,q\}}$ is $\mathbb{Z}_2$-graded: with $A^0$ being the usual even part of the Clifford algebra, and $A^1$ being the odd part. The same algebra can admit more than one grading: for example, the algebra of $2\times 2$ real matrices, $M_2(\mathbb{R})$, is generated over $\mathbb{R}$ by $\gamma_1,\gamma_2$ subject to the relations ${\gamma_1^2=\gamma_2^2=1}$ and ${\gamma_1 \gamma_2=-\gamma_2 \gamma_1}$. We can define the $\mathbb{Z}_2$-grading by making both $\gamma_1$ and $\gamma_2$ odd, in which case $\gamma_1 \gamma_2$ will be even, and the even part would be ${\mathbb{R}\langle \gamma_1 \gamma_2\rangle \cong \mathbb{C}}$; this corresponds to the identification ${M_2({\mathbb{R}})\cong{{\mathrm{Cl}}}_{0,2}}$. However, we could also define the grading by making $\gamma_1$ odd and $\gamma_2$ even, in which case the even part would be ${\mathbb{R}\langle \gamma_2 \rangle\cong \mathbb{R}\oplus \mathbb{R}}$; this corresponds to the identification ${M_2({\mathbb{R}})\cong{{\mathrm{Cl}}}_{1,1}}$. Given two $\mathbb{Z}_2$-graded algebras $A$ and $A'$ generated by homogeneous elements, the graded tensor product $A \hat{\otimes} A'$ is defined to be the $\mathbb{R}$-algebra generated by the generators of $A$ and $A'$ put together such that they satisfy their former relations, plus the following: every even generator of $A$ commutes with all the generators of $A'$, and vice versa, and for odd generators ${a\in A^1}$ and ${a'\in A'^1}$ we have ${aa'=-a'a}$. Specifically, the ${\mathbb{Z}}_2$-graded tensor product [@Vela2002Central] of two Clifford algebras is simply given by $$\label{cliffprod} {{\mathrm{Cl}}}_{p_1,q_1}\hat{\otimes}{{\mathrm{Cl}}}_{p_2,q_2}={{\mathrm{Cl}}}_{p_1+p_2,q_1+q_2}.$$ Given an algebra $A$, the algebra of $n\times n$ matrices with entries in $A$ is denoted by $M_n(A)$. Recall that by the renowned Artin-Wedderburn theorem, every semi-simple algebra decomposes uniquely as a direct sum of matrix algebras over division algebras. Two semi-simple algebras are said to be “Morita equivalent" if they decompose as direct sums of matrix algebras (of possibly different dimensions) over the same division algebras. For example, $\mathbb{R}\oplus \mathbb{C}$ is Morita equivalent to $M_2(\mathbb{R}) \oplus M_3({\mathbb{C}})$. When the algebras are also $\mathbb{Z}_2$-graded, we say the algebras are equivalent if they are equivalent and also their even parts are Morita equivalent. ### (${\mathbb{Z}}_2$-graded) Algebra Extensions {#sec:ext} By the Bott periodicity [@atiyah1964clifford], all real Clifford algebras [@cliff] are Morita equivalent to eight prescribed algebras; see Table \[tab:per\]. Their parametrization is a matter of choice, and we follow Kennedy and Zirnbauer \[\] to consider the algebra ${B=\langle{{\mathrm{Cl}}}_{q+6+1,d},\Pi(G)\rangle}$ for a given $q \in \{0,1,\dots,7\}$. The presentation of ${{\mathrm{Cl}}}_{q+6+1,d}$ we consider is the following: write ${\gamma_1,\gamma_2,\dots,\gamma_d}$ for the Dirac gamma matrices, ${x_1,\dots,x_{q+6}}$ for the non-spatial symmetries, and $x_0$ for the spectrally flattened mass matrix. These generators satisfy ${\gamma_i^2=1,~x_i^2=-1}$. We write $y_1,\dots,y_{q+7+d}$ for the generators $x_1,\dots,x_{q+6};x_0;\gamma_1,\dots,\gamma_d$, and we refer to the generators this way in statements which hold for all the generators. Let us look at a Hilbert space, $E$, which is a $B$-module and hence also a $B^0$-module. The choices of Hamiltonians are equivalent to defining a $B$-module structure on $E$ which is compatible with $B^0$, i.e., all homomorphisms in the category of $B^0$-algebras between $B$ and the endomorphisms of $E$ as a $B^0$-module [@Rowen2008]. This is the definition of the “${{\mathrm{Hom}}}$" functor, ${{\mathrm{Hom}}}_{B^0-\mathbf{alg}}(B,{{\mathrm{End}}}_{B^0}{E})$ and we thus associate it with the algebra extension $$\label{MapstoDef} (B^0\hookrightarrow B)\mapsto \lim_{\dim E\to\infty}{{\mathrm{Hom}}}_{B^0-\mathbf{alg}}(B,{{\mathrm{End}}}_{B^0}{E}),$$ where we take the stable limit [@kitaev2009periodic; @kennedy2016bott]. The known results for the Clifford algebra extension problems are thus formulated as $$\label{MapstoR} {{\mathrm{Cl}}}_{p+1,q}\mapsto {\mathcal{R}}_{p-q+2},$$ where ${{\mathcal{R}}_q={\mathcal{R}}_{q+8}}$. Here, we omit the explicit extension ${({{\mathrm{Cl}}}_{p,q}\hookrightarrow{{\mathrm{Cl}}}_{p+1,q})}$ since the Clifford algebras always satisfy ${{{\mathrm{Cl}}}_{p+1,q}^0={{\mathrm{Cl}}}_{p,q}}$. This classification scheme is reviewed in Appendix \[app:class-space\]; see Fig. \[fig:diag1\]. The eightfold Bott periodicity structure [@atiyah1964clifford] in Table \[tab:per\] follows from the relations [@cliff] $${{\mathrm{Cl}}}_{p+1,q+1}\cong M_2({{\mathrm{Cl}}}_{p,q}),{{\;\;\:}}{{\mathrm{Cl}}}_{p+8,q}\cong M_{16}({{\mathrm{Cl}}}_{p,q}).$$ And indeed, the algebraic structure of Eq. (\[MapstoDef\]) is invariant under the Morita equivalence of $B$ with $M_n(B)$. The complex Clifford algebras [@cliff], ${{\mathrm{Cl}}}_{q}$, are given by the complexification, ${{\mathbb{C}}\otimes_{\mathbb{R}}{{\mathrm{Cl}}}_{p,q}={{\mathrm{Cl}}}_{p+q}}$, and satisfy $$\label{MapstoC} {{\mathrm{Cl}}}_{q+1}\mapsto {\mathcal{C}}_{q},$$ where ${{\mathcal{C}}_q={\mathcal{C}}_{q+2}}$. Useful identities include, $$\label{RCH} {\mathbb{H}}\otimes_{\mathbb{R}}{\mathbb{H}}\cong M_4({\mathbb{R}}),{{\;\;\:}}{\mathbb{C}}\otimes_{\mathbb{R}}{\mathbb{H}}\cong M_2({\mathbb{C}}),{{\;\;\:}}{\mathbb{C}}\otimes_{\mathbb{R}}{\mathbb{C}}\cong{\mathbb{C}}^{{\oplus 2}}.$$ A particular consequence is that ${{\mathbb{C}}\otimes_{\mathbb{R}}{{\mathrm{Cl}}}_q\cong{{\mathrm{Cl}}}_q^{{\oplus 2}}}$, which enables one to directly read off the classifying spaces of the complex AZ classes (A and AIII) from the classifying spaces of the real AZ classes. ### Point Group Symmetries {#sec:pgs} ---------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Solids with either inversion, rotation, reflection, or rotoreflection symmetry. The generators of the symmetry are also specified.[]{data-label="fig:PiG"}](Ci_3D.png "fig:"){height="\fgya"} ![Solids with either inversion, rotation, reflection, or rotoreflection symmetry. The generators of the symmetry are also specified.[]{data-label="fig:PiG"}](C6_3D.png "fig:"){height="\fgya"} ![Solids with either inversion, rotation, reflection, or rotoreflection symmetry. The generators of the symmetry are also specified.[]{data-label="fig:PiG"}](Cs_3D.png "fig:"){height="\fgya"} ![Solids with either inversion, rotation, reflection, or rotoreflection symmetry. The generators of the symmetry are also specified.[]{data-label="fig:PiG"}](S6_3D.png "fig:"){height="\fgya"} Inversion Rotation Reflection Rotoreflection $G$ $I^2\!=\!1$ $(c_n)^n\!=\!1$ $\sigma^2\!=\!1$ $(s_{2n})^{2n}\!=\!1$ $\Pi(G)\!\!\!\!\!\!\!\!\!$ $\hat{I}^2\!=\!1$ $(\hat{c}_n)^n\!=\!-1$ $\hat{\sigma}^2\!=\!-1$ $(\hat{s}_{2n})^{2n}\!=\!-(-1)^{n}$ ---------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The point groups are generated by: inversions $I$, rotations $c_n$ by $\frac{2\pi}{n}$, reflections ${\sigma=Ic_2}$, and rotoreflections ${s_{2n}=c_{2n}\sigma_h}$, where $\sigma_h,\sigma_v$ are horizontal and vertical reflections. Importantly, one must take into account the fermionic nature of the electrons and hence the projective spinor representation $\Pi(G)$; see Eq. (\[spinor\]). This spinor representation is constructed from subgroups of the Pin group, which is a double cover of the orthogonal group \[just as the Spin group, ${\mathrm{Spin}(3)\cong\mathrm{SU}(2)}$, is a double cover of the special orthogonal group\]: $$\begin{array}{ccccc} \{\pm1\} & \hookrightarrow & \mathrm{SU}(2) & \twoheadrightarrow & \mathrm{SO}(3) \\ {\mathbin{\text{\rotatebox[origin=c]{-90}{$=$}}}}& & {\mathbin{\text{\rotatebox[origin=c]{-90}{$\hookrightarrow$}}}}& & {\mathbin{\text{\rotatebox[origin=c]{-90}{$\hookrightarrow$}}}}\\ \{\pm1\} & \hookrightarrow & \mathrm{Pin}_-(3) & \twoheadrightarrow & \mathrm{O}(3) \\ & & {\mathbin{\text{\rotatebox[origin=c]{-90}{$\twoheadrightarrow$}}}}& & {\mathbin{\text{\rotatebox[origin=c]{-90}{$\twoheadrightarrow$}}}}\\ & & \{\pm1\} & = & \{\pm1\} \end{array}$$ Within the Pin (and Spin) group, one obtains the fermionic property that a $4\pi$ rotation equals 1. The subgroups are known as double point groups, ${\hat{G}\subset\mathrm{Pin}_-(3)}$, such that ${\hat{G}/{\mathbb{Z}}_2=G}$. However, when taking into account the projective nature of the Hilbert space and constructing the spinor representation, this amounts to setting a $2\pi$ rotation to be ${(\hat{c}_n)^n=-1}$, and a double inversion to be ${\hat{I}^2=1}$. This is referred to as the spinor representation, $\Pi(G)$. To clarify things [@Ando2015Topological], we note that:\ (i) In $G$ a $2\pi$ rotation is $1$.\ (ii) In $\hat{G}$ a $2\pi$ rotation squares to $1$.\ (iii) In $\Pi(G)$ a $2\pi$ rotation is $-1$. For brevity, we shall use $\hat{c},\hat{s},\hat{\sigma}$ for the generators of both $\hat{G}$ and $\Pi(G)$. Moreover, one finds that a reflection satisfies ${\hat{\sigma}^2=I^2\hat{c}_2^2=-1}$, and that since the product of orthogonal reflections $\hat\sigma_h\hat\sigma_v$ is a $\pi$ rotation, they satisfy ${(\hat\sigma_h\hat\sigma_v)^2=-1}$. The rotoreflections, however, depend on the parity, as ${(\hat s_{2n})^{2n}=\hat c_{2n}^{2n}(\hat\sigma_h^2)^n=-(-1)^n}$ (see Fig. \[fig:PiG\]). The point group symmetries each correspond to one of the abstract groups ${\mathbb{Z}}_n$, ${{\mathrm{Dih}}}_n$, $A_4$, $S_4$, or their product with ${\mathbb{Z}}_2$, as presented in Table \[tab:pgs\]. Useful identities include, $$\label{prodgroup} {\mathbb{Z}}_{6}={\mathbb{Z}}_{3}\times{\mathbb{Z}}_2,{{\;\;\:}}{{\mathrm{Dih}}}_{2}={\mathbb{Z}}_2\times{\mathbb{Z}}_2,{{\;\;\:}}{{\mathrm{Dih}}}_{6}={{\mathrm{Dih}}}_{3}\times{\mathbb{Z}}_2.$$ ------------- ------------------------------------------ -------------------- ---------- Name Symbol Name Schön. Cyclic ${\mathbb{Z}}_n$ Rotational $C_n$ ${\mathbb{Z}}_{2n}$ Rotoreflection $S_{2n}$ Dihedral ${{\mathrm{Dih}}}_n$ Pyramidal $C_{nv}$ ${{\mathrm{Dih}}}_n$ Dihedral $D_{n}$ ${{\mathrm{Dih}}}_{2n}$ Antiprismatic $D_{nd}$ Alternating $A_4$ Chiral tetrahedral $T$ Symmetric $S_4$ Full tetrahedral $T_d$ $S_4$ Chiral octahedral $O$ ${\mathbb{Z}}_n\times{\mathbb{Z}}_2$ Dipyramidal $C_{nh}$ ${{\mathrm{Dih}}}_n\times{\mathbb{Z}}_2$ Prismatic $D_{nh}$ $A_4\times{\mathbb{Z}}_2$ Pyritohedral $T_h$ $S_4\times{\mathbb{Z}}_2$ Full octahedral $O_h$ ------------- ------------------------------------------ -------------------- ---------- : The abstract groups corresponding to each of the point group symmetries.[]{data-label="tab:pgs"} ### $\mathbb{Z}_2$-Graded Groups and their Representations {#sec:ring} A group $G$ is called $\mathbb{Z}_2$-graded [@vershik2008new] if there exists a group homomorphism ${G\to{\mathbb{Z}}_2}$, in this case, the kernel of the homomorphism is denoted $G^0$ and dubbed “the even part", while ${G^1=G\setminus G^0}$ is dubbed “the odd part". Similar to Sec. \[sec:graded\], one has ${G=G^0 \cup G^1}$, where ${g \cdot g' \in G^{i+j \pmod{2}}}$ for every ${g \in G^i}$ and ${g'\in G^j}$. The even part forms a normal subgroup, ${G^0\triangleleft G}$, while the odd part is not a group as it is not closed under multiplication. The simplest case is when $G\xrightarrow{0}{\mathbb{Z}}_2$, such that all element are even and ${G^0=G}$. Otherwise, $G\twoheadrightarrow{\mathbb{Z}}_2$, and by the “first isomorphism theorem", ${G^0\triangleleft G}$ is a normal subgroup and $$G/G^0={\mathbb{Z}}_2.$$ A ${\mathbb{Z}}_2$-graded representation [@vershik2008new] of a ${\mathbb{Z}}_2$-graded group, $G$, is a representation, $\rho$, into ${\mathbb{Z}}_2$-graded matrix algebras that preserve parity, i.e., the even/odd elements of ${G=G^0 \cup G^1}$ are represented by even/odd elements of ${A_\rho=A_\rho^0 \oplus A_\rho^1}$. When studying group representations, a key notion is that of the “group ring" [@Passman1977]. It is the algebra generated by a group $G$ over ${\mathbb{R}}$ (or over any other ring) and denoted ${\mathbb{R}}[G]$. It is defined to be the ${\mathbb{R}}$-algebra, $\bigoplus_{g\in G} {\mathbb{R}}g$, whose basis as a real vector space consists of the elements of $G$, and its multiplication table is given by $(r_1 g_1)(r_2 g_2)=(r_1 r_2)(g_1 g_2)$. Moreover, by construction, $$G^0\triangleleft G {{\;\;\:}}\Rightarrow{{\;\;\:}}{\mathbb{R}}[G^0]\hookrightarrow{\mathbb{R}}[G],$$ and a ${\mathbb{Z}}_2$ grading of $G$ induces a ${\mathbb{Z}}_2$ grading of ${\mathbb{R}}[G]$ such that ${\mathbb{R}}[G^0]$ forms the even part of ${\mathbb{R}}[G]$, i.e., ${{\mathbb{R}}[G]^0={\mathbb{R}}[G^0]}$ and ${{\mathbb{R}}[G]^1=\bigoplus_{g\in G^1} {\mathbb{R}}g}$. A highly useful property, which follows the Maschke and Artin-Wedderburn theorems, is that the group ring ${\mathbb{R}}[G]$ is isomorphic to a direct sum of the matrix algebras corresponding to each irreducible representation of $G$. We thus list the group rings of all point group symmetries: Write ${n=2c(n)+r(n)}$, where $$c(n)=\left\lfloor\frac{n-1}{2}\right\rfloor,{{\;\;\:}}r(n)=\begin{cases} 1 & n~\mathrm{odd},\\ 2 & n~\mathrm{even}. \end{cases}$$ The group rings, associated with the abstract groups corresponding to the point group symmetries in Table \[tab:pgs\], are each equivalent to one of the following cases, $$\begin{aligned} {\mathbb{R}}[{\mathbb{Z}}_n]&={\mathbb{R}}^{\oplus r(n)}\oplus{\mathbb{C}}^{\oplus c(n)},\\ {\mathbb{R}}[{{\mathrm{Dih}}}_n]&=({\mathbb{R}}^{{\oplus 2}})^{\oplus r(n)}\oplus M_2({\mathbb{R}})^{\oplus c(n)},\\ {\mathbb{R}}[A_4]&={\mathbb{R}}\oplus{\mathbb{C}}\oplus M_3({\mathbb{R}}),\\ {\mathbb{R}}[S_4]&={\mathbb{R}}^{{\oplus 2}}\oplus M_2({\mathbb{R}})\oplus M_3({\mathbb{R}})^{{\oplus 2}},\\ {\mathbb{R}}[G\times{\mathbb{Z}}_2]&={\mathbb{R}}[G]^{{\oplus 2}}. \end{aligned}$$ ### From Graded Tensor Products to Topological Invariants {#rgtp} By successively analysing all possible point group symmetries, we show that within the extended algebra of symmetries, ${B=\langle{{\mathrm{Cl}}}_{p+1,q},\Pi(G)\rangle}$, one can always make a change of variables and *redefine the action* of $G$ as a ${\mathbb{Z}}_2$-graded abstract group, ${G=G^0\cup G^1}$ (see Appendix \[app:pgf\]). We find elements of $B$ which satisfy ${\tilde{U}_{g}\tilde{U}_{g'}=\tilde{U}_{g\cdot g'}}$ such that the even part, $G^0$, acts trivially on ${{\mathrm{Cl}}}_{p+1,q}$; the odd elements, ${g\in G^1}$, act by ${\tilde{U}_g^{\phantom{1}}a\tilde{U}_g^{-1}=-a}$ on the odd elements of the Clifford algebra, ${a\in{{\mathrm{Cl}}}_{p+1,q}^1}$. These algebraic relations, by definition, bring the extended algebra of symmetries to a ${\mathbb{Z}}_2$-graded tensor product structure, ${B={{\mathrm{Cl}}}_{p+1,q}\hat\otimes{\mathbb{R}}[G]}$, (see Sec. \[sec:graded\]) where the grading of ${\mathbb{R}}[G]$ is induced by $G$. This presentation is much simpler than the original projective action, $\Pi(G)$, (see Sec. \[sec:paradigm\]) since the structure of $B$ is now determined by identifying the ${\mathbb{Z}}_2$-grading of $G$. If $G$ acts trivially on ${{\mathrm{Cl}}}_{p+1,q}$ then we have the simplest cases of ${B={\mathbb{R}}[G]\otimes{{\mathrm{Cl}}}_{p+1,q}}$. Otherwise, we must have a nontrivial grading $$G/G^0={\mathbb{Z}}_2$$ (see Sec. \[sec:ring\]). In order to determine the ${\mathbb{Z}}_2$-graded structure of the group ring corresponding to a specific point group symmetry, we use either ${G=G^0\times{\mathbb{Z}}_2}$ or one of the following possible gradings: $$\begin{gathered} {\mathbb{Z}}_{2n}/{\mathbb{Z}}_{n}={\mathbb{Z}}_2,{{\;\;\:}}{{\mathrm{Dih}}}_{2n}/{{\mathrm{Dih}}}_{n}={\mathbb{Z}}_2,\\ {{\mathrm{Dih}}}_n/{\mathbb{Z}}_n={\mathbb{Z}}_2,{{\;\;\:}}S_4/A_4={\mathbb{Z}}_2.\\ \end{gathered}$$ The grading is determined by identifying the correct even normal subgroup, ${G^0\triangleleft G}$, acting trivially on ${{\mathrm{Cl}}}_{p+1,q}$. This enables us to decompose the group ring ${\mathbb{R}}[G]$ according to irreducible ${\mathbb{Z}}_2$-graded representations of $G$, $${\mathbb{R}}[G]=\bigoplus\nolimits_{\rho} A_{\rho},{{\;\;\:}}{\mathbb{R}}[G^0]=\bigoplus\nolimits_{\rho} A^0_{\rho},$$ where $A^0_{\rho}$ is the even part of $A_{\rho}$, see Sec. \[sec:ring\]. Note, that for each ${\mathbb{Z}}_2$-graded representation, $\rho$, either $A_\rho$ is an *ungraded* representation of $G$, or $A_\rho^0$ is an *ungraded* representation of $G^0$. Hence, these representations may be used to label [@cotton2003chemical] $\rho$ (this is demonstrated in Sec. \[sec:Td\]). Since any ${\mathbb{Z}}_2$-graded real algebra is Morita equivalent to a direct sum of Clifford algebras, we may always write $${\mathbb{R}}[G]=\bigoplus\nolimits_{\rho^{\mathbb{R}}}[M_n({{\mathrm{Cl}}}_{p'',q''})]_{\rho^{\mathbb{R}}}\oplus\bigoplus\nolimits_{\rho^{\mathbb{C}}}[M_n({{\mathrm{Cl}}}_{q''})]_{\rho^{\mathbb{C}}},$$ where ${\{\rho^{{\mathbb{R}}}\}\cup\{\rho^{{\mathbb{C}}}\}=\{\rho\}}$ denote the irreducible ${\mathbb{Z}}_2$-graded representations for which $A_{\rho}$ is either real or complex as a ${\mathbb{Z}}_2$-graded algebra. This decomposition enables us to identify the algebraic structure of the extended algebra of symmetries, and we thus find a Clifford algebra structure corresponding to each irreducible ${\mathbb{Z}}_2$-graded representation $$\begin{aligned} B&={{\mathrm{Cl}}}_{p+1,q}\hat\otimes{\mathbb{R}}[G]\\ &=\bigoplus\nolimits_{\rho^{\mathbb{R}}}[M_n({{\mathrm{Cl}}}_{p'+1,q'})]_{\rho^{\mathbb{R}}}\oplus\bigoplus\nolimits_{\rho^{\mathbb{C}}}[M_n({{\mathrm{Cl}}}_{q'+1})]_{\rho^{\mathbb{C}}}.\nonumber\end{aligned}$$ Here, using the algebraic properties of Clifford algebras presented in Secs. \[sec:graded\] and \[sec:ext\], we find ${p'=p+p''}$ and ${q'=q+q''}$ for the real representations, and ${q'=p+q+q''}$ for the complex representations. This is demonstrated in Sec. \[sec:exam\]. In accordance with Sec. \[sec:ext\], the Clifford algebras corresponding to each ${\mathbb{Z}}_2$-graded representation $\rho$ maps to either a real or a complex classifying space, $$B\mapsto\prod\nolimits_{\rho^{\mathbb{R}}}[{\mathcal{R}}_{p'-q'+2}]_{\rho^{\mathbb{R}}}\times\prod\nolimits_{\rho^{\mathbb{C}}}[{\mathcal{C}}_{q'}]_{\rho^{\mathbb{C}}}.$$ The topological indices classifying materials with point group symmetry $G$, presented in Tables \[tab:p1\]-\[tab:m1\], are the topological invariants of these classifying spaces. Let us denote a Hamiltonian [@Schnyder2008Classification; @ryu2010topological] of AZ class $q$ in $d$ spatial dimensions by $\mathcal{H}_{q,d}^{{\mathbb{R}}}$ for the eight real AZ classes, and by $\mathcal{H}_{q,d}^{{\mathbb{C}}}$ for the two complex AZ classes. The Hamiltonian, $\mathcal{H}^{{\mathbb{R}},G}_{q,d}$, of a crystalline material with point group symmetry $G$ in real AZ class $q$, may be block-decomposed into irreducible ${\mathbb{Z}}_2$-graded representations $\{\rho\}$ of $G$: $$\label{blocks} \mathcal{H}^{{\mathbb{R}},G}_{q,d}=\bigoplus\nolimits_{\rho^{\mathbb{R}}}[\mathcal{H}^{{\mathbb{R}}}_{p'-q'+2+d,d}]_{\rho^{\mathbb{R}}}\oplus\bigoplus\nolimits_{\rho^{\mathbb{C}}}[\mathcal{H}^{{\mathbb{C}}}_{q'+d,d}]_{\rho^{\mathbb{C}}}.$$ Each block is equivalent to an AZ Hamiltonian which is determined by the Clifford algebras corresponding to the irreducible ${\mathbb{Z}}_2$-graded representation, $\rho$. Note, that a simpler analogous construction applies for the Hamiltonians, $\mathcal{H}^{{\mathbb{C}},G}_{q,d}$, of complex AZ classes. The standard topological invariants [@Schnyder2008Classification; @ryu2010topological; @Shiozaki2014Topology; @Chiu2016Classification] characterize the block Hamiltonians of each irreducible ${\mathbb{Z}}_2$-graded representation. These invariants are presented in Table \[tab:inv\]. $s$ $d$ even $d$ odd --------- ----- ------------------ ------------------- ----------------------- complex 0 ${\mathbb{Z}}$ Chern number winding number real 0,4 ${\mathbb{Z}}$ Chern number winding number 1,2 ${\mathbb{Z}}_2$ Fu-Kane invariant Chern-Simons integral : Topological invariants [@Schnyder2008Classification; @ryu2010topological; @Shiozaki2014Topology; @Chiu2016Classification] characterizing the Hamiltonians, $[\mathcal{H}^{{\mathbb{R}}/{\mathbb{C}}}_{s+d,d}]_{\rho}$, corresponding to irreducible ${\mathbb{Z}}_2$-graded representations; see Eq. (\[blocks\]).\[tab:inv\] $$\begin{aligned} &\begin{array}{|l|l||c|c|c||c|c|c||c|c|c} \hline \multicolumn{2}{|l||}{\mathrm{Sch\ddot{o}nflies}} & C_1 & C_{(2n)} & C_3 & C_{1h} & C_{(2n)h} & C_{3h} & C_{1v} & C_{(2n)v} & C_{3v} \\ \hline \multicolumn{2}{|l||}{\mathrm{HM}} & 1 & (2n) & 3 & \mathrm{m} & (2n)/\mathrm{m} & 3/\mathrm{m} & \mathrm{m} & (2n)\mathrm{mm} & 3\mathrm{m} \\ \hline\hline \mathrm{3D} & \mathrm{complex} & {\mathcal{C}}_{q+1} & {\mathcal{C}}_{q+1}^{2n} & {\mathcal{C}}_{q+1}^3 & {\mathcal{C}}_{q} & {\mathcal{C}}_{q}^{2n} & {\mathcal{C}}_{q}^3 & {\mathcal{C}}_{q} & {\mathcal{C}}_{q}^2\!\times\!{\mathcal{C}}_{q+1}^{n-1} & {\mathcal{C}}_{q}\!\times\!{\mathcal{C}}_{q+1} \\ \cline{2-11} & \mathrm{real} & {\mathcal{R}}_{q-3} & {\mathcal{R}}_{q-3}^2\!\times\!{\mathcal{C}}_{q+1}^{n-1} & {\mathcal{R}}_{q-3}\!\times\!{\mathcal{C}}_{q+1} & {\mathcal{R}}_{q-4} & {\mathcal{R}}_{q-4}^2\!\times\!{\mathcal{C}}_{q}^{n-1} & {\mathcal{R}}_{q-4}\!\times\!{\mathcal{C}}_{q} & {\mathcal{R}}_{q-4} & {\mathcal{R}}_{q-4}^2\!\times\!{\mathcal{R}}_{q-5}^{n-1} & {\mathcal{R}}_{q-4}\!\times\!{\mathcal{R}}_{q-5} \\ \hline\hline \mathrm{2D} & \mathrm{complex} & {\mathcal{C}}_{q} & {\mathcal{C}}_{q}^{2n} & {\mathcal{C}}_{q}^3 & {\mathcal{C}}_q^2 & {\mathcal{C}}_q^{4n} & {\mathcal{C}}_q^6 & {\mathcal{C}}_{q+1} & {\mathcal{C}}_{q+1}^2\!\times\!{\mathcal{C}}_{q}^{n-1} & {\mathcal{C}}_{q+1}\!\times\!{\mathcal{C}}_{q} \\ \cline{2-11} & \mathrm{real} & {\mathcal{R}}_{q-2} & {\mathcal{R}}_{q-2}^2\!\times\!{\mathcal{C}}_{q}^{n-1} & {\mathcal{R}}_{q-2}\!\times\!{\mathcal{C}}_{q} & {\mathcal{C}}_q & {\mathcal{C}}_q^{2n} & {\mathcal{C}}_q^3 & {\mathcal{R}}_{q-3} & {\mathcal{R}}_{q-3}^2\!\times\!{\mathcal{R}}_{q-4}^{n-1} & {\mathcal{R}}_{q-3}\!\times\!{\mathcal{R}}_{q-4} \\\hline \end{array}\\\\ &\begin{array}{l|l||c|c|c||c|c|c||c|c|c} \hline \multicolumn{2}{l||}{\mathrm{Sch\ddot{o}nflies}} & D_1 & D_{(2n)} & D_3 & D_{1h} & D_{(2n)h} & D_{3h} & D_{1d} & D_{2d} & D_{3d} \\ \hline \multicolumn{2}{l||}{\mathrm{HM}} & 2 & (2n)22 & 32 & \mathrm{mm}2 & (2n)/\mathrm{mmm} & \bar{6}\mathrm{m}2 & 2/\mathrm{m} & \bar{4}2\mathrm{m} & \bar{3}\mathrm{m} \\ \hline\hline \mathrm{3D} & \mathrm{complex} & {\mathcal{C}}_{q+1}^2 & {\mathcal{C}}_{q+1}^{n+3} & {\mathcal{C}}_{q+1}^{3} & {\mathcal{C}}_{q}^2 & {\mathcal{C}}_{q}^{n+3} & {\mathcal{C}}_{q}^{3} & {\mathcal{C}}_{q}^{2} & {\mathcal{C}}_{q}^{2}\!\times\!{\mathcal{C}}_{q+1} & {\mathcal{C}}_{q}^{3} \\ \cline{2-11} & \mathrm{real} & {\mathcal{R}}_{q-3}^2 & {\mathcal{R}}_{q-3}^{n+3} & {\mathcal{R}}_{q-3}^{3} & {\mathcal{R}}_{q-4}^2 & {\mathcal{R}}_{q-4}^{n+3} & {\mathcal{R}}_{q-4}^{3} & {\mathcal{R}}_{q-4}^{2} & {\mathcal{R}}_{q-4}^{2}\!\times\!{\mathcal{R}}_{q-3} & {\mathcal{R}}_{q-4}^{3} \\ \hline\hline \mathrm{2D} & \mathrm{complex} & {\mathcal{C}}_{q+1} & {\mathcal{C}}_{q+1}^2\!\times\!{\mathcal{C}}_{q}^{n-1} & {\mathcal{C}}_{q+1}\!\times\!{\mathcal{C}}_{q} & {\mathcal{C}}_{q} & {\mathcal{C}}_{q}^{2n} & {\mathcal{C}}_{q}^3 & {\mathcal{C}}_{q} & {\mathcal{C}}_{q+1}^2\!\times\!{\mathcal{C}}_{q} & {\mathcal{C}}_q^3 \\ \cline{2-11} & \mathrm{real} & {\mathcal{R}}_{q-3} & {\mathcal{R}}_{q-3}^2\!\times\!{\mathcal{R}}_{q-4}^{n-1} & {\mathcal{R}}_{q-3}\!\times\!{\mathcal{R}}_{q-4} & {\mathcal{R}}_{q-2} & {\mathcal{R}}_{q-2}^{2n} & {\mathcal{R}}_{q-2}^3 & {\mathcal{R}}_{q-4} & {\mathcal{R}}_{q-3}^2\!\times\!{\mathcal{R}}_{q-4} & {\mathcal{R}}_{q-4}\!\times\!{\mathcal{C}}_q \\\hline \end{array}\\\\ &\begin{array}{l|l||c|c|c||c|c|c|c|c|} \hline \multicolumn{2}{l||}{\mathrm{Sch\ddot{o}nflies}} & S_2 & S_4 & S_6 & T & T_h & T_d & O & O_h \\ \hline \multicolumn{2}{l||}{\mathrm{HM}} & \bar{1} & \bar{4} & \bar{3} & 23 & \mathrm{m}\bar{3} & \bar{4}3\mathrm{m} & 432 & \mathrm{m}\bar{3}\mathrm{m} \\ \hline\hline \mathrm{3D} & \mathrm{complex} & {\mathcal{C}}_{q} & {\mathcal{C}}_{q}^2 & {\mathcal{C}}_{q}^3 & {\mathcal{C}}_{q+1}^{4} & {\mathcal{C}}_{q}^{4} & {\mathcal{C}}_{q}^2\!\times\!{\mathcal{C}}_{q+1} & {\mathcal{C}}_{q+1}^{5} & {\mathcal{C}}_{q}^{5} \\ \cline{2-10} & \mathrm{real} & {\mathcal{R}}_{q-4} & {\mathcal{R}}_{q-4}\!\times\!{\mathcal{R}}_{q-2} & {\mathcal{R}}_{q-4}\!\times\!{\mathcal{C}}_{q} & {\mathcal{R}}_{q-3}^{2}\!\times\!{\mathcal{C}}_{q+1} & {\mathcal{R}}_{q-4}^{2}\!\times\!{\mathcal{C}}_{q} & {\mathcal{R}}_{q-4}^2\!\times\!{\mathcal{R}}_{q-5} & {\mathcal{R}}_{q-3}^{5} & {\mathcal{R}}_{q-4}^{5} \\ \hline\hline \mathrm{2D} & \mathrm{complex} & {\mathcal{C}}_q^2 & {\mathcal{C}}_{q}^{4} & {\mathcal{C}}_q^6 \\ \cline{2-5} & \mathrm{real} & {\mathcal{C}}_q & {\mathcal{R}}_{q-2}^{2}\!\times\!{\mathcal{C}}_{q} & {\mathcal{C}}_q^3 \\\cline{1-5} \end{array} \end{aligned}$$ Our classification scheme, which was presented in this section, is fully demonstrated within the following concrete examples. Examples {#sec:exam} ======== We now give three pedagogical examples demonstrating our technique. We then use these examples to demonstrate the construction of a model Hamiltonian manifesting our classification results. The analyses for all other point group symmetries are brought in Appendix \[app:pgf\]. Threefold Rotational Symmetry $C_3$ {#sec:c3} ----------------------------------- {height="\fgx"} Let us begin by considering a crystalline insulator (or superconductor) with a threefold rotational symmetry point group $C_3$ which is one of the simplest point group symmetries. As an abstract group, the symmetry group is given by the cyclic group ${G={\mathbb{Z}}_3}$, and we mark the generator of its projective fermionic representation by $\hat{c}$ such that $$\hat c^3=-1.$$ The action of this generator on the 3D space is given by a simple threefold rotation $$\hat c(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix})\hat c^{-1}=(\begin{smallmatrix}\cos\frac{2\pi}{3}&\sin\frac{2\pi}{3}\\-\sin\frac{2\pi}{3}&\cos\frac{2\pi}{3}\end{smallmatrix})(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix}).$$ We wish to find the structure of the algebra ${B=\langle{{\mathrm{Cl}}}_{q+7,3},\Pi({\mathbb{Z}}_3)\rangle}$ (and corresponding classifying space) generated by $\hat{c}$ as well as by the Dirac gamma matrices, $\gamma_1,\gamma_2,\gamma_3$, and the other non spatial Clifford algebra generators $\{\gamma\}\subset\{y\}$. In order to do so, we notice that $$\label{rotrel} \begin{aligned} e^{-\gamma_1\gamma_2\frac{2\pi}{6}}(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix})e^{\gamma_1\gamma_2\frac{2\pi}{6}} &=e^{-\gamma_1\gamma_2\frac{2\pi}{3}}(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix}) \\ &=(\begin{smallmatrix}\cos\frac{2\pi}{3}&\sin\frac{2\pi}{3}\\-\sin\frac{2\pi}{3}&\cos\frac{2\pi}{3}\end{smallmatrix})(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix}), \end{aligned}$$ where we used the anticommutation relations of the Dirac gamma matrices; see Eq. (\[gammapt\]). This prompts the definition of a new abstract generator $c$ such that $$c = e^{\gamma_1\gamma_2\frac{2\pi}{6}}\hat c.$$ This new abstract generator satisfies $$\label{c3rel} {{\;\;\:}}c y_i c^{-1}=y_i,{{\;\;\:}}c^3=1.$$ We see that $c$ is completely decoupled from the generators of the Clifford algebra, $y_i$, and thus get ${B={\mathbb{R}}[{\mathbb{Z}}_3]\otimes{{\mathrm{Cl}}}_{q+7,3}}$ and hence $$\label{resc3} ({\mathbb{R}}\oplus{\mathbb{C}})\otimes{{\mathrm{Cl}}}_{q+7,3}\mapsto{\mathcal{R}}_{q-3}\times{\mathcal{C}}_{q+1}.$$ Here, we have used the mapping in Sec. \[sec:ext\] to determine the classifying space. The classifying spaces of the complex AZ classes immediately follow by utilizing ${{\mathbb{C}}\otimes{{\mathrm{Cl}}}_{p,q}={{\mathrm{Cl}}}_{p+q}}$ and ${{\mathbb{C}}\otimes{{\mathrm{Cl}}}_{q}={{\mathrm{Cl}}}_{q}^{{\oplus 2}}}$, $$\label{resc3c} ({\mathbb{R}}\oplus{\mathbb{C}})\otimes{{\mathrm{Cl}}}_{q+10}\mapsto{\mathcal{C}}_{q+1}^3.$$ The topological indices classifying materials with $C_3$ threefold rotational symmetry, presented in Table \[tab:exc3\], are the topological invariants of these classifying spaces \[Eqs. (\[resc3\]) and (\[resc3c\])\]. ### Model Hamiltonians and Topological Invariants {#model-hamiltonians-and-topological-invariants .unnumbered} In order to construct a model Hamiltonian for these crystalline phases, one can use the model Hamiltonians of the tenfold-way classification with non-spatial symmetries [@Schnyder2008Classification; @ryu2010topological]. We denote the model Hamiltonian of AZ class $q$ in $d$ spatial dimensions by $\mathcal{H}_{q,d}^{{\mathbb{R}}}$ for the eight real AZ classes, and by $\mathcal{H}_{q,d}^{{\mathbb{C}}}$ for the two complex AZ classes; see Sec. \[rgtp\]. As a consequence of Eq. (\[c3rel\]) one can diagonalize the Hamiltonians for $C_3$ threefold rotational symmetry simultaneously with $c$. We thus write a model Hamiltonian, $\mathcal{H}_{q,3}^{{\mathbb{R}},C_3}$, in a block diagonal form, labelled by the eigenvalues of $c$: $$\mathcal{H}_{q,3}^{{\mathbb{R}},C_3}=\begin{pmatrix} \big[\mathcal{H}_{q,3}^{{\mathbb{R}}}\big]_{c=1} & 0 \\ 0 & \big[\mathcal{H}_{q,3}^{{\mathbb{C}}}\big]_{c=e^{\pm\frac{2\pi i}{3}}} \end{pmatrix}.$$ The complex AZ class yield a similar decomposition, $$\mathcal{H}_{q,3}^{{\mathbb{C}},C_3}=\begin{pmatrix} \big[\mathcal{H}_{q,3}^{{\mathbb{C}}}\big]_{c=1} & 0 & 0 \\ 0 & \big[\mathcal{H}_{q,3}^{{\mathbb{C}}}\big]_{c=e^{+\frac{2\pi i}{3}}} & 0 \\ 0 & 0 & \big[\mathcal{H}_{q,3}^{{\mathbb{C}}}\big]_{c=e^{-\frac{2\pi i}{3}}} \end{pmatrix}.$$ The topological invariants of each block are the topological invariants of the corresponding Hamiltonians [@Schnyder2008Classification; @ryu2010topological; @Shiozaki2014Topology; @Chiu2016Classification]; see Table \[tab:inv\]. Fourfold Rotoreflection Symmetry $S_{4}$ {#sec:s4} ---------------------------------------- ![A solid with the fourfold rotoreflection symmetry point group $S_4$ of tetragonal-disphenoidal crystals.\[fig:s4\]](S4_3D.png){height="\fgx"} Let us next consider a crystalline insulator (or superconductor) with a fourfold rotoreflection symmetry point group $S_4$. As an abstract group, the symmetry group is given by the cyclic group ${G={\mathbb{Z}}_4}$, and we mark the generator of its projective fermionic representation by $\hat{s}_4$ such that $$\hat s^{4}=-1.$$ The action of this generator on the 3D space is given by a fourfold rotoreflection $$\hat s(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix})\hat s^{-1}=(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix})(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix}),{{\;\;\:}}\hat s\gamma_3 \hat s^{-1}=-\gamma_3.$$ We wish to find the structure of the algebra ${B=\langle{{\mathrm{Cl}}}_{q+7,3},\Pi({\mathbb{Z}}_4)\rangle}$ (and corresponding classifying space) generated by $\hat{s}_4$ as well as by the Dirac gamma matrices, $\gamma_1,\gamma_2,\gamma_3$, and the other non spatial Clifford algebra generators $\{\gamma\}\subset\{y\}$. In an analogous manner to Eq. (\[rotrel\]) we define new abstract generators $s_4$ such that $$s = e^{\gamma_1\gamma_2\frac{\pi}{4}}\hat s\gamma_3.$$ This new abstract generator satisfies $$\label{s4rel} s y_i s^{-1}= -y_i,{{\;\;\:}}s^{4}=1.$$ Here, we used the equality ${(s^{2})^2=(-e^{-\gamma_1\gamma_2\frac{2\pi}{4}}\hat s^2)^2=1}$. Trying to analyse the algebraic structure, we find a central element ${t=s^2}$ satisfying ${t^2=1}$. Our algebra decomposes accordingly as a direct sum of two algebras, such that in one of them ${s^2=t=1}$ and in the other ${s^2=t=-1}$. In each of these algebras, the appropriate extra generator (${s^2=\pm1}$) is simply added to the generators, $y_i$, of the original Clifford algebra, ${{\mathrm{Cl}}}_{q+7,3}$, and we therefore get $${{\mathrm{Cl}}}_{q+7,3+1}\oplus{{\mathrm{Cl}}}_{q+7+1,3}\mapsto{\mathcal{R}}_{q-4}\times{\mathcal{R}}_{q-2},$$ where we have once again used the mapping in Sec. \[sec:ext\] to determine the classifying space. The classifying spaces of the complex AZ classes immediately follow by utilizing ${\mathbb{C}}\otimes{{\mathrm{Cl}}}_{p,q}={{\mathrm{Cl}}}_{p+q}$: $$\label{ress4c} {{\mathrm{Cl}}}_{q+11}\oplus{{\mathrm{Cl}}}_{q+11}\mapsto{\mathcal{C}}_{q}^2.$$ Moreover, the same results may be achieved by noticing that the relations in Eq. (\[s4rel\]) are exactly the relations of a ${\mathbb{Z}}_2$-graded tensor product ${B={{\mathrm{Cl}}}_{q+7,3}\hat{\otimes}{\mathbb{R}}[{\mathbb{Z}}_4]}$ where the “even" part of ${\mathbb{R}}[{\mathbb{Z}}_4]$, which commutes with the generators of ${{\mathrm{Cl}}}_{q+7,3}$, is generated by elements with the generator, $s$, occurring an even number of times, i.e., $s^2$ and $1$. These even elements generate the subalgebra ${\mathbb{R}}[{\mathbb{Z}}_2]$ of the cyclic group ${{\mathbb{Z}}_2\triangleleft {\mathbb{Z}}_4}$. By knowing the grading of the groups ${{\mathbb{Z}}_4/{\mathbb{Z}}_2={\mathbb{Z}}_2}$, we can find the grading of the group rings ${{\mathbb{R}}[{\mathbb{Z}}_2]\hookrightarrow{\mathbb{R}}[{\mathbb{Z}}_4]}$ as a direct sum of Clifford algebras $$\begin{matrix} {\mathbb{R}}[{\mathbb{Z}}_2]&=&{\mathbb{R}}\oplus{\mathbb{R}}\\ {\mathbin{\text{\rotatebox[origin=c]{-90}{$\hookrightarrow$}}}}&&{\mathbin{\text{\rotatebox[origin=c]{-90}{$\hookrightarrow$}}}}&&\\ {\mathbb{R}}[{\mathbb{Z}}_4]&=&{\mathbb{R}}^{{\oplus 2}}\oplus{\mathbb{C}}&=&{{\mathrm{Cl}}}_{0,1}\oplus{{\mathrm{Cl}}}_{1,0}. \end{matrix}$$ We can thus find its graded tensor product with any Clifford algebra $$\begin{aligned} {{\mathrm{Cl}}}_{p,q}\hat{\otimes}{\mathbb{R}}[{\mathbb{Z}}_4]&={{\mathrm{Cl}}}_{p,q}\hat{\otimes}({{\mathrm{Cl}}}_{0,1}\oplus{{\mathrm{Cl}}}_{1,0})\\&={{\mathrm{Cl}}}_{p,q+1}\oplus{{\mathrm{Cl}}}_{p+1,q}. \end{aligned}$$ And specifically for ${B={{\mathrm{Cl}}}_{q+7,3}\hat{\otimes}{\mathbb{R}}[{\mathbb{Z}}_4]}$ we find $$\label{ress4} {{\mathrm{Cl}}}_{q+7,3+1}\oplus{{\mathrm{Cl}}}_{q+7+1,3}\mapsto{\mathcal{R}}_{q-4}\times{\mathcal{R}}_{q-2}.$$ Such analyses would come in handy in other more complicated cases such as our final example. The topological indices classifying materials with $S_4$ fourfold rotoreflection symmetry, presented in Table \[tab:exs4\], are the topological invariants of these classifying spaces, Eqs. (\[ress4\]), (\[ress4c\]). ### Model Hamiltonians and Topological Invariants {#model-hamiltonians-and-topological-invariants-1 .unnumbered} In order to construct a model Hamiltonian for these crystalline phases, one can use the model Hamiltonians of the tenfold-way classification with non-spatial symmetries [@Schnyder2008Classification; @ryu2010topological], $\mathcal{H}_{q,d}^{{\mathbb{R}}/{\mathbb{C}}}$, for real/complex AZ class $q$ in $d$ spatial dimensions; see Sec. \[rgtp\]. As a consequence of Eq. (\[s4rel\]) one can diagonalize the Hamiltonians for $S_4$ fourfold rotoreflection symmetry simultaneously with $s^2$. We thus write a model Hamiltonian, $\mathcal{H}_{q,3}^{{\mathbb{R}},S_4}$, in a block diagonal form, labelled by the eigenvalues of $s^2$: $$\mathcal{H}_{q,3}^{{\mathbb{R}},S_4}=\begin{pmatrix} \big[\mathcal{H}_{q-1,3}^{{\mathbb{R}}}\big]_{s^2=+1} & 0 \\ 0 & \big[\mathcal{H}_{q+1,3}^{{\mathbb{R}}}\big]_{s^2=-1} \end{pmatrix}.$$ The complex AZ class yield a similar decomposition, $$\mathcal{H}_{q,3}^{{\mathbb{C}},S_4}=\begin{pmatrix} \big[\mathcal{H}_{q+1,3}^{{\mathbb{C}}}\big]_{s^2=+1} & 0 \\ 0 & \big[\mathcal{H}_{q+1,3}^{{\mathbb{C}}}\big]_{s^2=-1} \end{pmatrix}.$$ The topological invariants of each block are the topological invariants of the corresponding Hamiltonians [@Schnyder2008Classification; @ryu2010topological; @Shiozaki2014Topology; @Chiu2016Classification]; see Table \[tab:inv\]. Full Tetrahedral Symmetry $T_d$ {#sec:Td} ------------------------------- {height="\fgx"} Let us finally consider a crystalline insulator (or superconductor) with a full tetrahedral cubic symmetry point group $T_d$ which is one of the most intricate point group symmetries. As an abstract group, the symmetry group is given by the symmetric group ${G=S_4}$, and we mark the generators of its projective fermionic representation by $\hat{c}_3,\hat{s}_4$ such that $$\hat c_3^3=-1,{{\;\;\:}}\hat s_4^4=-1,{{\;\;\:}}(\hat{s}_4\hat c_3)^2=-1.$$ The action of these generators on the 3D space is given by a threefold rotation for $\hat{c}_3$ and by a fourfold rotoreflection for $\hat{s}_4$ such that $$\hat c_3^{\phantom{|}}\gamma_i\hat c_3^{-1}=\gamma_{i+1},{{\;\;\:}}\hat s_4^{\phantom{|}}\Big(\begin{smallmatrix}\gamma_1\\\gamma_2\\\gamma_3\end{smallmatrix}\Big)\hat s_4^{-1}=\Big(\begin{smallmatrix}-\gamma_2\\+\gamma_1\\-\gamma_3\end{smallmatrix}\Big).$$ We wish to find the structure of the algebra ${B=\langle{{\mathrm{Cl}}}_{q+7,3},\Pi(S_4)\rangle}$ (and corresponding classifying space) generated by $\hat{c}_3,\hat{s}_4$ as well as by the Dirac gamma matrices, $\gamma_1,\gamma_2,\gamma_3$, and the other non spatial Clifford algebra generators $\{\gamma\}\subset\{y\}$. In an analogous manner to Eq. (\[rotrel\]), we define new abstract generators $c_3,s_4$ such that $$c_3 = \tfrac{1+\gamma_1\gamma_2+\gamma_2\gamma_3+\gamma_3\gamma_1}{2}c_3,{{\;\;\:}}s_4 = e^{-\gamma_1\gamma_2\frac{\pi}{4}}\hat{s}_4\gamma_3.$$ These new abstract generators satisfy $$\label{Tdrel} \begin{gathered} c_3^{\phantom{|}} y_i c_3^{-1}=y_i,{{\;\;\:}}s_4^{\phantom{|}} y_i s_4^{-1}=-y_i,\\ c_3^3=1,{{\;\;\:}}s_4^4=1,{{\;\;\:}}(s_4 c_3)^2=1. \end{gathered}$$ As above, these relations are exactly the relations of a ${\mathbb{Z}}_2$-graded tensor product ${B={{\mathrm{Cl}}}_{q+7,3}\hat{\otimes}{\mathbb{R}}[S_4]}$ where the “even" part of ${\mathbb{R}}[S_4]$, which commutes with the generators of ${{\mathrm{Cl}}}_{q+7,3}$, is generated by elements with $s_4$ occurring an even number of times. These even elements generate the subalgebra ${\mathbb{R}}[A_4]$ of the alternating group ${A_4\triangleleft S_4}$. By knowing the grading of the groups ${S_4/A_4={\mathbb{Z}}_2}$, we can find the grading of the group rings ${{\mathbb{R}}[A_4]\hookrightarrow{\mathbb{R}}[S_4]}$ as a direct sum of Clifford algebras $$\begin{matrix} {\mathbb{R}}[A_4]&=&{\mathbb{R}}\oplus{\mathbb{C}}\oplus M_3({\mathbb{R}})\\ {\mathbin{\text{\rotatebox[origin=c]{-90}{$\hookrightarrow$}}}}&&{\mathbin{\text{\rotatebox[origin=c]{-90}{$\hookrightarrow$}}}}\\ {\mathbb{R}}[S_4]&=&{\mathbb{R}}^{{\oplus 2}}\oplus M_2({\mathbb{R}})\oplus M_3({\mathbb{R}})^{{\oplus 2}}\\ &&{\mathbin{\text{\rotatebox[origin=c]{-90}{$=$}}}}\\ &&{{\mathrm{Cl}}}_{0,1}\oplus{{\mathrm{Cl}}}_{0,2}\oplus(M_3({\mathbb{R}})\otimes{{\mathrm{Cl}}}_{0,1}). \end{matrix}$$ We can thus find its graded tensor product with any Clifford algebra $$\begin{aligned} {{\mathrm{Cl}}}_{p,q}\hat\otimes{\mathbb{R}}[S_4]&={{\mathrm{Cl}}}_{p,q}\hat\otimes({{\mathrm{Cl}}}_{0,1}\oplus{{\mathrm{Cl}}}_{0,2}\oplus(M_3({\mathbb{R}})\otimes{{\mathrm{Cl}}}_{0,1}))\\ &=(({\mathbb{R}}\oplus M_3({\mathbb{R}}))\otimes{{\mathrm{Cl}}}_{p,q+1})\oplus{{\mathrm{Cl}}}_{p,q+2}. \end{aligned}$$ And specifically for ${B={{\mathrm{Cl}}}_{q+7,3}\hat{\otimes}{\mathbb{R}}[S_4]}$ we find $$\begin{gathered} \label{resTd} (({\mathbb{R}}\oplus M_3({\mathbb{R}}))\otimes{{\mathrm{Cl}}}_{q+7,3+1})\oplus{{\mathrm{Cl}}}_{q+7,3+2}\\ \mapsto{\mathcal{R}}_{q-4}^2\times{\mathcal{R}}_{q-5},\end{gathered}$$ where we have once again used the mapping in Sec. \[sec:ext\] to determine the classifying space. The classifying spaces of the complex AZ classes immediately follow by utilizing ${\mathbb{C}}\otimes{{\mathrm{Cl}}}_{p,q}={{\mathrm{Cl}}}_{p+q}$: $$\label{resTdc} (({\mathbb{R}}\oplus M_3({\mathbb{R}}))\otimes{{\mathrm{Cl}}}_{q+11})\oplus{{\mathrm{Cl}}}_{q+12} \mapsto{\mathcal{C}}_{q}^2\times{\mathcal{C}}_{q+1}.$$ The topological indices classifying materials with $T_d$ full tetrahedral symmetry, presented in Table \[tab:exTd\], are the topological invariants of these classifying spaces, Eqs. (\[resTd\]), (\[resTdc\]). The similar analyses of Appendix \[app:pgf\] are used to determine the classifying spaces in Table \[tab:class\] and the topological invariants in Tables \[tab:p1\]-\[tab:m1\]. ### Model Hamiltonians and Topological Invariants {#model-hamiltonians-and-topological-invariants-2 .unnumbered} In order to construct a model Hamiltonian for these crystalline phases, one can use the model Hamiltonians of the tenfold-way classification with non-spatial symmetries [@Schnyder2008Classification; @ryu2010topological], $\mathcal{H}_{q,d}^{{\mathbb{R}}/{\mathbb{C}}}$, for real/complex AZ class $q$ in $d$ spatial dimensions; see Sec. \[rgtp\]. As a consequence of Eq. (\[Tdrel\]) one can write a model Hamiltonian, $\mathcal{H}_{q,3}^{{\mathbb{R}},T_d}$, in a block diagonal form, labelled by the irreducible representations of the subgroup $A_4$ generated by $c_3$ and $s_4^2$: $$\mathcal{H}_{q,3}^{{\mathbb{R}},T_d}=\begin{pmatrix} \big[\mathcal{H}_{q-1,3}^{{\mathbb{R}}}\big]_{A} & 0 & 0\\ 0 & \big[\mathcal{H}_{q-2,3}^{{\mathbb{R}}}\big]_{E} & 0\\ 0 & 0 & \big[\mathcal{H}_{q-1,3}^{{\mathbb{R}}}\big]_{T} \end{pmatrix}.$$ Here, we use the Mulliken symbols [@cotton2003chemical], $A$, $E$, and $T$, to identify the irreducible representations corresponding to the subalgebra ${{\mathbb{R}}[A_4]={\mathbb{R}}\oplus{\mathbb{C}}\oplus M_3({\mathbb{R}})}$. The complex AZ class yields a similar decomposition, $$\mathcal{H}_{q,3}^{{\mathbb{C}},T_d}=\begin{pmatrix} \big[\mathcal{H}_{q+1,3}^{{\mathbb{C}}}\big]_{A} & 0 & 0\\ 0 & \big[\mathcal{H}_{q,3}^{{\mathbb{C}}}\big]_{E} & 0\\ 0 & 0 & \big[\mathcal{H}_{q+1,3}^{{\mathbb{C}}}\big]_{T} \end{pmatrix}.$$ The topological invariants of each block are the topological invariants of the corresponding Hamiltonians [@Schnyder2008Classification; @ryu2010topological; @Shiozaki2014Topology; @Chiu2016Classification]; see Table \[tab:inv\]. In fact, the above examples in Secs. \[sec:c3\] and \[sec:s4\] may also be recast into irreducible representation form by similarly identifying the subalgebras ${{\mathbb{R}}[{\mathbb{Z}}_3]={\mathbb{R}}\oplus{\mathbb{C}}}$ and ${{\mathbb{R}}[{\mathbb{Z}}_2]={\mathbb{R}}\oplus{\mathbb{R}}}$ with the Mulliken symbols $\{A,E\}$ and $\{A,B\}$, respectively. Moreover, one may use analogous irreducible representation analyses of the other point groups (see Sec. \[sec:pgs\] and Ref. \[\]) in order to construct model Hamiltonians for all crystalline insulators and superconductors presented in Tables \[tab:p1\]-\[tab:m1\]; see Sec. \[rgtp\]. Discussion {#sec:discussion} ========== In this paper, we have presented a complete classification of bulk topological invariants of crystalline topological insulators and superconductors in all AZ symmetry classes protected by all 32 point group symmetries of 3D crystals as well as by all 31 symmorphic layer group symmetries of 2D crystals. The majority of phases found by our classification paradigm are indeed novel crystalline topological insulators and superconductors. However, this classification is not exhaustive as crystals in nature may also have a nonsymmorphic magnetic space/layer group symmetry [@graph]. In this discussion, we compare our results to those of previous works which have provided other classification schemes, emphasize the similarities and differences, and present some prospective ideas as to gaining a future exhaustive classification scheme of all topological phases of crystalline matter. Moreover, the experimental edge content signatures of many topological invariants are yet unclear, we thus also discuss the possible classification of edge content related to the invariants of crystalline topological phases. Magnetic Crystals {#sec:mag} ----------------- Although not directly calculated in this paper, it is relatively straightforward to generalize our results and classify topological phases protected by any of the 122 magnetic point group symmetries [@graph] (also known as double point groups) of magnetic crystals [@Shiozaki2014Topology; @Schindler2018Higher; @Sato2009Topological; @Mong2010Antiferromagnetic; @Mizushima2012Symmetry; @mizushima2013topological; @Fang2013Topological; @liu2013antiferromagnetic; @kotetes2013classification; @Fang2014New; @Zhang2015Topological; @Watanabe2018Structure; @lifshitz2005magnetic]. These crystals are invariant under elements of a group ${M=\langle N,(G\setminus N)1'\rangle}$ for some normal subgroup ${G/N={\mathbb{Z}}_2}$ of the crystallographic point group $G$. Here, we use ${1'^2=1}$ to be the symmetry action flipping the spin direction. Our methods can be directly extended to treat such cases; as an example we present the results for the simplest case of $C_{2n}'$ symmetry in Table \[tab:mag\]; see Appendix \[app:mag\]. Order-two magnetic point groups [@graph] such as $C_2'$ were previously treated by Morimoto and Furusaki \[\] and by Shiozaki and Sato \[\], and our results are in complete agreement. The special case of $C_4'$ was recently treated by Schindler *et al.* \[\] where a ${\mathbb{Z}}_2$ classification of the higher-order hinge states was found and is also in complete agreement with our results. We henceforth discuss such states in detail. Note, that a complete classification of the symmetry indicators of band structure topology of crystals with all 1651 magnetic space groups [@graph] in AZ classes A and AI was recently carried out in Ref. \[\]. $$\begin{array}{l||c|c|c} \mathrm{Sch\ddot{o}nflies} & C_2' & C_4' & C_6' \\ \hline \mathrm{HM} & 2' & 4' & 6' \\ \hline\hline T^2=-1 & {\mathcal{R}}_{4-d} & {\mathcal{R}}_{4-d}\times{\mathcal{R}}_{-d} & {\mathcal{R}}_{4-d}\times{\mathcal{C}}_{d} \\ \hline d=3 & {\mathbb{Z}}_2 & {\mathbb{Z}}_2 & {\mathbb{Z}}_2 \\ d=2 & {\mathbb{Z}}_2 & {\mathbb{Z}}_2 & {\mathbb{Z}}_2\times{\mathbb{Z}}\\ \hline\hline T^2=+1 & {\mathcal{R}}_{-d} & {\mathcal{R}}_{-d}\times{\mathcal{R}}_{4-d} & {\mathcal{R}}_{-d}\times{\mathcal{C}}_{d} \\ \hline d=3 & 0 & {\mathbb{Z}}_2 & 0 \\ d=2 & 0 & {\mathbb{Z}}_2 & {\mathbb{Z}}\end{array}$$ Edge States and Higher Order Topological Insulators and Superconductors ----------------------------------------------------------------------- Until recently, bulk-boundary correspondence was considered a defining hallmark of topological phases [@franz2013topological; @Kane2005Topological; @Kane2005Quantum; @FuKaneMele2007Topological; @Moore2007Topological; @Hsieh2009Observation; @Roy2009Topological; @Fu2007Topological; @Konig2007Quantum; @xia2009observation; @Chen2009Experimental]. Nevertheless, the recent discovery of higher-order topological insulators and superconductors drastically changed our understanding; some crystalline topological phases host no boundary modes [@Hughes2011Inversion; @Shiozaki2014Topology] while others host either boundary states [@Shiozaki2015Z2; @Khalaf2018Symmetry] or more exotic hinge or corner states [@parameswaran2017topological; @Benalcazar2017Quantized; @Benalcazar2017Electric; @Song2017d; @Langbehn2017Reflection; @Schindler2018Higher; @schindler2018bismuth; @xu2017topological; @Shapourian2018Topological; @lin2017topological; @Ezawa2018Higher; @Khalaf2018Higher; @Geier2018Second; @trifunovic2018higher; @fang2017rotation]. A full treatment of the surface states of 3D crystalline topological insulators in AZ class AII was recently carried out by Khalaf *et al.* \[\] for all space group symmetries [@graph]. Their analysis indicates that all surface states are in fact projections of the $\Gamma$-point Hamiltonian \[see Eq. (\[gammapt\])\], and indeed, when comparing with our results for all point group symmetries, we find the surface state indices to be subgroups of our bulk invariants. The classification of higher-order topological insulators and superconductors with order-two symmetries [@graph] was recently accomplished by Geier *et al.* \[\], Trifunovic and Brouwer \[\], and Khalaf \[\]; we believe it should be possible to combine their methods with ours to achieve a full classification of higher-order topological invariants for all crystalline symmetries. Note, that parallel work in the complex AZ class A by Okuma *et al.* \[\] achieved the classification of higher-order topological phases for all magnetic point group symmetries, and their bulk topological invariants in AZ class A are in complete agreement with our results for all point group symmetries. Full Brillouin zone structure, Symmetry indicators, and “Weak" Topological Invariants ------------------------------------------------------------------------------------- Even without crystalline symmetry, the strong bulk invariant of the tenfold-way classification [@kitaev2009periodic; @schnyder2009classification; @ryu2010topological; @Hasan2010Colloquium; @moore2010birth; @franz2013topological; @witten2015three] (see Table \[tab:per\]) is not the only possible topological invariant characterizing the material. Non-trivial topology may also occur along lower-dimensional surfaces(/curves) within the BZ torus $T^d$; these are known as “weak" topological insulators and superconductors [@FuKaneMele2007Topological; @Moore2007Topological]. In fact, in AZ class AII, for example, the cellular (CW-complex) decomposition of the torus ${T^3=e^0\cup3e^1\cup3e^2\cup e^3}$ gives rise [@kitaev2009periodic; @Kennedy2015Homotopy] to the three “weak" ${\mathbb{Z}}_2$ topological indices (see Table \[tab:per\]) $$\underbrace{{\mathbb{Z}}\ \times\ 0^3\ \times\ {\mathbb{Z}}_2^3}_{\mathrm{weak}}\ \times\underbrace{{\mathbb{Z}}_2}_{\mathrm{strong}}.$$ The introduction of the crystalline structure complicates this simple relation between the strong bulk invariant and the weak ones [@Varjas2017Space]. Using the elementary band representations approach it is relatively easy to get the complete symmetry indicators for the full BZ torus [@Dong2016Classification; @po2017symmetry; @bradlyn2017topological; @Watanabe2018Structure; @Bradlyn2018Band; @song2018quantitative; @Cano2018Building; @Vergniory2017Graph; @Ono2018Unified; @vergniory2018high]. However, one still has to explicitly evaluate the Berry phases through various surfaces and curves (and Berry phases thereof) to find the different topological phases sharing an elementary band representation. Using the K-theoretic approach [@freed2013twisted; @Shiozaki2017Topological; @shiozaki2018atiyah] one studies the equivariant symmetry group action on the BZ torus ($G$-CW complex). This approach was successfully utilized by Shiozaki, Sato, and Gomi \[\] and yielded a complete classification for the wallpaper groups in the complex AZ classes (A and AIII). Indeed, when comparing with our results for the symmorphic layer groups [@graph], $C_n,C_{nv}$, we find our bulk invariants to be subgroups of their full BZ torus K-groups. Defects and Higher Dimensional Systems -------------------------------------- Our paradigm is not restricted to point groups and can in fact be applied to classify crystalline topological phases in any spatial dimension, in particular, it can be applied to any of the 271 point groups (and 1202 magnetic point groups) of four-dimensional (4D) space. However, the immediate physical applicability of 4D crystals is less obvious than of 3D crystals, and so we leave this for prospecting future work. A much more immediately relevant topic is that of crystals with defects. It has long been noticed [@Teo2010Topological] that for the non-crystalline topological phases, all point, line, and surface defects may be easily incorporated into the classification schemes. This fact was explicitly shown to hold for crystalline materials with order-two [@graph] symmetries [@Shiozaki2014Topology] and had even also been formulated for the general crystalline case [@Benalcazar2014Classification; @Shiozaki2017Topological]. Following Refs. \[\], let us observe a ${(\delta-1)}$-dimensional defect. It is surrounded by a sphere $S^D$ of codimension ${D=d-\delta}$; let us parametrize it by the spatial coordinates ${\mathbf{r}\in {\mathbb{R}}^{D+1}}$ with ${||\mathbf{r}||=1}$. The crystalline symmetry $G$ leaves the defect invariant and hence acts separately on the momenta parallel to the defect ${\mathbf{k}_\parallel\in T^{d-D-1}}$ and on the momenta ${\mathbf{k}_\perp}$ conjugate to $\mathbf{r}$, $$U_g\mathcal{H}(\mathbf{k},\mathbf{r})U_g^{-1}=\mathcal{H}(O_{g\parallel}\mathbf{k}_\parallel,O_{g\perp}\mathbf{k}_\perp,O_{g\perp}\mathbf{r}),$$ where ${O_{g\parallel}\in\mathrm{O}(d-D-1)}$ and ${O_{g\perp}\in\mathrm{O}(D+1)}$ are orthogonal transformations; cf. Eq. (\[Og\]). The algebraic structure may be restored by setting ${M(\mathbf{r})=\bm{\gamma}'\cdot\mathbf{r}}$ with ${\{\gamma'_i,\gamma'_j\}=-2\delta_{ij}}$ and ${\{\gamma'_i,\gamma_j\}=0}$ such that ${M^2=-1}$; cf. Eq. (\[gammapt\]). Such analyses may be carried out in future works to classify the topological phases of defected crystals. “Fragile" Topological Phases ---------------------------- It was recently noticed that some disconnected elementary band representations corresponding to non-trivial topological phases are trivializable by addition of trivial occupied bands [@Po2018Fragile; @bouhon2018wilson; @bradlyn2018disconnected], these were dubbed “fragile" topological phases. It is often the case that in order to capture the topology of a macroscopically large number of bands, one introduces a stable-equivalence relation which disregards two phases as equivalent if they differ by addition of some trivial occupied or unoccupied bands. Such “fragile" phases are missed by this stable-equivalence relation. The existence of similar phases was noticed even without crystalline symmetries, e.g., in Hopf topological insulators [@Moore2008Topological; @Deng2013Hopf] and Hopf topological superconductors [@Kennedy2016Topological]. In order to generalize the results of our paper for such “fragile" phases, one needs to avoid taking the stable limit in Eq. (\[MapstoDef\]) and count the connected components of the resulting topological space. However, more analysis is required as in the unstable case [@kennedy2016bott]; not all bulk topologies are captured by a Dirac Hamiltonian (\[gammapt\]). Acknowledgements {#acknowledgements .unnumbered} ================ All symmetry solids in this paper are taken with permission from Ashcroft and Mermin [@ashcroft1976solid]. E.C. acknowledges the help of his PhD advisor E. Sela, and of the K-theory study group at Tel-Aviv University for providing the required background. A.C. thanks Perimeter Institute for Theoretical Physics for the hospitality in the Summer of 2018, during which most of the work on this paper was carried out. The authors thank B. A. Bernevig, T. Neupert, I. Le, R. Lifshitz, R. Ilan, A. Bouhon, and J. Hoeller for the fruitful discussions. Hierarchy of Symmetry Groups {#app:symmgraph} ============================ ![Hierarchy of symmetry group classes. []{data-label="fig:graph"}](FigApp1.png){width="\linewidth"} In this appendix, we present the hierarchical structure of point group classes. The hierarchical structure is presented as a graph in Fig. \[fig:graph\]. The intersection of every two adjacent symmetry group classes is given below them; e.g., the 2D point groups are all symmetry groups which are both wallpaper groups and symmorphic layer groups. Note, however, that there are some “accidental" isomorphisms and so, for example, the distinct symmorphic layer groups $C_{1h}$ and $C_{1v}$ are isomorphic as point groups and sometimes denoted $C_s$. Also note, that every class has a larger “magnetic" variant (e.g., magnetic layer groups) and we omit them from the graph for simplicity. In the paper we sometimes refer to “order-two" symmetries. An order-two symmetry, in any symmetry group class, is a symmetry which is generated by a single element $g$ such that ${g^2=1}$. However, its projective representation may be non trivial ${\hat{g}^2=\pm1}$, and it may also be realized by antiunitary operators such as time-reversal (e.g., ${\hat{g}=\hat{c}_2T}$) in the magnetic classes. Examples of order-two symmetries include $C_2,C_{1h},C_{1v},S_2,D_1$. The constituents of actions of a symmetry group in any of the classes are as follows: \(i) 2D point groups - rotations and inversions (and their combined actions, i.e., reflections and rotoreflections) of a 2D space. \(ii) Wallpaper groups - rotations, inversions, and translations (and their combined actions, e.g., glides) of a 2D space. \(iii) Symmorphic layer groups - rotations and inversions of a 2D surface in a 3D space. \(iv) Layer groups - rotations, inversions and translations of a 2D surface in a 3D space. \(v) Point groups - rotations and inversions of a 3D space. \(vi) Space groups - rotations, inversions, and translations of a 3D space. \(vii) Magnetic point groups - point groups with time-reversal. \(viii) Magnetic space groups - space groups with time-reversal. A symmetry group is dubbed nonsymmorphic if an element within it acts by a translation combined with any of the other constituents. From Graded Algebra Extensions to Classifying Spaces {#app:class-space} ==================================================== In this appendix, following Abramovici and Kalugin \[\], we give the algebraic derivation of the classifying spaces from algebra extensions. We wish to study the extension of ${\mathbb{Z}}_2$-graded ${\mathbb{R}}$-algebras, $$B^0\hookrightarrow B=B^0\oplus B^1,$$ graded by an element $x_0\in B^1$. Let us look at a $B$-module $E$, it is also a $B^0$-module. In how many ways can one define a $B$-module structure on $E$? i.e., define a map $\varphi\in{{\mathrm{Hom}}}_{B^0-\mathbf{alg}}(B,{{\mathrm{End}}}_{B^0}{E})$. Such an action would act as $\varphi(x_{0})$ on $E$ as a $B^0$-module, however since $E$ has a $B$-module structure than the most general action must be one may simply map $E$ to the category of $B$-modules, act by $x_{0}$, and map back, it is useful to write $$\begin{gathered} \varphi(x_{0})=V^{-1}\alpha(x_{0})V,\\ \alpha\in\frac{{{\mathrm{Hom}}}_{B^0-\mathbf{alg}}(B,{{\mathrm{End}}}_{B^0}{E})}{V^{-1}\varphi V\simeq\varphi},\\ V\in{{\mathrm{Aut}}}_{B^0}{E}.\end{gathered}$$ What are the choices of $\varphi$? Let us focus on the case where $B={{\mathrm{Cl}}}_{p+1,q}$ and $B^0={{\mathrm{Cl}}}_{p,q}$. If $B$ is simple (Morita equivalent to ${\mathbb{R}}, {\mathbb{C}}, {\mathbb{H}}$), then $E={\mathbb{R}}^k\otimes_{\mathbb{R}}\Lambda^2$ where $\Lambda={\mathbb{R}}^{2^{p+q}}$ and $\Lambda^2$ is a simple $B$-module; pick an action $\alpha_0\in{{\mathrm{Hom}}}_{B^0-\mathbf{alg}}(B,{{\mathrm{End}}}_{B^0}{\Lambda})$. Moreover, as a $B^0$-module either $E=({\mathbb{R}}^k\otimes_{\mathbb{R}}\Lambda)\oplus({\mathbb{R}}^k\otimes_{\mathbb{R}}\Lambda)$ or $E={\mathbb{R}}^{2k}\otimes_{\mathbb{R}}\Lambda$. Therefore, the only choice of $\alpha$ is $$\begin{aligned} \alpha\in&\frac{{{\mathrm{Hom}}}_{B^0-\mathbf{alg}}(B,{{\mathrm{End}}}_{B^0}{E})}{V^{-1}\varphi V\simeq\varphi} \nonumber\\ &=\frac{(\alpha_0\circ({{\mathrm{Aut}}}_{B^0}{B}))^{\oplus k}}{V^{-1}\varphi V\simeq\varphi}=[\alpha_0]^{\oplus k},\end{aligned}$$ where the last equality follows from Skolem–-Noether. On the other hand, this action is invariant under choices of $V\in{{\mathrm{Aut}}}_{B^0}{E}$ that commute with $\alpha_0(x_{0})$ which are just ${{\mathrm{Aut}}}_B{E}\triangleleft{{\mathrm{Aut}}}_{B^0}{E}$ so $$\varphi\in\frac{{{\mathrm{Aut}}}_{B^0}{E}}{{{\mathrm{Aut}}}_B{E}}.$$ This space is homotopic to an appropriate symmetric space of ${\mathcal{R}}_1,{\mathcal{R}}_2,{\mathcal{R}}_3,{\mathcal{R}}_5,{\mathcal{R}}_6,{\mathcal{R}}_7,{\mathcal{C}}_1$; see Table \[tab:per\]. If $B=A\oplus A$ on the other hand (Morita equivalent to ${\mathbb{R}}\oplus{\mathbb{R}}, {\mathbb{C}}\oplus{\mathbb{C}}, {\mathbb{H}}\oplus{\mathbb{H}}$), then clearly $E\simeq E_1\oplus E_2$ where $E_{1,2}={\mathbb{R}}^{k_{1,2}}\otimes_{\mathbb{R}}\Lambda$ and $\Lambda={\mathbb{R}}^{2^{p+q}}$ is a simple $A$-module; pick actions $\alpha_{k_1,k_2}\in{{\mathrm{Hom}}}_{B^0-\mathbf{alg}}(B,E_1\oplus E_2)$. One should thus consider all possible splittings $E=E_1\oplus E_2$ $$\alpha\in\frac{{{\mathrm{Hom}}}_{B^0-\mathbf{alg}}(B,{{\mathrm{End}}}_{B^0}{E})}{V^{-1}\varphi V\simeq\varphi}=\bigcup_{E_1\oplus E_2=E}[\alpha_{k_1,k_2}].$$ The options for the action of $x_{0}$ are thus $$\begin{aligned} \varphi\in&\bigcup_{E_1\oplus E_2=E}\frac{{{\mathrm{Aut}}}_{B^0}{E}}{{{\mathrm{Aut}}}_B(E_1\oplus E_2)} \nonumber\\ &=\bigcup_{E_1\oplus E_2=E}\frac{{{\mathrm{Aut}}}_{B^0}{E}}{{{\mathrm{Aut}}}_{B^0}{E_1}\times{{\mathrm{Aut}}}_{B^0}{E_2}}.\end{aligned}$$ This space is homotopic to an appropriate symmetric space of ${\mathcal{R}}_0,{\mathcal{R}}_4,{\mathcal{C}}_0$; see Table \[tab:per\]. Clifford Algebras {#app:cliff} ================= In this appendix we give some of some complex and real Clifford algebras, they are presented in Table \[tab:clifC\] and Table \[tab:clifR\]. $$\begin{array}{c||ccc} q & 0 & 1 & 2 \\\hline\hline {{\mathrm{Cl}}}_{q} & {\mathbb{C}}& {\mathbb{C}}^{{\oplus 2}}& M_2({\mathbb{C}}) \end{array}$$ $$\begin{array}{c||ccccccccc} p\setminus q & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\hline \mathrm{``-1"}& & {\mathbb{R}}& {\mathbb{C}}& {\mathbb{H}}& {\mathbb{H}}^{{\oplus 2}}& M_2({\mathbb{H}}) & M_4({\mathbb{C}}) & M_8({\mathbb{R}}) & M_8({\mathbb{R}})^{{\oplus 2}}\\\hline\hline 0 & {\mathbb{R}}& {\mathbb{R}}^{{\oplus 2}}& M_2({\mathbb{R}}) & M_2({\mathbb{C}}) & M_2({\mathbb{H}}) & M_2({\mathbb{H}})^{{\oplus 2}}& M_4({\mathbb{H}}) & M_8({\mathbb{C}}) & M_{16}({\mathbb{R}}) \\ 1 & {\mathbb{C}}& M_2({\mathbb{R}}) & M_2({\mathbb{R}})^{{\oplus 2}}& M_4({\mathbb{R}}) & M_4({\mathbb{C}}) & M_4({\mathbb{H}}) & M_4({\mathbb{H}})^{{\oplus 2}}& M_8({\mathbb{H}}) & M_{16}({\mathbb{C}}) \\ 2 & {\mathbb{H}}& M_2({\mathbb{C}}) & M_4({\mathbb{R}}) & M_4({\mathbb{R}})^{{\oplus 2}}& M_8({\mathbb{R}}) & M_8({\mathbb{C}}) & M_8({\mathbb{H}}) & M_8({\mathbb{H}})^{{\oplus 2}}& M_{16}({\mathbb{H}}) \\ 3 & {\mathbb{H}}^{{\oplus 2}}& M_2({\mathbb{H}}) & M_4({\mathbb{C}}) & M_8({\mathbb{R}}) & M_8({\mathbb{R}})^{{\oplus 2}}& M_{16}({\mathbb{R}}) & M_{16}({\mathbb{C}}) & M_{16}({\mathbb{H}}) & M_{16}({\mathbb{H}})^{{\oplus 2}}\\ 4 & M_2({\mathbb{H}}) & M_2({\mathbb{H}})^{{\oplus 2}}& M_4({\mathbb{H}}) & M_8({\mathbb{C}}) & M_{16}({\mathbb{R}}) & M_{16}({\mathbb{R}})^{{\oplus 2}}& M_{32}({\mathbb{R}}) & M_{32}({\mathbb{C}}) & M_{32}({\mathbb{H}}) \\ 5 & M_4({\mathbb{C}}) & M_4({\mathbb{H}}) & M_4({\mathbb{H}})^{{\oplus 2}}& M_8({\mathbb{H}}) & M_{16}({\mathbb{C}}) & M_{32}({\mathbb{R}}) & M_{32}({\mathbb{R}})^{{\oplus 2}}& M_{64}({\mathbb{R}}) & M_{64}({\mathbb{C}}) \\ 6 & M_8({\mathbb{R}}) & M_8({\mathbb{C}}) & M_8({\mathbb{H}}) & M_8({\mathbb{H}})^{{\oplus 2}}& M_{16}({\mathbb{H}}) & M_{32}({\mathbb{C}}) & M_{64}({\mathbb{R}}) & M_{64}({\mathbb{R}})^{{\oplus 2}}& M_{128}({\mathbb{R}}) \\ 7 & M_8({\mathbb{R}})^{{\oplus 2}}& M_{16}({\mathbb{R}}) & M_{16}({\mathbb{C}}) & M_{16}({\mathbb{H}}) & M_{16}({\mathbb{H}})^{{\oplus 2}}& M_{32}({\mathbb{H}}) & M_{64}({\mathbb{C}}) & M_{128}({\mathbb{R}}) & M_{128}({\mathbb{R}})^{{\oplus 2}}\\ 8 & M_{16}({\mathbb{R}}) & M_{16}({\mathbb{R}})^{{\oplus 2}}& M_{32}({\mathbb{R}}) & M_{32}({\mathbb{C}}) & M_{32}({\mathbb{H}}) & M_{32}({\mathbb{H}})^{{\oplus 2}}& M_{64}({\mathbb{H}}) & M_{128}({\mathbb{C}}) & M_{256}({\mathbb{R}}) \end{array}$$ Point Group Symmetry Classification {#app:pgf} =================================== In this appendix we derive the bulk invariants of all 32 point group symmetries of 3D crystals. This is done using the techniques demonstrated in Sec. \[sec:exam\]. Rotational Symmetry $C_n$ ------------------------- {height="\fg"} {height="\fg"} {height="\fg"} {height="\fg"} {height="\fg"} The symmetry group is ${G={\mathbb{Z}}_n}$ with generator $\hat{c}$ such that $$\hat c^n=-1.$$ It acts on the 3D space as $$\hat c(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix})\hat c^{-1}=(\begin{smallmatrix}\cos\frac{2\pi}{n}&\sin\frac{2\pi}{n}\\-\sin\frac{2\pi}{n}&\cos\frac{2\pi}{n}\end{smallmatrix})(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix}).$$ Notice that $$\begin{aligned} e^{-\gamma_1\gamma_2\frac{2\pi}{2n}}(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix})e^{\gamma_1\gamma_2\frac{2\pi}{2n}} &=e^{-\gamma_1\gamma_2\frac{2\pi}{n}}(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix}) \\ &=(\begin{smallmatrix}\cos\frac{2\pi}{n}&\sin\frac{2\pi}{n}\\-\sin\frac{2\pi}{n}&\cos\frac{2\pi}{n}\end{smallmatrix})(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix}). \end{aligned}$$ We set $$c = e^{\gamma_1\gamma_2\frac{2\pi}{2n}}\hat c,$$ which satisfies $${{\;\;\:}}c y_i c^{-1}=y_i,{{\;\;\:}}c^n=1.$$ We thus get ${B={\mathbb{R}}[{\mathbb{Z}}_n]\otimes{{\mathrm{Cl}}}_{q+7,3}}$ and hence $$({\mathbb{R}}^{\oplus r(n)}\oplus{\mathbb{C}}^{\oplus c(n)})\otimes{{\mathrm{Cl}}}_{q+7,3}\mapsto{\mathcal{R}}_{q-3}^{r(n)}\times{\mathcal{C}}_{q+1}^{c(n)}.$$ Here, we have used the mapping in Sec. \[sec:ext\] to determine the classifying spaces; we employ this mapping throughout this section in treatment of all point group symmetries. Rotoreflection Symmetry $S_{2n}$ -------------------------------- {height="\fg"} {height="\fg"} {height="\fg"} The symmetry group is ${G={\mathbb{Z}}_{2n}}$ with generator $\hat{s}$ such that $$\hat s^{2n}=-(-1)^n.$$ It acts on the 3D space as $$\hat s(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix})\hat s^{-1}=(\begin{smallmatrix}\cos\frac{2\pi}{2n}&\sin\frac{2\pi}{2n}\\-\sin\frac{2\pi}{2n}&\cos\frac{2\pi}{2n}\end{smallmatrix})(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix}),{{\;\;\:}}\hat s\gamma_3 \hat s^{-1}=-\gamma_3.$$ We set $$s = e^{\gamma_1\gamma_2\frac{2\pi}{4n}}\hat s\gamma_3,$$ which satisfies $${{\;\;\:}}s y_i s^{-1}= -y_i,{{\;\;\:}}s^{2n}=1.$$ Here we used the equality $s^{2n}=(-e^{-\gamma_1\gamma_2\frac{2\pi}{2n}}\hat s^2)^n=1$. For $n=1$ we add $s$ as a generator such that ${\mathbb{R}}\langle s\rangle={{\mathrm{Cl}}}_{0,1}$ and ${{{\mathrm{Cl}}}_{p,q}\hat{\otimes}{{\mathrm{Cl}}}_{0,1}={{\mathrm{Cl}}}_{p,q+1}}$, and get $${{\mathrm{Cl}}}_{q+7,3+1}\mapsto{\mathcal{R}}_{q-4}.$$ For $n=2$ we have a central element ${t=s^2}$ satisfying ${t^2=1}$. Our algebra decomposes accordingly as a direct sum of two algebras such that in one of them $s^2=t=1$ and in the other $s^2=t=-1$. Therefore, we get $${{\mathrm{Cl}}}_{q+7,3+1}\oplus{{\mathrm{Cl}}}_{q+7+1,3}\mapsto{\mathcal{R}}_{q-4}\times{\mathcal{R}}_{q-2}.$$ This can also be done by considering ${{\mathbb{Z}}_2\triangleleft{\mathbb{Z}}_4}$ such that the graded structure is $$\begin{matrix} {\mathbb{R}}[{\mathbb{Z}}_2]&=&{\mathbb{R}}\oplus{\mathbb{R}}\\ {\mathbin{\text{\rotatebox[origin=c]{-90}{$\hookrightarrow$}}}}&&{\mathbin{\text{\rotatebox[origin=c]{-90}{$\hookrightarrow$}}}}&&\\ {\mathbb{R}}[{\mathbb{Z}}_4]&=&{\mathbb{R}}^{{\oplus 2}}\oplus{\mathbb{C}}&=&{{\mathrm{Cl}}}_{0,1}\oplus{{\mathrm{Cl}}}_{1,0}, \end{matrix}$$ $$\begin{aligned} {{\mathrm{Cl}}}_{p,q}\hat{\otimes}{\mathbb{R}}[{\mathbb{Z}}_4]&={{\mathrm{Cl}}}_{p,q}\hat{\otimes}({{\mathrm{Cl}}}_{0,1}\oplus{{\mathrm{Cl}}}_{1,0})\\&={{\mathrm{Cl}}}_{p,q+1}\oplus{{\mathrm{Cl}}}_{p+1,q}. \end{aligned}$$ Such analyses would come in handy in other more complicated cases. For $n=3$ we use ${\mathbb{Z}}_6={\mathbb{Z}}_2\times{\mathbb{Z}}_3$ and split our algebra to ${{{\mathrm{Cl}}}_{p,q}\hat\otimes{\mathbb{R}}\langle s\rangle=({{\mathrm{Cl}}}_{p,q}\hat\otimes{\mathbb{R}}\langle s^3\rangle)\otimes{\mathbb{R}}\langle s^2\rangle}$ and get $$({\mathbb{R}}\oplus{\mathbb{C}})\otimes{{\mathrm{Cl}}}_{q+7,3+1}\mapsto{\mathcal{R}}_{q-4}\times{\mathcal{C}}_{q}.$$ Dipyramidal Symmetry $C_{nh}$ ----------------------------- {height="\fg"} {height="\fg"} {height="\fg"} {height="\fg"} {height="\fg"} The symmetry group is ${G={\mathbb{Z}}_n\times{\mathbb{Z}}_2}$ with generators $\hat{c},\hat{\sigma}_h$ such that $$\hat c^n=-1,{{\;\;\:}}\hat{\sigma}_h^2=-1.$$ It acts on the 3D space as $$\hat c(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix})\hat c^{-1}=(\begin{smallmatrix}\cos\frac{2\pi}{n}&\sin\frac{2\pi}{n}\\-\sin\frac{2\pi}{n}&\cos\frac{2\pi}{n}\end{smallmatrix})(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix}),{{\;\;\:}}\hat{\sigma}_h^{\phantom{|}}\gamma_3 \hat{\sigma}_h^{-1}=-\gamma_3.$$ We set $$c = e^{\gamma_1\gamma_2\frac{2\pi}{2n}}\hat c,{{\;\;\:}}\sigma_h = \hat{\sigma}_h\gamma_3,$$ which satisfy $$\begin{gathered} c y_i c^{-1}=y_i,{{\;\;\:}}\sigma_h^{\phantom{|}} y_i \sigma_h^{-1}=-y_i,{{\;\;\:}}c^n=1,{{\;\;\:}}\sigma_h^2=1. \end{gathered}$$ We add $\sigma_h$ as a generator and get ${{\mathbb{R}}[{\mathbb{Z}}_n]\otimes({{\mathrm{Cl}}}_{p,q}\hat\otimes{\mathbb{R}}[{\mathbb{Z}}_2])}$ and hence $$({\mathbb{R}}^{\oplus r(n)}\oplus{\mathbb{C}}^{\oplus c(n)})\otimes{{\mathrm{Cl}}}_{q+7,3+1}\mapsto{\mathcal{R}}_{q-4}^{r(n)}\times{\mathcal{C}}_{q}^{c(n)}.$$ Pyramidal Symmetry $C_{nv}$ --------------------------- {height="\fg"} {height="\fg"} {height="\fg"} {height="\fg"} {height="\fg"} The symmetry group is ${G={{\mathrm{Dih}}}_n}$ with generators $\hat{c},\hat{\sigma}_v$ such that $$\hat c^n=-1,{{\;\;\:}}\hat{\sigma}_v^2=-1,{{\;\;\:}}(\hat c\hat{\sigma}_v)^2=-1.$$ It acts on the 3D space as $$\hat c(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix})\hat c^{-1}=(\begin{smallmatrix}\cos\frac{2\pi}{n}&\sin\frac{2\pi}{n}\\-\sin\frac{2\pi}{n}&\cos\frac{2\pi}{n}\end{smallmatrix})(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix}),{{\;\;\:}}\hat{\sigma}_v^{\phantom{|}}\gamma_1 \hat{\sigma}_v^{-1}=-\gamma_1.$$ We set $$c = e^{\gamma_1\gamma_2\frac{2\pi}{2n}}\hat c,{{\;\;\:}}\sigma_v = \hat{\sigma}_v\gamma_1,$$ which satisfy $$\begin{gathered} c y_i c^{-1}=y_i,{{\;\;\:}}\sigma_v^{\phantom{|}} y_i \sigma_v^{-1}=-y_i,\\ c^n=1,{{\;\;\:}}\sigma_v^2=1,{{\;\;\:}}(c\sigma_v)^2=1. \end{gathered}$$ Here we used the equality $$\begin{aligned} c\sigma_v &= e^{\gamma_1\gamma_2\frac{2\pi}{2n}}\hat c\hat\sigma_v\gamma_1 = \hat c\hat\sigma_v\gamma_1 e^{\gamma_1\gamma_2\frac{2\pi}{2n}} \\ &= \hat\sigma_v e^{\gamma_1\gamma_2\frac{2\pi}{n}}\gamma_1 \hat c^{-1} e^{\gamma_1\gamma_2\frac{2\pi}{2n}} \\ &= \hat\sigma_v\gamma_1 \hat c^{-1} e^{-\gamma_1\gamma_2\frac{2\pi}{2n}} = \sigma_v c^{-1}. \end{aligned}$$ We find that the ring of ${{\mathbb{Z}}_n\triangleleft{{\mathrm{Dih}}}_n}$ commutes with ${{\mathrm{Cl}}}_{p,q}$. Let us look at the algebra structure $$\begin{matrix} {\mathbb{R}}[{\mathbb{Z}}_n]&=&{\mathbb{R}}^{\oplus r(n)}\oplus{\mathbb{C}}^{\oplus c(n)}\\ {\mathbin{\text{\rotatebox[origin=c]{-90}{$\hookrightarrow$}}}}&&{\mathbin{\text{\rotatebox[origin=c]{-90}{$\hookrightarrow$}}}}\\ {\mathbb{R}}[{{\mathrm{Dih}}}_n]&=&({\mathbb{R}}^{{\oplus 2}})^{\oplus r(n)}\oplus M_2({\mathbb{R}})^{\oplus c(n)}\\ &&{\mathbin{\text{\rotatebox[origin=c]{-90}{$=$}}}}\\ &&{{\mathrm{Cl}}}_{0,1}^{\oplus r(n)}\oplus {{\mathrm{Cl}}}_{0,2}^{\oplus c(n)}, \end{matrix}$$ $$\begin{aligned} {{\mathrm{Cl}}}_{p,q}\hat\otimes{\mathbb{R}}[{{\mathrm{Dih}}}_n]&={{\mathrm{Cl}}}_{p,q}\hat\otimes({{\mathrm{Cl}}}_{0,1}^{\oplus r(n)}\oplus {{\mathrm{Cl}}}_{0,2}^{\oplus c(n)})\\ &={{\mathrm{Cl}}}_{p,q+1}^{\oplus r(n)}\oplus{{\mathrm{Cl}}}_{p,q+2}^{\oplus c(n)}. \end{aligned}$$ We thus get $${{\mathrm{Cl}}}_{q+7,3+1}^{\oplus r(n)}\oplus{{\mathrm{Cl}}}_{q+7,3+2}^{\oplus c(n)}\mapsto{\mathcal{R}}_{q-4}^{r(n)}\times{\mathcal{R}}_{q-5}^{c(n)}.$$ Dihedral Symmetry $D_n$ ----------------------- {height="\fg"} {height="\fg"} {height="\fg"} {height="\fg"} {height="\fg"} The symmetry group is ${G={{\mathrm{Dih}}}_n}$ with generators $\hat{c}_n,\hat{c}_2$ such that $$\hat c_n^n=-1,{{\;\;\:}}\hat c_2^2=-1,{{\;\;\:}}(\hat c_n\hat{c}_2)^2=-1.$$ It acts on the 3D space as $$\hat c_n^{\phantom{|}}(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix})\hat c_n^{-1}=(\begin{smallmatrix}\cos\frac{2\pi}{n}&\sin\frac{2\pi}{n}\\-\sin\frac{2\pi}{n}&\cos\frac{2\pi}{n}\end{smallmatrix})(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix}),{{\;\;\:}}\hat{c}_2^{\phantom{|}}\gamma_{2,3} \hat{c}_2^{-1}=-\gamma_{2,3}.$$ We set $$c_n = e^{\gamma_1\gamma_2\frac{2\pi}{2n}}\hat c_n,{{\;\;\:}}c_2 = \hat{c}_2\gamma_2\gamma_3,$$ which satisfy $$\begin{gathered} c_n^{\phantom{|}} y_i c_n^{-1}=y_i,{{\;\;\:}}c_2^{\phantom{|}} y_i c_2^{-1}=y_i,\\ c_n^n=1,{{\;\;\:}}c_2^2=1,{{\;\;\:}}(c_n c_2)^2=1. \end{gathered}$$ We thus get ${B={\mathbb{R}}[{{\mathrm{Dih}}}_n]\otimes{{\mathrm{Cl}}}_{q+7,3}}$ and hence $$(({\mathbb{R}}^{{\oplus 2}})^{\oplus r(n)}\oplus M_2({\mathbb{R}})^{\oplus c(n)})\otimes{{\mathrm{Cl}}}_{q+7,3}\mapsto{\mathcal{R}}_{q-3}^{2r(n)+c(n)}.$$ Antiprismatic Symmetry $D_{nd}$ ------------------------------- {height="\fg"} {height="\fg"} {height="\fg"} The symmetry group is ${G={{\mathrm{Dih}}}_{2n}}$ with generators $\hat{s},\hat{\sigma}_v$ such that $$\hat s^{2n}=-(-1)^n,{{\;\;\:}}\hat{\sigma}_v^2=-1,{{\;\;\:}}(\hat s\hat{\sigma}_v)^2=-1.$$ It acts on the 3D space as $$\begin{gathered} \hat s(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix})\hat s^{-1}=(\begin{smallmatrix}\cos\frac{2\pi}{2n}&\sin\frac{2\pi}{2n}\\-\sin\frac{2\pi}{2n}&\cos\frac{2\pi}{2n}\end{smallmatrix})(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix}),\\ \hat s\gamma_3 \hat s^{-1}=-\gamma_3,{{\;\;\:}}\hat{\sigma}_v^{\phantom{|}}\gamma_1 \hat{\sigma}_v^{-1}=-\gamma_1. \end{gathered}$$ We set $$s = e^{\gamma_1\gamma_2\frac{2\pi}{2n}}\hat s\gamma_3,{{\;\;\:}}\sigma_v = \hat{\sigma}_v\gamma_1,$$ which satisfy $$\begin{gathered} s y_i s^{-1}=-y_i,{{\;\;\:}}\sigma_v^{\phantom{|}} y_i \sigma_v^{-1}=-y_i,\\ s^{2n}=1,{{\;\;\:}}\sigma_v^2=1,{{\;\;\:}}(s\sigma_v)^2=1. \end{gathered}$$ Here we used the equality $$\begin{aligned} s\sigma_v &= e^{\gamma_1\gamma_2\frac{2\pi}{4n}}\hat s\gamma_3\hat\sigma_v\gamma_1 = \hat s\gamma_3\hat\sigma_v\gamma_1 e^{\gamma_1\gamma_2\frac{2\pi}{4n}} \\ &=\hat\sigma_ve^{\gamma_1\gamma_2\frac{2\pi}{2n}}\gamma_1 \gamma_3\hat s^{-1} e^{\gamma_1\gamma_2\frac{2\pi}{4n}} \\ &= \hat\sigma_v\gamma_1 \gamma_3 \hat s^{-1} e^{-\gamma_1\gamma_2\frac{2\pi}{4n}} = \sigma_v s^{-1}. \end{aligned}$$ For ${n=1}$ we have ${D_{1d}=C_{2h}}$ by setting ${c = s\sigma_v},{\sigma_h = \sigma_v}$. For ${n=2}$ we have a central element ${t = s^2}$ with ${t^2=1}$, and our algebra decomposes accordingly as a direct sum of simple algebras such that in the first component (isomorphic to $D_{1d}$) we have $t=-1$, and in the second component (in which ${s^2=-1},{s\sigma_v=-\sigma_v s}$) we have $t=-1$. Therefore, we get $$({\mathbb{R}}^{{\oplus 2}}\otimes{{\mathrm{Cl}}}_{q+7,3+1})\oplus{{\mathrm{Cl}}}_{q+7+1,3+1} \mapsto{\mathcal{R}}_{q-4}^2\times{\mathcal{R}}_{q-3}.$$ This can also be done by considering ${{{\mathrm{Dih}}}_2\triangleleft{{\mathrm{Dih}}}_4}$ such that the graded structure is $$\begin{matrix} {\mathbb{R}}[{{\mathrm{Dih}}}_2]&=&{\mathbb{R}}^{{\oplus 2}}\oplus{\mathbb{R}}^{{\oplus 2}}&&\\ {\mathbin{\text{\rotatebox[origin=c]{-90}{$\hookrightarrow$}}}}&&{\mathbin{\text{\rotatebox[origin=c]{-90}{$\hookrightarrow$}}}}&&\\ {\mathbb{R}}[{{\mathrm{Dih}}}_4]&=&({\mathbb{R}}^{{\oplus 2}})^{{\oplus 2}}\oplus M_2({\mathbb{R}})&=&{{\mathrm{Cl}}}_{0,1}^{{\oplus 2}}\oplus{{\mathrm{Cl}}}_{1,1}, \end{matrix}$$ $$\begin{aligned} {{\mathrm{Cl}}}_{p,q}\hat{\otimes}{\mathbb{R}}[{\mathbb{Z}}_4]&={{\mathrm{Cl}}}_{p,q}\hat{\otimes}({{\mathrm{Cl}}}_{0,1}^{{\oplus 2}}\oplus{{\mathrm{Cl}}}_{1,1})\\&={{\mathrm{Cl}}}_{p,q+1}^{{\oplus 2}}\oplus{{\mathrm{Cl}}}_{p+1,q+1}. \end{aligned}$$ For ${n=3}$ we use ${{\mathrm{Dih}}}_6={\mathbb{Z}}_2\times{{\mathrm{Dih}}}_3$ and split our algebra to ${{{\mathrm{Cl}}}_{p,q}\hat\otimes{\mathbb{R}}\langle s,\sigma_v\rangle=({{\mathrm{Cl}}}_{p,q}\hat\otimes{\mathbb{R}}\langle s^3\rangle)\otimes{\mathbb{R}}\langle s^2,s\sigma_v\rangle}$ and get $$({\mathbb{R}}^{{\oplus 2}}\oplus M_2({\mathbb{R}}))\otimes{{\mathrm{Cl}}}_{q+7,3+1}\mapsto{\mathcal{R}}_{q-4}^3.$$ Prismatic Symmetry $D_{nh}$ --------------------------- {height="\fg"} {height="\fg"} {height="\fg"} {height="\fg"} {height="\fg"} The symmetry group is ${G={{\mathrm{Dih}}}_{n}\times{\mathbb{Z}}_2}$ with generators $\hat{c}_n,\hat{c}_2,\hat{\sigma}_h$ such that $$\begin{gathered} \hat c_n^{n}=-1,{{\;\;\:}}\hat{c}_2^2=-1,{{\;\;\:}}\hat{\sigma}_h^2=-1,\\ (\hat c_n\hat{c}_2)^2=-1,{{\;\;\:}}(\hat c_2\sigma_h)^2=-1. \end{gathered}$$ It acts on the 3D space as $$\begin{gathered} \hat c_n^{\phantom{|}}(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix})\hat c_n^{-1}=(\begin{smallmatrix}\cos\frac{2\pi}{n}&\sin\frac{2\pi}{n}\\-\sin\frac{2\pi}{n}&\cos\frac{2\pi}{n}\end{smallmatrix})(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix}),\\ \hat{c}_2^{\phantom{|}}\gamma_{2,3} \hat{c}_2^{-1}=-\gamma_{2,3},{{\;\;\:}}\hat{\sigma}_h^{\phantom{|}}\gamma_3 \hat{\sigma}_h^{-1}=-\gamma_3. \end{gathered}$$ We set $$c_n = e^{\gamma_1\gamma_2\frac{2\pi}{2n}}\hat c_n,{{\;\;\:}}c_2 = \hat{c}_2\gamma_2\gamma_3,{{\;\;\:}}\sigma_h = \hat{\sigma}_h\gamma_3,$$ which satisfy $$\begin{gathered} c_n^{\phantom{|}} y_i c_n^{-1}=y_i,{{\;\;\:}}c_2^{\phantom{|}} y_i c_2^{-1}=y_i,{{\;\;\:}}\sigma_h^{\phantom{|}} y_i \sigma_h^{-1}=-y_i,\\ c_n^{n}=1,{{\;\;\:}}c_2^2=1,{{\;\;\:}}\sigma_h^2=1,{{\;\;\:}}(c_nc_2)^2=1,{{\;\;\:}}(c_2\sigma_h)^2=1. \end{gathered}$$ We add $\sigma_h$ as a generator and get ${{\mathbb{R}}[{{\mathrm{Dih}}}_n]\otimes({{\mathrm{Cl}}}_{p,q}\hat{\otimes}{\mathbb{R}}[{\mathbb{Z}}_2])}$ and hence $$(({\mathbb{R}}^{{\oplus 2}})^{\oplus r(n)}\oplus M_2({\mathbb{R}})^{\oplus c(n)})\otimes{{\mathrm{Cl}}}_{q+7,3+1} \mapsto{\mathcal{R}}_{q-4}^{2r(n)+c(n)}.$$ {height="\fg"} {height="\fg"} {height="\fg"} {height="\fg"} {height="\fg"} Chiral Tetrahedral Symmetry $T$ ------------------------------- The symmetry group is ${G=A_4}$ with generators $\hat{c}_3,\hat{c}_2$ such that $$\hat c_3^3=-1,{{\;\;\:}}\hat c_2^2=-1,{{\;\;\:}}(\hat{c}_2\hat c_3)^3=-1.$$ It acts on the 3D space as $$\hat c_3^{\phantom{|}}\gamma_i\hat c_3^{-1}=\gamma_{i+1},{{\;\;\:}}\hat{c}_2^{\phantom{|}}\gamma_{1,2} \hat{c}_2^{-1}=-\gamma_{1,2}.$$ We set $$c_3 = \tfrac{1+\gamma_1\gamma_2+\gamma_2\gamma_3+\gamma_3\gamma_1}{2}c_3,{{\;\;\:}}c_2 = -\hat{c}_2\gamma_1\gamma_2,$$ which satisfy $$\begin{gathered} c_3^{\phantom{|}} y_i c_3^{-1}=y_i,{{\;\;\:}}c_2^{\phantom{|}} y_i c_2^{-1}=y_i,\\ c_3^3=1,{{\;\;\:}}c_2^2=1,{{\;\;\:}}(c_2 c_3)^3=1. \end{gathered}$$ Here we used the equality $$\begin{aligned} (c_2 c_3)^3 &= (\gamma_1\gamma_2 \hat c_2 \hat c_3)^3\\&=(\gamma_1\gamma_2)(-\gamma_2\gamma_3)(+\gamma_3\gamma_1)(\hat c_2 \hat c_3)^3=1. \end{aligned}$$ We thus get ${B={\mathbb{R}}[A_4]\otimes{{\mathrm{Cl}}}_{q+7,3}}$ and hence $$({\mathbb{R}}\oplus{\mathbb{C}}\oplus M_3({\mathbb{R}}))\otimes{{\mathrm{Cl}}}_{q+7,3}\mapsto{\mathcal{R}}_{q-3}^2\times{\mathcal{C}}_{q+1}.$$ Pyritohedral Symmetry $T_h$ --------------------------- The symmetry group is ${G=A_4\times{\mathbb{Z}}_2}$ with generators $\hat{c}_3,\hat{c}_2,\hat{I}$ such that $$\hat c_3^3=-1,{{\;\;\:}}\hat c_2^2=-1,{{\;\;\:}}(\hat{c}_2\hat c_3)^3=-1,{{\;\;\:}}\hat{I}^2=1.$$ It acts on the 3D space as $$\hat c_3^{\phantom{|}}\gamma_i\hat c_3^{-1}=\gamma_{i+1},{{\;\;\:}}\hat{c}_2^{\phantom{|}}\gamma_{1,2} \hat{c}_2^{-1}=-\gamma_{1,2},{{\;\;\:}}\hat I\gamma_i \hat I^{-1}=-\gamma_i.$$ We set $$c_3 = \tfrac{1+\gamma_1\gamma_2+\gamma_2\gamma_3+\gamma_3\gamma_1}{2}c_3,{{\;\;\:}}c_2 = -\hat{c}_2\gamma_1\gamma_2,{{\;\;\:}}I = \hat I \gamma_1\gamma_2\gamma_3,$$ which satisfy $$\begin{gathered} c_3^{\phantom{|}} y_i c_3^{-1}=y_i,{{\;\;\:}}c_2^{\phantom{|}} y_i c_2^{-1}=y_i,{{\;\;\:}}Iy_iI^{-1}=-y_i,\\ {{\;\;\:}}c_3^3=1,{{\;\;\:}}c_2^2=1,{{\;\;\:}}(c_2 c_3)^3=1,{{\;\;\:}}I^2=1. \end{gathered}$$ We add $I$ as a generator and get ${{\mathbb{R}}[A_4]\otimes({{\mathrm{Cl}}}_{p,q}\hat{\otimes}{\mathbb{R}}[{\mathbb{Z}}_2])}$ and hence $$({\mathbb{R}}\oplus{\mathbb{C}}\oplus M_3({\mathbb{R}}))\otimes{{\mathrm{Cl}}}_{q+7,3+1}\mapsto{\mathcal{R}}_{q-4}^2\times{\mathcal{C}}_{q}.$$ Full Tetrahedral Symmetry $T_d$ {#full-tetrahedral-symmetry-t_d} ------------------------------- The symmetry group is ${G=S_4}$ with generators $\hat{c}_3,\hat{s}_4$ such that $$\hat c_3^3=-1,{{\;\;\:}}\hat s_4^4=-1,{{\;\;\:}}(\hat{s}_4\hat c_3)^2=-1.$$ It acts on the 3D space as $$\hat c_3^{\phantom{|}}\gamma_i\hat c_3^{-1}=\gamma_{i+1},{{\;\;\:}}\hat s_4^{\phantom{|}}\Big(\begin{smallmatrix}\gamma_1\\\gamma_2\\\gamma_3\end{smallmatrix}\Big)\hat s_4^{-1}=\Big(\begin{smallmatrix}-\gamma_2\\+\gamma_1\\-\gamma_3\end{smallmatrix}\Big).$$ We set $$c_3 = \tfrac{1+\gamma_1\gamma_2+\gamma_2\gamma_3+\gamma_3\gamma_1}{2}c_3,{{\;\;\:}}s_4 = e^{-\gamma_1\gamma_2\frac{\pi}{4}}\hat{s}_4\gamma_3,$$ which satisfy $$\begin{gathered} c_3^{\phantom{|}} y_i c_3^{-1}=y_i,{{\;\;\:}}s_4^{\phantom{|}} y_i s_4^{-1}=-y_i,\\ c_3^3=1,{{\;\;\:}}s_4^4=1,{{\;\;\:}}(s_4 c_3)^2=1. \end{gathered}$$ Here we used the equality $$\begin{aligned} &(s_4 c_3)^2 \\&= -\tfrac{1-\gamma_1\gamma_2-\gamma_2\gamma_3-\gamma_3\gamma_1}{2}e^{-\gamma_1\gamma_2\frac{\pi}{4}} \hat s_4 \gamma_3 \hat c_3 e^{-\gamma_1\gamma_2\frac{\pi}{4}} \hat s_4 \gamma_3 \hat c_3 \\ &= -\tfrac{1-\gamma_1\gamma_2-\gamma_2\gamma_3-\gamma_3\gamma_1}{2}e^{-\gamma_2\gamma_3\frac{\pi}{4}}e^{-\gamma_1\gamma_2\frac{\pi}{4}}\gamma_2\gamma_3 \hat s_4 \hat c_3 \hat s_4 \hat c_3=1. \end{aligned}$$ We find that the ring of ${A_4\triangleleft S_4}$ commutes with ${{\mathrm{Cl}}}_{p,q}$. Let us look at the algebra structure $$\begin{matrix} {\mathbb{R}}[A_4]&=&{\mathbb{R}}\oplus{\mathbb{C}}\oplus M_3({\mathbb{R}})\\ {\mathbin{\text{\rotatebox[origin=c]{-90}{$\hookrightarrow$}}}}&&{\mathbin{\text{\rotatebox[origin=c]{-90}{$\hookrightarrow$}}}}\\ {\mathbb{R}}[S_4]&=&{\mathbb{R}}^{{\oplus 2}}\oplus M_2({\mathbb{R}})\oplus M_3({\mathbb{R}})^{{\oplus 2}}\\ &&{\mathbin{\text{\rotatebox[origin=c]{-90}{$=$}}}}\\ &&{{\mathrm{Cl}}}_{0,1}\oplus{{\mathrm{Cl}}}_{0,2}\oplus(M_3({\mathbb{R}})\otimes{{\mathrm{Cl}}}_{0,1}), \end{matrix}$$ $$\begin{aligned} {{\mathrm{Cl}}}_{p,q}\hat\otimes{\mathbb{R}}[S_4]&={{\mathrm{Cl}}}_{p,q}\hat\otimes({{\mathrm{Cl}}}_{0,1}\oplus{{\mathrm{Cl}}}_{0,2}\oplus(M_3({\mathbb{R}})\otimes{{\mathrm{Cl}}}_{0,1}))\\ &=(({\mathbb{R}}\oplus M_3({\mathbb{R}}))\otimes{{\mathrm{Cl}}}_{p,q+1})\oplus{{\mathrm{Cl}}}_{p,q+2}. \end{aligned}$$ We thus get $$\begin{gathered} (({\mathbb{R}}\oplus M_3({\mathbb{R}}))\otimes{{\mathrm{Cl}}}_{q+7,3+1})\oplus{{\mathrm{Cl}}}_{q+7,3+2}\\ \mapsto{\mathcal{R}}_{q-4}^2\times{\mathcal{R}}_{q-5}.\end{gathered}$$ Chiral Octahedral Symmetry $O$ ------------------------------ The symmetry group is ${G=S_4}$ with generators $\hat{c}_3,\hat{c}_4$ such that $$\hat c_3^3=-1,{{\;\;\:}}\hat c_4^4=-1,{{\;\;\:}}(\hat{c}_4\hat c_3)^2=-1.$$ It acts on the 3D space as $$\hat c_3^{\phantom{|}}\gamma_i\hat c_3^{-1}=\gamma_{i+1},{{\;\;\:}}\hat c_4^{\phantom{|}}(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix})\hat c_4^{-1}=(\begin{smallmatrix}+\gamma_2\\-\gamma_1\end{smallmatrix}).$$ We set $$c_3 = \tfrac{1+\gamma_1\gamma_2+\gamma_2\gamma_3+\gamma_3\gamma_1}{2}c_3,{{\;\;\:}}c_4 = e^{\gamma_1\gamma_2\frac{\pi}{4}}\hat{c}_4,$$ which satisfy $$\begin{gathered} c_3^{\phantom{|}} y_i c_3^{-1}=y_i,{{\;\;\:}}c_4^{\phantom{|}} y_i c_4^{-1}=y_i,\\ c_3^3=1,{{\;\;\:}}c_4^4=1,{{\;\;\:}}(c_4 c_3)^2=1. \end{gathered}$$ Here we used the equality $$\begin{aligned} (c_4 c_3)^2 &= -\tfrac{1-\gamma_1\gamma_2-\gamma_2\gamma_3-\gamma_3\gamma_1}{2}e^{\gamma_1\gamma_2\frac{\pi}{4}} \hat c_4 \hat c_3 e^{\gamma_1\gamma_2\frac{\pi}{4}} \hat c_4 \hat c_3 \\&= -\tfrac{1-\gamma_1\gamma_2-\gamma_2\gamma_3-\gamma_3\gamma_1}{2}e^{\gamma_2\gamma_3\frac{\pi}{4}}e^{\gamma_1\gamma_2\frac{\pi}{4}} \hat c_4 \hat c_3 \hat c_4 \hat c_3=1. \end{aligned}$$ We thus get ${B={\mathbb{R}}[S_4]\otimes{{\mathrm{Cl}}}_{q+7,3}}$ and hence $$({\mathbb{R}}^{{\oplus 2}}\oplus M_2({\mathbb{R}})\oplus M_3({\mathbb{R}})^{{\oplus 2}})\otimes{{\mathrm{Cl}}}_{q+7,3}\mapsto{\mathcal{R}}_{q-3}^5.$$ Full Octahedral Symmetry $O_h$ ------------------------------ The symmetry group is ${G=S_4\times{\mathbb{Z}}_2}$ with generators $\hat{c}_3,\hat{c}_4,\hat I$ such that $$\hat c_3^3=-1,{{\;\;\:}}\hat c_4^4=-1,{{\;\;\:}}(\hat{c}_4\hat c_3)^2=-1,{{\;\;\:}}\hat I^2=1.$$ It acts on the 3D space as $$\hat c_3^{\phantom{|}}\gamma_i\hat c_3^{-1}=\gamma_{i+1},{{\;\;\:}}\hat c_4^{\phantom{|}}(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix})\hat c_4^{-1}=(\begin{smallmatrix}+\gamma_2\\-\gamma_1\end{smallmatrix}),{{\;\;\:}}\hat I\gamma_i \hat I^{-1}=-\gamma_i.$$ We set $$c_3 = \tfrac{1+\gamma_1\gamma_2+\gamma_2\gamma_3+\gamma_3\gamma_1}{2}c_3,{{\;\;\:}}c_4 = e^{\gamma_1\gamma_2\frac{\pi}{4}}\hat{c}_4,{{\;\;\:}}I = \hat I \gamma_1\gamma_2\gamma_3,$$ which satisfy $$\begin{gathered} c_3^{\phantom{|}} y_i c_3^{-1}=y_i,{{\;\;\:}}c_4^{\phantom{|}} y_i c_4^{-1}=y_i,{{\;\;\:}}Iy_iI^{-1}=-y_i,\\ c_3^3=1,{{\;\;\:}}c_4^4=1,{{\;\;\:}}(c_4 c_3)^2=1,{{\;\;\:}}I^2=1. \end{gathered}$$ We add $I$ as a generator and get ${{\mathbb{R}}[S_4]\otimes({{\mathrm{Cl}}}_{p,q}\hat\otimes{\mathbb{R}}[{\mathbb{Z}}_2])}$ and hence $$({\mathbb{R}}^{{\oplus 2}}\oplus M_2({\mathbb{R}})\oplus M_3({\mathbb{R}})^{{\oplus 2}})\otimes{{\mathrm{Cl}}}_{q+7,3+1}\mapsto{\mathcal{R}}_{q-4}^5.$$ Symmorphic Layer Group Symmetry Classification {#app:layer} ============================================== $$\begin{array}{l||c|c||c|c|c||c|c|c||c|c|c|c} \mathrm{Sch\ddot{o}n.} & C_1 & C_i,S_2 & D_1 & C_{1v} & D_{1d} & C_2 & C_{1h} & C_{2h} & D_2 & C_{2v} & D_{1h} & D_{2h} \\ \hline\hline & {\mathcal{C}}_{q} & {\mathcal{C}}_{q}^2 & {\mathcal{C}}_{q+1} & {\mathcal{C}}_{q+1} & {\mathcal{C}}_{q} & {\mathcal{C}}_{q}^2 & {\mathcal{C}}_q^2 & {\mathcal{C}}_q^4 & {\mathcal{C}}_{q+1}^2 & {\mathcal{C}}_{q+1}^2 & {\mathcal{C}}_{q} & {\mathcal{C}}_{q}^2 \\ \hline\hline \mathrm{A} & {\mathbb{Z}}& {\mathbb{Z}}^2 & 0 & 0 & {\mathbb{Z}}& {\mathbb{Z}}^2 & {\mathbb{Z}}^2 & {\mathbb{Z}}^4 & 0 & 0 & {\mathbb{Z}}& {\mathbb{Z}}^2 \\ \hline \mathrm{AIII} & 0 & 0 & {\mathbb{Z}}& {\mathbb{Z}}& 0 & 0 & 0 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^2 & 0 & 0 \\ \hline\hline & {\mathcal{R}}_{q-2} & {\mathcal{C}}_{q} & {\mathcal{R}}_{q-3} & {\mathcal{R}}_{q-3} & {\mathcal{R}}_{q-4} & {\mathcal{R}}_{q-2}^2 & {\mathcal{C}}_q & {\mathcal{C}}_q^2 & {\mathcal{R}}_{q-3}^2 & {\mathcal{R}}_{q-3}^2 & {\mathcal{R}}_{q-2} & {\mathcal{R}}_{q-2}^2 \\ \hline\hline \mathrm{AI} & 0 & {\mathbb{Z}}& 0 & 0 & {\mathbb{Z}}& 0 & {\mathbb{Z}}& {\mathbb{Z}}^2 & 0 & 0 & 0 & 0 \\ \hline \mathrm{BDI} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \mathrm{D} & {\mathbb{Z}}& {\mathbb{Z}}& 0 & 0 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}& {\mathbb{Z}}^2 & 0 & 0 & {\mathbb{Z}}& {\mathbb{Z}}^2 \\ \hline \mathrm{DIII} & {\mathbb{Z}}_2 & 0 & {\mathbb{Z}}& {\mathbb{Z}}& 0 & {\mathbb{Z}}_2^2 & 0 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^2 & {\mathbb{Z}}_2 & {\mathbb{Z}}_2^2 \\ \hline \mathrm{AII} & {\mathbb{Z}}_2 & {\mathbb{Z}}& {\mathbb{Z}}_2 & {\mathbb{Z}}_2 & {\mathbb{Z}}& {\mathbb{Z}}_2^2 & {\mathbb{Z}}& {\mathbb{Z}}^2 & {\mathbb{Z}}_2^2 & {\mathbb{Z}}_2^2 & {\mathbb{Z}}_2 & {\mathbb{Z}}_2^2 \\ \hline \mathrm{CII} & 0 & 0 & {\mathbb{Z}}_2 & {\mathbb{Z}}_2 & {\mathbb{Z}}_2 & 0 & 0 & 0 & {\mathbb{Z}}_2^2 & {\mathbb{Z}}_2^2 & 0 & 0 \\ \hline \mathrm{C} & {\mathbb{Z}}& {\mathbb{Z}}& 0 & 0 & {\mathbb{Z}}_2 & {\mathbb{Z}}^2 & {\mathbb{Z}}& {\mathbb{Z}}^2 & 0 & 0 & {\mathbb{Z}}& {\mathbb{Z}}^2 \\ \hline \mathrm{CI} & 0 & 0 & {\mathbb{Z}}& {\mathbb{Z}}& 0 & 0 & 0 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^2 & 0 & 0 \end{array}$$ $$\begin{array}{l||c||c|c|c|c|c|c|c} \mathrm{Sch\ddot{o}n.} & C_1 & C_4 & S_4 & C_{4h} & D_4 & C_{4v} & D_{2d} & D_{4h} \\ \hline\hline & {\mathcal{C}}_{q} & {\mathcal{C}}_{q}^4 & {\mathcal{C}}_{q}^4 &{\mathcal{C}}_{q}^8 & {\mathcal{C}}_{q+1}^2\times{\mathcal{C}}_{q} & {\mathcal{C}}_{q+1}^2\times{\mathcal{C}}_{q} & {\mathcal{C}}_{q+1}^2\times{\mathcal{C}}_{q} & {\mathcal{C}}_{q}^4 \\ \hline\hline \mathrm{A} & {\mathbb{Z}}& {\mathbb{Z}}^4 & {\mathbb{Z}}^4 & {\mathbb{Z}}^8 & {\mathbb{Z}}& {\mathbb{Z}}& {\mathbb{Z}}& {\mathbb{Z}}^4 \\ \hline \mathrm{AIII} & 0 & 0 & 0 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^2 & {\mathbb{Z}}^2 & 0 \\ \hline\hline & {\mathcal{R}}_{q-2} & {\mathcal{R}}_{q-2}^2\times{\mathcal{C}}_{q} & {\mathcal{R}}_{q-2}^2\times{\mathcal{C}}_{q} &{\mathcal{C}}_{q}^4 & {\mathcal{R}}_{q-3}^2\times{\mathcal{R}}_{q-4} & {\mathcal{R}}_{q-3}^2\times{\mathcal{R}}_{q-4} & {\mathcal{R}}_{q-3}^2\times{\mathcal{R}}_{q-4} & {\mathcal{R}}_{q-2}^4 \\ \hline\hline \mathrm{AI} & 0 & {\mathbb{Z}}& {\mathbb{Z}}& {\mathbb{Z}}^4 & {\mathbb{Z}}& {\mathbb{Z}}& {\mathbb{Z}}& 0 \\ \hline \mathrm{BDI} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \mathrm{D} & {\mathbb{Z}}& {\mathbb{Z}}^3 & {\mathbb{Z}}^3 & {\mathbb{Z}}^4 & 0 & 0 & 0 & {\mathbb{Z}}^4 \\ \hline \mathrm{DIII} & {\mathbb{Z}}_2 & {\mathbb{Z}}_2^2 & {\mathbb{Z}}_2^2 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^2 & {\mathbb{Z}}^2 & {\mathbb{Z}}_2^4 \\ \hline \mathrm{AII} & {\mathbb{Z}}_2 & {\mathbb{Z}}_2^2\times{\mathbb{Z}}& {\mathbb{Z}}_2^2\times{\mathbb{Z}}& {\mathbb{Z}}^4 & {\mathbb{Z}}_2^2\times{\mathbb{Z}}& {\mathbb{Z}}_2^2\times{\mathbb{Z}}& {\mathbb{Z}}_2^2\times{\mathbb{Z}}& {\mathbb{Z}}_2^4 \\ \hline \mathrm{CII} & 0 & 0 & 0 & 0 & {\mathbb{Z}}_2^3 & {\mathbb{Z}}_2^3 & {\mathbb{Z}}_2^3 & 0 \\ \hline \mathrm{C} & {\mathbb{Z}}& {\mathbb{Z}}^3 & {\mathbb{Z}}^3 & {\mathbb{Z}}^4 & {\mathbb{Z}}_2 & {\mathbb{Z}}_2 & {\mathbb{Z}}_2 & {\mathbb{Z}}^4 \\ \hline \mathrm{CI} & 0 & 0 & 0 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^2 & {\mathbb{Z}}^2 & 0 \end{array}$$ $$\begin{array}{l||c||c|c|c|c|c} \mathrm{Sch\ddot{o}n.} & C_1 & C_6 & C_{3i},S_6 & D_3 & C_{3v} & D_{3d} \\ \hline\hline & {\mathcal{C}}_{q} & {\mathcal{C}}_{q}^3 & {\mathcal{C}}_{q}^6 & {\mathcal{C}}_{q+1}\times{\mathcal{C}}_{q} & {\mathcal{C}}_{q+1}\times{\mathcal{C}}_{q} & {\mathcal{C}}_q^3 \\ \hline\hline \mathrm{A} & {\mathbb{Z}}& {\mathbb{Z}}^3 & {\mathbb{Z}}^6 & {\mathbb{Z}}& {\mathbb{Z}}& {\mathbb{Z}}^3 \\ \hline \mathrm{AIII} & 0 & 0 & 0 & {\mathbb{Z}}& {\mathbb{Z}}& 0 \\ \hline\hline & {\mathcal{R}}_{q-2} & {\mathcal{R}}_{q-2}\times{\mathcal{C}}_{q} & {\mathcal{C}}_{q}^3 & {\mathcal{R}}_{q-3}\times{\mathcal{R}}_{q-4} & {\mathcal{R}}_{q-3}\times{\mathcal{R}}_{q-4} & {\mathcal{R}}_{q-4}\times{\mathcal{C}}_q \\ \hline\hline \mathrm{AI} & 0 & {\mathbb{Z}}& {\mathbb{Z}}^3 & {\mathbb{Z}}& {\mathbb{Z}}& {\mathbb{Z}}^2 \\ \hline \mathrm{BDI} & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \mathrm{D} & {\mathbb{Z}}& {\mathbb{Z}}^2 & {\mathbb{Z}}^3 & 0 & 0 & {\mathbb{Z}}\\ \hline \mathrm{DIII} & {\mathbb{Z}}_2 & {\mathbb{Z}}_2 & 0 & {\mathbb{Z}}& {\mathbb{Z}}& 0 \\ \hline \mathrm{AII} & {\mathbb{Z}}_2 & {\mathbb{Z}}_2\times{\mathbb{Z}}& {\mathbb{Z}}^3 & {\mathbb{Z}}_2\times{\mathbb{Z}}& {\mathbb{Z}}_2\times{\mathbb{Z}}& {\mathbb{Z}}^2 \\ \hline \mathrm{CII} & 0 & 0 & 0 & {\mathbb{Z}}_2^2 & {\mathbb{Z}}_2^2 & {\mathbb{Z}}_2 \\ \hline \mathrm{C} & {\mathbb{Z}}& {\mathbb{Z}}^2 & {\mathbb{Z}}^3 & {\mathbb{Z}}_2 & {\mathbb{Z}}_2 & {\mathbb{Z}}_2\times{\mathbb{Z}}\\ \hline \mathrm{CI} & 0 & 0 & 0 & {\mathbb{Z}}& {\mathbb{Z}}& 0 \end{array}$$ $$\begin{array}{l||c||c|c|c|c|c|c|c} \mathrm{Sch\ddot{o}n.} & C_1 & C_6 & C_{3h} & C_{6h} & D_6 & C_{6v} & D_{3h} & D_{6h} \\ \hline\hline & {\mathcal{C}}_{q} & {\mathcal{C}}_{q}^6 & {\mathcal{C}}_{q}^6 &{\mathcal{C}}_{q}^{12} & {\mathcal{C}}_{q+1}^2\times{\mathcal{C}}_{q}^2 & {\mathcal{C}}_{q+1}^2\times{\mathcal{C}}_{q}^2 & {\mathcal{C}}_{q}^3 & {\mathcal{C}}_{q}^6 \\ \hline\hline \mathrm{A} & {\mathbb{Z}}& {\mathbb{Z}}^6 & {\mathbb{Z}}^6 & {\mathbb{Z}}^{12} & {\mathbb{Z}}^2 & {\mathbb{Z}}^2 & {\mathbb{Z}}^3 & {\mathbb{Z}}^6 \\ \hline \mathrm{AIII} & 0 & 0 & 0 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^2 & 0 & 0 \\ \hline\hline & {\mathcal{R}}_{q-2} & {\mathcal{R}}_{q-2}^2\times{\mathcal{C}}_{q}^2 & {\mathcal{C}}_{q}^3 &{\mathcal{C}}_{q}^6 & {\mathcal{R}}_{q-3}^2\times{\mathcal{R}}_{q-4}^2 & {\mathcal{R}}_{q-3}^2\times{\mathcal{R}}_{q-4}^2 & {\mathcal{R}}_{q-2}^3 & {\mathcal{R}}_{q-2}^6 \\ \hline\hline \mathrm{AI} & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^3 & {\mathbb{Z}}^6 & {\mathbb{Z}}^2 & {\mathbb{Z}}^2 & 0 & 0 \\ \hline \mathrm{BDI} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \mathrm{D} & {\mathbb{Z}}& {\mathbb{Z}}^4 & {\mathbb{Z}}^3 & {\mathbb{Z}}^6 & 0 & 0 & {\mathbb{Z}}^3 & {\mathbb{Z}}^6 \\ \hline \mathrm{DIII} & {\mathbb{Z}}_2 & {\mathbb{Z}}_2^2 & 0 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^2 & {\mathbb{Z}}_2^3 &{\mathbb{Z}}_2^6 \\ \hline \mathrm{AII} & {\mathbb{Z}}_2 &{\mathbb{Z}}_2^2\times{\mathbb{Z}}^2 & {\mathbb{Z}}^3 & {\mathbb{Z}}^6 & {\mathbb{Z}}_2^2\times{\mathbb{Z}}^2 &{\mathbb{Z}}_2^2\times{\mathbb{Z}}^2 & {\mathbb{Z}}_2^3 &{\mathbb{Z}}_2^6 \\ \hline \mathrm{CII} & 0 & 0 & 0 & 0 & {\mathbb{Z}}_2^4 & {\mathbb{Z}}_2^4 & 0 & 0 \\ \hline \mathrm{C} & {\mathbb{Z}}& {\mathbb{Z}}^4 & {\mathbb{Z}}^3 & {\mathbb{Z}}^6 & {\mathbb{Z}}_2^2 & {\mathbb{Z}}_2^2 & {\mathbb{Z}}^3 & {\mathbb{Z}}^6 \\ \hline \mathrm{CI} & 0 & 0 & 0 & 0 & {\mathbb{Z}}^2 & {\mathbb{Z}}^2 & 0 & 0 \end{array}$$ In this appendix we briefly present the topological bulk invariants of the symmorphic layer groups using analogous derivation to the 3D point groups presented in Sec. \[sec:exam\] and Appendix \[app:pgf\]. Bulk topological invariants for all Altland-Zirnbauer symmetry classes are presented in Table \[tab:pp1\] through Table \[tab:mm1\]. All classifying spaces for all symmorphic layer group symmetries in all crystal systems are compactly presented in Table \[tab:class\]. The layer groups are given by omitting the $z$-direction $\gamma_3$ generator. Notice, that the equivalences of the 3D monoclinic and orthorhombic point groups no longer hold, i.e., $C_2\neq D_1,~C_{1h}\neq C_{1v},~C_{2h}\neq D_{1d},~C_{2v}\neq D_{1h}$. ### Rotational Symmetry $C_n$ This has no effect on the $z$ direction and is hence just shifted by 1 due to the lack of the $\gamma_3$ generator. We thus find $${\mathcal{R}}_{q-2}^{r(n)}\times{\mathcal{C}}_{q}^{c(n)}.$$ ### Rotoreflection Symmetry $S_{2n}$ One has to modify the $s$ generator to exclude $\gamma_3$ by $s'= e^{\gamma_1\gamma_2\frac{2\pi}{4n}}\hat s$. This however satisfies $s' y_i s'^{-1}= y_i$ and $s'^{2n}=(-1)^n$. For $n=1$ this is just a complex structure ${\mathbb{C}}={\mathbb{R}}\langle\sigma_h'\rangle$ and we get $B={{\mathrm{Cl}}}_{p+1,q}\otimes{\mathbb{C}}$ and hence $${\mathcal{C}}_{q}.$$ For $n=2$ we get $B={{\mathrm{Cl}}}_{p+1,q}\otimes{\mathbb{R}}[{\mathbb{Z}}_4]$ and hence $${\mathcal{R}}_{q-2}^2\times{\mathcal{C}}_{q}.$$ For $n=3$ we have ${\mathbb{R}}\langle s'\rangle={\mathbb{R}}\langle -s'^2\rangle\otimes{\mathbb{R}}\langle s'^3\rangle={\mathbb{C}}^{\oplus 3}$ and we get $B={{\mathrm{Cl}}}_{p+1,q}\otimes{\mathbb{C}}^{\oplus 3}$ and hence $${\mathcal{C}}_{q}^3.$$ ### Dipyramidal Symmetry $C_{nh}$ One has to modify the $\sigma_h$ generator to exclude $\gamma_3$ by $\sigma_h'=\hat{\sigma}_h$. This however satisfies $\sigma_h'^2=-1$ and $\sigma_h'y_i\sigma_h'^{-1}=y_i$. It thus adds a complex structure ${\mathbb{C}}={\mathbb{R}}\langle\sigma_h'\rangle$ to $C_{n}$ and we get $B={{\mathrm{Cl}}}_{p+1,q}\otimes{\mathbb{R}}[{\mathbb{Z}}_n]\otimes{\mathbb{C}}$ and hence $${\mathcal{C}}_{q}^{r(n)+2c(n)}.$$ ### Pyramidal Symmetry $C_{nv}$ This has no effect on the $z$ direction and is hence just shifted by 1 due to the lack of the $\gamma_3$ generator. We thus find $${\mathcal{R}}_{q-3}^{r(n)}\times{\mathcal{R}}_{q-4}^{c(n)}.$$ ### Dihedral Symmetry $D_n$ One has to modify the $c_2$ generator to exclude $\gamma_3$ by $c_2'=\hat{c_2}\gamma_2$. This, however, satisfies $c_2'^2=1$, $c_2'y_ic_2'^{-1}=y_i$, and $(c_n c'_2)^2=1$ which makes it equivalent to $C_{nv}$ by $c_2'\leftrightarrow\sigma_v$ and hence we find $${\mathcal{R}}_{q-3}^{r(n)}\times{\mathcal{R}}_{q-4}^{c(n)}.$$ ### Antiprismatic Symmetry $D_{nd}$ One has to modify the $s$ generator to exclude $\gamma_3$ by $s'= e^{\gamma_1\gamma_2\frac{2\pi}{4n}}\hat s$. This, however, satisfies $s' y_i s'^{-1}= y_i$ and $s'^{2n}=(-1)^n$ as well as $(s'\sigma_v)^2=1$. For $n=1$ and we get $B={{\mathrm{Cl}}}_{p+1,q}\hat\otimes{{\mathrm{Cl}}}_{0,2}$ and hence $${\mathcal{R}}_{q-4}.$$ For $n=2$ it is equivalent to $C_{4v}$ by $s'\leftrightarrow c$ and hence we find $${\mathcal{R}}_{q-3}^2\times{\mathcal{R}}_{q-4}.$$ For $n=3$ we have ${\mathbb{R}}\langle s',\sigma_v\rangle={\mathbb{R}}\langle -s'^2\rangle\otimes{\mathbb{R}}\langle s'^3,\sigma_v\rangle$ and we get $B=({{\mathrm{Cl}}}_{p+1,q}\hat\otimes{{\mathrm{Cl}}}_{0,2})\otimes({\mathbb{R}}\oplus{\mathbb{C}})$ and hence $${\mathcal{R}}_{q-4}\times{\mathcal{C}}_q.$$ ### Prismatic Symmetry $D_{nh}$ One has to modify both the $c_2$ and the $\sigma_h$ generators to exclude $\gamma_3$ by $c_2'=\hat{c_2}\gamma_2$ and $\sigma_h'=\hat{\sigma}_h$. These, however, satisfy $c_2'^2=1$, $\sigma_h'^2=-1$ and $c_2'y_ic_2'^{-1}=y_i$, $\sigma_h'y_i\sigma_h'^{-1}=y_i$ as well as $(c_n c'_2)^2=1$, $(c'_2\sigma'_h)^2=1$. We have ${\mathbb{R}}\langle c_n,c_2',\sigma_h'\rangle={\mathbb{R}}\langle\sigma_h'\rangle\hat{\otimes}{\mathbb{R}}\langle c_n,c_2'\rangle$, where ${\mathbb{R}}\langle\sigma_h'\rangle={\mathbb{C}}={{\mathrm{Cl}}}_{1,0}$ and ${\mathbb{R}}\langle c_n,c_2'\rangle={\mathbb{R}}[{{\mathrm{Dih}}}_n]={{\mathrm{Cl}}}_{0,1}^{\oplus r(n)}\oplus{{\mathrm{Cl}}}_{0,2}^{\oplus c(n)}$ is graded by ${\mathbb{Z}}_2={{\mathrm{Dih}}}_n/{\mathbb{Z}}_n$. We hence get ${{\mathrm{Cl}}}_{p,q}\otimes({{\mathrm{Cl}}}_{1,1}^{\oplus r(n)}\oplus{{\mathrm{Cl}}}_{1,2}^{\oplus c(n)})={{\mathrm{Cl}}}_{p,q}\otimes M_2({\mathbb{R}})^{\oplus r(n)+2c(n)}$ and find $${\mathcal{R}}_{q-2}^{r(n)+2c(n)}.$$ Magnetic Point Groups - an Example {#app:mag} ================================== In this appendix, we demonstrate the generalizability of our method for the treatment of crystalline topological insulators and superconductors with magnetic point group symmetries. Let us look at $C_{2n}'$ with generators ${\hat{c}'=\hat{c}T}$ such that $T^2=\epsilon_T$ and $$\hat{c}'^{2n}=-\epsilon_T^n.$$ It acts on the 3D space as $$\hat c'(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix})\hat c'^{-1}=-(\begin{smallmatrix}\cos\frac{2\pi}{2n}&\sin\frac{2\pi}{2n}\\-\sin\frac{2\pi}{2n}&\cos\frac{2\pi}{2n}\end{smallmatrix})(\begin{smallmatrix}\gamma_1\\\gamma_2\end{smallmatrix}).$$ We set $$c'= e^{\gamma_1\gamma_2\frac{2\pi}{2n}}\hat{c}',$$ which satisfies $$c' y_i c'^{-1}=-y_i,{{\;\;\:}}c'^{2n}=\epsilon_T^n.$$ The presence of charge conservation implies the existence of an “imaginary" generator ${J^2=-1}$ such that ${c'Jc'^{-1}=-J}$. We thus get ${B={{\mathrm{Cl}}}_{1,d}\hat{\otimes}{\mathbb{R}}\langle c',J\rangle}$. For ${n=1}$ we have ${{\mathbb{R}}\langle c',J\rangle={\mathbb{R}}\langle c',c'J\rangle}$ and find $$\begin{cases} {{\mathrm{Cl}}}_{1,d}\hat{\otimes}{{\mathrm{Cl}}}_{0,2}\mapsto{\mathcal{R}}_{-d}, & T^2=+1,\\ {{\mathrm{Cl}}}_{1,d}\hat{\otimes}{{\mathrm{Cl}}}_{2,0}\mapsto{\mathcal{R}}_{4-d}, & T^2=-1.\\ \end{cases}$$ For ${n=2}$ we split our algebra by the central element $c'^2$ and find $${{\mathrm{Cl}}}_{1,d}\hat{\otimes}({{\mathrm{Cl}}}_{0,2}\oplus{{\mathrm{Cl}}}_{2,0})\mapsto{\mathcal{R}}_{-d}\times{\mathcal{R}}_{4-d}.$$ For ${n=3}$ we have ${B={\mathbb{R}}\langle c'^2\rangle{\otimes}({{\mathrm{Cl}}}_{1,d}\hat{\otimes}{\mathbb{R}}\langle c'^3,c'^3J\rangle)}$ and thus find $$\begin{cases} ({\mathbb{R}}\oplus{\mathbb{C}}){\otimes}({{\mathrm{Cl}}}_{1,d}\hat{\otimes}{{\mathrm{Cl}}}_{0,2})\mapsto{\mathcal{R}}_{-d}\times{\mathcal{C}}_{d}, & T^2=+1,\\ ({\mathbb{R}}\oplus{\mathbb{C}}){\otimes}({{\mathrm{Cl}}}_{1,d}\hat{\otimes}{{\mathrm{Cl}}}_{2,0})\mapsto{\mathcal{R}}_{4-d}\times{\mathcal{C}}_{d}, & T^2=-1.\\ \end{cases}$$ As expected [@kennedy2016bott], in all cases, one may switch between ${\epsilon_T=\pm1}$ by tensoring with ${\mathbb{H}}$. [117]{} ifxundefined \[1\][ ifx[\#1]{} ]{} ifnum \[1\][ \#1firstoftwo secondoftwo ]{} ifx \[1\][ \#1firstoftwo secondoftwo ]{} ““\#1”” @noop \[0\][secondoftwo]{} sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{} @startlink\[1\] @endlink\[0\] @bib@innerbibempty [****, ()](\doibase 10.1103/RevModPhys.82.3045) [****,  ()](\doibase 10.1103/RevModPhys.83.1057) [****,  ()](\doibase 10.1103/RevModPhys.88.035005) [****, ()](\doibase 10.1103/RevModPhys.90.015001) [****,  ()](\doibase 10.1103/PhysRevLett.49.405) [****,  ()](\doibase 10.1103/PhysRevLett.51.51) [****,  ()](\doibase 10.1103/PhysRevLett.95.146802) [****,  ()](\doibase 10.1103/PhysRevLett.102.187001) [****,  ()](\doibase 10.1103/PhysRevB.78.195125) [****,  ()](\doibase 10.1103/PhysRevLett.95.226801) [****,  ()](\doibase 10.1126/science.1148047) [****,  ()](\doibase 10.1103/PhysRevLett.98.106803) [****,  ()](\doibase 10.1103/PhysRevB.75.121306) [****, ()](\doibase 10.1103/PhysRevB.79.195322) [****,  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--- abstract: 'We apply techniques from the field of computational mechanics to evaluate the statistical complexity of neural recording data in fruit flies. We connect statistical complexity to the flies’ level of conscious arousal, which is manipulated by general anaesthesia (isoflurane). We show that the complexity of even single channel time series data decreases under anaesthesia. The observed difference in complexity between the two states of conscious arousal increases as higher orders of temporal correlations are taken into account. In contrast to prior work, our results show that complexity differences can emerge at very short time scales and across broad regions of the fly brain without the need to saturate Markov order, thus heralding the macroscopic state of anaesthesia in a previously unforeseen manner. Furthering the links between physics, complexity science and neuroscience promotes the understanding of the physical basis that supports the level of conscious arousal in biological organisms.' author: - 'Roberto N. Mu[ñ]{}oz' - Angus Leung - 'Felix A. Pollock' - Dror Cohen - Bruno van Swinderen - Naotsugu Tsuchiya - Kavan Modi bibliography: - 'references.bib' title: Distinguishing states of conscious arousal using statistical complexity --- Introduction {#sec:Intro} ============ Complex phenomena are everywhere in the physical world. Typically, these emerge from simple interactions among elements in a network, such as atoms making up molecules or organisms in a society. Despite their diversity, it is possible to approach these subjects with a common set of tools, using numerical and statistical techniques to relate microscopic details to emergent macroscopic properties [@Thurner2018]. There has long been a trend of applying these tools to the brain, the archetypal complex system, and much of neuroscience is concerned with relating electrical activity in networks of neurons to psychological and cognitive phenomena [@CognitiveNeurosciences]. In particular, there is a growing body of experimental evidence [@Boly2013], that neural firing patterns can be strongly related to the level of conscious arousal in animals. In humans, level of consciousness varies from very low in coma and under deep general anaesthesia, to very high in fully wakeful states of conscious arousal [@Laureys2012]. With the current technology, precise discrimination between unconscious vegetative states and minimally conscious states are particularly challenging and remains a clinical challenge [@NeurologyOfConsciousness]. Therefore, substantial improvement in accuracy of determining such conscious states using neural recording data will have significant societal impacts. Towards such a goal, neural data has been analysed using various techniques and notions of *complexity* to try to find the most reliable measure of consciousness [@Engemann2018; @Sitt2014]. One of the most successful techniques to date in distinguishing levels of conscious arousal is the *perturbational complexity index* [@Massimini2005; @Casali2013; @Casarotto2016], which measures the neural activity patterns that follows a perturbation of the brain through magnetic stimulation. The evoked patterns are processed through a pipeline then finally summarised using Lempel-Ziv complexity [@Casali2013]. This method is inspired by a theory of consciousness, called *integrated information theory* (**IIT**) [@Tononi2004; @Tononi2016], which proposes that a high level of conscious arousal should be correlated with the amount of so-called *integrated information*, or the degree of differentiated integration in a neural system (see Ref. [@Oizumi2014] for details). While there are various ways to capture this essential concept [@Mediano2019; @Barrett2011], one way to interpret integrated information is as the amount of loss of information a system has on its own future or past states based on its current state, when the system is minimally disconnected [@Tegmark2016; @Oizumi2016PNAS; @Oizumi2016PLOS]. These complexity measures, inspired by IIT, are motivated by the fundamental properties of conscious phenomenology, such as informativeness and integratedness of any experience [@Tononi2004]. While there are ongoing efforts to accurately translate these phenomenological properties into mathematical postulates [@Oizumi2014], such translation often contains assumptions about the underlying process which are not necessarily borne out in reality. For example, the derived mathematical postulates in IIT assume Markovian dynamics, i.e., that the future evolution of a neural system is determined statistically by its present state [@Barrett2011]. Moreover, IIT requires computing the correlations across all possible divisions between subsystems, which makes it computationally very hard. Given the hierarchical causal influences in the brain, manifesting as oscillations across a range of frequencies and spatial regions [@Buzsaki2006], non-Markovian temporal correlations likely play a significant role in explaining any experimentally measurable behaviours, potentially including the level of conscious arousal. There is therefore, scope for applying more general notions of complexity to meaningfully distinguish macroscopic brain states that support consciousness. A conceptually simple approach to quantifying the complexity of time series data, such as the fluctuating potential in a neuron, is to construct the minimal model which statistically reproduces it. Remarkably, this minimal model, known as an *epsilon machine* ([$\epsilon\text{-machine}$]{}), can be found via a systematic procedure which has been developed within the field of computational mechanics [@CrutchPRL1989; @epsilonMachines2; @CrutcharXiv2017]. Crucially, [$\epsilon\text{-machines}$]{} account for multiple temporal correlations contained in the data and can be used to quantify the *statistical complexity* of a process – the minimal amount of information required to specify its state. As such they have been applied over various fields, ranging from neuroscience [@Haslinger2009; @Klinkner2006] and psychology [@CSSR2] to crystallography [@Varn2004] and ecology [@Boschetti2008], to the stock market [@Park2007]. Lastly, unlike IIT the [$\epsilon\text{-machine}$]{} analysis can be performed for data coming from a single channel. In this paper, we use the statistical complexity derived from an [$\epsilon\text{-machine}$]{} analysis of neural activity to distinguish states of conscious arousal in fruit flies (*D. melanogaster*) as measured in the presence or absence of general anaesthetics. We analyse neural data collected from flies under different concentrations of isoflurane [@CohenEneuro2016; @CohenEneuro]. By analysing signals from individual electrodes and disregarding spatial correlations, we find that statistical complexity distinguishes between the two states of conscious arousal through temporal correlations alone. In particular, as the degree of temporal correlations increases, the difference in complexity between the wakeful and anaesthetised states becomes larger. Before presenting these results in detail in Sec. \[sec:Complexity\] and discussing their implications in Sec. \[sec:discussion/conclusion\], we begin in the next section with a brief overview of the [$\epsilon\text{-machine}$]{} framework we will use for our analysis. Theory: $\epsilon$-Machines and statistical complexity {#sec:Background} ====================================================== To uncover the underlying statistical structure of neural activity that characterises a given conscious state, we treat the measured neural data, given by voltage fluctuations in time, as discrete time series. To analyse these time series, we use the mathematical tools of computational mechanics, which we outline in this section. We start a general discussion on the ways to use time series data to infer a model of a system while placing [$\epsilon\text{-machines}$]{} in this context. Next, we explain how we construct [$\epsilon\text{-machines}$]{} in practice. Finally, we show how this can be used to extract a meaningful notion of statistical complexity of a process. From time series to $\epsilon$-Machines {#sec:Bkg-eMs} --------------------------------------- In abstract terms, a discrete-time series is a sequence of symbols $\mathbf{r} = (r_0, \ldots, r_{k}, \ldots)$ that appear over time, one after the other [@Rabiner1989]. Each element of $\mathbf{r}$ corresponds to a symbol from a finite alphabet $\mathcal{A}$ observed at the discrete time step labelled by the subscript $k$. The occurrence of a symbol, at a given time step, is random in general and thus the process, which produces the time series, is stochastic [@DoobStochastic]. However, the symbols may not appear in a completely independent manner, i.e., the probability of seeing a particular symbol may strongly depend on symbols observed in the past. These temporal correlations are often referred to as *memory*, and they play an important role in constructing models that are able to predict the *future* behaviour of a given stochastic process [@Gu2012]. Relative to an arbitrary time $k$, let us denote the future and the past partitions of the complete sequence as $\mathbf{r} = ({\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}, \vec{r})$, where the past and the future are ${\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}} = (\ldots, r_{k-2},r_{k-1})$ and $\vec{r} = (r_{k}, r_{k+1}, \ldots)$ respectively. In general, for the prediction of the immediate future symbol $r_k$, knowledge of the past $\ell$ symbols ${\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\ell} :=(r_{k-\ell}, \ldots, r_{k-2}, r_{k-1})$, may be necessary. The number of past symbols we need to account for in order to optimally predict the future sequence is called the Markov order [@Gagniuc2017]. In general, the difficulty of modelling a time series increases exponentially with its Markov order. However, not all distinct pasts lead to unique future probability distributions, leaving room for compression in the model. In a seminal work, Crutchfield and Young showed the existence of a class of models, which they called $\epsilon$-machines, that are provably the optimal predictive models for a non-Markovian process under the assumption of statistical stationarity [@CrutchPRL1989; @epsilonMachines2]. Constructing the $\epsilon$-machine is achieved by partitioning sets of *partial* past observations ${\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\ell}$ into *causal states*. That is, two distinct sequences of partial past observations ${\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\ell}$ and ${\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\ell}'$ belong to the same causal state $S_i \in \mathcal{S}$, if the probability of observing a specific $\vec{r}$ given ${\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\ell}$ or ${\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\ell}'$ is the *same*; that is $$\begin{gathered} {\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\ell} \sim_\epsilon {\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\ell}' \quad\text{if}\quad P(\vec{r} \;|\; {\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\ell}) = P(\vec{r} \;|\; {\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\ell}'), \label{eq:equivRelation}\end{gathered}$$ where $\sim_\epsilon$ indicates that two histories correspond to the same causal state. The conditional probability distributions in Eq.  may always be estimated from a finite set of (statistically stationary) data via the naive maximum likelihood estimate, given by $P(r_k|{\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\ell}) =\nu(r_k,{\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\ell})/\nu({\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\ell})$, where $\nu(X)$ is the frequency of occurrence of sub-sequence $X$ in the data. We now discuss how to practically construct an [$\epsilon\text{-machine}$]{} for a given time series. Constructing [$\epsilon\text{-machines}$]{} with the CSSR algorithm {#sec:Bkg-cssr} ------------------------------------------------------------------- Several algorithms have been developed to construct [$\epsilon\text{-machines}$]{} from time series data [@Tino2001; @CrutchPRL1989; @Crutchfield1990]. Here, we briefly explain the *Causal State Splitting Reconstruction* **(CSSR)** algorithm [@CSSR2], which we use in this work to infer [$\epsilon\text{-machines}$]{} predicting the statistics of neural data we provide as input. The CSSR algorithm proceeds to iteratively construct sets of causal states accounting for longer and longer sub-sequences of symbols. In each iteration, the algorithm first estimates the probabilities $P(r_k|{\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\ell})$ of observing a symbol conditional on each length $\ell$ prior sequence and compares them with the distribution $P(r_k | \mathcal{S} = S_i)$ it would expect from the causal states it has so far reconstructed. If $P(r_k|{\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\ell}) = P(r_k | \mathcal{S} = S_i)$ for some causal state, then ${\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\ell}$ is identified with it. If the probability is found to be different for all existing $S_i$, then a new causal state is created to accommodate the sub-sequence. By constructing new causal states only as necessary, the algorithm guarantees a minimal model that describes the non-Markovian behaviour of the data (up to a given memory length), and hence the corresponding [$\epsilon\text{-machine}$]{} of the process. The CSSR algorithm compares probability distributions via the *Kolmogorov-Smirnov* **(KS)** test [@Massey1951; @Hollander2013]. The hypothesis that $P(r_k|{\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\ell})$ and $P(r_k | \mathcal{S} = S_i)$ are identical up to statistical fluctuations is rejected by the KS test at the significance level $\sigma$ when a distance $\mathcal{D}_{KS}$ [^1] is greater than tabulated critical values of $\sigma$ [@Miller1956]. In other words, $\sigma$ sets a limit on the accuracy of the history grouping by parametrising the probability that an observed history ${\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\ell}$ belonging to a causal state $S_i$, is mistakenly split off and placed in a new causal state $S_j$. Our analysis, in agreement with Ref. [@CSSR2], found that the choice of this value does not affect the outcome of CSSR within the tested range of $0.001 < \sigma < 0.01$. As a result, we set $\sigma = 0.005$ for the entirety of this study unless stated otherwise. As it progresses, the CSSR algorithm compares future probabilities for longer and longer sub-sequences, up to a maximum past history length of $\lambda$, which is the only important parameter that must be selected prior to running CSSR in addition to $\sigma$. A value of $\lambda$ is the largest size of memory we consider. If the considered time series is generated by a stochastic process of Markov order $\ell$, choosing $\lambda < \ell$ results in poor prediction because the inferred ${\ensuremath{\epsilon\text{-machine}}}{}$ cannot capture the long-memory structures present in the data. Despite this, the CSSR algorithm will still produce an [$\epsilon\text{-machine}$]{} that is consistent with the approximate future statistics of the process up to order-$\lambda$ correlations [@CSSR2]. Given sufficient data, choosing $\lambda \geq \ell$ guarantees convergence on the true ${\ensuremath{\epsilon\text{-machine}}}{}$. One important caveat to note is that the time complexity of the algorithm scales asymptotically as $\mathcal{O} (|\mathcal{A}|^{2\lambda+1})$, putting an upper limit to the longest history length that is computationally feasible to use. Furthermore, the finite length of the time series data implies an upper limit on an ‘acceptable’ value of $\lambda$. Estimating $P(r_k | {\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\lambda})$ requires sampling strings of length $\lambda$ from the finite data sequence. Since the number of such strings grows exponentially with $\lambda$, a value of $\lambda$ that is too long relative to the size $N$ of the data, will result in a severely under-sampled estimation of the distribution. A distribution $P(r_k | {\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}_{\lambda})$ that has been estimated from an under-sampled space is almost always never equal to $P(r_k | \mathcal{S} = S_i)$, resulting in the algorithm creating a new causal state for every string of length $\lambda$ it encounters. A bound for the largest permissible history length is $L(N) \geq \log_2 N/\log_2 |\mathcal{A}|$, where $L(N)$ denotes maximum length for a given data size of $N$ [@MartonAoP; @CoverThomas]. Once these considerations have been taken into account, the [$\epsilon\text{-machine}$]{} produced by the algorithm provides us with a meaningful quantifier of the complexity of the process generating the time series, as we now discuss. Statistical complexity of a process ----------------------------------- The output of the CSSR algorithm is the set of causal states and rules for transitioning from one state to another. That is, CSSR gives a Markov chain represented by a digraph [@CrutchPRL1989; @Gagniuc2017] $G(V,E)$ consisting of a set of vertices $v_i \in V$ and directed edges $\{i,j\} \in E$, e.g. Figs. \[fig:workflow\](c) and (d). Using these rules, one can find the probabilities $P(S_i)$ to find the [$\epsilon\text{-machine}$]{} in each of the causal states at a randomly chosen time. The Shannon entropy of this distribution quantifies the minimal number of bits of information required to optimally predict the future process; this measure, first introduced in Ref. [@CrutchPRL1989], is called the *statistical complexity*: $$\begin{gathered} C_{\mu} := H\left[\mathcal{S}\right] = -\sum_i P(S_i) \log P(S_i). \label{eq:statComplexity}\end{gathered}$$ In the next section, we describe the experimental and analysis methods and the results: the statistical complexity of the neural time series for the conscious arousal states of the fly corresponding to awake and anaesthetised conditions are significantly different. {width="\textwidth"} Experimental results and analysis {#sec:Complexity} ================================= Methods {#sec:Methods} ------- We analysed local field potential **(LFP)** data from the brains of awake and isoflurane-anaesthetised *D. melanogaster* (Canton S wild type) flies. Here, we briefly provide the essential experimental outline that is necessary to understand this paper. The full details of the experiment are presented in Refs. [@CohenEneuro; @CohenEneuro2016]. LFPs were recorded by inserting a linear silicon probe (Neuronexus 3mm-25-177) with 16 electrodes separated by 25 $\mu$m. The probe covered approximately half of the fly brain and recorded neural activity as illustrated in Fig.  \[fig:workflow\](a). A tungsten wire inserted into the thorax acted as the reference. The LFPs at each electrode were recorded for 18s while the fly was awake and 18s more after the fly was anaesthetised (isoflurane, 0.6% by volume, through an evaporator). Flies’ unresponsiveness during anaesthesia was confirmed by the absence of behavioural responses to a series of air puffs, and recovery was also confirmed after isoflurane gas was turned off [@CohenEneuro2016]. We used data sampled at 1kHz for the analysis [@CohenEneuro2016], and to obtain an estimate of local neural activity, the 16 electrodes were re-referenced by subtracting adjacent signals giving 15 channels which we parametrise as $c \in [1,15]$. Line noise was removed from the recordings, followed by linear de-trending and removing the mean. The resulting data is a fluctuating voltage signal, which is time-binned (1ms bins) and binarised by splitting over the median, leading to a time series, see Fig. \[fig:workflow\](b). For each of the 13 flies in our data set, we have 30 time series of length $N = 18,000$. They correspond to the 15 channels, labelled numerically from the central to peripheral region as depicted in Fig. \[fig:workflow\](a), and the two states of conscious arousal. Using the CSSR algorithm [@CSSR2], we construct [$\epsilon\text{-machines}$]{} for each of these time series as a function of maximum memory length within the range $\lambda \in [2,11]$. This is below the memory length $L(N) \sim 14$ beyond which we would be unable to reliably determine transition probabilities for a sequence of length $N$ (see Sec. \[sec:Bkg-cssr\]) [^2]. We record the resulting $3,900$ [$\epsilon\text{-machine}$]{} structures and their corresponding statistical complexities, and group them according to their respective level of conscious arousal, $\psi$, channel location, $c$, and maximum memory length, $\lambda$. Thus, statistical complexity $C_{\mu}$ is a function of the set of parameters $\{\psi,c,\lambda\}$ for each fly, $f$. We are principally interested in the difference in statistical complexity over states of conscious arousal $$\begin{gathered} \Delta C_{\mu} = C_{\mu}^{\text{wake}} - C_{\mu}^{\text{anaes}}\end{gathered}$$ for fixed values of $\{f, c, \lambda\}$. Positive values of $\Delta C_{\mu}$ indicate higher complexities observed in the wakeful state relative to the anaesthetised one. Finally, we use the notation $\langle \Delta C_{\mu} \rangle_x$ to denote taking an average of [$\Delta C_{\mu}$]{} over parameter set $x \in \{f, c, \lambda\}.$ To assess the significance each of the parameters $\psi$, $c$, and $\lambda$, or some combination of them, have on the response of $C_{\mu}$ across flies, we conduct a statistical analysis using linear mixed effects modelling [@Harrison2018lme] (**LME**). The LME analysis describes the response of statistical complexity by modelling it as a multidimensional linear regression of the form $$\begin{gathered} \mathcal{C} = {\ensuremath{\mathbf{F}}}{\ensuremath{\mbox{\boldmath$ \beta $}}} + {\ensuremath{\mathbf{R}}}{\ensuremath{\mathbf{b}}} + \mathcal{E}. \label{eq:glme}\end{gathered}$$ The resulting model in Eq.  consists of a family of equations where $\mathcal{C}$ is the vector allowing for different responses of $C_{\mu}$ for each specific fly, channel location, and level of conscious arousal. Memory length $\lambda$, channel location $c$, and state of conscious arousal $\psi$, are the parameters that $\mathcal{C}$ responds to. To account for variations in the response caused by interactions between parameters (e.g. between memory length and channel location), we include them in the model. The set of these parameters, and any combination between them $\mathcal{F}=\{\lambda, c, \psi, \lambda c, \lambda\psi, c\psi, \lambda c\psi\}$, are known as the *fixed effects* of Eq. , and are contained as elements within the matrix ${\ensuremath{\mathbf{F}}}$. The vector ${\ensuremath{\mbox{\boldmath$ \beta $}}}$, contains the regression coefficients describing the strength of each of the fixed effects $\mathcal{F}$. In addition to fixed effects affecting the response of statistical complexity in our experiment, we also take into account any variation in response caused by known, *random effects*. In particular, we expect stronger response variations to be caused by correlations occurring between the channels within a single fly, compared to between channels across flies. These random effects are contained as elements of the matrix ${\ensuremath{\mathbf{R}}}$, where the vector ${\ensuremath{\mathbf{b}}}$ are the regression coefficients describing their strengths. Finally, the vector $\mathcal{E}$ describes the normally-distributed *unknown*, random effects in the model. The regression coefficients contained in the vectors ${\ensuremath{\mbox{\boldmath$ \beta $}}}$ and ${\ensuremath{\mathbf{b}}}$, are obtained via maximum likelihood estimation such that $\mathcal{E}$ are minimised. The explicit form of Eq.  used in this analysis is detailed in the Appendix \[app:lme\]. With the full linear mixed effects model given by Eq. , we test the statistical significance of a fixed effect $\mathcal{F}$. This is accomplished by comparing the log-likelihood of the full model with all fixed effects, to the log-likelihood of a reduced model with the effect of interest removed [@Bates2015] (regression coefficients associated with the effect are removed). This comparison between the likelihood models is given by $\Lambda = 2 (h_{\text{full}} -h_{\text{reduced}})$, where $\Lambda$ is the likelihood ratio, $h_{\text{full}}$ is the log-likelihood of full model, and $h_{\text{reduced}}$ is the log-likelihood of the model with the effect of interest removed. Under the null hypothesis, when a fixed effect does not have any influence on $C_{\mu}$, i.e., the regression coefficients for the effect are vanishing, the likelihood ratio $\Lambda$, is $\chi^2$ distributed with degrees of freedom equal to the difference in the number of coefficients between the models. Therefore, we consider any fixed effect in the set $\mathcal{F}$ to have a statistically significant effect on $C_\mu$, if the probability of obtaining the likelihood ratio given the relevant $\chi^2$ distribution is less than 5% ($p<0.05$). Thus, for each significant effect we report the fixed effect being tested, i.e., an element of $\mathcal{F}$, the obtained likelihood ratio with its associated degrees of freedom ($\chi^2(d.o.f.)$), and the associated probability of obtaining the statistic under the null hypothesis ($p$). The LME and likelihood procedure are also repeated for [$\Delta C_{\mu}$]{} in order to find the significant interaction effects of the parameters. Here, we also model [$\Delta C_{\mu}$]{} as dependent on $\lambda$ and $c$ as in Eq. , but excluding the parameter $\psi$. Once the significant effects of memory length, level of conscious arousal, and channel location are characterised with our statistical analysis, we follow with post-hoc, one-sample two-tailed $t$-tests given by $$\begin{gathered} t = \frac{\langle \Delta C_{\mu}\rangle_f - \mu_0}{s_f / \sqrt{|f|}}, \label{eq:ttest}\end{gathered}$$ in order to examine the nature of interactions between $\lambda$ and $c$ on [$\Delta C_{\mu}$]{}. We set $\mu_0$ to be the value of $\langle \Delta C_{\mu} \rangle_f$ under the null hypothesis $(\mu_0 = 0)$, $s_f$ is the standard deviation of $\langle \Delta C_{\mu} \rangle_f$, and $|f| = 13$ is the sample size. We present the results of these analyses in the following sections. Results {#sec:results} ------- In order to observe the effects of isoflurane on neural complexity, we begin by visually inspecting the structure of the reconstructed [$\epsilon\text{-machines}$]{} for the two levels of conscious arousal. We take special interest in observing the differences in the characteristics of the two groups of [$\epsilon\text{-machines}$]{} heralding the two levels of conscious arousal. Here, memory length $\lambda$ plays an important role. At a given $\lambda$, the maximum number of causal states that may be generated scales according to $|\mathcal{A}|^{\lambda}$ [@CSSR2]. In our case, the alphabet is binary, $\mathcal{A} = \{0,1\}$. This greatly restricts the space of [$\epsilon\text{-machine}$]{} configurations available for short history lengths [@Johnson2010]. For $\lambda = 2$ we can observe up to four distinct configurations, which is unlikely to reveal the difference based on conscious states. Given the previous findings [@CohenEneuro], we generally expected that the data from the wakeful state would present more complexity than those from the anaesthetised state. Visual inspection of the directed graphs indeed suggests higher [$\epsilon\text{-machine}$]{} complexity during the wakeful state compared to the anaesthetised state, at a given set of parameters $\{f,c,\lambda\}$. In particular, the data from the anaesthetised state tended to result in a fewer number of causal states and overall reduced graph connectivity. Panels (c) and (d) of Fig. \[fig:workflow\] are examples of [$\epsilon\text{-machines}$]{} (channel 1 data recorded from fly 1, at maximum memory length $\lambda = 3$), where a simpler [$\epsilon\text{-machine}$]{} is derived from the data under the anaesthetised condition. ![Colour map of statistical complexity response averaged over $(n=13)$ flies $\langle C_{\mu} \rangle_f$, during wakefulness (left) and isoflurane (right), over channel location and memory length $\lambda$. Hatched cells on the right sub-figure, show regions where $C_{\mu}$ did not decrease under anaesthesia.[]{data-label="fig:Cmu-raw-response"}](DC_Lmax_channel_cond.pdf){width="\columnwidth"} Differentiating between two conscious arousal states by visual inspection quickly becomes impractical because of the large number of [$\epsilon\text{-machines}$]{}. Moreover, for large values of $\lambda$ the number of causal states is exponentially large and it becomes difficult to see the difference in two graphs. To overcome these challenges, we look at the statistical complexity $C_\mu$ to differentiate between conscious arousal states. To systematically determine the relationships between $C_\mu$ and the set of variables $\{c,f,\psi\}$ we employ the LME analysis outlined in Sec. \[sec:Methods\]. We first test whether $\lambda$ significantly affects $C_{\mu}$. We found $\lambda$ to indeed have a significant effect on $C_{\mu}$ ($\lambda$, ${\chi}^2(1)=443.64$, $p<10^{-16}$). Fig. \[fig:Cmu-raw-response\] shows that independent of the conscious arousal condition or channel location, $C_{\mu}$ increases with larger $\lambda$. This indicates that the Markov order of the neural data is much larger than the largest memory length ($\lambda=11$) we consider. Nevertheless, we have enough information to work with. We now seek to confirm if the complexity of [$\epsilon\text{-machines}$]{} during anaesthesia are reduced, as suggested from visual inspection. Our statistical analysis indicates that $C_{\mu}$ is not invariably reduced during anaesthesia ($\psi$, ${\chi}^2(1)=0.212$, $p=0.645$) at all levels of $\lambda$ and all channel locations. This means that $C_\mu$ cannot simply indicate the causal arousal state without some additional information about time ($\lambda$) or space ($c$). In addition, we find that neither $c$ alone nor $c\psi$ strongly effects $C_\mu$. However, we find significant reductions in complexity when either the level of conscious arousal or the channel location, interacts with memory length ($\lambda\psi$, ${\chi}^2(1)=14.63$, $p=1.31\times10^{-4}$) and (${c\lambda}$, $\chi^2(14)=42.876$, $p=8.97\times 10^{-5}$) respectively. Moreover, the three-way interaction also has a strong effect ($\lambda\psi c$, ${\chi}^2(14)=24.00$, $p=0.0458$). As the three-way interaction between $\lambda$, $\psi$, and $c$ complicates interpretation of their effects, we perform a second LME analysis where we model [$\Delta C_{\mu}$]{} instead of $C_{\mu}$, thus accounting for $\psi$ implicitly. In doing so, we now investigate whether the change in statistical complexity due to anaesthesia is affected by memory length $\lambda$ or channel location $c$. Using this model, we find a non-significant effect of $c$ on [$\Delta C_{\mu}$]{}, while a significant effect of $\lambda$ on [$\Delta C_{\mu}$]{} is seen ($\lambda$, ${\chi}^2(1)=20.97$, $p=4.65\times10^{-6}$), indicating that [$\Delta C_{\mu}$]{} overall changes with $\lambda$. Specifically, [$\Delta C_{\mu}$]{} tends to increase with larger $\lambda$ when ignoring channel location, as is evident in Fig. \[fig:cmuvslmax\]. Further, explaining our previous interaction between $\lambda$ and $\psi$, [$\Delta C_{\mu}$]{} was not clearly larger than $0$ for small memory length ($\lambda=2$; Fig. \[fig:cmuvslmax\]). This suggests that the information to differentiate between states of conscious arousal is contained in higher order correlations. We also find that the interaction between ${\lambda}$ and channel location has a significant effect on [$\Delta C_{\mu}$]{} ($\lambda c$, ${\chi}^2(14)=37.19$, $p=6.90\times10^{-4}$), indicating that the effect of $\lambda$ is not constant across channels. Given that [$\Delta C_{\mu}$]{} overall increases with $\lambda$, we considered that that the largest [$\Delta C_{\mu}$]{} should occur at the largest $\lambda$. Fig. \[fig:cmuvschannel\] examines [$\Delta C_{\mu}$]{} across channels at $\lambda=11$. Here, we see varying [$\Delta C_{\mu}$]{} with channel location, which is likely reflecting different rates at which [$\Delta C_{\mu}$]{} increases with $\lambda$ for each channel. To further break down the interaction between $\lambda$ and $c$, we perform a one sample $t$-tests at each value of memory length and channel location to find regions in the parameter space $(\lambda, c)$ where $C_{\mu}$ reliably differentiates wakefulness from anaesthesia across flies. We plot the $t$-statistic at each parameter combination in Fig. \[fig:heatmap\], outlining regions in the parameter space where [$\Delta C_{\mu}$]{} is significantly greater than $0$ (with $p<0.05$, uncorrected, two-tailed). We find that the majority of the significance map is directed towards positive values of the $t$-statistic. However, only a subset of $(\lambda,c)$ cells contain values which are significantly different from $0$. Interestingly, we observe that for $\lambda=2$, [$\Delta C_{\mu}$]{} is actually significantly negative, corresponding to greater complexity during anaesthesia, not during wakefulness. This marks $\lambda=2$ as anomalous relative to other levels of $\lambda$, and this reversal of the direction of the effect of anaesthesia likely contributed to the interaction between $\lambda$ and $\psi$. Disregarding $\lambda=2$, we find [$\Delta C_{\mu}$]{} to be significantly greater than $0$ for channels $1, 3, 5-7, 9, 10$, and $13$, at varying levels of $\lambda$. As expected from our reported interaction between $\lambda$ and $c$, we observe [$\Delta C_{\mu}$]{} to already be significantly greater than $0$ at small $\lambda$ for channels $5-7$, while [$\Delta C_{\mu}$]{} only becomes significantly greater at larger $\lambda$ for channels $1, 3, 9, 10$ and $13$. Further, other channels such as the most peripheral channel ($c=15$) do not have [$\Delta C_{\mu}$]{} significantly greater than $0$ at any $\lambda$. ![Difference in statistical complexity $\Delta C_{\mu} =C_{\mu}^{\text{wake}} - C_{\mu}^{\text{anaes}}$ of [$\epsilon\text{-machines}$]{} between states of conscious arousal, increases with memory length $\lambda$. Grey lines indicate complexity averages over channels per fly $(n=13)$, $\langle \Delta C_{\mu}\rangle_{c}$, while the blue line denotes the average over both channels and flies $\langle \Delta C_{\mu} \rangle_{c,f}$. Error bars are $95\%$ confidence intervals of the population.[]{data-label="fig:cmuvslmax"}](DCLmax-nolegend.eps){width="\columnwidth"} ![Difference in statistical complexity $\Delta C_{\mu}=C_{\mu}^{\text{wake}} - C_{\mu}^{\text{anaes}}$ at maximum memory length $\lambda = 11$, mapped throughout the fly brain. Grey and red lines indicate the result per fly and the average over $(n=13)$ flies, $\langle \Delta C_{\mu} \rangle_{f}$, respectively. Error bars corresponding to the $95\%$ confidence intervals over the sample of files.[]{data-label="fig:cmuvschannel"}](DCvsChannel-nolegend.eps){width="\columnwidth"} ![Colour map for t-scores on statistical complexity differences $\langle\Delta C_{\mu}\rangle_f = \langle C_{\mu}^{\text{wake}} - C_{\mu}^{\text{anaes}}\rangle_f$ over channel location and memory length $\lambda$. Dotted lines indicate the pixels that exceeded $p < 0.05$ (uncorrected).[]{data-label="fig:heatmap"}](DCtv2.pdf){width="\columnwidth"} Discussion {#sec:discussion/conclusion} ========== Discovering a reliable measure of conscious arousal in animals and humans remains one of the major outstanding challenges of neuroscience. The present study takes this challenge by connecting a complexity measure to the degree of conscious arousal. Our study here is a step forward to strengthening the link between physics, complexity science, and neuroscience. Here we have taken tools from the former and have applied them to a problem in the latter. Namely, we have studied the statistical complexity of neural recordings in the brains of flies over two states of conscious arousal: awake and anaesthetised. We have demonstrated that differences between these macroscopic conditions can be revealed by the statistical complexity of local electrical fluctuations in various brain regions. Specifically, we analysed the single-channel signals from electrodes embedded in the brain using the [$\epsilon\text{-machine}$]{} formalism, and quantified the statistical complexity, $C_{\mu}$, of the recorded data for 15 channels in 13 flies over two states of conscious arousal. We found the statistical complexity to be larger on average when a fly is awake than when the same fly is anaesthetised ($\Delta C_{\mu} > 0$; Figs. \[fig:cmuvslmax\], \[fig:cmuvschannel\], and \[fig:heatmap\]). We found that the measured difference in complexity is present across various brain regions (Fig. \[fig:heatmap\]), even at the non-trivial memory lengths ($\lambda>2$), continuing to grow as longer temporal correlations are taken into account, up to $\lambda=11$ that we tested. While Fig. \[fig:cmuvslmax\] showed a continued increase in the difference of statistical complexity, $\Delta C_{\mu}$, as a function of history length, $\lambda$, we did not pursue longer history lengths, due to limitations in the amount of the data and stability of the estimation of $C_{\mu}$. In addition to this general observation of increasing [$\Delta C_{\mu}$]{} over $\lambda$, we observed that some brain regions surprisingly discriminate the conscious arousal states with history length of only 3. One trivial explanation for this effect is that under anaesthesia, the required memory length is indeed $\lambda =2$, while the optimal $\lambda$ for awake is much larger. However, a quick observation of Fig. \[fig:Cmu-raw-response\] rules out this simple possibility; under both wakeful and anaesthetised states, $C_{\mu}$ continues to increase. It is likely however, that the tested range for $\lambda$ remains below the Markov order of the neural data; this is clearly indicated by the lack of a plateau in statistical complexity in Fig. \[fig:cmuvslmax\]. This suggests that we are far from saturating the Markov order of the process, and with more data we would be able to further distinguish between the two states. Future analyses with longer time series would also contribute to our understanding of the Markov order (maximum memory length) differences between the two states of conscious arousal Nevertheless, our results, in Figs. \[fig:cmuvslmax\] and \[fig:heatmap\], demonstrate that saturation of Markov order is not required for discrimination between conscious arousal states. This finding has a practical implication about the empirical utility of [$\epsilon\text{-machines}$]{}; even if the history length is too low the inferred [$\epsilon\text{-machine}$]{} and its statistical complexity can be useful. As we demonstrated in this study, the local information contained within a single channel, may contain the information about the global conscious states, that are believed to arise from interactions among many neurons. Theoretically, single channels can reflect the complexity of the multiple channels due to the concept of Sugihara causality [@Sugihara2012]. This arises due to any one region of the brain causally interacting with the rest of the brain, making the temporal correlation in a single channel time series contain information about the spatial correlations, i.e., information that would be contained in multiple channels. With this logic, Ref. [@Tajima2015] infers the complexity of the multi-channel interactions from a single channel temporal structure of the time series. This is often known as the backflow of information in non-Markovian dynamics [@BreuerPRL2009]. The surprising periodic structure of statistical complexity observed across channels in Fig. \[fig:Cmu-raw-response\], demonstrates an unexpected example of spatial effects present in our study – one that was not observed with conventional LFP analyses. While the origin of this effect may be attributed to polarity reversals of LFPs across brain lobes, the ultimate cause is unexplained for the moment. This observation provides a strong motivation for multi-channel analyses, as it is clear that causal history metrics revealing periodic spatial information structures, hidden from LFP signal processing techniques, may relate to information backflow. While we already find differences between conscious states in the single channel based [$\epsilon\text{-machine}$]{} analysis, it would be beneficial to extend the present analysis to the multi-channel scenario, in which [$\epsilon\text{-machine}$]{} can be contrasted with the methods of IIT [@Casali2013; @Casarotto2016; @Tononi2004; @Tononi2016; @Oizumi2014; @Mediano2019; @Barrett2011; @Tegmark2016; @Oizumi2016PNAS; @Oizumi2016PLOS]. Formal comparison of the distinguishing power of conscious states among proposed methods [@Engemann2018; @Sitt2014], will contribute to refine models and theories of consciousness. Our results can be informally compared with a previous study, where the *power spectra* of the same data in the frequency domain [@CohenEneuro] was analysed. Here, a principal observation was the power in low-frequency signals in central and peripheral regions, which was more pronounced in the central region (corresponding to channel 1-6 in this study). Our [$\epsilon\text{-machine}$]{} analysis here revealed that the region between periphery and centre (channels 5-7) showed most consistent difference in $C_{\mu}$ across history length $\lambda > 2$. Ultimately, the reason for this difference is due to our distinct approach, in so far as [$\epsilon\text{-machines}$]{} are provably the optimal predictive models of a large class of time series that take into account higher order correlations memory structure [@CrutchPRL1989; @epsilonMachines2]. Thus, our application of [$\epsilon\text{-machines}$]{} contrasts the power spectra analysis, by considering these higher order correlations for the very high-frequency signals, instead of only two-point correlations in both high- and low-frequency signals. Our multi-time analysis further reveals an interesting effect upon expanding the details for the anaesthetised [$\epsilon\text{-machine}$]{} example shown in Fig. \[fig:workflow\](c). When we examine the binary strings belonging to each causal state, we find a clear split between active (consecutive strings of ones) and inactive (consecutive strings of zeros) neural behaviour corresponding to the left and right hand sides of Fig. \[fig:em-with-histories\] respectively. Previous studies have demonstrated an increase in low-frequency LFP and EEG power for mammals and birds during sleep and anaesthesia, mediated by similar neural states of activity and inactivity known as ‘up’ and ‘down’ states [@Sarasso2015; @Lewis2012]. This phenomena remains to be observed in flies. While our study does not directly observe this slow oscillation between neural activity and quiescence, the active left and quiet right causal state structure of Fig. \[fig:em-with-histories\] may suggest an analogue for the ‘up’ and ‘down’ states for flies in the absence of increased low-frequency power. Future studies with more formal comparisons between power spectra and [$\epsilon\text{-machines}$]{} in theory and computer simulation may be a fruitful venue for further research for this comparison. ![[$\epsilon\text{-machine}$]{} for same channel, fly, conscious state as Fig. \[fig:workflow\](c), but with histories stored in each causal state explicitly stated. The sequences after the asterisk $*$ represent the sequence of symbol observations with the most recent observed symbol on the far right. Sequences collected within a causal state (grey circle) warrant significantly different future statistics to observed sequences in other causal states. The red lines emit a “1" upon transition, and blue lines emit “0"s.[]{data-label="fig:em-with-histories"}](sleep-c1-f1-L3-histories.pdf){width="\columnwidth"} Indeed, many definitions and measures of complexity have been proposed in the literature, see Ref. [@Edmonds1997] for a list. Moreover, there is a flow of ideas going the other way as well [@PhysRevLett.119.225301; @PhysRevA.97.052320; @QIIT]. Our interdisciplinary study opens up new possibilities; physics can improve its theoretical constructs through the application of tools to empirical data, while neuroscience can benefit from rigorous quantitative tools that have proven their physical basis across different spatio-temporal scales. Among those complexity measures, $C_{\mu}$ can be easily interpreted in terms of temporal structure [@whyCmu], as it has a direct relation to process predictability and memory requirements. One important property of $C_{\mu}$ is that it is zero for both deterministic and uniformly random processes, and it is maximum for stochastic processes with large memory effects (see Eq. ). When coupled with our results, we can conclude that anaesthetised brains become less structured and more random, and approaches a stochastic process with smaller memory capacity compared to the wakeful brains. Overall, our results suggest that measures of complexity, including [$\epsilon\text{-machines}$]{}, which have not been tested before in this context, might be able to identify further structures that are affected by anaesthesia at different spatial and temporal scales. It is also likely that applying a similar analysis to other data sets, in particular, human EEG data will lead to new discoveries regarding the relationship between consciousness and complexity that can be retrieved simply at the single channel level. RNM, FAP, NT, KM acknowledge support from Monash University’s Network of Excellence scheme and the Foundational Questions Institute grant on *Agency in the Physical World*. DC was funded by an Overseas JSPS Postdoctoral Fellowship. NT was funded by Australian Research Council Discovery Project grants (DP180104128, DP180100396). NT and CD were supported by a grant (TWCF0199) from Templeton World Charity Foundation, Inc. We thank Felix Binder, Alec Boyd, Mile Gu, Rhiannon Jeans, and Jayne Thompson for valuable comments. Appendix {#appendix .unnumbered} ======== Linear mixed-effects model {#app:lme} -------------------------- The main goal of the LME analysis we perform in this study is to determine the degree of contributions each and combinations of memory length ($\lambda$), channel location ($c$), and level of conscious arousal ($\psi$) have on statistical complexity $C_{\mu}$. LME accomplishes this by modelling statistical complexity as a general linear regression equation (Eq. ), whose response is predicted by the aforementioned parameters $\lambda$, $c$, and $\psi$. In this Appendix, we show the exact form of the linear regression equation used in this analysis, while referring to the terminology introduced in the methods (Sec. \[sec:Methods\]). We begin by restating Eq. , $\mathcal{C} = {\ensuremath{\mathbf{F}}}{\ensuremath{\mbox{\boldmath$ \beta $}}} + {\ensuremath{\mathbf{R}}}{\ensuremath{\mathbf{b}}} + \mathcal{E}$, which has the form of a general multidimensional linear equation. We will set aside the right hand side of the equality for now. On the left hand side, statistical complexity takes the form of a column vector $\mathcal{C}$. Each row corresponds to the unique response of $C_{\mu}$, at a specific selection of parameters. There is a general freedom of choice associated with the number of parameters one would like to assign to the elements $\mathcal{C}$. We index the rows with fly number $f$, channel location $c$, and the conscious arousal state $\psi$. That is, the $(i,j,k)$th element is $$\begin{gathered} [\mathcal{C}]_{(i,j,k)} = C^{(i,j,k)}_\mu.\end{gathered}$$ In other words, it is the $i$th fly’s $j$th channel in $k$th condition. Thus, $\mathcal{C}$ has length of $|f|\times|c|\times|\psi|=390$. Each $C_{\mu}$ in this vector is a function of $\lambda$. The matrix ${\ensuremath{\mathbf{F}}}$ introducing the set of fixed effects $\mathcal{F} = \{\lambda, c, \psi, \lambda c, \lambda \psi, c \psi, \lambda c \psi \}$ into the model (known in the context of general linear models as the *design matrix*) can then be represented as ${\ensuremath{\mathbf{F}}} = ({\ensuremath{\mathbf{F}}}_1,\dots, {\ensuremath{\mathbf{F}}}_{13})^T$, with each element corresponding to the design matrix of a specific fly. These individual fly response matrices can be explicitly expressed as $$\begin{gathered} {\ensuremath{\mathbf{F}}}_f = \begin{pmatrix} \vec{{\ensuremath{\mbox{\boldmath$ \lambda $}}}} & {\ensuremath{\mathbf{D}}} & \vec{{\ensuremath{\mbox{\boldmath$ \Psi $}}}}_W & \lambda{\ensuremath{\mathbf{D}}} & \lambda\vec{{\ensuremath{\mbox{\boldmath$ \Psi $}}}}_W & {\ensuremath{\mathbf{D}}}_{\Psi_W} & \lambda{\ensuremath{\mathbf{D}}}_{\Psi_W}\\ \vec{{\ensuremath{\mbox{\boldmath$ \lambda $}}}} & {\ensuremath{\mathbf{D}}} & \vec{{\ensuremath{\mbox{\boldmath$ \Psi $}}}}_A & \lambda{\ensuremath{\mathbf{D}}} & \lambda\vec{{\ensuremath{\mbox{\boldmath$ \Psi $}}}}_A & {\ensuremath{\mathbf{D}}}_{\Psi_A} & \lambda{\ensuremath{\mathbf{D}}}_{\Psi_A} \end{pmatrix},\end{gathered}$$ where $\vec{{\ensuremath{\mbox{\boldmath$ \lambda $}}}} = (\lambda,\ldots,\lambda)^{T}$ and $\vec{{\ensuremath{\mbox{\boldmath$ \Psi $}}}}_X = (\Psi_X,\ldots,\Psi_X)^{T}$ are column vectors of length $15$ containing the predictor variables of memory length and level of conscious arousal respectively, ${\ensuremath{\mathbf{D}}}$ is the $15\times 15$ identity matrix which “selects out" the channel of interest, ${\ensuremath{\mathbf{D}}}_{\Psi_X} = \text{diag}(\Psi_X,\ldots, \Psi_X)$ is the $15\times 15$ matrix which “selects out" the condition of interest correlated with the level of conscious arousal, where $$\begin{gathered} \Psi_{W (A)} = \begin{cases} 1 & \text{if $\psi=$ wakeful (anaesthetised)} \\ 0 & \text{otherwise}. \end{cases}\end{gathered}$$ In a similar fashion, the expression for the matrix containing the random effects ${\ensuremath{\mathbf{R}}}$ can be determined. For the case of our study, we only consider random effects arising due to correlations between channels within a specific fly. The result of this is an adjustment to the intercept of the linear model for each fly and channel combination. Therefore, the random effects matrix ${\ensuremath{\mathbf{R}}}$ is simply an identity matrix of dimension $390$. The accompanying elements of the random effects vector ${\ensuremath{\mathbf{b}}}$ consist of regression coefficients $b_{fc}$ describing the strength of each intercept adjustment. The execution of the LME analysis which included coefficient fitting, and log-likelihood estimations was facilitated by running `fitlme.m` in MATLAB R2108b. [^1]: The distance $\mathcal{D}_{KS} = \max | F(r_k | \mathcal{S} = S_i) - F(r_k | \unexpanded{{\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}}_{\ell})|$, where $F(r_k | \mathcal{S} = S_i)$ and $F(r_k | \unexpanded{{\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}}_{\ell})$ are cumulative distributions of $P(r_k | \mathcal{S} = S_i)$ and $P(r_k | \unexpanded{{\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{r}}}}}}}_{\ell})$ respectively. [^2]: $L(N) \sim 14$ only serves as a lower bound on $\lambda$, past which CSSR is guaranteed to return incorrect causal states for the neural data. In practice, this may occur at even lower memory lengths than this limit. We observe this effect marked by an exponential increase in the number of inferred causal states for $\lambda > 11$, and thus exclude these memory lengths from the study.

--- abstract: 'We revealed the detailed structure of a vastly extended H$\alpha$-emitting nebula (“H$\alpha$ nebula”) surrounding the starburst/merging galaxy NGC 6240 by deep narrow-band imaging observations with the Subaru Suprime-Cam. The extent of the nebula is $\sim$90 kpc in diameter and the total H$\alpha$ luminosity amounts to $L_{\rm H\alpha} \approx 1.6 \times 10^{42}$ erg s$^{-1}$. The volume filling factor and the mass of the warm ionized gas are $\sim$10$^{-4}$–10$^{-5}$ and $\sim$$5 \times 10^8$ $M_\odot$, respectively. The nebula has a complicated structure, which includes numerous filaments, loops, bubbles, and knots. We found that there is a tight spatial correlation between the H$\alpha$ nebula and the extended soft X-ray-emitting gas, both in large and small scales. The overall morphology of the nebula is dominated by filamentary structures radially extending from the center of the galaxy. A large-scale bi-polar bubble extends along the minor axis of the main stellar disk. The morphology strongly suggests that the nebula was formed by intense outflows – superwinds – driven by starbursts. We also found three bright knots embedded in a looped filament of ionized gas that show head-tail morphologies in both emission-line and continuum, suggesting close interactions between the outflows and star forming regions. Based on the morphology and surface brightness distribution of the H$\alpha$ nebula, we propose the scenario that three major episodes of starburst/superwind activities which were initiated $\sim$10$^2$ Myr ago formed the extended ionized gas nebula of NGC 6240.' author: - 'Michitoshi Yoshida, Masafumi Yagi, Youichi Ohyama, Yutaka Komiyama, Nobunari Kashikawa, Hisashi Tanaka, and Sadanori Okamura,' title: 'Giant H$\alpha$ nebula surrounding the starburst merger NGC 6240' --- Introduction ============ Galaxy mergers lead to intense star formation, i.e., starburst [@Mihos1994; @Barnes1998; @Schweizer1998; @Genzel2001; @Hopkins2008]. The gravitational perturbations and dynamical disturbances induced by the merging process remove angular momentum from the galactic interstellar medium (ISM), The ISM thus settles into the gravitational center of the system leading to starburst [@Teyssier2010; @Hopkins2013]. The collective effects of supernovae and stellar winds from massive stars within the starburst region drive large-scale outflows, or “superwinds” [e.g., @Veilleux2005]. Galaxy mergers and their successive processes, starbursts and superwinds, significantly alter the morphology, stellar population, gas content, and chemical abundance of the galaxy. Together with statistical investigations based on a large sample of mergers, detailed case studies of merging starburst galaxies provide us with important clues for understanding the above process and its impact on galaxy evolution. NGC 6240 is a well studied merger/starburst galaxy in the local universe [$z \approx 0.0245$; @Downs1993]. It has a highly disturbed morphology indicating that two spiral galaxies are in the process of colliding with each other. NGC 6240 is often considered as an ultra-luminous infrared galaxy (ULIRG) in the local universe, although its infrared luminosity [$\approx$$7\times10^{11}$ $L_{\odot}$: @Sanders1996] is slightly smaller than that of the criterion for ULIRG [$L_{\rm IR}$(8 – 1000 $\mu$m)$>10^{12}$ $L_{\odot}$; @Lonsdale2006]. This system includes double nuclei (the northern and southern nuclei; hereafter referred to as the n-nucleus and the s-nucleus) whose separation is $\sim$1 kpc. These nuclei are hard X-ray point sources suggesting that these are the colliding active galactic nuclei (AGNs) of the progenitor galaxies [@Komossa2003]. The s-nucleus is $\sim$3 times brighter than the n-nucleus in the 0.2–10 keV band [@Komossa2003]. It is well known that a bright optical emission line nebula is associated with NGC 6240. Early studies by various authors [e.g. @Heckman1987; @Armus1990; @Keel1990] suggested that this optical nebula is primarily excited by shock-heating induced by a starburst superwind. High spatial resolution [*Chandra*]{} observations have revealed that the central H$\alpha$ nebula (“butterfly nebula”) is spatially coincident with the soft X-ray-emitting hot gas surrounding the double nuclei [@Komossa2003; @Gerssen2004]. In addition, there is a highly extended ionized gas region surrounding NGC 6240. @Heckman1987 [@Heckman1990] detected faint H$\alpha$+\[\] emission filaments extending $\gtrsim$30 kpc from the center. @Veilleux2003 revealed a more extended optical emission line nebula approximately 80 kpc in diameter. An extended hot gas nebula of NGC 6240 has also been found in soft X-rays [@Komossa2003; @Huo2004]. A spatial correlation between the optical emission line nebula and the extended soft X-ray gas was identified by @Veilleux2003. Recently, @Nardini2013 showed the extent and detailed structure of the very extended (its total size reaches $\sim$100 kpc) soft X-ray-emitting gas. Although the central region of NGC 6240 has been studied many times, the detailed structure of the extended optical emission line gas is not yet understood. Here we present the results of deep optical narrow-band imaging of NGC 6240 using the Subaru Suprime-Cam. Our deep observations are the first to reveal the detailed structure of the extended H$\alpha$ emitting warm ionized gas surrounding this galaxy. We assumed cosmological parameters of ($h_0$, $\Omega_m$, $\Omega_\lambda$) = (0.705, 0.27, 0.73) [@Komatsu2011] and that the luminosity distance of NGC 6240 is 107 Mpc. The linear scale at the galaxy is 492 pc arcsec$^{-1}$ under this assumption. Observations ============ [lccc]{} N-A-L671 & 0.88 & 27.5 & 10 $\times$ 600 s\ $B$ & 0.91 & 28.5 & 5 $\times$ 300 s\ $R_{\rm C}$ & 0.76 & 27.7 & 5 $\times$ 180 s\ $i^\prime$ & 0.87 & 27.3 & 5 $\times$ 180 s\ \[tab:obs\] Narrow-band H$\alpha$+\[\] and broad-band $B$, $R_{\rm C}$, and $i^\prime$ imaging observations of NGC 6240 were made using the Suprime-Cam [@Miyazaki2002] attached to the Subaru Telescope on April 30 (UT) 2014. A journal of the observations is shown in Table \[tab:obs\]. We used the N-A-L671 narrow-band filter [@Yagi2007; @Yoshida2008] for H$\alpha$+\[\] imaging of the galaxy ($z \approx 0.0245$). The central wavelength and full-width at half maximum (FWHM) of the transmission curve of the filter were 6712 Å and 120 Å respectively. The total exposure time was 6,000 sec for N-A-L671. The sky was clear and photometric. Seeing was 07–09 on the observing night (Table \[tab:obs\]). Data Reduction ============== Data reduction was carried out in a standard manner. Overscan subtraction, flat-fielding, distortion correction, sky subtraction, and mosaicking were performed with custom-made Suprime-Cam data-reduction software, nekosoft [@Yagi2002]. During the overscan subtraction procedure, we also conducted cross-talk correction and gain correction between the CCDs [see @Yagi2012]. We used sky flats, which had been created from a number of object frames taken during the same observing run, for the flat-fielding correction of the $B$ and $R_{\rm C}$ band frames. For the N-A-L671 and $i^\prime$ frames, dome flat data were used for flat-fielding. We then extracted an area of 400 $\times$ 400 (197 kpc $\times$ 197 kpc) centered on the s-nucleus of NGC 6240 from each flat-fielded frame. After adjusting the positions and scales, the frames for each band were combined using a 3 sigma clipping method. The flux calibration was performed using the 9th Data Release of the Sloan Digital Sky Survey (SDSS) photometric catalog [@Ahn2012]. The details of the flux calibration are described in Appendix A and also in @Yagi2013. The FWHM of the point-spread function (PSF) was measured by fitting a two-dimensional (2D) Gaussian function to the brightness profiles of non-saturated star images for each band. The limiting surface brightness was estimated by photometry using randomly-sampled 2 arcsec apertures on the blank regions. The PSF FWHMs and limiting magnitudes of the final $B$, $R_{\rm C}$, $i^\prime$ and N-A-L671 images are listed in Table \[tab:obs\]. The reduced N-A-L671 image is shown in the right panel of Figure \[fig:R-H-alpha\]. To obtain a net (pure) H$\alpha$+\[\] image of NGC 6240, we subtracted the scaled $R_{\rm C}$ frame from the N-A-L671 frame. Scaling was performed using several unsaturated field stars common to both frames. Before subtraction, we convolved 2D double Gaussian functions to the $R_{\rm C}$ frame to adjust the PSFs of both of the frames. After subtraction, we multiplied the net H$\alpha$+\[\] image by a factor of 1.23 to correct the over-subtraction of the H$\alpha$+\[\] flux due to the contamination of the emission lines in the $R_{\rm C}$ frame (see Appendix B). In addition, we subtracted the net H$\alpha$+\[\] that was multiplied by a factor of 0.23 from the scaled $R_{\rm C}$ frame to create the emission-line-free $R_{\rm C}$ frame. The emission-line-free $R_{\rm C}$ image is shown in the left panel of Figure \[fig:R-H-alpha\]. The net H$\alpha$+\[\] image is shown in Figure \[fig:H-alpha\] using three different contrasts. The H$\alpha$+\[\] image thus created is affected by the positional variation of the emission line ratios in the nebula. We assumed constant emission line ratios over the entire region of the H$\alpha$ nebula in the above procedure. However, it is well known that there is a tendency for the flux ratios of the forbidden lines to the H$\alpha$ emission line to be large at larger distances from the center of NGC 6240 [@Heckman1990; @Keel1990; @Veilleux2003]. This causes the systematic underestimation of the net H$\alpha$+\[\] flux at the outer region of the nebula. In addition, the combination of the recession velocity variation of the nebula and the transmission curve of the N-A-L671 filter affects the detected H$\alpha$+\[\] flux. The error of the H$\alpha$+\[\] flux due to this effect was estimated to be $\approx$$\pm$5%; the variation of the transmission of the filter is $\approx$5% in the recession velocity range of the ionized gas nebula of NGC 6240 [@Heckman1990; @Keel1990; @Gerssen2004]. Results ======= Main stellar disk ----------------- The morphology of NGC 6240 in optical broad bands (e.g. $R_{\rm C}$ band image: Left panel of Figure \[fig:R-H-alpha\]) is complicated, but there are some characteristic features in it. The bright part of the system is elongated as long as $\sim$20 kpc with a position angle (PA) of $\sim$30$^{\circ}$ and the clear dust lane runs along this structure. In the southeastern side of this bright feature, a “banana”-shaped tail whose PA is $\sim$$-20^{\circ}$ is connected to the center. A “S”-shaped extended faint tidal feature surrounds these bright parts of NGC 6240 (Figure \[fig:R-H-alpha\]). From a morphological point of view, we suspect that the elongated bright part would be the remnant of the disk of the primary galaxy of this colliding system. We tried to extract the disk component from the central bright part by fitting an exponential disk to the $i^\prime$ band image. We used GALFIT [@Peng2002; @Peng2010] for profile fitting. We added four curved S$\acute{\rm e}$rsic components to fit residuals of the exponential disk fitting. In the fitting procedure, we excluded the circular region whose radius is 3at the center to avoid the influence of saturation and AGN components. In addition, we also excluded bright stars and blooming patterns around the galaxy. [cccc]{} 12.67 & 4.7 kpc & 28$^\circ$ & 62$^\circ$\ \[tab:expdisk\] The best fit parameters of the exponential disk are listed in Table \[tab:expdisk\]. This exponential disk component dominates in the $i^\prime$ band brightness; its brightness is more than half of the SDSS $i$ band magnitude, 11.99$\pm$0.05 [@Brown2014]. The other S$\acute{\rm e}$rsic components are 2–3 magnitude fainter than the disk, thus do not contribute to the continuum light significantly. @Bland1991 performed a Fabry-Perot spectroscopy of the central region of NGC 6240 and found a rotating disk component whose PA and inclination are $\sim$45$^\circ$ and $\sim$70$^\circ$, respectively. These values are roughly consistent with the parameters of the exponential disk we fitted. Thus we conclude that there still is a large stellar disk, which is probably the remnant of the primary galaxy of the merger, in the center of the system. We refer to this disk as the “main stellar disk” of NGC 6240. Morphology of H$\alpha$ nebula ------------------------------ The net H$\alpha$+\[\] image obtained in this study revealed the rich and complex structure of the optical emission line nebula of NGC 6240 (Figures \[fig:H-alpha\], \[fig:H-alpha-color\], and \[fig:colorimg\]). We refer to this nebula as the “H$\alpha$ nebula”. The H$\alpha$ nebula has a diameter of $\sim$90 kpc. The central “butterfly” nebula [e.g. @Gerssen2004] is surrounded by a relatively bright hourglass shaped region (“hourglass region”; light blue color in Figure \[fig:H-alpha-color\]; @Heckman1987). In addition to these well-known features, @Veilleux2003 also detected a faint optical emission line region extending out to over 70 kpc $\times$ 80 kpc around NGC 6240. Our deep Suprime-Cam observations are the first to reveal the detailed and complex structure of this extended component of the H$\alpha$ nebula. We applied an unsharp masking technique to the net H$\alpha$+\[\] image to enhance the filamentary structure of the H$\alpha$ nebula. We first masked bright stars and blooming patterns in the H$\alpha$+\[\] image. Second, we smoothed the masked image with a 35 $\times$ 35 pixels ($\approx$3.4 kpc $\times$ 3.4 kpc) median filter, then subtracted the smoothed image from the masked image. The unsharp masked image is shown in Figure \[fig:unsharp\]. It is clear that a number of filaments are running radially from the galactic center. A sketch of the structure of the H$\alpha$ nebula is shown in Figure \[fig:sketch\]. We identified several characteristic features in this nebula. In the following subsections, we describe these features in detail. ### West filament The most remarkable feature of the extended H$\alpha$ nebula is a bright, wiggled filament (“W-filament”) in the western region. It is prominent in the middle panel of Figure \[fig:H-alpha\] (see panel (c) of Figure \[fig:detail\] for the detailed structure). Although the W-filament was previously visible in the narrow-band H$\alpha$ images taken by @Heckman1987 and @Veilleux2003, its detailed structure and surrounding complex features were only revealed in our H$\alpha$+\[\] image. The bright part of the W-filament begins at 18 kpc west from the s-nucleus and extends to the west-southwest. The filament turns slightly west-northwest at $\sim$25 kpc and reaches $\sim$36 kpc from the s-nucleus. There is a bright cloud to the east of the bright region of the W-filament ($\sim$12 kpc west from the s-nucleus). We assume, from a morphological point of view, this could be a part of the W-filament (Figure \[fig:sketch\]). ### Northern complex of clouds and filaments On the northern side of the galaxy, there exists a complex of clouds and filaments (“N-complex”: Figure \[fig:sketch\] and panel (a) of Figure \[fig:detail\]), which extends along the PA $\sim$310$^\circ$ from $\sim$25 kpc north to $\sim$33 kpc north-northwest of the galaxy. The unsharp masked image indicates that the N-complex is the tip of a large curved structure connected to the hourglass region (Figure \[fig:unsharp\]). ### Southeastern and northwestern bubbles {#sec:bubble} A clear bubble-like structure is visible in the southeastern side of the galaxy (“SE-bubble”: Figure \[fig:sketch\] and panel (b) of Figure \[fig:detail\]). This faint bubble was unknown before this study. The size and PA of the SE-bubble are $\sim$28 kpc $\times$ 38 kpc and $\sim$135$^\circ$, respectively. On the other side of the galaxy, the N-complex and W-filament appear to form a large broken bubble whose size is comparable to the SE-bubble (“NW-bubble”). The PA of the NW-bubble axis is approximately $-$40$^\circ$. The axes of these two bubbles are almost perpendicular to the major axis of the main body of NGC 6240. The opening angle of the SE-bubble and the NW-bubble is $\sim$70$^\circ$–90$^\circ$ (Figures \[fig:unsharp\] and \[fig:sketch\]). ### Bright knots and western loops The western side of the H$\alpha$ nebula is much brighter and has a more complex structure than the eastern side. In particular, the W-filament and its surroundings have a rich structure consisting of many filaments and blobs. There are several bright knots (“BK1”–“BK3”: Figure \[fig:sketch\]) located in a loop-like filament extending from the central nebula towards the southwest (“W-loop1”: Figure \[fig:sketch\]). These knots are marginally resolved in our H$\alpha$+\[\] image; the FWHMs of the knots are $\approx$10–11. W-loop1 extends almost straight towards the southwest from the center, then turns to the north at $\sim$23 kpc drawing a loop-like trajectory (panel (d) of Figure \[fig:detail\]). One of the bright knots (BK3) is located at the tip of this loop. Another loop (“W-loop2”) is located at $\sim$35 kpc from the center (panel (e) of Figure \[fig:detail\]). The size and shape of W-loop2 are similar to those of W-loop1, but its surface brightness is much lower than that of W-loop1. ### Southern and southwestern filaments There are two streams extending along the north-south direction on the south side of the galaxy (“S-filaments” and “SW-filaments”: Figure \[fig:sketch\]; see also Figure \[fig:unsharp\]). The S-filaments consist of several filaments extending from the center towards the south. The SW-filaments consist of a number of filaments distributed from the region close to W-loop2 to $\sim$59 kpc southwest from the s-nucleus. The extension of the S-filaments is similar to but slightly offset from that of the southern tidal feature which is the southern end of the S-shaped morphology of the optical continuum emission of NGC 6240. There is no clear optical continuum counterpart to the SW-filaments. Physical parameters of H$\alpha$ nebula --------------------------------------- ### H$\alpha$ surface brightness profile and luminosity {#sec:lum} We performed multiple aperture photometry for the H$\alpha$ emission of NGC 6240. We selected the region further out than 140 ($\sim$70 kpc) from the s-nucleus for sky background estimation. Before performing photometry, bright stars and artifacts were masked and interpolated by linear functions around the patterns. We adopted circular apertures whose center was the brightest point of the H$\alpha$+\[\] image corresponding to the s-nucleus. We excluded the very center ($r <1$) region in this aperture photometry procedure, because the $R_{\rm C}$ band frame that was subtracted from the N-A-L671 frame to create the net H$\alpha$+\[\] image was saturated at the central 1 region of the galaxy. The contamination of the \[\]$\lambda\lambda$6548,6583 emission was corrected assuming $f_{\rm[N II]6583} / f_{\rm H\alpha} = 1$ [@Heckman1990; @Keel1990; @Veilleux2003] for the entire region of the nebula. In addition, we assumed that the foreground extinction is the same as the $R$ band Galactic extinction toward NGC 6240; $A_{\rm R}$ = 0.165 magnitude [@Schlafly2011]. The surface brightness profile of the H$\alpha$ emission is given in Figure \[fig:sb\]. It is clear that the H$\alpha$ surface brightness distribution indicates the core-halo structure of the nebula, which is similar to the case of the soft X-ray nebula [@Huo2004; @Nardini2013], whereas the core is more compact in H$\alpha$. The total H$\alpha$ flux of NGC 6240 was derived by integrating the fluxes in the regions at which the H$\alpha$ surface brightness is greater than 2$\sigma$ of the sky background. We assumed that the H$\alpha$ surface brightness within the central 1 region which we excluded in the aperture photometry is the same as that at 1 away from the s-nucleus. As mentioned above, we assumed that $f_{\rm[N II]6583} / f_{\rm H\alpha}$ $= 1$ for the whole region of the nebula. This ratio tends to increase toward the outer part of the H$\alpha$ nebula, reaching $\gtrsim$1.4 around the W-filament [@Veilleux2003]. Thus our method may underestimate the contribution of the \[\] emission in some parts of the nebula. However, since most of the bright regions of the nebula have almost constant $f_{\rm[N II]6583} / f_{\rm H\alpha}$ $\sim 1$ [@Keel1990; @Veilleux2003], the variation of this ratio does not significantly affect the estimation of the total H$\alpha$ flux. The resultant total H$\alpha$ flux $f_{\rm H\alpha}$ is $\approx$1.3 $\times$ 10$^{-12}$ erg s$^{-1}$ cm$^{-2}$. The H$\alpha$ luminosity $L_{\rm H\alpha}$ is $\approx$1.6 $\times$ 10$^{42}$ erg s$^{-1}$. ### Electron density, filling factor and mass {#sec:ed} [cccccc]{} 0 $-$ 1.7 & $4.5\times10^{-2}$ & $1.3\times10^7$ & $2.4\times10^{-1}$ & $1.6\times10^{-5}$ & $2.9\times10^{-5}$\ 1.7 $-$ 3.4 & $2.3\times10^{-2}$ & $1.1\times10^7$ & $1.2\times10^{-1}$ & $2.2\times10^{-5}$ & $2.7\times10^{-5}$\ 3.4 $-$ 7.4 & $6.7\times10^{-3}$ & $8.7\times10^6$ & $5.2\times10^{-2}$ & $7.9\times10^{-5}$ & $5.8\times10^{-5}$\ 7.4 $-$ 11 & $3.5\times10^{-3}$ & $8.4\times10^6$ & $2.8\times10^{-2}$ & $8.9\times10^{-5}$ & $6.5\times10^{-5}$\ 11 $-$ 15 & $2.4\times10^{-3}$ & $7.8\times10^6$ & $2.0\times10^{-2}$ & $1.1\times10^{-5}$ & $7.1\times10^{-5}$\ 15 $-$ 20 & $2.0\times10^{-3}$ & $7.7\times10^6$ & $1.4\times10^{-2}$ & $8.5\times10^{-5}$ & $4.9\times10^{-5}$\ 20 $-$ 27 & $1.6\times10^{-3}$ & $8.1\times10^6$ & $1.1\times10^{-2}$ & $7.5\times10^{-5}$ & $4.7\times10^{-5}$\ 27 $-$ 39 & $1.4\times10^{-3}$ & $7.2\times10^6$ & $0.7\times10^{-2}$ & $5.2\times10^{-5}$ & $2.5\times10^{-5}$\ \[tab:eden\] We derived the root mean square (rms) electron density $n_{\rm e, H\alpha}$(rms) of the H$\alpha$ nebula assuming Case B recombination. We did not, however, apply an extinction correction to this calculation. The surface brightness of H$\alpha$ $SB_{\rm H\alpha}$ is defined as $$SB_{\rm H\alpha} = \alpha_{\rm B, H\alpha} \cdot h \nu_{\rm H\alpha} \cdot {\rm EM} ,$$ where $\alpha_{\rm B, H\alpha}$ and EM are the photon production coefficient of H$\alpha$ in Case B recombination [@Osterbrock2006] and the emission measure, respectively. Assuming a fully-ionized pure hydrogen gas ($n_{\rm e} \approx n_{\rm p}$; $n_{\rm p}$ is proton density) and a constant density distribution along the line of sight, the EM can be written as ${\rm EM} = n_{\rm e, H\alpha}^2 l$, where $l$ is the line of sight length of the nebula. In the case of spherically symmetric nebula, $l = 2 \sqrt{R^2 - r^2}$, where $R$ and $r$ are the radius of the nebula and the projected distance from the nebula center, respectively. Thus, the $n_{\rm e, H\alpha}$(rms) is given by $$n_{\rm e, H\alpha} ({\rm rms}) = \sqrt{\frac{SB_{\rm H\alpha}}{\alpha_{\rm B, H\alpha} \cdot h \nu_{\rm H\alpha} \cdot l }} .$$ The radial profile of the $n_{\rm e, H\alpha}$(rms) distribution is shown in Figure \[fig:ne\]. The mass of the H$\alpha$ emitting gas $M_{\rm H\alpha}$ is given by $$M_{\rm H\alpha} = \int f_{\rm v, H\alpha}^{1/2} \cdot n_{\rm e, H\alpha} ({\rm rms})\; m_{\rm H} \; dV ,$$ where $f_{\rm v, H\alpha}$ and $m_{\rm H}$ are the volume filling factor of the H$\alpha$ emitting gas and the mass of a hydrogen atom, respectively. We estimated the volume filling factor $f_{\rm v, H\alpha}$ by the following procedure. First, considering the close spatial correlation between the H$\alpha$ nebula and the soft X-ray-emitting hot gas (see section \[compX\] below), we assumed that the H$\alpha$ gas and X-ray gas are locally coexisting, Then we considered the following two cases for the coexistent state; 1) a simple thermal pressure equilibrium case and 2) a ram pressure balance case. In the case 1), we calculated the local electron densities $n_{\rm e, H\alpha}({\rm local})$ of the H$\alpha$ emitting cloud assuming that the equation $n_{\rm e, X}({\rm rms}) \cdot f_{\rm v, X}^{1/2} \cdot T_{\rm X} = n_{\rm e, H\alpha}({\rm local}) \cdot T_{\rm H\alpha}$ holds over each radial zone in which @Nardini2013 provided the averaged electron densities $n_{\rm e, X}$ and temperatures $T_{\rm X}$ of the hot gas individually. Here, $f_{\rm v, X}$ is the volume filling factor of the hot gas. Because the optical emission line filaments of NGC 6240 show clear shock excitation nature in its emission line ratios [@Heckman1990; @Keel1990], we assumed that $T_{\rm H\alpha} = 10^5$ K (typical temperature of optical shock excitation gas [@Shull1979]) for all of the zones in this case. The case 2) is the one that shock excitation is responsible for the H$\alpha$ emission gas and X-ray emitting hot gas. We calculated the $n_{\rm e, H\alpha}({\rm local})$ assuming the ram pressure balance equation [@Spitzer1978], $n_{\rm e, X}({\rm rms}) \cdot f_{\rm v, X}^{1/2} \cdot V_{\rm s, X}^2 = n_{\rm e, H\alpha}({\rm local}) \cdot V_{\rm s, H\alpha}^2 $ , where $V_{\rm s, X}$ and $V_{\rm s, H\alpha}$ are the shock velocities at the shock front between the hot gas and warm gas, respectively. The shock velocity required to produce X-ray emitting gas with a temperature of $\sim$10$^7$ K is $\sim$1000 km s$^{-1}$ for an adiabatic shock [@Hollenbach1979]. @Heckman1990 found that the averaged \[\]$\lambda 6300$/H$\alpha$ and \[\]$\lambda\lambda$6717+6731/H$\alpha$ emission line intensity ratios of the optical filaments of NGC 6240 are $\sim$0.3 and $\sim$0.8, respectively. These values are consistent with shock models with the shock velocity $\sim$100 km s$^{-1}$ [e.g., @Shull1979]. Hence, we assume that $V_{\rm s, X}$ and $V_{\rm s, H\alpha}$ are $\sim$1000 km s$^{-1}$ and $\sim$100 km s$^{-1}$, respectively. We then derived $f_{\rm v, H\alpha}$ by $f_{\rm v, H\alpha} = ( n_{\rm e, H\alpha} ({\rm rms}) / n_{\rm e, H\alpha}({\rm local}) )^2$ for the both cases. In the case 1), $f_{\rm v, H\alpha}$ is given by $$\begin{gathered} f_{\rm v, H\alpha} \approx 10^{-4} \cdot \biggl( \frac{f_{\rm v, X}}{10^{-2}} \biggr) \biggl( \frac{T_{\rm H\alpha}}{10^5\; {\rm K}} \biggr)^2 \biggl( \frac{T_{\rm X}}{10^7\; {\rm K}} \biggr)^{-2}\\ \biggl( \frac{n_{\rm e, H\alpha}({\rm rms})}{10^{-1}\; {\rm cm^{-3}}} \biggr)^2 \biggl( \frac{n_{\rm e, X}({\rm rms})}{10^{-2}\; {\rm cm^{-3}}} \biggr)^{-2}. \label{eq:fv1}\end{gathered}$$ In the case 2), we obtain $$\begin{gathered} f_{\rm v, H\alpha} \approx 10^{-4} \cdot \biggl( \frac{f_{\rm v, X}}{10^{-2}} \biggr) \biggl( \frac{\eta}{10} \biggr)^{-4} \biggl( \frac{n_{\rm e, H\alpha}({\rm rms})}{10^{-1}\; {\rm cm^{-3}}} \biggr)^2\\ \biggl( \frac{n_{\rm e, X}({\rm rms})}{10^{-2}\; {\rm cm^{-3}}} \biggr)^{-2}, \label{eq:fv2}\end{gathered}$$ where $\eta = V_{\rm s, X} / V_{\rm s, H\alpha}$. The volume filing factor $f_{\rm v, X}$ is not well known, but the soft X-ray would be strongly enhanced at the upstream of the shocks in the wind [@Strickland2000; @Cooper2008; @Cooper2009] and the $f_{\rm v, X}$ of such shocked gas clouds is considerably smaller than 1. @Strickland2000 found that the typical $f_{\rm v, X}$ of the X-ray gas in a superwind is of order 10$^{-2}$. Assuming that $f_{\rm v, X} = 10^{-2}$, $T_{\rm H\alpha} = 10^5$ K, and $\eta = 10$, we found that the resultant volume filling factors are orders of 10$^{-4}$–10$^{-5}$ for the both cases (Table \[tab:eden\]). Although these values are much smaller than the typical $f_{\rm v}$ of regions or planetary nebulae in our Galaxy [$\sim$10$^{-2}$–10$^{-3}$; e.g. @Osterbrock2006], they are comparable to the $f_{\rm v}$ derived for outflow/wind gas by the observations of nearby starburst galaxies and AGNs [@Yoshida1993; @Robinson1994; @Cecil2001; @Sharp2010]. Under the assumption of spherically symmetric nebula, we obtained $M_{\rm H\alpha} \sim 5 \times 10^8$ $M_\odot$ using the $f_{\rm v, H\alpha}$ derived above. The spherical symmetry is, however, too simple assumption for the geometry of the emission line nebula. Hence, we next considered a more realistic geometry for the H$\alpha$ nebula; it has a filamentary structure and the emission line filaments occupy a fraction $\zeta$ ($\zeta < 1$) along the line of sight of the spherical nebula. In this case, the rms electron densities are increased by a factor of $\zeta^{-1/2}$. Since we think that the X-ray gas and H$\alpha$ gas spatially coexist, the line-of-sight occupation factor $\zeta$ is the same for the both gas. Thus the factor $\zeta$ vanishes in the calculation of the volume filling factor (see equations (\[eq:fv1\]) and (\[eq:fv2\])). On the other hand, the total volume of the nebula is decreased by a factor of $\zeta$. As a result, the $M_{\rm H\alpha}$ alters by a factor of $\zeta^{1/2}$. Then we obtained $M_{\rm H\alpha} \sim 5 \times 10^8 \cdot \zeta^{1/2}$ $M_\odot$ for this partly occupied filamentary nebula case. Comparison with X-ray and UV emission {#compX} ------------------------------------- We found that the H$\alpha$ nebula shows a very good spatial correlation with the highly extended soft X-ray-emitting hot gas nebula [@Huo2004; @Nardini2013], both in large and small scales. A tight spatial correlation between the H$\alpha$ and soft X-ray is well known for the butterfly nebula of NGC 6240 [@Komossa2003; @Gerssen2004]. The overall coincidence between the optical emission line gas and the extended soft X-ray gas of NGC 6240 has been previously pointed out by some authors [@Veilleux2003; @Huo2004]. Our deep observations, however, allow us to make a more detailed comparison between the hot and warm phase gases. We overlaid the contours of the H$\alpha$+\[\] image on the $\sim$150 ks deep soft X-ray image[^1] taken with [*Chandra*]{} [@Nardini2013] in Figure \[fig:X-alpha\]. The overall shape (a “diamond shape”) of the faint emission level of the soft X-ray image is well traced by the H$\alpha$+\[\] emission. On smaller scales, there are a number of common features between the X-ray and H$\alpha$+\[\], including the central butterfly nebula, the hourglass region, the N-complex, the SE-bubble, the W-filament, and W-loop2. There is also an indication of the counterpart of S-filaments in the X-ray image, but the southern tips of the filaments are located outside the X-ray-emitting gas. There seems to be no clear counterpart to the SW-filaments in the X-ray. A very tight spatial correlation between the warm ionized gas and the hot gas is maintained from the central region to the edge of the nebula of NGC 6240. We additionally compared the H$\alpha$ nebula image with ultraviolet (NUV and FUV) images of NGC 6240 taken with [*GALEX*]{}[^2] (Figure \[fig:uv-H-alpha\]). There also is a good spatial correlation between the H$\alpha$ and the UV images, particularly in the western side of the galaxy. The western UV emission of NGC 6240 is mostly overlapped with the western complex of filaments of the H$\alpha$ nebula. The W-filament and N-complex can be traced in the UV images (Figure \[fig:uv-H-alpha\]). The bright knots (BK1 – BK3) are also visible in the UV images, and the S-filaments are well traced by the UV emission. There also are faint UV counterparts for the SW-filaments. Discussion ========== Nature of the extended H$\alpha$ nebula --------------------------------------- ### Starburst-driven superwind We found that the overall morphology of the H$\alpha$ nebula is dominated by the radially-extended filamentary structure (Figure \[fig:unsharp\]). We also identified a large-scale bi-polar bubble (SE- and NW-bubble) along with prominent loop structures (W-loop1 and 2). These morphological characteristics indicate that most of the H$\alpha$ nebula consists of radially-outflowing gas – the starburst superwind. The close spatial coincidence between the H$\alpha$ nebula and the extended soft X-ray gas supports this idea. Many common features, including the SE-bubble, the N-complex, the W-filament, and W-loop2, are observed in the extended ionized gas of NGC 6240 in both H$\alpha$ and X-ray. A similar morphological correlation between the optical emission line region and the soft X-ray gas has been widely observed in starburst galaxies [@Strickland2000b; @Strickland2004; @Smith2001; @Cecil2002; @Veilleux2003; @Tullmann2006]. Such good spatial coincidence has been interpreted as an indication that the hot and warm phase gases are heated and excited via the same mechanism – probably from shocks in the outflowing gas – over the entire region of the nebula. Numerical simulations of superwinds have supported this scenario [@Strickland2000; @Cooper2008; @Cooper2009]. ### Star-forming bright knots @Heckman1990 obtained the optical spectra of two of the bright knots (BK1 and BK2), and found that they are active star-forming regions. Although there are no spectral data for BK3, we suggest that it is also a star-forming region because the continuum counterparts of the bright knots that show a blue color ($B - i^\prime$ (AB) $\sim -0.3$) are clearly seen in our broad-band images (Figure \[fig:knots\])[^3]. It appears that the bright knots are interacting with the outflowing filament, and the gas in the filament is intercepted by these knots, creating W-loop1. Tail-like structures are also observed in the broad-band images, in particular, the tails extending to the southwest are clearly visible in the $B$ band image (Figure \[fig:knots\]). Indications of the tails are also marginally seen in the $i^\prime$ band image. This feature is recognizable in a Hubble Space Telescope (HST) public image of NGC 6240[^4]. The head-tail morphology of the knots suggests that the outflow gas and condensed gas clouds in the main stellar disk interact with each other and that its ram pressure and/or shock may induce active star formation. The bright knots are also visible in the UV images supporting the view that these knots are the sites of active star formation. In all probability, the young stellar population observed in the tails of the bright knots was also formed by this process. The S-filaments are well traced by the UV emission. There also are faint UV counterparts for the SW-filaments. Because these filaments have either a weak or no spatial correlation with the X-ray emission, we consider that the S- and SW-filaments are not closely related to the starburst winds. These features are more likely to be star-forming tidal tails. ### Extended UV emission The UV emission associated with the optical filaments may partly be the precursor emission of a shock within the nebula. The extended UV emission associated with a superwind has been observed in nearby starburst galaxies including M82 and NGC 253 [e.g. @Hoopes2005]. In these galaxies, most of the UV emission is thought to be from the stellar continuum scattered by dust entrained in the wind [@Hoopes2005; @Hutton2014], because it is too bright to be provided by shock-heated or photoionized gas in the wind. It is well-known that NGC 6240 is a dusty system, thus it is possible that the extended UV emission of NGC 6240 is mostly derived from light scattered by dust. To estimate the contribution of dust scattering, we compared the UV intensity and H$\alpha$ intensity at the W-filament. We measured the UV and H$\alpha$ intensities in a 195 $\times$ 105 rectangular region at the W-filament (the center of the photometry region is located 54 west and 5 north from the s-nucleus), and determined that $I_{\rm NUV} =2.0 \times 10^{-14}$ erg s$^{-1}$ cm$^{-2}$, $I_{\rm FUV} =9.3 \times 10^{-15}$ erg s$^{-1}$ cm$^{-2}$, and $I_{\rm H\alpha} = 2.7 \times 10^{-15}$ erg s$^{-1}$ cm$^{-2}$ (assuming that $f_{\rm [NII]6583}/f_{\rm H\alpha} = 1.4$, @Veilleux2003). The average H$\alpha$ surface brightness in this region was $1.7 \times 10^{-17}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$. Both the $I_{\rm NUV}/I_{\rm H\alpha}$ and $I_{\rm FUV}/I_{\rm H\alpha}$ ratios – 7.4 and 3.4, respectively, – are slightly smaller than, but almost consistent with, values in the outflow filaments in M82 and NGC 253 at the same H$\alpha$ surface brightness level [@Hoopes2005]. The ratios are, however, still larger than the values predicted by the shock model. This indicates that the UV emission of NGC 6240 is too bright to be attributed to shock-heated or photoionized gas emission in the wind. Thus, some part of the UV emission is possibly light scattering by dust. Alternatively, this emission may reflect extended star-forming activity within the filaments. In this case, gas compression by the interaction of the outflowing gas and the interstellar medium in the main stellar disk may have induced star formation. @Nardini2013 found that the X-ray hot gas of NGC 6240 has a spatially-uniform sub-solar metal abundance whose abundance pattern is consistent with Type-II supernova enrichment. They discussed the age of the nebula using the thermal velocity dispersion estimated from the temperature of the hot gas, and concluded that the whole NGC 6240 nebula could not be formed by the recent activity of the nuclear starburst or AGNs. They suggested that extended continuous star formation from the early phase of the merging event would be responsible for the formation of the outer nebula. The extended UV emission and its spatial coincidence with the H$\alpha$ emission might support this scenario. Most of the H$\alpha$ emission of the H$\alpha$ nebula, however, probably originates from shocks in the outflow, because the extended ionized gas shows a large \[\]/H$\alpha$ intensity ratio (\[\]/H$\alpha$ $\gtrsim$1; @Keel1990 [@Veilleux2003]). Detailed polarimetry and spectroscopy of the extended nebula are necessary to qualitatively understand the contribution of dust-scattering and local star formation to the H$\alpha$ nebula. Ages and durations of superwinds {#sec:age} -------------------------------- [lccccccc]{} Butterfly & 4.5 & $1.4\times10^{8}\;\zeta^{1/2} $ & 700 & 25 & $8.4$ & $1.2\;\xi^{-1} \zeta^{1/2}$\ Hourglass & 13 & $5.0\times10^{8}\;\zeta^{1/2} $ & 700 & 25 & $24$ & $4.2\;\xi^{-1} \zeta^{1/2}$\ Outer & 45 & $1.0\times10^{9}\;\zeta^{1/2} $ & 700 & $\sim 10$ & $84$ & $\sim 20\;\xi^{-1} \zeta^{1/2}$\ \[tab:windparam\] We discuss the ages and durations of the starburst superwinds of NGC 6240. From a morphological point of view, we divided the H$\alpha$ nebula into three regions; the central butterfly nebula [@Heckman1987; @Keel1990; @Lira2002; @Gerssen2004], the hourglass region [@Heckman1987], and the outer filaments. The characteristic radii of the three regions are $\sim$4.5 kpc, $\sim$13 kpc, and $\sim$45 kpc, respectively. We assume that these three regions were formed in the past by different starburst/superwind activities, and estimate the ages of the regions and the durations of these activities. First, we estimate the outflow velocity of the winds and derive the dynamical ages of the regions. @Heckman1990 studied the kinematics of the optical ionized gas of the central region of NGC 6240 and found that the FWHM of the H$\alpha$ line exceeds 1000 km s$^{-1}$ within 4–5 kpc from the center [see also @Keel1990; @Gerssen2004]. They also found that some parts of the extended region have radial velocities of $\sim$500 km s$^{-1}$. We thus adopted a value of 500 km s$^{-1}$ as the line-of-sight component of the outflow velocity of the gas. Assuming that the typical inclination of the outflow is 45$^\circ$, we estimate the outflow velocity of the H$\alpha$ nebula to be $v_{\rm flow} \sim 700$ km s$^{-1}$. We also assume that the outflow velocity is constant over the whole nebula. The ages of the three regions are thus $\sim$8 Myr, $\sim$24 Myr, and $\sim$84 Myr, for the butterfly nebula, the hourglass region, and the outer filaments, respectively. Next, we estimate the durations of the superwind activities that formed the three regions based on the standard superwind model. The terminal velocity of a mass loading galactic wind is given by $$V_{\inf} \approx 3000 \times (\xi / \Lambda)^{1/2} \;\; {\rm km\; s^{-1}}, \label{eq:vinf}$$ where $\xi$ and $\Lambda$ are the thermalization efficiency and mass loading factor of the wind, respectively [@Veilleux2005]. Substituting $V_{\inf} = v_{\rm flow} = 700$ km s$^{-1}$ into the equation (\[eq:vinf\]), we find $\Lambda \approx 18 \xi$. Although the exact value of $\xi$ is unknown and is difficult to estimate, it would be expected to lie between 0.1 and 1 [@Strickland2000]. On the other hand, the mass-loss rate from the starburst region is scaled by the star formation rate (SFR) and given by $$\dot{M_\ast} = 0.26\; (SFR / {\rm M_\odot yr^{-1}}) \; \; {\rm M_\odot\; yr^{-1}}$$ [@Veilleux2005]. Then the duration of the superwind $t_{\rm wind}$ is calculated by $t_{\rm wind} \approx M_{\rm wind} / (\Lambda \times \dot{M_\ast})$, leading to $$t_{\rm wind} \approx 2.1 \times \biggl(\frac{M_{\rm wind}}{\rm 10^8\; M_\odot}\biggr) \biggl(\frac{\xi}{1.0}\biggr)^{-1} \biggl(\frac{SFR}{\rm 10\; M_\odot yr^{-1}}\biggr)^{-1} \;\; {\rm Myr} \label{eq:mw},$$ where $M_{\rm wind}$ is the total mass of the wind including the loaded mass. The SFRs are estimated as follows. Since the butterfly nebula was formed most recent star formation activities have played important roles in its formation. We found that there is a wide spread in the estimated values of the current SFR. A well-used equation for converting the far infrared luminosity to the SFR [@Kennicutt1998], $SFR$ $M_\odot$ yr$^{-1}$ $\approx 17$ $(L_{\rm FIR}/10^{11} {\rm erg~s^{-1}})$, gives us an SFR of $\sim$120 $M_\odot$ yr$^{-1}$. This value is clearly overestimated, because a significant fraction of the bolometric luminosity of NGC 6240 is known to originate from the AGNs and extended star formation other than the current nuclear starburst [@Lutz2003; @Engel2010; @Mori2014]. @Engel2010 found a SFR of $\sim$25 $M_\odot$ yr$^{-1}$ for the nuclear starburst based on the $K$ band luminosity. On the other hand, @Beswick2001 found a SFR of 83 $M_\odot$ yr$^{-1}$ based on the 1.4 GHz radio continuum observation. @Engel2010 noted that the discrepancy between the above two values arises from the difference in the star formation histories they used; @Beswick2001 assumed a constant star formation history over 20 Myr, whereas @Engel2010 adopted a merger-induced increasing star formation scenario. The morphology of the H$\alpha$ nebula suggests an intermittent star formation nature for the star-forming history of NGC 6240. In particular, the estimated age of the butterfly nebula is much shorter than the duration of the star formation assumed by @Beswick2001. We thus adopt a SFR of 25 $M_\odot$ yr$^{-1}$ for the butterfly nebula. The age of the hourglass region estimated above ($\sim$24 Myr) is almost consistent with the starburst age ($\sim$15–25 Myr) estimated by @Tecza2000. We thus interpret that the hourglass region was created by this starburst, and its SFR was 25 $M_\odot$ yr$^{-1}$, which is the value estimated by @Tecza2000 from the central red stellar population. We have no suitable observational basis for estimating the SFR for the outer filaments. Hence, for the outer filaments we tentatively assume a typical SFR from the local starburst galaxies, $\sim$10 $M_\odot$ yr$^{-1}$. The H$\alpha$ gas masses of the three regions were determined by integrating the H$\alpha$ gas mass in three concentric regions; $r$ $\leq$ 4.5 kpc, 4.5 kpc $<$ $r$ $\leq$ 13 kpc, and 13 kpc $<$ $r$ $\leq$ 45 kpc. The close morphological correlation between the H$\alpha$ emission and soft X-ray of NGC 6240 indicates that these two-phase gases are spatially associated with each other. Therefore, the mass of the soft X-ray gas must be taken into account to derive the total mass of the superwinds $M_{\rm wind}$. @Nardini2013 estimated the total mass of the soft X-ray gas as $\sim$10$^{10}$ $M_\odot$ with the assumption that the volume filling factor $f_{\rm v, X}$ is almost unity and a spherically symmetric geometry of the hot gas nebula. However, this assumption is not adequate for the superwind. As we described in section \[sec:ed\], $f_{\rm v, X}$ would be of order 10$^{-2}$. In addition, we introduced the line-of-sight occupation factor $\zeta$ to represent the filamentary structure of the nebula (see section \[sec:ed\]). Using these factors, we calculated the total wind mass $M_{\rm wind}$. The ages and durations of the superwinds that formed the three regions derived by the above calculations are listed in Table \[tab:windparam\]. The duration of the superwind is approximately one order of magnitude smaller than the age of each region. This suggests that the starburst/superwind activity has an intermittent nature as suggested by numerical simulations of galaxy mergers [@Mihos1996; @Hopkins2006; @Hopkins2013]. It should be noted that the duration of the superwind for the hourglass region (4 Myr) is consistent with the starburst duration ($\sim$5 Myr) estimated by @Tecza2000. In the above discussion, we ignored the AGN contribution to the wind evolution. Currently, there are double active nuclei at the center of NGC 6240 and they play a non negligible role in energetics in the central region of the galaxy [@Elston1990; @Komossa2003; @Feruglio2013]. The AGNs are, however, deeply embedded and still not active enough to form the extended hot and warm gas nebula surrounding the galaxy [@Sugai1997; @Mori2014]. Thus the most part of the nebula was probably formed by the starburst activity [@Heckman1990; @Keel1990; @Nardini2013]. Even in the vicinity of the nuclei, the influence of the AGNs to the surrounding excited gas region has not been clear yet [@Sugai1997; @Ohyama2000; @Engel2010; @Wang2014]. Therefore, we conclude that the starburst superwind is the prime mover in forming the extended nebula and the contribution of the AGNs would be negligible except for the central region (a part of the butterfly nebula). Starburst history of NGC 6240 ----------------------------- NGC 6240 is thought to be in the late stages of a galaxy merger [@Hopkins2006; @Engel2010]. Its morphology indicates that two gaseous spiral galaxies first encountered each other $\sim$1 Gyr ago and their nuclei are now in the final stages of coalescence. The H$\alpha$ nebula was formed by intermittent starburst events which occurred within the last 100 Myr. Here, we propose a possible scenario of the starburst history of NGC 6240 from the point of view of the formation of the H$\alpha$ nebula by starburst superwinds. A starburst which occurred $\sim$80 Myr ago created powerful bi-polar galactic bubbles; the NW- and SE-bubbles. At that time, the ISM in the merging system might be more abundant in the northwestern and western sides of the merging galaxies than in the eastern and southeastern sides. A combination of the intermittent nature of the starburst activity and the lopsided ISM distribution formed a rich structure of loops, filaments, or blobs in the western side of the system. The extended active star formation as indicated by the UV images was also preferentially induced in the western side as the wind propagated outward from the central starburst. The bright knots observed in the western side of the system were probably formed by collisions between the high-speed outflowing gas from the center and dense gas blobs. The ram pressure of the outflow served to highly condense the gas blobs and initiated active star formation. Approximately 20 Myr ago, a short duration starburst ($\sim$4–5 Myr) occurred at the center of the system. This starburst and the subsequent superwind formed the central stellar population of NGC 6240 and the hourglass region of the H$\alpha$ nebula. Active star formation ceased temporarily following the short duration burst described above. Several Myr ago, the concentration of gas by the continuing approach of the two nuclei induced an intense starburst that continues to the present; and also drove AGN activity. The SFR may be steadily increasing, and has currently reached $\sim$25 $M_\odot$ yr$^{-1}$ [@Engel2010]. This recent starburst and AGN activity formed various features including the butterfly nebula [@Heckman1990; @Keel1990; @Komossa2003; @Gerssen2004], the central extended hard X-ray gas [@Wang2014], the molecular outflow [@Tacconi1999; @Nakanishi2005; @Iono2007; @Engel2010; @Feruglio2013; @Scoville2015], and the intense H$_2$ and \[\] emission [@Tanaka1991; @Werf1993; @Sugai1997; @Ohyama2003]. Summary ======= We conducted unprecedented deep H$\alpha$ narrow-band imaging observations of NGC 6240 using the Subaru Suprime-Cam. The 6,000-sec Suprime-Cam observation was the first to reveal the detailed structure of a highly-extended H$\alpha$ emitting nebula (the H$\alpha$ nebula) whose size reaches to $\sim$90 kpc in diameter. This nebula was probably formed by large-scale energetic outflows — or superwinds — driven by starburst events induced by a galaxy–galaxy merger. In addition to the butterfly-shaped inner nebula, which has already been investigated by various authors, the outer nebula has a complex structure consisting of wiggled narrow filaments, loops, bubbles, and blobs. We found a huge bi-polar bubble (the SE-bubble plus the NW-bubble), which extends along the minor axis of the main stellar disk of NGC 6240. The total H$\alpha$ luminosity is $\approx$1.6 $\times$ 10$^{42}$ erg s$^{-1}$. We estimated the volume filling factor $f_{\rm v, H\alpha}$ of the H$\alpha$ nebula assuming a simple thermal pressure equilibrium and a ram pressure balance between the warm ionized gas and the soft X-ray gas. The $f_{\rm v, H\alpha}$ is of orders 10$^{-4}$–10$^{-5}$ for the both cases. We calculated the mass of the H$\alpha$ emitting gas to be $M_{\rm H\alpha}$ of $\sim$5 $\times 10^8$ $M_\odot$ using the $f_{\rm v, H\alpha}$. It is remarkable that the H$\alpha$ extent and morphology are quite similar to the X-ray (0.7–1.1 keV) emitting hot gas detected with a deep [*Chandra*]{} observation [@Nardini2013]. Such a close spatial coincidence between the optical emission line nebula and the hot gas is generally observed in starburst superwinds. This strongly suggests that the same physical mechanism, shock-heating, plays an important role in the excitation of both phases of the gas at the outer regions of the H$\alpha$ nebula. The H$\alpha$ nebula also has a close spatial correlation with the UV emission in the western region of the galaxy. Some bright star-forming knots are located in the western region. These facts suggest, therefore, that star-forming activity was induced by the interaction of the outflow and the ISM, and that the interaction was probably enhanced due to a lopsided gas distribution formed by the merger. The morphology and surface brightness distribution of the H$\alpha$ nebula suggest that there were probably three major episodes of starburst/superwind activities that formed the nebula. The most extended part of the nebula was initiated $\sim$80 Myr ago. Multiple intermittent starbursts in this era created many outflow filaments and clouds extending $\sim$40 kpc out from the center. At the same time, the outflowing gas interacted with the ISM and induced extended star formation in the main stellar disk. An intense short duration starburst then occurred $\sim$20 Myr ago, forming the hourglass region of the H$\alpha$ nebula. NGC 6240 entered a new active phase $\sim$1 Myr ago. The galaxy is now in the final stages of the merger and its activity is increasing. We thank the anonymous referee for the careful reading and constructive suggestions. We are grateful to the staff of the Subaru telescope for their kind help with the observations. This work was in part carried out using the data obtained by a collaborative study on the cluster Abell 1367. This study was in part carried out using the facilities at the Astronomical Data Analysis Center (ADAC), National Astronomical Observatory of Japan. This research made use of NASA’s Astrophysics Data System Abstract Service, NASA/IPAC Extragalactic Database, GALEX GR6 database, and SDSS DR9 database. This work was financially supported in part by Grant-in-Aid for Scientific Research No.23244030, No.15H02069 from the Japan Society for the Promotion of Science (JSPS), No.24103003 from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), and MOST grant 104-2112-M-001-034-. Ahn, C. P., et al. 2012, , 203, 21 Armus, L., et al. 1990, , 364, 471 Barnes, J. 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S., Bland-Hawthorn, J., & Cecil, G. 2003, , 126, 2185 Veilleux, S., et al. 2005, 43, 769 Wang, J., et al. 2014, , 781, 55 Yagi, M., et al. 2002, , 123, 66 Yagi, M., et al. 2007, , 660, 1209 Yagi, M. 2012, , 124, 1347 Yagi, M., et al. 2013, , 65, 22 Yoshida, M. & Ohtani, H. 1993, , 45, 407 Yoshida, M., et al. 2008, , 688, 918 A. Color conversion from SDSS to the Suprime-Cam filter system ============================================================== We adopted a color conversion from SDSS to the Suprime-Cam filter system with HPK CCDs as $$SDSS - Suprime = \sum c_i (SDSS color)^i,$$ where the coefficients and the color range of the stars are given in Table \[tab:conv\]. The coefficients are estimated using the method given in @Yagi2013, with the model quantum efficiency curve of the CCDs. We used stars of magnitude $17<r<21$ in SDSS DR9. The K-correct [@Blanton2007] v4 offset for SDSS[^5] was applied; $m_{AB}-m_{SDSS}=$ 0.012, 0.010 and 0.028 for $g$, $r$, and $i$, respectively. The number of stars is approximately 60 - 200, and the estimated rms error around the best fit ranged from 0.05 to 0.07. [cccccccccccccc]{} $g-B$ & $g-r$ & $-0.6<g-r<0.6$ & $-0.031$ & $-0.124$ & 0.070 & $-0.415$ & $-0.213$ & 0.881 & ... & ...\ $r-R$ & $r-i$ & $-0.6<r-i<0.6$ & 0.006 & 0.312 & $-0.064$ & $-0.152$ & 1.601 & $-1.262$ & $-7.990$ & 9.992\ $r-NA671$ & $r-i$ & $-0.6<r-i<0.3$ & 0.020 & 0.536 & 0.021 & $-0.381$ & $-0.722$ & ... & ... & ...\ $i-i$ & $r-i$ & $-0.6<r-i<1.8$ & $-0.006$ & 0.089 & 0.019 & $-0.016$ & ... & ... & ... & ...\ \[tab:conv\] B. Correction of the contaminated emission line flux for the net H$\alpha$+\[\] image ===================================================================================== The use of the scaled $R_{\rm C}$ frame as an off-band frame leads an over-subtraction of the continuum, because the H$\alpha$+\[\] emission is non-negligibly contaminated in the scaled $R_{\rm C}$ frame. The optical emission line nebula of NGC 6240 primarily consists of shock-heated gas [e.g. @Heckman1990; @Keel1990], thus the \[\]$\lambda$6300 and \[\]$\lambda\lambda$6717,6731 lines must also contribute towards contaminating the emission line flux in the $R_{\rm C}$ frame. Assuming that the emission line intensity ratio $f_{\rm [O I]+[S II]} / f_{\rm H\alpha+[N II]}$ is constant over the whole region of the nebula, we can write the H$\alpha$+\[\] flux in the net H$\alpha$+\[\] frame $f_{\rm H\alpha+[N II]}^\prime$ as $$f_{\rm H\alpha+[N II]}^\prime = (1 - \beta) f_{\rm H\alpha+[N II]}, \label{eq:f}$$ where $\beta$ is the contamination factor in the $R_{\rm C}$ frame, $f_{\rm H\alpha+[N II]}^\prime$ is the H$\alpha$+\[\] flux uncorrected for over-subtraction of line emission from the $R_{\rm C}$ frame, and $f_{\rm H\alpha+[N II]}$ is that corrected for this effect. Here, we assumed that the filter transmissions are the same for all the emission lines, for simplicity. We then derived the scaling factor $c$ in the following equation by subtracting the net H$\alpha$+\[\] frame multiplied by $c$ from the scaled $R_{\rm C}$ frame so that the central butterfly nebula component disappears in the subtracted frame, $$c\;f_{\rm H\alpha+[N II]}^\prime = \beta\; f_{\rm H\alpha+[N II]} \label{eq:c}.$$ Equations (\[eq:f\]) and (\[eq:c\]) give $f_{\rm H\alpha+[N II]} = (1+c) f_{\rm H\alpha+[N II]}^\prime$. We found that the best scaling factor $c$ was 0.23. Thus the correction factor for the H$\alpha$+\[\] flux of the net H$\alpha$+\[\] image is 1.23. [^1]: This X-ray image is an “adaptively smoothed” image [Figure 4 of @Nardini2013]; the smoothing scale changes with the signal-to-noise ratio of the original image. See also <http://chandra.harvard.edu/photo/2013/ngc6240/ngc6240_xray.jpg>, and compare the jpg image with our Figure \[fig:unsharp\] to inspect the spatial correlation between the H$\alpha$ nebula and the X-ray nebula in, more detail. [^2]: [*GALEX*]{} GR6 data release: <http://galex.stsci.edu/GR6/> [^3]: Although the $B$ and $i^\prime$ images in Figure \[fig:knots\] were not corrected for emission lines, we think that the most flux in these bands comes from the stellar continuum. In the $B$ band, there are several emission lines such as H$\beta$, H$\gamma$ and \[\]$\lambda$5007. We estimated that the contribution of these lines to the $B$ band flux is less than 10% assuming the ordinary Balmer decrement and $f_{\rm [OIII]}$/$f_{\rm H\beta}\approx 1$ [@Keel1990]. In the $i^\prime$ band, there is no strong emission lines. [^4]: <https://www.spacetelescope.org/images/potw1520a/> [^5]: <http://howdy.physics.nyu.edu/index.php/Kcorrect>

--- author: - 'Cheng Chen $^a$,' - 'Zhenwei Cui $^a$,' - 'Gang Li$^{b,c}$,' - 'Qiang Li$^{a,b}$,' - 'Manqi Ruan$^{b,c}$,' - 'Lei Wang$^a$,' - 'Qi-shu Yan$^d$' bibliography: - 'references.bib' title: '$\Hboson \xrightarrow {}\ee$ at CEPC: ISR effect with MadGraph' --- Introduction {#intr} ============ The amazing discovery of Higgs boson [@ref:1; @ref:2] in 2012 by the ATLAS and CMS experiments at the CERN LHC has made a considerable step in particle physics, opening doors to new physics search through Higgs portal. The up-to-date results indicate that it is highly Standard Model (SM) like [@cmshig; @lhcsub1; @lhcsub2; @lhcsub3; @lhcsub4; @lhcsub5]. However, many new physics models predict the Higgs couplings deviate from the SM at the percent level. Thus the percent or even sub-percent level precision becomes necessary for the future Higgs measurement program. With this consideration, a Higgs factory at $e^+e^-$ collider with high luminosity is best suited for this goal, due to its clean environment and relative lower cost. The Circular Electron-Positron Collider(CEPC) [@ref:3] is such a nice example, which is a proposed circular collider, designed to run around $240\sim250$ GeV with an instantaneous luminosity of 2 $\times$ $\rm{10}^{\rm{34}}$ $\rm{cm}^{\rm{-2}}$ $\rm{s}^{\rm{-1}}$, and will deliver 5 $\mathrm{ab}^{-1}$ of integrated luminosity during 10 years of operation. About $10^6$ Higgs events will be produced in a clean environment, which allows the measurement of the cross section of the Higgs production as well as its mass, decay width and branching ratios with precision much beyond those of hadron colliders. At CEPC with the center-of-mass energy of 250 GeV, the Higgs bosons are dominantly produced from $ZH$ process, where the Higgs boson is produced in association with a $Z$ boson. Major deay modes of the Higgs boson have been extensively studied in Refs [@ref:3; @Chen:2016zpw], such as the channel of $\Hboson \xrightarrow {}\Zboson\Zboson$, and $\Hboson \xrightarrow {}\gamma\gamma$ etc. In this study we are interested in a rare decay $\Hboson \xrightarrow {}\ee$. The Feynman diagram of $\Hboson \xrightarrow {}\ee$ is shown in Figure. \[fig:fmd\]. The SM prediction for the branching fraction ${\cal B}(\Hboson \xrightarrow {}\ee)$ is as tiny as approximately $5\times 10^{-9}$. However, in new physics scenario (see e.g. [@Altmannshofer:2015qra]), it can be enhanced significantly. Moreover, searching or measurement for $\Hboson \xrightarrow {}\ee$ together with $\mm$ and $\tt$, can be used to test the lepton universality of Higgs boson couplings. The two electrons from Higgs decay can be easily identified and their momentum can be precisely measured in the detector. The Higgsstrahlung events can then be reconstructed with the recoil mass method: $$m_{recoil}^{2}=s+m^{2}_{H}-2\cdot E_{H}\cdot~ \sqrt[]s ~,$$ where $\sqrt[]s$ is the center of mass energy, $m_{H}$ and $E_{H}$ are the mass and energy of the Higgs boson reconstructed by the two lepton four momentum. Therefore, the $ZH$ ($\Hboson \xrightarrow {}\ee$) events form a peak in the $M_{\rm{recoil}}$ distribution at the Z boson mass. With the recoil mass method, the $ZH$ events are selected without using the decay information of the Z boson. ![\[fig:fmd\] Example Feynman diagram.](fdiagram.eps){width="40.00000%"} A search has already been performed at CMS with RunI data [@Khachatryan:2014aep], with an upper limit of 0.19% placed on the branching fraction ${\cal B}(\Hboson \xrightarrow {}\ee)$. Studies through resonant s-channel $\ee \xrightarrow {} \Hboson$ have also been proposed at FCC-ee [@fccee] operating at a collison energy of 125 GeV, with sensitivies being able to reach down to 2 times SM prediction with 10 $\mathrm{ab}^{-1}$ of integrated luminosity, depending, however, on good controls on beam spread. This paper is organized as follows. Section 2 shows the ISR implementation in [[MadGraph]{}]{}. Section 3 describes the detector model, Monte Carlo (MC) simulation and samples used in the studies. Section 4 presents the measurements of $\Hboson \xrightarrow {}\ee$. The conclusion is summarized in Section 5. ISR implementation in MadGraph {#isrmg} ============================== The Initial State Radiation (ISR) is an important issue in high energy processes, especially for lepton colliders. ISR affects cross section significantly, for example, reduces the $ZH$ cross section by more than 10%. Following Whizard [@ref:4], we have implemented in [[MadGraph]{}]{}  the lepton ISR structure function that includes all orders of soft and soft-collinear photons as well as up to the third order in hard-collinear photons. Comparisons can be seen in Fig. \[fig:isr\] for $e^+e^- \rightarrow ZH$, from which one can see the good agreement between [[Whizard]{}]{}and [[MadGraph]{}]{}  with ISR included, on distributions of center-of-mass energy and Higgs transverse momentum. Similar checks have also been passed for other processes including for the process $e^+e^- \rightarrow W^+W^-$ and $W^+W^-Z$. ![\[fig:isr\] Comparisons plots on center-of-mass energy and Higgs transverse momentum, between [[Whizard]{}]{}  and [[MadGraph]{}]{}  with or without ISR effect included, for the process $e^+e^- \rightarrow ZH$.](COM.eps "fig:"){width="48.00000%"} ![\[fig:isr\] Comparisons plots on center-of-mass energy and Higgs transverse momentum, between [[Whizard]{}]{}  and [[MadGraph]{}]{}  with or without ISR effect included, for the process $e^+e^- \rightarrow ZH$.](PTH.eps "fig:"){width="48.00000%"} One should note that besides ISR, another macro effect at high luminosity electron-positron collider, beamstrahlung, also affects the cross section. In the storage ring the beamstrahlung effect makes the beam energy spread larger and reduces the center of mass energy [@ref:cepc_acc]. The effect, however, are found to be small at CEPC. Based on above progress, we are now able to generate signal samples in [[MadGraph]{}]{}  with ISR effect included, for $e^+e^- \rightarrow ZH$, together with the decay of $\Hboson \xrightarrow {}\ee$ at matrix element level, thanks to the convinience of [[MadGraph]{}]{}. Detector and Simulation {#isrmg} ======================= The analysis is performed on the MC samples simulated on the CEPC conceptual detector, which is based on the International Large Detector (ILD) [@ild1; @ild2] at the ILC [@ILC]. At CEPC, electron identification efficiency is expected to be over 99.5% for $p_{\rm{T}}$ larger than 10 GeV, and with excellent $p_{\rm{T}}$ resolution of $\sigma_{1/p_{\rm{T}}} = 2\times 10^{-5} \oplus 1\times 10^{-3}/(p_{\rm{T}}\sin\theta)$. More details can be checked in [@ref:3; @Chen:2016zpw]. For the signal process, $e^+e^- \rightarrow ZH$ with $\Hboson \xrightarrow {}\ee$, 50K events are generated by [[MadGraph]{}]{}  V2.3.3 with ISR effect included, with Higgs mass set to be 125 GeV. For the backgrounds, [[Whizard]{}]{}  V2.2.8 [@ref:4], are exploited as the event generator. All these samples are produced at the center-of-mass energy of 250 GeV. The major SM backgrounds, including all the 2-fermion processes($e^+e^-\rightarrow f\bar{f}$, where $f\bar{f}$ refers to all lepton and quark pairs except $t\bar{t}$) and 4-fermion processes($ZZ$, $WW$, $ZZ$ or $WW$, single $Z$, single $W$). The initial states radiation (ISR) and all possible interference effects are taken into account in the generation automatically. The classification for four fermions production, is referred to LEP [@ref:17], depending crucially on the final state. For example, if the final states consist of two mutually charge conjugated fermion pairs that could decay from both $WW$ and $ZZ$ intermediate state, such as $e^{+}e^{-}\nu_{e}\bar{\nu_{e}}$, this process is classified as “$ZZ$ or $WW$” process. If there are $e^{\pm}$ together with its parter neutrino and an on-shell $W$ boson in the final state, this type is named as “single $W$”. Meanwhile, if there are a electron-positron pair and a on-shell $Z$ boson in the final state, this case is named as “single $Z$”. More details about the CEPC samples set can be found in reference [@Mo:2015mza]. Signal and background samples are further interfaced with [[Pythia]{}]{} 6 [@pythia] for parton shower and hadronization, and then fully simulated with Mokka [@Mokka] and reconstructed with ArborPFA [@arbor]. Results ======= As mentioned in Section \[intr\], signal events can be extracted with recoil mass method without using the decay information of the Z boson decay. The detailed event selections are listed as following: at least one pair of electrons with opposite charge is required, with final state radiation photon in included in the electron momenta. The pair with invaraiant mass $M_{e^{+}e^{-}}$ closer to Higgs mass is selected in case of multi-combinations, and required then to satisfy $120<M_{e^{+}e^{-}}<130$GeV. The recoil mass $M_{recoil}$ of $e^{+}e^{-}$ is required to be greater than 90 GeV and less than 93 GeV, to be consistent with the Z-boson hypothesis. Fig. \[fig:pt\] shows signal and backgrounds distributions on various kinematic variables, where signal without ISR effect included are also superimposed for comparison. ![\[fig:pt\] Distributions of $p_{\rm{T}e^{+}e^{-}}$, $p_{\rm{Z}e^{+}e^{-}}$, $M_{e^{+}e^{-}}$ and $M_{recoil}$ for signals and backgrounds. Signals without ISR effect included are also superimposed for comparison.](Pt.eps "fig:"){width="40.00000%"} ![\[fig:pt\] Distributions of $p_{\rm{T}e^{+}e^{-}}$, $p_{\rm{Z}e^{+}e^{-}}$, $M_{e^{+}e^{-}}$ and $M_{recoil}$ for signals and backgrounds. Signals without ISR effect included are also superimposed for comparison.](Pz.eps "fig:"){width="40.00000%"} ![\[fig:pt\] Distributions of $p_{\rm{T}e^{+}e^{-}}$, $p_{\rm{Z}e^{+}e^{-}}$, $M_{e^{+}e^{-}}$ and $M_{recoil}$ for signals and backgrounds. Signals without ISR effect included are also superimposed for comparison.](inv.eps "fig:"){width="40.00000%"} ![\[fig:pt\] Distributions of $p_{\rm{T}e^{+}e^{-}}$, $p_{\rm{Z}e^{+}e^{-}}$, $M_{e^{+}e^{-}}$ and $M_{recoil}$ for signals and backgrounds. Signals without ISR effect included are also superimposed for comparison.](Recoil.eps "fig:"){width="40.00000%"} To suppress 2-fermions background, it is required that the difference between the two electrons’ azimuth angles should satisfy $\Delta\phi\ <$ 166$^{\circ}$. In addition, to suppress background from 4-fermions background, the transverse momentum of electron pair and the scalar sum over Z-direction momentum, are required to $46<p_{\rm{T}e^{+}e^{-}}<93$GeV and $-42<p_{\rm{Z}e^{+}e^{-}}<41$GeV, which can efficiently cut away $ZZ$ and single $Z$ backgrounds, as shown in Fig. \[fig:pt\]. Finally, requirements are set on polar angle of each lepton particle, $cos_{e^{+}}$ $\ge$ -0.07 and $cos_{e^{-}}$ $\le$ 0.14 , as the electrons from Higgs boson are more uniformly distributed as it is a scalar particle. The selections of each variable as mentioned above are determined by maximazing the significance $S/\sqrt{B}$, where S is the number of signal events passing all the selection criteria, and B is the number of the corresponding background events number. The cut chain table is shown in Table \[tab:example1\]. The background yields are scaled to 5000$fb^{-1}$ . The signal yields starts from 50K before any selection, and the final efficiency is about 7.1%. Category signal 2fermions single ZorW single Z single W ----------------------------------------------------- -------- ----------- ------------- ---------- ---------- -- -- -- -- -- total 50000 418194802 1259165 7913405 17190655 $N_{e^{+}}$ $\ge$ 1, $N_{e^{-}}$ $\ge$ 1 47418 36822471 978594 3480494 2260761 120 GeV $<\ M_{e^{+}e^{-}}\ <$ 130 GeV 34463 1954192 71193 126094 151950 90 GeV $<\ M_{recoil}\ <$ 93 GeV 12362 61089 3564 6954 7255 46 GeV $<\ p_{\rm{T}e^{+}e^{-}}\ <$ 63 GeV 8582 6816 1863 1861 3652 -42 GeV $<\ p_{\rm{Z}e^{+}e^{-}}\ <$ 41 GeV 8511 6372 1783 1750 3468 $\Delta\phi\ <$ 166$^{\circ}$ 7404 5131 1696 1651 3233 $cos_{e^{+}}$ $\ge$ -0.07, $cos_{e^{-}}$ $\le$ 0.14 3564 241 86 48 161 : Yields for backgrounds and signals at the CEPC with $\sqrt{s}=250$ GeV and integrated luminosity of $5000{\mbox{\ensuremath{~\text{fb}^\text{$-$1}}}}$.[]{data-label="tab:example1"} \ Category WW ZZ WWorZZ total background ----------------------------------------------------- ---------- --------- ---------- ------------------ -- -- -- -- -- -- total 49115769 4967152 21902983 520543931 $N_{e^{+}}$ $\ge$ 1, $N_{e^{-}}$ $\ge$ 1 640839 758732 814608 45756499 120 GeV $<\ M_{e^{+}e^{-}}\ <$ 130 GeV 26731 7593 55196 2392949 90 GeV $<\ M_{recoil}\ <$ 93 GeV 1783 1464 2434 84543 46 GeV $<\ p_{\rm{T}e^{+}e^{-}}\ <$ 63 GeV 868 682 1297 17039 -42 GeV $<\ p_{\rm{Z}e^{+}e^{-}}\ <$ 41 GeV 837 647 1247 16104 $\Delta\phi\ >$ 166$^{\circ}$ 702 566 1182 14161 $cos_{e^{+}}$ $\ge$ -0.07, $cos_{e^{-}}$ $\le$ 0.14 20 178 70 804 : Yields for backgrounds and signals at the CEPC with $\sqrt{s}=250$ GeV and integrated luminosity of $5000{\mbox{\ensuremath{~\text{fb}^\text{$-$1}}}}$.[]{data-label="tab:example1"} We have also exploited the Toolkit for Multivariate Analysis (TMVA) [@TMVA] for further background rejection, where the method of Boosted Decision Trees (BDT) is adopted and the selected variables for TMVA input are those as mentioned above. No significant improvement is found compared with the cut-based results, thus in this study, we provide only the latter. After the event selections as mentioned above, we perform a $\mu S + B$ fit (with $\mu$ as the signal strength) on CEPC simulated data which is essential purely background as the SM predicted $\Hboson \xrightarrow {}\ee$ branch ratio is too low. As shown in Fig. \[fig:fit\], an unbinned maximum likelihood fit is performed on $M_{e^{+}e^{-}}$ spectrum, in the region of 120 GeV to 130 GeV. The Higgs signal shape is described by a Crystal Ball function, while the background is represented by a second order Chebychev polynomial function, whose parameters are fixed to the values extracted from the background samples. By scanning over signal strength in the $\mu S + B$ fit, one can extract the dependence of negative log likelihood on it. The 95% confidence level upper limit on $\Hboson \xrightarrow {}\ee$ branch ratio can then be decided to be 0.024%. This corresponds to a signal yield of around 20, while from Figure \[fig:fit\], the background yield under the Higgs peak is near 200, and thus by naively couting, $S/\sqrt{B}\sim 1.4$ which supports the above result from the shape analysis. Finally we mention that checks with different background modelling have also been done. With e.g. third order Chebychev polynomial function, the result improves a bit while the fit goodness gets worse. ![\[fig:fit\] The invariant mass spectrum of $e^{+}e^{-}$ in the inclusive analysis. The dots with error bars represent data from CEPC simulation. The solid (blue) line indicates the fit. The dashed (red) shows the signal (assuming ${\cal B}(\Hboson \xrightarrow {}\ee)$=0.024%) and the long-dashed (green) line is the background.](fitresult.eps){width="70.00000%"} Summary and Conclusions {#talk} ======================= The CEPC is expected to play a crucial role in understanding Higgs boson properties. In this paper, a probe on $\Hboson \xrightarrow {}\ee$ at CEPC is investigated with full simulated Higgsstrahlung signal at 5 ab$^{\rm{-1}}$ integrated luminosity at 250 GeV center-of-mass energy. The upper limit at 95% confidence level on the production cross section times branching fraction for $e^+e^- \rightarrow ZH$ with $\Hboson \xrightarrow {}\ee$ are found to be 0.051 fb. This corresponds to an upper limit on the branching fraction of 0.024%. As a by-product, ISR effect has been implemented in [[MadGraph]{}]{}  to generate the signal process. Finally, we mention that with similar framework, measurements for $\Hboson \xrightarrow {}\mm$ together with $\tt$ at CEPC are being finalized [@cepccdr], which show similar or even improved accuracy compared with the results for HL-LHC [@ATLprojection; @CMS:2013xfa]. 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--- author: - | Fu-Tao Hu,Jun-Ming Xu[^1] \ \ [Department of Mathematics]{}\ [University of Science and Technology of China]{}\ [Hefei, Anhui, 230026, China]{} title: 'Total and paired domination numbers of toroidal meshes [^2]' --- > **Abstract**: Let $G$ be a graph without isolated vertices. The total domination number of $G$ is the minimum number of vertices that can dominate all vertices in $G$, and the paired domination number of $G$ is the minimum number of vertices in a dominating set whose induced subgraph contains a perfect matching. This paper determines the total domination number and the paired domination number of the toroidal meshes, i.e., the Cartesian product of two cycles $C_n$ and $C_m$ for any $n\ge 3$ and $m\in\{3,4\}$, and gives some upper bounds for $n, m\ge 5$. > > [**Keywords**]{}: combinatorics, total domination number, paired domination number, toroidal meshes, Cartesian product. > > [**AMS Subject Classification:** ]{} 05C25, 05C40, 05C12 Introduction ============ For notation and graph-theoretical terminology not defined here we follow [@x03]. Specifically, let $G=(V,E)$ be an undirected graph without loops, multi-edges and isolated vertices, where $V=V(G)$ is the vertex-set and $E=E(G)$ is the edge-set, which is a subset of $\{xy|\ xy$ is an unordered pair of $V \}$. A graph $G$ is [*nonempty*]{} if $E(G)\ne \emptyset$. Two vertices $x$ and $y$ are [*adjacent*]{} if $xy\in E(G)$. For a vertex $x$, denote $N(x)=\{y: xy\in E(G)\}$ be the [*neighborhood*]{} of $x$. For a subset $D\subseteq V(G)$, we use $G[D]$ to denote the subgraph of $G$ induced by $D$. We use $C_n$ and $P_n$ to denote a cycle and a path of order $n$, respectively, throughout this paper. A subset $D\subseteq V(G)$ is called a [*dominating set*]{} if $N(x)\cap D\ne \emptyset$ for each vertex $x\in V(G)\setminus D$. The [*domination number*]{} $\gamma(G)$ is the minimum cardinality of a dominating set. A thorough study of domination appears in [@hhs98a; @hhs98b]. A subset $D\subseteq V(G)$ of $G$ is called a [*total dominating set*]{}, introduced by Cockayne [*et al.*]{} [@cdh80], if $N(x)\cap D\ne \emptyset$ for each vertex $x\in V(G)$ and the [*total domination number*]{} of $G$, denoted by $\gamma_t(G)$, is the minimum cardinality of a total dominating set of $G$. The total domination in graphs has been extensively studied in the literature. A survey of selected recent results on this topic is given in [@h09] by Henning. A dominating set $D$ of $G$ is called to be [*paired*]{}, introduced by Haynes and Slater  [@hs95; @hs98], if the induced subgraph $G[D]$ contains a perfect matching. The [*paired domination number*]{} of $G$, denoted by $\gamma_p(G)$, is the minimum cardinality of a paired dominating set of $G$. Clearly, $\gamma(G)\le\gamma_t(G)\le \gamma_p(G)$ since a paired dominating set is also a total dominating set of $G$, and $\gamma_p(G)$ is even. Pfaff, Laskar and Hedetniemi [@plh83] and Haynes and Slater [@hs98] showed that the problems determining the total-domination and the paired-domination for general graphs are NP-complete. Some exact values of total-domination numbers and paired-domination numbers for some special classes of graphs have been determined by several authors. In particularly, $\gamma_t(P_n \times P_m)$ and $\gamma_p(P_n \times P_m)$ for $2\le m\le 4$ are determined by Gravier [@g02], and Proffitt, Haynes and Slater [@phs01], respectively. Use $G_{n,m}$ to denote the toroidal meshes, i.e., the Cartesian product $C_n \times C_m$ of two cycles $C_n$ and $C_m$. Klavžar and Seifter [@ks95] determined $\gamma(G_{n,m})$ for any $n\ge 3$ and $m\in\{3,4,5\}$. In this paper, we obtain the following results. $$\begin{array}{rl} &\gamma_{t}(G_{n,3})=\lceil \frac{4n}{5}\rceil;\\ &\gamma_{p}(G_{n,3})=\left\{ \begin{array}{ll} \lceil\frac{4n}{5}\rceil & {\rm if}\ n\equiv 0,2,4\,({\rm mod}\,5),\\ \lceil\frac{4n}{5}\rceil+1 & {\rm if}\ n\equiv 1,3\,({\rm mod}\,5);\\ \end{array}\right.\\ &\gamma_{t}(G_{n,4})=\gamma_{p}(G_{n,4})=\left\{ \begin{array}{ll} n & {\rm if}\ n\equiv 0\,({\rm mod}\,4),\\ n+1 & {\rm if}\ n\equiv 1,3\,({\rm mod}\,4),\\ n+2 & {\rm if}\ n\equiv 2\,({\rm mod}\,4). \end{array}\right. \end{array}$$ Preliminary results =================== In this section, we recall some definitions, notations and results used in the proofs of our main results. Throughout this paper, we assume that a cycle $C_n$ has the vertex-set $V(C_n)=\{1,\ldots,n\}$. Use $G_{n,m}$ to denote the toroidal meshes, i.e., the Cartesian product $C_n \times C_m$, which is a graph with vertex-set $V(G_{n,m})=\{x_{ij}|\ 1\leq i \leq n, 1\leq j \leq m\}$ and two vertices $x_{ij}$ and $x_{i'j\,'}$ being linked by an edge if and only if either $i=i'\in V(C_n)$ and $jj\,'\in E(C_m)$, or $j=j\,'\in V(C_m)$ and $ii'\in E(C_n)$. Let $Y_i=\{x_{ij}|\ 1\leq j \leq m\}$ for $1\leq i\leq n$, called a set of [*vertical vertices*]{} in $G_{n,m}$. In [@gs02], Gavlas and Schultz defined an efficient total dominating set, which is such a total dominating set $D$ of $G$ that $|N(v)\cap D|=1$ for every $v\in V(G)$. The related research results can be found in [@ds03; @gs02; @hx08]. \[lem2.1\][(Gavlas and Schult[@gs02])]{} If a graph $G$ has an efficient total dominating set $D$, then the edge-set of the subgraph $G[D]$ forms a perfect matching, and so the cardinality of $D$ is even, and $\{N(v): v\in D\}$ partitions $V(G)$. \[lem2.2\] Let $G$ be a $k$-regular graph of order $n$. Then $\gamma_t(G)\ge \frac{n}{k}$, with equality if and only if $G$ has an efficient total dominating set. Since $G$ is $k$-regular, each $v\in V(G)$ can dominate at most $k$ vertices. Thus $\gamma_t(G)\ge \frac{n}{k}$. It is easy to observe that the equality holds if and only if there exists a total dominating set $D$ such that $\{N(v): v\in D\}$ partitions $V(G)$, equivalently, $D$ is an efficient total dominating set. \[lem2.3\] $\gamma_{t}(G_{n,m})=\gamma_{p}(G_{n,m})=\frac{nm}{4}$ for $n,m\equiv \,0~({\rm mod}\,4)$. Let $D=\{x_{ij},x_{i(j+1)},x_{(i+2)(j+2)},x_{(i+2)(j+3)}: ~i,j\equiv \,1~({\rm mod}\,4)\}$, where $1\le i\le n$ and $1\le j\le m$. Figure \[f1\] is such a set $D$ in $G_{8,4}$. It is easy to see that $D$ is a paired dominating set of $G_{n,m}$ with cardinality $\frac{nm}{4}$. Thus, $\gamma_{p}(G_{n,m})\le \frac{nm}{4}$. (0,.5)(8.5,5) (1,1)[2pt]{}[11]{}(1,.7)[$x_{11}$]{} (1,2)[2pt]{}[12]{} (1,3)[2pt]{}[13]{} (1,4)[2pt]{}[14]{}(1.3,3.8)[$x_{14}$]{} (2,1)[2pt]{}[21]{}(2,.7)[$x_{21}$]{} (2,2)[2pt]{}[22]{} (2,3)[2pt]{}[23]{} (2,4)[2pt]{}[24]{}(2.3,3.8)[$x_{24}$]{} (3,1)[2pt]{}[31]{}(3,.7)[$x_{31}$]{} (3,2)[2pt]{}[32]{} (3,3)[2pt]{}[33]{} (3,4)[2pt]{}[34]{}(3.3,3.8)[$x_{34}$]{} (4,1)[2pt]{}[41]{}(4,.7)[$x_{41}$]{} (4,2)[2pt]{}[42]{} (4,3)[2pt]{}[43]{} (4,4)[2pt]{}[44]{}(4.3,3.8)[$x_{44}$]{} (5,1)[2pt]{}[51]{}(5,.7)[$x_{51}$]{} (5,2)[2pt]{}[52]{} (5,3)[2pt]{}[53]{} (5,4)[2pt]{}[54]{}(5.3,3.8)[$x_{54}$]{} (6,1)[2pt]{}[61]{}(6,.7)[$x_{61}$]{} (6,2)[2pt]{}[62]{} (6,3)[2pt]{}[63]{} (6,4)[2pt]{}[64]{}(6.3,3.8)[$x_{64}$]{} (7,1)[2pt]{}[71]{}(7,.7)[$x_{71}$]{} (7,2)[2pt]{}[72]{} (7,3)[2pt]{}[73]{} (7,4)[2pt]{}[74]{}(7.3,3.8)[$x_{74}$]{} (8,1)[2pt]{}[81]{}(8,.7)[$x_{81}$]{} (8,2)[2pt]{}[82]{} (8,3)[2pt]{}[83]{} (8,4)[2pt]{}[84]{}(8.3,3.8)[$x_{84}$]{} By Lemma \[lem2.2\], $\gamma_{t}(G_{n,m})\ge \frac{nm}{4}=n$. Since $\gamma_t(G_{n,m})\le \gamma_{p}(G_{n,m})$, $\gamma_{t}(G_{n,m})=\gamma_{p}(G_{n,m})=\frac{nm}{4}$. Total and paired domination number of $G_{n,3}$ =============================================== In this section, we determine the exact values of the total and the paired domination numbers of $G_{n,3}$, which can be stated the following theorem. \[thm3.1\] For any $n\geq 3$, $$\gamma_{t}(G_{n,3})=\left\lceil \frac{4n}{5}\right\rceil$$ and $$\gamma_{p}(G_{n,3})=\left\{ \begin{array}{ll} \lceil\frac{4n}{5}\rceil, & {\rm if}\ n\equiv 0,2,4\,({\rm mod}\,5);\\ \lceil\frac{4n}{5}\rceil+1,& {\rm if}\ n\equiv 1,3\,({\rm mod}\,5). \end{array}\right.$$ Let $D$ be a minimum total dominating set of $G_{n,3}$. First, we may assume that $|Y_i\cap D|\le 2$ for any $1\le i\le n$. Indeed, if $|Y_i\cap D|=3$ for some $i\notin\{1,n\}$, then the set $D'=(D\setminus \{x_{i1}, x_{i3}\})\cup \{x_{(i-1)2}, x_{(i+1)2}\}$ is also a total dominating set of $G_{n,3}$ with $|D'|=|D|$. Let $\alpha_k$ be the number of $i$’s for which $|Y_i\cap D|=k$ for $1\le i\le n$ and $0\le k\le 2$. Then we have $$\label{e3.1} \alpha_0+\alpha_1+\alpha_2=n.$$ Assume $|Y_i\cap D|=0$ for some $i\notin\{1,n\}$. At least one of $|Y_{i-1}\cap D|$ and $|Y_{i+1}\cap D|$ is 2 since the three vertices in $Y_i$ should be dominated by $D$, which means that $$\label{e3.2} 2\alpha_2-\alpha_0\ge 0.$$ If $|Y_i\cap D|=2$ for some $i$ with $1\le i\le n$, then the two vertices in $Y_i\cap D$ can dominate at most 7 vertices. Since any vertex $x\in D$ can dominate at most 4 vertices, we have $$\label{e3.3} 4\alpha_1+7\alpha_2\ge 3n.$$ The sum of (\[e3.1\]), (\[e3.2\]) and (\[e3.3\]) implies $$5\alpha_1+10\alpha_2\ge 4n,$$ and, hence, $$\label{e3.4} \gamma_{t}(G_{n,3})=|D|=\alpha_1+2\alpha_2\ge \left\lceil \frac{4n}{5}\right\rceil.$$ (0,.5)(11,4) (1,1)[2pt]{}[11]{}(1,.7)[$x_{11}$]{} (1,2)[2pt]{}[12]{}(1.3,1.8)[$x_{12}$]{} (1,3)[2pt]{}[13]{}(1.3,2.8)[$x_{13}$]{} (2,1)[2pt]{}[21]{}(2,.7)[$x_{21}$]{} (2,2)[2pt]{}[22]{}(2.3,1.8)[$x_{22}$]{} (2,3)[2pt]{}[23]{}(2.3,2.8)[$x_{23}$]{} (3,1)[2pt]{}[31]{}(3,.7)[$x_{31}$]{} (3,2)[2pt]{}[32]{}(3.3,1.8)[$x_{32}$]{} (3,3)[2pt]{}[33]{}(3.3,2.8)[$x_{33}$]{} (4,1)[2pt]{}[41]{}(4,0.7)[$x_{41}$]{} (4,2)[2pt]{}[42]{}(4.3,1.8)[$x_{42}$]{} (4,3)[2pt]{}[43]{}(4.3,2.8)[$x_{43}$]{} (5,1)[2pt]{}[51]{}(5,0.7)[$x_{51}$]{} (5,2)[2pt]{}[52]{}(5.3,1.8)[$x_{52}$]{} (5,3)[2pt]{}[53]{}(5.3,2.8)[$x_{53}$]{} (6,1)[2pt]{}[61]{}(6,0.7)[$x_{61}$]{} (6,2)[2pt]{}[62]{}(6.3,1.8)[$x_{62}$]{} (6,3)[2pt]{}[63]{}(6.3,2.8)[$x_{63}$]{} (7,1)[2pt]{}[71]{}(7,0.7)[$x_{71}$]{} (7,2)[2pt]{}[72]{}(7.3,1.8)[$x_{72}$]{} (7,3)[2pt]{}[73]{}(7.3,2.8)[$x_{73}$]{} (8,1)[2pt]{}[81]{}(8,.7)[$x_{81}$]{} (8,2)[2pt]{}[82]{}(8.3,1.8)[$x_{82}$]{} (8,3)[2pt]{}[83]{}(8.3,2.8)[$x_{83}$]{} (9,1)[2pt]{}[91]{}(9,.7)[$x_{91}$]{} (9,2)[2pt]{}[92]{}(9.3,1.8)[$x_{92}$]{} (9,3)[2pt]{}[93]{}(9.3,2.8)[$x_{93}$]{} (10,1)[2pt]{}[101]{}(10,0.7)[$x_{(10)1}$]{} (10,2)[2pt]{}[102]{}(10.5,1.8)[$x_{(10)2}$]{} (10,3)[2pt]{}[103]{}(10.5,2.8)[$x_{(10)3}$]{} To obtain the upper bounds of $\gamma_{t}(G_{n,3})$ and $\gamma_{p}(G_{n,3})$, we set $$D=\{x_{i2}: i\equiv\,1,2 \,({\rm mod}\,5)\} \cup \{x_{j1}, x_{j3}: j\equiv\,4 \,({\rm mod}\,5)\},$$ where $1\le i\le n$. See Figure \[f2\], where $D$ consists of bold vertices. If $n\not\equiv \,3 \,({\rm mod}\,5)$, then $D$ is a total dominating set and $\gamma_{t}(G_{n,3})\leq |D|=\lceil \frac{4n}{5}\rceil$. If $n\equiv \,3 \,({\rm mod}\,5)$, then $D\cup \{x_{n2}\}$ is a total dominating set and $\gamma_{t}(G_{n,3})\leq |D|+1=\lceil \frac{4n}{5}\rceil$. Combining these facts with (\[e3.4\]), we have that $\gamma_{t}(G_{n,3})=\lceil \frac{4n}{5}\rceil$. If $n\equiv \,0,2,4 \,({\rm mod}\,5)$, then $D$ is a paired dominating set and $\gamma_{p}(G_{n,3})\leq |D|=\lceil \frac{4n}{5}\rceil$. If $n\equiv \,1\,({\rm mod}\,5)$, then $D\cup \{x_{n1}\}$ is a paired dominating set and $\gamma_{p}(G_{n,3})\leq |D|+1=\lceil \frac{4n}{5}\rceil+1$. If $n\equiv \,3 \,({\rm mod}\,5)$, then $D\cup \{x_{n1},x_{n2}\}$ is a paired dominating set and $\gamma_{p}(G_{n,3})\leq |D|+2=\lceil \frac{4n}{5}\rceil+1$. Since $\gamma_p(G_{n,3})\ge \gamma_t(G_{n,3})$ and $\gamma_p(G_{n,3})$ is even, $\gamma_p(G_{n,3})=\lceil\frac{4n}{5}\rceil$ if $n\equiv \,0,2,4 \,({\rm mod}\,5)$, and $\gamma_p(G_{n,3})=\lceil\frac{4n}{5}\rceil+1$ if $n\equiv\,1,3 \,({\rm mod}\,5)$. The theorem follows. Total and paired domination number of $G_{n,4}$ =============================================== In this section, we determine the exact values of $\gamma_{t}(G_{n,4})$ and $\gamma_{p}(G_{n,4})$, the latter has been announced by Brešar, Henning and Rall [@bhr05], but without proofs. \[lem4.2\] $\gamma_{p}(G_{n,4})=\gamma_{t}(G_{n,4})=n+1$ for $n\equiv 1,3~\,({\rm mod}~4)$. For $n\equiv 1~\,({\rm mod}~4)$, let $$D=\{x_{i1},x_{i2},x_{(i+2)3},x_{(i+2)4}:\ i\equiv 1\,({\rm mod}\, 4), i\ne n\} \cup \{x_{n1},x_{n2}\}.$$ Then $D$ is a paired dominating set of $G_{n,4}$ with cardinality $n+1$. For $n\equiv 3~\,({\rm mod}~4)$, $D=\{x_{i1},x_{i2},x_{(i+2)3},x_{(i+2)4}:~i\equiv 1~\,({\rm mod}~4)\}$ is a paired dominating set of $G_{n,4}$ with cardinality $n+1$. Thus, $\gamma_{t}(G_{n,4})\le \gamma_{p}(G_{n,4})\le n+1$ for $n\equiv 1,3~\,({\rm mod}~4)$. By Lemma \[lem2.2\], $\gamma_{t}(G_{n,4})\ge \frac{4n}{4}=n$. Now, we prove $\gamma_{t}(G_{n,4})\ge n+1$. Suppose to the contrary that $\gamma_{t}(G_{n,4})=n$. By Lemma \[lem2.2\], $G_{n,4}$ has an efficient total dominating set $D'$. By Lemma \[lem2.1\], $|D'|=n$ is even, a contradiction. Therefore $\gamma_{t}(G_{n,4})>n$, and hence $\gamma_{p}(G_{n,4})=\gamma_{t}(G_{n,4})=n+1$. \[lem4.3\] $\gamma_{t}(G_{n,4})\le \gamma_{p}(G_{n,4})\le n+2$ for $n\equiv 2~\,({\rm mod}~4)$. Let $$D=\{x_{i1},x_{i2},x_{(i+2)3},x_{(i+2)4}:\ i\equiv 1\,({\rm mod}\,4), i\le n-2\} \cup \{x_{(n-1)1},x_{(n-1)2},x_{n1},x_{n2}\}.$$ Then $D$ is a paired dominating set of $G_{n,4}$ with cardinality $n+2$. Thus, $\gamma_{t}(G_{n,4})\le \gamma_{p}(G_{n,4})\le n+2$. To prove $\gamma_{t}(G_{n,4})\ge n+2$ for $n\equiv 2~\,({\rm mod}~4)$, we need the following notations and two lemmas. Let $H_i^j=Y_{i}\cup Y_{i+1} \cup\ldots \cup Y_{i+j-1}$, and let $G_i^j$ be the graph obtained from $G_{n,4}-H_i^j$ by adding the edge-set $\{x_{(i-1)k}x_{(i+j)k}:\ 1\le k\le 4\}$, where the subscripts are modulo $n$. Clearly, $G_i^j\cong G_{n-j,4}$. \[lem4.4\] Let $D$ be a total dominating set of $G_{n,4}$. Then $|D\cap H_i^4|\ge 4$ for any $i$ with $1\le i\le n$. Moreover, if there exists some $i$ with $1\le i\le n$ such that $|N(v)\cap D|=1$ for any vertex $v$ in $H_i^4$, then $D'=D\setminus (D\cap H_i^4)$ is a total dominating set of $G_i^4$. Without loss of generality, assume $i=2$. It can be easy verified to dominate 8 vertices in $Y_3\cup Y_4$, at least $4$ vertices are needed, and hence $|D\cap H_2^4|\ge 4$. We now show the second assertion. Suppose to the contrary that $D'$ is not a total dominating set of $G_2^4$. Then there is a vertex $u$ in $Y_1\cup Y_6$ such that it is not dominated by $D'$, that is, $N_{G_2^4}(u)\cap D'=\emptyset$. Without loss of generality assume $u=x_{11}$. Then $x_{21}\in D$ and $x_{61}\notin D$. Also $x_{41}\notin D$ since $|N(x_{31})\cap D|=1$. Since $x_{33}$ should be dominated by $D$ and $|N(x_{33})\cap D|=1$, only one of $x_{32}$, $x_{34}$, $x_{23}$, and $x_{43}$ belongs to $D$. If $x_{32}\in D$ or $x_{34}\in D$, then $|N(x_{31})\cap D|\ge 2$, a contradiction. If $x_{23}\in D$, then $|N(x_{22})\cap D|\ge 2$, a contradiction. Thus, $x_{43}\in D$. Since $x_{51}$ should be dominated by $D$, $x_{52}\in D$ or $x_{54}\in D$. But then $|N(x_{53})\cap D|\ge 2$, a contradiction. Thus, $D'=D\setminus (D\cap H_2^4)$ is a total dominating set of $G_i^4$. \[lem4.5\] Let $D$ be a total dominating set of $G_{n,4}$. If $x_{ij}$ is dominated by two vertices $u,v\in D$, then there exists a vertex $w$ in $H_{i-1}^2$ or $H_i^2$ such that $|N(w)\cap D|\ge 2$. Without loss of generality, let $i=j=2$. If $u,v\in Y_2$, then assume $u=x_{21}$, $v=x_{23}$ and, hence, $|N(x_{24})\cap D|\ge 2$. If one of $u$ and $v$ is in $Y_2$ and another is in $Y_1\cup Y_3$, then without loss of generality assume $u=x_{21}\in Y_2$ and $v=x_{32}\in Y_3$. And then $|N(x_{31})\cap D|\ge 2$. If one of $u$ and $v$ is in $Y_1$ and another is in $Y_3$, then without loss of generality assume $u=x_{12}\in Y_2$ and $v=x_{32}\in Y_3$. Since $x_{24}$ should be dominated by $D$, let $s\in N(x_{24})\cap D$. It is clearly that $N(s)\cap N(u)\ne \emptyset$ or $N(s)\cap N(v)\ne \emptyset$, which implies that there exists a vertex $w\notin \{u,v\}$ in $H_{1}^2\cup H_{2}^2$ such that $|N(w)\cap D|\ge 2$. \[lem4.6\] $\gamma_{t}(G_{n,4})=\gamma_{p}(G_{n,4})= n+2$ for $n\equiv 2~\,({\rm mod}~4)$. By Lemma \[lem4.3\], we only need to show $\gamma_{t}(G_{n,4})\ge n+2$. To this end, let $n=4k+2$. We proceed by induction on $k\ge 1$. It is easy to verify that $\gamma_{t}(G_{6,4})=8$ and $\gamma_{t}(G_{10,4})=12$. The conclusion is true for $k=1,2$. Assume that the induction hypothesis is true for $k-1$ with $k\ge 3$. Let $D$ be a minimum total dominating set of $G_{n,4}$, where $n=4k+2$ for $k\ge 3$. Assume to the contrary that $|D|\le n+1$. Since any vertex $u$ can dominate at most 4 vertices in $G_{n,4}$ and $|V(G_{n,4})|=4n$, there are at most four vertices such that each of them is dominated by at least two vertices in $D$. We now prove that there exists some $i\in\{1,2,\ldots, n\}$ such that $|N(v)\cap D|=1$ for any vertex $v\in H_i^4$. There is nothing to do if there are at most three vertices such that each of them is dominated by at least two vertices since $n\ge 14$. Now, assume there are exactly four vertices such that each of them is dominated by at least two vertices. By Lemma \[lem4.5\], there exists two integers $s$ and $t$ with $1\le s,t\le n$ such that two of the four vertices are in $H_s^2$ and the other two are in $H_t^2$. Therefore, there exists an integer $i$ with $1\le i\le n$ such that for any vertex $v\in Y_i$, $|N(v)\cap D|=1$ since $n\ge 14$. By Lemma \[lem4.4\], $|D\cap H_i^4|\ge 4$ and $D'=D\setminus (D\cap H_i^4)$ is a total dominating set of $G_i^4\cong G_{n-4,4}$. By the inductive hypothesis, $|D'|\ge \gamma_t(G_{n-4,4})\ge n-2$. It follows that $$n+1\ge |D|=|D\cap H_i^4|+|D'|\ge 4+n-2=n+2,$$ a contradiction, which implies that $\gamma_{t}(G_{n,4})=|D|\ge n+2$. By the induction principle, the lemma follows. We state the above results as the following theorem. \[thm4.1\] For any integer $n\ge 3$, $$\gamma_{t}(G_{n,4})=\gamma_{p}(G_{n,4})=\left\{ \begin{array}{ll} n, & {\rm if}\ n\equiv 0\,({\rm mod}\,4);\\ n+1,& {\rm if}\ n\equiv 1,3\,({\rm mod}\,4);\\ n+2,& {\rm if}\ n\equiv 2\,({\rm mod}\,4). \end{array}\right.$$ Upper bounds of $\gamma_{p}(G_{n,m})$ for $n, m\ge 5$ ===================================================== The values of $\gamma_{t}(G_{n,m})$ and $\gamma_{p}(G_{n,m})$ for $m\in\{3,4\}$ have been determined in the above sections, but their values for $m\ge 5$ have been not determined yet. In this section, we present their upper bounds. Since $\gamma_t(G)\le \gamma_p(G)$ for any graph $G$ without isolated vertices, we establish upper bounds only for $\gamma_{p}(G_{n,m})$ if we can not obtain a smaller upper bound of $\gamma_t(G_{n,m})$ than that of $\gamma_{p}(G_{n,m})$. \[lem5.1\] $\gamma_t(G_{n,m})\le \gamma_t(G_{n+1,m})$ and $\gamma_p(G_{n,m})\le \gamma_p(G_{n+1,m})$. Let $D$ be a minimum paired (total) dominating set of $G_{n+1,m}$. If $D\cap Y_{n+1}=\emptyset$, then $D$ is also a paired (total) dominating set of $G_{n,m}$, and hence $\gamma_{p}(G_{n,m})\leq |D|$ ($\gamma_{t}(G_{n,m})\leq |D|$). Assume $D\cap Y_{n+1}\ne\emptyset$ below. Let $A=\{j|\ x_{(n+1)j}\in D\}$ and $B=\{j|\ x_{nj}\in D\}$. Then $D'=(D\setminus Y_{n+1})\cup \{x_{(n-1)j}|\ j\in A\cap B\} \cup \{x_{nj}|\ j\in A\setminus B\}$ is a total dominating set of $G_{n,m}$ and $|D'|\leq |D|$. Therefore $\gamma_{t}(G_{n,m})\leq \gamma_t(G_{n+1,m})$. The vertex set $D'$ may not be a paired dominating set of $G_{n,m}$, that means, the induced subgraph $G$ by $D'$ in $G_{n,m}$ may contains odd connected components. Let $p$ be the number of odd connected components in $G$. It is clear that $|D'|\le |D|-p$ by the construction of $D'$ from $D$. Therefore, we can obtain $D''$ by adding at most $p$ vertices to $D'$ such that the induced subgraph by $D''$ in $G_{n,m}$ does not contain odd connected components. Then $D''$ is a paired dominating set of $G_{n,m}$, and hence $\gamma_{p}(G_{n,m})\leq |D''|\leq |D|$. \[thm5.1\] $\gamma_p(G_{n,m})\le 4\lceil\frac{n}{4}\rceil\lceil\frac{m}{4}\rceil$. Let $n=4a-i$ and $m=4b-j$ where $0\le i,j\le 3$. By Lemma \[lem2.3\], $\gamma_p(G_{4a,4b})=4ab =4\lceil\frac{n}{4}\rceil\lceil\frac{m}{4}\rceil$. By Lemma \[lem5.1\], $\gamma_p(G_{m,n})\le \gamma_p(G_{4a,4b}) =4\lceil\frac{n}{4}\rceil\lceil\frac{m}{4}\rceil$. For $n,m\ge 5$, let $m\equiv \,a~({\rm mod}\,4)$ and $n\equiv \,b~({\rm mod}\,4)$ where $0 \le a,b\le 3$. We will establish some better bounds of $\gamma_{t}(G_{n,m})$ and $\gamma_{p}(G_{n,m})$ than those in Theorem \[thm5.1\] for some special $a$ and $b$. Without loss of generality, we can assume $b\ge a$ since $G_{n,m}\cong G_{m,n}$. Let $$D_e=\{x_{ij},x_{i(j+1)},x_{(i+2)(j+2)},x_{(i+2)(j+3)}: ~i,j\equiv \,1~({\rm mod}\,4)\},$$ where $1\le i\le n-2$, $1\le j\le m-2$, and $n,m\ge 5$. $\gamma_{p}(G_{n,m})\le \frac{(n+1)m}{4}$ for $m\equiv \,0~({\rm mod}\,4)$ and $n\equiv \,1~({\rm mod}\,4)$. Let $D=D_e\cup \{x_{nj},x_{n(j+1)}: ~j\equiv \,1~({\rm mod}\,4)\}$, where $1\le j\le m-2$. Then, it is easy to see that $D$ is a paired dominating set of $G_{n,m}$ with cardinality $\frac{(n+1)m}{4}$. Thus, $\gamma_{p}(G_{n,m})\le \frac{(n+1)m}{4}$. \[thm5.3\] $\gamma_{t}(G_{n,m})\le \frac{(n+1)(m+1)}{4}$ and $\gamma_{p}(G_{n,m})\le \frac{(n+1)(m+1)}{4}+1$ for $m,n\equiv \,1~({\rm mod}\,4)$. Let $D=D_e\cup\{x_{nj},x_{n(j+1)},x_{(i+1)(m-1)},x_{(i+2)m}: ~i,j\equiv \,1~({\rm mod}\,4)\}\cup \{x_{nm}\}$, where $1\le i\le n-2$ and $1\le j\le m-2$. Then, it is easy to see that $D$ is a total dominating set of $G_{n,m}$ with cardinality $\frac{(n+1)(m+1)}{4}$, and $D\cup \{x_{n(m-1)}\}$ is a paired dominating set of $G_{n,m}$ with cardinality $\frac{(n+1)(m+1)}{4}+1$. Thus, $\gamma_{t}(G_{n,m})\le \frac{(n+1)(m+1)}{4}$ and $\gamma_{p}(G_{n,m})\le \frac{(n+1)(m+1)}{4}+1$. \[thm5.4\] $\gamma_{t}(G_{n,m})\le \frac{(n+1)(m+1)}{4}-3$ and $\gamma_{p}(G_{n,m})\le \frac{(n+1)(m+1)}{4}-2$ for $m\equiv \,1~({\rm mod}\,4)$ and $n\equiv \,3~({\rm mod}\,4)$. Let $D=(D_e\cup\{x_{(i+1)(m-1)},x_{(i+2)m} :~i\equiv \,1~({\rm mod}\,4)\})\setminus \{x_{n(m-2)},x_{nm}\}$, where $1\le i\le n-2$. Then, $D$ is a paired dominating set of $G_{n,m}$ with cardinality $\frac{(n+1)(m+1)}{4}-2$, and $D\setminus\{x_{2(m-1)}\}$ is a total dominating set of $G_{n,m}$ with cardinality $\frac{(n+1)(m+1)}{4}-3$. Thus, $\gamma_{t}(G_{n,m})\le \frac{(n+1)(m+1)}{4}-3$ and $\gamma_{p}(G_{n,m})\le \frac{(n+1)(m+1)}{4}-2$. $\gamma_{t}(G_{n,m})\le \frac{(n+2)(m+1)}{4}-3$ and $\gamma_{p}(G_{n,m})\le \frac{(n+2)(m+1)}{4}-2$ for $m\equiv \,1~({\rm mod}\,4)$ and $n\equiv \,2~({\rm mod}\,4)$. By Lemma \[lem5.1\], $\gamma_t(G_{n,m})\le \gamma_t(G_{n+1,m})$ and $\gamma_p(G_{n,m})\le \gamma_p(G_{n+1,m})$. The corollary follows from Theorem \[thm5.4\]. \[thm5.5\] $\gamma_{p}(G_{n,m})\le \frac{(n+2)(m+2)}{4}-6$ for $m,n\equiv \,2~({\rm mod}\,4)$. Let $D=(D_e\cup\{x_{i(m-2)},x_{i(m-1)},x_{(i+2)(m-1)},x_{(i+2)m}:\ i\equiv \,1~({\rm mod}\,4)\}\cup \{x_{(n-1)j},$ $x_{(n-1)(j+1)},x_{n(j+2)},x_{n(j+3)}:\ j\equiv \,1~({\rm mod}\,4)\}\cup \{x_{n(m-1)}\}) \setminus \{x_{1(m-2)},$ $x_{1(m-1)},x_{n(m-3)}\}$, where $1\le i\le n-2$ and $1\le j\le m-2$. Then $D$ is a paired dominating set of $G_{n,m}$ with cardinality $\frac{(n+2)(m+2)}{4}-6$. Thus, $\gamma_{p}(G_{n,m})\le \frac{(n+2)(m+2)}{4}-6$. [99]{} B. Brešar, M. A. Henning, and D. F. Rall, Paired-domination of Cartesian products of graphs and rainbow domination. Electronic Notes in Discrete Mathematics, 22 (2005), 233-237. E. J. Cockayne, R. M. Dawes, S. T. Hedetniemi, Total domination in graphs. Networks, 10 (1980), 211-219. I. J. Dejter and O. Serra, Efficient dominating sets in Cayley graphs. Discrete Applied Mathematics, 129 (2003), 319-328. H. Gavlas and K. Schultz, Efficient open domination. Electronic Notes in Discrete Mathematics, 11 (2002), 681-691. S. Gravier, Total domination number of grid graphs. Discrete Applied Mathematics, 121 (2002), 119-128. T. W. Haynes, S. T. Hedetniemi, P. J. Slater, Fundamentals of Domination in Graphs. Marcel Dekker, New York, 1998. T. W. Haynes, S. T. Hedetniemi, P. J. Slater (Eds.), Domination in Graphs: Advanced Topics. Marcel Dekker, New York, 1998. M. A. Henning, A survey of selected recent results on total domination in graphs. Discrete Mathematics, 309(1) (2009), 32-63. T. W. Haynes and P. J. Slater, Paired-domination and the paired-domatic number. Congresses Numerantium, 109 (1995), 65-72. T. W. Haynes and P. J. Slater, Paired-domination in graphs. Networks, 32 (1998), 199-206. J. Huang and J.-M. Xu, The bondage numbers and efficient dominations of vertex-transitive graphs. Discrete Mathematics, 308(4) (2008), 571-582. J. Pfaff, R.C. Laskar, S.T. Hedetniemi, NP-completeness of total and connected domination and irredundance for bipartite graphs, Technical Report 428, Clemson University, Dept. Math. Sciences, 1983. K. E. Proffitt, T. W. Haynes, and P. J. Slater, Paired-domination in grid graphs. Congresses Numerantium, 150 (2001), 161-172. S. Klavžar and N. Seifter, Dominating cartesian products of cycles. Discrete Applied Mathematics, 59 (1995), 129-136. J.-M. Xu, Theory and Application of Graphs. Kluwer Academic Publishers, Dordrecht/Boston/London, 2003. [^1]: Corresponding author: [email protected] [^2]: The work was supported by NNSF of China (No. 11071233).

--- author: - 'Stefan Boettcher and Bruno Gon[ç]{}alves' bibliography: - '/Users/stb/Boettcher.bib' title: Anomalous Diffusion on the Hanoi Networks --- Introduction ============ The study of anomalous diffusion is an integral part in the analysis of transport processes in complex materials [@Shlesinger84; @Bouchaud90; @Metzler04; @Bollt05; @Condamin07]. Random environments often slow transport significantly, leading to sub-diffusive behavior. Much attention has thus been paid to model sub-diffusion on designed structures with some of the trappings of disordered materials, exemplified by Refs. [@Ogielski85; @Huberman85; @Havlin87; @Sibani87; @Maritan93; @Anh05]. Even self-organized critical processes can be shown to evolve sub-diffusively, controlled by the memory of all past events [@BoPa2]. On the other hand, tracer particles in rapidly driven fluids may exhibit super-diffusive behavior [@Solomon93], typically modeled in terms of Lévy flights [@Shlesinger93; @Metzler04]. Both regimes are self-similar, fractal generalizations of ordinary diffusion. In this Letter we consider diffusion on two new networks, which yield interesting realizations of super-diffusive behavior. Both of these networks were introduced to explore certain aspects of small-world behavior [@SWPRL]. Their key distinguishing characteristic is their ability to mix a geometric backbone, i. e. a one-dimensional lattice, with small-world links in a non-random, hierarchical structure. In particular, these networks permit a smooth interpolation between finite-dimensional and mean-field properties, which is absent from the renormalization group (RG) due to Migdal and Kadanoff, for instance [@Plischke94]. The unusual structure of these networks recasts the RG into a novel form, where the equations are essentially those of a one-dimensional model in which the complex hierarchy enters at each RG-step as a (previously unrenormalized) source term. This effect is most apparent in the real-space RG for the Ising models discussed in Ref. [@SWPRL]. It is obscured in our dynamic RG treatment below, since these walks are always embedded on the lattice backbone. On the practical side, their regular, hierarchical structure allows for easily engineered implementations, say, to efficiently synchronize communication networks [@SWPRL]. Regarding diffusion, one of the networks proves to be merely an incarnation of a Weierstrass random walk found for Lévy flights [@Shlesinger93] with ballistic transport, while the other network shows highly non-trivial transport properties, very much unlike a Lévy flight, as revealed by our exact RG treatment. The fixed point equations are singular and exhibit a boundary layer [@BO]. It provides a tangible case of a singularity in the RG [@Griffiths78; @Maritan93] that is easily interpreted in terms of the physics. Generating Hanoi Networks {#generating} ========================= In the Tower-of-Hanoi problem [@Sedgewick04], disks of increasing size, labeled $i=1$ to $k$ from top to bottom, are stacked up and have to be moved in a Sisyphean task into a 2nd stack, disk-by-disk, while at no time a larger disk can be placed onto a smaller one. To this end, a 3rd stack is provided as overflow. First, disk 1 moves to the overflow and disk 2 onto the 2nd stack, followed by disk 1 on top of 2. Now, disk 3 can move to the overflow, disk 1 back onto disk 4, disk 2 onto 3, and 1 onto 2. Now we have a new stack of disks 1, 2, and 3 in prefect order, and only $k-3$ more disk to go! But note the values of disk-label $i$ in the sequence of moves: 1-2-1-3-1-2-1-4-1-2-1-3-1-2-1-5-..., and so on. Inspired by models of ultra-slow diffusion [@Ogielski85; @Huberman85], we create our networks as follows. First, we lay out this sequence on a $1d-$line of nearest-neighbor connected sites labeled from $n=1$ to $n=L=2^{k}-1$ (the number of moves required to finish the problem). In general, any site $n(\not=0)$ can be described uniquely by $$n=2^{i-1}(2j+1), \label{neq}$$ where $i$ is the label of the disk moved at step $n$ in the sequence above and $j=0,1,2\ldots$. To wit, let us further connect each site $n$ to the closest site $n'$ that is $2^i$ steps away and possesses the same value of $i$, both only having a site of value at most $i+1$ between them. According to the sequence, site $n=1$ (with $i=1$) is now also connected to $n'=3$, 5 to 7, 9 to 11, etc. For sites with $i=2$, site $n=2$ now also connects to $n'=6$, 10 to 14, 18 to 22, etc, and so on also for $i>2$. As a result, we get the network depicted in Fig. \[fig:3hanoi\] that we call HN3. Except at the boundary, each site now has three neighbors, left and right along the $1d$ ”backbone” and a 3rd link to a site $2^i$ steps away. If we further connect each site also to a fourth site $2^i$ steps in the other direction and allow $j=0,\pm1,\pm2,\ldots$, we obtain the network in Fig. \[fig:4hanoi\], called HN4, where each site now has four neighbors. ![Depiction of the planar “Tower-of-Hanoi” network HN3. Here, the $1d-$backbone of sites extends over $0<n<\infty$.[]{data-label="fig:3hanoi"}](hanoi3) ![Depiction of the “Tower-of-Hanoi” network HN4. Here, the $1d-$backbone of sites extends over $-\infty<n<\infty$. The site $n=0$, not covered by Eq. (\[neq\]), is special and is connected to itself here.[]{data-label="fig:4hanoi"}](hanoi4) These new “Tower-of-Hanoi” networks – a mix of local, geometric connections and “small-world”-like long-range jumps – has fascinating properties. It is recursively defined with obvious fractal features. A collection of the structural and dynamic features of HN3 and HN4 are discussed in Ref. [@SWPRL], such as results for Ising models and synchronization. Diffusion on the Hanoi Networks {#diffusion} =============================== To model diffusion on these networks, we study simple random walks with nearest-neighbor jumps along the available links, but using the one-dimensional lattice backbone as our metric to measure distances, which implies a fractal dimension of $d_f=1$. Embedded in that space, we want to calculate the non-trivial diffusion exponent $d_w$ defined by the asymptotic mean-square displacement $$\left\langle r^{2}\right\rangle \sim t^{2/d_{w}}. \label{MSDeq}$$ A more extensive treatment yielding also first-return probabilities is given elsewhere [@SWlong]. First, we consider a random walk on HN4. The “master-equation” [@Redner01] for the probability of the walker to be at site $n$, as defined in Eq. (\[neq\]), at time $t$ is given by $$\begin{aligned} {\cal P}_{n,t}&=& \frac{1-p}{2}\left[{\cal P}_{n-1,t-1}+{\cal P}_{n+1,t-1}\right]\nonumber\\ \nonumber\\ &&\quad+\frac{p}{2}\left[{\cal P}_{n-2^i,t-1}+{\cal P}_{n+2^i,t-1}\right], \label{eq:4RW}\end{aligned}$$ where $p$ is the probability to make a long-range jump. (Throughout this Letter, we considered $p$ uniform, independent of $n$ or $t$). A detailed treatment of this equation in terms of generating functions is quite involved and proved fruitless, as will be shown elsewhere [@SWlong]. Instead, we note that the long-time behavior is dominated by the long-range jumps, as discussed below for HN3. To simplify matters, we set $p=1/2$ here, although any other finite probability should lead to the same result. We make an “annealed” approximation, i. e., we assume that we happen to be at some site $n$ in Eq. (\[neq\]) with probability $1/2^i$, corresponding to the relative frequency of such a site, yet independent of update-time or history. This ignores the fact that in the network geometry a long jump of length $2^i$ can be followed *only* by another jump of that length or a jump of unit length, and that many intervening steps are necessary to make a jump of length $2^{i+1}$, for instance. Here, at each instant the walker jumps a distance $2^i$ left or right irrespectively with probability $1/2^i$, and we can write $$\begin{aligned} {\cal P}_{n,t} & = &\sum_{n'}T_{n,n'}{\cal P}_{n',t-1} \label{eq:Transfer}\end{aligned}$$ with $$\begin{aligned} T_{n,n'}&=&\frac{a-1}{2a}\sum_{i=0}^\infty a^{-i}\left(\delta_{n-n',b^i}+\delta_{n-n',-b^i}\right), \label{eq:T}\end{aligned}$$ where $a=b=2$. Eqs. (\[eq:Transfer\]-\[eq:T\]) are identical to the Weierstrass random walk discussed in Refs. [@Hughes81; @Shlesinger93] for arbitrary $1<a<b^2$. There, it was shown that $d_w=\ln(a)/\ln(b)$, which leads to the conclusion that $d_w=1$ in Eq. (\[MSDeq\]) for HN4, as has been predicted (with logarithmic corrections) on the basis of numerical simulations in Ref. [@SWPRL]. These logarithmic corrections are typical for walks with marginal recurrence, which typically occurs when $d_w=d_f$, such as for ordinary diffusion in two dimensions [@Bollt05]. For HN3, the master-equation in the bulk reads for $$\begin{aligned} {\cal P}_{n,t} & = & \frac{1-p}{2}\left[{\cal P}_{n-1,t-1}+{\cal P}_{n+1,t-1}\right]+p\,{\cal P}_{n',t-1},\nonumber\\ \nonumber\\ &&\quad n'=\begin{cases} n+2^{i},& j~ {\rm even,}\\ \\ n-2^{i},& j~ {\rm odd,}\end{cases} \label{eq:3RW}\end{aligned}$$ with $n$ as in Eq. (\[neq\]), and $p$ as before. In the RG [@Redner01; @Kahng89] solution of Eq. (\[eq:3RW\]), at each step we eliminate all odd sites, i. e., those sites with $i=0$ in Eq. (\[neq\]). As shown in Fig. \[fig:RG3RW\], the elementary unit of sites effected is centered at all sites $n$ having $i=1$ in Eq. (\[neq\]). We know that such a site $n$ is surrounded by two sites of odd index, which are mutually linked. Furthermore, $n$ is linked by a long-distance jump to a site also of type $i=1$ at $n\pm4$ in the neighboring elementary unit, where the direction does not matter here. The sites $n\pm2$, which are shared at the boundary between such neighboring units also have even index, but their value of $i\geq2$ is indetermined and irrelevant for the immediate RG step, as they have a long-distance jump to some sites $m_\pm$ at least eight sites away. Using a standard generating function [@Redner01], $$\begin{aligned} x_{n}(z)&=&\sum_{t=0}^{\infty}{\cal P}_{n,t}\,z^{t}, \label{eq:generator}\end{aligned}$$ yields for the five sites inside the elementary unit centered at $n$: $$\begin{aligned} x_{n}&=&a\left(x_{n-1}+x_{n+1}\right)\nonumber\\ &&\qquad+c\left(x_{n-2}+x_{n+2}\right)+p_{2}\,x_{n\pm4}, \nonumber\\ \nonumber\\ x_{n\pm1} & = & b\left(x_{n}+x_{n\pm2}\right)+p_{1}\,x_{n\mp1}, \label{eq:transformed3RW}\\ \nonumber\\ x_{n\pm2}&=&a\left(x_{n\pm1}+x_{n\pm3}\right)\nonumber\\ &&\qquad+c\left(x_{n}+x_{n\pm4}\right)+p_{2}\,x_{m_\pm}, \nonumber\end{aligned}$$ where we have absorbed the parameters $p$ and $z$ into general “hoping rates” that are initially $a^{(0)}=b^{(0)}=\frac{z}{2}(1-p)$, $c^{(0)}=0$, and $p_{1}^{(0)}=p_{2}^{(0)}=zp$. The RG update step consist of eliminating from these five equations those two that refer to an odd index, $n\pm1$. After some algebra, we obtain $$\begin{aligned} x_{n}&=&b'\left(x_{n-2}+x_{n+2}\right)+p_{1}'\,x_{n\pm4}, \nonumber\\ \nonumber\\ x_{n\pm2}&=&a'\left(x_{n}+x_{n\pm4}\right)\label{eq:RGafter3RW}\\ &&\qquad+c'\left(x_{n\mp2}+x_{n\pm6}\right)+p_{2}'\,x_{m_\pm}, \nonumber\end{aligned}$$ with $$\begin{aligned} a' & = & \frac{\left[ab+c\left(1-p_{1}\right)\right]\left(1+p_{1}\right)}{1-p_{1}^{2}-2ab},\nonumber\\ \nonumber\\ b' &=&\frac{ab+c\left(1-p_{1}\right)}{1-p_{1}-2ab},\nonumber\\ \nonumber\\ c' &=&\frac{abp_{1}}{1-p_{1}^{2}-2ab}, \label{eq:RG3RWfp}\\ \nonumber\\ p_{1}'&=&\frac{p_{2}\left(1-p_{1}\right)}{1-p_{1}-2ab},\nonumber\\ \nonumber\\ p_{2}'&=&\frac{p_{2}\left(1-p_{1}^{2}\right)}{1-p_{1}^{2}-2ab}.\nonumber \end{aligned}$$ If for all sites $l=n,n\pm2,n\pm4,\ldots$ in Eq. (\[eq:RGafter3RW\]) we further identify[^1] $x_{l}=C\,x_{l/2}'$, we note that the primed equations coincide with the unprimed ones in Eqs. (\[eq:transformed3RW\]). Hence, the RG recursion equations in (\[eq:RG3RWfp\]) are *exact* at any step $k$ of the RG, where unprimed quantities refer to the $k$th recursion and primed ones to $k+1$. ![Depiction of the (exact) RG step for random walks on HN3. Hopping rates from one site to another along a link are labeled at the originating site. The RG step consists of tracing out odd-labeled variables $x_{n\pm1}$ in the top graph and expressing the renormalized rates $(a',b',c',p_{1}',p_{2}')$ on the right in terms of the previous ones $(a,b,c,p_{1},p_{2})$ on the bottom. The node $x_{n}$, bridged by a (dotted) link between $x_{n-1}$ and $x_{n+1}$, is special as it *must* have $n=2(2j+1)$ and is to be decimated at the following RG step, justifying the designation of $p_{1}'$. Note that the original graph in Fig. \[fig:3hanoi\] does not have the green, dashed links with hopping rates $(c,c')$, which *emerge* during the RG recursion. []{data-label="fig:RG3RW"}](RG3RW "fig:") ![Depiction of the (exact) RG step for random walks on HN3. Hopping rates from one site to another along a link are labeled at the originating site. The RG step consists of tracing out odd-labeled variables $x_{n\pm1}$ in the top graph and expressing the renormalized rates $(a',b',c',p_{1}',p_{2}')$ on the right in terms of the previous ones $(a,b,c,p_{1},p_{2})$ on the bottom. The node $x_{n}$, bridged by a (dotted) link between $x_{n-1}$ and $x_{n+1}$, is special as it *must* have $n=2(2j+1)$ and is to be decimated at the following RG step, justifying the designation of $p_{1}'$. Note that the original graph in Fig. \[fig:3hanoi\] does not have the green, dashed links with hopping rates $(c,c')$, which *emerge* during the RG recursion. []{data-label="fig:RG3RW"}](RG3RW_after "fig:") ![Plot of the results from simulations of the mean-square displacement of random walks on HN3 displayed in Fig. \[fig:3hanoi\]. More than $10^7$ walks were evolved up to $t_{\rm max}=10^6$ steps to measure $\langle r^2\rangle_t$. The data is extrapolated according to Eq. (\[MSDeq\]), such that the intercept on the vertical axis determines $d_w$ asymptotically. The exact result from Eq. (\[eq:D-expo\]) is indicated by the arrow. []{data-label="fig:MSDextra"}](3H_MSDextra) ![Plot of the probability $P_F(\Delta t)$ of first returns to the origin after $\Delta t$ update steps on a system of unlimited size. Data was collected for three different walks on HN3 with $p=0.1$ (circles), $p=0.3$ (squares), and $p=0.8$ (diamonds). The data with the smallest and largest $p$ exhibit strong transient effects. The exact result in Eq. (\[eq:tau-expo\]), $\mu=1.234\ldots$, is indicated by the dashed line. []{data-label="fig:FR"}](P_F.eps) Solving Eqs. (\[eq:RG3RWfp\]) algebraically at infinite time \[which corresponds to the limit $z\nearrow1$, see Eq. (\[eq:generator\])\] and for $k+1\sim k\to\infty$ (by dropping the prime on all left-hand parameters), we – apparently – obtain only two fixed points at $a=b=1/2$ and $c=p_{1}=p_{2}=0$, and $a=b=c=0$ and $p_{1}=p_{2}=1$. The first fixed point corresponds to an ordinary $1d$ walk without long-range jumps, in the second there is no hopping along the $1d$-backbone at all and the walker stays *confined,* jumping back-and-forth within a single, long-range jump. Yet, both fixed points are *unstable* with respect to small perturbations in the initial parameters. Starting with any positive probability $p$ for long-range jumps, those dominate over the $1d$ walk at long times. Paradoxically, exclusive long-range jumps found at the 2nd fixed point lead to confinement, itself undermined by *any* positive probability to escape along the $1d-$line, allowing to reach even longer jumps. Instead, the process gets attracted to a third, stable fixed point hidden inside a singular *boundary layer*[@BO] in the renormalization group equations (\[eq:RG3RWfp\]) near the confined state. We have to account for the asymptotic boundary layer in Eqs. (\[eq:RG3RWfp\]) with the Ansatz $y\sim A_y\alpha^{-k}\to0$ for $y\in\{a,b,c,1-p_{1},1-p_{2}\}$, where $k\to\infty$ refers to the $k$th RG step. Choosing $A_a=1$, the other $A_y$’s and the eigenvalues $\alpha$ are determined *self-consistently*. The only eigenvalue satisfying the requirement $\alpha>1$ is $\alpha=2/\phi$. Here, $\phi=\left(\sqrt{5}+1\right)/2=1.6180\ldots$ is the legendary “golden ratio” [@Livio03] defined by Euclid [@Euclid]. Hence, every renormalization of network size, $L\to L'=2L$, has to be matched by a rescaling of hopping rates with $\alpha=2/\phi$ to keep motion along the $1d$-backbone finite and prevent confinement. Extending the analysis to include finite-time corrections (i. e., $1-z\ll1$), we extend the above Ansatz to $$\begin{aligned} y^{(k)}&\sim& A_{y}\alpha^{-k}\left\{1+\left(1-z\right)B_{y}\beta^{k}+\ldots\right\} \label{eq:Ansatz}\end{aligned}$$ for all $y\in\{a,b,c,1-p_{1},1-p_{2}\}$. In addition to the leading-order constants $A_{y}$ and $\alpha$, also the next-leading constants are determined self-consistently, and we extract uniquely $\beta=2\alpha$. Accordingly, time re-scales now as $$\begin{aligned} T &\to & T'=2\alpha T, \label{eq:Tscal}\end{aligned}$$ and we obtain from Eq. (\[MSDeq\]) with $T\sim L^{d_{w}}$ for the diffusion exponent for HN3 $$\begin{aligned} d_{w}&=&2-\log_2\phi=1.30576\ldots. \label{eq:D-expo}\end{aligned}$$ The result for $d_{w}$ is in excellent agreement with our simulations, as shown in Fig. \[fig:MSDextra\]. Using the methods from Ref. [@Redner01], a far more extensive treatment shows [@SWlong] that the exponent $\mu$ for the probability distribution, $P_F(\Delta t)\sim\Delta t^{-\mu}$, of first-return times $\Delta t$ is given by $$\begin{aligned} \mu&=&2-\frac{1}{d_w}=1.2342\ldots. \label{eq:tau-expo}\end{aligned}$$ The relation between $\mu$ and $d_w$ is typical also for Lévy flights [@Metzler04], and the result is again borne out by our simulations, see Fig. \[fig:FR\]. It is remarkable, though, that the more detailed analysis in Ref. [@SWlong] also shows that walks on HN3 are *not* uniformly recurrent, as the result of $d_w>d_f=1$ here would indicate. That calculation shows that only sites on the highest level of the hierarchy are recurrent. While all other sites do share the same exponent $\mu$ in Eq. (\[eq:tau-expo\]) for actual recurrences, they have a diminishing return probability with decreasing levels in the infinite system limit. This is clearly a consequence of walkers being nearly-confined to the highest levels of the hierarchy at long times, as expressed by the boundary layer. We finally contrast the behavior of HN4 discovered above with the analysis of HN3. Clearly, when long-range jumps are interconnected as in HN4, there is no confinement, the boundary layer disappears \[which would be similar to $\alpha=1$ in Eqs. (\[eq:Ansatz\]-\[eq:Tscal\]) for HN3\], and diffusion spreads ballistically, $d_{w}=1$. Our numerical studies, and the similarity to Weierstrass random walks [@Hughes81], further supports that $\mu$ for walks on HN4 is also given by Eq. (\[eq:tau-expo\]), leading to $\mu=1$. This scaling is again indicative of a marginally recurrent state and requires logarithmic corrections for proper normalization, as was observed in simulations [@SWPRL]. Conclusions =========== We conclude with two further considerations. First, in reference to the potential of these networks to interpolate between long-range and a finite-dimensional behavior that we invoked in the introduction, we just add the following illustrative remark: If the probability to undertake a long-distance jump would be distance-dependent in each level of the hierarchy, we can obtain immediately a new result for walks on HN4 in the annealed approximation above. Let $p$ vary with a power of the backbone-distance between sites, say $p\propto r^{-\sigma}$, then for each level $i$ of the hierarchy it is $r=r_i=2^i$, i. e. $p=p_i\propto2^{-i\sigma}$, and the weight to make a jump of length $2^i$ in Eq. (\[eq:T\]) is given by $a=2^{1+\sigma}$, leading to $d_w=1+\sigma$. As can be expected, the analysis of the Weierstrass walk breaks down for $\sigma\to1^-$, at which point the long-range jumps become irrelevant and we obtain the results for ordinary $1d$ diffusion. Hence, $0\leq\sigma\leq1$ interpolates analytically between long-range and one-dimensional behavior of the random walk on HN4. (In fact, the analysis formally can be extended to $0>\sigma>-1$, where the walk becomes non-recurrent and is dominated by high levels in the hierarchy. Yet, the annealed approximation that assumes free transitions between different levels of the hierarchy is bound to fail.) ![Plot of the mean first-passage times $\langle T\rangle$ for walks as a function of distance $r$ between starting and target site for HN3 (top) and HN4 (bottom). The data has been scaled according to Eq. (\[eq:condamin\] such that the data collapses asymptotically onto a line that only depends on the model but that is independent of system size $N$. This collapse is excellent for HN3, it is somewhat weaker for HN4. Although all system sizes lead to linear forms in $\ln(r)$, their slope apparently varies with $N$. This could be caused by logarithmic scaling corrections to the slope, or by the lack of asymptotic behavior at the available system sizes $N$. []{data-label="fig:mfpt"}](mfpt_HN3linear.eps) ![Plot of the mean first-passage times $\langle T\rangle$ for walks as a function of distance $r$ between starting and target site for HN3 (top) and HN4 (bottom). The data has been scaled according to Eq. (\[eq:condamin\] such that the data collapses asymptotically onto a line that only depends on the model but that is independent of system size $N$. This collapse is excellent for HN3, it is somewhat weaker for HN4. Although all system sizes lead to linear forms in $\ln(r)$, their slope apparently varies with $N$. This could be caused by logarithmic scaling corrections to the slope, or by the lack of asymptotic behavior at the available system sizes $N$. []{data-label="fig:mfpt"}](mfpt_HN4center.eps) Our final consideration concerns a recent proposal by Condamin et al [@Condamin07] for a very general scaling form for mean first-passage times $\langle T\rangle$ for walks as a function of distance $r$ between starting and target site on a graph (lattice, network, etc.) of $N$ sites. Based on $d_w-d_f$, Ref. [@Condamin07] determined that $$\begin{aligned} \langle T\rangle\sim N\begin{cases} A + B r^{d_w-d_f},& d_w>d_f,\\ A + B \ln(r),& d_w=d_f,\\ A-Br^{d_w-d_f},&d_w<d_f, \end{cases} \label{eq:condamin}\end{aligned}$$ for *fixed* constants $A,B$, independent of $N$ and $r$. Our networks provide a non-trivial set of exponents to explore these relations with simple simulations. In particular, HN3 with $d_w-d_f=0.30576$ provides an instance for a powerlaw-divergent mean first-passage time, while HN4 exactly probes the marginal case $d_w=d_f(=1)$ with a logarithmic divergence of $\langle T\rangle$. When plotting $\langle T\rangle/N$ as a function of $r^{d_w-d_f}$ or $\ln(r)$, resp., in Figs. \[fig:mfpt\] we indeed obtain a universal straight line over many orders of magnitude in $N$ and $r$, indicative of fixed $A,B$. We like to thank F. Family, S. Redner, S. Coppersmith, and M. Shlesinger for helpful discussions. We thank the referee for calling our attention to Ref. [@Condamin07]. [^1]: As we will show elsewhere [@SWlong], the constant $C$ is determined when initial and boundary conditions are considered, as is essential for the case of first transit and return times [@Redner01].

--- abstract: 'Whilst a straightforward consequence of the formalism of non-relativistic quantum mechanics, the phenomenon of quantum teleportation has given rise to considerable puzzlement. In this paper, the teleportation protocol is reviewed and these puzzles dispelled. It is suggested that they arise from two primary sources: 1) the familiar error of hypostatizing an abstract noun (in this case, ‘information’) and 2) failure to differentiate interpretation dependent from interpretation *in*dependent features of quantum mechanics. A subsidiary source of error, the *simulation fallacy*, is also identified. The resolution presented of the puzzles of teleportation illustrates the benefits of paying due attention to the logical status of ‘information’ as an abstract noun.' author: - | Christopher G. Timpson[^1]\ *Division of History and Philosophy of Science,*\ *School of Philosophy, University of Leeds,*\ *Leeds, LS2 9JT, UK* title: The Grammar of Teleportation --- Introduction ============ ‘The questions “What is length?", “What is meaning?", “What is the number one?" etc. produce in us a mental cramp. We feel that we can’t point to anything in reply to them and yet ought to point to something. (We are up against one of the great sources of philosophical bewilderment: a substantive makes us look for a thing that corresponds to it.)’ @wittgenstein:blue Quantum teleportation (@teleportation) is one of the singular fruits of the burgeoning field of quantum information theory. In this theory, one seeks to describe and make use of the distinctive possibilities for information processing and communication that quantum systems allow: quantum features like entanglement and non-commutativity are put to work. Teleportation is perhaps the most striking example of the use of entanglement in assisting communication and it illustrates vividly several of the general features associated with quantum information protocols, most notably the fact that entanglement (a characteristically quantum property) serves as an important resource, and that unknown quantum states cannot be cloned (@dieks [@wootters:zurek]). Although a straightforward consequence of the formalism of non-relativistic quantum mechanics, teleportation has given rise to some confusion and to a good deal of controversy. In this paper I review the main lines of controversy (Sections \[protocol\] and \[puzzles\]) and seek to dispell the confusion that has surrounded the interpretation of the protocol. I will suggest (Section \[dissolving\]) that puzzlement has generally arisen as a consequence of a familiar philosophical error—in fact the one that Wittgenstein famously warns us of in the *Blue Book*—that is, the error of assuming that every grammatical substantive is a referring term. Here the culprit is the word ‘information’. ‘Information’ is an abstract (mass) noun and hence does not refer to a spatio-temporal particular, to an entity or a substance[^2]. It follows that one should not be seeking in an information theoretic protocol—quantum or otherwise—for some particular, denoted by ‘the information’, whose path one is to follow, but rather concentrating on the physical processes by which the information is transmitted, that is, by which the end result of the protocol is brought about. Once this is recognised, I suggest, much of our confusion is dispelled. (A subsidiary source of difficulty—what I term the *simulation fallacy*—is also remarked upon.) With this clarification in place, the other major source of controversy is thrown into relief: just what *are* the physical processes by which teleportation is effected? This is, in fact, a relatively straightforward question; but it is a question that will find a different answer depending on what interpretation of quantum mechanics one wishes to adopt (Section \[interpretations\]) a point which has not been sufficiently recognised to date. The central theme of this paper is that the conceptual puzzles surrounding teleportation arise from thinking about information in the wrong way. The converse point holds too: the clarification of these puzzles clearly illustrates the value of recognising the logico-grammatical status of ‘information’ as an abstract noun. Let us begin by briefly reviewing the teleportation protocol[^3]. The quantum teleportation protocol {#protocol} ================================== In the teleportation protocol we consider two parties, Alice and Bob, who are widely separated, but each of whom possess one member of a pair of particles in a maximally entangled state. Alice is presented with a system in some unknown quantum state, and her aim is to transmit this state to Bob. In the standard example, Alice and Bob share one of the four Bell states (Table \[bellstates\]) and she is presented with a spin-1/2 system in the unknown state ${|\chi\rangle _{}}=\alpha{|{\!\uparrow}\rangle _{}}+\beta{|{\!\downarrow}\rangle _{}}$. $$\begin{array}{c} {|\phi^{+}\rangle _{}}=1/\sqrt{2}({|{\!\uparrow}\rangle _{}}{|{\!\uparrow}\rangle _{}}+{|{\!\downarrow}\rangle _{}}{|{\!\downarrow}\rangle _{}}),\\ {|\phi^{-}\rangle _{}}=1/\sqrt{2}({|{\!\uparrow}\rangle _{}}{|{\!\uparrow}\rangle _{}}-{|{\!\downarrow}\rangle _{}}{|{\!\downarrow}\rangle _{}}),\\ {|\psi^{+}\rangle _{}}=1/\sqrt{2}({|{\!\uparrow}\rangle _{}}{|{\!\downarrow}\rangle _{}}+{|{\!\downarrow}\rangle _{}}{|{\!\uparrow}\rangle _{}}),\\ {|\psi^{-}\rangle _{}}=1/\sqrt{2}({|{\!\uparrow}\rangle _{}}{|{\!\downarrow}\rangle _{}}-{|{\!\downarrow}\rangle _{}}{|{\!\uparrow}\rangle _{}}). \end{array}$$ By performing a suitable joint measurement on her half of the entangled pair and the system whose state she is trying to transmit (in this example, a measurement in the Bell basis), Alice can flip the state of Bob’s half of the entangled pair into a state that differs from ${|\chi\rangle _{}}$ by one of four unitary transformations, depending on what the outcome of her measurement was. If a record of the outcome of Alice’s measurement is then sent to Bob, he may perform the required operation to obtain a system in the state Alice was trying to send (Fig. [\[telep1\]]{}). ![Teleportation. A pair of systems is first prepared in an entangled state and shared between Alice and Bob, who are widely spatially separated. Alice also possesses a system in an unknown state ${|\chi\rangle _{}}$. Once Alice performs her Bell-basis measurement, two classical bits recording the outcome are sent to Bob, who may then perform the required conditional operation to obtain a system in the unknown state ${|\chi\rangle _{}}$. (Continuous black lines represent qubits, dotted lines represent classical bits. Time runs along the horizontal axis.)\[telep1\]](leedsfig1.eps) The end result of the protocol is that Bob obtains a system in the state ${|\chi\rangle _{}}$, with nothing that bears any relation to the identity of this state having traversed the space between him and Alice. Only two classical bits recording the outcome of Alice’s measurement were sent between them; and the values of these bits are completely random, with no dependence on the parameters $\alpha$ and $\beta$. Meanwhile, no trace of the identity of the unknown state remains in Alice’s region, as is required in accordance with the no-cloning theorem (the state of her original system will usually now be maximally mixed). The state has ‘disappeared’ from Alice’s region and ‘reappeared’ in Bob’s, hence the use of the term *teleportation* for this phenomenon. To fix the process in our minds, let’s review how the standard example goes. We begin with system 1 in the unknown state ${|\chi\rangle _{}}$ and with Alice and Bob sharing a pair of systems (2 and 3) in, say, the singlet state ${|\psi^{-}\rangle _{}}$. The total state of the three systems at the beginning of the protocol is therefore simply $$\label{start} {|\chi\rangle _{1}}{|\psi^{-}\rangle _{23}} = \frac{1}{\sqrt 2} \bigl(\alpha {|{\!\uparrow}\rangle _{1}} + \beta{|{\!\downarrow}\rangle _{1}}\bigr)\bigl({|{\!\uparrow}\rangle _{2}}{|{\!\downarrow}\rangle _{3}}-{|{\!\downarrow}\rangle _{2}}{|{\!\uparrow}\rangle _{3}}\bigr).$$ Notice that at this stage, the state of system 1 factorises from that of systems 2 and 3; and so in particular, the state of Bob’s system is independent of $\alpha$ and $\beta$. We may re-write this initial state in a suggestive manner, though: $$\begin{aligned} {|\chi\rangle _{1}}{|\psi^{-}\rangle _{23}} & = \frac{1}{\sqrt{2}}\biggl(\alpha{|{\!\uparrow}\rangle _{1}}{|{\!\uparrow}\rangle _{2}}{|{\!\downarrow}\rangle _{3}} + \beta{|{\!\downarrow}\rangle _{1}}{|{\!\uparrow}\rangle _{2}}{|{\!\downarrow}\rangle _{3}} - \alpha{|{\!\uparrow}\rangle _{1}}{|{\!\downarrow}\rangle _{2}}{|{\!\uparrow}\rangle _{3}} - \beta{|{\!\downarrow}\rangle _{1}}{|{\!\downarrow}\rangle _{2}}{|{\!\uparrow}\rangle _{3}}\biggr) \\ \begin{split} & = \frac{1}{2}\biggl({|\phi^{+}\rangle _{12}} \bigl(\alpha {|{\!\downarrow}\rangle _{3}} - \beta{|{\!\uparrow}\rangle _{3}}\bigr)+{|\phi^{-}\rangle _{12}} \bigl(\alpha {|{\!\downarrow}\rangle _{3}} + \beta{|{\!\uparrow}\rangle _{3}}\bigr) \\ & \phantom{{|\phi^{+}\rangle _{}} }+ {|\psi^{+}\rangle _{12}} \bigl(-\alpha {|{\!\uparrow}\rangle _{3}} + \beta{|{\!\downarrow}\rangle _{3}}\bigr) + {|\psi^{-}\rangle _{12}} \bigl(-\alpha {|{\!\uparrow}\rangle _{3}} - \beta{|{\!\downarrow}\rangle _{3}}\bigr)\biggr). \end{split} \label{rewrite1}\end{aligned}$$ The basis used is the set $$\{{|\phi^{\pm}\rangle _{12}}{|{\!\uparrow}\rangle _{3}},\,{|\phi^{\pm}\rangle _{12}}{|{\!\downarrow}\rangle _{3}},\, {|\psi^{\pm}\rangle _{12}}{|{\!\uparrow}\rangle _{3}}, \, {|\psi^{\pm}\rangle _{12}}{|{\!\downarrow}\rangle _{3}}\},$$ that is, we have chosen (as we may) to express the total state of systems 1,2 and 3 using an entangled basis for systems 1 and 2, even though these systems are quite independent. But so far, of course, all we have done is re-written the state in a particular way; nothing has changed physically and it is still the case that it is really systems 2 and 3 that are entangled and wholly independent of system 1, in its unknown state. Looking closely at (\[rewrite1\]) we notice that the relative states of system 3 with respect to particular Bell basis states for 1 and 2 have a very simple relation to the initial unknown state ${|\chi\rangle _{}}$; they differ from ${|\chi\rangle _{}}$ by one of four local unitary operations: $$\begin{gathered} \label{rewrite2} {|\chi\rangle _{1}}{|\psi^{-}\rangle _{23}} = \frac{1}{2}\biggl( {|\phi^{+}\rangle _{12}} \bigl(-i\sigma_{y}^{3}{|\chi\rangle _{3}}\bigr) + {|\phi^{-}\rangle _{12}} \bigl(\sigma_{x}^{3}{|\chi\rangle _{3}}\bigr) \\ + {|\psi^{+}\rangle _{12}} \bigl(-\sigma_{z}^{3}{|\chi\rangle _{3}}\bigr) + {|\psi^{-}\rangle _{12}} \bigl(-\mathbf{1}^{3}{|\chi\rangle _{3}}\bigr)\biggr), \end{gathered}$$ where the $\sigma_{i}^{3}$ are the Pauli operators acting on system 3 and $\mathbf{1}$ is the identity. To re-iterate, though, only system 1 actually depends on $\alpha$ and $\beta$; the state of system 3 at this stage of the protocol (its reduced state, as it is a member of an entangled pair) is simply the maximally mixed $1/2\,\mathbf{1}$. Alice is now going to perform a measurement. If she were simply to measure system 1 then nothing of interest would happen—she would obtain some result and affect the state of system 1, but systems 2 and 3 would remain in the same old state ${|\psi^{-}\rangle _{}}$. However, as she has access to both systems 1 and 2, she may instead perform a *joint* measurement, and now things get interesting. In particular, if she measures 1 and 2 in the Bell basis, then after the measurement we will be left with only one of the terms on the right-hand side of eqn. (\[rewrite2\]), at random; and this means that Bob’s system will have jumped instantaneously into one of the states $-i\sigma_{y}^{3}{|\chi\rangle _{3}},\, \sigma_{x}^{3}{|\chi\rangle _{3}},\, -\sigma_{z}^{3}{|\chi\rangle _{3}}$ or $-{|\chi\rangle _{3}}$, with equal probability. But how do things look to Bob? As he neither knows whether Alice has performed her measurement, nor, if she has, what the outcome turned out to be, he will still ascribe the same, original, density operator to his system—the maximally mixed state[^4]. No measurement on his system could yet reveal any dependence on $\alpha$ and $\beta$. To complete the protocol therefore, Alice needs to send Bob a message instructing him which of four unitary operators to apply $(i\sigma_{y},\,\sigma_{x},\,-\sigma_{z},\,\mathbf{-1})$ in order to make his system acquire the state ${|\chi\rangle _{}}$ with certainty; for this she will need to send two bits[^5]. With these bits in hand, Bob applies the needed transformation and obtains a system in the state ${|\chi\rangle _{}}$. Now of course, this quantum mechanical process differs from science fiction versions of teleportation in at least two ways. First, it is not *matter* that is transported, but simply the quantum state ${|\chi\rangle _{}}$; and second, the protocol is not instantaneous, but must attend for its completion on the arrival of the classical bits sent from Alice to Bob. Whether or not the quantum protocol approximates to the science fiction ideal, however, it remains a very remarkable phenomenon from the information-theoretic point of view[^6]. For consider what has been achieved. An unknown quantum state has been sent to Bob; and how else could this have been done? Only by Alice sending a quantum system *in* the state ${|\chi\rangle _{}}$ to Bob[^7], for she cannot determine the state of the system and send a description of it instead. (Recall, it is impossible to determine an unknown state of an individual quantum system. See @busch:observable for a nice review.) If, however, Alice did *per impossibile* somehow learn the state and send a description to Bob, then systems encoding that description would have to be sent between them. In this case something that *does* bear a relation to the identity of the state is transmitted from Alice to Bob, unlike in teleportation. Moreover, sending such a description would require a *very great deal* of classical information, as in order to specify a general state of a two dimensional quantum system, two *continuous* parameters need to be specified. The picture we are left with, then, is that in teleportation there has been a transmission of something that is inaccessible at the classical level (often loosely described as a transmission of *quantum* information); in the transmission this information has been in some sense disembodied; and finally, the transmission has been very efficient—requiring, apart from prior shared entanglement, the transfer of only two classical bits. Some information-theoretic aspects of teleportation --------------------------------------------------- ### Preamble The notion of information that is central to quantum information theory is that deriving from the seminal work of @shannon in communication theory. He introduced a measure of information $H(X)$ to characterise a source $X$ of messages which are produced from a fixed alphabet $\{x_{1},\ldots ,x_{n}\}$ whose elements occur with probability $p(x_{i})$. The Shannon information $H(X)$ measures in bits (classical two-state systems) the resources required to transmit all the messages that the source produces (Shannon’s *noiseless coding theorem* see Appendix \[elements\]). That is, it measures how much the messages from the source can be compressed. Shannon also introduced the *mutual information*, $H(X:Y)$, which indicates how much information it is possible to transmit over a noisy channel (intuitively: how much can be inferred about the input to a channel, given the output obtained). The Shannon information measure has many important applications in the quantum context[^8], for example, when considering the transmission of classical (Shannon) information over a channel consisting of quantum devices; but it is also possible to introduce an important and closely related concept—that of *quantum* information properly so-called. This was done by Schumacher in the early 1990s[^9]. Schumacher followed Shannon’s lead: consider a device—a *quantum source*—which, rather than outputting systems corresponding to elements of a classical alphabet, produces systems in particular quantum states $\rho_{x_{i}}$ with probabilities $p(x_{i})$. By reasoning analogous to Shannon’s, Schumacher showed that the output of this source could be compressed by an amount measured by the von Neumann entropy of the source (the *quantum* noiseless coding theorem, Appendix \[elements\]). We therefore have, analogously to the classical case, a notion of the amount of quantum information that the source produces: a measure of the minimum number of two-state quantum systems (*qubits*) required to encode the output of the source. While it is important to distinguish these technical notions of information from the everyday notion of information linked to knowledge, language and meaning (*pace* @dretske:1981, cf. @thesis Chpt.1) there is at least one interesting property held in common: in both the technical and everyday settings ‘information’ functions as an abstract noun, that is, as a term which does not serve to denote a kind of entity having a location in space and time. Briefly: in the latter case, because ‘information’ is a nominalization of the verb ‘to inform’; in the former, because an answer to the question ‘what is transmitted?’ will refer to an abstract type rather than a concrete thing; and because what is measured is a property of a source (compressibility) or a channel (capacity), not an amount of some stuff present in a message[^10]. ### Application to Teleportation There are two information-theoretic aspects of the teleporation protocol it is helpful to go into in somewhat more detail. The first concerns our reason for saying that a very large amount of information is required to specify the state that is teleported. As we have noted, in order to describe an arbitrary (pure) state of a two dimensional quantum system, it is necessary to specify two continuous parameters. A useful means of picturing this is via the Bloch sphere representation. The pure states of a two-state quantum system are in one-to-one correspondence with the points on the surface of the unit 3-sphere, and we may specify two real numbers (angles) to determine a point on the sphere. But why should doing this have associated with it an amount of information? If it is to do so we will need to imagine a classical information source that is selecting these pairs of angles with various probabilities; then a certain Shannon information may be ascribed to the process. Given a particular output of this information source, a quantum system is prepared in the state corresponding to the two angles selected. The quantum states prepared in this manner will then have associated with them a *specification information*[^11] given by the information of the source. Once a system has been prepared in some state in this way, it is presented to Alice, who may proceed to teleport the state to Bob. Rather than the pairs of angles being selected from their full, continuous, range of possible values, the surface of the sphere might be coarse-grained evenly to give a finite number of choices. One might pick the angles specifying the mid-point, say, of each small element of surface area to provide the finite set of pairs of angles to choose between. Loosely speaking this coarse-graining corresponds to considering angles only to a certain degree of accuracy. As this accuracy is increased (the choices made more finely grained), the number of bits required to specify the choice increases without bound. If our information source is selecting states to an arbitrarily high accuracy then, the specification information is unboundedly large. (On the other hand, if the information source is only selecting between a small number of distinct states, then the specification information is correspondingly small. From now on we will assume that unless otherwise stated, the unknown states to be teleported are selected from a suitable coarse-graining of the whole range of possible angles.) It is essential to note, however, that even if the specification information associated with the state that has been teleported to Bob is exceedingly large, the majority of this information is not accessible to him. This leads on to the second point. When one considers encoding classical information in quantum systems, it is necessary to distinguish between specification information and *accessible information*[^12]. The specification information refers to the information of the classical source that selects sequences of quantum states, the accessible information to the maximum amount of information that is available following measurements on the systems prepared in these states. Because of the existence of non-orthogonal states, specification and accessible information may differ; and as it is impossible to distinguish non-orthogonal states perfectly, the specification information associated with a string of systems selected from a source may be much greater than the accessible information. In teleportation, the systems are prepared near Alice before teleportation of their states to Bob. He may then perform various measurements to try and learn something. Call the information of the source selecting the states to be teleported by Alice $H(A)$; the mutual information $H(A:B)$ will determine the amount of classical information per system that Bob is able to extract by performing some measurement, $B$, following successful teleportation of the unknown state. The accessible information is given by the maximum over all decoding measurements of $H(A:B)$. A well known result due to @holevo—the Holevo bound—restricts the amount of information that Bob may acquire to a maximum of one bit of information per qubit, that is, to a maximum of one bit of information per successful run of the teleportation protocol. So this gives us the sense in which the very large amount of information that may be associated with the unknown state being teleported to Bob is largely inaccessible to him. Note that the amount of information that Bob may acquire from the teleported state is less than the amount of classical information—two bits—that Alice had to send to him during the protocol. This fact is of the utmost importance, for if the Holevo bound did not guarantee this, and Bob were able to extract more than two bits of information from his system, then teleportation would give rise to paradox (when embedded in a relativistic theory) as superluminal signalling would be possible[^13]. So the Holevo bound ensures that teleportation is not paradoxical, but it also means that teleportation, when considered as a mode of ordinary *classical* information transfer, is pretty inefficient, requiring two classical bits to be sent for every bit of information that Bob can extract at his end. The puzzles of teleportation {#puzzles} ============================ Let us return to the picture of teleportation that was sketched earlier. An unknown quantum state is teleported from Alice to Bob with nothing that bears any relation to the identity of the state having travelled between them. The two classical bits sent are quite insufficient to specify the state teleported; and in any case, their values are independent of the parameters describing the unknown state. The unboundedly large specification information characterizing the state—information that is inaccessible at the classical level—has somehow been disembodied, and then reincarnated at Bob’s location, as the quantum state first ‘disappears’ from Alice’s system and then ‘reappears’ with Bob. The conceptual puzzles that this process presents seem to cluster around two essential questions. First, how is *so much* information transported? And second, most pressingly, just *how* does the information get from Alice to Bob? Perhaps the prevailing view on how these questions are to be answered is the one that has been expressed by @jozsa:1998 [@jozsa:2003] and @penrose:1998. In their view, the classical bits used in the protocol evidently can’t be carrying the information, for the reasons we have just rehearsed; therefore the entanglement shared between Alice and Bob must be providing the channel down which the information travels. They conclude that in teleportation, an indefinitely large, or even infinite amount of information travels backwards in time from Alice’s measurement to the time at which the entangled pair was created, before propagating forward in time from that event to Bob’s performance of his unitary operation and the attaining by his system of the correct state. Teleportation seems to reveal that entanglement has a remarkable capacity to provide a hitherto unsuspected type of information channel, which allows information to travel backwards in time; and a very great deal of it at that. Further, since it is a purely quantum link that is providing the channel, it must be purely *quantum* information that flows down it. It seems that we have made the discovery that quantum information is a new *type* of information with the striking, and non-classical, property that it may flow backwards in time. The position is summarized succinctly by Penrose: ‘How is it that the *continuous* “information" of the spin direction of the state that she \[Alice\] wishes to transmit...can be transmitted to Bob when she actually sends him only two bits of discrete information? The only other link between Alice and Bob is the quantum link that the entangled pair provides. In spacetime terms this link extends back into the past from Alice to the event at which the entangled pair was produced, and then it extends forward into the future to the event where Bob performs his \[operation\]. Only *discrete* classical information passes from Alice to Bob, so the complex number ratio which determines the specific state being “teleported" must be transmitted by the *quantum* link. This link has a channel which “proceeds into the past" from Alice to the source of the EPR pair, in addition to the remaining channel which we regard as “proceeding into the future" in the normal way from the EPR source to Bob.There is no other physical connection.’ (@penrose:1998 [p.1928]) But one might feel, with good reason, that this explanation of the nature of information flow in teleportation is simply too outlandish. This is the view of @dh, who conclude instead that with suitable analysis, the message sent from Alice to Bob can, after all, be seen to carry the information characterizing the unknown state. The information flows from Alice to Bob hidden away, unexpectedly, in Alice’s message. This approach, and the question of what light it may shed on the notion of quantum information, has been considered in detail elsewhere (@nifpaper). Suffice it to say for present purposes that Deutsch and Hayden disagree with Jozsa and Penrose over the nature of quantum information and how it may flow in teleportation. One might adopt yet a third, and perhaps more prosaic response to the puzzles that teleportation poses. This is to adopt the attitude of the *conservative classical quantity surveyor*[^14]. According to this view, an amount of information cannot be said to have been transmitted to Bob unless it is accessible to him. But of course, as we noted above, the specification information associated with the state teleported to Bob is *not* accessible to him: he cannot determine the identity of the unknown state. On this view, then, the information associated with selecting some unknown state ${|\chi\rangle _{}}$ will not have been transmitted to Bob until an entire ensemble of systems in the state ${|\chi\rangle _{}}$ has been teleported to him, for it is only then that he may determine the identity of the state[^15]. To teleport a whole ensemble of systems, though, Alice will need to send Bob an infinite number of classical bits; and now there isn’t a significant disparity between the amount of information that has been explicitly sent by Alice and the amount that Bob ends up with. One needs to send a very large number of classical bits to have transmitted by teleportation the very large amount of information associated with selecting the unknown state. This approach does not seem to solve all our problems, however. Someone sympathetic to the line of thought espoused by Jozsa and Penrose can point out in reply that there still remains a mystery about *how* the information characterizing the unknown state got from Alice to Bob—the bits sent between them, recall, have no dependence on the identity of the unknown state. So while the approach of the conservative classical quantity surveyor may mitigate our worry to some extent over the first question, it does not seem to help with the second. Resolving (dissolving) the problem {#dissolving} ================================== Dwelling on the question of how the information characterizing the unknown state is transmitted from Alice to Bob has given rise to some conundrums. Should we side with Jozsa and Penrose and admit that quantum information may flow backwards in time down a channel constituted by shared entanglement? Or perhaps with Deutsch and Hayden, and agree that information should flow in a less outlandish fashion, but that quantum information may be squirrelled away in seemingly classical bits? Counting conservatively the amounts of information available after teleportation may make us less anxious about the load carried in a single run of the protocol, but the question still remains: how did the information, in the end, get to Bob? Should we just conclude that it is transported nonlocally in some way? But what might that mean? If the question ‘How does the information get from Alice to Bob?’ is causing us these difficulties, however, perhaps it might pay to look at the question itself rather more closely. In particular, let’s focus on the crucial phrase ‘the information’. Our troubles arise when we take this phrase to be referring to a particular, to some sort of substance or entity whose behaviour in teleportation it is our task to describe. The assumption common to the approaches of Deutsch and Hayden on the one hand, and Jozsa and Penrose on the other, is that we need to provide a story about how some *thing* denoted by ‘the information’ travels from Alice to Bob. Moreover, it is assumed that this supposed thing should be shown to take a spatio-temporally continuous path. But notice that ‘information’ is an abstract noun. This means that ‘the information’ certainly does *not* refer to a substance or to an entity. The shared assumption is thus a mistaken one, and is based on the error of hypostatizing an abstract noun. If ‘the information’ doesn’t introduce a particular, then the question ‘How does the information get from Alice to Bob?’ cannot be a request for a description of how some thing travels. It follows that the locus of our confusion is dissolved. But if it is a mistake to take ‘How does the information get from Alice to Bob?’ as a question about how some thing is transmitted, then what is its legitimate meaning, if any? It seems that the only legitimate use that can remain for this question is as a flowery way of asking: what are the physical processes involved in the transmission? Now *this* question is a perfectly straightforward one, even if, as we shall see (Section \[interpretations\]), the answer one actually gives will depend on the interpretation of quantum mechanics one adopts. But there is no longer a *conceptual* puzzle over teleportation. Once it is recognised that ‘information’ is an abstract noun, then it is clear that there is no further question to be answered regarding how information is transmitted in teleportation that goes beyond providing a description of the physical processes involved in achieving the aim of the protocol. That is all that ‘How is the information transmitted?’ can intelligibly mean; for there is not a question of information being a substance or entity that is transported, nor of ‘the information’ being a referring term. Thus, one does not face a double task consisting of a) describing the physical processes *by which* information is transmitted, followed by b) tracing the path of a ghostly particular, information. There is only task (a). The point should not be misunderstood: I am not claiming that there is no such thing as the transmission of information, but simply that one should not understand the transmission of information on the model of transporting potatoes, or butter, say, or piping water[^16]. The simulation fallacy {#simulation fallacy} ---------------------- Whilst paying due attention to the status of ‘information’ as an abstract noun provides the primary resolution of the problems that teleportation can sometimes seem to present us with, there is a secondary possible source of confusion that should be noted. This is what may be termed the *simulation fallacy*. Imagine that there is some physical process $\cal{P}$ (for example, some quantum-mechanical process) that would require a certain amount of communication or computational resources to be simulated classically. Call the classical simulation using these resources $\cal{S}$. The simulation fallacy is to assume that because it requires these classical resources to simulate $\cal{P}$ using $\cal{S}$, there are processes going on when $\cal{P}$ occurs that are physically equivalent to (are instantiations of) the processes that are involved in the simulation $\cal{S}$ itself (although these processes may be being instantiated using different properties in $\cal{P}$). In particular, when $\cal{P}$ is going on, the thought is that there must be, at some level, physical processes involved in $\cal{P}$ which correspond concretely to the evolution of the classical resources in the simulation $\cal{S}$. The fallacy is to read off features of the simulation as real features of the thing simulated[^17]. A familiar example of the simulation fallacy is provided by Deutsch’s argument that Shor’s factoring algorithm supports an Everettian view of quantum mechanics (@FoR [p.217]). The argument is that if factoring very large numbers would require greater computational resources than are contained in the visible universe, then how could such a process be possible unless one admits the existence of a very large number of (superposed) computations in Everettian parallel universes? A computation that would require a very large amount of resources if it were to be performed classically is explained *as* a process that consists of a very large number of classical computations. But of course, considered as an argument, this is fallacious. The fact that a very large amount of classical computation might be required to produce the same result as a quantum computation does not entail that the quantum computation consists of a large number of parallel classical computations[^18]. The simulation fallacy is also evident in the common claim that Bell’s theorem shows us that quantum mechanics is nonlocal, or the claim that the experimental violation of Bell inequalities means that the world must be nonlocal. Of course, what is in fact shown by these well-known results is that no local hidden variable model can simulate the predictions of quantum mechanics, nor provide a model for the experimentally observed correlations. But these facts about simulation don’t lead directly to facts about the simulated: the fact that any adequate hidden variable model must be nonlocal does not show that quantum mechanics is nonlocal (this, of course, is an interpretation dependent property), nor show the world to be nonlocal. While the question of what classical resources would be required to simulate a given quantum process is an indispensible guide in the search for interesting quantum information protocols and is vitally important for that reason, the simulation fallacy indicates that it is by no means a sure guide to ontology. With regard to teleportation, it is important to recognise the simulation fallacy in order to assuage any worries that might remain over the question ‘How does so much information get from Alice to Bob?’, and to undermine further the thought that teleportation must be understood as a flow of information. For the fact that it would take a very large number of classical bits to transmit the identity of an unknown state from Alice to Bob does not entail that in teleportation there is a real corresponding transmission of information, some physical process going on that instantiates, albeit in a different medium, the transport of this large amount of information[^19]. (Note that the flow of the hypostatized ‘quantum information’ of Jozsa and Penrose plays precisely this rôle: the analogue, in a different medium, of the transport of the large amount of classical information.) Equivalence from the point of view of information processing does not imply physical equivalence. Awareness of the simulation fallacy is particularly relevant when we consider the approach of the conservative classical quantity surveyor. Recall that the point of this approach is to deny that a large amount of information can be said to have been transported to Bob in teleportation until that information is actually available to him. However, it might be objected to this that after a single run of the teleportation protocol, the information characterizing the state is certainly present at Bob’s location, even if inaccessible to him, as a system *in* the unknown state is present[^20]. This contention would seem to rest on an argument of the following form: The only way the unknown state can appear at Bob’s location is if the information characterizing the state has actually been transported to Bob, hence on appearance of the state, the specification information associated with the state has indeed been transported to Bob’s location. (Crudely, if a system in the given state is present, then the information is present, as it takes this information to specify the state.) But such an argument needs to be treated with care, for the main premise appears to rest on the simulation fallacy. Just because it would take a large amount of information to specify a state doesn’t mean that we should conclude that this amount of information has been physically transported in teleportation when Bob’s system acquires the state. In any event, the simplest way to remain clear on whether or not, or in what way, information can be said to be present at Bob’s location following a single run of the teleportation protocol is to respect the distinction between the specification information associated with a system and the amount of information that may be said to be encoded or contained in the system. Once Bob’s system has acquired the state ${|\chi\rangle _{}}$ teleported by Alice, then his system has associated with it the same specification information, $H(A)$: *if* one were now *asked* to specify the state of Bob’s system, then this number of bits would be required, on average. This quantity of information is not encoded or contained in the system however. The mutual information $H(A:B)$ and the accessible information provide the relevant measures of how much information Bob’s system can be said to contain, for they govern the amount that may be decoded. But of course, as ‘information’ is an abstract noun, containing information is not containing some *thing*, however aethereal. The teleportation process under different interpretations {#interpretations} ========================================================= By reflecting on the logico-grammatical status of the term ‘information’ we have been able to replace the (needlessly) conceptually puzzling question of how the information gets from Alice to Bob in teleportation, with the simple, genuine, question of what the physical processes involved in teleportation are. While this may not, perhaps, be quite enough to still all the controversy that trying to understand teleportation has evoked, the controversy is now of a very familiar kind: it concerns what interpretation of quantum mechanics one adopts. For the detailed story one tells about the physical processes involved in teleporation will of course depend upon one’s interpretive stance. Two questions in particular will find different answers under different interpretations: first, is nonlocality involved in teleportation? and second, has anything interesting happened before Alice’s classical bits are sent to Bob and he performs the correct unitary operation? We will now see how some of these differences play out in the following familiar interpretations (the list of approaches considered is by no means exhaustive). Collapse interpretations: Dirac/von Neumann, GRW ------------------------------------------------ The natural place to begin is with the orthodox approach of @dirac and @vN in which there is a genuine process of collapse on measurement[^21]. (The vagueness over where, when, why and how this collapse takes place might be alleviated along lines suggested by @grw, perhaps.) If one has a genuine process of collapse then as noted long ago by @EPR[^22], one has action-at-a-distance. In the presence of entanglement, a measurement on one system can result in a real change to the possessed properties of another system, even when the two systems are widely separated. (Although, as is well known, these changes do not allow one to send signals superluminally—this is known as the *no-signalling theorem*[^23].) In teleportation, then, under a collapse interpretation, the effect of Alice’s Bell-basis measurement will be to prepare Bob’s system, at a distance, in one of four pure states which depend on the unknown state ${|\chi\rangle _{}}$, by using the nonlocal effect of collapse. It then only remains for Alice to send her two bits to Bob to tell him which (type of) state he now has in his possession. Under this interpretation, teleportation explicitly involves nonlocality, or action-at-a-distance; and it is precisely because of the nonlocal effect of collapse, preparing Bob’s system in a state that differs in one of only four ways from ${|\chi\rangle _{}}$, that a mere two classical bits need be sent by Alice in order for Bob’s system to acquire a state parameterised by two continuous values. It is enlightening to compare the effect of collapse in this scenario to that of a rigid rod held by two parties. Imagine that Alice wanted to let Bob know the value of a parameter that could take on values in the interval $[0,1]$. If they were each holding one end of a long rigid rod, then Alice could let Bob know the value she has in mind simply by moving her end of the rod along in Bob’s direction by a suitable distance. Bob, seeing how far his end of the rod moves, may infer the value Alice is thinking of[^24]. There is no mystery here about how the value of the continuous parameter is transmitted from Alice to Bob. Alice, by moving her end of the rod, moves Bob’s by a corresponding amount. In teleportation, the effect of collapse is somewhat analogous: Bob’s system is prepared, by the nonlocal effect of collapse, in a state that depends on the two continuous parameters characterizing ${|\chi\rangle _{}}$. As we have said, collapse allows a real change in the physical properties that a distant system possesses, if there was prior entanglement. Compare: pushing one end of a rigid rod axially leads to a change in the position of the far end. The nonlocal effect of collapse, which is here understood as a real physical process, is providing the main physical mechanism behind teleportation; and recall that once the physical mechanisms have been described (I have argued) there is no further question to be asked about how information is transmitted in the protocol. In a collapse interpretation, teleportation thus involves nonlocality, in the sense of action-at-a-distance, crucially. Also, something interesting certainly has happened once Alice performs her measurement and before she sends the two classical bits to Bob. There has been a real change in the physical properties of Bob’s system, as it acquires one of four pure states. (Although note that at this stage the probability distributions for measurements on Bob’s system will nonetheless not display any dependence on the parameters characterizing ${|\chi\rangle _{}}$, in virtue of the no-signalling theorem. It is only once the bits from Alice have arrived and Bob has performed the correct operation that measurements on his system will display a dependence on the parameters $\alpha$ and $\beta$.) No collapse and no extra values: Everett ---------------------------------------- It is possible to retain the idea that the wavefunction provides a complete description of reality while rejecting the notion of collapse; this way lies the Everrett interpretation (@everett)[^25]. The characteristic feature of the Everett interpretation is that the dynamics is always unitary; and no extra values are added to the description provided by the wavefunction in order to account for definite measurement outcomes. Instead, measurements are simply unitary interactions which have been chosen so as to correlate states of the system being measured to states of a measuring apparatus. Obtaining a definite value on measurement is then understood as the measured system coming to have a definite state (eigenstate of the measured observable) *relative to* the indicator states of the measuring apparatus and ultimately, relative to an observer[^26]. A treatment of teleportation in the Everettian context was given by @vaidman. @braunstein:irreversible provides a detailed discussion of the teleportation protocol within unitary quantum mechanics without collapse. With teleportation in an Everettian setting, and unlike teleportation under the orthodox account, it is clear that there will be no action-at-distance in virtue of collapse when Alice performs her measurement, for the simple reason that there is no process of collapse. Instead, the result of Alice’s measurement will be that Bob’s system comes to have definite relative states related to the unknown state ${|\chi\rangle _{}}$, with respect to the indicator states of the systems recording the outcome of Alice’s measurement (see Appendix \[nocollapse appendix\]). (It is argued in @erpart1 and @nifpaper that this does not amount to a new form of nonlocality.) Note, though, that at this stage of the protocol, the *reduced* state of every system involved will now be maximally mixed[^27]. As @braunstein:irreversible notes, this feature corresponds to the ‘disembodiment’ of the information characterizing the unknown state in the orthodox account of teleportation: following Alice’s measurement, all the systems involved in the protocol will have become fully entangled. Dependence on the parameters characterizing the unknown state will only be observable with a suitable *global* measurement, not for any local measurements. In particular, one can consider the correlations that now exist between the systems recording the outcome of Alice’s measurement and Bob’s system. Certain of the joint (and irreducible) properties of these spatially separated systems will depend on the identity of the unknown state. In this sense, the information characterizing ${|\chi\rangle _{}}$ might now be said to be ‘in the correlations’ between these systems. (This is the terminology Braunstein adopts.) Once Bob has been sent the systems recording the outcome of Alice’s measurement, however, he is able to disentangle his system from the other systems involved in the protocol. Its state will now factorise from the joint state of the other systems; and will in fact be the pure state ${|\chi\rangle _{}}$. Dependence on the parameters $\alpha$ and $\beta$ *will* finally be observable for local measurements once more, but this time, only at Bob’s location. In collapse versions of quantum mechanics, the nonlocal effect of collapse was the main physical mechanism underlying teleportation. In the no-collapse Everettian setting, the fundamental mechanism is provided by the fact that in the presence of entanglement, local unitary operations—in this case, Alice’s measurement—can have a non-trivial effect on the global state of the joint system. So, has anything significant happened at Bob’s location before Alice sends him the result of her measurement and he performs his conditional unitary operation? Well, arguably not: nothing has happened other than all of the systems involved in the protocol having become entangled, as a result of the various local unitary operations. No collapse, but extra values: Bohm ----------------------------------- The Bohm theory account provides us with an interesting intermediary view of teleportation, in which there is no collapse of the wavefunction, but nonlocality plays an interesting rôle. We shall follow the analysis of @maroney:hiley. The Bohm theory (@bohm:1952) is a nonlocal, contextual, deterministic hidden variable theory, in which the wavefunction $\Psi(\mathbf{x}_{1},\mathbf{x}_{2}\ldots\mathbf{x}_{n},t)$ of an $n$-body system evolves unitarily according to the Schrödinger dynamics, but is supplemented with definite values for the positions $\mathbf{x}_{1}(t),\mathbf{x}_{2}(t)\ldots\mathbf{x}_{n}(t)$ of the particles. Momenta are also defined according to $\mathbf{p}_{i}=\nabla_{i}S$, where $S$ is the phase of $\Psi$, hence a definite trajectory may be associated with a system, where this trajectory will depend on the many-body wavefunction (and thus, in general, on the positions and behaviour of all the other systems, however far away). If the initial probability distribution for particle positions is assumed to be given by $|\Psi|^{2}$, then the same predictions for measurement outcomes will be made as in ordinary quantum mechanics. For detailed presentations of the Bohm theory, see @bohmhiley and @holland:1995. The guiding effect of the wavefunction on the particle positions may also be understood in terms of a new *quantum potential* that acts on particles in addition to the familiar classical potentials. The quantum potential is given by $$Q(\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{n})=-\hbar^{2}\sum_{i=1}^{n}\frac{\nabla^{2}_{i}R}{2m_{i}R},$$ where $R$ is the amplitude of $\Psi$ and $m_{i}$ is the mass of the $i$-th particle. Among the ways in which this quantity differs from a classical potential is that it will in general give rise to a nonlocal dynamics (that is, in the presence of entanglement, the force on a given system will depend on the instantaneous positions of the other particles, no matter how far away); and it may be large even when the amplitude from which it is derived is small. @bohmhiley [§3.2] suggest that the quantum potential should be understood as an ‘information potential’ rather than a mechanical potential, as a way of accounting for its peculiar properties. The determinate values for position in the Bohm theory are usually understood as providing the definite outcomes of measurement[^28] that would appear to be lacking in a no-collapse version of quantum mechanics, in the absence of an Everett-style relativization. Following a measurement interaction, the wavefunction of the joint object-system and apparatus will have separated out (in the ideal case) into a superposition of non-overlapping wavepackets (on configuration space) corresponding to the different possible outcomes of measurement. The determinate values for the positions of the object-system and apparatus pointer variable will pick out a point in configuration space; and the outcome that is observed, or is made definite, is the one corresponding to the wavepacket whose support contains this point. The wavefunction for the total system remains as a superposition of all of the non-overlapping waverpackets, however. @bohmhiley introduce the notions of *active*, *passive* and *inactive* information to describe this feature of the theory. If $\Psi$ may be written as a superposition of non-overlapping wavepackets, then they suggest that the definite configuration point of the total system picks out one of these wavepackets (the one whose support contains the point) as active. The evolution of the point is determined solely by the wavepacket containing it; and in keeping with their conception of $Q$ as an information potential, the information associated with this wavepacket is said to be active. The information associated with the other wavepackets is termed either ‘passive’, or ‘inactive’. ‘Passive’, if the different wavepackets may in the future be made to overlap and interfere, ‘inactive’ if such interference would be a practical impossibility (as for example, if environmental decoherence has occurred in a measurement-type situation — this corresponds to the case of ‘effective collapse’ of the wavefunction). In their discussion of the teleportation protocol, Maroney and Hiley adopt the approach in which a definite spin vector is also associated with each spin 1/2 particle, in addition to its definite position. The idea is that with each system is associated an orthogonal set of axes (body axes) whose orientation is specified by a real three dimensional spin vector, $\mathbf{s}$, along with an angle of rotation about this vector; where these quantities are determined by the wavefunction[^29]. The analysis of teleportation then proceeds much as in the Everett interpretation, save that we may also consider the evolution of the determinate spin vectors associated with the various systems. Initially, system 1 in the unknown state ${|\chi\rangle _{}}$ will have some definite spin vector that depends on $\alpha$ and $\beta$, $\mathbf{s}(\alpha,\beta)$, while it turns out that if Alice and Bob share a singlet state, the spin vectors for their two systems will be zero (@bohmhiley [§10.6]). Now Alice performs her Bell-basis measurement. As in the Everettian picture, the effect of measurement is to entangle the systems being measured with systems recording the outcome of the measurement. But this is not the only effect, in the Bohm theory. The total wavefunction is now a superposition of four terms corresponding to the four possible outcomes of Alice’s measurement; and one of these four terms will be picked out by the definite position value of the measuring apparatus pointer variable. For each of these four terms taken individually, Bob’s system will be in a definite state related to the state ${|\chi\rangle _{}}$, thus with each will be associated a definite spin vector $\mathbf{s}^{j}(\alpha,\beta)$, $j=1,\ldots,4$, pointing in some direction. When one of the four terms is picked out as active, and the others rendered passive (or inactive), following Alice’s measurement, the spin vector for Bob’s system will change instantaneously from zero to one of the four $\mathbf{s}^{j}(\alpha,\beta)$ (@maroney:hiley). Thus in the Bohm theory, teleportation certainly involves nonlocality; and moreover, something very interesting does happen as soon as Alice has made her measurement. Bob’s system acquires a definite spin vector that depends on the parameters characterizing the unknown state, as a result of a nonlocal quantum torque (@maroney:hiley). Furthermore, there is a one in four chance that this spin vector will be the same as the original $\mathbf{s}(\alpha,\beta)$; and all this while the total state of the system remains uncollapsed, with all the particles entangled. Finally, as we have seen before, once Alice sends Bob systems recording the outcome of her measurement, he may perform the conditional unitary operation necessary to disentangle his system from the others, and leave his system in the state ${|\chi\rangle _{}}$. The spin vector of his system will now be $\mathbf{s}(\alpha,\beta)$ with certainty. ### A note on active information {#active information} The conclusion of @maroney:hiley and @hiley:1999 is that according to the Bohm theory, what is transferred from Alice’s region to Bob’s region in the teleportation protocol is the active information that is contained in the quantum state of the initial system. However questions may be raised about how apposite this description is. Let us label the pointer degree of freedom of the measuring apparatus by $x_{0}$. At the beginning of the teleportation protocol, the state of system 1 factorises from the entangled joint state of 2 and 3; and the state of the measuring apparatus will also factorise. Accordingly, the quantum potential will be given by a sum of separate terms: $$\label{qpotential before} Q(\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{3},x_{0}) = Q(\mathbf{x}_{1},\alpha,\beta) + Q(\mathbf{x}_{2},\mathbf{x}_{3}) + Q(x_{0}),$$ where it has been noted that the first term, the one that will determine the motion of system 1, depends on the parameters characterizing the unknown state[^30]. Once Alice performs her Bell basis measurement, however, all the systems become entangled; and the potential will be of the form: $$\label{qpotential middle} Q(\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{3},x_{0}) = Q(\mathbf{x}_{1},\mathbf{x}_{2},x_{0}) + Q(\mathbf{x}_{3},x_{0},\alpha,\beta)$$ The part of the quantum potential that will affect system 3 now depends on $\alpha$ and $\beta$. Finally, at the end of the protocol, systems 1, 2 and the measuring apparatus are left entangled; and system 3, in the pure state ${|\chi\rangle _{3}},$ factorises. The quantum potential then takes the form: $$\label{qpotential end} Q(\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{3},x_{0}) = Q(\mathbf{x}_{1},\mathbf{x}_{2},x_{0}) + Q(\mathbf{x}_{3},\alpha,\beta)$$ Maroney and Hiley say: ‘What we see clearly emerging here is that it is active information that has been transferred from particle 1 to particle 3 and that this transfer has been mediated by the nonlocal quantum potential.’(@maroney:hiley [p.1413]) ‘...it is the objective active information contained in the wavefunction that is transferred from particle 1 to particle 3.’ (@maroney:hiley [p.1414]) Note that the part of the potential that is active on system 3 will already have acquired a dependence on $\alpha$ and $\beta$ before the end of the protocol; that is, as soon as Alice has performed her measurement. So if active information depending on these parameters is transferred at all, it will have been transferred before the end of the protocol. However it is not until Alice has sent her message to Bob and he performs his conditional operation that the term $Q(\mathbf{x}_{3},\alpha,\beta)$ in eqn. (\[qpotential end\]) will take the same form as the initial $Q(\mathbf{x}_{1},\alpha,\beta)$. The difficulties for the stated conclusion arise when we consider more closely what is meant by ‘active information’. In @maroney:hiley [@hiley:1999], the connection is made with a different sense of the word ‘information’ than the ones we have considered so far. This is a sense that derives from the verb ‘inform’ under its branch I and II senses (Oxford English Dictionary), *viz.* to give form to, or, to give formative principle to (this latter, a Scholastic Latin offshoot). Thus ‘information’ as it appears in ‘active information’ and company, means the action of giving form to[^31]. ‘The information of $x$’ (read: The *in*-formation of $x$) means the action of giving form to $x$. Now, while we may understand what is meant by $Q$ being said to be an information potential—it is a potential that gives form to something, presumably the possible trajectories associated with particles (although note that the distinction with mechanical potentials is now blurred, as these give form to the possible trajectories too)—and may understand the term ‘active’ as picking out the part of the quantum potential that is shaping the actual trajectory in configuration space of the total system, it does not make sense to say that active information is transferred in teleportation. Because ‘information’ here refers to a particular action—the giving of a form to something—and an action is not a *thing* that can be moved[^32]. The same *type* of action may be taking place at two different places, or at two different times, but an action may not be moved from $A$ to $B$. Thus with ‘active information’ understood in the advertised way, all that can be said is that an action of the same *type* is being performed (by the quantum potential) on system 3 at the end of the teleportation protocol as was being performed on system 1 at the beginning, not that something has been transferred between the two. We may not, then, understand ‘transfer’ literally. When all is said and done, it is perhaps clearer simply to adopt the standard description and say that the quantum *state* of particle 1 has been ‘transferred’ in teleportation; that is (as a quantum state is a mathematical object and therefore cannot literally be moved about either), that system 3 has been made to acquire (is left in) the unknown state ${|\chi\rangle _{}}$. To sum up: it perhaps looked as if the Bohmian notion of active information might provide us with a sense of what is transported in teleportation if we insist that *information*, ‘the information in the wavefunction’, is, in a literal sense, transported. But this proves not to be the case. Ensemble and statistical viewpoints ----------------------------------- So far, in all the interpretations we have considered, the quantum state may describe individual systems. Let us close this section by looking briefly at approaches in which the state is taken only to describe *ensembles* of systems. We may broadly distinguish two such approaches. The first I will term an *ensemble* viewpoint. In this approach, the state is taken to represent a real physical property, but only of an ensemble. Following a measurement, the ensemble must be left in a *proper* mixture[^33], in order for there to be definite outcomes, i.e., the ensemble is left in an appropriate mixture of sub-ensembles, each described by a pure state (eigenstate of the measured observable). Thus there will be a real process of collapse, but only at the level of the ensemble, not for individual systems (which are not being described by a quantum state, if at all). The second approach I call a *statistical* interpretation. (This is the interpretation that would be adopted by instrumentalists, for example.) On this view, the quantum formalism merely describes the probabilities for measurement outcomes for ensembles, there is no description of individual systems and collapse does not correspond to any real physical process. On both these approaches, as the state is only associated with an ensemble, it is not until an entire ensemble has been teleported to Bob (that is, Alice has run the teleportation protocol on every member of an ensemble in the unknown state ${|\chi\rangle _{}}$) that he acquires something in the state ${|\chi\rangle _{}}$. An ensemble or statistical viewpoint thus makes a natural partner to conservative classical quantity surveying in teleportation. Under the statistical interpretation, there is clearly no nonlocality involved in teleportation, as there is no real process of collapse; and nothing of any interest has happened before the required classical bits are sent to Bob. (The no-signalling theorem entails that Alice’s measurement won’t affect the probability distributions for distant measurements.) The end result of the completed teleportation process is that Bob’s ensemble is ascribed the state ${|\chi\rangle _{}}$; where this merely means that the statistics one will expect for measurements on Bob’s ensemble are now the same as those one would have expected for measurements on the initial ensemble presented to Alice. The ensemble viewpoint presents a rather different picture, as it does involve a real process of collapse, even if only at the ensemble level. Let us suppose that Alice has performed the Bell basis measurement on her ensembles, but has not yet sent the ensemble of classical bits to Bob. The effect of this measurement will have been to leave Bob’s ensemble in a proper mixture composed of sub-ensembles in the four possible states a fixed rotation away from ${|\chi\rangle _{}}$. Thus there has been a nonlocal effect: that of preparing what was an improper mixture into a particular proper mixture, whose components depend on the parameters characterizing the unknown state. The use of the flock of classical bits that Alice sends to Bob is to allow him to separate out the ensemble he now has into four distinct sub-ensembles, on each of which he performs the relevant unitary operation, ending up with all four being described by the state ${|\chi\rangle _{}}$. Concluding remarks {#study concluding} ================== The aim of this paper has been to show how substantial conceptual difficulties can arise if one neglects the fact that ‘information’ is an abstract noun. This oversight seems to lie at the root of much confusion over the process of teleportation; and this gives us very good reason to pay attention to the logical status of the term. A few closing remarks should be made. Schematically, a central part of the argument has been of the following form: Puzzles arise when we feel the need to tell a story about how something travels from Alice to Bob in teleportation. In particular, it might be felt that this something needs to travel in a spatio-temporally continuous fashion; and one might accordingly feel pushed towards adopting something like the Jozsa/Penrose view. But if ‘the information’ doesn’t pick out a particular, then there is no thing to take a path, continuous or not, therefore the problem is not a genuine one, but an illusion. We can imagine a number of objections. A very simple one might take the following form: You have said that information is not a particular or thing, therefore it does not make sense to inquire how *it* flows (but only inquire about the means by which it is transmitted). But don’t we have a theory that quantifies information (*viz.* communication theory); and if we can say how much of something there is, isn’t that enough to say that we have a thing, or a quantity that can be located? This objection is dealt with quickly. Note that this form of argument will not work in general—one can say how much a picture might be worth in pounds and pence, for example, but this is not quantifying an amount of stuff, nor describing a quantity with a location—and it does not work in this particular case either. The Shannon information doesn’t quantify an amount of stuff that is present in a message, say, nor the amount of a certain quantity that is present at some spatial location. The Shannon information $H(X)$ describes a specific property of a *source* (not a message), namely, the amount of channel resources that would be required to transmit the messages the source produces. This is evidently not to quantify an amount of stuff, nor to characterize a quantity that has a spatial location. (The source certainly has a spatial location, but its information does not.) Or consider the mutual information. Loosely speaking, this quantity tells us about the amount we may be able to infer about some event or state of affairs from the obtaining of another event or state of affairs. But how much we may infer is not a quantity it makes sense to ascribe a spatial location to. Another objection might be as follows: You have suggested that it is a mistake to hypostatize information, to talk of it as a thing that moves about. How is this to be reconciled with some of the ways we often talk about information in physics, especially the example in relativity, where the most natural way of stating an important constraint is to say that relativity rules out the propagation of *information* faster than the speed of light? The response is that one can admit this mode of talking without it entailing a hypostatized conception of information. The constraint is that superluminal signalling is ruled out on pain of temporal loop paradoxes (@rindler [§7.ix]). What this means is that no *physical process* is permissible that would allow a signal to be sent superluminally and thus allow information to be transmitted superluminally. What are ruled out are certain types of physical processes, not, save as a metaphor, certain types of motion of information[^34]. A final objection that might be raised to support the line of thought that inclines one towards the Jozsa and Penrose conception of teleportation is just this: Well, don’t we after all require that information be propagated in a spatio-temporally continuous way? Even if this is not to be construed as a flow of stuff, or the passage of an entity? The response illustrates part of the value of noting the features of the term ‘information’ that have been emphasized here. The genuine question we face is: what are the physical processes that may be used to transmit information? Not the (obscure) question ‘How does information behave?’. Once we see what the question is clearly, then the answer, surely, is to be given by our best physical theory describing the protocol in question. To be sure, many of the most familiar classical examples we are used-to use spatio-temporally continuous changes in physical properties to transmit information (a prime example might be the use of radio waves), but it is up to physical theory to tell us about the nature of the processes we are using to transmit information in any given situation. And the examples we have found in entanglement assisted communication seem precisely to be examples in which *global* rather than local properties are being used to carry information; and there seems not to be a useful sense in which information is being carried in a spatio-temporally continuous way (although, see @nifpaper for further discussion of Deutsch and Hayden’s opposing view). It is not the nature of information that is at issue, but the nature of the physical objects and the physical properties we may use to transmit information. On a final note, the deflationary approach towards teleportation that I have advocated should be compared with what may be termed the ‘nihilist’ approach of @duwell:2003. While I am in broad sympathy with much of what Duwell has to say, we differ on some important points. Duwell also advocates the view that quantum information is not a substance, but reaches from this the strong conclusion that quantum information does not exist. From the current point of view this conclusion is unwarranted. Certainly, quantum information is not a substance or entity, but this does not mean that it doesn’t exist, it is just a reflection of the fact that ‘information’ is an abstract noun. ‘Beauty’ for example, is an abstract noun, but no one would want to conclude that there is no beauty in the world. Moreover, Duwell’s conclusion could only possibly be hyperbolical, for if classical information can be said to exist, then so too can quantum information; and contrapositively, if quantum information does not exist, then no more does classical information. The concept of classical information is given by Shannon’s noiseless coding theorem, the concept of quantum information, by the quantum noiseless coding theorem. As we are by now vividly aware, these are not concepts of material quantities or things. But rejecting the concept of quantum information would be akin to cutting off one’s nose to spite one’s face; and is by no means necessary in order to get a proper understanding of teleportation. Teleportation is not rendered unproblematic by trying to do without the notion of quantum information and facing the protocol equipped only with Shannon’s concept, but simply by resisting the temptation to hypostatize an abstract noun; and, having recognised the status of ‘information’ as an abstract noun, by realising that the only genuine question one faces is the relatively straightforward one of describing the physical processes by which information is transmitted. Acknowledgements {#acknowledgements .unnumbered} ================ Thanks are due to Jon Barrett, Harvey Brown and Peter Morgan for useful discussion, to Jane Timpson for the Figure; and to John Christie and Joesph Melia for asking some good questions. Elements of information theory {#elements} ============================== The Shannon information measure is defined as: $$H(X) = -\sum_{i=1}^{n} p(x_{i})\log p(x_{i});$$ logarithms to base 2. Its primary role is in characterising the degree to which the output of a source modelled as producing letters picked from a finite alphabet $\{ x_{1},\ldots,x_{n}\}$ with probabilities $p(x_{i})$ may be compressed. @shannon showed in his noiseless coding theorem that for very long messages of length $N$ produced by such a source, it was possible to encode them onto a string of $NH(X)$ bits, thus effecting a compression from $2^{N\log n}$ (the number of possible messages of length $N$ drawn from an alphabet of $n$ letters) to $2^{NH(X)}$. The value of $H(X)$ depends on the probability distribution $\{p(x_{i})\}$, in fact: $$0 \leq H(X) \leq \log n,$$ where the minimum is achieved when all but one of the $p(x_{i})$ are zero (the distribution is maximally peaked); and the maximum when the distribution is flat, $ \forall i\; p(x_{i}) = 1/n$. Thus, a source that always outputs the same letter has zero information (cf. footnote \[noinfo\]); and one that produces each letter with equal probability has maximum information for a given alphabet size. $H(X)$ may also be understood as a measure of uncertainty, that is, as a measure of how concentrated the probability distribution $\{p(x_{i})\}$ is (@jos). But this rôle is logically independent of its rôle in characterising information sources (*ibid.*), a fact which is rarely recognised and has accordingly given rise to considerable confusion. (*Ibid.* See @supposed for discussion.) A *communication channel* (to be precise, a *discrete memoryless* channel) may be characterised in terms of conditional probabilities $p(y_{j}|x_{i})$: given that input $x_{i}$ to the channel is prepared, what is the probability of getting output $y_{j}$? If the channel is noiseless, then the mapping from input to output states will be one-to-one; in the presence of noise, a given input may give rise to a spread of outputs with various probabilities. In his *noisy coding theorem*, @shannon proved the surprising result that even in the presence of noise, it is possible to send messages over a channel with a probability of error that tends to zero as the length of message, $N$, tends to infinity. Given an input distribution $p(x_{i})$ and the conditional probabilities $p(y_{j}|x_{i})$ characterising the channel, define the *mutual information* $H(X:Y)$: $$H(X:Y) = H(X) - H(X|Y),$$ where the ‘conditional entropy’ $H(X|Y) = \sum_{j}p(y_{j})\bigr(-\sum_{i} p(x_{i}|y_{j})\log p(x_{i}|y_{j})\bigr)$. It follows from the noisy coding theorem that the mutual information governs the rate at which it is possible to send information over a channel with input distribution $p(x_{i})$: It will be possible to send the output of any information source $W$ with an information $H(W)= H(X:Y)$ over the channel, with arbitrarily small error as $N$ is increased. The *capacity* of a channel is defined as the supremum over input distributions $p(x_{i})$ of the mutual information $H(X:Y)$. A quantum source (@qcoding) produces systems in signal states $\rho_{x_{i}}$ with probabilities $p(x_{i})$. A density operator $\rho=\sum_{i}p(x_{i})\rho_{x_{i}}$ may be associated with the output of this source. @qcoding considered quantum sources with pure signal states $\rho_{x_{i}}= {|x_{i}\rangle _{}}\!{\langle _{}x_{i}|}$. The von Neumann entropy $S(\rho)$ is defined as: $$S(\rho) = -\mathrm{Tr} \rho\log\rho.$$ The quantum noiseless coding theorem shows that for large $N$, the output of length $N$ of a source with signal states $\rho_{x_{i}}= {|x_{i}\rangle _{}}\!{\langle _{}x_{i}|}$ may be encoded onto a quantum system of $2^{NS(\rho)}$ dimensions, that is, using $NS(\rho)$ qubits. Teleportation in the absence of collapse {#nocollapse appendix} ======================================== The original @teleportation treatment of teleportation involved collapse following Alice’s measurement in order to pick out, probabilistically, a definite state of Bob’s system that depended in a fixed way on the identity of the unknown state. As we have seen, however, it is quite possible to treat the teleportation protocol in a no-collapse setting (@vaidman [@braunstein:irreversible; @mermin:teleportation]). It might prove helpful to see how this can work. Recall the pertinent expression of the initial state in eqn. (\[rewrite2\]): $$\begin{gathered} {|\chi\rangle _{1}}{|\psi^{-}\rangle _{23}} = \frac{1}{2}\biggl( {|\phi^{+}\rangle _{12}} \bigl(-i\sigma_{y}^{3}{|\chi\rangle _{3}}\bigr) + {|\phi^{-}\rangle _{12}} \bigl(\sigma_{x}^{3}{|\chi\rangle _{3}}\bigr) \\ + {|\psi^{+}\rangle _{12}} \bigl(-\sigma_{z}^{3}{|\chi\rangle _{3}}\bigr) + {|\psi^{-}\rangle _{12}} \bigl(-\mathbf{1}^{3}{|\chi\rangle _{3}}\bigr)\biggr). \end{gathered}$$ Alice is going to perform her Bell-basis measurement and she will need systems to record the outcome of that measurement. As there are four possible outcomes, she may use two qubits, call them $c$ and $d$, with an indicator basis $\{{|0\rangle _{}},{|1\rangle _{}}\!\}$. Her measurement interaction $U_{A}$ may be chosen (in the ideal case) so that: $$\label{measurement} \begin{split} {|\phi^{+}\rangle _{12}}{|0\rangle _{c}}{|0\rangle _{d}} & \mapsto {|\phi^{+}\rangle _{12}} {|0\rangle _{c}}{|0\rangle _{d}}, \\ {|\phi^{-}\rangle _{12}}{|0\rangle _{c}}{|0\rangle _{d}} & \mapsto {|\phi^{-}\rangle _{12}}{|0\rangle _{c}}{|1\rangle _{d}}, \\ {|\psi^{+}\rangle _{12}}{|0\rangle _{c}}{|0\rangle _{d}} & \mapsto {|\psi^{+}\rangle _{12}}{|1\rangle _{c}}{|0\rangle _{d}}, \\ {|\psi^{-}\rangle _{12}}{|0\rangle _{c}}{|0\rangle _{d}} & \mapsto {|\psi^{-}\rangle _{12}}{|1\rangle _{c}}{|1\rangle _{d}}. \end{split}$$ Before Alice’s measurement, then, the total state is ${|0\rangle _{c}}{|0\rangle _{d}}{|\chi\rangle _{1}}{|\psi^{-}\rangle _{23}}$. Note that relative to the states of $c$ and $d$, the state of system $1$ at Alice’s locale is definite, but the state of Bob’s system, $3$, is not. Being maximally mixed (half of a maximally entangled pair) it has no pure state of its own. When Alice performs her measurement on 1 and 2, the mapping (\[measurement\]) will apply and all the systems will now become entangled. The new total state is: $$\begin{gathered} {|\Psi\rangle _{\rm{tot}}} = \frac{1}{2}\biggl( {|0\rangle _{c}}{|0\rangle _{d}}{|\phi^{+}\rangle _{12}} \bigl(-i\sigma_{y}^{3}{|\chi\rangle _{3}}\bigr) + {|0\rangle _{c}}{|1\rangle _{d}}{|\phi^{-}\rangle _{12}} \bigl(\sigma_{x}^{3}{|\chi\rangle _{3}}\bigr) \\ + {|1\rangle _{c}}{|0\rangle _{d}}{|\psi^{+}\rangle _{12}} \bigl(-\sigma_{z}^{3}{|\chi\rangle _{3}}\bigr) + {|1\rangle _{c}}{|1\rangle _{d}}{|\psi^{-}\rangle _{12}} \bigl(-\mathbf{1}^{3}{|\chi\rangle _{3}}\bigr)\biggr). \end{gathered}$$ We can see that system 1 is certainly no longer in the state ${|\chi\rangle _{}}$, it is in fact now thoroughly entangled and has the reduced state $1/2\,\mathbf{1}$. But also, crucially, relative to definite measurement outcome states of systems $c$ and $d$, Bob’s system now has a definite state related to the initial ${|\chi\rangle _{}}$. As everything is entangled, though, the *reduced* state of his system is still simply $1/2\,\mathbf{1}$. We need to disentangle system 3 from all the others; and preferably in such a way that it is left in the unknown state ${|\chi\rangle _{}}$. To do this, Bob will need to perform a conditional unitary operation, that is, an operation on his system that depends on the states of the two qubits recording the outcome of Alice’s measurement. As he can only apply operations locally (the standard assumption) systems $c$ and $d$ will need to be transported to him before the operation may be performed. Once $c$ and $d$ are with Bob, he may perform the following conditional unitary transformation, $U_{B}$: $$U_{B} = P^{cd}_{{|00\rangle _{}}}\otimes (i\sigma_{y}^{3})\, +\, P^{cd}_{{|01\rangle _{}}}\otimes\sigma_{x}^{3}\, + \,P^{cd}_{{|10\rangle _{}}}\otimes(-\sigma_{z}^{3})\, +\, P^{cd}_{{|11\rangle _{}}}\otimes (-\mathbf{1}^{3}),$$ where $P^{cd}_{{|00\rangle _{}}}$ is the projector onto ${|0\rangle _{c}}{|0\rangle _{d}}$, and so on. This will have the effect of disentangling 3 and leaving it in the unknown state ${|\chi\rangle _{}}$, as desired (note that $U_{B}$ does not depend on the identity of ${|\chi\rangle _{}}$): $$U_{B}{|\Psi\rangle _{\rm{tot}}} = \frac{1}{2}\biggr({|\phi^{+}\rangle _{12}}{|0\rangle _{c}}{|0\rangle _{d}} + {|\phi^{-}\rangle _{12}}{|0\rangle _{c}}{|1\rangle _{d}} + {|\psi^{+}\rangle _{12}}{|1\rangle _{c}}{|0\rangle _{d}} + {|\psi^{-}\rangle _{12}}{|1\rangle _{c}}{|1\rangle _{d}}\biggl){|\chi\rangle _{3}}.$$ [69]{} urlstyle \[1\][doi:\#1]{} \[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} and \[\]: “”, **, ****, pp. . \[\]: “”, **, ****, p. , . \[\]: “”, **, ****, pp. , \[\]: “”, in , and (eds.), **, : , pp. , \[\]: **, : . , , , , and \[\]: “”, **, ****, pp. . \[\]: **, chapter , : , pp. . \[\]: “”, **, ****, pp. . and \[\]: **, : . , and \[\]: “”, **, ****, pp. . \[\]: “”, **, ****, pp. . and \[\]: “”, **, ****, pp. , . and \[\]: “”, **, ****, p. . \[\]: **, : , edition. \[\]: “”, in , and (eds.), **, : , pp. , . and \[\]: “”, **, ****, pp. , . \[\]: **, : . \[\]: **, : , edition. \[\]: **, : . \[\]: “”, **, ****, pp. , . and \[\]: “”, **, ****, pp. , . \[\]: “”, **, ****, pp. . \[\]: **, : , edition. \[\]: **, : . \[\]: “”, **, ****, pp. . \[\]: “”, **, ****, pp. . , and \[\]: “”, **, ****, pp. . \[\]: “”, **, ****, pp. . and \[\]: “”, **, ****, pp. . , and \[\]: “”, **, ****, pp. . , and \[\]: “”, **, ****, pp. . \[\]: “”, . \[\]: “”, . \[\]: “”, in , and (eds.), **, Vienna Circle Institute Yearbook, : , pp. . \[\]: “”, **, ****, pp. . \[\]: **, : , edition. \[\]: “”, in , , , and (eds.), **, : , pp. , . \[\]: “”, **, ****, pp. , . and \[\]: “”, **, ****, pp. . and \[\]: “”, **, ****, pp. . \[\]: “”, **, ****, p. , . \[\]: . \[\]: “”, **, ****, p. . \[\]: “”, **, ****, pp. . \[\]: **, chapter , : . \[\]: **, : , edition. \[\]: “”, **, ****, pp. . \[\]: “”, in (ed.), **, : , pp. . \[\]: “”, **, ****, pp. . \[\]: “”, **, ****, pp. . \[\]: “”, **, ****, pp. . \[\]: “”, **, ****, pp. , . and \[\]: **, : , edition. \[\]: “”, in (ed.), **, : , \[\]: “”, **, ****, pp. . \[\]: **, Ph.D. thesis, , . \[\]: “”, **, ****, pp. . \[\]: **, Ph.D. thesis, , . \[\]: “”, **, ****, pp. , . and \[\]: “”, in and (eds.), **, : , pp. , . and \[\]: “”, . \[\]: **, Ph.D. thesis, . \[\]: “”, in , and (eds.), **, volume , , pp. . \[\]: **, : , . \[\]: “”, **, ****, pp. , . \[\]: “”, **, ****, pp. . \[\]: “”, **, ****, pp. . \[\]: **, : . and \[\]: “”, **, ****, pp. . [^1]: `[email protected]` [^2]: For elaboration of this claim, see @thesis [Chpt.1]. The point is also made in @nifpaper. [^3]: Helpful discussions of further conceptual aspects of teleporation, in particular concerning the relation of teleportation to nonlocality, may be found in @hardy:disentangling, @jon1 and @cliftonpope. @mermin:teleportation also provides an interesting perspective. [^4]: Notice that an equal mixture of the four possible post-measurement states of his system results in the density operator $1/2\, \mathbf{1}$. [^5]: Two bits are clearly sufficient, for the argument that they are strictly necessary, see @teleportation Fig.2. [^6]: Interestingly, it can be argued that quantum teleporation is perhaps not so far from the sci-fi ideal as one might initially think. @vaidman suggests that if all physical objects are made from elementary particles, then what is distinctive about them is their form (i.e. their particular state) rather than the matter from which they are made. Thus it seems one could argue that *objects* really are teleported in the protocol. [^7]: Or by her sending Bob a system in a state explicitly related to ${|\chi\rangle _{}}$ (@park:1970). [^8]: Surprisingly, perhaps, this point has occasioned some controversy (@conceptualinadequacy). See @supposed for a detailed discussion of the applicability of the Shannon information in the quantum context. [^9]: Published as @qcoding, @schumacher:jozsa. [^10]: I will not seek to defend these views in more detail here. See @thesis Part 1 *passim* and Section 1.2.3 esp. for further discussion. [^11]: Cf. @qcoding, @supposed Section 4.1. [^12]: Cf. @qcoding. [^13]: The argument parallels the one given by @teleportation to the effect that two full classical bits are required in teleportation. In essence, if Bob were able to gain more than two bits of information in the protocol, then even if he were not to wait for Alice to send him the pair of bits each time and simply guessed their values instead, then some information would still get across. [^14]: A resolution along these lines, tied also to an ensemble view of the quantum state (*vide infra*) has been suggested by @jon1 and @petermorgan.\[ccqs ensemble\] [^15]: Note that we will need to adjust our scenario slightly to incorporate this view. In our initial set-up, the source $A$ selected a sequence of states which were then teleported one by one to Bob. Now we imagine instead that following some particular output of $A$, an entire ensemble of systems is prepared in the pure state associated with that output; then this ensemble of systems — all in the same unknown pure state — is teleported. This adjustment is required because in our initial set-up for the teleportation procedure, the only way in which an ensemble of systems all in the same state could be teleported to Bob would be by setting the information of the source $A$ to zero, with the tiresomely paradoxical result that Bob could now determine the state all right, but would gain no information by doing so.\[noinfo\] [^16]: Note that we do sometimes talk of a flow of information; and we do say of many physical quantities that are not entities or substances — for example, energy, heat — that they flow. But there is no analogy between the two cases, for what this latter description means is that the quantities in question obey a local conservation equation. It is not clear that it is at all intelligible to suggest that information should obey a local conservation equation. Certainly, the concept of quantity of information that is provided by the Shannon theory does not give us a concept of a quantity it makes sense to suggest might obey such an equation. (On this, see Section \[study concluding\] below.) [^17]: Note that it will not always be fallacious to take features of a simulation to correspond to features of the simulated — if the features in question are explicitly *analogues* of features of the system or process being simulated. One should thus distinguish between i) simulations that involve analogues and ii) functional ‘black-box’, or input-output simulations. [^18]: For further discussion of Deutsch’s conception of quantum ‘parallel processing’, see @steane and @horsman. [^19]: Nor, for example, does the fact that there are protocols in which the state of a qubit can be substituted for an arbitrarily large amount of classical information (@lucien:substituting) imply that this large amount of information is really there in the qubit. [^20]: It is for this reason that it is natural to marry conservative classical quantity surveying with an ensemble view of the quantum state (see footnote \[ccqs ensemble\]), for then this objection would not go through — when the two positions are conjoined, not only is the information characterizing the state not available until the whole ensemble is teleported, but neither has the *state* been teleported until the whole ensemble has been teleported to Bob. [^21]: One of the defining features of what I here term ‘orthodoxy’ is the adoption of the standard eigenstate-eigenvalue link for the ascription of definite values to quantum systems. See e.g. @bub:1997. [^22]: See @erpart1 for a recent discussion. [^23]: An early version of the no-signalling theorem, specialised to the case of spin 1/2 EPR-type experiments appears in @bohm. Later, more general versions are given by @tausk [@eberhard; @grw:no-signalling]. See also @shimony, @redhead:no-signalling. [^24]: Of course, in a relativistic setting, rigid bodies would not be permissible, although they are in non-relativistic quantum mechanics. This does not in any case affect the point of the analogy. [^25]: It should be noted that there have been a number of different attempts to develop Everett’s original ideas into a full-blown interpretation of quantum theory. The most satisfactory of these would appear to be an approach on the lines of Saunders and Wallace (@simon1 [@simon2; @simon3; @simonrelativism; @wallace:worlds; @wallace:structure]) which resolves the preferred basis problem and has made considerable progress on the question of the meaning of probability in Everett (on this, see in particular @deutsch:decision [@wallace:rationality]). [^26]: This is the case for ideal first-kind (non-disturbing) measurements. The situation becomes more complicated when we consider the more physically realistic case of measurements which are not of the first kind; in some cases, for example, the object system may even be destroyed in the process of measurement. What is important for a measurement to have taken place is that measuring apparatus and object system were coupled together in such a way that if the object system had been in an eigenstate of the observable being measured prior to measurement, then the subsequent state of the measuring apparatus would allow us to infer what that eigenstate was. In this more general framework the importance is not so much that the object system is left in a eigenstate of the observable relative to the indicator state of the measuring apparatus, but that we have definite indicator states relative to macroscopic observables. [^27]: This would not in general be the case if the initial entangled state were not maximally entangled, or if Alice’s measurement were not an ideal measurement; with these eventualities, the teleportation would be imperfect (fidelity less than 1). [^28]: Note, though, that measurement may not usually be understood as revealing pre-existing values in the Bohm theory. Interestingly, @bw:2005 have recently argued that these definite position values may not be so helpful in solving the measurement problem as is often supposed. [^29]: This is the approach to spin of @bst:1955. For a systematic presentation see @bohmhiley [§10.2-10.3] or @holland:1995 [Chpt. 9]. Other approaches to spin are possible, e.g., @bohmhiley [§10.4-10.5], @holland:1995 [Chpt. 10], or the ‘minimalism’ of @bell:1966 [@bell:1981], in which no spin values are added. [^30]: The component of the force on the $i$-th system due to the quantum potential is given by $m_{i}\ddot{\mathbf{x}}_{i}=-\nabla_{i}Q$ (@holland:1995 [§7.1.2]); therefore, only terms in the sum which depend on $\mathbf{x}_{i}$ will contribute to the motion of the $i$-th system. [^31]: Cf. OED ‘information’, sense 7. [^32]: On some accounts, an action is the bringing about of some event or state of affairs by an agent (@hyman:alvarez); on others, an action is an event (@davidson:actionsandevents). On no account is an action something which can intelligibly be said to be moved about. [^33]: See @d'Espagnat for this terminology, also @impsep. [^34]: The types of processes in question might not be identifiable without recourse to concepts of what would count as successful transmission of information, but this does not mean that one has to conceive of information as an entity or substance, just that one needs a concept of what it means to receive a signal from which one can learn something.

--- abstract: 'We have studied the structure and kinematics of the dense molecular gas in the Orion Molecular Cloud 1 (OMC1) region with the $\rm{N_2H^+}$ 3–2 line. The $6'' \times 9''$ ($\sim 0.7 \times 1.1$ pc) region surrounding the Orion KL core has been mapped with the Submillimeter Array (SMA) and the Submillimeter Telescope (SMT). The combined SMA and SMT image having a resolution of $\sim 5.4''''$ ($\sim 2300$ au) reveals multiple filaments with a typical width of $0.02$–$0.03$ pc. On the basis of the non-LTE analysis using the $\rm{N_2H^+}$ 3–2 and 1–0 data, the density and temperature of the filaments are estimated to be $\sim 10^7$ $\rm{cm^{-3}}$ and $\sim 15$–$20$ K, respectively. The core fragmentation is observed in three massive filaments, one of which shows the oscillations in the velocity and intensity that could be the signature of core-forming gas motions. The gas kinetic temperature is significantly enhanced in the eastern part of OMC1, likely due to the external heating from the high mass stars in M42 and M43. In addition, the filaments are colder than their surrounding regions, suggesting the shielding from the external heating due to the dense gas in the filaments. The OMC1 region consists of three sub-regions, i.e. north, west, and south of Orion KL, having different radial velocities with sharp velocity transitions. There is a north-to-south velocity gradient from the western to the southern regions. The observed velocity pattern suggests that dense gas in OMC1 is collapsing globally toward the high-mass star-forming region, Orion Nebula Cluster.' author: - 'Yu-Hsuan Teng' - Naomi Hirano title: Physical Conditions and Kinematics of the Filamentary Structure in Orion Molecular Cloud 1 --- Introduction {#sec:intro} ============ Filamentary structure has been commonly observed in star-forming clouds from parsec scale to sub-parsec scale [e.g. @1979ApJS...41...87S; @2014prpl.conf...27A]. The prevalence of filamentary structure indicates its persistence for a large fraction of the lifetime of a star-forming cloud. Therefore, it is believed that such structure plays an important role in star formation process, and provides clues about the evolution of star-forming clouds. In addition, by examining the low-mass star-forming clouds within 300 pc, @hub_filament found that all young stellar groups are associated with “hub-filament structure”, where the “hub” is a high column density region harboring young stellar groups, and the “filaments” are elongated structures with lower column density radiating from the hub. Such structure also exists in some distant regions that form high-mass stars, although its incidence in massive star-forming regions is still unclear [@hub_filament]. The Orion A molecular cloud, a large-scale filament with an integral-like shape, is the nearest high-mass star-forming region at a distance of 414 pc [@omc1_dist]. The Orion Molecular Cloud 1 (OMC1), residing at the center of the Orion A, is the most massive component ($>2200 M_\odot$) and the most active star-forming region in the Orion Molecular Cloud [@omc_fil]. Previous VLA observations in $\rm{NH_3}$ [@omc1_nh3] revealed a typical hub-filament structure in OMC1, in which several filaments radiate from the Orion KL. These filaments appear to be hierarchical: the large-scale filament is consist of narrower filaments in small scale. Recent high-resolution observation with ALMA in $\rm{N_2H^+}$ J=1–0 [@omc1_alma] resolved a total of 28 filaments with a FWHM of $\sim 0.02$–$0.05$ pc in OMC1, and the cores inside these small-scale filaments are possible sites for star formation. Therefore, studying the physical conditions and gas motions in OMC1 is likely the key to understand the evolution of hub-filament structure and its relation with star formation. We present our observations toward the OMC1 region in $\rm N_2H^+$ 3–2 with an angular resolution of $\sim 5.4''$ obtained using the Submillimeter Array (SMA) and the Submillimeter Telescope (SMT). Further analyses using the $\rm N_2H^+$ 1–0 data provided by @omc1_alma are also presented. The rotational transitions of $\rm N_2H^+$ are known to be good tracers of dense and quiescent gas. With a critical density of $\sim 10^6$ $\rm cm^{−3}$, which is higher than that of ammonia studied by @omc1_nh3, $\rm N_2H^+$ 3–2 can probe the dense gas inside the sub-parsec scale filaments that are directly related to star formation. In addition, $\rm N_2H^+$ is less affected by depletion even in the dense and cold environments, and is also less affected by dynamic processes such as outflows and expanding H[ II]{} regions. This study investigates the structure, physical conditions, and gas motions in the OMC1 region. We describe the details of our observations and data reduction in Section \[sec:obs\], and present the results in Section \[sec:result\]. The structural properties, physical conditions and gas kinematics are analyzed in Section \[sec:analysis\]. Finally, we discuss the implications of these results in Section \[sec:discussion\], and summarize our conclusions in Section \[sec:conclusion\]. Observations and Data Reduction {#sec:obs} =============================== 1.1 mm Observations with the SMA -------------------------------- Observations of the $\rm N_2H^+$ J=3–2, $\rm HCO^+$ J=3–2, and HCN J=3–2 lines together with the 1.1 mm continuum were carried out with the SMA on February 14, 20, and 24, 2014. A sub-compact configuration with six antennas in the array was used, providing baselines ranging from 9.476 m to 25.295 m. The shortest and longest uv distances are 5.6 $\rm k\lambda$ and 23.6 $\rm k\lambda$, repectively. The primary beam of the 6-m antennas has a size of $42''$ (HPBW), and the synthesized beam size is $5.53'' \times 5.25''$. The bandwidth was $4$ GHz per sideband, and the frequency resolution was 203 kHz that corresponds to the velocity resolution of $\sim 0.22$ $\rm km\ s^{-1}$ at the rest frequency of $\rm N_2H^+$ 3–2. Using 144 pointing mosaic with a Nyquist sampled hexagonal pattern, as shown in Figure \[fig:pointing\], the observed area covered $\sim 5' \times 7'$. In order to obtain uniform $uv$-coverage for 144 pointngs, we observed each pointing for 5 seconds and visited all pointings in a loop. Each of the 144 pointings was visited three times in each observing run, giving a total on-source integration time per pointing of 45 seconds. The visibility data were calibrated using the MIR/IDL software package.[^1] The gain calibrators were 0501-019 and 0607-085, the flux calibrator was Ganymede on Feb. 14 and Callisto on other two days, and the bandpass calibrator was 3C279. The image processing was carried out using the MIRIAD package [@miriad]. The image cube was generated with Briggs weighting with a robust parameter of 0.5, followed by a nonlinear joint deconvolution using the CLEAN-based algorithm, MOSSDI. The final data cube has a rms noise level of $\sim 0.5$ K for a $\sim 0.22$ $\rm km\ s^{-1}$ velocity channel. In this paper, we focus on the results and analyses of $\rm N_2H^+$ line. Results of $\rm HCO^+$ and HCN will be presented in a forthcoming paper. 1.1 mm Observations with the SMT -------------------------------- We simultaneously observed the $\rm N_2H^+$ J=3–2 and $\rm HCO^+$ J=3–2 lines on November 16 and 17, 2018 using the SMT of the Arizona Radio Observatory. We used the SMT 1.3 mm ALMA band 6 receiver and the filter-bank backend. The beam size is $28.45''$ in HPBW at the frequency of $\rm N_2H^+$ 3–2, and the main-beam efficiency is $0.71 \pm 0.05$. The On The Fly (OTF) mode was used in order to cover the mapping area of $6' \times 9'$ centered at R.A.(J2000) = $\rm 5^h 35^m 12^s.1$ and Decl.(J2000) = $-5^\circ 21' 15''.4$. The data were reduced with CLASS.[^2] The rms noise levels are $\sim 0.3$ K at a spectral resolution of 250 kHz ($\sim 0.27$ $\rm km\ s^{-1}$). SMA and SMT Data Combination ---------------------------- We combined the data cubes of the SMA and SMT by using the MIRIAD task *immerge*. The method of this task, known as feathering, merges linearly two images with different resolutions in their Fourier domain (i.e. spatial frequency). In the case of combining the single-dish and mosaicing data, *immerge* gives unit weight to the single-dish data at all spatial frequencies, and tapers the low spatial frequencies of the mosaicing data, so as to produce the gaussian beam of the combined data equal to that of the mosaicing data. Inputs of *immerge* include the “CLEANed” SMA image cube, SMT image cube, and the flux calibration factor. After checking the consistency in the flux scale of two input image cubes, we set the flux calibration factor to 1. The integrated intensity (moment 0) map and the intensity-weighted radial velocity (moment 1) map after combination are shown in Figure \[fig:obs\_combine\]. The angular resolution and the rms noise level of the combined map are $\sim 5.4''$ and $1.0$ $\rm K \cdot km\ s^{-1}$, respectively. $N_2H^+$ J=1–0 Observations with ALMA and IRAM 30-m --------------------------------------------------- To analyze the physical properties of the filamentary structure in high resolution, we use the $\rm N_2H^+$ 1–0 data cube provided by @omc1_alma, where the ALMA and IRAM 30-m data were combined.[^3] The ALMA + IRAM 30-m combined image cube has a circular beam with a size of $4.5''$ in FWHM, and a rms level of $25$ $\rm mJy\ beam^{-1}$ at a spectral resolution of $0.1$ $\rm km\ s^{-1}$. The observational details are described in @hacar_2017 [@omc1_alma]. Results {#sec:result} ======= {width="\linewidth"} {width="\linewidth"} Figure \[fig:obs\_combine\] presents the moment 0 (integrated intensity) and moment 1 (intensity-weighted radial velocity) maps of the SMA + SMT combined image. Figure \[fig:obs\_combine\]a shows that most of the emission comes from the filamentary structure having a typical FWHM of $0.02$–$0.03$ pc. Several high-intensity and clumpy structures can also be seen inside the filaments. Orion KL at R.A.(J2000) = $\rm 5^h 35^m 14^s.5$ and Decl.(J2000) = $-5^\circ 22' 30''$ is near the center of the maps. Different from observations in continuum or most other molecular lines, there is no significant $\rm{N_2H^+}$ emission from the Orion KL region due to the destruction of $\rm N_2H^+$ molecules in active regions. Figure \[fig:obs\_combine\]b reveals that the radial velocity distribution shows a trimodal pattern; the bright filaments to the north of Orion KL have a velocity range of $\sim$9–11 km s$^{-1}$ (hereafter, referred to as the northern region), the fainter filaments extending to the northwest are seen at $\sim$7–9 km s$^{-1}$ (western region), and the ones to the south of Orion KL are at $\sim$5–7 km s$^{-1}$ (southern region, also known as OMC1-South). These three regions with different velocities converge at the Orion KL region. The velocity difference between the northern and western region was also reported by @omc1_nh3 and @Monsch_2018 based on their NH$_3$ observations. A clearer analysis for the velocity transition among the three regions will be presented in Section \[subsec::fitting\]. In order to derive physical conditions in different sub-regions, we use the ALMA + IRAM 30-m image in $\rm N_2H^+$ 1–0 for analyses in Section \[subsec::non-lte\]. The high-resolution 3–2/1–0 ratio map (Figure \[fig:ratio\]) was made using the SMA + SMT image and the ALMA + IRAM 30-m image convolved to the same beam size as the SMA + SMT image. Figure \[fig:ratio\] reveals that the 3–2/1–0 ratio is overall higher in the eastern part of OMC1. In addition, the 3–2/1–0 ratio tends to be lower in the filament regions as compared to the surrounding non-filament regions. The typical line ratio in the filament regions is $\sim 1.0$ even in the eastern part of OMC1, while that of the non-filament region is $\sim 2.2$. As different ratios may imply different physical conditions, we determine the physical parameters of the filament and non-filament regions in Section \[subsec::non-lte\]. ![$\rm N_2H^+$ 3–2/1–0 intensity ratio map using 1–0 image observed with ALMA + IRAM 30-m [@omc1_alma]. Contour levels of $\rm N_2H^+$ 3–2 moment 0 are overlaid on the line ratio map.[]{data-label="fig:ratio"}](ratio_mom0_overlap_bar.pdf){width="\linewidth"} Analysis {#sec:analysis} ======== Structural Properties of OMC1 {#sec::structure} ----------------------------- {width="\linewidth"} {width="\linewidth"} \(b) Positions of the 10 cores identified using *2-D Clumpfind*. \[fig:structure\] ### Filament Identification Filament identification is done by applying the python package *FilFinder* [@filfinder] to the SMA + SMT combined moment 0 map. The *FilFinder* algorithm segments filamentary structure by using adaptive thresholding, which performs thresholding over local neighborhoods and allows for the extraction of structure over a large dynamic range. Input parameters for *FilFinder* include (1) global threshold—minimum intensity for a pixel to be included; (2) adaptive threshold—the expected full width of filaments for adaptive thresholding; (3) smooth size—scale size for removing small noise variations; (4) size threshold—minimum number of pixels for a region to be considered as a filament. We set 9 $\rm K \cdot km\ s^{-1}$ as the global threshold. To focus on filaments with length $\gtrsim 0.1$ pc, we set the size threshold as 400 square pixels, where the pixel size is 1 $\rm arcsec^2$. The adaptive threshold is set to 0.06 pc, which is approximately twice the width of the filaments, and the smooth size is set to 0.03 pc. We find the result matches better with identification by human eyes when the smooth size is set to $\sim 0.5$ times the adaptive threshold. The smaller we set the smooth size, the more short branches would be identified. On the other hand, a larger smooth size tends to make filaments connected since a larger regions of data are smoothed. In total, 11 filaments have been identified. Figure \[fig:structure\]a shows the result of filament identification, where the gray-scale image is the combined moment 0 map, and the colored lines are the identified major axes for each filament. The identified filaments and their lengths and widths are listed in Table \[tab:fil\_ident\], together with the physical properties described in Section \[subsec:core\_formation\]. Since the hyperfine components in $\rm N_2H^+$ 3–2 cannot be separated, the identification is based on the moment 0 map instead of 3-D datacube. We have estimated the typical FWHM of the identified filaments by fitting Gaussian to the several cuts perpendicular to the filaments. The FWHM of these filaments are $0.02$–$0.03$ pc, which is consistent with those identified in @omc1_alma based on the ALMA + IRAM 30-m observations in $\rm N_2H^+$ 1–0. @omc1_alma has identified the filaments in Orion A with a different algorithm based on the 3-D datacube. Since we identified the filaments in 2-D using the moment 0 map, the results are different from those of 3-D if there are multiple velocity components in the same line of sight. For example, our main filament and filament 6, which can be clearly seen in the moment 0 map are not identified by @omc1_alma. As shown in Figure \[fig:spec\]a and \[fig:spec\]b, these two filaments contain multiple velocity components along the line of sight, which could cause a difference between results of 2D- and 3D-based algorithms. Apart from these filaments, other OMC1 filaments show single velocity component in the spectra. Therefore, there is no significant difference in the results between 2-D and 3-D identifications. The moment 0 images in $\rm N_2H^+$ reveal three filaments with high-intensity clumpy cores, one of which is in the OMC1-South filament 8. We will refer to the two prominent filaments in the northern region as the *main filament* and the *east filament*, which are shown in blue and yellow, respectively, in Figure \[fig:structure\]a. The gas kinematics inside these filaments will be analyzed in Section \[subsec:kinematics\]. ---------- ----------------- -------------------- ---------------------- -------------------- ---------------------------- ---------------------- ------------------------ Filament Length $\rm Width_{FWHM}$ $\rm M_{lin}$ $\Delta v$ $\Delta v_{\rm nt}/c_s(T)$ $\rm M_{crit}$ $\rm M_{lin}/M_{crit}$ ($\pm 0.001$pc) ($\pm 0.001$pc) ($M_\odot\ pc^{-1}$) ($\rm km\ s^{-1}$) ($M_\odot\ pc^{-1}$) Main 0.349 $0.025$ 94.2–101.7 $1.0\pm0.2$ 1.5 119.7 0.79–0.85 East 0.365 $0.023$ 78.5–85.6 $0.9\pm0.3$ 1.3 103.8 0.76–0.83 1 0.159 $0.024$ 84.0 $0.6\pm0.2$ 0.85 66.0 1.27 2 0.100 $0.023$ 76.2 $0.7\pm0.3$ 1.0 76.9 0.99 3 0.279 $0.017$ 42.5 $1.3\pm0.6$ 1.9 177.6 0.24 4 0.303 $0.020$ 61.8 $1.1\pm0.3$ 1.6 137.3 0.45 5 0.155 $0.024$ 84.0 $1.0\pm0.5$ 1.5 119.7 0.70 6 0.124 $0.033$ 166.3 $1.1\pm0.3$ 1.6 137.3 1.21 7 0.175 $0.022$ 68.9 $1.0\pm0.4$ 1.5 119.7 0.58 8 0.237 $0.022$ 81.5–121.0 $2.0\pm0.9$ 2.95 371.6 0.22–0.33 9 0.159 $0.017$ 42.5 $1.0\pm0.3$ 1.5 119.7 0.36 ---------- ----------------- -------------------- ---------------------- -------------------- ---------------------------- ---------------------- ------------------------ ### Core Identification {#subsubsec::core_ident} We identify the high-intensity cores inside the filaments by using the 2-D version of *Clumpfind* [@clfind]. The *Clumpfind* algorithm contours input data with the values assigned by users, and then distinguishes each core region by those values. It starts from the highest contour level, and then works down through the lower levels, finding new cores and extending previously defined ones until the lowest contour level is reached. We use the combined SMA + SMT integrated intensity map as the input data, and set the contour levels as 50, 53.6, 57.2, 60.8, 64.4 $\rm K \cdot km\ s^{-1}$. The spacing of the contour levels are set constantly as $\Delta T = 2 T_{\rm rms} = 3.6$, which could lower the the percentage of false detection to $< 2\%$ as suggested in @clfind. Figure \[fig:structure\]b shows the positions of the 10 cores identified by *2-D Clumpfind*. Table \[tab:core\_ident\] lists the properties of these cores including position, peak flux, and effective radius ($\rm R_{\rm eff}$), which are all direct outputs of *2-D Clumpfind*. Note that the $R_{\rm eff}$ in *Clumpfind* is defined as $R_{\rm eff} = \sqrt{A/\pi}$, where $A$ is the area with emission above the criteria. In Table \[tab:core\_ident\], the linewidth ($\Delta v$) is determined by hyperfine spectral fitting (see Section \[subsec::fitting\]), the mass ($\rm M_{core}$) is derived by using the densities determined from non-LTE analysis (see Section \[subsec::non-lte\]), and the virial mass ($\rm M_{vir}$) is calculated from the linewidth using Equation \[def\_Mvir\] (see Section \[subsec:core\_formation\]). ------ ------------- ------------ ---------------------------- --------------- -------------------- ---------------- --------------- -- Core R.A. Decl. Peak flux $R_{\rm eff}$ $\Delta v$ $\rm M_{core}$ $\rm M_{vir}$ (J2000.0) (J2000.0) ($\rm K \cdot km\ s^{-1}$) (arcsec) ($\rm km\ s^{-1}$) ($M_\odot$) ($M_\odot$) 1 05:35:13.10 -5:24:12.4 84.3 8.9 3.5 14.4–45.6 46.1 2 05:35:13.70 -5:23:52.4 69.4 5.6 2.8 3.6–11.3 18.5 3 05:35:14.10 -5:23:41.4 66.5 4.7 1.6 2.0–6.4 5.0 4 05:35:15.37 -5:22:04.4 66.0 5.1 2.0 2.6–8.2 8.5 5 05:35:15.84 -5:20:38.4 63.4 3.5 0.9 0.9–2.8 1.2 6 05:35:16.90 -5:19:27.4 61.9 4.0 – 1.3–4.2 – 7 05:35:16.70 -5:19:34.4 60.8 4.4 – 1.7–5.3 – 8 05:35:16.84 -5:20:42.4 59.2 3.6 2.2 0.9–3.0 7.4 9 05:35:17.37 -5:19:17.4 58.7 4.2 – 1.5–4.7 – 10 05:35:15.77 -5:21:24.4 57.1 2.6 1.1 0.4–1.1 1.3 ------ ------------- ------------ ---------------------------- --------------- -------------------- ---------------- --------------- -- Hyperfine Spectral Fitting {#subsec::fitting} -------------------------- To determine the physical parameters, we conduct hyperfine spectral fitting on the combined SMA and SMT data for the regions with $S/N > 5$. We fit the $V_{\rm LSR}$, linewidths ($\Delta v$), excitation temperatures ($T_{\rm ex}$), and total opacities ($\tau_{\rm tot}$) under the assumption of local thermodynamic equilibrium (LTE). With the relative intensities among 16 main hyperfine components for the $\rm N_2H^+$ 3–2 line [@hfs_32], the opacity of each component is assumed as a Gaussian profile $$\begin{aligned} &&\tau_i(v) = \tau_i \exp \left[-4 \ln 2 \left( \frac{v - v_i - v_{\rm sys}}{\Delta v} \right)^2 \right] \label{eqn_tau_i} \end{aligned}$$ where $v_i$ is the velocity offset from the reference component and $v_{\rm sys}$ is the systemic velocity. Then, we obtain the optical depths of the multiplets as $$\begin{aligned} &&\hspace{-0.3in}\tau(v) = \tau_{\rm tot} \sum_{i=1}^{16} R_i \exp \left[-4 \ln 2 \left( \frac{v - v_i - v_{\rm sys}}{\Delta v} \right)^2 \right] \label{eqn_tau_v}\end{aligned}$$ where $R_i$ is the relative intensity for the $i$th hyperfine component, and $\tau_i = \tau_{\rm tot} R_i$. The brightness temperature at each pixel can be represented as $$\begin{aligned} &&T_{b}(v) = [J(T_{\rm ex}) - J(T_{\rm bg})] [1 - \exp(-\tau(v))] \label{Tb_lte}\end{aligned}$$ where $T_{\rm bg}$ is the cosmic background temperature (2.73 K), and $$\begin{aligned} &&J(T) = \frac{\frac{h \nu}{k}}{\exp \left( \frac{h \nu}{k T} \right) - 1} \label{def_J}\end{aligned}$$ Figure \[fig:spec\] presents the observed spectra obtained by averaging the $5'' \times 5''$ regions marked in Figure \[fig:structure\]a. The best-fit single-velocity components (in red) are overlaid on the spectra. Note that the multiple velocity components are only limited to the regions near core 6, 7, 9 and the northern part of filament 6, as indicated by the dashed green boxes in Figure \[fig:structure\]a. We applied the single-component fitting to the spectra of these regions, because two-component fitting requires too many parameters. Inside the regions with multiple components, the fitted $V_{\rm LSR}$ are still dominated by the major components, although the fitted linewidths are highly affected by secondary components. ---------------------------------------------------- ------------------------------------------------- ------------------------------------------------- -- {width="0.32\linewidth"} {width="0.32\linewidth"} {width="0.32\linewidth"} {width="0.32\linewidth"} {width="0.32\linewidth"} {width="0.32\linewidth"} ---------------------------------------------------- ------------------------------------------------- ------------------------------------------------- -- Using the centroid velocities determined from the hyperfine fitting, we illustrate the velocity distribution along right ascension and declination, respectively. Figure \[fig:radec2vel\]a shows the radial velocities for all the $\rm{N_2H^+}$ 3–2 components fitted in the northern and western regions. The figure reveals a sharp velocity transition near R.A.(J2000) = $\rm 5^h 35^m 14^s$, where a clear boundary can also be seen in Figure \[fig:obs\_combine\]b between the northern and western regions. Figure \[fig:radec2vel\]b shows the radial velocities of all three regions as a function of declination. While the velocities of the northern region are almost constant, those of the western region decrease from north to south and continuously connected to those of the southern region. Such a velocity decrease from north to south in OMC1 has been reported in @hacar_2017 [@omc1_alma], and interpreted as the gravitational collapse of OMC1. Figure \[fig:radec2vel\]b shows that the gas in the western region is accelerated toward OMC1-South. {width="\linewidth"} {width="\linewidth"} Comparing the fitting results with the observed spectra, we found that the observed satellite components are much brighter than those predicted by the LTE assumption (e.g. Figure \[fig:spec\]c–e). In addition, when the line is optically thin ($\tau_{\rm tot} \ll 1$), the fitting cannot constrain $T_{\rm ex}$ and $\tau_{\rm tot}$, because Equation \[Tb\_lte\] will be reduced to $$\begin{aligned} &&T_{b}(v) \simeq \tau_{\rm tot} \cdot T_{\rm ex} \label{eqn_Tmb_thin}\end{aligned}$$ indicating that the $T_{\rm ex}$ and $\tau_{\rm tot}$ can be determined arbitrarily. Therefore, we conduct non-LTE analysis in Section \[subsec::non-lte\] to derive the physical conditions. {width="\linewidth"} $N_2H^+$ RADEX Non-LTE Modeling {#subsec::non-lte} ------------------------------- Using RADEX [@radex], a non-LTE radiative transfer code, we construct spectra models in $\rm N_2H^+$ 3–2 and 1–0. The synthetic spectra are constructed with the equation $$\begin{aligned} &&T_{b}(v) = \Psi \left( \frac{\sum J(T_{\rm ex}^i)\,\tau_i(v)}{\sum \tau_i(v)} - J(T_{\rm bg}) \right) \left( 1 - e^{-\sum \tau_i(v)} \right) \nonumber \\ \label{Tb_non-lte}\end{aligned}$$ where $T_{\rm ex}^i$ and $\tau_i(v)$ represent the excitation temperature and optical depth for all hyperfine components in the 3–2 or 1–0 transitions, and $\Psi$ is the beam filling factor. In our models, the beam filling factors are assumed to be unity. We also construct an intensity ratio model by dividing the integrated spectra model in $\rm N_2H^+$ 3–2 with that in $\rm N_2H^+$ 1–0. The constructed models can be represented by a three-dimensional grid, where the three axes are $\rm H_2$ density ($n_{\rm H_2}$) ranging from $10^4$ to $10^9$ $\rm cm^{-3}$, kinetic temperature ($T_{\rm kin}$) ranging from $8$ to $60$ K, and the ratio of $\rm N_2H^+$ column density to linewidth ($N(\rm{N_2H^+})$/$\Delta v$) ranging from $5\times10^{7}$ to $5\times10^{8}$ $\rm s\ cm^{-3}$. The step sizes of the grid are 1 K for $T_{\rm kin}$ and 0.5 in decimal log scale for both $N(\rm N_2H^+)$ and $n_{\rm H_2}$. As $N(\rm{N_2H^+})$/$\Delta v$ is the input parameter in RADEX, the estimation of $N(\rm N_2H^+)$ varies among regions with different linewidths. Based on our fitting results, the linewidth in OMC1 could vary from $\sim$ 1 to 3 $\rm km\ s^{-1}$. Figure \[fig:non\_lte\] presents the constructed non-LTE model, where the gray-scale background shows the integrated spectra model in $\rm N_2H^+$ 3–2, and the contour levels show the 3–2/1–0 ratio model. By applying the observed integrated intensity and line ratio in this model, physical parameters ($n_{\rm H_2}$, $T_{\rm kin}$, and $N(\rm{N_2H^+})$/$\Delta v$) in different sub-regions can be constrained. For instance, the two bars on the first panel of Figure \[fig:non\_lte\] indicate the two possible solutions that satisfy the conditions with an intensity of 7–15 $\rm K \cdot km\ s^{-1}$ and a line ratio of $2.2 \pm 0.4$. The corresponding physical conditions of the left and right bars are $n_{\rm H_2} = 10^6$ $\rm cm^{-3}$ and $T_{\rm kin}$ = 45–60 K, and $n_{\rm H_2} = 3 \times 10^6$ $\rm cm^{-3}$ and $T_{\rm kin}$ = 21–30 K, respectively. ----------------------------------------- ---------------------------------- --------------------------- --------------------------- Core Regions Low Intensity Non-filament ($>50$ $\rm K \cdot km\ s^{-1}$) Regions Regions $n_{\rm H_2}$ ($\rm cm^{-3}$) $10^7$ or $3 \times 10^7$ $3 \times 10^6$ or $10^7$ $10^6$ or $3 \times 10^6$ $T_{\rm kin}$ (K) $19$–$23$ or $18$–$20$ $17$–$22$ or $13$–$16$ $>45$ or $21$–$30$ $N(N_2H^+)/\Delta v$ ($\rm s\ cm^{-3}$) $5\times10^{8}$ $1.5 \times 10^{8}$ $5\times10^{7}$ Intensity ($\rm K \cdot km\ s^{-1}$) 50–60 20–40 7–15 Typical Ratio $1 \pm 0.3$ $1 \pm 0.3$ $2.2 \pm 0.4$ ----------------------------------------- ---------------------------------- --------------------------- --------------------------- We use our SMA + SMT data in $\rm N_2H^+$ 3–2 together with the ALMA + IRAM 30-m data in $\rm N_2H^+$ 1–0 for the non-LTE analysis, where the scale size is $5.4''$ ($\sim0.01$ pc). Based on the 3–2 moment 0 map, we define three regions for analysis. The first region is defined as the core regions with integrated intensity 50–60 $\rm K \cdot km\ s^{-1}$ and a 3–2/1–0 ratio of $1 \pm 0.3$. The second region is the lower intensity regions inside the filaments (20–40 $\rm K \cdot km\ s^{-1}$) with a similar line ratio of $1 \pm 0.3$, and the third region is the non-filament region with low intensities (7–15 $\rm K \cdot km\ s^{-1}$) and a higher line ratio of $2.2 \pm 0.4$. The spectra averaged in these sub-regions are compared with the model spectra. The derived physical parameters for the non-LTE analysis together with the criteria of the three sub-regions are listed in Table \[tab:high-res\]. By comparing between the filament and non-filament regions, we find that the filament regions have a higher density of $\sim 10^7$ $\rm cm^{-3}$ and a lower temperature of $\sim 15$–$20$ K than the non-filament regions. As the major heating sources may come from outside the filaments, it is likely that the dense gas in the filaments could block the outer radiation, leading to a lower temperature in the filaments (see Section \[subsec:heating\] for further discussion). Inside the filaments, the core regions have a higher $\rm N_2H^+$ column density than the low intensity regions, if we assume similar linewidths in these regions. Also, the volume density of the core regions are generally higher than the low intensity regions. On the other hand, there is no significant difference in temperature between the core and the low intensity regions, although the derived $T_{\rm kin}$ is slightly higher in the core regions. Using the volume densities determined from the non-LTE analysis and the filament widths, we have estimated the masses of the cores and the line masses of the filaments under the assumption of uniform cylindrical filaments. We adopted this method instead of using the column density of N$_2$H$^+$ because the fractional abundance of N$_2$H$^+$ is highly uncertain in the regions close to Orion KL and because the inclination of each filament is also uncertain. The density is assumed to be $n_{\rm H_2} = 3\times10^6$ $\rm cm^{-3}$ for the filaments without cores. For those with cores, i.e. main filament, east filament, and filament 8, we use two possible core densities (i.e. $10^7$ or $\rm 3\times10^7 cm^{-3}$) determined in this section for the core regions and $n_{\rm H_2} = 3\times10^6$ $\rm cm^{-3}$ for the regions outside the cores. If we adopt the higher density solution, i.e. $10^7$ $\rm cm^{-3}$, for the low intensity regions of the filaments, the line masses of the filaments could be higher by a factor of $\sim$3. On the other hand, if we adopt the radial density profile of the isothermal cylinder with the same central density [e.g. @Mcrit_2], the line mass included in the radius of 0.01–0.015 pc (i.e. half of the filament width) becomes lower by a factor of two. Properties of the identified filaments and cores are summarized in Table \[tab:fil\_ident\] and Table \[tab:core\_ident\], respectively, and will be discussed in Section \[subsec:core\_formation\]. Gas Kinematics of the Filaments {#subsec:kinematics} ------------------------------- Characterizing the gas motion inside the filaments requires the studies of velocity structure. We investigate the radial velocity fields along both the major and minor axes of the filaments in OMC1, and compare our results with existing filament formation model and core formation model. Since the main filament and the east filament are identified with core fragmentation, we focus on the analyses of these two filaments. {width="\linewidth"} {width="\linewidth"} ### Minor-Axis Analysis {#subsubsec::minor} Systematic velocity gradients perpendicular to the filaments have been observed in the filaments of both low- and high-mass star-forming regions [e.g. @serpens_ngc1333; @schneider_2010; @beuther_2015]. Such velocity gradients across the filaments can be explained by the projection of gas accretion toward the filament axes [@serpens_ngc1333] or the rotation of filaments [@fil_rotate]. In order to assess whether the OMC1 filaments show similar features, we analyze the velocity fields along the minor axes. In the main and the east filaments, there is no systematic velocity gradient along the minor axes. However, local velocity gradients of $\sim 0.3$ $\rm km\ s^{-1}$ are observed in core 4 and 8 in the east filament. On the other hand, there is no significant velocity gradient in core 5 and 10. The velocity gradients in core 6, 7, and 9 in the northern part of the main filament are unclear due to the secondary velocity component along the line of sight. Figure \[fig:v\_minor\] shows Position–Velocity (P–V) diagrams across core 8 and core 4. The directions of velocity gradients across these cores are not consistent; the velocity increases from east to west in core 8, while it decreases in core 4. Such velocity gradients with different directions along the same filament have also been observed along the massive DR21 filament, although its origin is still unclear [@schneider_2010]. Using the effective radii determined in Section \[subsubsec::core\_ident\], i.e. $R_{\rm eff} = 5.1''$ and $3.6''$ for core 4 and core 8, respectively, the velocities at $R_{\rm eff}$ were determined from the spectral fitting. Then, the velocity gradient ($\nabla V$) along the east-western direction was estimated using the velocity difference at the core boundaries and the effective diameter. Assuming a linear velocity variation, core 8 and core 4 have $\nabla V = 17.2$ $\rm km\ s^{-1}\ pc^{-1}$ and $\nabla V = -11.3$ $\rm km\ s^{-1}\ pc^{-1}$, respectively. The origin of the observed velocity gradients with different directions are likely to be the local effects such as rotating motions or unresolved multiple components. If the observed velocity fields come from rotations, the rotational energy $E_{\rm rot}$ can be estimated under the assumption of solid-body rotation as [@rotation] $$\begin{aligned} &&E_{\rm rot} = \frac{1}{2} I \Omega^2 = \frac{1}{2} \left[\frac{2}{3} MR^2 \left( \frac{3-\alpha}{5-\alpha} \right)\right] \left(\frac{|\nabla V|}{\sin{i}}\right)^2 \label{rot_energy}\end{aligned}$$ where $i$ is the inclination angle of the rotational axis to the line of sight, and $\alpha$ indicates a power-law density profile of $\rho \propto r^{-\alpha}$. The gravitational energy can also be derived as $$\begin{aligned} &&E_{\rm grav} = -\frac{GM^2}{R} \left( \frac{3-\alpha}{5-2\alpha} \right) \label{grav_energy}\end{aligned}$$ To estimate the ratio of rotational to gravitational energy ($\beta_{\rm rot}$) for these cores, we assume a uniform density profile, i.e. $\alpha = 0$, and a random-averaged inclination angle of $\langle \sin i \rangle = \pi / 4$. It turns out that core 8 has $\beta_{\rm rot}$ ranging from 0.04 to 0.11, and core 4 has $\beta_{\rm rot}$ ranging from 0.02 to 0.06. This means that rotations of these cores are significant but not dominant in the energetics. ### Major-Axis Analysis {#subsubsec::major} {width="\linewidth"} {width="\linewidth"} The velocity fields along the major axes of the filaments are also analyzed. Using the central velocity ($V_{\rm LSR}$) and the peak intensity determined from the hyperfine spectral fitting (see Section \[subsec::fitting\]), we plot the intensity and velocity variations along the filament major axis in Figure \[fig:v\_major\], where the horizontal axes represent the offset (from north to south) in arcseconds, and the vertical axes show the peak intensity on the left and the $V_{\rm LSR}$ on the right. Analysis on the east filament (see Figure \[fig:v\_major\]a) reveals the oscillations in both intensity and velocity. It is found that there are positional shifts between the intensity and velocity peaks toward core 8 and 4. This is similar to the feature observed in two of the filaments in L1517 [@Hacar_Tafalla_2011]. The velocity oscillation along the filament can be related to the core-forming motions. If the gas flow is converging to the center of the core, its velocity with respect to the one at the core center is positive in one side and negative in the other side of the density peak. According to the kinematic model proposed in @Hacar_Tafalla_2011, where sinusoidal perturbations were assumed for both density and velocity, a $\lambda / 4$ phase shift between the two distributions is predicted. In Figure \[fig:v\_major\]a, we show the sinusoidal fits to the velocity oscillation in the east filament. In addition, the locations of the intensity peaks toward core 8 and 4 match well with the $\lambda / 4$ shift from the two corresponding sinusoidal peaks. Figure \[fig:v\_major\]b shows the intensity and velocity plot for the main filament. Although there is a secondary velocity component toward part of this filament, the velocities determined from the fitting represent the ones of the major component. In the main filament, the relations between the intensity and velocity variations are unclear. This is probably because of the evolutionary stage of the main filament. Previous 1.3 mm observations with the SMA [@sma_north_fil] revealed CO molecular outflows associated with some cores in the main filament, including core 5. It is therefore likely that some of the cores in the main filament have already harbor young protostars in the Class 0 stage. In contrast, no evidence of protostars have been observed in the east filament, and the positional shift between its intensity and velocity peaks may indicate that core formation is still ongoing in the east filament. As the evolutionary stage of different filaments can vary [@sf_fil_model], it is possible that the east filament is in an earlier evolutionary phase than the main filament with star formation signature. Discussion {#sec:discussion} ========== Filament and Core Properties {#subsec:core_formation} ---------------------------- As shown in Table \[tab:fil\_ident\], all three filaments with core fragmentation have line masses $\gtrsim 80$ $\rm M_\odot\ pc^{-1}$. In contrast, the filaments with lower line masses such as filament 3, 4, and 9 do not contain cores. However, three filaments having the line masses close to 80 $\rm M_\odot\ pc^{-1}$ (filament 1, 2, and 5) and filament 6 with the largest line mass do not contain any cores. Therefore, although line masses are often used as an indicator of the star formation stage of a filament [@Heitsch_2013; @Palmeirim_2013; @Li_2014], it may not be a conclusive discriminator, which is also stated in @serpens_ngc1333. To investigate the internal dynamics of the filaments, we derive the non-thermal velocity dispersion ($\Delta v_{\rm nt}$) by using the equation $$\begin{aligned} &&\Delta v_{\rm nt} = \sqrt{\frac{(\Delta v)^2}{8\ln{2}} - \frac{k \cdot T_{\rm kin}}{\mu(\rm{N_2H^+})}} \label{def_v_nt}\end{aligned}$$ where $\Delta v$ is the linewidth in FWHM obtained from the hyperfine fitting, and $\mu$ is the molecular mass. The non-thermal velocity dispersion can be compared with the thermal sound speed $$\begin{aligned} &&c_s(T) = \sqrt{\frac{k \cdot T_{\rm kin}}{\mu(\rm{H_2})}} \label{def_v_th}\end{aligned}$$ Then, the critical line mass for an infinite filament in hydrostatic equilibrium can be calculated as [@Mcrit_1; @Mcrit_2] $$\begin{aligned} &&M_{\rm crit} = \frac{2(\Delta v_{\rm eff})^2}{G} = \frac{2}{G} \left[c_s(T)^2 + (\Delta v_{\rm nt})^2 \right] \label{def_Mcrit}\end{aligned}$$ where $\Delta v_{\rm eff}$ is defined as the effective velocity dispersion considering both thermal and non-thermal effects. Based on the non-LTE results in Section \[subsec::non-lte\], we take $T_{\rm kin} = 20$ K for estimation, which leads to $c_s(T) = 0.287$ $\rm km\ s^{-1}$. In the case of the purely thermally-supported filament, $\rm M_{crit} = 38.4\ M_\odot\ pc^{-1}$. Due to the rather large non-thermal velocity dispersion, the critical line masses listed in the OMC1 filaments are $\sim$2–4 times larger than the purely thermally-supported case. Table \[tab:fil\_ident\] reveals that 7 of the 11 filaments have $\rm 0.5 \le M_{lin}/M_{crit} \le 1.5$, suggesting that most of the filaments are gravitationally bound. Filament 3, 4, and 9 having low line masses are found to have $\rm M_{lin}/M_{crit} < 0.5$ and thus may be gravitationally unbound. On the other hand, filament 8 has $\rm M_{lin}/M_{crit} < 0.5$ even though this filament contains cores. This filament resides in OMC1-South, where a cluster of young stellar objects (YSOs) have already been formed [@zapata_2004; @zapata_2006]. Due to the powerful outflows from these YSOs, the gas in this region is highly turbulent. Thus, filament 8 has the largest linewidth of $\sim 3$ $\rm km\ s^{-1}$ among all the filaments, leading to a high $\rm M_{crit}$ and a low $\rm M_{lin}/M_{crit}$ ratio. The low $\rm M_{lin}/M_{crit}$ value could imply that filament 8 is in the phase of disruption. Another possibility is the high external pressure in this region. If the filament is confined by external pressure, the filament is prone to fragmentation even though its line mass is smaller than the critical line mass . Since filament 8 in OMC1-South is adjacent to the Orion Nebula Cluster, the high external pressure from hot and diffuse gas in the cluster could lead the core formation in this filament. Our results show that the filaments in OMC1 have $\rm M_{lin}/M_{crit} < 1.5$, which is consistent with that of @omc1_alma based on their $\rm N_2H^+$ 1–0 data. However, even though the derived $\rm M_{lin}/M_{crit}$ ratios are similar, both the $\rm M_{lin}$ and $\rm M_{crit}$ determined from our data are higher than those of @omc1_alma by a factor of few. While they used $\rm N_2H^+$ intensities to estimate column densities of $\rm H_2$, [we use the volume densities of $\rm H_2$ derived from the non-LTE analysis. The empirical relation between N$_2$H$^+$ intensities to H$_2$ column densities used by @omc1_alma has large scatter. The inclination of each filament is also uncertain.]{} On the other hand, the uncertainty in inclination does not affect our estimation. However, the $\rm M_{lin}$ derived under our assumption of uniform density inside the filaments (excluding the cores) might be overestimated if the filaments have radial density profiles. For the difference in $\rm M_{crit}$, it is likely resulted from the difference between the linewidths observed in $\rm N_2H^+$ 3–2 ($\sim$1.0 km s$^{-1}$) and 1–0. This is partly because of the difference in the beamsize of our measurement ($5.4''$) and @omc1_alma ($4.5''$); the spectra observed with the larger beam contain the nonthermal motion in the larger area. The hyperfine components of 3–2, which is much more complicated than those of 1–0, introduce additional uncertainty in the derived linewidth if the relative intensities of the hyperfine components are different from the LTE values. Table \[tab:core\_ident\] shows that most of the cores have masses ($\rm M_{core}$) ranging from $1$–$10$ $\rm M_\odot$. Both the sizes and the masses of these cores are larger than those determined by @sma_north_fil, because multiple smaller-scale cores were revealed in their 1.3 mm continuum data using the SMA. These masses can also be compared with the masses inferred by the virial theorem. By assuming a uniform density in the cores, the virial mass ($\rm M_{vir}$) can be derived as $$\begin{aligned} &&M_{\rm vir} = \frac{5\ R_{\rm eff} (\Delta v)^2}{8\ G \ln{2}} \label{def_Mvir}\end{aligned}$$ where $R_{\rm eff}$ is an effective radius of the core, and $\Delta v$ is a typical linewidth. The linewidth $\Delta v$ was determined from the hyperfine fitting except for core 6, 7, and 9 having multiple velocity components along the line of sight. As shown in Table \[tab:core\_ident\], the measured core masses are similar to the calculated virial masses, indicating that most of the cores are gravitationally bound. One of the cores, core 5, already shows the signatures of star formation; toward this core, there is an infrared source and a clear bipolar outflow in CO [@sma_north_fil]. The velocity dispersion as well as $\rm M_{vir}$ are large in the southern cores i.e. core 1 and 2, because of the strong turbulence from the stellar activities in OMC1-South [@zapata_2006; @2018ApJ...855...24P]. External Heating from High-mass Stars {#subsec:heating} ------------------------------------- The $\rm N_2H^+$ 3–2/1–0 ratio map presented in Figure \[fig:ratio\] shows higher ratios in the eastern part of OMC1. From the non-LTE model shown in Figure \[fig:non\_lte\], regions with a higher intensity ratio generally indicates a higher kinetic temperature. This suggests that the eastern OMC1 has higher temperatures comparing with the remaining area. We find that the overall distribution of the high-ratio gas is similar to that of the CN and $\rm C_2H$ molecules presented in @ungerechts_1997 and @melnick_2011. Since CN and $\rm C_2H$ are sensitive to the presence of UV raidation [@fuente_1993; @stauber_2004], it is likely that the higher temperatures in the eastern OMC1 are caused by the UV heating from the high-mass stars in M42. For example, UV photons from the $\rm \theta^1$ Ori C at (R.A., Decl.) = ($\rm 5^h 35^m 16^s.5$,$-5^\circ 23' 22''.8$), which is southeast to the Orion KL, could be one of the major heating sources. In addition, the \[C[ I]{}\]/CO intensity ratio map in @orion_ci_mapping shows a ratio peak of $\sim 0.17$ around the position (R.A., Decl.) = ($\rm 5^h 35^m 20^s$,$-5^\circ 18' 30''$), implying that UV radiation also contributes to the heating of this region. Possible heating sources include the exciting star of M43—NU Ori at (R.A., Decl.) = ($\rm 5^h 35^m 31^s.0$, $-5^\circ 16' 12''$), which is northeast to the OMC1 region. The external heating scenario also explains the 3–2/1–0 ratios inside and outside the filaments. As shown in Figure \[fig:ratio\], heating features are seen only outside the filament regions, while temperatures inside the filaments remain significantly lower. @omc1_nh3 reported the variation of NH$_3$ $(J, K)$ = (2, 2) to (1, 1) line intensity ratio in the filaments; the higher ratio (i.e. higher temperature) gas appears between the emission peaks with lower ratio (i.e. lower temperature). Such a patchy heating pattern in the filaments does not appear in the N$_2$H$^+$. There is no significant difference in the $\rm N_2H^+$ 3–2/1–0 ratio between the core regions and the low intensity regions in the filaments. This is probably because the N$_2$H$^+$ lines having higher critical densities than the NH$_3$ trace the inner and denser part of the filaments where the gas is shielded well from the external radiation. Apart from external UV heating, local heating from Orion KL has been discussed on the basis of previous observations [@tang_2018; @bally_alma_co; @zapata_2011; @omc1_nh3]. However, since $\rm N_2H^+$ emission is missing toward Orion KL, local heating around the KL core ($\sim 100$ K) is not observed in our data. The 3–2/1–0 ratio at the position of core 4, $\sim 2.0$, is significantly higher than other regions of the east filament. This could be the effect of the local heating from Orion KL. Global Collapse of OMC1 {#subsec:collapse} ----------------------- Recent studies have shown that large-scale dynamical collapse are important in high-mass star-forming regions [e.g. @hartmann_2007; @hacar_2017]. The snapshots from the magneto-hydrodynamics (MHD) simulation of globally collapsing clouds presented in @schneider_2010 and @peretto_2013 reveal multiple filaments and sharp boundaries of radial velocity changes that separate the cloud into several regions, and these regions and filaments converge toward the massive core at the center. The radial velocity distribution of OMC1 shows the trimodal pattern centered near Orion KL with the sharp boundaries between the northern and western regions (Figure \[fig:radec2vel\]a) and the northern and southern regions (Figure \[fig:radec2vel\]b). In addition, the radial velocities in the western region monotonically decrease from north to south, and continue to the velocity gradient in the southern region (Figure \[fig:radec2vel\]b). This velocity gradient corresponds to the part of the V-shaped velocity structure centered around the OMC1-South, which is interpreted as the presence of accelerated gas motion inflowing toward the Orion Nebula Cluster [@hacar_2017]. Our results have revealed that the gas in the western and southern regions contributes to this inflow. Such a global collapse picture is also supported by the morphology of the magnetic field in OMC1. The magnetic field in OMC1 revealed by the B-Fields In Star-Forming Region Observations (BISTRO) survey using the James Clerk Maxwell Telescope (JCMT) shows a well-ordered U-shaped structure, which can be explained by the distortion of an initially cylindrically-symmetric magnetic field due to large-scale gravitational collapse [@jcmt_bistro]. Interestingly, the filaments in the northern and southern regions are almost perpendicular to the local magnetic field direction, while those in the western region such as filament 3, 4, and 5 are aligned along the magnetic field. This implies that the filaments in the western region are feeding material along the magnetic field lines to the Orion Nebula Cluster, as predicted from the MHD simulation of global collapse [@schneider_2010]. Conclusion {#sec:conclusion} ========== We conducted the $\rm N_2H^+$ 3–2 line observations toward the OMC1 region using the SMA and SMT. The SMA data are combined with the SMT data in order to recover the spatially extended emission. The filaments and cores in OMC1 have been identified, and their physical properties are derived. Using the $\rm N_2H^+$ 1–0 data provided by @omc1_alma, we conducted the non-LTE analysis, and determined the kinetic temperatures and $\rm H_2$ densities of the filaments and cores. We also examine the gas kinematics inside the two prominent (main and east) filaments and compare them with the filament/core formation models. The main results are summarized as the following: 1. The combined SMA and SMT image in $\rm N_2H^+$ 3–2 reveals multiple filamentary structure having typical widths of 0.02–0.03 pc. In total , 11 filaments and 10 cores are identified. Cores are found in the three filaments with the line masses $\gtrsim 80$ $\rm M_\odot\ pc^{-1}$. The masses of the cores are in the range of $1$–$10$ $\rm{M_\odot}$ except the most massive one with $>$14 $M_{\odot}$ in OMC1-South. Up to $\sim 65\%$ of the filaments are gravitationally bound, which could be current or future sites for star formation. 2. The result of non-LTE analysis shows that the kinetic temperature is enhanced in the eastern part of OMC1. This is probably because of the external heating from high-mass stars in M42 and M43 (e.g. $\theta^1$ Ori C and NU Ori). It is found that the filament regions with higher densities of $n_{\rm H_2}\sim 10^7$ $\rm cm^{-3}$ have lower temperatures ($T_{\rm kin}\sim$ 15–20 K) than their surrounding regions. The lower temperatures in the filaments can be explained by the shielding from the external heating by dense gas. 3. The moment 1 image reveals that OMC1 consists of three sub-regions with different radial velocities divided by the sharp velocity transitions. Three sub-regions intersect with each other at the location of Orion KL. The radial velocities in the western region monotonically decreases from north to south, and continue to that in the southern region. The observed velocity structure suggests the presence of global gas flow toward the Orion Nebula Cluster. Such a global collapse is also supported by the observed morphology of magnetic fields and large scale gas kinematics in the integral-shaped filament. 4. Non-thermal motion plays important roles in the OMC1 filaments. There is no systematic velocity gradient along the minor axes of the OMC1 filaments. Although the velocity gradient of $\sim 0.3$ $\rm km\ s^{-1}$ is observed in the east filament, the direction of the velocity gradient is different at different locations. 5. Two cores in the east filament show positional shifts between their intensity and velocity peaks along the filament major axis. It is possible that core formation is still ongoing in this filament. On the other hand, there is no such positional shift in the main filament. It is therefore likely that the east filament is in an earlier evolutionary phase than the main filament, which shows the signatures of star formation. We thank the staffs of the SMA and the SMT for the operation of our observations and their support in data reduction. We thank Dr. T.-H. Hsieh for helping us with the SMT observations. The SMA is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica. N. H. acknowledges a grant from the Ministry of Science and Technology (MoST) of Taiwan (MoST 108-2112-M-001-017). 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--- abstract: 'New Mersenne conjectures. The problems of simplicity, common prime divisors and free from squares of numbers $L(n) = 2^{2n}\pm2^n\pm1$ are investigated. Wonderful formulas $gcd $ for numbers $L (n) $ and numbers repunit are proved.' author: - '**Boris V. Tarasov**[^1]\' title: | **The concrete theory of numbers :\ New Mersenne conjectures. Simplicity and other wonderful properties of numbers $L(n) = 2^{2n}\pm2^n\pm1$.** --- Introduction ============ In present work we consider sequences of integers of the following kind : $$\begin{gathered} \textcolor[rgb]{0.00,0.00,0.50}{L_{1}(n) = 2^{2n} + 2^{n} + 1},\tag{$L_{1}$}\label{e:L1}\\ \textcolor[rgb]{0.00,0.00,0.50}{L_{2}(n) = 2^{2n} + 2^{n} - 1},\tag{$L_{2}$}\label{e:L2}\\ \textcolor[rgb]{0.00,0.00,0.50}{L_{3}(n) = 2^{2n} - 2^{n} + 1},\tag{$L_{3}$}\label{e:L3}\\ \textcolor[rgb]{0.00,0.00,0.50}{L_{4}(n) = 2^{2n} - 2^{n} - 1},\tag{$L_{4}$}\label{e:L4}\end{gathered}$$where $n\geq1$ is integer. For the numerical sequence, being the union of numerical sequences $L_{1}(n)$, $L_{2}(n)$, $L_{3}(n)$, $L_{4}(n)$, we use a designation $$\textcolor[rgb]{0.00,0.00,0.50}{L(n) = 2^{2n} \pm 2^{n} \pm 1},\label{e:L}$$ where $n\geq1$ is integer. The author is interested to research the new Mersenne conjectures, concerning numbers $L(n)$. The reviews concerning Mersenne numbers and new Mersenne conjectures are available here [@Graham; @WeissteinF; @WeissteinM; @Mersenne; @Bateman]. Will we use the following designations further : $(a,b)=gcd(a,b)$ is the greatest common divider of integers $a>0$, $b>0$. $p,q$ are odd prime numbers. $m \bot n \Longleftrightarrow m, n -$ are integers and $gcd(m,n)=1$(see[@Graham]). If it is not stipulated specially, the integer positive numbers are considered. We are interested by the following questions concerning numbers $L(n)$ : question of simplicity of numbers, question of the common divisors and question of freedom from squares. For numbers $L(n)$ two general simple statements are fair. These statements represent trivial consequences of the small Fermat’s theorem , submitted by the following comparison,\ for $k>0$(see[@Graham]) or chain of comparisons\ for $k>0$, $N\geq0$, $n$ of integers. \[S:Statement1\] If $L(l)\equiv0(mod\,p)$, where $l>0$ is integer, $p$ is prime number, then $$\textcolor[rgb]{0.00,0.00,0.50}{L((p-1)\cdot k+l)\equiv0(mod\,p)},$$ where $k\geq0$ is any integer. \[S:Statemen2\] If $L(l)\equiv0(mod\,p^{t})$, where $t>0$, $l>0$ is integer, $p$ is prime number, then $$\textcolor[rgb]{0.00,0.00,0.50}{L(p^{N+t}-p^{t-1}+l)\equiv0(mod\,p^{t})},$$ where $N\geq0$ is any integer. Numbers $L_{1}(n) = 2^{2n} + 2^{n} + 1$ ======================================= #### **1**. Let’s bring the simple statements concerning numbers $L_{1}(n)$. \[L:L1\] For numbers $L_{1}(n) = 2^{2n} + 2^{n} + 1$ following statements are\ fair : \(1) The first prime numbers $L_{1}(n)$ correspond to $n=1, 3, 9$.\ $L_{1}(1)\,=\,7$, $L_{1}(3)\,=\,73$, $L_{1}(9)\,=\,262657$. \(2) If $n\geq 2$ is an even number, then number $$\label{e:N1} \textcolor[rgb]{0.00,0.00,0.50}{L_{1}(n) \equiv 0(mod\,3)}$$ is composite. If $n\geq 1$ is an odd number, then $$\label{e:N2} \textcolor[rgb]{0.00,0.00,0.50}{L_{1}(n) \equiv 1\not\equiv 0(mod\,3)}$$ \(3) If $n>1$, $n\not\equiv 0(mod\,3)$, then number $$\label{e:N3} \textcolor[rgb]{0.00,0.00,0.50}{L_{1}(n) \equiv 0(mod\,7)}$$ is composite. Validity of congruences - obviously\ follows from trivial comparisons $2 \equiv -1(mod\,3)$, $2^3 \equiv 1(mod\,7)$. Thus, prime numbers $L_{1}(n)$ are probable only for $n = 3k$, where $k$ is odd number. \[L:L2\] Let $k>1$ is an integer. Then there will be a prime number $q>3$, such that number 2 on the module $q$ belongs to a index $3^k$. Let’s consider expression\ $A = 2^{3^k}-1 = (2^{3^{k-1}})^3-1 = (2^{3^{k-1}}-1)\cdot((2^{3^{k-1}})^2+2^{3^{k-1}}+1)$.\ Let $q$ is a prime number such that\ $B = (2^{3^{k-1}})^2+2^{3^{k-1}}+1\equiv0(mod\,q)$. If $q=3$, then $B \equiv 1\not\equiv 0(mod\,3)$. Hence $q>3$, $2^{3^k}-1\equiv0(mod\,q)$. Let’s assume, that\ $2^{3^l}-1\equiv0(mod\,q)$, where $k>l\geq0$. Let $d=k-1-l\geq0$, $3^d\geq1$.\ Then $(2^{3^l})^{3^d}\equiv1(mod\,q)$, $2^{3^{l+d}}\equiv1(mod\,q)$, $2^{3^{k-1}}\equiv1(mod\,q)$,\ $B = (2^{3^{k-1}})^2+2^{3^{k-1}}+1\equiv3\equiv0(mod\,q)$, $q=3$. Have received the contradiction. It is obvious, that $d=3^k$ is the least positive number, for which the comparison $2^{d}-1\equiv0(mod\,q)$ is feasible. #### **2**. In connection with the lemma 1 the research of prime divisors of numbers $L_{1}(n)$ is interesting, where $n=3^{k}t$, where $k\geq1$ is integer, $t\geq1$ is an odd number. Since only at such $n$ the numbers $L_{1}(n)$ can be *suspicious* on prime numbers. In the following theorem the interesting property $gcd$ for numbers $L_{1}(n)$ is proved. \[T:T1\] Let $k\geq0$, $k_{1}\geq0$, $k_{2}\geq0$ be integers; $t_{1}\geq1$, $t_{2}\geq1$ are odd numbers, $t_{1}\not\equiv0(mod\,3)$, $t_{2}\not\equiv0(mod\,3)$. Then the following statements are fair : $$\begin{gathered} \label{T1:1}\textcolor[rgb]{0.00,0.00,0.50}{gcd(L_{1}(3^{k}t_{1}),L_{1}(3^{k}t_{2}))=L_{1}(3^{k}gcd(t_{1},t_{2}))}.\\ \label{T1:2}\textcolor[rgb]{0.00,0.00,0.50}{gcd(L_{1}(3^{k_{1}}t_{1}),L_{1}(3^{k_{2}}t_{2})) = 1}\end{gathered}$$ at $k_{1}\neq k_{2}$. Let $t_{3}=gcd(t_{1},t_{2})$, $t_{1}=t_{3}d_{1}$, $t_{2}=t_{3}d_{2}$, where\ $(d_{1},d_{2})=1$, $d_{1}\geq1$, $d_{2}\geq1$ are odd numbers. 1\) Let’s prove equality . Let’s consider the following formulas $$\begin{aligned} \textcolor[rgb]{0.00,0.00,0.50}{A=L_{1}(3^{k}t_{1})=2^{2\cdot 3^{k}t_{1}} + 2^{3^{k}t_{1}} + 1,} &\textcolor[rgb]{0.00,0.00,0.50}{A(2^{3^{k}t_{1}}-1)=2^{3^{k+1}t_{3}d_{1}}-1},\notag\\ \textcolor[rgb]{0.00,0.00,0.50}{B=L_{1}(3^{k}t_{2})=2^{2\cdot 3^{k}t_{2}} + 2^{3^{k}t_{2}} + 1,} &\textcolor[rgb]{0.00,0.00,0.50}{B(2^{3^{k}t_{2}}-1)=2^{3^{k+1}t_{3}d_{2}}-1},\label{T1:3}\\ \textcolor[rgb]{0.00,0.00,0.50}{C=L_{1}(3^{k}t_{3})=2^{2\cdot 3^{k}t_{3}} + 2^{3^{k}t_{3}} + 1,} &\textcolor[rgb]{0.00,0.00,0.50}{C(2^{3^{k}t_{3}}-1)=2^{3^{k+1}t_{3}}-1}.\notag\end{aligned}$$ Then the following formulas are fair $$\begin{gathered} \textcolor[rgb]{0.00,0.00,0.50}{A(2^{3^{k}t_{1}}-1)=C(2^{3^{k}t_{3}}-1)[2^{3^{k+1}t_{3}(d_{1}-1)} + 2^{3^{k+1}t_{3}(d_{1}-2)} + \ldots}\\ \textcolor[rgb]{0.00,0.00,0.50}{\ldots + 2^{3^{k+1}t_{3}} + 1]},\label{T1:4}\end{gathered}$$ $$\begin{gathered} \textcolor[rgb]{0.00,0.00,0.50}{B(2^{3^{k}t_{2}}-1)=C(2^{3^{k}t_{3}}-1)[2^{3^{k+1}t_{3}(d_{2}-1)} + 2^{3^{k+1}t_{3}(d_{2}-2)} + \ldots}\\ \textcolor[rgb]{0.00,0.00,0.50}{\ldots + 2^{3^{k+1}t_{3}} + 1]}.\label{T1:5}\end{gathered}$$ Let $q>1$ is prime number such, that $$\textcolor[rgb]{0.00,0.00,0.50}{2^{3^{k}t_{1}}-1\equiv 0(mod\,q),\ C\equiv 0(mod\,q)}.\label{T1:6}$$ Then it follows from and , that $$\textcolor[rgb]{0.00,0.00,0.50}{2^{3^{k}t_{3}d_{1}}-1\equiv 0(mod\,q),\ 2^{3^{k+1}t_{3}}-1\equiv 0(mod\,q)}.$$ Let’s consider number $b=2^{3^{k}t_{3}}$. If $b\equiv 1(mod\,q)$, then it follows from and , that $C\equiv 3\equiv 0(mod\,q)$, i.e. $q=3$. Since $t_{3}$ is odd number, then it follows from lemma 1, that $C\not\equiv 0(mod\,3)$. We have come to the contradiction. Thus, $b\not\equiv 1(mod\,q)$. Then the number $b$ on the module $q$ belongs to index $l_{0}>1$, $b^{l_{0}}\equiv 1(mod\,q)$. As $b^{d_{1}}\equiv b^{3}\equiv 1(mod\,q)$, that $d_{1}\equiv 3\equiv 0(mod\,l_{0})$, $d_{1}\equiv 0(mod\,3)$. Have received the contradiction. We have proved, that $gcd(C,2^{3^{k}t_{1}}-1)=gcd(C,2^{3^{k}t_{2}}-1)=1$, hence, $gcd(A,B)\equiv 0(mod\,C)$. It is necessary to prove the opposite: if\ $d\mid gcd(A,B)$, then $d\mid C$, where $d>1$ an integer. Let’s assume, that there is an integer $d>1$ such, that $$\textcolor[rgb]{0.00,0.00,0.50}{A\equiv B\equiv 0(mod\,d),\ but\ gcd(C,d)=1}.\label{T1:7}$$ Then it follows from and , that $$\textcolor[rgb]{0.00,0.00,0.50}{2^{3^{k+1}t_{3}d_{1}}-1\equiv 0(mod\,d),\ 2^{3^{k+1}t_{3}d_{2}}-1\equiv 0(mod\,d)}.$$ Let $l_{0}>1$ is an index, to which the number $2$ belongs on the module $d$,\ $2^{l_{0}}-1\equiv 0(mod\,d)$. Then $(3^{k+1}t_{3})d_{1}\equiv (3^{k+1}t_{3})d_{2}\equiv 0(mod\,l_{0})$, $3^{k+1}t_{3}\equiv 0(mod\,l_{0})$. Then from $C(2^{3^{k}t_{3}}-1)\equiv 0(mod\,d)$,\ $2^{3^{k}t_{3}}-1\equiv 0(mod\,d)$, $A\equiv 3\equiv 0(mod\,d)$, $d=3$. Have received the contradiction. The equality is proved. 2\) Let’s prove equality . Let’s consider the following formulas $$\begin{aligned} \textcolor[rgb]{0.00,0.00,0.50}{A_{1}=2^{2\cdot 3^{k_{1}}t_{1}} + 2^{3^{k_{1}}t_{1}} + 1,} &\textcolor[rgb]{0.00,0.00,0.50}{A_{1}(2^{3^{k_{1}}t_{1}}-1)=2^{3^{k_{1}+1}t_{1}}-1,}\notag\\ \textcolor[rgb]{0.00,0.00,0.50}{A_{2}=2^{2\cdot 3^{k_{2}}t_{2}} + 2^{3^{k_{2}}t_{2}} + 1,} &\textcolor[rgb]{0.00,0.00,0.50}{A_{2}(2^{3^{k_{2}}t_{2}}-1)=2^{3^{k_{2}+1}t_{2}}-1,}\label{T1:8}\end{aligned}$$ where $k_{1}\neq k_{2}$, $k_{1}<k_{2}$. Let’s assume, that $q>1$ is a prime number such, that\ $A_{1}\equiv A_{2}\equiv 0(mod\,q)$, $2^{3^{k_{1}+1}t_{1}}\equiv 2^{3^{k_{2}+1}t_{2}}\equiv 1(mod\,q)$. Let $l_{0}>1$ is an index, to which the number $2$ belongs on the module $q$, $2^{l_{0}}-1\equiv 0(mod\,q)$. Then $3^{k_{1}+1}t_{1}\equiv 3^{k_{2}+1}t_{2}\equiv 0(mod\,l_{0})$. Since $k_{1}+1\leq k_{2}$, then\ $3^{k_{2}}t_{2}\equiv 0(mod\,l_{0})$, $A_{2}\equiv 3\equiv 0(mod\,q)$, $q=3$. Have received the contradiction. The theorem 1 is proved. If $n=3^{k}t$, where $k\geq1$ is integer, $t>1$ is an odd number, $t\not\equiv0(mod\,3)$, then $$\textcolor[rgb]{0.00,0.00,0.50}{L_{1}(3^{k}t)\equiv 0(mod\,L_{1}(3^{k}))}$$ is always composite number. The summary of the received results concerning numbers $L_{1}$. \[T:T2\] For numbers $L_{1}(n) = 2^{2n} + 2^{n} + 1$ the following statements are fair : \(1) The prime numbers $L_{1}(1)=7$, $L_{1}(3)=73$, $L_{1}(9)=262657$ are known. \(2) If $n\ne3^{k}$, where $k\geq0$ is an integer, then $L_{1}(n)$ is a composite number. From the identity $2^{3^{k+1}}-1=(2^{3^{k}}-1)[(2^{3^{k}})^{2}+2^{3^{k}}+1]$ the following equality is received for numbers $L_{1}$ $$\textcolor[rgb]{0.00,0.00,0.50}{2^{3^{k+1}}-1=L_{1}(1)\cdot L_{1}(3)\cdot\ldots\cdot L_{1}(3^{k}),}$$ where $k\geq0$ is integer. From the statement of the theorem 1 the property follows $$\textcolor[rgb]{0.00,0.00,0.50}{L_{1}(3^{i})\bot L_{1}(3^{j})}$$ at $i\ne j$. #### **3**. The numbers $L_{1}$ are not free from squares, that is confirmed by the following examples $L_{1}(7)\equiv 0(mod\,7^{2})$, $L_{1}(104)\equiv 0(mod\,13^{2})$,\ $L_{1}(114)\equiv 0(mod\,19^{2})$. The following theorem takes place: Let $k\geq0$, $n\not\equiv0(mod\,3)$ are integers. Then the comparison is fair $$\textcolor[rgb]{0.00,0.00,0.50}{L_{1}(7^{k}n)\equiv0(mod\,7^{k+1}).}$$ Let’s consider number $B=2^{7^{k}n}-1$. Since $n\not\equiv0(mod\,3)$, then\ $B\not\equiv0(mod\,7)$. Let $A=B\cdot L_{1}(7^{k}n)=2^{7^{k}3n}-1$. Let’s prove by induction on $k\geq0$, that $$\textcolor[rgb]{0.00,0.00,0.50}{2^{7^{k}3n}-1\equiv0(mod\,7^{k+1}).}\label{T3:1}$$ The case $k=0$ is obvious. Let’s make the inductive assumption, that for $k\leq m-1$ the comparison is fair. Let’s consider expression $$\begin{gathered} \notag \textcolor[rgb]{0.00,0.00,0.50}{2^{7^{m}3n}-1=(2^{7^{m-1}3n})^{7}-1=}\\ \textcolor[rgb]{0.00,0.00,0.50}{=(2^{7^{m-1}3n}-1)\cdot\sum_{i=0}^6(2^{7^{m-1}3n})^{i}\equiv0(mod\,7^{m}\cdot7)\equiv0(mod\,7^{m+1}).}\end{gathered}$$ Numbers $L_{2}(n) = 2^{2n} + 2^{n} - 1$ ======================================= First five prime numbers $L_{2}(1)=5$, $L_{2}(2)=19$, $L_{2}(3)=71$,\ $L_{2}(4)=271$, $L_{2}(6)=4159$. From **the statement 1** validity of the\ comparisons follows $$L_{2}(4k+1)\equiv0(mod\,5),\ L_{2}(10k+7)\equiv L_{2}(10k+8)\equiv0(mod\,11),$$ where $k\geq0$ is integer. The numbers $L_{2}(n)$ are not free from squares $L_{2}(68)\equiv 0(mod\,11^{3})$,\ $L_{2}(97)\equiv 0(mod\,11^{2})$. The author has checked up the following worthy to attention facts for numbers $L_{2}(n)$. 1\) For prime numbers $p\leq5003$ the prime numbers $L_{2}(p) = 2^{2p} + 2^{p} - 1$ exist only for $p=2$, $L_{2}(2)=19$; $p=3$, $L_{2}(3)=71$; $p=379$. 2\) If we consider numbers $L_{2}(2^{n}) = 2^{2^{n+1}} + 2^{2^{n}} - 1$, then prime numbers $L_{2}(2^{n})$ for $n\leq17$ exist at $n=1$, $L_{2}(2)=19$; $n=2$, $L_{2}(4)=271$; $n=4$, $L_{2}(16)=4295032831$. Numbers $L_{3}(n) = 2^{2n} - 2^{n} + 1$ ======================================= Trivial property of numbers $L_{3}(n)$ : if $n>0$ is an even number, then $$\textcolor[rgb]{0.00,0.00,0.50}{L_{3}(n) \equiv 1\not\equiv 0(mod\,3),\label{L3:1}}$$ if $n>0$ is an odd number, then $$\textcolor[rgb]{0.00,0.00,0.50}{L_{3}(n) \equiv 0(mod\,3).\tag{\ensuremath{\ref{L3:1}'}}\label{L3:2}}$$ From **the statement 1** validity of comparisons follows $$L_{3}(2(6k+1))\equiv L_{3}(2(6k+5))\equiv0(mod\,13),$$ where $k\geq0$ is integer. The numbers $L_{3}(n)$ are not free from squares\ $L_{3}(2\cdot13)\equiv L_{3}(10\cdot13)\equiv0(mod\,13^{2})$, $L_{3}(3\cdot19)\equiv 0(mod\,19^{2})$. In the following theorem the interesting property of $gcd$ for numbers $L_{3}$ is proved. \[T:T4\] Let $m\geq0$, $m_{1}\geq0$, $m_{2}\geq0$, $n>0$,\ $n_{1}>0$, $n_{2}>0$ are integers; $t_{1}\geq1$, $t_{2}\geq1$ are odd numbers,\ $t_{1}\not\equiv0(mod\,3)$, $t_{2}\not\equiv0(mod\,3)$. Then the statements are fair : $$\begin{gathered} \label{T4:1}\textcolor[rgb]{0.00,0.00,0.50}{gcd(L_{3}(3^{m}2^{n}t_{1}),L_{3}(3^{m}2^{n}t_{2}))= L_{3}(3^{m}2^{n}gcd(t_{1},t_{2})).}\\ \label{T4:2}\textcolor[rgb]{0.00,0.00,0.50}{gcd(L_{3}(3^{m_{1}}2^{n_{1}}t_{1}),L_{3}(3^{m_{2}}2^{n_{2}}t_{2})) = 1}\end{gathered}$$ for $m_{1}\neq m_{2}$ or $n_{1}\neq n_{2}$. Let $t_{3}=gcd(t_{1},t_{2})$, $t_{1}=t_{3}d_{1}$, $t_{2}=t_{3}d_{2}$, where\ $(d_{1},d_{2})=1$, $d_{1}\geq1$, $d_{2}\geq1$ are odd numbers. 1\) Let’s prove equality . Let’s consider the following formulas $$\begin{gathered} \textcolor[rgb]{0.00,0.00,0.50}{A=L_{3}(3^{m}2^{n}t_{1})=2^{2\cdot 3^{m}2^{n}t_{1}} - 2^{3^{m}2^{n}t_{1}} + 1,}\notag\\ \textcolor[rgb]{0.00,0.00,0.50}{A(2^{3^{m}2^{n}t_{1}}+1)=2^{3^{m+1}2^{n}t_{3}d_{1}}+1,}\notag\\ \textcolor[rgb]{0.00,0.00,0.50}{B=L_{3}(3^{m}2^{n}t_{2})=2^{2\cdot 3^{m}2^{n}t_{2}} - 2^{3^{m}2^{n}t_{2}} + 1,}\label{T4:3}\\ \textcolor[rgb]{0.00,0.00,0.50}{B(2^{3^{m}2^{n}t_{2}}+1)=2^{3^{m+1}2^{n}t_{3}d_{2}}+1,}\notag\\ \textcolor[rgb]{0.00,0.00,0.50}{C=L_{3}(3^{m}2^{n}t_{3})=2^{2\cdot 3^{m}2^{n}t_{3}} - 2^{3^{m}2^{n}t_{3}} + 1},\notag\\ \textcolor[rgb]{0.00,0.00,0.50}{C(2^{3^{m}2^{n}t_{3}}+1)=2^{3^{m+1}2^{n}t_{3}}+1.}\notag\end{gathered}$$ Then the formulas are fair $$\begin{gathered} \textcolor[rgb]{0.00,0.00,0.50}{A(2^{3^{m}2^{n}t_{1}}+1)=C(2^{3^{m}2^{n}t_{3}}+1)[2^{3^{m+1}2^{n}t_{3}(d_{1}-1)} -}\\ \textcolor[rgb]{0.00,0.00,0.50}{-2^{3^{m+1}2^{n}t_{3}(d_{1}-2)} + \ldots}\\ \textcolor[rgb]{0.00,0.00,0.50}{\ldots + 2^{3^{m+1}2^{n}t_{3}2} - 2^{3^{m+1}2^{n}t_{3}} + 1],}\label{T4:4}\end{gathered}$$ $$\begin{gathered} \textcolor[rgb]{0.00,0.00,0.50}{B(2^{3^{m}2^{n}t_{2}}+1)=C(2^{3^{m}2^{n}t_{3}}+1)[2^{3^{m+1}2^{n}t_{3}(d_{2}-1)} -}\\ \textcolor[rgb]{0.00,0.00,0.50}{-2^{3^{m+1}2^{n}t_{3}(d_{2}-2)} + \ldots}\\ \textcolor[rgb]{0.00,0.00,0.50}{\ldots + 2^{3^{m+1}2^{n}t_{3}2} - 2^{3^{m+1}2^{n}t_{3}} + 1].}\label{T4:5}\end{gathered}$$ Let $q>1$ is prime number such, that $$\textcolor[rgb]{0.00,0.00,0.50}{2^{3^{m}2^{n}t_{1}}+1\equiv 0(mod\,q),\ C\equiv 0(mod\,q).}\label{T4:6}$$ Then from and the comparisons follow $$\textcolor[rgb]{0.00,0.00,0.50}{2^{3^{m}2^{n}t_{3}d_{1}}+1\equiv 0(mod\,q),\ 2^{3^{m+1}2^{n}t_{3}}+1\equiv 0(mod\,q).}$$ Let’s consider number $b=2^{3^{m}2^{n}t_{3}}$. If $b\equiv 1(mod\,q)$, then\ $C=b^{2}-b+1\equiv 1\not\equiv0(mod\,q)$. If $b\equiv -1(mod\,q)$, then\ $C\equiv 3\equiv 0(mod\,q)$, $q=3$, but as $n>0$, that from follows, that\ $C\not\equiv 0(mod\,3)$. Have received the contradiction. Thus, $b^{2}\not\equiv 1(mod\,q)$. Let $l_{0}>1$ is an index, to which the number $b^{2}$ belongs on the module $q$, $(b^{2})^{l_{0}}\equiv 1(mod\,q)$. As $(b^{2})^{d_{1}}\equiv (b^{2})^{3}\equiv 1(mod\,q)$, that\ $d_{1}\equiv 3\equiv 0(mod\,l_{0})$, $d_{1}\equiv 0(mod\,3)$. Have received the contradiction.\ Have proved, that $gcd(A,B)\equiv 0(mod\,C)$. Let’s assume, that there is an integer $d>1$ such, that $$\textcolor[rgb]{0.00,0.00,0.50}{A\equiv B\equiv 0(mod\,d),\ but\ gcd(C,d)=1.}\label{T4:7}$$ Then it follows from and , that $$\textcolor[rgb]{0.00,0.00,0.50}{2^{3^{m+1}2^{n+1}t_{3}d_{1}}-1\equiv 0(mod\,d),\ 2^{3^{m+1}2^{n+1}t_{3}d_{2}}-1\equiv 0(mod\,d).}$$ Then $2^{3^{m+1}2^{n+1}t_{3}}-1\equiv 0(mod\,d)$, i.e. $$\textcolor[rgb]{0.00,0.00,0.50}{(2^{3^{m+1}2^{n}t_{3}}+1)\cdot(2^{3^{m+1}2^{n}t_{3}}-1)\equiv 0(mod\,d).}\label{T4:8}$$ If $gcd(2^{3^{m+1}2^{n}t_{3}}+1,d)=d_{0}>1$, then it follows from , that\ $2^{3^{m}2^{n}t_{3}}\equiv -1(mod\,d_{0})$, $A\equiv 3\equiv 0(mod\,d_{0})$, $d_{0}=3$. Have received the contradiction. Then from , and the comparisons follow\ $(2^{3^{m+1}2^{n}t_{3}}-1)\equiv 0(mod\,d)$, $2^{3^{m+1}2^{n}t_{3}d_{1}}+1\equiv 0\equiv 2(mod\,d)$. Have received the contradiction, since $d>1$ is odd number. The equality is proved. 2\) Let’s prove equality . Let’s consider the following formulas $$\begin{gathered} \textcolor[rgb]{0.00,0.00,0.50}{A_{1}=2^{2\cdot 3^{m_{1}}2^{n_{1}}t_{1}} - 2^{3^{m_{1}}2^{n_{1}}t_{1}} + 1,}\notag\\ \textcolor[rgb]{0.00,0.00,0.50}{A_{1}(2^{3^{m_{1}}2^{n_{1}}t_{1}}+1)=2^{3^{m_{1}+1}2^{n_{1}}t_{3}d_{1}}+1,}\label{T4:9}\\ \textcolor[rgb]{0.00,0.00,0.50}{A_{2}=2^{2\cdot 3^{m_{2}}2^{n_{2}}t_{2}} - 2^{3^{m_{2}}2^{n_{2}}t_{2}} + 1,}\notag\\ \textcolor[rgb]{0.00,0.00,0.50}{A_{2}(2^{3^{m_{2}}2^{n_{2}}t_{2}}+1)=2^{3^{m_{2}+1}2^{n_{2}}t_{3}d_{2}}+1.}\notag\end{gathered}$$ Let’s assume, that $q>1$ is prime number such, that\ $A_{1}\equiv A_{2}\equiv 0(mod\,q)$. Then $2^{3^{m_{1}+1}2^{n_{1}}t_{3}d_{1}}+1\equiv 2^{3^{m_{2}+1}2^{n_{2}}t_{3}d_{2}}+1\equiv 0(mod\,q)$. Let $l_{0}>1$ is an index, to which the number $2$ belongs on the module $q$. Then $3^{m_{1}+1}2^{n_{1}+1}t_{3}d_{1}\equiv 3^{m_{2}+1}2^{n_{2}+1}t_{3}d_{2}\equiv 0(mod\,l_{0})$, i.e. $$\textcolor[rgb]{0.00,0.00,0.50}{3^{m_{1}+1}2^{n_{1}+1}t_{3}\equiv 3^{m_{2}+1}2^{n_{2}+1}t_{3}\equiv 0(mod\,l_{0})}\label{T4:10}.$$ $2^{a}$) Let’s assume, that $m_{1}<m_{2}$. Then from we receive comparisons $$3^{m_{2}}2^{n_{2}+1}t_{3}\equiv 0(mod\,l_{0}),\ (2^{3^{m_{2}}2^{n_{2}}t_{3}}+1)\cdot(2^{3^{m_{2}}2^{n_{2}}t_{3}}-1)\equiv 0(mod\,q).$$ From the last comparison either $A_{2}\equiv 3\equiv 0(mod\,q)$, $q=3$, or\ $A_{2}\equiv 1\not\equiv0(mod\,q)$ follows. Have received the contradiction. $2^{b}$) Let’s assume, that $m_{1}=m_{2}$, $n_{1}<n_{2}$. Then from , we receive comparisons $$3^{m_{2}+1}2^{n_{2}}t_{3}\equiv 0(mod\,l_{0}),\ (2^{3^{m_{2}+1}2^{n_{2}}t_{3}d_{2}}+1)\equiv 0\equiv 2(mod\,q).$$ Have received the contradiction. The theorem 4 is proved. Let $n\geq1$ is an integer, $t>1$ is an odd number. Then the statements are fair : \(1) If $t\not\equiv0(mod\,3)$, then $$\textcolor[rgb]{0.00,0.00,0.50}{L_{3}(2^{n}t)\equiv 0(mod\,L_{3}(2^{n})).}$$ Besides, $1<L_{3}(2^{n})<L_{3}(2^{n}t)$, where $L_{3}(2^{n}t)$ is composite number. \(2) If $t\equiv0(mod\,3)$, then $$\textcolor[rgb]{0.00,0.00,0.50}{gcd(L_{3}(2^{n}t),L_{3}(2^{n}))=1.}$$ Let $m\geq0$, $n>0$ are integers; $t>1$ is an odd number, $t\not\equiv0(mod\,3)$. Then number $L_{3}(3^{m}2^{n}t)$ is always composite number. The summary of the received results concerning composite numbers $L_{3}$. \[T:T5\] For numbers $L_{3}$ the statements are fair : \(1) If $n\ne 3^{m}2^{n}$, where $m\geq0$, $n\geq0$ are integers, then number $L_{3}(n)$ is composite number. \(2) Prime numbers $L_{3}(1)=L_{3}(2^{0})=3$, $L_{3}(2)=L_{3}(2^{1})=13$, $L_{3}(4)=L_{3}(2^{2})=241$, $L_{3}(32)=L_{3}(2^{5})=18446744069414584321$ are known. For numbers $L_{3}(3^{m}2^{n})$, where $m\geq0$, $n>0$ are integers, the author has carried out the following check : 1\) Numbers $L_{3}(2^{n})$ at $6\leq n \leq 15$ is composite; 2\) Numbers $L_{3}(2\cdot3^{m})$ at $1\leq m \leq 8$ is composite; 3\) Numbers $L_{3}(3\cdot2^{n})$ at $1\leq n \leq 12$ is composite; 4\) Numbers $L_{3}(3^{2}\cdot2^{n})$ at $1\leq n \leq 11$ is composite; 5\) Numbers $L_{3}(3^{3}\cdot2^{n})$ at $1\leq n \leq 9$ is composite. Numbers $L_{4}(n) = 2^{2n} - 2^{n} - 1$ ======================================= From **the statement 1** validity of comparisons follows $$L_{4}(4k+3)\equiv0(mod\,5),\ L_{4}(10k+2)\equiv L_{4}(10k+3)\equiv0(mod\,11),$$ where $k\geq0$ is any integer. The numbers $L_{4}(n)$ are not free from squares, since $$\begin{gathered} L_{4}(13)\equiv L_{4}(42)\equiv L_{4}(123)\equiv0(mod\,11^{2})\notag,\\ L_{4}(52)\equiv L_{4}(119)\equiv 0(mod\,19^{2})\notag.\end{gathered}$$ Prime numbers $L_{4}(n)$ and $L_{4}(n+1)$ are named **the prime $L_{4}$ number-twins**. **The author has found 4 pairs of the prime $L_{4}$ number-twins up to $n\leq603$**, namely $L_{4}(1) = 1, L_{4}(2) = 11$ ; $L_{4}(4) = 239, L_{4}(5) = 991$ ;\ $L_{4}(9) = 261631, L_{4}(10) = 1047551$ ; $L_{4}(224) , L_{4}(225)$. Wonderful properties of $gcd$ insularity ======================================== Let $R_{n}\geq0$ is a sequence of integers, where $n>0$ is integer. $M$ is a subset of natural numbers.\ Let’s tell, that the sequence $R_{n}$ on set $M$ is isolated about $gcd$, if the condition is fair : $$\textcolor[rgb]{0.00,0.00,0.50}{gcd(R_{n},R_{m})=R_{gcd(n,m)}}$$ for $\forall n,m \in M$. Let $k\geq0$ is integer, then numbers $L_{1}$ on set $M_{k}=\{ 3^{k}\cdot t : t\geq1$ is an odd number, $t\not\equiv 0(mod\,3)\}$\ are isolated about $gcd$, i.e. $$\textcolor[rgb]{0.00,0.00,0.50}{gcd(L_{1}(n),L_{1}(m))=L_{1}(gcd(n,m))}\notag$$ for $\forall n,m \in M_{k}$. Let $k\geq0, l>0$ are integers, then numbers $L_{3}$ on set $M_{k,l}=\{ 3^{k}2^{l}\cdot t : t\geq1$ is an odd number, $t\not\equiv 0(mod\,3)\}$\ are isolated about $gcd$, i.e. $$\textcolor[rgb]{0.00,0.00,0.50}{gcd(L_{3}(n),L_{3}(m))=L_{3}(gcd(n,m))}\notag$$ for $\forall n,m \in M_{k,l}$. Let’s consider the generalized numbers repunit - integers of the following kind [@WeissteinM; @WeissteinR; @Dubner-1; @Dubner-2; @Granlund] : $$\label{GenRep1} \textcolor[rgb]{0.00,0.00,0.50}{M_{n}^{(b)} = (b^{n}-1)/(b-1),}$$ where $n\geq1$, $b\geq2$ are integers. $$\label{GenRep2} \textcolor[rgb]{0.00,0.00,0.50}{M_{n}^{+(b)} = (b^{n}+1)/(b+1),}$$ where $n\geq1$ is an odd number, $b\geq2$ is integer. For the generalized numbers repunit and the theorem takes place. \[T:T6\] Following formulas are fair : $$\label{e:Form1} \textcolor[rgb]{0.00,0.00,0.50}{gcd(M_{n}^{(b)},M_{m}^{(b)}) = M_{gcd(n,m)}^{(b)}},$$ where $n\geq1$, $b\geq2$ are integers. $$\label{e:Form2} \textcolor[rgb]{0.00,0.00,0.50}{gcd(M_{n}^{+(b)},M_{m}^{+(b)}) = M_{gcd(n,m)}^{+(b)}},$$ where $n\geq1$ is an odd number, $b\geq2$ is integer. 1\) Let $(n,m)=d\geq1$, where $n=n_{1}d$, $m=m_{1}d$, $(n_{1},m_{1})=1$. From definition equalities follow $$M_{n}^{(b)} = ((b^{d})^{n_{1}}-1)/(b-1)=M_{d}^{(b)}\cdot\{b^{d(n_1-1)}+\,\ldots\,+b^{2d}+b^d+1\},$$ $$M_{m}^{(b)} = ((b^{d})^{m_{1}}-1)/(b-1)=M_{d}^{(b)}\cdot\{b^{d(m_1-1)}+\,\ldots\,+b^{2d}+b^d+1\}.$$ Let $$A=b^{d(n_1-1)}+\,\ldots\,+b^{2d}+b^d+1,\ \ B=b^{d(m_1-1)}+\,\ldots\,+b^{2d}+b^d+1.$$ Let’s assume, that $A\equiv~B\equiv0(mod\,q)$, where $q>1$ is prime number.\ Let $b_{0}=b^{d}$. If $b_{0}\equiv1(mod\,q)$, then $n_{1}\equiv~m_{1}\equiv0(mod\,q)$. Have received the contradiction. Hence $b_{0}\not\equiv1(mod\,q)$, then there exists an index $d_{0}>1$, to which the number $b_{0}$ belongs on the module $q$ $$(b^d)^{d_{0}}\equiv1(mod\,q).$$ Then $n_{1}\equiv m_{1}\equiv0(mod\,d_{0})$. Have received the contradiction. 2\) Let $(n,m)=d\geq1$, where $n=n_{1}d$, $m=m_{1}d$ are odd numbers, $(n_{1},m_{1})=1$. From definition equalities follow $$M_{n}^{+(b)} = ((b^{d})^{n_{1}}+1)/(b+1) =$$ $$= M_{d}^{+(b)}\cdot\{b^{d(n_1-1)}-b^{d(n_1-2)}+\,\ldots\,+b^{2d}-b^d+1\},$$ $$M_{m}^{+(b)} = ((b^{d})^{m_{1}}+1)/(b+1) =$$ $$= M_{d}^{+(b)}\cdot\{b^{d(m_1-1)}-b^{d(m_1-2)}+\,\ldots\,+b^{2d}-b^d+1\}.$$ Let $$A=b^{d(n_1-1)}-b^{d(n_1-2)}+\,\ldots\,+b^{2d}-b^d+1,$$ $$B=b^{d(m_1-1)}-b^{d(m_1-2)}+\,\ldots\,+b^{2d}-b^d+1.$$ Let’s assume, that $A\equiv~B\equiv0(mod\,q)$, where $q>1$ is prime number.\ Let $b_{0}=b^{2d}$. If $b_{0}\equiv1(mod\,q)$, then either $b^{d}\equiv1(mod\,q)$, or\ $b^{d}\equiv-1(mod\,q)$. Then either $A\equiv1\not\equiv0(mod\,q)$, or\ $n_{1}\equiv~m_{1}\equiv0(mod\,q)$. Have received the contradiction. Hence $b_{0}\not\equiv1(mod\,q)$, then there exists an index $d_{0}>1$, to which the number $b_{0}$ belongs on the module $q$ $$(b^{2d})^{d_{0}}\equiv1(mod\,q).$$ Since $(b^{2d})^{n_{1}}\equiv (b^{2d})^{m_{1}}\equiv1(mod\,q)$, then $n_{1}\equiv m_{1}\equiv0(mod\,d_{0})$. Have received the contradiction. Let $\mathbb{P}$ is a set of all positive integers, $\mathbb{O}$ is a set of all odd numbers. \(1) Numbers $M_{n}^{(b)}$ on set $\mathbb{P}$ are isolated about $gcd$, i.e. $$\textcolor[rgb]{0.00,0.00,0.50}{gcd(M_{n}^{(b)},M_{m}^{(b)}) = M_{gcd(n,m)}^{(b)}}$$ for $\forall n,m \in \mathbb{P}$. \(2) Numbers $M_{n}^{+(b)}$ on set $\mathbb{O}$ are isolated about $gcd$, i.e. $$\textcolor[rgb]{0.00,0.00,0.50}{gcd(M_{n}^{+(b)},M_{m}^{+(b)}) = M_{gcd(n,m)}^{+(b)}}$$ for $\forall n,m \in \mathbb{O}$. The open problems of numbers $L(n)$ =================================== Author offers some open problems, as the unsolved tasks concerning numbers $L(n) = 2^{2n}\pm2^n\pm1$. ? The author has checked up, that the numbers $L_{1}(3^{k})$ for $k=3,4,5,6,7,\\8,9,10$ are composite ! ? ? ? ? ? Conclusion “The concrete theory of numbers” =========================================== It is necessary to explain the title of article “The concrete theory of numbers”. Having had a look in the Wladimir Igorewitsch Arnold foreword “From Fibonacci up to Erds” to the remarkable book of Ronald Graham, Donald Knuth and Oren Patashnik “The Concrete mathematics”[@Grekhem], it is possible to answer the question : **what is “the concrete theory of numbers” ?** **The theories come and leave, but natural series of numbers remains** and constantly generates new complicated problems. The new theories are again created for their decision. The process cannot be stopped **:)** The concrete theory of numbers created by titanic efforts of Pierre de Fermat and Leonhard Euler is an art to solve riddles of a natural series of numbers. It is enough to look to the tasks list [@Zadachi; @Ferma], which has been put and decided by Pierre de Fermat, or to get acquainted with tasks, which has been decided, investigated and propagandized by Waclaw Sierpinski [@Serpinskii],to be convinced - **a natural series of numbers doesn’t drowse, it is always ready to a human challenge!** Acknowledgement of gratitude ============================ The author expresses the deep gratitude to the creators of the calculator for number-theoretic researches. – GR/PARI CALCULATOR is free software [@GP/PARI]. GR/PARI CALCULATOR has opened to the author a door in the world of wonderful number-theoretic opportunities ! [9]{} *Teoriya chisel.-M.:Uchpedgiz,1939.* *Osnovy teorii chisel.-M.:Nauka,1981.* *250 zadach po elementarnoi teorii chisel.-M.:Prosveshenie,1968.*\ *Concrete Mathematics :A Foundation for Computer Science,2nd edition (Reading,Massachusetts:Addison-Wesley), 1994.* *Koncretnay matematika. Osnovanie informatiki :Per.s angl.-M.:Mir,1998.* *“Fermat Number”. From MathWorld–A Wolfram Web Resource. —http://mathworld.wolfram.com/FermatNumber.html/.\ 1999—2007 Wolfram Research,Inc.* *“Mersenne Number”. From MathWorld–A Wolfram Web Resource. —http://mathworld.wolfram.com/MersenneNumber.html/.\ 1999—2007 Wolfram Research,Inc.* *Repunit. From MathWorld–A Wolfram Web Resource. —http://mathworld.wolfram.com/Repunit.html/.\ 1999—2008 Wolfram Research,Inc.*\ *http://en.wikipedia.org/wiki/Mersenne conjectures/.* *Sequence A000225 in “The On-Line Encyclopedia of Integer Sequences.”* *—http://neves.suncloud.ru/task/fermat.htm* *is free software Version 2.3.0.\ —http://pari.math.u-bordeaux.fr/* *“The new Mersenne conjecture”. American Mathematical Monthly 96: 125-128.* *“Generalized Repunit Primes.” Math. Comput. 61, 927-930, 1993.* *“Primes of the Form $(b^n+1)/(b+1)$ .” J. Int. Sequences 3, No. 00.2.7, 2000. http://www.cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.html.* *“Repunits.” http://www.swox.com/gmp/repunit.html.* ——————————————————————— Institute of Thermophysics, Siberian Branch of RAS Lavrentyev Ave., 1, Novosibirsk, 630090, Russia E-mail: [email protected] ——————————————————————— Boris Vladimirovich Tarasov, independent researcher. Primary E-mail Address: [email protected] [^1]: Tarasov Boris V. The concrete theory of numbers: New Mersenne conjectures. Simplicity and other wonderful properties of numbers $L(n) = 2^{2n}\pm2^n\pm1$. MSC 11A51, MSC 11B83. **2008** **Tarasov Boris V.**

--- author: - 'Youichi [Yanase]{}[^1], Masahito [Mochizuki]{} and Masao [Ogata]{}' title: ' Multi-orbital Analysis on the Superconductivity in ${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$ ' --- Introduction ============ Since the discovery of High-[$T_{\rm c}$ ]{}superconductivity [@rf:bednortz] and heavy fermion superconductors [@rf:steglich], the mechanism of superconductivity induced by electron correlation has been one of the central issues in the condensed matter physics. In this study, recently discovered superconductor [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$ ]{} is analyzed in details. Immediately after the discovery of superconductivity in water-intercalated Cobalt oxides ${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$, [@rf:takada] both experimental [@rf:ong; @rf:sugiyama; @rf:chou; @rf:yoshimura; @rf:kobayashi; @rf:zheng; @rf:ishida; @rf:higemoto; @rf:motohashi; @rf:li; @rf:miyosi; @rf:uemura; @rf:kanigel; @rf:hdyang; @rf:lorenz; @rf:oeschler; @rf:sakurai] and theoretical [@rf:Atanaka; @rf:koshibae; @rf:baskaran; @rf:shastry; @rf:lee; @rf:ogata; @rf:ikeda; @rf:Ytanaka; @rf:honerkamp; @rf:kuroki; @rf:nisikawa; @rf:motrunich] studies have been performed extensively. While some controversial results exist, many experimental evidences for the non-$s$-wave superconductivity [@rf:sakurai] has been reported by NMR [@rf:yoshimura; @rf:kobayashi; @rf:zheng; @rf:ishida] and specific heat measurements. [@rf:hdyang; @rf:lorenz; @rf:oeschler] The characteristic behaviors in strongly correlated electron systems have been observed in the non-water-intercalated compounds. [@rf:chou; @rf:li; @rf:miyosi; @rf:ando] The existence of the magnetic phase [@rf:motohashi; @rf:ong; @rf:sugiyama] in ${\rm Na_{x}Co_{}O_{2}}$ with $x \sim 0.75$ also indicates an importance of electron correlation. These compounds have a layered structure like cuprate [@rf:bednortz] and ruthenate [@rf:maeno], and the two-dimensionality is enhanced by the water-intercalation. These circumstantial evidences indicate that [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$ ]{}is an unconventional superconductor induced by the electron correlation. The theoretical interests are turned on also by the symmetry of crystal structure. In contrast to the square lattice in cuprates and ruthenates, the layer is constructed from the triangular lattice of Co ions. Then, a novel symmetry of Cooper pairing is possible in principle. The $d$-wave superconductivity in cuprate superconductors and $p$-wave superconductivity in ruthenates have been established before. In addition to them, the spin triplet $f$-wave superconductivity and spin singlet $i$-wave one are possible from the analysis of pairing symmetry (see Table. I). The effect of frustration, which is characteristic in the spin system on the triangular lattice, has also attracted much attention. The RVB theory has been applied to the triangular lattice [@rf:baskaran; @rf:shastry; @rf:lee; @rf:ogata] and basically concluded the spin singlet $d$-wave superconductivity. Then, $d_{\rm x^{2}-y^{2}} \pm$ i$d_{\rm xy}$-wave symmetry is expected below [$T_{\rm c}$ ]{}owing to the six-fold symmetry of triangular lattice. However, the time-reversal symmetry breaking has not been observed until now. [@rf:higemoto] Some authors have pointed out the frustration of charge ordering for the electron filling $n=4/3$, [@rf:lee] and the $f$-wave superconductivity due to the charge fluctuation has been discussed. [@rf:Ytanaka; @rf:motrunich] Another interesting property of [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$ ]{}is the orbital degeneracy. The conduction band of this material mainly consists of three $t_{\rm 2g}$-orbitals in Co ions which hybridize with O2p-orbitals. Thus far, most of theoretical studies on the superconductivity have been performed on the basis of the single-orbital model. These investigations have successfully achieved microscopic understandings on the cuprate, organic and ruthenate superconductors. [@rf:yanasereview] However, we consider that the theoretical analysis including the orbital degeneracy is highly desired in order to understand a variety of superconductors including [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$ ]{}and heavy fermion compounds. The superconductivity in $d$-electron systems provides a favorable subject for the theoretical development along this line, because a simple electronic structure is expected compared to heavy fermion superconductors. Although Sr$_2$RuO$_4$ has been a precious compound in this sense, then the orbital degree of freedom is not important for the basic mechanism of superconductivity. [@rf:nomura; @rf:yanaseRuSO] In this study, we show that the orbital degeneracy plays an essential role in [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$ ]{}in contrast to the ruthenate superconductor. We conclude that [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$ ]{}is a typical multi-orbital superconductor in this sense. We adopt a perturbative method for the unconventional superconductivity, [@rf:yanasereview] which is a systematic approach for the electron correlation. Note that the spin fluctuation theory [@rf:moriyaAD] which is widely used for superconductivity is microscopically formulated in this method. It is expected that this approach is reliable from weak to intermediate coupling region. Before the discovery of [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$]{}, this method has been applied to the single-orbital triangular lattice model. Then, the $d$-wave, [@rf:vojta] $f$-wave [@rf:kuroki2001] and $p$-wave superconductivity [@rf:nisikawa2002] have been obtained. Some authors have applied this calculation to [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$]{}, and reported the spin singlet $d$- or $i$-wave superconductivity, [@rf:honerkamp] spin triplet $f$-wave superconductivity [@rf:kuroki] and nearly degeneracy between $d$- and $f$-wave superconductivity. [@rf:nisikawa] We consider that this puzzling problem should be resolved by the multi-orbital analysis involving the microscopic aspects of electronic structure. In this paper, we analyze a multi-orbital Hubbard model constructed from three Co $t_{\rm 2g}$-orbitals. This model appropriately reproduces the electronic structure obtained in the LDA calculation. [@rf:singh; @rf:pickett] The wave function of quasi-particles, which is neglected in the single-orbital Hubbard models, is appropriately taken into account in this multi-orbital model. We show that the momentum dependence of this wave function plays an essential role for the mechanism of superconductivity. We determine the most stable superconducting state with use of the perturbation theory. According to the results of second order perturbation (SOP), third order perturbation (TOP) and renormalized third order perturbation (RTOP) theories, it is concluded that the spin triplet $p$-wave or $f$-wave superconductivity is stable in the wide region of parameter space. The pairing interaction is closely related to the ferromagnetic character of spin susceptibility, although the pairing interaction is not simply described by the spin susceptibility like in the single-orbital model. [@rf:yanasereview] While the momentum dependence of spin susceptibility is usually not remarkable in the frustrating system, the ferromagnetic character clearly appears in the present case owing to the orbital degree of freedom. From a comparison with single-orbital Hubbard models, the important roles of orbital degeneracy are illuminated in §4.1. Alternatively, we propose a reduced two-orbital model including the $e_{\rm g}$-doublet in §4.2. It is shown that results for the superconductivity is appropriately reproduced in this simplified model. On the basis of the two-orbital model, we investigate the roles of vertex correction terms in §5. Then, we show that the vertex correction term, which significantly enhances the spin triplet pairing in Sr$_{2}$RuO$_{4}$, [@rf:nomura] is not important in case of [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$]{}. Thus, the superconducting instability is basically described within the SOP. Therefore, we first explain in details the results of SOP in §3, and discuss the reduced models in §4 and the role of vertex corrections in §5. Multi-orbital model =================== First, we construct a multi-orbital model for [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$]{}. [@rf:motiduki] We consider a two-dimensional model which represents the Co ions on the triangular lattice. Note that the superconductivity occurs when the two-dimensionality is enhanced by the water-intercalation. We also note that the conduction band mainly consists of Co [$t_{\rm 2g}$-orbitals]{}. [@rf:singh; @rf:pickett] Co ion is enclosed by an octahedron of oxygens and nearest neighbor Co ions share the edge of the octahedron. We describe the dispersion relation by using a tight-binding model and adopt a multi-orbital Hubbard Hamiltonian written as, $$\begin{aligned} && H_{3} = H_{0}+H_{{\rm I}}, \\ && H_{0} = \sum_{i,j,s} \sum_{a,b} t_{a,b,i,j} c_{i,a,s}^{\dag} c_{j,b,s}, \\ && H_{{\rm I}} = U \sum_{i} \sum_{a} n_{i,a,\uparrow} n_{i,a,\downarrow} + U' \sum_{i} \sum_{a>b} n_{i,a} n_{i,b} \nonumber \\ && \hspace{10mm} - {J_{{\rm H}}}\sum_{i} \sum_{a>b} (2 {\mbox{\boldmath$S$}}_{i,a} {\mbox{\boldmath$S$}}_{i,b} + \frac{1}{2} n_{i,a} n_{i,b}) \nonumber \\ && \hspace{10mm} + J \sum_{i} \sum_{a \neq b} c_{i,a,\downarrow}^{\dag} c_{i,a,\uparrow}^{\dag} c_{i,b,\uparrow} c_{i,b,\downarrow}. \label{eq:multi-orbital-model}\end{aligned}$$ The first term $H_{0}$ is a tight-binding Hamiltonian where $t_{a,b,i,j}$ are hopping matrix elements. Here, the indices $i$ and $j$ denote the sites in the real space and indices $a$ and $b$ denote the orbitals. We assign the $d_{\rm xy}$-, $d_{\rm yz}$- and $d_{\rm xz}$-orbitals to $a=1$, $a=2$ and $a=3$, respectively. The largest matrix element is the inter-orbital hopping through O2p-orbitals, which are $t_{1,2,i,j}$ for $j=i \pm$(+), $t_{2,3,i,j}$ for $j=i \pm$[ ]{}and $t_{1,3,i,j}$ for $j=i \pm$. We choose the lattice constant as a unit length and denote the unit vectors as =$(\sqrt3/2,-1/2)$ and =$(0,1)$ which are the basis of the triangular lattice. If we assume only the largest matrix elements, the system is regarded to be a superposition of the kagome lattice. [@rf:koshibae] However, the long range hopping through the O2p-orbitals and direct hopping between Co ions are necessary to reproduce the Fermi surface obtained in the LDA calculation. We take account of the matrix elements within third-nearest-neighbor sites according to the symmetry of orbitals and lattice. They are described by nine parameters from $t_1$ to $t_9$. The non-interacting Hamiltonian is described in the matrix representation, $$\begin{aligned} \label{eq:three-band-model-kinetic} && H_0 = \sum_{{\mbox{\boldmath$k$}},s} c_{{\mbox{\boldmath$k$}},s}^{\dag} \hat{H}({\mbox{\boldmath$k$}}) c_{{\mbox{\boldmath$k$}},s}, \\ && \hat{H}({\mbox{\boldmath$k$}}) = \left( \begin{array}{ccc} \varepsilon_{11}({\mbox{\boldmath$k$}}) & \varepsilon_{12}({\mbox{\boldmath$k$}}) & \varepsilon_{13}({\mbox{\boldmath$k$}})\\ \varepsilon_{21}({\mbox{\boldmath$k$}}) & \varepsilon_{22}({\mbox{\boldmath$k$}}) & \varepsilon_{23}({\mbox{\boldmath$k$}})\\ \varepsilon_{31}({\mbox{\boldmath$k$}}) & \varepsilon_{32}({\mbox{\boldmath$k$}}) & \varepsilon_{33}({\mbox{\boldmath$k$}})\\ \end{array} \right), \end{aligned}$$ where $c_{{\mbox{\boldmath$k$}},s}^{\dag}= (c_{{\mbox{\boldmath$k$}},1,s}^{\dag},c_{{\mbox{\boldmath$k$}},2,s}^{\dag},c_{{\mbox{\boldmath$k$}},3,s}^{\dag})$ is a vector representation of the Fourier transformed creation operators with spin $s$. The matrix elements are obtained as, $$\begin{aligned} \label{eq:e11} && \hspace{-10mm} \varepsilon_{11}({\mbox{\boldmath$k$}}) = 2 t_1 \cos k_1 + 2 t_2 (\cos k_2 +\cos k_3) \nonumber \\ && \hspace{-5mm} +2 t_4 (\cos(k_1 - k_3)+\cos(k_1-k_2)) + 2 t_5 \cos 2 k_1, \\ && \hspace{-10mm} \varepsilon_{22}({\mbox{\boldmath$k$}}) = 2 t_1 \cos k_2 + 2 t_2 (\cos k_1 +\cos k_3) \nonumber \\ && \hspace{-5mm} +2 t_4 (\cos(k_1 - k_2)+\cos(k_2 - k_3)) + 2 t_5 \cos 2 k_2, \\ && \hspace{-10mm} \varepsilon_{33}({\mbox{\boldmath$k$}}) = 2 t_1 \cos k_3 + 2 t_2 (\cos k_1 +\cos k_2) \nonumber \\ && \hspace{-5mm} +2 t_4 (\cos(k_1 - k_3)+\cos(k_2 - k_3)) + 2 t_5 \cos 2 k_3, \\ && \hspace{-10mm} \varepsilon_{12}({\mbox{\boldmath$k$}}) = 2 t_3 \cos k_3 + 2 t_6 \cos 2 k_3 +2 t_7 \cos(k_1 - k_3) \nonumber \\ && \hspace{0mm} + 2 t_8 \cos(k_2 - k_3) + t_9 \cos(k_1-k_2) - e_{\rm c}/3, \\ && \hspace{-10mm} \varepsilon_{13}({\mbox{\boldmath$k$}}) = 2 t_3 \cos k_2 + 2 t_6 \cos 2 k_2 +2 t_7 \cos(k_2 - k_3) \nonumber \\ && \hspace{0mm} + 2 t_8 \cos(k_1 - k_2) + t_9 \cos(k_1 - k_3) - e_{\rm c}/3, \\ && \hspace{-10mm} \varepsilon_{23}({\mbox{\boldmath$k$}}) = 2 t_3 \cos k_1 + 2 t_6 \cos 2 k_1 +2 t_7 \cos(k_1 - k_2) \nonumber \\ \label{eq:e23} && \hspace{0mm} + 2 t_8 \cos(k_1 - k_3) + t_9 \cos(k_2 - k_3) - e_{\rm c}/3, \end{aligned}$$ where $k_1=\sqrt{3}/2 k_{\rm x} - 1/2 k_{\rm y}$, $k_2=k_{\rm y}$ and $k_3=-k_1-k_2$. The parameter $e_{\rm c}$ represents the crystal field splitting of [$t_{\rm 2g}$-orbitals ]{}arising from the distortion of octahedron. A typical dispersion relation and Fermi surface are shown in Fig. 1. There is a hole pocket enclosing the $\Gamma$-point and six hole pockets near the K-points, which are consistent with LDA calculations. [@rf:singh; @rf:pickett] We choose the unit of energy as $t_{3}=1$ throughout this paper. Although $e_{\rm c}$ seems to be small, it is useful to use a non-degenerate $a_{\rm 1g}$-orbital and doubly-degenerate $e_{\rm g}$-orbitals. They are defined from the three [$t_{\rm 2g}$-orbitals ]{}as $$\begin{aligned} \label{eq:e1} |e_{\rm g}, 1> = \frac{1}{\sqrt{2}}(|{\rm xz}>-|{\rm yz}>), \\ \label{eq:e2} |e_{\rm g}, 2> = \frac{1}{\sqrt{6}}(2|{\rm xy}>-|{\rm xz}>-|{\rm yz}>), \\ \label{eq:a1g} |a_{\rm 1g}> = \frac{1}{\sqrt{3}}(|{\rm xy}>+|{\rm xz}>+|{\rm yz}>).\end{aligned}$$ The wave function of $a_{\rm 1g}$-orbital spreads along the [*c*]{}-axis, and those of $e_{\rm g}$-orbitals spread along the two-dimensional plane. We will show later that this representation is appropriate for understanding the mechanism of superconductivity (§4.2). ![(a) Fermi surfaces and (b) dispersion relation obtained from the tight-binding Hamiltonian. The dashed line in (a) shows the first Brillouin zone. The parameters are chosen to be $(t_1,t_2,t_3,t_4,t_5,t_6,t_7,t_8,t_9)= (0.08,0.16,1,0.24,-0.16,-0.04,0.16,0.16,-0.2)$. []{data-label="fig:fermisurface"}](Fig1tate.eps){width="7cm"} The hole pocket around the $\Gamma$-point in Fig. \[fig:fermisurface\](a) mainly consists of the $a_{\rm 1g}$-orbital and the six hole pockets near the K-points mainly consist of the $e_{\rm g}$-orbitals. Thus, we denote these Fermi surfaces as $a_{\rm 1g}$-Fermi surface and $e_{\rm g}$-Fermi surface, respectively. This nature of the Fermi surface is consistent with LDA calculations. [@rf:singh; @rf:pickett] Note that recent ARPES measurements [@rf:hasan; @rf:yang] for non-superconducting Na$_x$CoO$_{2}$ observed the $a_{\rm 1g}$-Fermi surface, but the $e_{\rm g}$-Fermi surface has not been found. Fermi surface of water-intercalated Na$_x$CoO$_{2}$ is not clear at present. Moreover, the valence of Co ion in superconducting materials is also under debate. [@rf:karppinen] Therefore, we investigate a wide region in the parameter space and study the possible pairing instability. It is one of the goals of this paper to study the relation between the electronic state and superconductivity. It will be shown that the superconductivity is hard to be stabilized when $e_{\rm g}$-Fermi surface vanishes. The second term $H_{{\rm I}}$ describes the short range Coulomb interactions which include the intra-orbital repulsion $U$, inter-orbital repulsion $U'$, Hund’s rule coupling ${J_{{\rm H}}}$ and pair hopping term $J$. The relations $U=U'+{J_{{\rm H}}}+J$ and ${J_{{\rm H}}}=J$ are satisfied in a simple estimation. Under these conditions, the interaction term $H_{{\rm I}}$ is invariant for the local unitary transformation between orbitals which will be used later. If these relations are violated, the symmetry of triangular lattice is artificially broken. Therefore, we impose these relations through this paper. Although possible roles of the long range Coulomb interaction have been investigated, [@rf:baskaran; @rf:Ytanaka; @rf:motrunich] we concentrate on the short range interaction in this paper. Note that previous studies based on a perturbative method for cuprates, organics and ruthenate have succeeded in identifying the dominant scattering process leading to the superconductivity. [@rf:yanasereview] This theory is complementary to the fluctuation theory which is represented by a random phase approximation (RPA) or fluctuation exchange approximation (FLEX). Generally speaking, the fluctuation theory will be appropriate in the vicinity of the magnetic or other instabilities, because the critical enhancement of the fluctuation is taken into account. On the other hand, the perturbation theory is more appropriate when the critical enhancement of any particular fluctuation is absent, because all terms in the same order are taken into account without any prejudice. We perform the second order perturbation as well as the third order perturbation in this paper. The results of FLEX study will be published elsewhere, [@rf:motiduki] where qualitatively consistent results are obtained. Second Order Perturbation ========================= Details of calculation and classification of pairing symmetry ------------------------------------------------------------- In this section, we investigate the superconducting instability by using the [$\acute{{\rm E}}$liashberg ]{}equation within the second order perturbation (SOP). The basic procedure has been explained in literatures [@rf:yanasereview] and the extension to multi-orbital model is straightforward. The [$\acute{{\rm E}}$liashberg ]{}equation is described by the Green function and the effective interaction. The latter is represented by an irreducible four point vertex in the particle-particle channel (Fig. 2(a)). The second order terms in the effective interaction are diagrammatically represented by Figs. 2(b-e). In case of the single-orbital Hubbard model, this term is simply expressed as $V(k,k')=U^{2}\chi_{0}(k-k')$ for spin singlet pairing and $V(k,k')=-U^{2}\chi_{0}(k-k')$ for spin triplet pairing, respectively, with a bare spin susceptibility $\chi_{0}(k-k')$. However, in the multi-orbital model, the four point vertex has indices of orbitals as $V_{abcd}(k,k')$ (see Fig. 2(a)), which is calculated from the possible combination of Coulomb interactions and Green functions. ![ (a) Diagrammatic representation of the effective interaction leading to the superconductivity. (b-e) The second order terms with respect to the Coulomb interactions (dashed lines). The solid line denotes the Green function having the indices of spin and orbital. []{data-label="fig:diagram"}](Fig2.eps){width="8cm"} In order to make the following discussions clear, we introduce a unitary transformation $\hat{U}({\mbox{\boldmath$k$}})=(u_{ij}({\mbox{\boldmath$k$}}))$ which diagonalizes $\hat{H}({\mbox{\boldmath$k$}})$, namely $$\begin{aligned} \label{eq:unitary} \hat{U}^{\dag}({\mbox{\boldmath$k$}}) \hat{H}({\mbox{\boldmath$k$}}) \hat{U}({\mbox{\boldmath$k$}}) = \left( \begin{array}{ccc} E_1({\mbox{\boldmath$k$}}) & 0 & 0\\ 0 & E_2({\mbox{\boldmath$k$}}) & 0\\ 0 & 0 & E_3({\mbox{\boldmath$k$}})\\ \end{array} \right).\end{aligned}$$ Here, we choose $E_1({\mbox{\boldmath$k$}}) \leq E_2({\mbox{\boldmath$k$}}) \leq E_3({\mbox{\boldmath$k$}})$. With use of these matrix elements, the matrix form of Green function characterized by orbitals $\hat{G}(k) = ({\rm i}\omega_{n} \hat{1} - \hat{H}({\mbox{\boldmath$k$}}))^{-1}$ is described as, $$\begin{aligned} \label{eq:Green-function} G_{ij}(k)=\sum_{\alpha=1}^{3} u_{i\alpha}({\mbox{\boldmath$k$}}) u_{j\alpha}({\mbox{\boldmath$k$}}) G_{\alpha}(k),\end{aligned}$$ where $G_{\alpha}(k)=\frac{1}{{\rm i}\omega_{n}- E_{\alpha}(\mbox{{\scriptsize \boldmath$k$}})}$. In the following, we denote the energy band described by the dispersion relation $E_3({\mbox{\boldmath$k$}})$ as $\gamma$-band. As we have shown in Fig. 1, the $\gamma$-band crosses the Fermi level, and the others are below the Fermi level. Therefore, the superconducting transition is induced by the Cooper pairing in the $\gamma$-band. In this case, the [$\acute{{\rm E}}$liashberg ]{}equation is written in terms of an effective interaction within the $\gamma$-band, $$\begin{aligned} \lambda_{\rm e} \Delta(k) = - \sum_{k'} V(k,k') |G_{3}(k')|^{2} \Delta(k'), \label{eq:eliashberg-equation}\end{aligned}$$ with $$\begin{aligned} \label{eq:effective-interaction} && \hspace{-8mm} V(k,k')=\sum_{abcd} u_{a3}({\mbox{\boldmath$k$}}) u_{b3}(-{\mbox{\boldmath$k$}}) V_{abcd}(k,k') u_{c3}({\mbox{\boldmath$k$}}') u_{d3}(-{\mbox{\boldmath$k$}}'). \nonumber \\\end{aligned}$$ The [$\acute{{\rm E}}$liashberg ]{}equation (eq. (\[eq:eliashberg-equation\])) is regarded to be an eigenvalue equation and $\lambda_{\rm e}$ represents the maximum eigenvalue. The superconducting transition temperature is determined by the criterion $\lambda_{\rm e}=1$. Here, we have ignored the normal self-energy which is important for a quantitative estimation of [$T_{\rm c}$]{}. However, qualitative nature of the superconductivity, such as the pairing symmetry and the pairing mechanism, is not affected in many cases including cuprates, ruthenates and organics. [@rf:yanasereview] This is highly expected in case of [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$]{}, unless the electronic structure is significantly affected by the normal self-energy. We will show that the volume of [$e_{\rm g}$-Fermi surface ]{}, which will be denoted as $n_{\rm e}$ below, is an important parameter for the pairing symmetry. Therefore, it is possible that the pairing symmetry is affected by the normal self-energy through the modification of $n_{\rm e}$. It is, however, expected that the following results are still valid even in this case by regarding the $n_{\rm e}$ modified by the normal self-energy as a relevant parameter. ------------------------------------------------------------------------------------- A$_1$ $s$-wave $1$ ------- ------------ ---------------------------------------------------------------- E$_2$ $d$-wave $ \sin \frac{\sqrt{3}}{2}k_{\rm x} \sin \frac{1}{2}k_{\rm y} $ A$_2$ $i$-wave $ \sin \frac{3\sqrt{3}}{2}k_{\rm x} \sin \frac{1}{2}k_{\rm y} +\sin \frac{\sqrt{3}}{2}k_{\rm x} \sin \frac{5}{2}k_{\rm y} -\sin \sqrt{3} k_{\rm x} \sin 2 k_{\rm y} $ E$_1$ $p$-wave $\sin \frac{\sqrt{3}}{2}k_{\rm x} \cos \frac{1}{2}k_{\rm y}$ B$_1$ $f_1$-wave $ \sin \frac{1}{2}k_{\rm y} (\cos \frac{\sqrt{3}}{2}k_{\rm x} - \cos \frac{1}{2}k_{\rm y}) $ B$_2$ $f_2$-wave $\sin \frac{\sqrt{3}}{2}k_{\rm x} (\cos \frac{\sqrt{3}}{2}k_{\rm x} - \cos \frac{3}{2}k_{\rm y}) $ ------------------------------------------------------------------------------------- : Classification of the pairing symmetry in the triangular lattice. The first column shows the irreducible representations of D$_6$ group. The second column shows the notation adopted in this paper. The $s$-wave, $p$-wave, [*etc*]{} are the counterparts of the isotropic system. The third column shows the typical wave function of Cooper pairs. Before showing the results, it is necessary to classify the pairing symmetry. The symmetry of Cooper pairs is classified into $s$-, $p$-, $d$-wave [*etc.*]{} in case of an isotropic system like $^3$He. For metals, the Cooper pairing is classified into the finite species according to the symmetry of crystals. [@rf:sureview] We show the classification in case of the triangular lattice in Table I. We denote “$s$-wave”, “$d$-wave” [*etc.*]{} in analogy with the isotropic case. While the $s$-, $d$- and $i$-wave are spin singlet pairings, the $p$-, $f_1$- and $f_2$-wave are spin triplet pairings. Note that there remains two-fold degeneracy in the $p$- and $d$-wave states, namely $p_{\rm x}$- and $p_{\rm y}$-wave, $d_{\rm xy}$- and $d_{\rm x^2-y^2}$-wave, respectively. The time-reversal-symmetry-breaking is expected below [$T_{\rm c}$ ]{} in the $d$-wave state, as discussed in the RVB theory. [@rf:baskaran; @rf:shastry; @rf:lee; @rf:ogata] On the contrary, time-reversal-symmetry is not necessarily broken in the $p$-wave case because there is an internal degree of freedom representing the direction of $S=1$, as discussed in Sr$_2$RuO$_4$. [@rf:yanaseRuSO] The eigenvalues of the [$\acute{{\rm E}}$liashberg ]{}equation, eq. (\[eq:eliashberg-equation\]) are classified according to the symmetry of Cooper pairs. The pairing symmetry corresponding to the largest eigenvalue is stabilized below [$T_{\rm c}$]{}. Hereafter, we ignore the possibility of $s$-wave pairing because the strong on-site repulsion will destabilize even the extended $s$-wave pairing. When the symmetry of crystal is lowered, some candidates in Table I are classified into the same irreducible representation. For example, the $d_{\rm xy}$-wave and $s$-wave symmetries are included in the same representation for the anisotropic triangular lattice. [@rf:Ytanakaorganic; @rf:kurokiorganic] However, we can ignore this possibility in the isotropic triangular lattice. Phase diagram of three-orbital model ------------------------------------ In order to search possible pairing symmetries in a phase diagram, we introduce two controlling parameters, $a$ and $n_{\rm e}$. Among the hopping matrix elements in eqs. (\[eq:e11\]-\[eq:e23\]), the largest one, namely $t_3$ is fixed to $1$ but the other matrix elements are chosen to be $$\begin{aligned} \label{eq:minor-matrix} && \hspace{-10mm} (t_1,t_2,t_4,t_5,t_6,t_7,t_8,t_9)= \nonumber \\ && \hspace{-5mm} a (0.1,0.2,0.3,-0.2,-0.05,0.2,0.2,-0.25). \end{aligned}$$ We choose this parameter set so that the dispersion relation obtained in the LDA calculation [@rf:singh; @rf:pickett] is appropriately reproduced when $a \sim 1$. In case of $a=0$, the system is regarded to be a superposition of kagome lattice, [@rf:koshibae] but we have to choose $a \geq 0.6$ in order to obtain a realistic Fermi surface. Thus, the parameter $a$ indicates a deviation from the kagome lattice. Although there are many choices of controlling the minor matrix elements, we have confirmed that the following results are qualitatively independent of the choice. As another controlling parameter, we use the hole number $n_{\rm e}$ in the $e_{\rm g}$-Fermi surface, which can be altered by adjusting the crystal field splitting $e_{\rm c}$. When we decrease $e_{\rm c}$, the energy of $e_{\rm g}$-orbitals is lowered and thus $n_{\rm e}$ decreases. We have confirmed that the value $n_{\rm e}$ is essential rather than the total electron number $n$ for the following results which are almost independent of the way to alter $n_{\rm e}$. Note that the total electron number is fixed as $n=5.33$ throughout this paper. ![Phase diagram for (a)$U'={J_{{\rm H}}}=J=U/3$ and (b)$U'=U/2$ and ${J_{{\rm H}}}=J=U/4$. The horizontal and vertical axes are described in the text. The solid line is the phase boundary obtained by the interpolation. []{data-label="fig:phasediagram3D"}](Fig3tate.eps){width="7cm"} We divide the first Brillouin zone into $128 \times 128$ lattice points and take 512 Matsubara frequencies. We have confirmed that the following results do not depend on the numerical details, qualitatively. In the following, the temperature is fixed to be $T=0.01$ unless we mention explicitly. It will be shown in Fig. 5 that the stable pairing symmetry is almost independent of the temperature. We fix $U=5$ and change the value of ${J_{{\rm H}}}=J$. Under the reasonable conditions $U=U'+2{J_{{\rm H}}}$ and $U'-{J_{{\rm H}}}> 0$, ${J_{{\rm H}}}=U/3$ is the maximum value of the Hund’s rule coupling. Figure. 3 shows the most stable pairing symmetry in the phase diagram of $a$ and $n_{\rm e}$ for two values of the interaction strength. As shown in Fig. 3(a), the spin triplet $p$-wave superconductivity is stabilized in the wide region of parameter space when ${J_{{\rm H}}}=U/3$. The $f_1$-wave superconductivity is also stabilized when $e_{\rm g}$-Fermi surface is very small or very large. For the values of $n_{\rm e}$ expected in the LDA calculation, namely $n_{\rm e}=0.1 \sim 0.3$, we obtain the $p$-wave superconductivity independent of the value of $a$. When the value of Hund’s rule coupling is decreased (Fig. 3(b)), the $f_1$-wave superconductivity becomes more stable. We see that in both cases the spin triplet superconductivity is stable. By definition, the $e_{\rm g}$-Fermi surface vanishes in case of $n_{\rm e}=0$. Then, it is difficult to determine the pairing state since the tendency to superconductivity is very weak independent of the pairing symmetry. On the other hand, the superconductivity is not significantly affected by the disappearance of $a_{\rm 1g}$-Fermi surface which occurs at $n_{\rm e}=0.67$. ![ $n_{\rm e}$-dependence of eigenvalues of [$\acute{{\rm E}}$liashberg ]{}equation. We choose $T=0.01$, $a=0.8$ and (a)$U'={J_{{\rm H}}}=J=U/3$ or (b)$U'=U/2$ and ${J_{{\rm H}}}=J=U/4$. []{data-label="fig:ne-dependence"}](Fig4tate.eps){width="7cm"} In order to make the situation clearer, we show the eigenvalues of [$\acute{{\rm E}}$liashberg ]{}equation for each pairing symmetry in Fig. 4. It is shown that the $p$- and $f_1$-wave superconductivity have nearly degenerate eigenvalues in a wide parameter range. If we assume the weak crystal field splitting $e_{\rm c} \sim 0$, we obtain $n_{\rm e} \sim 0.3$ which is consistent with LDA calculation. The eigenvalue in the $f_1$-wave symmetry shows a minimum around this value. As a result, the $p$-wave superconductivity is stable in this region. As the Hund’s rule coupling decreases, eigenvalues of both $p$- and $f_1$-wave symmetries increase, but that of the $f_1$-wave symmetry increases more rapidly (See also Fig. 7). Note that the eigenvalues for the $d$-wave, $i$-wave and $f_2$-wave states are very small compared to the $p$- and $f_1$-wave states. As is shown later, the $d$-wave state is stabilized when Hund’s rule coupling is very small (Figs. 7 and 8). ![ Temperature dependence of eigenvalues of [$\acute{{\rm E}}$liashberg ]{}equation. We choose $a=0.8$, $n_{\rm e}=0.238$, $U'=U/2$ and ${J_{{\rm H}}}=J=U/4$. []{data-label="fig:te-dependence"}](Fig5.eps){width="7cm"} We see that $\lambda_{\rm e}$ is still less than $1$ at $T=0.01$ (Fig. 4). Therefore, the pairing instability occurs at lower temperature. Fig. 5 shows the temperature dependence of $\lambda_{\rm e}$ at ${J_{{\rm H}}}=U/4$, $a=0.8$ and $n_{\rm e}=0.238$ where the maximum eigenvalue is $\lambda_{\rm e} \sim 0.7$ at $T=0.01$. Then, we obtain $\lambda_{\rm e}=1$ at [$T_{\rm c}$ ]{}$ = 0.0037$ for the $p$-wave symmetry. If we assume $t_{3}=200 $meV so that the total band width is $W=1.8$eV, [$T_{\rm c}$ ]{}$ = 0.0037$ corresponds to [$T_{\rm c}$ ]{}$ = 8$K consistent with experimental value. Furthermore, Fig. 5 clearly shows that most stable pairing symmetry is almost independent of temperature. This means that the phase diagram obtained at $T=0.01$ is very accurate. Another interesting result in Fig. 4 is that the maximum eigenvalue does not significantly depend on $n_{\rm e}$. Even if the size of $e_{\rm g}$-Fermi surface is remarkably reduced, the instability of superconductivity is not suppressed unless the $e_{\rm g}$-Fermi surface vanishes. This is mainly because the DOS of [$e_{\rm g}$-Fermi surface ]{}little depends on the value of $n_{\rm e}$. This is one of the characteristics of the two-dimensional system in the low density region. Note that the number of hole included in each hole pocket is very small as $n_{\rm e}/6 \sim 0.05$. Then, an analogy with the isotropic system like $^3$He is partly justified. This picture is important for the pairing mechanism as we will explain in §3.3. The $n_{\rm e}$-dependence of [$T_{\rm c}$ ]{}can be measured by varying the Na-content of [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$]{}. However, experimental results seems to be controversial. [@rf:schaak; @rf:milne] The eigenvalue rapidly decreases when [$e_{\rm g}$-Fermi surface ]{}vanishes. This result indicates that the [$e_{\rm g}$-Fermi surface ]{}plays an essential role for the superconductivity. This implication will be clearly confirmed in §4.2. Although the eigenvalues are very small, the $d$-wave symmetry seems to be most stable at $n_{\rm e}=0$. Then, the topology of Fermi surface is equivalent to the simple triangular lattice including only the nearest neighbor hopping. In this sense, our result at $n_{\rm e}=0$ is qualitatively consistent with the RVB theory based on the $t$-$J$ model in the triangular lattice, which shows the $d_{\rm x^{2}-y^{2}} \pm {\rm i} d_{\rm xy}$-wave superconductivity. [@rf:baskaran; @rf:shastry; @rf:lee; @rf:ogata] However, the used parameters are quite different. The $t$-$J$ model assumes $U/t > 8$, while $U/t=5$ in this paper. In the intermediate coupling region, the momentum dependence arising from the vertex correction is probably important when the SOP gives very small $\lambda_{\rm e}$. [@rf:yanasereview] In case of the simple triangular lattice, the lowest order vertex correction favors the $p$-wave state. [@rf:nisikawa2002] It should be stressed that the SOP gives much larger value of $\lambda_{\rm e}$ when [$e_{\rm g}$-Fermi surface ]{}exists, as shown in Fig. 4. ![ $a$-dependence of eigenvalues of [$\acute{{\rm E}}$liashberg ]{}equation. We choose the parameter ${J_{{\rm H}}}=U/3$ or ${J_{{\rm H}}}=U/4$. Here, we fix the parameter $e_{\rm c}=0$ instead of $n_{\rm e}$. Therefore, $n_{\rm e}$ slightly differs from $n_{\rm e}=0.33$ at $a=0.6$ to $n_{\rm e}=0.31$ at $a=1$. []{data-label="fig:a-dependence"}](Fig6.eps){width="7cm"} Fig. 6 shows the $a$-dependence of eigenvalues. It is shown that the eigenvalue monotonically increases with decreasing $a$. This variation is basically owing to the increase of the DOS. In case of $a=0.5$, almost flat band is realized around the [$e_{\rm g}$-Fermi surface]{}. Therefore, a steep increase of the eigenvalue leading to the remarkable enhancement of [$T_{\rm c}$ ]{}occurs toward $a=0.5$. We note that most important parameter for the appearance of flat band is the next nearest neighbor hoppings. Although by changing the parameter $a$, the nearest and third nearest neighbor hoppings vary simultaneously, these parameters play only quantitative roles. From Figs. 3-6, we see that the variable $a$ is important for the value of [$T_{\rm c}$]{}, while the variable $n_{\rm e}$ plays an essential role for determining the pairing symmetry. ![ ${J_{{\rm H}}}$-dependence of eigenvalues of [$\acute{{\rm E}}$liashberg ]{}equation. The parameters are chosen to be $a=0.8$ and $n_{\rm e}=0.31$ []{data-label="fig:j-dependence"}](Fig7.eps){width="7cm"} Before closing this subsection, let us discuss the possibility of $d$-wave superconductivity in case of the small Hund’s rule coupling. Fig. 7 shows the ${J_{{\rm H}}}$-dependence of eigenvalues for each pairing symmetry. It is shown that all eigenvalues increase with the decrease of Hund’s rule coupling. Among them, the eigenvalue in the $d$-wave symmetry increases most rapidly and the $d$-wave superconductivity is stabilized for ${J_{{\rm H}}}< U/12$. The phase diagram in the ${J_{{\rm H}}}$-$n_{\rm e}$ plane is shown in Fig. 8. We see that the $d$-wave superconductivity is more stable when $n_{\rm e}$ is small. ![Phase diagram in the ${J_{{\rm H}}}$-$n_{\rm e}$ plane at $a=0.8$. The solid line is the phase boundary obtained by the interpolation. []{data-label="fig:j-phase"}](Fig8.eps){width="8cm"} This stability of the $d$-wave pairing is basically owing to the large value of $U'$ which is comparable to $U$. The inter-orbital repulsion $U'$ couples to the charge and orbital excitations which contribute to the effective interactions equivalently in the singlet and triplet channels. Therefore, the difference between singlet and triplet superconductivity is reduced when $U'$ is large. In other words, the Hund’s rule coupling favors the spin triplet superconductivity, although the value of [$T_{\rm c}$ ]{}is reduced. However, we expect that the $d$-wave superconductivity is less stable if we include the higher order terms because higher order terms significantly enhance the spin excitation rather than the orbital and charge excitation. In other words, the role of $U'$ will be reduced in the higher order theory. This is confirmed by the FLEX calculation. [@rf:motiduki] Basic mechanism of superconductivity ------------------------------------ In order to clarify the basic mechanism of superconductivity, we study the momentum dependence of effective interaction $V(k,k')$ in the spin triplet channel. Figure 9 shows the ${\mbox{\boldmath$k$}}'$-dependence of $V(k,k')$ with ${\mbox{\boldmath$k$}}$ being fixed at the momentum shown by an arrow at which the order parameter in the $p$-wave symmetry takes maximum value. It is apparent that there is a strong attractive interaction between momenta included in the same hole pocket Fermi surface. This is the reason why the spin triplet superconductivity is favored. We can show that in case of ${J_{{\rm H}}}=U/3$, the effective interaction in the singlet channel has opposite sign to that in the triplet channel. This strong repulsive interaction remarkably suppresses the spin singlet superconductivity. ![ Contour plot of the effective interaction $V(k,k')$. The initial momentum ${\mbox{\boldmath$k$}}$ is shown in the figure. The horizontal and vertical axis show $k_{\rm x}'$ and $k_{\rm y}'$, respectively. Matsubara frequency is fixed to the lowest value $\omega_{\rm n}=\omega'_{\rm n}=\pi T$. The Fermi surface is simultaneously described by the thin solid line. The parameters are chosen to be $n_{\rm e}=0.31$, $a=0.8$, $U'=U/2$ and ${J_{{\rm H}}}=J=U/4$. []{data-label="fig:effectiveinteraction"}](Fig9.eps){width="8cm"} ![ (a) Momentum dependence of the static spin susceptibility at $a=0.8$. (b) Schematic figure for the classification of hole pockets. []{data-label="fig:kaitotal"}](Fig10tate.eps){width="7cm"} The microscopic origin of this momentum dependence can be understood as follows. First, we point out the ferromagnetic character of spin fluctuation. Fig. 10(a) shows the spin susceptibility which is estimated by the Kubo formula within the bubble diagram. It is clearly that the spin susceptibility has a trapezoidal peak around ${\mbox{\boldmath$q$}}=0$. Note that the ferromagnetic spin fluctuation has been expected in the LDA calculation [@rf:singh] and observed by the NMR measurement [@rf:ishida]. Owing to the ferromagnetic character of spin susceptibility, the attractive interaction in the same hole pocket is very strong and favors the spin triplet superconductivity. The ferromagnetic spin fluctuation is basically comes from the [$e_{\rm g}$-Fermi surface]{}. Each hole pocket gives rise to the ferromagnetic spin fluctuation like in the two-dimensional electron gas, which has a susceptibility with the trapezoidal structure. Actually, as shown in Fig. 10(a), when we increase the size of hole pockets by changing $n_{\rm e}$, the width of the trapezoidal peak around the $\Gamma$ point increases. Next, we illuminate the essential roles of the orbital degree of freedom. First, we point out that the ferromagnetic spin fluctuation is indeed induced by the orbital degree of freedom. In the multi-orbital model, the spin susceptibility is determined by the dispersion relation and the structure factor arising from the orbital degree of freedom. If we neglect the momentum dependence of structure factor as was done in the previous studies, [@rf:johannes; @rf:nisikawa] we obtain two peaks of spin susceptibility which are quit different from ours. One is located around the M point and the other is slightly removed from the $\Gamma$ point. However, we obtain the trapezoidal peak centered at the $\Gamma$ point by appropriately taking account of the structure factor. Thus, the frustration inherent in the triangular lattice is removed by the orbital degree of freedom which gives rise to the ferromagnetic spin fluctuation. Second, we point out that the roles of the orbital degree of freedom can be understood by considering the momentum dependence of the wave function which is expressed by the unitary matrix $\hat{U}({\mbox{\boldmath$k$}})$ in eq. (\[eq:unitary\]). This wave function indicates the orbital character of quasi-particles (see also §4). The structure factor of spin discussed above is also obtained by this wave function. Furthermore, the effective interaction $V(k,k')$ has another distinct property arising from this momentum dependence. As we have mentioned before, the [$e_{\rm g}$-Fermi surface ]{}mainly consists of the $e_{\rm g}$-doublet whose wave function is shown in eqs. (\[eq:e1\]) and (\[eq:e2\]). Furthermore, we find that the six hole pockets are divided into three pairs as is shown in Fig. 10(b). For example, more than $90$% of the weight of wave function in the Fermi surface “A” originates from the orbital $|e_{\rm g}, 1>$, while the other two pairs are dominated by respective linear combinations of $|e_{\rm g}, 1>$ and $|e_{\rm g}, 2>$. It is generally expected that the electron correlation between the same orbitals is stronger than that between the different orbitals. Actually, the effective interaction between different pairs “A”, “B” and “C” is significantly smaller than those between the same pairs, as shown in Fig. 9. This is the reason why the $p$- and $f_1$-wave superconductivities are stabilized with nearly degenerate eigenvalues as shown in Fig. 4. Which is more stable between $p$- and $f_1$-wave states depends on the coupling between different pairs of hole pockets, which is generally small as explained above. Note that if we apply the phenomenological theory on the ferromagnetic spin-fluctuation-induced superconductivity to [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$]{}, the $f_1$-wave superconductivity is much more stable rather than the $p$-wave superconductivity. The single band model leading to the ferromagnetic spin fluctuation [@rf:kuroki] also concludes the $f_1$-wave symmetry. However, the $p$-wave superconductivity can be stabilized in the present case owing to the orbital degeneracy. It should be noticed that the origin of trapezoidal peak of spin susceptibility around $\Gamma$ point is clearly understood by this momentum dependence of wave function. Although the wave functions are not orthogonal between different pairs of hole pockets, the matrix elements between them in calculating $\chi(q)$ are small. Therefore, in the zeroth order approximation, pairs of hole pockets are regarded to be decoupled from each other. Then, each hole pocket independently induces the trapezoidal peak of $\chi(q)$ as in the two-dimensional electron gas model. Another point to stabilize the superconductivity is the disconnectivity of the [$e_{\rm g}$-Fermi surface ]{}as discussed before the discovery of [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$]{}. [@rf:kuroki2001] Even in the anisotropic superconductivity such as $p$-wave or $f_1$-wave symmetry, the order parameter can take a same sign in each hole pocket, which stabilizes the superconductivity induced by the ferromagnetic spin fluctuation. Note that the difficulty of the ferromagnetic spin-fluctuation-induced superconductivity (superfluidity) has been discussed for $^3$He. [@rf:2dparamagnon] This difficulty is removed by the topological aspect of Fermi surface in case of [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$]{}. Momentum dependence of superconducting gap ------------------------------------------ Next, we show the momentum dependence of order parameter $\Delta({\mbox{\boldmath$k$}},{\rm i}\pi T)$ in Fig. 11. Although $\lambda_{\rm e}$ does not reach $1$ at $T=0.01$ (Fig. 5), it is generally expected that the amplitude $|\Delta({\mbox{\boldmath$k$}},{\rm i}\pi T)|$ shows the momentum dependence of superconducting gap below [$T_{\rm c}$ ]{}and determines the low energy excitation. We note that even if the superconducting instability is dominated by the [$e_{\rm g}$-Fermi surface]{}, the [$a_{\rm 1g}$-Fermi surface ]{}also contributes to the low energy excitations observed by NMR $1/T_1T$, specific heat and magnetic field penetration depth. Fig. 11(a) shows the order parameter in the $p$-wave symmetry. We choose the Hund’s rule coupling as ${J_{{\rm H}}}=U/3$ where the $p$-wave superconductivity is stabilized. Among the two degenerate $p_{\rm x}$- and $p_{\rm y}$-states, only the $p_{\rm y}$-state is shown. Because of the discontinuity of the [$e_{\rm g}$-Fermi surface]{}, the order parameter is node-less on the [$e_{\rm g}$-Fermi surface]{}, while it has nodes on the [$a_{\rm 1g}$-Fermi surface]{}. Since $p_x \hat{x} \pm p_y \hat{y}$, $p_x \hat{y} \pm p_y \hat{x}$ or $(p_x \pm {\rm i} p_y) \hat{z}$ states are expected below [$T_{\rm c}$]{}, the superconducting gap becomes $\sqrt{\Delta_{\rm x}(k)^{2}+\Delta_{\rm y}(k)^{2}}$, where $\Delta_{\rm x}(k)$ and $\Delta_{\rm y}(k)$ are the order parameters for $p_{\rm x}$- and $p_{\rm y}$-states, respectively. In this case, the superconducting gap does not vanish even on the [$a_{\rm 1g}$-Fermi surface]{}. But, we find a remarkable anisotropy of the superconducting gap on the [$a_{\rm 1g}$-Fermi surface ]{}which can explain the power-law behaviors of NMR $1/T_{1}T$ and so on, like in the case of Sr$_2$RuO$_4$. [@rf:nomuragap] However, we note that this is an accidental result. Fig. 11(b) shows the order parameter in the $f_1$-wave symmetry. We choose the Hund’s rule coupling as ${J_{{\rm H}}}=U/6$ where the $f_1$-wave superconductivity is most stable. We can see the clear six times alternation of the sign of order parameter. Also in this case, the [$e_{\rm g}$-Fermi surface ]{}is node-less and [$a_{\rm 1g}$-Fermi surface ]{}has line nodes. As we showed before for the magnetic penetration depth, [@rf:uemura] the combination of fully gaped [$e_{\rm g}$-Fermi surface ]{}and line nodes on [$a_{\rm 1g}$-Fermi surface ]{}gives an intermediate temperature dependence between $s$-wave and anisotropic superconductivity. ![Momentum dependence of order parameter (a) in the $p_{\rm y}$-wave symmetry, (b) in the $f_1$-wave symmetry and (c) in the $d_{\rm xy}$-wave symmetry. The parameters are chosen to be $a=0.8$ and $n_{\rm e}=0.31$. []{data-label="fig:wavefunction"}](Fig11tate.eps){width="7cm"} In Fig. 11(c) we show the order parameter in the $d_{\rm xy}$-wave state which is stabilized when ${J_{{\rm H}}}$ is very small, ${J_{{\rm H}}}=U/12$. The $d_{\rm xy} \pm {\rm i}d_{\rm x^{2}-y^{2}}$ state is expected below [$T_{\rm c}$ ]{}and both [$a_{\rm 1g}$-Fermi surface ]{}and [$e_{\rm g}$-Fermi surface ]{}are node-less in this case. The exponential behaviors in many quantities are expected unless some accidental situation occurs as in the $p$-wave state. Our calculation does not support such an accidental situation in the $d$-wave symmetry. It should be noticed that in all of the above cases we have shown, the amplitude of order parameter is large on the [$e_{\rm g}$-Fermi surface]{}, while it is small on the [$a_{\rm 1g}$-Fermi surface]{}. This result is expected from the fact that the [$e_{\rm g}$-Fermi surface ]{}is responsible for the pairing instability as discussed in §3.3. This point will be illuminated more clearly in the next section. Reduced Models ============== We have analyzed the possibility of unconventional superconductivity in [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$ ]{}on the basis of the three-orbital model. Because calculations for this model need much computational time, a simplified model appropriate for studying the superconductivity is highly desired for a future development in the theoretical field. In this section, we try to find an appropriate model from the comparison to the three-orbital model. We show that the two-orbital model is satisfactory for this purpose, while the single-orbital model is not. The essential origin of the results in §3 will be clarified by these trials. Failure of single-orbital Hubbard model --------------------------------------- Thus far, we have stressed some essential roles of the orbital degeneracy. They are illuminated by showing the failure of single-orbital model. Some authors have already studied single-orbital Hubbard models reproducing the LDA Fermi surface. [@rf:nisikawa; @rf:kuroki] In this paper, we try a single-orbital Hubbard model by keeping only the $\gamma$-band, [*i.e,*]{} the highest-energy eigenstates obtained in eq. (\[eq:unitary\]). Hamiltonian is expressed in the following way. $$\begin{aligned} H_{1} =\sum_{{\mbox{\boldmath$k$}},s} E_{3}({\mbox{\boldmath$k$}}) c_{{\mbox{\boldmath$k$}},s}^{\dag} c_{{\mbox{\boldmath$k$}},s} + U \sum_{i} n_{i,\uparrow} n_{i,\downarrow}. \label{eq:single-orbital-model}\end{aligned}$$ As has been shown in Fig. 1, the typical Fermi surface is reproduced in this model. Indeed, this is the minimal model describing the electron correlation in this material. However, as shown below, this model is inappropriate for the study of superconductivity because the results are qualitatively different from those in the multi-orbital model. We clarify the term “single-orbital Hubbard model” in order to avoid any confusion. In this paper, “single-orbital Hubbard model” suggests the single-band model including only the [*momentum independent*]{} interaction like eq. (\[eq:single-orbital-model\]). As is shown later, we can construct a single-band model in which the roles of orbital degeneracy are appropriately represented in the momentum dependence of interaction term. Thus, we distinguish “single-orbital Hubbard model” from ‘single-band model’’. ![Phase diagram of the single-orbital Hubbard model. The qualitatively different results from Fig. 3 indicate the failure of this model. []{data-label="fig:singlebandmodel"}](Fig12.eps){width="7cm"} In Fig. 12, we show the phase diagram obtained by the SOP applied to the single-orbital Hubbard model (eq. (\[eq:single-orbital-model\])). In the wide region of parameter space, the $d$-wave and $i$-wave superconductivities are stabilized instead of $p$-wave and $f_1$-wave states. The $f_1$-wave superconductivity competes with the $d$-wave one, but is stabilized only in a narrow region. The $p$-wave superconductivity is not stabilized in the whole parameter range. This difference arises from the disregard of the momentum dependence of wave function which is represented by $\hat{U}({\mbox{\boldmath$k$}})$. If we neglect the momentum dependence of $\hat{U}({\mbox{\boldmath$k$}})$ in eq. (\[eq:unitary\]), the three-orbital model is reduced to the single-orbital Hubbard model in eq. (\[eq:single-orbital-model\]). The difference of stable pairing state is apparent if we check the spin susceptibility $\chi(q)$. In the single-orbital Hubbard model, $\chi(q)$ is similar to that obtained in Ref. 31 and we do not clearly see the ferromagnetic tendency (see also the discussion in §3.3). As a result, the momentum dependence of the effective interaction is qualitatively different form that in the three-orbital model. This difference is partly improved by neglecting the $a_{\rm 1g}$-orbital like Ref. 30. Then, we obtain the nearly ferromagnetic spin fluctuation and spin triplet superconductivity. However, the coupling between different pairs of hole pocket Fermi surfaces (see Fig. 10(b)) is over-estimated, and therefore, the $f_1$-wave state is stabilized much more than the $p$-wave state. This is not consistent with the results in §3. We wish to stress again that the characteristic nature of orbital in each hole pocket Fermi surface induces the nearly degeneracy between the $p$-wave and $f_1$-wave states. This characteristic nature can not be taken into account in the single-orbital Hubbard model. Effective two-orbital model --------------------------- The results in the previous subsection show that the single-orbital Hubbard model is qualitatively inappropriate for studying the superconductivity. The important factor to be taken into account is the orbital character of quasi-particles on each Fermi surface. This is described by the momentum dependence of the unitary matrix $\hat{U}({\mbox{\boldmath$k$}})$ in eq. (\[eq:unitary\]). Considering these points, we propose a simplification of the three-orbital model in this subsection. The reduced model is an effective two-orbital model representing the $e_{\rm g}$-doublet. The simplification is performed by the following two steps. \(i) The $a_{\rm 1g}$-orbital is simply ignored. \(ii) The lower band below the Fermi level is ignored.\ The first step is justified because we find that the superconducting instability is dominated by the six hole pocket Fermi surfaces which mainly consist of the $e_{\rm g}$-orbitals. The second one is generally justified because the quasi-particles around the Fermi surface lead to the superconductivity. In order to perform the first step, we transform the basis of local orbitals. This is carried out by using the unitary transformation as, $$\begin{aligned} \label{eq:unitary-local} && \hspace{-20mm} (d_{{\mbox{\boldmath$k$}},1,s}^{\dag},d_{{\mbox{\boldmath$k$}},2,s}^{\dag},d_{{\mbox{\boldmath$k$}},3,s}^{\dag})= (c_{{\mbox{\boldmath$k$}},1,s}^{\dag},c_{{\mbox{\boldmath$k$}},2,s}^{\dag},c_{{\mbox{\boldmath$k$}},3,s}^{\dag}) \hat{U}_{\rm l}, \\ \hspace{-30mm} && \hat{U}_{\rm l} = \left( \begin{array}{ccc} \frac{1}{\sqrt{3}} & 0 & \frac{2}{\sqrt{6}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{6}} \\ \frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{6}} \\ \end{array} \right). \end{aligned}$$ The interaction term $H_{\rm I}$ in the Hamiltonian $H_{3}$ is invariant for this unitary transformation owing to the relations $U=U'+2 {J_{{\rm H}}}$ and ${J_{{\rm H}}}=J$. The non-interacting term is transformed as, $$\begin{aligned} \label{eq:transformed-non-interactiong-part} && \hspace{-10mm} H_0 = \sum_{{\mbox{\boldmath$k$}},s} d_{{\mbox{\boldmath$k$}},s}^{\dag} \hat{H}'({\mbox{\boldmath$k$}}) d_{{\mbox{\boldmath$k$}},s}, \\ && \hspace{-10mm} \hat{H}'({\mbox{\boldmath$k$}}) = \hat{U}_{\rm l}^{\dag} \hat{H}({\mbox{\boldmath$k$}}) \hat{U}_{\rm l}.\end{aligned}$$ The first step is performed by dropping the creation (annihilation) operator $d_{{\mbox{\boldmath$k$}},1,s}^{\dag}$ ($d_{{\mbox{\boldmath$k$}},1,s}$) which corresponds to the $a_{\rm 1g}$-orbital. As a result, the three-orbital model is reduced to the following two-orbital model. $$\begin{aligned} && \hspace{-10mm} H_{2} = \sum_{{\mbox{\boldmath$k$}},s} a_{{\mbox{\boldmath$k$}},s}^{\dag} \hat{h}({\mbox{\boldmath$k$}}) a_{{\mbox{\boldmath$k$}},s} + U \sum_{i} \sum_{a=1}^{2} n_{i,a,\uparrow} n_{i,a,\downarrow} \nonumber \\ && \hspace{-5mm} + U' \sum_{i} \sum_{a>b} n_{i,a} n_{i,b} - {J_{{\rm H}}}\sum_{i} \sum_{a>b} (2 {\mbox{\boldmath$S$}}_{i,a} {\mbox{\boldmath$S$}}_{i,b} + \frac{1}{2} n_{i,a} n_{i,b}) \nonumber \\ && \hspace{-5mm} + J \sum_{i} \sum_{a \neq b} a_{i,a,\downarrow}^{\dag} a_{i,a,\uparrow}^{\dag} a_{i,b,\uparrow} a_{i,b,\downarrow}. \label{eq:two-orbital-model}\end{aligned}$$ Here, we have introduced a $2 \times 2$ matrix $\hat{h}({\mbox{\boldmath$k$}})_{i,j}=\hat{H}'({\mbox{\boldmath$k$}})_{i+1,j+1}$ and two component vector $a_{{\mbox{\boldmath$k$}},s}^{\dag}=(d_{{\mbox{\boldmath$k$}},2,s}^{\dag},d_{{\mbox{\boldmath$k$}},3,s}^{\dag})$. Then, the Green function is described by a $2 \times 2$ matrix as $\hat{G}(k)=({\rm i}\omega_{n} \hat{1} - \hat{h}({\mbox{\boldmath$k$}}))^{-1}$, whose elements are expressed as $$\begin{aligned} \label{eq:2by2-Green-function} G_{ij}(k)=\sum_{\alpha=1}^{2} v_{i\alpha}({\mbox{\boldmath$k$}}) v_{j\alpha}({\mbox{\boldmath$k$}}) G_{\alpha}(k). \end{aligned}$$ Here, $v_{i\alpha}({\mbox{\boldmath$k$}})$ are components of the unitary matrix $\hat{V}^{\dag}({\mbox{\boldmath$k$}})$ which diagonalizes the matrix $\hat{h}({\mbox{\boldmath$k$}})$ $$\begin{aligned} \label{eq:unitary2by2} \hat{V}^{\dag}({\mbox{\boldmath$k$}}) \hat{h}({\mbox{\boldmath$k$}}) \hat{V}({\mbox{\boldmath$k$}}) = \left( \begin{array}{cc} e_1({\mbox{\boldmath$k$}}) & 0 \\ 0 & e_2({\mbox{\boldmath$k$}}) \\ \end{array} \right), \end{aligned}$$ with $e_1({\mbox{\boldmath$k$}})<e_2({\mbox{\boldmath$k$}})$. The diagonalized Green function is obtained as $G_{\alpha}(k)=\frac{1}{{\rm i}\omega_{n}- e_{\alpha}(\mbox{{\scriptsize \boldmath$k$}})}$. We show the dispersion relation $e_1({\mbox{\boldmath$k$}})$ and $e_2({\mbox{\boldmath$k$}})$ in Fig. 13. Apparently the band structure around the [$e_{\rm g}$-Fermi surface ]{}is unchanged by this simplification, while the [$a_{\rm 1g}$-Fermi surface ]{}vanishes. ![ Dispersion relation in the two-orbital model (solid lines). The parameters are chosen to be $a=0.8$ and $n_{\rm e}=0.36$. We have shown the dispersion relation of the $a_{\rm 1g}$-orbital which is obtained as $\hat{H}'({\mbox{\boldmath$k$}})_{11}-\mu$ (dashed line). []{data-label="fig:fermisurface2by2"}](Fig13.eps){width="7cm"} The second step is performed by ignoring the lower energy band, $e_1({\mbox{\boldmath$k$}})$. Then, the Green function is obtained as, $G_{ij}(k)=v_{i2}({\mbox{\boldmath$k$}}) v_{j2}({\mbox{\boldmath$k$}}) G_{2}(k)$. Owing to this procedure, the calculation becomes equivalent to that for a single band Hamiltonian with momentum-dependent interaction, $$\begin{aligned} && \hspace{-10mm} H_{L} =\sum_{{\mbox{\boldmath$k$}},s} e_{2}({\mbox{\boldmath$k$}}) c_{{\mbox{\boldmath$k$}},s}^{\dag} c_{{\mbox{\boldmath$k$}},s} \nonumber \\ && \hspace{-5mm} + \sum_{{\mbox{\boldmath$q$}},{\mbox{\boldmath$k$}}',{\mbox{\boldmath$k$}}} S({\mbox{\boldmath$q$}},{\mbox{\boldmath$k$}}',{\mbox{\boldmath$k$}}) c_{{\mbox{\boldmath$q$}}-{\mbox{\boldmath$k$}},\uparrow}^{\dag} c_{{\mbox{\boldmath$q$}}-{\mbox{\boldmath$k$}}',\downarrow}^{\dag} c_{{\mbox{\boldmath$k$}}',\downarrow} c_{{\mbox{\boldmath$k$}},\uparrow} \nonumber \\ && \hspace{-5mm} + \sum_{{\mbox{\boldmath$q$}},{\mbox{\boldmath$k$}}',{\mbox{\boldmath$k$}},\sigma} S'({\mbox{\boldmath$q$}},{\mbox{\boldmath$k$}}',{\mbox{\boldmath$k$}}) c_{{\mbox{\boldmath$q$}}-{\mbox{\boldmath$k$}},\sigma}^{\dag} c_{{\mbox{\boldmath$q$}}-{\mbox{\boldmath$k$}}',\sigma}^{\dag} c_{{\mbox{\boldmath$k$}}',\sigma} c_{{\mbox{\boldmath$k$}},\sigma}. \label{eq:long-range-model}\end{aligned}$$ The momentum dependent factors $S({\mbox{\boldmath$q$}},{\mbox{\boldmath$k$}}',{\mbox{\boldmath$k$}})$ and $S'({\mbox{\boldmath$q$}},{\mbox{\boldmath$k$}}',{\mbox{\boldmath$k$}})$ are expressed by the Coulomb interactions $U$, $U'$, ${J_{{\rm H}}}$ and $J$ and the wave function $v_{i2}({\mbox{\boldmath$k$}})$. If we neglect the momentum dependence of unitary matrix $\hat{V}^{\dag}({\mbox{\boldmath$k$}})$, the factor $S({\mbox{\boldmath$q$}},{\mbox{\boldmath$k$}}',{\mbox{\boldmath$k$}})$ becomes $U$ and $S'({\mbox{\boldmath$q$}},{\mbox{\boldmath$k$}}',{\mbox{\boldmath$k$}})=0$. Then, the model is exactly reduced to the single-orbital Hubbard model described by eq. (\[eq:single-orbital-model\]) with use of $e_{2}({\mbox{\boldmath$k$}})$ instead of $E_{3}({\mbox{\boldmath$k$}})$. We have discussed in §4.1 that this single-orbital Hubbard model is not appropriate. On the other hand, the Hamiltonian $H_{L}$ is appropriate because the roles of orbital degeneracy are taken into account in the momentum dependence of interaction. ![Eigenvalues of [$\acute{{\rm E}}$liashberg ]{}equation obtained in the effective two-orbital model. We choose the parameters $a=0.8$ and ${J_{{\rm H}}}=U/4$. []{data-label="fig:phasediagram2by2"}](Fig14.eps){width="7cm"} We find that the results for the superconductivity are almost the same between the Hamiltonian $H_{2}$ and $H_{L}$. In Fig. 14 we show the $n_{\rm e}$-dependence of eigenvalues of [$\acute{{\rm E}}$liashberg ]{} equation for the simplified model, $H_{L}$. We see that the increase of eigenvalues with $n_{\rm e}$ is steeper than that in Fig. 4. This is mainly owing to the increase of DOS. However, the relation between each pairing symmetry closely resembles. For example, the $p$-wave superconductivity is stable around $n_{\rm e}=0.2$, while the $f_1$-wave superconductivity is realized for larger values of $n_{\rm e}$. The eigenvalue for the spin singlet $d$-wave superconductivity is far below that for the spin triplet one. These results mean that the effective two-orbital model described by eq. (\[eq:two-orbital-model\]) or eq. (\[eq:long-range-model\]) appropriately reproduces the results in the three-orbital model. The fact that the step (1) is appropriate clearly means that the superconductivity is basically led by the [$e_{\rm g}$-Fermi surface]{}. The [$a_{\rm 1g}$-Fermi surface ]{}plays only a secondary role. Note that the eigenvalue of [$\acute{{\rm E}}$liashberg ]{}equation decreases owing to the step (1), mainly because the DOS in the [$e_{\rm g}$-Fermi surface ]{}decreases. We have confirmed that the step (2) slightly enhances the spin triplet superconductivity. Effects of Vertex Corrections in a Two-Orbital Model ==================================================== In this section, we study the effects of vertex corrections. Although it is desirable to study these effects in the three-orbital model, we use the effective two-orbital model whose validity has been demonstrated in §4.2, because of numerical difficulties. Generally speaking, the higher order terms may play an important role for the superconducting instability, since it is considered that most of unconventional superconductors are in the intermediate coupling region. For example, vertex correction which is not included in the RPA plays an important role to stabilize the spin triplet pairing in Sr$_2$RuO$_4$. [@rf:nomura] Therefore, it is an important issue to investigate the role of higher order corrections in the present model. ![Diagrammatic representation of the third order terms in the effective interaction. (a-f) correspond to the spin singlet channel or spin triplet channel with d-vector $d \parallel {\hat z}$. (c’-f’) correspond to the spin triplet channel with d-vector $d \perp {\hat z}$. []{data-label="fig:3rddiagram"}](Fig15.eps){width="8cm"} We apply the third order perturbation theory (TOP) and its renormalized version to the Hamiltonian $H_2$ (eq. (\[eq:two-orbital-model\])). We adopt this model instead of more simplified model $H_{\rm L}$ (eq. (\[eq:long-range-model\])) because the computational time is hardly reduced by the second step (ii) in §4.2. The parameter is chosen to be ${J_{{\rm H}}}=U/3$, where the interaction between electrons with same spin vanishes and thus the number of diagrams is much reduced. As discussed in §3.2, this region will be relevant rather than the region where the Hund’s rule coupling is small. Fig. 15 shows the diagrammatic representation of third order terms in the effective interaction. Figs. 15(a) and (b) are classified into the RPA terms and others are the vertex corrections. The present theory is invariant for the rotation of spin, since we do not take account of the spin-orbit interaction. Therefore, the result on the spin triplet pairing does not depend on the direction of $d$-vector. Note that two RPA terms cancel each other in case of the spin triplet pairing with $d \parallel z$. ![ Eigenvalues of [$\acute{{\rm E}}$liashberg ]{}equation in the third order perturbation theory. The thick solid line shows the maximum eigenvalue in the second order perturbation theory, which is classified into the $p$-wave symmetry. We do not show the eigenvalue in the $d$-wave symmetry because the tendency to superconductivity is very weak. We fix the parameters $a=0.6$, $n_{\rm e}=0.35$ and ${J_{{\rm H}}}=U/3$. []{data-label="fig:naivetop"}](Fig16.eps){width="7cm"} We numerically solve the [$\acute{{\rm E}}$liashberg ]{}equation within the TOP and show the eigenvalues in Fig. 16. We see that the $p$- and $f_2$-wave superconductivity are significantly stabilized for $U>4$, while the $f_1$-wave and spin singlet pairings are unfavored. However, as discussed below, we find that these results in the intermediate coupling region are fictitious. Within the third order terms in Fig. 15, dominant contributions for triplet channel come from the terms represented in Figs. 15(e’) and (f’), which include a particle-particle ladder. In contrast, the terms represented in Figs. 15(c’) and (d’) with a particle-hole ladder are negligible. As is well known in the Kanamori theory on the metallic ferromagnetism, [@rf:kanamori] the particle-particle ladder diagrams generally induce the screening of interaction as $U \rightarrow U(q)=U/(1+U\phi(q))$ where $\phi(q)$ is obtained by the particle-particle ladder diagram. If $q$-dependence of $U(q)$ is not important, this scattering process is incorporated by the renormalized coupling constant $\bar{U}$. In the above TOP calculation, only the lowest order term in the Kanamori-type correction was taken into account. Therefore, it is reasonable to think that the contributions from Figs. 15(e’) and (f’) can be suppressed if we include the higher order perturbation terms. ![ (a) Renormalization of the particle-particle ladder diagram. In (e”) and (f”), the renormalized particle-particle ladder is used instead of bare ladder. In the RTOP, we take account of the terms (e”) and (f”) instead of (e’) and (f’) in Fig. 15. []{data-label="fig:t-matrix"}](Fig17.eps){width="6cm"} In order to investigate this possibility, we perform a calculation of a renormalized TOP (RTOP), as shown in Fig. 17. The particle-particle ladder in Figs. 15(e’) and (f’) are replaced by the T-matrix shown in Fig. 17(a). As a result, the infinite order terms representing the screening effect are taken into account as in the Kanamori theory. By using the diagrams in Figs. 2, 15(c’,d’) and 17(e”,f”), we estimate the effective interaction and solve the [$\acute{{\rm E}}$liashberg ]{}equation. The obtained eigenvalues are shown in Fig. 18. It is apparent that the results of naive TOP is significantly altered by the renormalization and that the correction to the SOP is small. In particular, the $p$-wave superconductivity is slightly stable over the $f_1$-wave superconductivity. The nearly degeneracy between these states is also reproduced. The order parameter in each pairing symmetry is very similar to Fig. 11, although that in the naive TOP is remarkably different. We see that the eigenvalues are slightly reduced from the SOP, however the $U$-dependence is almost unchanged. These results are naturally interpreted if we consider that the vertex corrections basically work as a screening effect. Then, the second order perturbation theory is justified by regarding the interactions to be the renormalized ones. ![ Eigenvalues of [$\acute{{\rm E}}$liashberg ]{}equation in the renormalized third order perturbation theory for $p$-wave (circles) and $f_1$-wave (diamonds) symmetry. Note that the eigenvalue in the $f_2$-wave symmetry is very small. The thick solid and dashed lines show the eigenvalues in the second order perturbation theory for the $p$-wave and $f_1$-wave symmetry, respectively. The parameters are the same as in Fig. 16. []{data-label="fig:rtop"}](Fig18.eps){width="7cm"} Let us compare the present results to the case of high-[$T_{\rm c}$ ]{}cuprates and Sr$_2$RuO$_4$. For high-[$T_{\rm c}$ ]{}cuprates, the $d$-wave superconductivity is basically induced by the RPA terms and the vertex correction due to the particle-particle ladder diagrams effectively reduces the coupling constant. [@rf:yanasereview; @rf:bulut] Therefore, the situation is very similar to the present case, although there is a difference of singlet and triplet pairing. On the other hand, in case of Sr$_2$RuO$_4$, the effective interaction derived from the RPA terms has very weak momentum dependence, which does not work for the anisotropic pairing. However, the $q$-dependence of particle-particle ladder in TOP favors the spin triplet superconductivity. [@rf:nomura] Then, the naive discussion on the screening effect can not be applied. It has been confirmed that the qualitative results of TOP applied to Sr$_2$RuO$_4$ are not altered even when the renormalization of particle-particle ladder is taken into account. [@rf:nomuraforth] Thus, the basic mechanism of possible spin triplet superconductivity in [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$ ]{}is qualitatively different from that in Sr$_2$RuO$_4$. Discussions =========== In this paper, we have investigated the multi-orbital model for [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$ ]{}on the basis of the perturbation theory. The obtained results indicate a possibility of spin triplet superconductivity in this material, although the $d$-wave superconductivity is also stabilized in a part of parameter space. There are two candidates of spin triplet pairing; $p$-wave and $f$-wave superconductivity are nearly degenerate. Although the spin triplet superconductivity is one of the most interesting issues in the condensed matter physics, the microscopic theory remains in the developing stage. This is mainly owing to very few $d$-electron materials showing the spin triplet superconductivity. Although we see many candidates in the heavy fermion materials, the theoretical treatment is generally difficult for $f$-electron systems. Therefore, a discovery of spin triplet superconductor in transition metal oxides will lead to an important development in the microscopic understandings. Probably, most established spin triplet superconductor in $d$-electron systems is Sr$_2$RuO$_4$. [@rf:maeno] Therefore, we have provided detailed discussions on the comparison between Sr$_2$RuO$_4$ and [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$]{}. According to the results in this paper, [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$ ]{}provides a qualitatively different example from Sr$_2$RuO$_4$ in the following two points. First, the RPA terms give rise to the dominant scattering process leading to the spin triplet pairing. The spin excitation is clearly ferromagnetic and favorable for the spin triplet pairing. This is in sharp contrast to the case of Sr$_2$RuO$_4$ where the vertex corrections are essential for the $p$-wave pairing. In case of [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$]{}, the vertex corrections induce only the screening effect which is not important for the qualitative results. While the ferromagnetic spin-fluctuation-induced spin triplet superconductivity has been discussed from early years, the corresponding superconductivity has not been established until now. We expect that [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$ ]{}will be a first example realizing this mechanism. Second, the orbital degeneracy plays an essential role in case of [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$]{}. The conduction band in [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$ ]{}as well as that in Sr$_2$RuO$_4$ are basically described by three t$_{\rm 2g}$-orbitals. Although the single-orbital Hubbard model is an appropriate model for describing the pairing mechanism of Sr$_2$RuO$_4$, [@rf:nomura] such a simplification is qualitatively inappropriate for [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$]{}. The success of single-orbital Hubbard model for Sr$_2$RuO$_4$ is due to the electronic structure where the $\gamma$-band is basically described by the local $d_{\rm xy}$-orbital. The failure for [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$ ]{}is due to the fact that the [$e_{\rm g}$-Fermi surface ]{}can not be described by any individual local orbital. In other words, the hybridization term in the unperturbed Hamiltonian is large in case of [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$]{}, while it is negligible in Sr$_2$RuO$_4$ owing to the particular crystal symmetry. In this sense, [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$ ]{}will be a more typical example of the multi-orbital superconductor. Then, the momentum dependence of the wave function of quasi-particles essentially affects the effective interaction leading to the Cooper pairing. We have pointed out that the reduced two-orbital model is appropriate, instead of the failure of single-orbital model. This is because the Fermi surface in [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$ ]{}can be classified according to the local orbitals. Then, the superconductivity is basically triggered by the [$e_{\rm g}$-Fermi surface]{}. Since a portion of $a_{\rm 1g}$-orbital in the [$e_{\rm g}$-Fermi surface ]{}is less than 5%, this orbital is safely ignored. This situation is similar to the case of Sr$_2$RuO$_4$. However, the orbital degeneracy in $e_{\rm g}$-doublet can not be ignored in case of [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$]{}. From the above comparisons, we obtain the following empirical rules. \(1) When the RPA-terms are favorable for the anisotropic superconductivity, the non-RPA terms are not qualitatively important, and [*vice versa*]{}. \(2) When a part of Fermi surface is described by a few local orbitals, the simplification of microscopic model is possible. In particular, the second rule will be helpful for a future development of microscopic understanding on the multi-band superconductors. For example, several Fermi surfaces appear in heavy fermion materials. This fact as well as the 14-fold degeneracy in $f$-shell make the microscopic treatment difficult. However, it will be possible to obtain a simplified model by identifying the microscopic character of each Fermi surface. Thus far, we have discussed the superconductivity in [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$ ]{}induced by the electron-electron correlation and highlighted the possibility of spin triplet pairing. However, any clear experimental evidence for the symmetry of superconductivity has not been obtained up to now. Instead, we see some experimental observations which restrict the pairing state. For example, the absence of (or very small) coherence peak in NMR $1/T_{1}T$, [@rf:yoshimura; @rf:kobayashi; @rf:ishida; @rf:zheng] power-law temperature dependence of $1/T_{1}T$ [@rf:ishida; @rf:zheng] and specific heat, [@rf:hdyang; @rf:lorenz; @rf:oeschler] NMR Knight shift below [$T_{\rm c}$ ]{} [@rf:yoshimura; @rf:kobayashi; @rf:ishidaprivate; @rf:zhengprivate] and time-reversal symmetry observed in $\mu$SR [@rf:higemoto] should be cited, although a part of them are controversial. As for the results in this paper, spin triplet $p$- or $f_1$-wave superconductivity is consistent with the absence of coherence peak and with the power-law behaviors below [$T_{\rm c}$ ]{}. In both cases, the (quasi-)line nodes appear in the [$a_{\rm 1g}$-Fermi surface]{}. In case of the $p$-wave pairing, the time-reversal-symmetry observed in $\mu$SR indicates a $d$-vector parallel to the plane, namely $\hat{d}=p_{\rm x}\hat{x} \pm p_{\rm y}\hat{y}$ or $\hat{d}=p_{\rm x}\hat{y} \pm p_{\rm y}\hat{x}$. This direction of $d$-vector is consistent with the recent measurements of NMR Knight shift under the parallel field [@rf:kobayashi; @rf:ishidaprivate; @rf:zhengprivate] as well as macroscopic $H_{\rm c2}$, [@rf:chou; @rf:sasaki] if we assume that the $d$-vector is strongly fixed against the applied magnetic field. We note that the qualitatively different result has been obtained in the NMR Knight shift, [@rf:yoshimura] which is consistent with this pairing state if the $d$-vector is weakly fixed against the magnetic field. Although we have shown that the $d$-vector in Sr$_2$RuO$_4$ is very weakly fixed against the magnetic field, [@rf:yanaseRuSO] this is partly owing to the particular electronic structure of Sr$_2$RuO$_4$. Therefore, we expect that the anisotropy of $d$-vector is larger for [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$]{}. The symmetry breaking interaction leading to the anisotropy arises from the second order term with respect to the spin-orbit interaction for Sr$_2$RuO$_4$, while it arises from the first order term in case of [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$]{}. Therefore, it is possible that the $d$-vector is strongly fixed against the magnetic field in case of [${\rm Na_{x}Co_{}O_{2}} \cdot y{\rm H}_{2}{\rm O}$]{}. Quantitative estimations for the anisotropy will be one of the interesting future issues. On the other hand, the possibility of spin singlet superconductivity has not been denied up to now. Then, the absence of time-reversal symmetry breaking will be a issue to be resolved for $d$-wave pairing because the $d_{\rm x^{2}-y^{2}} \pm {\rm i} d_{\rm xy}$ state is expected so as to gain the condensation energy. The local distortion of triangular lattice or the feedback effect will be a candidate of the resolution. It seems that the $i$-wave superconductivity [@rf:kurokiprivate] is consistent with the present experimental results except for the very weak impurity effects. [@rf:yokoi] However, the microscopic mechanism leading to the pairing with $T_{\rm c}=5$K will be difficult for such a high angular momentum state. In our study, we have not found the stable $i$-wave state. 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--- abstract: 'The peak position, impact-parameter range, and optimal conditions for observing spiral scattering of relativistic particles in a uniformly bent crystal are estimated. The existence of spiral scattering with a square-root singularity is pointed out. In this case, the secondary process of volume capture to the channeling mode is absent and the conditions for observing this effect are most favorable.' author: - 'G.V.Kovalev [^1]' date: 'Dec. 26, 2008' title: Possibility of Observing Spiral Scattering of Relativistic Particles in a Bent Crystal --- The phenomenon of spiral scattering occurs due to the appearance of a negative logarithmic singularity of the classical deflection function $\chi(b)$ of a particle or light ray for a certain impact parameter $b=b_s$ [@FordWheeler_1959_1] . Resonance scattering is a quantum mechanical analog of spiral scattering [@berry_mount_1972]. However, resonance scattering includes a wider class of quantum-mechanical phenomena. In particular, it can appear in the scattering of fast particles by a cylindrical well (see, e.g., [@kal_kov_79]), whereas classical spiral scattering by such a potential is absent[@kov08_1]. ![ (a) Square-well potential shown by solid curve 1 and a smoothed well given by dashed curve 2. (b) The function $u(r)$ given by Eq. (3) for the (line 1) well, (line 2) smoothed well, and (line 3) barrier. (c) The deflection function $\alpha=\chi/2$. (d) The trajectory of a particle in the square well. (e) The spiral trajectory of a particle in the smoothed square well.[]{data-label="fig:CentralWells"}](CentralWells.eps) To illustrate this feature and determine the spiral-scattering boundaries, let us compare scattering by a cylindrical or spherical potential well of radius $ R$ and depth $-U_0$ and scattering by a well of the same depth and radius, but with a smoothed parabolic edge of width $d/2$ ($d << R$) (curves 1 and 2, respectively, in Fig. 1a): $$\begin{aligned} U(r)= - U_0 \frac{4}{d^2} \left \{\begin{array}{ll} 0,& \; R < r;\\ (r-R)^2,& \; (R-\frac{d}{2}) < r < R;\\ \frac{d^2}{4},& \; 0 < r < (R-\frac{d}{2}). \end{array} \right. \label{SmoothEdgePotential}\end{aligned}$$ The classical deflection function $$\begin{aligned} \chi(b)=2\alpha(b)=\pi-2b\int^{\infty}_{r_{o}}\frac{d r }{r \sqrt{r^2[1-\phi(r)]-b^2}}, \label{deflection_function}\end{aligned}$$ where $\phi(r)={2U(r) E}/({p_{\infty}^2 c^2})$ and $b, U(r), E, p_{\infty}$ are the impact parameter, centrally symmetric potential, total energy, and momentum of a particle at infinity, respectively, can be calculated for both cases, but the absence of spiral scattering for a square well can be seen directly in the plot (curve 1 in Fig. 1b) of the function $$\begin{aligned} u(r)=r \sqrt{1-\phi(r)}, \label{u_function}\end{aligned}$$ which has a finite step and a local minimum for $r = R$. Therefore, the derivative $u(R)^{'}$ is indeterminate at this point. Deflection function (2) is expressed in terms of function (3) as $$\begin{aligned} \chi(b)=\pi-2b\int^{\infty}_{r_{o}}\frac{d r }{r \sqrt{u(r)-b}\sqrt{u(r)+b}}, \label{deflection_function1}\end{aligned}$$ where, as before, the turning point $r_0$ is determined from the equation $u(r_0) = b$. In the case of the smoothed potential (curve 2), the function $u(r)$ has a smooth local minimum at the point rmin, at which $u'(rmin) = 0$. If the impact parameter coincides with this minimum, i.e., $$\begin{aligned} b_s = u(r_{min}), \label{SpiralImpact}\end{aligned}$$ both the radial velocity and radial acceleration of the particle become zero (for more detail, see \[1, 5\]) and the conditions for spiral scattering are implemented. Here, the function $u(r)$ in the vicinity of the minimum $r_{min}$ is represented in the form $$\begin{aligned} u(r)\approx u(r_{min}) + \frac{u(r_{min})^{''}}{2}(r-r_{min})^2, \label{u_functionAppr}\end{aligned}$$ while the local-minimum point $r_{min}$ determined from the condition $u'(r_{min}) = 0$ is specified by the equation $$\begin{aligned} 1-\phi(r_{min})-\frac{r_{min}}{2} \phi(r_{min})^{'} = 0. \label{gen_r_min_eq}\end{aligned}$$ This equation holds for any potential; for Eq. (1), it reduces to the simple form $$\begin{aligned} 2\hat{r}^2_{min}-3\hat{r}_{min}+1+\delta = 0. \label{r_min_eq}\end{aligned}$$ Here, $$\begin{aligned} \hat{r}=\frac{r}{R}, \;\; \hat{d}=\frac{d}{R}, \;\; \delta= \frac{\hat{d}^{2}}{4 |\phi_o|}, \;\; \phi_o=\frac{-2U_0 E}{p_{\infty}^2 c^2} \label{Notations}\end{aligned}$$ and $θ_L = \sqrt{|\phi_o|} $ is the Lindhard channeling angle. In the case of the potential given by Eq. (1), semi-channeling can occur when a particle moves near the potential edge and is successively reflected from its inner wall. Hereafter, a quantity with the symbol $\hat{}$ denotes the corresponding quantity without this symbol divided by $R$. Only one, physically meaningful, of the two solutions of Eq. (8) with the plus sign of the square root should be retained: $$\begin{aligned} \hat{r}_{min}=\frac{3}{4} + \frac{\sqrt{1-8\delta}}{4}. \label{r_min}\end{aligned}$$ Note the important condition that should be in the range$1- \hat{d}/2 \leq \hat{r}_{min} \leq 1$ (see Fig. 1); otherwise, the local minimum does not exist. The lower boundary $1- \hat{d}/2 = \hat{r}_{min}$ substituted into the left-hand part of Eq. (10) yields the critical parameter $\delta_c$ for the observation of the spiral scattering of relativistic particles by a smoothed well: $$\begin{aligned} \delta \leq \delta_c= \frac{\hat{d}(1 -\hat{d})}{2}. \label{criteria_delta}\end{aligned}$$ Since, according to Eq. (9), $|\phi_o|= \frac{\hat{d}^{2}}{4\delta }$, the corresponding criterion for the squared Lindhard angle $|\phi_{o,c}|$ can also be written as $$\begin{aligned} |\phi_{o}| \geq |\phi_{o,c}| = \frac{\hat{d}}{2(1 -\hat{d})}. \label{criteria_Squar_Lindhard}\end{aligned}$$ Next, since the smooth-edge region is small, $\hat{d} << 1$ (or $d << R$), Eqs. (11) and (12) reduce to the inequalities $\delta \leq \hat{d}/2$, $|\phi_{o}| \geq \hat{d}/2$, respectively. It is easy to see that these constraints are equivalent to the criterion $R \geq R_c= {p_{\infty}^2c^2 d}/(4U_0 E)=d/(2 \theta^{2}_{L})$. It was noted in \[4, 5\] that this criterion is equivalent to the Tsyganov criterion for the existence of the channeling effect in a bent crystal. Since $\hat{d} << 1$, $\delta << 1$ and Eq. (10) can be approximately written as $\hat{r}_{min} \cong 1-\delta$. For comparison, the quantity $\alpha(\hat{b})=\chi(\hat{b})/2$ for the square well in the range $0\leq\hat{b}\leq1$ is given by the expression (see Section 19 in \[6\] or Eq. (12) in \[5\]) $$\begin{aligned} \alpha(\hat{b}) = \arcsin(\frac{\hat{b}(\sqrt{1-\hat{b}^2}-\sqrt{1-\phi_{o}-\hat{b}^2})}{{\sqrt{1-\phi_{o}}}}). \label{deflection_Well}\end{aligned}$$ This corresponds to curve 1 in Fig. 1c. All deflection angles for the square well are negative with the maximum absolute value $$\begin{aligned} |\alpha(1)| = \arcsin(\frac{\sqrt{-\phi_{o}}}{{\sqrt{1-\phi_{o}}}}), \label{deflection_WellMIN}\end{aligned}$$ achieved near the well edge $\hat{b}=1$. Under the condition $\sqrt{|\phi_{o}|}<<1$, this angle is $|\alpha(1)| \approx \theta_{L}$. A typical trajectory of particles in such a potential is shown in Fig. 1d. In the case of the smoothed well given by Eq. (1), deflection function (2) is given by a lengthy formula with elliptical functions. However, if approximation (6) is used, the deflection function for the impact parameters $\hat{b}_s < \hat{b} < 1$ takes the form $$\begin{aligned} \alpha(\hat{b}) =\frac{\pi}{2}-\arcsin(\hat{b})-\sqrt{\frac{\hat{b}}{\hat{u}(\hat{r}_{min})^{''}}}*\nonumber \\ \frac{\ln(\frac{2(\hat{r}_0-\hat{r}_{min})}{(\sqrt{\hat{r}_0(1+\hat{r}_0-2\hat{r}_{min})}-\sqrt{(1-\hat{r}_0)(2 \hat{r}_{min}-\hat{r}_0)})^2})}{\sqrt{\hat{r}_{min}^2-(\hat{r}_0-\hat{r}_{min})^2}}, \label{deflection_EdgeWell1}\end{aligned}$$ where $\hat{r}_0=\hat{r}_{min} + \sqrt{2(\hat{b}-\hat{u}(\hat{r}_{min}))/\hat{u}(\hat{r}_{min})^{''}}$. The deflection function for the impact parameters $0 < \hat{b} < \hat{b}_s$ has the different form $$\begin{aligned} \alpha(\hat{b}) =\arcsin(\frac{\hat{b}}{A\sqrt{1+\hat{d}^2/(4\delta)}})- \arcsin(\hat{b})-\nonumber \\ \sqrt{\frac{\hat{b}}{C\hat{u}(\hat{r}_{min})^{''}}} \ln(\frac{2C+AB+2\sqrt{C}\sqrt{A^2+AB+C}}{A(2C+B+2\sqrt{C}\sqrt{1+B+C})}), \label{deflection_EdgeWell2}\end{aligned}$$ where $A=1-\hat{d}/2$, $B=-2\hat{r}_{min}$, $C=\hat{r}^2_{min}+2(\hat{u}(\hat{r}_{min})-\hat{b})/(\hat{u}(\hat{r}_{min})^{''})$. If $\hat{b} \rightarrow \hat{b}_s=\hat{u}(\hat{r}_{min})$, the deflection angles given by Eqs. (15) and (16) tend logarithmically to$-\infty$ (see curve 2 in Fig. 1c); i.e., the particle orbits the potential center by an angle exceeding maximum angle (14) for a square well. Under the condition $\hat{b} = \hat{b}_s$, spiral scattering appears; in this case, the relativistic particle does not have the outgoing branch and is located near the potential for an infinitely long time (see Fig. 1e) approaching the limit cycle $\hat{r} \rightarrow \hat{r}_{min}$. Only one trajectory satisfies the spiral-scattering criterion. Trajectories with impact parameters close to $\hat{b}_s$ have the usual ingoing and outgoing branches, but can follow the circle over an angle exceeding the Lindhard angle (14). Let the spiral-scattering range be defined as the impact-parameter range $\Delta b_s$ near $b = b_s$, in which the absolute values of negative angles (15) and (16) exceed maximum angle (14) for the square well (see Fig. 1c). Such a definition is useful since it distinguishes the spiral scattering from both refraction ranging within the angle given by Eq. (14) and volume reflection, which has a positive sign \[7\], but does not exceed the angle given by Eq. (14). The spiral scattering is also different from channeling (see below). The substitution of Eq. (10) into Eq. (5) yields the exact value for the spiral impact parameter $\hat{b}_s$ in the case of potential (1): $$\begin{aligned} \hat{b}_s=\frac{3+\sqrt{1-8\delta}}{4}\sqrt{1+\frac{(1-\sqrt{1-8\delta})^2}{16\delta}} . \label{SpiralImpactForSmoothEdgePotential}\end{aligned}$$ If $\delta << 1/8$, $\hat{b}_s\approx 1- \delta/2$. The right and left boundaries of the spiral-scattering range are determined as the roots of the transcendental equation $$\begin{aligned} \alpha(\hat{b}) = -\arcsin(\frac{\sqrt{|\phi_{o}|}}{{\sqrt{1+|\phi_{o}|}}}), \label{TransendentalEq}\end{aligned}$$ with $\alpha(\hat{b})$ given by Eqs. (15) and (16), respectively. By solving the simpler transcendental equation $c_1 \ln{(\Delta \hat{b}_{sr})} +c_2=0$ (where c1 and c2 are constants), which is obtained by retaining only the first terms in the expansions of all functions in Eq. (15) in the power series of $(\hat{b}-\hat{b}_s)$ in the right neighborhood of $\hat{b}_s$, the right part $\Delta \hat{b}_{sr}=\hat{b}_{r}-\hat{b}_{s}$ of the region $\Delta \hat{b}_s$ is determined in the form $$\begin{aligned} \Delta \hat{b}_{sr}=2\hat{u}(\hat{r}_{min})^{''}\delta^2 \exp\biggl (-\sqrt{\frac{\hat{u}(\hat{r}_{min})^{''}}{\hat{b}_{s}}}*\nonumber \\ \Bigl (\frac{\hat{d}}{\sqrt{\delta}} +2 \sqrt{1-\hat{b}_{s}^2} \Bigr)\biggr ). \label{RightBoundary}\end{aligned}$$ Hereafter, it is taken into account the conditions $|\phi_{o}|<<1$ and $\delta<<1/8$, which are certainly valid in the relativistic case. To calculate the left-hand part $\Delta \hat{b}_{sl}=\hat{b}_{s}-\hat{b}_{l}$ in the entire range given by Eq. (12), it is necessary to retain a few terms in the expansion of Eq. (16) in the power series of $(\hat{b}-\hat{b}_s)$, since the impact parameter $\hat{b}_s$ approaches the inflection point $\approx \hat{d}/4$ of the function $\hat{u}(\hat{r})$ with a variation in the parameter $|\phi_{o}|$. In the case where they are kept, the solution of the transcendental equation $c_1 \ln{(c_3 \Delta \hat{b}_{sl})}+ \Delta \hat{b}_{sl} +c_2=0$, obtained by retaining only first two terms in the series, has a solution expressed in terms of the Lambert function $W(x)$, which is defined through the relation $W(x) \exp{W(x)}=x$ \[8\], as $$\begin{aligned} \Delta \hat{b}_{sl}= c_1 W(\frac{\exp{(-\frac{c_2}{c_1})}}{c_1 c_3}), \label{LeftBoundary}\end{aligned}$$ where the parameters $c_1, c_2$ and $c_3$ are $$\begin{aligned} c_1=\frac{\sqrt{A^2(1+\phi_0)-\hat{b}^2_{s}}\sqrt{1-\hat{b}^2_{s}}}{(\sqrt{A^2(1+\phi_0)-\hat{b}^2_{s}}-\sqrt{1-\hat{b}^2_{s}})\sqrt{\hat{u}(\hat{r}_{min})^{''}}} \nonumber \\ c_2 =-\sqrt{A^2(1+\phi_0)-\hat{b}^2_{s}}\sqrt{1-\hat{b}^2_{s}} +\nonumber \\ + \frac{\hat{d}\sqrt{A^2(1+\phi_0)-\hat{b}^2_{s}}\sqrt{1-\hat{b}^2_{s}}}{2\sqrt{\delta (1+\phi_0)}(\sqrt{A^2(1+\phi_0)-\hat{b}^2_{s}}-\sqrt{1-\hat{b}^2_{s}})}\nonumber \\ c_3 =\frac{A}{2 \delta(\hat{d}/2-\delta)\hat{u}(\hat{r}_{min})^{''}}. \label{LeftBoundaryParam}\end{aligned}$$ The width of the impact-parameter region $\Delta \hat{b}_{s}$ where spiral scattering is significant is determined by the sum of Eqs. (19) and (20): $$\begin{aligned} \Delta \hat{b}_{s}=\Delta \hat{b}_{sr}+\Delta \hat{b}_{sl}. \label{Boundary}\end{aligned}$$ The second derivative $\hat{u}(\hat{r}_{min})^{''}$ in Eqs. (19) and (20) is expressed in terms of $\delta$ as $$\begin{aligned} \hat{u}(\hat{r}_{min})^{''}=\frac{4\hat{r}_{min}-3}{\delta \sqrt{1+\frac{(\hat{r}_{min}-1)^2}{\delta}}}=\frac{\sqrt{1-8\delta}}{\delta \sqrt{1+\frac{(1-\sqrt{1-8\delta})^2}{16\delta}}}, \label{2ndDerivative}\end{aligned}$$ and $\hat{u}(\hat{r}_{min})^{''} \approx \delta^{-1}$ for small $\delta$ values. ![Dimensionless spiral-scattering width $\Delta \hat{b}_{s}$ versus the parameter $|{\phi_{o}}|/|{\phi_{o,c}|}$. The solid curve is the numerical calculation by \[TransendentalEq\]. The circles show the contribution from $\Delta \hat{b}_{s}$ to $\Delta \hat{b}_{sl}$, according to \[LeftBoundary\]. The diamonds show the contribution from $\Delta \hat{b}_{s}$ to $\Delta \hat{b}_{sr}$, according to \[RightBoundary\].[]{data-label="fig:Width"}](Width.eps) In addition to approximate formulas (19) and (20) for the right and left boundaries of the spiral-scattering region, respectively, transcendental equations (18) were also solved numerically. The result is shown by the solid curve in Fig. 2. The numerical solution for the right-hand boundary (diamonds) coincides with high accuracy with Eq. (19). Formula (20) for the left-hand boundary (circles) provides an underestimated result, but the behavior of the curve is the same as predicted by the accurate numerical solution. In addition, the contribution from the right-hand part of the spiral-scattering range to the total width is much smaller than the contribution from the left-hand part, except for the vicinity of the critical point $|\phi_{o}|=|\phi_{o,c}|$. As a result, the logarithmic branches of the deflection function on the right and left sides of $b = b_s$ are not symmetric for the considered potential (curves 2 and 3 in Fig. 3). ![Deflection function $\alpha (\hat{b})$ for the potential given by \[SmoothEdgePotential\]: 1 - $\hat{d}=2\cdot10^{-4}$, $|\phi_{o}|=10^{-4}$, 2 - $\hat{d}=2\cdot10^{-4}$, $|\phi_{o}|=3\cdot10^{-4}$, 3 - $\hat{d}=2\cdot10^{-4}$, $|\phi_{o}|=2\cdot10^{-3}$. The horizontal line segments for each curve correspond to the refraction by the angle $\theta_{L}=\sqrt{|\phi_{o}|}$. The widths of these segments correspond to the spiral-scattering and deflection boundaries.[]{data-label="fig:Deflection"}](Deflection.eps) Note an important feature of the width of the spiral-scattering region. Let the potential parameters $R$, $U_0$, and $d$ be fixed and the relativistic particle energy decrease so that $|\phi_{o}|$ increases from critical value (12) to a few tens of $|\phi_{o,c}|$ without violating the condition $|\phi_{o}|<<1$. Then, the width $\Delta \hat{b}_{s}$ of the spiral-scattering region increases rapidly up to the maximum for $|\phi_{o}|\approx 3|\phi_{o,c}|$ and then decreases to zero. In this specific case, the normalized maximum value$\Delta \hat{b}_{s,max}/(\hat{d}/2)$ is $\approx 19 \%$ of the width $\hat{d}/2$ (see Fig. 2) and its distance from the critical energy is $|\phi_{o}|/|\phi_{o,c}|$. Thus, the position of the maximum on this scale is independent of the particle energy. Moreover, the normalized width $$\begin{aligned} \Delta \hat{b}_{s}/(\hat{d}/2)=f(|\phi_{o}|/|\phi_{o,c}|) \label{InvariantEq}\end{aligned}$$ is a universal function determined only by the shape of the potential. At the same time, the value of the parameter $|\phi_{o}|$ for which the maximum is reached evidently depends on the particle energy. However, the above-mentioned scaling invariance holds in this case. This feature is seen in Fig. 4, which presents the width of the spiral-scattering range for the ring potential given by Eq. (1) with the hollow core ($U(r)=0$ for $r< R-d/2$). This potential ensures maximum volume reflection in comparison with other ring potentials of the depth $–U_0$ and width $d$ constructed of two inversed parabolas. Thus, this potential provides a minimum estimate for the possible width of spiral scattering for an actual crystal. The value $\Delta \hat{b}_{s,max}/(\hat{d}/2)\approx 7 \%$ and position $|\phi_{o}|/|\phi_{o,c}|\approx 1.3$ of the maximum of the spiral scattering width are different in this case, but the scaling invariance holds. ![Solid curve is the dimensionless spiral-scattering width $\Delta \hat{b}_{s}/(\hat{d}/2)$ calculated numerically for the ring potential with a half-parabola by \[TransendentalEq\]. The circles show the contribution from $\Delta \hat{b}_{s}$ to $\Delta \hat{b}_{sl}$.The diamonds show the contribution from $\Delta \hat{b}_{s}$ to $\Delta \hat{b}_{sr}$.[]{data-label="fig:Width1"}](Width1.eps) Note that the $\alpha (\hat{b})$ plots for $\hat{d}$ and $|\phi_{o}|$ values other than those shown in Fig. 3 have the same form if $\alpha$ and $\hat{b}$ are normalized to $\theta_{L}$ and $\hat{d}/2$, respectively, due to the above mentioned scaling invariance. It is seen that the width of the spiral-scattering range for large $|\phi_{o}|$ values ($|\phi_{o}|=20 \cdot|\phi_{o,c}|$, curve 3 for the relatively low energies) decreases and tends to zero in the limit. In this case, the deflection function tends to Eq. (13) for the square-well scattering, for which the smoothness of the edge is not important and the deflection angles $\alpha (\hat{b})$ are no larger than $\theta_{L}$ (the angle $\chi (\hat{b})$ is no larger than $2 \theta_{L}$). The width of the spiral-scattering range, $\Delta \hat{b}_s$, increases with energy (with decreasing $|\phi_{o}|$). This width on curve 2 is maximal for $|\phi_{o}|=3|\phi_{o,c}|$. Then, the width $\Delta \hat{b}_s$ decreases. Curve 1 corresponds to the critical energy $|\phi_{o}|=|\phi_{o,c}|=\hat{d}/2$, for which channeling is already impossible since the effective well is absent. Meanwhile, spiral scattering exists for this energy. ![Function $\hat{u}(\hat{r})=\hat{r}\sqrt{1\pm\frac{\phi_0 (1-\cos(2 \pi (1-\hat{r})/\hat{d}))}{2}}$ for a crystal potential for various particle energies and fixed parameters $R$, $U_0$, $d$. $1$, $2$, $3$ - negatively charged particles with ($\hat{d}=0.002$, $\phi_0=0.001,\; 0.003,\; 0.01$); $1p$, $2p$, $3p$ - positively charged particles (the minus sign under the square root in $\hat{u}(\hat{r})$ ); $P_c$ are the inflection points of plots $1$ and $1p$ for the critical energies corresponding to the Tsyganov criterion.[]{data-label="fig:InflactionPoint"}](InflactionPoint.eps) The last feature is due to the existence of a singularity of the deflection function for any smooth periodic potential similar to that shown in Fig. 5 if the critical conditions corresponding to the equality in Eqs. (11) and (12) are exactly satisfied. However, in contrast to the logarithmic singularity considered by Ford and Wheeler \[1\] and given by expansion (6), the singularity type changes. In this case, an inflection point ( $P_c$ in Fig. 5) appears instead of the local minimum; i.e., the second derivative $u^{''}(r_{min})$ is also zero at this point and the series expansion of $u(r)$ in the vicinity of the minimum $r =r_{min}$ becomes $$\begin{aligned} u(r)\approx u(r_{min}) + \frac{u(r_{min})^{'''}}{6}(r-r_{min})^3. \label{u_functionAppr2}\end{aligned}$$ It is easy to see that integral (4) for $b=u(r_{min}) $ has the inverse square-root singularity $$\begin{aligned} \chi(b) \approx -\frac{1}{\sqrt{|u(r_{min})-b|}}, \label{deflection_function2}\end{aligned}$$ Hence, the spiral scattering and deflection should take place for $|\phi_{o}|=|\phi_{o,c}|$. Note that although the effective force is zero at the local minimum and inflection points, the spiral scattering and deflection by a potential without the centrifugal term always take place at the internal slope of the potential responsible for attraction (see the local minima in Fig. 5). These features hold also for a real crystal potential having one local minimum or inflection point per period under conditions (11) and (12). In this case, the position and maximum width are certainly different and, in addition, the width should be normalized to the crystal period $\hat{d}$. This yields the lower estimate for the maximum width $\Delta \hat{b}_s/\hat{d}\approx 3.5 \%$ (ring potential) and the upper estimate $\Delta \hat{b}_s/\hat{d}\approx 9.5 \%$ (potential with edge). However, the scaling invariance should also hold for these potentials. For experiments, this means that if, e.g., the crystal curvature radius $R$ is changed, then the particle energy $E$ for a given crystal potential can always be chosen so that the parameter $|\phi_{o}|$ satisfies the criterion for which the spiral-scattering width is maximal. At the same time, the bending radius $R$ maximizing the spiral-scattering width can always be pointed out for any relativistic particle energy $E$. It should be noted that the multiple scattering and dissipation processes in a real crystal do not eliminate the singularity associated with spiral scattering. This was demonstrated for nuclear reactions such as, e.g., $^{40}Ar$ + $^{232}Th$, in which deep inelastic processes were interpreted in terms of negative spiral-scattering angles with allowance for friction forces \[9\]. This is explained by the fact that friction forces deform particle trajectories, but the spiral-scattering range $\Delta \hat{b}_s$ remains unchanged. Owing to the absence of thermal vibrations and the low electron density in the region of local minima, the volume capture of negatively charged particles is much smaller and the spiral-scattering width is much greater than the respective quantities for positively charged particles. Hence, negatively charged particles deflected along the crystal bending by an angle larger than $2 \theta_{L}$ propagate in the spiral-scattering mode. The above estimates show that the fraction of negatively charged particles for the maximum spiral-scattering width is $3\%$ äî $9\%$. The preliminary experiments \[10, 11\] on the scattering of 180-GeV electrons in $Si$ show that a fairly large number of particles move at an angle larger than $>2 \theta_{L}=32$ µrad and do not have the peak typical of volume capture at the end of the angular size of the crystal. The local minima in the case of positively charged particles moving in weakly bent crystals are in the region of high electron densities and thermal fluctuations of potentials. Thus, the trapping of spiral particles to the channeling mode seems to be the most probable secondary process. However, the spiral scattering is also significant in this case since the motion along the crystal bending ensures the intense volume capture of the particles in the region $\Delta \hat{b}_s$. For the crystal bending equal to the critical radius $R_c$ when the channeled states are absent, the presence of particles with a rotation angle exceeding $2\theta_{L}$ can be undoubtedly interpreted as the presence of spiral scattering in this case and in the case of negatively charged particles. I am grateful to Andrea Mazzolari for attracting my attention to work \[10\]. [10]{} K. W. [Ford]{} and J. A. [Wheeler]{}. , 7, 259–286, (1959). M. V. Berry and K.E. Mount. , 35, 315–397, (1972). N. P. Kalashnikov and G. V. Kovalev. , 29(6), 302–307, (1979). G. V. Kovalev. , 87(2), 94–98, (2008). G. V. Kovalev. , 87(7), 349–353, (2008). L. D. Landau and E. M. Lifshitz. . Butterworth-Heinemann; 3rd edition, NY, (1976). A. M. Taratin and S. A. Vorobiev. , A 119(8), 425–428, (1987). J. Borwein, D. Bailey, and R. Girgensohn. . A K Peters, Natick, Massachusetts, (2004). J. Wilczynski. , 47B(6), 484–486, (1973). W Scandale. , page 32, (06.12.2007). W Scandale. , page 26, (29.10.2008). [^1]: e-mail: [email protected]

--- abstract: 'The *girth* of a finitely generated group $\Gamma$ is the supremum of the girth of Cayley graphs for $\Gamma$ over all finite generating sets. Let $\Gamma$ be a finitely generated subgroup of the mapping class group $\Mod({\Sigma})$, where ${\Sigma}$ is a compact orientable surface. Then, either $\Gamma$ is virtually abelian or it has infinite girth; moreover, if we assume that $\Gamma$ is not infinite cyclic, these alternatives are mutually exclusive.' address: | Department of Mathematics\ Temple University\ Philadelphia, PA 19122 author: - Kei Nakamura bibliography: - '../../../MyTeX/MyMaster.bib' title: | The girth alternative for\ mapping class groups --- Introduction {#Introduction} ============ Let ${\Sigma}$ be a compact orientable surface, which is possibly disconnected. The *mapping class group* of ${\Sigma}$, denoted by $\Mod({\Sigma})$, is the group of isotopy classes of $\Homeo^+({\Sigma})$, i.e. the group of orientation preserving homeomorphisms of ${\Sigma}$. Mapping class groups have been studied extensively in complex analysis, low-dimensional topology, and geometric group theory for more than a century. A fascinating aspect of the mapping class groups is that they share many properties with lattices in semi-simple Lie groups. One analogy between linear groups and mapping class groups can be seen in the following famous dichotomy regarding their subgroups, which has become known as the *Tits-alternative* for linear groups and mapping class groups respectively. \[tits-alt: linear\] Let $\Bbbk$ be a field, and let $\Gamma$ be a finitely generated subgroup of $GL(n,\Bbbk)$. Then, $\Gamma$ either contains a non-abelian free subgroup or is virtually solvable; moreover, these alternatives are mutually exclusive. \[tits-alt: mcg\] Let ${\Sigma}$ be a compact orientable surface, and let $\Gamma$ be a finitely generated subgroup of $\Mod({\Sigma})$. Then, $\Gamma$ either contains a non-abelian free subgroup or is virtually abelian; moreover, these alternatives are mutually exclusive. The dichotomy in the Tits alternative has been further investigated, and some refinements and variations are known for linear groups and mapping class groups; see, for example, the work of Margulis and Soĭfer on maximal subgroups of linear groups [@Margulis--Soifer:MaximalSubgrp] and the analogous result by Ivanov for mapping class groups [@Ivanov:Tits-Margulis-Soifer]. The purpose of this article is to demonstrate that the structural analogy between these groups can be reinterpreted in terms of the *girth* of finitely generated groups. Recall that the *girth* of a graph is the length of the shortest graph cycle, if any, in the graph. In [@Schleimer:Girth], Schleimer defined the *girth* of a finitely generated group $\Gamma$ to be the supremum of the girth of Cayley graphs of $\Gamma$ over all finite generating sets. By the observations of Schleimer in [@Schleimer:Girth] and the subsequent work of Akhmedov in [@Akhmedov:Girth1] and [@Akhmedov:Girth2], there appears to be a significant qualitative difference between groups with finite girth and groups with infinite girth. In particular, Akhmedov gave the following “*girth alternative”* for linear groups, which shows that the division essentially coincides with the dichotomy in the Tits alternative for linear groups. \[[@Akhmedov:Girth2]\] \[girth-alt: linear\] Let $\Bbbk$ be a field, and let $\Gamma$ be a finitely generated subgroup of $GL(n,\Bbbk)$. Then, $\Gamma$ is either a non-cyclic group with infinite girth or a virtually solvable group; moreover, these alternatives are mutually exclusive. Our main result is the following analogous girth alternative for mapping class groups $\Mod({\Sigma})$, showing that the division of subgroups into the ones with finite girth and the ones with infinite girth again coincides with the dichotomy in the Tits alternative. \[girth-alt: mcg\] Let ${\Sigma}$ be a compact orientable surface, and let $\Gamma$ be a finitely generated subgroup of $\Mod({\Sigma})$. Then, $\Gamma$ is either a non-cyclic group with infinite girth or a virtually abelian group; moreover, these alternatives are mutually exclusive. For *irreducible* subgroups of mapping class groups $\Mod({\Sigma})$, the result was independently proved by Yamagata [@Yamagata:Girth]; see §4 for the definition of irreducible subgroups. One difficulty in proving Theorem \[girth-alt: mcg\] is that we cannot simply pass the statement to a finite index normal subgroup; unfortunately, we don’t know that a group has infinite girth even if it has a finite index normal subgroup with infinite girth. This requires us to study the structure of $\Gamma$ carefully when ${\Sigma}$ is disconnected or when $\Gamma$ is *reducible*. In the general context of finitely generated groups, the existence of a non-abelian free subgroup in $\Gamma$ does not imply that $\Gamma$ has infinite girth. Hence, the girth alternative is not a mere consequence of the Tits alternative. As we shall see, for subgroups of mapping class groups, the girth alternative is a slightly more intricate manifestation of underlying structural properties that are responsible for the Tits alternative. A part of the arguments in the proof of Theorem \[girth-alt: mcg\] can also be used to show the girth alternative for convergence groups (see [@Nakamura:Thesis], [@Yamagata:Girth]) and subgroups of $\Out(F_n)$ that contain an irreducible element with irreducible powers (see [@Nakamura:Thesis]). Acknowledgement --------------- The results in this paper are contained in the Ph.D. dissertation by the author, submitted in 2008 to University of California, Davis [@Nakamura:Thesis]. The author would like to thank Dmitry Fuchs, Misha Kapovich, and his advisor Joel Hass for being the members of the dissertation committee, carefully reading the original exposition of this work, and providing insightful comments while the author was at University of California, Davis. The author would also like to thank David Futer and Igor Rivin for encouraging him to make this work publicly available in the present format. Outline ------- We present some results on the girth of finitely generated groups in §\[Criteria\]. The main tool used in the proof of Theorem \[girth-alt: mcg\] is the Infinite Girth Criterion (Proposition \[igc\]) in §\[Infinite Girth\], which reformulates and generalizes the work of Akhmedov in [@Akhmedov:Girth2]; see [@Yamagata:Girth] for a similar alternative reformulation. The essence of the proof of this criterion is a reminiscent of the classical *ping-pong argument*, which goes back to the work of Blaschke, Klein, Schottky, and Poincaré on Schottky groups in $\PSL(2,\R)$ and $\PSL(2,\C)$; see, for example, [@Klein:Ping-Pong]. In §\[Elements\] and §\[Subgroups\], we review the properties of elements and subgroups of mapping class groups from Thurston’s theory [@Thurston:Surfaces] and the proof of the Tits alternative [@Birman-Lubotzky-McCarthy:AbSol], [@Ivanov:Tits-Margulis-Soifer], [@McCarthy:Tits], [@Ivanov:MCGBook], and also prove a few facts that are necessary for the application of Infinite Girth Criterion in the proof of Theorem \[girth-alt: mcg\]. Finally, in §\[girth alternative\], we present the proof of Theorem \[girth-alt: mcg\]. Conventions ----------- For various reasons, the mapping class group $\Mod({\Sigma})$ of a connected surface ${\Sigma}$ *with boundary* is often defined in the literature to be the group of isotopy classes of $\Homeo^+({\Sigma},\partial {\Sigma})$, consisting of homeomorphisms which take each component of $\partial {\Sigma}$ to itself; that is *not* the convention we follow in this article. Our definition of $\Mod({\Sigma})$ as the group of isotopy classes of $\Homeo^+({\Sigma})$ implies that elements of $\Mod({\Sigma})$ may permute components of $\partial {\Sigma}$. This convention coincides with the ones used in [@Birman-Lubotzky-McCarthy:AbSol], [@McCarthy:Tits], [@Ivanov:Tits-Margulis-Soifer], and [@Ivanov:MCGBook], where Thurston’s theory on mapping class groups [@Thurston:Surfaces] was utilized in the studies of the structures of subgroups. One aspect of Thurston’s theory involves a process of cutting the surface ${\Sigma}$ along an essential multi-loop which is invariant under the action of a mapping class and obtaining the induced mapping class on the resulted surface, say ${\Sigma}_{\calC}$; this new mapping class generally permutes the components of ${\Sigma}_{\calC}$ and $\partial {\Sigma}_{\calC}$, and the theory is developed most naturally under the convention we adopt in this article. Girth of Finitely Generated Groups {#Criteria} ================================== The *girth* of a graph is the combinatorial length of the shortest cycle in the graph if there is a nontrivial cycle in the graph, and is set to be infinity if there is no cycle in the graph. Using the girth of Cayley graphs, Schleimer introduced in [@Schleimer:Girth] the notion of the *girth* of a finitely generated group. \[girth\] Let $\Gamma$ be a finitely generated group. Let $U(\Gamma, \calG)$ be the girth of the Cayley graph of $\Gamma$ with respect to a generating set $\calG$. The *girth* of the group $\Gamma$ is defined to be $U(\Gamma):=\sup_\calG\{U(\Gamma,\calG)\}$, where the supremum is taken over all finite generating sets $\calG$ of $\Gamma$. Throughout the rest of the article, a group $\Gamma$ is assumed to be finitely generated unless otherwise stated. Clearly, every finite group has finite girth and any free group has infinite girth. It is also easy to see that an abelian group has finite girth unless it is the infinite cyclic group. In this section, we discuss some criteria for the girth of a group to be finite or infinite in a general setting. The Infinite Girth Criterion (Proposition \[igc\]) in §\[Infinite Girth\] is employed as the main tool in §\[girth alternative\]. Criteria for Finite Girth {#Criteria: Finite Girth} ------------------------- Recall that a group $\Gamma$ is said to satisfy a law if there is a word $w(x_1, \cdots , x_n)$ on $n$ letters such that $w(\gamma_1, \cdots , \gamma_n)=1$ in $\Gamma$ for any $\gamma_1, \cdots , \gamma_n \in \Gamma$. Schleimer obtained an important criterion for a group to have finite girth in [@Schleimer:Girth]. \[law criteria\] A group $\Gamma$ has finite girth if it satisfies a law and is not infinite cyclic. Together with the proposition below, Theorem \[law criteria\] provides a significant portion of the class of groups that are known to have finite girth. \[schleimer criteria\] A group $\Gamma$ has finite girth if it satisfies one of the following conditions: - $\Gamma$ contains a finite-index subgroup with finite girth. - $\Gamma$ admits a finite-kernel surjection onto a group with finite girth. \[virtually solvable\] A virtually solvable group $\Gamma$ has finite girth, unless it is infinite cyclic. One may ask if amenable groups, other than the infinite cyclic one, always have finite girth. However, according to the work of Akhmedov in [@Akhmedov:Girth1 §2], the answer turns out to be negative. We also note his observation in [@Akhmedov:Girth1 §4] that certain groups constructed by Olshanskii in his book [@Olshanskii:RelationsBook] have finite girth but do not satisfy any law. Characterizing groups with finite girth appears to be a delicate and difficult task. See [@Akhmedov:Girth1 §5] for related questions. Criteria for Infinite Girth {#Infinite Girth} --------------------------- The proof of the Tits alternative for linear groups and mapping class groups, as well as for other classes of groups, use variations of *ping-pong* lemma [@Klein:Ping-Pong] to construct a free subgroup. The following formulation was given in [@Tits:FreeSubgrp]. \[fsc\] Let $\Gamma$ be a group acting on a set $X$. Suppose there exist elements $\sigma, \tau \in \Gamma$, subsets $U_\sigma, U_\tau \subset X$, and a point $x \in X$, such that 1. $\displaystyle{x \not \in U_\sigma \cup U_\tau}$, 2. $\displaystyle{\sigma^k ( \{x\} \cup U_\tau ) \subset U_\sigma }$ for all $k \in \Z-\{0\}$, and 3. $\displaystyle{\tau^k ( \{x\} \cup U_\sigma) \subset U_\tau }$ for all $k \in \Z-\{0\}$. Then, $\langle \sigma, \tau \rangle$ is a non-abelian free subgroup of $\Gamma$. For the proof of the above criterion, one merely observes inductively that any nontrivial reduced word in $\sigma^{\pm 1}$ and $\tau^{\pm 1}$ takes $x \in X-(U_\sigma \cup U_\tau)$ into $U_\sigma \cup U_\tau$ via the action of $\langle \sigma, \tau \rangle$, showing that the word cannot represent the identity element of $\Gamma$. This is the classical *ping-pong argument*. In the study of the girth of groups, a criterion for a group to have infinite girth is instrumental. Generalizing and reformulating the work of Akhmedov [@Akhmedov:Girth2], we give such a criterion in comparable generality as Proposition \[fsc\]; see [@Yamagata:Girth] for a similar alternative reformulation. \[igc\] Let $\Gamma$ be a group acting on a set $X$, with a finite generating set $\calG:=\{\gamma_1, \cdots , \gamma_n\}$. Suppose there exist elements $\sigma, \tau \in \Gamma$, subsets $U_\sigma, U_\tau \subset X$, and a point $x \in X$, such that 1. $\displaystyle{x \not \in (U_\sigma \cup U_\tau) \cup \bigcup_{\varepsilon=\pm1} \bigcup_{i=1}^n \gamma_i^\varepsilon(U_\sigma \cup U_\tau)}$, 2. $\displaystyle{\sigma^k \bigg( \{x\} \cup U_\tau \cup \bigcup_{\varepsilon=\pm1} \bigcup_{i=1}^n \gamma_i^\varepsilon(U_\tau) \bigg) \subset U_\sigma }$ for all $k \in \Z-\{0\}$, and 3. $\displaystyle{\tau^k \bigg( \{x\} \cup U_\sigma \cup \bigcup_{\varepsilon=\pm1} \bigcup_{i=1}^n \gamma_i^\varepsilon(U_\sigma) \bigg) \subset U_\tau }$ for all $k \in \Z-\{0\}$. Then, $\Gamma$ is a non-cyclic group with infinite girth. Clearly, $\sigma$, $\tau$, $U_\sigma$, $U_\tau$, and $x \in X$ in Proposition \[igc\] satisfy the conditions in Proposition \[fsc\]. Hence, $\langle \sigma, \tau \rangle < \Gamma$ must be a non-abelian free subgroup. Let $M$ be a positive integer, and we aim to find a new generating set $\hat{\calG}$ for $\Gamma$ such that $U(\Gamma, \hat{\calG}\, ) \geq M$. Let $P=\{ p_1, \cdots , p_n \}$ be a set of positive intergers such that $p_i>M$ for all $i$ and $|p_i-p_j|>M$ for all distinct $i, j$. Let $\hat \gamma_i:= \sigma^{p_i}\gamma_i \tau^{-p_i}$ for each $i$. The set $\hat{\calG}:=\{\sigma, \tau, \hat \gamma_1, \cdots , \hat \gamma_n \}$ clearly generates $\Gamma$. Let $w$ be a nontrivial reduced word in $\hat{\calG} \cup \hat{\calG}^{-1}$ with the length less than $M$ with respect to $\hat{\calG} \cup \hat{\calG}^{-1}$. We can write $w$ as $$w=u_1 \hat \gamma_{i_1}^{\varepsilon_1} u_2 \hat \gamma_{i_2}^{\varepsilon_2} \cdots \hat \gamma_{i_s}^{\varepsilon_s} u_{s+1}\\$$ where $\varepsilon_\ell \in \{ \pm1 \}$ and the subword $u_\ell=u_\ell(\sigma, \tau)$ is a (possibly empty) reduced word in $\{ \sigma^{\pm 1}, \tau^{\pm 1} \}$. If $u_\ell$ is an empty word and $i_{\ell-1} = i_\ell$ for some $\ell$, we must have $\varepsilon_{\ell-1}=\varepsilon_\ell$. For otherwise, a cancellation occurs and contradicts the assumption that $w$ is a reduced word in $\hat{\calG} \cup \hat{\calG}^{-1}$. Now, regarded as an element of $\Gamma$, $w$ can be expressed as $$\begin{aligned} w&=&u_1 \hat \gamma_{i_1}^{\varepsilon_1} u_2 \hat \gamma_{i_2}^{\varepsilon_2} \cdots \hat \gamma_{i_s}^{\varepsilon_s} u_{s+1}\\ &=&v_1 \gamma_{i_1}^{\varepsilon_1} v_2 \gamma_{i_2}^{\varepsilon_2} \cdots \gamma_{i_s}^{\varepsilon_s} v_{s+1}\end{aligned}$$ where $v_\ell=v_\ell(\sigma, \tau)$ is a reduced word in $\{ \sigma^{\pm 1}, \tau^{\pm 1} \}$ for $\beta_{\ell-1} u_\ell \alpha_\ell$ (with convention $\alpha_{s+1}=\beta_0=1$) and $$\alpha_\ell=\begin{cases} \sigma^{p_{i_\ell}} & \text{if $\varepsilon_\ell=+1$} \\ \tau^{p_{i_\ell}} & \text{if $\varepsilon_\ell=-1$} \end{cases} \;\;\;\;\;\; \beta_\ell=\begin{cases} \tau^{-p_{i_\ell}} & \text{if $\varepsilon_\ell=+1$} \\ \sigma^{-p_{i_\ell}} & \text{if $\varepsilon_\ell=-1$} \end{cases}$$ The idea of the proof is to apply the ping-pong argument to $w$ to show that $w$ cannot represent the identity element of $\Gamma$. Provided with suitable initial points in $X$, the ping-pong argument applies easily to the strings $v_\ell$. What we need to show is that we can pass each $\gamma^{\varepsilon_\ell}_{i_\ell}$ in the ping-pong argument; in other words, we need to check that $\gamma^{\varepsilon_\ell}_{i_\ell}$ takes the terminal point from the ping-pong rally $v_{\ell+1}$ to a suitable initial point for the ping-pong rally $v_\ell$. We will see that our choice of $p_i$ prevents excessive cancellations, and we can indeed pass each $\gamma^{\varepsilon_\ell}_{i_\ell}$ under the conditions (2) and (3) in the statement of the proposition. \[pass gamma\]For each $\ell$, $v_\ell$ is not an empty word. If $\varepsilon_\ell=+1$, then the last letter of $v_\ell$ is $\sigma^{\pm 1}$ and the first letter of $v_{\ell+1}$ is $\tau^{\pm 1}$. If $\varepsilon_\ell=-1$, then the last letter of $v_\ell$ is $\tau^{\pm 1}$ and the first letter of $v_{\ell+1}$ is $\sigma^{\pm 1}$. Let us show that, if $\varepsilon_\ell=+1$, then $v_\ell$ is a non-empty word ending with $\sigma^{\pm1}$. Since $\varepsilon_\ell=+1$, we have $\alpha_\ell= \sigma^{p_{i_\ell}}$ and $\beta_\ell= \tau^{-p_{i_\ell}}$. There are three cases to consider: (i) $u_\ell$ is an empty word; (ii) the last letter of $u_\ell$ is $\tau^{\pm 1}$; or (iii) the last letter of $u_\ell$ is $\sigma^{\pm 1}$. Case (i): If $u_\ell$ is empty, then $v_\ell$ is the reduced word for $\beta_{\ell-1} \alpha_\ell$, and thus we have $$v_\ell=\begin{cases} \sigma^{p_{i_\ell}} & \text{if $\ell=1$} \\ \tau^{-p_{i_{\ell-1}}} \sigma^{p_{i_\ell}} & \text{if $\ell \neq 1$ and $\varepsilon_{\ell-1}=+1$} \\ \sigma^{-p_{i_{\ell-1}}+p_{i_\ell}} & \text{if $\ell \neq 1$ and $\varepsilon_{\ell-1}=-1$} \\ \end{cases}$$ In the last subcase, since $\varepsilon_\ell=+1 \neq -1=\varepsilon_{\ell-1}$, we must have $i_\ell \neq i_{\ell-1}$ as noted before. Thus, we must have $|p_{i_\ell}-p_{i_{\ell-1}}|>M$, and it follows that $v_\ell$ is a nontrivial power of $\sigma$. In all subcases, $v_\ell$ is indeed a non-empty word ending with $\sigma^{\pm 1}$. Case (ii): If the last letter of $u_\ell$ is $\tau^{\pm 1}$, then there is no cancellation between $u_\ell$ and $\alpha_\ell= \sigma^{p_{i_\ell}}$ as a word in $\{ \sigma^{\pm 1}, \tau^{\pm 1} \}$. Hence, $v_\ell$ is again a non-empty word ending with $\sigma^{\pm 1}$. Case (iii): Finally, suppose the last letter of $u_\ell$ is $\sigma^{\pm 1}$. If $u_\ell$ is not a power of $\sigma$, then $u_\ell= \cdots \tau^{q} \sigma^{p}$ for some $q$ and $p$. So, $u_\ell \alpha_\ell= \cdots \tau^q \sigma^{p+p_{i_\ell}}$. Note that we must have $|p| < M$. For otherwise, the length of $u_\ell$ as a word in $\hat{\calG} \cup \hat{\calG}^{-1}$, and thus the length of $w$ as a word in $\hat{\calG} \cup \hat{\calG}^{-1}$, is at least $M$; this contradicts with the assumption on the length of $w$. Now, $|p|<M$ and $p_{i_\ell}>M$ together imply $p+p_{i_\ell} \neq 0$. Thus, the last letter of $v_\ell$ must be $\sigma^{\pm 1}$. If $u_\ell$ is a power of $\sigma$, say $u_\ell= \sigma^p$, then $$v_\ell=\begin{cases} \sigma^{p+p_{i_\ell}} & \text{if $\ell=1$} \\ \tau^{-p_{i_{\ell-1}}} \sigma^{p+p_{i_\ell}} & \text{if $\ell \neq 1$ and $\varepsilon_{\ell-1}=+1$} \\ \sigma^{-p_{i_{\ell-1}}+p+p_{i_\ell}} & \text{if $\ell \neq 1$ and $\varepsilon_{\ell-1}=-1$} \end{cases}$$ In the first two subcases, $v_\ell$ ends with a nontrivial power of $\sigma$, because $|p|<M$ and $p_{i_\ell} >M$ imply $p+p_{i_\ell} \neq 0$. In the third subcase, we must have $i_\ell \neq i_{\ell-1}$, and hence $|p_{i_\ell}-p_{i_{\ell-1}}|>M$. It now follows from $|p|<M$ that $-p_{i_{\ell-1}}+p+p_{i_\ell} \neq 0$, and $v_\ell$ is again a nontrivial power of $\sigma$. Thus, in all subcases, $v_\ell$ is again a non-empty word ending with $\sigma^{\pm 1}$ as desired. This concludes the proof that, if $\varepsilon_\ell=+1$, then $v_\ell$ is a non-empty word ending with $\sigma^{\pm 1}$. The analogous arguments show that, if $\varepsilon_\ell=+1$, then $v_{\ell+1}$ is a non-empty word beginning with $\tau^{\pm 1}$. The symmetric arguments show that, if $\varepsilon_\ell=-1$, then $v_\ell$ is a nonempty word ending with $\tau^{\pm1}$ and $v_{\ell+1}$ is a non-empty word beginning with $\sigma^{\pm1}$. \[nontrivial\] If the last letter of $v_{s+1}$ is $\sigma^{\pm1}$ and $y \in \{x\} \cup U_\tau \cup \bigcup_{\varepsilon=\pm1} \bigcup_{i=1}^n \gamma^{\varepsilon}_i(U_\tau)$, or if the last letter of $v_{s+1}$ is $\tau^{\pm1}$ and $y \in \{x\} \cup U_\sigma \cup \bigcup_{\varepsilon=\pm1} \bigcup_{i=1}^n \gamma^{\varepsilon}_i(U_\sigma)$, then $w(y) \in U_\sigma \cup U_\tau$. We will prove the claim by induction on $s$. If $s=0$, $w=v_1$ is merely a reduced word in $\sigma^{\pm1}$ and $\tau^{\pm1}$. In this case, $w(y) \in U_\sigma \cup U_\tau$ follows from the classical ping-pong argument as in the proof of Free Subgroup Criterion. Now, as the induction hypothesis, suppose that the claim is true for $s-1 \geq 0$, and let $w=v_1 \gamma_{i_1}^{\varepsilon_1} v_2 \gamma_{i_2}^{\varepsilon_2} \cdots \gamma_{i_s}^{\varepsilon_s} v_{s+1}$. Suppose $\varepsilon_s=+1$ for now, so that the first letter of $v_{s+1}$ is $\tau^{\pm1}$ by Claim \[pass gamma\]. Then, we have $v_{s+1}(y) \in U_\tau$ by the classical ping-pong argument, and we obtain $y':=\gamma_{i_s}v_{s+1}(y) \in \gamma_{i_s}(U_\tau)$. Now, also by Claim \[pass gamma\], the last letter of $v_s$ is $\sigma^{\pm1}$. Thus, applying the induction hypothesis to $w':=v_1 \gamma_{i_1}^{\varepsilon_1} v_2 \gamma_{i_2}^{\varepsilon_2} \cdots \gamma_{i_{s-1}}^{\varepsilon_{s-1}} v_s$ and $y'$, we see that $w(y)=w'(y') \in U_\sigma \cup U_\tau$. The $\varepsilon_s=-1$ case is analogous. Since $x \not \in U_\sigma \cup U_\tau$ by the assumption and $w(x) \in U_\sigma \cup U_\tau$ by Claim \[nontrivial\], it follows that $w$ cannot represent the identity element in $\Gamma$. Namely, any non-empty word in $\hat{\calG}$ that represents the identity element of $\Gamma$ must be of length at least $M$ with respect to $\hat{\calG}$. Hence, $U(\Gamma) \geq U(\Gamma; \hat{\calG}\,) \geq M$. Although every group that satisfies our Infinite Girth Criterion (Proposition \[igc\]) must contain a non-abelian free subgroup, generally groups containing non-abelian free subgroup need *not* have infinite girth. The groups constructed by Olshanskii [@Olshanskii:RelationsBook], which we mentioned in §\[Criteria: Finite Girth\], contain non-abelian free subgroups and have finite girth; indeed, the existence of such non-abelian free subgroups is precisely the reason why those groups do not satisfy any law. It is natural to ask how the property of having infinite girth behaves under homomorphisms. The following partial answer by Akhmedov [@Akhmedov:Girth2] plays an important role in §4. \[akhmedov criterion\] A group $\Gamma$ has infinite girth if it admits a surjection onto a non-cyclic group with infinite girth. As noted in §\[Introduction\], we do not know if the presence of a finite-index infinite-girth subgroup of $\Gamma$ guarantees that $\Gamma$ itself has infinite girth; indeed, it is suspected that is not the case in general (see [@Akhmedov:Girth2]). This is unfortunate, because we cannot pass to a finite-index subgroup to prove that a given group has infinite girth. In our proof of the girth alternative for subgroups of mapping class groups, we go around this obstacle by applying the above proposition in a suitable manner. Elements of Mapping Class Groups {#Elements} ================================ Let ${\Sigma}$ be a (not necessarily connected) compact orientable surface. A *multi-loop* on ${\Sigma}$ is a pair-wise disjoint collection of simple closed curves on ${\Sigma}$, and it is said to be *essential* if each component is not null-homotopic or peripheral. - An isotopy class $\calA$ of an essential (possibly empty) multi-loop on ${\Sigma}$ is said to be a *reduction system* for $\sigma \in \Mod({\Sigma})$ if $\sigma$ fixes $\calA$. - An element $\sigma \in \Mod({\Sigma})$ is said to be *reducible* if it admits a *non-empty* reduction system $\calA$; it is said to be *irreducible* otherwise. - An element $\sigma \in \Mod({\Sigma})$ is said to be *periodic* if its order is finite; it is said to be *aperiodic* otherwise. Nielsen studied $\Mod(T^2) \cong \SL(2,\Z)$ in terms of the dynamical properties of its action on the Teichmüller space $\Teich(T^2) \cong \Hyp^2$ and its boundary [@Nielsen:MCG], and he essentially showed that irreducible elements are precisely the *Anosov elements*, i.e. the elements that can be represented by *Anosov homeomorphisms* of $T^2$. The far-reaching generalization of Nielsen’s work, dealing with surfaces ${\Sigma}$ that admit complete hyperbolic metrics, was developed by Thurston [@Thurston:Surfaces]. Among other things, Thurston introduced the notion of *pseudo-Anosov homeomorphisms* of ${\Sigma}$ and showed that irreducible elements of $\Mod({\Sigma})$ are precisely the *pseudo-Anosov elements*, i.e. the elements that can be represented by pseudo-Anosov homeomorphisms of ${\Sigma}$. The Canonical Reduction of Reducible Elements {#Can-Red: Elts} --------------------------------------------- The notion of the the *reduction* of an element of $\Mod({\Sigma})$ appeared in Thurston’s theory, and it was further studied in [@Birman-Lubotzky-McCarthy:AbSol]. If $\calA$ is a reduction system for $\tau \in \Mod({\Sigma})$ and ${\Sigma}_{\calA}$ is the compactification of ${\Sigma}-\calA$, then $\tau$ induces a mapping class ${\operatorname{\rho}}_{\!\calA}(\tau) \in \Mod({\Sigma}_{\calA})$; this element ${\operatorname{\rho}}_{\!\calA}(\tau)$ is called the *reduction of $\tau$ along $\calA$*. Although the reduction is meant to be applied to reducible elements along non-empty reduction system, we allow a reduction system $\calA$ of an element $\tau$ to be empty for the logical convenience; if $\calA=\nil$, the reduction ${\operatorname{\rho}}_{\!\calA}(\tau)$ coincides with $\tau$. Given an element $\tau \in \Mod({\Sigma})$, we can always take some positive power $\tau^N$ so that $\tau^N$ takes each connected component of ${\Sigma}$ to itself and each component of $\partial {\Sigma}$ to itself. If the restrictions of $\tau^N$ to some components are periodic, we can take yet higher power $\tau^{N'}$ so that the restriction of $\tau^{N'}$ to each component is either trivial or aperiodic. - An element $\tau \in \Mod({\Sigma})$ is *adequately reduced* if there is some power $\tau^N$, taking each component of ${\Sigma}$ to itself and each component of $\partial {\Sigma}$ to itself, such that the restriction of $\tau^N$ to each component of ${\Sigma}$ is either (i) trivial or (ii) aperiodic and irreducible. - Given an element $\tau \in \Mod({\Sigma})$, a (possibly empty) reduction system $\calA$ of $\tau$ is said to be an *adequate reduction system* if the reduction ${\operatorname{\rho}}_{\!\calA}(\tau)$ is adequately reduced. By definition, an empty reduction system for an adequately reduced element is indeed an adequate reduction system. Thurston observed that every element $\tau \in \Mod(S)$ either is adequately reduced or has a non-empty adequate reduction system [@Thurston:Surfaces]. Generally, adequate reduction systems of $\tau$ are not unique. The crucial fact shown in [@Birman-Lubotzky-McCarthy:AbSol] is that there is a canonical choice of an adequate reduction system for each element $\tau \in \Mod({\Sigma})$; it is indeed the unique minimal adequate reduction system for the element. We call such system the *canonical reduction system* for $\tau$, and denote it by $\calC$ (the reference to $\tau$ should always be clear from the context). For a connected surface ${\Sigma}$, the canonical reduction system for $\tau \in \Mod({\Sigma})$ is empty if and only if $\tau$ is periodic or irreducible. More generally, for any surface ${\Sigma}$, which may be connected or disconnected, the canonical reduction system for $\tau \in \Mod({\Sigma})$ is empty if and only if $\tau$ is adequately reduced. Adequately reduced elements are more general than irreducible ones, but their important properties can be derived from those of irreducible ones. Pseudo-Anosov Elements for Connected Surfaces {#Pseudo-Anosov: Conn} --------------------------------------------- Suppose for now that ${\Sigma}$ is a compact orientable *connected* surface such that the interior of ${\Sigma}$ admits a complete hyperbolic metric, i.e. $\chi({\Sigma})<0$. We write $g({\Sigma})$ and $b({\Sigma})$ for the genus and the number of connected components of $\partial {\Sigma}$ respectively. Thurston introduced the space $\PML({\Sigma})$ of *projective measured laminations*, equipped with a topology which makes it homeomorphic to a sphere of dimension $6g({\Sigma})+2b({\Sigma})-7$. With respect to this topology, the mapping class group $\Mod({\Sigma})$ acts naturally by homeomorphisms. $\PML({\Sigma})$ compactifies the Teichmüller space of ${\Sigma}$, coherently with respect to the actions of $\Mod({\Sigma})$. An element $\sigma \in \Mod({\Sigma})$ is said to be *pseudo-Anosov* if the fixed-point set $\Fix(\sigma) \subset \PML({\Sigma})$ of the action of $\sigma$ on $\PML({\Sigma})$ consists of a pair of distinct measured laminations. The key property of a pseudo-Anosov element $\sigma$ is that its action on $\PML({\Sigma})$ exhibits the *north-south dynamics* with one of the fixed points as an attractor, denoted by $\scrL^+_\sigma$, and the other as a repeller, denoted by $\scrL^-_\sigma$. More precisely, for any pair of disjoint neighborhoods $U^+_\sigma$ of $\scrL_\sigma^+$ and $U^-_\sigma$ of $\scrL_\sigma^-$, we have $\sigma^{\pm N} \big( \PML({\Sigma})-U^\mp_\sigma \big) \subset U^\pm_\sigma$ respectively for all sufficiently large $N$. We remark that pseudo-Anosov elements are always aperiodic. Furthermore, as mentioned earlier, one of the main results in Thurston’s work [@Thurston:Surfaces] is that irreducible elements in $\Mod({\Sigma})$ are precisely pseudo-Anosov ones; see [@FLP:Thurston] for the detail. Pseudo-Anosov Elements for Disonnected Surfaces {#Pseudo-Anosov: Disconn} ----------------------------------------------- Let us now allow a compact orientable surface ${\Sigma}=\bigsqcup_{i=1}^{c({\Sigma})}{\Sigma}^i$ to be *disconnected*, where $c({\Sigma})$ denotes the number of connected components of ${\Sigma}$ and ${\Sigma}^i$ denotes each connected component. As in the case of connected surfaces, we will assume that the interior of ${\Sigma}$ admits a complete hyperbolic metric, i.e. $\chi({\Sigma}^i)<0$ for each ${\Sigma}^i$. Every element $\sigma \in \Mod({\Sigma})$ has a nontrivial power that takes each connected component ${\Sigma}^i$ to itself. An element of $\Mod({\Sigma})$ is said to be *pseudo-Anosov* if the restriction of such a power to each component ${\Sigma}^i$ is a pseudo-Anosov element in $\Mod({\Sigma}^i)$. It is easy to see that this notion of pseudo-Anosov elements is well-defined, and they are necessarily aperiodic. Furthermore, one can check that irreducible elements of $\Mod({\Sigma})$ are precisely pseudo-Anosov ones. With some care, the space $\PML({\Sigma})$ of projective measure laminations can be defined. As shown by Ivanov [@Ivanov:Tits-Margulis-Soifer] and McCarthy [@McCarthy:Tits], the space $\PML({\Sigma})$ turns out to be homeomorphic to the *join* of $\PML({\Sigma}^i)$. There is a natural action of $\Mod({\Sigma})$ on $\PML({\Sigma})$, but the description of this action is rather cumbersome. For example, the action of pseudo-Anosov element $\sigma \in \Mod({\Sigma})$ still exhibits dynamical properties similar to the connected case, with an attracting neighborhood $U^+_\sigma$ and a repelling neighborhood $U^-_\sigma \subset \PML({\Sigma})$, but neither of them can arise as a neighborhood of a single projective measured lamination any more; under the homeomorphism $\PML({\Sigma}) \cong \ast_{i=1}^{c({\Sigma})} \PML({\Sigma}^i)$, certain simplexes in the join $\ast_{i=1}^{c({\Sigma})} \PML({\Sigma}^i)$ play the roles of the attractor and the repeller. For our purposes, it is convenient to look at the action of $\Mod({\Sigma})$ on an alternative space. Following Ivanov [@Ivanov:MCGBook], we look at the natural action of $\Mod({\Sigma})$ on the space $$\PML^\sharp({\Sigma}):=\bigsqcup_{i=1}^{c({\Sigma})} \PML({\Sigma}^i)$$ instead. If an element $\sigma \in \Mod({\Sigma})$ takes each component ${\Sigma}^i \subset {\Sigma}$ to itself, it takes each component $\PML({\Sigma}^i) \subset \PML^\sharp({\Sigma})$ to itself under this action. If $\sigma$ is further assumed to be pseudo-Anosov, then its action on each component $\PML({\Sigma}^i) \subset \PML^\sharp({\Sigma})$ exhibits the honest north-south dynamics with fixed points $\scrL^\pm_{\sigma|_{{\Sigma}^i}} \in \PML({\Sigma}^i)$. Moreover, the points $\scrL^\pm_{\sigma|_{{\Sigma}^i}}$ are precisely the fixed points for the action of $\sigma$ on $\PML^\sharp({\Sigma})$. For any pair of disjoint neighborhoods $U^+_\sigma$ of $\big\{\scrL^+_{\sigma|_{{\Sigma}^i}} \big\}_{i=1}^{c({\Sigma})}$ and $U^-_\sigma$ of $\big\{\scrL^-_{\sigma|_{{\Sigma}^i}} \big\}_{i=1}^{c({\Sigma})}$, we have $\sigma^N \big( \PML^\sharp({\Sigma})-U^-_\sigma \big) \subset U^+_\sigma$ and $\sigma^{-N} \big( \PML^\sharp({\Sigma})-U^+_\sigma \big) \subset U^-_\sigma$ for all sufficiently large $N$. Subgroups of Mapping Class Groups {#Subgroups} ================================= Ivanov further generalized Thurston’s theory to subgroups of $\Mod({\Sigma})$ in order to study their structures [@Ivanov:MCGBook], and obtained a classification of subgroups that parallels the classification of elements of $\Mod({\Sigma})$. - An isotopy class $\calA$ of an essential (possibly empty) multi-loop is said to be a *reduction system* for $\Gamma$ if it is a reduction system for every element $\sigma \in \Gamma$, i.e. if every element $\sigma \in \Gamma$ fixes $\calA$. - A subgroup $\Gamma$ is said to be *reducible* if it admits a *non-empty* reduction system $\calA$; it is said to be *irreducible* otherwise. By definition, an element $\sigma \in \Mod({\Sigma})$ is reducible (resp. irreducible) if and only if the cyclic subgroup $\langle \sigma \rangle < \Mod({\Sigma})$ is reducible (resp. irreducible). More generally, the above definitions suggest that (i) the analogue of periodic elements are finite subgroups, (ii) the analogue of aperiodic reducible elements are infinite reducible subgroups, and (iii) the analogue of pseudo-Anosov elements are infinite irreducible subgroups. Torsion-Free Finite-Index Subgroups {#Mod-m} ----------------------------------- Before we proceed, let us first consider a useful family of subgroups of $\Mod({\Sigma})$. For each integer $m \geq 3$, we consider the natural homomorphisms $$\Mod({\Sigma}) \rightarrow \Aut(H_1({\Sigma};\Z)) \rightarrow \Aut(H_1({\Sigma};\Z/m\Z))$$ and let $\Mod_{(m)}({\Sigma})$ be the kernel of this composition of homomorphisms. $\Mod_{(m)}({\Sigma})$ is clearly a finite-index normal subgroup of $\Mod({\Sigma})$, and a classical theorem of Serre [@Serre:Rigidite] says that $\Mod_{(m)}({\Sigma})$ is torsion-free. For each subgroup $\Gamma < \Mod({\Sigma})$, we set $\Gamma_{(m)}:=\Gamma \cap \Mod_{(m)}({\Sigma})$; it is a torsion-free finite-index normal subgroup of $\Gamma$. Ivanov observed that elements of $\Mod_{(m)}({\Sigma})$ possess other useful properties. For every element $\sigma \in \Mod_{(m)}({\Sigma})$ and for every reduction system $\calA$ of $\sigma$, the following statements hold [@Ivanov:MCGBook §1.2]: - $\sigma$ takes each component of ${\Sigma}$ to itself, and each component of $\partial {\Sigma}$ to itself; - $\sigma$ takes each component of $\calA$ to itself with its orientation preserved; - hence, the reduction ${\operatorname{\rho}}_{\!\calA}(\sigma) \in \Mod({\Sigma}_{\calA})$ takes each component of ${\Sigma}_{\calA}$ to itself, and each component of $\partial {\Sigma}_{\calA}$ to itself. It follows that $\sigma$ can be restricted to each component of ${\Sigma}$, and ${\operatorname{\rho}}_{\!\calA}(\sigma)$ can be restricted to each component of ${\Sigma}_{\calA}$. Such restrictions have the following properties, which is much stronger than the aperiodicity of $\sigma \in \Mod_{(m)}({\Sigma})$ [@Ivanov:MCGBook §1.6]: - the restriction of $\sigma$ to each component of ${\Sigma}$ is trivial or aperiodic; - the restriction of ${\operatorname{\rho}}_{\!\calA}(\sigma)$ to each component of ${\Sigma}_{\calA}$ is trivial or aperiodic. As an immediate consequence, we have the following observation: for any subgroup $\Gamma < \Mod({\Sigma})$, the normal subgroup $\Gamma_{(m)} \lhd$ can be restricted to each component of ${\Sigma}$, and such restrictions are torsion-free. The properties of $\Gamma_{(m)}$ mentioned in the above paragraphs are used extensively in Ivanov’s proof of the analogue of the Tits alternative. In the proof of the analogue of *Margulis–Soĭfer theorem*, which is a significantly stronger theorem than the Tits alternative, Ivanov employed additional properties of the subgroup $\Gamma_{(m)}$ and its relationship to $\Gamma$ [@Ivanov:MCGBook §9]. We extract one of such properties as the following lemma. \[partition\] Let ${\Sigma}=\bigsqcup_{i=1}^{c({\Sigma})} {\Sigma}^i$ be a surface which is not necessarily connected. For every component ${\Sigma}^i$ and every element $\sigma \in \Gamma$, the restrictions of $\Gamma_{(m)}$ to ${\Sigma}^i$ and $\sigma({\Sigma}^i)$ are isomorphic. Hence, if the restriction of $\Gamma_{(m)}$ to components ${\Sigma}^i$ and ${\Sigma}^j$ are not isomorphic, then no element of $\Gamma$ can take ${\Sigma}^i$ to ${\Sigma}^j$. This elementary fact did not play any role in proving the Tits alternative for $\Mod({\Sigma})$, while it serves as a critical step in proving the Margulis–Soĭfer theorem for $\Mod({\Sigma})$. The proof of our main theorem too make use of the above lemma. The proof we present here is contained in [@Ivanov:MCGBook §9.10] where the lemma was stated for a special case. Since $\Gamma_{(m)} \lhd \Gamma$, the conjugate of an element $\tau \in \Gamma_{(m)}$ by an element $\sigma \in \Gamma$ must again belong to $\Gamma_{(m)}$; if $s$ and $t$ are homeomorphisms representing $\sigma$ and $\tau$ respectively, then $s \circ t \circ s^{-1}$ represents the element $\sigma \tau \sigma^{-1} \in \Gamma_{(m)}$. The restriction $\tau|_{{\Sigma}^i}$ is represented by $t|_{{\Sigma}^i}$, and the restriction $\sigma \tau \sigma^{-1}|_{\sigma({\Sigma}^i)}$ is represented by $s \circ (t|_{{\Sigma}^i}) \circ s^{-1}$. Thus, the conjugation by $\sigma$ defines a homomorphism that takes the restriction $\Gamma_{(m)}|_{{\Sigma}^i}$ to the restriction $\Gamma_{(m)}|_{\sigma({\Sigma}^i)}$. Clearly, this homomorphism is an isomorphism, with the inverse given by the conjugation by $\sigma^{-1}$. Canonical Reduction of Reducible Subgroups {#Can-Red: Subgrps} ------------------------------------------ The theory regarding the *reduction* of a subgroup of $\Mod({\Sigma})$ can be developed in essentially the same manner as it was done for an element of $\Mod({\Sigma})$, with suitable modifications. If $\calA$ is a reduction system for $\Gamma<\Mod({\Sigma})$, then the reduction ${\operatorname{\rho}}_{\!\calA}(\sigma) \in \Mod({\Sigma}_{\calA})$ is well-defined for all $\sigma \in \Mod({\Sigma})$, where ${\Sigma}_{\calA}$ is the compactification of ${\Sigma}-\calA$ as before. The assignment $\sigma \mapsto {\operatorname{\rho}}_{\!\calA}(\sigma)$ indeed defines the *reduction homomorphism* $${\operatorname{\rho}}_{\!\calA}: \Gamma \rightarrow \Mod({\Sigma}_{\calA})$$ whose kernel is a free-abelian group generated by Dehn twists along some components of $\calA$; the image ${\operatorname{\rho}}_{\!\calA}(\Gamma)$ is called the *reduction of $\Gamma$ along $\calA$*. As before, a reduction system $\calA$ of a subgroup $\Gamma$ is allowed to be empty; if $\calA=\nil$, the reduction $\Gamma_\calA$ coincides with $\Gamma$. Recall from §\[Can-Red: Elts\] that, for an element $\tau \in \Mod({\Sigma})$, taking a finite power $\tau^N$ was essential in understanding the reduction systems of $\tau$. Ivanov observed that, for a subgroup $\Gamma < \Mod({\Sigma})$, the correct analogue is passing to a finite-index normal subgroup $\Gamma' \lhd \Gamma$. To make this analogy apparent, we introduce the following notions, which did not appear explicitly in Ivanov’s exposition: - A subgroup $\Gamma < \Mod({\Sigma})$ is *adequately reduced* if there is a *finite-index normal subgroup* $\Gamma' \lhd \Gamma$, consisting of elements that take each component of ${\Sigma}$ to itself and each component of $\partial {\Sigma}$ to itself, such that the restriction of $\Gamma'$ to each component is either (i) trivial or (ii) infinite and irreducible. - Given a subgroup $\Gamma < \Mod({\Sigma})$, a (possibly empty) reduction system $\calA$ of $\Gamma$ is said to be an *adequate reduction system* if the reduction ${\operatorname{\rho}}_{\!\calA}(\Gamma)$ is adequately reduced. The work of Ivanov shows that for any subgroup $\Gamma < \Mod({\Sigma})$ there is a canonical choice of reduction system [@Ivanov:MCGBook §7.2-7.4], which is indeed the unique minimal adequate reduction system [@Ivanov:MCGBook §7.16 and §7.18]; we call this system the *canonical reduction system* for $\Gamma$, and denote it by $\calC$ (the reference to $\Gamma$ should always be clear from the context). Although we will not go into the detail of the definition of the canonical reduction system, we note that the definition is invariant under passing to finite-index normal subgroup. \[adeq-red and Gamma\_m\] Let ${\Sigma}=\bigsqcup_{i=1}^{c({\Sigma})}{\Sigma}^i$ be a compact orientable surface. For any subgroup $\Gamma < \Mod({\Sigma})$, the following are equivalent: - the canonical reduction system $\calC$ for $\Gamma$ is empty; - $\Gamma$ is adequately reduced; - some finite-index normal subgroups of $\Gamma$ are adequately reduced; - every finite-index normal subgroup of $\Gamma$ is adequately reduced; - for some integers $m \geq 3$, the restriction of $\Gamma_{(m)}$ to each component of ${\Sigma}$ is either (i) trivial or (ii) infinite and irreducible; - for every integer $m \geq 3$, the restriction of $\Gamma_{(m)}$ to each component of ${\Sigma}$ is either (i) trivial or (ii) infinite and irreducible. \(1) $\Leftrightarrow$ (2) follows from the fact that $\calC$ is the unique minimal adequate reduction system. (3) $\Rightarrow$ (2) and (2) $\Rightarrow$ (4) follows from the invariance of $\calC$ under passing to finite-index normal subgroup and the equivalence (1) $\Leftrightarrow$ (2). With the trivial implication (4) $\Rightarrow$ (3), we have the equivalence of (1)–(4). Since (6) $\Rightarrow$ (5) is trivial and (5) $\Rightarrow$ (2) follows from the definition, it remains to check the implication (2) $\Rightarrow$ (6). Suppose $\Gamma$ is adequately reduced, and take an integer $m \geq 3$. By the equivalence (2) $\Leftrightarrow$ (4), $\Gamma_{(m)}$ is adequately reduced, i.e. there exists a finite-index normal subgroup $\Gamma' \lhd \Gamma_{(m)}$ such that the restriction of $\Gamma'$ to each component is either trivial or irreducible. Recall that the restriction of $\Gamma_{(m)}$ to each component is torsion-free; hence, if the restriction of $\Gamma'$ to a component ${\Sigma}^i$ is trivial, then the restriction of $\Gamma_{(m)}$ must also be trivial. If the restriction of $\Gamma'$ to a component ${\Sigma}^j$ is irreducible, then clearly the restriction of the larger group $\Gamma_{(m)}$ to ${\Sigma}^j$ must also be irreducible. This completes the proof. Adequately Reduced Subgroups for Connected Surfaces {#Adeq-Red: Conn} --------------------------------------------------- Let ${\Sigma}$ be a compact orientable *connected* surface with $\chi({\Sigma})<0$, and let $\Gamma < \Mod({\Sigma})$ be an adequately reduced subgroup. $\Gamma$ is either a finite subgroup or an infinite irreducible subgroup. An infinite irreducible subgroup always contains a pseudo-Anosov element [@Ivanov:MCGBook §5.9 and §7.14]. Furthermore, if $\Gamma$ is infinite, irreducible, and not virtually infinite-cyclic, then $\Gamma_{(m)} \lhd \Gamma$ contains two pseudo-Anosov elements $\sigma$ and $\tau$ such that $\Fix(\sigma)\cap\Fix(\tau)=\nil$ in $\PML({\Sigma})$ [@Ivanov:MCGBook §5.12 and §7.15]. These facts were proved with the following lemmas, which we cite here for reference: \[constructing pseudo-Anosov for conn\] Let $\Gamma<\Mod({\Sigma})$, and suppose that $\sigma, \tau \in \Gamma_{(m)}$ are pseudo-Anosov elements such that $\Fix(\sigma) \cap \Fix(\tau)=\nil$ in $\PML({\Sigma})$. Then, the elements $\sigma^p\tau^q$ is pseudo-Anosov for all sufficiently large $p$ and $q$. \[[[@Ivanov:MCGBook §5.11]]{}\] \[fixed-point sets of pseudo-Anosov\] Let $\Gamma<\Mod({\Sigma})$, and suppose that $\sigma, \tau \in \Gamma_{(m)}$ are pseudo-Anosov elements. Then, $\Fix(\sigma)=\Fix(\tau)$ or $\Fix(\sigma)\cap\Fix(\tau)=\nil$ in $\PML({\Sigma})$. In order to study the girth of adequately reduced subgroups using the Infinite Girth Criterion, we need a slightly stronger fact than Ivanov’s results. The following theorem summarizes and refines Ivanov’s results. \[trichotomy: adeq-red for conn\] Fix an integer $m \geq 3$. Let ${\Sigma}$ be a connected surface with $\chi({\Sigma})<0$, and let $\Gamma < \Mod({\Sigma})$ be an adequately reduced subgroup. Then, $\Gamma$ satisfies exactly one of the following: - $\Gamma_{(m)} \lhd \Gamma$ is trivial, and $\Gamma$ is finite; - $\Gamma_{(m)} \lhd \Gamma$ is infinite-cyclic, and $\Gamma$ is virtually infinite-cyclic; - For any (possibly empty) finite collection $\varphi_1, \cdots , \varphi_n \in \Gamma_{(m)} \lhd \Gamma$ of pseudo-Anosov elements, there exists another pseudo-Anosov element $\psi \in \Gamma_{(m)} \lhd \Gamma$ such that $\Fix(\varphi_j)\cap\Fix(\psi)=\nil$ in $\PML({\Sigma})$ for all $j$. There exist finite irreducible subgroups which consist entirely of reducible elements [@Gilman:FiniteIrreducible]. Since ${\Sigma}$ is connected, Lemma \[adeq-red and Gamma\_m\] says that $\Gamma_{(m)}$ is either trivial or irreducible. If $\Gamma_{(m)}$ is trivial, $\Gamma$ is finite; if $\Gamma_{(m)}$ is infinite-cyclic, then $\Gamma$ is virtually infinite-cyclic. Hence, we assume $\Gamma_{(m)}$ is infinite, irreducilbe, and not infinite-cyclic. As we have already mentioned, there exist two pseudo-Anosov elements $\sigma$ and $\tau$ in $\Gamma_{(m)}$ such that $\Fix(\sigma)\cap\Fix(\tau)=\nil$ in $\PML({\Sigma})$ [@Ivanov:MCGBook §5.12]. We need to show that the slightly stronger condition (2) is satisfied. Suppose we are given a finite collection $\varphi_1, \cdots , \varphi_n \in \Gamma_{(m)} \lhd \Gamma$ of pseudo-Anosov elements. For each $\varphi_j$, we let $U^+_j, U^-_j \subset \PML({\Sigma})$ be attracting and repelling neighborhoods of $\varphi_j$ such that $U^+_j \cap U^-_j=\nil$; we set $U_j=U^+_j \cup U^-_j$. Similarly, we let $U^+_\sigma, U^-_\sigma, U^+_\tau, U^-_\tau \subset \PML({\Sigma})$ be attracting and repelling neighborhoods of $\sigma$, $\tau$ respectively such that $U^+_\sigma \cap U^-_\sigma=\nil$ and $U^+_\tau \cap U^-_\tau=\nil$; again, we set $U_\sigma=U^+_\sigma \cup U^-_\sigma$ and $U_\tau=U^+_\tau \cup U^-_\tau$. We can take these neighborhoods to be sufficiently small so that the following holds: - $U_j \cap U_k=\nil$ whenever $\Fix(\varphi_j) \cap \Fix(\varphi_k)=\nil$; - $U_j \cap U_\sigma=\nil$ whenever $\Fix(\varphi_j) \cap \Fix(\sigma)=\nil$; - $U_j \cap U_\tau=\nil$ whenever $\Fix(\varphi_j) \cap \Fix(\tau)=\nil$; - $U_\sigma \cap U_\tau=\nil$ (since $\Fix(\sigma) \cap \Fix(\tau)=\nil$). By the property of pseudo-Anosov elements, there exists a large enough integer $M$ such that $\sigma^{\pm m}(\PML({\Sigma})-U^\mp_\sigma) \subset U^\pm_\sigma$ and $\tau^{\pm m}(\PML({\Sigma})-U^\mp_\tau) \subset U^\pm_\tau$ for all $m$ with $m>M$. By Lemma \[constructing pseudo-Anosov for conn\], $\psi=\sigma^p \tau^q$ is pseudo-Anosov for all sufficiently large $p$ and $q$. Hence, taking such $p$ and $q$ greater than $M$, we obtain a pseudo-Anosov element $\psi=\sigma^p\tau^q$ such that $\psi(\PML({\Sigma})-U^-_\tau) \in U^+_\sigma \subset U_\sigma$ and $\psi^{-1}(\PML({\Sigma})-U^+_\sigma) \in U^-_\tau \subset U_\tau$. Furthermore, by the disjointness of $U_\sigma$ and $U_\tau$, we see that $\psi^\ell(\PML({\Sigma})-U^-_\tau) \in U^+_\sigma \subset U_\sigma$ and $\psi^{-\ell}(\PML({\Sigma})-U^+_\sigma) \in U^-_\tau \subset U_\tau$ for all positive integer $\ell$. Choose $\scrL \in \PML({\Sigma})-(U_\sigma \cup U_\tau)$ and consider a sequence $\scrL^\ell:=\psi^\ell(\scrL)$, $\ell \in \Z$. Note that $\scrL^\ell \in U^+_\sigma$ and $\scrL^{-\ell} \in U^-_\tau$ for all $\ell>0$. Since $\psi$ is pseudo-Anosov, we must have a convergence $\scrL^\ell \rightarrow \scrL^+_\psi$ and $\scrL^{-\ell} \rightarrow \scrL^-_\psi$ as $\ell \rightarrow \infty$. Thus, we see that $\scrL^+_\psi \in \overline{U^+_\sigma} \subset \overline{U_\sigma}$ and $\scrL^-_\psi \in \overline{U^-_\tau} \subset \overline{U_\tau}$. We claim that this $\Fix(\psi) \cap \Fix(\varphi_j)=\nil$ for each $j$. We first note that the assumption $\Fix(\sigma) \cap \Fix(\tau)=\nil$ and Lemma \[fixed-point sets of pseudo-Anosov\] imply that we have either $\Fix(\varphi_j) \cap \Fix(\sigma) =\nil$ or $\Fix(\varphi_j) \cap \Fix(\tau) = \nil$ for each $j$. By the choice of our neighborhoods $U_j$, $U_\sigma$, $U_\tau$, we thus have $U_j \cap \overline{U_\sigma}=\nil$ or $U_j \cap \overline{U_\tau}=\nil$ for each $i$. It follows that $\scrL^+_\psi \not \in \Fix(\varphi_j)$ or $\scrL^-_\psi \not \in \Fix(\varphi_j)$. Thus, by Lemma \[fixed-point sets of pseudo-Anosov\], we conclude that $\Fix(\psi) \cap \Fix(\varphi_j)=\nil$ for each $j$. Adequately Reduced Subgroups for Disconnected Surfaces {#Adeq-Red: Disconn} ------------------------------------------------------ We now allow a compact orientable surface ${\Sigma}=\bigsqcup_{i=1}^{c({\Sigma})} {\Sigma}^i$ to be *disconnected* surface, where $c({\Sigma})$ denotes the number of component of ${\Sigma}$ and ${\Sigma}^i$ denotes each component as before. Note that it is possible to have an infinite irreducible subgroup $\Gamma < \Mod({\Sigma})$ such that it restricts to some component ${\Sigma}^i$ as a finite irreducible subgroup of $\Mod({\Sigma}^i)$; there is no hope in finding a pseudo-Anosov element in such $\Gamma$. It turns out that this is essentially the only obstacle to finding a pseudo-Anosov element in $\Gamma$. Ivanov showed that, if $\Gamma_{(m)} \lhd \Gamma$ is also irreducible, i.e. the restriction of $\Gamma_{(m)}$ to each component is irreducible, then $\Gamma_{(m)} \lhd \Gamma$ contains a pseudo-Anosov element [@Ivanov:MCGBook §6.3]. Furthermore, when the restriction of $\Gamma_{(m)}$ to *every* component of ${\Sigma}$ is larger than an infinite-cyclic group, he showed that $\Gamma_{(m)} \lhd \Gamma$ contains two pseudo-Anosov elements $\sigma$ and $\tau$ such that $\Fix(\sigma) \cap \Fix(\tau)=\nil$ in $\PML^\sharp({\Sigma})$ [@Ivanov:MCGBook §6.4]. We need the analogue of Theorem \[trichotomy: adeq-red for conn\] for disconnected surfaces; this will be given below as Theorem \[trichotomy: adeq-red for disconn\] after a couple of lemmas. Generally, an adequately reduced group $\Gamma$ is a hybrid of three cases that appeared in Theorem \[trichotomy: adeq-red for conn\]. \[partition: adeq-red\] Fix an integer $m \geq 3$, and let $\Gamma < \Mod({\Sigma})$ be an adequately reduced subgroup. Consider a partition ${\Sigma}={\Sigma}^{[0]} \sqcup {\Sigma}^{[1]} \sqcup {\Sigma}^{[2]}$ defined by the following: - the subsurface ${\Sigma}^{[0]}$ is the union of all components ${\Sigma}^i$ such that the restriction of $\Gamma_{(m)}$ to ${\Sigma}^i$ is trivial; - the subsurface ${\Sigma}^{[1]}$ is the union of all components ${\Sigma}^i$ such that the restriction of $\Gamma_{(m)}$ to ${\Sigma}^i$ is infinite-cyclic; - the subsurface ${\Sigma}^{[2]}$ is the union of all components ${\Sigma}^i$ such that the restriction of $\Gamma_{(m)}$ to ${\Sigma}^i$ is nontrivial and non-cyclic. Then, every element $\sigma \in \Gamma$ preserves this partition, i.e. $\sigma({\Sigma}^{[\ell]})={\Sigma}^{[\ell]}$ for $\ell=0,1,2$; hence, the restriction of $\Gamma$ to ${\Sigma}^{[\ell]}$ is well-defined for $\ell=0,1,2$. This follows immediately from Lemma \[partition\]. \[two subgroups\] Fix an integer $m \geq 3$. For each $\ell$, let $\Gamma^{[\ell]}$ and $(\Gamma_{(m)})^{[\ell]}$ be the restrictions of $\Gamma$ and $\Gamma_{(m)}$ to ${\Sigma}^{[\ell]}$ respectively, and set $(\Gamma^{[\ell]})_{(m)}=\Gamma^{[\ell]} \cap \Mod_{(m)}({\Sigma}^{[\ell]})$. Then, $(\Gamma_{(m)})^{[\ell]}$ is a finite-index normal subgroup of $(\Gamma^{[\ell]})_{(m)}$. By definition, an element of $(\Gamma^{[\ell]})_{(m)}$ is a restriction of an element of $\Gamma$ that acts trivially on $H_1({\Sigma}^{[\ell]}; \Z/m\Z)<H_1({\Sigma}; \Z/m\Z)$. An element of $(\Gamma_{(m)})^{[\ell]}$ clearly satisfies this property since it is a restriction of an element of $\Gamma_{(m)}$ that acts trivially on the entire $H_1({\Sigma}; \Z/m\Z)$. Hence, $(\Gamma_{(m)})^{[\ell]}<(\Gamma^{[\ell]})_{(m)}<\Gamma^{[\ell]}$. Since $\Gamma_{(m)}$ is a finite-index normal subgroup of $\Gamma$, we see that $(\Gamma_{(m)})^{[\ell]}$ must be a finite-index normal subgroup of $\Gamma^{[\ell]}$. Hence, we conclude that $(\Gamma_{(m)})^{[\ell]}$ must also be a finite-index normal subgroup of the intermediate subgroup $(\Gamma^{[\ell]})_{(m)}$. In general, $(\Gamma_{(m)})^{[\ell]}$ is contained in $(\Gamma^{[\ell]})_{(m)}$ as a *proper* subgroup of $(\Gamma^{[\ell]})_{(m)}$. We also note that both $(\Gamma_{(m)})^{[\ell]}$ and $(\Gamma^{[\ell]})_{(m)}$ are torsion-free. \[trichotomy: adeq-red for disconn\] Fix an integer $m \geq 3$, and let $\Gamma < \Mod({\Sigma})$ be an adequately reduced subgroup. Suppose ${\Sigma}={\Sigma}^{[0]} \sqcup {\Sigma}^{[1]} \sqcup {\Sigma}^{[2]}$ is the partition given in Lemma \[partition: adeq-red\], and set $\Gamma^{[\ell]}$ and $(\Gamma^{[\ell]})_{(m)}$ as in Lemma \[two subgroups\]. Then, $\Gamma$ satisfies all of the following: - $(\Gamma^{[0]})_{(m)} \lhd \Gamma^{[0]}$ is trivial, and $\Gamma^{[0]}$ is finite; - $(\Gamma^{[1]})_{(m)} \lhd \Gamma^{[1]}$ is free-abelian, and $\Gamma^{[1]}$ is virtually free-abelian; - for any (possibly empty) finite collection $\varphi_1, \cdots, \varphi_n \in (\Gamma^{[2]})_{(m)} \lhd \Gamma^{[2]}$ of pseudo-Anosov elements, there exists another pseudo-Anosov element $\psi \in (\Gamma^{[2]})_{(m)} \lhd \Gamma^{[2]}$ such that $\Fix(\varphi_i) \cap \Fix(\psi)=\nil$ in $\PML^\sharp({\Sigma}^{[2]})$ for all $i$. It follows from the choice of ${\Sigma}^{[0]}$ in Lemma \[partition: adeq-red\] that $(\Gamma_{(m)})^{[0]}$ is trivial; hence, $(\Gamma^{[0]})_{(m)}$ must also be trivial, and $\Gamma^{[0]}$ must be finite. To see that $(\Gamma^{[1]})_{(m)}$ is free-abelian, we consider its restriction to each component of ${\Sigma}^{[1]}$. Since the restriction of $(\Gamma_{(m)})^{[1]}<(\Gamma^{[1]})_{(m)}$ to each component is infinite-cyclic by definition, the restriction of $(\Gamma^{[1]})_{(m)}$ to each component must be virtually infinite-cyclic. Theorem \[trichotomy: adeq-red for conn\] then implies that this restriction of $(\Gamma^{[1]})_{(m)}$ to each component must be infinite-cyclic. Hence, $(\Gamma^{[1]})_{(m)}$ must be free-abelian, and $\Gamma^{[1]}$ is virtually free-abelian. Now, we consider $\Gamma^{[2]}$ and $(\Gamma^{[2]})_{(m)}$. Note that it suffices to prove the statement (2) under the assumption ${\Sigma}={\Sigma}^{[2]}$, which allow us to reduce the cluttering of the notations. With this assumption, we have $\Gamma=\Gamma^{[2]}$ and $\Gamma_{(m)}=(\Gamma^{[2]})_{(m)}$. The restriction of $\Gamma_{(m)}$ to each component of ${\Sigma}$ is nontrivial and non-cyclic, and $\Gamma_{(m)}$ is infinite and irreducible by Lemma \[adeq-red and Gamma\_m\]. Hence, by Ivanov’s result [@Ivanov:MCGBook §6.4], we know that there exists two pseudo-Anosov elements $\sigma$ and $\tau$ in $\Gamma_{(m)}$ such that $\Fix(\sigma)\cap\Fix(\tau)=\nil$ in $\PML^\sharp({\Sigma})$. We need to show that the following slightly stronger statement holds: for any finite collection $\varphi_1, \cdots, \varphi_n \in \Gamma_{(m)} \lhd \Gamma$ of pseudo-Anosov elements, there exists another pseudo-Anosov element $\psi \in \Gamma_{(m)} \lhd \Gamma$ such that $\Fix(\varphi_j) \cap \Fix(\psi)=\nil$ in $\PML^\sharp({\Sigma})$ for all $j$. Recall that the group $\Gamma_{(m)}$ can be restricted to each component ${\Sigma}^i$ of ${\Sigma}$; in particular, the restrictions of pseudo-Anosov elements $\sigma, \tau, \varphi_j \in\Gamma_{(m)}$ to each component ${\Sigma}^i$ are again pseudo-Anosov. The action of $\Gamma_{(m)}$ on $\PML^\sharp({\Sigma})$ descends to the action of the restriction of $\Gamma_{(m)}$ to ${\Sigma}^i$ on the component $\PML({\Sigma}^i)$. The proof of Theorem \[trichotomy: adeq-red for conn\] applies to each component, and we will combine them to construct the desired $\psi \in \Gamma_{(m)}$. Suppose we are given a finite collection $\varphi_1, \cdots, \varphi_n \in \Gamma_{(m)} \lhd \Gamma$ of pseudo-Anosov elements. For each $\varphi_j$ and each component ${\Sigma}^i$, we let $U^{+,i}_j, U^{-,i}_j \subset \PML({\Sigma}^i)$ be attracting and repelling neighborhoods of $\varphi_j|_{{\Sigma}^i}$ such that $U^{+,i}_j \cap U^{-,i}_j=\nil$; we set $U^i_j=U^{+,i}_j \cup U^{-,i}_j$. Similarly, we let $U^{+,i}_\sigma, U^{-,i}_\sigma, U^{+,i}_\tau, U^{-,i}_\tau \subset \PML({\Sigma}^i)$ be attracting and repelling neighborhoods of $\sigma|_{{\Sigma}^i}$, $\tau|_{{\Sigma}^i}$ respectively such that $U^{+,i}_\sigma \cap U^{-,i}_\sigma=\nil$ and $U^{+,i}_\tau \cap U^{-,i}_\tau=\nil$; again, we set $U^i_\sigma=U^{+,i}_\sigma \cup U^{-,i}_\sigma$ and $U^i_\tau=U^{+,i}_\tau \cup U^{-,i}_\tau$. We can take these neighborhoods to be sufficiently small so that the following holds: - $U^i_j \cap U^i_k=\nil$ whenever $\Fix(\varphi_j|_{{\Sigma}^i}) \cap \Fix(\varphi_k|_{{\Sigma}^i})=\nil$; - $U^i_j \cap U^i_\sigma=\nil$ whenever $\Fix(\varphi_j|_{{\Sigma}^i}) \cap \Fix(\sigma|_{{\Sigma}^i})=\nil$; - $U^i_j \cap U^i_\tau=\nil$ whenever $\Fix(\varphi_j|_{{\Sigma}^i}) \cap \Fix(\tau|_{{\Sigma}^i})=\nil$; - $U^i_\sigma \cap U^i_\tau=\nil$ (since $\Fix(\sigma|_{{\Sigma}^i}) \cap \Fix(\tau|_{{\Sigma}^i})=\nil$ by assumption). By the property of pseudo-Anosov elements, there exists a large enough integer $M$ such that $(\sigma|_{{\Sigma}^i})^{\pm m}(\PML({\Sigma}^i)-U^{\mp,i}_\sigma) \subset U^{\pm,i}_\sigma$ and $(\tau|_{{\Sigma}^i})^{\pm m}(\PML({\Sigma}^i)-U^{\mp,i}_\tau) \subset U^{\pm,i}_\tau$ for all $i$ and for all $m>M$. Also, by Lemma \[constructing pseudo-Anosov for conn\], there exists large enough integers $P$ and $Q$ such that $\sigma^p\tau^q|_{{\Sigma}^i}=(\sigma|_{{\Sigma}^i})^p (\tau|_{{\Sigma}^i})^q$ is pseudo-Anosov for all $i$ and for all $p>P$ and $q>Q$; in other words, $\psi=\sigma^p \tau^q$ is pseudo-Anosov on ${\Sigma}$ for all $p>P$ and $q>Q$. Taking $p$ and $q$ greater than $M$, we obtain a pseudo-Anosov element $\psi=\sigma^p\tau^q$ such that $\psi|_{{\Sigma}^i}(\PML({\Sigma}^i)-U^{-,i}_\tau) \in U^{+,i}_\sigma \subset U^i_\sigma$ and $\psi^{-1}|_{{\Sigma}^i}(\PML({\Sigma}^i)-U^{+,i}_\sigma) \in U^{-,i}_\tau \subset U^i_\tau$. Furthermore, by the disjointness of $U^i_\sigma$ and $U^i_\tau$, we see that $\psi^\ell|_{{\Sigma}^i}(\PML({\Sigma}^i)-U^{-,i}_\tau) \in U^{+,i}_\sigma \subset U^i_\sigma$ and $\psi^{-\ell}|_{{\Sigma}^i}(\PML({\Sigma}^i)-U^{+,i}_\sigma) \in U^{-,i}_\tau \subset U^i_\tau$ for all $i$ and for all positive integer $\ell$. Choose $\scrL \in \PML({\Sigma}^i)-(U^i_\sigma \cup U^i_\tau)$ and consider a sequence $\scrL^\ell:=\psi^\ell(\scrL)$, $\ell \in \Z$. By the same arguments as in the proof of Theorem \[trichotomy: adeq-red for conn\], we see that $\scrL^\ell \rightarrow \scrL^{+,i}_\psi \in \overline{U^{+,i}_\sigma} \subset \overline{U^i_\sigma}$ and $\scrL^{-\ell} \rightarrow \scrL^-_\psi \in \overline{U^{-,i}_\tau} \subset \overline{U^i_\tau}$ as $\ell \rightarrow \infty$, and we deduce that $\Fix(\psi|_{{\Sigma}^i}) \cap \Fix(\varphi_j|_{{\Sigma}^i})=\nil$. Since this is true for each component ${\Sigma}^i$, we conclude that $\Fix(\psi) \cap \Fix(\varphi_j)=\nil$ in $\PML^\sharp({\Sigma})$. Girth Alternative {#girth alternative} ================= We now consider the girth of subgroups $\Gamma$ of a mapping class group $\Mod({\Sigma})$, where ${\Sigma}$ is a compact surface; we do not a priori assume that ${\Sigma}$ is connected. Our main result is that the dichotomy between the subgroups with infinite girth and the ones with finite girth indeed coincides with the structural dichotomy of the Tits alternative shown in [@Ivanov:Tits-Margulis-Soifer] and [@McCarthy:Tits]. In other words, we have the following *girth alternative*: Let ${\Sigma}$ be a compact orientable surface, and let $\Gamma$ be a finitely generated subgroup of $\Mod({\Sigma})$. Then, $\Gamma$ is either a non-cyclic group with infinite girth or a virtually free-abelian group; moreover, these alternatives are mutually exclusive. The girth alternative above reduces to the case where the interior of ${\Sigma}$ admits a complete hyperbolic metric. \[girth-alt: mcg hyp\] Let ${\Sigma}$ be a compact orientable surface whose interior admits a complete hyperbolic metric, and let $\Gamma$ be a finitely generated subgroup of $\Mod({\Sigma})$. Then, $\Gamma$ is either a non-cyclic group with infinite girth or a virtually free-abelian group; moreover, these alternatives are mutually exclusive. Let us first show that the general case, Theorem \[girth-alt: mcg\] in §\[Introduction\], follows from the hyperbolic case, Theorem \[girth-alt: mcg hyp\] above. Suppose for now that ${\Sigma}$ consists of copies of tori and a (possibly disconnected) surface that admits a complete hyperbolic metric. Since the mapping class groups of tori and one-punctured tori are isomorphic, we may replace the copies of tori in ${\Sigma}$ with the same number of copies of once-punctured tori. In turn, we now realize $\Gamma$ as a subgroup of the mapping class group of a hyperbolic surface; Theorem \[girth-alt: mcg\] follows from Theorem \[girth-alt: mcg hyp\] as desired. For the general case, let ${\Sigma}={\Sigma}' \sqcup {\Sigma}''$, such that ${\Sigma}'$ is the union of tori and hyperbolic components, and that ${\Sigma}''$ is the union of spheres, disks, and annuli. If the restriction $\Gamma'$ of $\Gamma$ to ${\Sigma}'$ is a non-cyclic group with infinite girth, then so is $\Gamma$ by Proposition \[akhmedov criterion\]. So, assume that $\Gamma'$ is virtually abelian and let $A' < \Gamma'$ be a finite-index abelian subgroup. Recall that the mapping class groups are trivial for the sphere, the disk, and the annulus; hence, the restriction of $\Gamma$ to ${\Sigma}''$ can only permute these components. Hence, the kernel $K$ of the restriction to ${\Sigma}''$ is a finite-index normal subgroup of $\Gamma$. Let $K'$ be the restriction of $K$ to ${\Sigma}'$; note that $K' \cap A'$ is a finite-index abelian subgroup of $K'$. Now, since elements of $K$ acts trivially on ${\Sigma}''$, we see that $K \cong K'$. Hence, $K$ contains a finite-index abelian subgroup isomorphic to $K' \cap A'$. As $K$ itself has finite index in $\Gamma$, we conclude that $\Gamma$ contains a finite-index abelian subgroup. Throughout the rest of this section, we will assume that the surface ${\Sigma}$ admits a complete hyperbolic metric. The proof for Theorem \[girth-alt: mcg hyp\] is the content of §\[Girth: Adeq-Red\] and §\[Girth: Red\], and will be given in the form of Propositions \[girth-alt: adeq-red for conn\], \[girth-alt: adeq-red for disconn\], and \[girth-alt: red\]. The main idea of the proof is that a pair of generators of free subgroups in the Tits alternative can be carefully chosen so that Infinite Girth Criterion (Proposition \[igc\]) can be applied to these elements. Proposition \[girth-alt: adeq-red for conn\], stated in a slightly different language, was independently proved by Yamagata [@Yamagata:Girth]. Girth of Adequately Reduced Subgroups {#Girth: Adeq-Red} ------------------------------------- We first consider an adequately reduced subgroup $\Gamma < \Mod({\Sigma})$, where ${\Sigma}$ is a *connected* surface. \[girth-alt: adeq-red for conn\] Let ${\Sigma}$ be a connected compact orientable surface with $\chi({\Sigma})<0$, and suppose that $\Gamma < \Mod({\Sigma})$ is an adequately reduced subgroup. Then, $\Gamma$ is either a non-cyclic group with infinite girth or a virtually infinite-cyclic group, or a finite group; moreover, these alternatives are mutually exclusive. In particular, if ${\Sigma}$ is not a pair of pants, $\Mod({\Sigma})$ has infinite girth. Choose an integer $m \geq 3$. Suppose $\Gamma$ is an adequately reduced subgroup of $\Mod({\Sigma})$, and let $\calG=\{ \gamma_1, \cdots , \gamma_n \}$ be a generating set of $\Gamma$. We assume that $\Gamma$ is infinite and not virtually infinite-cyclic, and aim to show that it has infinite girth. In this case, the statement (2) of Theorem \[trichotomy: adeq-red for conn\] must be satisfied. First, we know that there is a pseudo-Anosov element $\sigma \in \Gamma_{(m)} \lhd \Gamma$. For each $1 \leq j \leq n$ and $\varepsilon=\pm 1$, the conjugate $\gamma_j^\varepsilon \sigma \gamma_j^{-\varepsilon} \in \Gamma_{(m)} \lhd \Gamma$ is again a pseudo-Anosov element with $\Fix(\gamma_j^\varepsilon \sigma \gamma_j^{-\varepsilon})=\gamma_j^\varepsilon(\Fix(\sigma))$. Hence, applying the statement (2) of Theorem \[trichotomy: adeq-red for conn\] to the collection $\{\sigma\} \cup \{ \gamma_j^\varepsilon \sigma \gamma_j^{-\varepsilon} \; | \; 1 \leq j \leq n, \varepsilon=\pm1 \}$, we see that there is another pseudo-Anosov element $\tau \in \Gamma_{(m)} \lhd \Gamma$ such that $\Fix(\tau) \cap \Fix(\sigma)=\nil$ and $\Fix(\tau) \cap \Fix(\gamma_j^\varepsilon \sigma \gamma_j^{-\varepsilon})=\nil$ for all $1 \leq j \leq n$ and $\varepsilon=\pm 1$. Note that, if $U_\sigma$ is a neighborhood of $\Fix(\sigma)$, then $\gamma_j^\varepsilon(U_\sigma)$ is a neighborhood of $\Fix(\gamma_j^\varepsilon \sigma \gamma_j^{-\varepsilon})$ for each $1 \leq j \leq n$ and $\varepsilon=\pm1$. It then follows that there are small enough neighborhoods $U_\sigma \supset \Fix(\sigma)$ and $U_\tau \supset \Fix(\tau)$ such that $$U_\sigma \cap U_\tau=\nil \;\; \text{and} \;\; U_\tau \cap \gamma_j^\varepsilon(U_\sigma)=\nil$$ for each $1 \leq j \leq n$ and $\varepsilon=\pm1$, or equivalently $$U_\sigma \cap U_\tau=\nil \;\; \text{and} \;\; \gamma_j^\varepsilon(U_\tau) \cap U_\sigma=\nil$$ for each $1 \leq j \leq n$ and $\varepsilon=\mp1$. Now, since $\sigma$ and $\tau$ are pseudo-Anosov elements, we can take high enough powers $\tilde \sigma:=\sigma^N$ and $\tilde \tau:=\tau^N$ such that $$\tilde \sigma^k \big( \PML({\Sigma})-U_\sigma \big) \subset U_\sigma \;\; \text{and} \;\; \tilde \tau^k \big( \PML({\Sigma})-U_\tau \big) \subset U_\tau$$ for all non-zero integer $k$. In particular, we have $$\displaystyle{ \tilde \sigma^k \bigg( U_\tau \cup \bigcup_{\varepsilon=\pm1} \bigcup_{j=1}^n \gamma_j^\varepsilon(U_\tau) \bigg) \subset U_\sigma } \;\; \text{and} \;\; \displaystyle{ \tilde \tau^k \bigg( U_\sigma \cup \bigcup_{\varepsilon=\pm1} \bigcup_{j=1}^n \gamma_j^\varepsilon(U_\sigma) \bigg) \subset U_\tau }$$ for all non-zero integer $k$. Applying the Infinite Girth Criterion (Proposition \[igc\]) to $\tilde \sigma$, $\tilde \tau$, $U_\sigma$, $U_\tau$, and $$x \in \PML({\Sigma})-\bigg((U_\sigma \cup U_\tau) \cup \bigcup_{\varepsilon=\pm1} \bigcup_{j=1}^n \gamma_j^\varepsilon(U_\sigma \cup U_\tau)\bigg)$$ we conclude that $\Gamma$ must be a non-cyclic group with infinite girth. Now, we allow ${\Sigma}$ to be disconnected, and consider an adequately reduced subgroup $\Gamma < \Mod({\Sigma})$. The idea of the proof of girth alternative in this case is to consider the partition ${\Sigma}={\Sigma}^{[0]} \sqcup {\Sigma}^{[1]} \sqcup {\Sigma}^{[2]}$ from Lemma \[partition: adeq-red\], and restrict the group $\Gamma$ to ${\Sigma}^{[2]}$ when ${\Sigma}^{[2]} \neq \nil$. The non-emptiness of ${\Sigma}^{[2]}$ is the source of the existence of non-abelian free subgroup in the Tits alternative, as well as the infinite girth in the girth alternative. When ${\Sigma}^{[2]}$ is empty, the following lemma from the proof of the Tits alternative shows that $\Gamma$ must be virtually free-abelian. \[vAb: adeq-red\] Let $m \geq 3$ be an integer, and let $\Gamma$ be an adequately reduced group. $\Gamma$ is virtually free-abelian if and only if the restriction of $\Gamma_{(m)}$ to each component of ${\Sigma}$ is either trivial or infinite-cyclic, i.e. ${\Sigma}^{[2]}=\nil$ in the partition from Lemma \[partition: adeq-red\]. \[girth-alt: adeq-red for disconn\] Let ${\Sigma}=\bigsqcup_{i=1}^{c({\Sigma})} {\Sigma}^i$ be a (possibly disconnected) compact orientable surface with $\chi({\Sigma}^i)<0$ for all $i$, and suppose that $\Gamma < \Mod({\Sigma})$ is an adequately reduced subgroup. Then, $\Gamma$ is either a non-cyclic group with infinite girth or a virtually free-abelian group; moreover, these alternatives are mutually exclusive. Consider the decomposition ${\Sigma}={\Sigma}^{[0]} \sqcup {\Sigma}^{[1]} \sqcup {\Sigma}^{[2]}$ in Lemma \[partition: adeq-red\]. If ${\Sigma}^{[2]}=\nil$, then $\Gamma$ is virtually free-abelian by Lemma \[vAb: adeq-red\]. Hence, we may assume that ${\Sigma}^{[2]} \neq \nil$. Note that the restriction $\Gamma^{[2]}$ of $\Gamma$ to ${\Sigma}^{[2]}$ is the image of a homomorphism from $\Gamma$ into $\Mod({\Sigma}^{[2]})$. If $\Gamma^{[2]}$ is a non-cyclic group with infinite girth, $\Gamma$ itself must also be a noncyclic group with infinite girth by Proposition \[akhmedov criterion\]. Hence, we may as well assume that ${\Sigma}={\Sigma}^{[2]}$ and $\Gamma=\Gamma^{[2]}$. Choose $m \geq 3$. Let $\calG=\{\gamma_1, \cdots, \gamma_n\}$ be a generating set of $\Gamma$. By Theorem \[trichotomy: adeq-red for disconn\], (i) there is a pseudo-Anosov element $\sigma \in \Gamma_{(m)} \lhd \Gamma$, and (ii) there is another pseudo-Anosov element $\tau \in \Gamma_{(m)} \lhd \Gamma$ such that $\Fix(\tau) \cap \Fix(\sigma)=\nil$ and $\Fix(\tau) \cap \Fix(\gamma_j^\varepsilon \sigma \gamma_j^{-\varepsilon})$ for each $1 \leq j \leq n$ and $\varepsilon=\pm1$. Here, the conjugates $\gamma_j^\varepsilon \sigma \gamma_j^{-\varepsilon}$ are pseudo-Anosov elements with $\Fix(\gamma_j^\varepsilon \sigma \gamma_j^{-\varepsilon})= \gamma_j^\varepsilon(\Fix(\sigma))$ in $\PML^\sharp({\Sigma})$. The arguments identical to the proof of Proposition \[girth-alt: adeq-red for conn\] — with the space $\PML(\Sigma)$ replaced by $\PML^{\sharp}({\Sigma})$ — completes the proof of the proposition. Girth of Reducible Subgroups {#Girth: Red} ---------------------------- We now consider the girth of a reducible group $\Gamma$. We take the canonical reduction system $\calC$, and consider the canonical reduction ${\operatorname{\rho}}_{\calC}(\Gamma)<\Mod({\Sigma}_{\calC})$, for which the girth alternative holds by Proposition \[girth-alt: adeq-red for disconn\]. The following lemma, extracted from the proof of the Tits alternative [@Ivanov:MCGBook §8.9], characterizes virtually free-abelian subgroups $\Gamma$ in terms of its reduction ${\operatorname{\rho}}_{\calC}(\Gamma)$. \[[c.f. [@Ivanov:MCGBook §8.9]]{}\] \[vAb: red\] $\Gamma$ is virtually free-abelian if and only if the canonical reduction ${\operatorname{\rho}}_{\calC}(\Gamma)$ is virtually free-abelian. \[girth-alt: red\] Let ${\Sigma}=\bigsqcup_{i=1}^{c({\Sigma})}{\Sigma}^i$ be a compact orientable surface with $\chi({\Sigma}^i)<0$, and suppose that $\Gamma < \Mod({\Sigma})$ is a reducible subgroup. Then, $\Gamma$ is either a non-cyclic group with infinite girth or a virtually free-abelian group; moreover, these alternatives are mutually exclusive. Let $\calC$ be the canonical reduction system for $\Gamma$ and ${\operatorname{\rho}}_{\calC}$ be the corresponding reduction homomorphism. The reduction ${\operatorname{\rho}}_{\calC}(\Gamma)$ is adequately reduced, and hence by Proposition \[girth-alt: adeq-red for disconn\], ${\operatorname{\rho}}_{\calC}(\Gamma)$ is either a non-cyclic group with infinite girth or a virtually free-abelian group. If ${\operatorname{\rho}}_{\calC}(\Gamma)$ is a non-cyclic group with infinite girth, noting that ${\operatorname{\rho}}_{\calC}(\Gamma)$ is the image of the homomorphism ${\operatorname{\rho}}_{\calC}:\Gamma \rightarrow \Mod({\Sigma}_{\calC})$, we see that $\Gamma$ is also a non-cyclic group with infinite girth by Proposition \[akhmedov criterion\]. If ${\operatorname{\rho}}_{\calC}(\Gamma)$ is virtually free-abelian, so is $\Gamma$ by Lemma \[vAb: red\].

--- abstract: 'A permanental vector with a symmetric kernel and index $2$ is a squared Gaussian vector. The definition of permanental vectors is a natural extension of the definition of squared Gaussian vectors to nonsymmetric kernels and to positive indexes. The only known permanental vectors either have a positive definite kernel or are infinitely divisible. Are there some others? We present a partial answer to this question.' address: | Laboratoire de probabilités\ et modèles aléatoires\ Université Paris 6\ Case 188, 4 Place Jussieu\ 75252 Paris cedex 05\ France\ author: -   title: Permanental vectors with nonsymmetric kernels --- ./style/arxiv-general.cfg Introduction ============ A real-valued positive vector $\psi= (\psi_i, 1 \leq i \leq n)$ is a permanental vector if its Laplace transform satisfies for every $(\alpha_1, \alpha_2, \ldots,\alpha_n)$ in $\mathbb{R}^n_+$ $$\label{perm} \mathbb{E} \Biggl[\exp \Biggl\{-{1\over2} \sum _{i = 1}^n \alpha_i \psi _{i} \Biggr\} \Biggr] ={\vert}I + \alpha G {\vert}^{-\beta},$$ where $I$ is the $n\times n$-identity matrix, $\alpha$ is the diagonal matrix $\operatorname{Diag}((\alpha_i)_{1 \leq i \leq n})$, $G = (G(i,j))_{1\leq i,j \leq n}$ and $\beta$ is a fixed positive number. Such a vector $(\psi_i, 1 \leq i \leq n)$ is a permanental vector with kernel $(G(i,j),\allowbreak 1 \leq i,j \leq n)$ and index $\beta$. Note that the kernel of $\psi$ is not uniquely determined. Indeed any matrix $D G D^{-1}$ with $D$ $n\times n$-diagonal matrix with nonzero entries is a kernel for $\psi$. The matrices $G$ and $DGD^{-1}$ are said to be *diagonally equivalent*. But remark that $\psi$ also admits $G^t$ for kernel. More generally, the kernels of $\psi$ are said to be *effectively equivalent*. Vere-Jones has established a necessary and sufficient condition on the couple $(G, \beta)$ for the existence of such a vector. His criterion is reminded at the beginning of Section \[2\]. For $G$ $n\times n$-symmetric positive definite matrix and $\beta= 2$, (\[perm\]) is the Laplace transform of the vector $(\eta_1^2, \eta_2^2, \ldots, \eta_n^2)$ where $(\eta_1, \eta_2, \ldots, \eta_n)$ is a centered Gaussian vector with covariance $G$. The definition of permanental vectors hence represents an extension of the definition of squared Gaussian vectors. The question is: to which point? More precisely, one already knows two classes of matrices that satisfy Vere-Jones criterion: the symmetric positive definite matrices and the inverse $M$-matrices (a nonsingular matrix $A$ is a $M$-matrix if its off-diagonal entries are nonpositive and the entries of $A^{-1}$ are nonnegative). Up to effective equivalence, these are the only known examples of permanental kernels. The question becomes: Is there an irreducible permanental kernel that would not belong to any of this two classes? This two classes correspond respectively to vectors with the Laplace transform of a squared Gaussian vector to a positive power and to infinitely divisible permanental vectors. Infinitely divisible permanental processes are connected to local times of Markov processes thanks to Dynkin’s isomorphism theorem and its extensions (see [@EK2]). Besides, we have shown in [@E] that for permanental vectors, infinite divisibility and positive correlation are equivalent properties. In dimension one, obviously, the above two classes are identical and the answer is negative. One easily checks that a 2-dimensional permanental vector with index $2$ is a squared Gaussian couple. Moreover, Vere-Jones [@V1], solving a question raised by Lévy [@L], proved that a squared Gaussian couple is always infinitely divisible. Hence, in this case also the two classes are identical and the answer is negative. In dimension 3, the situation is different. Indeed, Kogan and Marcus [@KM] have shown that if the kernel of a $3$-dimensional permanental vector is not effectively equivalent to a symmetric matrix (in short, is not symmetrizable), then it is diagonally equivalent to an inverse $M$-matrix. Since there exist inverse $M$-matrices that are not symmetrizable, the two classes are not identical and have a nonempty intersection. But the answer to the above question remains negative. The case of dimension $d$ strictly greater than $3$ is still an open question. We show here that if the kernel of a $d$-dimensional permanental vector is strongly not symmetrizable, meaning that none of its principal submatrices of dimension $3$ is symmetrizable, then it is diagonally equivalent to the inverse of an $M$-matrix. The result presented below is actually a little stronger and suggests that the answer should still be negative in the general case. In other words, one might think that the permanental vectors with a kernel not effectively equivalent to a symmetric matrix, are always infinitely divisible. For a set of indexes $I$, we adopt the notation: $G_{I \times I} = (G(i,j))_{ (i,j) \in I \times I}$. \[T1\] For $d > 3$, let $\psi$ be a $d$-dimensional permanental vector with kernel $G$. Assume that there exists at most one subset $I$ of three indexes such that $G_{I \times I} $ is symmetrizable. Then $\psi$ is infinitely divisible. Theorem \[T1\] can also be stated as follows: *For $\psi= (\psi_i)_{1 \leq i \leq d}$ permanental vector of dimension $d>3$, assume that there exists at most three integers $1 \leq i_1, i_2,i_3 \leq d$ such that $(\psi_{i_1}, \psi_{i_2}, \psi _{i_3})$ has the Laplace transform of a squared Gaussian vector to some power. Then $\psi$ is infinitely divisible.* The proof of Theorem \[T1\] is given in Section \[2\]. Section \[1\] introduces the needed preliminaries and definitions. Section \[3\] presents some examples and remarks. Preliminaries {#1} ============= We remind first the necessary and sufficient condition established by Vere-Jones [@V] for a given matrix $K$ to be the kernel of a permanental vector. *A $n\times n$-matrix $K$ is the kernel of a permanental vector with index $\beta> 0$ if and only if*: *All the real eigenvalues of $K$ are nonnegative.* *For every $\gamma> 0$, set $K_{\gamma} = (I + \gamma K)^{-1} K$, then $K_{\gamma}$ is $\beta$-positive definite.* A $n\times n$-matrix $M = (M(i,j))_{1\leq i,j \leq n}$ is said to be $\beta$-positive definite if for every integer $m$, every (not necessarily distinct) $k_1, k_2,\ldots, k_m$ in $\{1,2,\ldots, n\}$ $$\mathrm{per}_{\beta} \bigl( \bigl(M(k_i,k_j) \bigr)_{1\leq i,j \leq m} \bigr) \geq0,$$ where for any $m\times m$-matrix $A = (A(i,j))_{1 \leq i,j \leq m}$, the quantity $\mathrm{per}_{\beta}(A)$ is defined as follows: $\mathrm{per}_{\beta}(A) = \sum_{\tau\in{\mathcal S}_m} \beta^{\nu(\tau)} \prod_{i = 1}^m A_{i,\tau(i)}$, with ${\mathcal S}_m$ the set of the permutations on $\{1,2,\ldots,m\}$, and $\nu(\tau)$ the signature of $\tau$. Note that the property of $\beta$-positive definiteness for a matrix $M$ is supported by an infinite family of matrices derived from $M$. The proposition below is just the regrouping of results of Kogan and Marcus on the three dimensional permanental kernels. For the sake of clarity, we explain where to find this results in [@KM]. Adopting their convention, $0$ is both positive and negative. \[effequ\] Two $d\times d$-matrices $A$ and $B$ are said to be effectively equivalent if for every $x$ in $\mathbb{R}^d\dvtx {\vert}I + x A{\vert}={\vert}I + x B{\vert}$. \[Dsym\] A squared matrix is symmetrizable if it is effectively equivalent to a symmetric matrix. \[P1\] Let $\psi$ be a 3-dimensional permanental vector with kernel $G = (G_{ij})_{1 \leq i,j \leq3}$. Then we have: $G$ is diagonally equivalent either to a matrix with all positive entries or to a matrix with all negative off-diagonal entries. If $G$ has all its off-diagonal entries strictly negative, then $G$ is diagonally equivalent to a symmetric matrix. If $G$ has all its off-diagonal entries strictly positive, then $G$ is either an inverse $M$-matrix or it is diagonally equivalent to a symmetric matrix. If $G$ has one or more zero off-diagonal entries, it is effectively equivalent to a symmetric matrix $\tilde{G}$ such that $\tilde{G}_{ij} = 0$ when $G_{ij}G_{ji} = 0$. Up to some misprints, (i) is Remark 2.1 in [@KM], which is a consequence of the fact that $G_{ij} G_{ji} \geq0$ for every $1 \leq i,j \leq3$. Indeed, for example, for $G$ with only positive entries $$\pmatrix{ G_{11} & G_{12} & G_{13} \cr G_{21} & G_{22}& G_{23} \cr G_{31} & G_{32} & G_{33} }\quad \mbox{and}\quad \pmatrix{ G_{11} & -G_{12} & -G_{13} \cr -G_{21} & G_{22}& G_{23} \cr -G_{31} & G_{32} & G_{33} }$$ are diagonally equivalent. is established in the first part of the proof of Lemma 4.1 in [@KM]. is established in the first part of the proof of Lemma 5.1 in [@KM]. When $G$ is diagonally equivalent to a matrix with all negative off-diagonal entries, this is a consequence of the second part of the proof of Lemma 4.1 in [@KM]. When $G$ is diagonally equivalent to a matrix with all positive off-diagonal entries, (iv) is a consequence of the last paragraph of the proof of Lemma 5.1 in [@KM] together with its Lemma 2.3 cleaned from a misprint. For the last sentence of Lemma 2.3 to be correct, the word “diagonally” should be replaced by “effectively.” Indeed Lemma 2.3 in [@KM], assuming that the two matrices $$\pmatrix{ 1 & 0 & c_2 \cr a_2 & 1& b_1 \cr c_1 & 0 & 1 }\quad \mbox{and}\quad \pmatrix{ 1 & 0 & \sqrt{c_1c_2} \cr 0 &1& 0 \cr \sqrt{c_1c_2} & 0 & 1 }$$ are permanental kernels, states that for $a_2b_1 c_1c_2 \neq 0$, they are diagonally equivalent. But they cannot be diagonally equivalent. They are effectively equivalent. We will use repeatedly the following lemma which is an elementary remark. \[L1\] For $A$ $n\times n$-matrix, the following points are equivalent: $A$ is diagonally equivalent to a symmetric matrix. For every couple $(D_1, D_2)$ of diagonal $n\times n$-matrices with strictly positive diagonal entries, $D_1AD_2$ is diagonally equivalent to a symmetric matrix. There exist two diagonal $n\times n$-matrices with strictly positive diagonal entries $D_1$ and $D_2$ such that $D_1 A D_2$ is diagonally equivalent to a symmetric matrix. \[R0\] In view of Proposition \[P1\], a permanental kernel of dimension $3$, $G = (G_{ij})_{1 \leq i,j \leq 3}$, is symmetrizable iff: - either $G$ has an off-diagonal entry equal to zero, - either $G$ has no zero off-diagonal entry and it is diagonally equivalent to a symmetric matrix with strictly positive entries. To check whether a $3\times3$-matrix $K$ without zero off-diagonal entry, is symmetrizable, one has first to check the existence of a signature matrix $\sigma$ (a diagonal matrix with ${\vert}\sigma_{ii}{\vert}= 1, 1\leq i \leq3$) such that: $\sigma K \sigma= ({\vert}K_{ij}{\vert})_{1 \leq i,j \leq3}$, and then check that $$\bigl{\vert}K(1,2) K(2,3) K(3,1) \bigr{\vert}= \bigl{\vert}K(2,1) K(1,3) K(3,1) \bigr{\vert}.$$ Proof of Theorem 1.1 {#2} ==================== *Step 1*: Assume that $d = 4$ and that $G$ has no symmetrizable $3 \times3$-principal submatrices, we show then that $\psi$ is infinitely divisible. Thanks to Remark \[R0\], we know that $G$ has no entry equal to $0$. Moreover, in view of (i) and (ii) of Proposition \[P1\], every $3\times3$-principal submatrix of $G$ has to be diagonally equivalent to a matrix with all positive entries. This means that for every subset of three indexes $I$, there exists $S_I$ from $I$ into $\{-1, +1\}$ such that $S_I(i) G(i,j) S_I(j) \geq0$, for $i,j \in I$. This leads to $$G(i,j)G(j,k)G(k,i) > 0\qquad \forall i,j,k \in\{1,2,3,4\}.$$ Since $G$ has no zero entry, this property implies the existence of $S$ from $\{1,2,3,4\}$ into $\{-1, +1\}$ such that: $S(i) G(i,j) S(j) > 0, \forall i,j \in\{1,2,3,4\}$. Since $\psi$ also admits for kernel $SGS$, we will assume from now that the entries of $G$ are all strictly positive. For $\sigma> 0$, consider the $3$-dimensional vector $\phi_{\sigma}$ with Laplace transform $$\label{LT} {\mathbb{E}[ \exp\{-{(1/2)} \sum_{j = 1}^{3}\lambda_j \psi_j\} \exp\{ -{(\sigma/2)} \psi_4\}] \over\mathbb{E}[ \exp\{-{(\sigma/ 2)} \psi _4\}]}.$$ This vector is a permanental vector with the same index as $\psi$ and admits for kernel $H({\sigma}, G)$ (see [@KM]) $$H({\sigma}, G) = \biggl(G(i,j) - {\sigma\over1 + \sigma G(4,4)} G(i,4) G(4,j) \biggr)_{ 1 \leq i,j \leq3}.$$ For which values of $\sigma$, is $H(\sigma, G)$ symmetrizable? For $\sigma> 0$, we have${\sigma\over1 + \sigma G(4,4)} < {1 \over G(4,4)}$. We set: $\Gamma= ({G(i,j)\over G(i,4)G(4,j)})_{1 \leq i,j \leq4}$. Making use of Lemma \[L1\], we are looking for the values of $c $ in $(0, {1 \over G(4,4)})$ such that $(\Gamma(i,j) - c)_{1 \leq i,j \leq 3}$ is symmetrizable. In view of Remark \[R0\], this can occur in two ways: - either $(\Gamma(i,j) - c)_{1 \leq i,j \leq3}$ has an off-diagonal entry equal to zero. - either $(\Gamma(i,j) - c)_{1 \leq i,j \leq3}$ has no zero off-diagonal entry and it is diagonally equivalent to a symmetric matrix. The first possibility is excluded because it would imply the existence of $i$ and $j$ distinct from $4$, such that: ${G(i,j)\over G(i,4) G(4,j)} < {1 \over G(4,4)}$. But since the entries of $G$ are all strictly positive, we know by assumption that $G_{\{i,j,4\} \times \{i,j,4\}}$ is an inverse $M$-matrix. This last property implies in particular that: $G(i,j) G(4,4) \geq G(i,4)G(4,j)$. This can bee seen by computing the inverse of $G_{\{i,j,4\} \times\{i,j,4\}}$ or by using Willoughby’s paper [@W]. We now study the second possibility. Since $(\Gamma(i,j) -c)_{1\leq i,j \leq3}$ has only strictly positive entries, we know, thanks to Lemma \[L1\](iii), that $(\Gamma(i,j) - c)_{1 \leq i,j \leq3}$ equivalent to a symmetric matrix if and only if$({\Gamma(i,j) - c \over(\Gamma(i,3) -c)(\Gamma(3,j) - c)})_{ 1 \leq i,j \leq3}$ is. Denote this last matrix by $A_c$. Since $A_c(i,3) = A_c(3,j) = {1\over\Gamma(3,3) - c}$ for every $1\leq i,j \leq3$, $A_c$ is diagonally equivalent to a symmetric matrix if and only if $(A_c(i,j))_{1 \leq i,j \leq2}$ is symmetric. This translates into $${\Gamma(1,2) - c \over(\Gamma(1,3) -c)(\Gamma(3,2) - c)} = {\Gamma (2,1) - c \over(\Gamma(2,3) -c)(\Gamma(3,1) - c)},$$ which means that $c$ must solve a polynomial equation with degree $3$. Hence, only the two following cases might occur: - either there are at most three distinct values for $c$ such that $(\Gamma(i,j) - c)_{1 \leq i,j \leq3}$ is diagonally equivalent to a symmetric matrix, - either for every value of $c$, $(\Gamma(i,j) - c)_{1 \leq i,j \leq3}$ is diagonally equivalent to a symmetric matrix. In the later case, one obtains in particular $(\Gamma(i,j))_{1 \leq i,j \leq3}$ is diagonally equivalent to a symmetric matrix. Thanks to Lemma \[L1\], this implies that $(G(i,j))_{1 \leq i,j \leq3}$ is diagonally equivalent to a symmetric matrix. But this is excluded by assumption. Consequently, except for at most three distinct values of $\sigma$, $H(\sigma, G)$ is not symmetrizable. Set now $G_{\sigma} = (I + \sigma G)^{-1} G$. We have shown (Proposition 3.2 in [@E]) that there exists a permanental vector $\psi_{\sigma}$ with the same index as $\psi$, admitting $G_{\sigma }$ for kernel and such that its Laplace transform satisfies $$\begin{aligned} &&\mathbb{E} \Biggl[ \exp\Biggl\{-{1\over2} \sum _{j = 1}^{4}\lambda_j \psi _{\sigma }(j) \Biggr\} \Biggr] \\ &&\qquad = \mathbb{E} \Biggl[{ \exp \{- {(\sigma/2)} \sum_{i = 1}^4 \psi (i) \} \over\mathbb{E}[\exp\{- {(\sigma/2)} \sum_{i = 1}^4 \psi(i) \} ]} \exp \Biggl\{- {1\over2} \sum_{j = 1}^{4} \lambda_j \psi(j) \Biggr\} \Biggr].\end{aligned}$$ Hence, thanks to (\[LT\]), it also satisfies $$\label{LT2} \mathbb{E} \Biggl[ \exp \Biggl\{-{1\over2} \sum _{j = 1}^{3}\lambda_j \psi _{\sigma }(j) \Biggr\} \Biggr] = c(\sigma) \mathbb{E} \Biggl[ \exp \Biggl\{- {1\over2} \sum_{j = 1}^{3}( \lambda_j + \sigma) \phi _{\sigma}(j) \Biggr\} \Biggr],$$ where $c(\sigma) = { \mathbb{E}[ \exp\{- {(\sigma/2)} \psi(4)\} ] \over\mathbb{E}[ \exp\{- {(\sigma/2)} \sum_{i = 1}^4 \psi(i)\} ]} $. We show now that $(\psi_{\sigma}(i))_{1 \leq i \leq3}$ admits a kernel without zero entry. To do so, we first adopt the notation: ${\underline G}_{\sigma} = (G_{\sigma}(i,j))_{1 \leq i,j \leq3}$. Note that (\[LT2\]) can be written as follows for every $x \in \mathbb{R}^3$: $${\vert}I + x {\underline G}_{\sigma}{\vert}= {{\vert}I + ( x + \sigma I) H_{\sigma}{\vert}\over{\vert}I + \sigma H_{\sigma}{\vert}}.$$ Besides denote by $R_{\alpha}$ the $\alpha$-resolvent of $H_{\sigma }\dvtx R_{\alpha} = (I + \alpha H_{\sigma})^{-1} H_{\sigma}$, then we have $$\begin{aligned} {\vert}I + x R_{\sigma}{\vert}&=&{\vert}I + x (I + \sigma H_{\sigma})^{-1} H_{\sigma}{\vert}={\vert}I + \sigma H_{\sigma}{\vert}^{-1}{\vert}I + (x + \sigma I) H_{\sigma}{\vert}\\ &=& {\vert}I +x {\underline G}_{\sigma}{\vert}.\end{aligned}$$ Consequently: ${\underline G}_{\sigma} $ and $R_{\sigma}$ are effectively equivalent. This implies that ${(\underline G}_{\sigma })^{-1}$ and $R_{\sigma}^{-1}$ are effectively equivalent. Assume that ${\underline G}_{\sigma}$ has an off-diagonal entry equal to zero. Then thanks to Proposition \[P1\](iv), ${\underline G}_{\sigma}$ is effectively equivalent to a symmetric matrix. This implies that ${(\underline G}_{\sigma})^{-1}$, and consequently $R_{\sigma}^{-1}$, is effectively equivalent to a symmetric matrix. Since $R_{\sigma}^{-1} = H_{\sigma}^{-1} + \sigma I$, we obtain that $(H_{\sigma}^{-1} + \sigma I)$ is effectively equivalent to a symmetric matrix, which easily implies that $H_{\sigma}^{-1}$, and then $H_{\sigma}$ must be effectively equivalent to a symmetric matrix. Except for at most three values of $\sigma$, this is not true. Consequently, we have obtained, except for at most three values of $\sigma$, that $(G_{\sigma}(i,j))_{1 \leq i,j \leq3}$ has no zero entry and is not effectively equivalent to a symmetric matrix. In view of Proposition \[P1\], the only possibility for $(G_{\sigma }(i,j))_{1 \leq i,j \leq3}$ is to be diagonally equivalent to an inverse $M$-matrix. Hence, there exists a function $S$ from $\{1,2,3\}$ into $\{-1, +1\} $ such that for every $i,j$ in $\{1,2,3\}$: $$S(i) G_{\sigma}(i,j) S(j) > 0,$$ which leads to $$\label{cle} G_{\sigma}(i,j) G_{\sigma}(j,k) G_{\sigma}(k,i) > 0$$ for every $i,j,k$ in $\{1, 2, 3 \}$. The choice of the three indexes $1$, $2$ and $3$, being arbitrary, we conclude that excepted for at most a finite number of values of $\sigma$, $G_{\sigma}$ has no zero entry and satisfies $$G_{\sigma}(i,j) G_{\sigma}(j,k) G_{\sigma}(k,i) > 0$$ for every $i,j,k$ in $\{1, 2, 3, 4 \}$. This last property implies that there exists a function ${S}_{\sigma }$ from $\{1,2,3,4\}$ into $\{-1, +1\}$ such that for every $i$, $j$ in $\{1,2,3,4\}$ $$\label{signe} {S}_{\sigma}(i) G_{\sigma}(i,j) {S}_{\sigma}(j) > 0.$$ For the three values of $\sigma$ that we have excluded, we still have (\[signe\]). Indeed assume that there exists such a value and that it is strictly positive. Denote it by $\alpha$. We know now that there exists $\varepsilon> 0$, such that for every $\sigma$ in $(\alpha, \alpha+ \varepsilon]$, $G_{\sigma}$ satisfies (\[signe\]). Note that $G_{\sigma}$ is the $(\sigma- \alpha)$-resolvent of $G_{\alpha}$: $$G_{\alpha} = G_{\sigma} \bigl( I - (\sigma- \alpha)G_{\sigma} \bigr)^{-1}.$$ For $(\sigma- \alpha)$ small enough, we have $G_{\alpha} = \sum_{k = 1}^{\infty}(\sigma- \alpha)^kG_{\sigma }^k$. Making use of (\[signe\]), one obtains for every $1\leq i,j \leq4$: $$S_{\sigma}G_{\alpha}S_{\sigma}(i,j) = \sum _{k = 1}^{\infty}(\sigma- \alpha)^kS_{\sigma}G_{\sigma}^kS_{\sigma}(i,j) = \sum_{k = 1}^{\infty}(\sigma- \alpha)^k(S_{\sigma}G_{\sigma}S_{\sigma })^k (i,j) > 0.$$ Obviously, (\[signe\]) implies that $G_{\sigma}$ is $\beta $-positive definite for every $\beta> 0$. Vere-Jones criteria allows then to conclude that $\psi$ is infinitely divisible. *Conclusion of Step 1*: We have actually established that if $G$ is a permanental kernel of dimension $4$, with no symmetrizable $3 \times3$-principal submatrices, then it is the kernel of an infinitely divisible permanental vector, and moreover, for every $\sigma> 0$, its $\sigma$-resolvent $G_{\sigma}$ has a no zero entry. *Step 2*: Define the claim $(R_n)$ as follows. $(R_n)$: If $G$ is a $n\times n$-square matrix without symmetrizable $3\times3$-principal submatrix, then $\psi$ is infinitely divisible and for every $\sigma>0$, its $\sigma$-resolvent $G_{\sigma}$ has no zero entry. We have just established $(R_4)$. Assume that $(R_n)$ is satisfied. We now establish $(R_{ n+1})$. First note that $G$ is diagonally equivalent to a matrix with only strictly positive entries. Indeed, using exactly the same argument as at the beginning of Step 1, one obtains $$G(i,j)G(j,k)G(k,i) > 0\qquad \forall i,j,k \in\{1,2,\ldots,n\}.$$ Similarly, as in Step 1, one concludes that there exists $S$ from $\{ 1,2,\ldots,n\}$ into $\{-1,+1\}$ such that $S(i) G(i,j) S(j) > 0, \forall i,j \in\{1,2,\ldots,n\}$. Hence, we can assume that all the entries of $G$ are strictly positive. Then consider the vector $(\phi_{\sigma}(i))_{1\leq i \leq n}$ with Laplace transform $$\label{LTH} {\mathbb{E}[ \exp\{-{(1/2)} \sum_{j = 1}^{n}\lambda_j \psi_j\} \exp\{ -{(\sigma/2)} \psi_{n+1}\}] \over\mathbb{E}[ \exp\{-{(\sigma /2)} \psi_{n+1}\}]}.$$ The vector $(\phi_{\sigma}(i))_{1\leq i \leq n}$ is a permanental vector admitting for kernel $H(\sigma, G)$ defined by $$H({\sigma}, G) = \biggl(G(i,j) - {\sigma\over1 + \sigma G(n+1,n+1)} G(i,n+1) G(n+1,j) \biggr)_{ 1 \leq i,j \leq n}.$$ We look for the values of $\sigma$ such that $H(\sigma, G)$ would have a $3\times3$-principal symmetrizable matrix. We set $\Gamma= ( {G(i,j)\over G(i,n+1) G(n+1,j)})_{1 \leq i,j \leq n+1}$. We hence look for the values of $c$ in $(0, {1\over G(n+1,n+1)})$ such that $(\Gamma (i,j) - c)_{1 \leq i,j \leq n}$ would have a symmetrizable $3\times 3$-principal submatrix. We fix $I$, a subset of three elements of $\{ 1,2,\ldots,n\}$. Similarly, as in the case $d = 4$, we know that $(\Gamma- c)_{I\times I}$ has no zero entry. The only way for $(\Gamma- c)_{I\times I}$ to be symmetrizable is to be diagonally equivalent to a symmetric matrix. Again as in Step 1, we have: - either $(\Gamma- c)_{I\times I}$ is symmetrizable for at most three distinct values of $c$, - either for every real $c$, $(\Gamma- c)_{I\times I}$ is diagonally equivalent to a symmetric matrix. In the second case, one obtains that $\Gamma_{I\times I}$, and consequently $G_{I\times I}$ thanks to Lemma \[L1\], is diagonally equivalent to a symmetric matrix, which is excluded by assumption. Consequently, for $\sigma$ outside of a finite set, $H(\sigma, G)$ does not contain any $3\times3$-principal symmetrizable matrix. Thanks to our assumption on the case $d= n$, we know that $\phi _{\sigma}$ is infinitely divisible and that for every $\alpha>0$, $R_{\alpha}$, the $\alpha$-resolvant of $H_{\sigma}$ has no zero entry. Besides there exists a permanental vector with the same index as $\phi_{\sigma}$, admitting $R_{\alpha}$ for kernel (see Proposition 3.2 in [@E]). Making use of Vere-Jones criteria, one easily shows that this permanental vector is infinitely divisible too. Setting $G_{\sigma} = (I+ \sigma G)^{-1} G$, we know (Proposition 3.2 in [@E]) that there exists a permanental vector $\psi_{\sigma }$ with the same index as $\psi$, admitting $G_{\sigma}$ for kernel and such that its Laplace transform satisfies $$\begin{aligned} &&\mathbb{E} \Biggl[ \exp \Biggl\{-{1\over2} \sum _{j = 1}^{n+1}\lambda_j \psi _{\sigma }(j) \Biggr\} \Biggr] \\ &&\qquad = \mathbb{E} \Biggl[{ \exp\{- {(\sigma/2)} \sum_{i = 1}^{n+1} \psi(i) \} \over\mathbb{E}[\exp\{- {(\sigma/2)} \sum_{i = 1}^{n+1} \psi (i) \}]} \exp \Biggl\{- {1\over2} \sum_{j = 1}^{n+1} \lambda_j \psi(j) \Biggr\} \Biggr].\end{aligned}$$ Hence, thanks to (\[LTH\]), it also satisfies $$\mathbb{E} \Biggl[ \exp \Biggl\{-{1\over2} \sum _{j = 1}^{n}\lambda_j \psi _{\sigma }(j) \Biggr\} \Biggr] = c(\sigma) \mathbb{E} \Biggl[ \exp \Biggl\{- {1\over2} \sum_{j = 1}^{n}( \lambda_j + \sigma) \phi _{\sigma}(j) \Biggr\} \Biggr],$$ where $c(\sigma) = { \mathbb{E}[ \exp\{- {(\sigma/2)} \psi (n+1)\}] \over\mathbb{E}[ \exp\{- {(\sigma/2)} \sum_{i = 1}^{n+1} \psi (i)\}]} $. Similarly, as in Step 1, one shows that the two $n\times n$-matrices $R_{\sigma}$ and $(G_{\sigma}(i,j))_{1 \leq i,j \leq n}$ are effectively equivalent. Note that for every $1 \leq i,j \leq n$: $G_{\sigma}(i,j)G_{\sigma }(j,i) = R_{\sigma}(i,j) R_{\sigma}(j,i)$. For $\sigma$ outside a finite set, $R_{\sigma}$ has no zero entry, and hence neither$(G_{\sigma}(i,j))_{1 \leq i,j \leq n}$. Moreover, we know also that $(\psi_{\sigma}(i))_{1 \leq i \leq n}$ is infinitely divisible. In particular for every triplet of indexes $i$, $j$ and $k$ in $\{ 1,2,\ldots,n\}$, $(\psi_{\sigma}(i), \psi_{\sigma}(j), \psi_{\sigma}(k))$ is infinitely divisible. Consequently, $(G_{\sigma})_{\{i,j,k\}\times\{i,j,k\}}$ is diagonally equivalent to an inverse $M$-matrix. The choice of the index $(n+1)$ being arbitrary, we actually obtain that for $\sigma$ outside of a finite set there exists a function ${S}_{\sigma}$ from $\{1,2,\ldots,n,n+1\}$ into $\{-1, +1\}$ such that for every $i$, $j$ in $\{1,2,\ldots,n,n+1\}$ $$\label{signe1} {S}_{\sigma}(i) G_{\sigma}(i,j) {S}_{\sigma}(j) > 0.$$ For $\sigma$ element of the finite set of excluded values, one shows that (\[signe1\]) is still true exactly as we did it for (\[signe\]) in Step 1. We conclude that for every $\sigma> 0$, $G_{\sigma}$ has no zero entry and is $\beta$-positive definite for every $\beta> 0$. Thanks to Vere-Jones criteria, $\psi$ is infinitely divisible and $(R_{n+1})$ is established. *Step 3*: Assume that $d= 4$ and that $G$ is such that the matrices$(G(i,j))_{i,j \in\{1,2,4\}}$, $(G(i,j))_{i,j \in\{1,3,4\}}$ and $(G(i,j))_{i,j \in\{2,3,4\}}$ are not symmetrizable. We show that $\psi$ is infinitely divisible and that for every $\sigma> 0$, its $\sigma$-resolvent $G_{\sigma}$ has no zero entry. First, note that according Remark \[R0\], the three matrices $(G(i,j))_{i,j \in\{1,2,4\}}$, $(G(i,j))_{i,j \in\{1,3,4\}}$ and $(G(i,j))_{i,j \in\{2,3,4\}}$ have no zero entry. Hence, $G$ has no zero entry. Since these three matrices are all diagonally equivalent to inverse of $M$-matrices, we can then easily establish the existence of $S$ from $\{1,2,3,4\}$ into $\{-1, +1\}$ such that: $S(i) G(i,j) S(j) > 0, \forall i,j \in\{1,2,3,4\}$. We can hence assume that the entries of $G$ are all strictly positive. We now make the notation for $H(\sigma, G)$ more precise, by writing $$H( \sigma, G, 4) = \biggl(G(i,j) - {\sigma\over1 + \sigma G(4,4)}G(i,4)G(4,j) \biggr)_{1 \leq i,j \leq3}.$$ Similarly, for any $k$ in $\{1,2,3,4\}$, $H(\sigma, G, k)$ is defined by $$H( \sigma, G, k) = \biggl(G(i,j) - {\sigma\over1 + \sigma G(k,k)}G(i,k)G(k,j) \biggr)_{ i,j \in\{1,2,3,4\}\setminus\{k\} }.$$ Making use of the argument developed in Step 1, we know that for each of the three matrices $H(\sigma, G, 3)$, $H(\sigma, G, 2)$ and $H(\sigma, G, 1)$, there are at most three distinct values of $\sigma $ for which they are not symmetrizable. Consequently, for $\sigma$ outside of a finite set, the three matrices $(G_{\sigma}(i,j))_{i,j \in\{1,2,4\}}$, $(G_{\sigma}(i,j))_{i,j \in\{1,3,4\}}$ and $(G_{\sigma}(i,j))_{i,j \in\{2,3,4\}}$ have no zero entry and are diagonally equivalent to inverse $M$-matrices. Setting $I_3 = \{1,2,4\}$, $I_2 = \{1,3,4\}$ and $I_1 = \{2,3, 4\}$, we hence know that there exist three functions $S_{3}, S_{2}$ and $S_{1}$ from respectively $I_3$, $I_2$ and $I_1$ into $\{-1, +1\}$ such that for every $p = 1,2$ or $3$, and every couple $(i,j)$ of $I_p$, we have $$S_p(i) G_{\sigma}(i,j) S_p(j) > 0.$$ To determine the sign of $G_{\sigma}(1,2) G_{\sigma}(2,3) G_{\sigma }(3,1)$, note that it has the same sign as $ S_3(1)S_3(2)\cdot S_1(2)S_1(3)\cdot S_2(3)S_2(1)$. But $S_3(1)S_2(1)$ has the same sign as $S_3(4)S_2(4)G_{\sigma}(4,1)^2$; $S_3(2)S_1(2)$ has the same sign as $S_3(4)S_1(4)G_{\sigma}(2,4)^2$ and $S_1(3)S_2(3)$ has the same sign as $S_1(4)S_2(4)G_{\sigma }(4,3)^2$. One obtains $$G_{\sigma}(1,2)\* G_{\sigma}(2,3) G_{\sigma}(3,1) > 0.$$ Consequently, for every $i$, $j$, $k$ in $\{1,2,3,4\}$ we have $$G_{\sigma}(i,j) G_{\sigma}(j,k) G_{\sigma}(k,i) > 0,$$ which leads to the existence of a function $S$ from $\{1,2,3,4\}$ to $\{ -1,+1\} $ such that for every $i$,$j$ in $\{1,2,3,4\}$: $$\label{signe2} S(i) G_{\sigma}(i,j) S(j) > 0.$$ For $\sigma$ element of the finite set of excluded values, one shows that (\[signe2\]) is still true exactly as we did it for (\[signe\]) in Step 1. We conclude that for every $\sigma>0$, $G_{\sigma}$ has no zero entry and that $\psi$ is infinitely divisible. *Step 4*: We assume that $G$ has exactly one symmetrizable principal $3\times3$-submatrix. Denote by $I$ the subset of the corresponding three distinct indexes. We show that $\psi$ is infinitely divisible and that for every $\sigma> 0$, its $\sigma $-resolvent $G_{\sigma}$ has no zero entry. Define the claim $(\tilde{R}_n)$ as follows. $(\tilde{R}_n)$: If $G$ is a $n\times n$-square matrix with exactly one symmetrizable $3\times3$-principal submatrix, then $\psi $ is infinitely divisible and for every $\sigma> 0$, its $\sigma $-resolvent $G_{\sigma}$ has no zero entry. We just established $(\tilde{R}_4)$. Assume that $(\tilde{R}_n)$ is satisfied we show now that $(\tilde {R}_{n+1})$ is satisfied. As in Step 3, one shows that we can assume that the entries of $G$ are strictly positive. Note that for every index $p$ in $\{1,2,\ldots, n+1\}$, $H(\sigma, G, p)$ is the kernel of a $n$-dimensional permanental vector. We still set $\Gamma=\break ({G(i,j) \over G(i,p) G(p, j)})_{1 \leq i,j \leq n+1}$. Fix $J$ subset of three elements of $\{1,2,\ldots, n+1\} \setminus \{p\}$. We look for the values $c$ in $(0, {1\over G(p,p)})$ for which $(\Gamma (i,j) - c)_{i,j \in J\times J}$ is symmetrizable. Unless $J = I$, we know, similarly as in Step 2, that $(\Gamma(i,j) - c)_{i,j \in J\times J}$ has no off-diagonal zero entry. Hence, for $J \neq I$, the only way for $(\Gamma(i,j) - c)_{i,j \in J\times J}$ to be symmetrizable is to be diagonally equivalent to a symmetric matrix. We know that: - either $(\Gamma(i,j) - c)_{i,j \in J\times J}$ is diagonally equivalent to a symmetric matrix for at most three distinct values of $c$, - either $(\Gamma(i,j) - c)_{i,j \in J\times J}$ is diagonally equivalent for every value of $c$. In the later case, one obtains $G_{J\times J}$ is symmetrizable, which implies that $J = I$. Consequently for every $p$, and every $\sigma$ outside of a finite set, $H(\sigma, G,p)$ contains at most one $3\times3$-symmetrizable principal submatrix. If there is none, then Step 2 tells us that the corresponding permanental vector is infinitely divisible and that for every $\alpha> 0$, its $\alpha$-resolvent has no zero entry. If $H(\sigma, G,p)$ has exactly one $3\times3$- symmetrizable principal submatrix, we obtain the same property thanks to $(\tilde{R}_n)$. Making use of the argument developed in Step 2, one shows that $\psi$ is infinitely divisible and for every $\sigma> 0$, its $\sigma $-resolvent has no zero. We have hence obtained $\tilde{R}_{n+1}$. This completes the proof of Theorem \[T1\]. Remarks and examples {#3} ==================== Theorem \[T1\] can be reformulated in terms of linear algebra as follows. *For $d > 3$, let $G$ be a $d\times d$-matrix with no zero entry such that at most one of its $3\times3$-principal submatrices is diagonally equivalent to a symmetric matrix. Assume that*: *all the real eigenvalues of $G$ are nonnegative*, *there exists $\beta> 0$, such that for every $\gamma> 0$, setting $G_{\gamma} = (I + \gamma G)^{-1} G$, $G_{\gamma}$ is $\beta $-positive definite*, *then $G$ is diagonally equivalent to an inverse $M$-matrix*. Assumptions (I) and (II) are necessary to obtain the conclusion. Indeed, consider the following nonsingular matrix borrowed from [@JS]: $$A = \pmatrix{ 1 & 0,10 & 0,40 & 0,30 \cr 0,40 &1 & 0,40 & 0,65 \cr 0,10 & 0,20 & 1 & 0,60 \cr 0,15 & 0,30 & 0,60 & 1 }.$$ It is not an inverse $M$-matrix, since $A^{-1}(2,3)$ is positive. But note that every $3\times3$-principal submatrix is an inverse $M$-matrix and is not symmetrizable. Consequently, $A$ is not the kernel of a permanental vector. The condition required by Theorem \[T1\] to obtain infinite divisibility, is sufficient and not necessary. Indeed, there exist nonsymmetrizable inverse $M$-matrices with more than one symmetrizable $3\times3$-principal submatrix. Here is a family of such matrices with dimension $4$: $$\Gamma= \pmatrix{ \Gamma(1,1) & a & a & \Gamma(4,4) \cr b &\Gamma(2,2) & e & \Gamma(4,4) \cr b & e & \Gamma(3,3) & \Gamma(4,4) \cr \Gamma(4,4) & \Gamma(4,4) & \Gamma(4,4) & \Gamma(4,4) },$$ with $\Gamma(i,i) > e$ for $i = 1,2,3$; $a,b, e > \Gamma(4,4)$ and $e > a,b$. For $a \neq b$, $\Gamma$ is not symmetrizable and has exactly two symmetrizable $3\times3$-principal submatrices, $\Gamma_{{\vert}_{\{1,2,3\}\times\{1,2,3\}}}$ and $\Gamma_{{\vert}_{\{2,3,4\}\times\{2,3,4\}}}$, and two nonsymmetrizable $3\times3$-principal submatrices. Here are now examples of matrices illustrating Theorem \[T1\]. $$\mbox{Set: } K = \pmatrix{ K(1,1) & e & a &K(4,4) \cr b & K(2,2) & a & K(4,4) \cr b & e & K(3,3)& K(4,4) \cr K(4,4)& K(4,4)& K(4,4)& K(4,4) },$$ with $a$, $b$ and $e$ positive: $K(i,i) > K(4,4)$ for $i = 1,2,3$; $K(i,i) > \sup\{a, b, e\}$ for $i = 1,2,3$; $\inf\{a,b,e\} > K(4,4)$. For $a$, $ b$, $e$ distinct, $K$ is not symmetrizable and $K_{\vert _{\{ 1,2,3\}\times\{1,2,3\}}}$ is its unique symmetrizable principal submatrix of order $3$. Moreover, the matrix $K$ is an inverse $M$-matrix. For permanental vectors with symmetrizable kernel, one might think that assuming the infinite divisibility of all its triplets would lead to the infinite divisibility of the vector itself. As it has been noticed in [@EK], the Brownian sheet provides a counter-example. Here is another one found in [@JS]. Indeed the following matrix $B$ is a $4\times4$-covariance matrix of a centered Gaussian vector $(\eta_1, \eta_2, \eta_3, \eta_4)$ such that for every triplet of distinct indexes $i$, $j$, $k$, $(\eta_i^2, \eta_j^2, \eta_k^2)$ is infinitely divisible, but $(\eta_1^2, \eta _2^2, \eta_3^3, \eta_4^3)$ is not infinitely divisible: $$B = \pmatrix{ 1 & 0,50 & 0,35 & 0,40 \cr 0,50 &1& 0,50 & 0,26 \cr 0,35 & 0,50 & 1 & 0,50 \cr 0,40 & 0,26 & 0,50 & 1 }$$ and $B^{-1}(2,4)$ is positive. Let $(G(i,j))_{1 \leq i,j \leq n}$ be the kernel of a permanental vector $\psi$. We assume that $G$ is nonsingular. For $\alpha$ in $[0,1]$, consider now the $2n\times2n$-matrix $H(\alpha )$ defined by $$H(\alpha) = {\left}[\matrix{ G & \alpha G \cr \alpha G & G } {\right}] .$$ The matrix $H(1)$ is the kernel of the vector $(\psi, \psi)$. The matrix $H(0)$ is the kernel of the permanental vector $(\psi, \tilde {\psi})$, where $\tilde{\psi}$ is an independent copy of $\psi$. If $G$ does not contain any symmetrizable $3\times 3$-principal submatrix, then for any $\alpha$ in $(0,1)$, $H(\alpha )$ is not the kernel of a permanental vector. For $x$ complex number, we have $$\begin{aligned} \operatorname{det} \bigl(H(\alpha) - xI \bigr) & = &\operatorname{det} {\left}[\matrix{ G - xI & \alpha G \cr \alpha G & G - xI} {\right}] \\ & =& \bigl{\vert}( G - x I)^2 - \alpha^2 G^2 \bigr{\vert}= \bigl{\vert}(1 + \alpha) G - xI \bigr{\vert}\bigl{\vert}(1- \alpha)G - xI \bigr{\vert},\end{aligned}$$ since $\alpha G$ and $(G - xI)$ commute. Hence, $H(\alpha)$ satisfies the first condition of Vere-Jones criterion of existence of a permanental vector. Moreover, for $\alpha< 1$, $H(\alpha)$ is not singular. By assumption, $G$ has no zero entry (if not it would contain a symmetrizable $3\times3$-principal submatrix) and thanks to Theorem \[T1\], it is hence diagonally equivalent to an inverse $M$-matrix. In particular, there exists a signature matrix $\sigma$ such that the entries of $\sigma G \sigma$ are all strictly positive. Consequently, we can assume that the entries of $H(\alpha)$ are all strictly positive. For $\alpha$ in $(0,1)$, $H(\alpha)$ is not an inverse $M$-matrix. Indeed, write $$H(\alpha) = H = {\left}[\matrix{ H_{11} & H_{12} \cr H_{21} & H_{22} } {\right}] ,$$ with $H_{11}= H_{22} = G$ and $H_{12}= H_{21} = \alpha G$, then $$H^{-1} = {\left}[\matrix{ (H/H_{22})^{-1} & -(H/H_{22})^{-1} H_{12} (H_{22})^{-1} \cr -H_{22}^{-1} H_{21} (H/H_{22})^{-1} & (H/H_{11})^{-1} } {\right}], $$ where $H/H_{11}$ is the Schur complement of $H_{11}$ in $H$ defined by $$H/H_{11} = H_{22} - H_{21} H_{22}^{-1} H_{12},$$ and similarly $H/H_{22}$ is the Schur complement of $H_{22}$ in $H$: $$H/H_{22} = H_{11} - H_{12} H_{22}^{-1} H_{21}.$$ Then, as it has been noticed by Johnson and Smith [@JS], $H$ is an inverse $M$-matrix iff: $H/H_{11}$ is an inverse $M$-matrix, $H/H_{22}$ is an inverse $M$-matrix, $(H_{22})^{-1} H_{21} (H/H_{22})^{-1} $ has nonnegative entries only, $(H/H_{22})^{-1} H_{12} (H_{22})^{-1}$ has nonnegative entries only. For $H = H(\alpha)$ with $\alpha$ in $[0,1)$, this criterion gives the following: \(i) and (ii): $(1 - \alpha^2) G$ is an inverse $M$-matrix. \(iii) and (iv): ${\alpha\over1 - \alpha^2} G^{-1}$ has only nonnegative entries. Hence unless $\alpha= 0$, $H(\alpha)$ is never an inverse $M$-matrix. Now assume that there exists a permanental vector admitting $H(\alpha)$ for kernel. Then we know that this permanental vector is not infinitely divisible. But as soon as $G$ does not contain any $3\times3$-covariance matrix, neither does $H(\alpha)$. Thanks to Theorem \[T1\], a permanental vector that would admit $H(\alpha)$ for kernel should be infinitely divisible. Hence, $H(\alpha)$ can not be the kernel of a permanental vector. Note that for $G$ symmetric positive definite matrix, we know that $H(\alpha)$ is still a covariance matrix. But the corresponding $2n$-dimensional permanental vector is never infinitely divisible, because Conditions (i) and (iii) above are always antagonistic. [10]{} (). . . (). . . (). . . (). . . (). . . (). . . (). . . (). . . (). . .

--- abstract: 'We apply a method of group averaging to states and operators appearing in (truncations of) the $\operatorname{\textrm{Spin}}(9) \times \operatorname{\textrm{SU}}(N)$ invariant matrix models. We find that there is an exact correspondence between the standard supersymmetric Hamiltonian and the $\operatorname{\textrm{Spin}}(9)$ average of a relatively simple lower-dimensional model.' author: - | Jens Hoppe$^a$ [^1]  , Douglas Lundholm$^a$ [^2]\ and Maciej Trzetrzelewski$^{a,b}$ [^3] title: | Spin(9) Average of SU(N) Matrix Models\ I. Hamiltonian --- Introduction ============ Due to its relevance to M-theory, reduced Yang-Mills theory, and membrane theory, considerable effort has been put into investigating the structure of Spin(9) $\times$ SU(N) invariant supersymmetric matrix models (see e.g. [@review] for a review). Despite this, a concrete knowledge of the conjectured zero-energy eigenfunction of the Hamiltonian $H$ is still lacking. In [@Hoppe3] a certain truncation of the Spin(9) invariant model was introduced, based on a coordinate split of $\mathbb{R}^9$ into $\mathbb{R}^7 \times \mathbb{R}^2$. The corresponding Hamiltonian $H_D = \{Q_D,Q_D^\dagger\}$, which is essentially just a set of supersymmetric harmonic oscillators, can be interpreted as a two-dimensional supersymmetric SU(N) matrix model with a seven-dimensional space of parameters. Recently, a deformation of the standard matrix model – based on the same coordinate split – was considered that produces a $G_2 \times U(1)$ invariant supersymmetric Hamiltonian $\tilde{H}$ in which $H_D$ plays a central role [@octonionic]. The explicit knowledge of the structure of $H_D$ and its eigenfunctions made it possible to prove in a straightforward manner that $\tilde{H}$ and $H$ have similar spectra. In this paper we calculate the Spin(9) average of the truncated Hamiltonian $H_D$ and find that it is essentially equal to the full supersymmetric Hamiltonian $H$. The correspondence is made exact by a slight modification of $H_D$. Motivated by this result, we also expect that the *wavefunctions* obtained by averaging the eigenfunctions of $H_D$ (or slight modifications of those) could be related to the Spin(9) invariant eigenfunctions of $H$. Calculating the average of such eigenstates, however, is a technically more difficult problem, to be addressed in a forthcoming paper [@avg_state]. Group averaging =============== First, let us define what we mean by group averaging (the notion is well-known in the literature; see e.g. [@Thiemann] and references therein for a general approach and various applications). Assume that we are given a unitary representation $U(g)$ of a compact Lie group $G$ acting on a complex separable Hilbert space $\mathcal{H}$. Then, given any state $\Psi \in \mathcal{H}$ and linear operator $A$ acting on $\mathcal{H}$, we define the corresponding $G$-averaged state $[\Psi\rangle_G$ resp. operator $[A]_G$ by $$[\Psi\rangle_G := \int_{g \in G} U(g)\Psi \ d\mu(g)$$ resp. $$[A]_G := \int_{g \in G} U(g)AU(g)^{-1} \ d\mu(g),$$ where $\mu$ denotes the unique *normalized* left- and right-invariant (Haar) measure on $G$. Due to the translation invariance of $\mu$, $[\Psi\rangle_G$ will be invariant under the action of $U(g)$, and $[A]_G$ will commute with $U(g)$. One can also extend the above definition to generalized (non-normalizable) states, e.g. Schwartz distributions $\psi \in \mathcal{D}'$, by taking[^4] $$\langle\psi]_G(\phi) := \int_{g \in G} \psi \left( U(g)\phi \right) \ d\mu(g)$$ for any test function $\phi \in \mathcal{D} = C_0^\infty(\Omega)$. The model and its group actions =============================== We are interested in the supersymmetric matrix model described by the Hilbert space $$\mathcal{H} = L^2(\mathbb{R}^{9n}) \otimes \mathcal{F}, \qquad \mathcal{F} = \mathop{\otimes}_{A=1}^{n} \mathcal{F}^{(A)} = \mathbb{C}^{2^{8n}}$$ and the Hamiltonian $$H = p_{sA}p_{sA} + \frac{1}{2}(f_{ABC}x_{sB}x_{tC})^2 + \frac{i}{2} x_{sC} f_{ABC} \gamma^s_{\alpha \beta} \operatorname{\boldsymbol{\theta}}_{\alpha A}\operatorname{\boldsymbol{\theta}}_{\beta B} = -\Delta + V + H_F,$$ where we sum over corresponding indices $s,t,\ldots=1,\ldots,9$, $A,B,\ldots=1,\ldots,n := N^2-1$, $\alpha,\beta,\ldots=1,\ldots,16$. $\gamma^s$ generate (a matrix representation of) the Clifford algebra over $\mathbb{R}^9$ acting irreducibly on $\mathbb{R}^{16}$, while $\operatorname{\boldsymbol{\theta}}_{\alpha A}$ generate the Clifford algebra over $\mathbb{R}^{16} \otimes \mathbb{R}^n$, i.e. $\{ \operatorname{\boldsymbol{\theta}}_{\alpha A} , \operatorname{\boldsymbol{\theta}}_{\beta B} \} = 2\delta_{\alpha\beta}\delta_{AB}$, acting irreducibly on $\mathcal{F}$. The coordinates $x_{sA}$, canonically conjugate to $p_{sA} = -i\partial_{sA}$, comprise a set of 9 traceless hermitian matrices $(X_1,\ldots,X_9) = \boldsymbol{X} \in \mathbb{R}^9 \otimes \mathbb{R}^n$, and we use the isomorphism $i \cdot \mathfrak{su}(N) \cong \mathbb{R}^n$ to map seamlessly between such a matrix $E$ and its coordinate representation $e_A$ in a basis where the $\operatorname{\textrm{SU}}(N)$ structure constants $f_{ABC}$ are totally antisymmetric. $H$ is invariant under the action of $\operatorname{\textrm{SU}}(N)$, where the corresponding representation on $\mathcal{H}$ is generated by the anti-hermitian operators $$\begin{aligned} \tilde{J}_A &=& iJ_A = if_{ABC}\Big( x_{sB}p_{sC} - \frac{i}{4}\operatorname{\boldsymbol{\theta}}_{\alpha B}\operatorname{\boldsymbol{\theta}}_{\alpha C} \Big) \\ &=& \sum_{B<C} f_{ABC} \Big( x_{sB}\partial_{sC} - x_{sC}\partial_{sB} + \frac{1}{2} \operatorname{\boldsymbol{\theta}}_{\alpha B}\operatorname{\boldsymbol{\theta}}_{\alpha C} \Big) = \tilde{L}_A + \tilde{M}_A,\end{aligned}$$ with $\tilde{L}_A$ and $\tilde{M}_A$ generating the representation of $\mathfrak{su}(N) \hookrightarrow \mathfrak{so}(n)$ on $L^2(\mathbb{R}^{9n})$ and $\mathcal{F}$, respectively. Furthermore, $H$ is also invariant under $\operatorname{\textrm{Spin}}(9)$, generated by $$\tilde{J}_{st} = iJ_{st} = i\Big( x_{sA}p_{tA} - x_{tA}p_{sA} - \frac{i}{8} \gamma^{st}_{\alpha\beta} \operatorname{\boldsymbol{\theta}}_{\alpha A}\operatorname{\boldsymbol{\theta}}_{\beta A} \Big) = \tilde{L}_{st} + \tilde{M}_{st},$$ with $$\tilde{L}_{st} = \sum_A \tilde{L}^{(A)}_{st} = x_{sA}\partial_{tA} - x_{tA}\partial_{sA}$$ and $$\tilde{M}_{st} = \sum_A \tilde{M}^{(A)}_{st} = \sum_{\alpha<\beta} \frac{1}{2} [{\textstyle \frac{1}{2}\gamma^{st}}]_{\alpha\beta} \operatorname{\boldsymbol{\theta}}_{\alpha A}\operatorname{\boldsymbol{\theta}}_{\beta A}.$$ Note that the spinor representation of $\operatorname{\textrm{Spin}}(9)$ is generated by $\frac{1}{2}\gamma^{st} := \frac{1}{4}[\gamma^{s},\gamma^{t}]$ acting by left multiplication on the Clifford algebra generated by the $\gamma$’s, i.e. left multiplication by the matrix $[\frac{1}{2}\gamma^{st}] \in \mathfrak{so}(16)$ acting on the spinor space $\mathbb{R}^{16}$. This action is in turn represented on the Fock space $\mathcal{F}^{(A)} = \mathbb{C}^{2^8}$ by the spinor representation of $\mathfrak{spin}(16) = \frac{1}{2} \cdot \mathfrak{so}(16)$. The full, exponentiated, action of $g = e^{\epsilon_{st}\frac{1}{2}\gamma^{st}} \in \operatorname{\textrm{Spin}}(9)$ on a state $\Psi \in \mathcal{H}$, i.e. a wavefunction $\Psi: \mathbb{R}^9 \otimes \mathbb{R}^n \to \mathcal{F}$, is then given by $$( U(g) \Psi )(\boldsymbol{X}) = ( e^{\epsilon_{st} \tilde{J}_{st}} \Psi )(\boldsymbol{X}) = e^{\epsilon_{st} \tilde{M}_{st}} \Psi(e^{\epsilon_{st} \tilde{L}_{st}} \boldsymbol{X}) = \tilde{R}_g \Psi(R_{g^{-1}}(\boldsymbol{X})),$$ where $\tilde{R}_g := e^{\epsilon_{st}\frac{1}{8}\gamma^{st}_{\alpha\beta} \operatorname{\boldsymbol{\theta}}_{\alpha A}\operatorname{\boldsymbol{\theta}}_{\beta A}}$ is the unitary representative of $g$ acting on $\mathcal{F}$, and $R_g(\boldsymbol{x}) := g\boldsymbol{x}g^{-1}$ is the corresponding rotation $R_g \in \operatorname{\textrm{SO}}(9)$ acting on vectors $\boldsymbol{x} = x_t\gamma_t \in \mathbb{R}^9$ considered as grade-1 elements of the Clifford algebra. This follows by considering the infinitesimal action on a function $f: \mathbb{R}^9 \to \mathbb{R}$, i.e. $(\tilde{L}_{st}f)(\boldsymbol{x}) = \operatorname{grad}f \cdot \tilde{L}_{st}(\boldsymbol{x})$, and (using $\boldsymbol{x} \cdot \boldsymbol{y} = \frac{1}{2}\{\boldsymbol{x},\boldsymbol{y}\}$) $$\begin{aligned} \tilde{L}_{st}\boldsymbol{x} &=& (x_s\partial_t - x_t\partial_s)(x_u\gamma_u) = x_s\gamma_t - x_t\gamma_s \\ &=& (\boldsymbol{x} \cdot \gamma_s) \gamma_t - (\boldsymbol{x} \cdot \gamma_t) \gamma_s = - \frac{1}{2}[\gamma^{st},\boldsymbol{x}].\end{aligned}$$ Consider now an operator of the form $$\mathcal{B} = [B(\boldsymbol{X})]_{\alpha\beta} \operatorname{\boldsymbol{\theta}}_{\alpha A} \wedge \operatorname{\boldsymbol{\theta}}_{\beta B}$$ where $B(\boldsymbol{X})$ is a symmetric $16 \times 16$ matrix and $A \wedge B := \frac{1}{2}[A,B]$. The infinitesimal action is $$[\tilde{J}_{st} , \mathcal{B}] = \left[ [\tilde{L}_{st}, B(\boldsymbol{X})] \right]_{\alpha\beta} \operatorname{\boldsymbol{\theta}}_{\alpha A} \wedge \operatorname{\boldsymbol{\theta}}_{\beta B} \ +\ [B(\boldsymbol{X})]_{\alpha\beta} [\tilde{M}_{st}, \operatorname{\boldsymbol{\theta}}_{\alpha A} \wedge \operatorname{\boldsymbol{\theta}}_{\beta B}],$$ which, using $$\begin{aligned} [\tilde{M}_{st}, \operatorname{\boldsymbol{\theta}}_{\alpha A} \wedge \operatorname{\boldsymbol{\theta}}_{\beta B}] &=& \frac{1}{8} \gamma^{st}_{\alpha'\beta'} [\operatorname{\boldsymbol{\theta}}_{\alpha'C} \wedge \operatorname{\boldsymbol{\theta}}_{\beta'C}, \operatorname{\boldsymbol{\theta}}_{\alpha A} \wedge \operatorname{\boldsymbol{\theta}}_{\beta B}] \\ &=& \frac{1}{2} (\gamma^{st}_{\epsilon \alpha} \operatorname{\boldsymbol{\theta}}_{\epsilon A} \wedge \operatorname{\boldsymbol{\theta}}_{\beta B} - \gamma^{st}_{\beta \epsilon} \operatorname{\boldsymbol{\theta}}_{\alpha A} \wedge \operatorname{\boldsymbol{\theta}}_{\epsilon B}),\end{aligned}$$ exponentiates to $$\begin{aligned} e^{\epsilon_{st} \tilde{J}_{st}} \mathcal{B} e^{-\epsilon_{st} \tilde{J}_{st}} &=& \left[ e^{\frac{1}{2}\epsilon_{st}\gamma^{st}} B(e^{-\frac{1}{2}\epsilon_{st}\gamma^{st}} \boldsymbol{X} e^{\frac{1}{2}\epsilon_{st}\gamma^{st}}) e^{-\frac{1}{2}\epsilon_{st}\gamma^{st}} \right]_{\alpha\beta} \operatorname{\boldsymbol{\theta}}_{\alpha A} \wedge \operatorname{\boldsymbol{\theta}}_{\beta B} \nonumber \\ &=& [ gB(R_g^{\operatorname{\textrm{T}}}(\boldsymbol{X}))g^{-1} ]_{\alpha\beta} \operatorname{\boldsymbol{\theta}}_{\alpha A} \wedge \operatorname{\boldsymbol{\theta}}_{\beta B}. \label{operator_action}\end{aligned}$$ Regarding the supersymmetry of $H$, it is for the following sufficient to know that there is a set of hermitian supercharge operators $\mathcal{Q}_\alpha$ such that $H = \mathcal{Q}_\alpha^2$ on the subspace of $\operatorname{\textrm{SU}}(N)$ invariant states, $\mathcal{H}_{\textrm{phys}}$, which is the physical Hilbert space of the theory. In order to arrive at a conventional Fock space formulation of the model it is necessary to make certain choices which break the explicit $\operatorname{\textrm{Spin}}(9)$ symmetry. After introducing a split of the coordinates into $(x',z)$, with $x'= (x_{j=1,\ldots,7}) \in \mathbb{R}^{7n}$, $z_A := x_{8A} + ix_{9A}$, and a representation of $\operatorname{\boldsymbol{\theta}}_{\alpha A}$ in terms of creation and annihilation operators $\lambda, \lambda^\dagger$, together with a suitable representation of $\gamma^s$ (see e.g. Appendix A of [@octonionic]), it is rather natural to single out a certain part of the supercharges, resulting in a truncation of $H$ to the Hamiltonian [@Hoppe3; @octonionic] $$\begin{aligned} H_D &=& -4 \bar{\partial}_z \cdot \partial_z + \bar{z} \cdot S(x') z + 2W(x')\lambda \lambda^\dagger = -\Delta_{89} + V_D + W_D,\end{aligned}$$ where each of these terms will be explained in the next section. This operator constitutes a set of $2n$ supersymmetric harmonic oscillators in $x_{8A}$ and $x_{9A}$ whose frequencies are the square root of the eigenvalues of the positive semidefinite matrix operator $S(x') = \sum_{j=1}^7 \operatorname{ad}_{X_j} \circ \operatorname{ad}_{X_j}$. Thus, $H_D$ can be considered as acting on a smaller Hilbert space over the [z]{}-coordinates, $$\mathpzc{h} = L^2(\mathbb{R}^{2n}) \otimes \mathcal{F},$$ with $x_j$ entering as parameters, and has with respect to $\mathpzc{h}$ the complete basis of eigenstates $$\label{eigenstates} \psi_{k,\sigma}(x',z) = \pi^{-\frac{n}{2}} (\det S(x'))^{\frac{1}{4}} H_k(x',z) e^{-\frac{1}{2}\bar{z}\cdot S(x')^{1/2} z} \xi_{x'}^\sigma.$$ $H_k(x',z)$ denote products of normalized Hermite polynomials in $S(x')^{\frac{1}{4}}x_{8}$ and $S(x')^{\frac{1}{4}}x_{9}$, while $\xi_{x'}^\sigma \in \mathcal{F}$, $\sigma \in \{0,1\}^{8n}$, form the basis of eigenvectors of $W_D$ (see [@Hoppe3; @octonionic] for details). As pointed out in [@octonionic], both $H_D$ and its nondegenerate eigenstates are SU(N) invariant (covariant) in the sense that they are unchanged under the simultaneous action of $\operatorname{\textrm{SU}}(N)$ on $\mathpzc{h}$ and the parameters $x_j$. The averaged Hamiltonian ======================== We would like to apply group averaging w.r.t. $G = \operatorname{\textrm{Spin}}(9)$ to the truncated Hamiltonian $H_D$ and its $\mathpzc{h}$-eigenstates (which are generalized states w.r.t. the full Hilbert space $\mathcal{H}$). Note that averaging the supercharge $Q_D$ corresponding to $H_D$ gives zero in the same way that, for the supercharges $\mathcal{Q}_\alpha$ corresponding to $H$ and transforming like spinors, $[\mathcal{Q}_\alpha]_G = [g_{\beta\alpha}\mathcal{Q}_\beta]_G = 0$, taking $g=-1$. Laplacian part -------------- The principal part of $H_D$ is the Laplace operator on $\mathbb{R}^{2} \otimes \mathbb{R}^n$, $$\Delta_{89} = \Delta_8 + \Delta_9 = \partial_{8A}\partial_{8A} + \partial_{9A}\partial_{9A}.$$ In order to average this operator, consider first $x_1^2 = (\boldsymbol{x} \cdot \gamma_1)^2$ in $\mathbb{R}^d$, for which $$\begin{aligned} \lefteqn{ [x_1^2]_{\operatorname{\textrm{Spin}}(d)} = \int_{g \in \operatorname{\textrm{Spin}}(d)} U(g) (\boldsymbol{x} \cdot \gamma_1)^2 U(g)^{-1}\ d\mu(g) } \\ &&= \int (R^{\operatorname{\textrm{T}}}_g(\boldsymbol{x}) \cdot \gamma_1)^2\ d\mu(g) = \frac{1}{d} \sum_{j=1}^d \int (\boldsymbol{x} \cdot R_{gh_j}(\gamma_1))^2\ d\mu(g) \\ &&= \frac{1}{d} \int \sum_{j=1}^d (\boldsymbol{x} \cdot R_{g}(\gamma_j))^2\ d\mu(g) = \frac{1}{d} \left[ |\boldsymbol{x}|^2 \right]_{\operatorname{\textrm{Spin}}(d)} = \frac{1}{d} |\boldsymbol{x}|^2,\end{aligned}$$ where we used the invariance of $\mu$ to insert $h_j \in \operatorname{\textrm{Spin}}(d)$ s.t. $R_{h_j}(\gamma_1) = \gamma_j$. Analogously, one finds $[\partial_1^2]_{\operatorname{\textrm{Spin}}(d)} = \frac{1}{d} \Delta_{\mathbb{R}^d}$. Hence, $$[\Delta_{89}]_{\operatorname{\textrm{Spin}}(9)} = \frac{2}{9} \Delta_{\mathbb{R}^{9n}}.$$ Potential part -------------- Denoting the norm in $i \cdot \mathfrak{su}(N)$ by $\|\cdot\|$, so that for such a matrix $E \leftrightarrow e_A$, $\|E\|^2 = e_Ae_A$, we have $$V_D = \bar{z}_A S(x')_{AA'} z_{A'} = \bar{z}_A f_{ABC}x_{jB} f_{A'B'C}x_{jB'} z_{A'} = \sum_{\genfrac{}{}{0pt}{}{a=8,9}{j=1,\ldots,7}} \|[X_a, X_j]\|^2.$$ Using that any pair $(\gamma_a,\gamma_j)$ of orthonormal vectors can be rotated into any other orthonormal pair $(\gamma_s,\gamma_t) = (R_h(\gamma_a),R_h(\gamma_j))$ by some $R_{h}$, $h \in \operatorname{\textrm{Spin}}(9)$, we find $$\begin{aligned} \lefteqn{ \left[ \|[X_a,X_j]\|^2 \right]_G = \int_{G} U(g) \|[\boldsymbol{X} \cdot \gamma_a, \boldsymbol{X} \cdot \gamma_j]\|^2 U(g)^{-1}\ d\mu(g) }\\ &&= \int \|[R^{\operatorname{\textrm{T}}}_g(\boldsymbol{X}) \cdot \gamma_a, R^{\operatorname{\textrm{T}}}_g(\boldsymbol{X}) \cdot \gamma_j]\|^2 \ d\mu(g) \\ &&= \int \|[\boldsymbol{X} \cdot R_{gh}(\gamma_a), \boldsymbol{X} \cdot R_{gh}(\gamma_j)]\|^2 \ d\mu(g) \\ &&= \int \|[\boldsymbol{X} \cdot R_g(\gamma_s), \boldsymbol{X} \cdot R_g(\gamma_t)]\|^2 \ d\mu(g) = \left[ \|[X_s,X_t]\|^2 \right]_G.\end{aligned}$$ Therefore, $$\begin{aligned} \left[ V_D \right]_G = \sum_{\genfrac{}{}{0pt}{}{a=8,9}{j=1,\ldots,7}} \left[ \|[X_a,X_j]\|^2 \right]_G = \frac{14}{36} \sum_{s<t} \left[ \|[X_s,X_t]\|^2 \right]_G = \frac{7}{18} [V]_G = \frac{7}{18} V.\end{aligned}$$ Fermionic part -------------- The fermionic part of $H_D$, given in terms of Fock space operators $\lambda_{\alpha' A} := \frac{1}{2}(\operatorname{\boldsymbol{\theta}}_{\alpha' A} + i\operatorname{\boldsymbol{\theta}}_{8+\alpha' \thinspace A})$, $\alpha' = 1,\ldots,8$, is [@Hoppe3] $$W_D = 2x_{jC}f_{CAB}( \delta_{\alpha' 8}\delta_{\beta' j} - \delta_{\alpha' j}\delta_{\beta' 8} )\lambda_{\alpha' A}\lambda_{\beta' B}^{\dagger}.$$ With our choice of representation of the $\gamma$ matrices (see Appendix A of [@octonionic]), we find $$W_D = i \sum_{\rho=8,16} x_{jC} f_{CAB} \gamma^j_{\rho\beta} \operatorname{\boldsymbol{\theta}}_{\rho A} \operatorname{\boldsymbol{\theta}}_{\beta B} = i x_{jC} f_{CAB} [P\gamma^j]_{\alpha\beta} \operatorname{\boldsymbol{\theta}}_{\alpha A} \operatorname{\boldsymbol{\theta}}_{\beta B},$$ where $P$ is a projection matrix s.t. $P_{8,8} = P_{16,16} = 1$ and zero otherwise. Furthermore, one can verify that $P$ can be written as a product of three commuting projectors of the form $\frac{1}{2}(1 \pm E_\mu)$, $E_\mu^2=1$, in the Clifford algebra: $$\label{P} P = \frac{1}{8}(1 - \gamma_1\gamma_2\gamma_3 I_7)(1 - \gamma_2\gamma_5\gamma_7 I_7)(1 - \gamma_3\gamma_6\gamma_7 I_7) = \frac{1}{8}(1 - CI_7),$$ where $I_7 := \gamma_1\gamma_2\gamma_3\gamma_4\gamma_5\gamma_6\gamma_7$, and $$C := \gamma^{123} + \gamma^{165} + \gamma^{246} + \gamma^{435} + \gamma^{147} + \gamma^{367} + \gamma^{257}$$ defines an octonionic structure. By choosing different signs for $E_\mu$ in the three projectors one obtains all $8 = 2^3$ projection matrices of that form. Also note that $\gamma_1$, $\gamma_5$, and $\gamma_6$ share a particular property in the expression . The action yields $$\begin{aligned} \lefteqn{ [W_D]_G = \int_{G} i (R^{\operatorname{\textrm{T}}}_g(\boldsymbol{X}) \cdot \gamma_j)_C f_{CAB} [gP\gamma^jg^{-1}]_{\alpha\beta} \operatorname{\boldsymbol{\theta}}_{\alpha A} \operatorname{\boldsymbol{\theta}}_{\beta B}\ d\mu(g) }\\ &&= \frac{1}{8} \sum_{p=1}^8 \int_{G} i (\boldsymbol{X} \cdot R_{gh_p}(\gamma_j))_C f_{CAB} [gh_pPh_p^{-1}g^{-1}R_{gh_p}(\gamma^j)]_{\alpha\beta} \operatorname{\boldsymbol{\theta}}_{\alpha A} \operatorname{\boldsymbol{\theta}}_{\beta B}\ d\mu(g),\end{aligned}$$ where we insert 8 different $h_p \in \operatorname{\textrm{Spin}}(7)$ such that $R_{h_p}(\gamma_j) = \sigma_{p,j}\gamma_j \ \forall j$, and $\sigma_{p,j} \in \{+,-\}$ are signs chosen so that $\sum_p h_pPh_p^{-1} = 1$, e.g. according to the following table: $$\begin{array}{c|ccccccc} p & \sigma_{p,1} & \sigma_{p,2} & \sigma_{p,3} & \sigma_{p,4} & \sigma_{p,5} & \sigma_{p,6} & \sigma_{p,7} \\ \hline 1 & + & + & + & + & + & + & + \\[-5pt] 2 & + & + & + & - & + & - & + \\[-5pt] 3 & + & + & + & - & - & + & + \\[-5pt] 4 & + & + & + & + & - & - & + \\[-5pt] 5 & - & + & + & - & + & + & + \\[-5pt] 6 & - & + & + & + & + & - & + \\[-5pt] 7 & - & + & + & + & - & + & + \\[-5pt] 8 & - & + & + & - & - & - & + \end{array}$$ This is possible with $h_p \in \operatorname{\textrm{Spin}}(7)$ (and not only Pin(7)) because $\gamma_4$ does not appear explicitly in except in $I_7$, which with the choice of signs above is invariant, i.e. $h_p I_7 h_p^{-1} = R_{h_p}(\gamma_1) R_{h_p}(\gamma_2) \ldots R_{h_p}(\gamma_7) = I_7$. Hence, $$\begin{aligned} \lefteqn{ [W_D]_G = }\\ &&= \frac{1}{8} \sum_{j=1}^7 \int_{G} i (\boldsymbol{X} \cdot R_g(\gamma_j))_C f_{CAB} \left[ g \left({\textstyle \sum_p \sigma_{p,j}^2 h_pPh_p^{-1}}\right) g^{-1}R_g(\gamma^j) \right]_{\alpha\beta}\! \operatorname{\boldsymbol{\theta}}_{\alpha A} \operatorname{\boldsymbol{\theta}}_{\beta B}\thinspace d\mu(g) \\ &&= \frac{1}{8} \sum_{j=1}^7 \int_{G} i (\boldsymbol{X} \cdot R_{gh'_j}(\gamma_j))_C f_{CAB} [R_{gh'_j}(\gamma^j)]_{\alpha\beta} \operatorname{\boldsymbol{\theta}}_{\alpha A} \operatorname{\boldsymbol{\theta}}_{\beta B}\ d\mu(g) \\ &&= \frac{1}{4} \frac{7}{9} [H_F]_G,\end{aligned}$$ again using some appropriately chosen $h_j' \in \operatorname{\textrm{Spin}}(9)$. Result ====== In total, we have $$[H_D]_G = [-\Delta_{89}]_G + [V_{D}]_G + [W_{D}]_G = -\frac{2}{9} \Delta_{\mathbb{R}^{9(N^2-1)}} + \frac{7}{2 \cdot 9} V + \frac{7}{4 \cdot 9} H_F.$$ The relative coefficients of the terms of the resulting operator do not match those of $H$. In fact, $[H_D]_G$ has a discrete spectrum on $\mathcal{H}_\textrm{phys}$ (contrary to $H$ whose spectrum covers the whole positive axis [@dWLN]). This can be seen by rescaling the coordinates by $(\sqrt{7}/2)^{1/3}$, obtaining up to a constant $$[H_D]_G \sim -\Delta + V + \kappa H_F = (1-\kappa)(-\Delta + V) + \kappa H \ge (1-\kappa)(-\Delta + V),$$ with $\kappa = \sqrt{7}/4 < 1$. The observation follows since $H$ is a positive operator (by supersymmetry) and $-\Delta + V$ has a purely discrete spectrum [@Simon-Luscher]. On the other hand, we can of course define a rescaled operator $$H_D' := -\frac{9}{2}\Delta_{89} + \frac{2 \cdot 9}{7}V_D + \frac{4 \cdot 9}{7}W_D$$ for which the average then is $[H_D']_{\operatorname{\textrm{Spin}}(9)} = H$. Unlike $H_D$ which is positive due to supersymmetry, $H_D'$ has energies tending to negative infinity in certain regions of the $x'$ parameter space (note that its $\mathpzc{h}$-eigenstates are still given by , but with a rescaled frequency $S$). However, considering the action on $\operatorname{\textrm{Spin}}(9) \times \operatorname{\textrm{SU}}(N)$ invariant states $\Psi = U(g)\Psi$, we have $$\begin{aligned} \lefteqn{ \langle \Psi, H_D' \Psi \rangle = \int \langle U(g^{-1})\Psi, H_D' U(g^{-1})\Psi \rangle \thinspace d\mu(g) } \nonumber \\ &&= \left\langle \Psi, \int U(g) H_D' U(g)^{-1} d\mu(g) \Psi \right\rangle = \langle \Psi, [H_D'] \Psi \rangle \nonumber \\ &&= \langle \Psi, H \Psi \rangle = \|\mathcal{Q}_\alpha \Psi\|^2 \geq 0. \label{quad_form_rel}\end{aligned}$$ Hence, we conclude that these quadratic forms coincide on the subspace $\mathcal{H}_\textrm{inv}$ of invariant states, so that $H_D'$ and $H$ are actually the same operator on that subspace[^5]. Furthermore, because a zero-energy state of $H$ must be $\operatorname{\textrm{Spin}}(9)$ invariant [@Hasler-Hoppe] it is therefore sufficient to check that it is annihilated by $H_D'$, i.e. that $$\left( -7\Delta_{89} + 4\bar{z} \cdot S(x') z + 16W(x')\lambda \lambda^\dagger \right)\Psi(x',z) = 0 \quad \forall x'.$$ Also note that the same holds for any linear combination, $(\alpha H + \beta H_D')\Psi = 0$. ### Acknowledgements {#acknowledgements .unnumbered} We would like to thank V. Bach, M. Björklund and J.-B. Zuber for discussions, as well as the Swedish Research Council and the Marie Curie Training Network ENIGMA (contract MRNT-CT-2004-5652) for financial support. [99]{} W. Taylor, *M(atrix) Theory: Matrix Quantum Mechanics as a Fundamental Theory,* Rev. Mod. Phys. 73 (2001) 419-462. J. Hoppe, *On the construction of zero energy states in supersymmetric matrix models III,* [arXiv:hep-th/9711033]{}. J. Hoppe, D. Lundholm, M. Trzetrzelewski, *Octonionic twists for supermembrane matrix models,* [arXiv:0803.1316]{}. J. Hoppe, D. Lundholm, M. Trzetrzelewski, *Spin(9) Average of SU(N) Matrix Models II. Eigenstates,* in preparation. T. Thiemann, *Modern canonical quantum general relativity,* Cambridge University Press, 2007. B. de Wit, W. Lüscher, H. Nicolai, *The supermembrane is unstable,* Nucl. Phys. B320 (1989) 135-159, A. Smilga, *Super Yang Mills quantum mechanics and supermembrane spectrum*, [*Proc. 1989 Trieste Conf.*]{} ed M. Duff, C. Pope and E. Sezgin (Singapore: World Scientific). B. Simon, *Some Quantum Mechanical Operators with Discrete Spectrum but Classically Continuous Spectrum,* Ann. Phys. 146 (1983) 209-220, M. Lüscher, *Some analytic results concerning the mass spectrum of Yang-Mills gauge theories on a torus,* Nucl. Phys. B219 (1983) 233-261. D. Hasler, J. Hoppe, *Zero Energy States of Reduced Super Yang-Mills Theories in $d+1 = 4,6$ and 10 dimensions are necessarily $Spin(d)$ invariant,* [arXiv:hep-th/0211226]{}. [^1]: e-mail: [email protected] [^2]: e-mail: [email protected] [^3]: e-mail: [email protected] [^4]: Note that the action of $G$ on distributions is given by $(U(g)\psi)(\phi) = \psi(U(g^{-1})\phi)$. [^5]: The reader who is worried about the unboundedness of the operators $H$ and $H_D'$ may consider the dense subspace $\mathcal{H}_\textrm{inv} \cap C^\infty_0$, where makes perfect sense, and then conclude that the Friedrichs extensions of $H_D'$ and $H$ on $\mathcal{H}_\textrm{inv}$ are equal.

--- abstract: 'Let $A$ and $B$ be unital $C^*$-algebras and let $H$ be a finite dimensional $C^*$-Hopf algebra. Let $H^0$ be its dual $C^*$-Hopf algebra. Let $(\rho, u)$ and $(\sigma, v)$ be twisted coactions of $H^0$ on $A$ and $B$, respectively. In this paper, we shall show the following theorem: We suppose that the unital inclusions $A\subset A\rtimes_{\rho, u}H$ and $B\subset B\rtimes_{\sigma, v}H$ are strongly Morita equivalent. If $A''\cap (A\rtimes_{\rho, u}H)={\mathbf C}1$, then there is a $C^*$-Hopf algebra automorphism $\lambda^0$ of $H^0$ such that the twisted coaction $(\rho, u)$ is strongly Morita equivalent to the twisted coaction $(({{\rm{id}}}_B \otimes\lambda^0 )\circ\sigma \, , \, ({{\rm{id}}}_B \otimes\lambda^0 \otimes\lambda^0 )(v))$ induced by $(\sigma, v)$ and $\lambda^0$.' address: - 'Department of Mathematical Sciences, Faculty of Science, Ryukyu University, Nishihara-cho, Okinawa, 903-0213, Japan' - 'Faculty of Education, Gunma University, 4-2 Aramaki-machi, Maebashi City, Gunma, 371-8510, Japan' - '*[E-mail address]{}: [[email protected]]{}*' - '*[E-mail address]{}: [[email protected]]{}*' author: - Kazunori Kodaka and Tamotsu Teruya title: 'Coactions of a finite dimensional $C^*$-Hopf algebra on unital $C^*$-algebras, unital inclusions of unital $C^*$-algebras and the strong Morita equivalence' --- Introduction {#sec:intro} ============ In the previous papers [@KT3:equivalence] and [@KT4:morita], we discussed the strong Morita equivalences for twisted coactions of a finite dimensional $C^*$-Hopf algebra on unital $C^*$-algebras and unital inclusions of unital $C^*$-algebras. In this paper, we shall discuss the relation between the strong Morita equivalence for twisted coactions of a finite dimensional $C^*$-Hopf algebra on unital $C^*$-algebras and the strong Morita equivalence for the unital inclusions of the unital $C^*$-algebras induced by the twisted coactions of the finite dimensional $C^*$-Hopf algebra on the unital $C^*$-algebras. Let us explain the problem in detail. Let $A$ and $B$ be unital $C^*$-algebras. Let $H$ be a finite dimensional $C^*$-Hopf algebra and $H^0$ its dual $C^*$-Hopf algebra. Let $(\rho, u)$ and $(\sigma, v)$ be twisted coactions of $H^0$ on $A$ and $B$, respectively. Then we can obtain the unital inclusions of unital $C^*$-algebras $A\subset A\rtimes_{\rho, u}H$ and $B\subset B\rtimes_{\sigma, v}H$. In the same way as in [@KT4:morita Example], we can see that if $(\rho, u)$ and $(\sigma ,v)$ are strongly Morita equivalent, then the unital nclusions $A\subset A\rtimes_{\rho, u}H$ and $B\subset B\rtimes_{\sigma, v}H$ are strongly Morita equivalent. In this paper, we shall discuss the inverse implication. Our main theorem is as follows: We suppose that the unital inclusions $A\subset A\rtimes_{\rho, u}H$ and $B\subset B\rtimes_{\sigma, v}H$ are strongly Morita equivalent in the sense of [@KT4:morita Definition 2.1]. If $A'\cap (A\rtimes_{\rho, u}H)={\mathbf C}1$, then there is a $C^*$-Hopf algebra automorphism $\lambda^0$ of $H^0$ such that the twisted coaction $(\rho, u)$ is strongly Morita equivalent to the twisted coaction $(({{\rm{id}}}_B \otimes\lambda^0 )\circ\sigma \, , \, ({{\rm{id}}}_B \otimes\lambda^0 \otimes\lambda^0 )(v))$ induced by $(\sigma, v)$ and $\lambda^0$. For a unital $C^*$-algebra $A$, let $M_n (A)$ be the $n\times n$-matrix algebra over $A$ and $I_n$ denotes the unit element in $M_n (A)$. We identify $M_n (A)$ with $A\otimes M_n ({\mathbf C})$. Let $A$ and $B$ be $C^*$-algebras and $X$ an $A-B$-bimodule. We denote its left $A$-action and right $B$-action on $X$ by $a\cdot x$ and $x\cdot b$ for any $a\in A$, $b\in B$, $x\in X$, respectively. Also, we denote by $\widetilde{X}$ the dual $B-A$-bimodule of $X$ and we denote by $\widetilde{x}$ the element in $\widetilde{X}$ induced by $x\in X$. Preliminaries {#sec:pre} ============= Let $H$ be a finite dimensional $C^*$-Hopf algebra. We denote its comultipication, counit and antipode by $\Delta$, $\epsilon$ and $S$, respectively. We shall use Sweedler’s notation, $\Delta(h)=h_{(1)}\otimes h_{(2)}$ for any $h\in H$ which suppresses a possible summation when we write comultiplications. We denote by $N$ the dimension of $H$. Let $H^0$ be the dual $C^*$-Hopf algebra of $H$. We denote its comultiplication, counit and antipode by $\Delta^0$, $\epsilon^0$ and $S^0$, respectively. There is the distinguished projection $e$ in $H$. We note that $e$ is the Haar trace on $H^0$. Also, there is the distinguished projection $\tau$ in $H^0$ which is the Haar trace on $H$. Since $H$ is finite dimensional, $H\cong\oplus_{k=1}^L M_{f_k}({\mathbf C})$ and $H^0 \cong \oplus_{k=1}^K M_{d_k }({\mathbf C})$ as $C^*$-algebras. Let $\{v_{ij}^k \, | \, k=1,2, \dots L , \, i, j=1,2,\dots, f_k \}$ be a system of matrix units of $H$. Let $\{w_{ij}^k \, | \, k=1,2,\dots,K ,\, i, j=1,2, \dots , d_k \}$ be a basis of $H$ satisfying Szymański and Peligrad’s [@SP:saturated Theorem 2.2,2], which is called a system of *comatrix units of $H$, that is, the dual basis of the system of matrix units of $H^0$. Also, let $\{\phi_{ij}^k \, | \, k=1,2,\dots,K, \, i,j=1,2,\dots,d_k \}$ and $\{\omega_{ij}^k \, | \, k=1,2,\dots, L, \, i, j=1,2,\dots,f_k \}$ be systems of matrix units and comatrix units of $H^0$, respectively. Let $A$ be a unital $C^*$-algebra. We recall the definition of a twisted coaction $(\rho, u)$ of $H^0$ on $A$ (See [@KT1:inclusion], [@KT2:coaction]). Let $\rho$ be a weak coaction of $H^0$ on $A$ and $u$ a unitary element in $A\otimes H^0 \otimes H^0$. Then we say that $(\rho, u)$ is a *twisted coaction of $H^0$ on $A$ if (1) $(\rho\otimes{{\rm{id}}})\circ\rho={{\rm{Ad}}}(u)\circ({{\rm{id}}}\otimes\Delta^0 )\circ\rho$, (2) $(u\otimes 1^0 )({{\rm{id}}}\otimes\Delta^0 \otimes{{\rm{id}}})(u)=(\rho\otimes{{\rm{id}}}\otimes{{\rm{id}}})(u) ({{\rm{id}}}\otimes{{\rm{id}}}\otimes\Delta^0 )(u)$, (3) $({{\rm{id}}}\otimes h\otimes\epsilon^0 )(u)=({{\rm{id}}}\otimes\epsilon^0 \otimes h)(u)=\epsilon^0 (h)1$ for any $h\in H$.** Let ${{\rm{Hom}}}(H, A)$ be the linear space of all linear maps from $H$ to $A$. Then by Sweedler [@Sweedler:Hopf pp67-70], it becomes a unital convolution $*$-algebra. Since $H$ is finite dimensional, ${{\rm{Hom}}}(H, A)$ is isomorphic to $A\otimes H^0$. For any element $x\in A\otimes H^0$, we denote by $\widehat{x}$ the element in ${{\rm{Hom}}}(H, A)$ induced by $x$. Similarly, we define ${{\rm{Hom}}}(H\times H, A)$. We identify $A\otimes H^0 \otimes H^0$ with ${{\rm{Hom}}}(H\times H, A)$. For any element $y\in A\otimes H^0 \otimes H^0$, we denote by $\widehat{y}$ the element in ${{\rm{Hom}}}(H\times H, A)$ induced by $y$. Furthermore, for a Hilbert $C^*$-bimodule $X$, let ${{\rm{Hom}}}(H, X)$ be the linear space of all linear maps from $H$ to $X$. We also identify ${{\rm{Hom}}}(H, X)$ with $X\otimes H^0$. For any element $x\in X\otimes H^0$, we denote by $\widehat{x}$ the element in ${{\rm{Hom}}}(H, X)$ induced by $x$. For a twisted coaction $(\rho, u)$, we can consider the twisted action of $H$ on $A$ and its unitary element $\widehat{u}$ defined by $$h\cdot_{\rho, u}x=\widehat{\rho(x)}(h)=({{\rm{id}}}\otimes h)(\rho(x))$$ for any $x\in A$, $h\in H$. We call it the twisted action induced by $(\rho, u)$. Let $A\rtimes_{\rho, u}H$ be the twisted crossed product by the twisted action of $H$ on $A$ induced by $(\rho, u)$. Let $x\rtimes_{\rho, u}h$ be the element in $A\rtimes_{\rho, u}H$ induced by elements $x\in A$, $h\in H$. Let $\widehat{\rho}$ be the dual coaction of $H$ on $A\rtimes_{\rho, u}H$ defined by $$\widehat{\rho}(x\rtimes_{\rho, u}h)=(x\rtimes_{\rho, u}h_{(1)})\otimes h_{(2)}$$ for any $x\in A$, $h\in H$. Let $E_1^{\rho, u}$ be the canonical conditional expectation from $A\rtimes_{\rho, u}H$ onto $A$ defined by $$E_1^{\rho, u}(x\rtimes_{\rho, u}h)=\tau(h)x$$ for any $x\in A$, $h\in H$. Let $\Lambda$ be the set of all triplets $(i, j, k)$, where $i, j=1,2,\dots,d_k$ and $k=1,2,\dots,K$ with $\sum_{k=1}^K d_k^2 =N$. Let $W_I=\sqrt{d_k}\rtimes_{\rho, u}w_{ij}^k$ for any $I=(i, j, k)\in\Lambda$. By [@KT1:inclusion Proposition 3.18], $\{(W_I^{\rho *}, W_I^{\rho})\}_{I\in \Lambda}$ is a quasi-basis for $E_1^{\rho, u}$. \[lem:pre\]With the above notations, for any $\psi\in H^0$, $$1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\psi =\sum_{I\in\Lambda}([\psi\cdot_{\widehat{\rho}}W_I^{\rho*}]\rtimes_{\widehat{\rho}}1^0 ) (1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\tau)(W_I^{\rho}\rtimes_{\widehat{\rho}}1^0 ) .$$ Since $\{(W_I^{\rho*}, W_I^{\rho})\}_{I\in\Lambda}$ is a quasi-basis for $E_1^{\rho, u}$ by [@KT1:inclusion Proposition 3.18], $$\sum_{I\in\Lambda}(W_I^{\rho*}\rtimes_{\widehat{\rho}}1^0 )(1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\tau) (W_I^{\rho}\rtimes_{\widehat{\rho}}1^0 )=1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}1^0 .$$ Hence for any $\psi\in H^0$, $$\begin{aligned} 1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\psi & = \sum_{I\in \Lambda}(1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\psi)(W_I^{\rho*}\rtimes_{\widehat{\rho}}1^0 ) (1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\tau)(W_I^{\rho}\rtimes_{\widehat{\rho}}1^0 ) \\ & =\sum_{I\in\Lambda}([\psi_{(1)}\cdot_{\widehat{\rho}}W_I^{\rho*}]\rtimes_{\widehat{\rho}}\psi_{(2)}) (1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\tau)(W_I^{\rho}\rtimes_{\widehat{\rho}}1^0 ) \\ & =\sum_{I\in\Lambda}([\psi\cdot_{\widehat{\rho}}W_I^{\rho*}]\rtimes_{\widehat{\rho}}1^0 ) (1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\tau)(W_I^{\rho}\rtimes_{\widehat{\rho}}1^0 ) .\end{aligned}$$ Therefore, we obtain the conclusion. A left coaction of a finite dimensional $C^*$-Hopf algebra on an equivalence bimodule {#sec:left} ===================================================================================== Let $A$ and $B$ be unital $C^*$-algebras and let $(\rho, u)$ and $(\sigma, v)$ be twisted coactions of $H^0$ on $A$ and $B$, respectively. Let $A\rtimes_{\rho, u}H$ and $B\rtimes_{\sigma, v}H$ be the twisted crossed products of $A$ and $B$ by $(\rho, u)$ and $(\sigma, v)$, respectively. We denote them by $C$ and $D$, respectively. Then we obtain unital inclusions $A\subset C$ and $B\subset D$ of unital $C^*$-algebras. We suppose that $A\subset C$ and $B\subset D$ are strongly Morita equivalent with respect to a $C-D$-equivalence bimodule $Y$ and its closed subspace $X$ in the sense of [@KT4:morita Definition 2.1]. Also, by [@KT4:morita Section 2], there is a conditional expectation $E^X$ from $Y$ onto $X$ with respect to $E_1^{\rho, u}$ and $E_1^{\sigma, v}$ satisfying Conditions (1)-(6) in [@KT4:morita Definition 2.2]. Furthermore, by [@KT4:morita Section 6], we can see that the unital inclusions $C\subset C_1$ and $D\subset D_1$ are strongly Morita equivalent with respect to the $C_1 -D_1$-equivalence bimodule $Y_1$ and its closed subspace $Y$, where $C_1 =C\rtimes_{\widehat{\rho}}H^0$ and $D_1 =D\rtimes_{\widehat{\sigma}}H^0$. As defined in [@KT4:morita], we define $Y_1$ as follows: We regard $C$ and $D$ as a $C_1 -A$-equivalence bimodule and a $D_1 -B$-equivalence bimodule in the usual way as in [@KT4:morita Section 4], respectively. Let $Y_1 =C\otimes _A X\otimes_B \widetilde{D}$. Let $E^Y$ be the conditional expectation from $Y_1$ onto $Y$ with respect to $E_2^{\rho, u}$ and $E_2^{\sigma, v}$ defined by $$E^Y (c\otimes x\otimes\widetilde{d})=\frac{1}{N}c\cdot x\cdot d^*$$ for any $c\in C$, $d\in D$, $x\in X$, where $E_2^{\rho, u}$ and $E_2^{\sigma, v}$ are the canonical conditional expectations from $C_1$ and $D_1$ onto $C$ and $D$ defined by $$E_2^{\rho, u}(c\rtimes_{\widehat{\rho}}\psi) =c\psi(e) , \quad E_2^{\sigma, v}(d\rtimes_{\widehat{\sigma}}\psi)=d\psi(e) .$$ for any $c\in C$, $d\in D$, $\psi\in H^0$, respectively. We regard $Y$ as a closed subspace of $Y_1$ by the injective linear map $\phi$ from $Y$ to $Y_1$ defined by $$\phi(y)=\sum_{I, J\in \Lambda}W_I^{\rho*}\otimes E^X (W_I^{\rho}\cdot y\cdot W_J^{\sigma*}) \otimes\widetilde{W_J^{\sigma*}}$$ for any $y\in Y$. In this section, we construct a left coaction of $H$ on $Y$ with respect to $(C, \widehat{\rho})$. First, we define the bilinear map $`` \triangleright "$ from $H^0 \times Y$ to $Y$ as follows: For any $\psi\in H^0$, $y\in Y$, $$\psi\triangleright y =NE^Y ((1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\psi)\cdot \phi(y)\cdot (1\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}\tau)) .$$ \[remark:calculus\]For any $\psi\in H^0$, $y\in Y$, $$\psi\triangleright y=\sum_{I\in \Lambda}[\psi\cdot_{\widehat{\rho}}W_I^{\rho*}] \cdot E^X (W_I^{\rho}\cdot y) .$$ Indeed, since $1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}} \tau$ is the Jones projection for the conditional expectation $E_1^{\rho, u}$, for any $c\in C$ $(1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\tau)\cdot c=E_1^{\rho, u}(c)$. Also, $(1\rtimes_{\sigma, v}\rtimes_{\widehat{\sigma}}\tau)\cdot d =E_1^{\sigma, v}(d)$ for any $d\in D$. Hence by Lemma \[lem:pre\], for any $\psi\in H^0$, $y\in Y$, $$\begin{aligned} &\psi\triangleright y \\ &=\sum_{I, J\in \Lambda}NE^Y ((1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\psi)\cdot W_I^{\rho*} \otimes E^X (W_I^{\rho}\cdot y\cdot W_J^{\sigma*} )\otimes\widetilde{W_J^{\sigma*}}\cdot (1\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}\tau)) \\ & =\sum_{I, I_1 , J \in\Lambda}NE^Y ([\psi\cdot_{\widehat{\rho}}W_{I_1}^{\rho*}] E_1^{\rho, u}(W_{I_1}^{\rho}W_I^{\rho*})\otimes E^X (W_I^{\rho}\cdot y\cdot W_J^{\sigma*}) \otimes E_1^{\sigma, v}(W_J^{\sigma*})^{\widetilde{}}) \\ & =\sum_{I, I_1 , J\in \Lambda}NE^Y ([\psi\cdot_{\widehat{\rho}}W_{I_1}^{\rho*}] \otimes E_1^{\rho, u}(W_{I_1}^{\rho}W_I^{\rho*})\cdot E^X (W_I^{\rho}\cdot y\cdot W_{J}^{\sigma*})\cdot E_1^{\sigma, v}(W_J^{\sigma})\otimes \widetilde{1}) \\ & =\sum_{I, I_1 , J\in\Lambda}[\psi\cdot_{\widehat{\rho}}W_{I_1}^{\rho*}]\cdot E^X (E_1^{\rho, u}(W_{I_1}^{\rho} W_I^{\rho*})W_I^{\rho}\cdot y\cdot W_J^{\sigma*}E_1^{\sigma, v}(W_J^{\sigma})) \\ & =\sum_{I_1 \in \Lambda}[\psi\cdot_{\widehat{\rho}}W_{I_1}^{\rho*}]\cdot E^X (W_{I_1}^{\rho}\cdot y).\end{aligned}$$ \[lem:fix\]With the above notations, for any $x\in X$, $\psi\in H^0$, $\psi\triangleright x=\epsilon^0 (\psi)x$. By Remark \[remark:calculus\], [@SP:saturated Theorem 2.2] and [@KT4:morita Lemma 6.9] $$\begin{aligned} \psi\triangleright x & =NE^Y ((1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\psi)\cdot\phi(x)\cdot (1\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}\tau)) \\ & =NE^Y ((1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\psi\tau)\cdot\phi(x)) \\ & =N\epsilon^0 (\psi)E^Y ((1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\tau)\cdot\phi(x)) =\epsilon^0 (\psi)\sum_{I\in\Lambda}[\tau\cdot_{\widehat{\rho}}W_I^{\rho*}]\cdot E^X (W_I^{\rho}\cdot x) \\ & =\epsilon^0 (\psi) ( \sum_{I\in\Lambda}E_1^{\rho, u}(W_I^{\rho*}E_1^{\rho, u}(W_I^{\rho})))\cdot x =\epsilon^0 (\psi)x .\end{aligned}$$ \[lem:definition1\]With the above notations, for any $a\in A$, $h\in H$, $\psi\in H^0$, $y\in Y$, $$\psi\triangleright(a\rtimes_{\rho, u}h)\cdot y=[\psi_{(1)}\cdot_{\widehat{\rho}}(a\rtimes_{\rho, u}h)]\cdot [\psi_{(2)}\triangleright y] .$$ For any $a\in A$, $h\in H$, $\psi\in H^0$ , $y\in Y$, $$\begin{aligned} & \psi\triangleright(a\rtimes_{\rho, u}h)\cdot y = NE^Y ((1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\psi)\cdot \phi((a\rtimes_{\rho, u}h)\cdot y)\cdot(1\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}\tau)) \\ & =NE^Y ((1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\psi)(a\rtimes_{\rho, u}h\rtimes_{\widehat{\rho}}1^0 ) \cdot \phi(y)\cdot(1\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}\tau)) \\ & =NE^Y (([\psi_{(1)}\cdot_{\widehat{\rho}}(a\rtimes_{\rho, u}h)]\rtimes_{\widehat{\rho}}\psi_{(2)}) \cdot\phi(y)\cdot(1\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}\tau)) \\ & =[\psi_{(1)}\cdot_{\widehat{\rho}}(a\rtimes_{\rho, u}h)] \cdot NE^Y ((1\rtimes_{\rho, u} 1\rtimes_{\widehat{\rho}}\psi_{(2)})\cdot\phi(y)\cdot (1\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}\tau)) \\ & =[\psi_{(1)}\cdot_{\widehat{\rho}}(a\rtimes_{\rho, u}h)]\cdot[\psi_{(2)}\triangleright y]\end{aligned}$$ by [@KT4:morita Lemma 6.1]. \[lem:definition2\]With the above notations, for any $\psi, \chi\in H^0$, $y\in Y$, $$\psi\triangleright[\chi\triangleright y]=\psi\chi\triangleright y .$$ By Remark \[remark:calculus\], for any $\psi, \chi\in H^0$, $y\in Y$, $$\psi\triangleright[\chi\triangleright y]=\psi\triangleright\sum_{I\in \Lambda}[\chi\cdot_{\widehat{\rho}}W_I^{\rho*}] \cdot E^X (W_I^{\rho}\cdot y) .$$ Since $\chi\cdot_{\widehat{\rho}}W_I^{\rho*}\in C$, by Lemma \[lem:definition1\], $$\psi\triangleright[\chi\triangleright y]=\sum_{I\in\Lambda}[\psi_{(1)}\cdot_{\widehat{\rho}}[\chi\cdot_{\widehat{\rho}} W_I^{\rho*}]]\cdot[\psi_{(2)}\triangleright E^X (W_I^{\rho}\cdot y)] .$$ Since $E^X (W_I^{\rho} \cdot y)\in X$, by Lemma \[lem:fix\] and Remark \[remark:calculus\] $$\begin{aligned} \psi\triangleright[\chi\triangleright y] & =\sum_{I\in\Lambda}[\psi_{(1)}\chi\cdot_{\widehat{\rho}}W_I^{\rho*}] \cdot\epsilon^0 (\psi_{(2)})E^X (W_I^{\rho}\cdot y) \\ & =\sum_{I\in \Lambda}[\psi\chi\cdot_{\widehat{\rho}}W_I^{\rho*}]\cdot E^X (W_I^{\rho}\cdot y) =\psi\chi\triangleright y .\end{aligned}$$ \[lem:definition3\]With the above notations, for any $\psi\in H^0$, $y, z\in Y$, $$\psi\cdot_{\widehat{\rho}}{}_C {\langle}y, z {\rangle}={}_C {\langle}\psi_{(1)}\triangleright y \, , \, S^0 (\psi_{(2)}^* )\triangleright z {\rangle}.$$ By Remark \[remark:calculus\] and [@KT4:morita Lemma 5.4], for any $\psi\in H^0$, $y, z\in Y$, $$\begin{aligned} & {}_C {\langle}\psi_{(1)}\triangleright y \, , \, S^0 (\psi_{(2)}^* )\triangleright z {\rangle}\\ & =\sum_{I, J\in\Lambda}{}_C {\langle}[\psi_{(1)}\cdot_{\widehat{\rho}}W_I^{\rho*}]\cdot E^X (W_I^{\rho}\cdot y) \, , \, [S^0 (\psi_{(2)}^* )\cdot_{\widehat{\rho}}W_J^{\rho*}]\cdot E^X (W_J^{\rho}\cdot z) {\rangle}\\ & =\sum_{I, J\in\Lambda}[\psi_{(1)}\cdot_{\widehat{\rho}}W_I^{\rho*}] {}_A {\langle}E^X (W_I^{\rho}\cdot y) \, , \, E^X (W_J^{\rho}\cdot z) {\rangle}[\psi_{(2)}\cdot_{\widehat{\rho}}W_J^{\rho}] \\ & =\sum_{I, J\in\Lambda}[\psi\cdot_{\widehat{\rho}}W_I^{\rho*} \, {}_A {\langle}E^X (W_I^{\rho}\cdot y) \, , \, E^X (W_J^{\rho}\cdot z ) {\rangle}W_J^{\rho}] \\ & =\sum_{I, J\in\lambda}[\psi\cdot_{\widehat{\rho}} \, {}_C {\langle}W_J^{\rho*}\cdot E^X (W_I^{\rho}\cdot y) \, ,\, W_J^{\rho*}\cdot E^X (W_J^{\rho}\cdot z ) {\rangle}] \\ & =\psi\cdot_{\widehat{\rho}}{}_C {\langle}y, \, z {\rangle}.\end{aligned}$$ \[prop:leftcoaction\]With the above notations, the linear map from $Y$ to $Y\otimes H$ induced by the bilinear map $ ``\triangleright"$ from $H^0 \times Y$ to $Y$ defined by $$\psi\triangleright y =NE^Y ((1\rtimes_{\rho, u}1\rtimes_{\widehat{\rho}}\psi)\cdot \phi(y)\cdot (1\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}\tau))$$ for any $y\in Y$, $\psi\in H^0$ is a left coaction of $H$ on $Y$ with respect to $(C, \widehat{\rho})$ satisfying that $$\psi\triangleright x = \epsilon^0 (\psi)x$$ for any $x\in X$, $\psi\in H^0$. By Lemmas \[lem:fix\], \[lem:definition1\], \[lem:definition2\] and \[lem:definition3\], it suffices to show that $1^0 \triangleright y=y$ for any $y\in Y$. Indeed, by Remark \[remark:calculus\] and [@KT4:morita Lemma 5.4] $$1^0 \triangleright y=\sum_{I\in \Lambda}W_I^{\rho*}\cdot E^X (W_I^{\rho}\cdot y)=y$$ for any $y\in Y$. We denote by $\mu$ the above left coaction induced by $``\triangleright"$. A coaction of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-algebra induced by a left coaction {#sec:coaction} ========================================================================================================== We use the same notations as in the previous section and also we suppose that the same assumptions as in the previous section. We note that $C=A\rtimes_{\rho, u}H$ and $D=B\rtimes_{\sigma, v}H$. We also note that $D$ is anti-isomorphic to ${}_C {\mathbf B}(Y)$, the $C^*$-algebra of all adjointable left $C$-modules maps on $Y$. We identify $D$ with ${}_C {\mathbf B}(Y)$. Hence for any $T, R\in {}_C {\mathbf B}(Y)$ we can write their product as follows: For any $y\in Y$, $$(TR)(y)=R(T(y)) .$$ For any $y, z\in Y$, let $\Theta_{y, z}$ be the rank-one left $C$-module map on $Y$ defined by $$\Theta_{y, z}(x)={}_C {\langle}x, z {\rangle}\cdot y$$ for any $x\in Y$. Then since $Y$ is of finite type in the sense of Kajiwara and Watatani [@KW1:bimodule], ${}_C {\mathbf B}(Y)$ is the linear span of the above rank-one left $C$-module maps on $Y$ by [@KW1:bimodule]. We define a bilinear map $``\rightharpoonup"$ from $H^0 \times D$ to $D$ so that $$\psi\rightharpoonup{\langle}y, z {\rangle}_D ={\langle}S^0 (\psi_{(1)}^* )\triangleright y \, , \, \psi_{(2)}\triangleright z {\rangle}_D$$ for any $\psi \in H^0$ and $y, z\in Y$. Hence since ${\langle}y, z {\rangle}_D =\Theta_{z, y}$ for any $y, z\in Y$, we define $``\rightharpoonup"$ as follows: For any $\psi\in H^0$, $y, z\in Y$, $$\psi\rightharpoonup \Theta_{y, z}=\Theta_{[\psi_{(2)}\triangleright y]\,, \,[S^0 (\psi_{(1)}^* )\triangleright z]} .$$ \[lem:identity\]With the above notations, for any $\psi\in H^0$ and $T\in {}_C {\mathbf B}(Y)$, $$[\psi\rightharpoonup T](x)=\psi_{(2)}\triangleright T(S^0 (\psi_{(1)})\triangleright x)$$ for any $x\in Y$. Since ${}_C {\mathbf B}(Y)$ is the linear span of the rank-one left $C$-module maps on $Y$, it suffices to show that $$[\psi\rightharpoonup \Theta_{y, z}](x)=\psi_{(2)}\triangleright \Theta_{y, z}(S^0 (\psi_{(1)})\triangleright x)$$ for any $\psi\in H^0$ and $x, y, z\in Y$. For any $\psi\in H^0$ and $x, y, z\in Y$, $$\begin{aligned} [\psi\rightharpoonup\Theta_{y, z}](x) & =\Theta_{[\psi_{(2)}\triangleright y] , [S^0 (\psi_{(1)}^* )\triangleright z]}(x) ={}_C {\langle}x \, , \, S^0 (\psi_{(1)}^* )\triangleright z {\rangle}\cdot [\psi_{(2)}\triangleright y] \\ & ={}_C {\langle}\psi_{(2)}S^0 (\psi_{(1)})\triangleright x \, , \, S^0 (\psi_{(3)}^* )\triangleright z {\rangle}\cdot [\psi_{(4)}\triangleright y] \\ & =[\psi_{(2)}\cdot_{\widehat{\rho}}{}_C {\langle}S^0 (\psi_{(1)})\triangleright x \, , \, z {\rangle}]\cdot [\psi_{(3)}\triangleright y] \\ & =\psi_{(2)}\triangleright ({}_C {\langle}S^0 (\psi_{(1)})\triangleright x \, ,\, z {\rangle}\cdot y) \\ & =\psi_{(2)}\triangleright \Theta_{y, z}(S^0 (\psi_{(1)})\triangleright x) .\end{aligned}$$ Hence we obtain the conclusion. \[lem:action\]With the above notations, the bilinear map $`` \rightharpoonup "$ from $H^0 \times D$ to $D$ is an action of $H^0$ on $D$. Let $T, R\in {}_C {\mathbf B}(Y)$ and $\psi, \chi \in H^0$. For any $y\in Y$, $$\begin{aligned} ([\psi_{(1)}\rightharpoonup T][\psi_{(2)}\rightharpoonup R])(y)) & =[\psi_{(2)}\rightharpoonup R]([\psi_{(1)}\rightharpoonup T](y)) \\ & =\psi_{(3)}\triangleright R(S^0 (\psi_{(2)})\triangleright[\psi_{(1)}\rightharpoonup T](y)) \\ & =\psi_{(4)}\triangleright R(S^0 (\psi_{(3)})\psi_{(2)}\triangleright T(S^0 (\psi_{(1)})\triangleright y)) \\ & =\psi_{(2)}\triangleright R(T(S^0 (\psi_{(1)})\triangleright y)) \\ & =\psi_{(2)}\triangleright(TR)(S^0 (\psi_{(1)})\triangleright y) \\ & =[\psi\rightharpoonup TR](y) .\end{aligned}$$ Hence we obtain that $\psi\rightharpoonup TR=[\psi_{(1)}\rightharpoonup T][\psi_{(2)}\rightharpoonup R]$. For any $y\in Y$, $$\begin{aligned} [\psi\rightharpoonup[\chi\rightharpoonup T]](y) & = \psi_{(2)}\triangleright[\chi\rightharpoonup T](S^0 (\psi_{(1)})\triangleright y) \\ & =\psi_{(2)}\triangleright\chi_{(2)}\triangleright T(S^0 (\chi_{(1)})\triangleright S^0 (\psi_{(1)})\triangleright y) \\ & =\psi_{(2)}\chi_{(2)}\triangleright T(S^0 (\psi_{(1)}\chi_{(1)})\triangleright y) \\ & =[\psi\chi\rightharpoonup T](y) .\end{aligned}$$ Hence we obtain that $\psi\rightharpoonup[\chi\rightharpoonup T]=\psi\chi\rightharpoonup T$. Let $I_Y$ be the identity map on $Y$. Then for any $y\in Y$ $$[\psi\rightharpoonup I_Y ](y)=\psi_{(2)}\triangleright I_Y (S^0 (\psi_{(1)})\triangleright y) =\epsilon^0 (\psi)y .$$ Hence $\psi\rightharpoonup I_Y =\epsilon^0 (\psi)I_Y$. Also, for any $y\in Y$, $[1^0 \rightharpoonup T](y)=T(y)$. Thus $1^0 \rightharpoonup T=T$. Furthermore, for any $y, z\in Y$, $$\begin{aligned} {}_C {\langle}[\psi\rightharpoonup T]^* (y) \, , \, z {\rangle}& ={}_C {\langle}y \, , \, [\psi\rightharpoonup T](z) {\rangle}\\ & ={}_C {\langle}y \, , \, \psi_{(2)}\triangleright T(S^0 (\psi_{(1)})\triangleright z) {\rangle}\\ & ={}_C {\langle}S^0 (\psi_{(3)}^* )\psi_{(4)}^* \triangleright y \, , \, \psi_{(2)} \triangleright T(S^0 (\psi_{(1)})\triangleright z) {\rangle}\\ & = S^0 (\psi_{(2)}^* )\cdot_{\widehat{\rho}}\,{}_C {\langle}\psi_{(3)}^* \triangleright y \, , \, T(S^0 (\psi_{(1)})\triangleright z) {\rangle}\\ & = S^0 (\psi_{(2)}^* )\cdot_{\widehat{\rho}}\,{}_C {\langle}T^* (\psi_{(3)}^* \triangleright y) \, , \, S^0 (\psi_{(1)})\triangleright z {\rangle}\\ & ={}_C {\langle}S^0 (\psi_{(3)}^* )\triangleright T^* (\psi_{(4)}^* \triangleright y) \, , \, \psi_{(2)}S^0 (\psi_{(1)}) \triangleright z {\rangle}\\ & ={}_C {\langle}S^0 (\psi_{(1)}^* )\triangleright T^* (\psi_{(2)}^* \triangleright y) \, , \, z {\rangle}\\ & ={}_C {\langle}[S^0 (\psi^* )\rightharpoonup T^* ](y) \, ,\, z {\rangle}.\end{aligned}$$ Hence we can see that $[\psi\rightharpoonup T]^* =S^0 (\psi^* )\rightharpoonup T^* $. Therefore we obtain the conclusion. By Lemma \[lem:action\], the map $``\rightharpoonup"$ is an action of $H^0$ on $D$. We denote by $\beta$ the coaction of $H$ on $D$ induced by the action $``\rightharpoonup"$ of $H^0$ on $D$. By the definition of the action $``\rightharpoonup"$, the left coaction $\mu$ of $H$ on $Y$ induced by the action $``\triangleright"$ is also a right coaction of $H$ on $Y$ with respect to $(D, \beta)$. Thus $\mu$ is a coaction of $H$ on $Y$ with respect to $(C, D, \widehat{\rho}, \beta)$. Hence we can see that $\widehat{\rho}$ is strongly Morita equivalent to $\beta$. By [@KT4:morita Section 4], the unital inclusions $C\subset C_1 (=C\rtimes_{\widehat{\rho}}H^0)$ and $D\subset D\rtimes_{\beta}H^0$ are strongly Morita equivalent with respect to the $C_1 -D\rtimes_{\beta}H^0$-equivalence bimodule $Y\rtimes_{\mu}H^0$ and its closed subspace $Y$. Since the unital inclusions $C\subset C_1$ and $D\subset D_1$ are also strongly Morita equivalent with respect to the $C_1 -D_1$-equivalence bimodule $Y_1$ and its closed subspace $Y$ by [@KT4:morita Corollary 6.3], we can see that the unital inclusions $D\subset D_1$ and $D\subset D\rtimes_{\beta}H^0 $ are strongly Morita equivalent with respect to the $D_1 -D\rtimes_{\beta}H^0$-equivalence bimodule $\widetilde{Y_1}\otimes_{C_1} (Y\rtimes_{\mu}H^0 )$ and its closed subspace $\widetilde{Y}\otimes_C Y\cong D$ by [@KT4:morita Proposition 2.2]. The exterior equivalence and the strong Morita equivalence {#sec:exterior} ========================================================== First, we shall present some basic properties on coactions of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-algebra and their exterior equivalence and strong Morita equivalence. Let $H$ and $K$ be finite dimensional $C^*$-Hopf algebras and let $H^0$ and $K^0$ be their dual $C^*$-Hopf algebras, respectively. Let $A$ and $B$ unital $C^*$-algebras and let $\pi$ be an isomorphism of $B$ onto $A$. Let $\lambda^0$ be a $C^*$-Hopf algebra isomorphism of $K^0$ onto $H^0$. Let $(\rho, u)$ be a coaction of $K^0$ on $B$ and let $\rho_{\pi, \lambda^0}$ be the homomorphism of $A$ to $A\otimes H^0$ defined by $$\rho_{\pi, \lambda^0}=(\pi\otimes \lambda^0 )\circ\rho\circ\pi^{-1} .$$ Let $u_{\pi, \lambda^0}$ be the unitary element in $A\otimes H^0 \otimes H^0$ defined by $$u_{\pi, \lambda^0}=(\pi\otimes\lambda^0 \otimes\lambda^0 )(u) .$$ \[lem:induced\]With the above notations, $(\rho_{\pi, \lambda^0} ,\, u_{\pi, \lambda^0})$ is a twisted coaction of $H^0$ on $A$. This is immediate by routine computations. We call the above pair $(\rho_{\pi, \lambda^0}, u_{\pi, \lambda^0})$ the twisted coaction of $H^0$ on $A$ induced by $(\rho, u)$ and $\pi$, $\lambda^0$. Let $(\sigma, v)$ be a coaction of $K^0$ on $B$ and $(\sigma_{\pi, \lambda^0}, v_{\pi, \lambda^0})$ the twisted coaction of $H^0$ on $A$ induced by $(\sigma, v)$ and $\pi$, $\lambda^0$. \[lem:exterior\]With the above notations, if $(\rho, u)$ is exterior equivalent $(\sigma, v)$, then $(\rho_{\pi, \lambda^0} ,\, u_{\pi, \lambda^0})$ is exterior equivalent to $(\sigma_{\pi, \lambda^0}, v_{\pi, \lambda^0})$. Let $\Delta_H^0$ and $\Delta_K^0$ be the comultiplications of $H^0$ and $K^0$, respectively. Since $(\rho, u)$ and $(\sigma, v)$ are exterior equivalent, there is a unitary element $w\in B\otimes K^0$ such that $$\sigma ={{\rm{Ad}}}(w)\circ \rho, \quad v=(w\otimes 1^0 )(\rho\otimes {{\rm{id}}}_{K^0})(w)u({{\rm{id}}}_B \otimes\Delta_K^0 )(w)^* .$$ Since $\rho=(\pi\otimes\lambda^0 )^{-1}\circ\rho_{\pi, \lambda^0}\circ\pi$ and $\sigma=(\pi\otimes\lambda^0 )^{-1}\circ\sigma_{\pi. \lambda^0 }\circ\pi$, $$(\pi\otimes\lambda^0 )^{-1}\circ\sigma_{\pi, \lambda^0}\circ\pi ={{\rm{Ad}}}(w)\circ(\pi\otimes\lambda^0 )^{-1}\circ\rho_{\pi, \lambda^0}\circ\pi .$$ Hence $$\sigma_{\pi, \lambda^0 }={{\rm{Ad}}}((\pi\otimes\lambda^0 )(w))\circ\rho_{\pi, \lambda^0 } .$$ Furthermore, $$\begin{aligned} v_{\pi, \lambda^0} =(\pi\otimes\lambda^0 \otimes\lambda^0 )(v) &=((\pi\otimes\lambda^0 )(w)\otimes 1^0 )(\pi\otimes\lambda^0 \otimes\lambda^0 )((\rho\otimes{{\rm{id}}}_{K^0})(w)) \\ & \times u_{\pi, \lambda^0 }(\pi\otimes\lambda^0 \otimes\lambda^0 )(({{\rm{id}}}_B \otimes\Delta_K^0 )(w^* )) .\end{aligned}$$ Since $(\lambda^0 \otimes\lambda^0 )\circ\Delta_K^0 =\Delta_H^0 \circ\lambda^0 $, $$(\pi\otimes\lambda^0 \otimes\lambda^0 )(({{\rm{id}}}_B \otimes\Delta_K^0 )(w^* )) =({{\rm{id}}}_A \otimes\Delta_H^0 )((\pi\otimes\lambda^0 )(w^* )) .$$ Since $w\in B\otimes H^0$, we can write that $w=\sum_i b_i\otimes\phi_i $, where $b_i \in B$, $\phi_i \in H^0$. Then $$(\pi\otimes\lambda^0 \otimes\lambda^0 )((\rho\otimes{{\rm{id}}}_{K^0})(w)) =\sum_i (\pi\otimes\lambda^0 )(\rho(b_i ))\otimes\lambda^0 (\phi_i ) .$$ On the other hand $$\begin{aligned} (\rho_{\pi, \lambda^0}\otimes{{\rm{id}}}_{H^0})((\pi\otimes\lambda^0 )(w)) & = (((\pi\otimes\lambda^0 )\circ\rho\circ\pi^{-1})\otimes{{\rm{id}}}_{H^0})((\pi\otimes\lambda^0 )(w)) \\ & =(((\pi\otimes\lambda^0 )\circ\rho\circ\pi^{-1})\otimes{{\rm{id}}}_{H^0})(\sum_i \pi(b_i )\otimes\lambda^0 (\phi_i )) \\ & =\sum_i ((\pi\otimes\lambda^0 )\circ\rho)(b_i )\otimes\lambda^0 (\phi_i ) \\ & =\sum_i (\pi\otimes\lambda^0 )(\rho(b_i ))\otimes\lambda^0 (\phi_i ) .\end{aligned}$$ Thus $$(\pi\otimes\lambda^0 \otimes\lambda^0 )((\rho\otimes{{\rm{id}}}_{K^0})(w)) =(\rho_{\pi, \lambda^0 }\otimes{{\rm{id}}}_{H^0}((\pi\otimes\lambda^0 )(w)) .$$ Hence $$\begin{aligned} v_{\pi, \lambda^0 } &= ((\pi\otimes\lambda^0 )(w)\otimes 1^0 )(\rho_{\pi, \lambda^0 }\otimes{{\rm{id}}}_{H^0 }) ((\pi\otimes\lambda^0 )(w))u_{\pi, \lambda^0} \\ & \times ({{\rm{id}}}_A \otimes\Delta_H^0 )((\pi\otimes\lambda^0 )(w^* )) .\end{aligned}$$Therefore, we obtain the conclusion. Let $(\rho, u)$ be a twisted coaction of $H^0$ on $B$ and let $\pi$ be an isomorphism of $B$ onto $A$. Let $$\rho_{\pi}=(\pi\otimes{{\rm{id}}})\circ\rho\circ\pi^{-1} , \quad u_{\pi}=(\pi\otimes{{\rm{id}}}_{H^0}\otimes{{\rm{id}}}_{H^0})(u) .$$ By Lemma \[lem:induced\], $(\rho_{\pi}, u_{\pi})$ is a twisted coaction of $H^0$ on $A$. \[lem:iso\]With the above notations, $(\rho_{\pi}, u_{\pi})$ is strongly Morita equivalent to $(\rho, u)$. Let $X_{\pi}$ be the $B-A$-equivalence bimodule induced by $\pi$, that is, $X_{\pi}=B$ as vector spaces and the left $B$-action on $X_{\pi}$ and the left $B$-valued inner product are defined in the evident way. We define the right $A$-action and the right $A$-valued inner product as follows: For any $a\in A$, $x, y\in X_{\pi}$, $$x\cdot a=x\pi^{-1}(a) , \quad {\langle}x, y {\rangle}_A =\pi(x^* y) .$$ Let $\nu$ be the linear map from $X_{\pi}$ to $X_{\pi}\otimes H^0$ defined by $\nu(x)=\rho(x)$ for any $x\in X_{\pi}$. Then $\nu$ is a twisted coaction of $H^0$ on $X_{\pi}$ with respect to $(B, A, \rho, u, \rho_{\pi}, u_{\pi})$. Indeed, for any $a\in A$, $b\in B$, $x, y\in X_{\pi}$, $$\begin{aligned} & \nu(b\cdot x) =\rho(b)\nu(x)=\rho(b)\cdot \nu(x) , \\ & \nu(x\cdot a) =\nu(x) ((\pi^{-1}\otimes{{\rm{id}}})\circ(\pi\otimes{{\rm{id}}})\circ\rho\circ\pi^{-1})(a) =\nu(x)\cdot\rho_{\pi}(a) , \\ & {}_{B\otimes H^0} {\langle}\nu(x), \nu(y) {\rangle}=\rho(xy^* )=\rho({}_B {\langle}x, y {\rangle}) , \\ & {\langle}\nu (x), \nu(y) {\rangle}_{A\otimes H^0 } =(\pi\otimes{{\rm{id}}})(\rho(x^* y))=\rho_{\pi}({\langle}x, y {\rangle}_A ), \\ & ({{\rm{id}}}\otimes\epsilon^0 )(\nu(x))= ({{\rm{id}}}\otimes\epsilon^0 )(\rho(x))=x , \\ & ((\nu\otimes{{\rm{id}}})\circ\nu)(x)=u({{\rm{id}}}\otimes\Delta^0 )(\rho(x))u^* =u\cdot ({{\rm{id}}}\otimes\Delta^0 )(\rho(x))\cdot u_{\pi}^* .\end{aligned}$$ Thus $\nu$ is a twisted coaction of $H^0$ on $X_{\pi}$ with respect to $(B, A, \rho, u, \rho_{\pi}, u_{\pi})$. Therefore we obtain the conclusion. \[remark:review\]Let $(\rho, u)$ be a twisted coaction of $H^0$ on $A$. By Lemma \[lem:iso\], [@Kodaka:equivariance Lemma 4.10] and [@KT2:coaction Theorem 3.3], we can see that $(\rho, u)$ is strongly Morita equivalent to $\widehat{\widehat{\rho}}$, the second dual coaction of $(\rho, u)$. This is obtained in [@KT3:equivalence the proof of Corollary 4.8]. Let $A$ and $H$, $H^0$ be as before. Let $(\rho, u)$ and $(\sigma, v)$ be twisted coactions of $H^0$ on $A$. Let $C=A\rtimes_{\rho, u}H$ and $D=A\rtimes_{\sigma, v}H$. We also regard $A$ as an $A-A$-equivalence bimodule in the usual way. We suppose that the unital inclusions $A\subset C$ and $A\subset D$ are strongly Morita equivalent with respect to a $C-D$-equivalence bimodule $Y$ and its closed subspace $A$, that is, we assume that the $A-A$-equivalence bimodule $A$ is included in $Y$ as a closed subspace. Furthermore, we suppose that $A'\cap C={\mathbf C}1$. Then by [@KT4:morita Lemma 10.3], $A' \cap D={\mathbf C}1$. Let $E_1^{\rho, u}$ and $E_1^{\sigma, v}$ be the canonical conditional expectations from $C$ and $D$ onto $A$ defined by $$E_1^{\rho, u}(a\rtimes_{\rho, u}h)=\tau(h)a , \quad E_1^{\sigma, v}(a\rtimes_{\sigma, v}h)=\tau(h)a$$ for any $a\in A$, $h\in H$. By [@KT4:morita Theorem 2.7], there are a conditional expectation $F^A$ of Watatani index-finite type from $D$ onto $A$ and a conditional expectation $G^A$ from $Y$ onto $A$ with respect to $E_1^{\rho, u}$ and $F^A$. Since $A' \cap D={\mathbf C}1$, by Watatani [@Watatani:index Proposition 4.1], $F^A =E_1^{\sigma, v}$. Hence $G^A$ is a conditional expectation from $Y$ onto $A$ with respect to $E_1^{\rho, u}$ and $E_1^{\sigma, v}$. By the discussions in [@KT4:morita Section 2] and the proof of Rieffel [@Rieffel:rotation Proposition 2.1], there is an isomorphism $\Psi$ of $D$ onto $C$ defined by $$\Psi(d)={}_C {\langle}1_A \cdot d \, , \, 1_A {\rangle}$$ for any $d\in D$, where $A$ is a closed subspace of $Y$ and the unit element in $A$ is regarded as an element in $Y$. Then for any $a\in A$ $$\Psi(a)={}_A {\langle}1_A \cdot a \, , \, 1_A {\rangle}={}_A {\langle}a , \, 1_A {\rangle}=a .$$ Also, for any $d\in D$ $$\begin{aligned} (E_1^{\rho, u}\circ\Psi)(d) & =E_1^{\rho, u}({}_C {\langle}1_A \cdot d \, , \, 1_A {\rangle}) ={}_A {\langle}G^A (1_A \cdot d) \, , \, 1_A {\rangle}={}_A {\langle}1_A \cdot E_1^{\sigma, v}(d) \, , \, 1_A {\rangle}\\ & =E_1^{\sigma, v}(d) .\end{aligned}$$ Thus, we obtain the following lemma: \[lem:equation\]With the above notations, we suppose that $A' \cap C={\mathbf C}1$. Then $\Psi$ is an isomorphism of $D$ onto $C$ satisfying that $$\Psi|_A ={{\rm{id}}}_A, \quad E_1^{\rho, u}\circ\Psi=E_1^{\sigma, v}$$ Let $\widehat{\rho}$ and $\widehat{\sigma}$ be the dual coactions of $H$ on $C$ and $D$ induced by the twisted coactions $(\rho, u)$ and $(\sigma, v)$, respectively and let $C_1 =C\rtimes_{\widehat{\rho}}H^0$ and $D_1 =D\rtimes_{\widehat{\sigma}}H^0$ the crossed products of $C$ and $D$ by the actions of $H^0$ on $C$ and $D$ induced by $\widehat{\rho}$ and $\widehat{\sigma}$, respectively. Similarly, we define $\widehat{\widehat{\rho}}$ and $\widehat{\widehat{\sigma}}$, $C_2 =C_1\rtimes_{\widehat{\widehat{\rho}}}H$ and $D_2 =D_1 \rtimes_{\widehat{\widehat{\sigma}}}H$, respectively. Let $E_2^{\rho, u}$ and $E_2^{\sigma, v}$ be the canonical conditional expectations from $C_1$ and $D_1$ onto $C$ and $D$ defined in the same way as above. Also, let $E_3^{\rho, u}$ and $E_3^{\sigma, v}$ be the canonical conditional expectations from $C_2$ and $D_2$ onto $C_1$ and $D_1$ defined in the same way as above, respectively. Using $E_1^{\rho, u}$ and $E_1^{\sigma, v}$, we regard $C$ and $D$ as right Hilbert $A$-modules, respectively. Let ${\mathbf B}_A (C)$ and ${\mathbf B}_A (D)$ be the $C^*$-algebras of all right $A$-module maps on $C$ and $D$, respectively. We note that any right module map in ${\mathbf B}_A (C)$ or ${\mathbf B}_A (D)$ is adjointable by Kajiwara and Watatani [@KW1:bimodule Lemma 1.10] since $C$ and $D$ are of finite index in the sense of [@KW1:bimodule]. For any $x, y\in C$, let $\theta_{x, y}^C$ be the rank-one $A$-module map on $C$ defined by $\theta_{x, y}^C (z)=x\cdot {\langle}y, z {\rangle}_A$ for any $z\in C$. Similarly, we define $\theta_{x,y}^D $ the rank-one $A$-module map on $D$ for any $x, y\in D$. Then ${\mathbf B}_A (C)$ and ${\mathbf B}_A (D)$ are the linear spans of the above rank-one $A$-module maps on $C$ and $D$, respectively. Furthermore, we define ${\mathbf B}_C (C_1 )$, ${\mathbf B}_D (D_1 )$ and $\theta_{x, y}^{C_1}$, $\theta_{x',y'}^{D_1}$ for $x, y\in C_1$, $x', y' \in D_1$. \[lem:equation2\]With the above notations and assumptions, there is an isomorphism $\widehat{\Psi}$ of $D_1$ onto $C_1$ satisfying that $$\widehat{\Psi}|_D =\Psi, \quad \Psi\circ E_2^{\sigma, v}=E_2^{\rho, u}\circ\widehat{\Psi}, \quad \widehat{\Psi}(1\rtimes_{\widehat{\sigma}}\tau)=1\rtimes_{\widehat{\rho}}\tau .$$ We note that $C_1 \cong{\mathbf B}_A (C)$ and $D_1 \cong {\mathbf B}_A (D)$, respectively. Then $\Psi$ can be regarded as a Hilbert $A-A$-bimodule isomorphism of $D$ onto $C$ since $E_1^{\rho, u}\circ\Psi=E_1^{\sigma, v}$. Let $\widehat{\Psi}$ be the map from ${\mathbf B}_A (D)$ to ${\mathbf B}_A (C)$ defined by $\widehat{\Psi}(T)=\Psi\circ T\circ\Psi^{-1}$ for any $T\in {\mathbf B}_A (D)$. By routine computations, $\widehat{\Psi}$ is an isomorphism of $D_1$ onto $C_1$. For any $x\in D$, let $T_x$ be the element in ${\mathbf B}_A (D)$ defined by $T_x (y)=xy$ for any $y\in D$. Then for any $x\in D$ and $z\in C$, $$\widehat{\Psi}(T_x )(z) =(\Psi\circ T_x \circ\Psi^{-1})(z) = \Psi(x\Psi^{-1}(z))=\Psi(x)z =T_{\Psi(x)}(z) .$$ Hence $\widehat{\Psi}|_D =\Psi$. Also, since $1\rtimes_{\widehat{\rho}}\tau$ and $1\rtimes_{\widehat{\sigma}}\tau$ are identified with $\theta_{1,1}^C$ and $\theta_{1,1}^D$, respectively. For any $z\in C$, $$\begin{aligned} \widehat{\Psi}(1\rtimes_{\widehat{\sigma}}\tau)(z) & =(\Psi\circ\theta_{1,1}^D \circ\Psi^{-1})(z) =\Psi (1\cdot {\langle}1, \, \Psi^{-1}(z) {\rangle}_A )=\Psi(E_1^{\sigma, v}(\Psi^{-1}(z))) \\ & =E_1^{\sigma, v}(\Psi^{-1}(z))=E_1^{\rho, u}(z)=\theta_{1,1}^C (z)=(1\rtimes_{\widehat{\rho}}\tau)(z) .\end{aligned}$$ Thus, $\widehat{\Psi}(1\rtimes_{\widehat{\sigma}}\tau)=1\rtimes_{\widehat{\rho}}\tau$. Furthermore, for any $x, y\in D$, $$\begin{aligned} (E_2^{\rho, u}\circ\widehat{\Psi})((x\rtimes_{\widehat{\sigma}}1^0 ) (1\rtimes_{\widehat{\sigma}}\tau)(y\rtimes_{\widehat{\sigma}}1^0 )) & =E_2^{\rho, u}((\Psi(x)\rtimes_{\widehat{\rho}}1^0 ) (1\rtimes_{\widehat{\rho}}\tau)(\Psi(y)\rtimes_{\widehat{\rho}}1^0 )) \\ & =\Psi(x)\tau(e)\Psi(y)=\Psi(xy)\tau(e) .\end{aligned}$$ On the other hand $$(\Psi\circ E_2^{\sigma, v})((x\rtimes_{\widehat{\sigma}}1^0 )(1\rtimes_{\widehat{\sigma}}\tau) (y\rtimes_{\widehat{\sigma}}1^0 ))=\Psi(x\tau(e)y) =\Psi(xy)\tau(e) .$$ Hence $E_2^{\rho, u}\circ\widehat{\Psi}=\Psi\circ E_2^{\sigma, v}$. Therefore, we obtain the conclusion. \[lem:equation3\]With the above notations and assumptions, there is an isomorphism $\widehat{\widehat{\Psi}}$ of $D_2$ onto $C_2$ satisfying that $$\widehat{\widehat{\Psi}}|_{D_1} =\widehat{\Psi}, \quad \widehat{\widehat{\Psi}}(1_{D_1}\rtimes_{\widehat{\widehat{\sigma}}}e) =1_{C_1}\rtimes_{\widehat{\widehat{\rho}}}e , \quad E_3^{\rho, u}\circ\widehat{\widehat{\Psi}}=\widehat{\Psi}\circ E_3^{\sigma, v} .$$ We can prove this lemma in the same way as in the proof of Lemma \[lem:equation2\] By Lemmas \[lem:equation2\] and \[lem:equation3\], $\widehat{\Psi}|_{A' \cap D_1}$ and $\widehat{\widehat{\Psi}}|_{D' \cap D_2}$ are isomorphisms of $A' \cap D_1$ and $D' \cap D_2$ onto $A' \cap C_1$ and $C' \cap C_2$, respectively. \[lem:Ciso\]With the above notations and assumptions, $A' \cap D_1 \cong H^0$ as $C^*$-algebras. Since $H^0 \cong 1\rtimes_{\widehat{\sigma}}H^0$ as $C^*$-algebras, it suffices to show that $A' \cap D_1 =1\rtimes_{\widehat{\sigma}}H^0$, where we identify $A$ with $A\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}1^0$. Let $x\in A' \cap D_1$. Then we can write that $x=\sum_i x_i \rtimes_{\widehat{\sigma}}\phi_i$, where $x_i \in D$ and $\{\phi_i \}_i$ is a basis of $H^0$. For any $a\in A$, $$\begin{aligned} (a\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}1^0 )x & =\sum_i (a\rtimes_{\sigma, v}1)x_i \rtimes_{\widehat{\sigma}}\phi_i , \\ x(a\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}1^0 ) & =\sum_i (x_i \rtimes_{\widehat{\sigma}}\phi_i ) (a\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}1^0 ) \\ & =\sum_i x_i [\phi_{i (1)}\cdot_{\widehat{\sigma}}a\rtimes_{\sigma, v}1]\rtimes_{\widehat{\sigma}}\phi_{i(2)} \\ & =\sum_i x_i (a\rtimes_{\sigma, v}1)\rtimes_{\widehat{\sigma}}\phi_i .\end{aligned}$$ Since $(a\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}1^0 )x =x(a\rtimes_{\sigma, v}1\rtimes_{\widehat{\sigma}}1^0 )$, $(a\rtimes_{\sigma, v}1)x_i =x_i (a\rtimes_{\sigma, v}1)$ for any $a\in A$ and $i$. Since $A' \cap D={\mathbf C}1$, $x_i \in {\mathbf C}1$ for any $i$. Therefore, $A' \cap D_1 \subset 1\rtimes_{\widehat{\sigma}}H^0$. The inverse inclusion is clear. Hence we obtain the conclusion. By Lemma \[lem:Ciso\], $\widehat{\Psi}|_{A' \cap D_1}$ can be regarded as a $C^*$-algebra automorphism of $H^0$. We denote it by $\lambda^0$. \[lem:Hopfiso\]With the above notations, $\lambda^0$ is a $C^*$-Hopf algebra automorphism of $H^0$. It suffices to show that for any $\phi\in H^0$, $$S^0 (\lambda^0 (\phi))=\lambda^0 (S^0 (\phi)), \quad \Delta^0 (\lambda^0 (\phi))=(\lambda^0 \otimes\lambda^0 )(\Delta^0 (\phi)), \quad \epsilon^0 (\lambda^0 (\phi))=\epsilon^0 (\phi) .$$ We note that for any $h\in H$, $\phi\in H^0$, $$\phi(h)=N^2 (E_2^{\rho, u}\circ E_3^{\rho, u})((1\rtimes_{\widehat{\rho}}\phi\rtimes_{\widehat{\widehat{\rho}}}1) (1\rtimes_{\widehat{\rho}}1^0 \rtimes_{\widehat{\widehat{\rho}}}e) (1\rtimes_{\widehat{\rho}}\tau\rtimes_{\widehat{\widehat{\rho}}}1) (1\rtimes_{\widehat{\rho}}1^0\rtimes_{\widehat{\widehat{\rho}}}h))$$ by easy computations. Indeed, $$\begin{aligned} & N^2 (E_2^{\rho, u}\circ E_3^{\rho, u})((1\rtimes_{\widehat{\rho}}\phi\rtimes_{\widehat{\widehat{\rho}}}1) (1\rtimes_{\widehat{\rho}}1^0 \rtimes_{\widehat{\widehat{\rho}}}e) (1\rtimes_{\widehat{\rho}}\tau\rtimes_{\widehat{\widehat{\rho}}}1) (1\rtimes_{\widehat{\rho}}1^0\rtimes_{\widehat{\widehat{\rho}}}h)) \\ & =N^2 (E_2^{\rho, u}\circ E_3^{\rho, u})((1\rtimes_{\widehat{\rho}}\phi\rtimes_{\widehat{\widehat{\rho}}}e) (1\rtimes_{\widehat{\rho}}\tau\rtimes_{\widehat{\widehat{\rho}}}h)) \\ & =N^2 (E_2^{\rho, u}\circ E_3^{\rho, u})((1\rtimes_{\widehat{\rho}}\phi)[e_{(1)}\cdot_{\widehat{\widehat{\rho}}} (1\rtimes_{\widehat{\rho}}\tau)]\rtimes_{\widehat{\widehat{\rho}}}e_{(2)}h) \\ & =N^2 E_2^{\rho, u}((1\rtimes_{\widehat{\rho}}\phi)[e_{(1)}\cdot_{\widehat{\widehat{\rho}}} (1\rtimes_{\widehat{\rho}}\tau)]) \tau' (e_{(2)}h) \\ & =N^2 E_2^{\rho, u}((1\rtimes_{\widehat{\rho}}\phi)(1\rtimes_{\widehat{\rho}}\tau_{(1)})\tau_{(2)}(e_{(1)})) \tau' (e_{(2)}h) \\ & =N^2 E_2^{\rho, u}(1\rtimes_{\widehat{\rho}}\phi\tau_{(1)})\tau_{(2)}(e_{(1)})\tau' (e_{(2)}h) \\ & =N^2 (\phi\tau_{(1)})(e' )\tau_{(2)}(e_{(1)})\tau' (e_{(2)}h) \\ & =N^2 (\phi\tau_{(1)})(e' )\tau_{(2)}(e_{(1)}h_{(2)}S(h_{(1)}))\tau' (e_{(2)}h_{(3)}) \\ & =N^2 (\phi\tau_{(1)})(e' )\tau_{(2)}(e_{(1)}h_{(2)})\tau_{(3)}(S(h_{(1)}))\tau' (e_{(2)}h_{(3)}) \\ & =N^2 (\phi\tau_{(1)})(e' )(\tau_{(2)}\tau' )(eh_{(2)})\tau_{(3)}(S(h_{(1)})) \\ & =N(\phi\tau_{(1)})(e' )\tau_{(2)}(S(h)) \\ & =N(\phi_{(1)}\tau_{(1)})(e' )(S^0 (\phi_{(3)})\phi_{(2)}\tau_{(2)})(S(h)) \\ & =N(\phi_{(1)}\tau_{(1)})(e' )S^0 (\phi_{(3)})(S(h_{(2)}))(\phi_{(2)}\tau_{(2)})(S(h_{(1)})) \\ & =N(\phi_{(1)}\tau)(e' S(h_{(1)}))S^0 (\phi_{(2)})(S(h_{(2)})) \\ & =S^0 (\phi)(S(h)) \\ & =\phi(h) ,\end{aligned}$$ where $e' =e$ and $\tau' =\tau$. We also note that $$\phi(h)=N^2 (E_2^{\sigma, v}\circ E_3^{\sigma, v}) ((1\rtimes_{\widehat{\sigma}}\phi\rtimes_{\widehat{\widehat{\sigma}}}1) (1\rtimes_{\widehat{\sigma}}1^0 \rtimes_{\widehat{\widehat{\sigma}}}e) (1\rtimes_{\widehat{\sigma}}\tau\rtimes_{\widehat{\widehat{\sigma}}}1^0 ) (1\rtimes_{\widehat{\sigma}}1^0 \rtimes_{\widehat{\widehat{\sigma}}}h))$$ for any $h\in H$, $\phi\in H^0$. Hence for any $h\in H$, $\phi\in H^0$, $$\begin{aligned} & \lambda^0 (\phi)(h) \\ & =N^2 (E_2^{\rho, u}\circ E_3^{\rho, u})(\widehat{\widehat{\Psi}} (1\rtimes_{\widehat{\sigma}}\phi\rtimes_{\widehat{\widehat{\sigma}}}1) (1\rtimes_{\widehat{\rho}}1^0 \rtimes_{\widehat{\widehat{\rho}}}e) (1\rtimes_{\widehat{\rho}}\tau\rtimes_{\widehat{\widehat{\rho}}}1) (1\rtimes_{\widehat{\rho}}1^0 \rtimes_{\widehat{\widehat{\rho}}}h)) \\ & =N^2 (E_2^{\rho, u}\circ E_3^{\rho, u}\circ\widehat{\widehat{\Psi}}) ((1\rtimes_{\widehat{\sigma}}\phi\rtimes_{\widehat{\widehat{\sigma}}}1) (1\rtimes_{\widehat{\sigma}}1^0 \rtimes_{\widehat{\widehat{\sigma}}}e) (1\rtimes_{\widehat{\sigma}}\tau\rtimes_{\widehat{\widehat{\sigma}}}1) \\ & \times\widehat{\widehat{\Psi}}^{-1}(1\rtimes_{\widehat{\rho}}1^0 \rtimes_{\widehat{\widehat{\rho}}}h)) \\ & =N^2 (\Psi\circ E_2^{\sigma, v}\circ E_3^{\sigma, v}) ((1\rtimes_{\widehat{\sigma}}\phi\rtimes_{\widehat{\widehat{\sigma}}}1) (1\rtimes_{\widehat{\sigma}}1^0 \rtimes_{\widehat{\widehat{\sigma}}}e) (1\rtimes_{\widehat{\sigma}}\tau\rtimes_{\widehat{\widehat{\sigma}}}1) \\ & \times\widehat{\widehat{\Psi}}^{-1}(1\rtimes_{\widehat{\rho}}1^0 \rtimes_{\widehat{\widehat{\rho}}}h)) \\ & =\Psi(\phi(\widehat{\widehat{\Psi}}^{-1}(h)))=\phi(\widehat{\widehat{\Psi}}^{-1}(h))\end{aligned}$$ by Lemmas \[lem:equation2\] and \[lem:equation3\]. Hence by the above equation, for any $h\in H$, $\phi\in H^0$, $$\begin{aligned} S^0 (\lambda^0 (\phi))(h) & =\lambda^0 (\phi)(S(h))=\overline{\lambda^0 (\phi)^* (h^* )} =\overline{\widehat{\Psi}(\phi^* )(h^* )}=\overline{\lambda^0 (\phi^* )(h^* )} \\ & =\overline{\phi^*(\widehat{\widehat{\Psi}}^{-1}(h^* ))}=\phi(S(\widehat{\widehat{\Psi}}^{-1}(h))) =S^0 (\phi)(\widehat{\widehat{\Psi}}^{-1}(h)) =\lambda^0 (S^0 (\phi))(h) .\end{aligned}$$ Thus $S^0 (\lambda^0 (\phi))=\lambda^0 (S^0 (\phi))$ for any $\phi\in H^0 $. Also, for any $h, l\in H$, $\phi\in H^0$, $$\begin{aligned} \Delta^0 (\lambda^0 (\phi))(h\otimes l) & = \lambda^0 (\phi)(hl)=\phi(\widehat{\widehat{\Psi}}^{-1}(hl)) =\Delta^0 (\phi)(\widehat{\widehat{\Psi}}^{-1}(h)\otimes\widehat{\widehat{\Psi}}^{-1}(l)) \\ & =\phi_{(1)}(\widehat{\widehat{\Psi}}^{-1}(h))\phi_{(2)}(\widehat{\widehat{\Psi}}^{-1}(l)) =\lambda^0 (\phi_{(1)})(h)\lambda^0 (\phi_{(2)})(l) \\ & =(\lambda^0 (\phi_{(1)})\otimes\lambda^0 (\phi_{(2)}))(h\otimes l) \\ & =(\lambda^0 \otimes\lambda^0 )(\Delta^0 (\phi))(h\otimes l) .\end{aligned}$$ Hence $\Delta^0 (\lambda^0 (\phi))=(\lambda^0 \otimes\lambda^0 )(\Delta^0 (\phi))$ for any $\phi\in H^0$. Furthermore, for any $\phi\in H^0 $, $$\epsilon^0 (\lambda^0 (\phi))=\lambda^0 (\phi)(1)=\phi(\widehat{\widehat{\Psi}}^{-1}(1))=\phi(1)=\epsilon(1) .$$ Therefore, we obtain the conclusion. \[lem:conjugate\]With the above notations, $$\widehat{\widehat{\rho}}\circ\widehat{\Psi}=(\widehat{\Psi}\otimes\lambda^0 )\circ\widehat{\widehat{\sigma}}$$ on $D_1$. For any $x\in D$, $$\begin{aligned} (\widehat{\widehat{\rho}}\circ\widehat{\Psi})(x\rtimes_{\widehat{\sigma}}1^0 ) &=\widehat{\widehat{\rho}} (\Psi(x)\rtimes_{\widehat{\rho}}1^0 )= (\Psi(x)\rtimes_{\widehat{\sigma}}1^ 0)\otimes 1^0 =\widehat{\Psi}(x\rtimes_{\widehat{\sigma}}1^0 )\otimes 1^0 \\ & =(\widehat{\Psi}\otimes\lambda^0 )((x\rtimes_{\widehat{\sigma}}1^0 )\otimes 1^0 ) =((\widehat{\Psi}\otimes\lambda^0 )\circ\widehat{\widehat{\sigma}})(x\rtimes_{\widehat{\sigma}}1^0 )\end{aligned}$$ by Lemma \[lem:equation2\]. Also, for any $\phi\in H^0$, $$\begin{aligned} (\widehat{\widehat{\rho}}\circ\widehat{\Psi})(1\rtimes_{\widehat{\sigma}}\phi) & = \widehat{\widehat{\rho}}(1\rtimes_{\widehat{\rho}}\lambda^0 (\phi))= (1\rtimes_{\widehat{\rho}}\lambda^0 (\phi)_{(1)})\otimes\lambda(\phi)_{(2)} \\ & =(1\rtimes_{\widehat{\rho}}\lambda^0 (\phi_{(1)}))\otimes\lambda^0 (\phi_{(2)}) =\widehat{\Psi}(1\rtimes_{\widehat{\sigma}}\phi_{(1)})\otimes\lambda^0 (\phi_{(2)}) \\ & =(\widehat{\Psi}\otimes\lambda^0 )((1\rtimes_{\widehat{\sigma}}\phi_{(1)})\otimes\phi_{(2)}) =((\widehat{\Psi}\otimes\lambda^0 )\circ\widehat{\widehat{\sigma}})(1\rtimes_{\widehat{\sigma}}\phi)\end{aligned}$$ by Lemma \[lem:Hopfiso\]. Therefore, we obtain that $\widehat{\widehat{\rho}}\circ\widehat{\Psi}=(\widehat{\Psi}\otimes\lambda^0 )\circ\widehat{\widehat{\sigma}}$. \[prop:key2\]Let $(\rho, u)$ and $(\sigma, v)$ be twisted coactions of $H^0$ on $A$. Let $C=A\rtimes_{\rho, u}H$ and $D=A\rtimes_{\sigma, v}H$. We suppose that there is an isomorphism $\Psi$ of $C$ onto $D$ satisfying that $$\Psi|_A ={{\rm{id}}}_A , \quad E_1^{\rho, u}\circ\Psi=E_1^{\sigma, v} ,$$ where $E_1^{\rho, u}$ and $E_1^{\sigma, v}$ are the canonical conditional expectations from $C$ and $D$ onto $A$ defined by $$E_1^{\rho, u}(a\rtimes_{\rho, u}h)=\tau(h)a , \quad E_1^{\sigma, v}(a\rtimes_{\sigma, v}h)=\tau(h)a$$ for any $a\in A$, $h\in H$, respectively. Furthermore, we suppose that $A' \cap C={\mathbf C}1$. Then there is a $C^*$-Hopf algebra automorphism $\lambda^0$ of $H^0$ such that $(\rho, u)$ is strongly Morita equivalent to the twisted coaction $(({{\rm{id}}}_A \otimes \lambda^0 )\circ \sigma,\, ({{\rm{id}}}_A \otimes \lambda^0 \otimes \lambda^0 )(v))$. By [@KT2:coaction Theorem 3.3], there are isomorphism of $\Phi_{\sigma}$ of $A\otimes M_N ({\mathbf C})$ onto $D_1$ and a unitary element $U_{\sigma}\in D_1 \otimes H^0$ such that $$\begin{aligned} {{\rm{Ad}}}(U_{\sigma})\circ\widehat{\widehat{\sigma}} & =(\Phi_{\sigma}\otimes{{\rm{id}}}_{H^0}) \circ(\sigma\otimes{{\rm{id}}}_{M_N ({\mathbf C})})\circ\Phi_{\sigma}^{-1} , \\ (\Phi_{\sigma}\otimes{{\rm{id}}}_{H^0}\otimes{{\rm{id}}}_{H^0})(v\otimes I_N) & =(U_{\sigma}\otimes 1^0 ) (\widehat{\widehat{\sigma}}\otimes{{\rm{id}}}_{H^0})(U_{\sigma})({{\rm{id}}}\otimes\Delta^0 )(U_{\sigma}^* ) ,\end{aligned}$$ where we identify $M_N ({\mathbf C})\otimes H^0$ with $H^0 \otimes M_N ({\mathbf C})$. Thus by Lemma \[lem:exterior\], $({{\rm{id}}}_{D_1}\otimes\lambda^0 )\circ\widehat{\widehat{\sigma}}$ is exterior equivalent to the twisted coaction $$\begin{aligned} (({{\rm{id}}}_{D_1}\otimes\lambda^0 )& \circ(\Phi_{\sigma}\otimes{{\rm{id}}}_{H^0 }) \circ(\sigma\otimes{{\rm{id}}}_{M_N ({\mathbf C})})\circ\Phi_{\sigma}^{-1} \, , \\ & ({{\rm{id}}}_{D_1}\otimes\lambda^0 \otimes\lambda^0 )((\Phi_{\sigma}\otimes{{\rm{id}}}_{H^0}\otimes{{\rm{id}}}_{H^0})(v\otimes I_N ))) ,\end{aligned}$$ where $\lambda^0 $ is the $C^*$-Hopf automorphism of $H^0$ defined before Lemma \[lem:Ciso\]. Since $$\begin{aligned} & ({{\rm{id}}}_{D_1}\otimes\lambda^0 ) \circ(\Phi_{\sigma}\otimes{{\rm{id}}}_{H^0 }) \circ(\sigma\otimes{{\rm{id}}}_{M_N ({\mathbf C})})\circ\Phi_{\sigma}^{-1} \\ & =(\Phi_{\sigma}\otimes{{\rm{id}}}_{H^0})\circ({{\rm{id}}}_{A\otimes M_N ({\mathbf C})}\otimes\lambda^0 ) \circ(\sigma\otimes{{\rm{id}}}_{M_N ({\mathbf C})})\circ\Phi_{\sigma}^{-1},\end{aligned}$$ and $$\begin{aligned} & ({{\rm{id}}}_{D_1}\otimes\lambda^0 \otimes\lambda^0 ) ((\Phi_{\sigma}\otimes{{\rm{id}}}_{H^0}\otimes{{\rm{id}}}_{H^0})(v\otimes I_N ))) \\ & =(\Phi_{\sigma}\otimes{{\rm{id}}}_{H^0}\otimes{{\rm{id}}}_{H^0}) (({{\rm{id}}}_{A\otimes M_N ({\mathbf C})}\otimes\lambda^0 \otimes\lambda^0 )(v\otimes I_N )) ,\end{aligned}$$ by Lemma \[lem:iso\], the twisted coaction $$(({{\rm{id}}}_{A\otimes M_N ({\mathbf C})}\otimes\lambda^0 ) \circ(\sigma\otimes{{\rm{id}}}_{M_N ({\mathbf C})}) \, , \, ({{\rm{id}}}_{A\otimes M_N ({\mathbf C})}\otimes\lambda^0 \otimes\lambda^0 )(v\otimes I_N ))$$ is strongly Morita equivalent to the twisted coaction $$\begin{aligned} (({{\rm{id}}}_{D_1}\otimes\lambda^0 )& \circ(\Phi_{\sigma}\otimes{{\rm{id}}}_{H^0 }) \circ(\sigma\otimes{{\rm{id}}}_{M_N ({\mathbf C})})\circ\Phi_{\sigma}^{-1} \, , \\ & ({{\rm{id}}}_{D_1}\otimes\lambda^0 \otimes\lambda^0 )((\Phi_{\sigma}\otimes{{\rm{id}}}_{H^0}\otimes{{\rm{id}}}_{H^0})(v\otimes I_N ))) ,\end{aligned}$$ Hence $({{\rm{id}}}_{D_1}\otimes\lambda^0 )\circ\widehat{\widehat{\sigma}}$ is strongly Morita equivalent to $$(({{\rm{id}}}_{A\otimes M_N ({\mathbf C})}\otimes\lambda^0 ) \circ(\sigma\otimes{{\rm{id}}}_{M_N ({\mathbf C})}) \, , \, ({{\rm{id}}}_{A\otimes M_N ({\mathbf C})}\otimes\lambda^0 \otimes\lambda^0 )(v\otimes I_N )) .$$ Thus by [@Kodaka:equivariance Lemma 4.10], $({{\rm{id}}}_{D_1}\otimes\lambda^0 )\circ\widehat{\widehat{\sigma}}$ is strongly Morita equivalent to $$(({{\rm{id}}}_A \otimes\lambda^0 )\circ\sigma \, , \, ({{\rm{id}}}_A \otimes\lambda^0 \otimes \lambda^0 )(v)) .$$ On the other hand, by Lemmas \[lem:exterior\] and \[lem:conjugate\], $\widehat{\widehat{\rho}}$ is strongly Moriat equivalent to $({{\rm{id}}}_{D_1}\otimes\lambda^0 )\circ\widehat{\widehat{\sigma}}$. Since $\widehat{\widehat{\rho}}$ is strongly Morita equivalent to $(\rho, u)$ by Remark \[remark:review\], $(\rho, u)$ is strongly Morita equivalent to $$(({{\rm{id}}}_A \otimes\lambda^0 )\circ\sigma \, , \, ({{\rm{id}}}_A \otimes\lambda^0 \otimes \lambda^0 )(v)) .$$ \[cor:key3\]Let $(\rho, u)$ and $(\sigma, v)$ be twisted coactions of $H^0$ on $A$. Let $C=A\rtimes_{\rho, u}H$ and $D=A\rtimes_{\sigma, v}H$. We regard $A$ as an $A-A$equivalent bimodule in the usual way. We suppose that the unital inclusions $A\subset C$ and $A\subset D$ are strongly Morita equivalent with respect to a $C-D$-equivalence bimodule $Y$ and its closed subspace $A$, that is, we assume that the $A-A$-equivalence bimodule $A$ is included in $Y$. Furthermore, we suppose that $A' \cap C={\mathbf C}1$. Then there is a $C^*$-Hopf algebra automorphism $\lambda^0$ of $H^0$ such that $(\rho, u)$ is strongly Morita equivalent to the twisted coaction $$(({{\rm{id}}}_A \otimes \lambda^0 )\circ\sigma ,\, ({{\rm{id}}}_A \otimes \lambda^0 \otimes \lambda^0 )(v)) .$$ This is immediate by Proposition \[prop:key2\] and the discussions before Lemma \[lem:equation\]. The main result {#sec:main} =============== In this section, we present the main result in the paper. We recall the previous discussions: Let $A$ and $B$ be unital $C^*$-algebras and $H$ a finite dimensional $C^*$-Hopf algebra with its dual $C^*$-Hopf algebra $H^0$. Let $(\rho, u)$ and $(\sigma, v)$ be twisted coactions of $H^0$ on $A$ and $B$, respectively. Le $C=A\rtimes_{\rho, u}H$ and $D=B\rtimes_{\sigma, v}H$ and let $C_1 =C\rtimes_{\widehat{\rho}}H^0$ and $D_1 =D\rtimes_{\widehat{\sigma}}H^0$, where $\widehat{\rho}$ and $\widehat{\sigma}$ are the dual coactions of $H$ on $C$ and $D$ induced by $(\rho, u)$ and $(\sigma, v)$, respectively. W suppose that the inclusions $A\subset C$ and $B\subset D$ are strongly Morita equivalent with respect to a $C-D$-equivalence bimodule $Y$ and its closed subspace $X$ in the sense of [@KT4:morita Definition 2.1]. Furthermore, we suppose that $A' \cap C={\mathbf C}1$. Then by Proposition \[prop:leftcoaction\] and Lemma \[lem:action\] there are a coaction $\beta$ of $H$ on $D$ and a coaction $\mu$ of $H$ on $Y$ such that $(C, D, Y, \widehat{\rho}, \beta, \mu, H)$ is a covariant system. Hence the coactions $\widehat{\rho}$ and $\beta$ of $H$ on $D$ are strongly Morita equivalent. Moreover, by the discussions at the end of Section \[sec:coaction\], the unital inclusions $D\subset D_1 (=D\rtimes_{\widehat{\sigma}}H^0 )$ and $D\subset D\rtimes_{\beta}H^0 $ are strongly Morita equivalent with respect to a $D_1 -D\rtimes_{\beta}H^0$-equivalence bimodule $W(=\widetilde{Y_1}\otimes_{C_1}(Y\rtimes_{\mu}H^0 ))$ and its closed subspace $D$, where we regard $D$ as a $D-D$-equivalence bimodule in the usual way and we can also see that $D$ is included in $W$ by the discussions at the end of Section \[sec:coaction\]. Since $A' \cap C={\mathbf C}1$, by [@KT4:morita Lemma 10.3] and $D' \cap D_1 ={\mathbf C}1$. Thus by Corollary \[cor:key3\], there is a $C^*$-Hopf algebra automorphism $\lambda$ of $H$ such that $\beta$ is strongly Morita equivalent to $({{\rm{id}}}_D \otimes\lambda)\circ\widehat{\sigma}$. Hence $\widehat{\rho}$ is strongly Morita equivalent to $({{\rm{id}}}_D \otimes\lambda)\circ\widehat{\sigma}$. Let $\lambda^0$ be the $C^*$-Hopf algebra automorphism of $H^0$ induced by $\lambda$, that is, we define $\lambda^0$ as follows: For any $\psi\in H^0$, $h\in H$, $$\lambda^0 (\psi)(h)=\psi(\lambda^{-1}(h)) .$$ Let $$\sigma_{\lambda^0}=({{\rm{id}}}_B \otimes\lambda^0 )\circ\sigma , \quad v_{\lambda^0}=({{\rm{id}}}_B \otimes\lambda^0 \otimes\lambda^0 )(v) .$$ Then by Lemma \[lem:induced\], $(\sigma_{\lambda^0 }, v_{\lambda^0})$ is a twisted coaction of $H^0$ on $B$. Let $D_{\lambda^0} =B\rtimes_{\sigma_{\lambda^0}, v_{\lambda^0}}H$. \[lem:iso3\]With the above notations, let $\pi$ be the map from $D$ to $D_{\lambda^0}$ defined by $$\pi(b\rtimes_{\sigma, v}h)=b\rtimes_{\sigma_{\lambda^0}, v_{\lambda^0}}\lambda(h)$$ for any $b\in B$, $h\in H$. Then $\pi$ is an isomorphism of $D$ onto $D_{\lambda^0}$ such that $$\widehat{\sigma_{\lambda^0}}\circ\pi=(\pi\otimes{{\rm{id}}}_{H})\circ({{\rm{id}}}_D \otimes\lambda)\circ\widehat{\sigma} ,$$ where $\widehat{\sigma_{\lambda^0}}$ is the dual coaction of $(\sigma_{\lambda^0}, v_{\lambda^0})$, a coaction of $H$ on $D_{\lambda^0}$. This is immediate by routine computations. \[thm:main\]Let $A$ and $B$ be unital $C^*$-algebras and $H$ a finite dimensional $C^*$-Hopf algebra with its dual $C^*$-Hopf algebra $H^0$. Let $(\rho, u)$ and $(\sigma, v)$ be twisted coactions of $H^0$ on $A$ and $B$, respectively. Let $C=A\rtimes_{\rho, u}H$ and $D=B\rtimes_{\sigma ,v}H$. We suppose that the unital inclusions $A\subset C$ and $B\subset D$ are strongly Morita equivalent. Also, we suppose that $A' \cap C={\mathbf C}1$. Then there is a $C^*$-Hopf algebra automorphism $\lambda^0 $ of $H^0$ such that the twisted coaction $(\rho, u)$ is strongly Morita equivalent to the twisted coaction $$(({{\rm{id}}}_B \otimes \lambda^0 )\circ\sigma , \, ({{\rm{id}}}_B \otimes\lambda^0 \otimes\lambda^0 )(v)) ,$$ induced by $(\sigma, v)$ and $\lambda^0$. By the discussions at the beginning of this section, the dual coaction $\widehat{\rho}$ of $(\rho, u)$ is strongly Morita equivalent to $({{\rm{id}}}_D \otimes\lambda)\circ\widehat{\sigma}$. Also, by Lemmas \[lem:iso\] and \[lem:iso3\], $({{\rm{id}}}_D \otimes\lambda)\circ\widehat{\sigma}$ is strongly Morita equivalent to $(({{\rm{id}}}_B \otimes \lambda^0 )\circ\sigma)\,\widehat{}$, the dual coaction of the twisted coaction $$(({{\rm{id}}}_B \otimes\lambda^0 )\circ\sigma , \, ({{\rm{id}}}\otimes\lambda^0 \otimes\lambda^0 )(v)) ,$$ where $\lambda^0$ is the $C^*$-Hopf algebra automorphism of $H^0$ induced by $\lambda$. Hence by [@KT3:equivalence Corollary 4.8], $(\rho, u)$ is strongly Morita equivalent to $$(({{\rm{id}}}_B \otimes\lambda^0 )\circ\sigma , \, ({{\rm{id}}}\otimes\lambda^0 \otimes\lambda^0 )(v)) ,$$ induced by $(\sigma, v)$ and $\lambda^0$. In the rest of the paper, we show the inverse implication of Theorem \[thm:main\]. \[lem:inverse1\]Let $A\subset C$ and $B\subset D$ be unital inclusions of unital $C^*$-algebras. We suppose that there is an isomorphism $\pi$ of $D$ onto $C$ such that $\pi|_B$ is an isomorphism of $B$ onto $A$. Then $A\subset C$ and $B\subset D$ are strongly Morita equivalent. Let$Y_{\pi}$ be the $D-C$-equivalence bimodule induced by $\pi$, that is, $Y_{\pi}=D$ as vector spaces and the left $D$-action on $Y_{\pi}$ and the left $D$-valued inner product are defined in the evident way. We define the right $C$-action and the right $C$-valued inner product as follows: For any $c\in C$ and $y, z\in Y_{\pi}$, $$y\cdot c=y\pi^{-1}(c), \quad {\langle}y, z {\rangle}_C=\pi(y^* z) .$$ Then $B$ is a closed subset of $Y_{\pi}$ and $Y_{\pi}$ and $B$ satisfy Conditions (1), (2) in [@KT4:morita Definition 2.1] by easy computations. Therefore, $A\subset C$ and $B\subset D$ are strongly Morita equivalent. Let $(\rho, u)$ be a twisted coaction of $H^0$ on $A$ and $\lambda^0$ a $C^*$-Hopf algebra automorphism of $H^0$. Let $(\rho_{\lambda^0}, \, u_{\lambda^0})$ be the twisted coaction of $H^0$ on $A$ induced by $(\rho, u)$ and $\lambda^0$, that is $$\rho_{\lambda^0}=({{\rm{id}}}\otimes\lambda^0 )\circ\rho, \quad u_{\lambda^0}=({{\rm{id}}}\otimes\lambda^0 \otimes\lambda^0 )(u) .$$ \[lem:inverse2\]With the above notations, the unital inclusions $A\subset A\rtimes_{\rho, u}H$ and $A\subset A\rtimes_{\rho_{\lambda^0}, u_{\lambda^0}}H$ are strongly Morita equivalent. Let $\pi$ be the map from $A\rtimes_{\rho, u}H$ to $A\rtimes_{\rho_{\lambda^0}, u_{\lambda^0}}H$ defined by $$\pi(a\rtimes_{\rho, u}h)=a\rtimes_{\rho_{\lambda^0}, u_{\lambda^0}}\lambda(h)$$ for any $a\in A$, $h\in H$, where $\lambda$ is the $C^*$-Hopf algebra automorphism of $H$ induced by $\lambda^0$, that is, $$\lambda^0 (\psi)(h)=\psi(\lambda^{-1}(h))$$ for any $\psi\in H^0$ and $h\in H$. By easy computations, $\pi$ is an isomorphism of $A\rtimes_{\rho, u}H$ onto $A\rtimes_{\rho_{\lambda^0}, u_{\lambda^0}}H$. Also, for any $a\in A$, $$\pi(a\rtimes_{\rho, u}1) =a\rtimes_{\rho_{\lambda^0}, u_{\lambda^0 }}\lambda(1) =a\rtimes_{\rho_{\lambda^0}, u_{\lambda^0}}\lambda(1)=a\rtimes_{\rho_{\lambda^0}, u_{\lambda^0}}1 .$$ Hence $\pi_A ={{\rm{id}}}_A$. Thus, by Lemma \[lem:inverse1\], we obtain the conclusion. \[cor:iff\]Let $A$ and $B$ be unital $C^*$-algebras and $H$ a finite dimensional $C^*$-Hopf algebra wit hits dual $C^*$-Hopf algebra $H^0$. 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--- abstract: 'This paper describes our work on demonstrating verification technologies on a flight-critical system of realistic functionality, size, and complexity. Our work targeted a commercial aircraft control system named Transport Class Model (TCM), and involved several stages: formalizing and disambiguating requirements in collaboration with domain experts; processing models for their use by formal verification tools; applying compositional techniques at the architectural and component level to scale verification. Performed in the context of a major NASA milestone, this study of formal verification in practice is one of the most challenging that our group has performed, and it took several person months to complete it. This paper describes the methodology that we followed and the lessons that we learned.' author: - Guillaume Brat - David Bushnell - Misty Davies - | \ Dimitra Giannakopoulou - 'Falk Howar [^1]' - Temesghen Kahsai bibliography: - 'biblio.bib' title: 'Verifying the Safety of a Flight-Critical System' --- [^1]: F. Howar did this work while at Carnegie Mellon University.

--- abstract: 'We show that the spectrum of radial pulsation modes in luminous red giants consists of both normal modes and a second set of modes with periods similar to those of the normal modes. These additional modes are the red giant analogues of the strange modes found in classical Cepheids and RR Lyrae variables. Here, we describe the behaviour of strange and normal modes in luminous red giants and discuss the dependence of both the strange and normal modes on the outer boundary conditions. The strange modes always appear to be damped, much more so than the normal modes. They should never be observed as self-excited modes in real red giants but they may be detected in the spectrum of solar-like oscillations. A strange mode with a period close to that of a normal mode can influence both the period and growth rate of the normal mode.' author: - | P. R. Wood$^{1}$[^1] and E. A. Olivier$^{2}$\ $^{1}$Research School of Astronomy and Astrophysics, Australian National University, Cotter Road, Weston Creek ACT 2611, Australia\ $^{2}$Physics Department, University of Western Cape, Private bag X17, Cape Town 7535, South Africa;\ South African Astronomicaly Observatory, PO Box 9, Observatory 7935, Cape Town, South Africa title: Strange pulsation modes in luminous red giants --- \[firstpage\] stars: AGB and post-AGB – stars: oscillations (including pulsations) – stars: variables: general. Introduction ============ Luminous red giant stars are known to exhibit periods of variation that fall on 7 or more roughly parallel period-luminosity sequences [e.g. @woo99; @ita04; @fra05; @sos07]. Some of these sequences are known to be due to different radial pulsation modes [@woo99; @sos07; @tak13]. While exploring the periods and stability of radial pulsation modes in luminous red giants, we encountered situations where we found two radial modes with identical pulsation periods. It turned out that there are two independent sets of radial pulsation modes occurring in luminous red giants, one set being the well-known normal modes and the second set being the strange modes encountered in classical Cepheid and RR Lyrae variables [@buc97; @buc01]. Related strange modes also appear in luminous main-sequence stars [@sai98]. Here we describe the behaviour of the strange modes in red giants and how they depend on the surface boundary conditions. The model calculations ====================== The study of radial pulsation in red giants requires construction of a static model and then an analysis of the linear nonadiabatic modes of pulsation. A pair of computer codes is required for these two steps. The codes used are based on those described by @fox82 and modified to include turbulent viscosity as described in @kel06 (although a turbulent viscosity parameter $\alpha_{\rm \nu} = 0.0$ was used in the present calculations). Convective energy transport is treated using mixing length theory: these models do not include turbulent pressure or the kinetic energy of turbulent motions. Opacities in the interior are from the OPAL project [@igl96] while in the outer layers we use the opacities of @mar09 which include a molecular component. The models use a composition X=0.73 and Z=0.008, which is appropriate for young to intermediate age stars in the Large Magellanic Cloud (LMC). A mixing length of 1.97 pressure scale heights was used (to reproduce the giant branch temperature given by @kam10 for the luminous O-rich stars in the populous intermediate age LMC cluster NGC1978). A problem with the study of red giants is that, unlike classical pulsating stars such as Cepheids and RR Lyraes, the atmosphere is not thin and selecting the position for the outer boundary is not straightforward. The outer boundary in the static models is placed according to two requirements. The mass zoning in the scheme used by @fox82 has radius $r$ and luminosity $L_r$ defined at zone boundaries $j$=1,...,N+1 where j=N+1 corresponds to the surface of the star. The gas pressure $P_{gas}$ and temperature $T$ are defined at zone centres $j+\frac{1}{2}$, $j$=1,...,N. @fox82 use a boundary condition $P_{\rm gas}$=0 at $j$=N+1. In this study, we do not set the gas pressure to zero at the surface. Instead, our first requirement at the surface is that $P_{\rm gas}$($j$=N+1)=0.9$P_{\rm gas}$($j$=N+$\frac{1}{2}$) so that there is a significant gas pressure at the surface. The factor 0.9 is arbitrary and is chosen so that the change in gas pressure across the surface zone is not too large. The second requirement is that the optical depth from the surface ($j$=N+1) to the centre of the outermost zone ($j$=N+$\frac{1}{2}$) is a pre-specified value $\tau_{\rm c}$. Given these boundary conditions and an initial guess at $R=r$($j$=N+1), the equations of hydrostatic structure are integrated in from the surface to the core, defined to be at $r = 0.15\,{\rm R}_{\odot}$. The mass of the core is defined by this inward integration. The outer radius $R$ is then found by iterating on $R$ until the core mass has the required value $M_{\rm core}$. For AGB stars, we obtain $M_{\rm core}$ from the core mass-luminosity relation of @woo81 while for RGB stars $M_{\rm core}$ comes from a fit to the core mass-luminosity relation of the evolutionary models of @ber08 for Z=0.008. At this stage, the radius and effective temperature of the model and the mass coordinates of each zone are defined: they are dependent on the input stellar mass ($M$), luminosity ($L$) and composition (as well as the input physics and mixing length parameters). The mechanical outer boundary condition in pulsation models ----------------------------------------------------------- The gas pressure at the surface ($M_{\rm r}=M$) of our models is not zero so we need to account for its variation as the defined stellar surface oscillates. Since we are typically studying up to 8 modes in each star, the frequencies of the higher overtones can approach or exceed the acoustic cutoff frequency at the surface. This means that we need to account for the possibility of running waves escaping through $M_{\rm r}=M$ into the layers above. In this study, we follow a slightly modified version of the approach described in the Appendix, part $b$, of @bak65. We let the radius and pressure variations above $r = R$ be given by $r=r_{0}(1+xe^{i \omega t})$ and $P_{\rm gas}=P_{\rm gas,0}(1+pe^{i \omega t})$, respectively, where the subscript 0 indicates the static value. We adopt the assumptions of @bak65 that the region above the surface ($M_{\rm r}=M$, $r = R$) of the star which influences the interior pulsation is relatively small in radius compared to $R$ and it is effectively isothermal so that the gas pressure scale height $H_0$ and the ratio $H_0/r \approx H_0/R$ are constant. With these approximations, the variation of $x$ with radius is given by $x \propto e^{\nu r_{0}/R}$ where $$\nu = \frac{1}{2h} \left\{ (1-4h)-\left[ (1-4h)^2-4h( \frac{4+3\sigma^2}{\gamma}-3 ) \right]^{\frac{1}{2}} \right\} .$$ Here, $h = H_0/R$ and $\sigma^2 = \omega_{\rm r}^2/(3GM/R^3)$, where $\omega_{\rm r}$ is the real part of $\omega$. Without making any further assumptions about $\nu$ (@bak65 assumed $h \ll 1$), the general relation between $p$ and $x$ at $r=R$ is $$p = -\gamma (\nu+3)x~~.$$ This is the mechanical boundary condition we use in our calculations. We adopted $\gamma = 1$, corresponding to isothermal oscillations in the outer layers. If the expression in square brackets is negative, then $\nu$ is complex and it corresponds to a running wave in the region above $r = R$. Setting the expression in square brackets to zero defines $\sigma$ to be the acoustic cutoff frequency $\sigma_{\rm ac}$ at the adopted outer boundary of the star. Note that unlike most stars which have sharp boundaries with $h \ll 1$, in extended red giants the outer boundary can be reasonably placed over a range of radii corresponding to moderately different $R$ values. Hence, in a given star, $\sigma_{\rm ac}$ can vary moderately depending on where the outer boundary is placed (see Figure \[nonad+ad\_p-logl\] for the value of the acoustic cutoff frequency relative to the frequency of radial pulsation modes in a typical case). The use of the boundary condition defined by Equations 1 and 2 gives a smooth transition from frequencies well below the acoustic cutoff frequency where $\nu$ is real (the condition usually assumed for pulsating stars) to frequencies above the acoustic cutoff frequency where $\nu$ is complex. Note that the value of $\nu$ we use is $\nu_{-}$ in the nomenclature of @bak65. This value of $\nu$ gives a finite pulsation amplitude at large distances above the stellar surface, and it corresponds to an outward propagating wave for frequencies above the acoustic cutoff frequency. Results ======= Normal and strange modes ------------------------ ![ The periods (top panel) and growth rates (bottom panel) for the first 8 radial modes in a series of AGB models with $M = 1.6\,{\rm M}_{\odot}$ plotted against luminosity. The growth rate is defined as $exp(-\omega_{\rm i}P)-1$ and it is the fractional growth in amplitude per period ($\omega_{\rm i}$ is the imaginary part of the complex eigenvalue $\omega$). A given mode is defined by its colour with the modal colour being the same in both panels. See the online version for the colours. []{data-label="nonad_p+gr-logl"}](gr+lgp-lgl_m1.6_modes_in_mode_order.eps){width="1.0\columnwidth"} Figure \[nonad\_p+gr-logl\] shows the period and growth rate for the first 8 radial pulsation modes in a star with $M = 1.6\,{\rm M}_{\odot}$ as the luminosity is varied. The optical depth to the centre of the outer zone is set to $\log \tau_{\rm c} = -3$. The maximum luminosity examined for these models with metallicity Z=0.008 is $\log L/{\rm L}_{\odot} = 4.3$. This is somewhat larger than the maximum observed luminosity of $\log L/{\rm L}_{\odot} \approx 4$ for stars with $M \approx 1.6\,{\rm M}_{\odot}$ and metallicity ${\rm Z} \approx 0.008$ in the the Magellanic Clouds [@kam10]. At the lowest luminosities ($\log L/L_{\odot} < 3.2$), the periods and growth rates of all modes behave smoothly as the luminosity changes. These modes correspond to the normal modes of radial pulsation. We refer to the normal fundamental mode as $P_0$, the normal first overtone as $P_1$, the normal second overtone as $P_2$ and so on. As the luminosity increases past $\log L/{\rm L}_{\odot} \approx 3.2$, a mode appears with a period equal to that of $P_7$. The damping rate of this new mode is much higher than that of the normal mode so that the complex eigenvalues $\omega$ of the two modes are very different. As the luminosity of the star increases, the period of the new mode increases more rapidly than that of the normal modes so that the period of the new mode successively equals that of $P_6$, $P_5$, $P_4$, $P_3$, $P_2$ and $P_1$. However, in each case of period equality, the growth rates and hence complex eigenvalues of the two modes are different. The new mode belongs to the group of modes known as strange modes. Their behaviour is explained lucidly in the paper by @buc97 who show that the strange modes are essentially surface modes with low interior amplitudes. Figure \[eigenfns\] shows the amplitude as a function of radius for a normal and strange mode of identical period (these two modes have $\log L/{\rm L}_{\odot} = 4.03$ and they lie at the position where the brown and green lines cross in Figure \[nonad\_p+gr-logl\]). It can be seen that the strange mode is indeed more concentrated to the stellar surface than the normal mode. In their analysis, @buc97 claimed that the periods of normal modes and a strange modes always avoided crossing but we see no need for this since the eigenvalues of both modes move continuously around the complex $\omega$ plane as separate quantities. ![ The amplitude of the radius perturbation eigenfunctions for a normal and strange mode of identical period plotted against the radius $r$. The amplitude is $(\delta_{\rm r}^2 + \delta_{\rm i}^2)^{\frac{1}{2}}$ where $\delta_{\rm r}$ and $\delta_{\rm i}$ are the real and imaginary parts of radius perturbation $\delta$, respectively. For nonadiabatic pulsation, the amplitude is not necessarily zero at a node. The bump in the curves at $r = 335\,{\rm R}_{\odot}$ is caused by the hydrogen ionization zone. []{data-label="eigenfns"}](compare_strange+normal_m1.6_l10636.eps){width="1.0\columnwidth"} However, avoided crossings do occur. Following $P_4$ (the cyan line) from low luminosities in Figure \[nonad\_p+gr-logl\], we see that near $\log L/{\rm L}_{\odot} = 3.97$ the normal mode $P_4$ comes close in period to a second strange mode (purple line). In this case, the two lines avoid crossing in period but they do cross in growth rate. The effect of this is to convert the normal mode into a strange mode and [*vica versa*]{}. In fact, as the luminosity increases further, the mode shown by the purple line undergoes another avoided crossing and converts back from normal mode characteristics to strange mode characteristics. At this time, this strange mode is the third strange mode at these luminosities (where we order the strange modes by decreasing period). There are also intermediate cases where strange and normal mode periods do cross while at the same time the growth rates are influenced by the other mode. The growth rates tend to be attracted to each other by mode interaction. Examples of this behaviour are the pink and purple modes that cross in period at $\log L/{\rm L}_{\odot} \approx 3.85$ and the blue and cyan modes that cross in period at $\log L/{\rm L}_{\odot} \approx 4.12$. We note that at the high luminosity end of the sequence of $1.6\,{\rm M}_{\odot}$ models, there seems to be a strict alternation between normal and strange modes as one moves to shorter periods. This behaviour does not seem to apply strictly at lower luminosities. We have found no cases where strange modes have positive growth rates. In fact, Figure \[nonad\_p+gr-logl\] shows that the strange modes are always more highly damped than the normal modes. Thus, when considering self-excited modes, we should only expect to see the normal modes of oscillation in real stars. The strange modes may, however, influence both the growth rate and period of normal modes in the case of near-resonance. It is possible that the strange modes could be stochastically excited by convective motions and thus become observable as part of the solar-like oscillation spectrum which has been detected in red giants at lower luminosity [e.g. @bed10]. The peak power of a strange mode in a solar-like oscillation power spectrum will be determined largely by the way in which the convective perturbations can couple to the mode in question. In addition, the peak power will decrease as the damping of the mode increases. An estimation of the relative amplitudes of stochastically excited normal and strange modes is beyond the scope of this paper. We note that the results in @ban13 show that modes in red giants which have periods $P > 10$ days have relatively large amplitudes, which suggests that these modes are self-excitated. On the other hand, the shorter period, lower amplitude modes appear to be stochastically excited. Thus in the luminous red giants considered here, which have $P > 10$ days for at least the first two modes, it may be difficult to find the signal of a very damped strange mode in the overall power spectrum where excited modes are likely to dominate. However, if the signal of a strange mode could be detected in a given star, its strange mode nature could possibly be determined by its frequency spacing relative to nearby radial ($\ell = 0$) modes. ![ Top panel: Similar to the top panel in Figure \[nonad\_p+gr-logl\] but also showing the acoustic cutoff period corresponding to the acoustic cutoff frequency $\sigma_{\rm ac}$ at the surface of the star (dashed black line). Bottom panel: The periods of the first 8 adiabatic radial modes. []{data-label="nonad+ad_p-logl"}](lgp-lgl_m1.6_ad+nonad_modes_in_mode_order+pac.eps){width="1.0\columnwidth"} Although strange modes and normal modes can have equal periods in the nonadiabatic case, for adiabatic pulsation the Sturm-Liouville theorem requires that the eigenvalues $\omega$ remain distinct. This is shown in Figure \[nonad+ad\_p-logl\]. However, strange mode behaviour still influences the periods to some extent as seen most prominently for $P_2$ and $P_3$ (the green and blue modes) near $\log L/{\rm L}_{\odot} \approx 4.15$. A full discussion of the origin of strange mode behaviour in the adiabatic case in given in @buc97. The effect of the position of the outer boundary ------------------------------------------------ ![ Similar to Figure \[nonad\_p+gr-logl\] except that all models have $L = 4000\,{\rm L}_{\odot}$ and the periods and growth rates are plotted against $\log \tau_{\rm c}$ rather than $\log L/{\rm L}_{\odot}$. []{data-label="nonad_p+gr-tau"}](gr+lgp-lgtau_m1.6_modes_in_mode_order.eps){width="1.0\columnwidth"} ![ Surface structure and eigenfunctions for models with $M = 1.6\,{\rm M}_{\odot}$ and $L = 4000\,{\rm L}_{\odot}$. Top panel: $\log \rho$ and $\log T$ plotted against $r$/R$_{\odot}$ for a model with $\log \tau_{\rm c} = -3$ (solid blue lines) and a model with $\log \tau_{\rm c} = -8$ (dashed red lines). The thick parts of the line correspond to regions that are convective. The green dotted line shows the ratio of the pressure scale height H$_0$ to $r$, with a vertical axis scale from 0 to 1. Bottom panel: the real part of the radius eigenfunctions for the first 4 modes of pulsation plotted against the radius $r$. Blue solid lines show the eigenfunctions for the model with $\log \tau_{\rm c} = -3$ while the red dashed lines show the eigenfunctions for the model with $\log \tau_{\rm c} = -8$. These complex eigenfunctions are normalized to 1.0 at the surface for both models. The green dashed lines show the eigenfunctions for the model with $\log \tau_{\rm c} = -8$ normalized to 1.0 at a radius which corresponds to the surface of the model with $\log \tau_{\rm c} = -3$. []{data-label="eigfn"}](compare_mod+eig_normalized.eps){width="1.0\columnwidth"} We now show how the placement of the outer boundary influences the pulsation periods of red giants. The periods of the eight lowest order modes in a star with $M = 1.6\,{\rm M}_{\odot}$ and $L = 4000\,{\rm L}_{\odot}$ are plotted against $\log \tau_{\rm c}$ in the Figure \[nonad\_p+gr-tau\]. Note that a decrease in $\tau_{\rm c}$ means that the surface radius is placed further out in the stellar atmosphere (see below). We restricted $\log \tau_{\rm c} < -2$ since for larger values of $\tau_{\rm c}$ the outer boundary is placed in a region where the temperature gradient starts to become significant and the periods of all modes become dependent of $\tau_{\rm c}$. It is clear that the periods of the normal modes are independent of $\log \tau_{\rm c}$ i.e. they are independent of the placement of the stellar surface. For the strange modes, the periods vary markedly with $\log \tau_{\rm c}$. This is consistent with the fact that the strange modes are predominantly surface modes largely confined to the region between the hydrogen ionization zone and the stellar surface [@buc97; @sai98]. The reason that the periods of the strange modes increase as the outer boundary is placed at larger radii is that the ratio $z_0$ of the radius of the hydrogen ionization zone to the stellar surface decreases. As shown in the toy models of @buc97 (see their Figure 13), decreasing $z_0$ causes the strange mode period to increase relative to the normal mode periods. Note also that it is the decrease in $z_0$ with luminosity in the sequence of 1.6M$_{\odot}$ models shown in Figure \[nonad\_p+gr-logl\] that causes the strange mode periods to increase faster with luminosity than the normal mode periods. The strange modes in Figure \[nonad\_p+gr-tau\] all have lower growth rates (larger damping rates) than the normal modes. They also cross the periods of the normal modes and in each case it can be seen that mode interaction influences the growth rate as the modes come in and out of resonance. The top panel of Figure \[eigfn\] shows the outer structure of a typical luminous red giant from the centre to the surface for two placements of the outer boundary, corresponding to $\log \tau_{\rm c} = -3$ and $\log \tau_{\rm c} = -8$. The model with $\log \tau_{\rm c} = -8$ has a considerably larger surface radius than the model with $\log \tau_{\rm c} = -3$. The physical structure of the two models is essentially indistinguishable at common radii. The eigenfunctions of the radius perturbation for the 4 lowest order modes are shown in the bottom panel of Figure \[eigfn\] for the two placements of the outer boundary. At first sight, the eigenfunctions of the corresponding modes in the two models with different surface radii look very different. However, this is not so. To compare the eigenfunctions with the same normalization, the eigenfunctions for the more extended model were multiplied by a complex constant which caused the eigenfunction to be normalized to a value of 1.0 at a radius corresponding to the surface radius of the smaller model. These transformed eigenfunctions are shown as green dashed lines in Figure \[eigfn\]. It can be seen that these transformed eigenfunctions are essentially indistinguishable from those of the less extended model (blue lines) at common radii. It is not clear to us where the outer boundary should be placed. Clearly, it should be at $\log \tau_{\rm c} < -2$ since the periods of all modes are affected if the outermost model point is deeper in the star than the surface point of the model with $\log \tau_{\rm c} = -2$. Since the periods of the normal modes are essentially independent of the position adopted for the stellar surface, we have adopted $\log \tau_{\rm c} = -3$ as the criterion by which our outer boundary is defined. This value of $\tau_{\rm c}$ also means that the growth rates of the normal modes are not greatly affected by mode interaction with strange modes. As seen in Figure \[nonad\_p+gr-tau\], these interactions can become significant for very small $\tau_{\rm c}$ values when the eigenfunctions extend high into the stellar atmosphere. The strong dependence of strange mode periods on the adopted surface radius means that if strange modes could be detected in the spectrum of solar-like oscillations in a red giant, then the detected periods could be used to determine the outer radius of the red giant as experienced by strange modes. Summary and conclusions ======================= We have shown that in luminous red giant stars, a series of radial strange modes exists in addition to the series of radial normal modes of pulsation. At high luminosities, strange modes can have periods as long as that of the first overtone for plausible luminosities, especially if an extended outer atmosphere in included in the calculations. The periods of the strange modes increase faster with $\log L/{\rm L}_{\odot}$ (and hence surface radius) than the periods of the normal modes. This means that normal and strange modes in a given star can have identical periods at certain luminosities. In some cases, avoided crossings in period occur leading to a given mode (identified by continuity as the luminosity is varied) changing back and forth between a normal and strange mode character as the luminosity changes. In cases where the modes cross in period, the growth rates of each mode is affected by the near-resonance condition. The strange modes are always damped, and more so than the normal modes. We should not expect to see self-excited strange modes in real stars but strange modes may be observed in the spectrum of solar-like oscillations. The periods and growth rates of normal modes may be influened by resonances with strange modes. Fortunately, the normal modes periods are essentially unaffected by the placement of the outer boundary. On the other hand, the strange mode periods increase as the outer boundary is placed at larger radii. Finally, we note that although these calculations were performed for radial modes, we expect that strange modes should also exist in the nonradial case. Acknowledgments {#acknowledgments .unnumbered} =============== PRW was partially funded during this research by the Australian Research Council Discovery grant DP1095368. He was also gratefully acknowledges funding from UWC and SAAO for travel and accommodation during a trip to South Africa where some of this work was done. We thank the anonymous referee for useful comments. 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--- abstract: 'We present high-resolution X-ray spectra from the young supernova remnant Cas A using a 70-ks observation taken by the [*Chandra*]{} High Energy Transmission Grating Spectrometer (HETGS). Line emission, dominated by Si and S ions, is used for high-resolution spectral analysis of many bright, narrow regions of Cas A to examine their kinematics and plasma state. These data allow a 3D reconstruction using the unprecedented X-ray kinematic results: we derive unambiguous Doppler shifts for these selected regions, with values ranging between $-2500$ and $+4000$[ kms$^{-1}$]{} and the typical velocity error less than 200[ kms$^{-1}$]{}. Plasma diagnostics of these regions, derived from line ratios of resolved He-like triplet lines and H-like lines of Si, indicate temperatures largely around 1 keV, which we model as O-rich reverse-shocked ejecta. The ionization age also does not vary considerably over these regions of the remnant. The gratings analysis was complemented by the non-dispersed spectra from the same dataset, which provided information on emission measure and elemental abundances for the selected Cas A regions. The derived electron density of X-ray emitting ejecta varies from 20 to 200[ cm$^{-3}$]{}. The measured abundances of Mg, Si, S and Ca are consistent with O being the dominant element in the Cas A plasma. With a diameter of 5, Cas A is the largest source observed with the HETGS to date. We, therefore, describe the technique we use and some of the challenges we face in the HETGS data reduction from such an extended, complex object.' author: - 'J. S. Lazendic, D. Dewey, N. S. Schulz, and C. R. Canizares' title: 'The Kinematic and Plasma Properties of X-ray Knots in Cassiopeia A from the Chandra HETGS ' --- Introduction ============ Cassiopeia A (Cas A, G111.7–2.1) is the youngest Galactic supernova remnant (SNR), believed to be a product of a SN explosion in $\sim 1670$ [e.g. @thorstensen01; @fesen06]. The SNR is the brightest object in the radio band, and is very bright in the X-ray band, providing a suitable target for detailed studies across a wide range of wavelengths. Cas A shows a wealth of phenomena important for studying the SN progenitor, its explosion mechanism, the early evolution of SNRs, and their impact on the interstellar medium. The remnant appears as a bright, clumpy ring of emission with a diameter of around 3 associated with the SNR ejecta, while the fainter emission from the SNR forward shock forms a filamentary ring with a diameter of 5. The almost perfectly circular appearance of the SNR is disrupted by the clear extension in the NE region of the SNR, called the jet, and a less obvious one in the SW called the counter-jet [e.g @fesen96; @hwang00]. Many of Cas A’s bright knots have been identified as shocked ejecta, still clearly visible due to the young age of the SNR. Observations in the optical band provided the first insights into the kinematics and chemical composition of these ejecta [e.g. @minkowski59; @peimbert71; @chevalier78]. The optical-emitting ejecta have been classified into two main groups: the fast-moving knots (FMKs), moving with speeds from 4000[ kms$^{-1}$]{} up to 15,000[ kms$^{-1}$]{}, and the slow-moving quasi-stationary floculi (QFS), moving with speeds less than 300[ kms$^{-1}$]{} [@kamper76]. The emission from FMKs, believed to be ejecta, is H-deficient (from the lack of H$\alpha$ emission) and dominated by forbidden O and S emission [e.g. @fesen01]. This lack of hydrogen in FMKs suggests that Cas A was a Type Ib supernova produced by the core-collapse of a Wolf-Rayet star [e.g. @woosley93]. The emission from QFSs is rich in N, which is the reason these knots are believed to originate from the circumstellar envelope released by the progenitor star and subsequently shocked (or “over-run”). While the available kinematic information from the optical observations of Cas A is based on observations of thousands of individual knots, X-ray observations are more dynamically important because X-rays probe a much larger fraction of the ejecta mass (more than 4[$M_{\odot}$]{}) than does the optical emission (accounted for less than 0.1[$M_{\odot}$]{}). Kinematics in the X-ray band have been studied using radial velocities [@markert83; @holt94; @hwang01; @willingale02] and proper motion [@vink98; @delaney03; @delaney04], but there is still a need for high-resolution spatial and spectral observations that will match the quality of results provided in the optical band. Here we present observations of Cas A with the [*Chandra*]{} High Energy Transmission Grating Spectrometer (HETGS). Cas A is one of the rare extended objects viable for grating observations due to its bright emission lines, narrow bright filaments and small bright clumps that standout well against the continuum emission. Observations and Data Analysis ============================== Cas A was observed with the HETGS on board the [*Chandra X-ray Observatory*]{} in May 2001 as part of the HETG guaranteed time observation program (ObsID 1046). The exposure was 70 ks, the roll angle was 86 West to North and the aim point was at [${\rm RA}={23}^{\rm h}{23}^{\rm m}{29}^{\rm s}$]{}, [${\rm Dec.}={+58}^{\circ}{48}\arcmin{59}\farcs$]{}[6]{}. The HETGS consists of two grating arms with different dispersion directions: 1) the medium-energy grating (MEG) arm which covers an energy range of 0.4-5.0 keV and has FWHM of 0.023 Å, and 2) the high-energy grating (HEG) arm which covers 0.9-10.0 keV band and has FWHM of 0.012 Å (for more details see e.g. @canizares05). The different dispersion directions and wavelength scales of the two arms provide a way to resolve potential spectral/spatial confusion problems [see Appendix A; also @dewey02]. The HETGS is used in conjunction with the Advanced CCD Imaging Spectrometer (ACIS-S). In addition to the dispersed images, a non-dispersed zeroth-order image from both gratings occupies the S3 chip at the aimpoint, while the dispersed photons are distributed across the entire S-array. The zeroth-order image, thus, has spatial and spectral resolution provided by the ACIS detector. The Cas A HETGS event file was re-processed using the standard procedures in CIAO[^1] software version 3.2.2, employing the latest calibration files. Processing of the data included removal of the pipeline afterglow correction (a significant fraction of the rejected events was from the source), correctly assigning ACIS pulse heights to the events and filtering the data for energy, status, and grade. To retain more valid events, the removal of artifacts (destreaking) from the S4 chip was done by requiring more than 5 events in a row in order to destreak it. The standard procedures in CIAO for gratings data assume that the source is point-like. We, therefore, used alternative software (written in IDL) that basically follows the steps of the CIAO threads for gratings spectra, but accounts for an extended, filament-like source during extraction of the PHA spectra and the corresponding spectral redistribution matrix files (RMFs). The RMFs are particularly important as they relate the photon energy scale to the detector dispersion scale of the gratings. We also used standard CIAO threads to create auxiliary response files (ARFs), which contain the information on telescopes effective area and the quantum efficiency as a function of energy averaged over time. The resulting ARFs were examined for bad columns, and the parts of the spectrum where the bad columns are present were ignored in the fitting procedure. Details of this “filament analysis” are presented in Appendix A. For the zeroth-order data, we extracted spectra and corresponding ARFs and CCD RMFs with the standard CIAO thread [acisspec]{}. For the background spectra we tried using the emission-free region on the S3 chip, the SNR regions surrounding our bright knots, and the regions from the ACIS blank-sky event files. We found no difference when using any of these spectra, so we decided to use the ACIS blank-sky events. The background spectra are extracted with region sizes a factor of 2 larger than the source spectra. The zeroth-order model fits were carried out binning the data to contain a minimum of 25 counts per bin. Results ======= Figure \[fig-casA\] shows the non-dispersed (zeroth-order) image of Cas A at the center of each panel and dispersed images in different energy bands that contain predominately He- and H-like ions, and in the case of Fe also Li- and Be-like ions: O+Ne+Fe-L (0.65–1.2 keV), Mg (1.25–1.55 keV), Si (1.72–2.25 keV), S (2.28–2.93 keV), Ar (2.96–3.20 keV), Ca (3.75-4.00 keV) and Fe-K (6.30–6.85 keV). The Si-band image also indicates the two dispersion axes of the MEG and HEG, which are rotated against each other by $\sim10\degr$. The Cas A X-ray spectrum is dominated by Si and S lines and these bands produce the clearest dispersed images. The dispersion angle $\theta$ defines the location of the dispersed photon and varies nearly linearly with wavelength $\lambda$, $sin \theta = m\lambda / p$, where $m$ is the order of dispersion and $p$ is the period of the grating. Thus, the dispersed images in the longer wavelengths (O+Ne+Fe-L band) are spread out furthest across the spectroscopy CCD array, while in the shorter wavelength (Fe-K band) the images stay tight to the zeroth-order. Because of this, the longer-wavelength images are more sensitive to velocity gradients and distortions are more prominent, as indicated by the larger smearing seen in the top two panels. The O+Ne+Fe-L image is additionally confused due to the many lines in this band [e.g., Fe-L lines and the O Lyman series; see @bleeker01], and a larger relative contribution by the continuum. Figure \[fig-regions\] shows the regions in the image of Cas A selected for this study. These regions have a morphology and brightness that allow the reliable measurement of line fluxes and centroids. They are spatially narrow along the dispersion direction (north-south) and sufficiently isolated above the local, extended background emission to provide a clear line profile. Figure \[fig-close-up\] shows close-ups of these regions. For each region we concentrated on the four strongest dispersed spectra: the MEG +1 and $-1$ orders and the HEG +1 and $-1$ orders. Because of their dominant emission we use the He- and H-like transitions of Si and S lines for the analysis of velocity and plasma structure in Cas A. For typical SNR plasma densities the He-like triplet of the Si and S line shows strong forbidden ($f$) and resonance lines ($r$) and a comparatively weaker intercombination line ($i$). We, therefore, jointly fit our four HETG spectra (MEG$\pm 1$ and HEG$\pm 1$) with a model consisting of 4 Gaussians (representing 3 He-like lines and 1 H-like line for each element) and a constant that accounts for the continuum level (for more details see Appendix A). Figure \[fig-hetg-si\] shows the four sets of Si spectra from region R1, the brightest of the 17 regions, and from region R9 which has the most prominent Si XIV line. Figure \[fig-hetg-s\] shows similarly S spectra from these two regions. In the fit we allow only the centroid of the $r$-line to be a free parameter, while the other three centroids are tied to it, so the relative centroids are fixed, but the template is allowed to slide. We also assume that a single velocity is present in each Cas A region, so all 4 Gaussians have frozen narrow widths. In other words, if there is no velocity structure in a region, the chosen Gaussian width would be the narrowest profile for that spatial structure. In addition, the flux ratio of the $f$-line to the $i$-line (so-called R-ratio) for the Si triplet is tied to be 2.63, suitable for the very low density regime of the Cas A plasma [e.g. @porquet01]. Similarly, the $f$-line flux is tied to be 1.75 times the $i$-line flux for the S triplet [e.g. @pradhan82]. To derive accurate Doppler shifts from measuring line centroids we used HEG and MEG spectra binned to 0.01 Å and 0.02 Å, respectively. Our Gaussian fits are not sensitive to measure velocity dispersion as done in the case of E0102–072 [@flanagan04], where topographical changes in the dispersed image between positive and negative orders were used to determine intrinsic bulk motions. The RMFs we use in the fitting of Cas A HETG spectra only include broadening due to spatial structure and do not include any further velocity structure degrees of freedom in the quantitative fitting. But we can qualitatively distinguish regions with more or less velocity structure present. Among the line profiles from our 17 regions we find two qualitative types: 1) a narrow profile, where the $r$ and $f$ lines are clearly resolved and shifted due to the region’s bulk motion along the line of sight, and 2) a smeared profile, where one of the grating orders appears broadened due to high-velocity motions within the region relative to its center of mass that causes distortion of the dispersed images along the dispersion direction. For each region the type of the line profile is indicated in Table \[tab-hetg\]; Figure \[fig-hetg-si\] shows an example of a double-peak profiles, and an example of a clearly smeared spectral profiles is shown in Figure \[fig-smeared\]. Line Dynamics - Doppler Shift Measurement ----------------------------------------- Results of Si line measurements and derived Doppler shift values are summarized in Table \[tab-hetg\]. The spatial distribution of Doppler velocities is shown in Figure \[fig-doppler\]. Derived values range between $-2500$[ kms$^{-1}$]{} to $+4000$[ kms$^{-1}$]{}. The measured velocity shifts for Si and S (not listed here) are similar (see Fig. \[fig-projected\]). This is not surprising since they arise from the same nucleosynthesis layer, and have been found to have the same spatial distribution [@hwang00; @willingale02]. The uncertainties in derived velocities depend, besides on the intrinsic energy resolution of the HETGS, on the number of counts detected in the line and the errors associated with estimates of the continuum contribution. For the Si line the statistical errors for the Doppler shift, based on fit confidence limits, range range within 200[ kms$^{-1}$]{} for all regions except for regions R10 and R17 which have errors of 650[ kms$^{-1}$]{} and 360[ kms$^{-1}$]{}, respectively. The HETGS derived velocities of our 17 Cas A regions are combined with their spatial location on the sky to graphically indicate their 3D location and velocities, as shown in Figure \[fig-projected\]. Our measured Doppler velocity is plotted on the y-axis and the x-axis value is the 2D sky radial displacement of the region from the expansion center of Cas A on the sky given by @reed95. A factor of $0.032\arcsec \pm 0.002\arcsec$ per [ kms$^{-1}$]{} is used to relate velocity to spatial location by minimizing (by eye) the shell width needed to enclose most of the data points (dotted lines.) The velocity center for the shells of +770[ kms$^{-1}$]{} is taken from @reed95. For a distance to Cas A of 3.4 kpc [@reed95] our factor corresponds to a fractional expansion rate of $(0.19\pm 0.01)$ % per year. The forward shock location at 153 and the mean reverse shock location at 95, as determined by @gotthelf01, is also indicated. Line Diagnostics - Flux Ratio Measurements ------------------------------------------ One advantage of the high-resolution grating spectra of our Cas A regions over lower-resolution CCD data is that we can investigate the plasma state of individual regions using individual emission lines. The He-like K-shell lines, like those of Si and S present in Cas A, are the dominant ion species for each element over a wide range of temperature [e.g. @paerels03]. From the He-like triplet, the ratio of the forbidden ($f$) and resonance ($r$) lines is a useful diagnostic for electron temperature [e.g., the G-ratio $=(f+i)/r$, @porquet01], especially since the lines are from the same ion, which reduces dependence on the relative ionization fraction. Lines from different ions of the same element are also useful because they eliminate the impact of uncertainties in abundance, so e.g. the ratio of the H-like to He-like Si lines in conjuction with the G-ratio of the He-like lines can give an accurate measurement of the progression of plasma ionization. To determine accurate Si line flux ratios from our data, we fixed the modelled line locations based on our nominal binning and fits described above, and then re-fitted HETG spectra using coarser-bin with modified errors. This procedure, described in Appendix A, is less sensitive to differences between the shape of the analysis RMF and the velocity-modified line shapes. The resulting Si $f/r$ and Si XIV/XIII line ratios are listed in Table \[tab-hetg\]. We also calculated the expected line ratios, employing the non-equilibrium ionization collisional plasma model with variable abundances, VNEI [@borkowski01] in XSPEC with vneivers version 1.1, which uses updated calculations of ionization fractions from @mazzotta98. We then used these model grids to map our measured line ratios to equivalent plasma temperature $kT_e$ and ionization timescale $\tau = n_e t$. Figure \[fig-line\_ratios\] shows the distribution of $kT_e$ and $\tau$ for the 17 Cas A regions. The point for region R6 falls outside the displayed ratio range because its Si XIV/XIII ratio has an extremely low value. The temperatures range between 0.4 to 5 keV, with the majority of regions having a temperature between 0.7 and 1.0 keV; the distribution of plasma temperature is shown more clearly in Figure \[fig-kT-tau\]. The ionization timescale ranges between [$10^{10}$]{} and 4[$\times 10^{11}$]{}[ cm$^{-3}$s]{}. The measured values for region R9 and R12 fall off of the grid in Figure \[fig-line\_ratios\] because of their extremely high $f/r$ line ratios. To assign $kT_e$ and $\tau$ values to region R9, we used the lower error limit which falls within the NEI grid. For region R12, we used a value within a 20–30% discrepancy of the lower limit. Thus, the derived values (e.g., $n_e$, $t_{\rm shock}$) for regions R6, R9 and R12 should be taken with some leeway. Zeroth-order Spectra - Density and Abundance Measurements {#sec-zeroth} --------------------------------------------------------- In addition to the dispersed data, we also use the non-dispersed data (from the central ACIS chip seen in Fig. \[fig-casA\]) to obtain information on the abundances and emission measure for the individual regions. Obtaining information on global plasma parameters such as electron density and elemental abundance requires comparing line to continuum in the spectra, and also knowing what are the contributions to the continuum. The dispersed spectra will have contribution from superimposed lines and continua from other image regions, although the superimposed lines would not have the right energies (see Appendix A for more details). Because of this confusion dispersed data are not reliable for measuring line to continuum ratio. Therefore, to determine these plasma parameters for each of the 17 regions, we fitted the non-dispersed zeroth-order spectra with a single VNEI plasma model [@borkowski01], which allows for varying elemental abundances. In these fits we fixed the $kT_e$ and $\tau$ values for each region according to the values derived from HETG line ratios. Cas A spectra, even of isolated features, are very complex and four spectral components have been identified [e.g. @delaney04]. These four spectral components are rarely present at the same location and most of @delaney04 knots show characteristics of a single type, with some showing mixed characteristics. To avoid complications due to varying column density ($N_{\rm H}$) and continuum levels we ignore the low-energy part of the spectrum and consider only the range between 1.1 and 8.0 keV encompassing K-shell lines of Mg, Si, S, Ar, Ca and Fe-K. An O-rich plasma is often employed when describing Cas A ejecta [e.g. @vink99; @hughes00; @laming03], since optical observations showed that ejecta in Cas A are deficient in H and rich in O and O-burning products [e.g. @chevalier78]. We, therefore, assume that O dominates the continuum emission, and that O provides many of the electrons instead of H and He, as in the typical solar abundance plasma. Thus, we fix the O abundance to be a factor of 1000 higher than the solar value. The results are, of course, not very sensitive to the exact O overabundance factor. We also fix $N_{\rm H}$ to 1.5[$\times 10^{22}$]{}[ cm$^{-2}$]{}, which is found to be an average value across the SNR [e.g. @vink96; @willingale02]. In the fit we allowed the normalization and the abundances of Mg, Si, S and Ca to vary. The Ar line is not included in the VNEI model (vneivers version 1.1), so we use a Gaussian to model the Ar emission. The prominent Fe-K XXV line is only present in regions R9 and R16, which is not surprising since they are located in the area rich with Fe called the “Fe cloud” [@hwang03]. We tried different approaches for handling the red-shift parameter $z$ in fitting these CCD data. We first froze the $z$ values to those derived from the HETGS data, but the resulting fits were not acceptable, showing a clear mismatch between data and model line peaks. These offsets are likely ACIS gain uncertainties and other calibration errors in the CCD response. Reasonable fits were obtained by manually adjusting the $z$ values separately for each data set; the values used varied between $-0.005$ and $+0.035$. Table \[tab-zo\] lists parameters measured from the zeroth-order spectral fits of the 17 regions. The first column lists the region emission volume $V_R$, which was taken to be a triaxial ellipsoid whose 2D-projected area corresponds to the spectral extraction region shown in Figure \[fig-close-up\] with radii $a$ and $b$. For the third axis along the line of sight we take the average value of the two observed axes, $ c = (a+b)/2 $; the volume of the region is then: $$V_R = {4\over 3}\pi \ a \ b \ c .$$ The second column lists the normalization factor (“norm”) used in the VNEI model, e.g., $$X_{\rm norm} \ \equiv \ { {10^{-14}} \over {(4 \pi d^2)} } \ \int n_e n_H \ dV .$$ Note that the tabulated norm here assumes that the model oxygen abundance is set to 1000. The rest of the columns list the abundance ratios with respect to oxygen for the elements Mg, Si, S and Ca. Using the equations given in Appendix \[sec:vnei\_to\_phys\] we derive and tabulate in Table \[tab-zo-param\] some relevant physical parameters for the regions based on these fit results: the region’s electron density, the total mass of plasma in the region, the time-since-shocked for the region $t_{\rm shock} = \tau / n_e $, in units of years, and finally the fraction of oxygen by mass per region. Discussion ========== Doppler measurements in Cas A ----------------------------- The measurement of Cas A Doppler shifts in the X-ray band was first conducted with the [*Einstein X-ray Observatory*]{} using the Focal Plane Crystal Spectrometer (FPCS) [@markert83]. The bulk velocities of two regions of Cas A, the SE and NW halves, were measured using the line centroids of the resolved Si and S triplets. These observations found a velocity broadening and asymmetry in the X-ray emitting material, with the NW region having more red-shifted emission, and the SE region of the remnant having more blue-shifted emission. @markert83 suggested that the asymmetry could be the result of an inclined ring-like distribution of Cas A material, possibly influenced by the distribution of the mass-loss material of the progenitor. Our HETGS spectra of individual Cas A filaments reconfirms this global asymmetry trend. The SE regions of the SNR appear to be mostly blue-shifted and the regions in the NW have the extreme red-shifted values. The asymmetry was also found with moderate spectral resolution of [*ASCA*]{} [@holt94]. The [*ASCA*]{} observations provided a velocity map on the spatial scale of 1 and were derived using the Si line centroids at the CCD resolution. Doppler velocity maps with a much finer spatial resolution were produced using [*Chandra*]{} [@hwang01] and [*XMM-Newton*]{} observations [@willingale02]. @hwang01 used the Si XIII resonance (1.865 keV) and Si XIV Ly$\alpha$ (2.006 keV) line centroids to derive the velocity shifts on a 4 spatial scale. While most of the SNR showed velocities between $-1500$ and $+1500$[ kms$^{-1}$]{}, extreme velocities of $-6000$[ kms$^{-1}$]{} were found in the SE region, including the region of our R1, R2, R9 and R13 for which HETGS data imply velocities between $-2500$ and $-1000$[ kms$^{-1}$]{}. This discrepancy is not too surprising. Although @hwang01 argued that they can ignore the ionization effects on the Si He$\alpha$ centroid to a reasonable approximation, they note that high spectral resolution measurements are more desirable to measure the energy shifts directly from resolved rather than blended lines. Indeed, their [*Chandra*]{} data do not resolve the Si XIII triplet, and even the Si XIV and Si XIII Ly$\alpha$ lines are not fully resolved, introducing some level of uncertainty in line centroid measurement. The most recent results from the 1 Ms Very Large Project (VLP) [*Chandra*]{} data show values more consistent with our HETGS data [@stage05], but even here absolute gain calibration accuracies for the CCD may only be good to 0.5% or 1500[ kms$^{-1}$]{}. [*XMM*]{} Doppler maps of Si, S and Fe lines by @willingale02 show a smaller range in velocity than the Si velocity map of @hwang01 and, thus, values that are at a glance closer to our values in Table \[tab-hetg\]. The [*XMM*]{} data have been binned to a $20\arcsec \times 20\arcsec$ spatial grid and the spectra were fit with two thermal NEI components representing the ejecta and the shocked component, each with a separate energy shift. The Doppler shift values derived in this way will depend on the ability of the NEI spectral model to predict the line blends combined with the uncertainties in the gain calibration of the detectors. In comparison to our results, it is obvious that some of the errors in their Doppler shift values are produced because of the spatial averaging over a variety of features with very different velocities. For example, in the NW region the Doppler velocity map of @willingale02 has a smooth distribution of red-shifted values with a 1000[ kms$^{-1}$]{} or so dispersion, whereas our regions R5 and R6 in that part of the SNR show significantly blue-shifted values of around $-1500$[ kms$^{-1}$]{}, as well as red-shifted velocities with 1000[ kms$^{-1}$]{} difference between regions R8 and R15. @willingale02 find that the velocity patterns for S are very similar to those for Si; our measured Si and S velocities agree with this. The Fe-K velocities, however, exhibit higher velocities than those of Si and S; this has also been found with the 1 Ms VLP [*Chandra*]{} observations of Cas A [@stage05]. Note that our two regions showing Fe-K emission, R9 and R16, are located in the middle (R9) and outside (R16) the range of other regions (see Fig. \[fig-projected\]). Unfortunately, our HETGS data do not have enough counts in the Fe-K band to measure shifts in the Fe energy; deeper HETGS observations might yield Fe-K velocity measurements with 500[ kms$^{-1}$]{} errors, provided the required narrow features are present. Probably the best demonstration of the magnitude of the difference between the CCD and HETGS measured Doppler shifts is given by comparing the estimated shock expansion rate in Cas A. @delaney03 derived forward shock expansion measurements of 0.21% per year using transverse velocity measurements of Cas A knots using [*Chandra*]{} CCD data. @delaney04 compared their transverse velocities (3100–3900[ kms$^{-1}$]{}) with that of @willingale02 (1000–1500[ kms$^{-1}$]{}) and @hwang01 (2000-3000[ kms$^{-1}$]{}) obtained from Doppler measurements using CCD spectra, and graciously ascribe mismatch to possible projection effects of the asymmetric remnant. However, our Doppler measurements imply an ejecta expansion of 0.19% per year, consistent with the @delaney03 and @delaney04 value. This supports their suggestion that there might be a dynamic coupling between forward shock and ejecta, and they are both part of one homologously expanding structure. Note that @vink98 derived an expansion rate of 0.2% per year by comparing the [*ROSAT*]{} and [*Einstein*]{} X-ray images from two different epochs. The region R17 is the only region outside our reverse-shock/forward-shock range and its location at a large radius from the expansion center puts it outside the nominal forward shock 3D radius as seen in Figure \[fig-projected\]. This is also a characteristic of the optical FMKs, which are found mostly in the NE part of the remnant, with many of them located along the jet. Our region R17 is located in the base of the jet region (see Fig. \[fig-regions\]), and its velocity indicates that it indeed might be part of the jet feature. Plasma properties ----------------- Beside the Doppler shifts, plasma temperature and ionization timescale are the two other plasma parameters derived directly from our HETGS analysis. Early observations of Cas A in X-rays identified two distinct plasma components — the cold component with temperature around 0.6 keV, and the hot component with temperature up to 4 keV [e.g. @becker79], and they have been confirmed with newer observations [e.g. @vink96; @willingale02]. The cooler component is associated with the reverse shock traveling through the expanding ejecta, and the hotter component is associated with the forward shock propagating into the circumstellar material. Plasma temperatures derived from our HETGS data have values mostly around 1 keV. Therefore, most of our regions have been heated up by a reverse shock propagating inward into the supernova ejecta. There is no significant pattern to the variation in the temperature across the SNR, as shown in Figure \[fig-kT-tau\]. The exceptions are regions R8, R10 and R17 which do have temperatures of 4–5 keV, several times higher than most other regions. Regions R8 and R10, which also share similar velocity, could, therefore, be associated with circumstellar material and the forward shock. Region R17 could be an X-ray counterpart of the optical FMKs found in Cas A jet. @willingale02 derived a map of ionization timescale for the cool plasma component in Cas A, which shows some variation across the surface of the SNR, but most of the SNR has $\tau$ values larger than [$10^{11}$]{} [ cm$^{-3}$s]{}. For our 17 regions we also find little variation, most of the regions have $\tau$ around a few [$10^{11}$]{}[ cm$^{-3}$s]{}. The exception are, again, only regions R8 and R10, as shown in Figure \[fig-kT-tau\], which have ionization timescale up to an order of magnitude smaller. Information on density and abundances for the 17 Cas A regions is derived in conjunction with the zeroth-order data. The zeroth-order spectra are reasonably fit with a single VNEI model with fixed temperature and ionization timescale, certainly well enough for our primary purposes here to establish relative abundances. An example spectrum is given in Figure \[fig-r9-zo\] for region R9. In the 1.1 and 8 keV range the spectrum shows a weak Mg lines, strong Si, S, Ar and Ca lines, and a weaker Fe-K line, and shows reasonable agreement between the data and the model for the continuum part of the spectrum. Note that a low-energy “up-turn” is also present due to emission from Fe-L lines and this part of the spectrum, below 1.1 keV, was not used in the fit. The Si XIII Ly$\beta$ line at 2.18 keV is generally under-fit in the spectra; this is likely due to several calibration issues each at the 10-20% level (e.g., Ir contamination is not included in the HRMA model, there are possible zeroth-order HETG calibration errors, CCD gain offsets are coupled with steep ARFs.) Future work using the higher-count Cas A VLP data set and including these calibration effects could usefully extract the He-like Ly$\beta$ line flux allowing associated diagnostic ratios [@porquet01] and possibly indicating charge-exchange processes [@pepino04]. Electron density for each of the regions, derived as described in Appendix B, is listed in Table \[tab-zo-param\]. The $n_e$ ranges from around 20 to 200[ cm$^{-3}$]{} and it seems to have higher values for the blue-shifted regions. A factor of 5 density difference has been suggested between the front and the back of Cas A [@reed95]. The time since individual Cas A regions have been shocked, given by $t_{\rm shock}=\tau/ n_e$, is also listed in Table \[tab-zo-param\]. These times vary significantly from region to region and it seems that the red-shifted regions have been shocked more recently compared to the regions on the front side which have generally larger $t_{\rm shock}$ values (regions R6, R9 and R12 have extreme values that should be taken with caution), as shown in Figure \[fig-tshock\]. However, these values could be the result of density differences between the front and the back side of the SNR. We do not find any correlation between $kT_e$ and $t_{\rm shock}$ or $n_e$ and $\tau$ that would indicate a possible electron-ion equilibration occurring in our 17 regions. The other parameters listed in Table \[tab-zo-param\] are the total mass and the fraction of O mass per region. These values should be taken with caution, because we assumed in our fit that continuum is dominated by O. Note also that our total mass estimate does not include Fe or Ar since they are not included in our NEI model, so the $M_{\rm total}$ values could be somewhat modified when these are included. The O mass fraction ranges from 0.82 to 0.97, thus confirming that, under our assumptions, O is the dominant element in these Cas A ejecta features. Cas A is an O-rich SNR, as indicated by optical observations, and this leads to the further assumption that perhaps there is also a lot of O in the regions between the bright Si knots that we see. For example, @laming03 estimate that a density enhancement of around 2 coupled with the presence of the Si-Ca metals would allow these knots to stand out even though surrounded with a pure oxygen plasma. This may be another reason that the O-band dispersed image appears so smeared out and that we do not detect clear O-lines in our HETGS data: there is O emission everywhere within the shocked ejecta region so we do not see discrete O filaments or clumps like those in the Si and S band images. In order to test the suggestion by @laming03 about the wide-spread presence of O emission, we make a few simple estimates. Table \[tab-zo-param\] shows the mass fraction of oxygen to be 0.82 in region R1. Under the assumption that the entire SNR volume within a radius of 100 and 130, for example, is filled with oxygen of the density projected for R1, the total mass of oxygen would amount to 85[$M_{\odot}$]{}. Earlier studies imply a mass of 4[$M_{\odot}$]{} for the entire ejecta at most [e.g. @vink96]. Thus, our results seem to either overestimate the oxygen density or the density of that particular region is not representative for such a large volume. Even if we further assume that the density in R1 is enhanced with respect to the ambient medium by the factor of 2 to 5 [@reed95], we would wind up with 40 to 16[$M_{\odot}$]{} of oxygen under uniform conditions. Introducing a density gradient to the ejecta, such as a rise in density towards the center of the remnant, gives an increase of at least another factor of 2. Therefore, such simple estimates do not seem to support the @laming03 assumption. Alternatively, @willingale03 derived a filling fraction of 0.009 for the ejecta component by assuming a pressure equilibrium between hot and cool plasma components, which lead to an electron density range of 40–90[ cm$^{-3}$]{}, similar to our values. Adding such a factor to our estimate above would then allocate much of the metals and oxygen into spaghetti-like filaments requiring a total mass of order 1[$M_{\odot}$]{}. This value represents a more reasonable contribution to a total oxygen mass of 2.6[$M_{\odot}$]{} suggested by @vink96. Summary ======= High-resolution HETGS observations from the young SNR Cas A yield unprecedented kinematical X-ray results for a few bright SNR regions. These observations show that high-resolution X-ray spectroscopy is catching up with that in optical, IR and UV bands in its ability to measure velocities and add a third dimension to the data. Unambiguous Doppler shifts are derived for these selected regions, with the SE region of the SNR showing mostly blue-shifted values reaching up to $-2500$[ kms$^{-1}$]{}, and the NW side of the SNR having extreme red-shifted values with up $+4000$[ kms$^{-1}$]{}. This global asymmetry is consistent with previous lower spatial or spectral resolution X-ray observations. From our Doppler measurements we derive ejecta expansion of 0.19% per year, supporting suggestion of @delaney04 that there might be a dynamic coupling between the forward shock, that is expanding at the same rate, and the ejecta. Plasma diagnostics using resolved Si He-like triplet lines and Si H-like line shows that most of the selected regions in Cas A have temperatures around 1 keV, consistent with reverse-shocked ejecta. However, two regions, R8 and R10, with significantly different temperature of $\sim4$ keV, might be part of the circumstellar material. Similarly, the ionization age does not vary considerably across the remnant, except for the two above mentioned regions, which have an order of magnitude lower ionization timescale. One of the regions, R17, is located outside the forward shock boundary and also has a high plasma temperature, both suggesting that this region is part of the Cas A NE jet feature. The analysis of HETGS data was complemented by the non-dispersed CCD ACIS spectra from the same observation, which allowed us to derive the electron density of the X-ray emitting ejecta and the elemental abundances of Mg, Si, S and Ar in our 17 regions. The electron density varies from 20 to 200[ cm$^{-3}$]{} and does not show any correlation with ionization timescale. The derived elemental abundances of Mg, Si, S and Ca are consistent with O being the dominant element in the Cas A plasma. We thank John Houck and John Davis for contributions in the planning of the Cas A HETGS observation, and Glenn Allen, Mike Stage, Kathy Flanagan, Tracy DeLaney, Dick Edgar, and Dan Patnaude for useful discussions. Support for this work was provided by NASA through the Smithsonian Astrophysical Observatory (SAO) contract SV3-73016 to MIT for support of the Chandra X-Ray Center and Science Instruments, operated by SAO for and on behalf of NASA under contract NAS8-03060. Filament Analysis {#sec:fil_analysis} ================= Filament Analysis Scheme ------------------------ The “filament analysis” we used for Cas A HETGS data is very similar to the standard analysis of dispersed grating data from a point source, in that it produces a one-dimensional PHA file with associated ARF and RMF for each grating and order. The two main additions to the standard processing that adapt it to extended, filament-like sources are described briefly below. First, adjustments to the event locations are applied to effectively straighten the filament-like source perpendicular to dispersion while retaining wavelength accuracy; specifically, the following steps are carried out. The shape of the source in zeroth-order is manually traced by a piecewise-linear path that is saved as a set of vertices. The vertex locations are then transformed into coordinates aligned with the dispersion and cross-dispersion direction of the particular spectra (HEG or MEG) being extracted, see top diagram in Figure \[fig-analysis\]. Each event, in both the zeroth-order and the dispersed orders, is then translated along the grating dispersion direction by an amount equal and opposite to the path offset at that same cross-dispersion location. This “shearing” causes the zeroth-order and any dispersed line-images of the feature of interest (FOI) to become narrower along the dispersion coordinate, see middle diagram of Figure \[fig-analysis\]. A PHA file of counts per dispersion bin is then created in the usual way by projecting the events along the dispersion axis, see bottom diagram Figure \[fig-analysis\]. Because the dispersed features are effectively narrowed, the ability to detect and resolve discrete lines from the feature is improved. Second, a companion response matrix, RMF, is created based on the observed, sheared zeroth-order events. For each wavelength in the RMF, a subset of events are selected using the available non-dispersive energy reported by the CCD detector and the expected response histogram is created and stored with appropriate offsets in the RMF. Operationally this is similar to the effect of the [rgsxsrc]{} convolution model available in XSPEC[^2] for use with XMM-Newton Reflection Grating Spectrometer data. This is just one approach to extended source grating analysis [@dewey02] and has the advantage of producing familiar PHA files which can be analyzed with standard software like ISIS [@houck02]. Note, however, that it is fundamentally an approximation to a fully multi-dimensional, spatial-spectral method and so will necessarily have limitations; some of these are implicit in the considerations and techniques described in the following sections. Treatment of Spectral Continua ------------------------------ In addition to line emission from the narrow FOI, there is also continuum emission from the feature, as well as line and continuum emission from other parts of the extended source. These give rise to a continuum component in the observed PHA spectrum which we discuss and estimate here. The observed count rate at a given location on the detector is given by a 3D integral over the source flux as a function of wavelength and position on the sky [see eq.(49) in @davis01]. This integral includes the grating response function, which has the property that the location of a dispersed photon includes a continuous dependence on the photon’s wavelength. Thus, for a given position on the detector there is a set of sky locations and corresponding wavelengths which all contribute detected counts in this same location. This is in contrast to the point-source case where the point source introduces a delta function in the integral and preserves a one to one mapping of source wavelength to dispersed location in each grating-order. The continuum counts seen within a bin in the PHA distribution file will then consist of the sum over the spatial regions along the dispersion axis weighted by their flux at the grating-equation-allowed wavelengths. Equivalently, the resulting spectrum includes not only the true continuum from the FOI, but the overlapping, shifted spectra from all regions along the dispersion axis. In practice, the integral may be effectively truncated, e.g., if the spatial extent of the source is moderate. In addition, even for very extended sources, the inherent energy resolution of the detector can be used to truncate the integral to a limited range in pulse-height, reducing the artificial continuum level. In this case, the observed continuum will be of order a factor of $R_g / R_{\rm CCD}$ greater than the true FOI continuum level, where $R_g$ and $R_{\rm CCD}$ are the effective resolving powers of the grating and order-sorting detector, respectively. Note that $R_g$ for a given feature varies like $1/E$ and $R_{\rm CCD}$ varies roughly like $\sqrt E$, where $E$ is the energy; thus, the $R_g/R_{\rm CCD}$ ratio varies approximately like $E^{-1.5}$ (or $\lambda^{1.5}$). This is a small variation, of order $\pm$12% over the 6–7 Å range for Si lines. As a demonstration of this, a simple MARX simulation was made consisting of a $\approx4.0\arcsec\times10\arcsec$ rectangle emitting at 1.865 keV embedded in a disk of emission 100in radius having a uniform spectrum from 1.2 to 2.8 keV. Figure \[fig-analysis-cont\] shows the CCD spectrum extracted from the zeroth-order (e.g., the rectangular region) and the spectrum extracted from the MEG dispersed first order. The effective grating resolving power here is $R_g \approx 75$ based on the source full width. For the order-sorting effective resolving power, $R_{\rm CCD}$, the dispersed extraction was performed with a wide pulse-height selection including $\pm 0.15\lambda$, giving $R_{\rm CCD} \approx 3.3$, based on the energy full width. Together these give $R_g / R_{\rm CCD} \approx 23.$ In the simulation the equivalent width, EW, of the line is 21.4 keV for the zeroth-order case, and 1.29 keV in the dispersed case; the continuum level is, therefore, $\approx 17$ times larger in the dispersed data set, which is of the order expected from the simple estimate. Velocity Effects and Fitting ---------------------------- The features we observe in Cas A show Doppler shifts of up to several thousand [ kms$^{-1}$]{}. For a simple bulk motion of the emitting region this just produces an overall Doppler shift to the line wavelength which can be readily measured using the filament analysis products. It is also possible that there are velocity variations within the filament itself and this can introduce some complications, especially when carrying out standard $\chi^2$-driven fitting. In order to study these velocity effects, simple MARX simulations were carried out using a filament source (resembling our region R1) consisting of two sets, upper and lower, of three closely spaced (by precisely 1) parallel line sources. The two sets were tilted and offset to produce a wide, kinky filament, shown in panel (a) in Figure \[fig-analysis-plots\]. In the MEG simulations of panels (a) and (b) all the 6 line sources making up the filament all have the same wavelength. These two panels show how the shearing of the filament analysis improves the resolution of the dispersed spectra. In simulation (c) the upper set of three lines have all been blue-shifted by 1000[ kms$^{-1}$]{} demonstrating the effect of velocity variation along the filament. This causes the dispersed projections to be additionally blurred, having a FWHM larger than the (unaffected) zeroth-order. A similar result would arise for the case of turbulent velocity broadening where there would be a range of velocities in all parts of the emitting feature. The simulation in panel (d) is perhaps the most pathological: here the velocity of the filament varies across the filament, along the dispersion direction. The zeroth-order continues to be unaffected by these velocity effects, and now not only do the the dispersed order projections differ from the zeroth-order, but they are different from each other as well [see HETGS observations of SNR E0102–072, @flanagan04]. The effect of these velocity produced changes in the dispersed line profiles is that the RMF created from the zeroth-order is not a good match when there is velocity structure in the feature. For purposes of fitting the line location, the generally peaked nature of both the RMF and the dispersed line peaks leads to reasonable centroid fitting and confidence ranges; however, the formal $\chi^2$ values can be large. The mismatch of the zero-velocity line shape used in our RMFs is more problematic when measuring the line fluxes. To reduce these mismatch effects we can fix the line locations in the model at the centroid values determined in our nominal fitting and then rebin the data and model to a coarser grid. We use binning of order 10 bins, with bin boundaries located in the regions between the lines. In this way the total counts in a line region are compared between model and data while ignoring line shape differences. Because of the large number of counts in each coarse bin, the errors are now dominated by the approximations of our RMFs and systematic calibration errors between the 4 spectra we are jointly fitting, HEG m=$\pm 1$ and MEG m=$\pm 1$. In place of the usual statistical error, we assign a constant error value to the bins of each coarse spectrum with a value of 4% of the maximum counts in a bin of that spectrum. This produces fits in which the fluxes of the lines are better estimated, as we confirm by examining the fits when data and coarser-fit model are re-plotted to our nominal binning, as demonstrated in Figure \[fig-coarse\]. Converting VNEI Parameters to Physical Units {#sec:vnei_to_phys} ============================================ This appendix summarizes the conversion of model parameter values into physically meaningful plasma quantities, e.g., the electron density, $n_e$, and the mass of each element present, $M(Z)$. A variety of models in the XSPEC library, including the VNEI model we use in this work, digest the properties of the emitting plasma into a normalization factor, $X_{\rm norm}$, and a set of relative elemental abundances, $X_A(Z)$. In addition to these, two other source parameters are needed: the source distance, $$d \ [{\rm cm}] = 3.1\times 10^{21} \ d_{\rm kpc} ,$$ and the volume of the emitting region, $$V_R \ [{\rm cm^3}] = V_{\rm as3} \ ( d {{\pi}\over{180}} {{1}\over{3600}} )^3 .$$ For convenience the conversion from values in kpc, $d_{\rm kpc}$, and arcseconds$^3$, $V_{\rm as3}$, are shown in these equations. The actual number fraction of the element $Z$ in the plasma is then given by: $$f(Z) = X_A(Z) \ A_{\rm model}(Z) \ / \ \sum X_A(Z) A_{\rm model}(Z) ,$$ where $A_{\rm model}(Z)$ is the reference “solar” number abundance ratio assumed by the model; in our case these are the @anders89 values. The parameters, $X_A(Z)$, are the usual relative abundance values as input in XSPEC models. Because we do not assume that hydrogen dominates the plasma, it is necessary for the electron density to be self-consistent with the densities and ionization states of the ions present. The ratio of electron $n_e$ to total ion density $n_i$ is then given by: $$({ {n_e} \over {n_i}}) \ = \ \sum Q(Z)\ f(Z) ,$$ where $Q(Z)$ is the average number of electrons stripped from the ions of element $Z$, in the range 0 to $Z$. Ideally $Q(Z)$ would be provided by the model; lacking direct access to it, however, we can make a simple approximation to the ionization state of the elements in our O-rich plasma by setting: $$Q(Z) \ = \cases{Z,& if $Z \leq 9$; \cr Z-2,& for $10 \leq Z \leq 16$; \cr Z-10,& if $Z \geq 17$. \cr }$$ The electron density can then be calculated as: $$n_e = \sqrt{ X_A(Z=1) \ ( n_e n_H ) \ ({ {n_e} \over {n_i}}) \ {{1}\over{f(Z=1)}} } ,$$ where $n_H$ is the hydrogen density and $( n_e n_H )$ is given by the usual normalization definition $$( n_e n_H ) \ = \ 4 \pi d^2 \ {10^{14}} \ X_{\rm norm} / V_R .$$ The $X_A(Z=1)$ factor is included in case the hydrogen model abundance is set to other than 1.0, e.g., to a small value like 1[$\times 10^{-9}$]{} for a pure metal plasma. The density of element $Z$ is then given by $$n(Z) \ = \ f(Z) \ n_e \ / \ ({ {n_e} \over {n_i}}) ,$$ and other quantities like the mass or mass fraction can be calculated in a straight forward way using the volume, $V$, and appropriate constants (1 amu = 1.66[$\times 10^{-24}$]{} g and [$M_{\odot}$]{}= 2[$\times 10^{33}$]{} g.) 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Grating dispersion axes and north direction are marked in the Si-band image. Note the smeared out dispersed-order images, especially in the O+Ne+Fe-L band, due to multiple lines and velocity shifts which, however, do not affect the central, zeroth-order image. []{data-label="fig-casA"}](fig1.eps){height="20cm"} ![ Zeroth-order Cas A image from this work. Regions used in the analysis of HETGS Cas A data. These regions are spatially narrow and they are isolated sufficiently above the local and extended background to provide a clear line profile for spectral fitting.[]{data-label="fig-regions"}](fig2.eps){height="12cm"} ![Close up of the regions used in the analysis of HETGS Cas A data. Note that the grayscale dynamic range is not the same for all the sub-plots. Some of the regions have been covered with two elliptical regions to follow the HETGS extraction path (see Appendix A).[]{data-label="fig-close-up"}](fig3.eps){height="25cm"} \ ![Doppler velocity values in [ kms$^{-1}$]{} for individual Cas A regions. The south-east SNR regions measured here are all blue-shifted, while extreme red-shifts are seen in the north-west of the SNR.[]{data-label="fig-doppler"}](fig7.eps){height="12cm"} ![3D Location and velocity of the regions. The line-of-sight velocity, as measured by Si lines and S lines, is plotted vs. the 2D projected distance on the sky from the nominal expansion center of @reed95. The Si velocity values (diamonds) are given with 90% confidence error bars; the velocity values for S (stars) are given just for the best-fit value. An expansion rate of 0.19 % per year relates the velocity and distance scales (see text). The inner and outer solid lines indicate the locations of the reverse shock (95) and the forward shock (153) in year 2000 and centered on the @reed95 velocity center (marked with triangle), ${\rm V_{center}}=+770$[ kms$^{-1}$]{}. Dotted lines at 102 and 130 are a guide to show the shell in which our regions generally lie. For reference, the location of the compact central object in Cas A [e.g. @chakrabarty00] is also shown; its line-of-sight velocity is unknown.[]{data-label="fig-projected"}](f8.ps){height="15cm"} ![Si line ratios overlaid onto a NEI grid. Upper panel shows the full range of values for all the regions (region R6 is located off the graph), and the lower panel zooms in on the cluster of values in the upper region of the graph. Temperature, $kT_e$, increases along the lines which go generally from upper-left to lower right. Ionization time, $\tau$, increases along the lines which are nearly horizontal at left and move to the upper-right ending at the high-$\tau$ limiting asymptote. A few $kT$ values are labeled in red, while some $\tau$ values are labeled in blue.[]{data-label="fig-line_ratios"}](f9a.eps "fig:"){height="10cm"} ![Si line ratios overlaid onto a NEI grid. Upper panel shows the full range of values for all the regions (region R6 is located off the graph), and the lower panel zooms in on the cluster of values in the upper region of the graph. Temperature, $kT_e$, increases along the lines which go generally from upper-left to lower right. Ionization time, $\tau$, increases along the lines which are nearly horizontal at left and move to the upper-right ending at the high-$\tau$ limiting asymptote. A few $kT$ values are labeled in red, while some $\tau$ values are labeled in blue.[]{data-label="fig-line_ratios"}](f9b.ps "fig:"){height="10cm"} ![Plot of plasma temperature $kT_e$ vs. ionization timescale $\tau=n_e t$ for 17 Cas A regions. The upper panel shows the full range of values, and the lower panel zooms in on the cluster of values. Three regions, R8, R10 and R17, show different properties from the rest. Region R6 is represented with two points, representing the mean value (lower left corner in the upper panel) and the upper limits (lower right corner in the upper panel).[]{data-label="fig-kT-tau"}](f10a.ps "fig:"){height="10cm"} ![Plot of plasma temperature $kT_e$ vs. ionization timescale $\tau=n_e t$ for 17 Cas A regions. The upper panel shows the full range of values, and the lower panel zooms in on the cluster of values. Three regions, R8, R10 and R17, show different properties from the rest. Region R6 is represented with two points, representing the mean value (lower left corner in the upper panel) and the upper limits (lower right corner in the upper panel).[]{data-label="fig-kT-tau"}](f10b.ps "fig:"){height="10cm"} ![ The time-since-shocked in years is indicated for each region in a similar plot to Fig. \[fig-projected\]. The red-shifted regions appear to have been shocked more recently compared to the regions on the front side which have longer $t_{\rm shock}$ values; the low Si XIV region, R6, with 0.5 yr is an exception.[]{data-label="fig-tshock"}](f12.ps){height="15cm"} ![Filament analysis schematic. A path is defined and used to straighten a filament-like feature seen in zeroth-order. A “shearing” process is carried out to translate each event location in the dispersion direction by an amount given by its cross-dispersion coordinate and the path. An effective RMF and corresponding PHA file are created and can be used in data analysis.[]{data-label="fig-analysis"}](f13.ps){height="8cm"} ![ Elevated continuum level in dispersed spectrum. The zeroth-order and MEG-first order spectra are shown for a simple MARX simulation of a monochromatic source embedded in a large region of continuum. The spectra are normalized to unit flux-in-the-line so that the equivalent widths, EW, of the lines serve as a measure of the continuum level. The EW for zeroth-order (left, dashed line) is 21.4 keV, whereas the dispersed spectrum has a reduced EW of 1.29 keV (right, dotted). Hence, relative to the line flux, the dispersed spectrum has an artificially higher continuum level due to overlapping continua from other locations along the dispersion axis.[]{data-label="fig-analysis-cont"}](f14.ps){height="8cm"} ![Velocity effects on dispersed line shapes. Each of the four panels here is from a simple MARX simulation and filament analysis, showing close-ups of the minus order (MEG $-1$), zeroth-order (0th), and plus-order (MEG +1) with event scatter-plots and 1D histograms. The top panel (a) shows the original, simulated filament without shearing applied. In (b) all events have been sheared using a simple path; the width of the feature has been narrowed to about 5 pixels wide. In (c) the upper-half of the feature is emitting at a wavelength blue-shifted by 1000[ kms$^{-1}$]{} with respect to the lower half. The zeroth-order is unaffected, but the first orders are blurred equally by this velocity variation in the cross-dispersion direction. In (d) the source has a velocity variation along the dispersion direction with blue-shift values of: 0, -500, and -1000[ kms$^{-1}$]{}. In this case the effect is to broaden the minus order (showing the three individual line sources used to simulate the wide filament) and narrow the plus order to the point that it is actually narrower than the zeroth-order itself; the zeroth-order is unaffected. []{data-label="fig-analysis-plots"}](fig15.eps){height="15cm"} \ \ [^1]: http://asc.harvard.edu/ciao/ [^2]:    http://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/manual/XSmodelRgsxsrc.html

--- abstract: 'An experimental verification of the inertial theorem is presented involving two hyperfine states of a trapped Ytterbium ion. The theorem generates an analytical solution for non-adiabaticlly driven systems ‘accelerated’ slowly, bridging the gap between the sudden and adiabatic limits. These solutions have shown to be stable to small deviations, both experimentally and theoretically. As a result, the inertial solutions pave the way to rapid quantum control of closed as well as open quantum systems. For large deviations from the inertial condition, the phase of the $SU(2)$ algebra solution remains accurate, involving inaccuracies dominated by an amplitude difference.' author: - 'Chang-Kang Hu' - Roie Dann - 'Jin-Ming Cui' - 'Yun-Feng Huang' - 'Chuan-Feng Li' - 'Guang-Can Guo' - 'Alan C. Santos' - Ronnie Kosloff title: Experimental Verification of the Inertial Theorem --- [^1] [^2] [*[Introduction]{}*]{}. A prerequisite for progress in contemporary quantum technology is a precise control of the quantum dynamics of the device [@cirac1995quantum; @monroe1995demonstration; @barreiro2011open; @rosi2013fast; @mandel2003coherent; @bloch2008many; @jaksch2000fast; @duan2001geometric; @jonathan2000fast; @nielsen2002quantum; @loss1998quantum; @kadowaki1998quantum; @finnila1994quantum; @brooke1999quantum; @venegas2018cross; @johnson2011quantum; @santoro2002theory; @rossnagel2016single; @pekola2015towards]. A vocabulary of control techniques has emerged which is universal, meaning it applies across a broad range of experimental platforms, such as NV-centers [@doherty2013nitrogen; @doherty2012theory; @bar2013solid], trapped ions [@cirac1995quantum; @haffner2008quantum; @kielpinski2002architecture], as well as Josephson devices [@makhlin2001quantum; @martinis2002rabi; @svetitsky2014hidden]. A primary example of a universal control scenario relies on the adiabatic theorem. This theorem implies that the control timescale is slow relative to the inverse square of the system’s spectral gap. The other extreme control scenario relies on the sudden limit where the control timescale is much faster than the system under consideration. For intermediate time scales a universal control paradigm is lacking. As a result, one has to rely on customised numerical schemes such as obtained by optimal control theory [@palao2002quantum; @glaser2015training]. The purpose of this paper is to demonstrate experimentally a universal quantum control scheme based on the inertial theorem [@dann2018inertial]. The analogous structure of the adiabatic and inertial solutions implies that such control may allow for possible applications in quantum information processing [@aharonov2008adiabatic; @farhi2000quantum; @farhi2001quantum; @childs2001robustness] and sensing [@perdomo2015quantum]. The demonstration utilizes the $SU(2)$ algebra realised by a quantum system composed of $^{171}$Yb$^+$ ion in a Paul trap. For a quantum control scheme to be generic it has to rely on simple principles which apply across platforms. The theory requires a formulation of a dynamical map $\Lambda_t$ from an initial to a final state $\hat \rho (t) = \Lambda_t \hat \rho (0) = \hat U \hat \rho(0) \hat U^{\dagger} $. We consider a control Hamiltonian which generates the dynamical map: $$\hat H(t) = \hat H_0 +\sum_j g_j(t) \hat G_j \label{eq:chamil}$$ where $\hat H_0$ is termed the drift Hamiltonian, $g_j(t)$ control fields and $\hat G_j$ control operator. The major obstacle to generate such a map from a time-dependent control Hamiltonian is the time-ordering operation, resulting from the fact that $[\hat H(t),\hat H(t')] \ne 0$. The adiabatic control circumvents this problem employing a slow drive $g_j(t)$ such that $[\hat H(t),\hat H(t')] \sim 0 $ [@messiah2003quantum; @comparat2009general; @mostafazadeh1997quantum; @sarandy2005adiabatic; @kato1950adiabatic]. At the other extreme, in the sudden limit the control is so powerful that it overshadows the dynamics generated by the drift Hamiltonian $\hat H_0$. The new control paradigm is based on the inertial theorem [@dann2018inertial], introducing an explicit solution of the dynamical map $\Lambda_t$. The inertial theorem is formulated employing a Lie algebra formed by a set of operators ${\cal G}$, closed to commutation relations. We assume that the operators in the control Hamiltonian Eq. (\[eq:chamil\]) $\hat H_0$ and $\hat G_j$ are members of the operator algebra. As a result, the Heisenberg equations of motion for ${\cal G}$ are closed [@alhassid1978connection]. Using the Liouville space, a vector space formed from the operators of ${\cal G}$ with the scalar product ${\left({\hat{G_i},\hat{G_j}}\right)}\equiv\text{tr}{\left({\hat{G_i}^{\dagger}\hat{G_j}}\right)}$, the Heisenberg equations of motion become $${\frac{d}{dt}}{\vec{v}} {\left({t}\right)} = {-i {\cal{M}}{\left({t}\right)}} {\vec{v}} {\left({t}\right)}~~, \label{Schro Liouv1}$$ where ${\cal{M}}$ is a $N$ by $N$ matrix with time-dependent elements and ${\vec{v}}$ is a vector of operators of size $N$ of the set $\{ {\cal G} \}$. The time ordering problem is now transformed to ${\cal{M}}{\left({t}\right)}$. To overcome this issue a combination of a time dependent operator basis and a control protocol $g_j(t)$ is employed such that the explicit time-dependence is factorised $${\cal{M}}{\left({t}\right)} = \Omega {\left({t}\right)} {\cal{B}}{\left({{\vec{\chi}}}\right)}~~. \label{eq:factorized}$$ Here, $\Omega {\left({t}\right)}$ is a time-dependent real function, and the matrix ${\cal{B}}{\left({{\vec{\chi}}}\right)}$ is a function of the constant parameters $\{\chi\}$. This decomposition has been obtained for $SU(2)$, $SU(3)$ and Heisenberg-Weil algebras. We conjecture that such decomposition is general. Once the decomposition is obtained, the dynamics can be expressed as $${\frac{d}{d \theta}}{\vec{v}} {\left({\theta}\right)} = {-i {\cal{B}} {\left({{\vec{\chi}}}\right)}} {\vec{v}} {\left({\theta}\right)}~~, \label{theta_motion}$$ here, $\theta\equiv \theta {\left({t}\right)}=\int_0^t dt'\, \Omega {\left({t'}\right)}$ is the scaled time. The solution of Eq. (\[theta\_motion\]) is obtained by diagonalizing ${\cal B}$, yielding $${\vec{v}} {\left({\theta}\right)} = \sum_k^N c_k {\vec{F_k}}{\left({{\vec{\chi}}}\right)} e^{-i {\lambda}_k \theta}~~, \label{constat v}$$ where ${\vec{F_k}}$ and $\lambda_k$ are eigenvectors and eigenvalues of ${\cal{B}}$ and $c_k$ are constant coefficients. Each eigenvector ${\vec{F}}_k$ corresponds to the eigenoperator $\hat F_k = \sum_j f_{kj}(t) \hat G_j$. The inertial theorem allows solving the dynamics beyond the restriction given by the decomposition Eq. (\[eq:factorized\]). For a slow change of ${\cal B}$, in analogy to the adiabatic theorem, the inertial theorem states that an eigenoperator of a slowly varying ${\cal B}$ is maintained while accumulating phase. The inertial solution obtains the form $${\vec{v}}{\left({\chi,\theta}\right)}=\sum_k c_k e^{-i\int_{\theta_0 }^{\theta}d\theta'{\lambda}_{k}}e^{i \phi_{k}}{\vec{F}}_{k}{\left({{\vec{\chi}}{\left({\theta}\right)}}\right)}~~, \label{eq:inetrial state1}$$ where ${\vec{F_k}}$ and $\lambda_k$ are eigenvectors and eigenvalues of ${\cal{B}}$ at normalized time $\theta$. The dynamical phase is $-i\int_{\theta_0}^{\theta}d\theta'{\lambda}_{k}$ with ${\lambda}_k ={\lambda}_k{\left({\theta}\right)}$, $\theta_0=\theta{\left({0}\right)}$, $\theta=\theta{\left({t}\right)}$ and the second exponent includes a new geometric phase $$\phi_k{\left({\theta}\right)}= i\int_{{\vec{\chi}} {\left({\theta_0}\right)}}^{{\vec{\chi}} {\left({ \theta}\right)}} d{\vec{\chi}}{\left({{\vec{G}}_{k}|\nabla_{{\vec{\chi}}} {\vec{F}}_{k}}\right)} ~~. \label{eq:phi1}$$ Here, ${\vec{G}}_{k}$ are the bi-orthogonal partners of ${\vec{F}}_{k}$ and the inertial parameter is defined as $$\Upsilon=\text{max}_{{\vec{\chi}}}{\left[{{\frac{{\left({{\vec{G}}_{k},\nabla_{{\vec{\chi}}}{\cal B}{\vec{F}}_{n}}\right)}}{{\left({{\lambda}_{n}-{\lambda}_{k}}\right)}^{2}}}{\left({{\frac{d{\vec{\chi}}}{d\theta}}}\right)}^2}\right]}\ll1~~~, \label{eq:parameter}$$ for all $n\neq k$. The inertial solution composed of the eigenoperators Eq. (\[eq:inetrial state1\]), holds for a slow variation of ${\vec{\chi}}$, $d{\vec{\chi}}/dt\ll1$, $\Upsilon\ll 1$. Physically, the condition on $d\chi/dt$, is associated with a slow ‘acceleration’ of the driving [@dann2018inertial]. In the adiabatic limit, decomposition Eq. is satisfied and the inertial solution converges to the adiabatic result. We will demonstrate the inertial solution in the context of the $SU(2)$ algebra. We now consider a Two-Level-System (TLS) which is a realization of the $SU(2)$ algebra. For the demonstration, we choose a dynamical map $\Lambda_t$ which varies the energy scale and controls the relation between energy and coherence in a non-periodic fashion. The control Hamiltonian reads $$\hat{H}{\left({t}\right)} = {\frac{1}{2}}{\left({\omega{\left({t}\right)} \hat \sigma_{z} + \varepsilon{\left({t}\right)} \hat \sigma_{x}}\right)}~~, \label{eq:Ham}$$ where the control protocol has the functional form $$\begin{aligned} \begin{array}{lcl} \omega{\left({t}\right)} &=& \Omega{\left({t}\right)}\cos{\left({\alpha{\left({t}\right)}t}\right)}\\ \varepsilon {\left({t}\right)} &=& \Omega{\left({t}\right)} \sin{\left({\alpha{\left({t}\right)}t}\right)} \label{eq:protocol} \end{array} ~~. \label{eq:omega eps}\end{aligned}$$ Here, the frequencies $\omega$ and ${\varepsilon}$ are the detuning and Rabi frequency, respectively. These define the generalized Rabi frequency $\Omega {\left({t}\right)}\equiv \sqrt{{\varepsilon}^{2}{\left({t}\right)}+\omega^{2}{\left({t}\right)}}$. To factorize the equation of motion we define a time-dependent operator basis ${\vec{v}} =\{ \hat{H},\hat{L},\hat{C},\hat{I}\}^T$, where $\hat{L}{\left({t}\right)}={\left({{\varepsilon}{\left({t}\right)}\hat{\sigma}_{z}-\omega{\left({t}\right)}\hat{\sigma}_{x}}\right)}/2$, $\hat{C}{\left({t}\right)}={\left({\Omega {\left({t}\right)}/2}\right)} \hat{\sigma}_{y}$ and $\hat{I}$ is the identity operator. Since $\hat I$ is a constant of motion a $3 \times 3$ vectors space is sufficient for the dynamical description. An external driving protocol which satisfies the factorization, Eq. (\[eq:factorized\]), requires a constant adiabatic parameter $\mu$. The adiabatic parameter has the form $|\mu{\left({t}\right)}|\sim\sum_{n\neq m} \frac{|{\left< E_{m}{\left({t}\right)} \right|}\dot{\hat{H}}{\left({t}\right)}{\left| E_{n}{\left({t}\right)} \right>}|}{{\left({E_{m}{\left({t}\right)} - E_{n}{\left({t}\right)}}\right)}^2}$, and for the Hamiltonian Eq. (\[eq:Ham\]) is defined as $$\mu \equiv {\frac{\dot{\omega}{\varepsilon}-\dot{{\varepsilon}}\omega}{\Omega^3}}~~. \label{eq:mu}$$ The matrix $\cal{B}{\left({{\vec{\chi}}}\right)}$ then obtains the form $${\cal{B}}{\left({\mu}\right)}\equiv i{\frac{\dot{\Omega}}{\Omega^{2}}}{\cal{I}}+{\cal{B}}'~~~, \label{eq:Bmodel1}$$ with $${\cal{B}}'{\left({\mu}\right)}\equiv i{\left[{\begin{array}{ccc} 0 & \mu & 0\\ -\mu & 0 & 1\\ 0 & -1 & 0 \end{array}}\right]}~~~, \label{eq:Bmodel}$$ where ${\cal{I}}$ is the $3 \times 3$ identity operator in Liouville space and ${\vec{\chi}}$ can be recognized as ${\vec{\chi}}=\mu$ for the $SU(2)$ model. Employing the inertial theorem for a slow change in the adiabatic parameter ($\dot{\mu}\ll1$) the dynamics of the system is described by Eq. see Appendix \[ap:inertail solution\] for more details. In the experiment we check the validity of the inertial solution by choosing a protocol associated with a linear change in the adiabatic parameter so that $\frac{d \mu}{dt}= \delta$ $$\mu{\left({t}\right)}= \mu{\left({0}\right)} + \delta \cdot t~~. \label{eq:lin mu}$$ Moreover, we consider a linear chirp of the protocol frequencies $$\alpha{\left({t}\right)}=\alpha{\left({0}\right)} +\gamma\cdot t~~. \label{eq:alpha}$$ Equations and determine the Rabi frequency, substituting into Eq. leads to $\Omega{\left({t}\right)} = -{\frac{\alpha{\left({0}\right)}+2\dot{\alpha}{\left({t}\right)} t}{\mu}}$. For this protocol, the frequencies $\omega {\left({t}\right)}$ and $\epsilon {\left({t}\right)}$ become $$\begin{aligned} \begin{array}{lcl} \omega{\left({t}\right)} &=&-{\frac{{\left({\alpha{\left({0}\right)}+2\gamma\cdot t}\right)}}{\mu{\left({0}\right)} +\delta \cdot t}}\cdot \cos{\left({{\left({\alpha{\left({0}\right)} +\gamma t}\right)}\cdot t}\right)}\\ {\varepsilon}{\left({t}\right)} &=& -{\frac{{\left({\alpha{\left({0}\right)}+2\gamma \cdot t}\right)}}{\mu{\left({0}\right)} +\delta \cdot t}}\cdot \sin{\left({{\left({\alpha{\left({0}\right)} +\gamma t}\right)}\cdot t}\right)} \end{array} ~~. \label{eq:protocol_exp}\end{aligned}$$ The quality of the inertial approximation is directly connected to the parameter $\delta$. For small $\delta$, the inertial approximation is satisfied, and the inertial solution to remains accurate. The dynamical map can be evaluated using the time-dependent control protocol, Eq. . We choose the initial condition ${\vec{v}}{\left({0}\right)}=\{\hat{H}{\left({0}\right)},0,0,1 \}$ which describes the system in the ground state (${\langle {\hat{H}{\left({0}\right)}} \rangle}=-\Omega{\left({0}\right)}/2$). For these conditions, we compare the experimental measured normalized energy, ${\langle {\hat{H}{\left({t}\right)}} \rangle}/{\langle {\hat{H}{\left({0}\right)}} \rangle}$, to the inertial solution, Eq. , and to a converged numerical calculation of Eq. , generated by the Hamiltonian Eq. (\[eq:Ham\]). The inertial and numerical solutions are given in terms of the vector in Liouville space ${\vec{v}} =\{ \hat{H},\hat{L},\hat{C},\hat{I}\}^T$. [*[Experimental setup]{}*]{}.The experimental analysis of the inertial solution employs a single Ytterbium ion $^{171}$Yb$^+$, trapped in a six needle Paul trap schematically shown in Fig. \[ExpSche\]. The two-level-system used in our study is encoded in the hyperfine energy levels of $^{171}$Yb$^+$ represented as ${{\left| 0 \right>} \equiv \,^{2}S_{1/2}\, {\left| F=0,m_{F}=0 \right>}}$ and ${{\left| 1 \right>} \equiv \,^{2}S_{1/2}\, {\left| F=1,m_{F}=0 \right>}}$ [@Olmschenk:07]. After Doppler cooling, the system is initialized in the state ${\left| 0 \right>}$ with a standard optical pumping process. The inertial protocols are obtained by driving the hyperfine qubit with a programmable Arbitrary Waveform Generator (AWG) [@Hu:18; @Hu-18-b]. This enables to implement the components $\hat{\sigma}_{z}$ and $\hat{\sigma}_{x}$ of the Hamiltonian in Eq. . The generalized Rabi frequency $\Omega(t)$ is implemented by a simultaneous control of the microwave amplitude [@Hu:18], and the detuning between microwave frequency $\omega_{0}$ and the transition frequency $\omega_{\text{hf}}$. Where $\omega_{\text{hf}}$ is the frequency between the states ${\left| 0 \right>}$ and ${\left| 1 \right>}$ of the ion. Utilizing the AWG the time-frequency protocols of the inertial solutions are implemented. For each experimental protocol the normalized energy as a function of time is evaluated ${\langle {\hat{H}{\left({t}\right)}} \rangle}/{\langle {\hat{H}{\left({0}\right)}} \rangle}$. The measurement procedure is performed via fluorescence detection using a $369.5$ nm laser, where the population of the ${\left| 1 \right>}$ state is measured with high fidelity [@Hu:18; @Hu-18-b]. As shown in Fig. \[ExpSche\], we can detect photons for the bright state ${\left| 1 \right>}$ and zero photons for the dark state ${\left| 0 \right>}$. The measurement fidelity is estimated to be $99.4\%$ [@Hu:18; @hu2019quantum]. ![Experimental apparatus and relevant Ytterbium energy levels used in our experiment, where we highlight the encoding of the two-level system used in our experimental implementation.[]{data-label="ExpSche"}](ExpSche.pdf) Figure \[fig:1\] presents the normalized energy as a function of time, comparing the experimental result (blue) to the inertial solution (red) and an exact numerical solution (black). The results show good agreement between the theoretical and the experimental results for small $\delta$, see panel (c) and (d), demonstrating the high accuracy of the inertial solution. ![The normalized energy as a function of time for the experimental result (blue), inertial solution (red) and numerical solution (dashed-black) for different values of $\delta$; (a) $\delta=- \alpha{\left({0}\right)}$, (b) $\delta=-0.05\cdot\alpha{\left({0}\right)}$, (c) $\delta=-0.01\cdot\alpha{\left({0}\right)}$, (d) $\delta=0.01\cdot\alpha{\left({0}\right)}$, (e) $\delta=0.05\cdot\alpha{\left({0}\right)}$, (f) $\delta=0.1\cdot\alpha{\left({0}\right)}$. The experimental parameters are: $\alpha{\left({0}\right)}=6\cdot2\pi KHz$, $\gamma= 50\cdot2\pi M(Hz)^2$ with $\mu{\left({0}\right)}=-1$. The varying values of $|\delta|=|d\mu/dt|$ are related to the quality of the inertial approximation, for slow change in $\mu$ the inertial approximation is satisfied, Cf. panels (c) and (d). Varying $\mu$ rapidly leads to the breakdown of the inertial theorem, see panels (a),(b),(e) and (f).[]{data-label="fig:1"}](graph_a.pdf "fig:"){width="23.50000%"} ![The normalized energy as a function of time for the experimental result (blue), inertial solution (red) and numerical solution (dashed-black) for different values of $\delta$; (a) $\delta=- \alpha{\left({0}\right)}$, (b) $\delta=-0.05\cdot\alpha{\left({0}\right)}$, (c) $\delta=-0.01\cdot\alpha{\left({0}\right)}$, (d) $\delta=0.01\cdot\alpha{\left({0}\right)}$, (e) $\delta=0.05\cdot\alpha{\left({0}\right)}$, (f) $\delta=0.1\cdot\alpha{\left({0}\right)}$. The experimental parameters are: $\alpha{\left({0}\right)}=6\cdot2\pi KHz$, $\gamma= 50\cdot2\pi M(Hz)^2$ with $\mu{\left({0}\right)}=-1$. The varying values of $|\delta|=|d\mu/dt|$ are related to the quality of the inertial approximation, for slow change in $\mu$ the inertial approximation is satisfied, Cf. panels (c) and (d). Varying $\mu$ rapidly leads to the breakdown of the inertial theorem, see panels (a),(b),(e) and (f).[]{data-label="fig:1"}](graph_b.pdf "fig:"){width="23.50000%"} ![The normalized energy as a function of time for the experimental result (blue), inertial solution (red) and numerical solution (dashed-black) for different values of $\delta$; (a) $\delta=- \alpha{\left({0}\right)}$, (b) $\delta=-0.05\cdot\alpha{\left({0}\right)}$, (c) $\delta=-0.01\cdot\alpha{\left({0}\right)}$, (d) $\delta=0.01\cdot\alpha{\left({0}\right)}$, (e) $\delta=0.05\cdot\alpha{\left({0}\right)}$, (f) $\delta=0.1\cdot\alpha{\left({0}\right)}$. The experimental parameters are: $\alpha{\left({0}\right)}=6\cdot2\pi KHz$, $\gamma= 50\cdot2\pi M(Hz)^2$ with $\mu{\left({0}\right)}=-1$. The varying values of $|\delta|=|d\mu/dt|$ are related to the quality of the inertial approximation, for slow change in $\mu$ the inertial approximation is satisfied, Cf. panels (c) and (d). Varying $\mu$ rapidly leads to the breakdown of the inertial theorem, see panels (a),(b),(e) and (f).[]{data-label="fig:1"}](graph_c.pdf "fig:"){width="23.50000%"} ![The normalized energy as a function of time for the experimental result (blue), inertial solution (red) and numerical solution (dashed-black) for different values of $\delta$; (a) $\delta=- \alpha{\left({0}\right)}$, (b) $\delta=-0.05\cdot\alpha{\left({0}\right)}$, (c) $\delta=-0.01\cdot\alpha{\left({0}\right)}$, (d) $\delta=0.01\cdot\alpha{\left({0}\right)}$, (e) $\delta=0.05\cdot\alpha{\left({0}\right)}$, (f) $\delta=0.1\cdot\alpha{\left({0}\right)}$. The experimental parameters are: $\alpha{\left({0}\right)}=6\cdot2\pi KHz$, $\gamma= 50\cdot2\pi M(Hz)^2$ with $\mu{\left({0}\right)}=-1$. The varying values of $|\delta|=|d\mu/dt|$ are related to the quality of the inertial approximation, for slow change in $\mu$ the inertial approximation is satisfied, Cf. panels (c) and (d). Varying $\mu$ rapidly leads to the breakdown of the inertial theorem, see panels (a),(b),(e) and (f).[]{data-label="fig:1"}](graph_d_1.pdf "fig:"){width="23.50000%"} ![The normalized energy as a function of time for the experimental result (blue), inertial solution (red) and numerical solution (dashed-black) for different values of $\delta$; (a) $\delta=- \alpha{\left({0}\right)}$, (b) $\delta=-0.05\cdot\alpha{\left({0}\right)}$, (c) $\delta=-0.01\cdot\alpha{\left({0}\right)}$, (d) $\delta=0.01\cdot\alpha{\left({0}\right)}$, (e) $\delta=0.05\cdot\alpha{\left({0}\right)}$, (f) $\delta=0.1\cdot\alpha{\left({0}\right)}$. The experimental parameters are: $\alpha{\left({0}\right)}=6\cdot2\pi KHz$, $\gamma= 50\cdot2\pi M(Hz)^2$ with $\mu{\left({0}\right)}=-1$. The varying values of $|\delta|=|d\mu/dt|$ are related to the quality of the inertial approximation, for slow change in $\mu$ the inertial approximation is satisfied, Cf. panels (c) and (d). Varying $\mu$ rapidly leads to the breakdown of the inertial theorem, see panels (a),(b),(e) and (f).[]{data-label="fig:1"}](graph_e.pdf "fig:"){width="23.50000%"} ![The normalized energy as a function of time for the experimental result (blue), inertial solution (red) and numerical solution (dashed-black) for different values of $\delta$; (a) $\delta=- \alpha{\left({0}\right)}$, (b) $\delta=-0.05\cdot\alpha{\left({0}\right)}$, (c) $\delta=-0.01\cdot\alpha{\left({0}\right)}$, (d) $\delta=0.01\cdot\alpha{\left({0}\right)}$, (e) $\delta=0.05\cdot\alpha{\left({0}\right)}$, (f) $\delta=0.1\cdot\alpha{\left({0}\right)}$. The experimental parameters are: $\alpha{\left({0}\right)}=6\cdot2\pi KHz$, $\gamma= 50\cdot2\pi M(Hz)^2$ with $\mu{\left({0}\right)}=-1$. The varying values of $|\delta|=|d\mu/dt|$ are related to the quality of the inertial approximation, for slow change in $\mu$ the inertial approximation is satisfied, Cf. panels (c) and (d). Varying $\mu$ rapidly leads to the breakdown of the inertial theorem, see panels (a),(b),(e) and (f).[]{data-label="fig:1"}](graph_f.pdf "fig:"){width="23.50000%"} When $|\delta|=|d\mu/dt|$ is increased, we witness the breakdown of the inertial solution. Figure \[fig:1\] panels (a)-(f) compare results for different values of $\delta$, varying from a large negative value ($\delta=-\alpha{\left({0}\right)}$) to the large positive value ($\delta=0.1\cdot \alpha{\left({0}\right)}$). For large $|\delta|$ (panels (a),(b),(e) and (f)) the deviations between the predicted normalized energy values of the inertial solution and the experimental results increase. The deviation between the theoretical and experimental results are first observed in amplitude of the energy oscillations, while the phase follows even for large $|\delta|$, see panel (a) with $\delta =-\alpha{\left({0}\right)}$. This behaviour can be rationalized by calculating the correction terms to the inertial solution. Gathering Eq. and , and defining ${\vec{v}}\equiv{\left({\Omega{\left({t}\right)}/\Omega{\left({0}\right)}}\right)}{\vec{u}}$, we obtain $${\frac{d {\vec{u}} {\left({\theta}\right)}}{d\theta}} =-i {\cal{B}}'{\left({\mu {\left({\theta}\right)}}\right)} {\vec{u}} {\left({\theta}\right)}~~. \label{eq: u theta}$$ Next, we define the instantaneous diagonalizing matrix of ${\cal{B}}'{\left({\mu}\right)}$, satisfying ${\cal{P}}^{-1}{\left({\mu}\right)}{\cal{B}}'{\left({\mu}\right)}{\cal{P}}{\left({\mu}\right)}={\cal{D}}{\left({\mu}\right)}$ and ${\vec{u}}{\left({\theta}\right)}={\cal{P}}{\left({\mu}\right)}{\vec{w}}{\left({\theta}\right)}$. The exact system dynamics can be expressed as $${\frac{d{\vec{w}}{\left({\theta}\right)}}{d\theta}} =-i{\cal{D}}{\vec{w}} {\left({\theta}\right)}- {\cal{P}}^{-1}{\frac{d{\cal{P}}}{d\theta}}{\vec{w}}{\left({\theta}\right)} \label{eq:w}$$ For a slow change in $\mu$, $\cal{B}'$ and consequently ${\cal{P}}$ varies slowly with respect to $\theta$. This property allows to neglect the second term in Eq. , which is qualitatively similar to the inertial approximation. The deviations from the exact solution are reflected by the term ${\cal{O}}{\left({\theta}\right)}={\cal{P}}^{-1}{\frac{{\cal{P}}}{d\theta}}$, leading to $${\cal{O}}= {\frac{2\mu}{1+\mu^2}}{\frac{d\mu}{d\theta}}{\cal{I}}+{\cal{S}}~~, \label{eq:O}$$ where $\cal{S}$ is given in Appendix \[ap:diviation\]. For the protocol Eq. , the dominant contribution comes from the first term in Eq. , changing the general scaling and with it the energy amplitude. The phase of the inertial solution is not affected even when $|d\mu/dt|=|\delta|$ is large. ![The distance ${\cal{D}}$ between the inertial solution and the exact numerical solution as a function of $\delta$ and time. For $\delta=0$ the inertial solution is exact at all times. For larger $|\delta|$ the distance increases almost linearly with time and $|\delta|$. []{data-label="fig:Fid"}](distance-2.png){width="40.00000%"} ![The inertial trajectory (red), exact numerical (blue) and adiabatic solutions (green straight line) in the ${\langle {\hat{H}} \rangle},{\langle {\hat{L}} \rangle},{\langle {\hat{C}} \rangle}$ coordinate space, for (a) $\delta=-0.01\cdot \alpha{\left({0}\right)}$ and (b) $\delta=-0.05\cdot \alpha {\left({0}\right)}$. []{data-label="fig:3D"}](new-0_01.png "fig:"){width="23.00000%"} ![The inertial trajectory (red), exact numerical (blue) and adiabatic solutions (green straight line) in the ${\langle {\hat{H}} \rangle},{\langle {\hat{L}} \rangle},{\langle {\hat{C}} \rangle}$ coordinate space, for (a) $\delta=-0.01\cdot \alpha{\left({0}\right)}$ and (b) $\delta=-0.05\cdot \alpha {\left({0}\right)}$. []{data-label="fig:3D"}](new-0_05.png "fig:"){width="23.00000%"} Figure \[fig:Fid\] shows the distance $\cal{D}$ between the inertial solution and the exact numerical result as a function of $\delta$ and time. $\cal{D}$ is defined as the Euclidean distance between the expectation values of the Liouville state vectors, $${\cal{D}}{\left({t}\right)}=\sqrt{\sum_i{\left({{\langle {v^i{\left({t}\right)}} \rangle}-{\langle {v_{num}^i{\left({t}\right)}} \rangle}^2}\right)}}~~.$$ where $v_i$ and $v_{num}^i$ are the $i$’th component of ${\vec{v}}$ (the inertial solution) and ${\vec{v}}_{num}$ (the exact numerical solution). When $\mu$ varies slowly ($\delta=-0.01$) the inertial solution remains exact, while for larger absolute values the numerical and inertial solutions deviate linearly in $\delta$ and time In Figure \[fig:3D\] we present the inertial, numerical and adiabatic trajectories for $\delta=-0.01,-0.05$ in the ${\langle {\hat{H}} \rangle},{\langle {\hat{L}} \rangle},{\langle {\hat{C}} \rangle}$ space. Such a representation serves as a complete description of the dynamics, demonstrating the large deviation between the adiabatic and inertial solutions. [*[Discussion]{}*]{}.The hyperfine levels of an Ytterbium ion $^{171}$Yb$^+$ in a Paul trap, are utilized to demonstrate the validity and breakdown of the inertial theorem. The theorem provides a family of non-adiabatic protocols that bridge the gap between the sudden and adiabatic limits [@dann2018inertial]. The experimental protocol involves a chirp in frequency and change in the generalized Rabi frequency, associated with a linear change in the adiabatic parameter. These protocols are designed to demonstrate the inertial solution, and its accuracy for protocols satisfying a slow change of the adiabatic parameter $\mu$. The experiments verify the theorem and the ability to perform inertial protocols. Moreover, as all experiments are influenced by various kinds of noise [@childs2001robustness], the accuracy achieved confirms the robustness of the inertial solution. This conclusion is supported by theoretical simulations which verify that the solution is stable to small deviations and noise. For a larger deviation from the inertial condition ($d{\vec{\chi}}/dt{\rightarrow}1$), Fig. \[fig:1\] panels (a), (b), (e) and (f), the error first appears in the amplitude while the phase of the inertial solution is still accurate. We prove this by analyzing correction to the inertial solution. In the $SU(2)$ algebra, the first order correction in $\theta$ to the phase vanishes, see the discussion beneath Eq. . Incorporating the amplitude correction into the inertial solution can lead to higher accuracy. The phase information can be utilized for parameter estimation beyond the inertial limit. The experimental validation of the inertial solution paves the way to rapid high precision control. This control can be extended to [*inertially*]{} driven open systems [@dann2018inertial], utilizing the non-adiabatic master equation [@dann2018time]. Such control can regulate the system entropy [@dann2018shortcut]. Moreover, the analogous structure of the inertial and adiabatic solutions and conditions ($d\mu/dt{\rightarrow}0$ and $\mu{\rightarrow}0$) implies an analogous application in quantum information processing [@aharonov2008adiabatic; @farhi2000quantum; @farhi2001quantum; @childs2001robustness]. Another important study concerns the applicability of the inertial theorem to highly oscillating fields which exhibit resonance phenomena. In such a regime, the adiabatic theorem can not be applied [@Hu-18-b], while the inertial theorem remains valid. This can be seen by taking the chirp frequency in Eq. as $\alpha{\left({t}\right)}=\Omega=const$, leading to $\mu=-1$. The derivation of the inertial solution remains valid in such regime and the solution is given by Eq. . A detailed analysis of these issues remain a subject of future research.\ We thank KITP for their hospitality, this research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958 and the Israel Science Foundation Grant No. 2244/14, the National Key Research and Development Program of China (No. 2017YFA0304100), National Natural Science Foundation of China (Nos. 61327901, 61490711, 11774335, 11734015, 11474268, 11374288, 11304305), Anhui Initiative in Quantum Information Technologies (AHY070000, AHY020100), Anhui Provincial Natural Science Foundation (No. 1608085QA22), Key Research Program of Frontier Sciences, CAS (No. QYZDY-SSWSLH003), the National Program for Support of Top-notch Young Professionals (Grant No. BB2470000005), the Fundamental Research Funds for the Central Universities (WK2470000026). A.C.S. wishes to acknowledge the partial financial support by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior and the Brazilian National Institute for Science and Technology of Quantum Information (INCT-IQ). Inertial solution {#ap:inertail solution} ================= We present a brief derivation of the inertial solution for a two-level-system, represented by the Hamiltonian of equation . The dynamics of such system can be conveniently described in terms of a time-dependent operator basis ${\vec{v}} =\{ \hat{H},\hat{L},\hat{C},\hat{I}\}^T$, defined in the discussion below Eq. . Such a vector of operators serves as a basis for the Liouville space representation. In the following we abuse the notation of ${\vec{v}}$, considering only the first three operators of ${\vec{v}}$. This is allowed as the identity is a constant of motion and does not affect the other basis operators. In Liouville space the dynamics are generated by the Heisenberg equation $${\frac{d}{dt}}{\vec{v}} {\left({t}\right)} = {\left({i {\left[{\hat{H}{\left({t}\right)},\bullet}\right]} +{\frac {\partial }{\partial t}}}\right)} {\vec{v}}{\left({t}\right)}~~, \label{dynamics Liouv}$$ which can be expressed in a vector matrix notation by $${\frac{1}{\Omega}}{\frac{d}{dt}}{\vec{v}}={\frac{\dot{\Omega}}{\Omega^{2}}}{\cal{I}}{\vec{v}}-i{\cal{B}}'{\vec{v}}~~,$$ where ${\cal{B}}'$ and $\mu$ are given in Eq. and . Defining the scaled time $\theta{\left({t}\right)}=\int_0^t{\Omega{\left({t'}\right)}dt'}$ and decomposing the system state as $${\vec{v}}{\left({t}\right)} = {\vec{u}}{\left({t}\right)} \exp{\int_0^t{{\frac{\dot{\Omega}}{\Omega}}dt'}}={\frac{\Omega{\left({t}\right)}}{\Omega{\left({0}\right)}}} {\vec{u}}{\left({t}\right)} \label{ap:v u eq}$$ leads to a time-independent equation for ${\vec{u}}{\left({\theta}\right)}$ $${\frac{d}{d\theta}}{\vec{u}}{\left({\theta}\right)}={\left[{\begin{array}{ccc} 0 & \mu & 0\\ -\mu & 0 & 1\\ 0 & -1 & 0 \end{array}}\right]}{\vec{u}}{\left({\theta}\right)}~~~. \label{eq:u_dynm}$$ For a constant adiabatic parameter $\mu$, we solve Eq. by diagaonalization and obtain a solution in terms of the basis of eigenoperators ${\vec{F}} =\{\hat{F}_1,\hat{F}_2,\hat{F}_3,\hat{I}\}^T$. The solution reads $${\vec{F}}{\left({t}\right)}=e^{-i{\cal{D}}\theta{\left({t}\right)}}{\vec{F}}{\left({ 0 }\right)}~~, \label{eq:F}$$ where ${\cal{D}}=\text{diag}{\left({0,\kappa,-\kappa}\right)}$ with $\kappa = \sqrt{1+\mu^2}$. The eigenoperators $\hat{F}_k$ are associated with the left eigenvectors of ${\cal{B}}$. The eigenoperators are calculated with the help of the diagonalization matrix $\cal{P}$: ${\vec{F}}_i=\sum_j {\cal{P}}^{-1}_{ij}{\vec{u}}_j$. In the ${\vec{v}}=\{\hat{H},\hat{L},\hat{C},\hat{I} \}$ basis the eigenoperators can be written as: ${\vec{F}}_1={\frac{\mu}{\kappa^{2}}}\{1,0,\mu,0\}^{T}$, ${\vec{F}}_2={\frac{1}{2\kappa^{2}}}\{-\mu,-i\kappa,1,0\}^{T}$ and ${\vec{F}}_3 ={\frac{1}{2\kappa^{2}}}\{-\mu,i\kappa,1,0\}^{T}$, corresponding to the eigenvalues ${\lambda}_1=0$ , ${\lambda}_2={\kappa}$, ${\lambda}_3 =-{\kappa}$. Any system observable can be expressed in terms of the eigenoperators $\hat{F}_k$, at initial time, and the exact evolution is then given by equation . The disadvantage of such a solution is the restriction to protocols obeying $\mu=\text{const}$. In order to relax this requirement and extend the solution to a broad class of protocols, the inertial theorem was developed. When the driving of the system satisfies a slow change in $\mu$, the inertial theorem can be employed to describe the system dynamics [@dann2018inertial]. The inertial solution is given by Eq. . The dynamics of any system observable is obtained by the following method: First, the scaled time $\theta$ is calculated by integration over the Generalized Rabi frequency $\Omega{\left({t}\right)}$, and the system observable at initial time is expanded in terms of the eigenoperators. Assuming a non-cyclic process in $\mu$, the geometric phase is neglected relative to the dynamical one. The integration over the time-dependent eigenvalues determines the dynamical phase and the solution is obtained by summing over the linear combination in Eq. . Making use of Eq. , and the definition of ${\vec{F}}_k$, the solution of the $SU(2)$ dynamics becomes (neglecting the geometric phase) $$\begin{gathered} v{\left({\theta{\left({t}\right)}}\right)}={\frac{\Omega{\left({t}\right)}}{\Omega{\left({0}\right)}}}{\cal{P}}{\left({\theta{\left({0}\right)}}\right)}e^{-i\int_{\theta_0}^{\theta{\left({t_{f}}\right)}}{D{\left({\theta'}\right)}d\theta'}}\\ \times{\cal{P}}^{-1}{\left({\theta{\left({0}\right)}}\right)}{\vec{v}}{\left({\theta{\left({0}\right)}}\right)}~~.\end{gathered}$$ Deviations from the exact solution {#ap:diviation} ================================== We derive the correction term for the inertial solution for an $SU(2)$ algebra with the protocol Eq. . Defining $\cal{P}$ as the diagonalizing matrix of ${\cal{B}}'$, Cf. Eq. , we obtain the exact dynamics for the vector ${\vec{w}} ={\cal{P}}{\vec{u}}$ $${\frac{d{\vec{w}}{\left({\theta}\right)}}{d\theta}} =-i{\cal{D}}{\vec{w}} {\left({\theta}\right)}+{\cal{O}}{\vec{w}} {\left({\theta}\right)}~~, \label{ap:w}$$ where ${\cal{O}}=-{\cal{P}}^{-1}{\frac{{\cal{P}}}{d\theta}}$. For the studied model the diagonalizing matrix of ${\cal{B}}'{\left({\mu}\right)}$, Eq. , obtains the form $${\cal{P}}={\frac{1}{2\kappa^2}}\left(\begin{array}{ccc} \frac{1}{\mu} & -\mu & -\mu\\ 0 & i\kappa & -i\kappa\\ 1 & 1 & 1 \end{array}\right)$$ Utilizing the identity ${\frac{d{\cal{P}}}{d\theta}}={\frac{1}{\Omega}}{\frac{d{\cal{P}}}{d t}}$ we obtain $${\cal{O}}={\frac{2\delta\mu}{\Omega\kappa^{2}}}{\cal{I}}+{\cal{S}}~~, \label{ap:O_final}$$ where $${\cal{S}}= {\frac{\delta}{2\Omega\kappa^{2}}}\left(\begin{array}{ccc} \frac{1}{\mu} & \mu & \mu\\ -\frac{1}{2\mu} & -\mu & 0\\ -\frac{1}{2\mu} & 0 & -\mu \end{array}\right)~~.$$ Solving the dynamics explicitly leads to $${\vec{w}}{\left({\theta}\right)}=e^{{\left({-i{\cal{D}+\cal{O}}}\right)}\theta}{\vec{w}}{\left({0}\right)}~~~.$$ Next, we utilize the Zassenhaus formula to obtain a solution up to first order in $\theta$ $${\vec{w}}{\left({\theta}\right)}\approx e^{-i{\cal{D}}\theta}e^{{\cal{O}}\theta}{\vec{w}}{\left({0}\right)}~~~.$$ The correction term to the inertial solution has real eigenvalues, and therefore does not influence the phase. 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--- abstract: 'We study properties concerning decomposition in cohomology by means of generalized-complex structures. This notion includes the $\mathcal{C}^\infty$-pure-and-fullness introduced by Li and Zhang in the complex case and the Hard Lefschetz Condition in the symplectic case. Explicit examples on the moduli space of the Iwasawa manifold are investigated.' address: - Istituto Nazionale di Alta Matematica - | Dipartimento di Matematica e Informatica “Ulisse Dini”\ Università di Firenze\ via Morgagni 67/A, 50134\ Firenze, Italy - | Departamento de Matemáticas – IUMA\ Universidad de Zaragoza\ Campus Plaza San Francisco, 50009\ Zaragoza, Spain author: - Daniele Angella - Simone Calamai - Adela Latorre title: 'On cohomological decomposition of generalized-complex structures' --- Introduction {#introduction .unnumbered} ============ The decomposition of harmonic forms on compact Kähler manifolds into bi-graded components is a strong result in Hodge theory. Therefore, one would also like to extend results on cohomological decompositions to weaker structures, maybe relying on their complex, symplectic, or Riemannian aspects. We recall, for example, the theory initiated by J.-L. Brylinski on Hodge theory for symplectic manifolds [@brylinski]. The analogies and differences between the results obtained in the complex and symplectic cases acquire a deeper meaning when they are framed into the generalized-complex setting. Tools from generalized-complex geometry have been recently used to develop a Hodge theory for SKT structures in [@cavalcanti-skt]. The aim of this note is to provide a notion of cohomological-decomposability on generalized-complex manifolds, showing its coherence with already-known notions for complex and symplectic structures. We consider that having a comparison frame between these two parallel cases may inspire further results, in either complex or symplectic geometry. Let $(X,\, J)$ be a compact complex manifold. There is a natural bi-graded subgroup of the de Rham cohomology of $X$, given by the image of the map $$H^{\bullet,\bullet}_{BC}(X) \;:=\; \frac{\ker{\partial}\cap\ker{\overline{{\partial}}}}{\operatorname{im}{\partial}{\overline{{\partial}}}} \longrightarrow H^\bullet_{dR}(X;{\mathbb{C}}) \;.$$ Observe that the surjectivity of this map would yield to a cohomological decomposition of de Rham cohomology related to the complex structure. In fact, the stronger property of the above map being an isomorphism is called the $\partial\overline\partial$-Lemma property. Furthermore, it should be noted that these subgroups also make sense in a more general framework: that of almost-complex structures $J$ on $X$. Indeed, it suffices to set $$H^{(p,q)}_{J}(X) \;:=\; \left\{ \left[\alpha\right] \in H^{p+q}_{dR}(X;{\mathbb{C}}) {\;:\;}\alpha\in\wedge^{p,q}_JX \right\} \;\subseteq\; H^{p+q}_{dR}(X;{\mathbb{C}}) \;.$$ In the integrable case, $H^{(p,q)}_{J}(X) = \operatorname{im}\left(H^{p,q}_{BC}(X)\to H^{p+q}_{dR}(X;{\mathbb{C}}) \right)$. Such subgroups have been studied by T.-J. Li and W. Zhang in [@li-zhang] when investigating symplectic cones on almost-complex manifolds. It is important to mention that, in general, these subgroups may not yield to a decomposition of the de Rham cohomology. In this sense, the result by T. Drǎghici, T.-J. Li, and W. Zhang in [@draghici-li-zhang Theorem 2.3] appears as a very specific property of compact $4$-dimensional manifolds: it states that any almost-complex structure $J$ on a compact $4$-dimensional manifold $X^4$ satisfies $$\begin{aligned} H^2_{dR}(X^4;{\mathbb{R}}) &=& \left\{ \left[\alpha\right]\in H^2_{dR}(X^4;{\mathbb{R}}) {\;:\;}J\alpha=\alpha \right\} \\[5pt] &\oplus & \left\{ \left[\alpha\right]\in H^2_{dR}(X^4;{\mathbb{R}}) {\;:\;}J\alpha=-\alpha \right\} \;.\end{aligned}$$ Another interesting example is the Iwasawa manifold $\mathbb{I}_3$, which is one of the simplest non-Kähler example of complex threefold [@fernandez-gray]. With respect to its natural holomorphically-parallelizable complex structure, the map $H^{\bullet,\bullet}_{BC}(\mathbb{I}_3) \to H^\bullet_{dR}(\mathbb{I}_3;{\mathbb{C}})$ is surjective (see [@angella-tomassini-1 Theorem 3.1] and §\[sec:iwasawa\]). The same holds when one endows the underlying differentiable manifold of $\mathbb{I}_3$ with the Abelian complex structure given in §\[sec:iwasawa\] (see [@latorre-ugarte]). Such examples show that this kind of decomposition is a strictly weaker property than the $\partial\overline\partial$-Lemma. Consider now a compact symplectic manifold $(X,\, \omega)$. A symplectic Hodge theory was proposed by J.-L. Brylinski in [@brylinski] and further results in this direction were obtained, among others, by O. Mathieu [@mathieu], D. Yan [@yan], V. Guillemin [@guillemin], and G. R. Cavalcanti [@cavalcanti-phd]. Recently, L.-S. Tseng and S.-T. Yau introduced and studied some symplectic cohomologies [@tseng-yau-1; @tseng-yau-2; @tseng-yau-3; @tsai-tseng-yau]. The group $$SH^\bullet_{BC}(X) \;:=\; \frac{\ker\operatorname{d}\cap\ker\operatorname{d}^\Lambda}{\operatorname{im}\operatorname{d}\operatorname{d}^\Lambda} \;,$$ also denoted by $H^\bullet_{\operatorname{d}+\operatorname{d}^\Lambda}$ in [@tseng-yau-1], plays the same role as the Bott-Chern cohomology for complex manifolds (here, $\operatorname{d}^\Lambda:=\left[\operatorname{d},-\iota_{\omega^{-1}}\right]$). As shown in Proposition \[prop:Cpf-sympl\], the surjectivity of the natural map $$SH^\bullet_{BC}(X) \longrightarrow H^\bullet_{dR}(X;{\mathbb{R}})$$ induced by the identity turns out to be equivalent to the property that every de Rham cohomology class admits a $\operatorname{d}$-closed, $\operatorname{d}^\Lambda$-closed representative; that is, the Brylinski conjecture [@brylinski Conjecture 2.2.7] holds. This is also equivalent to the Hard Lefschetz Condition [@mathieu Corollary 2], [@yan Theorem 0.1] and to the so-called $\operatorname{d}\operatorname{d}^\Lambda$-Lemma [@merkulov Proposition 1.4], [@guillemin], [@cavalcanti-phd Theorem 5.4]. (In fact, note that the spectral sequences associated to the bi-differential complex $\left( \wedge^\bullet X,\, \operatorname{d},\, \operatorname{d}^\Lambda \right)$ degenerate at the first level [@brylinski Theorem 2.3.1], [@fernandez-ibanez-deleon Theorem 2.5], in contrast to the complex case.) Generalized-complex geometry was introduced by N. Hitchin [@hitchin] and studied, among others, by his students M. Gualtieri [@gualtieri-phd; @gualtieri-annals] and G. R. Cavalcanti [@cavalcanti-jgp] (see also [@cavalcanti-impa]). It provides a unified framework for both symplectic and complex structures. In fact, any generalized-complex structure is locally equivalent to the product of the standard complex structure on ${\mathbb{C}}^k$ and the standard symplectic structure on ${\mathbb{R}}^{2n-2k}$; see [@gualtieri-phd Theorem 4.35], [@gualtieri-annals Theorem 3.6]. In this note, we study some results concerning decomposition in cohomology induced by generalized-complex structures $\mathcal{J}$ on a compact manifold $X$. More precisely, we consider the property that the natural map $$GH^\bullet_{BC}(X) \;:=\; \frac{\ker{\partial}\cap\ker{\overline{{\partial}}}}{\operatorname{im}{\partial}{\overline{{\partial}}}} \longrightarrow GH_{dR}(X)$$ is surjective (here, ${\partial}$ and ${\overline{{\partial}}}$ are the components of the exterior differential with respect to the graduation induced by $\mathcal{J}$ on the space of complex forms). In the special cases when $\mathcal{J}$ is induced by either a symplectic or a complex structure, we compare this property with the already-known notions (see Proposition \[prop:Cpf-sympl\] and Proposition \[prop:Cpf-cplx\]). As an explicit example, we study structures on the real nilmanifold underlying the Iwasawa manifold. In particular, we focus on the holomorphically-parallelizable and the Abelian complex structures mentioned above, which provide a cohomological decomposition in complex sense (see §\[sec:iwasawa\]). In fact, they induce a cohomological decomposition in generalized-complex sense, too. As observed in [@ketsetzis-salamon], they belong to two different components of the moduli space of left-invariant complex structures on the differentiable Iwasawa manifold. Nevertheless, G. R. Cavalcanti and M. Gualtieri showed in [@cavalcanti-gualtieri] that such structures can be connected by a path of generalized-complex structures, which are given as $\beta$-transform and $B$-transform of a generalized-complex structure $\rho$. We study the cohomological decomposition property of such $\rho$, proving that the natural map from the generalized-Bott-Chern to the generalized-de Rham cohomology is surjective (see §\[subsec:iwasawa-gencplx-path\]). Furthermore, we provide another curve of generalized-complex structures connecting these two complex structures, but arising as $\beta$-transform and $B$-transform of a curve $\left\{J_t\right\}_{t\in[0,1]}$ of almost-complex structures. However, we show that these $J_t$s do not satisfy cohomological decomposition in the sense of Li and Zhang. [*Acknowledgments.*]{} The authors are greatly indebted to Xiuxiong Chen, Adriano Tomassini, and Luis Ugarte for their constant support and encouragement. This work was in part initially conceived during the stay of the first author at Universidad de Zaragoza thanks to a grant by INdAM: the first author would like to thank the Departamento de Matemáticas for the warm hospitality. Thanks are also due to Magda Rinaldi for useful discussions. Preliminaries and notation ========================== In this section, we recall the main definitions and results in generalized-complex geometry, in order to fix the notation. See, e.g., [@cavalcanti-impa] and the references therein for more details. Generalized-complex structures ------------------------------ Let $X$ be a compact differentiable manifold of dimension $2n$. Consider the bundle $TX \oplus T^*X$ endowed with a natural symmetric pairing given by $$\begin{aligned} && {\left\langle X+\xi \,\middle|\, Y+\eta \right\rangle}\;:=\; \frac{1}{2}\,\left(\xi(Y)+\eta(X)\right) \;.\end{aligned}$$ A [*generalized-almost-complex structure*]{} on $X$, [@gualtieri-phd Definition 4.14], is a ${\left\langle {\text{\--}}\,\middle|\, {\text{\textdblhyphen}}\right\rangle}$-orthogonal endomorphism $\mathcal{J}\in \operatorname{End}(TX\oplus T^*X)$ such that $\mathcal{J}^2=-\operatorname{id}_{TX\oplus T^*X}$. Following [@gualtieri-phd §3.2], [@gualtieri-annals §2], consider the *Courant bracket* on $TX\oplus T^*X$, $$\begin{aligned} && \left[X+\xi,\, Y+\eta\right] \;:=\; \left[X,\, Y\right] + \mathcal{L}_X\eta - \mathcal{L}_Y\xi - \frac{1}{2}\, \operatorname{d}\left(\iota_X\eta-\iota_Y\xi\right) \;,\end{aligned}$$ and its associated Nijenhuis tensor for $\mathcal{J}\in\operatorname{End}\left(TX\oplus T^*X\right)$, $$\begin{aligned} \mathrm{Nij}_{\mathcal{J}} &:=& -\left[ \mathcal{J}\,{\text{\--}},\, \mathcal{J}\,{\text{\textdblhyphen}}\right] + \mathcal{J} \left[ \mathcal{J}\,{\text{\--}},\, {\text{\textdblhyphen}}\right] + \mathcal{J} \left[ {\text{\--}},\, \mathcal{J}\,{\text{\textdblhyphen}}\right] + \mathcal{J} \left[ {\text{\--}},\, {\text{\textdblhyphen}}\right] \;.\end{aligned}$$ (As a matter of notation, $\iota_{X}\in \operatorname{End}^{-1}\left(\wedge^\bullet X\right)$ denotes the interior product with $X\in \mathcal{C}^\infty(X;TX)$, and $\mathcal{L}_X:=\left[\iota_X,\, \operatorname{d}\right]\in \operatorname{End}^0\left(\wedge^\bullet X\right)$ denotes the Lie derivative along $X\in \mathcal{C}^\infty(X;TX)$.) A [*generalized-complex structure*]{} on $X$ is a generalized-almost-complex structure $\mathcal{J}\in \operatorname{End}(TX\oplus T^*X)$ such that $\mathrm{Nij}_{\mathcal{J}}=0$, [@gualtieri-phd Definition 4.14, Definition 4.18], [@gualtieri-annals Definition 3.1]. Graduation on forms ------------------- Generalized-complex structures provide a graduation on the space of complex differential forms [@gualtieri-phd §4.4], [@gualtieri-annals Proposition 3.8]. Consider a $2n$-dimensional differentiable manifold $X$ endowed with a generalized-almost-complex structure $\mathcal{J}$. Let $L$ be the $\operatorname{i}$-eigenspace of the ${\mathbb{C}}$-linear extension of $\mathcal{J}$ to $\left(TX \oplus T^*X\right)\otimes_{\mathbb{R}}{\mathbb{C}}$. Consider the complex rank $1$ sub-bundle $U$ of $\wedge^\bullet X \otimes_{\mathbb{R}}{\mathbb{C}}$ generated by a complex form $\rho$ whose Clifford annihilator is precisely $L=\left\{v\in \left(TX \oplus T^*X\right) \otimes_{\mathbb{R}}{\mathbb{C}}{\;:\;}v\cdot \rho=0\right\}$. Here, the operation denotes the Clifford action of $TX\oplus T^*X$ on the space of differential forms on $X$ with respect to ${\left\langle {\text{\--}}\,\middle|\, {\text{\textdblhyphen}}\right\rangle}$, i.e., $$\begin{array}{rcl} \operatorname{Cliff}\left(TX\oplus T^*X\right) \times \wedge^\bullet X &\longrightarrow & \wedge^{\bullet-1} X \oplus \wedge^{\bullet+1} X\\[2pt] \left( (X+\xi), \, \varphi \right) &\longmapsto & (X+\xi) \cdot \varphi \;:=\; \iota_X\varphi + \xi\wedge\varphi \end{array}$$ as well as its bi-${\mathbb{C}}$-linear extension. For each $k\in{\mathbb{Z}}$, define $$U^k \;:=\; \wedge^{n-k}\bar L \cdot U \;\subseteq\; \wedge X \otimes_{\mathbb{R}}{\mathbb{C}}\;.$$ By [@gualtieri-phd Theorem 4.3], [@gualtieri-annals Theorem 3.14], the condition $\mathrm{Nij}_{\mathcal{J}}=0$ is equivalent to the property $$\operatorname{d}U^\bullet \subset U^{\bullet+1} \oplus U^{\bullet-1} \;.$$ Therefore, one has [@gualtieri-phd §4.4], [@gualtieri-annals §3] $$\operatorname{d}\;=\; {\partial}+ {\overline{{\partial}}}\;,$$ where $$\begin{aligned} {\partial}\lfloor_{U^\bullet} &:=& \pi_{U^{\bullet+1}}\circ \operatorname{d}\lfloor_{U^\bullet} \colon \colon U^\bullet \to U^{\bullet+1} \;, \\[5pt] {\overline{{\partial}}}\lfloor_{U^\bullet} &:=& \pi_{U^{\bullet-1}}\circ \operatorname{d}\lfloor_{U^\bullet} \colon \colon U^\bullet \to U^{\bullet-1} \;.\end{aligned}$$ Now, let us recall the notion of $B$-field transform [@gualtieri-phd §3.3] and see how it may affect the initial graduation of forms for a given generalized-complex structure $\mathcal{J}$ defined on $X$. Consider a $\operatorname{d}$-closed $2$-form $B$, viewed as a map $TX\to T^*X$. Consider the generalized-complex structure given by $$\mathcal{J}^B \;:=\; \exp \left(-B\right) \, \mathcal{J} \, \exp B, \quad \text{ where } \quad \exp B \;=\; \left( \begin{array}{c|c} \operatorname{id}_{TX} & 0 \\ \hline B & \operatorname{id}_{T^*X} \end{array} \right) \;.$$ Then, the ${\mathbb{Z}}$-graduation is given by [@cavalcanti-jgp §2.3] $$U^{\bullet}_{\mathcal{J}^B} \;=\; \exp B \wedge U^\bullet_{\mathcal{J}}$$ and in particular, [@cavalcanti-jgp §2.3], $${\partial}_{\mathcal{J}^B} \;=\; \exp \left(-B\right) \, {\partial}_{\mathcal{J}} \, \exp B \qquad \text{ and } \qquad {\overline{{\partial}}}_{\mathcal{J}^B} \;=\; \exp \left(-B\right) \, {\overline{{\partial}}}_{\mathcal{J}} \, \exp B \;.$$ Complex and symplectic structures --------------------------------- Complex and symplectic structures can be seen as very special examples of generalized-complex structures. Consider a $2n$-dimensional differentiable manifold $X$ endowed with a generalized-complex structure $\mathcal{J}$. We recall that, [@gualtieri-phd §4.3], [@gualtieri-annals Definition 3.5], [@gualtieri-annals Definition 1.1], the *type* of $\mathcal{J}$ is given by the upper-semi-continuous function on $X$ defined by $$\mathrm{type}\left(\mathcal{J}\right) \;:=\; \frac{1}{2}\, \dim_{\mathbb{R}}\left( T^*X \cap \mathcal{J}T^*X \right) \;\in\; \left\{0, \ldots, n \right\} \;.$$ Points at which the type of the generalized-complex structure is locally constant are called *regular points*. A generalized Darboux theorem was proven by M. Gualtieri [@gualtieri-phd Theorem 4.35], [@gualtieri-annals Theorem 3.6]. More precisely, for any regular point with type equal to $k$, there is an open neighbourhood endowed with a set of local coordinates such that the generalized-complex structure is a $B$-field transform of the standard generalized-complex structure of ${\mathbb{C}}^{k}\times{\mathbb{R}}^{2n-2k}$. ### Symplectic structures Symplectic structures can be interpreted as generalized-complex structures of type $0$ [@gualtieri-phd Example 4.10]. Let $X$ be a compact $2n$-dimensional manifold endowed with a symplectic structure $\omega \in \wedge^2 X$. The form $\omega\in\wedge^2X$ might be viewed as the isomorphism $\omega \colon TX \to T^*X$, which gives rise to the generalized-complex structure $$\mathcal{J}_\omega \;:=\; \left( \begin{array}{c|c} 0 & -\omega^{-1} \\ \hline \omega & 0 \end{array} \right) \;.$$ The ${\mathbb{Z}}$-graduation on forms is given by [@cavalcanti-jgp Theorem 2.2] $$U^{n-\bullet} \;=\; \exp{\left(\operatorname{i}\omega\right)}\, \left(\exp{\left(\frac{\Lambda}{2\operatorname{i}}\right)} \left(\wedge^\bullet X \otimes_{\mathbb{R}}{\mathbb{C}}\right)\right) \;,$$ where $\Lambda := -\iota_{\omega^{-1}}$. By considering the isomorphism [@cavalcanti-jgp §2.2] $$\varphi\colon \wedge X \otimes_{\mathbb{R}}{\mathbb{C}}\longrightarrow \wedge X \otimes_{\mathbb{R}}{\mathbb{C}}\;, \quad\text{where}\quad \varphi(\alpha) \;:=\; \exp{\left(\operatorname{i}\omega\right)}\, \left(\exp{\left(\frac{\Lambda}{2\operatorname{i}}\right)}\, \alpha\right) \;,$$ one has [@cavalcanti-jgp Corollary 1] $$\varphi\left(\wedge^\bullet X\otimes_{\mathbb{R}}{\mathbb{C}}\right) \simeq U^{n-\bullet} \;,$$ but also $$\varphi \, \operatorname{d}\;=\; {\overline{{\partial}}}\, \varphi \quad\text{ and }\quad \varphi \, \operatorname{d}^{\Lambda} \;=\; -2\operatorname{i}\, {\partial}\, \varphi \;,$$ where $\operatorname{d}^\Lambda := \left[\operatorname{d},\,\Lambda\right]$, see [@koszul; @brylinski]. ### Complex structures Complex structures can be interpreted as generalized-complex structures of type $n$ [@gualtieri-phd Example 4.11, Example 4.25]. Let $X$ be a compact $2n$-dimensional manifold endowed with a complex structure $J\in\operatorname{End}(TX)$. The complex structure gives rise to the generalized-complex structure $$\mathcal{J}_J \;:=\; \left( \begin{array}{c|c} -J & 0 \\ \hline 0 & J^* \end{array} \right) \;\in\; \operatorname{End}\left(TX\oplus T^*X\right) \;,$$ where $J^*\in\operatorname{End}(T^*X)$ denotes the dual endomorphism of $J\in\operatorname{End}(TX)$. The ${\mathbb{Z}}$-graduation on forms is given by [@gualtieri-phd Example 4.25] $$U^\bullet_{\mathcal{J}_J} \;=\; \bigoplus_{p-q=\bullet}\wedge^{p,q}_JX \;.$$ Finally, note that ${\partial}= {\partial}_J$ and ${\overline{{\partial}}}= {\overline{{\partial}}}_J$ (see also [@gualtieri-phd Remark 4.26]). Generalized-complex subgroups of cohomologies ============================================= Let $\mathcal{J}$ be a generalized-almost-complex structure on the manifold $X$. Note that the differential $\operatorname{d}$ does not preserve the ${\mathbb{Z}}$-graduation $U^\bullet$. In fact, one can see that $GH_{dR}(X):=\frac{\ker\operatorname{d}}{\operatorname{im}\operatorname{d}}$ is not ${\mathbb{Z}}$-graded. Hence, following [@li-zhang], it is possible to force a ${\mathbb{Z}}$-graduation by studying the subgroups $$\left\{ GH_{\mathcal{J}}^{(k)}(X) \;:=\; \frac{\ker \operatorname{d}\cap\, U^k}{\operatorname{im}\operatorname{d}} \right\}_{k\in{\mathbb{Z}}} \;.$$ They are denoted by $HH^k(X)$ and called [*generalized cohomology*]{} by G. R. Cavalcanti in [@cavalcanti-phd Definition at page 72]. In the integrable case, one can consider the natural map $$GH^\bullet_{BC}(X) \;:=\; \frac{\ker {\partial}\cap \ker {\overline{{\partial}}}}{\operatorname{im}{\partial}{\overline{{\partial}}}}\, \longrightarrow\, GH_{dR}(X)$$ in such a way that, for any $k\in{\mathbb{Z}}$, $$GH^{(k)}_{\mathcal{J}}(X) \;=\; \operatorname{im}\left( GH^k_{BC}(X) \to GH_{dR}(X) \right) \;.$$ Note that $$\sum_{k\in{\mathbb{Z}}} GH_{\mathcal{J}}^{(k)}(X) \;\subseteq\; GH_{dR}(X) \;,$$ but in general, neither the sum is direct nor the inequality is an equality. Let $X$ be a compact manifold. A generalized-almost-complex structure $\mathcal{J}$ on $X$ is called - [*$\mathcal{C}^\infty$-pure*]{} if $$\bigoplus_{k\in{\mathbb{Z}}} GH_{\mathcal{J}}^{(k)}(X) \;\subseteq\; GH_{dR}(X) \;;$$ - [*$\mathcal{C}^\infty$-full*]{} if $$\sum_{k\in{\mathbb{Z}}} GH_{\mathcal{J}}^{(k)}(X) \;=\; GH_{dR}(X) \;;$$ - [*$\mathcal{C}^\infty$-pure-and-full*]{} if it is both $\mathcal{C}^\infty$-pure and $\mathcal{C}^\infty$-full, that is, $$\bigoplus_{k\in{\mathbb{Z}}} GH_{\mathcal{J}}^{(k)}(X) \;=\; GH_{dR}(X) \;.$$ Analogously to [@li-zhang Proposition 2.5] and [@angella-tomassini-1 Theorem 2.1], we have the following proposition, assuring that $\mathcal{C}^\infty$-fullness is sufficient to have $\mathcal{C}^\infty$-pure-and-fullness (compare also with [@cavalcanti-phd Proposition 4.1]). \[prop:Cf-Cpf\] Let $X$ be a compact manifold endowed with a generalized-complex structure $\mathcal{J}$. If $\mathcal{J}$ is $\mathcal{C}^\infty$-full, then it is also $\mathcal{C}^\infty$-pure, and hence it is $\mathcal{C}^\infty$-pure-and-full. Suppose that there exists $$\label{a-in-h-k} \mathfrak{a} \;\in\; GH^{(h)}_{\mathcal{J}}(X) \cap GH^{(k)}_{\mathcal{J}}(X)$$ with $h\neq k$. Let $\alpha^{h}\in U^h$ and $\alpha^{k}\in U^k$ such that $\mathfrak{a}=\left[\alpha^{h}\right]=\left[\alpha^{k}\right]$. By hypothesis, we have $$GH_{dR}(X) \;=\; \sum_{\ell\in{\mathbb{Z}}} GH^{(\ell)}_{\mathcal{J}}(X) \;.$$ Consider the Mukai pairing: $$\left( {\text{\--}}, {\text{\textdblhyphen}}\right) \colon \wedge^\bullet X\otimes{\mathbb{C}}\times \wedge^\bullet X\otimes{\mathbb{C}}\longrightarrow \mathcal{C}^\infty(X;{\mathbb{C}}) \;, \qquad \left( \varphi_1, \varphi_2 \right) \;:=\; \left( \sigma(\varphi_1)\wedge\varphi_2 \right)_{\text{top}} \;,$$ where $\sigma$ acts on decomposable forms as $\sigma(e_1\wedge\cdots\wedge e_\ell):=e_\ell\wedge\cdots\wedge e_1$ and $({\text{\--}})_{\text{top}}$ denotes the top-dimensional component. By [@cavalcanti-phd Proposition 2.2], one has that the previous pairing vanishes in $U^h \times U^k$ unless $h+k=0$, in which case it is non-degenerate. Furthermore, it can be seen that it induces a non-degenerate pairing in cohomology, $$\left( {\text{\--}}, {\text{\textdblhyphen}}\right) \colon GH_{dR}(X) \times GH_{dR}(X) \longrightarrow {\mathbb{C}}\;.$$ However, $$\left( \mathfrak{a} , GH_{dR}(X) \right) \;=\; \sum_{\ell\in{\mathbb{Z}}} \left( \mathfrak{a} , GH^{(\ell)}_{\mathcal{J}}(X) \right) \;=\; 0$$ due to . Thus, we can conclude that $\mathfrak{a}=0$. As a consequence, we have the following interpretation of $\mathcal{C}^\infty$-pure-and-fullness. \[prop:Cpf-BC-surj-dR\] A generalized-complex structure $\mathcal{J}$ on a compact manifold $X$ is $\mathcal{C}^\infty$-pure-and-full if and only if the natural map $$\bigoplus_{k\in{\mathbb{Z}}} GH^k_{BC}(X) \longrightarrow GH_{dR}(X)$$ induced by the identity is surjective. We recall that a generalized-complex manifold is said to [*satisfy the ${\partial}{\overline{{\partial}}}$-Lemma*]{}, [@cavalcanti-phd Definition at page 70], if $$\operatorname{im}{\partial}\cap \ker{\overline{{\partial}}}\;=\; \operatorname{im}{\partial}{\overline{{\partial}}}\;=\; \operatorname{im}{\overline{{\partial}}}\cap \ker{\partial}\;.$$ Note that compact generalized-complex manifolds satisfying the ${\partial}{\overline{{\partial}}}$-Lemma provide examples of $\mathcal{C}^\infty$-pure-and-full structures. In fact, by [@cavalcanti-phd Theorem 4.2], they are $\mathcal{C}^\infty$-full, and by Proposition \[prop:Cf-Cpf\], they are also $\mathcal{C}^\infty$-pure. Let $X$ be a compact manifold endowed with a generalized-complex structure $\mathcal{J}$. If it satisfies the ${\partial}{\overline{{\partial}}}$-Lemma, then $\mathcal{J}$ is $\mathcal{C}^\infty$-pure-and-full. Finally, we prove that $B$-transforms preserve $\mathcal{C}^\infty$-pure-and-fullness. \[B-trans\] Let $B$ be a $\operatorname{d}$-closed $2$-form on a compact manifold $X$. The generalized-complex structure $\mathcal{J}$ on $X$ is $\mathcal{C}^\infty$-pure-and-full if and only if its $B$-transform $\mathcal{J}_B$ is. Suppose that $\mathcal{J}$ is $\mathcal{C}^\infty$-pure-and-full. Let $\alpha \in GH_{dR}(X, \mathcal{J})$ and consider the $\operatorname{d}$-closed form $\exp(-B)\, \alpha$. By hypothesis, there exists a form $\gamma$ such that $$\exp(-B)\, \alpha \;=\; \sum_{k\in{\mathbb{Z}}} \alpha^{k}_{\mathcal{J}} + \operatorname{d}\gamma, \quad \text{ where } \quad \alpha^{k}_{\mathcal{J}} \;\in\; U^k_{\mathcal{J}} \cap \ker \operatorname{d}\;.$$ Therefore, $$\alpha \;=\; \sum_{k\in{\mathbb{Z}}} \exp(B)\, \alpha^{k}_{\mathcal{J}} + \operatorname{d}\left(\exp(B)\,\gamma\right) \;.$$ Since $\exp(B)\, \alpha^{k}_{\mathcal{J}} \in U^{k}_{\mathcal{J}_B} \cap \ker\operatorname{d}$ (see [@cavalcanti-computations §2]), we can conclude that also $\mathcal{J}_B$ is $\mathcal{C}^\infty$-pure-and-full. The converse follows noting that $\mathcal{J}$ is the $(-B)$-transform of $\mathcal{J}_B$. As generalized-complex-geometry provides a common framework for both complex and symplectic geometry, one would expect to recover existing concepts of $\mathcal{C}^\infty$-pure-and-fullness for these two special cases. We devote to this aim the following lines. Symplectic subgroups of cohomologies ------------------------------------ Let $X$ be a compact manifold endowed with a symplectic structure $\omega$. Denote $L\colon \wedge^\bullet X \ni \alpha \mapsto \alpha\wedge\omega \in \wedge^{\bullet+2}X$ and $\Lambda:=-\iota_{\omega^{-1}}$. Set also $P^\bullet := \ker \Lambda$. A counterpart of the Bott-Chern cohomology in the symplectic case was introduced and studied by S.-T. Yau and L.-S. Tseng [@tseng-yau-1; @tseng-yau-2; @tseng-yau-3; @tsai-tseng-yau]: $$SH^{\bullet}_{BC}(X) \;:=\; \frac{\ker\operatorname{d}\cap\ker\operatorname{d}^\Lambda}{\operatorname{im}\operatorname{d}\operatorname{d}^\Lambda} \;,$$ where $\operatorname{d}^\Lambda:=\left[\operatorname{d},\Lambda\right]$. Inspired by Proposition \[prop:Cpf-BC-surj-dR\], we will say that, for every $k\in{\mathbb{Z}}$, the symplectic structure $\omega$ is [*$\mathcal{C}^\infty$-pure-and-full at the $k$th stage in the sense of Brylinski*]{} [@brylinski] if the natural map $$SH^{k}_{BC}(X) \longrightarrow H_{dR}^{k}(X;{\mathbb{R}})$$ induced by the identity is surjective. When this holds at every stage, it means that every class in the de Rham cohomology admits a representative being both $\operatorname{d}$-closed and $\operatorname{d}^\Lambda$-closed: this property is known as [*satisfying the Brylinski conjecture*]{} [@brylinski Conjecture 2.2.7]. By [@mathieu Corollary 2], [@yan Theorem 0.1], [@merkulov Proposition 1.4], [@guillemin], [@cavalcanti-phd Theorem 5.4], it turns out that the following conditions are equivalent: - being $\mathcal{C}^\infty$-pure-and-full at every stage in the sense of Brylinski; - satisfying the Brylinski conjecture; - satisfying the Hard Lefschetz Condition; - satisfying the $\operatorname{d}\operatorname{d}^\Lambda$-Lemma. (Recall that a symplectic structure $\omega$ on a compact $2n$-dimensional manifold $X$ is said to satisfy the [*Hard Lefschetz Condition*]{} if, for any $k\in{\mathbb{Z}}$, the map $L^k \colon H^{n-k}_{dR}(X;{\mathbb{R}}) \to H^{n+k}_{dR}(X;{\mathbb{R}})$ is bijective. Recall also that it is said to satisfy the [*$\operatorname{d}\operatorname{d}^\Lambda$-Lemma*]{} if every $\operatorname{d}^\Lambda$-closed, $\operatorname{d}$-exact form is $\operatorname{d}\operatorname{d}^\Lambda$-exact too; that is, if the natural map $SH^{\bullet}_{BC}(X) \to H^{\bullet}_{dR}(X;{\mathbb{R}})$ is injective). Let us now show the explicit decomposition of $H^{\bullet}_{dR}(X;{\mathbb{R}})$ in this case. Consider the Lefschetz decomposition on the space of forms, $\wedge^\bullet X = \bigoplus_{2r+s=\bullet} L^rP^s$. Note that $\mathcal{C}^\infty$-pure-and-fullness at every stage in the sense of Brylinski means that the Lefschetz decomposition moves to cohomology. More precisely, one can define the following subgroups [@angella-tomassini-4] $$H^{(r,s)}_{\omega}(X) \;:=\; \left\{ \left[\alpha\right] \in H^\bullet_{dR}(X;{\mathbb{C}}) {\;:\;}\alpha\in L^r P^s \right\}$$ and consider [@tseng-yau-1] $$\begin{aligned} SH^{(r,s)}_{\omega}(X) &:=& L^r H^{(0,s)}_{\omega}(X) \\[5pt] &=& \operatorname{im}\left( L^r PH^s_{BC}(X) \to H^{2r+s}_{dR}(X;{\mathbb{R}}) \right) \\[5pt] &=& \left\{ L^r\left[\beta^{(s)}\right] \in H^{2r+s}_{dR}(X;{\mathbb{R}}) {\;:\;}\beta^{(s)} \in P^s \right\} \;\subseteq\; H^{2r+s}_{dR}(X;{\mathbb{R}}) \;,\end{aligned}$$ where $$PH^{\bullet}_{BC}(X) \;:=\; \frac{\ker\operatorname{d}\cap\ker\operatorname{d}^\Lambda\cap\, P^\bullet}{\operatorname{im}\operatorname{d}\operatorname{d}^\Lambda} \;.$$ By [@angella-tomassini-4 Remark 2.3], it can be seen that $\omega$ is $\mathcal{C}^\infty$-pure-and-full in the sense of Brylinski if and only if $$H^{\bullet}_{dR}(X;{\mathbb{R}}) \;=\; \bigoplus_{2r+s=\bullet} SH^{(r,s)}_{\omega}(X) \;.$$ Let us now compare the notions of $\mathcal{C}^\infty$-pure-and-fullness for a symplectic structure in the sense of Brylinski and for its induced generalized-complex structure. \[prop:Cpf-sympl\] Let $X$ be a compact manifold. Consider a symplectic structure $\omega$ on $X$, viewed as a generalized-almost-complex structure $\mathcal{J}$. Then $\mathcal{J}$ is $\mathcal{C}^\infty$-pure-and-full if and only if $\omega$ is $\mathcal{C}^\infty$-pure-and-full at every stage in the sense of Brylinski. It suffices to observe that, in view of [@cavalcanti-computations §2], $$\varphi \;:=\; \exp(\operatorname{i}\omega)\,\exp\left(\frac{\Lambda}{2\operatorname{i}}\right) \colon \wedge^{n-\bullet}X \stackrel{\simeq}{\longrightarrow} U^{\bullet}_{\mathcal{J}} \;.$$ Furthermore, $${\partial}_{\mathcal{J}} \;=\; -\frac{\operatorname{i}}{2}\, \varphi \circ \operatorname{d}^\Lambda \circ \varphi^{-1} \qquad \text{ and } \qquad {\overline{{\partial}}}_{\mathcal{J}} \;=\; \varphi \circ \operatorname{d}\circ \varphi^{-1} \;.$$ Then we have the commutative diagram $$\xymatrix{ \operatorname{Tot}SH^{\bullet}_{BC}(X) \ar[r]^\simeq_{\varphi} \ar@{->>}[d] & \operatorname{Tot}GH^\bullet_{BC}(X) \ar@{->>}[d] \\ \operatorname{im}\left(\operatorname{Tot}SH^{\bullet}_{BC}(X) \to \operatorname{Tot}H^\bullet_{dR}(X;{\mathbb{R}})\right) \ar[r]^{\varphi}_{\simeq} \ar@{^{(}->}[d] & \operatorname{im}\left( \operatorname{Tot}GH^\bullet_{BC}(X) \to GH_{dR}(X) \right) \ar@{^{(}->}[d] \\ \operatorname{Tot}H^\bullet_{dR}(X;{\mathbb{R}}) \ar@{=}[r] & GH_{dR}(X) \;. }$$ This concludes the proof. Concerning the notion of [*$\mathcal{C}^\infty$-pure-and-fullness in the sense of [@angella-tomassini-4]*]{}, that is, the property that $$H^{\bullet}_{dR}(X;{\mathbb{R}}) \;=\; \bigoplus_{2r+s=\bullet} H^{(r,s)}_{\omega}(X) \;,$$ we note that it is strictly weaker than the notion of $\mathcal{C}^\infty$-pure-and-fullness in the sense of Brylinski. In fact, by [@angella-tomassini-4 Theorem 2.6], every compact $4$-dimensional symplectic manifold is $\mathcal{C}^\infty$-pure-and-full in the sense of [@angella-tomassini-4]. On the other side, there are examples of such manifolds that do not satisfy the Hard Lefschetz Condition: hence, they are non-$\mathcal{C}^\infty$-pure-and-full in the sense of Brylinski. For example, consider non-tori nilmanifolds, [@benson-gordon Theorem A], see also [@bock Theorem 9.2]. For a higher-dimensional example, see the results by M. Rinaldi in [@magda-thesis]. Complex subgroups of cohomologies --------------------------------- In the almost-complex case, T.-J. Li and W. Zhang introduced and studied the notion of $\mathcal{C}^\infty$-pure-and-fullness in [@li-zhang] (see [@draghici-li-zhang; @angella-tomassini-1; @angella-tomassini-2] and the references therein for further results). More precisely, let $J$ be an almost-complex structure on the manifold $X$. For each $(p,q)\in{\mathbb{Z}}^2$, consider the subgroup $$H^{(p,q)}_{J}(X) \;:=\; \left\{ \left[\alpha\right] \in H^\bullet_{dR}(X;{\mathbb{C}}) {\;:\;}\alpha\in\wedge^{p,q}X \right\} \;.$$ Given $k\in{\mathbb{Z}}$, the almost-complex structure $J$ is called [*complex-$\mathcal{C}^\infty$-pure-and-full at the $k$th stage in the sense of Li and Zhang*]{} [@li-zhang] if $$\bigoplus_{p+q=k} H^{(p,q)}_{J}(X) \;=\; H^k_{dR}(X;{\mathbb{C}}) \;.$$ Now we would like to compare the notions of $\mathcal{C}^\infty$-pure-and-fullness for a complex structure in the sense of Li and Zhang and for its induced generalized-complex structure. Let $X$ be a compact manifold. Consider an almost-complex structure $J$ on $X$, viewed as a generalized-almost-complex structure $\mathcal{J}$. Then, for any $k\in{\mathbb{Z}}$, it holds $$GH^{(k)}_{\mathcal{J}} \;=\; \bigoplus_{p-q=k} H^{(p,q)}_J(X) \;.$$ Observe that in the almost-complex case the following equatily holds $$U^k_{\mathcal{J}} \;=\; \bigoplus_{p-q=k} \wedge^{p,q}_JX \;.$$ In general, the differential $\operatorname{d}$ of a generalized-complex structure can be decomposed as $$\operatorname{d}\;=\; A_{\mathcal{J}} + {\partial}_{\mathcal{J}} + {\overline{{\partial}}}_{\mathcal{J}} + \bar A_{\mathcal{J}} \colon U^k_{\mathcal{J}} \longrightarrow U^{k+2}_{\mathcal{J}} \oplus U^{k+1}_{\mathcal{J}} \oplus U^{k-1}_{\mathcal{J}} \oplus U^{k-2}_{\mathcal{J}} \,.$$ However, as $\mathcal{J}$ is actually an almost-complex structure one has $$\operatorname{d}\;=\; A_{J} + {\partial}_{J} + {\overline{{\partial}}}_{J} + \bar A_{J} \colon \wedge^{p,q}_JX \longrightarrow \wedge^{p+2,q-1}_JX \oplus \wedge^{p+1,q}_JX \oplus \wedge^{p,q+1}_JX \oplus \wedge^{p-1,q+2}_JX$$ and therefore, $$A_{\mathcal{J}} \;=\; A_{J} \;, \qquad {\partial}_{\mathcal{J}} \;=\; {\partial}_{J} \;, \qquad {\overline{{\partial}}}_{\mathcal{J}} \;=\; {\overline{{\partial}}}_{J} \qquad \text{ and } \qquad \bar A_{\mathcal{J}} \;=\; \bar A_{J} \;.$$ Take $[\alpha]\in GH^{(k)}_{\mathcal{J}}(X)$ with $\alpha = \sum_{p-q=k} \alpha^{(p,q)} \in U^k_{\mathcal{J}}$ and $\alpha^{(p,q)}\in\wedge^{p,q}_JX$. Then $A_{\mathcal{J}}\alpha={\partial}_{\mathcal{J}}\alpha={\overline{{\partial}}}_{\mathcal{J}}\alpha=\bar A_{\mathcal{J}}\alpha=0$, but also $A_{J}\alpha={\partial}_{J}\alpha={\overline{{\partial}}}_{J}\alpha=\bar A_{J}\alpha=0$. Thus, $\operatorname{d}\alpha^{(p,q)}=0$ for any $(p,q)$; that is, $\left[\alpha\right] = \sum_{p-q=k} \left[\alpha^{(p,q)}\right]$ where $\left[\alpha^{(p,q)}\right]\in H^{(p,q)}_{J}(X)$. The sum is obviously direct. As a consequence, we get the following. \[prop:Cpf-cplx\] Let $X$ be a compact manifold. Consider an almost-complex structure $J$ on $X$, viewed as a generalized-almost-complex structure $\mathcal{J}$. Then $\mathcal{J}$ is $\mathcal{C}^\infty$-pure-and-full if and only if $J$ is complex-$\mathcal{C}^\infty$-pure-and-full at every stage in the sense of Li and Zhang. Generalized-complex structures on the differential nilmanifold underlying Iwasawa {#sec:iwasawa} ================================================================================= The [*Iwasawa manifold*]{} is the complex nilmanifold defined by $$\mathbb{I}_3 \;:=\; \left. \mathbb{H}(3;{\mathbb{Z}}[\operatorname{i}]) \middle\backslash \mathbb{H}(3;{\mathbb{C}}) \right.$$ where $\mathbb{H}(3;{\mathbb{C}})$ is the $3$-dimensional *Heisenberg group* over $\mathbb{C}$, that is, $$\mathbb{H}(3;{\mathbb{C}}) \;:=\; \left\{ \left( \begin{array}{ccc} 1 & z^1 & z^3 \\ 0 & 1 & z^2 \\ 0 & 0 & 1 \end{array} \right) \in \mathrm{GL}(3;\mathbb{C}) {\;:\;}z^1,\,z^2,\,z^3 \in{\mathbb{C}}\right\} \;,$$ and $\mathbb{H}(3;{\mathbb{Z}}[\operatorname{i}]) := \mathbb{H}(3;{\mathbb{C}}) \cap \mathrm{GL}(3;{\mathbb{Z}}[\operatorname{i}])$. It is worth to remark that it constitutes one of the simplest examples of non-Kähler complex manifold (see, e.g., [@fernandez-gray; @nakamura]). In our case, we are interested in its underlying real nilmanifold that we will denote by $M=\Gamma\backslash G$. Following [@salamon], let $\mathfrak{g}=(0,0,0,0,13+42,14+23)$ be the real nilpotent Lie algebra naturally associated to $G$ (i.e., the differentiable Lie group underlying $\mathbb{H}(3;{\mathbb{C}})$). This notation means that $\mathfrak{g}^*$ admits a basis $\{e^k\}_{k=1}^6$ satisfying $$\left\{ \begin{array}{l} \operatorname{d}e^1 \;=\; \operatorname{d}e^2 \;=\; \operatorname{d}e^3 \;=\; \operatorname{d}e^4 \;=\; 0 \\[5pt] \operatorname{d}e^5 \;=\; e^{13} - e^{24} \\[5pt] \operatorname{d}e^6 \;=\; e^{14} + e^{23} \end{array} \right. \;,$$ where $e^{ij}:=e^i\wedge e^j$. These are known as the *structure equations*. By the Nomizu theorem, [@nomizu Theorem 1], the de Rham cohomology of $M$ can be computed by means of the associated Lie algebra $\mathfrak{g}$ (i.e., using the previous structure equations). More precisely, given the Riemannian metric $g:=\sum_{j=1}^{6} e^j \odot e^j$, the harmonic representatives of the de Rham cohomology are the following: $$\begin{aligned} H^0_{dR}(M;{\mathbb{R}}) &=& {\mathbb{R}}\left\langle \left[1\right] \right\rangle \;, \\[5pt] H^1_{dR}(M;{\mathbb{R}}) &=& {\mathbb{R}}\left\langle \left[e^1\right],\, \left[e^2\right],\, \left[e^3\right],\, \left[e^4\right] \right\rangle \;, \\[5pt] H^2_{dR}(M;{\mathbb{R}}) &=& {\mathbb{R}}\left\langle \left[e^{12}\right],\, \left[e^{34}\right],\, \left[e^{15}-e^{26}\right],\, \left[e^{25}+e^{16}\right],\, \right. \\[5pt] && \left. \left[e^{35}-e^{46}\right],\, \left[e^{45}+e^{36}\right],\, \left[e^{13}+e^{24}\right],\, \left[e^{23}-e^{14}\right] \right\rangle \;, \\[5pt] H^3_{dR}(M;{\mathbb{R}}) &=& {\mathbb{R}}\left\langle \left[e^{125}\right],\, \left[e^{126}\right],\, \left[e^{345}\right],\, \left[e^{346}\right],\, \right. \\[5pt] && \left. \left[e^{135}-e^{245}-e^{236}-e^{146}\right],\, \left[e^{235}+e^{145}+e^{136}-e^{246}\right],\, \right. \\[5pt] && \left. \left[-e^{135}+e^{236}-e^{146}-e^{245}\right],\, \left[-e^{136}-e^{235}+e^{145}-e^{246}\right],\, \right. \\[5pt] && \left. \left[e^{135}+e^{245}+e^{236}-e^{146}\right],\, \left[-e^{235}+e^{145}+e^{136}+e^{246}\right]\right\rangle \;, \\[5pt] H^4_{dR}(M;{\mathbb{R}}) &=& {\mathbb{R}}\left\langle \left[e^{1256}\right],\, \left[e^{3456}\right],\, \left[e^{2346}-e^{1345}\right],\, \left[e^{1346}+e^{2345}\right],\, \right. \\[5pt] && \left. \left[e^{1246}-e^{1235}\right],\, \left[e^{1236}+e^{1245}\right],\, \left[e^{2456}+e^{1356}\right],\, \left[e^{1456}-e^{2356}\right] \right\rangle \;, \\[5pt] H^5_{dR}(M;{\mathbb{R}}) &=& {\mathbb{R}}\left\langle \left[e^{23456}\right],\, \left[e^{13456}\right],\, \left[e^{12456}\right],\, \left[e^{12356}\right] \right\rangle \;, \\[5pt] H^6_{dR}(M;{\mathbb{R}}) &=& {\mathbb{R}}\left\langle \left[e^{123456}\right] \right\rangle \;.\end{aligned}$$ Any linear complex structure $J$ defined on $\mathfrak{g}$ gives rise to a complex structure on $M$ that will be called *left-invariant*. The Iwasawa manifold can be regarded as one of these structures, although there is an infinite family of them (see [@andrada-barberis-dotti; @couv] for a complete classification up to isomorphism). Let $\mathfrak{g}^{1,0}$ be the $i$-eigenspace of $J$ as an endomorphism on $\mathfrak{g}^*_\mathbb{C}:=(\mathfrak{g}\otimes_{\mathbb{R}}\mathbb{C})^*$. It is well-known that $J$ is a complex structure on $\mathfrak{g}$ if and only if $\operatorname{d}(\mathfrak{g}^{1,0})\subset\wedge^{2,0}(\mathfrak{g}^*_\mathbb{C})\oplus\wedge^{1,1}(\mathfrak{g}^*_\mathbb{C})$. There are two special types of complex structures that deserve our attention. - $J$ is said to be *holomorphically-parallelizable* if $\operatorname{d}(\mathfrak{g}^{1,0})\subset\wedge^{2,0}(\mathfrak{g}^*_\mathbb{C})$. In this case, $\mathfrak{g}$ can be endowed with a complex Lie algebra stucture and $M$ has a global basis of holomorphic vector fields. - $J$ is called *Abelian* if $\operatorname{d}(\mathfrak{g}^{1,0})\subset\wedge^{1,1}(\mathfrak{g}^*_\mathbb{C})$. In this case, it turns out that $\mathfrak{g}^{1,0}$ is actually an Abelian complex Lie algebra. From the general study accomplished in [@latorre-ugarte], one can conclude that there are only two left-invariant complex structures defined on $M$ which are $\mathcal{C}^\infty$-pure-and-full at every stage in the sense of Li and Zhang. They are precisely the holomorphically-parallelizable stucture $J_0$ corresponding to the Iwasawa manifold (as proven in [@angella-tomassini-1]) and the Abelian stucture $J_1$ in [@andrada-barberis-dotti Theorem 3.3]. In the following lines, we give the explicit decomposition of the de Rham cohomology groups for each of these structures. With respect to the basis $\{e^k\}_{k=1}^6$, the complex structure $J_0$ can be defined as (see [@nakamura]) $$J_0 e^1 \;=\; -e^2 \;, \qquad J_0 e^3 \;=\; -e^4 \;, \qquad J_0 e^5 \;=\; -e^6 \;.$$ Therefore, the forms $$\left\{ \begin{array}{rcl} \varphi_{0}^1 &:=& e^1 + \operatorname{i}e^2 \;\stackrel{\text{loc}}{=}\; \operatorname{d}z^1 \;, \\[5pt] \varphi_{0}^2 &:=& e^3 + \operatorname{i}e^4 \;\stackrel{\text{loc}}{=}\; \operatorname{d}z^2 \;, \\[5pt] \varphi_{0}^3 &:=& e^5 + \operatorname{i}e^6 \;\stackrel{\text{loc}}{=}\; \operatorname{d}z^3 - z^1\, \operatorname{d}z^2 \end{array} \right.$$ provide a left-invariant co-frame for the space of $(1,0)$-forms on $M$ with respect to $J_0$, with complex structure equations $$\left\{ \begin{array}{rcl} \operatorname{d}\varphi_{0}^1 &=& 0 \\[5pt] \operatorname{d}\varphi_{0}^2 &=& 0 \\[5pt] \operatorname{d}\varphi_{0}^3 &=& \varphi_{0}^1\wedge\varphi_{0}^2 \end{array} \right. \;.$$ As already stated, $J_0$ is complex-$\mathcal{C}^\infty$-pure-and-full at every stage in the sense of Li and Zhang [@angella-1 Theorem 3.1]. In fact, it is possible to see that $$\begin{aligned} H^1_{dR}(M;{\mathbb{C}}) &=& \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{0}^1\right],\, \left[\varphi_{0}^2\right] \right\rangle}_{H^{(1,0)}_{J_0}(M)} \,\ \oplus \,\ \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{0}^{\bar1}\right],\, \left[\varphi_{0}^{\bar2}\right] \right\rangle}_{H^{(0,1)}_{J_0}(M)} \;, \\[5pt] H^2_{dR}(M;{\mathbb{C}}) &=& \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{0}^{12}\right],\, \left[\varphi_{0}^{23}\right] \right\rangle}_{H^{(2,0)}_{J_0}(M)} \,\ \oplus \,\ \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{0}^{\bar1\bar3}\right],\, \left[\varphi_{0}^{\bar2\bar3}\right] \right\rangle}_{H^{(0,2)}_{J_0}(M)} \\[5pt] & \oplus & \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{0}^{1\bar1}\right],\, \left[\varphi_{0}^{1\bar2}\right],\, \left[\varphi_{0}^{2\bar1}\right],\, \left[\varphi_{0}^{2\bar2}\right] \right\rangle}_{H^{(1,1)}_{J_0}(M)} \;, \\[5pt] H^3_{dR}(M;{\mathbb{C}}) &=& \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{0}^{123}\right] \right\rangle}_{H^{(3,0)}_{J_0}(M)} \,\ \oplus \,\ \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{0}^{\bar1\bar2\bar3}\right] \right\rangle}_{H^{(0,3)}_{J_0}(M)} \\[5pt] & \oplus & \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{0}^{13\bar1}\right],\, \left[\varphi_{0}^{13\bar2}\right],\, \left[\varphi_{0}^{23\bar1}\right],\, \left[\varphi_{0}^{23\bar2}\right] \right\rangle}_{H^{(2,1)}_{J_0}(M)} \\[5pt] & \oplus & \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{0}^{1\bar1\bar3}\right],\, \left[\varphi_{0}^{1\bar2\bar3}\right],\, \left[\varphi_{0}^{2\bar1\bar3}\right],\, \left[\varphi_{0}^{2\bar2\bar3}\right] \right\rangle}_{H^{(1,2)}_{J_0}(M)} \;, \\[5pt] H^4_{dR}(M;{\mathbb{C}}) &=& \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{0}^{123\bar1}\right],\, \left[\varphi_{0}^{123\bar2}\right] \right\rangle}_{H^{(3,1)}_{J_0}(M)} \,\ \oplus \,\ \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{0}^{1\bar1\bar2\bar3}\right],\, \left[\varphi_{0}^{2\bar1\bar2\bar3}\right] \right\rangle}_{H^{(1,3)}_{J_0}(M)} \\[5pt] & \oplus & \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{0}^{13\bar1\bar3}\right],\, \left[\varphi_{0}^{13\bar2\bar2}\right],\, \left[\varphi_{0}^{23\bar1\bar3}\right],\, \left[\varphi_{0}^{23\bar2\bar3}\right] \right\rangle}_{H^{(2,2)}_{J_0}(M)} \;, \\[5pt] H^5_{dR}(M;{\mathbb{C}}) &=& \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{0}^{123\bar1\bar3}\right],\, \left[\varphi_{0}^{123\bar2\bar3}\right] \right\rangle}_{H^{(3,2)}_{J_0}(M)} \,\ \oplus \,\ \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{0}^{13\bar1\bar2\bar3}\right],\, \left[\varphi_{0}^{23\bar1\bar2\bar3}\right] \right\rangle}_{H^{(2,3)}_{J_0}(M)} \;, \\[5pt]\end{aligned}$$ where we have listed the harmonic representatives with respect to the Hermitian metric $g_0 := \sum_{j=1}^{3} \varphi_{0}^j \odot \bar\varphi_{0}^j$ and we have shortened, e.g., $\varphi_{0}^{1\bar1}:=\varphi_{0}^1\wedge \bar\varphi_{0}^1$. Following [@andrada-barberis-dotti Theorem 3.3], one can define the Abelian complex structure $J_1$ with respect to the basis $\{e^k\}_{k=1}^6$ by $$J_1 e^1 \;=\; -e^3 \;, \qquad J_1 e^2 \;=\; -e^4 \;, \qquad J_1 e^5 \;=\; -e^6 \;.$$ Then, the forms $$\tilde{\varphi}_{1}^1 \;:=\; e^1 + \operatorname{i}e^3 \;, \quad \tilde{\varphi}_{1}^2 \;:=\; e^2 + \operatorname{i}e^4 \;, \quad \tilde{\varphi}_{1}^3 \;:=\; e^5 + \operatorname{i}e^6$$ provide a left-invariant co-frame for the space of $(1,0)$-forms on $M$ with respect to $J_1$, with complex structure equations $$\left\{ \begin{array}{rcl} \operatorname{d}\tilde{\varphi}_{1}^1 &=& 0 \\[5pt] \operatorname{d}\tilde{\varphi}_{1}^2 &=& 0 \\[5pt] \operatorname{d}\tilde{\varphi}_{1}^3 &=& \frac{\operatorname{i}}{2}\,\tilde{\varphi}_1^{1}\wedge\tilde{\varphi}_1^{\bar{1}}-\frac{1}{2}\,\tilde{\varphi}_1^{1}\wedge\tilde{\varphi}_1^{\bar{2}} -\frac{1}{2}\,\tilde{\varphi}_1^{2}\wedge\tilde{\varphi}_1^{\bar{1}}-\frac{\operatorname{i}}{2}\,\tilde{\varphi}_1^{2}\wedge\tilde{\varphi}_1^{\bar{2}} \end{array} \right. \;.$$ Applying the change of basis $$\varphi_1^1=\tilde{\varphi}_1^1-\operatorname{i}\tilde{\varphi}_1^2, \qquad \varphi_1^2=\tilde{\varphi}_1^1+\operatorname{i}\tilde{\varphi}_1^2, \qquad \varphi_1^3=2\operatorname{i}\tilde{\varphi}_1^3,$$ we obtain $$\left\{ \begin{array}{rcl} \operatorname{d}\varphi_{1}^1 &=& 0 \\[5pt] \operatorname{d}\varphi_{1}^2 &=& 0 \\[5pt] \operatorname{d}\varphi_{1}^3 &=& - \varphi_{1}^2\wedge\bar\varphi_{1}^1 \end{array} \right. \;.$$ Observe that these last equations yield to the following equivalent definition of the complex structure $J_1$: $$J_1 e^1 \;=\; -e^2 \;, \qquad J_1 e^3 \;=\; e^4 \;, \qquad J_1 e^5 \;=\; e^6 \;.$$ As previously said, $J_1$ is complex-$\mathcal{C}^\infty$-pure-and-full at every stage in the sense of Li and Zhang (see [@latorre-otal-ugarte-villacampa Proposition 4] as regards to the first stage, see also [@latorre-ugarte]). In fact, one has $$\begin{aligned} H^1_{dR}(M;{\mathbb{C}}) &=& \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{1}^1\right],\, \left[\varphi_{1}^2\right] \right\rangle}_{H^{(1,0)}_{J_1}(M)} \,\ \oplus \,\ \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{1}^{\bar1}\right],\, \left[\varphi_{1}^{\bar2}\right] \right\rangle}_{H^{(0,1)}_{J_1}(M)} \;, \\[5pt] H^2_{dR}(M;{\mathbb{C}}) &=& \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{1}^{12}\right],\, \left[\varphi_{1}^{23}\right] \right\rangle}_{H^{(2,0)}_{J_1}(M)} \,\ \oplus \,\ \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{1}^{\bar1\bar2}\right],\, \left[\varphi_{1}^{\bar2\bar3}\right] \right\rangle}_{H^{(0,2)}_{J_1}(M)} \\[5pt] & \oplus & \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{1}^{1\bar1}\right],\, \left[\varphi_{1}^{2\bar2}\right],\, \left[\varphi_{1}^{1\bar3}\right],\, \left[\varphi_{1}^{3\bar1}\right] \right\rangle}_{H^{(1,1)}_{J_1}(M)} \;, \\[5pt] H^3_{dR}(M;{\mathbb{C}}) &=& \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{1}^{123}\right] \right\rangle}_{H^{(3,0)}_{J_1}(M)} \,\ \oplus \,\ \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{1}^{\bar1\bar2\bar3}\right] \right\rangle}_{H^{(0,3)}_{J_1}(M)} \\[5pt] & \oplus & \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{1}^{12\bar3}\right],\, \left[\varphi_{1}^{13\bar1}\right],\, \left[\varphi_{1}^{23\bar1}\right],\, \left[\varphi_{1}^{23\bar2}\right] \right\rangle}_{H^{(2,1)}_{J_1}(M)} \\[5pt] & \oplus & \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{1}^{1\bar1\bar3}\right],\, \left[\varphi_{1}^{1\bar2\bar3}\right],\, \left[\varphi_{1}^{2\bar2\bar3}\right],\, \left[\varphi_{1}^{3\bar1\bar2}\right] \right\rangle}_{H^{(1,2)}_{J_1}(M)} \;, \\[5pt] H^4_{dR}(M;{\mathbb{C}}) &=& \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{1}^{123\bar1}\right],\, \left[\varphi_{1}^{123\bar3}\right] \right\rangle}_{H^{(3,1)}_{J_1}(M)} \,\ \oplus \,\ \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{1}^{1\bar1\bar2\bar3}\right],\, \left[\varphi_{1}^{3\bar1\bar2\bar3}\right] \right\rangle}_{H^{(1,3)}_{J_1}(M)} \\[5pt] & \oplus & \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{1}^{13\bar1\bar3}\right],\, \left[\varphi_{1}^{23\bar2\bar3}\right],\, \left[\varphi_{1}^{12\bar2\bar3}\right],\, \left[\varphi_{1}^{23\bar1\bar2}\right] \right\rangle}_{H^{(2,2)}_{J_1}(M)} \;, \\[5pt] H^5_{dR}(M;{\mathbb{C}}) &=& \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{1}^{123\bar1\bar3}\right],\, \left[\varphi_{1}^{123\bar2\bar3}\right] \right\rangle}_{H^{(3,2)}_{J_1}(M)} \,\ \oplus \,\ \underbrace{{\mathbb{C}}\left\langle \left[\varphi_{1}^{13\bar1\bar2\bar3}\right],\, \left[\varphi_{1}^{23\bar1\bar2\bar3}\right] \right\rangle}_{H^{(2,3)}_{J_1}(M)} \;. \\[5pt]\end{aligned}$$ Again, we have listed the harmonic representatives with respect to the Hermitian metric $g_1 := \sum_{j=1}^{3} \varphi_{1}^j \odot \bar\varphi_{1}^j$ and we have shortened, e.g., $\varphi_{1}^{1\bar1}:=\varphi_{1}^1\wedge \bar\varphi_{1}^1$. As it is observed in [@ketsetzis-salamon Theorem 4.6, Theorem 1.3], the space of left-invariant oriented complex structures on $M$ has the homotopy type of the disjoint union of a point and a $2$-sphere. However, G. R. Cavalcanti and M. Gualtieri show in [@cavalcanti-gualtieri] that it is possible to connect these two disjoint components by means of a left-invariant generalized-complex structure $\rho$ of type $1$ on $M$. In fact, one can see that this structure precisely connects, up to $B$-transforms and $\beta$-transforms, our complex structures $J_0$ and $J_1$. As these are the only two left-invariant complex structures on $M$, up to equivalence, that are $\mathcal{C}^\infty$-pure-and-full as generalized-complex structures, it is natural to wonder what happens to $\rho$. Next, we prove that $\rho$ is actually $\mathcal{C}^\infty$-pure-and-full and we provide another manner of joining $J_0$ and $J_1$ by means of left-invariant almost-complex structures on $M$. In contrast, we show that this new path is not $\mathcal{C}^\infty$-pure-and-full in the sense of Li and Zhang. Connecting generalized-complex structure by Cavalcanti and Gualtieri {#subsec:iwasawa-gencplx-path} -------------------------------------------------------------------- Consider the generalized-complex structure $\mathcal{J}$ given by G. R. Cavalcanti and M. Gualtieri in [@cavalcanti-gualtieri §5]. Observe that it is defined by the canonical bundle with generator $$\rho \;:=\; \exp \left(\operatorname{i}\,\left(-e^{36}-e^{45}\right)\right) \wedge \left( e^1+\operatorname{i}\,e^2\right) \;.$$ Let us see that this structure is $\mathcal{C}^\infty$-pure-and-full by computing the subgroups $GH^{(k)}_{\mathcal{J}}(M) \;=\; \operatorname{im}\left( GH^k_{BC}(M) \to GH_{dR}(M) \right)$. Observe that these calculations can be done at the Lie algebra level as a consequence of [@nomizu] and [@angella-calamai]. Consider the fibration $$\xymatrix{ \left( \mathbb{T}^4,\, \omega \right) \ar@{^{(}->}[rr] && M \ar@{->>}[dd] \\ (e^3,e^4,e^5,e^6) \ar@{|->}[r] & (e^1,\ldots,e^6) \ar@{|->}[d] & \\ & (e^1,e^2) & \left( \mathbb{T}^2,\, J \right) }$$ where $$\omega \;:=\; -e^{36}-e^{45} \qquad \text{ and } \qquad J \colon e_1 \longmapsto e_2 \;.$$ We note that, for every $k\in{\mathbb{Z}}$, one has $$U^k \;=\; \bigoplus_{r+s=k} U^r_J \otimes U^s_\omega \;.$$ Therefore, we compute $$\begin{aligned} U^1_J &=& \left\langle e^1+\operatorname{i}\,e^2 \right\rangle \;, \\[5pt] U^0_J &=& \left\langle 1,\; e^{12} \right\rangle \;, \\[5pt] U^{-1}_J &=& \left\langle e^1-\operatorname{i}\,e^2 \right\rangle \;, \end{aligned}$$ and $$\begin{aligned} U^2_\omega &=& \left\langle 1-\operatorname{i}\,e^{36}-\operatorname{i}\,e^{45}-e^{3456} \right\rangle \;, \\[5pt] U^1_\omega &=& \left\langle e^{3}-\operatorname{i}\,e^{345},\; e^{4}+\operatorname{i}\,e^{346},\; e^{5}+\operatorname{i}\,e^{356},\; e^{6}-\operatorname{i}\,e^{456} \right\rangle \;, \\[5pt] U^0_\omega &=& \left\langle e^{34},\; e^{35},\; e^{46},\; e^{56},\; e^{36}-e^{45},\; 1+e^{3456} \right\rangle \;, \\[5pt] U^{-1}_\omega &=& \left\langle e^{3}+\operatorname{i}\,e^{345},\; e^{4}-\operatorname{i}\,e^{346},\; e^{5}-\operatorname{i}\,e^{356},\; e^{6}+\operatorname{i}\,e^{456} \right\rangle \;, \\[5pt] U^{-2}_\omega &=& \left\langle 1+\operatorname{i}\,e^{36}+\operatorname{i}\,e^{45}-e^{3456} \right\rangle \;.\end{aligned}$$ From these computations, we get the claim. In fact, it can be seen that: $$\begin{aligned} GH^{(3)}_{\mathcal{J}}(M) &=& {\mathbb{C}}\left\langle \left[e^1+\operatorname{i}\,e^2-\operatorname{i}\,e^{136}-\operatorname{i}\,e^{145}+e^{236}+e^{245}-e^{13456}-\operatorname{i}\,e^{23456}\right] \right\rangle \;, \\[10pt] GH^{(2)}_{\mathcal{J}}(M) &=& {\mathbb{C}}\left\langle \left[e^{13}+\operatorname{i}\,e^{23}-\operatorname{i}\,e^{1345}+e^{2345}\right],\; \left[1-\operatorname{i}\,e^{36}-\operatorname{i}\,e^{45}-e^{3456}\right],\; \right. \\[5pt] && \left. \left[e^{15}+\operatorname{i}\,e^{16}+\operatorname{i}\,e^{25}-e^{26}+\operatorname{i}\,e^{1356}+e^{1456}-e^{2356}+\operatorname{i}\,e^{2456}\right],\; \right. \\[5pt] && \left. \left[e^{12}-\operatorname{i}\,e^{1236}-\operatorname{i}\,e^{1245}-e^{123456}\right] \right\rangle \;, \\[10pt] GH^{(1)}_{\mathcal{J}}(M) &=& {\mathbb{C}}\left\langle \left[e^1+\operatorname{i}\,e^2+e^{13456}+\operatorname{i}\,e^{23456}\right],\; \left[e^3-\operatorname{i}\,e^{345}\right],\; \left[e^{4}+\operatorname{i}\,e^{346}\right],\; \right. \\[5pt] && \left. \left[e^{125}+\operatorname{i}\,e^{12356}\right],\; \left[e^{126}-\operatorname{i}\,e^{12456}\right],\; \left[e^{135}-e^{146}+\operatorname{i}\,e^{235}-\operatorname{i}\,e^{246}\right],\; \right. \\[5pt] && \left. \left[e^{1}-\operatorname{i}\,e^{2}-\operatorname{i}\,e^{136}-\operatorname{i}\,e^{145}-e^{236}-e^{245}-e^{13456}+\operatorname{i}\,e^{23456}\right],\; \right. \\[5pt] && \left. \left[e^{136}-e^{145}-2\,\operatorname{i}\,e^{146}+\operatorname{i}\,e^{236}-\operatorname{i}\,e^{245}+2\,e^{246}\right] \right\rangle \\[10pt] GH^{(0)}_{\mathcal{J}}(M) &=& {\mathbb{C}}\left\langle \left[e^{13}+\operatorname{i}\,e^{23}+\operatorname{i}\,e^{1345}-e^{2345}\right],\; \left[e^{34}\right],\; \left[1+e^{3456}\right],\; \left[e^{1235}\right],\; \right. \\[5pt] && \left. \left[e^{1256}\right],\; \left[e^{12}+e^{123456}\right],\; \left[e^{13}-\operatorname{i}\,e^{23}-\operatorname{i}\,e^{1345}-e^{2345}\right],\; \left[e^{35}-e^{46}\right],\; \right. \\[5pt] && \left. \left[e^{15}+\operatorname{i}\,e^{16}+\operatorname{i}\,e^{25}-e^{26}-\operatorname{i}\,e^{1356}-e^{1456}+e^{2356}-\operatorname{i}\,e^{2456}\right],\; \right. \\[5pt] && \left. \left[e^{15}-\operatorname{i}\,e^{16}-\operatorname{i}\,e^{25}-e^{26}+\operatorname{i}\,e^{1356}-e^{1456}+e^{2356}+\operatorname{i}\,e^{2456}\right] \right\rangle \;, \\[10pt] GH^{(-1)}_{\mathcal{J}}(M) &=& {\mathbb{C}}\left\langle \left[e^{1}-\operatorname{i}\,e^{2}+e^{13456}-\operatorname{i}\,e^{23456}\right],\; \left[e^{3}+\operatorname{i}\,e^{345}\right],\; \left[e^{4}-\operatorname{i}\,e^{346}\right],\; \right. \\[5pt] && \left. \left[e^{125}-\operatorname{i}\,e^{12356}\right],\; \left[e^{126}+\operatorname{i}\,e^{12456}\right],\; \left[e^{135}-e^{146}-\operatorname{i}\,e^{235}+\operatorname{i}\,e^{246}\right],\; \right. \\[5pt] && \left. \left[e^{1}+\operatorname{i}\,e^{2}+\operatorname{i}\,e^{136}+\operatorname{i}\,e^{145}-e^{236}-e^{245}-e^{13456}-\operatorname{i}\,e^{23456}\right],\; \right. \\[5pt] && \left. \left[e^{136}-e^{145}+2\,\operatorname{i}\,e^{146}-\operatorname{i}\,e^{236}+\operatorname{i}\,e^{245}+2\,e^{246}\right] \right\rangle \;, \\[10pt] GH^{(-2)}_{\mathcal{J}}(M) &=& {\mathbb{C}}\left\langle \left[e^{13}-\operatorname{i}\,e^{23}+\operatorname{i}\,e^{1345}+e^{2345}\right],\; \left[1+\operatorname{i}\,e^{36}+\operatorname{i}\,e^{45}-e^{3456}\right],\; \right. \\[5pt] && \left. \left[e^{15}-\operatorname{i}\,e^{16}-\operatorname{i}\,e^{25}-e^{26}-\operatorname{i}\,e^{1356}+e^{1456}-e^{2356}-\operatorname{i}\,e^{2456}\right] ,\; \right. \\[5pt] && \left. \left[e^{12}+\operatorname{i}\,e^{1236}+\operatorname{i}\,e^{1245}-e^{123456}\right] \right\rangle \;, \\[5pt] GH^{(-3)}_{\mathcal{J}}(M) &=& {\mathbb{C}}\left\langle \left[e^{1}-\operatorname{i}\,e^{2}+\operatorname{i}\,e^{136}+\operatorname{i}\,e^{145}+e^{236}+e^{245}-e^{13456}+\operatorname{i}\,e^{23456}\right] \right\rangle \;.\end{aligned}$$ Connecting almost-complex structures ------------------------------------ Let us start noting that the generalized-complex structure $\rho$ cannot be viewed as an almost-complex structure. For this aim, consider again the generalized-complex structure $\rho := \exp \left(\operatorname{i}\,\left(-e^{36}-e^{45}\right)\right) \wedge \left( e^1+\operatorname{i}\,e^2\right)$ by Cavalcanti and Gualtieri [@cavalcanti-gualtieri §5]. For each $t\in[0,1]$, take the $B$-field $$B_t \;:=\; \exp(\pi\,\operatorname{i}\,t)\,\left( e^{35}-e^{46} \right)$$ and the $\beta$-field $$\beta_t \;:=\; -\frac{1}{4}\, \exp(\pi\,\operatorname{i}\,t)\, \left( e_3-\operatorname{i}\,\exp(\pi\,\operatorname{i}\,t)\,e_4 \right)\,\left( e_5-\operatorname{i}\,\exp(\pi\,\operatorname{i}\,t)\,e_6 \right) \;.$$ One has $$\exp(-\beta_t)\,\exp(-B_t)\,\rho \;=\; \left( e^1+\operatorname{i}\,e^2 \right)\wedge\left( e^3+\operatorname{i}\,\exp(\pi\,\operatorname{i}\,t)\,e^4 \right)\wedge\left( e^5+\operatorname{i}\,\exp(\pi\,\operatorname{i}\,t)\,e^6 \right) \;.$$ The endomorphism $$K_t \colon \left\{\begin{array}{rcl} e^1 &\mapsto& - e^2 \\[5pt] e^3 &\mapsto& - \exp(\pi\,\operatorname{i}\,t)\,e^4 \\[5pt] e^5 &\mapsto& - \exp(\pi\,\operatorname{i}\,t)\,e^6 \end{array}\right.$$ is not an almost-complex structure for each $t\in[0,1]$. We now construct a curve of almost-complex structures on $M$ connecting the holomorphically-parallelizable structure $J_0$ and the Abelian complex structure $J_1$. Notice that, up to $\beta$-transforms and $B$-transform, it gives rise to a curve of generalized-almost-complex structures. We study $\mathcal{C}^\infty$-pure-and-fullness for the almost-complex structures. For $t\in [0,1]$, consider the almost-complex structure $$\label{eq:Jt} J_t \;:=\; \left( \begin{array}{cc|cc|cc} & \ 1 \ & & & & \\ \ -1 \ & & & & & \\ \hline & & & \cos(\pi\,t) & & \sin(\pi\,t) \\ & & -\cos(\pi\,t) & & \sin(\pi\,t) & \\ \hline & & & -\sin(\pi\,t) & & \cos(\pi\,t) \\ & & -\sin(\pi\,t) & & -\cos(\pi\,t) & \end{array} \right) \in \operatorname{End}(TM) \;.$$ Observe that the notation is coherent with the previous for $t=0$ and $t=1$. That is, $J_0$ coincides with the above holomorphically-parallelizable structure, and $J_1$, with the above Abelian complex structure. Consider $$\begin{aligned} \rho_t &:=& \left( e^1 + \operatorname{i}\,e^2 \right) \wedge \left( e^3 + \operatorname{i}\,\left( \cos(\pi\,t)\,e^4 + \sin(\pi\,t)\,e^6 \right) \right) \\[5pt] && \wedge \left( e^5 - \operatorname{i}\,\left( \sin(\pi\,t)\,e^4 - \cos(\pi\,t)\,e^6 \right) \right) \;. \end{aligned}$$ In an equivalent manner, we can write it as: $$\rho_t \;=\; \left( e^1 + \operatorname{i}\,e^2 \right) \wedge \left( -1 + \exp(\Xi) \wedge \exp(\operatorname{i}\,\omega_t) \right)$$ where $$\Xi \;:=\; e^{35} - e^{46} \quad \text{ and } \quad \omega_t \;:=\; \cos(\pi\,t)\,\left( e^{45} + e^{36} \right) - \sin(\pi\,t)\, \left( e^{34}+e^{56} \right) \;.$$ Take $\beta:=e_{35}\in\wedge^2TM$. Then $$\exp(\beta)\,\rho_t \;=\; \left(e^1+\operatorname{i}\,e^2\right) \wedge \exp(\Xi) \wedge \exp(\operatorname{i}\,\omega_t) \;.$$ Take also $B:=-\Xi \in \wedge^2M$. We obtain $$\exp(B)\exp(\beta)\,\rho_t \;=\; \left(e^1+\operatorname{i}\,e^2\right) \wedge \exp(\operatorname{i}\,\omega_t) \;.$$ By computing $\left( e^1 + \operatorname{i}\,e^2 \right) \wedge \left( e^1 - \operatorname{i}\,e^2 \right) \wedge \omega_t^{2} = -4\operatorname{i}\,e^{123456} \neq 0$, we have that $\exp(B)\exp(\beta)\,\rho_t$ yields to a generalized-almost-complex structure of type $1$ on $M$, due to [@gualtieri-phd Theorem 4.8]. We study complex-$\mathcal{C}^\infty$-pure-and-fullness in the sense of Li and Zhang for $J_t$. Define the following basis of $(1,0)$-forms with respect to $J_t$: $$\left\{ \begin{array}{l} \varphi_t^1 \;=\; e^1 + \operatorname{i}\, e^2 \\[5pt] \varphi_t^2 \;=\; e^3 + \operatorname{i}\, \left( \cos(\pi\,t)\, e^4 + \sin(\pi\,t)\, e^6 \right) \\[5pt] \varphi_t^3 \;=\; e^5 - \operatorname{i}\, \left( \sin(\pi\,t)\, e^4 - \cos(\pi\,t)\, e^6 \right) \end{array} \right. \;.$$ The structure equations can be expressed as $$\left\{ \begin{array}{lll} \operatorname{d}\varphi_t^1 &=& 0 \\[5pt] \operatorname{d}\varphi_t^2 &=& \frac{1}{4}\, \sin(\pi\,t)\, \left( \left( 1+\cos(\pi\,t) \right)\, \varphi_t^{12} - \sin(\pi\,t)\, \varphi_t^{13} \right. \\[5pt] && \ \left. + \left( 1-\cos(\pi\,t) \right)\, \varphi_t^{1\bar{2}} + \sin(\pi\,t)\, \varphi_t^{1\bar{3}} + \left( 1-\cos(\pi\,t) \right)\, \varphi_t^{2\bar{1}} \right. \\[5pt] && \ \left. \sin(\pi\,t)\, \varphi_t^{3\bar{1}} - \left( 1+\cos(\pi\,t) \right)\, \varphi_t^{\bar{1}\bar{2}} + \sin(\pi\,t)\, \varphi_t^{\bar{1}\bar{3}} \right) \\[5pt] \operatorname{d}\varphi_t^3 &=& \frac{1}{4}\, \left( \left( 1+\cos(\pi\,t) \right)^2\, \varphi_t^{12} - \sin(\pi\,t) \left( 1+\cos(\pi\,t) \right)\, \varphi_t^{13} \right. \\[5pt] && \ \left. \left( 1-\cos^2(\pi\,t) \right)\,\varphi_t^{1\bar{2}} + \sin(\pi\,t)\, \left( 1+\cos(\pi\,t) \right)\, \varphi_t^{1\bar{3}} \right. \\[5pt] && \ \left. - \left( 1-\cos(\pi\,t) \right)^2\, \varphi_t^{2\bar{1}} - \sin(\pi\,t)\, \left( 1-\cos(\pi\,t) \right)\, \varphi_t^{3\bar{1}} \right. \\[5pt] && \ \left. \left( 1-\cos^2(\pi\,t) \right)\, \varphi_t^{\bar{1}\bar{2}} - \sin(\pi\,t)\, \left( 1-\cos(\pi\,t) \right)\, \varphi_t^{\bar{1}\bar{3}} \right) \end{array} \right. \;.$$ By direct computation, it is possible to see that, for $t\in \{0,\, 1\}$, one has $$H_{J_t}^{(1,0)}(M) \;=\; {\mathbb{C}}\left\langle \left[ \varphi_t^1 \right],\; \left[ \varphi_t^2 \right] \right\rangle \qquad\text{and}\qquad H_{J_t}^{(0,1)}(M) \;=\; {\mathbb{C}}\left\langle \left[ \bar\varphi_t^{1} \right],\; \left[ \bar\varphi_t^{2} \right] \right\rangle \;,$$ whereas, for $t\in(0,1)$, $$H_{J_t}^{(1,0)}(M) \;=\; {\mathbb{C}}\left\langle \left[ \varphi_t^1 \right] \right\rangle \qquad\text{and}\qquad H_{J_t}^{(0,1)}(M) \;=\; {\mathbb{C}}\left\langle \left[ \bar\varphi_t^{1} \right] \right\rangle \;.$$ Therefore, the almost-complex structures in the interior of the path joining the holomorphically-parallelizable and the Abelian complex structures on $M$ are not complex-$\mathcal{C}^\infty$-full at the first stage in the sense of Li and Zhang. 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--- author: - Frédéric Bayart and Yanick Heurteaux title: On the Hausdorff dimension of graphs of prevalent continuous functions on compact sets --- Introduction ============ Let $d\ge 1$ and let $K$ be a compact subset in $\RR^d$. Denote by $\mathcal C(K)$ the set of continuous functions on $K$ with real values. This is a Banach space when equipped with the supremum norm, $\|f\|_\infty=\sup_{x\in K} |f(x)|$. The graph of a function $f\in\mathcal C(K)$ is the set $$\Gamma_f^K=\left\{ (x,f(x))\ ;\ x\in K\right\}\subset\RR^{d+1}.$$ It is often difficult to obtain the exact value of the Hausdorff dimension of the graph $\Gamma_f^K$ of a precised continuous function $f$. For example, a famous conjecture says that the Hausdorff dimension of the graph of the Weierstrass function $$f(x)=\sum_{k=0}^{+\infty}2^{-k\alpha}\cos(2^kx),$$ where $0<\alpha<1$, satisfies $$\dim_{\mathcal H}\left(\Gamma_f^{[0,2\pi]}\right)=2-\alpha.$$ This is the natural expected value, but, to our knowledge, this conjecture is not yet solved. If we add some randomness, the problem becomes much easier and Hunt proved in [@Hunt] that the Hausdorff dimension of the graph of the random Weierstrass function $$f(x)=\sum_{k=0}^{+\infty}2^{-k\alpha}\cos(2^kx+\theta_k)$$ where $(\theta_k)_{k\ge 0}$ is a sequence of independent uniform random variables is almost surely equal to the expected value $2-\alpha$. In the same spirit we can hope to have a generic answer to the following question: $$\begin{aligned} \textrm{``What is the Hausdorff dimension of the graph of a continuous function?''}\end{aligned}$$ Curiously, the answer to this question depends on the type of genericity we consider. If genericity is relative to the Baire category theorem, Mauldin and Williams proved at the end of the 80’s the following result: \[THMBAIRE\][*([@MW])*]{} For quasi-all functions $f\in \mathcal C([0,1])$, we have $$\dim_{\mathcal H} \left(\Gamma_f^{[0,1]}\right)=1.$$ This statement on the Hausdorff dimension of the graph is very surprising because it seems to say that a generic continuous function is quite regular. Indeed it is convenient to think that there is a deep correlation between strong irregularity properties of a function and large values of the Hausdorff dimension of its graph. This curious result seems to indicate that genericity in the sense of the Baire category theorem is not “the good notion of genericity” for this question. In fact, when genericity is related to the notion of prevalence (see Section \[SECPRE\] for a precise definition), Fraser and Hyde recently obtained the following result. \[THMFH\][*([@FH])*]{} Let $d\in\NN^*$. The set $$\left\{ f\in\mathcal C([0,1]^d)\ ;\ \dim_{\mathcal H}\left(\Gamma_f^{[0,1]^d}\right)=d+1\right\}$$ is a prevalent subset of $\mathcal C([0,1]^d)$. This result says that the Hausdorff dimension of the graph of a generic continuous function is as large as possible and is much more in accordance with the idea that a generic continuous function is strongly irrregular. The main tool in the proof of Theorem \[THMFH\] is the construction of a fat Cantor set in the interval $[0,1]$ and a stochastic process on $[0,1]$ whose graph has almost surely Hausdorff dimension 2. This construction is difficult to generalize in a compact set $K\not=[0,1]$. Nevertheless, there are in the litterature stochastic processes whose almost sure Hausdorff dimension of their graph is well-known. The most famous example is the Fractional Brownian Motion. Using such a process, we are able to prove the following generalisation of Theorem \[THMFH\]. \[THMMAIN\] Let $d\ge 1$ and let $K\subset {\mathbb{R}^{d}}$ be a compact set such that $\dim_{\mathcal H}(K)>0$. The set $$\left\{ f\in\mathcal C(K)\ ;\ \dim_{\mathcal H}\left(\Gamma_f^K\right)=\dim_{\mathcal H}(K)+1\right\}$$ is a prevalent subset of $\mathcal C(K)$. In this paper, we have decided to focus to the notion of Hausdorff dimension of graphs. Nevertheless, we can mention that there are also many papers that deal with the generic value of the dimension of graphs when the notion of dimension is for example the lower box dimension (see [@FF; @GJ; @HL; @Shaw]) or the packing dimension (see [@HP; @McClure]). The paper is devoted to the proof of Theorem \[THMMAIN\] and is organised as follows. In Section \[SECPRE\] we recall the basic facts on prevalence. In particular we explain how to use a stochastic process in order to prove prevalence in functional vector spaces. In Section \[SECW\], we prove an auxiliary result on Fractional Brownian Motion which will be the key of the main theorem. We finish the proof of Theorem \[THMMAIN\] in Section \[SECMAIN\]. Finally, in a last section, we deal with the case of $\alpha$-Hölderian functions. Prevalence {#SECPRE} ========== Prevalence is a notion of genericity who generalizes in infinite dimensional vector spaces the notion of “almost everywhere with respect to Lebesgue measure”. This notion has been introduced by J. Christensen in [@Chr72] and has been widely studied since then. In fractal and multifractal analysis, some properties which are true on a dense $G_\delta$-set are also prevalent (see for instance [@FJK], [@FJ06] or [@BH11b]), whereas some are not (see for instance [@FJK] or [@Ol10]). Let $E$ be a complete metric vector space. A Borel set $A\subset E$ is called *Haar-null* if there exists a compactly supported probability measure $\mu$ such that, for any $x\in E$, $\mu(x+A)=0$. If this property holds, the measure $\mu$ is said to be *transverse* to $A$.\ A subset of $E$ is called *Haar-null* if it is contained in a Haar-null Borel set. The complement of a Haar-null set is called a *prevalent* set. The following results enumerate important properties of prevalence and show that this notion supplies a natural generalization of “almost every” in infinite-dimensional spaces: - If $A$ is Haar-null, then $x+A$ is Haar-null for every $x\in E$. - If $\dim(E)<+\infty$, $A$ is Haar-null if and only if it is negligible with respect to the Lebesgue measure. - Prevalent sets are dense. - The intersection of a countable collection of prevalent sets is prevalent. - If $\dim(E)=+\infty$, compacts subsets of $E$ are Haar-null. In the context of a functional vector space $E$, a usual way to prove that a set $A\subset E$ is prevalent is to use a stochastic process. More precisely, suppose that $W$ is a stochastic process defined on a probability space $(\Omega,\mathcal F, \mathbb P)$ with values in $E$ and satisfies. $$\forall f\in E,\quad f+W\in A\quad\text{almost surely}.$$ Replacing $f$ by $-f$, we get that the law $\mu$ of the stochastic process $W$ is such that $$\forall f\in E,\quad \mu(f+A)=1.$$ In general, the measure $\mu$ is not compactly supported. Nevertheless, if we suppose that the vector space $E$ is also a Polish space (that is if we add the hypothesis that $E$ is separable), then we can find a compact set $Q\subset E$ such that $\mu(Q)>0$. It follows that the compactly supported probability measure $\nu=(\mu(Q))^{-1}\mu_{|Q}$ is transverse to $E\setminus A$. On the graph of a perturbed Fractional Brownian Motion {#SECW} ====================================================== In this section, we prove an auxilliary result which will be the key of the proof of Theorem \[THMMAIN\]. For the definition and the main properties of the Fractional Brownian Motion, we refer to [@Falc Chapter 16]. \[THMFBM\] Let $K$ be a compact set in ${\mathbb{R}^{d}}$ such that $\dim_{\mathcal H}(K)>0$ and $\alpha\in (0,1)$. Define the stochastic process in ${\mathbb{R}^{d}}$ $$\begin{aligned} \label{EQNPROCESS} W(x)=W^1(x_1)+\cdots+ W^d(x_d)\end{aligned}$$ where $W^1,\cdots,W^d$ are independent Fractional Brownian Motions starting from 0 with Hurst parameter equal to $\alpha$. Then, for any function $f\in\mathcal C(K)$ $$\dim_{\mathcal H}\left(\Gamma_{f+W}^K\right)\ge\min\left(\frac{\dim_{\mathcal H}(K)}{\alpha}\,,\, \dim_{\mathcal H}(K)+1-\alpha\right)\quad\textrm{almost surely}.$$ Let us remark that the conclusion of Theorem \[THMFBM\] is sharp. More precisely, suppose that $f=0$ and let $\varepsilon>0$. It is well known that the Fractional Brownian Motion is almost-surely uniformly $(\alpha-\varepsilon)$-Hölderian. It follows that the stochastic process $W$ is also uniformly $(\alpha-\varepsilon)$-Hölderian on $K$. It is then straightforward that the graph $\Gamma_W^K$ satisfies $$\dim_{\mathcal H}\left(\Gamma_W^K\right)\le \dim_{\mathcal H}(K)+1-(\alpha-\varepsilon)\quad\mbox{a.s.}.$$ On the other hand, the application $$\Phi\ :\ x\in K\longmapsto (x,W(x))\in{\mathbb R}^{d+1}$$ is almost-surely $(\alpha-\varepsilon)$-Hölderian. It follows that $$\dim_{\mathcal H}\left(\Gamma_W^K\right)\le \frac{\dim_{\mathcal H}(K)}{\alpha-\varepsilon}\quad\mbox{a.s.}.$$ The proof of Theorem \[THMFBM\] is based on the following lemma. \[LEMESP\] Let $s>0$, $\alpha\in (0,1)$ and $W$ be the process defined as in [*(\[EQNPROCESS\])*]{}. There exists a constant $C:=C(s)>0$ such that for any $\lambda\in\RR$, for any $x,y\in{\mathbb{R}^{d}}$, $$\mathbb E\left[\frac1{(\Vert x-y\Vert^2+(\lambda+W(x)-W(y))^2)^{s/2}}\right]\le C\left\{ \begin{array}{ll} \displaystyle \Vert x-y\Vert^{1-s-\alpha}&\textrm{ provided }s>1\\ \displaystyle \Vert x-y\Vert^{-\alpha s}&\textrm{ provided }s<1. \end{array} \right.$$ Observe that $W(x)-W(y)$ is a centered gaussian variable with variance $$\sigma^2=h_1^{2\alpha}+\cdots+h_d^{2\alpha}$$ where $h=(h_1,\cdots,h_d)=x-y$. Hölder’s inequality yields $$\Vert h\Vert^{2\alpha}\le\sigma^2\le d^{1-\alpha}\Vert h\Vert^{2\alpha}.$$ Now, $$\mathbb E\left[\frac1{(\Vert x-y\Vert^2+(\lambda+W(x)-W(y))^2)^{s/2}}\right]=\int\frac{e^{-u^2/(2\sigma^2)}}{(\Vert h\Vert^2+(\lambda+u)^2)^{s/2}}\frac{du}{\sigma\sqrt{2\pi}}.$$ Suppose that $s>1$. We get $$\begin{aligned} \mathbb E\left[\frac1{(\Vert x-y\Vert^2+(\lambda+W(x)-W(y))^2)^{s/2}}\right]&\le&\int\frac{du}{(\Vert h\Vert^2+(\lambda+u)^2)^{s/2}\sigma\sqrt{2\pi}}\\ &=&\int\frac{\Vert h\Vert\,dv}{(\Vert h\Vert^2+(\Vert h\Vert v)^2)^{s/2}\sigma\sqrt{2\pi}}\\ &\le&\Vert h\Vert^{1-s-\alpha}\frac1{\sqrt{2\pi}}\int\frac{dv}{(1+v^2)^{s/2}}\\ &:=& C\,\|x-y\|^{1-s-\alpha}.\end{aligned}$$ In the case where $0<s<1$, we write $$\begin{aligned} \mathbb E\left[\frac1{(\Vert x-y\Vert^2+(\lambda+W(x)-W(y))^2)^{s/2}}\right]&\le&\int\frac{e^{-v^2/2}}{(\lambda+\sigma v)^{s}}\frac{dv}{\sqrt{2\pi}}\\ &\le&\Vert h\Vert^{-\alpha s}\int\frac{e^{-v^2/2}}{(\gamma+v)^s}\frac{dv}{\sqrt{2\pi}}\end{aligned}$$ where $\gamma=\lambda\sigma^{-1}$. On the other hand, $$\begin{aligned} \int\frac{e^{-v^2/2}\,dv}{(\gamma+v)^s}=\int\frac{e^{-(v-\gamma)^2/2}\,dv}{v^s}&\le&\int_{-1}^1\frac{dv}{v^s}+\int_{\RR\setminus[-1,1]}\frac{e^{-(v-\gamma)^2/2}\,dv}{v^s}\\ &\le& \int_{-1}^1\frac{dv}{v^s}+\int_\RR e^{-x^2/2}\,dx\end{aligned}$$ which is a constant $C$ independent of $\gamma$ and $\alpha$. We are now able to finish the proof of Theorem \[THMFBM\]. We use the potential theoretic approach (for more details on the potential theoretic approach of the calculus of the Hausdorff dimension, we can refer to [@Falc Chapter 4]). Suppose first that $\dim_{\mathcal H}(K)>\alpha$ and let $\delta$ be a real number such that $$\alpha<\delta<\dim_{\mathcal H}(K).$$ There exists a probability measure $m$ on $K$ whose $\delta$-energy $I_\delta(m)$, defined by $$I_\delta(m)=\int\!\!\int_{K\times K}\frac{dm(x)\,dm(y)}{\Vert x-y\Vert^\delta}$$ is finite. Conversely, to prove that the Hausdorff dimension of the graph $\Gamma_{f+W}^K$ is at least $\dim_{\mathcal H}(K)+1-\alpha$, it suffices to find, for any $s<\dim_{\mathcal H}(K)+1-\alpha$, a measure $\mu$ on $\Gamma_{f+W}^K$ with finite $s$-energy. Let $(\Omega,\mathcal F, \mathbb P)$ be the probability space where are defined the Fractional Brownian Motions $W^1,\cdots,W^d$. For any $\omega\in\Omega$, define $m_\omega$ as the image of the measure $m$ on the graph $\Gamma_{f+W_\omega}^K$ via the natural projection $$x\in K\longmapsto (x,f(x)+W_\omega(x)).$$ Set $s=\delta+1-\alpha$ which is greater than 1. The $s$-energy of $m_\omega$ is equal to $$\begin{aligned} I_s(m_\omega)&=&\int\!\!\int_{\Gamma_{f+W_\omega}^K\times\Gamma_{f+W_\omega}^K}\frac{dm_\omega(X)\,dm_\omega(Y)}{\Vert X-Y\Vert^s}\\ &=&\int\!\!\int_{K\times K}\frac{dm(x)\,dm(y)}{\Big(\Vert x-y\Vert^2+\big(f(x)+W_\omega(x)-(f(y)+W_\omega(y))\big)^2\Big)^{s/2}}.\end{aligned}$$ Fubini’s theorem and Lemma \[LEMESP\] ensure that $$\begin{aligned} \mathbb E\left[I_s(m_\omega)\right]&=&\int\!\!\int_{K\times K}\mathbb E\left[ \frac{1}{\Big(\Vert x-y\Vert^2+\big((f(x)-f(y))+(W(x)-W(y))\big)^2\Big)^{s/2}}\right]dm(x)\,dm(y)\\ &\le&C\int\!\!\int_{K\times K}\Vert x-y\Vert^{1-s-\alpha}dm(x)\,dm(y)\\ &=&CI_\delta(m)\\ &<&+\infty.\end{aligned}$$ We deduce that for $\mathbb P$-almost all $\omega\in \Omega$, the energy $I_s(m_\omega)$ is finite. Since $s$ can be chosen arbitrary closed to $\dim_{\mathcal H}(K)+1-\alpha$, we get $$\dim_{\mathcal H}\left(\Gamma_{f+W_\omega}^K\right)\ge \dim_{\mathcal H}(K)+1-\alpha\quad \mbox{almost surely}.$$ In the case where $\dim_{\mathcal H}(K)\le\alpha$, we proceed exactly in the same way, except that we take any $\delta<\dim_{\mathcal H}(K)$ and we set $s=\frac\delta\alpha$ which is smaller than 1. We then get $$\dim_{\mathcal H}\left(\Gamma_{f+W}^K\right)\ge\frac{\dim_{\mathcal H}(K)}{\alpha}\quad\mbox{ almost surely}.$$ Proof of Theorem \[THMMAIN\] {#SECMAIN} ============================ We can now prove Theorem \[THMMAIN\]. Let $K$ be a compact set in ${\mathbb{R}^{d}}$ satisfying $\dim_{\mathcal H}(K)>0$. Remark first that for any function $f\in\mathcal C(K)$, the graph $\Gamma_f^K$ is included in $K\times\RR$. It follows that $$\dim_{\mathcal H}\left(\Gamma_f^K\right)\le \dim_{\mathcal H}(K\times \RR)= \dim_{\mathcal H}(K)+1.$$ Define $$G=\left\{ f\in\mathcal C(K);\ \dim_{\mathcal H}\left(\Gamma_f^K\right)=\dim_{\mathcal H}(K)+1\right\}.$$ Theorem \[THMFBM\] says that for any $\alpha$ such that $0<\alpha<\min(1,\dim_{\mathcal H}(K))$, the set $G_\alpha$ of all continuous functions $f\in\mathcal C(K)$ satisfying $\dim_{\mathcal H}\left(\Gamma_f^K\right)\ge\dim_{\mathcal H}(K)+1-\alpha$ is prevalent in $\mathcal C(K)$. Finally, we can write $$G=\bigcap_{n\ge 0}G_{\alpha_n}$$ where $(\alpha_n)_{n\ge 0}$ is a sequence decreasing to 0 and we obtain that $G$ is prevalent in $\mathcal C(K)$. It is an easy consequence of Ascoli’s theorem that the law of the process $W$ is compactly supported in $\mathcal C(K)$ (remember that $W$ is almost surely $(\alpha-\varepsilon)$-Hölderian). Then, we don’t need to use that $\mathcal C(K)$ is a Polish space to obtain Theorem \[THMMAIN\]. Let $K=[0,1]$ and $f\in \mathcal C([0,1])$. Theorem \[THMMAIN\] implies that the set $G\bigcap(f+G)$ is prevalent. We can then write $$f=f_1-f_2\quad\text{with}\quad\dim_{\mathcal H}\left(\Gamma_{f_1}^{[0,1]}\right)=2\quad\text{and}\quad\dim_{\mathcal H}\left(\Gamma_{f_2}^{[0,1]}\right)=2$$ where $f_1$ and $f_2$ are continuous functions. On the other hand, it was recalled in Theorem \[THMBAIRE\] that the set $$\tilde G=\left\{f\in\mathcal C([0,1])\ ;\ \dim_{\mathcal H}\left(\Gamma_f^{[0,1]}\right)=1\right\}$$ contains a dense $G_\delta$-set of $\mathcal C([0,1])$. It follows that any continuous function $f\in\mathcal C([0,1])$ can be written $$f=f_1-f_2\quad\text{with}\quad\dim_{\mathcal H}\left(\Gamma_{f_1}^{[0,1]}\right)=1\quad\text{and}\quad\dim_{\mathcal H}\left(\Gamma_{f_2}^{[0,1]}\right)=1$$ where $f_1$ and $f_2$ are continuous functions. We can then ask the following question: given a real number $\beta\in(1,2)$ can we write an arbitrary continuous function $f\in\mathcal C([0,1])$ in the following way $$f=f_1-f_2\quad\text{with}\quad\dim_{\mathcal H}\left(\Gamma_{f_1}^{[0,1]}\right)=\beta\quad\text{and}\quad\dim_{\mathcal H}\left(\Gamma_{f_2}^{[0,1]}\right)=\beta$$ where $f_1$ and $f_2$ are continuous functions? We do not know the answer to this question. The case of $\alpha$-Hölderian functions {#SECHOLDER} ======================================== Let $0<\alpha<1$ and let $\mathcal C^\alpha(K)$ be the set of $\alpha$-Hölderian functions in $K$ endowed with the standard norm $$\Vert f\Vert_\alpha=\sup_{x\in K}|f(x)|+\sup_{(x,y)\in K^2}\frac{|f(x)-f(y)|}{\| x-y\|^\alpha}.$$ It is well known that the Hausdorff dimension of the graph $\Gamma_f^K$ of a function $f\in\mathcal C^\alpha(K)$ satisfies $$\begin{aligned} \label{EQNCALPHA} \dim_{\mathcal H}\left(\Gamma_f^K\right)\le\min\left(\frac{\dim_{\mathcal H}(K)}{\alpha}\,,\,\dim_{\mathcal H}(K)+1-\alpha\right)\end{aligned}$$ (see for example the remark following the statement of Theorem \[THMFBM\]). It is then natural to ask if inequality (\[EQNCALPHA\]) is an equality in a prevalent set of $\mathcal C^\alpha(K)$. This is indeed the case as said in the following result. \[THMALPHA\] Let $d\ge 1$, $0<\alpha<1$ and $K\subset \RR^d$ be a compact set with strictly positive Hausdorff dimension. The set $$\left\{f\in\mathcal C^\alpha(k)\ ;\ \dim_{\mathcal H}\left(\Gamma_f^K\right)=\min\left(\frac{\dim_{\mathcal H}(K)}{\alpha}\,,\,\dim_{\mathcal H}(K)+1-\alpha\right)\right\}$$ is a prevalent subset of $\mathcal C^\alpha(K)$. This result generalizes in an arbitrary compact subset of $\RR^d$ a previous work of Clausel and Nicolay (see [@CN Theorem 2]). Let $\alpha<\alpha'<1$ and let $W$ be the stochastic process defined in Theorem \[THMFBM\] with Hurst parameter $\alpha'$ instead of $\alpha$. The stochastic process $W_{|K}$ takes values in $\mathcal C^\alpha(K)$. Moreover, if $\alpha<\alpha''<\alpha'$, the injection $$f\in\mathcal C^{\alpha''}(K)\longmapsto f\in\mathcal C^{\alpha}(K)$$ is compact. It follows that the law of the stochastic process $W_{|K}$ is compactly supported in $\mathcal C^{\alpha}(K)$ ($W$ is $\alpha''$-Hölderian). Then, Theorem \[THMFBM\] ensures that the set $$\left\{f\in\mathcal C^\alpha(K)\ ;\ \dim_{\mathcal H}\left(\Gamma_f^K\right)\ge\min\left(\frac{\dim_{\mathcal H}(K)}{\alpha'}\,,\,\dim_{\mathcal H}(K)+1-\alpha'\right)\right\}$$ is prevalent in $\mathcal C^\alpha(K)$. Using a sequence $(\alpha_n)_{n\ge 0}$ decreasing to $\alpha$, we get the conclusion of Theorem \[THMALPHA\]. [99.]{} Bayart F. and Heurteaux Y.: Multifractal analysis of the divergence of Fourier series: the extreme cases. arXiv:1110:5478, submitted (2011) Christensen J.P.R.: On sets of Haar measure zero in Abelian Polish groups. Israel J. Math. **13**, 255-260 (1972) Clausel M. and Nicolay S.: Some prevalent results about strongly monoH¨older functions. Nonlinearity **23**, 2101-2116 (2010) Falconer K.J.: Fractal geometry: Mathematical foundations and applications. Wiley (2003) Falconer K.J. and Fraser J.M.: The horizon problem for prevalent surfaces, preprint (2011) Fraser J.M. and Hyde J.T.: The Hausdorff dimension of graphs of prevalent continuous functions. arXiv:1104.2206 (2011) Fraysse A and Jaffard S.: How smooth is almost every function in a Sobolev space?. Rev. Mat. Iboamericana **22**, 663-682 (2006) Fraysse A., Jaffard S. and Kahane J.P.: Quelques propriétés génériques en analyse. (French) \[Some generic properties in analysis\]. C. R. Math. Acad. Sci. Paris **340**, 645-651 (2005) Gruslys V., Jonusas J., Mijovic V., Ng O., Olsen L. and Petrykiewicz I.: Dimensions of prevalent continuous functions, preprint (2010) Humke P.D. and Petruska G.: The packing dimension of a typical continuous function is 2. Bull. Amer. Math. Soc. (N.S.) **27**, 345-358 (1988-89) Hunt B.R.: The Hausdorff dimension of graphs of Weierstrass functions. Proc. Amer. Math. Soc. **126**, 791-800 (1998) Hyde J.T., Laschos V., Olsen L., Petrykiewicz I. and Shaw A.: On the box dimensions of graphs of typical functions, preprint (2010) Mauldin R.D. and Williams S.C.: On the Hausdorff dimension of some graphs. Trans. Amer. Math. Soc. **298**, 793-803 (1986) McClure M.: The prevalent dimension of graphs, Real Anal. Exchange **23**, 241-246 (1997) Olsen L.: Fractal and multifractal dimensions of prevalent measures. Indiana Univ. Math. Journal, **59**, 661-690 (2010) Shaw A.: Prevalence, M. Math Dissertation, University of St. Andrews (2010)

--- abstract: 'Growing observations of brown dwarfs have provided evidence for strong atmospheric circulation on these objects. Directly imaged planets share similar observations, and can be viewed as low-gravity versions of brown dwarfs. Vigorous condensate cycles of chemical species in their atmospheres are inferred by observations and theoretical studies, and latent heating associated with condensation is expected to be important in shaping atmospheric circulation and influencing cloud patchiness. We present a qualitative description of the mechanisms by which condensational latent heating influence the circulation, and then illustrate them using an idealized general circulation model that includes a condensation cycle of silicates with latent heating and molecular weight effect due to rainout of condensate. Simulations with conditions appropriate for typical T dwarfs exhibit the development of localized storms and east-west jets. The storms are spatially inhomogeneous, evolving on timescale of hours to days and extending vertically from the condensation level to the tropopause. The fractional area of the brown dwarf covered by active storms is small. Based on a simple analytic model, we quantitatively explain the area fraction of moist plumes, and show its dependence on radiative timescale and convective available potential energy. We predict that, if latent heating dominates cloud formation processes, the fractional coverage area by clouds decreases as the spectral type goes through the L/T transition from high to lower effective temperature. This is a natural consequence of the variation of radiative timescale and convective available potential energy with spectral type.' author: - 'Xianyu Tan and Adam P. Showman' bibliography: - 'draft.bib' title: Effects of latent heating on atmospheres of brown dwarfs and directly imaged planets --- INTRODUCTION ============ Observations of brown dwarfs (BDs) have shown increasing evidence of a vigorous circulation in their atmospheres . This evidence includes near-infrared brightness variability [@artigau2009; @radigan2012; @apai2013; @buenzli2014; @radigan2014; @wilson2014; @buenzli2015; @metchev2015; @yang2015; @yang2016; @cushing2016], chemical disequilibrium [@fegley1996; @saumon2006; @saumon2007; @hubeny2007; @stephens2009; @visscher2011; @zahnle2014] and surface patchiness [@crossfield2014]. Cloud disruption has been proposed to help explain properties of the L/T transition [@ackerman2001; @burgasser2002; @marley2010], and such patchiness is also likely responsible for the near-infrared brightness variability [@marley2015]. Nevertheless, the mechanism responsible for cloud disruption is yet unclear. Atmospheric circulation is expected to play a crucial role in controlling cloud coverage fraction, but the details remain poorly understood. A handful of directly imaged extrasolar giant planets (EGPs) exhibit similarities with BDs: the near-infrared colors, inference of dust and clouds, chemical disequilibrium in their atmospheres and fast spin [@hinz2010; @barman2011a; @barman2011b; @marley2012; @oppenheimer2013; @ingraham2014; @skemer2014; @snellen2014; @macintosh2015; @wagner2016]. Near-IR brightness variability has also recently been observed on directly imaged EGPs [@biller2015; @zhou2016]. From a meteorological point of view, the directly imaged EGPs resemble low-gravity versions of BDs, for which their atmospheric dynamical regime is characterized by fast rotation, vigorous convection and negligible external heating. Motivated by the observations, several studies have been conducted to explore the atmospheric dynamics of ultra cool objects (compared to stars). Local two-dimensional hydrodynamics simulations by [@freytag2010] showed that interactions between the convective interior and the stratified layer can generate gravity waves that propagate upward, and the breaking of these waves causes vertical mixing that leads to small-scale cloud patchiness. presented the first global model of brown dwarf dynamics for the convective interior, and showed that large-scale convection is dominated by the fast rotation. Using an analytic theory, they proposed that atmospheric circulation can be driven by atmospheric waves in the stably stratified upper atmosphere. Using a two-layer shallow-water model, showed that weak radiative dissipation and strong forcing favor large-scale zonal jets for brown dwarfs, whereas strong dissipation and weak forcing favor transient eddies and quasi-isotropic turbulence. Despite these studies, no global model that includes condensate cycles and clouds has yet been published for brown dwarfs. Clouds play a significant role in sculpting the temperature structure, spectra and brightness variations of brown dwarfs (see recent reviews of [@marley2015] and [@helling2014]). There is a pressing need to couple condensation cycles and clouds to global models to study how the circulation controls global cloud patchiness, and in turn how the condense cycle affects the circulation. In this paper, we propose the importance of latent heating on the atmospheric circulation and cloud patchiness of brown dwarfs by using an idealized general circulation model that includes a condensation cycle of silicate vapor. Latent heating is of paramount importance in Earth’s atmosphere [@emanuel1994]. For giant planets in our solar system whose atmospheres are likely analogous to brown dwarfs’, a long history of studies has shown the importance of latent heating in driving their atmospheric circulation (). demonstrated that large-scale latent heating from condensation of water can drive patterns of zonal (east-west) jet streams that resemble those on all four giant planets of the solar system: numerous zonal jets off the equator and a strong prograde equatorial jet on Jupiter and Saturn, and a three-jet pattern including retrograde equatorial flow and high-latitude prograde flow on Uranus and Neptune. Such models also exhibit episodic storms that qualitatively resemble those observed on Jupiter and Saturn. For brown dwarfs, the evidence for patchy clouds in controlling brightness variability and the L/T transition itself also suggests a strong role for an active condensate cycle, and latent heating may be similarly important for atmospheric circulation of BDs and directly imaged EGPs. Because temperature perturbations associated with (dry) convection at condensable pressure levels are generally small, the latent heating that accompanies the condensation of relevant chemical species can dominate the buoyancy in the layers where condensation occurs. The main point of this paper is to illustrate how latent heating modifies a circulation and influences cloud patchiness in the simplest possible context, so we intentionally exclude clouds, radiative transfer and detailed microphysics to allow a simpler environment in which to clarify the dynamical processes that are at play. Cloud microphysics processes are highly complex [@rossow1978], and significant prior work on the cloud microphysics issue (see a review by [@helling2014]), as well as parameterized cloud models [@allard2001; @ackerman2001; @tsuji2002; @cooper2003; @barman2011a] has been done for ultra cool atmospheres. We are well aware of the important feedback of clouds to atmospheres, and will leave it for future efforts. Also, to resemble the vigorous convection and the dynamics in radiative-convective boundary caused by convective perturbation, one needs a model that can properly treat both the convective interior and the overlying stably stratified layer. Therefore, we do not expect our current simulations to resemble the true atmospheres of brown dwarfs and directly imaged EGPs. The paper is organized as follows. We start out in Section \[mechanism\] by describing several important effects of latent heating on the atmosphere; in Section \[model\], we briefly introduce our idealized model that is used to illustrate the mechanisms described in Section \[mechanism\]; in Section \[results\], we show result of our simulations; finally in Section \[discussion\], we discuss our results and implications for observations, and draw conclusions. Effects of latent heating on atmospheres {#mechanism} ======================================== Conditional Instability {#instability} ----------------------- Most atmospheres of planets and ultra cool brown dwarfs have constituents that can condense. Due to atmospheric motion and diabatic heating/cooling, air parcels containing condensable species can undergo change of temperature and pressure, leading to condensation. The latent heating/cooling due to condensation/evaporation has important effects on the stability of atmospheres, which we summarize here; a more detailed discussion can be found, e.g., in Chapter 7 of @salby2012. For simplicity, we begin our discussion assuming the molecular weight is constant but return to this issue in a later subsection. It is well known that a rapidly ascending or descending dry air parcel follows a dry adiabatic lapse rate $$\frac{d\ln T}{d\ln p} = \frac{R}{c_p},$$ where $T$ is temperature, $p$ is pressure, $R$ is specific gas constant and $c_p$ is specific heat capacity of dry atmosphere. Similarly for a saturated air parcel mixed with condensable and non-condensable gases, it follows a moist adiabatic lapse rate[^1]: $$\frac{d\ln T}{d\ln p} = \frac{R_u+\frac{L_m\xi}{T}}{c_p+\frac{L_m^2\xi}{R_uT^2}},$$ where $L_m$ is the latent heat per mole, $c_p$ is the specific heat capacity per mole for the mixture, $R_u$ is the universal gas constant, and $\xi = p_{cond}/p_d$ is the molar mixing ratio of condensable gas over dry gas. Under normal conditions of most atmospheres, the dry adiabat is larger than the moist adiabat as long as $L_m > c_pT$. In the presence of two adiabats, the atmosphere can have different stability criteria. If the atmospheric lapse rate $\frac{d\ln T}{d\ln p}$ is larger than the dry adiabatic lapse rate $R/c_p$, the atmosphere will be absolutely unstable; if the lapse rate is smaller than the moist adiabatic lapse rate, the atmosphere will be absolutely stable; if the lapse rate is in between the dry and moist adiabatic lapse rate, the atmosphere is stable against dry convection but unstable to moist convection, which is referred to as *conditional instability*. Examples include the tropospheres of the Earth, Titan, and probably Jupiter and Saturn. How does an air parcel behave in a conditional unstable and unsaturated atmosphere? This is schematically illustrated in Figure \[schematic\]: initially starting from an arbitrary level below the condensation level, the ascending air parcel follows a dry adiabat until its relative humidity reaches 100%, and reaches the lifting condensation level (LCL). Afterward it will follow a moist adiabat, and then at some point it will reach the level of free convection (LFC) where it has a lower density than the environment and becomes positively buoyant. The air parcel then can freely convect to the top of a cumulus storm where its buoyancy diminishes and it stops ascending. In reality, because the atmospheric lapse rate may be stable to dry convection, some external lifting mechanism is needed for the air parcel to reach the LFC, and this is why storms do not occur everywhere and in every moment in Earth’s tropics even though the atmosphere is conditionally unstable. The amount of energy required to reach the LFC is referred to as convective inhibition (CIN). To initiate moist convection, either strong initial diabatic heating (e.g., in the case of Earth, heating of the surface by sunlight) or kinetic energy (e.g., forced lifting by atmospheric waves or other large-scale motion) is needed for the parcel to overcome the CIN. The CIN can act to limit the frequency of moist convection and preserve large convective available potential energy, which can be essential for the development of deep moist convection. One necessary condition for conditional instability is the stratification to dry convection in the troposphere. One mechanism to produce the tropospheric stratification is by latent heating and moist convection. As illustrated in Figure \[schematic\], the rising air that follows a moist adiabat in storms carries a higher entropy than where it is initiated. Mass continuity implies that the high-entropy surrounding atmosphere at the top of the storm must subside. During the subsidence, air continues to lose its entropy to space via IR radiation. Air closer to the ground has been subsiding for longer – and thus exhibits lower entropy – than air aloft. As a result, the entropy increases with height in the background temperature profile, which implies that the background temperature profile is stable to dry convection. On the other hand, the background temperature exhibits lower temperature than the moist adiabat, allowing moist instability. For giant planets, the tropospheric stratification has been inferred from observations for Jupiter [@flasar1986; @magalhaes2002; @reuter2007], and it has been demonstrated by numerical simulations that the stratification in Jupiter can result from latent heating of water condensation [@nakajima2000; @sugiyama2014]. Moist Convection on Controlling the Area Fraction of Moist Plumes ----------------------------------------------------------------- Vertical velocity within the moist convecting plume is much larger than the surrounding subsidence flow, as observed in Earth’s atmosphere. The fast upwelling velocity is driven by the large convective available potential energy (CAPE) in deep moist convection. CAPE is the amount of potential energy per unit mass available for the convection of a particular air parcel, and is essentially an integration of buoyancy with respect to height during the lifting of the parcel (e.g., [@emanuel1994], Chapter 6). In contrast to the buoyant ascending convective plumes, the subsiding air is stratified and not convecting, but it can gradually subside as described in Section \[instability\]. Because the radiative cooling time scale is generally long, the subsidence is generally slow. The asymmetry of the upwelling and downwelling vertical velocity results in a small area fraction of moist ascending plumes by the requirement of mass continuity. The area covered by clouds may not closely follow the area of moist plumes because cloud particles will be spread by the wind field near the cloud top, but this argument for moist plume area fraction can qualitatively explain the origin of patchiness of cumulus clouds [@lunine1987]. For brown dwarfs, patchy clouds deduced from near-IR brightness variability may qualitatively be explained by this mechanism. Molecular Weight Effect {#molecular} ----------------------- In atmospheres of gaseous giant planets and BDs, condensable species generally have a much higher molecular weight than the dominant dry constituent $\rm{H}_2$. Rainout of these condensates can decrease the density of air, and this effect can play an important role in atmospheric thermal structure and dynamics. For example, [@guillot1995] presented the idea that moist convection in giant planets may be inhibited due to molecular weight effect if the mixing ratio of water is substantially higher than solar abundance; [@Li2015] proposed that Saturn’s 20-to-30-year quasi-periodic planetary-scale storm is related to the molecular weight effect of water. In BDs and directly imaged EGPs, a local thin stably stratified layer associated with molecular weight gradients may exist right above the condensation level, due to the fact that the subsiding environmental air has experienced rainout, is thus relatively dry and has lower molecular mass. Whereas the air below the condensation level contains significant condensable vapor and has higher molecular mass. Therefore, a sharp gradient in molecular mass naturally exists near the condensation level, where molecular mass decreases with increasing altitude. This produces stratification to both dry and moist convection, contributing to CIN. Large CIN can suppress moist convection within (and above) the stable layer, leaving the atmosphere gradually cooling off towards its radiative equilibrium by thermal radiation. The radiative equilibrium temperature profile in the troposphere (for instance, on Earth and Jupiter) usually has a temperature lapse rate $d\ln T / d\ln p$ larger than the moist adiabatic lapse rate. Thus, the existence of CIN can help contribute to the accumulation of CAPE. This phenomenon has been shown by simulation on Jupiter as well [@nakajima2000]. By contribution to the accumulation of CAPE, the stratification from molecular weight effect also help to control the moist plume fraction via the mechanism discussed above. Large-scale Latent Heating on Atmospheric Dynamics {#latentheating} -------------------------------------------------- Moist convection provides a source of small-scale eddies, which can grow into large-scale eddies via an inverse energy cascade. The interactions among these eddies and the mean flow in a rapidly rotating sphere can produce zonally banded structure and vortices (, also see a review by @vasavada2005 for Jovian atmospheric dynamics). The latent heating can interact with the dynamics in many ways, and may produce organized clouds that can lead to cloud radiative feedback to the dynamics. The temperature perturbations by latent heating on isobaric surface can generate a wealth of waves that propagate upward to the stratosphere, driving circulation by the dissipation and breaking of these waves . MODEL ===== Here we summarize the key aspects of our model; for detailed implementation see . We solve the three-dimensional hydrostatic primitive equations using an atmospheric general circulation model (GCM), the MITgcm ([@adcroft2004], see also mitgcm.org). The horizontal momentum, hydrostatic equilibrium, continuity, thermodynamic energy and tracer equations in pressure coordinates are, respectively, $$\frac{d\mathbf{v}}{dt} + f\hat{k} \times \mathbf{v} + \nabla_p \Phi = 0, \label{eq.momentum}$$ $$\frac{\partial \Phi}{\partial p} = -\frac{1}{\rho}, \label{eq.hydro}$$ $$\nabla_p \cdot \mathbf{v} + \frac{\partial\omega}{\partial p} = 0, \label{eq.cont}$$ $$\frac{d\theta}{dt} = -\frac{\theta-\theta_{\rm{ref}}}{{\tau_{\rm{rad}}}} + \frac{L\theta}{c_p T}(\delta \frac{q-q_s}{\tau_{\rm{cond}}}), \label{thermal}$$ $$\frac{dq}{dt} = -\delta\frac{q-q_s}{\tau_{\rm{cond}}}+Q_{\rm{deep}}, \label{tracer}$$ where $\mathbf{v}$ is the horizontal velocity vector on isobars, $\omega=dp/dt$ is the vertical velocity in pressure coordinates, $f=2\Omega\sin \phi$ is the Coriolis parameter (here $\phi$ is latitude and $\Omega$ is the planetary rotation rate), $\Phi$ is geopotential, $\hat{k}$ is the local unit vector in the vertical direction, $\rho$ is density, $\nabla_p$ is the horizontal gradient in pressure coordinate, $d/dt=\partial/\partial t + \mathbf{v}\cdot\nabla_p + \omega\partial/\partial p$ is the material derivative, $\theta = T(\frac{p_0}{p})^{R/c_p}$ is the potential temperature, $p_0 = 1$ bar is a reference pressure, $\theta_{\rm{ref}}$ is the equilibrium potential temperature profile, ${\tau_{\rm{rad}}}$ is the radiative timescale, and $L$ is latent heat per mass for condensate. The ideal gas law is assumed for the equation of state for the atmosphere. The hydrostatic assumption used in the standard primitive equations solved in our model is a good approximation for large-scale flows in stratified atmospheres with large ratio of horizontal scale to vertical scale (e.g., see [@vallis2006], Chapter 2). In the atmospheres of brown dwarfs, the expected horizontal length scale of large-scale dynamics is $10^6 - 10^7$ meters, while the pressure scale height is $10^3 - 10^4$ meters. The aspect ratio of the atmosphere is on the order $10^2 - 10^3$, which is sufficient for the hydrostatic approximation to hold. The tracer $q$ is mass mixing ratio of condensable vapor to dry air, and $q_s$ is the local saturation vapor mass mixing ratio that is determined by saturation pressure function for specific condensable species. The “on-off switch” function $\delta$ controls the condensation: when $q>q_s$ then $\delta=1$ and vapor condenses over a characteristic timescale $\tau_{\rm{cond}}$ which is generally taken as $10^3$ sec, representative of a typical convective time; when $q \leq q_s$ then $\delta = 0$. Latent heating is immediately applied in the thermodynamic equation (Equation \[\[thermal\]\]) once condensation occurs. For simplicity, we include only one tracer, and choose enstatite vapor ($\rm{MgSiO_3}$) to represent silicate vapor in our brown dwarf models. Silicates are one of the most abundant condensates in the atmospheres of L/T dwarfs, and their condensation levels are closer to the photospheres than another dominant condensates – iron, and so silicates could have more influences on the photospheres of L/T dwarfs than iron. The saturation pressure function for $\rm{MgSiO_3}$ is adopted from [@ackerman2001]. Alternative saturation pressure functions for silicates are available, for example, see [@visscher2010]. Our study does not aim at precisely determining where condensation occurs but rather to explore dynamics driven by latent heating given a plausible condensation curve for a representative condensing species. The latent heat of silicates are similar in [@ackerman2001] and [@visscher2010], therefore, the detailed choice of the saturation T-P profile is not essential here. The saturation pressure function reads $$e_{\rm{s}} = \exp(25.37 - \frac{58663~\rm{K}}{T}) \quad \rm{bar},$$ which is shown by the dashed line in Figure \[f\_condensate\] assuming solar abundance for the mixing ratio of silicates. Here we assume that condensate will rain out immediately. The influence of rainout of condensable vapor on air density is properly included in the hydrostatic equilibrium equation where the density is affected by mean molecular weight. The replenishment term $Q_{\rm{deep}}$ crudely parameterizes evaporating precipitation and condensable species mixed upward from the deeper atmosphere. It takes the form $Q_{\rm{deep}} = (q_{\rm{deep}}-q)/\tau_{\rm{rep}}$, where $q_{\rm{deep}}$ is a specified abundance of condensable species in the deep atmosphere and $\tau_{\rm{rep}}$ is the replenish timescale which is typically taken $10^3$ sec. Both $\tau_{\rm{cond}}$ and $\tau_{\rm{rep}}$ are chosen to be short compared to dynamical timescales, and in this limit the dynamics should be independent of the two timescales. The $Q_{\rm{deep}}$ term is applied only at levels deeper than the condensation level. The radiation effects of the system are simplified by using the Newtonian cooling scheme (Equation \[\[thermal\]\]). For simplicity, the radiative timescale ${\tau_{\rm{rad}}}$ is taken to be constant through the atmosphere. In our application, the radiative equilibrium potential temperature $\theta_{\rm{ref}}$ is assumed spherically symmetric, and is characterized by two regimes, a nearly adiabatic deeper region and an isothermal upper region as $\theta_{\rm{ref}}(p)=[\theta_{\rm{adi}}^n(p)+\theta_{\rm{iso}}^n(p)]^{1/n}$, where $\theta_{\rm{adi}}$ represents the potential temperature of the nearly adiabatic lower layer, $\theta_{\rm{iso}}$ represents that of the isothermal upper layer, and $n$ is a smoothing parameter that we here set to 15. The equilibrium temperature profile is intended to crudely mimic the results from one-dimensional radiative-convection models, where the profile of the upper atmosphere approaches nearly isothermal and smoothly transitions to an adiabatic profile in the lower atmosphere (e.g., [@marley2002; @burrows2006; @morley2014]). Our equilibrium temperature structure is based on a gray radiative-convective calculation using the Rosseland-mean opacity table from [@freedman2014], and the radiative-convective boundary from our calculation is in good agreement with models using realistic opacities (e.g., [@tsuji2002]). The deep thermal structure is generally slightly unstable rather than strictly neutral to allow dry convective motions. We parameterize the temperature structure of the adiabatic deep region by $$\theta_{\rm{adi}}(p) = \theta_0 + \delta \theta \log \frac{p}{p_{bot}} \label{eq.theta}$$ where $\theta_0$ and $\delta \theta$ are constants and $ p_{bot}$ is the bottom pressure of the simulation domain. $\delta \theta$ is typically taken as 1 K for 1 times solar cases, which is qualitatively consistent with the argument in mixing length theory, but small enough to not affect the dynamics above the condensation level. Figure \[f\_condensate\] shows the equilibrium temperature and saturation vapor T-P profile for a typical T dwarf temperature regime, with silicates’ condensation T-P curve. The dotted line is the corresponding potential temperature $\theta_{\rm{ref}}$ profile, in which the adiabatic layer is characterized by nearly constant $\theta$ and the isothermal layer has an increasing $\theta$ with increasing altitude. Real moist convection involves the formation of cumulus clouds and thunderstorms on a length scale much smaller than that can be resolved by most general circulation models. There has been a long history of development for schemes that parameterize the effects of sub-grid-scale cumulus convection on large-scale flows resolved by global models (for a review see, e.g., @emanuel1993). These schemes are often complex, with concepts and parameterizations constrained by Earth’s atmosphere. It is not yet clear how relevant the specific parameterization of these schemes is to atmospheres of brown dwarfs and giant planets, so we do not include a moist convection sub-grid-scale scheme in our current model. As stated in , it is useful to first ascertain the effects of *large-scale* latent heating associated with the hydrostatic interactions of storms with the surroundings, as we pursue here. We include a weak linear damping of velocities similar to that of [@liu2013] at pressure larger than 50 bars to mimic the reduction of winds due to the Lorentz force and Ohmic dissipation at great depths where magnetic coupling may be important. This drag is deep and weak enough (with drag timescale of 100 days at the bottom) not to affect the dynamics above condensation level. We solve the equations of our global model on a sphere using the cube-sphere coordinate system [@adcroft2004; @showman2009]. For most of simulations, we assume a Jupiter radius, a five hour rotation period and 500 ${{\rm\,m\,s}^{-2}}$ surface gravity. The resolution in our nominal simulations is C128, which is equivalent to $0.7^{\circ}$ per grid longitudinally and latitudinally (i.e., an approximate resolution of $512\times256$ in longitude and latitude). The pressure domain in our model is from $0.01$ bar to $100$ bars, and it is divided into 55 layers with finer resolution on condensation layers as shown in Figure \[f\_condensate\]. The horizontal and vertical resolution is adequate to resolve the Rossby deformation radius which is the typical length scale of eddies expected on brown dwarfs, and the vapor partial pressure scale height above the condensation level, respectively. We do not include an explicit viscosity in our simulations, but a fourth-order Shapiro filter is added to the time derivative of $\mathbf{v}$ and $\theta$ to maintain numerical stability. RESULTS ======= A Typical T Dwarf {#tdwarf} ----------------- We begin by describing in detail a specific representative case for a typical T dwarf with a radiative timescale ${\tau_{\rm{rad}}}$ of $10^6$ sec, solar metallicity which is typical for field brown dwarfs (e.g., [@leggett2010]), and other parameters described in Section \[model\] (see also Figure \[f\_condensate\]). The spin-up time is about 1500 days for models with ${\tau_{\rm{rad}}}=10^6$ and $10^7$ sec and about 1000 days for model with ${\tau_{\rm{rad}}}=10^5$ sec. Figure \[f\_bd15-1\] shows a snapshot of a horizontal map of temperature and zonal (east-west) velocity at 1736 Earth days simulation time at 9.1 bars (upper row) near the condensation level. The simulation reached a statistically equilibrium state, where latent heating from the condensate cycle is statistically balanced by radiative cooling, and the upward transport of condensable vapor is balanced by rain out in storms. On the temperature map (left panel), the local red regions are storms with warm upwelling moist plumes, and they evolve on a timescale of hours to (Earth) days. The upper right panel in Figure \[f\_bd15-1\] shows the zonal wind map at the same pressure level, with yellow and red colors representing eastward velocity. Three eastward jets form near the equator, with maximum wind speed of about 40 ${{\rm\,m\,s}^{-1}}$. The jets are located where storms are generated, suggesting that jets are pumped by momentum transport associated with the storms. No jets form at mid-to-high latitudes, but velocity residuals manifest there, which are Rossby waves propagating northward and southward from the storm regions. The upwelling vertical motions are strongly suppressed near the stably stratified isothermal layer at $p<2$ bars, causing large horizontal velocity divergence; as a result, the wider spreading of the upwelling hot air produces the larger temperature perturbation patterns near the tropopause (lower-left panel in Figure \[f\_bd15-1\]), similar to simulations for the Jupiter model . Because the ascending air inside storms extend vertically from the storm base to the top near the tropopause, the locations of warm regions at 3.5 bars are generally correlated to those at about 9 bars. The horizontal zonal velocity map at the tropopause exhibits the similar multiple-jet configuration as that at 9 bars but with a larger maximum wind speed of about 90 ${{\rm\,m\,s}^{-1}}$. The larger horizontal velocity divergence near the tropopause causes more abundant turbulence and wave sources at this layer, and so stronger interactions of mean flow with turbulence and Rossby waves, generating stronger zonal flows. Storms occur mostly near the equator, with almost no storms in mid-to-high latitudes. Notice that in our model setup the equilibrium potential temperature profile $\theta_{\rm{ref}}$ is independent of latitude, such that we can exclude any latitudinal dependent forcing as a possible cause of banded structure seen in our simulations. Rather, any zonal banding, or latitude dependence of storms, must result from the latitudinal variation of $f$ and $\beta$ (where $\beta=d f/ d y$ is the gradient of Coriolis parameter with northward distance), which are the only sources that can introduce anisotropy in our simulations . Diagnosing the dynamical mechanism for the latitudinal dependent storms is difficult. Instead we offer speculation based on the fact that $f$ and $\beta$ are the only possible sources of anisotropy (and therefore of any latitude dependence). A possible reason is that the horizontal divergence of horizontal winds, $\nabla_p \cdot \mathbf{v}$, tends to be smaller at mid-to-high latitudes, implying smaller vertical velocities, which makes it more difficult to generate and maintain storms; the suppression of storms in turn further weakens vertical velocities by limiting horizontal temperature differences which are essential to drive horizontal divergence. There are two reasons that we expect small horizontal divergence at mid-to-high latitudes. First, winds tend to be more geostrophic (the balance between Coriolis and pressure gradient forces in the horizontal momentum equation \[\[eq.momentum\]\]) in higher latitudes where the Coriolis parameter $f$ is larger, and this can lead to a smaller horizontal divergence. At low latitudes, winds have larger ageostrophic components, which results in a larger horizontal divergence. Thus moist instability can be more easily triggered. The importance of rotation can be characterized by the Rossby number, $Ro=U/\mathcal{L}f$, where $U$ and $\mathcal{L}$ are the characteristic horizontal wind speed and horizontal length scale, respectively. If $Ro \ll 1$, winds are nearly geostrophic. We can quantitatively estimate the latitude above which the flow tends to be geostrophic by taking $U \sim 100{{\rm\,m\,s}^{-1}}$ and $\mathcal{L} \sim 10^6$ m, which are approximately the maximum relative velocity and the width of a local storm, respectively, and setting $Ro \sim 1$, and we have $\phi \sim 8^{\circ}$. This is qualitatively consistent with our simulations in which storms tend to clump inside $\pm 10^{\circ}$ latitudes (Figure \[f\_bd15-1\]). Second, even if flow were geostrophic, the horizontal divergence is $\nabla_p \cdot \mathbf{v} \sim \frac{v}{a\tan\phi}$ where $v$ is meridional (north-south) velocity and $a$ is the radius of the brown dwarf. The divergence in pure geostrophic flow comes from the gradient of $f$ with respect to latitudes. It is easy to see that even in geostrophic flow, horizontal divergence becomes smaller in higher latitudes and larger in lower latitudes. However, the argument here does not mean that there is no vertical motions in high latitudes in general situations. In fact, if one imposes an independent meridional temperature difference to the atmosphere, it is easy to generate vertical motion and overturning circulations in high latitudes (e.g., @williams2003 [@lian2008]). The difficulty in generating high-latitude storms here is that the horizontal temperature differences that would be required for vertical motions are not independently generated but can only come from the existence of vertical motion (and the associated latent heating). This additional sensitivity allows for the suppression of storms in situations where vertical motions tend to be smaller, as at high latitudes. Storms regions are buoyant, thereby causing ascending motion, which transports moist air upward from below, leading to condensation and latent heating — thereby maintaining the storms themselves. Therefore, storms are spatially well correlated with vapor mixing ratio and vertical velocities, as shown in the horizontal maps representing a local storm active area in Figure \[ttqv\]. The vertical relative vorticity $\zeta = \hat{k}\cdot \mathbf{\nabla}\times \mathbf{v}$ (lower two panels of Figure \[ttqv\]) measures the local spin of fluid in horizontal direction. If $\zeta$ has the same sign as the Coriolis parameter $f$, the storm is cyclonic, whereas the storm is anticyclonic if $\zeta$ and $f$ have the opposite sign. In the northern hemisphere, the base of storms (near 9 bars) generally have positive $\zeta$ and the top of storms (near 3.5 bar) generally have negative $\zeta$. The dynamical picture for a single storm is that: because of the latent heating, the lower density of the storm column causes a greater vertical spacing of isobars, i.e., constant pressure lines bow downward at the base of the storm and upward at the top of the storm. This causes a low-pressure center at the base of the storm and high pressure center at the top of the storm with respect to the surrounding environmental air at a given altitude. As a result, horizontal flow converges and diverges due to pressure gradient forces at the base and top of the storm, respectively. Meanwhile, the flow is accelerated by Coriolis force, which drives the flow to cyclonic at the base and anticyclonic at the top of the storms. Zonal Jets {#jets} ---------- Large-scale latent heating drives global atmospheric circulation and forms zonal jets in our simulations, as discussed in Section \[latentheating\]. The time-averaged zonal-mean zonal jet configuration from simulations with three different radiative timescales (${\tau_{\rm{rad}}}=10^5, 10^6$ and $10^7$ sec, with other parameters the same as the typical T dwarf in Section \[tdwarf\]) are shown in Figure \[f\_bd-zonal\]. The results are averaged over about 1000 days after the models being equilibrated. In general, two strong eastward subtropical jets form at about $\pm 12^{\circ}$ latitudes and weak jets form in mid-to-high latitudes, which are symmetric about the equator. At the equator, the equatorial jets are generally westward below the condensation level, and eastward equatorial jets appear just above the condensation level. The equatorial jet speed increases with height to the tropopause because of the baroclinic structure by latent heating, with its strength depending on radiative timescale. Here, baroclinic means that constant density surfaces are *not* aligned with constant pressure surfaces, whereas barotropic means that the two surfaces are aligned. The local maximum jet speed near the tropopause is caused by the strong dynamical perturbations from eddies generated at the tops of storms. Jets below the condensation level are generally weak, and the subtropical jets are presumably driven by the Coriolis force[^2] on the meridional circulation in the deep atmosphere that results from the circulation of the upper active layer [@haynes1991; @showman2006; @lian2008]. The jets extend into the upper stably stratified atmosphere. The circulation above the tropopause probably emerges from the absorption, dissipation and breaking of upward propagating waves that are generated at the tropopause . There have been extensive studies showing that the mechanical, wave-induced forcing is the dominant driver for stratospheric circulation on Earth despite the existence of equator-to-pole thermal forcing (see review by, for example, [@andrews1987; @haynes2005]). In conditions of our simulated atmospheres where isotropic equilibrium thermal structure is imposed, the wave-induced mechanical forcing should be responsible for the stratospheric circulation. In fact, we have observed upward propagating waves from levels perturbed by latent heating in our simulations, which supports our hypothesis. The jet structure exhibits differences with different radiative timescale ${\tau_{\rm{rad}}}$, as ${\tau_{\rm{rad}}}$ can affect the rate at which the characteristic horizontal temperature differences and dynamical perturbations are damped. First, the short-${\tau_{\rm{rad}}}$ model shows nearly barotropic jet structure, whereas relative high ${\tau_{\rm{rad}}}$ models show baroclinic structure. Second, the jet speed is generally larger for the relatively large-${\tau_{\rm{rad}}}$ model, presumably because there is more time for jets to pump up before dynamical perturbations are damped out; this relation has been formulated in using the quasi-geostrophic theory. The jet structure within about 2 – 3 pressure scale heights of the upper boundary for the model with ${\tau_{\rm{rad}}}=10^7$ sec (the lower panel in Figure \[f\_bd-zonal\], jets from $20^{\circ}$ to $50^{\circ}$ latitude) is likely affected by the upper boundary conditions. We have tested models with higher upper boundaries ($10^{-3}$ bar and less), and the dynamics deeper than about 3 pressure scale heights from the upper boundary remains almost the same as in the original model. We conclude that despite the imperfection of numerics near the upper boundary for model with ${\tau_{\rm{rad}}}=10^7$ sec, the dynamics presented here for the atmosphere below about 0.1 bar is physical. Area Fraction of Storms {#sec-fraction} ----------------------- The small area fraction of moist plumes has been visually shown in Figure \[f\_bd15-1\] and \[ttqv\] for a typical T dwarf model, in which the discretized warm areas only occupy a small fraction of the area in low latitudes where storms are active. Not only the area occupied by storms is small, the sizes of individual storms are also small, having diameter of around $2^{\circ}$ (about the length of 1700 km, or three grid cells) above the condensation level at around 10 bars for typical storms in all our models. As shown in Figure \[f\_bd15-1\], the storms slightly expand near the tropopause where flows experience stratification and expand laterally. Here we display a more quantitative measurement of the storm fraction in models with different radiative timescale ${\tau_{\rm{rad}}}$. The storms are defined roughly between the condensation level and the tropopause, and they should have both a saturated mixing ratio of vapor ($q\geq q_s$) and an upwelling velocity. Using the upward vertical velocity as an indicator for storms is reasonable in our case because, as will be shown below, the upward vertical velocities inside storms are much larger than the descending velocities outside storms. We tested the sensitivity of this criteria by choosing different numbers, for example, $q\geq 0.98q_s$ or $q\geq 1.02q_s$, and these different criteria do not affect the results. We define regions satisfying these two criteria as being inside storms. Regions not satisfying these criteria are defined as regions outside storms. We only count areas within about $\pm 9^{\circ}$ latitudes since this is the primary region where storms occur. We first count vertical velocities as a function of pressure inside and outside storms using instantaneous snapshots of vertical velocity field from simulations, then define the area fraction of storms as[^3] $\sigma_s(p) = |\omega_d(p)/ \omega_a(p)|$, where $\omega_d(p)$ and $\omega_a(p)$ are the spatially averaged vertical velocity outside storms and inside storms, respectively. Finally, $\sigma_s(p)$ is averaged over many snapshots at different simulation time over about 1000 days after the simulation equilibrates. The results are shown in Figure \[wsigma\] as a function of pressure for models with ${\tau_{\rm{rad}}}=10^5, 10^6$ and $10^7$ sec. In the left panel, the vertical velocity is in a modified log-pressure coordinate $-Hd(\ln p)/dt$, where $H$ is pressure scale height; this velocity is approximately equal to the vertical velocity in height coordinates. This is a standard way of representing vertical velocity in pressure coordinates (e.g., [@andrews1987]). Physically, the quantity $d(\ln p)/dt$ is the vertical velocity expressed in units of scale heights per second, with positive being downward. Multiplying by $-H$ converts this to the vertical velocity in $\rm{m~s^{-1}}$, with positive being upward. As long as the structure of isobars does not change rapidly with respect to $z$ over time, this quantity will be approximately equal to the vertical velocity in height coordinates. The magnitude of descending vertical velocities clearly decrease by order of magnitude with increasing ${\tau_{\rm{rad}}}$. The ascending velocities are similar for ${\tau_{\rm{rad}}}=10^5$ and ${\tau_{\rm{rad}}}=10^6$ sec, but they are a factor of $\sim 5$ smaller for ${\tau_{\rm{rad}}}=10^7$ sec. As a result, the area fraction of storms decreases orders of magnitude as ${\tau_{\rm{rad}}}$ increases. We quantitatively explore the mechanism controlling the area fraction in Section \[scalling\]. Enhanced Abundance of Condensate -------------------------------- Giant planets tend to have metal-rich atmospheres, having condensed out of the gas-depleted disks around preferentially metal-enriched host stars [@gonzalez1997]. In the context of our model, an enhanced metallicity means a greater abundance of silicate vapor, a higher latent heating and thus a stronger atmospheric circulation. We have ran models with three times solar abundance of silicate vapor, representing the possible conditions of directly imaged EGPs. Generally, the basic pattern of the condensation cycle and the zonal jet configuration are similar to the solar abundance models, but with larger temperature perturbations (proportional to the abundance of vapor), active storms occurring up to slightly higher latitudes and larger wind speeds. Figure \[m3t6\] shows the time-averaged zonal-mean zonal wind from a model with three times solar abundance (typical for heavy element abundances on Jupiter) and ${\tau_{\rm{rad}}}=10^6$ sec. The zonal jet structure is very similar to that of our model with solar abundance (middle panel of Figure \[f\_bd-zonal\]), except that the winds are enhanced by a factor of several. DISCUSSION and conclusions {#discussion} ========================== What Controls the Area Fraction of Moist Plumes? {#scalling} ------------------------------------------------ Here we construct a simple model to quantitatively understand the area fraction of storms shown in Section \[sec-fraction\] by constructing scaling relations for the governing equations (\[eq.momentum\]) – (\[thermal\]). The physical picture of the model comprises statistically steady storms and subsidence outside the storms. At its top, the storm center has high pressure relative to the surroundings which can drive an outward divergent flow; then the high-entropy air radiatively cools over time during the slow subsidence, reaching almost the same temperature as the environment at the condensation level by requirement of steady state. Here we ignore the density variations due to rainout of condensate. The area fraction of storms $\sigma_s$ is given by requirement of continuity $$\sigma_s \sim |\frac{\omega_d}{\omega_a}|, \label{eq.0}$$ where $|\omega_d| \ll |\omega_a|$. As presented in Section \[tdwarf\], storms mostly occur at low latitudes where the Rossby number is large ($\gtrsim 1$). Near the top of the storm, the horizontal force balance is primarily between advection and pressure gradient force. In low latitudes, the Coriolis force could still have a nontrivial magnitude compared to the advection force. Including the Coriolis force in our scaling induces only a mild correction to our final result (Equation \[\[fraction\]\]), but does not change our conclusion in this section. Therefore for the sake of a clearer illustration of the physical mechanism controlling the fractional area of storms, using the force balance between advection and pressure gradient is reasonable. Therefore the balance in horizontal momentum equation (\[eq.momentum\]) is $\mathbf{v}\cdot\nabla_p \mathbf{v} \sim -\nabla_p \Phi$, which to order of magnitude reads $$\frac{U^2}{\mathcal{L}} \sim \frac{\Delta\Phi}{\mathcal{L}}, \label{eq.1}$$ where $\Delta \Phi$ is the horizontal difference in gravitational potential between the top of the storm and its surroundings on a constant pressure surface. From hydrostatic equilibrium (Equation \[eq.hydro\]), we can estimate the pressure difference inside and outside the storm by integrating over the column: $$\Delta \Phi \sim R \delta \ln p \Delta T, \label{eq.2}$$ where $\delta \ln p$ is the vertical difference in log-pressure from the bottom to the top of the storm and $\Delta T$ is the characteristic horizontal temperature difference inside and outside the storm. From the continuity Equation (\[eq.cont\]), the horizontal divergence at the storm given by $\nabla_p \cdot \mathbf{v}\sim U/\mathcal{L}$, is balanced by vertical divergence of ascent inside the storms, This implies $$\frac{U}{\mathcal{L}} \sim \frac{\omega_a}{\delta p}, \label{eq.3}$$ where $\delta p$ is the difference in pressure from the bottom to the top of the storm. Combining Equation (\[eq.1\]), (\[eq.2\]) and (\[eq.3\]), and assuming constant vertical velocity, the ascending velocity $\omega_a$ can be estimated by $$\omega_a \sim \frac{\delta p \sqrt{R\Delta T \delta \ln p}}{\mathcal{L}}.$$ To estimate the descending velocity, we use the thermodynamic energy equation (\[thermal\]), and assume that vertical advection of the potential temperature is much larger than the horizontal advection. This is reasonable near the tropopause where vertical difference of potential temperature is much larger than the horizontal differences. We then can obtain the balance between radiative cooling and vertical advection, which to order of magnitude reads: $\omega_d \frac{\delta \theta}{\delta p} \sim \frac{\theta-\theta_{\rm{ref}}}{{\tau_{\rm{rad}}}}$, where $\delta \theta$ is the vertical difference in potential temperature outside storms between pressure levels corresponding to the bottom and the top of storms. Imagining a thermodynamics loop where air rises in storms and subsides in between storms, we expect that at the altitude of the storm top, the environmental air outside storms has just been detrained from the top of the storm, and therefore that the potential temperature of the storm air and environmental air are the same at the pressure of the storm top. Likewise, in a closed thermodynamic loop we expect that the potential temperature of environmental and storm air are equal at the storm bottom. To close the system, we assume that in a global-mean and steady state, the higher-entropy air descending from the top of storms radiates away most of its entropy gained from latent heating, and relaxes to nearly the reference temperature when the air reaches the bottom of storms, which implies $\delta \theta \sim \theta-\theta_{\rm{ref}}$. Assuming constant $\omega_d$, we can estimate the descending velocity as $$\omega_d \sim \frac{\delta p}{{\tau_{\rm{rad}}}}.$$ This equation simply states that the rate of descent is bottlenecked by the efficiency of radiation: in order for the air outside storms to descend over the vertical height of a storm, the air has to lose entropy (since the environment is stratified), and thus this descent must occur on timescales comparable to the radiative time constant. Finally, the area fraction can be obtained by substituting $\omega_d$ and $\omega_a$ into Equation (\[eq.0\]): $$\sigma_s \sim \frac{\mathcal{L}}{{\tau_{\rm{rad}}}\sqrt{R\Delta T \delta \ln p}}. \label{fraction}$$ This is essentially a timescale comparison, where $ \mathcal{L}/\sqrt{R\Delta T \delta \ln p}$ is the dynamical ascent timescale driven by CAPE and ${\tau_{\rm{rad}}}$ is the timescale driven by radiative dissipation. According to results in Section \[results\], taking $\mathcal{L}\sim 10^6$ m, $\Delta T \sim 2.5$ K, $\delta \ln p \sim 1.3$ and ${\tau_{\rm{rad}}}\sim 10^5, 10^6$ and $10^7$ sec, we have area fraction $\sigma_s \sim 10^{-1}, 10^{-2}$ and $10^{-3}$, respectively. Compared to results from simulations in the right panel of Figure \[wsigma\], our analytical model to order of magnitude agrees well with the maximum area fraction for different ${\tau_{\rm{rad}}}$. The area fraction from our numerical results show variation as a function of pressure which in general are off by a factor of a few to ten compared to our analytical model. Given the simplicity of the scaling theory, we can explain the order of magnitude decrement of area fraction with increasing ${\tau_{\rm{rad}}}$, illustrating the important regulation of radiation on the moist convection. Real cloud formation exhibits many complexities not accounted for in this simple scaling theory. For example, the intertropical convergence zone in Earth’s tropical region shows organized regions of vigorous cumulus convection, containing transient cloud clusters rather than simply regions of steady-state precipitation and mean updrafts. This is a result of interactions between local cumulus convection and large-scale atmospheric circulation ([@holton2012], Chapter 11). The large-scale latent heating scheme in our model does not represent the small-scale cumulus convection, but rather the hydrostatic interaction of the storms with their surroundings. To understand the interactions between sub-grid moist convection and large-scale flow, we need a better parameterization of moist convection in future studies. Also, radiative feedback by cloud particles can play an important role in the development of cumulus clouds. Our analysis here will be tested using more realistic models in future studies. Thermal Structure ----------------- The thermal structure of the atmosphere can be affected directly by latent heating via its effect on temperature, and indirectly by the molecular weight effect via introducing a stratification layer above the condensation level. The upper panel of Figure \[thermalstructure\] shows potential temperature $\theta$ as a function of pressure for air outside and inside of storms from the nominal simulation with radiative timescale ${\tau_{\rm{rad}}}= 10^6$ sec in Section \[results\], and the middle panel shows the corresponding virtual potential temperature $\theta_v$ profile. The virtual potential temperature $\theta_v$ is defined as $\theta_v=(\frac{1+q/\epsilon}{1+q})\theta$, where $\epsilon = m_v/m_d$ is the ratio of molecular mass of condensable species and the dry air, and $q$ is mass mixing ratio. It can be viewed as the theoretical potential temperature that a dry air parcel would have if the dry parcel has the same pressure and density as the moist air, so $\theta_v$ is a direct measurement of density. The solid line in the upper panel is the equilibrium background temperature profile prescribed by Equation (\[eq.theta\]). In the deep convective region, temperatures do not exactly follow the reference profile because dry motions tend to neutralize the thermal structure by having nearly constant $\theta$. However, due to rainout of condensates as shown in the lower panel of Figure \[thermalstructure\], the layer just above the condensation level is stratified, and is stable against dry convection. As a result, this thin layer just above the condensation level is not neutralized by dry motions. The stratification of this thin layer is better illustrated by looking at the virtual potential temperature $\theta_v$ profile in the middle panel, in which $\theta_v$ increases with height despite the fact that $\theta$ actually decreases with height. Note that increasing $\theta_v$ with height implies stratification against dry convection, accounting for both temperature and molecular weight gradients. The red circles represent (virtual) potential temperatures inside storms, and their profile is nearly close to a moist adiabat. Interestingly, at the bottom of the stratosphere, “overshooting” of moist plumes occurs, in which temperature inside storms (red circles) has lower temperature than surroundings. This suggests that ascending moist plumes penetrate into the stratosphere, inducing horizontal flow divergence described in Section \[tdwarf\]. Due to the dry subsidence, layers immediately above the condensation level outside storms are unsaturated, which leads to decreasing mean molecular weight with decreasing pressure. And this contributes to a strong stratified background environmental profile at pressures near the condensation level. Interestingly, it also results in a lower density of the background air than that of the moist upwelling plumes. It can be seen in the middle panel of Figure \[thermalstructure\], where air inside storms (red circle) has a lower $\theta_v$ than background air (blue triangle) right above the condensation level. Overall, the molecular weight effect is likely important in regulating the way storms occur and the way they interact with their environment. This is similar to regional moist convection simulations in atmospheres of giant planets where water has larger molecular weight than hydrogen-helium mixture [@nakajima2000; @sugiyama2014; @Li2015]. Indeed, simulations from models without molecular weight effect have more vigorous storm activities than those with molecular weight effect. However, how this regulation would work in a hydrostatic manner is unclear, and needs more diagnosis in future work. Implication for Observations ---------------------------- Due to the lack of radiative transfer and cloud particles, we are unable to directly compare the simulated variability to the observed near-IR variability. Still, our results have important implications for observations. Storms driven by moist instabilities can extend vertically over several pressure scale heights, reaching the photosphere. The vertical velocity inside the large-scale hydrostatic storms is high, and condensed particles can be lofted up to the storm top, forming cumulus clouds and inducing IR brightness variability. The storms can evolve on timescales of hours to days, and the cumulus clouds would be patchy due to the spatially inhomogeneous moist convection, and thus can help to explain patchy clouds inferred in the rapid evolving near-IR light curves and observationally inferred surface maps of brown dwarfs. Recently, @karalidi2015 and @karalidi2016 present retrieval surface temperature maps for a few brown dwarfs based on near-IR light curves. Interestingly, the deduced temperature anomaly patterns are much larger than the expected Rossby deformation radius ($\sim 10^7$ m). It may be caused by a cluster of storms over a large fraction of the globe similar to that shown in Figure \[ttqv\], which may produce a broad envelope of patchy clouds that represent as a single large spot. The L/T transition occurs over a narrow range of effective temperature accompanied with a *J*-band brightening (e.g., [@allard2001; @burrows2006; @saumon2008]), and its details remain poorly understood. Hypotheses include a change of sedimentation efficiency for condensates [@knapp2004] or that the cloud deck gradually becomes patchy as clouds form progressively deeper with increasing spectral type, allowing contributions from greater flux emitted from deeper levels [@ackerman2001; @burgasser2002; @marley2010]. However, the detailed mechanisms for cloud breaking during the L/T transition are yet unclear. Here we propose that, the area fraction of moist convection can help to support the idea of cloud breaking during the L/T transition. Moist convection occurs when large CAPE is available, that is, the condensation level should be much lower than the tropopause. This can also be quantified from Equation (\[fraction\]) that relatively large $\delta \ln p$ and long radiative time constant ${\tau_{\rm{rad}}}$ are needed to produce a small storm fraction. In the hotter L dwarfs, clouds first condense close to upper stratified atmosphere (e.g., [@tsuji2002; @burrows2006]), so moist convection can not happen, and cloud morphology may be dominated by the stratus clouds formed by more gradual processes such as transport by waves [@freytag2010] or large-scale atmospheric flow . For the cooler dwarfs near the L/T transition, the condensation level gradually sinks below the tropopause, moist convection thus can occur with increasing CAPE, producing patchy cumulus clouds. Also, as condensation level moves to a larger pressure, the radiative timescale ${\tau_{\rm{rad}}}$, which can be approximated by ${\tau_{\rm{rad}}}\sim \frac{p}{g}\frac{c_p}{4\sigma T^3}$ where $\sigma$ is the Stefan-Boltzmann constant, may become larger. According to our results, the larger ${\tau_{\rm{rad}}}$ can also decrease the storm area fraction. The changing of cloud patchiness during the L/T transition can be a natural consequence of the change of CAPE and radiative timescale with increasing spectral type. We predict that, if latent heating dominates cloud formation processes in atmospheres of BDs and directly imaged EGPs, the fractional coverage area of clouds gets smaller as the spectral type goes through the L/T transition from high to lower effective temperature. Future more realistic models are needed to test our hypothesis. Summary ------- Latent heating from condensation of various chemical species in brown dwarf atmospheres is important for shaping the atmospheric circulation and influencing cloud patchiness. We illustrated the dynamical mechanisms of latent heating using an idealized atmospheric circulation model that includes a condensation cycle of silicate vapor with the molecular weight effect included. For typical T dwarf models, zonal jets can be driven by large-scale latent heating. Temperature maps show inhomogeneous storm patterns, which evolve on timescales of hours to days and can extend vertically over a pressure scale height or more to the tropopause. The fractional area of the brown dwarf covered by active storms is small. Based on a simple analytic model, we quantitatively explain the fractional area of storms, and predict its dependence on radiative timescale and convective available potential energy. Our results have important implications for the observed near-IR variability and the cloud properties across the L/T transition. Further general circulation models with realistic clouds and radiative transfer are needed for better investigation of the global circulation. = We thank Xi Zhang for helpful discussion. This work was supported by NASA Headquarters under the NASA Earth and Space Science Fellowship Program and NSF grant AST 1313444 to APS. [^1]: In deriving this formula, the Clausius-Clapeyron equation for the saturation vapor pressure of the condensable species was used, assuming an ideal gas equation of state and that the condensate density is much greater than the gas density. This formula is applicable for full range of $\xi$, not limited to assumption of small mixing ratio of condensable gas. [^2]: Notice that because of the fast rotation, the Coriolis parameter $f$ has a large magnitude of about $10^{-4} ~\rm{s^{-1}}$ even at $10^{\circ}$ latitude. [^3]: This is a definition based on continuity argument, consistent with that defined in Section \[mechanism\].

--- author: - Amod Agashe and Mak Trifković title: '[Darmon points on elliptic curves over totally real fields]{}' --- =2em Introduction ============ Let $F$ be a number field and let $E$ be an elliptic curve over $F$ of conductor an ideal $N$ of $F$. We assume throughout that $F$ is totally real: in that case, it is known, under minor hypotheses, that there is a newform $f$ of weight $2$ on $\Gamma_0(N)$ over $F$ whose $L$-function coincides with that of $E$ (see, e.g., [@zhang-annals] or [@dar-rat §7.4]). We fix a quadratic extension $K/F$. When $K$ is totally complex, a classical theory produces a family of Heegner points of $E$, defined over ring class fields of $K$. The Galois action on them is given by a Shimura reciprocity law, and their heights relate to the derivative of the $L$-function of $E$ over $K$ at $1$ (see, e.g., [@zhang-annals] or [@dar-rat §7.5]). We assume therefore from now on that $K$ has at least one real place. The goal of the theory of Darmon points (earlier called Stark-Heegner points) is to extend to such $K$ the construction of Heegner points. In  [@dar-annals], Darmon started the theory in the case $F={{\bf{Q}}}$ and $K$ a totally real quadratic field. In [@greenberg], Greenberg generalizes this work to give a conjectural construction of points in the case where $F$ is arbitrary totally real quadratic field of narrow class number one, $E$ is semistable, $N \neq (1)$, the sign of the functional equation of $E$ over $K$ is $-1$, there is a prime dividing the conductor of $E$ that is inert in $K$, and the discriminant of $K$ is coprime to $N$. The techniques used are $p$-adic in nature. In [@gartner], Gartner generalizes the work of [@dar-rat Chap 8] and [@darlog] to give a construction of what he calls Darmon points in certain situations using archimedian techniques. The conditions under which Gartner’s construction works are a bit too technical to describe here, but we shall describe them in Section \[sec:gartner\]. All the constructions share a basic outline: one computes an archimedean or $p$-adic integral of the modular form associated to $E/F$, and plugs the resulting value into a Weierstrass of Tate parametrization of $E$ to produce the Darmon point. Let ${\mathcal{O}}\subseteq K$ be an ${\mathcal{O}}_F$-order such that ${\rm Disc}({\mathcal{O}}/{\mathcal{O}}_F)$ is coprime to $N$. In this article, we show that if the sign of the functional equation of $E$ over $K$ is $-1$, the discriminant of $K$ is coprime to $N$, and the part of $N$ divisible by primes that are inert in $K$ is square-free, then one can apply either the construction of Gartner or the construction of Greenberg (after removing the assumption that $F$ has narrow class number one, which we show how to do) to conjecturally associate to ${\mathcal{O}}$ a point that we call a Darmon point (actually there are choices in the construction, and so one gets several Darmon points). This point is intially defined over a transcendental extension of $K$, but we conjecture that the point is algebraic, defined over the narrow ring class field extension of $K$ associated to the order ${\mathcal{O}}$. This point comes with an action of the narrow class group of ${\mathcal{O}}$, and we state a conjectural Shimura receprocity law for this action. In Section \[sec:gg\] we recall and slightly modify the constructions of Greenberg and Gartner. In Section \[shdetails\] we show how one of the two constructions can be carried out under our hypotheses. We assume throughout this article that the reader is familiar with [@greenberg] and [@gartner]. The constructions of Gartner and Greenberg {#sec:gg} ========================================== In this section we discuss Gartner’s and Greenberg’s constructions. Gartner makes several assumptions that are sometimes not made explicit in [@gartner]; we clarify what hypotheses are needed in Gartner’s construction and also modify it a bit so that it can be unified better with Greenberg’s construction. We also show how to generalize Greenberg’s construction to remove the class number one assumption in [@greenberg]. Both constructions require the existence of a suitable quaternion algebra $B$ in order to use the Jacquet-Langlands correspondence. So let $B$ be a quaternion algebra over $F$. We will impose certain assumptions that $B$ (and other objects) will have to satisfy in each of Gartner’s or Greenberg’s constructions. In Section \[shdetails\] we shall explain when these assumptions are met. First, in either construction, one needs an embedding of $K$ into $B$. Recall that $B$ is said to be split at a place $v$ of $F$ if $B {\otimes}_F F_v$ is the matrix algebra, and ramified at $v$ otherwise. It is known that a quaternion algebra is determined up to isomorphism by the set of ramified places, which is finite of even cardinality. Conversely, for any finite set of places of even cardinality there is a quaternion algebra ramifying at these places. We say that a real place of $F$ splits in $K$ if there are two real places of $K$ lying over it, and we say that it is inert otherwise (such a place is usually said to be ramified, but we prefer to call it inert to avoid confusing ramification in $K$ with ramification of $B$).  \ [*Assumption A:*]{} Assume that there is an embedding of $q:K\hookrightarrow B$, i.e., that each place where $B$ ramifies, archimedean or not, is inert in $K$.\ Gartner’s construction {#sec:gartner} ---------------------- We now outline the construction of Gartner, along with some modifications. For details and proofs of the claims made below, please see [@gartner]. We try to use notation consistent with or similar to that in [@gartner] as much as possible. We start by listing the assumptions used in Gartner’s construction. Let $d$ denote the degree of $F$ over ${{\bf{Q}}}$ and let $\tau_1, \ldots, \tau_d$ denote the archimedian places of $F$.\ [*Assumption B1:* ]{}Suppose that there is exactly one archimedian place of $F$ where $B$ is split but which does not split in $K$.\ Without loss of generality, assume that the archimedian place of $F$ where $B$ is split but which does not split in $K$ is $\tau_1$. Let $r$ be the integer such that the archimedian places of $F$ that split in $K$ are $\tau_2, \ldots, \tau_r$; since $K$ is not a CM field, $r \geq 2$. By our Assumption A, $B$ necessarily splits at $\tau_1, \ldots, \tau_r$ and by Assumption B1, it necessarily ramifies at $\tau_{r+1}, \ldots, \tau_d$. If $S$ is a ring, then let $\widehat{S}$ denote $S{\otimes}_{{\bf{Z}}}\widehat{{{\bf{Z}}}}$. Let $R$ be an Eichler order of $B$.\ [*Assumption B2:*]{} Assume that $f$ corresponds to an automorphic form on $\widehat{R}$ under the Jacquet-Langlands correspondence.\ Let $b \in \widehat{B}^\times$. Let ${\mathcal{O}}\subseteq K$ be an ${\mathcal{O}}_F$-order such that ${\rm Disc}({\mathcal{O}}/{\mathcal{O}}_F)$ is coprime to $N$. In order to get a Darmon point in the narrow class field associated to the order ${\mathcal{O}}$, along with an action of ${{\rm Pic}}({\mathcal{O}})^+$, we make the following assumption (which is not made in [@gartner]):\ [*Assumption B3:*]{} Suppose that $q(K) \cap b \widehat{R}b^{-1} = q({\mathcal{O}})$, i.e., that $q$ is an optimal embedding of ${\mathcal{O}}$ into the order $B \cap b \widehat{R}b^{-1}$.\ We now start the construction. Let $G = {\rm Res}_{F/{{\bf{Q}}}} B^{\times}$ and let ${{\bf{A}}}_f$ denote the set of finite adeles over ${{\bf{Q}}}$. Let $H = \widehat{R}^\times$ and let ${\rm Sh}_H(G)$ denote the quaternionic Shimura variety whose complex valued points are given by $G({{\bf{Q}}}) \backslash ({{\bf{C}}}\setminus {{\bf{R}}})^r \times G({{\bf{A}}}_f)/H$. Let $b \in \widehat{B}^\times$. Let $T = {\rm Res}_{K/{{\bf{Q}}}}({{\bf{G}}}_m)$. The embedding $q$ induces an embedding of $T$ in $G$ that we again denote by $q$ for simplicity. Using the embeddings associated to $\tau_1, \ldots, \tau_r$, where $B$ splits, we get a natural action of $q(T({{\bf{R}}})^0)$ on $X = ({{\bf{C}}}\setminus {{\bf{R}}})^r$. Let $T^0$ be a fixed orbit of $q(T({{\bf{R}}})^0)$ whose projection to the first component of $X$ is a point (recall that $\tau_1$ is a complex place); we fix this point henceforth and denote it by $z_1$. Let $T_{q,b}$ denote the projection to ${\rm Sh_H}(G)({{\bf{C}}}) = G({{\bf{Q}}}) \backslash ({{\bf{C}}}\setminus {{\bf{R}}})^r \times G({{\bf{A}}}_f)/H$ of $T^0\times G({{\bf{A}}}_f)$.Then $T_{q,b}$ is a torus of dimension $r-1$.Note that our torus corresponds to the torus denoted ${\mathcal T}^0_b$ in Section 4.2 of [@gartner]; moreover Gartner actually works with a modified Shimura variety denoted ${\rm Sh}_H(G/Z, X)$ in loc. cit. However, the construction goes through mutatis mutandis with ${\rm Sh}_H(G)$ as well, which is what we shall do in this article. Thus our construction is a slightly modified version of that of Gartner. We are doing this modification to get a version of the (conjectural) Shimura receprocity law that is similar to that of Greenberg (Conjecture 3 in [@greenberg]). Using the theorem of Matsushima and Shimura [@matshi], one shows that there is an $r$-chain that we denote $\Delta_{q,b}$ (called $\Delta_b$ in [@gartner]) on ${\rm Sh}_H(G)$ whose boundary is an integral multiple of $T_{q,b}$. This uses Proposition 4.5 in [@gartner], whose proof assumes that the Shimura variety is compact, i.e., that $B$ is not the matrix algebra. If $B$ is the matrix algebra, then we are in the ATR setting dealt with in [@dar-rat Chap. 8] and [@darlog]; thus in this article, we are subsuming the ATR construction under Gartner’s construction(in fact, Gartner’s work was motivated by the ATR construction). Let $\phi$ denote the automorphic form on $H$ corresponding to $f$ under the Jacquet-Langlands correspondence (recall our Assumption B2). Analogous to the construction of the form denoted $\omega_\phi^\beta$ in [@gartner], we get a form that we denote $\omega_\phi$ on ${\rm Sh}_H(G)$ by taking $\beta$ to be the trivial character (we could allow $\beta$ to be an aribitrary character, but we are taking it to be the trivial character for the sake of simplicity and also to get an action of the narrow class group below). Assuming conjectures of Yoshida [@yoshida], the periods of $\omega_\phi$ form a lattice that is homothetic to a sublattice of the Neron lattice $\Lambda_E$ of $E$. Then the image of a suitable integer multiple of $\int_{\Delta_{q,b}} \omega_\phi$ is independent of the choice of the chain $\Delta_{q,b}$ made above. Let $\Phi: {{\bf{C}}}/\Lambda_E {\rightarrow}E({{\bf{C}}})$ denote the Weierstrass uniformization of $E$. Then the Darmon point $P_{q,b}$ in $E({{\bf{C}}})$ is defined as a suitable multiple of the image of $\int_{\Delta_{q,b}} \omega_\phi$ under $\Phi$ (our point corresponds to the point $P^\beta_b$ in [@gartner]). It is conjectured that the point $P_{q,b}$ in $E({{\bf{C}}})$ has algebraic coordinates. Let $\widehat{q}:\hat K\to\hat B$ denote the map obtained from $q$ by tensoring with $\widehat{F}$. Let $K_{{\bf{A}}}$ denote the ring of adèles of $K$. Denote by $a_f$ the non-archimedean part of $a \in K_{{\bf{A}}}$. Following Gartner, we define an action of $K_{{\bf{A}}}^\times$ on Darmon points $P_{q,b}$ by $$a * P_{q,b} = P_{(q, \widehat{q}(a_f) b)}.$$ An easy check shows that the new pair satisfies Assumption B3. Denote by $K_+$ the subset of elements of $K$ that are positive in all real embeddings. As usual, $K^\times$ is embedded into $K_{{\bf{A}}}^\times$ diagonally. We claim that the action above factors through $\widehat{{\mathcal{O}}}^\times (K {\otimes}_{{\bf{Q}}}{{\bf{R}}})^\times K_+^{\times}$, i.e., we have an action of $\widehat{{\mathcal{O}}}^\times \backslash K_{{\bf{A}}}^\times /(K {\otimes}_{{\bf{Q}}}{{\bf{R}}})^\times K_+^{\times}$, which is the narrow class group ${{\rm Pic}}({\mathcal{O}})^+$. To prove the invariance under the action of $\widehat{{\mathcal{O}}}$, note that by our condition above, $\widehat{q}(\widehat{{\mathcal{O}}}) \subseteq b \widehat{R}^\times b^{-1}$, so if $a \in \widehat{{\mathcal{O}}}$, then $\widehat{q}(a_f) = b r b^{-1}$ for some $r \in \widehat{R}$. Hence $\widehat{q}(a_f) b = b r b^{-1} b = b r$, and so $a * P_{q,b} = P_{(q, br )} = P_{q,b}$ since $\widehat{R}$ acts trivially on ${\rm Sh}_H(G)$. Next $(K {\otimes}_{{\bf{Q}}}{{\bf{R}}})^\times$ clearly acts trivially. It remains to show invariance under the action of $K_+^\times$. If $k \in K_+^\times$, then $k * P_{q,b} = P_{(q, \widehat{q}(k) b)}$. Let $x_0 \in T^0$. Then $T_{(q, q(k) b)}$ consists of images of points of the form $(y, q(k) b)$ in $({{\bf{C}}}\setminus {{\bf{R}}})^r \times G({{\bf{A}}}_f)$ such that $y = t x_0$ for some $t \in q(T({{\bf{R}}})^0)$. Letting $\pi$ denote the projection map from $({{\bf{C}}}\setminus {{\bf{R}}})^r \times G({{\bf{A}}}_f)$ to ${\rm Sh}_H(G)({{\bf{C}}}) = G({{\bf{Q}}}) \backslash ({{\bf{C}}}\setminus {{\bf{R}}})^r \times G({{\bf{A}}}_f)/H$, we have $\pi(y, q(k) b) = \pi((t x_0, q(k) b) = \pi(q(k^{-1}) t x_0, b) = \pi(t q(k^{-1}) x_0, b) $, as elements of $q(K)$ commute. Thus the point $P_{(q, \widehat{q}(k) b)}$ is obtained from the orbit with base point $q(k^{-1}) x_0$. Recall that the projection of $q(T({{\bf{R}}})^0)$ to the first component of $X$ is a point $z_1$, and thus $z_1$ is fixed by $q(K)$. In particular, the projection of the orbit of the point $q(k^{-1}) x_0$ to the first component of $X$ is again $z_1$. Moreover the projections of the orbits of the point $q(k^{-1}) x_0$ and of the point $x_0$ lie in the same connected component of each copy of ${{\bf{C}}}\setminus {{\bf{R}}}$ in $X$ since $k \in K_+$. In view of the last two sentences in [@gartner Prop 4.7], $P_{(q, \widehat{q}(k) b)} = P_{(q, b)}$, which finishes our proof of the claim. Thus we get an action of the narrow class group ${{\rm Pic}}({\mathcal{O}})^+$ on Darmon points $P_{q,b}$; we denote this action again by $*$. \[conj:gartner\] The point $P_{q,b}$ is defined over the narrow ring class field extension $K_{\mathcal{O}}^+$ of $K$ associated to the order ${\mathcal{O}}$. If $\alpha \in {{\rm Pic}}({\mathcal{O}})^+$, then ${{\rm rec}}(\alpha)(P_{q,b}) = \alpha * P_{q,b}$, where ${{\rm rec}}: {{\rm Pic}}^+ {\mathcal{O}}{\rightarrow}{\rm Gal}(K_{\mathcal{O}}^+/K)$ is the reciprocity isomorphism of class field theory. The following assumption is not needed for Gartner’s construction, but we shall mention it since it will be useful in Section \[shdetails\] (see also Remark \[rmk:assn\](i)).\ [*Assumption B4:*]{} Suppose that the finite primes where $B$ is ramified are exactly the primes that divide $N$ and are inert in $K$.\ Greenberg’s construction {#sec:greenberg} ------------------------ We now discuss the construction of Greenberg, and show how to remove assumption that $F$ has class number one made in [@greenberg]. For details of the construction, please see [@greenberg]. We start by listing the assumptions needed. Recall that $d$ denotes the degree of $F$ over ${{\bf{Q}}}$ and $\tau_1, \ldots, \tau_d$ denote the archimedian places of $F$. Since $K$ is not CM, there is at least one infinite place of $F$ that splits in $K$. Let $n$ denote the number of such places, and without loss of generality, assume that these places $\tau_1, \ldots, \tau_n$ (in the previous section, we wrote $r$ instead of $n$; we change notation to be consistent with or similar to that in [@greenberg] as much as possible). By Assumption A, $B$ is split at $\tau_1, \ldots, \tau_n$ and can be ramified or split at $\tau_{n+1}, \ldots, \tau_d$. However we insist:\ [*Assumption C1:*]{} Suppose that $\tau_{n+1}, \ldots, \tau_d$ are precisely the infinite primes where $B$ ramifies.\ [*Assumption C2:*]{} Suppose that there is a prime ideal ${{\mathfrak{p}}}$ of $F$ that exactly divides $N$ and is inert in $K$.\ [*Assumption C3:*]{} Suppose that the part of $N$ divisible by primes that are inert in $K$ is square-free and that the finite primes where $B$ is ramified are exactly the primes other than ${{\mathfrak{p}}}$ that divide $N$ and are inert in $K$.\ Let ${{\mathfrak{n}}}$ be the part of $N$ supported at primes other than ${{\mathfrak{p}}}$ that divide $N$ and where $B$ is split. For each ideal ${{\mathfrak{a}}}$ of ${\mathcal{O}}_F$ coprime to the discriminant of $B$, choose an Eichler order $R_0({{\mathfrak{a}}})$ in $B$ of level ${{\mathfrak{a}}}$ as in [@greenberg §2]. Let $R = R_0({{\mathfrak{n}}})$ be the Eichler order in $B$ of level ${{\mathfrak{n}}}$. As in Section \[sec:gartner\], we choose $b \in \widehat{B}^\times$ and impose the analog of Assumption B3:\ [*Assumption C4:*]{} Suppose that $q(K) \cap b \widehat{R} b^{-1} = q({\mathcal{O}})$, i.e., that $q$ is an optimal embedding of  ${\mathcal{O}}$ into the order $B \cap b \widehat{R} b^{-1}$.\ We remark that the assumptions made above are not exactly the assumptions made in [@greenberg], but suffice for the construction (e.g., the assumption made in [@greenberg] that the sign of the functional equation of $E$ over $K$ is $-1$ is used to show that a quaternion algebra $B$ satisfying Assumptions C1 and C3 exists). We now describe the construction. As in Section \[sec:gartner\], let $G = {\rm Res}_{F/{{\bf{Q}}}} B^{\times}$, let $H = \widehat{R}^\times$, and let ${\rm Sh}_H(G)$ denote the quaternionic Shimura variety whose complex points are given by $G({{\bf{Q}}}) \backslash ({{\bf{C}}}\setminus {{\bf{R}}})^n \times G({{\bf{A}}}_f)/H$, where $n$ is the number of real places of $F$ where $B$ splits ($n$ is the same as the $r$ in Section \[sec:gartner\]). Let $G({{\bf{R}}})^0$ denote the identity component of $G({{\bf{R}}})$ and let $G({{\bf{Q}}})^0 = G({{\bf{R}}})^0 \cap G({{\bf{Q}}})$. Let $C \subseteq \widehat{B}^\times$ be a system of representatives of the double cosets $G({{\bf{Q}}})^0 \backslash \widehat{B}^\times / H$. If $g \in \widehat{B}^\times$, then let $\Gamma'_g = gHg^{-1} \cap G({{\bf{Q}}})^0 \subseteq G({{\bf{R}}})^0$ and let $\Gamma_g$ denote the natural projection of the image of $\Gamma'_g$ in $PGL_2^+({{\bf{R}}})^n$ (the projection is obtained via the embeddings associated to the places $\tau_1, \ldots, \tau_n$ where $B$ splits). Let ${{{\mathfrak{H}}_{\scriptscriptstyle{2}}}}$ denote the upper half plane. Then ${\rm Sh}_H(G)({{\bf{C}}})$ is homeomorphic to the disjoint union of $\Gamma_g \backslash ({{{\mathfrak{H}}_{\scriptscriptstyle{2}}}})^n$ as $g$ ranges over elements of $C$. Greenberg assumes that the narrow class number of $F$ (and therefore of $B$) is one, in which case $C$ is a singleton set and $\Gamma_g = \Gamma_0({{\mathfrak{n}}})$. Greenberg’s construction uses group homology and cohomology for the group $\Gamma_0({{\mathfrak{n}}})$ with coefficients in various modules. When the narrow class number is not one, one has to replace the homology groups of $\Gamma_0({{\mathfrak{n}}})$ with the direct sum over $g \in C$ of the homology groups of $\Gamma_g$. As in Section \[sec:gartner\], we construct the torus $T_{q,b}$ in ${\rm Sh}_H(G)$. The inclusion of the torus in ${\rm Sh}_H(G)$ induces a map on the corresponding $n$-th homology groups. The image under this map of a generator of the $n$-th homology group of the torus gives an element of the $n$-th homology group of ${\rm Sh}_H(G)$(in fact, since the torus is connected, that element lies in one of direct summands in the homomlogy); this element replaces the element denoted $\Delta_\psi$ starting with Lemma 21 in [@greenberg] (note that in loc. cit., before Lemma 21 , $\Delta_\psi$ is considered to be an element of a homology group of $\Gamma_0({{\mathfrak{n}}})_\psi$, but starting with Lemma 21, $\Delta_\psi$ is considered to be an element of a homology group of $\Gamma_0({{\mathfrak{n}}})$). Greenberg also uses homology groups of $\Gamma_0({{\mathfrak{n}}})$ with coefficients in a module denoted ${\rm Div\ } {{{\mathcal{H}}}}_{{\mathfrak{p}}}^{\mathcal{O}}$ in loc. cit.; here, ${{{\mathcal{H}}}}_{{\mathfrak{p}}}^{\mathcal{O}}$ denotes a certain set of points, one corresponding to each optimal embedding of ${\mathcal{O}}$ in $R$ (see page 561 of loc. cit. for details). To generalize this construction, for each $g \in C$, we define ${{{\mathcal{H}}}}_{{{\mathfrak{p}}},g}^{\mathcal{O}}$ to be the analogous set of points, which is in bijection with the set of optimal embeddinga of ${\mathcal{O}}$ in $B \cap g \widehat{R}g^{-1}$. Then the homology groups $H_i(\Gamma_0({{\mathfrak{n}}}), {\rm Div\ }{{{\mathcal{H}}}}_{{\mathfrak{p}}}^{\mathcal{O}})$ get replaced by $\oplus_{g \in C} H_i(\Gamma_g, {\rm Div\ }{{{\mathcal{H}}}}_{{{\mathfrak{p}}},g}^{\mathcal{O}})$. On the group cohomology side, Greenberg considers cohomology groups of the groups $\Gamma_0({{\mathfrak{n}}})$ and $\Gamma_0({{\mathfrak{p}}}{{\mathfrak{n}}})$. The cohomology groups of $\Gamma_0({{\mathfrak{n}}})$ again get replaced by the direct sum as $g \in C$ of the cohomology groups of $\Gamma_g$, and the cohomology groups of $\Gamma_0({{\mathfrak{p}}}{{\mathfrak{n}}})$ get replaced similarly by taking $R = R_0({{\mathfrak{p}}}{{\mathfrak{n}}})$ (these replacements are especially needed to have the analog of Corollary 14(2) of [@greenberg], where the narrow class number assumption was used implicitly). As usual, let $K_{{\mathfrak{p}}}$ denote the completion of $K$ at ${{\mathfrak{p}}}$. With the changes above, the construction of Greenberg goes through mutatis mutandis to give a point in $E(K_{{\mathfrak{p}}})$ that we again denote $P_{q,b}$ (it is denoted $P_\psi$ in [@greenberg], where $b=1$). Note that while we are using the same notation $P_{q,b}$ as in Section \[sec:gartner\], it should be clear from the context which point we mean depending on which construction is used. We remark that for his construction, Greenberg assumes an analog of the conjecture of Mazur-Tate-Teitelbaum (conjecture 2 on p. 570 of loc. cit.), and we have to do the same. Just as in [@greenberg], one has an action of ${{\rm Pic}}({\mathcal{O}})^+$ on optimal embeddings of ${\mathcal{O}}$ in $b \widehat{R}^\times b^{-1}$, and thus on $P_{q,b}$; we denote this action by $*$ again. \[conj:greenberg\] The point $P_{q,b}$ is defined over the narrow ring class field extension $K_{\mathcal{O}}^+$ of $K$ associated to the order ${\mathcal{O}}$. If $\alpha \in {{\rm Pic}}({\mathcal{O}})^+$, then ${{\rm rec}}(\alpha)(P_{q,b}) = \alpha * P_{q,b}$, where ${{\rm rec}}: {{\rm Pic}}^+ {\mathcal{O}}{\rightarrow}{\rm Gal}(K_{\mathcal{O}}^+/K)$ is the reciprocity isomorphism of class field theory, as before. Note the similarity of the conjecture above to Conjecture \[conj:gartner\]. Choosing a suitable quaternion algebra {#shdetails} ====================================== Let G1 denote the set of assumptions A, B1, B2, B3, and B4, and let G2 denote the set of assumptions A, C1, C2, C3, and C4. If G1 is satisfies, we can carry out the construction of Gartner (as described in Section \[sec:gartner\]); if G2 holds, the construction of Greenberg (as described in Section \[sec:greenberg\]) works. \[thm:main\] (i) Suppose that $N$ is square-free. If either G1 or G2 hold, then the sign in the functional equation of $E$ over $K$ is $-1$.\ (ii) Suppose that the sign in the functional equation of $E$ over $K$ is $-1$ and the part of $N$ divisible by primes that are inert in $K$ is square-free. Then:\ (a) If there is an archimedian place of $F$ that is inert in $K$ (i.e., $K$ is not totally real), then one can find a quaternion algebra $B$ and an Eichler order $R$ such G1 holds, i.e., one can carry out the construction of Gartner (as described in Section \[sec:gartner\]; assuming the conjectures made in the construction).\ (b) If there is a prime dividing $N$ that is inert in $K$, then one can find a quaternion algebra $B$ and an Eichler order $R$ such G2 holds, i.e., one can carry out the construction of Greenberg (as described in Section \[sec:greenberg\]; assuming the conjectures made in the construction).\ (c) One can find a quaternion algebra $B$ and an Eichler order $R$ such that either G1 or G2 hold, i.e., one can carry out either the construction of Gartner (as described in Section \[sec:gartner\]) or the construction of Greenberg (as described in Section \[sec:greenberg\]) to construct a Darmon point (assuming the conjectures made in the constructions). Note that if the sign in the functional equation is $-1$, then the Birch and Swinnerton-Dyer conjecture predicts that $\text{rank } E(K)\ge 1$. If the rank is exactly 1, the Gross-Zagier formula would lead one to expect that (the trace to $E(K)$ of) the Darmon point has infinite order. In the rest of this section, we shall prove Theorem \[thm:main\]. To carry out the construction of Gartner or Greenberg, we need to find a suitable quaternion algebra $B$ and an Eichler order $R$ so that all the assumptions made in the construction are satisfied. We first list the restrictions, and then show when they can be met. The requirements are as follows:\ (i) Suppose that Assumption A holds: there is an embedding of $K$ in $B$. This happens if and only if each ramified place of $B$ is inert in $K$. Let ${{r_{\scriptscriptstyle{B}}}}$ denote the number of real places of $F$ where $B$ ramifies and ${{r_{\scriptscriptstyle{K}}}}$ the number of real places where $B$ is split but that are inert in $K$. The subscript thus indicates which of $B$ and $K$ is non-split, with the understanding that we write ${{r_{\scriptscriptstyle{B}}}}$ instead of $r_{\scriptscriptstyle{B,K}}$ since $B$ being non-split implies $K$ is non-split.\ (ii) Given $K$, the quantity ${{r_{\scriptscriptstyle{B}}}}+ {{r_{\scriptscriptstyle{K}}}}$ is decided, since it is the number of real places of $F$ that are not split in $K$.\ (iii) In Gartner’s construction, Assumption B1 says that ${{r_{\scriptscriptstyle{K}}}}= 1$, while for Greenberg’s construction, Assumption C1 says that ${{r_{\scriptscriptstyle{K}}}}= 0$ (see the statements just before the statement of Assumption C1).\ In either construction, one needs an Eichler order $R\subset B$ in order to apply the Jacquet-Langlands correspondence. Let $N^-$ denote the (finite part of the) discriminant of $B$, which is square-free by definition. Let $N^+$ and $N'$ be ideals of ${\mathcal{O}}_F$ such that $N^-$, $N'$ and $N^+$ are pairwise relatively prime. Let $R$ be the Eichler order of $B$ of level $N^+N'$, and put $\Gamma_0^B(N^+N')={\rm ker}(n: R^\times {\rightarrow}F^\times)$. The Jacquet-Langlands correspondence then says that $S_2(\Gamma_0^{B}(N^+ N'))=S_2(\Gamma_0(N^+ N'N^-))^{N^- -new}$ as modules over the Hecke algebra $\mathbb T={{\bf{C}}}[\{T_\ell\}_{\ell\nmid N^+ N' N^-}, \{U_p\}_{p|N^+}]$ (the indices $\ell$ and $p$ are ideals in $F$). The form associated to $E$ is in $S_2(\Gamma_0(N))$, so we get the following conditions on the level of the modular form and the discriminant of $B$:\ (iv) There is a factorization $N=N^+N'N^-$ into three pairwise coprime ideals. Here $N^-$ is square-free and divisible only by primes which are inert in $K$ (as $N^-$ is to serve as the discriminant ideal of $B$, the second assumption is necessary to satisfy (i)). In Greenberg’s construction, $N^-$ is the part of $N$ divisible by primes other than ${{\mathfrak{p}}}$ that are inert in $K$, and we take $N'={{\mathfrak{p}}}$. Such a factorization exists by Assumption C3. In Gartner’s construction, $N^-$ is the product of all prime divisors of $N$ that are inert in $K$, and $N'=1$.\ (v) By (i) and (iv), one sees that the (finite) primes where $B$ is allowed to ramify divide $N$. Let ${{f_{\scriptscriptstyle{B}}}}$ denote the number of primes dividing $N$ where $B$ is ramified and ${{f_{\scriptscriptstyle{K}}}}$ the number of primes dividing $N$ where $B$ splits but that are inert in $K$. Similar to (ii), given $K$ and $N$, the quantity ${{f_{\scriptscriptstyle{B}}}}+ {{f_{\scriptscriptstyle{K}}}}$ is independent of $B$, since it is the number of primes of $F$ dividing $N$ that are inert in $K$ (recall that we are assuming that $N$ is coprime to the discriminant of $K$, so no prime dividing $N$ ramifies in $K$).\ (vi) In Gartner’s construction, by the extra Assumption B4, ${{f_{\scriptscriptstyle{K}}}}= 0$, while in Greenberg’s construction, ${{f_{\scriptscriptstyle{K}}}}= 1$ by Assumptions C2 and C3.\ (vii) The total number of places where $B$ ramifies is even, so ${{r_{\scriptscriptstyle{B}}}}+ {{f_{\scriptscriptstyle{B}}}}$ has to be even. And conversely, if ${{r_{\scriptscriptstyle{B}}}}+ {{f_{\scriptscriptstyle{B}}}}$ is even then a $B$ exists (ignoring the other conditions).\ (viii) Let $R_b = B \cap b \widehat{R} b^{-1}$. One needs the existence of an optimal embedding ${\mathcal{O}}\hookrightarrow R_b$ (Assumption B3 for Gartner’s construction and Assumption C4 in Greenberg’s construction). Such an embedding exists if and only if it exists everywhere locally ([@vig] III.5.11), which happens if and only if all the primes dividing $N^-$ are inert in $K$ ([@vig] II.1.9), and all the primes dividing $N^+$ are split in $K$ ([@vig] sentence after II.3.2). In (iv), we already had the requirement that all the primes dividing $N^-$ are inert, so the only new requirement is that all the primes dividing $N^+$ are split. This requirement is already met in Greenberg’s construction (see (iv) and Assumption C3).\ We now prove part (i) of Theorem \[thm:main\]. Combining (iii) and (vi), in either construction, ${{r_{\scriptscriptstyle{K}}}}+ {{f_{\scriptscriptstyle{K}}}}=1$, which combined with (vii) implies that ${{r_{\scriptscriptstyle{B}}}}+ {{r_{\scriptscriptstyle{K}}}}+ {{f_{\scriptscriptstyle{B}}}}+ {{f_{\scriptscriptstyle{K}}}}$ is odd. But ${{r_{\scriptscriptstyle{B}}}}+ {{r_{\scriptscriptstyle{K}}}}+ {{f_{\scriptscriptstyle{B}}}}+ {{f_{\scriptscriptstyle{K}}}}$ is precisely the total number of places of $K$ where $E$ has a Weierstrass or Tate parametrization, which in turn is the exponent of $-1$ in the sign of the functional equation of the $L$-function of $E$. Thus the sign in the functional equation has to be $-1$. This proves part (i) of Theorem \[thm:main\]. We next prove part (ii) of Theorem \[thm:main\]. We shall give two proofs of part (c). In the first proof, we first try to see if the assumptions for Gartner’s construction are satisfied, and if not, we show that the assumptions for Greenberg’s construction hold. In the second proof, we reverse the process: we first try to satisfy the assumptions for Greenberg’s construction, and if we can’t, we show that the assumptions for Gartner’s construction are satisfied. In the process of proving part (c), we will prove parts (a) and (b). (Proof 1 of part (c) and proof of part (a)) We first show that if $K$ is not totally real, then we can apply Gartner’s construction; this will prove part (a). We start with the set of all $B$’s and $R$’s and we will impose restrictions on this set to satisfy (i)–(viii) (for Gartner’s construction). The main point is that as we impose the restrictions one by one, at each stage, there should be a choice of $B$ and $R$ left. Most of the restrictions in (i)–(viii) are about ramification of $B$ at various places. Now a quaternion algebra with specified ramifications at different places exists if and only if the number of places where it is ramified is even, which is condition (vii). We impose (i), and since (vii) can be satisfied while (i) holds, we have quaternion algebras $B$ satisfying (i) (this sort of argument will be used over and over again below, so we will not repeat the justification we gave in this sentence). Now the number of real places of $F$ that are inert in $K$ is ${{r_{\scriptscriptstyle{B}}}}+ {{r_{\scriptscriptstyle{K}}}}$, and so ${{r_{\scriptscriptstyle{B}}}}+ {{r_{\scriptscriptstyle{K}}}}$ is non-zero by our hypothesis. While ${{r_{\scriptscriptstyle{B}}}}+ {{r_{\scriptscriptstyle{K}}}}$ is decided by (ii) (independent of the $B$’s), we can restrict to $B$’s such that ${{r_{\scriptscriptstyle{K}}}}= 1$ (so that (iii) is satisfied) and let ${{r_{\scriptscriptstyle{B}}}}$ be decided by (ii). Next we restrict to the $B$’s for which ${{f_{\scriptscriptstyle{K}}}}= 0$ (so that (vi) is satisfied) and let ${{f_{\scriptscriptstyle{B}}}}$ be whatever it has to be according to (v); however, at this point, we have to check (vii): we cannot choose ${{r_{\scriptscriptstyle{B}}}}$ and ${{f_{\scriptscriptstyle{B}}}}$ both freely since their sum has to be even. Now ${{r_{\scriptscriptstyle{K}}}}+ {{r_{\scriptscriptstyle{B}}}}+ {{f_{\scriptscriptstyle{K}}}}+ {{f_{\scriptscriptstyle{B}}}}= 1 + {{r_{\scriptscriptstyle{B}}}}+ 0 + {{f_{\scriptscriptstyle{B}}}}$ is odd (this parity depends only on the sign of the functional equation of $L(E_{/K},s)$, hence is independent of the $B$’s), so ${{r_{\scriptscriptstyle{B}}}}+ {{f_{\scriptscriptstyle{B}}}}$ is even, and (vii) is satisfied, and we are OK. Thus we can take $N^-$ to be the part of $N$ divisible by primes that are inert in $K$ and restrict to those $B$’s for which the nonarchimedian places where $B$ ramifies are precisely the ones dividing $N^-$ (here we are using the hypothesis that the part of $N$ divisible by primes that are inert in $K$ is square-free). We take $N^+ = N/N^-$, so that (iv) and (viii) are satisfied (note that $N' = 1$) and choose an order $R$ of level $N^+$. Thus we can find a $B$ and an $R$ for which (i)–(viii) are satisfied for Gartner’s construction. If $K$ is totally real, then we claim that we can apply Greenberg’s construction. We start with the set of all $B$’s and $R$’s and impose (i). By (ii), ${{r_{\scriptscriptstyle{K}}}}+ {{r_{\scriptscriptstyle{B}}}}= 0$, so ${{r_{\scriptscriptstyle{K}}}}= {{r_{\scriptscriptstyle{B}}}}= 0$, and (iii) is satisfied (for Greenberg’s construction). Now ${{f_{\scriptscriptstyle{K}}}}+ {{f_{\scriptscriptstyle{B}}}}= {{r_{\scriptscriptstyle{K}}}}+ {{r_{\scriptscriptstyle{B}}}}+ {{f_{\scriptscriptstyle{K}}}}+ {{f_{\scriptscriptstyle{B}}}}$ is odd, and in particular, non-zero. So we may restrict to $B$’s such that ${{f_{\scriptscriptstyle{K}}}}= 1$ (then (vi) is satisfied) and let ${{f_{\scriptscriptstyle{B}}}}$ be whatever it needs to be to satisfy (v) (${{f_{\scriptscriptstyle{B}}}}$ will be even). Again, at this point, we have to check that there are $B$’s left satisfying the conditions above since by (vii), ${{r_{\scriptscriptstyle{B}}}}+ {{f_{\scriptscriptstyle{B}}}}$ has to be even; but this is true since ${{r_{\scriptscriptstyle{B}}}}=0$ and ${{f_{\scriptscriptstyle{B}}}}$ is even as mentioned above. Thus we take a prime ${{\mathfrak{p}}}$ that divides $N$ and is inert in $K$, and let $N^-$ be the product of all primes except ${{\mathfrak{p}}}$ that divide $N$ and are inert in $K$ (here we are using the hypothesis that the part of $N$ divisible by primes that are inert in $K$ is square-free). We take $N' = {{\mathfrak{p}}}$, $N^+ = N/ (N^- {{\mathfrak{p}}})$ so that (viii) is satisfied. Also, as mentioned above, (iv) is automatic for Greenberg’s construction. Thus we can find a $B$ and an $R$ for which (i)–(viii) are satisfied for Greenberg’s construction. (Proof 2 of part (c) and proof of part (b)) We first show that if $N$ is divisible by a prime that is inert in $K$, then we can apply Greenberg’s construction; this will prove part (b). As in Proof 1, we start with the set of all $B$’s and $R$’s and we impose restrictions on this set to satisfy (i)–(viii) (for Greenberg’s construction). The main point is that as we impose the restrictions one by one, at each stage, there should be a choice of $B$ and $R$ left. Most of the restrictions in (i)–(viii) are about ramification of $B$ at various places. Now a quaternion algebra with specified ramifications exists if and only if the number of places where it is ramified is even, which is condition (vii). We impose (i), and since (vii) can be satisfied while (i) holds, we have quaternion algebras $B$ satisfying (i) We pick a prime ${{\mathfrak{p}}}$ such that ${{\mathfrak{p}}}| N$ and ${{\mathfrak{p}}}$ is inert in $K$. We restrict to the $B$’s such that the (finite) primes where $B$ is ramified is precisely the set of primes except ${{\mathfrak{p}}}$ that divide $N$ and are inert in $K$. Thus $N^-$ is the product of all primes except for ${{\mathfrak{p}}}$ that divide $N$ and are inert in $K$. (so (v) is satisfied). We take $N^+ = N/ N^- {{\mathfrak{p}}}$ and $N' = {{\mathfrak{p}}}$; then (iv), (vi), and (viii) are satisfied (here we are using the hypothesis that the part of $N$ that is divisible by primes that are inert in $K$ is square-free). We further restrict to the $B$’s such that ${{r_{\scriptscriptstyle{K}}}}= 0$ (so that (iii) is satisfied) and let ${{r_{\scriptscriptstyle{B}}}}$ be decided by (ii); however, at this point, we have to check (vii): we cannot choose ${{r_{\scriptscriptstyle{B}}}}$ and ${{f_{\scriptscriptstyle{B}}}}$ both freely since their sum has to be even. Now ${{r_{\scriptscriptstyle{K}}}}+ {{r_{\scriptscriptstyle{B}}}}+ {{f_{\scriptscriptstyle{K}}}}+ {{f_{\scriptscriptstyle{B}}}}= 0 + {{r_{\scriptscriptstyle{B}}}}+ 1 + {{f_{\scriptscriptstyle{B}}}}$ is odd (this parity depends only on $E$ and $K$, and is independent of the $B$’s), so ${{r_{\scriptscriptstyle{B}}}}+ {{f_{\scriptscriptstyle{B}}}}$ is even, and (vii) is satisfied. Thus (i)–(viii) are satisfied for Greenberg’s construction. If the part of $N$ divisible by primes that are inert in $K$ is empty, then by (v), ${{f_{\scriptscriptstyle{K}}}}= {{f_{\scriptscriptstyle{B}}}}= 0$ (so (vi) is satisfied for Gartner’s construction), and we restrict to $B$’s that are not ramified at any (finite) prime (so $N^- = 1$). Now ${{r_{\scriptscriptstyle{K}}}}+ {{r_{\scriptscriptstyle{B}}}}= {{r_{\scriptscriptstyle{K}}}}+ {{r_{\scriptscriptstyle{B}}}}+ {{f_{\scriptscriptstyle{K}}}}+ {{f_{\scriptscriptstyle{B}}}}$ is odd by hypothesis, hence non-zero. We restrict to $B$’s such that ${{r_{\scriptscriptstyle{K}}}}= 1$ (so that (iii) is satisfied) and let ${{r_{\scriptscriptstyle{B}}}}$ be decided by (ii) (${{r_{\scriptscriptstyle{B}}}}$ will be even). Again, at this point, we have to check that there are $B$’s left satisfying the conditions above since by (vii), ${{r_{\scriptscriptstyle{B}}}}+ {{f_{\scriptscriptstyle{B}}}}$ has to be even; but this is true since ${{f_{\scriptscriptstyle{B}}}}=0$ and ${{r_{\scriptscriptstyle{B}}}}$ is even as mentioned above. We then take $N^- = N' = 1$, and $N^+ = N$, so that (iv) and (viii) are satified. Thus (i)–(viii) are satisfied for Gartner’s construction. \[rmk:assn\] (i) We made Asssumption B4 in Gartner’s construction (Section \[sec:gartner\]) in order to get part (i) of Theorem \[thm:main\]. Also, this assumption is a natural choice to be made in the construction anyway. If Assumption B4 is dropped, then part (ii) of Theorem \[thm:main\] is still true, and in particular, if the sign in the functional equation is $-1$ and the part of $N$ divisible by primes that are inert in $K$ is square-free, then one can carry out either the construction of Gartner or the construction of Greenberg to construct a Darmon point.\ (ii) There were choices for the quaternion algebra $B$ and the Eichler order $R$ in what we did above for the proof of part (ii) and there may be other ways of applying Greenberg’s or Gartner’s constructions than what we did. Also, if the sign in the functional equation is $-1$ and the part of $N$ divisible by primes that are inert in $K$ is square-free, and there is at least one real place and one prime dividing $N$ that are inert in $K$, then either of Greenberg’s or Gartner’s constructions can be carried out (by the first paragraphs of Proofs 1 and 2). It would be interesting to see if and how the Darmon points one gets by different choices (when available) are related. [Gar11]{} Henri Darmon, *Integration on [$H_p\times H$]{} and arithmetic applications*, Ann. of Math. (2) **154** (2001), no. 3, 589–639. [MR ]{}[1884617 (2003j:11067)]{} [to3em]{}, *Rational points on modular elliptic curves*, CBMS Regional Conference Series in Mathematics, vol. 101, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2004. [MR ]{}[2020572 (2004k:11103)]{} Henri Darmon and Adam Logan, *Periods of [H]{}ilbert modular forms and rational points on elliptic curves*, Int. Math. Res. Not. (2003), no. 40, 2153–2180. [MR ]{}[1997296 (2005f:11110)]{} J. Gartner, *Points de [D]{}armon et varietes de [S]{}himura*, Ph.D. thesis, Paris 6 (2011), available at\ [http://jerome.gaertner.free.fr/]{}. Matthew Greenberg, *Stark-[H]{}eegner points and the cohomology of quaternionic [S]{}himura varieties*, Duke Math. J. **147** (2009), no. 3, 541–575. [MR ]{}[2510743 (2010f:11097)]{} Yoz[ô]{} Matsushima and Goro Shimura, *On the cohomology groups attached to certain vector valued differential forms on the product of the upper half planes*, Ann. of Math. (2) **78** (1963), 417–449. [MR ]{}[0155340 (27 \#5274)]{} Marie-France Vign[é]{}ras, *Arithmétique des algèbres de quaternions*, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980. [MR ]{}[580949 (82i:12016)]{} Hiroyuki Yoshida, *On the zeta functions of [S]{}himura varieties and periods of [H]{}ilbert modular forms*, Duke Math. J. **75** (1994), no. 1, 121–191. [MR ]{}[1284818 (95d:11059)]{} Shouwu Zhang, *Heights of [H]{}eegner points on [S]{}himura curves*, Ann. of Math. (2) **153** (2001), no. 1, 27–147. [MR ]{}[1826411 (2002g:11081)]{}

--- abstract: 'This work offers a broad perspective on probabilistic modeling and inference in light of recent advances in *probabilistic programming*, in which models are formally expressed in Turing-complete programming languages. We consider a typical workflow and how probabilistic programming languages can help to automate this workflow, especially in the matching of models with inference methods. We focus on two properties of a model that are critical in this matching: its *structure*—the conditional dependencies between random variables—and its *form*—the precise mathematical definition of those dependencies. While the structure and form of a probabilistic model are often fixed *a priori*, it is a curiosity of probabilistic programming that they need not be, and may instead vary according to random choices made during program execution. We introduce a formal description of models expressed as programs, and discuss some of the ways in which probabilistic programming languages can reveal the structure and form of these, in order to tailor inference methods. We demonstrate the ideas with a new probabilistic programming language called *Birch*, with a multiple object tracking example.' author: - | Lawrence M. Murray\ Uppsala University - | Thomas B. Schön\ Uppsala University bibliography: - 'birch.bib' title: 'Automated learning with a probabilistic programming language: Birch' --- Introduction ============ Probabilistic approaches have become standard in system identification, machine learning and statistics, particularly in situations where the quantification of uncertainty or assessment of risk is paramount. A typical workflow proceeds through several stages, from experimental design, to data collection, to model development, to prior elicitation, to inference, to decision making. At least part of this workflow involves computer code. For the inference stage, this is often bespoke code tailored to a particular study: it couples the implementation of the model with the implementation of the chosen inference method. The model code is not easy to reuse with a different method, nor the method code with a different model. As data size and compute capacity increase, the complexity of models, and their implementations, increases too. Complex models arise in numerous fields, including nonparametrics, where there is an unbounded number of variables, in object tracking and phylogenetics, where data structures such as random finite sets and random trees appear, and in numerical weather prediction and oceanography, where specialized numerical methods are used for continuous-time systems and partial differential equations. For such complex models, bespoke implementations may involve nontrivial—even tedious—manual work, such as deriving the full conditionals of the posterior distribution, or calculating the gradients of a complex likelihood function, or tuning numerical methods for stability. The effort may need repeating if the model or inference method is later changed. A more scalable approach to implementation is desirable. Recognizing this, there has been a tradition of software that separates model specification from method implementation for the purposes of inference (e.g. WinBUGS [@Lunn2000], OpenBUGS [@Lunn2012], JAGS [@JAGS; @Plummer2003], Stan [@STAN], Infer.NET [@InferNET18], LibBi [@Murray2015], Biips [@Todeschini2014]). Typically, this software supports one predominant method but many possible models. The methods include Markov chain Monte Carlo (MCMC) methods such as the Gibbs sampler for WinBUGS, OpenBUGS and JAGS, and Hamiltonian Monte Carlo (HMC) for Stan, variational methods for Infer.NET, and Sequential Monte Carlo (SMC) methods for LibBi and Biips. The software provides a way to adapt the method for a large number of models and automate routine procedures, such as adaptation of Markov kernels in WinBUGS [@Spiegelhalter2003 p. 6], and automatic differentiation in Stan. It is typical for these languages to restrict the set of probabilistic models that can be expressed, in order to provide an inference method that works well for this restricted set. Stan, for example, works only with differentiable models using HMC, while LibBi works only with state-space models using SMC-based methods. Models outside of these sets may require more specialist tools. In phylogenetics, for example, RevBayes [@Hoehna2016] provides the particular modeling feature of *tree plates* to represent phylogenetic trees, for which specialized Markov kernels can be applied within MCMC. Naturally, methods have also become more complex to accommodate these more complex models and larger data sets. Modern Monte Carlo methods often nest multiple baseline algorithms, such as SMC within MCMC, as in particle MCMC [@Andrieu2010], or SMC within SMC, as in SMC$^{2}$ [@Chopin2013]. Data subsampling-based algorithms [@Bardenet2017] are becoming standard for dealing with large data sets. Monte Carlo samplers increasingly use gradient information, such as the Metropolis-adjusted Langevin algorithm (MALA) [@Roberts1998], HMC [@Neal2011a], and deterministic piecewise samplers [@Bouchard-Cote2017; @Bierkens2016; @Vanetti2017]. Various methods manipulate the stream of random numbers (e.g. [@Murray2013a; @Gerber2015; @Deligiannidis2016]) or potential functions (e.g. [@Whiteley2014; @DelMoral2015]) to improve estimates. Software has begun to address this complexity in methods, too. NIMBLE [@deValpine2017], for example, uses models similar to those of WinBUGS, OpenBUGS and JAGS, but provides manual customization of the Markov kernels used within MCMC. We see value in flexible tools that allow for the implementation of both complex models and complex methods, and in moving from one-to-many tools (one method, many models) to many-to-many tools (many methods, many models). To this end, we consider the potential of *probabilistic programming*: a programming paradigm that aims to accelerate workflow with new programming languages and software tools tailored for probabilistic modeling and inference. In particular, it aims to develop Turing-complete programming languages for model implementation, extending existing languages for model specification with programming concepts such as conditionals, recursion (loops), and higher-order functions, for greater expressivity. It aims to *decouple* the implementation of models and methods into modular components that can be reassembled and reused in multiple configurations in a many-to-many manner. It aims to automate the selection of an inference method for a given model, and to automate the tuning necessary for it to work well in practice. These goals remain aspirational, and an active area of research across disciplines including machine learning, statistics, system identification, artificial intelligence and programming languages. A number of probabilistic programming languages have been developed with such aims in recent years. Examples include Church [@Goodman2008], BLOG [@Milch2007], Venture [@Mansinghka2014], WebPPL [@Goodman2014], Anglican [@Tolpin2016], Figaro [@Pfeffer2016], Turing [@Ge2018], Edward [@Tran2016], and Pyro [@Pyro]. These all explore different approaches that reflect, in the first instance, the different problem domains to which they are orientated, and in the second instance, the relatively young age of the field. All of these languages are considered *universal* probabilistic programming languages—i.e. Turing-complete programming languages that admit arbitrary models rather than restricted sets. This is not to say, of course, that efficient inference is possible for all models that are admitted—but these languages can work well for a large class of models for which they do have efficient inference methods, they do provide useful libraries for implementing probabilistic models, and they may support the development or customization of inference methods from within the same language. We introduce a new universal probabilistic programming language called *Birch* ([www.birch-lang.org](www.birch-lang.org)), which implements the ideas presented in this work. Birch is an imperative language geared toward object-oriented and generic programming paradigms. It draws inspiration from several sources, notably from LibBi [@Murray2015]—for which it is something of a successor—but in moving from model specification language to universal probabilistic programming language it draws ideas from modern object-oriented programming languages such as Swift, too. Birch is Turing complete, with control flow statements such as conditionals and loops, support for unbounded recursion and higher-order functions, and dynamic memory management. Birch code compiles to C++14 code, providing ready access to the established ecosystem of C/C++ libraries available for scientific and numeric computing. A key component of Birch is its implementation of delayed sampling [@Murray2018], a heuristic to provide optimizations via partial analytical solutions to inference problems. While broadly applicable across problem domains, invariably the approach taken in Birch is flavored by the perspective of its developers, and so by applications in statistics, machine learning and system identification. This work is intended as a “big picture” perspective on the probabilistic workflow, and how new ideas in probabilistic programming can assist this. Birch is the concrete manifestation of these ideas. Throughout, we make use of the state-space model as a running example. While the ideas presented are not restricted to such models, they concretely illustrate some of the core concepts, and have numerous practical applications. In Section \[sec:models\] we introduce a formal description of the class of models considered in probabilistic programming. In Section \[sec:methods\] we introduce some methods of inference and consider some of the ways in which probabilistic programming languages can automate the many-to-many matching of models with inference methods. In Section \[sec:demonstration\] we introduce the Birch probabilistic programming language as a specific implementation of these ideas. In Section \[sec:worked-example\] we work through the concrete example of a multiple object tracking model—a state-space model with random size—and show how it is implemented in Birch, and the inference results obtained. Finally, we summarize in Section \[sec:discussion\]. Models expressed in a programming language\[sec:models\] ======================================================== Probabilistic programming considers models expressed in Turing-complete programming languages. Such models are usually referred to as *probabilistic programs*—qualified to *universal probabilistic programs* when one wishes to regard only the broadest class in terms of expressivity—and described in programming language nomenclature. Here, we adopt probabilistic nomenclature instead, to provide a more accessible treatment for the intended audience. Taking the lead from the term *graphical model*—a model expressed in a graphical language—we suggest that the term *programmatic model*—a model expressed in a programming language—might be more appropriate for this audience, and adopt this term throughout. Specifically, we avoid the use of the term *program* when referring only to a model implementation, as in ordinary usage one thinks of a computer program as combining the implementation of both a model and an inference method, which can cause confusion. The term can also be misleading given unrelated but similarly-named concepts in system identification, such as linear programs and stochastic programs. We follow the statistics convention of using uppercase letters to denote random variables (e.g. $V$) and lowercase letters to denote instantiations of them (e.g. $v$), with $v\in\mathbb{V}$. We then adopt measure theory notation to clearly distinguish between distributions (which we will ultimately simulate) and likelihood functions (which we will ultimately evaluate): the distribution of a random variable $V$ is denoted $p(\mathrm{d}v)$, while evaluation of an associated probability density function (pdf, for continuous-valued random variables) or probability mass function (pmf, for discrete-valued random variables) is denoted $p(v)$. Assume that we have a countably infinite set of random variables $\{V_{k}\}_{k=1}^{\infty}$, with a joint probability distribution over them, which has been implemented in code in some programming language. The only stochasticity available to the code is via these random variables. We execute the code, and as it runs it encounters a finite subset of the random variables in some order determined by that code. Denote this order by a permutation $\sigma$, with its (random) length denoted $|\sigma|$, defining a sequence $(V_{\sigma[k]})_{k=1}^{|\sigma|}$. The first element, $\sigma[1]$, is always the same. Each subsequent element, $\sigma[k]$, is given by a function of the random variables encountered so far, denoted $\mathrm{Ne}$ (for *next*), so that $\sigma[k]=\mathrm{Ne}(v_{\sigma[1]},\ldots,v_{\sigma[k-1]})$. This function $\mathrm{Ne}$ is implied by the code. Note that $\mathrm{Ne}$ is a deterministic function given preceding random variates, as there is no stochasticity available to the code except via these. This is also why $\sigma[1]$ is always the same: no source of stochasticity precedes it. As each random variable $V_{\sigma[k]}$ is encountered, the code associates it with a distribution $$V_{\sigma[k]}\sim p_{\sigma[k]}\left(\mathrm{d}v_{\sigma[k]}\mid\mathrm{Pa}(v_{\sigma[1]},\ldots,v_{\sigma[k-1]})\right),$$ where $\mathrm{Pa}$ (for *parents*) is a deterministic function of the preceding random variates, selecting from them a subset on which the distribution of $V_{\sigma[k]}$ depends. It is possible that the distribution $p_{\sigma[k]}$ also depends on exogenous factors such as user input; we leave this implicit to simplify notation. At some point the execution terminates, having established the distribution $$p_{\sigma}(\mathrm{d}v_{\sigma[1]},\ldots,\mathrm{d}v_{\sigma[|\sigma|]})=\prod_{k=1}^{|\sigma|}p_{\sigma[k]}\left(\mathrm{d}v_{\sigma[k]}\mid\mathrm{Pa}(v_{\sigma[1]},\ldots,v_{\sigma[k-1]})\right).$$ We will execute the code several times. The $n$th execution will be associated with the distribution $p_{\sigma_{n}}$, given by $$p_{\sigma_{n}}(\mathrm{d}v_{\sigma_{n}[1]},\ldots,\mathrm{d}v_{\sigma_{n}[|\sigma_{n}|]})=\prod_{k=1}^{|\sigma_{n}|}p_{\sigma_{n}[k]}\left(\mathrm{d}v_{\sigma_{n}[k]}\mid\mathrm{Pa}(v_{\sigma_{n}[1]},\ldots,v_{\sigma_{n}[k-1]})\right),$$ with $\sigma_{n}[k]=\mathrm{Ne}(v_{\sigma_{n}[1]},\ldots,v_{\sigma_{n}[k-1]})$. Subscript $n$ is used to denote execution-dependent variables. For different executions $n$ and $m$, it is possible for the number of random variables encountered ($|\sigma_{n}|$ and $|\sigma_{m}|$) to differ, for the sequences of random variables $(V_{\sigma_{n}[k]})_{k=1}^{|\sigma_{n}|}$ and $(V_{\sigma_{m}[k]})_{k=1}^{|\sigma_{m}|}$ to differ, and even for the two subsets of random variables $\{V_{\sigma_{n}[k]}\}_{k=2}^{|\sigma_{n}|}$ and $\{V_{\sigma_{m}[k]}\}_{k=2}^{|\sigma_{m}|}$ to be disjoint (recall that the first random variable to be encountered is always the same). In general, we should therefore assume that $p_{\sigma_{n}}$ and $p_{\sigma_{m}}$ are not the same, but rather components of a larger mixture. The above describes the class of models that we refer to as *programmatic models*. The permutation $\sigma_{n}$ reflects the fact that a program can make conditional choices during execution that are based on the simulation of random variables, and that these may lead to very different outcomes. Consider, for example, a model implementation that begins with a coin flip: on heads it executes one model, on tails some other model. Such an implementation represents a mixture of two models, but each execution can encounter only one. We are interested in two properties of a programmatic model from an inference perspective: 1. *structure*, by which we mean the factorization of the joint distribution $p_{\sigma_{n}}$ into conditional distributions $p_{\sigma_{n}[1]},\ldots,p_{\sigma_{n}[|\sigma_{n}|]}$, and 2. *form*, by which we mean the precise mathematical definition of the conditional distributions $p_{\sigma_{n}[1]},\ldots,p_{\sigma_{n}[|\sigma_{n}|]}$. Structure --------- The structure of a probabilistic model is typically defined as the dependency relationships between random variables. Popular model classes such as hidden Markov models (HMMs), state-space models (SSMs), Markov random fields, etc, encode particular structures for specialist purposes such as dynamical systems and spatial systems. Generalizing these, structure is perhaps most explicitly encoded by *graphical models* (see e.g. [@Jordan2004; @Bishop2007; @Koller2009]), where a probabilistic model is represented as a graph, with nodes as random variables, and edges encoding the relationships between them. Generic inference techniques such as the sum-product algorithm [@Pearl1988] make explicit use of the graph—and thus the structure of the model—to perform inference efficiently with respect to the number of computations required. We are interested in the same for programmatic models. For illustration, we can readily compare the class of programmatic models to the class of *directed graphical models*. Like all graphical models, the nodes of a directed graphical model represent random variables. The edges are directed and represent a conditional dependency relationship from a *parent* at the tail of the arrow, to a *child* at the head of the arrow. The entire graph must be acyclic. We can represent directed graphical models as programmatic models within the formal definition above. The nodes and edges are known *a priori* and establish a joint distribution, over $K$ random variables, of: $$p(\mathrm{d}v_{1},\ldots,\mathrm{d}v_{K})=\prod_{k=1}^{K}p_{k}(\mathrm{d}v_{k}\mid\mathrm{pa}_{k}),$$ where we use $\mathrm{pa}_{k}$ to denote the set of parents of $V_{k}$ under the graph. The model implementation in code must necessarily encounter the nodes in a valid topological ordering given the edges. On each execution $n$, the same finite set of random variables $\{V_{k}\}_{k=1}^{K}$ is encountered. A directed edge from $V_{i}$ to $V_{j}$ indicates that, if $V_{j}$ is the $k$th random variable to be encountered, then $V_{i}$ must necessarily have been encountered already, and $v_{i}\in\mathrm{Pa}(v_{\sigma_{n}[1]},\ldots,v_{\sigma_{n}[k-1]})$. The conditional distribution assigned to any $V_{k}$ is always the same $p_{k}(\mathrm{d}v_{k}\mid\mathrm{pa}_{k})$. Consequently, the execution establishes the distribution: $$\begin{aligned} p_{\sigma_{n}}(\mathrm{d}v_{\sigma_{n}[1]},\ldots,\mathrm{d}v_{\sigma_{n}[|\sigma_{n}|]}) & =\prod_{k=1}^{|\sigma_{n}|}p_{\sigma_{n}[k]}\left(\mathrm{d}v_{\sigma_{n}[k]}\mid\mathrm{Pa}(v_{\sigma_{n}[1]},\ldots,v_{\sigma_{n}[k-1]})\right)\nonumber \\ & =\prod_{k=1}^{K}p_{k}(\mathrm{d}v_{k}\mid\mathrm{pa}_{k})\nonumber \\ & =p(\mathrm{d}v_{1},\ldots,\mathrm{d}v_{K}).\label{eq:directed-graphical-model}\end{aligned}$$ This is to say that, while executions may encounter the random variables in different orders, according to how the directed graphical model has been implemented, each execution will always encounter the same finite subset of $K$ variables, and establish the same structure and form. If this is not the case, then the code must not be a correct implementation of the directed graphical model that was given. This motivates the difficulty of dealing with a programmatic model: unlike for a directed graphical model, the structure of a programmatic model is not known *a priori*. If a programmatic model were expressed as a graph, the nodes of the graph would not be known until revealed through the function $\mathrm{Ne}$, and only at that same time would incoming edges to the node be revealed through the function $\mathrm{Pa}$. The model must be executed to discover its structure. But, furthermore, the nodes and edges may differ between executions, so it is not simply a matter of executing the program once to determine the complete structure. We should not get too caught up on the general at the expense of the useful, however. Directed graphical models are useful, and many models used in practice can be expressed this way—it is those that cannot be expressed this way that motivate the more general treatment of programmatic models. Probabilistic models tend not be entangled assemblies of random variables connected in arbitrary ways, but rather arranged in recursive substructures such as chains, grids, stars, trees, and layers. We refer to these as structural *motifs* (see Figure \[fig:motifs\]). These motifs occur so frequently in probabilistic models that there have been attempts to automatically learn model structure based on them [@Ellis2013]. For example, the chain motif is the dominant feature of HMMs and SSMs, the grid motif that of spatial models, the tree motif that of phylogenetic trees or stream networks, the layer motif that of neural networks. The star motif occurs whenever there are repeated observations of some system, for example repeated time series observations of the same dynamical system, where the parameters are identified across time series, or multiple individuals in a medical study sharing common parameters. Parameters with global influence are also a common occurrence, determining the conditional probability distributions over variables in a chain or grid, for example. ![Structural motifs that occur frequently in probabilistic models, each with a global parameter. Clockwise from top-left: star, chain, tree, layer, grid.\[fig:motifs\]](Star){width="90.00000%"} ![Structural motifs that occur frequently in probabilistic models, each with a global parameter. Clockwise from top-left: star, chain, tree, layer, grid.\[fig:motifs\]](Chain){width="90.00000%"} ![Structural motifs that occur frequently in probabilistic models, each with a global parameter. Clockwise from top-left: star, chain, tree, layer, grid.\[fig:motifs\]](Tree){width="90.00000%"} ![Structural motifs that occur frequently in probabilistic models, each with a global parameter. Clockwise from top-left: star, chain, tree, layer, grid.\[fig:motifs\]](Layer){width="90.00000%"} ![Structural motifs that occur frequently in probabilistic models, each with a global parameter. Clockwise from top-left: star, chain, tree, layer, grid.\[fig:motifs\]](Grid){width="90.00000%"} These motifs can be used to characterize a model, especially for the purposes of selecting inference methods. While the nature of a programmatic model is that its structure may change between executions, it may be the case that a particular motif persists between executions, and that this is known *a priori*. If we can characterize this motif, we can leverage specialized inference methods for it, while reserving generalized inference methods for the remainder of the structure. We return to this idea in Section \[sec:demonstration\]. Form ---- The form of a probabilistic model refers to the mathematical definition of its distributions. In the case of programmatic models, this refers to the conditional probability distributions $p_{\sigma_{n}[1]},\ldots,p_{\sigma_{n}[|\sigma_{n}|]}$. In many cases these are common parametric forms such as Gaussian, Poisson, binomial, beta and gamma distributions, with parameters given by the parents of the random variable. Parametric distributions such as these are readily simulated using standard algorithms (see e.g. [@Devroye1986]), and admit either a pdf or pmf that can be evaluated, and perhaps differentiated with respect to its parameters. More difficult forms are those that can be simulated or evaluated but not both. For example, a nonlinear diffusion process defines a distribution $p(\mathrm{d}x(t+\Delta t)\mid x(t))$ that can be simulated (at least numerically), but it may be prohibitively expensive to evaluate the associated pdf $p(x(t+\Delta t)\mid x(t))$ for given values $x(t+\Delta t)$ and $x(t)$, or this may have no closed form. Conversely, the classic Ising model is defined as a product of potentials that readily permits evaluation of the likelihood of any state, but requires expensive iterative computations to simulate. Form may also carry information across structure. For example, where the form of the parents of a random variable is conjugate to the form of that random variable, a conjugate prior relationship is established. In such cases, analytical marginalization and conditioning optimizations may be possible within an inference method. Example ------- We demonstrate how these abstract ideas apply to the concrete case of an SSM. The SSM consists of a latent process $(X_{t})_{t=1}^{T}$, observed process $(Y_{t})_{t=1}^{T}$, and optional parameters $\Theta$. The joint probability distribution is given by: $$p\left(\mathrm{d}y_{1:T},\mathrm{d}x_{1:T},\mathrm{d}\theta\right)=\underbrace{p\left(\mathrm{d}\theta\right)}_{\text{parameter}}\underbrace{p\left(\mathrm{d}x_{1}\mid\theta\right)}_{\text{initial}}\prod_{t=2}^{T}\underbrace{p\left(\mathrm{d}x_{t}\mid x_{t-1},\theta\right)}_{\textrm{transition}}\prod_{t=1}^{T}\underbrace{p\left(\mathrm{d}y_{t}\mid x_{t},\theta\right)}_{\text{observation}},\label{eq:SSMjoint}$$ where we have assigned common names to the various conditional probability distributions that make up this joint. There are alternative representations. In the engineering literature, it is common to represent dependencies between the random variables using deterministic functions plus noise: \[eq:SSMengineering\] $$\begin{aligned} x_{t} & =f(x_{t-1},\theta)+\xi_{t}, & y_{t} & =g(x_{t},\theta)+\zeta_{t}.\end{aligned}$$ Here, the parameter and initial distributions are as per , but now the transition and observation distributions are expressed via the (possibly nonlinear) functions $f$ and $g$, plus independent—often Gaussian—noise terms $\xi_{t}$ and $\zeta_{t}$: $$\begin{aligned} \underbrace{p(\mathrm{d}x_{t}\mid x_{t-1},\theta)}_{\text{transition}} & =p_{\xi_{t}}(\mathrm{d}x_{t}-f(x_{t-1},\theta))\\ \underbrace{p(\mathrm{d}y_{t}\mid x_{t},\theta)}_{\text{observation}} & =p_{\zeta_{t}}(\mathrm{d}y_{t}-g(x_{t},\theta)),\end{aligned}$$ where $p_{\xi_{t}}$ denotes the distribution of the process noise $\xi_{t}$, and $p_{\zeta_{t}}$ the distribution of the observation noise $\zeta_{t}$. This is a mathematical description of the standard SSM. To understand it as a programmatic model, denote the set of random variables as $\{V_{k}\}_{k=1}^{2T+1}=\{\Theta,X_{1},\ldots,X_{T},Y_{1},\ldots,Y_{T}\}$. In this case the set is finite (or, equivalently, the infinite complement of the set is never encountered). An implementation of the model in Birch may look like the following (variable and function declarations have been removed for brevity): ``` {mathescape=""} $\theta$ $\sim$ Uniform(0.0, 1.0); x[1] $\sim$ Gaussian(0.0, 1.0); y[1] $\sim$ Gaussian(g(x[1], $\theta$), 0.1); for t in 2..T { x[t] $\sim$ Gaussian(f(x[t-1], $\theta$), 1.0); y[t] $\sim$ Gaussian(g(x[t], $\theta$), 0.1); } ``` Recall that $\mathrm{Ne}$ is the function that denotes the next random variable to be encountered given the values of those encountered so far, and $\mathrm{Pa}$ is the function that denotes the parents of that random variable, given the same values. If we think through executing the above code line-by-line we see that, for example, $\mathrm{Ne}(\theta,x_{1:t-1},y_{1:t-1})=X_{t}$ and $\mathrm{Pa}(\theta,x_{1:t-1},y_{1:t-1})=\{\theta,x_{t-1}\}$; also $\mathrm{Ne}(\theta,x_{1:t},y_{1:t-1})=Y_{t}$ and $\mathrm{Pa}(\theta,x_{1:t},y_{1:t-1})=\{\theta,x_{t}\}$. For some execution $n$, the order of random variables encountered, $\sigma_{n}$, will be: $$\begin{aligned} \sigma_{n}[1] & =1 & \text{i.e. } & \Theta\\ \sigma_{n}[2] & =2 & \text{i.e. } & X_{1}\\ \sigma_{n}[3] & =T+2 & \text{i.e. } & Y_{1}\\ \sigma_{n}[5] & =3 & \text{i.e. } & X_{2}\\ \sigma_{n}[6] & =T+3 & \text{i.e. } & Y_{2}\\ \ldots\text{etc}\end{aligned}$$ Now consider an alternative implementation: ``` {mathescape=""} $\theta$ $\sim$ Uniform(0.0, 1.0); x[1] $\sim$ Gaussian(0.0, 1.0); for t in 2..T { x[t] $\sim$ Gaussian(f(x[t-1], $\theta$), 1.0); } for t in 1..T { y[t] $\sim$ Gaussian(g(x[t], $\theta$), 0.1); } ``` This expresses precisely the same mathematical model, but in a different programmatic form. When the code is executed, the order in which the random variables are encountered is different to the previous example. We can readily write down the new $\mathrm{Ne}$, and the resulting order is given by the trivial permutation $\sigma_{n}[k]=k$. The function $\mathrm{Pa}$ differs because the permutation does, but it establishes the same parent relationships as before. This is not surprising: an SSM can be represented as a directed graphical model, so that the joint distribution $p_{\sigma_{n}}\left(\mathrm{d}y_{1:T},\mathrm{d}x_{1:T},\mathrm{d}\theta\right)$ associated with each execution $n$ is always the same joint distribution $p\left(\mathrm{d}y_{1:T},\mathrm{d}x_{1:T},\mathrm{d}\theta\right)$ that appears in (\[eq:SSMjoint\]), as explained in (\[eq:directed-graphical-model\]). This is a simple example. One can imagine more complex code that includes conditionals (e.g. `if` statements and `while` loops) that may cause only a subset of the random variables to be encountered on any single execution. The random variables may even be encountered in different orders, or may be countably infinite (rather than finite) in number. SSMs that exhibit this complexity occur in, for example, multiple object tracking. We provide such an example in Section \[sec:worked-example\]. Inference methods for programmatic models\[sec:methods\] ======================================================== We wish to infer the conditional distribution of one set of random variables, given values assigned to some other set. In a Bayesian context, this amounts to inferring the posterior distribution. For this purpose, we partition $\{V_{k}\}_{k=1}^{\infty}$ into two disjoint sets: the *observed* set $O\subseteq\mathcal{P}(\mathbb{N})$ (where $\mathcal{P}$ denotes the power set) containing the indices of all those random variables for which a value has been given, and the *latent* set $L\subseteq\mathcal{P}(\mathbb{N})$ with all other indices, so that $L=\mathbb{N}\setminus O$. We then clamp the observed random variables to have the given values, i.e. $V_{O}=v_{O}$, where we use subscript $O$ to select a subset rather than a single variable. Inference involves computing the posterior distribution: $$\begin{aligned} \underbrace{p(\mathrm{d}v_{L}\mid v_{O})}_{\text{posterior}} & =\frac{\overbrace{p(v_{O}\mid v_{L})}^{\text{likelihood}}\overbrace{p(\mathrm{d}v_{L})}^{\text{prior}}}{\underbrace{p(v_{O})}_{\text{evidence}}},\label{eq:posterior}\end{aligned}$$ which decomposes into likelihood, prior and evidence terms as annotated. Having obtained the posterior distribution, we may be interested in estimating the posterior expectation of some test function of interest, say $h(V_{L})$: $$\mathbb{E}_{p}[h(V_{L})\mid v_{O}]=\int_{\mathbb{V}_{L}}h(v_{L})p(\mathrm{d}v_{L}\mid v_{O}),$$ and subsequently making decisions based on this result. A particular execution $n$ of the model code may encounter some subset of the variables in $O$ and $L$, which we denote $O_{n}$ and $L_{n}$. The distribution that results from the execution is then: $$\begin{aligned} p_{\sigma_{n}}(\mathrm{d}v_{L_{n}}\mid v_{O_{n}})\propto & \prod_{k\in L_{n}}p_{\sigma_{n}[k]}\left(\mathrm{d}v_{\sigma_{n}[k]}\mid\mathrm{Pa}(v_{\sigma_{n}[1]},\ldots,v_{\sigma_{n}[k-1]})\right)\times\label{eq:prior}\\ & \prod_{k\in O_{n}}p_{\sigma_{n}[k]}\left(v_{\sigma_{n}[k]}\mid\mathrm{Pa}(v_{\sigma_{n}[1]},\ldots,v_{\sigma_{n}[k-1]})\right).\label{eq:likelihood}\end{aligned}$$ As before, different executions $m$ and $n$ may yield different distributions, which may be interpreted as different components of a mixture. In this case, this mixture is the posterior distribution. A baseline method for inference is importance sampling from the prior. The model code is executed: when encountering a random variable in $L$ it is simulated from the prior, and when encountering a random variable in $O$ a cumulative weight is updated by multiplying in the likelihood of the given value. We have then simulated from the first product (\[eq:prior\]), and assigned a weight according to the second product (\[eq:likelihood\]). Use of the prior distribution as a proposal is likely to produce estimates with high variance. We can improve upon this in a number of ways, such as by maintaining multiple executions simultaneously and selecting from amongst them in a resampling step, producing a particle filter [@Gordon1993]. The only limitation here is the alignment of resampling points between multiple executions in order that resampling is actually beneficial—each execution may encounter observations in different orders. But because importance sampling and the most basic particle filters—up to alignment—require only forward simulation of the prior and pointwise evaluation of the likelihood, they are unaffected by many of the complexities of programmatic models, and so particularly suitable for inference. For this reason they have become a common choice for inference in probabilistic programming (see e.g. [@Wood2014; @Paige2014a; @Mansinghka2014]). Various optimizations are available, such as attaching alternative proposal distributions $q_{\sigma_{n}[k]}$ to random variables in $L$, or marginalizing out one or more of these. The manual use of these optimizations is well understood (see [@Doucet2011] for a review), although a key ingredient of probabilistic programming is to automate their use (see e.g. [@Perov2015; @Murray2018]). For a tutorial introduction to the use of the particle filter for nonlinear system identification we refer to [@SchonLDWNSD:2015; @SchonSML:2018]. Because the structure of a programmatic model may change between executions, MCMC methods can be difficult to apply. Markov kernels on programmatic models are, in general, transdimensional, so that techniques such as reversible jump [@Green1995] are necessary. There are approaches to automating the design of reversible jump kernels that can work well in practice (see e.g. [@Wingate2011]). Random-walk Metropolis–Hastings kernels and more-recent gradient-based kernels do not support transdimensional moves, but might still be applied to the full conditional distribution of a set of random variables within, for example, a Gibbs sampler. Particular structural motifs may suggest particular inference methods. For example, the chain motif suggests the use of specialized Bayesian filtering methods, while tree and grid motifs are conducive to divide-and-conquer methods [@Lindsten2017]. Within probabilistic programming, recent attempts have been made to match inference methods to model substructures, using both manual and automated techniques (see e.g. [@Mansinghka2014; @Ge2018; @Pfeffer2018; @Mansinghka2018]). The precise choice of inference method depends not only on structure, but also on form. For example, while the chain motif suggests the use of a Bayesian filtering method based on structure, the precise choice of filter depends on form: the Kalman filter for linear-Gaussian forms, the forward-backward algorithm on HMMs for discrete forms, the particle filter otherwise. In all cases the structure is the same, but the form differs. Recognizing this within program code requires compiler or library support. Preferably, the choice of inference method based on structure and form is automated, and ideally by the programming language compiler [@Lunden2018], which has full access to the abstract syntax tree of the model code to inspect structure and form. There are fundamental limits to what can be known at compile time, however. In the general case, at least some structure and form is unknown until the program is run. For example, it is not possible to bound the trip count of a loop at compile time [@Nori2014] if this is a stochastic quantity with unbounded support. The optimal inference method for a problem may also depend on posterior properties, such as correlations between random variables, that—by definition—are unknown *a priori*. In such situations it may be necessary to require manual hints provided by the programmer, or to use dynamic mechanisms that adapt the inference method during execution. The *delayed sampling* [@Murray2018] heuristic is an example of the latter. Delayed sampling works for programmatic models in a similar way to the sum-product algorithm [@Pearl1988] for graphical models. By keeping a graph of relationships between random variables as they are encountered by a program, and delaying the simulation of latent variables for as long as possible, it opens opportunities for analytical optimizations based on conjugate prior and other relationships. This includes analytical conditioning, variable elimination, Rao–Blackwellization, and locally-optimal proposals. It is a heuristic in the sense that it must make myopic decisions based on the current state of the running program, without knowledge of its future execution, so that it may miss potential optimizations. Delayed sampling works through the control flow statements of a Turing-complete programming language, such as conditionals and loops, but does not attempt to marginalize over multiple branches in a single execution. Example ------- For the SSM, the task is to infer the latent variables $X_{1:T}$ and $\Theta$ given observations $Y_{1:T}=y_{1:T}$. In a Bayesian context, this is to infer the posterior distribution $$\begin{aligned} p(\mathrm{d}x_{1:T},\mathrm{d}\theta\mid y_{1:T})=p(\mathrm{d}x_{1:T}\mid\theta,y_{1:T})p(\mathrm{d}\theta\mid y_{1:T}).\label{eq:SSMposterior}\end{aligned}$$ The first factor in  provides information about the states. For $t=1,\ldots,T$, its marginals $p(\mathrm{d}x_{t}\mid\theta,y_{1:T})$ are called the filtering distributions, and are the target of Bayesian filtering methods. The second factor is the target of parameter estimation methods. We may be interested in obtaining the posterior distribution over parameters, or obtaining the maximum likelihood estimate instead by solving the optimization problem $$\begin{aligned} \widehat{\theta}=\mathrm{argmax}_{\theta}\,p(y_{1:T}\mid\theta).\label{eq:SSMml}\end{aligned}$$ In either case the central object of interest is the likelihood $p(y_{1:T}\mid\theta)$. By repeated use of conditional probabilities this can be written $$\begin{aligned} p(y_{1:T}\mid\theta)=\prod_{t=1}^{T}p(y_{t}\mid y_{1:t-1},\theta),\label{eq:SSMlikelihood1}\end{aligned}$$ with the convention that $y_{1:0}=\emptyset$. The terms in the likelihood are recursively computed via marginalization as $$\begin{aligned} p(y_{t}\mid y_{1:t-1},\theta) & =\int p(y_{t}\mid x_{t},\theta)p(x_{t}\mid y_{1:t-1},\theta)\,\mathrm{d}x_{t},\end{aligned}$$ so that we obtain $$\begin{aligned} p(y_{1:T}\mid\theta)=\prod_{t=1}^{T}\int p(y_{t}\mid x_{t},\theta)p(x_{t}\mid y_{1:t-1},\theta)\,\mathrm{d}x_{t}.\label{eq:SSMlikelihood2}\end{aligned}$$ There are numerous ways to compute this likelihood, but for this particular structure, the family of Bayesian filtering methods is ideal. Within this family, the preferred method depends on the form of the model. Recall Code \[code:model\]. In general, the functions $f$ and $g$ must be considered nonlinear, and for such cases the particle filter is the Bayesian filtering method of choice. Now consider the code that results from the particular choice $g(x_{t})=x_{t}$: ``` {mathescape=""} $\theta$ $\sim$ Uniform(0.0, 1.0); x[1] $\sim$ Gaussian(0.0, 1.0); y[1] $\sim$ Gaussian(x[1], 0.1); for t:Integer in 2..10 { x[t] $\sim$ Gaussian(f(x[t-1], $\theta$), 1.0); y[t] $\sim$ Gaussian(x[t], 0.1); } ``` A particle filter can still be used, but it is possible to improve its performance with Rao–Blackwellization and a locally-optimal proposal (see e.g. [@Doucet2011]), by using the local conjugacy between the Gaussian transition and Gaussian observation models. If we make the further choice that $f(x_{t-1},\theta)=\theta x_{t-1}$ we have: ``` {mathescape=""} $\theta$ $\sim$ Uniform(0.0, 1.0); x[1] $\sim$ Gaussian(0.0, 1.0); y[1] $\sim$ Gaussian(x[1], 0.1); for t:Integer in 2..10 { x[t] $\sim$ Gaussian($\theta$*x[t-1], 1.0); y[t] $\sim$ Gaussian(x[t], 0.1); } ``` On inspection, it is clear that this is now a linear-Gaussian SSM. In this case we would like to choose the Kalman filter, which is the optimal Bayesian filtering method for such a form. Note the important distinction here: the *structure* of the model is the same in all cases—that of the SSM—and suggests the use of a Bayesian filtering method over other means to compute the likelihood. But the *form* of the model differs, and it is these various forms that suggests the specific Bayesian filtering method to use: the particle filter, the particle filter with various optimizations, or the Kalman filter. Implementation in Birch\[sec:demonstration\] ============================================ In Birch, models are ideally implemented by specifying the joint probability distribution. Where possible, this means that the code for the model does not distinguish between latent and observed random variables. Instead, the value of any random variable can be clamped at runtime to establish, from the joint distribution, particular conditional distributions of interest. Methods are also implemented in the Birch language, rather than being external to the system. We are particularly interested in considering the structure and form of a model when choosing a method for inference. This requires some interface that allows the method implementation to query the model implementation, and perhaps manipulate its execution. This becomes complicated with the random structure of programmatic models. Birch uses a twofold approach. Firstly, meta-programming techniques are used to represent fine-grain structure and form within the various mathematical expressions that make up a model. Secondly, the model programmer has a range of classes available from which they can implement their model to explicitly describe coarse-grain structure. We deal with each of these in turn. Fine-grain structure and form ----------------------------- Birch adopts the meta-programming technique of *expression templates* to represent mathematical expressions involving random variables. When such expressions are encountered, they are not evaluated immediately, but rather arranged into a data structure for later evaluation. In the meantime, it is possible to inspect and modify that data structure to facilitate optimizations based on lazy or reordered evaluation. This provides some of the power of a compiler to inspect structure and form via an abstract syntax tree, but within the language itself. Expression templates are common in linear algebra libraries such as Boost uBLAS and Eigen [@Guennebaud2010], where they are used to omit unnecessary memory allocations, reads and writes, and so produce more efficient code. They are also used in reverse-mode automatic differentiation to compute the gradient of a function, such as in Stan for HMC, and to implement delayed sampling. At this stage, they are primarily used in Birch for this latter purpose. Recall Code \[code:model\]; declarations were omitted but might be as follows: ``` {mathescape=""} $\theta$:Real; x:Real[T]; y:Real[T]; ``` This declares `\theta`, and the arrays `x` and `y`, as being ordinary variables of type `Real`. We can instead declare them as random variates of type `Real` like so: ``` {mathescape=""} $\theta$:Random<Real>; x:Random<Real>[T]; y:Random<Real>[T]; ``` Various mathematical functions and operators are overloaded for this type `Random<Real>`, to construct a data structure for lazy evaluation, rather than eager evaluation. When Code \[code:model\] is executed using these random types, a data structure such as that in Figure \[fig:ExpressionTemplate\] is constructed. This represents all the functions that are called, and their arguments, without evaluating them until necessary. As the functions $f$ and $g$ are opaque, there is little of value to discover here at first. If, however, we were to code the model with $f(x_{t-1},\theta)=\theta x_{t}$ and $g(x_{t})=x_{t}$, as in Code \[code:linear-gaussian\], we would obtain the data structure in Figure \[fig:ExpressionTemplate2\]. Now, Birch identifies a chain of Gaussian random variables for which it is able to marginalize and condition forward analytically, using closed form solutions with which it has been programmed. This path is annotated in Figure \[fig:ExpressionTemplate2\] (it is precisely the $M$-path used in delayed sampling [@Murray2018]). Ultimately, the computations performed are identical to running a Kalman filter forward, followed by recursively sampling backward. This utilizes structure and form to yield an exact sample from the posterior distribution, using a method that is more efficient that other means. ![A data structure describing Code \[code:model\] after four iterations. Shaded circles represent literals and observed random variables. Empty circles represent latent random variables. Rectangles indicate function calls, with inbound edges denoting their arguments and outbound edges their result. The node $\Theta$ has been repeated for clarity.\[fig:ExpressionTemplate\]](ExpressionTemplate){width="100.00000%"} ![As Figure \[fig:ExpressionTemplate\] but for Code \[code:linear-gaussian\]. Here, the delayed sampling mechanism in Birch recognizes the linear relationships between a chain of Gaussian random variables (highlighted), and can apply appropriate optimizations based on analytical marginalization and conditioning.\[fig:ExpressionTemplate2\]](ExpressionTemplate2){width="100.00000%"} Coarse-grain structure ---------------------- Birch provides abstract classes for various structural motifs and model classes. These automate some of the task of assembling a model by encapsulating boilerplate code, and provide crucial information on coarse-grain structure that can be used by an inference method. Consider, again, Code \[code:model\]. A more complete implementation, as a class, may look something like this: ``` {mathescape=""} class Example < Model { a:Random<Real>; x:Random<Real>[10]; y:Random<Real>[10]; fiber simulate() -> Real { a $\sim$ Uniform(0.0, 1.0); x[1] $\sim$ Gaussian(0.0, 1.0); y[1] $\sim$ Gaussian(x[1], 0.1); for t:Integer in 2..10 { x[t] $\sim$ Gaussian(a*x[t-1], 1.0); y[t] $\sim$ Gaussian(x[t], 0.1); } } } ``` This declares a new class called `Example` that inherits from the class `Model`, provided by the Birch standard library. At this stage, classes in Birch use a limited subset of the functionality of the classes in C++ to which they are compiled. They support single but not multiple inheritance, and all member functions are virtual. Code \[code:more-complete-implementation\] makes use of a *fiber* (also called a *coroutine*). Intuitively, this is a function for which execution can be paused and resumed. Each time execution is paused, the fiber *yields* a value to the caller, in a manner analogous to the way that a function *returns* a value to its caller—although a fiber may yield many times while a function returns only once. Like member functions, member fibers are virtual in Birch. The `Example` class in Code \[code:more-complete-implementation\] declares some member variables to represent the random variables of the model, then specifies the joint probability distribution over them by overriding the `simulate` fiber inherited from `Model`. This fiber simply simulates the model forward, but does so incrementally via implicit yields in the $~$ statements, which have a particular behavior. If the random variable on the left has not yet been assigned a value, the $~$ statement associates it with the distribution on the right, so that a value can be simulated for it later. If, on the other hand, the random variable on the left has previously been assigned a value, the $~$ statement computes its log-likelihood under the distribution on the right and yields the result. This yield value is always of type `Real`, as shown in the fiber declaration. This forms the most basic interface by which an inference method can incrementally evaluate likelihoods and posteriors, and even condition by assigning values to random variables in the model before executing the `simulate` fiber. It also represents the ideal of writing code to specify the joint distribution, while assigning values to variables before execution to simulate from a conditional distribution instead. It is clear that the model represented by Code \[code:more-complete-implementation\] has the structure of an SSM as in (\[eq:SSMjoint\]). A compiler with sophisticated static analysis may recognize this. Lacking such sophistication, it is the responsibility of the programmer to provide some hints. The choice to inherit from the `Model` class provides no such hints—the model is essentially a black box. Alternatives do provide hints. The Birch standard library provides a generic class called` StateSpaceModel` that itself inherits from `Model`, but reveals more information about structure to an inference method, and handles the boilerplate code for such structure. `StateSpaceModel` is a generic class that takes a number of additional type arguments to specify the type of the parameters, state variables, and observed variables. A class that inherits from `StateSpaceModel` should also override the `parameter`, `initial`, `transition`, and `observation` member fibers to describe the components of the model. These four fibers replace the `simulate` member fiber that is overridden when inheriting directly from `Model`. The implementation may look something like Code \[more-complete-implementation-structured\]. Type arguments are given to `StateSpaceModel<...>` in the inheritance clause to indicate that the parameters, state variables and observed variables are all of type `Random<Real>`. The `x’` that appears in the `transition` fiber is simply a name; the prime is a valid character for names, motivated by bringing the representation in code closer to the representation in mathematics. ``` {mathescape=""} class Example < StateSpaceModel<Random<Real>,Random<Real>,Random<Real>> { fiber parameter(a:Random<Real>) -> Real { a $\sim$ Uniform(0.0, 1.0); } fiber initial(x:Random<Real>, a:Random<Real>) -> Real { x $\sim$ Gaussian(0.0, 1.0); } fiber transition(x':Random<Real>, x:Random<Real>, a:Random<Real>) -> Real { x' $\sim$ Gaussian(a*x, 1.0); } fiber observation(y:Random<Real>, x:Random<Real>, a:Random<Real>) -> Real { y $\sim$ Gaussian(x, 0.1); } } ``` The model structure defined through the `StateSpaceModel` class is a straightforward extension of the sorts of interfaces that existing software provides. LibBi [@Murray2015], for example, is based entirely on the SSM structure, and the user implements their model by writing code blocks with the same four names as the fibers above. But while LibBi supports only one interface, for SSMs, Birch can support many interfaces, for many model classes, with an object-oriented approach. Example in Birch\[sec:worked-example\] ====================================== We demonstrate the above ideas on a multiple object tracking problem (see e.g. [@Vo2015] for an overview). Such problems arise in application domains including air traffic control, space debris monitoring (e.g. [@Jones2015]), and cell tracking (e.g. [@Ulman2017]). In all of these cases, some unknown number of objects appear, move, and disappear in some physical space, with noisy sensor measurements of their positions during this time. The task is to recover the object tracks from these noisy sensor measurements. The model to be introduced is an SSM within the class of programmatic models described in Section \[sec:models\]. The size of the latent state $X_{t}$ varies according to the number of objects, which is unknown *a priori*, and furthermore changes in time as objects appear and disappear. Similarly, the size of the observation $Y_{t}$ varies according to the number of objects and, in addition, rates of detection and noise. While it is straightforward to simulate from the joint distribution $p(\mathrm{d}x_{1:T},\mathrm{d}y_{1:T})$, simulation of the posterior distribution $p(\mathrm{d}x_{1:T}\mid y_{1:T})$ is complicated by a *data association* problem: the random matching of observations to objects, which includes both missing and spurious detections (clutter). That is, the structure of the relationships between the components of $X_{t}$ and $Y_{t}$ is not fixed. For $M$ observations and $K$ detected objects, there are $(M+1)^{(K)}$ (rising factorial) possible associations of equal probability under the prior. Naive inference with forward simulation of this association, as in a bootstrap particle filter, will not scale beyond a small number of objects and observations. Optimizations that instead leverage the structure and form of the model are necessary. For this example, we work on a two-dimensional rectangular domain with lower corner $l=(-10,-10)$ and upper corner $u=(10,10)$. The model is described—and ultimately implemented—in a hierarchical way, by first specifying a model for single objects, then combining several of these into a model for multiple objects. Single object model ------------------- The single object model describes the dynamics and observations (including missing detections) of a single object. The state of that object is represented by a six-dimensional vector giving its position, velocity and acceleration in the two-dimensional space. These evolve according to a linear-Gaussian SSM. Using superscripts to denote object $i$, the initial and transition models are given by: $$\begin{aligned} X_{0}^{i} & \sim\mathcal{N}(\mu_{0}^{i},M), & X_{t}^{i}\sim\mathcal{N}(Ax_{t-1}^{i},Q),\end{aligned}$$ where $\mu_{0}^{i}$ has its position component drawn uniformly on the domain $[l,u]$, with its velocity and acceleration components set to zero, and $M$, $A$ and $Q$ are the matrices: $$\begin{aligned} M & =\left(\begin{array}{ccc} 5I & 0 & 0\\ 0 & 0.1I & 0\\ 0 & 0 & 0.01I \end{array}\right) & A & =\left(\begin{array}{ccc} I & I & 0.5I\\ 0 & I & I\\ 0 & 0 & I \end{array}\right) & Q & =\left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0.01I \end{array}\right),\end{aligned}$$ with $I$ the $2\times2$ identity matrix and $0$ the $2\times2$ zero matrix. At time $t$, the object may or may not be detected. This is indicated by the variable $D_{t}^{i}$, which takes value 1 with probability $\rho$ to indicate detection, and value 0 otherwise. If detected, the observation is of position only: $$\begin{aligned} D_{t}^{i} & \sim\mathrm{Bernoulli}(\text{\ensuremath{\rho}}), & Y_{t}^{i} & \sim\mathcal{N}(Bx_{t}^{i},R),\end{aligned}$$ with matrices: $$\begin{aligned} B & =\left(\begin{array}{ccc} I & 0 & 0\end{array}\right), & R & =0.1I.\end{aligned}$$ Multiple object model --------------------- The multiple object model mediates several single object models, including their appearance and disappearance, and clutter. At time $t$, the number of new objects to appear, $B_{t}$, is distributed as: $$B_{t}\sim\mathrm{Poisson}(\lambda)$$ for some parameter $\lambda$. The lifetime of each object, $S^{i}$, is distributed as: $$S^{i}\sim\mathrm{Poisson}(\tau)$$ for some parameter $\tau$. In practice this is implemented as a probability of disappearance at each time step. If object $i$ has been present for $s^{i}$ time steps, its probability of disappearing on the next is given by $\Pr[S^{i}=s^{i}]/\Pr[S^{i}\geq s^{i}]$, with these probabilities easily computed under the Poisson distribution. At time $t$, some number of spurious observations (clutter) occur that are not associated with an object. Their number is denoted $C_{t}$, distributed as $$C_{t}-1\sim\mathrm{Poisson}(\mu)\label{eq:clutter}$$ for some parameter $\mu$. The position of each is uniformly distributed on the domain $[l,u]$. That there is at least one spurious observation at each time merely simplifies the implementation slightly—we revisit this point in Section \[sec:implementation\] below. Inference method\[sec:inference\] --------------------------------- Inference can leverage the structure and form of the model. The structure is that of an SSM, with random latent state size and random observation size. Within this SSM is further structure, as each of the single objects follows an SSM independently of the others. The form of those inner SSMs is linear-Gaussian, suggesting the use of a Kalman filter for optimal tracking, while the outer SSM requires the use of a particle filter. The inference problem is further complicated by data association, specifically matching detected objects with given observations. This is handled in the multiple object model, with a specific proposal distribution used within the particle filter to propose associations of high likelihood. Let $O_{t}^{i}$ denote the index of the observation associated with object $i$ at time $t$. For an object $i$ that is not detected ($D_{t}^{i}=0$) we set $O_{t}^{i}=0$. If there are $N_{t}$ number of objects, of which $K_{t}=\sum_{i=1}^{N_{t}}d_{t}^{i}$ are detected, and $M_{t}$ number of observations, the prior distribution over associations is uniform on the $(M_{t}+1)^{(K_{t})}$ possibilities: $$p(\mathrm{d}o_{t}^{1:N_{t}}\mid d_{t}^{1:N_{t}})=\prod_{i=1}^{N_{t}}p(\mathrm{d}o_{t}^{i}\mid o_{t}^{1:i-1},d_{t}^{i}),$$ where the pmfs associated with the conditional distributions on the right are: $$p(O_{t}^{i}=j\mid o_{t}^{1:i-1},d_{t}^{i})=\begin{cases} \frac{\mathds{1}[j\notin\{o_{t}^{1},\ldots,o_{t}^{i-1}\}]}{\sum_{m=1}^{M_{t}}\mathds{1}[m\notin\{o_{t}^{1},\ldots,o_{t}^{i-1}\}]} & \text{\text{if }}d_{t}^{i}=1\text{ and }j\in\{1,\ldots,M_{t}\},\\ 1 & \text{if }d_{t}^{i}=0\text{ and }j=0,\\ 0 & \text{otherwise.} \end{cases}$$ Here, $\mathds{1}$ denotes the indicator function. These expressions simply limit the uniform distribution to the correct domain: as long as all $O_{t}^{1:N_{t}}$ are in the support and the nonzero $O_{t}^{1:N_{t}}$ (corresponding to detected objects) are distinct, the probability is uniformly $1/(M_{t}+1)^{(K_{t})}$. The proposal distribution is to iterate through the detected objects in turn, choosing for each an associated observation in proportion to its likelihood, excluding those observations already associated. Thus, we have: $$q(\mathrm{d}o_{t}^{1:N_{t}}\mid d_{t}^{1:N_{t}})=\prod_{i=1}^{N_{t}}q(\mathrm{d}o_{t}^{i}\mid o_{t}^{1:i-1},d_{t}^{i}),$$ where the pmfs associated with the conditional distributions on the right are: $$q(O_{t}^{i}=j\mid o_{t}^{1:i-1},d_{t}^{i})=\begin{cases} \frac{p(y_{t}^{j}\mid x_{t}^{i})\mathds{1}[j\notin\{o_{t}^{1},\ldots,o_{t}^{i-1}\}]}{\sum_{m=1}^{M_{t}}p(y_{t}^{m}\mid x_{t}^{i})\mathds{1}[m\notin\{o_{t}^{1},\ldots,o_{t}^{i-1}\}]} & \text{\text{if }\ensuremath{d_{t}^{i}}=1}.\\ 1 & \text{if }d_{t}^{i}=0\text{ and }j=0\\ 0 & \text{otherwise.} \end{cases}$$ The delayed sampling heuristic within Birch automatically applies a further optimization to this. As each object follows a linear-Gaussian SSM, the $X_{t}^{i}$ are marginalized out using a Kalman filter, so that the proposal becomes: $$q^{*}(O_{t}^{i}=j\mid o_{t}^{1:i-1},d_{t}^{i})=\begin{cases} \frac{\phi(y_{t}^{j};\,\hat{y}_{t}^{i},\hat{\Sigma}_{t}^{i})\mathds{1}[j\notin\{o_{t}^{1},\ldots,o_{t}^{i-1}\}]}{\sum_{m=1}^{M_{t}}\phi(y_{t}^{m};\,\hat{y}_{t}^{i},\hat{\Sigma}_{t}^{i})\mathds{1}[m\notin\{o_{t}^{1},\ldots,o_{t}^{i-1}\}]} & \text{\text{if }\ensuremath{d_{t}^{i}}=1}.\\ 1 & \text{if }d_{t}^{i}=0\text{ and }j=0\\ 0 & \text{otherwise,} \end{cases}$$ where $\phi$ is the pdf of the multivariate normal distribution, with $\hat{y}_{t}^{i}$ the mean and $\hat{\Sigma}_{t}^{i}$ the covariance of $Y_{t}^{i}$ as predicted by the Kalman filter for the $i$th object. At time $t$, then, it is straightforward to propose a data association $o_{t}^{1:N_{t}}$ from $q$ (or $q^{*}$) and, in the usual importance sampling fashion, weight with the ratio: $$\begin{aligned} \frac{p(o_{t}^{1:N_{t}}\mid d_{t}^{1:N_{t}})}{q(o_{t}^{1:N_{t}}\mid d_{t}^{1:N_{t}})} & =\frac{(M_{t}+1)^{(K_{t})}}{\prod_{i=1}^{N_{t}}q(o_{t}^{i}\mid o_{t}^{1:i-1},d_{t}^{i})}.\end{aligned}$$ Unassociated observations at the end of the procedure are considered clutter. Implementation\[sec:implementation\] ------------------------------------ The model is implemented in Birch in a modular fashion. It consists of three classes: one for the parameters, one for the single object model, and one for the multiple object model. A class called `Global` is declared with a member variable for each of the parameters of the model, given in Code \[code:global\]. ``` {mathescape=""} class Global { l:Real[_]; // lower corner of domain of interest u:Real[_]; // upper corner of domain of interest d:Real; // probability of detection M:Real[_,_]; // initial value covariance A:Real[_,_]; // transition matrix Q:Real[_,_]; // state noise covariance B:Real[_,_]; // observation matrix R:Real[_,_]; // observation noise covariance $\lambda$:Real; // birth rate $\mu$:Real; // clutter rate $\tau$:Real; // track length rate } ``` The implementation of the single object model is given in Code \[code:single-object-model\]. This declares a class called `Track` that, as before, inherits from the class `StateSpaceModel` in the Birch standard library. The parameter type is `Global`, while the state and observation types are both `Random<Real[_]>`. The `initial`, `transition`, and `observation` member fibers are overridden to specify the model. An unfamiliar operator appears in the code: the meaning of $<\sim$ is to simulate from the distribution on the right and assign the value to the variable on the left. ``` {mathescape=""} class Track < StateSpaceModel<Global,Random<Real[_]>,Random<Real[_]>> { t:Integer; // starting time of the track fiber initial(x:Random<Real[_]>, $\theta$:Global) -> Real { auto $\mu$ <- vector(0.0, 3*length($\theta$.l)); $\mu$[1..2] $<\sim$ Uniform($\theta$.l, $\theta$.u); x $\sim$ Gaussian($\mu$, $\theta$.M); } fiber transition(x':Random<Real[_]>, x:Random<Real[_]>, $\theta$:Global) -> Real { x' $\sim$ Gaussian($\theta$.A*x, $\theta$.Q); } fiber observation(y:Random<Real[_]>, x:Random<Real[_]>, $\theta$:Global) -> Real { d:Boolean; d $<\sim$ Bernoulli($\theta$.d); // is the track detected? if d { y $\sim$ Gaussian($\theta$.B*x, $\theta$.R); } } } ``` The implementation of the multiple object model is given in Code \[code:multiple-object-model\]. This declares a class called `Multi` that, again, inherits from the class `StateSpaceModel` in the Birch standard library. The parameter type is given as `Global`, the state type as a `List` of `Track` objects, and the observation type as a `List` of `Random<Real[_]>` objects; `List` is a generic class provided by the standard library. The `transition` and `observation` member fibers are overridden to specify the model. The `observation` fiber is complicated by the data association problem. Recall, as in (\[eq:clutter\]), that there is always at least one point of clutter; an empty list of observations can therefore be interpreted as missing observations to be simulated, rather than present observations to be conditioned upon. When observations are present, the code defers to an alternative `association` member fiber, detailed below. When missing, they are simulated. This is an instance where it is not possible to write a single piece of code that specifies the joint distribution and covers both cases. ``` {mathescape=""} class Multi < StateSpaceModel<Global,List<Track>,List<Random<Real[_]>>> { t:Integer <- 0; // current time fiber transition(x':List<Track>, x:List<Track>, $\theta$:Global) -> Real { t <- t + 1; /* move current objects */ auto track <- x.walk(); while track? { $\rho$:Real <- pmf_poisson(t - track!.t - 1, $\theta$.$\tau$); R:Real <- 1.0 - cdf_poisson(t - track!.t - 1, $\theta$.$\tau$) + $\rho$; s:Boolean; s $<\sim$ Bernoulli(1.0 - $\rho$/R); // does the object survive? if s { track!.step(); x'.pushBack(track!); } } /* birth new objects */ N:Integer; N $<\sim$ Poisson($\theta$.$\lambda$); for n:Integer in 1..N { track:Track; track.t <- t; track.$\theta$ <- $\theta$; track.start(); x'.pushBack(track); } } fiber observation(y:List<Random<Real[_]>>, x:List<Track>, $\theta$:Global) -> Real { if !y.empty() { // observations given, use data association association(y, x, $\theta$); } else { N:Integer; N $<\sim$ Poisson($\theta$.$\mu$); for n:Integer in 1..(N + 1) { clutter:Random<Real[_]>; clutter $<\sim$ Uniform($\theta$.l, $\theta$.u); y.pushBack(clutter); } } } } ``` Finally, we show the `association` member fiber, which is the most difficult part of the model and inference method. This is given in Code \[code:association\]. A number of new language features appear. First, the `yield` statement is used to yield a value from a fiber, much like the `return` statement is used to return a value from a function. In previous examples, yielding from fibers has been implicit, via $~$ operators, rather than explicit, via `yield` statements. Here, explicit yields are used to correct weights for the data association proposal described in Section \[sec:inference\]. Another unfamiliar operator appears: the meaning of $\sim>$ is to compute the pdf or pmf of the value of the variable on the left under the distribution on the right, and implicitly yield its logarithm. The symmetry with the previously-introduced $<\sim$ operator is intentional: these two operators work as a pair for simulation and observation. Other than this, the code makes use of the interface to the `Random<Real[_]>` class to query for detection and compute likelihoods. ``` {mathescape=""} fiber association(y:List<Random<Real[_]>>, x:List<Track>, $\theta$:Global) -> Real { K:Integer <- 0; // number of detections auto track <- x.walk(); while track? { if track!.y.back().hasDistribution() { /* object is detected, compute proposal */ K <- K + 1; q:Real[y.size()]; n:Integer <- 1; auto detection <- y.walk(); while detection? { q[n] <- track!.y.back().pdf(detection!); n <- n + 1; } Q:Real <- sum(q); /* propose an association */ if Q > 0.0 { q <- q/Q; n $<\sim$ Categorical(q); // choose an observation yield track!.y.back().realize(y.get(n)); // likelihood yield -log(q[n]); // proposal correction y.erase(n); // remove the observation for future associations } else { yield -inf; // detected, but all likelihoods (numerically) zero } } /* factor in prior probability of hypothesis */ yield -lrising(y.size() + 1, K); // prior correction } /* clutter */ y.size() - 1 $\sim>$ Poisson($\theta$.$\mu$); auto clutter <- y.walk(); while clutter? { clutter! $\sim>$ Uniform($\theta$.l, $\theta$.u); } } ``` Results ------- The model is simulated for 100 time steps to produce the ground truth and data set shown in the top left of Figure \[fig:multiple-object-tracking\]. Using Birch, we then run a particle filter several times to produce the remaining plots in Figure \[fig:multiple-object-tracking\]. The delayed sampling feature of Birch ensures that, within each particle of the particle filter, a separate Kalman filter is applied to each object. Each run uses 32768 particles, from which a single path is drawn as a posterior sample, to be weighted by the normalizing constant estimate obtained from the particle filter. Generally, these posterior samples show good alignment with the ground truth. The longer tracks in the posterior samples correspond to similar tracks in the ground truth. Some shorter tracks in the ground truth are missing in the posterior samples (for example, the short track in the top right of the ground truth that appears at time 97), and conversely, some spurious tracks that do not appear in the ground truth appear sporadically in some posterior samples. These spurious tracks should be expected: the prior puts positive probability on objects being detected very few times—or not at all—in their lifetime. ![Phase plots for the multiple object tracking example, showing ground truth (top left) and posterior samples (others) with associated log-weights (left to right, top to bottom) of -7868.9, -7864.6, -7873.6, -7865.4, -7870.3. Lines represent object tracks. Points along the lines represent the position at each time. The larger point at the start of each line denotes the birth position, labelled with the birth time. Grey dots represent observations; those circled are associated with an object track, while others are classified as clutter.\[fig:multiple-object-tracking\]](simulation){width="100.00000%"} ![Phase plots for the multiple object tracking example, showing ground truth (top left) and posterior samples (others) with associated log-weights (left to right, top to bottom) of -7868.9, -7864.6, -7873.6, -7865.4, -7870.3. Lines represent object tracks. Points along the lines represent the position at each time. The larger point at the start of each line denotes the birth position, labelled with the birth time. Grey dots represent observations; those circled are associated with an object track, while others are classified as clutter.\[fig:multiple-object-tracking\]](filter1){width="100.00000%"} ![Phase plots for the multiple object tracking example, showing ground truth (top left) and posterior samples (others) with associated log-weights (left to right, top to bottom) of -7868.9, -7864.6, -7873.6, -7865.4, -7870.3. Lines represent object tracks. Points along the lines represent the position at each time. The larger point at the start of each line denotes the birth position, labelled with the birth time. Grey dots represent observations; those circled are associated with an object track, while others are classified as clutter.\[fig:multiple-object-tracking\]](filter2){width="100.00000%"} ![Phase plots for the multiple object tracking example, showing ground truth (top left) and posterior samples (others) with associated log-weights (left to right, top to bottom) of -7868.9, -7864.6, -7873.6, -7865.4, -7870.3. Lines represent object tracks. Points along the lines represent the position at each time. The larger point at the start of each line denotes the birth position, labelled with the birth time. Grey dots represent observations; those circled are associated with an object track, while others are classified as clutter.\[fig:multiple-object-tracking\]](filter3){width="100.00000%"} ![Phase plots for the multiple object tracking example, showing ground truth (top left) and posterior samples (others) with associated log-weights (left to right, top to bottom) of -7868.9, -7864.6, -7873.6, -7865.4, -7870.3. Lines represent object tracks. Points along the lines represent the position at each time. The larger point at the start of each line denotes the birth position, labelled with the birth time. Grey dots represent observations; those circled are associated with an object track, while others are classified as clutter.\[fig:multiple-object-tracking\]](filter5){width="100.00000%"} ![Phase plots for the multiple object tracking example, showing ground truth (top left) and posterior samples (others) with associated log-weights (left to right, top to bottom) of -7868.9, -7864.6, -7873.6, -7865.4, -7870.3. Lines represent object tracks. Points along the lines represent the position at each time. The larger point at the start of each line denotes the birth position, labelled with the birth time. Grey dots represent observations; those circled are associated with an object track, while others are classified as clutter.\[fig:multiple-object-tracking\]](filter5){width="100.00000%"} Summary\[sec:discussion\] ========================= Probabilistic programming is a relatively young field that seeks to accelerate the workflow of probabilistic modeling and inference using new probabilistic programming languages. A key concept is to decouple the implementation of models and inference methods, using various techniques to match them together for efficient inference, preferably in an automated way. In this work, we have provided a definition of the class of *programmatic models*, with an emphasis on *structure* and *form*. The class reflects the nature of models expressed in programming languages where, in general, structure and form are not known *a priori*, but may instead depend on random choices made during program execution. We have discussed some of the complexity that this brings to inference, especially that different executions of the model code may encounter different sets of random variables. SMC methods have issues around alignment for resampling, where different particles may encounter observations in different orders. MCMC methods encounter issues around Markov kernels needing to be transdimensional. Nevertheless, we argue that persistent substructure is common, can be represented by structural motifs, and can be utilized in implementation to match models with appropriate inference methods. We have shown the particular implementation of these ideas in the universal probabilistic programming language Birch. Here, expression templates are used to explore fine-grain structure and form, while class interfaces reveal coarse-grain structure. The expression templates in particular are important to enable analytical optimizations via the delayed sampling heuristic. Finally, we have shown a multiple object tracking example, where the model involved resides in the class of programmatic models, featuring a random number of latent variables, random number of observed variables, and random associations between them. Nonetheless, the model exhibits a clear structure and form: multiple linear-Gaussian SSMs for single objects, within a single nonlinear SSM for multiple objects. The delayed sampling heuristic provided by Birch automatically adapts the inference method to a particle filter for the multiple object model, with each particle within that filter using a Kalman filter for each single object model. Acknowledgements ================ This research was financially supported by the Swedish Foundation for Strategic Research (SSF) via the project *ASSEMBLE* (contract number: RIT15-0012). The authors thank Karl Granström for helpful discussions around the multiple object tracking example.

--- abstract: 'A major outstanding question regarding the formation of planetary systems is whether wide-orbit giant planets form differently than close-in giant planets. We aim to establish constraints on two key parameters that are relevant for understanding the formation of wide-orbit planets: 1) the relative mass function and 2) the fraction of systems hosting multiple companions. In this study, we focus on systems with directly imaged substellar companions, and the detection limits on lower-mass bodies within these systems. First, we uniformly derive the mass probability distributions of known companions. We then combine the information contained within the detections and detection limits into a survival analysis statistical framework to estimate the underlying mass function of the parent distribution. Finally, we calculate the probability that each system may host multiple substellar companions. We find that 1) the companion mass distribution is rising steeply toward smaller masses, with a functional form of $N\propto M^{-1.3\pm0.3}$, and consequently, 2) many of these systems likely host additional undetected sub-stellar companions. Combined, these results strongly support the notion that wide-orbit giant planets are formed predominantly via core accretion, similar to the better studied close-in giant planets. Finally, given the steep rise in the relative mass function with decreasing mass, these results suggest that future deep observations should unveil a greater number of directly imaged planets.' author: - 'Kevin Wagner$^{\star}$, Dániel Apai, & Kaitlin M. Kratter' title: 'On the Mass Function, Multiplicity, and Origins of Wide-Orbit Giant Planets' --- Introduction ============ Recent high-contrast imaging surveys of nearby stars have began to unveil a population of wide-orbit ($a\geq8$ AU) giant companions that are unlike anything found in our solar system. These companions are typically at least twice the mass of Jupiter and twice its orbital separation. Some objects are within bounds of being planetary companions (e.g., @Marois2008, @Lagrange2010, @Rameau2013, @Macintosh2015, @Chauvin2017, @Keppler2018), while other, yet more massive objects, are among the class of brown dwarfs and low-mass stars (e.g., @Metchev2006, @Kuzuhara2013, @Konopacky2016, @Milli2017, and others). As an ensemble, these wide-orbit companions enable us to study the formation of outer planetary systems in a way that is similar and complementary to the prevalent studies of inner planetary systems.[^1] Two main mechanisms for the formation of wide-orbit giant companions within protoplanetary disks have been suggested and explored: 1) top-down formation by gravitational disk instability (GI: e.g., @Boss1997), and 2) bottom-up formation by core accretion (CA: e.g., @Pollack1996). While both mechanisms may plausibly operate within protoplanetary disks, they are expected to produce very different signatures in the statistics of companion properties (for example, see the population synthesis studies of @Mordasini2009, @Forgan2018, @Mul2018, and others). For companions not born in disks, collapse within the protostellar core phase is a plausible option, and the distribution of companion masses would likely resemble the low-mass end of the stellar initial mass function (e.g., @Kroupa2001, @Chabrier2003). The criteria of being less than the $\sim$13 M$_{\rm Jup}$ deuterium burning limit is a commonly used dividing line between planets and brown dwarf companions. This is often scrutinized, in part because it is not a formation-motivated definition. In this study we will treat both classes of objects uniformly. We will refer to objects from both categories as “wide-orbit companions", while for simplicity we will frequently refer to those beneath the deuterium burning limit as “planets", and those above this limit as “brown dwarfs". We make no further distinction in our definitions based on orbital configuration. With these definitions in mind, we now turn to briefly summarize the physical processes of GI and CA, focusing on their expected contributions to the relative frequency of planets and brown dwarf companions, and expected fractions of systems with multiple companions. Theoretical simulations suggest that GI typically produces very massive companions, and operates more easily at larger separations (e.g., @Matzner2005, @Rafikov2005, @Clarke2009, @Kratter2010). As a result, the majority of GI-born companions are likely massive enough to be classified as brown dwarfs and low-mass stars, with the process yielding a small (but perhaps detectable) fraction of planetary-mass companions. This is due to the fact that the process must begin very early, while enough mass exists in the disks to trigger the instability. In turn, this causes the majority of objects formed by GI to grow rapidly in mass, given the availability of material at such young ages (e.g., @Kratter2010, @Forgan2013, @Forgan2018). Still, it is possible that some planetary mass companions may originate from disk-born GI, which would be evident in a lower (or consistent) frequency of giant planets compared to brown dwarfs. While it is plausible that GI could produce multiple companions in the same system, overall the mechanism is expected to yield a low multiplicity fraction [@Forgan2018]. On the other hand, formation of giant planets via CA involves much longer timescales, primarily limited by the time required for the solid core to grow above the critical mass to trigger runaway gas accretion [@Bodenheimer1986]. Typically, $\sim$10 M$_{Earth}$ is considered as the critical core mass, although recent work has shown that smaller masses (down to several M$_{Earth}$) are sufficient to trigger runaway gas accretion at larger disk radii (@Piso2014). Additional factors, such as pebble accretion (e.g., @Ormel2010, @Lambrechts2012), may help to accelerate solid core growth. Nevertheless, the growth of a massive core and subsequent accumulation of a gaseous envelope is in contest with the dispersal of the gaseous protoplanetary disk ($\lesssim$10 Myr; @Ercolano2017). The late formation of the cores within a disk rapidly declining in mass limits the availability of accretable gas and, thus, the probability of the formation of super-Jupiters and brown dwarfs. As a result, and contrary to the GI scenario, CA is expected to produce a much higher relative frequency of lower mass planets compared to super-Jupiters and brown dwarfs. Furthermore, the fraction of systems hosting multiple giant companions is much higher for close-in planets formed via CA. [@Knutson2014] studied 51 systems containing giant planets of 1-13 M$_{\rm Jup}$ at orbital separations between 1-20 AU and found that the occurrence rate of additional massive outer companions is 51$\pm$10%. Similarly, the fraction of planetary systems hosting confirmed multiple planets is $\geq$21.8% and $\geq$24.3%, for detection via transit and radial velocity, respectively. The lower limit is the confirmed fraction of multiple systems, and the true fraction of multiple systems among these is likely even higher, considering that additional planets may exist that are either non-transiting or of sufficiently long-period to be non-detected.[^2] If the wide-orbit giant planets also formed via CA, we might expect a significant fraction of these systems to host multiple giant companions. The prevalence of binary and multi-star systems is evidence that protostellar cores are frequently subject to fragmentation. However, as the evolving system continues to decline in mass, the probability that GI will occur in the disk stage is diminished. Likewise, the resulting companion mass is also limited by the availability of material at later times, with the most likely outcome of disk-born GI being a companion in the brown dwarf regime. The paucity of such companions to main-sequence stars confirms this general picture. This trend, known as the “brown dwarf desert", was initially identified among close-in companions (e.g., @Duq1991). The same trend has also been identified in the low occurrence rates of wide-orbit brown dwarf companions (e.g., @McCarthy2004, @Kraus2011, @Vigan2017), although the effect is not as extreme, with a few percent of stars hosting such companions [@Metchev2009]. Similarly, relatively few directly imaged giant planets have been discovered. For this reason, there is an on-going debate over the dominant formation pathway of these objects, with arguments in favor of both GI$-$ and CA-like processes. While significant difficulty remains in determining the formation pathway of a particular object, the dominant formation mechanism for an ensemble of objects can be revealed by the form of their relative mass function. If the mass distribution reveals a higher relative frequency of lower-mass objects, this would indicate that similar CA-like planet formation processes occur within the inner and outer regions. On the other hand, if the mass function is relatively flat, or rising toward higher masses, this would indicate GI as the dominant formation mechanism. Likewise, insight may be gained by examining the possibility that a significant fraction of systems may host multiple giant companions, as CA is expected to produce a significant fraction of such systems. Here, we aim to constrain the relative mass function and multiplicity of directly imaged wide-orbit giant companions by applying the class of statistical methods that were developed for analyzing censored data. These methods, often referred to as “survival-analysis", are well-vetted in medical and risk management industries, and are becoming increasingly popular in astronomy. In the case of directly imaged companions, the data comprise a set of detected objects with estimated masses[^3], and a population of lower mass objects that are possibly present, but non-detected, that are included as upper mass limits. In particular, we will utilize the Kaplan-Meier maximum likelihood estimator (@Kaplan1958, @Feigelson1985) to estimate the cumulative distribution of the underlying parent population. In this study, we focus on systems with known wide-orbit substellar companions. With this approach, we isolate the question of what is the *relative* mass function of wide-orbit companions in systems where they have been identified, from the question of whether these systems are exceptional (i.e., having an overall low occurrence rate). This approach is different from, but complementary to the conventional occurrence rate studies, in which a mass function is typically assumed and then contrasted with the observed detection rate. In the conventional approach, the mass function and the occurrence rates are degenerate, and both are unknown (e.g., @Kasper2007, @Biller2013, @Brandt2014, @Nielsen2013, @Galicher2016, @Reggiani2016, @Vigan2017, @Stone2018, @Nielsen2019). We begin by assembling a list of known companions in §2. We then derive their mass and upper mass limit probability distributions in §2.1. We describe how these mass measurements and mass limits are incorporated into the survival analysis framework in §2.2. In §3, we present our results, namely the relative mass function of giant companions in §3.1, and the associated multiplicity probabilities in §3.2. We explore the effects of various model assumptions in §3.3 & §3.4, and show results for a selection of relevant subsamples in §3.5. Finally, we provide a brief discussion of the results and a critical assessment of the weaknesses of our approach in §4, and conclude by summarizing our findings in §5. Sample of Companions ==================== We assembled a list companions from the literature, beginning with the list on `exoplanet.eu` [@Schneider2011] as of 2018 November 23 with the selection criteria of being discovered by direct imaging. While this list is often scrutinized, in particular for containing objects beyond the deuterium burning limit as “planets", for our purposes this is desirable, as we aim to constrain the mass distribution across the planet to brown dwarf mass threshold. To our knowledge, this list is complete with respect to companions that have been directly detected in high-contrast imaging with mass estimates in the sub-stellar range, which we verified by cross-checking against the NASA Exoplanet Archive.[^4] We excluded planetary-mass companions around white dwarf and brown dwarf hosts, and also planet candidates that have been interpreted as potential disk features (e.g., the planet candidates around Fomalhaut: @Janson2012, @Lawler2015, HD 169142: [@Ligi2018], and LkCa 15: @Thalmann2015, @Mendigutia2018). This criteria resulted in a list of 57 companions, whose properties are given in Appendix A. The host stars among this sample, and the properties of their companions, are highly diverse. The host stars range from ages of a few Myr to several Gyr, and display spectral types spanning late-M to early-A types ($\sim$0.2$-$3 M$_{\odot}$ for main sequence stars), which reflects the diversity of selection criteria among the original surveys. Furthermore, their companions have estimated masses ranging from $\sim$2 M$_{\rm Jup}$ to the minimum hydrogen burning mass, and occupy orbital ranges of 8 AU to several thousand AU (estimated from their projected separations, in most cases). To reduce potential effects of including such a variety of orbital properties and host star mass, we restricted our initial analysis to companions whose projected separation is $\leq$100 AU, and with host stars of spectral type A0-K8 such that the mass ratio is $\lesssim$0.01 for a 5 M$_{Jup}$ planet around the least massive stars. This resulted in a list of 23 companions. We refer to this population as “sub-sample A", or our primary sample, and will discuss results obtained from this population unless otherwise noted. In §3.5 we will relax these criteria and examine the full sample, and will also examine select subsamples to utilize the full diversity among our sample to search for trends in companion properties. Conversion of Photometry to Masses and Upper Mass Limits -------------------------------------------------------- For each companion, we compute its mass probability distribution via a Monte Carlo (MC) simulation drawing from Gaussian priors on age, distance, and photometry, as reported in the literature. We convert these properties into mass estimates via the evolutionary tracks of [@Baraffe2003] for our primary analysis, and explore other models (including dusty photospheres, and “cold-start" initial conditions) in §3.4. In general, we utilize the most sensitive measurements currently available for upper limits of additional companions. We consider detection limits only in the outer regions, which are well-matched to the wide-orbit population that is the focus of this study. In these regions (typically $\gtrsim$0$\farcs$5), the sensitivity is not primarily limited by speckle noise from the central star, and instead is limited by thermal background and other spatially homogeneous sources of noise. When sensitivity between photometric bands is comparable, we utilize the longest wavelength data available (typically either $Ks$ or $L^{\prime}$) since these are less likely to be affected by differences in molecular absorption. For systems without published detection limits, we estimated 5$\sigma$ detection limits by assuming that the photometric uncertainty on the known companions corresponds to the $\sim$1$\sigma$ noise level.[^5] We tested this method of estimating detection limits on companions with published limits and found that this method consistently over-estimates the upper limits due to the fact that the photometric uncertainty also incorporates photon noise from the detection (which can be quite high), whereas a true detection limit would be dominated by background noise terms. In other words, these are likely conservative estimates on the detection limits within these systems. We use the most up-to-date age ranges available throughout the literature, and assume a Gaussian profile within this range.[^6] Likewise, we utilize the most up-to-date distance measurements available, which typically come from *Gaia DR2* [@GDR2]. Given the sparse and non-uniform sampling of the evolutionary model grids, we must interpolate between the points in order to generate solutions at arbitrary masses and ages (though still within the bounds of the grids, which for the @Baraffe2003 models is 0.5-100 M$_{\rm Jup}$ and 1 Myr to 10 Gyr). We adopted a bi-linear interpolation scheme, and also tested the output of cubic interpolation. We found similar results in both cases (mass probability distributions of similar mean and width), and choose to retain the simpler bi-linear interpolation for the proceeding analysis. We show the cumulative mass function of the detected objects around A0-K8 stars and with projected separations $\leq$100 AU in Fig 1. The individual mass probability distributions for the detected companions, and the mass detection limit probability distributions are shown in Appendix B, along with objects not included in the primary sample. Overall, the cumulative distribution shows a steeper slope toward lower masses, although an important (and non-physical) feature of this cumulative mass function is that it drops to zero below $\sim$2-3 M$_{\rm Jup}$, which simply reflects an observational bias arising from the difficulty of detecting such low-mass companions. In the next subsection, we estimate the correction to this distribution at small masses. Survival Analysis: Estimating the Underlying Cumulative Mass Probability Distribution ------------------------------------------------------------------------------------- The class of statistical methods that has been developed for dealing with censored data$-$e.g., data containing both detections and detection limits$-$is frequently referred to as “survival analysis". While many works are devoted to exploring these methods in detail, we refer the interested reader to [@Feigelson1985], which recasts the typical formulation from a context of right-censored data (involving lower limits, or in the name-sake problem, survival times) to a context of left-censored data (involving upper limits), which is applicable for our data-set, and in general for most astrophysical contexts. We utilize the Kaplan-Meier (KM) maximum likelihood estimator (@Kaplan1958, @Feigelson1985), which approximates the cumulative distribution of the underlying parent population from which the censored data were drawn. Specifically, we utilize the form given in Section 2, Equations 1-8 of [@Feigelson1985] for a sample containing indistinct measures (in this case, multiple objects within the same mass bin). The general form of the KM estimator is a monotonic increasing function whose value only changes at the values of uncensored measurements, with the size of the jumps being determined by the combination of the censored and uncensored measures. In this way, the KM estimator provides an estimated correction for the observational bias at low masses by including the information contained within the detection limits. While the result is likely closer to reality than considering merely the detections alone, it remains an approximation since the true masses of the undetected companions, and the number of such companions that actually exist beneath the detection limits, remain unknown (this is a topic of further discussion in §3.1 and §4.1). While the measurements originate from (often very) different data-sets, the end products are the same: namely photometry of detected sources and photometric detection limits on additional sources. By uniformly converting these measurements into estimated masses and mass detection limits, we eliminate the possibility that differing methods of converting the original measurements into estimated masses may bias our results. There exists the possibility that differing strategies for estimating photometric sensitivity may lead to different results (e.g., @Mawet2014). However, these effects are most prevalent at small separations, whereas we consider detection limits only in the outer regions, in which the sensitivity does not vary significantly with angular separation, and in which the noise is approximately Gaussian. Any remaining differences in the original data reductions are likely to enter as random errors, and on average should not bias our results. To incorporate the measurement uncertainties, we compute the survival function via an MC simulation of 1,000 trials, wherein each trial we calculate the KM estimator by randomly drawing a mass and upper mass limit from each companion’s probability distributions.[^7] We then average the cumulative distributions together, which is shown in Fig. 2 along with the cumulative distributions of 100 randomly selected MC trials. We split the distribution into six mass bins (3-7, 7-13, 13-20, 20-30, 30-45, and 45-65 M$_{\rm Jup}$), and within each bin fit a linear model.[^8] We repeat this analysis on each of the 1,000 MC trials to establish uncertainties on the relative frequency of each mass bin. The relative slopes of the linear fits provides an estimate of the relative frequency of companions within these mass bins, which is the topic of the following section. Results ======= The Wide-Orbit Planetary Mass Function -------------------------------------- The derivative of the cumulative mass function provides the relative mass function, which we henceforth refer to as the companion mass function (CMF). In Fig. 3, we show the result derived from the survival analysis-generated cumulative mass function (blue points), alongside the result derived from the cumulative probability distribution of the detections alone (red points). In both cases, the distribution drops steeply between the first three mass bins (3-20 M$_{\rm Jup}$), and is relatively flat at higher masses, with an approximate functional form of $N \propto M^{-1.3\pm 0.3}$. The similarity between the two distributions is due to the fact that many of the detection limits are lower than the minimum mass of 3 M$_{\rm Jup}$ considered here, which mostly shifts the cumulative distribution upward without affecting its overall shape. Nevertheless, some detection limits are higher than 3 $M_{\rm Jup}$, which can be seen as a higher relative frequency of 3-7 and 7-13 M$_{\rm Jup}$ objects in the distribution resulting from the survival analysis. The magnitude of this difference is impacted by our assumption that each detection limit corresponds to a *single* object whose mass is beneath the detection limit. If *multiple* companions exist within any of these systems that are beneath the detection limits, that would increase the frequency of planetary-mass companions even further. On the other extreme, if there is not a single companion beneath the detection limits among any of these systems, then the distribution would follow that of the detected objects alone, which sets a lower limit to the change of the slope across the CMF. Given this bottom-heavy CMF, we expect that more non-detected companions exist in the lower-mass range, and thus in the proceeding sections we utilize the results derived from the survival analysis. Comparison to Other CMFs ------------------------ In Fig. 4 we compare the observed CMF of wide-orbit companions to simulated CMFs from theoretical models of companions produced through solely CA [@Mordasini2009] and by GI [@Forgan2018]. We also compare the results to the CMF of inner planets discovered by RV surveys [@Schneider2011]. The CA population synthesis of [@Mordasini2009], and the relative frequency of close-in giant planets as derived from RV surveys[^9] are in good agreement with the data (unreduced $\chi^2 \sim$ 5-6 in both cases). A similar population synthesis model with GI as the dominant formation mechanism [@Forgan2018] does not match the data, as it follows a relatively flat distribution ($\chi^2 \gtrsim$ 100). However, when the GI model is re-normalized to fit only the highest three mass bins (20-65 M$_{\rm Jup}$), we see that this model does a fair job at matching the high-mass end of the distribution ($\chi^2 \sim$ 1 with respect to only these points), while contributing less significantly to the relative abundance of planetary mass companions (roughly 6% of the 3-7 M$_{\rm Jup}$ bin, and 14% of the 7-13 M$_{\rm Jup}$ bin). Similarly, we compared the results to a low-mass stellar initial mass function (IMF; @Kroupa2001). The result is essentially the same as for the GI model, which is to be expected since both distributions are relatively flat. While the stellar IMF is a poor fit to the distribution throughout the complete range of masses ($\chi^2 \sim$ 20), it provides an equivalent match to the higher-mass objects as the GI model. Multiplicity Probabilities -------------------------- The second aim of our study is to assess the fraction of these systems that could host multiple wide-orbit giant companions. So far HR 8799 and HIP79930 are the only examples systems with multiple companions that have been detected, although other candidate multiple systems exist. HR 8799 is a remarkable system containing *four* super-Jupiters between 10-70 AU (@Marois2008, @Marois2010). HIP73990 is also a remarkable case, as it hosts two brown dwarfs at projected separations of $\sim$18 and 28 AU [@Hinkley2015]. Since the actual number of observed multiples is low, the question of multiplicity essentially becomes whether the (apparently single) systems are in fact compatible with hosting additional companions that are beneath the detection limits. To address this possibility, we begin with the assumption that each system hosts an additional companion whose mass is independent of other bodies in the system. In reality, systems hosting one wide-orbit giant companion may be more (or less) likely to host additional companions. In the following, we explore the simplest scenario in which the masses are independent. If future, deep searches fail to reveal more companions within the known systems, then this could be taken as evidence that giant companion formation inhibits the potential for forming a second companion. On the other hand, if more systems are discovered to be multiples than predicted here, this would suggest that systems that are able to form a single giant companion have a higher likelihood of forming multiple such companions. We also assume that the hypothetical second companion exists within the semi-major axis range corresponding to that in which the upper mass limit was defined, which for simplicity can always be taken to be external to the known companion. We then perform an MC simulation of 1,000 trials, wherein each trial we randomly draw a second companion mass from the CMF, and an upper mass detection limit from the probability distributions derived in §2.1. We approximate an upper limit to the probability that each system hosts an additional wide-orbit super-Jovian companion by the fraction of trials resulting in a randomly drawn companion whose mass is beneath the detection limit for that system and $\geq$2 M$_{\rm Jup}$. [cccc]{} 51 Eri & 34.1 & 11.6 & 3.97\ GJ 504 & 77.9 & 60.7 & 47.3\ GJ 758 & 85.6 & 73.3 & 62.7\ HD 1160 & 79.1 & 62.6 & 49.5\ HD 19467 & 94.1 & 88.5 & 83.3\ HD 206893 & 84.2 & 70.9 & 59.7\ HD 4113 & 89.3 & 79.7 & 71.2\ HD 95086 & 40.9 & 16.7 & 6.84\ HD 984 & 76.2 & 58.1 & 44.2\ HIP 65426 & 53.3 & 28.4 & 15.1\ HIP 73990 & 100. & 79.6 & 63.4\ HIP 74865 & 74.3 & 55.2 & 41.0\ HR 2562 & 68.3 & 46.6 & 31.9\ HR 3549 & 60.8 & 37.0 & 22.5\ HR 8799 & 100. & 100. & 100.\ PDS 70 & 65.7 & 43.2 & 28.4\ PZ Tel & 48.4 & 23.4 & 11.3\ $\beta$ Pic & 22.5 & 5.06 & 1.14\ $\kappa$ And & 40.8 & 16.6 & 6.79\ \ Mean & 68.2% & 50.4% & 39.5%\ These probabilities are given in Table 1, along with the probabilities that each system may host two and three additional companions via the same reasoning. On average, these systems have a 68.2% probability of hosting a second wide-orbit giant companion ($\geq$2 M$_{\rm Jup}$) drawn from the CMF. Likewise, on average there is a 50.4% probability of hosting three such planets, and a 39.5% probability of hosting four planets, like the HR 8799 system. The frequency of similar systems with multiple super-Jupiter companions will be a topic of discussion in §4.6. These results are likely overly optimistic about the fraction of multiple systems, and could instead be considered as upper limits to the probability of hosting additional wide-orbit giant companions. In particular, semi-major axis effects also likely play a significant role in the probability that additional companions may exist. For instance, by considering requirements for dynamical stability of multiple orbiting bodies, these probabilities may be reduced further. While it is possible to increase the complexity of this analysis to include such effects, this simple analysis is revealing enough for our purposes: the observed bottom-heavy CMF in combination with the available detection limits suggests that some of these systems are likely hosting wide-orbit planetary mass companions that have not yet been detected. Exploration of Model Assumptions -------------------------------- The above results were derived under the assumptions inherent in the [@Baraffe2003] models: namely that planets and brown dwarfs retain all of their initial entropy (representative of a “hot-start" formation scenario) and have clear atmospheres. Now, we explore the effect of relaxing those assumptions. In §3.4.1 we explore the effect of planets forming with a variety of initial entropy conditions (representative of a “cold-start" scenario). In §3.4.2, we explore the effect of allowing companions to retain a significant fraction of dust in their atmospheres by utilizing the grid of models presented in [@Chabrier2000]. ### Hot vs. Cold-start Planets The initial luminosity of young giant planets remains an open question of giant planet formation (e.g., @Marley2007, @Fortney2008, @Spiegel2012, @Mordasini2017). The uncertainty primarily lies in how much energy is radiated away from the in-falling material at the accretion shock boundary [@Marleau2017]. The radiative efficiency is not clearly predicted from simulations, and while some young companions display an observable accretion luminosity (e.g., @Zhou2014, @Close2014, @Sallum2015, @Wagner2018), significant difficulty remains in establishing a radiative efficiency from these limited observations. The limiting conditions in which the shocked gas radiates 0% and 100% of its kinetic energy at the shock boundary lead to the classical “hot" and “cold" start models (e.g., @Baraffe2003, @Marley2007). The reality is likely somewhere in between, and there probably exists a spread in initial luminosities for a given mass (@Mordasini2017, @Berardo2017). At higher masses, the hot and cold-start models converge as deuterium burning begins to contribute to the overall luminosity. At later ages, the hot and cold-start models also converge, with lower-mass tracks converging within a few tens of millions of years, and the $\sim$10 M$_{\rm Jup}$ tracks taking the longest to converge at a few hundred million years [@Spiegel2012]. Our choice of utilizing the hot-start models of [@Baraffe2003] in the preceding analysis is motivated by the fact that for most of the mass range considered here, the hot and cold-start models are in good agreement (@Spiegel2012, @Mordasini2017). Our choice was further motivated by several indications suggesting that planets should form with initial luminosities close to those of the the hot-start models. For example, the planets in HR 8799 have dynamical masses that agree very well with the hot-start mass estimates (@Snellen2018, @Wang2018), and for $\beta$ Pictoris the presence of the disk is inconsistent with cold-start estimates for the planet’s mass (@Lagrange2010). Furthermore, recent simulations [@Mordasini2017] have shown that, while the accretion shock may radiate away a significant amount of energy, continued accretion of planetesimals during this phase will lead to a luminosity-core-mass effect, whereby higher mass cores lead to higher initial luminosities. This effect causes the initial luminosities of cold-start planets to become comparable to those of the hot-start models with no core-mass effect. Nevertheless, this assumption may have an important effect on our final results, and deserves exploration. While cold-start evolutionary grids exist (e.g., the models of @Spiegel2012), they remain significantly unconstrained due to the uncertainty in initial conditions and subsequent accretion history. Instead, we choose to employ a simple prescription to scale the luminosity of the hot-start evolutionary grids as an approximation of a cold-start case for objects beneath the deuterium burning limit. In computing the companion mass and upper mass limit probability distributions of objects, we assume a minimum efficiency representative of energy transfer during the accretion of the gaseous envelope. This enters as a numerical scaling factor in the temperature of the object, which we use to scale the luminosity by the corresponding fourth power of the change in temperature from the hot-start evolutionary grids. We assume a uniform distribution between this minimum efficiency and unity in the MC trials to assemble the mass probability distributions. This is representative of planets that formed in a variety of conditions, and is consistent with some fraction of planets attaining initial luminosities that are close to the hot-start predictions. In Fig. 5, we show an example of the CMFs obtained for a minimum efficiency of 50% corresponding to minimum luminosities that are 6% of the maximum for a given mass. Assuming that planets form colder tends to move planets from the 3-7 M$_{Jup}$ bin to the 7-13 M$_{Jup}$ bin, but the latter are not moved beyond the deuterium burning limit due to convergence with the hot-start tracks at higher masses (@Spiegel2012, @Mordasini2017). In other words, under any assumption of initial planet entropy the result is still a bottom-heavy mass function. The general trend remains the same, with planetary mass companions being much more frequent than brown dwarfs. Similarly, the average upper limits on the probability that each system may host multiple companions drawn from the CMF are only slightly lower, with $\lesssim$48%, $\lesssim$28%, and $\lesssim$18% average probability for hosting a double, triple, and quadruple system, respectively. ### Dusty vs. Clear Atmospheres A second assumption that may impact the results via conversion of photometry to mass is the choice of clear vs. dusty model atmospheres in the evolutionary grids. If an object has a significant fraction of dust (or clouds) in its photosphere, it will appear redder than the model predictions for clear atmospheres, and thus lead to a different interpretation of its mass. In the preceding sections, we utilized the clear model atmospheres of the `COND` grid [@Baraffe2003]. Now, we explore the effect of including dusty/cloudy model atmospheres of the `DUSTY` grid [@Chabrier2000] for applicable objects. We note that some objects, particularly T-dwarfs such as 51 Eri b [@Macintosh2015], fail to be matched by this grid, which is more limited at lower masses and older ages. For these objects, we retain the mass distribution estimated from the `COND` grid, which is physically motivated by the fact that T-dwarfs (by definition) display primarily clear atmospheres. The results are shown in Fig. 6. We find a consistent result: the general trend is a CMF that is rising toward smaller masses, which verifies that the assumed dust content of the companions’ photospheres does not significantly impact our results and conclusions. Exploration of Select Sub-samples --------------------------------- In the preceding subsection, we have shown that the form of the CMF and limits on the fraction of systems hosting multiple wide-orbit giant companions are valid independent of model assumptions on initial planet luminosity and atmospheric dust content. However, the objects that we included in the preceding analysis were restricted to those within 100 AU of A0-K8 stars. While these choices were physically motivated, the effect of these assumptions deserves attention. For this purpose, we examine the effect of relaxing sample restrictions on spectral type, on orbital separation, and their combination. [ccccc]{} A & A0-K8 & 8$-$100 AU & 23 & 3.1-3.4\ B & A0-M8 &8$-$100 AU & 28 & 3.5\ C & A0-K8 & $\geq$8 AU & 37 & 3.5\ D & A0-M8 & $\geq$8 AU & 57 & 3.5\ E & A0-M8 & $\geq$100 AU & 28 & 3.5\ F & K0-M8 & $\geq$8 AU & 27 & 3.5\ In addition, we explore several other subsamples, in pursuit of searching for potential differences in the CMF interior and exterior to 100 AU, and around hosts of different spectral type. These subsamples are listed in Table 2. We have so far focused on sub-sample A, which includes most of the objects typically considered to be directly imaged planets. This group has the added benefit of being partially isolated to effects of differing host mass and orbital configuration. Sub-samples B and C gradually relax these added selection criteria, by removing the spectral type criteria in B, and the projected separation criteria in C. Sub-sample D removes all selection criteria, and in other words includes all of the directly imaged substellar companions known to our study. Sub-sample E focuses on companions on *very* wide-orbits ($\gtrsim$100 AU), and sub-sample F focuses on companions around late-type (and presumably low-mass) hosts. In our primary sample we did not include companions around M-stars. First, we explore the effect of including these companions (Sub-sample B). This search resulted in 28 companions, and is shown as the light blue line in Fig. 7, panel B. The result of including these additional companions produces only negligible effects on the companion mass distribution$-$namely a small decrease in the slope toward lower masses, although the effect is within the 1-$\sigma$ uncertainties. Instead, we also tried relaxing the separation criteria (Sub-sample C), but again exclude the companions around hosts of spectral type M. This search resulted in 37 companions, and is shown in the magenta line in Fig. 7, panel C. The distribution is still peaked toward planetary masses, with a $\sim$1$\sigma$ decrease in the frequency of the lowest mass bin with respect to the primary sample. Relaxing all selection criteria resulted in 57 companions (Sub-sample D), and is shown in the blue line in Fig. 7, panel D. While the distribution matches the others at higher masses, this sample shows a tentative peak at $\sim$10M$_{\rm Jup}$, while the frequency of the 3-7 M$_{\rm Jup}$ is consistent with the frequency of higher mass brown dwarfs. With respect to the primary sample, the relative frequency of the 3-7 M$_{\rm Jup}$ bin is reduced by $\sim$2$\sigma$. To explore the properties of very wide companions (wider than the 100 AU maximum considered earlier), we relax the selection criteria on spectral type, and focus on companions *external* to 100 AU. This search resulted in 28 companions, and is represented in the orange line in Fig. 7, panel E. The behavior of the previous cases is enhanced, with the frequency of the lowest mass bin similarly reduced. This suggests that the similar behavior seen in the previous two samples is reflective of this very wide-orbit population. In this case, the frequency of 13-20 $M_{\rm Jup}$ is also enhanced by $\sim$2-$\sigma$ with respect to the CMF of the primary sample. In the final sub-sample, we examine K and M-type hosts, to explore differences around very late-type stars. This search resulted in 27 companions, and is represented in the purple line in Fig. 7, panel F. This sample shows a CMF that is similar to the previous case, with an enhancement only in the 7-13 M$_{\rm Jup}$ and 13-20 M$_{\rm Jup}$ bins, and a similar reduction of frequency of the 3-7 M$_{\rm Jup}$ bin. ### Comparison of the Considered Samples Now, we explore the differences between the CMFs for the various subsamples. To compare all of the subsamples simultaneously, it is convenient to re-normalize the samples compared to one that is selected as a “standard" distribution. We choose to normalize by the A-M, $\geq$100 AU population: i.e., the very wide-orbit population, containing companions around any spectral type host (sub-sample E). This is a desirable choice, as it enables differences to be easily identified for populations containing a) companions on orbits consistent with typical sizes of protoplanetary disks ($\lesssim$100 AU), and b) companions around hosts of specific spectral types. The result is shown in Figure 8. The most significant variation exists in the 3-7 M$_{\rm Jup}$ bin, with a strong a strong enhancement for planets of this mass among the samples restricted to within 100 AU. These show nearly an order of magnitude enhancement of 3-7 M$_{\rm Jup}$ planets compared to the other samples. Similarly, there is also an enhancement of the 3-7 $M_{\rm Jup}$ planets around early spectral type (A-K) hosts when orbital restrictions are relaxed. These effects are possibly further revealing of the formation mechanisms responsible for the various subsamples, and for the ensemble, which will be discussed in the next section. Discussion ========== The primary aim of our study was to investigate the relative mass function of giant planets and brown dwarfs (the planetary mass function, or CMF). The second aim of our study was to utilize this CMF to assess the probability that each system may in fact host multiple companions drawn independently from the same mass distribution. Here, we discuss those results in the context of predictions from various mechanisms that could have formed this population, and in the context of the results of other exoplanet surveys. In §3.1 we presented the CMF, which from the detected objects alone reveals a higher relative frequency of lower mass objects. The distribution becomes even more bottom heavy when incorporating information in the mass detection limits (§2.2). In §3.2 we compared the observed CMF to predictions from population syntheses, and to the observed CMF of the RV planets, and found a good agreement with both the CA predicted CMF and that of the RV population. A similar population synthesis-derived CMF representative of the GI scenario (followed by tidal downsizing for the lowest mass objects) does not match the observed form of the CMF, as it predicts a much lower frequency of planetary-mass companions relative to brown dwarf companions than is actually observed. In §3.2 we showed that the systems among our primary sample have (on average) a 68.2% probability of hosting an additional (typically undetected) giant planet ($\geq$2 M$_{\rm Jup}$) whose mass is drawn independently from the same CMF. This is also in line with the predictions of a high fraction of systems with multiple companions resulting from the CA formation scenario. These simple results point strongly toward a CA origin for the wide-orbit giant planets, as GI is expected to produce a relatively flat CMF (similar to the stellar IMF at low masses) and a corresponding low fraction of systems hosting multiple wide-orbit substellar companions To illustrate the robustness of this conclusion, we now turn to a critical assessment of our approach, considering its handling of observational biases, and its limitations. Finally, we discuss the general applicability of our results, and bearing the similarity of the CMF for wide-orbit and close-in giant planets, we argue for a general form of the CMF for planets within 100 AU. Observational Biases -------------------- We attempted to account for the observational biases by utilizing the statistical methods of survival analysis (as described in §2.2), which enables information contained within the detection limits to be incorporated into the derived CMF. In this analysis, we assumed that each detection limit corresponds to one non-detected object. This choice was motivated in part by simplicity, but also for physical reasons. Given that the shape of the CMF from the detections alone points strongly toward a bottom-heavy CMF, and thus a CA origin, it is reasonable to speculate that a high fraction of these systems hosts one or more additional companions of similar mass (perhaps close to 50% as in @Knutson2014). In reality, the average number of additional giant companions among these systems is very likely greater than zero, and of order unity, but is difficult to constrain further at present. Given that the detection limits typically correspond to planetary masses, it is not likely that unaccounted for observational biases would alter the inferred bottom-heavy form of the CMF. By assuming that the detection limits correspond to *any* companions that actually exist, or none at all, we still arrive at a bottom-heavy CMF. As discussed in §3.1, the inferred CMF is increasingly bottom-heavy as we assume that the detection limits correspond to a larger number of objects that actually exist, but whose masses are beneath the detection limits. In order to change this picture, many brown dwarf companions above the survey detection limits would have necessarily gone unreported, which is not likely given that these were targeted specifically by past surveys (e.g., @Metchev2009, @Brandt2014, @Galicher2016, @Vigan2017, @Stone2018, @Nielsen2019). Limitations and Generality of the Results ----------------------------------------- One potential limitation of our approach, which focuses on systems that host directly imaged giant planets and brown dwarfs, is that these results are only necessarily applicable to systems hosting such companions, which may themselves be exceptional. In other words, it is possible that the CMF among these systems differs from that of the average star. Here we present counter-arguments to this point, and suggest that these results are likely to be generally applicable. As a first consideration, these results may not be generally applicable if the systems considered here are non-representative of the general population (aside from the obvious and potentially exceptional property of hosting wide-orbit planets and brown dwarfs). Indeed, most of these systems share some similar properties, such as age (preferably young systems), and proximity, which are characteristics required to detect low-mass companions. Unless we reside in a particularly special location in the galaxy, there is no reason to expect that the nearby systems are different from the general population, so we can likely disregard the property of proximity. Youth, however, may play a role in causing the planets observed within the systems considered here to be exceptional. In particular, planets may experience significant orbital migration early in their lives. This is unlikely to bias our general results given that the median age of the systems considered here is 30 Myr, which is $\sim 10\times$ older than the typical disk lifetime. Nevertheless, some of the systems are young enough that they may still experience significant migration effects. In the case of outward migration, unless the companions are often ejected, they would still appear in the wide-orbit population, and have been considered as targets of this study. On the other hand, in the case that planets migrate inward, we may expect the mass functions of the inner and outer giant planets to be similar, which is indeed what is observed. Thus, it appears that youth may also be ruled out as a property that may cause our results to not be generally valid. Given the similarity in the CMF, we suggest that planets discovered by the direct imaging and RV surveys, constituting outer and inner planets, respectively, are drawn from the same distribution. This could be taken as evidence that the inner population forms first at wide-orbits and subsequently migrates inward (or vice versa), or that both populations formed *in situ* via similar processes. However, the picture is somewhat different at higher masses. While the inner and outer CMFs are of an overall bottom-heavy form, there are discernible differences among the relative frequency of brown dwarfs. The primary difference between the inner and outer CMFs is that there is a significant excess of brown dwarfs among the wide-orbit, directly imaged population compared to the mass function derived from RV surveys, and to the CMF predicted from CA population syntheses. This may be explained by a scenario in which GI is active in a minority of systems, and only in the outer regions, with the result typically being a wide-orbit brown dwarf or low mass stellar companion (as predicted by @Kratter2010, etc.). A Turn Over in the Wide-Orbit CMF? ---------------------------------- The wide-orbit CMF is best described by a distribution that is increasing sharply toward lower masses, similar to the form of the CMF of close-in planets. However, microlensing and transit surveys suggest that this behavior does not extend to arbitrarily low masses, and that a most likely companion-to-host mass-ratio exists. [@Suzuki2016] examined 22 planetary microlensing events from the MOA survey and inferred that the CMF follows a broken power law form with a peak at mass ratios of $q \sim$10$^{-4}$ (between Earth and Neptune’s mass for M-F stars). [@Udalski2018] confirmed this trend in the eight planetary-mass microlensing detections of the OGLE survey and suggested a peak at $q \sim$1.7$\times 10^{-4}$. [@Pascucci2018] performed a similar analysis for the *Kepler* planets, and found a break occurring at 2-3$\times$10$^{-5}$ that is universal among spectral types M-F. This break occurs at a mass ratio that is a few times lower than that for the microlensing planets, which are typically on wider orbits ($\gtrsim$1 AU) compared to those discovered by *Kepler*, suggesting that the location of the peak in the CMF may shift toward higher mass ratios with increasing orbital separation. While our sample consists exclusively of objects on wide-orbits ($\geq$8 AU), we do not resolve a peak in the CMF, which is likely due to the fact that we are limited to much higher mass ratios above $q \gtrsim 10^{-3}$. Given the similarity between the CMFs of the wide-orbit and close-in planets (e.g., @Malhotra2015, @Pascucci2018, @Fernandes2018), we may speculate that a similar break exists at lower mass ratios for the wide-orbit population, although this remains to be confirmed. We note that in subsamples D, E, and F, we do observe a tentative peak in the CMF at $\sim$10 M$_{\rm Jup}$, or $q\sim0.01-0.05$. This peak occurs at much higher mass ratios than the peak in the CMF inferred from transit and microlensing surveys, and is within the range of mass ratios representative of stellar binaries. Given that these subsamples consist of primarily objects on very wide-orbits ($\geq$100 AU) and/or around very late spectral types, this behavior is likely due to vastly different formation and evolution processes than those that give rise to the break at $q\sim$10$^{-4}$ for close-in planets. Identifying the Dominant Formation Mechanism as a Function of Mass ------------------------------------------------------------------ These results highlight an important difficulty in establishing a formation-motivated definition of what constitutes a “planet" at high-masses$-$namely, that it is impossible to completely determine how a given object has formed from knowledge of solely its mass. While not a complete determination, the fact that the CMF is a superposition of a CA-like distribution and a GI-like distribution enables us to assign a probability that an object of a given mass formed via one of these two mechanisms (similar to @Reggiani2016). Most objects beneath $\lesssim$10-20 M$_{\rm Jup}$ are representative of a CMF that is rising steeply toward lower masses, and thus likely originated via a CA-like formation process within a protoplanetary disk. Above $\gtrsim$10-20 M$_{\rm Jup}$, the opposite is true: most objects are representative of a flat CMF, similar to predictions from disk-born GI and the stellar IMF at low masses (e.g., @Kroupa2001), and thus were likely born in a manner incorporating a rapid initial hydrodynamic collapse akin to star formation. However, between 10-20 $M_{\rm Jup}$ there is apparently a similar likelihood of forming via either mechanism. This is similar to the findings of [@Schlaufman2018], who examined close in ($\leq$0.1 AU) transiting planets and brown dwarfs with Doppler inferred masses and found that bodies $\lesssim$10 M$_{\rm Jup}$ preferentially orbit solar-type dwarf stars with enhanced metalicity, while the same is not true for higher mass companions. Our study followed a different approach: we examined the relative mass function of wide-orbit companions ($\gtrsim$8 AU), and compared this to predictions from population synthesis models. We considered the point at which the synthetic mass distributions of CA and GI formed objects[^10] intersected to be the dividing line between planets and brown dwarfs. Despite following a vastly different approach, we found a similar result, which supports the notion that objects on either side of $\sim$10-20 M$_{\rm Jup}$ primarily form via distinct physical processes. Exploration of Sub-samples: Evidence for Different Populations of Companions ---------------------------------------------------------------------------- One possibility with the tools that we have developed for this study is to explore the differences in the relative mass function across different subsamples, which we have described in §3.5. We explored several subsamples to investigate the effect of our initial assumptions of not including M-stars, and not including companions exterior to 100 AU in our primary analysis. This was essentially a consistency check, to ensure that these assumptions did not significantly impact our main results, which we verified in §3.5. Additionally, we explored several subsamples for the purpose of investigating whether the CMF may vary interior/exterior to 100 AU, and around stars of different spectral types. We found that, in general, samples of companions interior to 100 AU show a more bottom-heavy mass function than samples restricted to companions exterior to 100 AU, and to hybrid populations that do not discriminate based on orbital separation.[^11] In other words, giant planets appear more frequently at smaller separations, although our data are not sensitive enough in the inner regions to resolve a turn-over in the distribution. From transit and RV planets, [@Fernandes2018] found a peak in the relative frequency of giant planets as a function of orbital separation at $\sim$2 AU, which is 4$\times$ smaller than the separation of our closest-in companion, $\beta$ Pic b. This implies that the relative frequency of giant planets continues to increase interior to the inner working angles of current high-contrast imaging facilities. Likewise, we found that earlier spectral type hosts tend to have more bottom-heavy mass functions than later spectral type hosts. This is in contrast to the rocky planets, which occur more frequently around lower mass stars (e.g., @Mulders2015), but which the observations considered here are insensitive to. The relative lack of 3-7 M$_{\rm Jup}$ planets around late spectral type hosts may reflect the unavailability of gas for giant planet formation within the protoplanetary disks around such low mass stars at later ages. In that case, the population of 7-13 M$_{\rm Jup}$ and higher mass companions could possibly represent either the population of companions born early on in the minority of gravitationally unstable disks around low mass stars, and/or those born even earlier as the low-mass end of binary star formation from turbulent fragmentation. An Order-of-Magnitude Assessment of The Frequency of HR 8799bcde-like Systems ----------------------------------------------------------------------------- A second possible avenue to explore following these results is the nature of an intriguing system among those considered here: HR 8799 (@Marois2008, @Marois2010). This system is remarkable among those with directly imaged planets as one of the few systems with (detected) multiple planets, and the only known system with four wide-orbit super-Jupiters. On a superficial level, this is consistent with expectations assuming that companions are drawn independently from the CMF representative of the CA scenario: each of its four known companions are of planetary mass, which is the most likely incarnation of a system with such a large number of companions given their higher relative frequency.[^12] Additionally, one may argue that such systems are likely not atypical given its close proximity to Earth ($\sim$40 pc), unless the density of such systems is for some reason higher than average at our present galactic position. We can make some very basic (order of magnitude) estimates on the frequency of such systems based on the results presented in this study. We must first make an additional assumption about the absolute occurrence rates of substellar objects, since we have presented only relative mass functions. This is easier at the high mass end, given the incompleteness of existing surveys at lower masses. Surveys sensitive to such companions (e.g., @Metchev2009, @Galicher2016, @Vigan2017, @Stone2018, @Nielsen2019) have shown that their frequency is on the order of a few percent, while the CMF presented here suggests approximately an order of magnitude more 3-7 M$_{\rm Jup}$ planets, comparable to those in HR 8799. Taking a pessimistic frequency of systems hosting a high mass brown dwarf to be $\sim$1%, then nearly $\sim$10% of systems would host a 3-7 M$_{\rm Jup}$ planet. Converting this into a probability that a single system will form multiple planets of this mass is more complicated, but as a simple approximation we will assume that the masses of the planets are drawn independently from the CMF. Thus, we arrive at a frequency of order $\sim$10$^{-4}$ for HR 8799-like systems. Within the 10 pc volume around the sun, there are roughly 400 known stars. Current instrumentation could detect an HR 8799-like system (a young system with multiple wide-orbit super-Jupiters) at a distance of $\sim$100 pc. Assuming the same stellar density as in the solar neighborhood, this volume contains roughly 4$\times$10$^{5}$ stars. Thus, if the occurrence rate of HR 8799 is roughly one in ten thousand stars, then there should be approximately forty such systems within 100 pc of Earth (though not all of these will be young). In other words, it is reasonable to expect that such a planetary system should exist at its proximity to the sun. However, this reasoning has not yet taken into account the fraction of stars that are young, and hence around which we could discover an HR 8799-like system. Assuming that the overall star formation rate is approximately flat in the galaxy, then the age distribution of stars (neglecting spectral type evolution) is also approximately flat, and consequently most stars will be quite old. For example, assuming an age distribution of 1 Myr to 10 Gyr results in only 0.4% that are 40 Myr or younger, which in turn results in $\sim$1,600 young stars within 100 pc, and a roughly 16% chance of observing such a system. While not impossible, the chance of this occurrence is still small enough to cause us to reconsider our initial assumptions. One possibility is that the density of young stars in the solar neighborhood may be higher than the galactic average. This is consistent with our position within the local bubble$-$a structure that is thought to represent multiple supernova explosions approximately 10-20 Mya and possibly related to the formation of the Gould belt 30-60 Mya (@Berg2002). This same event likely triggered further star formation, leading to an increased density of nearby young systems. If the local density of young stars is a few times higher than the galactic average, then we are left with a probability close to 100% of detecting an HR 8799-like system. Additionally, if the fraction of wide-orbit brown dwarf companions to nearby stars is closer to a few percent, instead of the pessimistic 1% assumed previously, this could further raise the probability of detecting an HR 8799-like system. Finally, it is possible that our initial assumption that planet masses are drawn independently from the CMF is incorrect. If systems hosting one super-Jupiter are in fact more likely to host an additional wide-orbit giant planet, this would further raise the probability of detecting an HR 8799-like system at its observed proximity and age. The latter scenario is supported, but not confirmed, by the result that a large fraction of the stars in our primary sample could host a second (typically undetected) wide-orbit planet more massive than 2 M$_{\rm Jup}$. This analysis is highly over-simplified, but is useful as a sanity check on our results, and as an estimated frequency of HR 8799-like systems. If the CMF that we uncovered predicted no chance of detecting a system consisting of four super-Jupiters, or if it instead predicted that we should have detected many such systems, this would be reason to doubt the results. As it stands, the sanity check is consistent with the observations. While HR 8799 is likely a rare outcome of the planet formation process (on the order of one such system per ten-thousand stars), it is reasonable that one such young system exists relatively close to the sun. Summary and Conclusions ======================= 1\) We computed the relative mass distribution for companions within 100 AU around BAFGK stars. We found a steep function with a rising slope toward lower masses, in line with predictions for a core accretion formed population, and in remarkable agreement with the CMF of the inner planets detected by RV surveys. 2\) We estimated the probability that each system may host multiple wide-orbit giant companions drawn this distribution, and found that, on average, the systems considered here have a $\lesssim$68.2% probability of hosting at least one additional wide-orbit giant companion whose mass is $\geq$2 M$_{\rm Jup}$. 3\) We verified that the above results are valid independent of model assumptions on initial planet luminosity (i.e., hot vs. cold start initial conditions), atmospheric dust content, and sample selection criteria (companions with projected separations $\leq$100 AU, excluding M-stars). 4\) We suggested that these results are consistent with a scenario in which CA is the primary mechanism at forming companions less massive than $\sim$10$-$20 $M_{Jup}$, and that GI is the primary mechanism at forming higher mass companions. 5) We explored the CMF of select subsamples, and find an enhanced population of super-Jupiters interior to 100 AU and around early-type hosts. 6\) As a sanity check, we estimated the frequency that these results would imply for HR 8799-like systems, and calculate the probability of detecting such a young system hosting multiple super-Jupiters with its proximity to the sun. We found that while HR 8799 is likely rare ($\sim 10^{-4}$ occurrence rate), it is reasonable that one such system has been discovered. 7\) Our analysis suggests that future deep observations of these and other targets should uncover a greater number of directly imaged planets, as the relative frequency of planets increases rapidly with decreasing mass. Acknowledgments =============== The authors acknowledge their sincere thanks to Jordan M. Stone and Thayne Currie for their insightful comments on an earlier version of this manuscript. The results reported herein benefited from collaborations and/or information exchange within NASA’s Nexus for Exoplanet System Science (NExSS) research coordination network sponsored by NASA’s Science Mission Directorate. KRW is supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. 2015209499. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. 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[lcccccccccccc]{} [1]{} & [1RXS1609b]{} & [Ks]{} & [16.2]{} & [0.180]{} & [22.0]{} & [140.]{} & [1.30]{} & [5.00]{} & [2.00]{} & [307.]{} & [7.00]{}\ [2]{} & [2M0103-55b]{} & [L$^{\prime}$]{} & [12.7]{} & [0.100]{} & [17.0]{} & [47.2]{} & [3.10]{} & [35.0]{} & [15.0]{} & [84.0]{} & [9.00]{}\ [3]{} & [2M0122-24b]{} & [Ks]{} & [14.0]{} & [0.110]{} & [21.7]{} & [33.9]{} & [0.0860]{} & [120.]{} & [10.0]{} & [49.1]{} & [9.00]{}\ [4]{} & [2M0219-39b]{} & [Ks]{} & [13.8]{} & [0.100]{} & [14.7]{} & [40.1]{} & [0.190]{} & [35.0]{} & [5.00]{} & [160.]{} & [9.00]{}\ [5]{} & [2M2236+47b]{} & [Ks]{} & [17.4]{} & [0.0400]{} & [19.1\*]{} & [69.7]{} & [0.160]{} & [120.]{} & [10.0]{} & [258.]{} & [7.00]{}\ [6]{} & [2M2250+23b]{} & [Ks]{} & [14.9]{} & [0.0400]{} & [16.6]{} & [57.3]{} & [0.140]{} & [165.]{} & [35.0]{} & [510.]{} & [9.00]{}\ [7]{} & [51Erib]{} & [Ks]{} & [17.5]{} & [0.140]{} & [19.5]{} & [29.8]{} & [0.120]{} & [20.0]{} & [6.00]{} & [13.4]{} & [2.00]{}\ [8]{} & [ABPicb]{} & [Ks]{} & 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[0.0800]{} & [4000]{} & [1800]{} & [43.9]{} & [4.00]{}\ [16]{} & [GJ570b]{} & [Ks]{} & [15.3]{} & [0.170]{} & [15.6\*]{} & [5.88]{} & [0.00294]{} & [6000]{} & [4000]{} & [1520]{} & [7.00]{}\ [17]{} & [GJ758b]{} & [L$^{\prime}$]{} & [16.0]{} & [0.190]{} & [16.0\*]{} & [15.6]{} & [0.00540]{} & [4700]{} & [4000]{} & [28.1]{} & [6.00]{}\ [18]{} & [GQLupb]{} & [L$^{\prime}$]{} & [11.7]{} & [0.100]{} & [12.6]{} & [152.]{} & [1.10]{} & [2.00]{} & [1.00]{} & [106.]{} & [7.00]{}\ [19]{} & [GSC6214-21b]{} & [Ks]{} & [14.9]{} & [0.100]{} & [15.8\*]{} & [109.]{} & [0.510]{} & [5.00]{} & [2.00]{} & [241.]{} & [8.00]{}\ [20]{} & [GUPScb]{} & [Ks]{} & [17.7]{} & [0.0300]{} & [21.6]{} & [47.6]{} & [0.160]{} & [100.]{} & [30.0]{} & [2000]{} & [9.00]{}\ [21]{} & [HD106906b]{} & [L$^{\prime}$]{} & [14.6]{} & [0.100]{} & [16.7]{} & [103.]{} & [0.400]{} & [13.0]{} & [2.00]{} & [734.]{} & [3.00]{}\ [22]{} & [HD1160b]{} & [Ks]{} & [14.0]{} & [0.120]{} & [19.0]{} & [126.]{} & [1.20]{} & [165.]{} & [135.]{} & [96.9]{} & [0.00]{}\ [23]{} & [HD19467b]{} & [Ks]{} & [18.0]{} & [0.0900]{} & [19.0\*]{} & [32.0]{} & [0.0400]{} & [7300]{} & [2700]{} & [52.8]{} & [5.00]{}\ [24]{} & [HD203030b]{} & [Ks]{} & [16.2]{} & [0.100]{} & [17.1\*]{} & [39.3]{} & [0.100]{} & [90.0]{} & [60.0]{} & [468.]{} & [6.00]{}\ [25]{} & [HD206893b]{} & [L$^{\prime}$]{} & [13.4]{} & [0.160]{} & [14.8]{} & [40.8]{} & [0.100]{} & [325.]{} & [275.]{} & [20.4]{} & [3.00]{}\ [26]{} & [HD284149b]{} & [Ks]{} & [14.3]{} & [0.0400]{} & [19.1]{} & [118.]{} & [0.710]{} & [32.5]{} & [17.5]{} & [437.]{} & [4.00]{}\ [27]{} & [HD4113c]{} & [Ks]{} & [19.7]{} & [0.120]{} & [20.4]{} & [41.9]{} & [0.0900]{} & [4800]{} & [1500]{} & [21.8]{} & [5.00]{}\ [28]{} & [HD95086b]{} & [Ks]{} & [18.8]{} & [0.300]{} & [20.8]{} & [86.4]{} & [0.200]{} & [17.0]{} & [4.00]{} & [53.6]{} & [2.00]{}\ [29]{} & [HD984b]{} & [Ks]{} & [12.2]{} & [0.0400]{} & [16.1]{} & [45.9]{} & [1.03]{} & [115.]{} & [85.0]{} & [9.18]{} & [3.00]{}\ [30]{} & [HII1348b]{} & [Ks]{} & [14.9]{} & [0.100]{} & [15.7\*]{} & [143.]{} & [0.996]{} & [113.]{} & [12.5]{} & [157.]{} & [7.00]{}\ [31]{} & [HIP64892b]{} & [Ks]{} & [13.6]{} & [0.100]{} & [20.8]{} & [125.]{} & [1.40]{} & [20.0]{} & [11.0]{} & [150.]{} & [0.00]{}\ [32]{} & [HIP65426b]{} & [Ks]{} & [16.6]{} & [0.300]{} & [18.8]{} & [109.]{} & [0.710]{} & [14.0]{} & [4.00]{} & [90.6]{} & [1.00]{}\ [33]{} & [HIP73990b]{} & [L$^{\prime}$]{} & [13.3]{} & [0.400]{} & [13.8]{} & [111.]{} & [0.790]{} & [15.0]{} & [5.00]{} & [17.7]{} & [2.00]{}\ [34]{} & [HIP73990c]{} & [L$^{\prime}$]{} & [13.2]{} & [0.450]{} & [13.8]{} & [111.]{} & [0.790]{} & [15.0]{} & [5.00]{} & [27.7]{} & [2.00]{}\ [35]{} & [HIP74865b]{} & [L$^{\prime}$]{} & [12.8]{} & [0.300]{} & [14.3]{} & [124.]{} & [0.940]{} & [15.0]{} & [5.00]{} & [24.7]{} & [3.00]{}\ [36]{} & [HIP77900b]{} & [Ks]{} & [14.0]{} & [0.01000]{} & [17.3\*]{} & [151.]{} & [2.69]{} & [5.00]{} & [1.00]{} & [3300]{} & [9.00]{}\ [37]{} & [HIP78530b]{} & [Ks]{} & [14.2]{} & [0.0400]{} & [15.9\*]{} & [137.]{} & [1.50]{} & [5.00]{} & [1.00]{} & [618.]{} & [0.00]{}\ [38]{} & [HNPegb]{} & [Ks]{} & [15.1]{} & [0.0300]{} & [17.2\*]{} & [18.1]{} & [0.0200]{} & [350.]{} & [50.0]{} & [783.]{} & [4.00]{}\ [39]{} & [HR2562b]{} & [Ks]{} & [16.6]{} & [0.140]{} & [20.1]{} & [33.6]{} & [0.300]{} & [475.]{} & [275.]{} & [20.2]{} & [3.00]{}\ [40]{} & [HR3549b]{} & [Ks]{} & [15.1]{} & [0.100]{} & [22.0]{} & [95.4]{} & [0.810]{} & [125.]{} & [25.0]{} & [85.8]{} & [0.00]{}\ [41]{} & [HR7329b]{} & [Ks]{} & [11.9]{} & [0.0600]{} & [13.2\*]{} & [48.2]{} & [0.480]{} & [20.0]{} & [10.0]{} & [193.]{} & [0.00]{}\ [42]{} & [HR8799b]{} & [L$^{\prime}$]{} & [15.7]{} & [0.120]{} & [18.7]{} & [41.3]{} & [0.100]{} & [30.0]{} & [10.0]{} & [70.9]{} & [2.00]{}\ [43]{} & [HR8799c]{} & [L$^{\prime}$]{} & [14.8]{} & [0.0900]{} & [18.7]{} & [41.3]{} & [0.100]{} & [30.0]{} & [10.0]{} & [39.3]{} & [2.00]{}\ [44]{} & [HR8799d]{} & [L$^{\prime}$]{} & [14.8]{} & [0.140]{} & [18.7]{} & [41.3]{} & [0.100]{} & [30.0]{} & [10.0]{} & [27.1]{} & [2.00]{}\ [45]{} & [HR8799e]{} & [L$^{\prime}$]{} & [14.9]{} & [0.160]{} & [18.7]{} & [41.3]{} & [0.100]{} & [30.0]{} & [10.0]{} & [16.3]{} & [2.00]{}\ [46]{} & [PDS70b]{} & [L$^{\prime}$]{} & [14.5]{} & [0.420]{} & [14.9]{} & [113.]{} & [0.520]{} & [5.40]{} & [1.00]{} & [22.1]{} & [7.00]{}\ [47]{} & [PZTelb]{} & [Ks]{} & [11.5]{} & [0.0700]{} & [18.4]{} & [47.1]{} & [0.130]{} & [24.0]{} & [2.00]{} & [23.5]{} & [6.00]{}\ [48]{} & [ROXs12b]{} & [L$^{\prime}$]{} & [12.6]{} & [0.0900]{} & [13.6\*]{} & [137.]{} & [0.750]{} & [6.50]{} & [3.50]{} & [233.]{} & [8.00]{}\ [49]{} & [ROXs42Bb]{} & [L$^{\prime}$]{} & [14.1]{} & [0.0900]{} & [15.3\*]{} & [144.]{} & [1.50]{} & [2.00]{} & [1.00]{} & [159.]{} & [8.00]{}\ [50]{} & [ROSS458ABc]{} & [Ks]{} & [16.5]{} & [0.0300]{} & [18.5\*]{} & [11.5]{} & [0.0200]{} & [295.]{} & [145.]{} & [1170]{} & [8.00]{}\ [51]{} & [SR12ABc]{} & [L$^{\prime}$]{} & [13.1]{} & [0.0800]{} & [14.2\*]{} & [112.]{} & [5.17]{} & [5.25]{} & [4.75]{} & [977.]{} & [8.00]{}\ [52]{} & [TWA5A(AB)b]{} & [L$^{\prime}$]{} & [12.1]{} & [0.100]{} & [12.9]{} & [49.4]{} & [0.140]{} & [10.0]{} & [5.00]{} & [98.8]{} & [9.00]{}\ [53]{} & [Usco1610-19b]{} & [Ks]{} & [12.7]{} & [0.01000]{} & [18.2]{} & [144.]{} & [7.25]{} & [5.00]{} & [1.00]{} & [833.]{} & [9.00]{}\ [54]{} & [Usco1612-18b]{} & [Ks]{} & [13.2]{} & [0.01000]{} & [18.2]{} & [158.]{} & [7.60]{} & [5.00]{} & [1.00]{} & [475.]{} & [9.00]{}\ [55]{} & [BetaCirb]{} & [Ks]{} & [13.2]{} & [0.0400]{} & [16.5]{} & [28.4]{} & [0.330]{} & [435.]{} & [65.0]{} & [6200]{} & [1.00]{}\ [56]{} & [BetaPicb]{} & [Ks]{} & [12.5]{} & [0.130]{} & [25.0]{} & [19.8]{} & [0.150]{} & [14.0]{} & [6.00]{} & [8.69]{} & [1.00]{}\ [57]{} & [KappaAndb]{} & [L$^{\prime}$]{} & [13.1]{} & [0.0900]{} & [17.4]{} & [51.6]{} & [0.500]{} & [35.0]{} & [15.0]{} & [56.8]{} & [0.00]{}\ Appendix B: Probability Distributions of Companion Masses and Detection Limits ============================================================================== In Fig. B1 we present probability distributions for companion masses and upper mass limits on additional companions that are described in §2.1. Many distributions show double-peaked and more complicated distributions, which is an artifact of degenerate mass-age-luminosity solutions within the measurement uncertainties. This effect is particularly prevalent for objects with large age uncertainties, as the spectral type transitions (representative of different cloud properties and atmospheric chemical abundances) result in non-linear evolution of photometric brightness with age [@Burrows2006]. [^1]: For a recent review of directly imaged planetary mass companions, see [@Bowler2016]. [^2]: Data obtained from exoplanet.eu [@Schneider2011] on January 4, 2019. [^3]: The masses are typically estimated via the combination of the system’s age and distance, the companion’s photometry, and a model grid that describes the mass-luminosity-age evolution. [^4]: http://exoplanetarchive.ipac.caltech.edu [^5]: The formula used for this conversion, and the systems for which it has been applied, can be found in Table A1. [^6]: Where necessary, we convert asymmetric uncertainties into a symmetric age range. [^7]: In this way, we implicitly assume that each detection limit corresponds to one non-detected object. On average, this is likely a reasonable approximation, and we will discuss the effect of altering this assumption in §3.1 and §4. [^8]: These mass bins are selected to roughly coincide with the inflection points in the distribution. The exact selection of mass bins does not significantly affect the results. [^9]: Data obtained from exoplanet.eu on November 7, 2018. [^10]: With the latter scaled to the higher mass brown dwarfs. [^11]: We note also that planets on $\gtrsim$100 AU orbits could be captured free-floating planets. For predictions regarding the characteristics of such a population, see [@Perets2012]. [^12]: Planetary masses are also expected for multiple companions on wide-orbits within the same system considering the requirements for dynamical stability (@Fabrycky2010).

--- abstract: 'In this paper, we derive a probabilistic registration algorithm for object modeling and tracking. In many robotics applications, such as manipulation tasks, nonvisual information about the movement of the object is available, which we will combine with the visual information. Furthermore we do not only consider observations of the object, but we also take space into account which has been observed to not be part of the object. Furthermore we are computing a posterior distribution over the relative alignment and not a point estimate as typically done in for example [*Iterative Closest Point*]{} (ICP). To our knowledge no existing algorithm meets these three conditions and we thus derive a novel registration algorithm in a Bayesian framework. Experimental results suggest that the proposed methods perform favorably in comparison to PCL [@rusu11] implementations of feature mapping and ICP, especially if nonvisual information is available.' author: - 'Manuel Wüthrich, Peter Pastor, Ludovic Righetti, Aude Billard, Stefan Schaal[^1][^2][^3][^4]' bibliography: - 'paper.bib' title: '**Probabilistic Depth Image Registration incorporating Nonvisual Information** ' --- [100mm]{}(.,-10.5cm) [0.8]{} [2012 IEEE International Conference on\ Robotics and Automation (ICRA)\ May 14-18, 2012. Saint Paul, USA]{} INTRODUCTION ============ In this paper we will focus on the scenario where the camera is fixed and only the object is manipulated. While the object is being moved, a 3D camera gathers depth images of the object in different orientations and positions. Let us denote two such images as image $A$ and image $B$. The core problem considered in this paper is to estimate the rigid body transformation ${\boldsymbol{T}}$ the object has undergone between the acquisitions of these two images. Segmentation is not the focus of this work, we employ existing algorithms [@rusu11] to determine whether a pixel in the depth image belongs to the object or to the background. A great deal of work has been done in this research area in the past years. In [@cui10] an algorithm is presented which creates 3D models of objects while the camera or the object is moved. However, the point clouds have to be approximately aligned initially and the model is created off line by optimizing the alignment of all images simultaneously. Our method is more general in the sense that point clouds do not have to be approximately aligned. However, task-specific assumptions like that can be introduced to significantly reduce the computational time for finding the optimal alignment. A lot of very promising work, such as [@krainin11; @li09], has been published in the last years about scanning objects while they are being held by the robot. We however want to treat a more general case where we do not assume that the object is already grasped or can be grasped in a straightforward manner. In [@rusu09] models are constructed by mapping shape primitives to the point clouds with promising results. In this work however we try to make as few assumptions as possible about the shape of the object and thus exclude the use of models or shape primitives. Among the most popular algorithms that tackle the registration problem are Iterative Closest Point (ICP) and feature mapping algorithms and combinations of both [@dai11; @liu02; @rusu08; @fukai11; @rusu09_fpfh]. We will compare the proposed method with these two approaches. ICP has been proven to converge to a local minima [@ICP:92]. In the scenario considered in this paper, an object can move very fast and therefore, point clouds of two subsequent images are not necessarily approximately aligned. This problem is usually tackled by initially aligning point clouds using a feature mapping algorithm [@dai11; @rusu08; @fukai11]. These methods perform well, if different parts of the object can easily be distinguished. For objects with a homogeneous texture, color or local shape, feature matching can be problematic. Furthermore if the quality of the features degrades with the quality of the point cloud, noisy data can cause problems. Often in robotics there is a great deal of nonvisual information about the transformation of the object available. In our scenario, this information can for example be that an object is pushed on a table and the movement will therefore be in a plane. If it is held by a robot, we approximately know how the object will move. This kind of information can certainly be incorporated in ICP and feature mapping algorithms, but they are not originally designed to do so. ICP and feature mapping algorithms commonly optimize a cost function that is only dependent on the relative alignment between two point clouds. In our proposed method, we take into account the space which has been observed to not contain any part of the object. Our results suggest that taking this information into account leads to more robust registration results. Introducing visibility constraints has previously been shown to help in estimating the occluded shape of an unknown object [@bohg:icra11]. Finally, feature mapping and ICP algorithms usually return a point estimate of the transformation and a fitness. It can however be preferable to have a more differentiated estimate of the transformation in form of a probability distribution over the 6 parameters of the transformation. This allows us to express, for example, that we are certain about the rotation around axis $x$ but uncertain about the translation in $y$ etc. In the results section we will show an example of the use of a probability distribution as result. To our knowledge, there is no registration algorithm that combines the three mentioned points: 1. Cost function based on visibility constraints. 2. Output of a posterior distribution over the estimated object pose change. 3. Straightforward incorporation of task-relevant nonvisual information. In the next section, we will derive the proposed registration algorithm in a Bayesian framework. In the result section, we show that under certain conditions that are quite common in the scenario of object model learning and tracking, our algorithm outperforms implementations of ICP and feature mapping methods. DERIVATION ========== Incorporated Information ------------------------ An overview of all the information we will make use of can be seen in Fig. \[fig:variables\]. The input data $D$ consists of the visual information $V$ and the nonvisual information $N$. ![Overview of the variables[]{data-label="fig:variables"}](variables.pdf) ### Nonvisual Information In the context of a robotic manipulation task often a great deal of nonvisual information about the movement of an object is available. $N$ can contain for example the information that the object will be moved on a table, that the robot has poked it with a certain movement or that the object is being held by the robot, and we thus know how it has moved approximately.\ ### Visual Information We divide the visual information into two types (see Fig. \[fig:concept\]). Firstly there are surface patches which are observed by the depth camera, from now on referred to as patches $P$. These patches can be represented as a point cloud and are thus the only information used by ICP and by most feature mapping algorithms.\ ![Two types of visual information: Surface patches $P$ and mask $M$.[]{data-label="fig:concept"}](concept.pdf) There is however another very important piece of information. No part of the object is inside the green area in Fig. \[fig:concept\], this area defines thus a mask $M$ for the object.\ We will always register two depth images, $A$ and $B$, at a time, therefore we have of course the masks, $M_A$ and $M_B$, as well as the patches, $P_A$ and $P_B$, from each image (see Fig. \[fig:variables\]). Parametrization --------------- ### Coordinate system Given that we will work with depth images we choose a suitable parametrization assuming the pinhole model for the camera. The first two parameters, $w$ and $h$, are chosen to be the projections of a 3D point onto a virtual image plane given a focal length of 1m, see Fig. \[fig:coordinates\]. ![Schematic representation of eight pixels acquired by the depth camera. The coordinates $r$ and $w$ are represented, while $h$ would be perpendicular to the image plane.[]{data-label="fig:coordinates"}](coordinates.pdf) The third parameter $r$ is the depth of the 3D point. These coordinates will be called ray coordinates. They are derived from Cartesian coordinates as follows: $$\begin{aligned} w &= \frac{x}{z}, ~ h = \frac{y}{z}, ~r = \sqrt{x^2 + y^2 + z^2}\end{aligned}$$ ### Rigid body transformation The rigid body transformation ${\boldsymbol{T}}$ has six independent parameters ${\boldsymbol{T}}=(T_1,...,T_6)^{\top}$. The parametrization can be chosen to be whatever is convenient for a given application. Measurement Error ----------------- Due to measurement errors in the camera, an observed 3D point ${\boldsymbol{p}}$ will not exactly correspond to the true point ${\boldsymbol{s}}$ on the object surface. As a measurement model $p({\boldsymbol{p}}|{\boldsymbol{s}})$ we use a normal distribution in ray coordinates. $$\begin{aligned} p({\boldsymbol{p}}|{\boldsymbol{s}}) &= \mathcal{\mathcal{N}}({\boldsymbol{s}}|{\boldsymbol{p}}, L) \label{eq:emissionprob}\end{aligned}$$ The covariance matrix $L$ is camera specific. The only assumption we make in our derivation is that the covariance matrix is such that $p({\boldsymbol{p}}|{\boldsymbol{s}})$ can be reasonably well approximated as being constant within a pixel. This assumption is sensible because the depth camera is not able to distinguish between points within the range of one pixel.\ Furthermore, assuming that $p({\boldsymbol{p}})$ and $p({\boldsymbol{s}})$ are uniform in the range of the depth camera, we have $p({\boldsymbol{s}}|{\boldsymbol{p}}) = p({\boldsymbol{p}}|{\boldsymbol{s}})$.\ Derivation ---------- Our objective is to express $p({\boldsymbol{T}}|D)$, the probability distribution over the transformation ${\boldsymbol{T}}$ the object has undergone, given all the available data $D$. Applying Bayes we have$$\begin{aligned} p({\boldsymbol{T}}|D) &= p({\boldsymbol{T}}|N,P,M)\\ &=\frac{ p(M|N,P,{\boldsymbol{T}}) p({\boldsymbol{T}}|N,P)}{\int p(M|N,P,{\boldsymbol{T}}) p({\boldsymbol{T}}|N, P)d{\boldsymbol{T}}} \label{eq:pTDD}\end{aligned}$$ In $p(M|N,P,{\boldsymbol{T}})$, given the transformation ${\boldsymbol{T}}$, the mask $M$ does not depend on the nonvisual information $N$ and we thus have $$\begin{aligned} p(M|N,P,{\boldsymbol{T}}) &=p(M|P,{\boldsymbol{T}})\\ &= p(M_A, M_B|P_A, P_B, {\boldsymbol{T}})\end{aligned}$$We assume $M_A$ and $M_B$ to be independent because the mask observed in one image does not give us any useful information about the mask observed in the other image.$$\begin{aligned} p(M|N,P,{\boldsymbol{T}}) &= p(M_A|P_A, P_B, {\boldsymbol{T}}) p(M_B|P_A, P_B, {\boldsymbol{T}}) \end{aligned}$$ As $M_A$ and $P_A$ are from the same image, the object $P_A$ will necessarily be respected in the mask $M_A$. $P_A$ does thus not add any information to the first term and can be removed. Similarly, for the second term we can omit $P_B$.$$\begin{aligned} p(M|N,P,{\boldsymbol{T}})&= p(M_A|P_B, {\boldsymbol{T}}) p(M_B|P_A, {\boldsymbol{T}}) \end{aligned}$$ It is reasonable to assume that the priors $p(M|{\boldsymbol{T}})$ and $p(P|{\boldsymbol{T}})$ are uniform because we do not have any prior information about the distribution of the points and the mask. Applying of Bayes’ rule, we thus have$$\begin{aligned} \addtolength{\fboxsep}{5pt} p(M|N,P,{\boldsymbol{T}})=k p(P_B| M_A, {\boldsymbol{T}}) p(P_A| M_B, {\boldsymbol{T}})\end{aligned}$$ with k being a constant. Inserting this result into Eq. \[eq:pTDD\] we obtain $$\begin{aligned} p({\boldsymbol{T}}|D) &=\frac{ p(P_B|M_A,{\boldsymbol{T}}) p(P_A|M_B,{\boldsymbol{T}})p({\boldsymbol{T}}|N, P)}{\int p(P_B|M_A,{\boldsymbol{T}}) p(P_A|M_B,{\boldsymbol{T}})p({\boldsymbol{T}}|N, P)d{\boldsymbol{T}}}\notag\end{aligned}$$ Finding this distribution is intractable, but for most purposes we do not need the distribution itself, we only use it for evaluating expectations. We thus need to find the expectation of a function $f({\boldsymbol{T}})$ expressing a property of ${\boldsymbol{T}}$ required for a given application. If $f$ is for example identity ($f({\boldsymbol{T}}) = {\boldsymbol{T}}$), then $E(f({\boldsymbol{T}})) = E({\boldsymbol{T}})$, or if $f({\boldsymbol{T}}) = ({\boldsymbol{T}} - E({\boldsymbol{T}})) ({\boldsymbol{T}} - E({\boldsymbol{T}}))^\top$ then $E(f({\boldsymbol{T}}))$ is the covariance matrix. The expectation of a function of ${\boldsymbol{T}}$ is $$\begin{aligned} &E(f({\boldsymbol{T}})) =\\ &\int \frac{ p(P_B|M_A,{\boldsymbol{T}}) p(P_A|M_B,{\boldsymbol{T}})p({\boldsymbol{T}}|N, P)}{\int p(P_B|M_A,{\boldsymbol{T}}) p(P_A|M_B,{\boldsymbol{T}})p({\boldsymbol{T}}|N, P)d{\boldsymbol{T}}}f({\boldsymbol{T}})d{\boldsymbol{T}} \notag\\ \Aboxed{ &E(f({\boldsymbol{T}})) \approx \sum_{l=1}^L w^{(l)} f({\boldsymbol{T}}^{(l)})}\label{eq:sampling} \end{aligned}$$ Where the samples ${\boldsymbol{T}}^{(l)}$ are drawn from $p({\boldsymbol{T}}|N,P)$. The sampling weights $w^{(l)}$ are defined by $$\begin{aligned} \Aboxed{&w^{(l)} = \frac{p(P_B| M_A, {\boldsymbol{T}}^{(l)}) p(P_A| M_B, {\boldsymbol{T}}^{(l)})}{\sum_{m=1}^Lp(P_B| M_A, {\boldsymbol{T}}^{(m)}) p(P_A| M_B, {\boldsymbol{T}}^{(m)})} } \label{eq:sampling2} \end{aligned}$$ We thus have represented $p({\boldsymbol{T}}|D)$ by a set of samples $\{{\boldsymbol{T}}^{(l)}\}$ and the corresponding weights $\{w^{(l)}\}$. The samples are drawn from $p({\boldsymbol{T}}|N,P)$, in other words, we will create a distribution, from which it is possible to sample, taking into account the nonvisual information as well as the observed surface patches. This distribution will be defined independently for a given application, an example is discussed in the results section.\ The terms $p(P_A|M_B,{\boldsymbol{T}})$ and $p(P_B|M_A,{\boldsymbol{T}})$ determine the weight of a given sample. The first one expresses the likelihood of ${\boldsymbol{T}}$ given the patches observed in $A$ and the mask observed in $B$. It essentially states that the transformation ${\boldsymbol{T}}$ has to be such that the patches observed in $A$ fit into the mask observed in $B$. Conversely the second term assures that the patches from $B$ fit into the mask from $A$.\ Now we will express $p(P_A|M_B,{\boldsymbol{T}})$. $P_A$ is the set of all the surface patches observed in image $A$ and can be expressed as a set of points $\{{\boldsymbol{a}}_{1}, {\boldsymbol{a}}_{2}, ... , {\boldsymbol{a}}_n\}$ . Similarly we have $P_B = \{{\boldsymbol{b}}_{1}, {\boldsymbol{b}}_{2}, ... , {\boldsymbol{b}}_{m}\}$. We can now write $$\begin{aligned} p(P_B| M_A, {\boldsymbol{T}}) &= p({\boldsymbol{b}}_{1}, {\boldsymbol{b}}_{2}, ... , {\boldsymbol{b}}_{m}| M_A,{\boldsymbol{T}}) \end{aligned}$$ Given the mask $M_A$ observed in image A, we look at the points $P_B$ observed in image $B$ as independent observations: $$\begin{aligned} \Aboxed{p(P_B| M_A, {\boldsymbol{T}}) &= \prod_{j = 1}^m p({\boldsymbol{b}}_j| M_A,{\boldsymbol{T}}) \label{eq:P_B}}\end{aligned}$$ After the derivation in Appx. \[ap:der\] we have\ \ with $D, {\boldsymbol{v}}, K_2, Z$ defined in Appx. \[ap:der\]. The second term in Eq. \[eq:sampling2\], $ p(P_A| M_B, {\boldsymbol{T}})$, can be expressed analogously. Discussion ---------- The first term in Eq. \[eq:pbd\] is a Gaussian over the parameters $w,h$ with mean $(w_i, h_i)^\top$. This term accounts for the fact that the closer $[{\boldsymbol{b}}_j]_A$ is to a pixel $i$, the likelier it is that the point which has been observed at ${\boldsymbol{b}}_j$ in image $B$ is observed in pixel $i$ in image $A$. The second term goes to zero if the depth of $[{\boldsymbol{b}}_j]_A$ is smaller than the depth at the pixel where it is projected on in image $A$, which is necessary in order to respect the mask $M_A$.\ Given that $p(P_B| M_A, {\boldsymbol{T}})$ (see Eq. \[eq:P\_B\]) is the product of all $p({\boldsymbol{b}}_j| M_A,{\boldsymbol{T}})$, it is zero if any $p({\boldsymbol{b}}_j| M_A,{\boldsymbol{T}})$ is zero. This result is illustrated in Fig. \[fig:principal\], all of the red points have to be inside the blue area. Implementation ============== The only parameter that has to be determined for our algorithm is the covariance matrix of the camera uncertainty (Eq. \[eq:emissionprob\]). This is however not a parameter that has to be optimized, it represents a meaningful quantity and should be estimated for the depth camera that is used. For our experiments with the Kinect camera we estimated the covariance matrix to be isotropic with $\sigma = 0.002$, which corresponds approximately to the resolution in ray coordinates of the Kinect. These values are a very rough estimation of the properties of the Kinect, but they prove to work well in the experiments.\ The core of the algorithm looks as follows: - For K samples - Sample from $p({\boldsymbol{T}}|N,P)$ - For all points in $B$ - if $p({\boldsymbol{b}}_j| M_A,{\boldsymbol{T}})$ is zero, sample a new transform - else $p(P_B| M_A, {\boldsymbol{T}}^{(l)}) ~*= p({\boldsymbol{b}}_j| M_A,{\boldsymbol{T}})$ - Do the same for points in $A$ - Given all the $p(P_A| M_B, {\boldsymbol{T}}^{(l)})$ and $p(P_B| M_A, {\boldsymbol{T}}^{(l)})$ we can compute the covariance matrix and the mean of ${\boldsymbol{T}}$ according to Eq. \[eq:sampling\] and Eq. \[eq:sampling2\]. Results ======= As mentioned in the introduction, the algorithms we want to compare against are ICP and feature mapping. We used the implementations in the Point Cloud Library (PCL) of these algorithms for our evaluation. We employed FPFH features which are described in [@rusu09_fpfh].\ \[fig:objects\] {width="45.00000%"} Our dataset consists of three objects, a box, a flashlight and a tube. Our dataset is small, the three objects however have a big variety in shape as seen in Fig. \[fig:objects\], and therefore this evaluation gives us a reasonable idea about the performance of our algorithm. Admittedly a broader evaluation will be necessary for a more precise assessment of the performance. In the associated video the algorithm is applied to a series of different objects on a tabletop [^5]. Each of the three objects has been rotated in steps of about $25^o$ and translated by a few $cm$ 14 times on a tabletop. At each step we acquire a depth image and measure the object’s exact position and orientation which will be used as ground truth. For evaluation we will align each image to the next, which gives a total of 13 alignments per object. We compare our algorithm to ICP, feature mapping and feature mapping with subsequent ICP. We use the implementations of these algorithms in the Point Cloud Library (PCL) [@rusu11]. The feature mapping algorithm uses Fast Point Feature Histograms (FPFH) as shape features [@rusu09_fpfh]. We used these algorithms to our best knowledge and implemented them as suggested in tutorials of PCL. We do not claim that the performance we measure here for ICP and feature mapping is the maximum that can be achieved with these algorithms, but it serves as a good point of comparison for our new algorithm. Evaluation of Alignment Performance without Nonvisual Information ----------------------------------------------------------------- In order to obtain a general estimate of the alignment performance of our algorithm we only make very general assumptions for the sampling distribution $p({\boldsymbol{T}}|N,P_A,P_B)$. We will assume that we have no information $N$, we do thus *not* use the information that object has only been translated and rotated on a table top. We only assume that the center of mass of $P_A$ will be no further than 4cm from the center of mass of $P_B$ in the aligned images and that the object will not be rotated by more than 50 degrees at a time. Note that these assumptions leave a very big search space open, and, therefore, we have to draw a very large amount of samples – about 100 million – and the algorithm is thus slow and takes about 30 seconds per image. ICP took about 1 second and feature mapping took about 5. In practice however we will have much stronger sampling distributions which will accelerate our algorithm considerably. ![Boxplot of alignment error for different algorithms.[]{data-label="fig:boxplot"}](boxplot.pdf){width="40.00000%"} In Fig. \[fig:boxplot\] we present the box-plots of the alignment error in degree of the four algorithms for all the objects. Our algorithm performs favorably compared to these implementations of ICP and feature mapping. We will now try to investigate how this advantage emerges. ![Flashlight aligned by ICP. Alignment error = $32^o$.[]{data-label="fig:icp_flashlight"}](icp_flashlight.png){width="40.00000%"} Fig. \[fig:icp\_flashlight\] shows an alignment performed by ICP with an error of $32^{\circ}$. The top image shows the aligned point clouds. The two bottom images represent the information about the mask. The left one illustrates $p([{\boldsymbol{b}}_j]_A|M_A,{\boldsymbol{T}})$. In the blue area the object has been observed, in the gray area background has been observed, and in the black area no observation has been made. The red points represent the points observed in $B$ projected into image $A$, $[{\boldsymbol{b}}_j]_A$. The result of our derivation suggests that the red points can only be in the blue or black area. If a point $[{\boldsymbol{b}}_j]_A$ is located on a pixel ${\boldsymbol{a}}_j$ in the blue area its distance to the camera $r$ has to be approximately equal or larger than the depth measured at ${\boldsymbol{a}}_j$. If the point is located in the black area its distance to the camera can be arbitrary because no depth has been measured at the corresponding pixel. ICP however only uses the information contained in the point clouds, which are quite sparse in the considered images. Looking at the top image of Fig. \[fig:icp\_flashlight\] it does not surprise that ICP performs poorly on this data. If we look at the two bottom images however we can see that many of the projected points are in front of the background. Taking this information into account we thus know that this alignment is not correct. ![Flashlight aligned by new algorithm. Alignment error = $2.1^o$.[]{data-label="fig:alpha_flashlight"}](alpha_flashlight.png){width="40.00000%"} In Fig. \[fig:alpha\_flashlight\] the alignment of the same two images by our algorithm is shown. Even though the point clouds are quite sparse it has performed well thanks to the information about the mask. Fig. \[fig:icp\_fm\_box\] illustrates a problem of a different nature that occurred with feature mapping and ICP. The problem here is that this box, looking only at the point clouds, allows different alignments. The red point cloud should be rotated about $90^\circ$ to the left. The box fortunately is a little bit broader than wide. Taking the mask into account we can thus resolve this ambiguity. In the small image on the right on the bottom we see that many blue points are in front of the background which enables our algorithm to discard this alignment. Our algorithm aligned these images with an error of $2.7^{\circ}.$ These results illustrate that taking the mask into account can resolve important problems. ![Box aligned by feature mapping followed by ICP. Alignment error = $86^o$.[]{data-label="fig:icp_fm_box"}](icp_box.png){width="40.00000%"} Evaluation of Alignment Performance with Nonvisual Information -------------------------------------------------------------- Now we will show the benefits of taking nonvisual information $N$ about the transformation into account. The dataset we have been working on consists of translations and rotations on a tabletop. Before we did not make use of this information. Now we include this information in the sampling distribution of our algorithm. We thus only sample from translation and rotations in the plane of the table. Of course our search space is much smaller now, and therefore we need less samples. The computational time is reduced to about 0.5 seconds per alignment. ![Boxplot of alignment error for different algorithms.[]{data-label="fig:prior"}](boxplot_prior.pdf){width="40.00000%"} The additional information of course also contributes to the alignment performance as we can see in Fig. \[fig:prior\]. There may be ways to make use of nonvisual information in ICP and feature mapping algorithms as well, it does however not emerge naturally and we did not try to do so. ![Very sparse point clouds of flashlight aligned with help of nonvisual information. Alignment error = $6.3^o$.[]{data-label="fig:prior"}](prior.png){width="40.00000%"} In Fig. \[fig:prior\] two very sparse point clouds are shown. Even for a human it is hard to tell how these should be aligned. ICP, feature mapping and feature mapping with subsequent ICP all aligned these point clouds with an error of at least $51^\circ$. Our algorithm without the table prior produced an error of $11^\circ$. With the prior however these points are aligned with an error of only $6.3^\circ$. This example illustrates that the combination of nonvisual information and the information from the mask can be complementary. Even this point cloud of very bad quality has been aligned almost correctly. Evaluation of Alignment Performance with Loop-closure ----------------------------------------------------- We argued that as output of the alignment we prefer a probability distribution to a point estimate. As an example why this is useful we will merge all the point clouds of the box, aligned by our algorithm with the table prior, into one point cloud, as shown in Fig. \[fig:opt\]. Frame 1 has been aligned to image 2, image 2 to image 3 and so on. Between the first and the last image the object has been rotated by around $360^\circ$, we can thus close the loop and align the last to the first image. Therefore we now have redundant information about the transformation of each image, and can optimize these transformations. This optimization is problematic if we only have a point estimate of each transformation. Fortunately however we can compute the mean as well as the covariance matrix of each transformation. We can thus estimate the probability of each transformation assuming that its distribution is normal with covariance matrix and mean as computed by our algorithm. We numerically maximize the joint probability of all the transformations using the graph optimization algorithm described in [@g2o]. ![The transformations on the right have been optimized, on the left not.[]{data-label="fig:opt"}](optimization.png){width="45.00000%"} The difference between the transformations which are optimized and the ones which are not is illustrated in Fig. \[fig:opt\]. This illustrates one of the advantages of having a posterior distribution rather than a point estimate. Conclusion and future work ========================== The results of our evaluation are promising, but for a full assessment of the performance of our novel algorithm many more experiments are necessary. The derivation of this algorithm is general and does not assume in any way that the object is on top of a table. Note that we only used this information where we explicitly mentioned it. It might however be favorable for the performance of our algorithm because the depth camera always manages to measure the depth on pixels which are on the table top. This gives us a lot of information about the mask. The next step will be to measure the performance of the algorithm in other cases, such as when the object is held by the robot hand. The core of our algorithm is sampling, it can thus easily be parallelized or even implemented on a GPU in order to reduce the computational time. In our sampling distribution $p({\boldsymbol{T}}|N,P_A,P_B)$ we have barely used the information coming from $P_A,P_B$. This information is not very important if we already have a good idea how the object has moved given $N$. If we have however no nonvisual information about how the object has moved we can make assumptions based on $P_A,P_B$. These assumptions are specific for a given application. If we know for example that we will observe only objects that are much longer in one dimension than in the others, then we can assume that the first Principal Component of $A$ is approximately aligned with the first Principal Component of $B$. Another possibility is to employ features. If we compute features for each point in $A$ and $B$ we can create a set of possible matches. Then we can sample from these matches, three at a time, which gives us a sample for ${\boldsymbol{T}}$. We might however inherit problems of feature mapping algorithms. When we ran the algorithm on the robot we moved its arms manually. This was of course only for evaluation, a possible application of the algorithm is to be used in the context of manipulation tasks. There are numerous possibilities, such as using the algorithm in a grasping pipeline. If the robot encounters, for example, an object which does not have an obvious associated grasp observing it from only one side, we can start poking it with actions that minimize the uncertainty in the alignment. While the object is being moved around, our algorithm tracks it and completes a model. The more information we gain, the more likely are we to select the correct grasp. In summary, we can say that there are many applications and possible extensions for this algorithm. Its most important feature is that due to its general formulation, it can make use of all the information available in a given case. Derivation of $p({\boldsymbol{b}}| M_A,{\boldsymbol{T}})$ {#ap:der} --------------------------------------------------------- As explained in Appx. \[ap:coord\], the transformation from ray coordinates in image $A$ to ray coordinates in image $B$ is not linear, it can however be approximated linearly around a point. Point ${\boldsymbol{b}}$ is expressed in ray coordinates in image $B$ and ${\boldsymbol{a}}$ is expressed in ray coordinates in image $A$. $$\begin{aligned} p({\boldsymbol{b}}| M_A,{\boldsymbol{T}}) &=\int\limits_{-{\boldsymbol{\infty}}}^{{\boldsymbol{\infty}}} p({\boldsymbol{b}}|{\boldsymbol{a}},{\boldsymbol{T}}) p({\boldsymbol{a}}|M_A)d{\boldsymbol{a}}\label{eq:pbj}\end{aligned}$$ $p({\boldsymbol{b}}|{\boldsymbol{a}},{\boldsymbol{T}})$ expresses the probability distribution over the position of a point ${\boldsymbol{b}}$ observed in $B$ given that we have observed the same point at ${\boldsymbol{a}}$ in $A$. With ${\boldsymbol{s}}$ being the underlying point, expressed in ray coordinates in image $A$, we can write $$\begin{aligned} p({\boldsymbol{b}}|{\boldsymbol{a}},{\boldsymbol{T}}) = \int\limits_{-{\boldsymbol{\infty}}}^{{\boldsymbol{\infty}}} p({\boldsymbol{b}}|{\boldsymbol{s}},{\boldsymbol{T}}) p({\boldsymbol{s}}|{\boldsymbol{a}},{\boldsymbol{T}}) d{\boldsymbol{s}}~~~\text{and inserting Eq.~\ref{eq:emissionprob} we obtain}~~~ = \int\limits_{-{\boldsymbol{\infty}}}^{{\boldsymbol{\infty}}} \mathcal{N}({\boldsymbol{b}}|[{\boldsymbol{s}}]_B, L) \mathcal{N}({\boldsymbol{s}}|{\boldsymbol{a}}, L) d{\boldsymbol{s}}\end{aligned}$$ Given that $[{\boldsymbol{s}}]_B$ is only relevant in the neighborhood of ${\boldsymbol{b}}$ we can replace $[{\boldsymbol{s}}]_B$ by its linear approximation around ${\boldsymbol{b}}$ obtained in Appx. \[ap:coord\]: $$\begin{aligned} & p({\boldsymbol{b}}|{\boldsymbol{a}},{\boldsymbol{T}}) = \int\limits_{-{\boldsymbol{\infty}}}^{{\boldsymbol{\infty}}} \mathcal{N}({\boldsymbol{b}}|{\boldsymbol{b}} + Q_BRQ_A^{-1} ({\boldsymbol{s}}-[{\boldsymbol{b}}]_A), L) \mathcal{N}({\boldsymbol{s}}|{\boldsymbol{a}}, L) d{\boldsymbol{s}}\\ \Aboxed{& p({\boldsymbol{b}}|{\boldsymbol{a}},{\boldsymbol{T}}) = K_1 e^{-\frac{1}{2}({\boldsymbol{a}}-[{\boldsymbol{b}}]_A)^{\top}\Lambda({\boldsymbol{a}}-[{\boldsymbol{b}}]_A)}}\\ & K_1 = \frac{1}{(2\pi)^{3/2}|L+Q_BRQ_A^{-1} L Q_A^{{-1}^{\top}}R^{\top}Q_B^{\top}|^{1/2}},~~\Lambda^{-1} = Q_A R^{-1}Q_B^{-1}LQ_B^{\top^{-1}}RQ_A^{^{\top}} +L \end{aligned}$$ As explained in the assumptions section, the whole term inside the integral of Eq. \[eq:pbj\] can be approximated as being constant within the range of a pixel. The integral over $w$ and $h$ thus becomes a sum over the number of pixels $n$: $$\begin{aligned} p({\boldsymbol{b}}| M_A,{\boldsymbol{T}}) & \propto \sum_{i = 1}^{n} \int\limits_{-\infty}^{\infty} p({\boldsymbol{b}}|w_i,h_i,r) p(w_i,h_i,r|M_A)dr\end{aligned}$$ $p(w_i,h_i,r|M_A)$ is the probability distribution over the observation of ${\boldsymbol{b}}$ in $A$, given the mask $M_A$. This probability distribution is equal to zero in the green area of Fig. \[fig:concept\] because we know that no part of the object has been observed there. Everywhere else it is uniform because we have no further information, considering only the mask. The green area, for a pixel $(w_i, h_i)$, corresponds to the range between the camera and the depth measured at the aforesaid pixel $(r_i)$. Therefore the probability distribution is equal to zero for $r<r_i$ and uniform for $r \geq r_i$. This can easily be translated into limits for the integral, and we obtain $$\begin{aligned} p({\boldsymbol{b}}| M_A,{\boldsymbol{T}}) & \propto \sum_{i = 1}^{n} \int\limits_{r_i}^{\infty} p({\boldsymbol{b}}|w_i,h_i,r) dr\end{aligned}$$ We can now integrate and obtain $$\begin{aligned} \Aboxed{p({\boldsymbol{b}}| M_A,{\boldsymbol{T}}) &\propto K_2\sum_{i = 1}^{n} e^{-\frac{1}{2} ([{\boldsymbol{b}}]_A - {\boldsymbol{a}}_i)_{w,h}^{\top}D([{\boldsymbol{b}}]_A - {\boldsymbol{a}}_i)_{w,h}} (1+erf({\boldsymbol{v}}^{\top} ([{\boldsymbol{b}}]_A - {\boldsymbol{a}}_i))}\\ \Lambda^{-1} &= Q_A R^{-1}Q_B^{-1}LQ_B^{{\boldsymbol{T}}^{-1}}RQ_A^{^{\top}} +L, ~~K_2 = \frac{1}{|L+Q_BRQ_A^{-1} L Q_A^{{-1}^{\top}}R^{\top}Q_B^{\top}|^{1/2}}\\ D &= \frac{1}{\Lambda_{33}} \begin{bmatrix} \Lambda_{11}\Lambda_{33}-\Lambda_{31}^2 & \Lambda_{33}\Lambda_{21}-\Lambda_{31}\Lambda_{32}\\ \Lambda_{33}\Lambda_{21}-\Lambda_{31}\Lambda_{32} & \Lambda_{22}\Lambda_{33}-\Lambda_{32}^2\end{bmatrix}, ~~ {\boldsymbol{v}} = \frac{1}{\sqrt{2 \Lambda_{33}}} \begin{bmatrix} \Lambda_{31}\\ \Lambda_{32}\\ \Lambda_{33} \end{bmatrix}\end{aligned}$$ Linear approximation to ray coordinate transformation {#ap:coord} ----------------------------------------------------- We want to linearly approximate the transformation from ray coordinates in image $A$, to ray coordinates in image $B$, around the point ${\boldsymbol{b}}$, which is defined in ray coordinates in image $B$. With ${\boldsymbol{s}}$ defined in image $A$, it is straightforward to show that $$\begin{aligned} \Aboxed{[{\boldsymbol{s}}]_B &\approx {\boldsymbol{b}} + Q_B R Q_A^{-1} ({\boldsymbol{s}}-[{\boldsymbol{b}}]_A)} ~~\text{with} ~~Q_A = \begin{bmatrix} \frac{\partial w}{\partial x}&\frac{\partial w}{\partial y}&\frac{\partial w}{\partial z}\\ \frac{\partial h}{\partial x}&\frac{\partial h}{\partial y}&\frac{\partial h}{\partial z}\\ \frac{\partial r}{\partial x}&\frac{\partial r}{\partial y}&\frac{\partial r}{\partial z}\\ \end{bmatrix}_{[{\boldsymbol{b}}]_{A} } ~\text{and}~~ Q_B = \begin{bmatrix} \frac{\partial w}{\partial x}&\frac{\partial w}{\partial y}&\frac{\partial w}{\partial z}\\ \frac{\partial h}{\partial x}&\frac{\partial h}{\partial y}&\frac{\partial h}{\partial z}\\ \frac{\partial r}{\partial x}&\frac{\partial r}{\partial y}&\frac{\partial r}{\partial z}\\ \end{bmatrix}_{{\boldsymbol{b}} }\end{aligned}$$ [^1]: This work has been done at University of Southern California (USC) and has been assigned by École Polytechnique Fédérale de Lausanne (EPFL) [^2]: M. Wuthrich is with the Faculty of Micro Engineering, EPFL [[email protected]]{} [^3]: P. Pastor, L. Righetti and S. Schaal are with the Computational Learning and Motor Control Lab (CLMC), USC [^4]: A. Billard is with the Learning Algorithms and Systems Laboratory (LASA), EPFL [^5]: http://youtu.be/oWiNbItu2yM

--- abstract: 'Multiple antenna systems have been extensively used by standards designing multi-gigabit communication systems operating in bandwidth of several GHz. In this paper, we study the use of transmitter (Tx) beamforming techniques to improve the performance of a MIMO system with a low precision ADC. We motivate an approach to use eigenmode transmit beamforming (which imposes a diagonal structure in the complete MIMO system) and use an eigenmode power allocation which minimizes the uncoded BER of the finite precision system. Although we cannot guarantee optimality of this approach, we observe that even low with precision ADC, it performs comparably to full precision system with no eigenmode power allocation. For example, in a high throughput MIMO system with a finite precision ADC at the receiver, simulation results show that for a $\frac{3}{4}$ LDPC coded $2\times 2$ MIMO OFDM 16-QAM system with 3-bit precision ADC at the receiver, a BER of $10^{-4}$ is achieved at an SNR of $26$ dB. This is $1$ dB better than that required for the same system with full precision but equal eigenmode power allocation.' author: - - bibliography: - 'IEEEabrv.bib' - 'ref\_journal.bib' title: Transmit Beamforming for MIMO Communication Systems with Low Precision ADC at the Receiver --- Introduction ============ Several standards designing muliGigabit communication systems (for example, IEEE 802.11ad and IEEE 802.15.3c) use multiple antennas at both the transmitter and receiver to boost up the data rates in the range of several Gbps. This gives rise to multiple input multiple output (MIMO) channel configurations. Almost all communication system with MIMO channels implement their receiver operations in the digital domain and thus analog to digital converters (ADC) becomes a critical component for such systems. Most communication system use ADCs with a precision of 6–8 bits per sample. However, high precision ADCs operating at sampling rates of several giga-samples-per second are extremely power hungry and expensive ([@ADCsurvey2008; @walden; @burmann]). Consequently, for designing communication systems requiring such high speed sampling, ADC becomes a bottleneck. We would like to highlight that a similar problem does exist for digital-to-analog (DAC) conversion (DAC) at the transmitter. However, we assume that the transmitter has significantly more power resources compared to the receiver and we focus only on the ADC problem. An example of such a scenario is when a handheld device downloads high definition content from an access point but uploads at normal speeds. A naive method to reduce the power consumption at the receiver is to use an ADC with a low bit precision (1-4 bits per sample). However, this can lead to serious performance degradation (see Fig. \[fig:performance\]). In the remainder of this section, we survey some of the previous works and highlight our contribution to improve the performance of a MIMO communication system when a low bit precision ADC is used at the receiver. We also set down some notational conventions at the end of this section. Prior Work ---------- Recently, there has been significant effort to address the ADC bottleneck and implement receivers for multi-Gbps single input single output (SISO) communication systems using low precision ADC at the receivers. Typically in SISO OFDM systems, equal transmit power subcarrier (ETSP) power is used. However, due to wide channel gain variations across the subcarriers, use of low precision ADC at the receiver results in loss of information from the weak carriers. As a result, the inter-carrier interference is not be canceled and we get an error floor (see Fig. 1 in [@tapan_vtc]). In [@tapan_vtc], we suggest a transmitter based technique for subcarrier interference management using subcarrier power allocation to ameliorate the error floor. In [@tapan_tcom], we further extend this work to find an optimal power allocation to minimize the error at the receiver. Using this optimal scheme, we observe that using a $3$–bit precision ADC at the receiver of $\frac{7}{8}$ LDPC coded $16$–QAM OFDM system, we can achieve a $2$ dB improvement in the performance compared to ETSP based OFDM. In this paper, we extend our earlier work to MIMO OFDM system. There is considerable literature in the design of transmitter-receiver (Tx-Rx) beamforming (joint or otherwise)[^1] techniques which optimizes a certain performance metric like mean square error (MSE), signal-to-interference noise ration (SINR), bit error rate (BER), transmit power etc. (see for example [@scaglione2002optimal; @ding2003minimum; @palomar2003joint; @wang2000wireless] and references therein) for full precision MIMO receivers. However, to the best of our knowledge, there has been no work in the design of Tx-Rx beamformers for MIMO receivers for a finite precision ADC. In this paper, using ideas similar to the use of sub-carrier power allocation for OFDM systems to improve the performance of a low precision receiver ([@tapan_vtc], [@tapan_tcom]), we use Tx-beamforming methods to achieve full precision performance for a MIMO receiver with a low precision ADC. Our Contribution ---------------- Exact expressions for BER for finite precision MIMO systems are fairly complicated and not amenable to finding closed form expression or computationally efficient algorithms for optimal Tx-beamformers. Instead, we impose a specific structure on the Tx-beamformer which transmits on the eigenmodes and diagonalizes the overall system. Although the [*optimality of diagonalization*]{} property cannot be proved in general for a BER minimization criteria, we motivate this property from the existence of similar property for MSE minimization criteria. This greatly simplifies the optimization problem and reduces it to a eigenmode power allocation problem. For such a diagonal structure, we compute exact expression for the uncoded BER of the MIMO-OFDM system with finite precision ADC at the receiver (Proposition 1, part 1). Using this expression of uncoded BER, we obtain a eigenmode power allocation (OEPA) which minimizes it (Proposition 1, part 2). We also propose a useful closed form approximately optimal eigenmode power allocation which can be easily used in practical system without significant increase in computational or storage requirements. We use simulations to illustrate the improvement in the performance using our power allocation stream with a low precision ADC at the receiver. As suggested in [@wpan], we use the Saleh Valenzuela (SV) to model the channel. We find that for a $\frac{3}{4}$ LDPC coded $2\times 2$ MIMO OFDM 16-QAM system with 3-bit precision at the receiver, our method requires $1$ dB less power compared to the traditional full precision system with equal eigenmode power allocation (EEPA) to achieve a BER of $10^{-4}$. On the other hand, a 3-bit system with EEPA has an error floor of $10^{-2}$. Notation -------- We use small case bold face letter to represent vectors and small case italics letters to represent scalars. Upper case bold face letter are used to represent matrices. The superscripts $[\cdot]^\dagger$ and $[\cdot]^T$ are used to denote conjugate transpose and transpose, respectively. We use $\mathbf{X}=\text{diag}\left(\mathbf{X}_1\ldots\mathbf{X}_L\right)$ to represent a block diagonal matrix where each block is $\mathbf{X}_i$, $1\leq i\leq L$. We use $\mathbf{I}$ to denote an identity matrix. The dimension of the identity matrix follows from the context. System Description ================== MIMO Channel model ------------------ For each single input single output channel between transmit antenna $i$ and receive antenna $j$, we consider an independent ISI channel in which the resolved multipath components are grouped into $L_c$ clusters, each having $L_b$ rays. The time domain channel impulse response is given by $$h(t)=\sum_{c=0}^{H_{c}-1}\sum_{b=0}^{H_b-1}g_{c,b}\delta\left(t -T_{c}-\tau_{c,b}\right), \label{channel}$$ where $g_{c,b}$ is the tap weight of the $b$-th ray of the $c$-th cluster, $T_c$ is the delay of $c$-th cluster, $\tau_{c,b}$ is the delay of the $b$-th ray relative to $T_c$ and $\delta(\cdot)$ is the dirac delta function. For simplicity of notation, we do not show the dependence on $i$ and $j$. Most standards like IEEE 802.15.3c which design communication system over a wideband channel, the Saleh-Valenzuela (S-V) model is the most popular model which characterizes the statistical properties of the parameters in . According to this model, $$\begin{aligned} f\left(T_c|T_{c-1}\right)&=\Lambda\exp\left[-\Lambda\left(T_c-T_{c-1}\right)\right],\quad c>0\\ f\left(\tau_{c,b}|\tau_{c,\left(b-1\right)}\right)&=\lambda\exp\left[-\lambda\left(\tau_{c,b}-\tau_{c,\left(b-1\right)}\right)\right],\quad b>0\end{aligned}$$ where $\Lambda$ and $\lambda$ are the cluster and ray arrival rate, respectively, and $f(\cdot)$ is the probability density function. Also, the mean square power of the tap weights are $$\text{E}[|g_{c,b}|^2]=\text{E}[|g_{0,0}|^2]\exp\left(-\frac{T_c}{\Gamma}\right)\exp\left(-\frac{\tau_{c,b}}{\gamma}\right),$$where $\Gamma$ and $\gamma$ are the cluster and ray decay rates, respectively. Since we are in the wideband regime, the distribution of the channel taps is modeled by a lognormal distribution [@ch_par; @ch_par1]. The receiver implements a front end filter of sufficient bandwidth and then samples the received analog signal uniformly. We assume that all the channel response vectors have length $L$ Signal model for a MIMO-OFDM channel with a finite precision receiver --------------------------------------------------------------------- {width="3.8in"} We consider a communications system with $n_T$ transmit antennas and $n_R$ receiver antennas which gives rise to a MIMO channel. In case of flat faded channels, the MIMO channel is represented by a channel matrix, where any entry $(i,j)$ of the matrix is channel gain between antenna $i$ and antenna $j$. In case of MIMO frequency selective channel, a multicarrier scheme is often used and each transmit antenna has an OFDM modulator and each receiver antenna has an OFDM demodulator (this can be assumed without any loss in capacity as showed in [@wang2000wireless; @raleigh1998spatio]). A detailed explanation of single input single output (SISO) OFDM can be found in [@goldsmith2005wireless] and we omit several details details here. Let $\mathbf{u}_i\in\mathbb{C}^{N\times 1}$ be the frequency domain vector to be transmitted at antenna $i$. Define $\mathbf{u}:=\left[\mathbf{u}_1\ldots\mathbf{u}_{n_T}\right]^T$ and $L:=\min(n_T,n_R)$. We assume a carrier–cooperative Tx–beamformer $\mathbf{B}\in \mathbb{C}^{n_TN\times LN}$ which allows for cooperation between different subcarriers while designing $\mathbf{B}$. The vector $\mathbf{u}$ is given by $$\mathbf{u}=\mathbf{B}\mathbf{x}, \label{beamforming}$$where $\mathbf{x}=\left[\mathbf{x}_1^T\ldots \mathbf{x}_L^T\right]^T\in\mathbb{C}^{LN}$ is the data vector to be communicated. We assume w.l.o.g. $\text{E}[\mathbf{x}\mathbf{x}^\dagger]=\mathbf{I}$. Let $\mathbf{F}$ be a block diagonal matrix of size $Nn_T\times Nn_T$ where each block is the N-point discrete Fourier matrix $\mathbf{F_N}$. The time domain transmitted vector from any antenna $i$ is given by $$\mathbf{s}_i=\mathbf{F}^\dagger_N\mathbf{u}_i \label{DFT}.$$ Thus we can define a vector $\mathbf{s}:=\left[\mathbf{s}_1^T\ldots\mathbf{s}_{n_T}^T\right]^T=\mathbf{F}^\dagger\mathbf{u}$. The total power constraint at transmitter can be expressed as $$\text{E}\left[||\mathbf{s}||^2\right]=Tr\left(\mathbf{B}^\dagger\mathbf{B}\right)\leq NL. \label{power_constraint}$$ At the receiver, the analog samples are down converted and discretized. If the discretization is done at full precision, the received vector $\mathbf{r}_j\in\mathbb{C}^N$ (after removing the cyclic prefix) is given by $$\mathbf{r}_j=\mathbf{C}_j\mathbf{s}+\mathbf{w}_j, \label{received_antenna_unquant}$$where $\mathbf{w}_j$ is additive zero mean Gaussian noise with covariance matrix $\xi^2\mathbf{I}$ and we define $\mathbf{C}_j:=[\mathbf{C}_{j,1}\ldots\mathbf{C}_{j,n_T}]$ and $\mathbf{C}_{j,i}$ represents the time domain SISO channel between transmit antenna $i$ and receive antenna $j$. The construction of the OFDM symbol forces the matrix $\mathbf{C}_{i,j}$ to be a circulant matrix. However, in practical systems, the discretization is done with finite precision. Let $A(\cdot)$ be the map which represents the analog-to-digital conversion. The ADC is defined by two parameters. $Resolution$ $b$: If the resolution is $b$ bits, then the real and imaginary parts are each quantized to $2^b$ levels. $Range$: We assume that $A(\cdot)$ has a constant range of $(-1 ,+1 )$. If the sampled signal exceeds this range, then it is clipped. In practice, an AGC block, with gain $G$ is used prior to the quantization to ensure that clipping occurs with low probability. In all our simulations, we use a uniform mid-point quantizer with range $(-1,1)$ and resolution $b$: $$\begin{split} A(x)&=\text{sign}\left(x\right)\left(\frac{1}{2^{b-1}}\lfloor 2^{b-1} |x|\rfloor+\frac{1}{2^b}\right),\quad |x|\leq 1.\\ &=\text{sign}\left(x\right)\left(1-\frac{1}{2^b}\right),\quad \text{otherwise}. \end{split} \label{eq:quant}$$where $\lfloor z \rfloor$ is the largest integer lesser than $z$. Then the received vector (after removing the cyclic prefix) at antenna $j$ is given by $ \mathbf{r}^q_j=A\left(\mathbf{C}_j\mathbf{s}+\mathbf{w}_j\right), \label{received_antenna} $where $A(\cdot)$ is applied elementwise. Defining $\mathbf{r}^q:=\left[\mathbf{r}^{q^T}_1\ldots\mathbf{r}^{q^T}_{n_R}\right]^T$, and $\mathbf{w}:=\left[\mathbf{w}^T_1\ldots\mathbf{w}^T_{n_R}\right]^T$ $\mathbf{C}=\left[\mathbf{C}_1\ldots\mathbf{C}_{n_R}\right]^T$, we can write $$\mathbf{r}^q=A\left(\mathbf{C}\mathbf{s}+\mathbf{w}\right) \label{quant_received}$$ [**Modeling quantization noise:**]{} Due to the quantizer nonlinearity, analyzing an OFDM system with finite precision quantization becomes intractable. A simple heuristic is to model the quantization noise as additive and independent (see pseudo quantization noise model in Chapter 4 of [@q_noise]). It is shown in [@dardari] that the PQN model is a valid model for quantization of OFDM signal only for a certain range of AGC. A description of the AGC calibration to ensure that the PQN model is valid is explained in [@tapan_tcom]. As per this description, if $$G^2=\frac{Nn_R\alpha}{\text{E}[||\mathbf{r}||^2]}, \label{AGC}$$ for a suitably chosen $\alpha$, where $\mathbf{r}=\left[\mathbf{r}_1^T\ldots \mathbf{r}_{n_RN}^T\right]^T$, the PQN model is a reasonable model for the quantization error. Using this model for quantization error, we can write $$\mathbf{r}^q_j=\mathbf{C}_j\mathbf{s}+\mathbf{w}_j+\mathbf{q}_j .\label{received}$$where $\mathbf{q}_j\in\mathbb{C}^{N\times 1}$ is the additive zero mean uniformly distributed quantization noise with covariance matrix $\frac{1}{G^2}\frac{2^{-2b}}{6}\mathbf{I}$. The receiver further transforms the received vector into the frequency domain by applying a $N$-point DFT, $$\mathbf{v}_j=\mathbf{F}_N\mathbf{r}^q_j.$$ Combining , and along with the fact that $\mathbf{D}_{ji}:=\mathbf{F}_N\mathbf{C}_{ji}\mathbf{F}_N^\dagger$ is a diagonal matrix, we get $$\mathbf{v}_j=\mathbf{D}_j\mathbf{Bx}+\mathbf{F}_N\mathbf{w}_j+\mathbf{F}_N\mathbf{q}_j,$$ where $\mathbf{D}_j=\left[\mathbf{D}_{j1}\ldots \mathbf{D}_{jn_T}\right]$. Concatenating the frequency domain vectors from all $n_R$ antennas, we can write $$\mathbf{v}=\mathbf{DBx}+\mathbf{Fw}+\mathbf{Fq},$$ where $\mathbf{D}:=\left[\mathbf{D}_1\ldots \mathbf{D}_{n_R}\right]^T$, $\mathbf{v}:=\left[\mathbf{v}^T_1\ldots \mathbf{v}^T_{n_R}\right]^T$ and $\mathbf{q}:=\left[\mathbf{q}^T_1\ldots \mathbf{q}^T_{n_R}\right]^T$. Let $\mathbf{A}$ be the Rx-beamformer. Again assuming cooperation among different carriers, the vector $\mathbf{v}$ is linearly transformed as $$\hat{\mathbf{x}}=\mathbf{A}^\dagger\mathbf{v}=\mathbf{A}^\dagger\mathbf{DBx}+\mathbf{A}^\dagger\mathbf{Fw}+\mathbf{A}^\dagger\mathbf{Fq}. \label{final}$$ The statistic $\hat{\mathbf{x}}$ is used as a statistic to decode the data vector $\mathbf{x}$. Optimal Tx-beamforming for MIMO systems for a specified Rx-beamformer ===================================================================== Often in downlink systems where the receivers have limited resources, it is beneficial to have a pre-specified linear receivers which depend only on the channel and not on the Tx-beamformer. For such systems systems, the goal is to design Tx-beamformers which optimizes a suitable metric. For most communication systems, the ultimate metric which we desire to optimize is the bit error rate. For the MIMO system with $\hat{L}=\min\left(L,\text{rank}\left(D^\dagger D\right)\right)$ substreams, the average BER can be defined as $$\text{BER}=\frac{1}{\hat{L}}\sum_{l=1}^{\hat{L}}\text{BER}_l$$ where $\text{BER}_l$ is the bit error rate for the $l$th substream. For a M-QAM constellation, a first order approximation of $\text{BER}_l$ can be expressed as a function of the expected signal-to-interference noise ratio (SINR) on the $l$th substream as $$\text{BER}_l\approx\frac{1}{\log_2M}\left(1-\frac{1}{\sqrt{M}}\right)Q\left(g_M\text{SINR}_l\right). \label{BER}$$ where $g_M=\frac{3}{M-1}$ and $Q(\cdot)$ is the tail probability of normal random distribution ([@palomar2003joint]). Using , we can write $$\text{SINR}_l=\frac{|\mathbf{a}_l^\dagger\mathbf{Db}_l|^2}{\mathbf{a}^\dagger_l\left(\xi^2\mathbf{I}+\xi_q^2\mathbf{I}+\sum_{k\neq l}\mathbf{Db}_k\mathbf{b}_k^\dagger\mathbf{D}^\dagger\right)\mathbf{a}_l}$$ where $\xi_q^2=\frac{1}{G^2} \frac{2^{-2b}}{6}$ and $\mathbf{b}_l$ and $\mathbf{a}_l$ are the $l$th column vectors of $\mathbf{A}$ and $\mathbf{B}$, respectively. Thus a BER minimizing criteria to design the Tx-beamformer can be written as $$\begin{split} \min_{\hat{\mathbf{B}}\in \mathbb{C}^{LN\times Nn_T}}\text{BER}, \quad \text{subject to}\\ Tr\left(\hat{\mathbf{B}}^\dagger\hat{\mathbf{B}}\right)\leq NL. \label{BER_criteria}\end{split}$$ A standard method to solve such problems is to use the Lagrange multiplier method. However, this method does not provide any closed form solution to compute $\mathbf{B}$. Instead a set of matrix fixed point equations are obtained, the solution of which is difficult to compute over a large search space. In view of the space constraints, we do not write down this equations in this draft. For example, in a $2\times 2$ MIMO OFDM system with $128$ sub-carriers, we have $256\times 256$ variables to be optimized. For full precision systems, several techniques have been used to get around the intractability of BER expressions e.g. using Chernoff bounds for the $Q(\cdot)$ function or maximizing the minimum of the SINR over all substreams. One other popularly used method is to minimize the mean square error (MSE) between $\mathbf{x}$ and $\hat{\mathbf{x}}$. We would like to point out that there exists a explicit analytical relationship between SINR and MSE only when optimal Weiner filters are used as beamformers at the receiver. Therefore, maximizing the SINR is equivalent to minimizing the MSE only for jointly designing Tx-Rx beamformers. Although this techniques do not necessarily guarantee a closed form expression for the [*optimal*]{} Tx-beamformer, it often helps in designing simpler algorithms. As representative example, we explain one such method which minimizes the MSE. [**Minimizing the MSE criteria**]{}: From , the $\text{MSE}$ can be expressed as $$\text{MSE}=\text{E}\left[\|\mathbf{x}-\hat{\mathbf{x}}\|^2\right]=\|\mathbf{A}^\dagger\mathbf{DB}-\mathbf{I}\|^2_F+\left(\xi^2+\xi_q^2\right)\|\mathbf{A}\|_F^2,$$ where $\|\cdot\|_F$ denotes the Frobenius norm. For the moment we consider the full precision case i.e. $\xi_q^2=0$. Then, if the channel is perfectly known at the receiver and the transmitter, the design criteria to find the optimal $\mathbf{B}$ is $$\begin{split} \min_{\hat{\mathbf{B}}\in\mathbb{C}^{Nn_T\times NL}}&\|\mathbf{A}^\dagger\mathbf{DB}-\mathbf{I}\|^2_F,\quad \text{subject to}\\ \label{MMSE} Tr\left(\hat{\mathbf{B}}^\dagger\hat{\mathbf{B}}\right)&\leq NL.\end{split}$$ Define $\bar{\mathbf{D}}:=\mathbf{A}^\dagger\mathbf{D}$. Let $\mathbf{U}_{\bar{D}}\mathbf{\Delta}_{\bar{D}}\mathbf{V}_{\bar{D}}^\dagger$ and $\mathbf{U}_{B}\mathbf{\Delta}_{B}\mathbf{V}_{B}^\dagger$ be the singular value decompositions of $\bar{\mathbf{D}}$ and $\mathbf{B}$, respectively. Then optimality (with respect to ) is achieved when $\mathbf{U}_B=\mathbf{V}_{\bar{D}}$ and $\mathbf{V}_B=\mathbf{U}_{\bar{D}}$. This lemma is proved as a part of Theorem 1 in [@wang2009worst]. Consequently, this proves the optimality of eigen mode transmission and hence the optimality of the diagonal structure of the complete channel described by the matrix $\mathbf{A}^\dagger\mathbf{DB}$. This reduces the complicated matrix optimization problem into a scalar power allocation problem, where the diagonal elements of the matrix $\mathbf{\Delta}_B$ gives the power allocated on the eigen modes. Using the above proposition, the optimization problem in simplifies to $$\begin{split} \min_{\Delta_{B,1},\ldots,\Delta_{B,NL}}&\sum_{l=1}^{NL}\left(\Delta_{B,l}\Delta_{\bar{D},l}-1\right)^2,\quad \text{subject to}\\ \sum_{l=1}^{NL}\Delta_{B,l}^2&\leq NL,\end{split} \label{MMSE}$$ where $\{\Delta_{B,l}\}$ and $\{\Delta_{\bar{D},l}\}$ are the diagonal elements of $\mathbf{\Delta}_{B}$ and $\mathbf{\Delta}_{\bar{D}}$, respectively. A simpler approach to design Tx-beamformers using the BER critieria {#section:sub-optimal} =================================================================== As proved in Lemma 1, the diagonal structure is optimal while using minimum MSE as the criterion for designing TX-beamformers for pre-specified Rx-beamformers. This simplification of the problem makes it more amenable for obtaining closed form approximations or designing faster algorithms. However, there does not exist a general [*optimality of diagonalization*]{} result for the BER or SINR criteria (diagonalization is optimal only when the Rx-beamformer is an optimal Weiner filter). Instead to utilize the useful properties of the diagonal structure, we suggest the following heuristic approach to design Tx-beamformers. 1. For the specified linear Rx-beamformer $\mathbf{A}$, find the Tx-beamformer such that $\mathbf{A}^\dagger\mathbf{DB}$ is diagonalized (as suggested in Lemma 1). 2. Use this diagonal structure to obtain an expression of the average BER (which is the average of BER on each parallel sub-channel). 3. Compute the [*optimal*]{} eigen mode power allocation by minimizing the average BER. As an example of the application of this approach, we consider a MIMO system where $\mathbf{A}=\mathbf{V}_D$ and $\mathbf{U}_D\mathbf{\Delta}_D\mathbf{V}_D$ is the singular value decomposition (SVD) of $\mathbf{D}$. According to part 1) of the method described above, we can use Lemma 1 to impose a diagonal structure on the complete MIMO system. This gives $\mathbf{B}=\sqrt{\mathbf{P}}\mathbf{U}_D$ where $\sqrt{\mathbf{P}}=\text{diag}\left(\sqrt{P_1}\ldots \sqrt{P_{NL}}\right)$ is the eigen power allocation to be determined. Such SVD based systems are often used in very high throughput systems (which is the main motivation of our work) which try to maximize the multiplexing gain. Without loss of generality, we assume all the singular values of $\mathbf{D}$ to be positive (If the matrix $\mathbf{D}$ has singular values to be zero, we remove that [*parallel*]{} channel from the system model). Since $\mathbf{U}_D$ is unitary, the power constraint is satisfied if $\text{Tr}\left(\mathbf{P}\right)\leq NL$. Under this structural assumptions, can be written as $$\hat{\mathbf{x}}=\mathbf{P}\mathbf{\Delta_D}\mathbf{x}+\bar{\mathbf{w}}+\bar{\mathbf{q}}, \label{svd_final}$$ where $\bar{\mathbf{w}}=\mathbf{V}_D^\dagger\mathbf{Fw}$ and $\bar{\mathbf{q}}=\mathbf{V}_D^\dagger\mathbf{Fq}$. Since $\mathbf{V}$ and $\mathbf{F}$ are unitary, $\text{E}[\bar{\mathbf{w}}\bar{\mathbf{w}}^\dagger]=\xi^2\mathbf{I}$. Using the asymptotic normality results in [@shomorony_asymptotic_normality], we can model $\bar{\mathbf{q}}$ to be a a zero mean Gaussian vector with covariance matrix $\xi_q^2\mathbf{I}$ [^2], where $\xi_q^2=\frac{1}{G^2} \frac{2^{-2b}}{6}$. Using the model , we have the following proposition Under the preceding assumptions, the following statement holds true 1. The uncoded BER for a $M-$QAM OFDM communication system with a $n_T\times n_R$ MIMO channel (parallelized into $L=\min(n_T,n_R)$ independent channels as described in the preceding discussion) and eigen power allocation $\mathbf{P}$ is given by $$BER=\frac{4S-O(S^2)}{\log_2M}$$where $$\begin{split} S&=\left(1-\frac{1}{\sqrt{M}}\right) \frac{1}{LN}\\ &\times \sum_{k=1}^{LN}Q\left(\sqrt{g_M\frac{P_k|\Delta_{D,k}|^2}{(c+1)\xi^2+\xi_q^2}}\right), \end{split}$$ $$\begin{split} g_M&=\frac{3}{M-1}, c=\frac{2^{-2b}}{6\alpha} \\ \text{and }& \xi_q^2= c\left(\frac{1}{LN}\sum_{j=0}^{LN}P_j|\Delta_{D,j}|^2\right).\end{split}$$ and $\{\Delta_{D,k}\}$ are the singular values of $\mathbf{D}$. 2. An optimal eigenmode power allocation $\mathbf{P}^{(b)}$ (OEPA) which minimizes $S$ defined in Part 1) is $$P_k^{(b)}=\frac{\left(\xi^2+\xi_q^2\right)W\left(\frac{g_M|\Delta_{D,k}|^4}{\left((c+1)\xi^2+\xi_q^2\right)^2\left(\Omega^2+|\Delta_{D,k}|^2a^2\right)}\right)}{g_M|\Delta_{D,k}|^2},\label{optimal} $$ where $W(\cdot)$ is the principal value Lambert function [@lambert], $$a=c\sum_{j=1}^{LN}\frac{\sqrt{P^{(b)}_j|\Delta_{D,j}|^2}\exp\left(-\frac{g_MP_j^{b}|\xi_j|^2}{\xi^2+\xi_q^2}\right)}{\left(\xi^2+\xi_q^2\right)^\frac{3}{2}}$$ and $\Omega$ is chosen to satisfy the power constraint . The Lambert function $W(\cdot)$ is defined as inverse function of $ f(w)=w\exp(w). $ [**Discussion**]{}: The proof of part 1) and part 2) are on similar lines to the proof of part 1) and part 2) of Proposition 1 in [@tapan_tcom]. Computing OEPA using is computationally expensive and we propose the following approximate OEPA (AOEPA). $$\tilde{P}_{k}^{(\infty)}= \frac{\frac{ W\left(\frac{g_M|\Delta_{D,k}|^4}{\xi^4 }\right)}{|\Delta_{D,k}|^2}}{\sum_{j=1}^{LN}\frac{ W\left(\frac{g_M|\Delta_{D,j}|^4}{\xi^4}\right)}{|\Delta_{D,j}|^2}},\quad \forall k, \label{eq:approximation_infinity}$$ The motivation of the approximation follows from the discussion in Section III.C of [@tapan_tcom]. [**Remark:**]{} The ultimate goal is to minimize coded BER but for analytical tractability we have worked with uncoded BER. In the next section, we show using simulations that the proposed power allocation also improves coded BER performance. Simulation Results ================== In this section, we present simulation results which highlights the improvement in the performance when using the Tx-Rx beamforming scheme presented in Section \[section:sub-optimal\]. For carrying out the simulations, the values of the parameters of the OFDM symbol and the MIMO channel model are summarized in Table \[table:parameter\]. Number of transmit antennas $n_T$ $2$ -------------------------------------- ------------ ------------------- Number of receive antennas $n_R$ $2$ Name of parameter Symbol Value Number of subcarriers $N$ $512$ OFDM symbol duration $T_s$ $204.8$ ns Length of cyclic prefix $L$ 64 Cluster arrival rate $\Lambda$ $0.037$ $ns^{-1}$ Ray arrival rate $\lambda$ $0.641$ $ns^{-1}$ Cluster decay rate $\Gamma$ $21.1$ $ns$ Ray decay rate $\gamma$ $8.85$ $ns$ Cluster lognormal standard deviation $\sigma_c$ $3.01$ dB Ray lognormal standard deviation $\sigma_r$ $7.69$ dB Mean number of clusters $L_c$ $3$ Mean number of rays $L_r$ $5$ : [Parameter values used in the simulation.]{} \[table:parameter\] We consider a high throughput $2\times 2$ MIMO system communication system which is parallelized using SVD described in previous section We highlight four possible scenarios for such a system: 1) EEPA with full precision ADC, 2) EEPA with a 3-bit precision ADC, 3) MSE minimizing eigenmode power allocation (which we call MMSE-PA) obtained by solving with a 3-bit precision ADC, and 4) AOEPA given by with 3-bit precision ADC. From Fig. \[fig:performance\], we see that for a $\frac{3}{4}$-rate low density parity check code (LDPC) coded $2\times 2 $ MIMO OFDM system with 3-bit receiver, AOEPA achieves a BER of $10^{-4}$ at an SNR of $26$ dB compared to $27$ dB required with full precision with EEPA. On the other hand, a 3-bit system with EEPA requires has an error floor of $10^{-2}$. Similarly from Fig. \[fig:performance1\], for a $\frac{1}{2}$-rate $2\times 2 $ MIMO OFDM system, 3-bit AOEPA requires $1$ dB less power than EEPA full precision system. The comparison of MSE minimizing eigen mode power allocation (MMSE-PA) with AOEPA provides a justification for our method. Even though the MSE criteria has the [*optimality of diagonalization*]{} property while BER criteria is not guaranteed of such property, optimal eigenmode power allocation which minimizes the MSE performs worse (it requires 30 dB to achieve BER of $10^{-4}$) compared to AOEPA. This justifies our approach to impose a diagonalizing structure on the MIMO system, computing the average BER (which is easier to calculate) and thereafter computing the eigenmode power allocations which minimize the BER. ![Coded BER performance for a rate $\frac{3}{4}-$rate LDPC coded $2\times 2$ high throughput MIMO OFDM system for EEPA, AOEPA and MMSE-PA.[]{data-label="fig:performance"}](MIMO_34){width="3.4in"} ![Coded BER performance for a rate $\frac{1}{2}-$rate LDPC coded $2\times 2$ high throughput MIMO OFDM system for EEPA, AOEPA and MMSE-PA.[]{data-label="fig:performance1"}](MIMO_12){width="3.4in"} Conclusion ========== In this paper, we investigate a Tx beamforming approach to improve performance of a MIMO-OFDM system with a low precision ADC at the receiver. We use Lemma 1 as a motivation and impose the structure emerging out of it to find a Tx-beamformer which minimizes the BER criteria. The primary reason for imposing this structure is reduction of the dimensionality of the space we are optimizing over. Due to this structure, the beamformer transmits on the eigenmodes of the channel and power allocation on each eigenmode is the variable to be optimized. For this structure, we compute the uncoded BER and find a eigenmode power allocation which minimizes the BER. We show that this eigenmode power allocation yields good performance compared to traditional systems with EEPA when low precision ADC are used at the receivers. In fact, our scheme achieves a performance which is comparable to that of full precision traditional systems. As a part of future work, we would like to investigate the performance of our method for other commonly used receivers, [*viz.*]{} zero-forcing (ZF), minimum mean square error (MMSE) and matched filters (MF). We would also like to gain further analytical insights into the [*optimality of diagonalization*]{} property for other design criteria. [^1]: Classical beamforming often refers to a single beamvector at the transmitter. However, we consider a more generalized beamforming with multiple beamvectors. Some authors prefer to use the terms precoder and equalizer instead of Tx-Rx beamformers. For the sake of consistency, we will use the beamforming terminology. [^2]: Here we assume that any two elements of vector $\bar{\mathbf{q}}$ are uncorrelated. This is not strictly true. However, this gives us simpler analytical expressions and at the same time gives accurate analytical predictions.

--- abstract: 'A recent global analysis of direct photon production at hadron collider and fixed target experiments has noted a disturbing trend of disagreement between next-to-leading-order (NLO) calculations and data. The conjecture has been made that the discrepancy is due to explicit multiple parton emission effects which are not accounted for in the theoretical calculations. We investigate this problem by merging a NLO calculation of direct photon production with extra multiple parton emissions via the parton shower (PS) algorithm. Our calculation maintains the integrity of the underlying NLO calculation while avoiding ambiguities due to double counting of multiple parton emissions. We find that the NLO+PS calculation can account for much of the theory/CDF data discrepancy at $\sqrt{s}=1.8$ TeV. It can also account for much of the theory/UA2 discrepancy if a very large virtuality is assumed to initiate the initial state parton shower. For lower energy data sets ([*e.g.*]{} $\sqrt{s}< 63$ GeV), NLO+PS calculations alone cannot account for the data/theory discrepancy, so that some additional non-perturbative $k_T$ smearing is needed.' address: - ' $^1$Department of Physics, Florida State University, Tallahassee, FL 32306 USA ' - ' $^2$Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242 USA ' author: - Howard Baer$^1$ and Mary Hall Reno$^2$ title: | MULTIPLE PARTON EMISSION EFFECTS\ IN NEXT-TO-LEADING-ORDER\ DIRECT PHOTON PRODUCTION --- Introduction ============ Direct photon production[@halzen; @cont; @co; @bbf; @rmp] in hadronic collisions has long been recognized as an important testing ground for perturbative QCD since many of the ambiguities involved in measuring jets are not present when analyzing photons. Direct photon production in lowest order QCD takes place via annihilation and Compton scattering Feynman graphs. Since the Compton graph involves initial state gluon-quark scattering, measurements of direct photon events can serve as important constraints in the determination of the gluon parton distribution function[@rmp]. For such a program to proceed, the greater precision involved in next-to-leading order (NLO) QCD calculations for the hard scattering are used. NLO calculations for parton+parton$\rightarrow \gamma X$ have been performed both analytically[@aurenche] and in a Monte Carlo framework[@boo]. A recent global analysis of direct photon production in hadron collisions has noted a discrepancy between NLO calculations and a large array of data for the transverse momentum $p_T$ distributions of the photon[@cteq]. Characteristically, in both fixed target and collider experiments, there is an experimental excess of photons at low transverse momentum. Several possible explanations have been put forth to resolve the discrepancy. These include [*i*]{}) improved (NLO) treatment of bremsstrahlung contributions[@gluck] and isolation criteria[@vv], [*ii*]{}) modifying gluon distribution functions and QCD scale choices to improve the data/theory agreement[@vv], or usage of alternative parton distribution functions (PDF’s)[@quack], and [*iii*]{}) invoking additional partonic $k_T$ smearing effects[@cteq]. The latter case comes in two different guises: extra partonic $k_T$ can come from non-perturbative effects from parton binding and intrinsic transverse momentum, or from additional hard multiple parton emissions which can be calculated or modeled in perturbative QCD. The non-perturbative effects are generally implemented as Gaussian smearing in an attempt to match the data. The perturbative multiple gluon emission effects can be implemented via even higher (but fixed) order perturbative calculations, via multiple gluon resummation techniques, or via the parton shower (PS) algorithm[@fw; @backsh]. The resummation and PS approaches both involve approximate [*all orders*]{} perturbative QCD effects. In this paper, we explore the extent to which the direct photon data can be explained by combining a NLO QCD calculation with multiple parton emission via the parton shower algorithm. In doing so, we follow generally the prescription outlined in Ref. [@baerreno], where NLO $W$ and $Z$ boson production were merged with parton showers. In these calculations, Owens’ method of phase space slicing is used to evaluate the NLO cross sections[@owens]. This method lends itself to a direct implementation of parton showers wherein a potential problem of double counting multiple parton emissions can be avoided. We show that our implementation of showering with the NLO QCD calculation yields an excess of events at low $p_T$ relative to the unshowered NLO result at the Fermilab Tevatron and CERN S$p\bar p$S energies, qualitatively accounting for the discrepancy between theory and experiment. Additional non-perturbative smearing is required for lower energies characteristic of the CERN ISR or fixed target experiments. Calculational Procedure ======================= Central to our calculation of direct photon production is the numerical integration of phase space via Monte Carlo methods[@boo]. One begins by evaluating the ${\cal O}(\alpha\alpha_s )$ and ${\cal O}(\alpha\alpha_s^2 )$ direct photon production subprocess Feynman graphs, including bremsstrahlung corrections to $q\bar{q}\to q\bar{q}$, [*etc.*]{} Dimensional regularization is used here for ultraviolet, soft and collinear singularities. The four-momenta for the $2\to 2$ subprocess are labeled according to, for instance, $g(p_1)+q(p_2)\to \gamma (p_3)+q(p_4)$; similarly, for $2\to 3$ subprocesses, we use $g(p_1)+q(p_2)\to \gamma (p_3)+q(p_4)+g(p_5)$, [*etc.*]{} Ultraviolet singularities are renormalized using the $\overline{\rm MS}$ prescription[@aurenche]. Collinear singularities are factorized and then absorbed into parton distribution functions (PDF’s) or fragmentation functions. Soft singularities are canceled between $2\to 3$ graphs and $2\to 2$ graphs. At this point, all cross section contributions are finite, so that numerical predictions can be made. What is peculiar to the Monte Carlo method of NLO calculation used here is that the phase space integrations are done partly analytically, and partly numerically. The boundary between numerical and analytical methods is chosen by selecting two theoretical cutoffs to demarcate the collinear and soft regimes. If any invariant quantity $t_{ij}\equiv (p_i-p_j)^2$ from the $2\to 3$ subprocess has a value $|t_{ij}|<\delta_c s_{12}$, where $s_{ij} =(p_i+p_j)^2$, then one is in the collinear regime. In this regime, the matrix element squared is evaluated in the leading pole approximation and the integration near the collinear pole is done analytically. The cross section contribution is [*de facto*]{} $2\to 2$, and it is combined with the leading order and virtual contributions to the $2\to 2$ subprocesses. If a final state gluon energy (in the subprocess rest frame) has value $E_g<\delta_s \sqrt{s_{12}}/2$, then one is in the soft regime. The integrations of the squared matrix elements are performed analytically using the soft gluon approximation, and combined with contributions from $2\to 2$ subprocesses. The total $2\to 2$ results, after factorization, are finite, but depend on $\delta_s$ and $\delta_c$, such that the soft and collinear singularities are recovered in the $\delta\to 0$ limit. The remaining phase space integrations are performed via Monte Carlo. This allows easy binning of any desired observables and allows for the simple evaluation of the effect of experimental cuts on the NLO prediction[@boo; @owens]. The $2\to 3$ contributions are all positive definite over phase space, but are also singular as $\delta_s\to 0$ or $\delta_c\to 0$. The $2\to 2$ contributions compensate the $2\to 3$ contributions and result in a total cross section which is independent of $\delta_s$ and $\delta_c$ over a wide range of values[@boo]. The expressions for all $2\to 2$ and $2\to 3$ processes in direct photon production, through NLO, are compiled in Ref. [@boo]. This is the starting point of our evaluation of the transverse momentum of the direct photon using a merger of NLO QCD and parton showers. The PS algorithm combines the simplified collinear dynamics, represented by the $Q^2$ evolution of parton distribution functions and fragmentation functions, with the exact kinematics of multiple parton emission[@fw; @backsh]. As implemented here, no additional weights to the integral are included with parton showers, as the $Q^2$ evolved distribution functions and fragmentation functions are used in evaluating the differential cross section. For the direct photon transverse momentum distribution, initial rather than final state showering is most important. Using a backward shower algorithm[@backsh], the initial state showers are evolved backward from a starting virtuality $t_v$. The kinematics of the multiple partons in the initial state shower result in transverse momenta for the partons participating in the hard scattering, effectively boosting the direct photon transverse momentum relative to the collinear approximation of the kinematics. In practice, the parton shower is cutoff at some low $t_{\rm min}$ value where perturbative QCD is still valid, but where the multiple emissions no longer become resolvable. In all the results described below, we set $t_{\rm min}=5$ GeV$^2$. Different prescriptions have been worked out for modeling final state showers[@fw] as opposed to initial state (backward) showers[@backsh]. At this stage, in a full simulation, the explicit parton emissions would be combined with a hadronization model which converts the partons into detectable particles. Our calculation does not include hadronization. The inclusion of hadronization should not alter our conclusion that multiple parton emission in the initial state can qualitatively account for the discrepancy between theory and experiment in direct photon production. While the PS prescription for LL calculations is straightforward, the prescription for merging PS with NLO calculations is not. One problem is that the shower emission from a $2\to 2$ subprocess may be double counted by the exact emission of an extra parton in the $2\to 3$ subprocess. Another problem is that, to be consistent, NLO dynamics should be used to govern the parton shower development. We use initial and final state shower algorithms consistent with LL dynamics, although we use the NLO parton distribution functions in our calculation of initial state shower probabilities. Consequently, our calculation is not consistent to NLO: the PS algorithm here should be regarded only as a parametrization of a fully consistent NLO PS program. From a practical standpoint, the error induced by using only collinear dynamics in the PS algorithm in the first place should be far larger than the error induced by neglecting NLO corrections to the underlying collinear shower dynamics. Our goal here is to demonstrate that multiple parton emissions may be responsible for the discrepancy between data and theory at low transverse momentum. To avoid the double counting problem, we restrict shower development to the $2\to 3$ subprocesses in which all momentum vectors are large and well separated. One can view a Monte Carlo NLO calculation as a sort of truncated parton shower, with only a single extra parton emission, but which is performed exactly to ${\cal O}(\alpha\alpha_s^2 )$. In this case, the $2\to 2$ contributions, which include various soft and collinear terms for which the starting shower virtuality would be tiny, would never shower. If the starting shower virtuality is appropriately chosen for the $2\to 3$ subprocesses, then only energetic, well-separated 3-body final states will develop a parton shower. Thus, the third parton of the $2\to 3$ subprocess can be viewed as the first of the potentially multiple emissions, but which is performed using exact instead of collinear dynamics. In our calculation of direct photon production, we have started with the NLO calculation of Ref. [@boo] merged with the PS along the lines of the preceding discussion. Our computer program generates $2\to 2$ subprocesses, which frequently have negative weights, along with $2\to n$ processes, with positive definite weights, but where $n\ge 3$. Crucial to our calculation is the stipulation of the starting virtualities for the parton shower. A naive choice of starting virtuality $t_v$, such as $|t_v|=n p_T^2(\gamma )$, (with $n\sim 1$) does not ensure that the 3-parton final state is well separated. This choice leads to large amounts of showering even for soft or collinear configurations. One example of allowed showering with $|t_v|=n p_T^2$ is a high $p_T$ photon recoiling against two nearly collinear partons, with $|t_{45}|>\delta_c s_{12}$ but still small. This is a region of phase space where the $2\rightarrow 2$ and $2\rightarrow 3$ contributions at a specific $p_T(\gamma )$ may cancel. Since showering is implemented only in the $2\rightarrow 3$ processes and may result in a boosted $p_T(\gamma)$ for the $2\to 3$ contribution, the required cancellation may not occur. This introduces a dependence on $\delta_c$ (and $\delta_s$ for other configurations) which is unphysical. In our procedure for merging NLO with PS, we minimize (but never completely eliminate) the dependence of results on variations of parameters. To minimize the dependence of results on $\delta_s$ and $\delta_c$, we set the starting virtuality for initial state partons to $|t_v|=c_v {\rm min}(|t_{ij}|,s_{ij})$ for $i,j=1-5$, namely, the minimum of all invariants formed by the five momenta in the $2\rightarrow 3$ process, up to a multiplicative constant $c_v$. With this prescription, any nearly soft or collinear emissions in the $2\to 3$ subprocess will result in small starting virtualities, and a small probability to shower. Only energetic, well separated $2\to 3$ subprocesses will develop a significant parton shower in the initial state. The final state showers are initiated with starting virtuality $s_{12}$. Final state showers do not change $p_T(\gamma )$ relative to the unshowered calculation; they can, however, affect the number of final state photons passing the isolation cut. Calculational Results and Comparison with Data ============================================== Direct photon production data from a variety of fixed target and collider experiments have been tabulated as a function of $x_T(\gamma )={{2p_T(\gamma )}/{\sqrt{s}}}$ in two recent studies[@cteq; @vv]. To compare against NLO calculations, it has proven convenient to plot the quantity ${({\rm Data-Theory})/{\rm Theory}}$. Thus, data in perfect agreement with theory would lie along the $y=0$ horizontal line. In Ref. [@cteq], a common trend amongst the various experimental data sets was noticed, when compared against NLO QCD. For almost all data sets tabulated, the low $x_T(\gamma )$ range was underestimated by the theory (NLO QCD). In Ref. [@vv], the authors were able to improve somewhat the data [*vs.*]{} theory discrepancy by adjusting independently the factorization and renormalization scales. Nevertheless, the discrepancy between data and theory persists. In Fig. 1[*a*]{}, we show ${\rm({Data-NLO})/{NLO}}$ [*vs.*]{} $x_T(\gamma )$ for data from the CDF experiment[@cdf] at the Fermilab Tevatron, using $p\bar p$ collisions at $\sqrt{s}=1.8$ TeV. The data points are taken from Ref. [@cteq], where the NLO distributions are calculated using the CTEQ2M PDF’s[@pdf] evaluated at the renormalization/factorization scale $\mu =p_T(\gamma )$. The large enhancement of data over theory can be seen below $x_T(\gamma )\sim 0.05$, which corresponds to $p_T(\gamma )\alt 45$ GeV at the Tevatron. Our calculation employs the same scale choices as Ref. [@cteq], but updated CTEQ3M PDF’s[@cteq3]. In keeping with CDF cuts, we require the photon pseudorapidity $|\eta (\gamma )| <0.9$, and a photon isolation cut which requires that the sum of energy, projected transverse to the beam axis, ($E_T^i$) of parton $i$ within a cone of size $\Delta R=\sqrt{(\Delta\eta)^2+(\Delta\phi)^2}=0.7$ satisfy $$\sum_i E_T^i\Biggr| _{\Delta R=0.7}<2\ {\rm GeV}.$$ These two cuts are also used in Figs. 2 and 3 below. To minimize differences due to parton distribution choices, [*etc.*]{}, rather than comparing the data to our NLO calculation merged with parton showers (NLO$\oplus$PS), we show the effect of showering as an excess or deficit relative to the unshowered NLO calculation. In Figs. 1[*b*]{} and 1[*c*]{}, we show the relative $x_T(\gamma )$ distributions (NLO$\oplus$PS-NLO)/NLO where the initial state virtuality is chosen with $c_v=4$. In our calculation, we have run for subprocess photon $p_T(\gamma )>4$ GeV, since the matrix elements are singular as $p_T(\gamma )\to 0$; the results do not change noticeably if instead we use $p_T(\gamma )>2$ GeV. Fig. 1[*b*]{} employs $\delta_s=10 \delta_c=0.1$, and Fig. 1[*c*]{} has $\delta_s=10 \delta_c=0.02$. We see in Figs. 1$b$ and 1$c$ that the incorporation of the PS has led to an enhancement of the relative $x_T(\gamma )$ distributions at $x_T(\gamma )\sim 0.02$ of about $30-40\%$, and hence is in accord with the data for the low range of $x_T(\gamma )$. The enhancement has been traced to the fact that a small fraction of the large population of very low $x_T(\gamma )$ photons gets boosted up to higher energies by recoiling against the multiple parton emissions. Although the enhancement at low $x_T(\gamma )$ from the NLO$\oplus$PS calculation is similar for the two cases, the large relative $x_T(\gamma )$ distributions show a deficit of $10-20\%$. The high $x_T(\gamma )$ deficit is due to the effect of the photon isolation cut. For very high energy events, there can still exist significant shower virtualities for events with quasi-soft or collinear partons, which introduces a slight dependence on $\delta_s$ and $\delta_c$. There is some enhancement in showering for very high energy events, which leads to fewer isolated photons, and a net diminution of signal due to the isolation cut. If we modify the initial shower virtuality magnitude by varying $c_v$, we find that a choice of $c_v\sim 1$ results in modest enhancements of the low $x_T(\gamma )$ region by only $\sim 10\%$. Choosing $c_v$ as high as $c_v\simeq 9$ yields enhancements typically around $80\%$. Also, we have investigated how the results change by changing the initial state shower cutoff virtuality choice from $t_{\rm min}=5$ GeV$^2$ to $t_{\rm min}=3$ GeV$^2$. The latter variation yields typically a $20\%$ effect. In spite of these various uncertainties, the overall qualitative trend of enhanced cross section at $x_T(\gamma )\alt 0.06$ persists in all the cases we have examined. In Ref. [@cteq], it was noted that an ad-hoc Gaussian smearing of the subprocess $p_T$ led to improved agreement between theory and data. In Fig. 1[*d*]{}, we additionally introduce Gaussian smearing (GS) to both $2\to 2$ and $2\to 3$ processes, with average transverse momentum zero and width $\sigma =1$ GeV. The overall enhancement of the NLO$\oplus$PS at $x_T(\gamma )\sim 0.02$ remains, but with some slight additional enhancement for NLO$\oplus$PS$\oplus$GS at even lower $x_T(\gamma )$ values. The small effect of the Gaussian smearing at CDF is not surprising since the average boost generated by the PS algorithm is $\sim 2.5$ GeV. In Fig. 2[*a*]{}, we show data from the UA2 experiment[@ua2] ($p\bar p$ collisions at $\sqrt{s}=630$ GeV) compared with NLO QCD, for scale choice $\mu =p_T(\gamma )/2$. Here we use a photon $p_T$ cutoff of $p_T(\gamma )>2$ GeV. Again, we see that data exceeds theory by $\sim 40\%$, although this time for $x_T(\gamma )\sim 0.05$ (corresponding to $p_T(\gamma )\sim 16$ GeV). In Fig. 2[*b*]{}, we plot the NLO$\oplus$PS result, using again the initial state virtuality choice $c_v=4$, and for $\mu=p_T(\gamma)/2$ and $\delta_s=10\delta_c=0.02$. Our merged NLO$\oplus$PS calculation gives an enhancement of $\sim 20\%$ above NLO results for $x_T(\gamma )\sim 0.05$. Although the CDF and UA2 calculations start with similar virtualities, the relatively higher value of Feynman-$x$ in the UA2 case leads to lesser amounts of initial state PS radiation. This can be offset to some extent by choosing a higher starting virtuality, $c_v=9$, shown in Fig. 2[*c*]{}. The increase in virtuality leads to a rise in our calculation to about $40\%$ above NLO expectations, in accord with the data. Finally, in Fig. 2[*d*]{}, we include as well the Gaussian smearing, which leads to some additional enhancement at low $x_T(\gamma )$. Finally, we turn to much lower energy $pp$ collider results from experiments at the CERN ISR at $\sqrt{s}=63$ GeV. In Fig. 3$a$, we show the data from the R806 experiment[@isr] compared with NLO QCD for $\mu=p_T(\gamma)/2$. Using the same scale $\mu$, and including parton showers, we show in Figs. $3b$ and $3c$, the comparison (NLO$\oplus$PS-NLO)/NLO for $c_v=4$ and $c_v=9$, respectively. We have lowered the photon $p_T$ cutoff here to $p_T(\gamma )>1$ GeV. Because of the large values of parton $x$ and small virtualities, at this energy, there is very little showering, so that perturbative multiple parton emission as described by the PS algorithm cannot explain the data/theory discrepancy. However, Gaussian smearing on the order of 1 GeV can be a large effect at this energy, where $x_T(\gamma)=0.1-0.4$ corresponds to $p_T(\gamma )=3-13$ GeV. In Fig. $3d$, we invoke as usual the $\sigma\sim 1$ GeV Gaussian smearing of the subprocess transverse momentum. In this case, the smearing can move the low $x_T(\gamma )$ theoretical prediction into rough agreement with the data. Summary and Conclusions ======================= In summary, we have investigated the effects of multiple parton emissions on direct photon production in hadronic collisions by merging the PS technique with NLO QCD. For experiments at very high energy ([*e.g.*]{} UA2 and CDF), the extra $k_T$ smearing of the hard scattering subprocess induced by the multiple parton emissions can cause some of the relatively numerous low $p_T$ photons from NLO QCD to be boosted to higher $p_T$ values. Such an effect causes a shift in the predicted $x_T(\gamma )$ distribution, thereby [*improving*]{} the agreement between theory and experiment. Our results cannot be interpreted as a QCD prediction due to the many uncertainties in the PS algorithm, and in our merging procedure. Amongst these uncertainties are the nature of the PS algorithm itself, and the prescription for initial and cutoff virtualities in the PS. On the other hand, our results can be interpreted as an existence proof that higher order effects (particularly from multiple parton emission) can account for the theory [*vs.*]{} data discrepancy. Other groups[@vv; @quack] have noted that the theory [*vs.*]{} data discrepancy can be resolved in NLO QCD mainly by using modified parton distribution functions. We comment that our result of an appropriately shifted $x_T(\gamma )$ distribution will obtain for any choice of PDF’s or hard scattering scale choices, as long as sufficient parton showering can be produced. Since hard scattering processes in nature are of course [*all-orders*]{} processes, one would expect at some level a discrepancy between data and fixed order QCD to occur. Our results show that this may already be the case for the direct photon $x_T(\gamma )$ distributions. 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$\bigskip$ Don Hadwin and Junhao Shen Mathematics Department, University of New Hampshire, Durham, NH, 03824 email: [email protected] and [email protected] $\bigskip$ **Abstract:** We introduce a new free entropy invariant, which yields significant improvements of most of the applications of free entropy to finite von Neumann algebras, including those with Cartan subalgebras, simple masas, property $T,$ property $\Gamma,$ nonprime factors, and thin factors. Introduction ============ The theory of free probability and free entropy was introduced by Voiculescu in 1980’s. In his papers [@V2] [@V3], Voiculescu introduced the concept of free entropy dimension and used it to provide the first example of II$_{1}$ factor that does not have Cartan subalgebras, which solves a long-standing open problem. Later Ge in [@Ge2] showed that the free group factors are not prime, i.e., are not a tensor product of two infinite-dimensional von Neumann algebras. This also answers a very old open question. In [@GS2], Ge and the second author computed free entropy dimension for a large class of finite von Neumann algebras including some II$_{1}$ factors with property $T$. Here we introduce a new invariant, the upper free orbit-dimension of a finite von Neumann algebra, which is closely related to Voiculescu’s free entropy dimension. Suppose that $\mathcal M$ is a von Neumann algebra with a tracial state $\tau$. Roughly speaking, if $x_{1},\ldots,x_{n}$ generates $\mathcal{M},$ Voiculescu’s free entropy dimension $\delta_{0}\left( x_{1},\ldots ,x_{n}\right) $ is obtained by considering the covering numbers of certain sets by $\omega$-balls, and letting $\omega$ approach $0$. The upper free orbit-dimension $\mathfrak{K}_{2}(x_{1},\ldots,x_{n})$ is obtained by considering the covering numbers of the same sets by $\omega$-neighborhoods of unitary orbits (see the definitions in section 2), and taking the supremum over $\omega,$ $0<\omega<1$. It is easily shown that $$\delta_{0}\left( x_{1},\ldots,x_{n}\right) \leq1+\mathfrak{K}_{2}(x_{1},\ldots,x_{n})$$ always  holds.  Most  of  the  important  applications involving  $\delta_{0}$  involve   showing\ $\delta_{0}\left( x_{1},\ldots,x_{n}\right) \leq1,$ while we see that it is much easier to show $\mathfrak{K}_{2}(x_{1},\ldots,x_{n})=0$. The upper free orbit-dimension has many useful properties, mostly in the case when $\mathfrak{K}_{2}(x_{1},\ldots,x_{n})=0.$ The key property is that if $\mathfrak{K}_{2}(y_{1},\ldots,y_{p})=0$ for some generating set for $\mathcal{M},$ then $\mathfrak{K}_{2}(x_{1},\ldots,x_{n})=0$ for every generating set. This fact allows us to show that the class of finite von Neumann algebras $\mathcal{M}$ with $\mathfrak{K}_{2}(\mathcal{M})=0$ is closed under certain operations that enlarge the algebra: 1. If $\mathfrak{K}_{2}(\mathcal{N}_{1})=\mathfrak{K}_{2}(\mathcal{N}_{2})=0$ and $\mathcal{N}_{1}\cap\mathcal{N}_{2}$ is diffuse, then $\mathfrak{K}_{2}(\left( \mathcal{N}_{1}\cup\mathcal{N}_{2}\right) ^{\prime\prime})=0.$ 2. If $\ \mathcal{M}=\{\mathcal{N},u\}^{\prime\prime}$ where $\mathcal{N}$ is a von Neumann subalgebra of $\ \mathcal{M}$ with $\mathfrak{K}_{2}(\mathcal{N})=0$ and $u$ is a unitary element in $\ \mathcal{M}$ satisfying, for a sequence $\left\{ v_{n}\right\} $ of Haar unitary elements in $\mathcal{N}$, dist$_{\left\Vert {}\right\Vert _{2}}\left( uv_{n}u^{\ast },\mathcal{N}\right) \rightarrow0$, then $\mathfrak{K}_{2}(\ \mathcal{M})=0$. 3. If $\{\mathcal{N}_{i}\}_{i=1}^{\infty}$ is an ascending sequence of von Neumann subalgebras of $\mathcal{M}$ such that $\mathfrak{K}_{2}(\mathcal{N}_{i})=0$ for all $i\geq1$ and $\mathcal{M}=\overline{\cup _{i}\mathcal{N}_{i}}^{SOT}$, then $\mathfrak{K}_{2}(\mathcal{M})=0$. Using these closure operations as building blocks, and the easily-proved fact that $\mathfrak{K}_{2}(\mathcal{M})=0$ whenever $\mathcal{M}$ is hyperfinite, we can can show that $\mathfrak{K}_{2}(\mathcal{M})=0$ for a large class of von Neumann algebras. As a corollary we recapture most of the old results. In particular, we extend results in [@V3], [@Ge2], [@GS2], [@GS1], [@GePopa], [@H], [@V4], [@Ge1], [@Dyk]. Definitions =========== Let $\mathcal{M}_{k}(\mathbb{C})$ be the $k\times k$ full matrix algebra with entries in $\mathbb{C}$, and $\tau_{k}$ be the normalized trace on $\mathcal{M}_{k}(\mathbb{C})$, i.e., $\tau_{k}=\frac{1}{k}Tr$, where $Tr$ is the usual trace on $\mathcal{M}_{k}(\mathbb{C})$. Let $\mathcal{U}(k)$ denote the group of all unitary matrices in $\mathcal{M}_{k}(\mathbb{C})$. Let $\mathcal{M}_{k}(\mathbb{C})^{n}$ denote the direct sum of $n$ copies of $\mathcal{M}_{k}(\mathbb{C})$. Let $\Vert\cdot\Vert_{2}$ denote the trace norm induced by $\tau_{k}$ on $\mathcal{M}_{k}(\mathbb{C})^{n}$, i.e., $$\Vert(A_{1},\ldots,A_{n})\Vert_{2}^{2}=\tau_{k}(A_{1}^{\ast}A_{1})+\ldots +\tau_{k}(A_{n}^{\ast}A_{n})$$ for all $(A_{1},\ldots,A_{n})$ in $\mathcal{M}_{k}(\mathbb{C})^{n}$. For every $\omega>0$, we define the $\omega$-ball $Ball(B_{1},\ldots ,B_{n};\omega)$ centered at $(B_{1},\ldots,B_{n})$ in $\mathcal{M}_{k}(\mathbb{C})^{n}$ to be the subset of $\mathcal{M}_{k}(\mathbb{C})^{n}$ consisting of all $(A_{1},\ldots,A_{n})$ in $\mathcal{M}_{k}(\mathbb{C})^{n}$ such that $\Vert(A_{1},\ldots,A_{n})-(B_{1},\ldots,B_{n})\Vert_{2}<\omega.$ For every $\omega>0$, we define the $\omega$-orbit-ball $\mathcal{U}(B_{1},\ldots,B_{n};\omega)$ centered at $(B_{1},\ldots,B_{n})$ in $\mathcal{M}_{k}(\mathbb{C})^{n}$ to be the subset of $\mathcal{M}_{k}(\mathbb{C})^{n}$ consisting of all $(A_{1},\ldots,A_{n})$ in $\mathcal{M}_{k}(\mathbb{C})^{n}$ such that there exists some unitary matrix $W$ in $\mathcal{U}(k)$ satisfying $$\Vert(A_{1},\ldots,A_{n})-(WB_{1}W^{\ast},\ldots,WB_{n}W^{\ast})\Vert _{2}<\omega.$$ Let $\mathcal{M}$ be a von Neumann algebra with a tracial state $\tau$, and $x_{1},\ldots,x_{n}$ be elements in $\mathcal{M}$. We now define our new invariants. For any positive $R$ and $\epsilon$, and any $m,k$ in $\mathbb{N}$, let $\Gamma_{R}(x_{1},\ldots,x_{n};m,k,\epsilon)$ be the subset of $\mathcal{M}_{k}(\mathbb{C})^{n}$ consisting of all $(A_{1},\ldots,A_{n})$ in $\mathcal{M}_{k}(\mathbb{C})^{n}$ such that $\Vert A_{j}\Vert\leq R$, $1\leq j\leq n$, and $$|\tau_{k}(A_{i_{1}}^{\eta_{1}}\cdots A_{i_{q}}^{\eta_{q}})-\tau(x_{i_{1}}^{\eta_{1}}\cdots x_{i_{q}}^{\eta_{q}})|<\epsilon,$$ for all $1\leq i_{1},\ldots,i_{q}\leq n$, all $\eta_{1},\ldots,\eta_{q}$ in $\{1,\ast\}$, and all $q$ with $1\leq q\leq m$. For $\omega>0$, we define the $\omega$-orbit covering number $\nu(\Gamma _{R}(x_{1},\ldots,x_{n};m,k,\epsilon),\omega)$ to be the minimal number of $\omega$-orbit-balls that cover $\Gamma_{R}(x_{1},\ldots,x_{n};m,k,\epsilon)$ with the centers of these $\omega$-orbit-balls in $\Gamma_{R}(x_{1},\ldots,x_{n};m,k,\epsilon)$. Now we define, successively, $$\begin{aligned} \frak K (x_1,,\ldots, x_n;\omega,R) &= \inf_{m\in \Bbb N, \epsilon>0}\limsup_{k\rightarrow \infty} \frac {\log(\nu(\Gamma_R(x_1,\ldots, x_n;m,k,\epsilon),\omega))}{-k^2\log\omega} \\ \frak K (x_1,,\ldots, x_n;\omega) &= \sup_{R>0} \frak K (x_1,,\ldots, x_n;\omega,R)\\ \frak K_1(x_1,,\ldots, x_n ) &= \limsup_{\omega\rightarrow 0}\frak K (x_1,,\ldots, x_n;\omega)\\ \frak K_2(x_1,,\ldots, x_n ) &= \sup_{0<\omega<1}\frak K (x_1,,\ldots, x_n;\omega), \end{aligned}$$ where $\mathfrak{K}_{1}(x_{1},,\ldots,x_{n})$ is called the [*free orbit-dimension*]{} of  $x_{1},\ldots,x_{n}$ and $\mathfrak{K}_{2}(x_{1},,\ldots,x_{n})$ is called the [*upper free orbit-dimension*]{} of $x_{1},\ldots x_{n} $. In the spirit as in Voiculescu’s definition of free entropy dimension, we shall also define free orbit-dimension and upper free orbit-dimension of $x_{1},\ldots,x_{n}$ in the presence of $y_{1},\ldots,y_{p}$ for all $x_{1},\ldots,x_{n},y_{1},\ldots,y_{p}$  in  the  von  Neumann algebra  $\mathcal{M}$  as  follows. Let $\Gamma_{R}(x_{1},\ldots,x_{n}:y_{1},\ldots,y_{p};m,k,\epsilon)$ be the image of the projection of $\Gamma _{R}(x_{1},\ldots,x_{n},y_{1},\ldots,y_{p};m,k,\epsilon)$ onto the first $n$ components, i.e., $$(A_{1},\ldots,A_{n})\in\Gamma_{R}(x_{1},\ldots,x_{n}:y_{1},\ldots ,y_{p};m,k,\epsilon)$$ if there are elements $B_{1},\ldots,B_{p}$ in $\mathcal{M}_{k}(\mathbb{C})$ such that $$(A_{1},\ldots,A_{n},B_{1},\ldots,B_{p})\in\Gamma_{R}(x_{1},\ldots,x_{n},y_{1},\ldots,y_{p};m,k,\epsilon).$$ Then we define, successively, $$\begin{aligned} \frak K (x_1,&\ldots, x_n:y_1,\ldots,y_p;\omega,R)\\ & \qquad \qquad = \inf_{m\in \Bbb N, \epsilon>0}\limsup_{k\rightarrow \infty} \frac {\log(\nu(\Gamma_R(x_1,\ldots, x_n:y_1,\ldots,y_p;m,k,\epsilon),\omega))}{-k^2\log\omega} \\ \frak K (x_1,&\ldots, x_n:y_1,\ldots,y_p;\omega ) = \sup_{R>0} \frak K (x_1, \ldots, x_n:y_1,\ldots,y_p;\omega,R)\\ \frak K_1(x_1,&\ldots, x_n:y_1,\ldots,y_p ) = \limsup_{\omega\rightarrow 0}\frak K (x_1,\ldots, x_n:y_1,\ldots,y_p;\omega)\\ \frak K_2(x_1,&\ldots, x_n :y_1,\ldots,y_p) = \sup_{0<\omega<1}\frak K (x_1,\ldots, x_n:y_1,\ldots,y_p;\omega). \end{aligned}$$ Suppose $\mathcal{M}$ is a finitely generated von Neumann algebra with a tracial state $\tau$. Then the *free orbit-dimension* $\mathfrak{K}_{1}(\mathcal{M})$ of $\mathcal{M}$ is defined by $$\mathfrak{K}_{1}(\mathcal{M})=\sup\{\mathfrak{K}_{1}(x_{1},\ldots ,x_{n})\ |\ \text{$x_{1},\ldots,x_{n}$ generate }\mathcal{M}\text{ as a von Neumann algebra}\},$$ and the *upper free orbit-dimension* $\mathfrak{K}_{2}(\mathcal{M})$ of $\mathcal{M}$ is defined by $$\mathfrak{K}_{2}(\mathcal{M})=\sup\{\mathfrak{K}_{2}(x_{1},\ldots ,x_{n})\ |\ \text{$x_{1},\ldots,x_{n}$ generate }\mathcal{M}\text{ as a von Neumann algebra}\},$$ Here, we quote a useful proposition from [@DH] Suppose $\mathcal M$ is a hyperfinite von Neumann algebra with a tracial state $\tau$. Suppose that $x_1,\ldots,x_n$ is a family of generators of $\mathcal M$. Then, for every $\omega>0$, $R>\max_{1\le j\le n}\|x_j\|$, there are a positive integer $m$ and a positive number $\epsilon$ such that the following hold: for all $k\ge 1$, if $A_1,\ldots, A_n, B_1,\ldots, B_n$ in $\mathcal M_k(\Bbb C)$ satisfying, (a) $ 0\le \|A_j\|, \|B_j\|\le R$ for all $1\le j\le n$; (b) $$\begin{aligned} &|\tau_k(A_{i_1}^{\eta_1}\cdots A_{i_p}^{\eta_p})- \tau (x_{i_1}^{\eta_1}\cdots x_{i_p}^{\eta_p})|<\epsilon\\ & |\tau_k(B_{i_1}^{\eta_1}\cdots B_{i_p}^{\eta_p})- \tau (x_{i_1}^{\eta_1}\cdots x_{i_p}^{\eta_p})|<\epsilon, \end{aligned}$$ for all $1\le i_1,\ldots, i_p\le n,$ $ \{\eta_{j}\}_{j=1}^p\subset \{*, 1\}$ and $1\le p\le m$, then there exists a unitary matrix $U$ in $\mathcal U(k)$ such that $$\sum_{j=1}^n \|U^*A_jU-B_j\|_2 <\omega.$$ Key Properties of $\mathfrak{K}_{2}$ ==================================== Let $x_{1},\ldots,x_{n}$ be self-adjoint elements in a von Neumann algebra $\mathcal{M}$ with a tracial state $\tau$. Let $\delta_{0}(x_{1},\ldots ,x_{n})$ be Voiculescu’s free entropy dimension. Then $$\delta_{0}(x_{1},\ldots,x_{n})\leq\mathfrak{K}_{1}(x_{1},\ldots,x_{n})+1\leq\mathfrak{K}_{2}(x_{1},\ldots,x_{n})+1.$$ The first inequality follows from Theorem 14 in [@DH], and the second inequality is obvious. Let $x_{1},\ldots,x_{n},y_{1},\ldots,y_{p}$ be elements in a von Neumann algebra $\mathcal{M}$ with a tracial state $\tau$. If $y_{1},\ldots,y_{p}$ are in the von Neumann subalgebra generated by $x_{1},\ldots,x_{n}$ in $\mathcal{M}$, then, for every $0<\omega<1,$ $$\mathfrak{K}(x_{1},\ldots,x_{n};\omega)=\mathfrak{K}(x_{1},\ldots,x_{n}:y_{1},\ldots,y_{p};\omega)\mathfrak{.}$$ ** ** It is a straightforward adaptation of the proof of Prop. 1.6 in [@V3] (see also Lemma 5 in [@DH]). Given $R>\max_{1\leq j\leq p}\Vert y_{j}\Vert $, $m\in \mathbb{N}$ and $\epsilon >0$, we can find $m_{1}\in \mathbb{N}$ and $\epsilon _{1}>0$ such that, for all $k\in \mathbb{N}$,$$\begin{aligned} \Gamma _{R}(x_{1},\ldots ,x_{n};m_{1},k,\epsilon _{1})&\subset \ \Gamma_{R}(x_{1},\ldots ,x_{n}:y_{1},\ldots ,y_{p};m,k,\epsilon )\\ & \subset \ \Gamma _{R}(x_{1},\ldots ,x_{n};m,k,\epsilon ). \end{aligned}$$ Hence $$\begin{aligned} \nu(\Gamma_R(x_1,\ldots,x_n;m_1,k,\epsilon_1),\omega) &\le\nu( \Gamma_R (x_1,\ldots,x_n: y_1,\ldots,y_p;m,k,\epsilon),\omega)\\ & \le\nu(\Gamma_R(x_1,\ldots,x_n;m,k,\epsilon),\omega), \end{aligned}$$for all $0<\omega <1$. The rest follows from the definitions. The following key theorem shows that, in some cases, the upper free orbit-dimension $\mathfrak{K_{2}}$ is a von Neumann algebra invariant, i.e., it is independent of the choice of generators. Suppose $\mathcal{M}$ is a von Neumann algebra with a tracial state $\tau$ and is generated by a family of elements $\{x_{1},\ldots,x_{n}\}$ as a von Neumann algebra. If $$\mathfrak{K}_{2}(x_{1},\ldots,x_{n})=0,$$ then $$\mathfrak{K}_{2}(\mathcal{M})=0.$$ ** ** Suppose that $y_{1},\ldots,y_{p}$ are elements in $\mathcal{M}$ that generate $\mathcal{M}$ as a von Neumann algebra. For every $0<\omega<1$, there exists a family of noncommutative polynomials $\psi_{i}(x_{1},\ldots,x_{n})$, $1\leq i\leq p$, such that $$\sum_{i=1}^{p}\Vert y_{i}-\psi_{i}(x_{1},\ldots,x_{n})\Vert_{2}^{2}<\left( \frac{\omega}{4}\right) ^{2}.$$ For such a family of polynomials $\psi_{1},\ldots,\psi_{p}$, and every $R>0$ there always exists a constant $D\geq1$, depending only on $R,\psi_{1},\ldots,\psi_{n}$, such that $$\left ( \sum_{i=1}^{p}\Vert\psi_{i}(A_{1},\ldots,A_{n})-\psi_{i}(B_{1},\ldots ,B_{n})\Vert_{2}^{2}\right )^{1/2}\leq D \Vert(A_{1},\ldots,A_{n})-(B_{1},\ldots ,B_{n})\Vert_{2},$$ for all $(A_{1},\ldots,A_{n}),(B_{1},\ldots,B_{n})$ in $\mathcal{M}_{k}(\mathbb{C})^{ {n}}$, all $k\in\mathbb{N}$, satisfying $\Vert A_{j}\Vert, \Vert B_{j}\Vert\leq R,$ for $1\leq j\leq n.$ For $R> 1, m$ sufficiently large, $\epsilon$ sufficiently small and $k$ sufficiently large, every $(H_{1},\ldots,H_{p},A_{1},\ldots,A_{n})$ in $\Gamma_{R}(y_{1},\ldots,y_{p},x_{1},\ldots,x_{n};m,k,\epsilon)$ satisfies $$\left (\sum_{i=1}^{p}\Vert H_{i}-\psi_{i}(A_{1},\ldots,A_{n})\Vert_{2}^{2}\right )^{1/2}\leq \frac{\omega}{4} .$$ It is obvious that such an $(A_{1},\ldots,A_{n})$ is also in $\Gamma_{R}(x_{1},\ldots,x_{n};m,k,\epsilon)$. On the other hand, by the definition of the orbit covering number, we know there exists a set $\{ \mathcal{U}(B_{1}^{\lambda },\ldots,B_{n}^{\lambda};\frac{\omega}{4D}) \}_{\lambda\in\Lambda_{k}}$ of $\frac{\omega}{4D}$-orbit-balls that cover $\Gamma_{R}(x_{1},\ldots ,x_{n};m,k,\epsilon)$ with the cardinality of $ \Lambda_{k} $ satisfying $|\Lambda_{k}|=\nu(\Gamma_{R}(x_{1},\ldots ,x_{n};m,k,\epsilon),\frac{\omega}{4D}).$  Thus  for  such  $(A_{1},\ldots,A_{n})$ in $\Gamma_{R}(x_{1},\ldots,x_{n};m,k,\epsilon)$, there exists some $\lambda\in\Lambda_{k}$ and $W\in\mathcal{U}(k)$ such that $$\Vert(A_{1},\ldots,A_{n})-(WB_{1}^{\lambda}W^{\ast},\ldots,WB_{n}^{\lambda }W^{\ast})\Vert_{2}\leq\frac{\omega}{4D}.$$ It follows that $$\sum_{i=1}^{p}\Vert H_{i}-W\psi_{i}(B_{1}^{\lambda},\ldots,B_{n}^{\lambda })W^{\ast}\Vert_{2}^{2}=\sum_{i=1}^{p}\Vert H_{i}-\psi_{i}(WB_{1}^{\lambda }W^{\ast},\ldots,WB_{n}^{\lambda}W^{\ast})\Vert_{2}^{2}\leq\left( \frac{\omega}{2}\right) ^{2},$$ for some $\lambda\in\Lambda_{k}$ and $W\in\mathcal U(k),$ i.e., $$(H_{1},\ldots,H_{p})\in\mathcal{U}(\psi_{1}(B_{1}^{\lambda},\ldots ,B_{n}^{\lambda}),\ldots,\psi_{p}(B_{1}^{\lambda},\ldots,B_{n}^{\lambda });\omega).$$ Hence, by the definition of the free orbit-dimension, we get $$\begin{aligned} 0&\le \frak K(y_1,\ldots,y_p: x_1,\ldots,x_n;\omega,R)\le \inf_{m\in \Bbb N, \epsilon>0}\limsup_{k\rightarrow \infty} \frac {\log(|\Lambda_k|)}{-k^2\log\omega}\\ &= \inf_{m\in \Bbb N, \epsilon>0}\limsup_{k\rightarrow \infty} \frac {\log(\nu(\Gamma_R(x_1,\ldots, x_n ;m,k,\epsilon),\frac \omega{4D}))}{-k^2\log\omega}\\ & = 0, \end{aligned}$$ since $\mathfrak{K}_{2}(x_{1},\ldots,x_{n})=0$. Therefore $\mathfrak{K} (y_{1},\ldots,y_{p}:x_{1},\ldots,x_{n};\omega)=0$. Now it follows from Lemma 2 that $$\mathfrak{K} (y_{1},\ldots,y_{p};\omega)=\mathfrak{K} (y_{1},\ldots ,y_{p}:x_{1},\ldots,x_{n};\omega)=0;$$ whence $\mathfrak{K}_{2}(y_{1},\ldots,y_{p})=0$ and $\mathfrak{K}_{2}( \mathcal{M})=0$. If $\mathcal{M}$ is a hyperfinite von Neumann algebra with a tracial state $\tau$, then $\mathfrak{K}_{2}( \mathcal{M})=0$. When $\mathcal{M}$ is an abelian von Neumann algebra, the result follows from [@V2 Lemma 4.3]. Generally, it is a direct consequence of Proposition 1, that, for each $0<\omega<1,$$$\nu\left( \Gamma_{R}\left( x_{1},\ldots,x_{n},m,\varepsilon,k\right) ,\omega\right) =1$$ whenever $m$ is sufficiently large and $\varepsilon$ is sufficiently small. The proof of next theorem, being a slight modification of that of Theorem 1, will be omitted. Suppose that $\mathcal{M}$ is a finitely generated von Neumann algebra with a tracial state $\tau$. Suppose that $\{\mathcal{N}_{i}\}_{i=1}^{\infty}$ is an ascending sequence of von Neumann subalgebras of $\mathcal{M}$ such that $\mathfrak{K}_{2}(\mathcal{N}_{i})=0$ for all $i\geq1$ and $\mathcal{M}=\overline{\cup_{i}\mathcal{N}_{i}}^{SOT}$. Then $\mathfrak{K}_{2}( \mathcal{M})=0$. A unitary matrix $U$ in $\mathcal{M}_{k}(\mathbb{C})$ is a *Haar unitary matrix* if $\tau_{k}(U^{m})=0$ for all $1\leq m<k$ and $\tau_{k}(U^{k})=1$. The proof of following lemma can be found in [@GS2] ( see also [@V4]). For the sake of completeness, we also sketch its proof here. Let $V_{1},V_{2}$ be two Haar unitary matrices in $\mathcal{M}_{k}(\mathbb{C})$. For every $\delta>0$, let $$\Omega(V_{1},V_{2};\delta)=\{U\in\mathcal{U}(k)\ |\ \Vert UV_{1}-V_{2}U\Vert_{2}\leq\delta\}.$$ Then, for every $0< \delta<r$, there exists a set $\{Ball (U_{\lambda}; \frac {4\delta} r)\}_{\lambda \in\Lambda}$ of $\frac{4\delta}{r}$-balls in $\mathcal{U}(k)$ that cover $\Omega(V_{1},V_{2};\delta)$ with the cardinality of $\Lambda$ satisfying $|\Lambda|\leq\left( \frac{3r}{2\delta}\right) ^{4rk^{2}}$. ** ** Let $D$ be a diagonal unitary matrix, $diag(\lambda _{1},\ldots ,\lambda _{k})$, where $\lambda _{j}$ is the $j$-th root of unity $1$. Since $V_{1},V_{2}$ are Haar unitary matrices, there exist $W_{1},W_{2}$ in $\mathcal{U}(k)$ such that $V_{1}=W_{1}DW_{1}^{\ast }$ and $V_{2}=W_{2}DW_{2}^{\ast }$. Let $\tilde{\Omega}(\delta )=\{U\in \mathcal{U}(k)\ |\ \Vert UD-DU\Vert _{2}\leq \delta \}.$ Clearly $\Omega (V_{1},V_{2};\delta )=\{W_{2}^{\ast }UW_{1}|U\in \tilde{\Omega}(\delta)\}$; whence $\tilde{\Omega}(\delta)$ and $\Omega (V_{1},V_{2};\delta )$ have the same covering numbers. Let $\{e_{st}\}_{s,t=1}^{k}$ be the canonical system of matrix units of $ \mathcal{M}_{k}(\mathbb{C})$. Let $$\begin{aligned} \mathcal S_1 = span \{e_{st} \ | \ |\lambda_s-\lambda_t|< r \} \qquad \mathcal S_2=M_k(\Bbb C) \ominus S_1. \end{aligned}$$For every $U=\sum_{s,t=1}^{k}x_{st}e_{st}$ in $\tilde{\Omega}(\delta )$, with $x_{st}\in \mathbb{C}$, let $T_{1}=\sum_{e_{st}\in \mathcal{S}_{1}}x_{st}e_{st}\in \mathcal{S}_{1}$ and $T_{1}=\sum_{e_{st}\in \mathcal{S}_{2}}x_{st}e_{st}\in \mathcal{S}_{2}$. But $$\begin{aligned} \delta ^{2}&\geq \Vert UD-DU\Vert _{2}^{2}=\sum_{s,t=1}^{k}|(\lambda _{s}-\lambda _{t})x_{st}|^{2}\geq \sum_{e_{st}\in \mathcal{S}_{2}}|(\lambda _{s}-\lambda _{t})x_{st}|^{2}\\ &\geq r^{2}\sum_{e_{st}\in \mathcal{S}_{2}}|x_{st}|^{2}=r^{2}\Vert T_{2}\Vert _{2}^{2}. \end{aligned}$$Hence $\Vert T_{2}\Vert _{2}\leq \frac{\delta }{r}$. Note that $\Vert T_{1}\Vert _{2}\leq \Vert U\Vert _{2}=1$ and $dim_{\mathbb{R}}{}\mathcal{S}_{1}\leq 4rk^{2}.$ By standard arguments on covering numbers, we know that $\tilde{\Omega}(\delta)$ can be covered by a set $\{Ball(A^{\lambda };\frac{2\delta }{r})\}_{\lambda \in \Lambda }$ of $\frac{2\delta }{r}$-balls in $ \mathcal{M}_{k}(\mathbb{C})$ with $|\Lambda |\leq \left( \frac{3r}{2\delta }\right) ^{4rk^{2}}.$ Because $\tilde{\Omega}(\delta )\subset \mathcal{U}(k)$, after replacing $A^{\lambda }$ by a unitary $U^{\lambda }$ in $Ball(A^{\lambda },\frac{2\delta }{r})$, we obtain that the set $\{Ball (U_{\lambda };\frac{4\delta }{r} )\}_{\lambda \in \Lambda }$ of $\frac{4\delta }{r}$-balls in $\mathcal{U}(k)$ that cover $\tilde{\Omega}(\delta )$ with the cardinality of $\Lambda $ satisfying $|\Lambda |\leq \left( \frac{3r}{2\delta }\right) ^{4r k^{2}}$. The same result holds for $\Omega (V_{1},V_{2};\delta )$. Suppose that $\mathcal{M}$ is a diffuse von Neumann algebra with a tracial state $\tau$. Then a unitary element $u$ in $\mathcal{M}$ is called a *Haar unitary* if $\tau(u^{m})=0$ when $m\neq0$. Suppose $\mathcal{M}$ is a diffuse von Neumann algebra with a tracial state $\tau$. Suppose $\mathcal{N}$ is a diffuse von Neumann subalgebra of $\mathcal{M}$ and $u$ is a unitary element in $\mathcal{M}$ such that $\mathfrak{K}_{2}(\mathcal{N})=0$ and $\{\mathcal{N},u\}$ generates $\mathcal{M}$ as a von Neumann algebra. If there exist Haar unitary elements $v_{1},v_{2},\ldots$ and $w_{1},w_{2},\ldots$ in $\mathcal{N}$ such that $\left\Vert v_{n}u-uw_{n} \right\Vert _{2}\rightarrow0$, then $\mathfrak{K}_{2}( \mathcal{M})=0$. In particular, if there are Haar unitary elements $v,w$ in $\mathcal{N},$ such that $vu=uw$, then $\mathfrak{K}_{2}\left( \mathcal{M}\right) =0.$ ** ** Suppose that $\{x_{1},\ldots,x_{n}\}$ is a family of generators of $\mathcal{N}$. Then we know that $\{x_{1},\ldots,x_{n}, u\}$ is a family of generators of $\mathcal{M}$. For every $0<\omega<1$, $0<r<1$, there exist an integer $p>0$ and two Haar unitary elements $v_{p}, w_{p}$ in $\mathcal{N}$ such that $$\|v_{p}u-uw_{p}\|_{2}< \frac{r\omega}{65}.$$ Note that $\{x_{1},\ldots,x_{n}, v_{p},w_{p}\} $ is also a family of generators of $\mathcal{N}$. For $R>1 $, $m\in\mathbb{N}$, $\epsilon>0$ and $k\in\mathbb{N}$, by the definition of the orbit covering number, there exists a set $\{\mathcal{U}(B_{1}^{\lambda},\ldots,B_{n}^{\lambda}, V^{\lambda},W^{\lambda};\frac {r\omega}{64})\}_{\lambda\in\Lambda_{k}}$ of $\frac{r\omega}{64} $-orbit-balls in $\mathcal{M}_{k}(\mathbb{C})^{n+2}$ that cover $\Gamma_{R}(x_{1},\ldots,x_{n}, v_{p}, w_{p};m,k,\epsilon)$, where the cardinality of $\Lambda$ satisfies $|\Lambda_{k}|=\nu(\Gamma_{R}(x_{1},\ldots,x_{n}, v_{p}, w_{p};m,k,\epsilon),\frac{r\omega}{64}).$ When $m$ is sufficient large, $\epsilon$ is sufficient small, by Proposition 1 we can assume that all $V^{\lambda},W^{\lambda}$ are Haar unitary matrices in $\mathcal{M}_{k}(\mathbb{C})$. For $m$ sufficiently large and $\epsilon$ sufficiently small, when $(A_{1},\ldots,A_{n}, V, W,U)$ is contained in $\Gamma_{R}(x_{1},\ldots,x_{n}, v_{p}, w_{p},u;m,k,\epsilon)$ then, by Proposition 1, there exists a unitary element $U_{1}$ in $\mathcal{U}(k)$ so that $$\|U_{1}-U\|_{2} <\frac{r\omega}{64}\qquad\ \text{ and } \ \qquad\Vert VU_{1}-U_{1}W\Vert_{2}<\frac{r\omega}{64}.$$ It is easy to see that $(A_{1},\ldots,A_{n},V, W)$ is also in $\Gamma_{R}(x_{1},\ldots,x_{n}, v_{p},w_{p};m,k,\epsilon)$. Since $\Gamma_{R}(x_{1},\ldots,x_{n}, v_{p}, w_{p};m,k,\epsilon)$ is covered by the set $\{\mathcal{U}(B_{1}^{\lambda},\ldots,B_{n}^{\lambda},V^{\lambda},W^{\lambda };\frac{r\omega}{64})\}_{\lambda\in\Lambda_{k}}$ of $\frac{r\omega}{64} $-orbit-balls, there exist some $\lambda\in\Lambda_{k}$ and $X\in \mathcal{U}(k)$ such that $$\Vert(A_{1},\ldots,A_{n},V,W)-(XB_{1}^{\lambda}X^{\ast},\ldots,XB_{n}^{\lambda}X^{\ast},XV^{\lambda}X^{\ast},XW^{\lambda}X^{\ast})\Vert_{2}\leq\frac{r\omega}{64}.$$ Hence, $$\Vert V^{\lambda}X^{\ast}U_{1}X-X^{\ast}U_{1}XW^{\lambda}\Vert_{2}=\Vert XV^{\lambda}X^{\ast}U_{1}-U_{1}XW^{\lambda}X^{\ast}\Vert_{2}\leq\frac{r\omega }{16}.$$ Note that $V^{\lambda},W^{\lambda}$ were chosen to be Haar unitary matrices in $\mathcal{M}_{k}(\mathbb{C})$. From Lemma 3, it follows that there exists a set $\{Ball (U_{\lambda,\sigma};\frac{\omega}{4})\}_{\sigma\in\Sigma_{k}}$ of $\frac{\omega}{4} $-balls in $\mathcal{U}(k)$ that cover $\Omega(V^{\lambda},W^{\lambda};\frac{r\omega}{16})$ with $|\Sigma_{k}|\leq\left( \frac{24}{\omega}\right) ^{4rk^{2}}$, i.e., there exists some $U_{\lambda,\sigma}$ in $\{U_{\lambda ,\sigma}\}_{\sigma\in\Sigma_{k}}$ such that $$\Vert X^{\ast}U_{1}X-U_{\lambda,\sigma}\Vert_{2}=\Vert U_{1}-XU_{\lambda ,\sigma}X^{\ast}\Vert_{2}\leq\frac{\omega}{4}.$$ Thus for such an $(A_{1},\ldots,A_{n},V, W,U)$ in $\Gamma_{R}(x_{1},\ldots,x_{n}, v_{p}, w_{p},u;m,k,\epsilon)$, there exists some $(B_{1}^{\lambda},\ldots,B_{n}^{\lambda},V^{\lambda},W^{\lambda})$ and $U_{\lambda,\sigma}$ such that $$\Vert(A_{1},\ldots,A_{n}, U)-(XB_{1}^{\lambda}X^{\ast},\ldots,XB_{n}^{\lambda }X^{\ast}, XU_{\lambda,\sigma}X^{\ast})\Vert_{2}\leq\frac\omega2,$$ for some $X\in\mathcal{U}(k)$, i.e., $$(A_{1},\ldots,A_{n}, U)\in\mathcal{U}(B_{1}^{\lambda},\ldots,B_{n}^{\lambda}, U_{\lambda,\sigma};\omega).$$ Hence, by the definition of the free orbit-dimension, we have shown $$\begin{aligned} 0\le \frak K(x_1, \ldots,x_n, u: v_p,w_p;\omega,R) &\le \inf_{m\in \Bbb N, \epsilon>0}\limsup_{k\rightarrow \infty} \frac {\log(|\Lambda_k||\Sigma_k|)}{-k^2\log\omega}\\ &\le \inf_{m\in \Bbb N, \epsilon>0}\limsup_{k\rightarrow \infty} \left (\frac {\log(|\Lambda_k| )}{-k^2\log\omega} + \frac {\log \left ( \frac {24} { \omega}\right )^{4rk^2}}{-k^2\log\omega}\right )\\ &\le 0+ 4r \cdot \frac {\log 24 -\log\omega}{-\log\omega}, \end{aligned}$$ since $\mathfrak{K}_{2}(x_{1},\ldots,x_{n}, v_{p}, w_{p})\leq\mathfrak{K}_{2}(\mathcal{N})=0$. Thus, by Lemma 2, $$\begin{aligned} 0\le \frak K(x_1, \ldots,x_n, u ;\omega) =\frak K(x_1, \ldots,x_n, u: v_p,w_p;\omega) \le 4r \cdot \frac {\log 24 -\log\omega}{-\log\omega}. \end{aligned}$$ Because $r$ is an arbitrarily small positive number, we have $\mathfrak{K} (x_{1},\ldots,x_{n}, u;\omega)=0$; whence, $\mathfrak{K}_{2} (x_{1},\ldots,x_{n}, u )=0$. By Theorem 1, $\mathfrak{K}_{2}( \mathcal{M})=0$. Using the results in [@DH Theorem 18], the preceding theorem can be easily extended as follows. Suppose $\mathcal{M}$ is a von Neumann algebra with a tracial state $\tau$. Suppose $\mathcal{N}$ is a von Neumann subalgebra of $\mathcal{M}$ and $a$ is an element in $\mathcal{M}$ such that $\mathfrak{K}_{2}(\mathcal{N})=0$, and $\{\mathcal{N},a\}$ generates $\mathcal{M}$ as a von Neumann algebra. If there exist two normal operators $b_{1},b_{2}$ in $\mathcal{N}$ such that $b_{1}$, $b_{2}$ have no common eigenvalues and $ab_{1}=b_{2}a$, then $\mathfrak{K}_{2}( \mathcal{M})=0$. Suppose $\mathcal{M}$ is a von Neumann algebra with a tracial state $\tau$. Suppose $\mathcal{M}$ is generated by von Neumann subalgebras $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ of $\mathcal{M}$. If $\mathfrak{K}_{2}(\mathcal{N}_{1})=\mathfrak{K}_{2}(\mathcal{N}_{2})=0$ and $\mathcal{N}_{1}\cap\mathcal{N}_{2}$ is a diffuse von Neumann subalgebra of $\mathcal{M}$, then $\mathfrak{K}_{2}( \mathcal{M})=0$. ** ** Suppose that $\{x_{1},\ldots,x_{n}\}$ is a family of generators of $\mathcal{N}_{1}$ and $\{y_{1},\ldots,y_{p}\}$ a family of generators of $\mathcal{N}_{2}$. Since $\mathcal{N}_{1}\cap\mathcal{N}_{2}$ is a diffuse von Neumann subalgebra, we can find a Haar unitary $u$ in $\mathcal{N}_{1}\cap\mathcal{N}_{2}$. For every $R>1+\max_{1\leq i\leq n,1\leq j\leq p}\{\Vert x_{i}\Vert,\Vert y_{j}\Vert\}$, $0<\omega<\frac{1}{2n}$, $0<r<1$ and $m\in\mathbb{N}$, $\epsilon>0$, $k\in\mathbb{N}$, there exists a set $\{\mathcal{U}(B_{1}^{\lambda},\ldots,B_{n}^{\lambda},U_{\lambda};\frac{r\omega}{24R})\}_{\lambda\in\Lambda_{k}}$ of $\frac{r\omega}{24R}$-orbit-balls in $\mathcal{M}_{k}(\mathbb{C})^{n+1}$ covering $\Gamma_{R}(x_{1},\ldots ,x_{n},u;m,k,\epsilon)$ with $|\Lambda_{k}|=\nu(\Gamma_{R}(x_{1},\ldots ,x_{n},u;m,k,\epsilon),\frac{r\omega}{24R})$. Also there exists a set $\{\mathcal{U}(D_{1}^{\sigma},\ldots,D_{p}^{\sigma },U_{\sigma};\frac{r\omega}{24R})\}_{\sigma\in\Sigma_{k}}$ of $\frac{r\omega }{24R}$-orbit-balls in $\mathcal{M}_{k}(\mathbb{C})^{p+1}$ that cover $\Gamma_{R}(y_{1},\ldots,y_{p},u;m,k,\epsilon)$ with $|\Sigma_{k}|=\nu (\Gamma_{R}(y_{1},\ldots,y_{p},u; m,k,\epsilon),\frac{r\omega}{24R})$. When $m$ is sufficiently large and $\epsilon$ is sufficiently small, by Proposition 1 we can assume all $U_{\lambda}$, $U_{\sigma}$ to be Haar unitary matrices in $\mathcal{M}_{k}(\mathbb{C})$. For each $(A_{1},\ldots,A_{n},C_{1},\ldots,C_{p},U)$ in $\Gamma_{R}(x_{1},\ldots,x_{n},y_{1},\ldots,y_{p},u;m,k,\epsilon)$, we know that $(A_{1},\ldots,A_{n},U)$ is contained in $\Gamma_{R}(x_{1},\ldots,x_{n},u;m,k,\epsilon)$ and $(C_{1},\ldots,C_{p},U)$ is contained in $\Gamma_{R}(y_{1},\ldots,y_{p},u;m,k,\epsilon)$. Note $\Gamma_{R}(x_{1},\ldots,x_{n},u;m,k,\epsilon)$ is covered by the set $\{\mathcal{U}(B_{1}^{\lambda},\ldots,B_{n}^{\lambda },U_{\lambda};\frac{r\omega}{24R})\}_{\lambda\in\Lambda_{k}}$ of $\frac{r\omega}{24R}$-orbit-balls and $\Gamma_{R}(y_{1},\ldots,y_{p},u;m,k,\epsilon)$ is covered by the set $\{\mathcal{U}(D_{1}^{\sigma},\ldots,D_{p}^{\sigma},U_{\sigma};\frac{r\omega}{24R})\}_{\sigma\in\Sigma_{k}}$ of $\frac{r\omega}{24R}$-orbit-balls. Hence, there exist some $\lambda \in\Lambda_{k}$, $\sigma\in\Sigma_{k}$ and $W_{1},W_{2}$ in $\mathcal{U}\left( k\right) $ such that $$\begin{aligned} &\|(A_1,\ldots,A_n,U) - (W_1B_1^\lambda W_1^*, \ldots, W_1B_n^\lambda W_1^*, W_1U_\lambda W_1^*)\|_2 \le \frac {r\omega}{24R}\\ & \|( C_1,\ldots,C_p, U)-(W_2D_1^\sigma W_2^*, \ldots, W_2D_p^\sigma W_2^*, W_2U_\sigma W_2^*)\|_2\le \frac {r\omega}{24R}. \end{aligned}$$ Hence, $$\Vert W_{2}^{\ast}W_{1}U_{\lambda}-U_{\sigma}W_{2}^{\ast}W_{1}\Vert_{2}=\Vert W_{1}U_{\lambda}W_{1}^{\ast}-W_{2}U_{\sigma}W_{2}^{\ast}\Vert_{2}\leq \frac{r\omega}{12R}.$$ From our assumption that $U_{\lambda},U_{\sigma}$ are Haar unitary matrices in $\mathcal{M}_{k}(\mathbb{C})$, by Lemma 3 we know that there exists a set $\{Ball (U_{\lambda\sigma\gamma}; \frac{\omega} {3R})\}_{\gamma\in\mathcal{I}_{k}}$ of $\frac{\omega} {3R}$-balls in $\mathcal{U}(k)$ that cover $\Omega(U_{\lambda},U_{\sigma };\frac{r\omega}{12R})$ with the cardinality of $\mathcal{I}_{k}$ never exceeding $\left( \frac{18R}{\omega}\right) ^{4rk^{2}}.$ Then there exists some $\gamma\in\mathcal{I}_{k}$ such that $\Vert W_{2}^{\ast}W_{1}-U_{\lambda\sigma\gamma}\Vert_{2}\leq\frac{\omega}{3R}$. This in turn implies $$\begin{aligned} \|(A_1,\ldots, A_n,C_1,\ldots,C_p, U) - & (W_2U_{\lambda\sigma\gamma}B_1^\lambda U_{\lambda\sigma\gamma}^*W_2^*, \ldots, W_2U_{\lambda\sigma\gamma}B_n^\lambda U_{\lambda\sigma\gamma}^*W_2^*, \\ & \quad\qquad \quad\quad W_2D_1^\sigma W_2^*, \ldots, W_2D_p^\sigma W_2^*,W_2U_\sigma W_2^* )\|_2\le n\omega\end{aligned}$$ for some $\lambda\in\Lambda_{k},\sigma\in\Sigma_{k},\gamma\in\mathcal{I}_{k}$ and $W_{2}\in\mathcal{U}(k)$, i.e., $$(A_{1},\ldots,A_{n},C_{1},\ldots,C_{p},U)\in\mathcal{U}(U_{\lambda\sigma \gamma}B_{1}^{\lambda}U_{\lambda\sigma\gamma}^{\ast},\ldots,U_{\lambda \sigma\gamma}B_{n}^{\lambda}U_{\lambda\sigma\gamma}^{\ast},D_{1}^{\sigma },\ldots,D_{p}^{\sigma},U_{\sigma};2n\omega).$$ Hence, by the definition of the free orbit-dimension we get $$\begin{aligned} \frak K(x_1,\ldots,x_n,&y_1,\ldots,y_p,u;2n\omega, R) \le \inf_{m\in \Bbb N, \epsilon>0}\limsup_{k\rightarrow \infty} \frac {\log(|\Lambda_k||\Sigma_k||\mathcal I_k|)}{-k^2\log(2n\omega)}\\ &\le \inf_{m\in \Bbb N, \epsilon>0}\limsup_{k\rightarrow \infty}\left ( \frac {\log(|\Lambda_k|)} {-k^2\log(2n\omega)} + \frac {\log(|\Sigma_k|)}{-k^2\log(2n\omega)} + \frac {\log(|\mathcal I_k|)}{-k^2\log(2n\omega)}\right )\\ &\le 0+\inf_{m\in \Bbb N, \epsilon>0}\limsup_{k\rightarrow \infty} \frac {\log \left ( \frac {18R} \omega \right )^{4rk^2} }{-k^2\log(2n\omega)}\\ &\le 4r\cdot \frac {\log (18R)-\log \omega }{- \log(2n\omega)}, \end{aligned}$$ since $\mathfrak{K}_{2}(N_{1})=\mathfrak{K}_{2}(N_{2})=0$. Since $r$ is an arbitrarily small positive number, we get that $\mathfrak{K} (x_{1},\ldots,x_{n},y_{1},\ldots,y_{p},u;2n\omega, R)=0$; whence $\mathfrak{K}_{2}(x_{1},\ldots,x_{n},y_{1},\ldots,y_{p},u)=0$. By Theorem 1, $\mathfrak{K}_{2}( \mathcal{M})=0$. Applications ============ In this section, we discuss a few applications of the results from the last section. Let $L(F_{n})$ denote the free group factor on $n$ generators. By Voiculescu’s fundamental result in [@V2], we know $\delta_{0}(L(F_{n}))\geq n$, where $\delta_{0}$ is Voiculescu’s free entropy dimension. By combining Theorem 1, 2, 3, 4, 5 and 6, we can easily obtain the results in [@V3], [@Ge2], [@GS2], [@GS1], and [@V4]. Here are a few sample improvements. The following lemma can be proved using Theorem 5.3 of [@ChSm]. If $\mathcal{M}$ is a II$_{1}$ factor with property $\Gamma$ with the tracial state $\tau$, then there are a hyperfinite II$_1$ factor $\mathcal R$ and a sequence $\left\{ u_{n}\right\} $ of Haar unitary elements of $\mathcal{R}$ such that$$\left\Vert u_{n}x-xu_{n}\right\Vert _{2}\rightarrow0$$ for every $x\in\mathcal{M}.$ If $\mathcal{M}$ is a II$_{1}$ factor with property $\Gamma$, then $\mathfrak{K}_{2}(\mathcal{M})=0.$ Choose a hyperfinite II$_1$ factor $\mathcal R$ and a sequence of Haar unitary elements $ u_{1},u_{2},\ldots $ in $\mathcal R$ such that $\lim_{n\rightarrow \infty}\| xu_n-u_nx\|_2=4$ for every $x$ in $\mathcal M$. Since $\mathcal{R}$ is hyperfinite, $\mathfrak{K}_{2}\left( \mathcal{R}\right) =0.$ If $\left\{ v_{1},v_{2},\ldots\right\} $ is a sequence of Haar unitaries that generate $\mathcal{M},$ it inductively follows from Theorem 4 that, for each $n\geq1$$$\mathfrak{K}_{2}\left( \left( \mathcal{R}\cup\left\{ v_{1},\ldots ,v_{n}\right\} \right) ^{\prime\prime}\right) =0.$$ Whence, by Theorem 3, $\mathfrak{K}_{2}\left( \mathcal{M}\right) =0.$ A maximal abelian self-adjoint subalgebra (or, masa) $\mathcal{A}$ in a II$_{1}$ factor $\mathcal{M}$ is called a *Cartan subalgebra* if the *normalizer algebra* of $\mathcal{A},$ $$\mathcal{N}_{1}\left( \mathcal{A}\right) =\left\{ u\in\mathcal{U}\left( \mathcal{M}\right) :u^{\ast}\mathcal{A}u\subset\mathcal{A}\right\} ^{\prime\prime}$$ equals $\mathcal{M}$. We define $\mathcal{N}_{k+1}\left( \mathcal{A}\right) =\mathcal{N}_{1}\left( \mathcal{N}_{k}\left( \mathcal{A}\right) \right) $ for $k\geq1$, and $\mathcal{N}_{\infty}\left( \mathcal{A}\right) =\left( \bigcup_{1\leq k<\infty}\mathcal{N}_{k}\left( \mathcal{A}\right) \right) ^{\prime\prime}.$ The following is a direct consequence of Theorems 4 and 3. Suppose $\mathcal{M}$ is a type II$_{1}$ factor, and $\mathcal{A}$ is a diffuse von Neumann subalgebra with $\mathfrak{K}_{2}\left( \mathcal{A}\right) =0$. If $\mathcal{M}=\mathcal{N}_{k}\left( \mathcal{A}\right) $ for some $k,1\leq k\leq\infty,$ then $\mathfrak{K}_{2}\left( \mathcal{M}\right) =0$, and $\delta_{0}\left( \mathcal{M}\right) \leq1$. Many important applications of free entropy to finite von Neumann algebras (nonprime factors, some II$_{1}$ factors with property $T$) are consequences of a result of L. Ge and J. Shen [@GS2], which states that if $\mathcal{M}$ is a II$_{1}$ von Neumann algebra generated by a sequence of Haar unitary elements $\{u_{i}\}_{i=1}^{\infty}$ in $\mathcal{M}$ such that each $u_{i+1}u_{i}u_{i+1}^{\ast}$ is in the von Neumann subalgebra generated by $\{u_{1},\ldots,u_{i}\}$ in $\mathcal{M}$, then $\delta_{0}(\mathcal{M})\leq1$. This result is an easy consequence of Theorem 4. Here is a sample of a result that is stronger. Suppose $\mathcal{M}$ is a factor of type II$_{1}$ that is generated by a family $\left\{ u_{ij}:1\leq i,j<\infty\right\} $ of Haar unitary elements in $\mathcal{M}$ such that 1. for each $i,j$, $u_{i+1,j}u_{ij}u_{i+1,j}^{\ast}$ is in the von Neumann subalgebra generated by $\{u_{1j},\ldots,u_{ij}\};$ and 2. for each $j\geq1$, $\left\{ u_{1j},u_{2j},\ldots\right\} \bigcap\left\{ u_{1,j+1},u_{2,j+1},\ldots\right\} \neq\varnothing.$ Then $\mathfrak{K}_{2}(\mathcal{M})=0$, $\delta_{0}(\mathcal{M})\leq1$. Thus $\mathcal{M}$ is not \*-isomorphic to any $L(F(n))$ for $n\geq2$. ** ** Many new examples can be obtained by using the preceding corollary. For example, suppose that $G$ is a group generated by elements $a,b,c$ such that $ab^{2}=b^{3}a$ and $ac^{2}=c^{3}a$. The group von Neumann algebra associated with $G$ is a type II$_{1}$ factor, and the preceding corollary easily implies that $\mathfrak{K}_{2}(L(G))=0$ and $\delta _{0}(L(G))\leq1.$ The next two corollaries follows directly from Corollary 3. Suppose $\mathcal M$ is a nonprime II$_1$ factor, i.e. $\mathcal M\simeq \mathcal N_1\otimes \mathcal N_2$ for some II$_1$ subfactors $ \mathcal N_1, \mathcal N_2$. Then $\mathfrak{K}_{2}( \mathcal{M})=0$, $\delta_{0}( \mathcal{M})\leq1$. Thus $\mathcal{M}$ is not \*-isomorphic to any $L(F(n))$ for $n\geq2$. If $\mathcal{M}=L(SL(\mathbb{Z},2m+1))$ is the group von Neumann algebra associated with $SL(\mathbb{Z},2m+1)$ (the special linear group with integer entries) for $m\geq1$, then $\mathfrak{K}_{2}( \mathcal{M})=0$, $\delta_{0}( \mathcal{M})\leq1$. Thus $\mathcal{M}$ is not \*-isomorphic to any $L(F(n))$ for $n\geq2$. In [@GePopa] L. Ge and S. Popa defined a type II$_{1}$ factor to be *weakly* $n$*-thin*, if it contains hyperfinite subalgebras $\mathcal{R}_{0},\mathcal{R}_{1}$ and $n$ vectors $\xi_{1},\ldots,\xi_{n}$ in $L^{2}\left( \mathcal{M},\tau\right) $ such that $\mathcal{M}=\overline {span}^{\left\Vert { \cdot }\right\Vert _{2}}\left( \mathcal{R}_{0}\{\xi_{1},\ldots,\xi_{n}\}\mathcal{R}_{1}\right) .$ They showed that $L(F_{m})$ is not weakly $n$-thin for $m> 2+2n$. Motivated by these facts, we have the following definition. A type II$_{1}$ factor $\mathcal{M}$ with the tracial state $\tau$ is *weakly* $\mathfrak{K}$*-thin* (or, respectively, *weakly* $n$*-*$\mathfrak{K}$*-thin*) if there exist von Neumann subalgebras $\mathcal{N}_{0},\mathcal{N}_{1}$ of $\mathcal{M}$ with $\mathfrak{K}_{2}\left( \mathcal{N}_{0}\right) =\mathfrak{K}_{2}\left( \mathcal{N}_{1}\right) =0$ and a vector $\xi$ $($or, respectively, $n$ vectors $\xi _{1},\ldots,\xi_{n}$$)$ in $L^{2}\left( \mathcal{M},\tau\right) $ such that $\overline{span}^{\Vert\cdot\Vert_{2}}\left( \mathcal{N}_{0}\xi \mathcal{N}_{1}\right) =L^{2}\left( \mathcal{M},\tau\right) $ $($or, respectively, $\overline{span}^{\Vert\cdot\Vert_{2}}\mathcal{N}_{0}\{\xi _{1},\ldots,\xi_{n}\}\mathcal{N}_{1}=L^{2}\left( \mathcal{M},\tau\right) $$)$. Suppose that $\mathcal{M}$ is a finitely generated *weakly* $n$*-*$\mathfrak{K}$*-thin* type II$_{1}$ factor with a tracial state $\tau$. Then $\mathfrak{K}_{1}( \mathcal{M})\leq1+2n$ and $\delta_{0}( \mathcal{M})\leq2+2n.$ Thus $\mathcal{M}$ is not \*-isomorphic to $L(F_{m})$ for $m> 2+2n$. ** ** Suppose $x_{1},\ldots,x_{p}$ is a family of self-adjoint elements in $\mathcal{M}$ that generate $\mathcal{M}$ as a von Neumann algebra. Note there exist von Neumann subalgebras $\mathcal{N}_{0},\mathcal{N}_{1}$ of $\mathcal{M}$ with $\mathfrak{K}_{2}(\mathcal{N}_{0})=\mathfrak{K}_{2}(\mathcal{N}_{1})=0$ and $n$ vectors $\xi_{1},\ldots,\xi_{n}$ in $L^{2}( \mathcal{M},\tau)$ such that $\overline{span}^{\Vert\cdot\Vert_{2}}\mathcal{N}_{0}\{\xi_{1},\ldots,\xi_{n}\}\mathcal{N}_{1}=L^{2}( \mathcal{M},\tau)$. We can choose self-adjoint elements $y_{1},y_{2},\ldots ,$ $y_{2n-1},y_{2n}$ in $\mathcal{M}$ to approximate $Re\xi_{1},Im\xi_{1},\ldots, Re\xi_{n},Im\xi_{n}$, respectively. Hence, for any positive $\omega<1$, there are a positive integer $N$, elements $\{a_{i,j,l}\}_{1\leq i\leq p,1\leq j\leq N,1\leq l\leq2n}$ in $\mathcal{N}_{0}$, $\{b_{i,j,l}\}_{1\leq i\leq p,1\leq j\leq N,1\leq l\leq2n}$ in $\mathcal{N}_{1}$, and self-adjoint elements $y_{1},\ldots,y_{2n}$ in $\mathcal{M}$ such that $$\sum_{i=1}^{p}\Vert x_{i}-\sum_{j=1}^{N}\sum_{l=1}^{2n}a_{i,j,l}y_{l}b_{i,j,l}\Vert_{2}^{2}\leq\left( \frac{\omega}{8}\right) ^{2}.$$ Without loss of generality, we can assume that $\{a_{i,j,l}\}_{1\leq i\leq p,1\leq j\leq N,1\leq l\leq n}$ generates $\mathcal{N}_{0}$ and $\{b_{i,j,l}\}_{1\leq i\leq p,1\leq j\leq N,1\leq l\leq n}$ generates $\mathcal{N}_{1}$ as von Neumann algebras. Otherwise we should add generators of $\mathcal{N}_{0}$, $\mathcal{N}_{1}$ into the families. Let $a$ be $\max_{1\leq i\leq p}\{\Vert x_{i}\Vert_{2}\}+2$. From now on the sequence $z_{1},\ldots,z_{s},\ldots,z_{t}$ is denoted by $(z_{s})_{s=1,\ldots,t}$ or $(z_{s})_{s}$ if there is no confusion arising from the range of index, where $z_{s}$ is an element in $\mathcal{M}$ or a matrix in $\mathcal{M}_{k}(\mathbb{C})$. For $R>a$, define mapping $\psi:( \mathcal{M}_{k}(\mathbb{C})^{N})^{2n}\times\mathcal{M}_{k}(\mathbb{C})^{2n}\times( \mathcal{M}_{k}(\mathbb{C})^{N})^{2n}\rightarrow\mathcal{M}_{k}(\mathbb{C})$ as follows, $$\psi((D_{j,l})_{jl},(E_{l})_{l},(F_{j,l})_{jl})=\sum_{j=1}^{N}\sum_{l=1}^{2n}D_{j,l}E_{l}L_{j,l}. $$ Let $( \mathcal{M}_{k}(\mathbb{C}))_{R}$ be the collection of all $A$ in $\mathcal{M}_{k}(\mathbb{C})$ such that $\Vert A\Vert\leq R$. Then there always exists a constant $D>1$, not depending on $k$, such that $$\begin{aligned} \Vert &\left( \psi( (A_{1,j,l}^{(1)})_{jl},(Y_{l})_{l},(B_{1,j,l}^{(1)})_{jl} ) ,\ldots,\psi( (A_{p,j,l}^{(1)})_{jl},(Y_{l})_{l},(B_{p,j,l}^{(1)})_{jl} ) \right) \\ & \qquad \qquad -\left( \psi( (A_{1,j,l}^{(2)})_{jl},(Y_{l})_{l},(B_{1,j,l}^{(2)})_{jl} ) ,\ldots,\psi( (A_{p,j,l}^{(2)})_{jl},(Y_{l})_{l},(B_{p,j,l}^{(2)})_{jl} ) \right) \Vert_{2}\nonumber\\ & \quad\leq D\Vert\left( (A_{i,j,l}^{(1)})_{ijl},(B_{i,j,l}^{(1)})_{ijl}\right) -\left( (A_{i,j,l}^{(2)})_{ijl},(B_{i,j,l}^{(2)})_{ijl}\right) \Vert_{2},\nonumber\end{aligned}$$ for all $$\left\{ A_{i,j,l}^{(1)},Y_{l},B_{i,j,l}^{(1)},A_{i,j,l}^{(2)},B_{i,j,l}^{(2)}\right\} _{i,j,l}\subset(\mathcal{M}_{k}(\mathbb{C}))_{R}\qquad\forall k\in N.$$ For $m$ sufficiently large, $\epsilon$ sufficiently small and $k$ sufficiently large, if $$\left( X_{1},\ldots,X_{p},(A_{i,j,l})_{ijl},(Y_{l})_{l},(B_{i,j,l})_{ijl}\right) \in\Gamma_{R}(x_{1},\ldots,x_{p},(a_{i,j,l})_{ijl},(y_{l})_{l},(b_{i,j,l})_{ijl};k,m,\epsilon),$$ then $$\begin{aligned} \Vert(X_{1},\ldots,X_{p})- & (\psi((A_{1,j,l})_{jl},(Y_{l})_{l},(B_{1,j,l})_{jl}),\ldots,\psi((A_{p,j,l})_{jl},(Y_{l})_{l},(B_{p,j,l})_{jl}))\Vert_{2}\\ & \qquad =(\ \sum_{i=1}^{p}\Vert X_{i}-\sum_{j=1}^{N}\sum_{l=1}^{2n}A_{i,j,l}Y_{l}B_{i,j,l}\Vert_{2}^{2}\ )^{1/2}\leq \frac{\omega}{8} ,\nonumber\end{aligned}$$ and $$\begin{aligned} &( (A_{i,j,l})_{ijl}) \in \Gamma_R( (a_{i,j,l})_{ijl} ; k,m,\epsilon),\quad \text {and} \quad ( (B_{i,j,l})_{ijl}) \in \Gamma_R( (b_{i,j,l})_{ijl} ; k,m,\epsilon). \end{aligned}$$ On the other hand, from the definition of the orbit covering number, it follows there exists a set $\{\mathcal{U}((A_{ijl}^{\lambda})_{ijl};\frac{\omega}{16D})\}_{\lambda \in\Lambda_{k}}$, or $\{\mathcal{U}((B_{ijl}^{\sigma})_{ijl};\frac{\omega }{16D})\}_{\sigma\in\Sigma_{k}}$, of $\frac{\omega}{16D}$-orbit-balls that cover $\Gamma_{R}((a_{i,j,l})_{ijl};k,m,\epsilon)$, or $\Gamma_{R}((b_{i,j,l})_{ijl};k,m,\epsilon)$ respectively, with $$|\Lambda_{k}|=\nu(\Gamma_{R}((a_{i,j,l})_{ijl};k,m,\epsilon),\frac{\omega }{16D}), \quad|\Sigma_{k}|=\nu(\Gamma_{R}((b_{i,j,l})_{ijl};k,m,\epsilon ),\frac{\omega}{16D}).$$ Therefore for such sequence $((A_{i,j,l})_{ijl},(B_{i,j,l})_{ijl})$, there exist some $\lambda\in\Lambda_{k}$, $\sigma\in\Sigma_{k}$ and $W_{1},W_{2}$ in $\mathcal{U}\left( k\right) $ such that $$\begin{aligned} \Vert\left( (A_{i,j,l})_{ijl},(B_{i,j,l})_{ijl}\right) -((W_{1}A_{i,j,l}^{\lambda}W_{1}^{\ast})_{ijl},(W_{2}B_{i,j,l}^{\sigma}W_{2}^{\ast })_{ijl})\Vert_{2}\leq\frac{\omega}{8D}.\end{aligned}$$ Thus, from (4.1), (4.2) and (4.3), it follows that $$\begin{aligned} & \Vert(X_{1},\ldots,X_{p}) -\left( \psi((W_{1}A_{1,j,l}^{\lambda}W_{1}^{*})_{jl},(Y_{l})_{l},(W_{2}B_{1,j,l}^{\sigma}W_{2}^{*})_{jl}),\right. \\ & \qquad\qquad\qquad\qquad\qquad\qquad\ldots\left. ,\psi((W_{1}A_{p,j,l}^{\lambda}W_{1}^{*})_{jl},(Y_{l})_{l},(W_{2}B_{p,j,l}^{\sigma}W_{2}^{*})_{jl})\right) \Vert_{2} \nonumber\\ & \qquad\qquad=\left ( \ \sum_{1\leq i\leq p}\Vert X_{i}-\sum_{j=1}^{N}\sum_{l=1}^{2n}W_{1}A_{i,j,l}^{\lambda}W_{1}^{\ast}Y_{l}W_{2}B_{i,j,l}^{\sigma}W_{2}^{\ast}\Vert_{2}^{2}\ \right )^{1/2}\leq \frac{\omega}{4} .\nonumber\end{aligned}$$ Hence $$\begin{aligned} \left (\sum_{1\leq i\leq p}\Vert W_{1}^{\ast}X_{i}W_{1}-\sum_{j=1}^{N}\sum_{l=1}^{2n}\left( A_{i,j,l}^{\lambda}W_{1}^{\ast}Y_{l}W_{2}B_{i,j,l}^{\sigma }\right) W_{2}^{\ast}W_{1}\Vert_{2}^{2}\right )^{1/2}\leq \frac{\omega}{4}.\end{aligned}$$ By a result of Szarek, there exists a $\frac{\omega}{4ap}$-net $\{U_{\gamma }\}_{\gamma\in_{k}}$ in $\mathcal{U}(k)$ that cover $\mathcal{U}(k)$ with respect to the uniform norm such that the cardinality of $\mathcal{I}_{k}$ does not exceed $(\frac{4apC}{\omega})^{k^{2}}$, where $C$ is a universal constant. Thus $\Vert W_{2}^{\ast}W_{1}-U_{\gamma}\Vert\leq\frac{\omega}{4ap},$ for some $\gamma\in\mathcal{I}_{k}$. Because of (4.5), we know $$\begin{aligned} \Vert\sum_{j=1}^{N}\sum_{l=1}^{2n}A_{i,j,l}^{\lambda}W_{1}^{\ast}Y_{l}W_{2}B_{i,j,l}^{\sigma}\Vert_{2}\leq\Vert X_{i}\Vert_{2}+\omega<a.\end{aligned}$$ From (4.5) and (4.6), we have $$\begin{aligned} \left (\sum_{1\leq i\leq p}\Vert W_{1}^{\ast}X_{i}W_{1}-\left( \sum_{j=1}^{N}\sum_{l=1}^{2n}A_{i,j,l}^{\lambda}W_{1}^{\ast}Y_{l}W_{2}B_{i,j,l}^{\sigma }\right) U_{\gamma}\Vert_{2}^{2}\right )^{1/2}\leq \frac{\omega}{2} $$ Define a linear mapping $\Psi_{\lambda\sigma\gamma}: \mathcal{M}_{k}(\mathbb{C})^{2n}\mathbb{\rightarrow} \mathcal{M}_{k}\mathbb{(C})^{p} $ as follows; $$\begin{aligned} \Psi_{ \lambda\sigma\gamma} (S_1, \ldots, S_{2n}) = \left ( \frac 1 2\sum_{j=1}^N\sum_{l=1}^{2n} \left (A_{i,j,l}^{\lambda} S_l B_{i,j,l}^{\sigma}\right )U_\gamma +\left (\left (A_{i,j,l}^{\lambda} S_l B_{i,j,l}^{\sigma}\right )U_\gamma \right)^*\right )_{i=1,\ldots,p}.\end{aligned}$$ Let $\mathfrak{F}_{\lambda\sigma\gamma}$ be the range of $\Psi_{\lambda \sigma\gamma}$ in $\mathcal{M}_{k}(\mathbb{C})^{p} $. It is easy to see that $\mathfrak{F}_{\lambda\sigma\gamma}$ is a real-linear subspace of $\mathcal{M}_{k}(\mathbb{C})^{p} $ whose real dimension does not exceed $2nk^{2}$. Therefore the bounded subset $$\begin{aligned} \{(H_{1},\ldots,H_{p})\in\mathfrak{F}_{\lambda\sigma\gamma}\ |\ \Vert (H_{1},\ldots,H_{p})\Vert_{2}\leq ap\}\end{aligned}$$ of $\mathcal{M}_{k}(\mathbb{C}){^{p}}$ can be covered by a set $\{(H_{1}^{\lambda\sigma\gamma,\rho},\ldots H_{p}^{\lambda\sigma\gamma,\rho})\}_{\rho\in\mathcal{S}_{k}}$ of $\omega$-balls with the cardinality of $\mathcal{S}_{k}$ satisfying $|\mathcal{S}_{k}|\leq(\frac{3ap}{\omega })^{2nk^{2}}.$ But we know from (4.6) that $$\begin{aligned} & \left \|\left( \frac1 2\sum_{j=1}^{N}\sum_{l=1}^{2n} \left( A_{i,j,l}^{\lambda }W_{1}^{*} Y_{l} W_{2}B_{i,j,l}^{\sigma}\right) U_{\gamma}+ \left( \left( A_{i,j,l}^{\lambda}W_{1}^{*} Y_{l} W_{2}B_{i,j,l}^{\sigma}\right) U_{\gamma }\right) ^{*} \right) _{i=1,\ldots, p}\right \|_{2} \\ & =\left (\sum_{i=1}^{p}\|\frac1 2\sum_{j=1}^{N}\sum_{l=1}^{2n} \left( A_{i,j,l}^{\lambda}W_{1}^{*} Y_{l} W_{2}B_{i,j,l}^{\sigma}\right) U_{\gamma}+ \left( \left( A_{i,j,l}^{\lambda}W_{1}^{*} Y_{l} W_{2}B_{i,j,l}^{\sigma }\right) U_{\gamma}\right) ^{*}\|_{2}^{2}\right )^{1/2}\nonumber\\ & < ap ,\nonumber\end{aligned}$$ and from (4.7) we have $$\begin{aligned} \|( & W_{1}^{*}X_{1}W_{1}, \ldots, W_{1}^{*}X_{p}W_{1})- \Psi_{\lambda \sigma\gamma}(W_{1}^{*} Y_{1} W_{2},\ldots,W_{1}^{*} Y_{2n} W_{2})\|_{2}\\ & =\|(W_{1}^{*}X_{1}W_{1}, \ldots, W_{1}^{*}X_{p}W_{1})-\nonumber\\ & \qquad\left( \frac1 2\sum_{j=1}^{N}\sum_{l=1}^{2n} \left( A_{i,j,l}^{\lambda}W_{1}^{*} Y_{l} W_{2}B_{i,j,l}^{\sigma}\right) U_{\gamma}+ \left( \left( A_{i,j,l}^{\lambda}W_{1}^{*} Y_{l} W_{2}B_{i,j,l}^{\sigma}\right) U_{\gamma}\right) ^{*} \right) _{i=1,\ldots, p} \|_{2}\nonumber\\ & \le\omega.\nonumber\end{aligned}$$ Thus, from (4.8), (4.9) and (4.10), there exists some $\rho\in\mathcal{S}_{k}$ such that $$\Vert(W_{1}^{\ast}X_{1}W_{1},\ldots,W_{1}^{\ast}X_{p}W_{1})-(H_{1}^{\lambda\sigma\gamma,\rho},\ldots H_{p}^{\lambda\sigma\gamma,\rho})\Vert _{2} \leq2\omega.$$ By the definition of the free orbit-dimension, we know that $$\begin{aligned} \frak K&(x_1,\ldots, x_p: (a_{ijl})_{ijl}, (y_l)_{l}, (b_{ijl})_{ijl};4\omega,R) \le \inf_{m\in \Bbb N, \epsilon>0}\limsup_{k\rightarrow \infty} \frac {\log(|\Lambda_k||\Sigma_k||\mathcal I_k||\mathcal S_k|)}{-k^2\log(4\omega)}\\ &\le \inf_{m\in \Bbb N, \epsilon>0}\limsup_{k\rightarrow \infty} \left (\frac {\log |\Lambda_k| }{-k^2\log(4\omega)}+\frac {\log |\Sigma_k| }{-k^2\log(4\omega)}+\frac {\log(\frac {4apC} \omega)^{k^2}(\frac {3ap} \omega)^{2nk^2} )}{-k^2\log(4\omega)} \right )\\ &= 0+0 + \frac {\log( 4\cdot (3ap)^{2n}\cdot apC)-(2n+1)\log \omega } {- \log(4\omega)}, \end{aligned}$$ since $\mathfrak{K}_{2}(\mathcal{N}_{0})=\mathfrak{K}_{2}(\mathcal{N}_{1})=0$. Thus, by Lemma 2 $$\begin{aligned} 0\le \frak K (x_1,\ldots, x_p ;4\omega )&=\frak K (x_1,\ldots, x_p: (a_{ijl})_{ijl}, (y_l)_{l}, (b_{ijl})_{ijl};4\omega) \\ &\le \frac {\log( 4\cdot (3ap)^{2n}\cdot apC)-(2n+1)\log \omega } {- \log(4\omega)}. \end{aligned}$$ By the definition of the free orbit-dimension, we obtain $$\mathfrak{K}_{1}(x_{1},\ldots,x_{p}) \leq\limsup_{\omega\rightarrow0} \frac{\log( 4\cdot(3ap)^{2n}\cdot apC)-(2n+1)\log\omega} {- \log(4\omega)} \leq1+2n.$$ Hence, $\mathfrak{K}_{1}( \mathcal{M})\leq1+2n$  and  $\delta_{0}( \mathcal{M})\leq2+2n$. The mapping $a\mapsto a^{\ast}$ extends from $\mathcal{M}$ to a unitary map on $L^{2}\left( \mathcal{M},\tau\right) ,$ so for $\xi\in L^{2}\left( \mathcal{M},\tau\right) ,$ it makes sense to talk about ${Re}\xi=\left( \xi+\xi^{\ast}\right) /2$ and ${Im}\xi=\left( \xi-\xi^{\ast}\right) /2i.$ In particular, it makes sense to talk about self-adjoint elements of $L^{2}\left( \mathcal{M},\tau\right) .$ If we have $\overline{span}^{\Vert\cdot\Vert_{2}}\mathcal{N}_{0}\{\xi_{1},\ldots,\xi_{n}\}\mathcal{N}_{1}=L^{2}\left( \mathcal{M},\tau\right) $ with $\xi_{1},\ldots,\xi_{n}$ self-adjoint elements in $L^{2}\left( \mathcal{M},\tau\right) ,$ the proof of Theorem 7 yields $\mathfrak{K}_{1}(\mathcal{M})\leq1+n$ and $\delta_{0}( \mathcal{M})\leq2+n.$ Combining Theorem 7 and the preceding remark with Theorem 3, we have the following corollaries (see also [@Ge1] and [@GePopa]). $L(F_{n})$ has no simple maximal abelian self-adjoint subalgebra for $n\geq4 $. $L(F_{n})$ is not a $\mathfrak{K}$-thin factor for $n\geq4$. ** ** Another corollary of Theorem 7 is as follows. Suppose $\mathcal{M}$ is a II$_{1}$ factor with a tracial state $\tau$. Suppose that $\mathcal{N}$ is a subfactor of $\mathcal{M}$ with finite index, i.e., $[ \mathcal{M}:\mathcal{N}]=r<\infty$. If $\mathfrak{K}_{2}(\mathcal{N})=0$,  then $\mathfrak{K}_{1}( \mathcal{M})\leq2[r]+3$ and $\delta_{0}( \mathcal{M})\leq2[r]+4$ where $[r]$ is the integer part of $r$. [99]{} D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory II," Invent. Math., [ 118]{} (1994), 411-440. D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory III: The absence of Cartan subalgebras," Geom. Funct. Anal. [ 6]{} (1996) 172–199. L. Ge, Applications of free entropy to finite von Neumann algebras," II. Ann. of Math. (2) 147 (1998), no. 1, 143–157. L. Ge, J. Shen, On free entropy dimension of finite von Neumann algebras," Geom. Funct. Anal. 12 (2002), no. 3, 546–566. L. Ge, J. Shen, Free entropy and property $T$ factors," Proc. Natl. Acad. Sci. USA 97 (2000), no. 18, 9881–9885 (electronic). L. Ge, S. Popa, On some decomposition properties for factors of type $\mathrm{II}_{1}$," Duke Math. J. 94 (1998), no. 1, 79–101. D. Hadwin, Free entropy and approximate equivalence in von Neumann algebras, Operator algebras and operator theory (Shanghai, 1997), 111–131, Contemp. Math., 228, Amer. Math. Soc., Providence, RI, 1998. D. Voiculescu, Free entropy dimension $\leq1$ for some generators of property $T$ factors of type $\mathrm{II}_{1}$," J. Reine Angew. Math. 514 (1999), 113–118. L. Ge, Applications of free entropy to finite von Neumann algebras," Amer. J. Math. 119 (1997), no. 2, 467–485. K. Dykema, Two applications of free entropy," Math. Ann. 308 (1997), no. 3, 547–558. M. Dostál, D. Hadwin, An alternative to free entropy for free group factors," International Workshop on Operator Algebra and Operator Theory (Linfen, 2001). Acta Math. Sin. (Engl. Ser.) 19 (2003), no. 3, 419–472. D. Voiculescu, Circular and semicircular systems and free product factors," Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989), 45–60, Progr. Math., 92, Birkhauser Boston, MA, 1990. “Hochschild cohomology of factors with property $\Gamma$," Ann. of Math. (2) 158 (2003), no. 2, 635–659.

--- abstract: | We extend the concept of Wigner-Yanase-Dyson skew information to something we call “metric adjusted skew information” (of a state with respect to a conserved observable). This “skew information” is intended to be a non-negative quantity bounded by the variance (of an observable in a state) that vanishes for observables commuting with the state. We show that the skew information is a convex function on the manifold of states. It also satisfies other requirements, proposed by Wigner and Yanase, for an effective measure-of-information content of a state relative to a conserved observable. We establish a connection between the geometrical formulation of quantum statistics as proposed by Chentsov and Morozova and measures of quantum information as introduced by Wigner and Yanase and extended in this article. We show that the set of normalized Morozova-Chentsov functions describing the possible quantum statistics is a Bauer simplex and determine its extreme points. We determine a particularly simple skew information, the “$ \lambda $-skew information,” parametrized by a $ \lambda\in (0,1], $ and show that the convex cone this family generates coincides with the set of all metric adjusted skew informations.\ Key words: Skew information, convexity, monotone metric, Morozova-Chentsov function, $ \lambda $-skew information. author: - Frank Hansen date: | July 22, 2006\ [(Revised February 20, 2008)]{} title: Metric adjusted skew information --- Introduction ============ In the mathematical model for a quantum mechanical system, the physical observables are represented by self-adjoint operators on a Hilbert space. The “states” (that is, the “expectation functionals” associated with the states) of the physical system are often “modeled” by the unit vectors in the underlying Hilbert space. So, if $ A $ represents an observable and $ x\in H $ corresponds to a state of the system, the expectation of $ A $ in that state is $ (Ax\mid x). $ For what we shall be proving, it will suffice to assume that our Hilbert space is finite dimensional and that the observables are self-adjoint operators, or the matrices that represent them, on that finite dimensional space. In this case, the states can be realized with the aid of the trace (functional) on matrices and an associated “density matrix”. We denote by $ \operatorname{Tr}(B) $ the usual trace of a matrix $ B $ (that is, $ \operatorname{Tr}(B) $ is the sum of the diagonal entries of $ B). $ The expectation functional of a state can be expressed as $ \operatorname{Tr}(\rho A), $ where $ \rho $ is a matrix, the density matrix associated with the state, and “$ \operatorname{Tr}(\rho A) $” is the trace of the product $ \rho A $ of the two matrices $ \rho $ and $ A. $ (Henceforth, we write “$ \operatorname{Tr}\rho A $” omitting the parentheses when they are clearly understood.) In [@kn:wigner:1952], Wigner noticed that the obtainable accuracy of the measurement of a physical observable represented by an operator that does not commute with a conserved quantity (observable) is limited by the “extent” of that non-commutativity. Wigner proved it in the simple case where the physical observable is the $ x $-component of the spin of a spin one-half particle and the $ z $-component of the angular momentum is conserved. Araki and Yanase [@kn:araki:1960] demonstrated that this is a general phenomenon and pointed out, following Wigner’s example, that under fairly general conditions an approximate measurement may be carried out. Another difference is that observables that commute with a conserved additive quantity, like the energy, components of the linear or angular momenta, or the electrical charge, can be measured easily and accurately by microscopic apparatuses (the analysis is restricted to one conserved quantity), while other observables can be only approximately measured by a macroscopic apparatus large enough to superpose sufficiently many states with different quantum numbers of the conserved quantity. Wigner and Yanase [@kn:wigner:1963] proposed finding a measure of our knowledge of a difficult-to-measure observable with respect to a conserved quantity. The quantum mechanical entropy is a measure of our ignorance of the state of a system, and minus the entropy can therefore be considered as an expression of our knowledge of the system. This measure has many attractive properties but does not take into account the conserved quantity. In particular, Wigner and Yanase wanted a measure that vanishes when the observable commutes with the conserved quantity. It should therefore not measure the effect of mixing in the classical sense as long as the pure states taking part in the mixing commute with the conserved quantity. Only transition probabilities of pure states “lying askew” (to borrow from the introduction of [@kn:wigner:1963]) to the eigenvectors of the conserved quantity should give contributions to the proposed measure. Wigner and Yanase discussed a number of requirements that such a measure should satisfy in order to be meaningful and suggested, tentatively, the skew information defined by $$I(\rho,A)=-{\textstyle\frac{1}{2}} \operatorname{Tr}\bigl([\rho^{1/2}, A]^2\bigr),$$ where $ [C,D] $ is the usual “bracket notation” for operators or matrices: $ [C,D]=CD-DC, $ as a measure of the information contained in a state $ \rho $ with respect to a conserved observable $ A. $ It manifestly vanishes when $ \rho $ commutes with $ A, $ and it is homogeneous in $ \rho. $ The requirements Wigner and Yanase discussed, all reflected properties considered attractive or even essential. Since information is lost when separated systems are united such a measure should be decreasing under the mixing of states, that is, be convex in $ \rho. $ The authors proved this for the skew information, but noted that other measures may enjoy the same properties; in particular, the expression $$-{\textstyle\frac{1}{2}}\operatorname{Tr}[\rho^p,A][\rho^{1-p},A]\qquad 0<p<1$$ proposed by Dyson. Convexity of this expression in $ \rho $ became the celebrated Wigner-Yanase-Dyson conjecture which was later proved by Lieb [@kn:lieb:1973:1]. (See also [@kn:hansen:2006:3] for a truly elementary proof.) The measure should also be additive with respect to the aggregation of isolated subsystems and, for an isolated system, independent of time. These requirements are discussed in more detail in section \[subsection: measures of quantum information\]. They are easily seen to be satisfied by the skew information. In the process that is the opposite of mixing, the information content should decrease. This requirement comes from thermodynamics where it is satisfied for both classical and quantum mechanical systems. It reflects the loss of information about statistical correlations between two subsystems when they are only considered separately. Wigner and Yanase conjectured that the skew information also possesses this property. They proved it when the state of the aggregated system is pure[^1]. The aim of this article is to connect the subject of measures of quantum information as laid out by Wigner and Yanase with the geometrical formulation of quantum statistics by Chentsov, Morozova and Petz. The Fisher information measures the statistical distinguishability of probability distributions. Let $ {\cal P}_n=\{p=(p_1,\dots,p_n)\mid p_i>0 \} $ be the (open) probability simplex with tangent space $ T{\cal P}_n. $ The Fisher-Rao metric is then given by $$M_p(u,v)=\sum_{i=1}^n \frac{u_i v_i}{p_i}\qquad u,v\in T{\cal P}_n.$$ Note that $ u=(u_1,\dots,u_n)\in T{\cal P}_n $ if and only if $ u_1+\dots+u_n=0, $ but that the metric is well-defined also on $ \mathbf R^n. $ Chentsov proved that the Fisher-Rao metric is the unique Riemannian metric contracting under Markov morphisms [@kn:censov:1982]. Since Markov morphisms represent coarse graining or randomization, it means that the Fisher information is the only Riemannian metric possessing the attractive property that distinguishability of probability distributions becomes more difficult when they are observed through a noisy channel. Chentsov and Morozova extended the analysis to quantum mechanics by replacing Riemannian metrics defined on the tangent space of the simplex of probability distributions with positive definite sesquilinear (originally bilinear) forms $ K_\rho $ defined on the tangent space of a quantum system, where $ \rho $ is a positive definite state. Customarily, $ K_\rho $ is extended to all operators (matrices) supported by the underlying Hilbert space, cf. [@kn:petz:1996:2; @kn:hansen:2006:2] for details. Noisy channels are in this setting represented by stochastic (completely positive and trace preserving) mappings $ T, $ and the contraction property by the monotonicity requirement $$K_{T(\rho)}(T(A),T(A))\le K_\rho(A,A)$$ is imposed for every stochastic mapping $ T:M_n(\mathbf C)\to M_m(\mathbf C). $ Unlike the classical situation, it turned out that this requirement no longer uniquely determines the metric. By the combined efforts of Chentsov, Morozova and Petz it is established that the monotone metrics are given on the form $$\label{Morozova-Chentsov function} K_\rho(A,B)=\operatorname{Tr}A^* c(L_\rho,R_\rho) B,$$ where $ c $ is a so called Morozova-Chentsov function and $ c(L_\rho,R_\rho) $ is the function taken in the pair of commuting left and right multiplication operators (denoted $ L_\rho $ and $ R_\rho $ respectively) by $ \rho. $ The Morozova-Chentsov function is of the form $$c(x,y)=\frac{1}{y f(x y^{-1})}\qquad x,y >0,$$ where $ f $ is a positive operator monotone function defined in the positive half-axis satisfying the functional equation $$\label{functional equation} f(t)=tf(t^{-1})\qquad t>0.$$ The function $$f(t)=\frac{t+2\sqrt{t}+1}{4}\qquad t>0$$ is clearly operator monotone and satisfies (\[functional equation\]). The associated Morozova-Chentsov function $$c^{WY}(x,y)=\frac{4}{(\sqrt{x}+\sqrt{y})^2}\qquad x,y>0$$ therefore defines a monotone metric $$K_\rho^{WY}(A,B)=\operatorname{Tr}A^* c^{WY}(L_\rho,R_\rho) B,$$ which we shall call the Wigner-Yanase metric. The starting point of our investigation is the observation by Gibilisco and Isola [@kn:gibilisco:2003] that $$I(\rho,A)={\textstyle\frac{1}{8}}\operatorname{Tr}i[\rho, A] c^{WY}(L_\rho,R_\rho) i[\rho, A].$$ There is thus a relationship between the Wigner-Yanase measure of quantum information and the geometrical theory of quantum statistics. It is the aim of the present article to explore this relationship in detail. The main result is that all well-behaved measures of quantum information - including the Wigner-Yanase-Dyson skew informations - are given in this way for a suitable subclass of monotone metrics. Regular metrics --------------- We say that a symmetric monotone metric [@kn:morozova:1990; @kn:petz:1996] on the state space of a quantum system is regular, if the corresponding Morozova-Chentsov function c admits a strictly positive limit $$m(c)=\lim_{t\to 0} c(t,1)^{-1}.$$ We call $ m(c) $ the metric constant. We also say, more informally, that a Morozova-Chentsov function $ c $ is regular if $ m(c)>0. $ The function $ f(t)=c(t,1)^{-1} $ is positive and operator monotone on the positive half-line and may be extended to the closed positive half-line. Thus the metric constant $ m(c)=f(0). $ 1em\ Let $ c $ be the Morozova-Chentsov function of a regular metric. We introduce the metric adjusted skew information $ I^c_\rho(A) $ by setting $$\label{formula: metric adjusted skew information} \begin{array}{rl} I^c_\rho(A)&=\frac{m(c)}{2}\displaystyle K_\rho^c(i[\rho, A], i[\rho, A])\\[2ex] &=\frac{m(c)}{2}\displaystyle\operatorname{Tr}i[\rho, A] c(L_\rho,R_\rho)i[\rho, A] \end{array}$$ for every $ \rho\in\mathcal M_n $ (the manifold of states) and every self-adjoint $ A\in M_n(\mathbf C). $ Note that the metric adjusted skew information is proportional to the square of the metric length, as it is calculated by the symmetric monotone metric $ K_\rho^c $ with Morozova-Chentsov function $ c, $ of the commutator $ i[\rho, A], $ and that this commutator belongs to the tangent space of the state manifold $ \mathcal M_n. $ Metric adjusted skew information is thus a non-negative quantity. If we consider the WYD-metric with Morozova-Chentsov function $$c^{WYD}(x,y)=\frac{1}{p(1-p)}\cdot\frac{(x^p-y^p)(x^{1-p}-y^{1-p})}{(x-y)^2} \qquad 0<p<1,$$ then the metric constant $ m(c^{WYD})=p(1-p) $ and the metric adjusted skew information $$\begin{array}{rl} I^{c^{WYD}}_\rho(A)&=\frac{p(1-p)}{2}\displaystyle\operatorname{Tr}i[\rho, A] c^{WYD}(L_\rho,R_\rho) i[\rho,A]\\[2ex] &=-\frac{1}{2}\operatorname{Tr}[\rho^p,A][\rho^{1-p},A] \end{array}$$ becomes the Dyson generalization of the Wigner-Yanase skew information[^2]. The choice of the factor $ m(c) $ therefore works also for $ p\ne 1/2. $ It is in fact a quite general construction, and the metric constant is related to the topological properties of the metric adjusted skew information close to the border of the state manifold. But it is difficult to ascertain these properties directly, so we postpone further investigation until having established that $ I^c_\rho(A) $ is a convex function in $ \rho. $ Since the commutator $ i[\rho,A]=i(L_\rho-R_\rho)A $ we may rewrite the metric adjusted skew information as $$\label{representation of metric adjusted skew information} \begin{array}{rl} I^c_\rho(A)&=\frac{m(c)}{2}\displaystyle\operatorname{Tr}A(i(L_\rho-R_\rho))^* c(L_\rho,R_\rho) i(L_\rho-R_\rho) A\\[2ex] &=\frac{m(c)}{2}\displaystyle\operatorname{Tr}A\, \hat c(L_\rho,R_\rho)A, \end{array}$$ where $$\hat c(x,y)=(x-y)^2 c(x,y)\qquad x,y>0.$$ Before we can address these questions in more detail, we have to study various characterizations of (symmetric) monotone metrics. Characterizations of monotone metrics ===================================== \[theorem: canonical representation for g\] A positive operator monotone decreasing function $ g $ defined in the positive half-axis and satisfying the functional equation $$\label{functional equation for g} g(t^{-1})=t\cdot g(t)$$ has a canonical representation $$\label{canonical representation for g} g(t)=\int_0^1\left(\frac{1}{t+\lambda}+\frac{1}{1+t\lambda}\right)d\mu(\lambda),$$ where $ \mu $ is a finite Borel measure with support in $ [0,1]. $ The function $ g $ is necessarily of the form $$g(t)=\beta+\int_0^\infty \frac{1}{t+\lambda}\, d\mu(\lambda),$$ where $ \beta\ge 0 $ is a constant and $ \mu $ is a positive Borel measure such that the integrals $ \int(1+\lambda^2)^{-1} d\mu(\lambda) $ and $ \int \lambda(1+\lambda^2)^{-1}d\mu(\lambda) $ are finite, cf. [@kn:hansen:2006:1 Page 9]. We denote by $ \tilde\mu $ the measure obtained from $ \mu $ by removing a possible atom in zero. Then, by making the transformation $ \lambda\to \lambda^{-1}, $ we may write $$\begin{array}{rl} g(t)&=\displaystyle\beta+\frac{\mu(0)}{t}+\int_0^\infty\frac{1}{t+\lambda}\,d\tilde\mu(\lambda)\\[3ex] &=\displaystyle\beta+\frac{\mu(0)}{t}+\int_0^\infty\frac{1}{t+\lambda^{-1}}\cdot\frac{1}{\lambda^2}\,d\tilde\mu(\lambda^{-1})\\[3ex] &=\displaystyle\beta+\frac{\mu(0)}{t}+\int_0^\infty\frac{1}{1+t\lambda}\,d\nu(\lambda), \end{array}$$ where $ \nu $ is the Borel measure given by $ d\nu(\lambda)=\lambda^{-1}d\tilde\mu(\lambda^{-1}). $ Since $ g $ satisfies the functional equation (\[functional equation for g\]) we obtain $$\beta+\mu(0)t+\int_0^\infty\frac{1}{1+t^{-1}\lambda}\,d\nu(\lambda)=t\beta+\mu(0)+\int_0^\infty\frac{t}{t+\lambda}\,d\tilde\mu(\lambda).$$ By letting $ t\to 0 $ and since $ \nu $ and $ \tilde\mu $ have no atoms in zero, we obtain $ \beta=\mu(0) $ and consequently $$\int_0^\infty\frac{1}{t+\lambda}\,d\nu(\lambda)=\int_0^\infty\frac{1}{t+\lambda}\,d\tilde\mu(\lambda)\qquad t>0.$$ By analytic continuation we realize that both measures $ \nu $ and $ \tilde\mu $ appear as the representing measure of an analytic function with negative imaginary part in the complex upper half plane. They are therefore, by the representation theorem for this class of functions, necessarily identical. We finally obtain $$\begin{array}{rl} g(t)&\displaystyle=\beta+\frac{\beta}{t}+\int_0^\infty\frac{1}{t+\lambda}\,d\tilde\mu(\lambda)\\[3ex] &\displaystyle=\beta+\frac{\beta}{t} +\int_0^1\frac{1}{t+\lambda}\,d\tilde\mu(\lambda)+\int_0^1 \frac{1}{t+\lambda^{-1}}\cdot\frac{1}{\lambda^2}\,d\tilde\mu(\lambda^{-1})\\[3ex] &\displaystyle=\beta+\frac{\beta}{t}+\int_0^1\frac{1}{t+\lambda}\,d\tilde\mu(\lambda)+\int_0^1 \frac{1}{1+t\lambda}\,d\nu(\lambda)\\[3ex] &\displaystyle=\beta+\frac{\beta}{t}+\int_0^1\left(\frac{1}{t+\lambda}+\frac{1}{1+t\lambda}\right)d\tilde\mu(\lambda)\\[3ex] &\displaystyle=\int_0^1\left(\frac{1}{t+\lambda}+\frac{1}{1+t\lambda}\right)d\mu(\lambda). \end{array}$$ The statement follows since every function of this form obviously is operator monotone decreasing and satisfy the functional equation (\[functional equation for g\]). We also realize that the representing measure $ \mu $ is uniquely defined. \[remark: construction of the measure\] Inspection of the proof of Theorem \[theorem: canonical representation for g\] shows that the Pick function $ -g(x)=-c(x,1) $ has the canonical representation $$-g(x)=-g(0) + \int_{-\infty}^0 \frac{1}{\lambda - t}\, d\mu(-\lambda).$$ The representing measure therefore appears as $ 1/\pi $ times the limit measure of the imaginary part of the analytic continuation $ -g(z) $ as $ z $ approaches the closed negative half-axis from above, cf. for example [@kn:donoghue:1974]. The measure $ \mu $ in (\[canonical representation for g\]) therefore appears as the image of the representing measure’s restriction to the interval $ [-1, 0] $ under the transformation $ \lambda\to -\lambda. $ We define, in the above setting, an equivalent Borel measure $ \mu_g $ on the closed interval $ [0,1] $ by setting $$d\mu_g(\lambda)=\frac{2}{1+\lambda}\, d\mu(\lambda)$$ and obtain: A positive operator monotone decreasing function $ g $ defined in the positive half-axis and satisfying the functional equation (\[functional equation for g\]) has a canonical representation $$\label{variant canonical representation for g} g(t)=\int_0^1 \frac{1+\lambda}{2}\left(\frac{1}{t+\lambda}+\frac{1}{1+t\lambda}\right)d\mu_g(\lambda),$$ where $ \mu_g $ is a finite Borel measure with support in $ [0,1]. $ The function $ g $ is normalized in the sense that $ g(1)=1, $ if and only if $ \mu_g $ is a probability measure. A Morozova-Chentsov function $ c $ allows a canonical representation of the form $$\label{canonical representation for c} c(x,y)=\int_0^1 c_\lambda(x,y)\, d\mu_c(\lambda)\qquad x,y>0,$$ where $ \mu_c $ is a finite Borel measure on $ [0,1] $ and $$\label{extreme Morozova-Chentsov functions} c_\lambda(x,y)=\frac{1+\lambda}{2}\left(\frac{1}{x+\lambda y}+\frac{1}{\lambda x+y}\right)\qquad \lambda\in[0,1].$$ The Morozova-Chentsov function $ c $ is normalized in the sense that $ c(1,1)=1 $ (corresponding to a Fisher adjusted metric), if and only if $ \mu_c $ is a probability measure. A Morozova-Chentsov function is of the form $ c(x,y)=y^{-1} f(xy^{-1})^{-1}, $ where $ f $ is a positive operator monotone function defined in the positive half-axis and satisfying the functional equation $ f(t)=tf(t^{-1}). $ The function $ g(t)=f(t)^{-1} $ is therefore operator monotone decreasing and satisfies the functional equation (\[functional equation for g\]). It is consequently of the form (\[variant canonical representation for g\]) for some finite Borel measure $ \mu_g. $ Since also $ c(x,y)=y^{-1} g(xy^{-1}) $ the assertion follows by setting $ \mu_c=\mu_g. $ We have shown that the set of normalized Morozova-Chentsov functions is a Bauer simplex, and that the extreme points exactly are the functions of the form (\[extreme Morozova-Chentsov functions\]). We exhibit the measure $ \mu_c $ in the canonical representation (\[canonical representation for c\]) for a number of Morozova-Chentsov functions. - The Wigner-Yanase-Dyson metric with (normalized) Morozova-Chentsov function $$c(x,y)=\frac{1}{p(1-p)}\cdot\frac{(x^p-y^p)(x^{1-p}-y^{1-p})}{(x-y)^2}$$ is represented by $$d\mu_c(\lambda)=\frac{2\sin p\pi}{\pi p (1-p)}\cdot\frac{\lambda^p + \lambda^{1-p}}{(1+\lambda)^3}\,d\lambda$$ for $ 0<p<1. $ The Wigner-Yanase metric is obtained by setting $ p=1/2 $ and it is represented by $$d\mu_c(\lambda)=\frac{16\lambda^{1/2}}{\pi (1+\lambda)^3}\, d\lambda.$$ - The Kubo metric with (normalized) Morozova-Chentsov function $$c(x,y)=\frac{\log x-\log y}{x-y}$$ is represented by $$d\mu_c(\lambda)=\frac{2}{(1+\lambda)^2}\,d\lambda.$$ - The increasing bridge with (normalized) Morozova-Chentsov functions $$c_\gamma(x,y)=x^{-\gamma}y^{-\gamma}\left(\frac{x+y}{2}\right)^{2\gamma-1}$$ is represented by $$\left\{\begin{array}{rll} \mu_c&=\delta(\lambda-1) &\gamma=0\\[1ex] d\mu_c(\lambda)&\displaystyle=\frac{2\sin\gamma\pi}{(1+\lambda)\pi} \lambda^{-\gamma} \left(\frac{1-\lambda}{2}\right)^{2\gamma-1}d\lambda\qquad &0<\gamma<1\\[2ex] \mu_c&\displaystyle=\delta(\lambda) &\gamma=1, \end{array}\right.$$ where $ \delta $ is the Dirac measure with unit mass in zero. We calculate the measures by the method outlined in Remark \[remark: construction of the measure\]. 1\. For the Wigner-Yanase-Dyson metric we therefore consider the analytic continuation $$-g(r e^{i\phi})=-c(r e^{i\phi}, 1)= \frac{-1}{p(1-p)}\cdot\frac{(r^p e^{ip\phi}-1)(r^{1-p} e^{i(1-p)\phi}-1)}{(r e^{i\phi}-1)^2}$$ where $ r>0 $ and $ 0<\phi<\pi. $ We calculate the imaginary part and note that $ r\to-\lambda $ and $ \phi\to\pi $ for $ z\to \lambda<0. $ We make sure that the representing measure has no atom in zero and obtain the desired expression by tedious but elementary calculations. 2\. For the Kubo metric we consider $$-g(x)=-c(x,1)=-\frac{\log x}{x-1}$$ and calculate the imaginary part $$-\Im g(re^{i\phi})=\frac{2r\log r \sin\phi + \phi -\phi r \cos\phi}{r^2-2r\cos\phi +1}$$ of the analytic continuation. It converges towards $ \pi/(1-\lambda) $ for $ z\to\lambda<0 $ and towards $ \pi/2 $ for $ z=re^{i\pi}\to 0. $ The representing measure has therefore no atom in zero, and $ d\mu(\lambda)=d\lambda/(1+\lambda) $ which may be verified by direct calculation. 3\. For the increasing bridge we consider $$-g_\gamma(x)=-c_\gamma(x,1)=-x^{-\gamma}\left(\frac{x+1}{2}\right)^{2\gamma-1}$$ and calculate the imaginary part $$-\Im g_\gamma(re^{i\phi})=-r^{-\gamma}r_1^{2\gamma-1}\exp i(-\gamma\phi+(2\gamma-1)\theta)$$ of the analytic continuation, where $$r_1={\textstyle\frac{1}{2}}(r^2+2r\cos\phi+1)^{1/2}\quad\text{and}\quad\theta=\arctan\frac{rsin\phi}{1+r\cos\phi}.$$ We first note that $ \theta=\pi/2 $ and $ r_1=(r \sin\phi)/2 $ for $ \lambda=-1, $ and that $ \theta\to 0 $ and $ r_1\to (1+\lambda)/2 $ for $ -1<\lambda\le 0. $ The statement now follows by examination of the different cases. In the reference [@kn:hansen:2006:2] we proved the following exponential representation of the Morozova-Chentsov functions. \[theorem: set of Morozova-Chentsov functions\] A Morozova-Chentsov function $ c $ admits a canonical representation $$\label{canonical representation of c in terms of h} c(x,y)=\frac{C_0}{x+y} \exp\int_0^1\frac{1-\lambda^2}{\lambda^2+1}\cdot \frac{x^2+y^2}{(x+\lambda y)(\lambda x +y)}h(\lambda)\,d\lambda$$ where $ h:[0,1]\to[0,1] $ is a measurable function and $ C_0 $ is a positive constant. Both $ C_0 $ and the equivalence class containing $ h $ are uniquely determined by $ c. $ Any function $ c $ on the given form is a Morozova-Chentsov function. We exhibit the constant $ C_0 $ and the representing function $ h $ in the canonical representation (\[canonical representation of c in terms of h\]) for a number of Morozova-Chentsov functions. - The Wigner-Yanase-Dyson metric with Morozova-Chentsov function $$c(x,y)=\frac{1}{p(1-p)}\cdot\frac{(x^p-y^p)(x^{1-p}-y^{1-p})}{(x-y)^2}$$ is represented by $$C_0=\frac{\sqrt{2}}{p(1-p)} \left(1-\cos p\frac{\pi}{2}\right)^{1/2} \left(1-\cos(1-p)\frac{\pi}{2}\right)^{1/2}$$ and $$h(\lambda)=\frac{1}{\pi}\arctan\frac{(\lambda^p + \lambda^{1-p})\sin p\pi}{1-\lambda-(\lambda^p - \lambda^{1-p})\cos p\pi}\qquad 0<\lambda<1,$$ for $ 0<p<1. $ Note that $ 0\le h\le 1/2. $ The Wigner-Yanase metric is obtained by setting $ p=1/2 $ and is represented by $$C_0=4(\sqrt{2}-1)$$ and $$h(\lambda)=\frac{1}{\pi}\arctan\frac{2\lambda^{1/2}}{1-\lambda}\qquad 0<\lambda<1.$$ - The Kubo metric with Morozova-Chentsov function $$c(x,y)=\frac{\log x-\log y}{x-y}$$ is represented by $$C_0=\frac{\pi}{2}\quad\text{and}\quad h(\lambda)=\frac{1}{2}-\frac{1}{\pi}\arctan\left(-\frac{\log\lambda}{\pi}\right).$$ Note that $ 0\le h\le 1/2. $ - The increasing bridge with Morozova-Chentsov functions $$c_\gamma(x,y)=x^{-\gamma}y^{-\gamma}\left(\frac{x+y}{2}\right)^{2j-1}$$ is represented by $$C_0=2^{1-\gamma}\quad\text{and}\quad h(\lambda)=\gamma,\qquad 0\le\gamma\le 1.$$ Setting $ \gamma=0, $ we obtain that the Bures metric with Morozova-Chentsov function $ c(x,y)=2/(x+y) $ is represented by $ C_0=2 $ and $ h(\lambda)=0. $ The analytic continuation of the operator monotone function $ g(x)=\log f(x) $ into the upper complex plane, where $ f(x)=c(x,1)^{-1} $ is the operator monotone function representing [@kn:petz:1996:2] the Morozova-Chentsov function, has bounded imaginary part. The representing measure of the Pick function $ g $ is therefore absolutely continuous with respect to Lebesgue measure. Since $ f $ satisfies the functional equation $ f(t)=t f(t^{-1}) $ we only need to consider the restriction of the measure to the interval $ [-1, 0], $ and the function $ h $ appears [@kn:hansen:2006:2] as the image under the transformation $ \lambda\to -\lambda $ of the Radon-Nikodym derivative. In the same reference it is shown that the constant $ C_0=\sqrt{2} e^{-\beta} $ where $ \beta=\Re\log f(i). $ 1\. For the Wigner-Yanase-Dyson metric the corresponding operator monotone function $$f(x)=\frac{1}{c(x,1)}=p(1-p)\frac{(x-1)^2}{(x^p-1)(x^{1-p}-1)}$$ and we calculate by tedious but elementary calculations $$\lim_{z\to\lambda}\Im\log f(z)=-\frac{1}{2i}\log H\qquad \lambda\in(-1,0),$$ where $$H=\frac{N} {((-\lambda)^{2p}-2(-\lambda)^p\cos p\pi+1)((-\lambda)^{2(1-p)}-2(-\lambda)^{(1-p)}\cos(1-p)\pi+1)}$$ and $$\begin{array}{rl} N&=(-\lambda)^2 + 2(-\lambda)^{1+p} e^{ip\pi}+(-\lambda)^{2p} e^{2ip\pi}-2(-\lambda)^{2-p} e^{-ip\pi}\\[1ex] &+\,4\lambda - 2(-\lambda)^p e^{ip\pi}+(-\lambda)^{2(1-p)} e^{-2ip\pi} + 2(-\lambda)^{1-p} e^{-ip\pi} + 1 \end{array}$$ happens to be the square of the complex number $$(1+\lambda) - ((-\lambda)^p - (-\lambda)^{1-p})\cos p\pi -i((-\lambda)^p + (-\lambda)^{1-p})\sin p\pi$$ with positive real part and negative imaginary part. Since $ H $ has modulus one we can therefore write $$H=e^{-2i\theta}\qquad \lambda\in(-1,0),$$ where $ 0<\theta<\pi/2 $ and $$\tan\theta=\frac{((-\lambda)^p + (-\lambda)^{1-p})\sin p\pi}{1+\lambda-((-\lambda)^p - (-\lambda)^{1-p})\cos p\pi}$$ which implies the expression for $ h. $ The constant $ C_0 $ is obtained by a simple calculation. 2\. For the Kubo metric the corresponding operator monotone function $$f(x)=\frac{1}{c(x,1)}=\frac{x-1}{\log x}$$ and we obtain by setting $ z=r e^{i\phi} $ and $ z-1=r_1 e^{i\phi_1} $ the expression $$\Im\log f(z)=\frac{1}{2i}\left(\log\frac{\log r - i\phi}{\log r +i\phi} +2i\phi_1\right)\qquad 0<\phi<\phi_1<\pi.$$ Since $$\log\frac{\log r - i\phi}{\log r +i\phi}\to \log\frac{\log(-\lambda) - i\pi}{\log(-\lambda) +i\pi}$$ for $ z\to\lambda\in(-1,0) $ and $$\frac{\log(-\lambda) - i\pi}{\log(-\lambda) +i\pi}=e^{-2i\theta}\quad\text{where}\quad\tan\theta=\frac{\pi}{\log(-\lambda)}$$ we obtain $$\lim_{z\to\lambda}\Im\log f(z)=\pi-\theta\qquad\frac{\pi}{2}<\theta<\pi,$$ and thus $$h(\lambda)=1-\frac{1}{\pi}\arctan\frac{\pi}{\log\lambda}.$$ The constant $ C_0 $ is obtained by a straightforward calculation. 3\. The statement for the increasing bridge was proved in [@kn:hansen:2006:2]. Convexity statements ==================== Every Morozova-Chentsov function $ c $ is operator convex, and the mappings $$(\rho,\delta)\to \operatorname{Tr}A^* c(L_\rho,R_\delta) A$$ and $$\rho\to K_\rho^c(A,A)$$ defined on the state manifold are convex for arbitrary $ A\in M_n(\mathbf C). $ Let $ c $ be a Morozova-Chentsov function. Since inversion is operator convex, it follows from the representation given in (\[canonical representation for c\]) that $ c $ as a function of two variables is operator convex. The two assertions now follows from [@kn:hansen:2006:3 Theorem 1.1]. \[lemma: operator convex functions\] Let $ \lambda\ge 0 $ be a constant. The functions of two variables $$f(t,s)=\frac{t^2}{t+\lambda s}\quad\mbox{and}\quad g(t,s)=\frac{ts}{t+\lambda s}$$ are operator convex respectively operator concave on $ (0,\infty)\times(0,\infty). $ The first statement is an application of the convexity, due to Lieb and Ruskai, of the mapping $ (A,B)\to AB^{-1}A. $ Indeed, setting $$C_1=A_1\otimes I_2+\lambda I_1\otimes B_1\quad\mbox{and}\quad C_2=A_2\otimes I_2+\lambda I_1\otimes B_2$$ we obtain $$\begin{array}{l} f(t A_1+(1-t)A_2, t B_1+(1-t)B_2)\\[2ex] =\displaystyle \Bigl((t A_1+(1-t)A_2)\otimes I_2\Bigr) (t C_1+(1-t) C_2 )^{-1} \Bigl((t A_1+(1-t)A_2)\otimes I_2\Bigr)\\[2ex] \le t (A_1\otimes I_2) C_1^{-1} (A_1\otimes I_2) + (1-t)(A_2\otimes I_2) C_2^{-1} (A_2\otimes I_2)\\[2ex] =t f(A_1,B_1)+(1-t)f(A_2,B_2)\qquad t\in[0,1]. \end{array}$$ The second statement is a consequence of the concavity of the harmonic mean $$H(A,B)=2(A^{-1}+B^{-1})^{-1}.$$ Indeed, we may assume $ \lambda>0 $ and obtain $$\begin{array}{l} g(t A_1+(1-t)A_2, t B_1+(1-t)B_2)\\[1.5ex] \displaystyle=\frac{1}{2} H\Bigl(t(\lambda^{-1}A_1\otimes I_2)+ (1-t)(\lambda^{-1}A_2\otimes I_2), t(I_1\otimes B_1)+(1-t)(I_1\otimes B_2)\Bigr)\\[2ex] \displaystyle\ge t\frac{1}{2} H(\lambda^{-1}A_1\otimes I_2,I_1\otimes B_1)+ (1-t)\frac{1}{2} H(\lambda^{-1}A_2\otimes I_2,I_1\otimes B_2)\\[2ex] =t g(A_1,B_1)+(1-t)g(A_2,B_2) \end{array}$$ for $ t\in(0,1]. $ \[c hat is operator convex\] Let $ c $ be a Morozova-Chentsov function. The function of two variables $$\hat c(x,y)=(x-y)^2 c(x,y)\qquad x,y>0$$ is operator convex. A Morozova-Chentsov function $ c $ allows the representation (\[canonical representation for c\]) where $ \mu $ is some finite Borel measure with support in $ [0,1]. $ Since $$\frac{(x-y)^2}{x+\lambda y}=\frac{x^2+y^2-2xy}{x+\lambda y}$$ by Lemma \[lemma: operator convex functions\] is a sum of operator convex functions the assertion follows. \[proposition: decomposition of c\] Let $ c $ be a regular Morozova-Chentson function. We may write $ \hat c(x,y)=(x-y)^2 c(x,y) $ on the form $$\label{formula: decomposition for c hat} \hat c(x,y)=\frac{x+y}{m(c)}-d_c(x,y)\qquad x,y>0,$$ where the positive symmetric function $$\label{the function d representing a regular metric} d_c(x,y)=\int_0^1 xy\cdot c_\lambda(x,y)\frac{(1+\lambda)^2}{\lambda}\,d\mu_c(\lambda)$$ is operator concave in the first quadrant, and the finite Borel measure $ \mu_c $ is the representing measure in (\[canonical representation for c\]) of the Morozova-Chentsov function $ c. $ In addition, we obtain the expression $$\label{skew information in terms of d} \begin{array}{rl} I^c_\rho(A)&=\frac{m(c)}{2}\displaystyle\operatorname{Tr}A\hat c(L_\rho,R_\rho)A\\[2ex] &=\operatorname{Tr}\rho A^2-\frac{m(c)}{2}\displaystyle\operatorname{Tr}A\, d_c(L_\rho,R_\rho)A \end{array}$$ for the metric adjusted skew information. We first notice that $$\label{integral formula for the metric constant} \int_0^1 \frac{(1+\lambda)^2}{2\lambda}\,d\mu_c(\lambda)=\lim_{t\to 0} c(t,1)=\frac{1}{m(c)}$$ and obtain $$\begin{array}{rl} d_c(x,y)&\displaystyle=\frac{x+y}{m(c)}-\hat c(x,y)\\[3ex] &\displaystyle=\frac{x+y}{m(c)}-(x-y)^2 c(x,y)\\[3ex] &\displaystyle=(x+y)\int_0^1\frac{(1+\lambda)^2}{2\lambda}\,d\mu_c(\lambda) -(x-y)^2\int_0^1 c_\lambda(x,y)\,d\mu_c(\lambda)\\[3ex] &\displaystyle=\int_0^1\left((x+y)\frac{(1+\lambda)^2}{2\lambda}-(x-y)^2 c_\lambda(x,y)\right)\,d\mu_c(\lambda). \end{array}$$ The asserted expression of $ d_c $ then follows by a simple calculation and the definition of $ c_\lambda(x,y) $ as given in (\[extreme Morozova-Chentsov functions\]). The function $ d_c $ is operator concave in the first quadrant by Proposition \[c hat is operator convex\]. We call the function $ d_c $ defined in (\[the function d representing a regular metric\]) the representing function for the metric adjusted skew information $ I^c_\rho(A) $ with (regular) Morozova-Chentsov function $ c. $ We introduce for $ 0<\lambda\le 1 $ the $ \lambda $-skew information $ I_\lambda(\rho, A) $ by setting $$I_\lambda(\rho, A)=I^{c_\lambda}_\rho(A).$$ The metric is regular with metric constant $ m(c_\lambda)=2\lambda(1+\lambda)^{-2} $ and the representing measure $ \mu_{c_\lambda} $ is the Dirac measure in $ \lambda. $ The representing function for the metric adjusted skew information is thus given by $$d_{c_\lambda}(x,y)=xy\cdot c_\lambda(x,y) \frac{(1+\lambda)^2}{\lambda} =\frac{m(c_\lambda)}{2}\, xy\cdot c_\lambda(x,y).$$ If we set $$f_\lambda(x,y)=xy\cdot c_\lambda(x,y)=\frac{1+\lambda}{2}\left(\frac{xy}{x+\lambda y}+\frac{xy}{\lambda x+y}\right) \qquad x,y>0,$$ we therefore obtain the expression $$I_\lambda(\rho, A)=\operatorname{Tr}\rho A^2 - \operatorname{Tr}A f_\lambda(L_\rho, R_\rho) A$$ for the $ \lambda $-skew information. Let $ c $ be a regular Morozova-Chentsov function. The metric adjusted skew information may be written on the form $$I^c_\rho(A)={\textstyle\frac{m(c)}{2}}\int_0^1 I_\lambda(\rho, A) \frac{(1+\lambda)^2}{\lambda}\, d\mu_c(\lambda),$$ where $ \mu_c $ is the representing measure and $ m(c) $ is the metric constant. By applying the expressions in (\[skew information in terms of d\]) and (\[the function d representing a regular metric\]) together with the observation in (\[integral formula for the metric constant\]) we obtain $$\begin{array}{rl} I^c_\rho(A)&=\displaystyle\operatorname{Tr}\rho A^2 -{\textstyle\frac{m(c)}{2}}\int_0^1 \operatorname{Tr}A f_\lambda(L_\rho,R_\rho) A\frac{(1+\lambda)^2}{\lambda}\, d\mu_c(\lambda)\\[2ex] &=\frac{m(c)}{2}\displaystyle (\operatorname{Tr}\rho A^2 - \operatorname{Tr}Af_\lambda(L_\rho,R_\rho)A) \frac{(1+\lambda)^2}{\lambda}\, d\mu_c(\lambda) \end{array}$$ and the assertion follows. Measures of quantum information {#subsection: measures of quantum information} ------------------------------- The next result is a direct generalization of the Wigner-Yanase-Dyson-Lieb convexity theorem. Let $ c $ be a regular Morozova-Chentsov function. The metric adjusted skew information is a convex function, $ \rho\to I^c_\rho(A), $ on the manifold of states for any self-adjoint $ A\in M_n(\mathbf C). $ The function $ \hat c(x,y)=(x-y)^2 c(x,y) $ is by Proposition \[c hat is operator convex\] operator convex. Applying the representation of the metric adjusted skew information given in (\[representation of metric adjusted skew information\]), the assertion now follows from [@kn:hansen:2006:3 Theorem 1.1]. The above proof is particularly transparent for the Wigner-Yanase-Dyson metric, since the function $$\begin{array}{rl} \hat c^{WYD}(\lambda,\mu)&\displaystyle=\frac{1}{p(1-p)}(\lambda^p-\mu^p)(\lambda^{1-p}-\mu^{1-p})\\[2ex] &\displaystyle=\frac{1}{p(1-p)}(2-\lambda^p\mu^{1-p}-\lambda^{1-p}\mu^p) \end{array}$$ is operator convex by the simple argument given in [@kn:hansen:2006:3 Corollary 2.2]. Wigner and Yanase [@kn:wigner:1963] discussed a number of other conditions which a good measure of the quantum information contained in a state with respect to a conserved observable should satisfy, but noted that convexity was the most obvious but also the most restrictive and difficult condition. In addition to the convexity requirement an information measure should be additive with respect to the aggregation of isolated systems. Since the state of the aggregated system is represented by $ \rho=\rho_1\otimes\rho_2 $ where $ \rho_1 $ and $ \rho_2 $ are the states of the systems to be united, and the conserved quantity $ A=A_1\otimes 1 + 1\otimes A_2 $ is additive in its components, we obtain $$[\rho,A]=[\rho_1, A_1]\otimes \rho_2 + \rho_1\otimes [\rho_2, A_2].$$ Inserting $ \rho $ and $ A, $ as above, in the definition of the metric adjusted skew information (\[formula: metric adjusted skew information\]), we obtain $$\begin{array}{rl} I^c_\rho(A)&=\frac{m(c)}{2}\displaystyle\operatorname{Tr}\Bigl(i[\rho_1, A_1]\otimes \rho_2 + \rho_1\otimes i[\rho_2, A_2]\Bigr)\\[1.5ex] &\hskip 3em c(L_{\rho_1},R_{\rho_1})\otimes c(L_{\rho_2},R_{\rho_2})\Bigl(i[\rho_1, A_1]\otimes \rho_2 + \rho_1\otimes i[\rho_2, A_2]\Bigr). \end{array}$$ The cross terms vanish because of the cyclicity of the trace, and since $ \rho_1 $ and $ \rho_2 $ have unit trace we obtain $$I^c_\rho(A)=I^c_{\rho_1}(A_1)+I^c_{\rho_2}(A_2)$$ as desired. The metric adjusted skew information for an isolated system should also be independent of time. But a conserved quantity $ A $ in an isolated system commutes with the Hamiltonian $ H, $ and since the time evolution of $ \rho $ is given by $ \rho_t=e^{itH}\rho e^{-itH} $ we readily obtain $$I^c_{\rho_t}(A)=I^c_\rho(A)\qquad t\ge 0$$ by using the unitary invariance of the metric adjusted skew information. The variance $ \operatorname{Var}_\rho(A) $ of a conserved observable $ A $ with respect to a state $ \rho $ is defined by setting $$\operatorname{Var}_\rho(A)=\operatorname{Tr}\rho A^2 - (\operatorname{Tr}\rho A)^2.$$ It is a concave function in $ \rho. $ \[theorem: the skew information is bounded by the variance\] Let $ c $ be a regular Morozova-Chentsov function. The metric adjusted skew information $ I^c(\rho,A) $ may for each conserved (self-adjoint) variable $ A $ be extended from the state manifold to the state space. Furthermore, $$I^c_\rho(A)=\operatorname{Var}_\rho(A)$$ if $ \rho $ is a pure state, and $$0\le I^c_\rho(A)\le\operatorname{Var}_\rho(A)$$ for any density matrix $ \rho. $ We note that the representing function $ d $ in (\[the function d representing a regular metric\]) may be extended to a continuous operator concave function defined in the closed first quadrant with $ d(t,0)=d(0,t)=0 $ for every $ t\ge 0, $ and that $ d(1,1)=2/m(c). $ Since a pure state is a one-dimensional projection $ P, $ it follows from the representation in (\[representation of metric adjusted skew information\]) and the formula (\[formula: decomposition for c hat\]) that $$\begin{array}{rl} I^c_P(A)&=\frac{m(c)}{2}\displaystyle\operatorname{Tr}\left(\frac{APA+AAP}{m(c)}-d(1,1) APAP\right)\\[2ex] &\displaystyle=\operatorname{Tr}PA^2 - \operatorname{Tr}(PAP)^2\\[2ex] &=\operatorname{Tr}PA^2-(\operatorname{Tr}PA)^2\\[2ex] &=\operatorname{Var}_P(A). \end{array}$$ An arbitrary state $ \rho $ is by the spectral theorem a convex combination $ \rho=\sum_i \lambda_i P_i $ of pure states. Hence $$I^c_\rho(A)\le\sum_i \lambda_i I^c_{P_i}(A)=\sum_i \lambda_i \operatorname{Var}_{P_i}(A)\le\operatorname{Var}_\rho(A),$$ where we used the convexity of the metric adjusted skew information and the concavity of the variance. The metric adjusted correlation ------------------------------- We have developed the notion of metric adjusted skew information, which is a generalization of the Wigner-Yanase-Dyson skew information. It is defined for all regular metrics (symmetric and monotone), where the term regular means that the associated Morozova-Chentsov functions have continuous extensions to the closed first quadrant with finite values everywhere except in the point $ (0,0). $ Let $ c $ be a regular Morozova-Chentsov function, and let $ d $ be the representing function (\[the function d representing a regular metric\]). The metric adjusted correlation is defined by $$\operatorname{Corr}^c_\rho(A, B)=\operatorname{Tr}\rho A^*B-{\textstyle\frac{m(c)}{2}}\operatorname{Tr}A^*\, d(L_\rho,R_\rho)B$$ for arbitrary matrices $ A $ and $ B. $ Since $ d $ is symmetric, the metric adjusted correlation is a symmetric sesqui-linear form which by (\[skew information in terms of d\]) satisfies $$\operatorname{Corr}^c_\rho(A, A)=I^c_\rho(A)\qquad\text{for self-adjoint\,} A.$$ The metric adjusted correlation is not a real form on self-adjoint matrices, and it is not positive on arbitrary matrices. Therefore, Cauchy-Schwartz inequality only gives a bound $$\label{correlation inequality} |\Re\operatorname{Corr}^c_\rho(A, B)|\le I^c_\rho(A)^{1/2} I^c_\rho(A)^{1/2} \le\operatorname{Var}_\rho(A)^{1/2}\cdot \operatorname{Var}_\rho(B)^{1/2}$$ for the real part of the metric adjusted correlation. However, since $$\operatorname{Corr}^c_\rho(A, B)-\operatorname{Corr}^c_\rho(B, A)=\operatorname{Tr}\rho[A,B]\qquad A^*=A,\, B^*=B,$$ we obtain $$\frac{1}{2}|\operatorname{Tr}\rho[A,B]|=|\Im\operatorname{Corr}^c_\rho(A, B)|$$ for self-adjoint $ A $ and $ B. $ The estimate in (\[correlation inequality\]) can therefore not be used to improve Heisenberg’s uncertainty relations[^3]. The variant bridge ------------------ The notion of a regular metric seems to be very important. We note that the Wigner-Yanase-Dyson metrics and the Bures metric are regular, while the Kubo metric and the maximal symmetric monotone metric are not. The continuously increasing bridge with Morozova-Chentsov functions $$c_\gamma(x,y)=x^{-\gamma}y^{-\gamma}\left(\frac{x+y}{2}\right)^{2j-1}\qquad 0\le\gamma\le 1$$ connects the Bures metric $ c_0(x,y)=2/(x+y) $ with the maximal symmetric monotone metric $ c_1(x,y)=2xy/(x+y). $ Since the Bures metric is regular and the maximal symmetric monotone metric is not, any bridge connecting them must fail to be regular at some point. However, the above bridge fails to be regular at any point $ \gamma\ne 0. $ A look at the formula (\[canonical representation of c in terms of h\]) shows that a symmetric monotone metric is regular, if and only if $ \lambda^{-1} $ is integrable with respect to $ h(\lambda)\,d\lambda. $ We may obtain this by choosing for example $$h_p(\lambda)=\left\{\begin{array}{lrl} 0,\quad &\lambda &<1-p\\[1ex] p, &\lambda&\ge 1-p \end{array}\right.\qquad 0\le p\le 1$$ instead of the constant weight functions. Since $$\int\frac{(\lambda^2-1)(1+t^2)}{(1+\lambda^2)(\lambda+t)(1+\lambda t)}\, d\lambda=\log\frac{1+\lambda^2}{(\lambda+t)(1+\lambda t)}$$ we are by tedious calculations able to obtain the expression $$f_p(t)=\frac{1+t}{2}\left(\frac{4(1-p+t)(1+(1-p)t)}{(2-p)^2 (1+t)^2}\right)^p\qquad t>0$$ for the normalized operator monotone functions represented by the $ h_p(\lambda) $ weight functions [@kn:hansen:2006:2 Theorem 1]. The corresponding Morozova-Chentsov functions are then given by $$\label{variant increasing bridge} c_p(x,y)=\frac{(2-p)^{2p}}{(x+(1-p)y)^p ((1-p)x+y)^p}\left(\frac{x+y}{2}\right)^{2p-1}$$ for $ 0\le p\le 1. $ The weight functions $ h_p(\lambda) $ provides a continuously increasing bridge from the zero function to the unit function. But we cannot be sure that the corresponding Morozova-Chentsov functions are everywhere increasing, since we have adjusted the multiplicative constants such that all the functions $ f_p(t) $ are normalized to $ f_p(1)=1. $ However, since by calculation $$\frac{\partial}{\partial p} f_p(t)= \frac{-2p^2(1-t)^2}{(2-p)^3(1+t)} \left(\frac{4(1-p+t)(1+(1-p)t)}{(2-p)^2 (1+t)^2}\right)^{p-1}<0,$$ we realize that the representing operator monotone functions are decreasing in $ p $ for every $ t>0. $ In conclusion, we have shown that the symmetric monotone metrics given by (\[variant increasing bridge\]) provides a continuously increasing bridge between the smallest and largest (symmetric and monotone) metrics, and that all the metrics in the bridge are regular except for $ p=1. $ [10]{} H. Araki and M.M. Yanase. Measurement of quantum mechanical operators. , 120:662–626, 1960. N.N. Censov. , volume 53. Amer. Math. Soc., Providence, 1982. W. Donoghue. . , [Berlin, Heidelberg, New York]{}, 1974. P. Gibilisco and T. Isola. igner-anase information on quantum state space: he geometric approach. , 44:3752–3762, 2003. P. Gibilisco and T. Isola. Uncertainty principle and quantum isher information. , 59:147–159, 2007. F. Hansen. Characterizations of symmetric monotone metrics on the the state space of quantum systems. , 6:597–605, 2006. F. Hansen. Extensions of Lieb’s concavity theorem. , 124:87–101, 2006. F. Hansen. Trace functions as aplace transforms. , 47:043504, 2006. F. Hansen. The igner-anase entropy is not subadditive. , 126:643–648, 2007. H. Hasegawa and D. Petz. Non-commutative extension of the information geometry . In O. Hirota, editor, [*Quantum Communication and Measurement*]{}, pages 109–118. Plenum, New York, 1997. W. Heisenberg. ber den anschaulichen nhalt der quantummechanischen inematik und echanik. , 43:172–198, 1927. H. Kosaki. Matrix trace inequalities related to uncertainty principle. , 16:629–645, 2005. A. Lesniewski and M.B. Ruskai. Monotone iemannian metrics and relative entropy on non-commutative probability spaces. , 40:5702–5724, 1999. E. Lieb. Convex trace functions and the igner-anase-yson conjecture. , 11:267–288, 1973. S. Luo. Wigner-anase skew information and uncertainty relations. , 91:180403, 2003. E.A. Morozova and N.N. Chentsov. arkov invariant geometry on state manifolds (ussian). , 36:69–102, 1990. Translated in J. Soviet Math. 56:2648-2669, 1991. D. Petz. Geometry of canonical correlation on the state space of a quantum system. , 35:780–795, 1994. H. Hasegawa and D. Petz. On the iemannian metric of $ \alpha $-entropies of density matrices. , 38:221–22, 1996. D. Petz. Monotone metrics on matrix spaces. , 244:81–96, 1996. D. Petz and C. Sud[á]{}r. Geometries of quantum states. , 37:2662–2673, 1996. E.P. Wigner. Die essung quantenmechanischer peratoren. , 133:101–108, 1952. E.P. Wigner and M.M. Yanase. Information contents of distributions. , 49:910–918, 1963. K. Yanagi, S. Furuichi, and K. Kuriyama. A generalized skew information and uncertainty relation. , 51:4401–4404, 2005. Frank Hansen: Department of Economics, University of Copenhagen, Studiestraede 6, DK-1455 Copenhagen K, Denmark. [^1]: We subsequently demonstrated [@kn:hansen:2007:2] that the conjecture fails for general mixed states. [^2]: Hasegawa and Petz proved in [@kn:petz:1996:1] that the function $ c^{WYD} $ is a Morozova-Chentsov function. They also proved that the Wigner-Yanase-Dyson skew information is proportional to the (corresponding) quantum Fisher information of the commutator $ i[\rho,A]. $ [^3]: In the first version of this paper, which appeared on July 22 2006, the estimation in (\[correlation inequality\]) was erroneously extended to the metric adjusted skew information itself and not only to the real part, cf. also Luo [@kn:luo:2003] and Kosaki [@kn:kosaki:2005]. The author is indebted to Gibilisco and Isola for pointing out the mistake.

--- abstract: | SAMP is a messaging protocol that enables astronomy software tools to interoperate and communicate. IVOA members have recognised that building a monolithic tool that attempts to fulfil all the requirements of all users is impractical, and it is a better use of our limited resources to enable individual tools to work together better. One element of this is defining common file formats for the exchange of data between different applications. Another important component is a messaging system that enables the applications to share data and take advantage of each other’s functionality. SAMP supports communication between applications on the desktop and in web browsers, and is also intended to form a framework for more general messaging requirements. author: - | M. Taylor ([email protected])\ T. Boch ([email protected])\ M. Fitzpatrick ([email protected])\ A. Allan ([email protected])\ J. Fay ([email protected])\ L. Paioro ([email protected])\ J. Taylor ([email protected])\ D. Tody ([email protected]) title: 'SAMP — Simple Application Messaging Protocol' --- Status of this Document {#status-of-this-document .unnumbered} ======================= This document has been produced by the IVOA Applications Working Group. It has been reviewed by IVOA Members and other interested parties, and has been endorsed by the IVOA Executive Committee as an IVOA Recommendation. It is a stable document and may be used as reference material or cited as a normative reference from another document. IVOA’s role in making the Recommendation is to draw attention to the specification and to promote its widespread deployment. This enhances the functionality and interoperability inside the Astronomical Community. Comments, questions and discussions relating to this document may be posted to the mailing list of the SAMP subgroup of the Applications Working Group, <[email protected]>. Supporting material and further discussion may be found at <http://www.ivoa.net/samp/>. Changes since earlier versions may be found in Appendix \[sect:changes\]. Introduction ============ Non-Technical Preamble and Position in IVOA Architecture {#sect:nonTechPreamble} -------------------------------------------------------- SAMP, the Simple Application Messaging Protocol, is a standard for allowing software tools to exchange control and data information, thus facilitating tool interoperability, and so allowing users to treat separately developed applications as an integrated suite. An example of an operation that SAMP might facilitate is passing a source catalogue from one GUI application to another, and subsequently allowing sources marked by the user in one of those applications to be visible as such in the other. The protocol has been designed, and implementations developed, within the context of the International Virtual Observatory Alliance (IVOA), but the design is not specific either to the Virtual Observatory (VO) or to Astronomy. It is used in practice for both VO and non-VO work with astronomical tools, and is in principle suitable for non-astronomical purposes as well. The SAMP standard itself is neither a dependent, nor a dependency, of other VO standards, but it provides valuable glue between user-level applications which perform different VO-related tasks, and hence contributes to the integration of Virtual Observatory functionality from a user’s point of view. Figure \[fig:ivoa-archi\] illustrates SAMP in the context of the IVOA Architecture [@architecture]. Most existing tools which operate in the User Layer of this architecture provide SAMP interoperability. ![IVOA Architecture diagram [@architecture]. The SAMP protocol appears in the “Using” region.[]{data-label="fig:ivoa-archi"}](ivoa-archi) The semantics of messages that can be exchanged using SAMP are defined by contracts known as MTypes (message-types), which are defined by developer agreement outside of this standard. The list of MTypes used for common astronomical and VO purposes can be found near <http://www.ivoa.net/samp/>; many of these make use of standards from elsewhere in the IVOA Architecture, including VOTable, VOResource, Simple Spectral Access, UCD and Utype. History ------- SAMP, the Simple Application Messaging Protocol, is a direct descendent of the PLASTIC protocol, which in turn grew — in the European VOTech framework — from the interoperability work of the Aladin [@2000A+AS..143...33B] and VisIVO [@2007HiA....14..622B] teams. We also note the contribution of the team behind the earlier XPA protocol [@xpa]. For more information on PLASTIC’s history and purpose see the IVOA Note [*PLASTIC — a protocol for desktop application interoperability*]{} [@plastic] and the PLASTIC SourceForge site [@plastic-sf]. SAMP has similar aims to PLASTIC, but incorporates lessons learnt from two years of practical experience and ideas from partners who were not involved in PLASTIC’s initial design. Broadly speaking, SAMP is an abstract framework for loosely-coupled, asynchronous, RPC-like and/or event-based communication, based on a central service providing multi-directional publish/subscribe message brokering. The message semantics are extensible and use structured but weakly-typed data. These concepts are expanded on below. It attempts to make as few assumptions as possible about the transport layer or programming language with which it is used. It also defines a “Standard Profile” which specifies how to implement this framework using XML-RPC [@xmlrpc] as the transport layer. The result of combining this Standard Profile with the rest of the SAMP standard is deliberately similar in design to PLASTIC, and this has been largely successful in its intention of enabling PLASTIC applications to be modified to use SAMP instead without great effort. More recently (version 1.3) an additional “Web Profile” has been introduced, in order to facilitate use of SAMP from web applications. Requirements and Scope ---------------------- SAMP aims to be a simple and extensible protocol that is platform- and language-neutral. The emphasis is on a simple protocol with a very shallow learning curve in order to encourage as many application authors as possible to adopt it. SAMP is intended to do what you need most of the time. The SAMP authors believe that this is the best way to foster innovation and collaboration in astronomy applications. It is important to note therefore that SAMP’s scope is reasonably modest; it is not intended to be the perfect messaging solution for all situations. In particular SAMP itself has no support for transactions, security, or guaranteed message delivery or integrity. However, by layering the SAMP architecture on top of suitable messaging infrastructures such capabilities could be provided. These possibilities are not discussed further in this document, but the intention is to provide an architecture which is sufficiently open to allow for such things in the future with little change to the basics. Types of Messaging {#sect:typeOfMsging} ------------------ SAMP is currently targetted at inter-application desktop messaging with the idea that the basic framework presented here is extensible to meet future needs, and so it is beyond the scope of this document to outline the many types of messaging systems in use today (these are covered in detail in many other documents). While based on established messaging models, SAMP is in many ways a hybrid of several basic messaging concepts; the protocol is however flexible enough that later versions should be able to interact fairly easily with other messaging systems because of the shared messaging models. The messaging concepts used within SAMP include: Publish/Subscribe Messaging: : A publish/subscribe (pub/sub) messaging system supports an event driven model where information producers and consumers participate in message passing. SAMP applications “publish” a message, while consumer applications “subscribe” to messages of interest and consume events. The underlying messaging system routes messages from producers to consumers based on the message types in which an application has registered an interest. Point-to-Point Messaging: : In point to point messaging systems, messages are routed to an individual consumer which maintains a queue of “incoming” messages. In a traditional message queue, applications send messages to a specified queue and clients retrieve them. In SAMP, the message system manages the delivery and routing of messages, but also permits the concept of a directed message meant for delivery to a specific application. SAMP does not, however, guarantee the order of message delivery as with a traditional message queue. Event-based Messaging: : Event-based systems are systems in which producers generate events, and in which messaging middleware delivers events to consumers based upon a previously specified interest. One typical usage pattern of these systems is the publish/subscribe paradigm, however these systems are also widely used for integrating loosely coupled application components. SAMP allows for the concept that an “event” occurred in the system and that these message types may have requirements different from messages where the sender is trying to invoke some action in the network of applications. Synchronous vs. Asynchronous Messaging: : As the term is used in this document, a “synchronous” message is one which blocks the sending application from further processing until a reply is received. However, SAMP messaging is based on “asynchronous” message and response in that the delivery of a message and its subsequent response are handled as separate activities by the underlying system. With the exception of the synchronous message pattern supported by the system, sending or replying to a message using SAMP allows an application to return to other processing while the details of the delivery are handled separately. About this Document ------------------- This document contains the following main sections describing the SAMP protocol and how to use it. Section \[sect:architecture\] covers the requirements, basic concepts and overall architecture of SAMP. Section \[sect:apis\] defines abstract (i.e. independent of language, platform and transport protocol) interfaces which clients and hubs must offer to participate in SAMP messaging, along with data types and encoding rules required to use them. Section \[sect:profile\] explains how the abstract API can be mapped to specific network operations to form an interoperable messaging system, and defines the “Standard Profile”, based on XML-RPC, which gives a particular set of such mappings suitable for general purpose desktop applications. Section \[sect:webprofile\] defines the “Web Profile”, an alternative mapping suitable for web applications. Section \[sect:mtypes\] describes the use of the MType keys used to denote message semantics, and outlines an MType vocabulary. The key words “MUST”, “MUST NOT”, “REQUIRED”, “SHALL”, “SHALL NOT”, “SHOULD”, “SHOULD NOT”, “RECOMMENDED”, “MAY”, and “OPTIONAL” in this document are to be interpreted as described in RFC 2119 [@rfc2119]. Architectural Overview {#sect:architecture} ====================== This section provides a high level view of the SAMP protocol. Nomenclature ------------ In the text that follows these terms are used: Hub: : A broker service for routing SAMP Messages. Client: : An application that talks to a Hub using SAMP. May be a Sender, Recipient, or both. Sender: : A Client that sends a SAMP Message to one or more Recipients via the Hub. Recipient: : A Client that receives a SAMP Message from the Hub. This may have originated from another Client or from the Hub itself. Message: : A communication sent from a Sender to a Recipient via a SAMP Hub. Contains an MType and zero or more named parameters. May or may not provoke a Response. Response: : A communication which may be returned from a Recipient to a Sender in reply to a previous Message. A Response may contain returned values and/or error information. In the terminology of this document, a Response is not itself a Message. A Response is also known as a Reply in this document. MType: : A string defining the semantics of a Message and of its arguments and return values (if any). Every Message contains exactly one MType, and a Message is only delivered to Clients subscribed to that MType. Subscription: : A Client is said to be Subscribed to a given MType if it has declared to the Hub that it is prepared to receive Messages with that MType. Callable Client: : A Client to which the Hub is capable of performing callbacks. Clients are not obliged to be Callable, but only Callable Clients are able to receive Messages or asynchronous Responses. Broadcast: : To send a SAMP Message to all Subscribed Clients excluding the Sender. Profile: : A set of rules which map the abstract API defined by SAMP to a set of I/O operations which may be used by Clients to send and receive actual Messages. Messaging Topology ------------------ SAMP has a hub-based architecture (see Figure \[fig:samp-archi\]). The hub is a single service used to route all messages between clients. This makes application discovery more straightforward in that each client only needs to locate the hub, and the services provided by the hub are intended to simplify the actions of the client. A disadvantage of this architecture is that the hub may be a message bottleneck and potential single point of failure. The former means that SAMP may not be suitable for extremely high throughput requirements; the latter may be mitigated by an appropriate strategy for hub restart if failure is likely. ![The SAMP hub architecture[]{data-label="fig:samp-archi"}](samp-archi) Note that the hub is defined as a service interface which may have any of a number of implementations. It may be an independent application running as a daemon, an adapter interface layered on top of an existing messaging infrastructure, or a service provided by an application which is itself one of the hub’s clients. The Lifecycle of a Client ------------------------- A SAMP client goes through the following phases: 1. Determine whether a hub is running by using the appropriate hub discovery mechanism. 2. If so, use the hub discovery mechanism to work out how to communicate with the hub. 3. Register with the hub. 4. Store metadata such as client name, description and icon in the hub. 5. Subscribe to a list of MTypes to define messages which may be received. 6. Interrogate the hub for metadata of other clients. 7. Send and/or receive messages to/from other clients via the hub. 8. Unregister with the hub. Phases 4–7 are all optional and may be repeated in any order. By subscribing to the MTypes described in Section \[sect:hub-mtypes\] a client may, if it wishes, keep track of the details of other clients’ registrations, metadata and subscriptions. The Lifecycle of a Hub ---------------------- A SAMP hub goes through the following phases: 1. Locate any existing hub by using the appropriate hub discovery mechanism. 1. Check whether the existing hub is alive. 2. If so, exit. 2. If no hub is running, or a hub is found but is not functioning, write/overwrite the hub discovery record and start up. 3. Await client registrations. When a client makes a legal registration, assign it a public ID, and add the client to the table of registered clients under the public ID. Broadcast a message announcing the registration of a new client. 4. When a client stores metadata in the hub, broadcast a message announcing the change and make the metadata available. 5. When a client updates its list of subscribed MTypes, broadcast a message announcing the change and make the subscription information available 6. When the hub receives a message for relaying, pass it on to appropriate recipients which are subscribed to the message’s MType. Broadcast messages are sent to all subscribed clients except the sender, messages with a specified recipient are sent to that recipient if it is subscribed. 7. Await client unregistrations. When a client unregisters, broadcast a message announcing the unregistration and remove the client from the table of registered clients. 8. If the hub is unable to communicate with a client, it may unregister it as described in phase 7. 9. When the hub is about to shutdown, broadcast a message to all subscribed clients. 10. Delete the hub discovery record. Phases 3–8 are responses to events which may occur multiple times and in any order. The MTypes broadcast by the hub to inform clients of changes in its state are given in Section \[sect:hub-mtypes\]. Readers should note that, given this scheme, race conditions may occur. A client might for instance try to register with a hub which has just shut down, or attempt to send to a recipient which has already unregistered. Specific profiles MAY define best-practice rules in order to best manage these conditions, but in general clients should be aware that SAMP’s lack of guaranteed message delivery and timing means that unexpected conditions are possible. Message Delivery Patterns {#sect:delivery-outline} ------------------------- Messages can be sent according to three patterns, differing in whether and how a response is returned to the sender: 1. Notification 2. Asynchronous Call/Response 3. Synchronous Call/Response The Notification pattern is strictly one-way while in the Call/Response patterns the recipient returns a response to the sender. If the sender expects to receive some useful data as a result of the receiver’s processing, or if it wishes to find out whether and when the processing is completed, it should use one of the Call/Response variants. If on the other hand the sender has no interest in what the recipient does with the message once it has been sent, it may use the Notification pattern. Notification, since it involves no communication back from the recipient to the sender, uses fewer resources. Although typically “event”-type messages will be sent using Notify and “request-for-information”-type messages will be sent using Call/Response, the choice of which delivery pattern to use is entirely distinct from the content of the message, and is up to the sender; any message (MType) may be sent using any of the above patterns. Apart from the fact of returning or not returning a response, the recipient SHOULD process messages in exactly the same way regardless of which pattern is used. From the receiver’s point of view there are only two cases: Notification and Asynchronous Call/Response. However, the hub provides a convenience method which simulates a synchronous call from the sender’s point of view. The purpose of this is to simplify the use of the protocol in situations such as scripting environments which cannot easily handle callbacks. However, it is RECOMMENDED to use the asynchronous pattern where possible due to its greater robustness. Extensible Vocabularies {#sect:vocab} ----------------------- At several places in this document structured information is conveyed by use of a controlled but extensible vocabulary. Some examples are the client metadata keys (Section \[sect:app-metadata\]), message encoding keys (Section \[sect:msg-encode\]) and MType names (Section \[sect:mtypes\]). Wherever this pattern is used, the following rules apply. This document defines certain well-known keys with defined meanings. These may be OPTIONAL or REQUIRED as documented, but if present MUST be used by clients and hubs in the way defined here. All such well-known keys start with the string “[samp.]{}”. Clients and hubs are however free to introduce and use non-well-known keys as they see fit. Any string may be used for such a non-standard key, with the restriction that it MUST NOT start with the prefix “[samp.]{}”. The prefix “[x-samp.]{}” has a special meaning as described below. The general rule is that hubs and clients encountering keys which they do not understand SHOULD ignore them, propagating them to downstream consumers if appropriate. As far as possible, where new keys are introduced they SHOULD be such that applications ignoring them will continue to behave in a sensible way. Hubs and clients are therefore able to communicate information additional to that defined in the current version of this document without disruption to those which do not understand it. This extensibility may be of use to applications which have mutual private requirements outside the scope of this specification, or to enable experimentation with new features. If the SAMP community finds such experiments useful, future versions of this document may bring such functionality within the SAMP specification itself by defining new keys in the “[samp.]{}” namespace. The ways in which these vocabularies are used means that such extensions should be possible with minimal upheaval to the existing specification and implementations. Non-well-known keys (those outside of the “[samp]{}” namespace) fall into two categories: those which are candidates for future incorporation into the SAMP standard as well-known, and those which are not. If developers are experimenting with keys which they hope or believe may be incorporated into the SAMP standard as well-known at some time in the future, they may use the special namespace “[x-samp]{}”. If a future version of the standard does incorporate such a key as well-known, the prefix is simply changed from “[x-samp.]{}” to “[samp.]{}”. Consumers of such keys SHOULD treat keys which differ only in the substitution of the prefix “[samp.]{}” for “[x-samp.]{}” or vice versa as if they have identical semantics, so for instance a client application should treat the value of a metadata item with key “[x-samp.a.b]{}” in exactly the same way as one with key “[samp.a.b]{}”. The “[samp]{}” and “[x-samp]{}” form of the same key SHOULD NOT be presented in the same map. If both are presented together, the “[samp]{}” form MAY be considered to take precedence, though any reasonable behaviour is permitted. This scheme makes it easy to introduce new well-known keys in a way which neither makes illicit use of the reserved “[samp.]{}” namespace nor requires frequent updates to the SAMP standard, and which places a minimum burden on application developers. Lists of keys in the “[x-samp]{}” namespace under discussion may be found near <http://www.ivoa.net/samp/>. Use of Profiles {#sect:profiles} --------------- The design of SAMP is based on the abstract interfaces defined in Section \[sect:apis\]. On its own however, this does not include the detailed instructions required by application developers to achieve interoperability. To achieve that, application developers must know how to map the operations in the abstract SAMP interfaces to specific I/O (in most cases, network) operations. It is these I/O operations which actually form the communication between applications. The rules defining this mapping from interface to I/O operations are what constitute a SAMP “Profile” (the term “Implementation” was considered for this purpose, but rejected because it has too many overlapping meanings in this context). There are two ways in which such a Profile can be specified as far as client application developers are concerned: 1. By describing exactly what bytes are to be sent using what wire protocols for each SAMP interface operation 2. By providing one or more language-specific libraries with calls which correspond to those of the SAMP interface Although either is possible, SAMP is well-suited for approach (1) above given a suitable low-level transport library. This is the case since the operations are quite low-level, so client applications can easily perform them without requiring an independently developed SAMP library. This has the additional advantages that central effort does not have to be expended in producing language-specific libraries, and that the question of “unsupported” languages does not arise. Splitting the abstract interface and Profile descriptions in this way separates the basic design principles from the details of how to apply them, and it opens the door for other Profiles serving other use cases in the future. This document defines two profiles along the lines of (1) above. The Standard Profile (Section \[sect:profile\]) which dates from the first version of this document, is suitable for desktop applications, while the Web Profile (Section \[sect:webprofile\]), introduced at SAMP version 1.3, is suitable for web (browser-based) applications. A client author will usually only need to implement SAMP communications using a single profile. Hub implementations should ideally implement all known profiles; in this way clients using different profiles can communicate transparently with each other via a hub which mediates between them. Since the different profiles are based on the same abstract interface (Section \[sect:apis\]), such mediation will not lead to loss or distortion of the communications. Security Considerations {#sect:security} ----------------------- SAMP enables inter-process communications including the capability for one client to cause execution of code by another client. This raises the possibility of an unprivileged client performing privileged actions in virtue of its SAMP-enabled interoperation. Whether this is problematic in practice depends on two things: first the identities of the interoperating clients (whether they all share similar levels of privilege or trust) and second the semantics of the messages (the nature of the code that may be executed remotely, and particularly how it can be parameterised). In the case that untrusted clients can cause execution of potentially damaging code by trusted clients, there is a serious security issue. The trustedness of registered clients is determined by the profile or profiles operated by the hub at a given time (Section \[sect:profiles\]), since the extent to which registered clients are trusted may differ between different profiles. Clients registering via the Standard Profile in its usual configuration can be assumed all to be owned by the same user and hence to have the same privileges (Section \[sect:std-security\]), but Web Profile clients usually have only limited access privileges outside of the interoperability granted by SAMP (Section \[sect:web-security\]). In most cases profiles will, in virtue of their definition or at least of their implementation, provide reasonable assurance that registered clients are unlikely to be hostile. However for clients which may be run in a general SAMP context, it is wise not to expose via SAMP mechanisms unrestricted access to sensitive resources. In particular, it is recommended not to introduce MTypes which can be made to execute arbitrary code (inviting injection attacks), or to declare metadata which reveals sensitive information. As an alternative approach, it may be appropriate in certain usage scenarios to ensure that only a restricted secure profile is running. Abstract APIs and Data Types {#sect:apis} ============================ Hub Discovery Mechanism {#sect:hub-discovery} ----------------------- In order to keep track of which hub is running, a hub discovery mechanism, capable of yielding information about how to determine the existence of and communicate with a running hub, is needed. This is a Profile-specific matter and specific prescriptions are described in Sections \[sect:lockfile\] (Standard Profile) and \[sect:web-httpd\] (Web Profile). Communicating with the Hub -------------------------- The details of how a client communicates with the hub are Profile-specific. Specific prescriptions are described in Sections \[sect:profile\] (Standard Profile) and \[sect:webprofile\] (Web Profile). SAMP Data Types {#sect:samp-data-types} --------------- For all hub/client communication, including the actual content of messages, SAMP uses three conceptual data types: 1. [string]{} — a scalar value consisting of a sequence of characters; each character is an ASCII character with hex code 09, 0a, 0d or 20–7f 2. [list]{} — an ordered array of data items 3. [map]{} — an unordered associative array of key-value pairs, in which each key is a [string]{} and each value is a data item These types can in principle be nested to any level, so that the elements of a list or the values of a map may themselves be strings, lists or maps. There is no reserved representation for a null value, and it is illegal to send a null value in a SAMP context even if the underlying transport protocol permits this. However a zero-length string or an empty list or map may, where appropriate, be used to indicate an empty value. Although SAMP imposes no maximum on the length of a string, particular transport protocols or implementation considerations may effectively do so; in general, hub and client implementations are not expected to deal with data items of unlimited size. General purpose MTypes SHOULD therefore be specified so that bulk data is not sent within the message or response. In general it is preferred to define a message parameter or result element as the URL or filename of a potentially large file rather than as the inline text of the file itself. SAMP defines no formal list of which URL protocols are permitted in such cases, but clients which need to dereference such URLs SHOULD be capable of dealing with at least the “http” and “file” schemes. “https”, “ftp” and other schemes are also permitted, but when sending such URLs, consideration should be given to whether receiving clients are likely to be able to dereference them. At the protocol level there is no provision for typing of scalars. Unlike many Remote Procedure Call (RPC) protocols SAMP does not distinguish syntactically between strings, integers, floating point values, booleans etc. This minimizes the restrictions on what underlying transport protocols may be used, and avoids a number of problems associated with using typed values from weakly-typed languages such as Python and Perl. The practical requirement to transmit these types is addressed however by the next section. Scalar Type Encoding Conventions {#sect:scalar-types} -------------------------------- Although the protocol itself defines [string]{} as the only scalar type, some MTypes will wish to define parameters or return values which have non-string semantics, so conventions for encoding these as [string]{}s are in practice required. Such conventions only need to be understood by the sender and recipient of a given message and so can be established on a per-MType basis, but to avoid unnecessary duplication of effort this section defines some commonly-used type encoding conventions. We define the following BNF productions: <digit> ::= "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9" <digits> ::= <digit> | <digits> <digit> <float-digits> ::= <digits> | <digits> "." | "." <digits> | <digits> "." <digits> <sign> ::= "+" | "-" With reference to the above we define the following type encoding conventions: - `<SAMP int> ::= [ <sign> ] <digits>`\ An integer value is encoded using its decimal representation with an OPTIONAL preceding sign and with no leading, trailing or embedded whitespace. There is no guarantee about the largest or smallest values which can be represented, since this will depend on the processing environment at decode time.\ - `<SAMP float> ::= [ <sign> ] <float-digits>`\ ` [ "e" | "E" [ <sign> ] <digits> ]`\ A floating point value is encoded as a mantissa with an OPTIONAL preceding sign followed by an OPTIONAL exponent part introduced with the character “[e]{}” or “[E]{}”. There is no guarantee about the largest or smallest values which can be represented or about the number of digits of precision which are significant, since these will depend on the processing environment at decode time.\ - `<SAMP boolean> ::= "0" | "1"`\ A boolean value is represented as an integer: zero represents false, and any other value represents true. 1 is the RECOMMENDED value to represent true. The numeric types are based on the syntax of the C programming language, since this syntax forms the basis for typed data syntax in many other languages. There may be extensions to this list in future versions of this standard. Particular MType definitions may use these conventions or devise their own as required. Where the conventions in this list are used, message documentation SHOULD make it clear using a form of words along the lines “this parameter contains a [*SAMP int*]{}”. Registering with the Hub {#sect:registration} ------------------------ A client registers with the hub to: 1. establish communication with the hub 2. advertise its presence to the hub and to other clients 3. obtain registration information The registration information is in the form of a [map]{} containing data items which the client may wish to use during the SAMP session. The hub MUST fill in values for the following keys in the returned [map]{}: [samp.hub-id]{} : — The client ID which is used by the hub when it sends messages itself (rather than forwarding them from other senders). For instance, this ID will be used when the hub sends the [samp.hub.event.shutdown]{} message. [samp.self-id]{} : — The client ID which identifies the registering client. These keys form part of an extensible vocabulary as explained in Section \[sect:vocab\]. In most cases a client will not require either of the above IDs for normal SAMP operation, but they are there for clients which do wish to know them. Particular Profiles may require additional entries in this map. Immediately following registration, the client will typically perform some or all of the following OPTIONAL operations: - supply the hub with metadata about itself, using the [declareMetadata()]{} call - tell the hub how it wishes the hub to communicate with it, if at all (the mechanism for this is profile-dependent, and it may be implicit in registration) - inform the hub which MTypes it wishes to subscribe to, using the [declareSubscriptions()]{} call Application Metadata {#sect:app-metadata} -------------------- A client may store metadata in the form of a [map]{} of key-value pairs in the hub for retrieval by other clients. Typical metadata might be the human-readable name of the application, a description and a URL for its icon, but other values are permitted. The following keys are defined for well-known metadata items: [samp.name]{} : — A one word title for the application. [samp.description.text]{} : — A short description of the application, in plain text. [samp.description.html]{} : — A description of the application, in HTML. [samp.icon.url]{} : — The URL of an icon in png, gif or jpeg format. [samp.documentation.url]{} : — The URL of a documentation web page. All of the above are OPTIONAL, but [samp.name]{} is strongly RECOMMENDED. These keys form the basis of an extensible vocabulary as explained in Section \[sect:vocab\]. MType Subscriptions {#sect:subscription} ------------------- As outlined above, an MType is a string which defines the semantics of a message. MTypes have a hierarchical form. Their syntax is given by the following BNF: <mchar> ::= [0-9A-Za-z] | "-" | "_" <atom> ::= <mchar> | <atom> <mchar> <mtype> ::= <atom> | <mtype> "." <atom> Examples might be “[samp.hub.event.shutdown]{}” or “[file.load]{}”. A client may [*subscribe*]{} to one or more MTypes to indicate which messages it is willing to receive. A client will only ever receive messages with MTypes to which it has subscribed. In order to do this it passes a subscriptions [map]{} to the hub. Each key of this map is an MType string to which the client wishes to subscribe, and the corresponding value is a map which may contain additional information about that subscription. Currently, no keys are defined for these per-MType maps, so typically they will be empty (have no entries). The use of a map here is to permit experimentation and perhaps future extension of the SAMP standard. As a special case, simple wildcarding is permitted in subscriptions. The keys of the subscription map may actually be of the form `<msub>`, where <msub> ::= "*" | <mtype> "." "*" Thus a subscription key “[file.event.\*]{}” means that a client wishes to receive any messages with MType which begin “[file.event.]{}”. This does not include “[file.event]{}”. A subscription key “” subscribes to all MTypes. Note that the wildcard “” character may only appear at the end of a subscription key, and that this indicates subscription to the entire subtree. More discussion of MTypes, including their semantics, is given in Section \[sect:mtypes\]. Message Encoding {#sect:msg-encode} ---------------- A message is an abstract container for the information we wish to send to another application. The message itself is that data which should arrive at the receiving application. It may be transmitted along with some external items (e.g. sender, recipient and message identifiers) required to ensure proper delivery or handling. A message is encoded for SAMP transmission as a [map]{} with the following REQUIRED keys: [samp.mtype]{} : — A [string]{} giving the MType which defines the meaning of the message. The MType also, via external documentation, defines the names, types and meanings of any parameters and return values. MTypes are discussed in more detail in Section \[sect:mtypes\]. [samp.params]{} : — A [map]{} containing the values for the message’s named parameters. These give the data required for the receiver to act on the message, for instance the URL of a given file. The names, types and semantics of these parameters are determined by the MType. Each key in this map is the name of a parameter, and the corresponding value is that parameter’s value. These keys form the basis of an extensible vocabulary as explained in Section \[sect:vocab\]. Response Encoding {#sect:response-encode} ----------------- A response is what may be returned from a recipient to a sender giving the result of processing a message (though in the case of the Notification delivery pattern, no such response is generated or returned). It may contain MType-specific return values, or error information, or both. A response is encoded for SAMP transmission as a [map]{} with the following keys: [samp.status]{} : (REQUIRED) — A [string]{} summarising the result of the processing. It may take one of the following defined values: [samp.ok]{}: : Processing successful. The [samp.result]{}, but not the [samp.error]{} entry SHOULD be present. [samp.warning]{}: : Processing partially successful. Both [samp.result]{} and [samp.error]{} entries SHOULD be present. [samp.error]{}: : Processing failed. The [samp.error]{}, but not the [samp.result]{} entry SHOULD be present. These values form the basis of an extensible vocabulary as explained in Section \[sect:vocab\]. [samp.result]{} : (REQUIRED in case of full or partial success) — A [map]{} containing the values for the message’s named return values. The names, types and semantics of these returns are determined by the MType. Each key in this map is the name of a return value, and the corresponding value is the actual value. Note that even for MTypes which define no return values, the value of this entry MUST still be a [map]{} (typically an empty one). [samp.error]{} : (REQUIRED in case of full or partial error) — A [map]{} containing error information. The following keys are defined for this map: [samp.errortxt]{} : (REQUIRED) — A short string describing what went wrong. This will typically be delivered to the user of the sender application. [samp.usertxt]{} : (OPTIONAL) — A free-form string containing any additional text an application wishes to return. This may be a more verbose error description meant to be appended to the [samp.errortxt]{} string, however it is undefined how this string should be handled when received. [samp.debugtxt]{} : (OPTIONAL) — A longer string which may contain more detail on what went wrong. This is typically intended for debugging purposes, and may for instance be a stack trace. [samp.code]{} : (OPTIONAL) — A string containing a numeric or textual code identifying the error, perhaps intended to be parsable by software. Values beginning “[samp.]{}” are reserved. These keys form the basis of an extensible vocabulary as explained in Section \[sect:vocab\]. These keys form the basis of an extensible vocabulary as explained in Section \[sect:vocab\]. In most cases, such responses will be generated by a Recipient client and forwarded by the Hub to the Sender. In some cases however the hub may pass to the sender an error response it has generated itself on behalf of the recipient. In particular, if the hub determines that no response will ever be received from the recipient (perhaps because the recipient has unregistered without replying) the hub MAY generate and forward a response with [samp.status=samp.error]{} and the [samp.code]{} key in the [samp.error]{} structure set to “[samp.noresponse]{}”. Clients SHOULD NOT generate such [samp.code=samp.noresponse]{} responses themselves. Sending and Receiving Messages {#sect:delivery} ------------------------------ As outlined in Section \[sect:delivery-outline\], three messaging patterns are supported, differing according to whether and how the response is returned to the sender. For a given MType there may be a messaging pattern that is most typically used, but there is nothing in the protocol that ties a particular MType to a particular messaging pattern; any MType may legally be sent using any delivery pattern. From the point of view of the sender, there are three ways in which a message may be sent, and from the point of view of the recipient there are two ways in which one may be received. These are described as follows. Notification: : In the notification pattern, communication is only in one direction: 1. The sender sends a message to the hub for delivery to one or more recipients. 2. The hub forwards the message to those requested recipients which are subscribed. 3. No reply from the recipients is expected or possible. Notifications can be sent to a given recipient or broadcast to all recipients. The notification pattern for a single recipient is illustrated in Figure \[fig:notification\]. ![Notification pattern[]{data-label="fig:notification"}](samp-notification) Asynchronous Call/Response: : In the asynchronous call pattern, [*message tags*]{} and [*message identifiers*]{} are used to tie together messages and their replies: 1. The sender sends a message to the hub for delivery to one or more recipients, supplying along with the message a tag string of its own choice, [*msg-tag*]{}. In return it receives a unique identifier string, [*msg-id*]{}. 2. The hub forwards the message to the appropriate recipients, supplying along with the message an identifier string, [*msg-id*]{}. 3. Each recipient processes the message, and sends its response back to the hub along with the ID string [*msg-id*]{}. 4. Using a callback, the hub passes the response back to the original sender along with the ID string [*msg-tag*]{}. The sender is free to use any value for the [*msg-tag*]{}. There is no requirement on the form of the hub-generated [*msg-id*]{} (it is not intended to be parsed by the recipient), but it MUST be sufficient for the hub to pair messages with their responses reliably, and to pass the correct [*msg-tag*]{} back with the response to the sender[^1]. In most cases the sender will not require the [*msg-id*]{}, since the [*msg-tag*]{} is sufficient to match calls with responses. For this reason, the sender need not retain the [*msg-id*]{} and indeed need not wait for it, avoiding a hub round trip at send time. The only case in which the sender may require the [*msg-id*]{} is if it needs to communicate later with the recipient about the message that was sent, for instance as part of a progress report. Asynchronous calls may be sent to a given recipient or broadcast to all recipients. In the latter case, the sender SHOULD be prepared to deal with multiple responses to the same call. The asynchronous pattern is illustrated in Figure \[fig:async\]. ![Asynchronous Call/Response pattern[]{data-label="fig:async"}](samp-asynchronous) Synchronous Call/Response : A synchronous utility method is provided by the hub, mainly for the convenience of environments where dealing with asynchronicity might be a problem. The hub will provide synchronous behaviour to the sender, interacting with the receiver in exactly the same way as for the asynchronous case above. 1. The sender sends a message to the hub for delivery to a given recipient, optionally specifying as well a maximum time it is prepared to wait. The sender’s call blocks until a response is available. 2. The hub forwards the message to the recipient, supplying along with the message an ID string, [*msg-id*]{}. 3. The recipient processes the message, and sends its response back to the hub along with the ID string [*msg-id*]{}. 4. The hub passes back the response as the return value from the original blocking call made by the sender. If no response is received within the sender’s specified timeout the blocking call will terminate with an error. The hub is not guaranteed to wait indefinitely; it MAY in effect impose its own timeout. There is no broadcast counterpart for the synchronous call. This pattern is illustrated in Figure \[fig:sync\]. ![Synchronous Call/Response pattern[]{data-label="fig:sync"}](samp-synchronous) Note that the two different cases from the receiver’s point of view, [*Notification*]{} and [*Call/Response*]{}, differ only in whether a response is returned to the hub. In other respects the receiver SHOULD process the message in exactly the same way for both patterns. Although it is REQUIRED by this standard that client applications provide a Response for every Call that they receive, there is no way that the hub can enforce this. Senders using the Synchronous or Asynchronous Call/Response patterns therefore should be aware that badly-behaved recipients might fail to respond, leading to calls going unanswered indefinitely. The timeout parameter in the Synchronous Call/Response pattern provides some protection from this eventuality; users of the Asynchronous Call/Response pattern may or may not wish to take their own steps. Operations a Hub Must Support {#sect:hubOps} ----------------------------- This section describes the operations that a hub MUST support and the associated data that MUST be sent and received. The precise details of how these operations map onto method names and signatures is Profile-dependent. The mapping for the Standard Profile is described in Section \[sect:mappingXMLRPC\], and for the Web Profile in Section \[sect:webXMLRPC\]. - `map reg-info = register()`\ Method called by a client wishing to register with the hub. The form of [reg-info]{} is given in Section \[sect:registration\]. Note that the form of this call may vary according to the requirements of the particular Profile in use. For instance authentication tokens may be passed in one or both directions to complete registration.\ - `unregister()`\ Method called by a client wishing to unregister from the hub.\ - `declareMetadata(map metadata)`\ Method called by a client to declare its metadata. May be called zero or more times to update hub state; the most recent call is the one which defines the client’s currently declared metadata. The form of the [metadata]{} map is given in Section \[sect:app-metadata\].\ - `map metadata = getMetadata(string client-id)`\ Returns the metadata information for the client whose public ID is [client-id]{}. The form of the [metadata]{} map is given in Section \[sect:app-metadata\].\ - `declareSubscriptions(map subscriptions)`\ Method called by a callable client to declare the MTypes it wishes to subscribe to. May be called zero or more times to update hub state; the most recent call is the one which defines the client’s currently subscribed MTypes. The form of the [subscriptions]{} map is given in Section \[sect:subscription\].\ - `map subscriptions = getSubscriptions(string client-id)`\ Returns the subscribed MTypes for the client whose public ID is [client-id]{}. The form of the [subscriptions]{} map is given in Section \[sect:subscription\].\ - `list client-ids = getRegisteredClients()`\ Returns the list of public ids of all other registered clients. The caller’s ID ([samp.self-id]{} from Section \[sect:registration\]) is not included, but the hub’s ID ([samp.hub-id]{} from Section \[sect:registration\]) is.\ - `map client-subs = getSubscribedClients(string mtype)`\ Returns a map with an entry for all other registered clients which are subscribed to the MType [mtype]{}. The key for each entry is a subscribed client ID, and the value is a (possibly empty) [map]{} providing further information on its subscription to [mtype]{} as described in Section \[sect:subscription\]. An entry for the caller is not included, even if it is subscribed. [mtype]{} MUST NOT include wildcards.\ - `notify(string recipient-id, map message)`\ Sends a message using the Notification pattern to a given recipient. The form of the [message]{} map is given in Section \[sect:msg-encode\]. An error results if the recipient is not subscribed to the message’s MType.\ - `list recipient-ids = notifyAll(map message)`\ Sends a message using the Notification pattern to all other clients which are subscribed to the message’s MType. The form of the [message]{} map is given in Section \[sect:msg-encode\]. The return value is a [list]{} of the client IDs of the clients to which an attempt to send the message is made.\ - `string msg-id = call(string recipient-id, string msg-tag,`\ ` map message)`\ Sends a message using the Asynchronous Call/Response pattern to a given recipient. The form of the [message]{} map is given in Section \[sect:msg-encode\]. An error results if the recipient is not subscribed to the message’s MType, or if the invoking client is not Callable.\ - `map calls = callAll(string msg-tag, map message)`\ Sends a message using the Asynchronous Call/Response pattern to all other clients which are subscribed to the message’s MType. The form of the [message]{} map is given in Section \[sect:msg-encode\]. The returned value is a [map]{} in which the keys are the client IDs of clients to which an attempt to send the message is made, and the values are the associated [msg-id]{} strings. An error results if the invoking client is not Callable.\ - `map response = callAndWait(string recipient-id,`\ ` map message, string timeout)`\ Sends a message using the Synchronous Call/Response pattern to a given recipient. The forms of the [message]{} and [response]{} maps are given in Sections \[sect:msg-encode\] and \[sect:response-encode\]. The [timeout]{} parameter is interpreted as a [*SAMP int*]{} (Section \[sect:scalar-types\]) giving the maximum number of seconds the client wishes to wait. If the response takes longer than that to arrive this method SHOULD terminate anyway with an error (it MUST not return a [response]{} indicating error). Any response arriving from the recipient after that will be discarded. If [timeout]{}$<=0$ then no artificial timeout is imposed. An error results if the recipient is not subscribed to the message’s MType.\ - `reply(string msg-id, map response)`\ Method called by a client to send its response to a given message. The form of the [response]{} map is given in Section \[sect:response-encode\]. Of these operations, only [callAndWait()]{} involves blocking communication with another client. The others SHOULD be implemented in such a way that clients can expect them to complete, and where appropriate return a value, on a timescale short compared to user response time. Operations a Callable Client Must Support {#sect:clientOps} ----------------------------------------- This section lists the operations which a client MUST support in order to be classified as callable. The hub uses these operations when it wishes to pass information to a callable client. Note that callability is OPTIONAL for clients; special (Profile-dependent) steps may be required for a client to inform the hub how it can be contacted, and thus become callable. Clients which are not callable can send messages using the Notify or Synchronous Call/Response patterns, but are unable to receive messages or to use Asynchronous Call/Response, since these operations rely on client callbacks from the hub. The precise details of how these operations map onto method names and signatures is Profile-dependent. The mapping for the Standard Profile is given in Section \[sect:mappingXMLRPC\] and for the Web Profile in Section \[sect:web-callable\]. - `receiveNotification(string sender-id, map message)`\ Method called by the hub when dispatching a notification to its recipient. The form of the [message]{} map is given in Section \[sect:msg-encode\].\ - `receiveCall(string sender-id, string msg-id, map message)`\ Method called by the hub when dispatching a call to its recipient. The client MUST at some later time make a matching call to [reply()]{} on the hub. The form of the [message]{} map is given in Section \[sect:msg-encode\].\ - `receiveResponse(string responder-id, string msg-tag,`\ ` map response)`\ Method used by the hub to dispatch to the sender the response of an earlier asynchronous call. The form of the [response]{} map is given in Section \[sect:response-encode\]. Error Processing {#sect:faults} ---------------- Errors encountered by clients when processing Call/Response-pattern messages themselves (in response to a syntactically legal [receiveCall()]{} operation) SHOULD be signalled by returning appropriate content in the response map sent back in the matching [reply()]{} call, as described in Section \[sect:response-encode\]. In the case of failed calls of the operations defined in Sections \[sect:hubOps\] and \[sect:clientOps\], for instance syntactically invalid parameters or communications failures, hubs and clients SHOULD where possible use the usual error reporting mechanisms of the transport protocol in use. Where it is problematic or impossible to use the transport protocol’s error reporting mechanisms, in the case of a Call/Response pattern message, the hub MAY signal errors by generating and passing back to the sender a suitable response map as described in Section \[sect:response-encode\]. Standard Profile {#sect:profile} ================ Section \[sect:apis\] provides an abstract definition of the operations and data structures used for SAMP messaging. As explained in Section \[sect:profiles\], in order to implement this architecture some concrete choices about how to instantiate these concepts are required. This section gives the details of a SAMP Profile based on the XML-RPC specification [@xmlrpc]. Hub discovery is via a lockfile in the user’s home directory. XML-RPC is a simple general purpose Remote Procedure Call protocol based on sending XML documents using HTTP POST (it resembles a very lightweight version of SOAP). Since the mappings from SAMP concepts such as API calls and data types to their XML-RPC equivalents is very straightforward, it is easy for application authors to write compliant code without use of any SAMP-specific library code. An XML-RPC library, while not essential, will make coding much easier; such libraries are available for many languages. Data Type Mappings {#sect:profile-typemap} ------------------ The SAMP argument and return value data types described in Section \[sect:samp-data-types\] map straightforwardly onto XML-RPC data types as follows: SAMP type XML-RPC element ------------ --- ----------------- [string]{} — [list]{} — [map]{} — The children of and elements themselves contain children of type , or . Note that other XML-RPC scalar types (, etc) are not used; even where the semantic sense of a value matches one of those types it MUST be encoded as an XML-RPC . API Mappings {#sect:mappingXMLRPC} ------------ The operation names in the SAMP hub and client abstract APIs (Sections \[sect:hubOps\] and \[sect:clientOps\]) very nearly have a one to one mapping with those in the Standard Profile XML-RPC APIs. The Standard Profile API MUST be implemented as described in Sections \[sect:hubOps\] and \[sect:clientOps\] with the following REQUIRED adjustments: 1. The XML-RPC method names (i.e. the contents of the XML-RPC elements) are formed by prefixing the hub and client abstract API operation names with “[samp.hub.]{}” or “[samp.client.]{}” respectively. 2. The [register()]{} operation takes the following form: - `map reg-info = register(string samp-secret)` The argument is the [samp-secret]{} value read from the lockfile (see Section \[sect:lockfile\]). The returned [reg-info]{} map contains an additional entry with key [samp.private-key]{} whose value is a string generated by the hub. 3. [*All*]{} other hub and client methods take the [private-key]{} as their first argument. 4. A new method, [setXmlrpcCallback()]{} is added to the hub API. - `setXmlrpcCallback(string private-key, string url)` This informs the hub of the XML-RPC endpoint on which the client is listening for calls from the hub. The client is not considered Callable unless and until it has invoked this method. 5. Another new method, [ping()]{} is added to the hub API. This may be called by registered or unregistered applications (as a special case the [private-key]{} argument may be omitted), and can be used to determine whether the hub is responding to requests. Any non-error return indicates that the hub is running. The [private-key]{} string referred to above serves two purposes. First it identifies the client in hub/client communications. Some such identifier is required, since XML-RPC calls have no other way of determining the sender’s identity. Second, it prevents application spoofing, since the private key is never revealed to other applications, so that one application cannot pose as another in making calls to the hub. The usual XML-RPC fault mechanism is used to respond to invalid calls as described in Section \[sect:faults\]. The XML-RPC ’s element SHOULD contain a user-directed message as appropriate and the value has no particular significance. Lockfile and Hub Discovery {#sect:lockfile} -------------------------- Hub discovery is performed by examining a lockfile to determine hub connection parameters, specifically the XML-RPC endpoint at which the hub can be found, and a “secret” token which affords some measure of security, given suitable restrictions on the lockfile’s readability (see Section \[sect:std-security\]). To discover the hub, a client must therefore: 1. Determine where to find the lockfile (\[sect:lockfileLoc\]) 2. Read the lockfile to obtain the hub connection parameters (\[sect:lockfileText\]) ### Lockfile Location {#sect:lockfileLoc} The default location of the lockfile is the file named “[.samp]{}” in the user’s home directory. However the content of the environment variable named SAMP\_HUB can be used to override this default. The value of the SAMP\_HUB environment variable is of the form `<samphub-value>`, as defined by the following BNF production: <samphub-value> ::= <hub-location> <hub-location> ::= <stdlock-prefix> <stdlock-url> <lockurl-prefix> ::= "std-lockurl:" <stdlock-url> ::= (any URL) The `<stdlock-url>` will typically, but not necessarily, be a file-type URL (as described in RFC 1738, section 3.10 [@rfc1738]). So for instance to indicate that the lockfile to be used will be the file “[/tmp/samp1]{}”, you would set SAMP_HUB=std-lockurl:file:///tmp/samp1 Although no other form of the `<hub-location>` value is defined here, the intention is that the SAMP\_HUB environment variable MAY be used with prefixes other than “[std-lockurl:]{}” to indicate use of other, non-Standard, profiles. Issues may in future arise related to the need to indicate multiple profiles or profile variants at once; the impact of this requirement on the syntax and semantics of the SAMP\_HUB variable is for now deferred. To locate the lockfile therefore, a Standard Profile-compliant client MUST determine whether an environment variable named SAMP\_HUB exists; if so, the client MUST examine the variable’s value; if the value begins with the prefix “[std-lockurl:]{}” the client MUST interpret the remainder of the value as a URL whose content is the text of the lockfile to be used for hub discovery. If no SAMP\_HUB environment variable exists, the client MUST use the file “.samp” in the user’s home directory as the lockfile to be used for hub discovery. If the variable exists, but its value begins with a different prefix, the client MAY interpret that in some non-Standard way for hub discovery. Rules for a Standard Profile-compliant hub to use when writing lockfiles are similar, but if a hub is unable or unwilling to write a lockfile such that it can be read using the above procedure, it MUST signal an error at the startup and then abort. For practical reasons, a hub will probably only be able to write a lockfile indicated by a [file]{}-type URL, not for instance an arbitrary [http]{}-type one. Lockfiles SHOULD be created with appropriate access restrictions as discussed in Section \[sect:std-security\]. The existence or readability of the lockfile MAY be taken (e.g. by a hub deciding whether to start or not) to indicate that a hub is running. However it is RECOMMENDED to attempt to contact the hub at the given XML-RPC URL (e.g. by calling [ping()]{}) to determine whether it is actually alive. The “home directory” referred to above is a somewhat system-dependent concept: we define it as the value of the [HOME]{} environment variable on Unix-like systems and as the value of the [USERPROFILE]{} environment variable on Microsoft Windows[^2]. “Environment variable” is itself potentially a system-dependent concept, but it is clear how to interpret it for all platforms on which we currently expect SAMP to be used, so no further explanation is provided here. In version 1.11 of the standard, the lockfile was always in the “[.samp]{}” file in the user’s home directory. The option of setting the SAMP\_HUB environment variable to override this has been introduced to allow more flexibility; for instance one user can run multiple unconnected hubs, or multiple users can share the same hub. If no SAMP\_HUB environment variable is defined, client and hub behaviour is exactly as in version 1.11. ### Security Considerations {#sect:std-security} The hub SHOULD normally create the lockfile with file permissions which allow only its owner to read it. This provides a measure of security in that only processes with the same privileges as the hub process (hence presumably running under the same user ID) will be able to register with the hub, since only they will be able to provide the secret token, obtained from the lockfile, which is required for registration. Thus under normal circumstances all Standard Profile clients can be presumed to be running with the same level of trust, so that no security issues of the type discussed in Section \[sect:security\] arise. If the lockfile is made available in some way other than an owner-only readable file, for instance via an unprotected [http]{}-type URL in order to facilitate use of the same hub by multiple users on different hosts, there is a potential security risk. In that case, protection through an authentication and/or authorization mechanism might be adopted by the hub implementations, for instance exploiting the TLS cryptographic protocol [@rfc2246]. ### Lockfile Content {#sect:lockfileText} The format of the lockfile is given by the following BNF productions: <file> ::= <lines> <lines> ::= <line> | <lines> <line> <line> ::= <line-content> <EOL> | <EOL> <line-content> ::= <comment> | <assignment> <comment> ::= "#" <any-string> <assignment> ::= <name> "=" <any-string> <name> ::= <token-string> <token-string> ::= <token-char> | <token-string> <token-char> <any-string> ::= <any-char> | <any-string> <any-char> <EOL> ::= "\r" | "\n" | "\r" "\n" <token-char> ::= [a-zA-Z0-9] | "-" | "_" | "." <any-char> ::= [\x20-\x7f] The only parts which are significant to SAMP clients/hubs are (a) existence of the file and (b) lines. A legal lockfile MUST provide (in any order) unique assignments for the following tokens: [samp.secret]{} : — An opaque text string which must be passed to the hub to permit registration. [samp.hub.xmlrpc.url]{} : — The XML-RPC endpoint for communication with the hub. [samp.profile.version]{} : — The version of the SAMP Standard Profile implemented by the hub (“” for the version described by this document). These keys form the basis of an extensible vocabulary as explained in Section \[sect:vocab\]. Other blank, comment or assignment lines may be included as desired. An example lockfile might therefore look like this: > \# SAMP lockfile written 2011-12-22T05:30:01\ > \# Required keys:\ > samp.secret=734144fdaab8400a1ec2\ > samp.hub.xmlrpc.url=http://andromeda.star.bris.ac.uk:8001/xmlrpc\ > samp.profile.version=\ > \# Info stored by hub for some private reason:\ > com.yoyodyne.hubid=c80995f1 ### Hub Discovery Sequences The hub discovery sequences are therefore as follows: - Client startup: - Determine hub existence as above - If no hub, client MAY start its own hub - Acquire [samp.secret]{} value from lockfile - If pre-existing or own hub is running, call [register()]{} and zero or more of [setXmlrpcCallback()]{}, [declareMetadata()]{}, [declareSubscriptions()]{} - Hub startup: - Determine hub existence as above - If hub is running, exit - Otherwise, start up XML-RPC server - Write lockfile containing mandatory assignments including XML-RPC endpoint, using appropriate access restrictions - Hub shutdown: - Remove lockfile (it is RECOMMENDED to first check that this is the lockfile written by self) - Notify candidate clients that shutdown will occur - Shut down services A hub implementation SHOULD make its best effort to perform the shutdown sequence above even if it terminates as a result of some error condition. Note that manipulation of a file is not atomic, so that race conditions are possible. For instance a client or hub examining the lockfile may read it after it has been created but before it has been populated with the mandatory assignments, or two hubs may look for a lockfile simultaneously, not find one, and both decide that they should therefore start up, one presumably overwriting the other’s lockfile. Hub and client implementations should be aware of such possibilities, but may not be able to guarantee to avoid them or their consequences. In general this is the sort of risk that SAMP and its Standard Profile are prepared to take — an eventuality which will occur sufficiently infrequently that it is not worth significant additional complexity to avoid. In the worst case a SAMP session may fail in some way, and will have to be restarted manually. Examples -------- Here is an example in pseudo-code of how an application might locate and register with a hub, and send a message requiring no response to other registered clients. # Locate and read the lockfile. string hubvar-value = readEnvironmentVariable("SAMP_HUB"); string lock-location = getLockfileLocation(hubvar-value); map lock-info = readLockfile(lock-location); # Extract information from lockfile to locate and register with hub. string hub-url = lock-info.getValue("samp.hub.xmlprc.url"); string samp-secret = lock-info.getValue("samp.secret"); # Establish XML-RPC connection with hub # (uses some generic XML-RPC library) xmlrpcServer hub = xmlrpcConnect(hub-url); # Register with hub. map reg-info = hub.xmlrpcCall("samp.hub.register", samp-secret); string private-key = reg-info.getValue("samp.private-key"); # Store metadata in hub for use by other applications. map metadata = ("samp.name" -> "dummy", "samp.description.text" -> "Test Application", "dummy.version" -> "0.1-3"); hub.xmlrpcCall("samp.hub.declareMetadata", private-key, metadata); # Send a message requesting file load to all other # registered clients, not wanting any response. map loadParams = ("filename" -> "/tmp/foo.bar"); map loadMsg = ("samp.mtype" -> "file.load", "samp.params" -> loadParams); hub.xmlrpcCall("samp.hub.notifyAll", private-key, loadMsg); # Unregister hub.xmlrpcCall("samp.hub.unregister", private-key); The first few XML-RPC documents sent over the wire for this exchange would look something like the following. The registration call from the client to the hub: POST /xmlrpc HTTP/1.0 User-Agent: Java/1.5.0_10 Content-Type: text/xml Content-Length: 189 <?xml version="1.0"?> <methodCall> <methodName>samp.hub.register</methodName> <params> <param><value><string>734144fdaab8400a1ec2</string></value></param> </params> </methodCall> which leads to the response: HTTP/1.1 200 OK Connection: close Content-Type: text/xml Content-Length: 464 <?xml version="1.0"?> <methodResponse> <params><param><value><struct> <member> <name>samp.private-key</name> <value><string>client-key:1a52fdf</string></value> </member> <member> <name>samp.hub-id</name> <value><string>client-id:0</string></value> </member> <member> <name>samp.self-id</name> <value><string>client-id:4</string></value> </member> </struct></value></param></params> </methodResponse> The client can then declare its metadata: the response to this call has no useful content so can be ignored or discarded. POST /xmlrpc HTTP/1.0 User-Agent: Java/1.5.0_10 Content-Type: text/xml Content-Length: 600 <?xml version="1.0"?> <methodCall> <methodName>samp.hub.declareMetadata</methodName> <params> <param><value><string>app-id:1a52fdf-2</string></value></param> <param><value><struct> <member> <name>samp.name</name> <value><string>dummy</string></value> </member> <member> <name>samp.description.text</name> <value><string>Test application</string></value> </member> <member> <name>dummy.version</name> <value><string>0.1-3</string></value> </member> </struct></value></param> </params> </methodCall> The message itself is sent from the client to the hub as follows: POST /xmlrpc HTTP/1.0 User-Agent: Java/1.5.0_10 Content-Type: text/xml Content-Length: 523 <?xml version="1.0"?> <methodCall> <methodName>samp.hub.notifyAll</methodName> <params> <param><value><string>app-id:1a52fdf-2</string></value></param> <param><value><struct> <member> <name>samp.mtype</name> <value>file.load</value> </member> <member> <name>samp.params</name> <value><struct> <name>filename</name> <value>/tmp/foo.bar</value> </struct></value> </member> </struct></value></param> </params> </methodCall> Again, there is no interesting response. Web Profile {#sect:webprofile} =========== This section defines the SAMP Web Profile which allows web applications to communicate with a SAMP hub. A [*web application*]{} in this context is code which is downloaded by a web browser from a remote server, usually as part of a web page, and which then runs from within that browser. The most common platforms (browser-based runtime environments) for such applications are currently JavaScript (a.k.a. JScript, ECMAScript), Java applets, Adobe Flash, and Microsoft Silverlight. For security reasons, these runtime environments run the web applications that they host inside a secure “sandbox”, which imposes restrictions on access to resources, making it impossible to use the Standard Profile defined in Section \[sect:profile\]. Java applets provide a client-controlled cross-browser mechanism, based on code signing, for circumventing these restrictions, but the others do not. Section \[sect:web-overview\] gives an illustrative overview of the way the Web Profile achieves its communication requirements, with comparison to the Standard Profile. Section \[sect:web-hub\] describes in detail how the Web Profile hub is implemented in order to provide the functionality defined by the SAMP abstract hub and client APIs (Sections \[sect:hubOps\] and \[sect:clientOps\]). Section \[sect:web-client\] outlines the steps that a Web Profile client must take to locate and communicate with the hub. The important topic of the security implications of this scheme, and measures which hub implementations can take in view of these, is covered separately in Section \[sect:web-security\]. Overview and Comparison with Standard Profile {#sect:web-overview} --------------------------------------------- The Web Profile is based on the Standard Profile (Section \[sect:profile\]), but with some modifications which allow clients to overcome the restrictions imposed by the browser sandbox. Browser restrictions present four main problems for a web-based SAMP client: hub discovery, outward hub communication, inward hub communication and use of third-party URLs. These are solved in the Web Profile by use of a well-known port, use of standard and de facto cross-origin access techniques, reversed HTTP communication, and URL proxying. These solutions are described, with comparison to the approaches used by the Standard Profile, in the following subsections. ### Hub Discovery A Standard Profile client locates the hub by reading a “lockfile” at a well-known location in the filesystem, which provides the HTTP endpoint at which the hub XML-RPC server is listening and a token which the client must present in order to register. Web applications have no access to the local filesystem and so are unable to read such a lockfile. In the Web profile, the hub HTTP server listens instead on a well-known port on the local host. The hub will apply some security measures at registration time (Section \[sect:web-sec-reg\]), but they are not based on presentation of a secret token. Note that since this well-known port number is fixed, it is not possible for more than one Web Profile hub to run on the same host. The Web Profile Hub and corresponding web browser MUST run on the same host, and SHOULD always be run by the same user. For a web client to be able to access this well-known port at all, the cross-origin techniques discussed in the next section are required. ### Outward Communications In the Standard Profile, all hub communication is done using the HTTP-based XML-RPC protocol [@xmlrpc], usually to a port on the local host. This is problematic for web-based clients, since so-called “cross-origin” or “cross-domain” policies enforced by browsers restrict HTTP access under normal circumstances so that web applications may [*only*]{} make HTTP requests to URLs at their own [*Origin*]{} [@origin], that is to URLs on the server from which the web application itself was downloaded. This deliberately excludes access to a server on the local host, which is where the SAMP hub is likely to reside. Since cross-origin access is a common requirement for web-based clients, and it is not always in conflict with the security concerns of servers, a number of platform-dependent but widely-used mechanisms have been implemented in browser technology which allow a sandboxed client to talk to an HTTP server which has explicitly opted in for such cross-origin communications. A Web Profile hub will implement one or more of these cross-origin workarounds (Section \[sect:web-httpd\]) and so permit Web Profile clients running in the relevant browser runtime environment(s) to make HTTP requests to itself, thereby allowing client-to-hub XML-RPC calls. ### Inward Communications If it wishes to receive as well as send messages, and also to make asynchronous calls, a SAMP client must declare itself [*Callable*]{}, by providing the Hub with a profile-dependent means to invoke the client API defined in Section \[sect:clientOps\]. In the Standard Profile a client declares itself Callable by providing to the Hub an HTTP endpoint to which the Hub may make XML-RPC requests. Thus, the client must itself run a publicly accessible HTTP server in order to be callable. Running an HTTP server is typically not within the capabilities of a web application. In the Web Profile, hub-to-client communication is effected by reversing the direction of the XML-RPC calls, and hence of the HTTP requests. Instead of the client running a server which listens for incoming messages from the Hub, the Hub maintains a queue of messages destined for the client, and the client polls the Hub to find out if any are available. The client may either make periodic short-timeout requests to the hub, or make a long-timeout (“long poll”) request which will return early if and when one or more messages are available. This effects inward communications using only the same outward HTTP capability discussed in the previous section. ### Third-Party URLs Although it is not fundamental to the SAMP protocol itself, many SAMP MTypes are defined in such a way that a receiving client must retrieve data from a URL external to the SAMP client-hub system in order to act on them. For instance the [table.load.votable]{} MType has an argument named “[url]{}”, whose value is the location of the VOTable document to be loaded. Such URLs may point to the local filesystem, to a server run by the sending client, or to some other web server internal or external to the host on which the SAMP communications are taking place. Similar considerations apply to some of the client metadata items (Section \[sect:app-metadata\]), for instance [samp.icon.url]{}. In any of these cases, it is likely that a browser-based client will be blocked by the browser’s cross-origin policy from accessing the content of the resource in question. The Web Profile therefore mandates that the Hub must provide to registered clients a mechanism for translating arbitrary URLs into cross-origin-accessible URLs with the same content as the specified resource. Since a hub must already be providing a cross-origin capable HTTP service accessible from the web client, it can use the same mechanism to operate a service which proxies external resources in a cross-origin capable way. Hub Behaviour {#sect:web-hub} ------------- This section specifies in detail the services that a SAMP hub must provide in order to implement the SAMP Web Profile. The Web Profile is based on client-to-hub XML-RPC calls, with the hub residing at a well-known port, and some special measures for allowing cross-origin requests. In most ways it resembles the Standard Profile (Section \[sect:profile\]), but there are some differences. ### Data Type Mappings {#data-type-mappings} SAMP argument and return value data types are encoded into XML-RPC exactly as for the Standard Profile (Section \[sect:profile-typemap\]). ### API Mappings {#sect:webXMLRPC} The operation names in the SAMP hub API very nearly have a one to one mapping with those in the Web Profile XML-RPC API. The Web Profile Hub API MUST be implemented as described in Section \[sect:hubOps\], with a number of REQUIRED adjustments. These are summarised as follows, and described in more detail later. 1. The XML-RPC method names (i.e. the contents of the XML-RPC elements) are formed by prefixing the hub abstract API operation names with “[samp.webhub.]{}”. For brevity, this prefix is not written in the rest of this document, but it is to be understood on all hub API XML-RPC calls. 2. The [register]{} operation takes the following form (Section \[sect:web-registration\]): - [map reg-info = register(map identity-info)]{} The [identity-info]{} is a map containing at least a declared application name supplied by the registering application to indicate its identity. 3. The [reg-info]{} map returned from the [register]{} method MUST contain two entries additional to those mandated by the hub API (Section \[sect:web-registration\]): [samp.private-key]{}: : used as the first argument of all hub API XML-RPC calls [samp.url-translator]{}: : used for translation of foreign URLs for cross-origin accessibility 4. [*All*]{} hub methods other than [register]{} take the [private-key]{} as their first argument, except where otherwise noted ([ping]{}). For brevity, this argument is not written in the rest of this document, but it is to be understood on all hub API calls. 5. Two new methods are added to the hub API to support reversed callbacks (Section \[sect:web-callable\]): - [allowReverseCallbacks(string allow)]{} - [map pullCallbacks(string timeout)]{} 6. Another new method is added to the hub API: - [ping()]{} This may be called by registered or unregistered applications (as a special case the [private-key]{} argument may be omitted), and can be used to determine whether the hub is responding to requests. Any non-error return indicates that the hub is running. ### Hub HTTP Server {#sect:web-httpd} Communications are XML-RPC calls [@xmlrpc] from the client to the Hub. XML-RPC works using POSTs to an HTTP server. The Web Profile hub HTTP server resides on the well-known port 21012, so that clients know where to find it on the local host. The XML-RPC endpoint for Web Profile requests is at the root of that server, so that web clients can access it by POSTing to the URL “[http://localhost:21012/]{}”. In general, web applications operate inside a browser-enforced sandbox that prevents them from accessing cross-origin resources, including HTTP-based ones served from the local host. However there are a number of ways in which an HTTP server can elect to permit access from browser-based clients. In order to be useful a Web Profile hub must implement at least one of these “cross-origin workarounds”. The following cross-origin workarounds are known to exist, and can be considered for use by Web Profile hub HTTP servers: Cross-Origin Resource Sharing: : CORS [@cors] is a W3C standard which works by manipulation of the HTTP Origin header and related headers by the browser runtime environment and the HTTP server, allowing the HTTP server to grant cross-domain access from clients with some or all Origins. CORS forms part of the XmlHttpRequest Level 2 standard [@xhr2], which is implemented by, at least, Chrome v2.0+, Firefox v3.5+ and Safari v4.0+. Microsoft’s IE8+ implements CORS via its own non-standard XDomainRequest object. This standard belongs to the loose HTML5 family of technologies, and it is likely that support will become wider in the future. A Web Profile hub HTTP server can grant unrestricted access to CORS-aware web applications by following the instructions in the CORS standard to enable both [*simple*]{} and [*preflight*]{} requests from clients with any Origin. Flash cross-domain policy: : Adobe’s Flash browser plugin makes use of a resource named “[crossdomain.xml]{}”, which, if present on an external HTTP server, is taken to indicate willingness to serve cross-domain requests [@flash-crossdomain]. This has emerged as something of a de facto standard, and the crossdomain file is honoured by Silverlight and unsigned Java Applets/WebStart applications[^3] as well as for Flash applications. A Web Profile hub HTTP server can grant unrestricted access to Flash-like web applications by serving a resource named “[/crossdomain.xml]{}” with a Content-Type header of “[text/x-cross-domain-policy]{}” and content like: <?xml version="1.0"?> <!DOCTYPE cross-domain-policy SYSTEM "http://www.adobe.com/xml/dtds/cross-domain-policy.dtd"> <cross-domain-policy> <site-control permitted-cross-domain-policies="all"/> <allow-access-from domain="*"/> <allow-http-request-headers-from domain="*" headers="*"/> </cross-domain-policy> Silverlight cross-domain policy: : Microsoft’s Silverlight environment will take note of Flash-style [crossdomain.xml]{} files, so the above measure ought to permit Silverlight clients to access a compliant HTTP server. However, Silverlight has its own cross-domain policy mechanism [@silverlight-crossdomain], which may be implemented in addition. A Web Profile hub HTTP server can grant unrestricted access to Silverlight web applications by serving a resource named “[/clientaccesspolicy.xml]{}” with a Content-Type header of “[text/xml]{}” and content like: <?xml version="1.0"?> <access-policy> <cross-domain-access> <policy> <allow-from> <domain uri="http://*"/> </allow-from> <grant-to> <resource path="/" include-subpaths="true"/> </grant-to> </policy> </cross-domain-access> </access-policy> If the hub implements these cross-origin workarounds it is believed that cross-origin access, hence Web Profile SAMP access, can be provided from nearly all browsers. Most modern browsers support CORS for JavaScript, nearly all others support Flash, and it is possible for JavaScript applications to make use of Flash libraries for their SAMP communications[^4]. Maximum interoperability therefore can be achieved by implementing all of these, or at least CORS and Flash, in the Web Profile HTTP server. There are however security implications of which ones to implement, discussed in Section \[sect:web-sec-confirm\]. In the usual browser-hub configuration, web applications will always seek the Web Profile HTTP server on the local host. Since no legitimate use of the Web Profile HTTP server is expected from non-local hosts, it is therefore strongly RECOMMENDED for security reasons that the Web Profile HTTP server refuses HTTP requests from external hosts with a 403 Forbidden status. This recommendation and possible exceptions to it are discussed further in Section \[sect:web-sec-host\]. ### Registration {#sect:web-registration} In order to request registration with the Web Profile, a client needs to invoke the following XML-RPC method: map register(map identity-info) The [identity-info]{} map provides information identifying the registering application which can inform the hub’s decision about whether to allow registration. It has the following REQUIRED entry: [samp.name]{} : — A [string]{} giving the name of the application wishing to register, in a form that can be presented to the user. This SHOULD be the same as the value of the [samp.name]{} key in the application metadata as described in Section \[sect:app-metadata\]. Particular implementations or future versions of this standard may specify additional required or optional entries to this map. The hub will accept or reject the registration based on the contents of the [identity-info]{} map, available information from the HTTP connection carrying the XML-RPC call, user confirmation, and the hub’s own security policy, as discussed in \[sect:web-sec-reg\]. The [register]{} XML-RPC request will not return until the hub has decided whether to accept registration. This decision may involve user interaction and hence take a significant amount of time. The likely timescales mean that an HTTP timeout is possible but not very probable; in case of a timeout, registration fails. If registration is accepted, the hub MUST return to the client a SAMP map containing the entries mandated by Section \[sect:registration\] and also the following entries: [samp.private-key]{}: : The value of this key is a string which identifies the registered client. This string SHOULD be difficult for third parties to guess. This arrangement is the same as for the Standard Profile (Section \[sect:mappingXMLRPC\]) [samp.url-translator]{}: : The value of this key is a string which forms the base for a URL proxying service, used as described in Section \[sect:web-urltrans\] If registration is rejected, the hub MUST return to the client an XML-RPC Fault, which SHOULD have a suitably explanatory [faultString]{}. ### Callable Clients {#sect:web-callable} In order to be able to receive communications (incoming messages and asynchronous call replies) [*from*]{} the hub, the Web Profile provides for the client to be able to poll the hub server for any messages or replies which are ready for receipt. In this way, such communications are pulled by the client rather than being pushed by the hub, so that no server component is required on the client side. Two hub methods are provided to implement this: - [allowReverseCallbacks(string allow)]{} - [list pullCallbacks(string timeout-secs)]{} Both these methods, like the others in the interface, are named with the [samp.webhub.]{} prefix and take the [private-key]{} as an additional first argument. The [allow]{} argument of [allowReverseCallbacks]{} is a [*SAMP boolean*]{} (“0” for false or “1” for true), and the [timeout-secs]{} argument of [pullCallbacks]{} is a [*SAMP int*]{} (see Section \[sect:scalar-types\]). If a client intends at some time in the future to poll for callbacks it MUST invoke [allowReverseCallbacks]{} with a true argument. If at some later point it decides that it will remain registered but will never poll for callbacks again it SHOULD invoke [allowReverseCallbacks]{} with a false argument (most clients will never make this second call). The client becomes [*Callable*]{} only when it has invoked this method with a true argument. Having invoked [allowReverseCallbacks]{} with a true argument, the client SHOULD periodically invoke [pullCallbacks]{} whose return value gives the details of any callbacks ready for dispatch to the client. The [timeout-secs]{} parameter is the maximum number of seconds the client wishes to wait for a response. When the method is called, the hub SHOULD wait until at least one callback is available, and at that point SHOULD return any pending callbacks. If the elapsed time since [pullCallbacks]{} was received exceeds the number of seconds given by the [timeout-secs]{} argument, the hub SHOULD return with an empty list of callbacks. A client may therefore make a non-waiting poll by using a [timeout-secs]{} argument of 0. The hub MAY return with an empty list of callbacks before the given timeout has elapsed, for instance if it reaches an internal timeout limit. The hub MAY discard pending messages before they have been polled for by the client, for instance to avoid excessive usage of resources to store them. If a [receiveCall]{} for an Asynchronous Call/Response-pattern message is discarded in this way, the hub SHOULD inform the sender by passing back a [samp.code=samp.noresponse]{}-type error response, as described in Section \[sect:response-encode\]. The format of the returned value from [pullCallbacks]{} is a [list]{} of elements each of which is a [map]{} representing a callback corresponding to one of the methods in the SAMP client API (Section \[sect:clientOps\]). Each of these callbacks is encoded as a [map]{} with the following REQUIRED keys: [samp.methodName]{} : — The client API method name for the callback. Its value is a [string]{} taking one of the values “[receiveNotification]{}”, “[receiveCall]{}” or “[receiveResponse]{}”. [samp.params]{} : — A [list]{} of the parameters taken by the client API method in question, as documented in Section \[sect:clientOps\]. These items correspond to the elements present in an XML-RPC call. Here is an example of a call to [pullCallbacks]{}. The client POSTs an XML-RPC call which requests any callbacks which are currently pending or which become available during the next 600 seconds: POST / Host: localhost:21012 User-Agent: Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.9.2.11) Gecko/20101028 Red Hat/3.6-2.el5 Firefox/3.6.11 Referer: http://www.star.bris.ac.uk/~mbt/websamp/sample.html Content-Length: 284 Content-Type: text/plain; charset=UTF-8 Origin: http://www.star.bris.ac.uk <?xml version='1.0'?> <methodCall> <methodName>samp.webhub.pullCallbacks</methodName> <params> <param> <value><string>wk:1_fjlyrdtwtigfqhnwkqokqpbq</string></value> </param> <param> <value><string>600</string></value> </param> </params> </methodCall> The response, which is returned by the hub after some delay between 0 and 600 seconds, specifies a [receiveCall]{} operation that the client should respond to: 200 OK Content-Length: 1444 Content-Type: text/xml Access-Control-Allow-Origin: http://www.star.bris.ac.uk <?xml version='1.0' encoding='UTF-8'?> <methodResponse> <params> <param> <value> <array> <data> <value> <struct> <member> <name>samp.methodName</name> <value>samp.webclient.receiveCall</value> </member> <member> <name>samp.params</name> <value> <array> <data> <value>hub</value> <value>hub_A_cc55_Ping-tag</value> <value> <struct> <member> <name>samp.mtype</name> <value>samp.app.ping</value> </member> <member> <name>samp.params</name> <value> <struct> </struct> </value> </member> </struct> </value> </data> </array> </value> </member> </struct> </value> </data> </array> </value> </param> </params> </methodResponse> Some of the HTTP headers in the outgoing request in this example have been added outside of the client’s control by the browser runtime environment. In particular the [Origin]{} inserted by the browser, and the [Access-Control-Allow-Origin]{} provided in response by the Hub, indicate that CORS negotiation [@cors] is in operation here to allow cross-origin access. ### URL Translation {#sect:web-urltrans} In order that sandboxed clients are able to obtain the content of URLs from foreign domains, as is often required by SAMP interoperation, the hub provides a service which is able to dereference general URLs. At registration time, as described in Section \[sect:web-registration\], one of the values provided to the registering client is that of the [samp.url-translator]{} key. This is a partial URL which, when another URL [*u1*]{} is appended to it, will return the same content as [*u1*]{} from an HTTP GET request. If [*u1*]{} is a syntactically legal URL according to RFC 2396 [@rfc2396], no additional encoding needs to be performed on it by the client prior to the concatenation. A sample of ECMAScript code using this facility might look something like this: var url_trans = reg_info["samp.url-translator"]; var u1 = msg["samp.params"]["url"]; // base URL received from message var u2 = url_trans + u1; // URL ready for retrieval The partial translator URL might typically be implemented as a URL pointing to the same HTTP server in which the hub is hosted, with an empty query part. The content of URLs accessed in this way SHOULD be available under the same cross-origin arrangements described in Section \[sect:web-httpd\]. For security reasons the hub SHOULD ensure that this facility can only be used by registered clients, for instance by embedding the private key in the URL. Thus a translator URL might look something like > [http://localhost:21012/translator]{}/[*client-private-key*]{}[?]{} The URL translation service SHOULD in general write an HTTP response with HTTP headers appropriate for the resource being served, in accordance with the HTTP version in use (e.g. [@rfc2616]). Where the content type of a resource is not known (which is typical if that resource is backed by a file rather than an HTTP URI) the HTTP Content-Type header MAY be omitted. For security reasons, such a hub URL translation service MAY refuse access to certain resources, as discussed in Section \[sect:web-sec-urls\]. Client Behaviour {#sect:web-client} ---------------- The steps that a client must take to register with a Web Profile hub and participate in two-way SAMP communications are as follows: 1. Prepare to make XML-RPC communications with the XML-RPC endpoint [http://localhost:21012/]{}. Web applications will need to do this using a client which supports one of the cross-origin workarounds described in Section \[sect:web-httpd\] and supported by the Web Profile hub. 2. Call the [register]{} XML-RPC method supplying a short application name and possibly other information in the [identity-info]{} argument. If this succeeds (returns a non-Fault XML-RPC response), the client is registered. 3. If the client wishes to receive as well as send communications (to be [*Callable*]{}), first call [allowReverseCallbacks]{} and then periodically call [pullCallbacks]{}. Call [declareSubscriptions]{} as required. 4. Act on retrieved callbacks as required. If any MType argument or return value is a URL, prefix it with the value of the [samp.url-translator]{} entry from the registration map before dereferencing it. 5. Send SAMP messages etc as required. 6. Unregister when no further SAMP activity is required, either because the user requests disconnection or on page unload or a similar event. Security Considerations {#sect:web-security} ----------------------- Web browsers implement cross-origin access restrictions in order to prevent web applications from activity on a local host which presents a security risk, for instance reading and writing local files. This means that, at least in principle, a user can visit a web page without worrying about security issues, in a way which is not the case if they download and install an application to run outside a browser. The Web Profile described in the preceding subsections however relies on neutralising these security measures to some extent. Although it only affects access to a single resource, the HTTP server on which the Web Profile hub resides, it is potentially serious since the services provided by the hub can expose sensitive resources. Section \[sect:web-sec-analysis\] below presents an analysis of the risks, Sections \[sect:web-sec-reg\] and \[sect:web-sec-behave\] outline how they may be mitigated, and Section \[sect:web-sec-summary\] summarises the security status of Web Profile hub deployments in practice. ### Risk Analysis {#sect:web-sec-analysis} Implementation in the Web Profile of one or more of the sandbox-defeating cross-origin workarounds described in Section \[sect:web-httpd\] allows an untrusted, hence potentially hostile, web application to make HTTP requests to the Web Profile SAMP hub HTTP server. In the first instance, there is only one potentially sensitive action that this access permits: attempting to register with the SAMP hub. If the registration attempt is denied, the web application can perform no useful or potentially dangerous operations (except for a denial of service attack, which sandboxed web applications are capable of in any case). If the registration is granted, the client can perform two classes of sensitive actions: first, exchange SAMP messages with other clients, and second, use the hub’s URL translation service to access cross-domain URLs which would normally be blocked by the browser. In order to protect against security breaches related to the Web Profile therefore, two lines of defence may be established: first, exercise control over which web applications are permitted to register, and second, restrict the actions that registered applications are permitted to take. These options are explored in the following sections, \[sect:web-sec-reg\] and \[sect:web-sec-behave\] respectively. ### Registration Restrictions {#sect:web-sec-reg} A running Web Profile implementation may receive requests to register from any web application running in a local browser, and even some clients in other categories. Since not all such applications may be trustworthy, the Web Profile SHOULD exercise careful control over which ones are permitted to register. A Web Profile implementation is permitted to make such decisions in accordance with whatever security policy it deems appropriate, but it is RECOMMENDED that at least the restrictions described in the following subsections are considered: restricting requests to the local host (Section \[sect:web-sec-host\]), requiring explicit user confirmation (Section \[sect:web-sec-confirm\]) and attempting client authentication (Section \[sect:web-sec-auth\]). #### Local Host Restriction {#sect:web-sec-host} As strongly RECOMMENDED in Section \[sect:web-httpd\], registration requests, and in fact all access to the hub HTTP server, SHOULD under normal circumstances only be permitted from the local host. This blocks registration attempts from web or non-web applications on the internet at large. Given this restriction, the only applications which may attempt to register with a hub run by user U are therefore: 1. web applications running in a browser run by user U on the local host 2. non-web applications run by user U on the local host 3. web or non-web applications run by users other than U on the local host Type 1 are the applications that the Web Profile is designed to serve. Type 2 are not what the Web Profile is designed for, since they could use the Standard Profile instead, but they already have user privileges so present no additional security risk. Type 3 are potentially problematic, if the host in question is a multi-user machine, since they may result in a different user who is already able to run processes on the local host acquiring access to the hub-owner’s resources (e.g. private files). In practice the User Confirmation step (Section \[sect:web-sec-confirm\]) should serve to distinguish type 3 from legitimate (type 1) requests, and the behaviour restrictions described in Section \[sect:web-sec-behave\] will limit any potential damage. There may be circumstances under which it is appropriate to relax this local host restriction, for instance to enable collaboration with a known external host not capable of Standard Profile communication, such as a mobile device operated by the hub user. However, it is RECOMMENDED that Web Profile implementations at least restrict access to the local host in their default configuration, and if access is permitted to external hosts it is only by explicit user request, and to a named host or list of hosts. Opening the well-known Web Profile hub server port to the internet at large would invite denial of service and perhaps phishing attacks in which the user is exposed to unwanted SAMP registration requests. #### User Confirmation {#sect:web-sec-confirm} It is strongly RECOMMENDED that the Hub requires explicit confirmation from the user before any Web Profile application is allowed to register. This will normally take the form of the Hub popping up a dialogue window which requires the user to click “OK” or similar for registration to proceed. An implication of this is that the Web Profile hub must have access to the same visual display on which the browser is running, which almost certainly means the hub and the browser are run by the same user. When enquiring about authorization the hub should make clear to the user the security implications of accepting the registration request, and should also present to the user any known information about the application attempting to register. Unfortunately, little such information is guaranteed to be available. The name declared by the application as part of its registration request will be present, but the application is free to declare any name, perhaps a misleading one. Certain HTTP headers on the incoming request may also be relevant: the “Origin” header [@origin] will be present for requests originating from CORS, and the “Referer” header [@rfc2616 section 14.36] may be provided, though its presence and reliability is dependent on the combination of browser, platform and cross-origin workaround. Note that use of non-CORS options might on some browser/plugin platforms permit faking of HTTP headers[^5], so that if the Web Profile HTTP server implements one of the non-CORS options alongside CORS this may reduce the reliability of header information even from HTTP requests which (apparently) originate from CORS. These headers should therefore be used with care. Since only the name, which may be chosen at will by the registering application, is guaranteed present, this looks on the face of it like a poor basis on which to accept or reject registration by a potentially hostile web application. However, in practice the timing of the request presentation provides the most useful information about the identity and credibility of the request. A user will only see such a popup dialogue at the time that a web application attempts to register with SAMP. This will normally be immediately following a deliberate user browser action like opening, or clicking a “Register” button on, a web page. If the user trusts the web page he has just interacted with, he can trust the application within it, and should hence authorize registration. If the user does not trust the web page he has just interacted with, or if the popup appears at a time when no obvious action has been taken to trigger a SAMP registration, then the user should deny registration. This pattern of user interaction, requiring authorization based on the timing of actions in a browser, is both intuitive and familiar to users; for instance it is used when launching a signed Java applet or Java WebStart application. #### Client Authentication {#sect:web-sec-auth} As an additional security measure it would be desirable to make a reliable identification of the author of a web application by examining an associated digital certificate, with reference to a list of trusted certificate authorities. If a certificate reliably associated with the application could be obtained, this additional information could be presented to the user or used automatically by the hub to inform the decision about whether to accept or reject the registration request. Unfortunately however the content of the actual application is not available to the Hub at registration time, so signing the application code will not in itself help. The Web Profile does not at present therefore make any recommendation concerning client authentication. Implementations may however wish to attempt some level of authentication, perhaps by somehow associating a certificate with the web client’s URL or Origin using the HTTP (or HTTPS) request headers noted in Section \[sect:web-sec-confirm\], or by use of additional credentials passed in the [identity-info]{} map. ### Behaviour Restrictions {#sect:web-sec-behave} Given the restrictions on client registration recommended by Section \[sect:web-sec-reg\], there is a reasonable expectation that clients registered with the Web Profile will be trustworthy. However, the possibility remains that user carelessness or some phishing-like attack might lead to registration of hostile clients, and so Web Profile implementations may additionally restrict the behaviour of registered clients. In general, a Web Profile hub implementation MAY impose such restrictions as it sees fit, based on its chosen security policy. This may lead to the inability of some Web Profile clients to perform some legitimate SAMP operations; in such cases the hub SHOULD signal that fact to the client using an appropriate error mechanism. Restrictions may be applied as described in the following subsections: restricting the MTypes that may be sent (Section \[sect:web-sec-mtypes\]), and restricting the scope of the URL translation service (Section \[sect:web-sec-urls\]). #### MType Restrictions {#sect:web-sec-mtypes} The SAMP standard imposes no restriction on the semantics of MTypes, so SAMP can in principle be used to send messages which exercise the privileges available to other SAMP clients in arbitrary ways. In practice, most SAMP MTypes are fairly harmless; a typical result is loading an image into an image viewer. While hostile abuse of such a capability could be annoying, it does not consitute a serious security concern. However one might imagine an MType that intentionally or unintentionally allowed execution of arbitrary scripting operations within the context of a connected client, and hostile abuse of such a facility could easily result in theft of or damage to data, or in other serious security breaches. With this in mind, Web Profile hub implementations MAY impose some restrictions on the MTypes that registered clients are permitted to send, via for instance some per-MType whitelisting or blacklisting mechanism. Given the open-ended nature of the MType vocabulary, a whitelisting approach may be most appropriate. The hub MAY also restrict MTypes that Web Profile registered clients are permitted to receive, though it is harder to imagine exploits based on message receipt. Hubs may implement such message blocking either by hiding blocked subscriptions from other clients as appropriate, or by refusing to forward messages corresponding to blocked subscriptions. In the latter case a communication failure should be signalled by responding with an XML-RPC fault. #### URL Restrictions {#sect:web-sec-urls} As explained in Section \[sect:web-urltrans\], the Web Profile provides a service for proxying arbitrary URLs, so that web clients can access data referenced by URL in SAMP messages or metadata, which sandbox-imposed cross-origin restrictions would otherwise block them from reading. This capability is essential for worthwhile use of many common SAMP MTypes. However, it is also open to abuse, for instance a hostile client might request to read [file:///etc/passwd]{} or some HTTP URL on the local host or network which is restricted to local access. Web Profile implementations therefore MAY impose such restrictions as they see fit on the use of the URL translation service provided to web clients, in order to prevent such abuse. Blocking all access to resources which are local ([file:]{} or [http://localhost/]{}) is too strict to be useful, since the URLs referenced in SAMP messages very often fall into this category. An appropriate policy might be to proxy only URLs which a web client is known to have some legitimate SAMP-based reason to access, namely those which have previously appeared in the metadata declared by, or in a message or response originating from, some other client. In consideration of the fact that web clients might be able to provoke other clients to emit a chosen URL, or might cooperate between themselves, such a list of permitted values SHOULD be further restricted to those URLs which first appeared in a metadata or message content or response map from a trusted (i.e. non-web) client. Since the hub in general lacks the relevant semantic knowledge there is no foolproof way to identify URLs in metadata or messages, but checking for syntactically suitable map values (e.g. [(http|https|ftp|file)://.\*]{}) is likely to be good enough for this purpose. Where the Web Profile implementation declines a given URL proxy request, it MUST respond with a 403 Forbidden HTTP response. It is also RECOMMENDED that proxied HTTP access is limited to the “safe” HTTP methods GET and optionally HEAD [@rfc2616 section 9.1.1], and that user credentials (cookies, authentication etc) are not propagated. Requests using unsupported HTTP methods MUST be met with a 405 Method Not Allowed response. ### Security Summary {#sect:web-sec-summary} The basic mechanics of the Web Profile present significant security risks for a host on which it runs. This section has described how security-conscious implementations of the Profile can mitigate those risks. Following the recommendations from Section \[sect:web-sec-reg\] on when to permit registration provides a reasonable assurance that registered clients will be trustworthy, and in particular guarantees that clients can only register with explicit authorization from a human user. Following the recommendations from Section \[sect:web-sec-behave\] about permitted behaviour of registered clients ensures that even if a hostile client is allowed to register it is unlikely to be able to do significant damage. By combining these measures it is believed that the level of risk associated with running a Web Profile, while it would not be appropriate for instance for financial transactions, is no greater than that encountered on a regular basis by use of the web in general. The mitigation measures are presented as (in some cases strong) RECOMMENDations and suggestions rather than REQUIREments, in order to allow implementations to experiment with the most appropriate configurations, which may change as a result of emerging technology and common usage patterns. Such experimentation and further consideration may result in some modification of the protocol or documentation of best practice in future versions of this document or elsewhere. MTypes: Message Semantics and Vocabulary {#sect:mtypes} ======================================== A message contains an MType string that defines the semantic meaning of the message, for example a request for another application to load a table. The concept behind the MType is similar to that of a UCD [@ucd] in that a small vocabulary is sufficient to describe the expected range of concepts required by a messaging system within the current scope of the SAMP protocol. Developers are free to introduce new MTypes for use within applications without restriction; new MTypes intended to be used for Hub messaging or other administrative purposes within the messaging system should be discussed within the IVOA for approval as part of the SAMP standard. The Form of an MType -------------------- MType syntax is formally defined in Section \[sect:subscription\]. Like a UCD, an MType is made up of [*atoms*]{}. These are not only meaningful to the developer, but form the central concept of the message. Because the capabilities one application is searching for are loosely coupled with the details of what another may provide, there is not a rigorous definition of the [*behavior*]{} that an MType must provoke in a receiver. Instead, the MType defines a specific semantic message such as “display an image”, and it is up to the receiving application to determine how it chooses to do the display (e.g. a rendered greyscale image within an application or displaying the image in a web browser might both be valid for the recipient and faithful to the meaning of the message). The ordering of the words in an MType SHOULD normally use the object of the message followed by the action to be performed (or the information about that object). For example, the use of “[image.display]{}” is preferred to “[display.image]{}” in order to keep the number of top-level words (and thus message classes) like ‘image’ small, but still allow for a wide variety of messages to be created that can perform many useful actions on an image. If no existing MType exists for the required purpose, developers can agree to the use of a new MType such as “[image.display.extnum]{}” if, e.g., the ability to display a specific image extension number warrants a new MType. The Description of an MType {#sect:mtype-doc} --------------------------- In order that senders and recipients can agree on what is meant by a given message, the meaning of an MType must be clearly documented. This means that for a given MType the following information must be available: 1. The MType string itself 2. A list of zero or more named parameters 3. A list of zero or more named returned values 4. A description of the meaning of the message For each of the named parameters, and each of the returned values, the following information must be provided: - name - data type ([map]{}, [list]{} or [string]{} as described in Section \[sect:samp-data-types\]) and if appropriate scalar sub-type (see Section \[sect:scalar-types\]) - meaning - whether it is OPTIONAL (considered REQUIRED unless stated otherwise) - OPTIONAL parameters MAY specify what default will be used if the value is not supplied Together, this is much the same information as should be given for documentation of a public interface method in a weakly-typed programming language. The parameters and return values associated with each MType form extensible vocabularies as explained in Section \[sect:vocab\], except that there is no reserved “[samp.]{}” namespace. Note that it is possible for the MType to have no returned values. This is actually quite common if the MType does not represent a request for data. It is not usually necessary to define a status-like return value (success or failure), since this information can be conveyed as the value of the [samp.status]{} entry in the call response as described in Section \[sect:response-encode\]. MType Vocabulary: Extensibility and Process ------------------------------------------- The set of MTypes forms an extensible vocabulary along the lines of Section \[sect:vocab\]. The relatively small set of MTypes in the “[samp.]{}” namespace is defined in Section \[sect:admin-mtypes\] of this document, but applications will need to use a wider range of MTypes to exchange useful information. Although clients are formally permitted to define and use any MTypes outside of the reserved “[samp.]{}” namespace, for effective interoperability there must be public agreement between application authors on this unreserved vocabulary and its semantics. Since addition of new MTypes is expected to be ongoing, MTypes from this broader vocabulary will be documented outside of this document to avoid the administrative overhead and delay associated with the IVOA Recommendation Track [@docstd]. At time of writing, the procedures for maintaining the list of publicly-agreed MTypes are quite informal. These procedures remain under review, however the current list and details of best practice for adding to it are, and will remain, available in some form from the URL <http://www.ivoa.net/samp/>. Core MTypes {#sect:admin-mtypes} ----------- This section defines those MTypes currently in the “[samp.]{}” hierarchy. These are the “administrative”-type MTypes which are core to the SAMP architecture or widely applicable to SAMP applications. ### Hub Administrative Messages {#sect:hub-mtypes} The following MTypes are for messages which SHOULD be broadcast by the hub in response to changes in hub state. By subscribing to these messages, clients are able to keep track of the current set of registered applications and of their metadata and subscriptions. In general, non-hub clients SHOULD NOT send these messages. [\ ]{} [\ ]{} [\ ]{} [\ ]{}The hub SHOULD broadcast this message just before it exits. It SHOULD also send it to clients who are registered using a given profile if that profile is about to shut down, even if the hub itself will continue to operate. The hub SHOULD make every effort to broadcast this message even in case of an exit due to an error condition. [\ ]{} [\ ]{} Public ID of newly registered client [\ ]{} [\ ]{}The hub SHOULD broadcast this message every time a client successfully registers. [\ ]{} [\ ]{} public ID of unregistered client [\ ]{} [\ ]{}The hub SHOULD broadcast this message every time a client unregisters. [\ ]{} [\ ]{} public ID of client declaring metadata new metadata declared by client [\ ]{} [\ ]{}The hub SHOULD broadcast this message every time a client declares its metadata. The [metadata]{} argument is exactly as passed using the [declareMetadata()]{} method. [\ ]{} [\ ]{} public ID of subscribing client new subscriptions declared by client [\ ]{} [\ ]{}The hub SHOULD broadcast this message every time a client declares its subscriptions. The [subscriptions]{} argument is exactly as passed using the [declareSubscriptions()]{} method, and hence may contain wildcarded MType strings. [\ ]{} [\ ]{} (OPTIONAL) Short text message indicating the reason that the disconnection is being forced [\ ]{} [\ ]{}The hub SHOULD send this message to a client if the hub intends to disconnect that client forcibly. This indicates that no further communication from that client is welcome, and any such attempts may be expected to fail. The hub may wish to disconnect clients forcibly as a result of some hub timeout policy or for other reasons. ### Client Administrative Messages The following messages are generic messages defined for client use. [\ ]{} [\ ]{} [\ ]{} [\ ]{}Diagnostic used to indicate whether an application is currently responding. No “status”-like return value is defined, since in general any response will indicate aliveness, and the normal [samp.status]{} key in the response may be used to indicate any abnormal state. [\ ]{} [\ ]{} Textual indication of status [\ ]{} [\ ]{}General purpose message to indicate application status. [\ ]{} [\ ]{} [\ ]{} [\ ]{}Indicates that the sending application is going to shut down. Note that sending this message is not a substitute for unregistering with the hub — registered clients about to shut down SHOULD always explicitly unregister. [\ ]{} [\ ]{} Message ID of a previously received message Textual indication of progress (OPTIONAL) SAMP float value giving the approximate percentage progress (OPTIONAL) SAMP float value giving the estimated time to completion in seconds [\ ]{} [\ ]{}Reports on progress of a message previously received by the sender of this message. Such progress reports MAY be sent at intervals between the receipt of the message and sending a reply. Note that the [msg-id]{} of the earlier message must be passed to identify it — the sender of the earlier message (the recipient of this one) will have to have retained it from the return value of the relevant [call\*()]{} method to match progress reports with requests. Changes between PLASTIC and SAMP ================================ In order to facilitate the transition from PLASTIC to SAMP from an application developer’s point of view, we summarize in this Appendix the main changes. In some cases the reasons for these are summarized as well. Language Neutrality: : PLASTIC contained some Java-specific ideas and details, in particular an API defined by a Java interface, use of Java RMI-Lite as a transport protocol option, and a lockfile format based on java Property serialization. No features of SAMP are specific to, or defined with reference to, Java (or to any other programming language). Profiles: : The formal notion of a SAMP Profile replaces the choices of transport protocol in PLASTIC. Nomenclature: : Much of the terminology has changed between PLASTIC and SAMP, in some cases to provide better consistency with common usage in messaging systems. There is not in all cases a one-to-one correspondence betweeen PLASTIC and SAMP concepts, but a partial translation table is as follows: PLASTIC SAMP ------------------------ --------------------------- message MType support a message subscribe to an MType registered application client synchronous request synchronous call/response asynchronous request notification MTypes: : In PLASTIC message semantics were defined using opaque URIs such as [ivo://votech.org/hub/event/HubStopping]{}. SAMP replaces these with a vocabulary of structured MTypes such as [samp.hub.event.shutdown]{}. Asynchrony: : Responses from messages in PLASTIC were returned synchronously, using blocking methods at both sender and recipient ends. As well as inhibiting flexibility, this risked timeouts for long processing times at the discretion of the underlying transport. The basic model in SAMP relies on asynchronous responses, though a synchronous façade hub method is also provided for convenience of the sender. Client toolkits may also wish to provide client-side synchronous façades based on fully asynchronous messaging. Registration: : In PLASTIC clients registered with a single call which acquired a hub connection and declared callback information, message subscriptions, and some metadata. In SAMP, these four operations have been decomposed into separate calls. As well as being tidier, this offers benefits such as meaning that the subscriptions and metadata can be updated during the lifetime of the connection. Client Metadata: : PLASTIC stored some application metadata (Name) in the hub and provided acess to others (Description, Icon URL, …) using custom messages. SAMP stores it all in the hub providing better extensibility and consistency as well as improving metadata provision for non-callable applications and somewhat reducing traffic and burden on applications. Named Parameters: : The parameters for PLASTIC messages were identified by sequence (forming a list), while the parameters for SAMP MTypes are identified by name (forming a map). As well as improving documentability, this makes it much more convenient to allow for optional parameters or to introduce new ones. The same arrangement applies to return values. Recipient Targetting: : PLASTIC featured methods for sending messages to all or to an explicit list of recipients. In practice the list variants were rarely used except to send to a single recipient. SAMP has methods for sending to all or to a single recipient. Typing: : Data types in PLASTIC were based partly on Java and partly on XML-RPC types. There was not a one-to-one correspondence between types in the Java-RMI transport and the XML-RPC one, which encouraged confusion. Parameter types included integer, floating point and boolean as well as string, which proved problematic to use correctly from some weakly-typed languages. SAMP uses a more restricted set of types (namely string, list and map) at the protocol level, along with some auxiliary rules for encoding numbers and booleans as strings. Lockfile: : The lockfile in SAMP’s standard profile is named [.samp]{}, its format is defined explicitly rather than with reference to Java documentation, and there is better provision for its location in a language-independent way on MS Windows systems. In many cases however, the same lockfile location/parsing code will work for both SAMP and PLASTIC except for the different filenames (“.samp” vs. “.plastic”). Public/Private ID: : In PLASTIC a single, public ID was used to label and identify applications during communications directed to the hub or to other applications. This meant that applications could easily, if they wished, impersonate other applications. The practice in SAMP is to use different IDs for public labelling and private identification, which means that such “spoofing” is no longer a danger. Errors: : SAMP has provision to return more structured error information than PLASTIC did. Extensibility: : Although PLASTIC was in some ways extensible, SAMP provides more hooks for future extension, in particular by pervasive use of the [*extensible vocabulary*]{} pattern. Change History {#sect:changes} ============== Changes to SAMP between Working Draft version 1.0 (2008-06-25) and Recommendation version 1.11 (2009-04-21): - Return values of [callAll]{} and [notifyAll]{} operations changed; they now return information about clients receiving the messages (Section \[sect:hubOps\]). - Characters allowed in [string]{} type restricted to avoid problems transmitting over XML; was 0x01–0x7f, now 0x09, 0x0a, 0x0d, 0x20–0x7f (Section \[sect:samp-data-types\]). - New hub administrative message [samp.hub.disconnect]{} (Section \[sect:hub-mtypes\]). - Empty placeholder appendix on SAMP/PLASTIC interoperability removed. - Wording clarified and made more explicit in a few places. - Typos fixed, including incorrect BNF in Section \[sect:subscription\]. - Author list re-ordered. - Editorial changes and clarifications following RFC period. - MType Vocabulary section now directs readers to [http://www.ivoa.net/samp/]{} to find current MType list and process. Changes to SAMP between Recommendation version 1.11 (2009-04-21) and version 1.2 (2010-12-16): - Use of new SAMP\_HUB environment variable lockfile location option documented in section \[sect:lockfile\]. - Added Non-Technical Preamble section \[sect:nonTechPreamble\] as per agreement for all new/revised IVOA documents. Changes to SAMP between Recommendation version 1.2 (2010-12-16) and version 1.3 (2012-04-11): - Add a new Section \[sect:webprofile\] on the Web Profile. Minor changes in the rest of the document noting the existence of this new Profile. - Add a new Section \[sect:security\] discussing security issues in general, with reference to their particular consideration for both Standard and Web Profiles. The discussion of Standard Profile security is moved to its own new Section \[sect:std-security\]. - MType syntax declaration in Section \[sect:subscription\] now permits upper-case letters (for consistency with actual usage). - Sections \[sect:response-encode\] and \[sect:faults\] now note that the hub is permitted to generate and forward an error response on behalf of a client under some circumstances. The [samp.code=samp.noresponse]{} code is reserved for this purpose. - Section \[sect:vocab\] now reserves a namespace “[x-samp]{}” for keys in an extensible vocabulary which are proposed for possible future introduction into this standard. - A comment has been added to Section \[sect:samp-data-types\] concerning recommended protocols for use with URLs in messages. [99]{} C. Arviset et al., “[IVOA Architecture](http://www.ivoa.net/Documents/Notes/IVOAArchitecture/index.html)”, IVOA Note, 2010 F. [Bonnarel]{}, P. [Fernique]{}, O. [Bienaym[é]{}]{}, D. [Egret]{}, F. [Genova]{}, M. [Louys]{}, F. [Ochsenbein]{}, M. [Wenger]{}, and J. G. [Bartlett]{}, “The ALADIN interactive sky atlas. A reference tool for identification of astronomical sources”, [*A&AS*]{}, 143:33–40, 2000 U. [Becciani]{}, M. [Comparato]{}, A. [Costa]{}, C. [Gheller]{}, B. [Larsson]{}, F. [Pasian]{}, and R. [Smareglia]{}. “VisIVO: an interoperable visualisation tool for Virtual Observatory data”, [*Highlights of Astronomy*]{}, 14:622–622, 2007 <http://hea-www.harvard.edu/RD/xpa/> J. [Taylor]{}, T. [Boch]{}, M. [Comparato]{}, M. [Taylor]{}, and N. [Winstanley]{}. “[PLASTIC — a protocol for desktop application interoperability](http://ivoa.net/Documents/latest/PlasticDesktopInterop.html)”, IVOA Note, 2006 <http://plastic.sourceforge.net/> <http://www.xmlrpc.com/> S. Bradner, [RFC 2119](http://www.rfc-editor.org/rfc/rfc2119.txt): “Key words for use in RFCs to Indicate Requirement Levels”, IETF Request For Comments, 1997 T. Berners-Lee, L. Masinter, M. McCahill, [RFC 1738](http://www.rfc-editor.org/rfc/rfc1738.txt): “Uniform Resource Locators (URL)”, IETF Request For Comments, 1994 T. Dierks, C. Allen, [RFC 2246](http://www.rfc-editor.org/rfc/rfc2246.txt): “The TLS Protocol”, IETF Request For Comments, 1999 A. van Kesteren (Ed.), “[Cross-Origin Resource Sharing](http://www.w3.org/TR/cors/)”, W3C Working Draft, 2010 A. van Kesteren (Ed.), “[XMLHttpRequest Level 2](http://www.w3.org/TR/XMLHttpRequest2/)”, W3C Working Draft, 2012 Adobe Flash cross-domain policy, <http://www.adobe.com/devnet/articles/crossdomain_policy_file_spec.html> Microsoft Silverlight cross-domain policy, <http://msdn.microsoft.com/en-us/library/cc645032(VS.95).aspx> T. Berners-Lee et al., [RFC 2396](http://www.rfc-editor.org/rfc/rfc2396.txt): “Uniform Resource Identifiers (URI): Generic Syntax”, IETF Request For Comments, 1998 R. Fielding et al., [RFC 2616](http://www.w3.org/Protocols/rfc2616/rfc2616.html): “Hypertext Transfer Protocol – HTTP/1.1”, IETF Request For Comments, 1999 D. Eastlake et al., [RFC 3275](http://www.rfc-editor.org/rfc/rfc3275.txt): “XML-Signature Syntax and Processing”, IETF Request For Comments, 2002 A. Barth, “[The Web Origin Concept](http://tools.ietf.org/html/draft-abarth-origin)”, IETF Draft, 2010 A. Preite Martinez et al. “[An IVOA Standard for Unified Content Descriptors](http://www.ivoa.net/Documents/latest/UCD.html)”, IVOA Recommendation, 2007 R. J. Hanisch et al. “[IVOA Document Standards](http://www.ivoa.net/Documents/latest/DocStd.html)”, IVOA Recommendation, 2010 [^1]: One way a hub might implement this is to generate [*msg-id*]{} by concatenating the sender’s client ID and the [*msg-tag*]{}. When any response is received the hub can then unpack the accompanying [*msg-id*]{} to find out who the original sender was and what [*msg-tag*]{} it used. In this way the hub can determine how to pass each response back to its correct sender without needing to maintain internal state concerning messages in progress. Hub and client implementations may wish to exploit this freedom in assigning message IDs for other purposes as well, for instance to incorporate timestamps or checksums. [^2]: Note to Java developers: contrary to what you might expect, the [user.home]{} system property on Windows does [*not*]{} give you the value of [USERPROFILE]{}. See <http://bugs.sun.com/bugdatabase/view_bug.do?bug_id=4787931>. [^3]: Support for the crossdomain.xml file is reportedly implemented in Java v1.6.0\_10 and later, see <http://bugs.sun.com/bugdatabase/view_bug.do?bug_id=6676256>. [^4]: See for instance the flXHR library at <http://flxhr.flensed.com/>. [^5]: See for example <http://secunia.com/advisories/22467/>, which refers to a Flash version from 2006. Hopefully browsers and plugins in current use do not contain such vulnerabilities, but an assurance of this is beyond the scope of this document.

--- abstract: 'Optical stochastic cooling (OSC) is expected to enable fast cooling of dense particle beams. Transition from microwave to optical frequencies enables an achievement of stochastic cooling rates which are orders of magnitude higher than ones achievable with the classical microwave based stochastic cooling systems. A subsytem critical to the OSC scheme is the focusing optics used to image radiation from the upstream “pickup" undulator to the downstream “kicker" undulator. In this paper, we present simulation results using wave-optics calculation carried out with the [Synchrotron Radiation Workshop]{} (SRW). Our simulations are performed in support to a proof-of-principle experiment planned at the Integrable Optics Test Accelerator (IOTA) at Fermilab. The calculations provide an estimate of the energy kick received by a 100-MeV electron as it propagates in the kicker undulator and interacts with the electromagnetic pulse it radiated at an earlier time while traveling through the pickup undulator.' address: - | Department of Physics and Northern Illinois Center for Accelerator & Detector Development,\ Northern Illinois University, DeKalb, IL 60115, USA - 'Fermi National Accelerator Laboratory, Batavia, IL 60510, USA' author: - 'M.B. Andorf' - 'V.A. Lebedev' - 'P. Piot' - 'J. Ruan' title: | Wave-Optics Modeling of the Optical-Transport Line for\ Passive Optical Stochastic Cooling --- beam-cooling technique ,electron-laser interaction ,undulator radiation ,beam dynamics Introduction ============ The optical stochastic cooling (OSC) is similar to the microwave-stochastic cooling. It relies on a time dependent signal to carry information on the beam distribution and apply the corresponding cooling force [@OSC_Mikhailichenko; @OSC_Zolotorev]; see Figure \[fig0\]. In OSC a particle radiates an electromagnetic wave while passing through an undulator magnet \[henceforth referred to as the pickup undulator (PU)\]. ![[Overview of the passive-OSC insertion beamline. The labels “Q$_i$", and “B$_i$" respectively refer to the quadrupole and dipole magnets and “$f_i$" represent the optical lenses. The solid blue (resp. ondulatory red) line gives the electron (resp. light) trajectory.]{}[]{data-label="fig0"}](OSCfigure.pdf){width="45.00000%"} The radiation pulse passes through a series of lenses and an optical amplifier and is imaged at the location of a downstream undulator magnet dubbed as kicker undulator (KU). The particle beam propagates through a bypass chicane (B$_1$, B$_2$, B$_3$, B$_4$) which provides an energy-dependent path length (i.e. time of flight) as well as a path length variation due betatron oscillations. The chicane also provides the space to house the optical components necessary for the optical-pulse manipulation and amplification. The imaged PU-radiation field and the particle that radiated it copropagate in the KU resulting in an energy exchange between them. When the time of arrival is properly selected a corrective energy kick is applied resulting in damping of the particles synchrotron oscillations as the process is repeated over many turns in a circular accelerator. If the KU is located in a dispersive section the corrective kick can also yield transverse cooling in the dispersive plane. Furthermore if the horizontal and vertical degrees of freedom are coupled outside of the cooling insertion the OSC can provide 6D phase-space particle cooling. Although the nominal OSC scheme discussed in most of the literature involves an optical amplifier, the experiment planned in the 100-MeV IOTA electron ring at Fermilab [@IOTA; @IOTA2] will not incorporate an optical amplifier in its first phase. This latter version of OSC is referred to as passive OSC (POSC) and it is considered throughout this paper. The nomination (amplified) OSC scheme will be implemented in a subsequent stage [@andorfAMPLI].\ A comprehensive treatment of the OSC can be found in Ref. [@OSC_val] where the kick amplitude is computed semi-analytically by considering a single focusing lens placed between the two undulators separated by a distance much larger than their length. In doing so the depth of field associated with the finite length of the undulators is suppressed. Although theoretically convenient, this focusing scheme is not practical and a three-lens configuration is instead adopted with focal lengths $f_i$ and distances $L_i$ fulfilling [@OSC_val] $$\begin{aligned} f_1=L_2 &\mbox{~and~} &f_2=-\dfrac{L_2^2}{2(L_1-L_2)} , \end{aligned}$$ where the parameters are defined in Fig. \[fig0\]. The resulting transfer matrix between the KU and PU defined in the position-divergence coordinate system $\pmb X =(x,x')$ is $M_{KU\rightarrow PU} =-I$, where $I$ is the $2\times 2$ identity matrix. The three-lens telescope configuration supports a longitudinal point-to-point imaging between the PU and KU while also flipping the transverse coordinate w.r.t. the horizontal kicker axis. Correspondingly the telescope addresses the depth-of-field issue and the results derived for a single lens are directly applicable. The parameters of the optical telescope and undulators (the PU and KU are identical) are listed in Tab. \[tab:TLT\]. Note that both undulators are providing a vertical magnetic field $\pmb B=B\hat{y}$ so that the oscillatory trajectory lies in the $(x,z)$ plane. The undulator radiation wavelength depends on the angle as: $\lambda_r=\frac{\lambda_u}{2\gamma^2}\left[ 1+\frac{K_u^2}{2}+(\gamma\theta)^2\right]$ where the parameters are defined in Tab. \[tab:TLT\] and $\theta$ is the observation angle w.r.t the electron direction. Specifically, we defined the on-axis resonant wavelength as $\lambda_0\equiv\lambda_r(\theta=0)$. parameter, symbol value units -------------------------------------- -------- -------- drift $L_1$ 143 cm focal length $f_1$ 143 cm drift $L_2$ 32 cm focal length $f_2$ -4.61 cm angular acceptance $\gamma \theta_m$ 0.8 undulator parameter $K_u$ 1.038 undulator length $L_u$ 77.4 cm undulator period, $\lambda_u$ 11.057 cm number of periods, $N_u$ 7 on-axis wavelength, $\lambda_0$ 2.2 $\mu$m electron Lorentz factor, $\gamma$ 195.69 : Parameters for the optical telescope and undulators for the proposed POSC experiment at IOTA. \[tab:TLT\] []{data-label="param"} Single-lens focusing ===================== A wave-optics model of single-lens focusing was implemented in the [Synchrotron Radiation Workshop]{} (SRW) program [@SRW_chubar; @SRW_chubar2] to benchmark our numerical implementation with the analytical model obtained for a single lens configuration [@OSC_val]. ![Computed suppression factor $F_h(K_u,\gamma \theta_m)$ (dashed traces, left axis scale) and energy loss (solid traces) of the particle passing through one undulator (right axis scale) as a function of the undulator parameter $K_u$ for different angular acceptances of the lens ($\gamma\theta_m$) (a). Comparison of the electric field at the focus on a single lens analytically computed (solid traces) and simulated with SRW (symbol with dashed traces) for the same cases of angular acceptance (b). []{data-label="fig1"}](Figure_2.pdf){width="50.00000%"} Considering the case of POSC, taking $K_u \ll 1$, and assuming an infinite numerical aperture of the focusing lens, the on axis electric field amplitude imaged in the KU is given by $$\begin{aligned} E_x(x=y=0)=\frac{4}{3} e K k_u^2\gamma^3, \label{small_K_field_amp}\end{aligned}$$ where $k_u\equiv 2\pi/\lambda_u$ ($\lambda_u$ is the undulator period) and $\gamma$ the Lorentz factor. The transverse velocity of the particle is $v_x=\dfrac{K c}{\gamma}\sin(k_uz)$ and the kick amplitude is approximately $$\begin{aligned} \Delta {\cal E} =e\int_{0}^{L_u} \dfrac{E_xK_u}{\gamma}\sin^2{(k_u z)}dz = \dfrac{eE_xK_u L_{u}}{2\gamma}, \end{aligned}$$ where $L_{u}$ is the undulator length. Combining the latter equation with Eq. \[small\_K\_field\_amp\] yields $$\begin{aligned} \Delta {\cal E}=\frac{2 \pi}{3}(eK\gamma)^2 k_u N_u, \label{small_K_kick}\end{aligned}$$ where $N_u$ is the number of undulator periods. Intuitively Eq. \[small\_K\_kick\] is just equal to the total energy loss as the electron travels through one undulator. When $K_u$ is increased (thereby resulting in an increased angular deflection) and the finite angular acceptance of the lens, $\theta_m$, taken into account, the on-axis electric field $E_x(x=y=0)$ in the KU is reduced by a factor $F_h(K_u,\gamma \theta_m)\leq1 $. The expression of $F_h(K_u,\gamma \theta_m)$ is derived in [@OSC_val] and its dependence on $K_u$ appears in Fig. \[fig1\] for three cases of $\gamma \theta_m$. There is an additional efficiency factor, $F_u(\kappa_u)=J_0(\kappa_u)-J_1(\kappa_u)$, which accounts the effect of the longitudinal oscillation \[given by $\frac{K_u^2}{8\gamma^2k_u}\sin(2k_u z)$\] of the particle in the KU where $\kappa_u\equiv K_u^2/4(1+K_u^2/2)$. The kick amplitude from Eq. \[small\_K\_kick\] is thus reduced by the factor of $F_h(K_u,\theta_m\gamma)\times F_u(\kappa_u)$.\ The simulation in SRW are performed in the frequency domain: the field frequency components within the PU-radiation bandwidth are propagated and the field amplitude in the time domain inside the KU is computed [@light_optics_ipac16]. This is first done for the case of a single focusing lens using $L_u$ and $\lambda_o$ from Tab. \[tab:TLT\], but varying $N_u$ and other parameters appropriately. For this benchmarking simulation, the distance between the PU and KU centers is taken to be $L_t=19.5$ m (i.e. $L_t\gg L_u$) in order to suppress the depth-of-field effect and the focal length of the lens is $f=L_t/2$. The simulated value for $E_x(K,\gamma \theta_m)$ are found to be in excellent agreement (relative difference below 5%) as shown in Fig. \[fig1\]. Imaging with a three-lens telescope =================================== We now focus on the imaging scheme proposed for the POSC experiment at IOTA with parameters summarized in Tab. \[tab:TLT\]. The point-to-point imaging of the KU radiation in the PU is accomplished with a three-lens telescope. First, the field amplitude at the KU longitudinal center is compared with the expected value from theory: using Eq. \[small\_K\_field\_amp\] and $F_h(1.038,0.8)=0.25$ yields $E_x=11.8$ V/m while SRW gives 10.9 V/m corresponding to a relative discrepancy $<7\%$. The kick amplitude using Eq. \[small\_K\_kick\] and $F_u(0.18)=0.91$ yields $\Delta {\cal E}=22$ meV while directly computing the kick in the same way with SRW yields a value of $20.1$ meV. Therefore the agreement between theoretical predictions and numerical simulations is reasonable as was already observed in the previous Section. It should be noted that with SRW the longitudinal and transverse dependence of the electric field neglected in theory can also be accounted. The latter of which is from the effective aperture of the outer lenses being less at the edges of the undulator than they are at the center. To find the kick value from SRW, the time-domain field was computed along the kicker every 3.2 mm. The average forward velocity of the particle is $\langle v_z\rangle \equiv \bar{\beta}c=c\beta(1-K_u^2/4\gamma^2)$ where $c$ denotes the velocity of light. Therefore as the particle advances through the kicker it falls back relative to the radiation packet by an amount $$\begin{aligned} \delta_t = \frac{z_l(1-\bar{\beta})+\frac{K_u^2}{8\gamma^2k_u}\sin{(2k_u z)}}{c}, \end{aligned}$$ with $z_l$ the location of radiation packet in the KU referenced to its entrance. The latter equation, which also accounts the electron’s longitudinal oscillatory motion, is used to compute the electric field $E_x(x,z)$ experienced by the electron as it propagates through the KU. The change in energy is then obtained via the numerical integration of: $$\begin{aligned} \Delta {\cal E} = \int_{z=0}^{z=L_u} v_x E_x(x,z) dz.\end{aligned}$$ ![Transverse electric field $E_x(x,z)$, experienced by a 100-MeV electron as it moves along the KU. The white trace represents the trajectory of the electron passing through the KU (a). Corresponding evolution of the energy change along the KU for an electron phased w.r.t. the E field such to maximize its kick (b). []{data-label="fig2"}](Figure_3.pdf){width="49.50000%"} It is also being tacitly assumed that the arrival time of the particle is such that $E_x(x,z)$ maximizes the kick. A plot of the electric field in the undulator mid plane $E_x(x,y=0,z)$ appears in Fig. \[fig2\](a) with the trajectory of the electron overlapped. The corresponding evolution of the electron energy along the KU is displayed in Fig. \[fig2\](b). The kick amplitude is found to be 18 meV. A reduction of 10.4 % comes from the longitudinal dependence of the field amplitude along the KU. The maximum transverse displacement of the particle in the KU is 93 $\mu$m allowing the particle to experience electric field values reduced by $\sim 5$ % w.r.t. to the maximum on-axis value. Such an effect reduces the kick by only 1.1 %. This is expected since the instantaneous energy transfer to the particle is proportional to $v_x$ which attains its maximum value on axis. As the electron’s transverse offset increases the velocity decreases to eventually vanishes when the electron reaches its maximum offset. Such a dependence of the velocity $v_x(x)$ mitigates the impact of the off-axis field reduction. Furthermore for the particle receiving the largest energy kick the phase of the wave (as seen in the co-moving frame of the particle) is such that the field is zero when the particle is the farthest off axis.\ Our simulations also allow for the kick to be computed as a function of $\tau$ the delay relative to a reference particle as displayed in Fig. \[fig3\]. ![Energy change as a function of the electron delay $\tau$ (solid trace). The value $\tau=0$ corresponds to the case of a reference electron which does not experience a net energy change. The dashed line is the envelope of the kick $w(\tau)$ approximated as a Gaussian function. []{data-label="fig3"}](Figure_4.pdf){width="48.00000%"} The envelope, $w(\tau)$, of the kick is approximately Gaussian with a RMS duration $\sigma_{\tau}=13.5$ fs. A common approximation to the pulse length is $t_l=N_u\lambda_0/c$ corresponding to 51.3 fs for the undulator parameters foreseen at IOTA. Since the telescope focuses light from one location in the PU to the corresponding location in the KU, the shape of the wave packet modulates while propagating through the KU. This modulation reduces the effective length of the wave packet at any particular location in the KU. Since the transverse dimensions of the wave packet ($\approx 520$ $\mu$m for the half-waist) are larger than the transverse beam size, the wave packet can be thought of as slicing the beam only along the longitudinal direction. Considering a bunch of $N$ electrons and taking the bunch density to be constant over the length of the wave packet the number of particles within a “sampling" slice can be approximated as $$\begin{aligned} N_s \simeq \frac{cN}{l_b}\int w(\tau)d\tau=\dfrac{Nc\sigma_{\tau}\sqrt{2\pi}}{l_b}\end{aligned}$$ where $l_b$ is the bunch length. The expected bunch length in IOTA during the OSC experiment, prior to cooling, is 14.2 cm giving $N_s/N$=7.1x10$^{-5}$. In IOTA intrabeam scattering is the major limitation on the number of particles per bunch. Conclusion ========== We used a wave-optics software, SRW, to investigate the resultant energy exchange in a POSC scheme. We compared our simulation results to the semi-analytic theory developed in [@OSC_val] and found agreement better than $ 5 \%$ for a range of $K$ values and angular acceptances of the focusing lens. The benchmarked model was used to compute the expected kick amplitude for the POSC proof-of-principle experiment planned at the IOTA ring at Fermilab. It was especially found the decreasing of the effective aperture for points away from the kicker center reduces the energy kick by approximately $10 \%$. In addition the transverse dependence of the field experienced as the particle follows an oscillatory trajectory in the KU was found to have an insignificant effect on the energy-kick amplitude. So far our calculations neglect reflective losses and dispersion from the lenses. Accounting and compensating for dispersion is the subject of on going optimization of the optical transport for POSC in IOTA. Acknowledgments =============== This work was supported by the US Department of Energy (DOE) under contract DE-SC0013761 to Northern Illinois University. Fermilab is managed by the Fermi Research Alliance, LLC for the U.S. Department of Energy Office of Science Contract number DE-AC02-07CH11359. [99]{} A.A.Mikhailichkenko, M.S. Zolotorev, [Optical stochastic cooling](https://doi.org/10.1103/PhysRevLett.71.4146) Phys. Rev. Lett. 71, 4146 (1993). M. S. Zolotorev, A. A. Zholents, [Transit-time method of optical stochastic cooling](https://doi.org/10.1103/PhysRevE.50.3087), Phys. Rev. E [**50**]{}, 3087 (1994) S. Antipov et al., [IOTA (Integrable Optics Test Accelerator): facility and experimental beam physics program,](http://iopscience.iop.org/article/10.1088/1748-0221/12/03/T03002/meta;jsessionid=282693D8B872C2E4F86A2567E63EF285.c2.iopscience.cld.iop.org) Journal of Instrumentation (JINST) [**12**]{}, T03002 (2017). V. Lebedev, and A.L. Romanov, [Optical stochastic cooling at the IOTA ring,](http://accelconf.web.cern.ch/AccelConf/cool2015/papers/wewaud03.pdf) Proceedings of COOL2015, Newport News VA, p 123-127 (2015). M. B. 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--- abstract: 'We derive a Feynman-Hellmann theorem relating the second-order nucleon energy shift resulting from the introduction of periodic source terms of electromagnetic and isovector axial currents to the parity-odd nucleon structure function $F_3^N$. It is a crucial ingredient in the theoretical study of the $\gamma W$ and $\gamma Z$ box diagrams that are known to suffer from large hadronic uncertainties. We demonstrate that for a given $Q^2$, one only needs to compute a small number of energy shifts in order to obtain the required inputs for the box diagrams. Future lattice calculations based on this approach may shed new light on various topics in precision physics including the refined determination of the Cabibbo-Kobayashi-Maskawa matrix elements and the weak mixing angle.' author: - 'Chien-Yeah Seng$^{a}$' - 'Ulf-G. Meißner$^{a,b,c}$' title: 'Toward a First-Principles Calculation of Electroweak Box Diagrams' --- The electroweak box diagrams involving the exchange of a photon and a heavy gauge boson ($W^\pm/Z$) between a lepton and a hadron (see Fig. \[fig:box\]) represent an important component in the standard model (SM) electroweak radiative corrections that enter various low-energy processes such as semileptonic decays of hadrons and parity-violating lepton-hadron scatterings. These are powerful tools in extractions of SM weak parameters. The precise calculations of such diagrams are, however, extremely difficult because they are sensitive to the loop momentum $q$ at all scales and include contributions from all possible virtual hadronic intermediate states which properties are governed by quantum chromodynamics (QCD) in its nonperturbative regime. Hence, they are one of the main sources of theoretical uncertainty in the extracted weak parameters such as the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements [@Hardy:2014qxa; @Hardy:2018zsb] and the weak mixing angle [@Kumar:2013yoa] at low scale. Modern treatments of the box diagrams are based on the pioneering work by Sirlin [@Sirlin:1977sv] in the late 1970s that separates the diagrams into “model-independent" and “model-dependent" terms, of which the former can be reduced to known quantities by means of current algebra. The model-dependent terms, on the other hand, consist of the interference between the electromagnetic and the axial weak currents, and are plagued with large hadronic uncertainties at $Q^2\lesssim1$ GeV$^2$. Earlier attempts to constrain these terms include varying the infrared cutoff [@Marciano:1982mm; @Marciano:1983ss; @Marciano:1985pd; @Bardin:2001ii] and the use of interpolating functions [@Marciano:2005ec], but all these methods suffer from nonimprovable theoretical uncertainties. The recent introduction of dispersion relations in treatments of the $\gamma Z$ [@Gorchtein:2008px; @Gorchtein:2011mz; @Blunden:2011rd; @Rislow:2013vta] and $\gamma W$ [@Seng:2018yzq; @Seng:2018qru; @Gorchtein:2018fxl] boxes provides a better starting point to the problem by expressing the loop integral in terms of parity-odd structure functions. Since the latter depend on on-shell intermediate hadronic states, one could in principle relate them to experimental data. Unfortunately, at the hadronic scale such data either do not exist or belong to a separate isospin channel which can only be related to our desired structure functions within a model. First-principles calculations of the parity-odd structure functions from lattice QCD have not yet been thoroughly investigated, and are expected to be challenging due to the existence of multihadron final states. Moreover, most of the recent developments in the lattice calculation of parton distribution functions (see, e.g. Ref. [@Lin:2017snn]) do not apply here because their applicability is restricted to large $Q^2$. But at the same time we also observe an encouraging development in the application of the Feynman-Hellmann theorem (FHT) [@Feynman:1939zza; @Hellmann], where external source terms are added to the Hamiltonian, and the required hadronic matrix elements of the source operator could be related to the energy shift of the corresponding hadron which is easier to obtain on lattice as it avoids the calculation of complicated (and potentially noisy) contraction diagrams. A nonzero momentum transfer $\vec{q}$ can also be introduced by adopting a periodic source term. Such a method shows great potential in the calculation of hadron electromagnetic form factors [@Chambers:2017tuf], Compton scattering amplitude [@Agadjanov:2016cjc; @Agadjanov:2018yxh], parity-even nucleon structure functions [@Chambers:2017dov], and hadron resonances [@RuizdeElvira:2017aet]. Furthermore, it does not involve any operator product expansion so its applicability is not restricted to large $Q^2$. Based on the developments above, we propose in this Letter a new method to study the $\gamma W$/$\gamma Z$ boxes, namely to compute a generalized parity-odd forward Compton tensor on the lattice through the second-order nucleon energy shift upon introducing two periodic source terms, and solve for the moments of $F_3$ through a dispersion relation. We will demonstrate that for a given $Q^2$, the calculation of a few energy shifts already provides sufficient information about the integrand of the box diagrams, and such a calculation is completely executable with the computational power in the current lattice community. When data are accumulated for sufficiently many values of $Q^2$ at the hadronic scale, one will eventually be able to remove the hadronic uncertainties in the electroweak boxes and provide a satisfactory solution to this long-lasting problem in precision physics. ![Direct and crossed box diagrams. Single and double lines represent a lepton and a hadron, respectively. The blob represents hadronic excitations and the wiggly lines denote the gauge bosons. \[fig:box\]](boxdiagram){width="0.7\linewidth"} We start by defining the electromagnetic current and the isovector axial current [(here we neglect the strange current just to simplify our discussions of the two examples below, but it is not a necessary approximation) ]{}: $$\begin{aligned} J_{em}^\mu&=&(2/3)\bar{u}\gamma^\mu u-(1/3)\bar{d}\gamma^\mu d\nonumber\\ J_A^\mu&=&\bar{u}\gamma^\mu\gamma_5 u-\bar{d}\gamma^\mu\gamma_5 d.\end{aligned}$$ The spin-independent, parity-odd nucleon structure function $F_3^N$ ($N=p,n$) can be defined through the hadronic tensor: $$\begin{aligned} W^{\mu\nu}_N(p,q)&=&\frac{1}{4\pi}\int d^4xe^{iq\cdot x}\left\langle N(\vec{p})\right|[J_{em}^\mu(x),J_A^\nu(0)]\left|N(\vec{p}) \right\rangle\nonumber\\ &=&-\frac{i\varepsilon^{\mu\nu\alpha\beta}q_\alpha p_\beta}{2p\cdot q}F_3^N(x_B,Q^2),\end{aligned}$$ where $x_B=Q^2/(2p\cdot q)$ is the Bjorken variable which lies between $-1$ and $1$, and $\varepsilon^{0123}=-1$. We stress that it is more natural to include negative values of $x_B$ because in a dispersion relation involving $F_3^N(x_B,Q^2)$, $x_B$ acts as the integration variable in the Cauchy integral that could lie on both the positive and the negative real axes. Notice that the spin label in the nucleon states are suppressed for simplicity. From that we may define the so-called “first Nachtmann moment" of $F_3^N$ as [@Nachtmann:1973mr; @Nachtmann:1974aj] $$M_1[F_3^{N}]=\int_0^1dx\Pi(x,Q^2)F_3^{N}(x,Q^2),$$ where $$\Pi(x,Q^2)=\frac{4}{3}\frac{1+2\sqrt{1+4m_N^2x^2/Q^2}}{(1+\sqrt{1+4m_N^2x^2/Q^2})^2}$$ and $m_N$ is the nucleon mass. To see the physical relevance of the definitions above, we shall briefly discuss some recent progress in the study of two weak processes that play central roles in low-energy precision tests of SM: First, superallowed nuclear $\beta$ decays represent the best avenues for the measurement of the CKM matrix element $V_{ud}$ as the corresponding weak nuclear matrix element is protected at tree level by the conserved vector current. With the inclusion of higher-order corrections one obtains [@Hardy:2014qxa]: $$|V_{ud}|^2=\frac{2984.432(3)\:\mathrm{s}}{\mathcal{F}t(1+\Delta_R^V)}$$ where $\mathcal{F}t$ is [the product between the half-life $t$ and the statistical function $f$, but]{} modified by nuclear-dependent corrections. $\Delta_R^V$ represents the nucleus-independent radiative correction. The main theoretical uncertainty of $|V_{ud}|$ comes from $\Delta_R^V$, which in turn acquires its largest uncertainty from the interference between the isosinglet electromagnetic current and the axial charged weak current in the $\gamma W$ box diagram. The latter can be expressed as: $$\left(\Delta_R^V\right)_{\gamma W}^{VA}=\int_0^\infty\frac{dQ^2}{Q^2}\frac{3\alpha}{\pi}\frac{M_W^2}{M_W^2+Q^2}M_1[F_3^{(0)}],$$ where $F_3^{(0)}=-(1/4)(F_3^p-F_3^n)$ through isospin symmetry. A recent determination of $\Delta_R^V$ based on a dispersion relation and neutrino scattering data gives 0.02467(22) [@Seng:2018yzq], which lies significantly above the previous sate-of-the-art result of 0.02361(38) [@Marciano:2005ec] and leads to an apparent violation of the first-row CKM unitarity at the level of 4$\sigma$ that calls for an immediate resolution. Besides, scrutinizing the problems in $V_{ud}$ will also lead to a better determination of $V_{us}$, because one of the main measuring channels of the latter, the $K\to\mu\nu(\gamma)$ decay, probes the ratio $|V_{us}|/|V_{ud}|$. Second, we look at parity-odd $ep$ scattering. The measurement of the proton weak charge $Q_W^p$ in the almost-forward elastic $ep$ scattering is a powerful probe of the physics beyond SM due to the accidental suppression of its tree-level value $1-4\sin^2\theta_W$, with $\theta_W$ the weak mixing angle. After including one-loop electroweak radiative corrections, the quantity reads [@Erler:2003yk] $$\begin{aligned} Q_W^p&=&(1+\Delta \rho+\Delta_e)[1-4\sin^2\theta_W(0)+\Delta_e']\nonumber\\ &&+\Box_{WW}+\Box_{ZZ}+\Box_{\gamma Z},\end{aligned}$$ among which $\Box_{\gamma Z}$ represents the contribution from the $\gamma Z$ box that bears the largest hadronic uncertainty. In the limit of vanishing beam energy, it takes the following form: $$\Box_{\gamma Z}=\int_0^\infty \frac{dQ^2}{Q^2}\frac{3\alpha}{2\pi}v_e\frac{M_Z^2}{M_Z^2+Q^2}M_1[F_3^{\gamma Z}],$$ where $v_e$ is the electron weak charge and $F_3^{\gamma Z}=-F_3^p$. [A recent]{} estimation of $\Box_{\gamma Z}$ reads $0.0044(4)$ [@Blunden:2011rd]. In view of the upcoming P2 experiment at the Mainz Energy-Recovering Superconducting Accelerator (MESA) that aims for the measurement of $\sin^2\theta_W$ to a precision of $0.15\%$ [@Becker:2018ggl], it is necessary for a revisit of the $\gamma Z$ box to proceed coherently with $\gamma W$ in order to ensure there is no unaccounted systematics as recently discovered in the latter [@Seng:2018yzq; @Seng:2018qru]. From the two examples above one sees that the object of interest is the first Nachtmann moment of $F_3^N$, which probes different on-shell intermediate states at different $Q^2$. The analysis of the data accumulated for an analogous parity-odd structure function $F_3^{WW}$ resulting from the interference between the vector and axial charged weak current in inclusive $\nu p/\bar{\nu}p$ scattering indicates that (1) at $Q^2< 0.1$ GeV$^2$ the first Nachtmann moment is saturated by the contribution from the elastic intermediate state and the lowest nucleon resonances [@Bolognese:1982zd] of which sufficient data are available, and (2) at $Q^2>2$ GeV$^2$ it is well described by a parton model with well-known perturbative QCD corrections [@Kataev:1994ty; @Kim:1998kia] [(see also, Sec. IV of Ref. [@Seng:2018qru] for a detailed description of the dominant physics that takes place at different $Q^2$)]{}. On the other hand, multihadron intermediate states dominate at $Q^2\lesssim 1$ GeV$^2$, and a first-principles theoretical description at this region is absent so far. Although there are attempts to relate, say, $F_3^{(0)}$ to the measured $F_3^{WW}$ in this region, such a relation is only established within a model because it belongs to different isospin channels. Therefore, the goal of this Letter is to outline a method that allows for a reliable first-principles calculation of $M_1[F_3^N]$ at $Q^2\lesssim 1$ GeV$^2$. To achieve this goal, we consider the following generalized forward Compton tensor: $$\begin{aligned} T_N^{\mu\nu}(p,q)&=&\int d^4xe^{iq\cdot x}\left\langle N(\vec{p})\right|T\left\{J_{em}^\mu(x)J_A^\nu(0)\right\} \left|N(\vec{p})\right\rangle\nonumber\\ &=&-\frac{i\varepsilon^{\mu\nu\alpha\beta}q_\alpha p_\beta}{2p\cdot q}T_3^N(\omega,Q^2),\label{eq:Compton}\end{aligned}$$ where $\omega=1/x_B=2p\cdot q/Q^2$, and time-reversal invariance requires $T_3^N(\omega,Q^2)$ to be an odd function of $\omega$. Unlike the structure function, here we do not require the intermediate states to stay on shell, so one could have $|\omega|<1$. A dispersion relation exists between $T_3^N$ and $F_3^N$: $$T_3^N(\omega,Q^2)=-4i\omega\int_0^1dx\frac{F_3^N(x,Q^2)}{1-\omega^2x^2}. \label{eq:dispersion}$$ Therefore, if one is able to compute $T_3^N(\omega,Q^2)$ at several points of $\omega$ below the elastic threshold, then one could extract useful information about the structure function $F_3^N$ through Eq. . Our approach is to make use of the second-order FHT that relates the second derivative of the nucleon energy upon the introduction of periodic source terms to $T_3^N$ below threshold. Let us first state our result here. We define the momentum transfer $q^\mu=(0,q_x,q_y,q_z)$ so that $Q^2=\vec{q}^2$ and $\omega=-2\vec{p}\cdot\vec{q}/\vec{q}^2$, and throughout this work we impose the off-shell condition, i.e. $|\omega|=2|\vec{p}\cdot\vec{q}|/\vec{q}^2<1$. We consider the addition of two external source terms to the ordinary QCD Hamiltonian (we choose $\mu=2$ and $\nu=3$ to be definite): [ $$\begin{aligned} H_\lambda(t)&=&H_0(t)+2\lambda_1\int d^3x\cos(\vec{q}\cdot\vec{x})J_{em}^2(\vec{x},t)\nonumber\\ &&-2\lambda_2\int d^3x\sin(\vec{q}\cdot\vec{x})J_A^3(\vec{x},t).\label{eq:Hlambda}\end{aligned}$$ ]{} The unperturbed nucleon energy with momentum $\vec{p}$ is simply $E_N(\vec{p})=\sqrt{m_N^2+\vec{p}^2}$. After the introduction of the external source terms, this energy becomes $E_{N,\lambda}(\vec{p})$. We remind the readers that, since the source terms break translational symmetry, the nucleon eigenstate with energy $E_{N,\lambda}(\vec{p})$ is no longer a momentum eigenstate. The second-order FHT states that: $$\left(\frac{\partial^2E_{N,\lambda}(\vec{p})}{\partial\lambda_1\partial\lambda_2}\right)_{\lambda=0}=\frac{iq_x}{Q^2\omega} T_3^N(\omega,Q^2).\label{eq:FHtheorem}$$ One could then express the amplitude $T_3^N$ in terms of the dispersion integral  to obtain $$\left(\frac{\partial^2E_{N,\lambda}(\vec{p})}{\partial\lambda_1\partial\lambda_2}\right)_{\lambda=0}=\frac{4q_x}{Q^2} \int_0^1dx\frac{F_3^N(x,Q^2)}{1-\omega^2x^2}~,\label{eq:central}$$ which is the central result of this Letter. For later convenience, we define the function $\Lambda(x,\omega)=1/(1-\omega^2x^2)$. Below we shall outline a proof of Eq.  based on the Euclidean path integral, which is closely connected to standard treatments in lattice QCD [@Bouchard:2016heu] [(we also refer interested readers to Ref. [@Agadjanov:2018yxh] that contains all details of an almost identical derivation for the case of the parity-even Compton amplitude)]{}. Throughout, Euclidean quantities will be labeled by a subscript $\mathbb{E}$. Also, if a quantity is supposed to be affected by the source terms but appears without a subscript $\lambda$, that implies its limit at $\lambda_1,\lambda_2\rightarrow 0$. First, the existence of extra source terms in Eq.  implies a shift of the Euclidean action: $$\begin{aligned} S_{\mathbb{E},\lambda}&=&S^0_\mathbb{E}+2\lambda_1\int d^4x_\mathbb{E}\cos(\vec{q}\cdot\vec{x})J_{em}^2(x_\mathbb{E})\nonumber\\ &&-2\lambda_2\int d^4x_\mathbb{E}\sin(\vec{q}\cdot\vec{x})J_A^3(x_\mathbb{E}).\end{aligned}$$ Next, we define a two-point correlation function: $$C_\lambda^N(\vec{p},t_\mathbb{E})=\int d^3xe^{-i\vec{p}\cdot\vec{x}}\left\langle\Omega_\lambda\right|T\{\chi_N(\vec{x},t_\mathbb{E})\chi_N^\dagger(0)\}\left|\Omega_\lambda\right\rangle,$$ with $t_\mathbb{E}>0$. Here, $\chi_N$ is an interpolating operator that possesses the same quantum numbers as the nucleon $N$. We remind the readers that a time-ordered correlation function [ of arbitrary operators $O_i$]{} with respect to the [vacuum state $\left|\Omega_\lambda\right\rangle$]{} can be expressed in terms of a Euclidean path integral: $$\begin{aligned} &&\left\langle\Omega_\lambda\right|T\{O_1(t_{1\mathbb{E}})...O_n(t_{n\mathbb{E}})\}\left|\Omega_\lambda\right\rangle\nonumber\\ &=&\frac{1}{Z_{\mathbb{E},\lambda}}\int D\phi\:O_1(t_{1\mathbb{E}})...O_n(t_{n\mathbb{E}})e^{-S_{\mathbb{E},\lambda}},\end{aligned}$$ [with $Z_{\mathbb{E},\lambda}$ the Euclidean partition function.]{} Based on the asymptotic behavior of $C_\lambda^N$, we define an “effective energy": $$E^{\rm eff}_{N,\lambda}(\vec{p};t_\mathbb{E},\tau_\mathbb{E})=\frac{1}{\tau_\mathbb{E}}\ln\left( \frac{C_\lambda^N(\vec{p},t_\mathbb{E})}{C_\lambda^N(\vec{p},t_\mathbb{E}+\tau_\mathbb{E})}\right),$$ that reduces to the nucleon energy in the large-time limit (which is only true when $|\omega|<1$): $$\lim_{t_\mathbb{E}\rightarrow \infty} E^{\rm eff}_{N,\lambda}(\vec{p};t_\mathbb{E},\tau_\mathbb{E})=E_{N,\lambda}(\vec{p}).$$ Therefore, one may obtain the partial derivatives of $E_{N,\lambda}(\vec{p})$ with respect to $\lambda_i$ through the partial derivatives of $E_{N,\lambda}^{\rm eff}$. An advantage in doing so is that one could see explicitly that the “vacuum matrix elements", i.e. terms with $\partial Z_{\mathbb{E},\lambda}/\partial\lambda_i$, do not contribute. We find that the first derivative vanishes: $$\left(\frac{\partial E_{N,\lambda}(\vec{p})}{\partial\lambda_i}\right)_{\lambda=0}=\lim_{t_\mathbb{E}\rightarrow 0}\left(\frac{\partial E_{N,\lambda}^{\rm eff}(\vec{p};t_\mathbb{E},\tau_\mathbb{E})}{\partial\lambda_i}\right)_{\lambda=0}=0.$$ The underlying reason is simple: the external source terms induce a momentum shift of $\pm \vec{q}$ upon each insertion; therefore, according to usual perturbation theory, the linear energy shift is proportional to $ \left\langle\vec{p}\right|\left.\vec{p}\pm\vec{q}\right\rangle =0$ for $\vec{q}\neq 0$. We are interested in the second derivative of $E_{N,\lambda}(\vec{p})$ which reads $$\left(\frac{\partial^2E_{N,\lambda}(\vec{p})}{\partial\lambda_1\partial\lambda_2}\right)_{\lambda=0}\!\!\!\!\!\!=\lim_{t_\mathbb{E}\rightarrow \infty} \frac{1}{\tau_\mathbb{E}}\left[\frac{R(\vec{p},\vec{q},t_\mathbb{E})}{C^N(\vec{p},t_\mathbb{E})}-(t_\mathbb{E}\rightarrow t_\mathbb{E} +\tau_\mathbb{E})\right]$$ where $$\begin{aligned} &&R(\vec{p},\vec{q},t_\mathbb{E})=\int d^3xe^{-i\vec{p}\cdot\vec{x}}\times\nonumber\\ &&\!\!\!\!\!\!\!\!\left\langle\Omega\right|T\left\{\chi_N(\vec{x},t_\mathbb{E})\chi_N^\dagger(0)\left(\frac{\partial S_{\mathbb{E},\lambda}}{\partial\lambda_1}\right)\left(\frac{\partial S_{\mathbb{E},\lambda}}{\partial\lambda_2}\right)\right\} \left|\Omega\right\rangle~.\end{aligned}$$ One then splits the time-ordered product in $R(\vec{p},\vec{q},t_\mathbb{E})$ into different time regions, and finds that at large $t_\mathbb{E}$ the dominant piece is the one with the two currents sandwiched between $\chi_N$ and $\chi_N^\dagger$. We may then insert two complete sets of states between the interpolating operators and the current product, and since the off-shell condition ensures $E_N(\vec{p}\pm 2\vec{q})>E_N(\vec{p})$, we find that the dominant piece consists of a time-ordered nucleon matrix element with the same momentum $\vec{p}$ in the initial and final states. We therefore isolate this piece and make use of the following identity: $$\begin{gathered} \int_0^{t_\mathbb{E}}dy^4_\mathbb{E}\int_0^{t_\mathbb{E}}dz^4_\mathbb{E}\left\langle N(\vec{p})\right|T\{J_{em}^2(\vec{y},y_\mathbb{E}^4-z_\mathbb{E}^4)J_A^3(0)\} \left|N(\vec{p})\right\rangle\\ \rightarrow t_\mathbb{E}\int_{-\infty}^{\infty}dy^4_\mathbb{E}\left\langle N(\vec{p})\right|T\{J_{em}^2(\vec{y},y_\mathbb{E}^4)J_A^3(0)\} \left|N(\vec{p})\right\rangle\end{gathered}$$ to obtain: $$\begin{aligned} &&E_N(\vec{p})\left(\frac{\partial^2E_{N,\lambda}(\vec{p})}{\partial\lambda_1\partial\lambda_2}\right)_{\lambda=0}=-\int d^4y_\mathbb{E} \sin(\vec{q}\cdot\vec{y})\times\nonumber\\ &&\left\langle N(\vec{p})\right|T\{J_{em}^2(y_\mathbb{E})J_A^3(0)\}\left|N(\vec{p})\right\rangle.\end{aligned}$$ We can now switch back to the Minkowskian space time through a Wick rotation: $\int d^4y_\mathbb{E}\to i\int d^4y $. Finally, we substitute the result into Eq.  and make use of crossing symmetry $T_3(-\omega,Q^2)=-T_3(\omega,Q^2)$ to arrive at Eq. . This completes the proof. Now let us discuss the practical use of Eq. . Ideally, it allows for a reconstruction of the full structure function $F_3^N(x,Q^2)$ by calculating the second-order energy shift at $n\gg 1$ discrete points of $\omega$: we simply discretize the dispersion integral to obtain a matrix equation: $$\left(\frac{\partial^2E_{N,\lambda}(\vec{p})}{\partial\lambda_1\partial\lambda_2}\right)_{\lambda=0}(\omega_i,Q^2)\approx \sum_{j=1}^n A_{\omega_i,x_j}F_3^N(x_j,Q^2),\label{eq:matrixeq}$$ and notice that the matrix $A$ does not possess any singularity with $\omega$ below the elastic threshold. We may then invert $A$ to obtain $F_3^N(x,Q^2)$ at the discrete points $\{x_j\}$. However, such an approach is accurate only when $n$ is large, which is difficult to achieve with the current lattice computational power when $Q^2\lesssim 1$ GeV$^2$. To see this, one first recalls that any momentum in a finite lattice can only take discrete values: $\vec{k}=(2\pi/L)(n_{kx},n_{ky},n_{kz})$, with $L$ is the spatial lattice size and $\{n_{kx},n_{ky},n_{kz}\}$ are integers. The requirements that $Q^2=\vec{q}^2\lesssim 1$ GeV$^2$ and $|\omega|=2|\vec{p}\cdot\vec{q}|/\vec{q}^2<1$ imply two conditions: $$\begin{aligned} \frac{4\pi^2}{L^2}(n_{qx}^2+n_{qy}^2+n_{qz}^2)&\lesssim &1\:\mathrm{GeV}^2\\ \frac{2|n_{px}n_{qx}+n_{py}n_{qy}+n_{pz}n_{qz}|}{n_{qx}^2+n_{qy}^2+n_{qz}^2}&<&1.\label{eq:condition2}\end{aligned}$$ In particular, with a fixed choice of $\vec{q}$, the second condition determines the allowed discrete values of $\omega$ at which the nucleon energy can be extracted on lattice. To understand how low in $Q^2$ one can probe, we consider a typical lattice setup: the configuration cA2.09.48 from the ETM Collaboration that features a spatial lattice size of $48\times 0.0931~\mathrm{fm}\approx 4.47~\mathrm{fm}$ [@Liu:2016cba]. For such a configuration, we get $Q^2\approx 0.38$ GeV$^2$ with the choice $\vec{q}=(2\pi/L)(2,1,0)$, but Eq.  restricts the number of allowed $|\omega|$ to three: 0, 2/5, and 4/5. Such a small amount is obviously insufficient to perform the matrix inversion of Eq.  to any satisfactory level of accuracy. Fortunately, in studies of the electroweak boxes we do not need the full $F_3^N(x,Q^2)$ as a function of $x$, but rather its first Nachtmann moment. Therefore, the real question is whether one could form a linear combination of the functions $\{\Lambda(x,\omega_i)\}$ that appear in the dispersion integral  with all allowed values of $\omega_i$ to approximate the function $\Pi(x,Q^2)$ to a satisfactory level, especially at small $x$ [(because apart from the known, isolated elastic contribution at $x=1$, $F_3^N(x,Q^2)$ is non-zero only at $x<x_\pi=Q^2/(2m_NM_\pi+M_\pi^2+Q^2)$, with $M_\pi$ the pion mass)]{}. As a proof of principle, let us still consider the example above. We define the following linear combination: $$\Lambda_\mathrm{tot}(x)=a\Lambda(x,0)+b\Lambda(x,2/5)+c\Lambda(x,4/5),\label{eq:Lambdatot}$$ and fit the parameters $\{a,b,c\}$ to match $\Pi(x,Q^2)$ at $Q^2\approx 0.38$ GeV$^2$. We find that they come to a good agreement at $x<0.9$ with the choice $a=7.82446$, $b=-7.58605$ and $c=0.734787$, as shown in Fig. \[fig:fit\]. That means we could obtain a very good approximation to $M_1[F_3^N]$ by adding the values of $(Q^2/4q_x)\times(\partial_{\lambda_1}\partial_{\lambda_2}E_{N,\lambda})_{\lambda=0}$ calculated at $\omega=0,2/5,4/5$ with the weighting coefficients $\{a,b,c\}$ respectively. We shall also discuss the efficiency of this procedure for different values of $Q^2$: with the same $L$, at larger $Q^2$ one has more available values of $\omega$ and the global fitting to $\Pi(x,Q^2)$ will be even better; this is encouraging because $Q^2>0.38$ GeV$^2$ already fully covers the so-called “intermediate distances" in Ref. [@Marciano:2005ec] that contain most of the hadronic uncertainties. On the other hand, at smaller $Q^2$ (such as $Q^2$=0.1 GeV$^2$) the allowed values of $\omega$ are less so one is not able to reproduce $M_1[F_3^N]$ with the same accuracy. The readers, however, should not be discouraged because (1) $\omega=0$ is always an accessible point, which gives the first [*Mellin*]{} moment of $F_3^N$ according to Eq. . This will provide important constraints for model parameterizations of the residual multihadron contributions to $F_3^N(x,Q^2)$ at small $Q^2$, and (2) future efforts in the increase of the lattice size (see, e.g. Ref. [@Luscher:2017cjh]) will then allow for a precise calculation of $M_1[F_3^N]$ at smaller $Q^2$ with our proposed method. ![Comparison between the function $\Pi(x,Q^2)$ at $Q^2=0.38$ GeV$^2$ (blue solid line) and $\Lambda_\mathrm{tot}(x)$ (red dashed line) defined in Eq. . \[fig:fit\]](fit){width="0.8\linewidth"} We shall end by commenting on the required level of precision for lattice calculations. We take the $\gamma W$ box as an example: in Ref. [@Seng:2018qru], the contribution from multiparticle intermediate states at [$Q^2\sim 1$ GeV$^2$]{} to $\Delta_R^V$ is estimated to be $(\alpha/\pi)\times(0.48\pm0.07)$ through a simple Regge-exchange model, with a $\sim 15\%$ error coming from the $\nu p/\bar{\nu}p$ scattering data. Possible systematic errors due to the simplicity of the model itself are not accounted for. In this sense, a successful lattice calculation of the second-order nucleon energy shift at a few points of $\omega$ with a precision level of $15\%$ will already be able to match the [*[precision]{}*]{} of the model and start to challenge its [*[accuracy]{}*]{}. [This is completely executable with current lattice techniques as a similar calculation for parity-even structure functions has already been performed in Ref. [@Chambers:2017dov] with a 10% overall projected error]{}. Also, by employing a larger $L$ one is able to probe smaller $Q^2$, and when sufficiently many points of $M_1[F_3^N]$ between $0.1$ GeV$^2<Q^2<1$ GeV$^2$ are determined, one will be able to reduce the hadronic uncertainties in the $\gamma W$ and $\gamma Z$ boxes to a level compatible with current and future precision experiments. [**Acknowledgements**]{} – The authors thank Akaki Rusetsky, Gerrit Schierholz and Mikhail Gorchtein for inspiring discussions. This work is supported in part by the DFG (Grant No. TRR110) and the NSFC (Grant No. 11621131001) through the funds provided to the Sino-German CRC 110 “Symmetries and the Emergence of Structure in QCD", by the Alexander von Humboldt Foundation through a Humboldt Research Fellowship (CYS), by the Chinese Academy of Sciences (CAS) President’s International Fellowship Initiative (PIFI) (grant no. 2018DM0034) (UGM) and by VolkswagenStiftung (grant no. 93562) (UGM). [31]{} natexlab\#1[\#1]{} bibnamefont \#1[\#1]{} bibfnamefont \#1[\#1]{} citenamefont \#1[\#1]{} url \#1[`#1`]{} urlprefix \[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ****, (), . , in ** (), . , , , , ****, (), . , ****, (), . , ****, (). , ****, (), . , ****, (). , , , , ****, (), . , ****, (), . , ****, (), . , , , ****, (), . , , , ****, (), . , ****, (), . , , , , ****, (), . , , (), . (), . , ****, (), . R. P. Feynman, Phys. Rev.  [**56**]{}, 340 (1939). H. Hellmann, [*Einführung in die Quantenchemie,*]{} Deuticke, Leipzig und Wien (1937). (), ****, (), . , , , ****, (), . , , , ****, (), . , , , , , , , , , , ****, (), . J. Ruiz de Elvira, U.-G. Mei[ß]{}ner, A. Rusetsky and G. Schierholz, Eur. Phys. J. C [**77**]{}, no. 10, 659 (2017). , ****, (). , ****, (). , , , ****, (), . , ****, (), . , , , , , (), ****, (). , in ** (), pp. , , <https://inspirehep.net/record/37980/files/arXiv:hep-ph_9405254.pdf>. , ****, (), . , , , , , ****, (), . L. Liu [*et al.*]{}, Phys. Rev. D [**96**]{} (2017) no.5, 054516. M. Lüscher, EPJ Web Conf.  [**175**]{} (2018) 01002.

--- author: - | M.I. Ostrovskii\ Department of Mathematics and Computer Science\ St. John’s University\ 8000 Utopia Parkway\ Queens, NY 11439\ USA\ e-mail: [[email protected]]{} - | \ V.S. Shulman\ Department of Mathematics\ Vologda State Technical University\ 15 Lenina street\ Vologda 160000\ RUSSIA\ e-mail: [shulman\[email protected]]{} - | \ L. Turowska\ Department of Mathematical Sciences\ Chalmers University of Technology and University of Gothenburg\ SE-41296, Gothenburg\ SWEDEN\ e-mail: [[email protected]]{} title: Unitarizable representations and fixed points of groups of biholomorphic transformations of operator balls --- [**Abstract.**]{} We show that the open unit ball of the space of operators from a finite dimensional Hilbert space into a separable Hilbert space (we call it “operator ball”) has a restricted form of normal structure if we endow it with a hyperbolic metric (which is an analogue of the standard hyperbolic metric on the unit disc in the complex plane). We use this result to get a fixed point theorem for groups of biholomorphic automorphisms of the operator ball. The fixed point theorem is used to show that a bounded representation in a separable Hilbert space which has an invariant indefinite quadratic form with finitely many negative squares is unitarizable (equivalent to a unitary representation). We apply this result to find dual pairs of invariant subspaces in Pontryagin spaces. In the appendix we present results of Itai Shafrir about hyperbolic metrics on the operator ball. [**Keywords.**]{} Hilbert space, bounded representation, unitary representation, hyperbolic space, fixed point, normal structure, biholomorphic transformation, indefinite quadratic form [**2000 Mathematics Subject Classification:**]{} 47H10, 47B50, 22D10 Introduction {#S:Introduction} ============ Let $K,H$ be Hilbert spaces; by $L(K,H)$ we denote the Banach space of all linear bounded operators from $K$ to $H$. We will denote the open unit ball of $L(K,H)$ by ${\mathcal{B}}$ and call it [*operator ball*]{}. We say that a subset $M$ of ${\mathcal{B}}$ is [*separated from the boundary*]{} if it is contained in a ball $r{\mathcal{B}}$, for some $r\in [0, 1)$. A group $G$ of transformations of ${\mathcal{B}}$ will be called [*elliptic*]{} if all its orbits are separated from the boundary (this terminology goes back to [@Helt]). We call $G$ [*equicontinuous*]{} if, for each $\varepsilon>0$ there is $\delta>0$ such that if $A,B\in {\mathcal{B}}$ and $\|A-B\|< \delta$ then $\|g(A)-g(B)\|< \varepsilon$ for all $g\in G$. This condition can be also called [*global equicontinuity*]{} because it is possible also to consider equicontinuity in a point. Since ${\mathcal{B}}$ is a bounded open set of a Banach space, one may consider holomorphic maps from ${\mathcal{B}}$ to Banach spaces. We will deal with invertible holomorphic maps from ${\mathcal{B}}$ onto ${\mathcal{B}}$; such maps are called [*biholomorphic automorphisms*]{} of ${\mathcal{B}}$. Our aim is to prove that if one of the spaces $K, H$ is finite-dimensional and the other is separable, then any elliptic group of biholomorphic automorphisms of ${\mathcal{B}}$ has a common fixed point. More precisely we will prove the following result. \[main\] Let $\dim K < \infty$ and $H$ be separable. For a group $G$ of biholomorphic automorphisms of ${\mathcal{B}}$, the following statements are equivalent: [(i)]{} $G$ is elliptic on ${\mathcal{B}}$; [(ii)]{} at least one orbit of $G$ is separated from the boundary; [(iii)]{} $G$ is equicontinuous; [(iv)]{} $G$ has a common fixed point in ${\mathcal{B}}$. The assumption $\dim K < \infty$ is essential, some of the results of this paper are known to fail without it, see, for example, the last paragraph of Section \[S:orthog\]. As for separability of $H$, it is just a technical convenience, our approach works for non-separable $H$ also, with a bit more complicated proofs. The result will be applied to the orthogonalization (or similarity) problem for bounded group representations on Hilbert spaces. This problem can be formulated as follows. Let $\pi$ be a representation of a group $G$ on a Hilbert space ${\mathcal{H}}$. Under which conditions there is an invertible operator $V$ such that the representation $\sigma$ of $G$, defined by the formula $\sigma(g) = V\pi(g)V^{-1}$, is unitary? Clearly a necessary condition is the boundedness of $\pi$: $\sup_{g\in G}\|\pi(g)\|< \infty$. In general it is not sufficient. Some sufficient conditions (on $G$ or $\pi$) are known, see the book [@Pis01]. We will show that a bounded representation $\pi$ of a group $G$ on a Hilbert space ${\mathcal{H}}$ is similar to a unitary representation if it preserves a quadratic form $\eta$ with finite number of negative squares. The last condition means that $\eta(x) = \|Px\|^2 - \|Qx\|^2$ and $P,Q$ are orthogonal projections in ${\mathcal{H}}$ with $P+Q = 1$ and $\dim(Q{\mathcal{H}}) < \infty$. As a consequence we obtain that each bounded group of $J$-unitary operators on a Pontryagin space $\Pi_k$ has an invariant dual pair of subspaces. In other words the space can be decomposed into $J$-orthogonal direct sum $H_+ + H_-$ of positive and negative subspaces which are invariant for all operators in the group. The proof of Theorem \[main\] is based on the analysis of the structure of the operator ball as a metric space with respect to the Carathéodory distance (see Chapters 4 and 5 of [@Vesent]). It was proved by Shafrir [@Sha] that ${\mathcal{B}}$ is a hyperbolic space with respect to this distance. Since [@Sha] is not easily accessible, we present a proof of this result in an appendix to our paper, with the kind permission of the author. We will show that ${\mathcal{B}}$ has a restricted form of a [*normal structure*]{} if $\dim(K) < \infty$. In the case where $K$ is one-dimensional Theorem \[main\] was obtained in [@Shul-80]; a transparent proof can be found in [@Kiss-Shul Section 23]. Hyperbolic spaces ================= In our definition of hyperbolic spaces we follow fixed point theory literature (see e.g. [@RS90], [@RZ01]). In geometric literature (see e.g. [@BH99]) hyperbolic spaces are defined differently. By a [*line*]{} in a metric space $({\mathcal{X}},\rho)$ we mean a subset of ${\mathcal{X}}$ which is isometric to the real line $\mathbb{R}$ with its usual metric (in the literature lines are also called [*metric lines*]{} or [*geodesic lines*]{}). Let $({\mathcal{X}},\rho)$ be a metric space with a distinguished set ${\mathcal{M}}$ of lines. We say that ${\mathcal{X}}$ is a [*hyperbolic space*]{} if the following conditions are satisfied: [**(1)**]{} (Uniqueness of a distinguished line through a given pair of points) For each $x,y\in {\mathcal{X}}$, there is exactly one line $\ell\in{\mathcal{M}}$ containing both $x$ and $y$. [**(2)**]{} (Convexity of the metric) To state the condition (see ) we need to introduce some more definitions and notation. The [*segment*]{} $[x,y]$ is defined as the part of the line $\ell\in{\mathcal{M}}$ containing both $x$ and $y$, consisting of all $z\in\ell$ satisfying $$\label{otrezok} \rho(x,y) = \rho(x,z)+\rho(z,y).$$ We write $$\label{otr1} z=(1-t)x\oplus ty$$ if $z\in[x,y]$, $\rho(z,x) = t\rho(x,y)$, and $\rho(z,y) = (1-t)\rho(x,y)$ (where $t\in[0,1]$). The convexity condition is: $$\label{hyperb1} \rho\left(\frac{1}{2}x\oplus\frac{1}{2}y,\frac{1}{2}x\oplus\frac{1}{2}z\right)\le \frac{1}{2}\rho(y,z).$$ Hyperbolic spaces satisfy also the following stronger form of the condition (\[hyperb1\]): $$\label{hyperb2} \rho((1-t)x\oplus ty,(1-t)w\oplus tz)\le (1-t)\rho(x,w)+t\rho(y,z).$$ (To get from we observe that, if for some value of $t$ we have the inequalities $\rho((1-t)x\oplus ty,(1-t)x\oplus tz)\le t\rho(y,z)$ and $\rho((1-t)x\oplus tz,(1-t)w\oplus tz)\le (1-t)\rho(x,w)$, then, by the triangle inequality, we have for that value of $t$. Using this observation repeatedly we prove the inequalities from this paragraph for $t$ of the form $\frac{k}{2^n}$ $(k\in\mathbb{N}, 1\le k\le 2^n)$. Then we use continuity.) A subset $C{\subset}{\mathcal{X}}$ is called [*convex*]{} if $x,y\in C$ implies $[x,y]{\subset}C$. Sometimes we say [*$\rho$-convex*]{} instead of convex, to avoid confusion with other natural notions of convexity for the same set. We use the notation $E_{a,r}$ for $\{x\in {\mathcal{X}}:~\rho(a,x)\le r\}$ and call such sets [*closed balls*]{}. The condition implies that in a hyperbolic space all closed balls are convex. Normal structure ================ Let $M$ be a subset in a metric space $({\mathcal{X}},\rho)$. The [*diameter*]{} of $M$ is defined by $$\label{diam} \diam M = \sup\{\rho(x,y): x,y\in M\}.$$ A point $a\in M$ is called [*diametral*]{} if $$\sup\{\rho(a,x):x\in M\} = \diam M.$$ A hyperbolic space ${\mathcal{X}}$ is said to have [*normal structure*]{} if every convex bounded subset of ${\mathcal{X}}$ with more than one element has a non-diametral point. This notion goes back to Brodskii and Milman [@BM48] who proved that uniformly convex Banach spaces (which are hyperbolic spaces) have normal structure. Takahashi [@Tak] introduced and studied normal structure in somewhat more general context. See [@BL00 Chapter 3] for a nice account on those aspects of fixed point theory which are related to the geometry of Banach spaces. \[perif\] Let $M$ be a separable bounded convex subset of a hyperbolic space ${\mathcal{X}}$ and $\alpha$ be the diameter of $M$. If all points of $M$ are diametral then $M$ contains a sequence $\{a_n\}$ with the property: $\lim_{n\to \infty} \rho(a_n,x) = \alpha$ for each $x\in M$. Let $\{c_n\}$ be a dense sequence in $M$. We define a sequence $\{b_n\}$ of “centers of mass” by the following rule: $b_1 = c_1$, $b_{n+1} = \frac{n}{n+1}b_n\oplus \frac{1}{n+1}c_{n+1}$. By convexity of $\rho$ we have $$\label{mean} \rho(x,b_n) \le \frac{1}{n}\sum_{k=1}^n \rho(x,c_k)$$ for all $n\in \mathbb{N}$. Indeed for $n=1$ this is obvious. If it is true for some $n$, then $\rho(x,b_{n+1})\le \frac{1}{n+1}\rho(x,c_{n+1}) + \frac{n}{n+1}\rho(x,b_n) \le \frac{1}{n+1}\rho(x,c_{n+1}) + \frac{n}{n+1}\frac{1}{n}\sum_{k=1}^n \rho(x,c_k) = \frac{1}{n+1}\sum_{k=1}^{n+1} \rho(x,c_k)$. By convexity of $M$ we have $b_n\in M$ for each $n\in\mathbb{N}$. Our assumption implies that $b_n$ is diametral, hence there is a point $a_n\in M$ with $\rho(b_n,a_n) \ge (1- \frac{1}{n^2})\alpha$. It follows that $(1- \frac{1}{n^2})\alpha \le \frac{1}{n}\sum_{k=1}^n \rho(a_n,c_k)$. If $\rho(a_n,c_j) < (1-\frac{1}{n})\alpha$, for some $j\le n$, then $\frac{1}{n}\sum_{k=1}^n \rho(a_n,c_k)< \frac1n(1-\frac{1}{n})\alpha +\frac{n-1}n\alpha=(1- \frac{1}{n^2})\alpha$, a contradiction. Hence $\rho(a_n,c_j) \ge (1-\frac{1}{n})\alpha$ for $j\le n$. This shows that $\lim_{n\to\infty}\rho(a_n,c_j)=\alpha$ for each fixed $j$. Since the sequence $\{c_j\}$ is dense in $M$, the lemma is proved. The invariant distance in the operator ball {#S:distance} =========================================== Recall that $K,H$ denote Hilbert spaces and ${\mathcal{B}}$ is the open unit ball of $L(K,H)$. For $A,X\in {\mathcal{B}}$ set $$\label{mobius} M_A(X) = (1-AA^*)^{-1/2}(A+X)(1+A^*X)^{-1}(1-A^*A)^{1/2}.$$ Clearly all $M_A$ are holomorphic on ${\mathcal{B}}$. They are called [*the Möbius transformations*]{}. It can be proved that $M_A^{-1} = M_{-A}$ (see [@harris], Theorem 2). Hence each Möbius transformation is a biholomorphic automorphism of ${\mathcal{B}}$. Since $M_A(0) = A$ the group of all biholomorphic automorphisms is transitive on ${\mathcal{B}}$. We set $$\label{dist0} \rho(A,B) = \tanh^{-1}( ||M_{-A}(B)||).$$ It is easy to see that $\rho$ coincides with the Carathéodory distance $c_{{\mathcal{B}}}$ in ${\mathcal{B}}$. Indeed, by [@Vesent Theorem 4.1.8], $c_{{\mathcal{B}}}(0,B) = \tanh^{-1}(\|B\|)$ (this holds for the unit ball of every Banach space). Since $c_{{\mathcal{B}}}$ is invariant and $M_A$ sends $A$ to $0$ we get: $$\label{dist} c_{{\mathcal{B}}}(A,B) = \tanh^{-1} ||M_{-A}(B)|| = \rho(A,B).$$ Hence $\rho$ is invariant with respect to biholomorphic automorphisms. I. Shafrir [@Sha] proved that the space $({\mathcal{B}},\rho)$ is hyperbolic. We present a proof of this result in the appendix. A set in ${\mathcal{B}}$ is called [*bounded*]{} if it is contained in some $\rho$-ball, or equivalently in a multiple $r{\mathcal{B}}$ of the operator ball with $r<1$. So a set is bounded if and only if it is separated from the boundary of ${\mathcal{B}}$ in the sense of Section \[S:Introduction\]. The following lemma is a special case of a more general result proved in [@Vesent Theorem IV.2.2]. \[twomet\] On any bounded set the hyperbolic metrics is equivalent to the operator norm. [[WOT]{}]{}-topology ==================== As before, let ${\mathcal{B}}$ be the unit ball of the space of operators from $K$ to $H$. We suppose that $K$ is finite-dimensional, $\dim K = n$, and that $H$ is separable. We consider biholomorphic maps on ${\mathcal{B}}$. By [[WOT]{}]{}  we denote the weak operator topology (see [@DS58 p. 476]). Because of the separability, the restriction of this topology to ${\mathcal{B}}$ is metrizable, so in our arguments we may consider only sequences, not nets. \[WOT\] If $K$ is finite-dimensional and $H$ is separable then all biholomorphic maps of ${\mathcal{B}}$ are ${{\rm WOT}}$-continuous. Let us firstly show that all Möbius transforms $M_B$ are ${{\rm WOT}}$-continuous (this was noticed and used already in the paper of Krein [@Krein]). Indeed let $B\in {\mathcal{B}}$ be fixed, then the map $\varphi: X\mapsto 1+B^*X$ from $({\mathcal{B}},{{\rm WOT}})$ to $(L(K,K),{{\rm WOT}})$. Moreover, since $K$ is finite-dimensional, $\varphi$ remains continuous if instead of [[WOT]{}]{}  we endow $L(K,K)$ with its norm topology. The map $T\to T^{-1}$ is norm continuous on the group of invertible operators on $K$. Hence the map $\psi: X\mapsto (1+B^*X)^{-1}$ is continuous from $({\mathcal{B}},{{\rm WOT}})$ to $L(K,K)$ with its norm topology. It follows that the map $\omega: X\to (X+B)(1+B^*X)^{-1}$ is continuous from $({\mathcal{B}}, {{\rm WOT}})$ to $({\mathcal{B}}, {{\rm WOT}})$. Indeed, if $X_n\to X$, then $\omega(X_n)-\omega(X) = (X_n +B)(\psi(X_n)-\psi(X)) + (X_n - X)\psi(X)$, where $\psi$ was defined above. The first summand tends to zero in norm while the second one tends to zero in ${{\rm WOT}}$. By a result of Harris [@Har], if a biholomorphic map of ${\mathcal{B}}$ preserves the point $0$, then it coincides with the restriction to ${\mathcal{B}}$ of an isometric linear map $h: L(K,H)\to L(K,H)$. Since $K$ is finite-dimensional, the [[WOT]{}]{}-topology on $L(K,H)$ coincides with the weak topology (indeed $L(K,H)$ is linearly homeomorphic to the direct sum of $n$ copies of $H$); since any bounded linear map is weakly continuous, $h$ is [[WOT]{}]{}-continuous. On the other hand, if $\varphi$ is a biholomorphic map of ${\mathcal{B}}$ and $A = \varphi(0)$ then $\psi = M_{-A}\circ \varphi$ is a biholomorphic map preserving $0$. Hence $\psi$ is an isometric linear map and $\varphi = M_{-A}^{-1}\circ \psi = M_A\circ \psi$ is a composition of two [[WOT]{}]{}-continuous maps. Thus $\varphi$ is [[WOT]{}]{}-continuous. \[loccompact\] If $\dim K < \infty$ and $H$ is separable, then each ball $E_{A,r}$ is [[WOT]{}]{}-compact. Since there is a Möbius transform that maps $E_{A,r}$ onto $E_{0,r}$, and since all Möbius transforms are [[WOT]{}]{}-continuous, it suffices to consider the case $A = 0$. But $E_{0,r}$ is a usual closed operator ball; its [[WOT]{}]{}-compactness follows from the Banach-Alaoglu theorem. Restricted normal structure of ${\mathcal{B}}$ ============================================== The purpose of this section is to show that in the case when $\dim K <\infty$ and $H$ is separable, the (open) operator ball ${\mathcal{B}}$ with the metric has a restricted form of normal structure in the sense that [[WOT]{}]{}-compact $\rho$-convex subsets in it have non-diametral points. As we already mentioned ${\mathcal{B}}$ with the metric is a hyperbolic space (see Section \[S:appendix\]). Our assumptions on $K$ and $H$ imply that ${\mathcal{B}}$ is separable in the norm-topology and hence, by Lemma \[twomet\], with respect to $\rho$. \[T:non-diam\] Let $K$ be finite dimensional and $H$ be separable. Let $M$ be a weakly compact, $\rho$-convex subset of ${\mathcal{B}}$ endowed with its hyperbolic metric. If $M$ is not a singleton, then $M$ contains a non-diametral point. Let $\alpha=\diam M>0$. Assume the contrary, that is, all points in $M$ are diametral. By Lemma \[perif\], there is a sequence $\{A_n\}$ in $M$ such that $\lim_{n\to \infty} \rho(A_n,X) = \alpha$ for each $X\in M$. Since $M$ is weakly compact, the sequence $\{A_n\}_{n=1}^\infty$ contains a weakly convergent subsequence. Let $W$ be its limit, we have $W\in M$ (since $M$ is weakly compact). Throughout this proof we will not change our notation after passing to a subsequence. Since $W\in M$ we get $$\label{E:limrho}\lim_{n\to\infty}\rho(W,A_n)=\alpha.$$ We will get a contradiction by proving $$\label{E:gamma}\sup_{n,m}\rho(A_n,A_m)>\alpha.$$ We may assume without loss of generality that $W=0$ (we can consider a Möbius transformation which maps $W$ to $0$, it is a $\rho$-isometry and weak homeomorphism). Let $\beta=\tanh\alpha$. Then leads to $\lim_{n\to\infty}||A_n||=\beta$ and it suffices to show that $$\sup_{n,m}||M_{A_m}(-A_n)||>\beta.$$ Since $K$ is finite dimensional and $A_n\in L(K,H)$, we can select a strongly convergent subsequence in the sequence $\{A_n^*A_n\}$. Assume that $A_n^*A_n\to P$, where $P\in L(K,K)$. It is clear that $P\ge 0$ and $\|P\| = \beta^2$. Choose $\varepsilon> 0$ and fix a number $m$ with $\|A_m^*A_m - P\|<\varepsilon $. For brevity, denote $A_m^*A_m$ by $Q$. We prove that $\lim_{n\to \infty}\|M_{A_m}(-A_n)\| > \beta$ if $\varepsilon>0$ is small enough. By the definition, $$\label{mobius2} M_{A_m}(-A_n) = (1-A_mA_m^*)^{-1/2}(A_m-A_n)(1-A_m^*A_n)^{-1}(1-A_m^*A_m)^{1/2}.$$ Since $A_m^*$ is of finite rank $A_m^*A_n \to 0$ in the norm topology. Hence $\lim_{n\to \infty}\|M_{A_m}(-A_n)\| = \lim_{n\to \infty}\|T_n\|$ where $$T_n = (1-A_mA_m^*)^{-1/2}(A_m-A_n)(1-A_m^*A_m)^{1/2} = A_m - (1-A_mA_m^*)^{-1/2}A_n(1-A_m^*A_m)^{1/2}.$$ It follows from the identity $$(1-t)^{-1/2} - 1 = \frac{t}{(1-t)(1+(1-t)^{-1/2})}$$ that the operator $(1-A_mA_m^*)^{-1/2}$ is a finite rank perturbation of the identity operator. Since $A_n\to 0$ in [[WOT]{}]{}, we obtain that $\|T_n-S_n\|\to 0$, where $S_n = A_m - A_n(1-A_m^*A_m)^{1/2}$. Denote $A_n(1-A_m^*A_m)^{1/2}$ by $B_n$. Since $B_n\to 0$ in [[WOT]{}]{}, the sequence $$(A_m - B_n)^*(A_m-B_n) - A_m^*A_m - B_n^*B_n = - A_m^*B_n - B_n^*A_m$$ tends to zero in norm topology. Furthermore, $$B_n^*B_n = (1-Q)^{1/2}A_n^*A_n(1-Q)^{1/2}$$ tends in norm topology to $(1-Q)^{1/2}P(1-Q)^{1/2}$. Therefore $$(A_m - B_n)^*(A_m-B_n) \to Q + (1-Q)^{1/2}P(1-Q)^{1/2}.$$ Since $\|P-Q\| <\varepsilon $, we have that $$\|Q + (1-Q)^{1/2}P(1-Q)^{1/2} - (Q +(1-Q)Q)\| <\varepsilon.$$ The inequalities $$\beta^2 - \varepsilon\le \|Q\|\le \beta^2$$ imply $$\|Q + (1-Q)Q\| \ge 2\beta^2 - \beta^4 - 2\varepsilon,$$ whence $$\lim_{n\to\infty}||S_n^*S_n||=\lim_{n\to\infty} \|(A_m - B_n)^*(A_m-B_n)\|\ge 2\beta^2 - \beta^4 - 3\varepsilon > \beta^2,$$ if $\varepsilon$ is sufficiently small. Fixed points ============ The main purpose of this section is to establish the existence of a common fixed point for an elliptic group $G$ of biholomorphic transformations of the operator ball ${\mathcal{B}}$. As is shown in Section \[S:appendix\] a biholomorphic transformation of ${\mathcal{B}}$ is a bijective isometric transformation of the metric space $({\mathcal{B}},\rho)$ which maps the set ${\mathcal{M}}$ onto itself (and hence segments onto segments). \[L:elliptic\] If $G$ is an elliptic group of biholomorphic transformations of ${\mathcal{B}}$, then there is a non-empty [[WOT]{}]{}-compact $\rho$-convex $G$-invariant subset of ${\mathcal{B}}$. Let $A\in {\mathcal{B}}$ be such that the orbit $G(A): = \{g(A):g\in G\}$ is bounded. Therefore $G(A)$ is contained in some closed ball $E_{a,r}$. Let $M$ be the intersection of all closed balls containing $G(A)$. It is clear that this intersection is non-empty (it contains $G(A)$), [[WOT]{}]{}-compact and $\rho$-convex (as an intersection of [[WOT]{}]{}-compact $\rho$-convex sets). It remains to check that it is $G$-invariant. To see this it suffices to observe that each element $g\in G$ maps the set of balls containing $G(A)$ bijectively onto itself. \[single\] Let $G$ be an elliptic group of biholomorphic transformations of ${\mathcal{B}}$. Let $M$ be a minimal [[WOT]{}]{}-compact $\rho$-convex $G$-invariant subset in $({\mathcal{B}},\rho)$. Then $M$ is a singleton. We use the approach suggested in [@BM48]. Assume the contrary, let $\diam M=\alpha>0$. By Theorem \[T:non-diam\] $M$ contains a non-diametral point $N$, so that $M\subset\{A:~\rho(A,N)\le\delta\}$ for some $\delta<\alpha$. Consider the set $$O=\bigcap_{B\in M}E_{B,\delta}.$$ The set $O$ is non-empty because $N\in O$. The set $O$ is weakly compact and $\rho$-convex since each of the balls $E_{B,\delta}$ is weakly compact and $\rho$-convex. The set $O$ is a proper subset of $M$ since $M$ has diameter $\alpha>\delta$. Since $G$ is a group of isometric transformations and $M$ is invariant under each element of $G$, the action of $G$ on $M$ is by isometric bijections. Therefore $O$ is $G$-invariant. We get a contradiction with the minimality of $M$. The implication (i) $\Rightarrow$ (ii) is obvious. On the other hand if $G(X_0)$ is separated from the boundary, for some $X_0\in {\mathcal{B}}$ then $\sup_{g\in G}\rho(0,g(X_0))< \infty$ whence, for each $X\in {\mathcal{B}}$, $\sup_{g\in G}\rho(0,g(X)) \le \sup_{g\in G}(\rho(0,g(X_0)) + \rho(g(X_0),g(X))) = \sup_{g\in G}(\rho(0,g(X_0)) + \rho(X_0,X)) < \infty$. This means that the orbit $G(X)$ is separated from the boundary. We proved that (i) $\Leftrightarrow$ (ii). The implication (i) $\Rightarrow$ (iv) can be derived from Lemmas \[L:elliptic\] and \[single\] as follows. It is clear that families of [[WOT]{}]{}-compact $\rho$-convex $G$-invariant sets with the finite intersection property have non-empty intersections which are also [[WOT]{}]{}-compact $\rho$-convex and $G$-invariant. Therefore, by the Zorn Lemma, there is a minimal non-empty [[WOT]{}]{}-compact $\rho$-convex $G$-invariant set $M_0$. By Lemma \[single\], $M_0$ is a singleton and (iv) is proved. If (iv) is true and $A$ is a fixed point of $G$ then, $G_1 = M_{-A}GM_{A}$ is a group of biholomorphic maps of ${\mathcal{B}}$ preserving $0$. Hence it consists of restrictions to ${\mathcal{B}}$ of isometric linear maps (see the beginning of Section \[S:distance\] in this connection). Thus $G_1$ is equicontinuous. Note that each Möbius transform is a Lipschitz map: $\|M_A(X)-M_A(Y)\|\le C\|X-Y\|$ for each $X,Y\in {\mathcal{B}}$, where the constant $C>0$ depends on $A$. Indeed setting $F(X) = (A+X)(1+A^*X)^{-1}$ and $D = (1- \|A\|)^{-1}$ we have $$\begin{split}\|F(X) -F(Y)\|&= \|(A+X)((1+A^*X)^{-1}- (1+A^*Y)^{-1}) + (X-Y)(1+A^*Y)^{-1}\|\\ & = \|(A+X)(1+A^*X)^{-1}A^*(Y-X)(1+A^*Y)^{-1}+(X-Y)(1+A^*Y)^{-1}\|\\ &\le 2D^2\|X-Y\| + D\|X-Y\|\le 3D^2\|X-Y\|.\end{split}$$ Hence $$\|M_A(X) -M_A(Y)\| = \|(1-AA^*)^{-1/2}(F(X) - F(Y))(1-A^*A)^{1/2}\|$$ $$\le D^{\frac12}\|F(X)-F(Y)\|\le 3D^{\frac52}\|X-Y\|.$$ Since $G = M_AG_1M_{-A}$ and the maps $M_A$, $M_{-A}$ are Lipschitz, $G$ is also equicontinuous. We proved that (iv) $\Rightarrow$ (iii). Let now (iii) hold, we have to prove (ii). We will show that the orbit of $0$ is separated from the boundary. Assuming the contrary we get that for any $\delta>0$ there is $g\in G$ with $\|g(0)\|>1-\delta$. Let $A = g(0)$; we may assume that $\delta< 1/2$ so $\|A\|> 1/2$. By the already mentioned result of [@Har], $g = M_A\circ h$ where $h$ is a linear isometry. Let $P$ be the spectral projection of $T = A^*A$ corresponding to the eigenvalue $\|T\|= \|A\|^2$ (recall that $T$ is an operator in a finite dimensional space). Then $$\|(1-T)P\| = 1-\|T\| \le 2(1- \|A\|) < 2\delta.$$ Set $X_1 = 0$, $X_2 = h^{-1}(\frac{1}{2}AP)$. Then $\|X_2-X_1\| = \frac{1}{2}\|AP\| = \|A\|/2 > 1/4$. On the other hand $$\begin{split} \|g(X_2)-g(X_1)\|& = \left\|M_A\left(\frac{1}{2}AP\right) - M_A(0)\right\|\\ &= \left\|(1-AA^*)^{-1/2}\left(\frac{1}{2}AP+A\right)\left(1+\frac{1}{2}A^*AP\right)^{-1}(1-A^*A)^{1/2}-A\right\|\\ &= \left\|A(1-T)^{-1/2}\left(\frac{1}{2}P+1\right)\left(1+\frac{1}{2}TP\right)^{-1}(1-T)^{1/2} - A\right\|\\ &= \left\|A\left(\frac{1}{2}P+1\right)\left(1+\frac{1}{2}TP\right)^{-1} - A\right\| = \left\|\frac{1}{2}A(1-T)P\left(1+\frac{1}{2}TP\right)^{-1}\right\|\\ & \le \frac{1}{2}\|A\|\|(1-T)P\| < \frac{1}{2}2\delta = \delta.\end{split}$$ This contradicts to the assumption of equicontinuity. Indeed for each $\delta$ we get points $Y_i = g(X_i)$ with $\|Y_1-Y_2\|< \delta$ and $\|g^{-1}(Y_1) - g^{-1}(Y_2)\| > 1/4$. Thus (ii) holds. Orthogonalization {#S:orthog} ================= \[orth\] If a bounded representation $\pi$ of a group $G$ on a Hilbert space ${\mathcal{H}}$ preserves a quadratic form $\eta$ with finite number of negative squares then it is similar to a unitary representation. By our assumptions, ${\mathcal{H}}= H_1\oplus H_2$, $\dim(H_2) < \infty$, and $\eta(x) = \|Px\|^2 - \|Qx\|^2$ where $P,Q$ are the projections onto $H_1$ and $H_2$ respectively. We write $H_1 = H$ and $H_2 = K$, for brevity. We will relate to each invertible operator $T$ on ${\mathcal{H}}$ preserving the form $\eta$ a biholomorphic map $w_T$ of ${\mathcal{B}}$ in such a way that $$\label{comp} w_{T_1T_2} = w_{T_1}\circ w_{T_2}.$$ Let us call a subspace $L$ of ${\mathcal{H}}$ [*positive*]{} ([*negative*]{}) if $\eta(y) > 0$ (respectively $\eta(y) < 0$) for all non-zero $y\in L$. Since each negative subspace $L$ is finite-dimensional, there is $\varepsilon > 0$ such that $$\eta(y) \le - \varepsilon\|y\|^2 \text{ for all non-zero }y\in L.$$ The supremum of all such $\varepsilon$ is called the [*degree of negativeness*]{} of $L$ and is denoted by $\epsilon(L)$. For each operator $A\in {\mathcal{B}}$, the set $$L(A) = \{Ax\oplus x: x\in K\}$$ is a negative subspace of ${\mathcal{H}}$. Furthermore the condition $$\eta(y) \le - \varepsilon \|y\|^2, \text{ for all }y\in L(A)$$ means that $$- \|x\|^2 + \|Ax\|^2 \le -\varepsilon (\|x\|^2 + \|Ax\|^2)$$ for all $x\in K$. That is $$\varepsilon\le \frac{1-\|A\|^2}{1+\|A\|^2}.$$ It follows that the degree of negativeness of $L(A)$ is related to $\|A\|$ by the equality $$\label{degree2} \varepsilon(L(A)) = \frac{1-\|A\|^2}{1+\|A\|^2}.$$ Since $\dim(L(A)) = \dim(K)$, $L(A)$ is a maximal negative subspace in ${\mathcal{H}}$. Indeed if some subspace $M$ of ${\mathcal{H}}$ strictly contains $L(A)$ then its dimension is greater than codimension of $H$, whence $M\cap H \neq \{0\}$. But all non-zero vectors in $H$ are positive. Conversely, each maximal negative subspace $Q$ of ${\mathcal{H}}$ coincides with $L(A)$, for some $A\in {\mathcal{B}}$. Indeed, since $Q\cap H = \{0\}$, there is an operator $A: K\to H$ such that each vector of $Q$ is of the form $Ax\oplus x$. Since $Q$ is negative, we have $\eta(Ax\oplus x)=\|Ax\|^2-\|x\|^2<0$, and therefore $\|A\| < 1$, so $A\in {\mathcal{B}}$. Thus $Q\subset L(A)$; and, by maximality, $Q = L(A)$. It is easy to see that the map $A\to L(A)$ from ${\mathcal{B}}$ to the set ${\mathcal{E}}$ of all maximal negative subspaces is injective and therefore bijective. Now we can define $w_T$. Indeed, if a subspace $L$ of ${\mathcal{H}}$ is maximal negative then its image $TL$ under $T$ is also maximal negative (because $T$ is invertible and preserves $\eta$). Hence, for each $A\in {\mathcal{B}}$, there is $B\in {\mathcal{B}}$ such that $L(B) = TL(A)$. We let $w_T(A)=B$. The equality (\[comp\]) follows easily because $L(w_{T_1}(w_{T_2}(A))) = T_1L(w_{T_2}(A)) = T_1T_2L(A) = L(w_{T_1T_2}(A))$ and the map $A\to L(A)$ is injective. Our next goal is to check that $w_T$ is biholomorphic. Since $w_T^{-1} = w_{T^{-1}}$ it suffices to show that $w_T$ is holomorphic. Let $T = (T_{ij})_{i,j=1}^2 $ be the matrix of $T$ with respect to the decomposition ${\mathcal{H}}= H_1\oplus H_2$. Then $T(Ax\oplus x) = (T_{11}Ax + T_{12}x)\oplus(T_{21}Ax + T_{22}x)$. Since $T(Ax\oplus x) \in L(w_T(A))$, we conclude that $$w_T(A)(T_{21}Ax + T_{22}x) = T_{11}Ax + T_{12}x.$$ Thus $$\label{fraclin} w_T(A) = (T_{11}A + T_{12})(T_{21}A + T_{22})^{-1}.$$ This shows that $w_T$ is a holomorphic map on ${\mathcal{B}}$. Suppose now that $\pi$ is a bounded representation of a group $G$ on ${\mathcal{H}}$ preserving $\eta$. Then $W = \{w_{\pi(g)}: g\in G\}$ is a group of biholomorphic maps of ${\mathcal{B}}$. Moreover since $\pi$ is bounded, the group $W$ is elliptic. To see this, note that for each negative subspace $L$, one has $$\eta(y)\le -\varepsilon(L)\|y\|^2 \text{ for all }y\in L.$$ If $T$ is an invertible operator preserving $\eta$ then $T^{-1}x\in L$, for each $x\in TL$, whence $$\eta(x)= \eta(T^{-1}x) \le -\varepsilon(L)\|T^{-1}x\|^2 \le -\varepsilon(L)\|T\|^{-2}\|x\|^2.$$ Thus $$\varepsilon(TL)\ge \varepsilon(L)\|T\|^{-2}.$$ For $L=L(A)$, $TL = L(w_T(A))$. This gives $$\frac{1-\|w_T(A)\|^2}{1+\|w_T(A)\|^2} \ge \|T\|^{-2} \frac{1-\|A\|^2}{1+\|A\|^2}$$ if one takes into account (\[degree2\]). Thus, if $\|\pi(g)\|\le C$ for all $g\in G$, then $$\frac{1-\|w_{\pi(g)}(A)\|^2}{1+\|w_{\pi(g)}(A)\|^2} \ge C^{-2} \frac{1-\|A\|^2}{1+\|A\|^2}.$$ Therefore $$1-\|w_{\pi(g)}(A)\|^2 \ge C^{-2} \frac{1-\|A\|^2}{1+\|A\|^2}$$ and $$\sup_{g\in G}\|w_{\pi(g)}(A)\| < 1$$ for each $A\in {\mathcal{B}}$. By Theorem \[main\], there is $D\in {\mathcal{B}}$ with $w_{\pi(g)}(D) = D$ for all $g\in G$. Hence $\pi(g)L(D) = L(D)$ for all $g\in G$. Let $U$ be an operator on ${\mathcal{H}}$ with the matrix $(U_{ij})$ where $U_{11} = (1_H-DD^*)^{-1/2}$, $U_{12} = - D(1_K-D^*D)^{-1/2}$, $U_{21} = -D^*(1_H-DD^*)^{-1/2}$, $U_{22} = (1_K-D^*D)^{-1/2}$. It can be checked that $U$ preserves $\eta$ and maps $L(D)$ onto $K$. Then all operators $\tau(g) = U\pi(g)U^{-1}$ preserve $\eta$, and the subspace $K$ is invariant for them. It follows that $H$ is also invariant for operators $\tau(g)$. Hence these operators preserve the scalar product on ${\mathcal{H}}$. Thus $g\mapsto \tau(g)$ is a unitary representation similar to $\pi$. It should be noted that Theorem \[orth\] does not extend to the case when $\eta$ has infinite number of negative (and positive) squares, that is, to the case that both $H_1$ and $H_2$ are infinite-dimensional [@Shul-77]. $J$-unitary operators on Pontryagin spaces ========================================== The Pontryagin space is a linear space ${\mathcal{H}}$ supplied with an indefinite scalar product $x,y\to [x,y]$ which has a finite number of negative squares. More precisely this means that one can choose a usual scalar product $x,y \to (x,y)$ with respect to which ${\mathcal{E}}$ is a Hilbert space and $[x,y] = (Jx,y)$, where $J$ is a selfadjoint involutive operator on this Hilbert space with $\text{rank} (1-J) < \infty$. An invertible operator $T$ on ${\mathcal{E}}$ is called $J$-[*unitary*]{} if $[Tx,Ty] = [x,y]$ for all $x,y\in {\mathcal{E}}$. It should be noted that the terminology does not seem to be successful because the choice of the operator $J$ and the corresponding scalar product is not unique while the set of $J$-unitary operator is completely determined by the original indefinite scalar product $[\cdot,\cdot]$. However, this terminology is widely used (see, for example, [@Az-Iohv], [@Kiss-Shul] and references therein). It is important that all scalar products defining $[\cdot,\cdot]$ via $J$-operators are equivalent, so one can speak, for example, about boundedness of a set of operators, without indicating which scalar product is chosen. A subspace $X\subset {\mathcal{E}}$ is called [*positive*]{} ([*negative*]{}) if $[x,x] > 0$ (respectively $[x,x] < 0$) for all $x\in X$. A [*dual pair of subspaces*]{} in ${\mathcal{E}}$ is a pair $Y,Z$, where $Y$ is a positive subspace, $Z$ is a negative subspace and $Y+Z = {\mathcal{E}}$. The study of dual pairs invariant for a given set of $J$-unitary operators was started by Sobolev and intensively developed by Pontryagin, Krein, Phillips, Naimark and other prominent mathematicians. The previous theorem on the orthogonalization of representations implies the following result. \[Pontr\] A group of $J$-unitary operators on a Pontryagin space has an invariant dual pair if and only if it is bounded. Choose a scalar product $(\cdot,\cdot)$ and the corresponding operator $J$. Denote by ${\mathcal{H}}$ the Hilbert space $({\mathcal{E}},(\cdot , \cdot))$. Since $J$ is an Hermitian involutive operator, there are orthogonal subspaces $H$, $K$ of ${\mathcal{H}}$ such that $J = P_H - P_K$. By our assumption on $J$, the subspace $K$ is finite-dimensional. Let $G$ be a group of $J$-unitary operators. If it is bounded, then the identity map can be regarded as a bounded representation of $G$ on ${\mathcal{H}}$. Moreover it preserves the form $\eta(x) = [x,x]$. Since it has a finite number of negative squares, Theorem \[orth\] implies that there is an invertible operator $V$ such that the representation $\tau(g) = T^{-1}gT$ is unitary. It follows from [@Kiss-Shul Theorem 5.8] that $G$ has an invariant dual pair of subspaces. For completeness we include the proof of this fact. Passing to adjoints in the equality $T\tau(g) = gT$ and taking into account that $g^* = Jg^{-1}J$, $\tau(g)^* = \tau(g^{-1})$ we obtain that $\tau(g^{-1})T^* = T^*Jg^{-1}J$. Using this identity for $g$ instead of $g^{-1}$ and multiplying both sides by $JT$ we get: $$\tau(g)T^*JT = T^*JgJJT = T^*JgT = T^*JT\tau(g).$$ Thus the invertible selfadjoint operator $R = T^*JT$ commutes with the group $\tau(G)$ of unitary operators. It follows that its spectral subspaces $H_1$ and $K_1$ corresponding to positive and negative parts of spectrum are invariant for $\tau(G)$. Note that $(Rx,x) > 0$ for $x\in T^{-1}H\backslash\{0\}$ and $(Rx,x) < 0$ for $x\in T^{-1}K\backslash\{0\}$. It follows that $\dim K_1 = \dim K$. Now the subspaces $H_2 = TH_1$ and $K_2 = TK_1$ form an invariant dual pair for $G$. The converse implication is simple. If $G$ has an invariant dual pair $H,K$ then the scalar product $(h_1+k_1,h_2+k_2) = [h_1,h_2] - [k_1,k_2]$ is invariant for $G$. Thus $G$ is a group of unitary operators on ${\mathcal{H}}= ({\mathcal{E}}, (\cdot,\cdot))$, hence it is bounded. As a consequence we obtain the following result proved in [@Shul-77]: A $J$-symmetric representation of a unital $C^*$-algebra on a Pontryagin space is similar to a ${}^*$-representation. For a proof it suffices to notice that restricting the representation to the unitary group of the $C^*$-algebra we obtain a bounded group of $J$-unitary operators. Appendix: Hyperbolicity of ${\mathcal{B}}$ (after Itai Shafrir) {#S:appendix} =============================================================== For any bounded domains $D_1,D_2$ of complex Banach spaces we denote by $Hol(D_1,D_2)$ the set of all holomorphic maps from $D_1$ to $D_2$. If $D_1 = D_2 = D$ then $Hol(D_1,D_2)$ is a semigroup with respect to the composition, and by $Aut(D)$ we denote the set of all its invertible elements (biholomorphic automorphisms of $D$). The group $Aut({\mathcal{B}})$ acts transitively on ${\mathcal{B}}$. Indeed, for each $A\in {\mathcal{B}}$ the Möbius transform $M_A$ is biholomorphic and sends $0$ to $A$ As usually the Carathéodory metric on ${\mathcal{B}}$ is defined by the equality: $$c_{{\mathcal{B}}}(A,B) = \sup\{\omega(f(A),f(B)): f\in Hol({\mathcal{B}},\Delta)\}$$ where $\Delta$ is the unit disk and $\omega$ is the Poincaré distance: $$\omega(z_1,z_2) = \tanh^{-1}\left|\frac{z_1-z_2}{1-\overline{z_1}z_2}\right|.$$ As it was mentioned in Section 4, $c_{{\mathcal{B}}}$ coincides with the metric $\rho$ defined by the formula (\[dist0\]). Clearly $c_{{\mathcal{B}}}$ is invariant under biholomorphic maps of ${\mathcal{B}}$. We shall prove that ${\mathcal{B}}$ is a hyperbolic space with respect to this metric. Furthermore the differential Carathéodory metrics on ${\mathcal{B}}$ is defined by $$\label{metr} \alpha(A,V) = \sup_{f\in Hol({\mathcal{B}},\Delta)}\frac{|{\mathcal{D}}f(A)V|}{1-|f(A)|^2}$$ for all $A\in {\mathcal{B}}, V\in L(K,H)$, where ${\mathcal{D}}f(A)$ is the differential of $f$ in $A$ (see [@Vesent], where $\alpha$ is denoted by $\gamma_{{\mathcal{B}}}$). \[differential\] For each $A\in{\mathcal{B}}$, $V\in L(K,H)$ $$\label{diff} {\mathcal{D}}M_B(A)V = (1-BB^*)^{1/2}(1+AB^*)^{-1}V(1+B^*A)^{-1}(1-B^*B)^{1/2}.$$ In particular, $${\mathcal{D}}M_B(0)V = (1-BB^*)^{1/2}V(1-B^*B)^{1/2}.$$ By definition, $M_B(X) = (1-BB^*)^{-1/2}(B+X)(1+B^*X)^{-1}(1-B^*B)^{1/2}$. We have to calculate the coefficient $c$ of $t$ in the Taylor decomposition of the function $t\to M_B(A+tV)$. For this, note that if $P$ is an invertible operator then $(P+tQ)^{-1} = P^{-1} -tP^{-1}QP^{-1} + o(t)$. It follows immediately that $$\begin{split}c &=(1-BB^*)^{-1/2}(V(1+B^*A)^{-1} - (B+A)(1+B^*A)^{-1}B^*V(1+B^*A)^{-1})(1-B^*B)^{1/2}\\ &=(1-BB^*)^{-1/2}(1-(B+A)(1+B^*A)^{-1}B^*)V(1+B^*A)^{-1}(1-B^*B)^{1/2}\\ &=(1-BB^*)^{-1/2}(1-(B+A)B^*(1+ AB^*)^{-1})V(1+B^*A)^{-1}(1-B^*B)^{1/2}\\ &=(1-BB^*)^{-1/2}((1+AB^*-(B+A)B^*)(1+AB^*)^{-1})V(1+B^*A)^{-1}(1-B^*B)^{1/2}\\ &=(1-BB^*)^{1/2}(1+AB^*)^{-1}V(1+B^*A)^{-1}(1-B^*B)^{1/2}.\end{split}$$ \[alpha\] $\alpha(A,V) = \|(1-AA^*)^{-1/2}V(1-A^*A)^{-1/2}\|$ for all $A\in {\mathcal{B}}$ and $V\in L(K,H)$. By [@Vesent Lemma V.1.5] $$\alpha(0,V)= \|V\|.$$ Let now $A$ be arbitrary. Then by [@Vesent Proposition V.1.2] $$\alpha(M_A(0),\mathcal D M_A(0)X)=\alpha(0,X)=||X||.$$ On the other hand, by Lemma \[differential\], $$\alpha(M_A(0),\mathcal D M_A(0)X)=\alpha(A, (1-AA^*)^{1/2}X(1-A^*A)^{1/2}).$$ Setting now $V=(1-AA^*)^{1/2}X(1-A^*A)^{1/2}$, we obtain $X=(1-AA^*)^{-1/2}V(1-A^*A)^{-1/2}$ and hence $\alpha(A,V)=||(1-AA^*)^{-1/2}V(1-A^*A)^{-1/2})||$. For any bounded operator $D$, set $D^{(1)} = D$, $D^{(3)} = DD^*D$, $D^{(5)} = DD^*DD^*D,\ldots,$ $D^{(2k+1)} = (DD^*)^kD$. Let $$\label{Th} \text{Th }D = \sum_{n=0}^{\infty}a_{2n+1}D^{(2n+1)}$$ where $a_j$ are the Taylor coefficients of $\tanh t$, i.e., $\tanh t = \sum_{n=0}^{\infty}a_{2n+1}t^{2n+1}$. It follows from the definition that $\text{Th }D = \tanh(D)$ if $D$ is selfadjoint. If $D = J|D|$ is the polar decomposition of $D$ (that is, $|D|=(D^*D)^{1/2}$ and $J$ is a partial isometry such that $(J^*J)|D| = |D| (J^*J) = |D|$), then $$D^{(2n+1)} = J|D|^{2n+1},$$ and hence $$\text{Th }D = J\tanh |D|.$$ On the other hand, we can write $D = |D^*|J$, where $|D^*| = J|D|J^*=(DD^*)^{1/2}$, therefore $$\text{Th }D = (\tanh |D^*|)J.$$ For the space $({\mathcal{B}},\rho)$ we define the set ${\mathcal{M}}$ of lines as follows: for $A\in {\mathcal{B}}$, $D\in \partial {\mathcal{B}}$ (i.e., $||D||=1$) we let $$\label{line } \gamma_{A,D} = \{\gamma_{A,D}(t):=M_A(\text{Th}(tD)):t\in \mathbb{R}\}$$ and set $${\mathcal{M}}=\{\gamma_{A,D}:A\in{\mathcal{B}},D\in \partial {\mathcal{B}}\}.$$ \[metricline\] $\gamma_{A,D}$ is a metric line. It is enough to show that $\rho(\gamma_{A,D}(s),\gamma_{A,D}(t))=|s-t|$. Since $\rho$ is invariant with respect to $M_A$ we can assume that $A=0$. We have $\rho(\gamma_{0,D}(s),\gamma_{0,D}(t))=\tanh^{-1}||M_B(\text{Th}(tD))||$, where $B=-\text{Th}(sD)$. Using polar decomposition $D = J|D|$ we have that $\text{Th}(tD) = J \tanh(t|D|)$, $\text{Th }(tD)^*\text{Th }(sD)=\tanh (t|D|)\tanh (s|D|)$, whence $$\begin{aligned} M_B(\text{Th}(tD)) &=& (1-\text{Th}(sD)\text{Th}(sD)^*)^{-1/2}(\text{Th}(tD)-\text{Th}(sD))\\ &&(1-\text{Th}(sD)^*\text{Th}(sD))^{-1} (1-\text{Th}(sD)^*\text{Th}(sD))^{1/2}\\ &=& J(1-\tanh^2(s|D|))^{-1/2}J^*J(\tanh(t|D|) - \tanh(s|D|))\\ &&(1-\tanh(s|D|)\tanh(t|D|))^{-1}(1-\tanh^2(s|D||))^{1/2}\\ &=& J \tanh((t-s)|D|) = \text{Th}((t-s)D)\end{aligned}$$ giving the statement. We have to prove that $\gamma_{A,D}(t)$ is a [*metric curve*]{}, in the sense that the metric of its derivation equals $1$, i.e., $\alpha(\gamma(t),\gamma^{\prime}(t)) = 1$. Let $\gamma(t) = \text{\rm Th}(tD)$, $D\in \partial {\mathcal{B}}$. Then $$\label{dif eq } \gamma^{\prime}(t) = D - \gamma(t)D^*\gamma(t).$$ We have $\gamma(t)=J\tanh(t|D|)$ and $$\gamma'(t)=J|D|\cosh(t|D|)^{-2}= D(\cosh(t|D|)^{-2}.$$ On the other hand $$D-\gamma(t)D^*\gamma(t)=D-J\tanh(t|D|)|D|J^*J\tanh(t|D|)=D-D\tanh^2(t|D|)=D(\cosh(t|D|))^{-2}$$ giving (\[dif eq \]). \[met\] Let $\gamma(t)=\text{\rm Th}(tD)$, $D\in\partial{\mathcal{B}}$. Then $$\label{metrcurve} (1-\gamma\gamma^*)^{-1/2}(D-\gamma D^*\gamma)(1-\gamma^*\gamma)^{-1/2} = D.$$ Setting $D = J|D|$, we have $$\gamma^*\gamma = \text{Th}(tD)^*\text{Th}(tD) = \tanh(t|D|)J^*J \tanh(t|D|) = \tanh^2(t|D|).$$ Furthermore $$\gamma\gamma^* = \tanh(t|D^*|)JJ^*\tanh(t|D^*|) = \tanh^2(t|D^*|).$$ Since $$D|D| = |D^*|D,$$ we have that $$D f(|D|) = f(|D^*|)D$$ for any bounded Borel function $f$. Taking $f(x) = 1-\tanh^2(tx)$, we get $$D(1-\gamma^*\gamma) = (1-\gamma\gamma^*)D.$$ Next $$D^*\gamma = \gamma^*D$$ because $D^*\text{Th}(tD) = D^*J \tanh(t|D|) = |D| \tanh(t|D|)$ is selfadjoint. Hence $$\begin{aligned} (1-\gamma\gamma^*)^{-1/2}(D-\gamma D^*\gamma)(1-\gamma^*\gamma)^{-1/2} = (1-\gamma\gamma^*)^{-1/2}(1-\gamma\gamma^*)D(1-\gamma^*\gamma)^{-1/2} \\ =(1-\gamma^*\gamma)^{1/2}(1-\gamma^*\gamma)^{-1/2}D = D.\end{aligned}$$ \[C:metrcurve\] Let $\gamma(t)=\gamma_{A,D}(t)$, $A\in{\mathcal{B}}$, $D\in\partial {\mathcal{B}}$. Then $$\alpha(\gamma(t),\gamma^{\prime}(t)) = 1.$$ It suffices to prove this for $A = 0$, since $\alpha(F(X),{\mathcal{D}}F(X)V)=\alpha(X,V)$ for any $F\in Aut({\mathcal{B}})$, $X\in{\mathcal{B}}$, $V\in L(H,K)$ (see [@Vesent Proposition V.1.2]), and hence $$\begin{aligned} \alpha(M_A(\text{Th}(tD)), (M_A(\text{Th}(tD)))')&=&\alpha(M_A(\text{Th}(tD)),{\mathcal{D}}M_A(\text{Th}(tD))(\text{Th}(tD))')\\ &=&\alpha(\text{Th}(tD),(\text{Th}(tD))').\end{aligned}$$ Assume therefore $\gamma(t)=\text{Th}(tD)$. By Lemma \[alpha\] and \[met\] we have $$\alpha(\gamma(t),\gamma^{\prime}(t)) = \|(1-\gamma\gamma^*)^{-1/2}(D-\gamma D^*\gamma)(1-\gamma^*\gamma)^{-1/2}\| = \|D\| = 1.$$ The next step is to prove that the family ${\mathcal{M}}$ of all lines is invariant with respect to the biholomorphic maps of ${\mathcal{B}}$. \[invbihol\] Let $\eta(t) = M_A(\gamma(t))$ where $\gamma(t) = \text{\rm Th}(tD)$. Then, for each biholomorphic map $h:{\mathcal{B}}\to {\mathcal{B}}$, the curve $h(\eta(t))$ belongs to the family ${\mathcal{M}}$. By [@harris Theorems 3 and 4], there is a linear isometry $L$ of the space $L(K,H)$ to itself satisfying the condition $$\label{multipl} L(AB^*A) = L(A)L(B)^*L(A)\text{ for all } A,B\in L(K,H)$$ and such that $$h = M_{h(0)}\circ L = L\circ M_{-h(0)}.$$ It follows from (\[multipl\]) (see a remark after [@harris Corollary  5]) that $$L\circ M_A = M_{L(A)}\circ L$$ for all $A\in{\mathcal{B}}$. So it suffices to consider the cases $h=L$ and $h = M_B$. Let us firstly prove that $L(\eta(t))\in {\mathcal{M}}$. Indeed, $$\begin{split}L(\eta(t)) &= L(M_A(\gamma(t)) = M_{L(A)}(L(\gamma(t)) \\&=M_{L(A)}(L(\text{Th}(tD))) = M_{L(A)}(\text{Th}(tL(D)))\in {\mathcal{M}}.\end{split}$$ Now we have to prove that $M_B(\eta(t))\in {\mathcal{M}}$. Applying [@harris Theorems 3 and 4] to $h(x)= M_B(M_A(x))$ we get a linear isometry $L$ satisfying (\[multipl\]) and such that $$M_B\circ M_A = M_C\circ L$$ where $C = h(0)=M_B(A)$. Thus $$M_B(\eta(t)) = M_B(M_A(\gamma(t)) = M_C(L(\gamma(t)))=M_C(\text{Th}(tL(D))) \in {\mathcal{M}}.$$ Our next goal is to show that for each $A,B\in{\mathcal{B}}$ there is a unique line in ${\mathcal{M}}$ which passes through $A$, $B$. \[viaA\] The set of all lines in ${\mathcal{M}}$ that go through $A$ is $\{\gamma_{A,D}: D\in \partial({\mathcal{B}})\}$. It suffices to assume that $A = 0$. Suppose that a line $\gamma(t) = M_B({\rm Th}(tD))$ goes through $0$, i.e., $\gamma(s) = 0$ for some $s\in{\mathbb R}$. Then clearly $B = - \text{Th}(sD)$. Using the arguments from the proof of Proposition \[metricline\] we obtain $\gamma(t)= \text{Th}((t-s)D).$ Thus $\gamma=\gamma_{0,D}$. \[unique\] For each $A,B\in {\mathcal{B}}$, there is a unique line in ${\mathcal{M}}$ that passes through them. We may assume that $A = 0$. Let $B=J|B|$ be the polar decomposition of $B$ and let $C=\tanh^{-1}|B|/t_0$ for $t_0>0$ be such that $||C||=1$. Then for $D=JC$ the line $\gamma_{0,D}$ passes through $0$ and $B$. If there are two lines, $\gamma_{0,D_1}$ and $\gamma_{0,D_2}$, going through $B$ then by the above lemma, $B = \text{Th}(tD_1) =\text{Th}(sD_2)$ for some $t$, $s\in{\mathbb R}$. We may suppose that $t$, $s > 0$. Taking polar decompositions of $D_1=J_1|D_1|$ and $D_2=J_2|D_2|$ we see that $J_1=J_2$ and $\tanh(t|D_1|)=\tanh(s|D_2|)$, which imply that $t|D_1|=s|D_2|$. But this clearly shows that the lines coincide. \[inequality\] $$\label{ineq} \|A\|\le \|(1-BB^*)^{-1/2}(A - BA^*B)(1-B^*B)^{-1/2}\|.$$ for each $A$, $B\in{\mathcal{B}}$. Consider the polar decomposition $B=J|B|$. Then $|B^*|=(BB^*)^{1/2}=J|B|J^*$. Let $P=\tanh^{-1}(|B^*|)$, and $Q=\tanh^{-1}(|B|)$. Then $$(1-BB^*)^{-1/2}(A - BA^*B)(1-B^*B)^{-1/2}=(\cosh P)A(\cosh Q)- (\sinh P)JAJ^*(\sinh Q).$$ For any $\varepsilon>0$, there are unit vectors $x,y$ such that $$((\cosh P)A(\cosh Q)x,y)\ge \|(\cosh P)y\|\|A\|\|(\cosh Q)x\| -\varepsilon.$$ Since $\|(\cosh P)y\|^2-\|(\sinh P)y\|^2 = \|y\|^2$, and $\|(\cosh Q)x\|^2-\|(\sinh Q)x\|^2 = \|x\|^2$ one can find numbers $a,b$ such that $\|(\sinh P)y\|= \sinh b$, $\|(\cosh P)y\|= \cosh b$, $\|(\sinh Q)x\|= \sinh a$, $\|(\cosh Q)x\|= \cosh a$. Hence $$\begin{aligned} &&\|(\cosh P)A(\cosh Q)- (\sinh P)JAJ^*(\sinh Q)\| \\ &&\ge (((\cosh P)A(\cosh Q)- (\sinh P)JAJ^*(\sinh Q))x,y) \\ &&\ge (\cosh b)(\cosh a)\|A\| - \varepsilon - (\sinh b)(\sinh a)\|A\|\\ &&\ge \cosh (b-a)\|A\| - \varepsilon\ge \|A\| - \varepsilon,\end{aligned}$$ giving the statement. \[lem-hyperb\] Let us consider two lines: $\gamma(t) = M_A(\text{\rm Th}(tC))$, $\eta(t) = M_A(\text{\rm Th}(tD))$. Then $$\label{hyp} 2\rho(\gamma(s),\eta(s))\le \rho(\gamma(2s),\eta(2s))$$ for each $s>0$. Since $\rho$ is invariant with respect to the transformations $M_A$ we may assume $A = 0$. Let $C(t)$ be a curve $\gamma_{B,E}(t)$ which joins $\gamma(2s)$ with $\eta(2s)$, we assume that $C(0) = \gamma(2s)$, $C(t_0) = \eta(2s)$ for some $t_0>0$ (such curve exists by Corollary \[unique\]). Define now a new curve $C_1$ by $$C_1 = \text{Th}\left(\frac{1}{2}\text{Th}^{-1}C\right).$$ Then $C_1(0)=\gamma(s)$, $C_1(t_0)=\eta(s)$ and $$\label{tanh} C(t)=2C_1(t)(1+C_1(t)^*C_1(t))^{-1}.$$ As usually we denote by $L(C_1)$ the length of the curve $C_1$: $L(C_1) = \int_0^{t_0}\alpha(C_1(t),C_1^{\prime}(t))dt$. If we could show that $$\label{length} L(C)\ge 2L(C_1)$$ for all curves $C$, $C_1$ satisfying (\[tanh\]) then we would obtain that $$\rho(\gamma(2s),\eta(2s)) = L(C)\ge 2L(C_1) \ge 2\rho(\gamma(s),\eta(s))$$ (the first equality follows from Proposition \[metricline\] and Proposition \[C:metrcurve\], the last inequality holds because the length of any curve is not smaller then the distance between its ends). Thus our goal is the inequality (\[length\]). It suffices to show that $$\label{alphaineq} 2\alpha(C_1(t),C_1^{\prime}(t))\le \alpha(C(t),C^{\prime}(t)).$$ Since $$2C_1 = C(1+C_1^*C_1),$$ we have $$C^{\prime}(1+C_1^*C_1) + C(C_1^{\prime *}C_1+C_1^*C_1^{\prime}) = 2C_1^{\prime},$$ whence $$\label{Cprime} C^{\prime} = ((2-CC_1^*)C_1^{\prime} - CC_1^{\prime *}C_1)(1+C_1^*C_1)^{-1}.$$ Since $$2-CC_1^* = 2-2C_1(1+C_1^*C_1)^{-1}C_1^* = 2(1-C_1C_1^*(1+C_1C_1^*)^{-1}) = 2(1+C_1C_1^*)^{-1},$$ substituting this into (\[Cprime\]) we obtain $$C^{\prime} = 2((1+C_1C_1^*)^{-1}C_1^{\prime} - C_1(1+C_1^*C_1)^{-1}C_1^{\prime *}C_1)(1+C_1^*C_1)^{-1}$$ $$= 2(1+C_1C_1^*)^{-1}(C_1^{\prime} - C_1C_1^{\prime *}C_1)(1+C_1^*C_1)^{-1}.$$ Now it follows from Lemma \[alpha\] that the inequality (\[length\]) is equivalent to the following $$\label{reform}\begin{split} &\|(1-C_1C_1^*)^{-1/2}C_1^{\prime}(1-C_1^*C_1)^{-1/2}\|\\ &\le\|(1-CC^*)^{-1/2}(1+C_1C_1^*)^{-1}(C_1^{\prime}-C_1C_1^{\prime *}C_1)(1+C_1^*C_1)^{-1}(1-C^*C)^{-1/2}\|. \end{split}$$ But $$\begin{split}& 1-CC^* = 1-4C_1(1+C_1^*C_1)^{-2}C_1^* = 1- 4 C_1C_1^*(1+C_1C_1^*)^{-2}\\ &=((1+C_1C_1^*)^2-4C_1C_1^*)(1+C_1C_1^*)^{-2} = (1-C_1C_1^*)^2(1+C_1C_1^*)^{-2}.\end{split}$$ Similarly $$(1-C^*C)^{-1/2} = (1+C_1^*C_1)(1-C_1^*C_1)^{-1}.$$ It follows now that (\[reform\]) is equivalent to the inequality $$\label{newineq} \|(1-C_1C_1^*)^{-1/2}C_1^{\prime}(1-C_1^*C_1)^{-1/2}\| \le \|(1-C_1C_1^*)^{-1}(C_1^{\prime }-C_1C_1^{\prime *}C_1)(1-C_1^*C_1)^{-1}\|.$$ But (\[newineq\]) follows from Lemma \[inequality\] by substituting $B = C_1$ and $A = (1-C_1C_1^*)^{-1/2}C_1^{\prime}(1-C_1^*C_1)^{-1/2}$ into inequality (\[ineq\]). 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--- abstract: 'Parseval frames can be thought of as redundant or linearly dependent coordinate systems for Hilbert spaces, and have important applications in such areas as signal processing, data compression, and sampling theory. We extend the notion of a Parseval frame for a fixed Hilbert space to that of a moving Parseval frame for a vector bundle over a manifold. Many vector bundles do not have a moving basis, but in contrast to this every vector bundle over a paracompact manifold has a moving Parseval frame. We prove that a sequence of sections of a vector bundle is a moving Parseval frame if and only if the sections are the orthogonal projection of a moving orthonormal basis for a larger vector bundle. In the case that our vector bundle is the tangent bundle of a Riemannian manifold, we prove that a sequence of vector fields is a Parseval frame for the tangent bundle of a Riemannian manifold if and only if the vector fields are the orthogonal projection of a moving orthonormal basis for the tangent bundle of a larger Riemannian manifold.' address: - 'Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin TX 78712-0257' - 'Department of Mathematics, Pomona College, Claremont, CA 92711' - 'Department of Mathematics, Case Western Reserve University, Cleveland, OH 44106' - 'Department of Mathematics, Pomona College, Claremont, CA 91711' author: - 'D. Freeman, D. Poore, A. R. Wei, M. Wyse' title: Moving Parseval frames for vector bundles --- Introduction {#S:1} ============ Frames for Hilbert spaces are essentially redundant coordinate systems. That is, every vector can be represented as a series of scaled frame vectors, but the series is not unique. Though this redundancy is not necessary in a coordinate system, it can actually be very useful. In particular, frames have played important roles in modern signal processing after originally being applied in 1986 by Daubechies, Grossmann, and Meyer [@DGM]. Besides being important for their real world applications, frames are also interesting for both their analytic and geometric properties [@BF],[@BCPS][@DFKLOW],[@HL] as well as their connection to the famous Kadison-Singer problem [@CCLV],[@W]. A sequence of vectors $(x_i)$ in a Hilbert space $H$ is called a [*frame*]{} for $H$ if there exists constants $A,B>0$ such that $$A\|x\|^2\leq\sum |\langle x_i,x\rangle|^2\leq B\|x\|^2\qquad\textrm{ for all } x\in H.$$ The constants $A,B$ are called the [*frame bounds*]{}. The frame is called [*tight*]{} if $A=B$, and is called [*Parseval*]{} if $A=B=1$. The name Parseval was chosen because $A=B=1$ if and only if the frame satisfies Parseval’s identity. That is, a sequence of vectors $(x_i)$ in a Hilbert space $H$ is a Parseval frame for $H$ if and only if $$\sum \langle x_i,x\rangle x_i=x \textrm{ for all }x\in H.$$ This useful reconstruction formula follows from the dilation theorem of Han and Larson [@HL]. They proved that if $(x_i)$ is a Parseval frame for a Hilbert space $H$, then $(x_i)$ is the orthogonal projection of an orthonormal basis for a larger Hilbert space which contains $H$ as a subspace. It is easy to see that the orthogonal projection of an orthonormal basis is a Parseval frame, and thus the dilation theorem characterizes Parseval frames as orthogonal projections of orthonormal bases. In differential topology and differential geometry, the word frame has a different meaning. A moving frame for the tangent bundle of a smooth manifold is essentially a basis for the tangent space at each point in the manifold which varies smoothly over the manifold. In other words, a moving frame for the tangent bundle of an $n$-dimensional smooth manifold is a set of $n$ linearly independent vector fields. These two different definitions for the word “frame”, naturally lead one to question how they are related. We will combine the concepts by studying Parseval frames which vary smoothly over a manifold, which we formally define below. \[frame\] Let $\pi:E\rightarrow M$ be a rank n-vector bundle over a smooth manifold $M$ with a given inner product $\langle\cdot,\cdot\rangle$. Let $k\geq n$, and $f_i:M\rightarrow E$ be a smooth section of $\pi$ for all $1\leq i\leq k$. We say that $(f_i)_{i=1}^k$ is a [*moving Parseval frame*]{} for $\pi$ if $(f_i(x))_{i=1}^k$ is a Parseval frame for the fiber $\pi^{-1}(x)$ for all $x\in M$. That is, for all $x\in M$, $$y=\sum_{i=1}^k \langle y,f_i(x)\rangle f_i(x)\qquad\textrm{ for all }y\in\pi^{-1}(x).$$ A Parseval frame for a fixed Hilbert space can be constructed by projecting an orthonormal basis, and thus the natural way to construct moving Parseval frames is to project moving orthonormal bases. For instance, the two-dimensional sphere $S^2$ does not have a nowhere-zero vector field, and hence cannot have a moving orthonormal basis for its tangent space. However, if we consider $S^2$ as the unit sphere in ${\mathbb{R}}^3$ and $(e_i)_{i=1}^3$ as the standard unit vector basis for ${\mathbb{R}}^3$, then at each point $p\in S^2$ we may project $(e_i)_{i=1}^3$ onto the tangent space $T_p(S^2)$, giving us a moving Parseval frame of three vectors for $T S^2$. As every vector bundle over a para-compact manifold is a subbundle of a trivial bundle, we may project the basis for the trivial bundle onto the sub-bundle and obtain that every vector bundle over a para-compact manifold has a moving Parseval frame. Thus in contrast to moving bases, we have that moving Parseval frames always exist. The natural general questions to consider are then: When do moving Parseval frames with particular structure exist? How do theorems about Parseval frames generalize to the vector bundle setting?, and How can we construct nice moving Parseval frames for vector bundles in the absence of moving bases? Our main results are the following theorems which extend the dilation theorem of Han and Larson to the context of vector bundles. The proofs will be given in Section \[S:3\]. \[VB\] Let $\pi_1:E_1\rightarrow M$ be a rank $n$ vector bundle over a paracompact manifold $M$ with a moving Parseval frame $(f_i)_{i=1}^k$. There exists a rank $k-n$ vector bundle $\pi_2:E_2\rightarrow M$ with a moving Parseval frame $(g_i)_{i=1}^k$ so that $(f_i\oplus g_i)_{i=1}^k$ is a moving orthonormal basis for the vector bundle $\pi_1\oplus\pi_2:E_1\oplus E_2\rightarrow M$. If $M$ is a Riemannian manifold with a moving Parseval frame for $TM$, we may apply Theorem \[VB\] to obtain a vector bundle containing $TM$ with a moving orthonormal basis which projects to the moving Parseval frame. However, if we start with a moving Parseval frame for a tangent bundle, we want to end up with a moving orthonormal basis for a larger tangent bundle which projects to the moving Parseval frame. This way we would remain in the class of tangent bundles, instead of general vector bundles. The following theorem states that we can do this. \[T:main\] Let $M^n$ be an $n$-dimensional Riemannian manifold and $(f_i)_{i=1}^k$ be a moving Parseval frame for $TM$ for some $k\geq n$. There exists a $k$-dimensional Riemannian manifold $N^k$ with a moving orthonormal basis $(e_i)_{i=1}^k$ for $TN$ such that $N^k$ contains $M^n$ as a submanifold and $P_{T_x M}e_i(x)=f_i(x)$ for all $x\in M^n$ and $1\leq i\leq k$, where $P_{T_x M}$ is orthogonal projection from $T_x N$ onto $T_x M$. Though the concept of a moving Parseval frame seems natural to consider, we are aware of only one paper on the subject. In 2009, P. Kuchment proved in his Institute of Physics select paper that particular vector bundles over the torus, which arise in mathematical physics, have natural moving Parseval frames but do not have moving bases [@K]. The relationship between frames for Hilbert spaces and manifolds was also considered in a different context by Dykema and Strawn, who studied the manifold structure of collections of Parseval frames under certain equivalent classes [@DySt]. We will use the term [*inner product*]{} on a vector bundle $\pi:E\rightarrow M$ to mean a positive definite symmetric bilinear form. All of our theorems will concern vector bundles with a given inner product. In the case that our vector bundle is the tangent bundle of a Riemannian manifold, we will take the inner product to be the Riemannian metric. For terminology and background on vector bundles and smooth manifolds see [@L], for terminology and background on frames for Hilbert spaces see [@C] and [@HKLW]. The majority of the research contained in this paper was conducted at the 2009 Research Experience for Undergraduates in Matrix Analysis and Wavelets organized by Dr David Larson. The first author was a research mentor for the program, and the second, third, and fourth authors were participants. We sincerely thank Dr Larson for his advice and encouragement. Preliminaries and Examples {#S:2} ========================== Our goal is to study moving Parseval frames and extend theorems about fixed Parseval frames for Hilbert spaces to moving Parseval frames for vector bundles. To do this, we will first need to define some notation and recall some useful characterizations of Parseval frames for Hilbert spaces in terms of matrices. For $F=(f_i)_{i=1}^k \in \oplus_{i=1}^k{\mathbb{R}}^n$ and $(u_i)_{i=1}^n$ a fixed orthonormal basis for ${\mathbb{R}}^n$, we denote $[F]_{n\times k}$ to be the matrix whose column vectors with respect to the basis $(u_i)_{i=1}^n$ are given by $(f_i)_{i=1}^k$. For $F=(f_i)_{i=1}^k \in \oplus_{i=1}^k{\mathbb{R}}^n$, $G=(f_i)_{i=1}^k \in \oplus_{i=1}^k{\mathbb{R}}^m$, we define $F \oplus G=(f_i\oplus g_i)_{i=1}^k \in \oplus_{i=1}^k{\mathbb{R}}^{n+m}$. If $k>n$, $(u_i)_{i=1}^n$ is a fixed orthonormal basis for ${\mathbb{R}}^n$ and $(u_i)_{i=n+1}^k$ is a fixed orthonormal basis for ${\mathbb{R}}^{k-n}$, then the matrix $[F \oplus G]_{(n+m)\times k}$ given with respect to $(u_i)_{i=1}^k$ will be formed by appending the column vectors $(g_i)_{i=1}^k$ to the column vectors $(f_i)_{i=1}^k$. In other words, $[F \oplus G]_{(n+m)\times k} = {\begin{pmatrix}}f_1 & \cdots & f_k \\ g_1 & \cdots & g_k {\end{pmatrix}}$. This matrix framework allows us to provide a simple proof of the Han-Larson dilation theorem for Parseval frames for ${\mathbb{R}}^n$. In a later section we will extend this proof to vector bundles. [@HL]\[HL dilation\] If $k>n$, and $(f_i)_{i=1}^k$ is a Parseval frame for ${\mathbb{R}}^n$, then there exists a Parseval frame $(g_i)_{i=1}^k$ for ${\mathbb{R}}^{k-n}$ such that $(f_i\oplus g_i)_{i=1}^k$ is an orthonormal basis for ${\mathbb{R}}^n\oplus{\mathbb{R}}^{k-n}$. We denote the unit vector basis for ${\mathbb{R}}^n$ by $(u_i)_{i=1}^n$. Let $F=(f_i)_{i=1}^k$ and let $[F]_{n\times k}$ be the matrix whose column vectors with respect to $(u_i)_{i=1}^n$ are given by $(f_i)_{i=1}^k$. If $1\leq p,q\leq n$, then the inner product of the $p$th row of $T$ with the $q$th row of $T$ is given by $\sum_{i=1}^k\langle f_i,u_p \rangle\langle f_i,u_q \rangle$. We now use the following equality. $$\begin{aligned} 2=\sum_{i=1}^k\langle f_i,u_p+u_q\rangle^2=&\sum_{i=1}^k\langle f_i,u_p\rangle^2+\sum_{i=1}^k\langle f_i,u_q\rangle^2+2\sum_{i=1}^k\langle f_i,u_p\rangle\langle f_i,u_q\rangle\\ =&2+2\sum_{i=1}^k\langle f_i,u_p\rangle\langle f_i,u_q\rangle\end{aligned}$$ Thus we have that $\sum_{i=1}^k\langle f_i,u_p \rangle\langle f_i,u_q \rangle=0$, and hence the rows of $[F]_{n\times k}$ are orthonormal. We can thus choose $G=(g_i)_{i=1}^k\subset {\mathbb{R}}^{k-n}$ such that the rows of $[F\oplus G]_{k\times k}$ are orthonormal. Thus the column vectors $(f_i\oplus g_i)_{i=1}^k$ of $[F\oplus G]_{k\times k}$ form an orthonormal basis for ${\mathbb{R}}^n\oplus{\mathbb{R}}^{k-n}$. We have that $(g_i)_{i=1}^k$ must be a Parseval frame for ${\mathbb{R}}^{k-n}$ as it is the orthogonal projection of the orthonormal basis $(f_i\oplus g_i)_{i=1}^k$. As shown in the proof of Theorem \[HL dilation\], a sequence of vectors $F=(f_i)_{i=1}^k\subset {\mathbb{R}}^n$ is a Parseval frame for ${\mathbb{R}}^n$ if and only if the matrix $[F]_{n\times k}$ has orthonormal rows. The dilation theorem gives that Parseval frames are exactly orthogonal projections of orthonormal bases. It is then immediate that the orthogonal projection of a moving orthonormal basis is a moving Parseval frame. \[projection thm\] Let $k\geq n$ and let $\pi: E\rightarrow M$ be a rank $k$ vector bundle with an inner product $\langle\cdot,\cdot\rangle$ and moving orthonormal basis $(e_i)_{i=1}^k$. If $\pi|_{E_0}:E_0\rightarrow M$ is a rank $n$ sub-bundle, then $(P_{E_0}e_i)_{i=1}^k$ is a moving Parseval frame for $\pi|_{E_0}:E_0\rightarrow M$, where $P_{E_0}(e_i(x))$ is the orthogonal projection of $e_i(x)$ onto the fiber $\pi|_{E_0}^{-1}(x)$ for all $x\in M$. As $\pi|_{E_0}:E_0\rightarrow M$ is a subbundle of $\pi: E\rightarrow M$, we have that $P_{E_0}:E\rightarrow E_0$ is continuous. Furthermore, for all $1\leq i\leq k$, we have that $\pi|_{E_0}(P_{E_0}(e_i(x)))=x$ for all $x\in M$. Thus $P_{E_0}e_i$ is a section of $\pi|_{E_0}:E_0\rightarrow M$ for all $1\leq i\leq k$. $(P_{E_0}e_i(x))_{i=1}^k$ is a Parseval frame for $\pi|_{E_0}^{-1}(x)$ for all $x\in M$, as it is the orthogonal projection of an orthonormal basis. Thus $(P_{E_0}e_i(x))_{i=1}^k$ is a moving Parseval frame for $\pi|_{E_0}:E_0\rightarrow M$. By applying Theorem \[projection thm\] to the tangent bundle of a smooth manifold, we obtain the following corollary for Riemannian manifolds. \[projection cor\] Let $k\geq n$ and let $N$ be a $k$-dimensional Riemannian manifold with a moving orthonormal basis $(e_i)_{i=1}^k$ for its tangent bundle $TN$. If $M\subset N$ is a smooth sub-manifold, then $(P_{TM}e_i)_{i=1}^k$ is a moving Parseval frame for $TM$, where $P_{TM}(e_i(x))$ is the orthogonal projection of $e_i(x)\in T_x N$ onto $T_{x}M$ for all $x\in M$ and $1\leq i\leq k$. Let $\pi:TN\rightarrow N$ be the tangent bundle for $N$. Then $(e_i|_M)_{i=1}^k$ is a moving orthonormal basis for the vector bundle $\pi|_{\pi^{-1}(M)}:\pi^{-1}(M)\rightarrow M$, which contains $TM$ as a sub-bundle. We may thus apply Theorem \[projection thm\]. For example, the two dimensional sphere $S^2$ does not have a moving basis for its tangent space, as it does not have a nowhere zero vector field. However, if we consider ${\mathbb{R}}^3$ to be a Riemanian manifold with the Riemannian metric given by the dot product, then $(e_i)_{i=1}^3$ is a moving orthonormal basis for $T{\mathbb{R}}^3$, where $e_1(x,y,z)=(1,0,0)$, $e_2(x,y,z)=(0,1,0)$, and $e_3(x,y,z)=(0,0,1)$ for all $(x,y,z)\in {\mathbb{R}}^3$. We can then project $(e_i)_{i=1}^3$ onto the tangent bundle of the unit sphere to obtain a moving Parseval frame $(f_i)_{i=1}^3$ for $TS^2$. In this case, $(f_i)_{i=1}^3$ will be defined by $f_1(x,y,z)=(1-x^2,-xy,-xz)$, $ f_2(x,y,z)=(-xy,1-y^2,-yz)$, and $f_3(x,y,z)=(-xz,-yz,1-z^2)$ for all $(x,y,z)\in S^2$. If $M$ is an $n$ dimensional smooth manifold, and $\phi:M\rightarrow N$ is an embedds into a $k$ dimensional Riemannian manifold $N$ with a moving orthonormal basis for $TN$, then we can project the moving orthonormal basis onto $T\phi(M)$ and then pull it back to obtain a moving Parseval frame for $TM$ of $k$ vectors. Furthermore, this may be done if $\phi$ is only an immersion instead of an embedding. The Whitney immersion theorem gives that for all $n\geq 2$, every $n$ dimensional paracompact smooth manifold immerses in ${\mathbb{R}}^{2n-1}$. Thus every $n$ dimensional smooth manifold has a moving Parseval frame for its tangent bundle of $2n-1$ vectors. When considering $n=2$, we have that ${\mathbb{R}}^{2}$, the cylinder and the torus are the only two dimensional manifolds with continuous moving basis for its tangent bundle. However, every two dimensional paracompact smooth manifold has a moving Parseval frame of three vectors obtained by immersing the manifold in ${\mathbb{R}}^3$. Unfortunately, obtaining a moving Parseval frame in this way often does not lend us much intuition about the space in question. We present here an intuitive moving Parseval frame for the tangent bundles of the M$\ddot{o}$bius strip and Klein bottle which cannot be obtained by immersing in ${\mathbb{R}}^3$ with the usual orthonormal basis, but which reflects the topology of the surface. We represent the M$\ddot{o}$bius strip and Klein bottle in the standard way with the square $[0,1]\times[0,1]$, where we identify the top and bottom according to $(x,1)\equiv (1-x,0)$ for all $0\leq x\leq 1$, and for the Klein bottle we identify the sides according to $(1,y)\equiv (0,y)$ for all $0\leq y\leq1$, as seen in Figure 1. (14,4.5) (2.1,0)[(1,0)[0]{}]{} (0,0)[(1,0)[4]{}]{} (1.9,4)[(-1,0)[0]{}]{} (0,4)[(1,0)[4]{}]{} (0,0)[(0,1)[4]{}]{} (4,0)[(0,1)[4]{}]{} (8.1,0)[(1,0)[0]{}]{} (6,0)[(1,0)[4]{}]{} (7.9,4)[(-1,0)[0]{}]{} (6,4)[(1,0)[4]{}]{} (6,2.3)[(0,1)[0]{}]{} (6,2)[(0,1)[0]{}]{} (6,0)[(0,1)[4]{}]{} (10,2)[(0,1)[0]{}]{} (10,2.3)[(0,1)[0]{}]{} (10,0)[(0,1)[4]{}]{} (12,0)[(1,0)[1]{}]{} (13.2,0) [$\hat{i}$]{} (12,0)[(0,1)[1]{}]{} (13,1.4) [$\hat{j}$]{} For all $(x, y)\in[0,1]\times[0,1]$, let $$f_1(x,y) = (cos(\pi y),0)\quad f_2(x,y) = (sin(\pi y),0)\quad f_3(x,y) = (0,1).$$ It is easy to see that $(f_i)_{i=1}^3$ is a moving Parseval frame for both the M$\ddot{o}$bius strip and the Klein bottle, which naturally shows the twist in their topology. Given a vector bundle $\pi_1:E_1\rightarrow M$, it is a classic problem in differential topology to find a vector bundle $\pi_2:E_2\rightarrow M$ so that $\pi_1\oplus\pi_2:E_1\oplus E_2\rightarrow M$ has a moving basis. This is of course closely related to our work. Before proving Theorem \[VB\], we need to show that our condition that $\pi_1\oplus\pi_2:E_1\oplus E_2\rightarrow M$ has an orthonormal basis which projects to a given Parseval frame for $\pi_1:E_1\rightarrow M$ is in fact stronger in general than the condition that $\pi_1\oplus\pi_2:E_1\oplus E_2\rightarrow M$ simply has a basis. Thus the dilation theorems for moving Parseval frames do not follow as corollaries from known results in differential topology. This will be illustrated by the following simple example. \[Sphere example\] We define a moving Parseval frame $(f_i)_{i=1}^3$ for the vector bundle $S^2\times{\mathbb{R}}$ by $f_1\equiv1$ and $f_2\equiv f_3\equiv0$. The normal bundle to $TS^2\subset T{\mathbb{R}}^3$ is simply $S^2\times{\mathbb{R}}$, and thus $(S^2\times{\mathbb{R}})\oplus TS^2\cong S^2\times{\mathbb{R}}^3$ has a moving basis. However, we claim that there does not exist a moving basis $(e_i)_{i=1}^3$ for $(S^2\times{\mathbb{R}})\oplus TS^2$ such that $P_{S^2\times{\mathbb{R}}}e_i=f_i$ for all $i=1,2,3$. Indeed, if $P_{S^2\times{\mathbb{R}}}e_2=f_2=0$ then $P_{TS^2}e_2$ is nowhere zero. However, $S^2$ does not have a nowhere zero vector field, and thus we have a contradiction. We have a case of two vector bundles $\pi_1:E_1\rightarrow M$ and $\pi_2:E_2\rightarrow M$ and a moving Parseval frame $(f_{i})_{i=1}^k$ for $\pi_1$ which does not dilate to a moving basis for $\pi_1\oplus\pi_2:E_1\oplus E_2\rightarrow M$, even though $\pi_1\oplus\pi_2:E_1\oplus E_2\rightarrow M$ has a moving basis of $k$ vectors. This motivates the following question. What properties of a moving Parseval frame $(f_i)_{i=1}^k$ for a vector bundle $\pi_1:E_1\rightarrow M$ would guarantee that if $\pi_1\oplus\pi_2:E_1\oplus E_2\rightarrow M$ has a moving basis of $k$ vectors, then $\pi_1\oplus\pi_2:E_1\oplus E_2\rightarrow M$ has a moving orthonormal basis which projects to $(f_i)_{i=1}^k$? The following theorem answers this question when $k=n+1$, where $n$ is the rank of the vector bundle $\pi_1$. \[T:n+1\] Let $(f_i)_{i=1}^{n+1}$ be a moving Parseval frame for a rank $n$ vector bundle $\pi_1:E_1\rightarrow M$. If $\pi_2:E_2\rightarrow M$ is a rank 1 vector bundle such that $\pi_1\oplus\pi_2:E_1\oplus E_2\rightarrow M$ has a moving basis, then $\pi_2: E_2\rightarrow M$ has a moving Parseval frame $(g_i)_{i=1}^{n+1}$ such that $(f_i\oplus g_i)_{i=1}^{n+1}$ is a moving orthonormal basis for $\pi_1\oplus\pi_2:E_1\oplus E_2\rightarrow M$. If $\pi_1\oplus\pi_2:E_1\oplus E_2\rightarrow M$ has a moving orthonormal basis $(e_i)_{i=1}^{n+1}$, then the determinant of an operator or matrix with respect to $(e_i)_{i=1}^{n+1}$ varies smoothly over $\pi_1\oplus\pi_2:E_1\oplus E_2\rightarrow M$. If $T$ is an operator or matrix, we will denote $\det_e(T)$ to be the determinant of $T$ with respect to $(e_i)_{i=1}^{n+1}$. We will first prove the result locally, and then we will show that our local choice can actually be made globally. By Lemma \[LemmaLocal\], for each $x\in M$ there exists ${\varepsilon}_x>0$ and a smoothly varying frame $(g_{x,i})_{i=1}^{n+1}$ for $\pi_2|_{\pi_2^{-1}(B_{{\varepsilon}_x}(x))}$ such that $(f_i\oplus g_{x,i})_{i=1}^{n+1}$ is a moving orthonormal basis for $\pi_1|_{\pi_1^{-1}(B_{{\varepsilon}_x}(x))}\oplus\pi_2|_{\pi_2^{-1}(B_{{\varepsilon}_x}(x))}$. For each $x,y\in M$, we denote $[f_i(y)\oplus g_{x,i}(y)]_{k\times k}$ to be the matrix with respect to the basis $(e_i(x))_{i=1}^{n+1}$ whose column vectors are $(f_i(y)\oplus g_{x,i}(y))_{i=1}^{n+1}$. It is easy to see that $(f_i(y)\oplus g_{x,i}(y))_{i=1}^{n+1}$ is an orthonormal basis if and only if $(f_i(y)\oplus -g_{x,i}(y))_{i=1}^{n+1}$ is an orthonormal basis. Thus without loss of generality, we may assume that $(g_{x,i})_{i=1}^{n+1}$ has been chosen such that $\det_e[f_i(x)\oplus g_{x,i}(x)]_{k\times k}=1$ for all $x\in X$, and hence $\det_e[f_i(y)\oplus g_{x,i}(y)]_{k\times k}=1$ for all $x\in X$ and $y\in B_{{\varepsilon}_x}(x)$ as $\det_e$ is continuous. As the span of $(f_i(y)\oplus 0)_{i=1}^{n+1}$ has co-dimension 1, there is exactly one choice for $(g_{x,i}(y))_{i=1}^{n+1}$ such that $(f_i(y)\oplus g_{x,i}(y))_{i=1}^{n+1}$ is orthonormal and $\det_e[f_i(y)\oplus g_{x,i}(y)]_{k\times k}=1$. Thus our locally smooth choice was unique, and hence is smooth globally. Proofs of Dilation Theorems {#S:3} =========================== We denote the set of all Parseval frames of $k$ vectors for ${\mathbb{R}}^n$ by ${\mathcal{P}_{k,n}}$. Specifically, ${\mathcal{P}_{k,n}}=\{(f_i)_{i=1}^k\in\oplus_{i=1}^k{\mathbb{R}}^n: (f_i)_{i=1}^k\textrm{ is a Parseval frame for }{\mathbb{R}}^n\}$. In order to study moving Parseval frames over smooth manifolds, we need to first establish that ${\mathcal{P}_{k,n}}$ itself is a smooth manifold. For every $k\geq n$, the set ${\mathcal{P}_{k,n}}$ is a smooth submanifold of $\oplus_{i=1}^k{\mathbb{R}}^n$ of dimension $kn-n(n+1)/2$. If ${\bf F}=(f_i)_{i=1}^k\in \oplus_{i=1}^k{\mathbb{R}}^n$ then the positive self-adjoint operator defined by $S_{\bf F}(x)=\sum_{i=1}^k\langle x,f_i\rangle f_i$ for all $x\in{\mathbb{R}}^n$ is called the frame operator for $(f_i)_{i=1}^k$. It is clear that the map $\phi: \oplus_{i=1}^k{\mathbb{R}}^n\rightarrow B({\mathbb{R}}^n)$ given by $\phi({\bf F})=S_{\bf F}$ is smooth and ${\mathcal{P}_{k,n}}=\phi^{-1}(Id)$. The set of self-adjoint operators on ${\mathbb{R}}^n$ is naturally diffeomorphic to ${\mathbb{R}}^{n(n+1)/2}$ as seen by fixing a basis and representing the self-adjoint operators by symmetric matrices. The positive definite self-adjoint operators are an open subset of the self-adjoint operators and thus form a smooth manifold. By Sard’s theorem there exists a positive definite self-adjoint operator $A$ which is a regular value of $\phi$, and thus $\phi^{-1}(A)$ is a smooth submanifold of $\oplus_{i=1}^k{\mathbb{R}}^n$. We have that $\phi^{-1}(A)$ has dimension $kn-n(n+1)/2$ as the manifold of positive definite self adjoint operators has dimension $n(n+1)/2$ and ${\mathbb{R}}^{kn}$ has dimension $kn$. We define a diffeomorphism $\psi_A: \oplus_{i=1}^k{\mathbb{R}}^n\rightarrow \oplus_{i=1}^k{\mathbb{R}}^n$ by $\psi_A((f_i)_{i=1}^k)=(A^{-1/2}f_i)_{i=1}^k$. As $A$ is self adjoint we have that $$\phi\circ\psi_A ({\bf F})(x)=\sum_{i=1}^k\langle x,A^{-\frac{1}{2}}f_i\rangle A^{-\frac{1}{2}}f_i=A^{-\frac{1}{2}}\sum_{i=1}^k\langle A^{-\frac{1}{2}}x,f_i\rangle f_i=A^{-\frac{1}{2}}\phi({\bf F})A^{-\frac{1}{2}}x.$$ Thus $\phi\circ\psi_A ({\bf F})=A^{-\frac{1}{2}}\phi({\bf F})A^{-\frac{1}{2}}$, and hence $\psi_A(\phi^{-1}A)=\phi^{-1}(Id)$. We conclude that ${\mathcal{P}_{k,n}}=\phi^{-1}(Id)$ is diffeomorphic to $\phi^{-1}(A)$ and is hence a smooth submanifold of $\oplus_{i=1}^k{\mathbb{R}}^n$ of dimension $kn-n(n+1)/2$. We note that the same proof gives that the set of frames of $k$-vectors in ${\mathbb{R}}^n$ with a given invertible frame operator is a smooth sub-manifold of $\oplus_{i=1}^k{\mathbb{R}}^n$. We now prove that locally, we can smoothly choose complementary frames for Parseval frames. Note that for $k\geq1$, ${\mathcal{P}_{k,k}}$ is the collection of all orthonormal bases for ${\mathbb{R}}^k$. \[LemmaLocal\] For every $k\geq n$ and $F \in {\mathcal{P}_{k,n}}$ there exists some $\varepsilon > 0$ and a smooth map $\phi:B_{\varepsilon}(F)\cap{\mathcal{P}_{k,n}}\rightarrow{\mathcal{P}_{k,k-n}}$ such that $G \oplus \phi(G) \in {\mathcal{P}_{k,k}}$ for all $G\in B_{\varepsilon}(F)\cap{\mathcal{P}_{k,n}}$. We choose $H\in{\mathcal{P}_{k,k-n}}$ such that $F\oplus H\in{\mathcal{P}_{k,k}}$ and fix an orthonormal basis for ${\mathbb{R}}^n$. As mentioned earlier, this is equivalent to the matrix $[F\oplus H]_{k\times k}$ being unitary. The set of invertible matrices is open, and thus there exists ${\varepsilon}>0$ such that $[G\oplus H]_{k\times k}$ is invertible for all $G\in B_{\varepsilon}(F)\cap{\mathcal{P}_{k,n}}$. For each $G\in B_{\varepsilon}(F)$, we apply the Gram-Schmidt procedure to the rows of $[G\oplus H]_{k\times k}$, where the procedure is applied to the rows of $[G]_{n\times k}$ before the rows of $[H]_{(k-n)\times k}$. As $G\in{\mathcal{P}_{k,n}}$, the rows of $[G]_{n\times k}$ are orthonormal. Thus the Gram-Schmidt procedure when applied to $[G\oplus H]_{k\times k}$ will leave the rows contained in $[G]_{n\times k}$ fixed, and hence the matrix resulting from applying the Gram-Schmidt procedure will be of the form $[G\oplus\phi(G)]_{k\times k}$ for some $\phi(G)\in{\mathcal{P}_{k,k-n}}$. Furthermore, the map $\phi:{\mathcal{P}_{k,n}}\rightarrow{\mathcal{P}_{k,k-n}}$ is smooth as the Gram-Schmidt procedure is smooth when applied to the rows of any set of invertible matrices. The frame given in Example \[Sphere example\] shows that Lemma \[LemmaLocal\] is not true globally for $k=3$ and $n=1$. However, the proof of Theorem \[T:n+1\] gives that Lemma \[LemmaLocal\] is true globally for $k=n+1$. We are now ready to prove Theorem \[VB\]. Let $\pi_1:E_1\rightarrow M$ be a rank $n$ vector bundle over a paracompact manifold $M$ with a moving Parseval frame $(f_i)_{i=1}^k$. By Lemma \[LemmaLocal\], we can locally choose a complementary moving Parseval frame. That is, for each $x\in M$, there exists ${\varepsilon}_x>0$ and a moving Parseval frame $(g_{x,i})_{i=1}^k$ for the trivial vector bundle $\pi_{x}:B_{{\varepsilon}_x}(x)\times{\mathbb{R}}^{k-n}\rightarrow B_{{\varepsilon}_x}(x)$ such that $(f_i\oplus g_i)_{i=1}^k$ is a moving orthonormal basis for the vector bundle $\pi_1|_{\pi_1^{-1}(B_{{\varepsilon}_x}(x))}\oplus\pi_{x}$. The collection of sets $\{B_{{\varepsilon}_x}(x)\}_{x\in M}$ is an open cover of $M$, and thus there is a partition of unity $\{\psi_a\}_{a\in A}$ subordinate to a locally finite open refinement $\{U_a\}_{a\in A}$. We thus have for each $a\in A$ a moving Parseval frame $(g_{a,i})_{i=1}^k$ for the trivial vector bundle $\pi_a:U_a\times{\mathbb{R}}^{k-n}\rightarrow U_a$ such that $(f_i\oplus g_{a,i})_{i=1}^k$ is a moving orthonormal basis for $\pi_1|_{\pi_1^{-1}(U_a)}\oplus \pi_a$. We use the partition of unity to extend $(g_{a,i})_{i=1}^k$ to all of $M$ by defining $g_i=\bigoplus_{a\in A}\psi_a^{1/2} g_{a,i}$ for all $1\leq i\leq n$, where we set $g_{a,i}(x)=0$ if $x\not\in U_a$. Thus $g_i$ is a smooth section of the trivial vector bundle $\pi_t:M\times\oplus_{a\in A}{\mathbb{R}}^{k-n}\rightarrow M$ for all $1\leq i\leq k$. The following simple calculations show that $(f_i(x)\oplus g_i(x))_{i=1}^k$ is an orthonormal set of vectors in $\pi_1^{-1}(x)\oplus\bigoplus_{a\in A} {\mathbb{R}}^{k-n}$ for all $x\in M$. For all $1\leq i,j\leq k$, we have the following calculation. $$\begin{aligned} \langle f_i(x)\oplus g_i(x),f_j(x)\oplus g_j(x)\rangle&=\langle f_i(x),f_j(x)\rangle+\sum_{a\in A}\psi_a \langle g_{a,i}(x), g_{a,j}(x)\rangle\\ &=\sum_{a\in A}\psi_a \big(\langle f_i(x),f_j(x)\rangle+\langle g_{a,i}(x), g_{a,j}(x)\rangle\big)\\ &=\sum_{a\in A}\psi_a \langle f_i(x)\oplus g_{a,i}(x),f_j(x)\oplus g_{a,j}(x)\rangle\\ &=\sum_{a\in A}\psi_a \delta_{i,j}=\delta_{i,j}\end{aligned}$$ Thus, $(f_i\oplus g_i)_{i=1}^k$ is a sequence of smooth orthonormal sections of $\pi_1\oplus \pi_t$, and hence $E:=span_{1\leq i\leq k,x\in M} f_i(x)\oplus g_i(x)$ is a smooth manifold and we have an induced vector bundle $\pi_E:E\rightarrow M$. We now show that $(f_j)_{1\leq j\leq k}$ being a moving Parseval frame implies that $span_{1\leq j\leq k} f_j(x)\oplus0\subset span_{1\leq j\leq k} f_j(x)\oplus g_j(x)$ for all $x\in M$. Indeed, if $x\in M$ and $y\in\pi_1^{-1}(x)$ then we calculate the following. $$\|P_{\pi_E^{-1}(x)}y\oplus0\|^2=\sum_{i=1}^k<y\oplus0,f_i\oplus g_i>^2 =\sum_{i=1}^k<y,f_i>^2=\|y\|^2$$ Which implies that $y\oplus0=P_{\pi_E^{-1}(x)}y\oplus0$. Thus we conclude that $span_{1\leq j\leq k} f_j(x)\oplus0\subset span_{1\leq j\leq k} f_j(x)\oplus g_j(x)$, and hence $p_1:E_1\rightarrow M$ is a sub-bundle of $\pi_E:E\rightarrow M$. It is then possible to define $\pi_2:E_2\rightarrow M$ as the orthogonal bundle of $\pi_1:E_1\rightarrow M$ in $\pi_E:E\rightarrow M$. We have that $f_i=P_{E_1}f_i\oplus g_i$ and thus by definition $g_i=P_{E_2} f_i\oplus g_i$. As the orthogonal projection of a moving orthonormal basis onto a sub-bundle, $(g_i)_{i=1}^\infty$ is a moving Parseval frame for $E_2$ and is a complementary frame for $(f_i)_{i=1}^\infty$. The above proof takes an approach using local coordinates, and then combines the pieces using a partition of unity. As this is a common technique in differential topology, the above construction is valuable in that it could potentially be combined with other proofs and constructions. We present as well a second proof which avoids local coordinates and is based on the original proof of the dilation theorem of Han and Larson. Let $\pi_1:E_1\rightarrow M$ be a rank $n$ vector bundle over a paracompact manifold $M$ with a moving Parseval frame $(f_i)_{i=1}^k$. Let $\pi:M\times{\mathbb{R}}^k\rightarrow M$ denote the trivial rank $k$ vector bundle over $M$, and let $(e_i)_{i=1}^k$ be the moving unit vector basis for $\pi:M\times{\mathbb{R}}^k\rightarrow M$. We define a bundle map $\theta: E_1\rightarrow M\times{\mathbb{R}}^k$ over $M$ by $\theta(y)=\sum_{i=1}^k \langle y , f_i(\pi_1(y))\rangle e_i(\pi_1(y))$. As $(f_i(\pi_1(y))_{i=1}^k$ is a Parseval frame for the fiber containing $y$, we have that $\|y\|^2=\sum_{i=1}^k \langle y , f_i(\pi_1(y))\rangle^2=\|\theta(y)\|^2$. Hence, $\theta|_{\pi_1^{-1}(x)}$ is a linear isometric embedding of the fiber for $\pi_1^{-1}(x)$ into the fiber $\pi^{-1}(x)$ all $x\in M$. Thus, for convenience, we may identify the bundle $\pi_1:E_1\rightarrow M$ with $\pi|_{\theta(E_1)}:\theta(E_1)\rightarrow M$. In particular, we have that $$\label{eq1} \langle y,e_i(\pi(y))\rangle=\langle y,f_i(\pi(y))\rangle\textrm{ for all }y\in E_1.$$ Let $P_1:M\times{\mathbb{R}}^k\rightarrow E_1$ be orthogonal projection, that is, $P_1$ is the bundle map such that $P_1(y)$ is the orthogonal projection of $y$ onto the fiber $\pi_1^{-1}(\pi(y))$. We now show that $P_1\circ e_i=f_i$ for all $1\leq i\leq k$. We let $x\in M$, $y\in\pi_1^{-1}(x)$ and $1\leq i\leq k$, and obtain $$\begin{aligned} \langle y, P_1(e_i(x))\rangle&= \langle y, e_i(x)\rangle\qquad\textrm{ as }y\in E_1\\ &=\langle y, f_i(x)\rangle\qquad\textrm{ by the definition of }\theta\textrm{ as }x=\pi_1(y)\\\end{aligned}$$ Thus $\langle y, P_1(e_i(x))\rangle=\langle y, f_i(x)\rangle$ for all $y\in\pi_1^{-1}(x)$, and hence $P_1(e_i(x))= f_i(x)$ for all $x\in M$. We now consider the case where $M$ is a smooth paracompact manifold with a moving Parseval frame for its tangent bundle $TM$. We may apply Theorem \[VB\] to obtain a vector bundle containing $TM$ with a moving orthonormal basis which projects to the moving Parseval frame. However, we want the moving orthonormal basis to be for a larger tangent bundle and not just a general vector bundle. To do this, we will show that actually the total space of the vector bundle given by Theorem \[VB\] will be a Riemannian manifold with an orthonormal basis which projects to the given Parseval frame for $TM$. We denote $\pi_1:TM\rightarrow M$ to be the tangent bundle. By Theorem \[VB\] there exists a rank $(k-n)$ vector bundle $\pi_2:N\rightarrow M$ with a moving Parseval frame $(g_i)_{i=1}^k$ so that $(f_i\oplus g_i)_{i=1}^k$ is a moving orthonormal basis for the vector bundle $\pi_1\oplus\pi_2$. The manifold $N$ has dimension $k$, as $M$ has dimension $n$ and the vector bundle $\pi_2:N\rightarrow M$ has rank $k-n$. For $p\in N$ and $\gamma:{\mathbb{R}}\rightarrow N$ such that $\gamma(0)=p$, we have the differential $D_\gamma\in TN$ defined by $$D_\gamma(f)=\frac{d}{dt}f(\gamma(t))|_{t=0},$$ for each smooth real valued $f$ defined on an open neighborhood of $p$. We define a smooth map $\Theta: N\times N\rightarrow N$ by $$\Theta(p,q)=\sum_{i=1}^k\langle g_i(\pi_2(p)),p\rangle g_i(\pi_2(q))\qquad\textrm{ for all }p,q\in N.$$ The map $\Theta$ has been constructed so that if $p$ and $q$ are contained in the same fiber of $\pi_2:N\rightarrow M$, i.e. $\pi_2(q)=\pi_2(p)$, then $\Theta(p,q)=p$. Note that if $q_0,q_1\in N$ such that $\pi_2(q_0)=\pi_2(q_1)$, then $\Theta(p,q_0)=\Theta(p,q_1)$. As $\pi_2(\Theta(p,q))=\pi_2(q)$, we thus have that $\Theta(p,q)=\Theta(p,\Theta(p,q))$ for all $p,q\in N$. We use $\Theta$ to define a smooth map $\psi:TN\rightarrow TN$ by setting for each smooth $\gamma:{\mathbb{R}}\rightarrow N$, $\psi(D_\gamma)=D_{\gamma}-D_{\Theta(\gamma(0),\gamma)}$. In other words, if $\gamma:{\mathbb{R}}\rightarrow N$ is smooth and $f$ is a smooth real valued function defined on an open neighborhood of $\gamma(0)$, then $$\psi(D_\gamma)(f)=\frac{d}{dt}f(\gamma(t))|_{t=0}-\frac{d}{dt}f\left(\sum_{i=1}^k\left\langle g_i(\pi_2(\gamma(0))),\gamma(0)\right\rangle g_i(\pi_2(\gamma(t)))\right)|_{t=0}.$$ Note that if $D_{\gamma_0}=D_{\gamma_1}$ then $\psi(D_{\gamma_0})=\psi(D_{\gamma_1})$, and that $\psi(a D_{\gamma_0}+ D_{\gamma_1})=a \psi(D_{\gamma_0})+\psi(D_{\gamma_1})$ for all $a\in {\mathbb{R}}$ and smooth $\gamma_0,\gamma_1:{\mathbb{R}}\rightarrow N$. Furthermore, if $\pi_2(\gamma)$ is constant, then $\psi(D_\gamma)=D_\gamma$. For each $p\in N$, we define a linear operator $\Phi:T_pN\rightarrow T_{\pi_2(p)}M$ by $\Phi(D_\gamma)=D_{\pi_2\circ\gamma}$. Then $\Phi(\psi(\gamma))=0$ for all smooth $\gamma:{\mathbb{R}}\rightarrow N$, as $$\Phi(\psi(\gamma))=D_{\pi_2\circ\gamma}-D_{\pi_2(\Theta(\gamma(0),\gamma))}=D_{\pi_2\circ\gamma}-D_{\pi_2\circ\gamma}=0.$$ For each $q\in \pi^{-1}_2(\pi_2(p))$ we define $D_{q}$ as the derivative at $p$ in the direction of $q$. That is, $D_{q}(f)=\frac{d}{dt}f(p+tq)|_{t=0}$, for each smooth real valued $f$ defined on an open neighborhood of $p$. We have that $\psi(D_q)=D_q$ for all $q\in\pi^{-1}_2(\pi_2(p))$ as $\pi_2(p+tq)$ is constant with respect to $t$. Thus $\{D_q\}_{\pi_2(q)=\pi_2(p)}=\psi(T_pN)=\Phi^{-1}(0)$ as $\{D_q\}_{\pi_2(q)=\pi_2(p)}\subseteq\psi(T_pN)\subseteq\Phi^{-1}(0)$ and both spaces $\{D_q\}_{\pi_2(q)=\pi_2(p)}$ and $\Phi^{-1}(0)$ are $(k-n)$-dimensional. For each smooth real valued $f$ defined on an open neighborhood of $p$, we denote $q_\gamma$ to be the unique vector in $\pi_2^{-1}(\pi_2(p))$ such that $D_{q_\gamma}=\psi(D_\gamma)$. We define a smooth bundle map $\phi: TN\rightarrow E(\pi_1\oplus\pi_2)$ by for $\gamma:{\mathbb{R}}\rightarrow N$, we set $\phi(D_\gamma)=D_{\pi_2\circ\gamma}\oplus q_\gamma$. Note that for each $p\in N$, $\phi|_{T_pN}:T_pN\rightarrow T_{\pi_2(p)}M$ is an isomorphism as $D_{\pi_2\circ\gamma}=0$ if and only if $D_\gamma=D_{q_\gamma}$. Thus $\phi$ induces a Riemannian metric $\langle\cdot,\cdot\rangle_N$ on $TN$ by $\langle f,g\rangle_N=\langle \phi(f),\phi(g)\rangle$. We have that $(f_i\oplus g_i)_{i=1}^k$ is a moving orthonormal basis for $\pi_1\oplus \pi_2$, and hence $(\phi^{-1}(f_i\oplus g_i))_{i=1}^k$ is a moving orthonormal basis for $TN$. If $p\in M$ and $\gamma:{\mathbb{R}}\rightarrow M\subset N$ such that $\gamma(0)=p$ then $\phi(D_\gamma)=D_\gamma\oplus0$. Thus $\phi^{-1}(f_i(p)\oplus 0)=f_i(p)$ for all $1\leq i\leq k$ and $p\in M$, and hence $f_i=P_{T_pM}\phi^{-1}(f_i\oplus g_i)$ for all $1\leq i\leq k$. We may apply Theorem \[T:main\] to obtain the following corollary, where we call a smooth manifold parallelizable if it has a moving basis for its tangent bundle. Let $M$ be a paracompact smooth manifold. If $M$ immerses in a $k$ dimensional parallelizable paracompact smooth manifold, then $M$ embeds in a $k$ dimensional parallelizable paracompact smooth manifold. Assume that $M$ immerses in a parallelizable smooth manifold $N$. We may assign a Riemannian metric to $N$ such that the moving basis for $TN$ is an orthonormal basis. Projecting this orthonormal basis then pulling back to $TM$ gives a moving Parseval frame for $TM$ of $k$ vectors. Thus $M$ embeds in a $k$ dimensional parallelizable smooth manifold by Theorem \[T:main\]. Open problems {#S:4} ============= The manifold $N$ constructed in Theorem \[T:main\] is the total space of a vector bundle, and is hence not compact. We thus have the following question: Let $M^n$ be a smooth $n$-dimensional compact Riemannian manifold and $(f_i)_{i=1}^k$ be a moving Parseval frame for $TM$ for some $k\geq n$. Does there exists a smooth $k$-dimensional compact Riemannian manifold $N^k$ which has a moving orthonormal basis $(e_i)_{i=1}^k$ such that $N^k$ contains $M^n$ as a submanifold and $P_{T_x M}e_i(x)=f_i(x)$ for all $x\in M^n$ and $1\leq i\leq k$? If, $(f_i)_{i=1}^k$ is a Parseval frame for ${\mathbb{R}}^n$ and $k> m> n$, then there exists a Parseval frame $(h_i)_{i=1}^k$ for ${\mathbb{R}}^n\oplus{\mathbb{R}}^{m-n}$ such that $P_{{\mathbb{R}}^n}(h_i)=f_i$ for all $1\leq i\leq m$. Thus instead of dilating all the way to an orthonormal basis for a $k$-dimensional Hilbert space, it is possible to dilate to a Parseval frame for a $m$-dimensional Hilbert space. This motivates the following question in the vector bundle setting. Let $k>m>n$ be integers, and let $\pi_1:E_1\rightarrow M$ be a rank $n$ vector bundle over a paracompact manifold $M$ with a moving Parseval frame $(f_i)_{i=1}^k$. Does there exists a rank $m-n$ vector bundle $\pi_2:E_2\rightarrow M$ with a moving Parseval frame $(g_i)_{i=1}^k$ so that $(f_i\oplus g_i)_{i=1}^k$ is a moving Parseval frame for the vector bundle $\pi_1\oplus\pi_2:E_1\oplus E_2\rightarrow M$? In [@BF], it is proven that for every natural numbers $k\geq n$, there exists a tight frame $(f_i)_{i=1}^k$ of ${\mathbb{R}}^n$ such that $\|f_i\|=1$ for all $1\leq i\leq k$, which they call a finite unit tight frame or FUNTF. An explicit construction for FUNTFs is given in [@DFKLOW], and they are further studied in [@BC],[@CFM]. FUNTFs are also of interest for applications in signal processing, as they minimize mean squared error under an additive noise model for quantization [@GK]. For a vector bundle to have a moving FUNTF, it is necessary that it have a nowhere zero section. Thus we have the following question concerning moving FUNTFs. Let $\pi:E\rightarrow N$ be a rank $n$ vector bundle over a paracompact manifold $N$ such that $\pi$ has a nowhere zero section. For what $k\geq n$ does $\pi$ have a moving tight frame $(f_i)_{i=1}^k$ such that $\|f_i(x)\| = 1$ for all $x\in N$ and $1\leq i \leq k$? These questions are general and potentially difficult, and so solutions for certain cases would still be valuable. For instance, for general values of $k$ and $n$, it is unknown if the collection of FUNTFs are connected [@DySt]. A moving FUNTF of $k$ sections for a rank $n$ vector bundle over the circle can be thought of as a path in the collection of FUNTFs of $k$ vectors for ${\mathbb{R}}^n$. Thus, knowing whether or not a rank $n$ vector bundle over the circle has a FUNTF of $k$ vectors will give insight into the problem of determining the connected components of the collection of FUNTFs of $k$ vectors for ${\mathbb{R}}^n$. [99]{} J. J. Benedetto and M. 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--- abstract: 'We propose to realize a two-dimensional chiral topological superconducting (TSC) state from the quantum anomalous Hall plateau transition in a magnetic topological insulator thin film through the proximity effect to a conventional $s$-wave superconductor. This state has a full pairing gap in the bulk and a single chiral Majorana mode at the edge. The optimal condition for realizing such chiral TSC is to have inequivalent superconducting pairing amplitudes on top and bottom surfaces of the doped magnetic topological insulator. We further propose several transport experiments to detect the chiral TSC. One unique signature is that the conductance will be quantized into a half-integer plateau at the coercive field in this hybrid system. In particular, with the point contact formed by a superconducting junction, the conductance oscillates between $e^2/2h$ and $e^2/h$ with the frequency determined by the voltage across the junction. We close by discussing the feasibility of these experimental proposals.' author: - Jing Wang - Quan Zhou - Biao Lian - 'Shou-Cheng Zhang' title: 'Chiral topological superconductor and half-integer conductance plateau from quantum anomalous Hall plateau transition' --- Introduction ============ The search for topological states of matter has become a central focus in condensed matter physics. Chiral topological superconductors (TSC) in two-dimensions (2D) with an odd-integer Chern number are predicted to host a Majorana zero mode in the vortex core, which obeys non-Abelian statistics [@read2000; @ivanov2001] and has potential applications in topological quantum computation [@nayak2008]. A chiral TSC with Chern number $\mathcal{N}$ breaks time-reversal symmetry, and has a full pairing bulk gap and $\mathcal{N}$ topologically protected gapless chiral Majorana edge modes (CMEMs), which can be viewed as a superconducting analogy of the quantum Hall (QH) state [@volovik1988; @qi2009; @schnyder2008]. As a minimal topological state in 2D, the $\mathcal{N}=1$ chiral TSC is of particular interest, as its edge state has only half the degrees of freedom of the QH state with Chern number $\mathcal{C}=1$. Intensive efforts have been made to search for the chiral TSC in 2D [@fu2008; @fu2009a; @sato2009; @sau2010; @alicea2010; @qi2010b; @ojanen2014; @li2015; @mackenzie2003; @raghu2010; @wangqh2013], however, it has not yet been confirmed in experiments. In principle, a QH state with Chern number $\mathcal{C}$ in proximity with an $s$-wave superconductor (SC) can be naturally viewed as a chiral TSC with even number $\mathcal{N}=2\mathcal{C}$ CMEMs. Therefore, it is theoretically possible to realize a chiral TSC with odd number of CMEMs near a QH plateau transition [@qi2010b]. However, the strong magnetic field required in a QH state will severely hinder the superconducting proximity. Instead, the quantum anomalous Hall (QAH) state has a finite Chern number $\mathcal{C}$ in the absence of an external magnetic field [@thouless1982; @haldane1988], which has been theoretically predicted in magnetic topological insulators (TIs) with ferromagnetic (FM) ordering [@hasan2010; @qi2011; @qi2008; @liu2008; @li2010; @yu2010; @wang2013a; @wang2013b; @wang2014b; @onoda2003; @biswas2010] and experimentally realized (for $\mathcal{C}=\pm1$) in both Cr-doped [@chang2013b; @checkelsky2014; @kou2014; @bestwick2015; @kandala2015] and V-doped [@chang2015] (Bi,Sb)$_2$Te$_3$ magnetic TI thin films. More recently, a new zero-plateau QAH state with $\mathcal{C}=0$ and the plateau transitions among $\mathcal{C}=\pm1,0$ states have been theoretically predicted [@wang2014a] and experimentally observed [@fengy2015; @kou2015]. Without requiring a large external magnetic field, the plateau transition from the $\mathcal{C}=\pm1$ QAH to the zero-plateau $\mathcal{C}=0$ state is a unique parent system for realizing a $\mathcal{N}=\pm1$ chiral TSC. In this paper, we propose to realize the $\mathcal{N}=\pm1$ chiral TSC in a magnetic TI near the QAH plateau transition via the proximity effect to an $s$-wave SC. The optimal condition for realizing the chiral TSC is to have *inequivalent* SC pairing amplitudes on top and bottom surfaces of the doped magnetic TI. We then propose several transport experiments to detect this chiral TSC. Generally, the conductance could be quantized into a half-integer plateau at the coercive field in this hybrid system (Fig. \[fig1\]), as a signature of the neutral CMEM backscattering. In particular, with a point contact formed by a SC junction (Fig. \[fig4\]), the conductance oscillates with a frequency determined by the voltage across the junction. Lastly, we briefly discuss the temperature dependence on the transmission of CMEM and the feasibility of these experimental proposals. The organization of this paper is as follows. After this introductory section, Sec. II describes the effective model for the SC proximity effect of the QAH state in a magnetic TI thin film. Section III presents the results on the phase diagram, edge transport and experimental proposals on point contacts. Section IV presents discussion on the feasibility of experimental realization of chiral TSC in a magnetic TI. Section V concludes this paper. Some auxiliary materials are relegated to appendixes. ![(color online). The hybrid QAH-SC device. In region II, a chiral TSC state is induced through the proximity effect to an $s$-wave SC layer on top of the QAH in magnetic TI. A back-gate voltage $V_{\mathrm{bg}}$ is applied to control the Fermi level in region II. Voltages $V_1$ and $V_2$ are applied on leads 1 and 2, respectively. The SC layer is grounded through a lead in its bulk.[]{data-label="fig1"}](fig1){width="3.3in"} Model ===== To start, we consider the SC proximity effect of the QAH state in a magnetic TI thin film with FM order. Without the proximity effect, the low energy physics of the system only consists of the Dirac-type surface states (SS) [@wang2014a]. The 2D effective Hamiltonian is $\mathcal{H}_0=\sum_{\mathbf{k}}\psi^{\dag}_{\mathbf{k}}H_0(\mathbf{k})\psi_{\mathbf{k}}$, with $\psi_{\mathbf{k}}=(c^t_{\mathbf{k}\uparrow}, c^t_{\mathbf{k}\downarrow},c^b_{\mathbf{k}\uparrow}, c^b_{\mathbf{k}\downarrow})^T$ and $$\label{QAH} H_0(\mathbf{k})=k_y\sigma_x\widetilde{\tau}_z-k_x\sigma_y\widetilde{\tau}_z+m(k)\widetilde{\tau}_x+\lambda\sigma_z,$$ where $c_{\mathbf{k}\sigma}$ annihilates an electron of momentum $\mathbf{k}$ and spin $\sigma=\uparrow, \downarrow$, and superscripts $t$ and $b$ denote SS in the top and bottom layers, respectively. $\sigma_i$ and $\widetilde{\tau}_i$ ($i=x,y,z$) are Pauli matrices for spin and layer, respectively. $\lambda$ is the exchange field along $z$ axis induced by the FM ordering. Here $\lambda\propto\langle S\rangle$ with $\langle S\rangle$ being the mean field expectation value of the local spin, and the value of $\lambda$ can be changed during the magnetization reversal process in magnetic TIs. $m(k)=m_0+m_1(k_x^2+k_y^2)$ describes the hybridization between the top and bottom SS. The Chern number of the system is $\mathcal{C}=\lambda/|\lambda|$ for $|\lambda|>|m_0|$, and $\mathcal{C}=0$ for $|\lambda|<|m_0|$. Correspondingly, the system has $|\mathcal{C}|$ chiral edge state [@wang2014a]. In proximity to an $s$-wave SC, a finite pairing amplitude is induced in the QAH system. The Bogoliubov-de Gennes (BdG) Hamiltonian becomes $\mathcal{H}_{\mathrm{BdG}}=\sum_{\mathbf{k}}\Psi^{\dag}_{\mathbf{k}}H_{\mathrm{BdG}}\Psi_{\mathbf{k}}/2$, where $\Psi_{\mathbf{k}}=[(c^t_{\mathbf{k}\uparrow}, c^t_{\mathbf{k}\downarrow}, c^b_{\mathbf{k}\uparrow}, c^b_{\mathbf{k}\downarrow}), (c^{t\dag}_{-{\mathbf{k}}\uparrow}, c^{t\dag}_{-{\mathbf{k}}\downarrow}, c^{b\dag}_{-{\mathbf{k}}\uparrow}, c^{b\dag}_{-{\mathbf{k}}\downarrow})]^T$ and $$\label{BdG} \begin{aligned} H_{\mathrm{BdG}} &= \begin{pmatrix} H_0(\mathbf{k})-\mu & \Delta_{\mathbf{k}}\\ \Delta_{\mathbf{k}}^\dag & -H_0^*(-\mathbf{k})+\mu \end{pmatrix}, \\ \Delta_{\mathbf{k}} &= \begin{pmatrix} i\Delta_1\sigma_y & 0\\ 0 & i\Delta_2\sigma_y \end{pmatrix}. \end{aligned}$$ Here $\mu$ is chemical potential, $\Delta_1$ and $\Delta_2$ are pairing gap functions on top and bottom SS, respectively. In a simple case for $\mu=0$ and $\Delta_1=-\Delta_2=\Delta$, a basis transformation [@basis_note] decouples the BdG Hamiltonian into two models with opposite chirality, and $$\label{decoupled} H_{\mathrm{BdG}} = \begin{pmatrix} H_+(\mathbf{k}) & 0\\ 0 & H_-(\mathbf{k}) \end{pmatrix},$$ where $H_\pm(\mathbf{k})=k_y\sigma_x\mp k_x\sigma_y\varsigma_z+(m(k)\pm\lambda)\sigma_z\varsigma_z\mp\Delta\sigma_y\varsigma_y$ with $\varsigma_{x,y,z}$ the Pauli matrices in Nambu space. The topological property of $H_+$ is clearly seen by a further basis transformation into a block diagonal form: $$H_+(\mathbf{k})=\begin{pmatrix} h_+(\mathbf{k}) & 0\\ 0 & -h^*_-(-\mathbf{k}) \end{pmatrix},$$ where $h_\pm(\mathbf{k})=k_y\sigma_x-k_x\sigma_y+(m(k)+\lambda\pm|\Delta|)\sigma_z$ characterizes a $p_x \pm ip_y$ SC [@read2000; @fu2008]. The BdG Chern number of $h_\pm(\mathbf{k})$ depends only on the sign of mass $m(k)+\lambda\pm|\Delta|$ at the $\Gamma$ point [@wang2014a]. Therefore, the Chern number of $H_+(\mathbf{k})$ is $\mathcal{N}_+=-2$ for $|\Delta|<-m_0-\lambda$, $\mathcal{N}_+=-1$ for $|\Delta|>|m_0+\lambda|$ and $\mathcal{N}_+=0$ for $|\Delta|<m_0+\lambda$. Similarly, the Chern number of $H_-(\mathbf{k})$ is $\mathcal{N}_-=2$ for $|\Delta|<\lambda-m_0$, $\mathcal{N}_-=1$ for $|\Delta|>|m_0-\lambda|$ and $\mathcal{N}_-=0$ for $|\Delta|<m_0-\lambda$. The total Chern number of the system is then $\mathcal{N}=\mathcal{N}_++\mathcal{N}_-$. Fig. \[fig3\]a shows the phase diagram of the system. The phase boundaries are determined by $\Delta\pm(m_0\pm\lambda)=0$, which reduce to the critical points $\lambda=\pm|m_0|$ between the $\mathcal{C}=\pm1$ QAH and the zero plateau normal insulator (NI) for $\Delta=0$. An infinitesimal SC gap drives the QAH phase into a $\mathcal{N}=\pm2$ TSC. More importantly, the $\mathcal{N}=\pm1$ TSC state emerges in the neighborhood of the transition between the QAH phase and NI phase. ![Phase diagram of the QAH-SC hybrid system with typical parameters. (a) $\Delta_1=\Delta$, $\Delta_2=0$, $\mu=0$. (b) $\Delta_1=\Delta_2=\Delta$, $\mu=0$. (c) $\Delta_1=\Delta$, $\Delta_2=0$, $\mu=0.7$. (d) $\Delta_1=-\Delta_2=\Delta$, $\mu=0.7$. Here $\Delta_{1}$, $\Delta_2$, $\mu$ are in the units of $|m_0|$.[]{data-label="fig2"}](fig2){width="3.3in"} Results ======= Phase diagram ------------- Now we turn to the optimal condition for realizing the $\mathcal{N}=\pm1$ TSC. First, consider the phase diagram for $\mu=0$ and general values of $\Delta_1$ and $\Delta_2$. The phase boundaries are determined by the bulk BdG gap closing in Eq. (\[BdG\]). Assuming $\Delta_2=\alpha\Delta_1$ and $\alpha$ is real, the phase boundaries are given by $\mp(1-\alpha)\Delta_1\lambda+\lambda^2=m_0^2+\alpha\Delta_1^2$, as shown in Fig. \[fig2\]. For $\Delta_1=\Delta_2$, the Chern number jumps directly from $\mathcal{N}=\pm2$ to $\mathcal{N}=0$, and $\mathcal{N}=\pm1$ TSC phases disappear due to accidental particle-hole symmetry in $H_0$ with $\mu=0$. As $\Delta_2$ decreases, the $\mathcal{N}=\pm1$ TSC phase space emerges and becomes the widest at $\Delta_2=0$. In particular when $\Delta_1\Delta_2<0$, a helical TSC phase with helical Majorana edge states emerges on the $\lambda=0$ line (Fig. \[fig3\]a). The general case for complex $\alpha$ is studied in Appendix A, where the topology of phase diagram remains unchanged. Next, for the case $\mu\neq0$, which corresponds to the SC proximity effect of a *doped* or *electrically gated* QAH system, the proximity effect is effectively enhanced by the finite density of states at the Fermi level [@qi2010b]. As shown in Fig. \[fig2\], the phase space of $\mathcal{N}=\pm1$ TSC near the $\Delta=0$ axis enlarges from $\mu=0$ to $\mu\neq0$. Therefore, the optimal condition for $\mathcal{N}=\pm1$ TSC is $\mu\neq0$ and $\Delta_2=0$. This leads us to design the transport device in Fig. \[fig1\]. The $s$-wave SC is only grown on top of the magnetic TI in region II to ensure the proximity pairing gap of the top SS is larger than that of the bottom SS, while the Fermi level can be tuned by the back-gate. The size of the SC layer should be *larger* than the back-gate electrode so that there is no metallic regions in the device. Similarly, one can also employ another device geometry by using a global back-gate and two top-gates in region I and III, to tune the Fermi levels in region I, II, and III separately. ![(color online). (a) Phase diagram of the QAH-SC hybrid system for $\mu=0$ and $\Delta_1=-\Delta_2\equiv\Delta$. Only $\Delta\ge0$ is shown. (b) Without SC proximity effect, the $\sigma_{xy}=-1\rightarrow0\rightarrow1$ QAH plateau transition occurs at the coercivity when the magnetization flips. (c) With SC proximity effect to region II in hybrid device Fig. \[fig1\], $\sigma_{12}$ shows plateau transition $1\rightarrow1/2\rightarrow0\rightarrow1/2\rightarrow1$ in the hysteresis loop. The half-integer plateau in $\sigma_{12}$ manifests the $\mathcal{N}=1$ TSC. (d)-(j) The edge transport configuration at A, B, C, D, C$'$, B$'$ and A$'$ in (c). There is no backscattering for $\mathcal{N}=\pm2$ TSC in (d),(j), and Majorana backscattering for $\mathcal{N}=\pm1$ TSC in (e),(i). Red and blue arrows represent $(c\pm c^\dag)$ CMEMs, respectively. NSC: normal, topologically trivial SC.[]{data-label="fig3"}](fig3){width="3.3in"} Edge transport and half-plateau ------------------------------- To identify the $\mathcal{N}=1$ TSC in the QAH-SC hybrid system, one can probe the neutral Majorana nature of CMEM or trap the vortex core zero mode. Several methods have been proposed to measure the Majorana fermions [@fu2009a; @tanaka2009a; @fu2009b; @akhmerov2009; @law2009; @lutchyn2010; @chung2011]. Here, we base our discussion on a recent proposal studying the CMEM backscattering [@chung2011]. The basic setup is shown in Fig. \[fig1\], consisting of a magnetic TI in proximity with a grounded top SC layer in region II and two current leads at the corners. When the magnetic domains of magnetic TI are aligned in the same direction, the magnetic TI is in a QAH state with a single chiral edge state propagating along the sample boundary. During the flipping of the magnetic domains at the coercive field, $\lambda$ decreases and the magnetic TI enters the NI with a zero-plateau in Hall conductance $\sigma_{xy}$ over a finite range of magnetic field [@wang2014a; @fengy2015; @kou2015], as shown in Fig. \[fig3\]b. Either perpendicular or in plane external magnetic field could induce such plateau transition [@kou2015]. When the SC proximity effect is sufficiently strong, the superconducting region II experiences the BdG Chern number variation $\mathcal{N}=-2\rightarrow-1\rightarrow0\rightarrow1\rightarrow2$ as $\lambda$ decreases in the hysteresis loop (dashed line in Fig. \[fig3\]a). Therefore, the transport setup Fig. \[fig1\] is a QAH/NI-TSC/NSC-QAH/NI junction. As we will discuss in details below, the edge transport features of the junction uniquely convey the topological properties of the SC in region II. The QAH edge state can be viewed as two CMEMs since a $\mathcal{C}=1$ QAH state is topologically equivalent to a $\mathcal{N}=2$ TSC. Therefore, in the case of QAH$_{\mathcal{C}=1}$-TSC$_{\mathcal{N}=2}$-QAH$_{\mathcal{C}=1}$ junction (Fig. \[fig3\]j), the edge current will be perfectly transmitted. By contrast, if the junction is QAH$_{\mathcal{C}=1}$-TSC$_{\mathcal{N}=1}$-QAH$_{\mathcal{C}=1}$ (Fig. \[fig3\]i), the chiral edge state in the QAH region separates into two CMEMs at the TSC boundary [@fu2009a; @akhmerov2009]. One CMEM is perfectly transmitted, while the other is totally reflected. The edge transport of the junction is governed by the generalized Landauer-Büttiker formalism, which includes the contributions from both the normal scattering and Andreev scattering [@anantram1996; @entin2008]. The general relationship between current and voltage on lead 1 and 2 shown in Fig. \[fig1\] is $I_1=(e^2/h)[(1-\mathcal{R}+\mathcal{R}_A)(V_1-V^0_{\mathrm{sc}})-(\mathcal{T}'-\mathcal{T}'_A)(V_2-V^0_{\mathrm{sc}})]$, and $I_2=(e^2/h)[(1-\mathcal{R}'+\mathcal{R}'_A)(V_2-V^0_{\mathrm{sc}})-(\mathcal{T}-\mathcal{T}_A)(V_1-V^0_{\mathrm{sc}})]$. Here $V^0_{\mathrm{sc}}=0$ is the voltage of the grounded SC layer, $I_1$ and $I_2$ are currents flowing into leads 1 and 2, respectively. $\mathcal{R}$, $\mathcal{T}$, $\mathcal{R}_A$ and $\mathcal{T}_A$ are the normal reflection, normal transmission, Andreev reflection and Andreev transmission probabilities for an electron injected from the left, while $\mathcal{R}'$, $\mathcal{T}'$, $\mathcal{R}'_A$, and $\mathcal{T}'_A$ are for an electron coming from the right. The two-terminal conductance is then defined as $\sigma_{12}\equiv I/(V_1-V_2)=(I_1-I_2)/2(V_1-V_2)$. For the QAH$_{\mathcal{C}=1}$-TSC$_{\mathcal{N}=1}$-QAH$_{\mathcal{C}=1}$ junction in Fig. \[fig3\]i, the probabilities of normal scattering and Andreev scattering are equal [@chung2011], and we have $\mathcal{R}=\mathcal{R}_A=\mathcal{T}=\mathcal{T}_A=\mathcal{R}'=\mathcal{R}'_A=\mathcal{T}'=\mathcal{T}'_A=1/4$, resulting in a half-quantized conductance $$\sigma_{12}=\frac{e^2}{h}(\mathcal{T}+\mathcal{R}_A)=\frac{e^2}{2h}.$$ Moreover, since the SC layer is not floating but grounded, the quantized net incoming current $I_{\text{SC}}=(V_1+V_2)e^2/h$ will be flowing from the SC layer to ground. Here we point out that the supercurrent due to the phase fluctuation of SC order parameter may give a small correction to conductance, which scales as $(\ell/L)^3$, where $\ell$ is the width of CMEM, and $L$ is the size of SC. For an estimation, $\ell\sim0.5~\mu$m, therefore such correction is neglible for $L>50~\mu$m. In contrast, the $\mathcal{N}=2$ TSC junction in Fig. \[fig3\]j exhibits a quantized conductance $\sigma_{12}=e^2/h$ [@chung2011]. The entire plateau transition of $\sigma_{12}$ in the hybrid junction device is shown in Fig. \[fig3\]c. In correspondence to the QAH plateau transition of $\sigma_{xy}$ in Fig. \[fig3\]b, $\sigma_{12}$ also exhibits plateaus quantized at $e^2/h$ and $0$ when region II is $\mathcal{N}=\pm2$ TSC and $\mathcal{N}=0$ NSC, respectively. In addition, an intermediate half-quantized plateau at $e^2/2h$ could occur at the coercivity under the condition $|\Delta|+|m_0|>|\lambda|>|m_0|$, which is a unique signature of the $\mathcal{N}=\pm1$ TSC in region II. We emphasize that a plateau usually indicates a stable phase instead of a fine-tuned state. The size of backscattering region is not necessarily mesoscopic. In fact, the size $L$ of the TSC region sets a temperature scale $k_BT_{\text{int}}\sim v_M/L$, above which the interference effect vanishes due to thermal averaging, where $v_M$ is the Fermi velocity of CMEM. For an estimation, $L\sim200~\mu$m, $v_M\sim2.0$ eV Å, $T_{\text{int}}\sim10$ mK. Therefore, the half-plateau is robust at large $L$ and finite temperature $T>T_{\text{int}}$. The plateau transitions and corresponding edge transport configuration in the hysteresis loop are illustrated in Fig. \[fig3\]c-j. In particular, *four* $1/2$-plateaus occur around the critical magnetic fields $\pm H_1^*$ and $\pm H_2^*$ shown in Fig. \[fig3\]c. Point contact ------------- Another useful transport configuration is a point contact formed by two SC islands which allow the transmission of CMEMs, as shown in Fig. \[fig4\]a. A voltage $V_{\mathrm{sc}}$ is applied onto island TSC$_1$, while TSC$_2$ is grounded. If either TSC$_1$ or TSC$_2$ is a $\mathcal{N}=2$ TSC, the edge current will be perfectly transmitted. Non-trivial physics occurs when both TSC$_1$ and TSC$_2$ are $\mathcal{N}=1$ TSC. An incident edge electron from $b_1$ splits into two CMEMs, one is perfectly transmitted along the edge, while the other is scattered at the point contact with transmission amplitude $t$, which depends on the phase difference $\delta\phi\equiv\phi_1-\phi_2$ of two TSCs (see Appendix B). The $I$-$V$ relation in this geometry is $I_1=(e^2/h)[(1-\mathcal{R}+\mathcal{R}_A)(V_1-V_{\mathrm{sc}})-(\mathcal{T}'-\mathcal{T}'_A)V_2]$, and $I_2=(e^2/h)[(1-\mathcal{R}'+\mathcal{R}'_A)V_2-(\mathcal{T}-\mathcal{T}_A)(V_1-V_{\mathrm{sc}})]$. where $\mathcal{R}=\mathcal{R}_A=\mathcal{R}'=\mathcal{R}'_A=r^2/4$, $\mathcal{T}=\mathcal{T}'=(1+t)^2/4$, $\mathcal{T}_A=\mathcal{T}'_A=(1-t)^2/4$, $r$ is reflection amplitude and $r^2+t^2=1$. Therefore, $I=e^2(1+t)(V_1-V_2-V_{\mathrm{sc}})/2h$. Note that the current is proportional to the tunneling amplitude $t$, not the tunneling probability. If $V_{\mathrm{sc}}=0$, we have $\sigma_{12}=(1+t)e^2/2h$, which directly measures $t$ of the neutral CMEMs. A finite $V_{\text{sc}}$ leads to a time dependent $\delta\phi$, which in turn affects $t$. A simple tunneling model for the CMEM is (also see Appendix B) $$\label{tunnel_model} H_{\text{tunnel}} = i\sigma_z\partial_x-\kappa(x)\sin(\delta\phi/2-\phi_0)\sigma_y,$$ where $\kappa(x)$ is nonzero in a finite interval, and the basis is the CMEMs $(\gamma_1,\gamma_2)$ shown in Fig. \[fig4\]a. The transmission amplitude $t$ at zero-energy in this model is $t(\delta\phi)=1/\cosh[\xi\sin(\delta\phi/2-\phi_0)]$, where $\xi=\int dx\kappa(x)/2$. Within this model, $t$ is purely real. With a fixed $V_{\mathrm{sc}}$ across the point contact, $\delta\phi$ varies linearly with time $\tau$ with a slope $d\delta\phi/d\tau=2eV_{\mathrm{sc}}/\hbar$. We can define a new conductance $$\sigma_{12}'\equiv \frac{I}{V_1-V_2-V_{\mathrm{sc}}}=\frac{e^2}{2h}\left[1+t(\delta\phi)\right],$$ which is a periodic function in time with the Josephson junction frequency $f=2eV_{\mathrm{sc}}/h$. Fig. \[fig4\]b shows $\sigma_{12}'$ as a function of time for different values of $\xi$. The time oscillation shape of $\sigma_{12}'$ are different for a weakly coupled point contact (small $\xi$) and a strongly coupled one (large $\xi$). However, $\sigma_{12}'$ always oscillates between $e^2/2h$ and $e^2/h$, since there is always at least one perfectly transmitted CMEM, which is also a unique feature of the $\mathcal{N}=1$ TSC state. ![(a) The point contact configuration of two SC islands with SC phases $\phi_1$ and $\phi_2$, across which the reflection and transmission amplitudes of the CMEMs are $r$ and $t$. (b) The conductance $\sigma_{12}'$ as a function of $\tau$ for different coupling strengths $\xi$. A dc current flows between $a_1$ and $a_2$, an ac voltage between them is measured, with frequency $f=2eV_{\text{sc}}/h$. []{data-label="fig4"}](fig4){width="3.3in"} Temperature dependence ---------------------- We further consider the temperature dependence of the above CMEM transmission (see Appendix D). It is straightforward to see by a dimensional counting that $t(\delta\phi)$ in the above free Majorana fermion model is marginal, therefore it remains constant at low temperature $T$. When the leading four-fermion interaction (irrelevant) is included, the tunneling amplitude acquires a weak temperature dependence. For $V_{\text{sc}}=0$, in this case $\sigma_{12}'=\sigma_{12}$, the renormalization group analysis gives a power-law correction $\delta t\sim-\lambda^2_{p}T^6$ to $t$, where $\lambda_p$ is the bare fermion interaction strength. The conductance $\sigma_{12}'\propto(1+t)$ will therefore decrease as $T$ increases. This perturbative result is no longer valid above a characteristic temperature of $T_c\sim \lambda_{p}^{-1/3}$, when the correction $\delta t$ is comparable to $t$. For higher temperature $T_c<T\ll|\Delta|$, $t$ will flow towards $0$, and the two TSC islands will behave like a single connected TSC analogous to that shown in Fig. \[fig1\]. In this regime, one can formulate a similar point-contact tunneling model between the left and right edges of the new TSC as in Eq. (\[tunnel\_model\]), but with an additional vortex tunneling through the bulk TSC. At high temperature, the leading contribution to $t$ then comes from the vortex tunneling, which leads to $t\sim\lambda^2_{\sigma} T^{-7/4}$, where $\lambda_{\sigma}$ is the bare vortex tunneling strength. Therefore, the half-quantized plateau in $\sigma_{12}$ remains robust in the high temperature regime $T_c<T\ll |\Delta|$. Discussion and experimental realization ======================================= Finally, we discuss the feasibility of our proposals. Experimentally, to observe the $\mathcal{N}=\pm1$ chiral TSC and all of the four half-quantized conductance plateaus, a good proximity effect between SC and magnetic TI is necessary. Moreover, the critical field $H^{\perp}_{c}$ of SC should be larger than the coercivity $H^*_{1,2}$ in magnetic TI. From Ref. , the estimated $H_1^*\sim0.05$ T and $H_2^*\sim0.2$ T. The candidate SC materials are Nb and NbSe$_2$. The bulk Nb is a type I SC with $T_{\text{sc}}=9.6$ K and $H^{\perp}_c\sim0.2$ T, while a thin film Nb becomes a type II SC with upper critical field $H^{\perp}_{c2}\sim1$ T. NbSe$_2$ is a type II SC and shows good proximity effect with Bi$_2$Se$_3$ [@wangmx2012] even at $4.2$ K and $0.4$ T, where the proximity effect induced SC gap is $\Delta\sim0.5$ meV. The width of the CMEM $\ell$ can be estimated as $v_F/\Delta\sim0.52~\mu$m, where the Fermi velocity $v_F\sim2.6$ eV Å [@chang2013b]. For a typical junction voltage $V_{\text{sc}}\sim1~\mu$V, $f\sim0.48$ GHz, which is easily accessible in experiments. Conclusion ========== In summary, we propose to realize the $\mathcal{N}=\pm1$ chiral TSC in a magnetic TI near the QAH plateau transition via the proximity effect to an $s$-wave SC. We show that inequivalent SC pairing amplitude on top and bottom surfaces in doped magnetic TIs will optimize the $\mathcal{N}=\pm1$ chiral TSC phases. Several edge transport measurements have been proposed to identify such $\mathcal{N}=1$ TSC in the QAH-SC hybrid system. In particular, the conductance could be quantized into a half-integer plateau at the coercive field in this hybrid system, as a unique signature of the neutral CMEM backscattering. We emphasize that such an experiment can work at reasonable temperature and does not depend on the interference effect of CMEM. We hope the theoretical work here can aid the search for chiral TSC phases in hybrid systems. We thank David Goldhaber-Gordon and Andre Broido for useful comments on the draft. This work is supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under Contract No. DE-AC02-76SF00515 and in part by FAME, one of six centers of STARnet, a Semiconductor Research Corporation program sponsored by MARCO and DARPA. Phase diagram under complex $\alpha=\Delta_2/\Delta_1$ ====================================================== In the paper we have only considered the case $\alpha=\Delta_2/\Delta_1$ is real. In general, in the absence of time reversal symmetry (as is in our model), $\alpha=|\alpha|e^{i\phi_\alpha}$ is complex. Correspondingly, the phase diagram will be modified quantitatively, but the topology of the phase boundaries remains unchanged compared to those shown in Fig. \[fig2\] of the paper. By a proper choice of basis we can always set $\Delta_1=\Delta$ real. As an illustrative example, we consider here the case $|\alpha|=1$, namely $\alpha=\Delta_2/\Delta_1=e^{i\phi_\alpha}$. Via a unitary transformation $(c^t_{\mathbf{k}\uparrow}, c^t_{\mathbf{k}\downarrow}, c^b_{\mathbf{k}\uparrow}, c^b_{\mathbf{k}\downarrow})\rightarrow (c^t_{\mathbf{k}\uparrow}, c^t_{\mathbf{k}\downarrow}, e^{i\phi_\alpha/2}c^b_{\mathbf{k}\uparrow}, e^{i\phi_\alpha/2}c^b_{\mathbf{k}\downarrow})$, $\Delta_2$ is transformed into a real number $\Delta_2'=\Delta_1=\Delta$, while the hybridization $m(k)$ between the top and bottom SS becomes a complex number $e^{-i\phi_\alpha/2}m(k)$. Therefore, we can always set two of the three parameters $\Delta_1$, $\Delta_2$ and $m_0$ to real numbers. Diagonalizing the BdG Hamiltonian $H_{\text{BdG}}$ yields the energy spectrum $E^2=k^2+\Big[\lambda\pm\sqrt{\left[m(k)\sin(\phi_\alpha/2)\pm\Delta\right]^2+m(k)^2\cos^2\left(\phi_\alpha/2\right)}\Big]^2$. The phase boundaries are given by the gap closing of the energy spectrum: $$\lambda\pm\sqrt{(m_0\sin(\phi_\alpha/2)\pm\Delta)^2+m_0^2\cos^2\left(\phi_\alpha/2\right)}=0,$$ namely, the following hyperbolas: $$\lambda^2-\Big(\Delta\pm m_0\sin(\phi_\alpha/2)\Big)^2=m_0^2\cos^2\left(\phi_\alpha/2\right).$$ The phase diagram is shown in Fig. \[fig5\]. As one can see, the topology of the phase diagram does not change much. In particular, when $\phi_\alpha=0$ and $\pi$, the phase diagram is as indicated in Fig. 2b and Fig. 3a of the paper, respectively. ![The phase diagram for $\Delta_2=e^{i\phi_\alpha}\Delta_1$ and $\mu=0$. When $\phi_\alpha=0$, the $\mathcal{N}=\pm1$ TSC phases disappear, while when $\phi_\alpha=\pi$, the phase spaces of $\mathcal{N}=1$ and $\mathcal{N}=-1$ TSC touch each other, as indicated in Fig. 2b and Fig. 3a of the main text, respectively.[]{data-label="fig5"}](fig5){width="3.3in"} Derivation of the effective tunneling Hamiltonian ================================================= Without loss of generality, consider the case $|\Delta|>\lambda-m_0>0$. The QAH has Chern number $\mathcal{C}=1$ and the SC in region II has BdG Chern number $\mathcal{N}=1$, both of which come from the lower block $H_{-}(\mathbf{k})$ of the BdG Hamiltonian $H_{\text{BdG}}$. When the pairing amplitude of the superconductor is $\Delta=|\Delta|e^{i\phi}=\Delta_1=-\Delta_2$ with a phase $\phi$, $H_{-}(\mathbf{k})$ can be rewritten as $$\begin{aligned} H_-(\mathbf{k})&= \begin{pmatrix} h'_+(\mathbf{k}) & 0 \\ 0 & -h'^*_-(-\mathbf{k}) \end{pmatrix}, \\ h'_\pm(\mathbf{k})&= \begin{pmatrix} m(k)-\lambda\pm|\Delta| & -ik_x \pm k_y \\ ik_x\pm k_y & -m(k)+\lambda\mp|\Delta| \end{pmatrix},\end{aligned}$$ under the following new basis $\frac{1}{\sqrt{2}}(e^{-i\phi/2}c_{\mathbf{k}\downarrow}+e^{i\phi/2}c^\dag_{-\mathbf{k}\uparrow}, e^{-i\phi/2}c_{\mathbf{k}\uparrow}+e^{i\phi/2}c^\dag_{-\mathbf{k}\downarrow}, -e^{-i\phi/2}c_{\mathbf{k}\downarrow}+e^{i\phi/2}c^\dag_{-\mathbf{k}\uparrow}, -e^{-i\phi/2}c_{\mathbf{k}\uparrow}+e^{i\phi/2}c^\dag_{-\mathbf{k}\downarrow})$, where we have used the notation $$\begin{aligned} c_{\mathbf{k}\uparrow}=\frac{c^t_{\mathbf{k}\uparrow}-c^b_{\mathbf{k}\uparrow}}{\sqrt{2}},\end{aligned}$$ and $$\begin{aligned} c_{\mathbf{k}\downarrow}=\frac{c^t_{\mathbf{k}\downarrow}+c^b_{\mathbf{k}\downarrow}}{\sqrt{2}}.\end{aligned}$$ The Majorana edge state between the QAH (where $|\Delta|=0$) and the TSC (where $|\Delta|>\lambda-m_0>0$) is given by $h'_+(\mathbf{k})$. As shown in Fig. \[fig4\] of the paper, the lower TSC$_1$ and the upper TSC$_2$ have superconducting phases $\phi_1$ and $\phi_2$ respectively. For simplicity, we shall approximate $m(k)$ as $m_0$, which does not change the topological physics. If the upper edge of the lower TSC$_1$ is set as $y=0$, the Hamiltonian of the corresponding Majorana edge state can be derived as $$H_1=\int dx \ i\gamma_1(x)\partial_x\gamma_1(x),$$ where $$\begin{aligned} \gamma_1(x) &=& \frac{e^{-i\phi_1/2}c_1(x)+e^{i\phi_1/2}c^\dag_1(x)}{\sqrt{2}}, \\ c_1(x) &=& \int_{-\infty}^{\infty}e^{(|\Delta|\Theta(-y)+m_0-\lambda)y}\left[e^{i\pi/4}c_{\uparrow}(x,y)\right. \nonumber \\ &&\left.+e^{-i\pi/4}c_{\downarrow}(x,y)\right]dy,\end{aligned}$$ with $\Theta(y)$ defined as the Heaviside function. Similarly, the lower edge of the upper TSC$_2$ at $y=y_0>0$ has a low energy Hamiltonian $$H_2=-\int dx \ i\gamma_2(x)\partial_x\gamma_2(x),$$ where $$\begin{aligned} \gamma_2(x) &=& \frac{e^{-i\phi_2/2}c_2(x)+e^{i\phi_2/2}c^\dag_2(x)}{\sqrt{2}}, \\ c_2(x) &=& \int_{-\infty}^{\infty}e^{(\lambda-m_0-|\Delta|\Theta(y-y_0))y}\left[e^{-i\pi/4}c_{\uparrow}(x,y)\right. \nonumber \\ &&\left.+e^{i\pi/4}c_{\downarrow}(x,y)\right]dy.\end{aligned}$$ We shall assume the point contact extends in the interval $0<x<L$, and the two edges have a nonzero hopping and pairing term: $$\begin{aligned} H_{I} &=& -\int_0^L dx \left[J_hc_1^\dag(x)c_2(x)\right. \nonumber \\ &&\left.+J_p(\Delta_1^*+\Delta_2^*) c_1(x)c_2(x)+\text{h.c.}\right],\end{aligned}$$ where $\Delta_{1,2}=|\Delta|e^{i\phi_{1,2}}$. When projected into the low energy Hilbert space of $\gamma_1$ and $\gamma_2$ via the substitutions $$c_1\rightarrow e^{i\phi_1/2}\gamma_1/\sqrt{2}, \ \ c_2\rightarrow e^{i\phi_2/2}\gamma_2/\sqrt{2},$$ this term becomes: $$\begin{aligned} H_{I} &= 2\int_0^L dx\ i\kappa(x)\sin\left(\frac{\delta\phi}{2}-\phi_0\right)\gamma_1(x)\gamma_2(x) \nonumber \\ &= 2\int_0^L dx\ i\lambda(x)\gamma_1(x)\gamma_2(x),\end{aligned}$$ where $$\begin{aligned} \delta\phi &=\phi_1-\phi_2, \\ \kappa(x) &=\left|J_h/2+i\text{Im}(J_p)\right|, \\ \phi_0 &=\arg\left[J_h+i2\text{Im}(J_p)\right].\end{aligned}$$ For simplicity we have defined $$\begin{aligned} \lambda(x)\equiv\kappa(x)\sin\left(\frac{\delta\phi}{2}-\phi_0\right).\end{aligned}$$ The total tunneling Hamiltonian is then $H_{\text{tunnel}}=H_1+H_2+H_I$ as given in Eq. (\[tunnel\_model\]) of the paper. The eigenwavefunction $\psi=(\eta_1,\eta_2)^T$ at energy $E$ can then be obtained by solving the following Shrödinger equation: $$\begin{pmatrix} i\partial_x & i\lambda(x) \\ -i\lambda(x) & -i\partial_x \end{pmatrix} \begin{pmatrix} \eta_1 \\ \eta_2 \end{pmatrix} =E\begin{pmatrix} \eta_1 \\ \eta_2 \end{pmatrix}$$ The solution for a wave incident from $x=-\infty$ with momentum $k$ is $E=k$, and $$\begin{aligned} &\left(\eta_1(x), \eta_2(x)\right) \nonumber \\ &=\left\{ \begin{array}{l@{\;\quad\;}l} \left(e^{-ikx}, \frac{\lambda\sinh(\sqrt{\lambda^2-k^2}L)}{\mathcal{G}[L]}e^{ikx}\right) & (x\le0)\\ \left(\frac{\mathcal{G}[L-x]}{\mathcal{G}[L]}, \frac{\lambda\sinh[\sqrt{\lambda^2-k^2}(L-x)]}{\mathcal{G}[L]}\right) & (0<x\le L)\\ \left(\frac{\sqrt{\lambda^2-k^2}e^{-ikx}}{\mathcal{G}[L]},0\right) & (x>L) \end{array} \right.\end{aligned}$$ where function $\mathcal{G}[x]=\sqrt{\lambda^2-k^2}\cosh(\sqrt{\lambda^2-k^2}x)-ik\sinh(\sqrt{\lambda^2-k^2}x)$. At low energies $k\ll\lambda$, the wavefunction can be approximately written as $$\begin{aligned} &\left(\eta_1(x),\ \eta_2(x)\right) \nonumber \\ &=\frac{1}{\cosh\lambda L}\left( \cosh\left[\int_x^\infty\lambda(x')dx'\right], \sinh\left[\int_x^\infty\lambda(x')dx'\right]\right),\end{aligned}$$ from which the transmission and reflection amplitudes can be extracted out as $$\begin{aligned} t &= \frac{1}{\cosh\left(\int dx\lambda(x)\right)}=\frac{1}{\cosh\left[\xi\sin(\delta\phi/2-\phi_0)\right]}, \\ r &= \tanh\left(\int dx\lambda(x)\right)=\tanh \left[\xi\sin(\delta\phi/2-\phi_0)\right],\end{aligned}$$ where $\xi=\int dx\kappa(x)$. Note that $t$ is always real and positive at low energies. For scattering at a finite energy $E=k$, the transmission amplitude $t$ is generally complex. S-matrix and conductance in general Josephson junction setup ============================================================ Here we formulate the scattering matrix of edge states in the setup of Fig. 4a, and derive the conductance $\sigma_{12}'$. The edge fermions at four ends of the sample are denoted by $a_{1,2}$ and $b_{1,2}$ as shown in Fig. 4a. With transmission coefficient $t$ and reflection coefficient $r$ at the point contact, the scattering matrix $S$ due to the point contact is $$\begin{aligned} &\begin{pmatrix} a_{1,\mathbf{k}}+a^{\dagger}_{1,-\mathbf{k}} \\ a_{1,\mathbf{k}}-a^{\dagger}_{1,-\mathbf{k}} \\ a_{2,\mathbf{k}}+a^{\dagger}_{2,-\mathbf{k}} \\ a_{2,\mathbf{k}}-a^{\dagger}_{2,-\mathbf{k}} \end{pmatrix} =S\begin{pmatrix} b_{1,\mathbf{k}}+b^{\dagger}_{1,-\mathbf{k}} \\ b_{1,\mathbf{k}}-b^{\dagger}_{1,-\mathbf{k}} \\ b_{2,\mathbf{k}}+b^{\dagger}_{2,-\mathbf{k}} \\ b_{2,\mathbf{k}}-b^{\dagger}_{2,-\mathbf{k}} \end{pmatrix} \nonumber \\ &=\begin{pmatrix} r & 0 & t & 0\\ 0 & 0 & 0 & 1\\ t^* & 0 & -r^* & 0\\ 0 & 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} b_{1,\mathbf{k}}+b^{\dagger}_{1,-\mathbf{k}} \\ b_{1,\mathbf{k}}-b^{\dagger}_{1,-\mathbf{k}} \\ b_{2,\mathbf{k}}+b^{\dagger}_{2,-\mathbf{k}} \\ b_{2,\mathbf{k}}-b^{\dagger}_{2,-\mathbf{k}} \end{pmatrix}.\end{aligned}$$ Upon basis transformation from Majorana fermions to charged fermions on QAH edges, we have $$\begin{aligned} \begin{pmatrix} a_{1,\mathbf{k}} \\ a^{\dagger}_{1,-\mathbf{k}} \\ a_{2,\mathbf{k}} \\ a^{\dagger}_{{2,-\mathbf{k}}} \end{pmatrix} &= \frac{1}{2}\begin{pmatrix} r & r & t+1 & t-1\\ r & r & t-1 & t+1\\ t^*+1 & t^*-1 & -r^* & -r^*\\ t^*-1 & t^*+1 & -r^* & -r^* \end{pmatrix} \begin{pmatrix} b_{1,\mathbf{k}} \\ b^{\dagger}_{1,-\mathbf{k}} \\ b_{2,\mathbf{k}} \\ b^{\dagger}_{2,-\mathbf{k}} \end{pmatrix},\end{aligned}$$ based on which the normal/Andreev transmission/reflection probabilities are given as $\mathcal{T}=|t+1|^2/4$, $\mathcal{T}_A=|t-1|^2/4$, and $\mathcal{R}=\mathcal{R}_A=|r|^2/4$. According to the generalized Landauer-Büttiker formula, the conductance defined in the main text is $$\sigma_{12}'=\frac{1+\mathrm{Re}(t)}{2}\frac{e^2}{h}.$$ Note that the conductance $\sigma_{12}'$ merely depends on the real part of Majorana transmission coefficient $t$, physically it is due to the fact that charged fermions are treated as combinations of Majorana fermions with transmissions $t$ and perfect transmission $1$. Temperature dependence and renormalization group analysis ========================================================= In this section we analyze the temperature dependence of Majorana transmission coefficient $t$ by renormalization group technique in detail [@kane1992; @kane2007]. Specifically, we focus on its real part $\mathrm{Re}(t)$, since it is directly related to the conductance $\sigma_{12}$. Our starting point is the action for the model in Eq. (\[tunnel\_model\]) of the paper, $$\begin{aligned} \mathcal{S}_0=&\int d\tau \int dx[\gamma_1 i(\partial_{\tau}+\partial_x) \gamma_1 +\gamma_2 i(\partial_{\tau}-\partial_x) \gamma_2 \nonumber \\ &+2\xi\delta(x)\sin(\delta\phi/2-\phi_0) i\gamma_1\gamma_2].\end{aligned}$$ Since the Majorana tunneling occurs locally at $x=0$, the scaling dimension of the tunneling strength $\xi$ vanishes, i.e. $[\xi]=0$. Therefore, $\xi$ is invariant when the temperature $T$ of the system changes, and so does the transmission coefficient $t$. The temperature dependence of $t$ comes from higher irrelevant terms at the point contact. The leading irrelevant term is a four fermion interaction of the following form: $$H_p=\int dx\lambda_{p}\delta(x)\gamma_1\partial_x\gamma_1\gamma_2\partial_x\gamma_2.$$ It represents the tunneling of one pair of Majorana fermions from one edge to the other. The scaling dimension of $\lambda_p$ is $[\lambda_p]=-3$, hence it is irrelevant and scales as $\lambda_p^{\text{eff}}\sim\lambda_p T^3$ when $T\rightarrow 0$. Increasing the temperature $T$ will enhance the effective interaction strength $\lambda_p$, which affects the transmission coefficient $t$. The contribution of $H_p$ to the transmission coefficient $t$ can be calculated perturbatively as follows. Suppose both $\xi$ and $\lambda_p$ are small, so that perturbation theory can be used. We shall regard $H_I=2i\xi\delta(x)\sin(\delta\phi/2-\phi_0)\gamma_1\gamma_2$ and $H_p$ given above as the perturbation. Consider an in-state $|i\rangle=\gamma_{1,-k}|\Omega\rangle$ of Majorana fermion $\gamma_1$, and a transmitted out-state $|f\rangle=\gamma_{1,-k'}|\Omega\rangle$, where $|\Omega\rangle$ is the system ground state. The transmission coefficient $t$ is then given by $$t\approx\left\langle f\right|T_\tau e^{-i\int_{-\infty}^\infty (H_I+H_p) d\tau}\left|i\right\rangle,$$ where $T_\tau$ stands for the time ordering. The zero-order $t^{(0)}$ is simply $\delta_{kk'}$. The first-order contribution $t^{(1)}$ is $$t^{(1)}= \left\langle f\right|-i\int (H_{I}+H_p)d\tau\left|i\right\rangle.$$ Since $H_I$ is odd in $\gamma_1$ and $\gamma_2$, its first-order contribution vanishes. The second term of $H_p$ is purely imaginary and therefore does not contribute to the conductance $\sigma_{12}'$. The second-order correction $$t^{(2)}\sim -\frac{1}{2}\left\langle f\right|T_\tau\int(H_I+H_p)(\tau)(H_I+H_p)(\tau')d\tau d\tau'\left|i\right\rangle.$$ The $H_I^2$ term gives a constant contribution $\sim -\xi^2\sin^2(\delta\phi/2-\phi_0)$, in agreement with calculations in Appendix B. The cross term $H_IH_p$ vanishes because it is odd in $\gamma_1$ and $\gamma_2$. The $H_p^2$ term results in a temperature dependent correction to the real part of transmission coefficient $t$ as $$\delta\text{Re}(t)\sim-\delta_{kk'}\left(\lambda_{p}^{\text{eff}}\right)^2=-\delta_{kk'}\lambda_p^2 T^6.$$ Therefore, the transmission coefficient $t$ generically decreases as temperature $T$ increases. When the temperature $T$ is above a characteristic temperature $T_c\sim\lambda_{p}^{-1/3}$, the interaction $\lambda_p$ at the point contact dominates, so that $t$ becomes small and $r$ becomes large. In this case, the above perturbative treatment is no longer valid. However, this case can be effectively viewed as a breaking up of original Majorana edge states $\gamma_1$ and $\gamma_2$ and a remerge of them into two new Majorana edge states $\psi_1$ and $\psi_2$ on the left and right of the point contact, and of the two TSCs merging into a single TSC. In the temperature range $T_c\ll T\ll |\Delta|$, we can do a perturbation calculation about the high temperature fixed point before the superconducting phase is destroyed. This scenario is very similar with our setup in Fig. 1a, except that the two edges are brought together at the point contact. Since the region between the edges in this case is a SC, there are both fermion tunnelings and vortex tunnelings between edges [@fendly2007]. The effective action for this point contact is $$\begin{aligned} \mathcal{S}'=& \int d\tau \int dy\left[\psi_1 i(\partial_{\tau}+v_m \partial_y) \psi_1 +\psi_2 i(\partial_{\tau}-v_m \partial_y) \psi_2\right. \nonumber \\ &\left.+\lambda_\psi \delta(y) i\psi_1\psi_2 +\lambda_\sigma \delta(y) \sigma_1\sigma_2\right],\end{aligned}$$ where $\sigma_1$ and $\sigma_2$ are the vortex operators on edges with a scaling dimension $[\sigma_1]=[\sigma_2]=1/16$. Dimension counting renders $[\lambda_\psi]=0$ and $[\lambda_\sigma]=7/8$, so the vortex-vortex tunneling is the most relevant. Therefore, at a high temperature $T$, the vortex-vortex tunneling term gives the temperature dependence of transmission coefficient $t$ $$t\sim\lambda^{2}_\sigma T^{-7/4}.$$ The power-law relation is valid above a characteristic temperature $T'_c\sim \lambda^{8/7}_\sigma$, provided the SC gap $|\Delta|$ is much higher. In fact, this confirms the robustness of the half-quantized plateau. For in the setup with reasonable finite temperature, the edges are far away from each other, so the tunneling strengths including $\lambda_{\sigma}$ are sufficiently tiny, resulting in an extremely low $T_c'$. [52]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](\doibase 10.1103/PhysRevB.61.10267) [****,  ()](\doibase 10.1103/PhysRevLett.86.268) [****,  ()](\doibase 10.1103/RevModPhys.80.1083) @noop [****,  ()](\doibase 10.1103/PhysRevLett.102.187001) [****,  ()](\doibase 10.1103/PhysRevB.78.195125) [****, ()](\doibase 10.1103/PhysRevLett.100.096407) [****, ()](\doibase 10.1103/PhysRevLett.102.216403) [****,  ()](\doibase 10.1103/PhysRevLett.103.020401) [****,  ()](\doibase 10.1103/PhysRevLett.104.040502) [****,  ()](\doibase 10.1103/PhysRevB.81.125318) [****,  ()](\doibase 10.1103/PhysRevB.82.184516) [****,  ()](\doibase 10.1103/PhysRevLett.114.236803) @noop [****, ()](\doibase 10.1103/RevModPhys.75.657) [****, ()](\doibase 10.1103/PhysRevLett.105.136401) @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevLett.49.405) [****,  ()](\doibase 10.1103/PhysRevLett.61.2015) [****, ()](\doibase 10.1103/RevModPhys.82.3045) [****,  ()](\doibase 10.1103/RevModPhys.83.1057) [****,  ()](\doibase 10.1103/PhysRevB.78.195424) [****,  ()](\doibase 10.1103/PhysRevLett.101.146802) @noop [****,  ()]{} [****,  ()](\doibase 10.1126/science.1187485) [****,  ()](\doibase 10.1103/PhysRevLett.111.136801) [****,  ()](\doibase 10.1103/PhysRevLett.111.086803) @noop [****,  ()](\doibase 10.1103/PhysRevLett.90.206601) [****,  ()](\doibase 10.1103/PhysRevB.81.233405) [****,  ()](\doibase 10.1126/science.1234414) @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevLett.113.137201) [****,  ()](\doibase 10.1103/PhysRevLett.114.187201) @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevB.89.085106) @noop @noop @noop [****,  ()](\doibase 10.1103/PhysRevLett.103.107002) [****,  ()](\doibase 10.1103/PhysRevB.79.161408) [****, ()](\doibase 10.1103/PhysRevLett.102.216404) [****,  ()](\doibase 10.1103/PhysRevLett.103.237001) [****,  ()](\doibase 10.1103/PhysRevLett.105.077001) [****,  ()](\doibase 10.1103/PhysRevB.83.100512) [****, ()](\doibase 10.1103/PhysRevB.53.16390) [****,  ()](\doibase 10.1103/PhysRevB.78.224510) [****,  ()](\doibase 10.1126/science.1216466) [****, ()](\doibase 10.1103/PhysRevB.46.15233) “,” in [**](\doibase 10.1002/9783527617258.ch4) (, ) pp.  [****, ()](\doibase 10.1103/PhysRevB.75.045317)

--- abstract: 'Supernovae (SNe) exploding in a dense circumstellar medium (CSM) are hypothesized to accelerate cosmic rays in collisionless shocks and emit GeV $\gamma$ rays and TeV neutrinos on a time scale of several months. We perform the first systematic search for $\gamma$-ray emission in [*Fermi*]{}LAT data in the energy range from $100\,$MeV to $300\,$GeV from the ensemble of 147 SNe Type IIn exploding in dense CSM. We search for a $\gamma$-ray excess at each SNe location in a one year time window. In order to enhance a possible weak signal, we simultaneously study the closest and optically brightest sources of our sample in a joint-likelihood analysis in three different time windows (1year, 6months and 3months). For the most promising source of the sample, SN2010jl (PTF10aaxf), we repeat the analysis with an extended time window lasting 4.5 years. We do not find a significant excess in $\gamma$ rays for any individual source nor for the combined sources and provide model-independent flux upper limits for both cases. In addition, we derive limits on the $\gamma$-ray luminosity and the ratio of $\gamma$-ray-to-optical luminosity ratio as a function of the index of the proton injection spectrum assuming a generic $\gamma$-ray production model. Furthermore, we present detailed flux predictions based on multi-wavelength observations and the corresponding flux upper limit at $95\%$ confidence level (CL) for the source SN2010jl (PTF10aaxf).' bibliography: - 'SN\_paper.bib' title: 'Search for Early Gamma-ray Production in Supernovae Located in a Dense Circumstellar Medium with the [*Fermi*]{}LAT' --- Keywords. Methods: data analysis; cosmic rays, gamma rays, supernova

--- abstract: 'A general scalar-tensor theory of gravity carries a conserved current for a trace free minimally coupled scalar field, under the condition that the potential $V(\phi)$ of the nonminimally coupled scalar field is proportional to the square of the parameter $f(\phi)$ that is coupled with the scalar curvature $R$. The conserved current relates the pair of arbitrary coupling parameters $f(\phi)$ and $\omega(\phi)$, where the latter is the Brans-Dicke coupling parameter. Thus fixing up the two arbitrary parameters by hand, it is possible to explore the symmetries and the form of conserved currents corresponding to standard and many different nonstandard models of gravity.' author: - Abhik Kumar Sanyal title: Scalar tensor theory of gravity carrying a conserved current --- -8mm -6mm -11mm Dept. of Physics, Jangipur College, Murshidabad, India - 742213\ and\ Relativity and Cosmology Research Centre\ Dept. of Physics, Jadavpur University\ Calcutta - 700032, India\ e-mail : [email protected]\ PACS 04.50.+h **[Introduction]{}** ==================== The importance of scalar-tensor theory of gravity has always increased since the advent of the Brans-Dicke [@b:d] theory of gravity, which was originally introduced to incorporate Mach principle. Later, Brans-Dicke field has been found to arise even from higher dimensional theories, like superstring theories [@a:w]. Brans-Dicke theory leads to Einstein’s theory in the limit $\omega \rightarrow \infty$ and so $\omega$, the Brans-Dicke function, is constrained by classical tests of general relativity. The light deflection and the time delay experiments demand $\omega > 500$, while the bounds on the anisotropy of the microwave background radiation demands $\omega < 30$. Hence, a viable model of the scalar-tensor theory of gravity requires $\omega$ to be a function of time and that too via the nonminimally coupled scalar field $\phi$. It has been observed that [@s:a] an asymptotic negative value of $\omega(\phi)$ leads to late time acceleration of the universe in Brans-Dicke cosmology. Induced theory of gravity [@a:m] on the other hand, appeared in an attempt to construct a gravitational theory consistent with quantum field theory in curved space time. It identifies the scalar field with the inflaton and has been found to be a strong candidate in several unified theories [@g:s]. Low energy effective action of string theory [@g:s] is also a nonminimally coupled scalar tensor theory of gravity, that contains a scalar field called dilaton. All these theories lead mostly to power law inflation [@k:k]. Further a scalar-tensor theory of gravity containing a coupling parameter in the form $f(\phi) = 1-\zeta\phi^2$, where, $f(\phi)$ is the parameter that is coupled with the scalar curvature $R$, is found to overcome the graceful exit problem and the problem of density perturbation for arbitrary large negative value of $\zeta$ [@f:u]. Recently, it has been observed [@a:d] that some string inspired scalar-tensor theories of gravitation [@g:v] lead to quintessence [@q:e]. Due to such growing interests of scalar-tensor theories of gravitation, it is required to study the theory in some more detail. All the above nonstandard theories of gravity which appeared in different context, contain the coupling parameters $f(\phi)$and $\omega(\phi)$ together with an arbitrary form of the potential $V(\phi)$. For a general form of scalar-tensor theories of gravity, these coupling parameters and the form of the potential are not known a-priori. It becomes less difficult to handle such theories if there exists some form of symmetry such that the corresponding conserved current somehow relates the coupling parameters. With this motivation, de Ritis et-al [@r:e] for the first time proposed that, if one demands the existence of Noether symmetry corresponding to such type of action, it might fix up the form of the coupling parameters and the potential. Indeed it does, if one of the two arbitrary parameters is considered. It has been observed [@a:b] however, that Noether symmetry often leads to some unpleasent features viz., it makes the Lagrangian degenerate, as the Hessian determinant $W = |\Sigma_{i,j}(\frac{\partial^2 L}{\partial\dot{q_{i}}\partial\dot{q_{j}}})| = 0$, and the effective Newtonian gravitational constant negative. Further, Noether theorem has So far been applied only in the cosmological models and it is not known how the existence of Noether symmetry for a nonstandard gravitational action can be claimed keeping the space-time arbitrary. Whatsoever, in the situations mentioned above [@a:b], some other forms of symmetry were found to be present that are free from such unpleasent features. Those symmetries were found right from the field equations, rather from the invariance of the action under some suitable transformations, as is required to find Noether symmetry . In view of the presence of such dynamical symmetries [@a:b] disussed above, here we are motivated to find such metric independent symmetry and the corresponding conserved current for a general form of scalar-tensor theory of gravity. In the following section we have achieved in finding such a conserved current for a trace free minimally coupled scalar field under the condition that the potential $V(\phi)$ of the nonminimally coupled field $\phi$ should be related to the parameter $f(\phi)$, that couples with the curvature scalar $R$. It is noteworthy that a trace free matter field corresponds to the radiation dominated era in the context of homogeneous cosmology. The conserved current found in the process relates the two coupling parameters considered here. Thus, such symmetry exists for all types of standard and nonstandard (scalar-tensor) theories of gravity. In section 3, we have made different choices of the coupling parameters and thus explored the forms of the conserved current in standard and some nonstandard models. Thus we have been able to study the form of symmetry of different theories of gravitation in a single frame-work. Section 4 is devoted to explore symmetry of axion-dilaton string effective action [@k:l]. **[Action with a conserved current]{}** ======================================= The gravitational action nonminimally coupled to a scalar field $\phi$ can be expressed in the following general form,- A =  d\^4x, where, $R$ is the Ricci scalar, $\sqrt{-g}$ is the determinant of the metric of the four-space, $f(\phi)$ and $\omega(\phi)$ are coupling parameters while $\chi$ corresponds to a minimally coupled scalar field. The coupling constant $k=16\pi G$. The above action (1) corresponds to all types of scalar-tensor theories of gravity other than dilaton-axionic one [@k:l], that we shall take up in section 4. The field equations corresponding to action (1) are, $$f(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R)+f^{;\alpha}_{;\alpha}g_{\mu\nu}-f_{;\mu ;\nu}-\frac{\omega}{\phi}\phi_{,\mu}\phi_{,\nu}+ \frac{1}{2}g_{\mu\nu}(\frac{\omega}{\phi}\phi_{,\alpha}\phi^{,\alpha}+V(\phi))$$ =\[\_[,]{}\_[,]{}-g\_(\_[,]{}\^[,]{}+U())\]=T\_ Rf’+2\^[;]{}\_[;]{}+(-)\^[,]{}\_[,]{}-V’() \^[;]{}\_[;]{}+= 0 where, f\^[;]{}\_[;]{}=f”\^[,]{}\_[,]{}+f’\^[;]{}\_[;]{} In the above $T_{\mu\nu}$ is the energy-momentum tensor corresponding to the minimally coupled scalar field $\chi$ and dash $(')$ represents derivative with respect to $\phi$. Now the trace of equation (2) is, Rf-3f\^[;]{}\_[;]{}-\^[,]{}\_[,]{}-2V = (\^[,]{}\_[,]{}+4U)=-T\^\_ Hence we can eliminate the curvature scalar $R$, in view of equations (3) and (6) and thus obtain the following relation, (3f’\^2+2f)’\^[,]{}\_[,]{}+2(3f’\^2+2f)\^[;]{}\_[;]{}+2f’V-fV’=-f’k(\^[,]{}\_[,]{}+4U)=f’k T\^\_ which can finally be expressed as \_[;]{}-()’=-(\^[,]{}\_[,]{}+4U)=T\^\_ In view of the above equation (8) we can make the following statement that there exists a conserved current $J^{\mu}$, where J\^\_[;]{}=\[(3f’\^2+2f)\^\^[;]{}\]\_[;]{}, corresponding to the general form of scalar tensor theory of gravitational action (1), provided the trace of the energy-momentum tensor $T^{\mu}_{\mu}$ of the minimally coupled scalar field $(\chi)$ vanishes and the potential $V(\phi)$ for the nonminimally coupled scalar field $(\phi)$ is either zero or proportional to the square of the coupling parameter $f(\phi)$. Now, in the context of cosmology one breaks the general covariance and splits the space-time into space and time components by defining time-like hypersurfaces. Hence, one can transform the volume integral into the surface integral as, \_[V]{} J\^\_[;]{}d\^4 x = \_ J\^d\_. As a result, the above conservation principle implies that, on every time-like hypersurface $\int_{\Sigma} J^{\mu}d\Sigma_{\mu}=0$. Further, since J\^\_[;]{}=\[(3f’\^2+2f)\^\^[;]{}\]\_[;]{}=\[(3f’\^2+2f)\^\^[,]{}\]\_[,]{} = 0, therefore, in the context of homogeneous cosmology, = \[(3f’\^2+2f)\^\] = 0, where, dot represents time derivative and $J^{0}$ is the time component of the current density. Thus, the conservation principle demands that, (3f’\^2+2f)\^ is an integral of motion under the conition already stated, viz., $T^{\mu}_{\mu}= 0$ and $V(\phi)=\lambda f^2(\phi)$ or is zero, $\lambda$ being a constant. In the context of homogeneous cosmology, vanishing of the trace of the matter field either indicates radiation dominated era, ie. $\rho = 3p$ where, the energy density, $\rho = \frac{1}{2}\dot \chi^2 + U(\chi)$ and the pressure, $p = \dot \chi^2 - U(\chi)$, or the vacuum $\rho = p = 0$. **[Different forms of scalar tensor theory of gravity carrying a conserved current]{}** ======================================================================================= We have observed that the coupling parameters are not fixed in the process of finding the conserved current. There only exists a pair of relations amongst them, in view of the conserved current (9), which relates $\omega(\phi)$ with $f(\phi)$ and the relation between the $f(\phi)$ and the potential $V(\phi)$ in the form $V(\phi) = f^2(\phi)$. Therefore, fixing such parameters by hand it is possible to study different situations. This is done in this section. To understand the situations thus arise, we shall often refer to homogeneous cosmological models, viz., the isotropic Robertson-Walker space time given by ds\^2=-dt\^2+a(t)\^2(d\^2+f\^2()(d\^2+sin\^2  d\^2)), and anisotropic axially symmetric Kantoski-Sachs space time given by ds\^2=-dt\^2+a\^2(t)dr\^2+b\^2(t)(d\^2+sin\^2  d\^2). Einstein’s theory, $f =$ constant and $\omega = k\frac{\phi}{2}$. ----------------------------------------------------------------- This case corresponds for $f = 1$ to the Einstein’s theory of gravity with a pair of minimally coupled scalar fields $\phi$ and $\chi$, with the potential $V(\phi)= \lambda$, a constant. Thus the corresponding action A =  d\^4x. carries a conserved current $J^{\mu}$, such that J\^\_[;]{}= \^[;]{}\_[;]{} = 0, in the absence of $\chi$-field or if the trace of the energy-momentum tensor corresponding to $\chi$-field vanishes ($T^{\mu}_{\mu} = 0$). In the context of homogeneous cosmology, the integral of motion is $\sqrt{g}\dot \phi$ which in the Robertson-Walker space-time reads $a^3 \dot \phi =$ constant. Since the potential $U(\chi)$ still remains arbitrary, so one has the liberty to consider the $\chi$-field as the quintessence field with a potential in the form $U(\chi) = \frac{\beta}{\chi^{\beta}}$, where, $\beta$ is a constant. Further, if the $\phi$-field is now treated as the perfect fluid source, then with the choice $$\frac{1}{2}\dot \phi^2 + V = \rho_{\phi}$$ and $$\frac{1}{2}\dot \phi^2 - V = p_{\phi}$$ where, $\rho_{\phi}$ and $p_{\phi}$ are the energy density and the pressure of the perfect fluid, we know that for $V = \lambda=0$ the above situation leads to stiff fluid equation of state $\rho_{\phi} = p_{\phi}$, for which we recover the well known result, viz., $\rho_{\phi} a^6=$ constant. On the other hand, if $V = \lambda \ne 0$, the same integral of motion $a^3 \dot\phi$ exists for the equation of state $\rho_{\phi} - p_{\phi} = 2\lambda$. Finally if $\dot \phi^2$ is small enough, ie., under slow roll approximation, $\lambda$ acts as cosmological constant for which the equation of state is $\rho_{\phi} + p_{\phi} = 0$. Brans-Dicke theory, $f = \phi$ and $\omega =$ constant. ------------------------------------------------------- In this situation $V = \lambda \phi^2$, and the action takes the form, A =  d\^4x, which reduces exactly to the action for Brans-Dicke scalar tensor theory of gravity minimally coupled to a matter field, provided $V = \lambda \phi^2 = 0$. Equation (8) now takes the following form, \^[;]{}\_[;]{}= T\^\_. Thus we again recover the well known result and thus conclude that if the trace of the matter field vanishes, Brans-Dicke action admits a conserved current $J^{\mu}$, where, $J^{\mu}_{;\mu} = \phi^{;\mu}_{;\mu}= 0$, even in the presence of a potential in the form $V = \lambda \phi^2$ of the Brans-Dicke field. We remember that Brans-Dicke originally replaced $G$ by $\phi^{-1}$, hence, in the above action (18) we have deliberately introduced a new coupling constant $k_{1} = 16\pi$. Induced theory of gravity, $f = \epsilon \phi^2$ and $\omega = \frac{\phi}{2}$. ------------------------------------------------------------------------------- Here, the potential is $V = \lambda \phi^4$ and the action is, A =  d\^4x. The above action (20) for induced theory of gravity thus admits a conserved current $J^{\mu}$ such that, $J^{\mu}_{;\mu} = (\phi \phi^{;\mu})_{;\mu}= 0$, if the $\chi$-field is trace free, provided $\epsilon \ne -\frac{1}{12}$. In the context of homogeneous cosmology the conservation principle reads, () = 0. In an attempt to find the forms of the coupling parameters and the potential by demanding the existance of Noether symmetry we have earlier observed that both in isotropic Robertson-Walker and anisotropic Kantowski-Sachs space-times [@a:b] Noether symmetry exists for $\epsilon = -\frac{1}{12}$, which makes the Newtonian gravitational constant negative and thus unphysical. It also makes the Lagrangian degenerate in the absense of the $\chi$-field, since Hessian determinant $W = |\Sigma_{i,j}(\frac{\partial^2 L}{\partial \dot q_{i}\partial \dot q_{j}})|$ vanishes. However, in that context, we have for the first time observed the existence of other conserved quantities found in view of the field equations, viz., $a^3 \phi \dot\phi =$ constant in the Robertson-Walker space-time and $ab^2 \phi \dot\phi =$ constant in anisotropic Kantowski-Sachs space-times, mentioned above. Thus, we recover the same results here too and hence conclude that induced theory of gravity admits above integral of motions given in (21), in vacuum or in the radiation dominated era provided, the induced field has got a potential in the form, $V(\phi) = \lambda \phi^4$. String effective action $(3f'^2+2f\frac{\omega}{\phi})' = \beta(3f'^2+2f\frac{\omega}{\phi})$. ---------------------------------------------------------------------------------------------- Under the above assumption, $f(\phi)$ and $\omega(\phi)$ are related in the following manner, 3f’\^2+2f=e\^, where, $\alpha$ and $\beta$ are constants. Thus, there exists a conserved current $J^{\mu}$, such that J\^\_[;]{} = (e\^\^[;]{})\_[;]{} = 0. One can fix up $\omega(\phi)$ by choosing $f(\phi)$ in the form $$f(\phi) = n e^{\frac{\beta}{2}\phi}$$ where, $n$ is a constant. As a result, $\omega(\phi)$ takes the form $$\omega(\phi) = \frac{2\alpha-3n^2 \beta^2}{4n}\phi e^{\frac{\beta}{2}\phi}.$$ Thus, the corresponding action is, A = n d\^4x. For $\beta > 0$ and a decaying $\phi$-field, the above action in the cosmological context, asymptotically goes over to the Einstein’s action with a cosmological constant, which is minimally coupled to a scalar field $\chi$, that acts as a perfect fluid source. Thus the action (24) asymptotically leads to de-Sitter universe in theabsence of the $\chi$-field. However, the action (24) reduces to the Gravi-dilaton string effective action minimally coupled to the scalar field $\chi$, for $\beta = -2$ and $\alpha = 4n^2$, given by Gasperini and Veneziano [@a:d] as, A = n d\^4x. In the above action (25), $n = \frac{M^2_{s}}{2}$ is the fundamental string length parameter. Hence, the Gravi-dilaton string effective action admits an integral of motion $a^3 e^{-\phi}\dot \phi^2$ in the Robertson-Walker space-time and $ab^2 e^{-\phi}\dot \phi^2$ in Kantowski-Sachs space-time in the radiation dominated era. It is to be mentioned that Gasperini [@a:d] has shown that such an action leads to quintessential effects. Late time acceleration of the universe may also be realized in view of the following action, different from the above Gravi-dilaton string effective action, carrying the same conserved current (23), viz., A =  d\^4x, where, we have chosen $f(\phi) = \phi^n$. Thus, $V(\phi) = \Lambda \phi^{2n}$, and =. The above form (27) of $\omega(\phi)$ admits a sign flip and so $\phi$-field acts as exotic matter. For $n = 0$, $\omega = \frac{\alpha}{2}\phi e^{\beta\phi}$, and hence there is no sign flip. However, for $n = 1$, $\omega = \frac{\alpha}{2} e^{\beta\phi} - \frac{3}{2}$ and for $n = 2$, $\omega = \frac{\alpha}{2} \frac{e^{\beta\phi}}{\phi} - 6\phi$, and so on. For large positive $\beta$ and for a decaying $\phi$-field, the first term of the last two expressions of $\omega$ falls of rapidly and hence $\omega < 0$ asymptotically, as a result, late time acceleration may be realized. For $n > 1$, $\omega$ finally vanishes, which may cause a future deceleration of the universe. The result is more pronounced for $n = -1$, for which $\omega = \frac{\alpha}{2}\phi^2 e^{\beta\phi} - \frac{3}{2\phi^2}.$ Nonminimal coupling $3f'^2+2f\frac{\omega}{\phi} = f^2_{0} =$ constant. ----------------------------------------------------------------------- Despite the standard forms of the scalar tensor theory of gravity investigated so far, we can also study many other nonminimally coupled theories carrying a conserved current. Some of these are explored in this subsection. Under the above choice the conserved current $J^{\mu}$ in view of equation (9) is such that J\^\_[;]{} = \^[;]{}\_[;]{} = 0. We shall now choose either $\omega(\phi)$ or $f(\phi)$ to fix up the other along with the potential $V(\phi)$. [**[Case 1. $\omega = \frac{\phi}{2}$]{}.**]{} As a result of such a choice, f() = f\_[0]{}\^2 - ,     V() = (f\_[0]{}\^2 - )\^2. Thus, in the cosmological context, if $\phi$ falls off with time then the signature of the effective gravitational constant flips from a negative to positive value. To restrict $f$ to a positive value throughout, $\phi^2$ should be restricted to, $\phi^2 < 12 f_{o}^2$. [**[Case 2. $f = \phi^n, n$]{}**]{} being a constant. Hence, V() = \^[2n]{},   = Now, for $n = 0$, $f = 1, V = \lambda =$ constant, and $\omega = \frac{f_{0}^2}{2}\phi$. Thus for $f_{0}^2 = k$, we recover Einstein’s gravity with a pair of nonminimally coupled fields, which has been studied in subsection 3.1. For, $n = 1$, $f(\phi) = \phi, V(\phi) = \lambda\phi^2$ and $\omega = \frac{f_{0}^2 - 3}{2}$ and hence we recover Brans-Dicke theory of gravity, which has been studied in subsection 3.2. For $n = 2$, $f(\phi) = \phi^2, V(\phi) = \lambda\phi^4$ and $\omega = \frac{f_{0}^2 - 12\phi^4}{2\phi}$. In this situation, a sign flip of $\omega(\phi)$ from negative to positive value for decaying $\phi$ field is observed and as a result, in the cosmological context, $\omega(\phi)$ turns out to be indefinitely large asymptotically. For, higher values of $n$ situation does not alter and the qualitative features remain the same. For $n = -1$, $f(\phi) = \phi^{-1}, V(\phi) = \lambda\phi^{-2}$ and $\omega = \frac{f_{0}^2\phi^4 - 3}{2\phi^2}$. Thus, the action (1) now takes the following form, A = d\^4x, where, $L_{m}$ indicates matter lagrangian. This action has got some interesting features. In the context of homogeneous cosmology, this action admits an integral of motion $\sqrt{-g}\dot\phi$, in the radiation era. The form of the potential $V(\phi)$ indicates that the $\phi$-field acts as ’Quintessent’ field. Indeed it is so, since for a time decaying $\phi$-field the sign of $\omega$ flips from a positive to negative value. Further, $\frac{R}{\phi}$ tends to remain finite asymptotically. Thus it appears that late time acceleration of the Universe can be explained in view of the above action. **[Axion-Dilaton string effective action carrying a conserved current.]{}** =========================================================================== To study the symmetry of axion-dilaton string effective action we consider the following general form of such an action, A =  d\^4x, where, we have introduced yet another coupling parameter $h(\phi)$. The above action reduces to the axion-dilaton string action for $f = $ constant and $\frac{\omega}{\phi} = \frac{1}{2}$. Proceeding in a similar fashion as is done in section 2, we can construct the following equation that is complementary to equation (8), viz., \_[;]{}-()’=-. We thus make the following statement that there exists a conserved current $J^{\mu}$ corresponding to the action (32), such that, J\^\_[;]{}=\[(3f’\^2+2f)\^\^[;]{}\]\_[;]{}, provided, $$V(\phi) = \lambda f^2 (\phi),~~h(\phi) = k f(\phi),~~U(\chi) = 0,$$ where, $\lambda$ and $k$ are constants. The action that admits such a conserved current, is given by, A =  d\^4x, Hence, we observe that a conserved current exists here too but under much restrictive condition. It is noteworthy that all the coupling parameters $h(\phi), \omega(\phi)$, and the potentials $V(\phi), U(\chi)$ are now fixed, once $f(\phi)$ is fixed up by hand. However, it is not possible to recover dialton-axionic form of the action, since, for $f =$ constant, $h(\phi)$ also becomes a constant, and $\chi$-field turns out to a minimally coupled one. Therefore we understand that dilaton-axion string field action does not admit a symmetry in the classical context, though it admits some symmetry [@j:m] in the quantum level. Nevertheless, it is interesting to study this situation also. In the context of homogeneous cosmology there exist an integral of motion $g \dot \phi^2 \frac{\omega}{\phi}$ under this situation, and thus $\phi$-field still behaves as a nonminimally coupled one. **[Concluding remarks]{}** ========================== The existence of a general form of symmetry and the corresponding conserved current for the standard and some nonstandard theories of gravity has been found for a trace free minimally coupled scalar field under the condition that the potential and one of the coupling parameters are related as $V(\phi) = f^2 (\phi)$. The conserved current relates the two coupling parameters $f(\phi),~ \omega(\phi)$ with the scalar field $\phi$ and the scale factor, in the context of homogeneous cosmology. Thus choosing the coupling parameters by hand we have explored symmetries of different standard and nonstandard models. The existence of the integral of motion in the cosmological context makes it easier to handle the field equation for studying exact solutions. The symmetry thus found is not a result of the invariance of action under some suitable transformation and so is not an artifact of Noether’s theorem. It has been also observed that dilaton-axionic string field action does not admit such symmetry. [20]{} C.Brans and R.H.Dicke, Phys.Rev. 124, 925 (1961). L.F.Abott and M.B.Wise, Ncl.Phys. B325, (1989). S.Sen and A.A.Sen, Phys.Rev.D 63, 124006 (2001). S.Adler, Rev.Mod.Phys. 54, 729 (1982). M.B.Green, J.Schwartz and E.Witten, Superstring theory, Cambridge Univ. Press, Cambridge, Ma (1987) and A.Zee, Phys.Rev.Lett. 42, 417 (1979). S.Kalara, N.Kaloper and K.A.Olive, Nucl.Phys. B341, 252 (1990). R.Fakir and W.G.Unruh, Phys.Rev.D 41, 1783 (1990) and R.Fakir, S.Habib and W.G.Unruh, Ap.J., 394, 396 (1992). M.Gasperini, arXiv:gr-qc/0105082, Phys.Rev.D (2001) and M.Gasperini and G.Veneziano, arXiv:hep-th/0207130, Phys.Rept. 373, 1 (2003). M.Gasperini and G.Veneziano, Astropart.Phys. 1, 317 (1993). R.R.Caldwell, R.Dave and P.J.Steinhardt, Phys.Rev.Lett. 80, 582 (1998), and J.A.Freeman and I.Vaga, Phys.Rev.D 57, 4642,(1998). R.de Ritis etal, Phys.Rev.D 42, 1091 (1990). A.K.Sanyal and B.Modak, Class.Quan.Gravit. 18, 3767 (2001), A.K.Sanyal, arXiv:gr-qc/0107053, Phys.Lett. B 524, 177 (2002) and A.K.Sanyal, C.Rubano and E.Piedipalumbo, Gen.Rel.Grav. 35, 1617 (2003). R.Kallosh, A.Linde, S.Prokushkin and M.Shmakova, arXiv:hep-th/0208156, Phys.Rev.D 66,123505 (2002). J.Maharana, arXiv:hep-th/0207059, Phys.Lett.B 549, 7 (2002).

--- author: - | Matt Visser[^1]\ School of Mathematical and Computing Sciences, Victoria University of Wellington, New Zealand.\ E-mail: title: 'Physical wavelets: Lorentz covariant, singularity-free, finite energy, zero action, localized solutions to the wave equation' --- Ł Motivation ========== While the particle physics community has for some time made extensive use of extended field configurations such as solitons, instantons, and sphalerons, no direct use has yet been made of the quite extensive literature on “localized wave” configurations developed by the engineering, optics, and mathematics communities. (For selected references see [@zed; @tippet; @LG; @LZG; @Kaiser; @Lekner].) These localized waves are classical solutions of the wave equation that are partially localized in space or time, this localization generally coming at a cost such as infinite total energy and/or instability (leading to dispersion or diffraction). The catalogue of known localized waves is large and growing [@key], but most of the known examples are not in a form that would be easy to apply to particle physics problems. In this article I will exhibit a particularly simple “physical wavelet” that is more promising from a particle physics standpoint. It satisfies the properties that: - It is a localized wave that solves the wave equation. - It is a Lorentz covariant classical field configuration that lives in physical Minkowski space. - The field is everywhere finite and nonsingular, and has quadratic falloff in both space and time. - The total energy is finite, depending on the peak field and the width of the pulse. - The total action is zero. These physical wavelets can be constructed for both complex and real scalar fields. Extending these idea to the Maxwell and Yang-Mills fields is straightforward. The simplest case is that of the complex scalar field and it is to that case that I first turn. Complex scalar field ==================== Let $\eta_{ab} = \hbox{diag}[+1,-1,-1,-1]$ be the Minkowski metric \[particle physics signature\], let $x_0$ be an arbitrary 4-vector, and let $\zeta^a$ be arbitrary timelike 4-vector, then $$\phi(x) = - {\phi_0 \; (\eta_{ab} \; \zeta^a \; \zeta^b) \over \eta_{ab} \;[x^a-x_0^a-i\zeta^a] \; [x^b-x_0^b-i \zeta^b] } \label{E:mother}$$ is a Lorentz covariant, finite energy, zero-action solution of the d’Alembertian wave equation $\Delta \phi=0$. The “center” of the pulse is at $x_0$ and its “width” is $a = \sqrt{\eta_{ab} \; \zeta^a \; \zeta^b}$. The field is everywhere finite and in fact $$|\phi(x)| \leq |\phi_0|.$$ To see this, use the fact that $\zeta$ is timelike. Then, using the manifest Lorentz covariance of the field configuration, we can without loss of generality first translate $x_0\to 0$, and then go into the zero-momentum frame where $$\zeta^a = (a,0,0,0).$$ Then the field configuration is $$\phi(x) = - {\phi_0 \; a^2 \over [t-ia]^2 - x^2-y^2-z^2 }.$$ That is $$\phi(x) = {\phi_0 \; a^2 \over r^2-t^2+a^2+2iat }.$$ Once written in this form it is a straightforward exercise to verify that the wave equation is satisfied. To see that the field is everywhere bounded note $$\begin{aligned} |\phi|^2 &=& {|\phi_0|^2 \; a^4 \over (r^2-t^2+a^2)^2 + 4 a^2 t^2} = {|\phi_0|^2 \; a^4 \over (r^2+t^2+a^2)^2 - 4 r^2 t^2} \leq \nonumber\\ & \leq & {|\phi_0|^2 \; a^4 \over (r^2+t^2+a^2)^2 - (r^2 +t^2)^2} = {|\phi_0|^2 \; a^4 \over a^4 + 2a^2(r^2+t^2)} \leq |\phi_0|^2.\end{aligned}$$ [From]{} the penultimate inequality we also derive $$|\phi|^2 \leq {1\over2} \; |\phi_0|^2 \; { a^2 \over r^2+t^2},$$ demonstrating the promised quadratic falloff in both space and time. Indeed for fixed $t$ the magnitude of the field is maximized when $$r^2 = \max \{ t^2-a^2, 0 \},$$ showing that the configuration disperses to spatial infinity at both $t\to\pm\infty$. To calculate the 4-momentum, we remind the reader that the stress-energy tensor for a massless complex scalar is $$T_{ab} = {1\over2}\left[\phi_a^* \; \phi_b+ \phi_a \; \phi_b^*\right] - {1\over2} \eta_{ab} |\nabla \phi|^2.$$ Then $$\begin{aligned} \nabla_a T^{ab} &\equiv& {1\over2} \Delta \phi^* \; \nabla_b \phi + {1\over2} \nabla^a \phi^* \;\nabla_a \nabla_b \phi + {1\over2} \Delta \phi \; \nabla_b \phi^* + {1\over2} \nabla^a \phi \;\nabla_a \nabla_b \phi^* \nonumber \\ &&\qquad\qquad -{1\over2} \nabla^b \phi^* \;\nabla_a \nabla_b \phi^* -{1\over2} \nabla^a \phi^* \;\nabla_a \nabla_b \phi, \\ &\equiv & {1\over2} \left[ \Delta \phi^* \; \nabla_b \phi + \Delta \phi \; \nabla_b \phi^* \right],\end{aligned}$$ which vanishes by the equations of motion. But this means that $$P^\mu = \oint T^{ab} \; \d \Sigma_b$$ is a conserved quantity, the 4-momentum of the configuration, which is independent of the particular spacelike hypersurface $\Sigma$ chosen to do the integration. By simple dimensional analysis $$P^a = C \; |\phi_0|^2 \zeta^a,$$ where $C$ is a dimensionless number to be calculated. \[Note that $\zeta^a$ has the dimensions of a position vector — a distance.\] The energy density is $$\rho = {1\over2} \left[ |\partial_t \phi|^2 + |\partial_r \phi|^2 \right],$$ and in the zero-momentum frame is easily calculated to be $$\rho = { 2 a^4 |\phi_0|^2 (r^2+t^2+a^2) \over (r^2-t^2+a^2+2iat)^2(r^2-t^2+a^2-2iat)^2. }$$ For arbitrary $t$ this integrates to $${\mathcal E} = \oint \d^3 r \rho = \int_0^\infty 4\pi\; r^2 \rho = {1\over2} \pi^2 |\phi_0|^2 a.$$ (This is independent of $t$ as it should be.) This is the invariant mass of the field configuration. By spherical symmetry, the total momentum is zero. Thus, for any timelike $\zeta^a$ $$P^a = {1\over2} \pi^2 |\phi_0|^2 \; \zeta^a.$$ Furthermore the Lagrangian is $$\L = {1\over2} \left[ |\partial_t \phi|^2 - |\partial_r \phi|^2 \right],$$ which evaluates (in the zero-momentum frame) to $$\L = { 2 a^4 |\phi_0|^2 (t^2+a^2-r^2) \over (r^2-t^2+a^2+2iat)^2(r^2-t^2+a^2-2iat)^2 }.$$ It is easy to check that $$\oint \d^4 x \; \L = 0,$$ so that the configuration is zero action. In summary, what we have is a Lorentz covariant, singularity-free, finite energy, zero action, exact localized solution to the d’Alembertian equation. In many ways this configuration has more right to be called an “instanton” than do the instantons of QFT; those instantons live in Euclidean signature. This field configuration lives in real physical time. Now the fact that there are finite energy solutions to the wave equation is not a surprise; that these finite energy solutions can coalesce, bounce, and disperse without producing field singularities is more interesting. One way of guessing that the field configuration in equation (\[E:mother\]) is worth investigating is the following: It is easy to convince oneself that in 4 Euclidean dimensions the solution to Laplace’s equation with a delta function source at the origin is $$\phi(x) \propto {1\over x^2+y^2+z^2+t^2}.$$ Thus in (3+1) Lorentzian dimensions the \[singular\] solution to the wave equation with a delta function source at the origin is $$\phi(x) \propto {1\over x^2+y^2+z^2-t^2}.$$ If the source is now moved to a real position $x_0^a$ we have $$\phi(x) \propto {1\over (x-x_0)^2+(y-y_0)^2+(z-z_0)^2-(t-t_0)^2}.$$ which is still a singular field configuration. Finally, move the source away from physical Minkowski space to the complex position $x_0^a - i\zeta^a$, then $$\phi(x) \propto {1\over (x-x_0+i\zeta^1)^2+(y-y_0+i\zeta^2)^2+(z-z_0+i\zeta^3)^2-(t-t_0+i\zeta^0)^2},$$ which is essentially equation (\[E:mother\]) above. This style of approach has been particularly advocated by Kaiser [@Kaiser]. As we have just seen, if $\zeta$ is timelike the resulting field configuration is singularity free. However, for null and spacelike $\zeta^a$, while the field is still a solution of the wave equation, the field is not bounded. Because of the singularities the energy and action integrals then diverge. Details are deferred for now and will be presented in sections \[S:null\] and \[S:spacelike\] below. One should also note that in the optics and engineering literature the most commonly used notations are not manifestly Lorentz covariant. Thus it is common to see expressions such as $$\phi \propto {1\over x^2+y^2+[b_1-i(z+t)]\;[b_2+i(z-t)]} \label{E:noncovariant}$$ (see, for instance, [@Lekner]) whose Lorentz transformation properties are less than obvious — in fact this field configuration is equivalent to equation (\[E:mother\]) with the identification $$\zeta^a = -\left({b_1+b_2\over2},0,0, {b_1-b_2\over2}\right); \qquad ||\zeta|| = b_1 b_2;$$ Real scalar field ================= By taking real and imaginary parts of the complex solution above we can write down two solutions for the real scalar field. Namely $$\phi_1 = {\phi_0 \; (\eta_{ab} \; \zeta^a \; \zeta^b) \; \left\{\eta_{ab} \;[x^a-x_0^a] \; [x^b-x_0^b] - \eta_{ab} \; \zeta^a \; \zeta^b \right\} \over (\eta_{ab} \;[x^a-x_0^a] \; [x^b-x_0^b] - \eta_{ab} \; \zeta^a \; \zeta^b )^2 + 4 (\eta_{ab} \;[x^a-x_0^a] \; \zeta^b)^2};$$ $$\phi_2 = {2 \phi_0 \; (\eta_{ab} \; \zeta^a \; \zeta^b) \; \left\{\eta_{ab} \;[x^a-x_0^a] \; \zeta^b \right\} \over (\eta_{ab} \;[x^a-x_0^a] \; [x^b-x_0^b] - \eta_{ab} \; \zeta^a \; \zeta^b )^2 + 4 (\eta_{ab} \;[x^a-x_0^a] \; \zeta^b)^2}.$$ As previously, we can without loss of generality translate $x_0\to0$ and go to the zero-momentum frame $\zeta^a=(a,0,0,0)$, then $$\phi_1 = {\phi_0 \; a^2 \; \left\{ t^2-r^2-a^2 \right\} \over (t^2-r^2-a^2)^2 + 4 a^2 t^2};$$ $$\phi_2 = {\phi_0 \; a^2 \; 2 a t \over (t^2-r^2-a^2)^2 + 4 a^2 t^2}.$$ The stress-energy for a real scalar field simplifies $$T_{ab} = \phi_a \; \phi_b - {1\over2} \eta_{ab} |\nabla \phi|^2.$$ Then $$\begin{aligned} \nabla_a T^{ab} &\equiv& \Delta \phi \; \nabla_b \phi + \nabla^a \phi \;\nabla_a \nabla_b \phi - \nabla^b \phi \;\nabla_a \nabla_b \phi \\ &\equiv & \Delta \phi \; \nabla_b \phi,\end{aligned}$$ which vanishes by the equations of motion. The calculation for the energy-momentum 4-vector now yields: $$P^a_1 = {1\over4} \pi^2 |\phi_0|^2 \; \zeta^a = P^a_2.$$ The action integral for both of these field configurations is still zero. \[$\oint \,\d^3 x \,\d t \; {\cal L}(\phi_{1,2}) = 0$.\] Maxwell field ============= A similar construction can be performed for the Maxwell field. There are a number of choices one could make, and I will simply pick one that leads to a relatively simple field configuration. Start by picking a timelike 4-velocity $V^a$ and a spacelike unit vector $m^a$ orthogonal to it. Now adopt the ansatz $$A^a = \left\{ V^a \; m^b - m^a \; V^b \right\} \nabla_b \psi.$$ Here one is automatically in Lorenz gauge [@Lorenz], $\nabla_a A^a = 0$, and the Maxwell equations reduce to the wave equation $\Delta\psi=0$ for the scalar potential $\psi$. The field configuration is manifestly Lorentz covariant and we can without loss of generality go to the inertial frame where $$V^a=(1,0,0,0); \qquad m^a= (0; \vec m).$$ In this inertial frame $$A_a = (\varphi,\vec A) = \left( [\vec m\cdot\vec\nabla] \psi, \vec m\; \dot\psi\right),$$ where $\vec m$ is a constant unit vector. We shall soon see that this is tantamount to working in the zero-momentum frame of the field configuration. The electric field is $$\vec E = - \vec\nabla\left([\vec m\cdot\vec\nabla] \psi\right) + \vec m \; \ddot \psi = - \vec\nabla\left([\vec m\cdot\vec\nabla] \psi\right) + \vec m \; \nabla^2 \psi = \vec\nabla\times(\vec\nabla\times[\vec m\; \psi]),$$ where we have used the wave equation for $\psi$. The magnetic field is $$\vec B = \vec\nabla\times(\vec m\; \dot\psi) = - \vec m \times \vec \nabla \dot\psi.$$ The energy density and momentum flux (Poynting vector) are $$\rho = {1\over2} [\vec E^2+\vec B^2]; \qquad \vec S = \vec E \times \vec B.$$ To evaluate total energy and momentum a useful integration by parts (subject to suitable falloff at spatial infinity) is $$\begin{aligned} \oint \d^3 x \; (\vec\nabla \times \vec X_1)\cdot(\vec\nabla \times \vec X_2) &=& \oint \d^3 x \; (\vec X_1 \times \vec\nabla )\cdot(\vec\nabla \times \vec X_2) \\ &=& \oint \d^3 x \; \vec X_1 \cdot \left[ \vec\nabla \times (\vec\nabla \times \vec X_2) \right] \\ &=& \oint \d^3 x \; \vec X_1 \cdot \left[ -\vec\nabla^2 \vec X_2 + \vec\nabla ( \vec\nabla\cdot X_2) \right] \\ &=& - \oint \d^3 x \; \left\{ \vec X_1 \cdot \vec\nabla^2 \vec X_2 + (\vec\nabla\cdot \vec X_1) ( \vec \nabla \cdot\vec X_2) \right\}.\end{aligned}$$ Consequently $$\begin{aligned} \oint \d^3 x \; \vec E^2 &=& - \oint \d^3 x \left\{ [\vec\nabla\times(\vec m\psi)] \nabla^2 [\vec\nabla\times(\vec m\psi)] \right\} \\ &=& + \oint \d^3 x \left\{ (\vec m\psi) \nabla^2 \nabla^2 (\vec m\psi) + ( \vec \nabla \cdot[\vec m\psi]) \nabla^2 ( \vec \nabla \cdot[\vec m\psi]) \right\} \\ &=& + \oint \d^3 x \left\{ \psi \nabla^2 \nabla^2 \psi + ( \vec m \cdot \vec \nabla\psi) \nabla^2 ( \vec m \cdot \vec \nabla\psi) \right\}.\end{aligned}$$ Now let us assume the potential $\psi$ is spherically symmetric $\psi(r,t)$. Then averaging over angular variables is the same as averaging over orientations of the unit vector $\vec m$ and under the angular integral we can effectively replace $$m_i \; m_j \to {1\over3} \delta_{ij}.$$ Consequently $$\begin{aligned} \oint \d^3 x \; \vec E^2 &=&+ \oint \d^3 x \left\{ \nabla^2 \psi \nabla^2 \psi + {1\over3} ( \vec \nabla\psi) \cdot \nabla^2 ( \vec \nabla\psi) \right\} \\ &=& + \oint \d^3 x \left\{ (\nabla^2 \psi)^2 - {1\over3} (\nabla^2 \psi)^2 \right\} \\ &=& + {2\over3} \oint \d^3 x \left\{ (\nabla^2 \psi)^2 \right\}.\end{aligned}$$ Similarly $$\begin{aligned} \oint \d^3 x \; \vec B^2 &=&- \oint \d^3 x \left\{ (\vec m\dot\psi) \cdot \nabla^2 (\vec m\dot\psi) + ( \vec m \cdot \vec \nabla\dot\psi) \; ( \vec m \cdot \vec \nabla\dot\psi) \right\}.\end{aligned}$$ Again invoking spherical symmetry for $\psi$ we have $$\begin{aligned} \oint \d^3 x \; \vec B^2 &=&- \oint \d^3 x \left\{ \dot\psi \nabla^2 \dot\psi + {1\over3} (\vec \nabla\dot\psi)\cdot(\vec \nabla\dot\psi) \right\} \\ &=&+ {2\over3} \oint \d^3 x \left\{ (\vec \nabla\dot\psi)\cdot(\vec \nabla\dot\psi) \right\}.\end{aligned}$$ Therefore $${\mathcal E} = \oint \d^3 x \; \rho = {1\over3} \oint \d^3 x \left\{ (\nabla^2 \psi)^2 + (\vec \nabla\dot\psi)\cdot(\vec \nabla\dot\psi) \right\}, \label{E:energy1}$$ so the total energy is sensibly positive. Before making further choices regarding the potential $\psi$, consider the total momentum $$\vec \wp = \oint \d^3 x \; \vec S = \oint \d^3 x \; \left[\vec\nabla\times(\vec\nabla\times[\vec m\; \psi])\right] \times \left[ \vec\nabla\times(\vec m\; \dot\psi) \right].$$ Integration by parts implies $$\vec \wp = \oint \d^3 x \left\{ (\vec m\; \dot\psi) \cdot \nabla^2 (\vec\nabla\times[\vec m\; \psi]) \right\} = -\oint \d^3 x \left\{ \dot\psi \vec m \cdot [\vec m \times \vec\nabla (\nabla^2 \psi)] \right\} = 0,$$ so that the net momentum is zero. For the action, an integration by parts together with the equations of motion yields $$\oint \d^4 x \; {1\over2} \left[ \vec E^2 - \vec B^2 \right] = {1\over3} \oint \d^3 x \left\{ (\nabla^2 \psi)^2 - (\vec \nabla\dot\psi)\cdot(\vec \nabla\dot\psi) \right\} = 0.$$ So we still have a zero action solution. Up to now the potential $\psi$ has only needed to be spherically symmetric and to satisfy the wave equation (plus some falloff constraint at spatial infinity to allow the integration by parts). The general solution to the wave equation in spherical symmetry is $$\psi(r,t) = {1\over r} \left[ f(r+t) - f(r-t) \right].$$ Returning to the energy ${\mathcal E}$ as given in equation (\[E:energy1\]), a further integration by parts, together with the wave equation for $\psi$ yields the computationally convenient form $${\mathcal E} = {1\over3} \oint \d^3 x \left\{ \partial_t^2 \psi \; \partial_t^2 \psi - \partial_t\psi \; \partial_t^3 \psi \right\}.$$ Whence $$\begin{aligned} {\mathcal E} &=& 4\pi \int_0^\infty \d r \Big\{ (\partial_t^2[ f(r+t) - f(r-t) ] )^2 \\ &&\qquad \qquad - (\partial_t[ f(r+t) - f(r-t) ] ) \; (\partial_t^3[ f(r+t) - f(r-t) ] ) \Big\} \nonumber \\ &=& 4\pi \int_0^\infty \d r \Big\{ (\partial_r^2[ f(r+t) - f(r-t) ] )^2 \\ &&\qquad\qquad - (\partial_r[ f(r+t) + f(r-t) ] ) \; (\partial_r^3[ f(r+t) + f(r-t) ] ) \Big\} \nonumber \\ &=& 4\pi \int_0^\infty \d r \Big\{ (\partial_r^2[ f(r+t) - f(r-t) ] )^2 \\ &&\qquad\qquad + (\partial_r^2[ f(r+t) + f(r-t) ] )^2 \Big\} \nonumber \\ &=& 8\pi \int_0^\infty \d r \left\{ [\partial_r^2 f(r+t) ]^2 + [\partial_r^2 f(r-t) ]^2 \right\} \\ &=& 8\pi \int_{-\infty}^{+\infty} \d s \; [\partial_s^2 f(s) ]^2.\end{aligned}$$ Which verifies that the energy is constant in a model-independent manner. Furthermore if $f(s)$ is smooth and satisfies suitable falloff conditions at $s\to\pm\infty$ then the wavelet will be nonsingular and of finite energy. To obtain a specific example it only remains to finish the complete specification of the potential $\psi(r,t)$. One particularly simple choice is to take one of the real scalar wavelets of the previous section $\psi\to\phi_{1,2}$, with $\zeta^a = ||\zeta|| \; V^a = a\; V^a$. Since the electric and magnetic fields are now specified in terms of derivatives of a smooth bounded function, the electromagnetic field is similarly smooth and bounded. For the energy, write it in the form $${\mathcal E} = {1\over3} \oint \d^3 x \left\{ \psi \; \partial_t^4 \psi - \partial_t\psi \; \partial_t^3 \psi \right\}.$$ Whence, for either $\psi\to\phi_{1,2}$, an integration carried out at arbitrary $t$ yields the time-independent quantity $${\mathcal E}_{1,2} = {1\over2} \;{\phi_0^2\over a}.$$ The calculation has for convenience been carried out in the zero-momentum frame of the wavelet. In a general Lorentz frame we would have $$P^a_{1,2} = {1\over2}\; {\phi_0^2\over a} \;\; V^a = {1\over2}\; {\phi_0^2\over a^2} \;\; \zeta^a.$$ Thus this field configuration, as for the scalar case, is Lorentz covariant, bounded, finite energy and zero action. The vector nature of the Maxwell field has added technical complications, but there is no real change in basic principles. In closing this section, I should mention one other wavelet that is particularly attractive. If one makes use of the pseudo-differential operator $(\nabla^2)^{-1/2}$ one could write $$\psi = (\nabla^2)^{-1/2} \; \phi_{1,2} = {1\over\Gamma(-1/2)} \int_0^\infty {\d t\over t^{3/2}} \exp[-t \nabla^2] \; \phi_{1,2}.$$ With this choice of $\psi$ the energy integral simplifies $${\mathcal E}_{1,2} = {1\over3} \oint \d^3 x \left\{ (\partial_t\phi_{1,2})^2 + (\nabla \phi_{1,2})^2 \right\} = {1\over 6} \; \pi^2 \; |\phi_0^2| \; a.$$ The price paid for making the energy look simple is that the electric and magnetic fields are much more complicated to calculate. Because the optics and engineering literature generally does not use manifestly Lorentz covariant notation, it can be very time consuming to calculate the 4-momentum of a specific pulse. Indeed only very recently [@Lekner] has Lekner provided specific and explicit computations of both ${\mathcal E}$ and $\vec\wp$ (as well as the angular momentum) for a pulse similar to that considered above. As expected (once one has the covariant perspective advocated in this article) $||\vec\wp|| < {\mathcal E}$, indicating the existence of a zero-momentum frame for the pulses of this type [@Lekner]. Yang-Mills field ================ Once we have the Maxwell wavelet above, a Yang–Mills wavelet is straightforward, indeed trivial. Let $\mathbf{\Lambda}$ be any constant matrix in the center of the gauge group and set $\mathbf{A}^a = A^a \; \mathbf{\Lambda}$. The construction is so simple that there is really nothing extra beyond the Maxwell wavelet considered above. Null $\zeta$ {#S:null} ============ Let us now return to the original scalar complex wavelet. Suppose the 4-vector $\zeta$ is null. Then because the numerator vanishes identically the original definition above gives $\phi\equiv 0$. We should at a minimum change our field definition to read $$\phi(x) = - {\psi_0 \over \eta_{ab} \;[x^a-x_0^a-i\zeta^a] \; [x^b-x_0^b-i \zeta^b] }.$$ Then without loss of generality we go into the frame $$\zeta^a = (a,0,0,a),$$ and then $$\phi(x) = - {\psi_0 \over [t-ia]^2 - x^2-y^2-[z-ia]^2 }.$$ That is $$\phi(x) = {\psi_0 \over r^2-t^2+2ia(t-z) }.$$ Note $$|\phi(x)| = {\psi_0 \over \sqrt{(r^2-t^2)^2 +4 a^2 (t-z)^2 } }.$$ The denominator now vanishes when $z-t$ and $x=y=0$, so that the field is divergent on the beam axis. Suppose we write $R = \sqrt{x^2+y^2}$ then $$\phi(x) = {\psi_0 \over R^2+ z^2-t^2+2ia(t-z) }.$$ So we see that the field drops off as $1/R^2$ as we move away from the beam axis. \[And more critically, the field blows up as $1/R^2$ as we approach the beam axis.\] Attempts at calculating the energy and action now lead to divergent integrals. In other words, despite the fact that it still solves the wave equation, for null $\zeta$ this is not a particularly useful field configuration. Spacelike $\zeta$ {#S:spacelike} ================= Foe the complex scalar wavelet, suppose the 4-vector $\zeta$ is spacelike. Then without loss of generality we go into the infinite velocity frame where $$\zeta^a = (0,0,0,a),$$ and then $$\phi(x) = {\phi_0 \; a^2 \over t^2 - x^2-y^2-[z-ia]^2 }.$$ That is $$\phi(x) = -{\phi_0 \; a^2 \over r^2-t^2-a^2-2iaz }.$$ Note $$|\phi(x)| = {\phi_0 \; a^2 \over \sqrt{(r^2-t^2-a^2)^2 +4 a^2 z^2 } }.$$ The denominator now vanishes when $z=0$ and $x^2+y^2=a^2+t^2$. That is, the field is divergent on a time-dependent circle orthogonal to the beam axis. There is a short distance singularity as one approaches this circle, and the energy and action integrals diverge. In other words, despite the fact that it still solves the wave equation, for spacelike $\zeta$ this is not a useful field configuration. Discussion ========== The physical wavelet discussed in this article is important because it represents a qualitatively different extended field configuration of a type not normally encountered in particle physics. The wavelet is neither a soliton, nor an instanton, nor a sphaleron though it shares properties with all three of these extended objects: - Like the soliton it lives in physical time (Minkowski space, not Euclidean space), and possesses a well-defined 4-velocity. - Like the instanton it “dies away” in the infinite past and future. - Like the instanton it possesses a continuously adjustable scale parameter. - Like the sphaleron it is unstable to dispersal. Because the wavelet fields are bounded and finite energy, wavelet configurations [*will*]{} be classically excited at any finite temperature. Because the wavelet configuration has zero action, arbitrarily complicated combinations of these physical wavelets can be added to the field configurations appearing in Feynman’s path integral without modifying the phase — quantum mechanically there is no “cost” in adding these configurations to the Lorentzian path integral and they [*will*]{} contribute. Other “localized waves” might be interesting in specific applications but the particular example discussed in this article is important because of its extreme simplicity and pleasant behaviour. I wish to thank John Lekner for stimulating my interest in these issues. I also wish to thank Damien Martin for bringing the whole Lorenz/Lorentz issue to my attention. [99]{} R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations”, J. Math. Phys. [**26**]{} (1985) 861–863. M. K. Tippet and R. W. Ziolkowski, “A bidirectional wave transformation of the cold plasma equations”, J. Math. Phys. [**32**]{} (1991) 488–492. J. Lu and J. F. Greenleaf, “Nondiffracting X waves — exact solutions to free-space scalar wave equation and their finite aperture realizations”, IEEE Transactions on ultrasonics, ferroelectrics, and frequency control, [**39**]{} (1992) 19–31. J. Lu, H. Zou, and J. F. Greenleaf, “A new approach to obtain limited diffraction beams”, IEEE Transactions on ultrasonics, ferroelectrics, and frequency control, [**42**]{} (1995) 850–853. G. Kaiser, “Physical wavelets and their sources: Real physics in complex spacetime,” arXiv:math-ph/0303027. J. Lekner, “Electromagnetic pulses which have a zero momentum frame”, submitted to J. Opt. A, 21 November 2002; arXiv:physics/0304022. Field configurations similar to the one considered in this article may variously be encountered under names such as: “localized waves”, “focus wave modes”, “pulses”, “X-waves”, “limited diffraction beams”, “wavelets”, and “physical wavelets”. The Lorenz gauge was apparently first used by the Danish physicist Ludwig Lorenz (1829-1891), though it is commonly misattributed to the Dutch physicist Hendrik Antoon Lorentz (1853-1928).\ L. Lorenz, “On the Identity of the Vibrations of Light with Electrical Currents”, Philos. Mag. [**34**]{} (1867) 287-301.\ J. van Bladel, “Lorenz or Lorentz?”, IEEE Antennas Prop. Mag. [**33**]{} (1991) 69. [^1]: Research supported by the Marsden Fund administered by the Royal Society of New Zealand

--- abstract: 'The rotating Ayón-Beato-García (ABG) black holes, apart from mass ($M$) and rotation parameter ($a$), has an additional charge $Q$ and encompassed the Kerr black hole as particular case when $Q=0$. We demonstrate the ergoregions of rotating ABG black holes depend on both rotation parameter $a$ and charge $Q$, and the area of the ergoregions increases with increase in the values of $Q$, when compared with the Kerr black hole and the extremal regular black hole changes with the value of $Q$. Ban[ã]{}dos, Silk and West (BSW) demonstrated that an extremal Kerr black hole can act as a particle accelerator with arbitrarily high center-of-mass energy ($E_{CM}$) when the collision takes place at any point in the ergoregion and thus in turn provides a suitable framework for Plank-scale physics. We study the collision of two general particles with different masses falling freely from rest in the equatorial plane of a rotating ABG black hole near the event horizon and find that the $E_{CM}$ of two colliding particles is arbitrarily high when one of the particles take a critical value of angular momentum in the extremal case, whereas for nonextremal case $E_{CM}$ for a pair of colliding particles is generically divergent at the inner horizon, and explicitly studying the effect of charge $Q$ on the $E_{CM}$ for ABG black hole. In particular, our results in the limit $Q\rightarrow 0$ reduce exactly to *vis-$\grave{a}$-vis* those of the Kerr black hole.' author: - Fazlay Ahmed - Muhammed Amir - 'Sushant G. Ghosh' title: 'Particle acceleration of two general particles in the background of rotating Ayón-Beato-García black holes' --- Introduction {#intro} ============ Banãdos, Silk and West [@Banados:2009pr] have proposed that the collision of two particles, say dark matter particles, falling from rest at infinity into the Kerr black hole [@Kerr:1963ud] can produce an infinitely large center-of-mass energy ($E_{CM}$) when collision takes place in the vicinity of the event horizon, with the black hole maximally spinning, and one of the particle have critical angular momentum. This mechanism of particle acceleration by a black hole is particularly called BSW mechanism, which is interesting from the viewpoint of theoretical physics because new Planck scale physics is possible in the vicinity of the black holes. Further, the extremal Kerr black hole surrounded by dark matter could be regarded as a Planck scale collider, which might help us to explain the astrophysical phenomena, such as the the active galactic nuclei and gamma ray bursts. Hence, the BSW mechanism about the collision of two particles near a rotating black hole has attracted significant attention [@Berti:2009bk; @Jacobson:2009zg; @Lake:2010bq; @Wei:2010gq; @Mao:2010di; @Grib:2010dz; @Liu:2010ja; @Grib:2010xj; @Zhu:2011ae; @Hussain:2012zza; @11; @12; @Igata:2012js; @Zaslavskii:2012fh] (see also [@Harada:2011xz], for a review). BSW phenomena is not only done for Kerr black holes but also for Kerr-Newman black holes [@2], Regular Black holes [@Ghosh:2014mea; @Pradhan:2014oaa; @Amir:2015pja; @Ghosh:2015pra], higher dimensional black holes [@Tsukamoto:2013dna] and naked singularities [@4; @5; @5a; @Patil:2011ya; @Patil:2011uf]. In particular for the Kerr-Newman black hole, the $E_{CM}$ of collision depends not only on the spin $ a $ but also on the charge $ Q $ of the black hole. Lake [@Lake:2010bq] analyzed the $E_{CM}$ of the collision occurring at the inner horizon of the non-extremal Kerr black hole and found that $E_{CM}$ is unlimited. Grib and Pavlov [@Grib:2010xj] showed that the $E_{CM}$ for two particles collision can be unlimited even in the non-maximal rotation. Later, Zaslavskii [@Zaslavskii:2010jd; @Zaslavskii:2010pw] demonstrated that an acceleration of the particles by BSW method is a universal property of rotating black holes. Recently, Harada and Kimura [@Harada:2011xz] generalized the BSW analysis of two colliding particles to general geodesics in the Kerr black hole to show an arbitrarily high $E_{CM}$ can occur near the horizon of maximally rotating black holes. Further, the subject of particle acceleration for two different masses and energetic particles is extended for a class of black holes [@Liu:2011wv; @Amir:2016nti]. The analysis is also valid to the collision of particles in non-equatorial motion for Kerr black holes [@6] and Kerr-Newman black holes [@Liu:2011wv]. The general explanation of BSW phenomenon is proposed in [@9]. It turns out that the horizon structure of the rotating regular black hole is complicated as compared to the Kerr black hole, which depend on the mass and spin of the black hole and on an additional deviation parameter that measures potential deviations from the Kerr metric, and includes the Kerr metric as the special case if this deviation parameter vanishes. Further, these regular black holes are very important as astrophysical black holes, like Cygnus X-1 or Sgr$ A^* $, although suppose to be like the Kerr black hole [@32; @33], but the definite nature of astrophysical black hole still need to be tested, and it may deviate from the standard Kerr black hole. More recently, the BSW mechanism when applied to an extremal regular black holes [@Ghosh:2014mea; @Pradhan:2014oaa], also lead to divergence of the $E_{CM}$. Hence, the BSW mechanism should be suitably adapted for the rotating regular black hole, which has very complicated horizon structure as compared to the Kerr black hole and can have extremal black holes depending on the additional parameter [@Ghosh:2014mea; @Amir:2015pja; @Ghosh:2015pra]. The rotating regular Ayón-Beato-García (ABG) black hole [@Bambi:2013ufa] is an exact solution of Einstein’s equations coupled to nonlinear electrodynamics. The rotating ABG black holes are axisymmetric, asymptotically flat, and depend on the mass ($M$) and spin ($a$) of the black hole as well as on a charge ($ Q $) that measures potential deviations from the Kerr metric and includes the Kerr metric as the special case if this deviation parameter vanishes. The main goal of this paper is to discuss the detailed behavior of the $E_{CM}$ for two particles with different rest masses $ m_1 $ and $ m_2 $, falling freely from rest at infinity in the background of a rotating ABG black hole and calculate the $E_{CM}$. We go on to show that the $E_{CM}$ near the horizon of an extremal ABG black hole is arbitrarily high when one of the two particles acquire the critical angular momentum and also high at the inner horizon. We also explicitly show the effect of the parameter $ Q $ on BSW mechanism and ergoregion of the black hole. The paper is organized as follows. In Sec. \[sptm\], we shall discuss horizons of the rotating ABG spacetime and also demonstrate the effect of parameter $Q$ on ergoregion. In Sec. \[geqm\], we will discuss the equations of motion for a particle in the background of a rotating ABG black hole. The calculation of expression for $E_{CM}$ of the collision for two general particles and their properties is the subject of Sec. \[cme\] and we conclude in the Sec. \[con\]. Rotating Ayón-Beato-García Black Holes {#sptm} ====================================== The spherically symmetric ABG black hole is an exact regular solution of general relativity with nonlinear electrodynamics field as a source, and it satisfies the weak energy condition. The gravitational field of ABG solution is described by the metric [@AyonBeato:1998ub]: $$\begin{aligned} \label{abg} ds^2 &=& -f(r)dt^2+\frac{1}{f(r)}d r^2+ r^2 d \Omega^2,\end{aligned}$$ with $$\begin{aligned} f(r)=1-\frac{2 M r^2}{(r^2+Q^2)^{3/2}}+\frac{Q^2 r^2}{(r^2+Q^2)^{2}},\end{aligned}$$ and $d \Omega^2=d \theta^2+\sin^2 \theta d \phi^2$. The parameter $M$ and $Q$ are respectively, mass and electric charge. The associated electric field is $$\begin{aligned} E=Q r^4\left[\frac{r^2-5Q^2}{(r^2+Q^2)^4}+\frac{15}{2}\frac{M}{(r^2+Q^2)^{7/2}} \right].\end{aligned}$$ Note that the ABG black hole is asymptotically go over to Reissner-Nordst[ö]{}rm black hole $$\begin{aligned} f(r) &=& 1-\frac{2M}{r}+\frac{Q^2}{r^2}+O(1/r^3), \nonumber \\ E &=& \frac{Q}{r^2}+O(1/r^3).\nonumber\end{aligned}$$ This solution corresponds to a regular charged black hole when $|Q|\leq 0.6M$, with the curvature invariant and electric field regular everywhere including at $r=0$ [@AyonBeato:1998ub]. The rotating ABG black hole metric obtained in [@Toshmatov:2014nya], beginning with the metric (\[abg\]) and applying the Newman-Janis transformation, they constructed a rotating ABG metric. The gravitational field of rotating ABG spacetime is described by a metric which in the Boyer-Lindquist coordinates (with $G=c=1$) [@Toshmatov:2014nya] reads: $$\begin{aligned} \label{metric} ds^{2} &=& -f(r,\theta)dt^{2} - 2 a \sin^{2}\theta (1-f(r,\theta))d\phi dt +\frac{\Sigma}{\Delta}dr^2 \nonumber \\ & + & \Sigma d \theta^{2} + \sin^{2}\theta [\Sigma-a^{2}(f (r,\theta)-2)\sin^{2}\theta]d \phi^{2},\end{aligned}$$ where, $\Sigma = r^2 + a^{2}\cos^{2}\theta, \;\;\;\; \Delta=\Sigma f(r,\theta) +a^{2}\sin^{2}\theta$, $a$ is a rotation parameter and the function $f(r, \theta)$ is given by $$\begin{aligned} \label{f} f(r,\theta)=1-\frac{2 M r \sqrt{\Sigma} }{(\Sigma +Q^2)^{3/2}}+\frac{Q^2 \Sigma}{(\Sigma +Q^2)^2}.\end{aligned}$$ The ABG metric (\[metric\]) is a regular rotating charged black hole [@Ghosh:2014mea; @Toshmatov:2014nya], which go over to Kerr black holes [@Kerr:1963ud] when $Q=0$. When $a=0$, it reduces to the ABG black hole [@AyonBeato:1998ub], and for $a= Q =0$, it reduces to the Schwarzschild black hole [@schw]. The invariant Ricci scalar ($R_{ab} R^{ab}$ ) and Kretschmann scalar ($R_{abcd} R^{abcd}$ ) are suppose to be regular everywhere including at $r=0$ [@Ghosh:2014mea]. The metric (\[metric\]) becomes singular if $\Sigma=0$ or $\Delta=0$ whereas $\Sigma=0$ is the curvature singularity and $\Delta=0$ is the coordinate singularity. $\Delta=0$ gives horizons of the rotating ABG black holes [@Ghosh:2014mea]. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- {width="0.245\linewidth"} {width="0.245\linewidth"} {width="0.245\linewidth"} {width="0.245\linewidth"} ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- Horizons and ergoregions of rotating ABG black holes ---------------------------------------------------- In this section, we would like to study the properties of an ergoregion of a rotating ABG black hole which is the region between the static limit surface and the event horizon. In fact, the ergoregion plays an important role in the astrophysics for a possible observational effects, since in this region the Hawking radiation can be analyzed. The ergoregion is also important due to energetics of black holes and also Penrose process [@Penrose:1971uk]. The static limit surface ($r_{+}^{sls}$) is also known as infinite redshift surface, where the time-translation Killing vector becomes null. Another property of the static limit surface is that the time-like geodesics becomes space-like after crossing the static limit surface. The static limit surface of a rotating ABG black hole can be calculated by solving the equation $g_{tt}= f(r,\theta)=0$. In the case of $Q=0$, it represents the static limit surface of a Kerr black hole. The metric has a coordinate singularity at $\Delta=0$. Horizons of the metric (\[metric\]) are given by $g^{rr}=\Delta=0$, i.e., $$\label{eh} \Sigma f(r, \theta) + a^2 \sin^{2} \theta = 0.$$ An ergoregion is located outside an event horizon, which has an oblate shape and touches the event horizon at poles and it has a highest radius at the equator. It turns out that Eq. (\[eh\]) admits two positive roots which take different values with charge $Q$. The two roots corresponding to the horizons of the black hole. Let us assume that $r^{EH}_{+}$ and $r^{CH}_{-}$ define respectively, the outer horizon (event horizon) and inner horizon of the black hole, and when they are equal $r^{EH}_{+}= r^{CH}_{-} = r_{ex}$, corresponds to an extremal black holes with degenerate horizons. Inside the ergoregion an object can not remain static, but it moves in the direction of black hole spin. In ergoregion, an object can enter and escape, and we can extract energy and mass via the Penrose process [@Penrose:1971uk]. \[Table A\] ------- -------------- --------------- -------------- -------------- --------------- -------------- -------------- --------------- -------------- -------------- --------------- -------------- $ Q $ $r^{EH}_{+}$ $r_{+}^{sls}$ $\delta^{a}$ $r^{EH}_{+}$ $r_{+}^{sls}$ $\delta^{a}$ $r^{EH}_{+}$ $r_{+}^{sls}$ $\delta^{a}$ $r^{EH}_{+}$ $r_{+}^{sls}$ $\delta^{a}$ 0.0 1.86603 1.93541 0.06938 1.80000 1.90554 0.10554 1.71414 1.86891 0.15477 1.60000 1.82462 0.22462 0.1 1.85116 1.92196 0.07080 1.78371 1.89163 0.10792 1.69558 1.85440 0.15882 1.57732 1.80932 0.23200 0.2 1.80488 1.88037 0.07549 1.73257 1.84856 0.11599 1.63641 1.80935 0.17294 1.50259 1.76161 0.25902 0.3 1.72097 1.80640 0.08543 1.63779 1.77155 0.13376 1.52186 1.72821 0.20635 1.33877 1.67476 0.33599 0.4 1.58263 1.68988 0.10725 1.47161 1.64881 0.17720 1.28210 1.59665 0.31455 - 1.53012 - ------- -------------- --------------- -------------- -------------- --------------- -------------- -------------- --------------- -------------- -------------- --------------- -------------- : Table for different values of $a$ and $Q$ for ABG black hole. Parameter $\delta^{a}$ is the region between static limit surface and event horizon ($\delta^{a}=r_{+}^{sls}-r^{EH}_{+}$). ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- {width="0.245\linewidth"} {width="0.245\linewidth"} {width="0.245\linewidth"} {width="0.245\linewidth"} {width="0.230\linewidth"} {width="0.245\linewidth"} {width="0.245\linewidth"} {width="0.245\linewidth"} {width="0.245\linewidth"} {width="0.245\linewidth"} {width="0.245\linewidth"} {width="0.245\linewidth"} {width="0.245\linewidth"} {width="0.245\linewidth"} {width="0.245\linewidth"} {width="0.245\linewidth"} {width="0.245\linewidth"} {width="0.245\linewidth"} {width="0.245\linewidth"} {width="0.245\linewidth"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- How the charge parameter $Q$ affects the shape of the ergoregion is shown in Fig. \[figB\], when compared with the Kerr black hole’s ergoregion (cf. Fig. \[figA\]). The numerical solution of $g_{tt} =0$ and $g^{rr} =0$ are summarized in Table \[Table A\]. Fig. \[figB\] shows that the ergoregion area of the ABG black hole increases with increase in the value of $Q$ as well as $a$. The outer horizon and static limit surface coincides at the poles (Fig. \[figB\]) as in the case of Kerr black hole (Fig. \[figA\]). It should be pointed out that when $Q>Q_{C}$, for a given parameter $a$, the horizons get disconnected (cf. Fig. \[figB\]). The static limit surface becomes more oblate with increase in the value of $Q$. Equations of motion in the background of a rotating ABG black hole {#geqm} ================================================================== Next, we calculate the equations of motion of the particle in the background of rotating ABG black hole, which are essential to study the collision of particles. Let us consider a particle of rest mass $m$ falling from rest at infinity on the equatorial plane ($\theta = \pi/2$) of a rotating ABG black hole. The motion of a particle is determined by the Lagrangian $$\begin{aligned} \mathcal{L} = \frac{1}{2} g_{\mu \nu} u^{\mu} u^{\nu},\end{aligned}$$ where $u^{\mu}$ is the four-velocity. The metric (\[metric\]) has two conserved quantities, namely, energy $E$ and angular momentum $L$, respectively, correspond to the timelike Killing vector $\xi^a=(\partial/\partial t)^a$ and axial Killing vector $\chi^a=(\partial/\partial \phi)^a$, given [@6] $$\begin{aligned} \label{el} E = -g_{ab} \xi^a u^b, \quad L = g_{ab} \chi^a u^b,\end{aligned}$$ which yields $$\begin{aligned} E = -g_{tt} u^t -g_{t \phi} u^{\phi}, \quad L = g_{t \phi} u^t +g_{\phi \phi} u^{\phi}. \nonumber\end{aligned}$$ We obtain the four-velocities of the particle by solving the above equations, $$\begin{aligned} \label{u^t} u^t &=& -\frac{1}{r^2} \Big[a (aE - L) - \frac{r^2 + a^2}{\Delta} \mathcal{P} \Big],\end{aligned}$$ $$\begin{aligned} \label{u^Phi} u^{\phi} &=& -\frac{1}{r^2} \Big[(aE - L ) - \frac{a}{\Delta} \mathcal{P} \Big], \end{aligned}$$ and the normalization condition of the four-velocity $u_{\mu}u^{\mu}=-m^2$, yields $$\label{u^r} u^{r} = \pm \frac{1}{r^2} \sqrt{\mathcal{P}^2 -\Delta \left[m^2 r^2 + (L-a E)^2 \right]},$$ where positive and negative sign correspond to the outgoing and incoming geodesics, respectively and $\mathcal{P} = (r^2 + a^2)E - aL $. The above equations have same expressions that of Kerr black hole, but now $\Delta$ is modified which includes charge $Q$. However, to obtain the trajectories of a particle, we need the angular momentum range of the particle and it can be calculated by using the circular orbit conditions, $$\begin{aligned} \label{lim} V_{eff}=0, \;\;\;\text{and}\;\;\; \frac{dV_{eff}}{dr}=0,\end{aligned}$$ where $V_{eff}$ (effective potential) is $$V_{eff} = -\frac{[(r^2 + a^2)E -La]^2 -\Delta [m^{2} r^2 + (L-a E)^2]}{2 r^4}.$$ It determines the allowed and prohibited regions of the particle trajectory around the ABG black hole. If $V_{eff} \leq 0$, then this is an allowed region and for $V_{eff} > 0$, the motion is prohibited. In Fig. \[figPot\], we have shown the behaviour of $V_{eff}$ with radius $r$. --------------------------------------------------------------------------------------- -- -- -- {width="0.48\linewidth"} {width="0.48\linewidth"} --------------------------------------------------------------------------------------- -- -- -- Since, the geodesics are timelike, i.e., $dt/d\tau \geq 0$, then using Eq. (\[u\^t\]) we get $$\begin{aligned} \frac{1}{r^2}[-a(a E-L)+(r^2+a^2)\frac{\mathcal{P}}{\Delta}]\geq 0,\end{aligned}$$ at horizon, the above condition on simplification leads to $$\begin{aligned} E-\Omega_H L \geq 0, \quad \Omega_H = \frac{a}{(r^{EH}_+)^2+a^2}.\end{aligned}$$ \[Table B\] $ Q $ $ a = a_{ex} $ $ r=r_{ex} $ $ L_{1} $ $ L_{2} $ ------- ---------------- -------------- ----------- ----------- 0 1 1 2.0 -4.82842 0.15 0.955870457 1.02859 2.06269 -4.78476 0.25 0.880328799 1.06201 2.16152 -4.70709 0.35 0.769966648 1.08805 2.30749 -4.58729 0.45 0.621056295 1.09518 2.55058 -4.41379 : Table for the range of the angular momentum in extremal cases of a rotating ABG black hole ($M=1$). The critical angular momentum of the particle is defined by $L_C = E / \Omega_H$, where $\Omega_H$ is the angular velocity at the horizon of the ABG black hole. We have shown the values of critical angular momentum for the extremal cases of rotating ABG black hole in Table \[Table B\]. It is known that the particle trajectory depends on the values of angular momentum which is different for different values of angular momentum. If angular momentum of the particle is larger than the critical angular momentum, i.e., $L>L_C$, then the particle will never fall into the black hole. When the angular momentum of particle is smaller then the critical angular momentum $L<L_C$, then the particle will always fall into the black hole. Moreover, when the particle’s angular momentum is equal to the critical angular momentum $L=L_C$, then the particle reaches up to the horizon and collision of two particle takes place. Center-of-mass energy of two particles in the rotating ABG spacetime {#cme} ==================================================================== After calculating the equations of motion of a particle, and checked the conditions for the particle to reach up to the horizon, now we are in a position to calculate the center-of-mass energy ($E_{CM}$) of two different mass particles. Let us consider the two particles with rest masses $m_{1}$ and $ m_{2}$ ($m_{1}\neq m_{2}$) moving in the equatorial plane ($\theta = \pi/2$) of rotating ABG black hole. These particles are coming from rest at infinity towards the black hole and collide in the vicinity of an event horizon. The collision takes place in the center-of-mass frame. The two particles which are involved in the collision have different angular momentum, i.e., $L_{1}$, $L_{2}$ and energy $E_1$, $E_2$. The four-momentum of the $i^{th}$ particle is given by [@Banados:2009pr] $$p^{\mu}_{i}=m_{i} u^{\mu}_{i},$$ where $i=1,2$ and $m_{i}$ and $u^{\mu}_{i}$ are corresponding to the rest mass and four-velocity of the $i^{th}$ particle. The total four-momentum of two colliding particles is $$P^{\mu}_{t} = P^{\mu}_{(1)} + P^{\mu}_{(2)}.$$ Hence, the $E_{CM}$ for the collision of two different mass particles is given by [@6] $$\begin{aligned} \label{pmu} E^{2}_{CM} = -P^{\mu}_{t} P_{t \mu} = -(P^{\mu}_{(1)} + P^{\mu}_{(2)})(P_{(1)\mu} + P_{(2)\mu}) \nonumber\\ = -\left(m_1 u^{\mu}_{(1)}+ m_2 u^{\mu}_{(2)}\right)\left(m_1 u_{(1)\mu}+m_2 u_{(2)\mu} \right). \end{aligned}$$ After simplifying and using the condition $u^{\mu}_{(i)} u_{(i) \mu} =-1$, in Eq. (\[pmu\]), we obtain the formula for $E_{CM}$ is $$\label{formula} \frac{E_{CM}^2}{2 m_{1} m_{2}} = 1+\frac{(m_{1} - m_{2})^2}{2 m_{1} m_{2}} - g_{\mu \nu} u^{\mu}_{(1)} u^{\nu}_{(2)}.$$ By substituting the values of $g_{\mu \nu}$, $u^{\mu}_{(1)}$ and $u^{\nu}_{(2)}$ from Eqs. (\[metric\]), (\[u\^t\]), (\[u\^Phi\]), and (\[u\^r\]) into Eq. (\[formula\]), therefore the $E_{CM}$ of rotating ABG black hole takes the following form: $$\begin{aligned} \label{ecm} \frac{E_{CM}^2}{2 m_{1} m_{2}} &=& \frac{(m_{1}-m_{2})^{2}}{2 m_{1} m_{2}}+ \frac{1}{r^2(r^2 f(r) + a^2)} \Big[ a (f(r)-1)(L_{1} E_{2} + L_{2} E_{1})r^2 \nonumber\\ && -a^2((f(r)-2)E_1E_2-1)r^2 - L_{1} L_{2} f(r)r^2+ (E_1E_2+ f(r))r^4 \nonumber \\ && -\sqrt{-r^2 [\left((f(r)-2)E^2_{1}+m_1^2\right)a^2-2a(f(r)-1)E_{1}L_{1}-r^2E_1^{2}+f(r)(L^2_{1}+r^2 m_1^2)]} \nonumber \\ && \times \sqrt{-r^2 [\left((f(r)-2)E^2_{2}+m_2^2\right)a^2-2a(f(r)-1)E_{2}L_{2}-r^2E_2^{2}+f(r)(L^2_{2}+r^2 m_2^2)]} \Big].\nonumber \\\end{aligned}$$ We have considered the motion of the particles in equatorial plane ($\theta = \pi/2$), where $f(r, \theta)= f(r)$. It is clear from Eq. (\[ecm\]), that $E_{CM}$ depends on charge $Q$ as well as rotation parameter $a$. If we change the value of charge $Q$ or rotation parameter $a$, then the $E_{CM}$ must be change. We analyze numerically the behavior of $E_{CM}$ with radius $r$. We plot Eq. (\[ecm\]) for different combinations of rotation parameter $a$, electric charge $Q$, angular momentum $L_{1}$ and $L_{2}$, and colliding particle masses $m_1$ and $m_2$. We observe from the Fig. \[fig3\] that the $E_{CM}$ would be very large if one of the colliding particle has critical value of angular momentum and collision takes place in the vicinity of the event horizon. Keep in mind that Eq. (\[ecm\]) is valid for two different massive particles. {width="0.48\linewidth"} {width="0.48\linewidth"} {width="0.48\linewidth"} {width="0.48\linewidth"} If we consider the limit $Q \rightarrow 0$ , then it follows from Eq. (\[ecm\]) that $$\begin{aligned} \label{ecm2} \frac{E_{CM}^2}{2 m_{1} m_{2}}(Q \rightarrow 0) &=& \frac{(m_{1}- m_{2})^{2}}{2 m_{1} m_{2}}+\frac{1}{r(r^2-2 M r+a^2)} \Big[a^2((2M+r)E_1 E_2 +r) \nonumber\\ &&-2 a M(L_1 E_2+L_2 E_1)-L_1 L_2 (-2M+r)+(-2M+r(1+E_1 E_2))r^2 \nonumber\\ && -\sqrt{r(r^2+a^2)(E^2_{1}-m_1^2)+2M(a E_{1}-L_{1})^2 - L_{1}^2 r + 2M r^2 m_1^2} \nonumber \\ && \times \sqrt{r(r^2+a^2)(E^2_{2}-m_2^2)+2M(a E_{2}-L_{2})^2 - L_{2}^2r + 2 M r^2 m_2^2}\Big],\end{aligned}$$ which is the center-of-mass energy when two particles of different masses collide near the Kerr black hole [@Harada:2011xz]. It can be analyze that when $m_{1}=m_{2}=m_0$ and $E_1=E_2=E=1$, then Eq. (\[ecm\]) reduces to $$\begin{aligned} \label{ecm1} \frac{E_{CM}^2}{2 m_{0}^2} &=&\frac{1}{r^2(r^2 f(r)+a^2)}\Big[a(f(r)-1) (L_{1}+L_{2})r^2 -a^2 (f(r)-3)r^2 - L_{1}L_{2}f(r)r^2 \nonumber\\ && + (1+f(r))r^4 - \sqrt{-r^2[(f(r)-1)a^2-2a(f(r)-1)L_{1}-r^2+f(r)(L_{1}^{2}+r^2)]} \nonumber\\ && \times \sqrt{-r^2[(f(r)-1)a^2-2a(f(r)-1)L_{1}-r^2+f(r)(L_{1}^{2}+r^2)]}\Big],\end{aligned}$$ which is similar to the $E_{CM}$ of rotating ABG black hole for two equal mass particles [@Ghosh:2014mea]. Again, if $Q \rightarrow 0$ and $m_{1}=m_{2}=m_0$, and $E_1=E_2=E=1$, then Eq. (\[ecm\]) transform into the $E_{CM}$ of Kerr black hole [@Banados:2009pr]: $$\begin{aligned} \label{ecm3} {E_{CM}^2}(Q \rightarrow 0) &=&\frac{2 m_0^{2}}{r(r^2 -2r +a^2)}\Big[2 a^2 (1+r)-2a (L_{1}+L_{2})-L_{1} L_{2}(-2+r) +2(-1+r)r^2 \nonumber\\ && -\sqrt{2(a-L_{1})^2- L_{1}^2 r+2 r^2} \sqrt{2(a-L_{2})^2- L_{2}^2 r+2 r^2} \Big].\end{aligned}$$ Hence, we can say that the $E_{CM}$ of two different mass particles in the background of rotating ABG black hole spacetime is the generalization of $E_{CM}$ of Kerr black hole. Now we discuss the $E_{CM}$ of two colliding particles for extremal and nonextremal black hole. {width="0.48\linewidth"} {width="0.48\linewidth"} {width="0.48\linewidth"} {width="0.48\linewidth"} We are interested in the near horizon collision of two different mass particles in case of extremal black hole, i.e., $r \rightarrow r_{ex}$. We analyze the behavior of $E_{CM}$ with radius $r$, and plot it in Fig. \[fig3\], which show that the $E_{CM}$ due to the collision of two different mass particles is infinite when one of the particles have a critical angular momentum (for a suitable choice of rotation parameter $a$ and charge $Q$). From the graphical representation of $E_{CM}$, we have seen that $E_{CM}$ is infinite for $L_1=2.0,2.16152,2.30749,2.55058$ corresponding to $Q=0.0,0.25,0.35,0.45$ and for all other values of angular momentum, it remains finite. After studied the near horizon collision in extremal case, now we extend it for nonextremal black holes. We can see the behavior of $E_{CM}$ vs $r$ for nonextremal black hole from Fig. \[fig4\] for different charge $Q$ and different mass. From Fig. \[fig4\], we can conclude that $E_{CM}$ would be infinite, if collision takes place at the inner horizon of the nonextremal black hole. If the particles collide at the outer horizon of the black hole, then the $E_{CM}$ will remain finite. So, for getting a very high energy, particle collision should happen at the inner horizon. Also, in Fig. \[fig5\], we have shown the effect of $Q$ on the $E_{CM}$. Conclusion {#con} ========== The gravitational collapse of sufficiently massive star leads to the formation of spacetime singularities is a quite common phenomenon in general relativity as predicated by famous singularity theorems [@he]. However, there is a belief that these spacetime singularities do not exist in Nature, but that they are artefact of the classical general relativity. On the other hand, the cosmic censorship conjecture asserts that these singularities can not be seen by an external observer [@rp]. However, the conjecture does not ruled out the possibility of regular black holes. In this paper, we consider the rotating ABG black holes, which can be written in Kerr-like form in Boyer-Lindquist coordinates with mass $M$ and has an additional parameter charge ($Q$) that measures potential deviations from the Kerr metric and includes the Kerr metric as the special case in the absence of the charge ($Q=0$). This special solution is a solution of general relativity coupled to nonlinear electrodynamics that is of Petrov type D and it is singularity free. We have studied in detailed the horizon properties including the ergoregion of rotating ABG black hole and also show how the charge parameter affects the ergoregion. It turns out that area of ergoregion depends both on the value of charge $Q$ and parameter $a$, and the area of ergoregion increase with either of the two parameters. We have also studied the collision of two different rest masses particles in the equatorial plane of rotating ABG black holes and also obtained an expression for the $E_{CM}$ for the particles. We demonstrate that the $E_{CM}$ depends not only on the rotation parameter $a$ but also on the charge $Q$ of the black holes. This work is the generalization of a previous work [@Ghosh:2014mea], where the analysis was restricted to the particles of the same rest mass particles moving in equatorial plane. Applying this general expressions, we realize that an infinite amount of energy can be obtained when the collision of the particles takes place in the vicinity of an event horizon of an extremal black hole and also on the inner horizon for the nonextremal black hole. Furthermore, we analyze the dependence of $E_{CM}$ on charge $Q$, and explicitly show the effect of $Q$ on the $E_{CM}$. 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--- abstract: 'In this paper we explore extensions of the Minimal Supersymmetric Standard Model involving two $SU(2)_L$ triplet chiral superfields that share a superpotential Dirac mass yet only one of which couples to the Higgs fields. This choice is motivated by recent work using two singlet superfields with the same superpotential requirements. We find that, as in the singlet case, the Higgs mass in the triplet extension can easily be raised to $125\,{~\text{GeV}}$ without introducing large fine-tuning. For triplets that carry hypercharge, the regions of least fine tuning are characterized by small contributions to the $\mathcal T$ parameter, and light stop squarks, $m_{\tilde t_1} \sim 300-450\,{~\text{GeV}}$; the latter is a result of the $\tan\beta$ dependence of the triplet contribution to the Higgs mass. Despite such light stop masses, these models are viable provided the stop-electroweakino spectrum is sufficiently compressed.' author: - 'C. Alvarado[^1]' - 'A. Delgado[^2]' - 'A. Martin[^3]' - 'and B. Ostdiek[^4]' bibliography: - 'MyBib.bib' title: Dirac Triplet Extension of the MSSM --- Introduction {#sec:intro} ============ The Minimal Supersymmetric Standard Model (MSSM) sets $m_{Z}$ as the upper bound of the tree-level mass of the lightest [*CP*]{} even scalar in the spectrum. Since this particle is commonly identified with the Standard Model Higgs boson, either large one-loop corrections due to heavy third family squarks or a high degree of stop mixing are necessary to push $m_{h}$ up to the observed value of $\sim 125 {~\text{GeV}}$ [@Aad:2012tfa; @Chatrchyan:2012ufa]. Either of these two requirements on the stops introduces sub-percent fine tuning [@Hall:2011aa]. This occurs because both effects radiatively induce large corrections to the soft mass of the Higgs field $m_{H_u}^2$, which must be canceled off in order stabilize the electroweak scale. In this sense, the observation of the Higgs with a 125 GeV mass makes the MSSM alarmingly fine-tuned, independent of the fact that we have not yet discovered any supersymmetric particles. A variety of techniques have been proposed to avoid such a heavy stop spectrum. The simplest possibilities are to extend the MSSM gauge group or field content, respectively modifying the $D$- and $F$-terms of the Higgs potential [@Cvetic:1997ky; @Ellwanger:2009dp; @Delgado:2010cw]. While the former necessarily alters the quartic terms in a manner dictated by the gauge group, the later relies on raising the quartic coupling of the Higgses via the inclusion of extra superpotential couplings. A class of well-known models based on this effect is the Next-to-Minimal Supersymmetric Standard Model (NMSSM) which adds a gauge singlet field $S$ to the MSSM. Although capable of rendering the correct Higgs mass, the NMSSM does so by decoupling the scalar part of the singlet superfield. However, the soft mass of the singlet feeds back into $m^2_{H_u}, m^2_{H_d}$ at one loop via the renormalization group equations. Large singlet masses therefore can lead to large corrections to $m_{H_{u,d}}^2$, so the NMSSM solution to the Higgs mass comes at the expense of substantial tine tuning. The authors of [@Lu:2013cta] extended the NMSSM with a second singlet $\bar{S}$ which does not couple to the Higgs doublets yet has a superpotential mass term with $S$: $$W=W_{\text{Yukawa}}+(\mu+\lambda S)H_{u}H_{d}+MS\bar{S},$$ where $W_{\text{Yukawa}}$ stands for the usual MSSM Yukawa terms; due to the Dirac mass term between the singlets, the model was dubbed the DiracNMSSM. The tree level Higgs mass squared in this setup is modified, receiving a positive contribution that depends on the $\bar S$ soft mass, and a negative contribution that depends on the $S$ soft mass. Including the one-loop correction from stop loops (see for example Ref [@Carena:2011aa]), the resulting Higgs mass is $$\begin{aligned} m^2_h = &m_Z^2 \cos^2(2\beta) + (\text{stop loops}) \nonumber \\ & + \lambda^2 v^2 \sin^2(2\beta)\left(\frac{m_{\bar{S}}^2}{M^2+m_{\bar{S}}^2}\right) - \frac{\lambda^2v^2}{M^2+m_S^2}\left|A_{\lambda} \sin(2\beta) -2\mu^* \right|^2. \label{eqn:singletModel}\end{aligned}$$ To efficiently raise the Higgs mass, one takes advantage of the positive term while trying to keep the negative term as small as possible. The positive term is increased by taking the soft mass of the non coupled singlet – $m^2_{\bar S}$ – to be much larger than the supersymmetric mass term, $M$. If $M$ is also larger than $\lambda^2v^2$ then the negative term is minimized. While large singlet masses in the NMSSM come hand-in-hand with increased tuning, this does not happen here. Specifically, the authors of [@Lu:2013cta] showed that the mass of $\bar{S}$ can be raised almost indefinitely without introducing fine tuning – a clear violation of the conventional wisdom that increases to the Higgs mass require new light states. As explained in [@Lu:2013cta], the keys to this behavior are the Dirac mass term between $S$ and $\bar S$ and the absence of couplings of $\bar{S}$ with $H_{u}, H_{d}$. A detailed study of the DiracNMSSM was performed in Ref. [@Kaminska:2014wia], taking into account all corrections at one loop order and dominant two-loop corrections. Beyond fine tuning, constraints from SUSY searches and dark matter were also applied. One disadvantage of the original DiracNMSSM is that the singlet contribution to $m^2_h$ has the same $\tan\beta$ dependence as in the NMSSM. Specifically, the singlet piece is largest at low $\tan\beta$, exactly the region where the MSSM tree level Higgs mass vanishes. This can be overcome, but requires sizable coupling of the singlet to Higgses. In this paper, we examine the effects of replacing the singlets in the DiracNMSSM with triplets under $SU(2)_L$, maintaining the key features of the Dirac mass and with only one triplet coupled to the Higgses. Triplet extensions of the MSSM have been studied extensively [@Espinosa:1991wt; @FelixBeltran:2002tb; @DiChiara:2008rg; @Agashe:2011ia; @Basak:2012bd; @Delgado:2012sm; @Delgado:2013zfa; @Bandyopadhyay:2013lca; @Kang:2013wm; @deBlas:2013epa]; they offer richer phenomenology than singlet extensions, but they are also more constrained. Specifically, the neutral components of the triplets generically acquire vacuum expectation values (vev), causing tension with electroweak precision observables [@Khandker:2012zu; @Englert:2013zpa][^5]. Nonetheless, triplets offer appealing features when compared with singlets, especially in the context of the DiracNMSSM: (i) more variety due to two possible hypercharge assignments, $Y=0$ or $Y=\pm1$, and (ii) triplets with hypercharge must be included in pairs for anomaly cancellation and can only have Dirac-type superpotential mass. The rest of the paper is organized as follows. In Sec. \[sec:model\] we introduce the key superpotential interactions and give the correction to the Higgs mass for both the $Y=0$ and $Y=\pm1$ triplet models. Next, in Sec. \[sec:ft\] we analytically study the various sources of fine tuning, pinpointing the dependence of each term on the triplet parameters. This is followed up by a discussion of the precision electroweak $\mathcal{T}$ parameter. From this discussion, it will be clear that the $Y = \pm 1$ model works better at raising the mass of the Higgs, avoiding fine tuning, and staying within the electroweak precision constraints. In Sec. \[sec:Num\] we perform a numerical study, focusing on the $Y = \pm 1$ scenario. As one of the primary differences between singlets and triplets is the existence of additional charged and potentially light fermions, in Sec. \[sec:TripPheno\] we review the phenomenology of ‘exotic’ states, examining both direct production and indirect effects such as altered stop decays. Finally, conclusions are drawn in Sec. \[sec:conclusions\]. The Models {#sec:model} ========== There are two signature features in the DiracNMSSM [@Lu:2013cta], a Dirac mass term between two strictly different superfields, and the fact that only one of the two singlets couples to the Higgs doublets. The extension explored here, where a pair of triplets take the role of the singlets should maintain both properties. With this in mind, we define $\Sigma_1$ to be a $SU(2)_{L}$ triplet chiral superfield which couples to the Higgses in the superpotential, and a define a second triplet $\Sigma_{2}$, which does not. This is not the most general superpotential allowed by the symmetries of the model but we follow the setup of the original DiracNMSSM, in any case the choice is radiatively stable since superpotential couplings can not be generated via radiative corrections[^6]. With the inclusion of the triplets $\Sigma_{1,2}$ the superpotential is enlarged to $$W=\mu H_{u}\cdot H_{d}+\mu_{\Sigma}\text{Tr}(\Sigma_{1}\cdot \Sigma_{2})+W_{H-\Sigma}+W_{\text{Yukawa}} \label{eqn_SuperPotential}$$ where the isospin product employs the convention $a\cdot b\equiv a_{i}\varepsilon_{ij}b_{j}$ with $\varepsilon_{21}=-\varepsilon_{12}=-1$. The parameter $\mu_{\Sigma}$ is a supersymmetric Dirac mass for the triplets, $W_{H-\Sigma}$ couples $H_{u,d}$ with $\Sigma_{1}$ in a way specified by the hypercharge assignments of the triplets, and $W_{\text{Yukawa}}$ represents the standard MSSM Yukawa couplings. We will analyze the cases $Y=0$ and $Y=\pm 1$ for the hypercharge of the triplets[^7]. When the triplets have hypercharge $Y=0$, they can couple to a combination of $H_u H_d$. This case should be seen as a simple extension of the singlet DiracNMSSM scenario, as the couplings take the same form up to factors of $\sqrt{2}$ coming from the normalization of the triplets. On the other hand, triplets with a hypercharge $Y=\pm1$ can only couple to $H_d^2$ or $H_u^2$. We examine the case where $H_u$ couples to the triplet but $H_d$ does not, since the latter will only generate an increased Higgs mass for the unphysical region of $\tan\beta<1$. Both triplet scenarios contain charged scalars and fermions that are absent in the singlet DiracNMSSM. While potentially interesting at colliders, these extra states have minimal impact on the Higgs mass or fine tuning, so we will largely ignore them here. Comments on the phenomenology of the extra states can be found in Sec. \[sec:TripPheno\]. $Y=0$ case ---------- Triplets with hypercharge $Y=0$ couple to both $H_u$ and $H_d$ and are a simple extension to the singlet case studied in [@Lu:2013cta]. The superpotential is given by Eq. (\[eqn\_SuperPotential\]) with $$W_{H-\Sigma}=\lambda H_{d}\cdot \Sigma_{1}H_{u}.$$ Forming the scalar potential, the superpotential terms are accompanied by the soft terms $$\Delta V_{\text{soft}} =m^2_{T} \text{Tr}{|\Sigma_1|^2} + m^2_{\chi} \text{Tr}{|\Sigma_2|^2} + \left( \lambda A_{\lambda} H_d \cdot \Sigma_1 H_u + \mu_{\Sigma} B_{\Sigma} \text{Tr}{(\Sigma_1 \cdot \Sigma_2)} + \text{h.c.} \right), \label{eq:vsoft}$$ and the usual $SU(2)_{L}$ and $U(1)_{Y}$ $D$-terms. Here, $m_{T,\chi}$ are the triplet soft masses, $A_{\lambda}$ and $B_{\Sigma}$ are the trilinear and bilinear soft couplings respectively. While it is possible to give $Y=0$ triplets a non-Dirac supersymmetric mass, we ignore this possibility here as we are particularly interested in the effects of Dirac masses. Focusing on the *CP* even scalar sector of the theory, the sole difference between the triplet and singlet MSSM extensions are factors of $\sqrt{2}$ coming from the normalization of the triplet. The full *CP* even scalar potential for this scenario is shown in Appendix \[sec:appY0\]. Isospin triplets can potentially disrupt electroweak precision tests unless their vevs remain small. A simple way to mitigate the size of the triplet vevs is to take the scalar triplets to be heavier than the Higgses. In this limit, which we will assume throughout, the scalar triplets can be integrated out and are effectively replaced by combinations of lighter fields: $$\begin{aligned} \Sigma_{1,\text{neut}} \equiv T^0 & \rightarrow &\frac{\lambda}{\sqrt{2}} \frac{\mu (|H_u^0|^2 + |H_d^0|^2) - A_{\lambda} H_u^{0*} H_d^{0*}}{\mu_{\Sigma}^2 + m_T^2}+\mathcal{O}\left( \dfrac{1}{D_{T}^{2}},\dfrac{1}{D_{T}D_{\chi}},\dfrac{1}{D_{\chi}^{2}} \right) \label{eqn:defTy0} \\ \Sigma_{2,\text{neut}} \equiv \chi^0 & \rightarrow& \frac{\lambda \mu_{\Sigma}}{\sqrt{2}} \frac{H_u^0 H_d^0}{\mu_{\Sigma}^2 + m_{\chi}^2}++\mathcal{O}\left( \dfrac{1}{D_{T}^{2}},\dfrac{1}{D_{T}D_{\chi}},\dfrac{1}{D_{\chi}^{2}} \right). \label{eqn:defCy0}\end{aligned}$$ where $D_{T,\chi}\equiv \mu_{\Sigma}^{2}+m_{T,\chi}^{2}$. The resulting effective potential for the Higgses can be found in Eq. . From the effective potential, we can read off the modified tree-level *CP*-even scalar mass matrices. Taking the decoupling limit for simplicity and adding the one-loop stop contribution to lightest tree-level mass eigenvalue, we find the Higgs mass: $$\begin{aligned} m_{h}^{2} &=& m_Z^2 \cos^2(2\beta) + (\text{stop loops})+\frac{v^2 \lambda^2}{2}\sin^2(2\beta) \frac{m^2_{\chi}}{\mu^2_{\Sigma}+m^2_{\chi}} \nonumber \\ &&- \frac{v^2\lambda^2}{2} \frac{\left|2 \mu^* -A_{\lambda} \sin(2\beta)\right|^2}{\mu^2_{\Sigma}+m^2_{T}}. \label{eqn_HiggsY0}\end{aligned}$$ The expression above, with a positive (negative) piece that depends on the uncoupled (coupled) triplet soft mass is clearly reminiscent of the singlet DiracNMSSM, Eq. . As in the singlet case, the interplay between the two terms plays an important role in the fine tuning of the model. $Y=\pm1$ case ------------- Given that the superpotential should conserve hypercharge and be holomorphic, a supersymmetric mass term for a triplet with hypercharge $Y=1$ can only be included if there is a second triplet with $Y=-1$. Anomaly cancellation also rests on introducing hypercharge triplets in vector-like pairs. As in the $Y= 0$ scenario above, we assume $\Sigma_1$ is the triplet with superpotential couplings to the Higgses. Depending on its hypercharge $\Sigma_1$ will only be able to couple either to $H_u^2$ or $H_d^2$, which is distinct from the $Y=0$ setup. To get the largest impact from the triplet-Higgs coupling, we want it to couple as much as possible to the physical Higgs boson. At large $\tan\beta$ and large $m_A$, the Higgs boson resides primarily in $H_u$, therefore we assign $Y = -1$ to $\Sigma_1$, permitting the interaction $$W_{H-\Sigma}=\lambda H_{u}\cdot \Sigma_{1}H_{u}.$$ The second triplet $\Sigma_2$ (now with hypercharge Y = 1) has no superpotential couplings. The soft terms are as in Eq. (\[eq:vsoft\]) with the same modification to the $A_{\lambda}$ term as in the superpotential, and the complete *CP* even scalar potential is given in Appendix \[sec:appY1\]. When the triplet scalars are integrated out in this scenario, the neutral components are replaced by: $$\begin{aligned} \Sigma_{1,\text{neut}} \equiv T^0 & \rightarrow & \frac{\lambda \left(A_{\lambda} H_u^{0*}H_u^{0*}-2 \mu H_u^{0*} H_d^0 \right)}{\mu_{\Sigma}^2 + m_T^2}+\mathcal{O}\left( \frac{1}{D_\chi^2},\frac{1}{D_\chi D_T},\frac{1}{D_T^2} \right) \label{eqn:defTY1} \\ \Sigma_{2,\text{neut}} \equiv \chi^0 & \rightarrow& \frac{-\lambda \mu_{\Sigma} H_u^0 H_u^0}{\mu_{\Sigma}^2 +m_{\chi}^2}+\mathcal{O}\left( \frac{1}{D_\chi^2},\frac{1}{D_\chi D_T},\frac{1}{D_T^2} \right). \label{eqn:defCY1}\end{aligned}$$ Working with the effective Higgs potential and proceeding as in the $Y = 0$ case, we find the decoupling-limit Higgs mass to be $$\begin{aligned} m_{h}^{2} &=& m_Z^2 \cos^2(2\beta) + (\text{stop loops}) + 4 v^2 \lambda^2 \sin^4(\beta)\left( \dfrac{ m_{\chi}^2}{\mu_{\Sigma}^2+m^2_{\chi}} \right) \nonumber\\ &&-\dfrac{v^2 \lambda^2 \sin^2{(2\beta)}}{\mu^2_{\Sigma} +m^2_{T}}\left|2\mu^* - A_{\lambda} \tan{(\beta)}\right|^2 . \label{eqn_HiggsY1}\end{aligned}$$ Comparing $m^2_h$ in the two models, Eqs. and , we see similar features. In both models there is a positive contribution to the Higgs mass proportional to $m^2_{\chi}/(\mu^2_{\Sigma}+m^2_{\chi})$. This is maximized when $m_\chi^2 \gg \mu_{\Sigma}^2$, and goes to zero when $m_{\chi}^2 \ll \mu_{\Sigma}^2$, so the Higgs mass is increased the most by decoupling the scalar part of $\Sigma_2$. In Section \[sec:ft\] we will show that the decoupling of $m_{\chi}^{2}$ barely affects the fine tuning. The amplitude and $\tan \beta$ dependence of the positive term is different for the $Y=0$ triplets and the $Y=\pm1$ triplets, $$C_0(\beta)=\frac{v^2 \lambda^2}{2} \sin^2(2\beta)$$ for $Y=0$ and $$C_1(\beta)= 4 v^2 \lambda^2 \sin^4 \beta.$$ for $Y=1$. $C_0$ is maximized when $2\beta=\pi/2$, or $\tan\beta=1$. However, $C_1$ is maximal as $\beta \rightarrow \pi/2$, or $\tan\beta \rightarrow \infty$. As the $\tan\beta$ dependence of $C_{1}$ aligns with that of the MSSM, the size of the triplet contributions to the Higgs mass do not need to be as large, leading to smaller values of $\lambda$ in the $Y=\pm1$ model. Equations and also have a term which acts to lower $m^2_h$. The negative terms depend on the mass of $\Sigma_1$, the triplet which couples to the doublets. A large soft mass for $\Sigma_1$ decreases the absolute value of the negative term, raising the Higgs mass. However, $m^2_T$ also enters into the radiative corrections of the Higgs soft masses, so the $m_T$ value that minimizes the fine tuning is less clear cut and is best tackled numerically. Both of the negative terms also contain a factor which depends on the difference between $\mu$ and $A_{\lambda}$, $\left| 2 \mu^* - A_{\lambda} \sin(2\beta) \right| ^2$ for the $Y=0$ case and $\left| 2 \mu^* - A_{\lambda} \tan \beta \right|^2$ for $Y=\pm1$ respectively. The same expressions appear in the effective triplet vevs, Eq.(\[eqn:defTy0\], \[eqn:defCy0\]) or Eq.(\[eqn:defTY1\],\[eqn:defCY1\]) after the Higgs doublets acquire vacuum expectation values. The $\mathcal{T}$ parameter is tightly constrained by precision electroweak measurements, however, the fact that the same expressions appear in the Higgs mass and the triplet effective vevs implies that regions with the smallest negative contribution to the Higgs mass are also the regions with the smallest $\mathcal{T}$ parameter. Having shown how the Higgs mass is altered in the two Dirac Triplet scenarios and identified key parameters, we now move on to study the fine tuning. Fine tuning calculations and $\mathcal T$ parameter {#sec:ft} =================================================== Equations and show that decoupling the soft mass of $\Sigma_2$ leads to a maximal increase in the Higgs mass. Ordinarily, the introduction of large scalar masses to correct the Higgs mass increases the fine tuning. In the next subsection we show that this is not the case for this model; the fact that $\Sigma_2$ does not couple to the doublets allows it to be decoupled with small effects on the fine tuning, as in the original DiracNMSSM model. Beyond the fine tuning of the Higgs mass, triplet models are also constrained by the $\mathcal{T}$ parameter, which we examine more closely in Section \[subsec:T\]. Fine tuning of $m^2_{H_u}$ {#subsec:MH2} -------------------------- We adopt the definition of fine tuning of [@Lu:2013cta], $$\Delta = \frac{2}{m^2_h} \text{max}\left(m^2_{H_u}, m^2_{H_d}, \frac{d m^2_{H_u}}{d\log{(u)}} L, \frac{d m^2_{H_d}}{d\log{(u)}} L, \delta m_{H_{u}^{0}}^{2}, \mu B_{\mu,\text{eff}} \right) \label{eqn:fine-tuning}$$ where $L\equiv \log(\Lambda/m_{\widetilde{t}})$ accounts for the running to the SUSY breaking scale, $\log(u)$ is the running scale and $\delta m_{H_{u}^{0}}^{2}$ is the one-loop finite threshold correction from the triplets; following [@Lu:2013cta], we set $L=6$. Although we use the same definition for $\Delta$ that was used for the singlet model, we expect the triplet case to be slightly different due to larger triplet-Higgs couplings (coming from the normalization of the triplets) and the different hypercharge possibilities. Putting all of the components together and taking the maximum contribution is best done numerically. However, before launching into numerics, in this section we examine each of the different components of $\Delta$ to get a better feeling for their relative importance and to see how they depend on the triplet parameters. The first entries in $\Delta$ are $m^2_{H_u}$ and $m^2_{H_d}$, the tree-level soft masses for the Higgs doublets. These are not free parameters, rather they are set by the requirement that electroweak symmetry is broken at the minimum of the scalar potential (see Eq.  and for $Y=0$ and and for $Y=\pm1$). In solving the minimization conditions, $m^2_{H_u}$ and $m^2_{H_d}$ inherit a complicated dependence on the triplet parameters that is difficult to generalize. As these entries are typically subdominant in $\Delta$, we do not attempt to tease out the triplet parameter dependence analytically. The next components of $\Delta$ are $\frac{d m^2_{H_u}}{d\log{(u)}} L, \frac{d m^2_{H_d}}{d\log{(u)}} L$, the radiative corrections to the Higgs soft masses. While nominally one-loop effects, these radiative pieces have the potential to be important because they depend quadratically on the masses of heavy particles (stops, triplets, etc.) – objects that do not appear or are subdominant in the tree level Higgs potential. Additionally, the radiative effects are enhanced by $L$, the logarithm that encapsulates the running of soft masses down from the supersymmetry mediation scale. As a result, these radiative pieces are often the largest component of $\Delta$. To see how the triplet parameters enter, we need the renormalization group equations (RGE) governing the evolution of $m^2_{H_u}, m^2_{H_d}$: $$(Y=0) ~~~ \left\{ \begin{matrix} 16 \pi^2 \frac{d m^2_{H_u}}{dt} \supset 6 h_t^2 \left(m^2_{Q_3} + m^2_{U_3} + m^2_{H_u} \right) + 6 \lambda^2 \left(m^2_{H_u} + m^2_{H_d} + m^2_{T} + A_{\lambda}^2 \right)\\ 16 \pi^2 \frac{d m^2_{H_d}}{dt} \supset 6 h_b^2 \left(m^2_{Q_3} + m^2_{D_3} + m^2_{H_d} \right) + 6 \lambda^2 \left(m^2_{H_u} + m^2_{H_d} + m^2_{T} + A_{\lambda}^2 \right) \end{matrix} \right. \label{eqn:HuRGEY0}$$ and $$(Y=\pm1) ~~~ \left\{ \begin{matrix} 16\pi^2 \frac{dm^2_{H_u}}{dt} \supset 6 h_t^2 \left( m^2_{Q_3} + m^2_{U_3} + m^2_{H_u} \right) + 6 \lambda^2 \left( 2 m^2_{H_u} + m^2_{T} + A_{\lambda}^2 \right) \\ 16\pi^2 \frac{d m^2_{H_d}}{dt}\supset 6 h_b^2 \left(m^2_{Q_3} + m^2_{D_3} + m^2_{H_u} \right) \end{matrix} \right. . \label{eqn:HdRGEY1}$$ The large top Yukawa, $h_t$ and the dependence on the stop masses needed in the MSSM to raise the Higgs mass are what drives the fine tuning. In the triplet scenario, the extra contributions to the (tree level) Higgs mass from the triplets permits lighter stops and allows for a less tuned model. The key difference between the DiracNMSSM and the traditional NMSSM is that the mass of the uncoupled state does not feed into the Higgs soft masses at loop level. This same behavior is reproduced in Eq (\[eqn:HuRGEY0\]) nor (\[eqn:HdRGEY1\]) , neither of which depends on $m_{\chi}$, the mass of $\Sigma_2$. As a result, large $m_{\chi}$ – and thereby large positive contributions to the Higgs mass – are permitted without giving rise to fine tuning. The soft mass of $\Sigma_1$ and the trilinear soft term $A_{\lambda}$ enter into the running of $m^2_{H_u}, m^2_{H_d}$, so in principle large values for them would increase $\Delta$. However, both $m_T^2$ and $A_{\lambda}^2$ enter into the beta functions multiplied by $\lambda^{2}$, hence a smaller $\lambda$ would permit these two quantities to take moderate values without dominating the fine tuning. Following the radiative piece in $\Delta$ is the threshold correction $\delta m_{H_{u}^{0}}^{2}$, the finite contribution to $m^2_{H_u}$ that emerges when heavy fields are integrated out. The threshold terms are important as they are the only place where the soft mass of the uncoupled triplet $m_{\chi}^{2}$ (or the non-coupling singlet, in the model of Ref. [@Lu:2013cta]) enters into the fine tuning. The threshold corrections, presented in full in Appendix \[sec:AppFTC\], depend on the soft masses of both triplets. However, since $m_T$ also appears in the (log-enhanced) RGE part of the tuning discussed above, keeping $m_T$ small minimizes the tuning. With $m_T$ kept small, the threshold correction is well approximated by the $\Sigma_2$ piece alone: $$\begin{aligned} (Y=0):~~~~ \delta m_{H_u^0}^2 & \simeq \frac{3}{2}\frac{\lambda^2 \mu_{\Sigma}^2}{16\pi^2} \log\frac{m^2_{\chi} + \mu_{\Sigma}^2}{\mu_{\Sigma}^2} \text{ and} \\ (Y=\pm1):~~~~ \delta m_{H_u^0}^2 & \simeq 6 \frac{ \lambda^2 \mu_{\Sigma}^2}{16 \pi^2} \log \frac{m^2_{\chi} + \mu_{\Sigma}^2}{\mu_{\Sigma}^2}. \label{eqn:thresholdCorrection} \end{aligned}$$ If $\mu_{\Sigma}^2 \gtrsim m_{\chi}^2$, there is little fine tuning from the threshold correction. We saw in Sec. \[sec:model\] that the most interesting parameter space – where the triplet contribution to the Higgs mass is large and positive – occurs when $m_{\chi}^2 \gg \mu^2_{\Sigma}$. For this hierarchy of parameters, the threshold contribution can be non-negligible, though only when $\mu^2_{\Sigma}$ is large (compared to $m_h$) as well. The final component of $\Delta$ is the dependence on $\mu$ and $B_{\mu}$. For the triplet scenario with hypercharge, this component of the tuning is identical to the MSSM. Triplets without hypercharge are slightly more complex, since the effective triplet vevs shift $\mu$ and $B_{\mu}$ from their MSSM values. The shifted values are given by $$\begin{aligned} \mu_{eff} &= \mu - \frac{\sqrt{2}}{2}\lambda \left\langle T^0 \right \rangle \text{ and} \label{eqn:muEff} \\ \mu B_{\mu,\text{eff}} &= \mu B_{\mu} - \frac{\lambda}{\sqrt{2}} \left(A_{\lambda} \left\langle T^0 \right \rangle + \mu_{\Sigma} \left \langle \chi^0 \right \rangle \right ). \label{eqn:BEff}\end{aligned}$$ Though not usually the dominant component in $\Delta$, these contributions to the fine tuning measure are inevitable as $\mu$ and $B_{\mu}$ enter directly into the tree-level mass matrix of the Higgs. After considering the individual components of the fine tuning measure, we are now ready for a full numerical study of the tuning over a range of triplet parameters. Before doing so, we first examine how the $\mathcal{T}$ parameter constrains the available parameter space. Constraints from the $\mathcal{T}$ parameter {#subsec:T} -------------------------------------------- Electroweak scalar triplets that acquire vacuum expectation values notoriously spoil the relation between $m_W$ and $m_Z$. This mass ratio is more commonly expressed as the $\mathcal T$ parameter $$\alpha \mathcal{T} = \frac{m_W^2}{m_Z^2 \cos^2 \theta_W}-1.$$ The authors of [@Baak:2012kk; @Baak:2014ora] used data from $Z$ pole measurements [@ALEPH:2005ab], the running quark masses [@Beringer:1900zz], the five-quark hadronic vanuum polarization contribution to $\alpha\left(M_Z^2\right)$, $\Delta\alpha_{\text{had}^{(5)}} \left(M_Z^2\right)$ [@Davier:2010nc], the mass and width of the $W$ [@Beringer:1900zz], top quark mass [@ATLAS:2014wva], and Higgs mass measurements [@Aad:2014aba; @CMS:2014ega] to preform a global fit of electroweak data. A value of $\mathcal{T} = 0.09\pm 0.13$ gives the best fit of the data if all of the oblique parameters are allowed to float[^8]. Forcing the (tree-level) triplet contributions to the $\mathcal T$ parameter to lie within the 1-$\sigma$ uncertainty, we can derive a bound on the triplet model parameters. In an effective theory where we have integrated out the triplets, there are no triplet fields around to get vevs, but the $\mathcal T$ contributions are still present in the form of higher dimensional operators. Specifically, after integrating out the triplets, the kinetic term for the $\Sigma_i$ becomes (schematically) $$\left| D_{\mu} \Sigma_i \right|^2 \xrightarrow[\text{integrated out}]{\Sigma} \frac{1}{\Lambda^2} \left| H D_{\mu} H \right|^2,$$ which, once the Higgses are set to their vevs, contributes differently to the $W$ and $Z$ mass. The operator is intentionally left vague, as the actual combinations of the $H_u$ and $H_d$ and the mass scale $\Lambda$ are different for each triplet. For the triplets with $Y=0$, this operator contributes to $\mathcal{T}$ by $$\mathcal{T}_{Y=0}=\dfrac{1}{\alpha} ~ \dfrac{4\bigl( \langle \chi^{0}\rangle^{2}+\langle T^{0}\rangle^{2} \bigr)}{v^{2}-4\bigl( \langle \chi^{0}\rangle^{2}+\langle T^{0}\rangle^{2} \bigr)} \label{eqn:TparamY0}$$ where $\langle T^0 \rangle$ and $\langle \chi^0 \rangle$ are the values of equation and after the doublets have developed vevs – what we dub ‘effective vevs’ for the triplets. The effective vevs are approximately given by $$\left\langle T^0 \right\rangle_{Y=0} \approx \dfrac{v^2 \lambda}{2\sqrt{2}} \dfrac{2 \mu^* - A_{\lambda}\sin(2\beta)}{\mu^2_{\Sigma}+m^2_{T}} \text{ and} ~~~~ \left\langle \chi^0 \right \rangle_{Y=0} \approx -\dfrac{v^2 \lambda}{2\sqrt{2}} \dfrac{\mu_{\Sigma} \sin(2\beta)}{\mu^2_{\Sigma} + m^2_{\chi}}, \label{eqn_vev0}$$ up to higher order terms in $1/(\mu_{\Sigma}^{2}+m_{T,\chi}^{2})$. For the case with hypercharge, the $\mathcal{T}$ parameter takes the form $$\mathcal{T}_{Y=\pm1} = -\frac{1}{\alpha}~ \frac{2 \left(\langle \chi^0 \rangle^2 + \langle T^0 \rangle^2 \right) }{v^2} \label{eqn_Y1}$$ with $\langle T^0 \rangle$ and $\langle \chi^0\rangle$ now coming from Eq. and once the doublets acquired the vevs, $$\left\langle T^0 \right \rangle_{Y=-1} \approx -\dfrac{v^2\lambda}{2} \dfrac{ \sin(2\beta) \left( 2 \mu^{*} -A_{\lambda} \tan{(\beta)} \right)}{ \mu^2_{\Sigma}+m^2_{T} }\text{ and} ~~~~ \left\langle \chi^0 \right\rangle_{Y=1} \approx v^2 \dfrac{ - \lambda \mu_{\Sigma} \sin^2{(\beta)}}{ \mu^2_{\Sigma}+m^2_{\chi} }. \label{eqn_vev1}$$ Inspecting these equations, we can identify several parameter combinations that dictate the size of the $\mathcal{T}$ parameter. - $m_{\chi}^{2}$: The effective vev ${\langle}\chi^0{\rangle}\rightarrow 0$ in the limit of large $m_{\chi}$. In order to effectively raise the Higgs mass, we want $m_{\chi}^2 \gg \mu^2_{\Sigma}$. Large $m_{\chi}$ also does not add to the fine tuning (see previous subsection), so large $m_{\chi}$ is preferred for both the fine tuning and the $\mathcal{T}$ parameter. - $m_{T}^{2}$: Similarly, the effective vev ${\langle}T^0 {\rangle}\rightarrow 0$ in the limit of large $m_{T}$. A large value for $m_{T}$ also reduces the negative term in the Higgs mass squared equations. However, $m^2_T$ enters into the tuning from the RGE running terms and can quickly dominate the fine tuning. - $\mu_{\Sigma}$: Both ${\langle}\chi^0 {\rangle}\rightarrow 0$ and ${\langle}T^0 {\rangle}\ \rightarrow 0$ for large $\mu_{\Sigma}$. This is not desired as it decreases the triplet contribution to the Higgs mass and removes any interesting phenomenology of extra light states. - $\lambda$: The $\mathcal{T}$ parameter goes as $\lambda^2$. The fact that the $Y=\pm1$ model can easily get the correct Higgs mass for lower values of $\lambda$ implies that the model with hypercharge will not be as constrained by the $\mathcal{T}$ parameter for fixed stop masses. - $\mu$ *and* $A_{\lambda}$: One could also have a cancelation between the $\mu$ and $A_{\lambda}$ terms. This would be a cancellation between a supersymmetric term and a soft term, which is in itself unnatural. - $\tan \beta$: The triplets with hypercharge $Y=\pm1$ have an extra dependence on $\sin(2\beta)$ in ${\langle}T^0{\rangle}$. At large values of $\tan\beta$, this goes to 0. Large values of $\tan\beta$ were already preferred for $Y=\pm1$ in order to raise the Higgs mass as much as possible. The $Y=0$ model is not as lucky. Considering these points, in particular the $\lambda$ and $\tan\beta$ dependence, it is clear that the $\mathcal{T}$ parameter is more constraining on the $Y=0$ model. In addition, for fixed triplet-Higgs coupling $\lambda$, the triplet contribution to the Higgs mass in the $Y = 0$ model is smaller than in the singlet DiracNMSSM scenario because of the $\sqrt 2$ factor in the normalization of the neutral components. As this scenario suffers in fine tuning and the $\mathcal{T}$ parameter without the promise of interesting phenomena, we choose to ignore the $Y=0$ Dirac triplet model for the rest of the paper and focus our numerical and phenomenological study on $Y \ne 0$. Lastly, we point out that ${\langle}T^0 {\rangle}$ and ${\langle}\chi^0{\rangle}$ contribute to the $\mathcal{T}$ parameter at tree level, and to order $\lambda^2$. To be consistent, we have also calculated the one-loop fermionic contributions to the $\mathcal{T}$ parameter to order $\lambda^2$. Because the triplet fermions are Dirac particles, and the mixing to order $\lambda^2$ keeps the entire triplet representation the same mass, there is no contribution to the $\mathcal{T}$ parameter at order $\lambda^2$. Numerical study: $Y = \pm 1$ {#sec:Num} ============================ The analytical expressions of the last section allowed us to determine the overall scheme needed to minimize fine tuning and yet maximize the triplet contributions to the Higgs mass. Focusing entirely on the $Y = \pm 1$ scenario, the preferred regions are large $\tan\beta$, large $m_{\chi}$, and small values for $m_{T}$ and the stop masses. The coupling $\lambda$ needs to be large enough to raise the Higgs mass without being so large as to induce large triplet vevs. While there are multiple free parameters at hand, we wish to keep our numerical analysis both detailed and manageable. For this reason, we limit the parameters we vary to two scans, one over $\lambda$ and $m_T$ and the other over $\mu_{\Sigma}$ and $m_{\chi}$. The other parameters are fixed to benchmark values shown in Table. \[tab\_BenchmarkFT\].\ -------------------------- --------------------------------- ----------------- $\tan{\beta} = 10$ $m_A = 300{~\text{GeV}}$ $A_t=0$ $\mu = 250{~\text{GeV}}$ $B_{\Sigma} = 100{~\text{GeV}}$ $A_{\lambda}=0$ -------------------------- --------------------------------- ----------------- : Benchmark parameter values for the calculation of the fine tuning variables for the $Y=\pm1$ model. For simplicity, the gaugino masses and all squark/slepton masses other than the stop are assumed to be decoupled.[]{data-label="tab_BenchmarkFT"} The values for the fixed parameters in Table \[tab\_BenchmarkFT\] are motivated by several considerations. First, since the Higgs mass contribution, fine tuning, and $\mathcal T$ parameter are improved at large $\tan{\beta}$, we select $\tan\beta = 10$ as a representative value. Next, the scalar masses $m_A$ and $B_{\Sigma}$ play little role in our results, so they are good parameters to fix. The mass $m_A$ enters into the Higgs mass matrix, however as we always assume the decoupling limit it has little effect (so long as the value we choose is large enough to justify the decoupling limit). Similarly, the soft parameter $B_{\Sigma}$ mixes the scalars from $\Sigma_1$ and $\Sigma_2$. This mixing does not change our results, but complicates the translation between the scalar mass eigenstates and the Lagrangian parameters. Therefore we select a small $B_{\Sigma}$ for simplicity. The effective vev ${\langle}T^0 {\rangle}$ (and therefore the $\mathcal{T}$ parameter) depend on the difference between $\mu$ and $A_{\lambda}$, however, this term is suppressed at large values of $\tan\beta$. Varying $A_{\lambda}$ over a moderate range of values, we find the fine tuning does not change much. Therefore, we set $A_{\lambda}$ to 0 (together with $A_{t}$ for consistency), a choice that fits well within gauge mediated SUSY breaking scenarios [@Dine:1981gu; @Nappi:1982hm; @AlvarezGaume:1981wy; @Dine:1993yw; @Dine:1994vc; @Dine:1995ag]. The last parameter we fix is $\mu$. Since we have decoupled/ignored the wino, the chargino mass is set by $\mu$, thus the existing LEP2 bound [@lepii] on charginos sets a lower bound of $\mu \gtrsim 100\,{~\text{GeV}}$. High $\mu$ values are also disfavored by fine tuning, so we therefore pick an intermediate value of $\mu = 250\,{~\text{GeV}}$ for our benchmark. The contribution to the tuning for this choice $\Delta(\mu)=8.47$; as this value is independent of the rest of the spectrum, $\Delta(\mu)$ should be regarded of as the minimum tuning possible according to our measure. From the fine tuning perspective alone, a value of $\mu$ closer to the LEP2 bound would be better. However, as we will detail in section \[sec:TripPheno\], $\mu$ also plays a role in stop phenomenology. To study the fine tuning, we scan over the remaining triplet parameters, the coupling $\lambda$, the Dirac mass, $\mu_{\Sigma}$, and the soft masses, $m_{\chi}$ and $m_T$. Once values for these are chosen, the triplet contribution to the Higgs mass is known (see Eq.) and the stops are the only part left to enforce $m_h=125{~\text{GeV}}$. As the stop contribution to the Higgs mass depends on the masses of both stops, we must make some assumptions in order to extract the values. We study two different assumptions: 1. Left and right-handed stop have the same mass. ($m_{\tilde{Q}_3}=m_{\tilde{u}^c_3}$) 2. The right-handed stop is used to set the Higgs mass while the left-handed stop is set to 800 GeV, which is above the most stringent LHC limits [@Aad:2012xqa; @Aad:2014qaa; @Aad:2012uu; @Aad:2014nra; @Aad:2014bva; @Aad:2012ywa; @Chatrchyan:2012lia; @Chatrchyan:2013xna; @Chatrchyan:2014lfa; @CMS:2014yma; @CMS:2014wsa]. Next, we use SuSpect2 [@Djouadi:2002ze] to find the mass of the Higgs in the MSSM for the benchmark values and a given set of stop masses. The final Higgs mass squared is then the result of adding the MSSM part and the triplet contribution in quadrature. $$m_h^2 \equiv (125.5{~\text{GeV}})^2 = m_h^2 (\text{MSSM}) + m_h^2 (\text{Triplet}).$$ We vary the value of the stop mass until this relationship is achieved. Then, once the stop mass is known, we can calculate the fine tuning defined in Eq. . Knowing that the triplet contribution to the Higgs mass is largest when $m_{\chi} \gg \mu_{\Sigma}$, we first choose to fix $$m_{\chi}=10 {~\text{TeV}}~~~\text{ and } ~~~ \mu_{\Sigma}=300{~\text{GeV}}$$ and scan over value of $\lambda$ and $m_T$. The left panels of Fig. \[fig:ftLambda\] show the values of the stop soft masses that are needed in order to set the correct Higgs mass; in Fig. \[fig:finetuningTB10BothEqualLambda\], both stop soft masses are equal, while in Fig. \[fig:finetuningTB10ChangeRightLambda\] the left-handed soft mass is fixed at $800{~\text{GeV}}$ and the right-handed soft mass is indicated by the contours. The triplets do not affect the Higgs mass in the MSSM limit that $\lambda\rightarrow0$, so very large stop masses are needed. As $\lambda$ is increased from zero, the necessary stop mass decreases. If $\lambda \gtrsim 0.35$, the triplet $F$-terms generate a Higgs mass that is alway greater than observed value. These regions are marked in green in the figures. The soft mass $m_T$ only affect the mass of the Higgs through the negative term in Eq. (\[eqn\_HiggsY1\]). For large values of $\tan\beta$, this term is negligible. The fine tuning is calculated at each point once the stop masses have been obtained. Contours of $\Delta$ are shown in the right panels of Fig. \[fig:ftLambda\]. The white, pink, and blue regions represent a fine tuning of $\Delta \le 100$, $100 < \Delta \le 1000$, and $\Delta > 1000$ respectively. The RGE running part of the fine tuning measure is dominant and depends on $h_t^2 (m^2_{Q_3} + m^2_{U_3})$ and $\lambda^2 m_T^2$. Increasing $\lambda$ lowers the stop masses, decreasing the fine tuning until $\lambda^2 m_T^2$ is comparable to $h_t^2 (m^2_{Q_3} + m^2_{U_3})$. As such, a small value of the soft mass is preferred for fine tuning, although the $\mathcal{T}$ parameter can cause issues if $m_T$ is too light. ![The left panels show contours of the stop soft mass needed in order to raise the Higgs mass to he observed value when $\mu_{\Sigma}=300{~\text{GeV}}$, $m_{\chi}=10{~\text{TeV}}$ and $\tan\beta=10$. In () both stops have the same mass while () only changes the right-handed soft mass and keeps the left-handed stop at $800{~\text{GeV}}$. The right panels show the corresponding contours of fine tuning. The dark red region marks where the vevs of the triplets cause too-large contributions to the $\mathcal{T}$ parameter. The orange region supposes an improvement in the measured $\mathcal{T}$ parameter by an order of magnitude.[]{data-label="fig:ftLambda"}](Stops_MuSig_300_mchi_10000_Both.pdf "fig:"){width="0.45\linewidth"} ![The left panels show contours of the stop soft mass needed in order to raise the Higgs mass to he observed value when $\mu_{\Sigma}=300{~\text{GeV}}$, $m_{\chi}=10{~\text{TeV}}$ and $\tan\beta=10$. In () both stops have the same mass while () only changes the right-handed soft mass and keeps the left-handed stop at $800{~\text{GeV}}$. The right panels show the corresponding contours of fine tuning. The dark red region marks where the vevs of the triplets cause too-large contributions to the $\mathcal{T}$ parameter. The orange region supposes an improvement in the measured $\mathcal{T}$ parameter by an order of magnitude.[]{data-label="fig:ftLambda"}](FT_MuSig_300_mchi_10000_Both.pdf "fig:"){width="0.45\linewidth"} ![The left panels show contours of the stop soft mass needed in order to raise the Higgs mass to he observed value when $\mu_{\Sigma}=300{~\text{GeV}}$, $m_{\chi}=10{~\text{TeV}}$ and $\tan\beta=10$. In () both stops have the same mass while () only changes the right-handed soft mass and keeps the left-handed stop at $800{~\text{GeV}}$. The right panels show the corresponding contours of fine tuning. The dark red region marks where the vevs of the triplets cause too-large contributions to the $\mathcal{T}$ parameter. The orange region supposes an improvement in the measured $\mathcal{T}$ parameter by an order of magnitude.[]{data-label="fig:ftLambda"}](Stops_MuSig_300_mchi_10000.pdf "fig:"){width="0.45\linewidth"} ![The left panels show contours of the stop soft mass needed in order to raise the Higgs mass to he observed value when $\mu_{\Sigma}=300{~\text{GeV}}$, $m_{\chi}=10{~\text{TeV}}$ and $\tan\beta=10$. In () both stops have the same mass while () only changes the right-handed soft mass and keeps the left-handed stop at $800{~\text{GeV}}$. The right panels show the corresponding contours of fine tuning. The dark red region marks where the vevs of the triplets cause too-large contributions to the $\mathcal{T}$ parameter. The orange region supposes an improvement in the measured $\mathcal{T}$ parameter by an order of magnitude.[]{data-label="fig:ftLambda"}](FT_MuSig_300_mchi_10000.pdf "fig:"){width="0.45\linewidth"} At each point in the scan we calculate the effective triplet vevs and their contribution to the $\mathcal{T}$ parameter. The red regions show where the triplet contributions to $\mathcal{T}$ are larger than the 0.13 1-$\sigma$ uncertainty [@Baak:2014ora]. We also mark in orange what could be excluded by a new precision study of the $Z$-pole if the uncertainty on the $\mathcal{T}$ parameter were decreased by an order of magnitude. Fig. \[fig:ftLambda\] has the soft mass of $\Sigma_2$ decoupled ($m_{\chi}=10{~\text{TeV}}$), so ${\langle}\chi^0{\rangle}$ is negligible and $\mathcal{T}$ is only affected by ${\langle}T^0 {\rangle}$. The large value of $\tan\beta$ suppresses ${\langle}T^0 {\rangle}$ so the current $\mathcal{T}$ bounds can only exclude $m_T < 200 {~\text{GeV}}$ at the largest allowed values of $\lambda$. An improved measurement brings the exclusion to values of $\lambda$ as low as 0.1 and soft masses as large as $500{~\text{GeV}}$. The vev ${\langle}T^0 {\rangle}$ is proportional to $1/(\mu_{\Sigma}^2+m_T^2)$, so the reach of this exclusion region is strongly dependent on the value of $\mu_{\Sigma}$ as well, which has been kept fixed up to this point. Before discussing the differences between the two different stop assumptions, we scan over $\mu_{\Sigma}$ and $m_{\chi}$ to understand how these affect the Higgs mass, fine tuning, and $\mathcal{T}$. We chose the point $$\lambda=0.25 ~~~\text{ and }~~~ m_T = 800~{~\text{GeV}},$$ which in the first scan lies close to the smallest fine tuned contour and is beyond the reach of the improved $\mathcal{T}$ exclusion. Figure \[fig:ft\] shows the results of the second scan again with the stop masses in the left panels and the shaded regions the same as in Fig. \[fig:ftLambda\]. The triplet contribution to $m_h^2$ is proportional to $m_{\chi}^2/(\mu_{\Sigma}^2+m_{\chi}^2)$. Larger values of $m_{\chi}$ decrease the stop masses while larger $\mu_{\Sigma}$ decouples the effect of the triplets and forces larger stop masses. Lines of constant stop mass run along the diagonal. The right panels of Fig. \[fig:ft\] show the corresponding fine tuning measure. Over most of the parameter space, the fine tuning contours follow the stop mass contours which implies that the RGE running term is dominating the fine tuning. This is not the case in the upper right part of the plots for large values of $m_{\chi}$ and $\mu_{\Sigma}$. In these regions the finite threshold correction piece of the fine tuning dominates. This term is never dominant for $\mu_{\Sigma} \lesssim 1~{~\text{TeV}}$ or $m_{\chi} \lesssim 10{~\text{TeV}}$. The $\mathcal{T}$ parameter constrains more of the parameter space in this scan. In this case, $m_T$ is large so ${\langle}T^0 {\rangle}$ does not contribute much to $\mathcal{T}$. Instead, $\mathcal{T}$ is controlled by ${\langle}\chi^0 {\rangle}$ which is proportional to $\mu_{\Sigma}/(\mu_{\Sigma}^2 + m^2_{\chi})$. Keeping the triplet contributions to $\mathcal{T}$ within the 1-$\sigma$ uncertainty excludes out to $\mu_{\Sigma} \le 1.1{~\text{TeV}}$ for $m_{\chi} \lesssim 800{~\text{GeV}}$. The orange region again shows what could be excluded if the uncertainty were improved by an order of magnitude. This may be the best method for explicitly excluding parameter space and reaches out to $\mu_{\Sigma} \le 1.5{~\text{TeV}}$ for $m_{\chi} \lesssim1.2{~\text{TeV}}$. Having a low value for $\mu_{\Sigma}$ allows for a large triplet contribution to the Higgs mass without the need to worry about the finite threshold correction term in the fine tuning. In this region, the $\mathcal{T}$ parameter forces $m_{\chi}$ to large values to decrease ${\langle}\chi^0 {\rangle}$. This in turn *increases* the triplet contribution to the Higgs mass, lowering the fine tuning. ![Analogous panels to Fig.\[fig:ftLambda\], this time with varying $\mu_{\Sigma}$ and $m_{\chi}$ for fixed $\lambda=0.25$ and $m_{T}=800{~\text{GeV}}$. In section \[sec:TripPheno\], we study the phenomenology of the dashed green line.[]{data-label="fig:ft"}](ChangeBoth_TB10_NoCancel_Stops "fig:"){width="0.45\linewidth"} ![Analogous panels to Fig.\[fig:ftLambda\], this time with varying $\mu_{\Sigma}$ and $m_{\chi}$ for fixed $\lambda=0.25$ and $m_{T}=800{~\text{GeV}}$. In section \[sec:TripPheno\], we study the phenomenology of the dashed green line.[]{data-label="fig:ft"}](ChangeBoth_TB10_NoCancel "fig:"){width="0.45\linewidth"} ![Analogous panels to Fig.\[fig:ftLambda\], this time with varying $\mu_{\Sigma}$ and $m_{\chi}$ for fixed $\lambda=0.25$ and $m_{T}=800{~\text{GeV}}$. In section \[sec:TripPheno\], we study the phenomenology of the dashed green line.[]{data-label="fig:ft"}](OnlyRight_TB10_NoCancel_Stops "fig:"){width="0.45\linewidth"} ![Analogous panels to Fig.\[fig:ftLambda\], this time with varying $\mu_{\Sigma}$ and $m_{\chi}$ for fixed $\lambda=0.25$ and $m_{T}=800{~\text{GeV}}$. In section \[sec:TripPheno\], we study the phenomenology of the dashed green line.[]{data-label="fig:ft"}](OnlyRight_TB10_NoCancel "fig:"){width="0.45\linewidth"} Having discussed how the fine tuning depends on the triplet parameters, we now examine the effects of the different stop assumptions. The general results apply to both scans, but we focus only on the second scan, with $\lambda$ and $m_T$ fixed. The stop contribution to the Higgs mass depends on the geometric mean of the stop masses. At $\mu_{\Sigma}= m_{\chi} = 10{~\text{TeV}}$, the geometric mean of the stops needs to be around $800{~\text{GeV}}$. In this case, both assumptions for choosing the stop mass give $m_{\tilde{Q}_3}=m_{\tilde{u}_3^c}=800{~\text{GeV}}$ and the corresponding measure of fine tuning is around 50. Lowering the value of $\mu_{\Sigma}$ increases the triplet contributions to the Higgs mass and decreases the stop masses and fine tuning. The minimum stop mass (still along $m_{\chi}=10{~\text{TeV}}$) is reached when $\mu_{\Sigma} \le 2 {~\text{TeV}}$. When both stop soft masses are simultaneously changed, they take on a minimum mass of around $450{~\text{GeV}}$. The minimum fine tuning is then $\Delta\sim9$. On the other hand, when only changing the right-handed soft mass, it needs to be even lighter. Its minimum soft mass is around 260 GeV which gives a fine tuning of 17. Although one stop mass is lighter, the RGE running (and thus the tuning) are worse because the left-handed mass is still at $800{~\text{GeV}}$. We have marked the line $m_{\chi}=10{~\text{TeV}}$ with a green dashed line and will study the phenomenology along this line in more detail in the next section. The benchmark values that we have used allow for quite low values of fine tuning for both assumptions about the stop masses. This low fine tuning comes at the cost of having light stops. In fact, for stop mass assumptions, the minimum stop mass achieved is well below the 750-800 GeV LHC limits [@Aad:2012xqa; @Aad:2014qaa; @Aad:2012uu; @Aad:2014nra; @Aad:2014bva; @Aad:2012ywa; @Chatrchyan:2012lia; @Chatrchyan:2013xna; @Chatrchyan:2014lfa; @CMS:2014yma; @CMS:2014wsa]. In the next section we will show that these searches do not exclude all of our regions of low fine tuning. However, it does raise the question about how the model can deal with LHC SUSY searches and what other signatures to search for. Although the triplet scalars need to be heavy, their fermion counterparts – the [*tripletinos*]{} – with mass $\sim \mu_{\Sigma}$, can be light enough to be reachable by the LHC. In the next section we briefly explore the phenomenology of the tripletinos at the LHC. We will examine both the direct constraints on these particles and how tripletinos affect the decay of the stops. Triplet Fermion Phenomenology {#sec:TripPheno} ============================= (Lack of) Constraints on Tripletinos {#sec:TripConstraints} ------------------------------------ The $Y=1$ triplets contain neutral, $\pm1$ and $\pm2$ charged fermions. The neutral and singly-charged fermions mix with the neutralinos and charginos, respectively (the mass matrices of the fermions are shown in Appendix \[sec:AppMixing\]). The doubly-charged states, on the other hand, do not mix with SM particles. One might expect that strong bounds would exist for such exotic states. The tripletinos, however, are good at hiding. 1. Direct Searches The charge $\pm 1, 0$ tripletinos are subject to MSSM electroweakino searches, which currently exclude regions where the LSP mass is less than around 150 GeV if there are no light sleptons [@Aad:2014vma; @CMS:2013dea]. These searches are most powerful if the LSP is light and if there is a large separation between the mass of the LSP and the mass of the rest of the other states. As a result, these conventional searches fail for quasi-degenerate electroweakino spectra, such as one expects in a pure Higgsino scenario or with a Higgsino-tripletino admixture. Another possibility is to look for disappearing tracks [@CMS:2014gxa] or long-lived charged particles [@Aad:2013pqd; @Chatrchyan:2013oca], though these approaches require a level of degeneracy that is atypical in the region of tripletino-Higgsino parameter space we are interested in. One potential avenue is a search focusing on the doubly charged tripletinos and $\mu_{\Sigma} < \mu$. The (lighter) mass eigenstates are then given by $$\begin{aligned} m_{\tilde{\chi}^{++}} &=\mu_{\Sigma}, \\ m_{\tilde{\chi}^{+}} &= \mu_{\Sigma} \left(1-\frac{1}{2}\frac{\lambda^2 v^2}{\mu^2} (1-\cos(2\beta) \right), ~~\text{and} \\ m_{\tilde{\chi}^0} &= \mu_{\Sigma} \left(1-\frac{\lambda^2 v^2}{\mu^2} (1-\cos(2\beta) \right). \end{aligned}$$ For benchmark parameters $\mu=250{~\text{GeV}}, \lambda = 0.25$ and taking $\mu_{\Sigma}=150~{~\text{GeV}}$, the masses are 150, 145.5, and 141 GeV respectively. The pair production cross section of the doubly charged state at the LHC is $1.05$ ($2.48$) pb for the LHC at $8$ ($14$) TeV. These decay down to the neutral state through $W^{\pm}$ bosons. Although the decay products will be soft and hard to detect, the signal has 4 $W^{\pm}$ bosons which can decay leptonically. A dedicated search is beyond the scope of this paper, but the relatively large cross section along with the clean final state could motivate a search for the doubly charged particles – recoiling off a hard, initial-state jet for triggering purposes. 2. Oblique parameters Triplet fermions have the potential to generate a loop level contribution to the $\mathcal T$ parameter. However, at $O(\lambda^2)$ we find this contribution to be zero due to the Dirac nature of the tripletinos and the near degeneracy of the states. We calculated this using mass insertions to account Higgsinos-tripletino mixing, as well as in an effective theory where the Higgsinos were integrated out. In both cases the vacuum polarization amplitudes $\Pi^{11}(0)$ and $\Pi^{33}(0)$ are non-zero, but their difference is zero. 3. Higgs observables The addition of $SU(2)_L$ triplets to the content of the MSSM adds more charged particles which couple to the Higgs and could affect the decay of $h\rightarrow \gamma\gamma$. Unlike more traditional triplet extensions [@Delgado:2012sm; @Delgado:2013zfa; @Kang:2013wm; @Arina:2014xya] only one of the triplets couples to Higgses, and in the $Y=\pm1$ Dirac Triplet extension of the MSSM, the partial width is not affected to lowest order. The only way that the triplets in this model play a role in the diphoton rate is allowing for lower stop masses which affect both the production and the decay of the Higgs [@Carena:2011aa; @Carena:2013iba]. Moving to direct production at the LHC, the triplet fermions are hard to detect due to the small mass splitting. Giving the triplets a Dirac mass and having only one triplet couple to the doublet makes their presence hard to find in sensitive loop level processes too. The effects of the triplets can still be seen in the efficient raising of the Higgs mass leading to light stops. If the triplet fermions happen to be lighter than the stops it would be possible use stop decays to observe the triplet fermions. Stop Decays {#subsec:StopDecays} ----------- We have seen that the inclusion of $Y = \pm 1$ triplets with interactions inspired by the DiracNMSSM – namely where only one triplet couples to Higgses – leads to light stops. While nice from a fine-tuning perspective, light stops are constrained by the LHC, so we must make sure these ‘natural’ scenarios are not ruled out by experimental searches. As we illustrate in this section, the phenomenology of the stops depends on the hierarchy of $\mu$ and $\mu_{\Sigma}$ and whether the lightest stop is left or right-handed. In all four scenarios we sketch out the viable parameter space. In most circumstances, we find that compressed spectra are required to avoid LHC limits, such that larger values of $\mu$ are necessary; this a posteriori motivates our benchmark choice $\mu = 250\,{~\text{GeV}}$. To anchor our phenomenology study, we fix $\lambda=0.25$, $m_T=800{~\text{GeV}}$ $m_{\chi} = 10{~\text{TeV}}$, and vary $\mu_{\Sigma}$ (all other parameters are taken from Table \[tab\_BenchmarkFT\]). This parameter slice is indicated by the green dashed line in Figs. \[fig:finetuningTB10BothEqual\] and \[fig:finetuningTB10ChangeRight\] and is characterized by low fine-tuning. The spectrum of the charginos, neutralinos and stops along this line is shown below in Fig. \[fig:Spectrum\]. The solid colored lines show the chargino/neutralino masses; the sharp feature at $\mu_{\Sigma} \sim \mu = 250\,{~\text{GeV}}$ corresponds to where the composition of the lightest $\tilde{\chi}^0_i, \tilde{\chi}^+$ shifts from primarily tripletino to primarily Higgsino. ![Spectrum of the stops, neutralinos and charginos. The Higgsino mass parameter $\mu=250{~\text{GeV}}$ while the triplet mass is along the horizontal axis. Two methods of choosing the stop mass are shown. The solid black line labelled $\tilde{t}_{1,2}$ marks changing both the left and right soft masses simultaneously. The dashed lines keep the left-handed soft mass at $800{~\text{GeV}}$ and use the right-handed mass to set the Higgs mass.[]{data-label="fig:Spectrum"}](LowSpectrum_stop_fermions_v2){width="0.45\linewidth"} The black lines in Figure \[fig:Spectrum\] indicate the stop spectra for both stop selection choices (see Sec. [\[sec:Num\]]{}). The solid line corresponds to changing both the left and the right-handed soft masses simultaneously. The dashed line, labeled $\tilde{t}_1$, and the dotted line, labelled $\tilde{t}_2$ mark the masses of the two stops when the left-handed soft mass is set to $800\,{~\text{GeV}}$ and the right-handed mass moves to accommodate the Higgs mass. The next ingredient in the stop phenomenology is the branching ratio. Using the same set of parameters as in Fig. \[fig:Spectrum\], we plot the branching ratio below in Fig. \[fig:BranchingRatio\] for both stop scenarios. In the branching ratio calculations we only keep the two-body final states. ![Branching ratios of the stops when only considering 2-body decays. The bino and wino have been completely decoupled, only leaving the Higgsino and tripletino for the stop decays. The left panel has the left-handed stop mass set to $800{~\text{GeV}}$ and uses the right-handed mass to raise the Higgs mass. The right panel has both soft masses change to set the Higgs mass. The Higgsino mass is $\mu=250{~\text{GeV}}$. []{data-label="fig:BranchingRatio"}](Branching_caseIVA_Ronly "fig:"){width="0.45\linewidth"} ![Branching ratios of the stops when only considering 2-body decays. The bino and wino have been completely decoupled, only leaving the Higgsino and tripletino for the stop decays. The left panel has the left-handed stop mass set to $800{~\text{GeV}}$ and uses the right-handed mass to raise the Higgs mass. The right panel has both soft masses change to set the Higgs mass. The Higgsino mass is $\mu=250{~\text{GeV}}$. []{data-label="fig:BranchingRatio"}](Branching_caseIVA_LR "fig:"){width="0.45\linewidth"} Both sets of branching ratios show a feature at $\mu_{\Sigma} \sim 250\,{~\text{GeV}}$ where the character of the electroweakinos changes. For light right-handed stops (left panel of \[fig:BranchingRatio\]) the branching fraction for $\tilde t_1 \to b\,\chi^+_1$ is $\sim100\%$ over a wide range of $\mu_{\Sigma}$ because the triplet states do not couple directly to the stops and the stop mass in this scenario is nearly the same mass as our benchmark Higgsino (the LSP) mass. In the right panel, where both left and right-handed stops have the same mass, there is more variety in the branching ratios because the stops are heavy enough to undergo both $\tilde t \to t\, \chi^0_i$ and $\tilde t \to b\, \chi^+_i$ decays. From Figs. \[fig:Spectrum\] and \[fig:BranchingRatio\], we can see the phenomenology naturally splits up into four categories, $\mu < \mu_{\Sigma}, \mu > \mu_{\Sigma}$ for either $m_{\tilde t_1} \ll m_{\tilde t_2}$ or $m_{\tilde t_1} \cong m_{\tilde t_2}$, which we discuss in more detail below:\ [*Case $m_{\tilde t_1} \ll m_{\tilde t_2}$:*]{} Here the left-handed stop mass is fixed to $800\,{~\text{GeV}}$ and the right-handed stop mass is variant to satisfy the Higgs mass. For $\mu_{\Sigma} \lesssim 2\,{~\text{TeV}}, m_{\tilde t_1} \sim 300\,{~\text{GeV}}$. - $\mu_{\Sigma} > \mu$: Here the tripletinos play little role, and the low energy states are simply stops and Higgsinos. These scenarios are tightly constrained unless the Higgsino mass $\mu$ is nearly the same as the stop mass and the only two-body decay mode is $\tilde t \to b\,\chi^{+}_1$. As $\mu$ approaches $m_{\tilde t_1}$, the $b$ and subsequent $\chi^+_1$ decay products become soft and conventional stop searches become inefficient. For $m_{\tilde t_1} = 300\,{~\text{GeV}}$, a Higgsino mass of $\mu \gtrsim 180\,{~\text{GeV}}$ is needed [@Aad:2014nra; @CMS:2014yma; @CMS:2014wsa; @Kribs:2013lua] to avoid current LHC bounds. - $\mu_{\Sigma} < \mu$: In this case the tripletinos are lighter than the Higgsinos, so stop decays proceed in two steps; stop decaying to Higgsino, then Higgsino decaying to tripletino. The visibility of this setup depends on the $\mu_{\Sigma} - \mu$ difference. If the two scales are sufficiently separated, the Higgsino decays are energetic and will be picked up by standard stop searches, regardless of how degenerate $\mu$ and $m_{\tilde t_1}$. Therefore, for this scenario to be viable, all three scales $m_{\tilde t_1}, \mu$ and $\mu_{\Sigma}$ must be nearby; for the benchmark value $\mu = 250\,{~\text{GeV}}$, we estimate $\mu_{\Sigma} \gtrsim 200\,{~\text{GeV}}$ is required. [*Case $m_{\tilde t_1} \sim m_{\tilde t_2}$:*]{} In this case, the stop masses are changed together to accommodate the Higgs mass. The stops have a mass of around $450{~\text{GeV}}$ for $\mu_{\Sigma}\lesssim 2\,{~\text{TeV}}$. For larger $\mu_{\Sigma}$, the triplet contribution to $m^2_h$ shrinks and the stops quickly increase in mass. - $\mu_{\Sigma} > \mu$: The stop now has phase space to decay through a top quark and does so around 30$\%$ of the time. Searches for this mode include the leptonic decays and all hadronic decays [@Aad:2014bva; @Aad:2012ywa; @Chatrchyan:2012lia; @Chatrchyan:2014lfa]. For a stop mass of 450 GeV, the limits extend up to an LSP mass of around 220 GeV, thus our model with $\mu=250{~\text{GeV}}$ survives. However, a left-handed stop implies a left-handed sbottom of similar mass. The sbottom searches are very effective for this sort of the spectrum and place constraints on the sbottom up to a mass of $\sim700{~\text{GeV}}$ for an LSP mass of $250{~\text{GeV}}$ [@Aad:2013ija; @CMS:2014nia]. The sbottom (and stop) mass is raised above $700{~\text{GeV}}$ when the triplet effects are decoupled with $\mu_{\Sigma} > 10{~\text{TeV}}$. In the large region of parameter space where the sbottoms are $450{~\text{GeV}}$, in order to be viable, the LSP mass ($\mu$ in this case) must be raised to $\sim300{~\text{GeV}}$. - $\mu_{\Sigma} < \mu$: In this region, all stops and bottoms first decay to Higgsino plus $b/t$, with the Higgsino subsequently decaying to tripletino. The sbottom searches can again be useful, but one potential caveat is that the sbottom decays in our scenario are quite busy, containing extra objects from the Higgsino decay. These final states may be inefficient in sbottom searches such as [@CMS:2014nia] which explicitly veto events with leptons or with more than two jets. The extent to which this scenario can evade the sbottom searches without being collected by another search requires a dedicated analysis, though it is possible that a region window near $\mu_{\Sigma} \sim\mu$ exists undetected by current stop or sbottom searches. Summarizing, the light stops that are a consequence of this triplet extension are safe from current LHC bounds if the spectrum is sufficiently squeezed. For $m_{\tilde t_1} \ll m_{\tilde t_2}$ (light right-handed stop), the benchmark ($\mu = 250{~\text{GeV}}$) scenario is safe provided $\mu_{\Sigma} > 200{~\text{GeV}}$. For degenerate left and right-handed stops, the bounds are more stringent and are driven by sbottom searches. For the benchmark set of parameters to be safe, either the entire stop spectrum must be raised to $\gtrsim 700{~\text{GeV}}(\mu_{\Sigma} > 10{~\text{TeV}})$ or the Higgsinos and tripletinos must be made more degenerate with the stops, $\mu_{\Sigma} \sim \mu \gtrsim 300{~\text{GeV}}$. Continued searches for stop and sbottom squarks will place tighter constraints on the model if no sparticle is found. These stop limits may be alleviated, for example by lowering $\lambda$ or raising $\mu$, though at the expense of increased fine tuning. Discussion and conclusion {#sec:conclusions} ========================= We have examined extensions of the MSSM by two $SU(2)_L$ triplets where only one triplet is permitted to couple to the Higgs doublets. While not generic, this setup is radiatively stable and has the property – first pointed out in the DiracNMSSM [@Lu:2013cta] using singlets – that large, $\gtrsim\,\text{few}\,{~\text{TeV}}$ soft masses for the uncoupled field generate tree level contributions to the Higgs mass without the price of increased fine tuning. Triplet extensions can either have $Y = 0$ or $Y = \pm 1$, we have a studied the Higgs mass contributions, fine tuning, and $\mathcal T$-parameter constraints for both cases. Triplets with nonzero hypercharge are well-suited to this scenario as they must appear in pairs and can only have Dirac-type superpotential masses. For $Y = \pm 1$ scenarios, we find $m_h = 125\,{~\text{GeV}}$ can be achieved with fine tuning as small as one part in ten (according to the same fine tuning measure used in [@Lu:2013cta]). We find that the least tuned regions of parameter space coincide with regions where the $\mathcal T$-parameter constraint – usually a thorn in the side of triplet models – is not an issue. The smallness of the $\mathcal T$-parameter is a consequence the $\tan\beta$ dependence of the triplet-Higgs interaction, aided by the fact that the uncoupled triplet soft mass can be very large ($\gtrsim {~\text{TeV}}$). The least tuned regions also have light stop spectra, either $m_{\tilde t_1} \sim300\,{~\text{GeV}}$ or $m_{\tilde t_1} \sim 450\,{~\text{GeV}}$ depending on whether only one stop is light or both. Such light stops are running out of hiding places at the the LHC. In order to remain undetected, the stops must be fairly degenerate with the LSP, $m_{\tilde t_1} - m_{LSP} \lesssim 100\,{~\text{GeV}}$, though the details of the bounds depend on the hierarchy of the triplet Dirac mass $\mu_{\Sigma}$ and the Higgsino mass $\mu$, as well as on the handedness of the lightest stop; scenarios with light right-handed stops are less constrained than with left-handed. In addition to light stops, the charged and neutral fermionic components of the triplets, the tripletinos, may be light. In the parameter space of interest for the purposes of raising the Higgs mass, these triplets are unconstrained by existing LHC searches. This stealthiness is due to the small splitting among the triplet states and because the tripletinos only couple to Higgs and gauge bosons at tree level. Finally, for certain triplet parameters – for example $\mu_{\Sigma} \sim m_{\chi} \sim 2\,{~\text{TeV}}$ for the parameter set in Fig. \[fig:finetuningTB10ChangeRight\], the $\mathcal T$-parameter contribution from the triplet sector may be within the reach of future precision electroweak studies. Acknowledgments {#acknowledgments .unnumbered} --------------- The work of AD was partially supported by the National Science Foundation under Grant No. PHY-1215979, and the work of AM was partially supported by the National Science Foundation under Grant No. PHY-1417118. Potential for the $Y=0$ triplets {#sec:appY0} ================================ In the following two appendices we list the effective potential, the expressions for the soft masses in terms of the model parameters (via the minimization conditions) and the change in the Higgs mass coming from the triplet sector. It must be emphasized that all of these are tree-level quantities that will receive loop corrections. For the model involving two $Y=0$ triplets, the triplet fields are given by $$\begin{aligned} \Sigma_{1} &= \begin{pmatrix} T^{0}/\sqrt{2} & -T_{2}^{+} \\ T_{1}^{-} & -T^{0}/\sqrt{2} \end{pmatrix} \text{ and}\\ \Sigma_{2} & = \begin{pmatrix} \chi^0/\sqrt{2} & -\chi^+_2 \\ \chi^-_1 & -\chi^0/\sqrt{2} \end{pmatrix}. \end{aligned}$$ The only change in the superpotential from the MSSM is $$W \supset \lambda H_u \cdot \Sigma_2 H_d.$$ Expanding the neutral scalar potential including the soft terms leads to $$\begin{aligned} V _{\text{neutral}} &= m^2_{H_u}|H_u^0 |^2 + m^2_{H_d^0}|H_d^0|^2 + m^2_{\chi}|\chi^0|^2 + m^2_{T}|T^0|^2 \notag \\ & +\left| \frac{\lambda}{\sqrt{2}} H_d^0 T^0 - \mu H_d^0 \right|^2 + \left| \frac{\lambda}{\sqrt{2}} H_u^0 T^0 - \mu H_u^0 \right|^2 \notag \\ &+ \left| \mu_{\Sigma} \chi^0 + \frac{\lambda}{\sqrt{2}} H_d^0 H_u^0 \right|^2 + \left|\mu_{\Sigma} T^0 \right|^2 + \frac{g^2 + g^{\prime 2}}{8} (|H^0_d|^2 - |H^0_u|^2)^{2} \notag \\ &+ \left(\mu_{\Sigma} B_{\Sigma} \chi^0T^0 + B_{\mu} \mu H_d^0 H_u^0 + \frac{A_{\lambda}\lambda}{\sqrt{2}} H_d^0 H_u^0 \chi^0 + \text{h.c.} \right). \label{eqn:NeutralPotentialY0}\end{aligned}$$ The heavy triplet scalars are then integrated out, leading to an effective potential of $$\begin{aligned} V_{\text{eff}} &\supset \left(m^2_{H_u} + |\mu|^2\right)|H_u^{0}|^2 + \left(m^2_{H_d} + |\mu|^2\right)|H_d^{0}|^2 \notag \\ &+ \frac{m_Z^2}{4 v^2} (|H_d^0|^{2}-|H_u^0|^{2})^2 - \left(B_{\mu} \mu H_d^0 H_u^0 + \text{ h.c.}\right) \notag \\ &+ \frac{\left| \lambda H_d^0 H_u^0 \right|^2}{2} \left(1- \frac{\mu_{\Sigma}^2}{\mu_{\Sigma}^2 + m^2_{\chi}} \right) \notag \\ &- \frac{\lambda^2}{2 (\mu_{\Sigma}^2 + m^2_{T})} \left|A_{\lambda} H_d^0 H_u^0 -\mu \left( |H_u^0|^2 + |H_d^0|^2 \right) \right|^2 + (\text{higher order}) \label{eqn:VintoutY0} .\end{aligned}$$ Terms of order $O(D_{\chi}^{-2},D_{T}^{-2},D_{\chi}^{-1}D_{T}^{-1})$ and higher inverse powers have been neglected, where $D_{\chi,T}\equiv (\mu_{\Sigma}^{2}+m_{\chi,T}^{2})$. The conditions needed to achieve EWSB at the minimum of this potential are $$\begin{aligned} m^2_{H_u} &=& -|\mu|^2 + \frac{m^2_Z}{2} \cos(2\beta) + m^2_A \cos^2 \beta -\frac{\lambda^2v^2}{2} \cos^2 \beta {\nonumber \\ }& &+ \frac{v^2 \lambda^2}{2} \frac{- 4 |\mu|^2 - A_{\lambda}(\mu + \mu^*)(\cos(2\beta)-2)\cot\beta - 2 A_{\lambda}^2 \cos^2 \beta}{\mu_{\Sigma}^2 + m^2_{T}} {\nonumber \\ }&& - \mu_{\Sigma}^2 v^2 \lambda^2 \frac{\cos^2 \beta}{\mu_{\Sigma}^2 + m^2_\chi} \text{, and} \label{eqn:mincondHu} \\ m^2_{H_d} &=& -|\mu|^2 -\frac{m^2_Z}{2} \cos(2\beta) + m^2_A \sin^2 \beta - \frac{\lambda^2 v^2}{2} \sin^2\beta {\nonumber \\ }&& + \frac{v^2 \lambda^2}{2} \frac{-4 |\mu|^2 + A_{\lambda} (\mu + \mu^*)(2+\cos(2\beta))\tan\beta - A_{\lambda}^2 \sin^2 \beta} {\mu_{\Sigma}^2 + m^2_{T}} {\nonumber \\ }&& - \mu_{\Sigma}^2 v^2 \lambda^2 \frac{\sin^2\beta}{\mu_{\Sigma}^2 + m^2_{\chi}} \label{eqn:mincondHd}\end{aligned}$$ The corresponding shift in the MSSM physical Higgs mass in the decoupling limit $$\Delta m_{h}^{2}=\frac{v^2 \lambda^2}{2}\sin^2(2\beta) \frac{m^2_{\chi}}{\mu^2_{\Sigma}+m^2_{\chi}} -\frac{v^2\lambda^2}{2} \frac{\left|2 \mu^* -A_{\lambda} \sin(2\beta)\right|^2}{\mu^2_{\Sigma}+m^2_{T}}.$$ Potential for the $Y=\pm1$ triplets {#sec:appY1} =================================== Now we examine the model where the triplets have hypercharge $Y=\pm1$, which can then be expressed as $$\begin{aligned} \Sigma_{1} &= \left(\begin{array}{cc} T^{-}/\sqrt{2} & -T^{0} \\ T^{--} & -T^{-}/\sqrt{2} \end{array}\right) \text{ and} \\ \Sigma_{2}&= \left(\begin{array}{cc} \chi^{+}/\sqrt{2} & -\chi^{++} \\ \chi^{0} & -\chi^{+}/\sqrt{2} \end{array} \right). \end{aligned}$$ The superpotential is modified from the MSSM with $$W \supset \lambda H_u \cdot \Sigma_1 H_u$$ The neutral potential is then given by $$\begin{aligned} V_{\text{neutral}} &= & m^2_{H_u} \left| H_u \right|^2 + m^2_{H_d} \left| H_d \right|^2 + m_{\chi}^2 {|\chi^0|^2} + m_{T}^2 {|T^0|^2} {\nonumber \\ }&&+ \left|2 \lambda H_u^0 T^0 + \mu H_d^0 \right|^2 +\left|\mu H_u^0\right|^2 +\left|\mu_{\Sigma} T^0\right|^2 + \left| \mu_{\Sigma} \chi^0 + \lambda H_u^0 H_u^0 \right|^2 {\nonumber \\ }&&+ \frac{g^2 + g^{\prime~2}}{8} \left( H_d^0 H_d^{0*} - H_u^0 H_u^{0*} + 2 T^0 T^{0*} - 2\chi^0 \chi^{0*} \right)^2 {\nonumber \\ }& &+ \left( - \lambda A_{\lambda} H_u^0 H_u^0 T^0 - \mu B_{\mu} H_d^0 H_u^0 - \mu_{\Sigma} B_{\Sigma} T^0 \chi^0 + \text{h.c.} \right)\end{aligned}$$ The heavy triplets are integrated out, leaving an effective potential of $$\begin{aligned} V_{\text{eff,neut}} &=& \left(m^2_{H_u} + \mu^2 \right) |H_u^0|^2 + \left(m^2_{H_d} + \mu^2 \right) |H_d^0|^2 {\nonumber \\ }&&+ \frac{m_Z^2}{4 v^2} \left( |H_d^0|^2 - |H_u^0|^2 \right)^2 -\left( \mu B_{\mu} H_d^0 H_u^0 +\text{h.c.}\right) {\nonumber \\ }&&+ \lambda^2 |H_u^0H_u^0|^2 \left(1 - \frac{2A_{\lambda}^2}{\mu_{\Sigma}^2 +m^2_{T}} - \frac{2\mu_{\Sigma}^2}{\mu_{\Sigma}^2 +m^2_{\chi}} \right){\nonumber \\ }&& -8 |H_u^0|^2 |H_d^0|^2 \lambda^2 \mu^2 \frac{1}{\mu_{\Sigma}^2+m^2_{T}}{\nonumber \\ }&& + \frac{4 \lambda^2 A_{\lambda}}{\mu^2_{\Sigma} + m^2_{T}} \left(\mu^* H_u^0 H_u^{0*} H_u^{0*} H_d^{0*} + \text{h.c.} \right) + \mathcal{O}(\frac{1}{D_\chi^2},\frac{1}{D_\chi D_T}, \frac{1}{D_T^2}). \label{eqn:VintoutY1}\end{aligned}$$ The minimization conditions are given by $$\begin{aligned} m^2_{H_u} &=&- |\mu|^2 +\frac{1}{2}m^2_Z \cos(2\beta) +m^2_A \cos^2(\beta) - 2 v^2 \lambda^2 \sin^2(\beta) \nonumber \\ &&+ 2 \sin^2(\beta) \frac{\mu^2_{\Sigma} v^2 \lambda^2 }{\mu^2_{\Sigma} + m^2_{\chi}} +2 v^2\lambda^2 \sin^2(\beta) \frac{A_{\lambda}^2 + 2 \mu^2 \cot^2(\beta) + 2 A_{\lambda} \mu \cot(\beta) }{\mu^2_{\Sigma}+m^2_{T}} \label{eqn:mincondHuY1}\\ m^2_{H_d} &=& -|\mu|^2 -\frac{1}{2}m^2_Z \cos(2\beta) - m^2_A \sin^2(\beta) + 4 \frac{|\mu|^2 v^2 \lambda^2 \sin^2(\beta)}{\mu^2_{\Sigma}+m^2_{T}}. \label{eqn:mincondHdY1}\end{aligned}$$ This leads to a shift in the Higgs mass in the decoupling limit of $$\Delta m_{h}^{2}=4 v^2 \lambda^2 \sin^4(\beta)\left( \dfrac{ m_{T_1}^2}{\mu_{\Sigma}^2+m^2_{\chi}} \right) -\dfrac{4 v^2 \lambda^2 \sin^2{(\beta)}}{\mu^2_{\Sigma} +m^2_{T}}\left|2\mu^* \cos {(\beta)} - A_{\lambda} \sin{(\beta)}\right|^2.$$ Finite threshold correction {#sec:AppFTC} =========================== The threshold correction arises when the heavy triplet fields are integrated out. The one loop contribution is given by $$\begin{aligned} \delta m_{H_u^0}^2 &= \frac{8(\lambda^2 + \frac{\lambda^2}{2}) \mu_{\Sigma}^2 }{16\pi^2} \left(\frac{2}{\epsilon} - \gamma + 1 + \log 4\pi - \log(\mu_{\Sigma}^2) \right) \notag \\ & + \left(4 \lambda^2 \mu{\Sigma}^2 + 2 \lambda^2 \mu_{\Sigma}^2 \right)\frac{1}{16\pi^2} \left( -\frac{2}{\epsilon} +\gamma -1 -\log4\pi + \log(m_{\chi}^2 + \mu_{\Sigma}^2) \right) \notag \\ &+ \lambda^2 (4 + 2) \left(\mu_{\Sigma}^2+m_{T}^2 \right) \frac{1}{16\pi^2} \left(-\frac{2}{\epsilon} +\gamma -1 -\log4\pi + \log(m_{T}^2 + \mu_{\Sigma}^2) \right) \notag \\ =& \frac{12 \lambda^2 \mu_{\Sigma}^2}{16 \pi^2} \left(\frac{1}{2} \log (m^2_{\chi} + \mu_{\Sigma}^2) +\frac{1}{2} \log (m^2_{T} + \mu_{\Sigma}^2) - \log(\mu_{\Sigma}^2) \right) \notag \\ &+ \frac{6 \lambda^2 m^2_{T}}{16\pi^2} \left(-\frac{2}{\epsilon} +\gamma -1 -\log4\pi + \log(m_{\chi}^2 + \mu_{\Sigma}^2) \right) \notag \\ =& \frac{6 \lambda^2 \mu_{\Sigma}^2}{16 \pi^2} \left(\log \frac{(m^2_{\chi} + \mu_{\Sigma}^2)}{\mu_{\Sigma}^2} + \log \frac{(m^2_{T} + \mu_{\Sigma}^2)}{\mu_{\Sigma}^2} \right) + \frac{6 \lambda^2 m^2_{T}}{16\pi^2} \left(-\frac{2}{\epsilon} +\gamma -1 -\log4\pi + \log(m_{T}^2 + \mu_{\Sigma}^2) \right).\end{aligned}$$ We are only interested in the finite piece. Neutralino and chargino mixing in $Y=\pm1$ {#sec:AppMixing} ========================================== The $Y=\pm1$ mixing matrix for the neutralinos in the basis $\psi^0=\left( \widetilde{B},\widetilde{W}^{0},\widetilde{H_{d}^{0}},\widetilde{H_{u}^{0}},\widetilde{T}^{0},\widetilde{\chi}^{0} \right)$ is given by $$\begin{aligned} {\mathcal{L}}_{\text{Neutralino Mass}} &=& -\frac{1}{2} (\psi^0)^T \mathbf{M}_{\tilde{N}} \psi^0 + \text{c.c.} \\ \mathbf{M}_{\tilde{N}} &=& \begin{pmatrix} M_{1} & 0 & -c_{\beta}s_{W}m_{Z} & s_{\beta}s_{W}m_{Z} & -\sqrt{2}g'v_{T} & \sqrt{2}g'v_{\chi} \\ 0 & M_{2} & c_{\beta}c_{W}m_{Z} & -s_{\beta}c_{W}m_{Z} & -\sqrt{2}gv_{T} & \sqrt{2}gv_{\chi} \\ -c_{\beta}s_{W}m_{Z} & c_{\beta}c_{W}m_{Z} & 0 & -\mu & 0 & 0 \\ s_{\beta}s_{W}m_{Z} & -s_{\beta}c_{W}m_{Z} & -\mu & -2v_{T}\lambda & -2v\lambda s_{\beta} & 0 \\ -\sqrt{2}g'v_{T} & -\sqrt{2}gv_{T} & 0 & -2v\lambda s_{\beta} & 0 & -\mu_{\Sigma} \\ \sqrt{2}g'v_{\chi} & \sqrt{2}gv_{\chi} & 0 & 0 & -\mu_{\Sigma} & 0 \end{pmatrix}, \nonumber \label{neutralinoMixY1}\end{aligned}$$ where $c_{\beta}$, $s_{\beta}$, $c_W$, and $s_W$ represent the cosine or sine of beta or $\theta_W$. The triplets add one chargino. Using the basis $\psi^{\pm} = \left( \widetilde{W}^{+},\widetilde{H}_{u}^{+},\widetilde{\chi}^{+}, \widetilde{W}^-, \widetilde{H}_d^-, \widetilde{T}^- \right)$, the chargino mass matrix is $${\mathcal{L}}_{\text{Chargino Mass}}=-\frac{1}{2} (\psi^{\pm})^T \mathbf{M}_{\tilde{C}} \psi^{\pm}, {\nonumber \\ }$$ where $$\mathbf{M}_{\tilde{C}}= \begin{pmatrix} \mathbf{0} & \mathbf{X}^T \\ \mathbf{X} & \mathbf{0} \end{pmatrix} \nonumber, \\$$ and $$\mathbf{X}= \begin{pmatrix} M_{2} & gvs_{\beta} & -\sqrt{2}gv_{\chi} \\ gvc_{\beta} & \mu & 0 \\ -\sqrt{2}gv_{T} & \sqrt{2}\lambda vs_{\beta} & \mu_{\Sigma} \end{pmatrix}.\label{eqn:charginoMixY1}$$ Finally, the doubly-charged fermion mass matrix is $${\mathcal{L}}_{\text{Doubly Charged}} = -\frac{1}{2}\begin{pmatrix} \widetilde{\chi}^{++} & \widetilde{T}^{--} \end{pmatrix} \begin{pmatrix} 0 & -\mu_{\Sigma} \\ -\mu_{\Sigma} & 0 \end{pmatrix} \begin{pmatrix} \widetilde{\chi}^{++} \\ \widetilde{T}^{--} \end{pmatrix}.$$ [^1]: E-mail: [email protected] [^2]: E-mail: [email protected] [^3]: E-mail: [email protected] [^4]: E-mail: [email protected] [^5]: Triplet extension which preserve custodial symmetry, such as the Supersymmetric Custodial Triplet Model, allow for large triplet vevs (and light scalars) without tension from electroweak precision observables [@Cort:2013foa; @Garcia-Pepin:2014yfa; @Delgado:2015aha]. [^6]: One could also use a spurion analysis of an extra broken symmetry which would suppress the unwanted couplings [@Lu:2013cta]. [^7]: These are the only possibilities that simultaneously permit a Dirac mass term and supply extra neutral scalars to raise $m_{h}^{2}$. [^8]: If the $U$ parameter is fixed to $U=0$, the best fit is $\mathcal{T}=0.10\pm0.07$.

--- abstract: 'Work by the Zürich school of causal (Epstein–Glaser) renormalization has shown that renormalizability in the presence of massless or massive gauge fields (as primary entities) explains gauge invariance and, in some instances, the presence of a Higgs-like particle, without need for a Brout–Englert–Higgs–Guralnik–Hagen–Kibble (BEHGHK) mechanism. We review that work, in a pedagogical vein, with a pointer to go beyond.' author: - | José M. Gracia-Bondía\ Departamento de Física Teórica,\ Universidad de Zaragoza, Zaragoza 50009, Spain\ and\ Departamento de Física,\ Universidad de Costa Rica, San Pedro 2060, Costa Rica title: On the causal gauge principle --- Introduction ============ By now spontaneous symmetry breaking (SSB) of local symmetry is a well-established paradigm of high-energy physics. At the end of the 60s and beginning of the 70s, it allowed the incorporation of (electro)weak interactions into the framework of renormalizable field theory. In connection with the contemporaneous rise of the Standard Model (SM), it enjoys immense historical success. However, allusion to unsatisfactory or mysterious aspects of the Higgs sector of the SM does pop up in the literature —see for instance [@AH04 Sect. 22.10]. The Higgs self-coupling terms are completely ad-hoc, unrelated to other aspects of the theory, and do not seem to constitute a gauge interaction. Moreover they raise the hierarchy problem [@ModernosPragmaticos Ch. 11]. The most frequent interpretation of the BEHGHK mechanism clashes with cosmology [@SolDedo]. Debate on the proper interpretation of the mechanism (whether the symmetry is “broken” or just “hidden”, whether the Higgs field truly has a non-zero vacuum expectation value (VEV) or not [@LosChinos], and so on) seems endless. This breds some skepticism, even among earlier and doughty practitioners. At the end of his Nobel lecture [@TiniDixit], Veltman chose to declare: *“While theoretically the use of spontaneous symmetry breakdown leads to renormalizable Lagrangians, the question of whether this is really what happens in Nature is entirely open”.* Indeed, since the *deus ex machina* fields involved in broken or hidden symmetry are unobservable, the status question for the BEHGHK contraption cannot be resolved by the likely sighting of the Higgs particle in the LHC.[^1] The subject has also been obscured all along by theoretical prejudice. In the SM the Higgs field carries the load of giving masses to *all* matter and force fields. For instance, it is said that mass terms for the vector bosons are incompatible with gauge invariance. It ain’t so: such mass terms fit in gauge theory by use of Stückelberg fields [@Altabonazo; @Felicitas]. Skepticism would be idle, nevertheless, in the absence of alternative theoretical frameworks. Assuming an agnostic stance, we pose the question: is it possible to formulate the main results of flavourdynamics, and to frame suggestions of new physics, without recourse to unobservable processes? In tune with the phenomenological SM Lagrangian [@SlimKilian], this amounts to regard massive vector bosons (MVB) as fundamental entities. So let us stop pretending we know the origins of mass. Higgs-like scalar fields will still come in handy for either renormalizability or unitarity; however, their gauge variations need not be the conventional ones. Fermions can be assigned Dirac masses, and couplings with the scalar field proportional to those; this contradicts in no way the chiral nature of their interactions in the SM. An approach with the mentioned traits is already found in the literature in the work by Scharf, Dütsch and others, under the label of the “quantum gauge invariance” principle. A few references to it are [@PGI-EW-I; @PGI-EW-II; @CabezondelaSal; @PepinsFriend] and mainly the book [@Zurichneverdies]. The “quantum Noether principle” of [@HurthS1; @HurthS2] coincides essentially with it. Both are based on the rigorous causal scheme for renormalization [@PastMasters] by Epstein and Glaser (EG). Henceforth we refer to the approach as causal gauge invariance (CGI). The usual plan of the article is found at the end of the next section, when the stakes hopefully have been made clearer. Overview of the CGI method ========================== The spirit of CGI is very much that of the [@LuisDixit]. Let $s$ denote the nilpotent BRS operation. To realize gauge symmetry, one should incorporate BRS symmetry ab initio in a “quantum” Lagrangian ${\mathcal{L}}$, such that (very roughly speaking) $s{\mathcal{L}}\sim0$, and proceed to build from there. We do this for MVBs. The starting point for the analysis is the Bogoliubov–Epstein–Glaser functional scattering matrix on Fock space, in the form of a power series: $${\mathbb{S}}(g) = 1 + \sum_{n=1}^\infty\frac{i^n}{n!}\,\int dx_1\ldots dx_n\, T_n(x_1, \ldots, x_n)\,g(x_1) \cdots g(x_n). \label{eq:dance-with-her}$$ The coupling constants of the model are replaced by test functions —we wrote just one of them for simplicity. The theory is then constructed basically by using causality and Poincaré invariance to recursively determine the form of the time-ordered products $T_n$ from the $T_m$ with $m<n$; in this sense the procedure is inverse to the “cutting rules”. Only those fields should appear in $T_n$ that already are present in $T_1$. The procedure yields a finite perturbation theory without regularization; ultraviolet divergences are avoided by proper definition of the $n$-point functions as distributions. (Ultimately one would be interested in the adiabatic limit $g(x)\uparrow g$. This is delicate, however, due to infrared problems. We look at the theory before that limit is taken.) With the proviso that two forms of $T_n$ are equivalent if they differ by $s$-coboundaries, CGI is formulated by the fact that $sT_n$ must be a divergence. Roughly speaking, we must have $$\begin{aligned} sT_n(x_1,\ldots,x_n) &= i\sum_{l=1}^{l=n} \operatorname{T}\bigl[T_1(x_1),\ldots,{\partial}_l{\cdot}Q(x_l),\ldots,T_1(x_n)\bigr], {\nonumber}\\ &=: i\sum_{l=1}^{l=n}{\partial}_l{\cdot}Q_n(x_1,\ldots,x_n). \label{eq:gold-mine}\end{aligned}$$ for vectors $Q_n$, called $Q$-vertices, with ${\partial}_l$ denoting the partial divergences with respect to the $x_l$ coordinates and $\operatorname{T}$ a time-ordering operator. In this way renormalization and gauge invariance are linked in the EG scheme. (We said “roughly” because  suggests that $\operatorname{T}$ and spacetime derivatives commute, which is not generally the case for on-shell fields.) Note that $T_1$ only contains the first-order part of the Lagrangian. Nevertheless, already the first order condition $$sT_1 = i{\partial}{\cdot}Q_1$$ constrains significantly the form of the Lagrangian. Later on, we show leisurely how the CGI method works for tree graphs belonging to $T_2$. This is almost all what is required for the purposes of this paper: for ordinary gauge theories, the treatment of $T_3$ is pretty simple, and higher orders not needed at all. Keep in mind that one works here with *free* fields. Interacting fields can be arrived at in the Epstein–Glaser procedure, somewhat a posteriori, using their definition by Bogoliubov as logarithmic functional derivatives of ${\mathbb{S}}(g)$ with respect to appropriate sources. Their gauge variations resemble more those of standard treatments; but we do not use them. Thus $s$ “sees” only the (massive or massless) gauge fields, and the attending (anti-)ghost and Stückelberg fields. This is why everything flows from the quantum gauge structure of the boson sector. Coupling to fermions, which ought not be organized in multiplets a priori, comes almost like an afterthought. As it turns out, the procedure is quite restrictive, and in particular only a few models for MVB theory pass muster. These exhibit very definite mass and interaction patterns, in particular quartic self-interaction for the scalar particles. We next compile the results, according to [@Zurichneverdies]. Consider a model with $t$ intermediate vector bosons $A_a$ in all, of which any may be in principle massive or massless. Let us say there are $r$ massive ones with masses $m_a, 1\le a\le r$ and $s$ massless ones, and $t =r+s$. We assume there is *one* (at most) physical scalar particle $H$ of mass $m_H$: *entia non sunt multiplicanda praeter necessitatem*. The BRS extension of the Wigner representation theory for MVBs requires Stückelberg fields $B_a$ [@CabezondelaSal], beyond the fermionic ghosts $u_a,{{\tilde u}}_a$; in case $A_a$ is massless, we of course let $B_a$ drop out. Adopting the Feynman gauge, the gauge variations are as follows: $$\begin{aligned} sA_a^\mu(x) &= i{\partial}^\mu u_a(x); \nonumber \\ sB_a(x) &= im_au_a(x); \nonumber \\ su_a(x) &= 0; \nonumber \\ s{{\tilde u}}_a(x) &= -i\big({\partial}{\cdot}A_a(x) + m_aB_a(x)\big). \nonumber \\ sH(x) &= 0. \label{eq:begging-for-trouble}\end{aligned}$$ This operator is nilpotent on-shell. The total bosonic interaction Lagrangian, in a notation close to that of [@Zurichneverdies], is of the form $${\mathcal{L}}_{\mathrm{int}} = gT_1 + \frac{g^2\,T_2}2, \label{eq:romeros-somos}$$ where $g$ is an overall dimensionless coupling constant; $$T_1 = f_{abc}\big(T^1_{1abc} + T^2_{1abc} + T^3_{1abc} + T^4_{1abc}\big) + C\big(T^5_1 + T^6_1 + T^7_1 + T^8_1 + T^9_1\big)$$ includes the cubic couplings, and $$T_2 = T^1_2 + T^2_2 + T^3_2 + T^4_2 + T^5_2 + T^6_2 + T^7_2$$ includes the quartic ones. The list of cubic couplings not involving $H$ is given by: $$\begin{aligned} T^1_{1abc} &= \bigl[A_a{\cdot}(A_b{\cdot}{\partial})A_c - u_b(A_a{\cdot}{\partial}{{\tilde u}}_c)\bigr]; {\nonumber}\\ T^2_{1abc} &= \frac{m_b^2 + m_c^2 - m_a^2}{4m_bm_c} \bigl[B_b(A_a{\cdot}{\partial}B_c) - B_c(A_a{\cdot}{\partial}B_b)\bigr]; {\nonumber}\\ T^3_{1abc} &= \frac{m_b^2 - m_a^2}{2m_c}(A_a{\cdot}A_b)B_c; {\nonumber}\\ T^4_{1abc} &= \frac{m_a^2 + m_c^2 - m_b^2}{2m_c}{{\tilde u}}_au_bB_c; \label{eq:abyssus-abyssum-invocat}\end{aligned}$$ The list of cubic couplings of the Higgs-like particle is: $$\begin{aligned} T^5_1 &= m_a[B_a(A_a{\cdot}{\partial}H) - H(A_a{\cdot}{\partial}B_a)]; \nonumber \\ T^6_1 &= m_a^2(A_a{\cdot}A_a) H; \nonumber \\ T^7_1 &= -m_a^2{{\tilde u}}_au_a H; \nonumber \\ T^8_1 &= -{\tfrac{1}{2}}m_H^2 B_a^2 H; \nonumber \\ T^9_1 &= -{\tfrac{1}{2}}m_H^2 H^3. \label{eq:dente-superbo}\end{aligned}$$ Remarks: in  and  we sum over repeated indices; the $f_{abc}$ are completely skewsymmetric in their three indices, and fulfil the Jacobi identity; $T^1_{1.}$ yields the cubic part in the classical Yang–Mills Lagrangian; $C$ is a constant independent of $a$. The dimension of the Lagrangian must be $M^4$ in natural units, and the boson field dimension in our formulation is 1 for *both* spins: the dimension of $C$ is $M^{-1}$. Note the diagonality of the couplings of the Higgs-like particle. Crossed terms like $(A_a{\cdot}A_b)H$ for $a\ne b$, and others like $B_aB_bB_c,B_aH^2\ldots$, that could be envisaged, are held to vanish by CGI. The list of quartic couplings: $$\begin{aligned} T^1_2 &= -{\tfrac{1}{2}}f_{abc}f_{ade}(A_b{\cdot}A_d)(A_c{\cdot}A_e); {\nonumber}\\ T^2_2 &= \bigg[\frac{(m_d^2 + m_e^2 - m_a^2)(m_c^2 + m_e^2 - m_b^2)} {8m_dm_cm_e^2}f_{ade}f_{bce} + c \leftrightarrow d {\nonumber}\\ &+ {\tfrac{1}{2}}C^2m_am_b{\delta}_{ad}{\delta}_{bc} + c \leftrightarrow d \bigg] {\times}(A_a{\cdot}A_b)B_cB_d; {\nonumber}\\ T^3_2 &= -{\tfrac{1}{4}}C^2 m_H^2 B^2_aB^2_b {\quad\hbox{irrespective of $a,b\le r$;}\quad} {\nonumber}\\ T^4_2 &= Cf_{abc}\frac{m_b^2 - m_a^2}{m_c}(A_a{\cdot}A_b)B_c H; {\nonumber}\\ T^5_2 &= C^2m_a^2(A_a{\cdot}A_a)H^2; {\nonumber}\\ T^6_2 &= -{\tfrac{1}{2}}C^2 m_H^2 B^2_a H^2 {\quad\hbox{irrespective of $a\le r$;}\quad} {\nonumber}\\ T^7_2 &= -{\tfrac{1}{4}}C^2 m_H^2 H^4. \label{eq:hoc-erat-in-votis}\end{aligned}$$ Every coefficient of the interaction Lagrangian is in principle determined in terms of the $f_{abc}$ and the pattern of masses. We are not through, because CGI implies *constraints*, in general non-linear and extremely restrictive, on *allowed patterns* of masses for the gauge fields. But we may anticipate a few more comments. The first term $T^1_2$ in  just yields the quartic part in the classical Yang–Mills Lagrangian, as expected. In case all the $A_a$ are massless, there is no need to add physical or unphysical scalar fields for renormalizability, and only $T^1_1$ and $T^1_2$ survive in the theory; they of course coincide respectively with the first and second order part of the usual Yang–Mills Lagrangian. In particular, CGI gives rise to gluodynamics. (It must be said, though, that the physical equivalence of couplings differing in a divergence is less compelling in this case, since there is no asymptotic limit for the Bogoliubov–Epstein–Glaser ${\mathbb{S}}(g)$-matrix; CGI offers no tools to deal with this infrared problem.) Remarkably, with independence of the masses, CGI unambiguously leads to generalized Yang–Mills theories on reductive Lie algebras; apparently this was realized first by Stora [@ESItalk]. The plan of the rest of the article is as follows. Notice that the case $r = 1,s = 0$ leads to an abelian model in which all the terms with the Higgs-like field $H$ survive. We use this example in Section 3 to illustrate in some detail —missing in [@Zurichneverdies]— how the second-order condition determines the couplings. Section 4 deals with *three* gauge fields —there are no models with two gauge fields to speak of, since ${\mathfrak{u}}(1)\oplus{\mathfrak{u}}(1)$ is the only two-dimensional reductive Lie algebra. For that we need to invoke the mentioned mass relations (reference [@Zurichneverdies] unfortunately contains misprints in this respect). Section 5 elaborates on the reconstruction of the SM in CGI, looking at the fermion sector as well. The paper ends with a discussion. The abelian model ================= Consider a theory with a neutral gauge field $A$ of mass $m$ and a *physical* neutral scalar field $H$ of mass $m_H$, and basic coupling $AAH$. (0,0) node [$\bullet$]{} node\[above right\] [$g$]{} – (2,0) node \[above left\] [$H$]{}; (0,0) – (120:2) node\[below left\] [$A$]{}; (0,0) – (240:2) node\[above left\] [$A$]{}; Since massive quantum electrodynamics is known to be renormalizable without an extra scalar field, this is perhaps not very interesting; but our aim here is merely showing the workings of the causal gauge principle. The first-order analysis ------------------------ For $T_1$, take the most general Ansatz containing cubic terms in the fields and leading to a renormalizable theory. With the benefit of hindsight, we write down on the first line the terms destined to survive: $$\begin{aligned} T_1/m &= (A{\cdot}A)H + b{{\tilde u}}uH + c\big(H(A{\cdot}{\partial}B) - B(A{\cdot}{\partial}H)\big) + dB^2H + eH^3 {\nonumber}\\ &+ a(A{\cdot}A)B + b_2{{\tilde u}}uB + b_3u(A{\cdot}{\partial}{{\tilde u}}) + d_1B^3 + d_3BH^2. \label{eq:few-are-the-chosen}\end{aligned}$$ The factor $m$ is natural according to our previous discussion on dimensions. The symmetric combination $HA{\cdot}{\partial}B+BA{\cdot}{\partial}H$ has been excluded for the following reason: $$A{\cdot}(B{\partial}H + H{\partial}B) = {\partial}.(BHA) - ({\partial}{\cdot}A)BH,$$ and in view of , the $({\partial}{\cdot}A)BH$ term is $s$-exact apart from terms of already present in . Concretely, $$s({{\tilde u}}BH) = -({\partial}{\cdot}A)BH - mB^2H - m{{\tilde u}}uH.$$ We calculate next $sT_1/m$ in  and obtain for the first group of terms: $$\begin{aligned} &2{\partial}{\cdot}(uHA) - 2u({\partial}{\cdot}A)H - 2uA{\cdot}{\partial}H - bu({\partial}{\cdot}A)H {\nonumber}\\ &- bmu BH + c{\partial}{\cdot}u(H{\partial}B - B{\partial}H) {\nonumber}\\ & + cm[HA{\cdot}({\partial}u) - u A{\cdot}{\partial}H] + 2dmuBH. \label{eq:troppo-lavoro}\end{aligned}$$ We have used $s(uC)=-usC$ for any $C$. In detail: $$-is(A{\cdot}AH) = ({\partial}u{\cdot}A)H = 2\big[{\partial}{\cdot}(uHA) - u({\partial}{\cdot}A)H - u A{\cdot}{\partial}{\cdot}H].$$ Next $$-is({{\tilde u}}uH) = -u({\partial}{\cdot}A)H - muBH.$$ Next $$-is(A{\cdot}(H{\partial}B - B{\partial}H)) = {\partial}u{\cdot}(H{\partial}B - B{\partial}H) + m[HA{\cdot}({\partial}u) - uA{\cdot}{\partial}H].$$ Finally $-is(B^2H)=2muBH$. Similarly, for the second group of terms we obtain: $$\begin{aligned} &2a{\partial}{\cdot}(uBA) - 2au({\partial}{\cdot}A)B - 2auA{\cdot}{\partial}B + amuA{\cdot}A {\nonumber}\\ & - b_2u({\partial}.A)B - b_2mu B^2 + b_3\big({\partial}u{\cdot}u\, {\partial}{{\tilde u}}+ uA{\cdot}{\partial}({\partial}{\cdot}A + mB)\big) {\nonumber}\\ &+ 3d_1muB^2 + d_3muH^2.\end{aligned}$$ All terms of that group are excluded because their contributions to $sT_1$ are not pure divergences. For instance, the first one corresponds to the term in $uA{\cdot}A$, that can be canceled only by setting $a=0$. On the other hand, the second term in the second line in  can be recast as $${\partial}{\cdot}\bigl(u(H{\partial}B - B{\partial}H)\bigr) + (m^2 - m_H^2)uBH.$$ For the following terms we have $$A{\cdot}({\partial}u)H - uA{\cdot}{\partial}H = {\partial}{\cdot}(uHA) - u({\partial}{\cdot}A)H - 2u A{\cdot}{\partial}H.$$ In all, $$\begin{aligned} &-isT_1/m = {\partial}{\cdot}(C + D) - (2 + cm + b)u({\partial}{\cdot}A)H \\ &- (2 + 2cm)u(A{\cdot}{\partial}H) + \big(2dm - bm + c(m^2 - m_H^2)\big)uBH;\end{aligned}$$ with the vectors $C,D$ given by $C:=(2+cm)uHA;\,D:=cu(H{\partial}B-B{\partial}H)$. The terms that are not a divergence must cancel. This at once leads to: $$c = -\frac{1}{m}; \; b = -1; \; d = -\frac{m_H^2}{2m^2}; {\quad\hbox{thus}\quad} C = uHA; \quad D=\frac{-u}{m}(H{\partial}B - B{\partial}H).$$ In summary, we have obtained the cubic couplings in the Lagrangian: (0,0) node [$\bullet$]{} node\[above right\] [$gm$]{} – (2,0) node \[above left\] [$H$]{}; (0,0) – (120:2) node\[below left\] [$A$]{}; (0,0) – (240:2) node\[above left\] [$A$]{}; (0,0) node [$\bullet$]{} node\[above right\] [$-gm$]{} – (2,0) node \[above left\] [$H$]{}; (0,0) – (120:2) node\[below left\] [$u$]{}; (0,0) – (240:2) node\[above left\] [$\tilde u$]{}; (0,0) node [$\bullet$]{} node\[above right\] [$\frac{-gm_H^2}{2m}$]{} – (2,0) node \[above left\] [$H$]{}; (0,0) – (120:2) node\[below left\] [$B$]{}; (0,0) – (240:2) node\[above left\] [$B$]{}; (0,0) node [$\bullet$]{} node\[above right\] [$g$]{} – (2,0) node \[above left\] [${\partial}H$]{}; (0,0) – (120:2) node\[below left\] [$A$]{}; (0,0) – (240:2) node\[above left\] [$B$]{}; (0,0) node [$\bullet$]{} node\[above right\] [$-g$]{} – (2,0) node \[above left\] [${\partial}B$]{}; (0,0) – (120:2) node\[below left\] [$A$]{}; (0,0) – (240:2) node\[above left\] [$H$]{}; To this we should add the $H^3$ coupling, whose coefficient is still indeterminate: (0,0) node [$\bullet$]{} node\[above right\] – (2,0) node \[above left\] [$H$]{}; (0,0) – (120:2) node\[below left\] [$H$]{}; (0,0) – (240:2) node\[above left\] [$H$]{}; Moreover: $$sT_1 = i{\partial}{\cdot}Q_1 {\quad\hbox{with}\quad} Q_1 = muHA - u(H{\partial}B - B{\partial}H). \label{eq:la-madre-del-cordero}$$ The second-order analysis ------------------------- The next step is less trivial. Equation  certainly makes sense outside the diagonals, for then the $\operatorname{T}$ product is calculated like an ordinary product. But the extension to the diagonals, which is simply $x_1=x_2$ for $n=2$, can produce local correction terms. At this order, the advanced and retarded products are given by: $$\begin{aligned} A_2(x_1,x_2) &= T_2(x_1,x_2) - T_1(x_1)T_1(x_2); \nonumber \\ R_2(x_1,x_2) &= T_2(x_1,x_2) - T_1(x_2)T_1(x_1); \label{eq:sleeping-dog}\end{aligned}$$ Here $T_2(x_1,x_2)$ is still unknown, but it is clear that $A_2$ will have support on the past light cone of $x_2$, and $R_2$ on its future light cone; hence the nomenclature. Consider then $D_2(x,y):= \big(R_2-A_2\big)(x,y)=[T_1(x),T_1(y)]$, whose support is within the light cone (we say $D_2$ is causal). We have thus $$\begin{aligned} sD_2(x,y) &= [sT_1(x), T_1(y)] + [T_1(x), sT_1(y)] \nonumber \\ &= i{\partial}_x[Q_1(x), T_1(y)] + i{\partial}_y[T_1(x), Q_1(y)]; \label{eq:salta-la-liebre}\end{aligned}$$ so that $D_2$ moreover *is* gauge-invariant. The crucial step in EG renormalization is the *splitting* of $D_2$ into the retarded part $R_2$ and the advanced part $A_2$; once this is done, $T_2$ is found at once from . The issue is how to preserve gauge invariance in this distribution splitting. For this, we split $D_2$ and the commutators —without the derivatives— in the previous equation; then gauge invariance: $$sR_2(x,y) = i{\partial}_x R_{2/1}(x,y) + i{\partial}_y R_{2/2}(x,y)$$ can only be (and is) violated for $x=y$, that is, by local terms in ${\delta}(x-y)$. However, if in turn local renormalization terms $N_2,N_{2/1},N_{2/2}$ can be found in such a way that $$s(R_2(x,y) + N_2(x,y)) = i{\partial}_x(R_{2/1} + N_{2/1}) + i{\partial}_y(R_{2/2} + N_{2/2}),$$ with an obvious notation, then CGI to second order holds. To the purpose we consider only tree diagrams. In view of , we systematically proceed to study the divergences coming from cross-terms between  and $$T_1 = m\bigl[(A{\cdot}A)H + u{{\tilde u}}H - \frac{1}{m}A{\cdot}\bigl(H{\partial}B - B{\partial}H\bigr) - \frac{m_H^2}{2m^2}\,B^2H + eH^3\bigr]. \label{eq:donya-toda}$$ Factors containing derivatives give rise to normalization contributions after distribution splitting. The most difficult part of the coming calculation asks for divergences of terms with commutators $[{\partial}^\mu B(x), {\partial}^\nu B(y)]$ and $[{\partial}^\mu H(x), {\partial}^\nu H(y)]$. Following [@Michael], we look at Section 4 in [@PGI-EW-I] in order to prepare the computation. There, for general functions $F,E$ we find the formulas: $$\begin{aligned} &{\partial}^x_\mu[F(x)E(y){\delta}(x - y)] + {\partial}^y_\mu[F(y)E(x){\delta}(x - y)] \nonumber \\ &= {\partial}_\mu F(x)\,E(x){\delta}(x - y) + F(x)\,{\partial}_\mu E(x){\delta}(x - y) \label{eq:villanous} \\ {\quad\hbox{and}\quad} & F(x)E(y){\partial}^x_\mu{\delta}(x - y) + F(y)E(x){\partial}^y_\mu{\delta}(x - y) \nonumber \\ &= F(x)\,{\partial}_\mu E(x){\delta}(x - y) - {\partial}_\mu F(x)\,E(x){\delta}(x - y). \label{eq:amorphous}\end{aligned}$$ We may prove both from the following observation: since $$F(x)E(y){\delta}(x - y) = F(x)E(x){\delta}(x - y),$$ it must be that $${\partial}^x_\mu\big(F(x)E(y){\delta}(x - y)\big) = {\partial}^x_\mu\big(F(x)E(x){\delta}(x - y)\big);$$ which forces $$E(y){\partial}^x_\mu{\delta}(x - y) = E(x){\partial}^x_\mu{\delta}(x - y) + {\partial}_\mu E(x){\delta}(x - y). \label{eq:penguins}$$ Now, $$\begin{aligned} &{\partial}^x_\mu[F(x)E(y){\delta}(x - y)] + {\partial}^y_\mu[F(y)E(x){\delta}(x - y)] \\ &= {\partial}_\mu F(x)\,E(x){\delta}(x - y) + F(x)E(y){\partial}^x_\mu{\delta}(x - y) \\ &+ {\partial}_\mu F(x)\,E(x){\delta}(x - y) - F(y)E(x){\partial}^x_\mu{\delta}(x - y) \\ &= {\partial}_\mu F(x)\,E(x){\delta}(x - y) + F(x)E(y){\partial}^x_\mu{\delta}(x - y) \\ &- F(x)E(x){\partial}^x_\mu{\delta}(x - y) = {\partial}_\mu F(x)\,E(x){\delta}(x - y) + F(x)\,{\partial}_\mu E(x){\delta}(x - y);\end{aligned}$$ where we have used  twice. Analogously, $$\begin{aligned} &F(x)E(y){\partial}^x_\mu{\delta}(x - y) + F(y)E(x){\partial}^y_\mu{\delta}(x - y) = F(x)E(x){\partial}^x_\mu{\delta}(x - y) \\ &+ F(x){\partial}_\mu E(x){\delta}(x - y) - F(y)E(x){\partial}^x_\mu{\delta}(x - y) \\ &= F(x)\,{\partial}_\mu E(x){\delta}(x - y) - {\partial}_\mu F(x)\,E(x){\delta}(x - y),\end{aligned}$$ using  twice again. We finally start the advertised computation. Coming from respectively the second term of $Q_1(x)$ in  and third of $T_1(y)$ in , now we find for $i[Q_1(x), T_1(y)]$: $$\begin{aligned} &iu(x)H(x)[{\partial}^\mu B(x), {\partial}^\nu B(y)]A_\nu(y)H(y) \\ &= u(x)H(x)A_\nu(y)H(y){\partial}^\mu_x{\partial}^\nu_y D(x - y).\end{aligned}$$ The identity $[B(x), B(y)]=-iD(x-y)$ for scalar fields has been employed. Next we need to tackle the divergence of the splitting of ${\partial}^\mu_x{\partial}^{\alpha}_yD$. Splitting of the Jordan–Pauli propagator $D$ gives rise to the retarded propagator $D^{\rm ret}$. Now, each derivation increases by one the singular order of a distribution. Thus, although ${\partial}_x^\mu{\partial}_y^\nu D^{\rm ret}$ is a well-defined distribution, its singular order is $-2+2=0$, therefore allowing a normalization term in the split distribution: $${\partial}_x^\mu{\partial}_y^\nu D^{\rm ret}(x-y) \to {\partial}_x^\mu{\partial}_y^\nu D^{\rm ret}(x - y) + C_Bg^{\mu\nu}{\delta}(x - y).$$ After applying ${\partial}_\mu$, simply from $${\partial}^x_\mu{\partial}_x^\mu D^{\rm ret}(x-y) = - m^2D^{\rm ret}(x - y) + {\delta}(x - y),$$ the total singular part is of the form $$C_B{\partial}^\nu_x[F(x)E(y)\,{\delta}(x - y)] + F(x)E(y){\partial}^\nu_y{\delta}(x - y), {\quad\hbox{with}\quad} F = uH; \; E = HA_\nu.$$ Adding the term with $x$ and $y$ interchanged, and using the identities  and , it comes finally the short rule for this kind of singular term: $$F(x)E(y){\partial}^\mu_x{\partial}^\nu_yD(x - y) \to [(C_B + 1)({\partial}^\nu F)E + (C_B - 1)F\,{\partial}^\nu E]{\delta}(x - y).$$ Therefore we obtain in the end $$\begin{aligned} &\quad (C_B + 1)\big[H^2(A{\cdot}{\partial}u) + uH(A{\cdot}{\partial}H)\big]{\delta}(x - y) \label{eq:bitter-end} \\ &+ (C_B - 1)\big[uH^2({\partial}{\cdot}A) + uH(A{\cdot}{\partial}H)\big]{\delta}(x - y). \nonumber\end{aligned}$$ By the same token, coming now from respectively the third and fourth terms in $Q_1(x)$ and $T_1(y)$, and performing entirely similar operations, we obtain $$\begin{aligned} &\quad (C_H + 1)\big[B^2(A{\cdot}{\partial}u) + uB(A{\cdot}{\partial}B)\big]{\delta}(x - y) \label{eq:RIP} \\ &+ (C_H - 1)\big[uB^2({\partial}{\cdot}A) + uB(A{\cdot}{\partial}B)\big]{\delta}(x - y). \nonumber\end{aligned}$$ There is no good reason for $C_H\ne C_B$; see further on. There are no singular contributions from the first term in $Q_1(x)$. The second term there will contribute for the commutators with the fourth and fifth terms in $T_1(y)$. Concretely, there is the term $$\begin{aligned} &-iu(x)H(x)[{\partial}^\mu B(x), B(y)]A_\nu(y){\partial}^\nu H(y) \\ &= -u(x)H(x)A_\nu(y){\partial}^\nu H(y){\partial}^\mu_x D(x-y),\end{aligned}$$ plus the analogous one in $[T_1(x), Q_1(y)]$. We are led to the singular part $$-2uH(A{\cdot}{\partial}H){\delta}(x-y). \label{eq:some-trouble-four}$$ The short rule here is ${\partial}^\mu D\to2{\delta}$. Next, we obtain $$\begin{aligned} &\frac{im^2_H}m u(x)H(x)[{\partial}^\mu B(x), B(y)]B(y)H(y) \\ &= \frac{m^2_H}m u(x)H(x)B(y)H(y){\partial}^\mu_x D(x-y),\end{aligned}$$ leading to the singular part $$\frac{2m^2_H}m uBH^2{\delta}(x-y). \label{eq:some-trouble-six}$$ From the last term in $Q_1$, combining with the first term in $T_1(y)$, we obtain in all the singular part $$2muB(A{\cdot}A){\delta}(x-y). \label{eq:some-trouble-seven}$$ Combining both third terms, we consider $$\begin{aligned} &-iu(x)B(x)[{\partial}^\mu H(x), H(y)]A_\nu(y){\partial}^\nu B(y) \\ &= -u(x)B(x)A_\nu(y){\partial}^\nu B(y){\partial}^\mu_x D_{m_H}(x-y).\end{aligned}$$ We have in all the singular part: $$-2uB(A{\cdot}{\partial}B){\delta}(x-y). \label{eq:some-trouble-eight}$$ Coming from respectively the third term in $Q$ and the fifth term in $T_1$, there is the commutator $$\begin{aligned} &\frac{-im_H^2}{2m}u(x)B(x)[{\partial}^\mu H(x), H(y)]B^2(y) \\ &= -\frac{m_H^2}m u(x)B(x)B^2(y) {\partial}^\mu_xD_{m_H}(x-y).\end{aligned}$$ After collecting the similar term and taking the divergences, this leads to $$-\frac{m_H^2}{m^3}u B^3{\delta}(x-y). \label{eq:some-trouble-ten}$$ Coming respectively from the third and sixth term, there is the commutator $$\begin{aligned} &3imeu(x)B(x)[{\partial}^\mu H(x), H(y)]H^2(y) \\ &= 3emu(x)B(x)H^2(y){\partial}^\mu_x D_{m_H}(x-y).\end{aligned}$$ After taking the divergences, this leads to a total singular part $$6emuBH^2{\delta}(x-y). \label{eq:some-trouble-eleven}$$ Next we list all possible normalization terms. Among them, the two first ones are coming from second-order tree graphs with two derivatives on the inner line. In other words, they come from $s[T_1(x), T_1(y)]$. Indeed, in this causal distribution, combining the third terms in the expression of $T_1$, there appears the term $$\begin{aligned} & iA_\mu(x)H(x)[{\partial}^\mu B(x), {\partial}^\nu B(y)]A_\nu(y) H(y) \\ &= A_\mu(x)H(x)A_\nu(y)H(y){\partial}_x^\mu{\partial}_x^\nu D(x - y).\end{aligned}$$ This leads us to a normalization term $C_B(A{\cdot}A)H^2 {\delta}(x-y)$. By the same token, the reader may verify that combining the fourth terms in the expression of $T_1$ there appears the normalization term $C_H(A{\cdot}A)B^2 {\delta}(x-y)$. However, any term of the same form, compatible with Poincaré covariance, discrete symmetries, ghost number and power counting represents in principle a legitimate normalization. Thus we introduce the list of (re)normalization terms we need: $$\begin{aligned} N_2^1 &= C_B(A{\cdot}A)H^2{\delta}(x - y); \\ N_2^2 &= C_H(A{\cdot}A)B^2{\delta}(x - y); \\ N_2^3 &= -\frac{m_H^2}{4m^2}B^4{\delta}(x - y); \\ N_2^4 &= \biggl(\frac{m_H^2}{m^2} + 3e\biggr) B^2H^2{\delta}(x - y).\end{aligned}$$ In view of  they generate new couplings. There is also a $N_2^5$ term in $H^4$, that we omit for now. For convenience, we have anticipated the coefficients in $N_2^3,N_2^4$, which are of the second class. The normalization terms amount to new vertices with four external legs. We compute the coboundaries: $$\begin{aligned} sN_2^1 &= 2C_BH^2(A{\cdot}{\partial}u){\delta}(x - y); \\ sN_2^2 &= 2C_H[B^2(A{\cdot}{\partial}u) + muB(A{\cdot}A)]{\delta}(x - y); \\ sN_2^3 &= -\frac{m_H^2}m uB^3{\delta}(x - y); \\ sN_2^4 &= \Bigl(\,\frac{2m_H^2}m + 6em\Bigr) uBH^2{\delta}(x - y).\end{aligned}$$ The cancellation now is easy to obtain: let $C_B=C_H=1$. This means that we have only to worry about the first two terms in  and similarly in . Now, respectively the term  cancels the second one in  and the term  cancels the second one in . The two remaining terms in  and , together with , , and  are exactly accounted for thanks to the normalization summands. Therefore we have determined $T_1$ and $T_2$, except that $e$ still remains indeterminate. But please read on. Higher-order analysis --------------------- For the higher-order analysis, it is convenient to have the expansion of the inverse ${\mathbb{S}}$-matrix: $${\mathbb{S}}^{-1}(g) =: 1 + \sum_1^\infty\frac{i^n}{n!}\int d^4x_1\dots\int d^4x_n\, {{\overline}T}_n(x_1,\dots,x_n)\,g(x_1)\dots g(x_n).$$ For instance, the second order term ${{\overline}T}_2(x_1,x_2)$ in the expansion of ${\mathbb{S}}^{-1}(g)$ is given by $${{\overline}T}_2(x_1,x_2) = -T_2(x_1,x_2) + T_1(x_1)T_1(x_2) + T_1(x_2)T_1(x_1).$$ Then, say, $$\begin{aligned} A_3(x_1,x_2,x_3) &= {{\overline}T}_1(x_1)T_2(x_2,x_3) + {{\overline}T}_1(x_2)T_2(x_1,x_3) + {{\overline}T}_2(x_1,x_2)T_1(x_3) \nonumber \\ &+ T_3(x_1,x_2,x_3); \nonumber \\ R_3(x_1,x_2,x_3) &= T_1(x_3){{\overline}T}_2(x_1,x_2) + T_2(x_1,x_3){{\overline}T}_1(x_2) + T_2(x_2,x_3){{\overline}T}_2(x_1) \nonumber \\ &+ T_3(x_1,x_2,x_3).\end{aligned}$$ Just as before, $D_3:=R_3-A_3$ depends only on known quantities, is causal in $x_3$ and is gauge invariant. Splitting it, we can calculate $T_3$. We refer to [@Zurichneverdies] for the outcome of the analysis in our case, which turns out to be quite simple. The the missing cubic term is given by: (0,0) node [$\bullet$]{} node\[above right\] [$\frac{-gm_H^2}{2m}$]{} – (2,0) node \[above left\] [$H$]{}; (0,0) – (120:2) node\[below left\] [$H$]{}; (0,0) – (240:2) node\[above left\] [$H$]{}; Also, it is seen that we need the new normalization term $N_2^5=fH^4{\delta}$. One finds $f=-m_H^2/4m^2$, and we are home. Summary of the abelian model ---------------------------- Thus we write down the final (interaction) Lagrangian associated to the abelian theory of the previous section. There are two physical fields $A^\mu,H$, of respective masses $m,m_H$, and an assortment of ghosts $u,{{\tilde u}},B$, which in our Feynman gauge all possess mass $m$. We obtained six cubic couplings (proportional to $g$) and five quartic ones (proportional to $g^2$). It is remarkable that CGI generates the latter from the former. Only four terms out of the eleven involve couplings exclusively among the physical fields. $$\begin{aligned} {\mathcal{L}}_{\rm int}(x) &= gm(A{\cdot}A)H - gm{{\tilde u}}uH + gB(A{\cdot}{\partial}H) \\ &- gH(A{\cdot}{\partial}B) - \frac{gm_H^2}{2m}H^3 - \frac{gm_H^2}{2m}B^2H \\ &+ \frac{g^2}2(A{\cdot}A)H^2 + \frac{g^2}2(A{\cdot}A)B^2 \\ & - \frac{g^2m_H^2}{8m^2}H^4 - \frac{g^2m_H^2}{4m^2}H^2B^2 - \frac{g^2m_H^2}{8m^2}B^4.\end{aligned}$$ With $C=1/m$ this tails down perfectly with  together with ,  and . By construction the total Lagrangian is BRS invariant in the sense defined here. (It has been proved recently in a rigorous way [@Michael05] in the EG framework for interacting fields that “classical” BRS invariance implies gauge invariance for all tree graphs at all orders.) We exhibit the quartic interaction vertices graphically. (0,0) node [$\bullet$]{} node\[right=5pt\] [$\frac{g^2}2$]{} – (45:2) node \[below right\] [$H$]{}; (0,0) – (135:2) node \[below left\] [$A$]{}; (0,0) – (-45:2) node \[above right\] [$H$]{}; (0,0) – (-135:2) node \[above left\] [$A$]{}; (0,0) node [$\bullet$]{} node\[right=5pt\] [$\frac{g^2}2$]{} – (45:2) node \[below right\] [$B$]{}; (0,0) – (135:2) node \[below left\] [$A$]{}; (0,0) – (-45:2) node \[above right\] [$B$]{}; (0,0) – (-135:2) node \[above left\] [$A$]{}; (0,0) node [$\bullet$]{} node\[right=3pt\] [$-\frac{g^2m_H^2}{8m^2}$]{} – (45:2) node \[below right\] [$H$]{}; (0,0) – (135:2) node \[below left\] [$H$]{}; (0,0) – (-45:2) node \[above right\] [$H$]{}; (0,0) – (-135:2) node \[above left\] [$H$]{}; (0,0) node [$\bullet$]{} node\[right=3pt\] [$-\frac{g^2m_H^2}{8m^2}$]{} – (45:2) node \[below right\] [$B$]{}; (0,0) – (135:2) node \[below left\] [$B$]{}; (0,0) – (-45:2) node \[above right\] [$B$]{}; (0,0) – (-135:2) node \[above left\] [$B$]{}; (0,0) node [$\bullet$]{} node\[right=3pt\] [$-\frac{g^2m_H^2}{4m^2}$]{} – (45:2) node \[below right\] [$H$]{}; (0,0) – (135:2) node \[below left\] [$H$]{}; (0,0) – (-45:2) node \[above right\] [$B$]{}; (0,0) – (-135:2) node \[above left\] [$B$]{}; Notice that the purely scalar couplings are $$\begin{aligned} &-g\frac{m_H^2}{2m}H(B^2 + H^2) -g^2\frac{m_H^2}{8m^2}(B^2 + H^2)^2 \\ &= -\,\frac{g^2m_H^2}{8m^2}(B^2 + H^2)\big(B^2 + H^2 + \frac{4m}{g}H\big).\end{aligned}$$ Performing now an asymptotic analysis (that is, taking the Stückelberg field $B=0$) it becomes $$-\,\frac{g^2m_H^2}{8m^2}\big(H^4 + \frac{4m}{g}H^3\big).$$ Three MVBs ========== Let us now seek all gauge theories with *three* gauge fields. The only interesting Lie algebra entering the game is $${\mathfrak{g}}= \mathfrak{su}(2);$$ in this case obviously total antisymmetry implies the Jacobi identity. The case $m_1=m_2=m_3=0$ is certainly possible, and then neither scalar Higgs nor Stückelberg fields are necessary. The simplest of the *mass relations* we referred to in Section 2 is the following: if $f_{abc}\ne0$ and $m_a=0$, then necessarily $m_b=m_c$. We see at once from this that if $m_1=0$ must be $m_2=m_3$: the case $m_1=m_2=0,m_3\ne0$ is downright impossible. The only other mass relation one needs to check to verify that models with two or three MVBs and one Higgs-like field are correct in our sense is $$\begin{aligned} 4C^2m_b^2m_a^2 &= 2(m_a^2 + m_b^2) \sum_{d:m_d=0}\! (f_{abd})^2 {\nonumber}\\ &+ \sum_{k:m_k\neq 0}\! \frac{(f_{abk})^2}{m_k^2} \bigl[ (m_a^2 + m_b^2 + m_k^2)^2 - 4(m_a^2 m_b^2 + m_k^4) \bigr]. \label{eq:mother-lode}\end{aligned}$$ With $C^{-1}=\pm m_2$, the model with the mass pattern $m_2=m_3\ne0, m_1=0$ passes muster. If we assume that all masses are different from zero, then necessarily $m_1=m_2=m_3$. Indeed, equation  implies $$4 m_a^2 m_b^2 m_c^2 C^2 = \bigl[ (m_a^2 + m_b^2 + m_c^2)^2 - 4(m_a^2 m_b^2 + m_c^4) \bigr]$$ where $(a,b,c)$ is any permutation of $(1,2,3)$. Therefore, $$m_1^2 m_2^2 + m_3^4 = m_2^2 m_3^2 + m_1^4 = m_3^2 m_1^2 + m_2^4.$$ This yields $$\begin{aligned} (m_1^2 m_2^2 + m_3^4) - (m_2^2 m_3^2 + m_1^4) &= (m_3^2 - m_1^2)(m_1^2 - m_2^2 + m_3^2) = 0, \\ (m_2^2 m_3^2 + m_1^4) - (m_3^2 m_1^2 + m_2^4) &= (m_1^2 - m_2^2)(m_2^2 - m_3^2 + m_1^2) = 0,\end{aligned}$$ whose only all-positive solution is $m_1 = m_2 = m_3 =: m$; and then $4m^6 C^2 = m^4$ yields $C^{-1}=\pm2m$. Physically, the two cases just examined correspond respectively to the Georgi–Glashow model of electroweak interactions without neutral currents; and to the $\mathfrak{su}(2)$ Higgs–Kibble model. Reference [@CabezondelaSal] claims that more than one Higgs-like particle for the $\mathfrak{su}(2)$ Higgs–Kibble model is not allowed. It is well known that the first mass pattern obtained here is arrived at by SSB when the Higgs sector is chosen to be a $SU(2)$ isovector; and the second one when it is a complex doublet. But in our derivation SSB played no role. The Weinberg–Salam model within CGI =================================== Scharf and coworkers (see references in the introduction) followed a “deductive” approach to the SM, with the only assumption that $m_1$, $m_2$, $m_3$ are all positive, plus existence of the photon, that is, $m_4 = 0$. There is no point in repeating that. Suffice to say that a structure constant like $f_{124}$ is found to be non-zero, thus $m_1=m_2$; and also the mass constraints imply $m_3>m_1$. Defining $$\cos {\theta_{\mathrm{W}}}:= m_1/m_3,$$ it is possible now to take for the non-zero structure constants $$|f_{123}| = \cos {\theta_{\mathrm{W}}}{\quad\hbox{and}\quad} |f_{124}| = \sin {\theta_{\mathrm{W}}}.$$ With this, simply bringing  together with equations with ,  and , one retrieves the boson part of the SM Lagrangian, as given for example in [@PapaTomate]. Thus it appears that the ordinary version of the Higgs sector for the gauge group $SU(2){\times}U(1)\simeq U(2)$ is “chosen” by CGI. Of course, one can argue for it from other considerations within the SSB framework, or refer to experiment. We comment in the final discussion on the problem of determining which patterns of broken symmetry are allowed in CGI for general gauge groups. Coupling to matter ------------------ Things stay interesting when considering the fermion sector. The basic interaction between carriers and matter in a gauge theory is of the form $$g(b^aA_{a_\mu}{\overline}\psi{\gamma}^\mu\psi + {b'}^aA_{a_\mu} {\overline}\psi{\gamma}^\mu{\gamma}^5\psi),$$ with ${\overline}\psi$ the Dirac adjoint spinor and $b,b'$ appropriate coefficients. In dealing with the SM our fermions are the known ones, fulfilling as free fields the Dirac equation: we do not assume chiral fermions *ab initio*. Their gauge variation is taken to be zero. Thus for the SM one makes the Ansatz $$\begin{aligned} T_1^F &= b_1W_\mu^+\bar e{\gamma}^\mu\nu + b'_1W_\mu^+\bar e{\gamma}^\mu{\gamma}^5\nu + b_2W_\mu^-\bar\nu{\gamma}^\mu e + b'_2W_\mu^-\bar\nu{\gamma}^\mu{\gamma}^5 e {\nonumber}\\ &\quad + b_3Z_\mu\bar e{\gamma}^\mu e + b'_3Z_\mu\bar e{\gamma}^\mu{\gamma}^5 e + b_4Z_\mu\bar\nu{\gamma}^\mu\nu + b'_4Z_\mu\bar\nu{\gamma}^\mu{\gamma}^5\nu {\nonumber}\\ &\quad + b_5A_\mu\bar e{\gamma}^\mu e + b'_5A_\mu\bar e{\gamma}^\mu{\gamma}^5 e + b_6A_\mu\bar\nu{\gamma}^\mu\nu + b'_6A_\mu\bar\nu{\gamma}^\mu{\gamma}^5\nu {\nonumber}\\ &\quad + c_1B^+\bar e\nu + c'_1B^+\bar e{\gamma}^5\nu + c_2B^-\bar\nu e + c'_2B^-\bar\nu{\gamma}^5 e {\nonumber}\\ &\quad + c_3B_Z\bar e e + c'_3B_Z\bar e{\gamma}^5 e + c_4B_Z\bar\nu\nu + c'_4B_Z\bar\nu{\gamma}^5\nu {\nonumber}\\ &\quad + c_5H\bar\nu\nu + c'_5H\bar\nu{\gamma}^5\nu + c_6H\bar e e + c'_6H\bar e{\gamma}^5 e. \label{eq:canto-en-los-dientes}\end{aligned}$$ Here $e$ stands for an electron, muon or neutrino or a (suitable combination of) quarks $d,s,b$; and $\nu$ for the neutrinos or the quarks $u,c,t$; the charge difference is always minus one. For instance in the “vertex” $W_\mu^+\bar e{\gamma}^\mu\nu$ a “positron” exchanges a $W^+$ boson and becomes a “neutrino”. Charge is conserved in each term. The method to determine the coefficients in  *remains the same*; only, it is simpler in practice. We limit ourselves to a few remarks. The direct equation $$sT_1^F = i{\partial}{\cdot}Q^F_1$$ already allows to determine $c_1,c'_1,c_2,c'_2,c_3,c'_3,c_4,c'_4$, as well as the vanishing of $b'_5$ and $b'_6$, assuming nonvanishing fermion masses. (For $\nu$ representing a true neutrino, we expect the term with coefficient $b'_6$ to vanish anyway, since the photon should not couple to uncharged particles. The same is true for $b_6$.) Thus the photon has no axial-vector couplings, “because” there is no Stückelberg field for it, that is, because it is massless. The reader will have no trouble in finding the explicit form of $Q^F_1$, that can be checked with [@Zurichneverdies Eq. 4.7.4]. At second order, one needs to take into account the interplay of contractions between $Q_1$ and $T_1^F$, as well as the “purely fermionic” ones between $Q^F_1$ and $T_1^F$. There are no contractions between $Q^F_1$ and $T_1$, since the former does not contain derivatives. Also, *no new normalization terms* with fermionic fields may be forthcoming in $sN_2$ or ${\partial}_x{\cdot}N_{2/1}, {\partial}_y{\cdot}N_{2/2}$, since a term $\sim{\varphi}_1{\varphi}_2\bar\psi\psi{\delta}$ would be nonrenormalizable by power counting: the only way to cancel local terms is that the coefficient of every generated Wick monomial add up to zero. At the end of the day, the physical Higgs couplings are proportional to the mass, and *chirality* of the interactions is a *consequence* of CGI [@PGI-EW-I; @PGI-EW-II]. For leptons it yields: $$\begin{aligned} T_1^F &= \frac1{2\sqrt2}W_\mu^+\bar e{\gamma}^\mu(1 \pm {\gamma}_5)\nu + \frac1{2\sqrt2}W_\mu^-\bar\nu{\gamma}^\mu(1 \pm {\gamma}_5)e + \frac1{4\cos{\theta_{\mathrm{W}}}}Z_\mu\bar e{\gamma}^\mu(1 \pm {\gamma}_5)e \\ &\quad - \sin{\theta_{\mathrm{W}}}\tan{\theta_{\mathrm{W}}}Z_\mu\bar e{\gamma}^\mu e -\frac1{4\cos{\theta_{\mathrm{W}}}}Z_\mu\bar\nu{\gamma}^\mu(1 \pm {\gamma}_5)\nu + \sin{\theta_{\mathrm{W}}}A_\mu\bar e{\gamma}^\mu e \\ &\quad + i\frac{m_e - m_\nu}{2\sqrt{2}m_{\mathrm{W}}}B^+\bar e\nu \pm i\frac{m_e + m_\nu}{2\sqrt{2}m_{\mathrm{W}}}B^+\bar e{\gamma}^5\nu -i\frac{m_e - m_\nu}{2\sqrt{2}m_{\mathrm{W}}}B^-\bar\nu e \\ &\quad \pm i\frac{m_e + m_\nu}{2\sqrt{2}m_{\mathrm{W}}}B^-\bar\nu{\gamma}^5 e \pm i \frac{m_e}{2m_{\mathrm{W}}}B_Z\bar e e \pm i \frac{m_\nu}{2m_{\mathrm{W}}}B_Z\bar e{\gamma}^5 \\ &\quad + \frac{m_\nu}{2m_{\mathrm{W}}}H\bar\nu\nu + \frac{m_e}{2m_{\mathrm{W}}}H\bar e e,\end{aligned}$$ as it should. In summary we have recovered the SM, with its rationale upside-down. Discussion ========== People define $e=g\sin{\theta_{\mathrm{W}}};\,g'=g\tan{\theta_{\mathrm{W}}}$. Therefore, $$\sec{\theta_{\mathrm{W}}}= \frac{\sqrt{g^2 + g'{}^2}}{g},$$ and selecting the chiral projector and apart from the standard factors, the effective coupling of the term in $W_\mu^+\bar e{\gamma}^\mu\nu$ and conjugate is $g$; that of the term $A_\mu\bar e{\gamma}^\mu e$ is $e$; that of the term $Z_\mu\bar\nu{\gamma}^\mu\nu$ is $-\sqrt{g^2 + {g'}^2}/2g$; and so on. Thus one can artfully write things as if $g,g'$ are two different coupling constant associated to the emerging representation of the gauge group. But we have seen that the coefficients come from the pattern of masses, which in our viewpoint is fixed by nature. In order to bring home the point, let us make the *Gedankenexperiment* of building the SM from the Georgi–Glashow model, by adding a vector boson, sitting on an invariant abelian subgroup. Implicitly we allow for two different coupling constants (plus mixing of the old photon and the new MVB). But in that case there is no reason for $m_Z>m_W$. It is more natural to assume that the SM stems from the Higgs–Kibble model, keeping one coupling constant, whereby two of the three masses are moderately “pulled down” by mixing with the new photon. This goes to the heart of the experimental situation; other weak isospin values do not enter the game. In other words, no support comes from our quarter to the idea that the SM as it stands is “imperfectly unified”. The argument is bolstered by the fact that the true group of the electroweak interaction is $U(2)$, not $SU(2){\times}U(1)$.[^2] In usual presentations of the SM the $U(2)$ symmetry is said to be “broken”, among other reasons, because there is only one conserved quantity, electric charge, instead of four. In CGI the interaction appears to respect the $U(2)$ symmetry. But of course symmetry is broken already at the level of the free Lagrangian, due to different masses (the residual symmetry $m_1=m_2$ goes in hand with electric charge conservation). This is to say that not all bases of the Lie algebra are equivalent, since there is a natural basis dictated by the pattern of masses. The role of the mass constraints is precisely to pick out this basis.[^3] Let us recapitulate. CGI is a tool for the actual construction of Lagrangians. We limited ourselves to polynomial couplings. At first order in the coupling constant, CGI fixes some of the couplings of the vector bosons and ghost (fermionic and bosonic) fields. At second order, it requires additional quartic couplings, as well as some extra ingredient, which here is made out of physical scalars or Higgs-like fields. Third-order invariance goes on to fix the remaining couplings of the Higgs-like fields. One obtains in that way potentials of the symmetry-breaking kind, although SSB does not enter the picture. Ockham’s razor, already invoked in Section 2 in relation with the number of Higgs fields, seems even more pertinent here. On the historical side, it is difficult to imagine the development of electroweak unification during the sixties without the SSB crutch. Massive vector bosons were beyond the pale then. The only contemporaneous article (still instructive today) I know of, willing and eager to start from them as fundamental entities is [@Samizdat]; it did not have enough impact. Around ten years later, after the invention of SSB, cogent arguments based on tree unitarity —see [@CornwallLevinTiktopoulosPRD-10] and references therein— weighed in favour of the phenomenological outcome of gauge theories with broken symmetry, plus abelian mass terms for invariant abelian subgroups. This is basically what CGI constructs. Since they lead to the same phenomenological Lagrangian, there seems to be no way as yet —within ordinary particle physics, at least— to distinguish between the SM as presented in textbooks and its causal version. This is good, because it shows that CGI is solidly anchored in physics. It is also bad: “a difference, to be a difference, has to make a difference”. Still, a constructive CGI program was in principle attractive because the apparent severity of the constraints on the masses of the gauge fields. Ambauen and Scharf [@FortunaJuvet] argued that the $SU(5)$ grand unification model by Georgi and Glashow with its standard pattern of Higgs fields [@Georgi99 Chap. 18], is not causally gauge invariant; and the situation in this respect for a while was murky. However, a systematic comparison between CGI and the general theory of broken local symmetries [@Ling-FongLi] has been performed recently [@VierMaenner], and the contention of [@FortunaJuvet] that there might be contradiction between causal gauge invariance and some grand unified models has been laid to rest. Acknowledgments {#acknowledgments .unnumbered} --------------- I am most grateful for discussions to Luis J. Boya, Florian Scheck and Joseph C. Várilly. Special thanks are due to Michael Dütsch, who patiently explained to me aspects of the gauge principle according to the Zürich school. I acknowledge support from CICyT, Spain, through grant FIS2005–02309. [99]{} I. J. R. Aitchison and A. J. G. Hey, *Gauge theories in particle physics: QCD and the electroweak theory*, IOP Publishing, Bristol, 2004. C. Burgess and G. Moore, *The Standard Model: a primer*, Cambridge University Press, Cambridge, 2007. M. J. G. Veltman, Phys. Rev. Lett. [**34**]{} (1975) 777. H. Cheng and E-C. Tsai, Phys. Rev. D [**40**]{} (1989) 1246. M. J. G. Veltman, Rev. Mod. Phys. [**72**]{} (2000) 341. J. Earman, Philos. Sci. [**71**]{} (2004) 1227. H. Lyre, Intl. Studies Philos. Sci. **22** (2008) 119. H. Ruegg and M. Ruiz-Altaba, Int. J. Mod. Phys. A [**19**]{} (2004) 3265. J. M. Gracia-Bondía, “BRS invariance for massive boson fields”, to appear in the Proceedings of the Summer School “Geometrical and topological methods for quantum field theory”, Cambridge University Press, Cambridge, 2010; hep-th/0808.2853. W. Kilian, *Electroweak symmetry breaking: the bottom-up approach*, Springer, New York, 2003. M. Dütsch and G. Scharf, Ann. Phys. (Leipzig) [**8**]{} (1999) 359. A. Aste, G. Scharf and M. Dütsch, Ann. Phys. (Leipzig) [**8**]{} (1999) 389. M. Dütsch and B. Schroer, J. Phys. A [**33**]{} (2000) 4317. D. R. Grigore, J. Phys. A [**33**]{} (2000) 8443. G. Scharf, *Quantum gauge theories. A true ghost story*, Wiley, New York, 2001. T. Hurth and K. Skenderis, Nucl. Phys. B [**541**]{} (1999) 566. T. Hurth and K. Skenderis, in *New developments in quantum field theory*, P. Breitenlohner, D. Maison and J. Wess, eds., Springer, Berlin, 2000; pp. 86–105. H. Epstein and V. Glaser, Ann. Inst. Henri Poincaré [**XIXA**]{} (1973) 211. L. Alvarez-Gaumé and L. Baulieu, Nucl. Phys. B [**212**]{} (1983) 255. R. Stora, “Local gauge groups in quantum field theory: perturbative gauge theories”, talk given at the workshop “Local quantum physics”, Erwin Schrödinger Institute, Vienna, 1997. M. Dütsch, private communication. M. Dütsch, Ann. Phys. (Leipzig) [**14**]{} (2005) 438. M. Veltman, *Diagrammatica*, Cambridge University Press, Cambridge, 1994. F. Scheck, *Leptons, hadrons and nuclei*, North-Holland, Amsterdam, 1983. H. Georgi and S. L. Glashow, Phys. Rev. D [**6**]{} (1972) 2977. V. I. Ogievetskij and I. V. Polubarinov, Ann. Phys. (New York) [**25**]{} (1963) 358. J. M. Cornwall, D. N. Levin and G. Tiktopoulos, Phys. Rev. D [**10**]{} (1974) 1145. M. Ambauen and G. Scharf, “Violation of quantum gauge invariance in Georgi–Glashow $SU(5)$”, hep-th/0409062. H. Georgi, *Lie algebras in particle physics*, Westview Press, Boulder, 1999. Ling-Fong Li, Phys. Rev. D [**9**]{} (1974) 1723. M. Dütsch, J. M. Gracia-Bondía, F. Scheck and J. C. Várilly, “Quantum gauge models without classical Higgs mechanism”, hep-th/1001.0932. [^1]: This situation has recently called the attention of knowledgeable philosophers of science [@HombreOreja; @NoClothesKing]: in epistemological terms, they argue that the mechanism had heuristic value in the context of discovery; but much less so in the context of justification. [^2]: To our knowledge, this was noticed first in [@Florian]. [^3]: In the early seventies, speculations on fermionic patterns of masses from SSB were rife —see for instance [@GG]. They have been since all but abandonded. They might perhaps become a respectable subject of study again in CGI. By now we may only say that differences between fermion masses are related to differences between boson masses in that (disregarding family mixing) models in which all bosons share the same mass would entail identity of all fermion masses as well.

--- abstract: 'We show that on a derived Artin $N$-stack, there is a canonical equivalence between the spaces of $n$-shifted symplectic structures and non-degenerate $n$-shifted Poisson structures.' author: - 'J.P.Pridham' bibliography: - 'references.bib' title: 'Shifted Poisson and symplectic structures on derived $N$-stacks' --- [^1] Introduction {#introduction .unnumbered} ============ In [@PTVV], the notion of an $n$-shifted symplectic structure for a derived Artin stack was introduced. The definition of an $n$-Poisson structure was also sketched, and an approach suggested to prove equivalence of $n$-shifted symplectic structures and non-degenerate $n$-Poisson structures, by first establishing deformation quantisation for symplectic structures. The Darboux theorems of [@BBBJdarboux; @BouazizGrojnowski] imply the existence of shifted Poisson structures locally on derived Deligne–Mumford stacks, but not globally. We prove existence of shifted Poisson structures by a direct approach, not involving deformation quantisation or Darboux theorems. The key new notion is that of compatibility between an $n$-shifted pre-symplectic (i.e. closed, possibly degenerate) structure $\omega$ and an $n$-Poisson structure $\pi$. In the case of unshifted structures on underived manifolds, compatibility simply says that the associated maps between tangent and cotangent bundles satisfy $$\pi^{\sharp} \circ \omega^{\sharp} \circ \pi^{\sharp}=\pi^{\sharp},$$ which ensures that $\omega^{\sharp}=(\pi^{\sharp})^{-1}$ whenever $\pi$ is non-degenerate. We demonstrate that this notion admits a natural generalisation incorporating shifts and higher coherence data. We expect that this notion of compatibility between Poisson and pre-symplectic structures is the same as in Definition 1.4.14 of the contemporaneous treatment [@CPTVV], whose proof also does not involve deformation quantisation. Our notion of compatibility is extended in [@DQvanish; @DQnonneg; @DQLag] from Poisson structures to quantisations. Our first main observation is that there is a canonical global section $\sigma$ of the tangent space $T\cP(X,n)$ of the space $\cP(X,n)$ of $n$-Poisson structures, given by differentiating the $\bG_m$-action on the differential graded Lie algebra of polyvectors. For an unshifted Poisson structure on a smooth underived scheme, this just maps $\pi$ to itself. In general, if we write $\pi = \sum_{r \ge 2} \pi_r$ with $\pi_r$ an $r$-vector, then $$\sigma(\pi) = \sum_{r \ge 2} (r-1)\pi_r.$$ For any $n$-shifted Poisson structure $\pi$, contraction gives a multiplicative map $\mu(-,\pi)$ from de Rham to Poisson cohomology, so any $n$-shifted pre-symplectic structure $\omega$ defines another global section $\mu(\omega,-)$ of the tangent space of the space of Poisson structures. In the unshifted case on a smooth scheme, the associated map from the cotangent space to the tangent space is simply given by $$\mu(\omega, \pi)^{\sharp}= \pi^{\sharp} \circ \omega^{\sharp} \circ \pi^{\sharp}.$$ We then say that a pair $(\omega, \pi)$ is compatible if $\mu(\omega,\pi) \simeq \sigma(\pi)$. More formally, the space of compatible pairs is the homotopy limit of the diagram $$\xymatrix@1{ \PreSp(X,n) \by \cP(X,n) \ar@<0.5ex>[r]^-{\mu} \ar@<-0.5ex>[r]_-{\sigma} & T\cP(X,n) \ar[r] & \cP(X,n)}.$$ In the unshifted case, this amounts to seeking compatible pairs in the sense above. Poisson structures are functorial with respect to étale morphisms, so the notions above are readily defined for derived DM $N$-stacks. For derived Artin $N$-stacks, the formulation of Poisson structures is more subtle: we show that a derived Artin $N$-stack admits an étale cover by derived stacks coming from commutative bidifferential bigraded algebras, and put Poisson structures on them. These algebras perform the same role as the formal affine derived stacks of [@CPTVV], and we refer to them as stacky CDGAs. We then show that: 1. If $\pi$ is non-degenerate, then every compatible pre-symplectic structure $\omega$ is also non-degenerate, hence symplectic, and the space of such structures is contractible. 2. If $\omega$ is symplectic, then the space of *non-degenerate* compatible Poisson structures is contractible. Thus the spaces of $n$-shifted symplectic structures and non-degenerate $n$-Poisson structures are both weakly equivalent to the space of non-degenerate compatible pairs $(\omega, \pi)$. These results have recently been obtained in [@CPTVV], using a very different method to formulate compatibility. The structure of the paper is as follows. Section \[affinesn\] addresses the case of a single, fixed, derived affine scheme. The compatibility operator $\mu$ is introduced in Definition \[mudef\], and the key technical result is Lemma \[keylemma\], which shows how $\mu$ interpolates between the de Rham differential and the Schouten–Nijenhuis bracket. §\[towersn\] shows how to realise the spaces of pre-symplectic structures, Poisson structures and compatible pairs as towers of homotopy fibres. This uses the obstruction theory associated to pro-nilpotent $L_{\infty}$-algebras, regarding the construction as a generalised deformation problem. The correspondence between symplectic and non-degenerate Poisson structures in the affine case is then given in Corollary \[compatcor2\]. Section \[DMsn\] then applies these results to derived stacks. To date, the only proof that the cotangent complex governs deformations of a (derived) algebraic stack (rather than just morphisms to it from a fixed object) uses simplicial resolutions by affine schemes (cf. [@aoki Theorem 1.2], [@stacks2 Theorem \[stacks2-deformstack\]]). Since the construction of Poisson structures is a kind of generalised deformation problem, it is unsurprising that it can be studied using such resolutions. For derived Deligne–Mumford $N$-stacks, we look at spaces of structures on the diagrams given by suitable simplicial resolutions, and show that they are independent of the resolution; the key being étale functoriality. Theorem \[DMthm\] then establishes an equivalence between the spaces of $n$-symplectic and non-degenerate $n$-Poisson structures. In section \[Artinsn\], these results are extended to derived Artin $N$-stacks. Given a simplicial resolution by derived affines, we can form an associated stacky CDGA, and indeed a simplicial resolution by stacky CDGAs (Corollary \[gooddescent\]). The morphisms in this resolution resemble étale maps, allowing a suitably functorial generalisation of the results of §\[affinesn\] to stacky CDGAs. The main result is then Theorem \[Artinthm\], which establishes an equivalence between the spaces of $n$-symplectic and non-degenerate $n$-Poisson structures on derived Artin $N$-stacks. I would like to thank Victor Ginzburg for suggesting a simplification to the proofs of Lemma \[keylemma\] and Proposition \[compatP1\]. I would also like to thank the anonymous referees for helpful suggestions and comments. Compatible structures on derived affines {#affinesn} ======================================== For the purposes of this section, we will fix a graded-commutative algebra $R=R_{\bt}$ in chain complexes over $\Q$, and a cofibrant graded-commutative $R_{\bt}$-algebra $A=A_{\bt}$. We will denote the differential on $A$ by $\delta$. In particular, the cofibrancy condition holds whenever the underlying morphism $R_{\#} \to A_{\#}$ of graded commutative algebras is freely generated in non-negative chain degrees. It ensures that the module $\Omega^1_{A/R}$ of Kähler differentials is a model for the cotangent complex of $A$ over $R$. For the purposes of this section, this latter property is all we need, so we could relax the cofibrancy condition slightly to include morphisms which are ind-smooth rather than freely generated. We define $\Omega^p_{A/R}:= \Lambda_A^p \Omega^1_{A/R}$, and we denote its differential (inherited from $A$) by $\delta$. There is also a de Rham cochain differential $\Omega^p_{A/R} \to \Omega^{p+1}_{A/R}$, which we denote by $d$. For a chain (resp. cochain) complex $M$, we write $M_{[i]}$ (resp. $M^{[j]}$) for the complex $(M_{[i]})_m= M_{i+m}$ (resp. $(M^{[j]})^m = M^{j+m}$). Given $A$-modules $M,N$ in chain complexes, we write $\HHom_A(M,N)$ for the cochain complex given by $$\HHom_A(M,N)^i= \Hom_{A_{\#}}(M_{\#},N_{\#[-i]}),$$ with differential $\delta f= \delta_N \circ f \pm f \circ \delta_M$, where $V_{\#}$ denotes the graded vector space underlying a chain complex $V$. Shifted Poisson structures {#poisssn} -------------------------- ### Polyvector fields {#polsn} \[poldef\] Define the cochain complex of $n$-shifted polyvector fields on $A$ by $$\widehat{\Pol}(A,n):= \HHom_A(\CoS_A((\Omega^1_{A/R})_{[-n-1]}),A),$$ with graded-commutative multiplication following the usual conventions for symmetric powers. \[Here, $\CoS_A^p(M) =\Co\Symm^p_A(M)= (M^{\ten_A p})^{\Sigma_p}$ and $\CoS_A(M) = \bigoplus_{p \ge 0}\CoS_A^p(M)$.\] The Lie bracket on $\Hom_A(\Omega^1_{A/R},A)$ then extends to give a bracket (the Schouten–Nijenhuis bracket) $$[-,-] \co \widehat{\Pol}(A,n)\by \widehat{\Pol}(A,n)\to \widehat{\Pol}(A,n)^{[-1-n]},$$ determined by the property that it is a bi-derivation with respect to the multiplication operation. Thus $\widehat{\Pol}(A,n)$ has the natural structure of a $P_{n+2}$-algebra (i.e. an $(n+1)$-shifted Poisson algebra), and in particular $\widehat{\Pol}(A,n)^{[n+1]}$ is a differential graded Lie algebra (DGLA) over $R$. Note that the cochain differential $\delta$ on $\widehat{\Pol}(A,n)$ can be written as $[\delta,-]$, where $\delta \in \widehat{\Pol}(A,n)^1$ is the element defined by the derivation $\delta$ on $A$. Strictly speaking, $\widehat{\Pol}$ is the complex of multiderivations, as polyvectors are usually defined as symmetric powers of the tangent complex. The two definitions agree (modulo completion) whenever the tangent complex is perfect, and Definition \[poldef\] is the more natural object when the definitions differ. \[Fdef\] Define a decreasing filtration $F$ on $\widehat{\Pol}(A,n)$ by $$F^i\widehat{\Pol}(A,n):= \HHom_A( \bigoplus_{j \ge i} \CoS_A^j((\Omega^1_{A/R})_{[-n-1]}),A);$$ this has the properties that $\widehat{\Pol}(A,n)= \Lim_i \widehat{\Pol}(A,n)/F^i$, with $[F^i,F^j] \subset F^{i+j-1}$, $\delta F^i \subset F^i$, and $F^i F^j \subset F^{i+j}$. Observe that this filtration makes $F^2\widehat{\Pol}(A,n)^{[n+1]}$ into a pro-nilpotent DGLA. ### Poisson structures \[mcPLdef\] Given a DGLA $L$, define the the Maurer–Cartan set by $$\mc(L):= \{\omega \in L^{1}\ \,|\, \delta\omega + \half[\omega,\omega]=0 \in \bigoplus_n L^{2}\}.$$ Following [@hinstack], define the Maurer–Cartan space $\mmc(L)$ (a simplicial set) of a nilpotent DGLA $L$ by $$\mmc(L)_n:= \mc(L\ten_{\Q} \Omega^{\bt}(\Delta^n)),$$ where $$\Omega^{\bt}(\Delta^n)=\Q[t_0, t_1, \ldots, t_n,\delta t_0, \delta t_1, \ldots, \delta t_n ]/(\sum t_i -1, \sum \delta t_i)$$ is the commutative dg algebra of de Rham polynomial forms on the $n$-simplex, with the $t_i$ of degree $0$. Given an inverse system $L=\{L_{\alpha}\}_{\alpha}$ of nilpotent DGLAs, define $$\mc(L):= \Lim_{\alpha} \mc(L_{\alpha}) \quad \mmc(L):= \Lim_{\alpha} \mmc(L_{\alpha}).$$ Note that $\mc(L)= \mc(\Lim_{\alpha}L_{\alpha})$, but $\mmc(L)\ne \mmc(\Lim_{\alpha}L_{\alpha}) $. \[poissdef\] Define an $R$-linear $n$-shifted Poisson structure on $A$ to be an element of $$\mc(F^2 \widehat{\Pol}(A,n)^{[n+1]}),$$ and the space $\cP(A,n)$ of $R$-linear $n$-shifted Poisson structures on $A$ to be given by the simplicial set $$\cP(A,n):= \mmc( \{F^2 \widehat{\Pol}(A,n)^{[n+1]}/F^{i+2}\}_i).$$ Also write $\cP(A,n)/F^{i+2}:= \mmc(F^2 \widehat{\Pol}(A,n)^{[n+1]}/F^{i+2})$, so $\cP(A,n)= \Lim_i \cP(A,n)/F^{i+2}$. Observe that elements of $\cP_0(A,n)= \mc(F^2 \widehat{\Pol}(A,n)^{[n+1]})$ consist of infinite sums $\pi = \sum_{i \ge 2}\pi_i$ with $$\pi_i \co \CoS_A^i((\Omega^1_{A/R})_{[-n-1]}) \to A_{[-n-2]}$$ satisfying $\delta(\pi_i) + \half \sum_{j+k=i+1} [\pi_j,\pi_k]=0$. This is precisely the condition which ensures that $\pi$ defines an $L_{\infty}$-algebra structure on $A_{[-n]}$. This then makes $A$ into a $\hat{P}_{n+1}$-algebra in the sense of [@melaniPoisson Definition 2.9]. Equivalently, this is an algebra for the operad $\Com \circ (L_{\infty}[-n])$ via a distributive law $ (L_{\infty}[-n])\circ\Com \to \Com \circ (L_{\infty}[-n])$. It is important to remember that $ \cP(-,n)$ is just one explicit presentation of the right-derived functor of $\cP_0(-,n) $. When we need to compare chain and cochain complexes, we silently make use of the equivalence $u$ from chain complexes to cochain complexes given by $(uV)^i := V_{-i}$. On suspensions, this has the effect that $u(V_{[n]}) = (uV)^{[-n]}$. We say that an $n$-shifted Poisson structure $\pi = \sum_{i \ge 2}\pi_i $ is non-degenerate if $\pi_2 \co \CoS_A^2((\Omega^1_{A/R})_{[-n-1]}) \to A_{[-n-2]}$ induces a quasi-isomorphism $$\pi_2^{\sharp}\co (\Omega^1_{A/R})_{[-n]} \to \HHom_A(\Omega^1_{A/R},A)$$ and $\Omega^1_{A/R} $ is perfect. Define $\cP(A,n)^{\nondeg}\subset \cP(A,n)$ to consist of non-degenerate elements — this is a union of path-components. ### The canonical tangent vector $\sigma$ The space $\cP(A,n) $ admits an action of $\bG_m(R_0)$, which is inherited from the scalar multiplication on $\widehat{\Pol}(A,n)$ in which $\Hom_A(\CoS_A^p((\Omega^1_{A/R})_{[-n-1]}),A)$ is given weight $p-1$. Differentiating this action gives us a global tangent vector on $\cP(A,n) $, as follows. Take $\eps$ to be a variable of degree $0$, with $\eps^2=0$. \[Tpoissdef\] Define the tangent spaces $$\begin{aligned} T\cP(A,n)&:=& \mmc( \{F^2 \widehat{\Pol}(A,n)^{[n+1]}\ten_{\Q} \Q[\eps]/F^{i+2}\}_i)\\ T\cP(A,n)/F^{i+2}&:=& \mmc( F^2 \widehat{\Pol}(A,n)^{[n+1]}\ten_{\Q} \Q[\eps]/F^{i+2}).\end{aligned}$$ These are simplicial sets over $\cP(A,n)$ (resp. $\cP(A,n)/F^{i+2}$), fibred in simplicial abelian groups. \[cPdef\] Given $\pi \in \cP_0(A,n)$, observe that $\delta+[\pi,-]$ defines a square-zero derivation on $\widehat{\Pol}(A,n) $, and denote the resulting complex by $$\widehat{\Pol}_{\pi}(A,n).$$ The product and bracket on polyvectors make this a $P_{n+2}$-algebra, and it inherits the filtration $F$. Given $\pi \in \cP_0(A,n)/F^p$, we define $\widehat{\Pol}_{\pi}(A,n)/F^p$ similarly. This is a CDGA, and $ F^1\widehat{\Pol}_{\pi}(A,n)/F^p$ is a $P_{n+2}$-algebra, because $F^i\cdot F^j \subset F^{i+j}$ and $[F^i,F^j] \subset F^{i+j-1}$. Note that $\widehat{\Pol}_{\pi}(A,n)$ is just the natural $n$-shifted analogue of the complex computing Poisson cohomology. The following is an instance of a standard result on square-zero extensions of DGLAs: The fibre $T_{\pi}\cP(A,n)$ of $T\cP(A,n)$ over $\pi \in \cP(A,n)$ is canonically homotopy equivalent to the Dold–Kan denormalisation of the good truncation $\tau^{\le 0} (F^2\widehat{\Pol}_{\pi}(A,n)^{[n+2]})$. In particular, its homotopy groups are given by $$\pi_iT_{\pi}\cP(A,n)= \H^{n+2-i}(F^2 \widehat{\Pol}(A,n), \delta +[\pi,-]).$$ The corresponding statements for $T_{\pi}\cP(A,n)/F^{i+2}$ also hold. Now observe that the map $$\begin{aligned} \sigma \co F^2 \widehat{\Pol}(A,n)^{[n+1]} &\to F^2 \widehat{\Pol}(A,n)^{[n+1]}\ten_{\Q} \Q[\eps]\\ \sum_{i \ge 2} \alpha_i &\mapsto \sum_{i \ge 2} (\alpha_i+ (i-1)\alpha_i\eps)\end{aligned}$$ is a morphism of filtered DGLAs, for $\alpha_i \co \CoS_A^i(\Omega^1_{A/R}[n+1]) \to A$. This can be seen either by direct calculation or by observing that $\sigma$ is the differential of the $\bG_m$-action on $\Pol$. \[sigmadef\] Define the canonical tangent vector $\sigma \co \cP(A,n) \to T\cP(A,n)$ on the space of $n$-shifted Poisson structures by applying $\mmc$ to the morphism $\sigma$ of DGLAs. Explicitly, this sends $\pi= \sum \pi_i $ to $\sigma(\pi)=\sum_{i \ge 2} (i-1)\pi_i \in T_{\pi}\cP(A,n)$, which thus has the property that $\delta \sigma(\pi) +[\pi, \sigma(\pi)]=0$. The map $\sigma$ preserves the cofiltration in the sense that it comes from a system of maps $\sigma \co \cP(A,n)/F^{i+2} \to T\cP(A,n)/F^{i+2} $. Shifted pre-symplectic structures {#prespsn} --------------------------------- \[DRdef\] Define the de Rham complex $\DR(A/R)$ to be the product total cochain complex of the double complex $$A \xra{d} \Omega^1_{A/R} \xra{d} \Omega^2_{A/R}\xra{d} \ldots,$$ so the total differential is $d \pm \delta$. We define the Hodge filtration $F$ on $\DR(A/R)$ by setting $F^p\DR(A/R) \subset \DR(A/R)$ to consist of terms $\Omega^i_{A/R}$ with $i \ge p$. Properties of the product total complex ensure that a map $f \co A \to B$ induces a quasi-isomorphism $\DR(A/R) \to \DR(B/R)$ whenever the maps $\Omega^p_{A/R} \to \Omega^p_{B/R} $ are quasi-isomorphisms, which will happen whenever $f$ is a weak equivalence between cofibrant $R$-algebras. The complex $\DR(A/R)$ has the natural structure of a commutative DG algebra over $R$, filtered in the sense that $F^iF^j \subset F^{i+j}$. \[presymplecticdef\] Define an $n$-shifted pre-symplectic structure $\omega$ on $A/R$ to be an element $$\omega \in \z^{n+2}F^2\DR(A/R).$$ Explicitly, this means that $\omega$ is given by an infinite sum $\omega = \sum_{i \ge 2} \omega_i$, with $\omega_i \in (\Omega^i_{A/R})_{i-n-2}$ and $d\omega_i = \delta \omega_{i+1}$. Define an $n$-shifted symplectic structure $\omega$ on $A/R$ to be an $n$-shifted pre-symplectic structure $\omega$ for which the component $\omega_2 \in \z^n\Omega^2_{A/R}$ induces a quasi-isomorphism $$\omega_2^{\sharp} \co \Hom_A(\Omega^1_{A/R},A) \to (\Omega^1_{A/R})_{[-n]}.$$ and $\Omega^1_{A/R} $ is perfect as an $A$-module. Now, we can regard $F^2\DR(A/R)^{[n+1]}$ as a filtered DGLA with trivial bracket. This has the property that $\mc( F^2\DR(A/R)^{[n+1]})= \z^{n+2}F^2\DR(A/R)$. We therefore make the following definition: \[PreSpdef\] Define the space of $n$-shifted pre-symplectic structures on $A/R$ to be the simplicial set $$\PreSp(A,n):= \mmc( \{F^2\DR(A/R)^{[n+1]}/F^{i+2}\}_i)$$ Also write $\PreSp(A,n)/F^{i+2}:= \mmc(F^2 \DR(A/R)^{[n+1]}/F^{i+2})$, so $ \PreSp(A,n)= \Lim_i \PreSp(A,n)/F^{i+2}$. Set $\Sp(A,n) \subset \PreSp(A,n)$ to consist of the symplectic structures — this is a union of path-components. Note that $\PreSp(A,n)/F^{i+2}$ is canonically weakly equivalent to the Dold–Kan denormalisation of the complex $\tau^{\le 0}(F^2\DR(A)^{[n+2]}/F^{i+2})$ (and similarly for the limit $ \PreSp(A,n)$), but the description in terms of $\mmc$ will simplify comparisons. Compatible pairs {#compsn} ---------------- We will now develop the notion of compatibility between a pre-symplectic structure and a Poisson structure. Analogous results for the unshifted, underived $\C^{\infty}$ context can be found in [@KosmannSchwarzbachMagriPN Proposition 6.4], and for the $\Z/2$-graded $\C^{\infty}$ context in [@KhudaverdianVoronov]. \[mudef\] Given $\pi \in (F^2\widehat{\Pol}(A,n)/F^{p})^{n+2}$, define $$\mu(-,\pi) \co \DR(A/R)/F^{p} \to \widehat{\Pol}(A,n)/F^{p}$$ to be the morphism of graded $A$-algebras given on generators $ \Omega^1_{A/R}$ by $$\mu(a df, \pi):= \pi \lrcorner (a df)= a[\pi,f].$$ Given $b \in (F^2\widehat{\Pol}(A,n)/F^{p})$, we then define $$\nu(-, \pi, b) \co \DR(A/R)/F^{p} \to \widehat{\Pol}(A,n)/F^{p}$$ by setting $\mu(\omega,\pi +b\eps)= \mu(\omega,\pi)+ \nu(-, \pi, b)\eps$ for $\eps^2=0$. More explicitly, $\nu(-, \pi, b)$ is the $A$-linear derivation with respect to the ring homomorphism $ \mu(-,\pi) $ given on generators $ \Omega^1_{A/R}$ by $$\nu(a df, \pi, b):= b \lrcorner (a df)= a[b,f].$$ To see that these are well-defined in the sense that they descend to the quotients by $F^{p}$, observe that because $\pi \in F^2$, contraction has the property that $$\mu(\Omega^1, \pi) \subset F^1, \quad \nu(\Omega^1, \pi, b) \subset F^1,$$ it follows that $\mu(F^p, \pi)\subset F^p$, $\nu(F^p,\pi,b) \subset F^p$ by multiplicativity. Explicitly, when $\phi = a df_1 \wedge \ldots \wedge d f_p$, the operations are given by $$\begin{aligned} \mu(\phi, \pi) &=a[\pi,f_1]\ldots [\pi,f_p,],\\ \nu(\phi, \pi, b)&= \sum_i \pm a[\pi,f_1]\ldots [b,f_i] \ldots [\pi,f_p].\end{aligned}$$ \[keylemma\] For $\omega \in \DR(A)/F^p$ and $\pi \in F^2\widehat{\Pol}(A,n)^{n+2}/F^p$, we have $$\begin{aligned} [\pi,\mu(\omega, \pi)] = \mu(d\omega, \pi) + \half \nu(\omega, \pi, [\pi,\pi]),\\ \delta_{\pi}\mu(\omega, \pi) = \mu(D\omega, \pi) + \nu(\omega, \pi,\kappa(\pi)),\end{aligned}$$ where $\delta_{\pi}= [\delta + \pi,-]$ is the differential on $T_{\pi}\cP(A,n)/F^{p}$, with $D= d \pm \delta$ the total differential on $(F^2\DR(A)/F^p) $ and $ \kappa(\pi)=[\delta, \pi] + \half [\pi,\pi]$. For fixed $\pi$ and varying $\omega$, all expressions are derivations with respect to $\mu(-, \pi)$, so it suffices to check this expression in the cases $p=0$ and $p=1$, $\omega = df$. In these cases, we have $$\begin{aligned} [\pi, \mu(a,\pi)] = [\pi, a] &= \mu(da, \pi),\\ \delta_{\pi}\mu(a, \pi) &= \mu(D a, \pi),\\ [\pi,\mu(df, \pi)]= [\pi, [\pi,f]]&= \half \nu(df, \pi, [\pi,\pi]),\\ \delta_{\pi}\mu(df, \pi)= \delta_{\pi}[\pi,f]&= \nu(df, \pi, [\delta, \pi] + \half [\pi,\pi]).\end{aligned}$$ Because $\nu(a, \pi, [\pi,\pi])=0 $ ($\nu$ being $A$-linear) and $ddf=0$, this gives the required results. In particular, this implies that when $\pi$ is Poisson, $\mu(-,\pi)$ defines a map from de Rham cohomology to Poisson cohomology. \[mulemma\] There are maps $$(\pr_2 + \mu\eps) \co \PreSp(A,n)/F^{p} \by \cP(A,n)/F^{p} \to T\cP(A,n)/F^{p}$$ over $\cP(A,n)/F^{p}$ for all $p$, compatible with each other. In particular, we have $$(\pr_2 + \mu\eps)\co \PreSp(A,n) \by \cP(A,n) \to T\cP(A,n).$$ For $\omega \in \PreSp(A,n)_0$, $\pi \in \cP(A,n)_0$, Lemma \[keylemma\] shows that $\mu(\omega, \pi) \in T_{\pi}\cP(A,n)/F^{p}$. Replacing $A,R$ with $A\ten \Omega^{\bt}(\Delta^m), R\ten \Omega^{\bt}(\Delta^m)$ then shows that the statement also holds on the $m$th level of the simplicial set. An alternative approach to proving Lemma \[mulemma\] is to observe that $\mu$ defines a filtered $L_{\infty}$-morphism $$F^2 \DR(A/R)^{[n+1]} \by F^2 \widehat{\Pol}(A,n)^{[n+1]} \xra{\pr_2 + \mu \eps} F^2 \widehat{\Pol}(A,n)^{[n+1]}[\eps]$$ (with respect to the filtration $F$), and then to apply the functor $\mmc$. \[compatex1\] If $\omega$ and $\pi$ are pre-symplectic and Poisson, with $\omega_i=0$ for $i >2$ and $\pi_i=0$ for $i>2$, then observe that $ \mu(\omega, \pi) $ induces the morphism $$\mu(\omega, \pi)^{\sharp} \co (\Omega^1_{A/R})_{[-n]}\to \Hom_A(\Omega^1_{A/R},A)$$ given by $$\mu(\omega, \pi)^{\sharp} = \pi^{\sharp} \circ \omega^{\sharp} \circ \pi^{\sharp}.$$ We say that an $n$-shifted pre-symplectic structure $\omega$ and an $n$-Poisson structure $\pi$ are compatible (or a compatible pair) if $$[\mu(\omega, \pi)] = [\sigma(\pi)] \in \H^{n+2}(F^2\widehat{\Pol}_{\pi}(A,n)) =\pi_0T_{\pi}\cP(A,n),$$ where $\sigma$ is the canonical tangent vector of Definition \[sigmadef\]. \[compatex2\] Following Example \[compatex1\], if $\omega_i=0$ for $i >2$ and $\pi_i=0$ for $i>2$, then $(\omega, \pi)$ are a compatible pair if and only if the map $$\pi^{\sharp} \circ \omega^{\sharp} \circ \pi^{\sharp} \co (\Omega^1_{A/R})_{[-n]}\to \Hom_A(\Omega^1_{A/R},A)$$ is homotopic to $\pi^{\sharp}$, because $\sigma(\pi)=\pi$ in this case. In particular, if $\pi$ is non-degenerate, this means that $\omega$ and $\pi$ determine each other up to homotopy. \[compatnondeg\] If $(\omega, \pi)$ is a compatible pair and $\pi$ is non-degenerate, then $\omega$ is symplectic. Even when the vanishing conditions of Example \[compatex1\] are not satisfied, we still have $$\pi^{\sharp}_2 \circ \omega^{\sharp}_2 \circ \pi^{\sharp}_2 \simeq \pi^{\sharp}_2,$$ so if $\pi^{\sharp}_2$ is a quasi-isomorphism, then $\omega^{\sharp}_2$ must be its homotopy inverse. \[vanishingdef\] Given a simplicial set $Z$, an abelian group object $A$ in simplicial sets over $Z$, and a section $s \co Z \to A$, define the homotopy vanishing locus of $s$ to be the homotopy limit of the diagram $$\xymatrix@1{ Z \ar@<0.5ex>[r]^-{s} \ar@<-0.5ex>[r]_-{0} & A \ar[r] & Z}.$$ We can write this as a homotopy fibre product $Z \by_{(s,0), A \by^h_Z A}^hA$, for the diagonal map $A \to A \by^h_Z A$. When $A$ is a trivial bundle $A = Z \by V$, for $V$ a simplicial abelian group, note that the homotopy vanishing locus is just the homotopy fibre of $s \co Z \to V$ over $0$. \[compdef\] Define the space $\Comp(A,n)$ of compatible $n$-shifted pairs to be the homotopy vanishing locus of $$\mu - \sigma \co \PreSp(A,n) \by \cP(A,n) \to \pr_2^*T\cP(A,n) =\PreSp(A,n) \by T\cP(A,n).$$ We define a cofiltration on this space by setting $ \Comp(A,n)/F^{p}$ to be the homotopy vanishing locus of $$\mu - \sigma \co \PreSp(A,n)/F^{p} \by \cP(A,n)/F^{p} \to \pr_2^*T\cP(A,n)/F^{p}.$$ We can rewrite $\Comp(A,n)$ as the homotopy limit of the diagram $$\xymatrix@1{ \PreSp(A,n) \by \cP(A,n) \ar@<0.5ex>[r]^-{(\pr_2 + \mu\eps)} \ar@<-0.5ex>[r]_-{ \pr_2 +\sigma\pr_2\eps} & T\cP(A,n) \ar[r] & \cP(A,n) }$$ of simplicial sets. In particular, an object of this space is given by a pre-symplectic structure $\omega$, a Poisson structure $\pi$, and a homotopy $h$ between $\mu(\omega,\pi)$ and $\sigma(\pi)$ in $T_{\pi}\cP(A,n)$. Define $\Comp(A,n)^{\nondeg} \subset \Comp(A,n)$ to consist of compatible pairs with $\pi$ non-degenerate. This is a union of path-components, and by Lemma \[compatnondeg\] has a natural projection $$\Comp(A,n)^{\nondeg}\to \Sp(A,n)$$ as well as the canonical map $$\Comp(A,n)^{\nondeg} \to\cP(A,n)^{\nondeg}.$$ \[compatP1\] The canonical map $$\begin{aligned} \Comp(A,n)^{\nondeg} \to \cP(A,n)^{\nondeg} \end{aligned}$$ is a weak equivalence. For any $\pi \in \cP(A,n)$, the homotopy fibre of $\Comp(A,n)^{\nondeg} $ over $\pi$ is just the homotopy fibre of $$\mu(-,\pi) \co \PreSp(A,n) \to T_{\pi}\cP(A,n)$$ over $\sigma(\pi)$. The map $\mu(-,\pi) \co \DR(A/R) \to (\widehat{\Pol}(A,n), \delta_{\pi})$ is a morphism of complete filtered CDGAs by Lemma \[keylemma\], and non-degeneracy of $\pi_2$ implies that we have a quasi-isomorphism on the associated gradeds $\gr_F$. We therefore have a quasi-isomorphism of filtered complexes, so we have isomorphisms on homotopy groups: $$\begin{aligned} \pi_j\PreSp(A,n) &\to& \pi_jT_{\pi}\cP(A,n)\\ \H^{n+2-j}(F^2 \DR(A/R)) &\to& \H^{n+2-j}(F^2\widehat{\Pol}(A,n), \delta_{\pi}).\end{aligned}$$ For an earlier analogue of this result, see [@KhudaverdianVoronov], which takes $\delta=0$ and works in the $\Z/2$-graded (rather than $\Z$-graded) context, allowing $\pi=\sum_{i\ge 0} \pi_i$ to have constant and linear terms. It that setting, they show that for $\pi$ non-degenerate, there is a unique solution $\omega$ of the equation $\mu(\omega, \pi) = \sigma(\pi)$, given by Legendre transformations. The obstruction tower {#towersn} ---------------------- We will now show that the tower $\Comp(A,n) \to \ldots \to \Comp(A,n)/F^{i+2} \to \ldots \to \Comp(A,n)/F^2 $ does not contain nearly as much information as first appears. ### Small extensions and obstructions Say that a surjection $L \to M$ of DGLAs is small if the kernel $I$ satisfies $[I,L]=0$. \[obslemma\] Given a small extension $e \co L \to M$ of DGLAs with kernel $I$, there is a sequence $$%0 \to \z^1(I) \lcirclearrowright \mc(L) \to \mc(M) \xra{o_e} \H^2(I) \pi_0\mmc(L) \xra{e} \pi_0\mmc(M) \xra{o_e} \H^2(I)$$ of sets, exact in the sense that $o_e^{-1}(0)$ is the image of $e$. This is well-known. The obstruction map $o_e $ is given by $$o_e(\omega):= d_L\tilde{\omega} + \half [\tilde{\omega},\tilde{\omega}]% [\sum_{n \ge 1} [\tilde{\omega}, \ldots, \tilde{\omega}]_n/n!],$$ for any lift $\tilde{\omega} \in L^1$ of $\omega \in \mc(M)$. \[obsDGLA\] For any small extension $e \co L \to M$ of DGLAs with kernel $I$, there is an obstruction map $\ob_e\co \mmc(M) \to \mmc(I^{[1]})$ in the homotopy category of simplicial sets, with homotopy fibre $\mmc(L)$. This is essentially the same as [@ddt1 Theorem \[ddt1-robs\]]. Set $M'$ to be the mapping cone of $I \to L$, with DGLA structure given by setting $[M',I^{[1]}]=0$. Then we have a surjection $M' \to M \oplus I^{[1]}$ of DGLAs with kernel $I$. Since the map $M' \to M$ is a small extension with acyclic kernel, it follows that $\mmc(M') \to \mmc(M)$ is a trivial fibration, hence a weak equivalence. The obstruction map is then given by $\mmc(M') \to \mmc(I^{[1]})$, the homotopy fibre being $\mmc(L)$, as required. \[obsles\] Let $A$ be an abelian group object over a simplicial set $Z$, with homotopy fibre $A_z$ over $z \in Z$. If $s \co Z \to A$ is a section with homotopy vanishing locus $Y$, there is a long exact sequence $$\xymatrix@R=0ex{ \cdots \ar[r] & \pi_i(Y,z) \ar[r]& \pi_i(Z,z) \ar[r]^-s& \pi_i(A_z,0) \ar[r] &\pi_{i-1}(Y,z) \ar[r] &\cdots\\ \cdots \ar[r]& \pi_0Y \ar[r]& \pi_0Z \ar[r]^-s& \pi_0(A_?), }$$ of groups and sets, where the final map sends $z$ to $s(z) \in \pi_0(A_s)$ Since $Y$ is the homotopy fibre product $$Z \by_{(s,0), (A\by^h_ZA)} A,$$ the long exact sequence of homotopy gives $$\ldots \to \pi_i(Y,z) \to \pi_i(Z,z) \by \pi_i(A,z) \to \pi_i(A\by^h_ZA,z)\to \ldots.$$ Since $(A\by^h_ZA)\by^h_A\{z\}= A_z$, this simplifies to $$\ldots \to \pi_i(Y,z) \to \pi_i(Z,z) \to \pi_i(A_z,z).$$ Combining Propositions \[obsDGLA\] and \[obsles\] gives: \[obsDGLAcor\] Given a small extension $e \co L \to M$ of DGLAs with kernel $I$, there is a canonical long exact sequence $$\xymatrix@R=0ex{ \cdots \ar[r]^-{e_*}&\pi_i\mmc(M) \ar[r]^-{o_e}& \H^{2-i}(I) \ar[r] &\pi_{i-1}\mmc(L)\ar[r]^-{e_*}&\cdots\\ \cdots \ar[r]^-{e_*}&\pi_1\mmc(M) \ar[r]^-{o_e}& \H^1(I) \ar[r] &\pi_0\mmc(L) \ar[r]^-{e_*}& \pi_0\mmc(M) \ar[r]^-{o_e}& \H^{2}(I). }$$ ### Towers of obstructions The following is the long exact sequence of cohomology: \[DRobs\] For each $p$, there is a canonical long exact sequence $$\begin{aligned} \ldots\to \H_{p+i-n-2}(\Omega^{p}_{A/R}) \to \pi_i(\PreSp/F^{p+1}) \to \pi_i(\PreSp/F^{p}) \to \H_{p+i-n-3}(\Omega^{p}_{A/R}) \to\ldots \end{aligned}$$ of homotopy groups, where $\PreSp=\PreSp(A,n)$. \[Mdef\] Given a compatible pair $(\omega, \pi) \in \Comp(A,n)/F^3$ and $p \ge 0$, define the cochain complex $ M(\omega,\pi,p) $ to be the cocone of the map $$\begin{aligned} &(\Omega^{p}_{A/R})^{[n-p+1]} \oplus \HHom_A(\CoS_A^{p}((\Omega^1_{A/R})_{[-n-1]}),A)^{[n+1]}\\ &\to \HHom_A(\CoS_A^{p}((\Omega^1_{A/R})_{[-n-1]}),A)^{[n+1]}\end{aligned}$$ given by combining $$\Symm^{p}(\pi^{\sharp}) \co (\Omega^{p}_{A/R})^{[-p]}\to \HHom_A(\CoS_A^{p}((\Omega^1_{A/R})_{[-n-1]}),A)$$ with the map $$\nu(\omega, \pi) - (p-1) \co \HHom_A(\CoS_A^{p}((\Omega^1_{A/R})_{[-n-1]}),A) \to \HHom_A(\CoS_A^{p}((\Omega^1_{A/R})_{[-n-1]}),A),$$ where $$\nu(\omega, \pi)(b):= \nu(\omega, \pi, b).$$ Because $\omega$ and $\pi$ lie in $\gr_F^2$, the description of Definition \[mudef\] simplifies to give $$\nu(a df_1\wedge df_2, \pi)(b)= \pm a[\pi,f_1] [b,f_2] \pm a[b,f_1] [\pi,f_2].$$ \[nondegtangent\] If $\pi$ is non-degenerate, then the projection $$M(\omega,\pi,p) \to (\Omega^{p}_{A/R})^{[n-p+1]}$$ is a quasi-isomorphism. On $\HHom_A((\Omega^1_{A/R})_{[-n-1]},A)$ (the case $p=1$), observe that the map $\nu(\omega, \pi)$ is just given by $\pi^{\sharp} \circ \omega^{\sharp}$. Moreover, contraction with a $1$-form defines a derivation with respect to the commutative multiplication on polyvectors, so $\nu(\omega, \pi)$ is the derivation $$\nu(\omega, \pi)\co \widehat{\Pol}(A,n) \to \widehat{\Pol}(A,n)$$ given on generators by $\pi^{\sharp} \circ \omega^{\sharp}$. If $\pi$ is non-degenerate, then (by compatibility) $\pi^{\sharp} \circ \omega^{\sharp}$ is homotopic to the identity map, and thus $ \gr_F^p\nu(\omega, \pi)$ is homotopic to $p$. In particular, $\gr_F^p\nu(\omega, \pi) - (p-1) $ is a quasi-isomorphism in this case, so the projection $ M(\omega,\pi,p) \to (\Omega^{p}_{A/R})^{[n-p+1]} $ is a quasi-isomorphism. If we apply Corollary \[obsDGLAcor\] to the small extensions $\gr^p_FL \to L/F^{p+1}\to L/F^p$ for the DGLA of polyvectors, and taking the long exact sequence of homotopy groups, we get: \[compatobs\] For each $p \ge 3$, there is a canonical long exact sequence $$\xymatrix@C=1ex{ \vdots \ar[d]^-{e_*} && \vdots \ar[d]^-{e_*} &&\\ \pi_i(\Comp(A,n)/F^{p}) \ar[d]^-{o_e}&&\pi_1 (\Comp(A,n)/F^{p}) \ar[d]^-{o_e}&& \pi_0(\Comp(A,n)/F^{p}) \ar[d]^-{o_e} \\ \H^{2-i}M(\omega_2,\pi_2,p) \ar[d]&& \H^1 M(\omega_2,\pi_2,p) \ar[d] && \H^2 M(\omega_2,\pi_2,p) \\ \pi_{i-1}(\Comp(A,n)/F^{p+1}) \ar[d]^-{e_*}&&\pi_0(\Comp(A,n)/F^{p+1}) %\ar@(d,u)[uurr]|{e_*} \ar `d[dr] `r[r]^{e_*} `[uuur] `r[uuurr] [uurr] && \\ \vdots \ar@{-->} `r[ur] `u[uuuurr] [uuuurr] &&&& }$$ of homotopy groups and sets, where $\pi_i$ indicates the homotopy group at basepoint $(\omega, \pi)$, and the target of the final map is understood to mean $${o_e}(\omega, \pi) \in \H^2 M(\omega_2,\pi_2,p).$$ Proposition \[obsDGLA\] gives fibration sequences $$\begin{aligned} \cP(A,n)/F^{p+1} \to &\cP(A,n)/F^{p} \to \mmc (\gr_F^{p}\widehat{\Pol}(A,n)^{[n+2]} )\\ T\cP(A,n)/F^{p+1} \to &T\cP(A,n)/F^{p} \to \mmc (\gr_F^{p}\widehat{\Pol}(A,n)^{[n+2]}[\eps] )\\ \PreSp(A,n)/F^{p+1} \to &\PreSp(A,n)/F^{p} \to \mmc ((\Omega^{p}_{A/R})^{[n+2-p]}),\end{aligned}$$ for $\eps^2=0$. We can regard these as homotopy vanishing loci for sections of trivial bundles. Combined with the description of $\Comp(A,n)$ as a homotopy vanishing locus, this gives $\Comp(A,n)/F^{p+1}$ as a homotopy vanishing locus on $ \Comp(A,n)/F^{p}$. In more detail, we pull the sequences above back along $$(i_1, i_2) \co \Comp(A,n)/F^{p} \to \PreSp(A,n)/F^{p} \by \cP(A,n)/F^{p},$$ giving a morphism $$\begin{aligned} \Comp(A,n)/F^{p}\by \mc (\gr_F^{p}\widehat{\Pol}(A,n)^{[n+2]} \oplus (\Omega^{p}_{A/R})^{[n+2-p]})\\ \xra{\gr^p_F(\mu-\sigma)} \Comp(A,n)/F^{p}\by \mc(\gr_F^{p}\widehat{\Pol}(A,n)^{[n+2]} )\end{aligned}$$ of trivial bundles on $ \Comp(A,n)/F^{p}$. If we write $N$ for the homotopy kernel of this map, we obtain a bundle over $\Comp(A,n)/F^{p}$ equipped with a section $s$ whose homotopy vanishing locus is $\Comp(A,n)/F^{p+1} $. It therefore remains to describe the tangents of the maps $\mu, \sigma$. Since $\sigma$ is linear on $\gr_F^{p}$, it is its own tangent, equal to $(p-1)$. To calculate the tangent of $\mu$, take $e$ with $e^2=0$, $a \in F^{p}\DR(A/R)$ and $b \in F^{p}\widehat{\Pol}(A,n)$. Then $$\begin{aligned} \mu(\omega+ ae, \pi+be) - \mu(\omega, \pi) &= \mu(ae, \pi+be)+\mu(\omega, \pi+be) - \mu(\omega, \pi)\\ &=\mu(a,\pi)e + \nu(\omega,\pi,b)e, \end{aligned}$$ because $e^2=0$. Since terms in $F^{p+1}$ vanish, and $\mu$ preserves the filtration $F^{*+2}$, the only contributions left are $$\mu(a,\pi_2)e + \nu(\omega_2,\pi_2,b)e.$$ At a point $(\omega, \pi)\in \Comp(A,n)/F^{p}$, the homotopy fibre $N_{\omega, \pi}$ of the bundle $N$ above is therefore just $\mmc(M(\omega_2,\pi_2,p)^{[1]})$. Then $\Comp(A,n)/F^{p+1}$ is the homotopy vanishing locus of the obstruction map (which takes values in $N$), giving the long exact sequence of homotopy groups from Proposition \[obsles\]. The equivalence --------------- \[compatcor1\] The canonical map $$\begin{aligned} \Comp(A,n)^{\nondeg} \to (\Comp(A,n)/F^3)^{\nondeg} \by^h_{(\PreSp(A,n)/F^3)} \PreSp(A,n) \end{aligned}$$ is a weak equivalence. Lemma \[nondegtangent\] shows that for non-degenerate $\omega_2$, the map $$\begin{aligned} M(\omega_2,\pi_2,p)\to(\Omega^{p}_{A/R})^{[n-p+1]}\end{aligned}$$ is a quasi-isomorphism. Propositions \[DRobs\] and \[compatobs\] thus combine to show that the maps $$\begin{aligned} \Comp(A,n)^{\nondeg}/F^{p+1} \to (\Comp(A,n)^{\nondeg}/F^{p})\by^h_{(\PreSp(A,n)/F^{p})}(\PreSp(A,n)/F^{p+1})\end{aligned}$$ are weak equivalences for all $p \ge 3$. We then just take the limit over all $p$. \[level0prop\] The canonical map $$\begin{aligned} \Comp(A,n)^{\nondeg}/F^3 &\to& \Sp(A,n)/F^3 \end{aligned}$$ is a weak equivalence. We may apply Proposition \[obsles\] to the definition of $\Comp$ as a homotopy vanishing locus, giving a long exact sequence $$\xymatrix@C=1ex { \vdots \ar[d] && %\vdots \ar[d] \\ \pi_i(\Comp(A,n)/F^3 ,(\omega, \pi)) \ar[d] && \pi_0(\Comp(A,n)/F^3) \ar[d]\\ \pi_i(\PreSp(A,n)/F^3, \omega) \by \pi_i( \cP(A,n)/F^3, \pi) \ar[d]^{\mu-\sigma} && \pi_0(\PreSp(A,n)/F^3) \by \pi_0(\cP(A,n)/F^3) \ar[d]^{\mu-\sigma}\\ \pi_i(T\cP(A,n)/F^3 ,0) %\ar[d] \ar@{-->} `d[dr] `r[r] `[uuur] `r[uuurr] [uurr] && \pi_0(T_{\omega, \pi}\cP(A,n)/F^3) \\ %\vdots %&&\ar@(d,u)[ruuuu] %\ar@{-->} `r[ur] `u[uuuurr] [uuuurr] && }$$ of groups and sets. We just have $$\begin{aligned} \pi_i(\PreSp(A,n)/F^3, \omega)&=& \H_{i-n}(\Omega^{2}_{A/R})\\ \pi_i(\cP(A,n)/F^3)&=& \H^{n+2-i}(\gr_F^2\widehat{\Pol}(A,n))\\ \pi_i(T_{\omega,\pi} \cP(A,n)/F^3)&=& \H^{n+2-i}(\gr_F^2\widehat{\Pol}(A,n)),\end{aligned}$$ with the map $\mu -\sigma$ given as in Definition \[Mdef\] by combining $ \Symm^{2}(\pi^{\sharp}) \co (\Omega^{2}_{A/R})_{[2]}\to \HHom_A(\CoS_A^{2}((\Omega^1_{A/R})_{[-n-1]}),A)$ with the map $\nu(\omega, \pi) - 1$. As in Lemma \[nondegtangent\], when $\pi$ is non-degenerate, the map $\gr_F^2\nu(\omega, \pi) - 1 $ is a quasi-isomorphism, inducing isomorphisms $$\begin{aligned} \pi_i(\Comp(A,n)/F^3 ,(\omega, \pi)) \cong \pi_i(\PreSp(A,n)/F^3, \omega)\end{aligned}$$ for all $i>0$. For $i=0$, the argument of Example \[compatex2\] works equally well modulo $F^3$; combined with the exact sequence above, it shows that the locus $$\begin{aligned} \pi_0(\Comp(A,n)/F^3)^{\nondeg} \into \pi_0(\PreSp(A,n)/F^3)= \H^n(\Omega^2_{A/R})\end{aligned}$$ corresponds to the non-degenerate elements $ \pi_0(\Sp(A,n)/F^3)$. \[compatcor2\] The canonical maps $$\begin{aligned} \Comp(A,n)^{\nondeg} &\to& \Sp(A,n) \\ \Comp(A,n)^{\nondeg} &\to& \cP(A,n)^{\nondeg} \end{aligned}$$ are weak equivalences. For the first map, just combine Corollary \[compatcor1\] with Proposition \[level0prop\]. The second map is given by Proposition \[compatP1\]. Compatible structures on derived DM stacks {#DMsn} ========================================== We will now study symplectic and Poisson structures on derived $N$-stacks. Rather than studying a single CDGA as in Corollary \[compatcor2\], will will establish a similar result for strings of étale morphisms between non-degenerate structures. Following the philosophy of [@stacks2], we can represent a derived Artin or DM $N$-stack $\fX$ as a simplicial derived affine scheme $X_{\bt}$ satisfying various conditions. Thus $X_{\bt}$ is $\Spec O(X)^{\bt}$ for a cosimplicial diagram $O(X)^{\bt}$ of CDGAs. For a derived DM stack, the morphisms in this diagram are étale, and descent results will show that Poisson structures are independent of the choice of resolution $X_{\bt}$. A modified construction will be given for derived Artin stacks in §\[Artinsn\]. Write $dg\CAlg(\Q)$ for the category of chain complexes over $\Q$ with graded-commutative multiplication. We will refer to its objects as chain CDGAs (in contrast with the more common, but equivalent, cochain CDGAs). Given $R \in dg\CAlg(\Q) $, we write $dg\CAlg(R) $ for the category of graded-commutative $R$-algebras in chain complexes. Throughout this section, we fix $R \in dg\CAlg(\Q)$. Compatible structures on diagrams {#DMdiagramsn} --------------------------------- ### Definitions Given a small category $I$, an $I$-diagram $A$ of chain CDGAs over $R$, and $A$-modules $M,N$ in $I$-diagrams of chain complexes, we can define the cochain complex $\HHom_A(M,N)$ to be the equaliser of the obvious diagram $$\prod_{i\in I} \HHom_{A(i)}(M(i),N(i)) \implies \prod_{f\co i \to j \text{ in } I} \HHom_{A(i)}(M(i),f_*N(j)).$$ All the constructions of §\[poisssn\] then adapt immediately; in particular, we can define $$\widehat{\Pol}(A,n):= \HHom_A(\Co\Symm_A((\Omega^1_{A/R})_{[-n-1]}),A),$$ leading to a space $\cP(A,n)$ of Poisson structures. In order to ensure that this has the correct homological properties, we now consider categories of the form $[m]= (0 \to 1 \to \ldots \to m)$. \[calcTOmegalemma\] If $A$ is an $[m]$-diagram in chain CDGAs over $R$ which is cofibrant and fibrant for the injective model structure (i.e. each $A(i)$ is cofibrant and the maps $A(i) \to A(i+1)$ are surjective), then $\HHom_A(\CoS_A^k\Omega^1_A,A)$ is a model for the derived $\Hom$-complex $\oR\HHom_A(\oL\CoS_A^k\oL\Omega^1_A,A)$, and $ \HHom_A(A,\Omega^p_A) \simeq \ho\Lim_i \oL\Omega^p_{A(i)}$. Because $A(i)$ is cofibrant, $\Omega^p_{A(i)}$ is cofibrant as an $A(i)$-module, so the complex $\oR\HHom_A(\oL\CoS_A^k\oL\Omega^1_A,A)$ is the homotopy limit of the diagram $$\HHom_{A(0)}(C(0),A(0))\to \HHom_{A(0)}(C(0),A(1)) \la \HHom_{A(1)}(C(1),A(1)) \to \ldots,% \la \HHom_{A(m)}(C(m),A(m)),$$ where we write $C(i) := \CoS_{A(i)}^k\Omega^1_{A(i)}$. Since the maps $A(i) \to A(i+1)$ are surjective, the maps $\HHom_{A(i)}(\CoS_{A(i)}^k\Omega^1_{A(i)},A(i))\to \HHom_{A(i)}(\Omega^p_{A(i)},A(j))$ are so, and thus the homotopy limit is calculated by the ordinary limit, which is precisely $\HHom_A(\CoS_{A}^k\Omega^1_A,A) $. Because $[m]$ has initial object $0$, $ \HHom_A(A,\Omega^p_A) \cong \HHom_{A(0)}(A(0), \Omega^p_{A(0)})= \Omega^p_{A(0)}$. For the same reason, $ \ho\Lim_i \Omega^p_{A(i)}\simeq \Omega^p_{A(0)}$. For such diagrams, we can also adapt all the constructions of §\[prespsn\] immediately by taking inverse limits, so setting $$\PreSp(A,n):= \PreSp(A(0),n)= \Lim_{i\in [m]} \PreSp(A(i),n)$$ for any $[m]$-diagram $A$ of chain CDGAs over $R$. We then adapt the constructions of §\[compsn\], defining $$\mu \co \PreSp(A,n) \by \cP(A,n) \to T\cP(A,n)$$ by setting $\mu(\omega, \Delta)(i):= \mu(\omega(i), \Delta(i)) \in T\cP(A(i),n)$ for $i \in [m]$, and letting $ \Comp(A,n)$ be the homotopy vanishing locus of $$(\mu - \sigma) \co \PreSp(A,n) \by \cP(A,n) \to \pr_2^*T\cP(A,n).$$ The obstruction functors and their towers from §\[towersn\] also adapt immediately to $[m]$-diagrams, giving the obvious analogues of the obstruction spaces defined in terms of $$\HHom_A(\CoS^p_A((\Omega^1_{A/R})_{[-n-1]}),A) \quad\text{ and }\quad \Omega^p_{A(0)}.$$ ### Functors and descent {#descentsn} \[calcTlemma2\] If $D=(A\to B)$ is a fibrant cofibrant $[1]$-diagram in $dg\CAlg(R)$ which is formally étale in the sense that the map $$\Omega_{A}^1\ten_{A}B \to \Omega_{B}^1$$ is a quasi-isomorphism, then the map $$% \HHom_D(\CoS_D^k((\Omega^1_D)_{[-n-1]}),D) \to \HHom_{A}(\CoS_{A}^k((\Omega^1_{A})_{[-n-1]}),A), \HHom_D(\CoS_D^k\Omega^1_D,D) \to \HHom_{A}(\CoS_{A}^k\Omega^1_{A},A),$$ is a quasi-isomorphism. This follows immediately from the proof of Lemma \[calcTOmegalemma\], using the quasi-isomorphism $$%\HHom_{B}(\CoS_{B}^k((\Omega^1_{B})_{[-n-1]}),B)\to \HHom_{A}(\CoS_{A}^k((\Omega^1_{A})_{[-n-1]}),B). \HHom_{B}(\CoS_{B}^k\Omega^1_{B},B)\to \HHom_{A}(\CoS_{A}^k\Omega^1_{A},B).$$ For a similar result, see [@CPTVV Lemma 1.4.13]. Write $dg\CAlg(R)_{c, \onto}\subset dg\CAlg(R) $ for the subcategory with all cofibrant chain CDGAs over $R$ as objects, and only surjective morphisms. We already have functors $\PreSp(-,n)$ and $\Sp(-,n)$ from $dg\CAlg(R)$ to $s\Set$, the category of simplicial sets, mapping quasi-isomorphisms in $dg\CAlg(R)_{c}$ to weak equivalences. Poisson structures are only functorial with respect to formally étale morphisms, in an $\infty$-functorial sense which we now make precise. Observe that, writing $F$ for any of the constructions $\cP(-,n)$, $\Comp(-,n)$, $\PreSp(-,n)$, and the associated filtered and graded functors, applied to $[m]$-diagrams in $dg\CAlg(R)_{c, \onto}$, we have: \[Fproperties\] 1. the maps from $F(A(0)\to \ldots \to A(m))$ to $$F(A(0)\to A(1))\by_{F(A(1))}^h F(A(1)\to A(2))\by^h_{F(A(2))}\ldots \by_{F(A(n-1))}^hF(A(n-1) \to A(n))$$ are weak equivalences; 2. if the $[1]$-diagram $A \to B$ is a quasi-isomorphism, then the natural maps from $F(A \to B)$ to $F(A)$ and to $F(B)$ are weak equivalences. 3. if the $[1]$-diagram $A \to B$ is formally étale, then the natural map from $F(A \to B)$ to $F(A)$ is a weak equivalence. These properties follow from Lemmas \[calcTOmegalemma\] and \[calcTlemma2\], together with the obstruction calculus of §\[towersn\] extended to diagrams. Property \[Fproperties\].1 ensures that the simplicial classes $\coprod_{ A \in B_m dg\CAlg(R)_{c, \onto}} F(A)$ fit together to give a complete Segal space $\int F$ over the nerve $Bdg\CAlg(R)_{c, \onto} $. Taking complete Segal spaces [@rezk §6] as our preferred model of $\infty$-categories: \[LintFdef\] Define $\oL dg\CAlg(R)_{c, \onto}$, $\oL dg\CAlg(R)$, $\oL\int F$, and $\oL s\Set$ to be the $\infty$-categories obtained by localising the respective categories at quasi-isomorphisms or weak equivalences. Property \[Fproperties\].2 ensures that the homotopy fibre of $\oL\int F \to \oL dg\CAlg(R)_{c, \onto}$ over $A$ is still just the simplicial set $F(A)$, regarded as an $\infty$-groupoid. Since the surjections in $dg\CAlg(R)$ are the fibrations, the inclusion $\oL dg\CAlg(R)_{c, \onto} \to \oL dg\CAlg(R)$ is a weak equivalence, and we may regard $\oL\int F $ as an $\infty$-category over $\oL dgCAlg(R) $. Furthermore, Property \[Fproperties\].3 implies that the $\infty$-functor $\oL\int F \to \oL dg\CAlg(R) $ is a co-Cartesian fibration when we restrict to the subcategory $\oL dg\CAlg(R)^{\et} $ of homotopy formally étale morphisms, giving: \[inftyFdef\] When $F$ is any of the constructions above, we define $$\oR F \co \oL dg\CAlg(R)^{\et} \to \oL s\Set$$ to be the $\infty$-functor whose Grothendieck construction is $\oL\int F $. In particular, the observations above ensure that $$(\oR F)(A) \simeq F(A)$$ for all cofibrant chain CDGAs $A$ over $R$. An immediate consequence of Corollary \[compatcor2\] is that the canonical maps $$\begin{aligned} \oR\Comp(-,n)^{\nondeg} &\to& \oR\Sp(-,n) \\ \oR\Comp(-,n)^{\nondeg} &\to& \oR\cP(-,n)^{\nondeg} \end{aligned}$$ are natural weak equivalences of $\infty$-functors on $\oL dg\CAlg(R)^{\et}$. Derived $N$-hypergroupoids {#hgpdsn} -------------------------- ### Background We now require our chain CDGA $R$ over $\Q$ to be concentrated in non-negative chain degrees, and write $dg_+\CAlg(R)\subset dg\CAlg(R) $ for the full subcategory of chain CDGAs which are concentrated in non-negative chain degrees. We denote the opposite category to $dg_+\CAlg(R) $ by $DG^+\Aff_R$. Write $sDG^+\Aff_R$ for the category of simplicial diagrams in $DG^+\Aff_R $. A morphism in $DG^+\Aff_R $ is said to be a fibration if it is given by a cofibration in the opposite category $dg_+\CAlg(R)$ (which in turn just means that it is a retract of a quasi-free map). As in [@hag2], a morphism $f:A \to B$ in $dg_+\CAlg(R)$ is said to be smooth (resp. étale) if $\H_0(f): \H_0A \to \H_0B$ is smooth, and the maps $\H_n(A)\ten_{\H_0(A)}\H_0(B) \to \H_n(B)$ are isomorphisms for all $n$. The associated map $\Spec B \to \Spec A$ in $ DG^+\Aff_R$ is said to be surjective if $\Spec \H_0B \to \Spec \H_0A$ is so. For $m \ge 0$, the combinatorial $m$-simplex $\Delta^m \in s\Set$ is characterised by the property that $\Hom_{s\Set}(\Delta^m, K) \cong K_m$ for all simplicial sets $K$. Its boundary $\pd\Delta^m \subset \Delta^m$ is given by $\bigcup_{i=0}^m\pd^i(\Delta^{m-1})$ (taken to include the case $\pd\Delta^0=\emptyset$), and for $m \ge 1$ the $k$th horn $\Lambda^{m,k}$ is given by $\bigcup_{\substack{0 \le i \le m\\ i \ne k}}\pd^i(\Delta^{m-1})$. \[mn\] Given a simplicial set $K$ and a simplicial object $X_{\bt}$ in a complete category $\C$, we follow [@sht Proposition VII.1.21] in defining the $K$-matching object in $\C$ by $$M_KX:= \Hom_{s\Set}(K, X).$$ Note that for finite simplicial sets $K$, the matching object $M_KX$ still exists even if $\C$ only contains finite limits. Explicitly, the matching object $M_{\pd \Delta^m}(X)$ is given by the equaliser of a diagram $$\prod_{0\le i \le m} X_{m-1} \implies \prod_{0\le i<j \le m} X_{m-2},$$ while the partial matching object $ M_{\Lambda^m_k} (X)$ is given by the equaliser of a diagram $$\prod_{\substack{0\le i \le m\\i \ne k}} X_{m-1} \implies \prod_{\substack{0\le i<j \le m\\i,j \ne k}} X_{m-2}.$$ We now recall some results from [@stacks2]. Given $Y \in sDG^+\Aff_R$, a DG Artin (resp. DM) $N$-hypergroupoid $X$ over $Y$ is a morphism $X \to Y$ in $sDG^+\Aff_R$ for which: 1. the matching maps $$X_m \to M_{\pd \Delta^m} (X)\by_{M_{\pd \Delta^m} (Y)}Y_m$$ are fibrations for all $m\ge 0$; 2. the partial matching maps $$X_m \to M_{\Lambda^m_k} (X)\by_{M^h_{\Lambda^m_k} (Y)}^hY_m$$ are smooth (resp. étale) surjections for all $m \ge 1$ and $k$, and are weak equivalences for all $m>N$ and all $k$. A morphism $X\to Y$ in $sDG^+\Aff_R$ is a trivial DG Artin (resp. DM) $N$-hypergroupoid if and only if the matching maps $$X_m \to M_{\pd \Delta^m} (X)\by_{M_{\pd \Delta^m} (Y)}Y_m$$ of Definition \[mn\] are surjective smooth (resp. étale) fibrations for all $m$, and are weak equivalences for all $m\ge n$. The following is [@stacks2 Theorem \[stacks2-bigthm\] and Corollary \[stacks2-Dequivcor\]], as spelt out in [@stacksintro Theorem \[stacksintro-dbigthm\]]: \[dbigthm\] The $\infty$-category of strongly quasi-compact $N$-geometric derived Artin (resp. DM) stacks over $R$ is given by localising the category of DG Artin (resp.) $N$-hypergroupoids over $R$ at the class of trivial relative DG Artin (resp. DM) $N$-hypergroupoids. Given a DG Artin (resp. DM) $N$-hypergroupoid $X$, we denote the associated $N$-geometric derived Artin (resp. DM) stack by $X^{\sharp}$. Given $X \in sDG^+\Aff_R$, define $cdg\Mod(X)$ to be the category of $O(X)$-modules in cosimplicial diagrams of chain complexes. We say that a morphism $M \to N$ in $cdg\Mod(X)$ is a weak equivalence if it induces quasi-isomorphisms $M^i \to N^i$ of chain complexes in each cosimplicial level. The following is [@stacks2 Proposition \[stacks2-qcohequiv\] and Remarks \[stacks2-hcartrks\]]: \[qcohequiv\] For a DG Artin $N$-hypergroupoid $X$, the $\infty$-category of quasi-coherent complexes (in the sense of [@lurie §5.2]) on the $n$-geometric derived stack $X^{\sharp}$ is equivalent to the localisation at weak equivalences of the full subcategory $dg\Mod(X)_{\cart}$ of $cdg\Mod(X)$ consisting of modules $M$ which are homotopy-Cartesian in the sense that the maps $$\pd^i\co \oL\pd_i^*M^{m-1} \to M^m$$ are quasi-isomorphisms for all $i$ and $m$. \[ulinedef\] We make $cdg\Mod(X)$ into a simplicial category by setting (for $K \in s\Set$) $$(M^K)^n: = (M^n)^{K_n},$$ as an $O(X)^n$-module in chain complexes. This has a left adjoint $M \mapsto M\ten K$. Given $M \in cdg\Mod(X)$, define $\uline{M} \in (cdg\Mod(X))^{\Delta}$ to be the cosimplicial diagram given in cosimplicial level $n$ by $M\ten \Delta^n$. Combining [@stacks2 Definition \[stacks2-cotdef\] and Corollary \[stacks2-loopcot\]] gives: \[cotdef\] For a DG Artin $N$-hypergroupoid $X$, define the cotangent complex $\bL^{X/S} \in cdg\Mod(X)$ by $\bL^{X/S}:= \Tot N_c^{\le N}\underline{\Omega(X/S)}$, where $N_c$ denotes cosimplicial conormalisation, and $\Tot$ is the direct sum total functor from cochain chain complexes to chain complexes. By [@stacks2 Lemma \[stacks2-Lcart\] and Corollary \[stacks2-cotgood\]], this is homotopy-Cartesian and recovers the usual cotangent complexes in derived algebraic geometry under the correspondence of Proposition \[qcohequiv\]. ### Poisson and symplectic structures Take a DG Deligne–Mumford $N$-hypergroupoid $X$ over $R$. Because the face maps $\pd^i \co \Delta^{m-1} \to \Delta^m$ are weak equivalences, the morphisms $\pd_i\co X_m \to X_{m-1}$ are all étale. Since $\sigma_i \co X_{m-1} \to X_m$ has left inverse $\pd_i$, it follows that $\sigma_i$ is also étale. Therefore $O(X)$ is a functor from $\Delta^{\op}$ to $dg\CAlg(R)^{\et}$. \[inftyFXdef\] For any of the functors $F$ in Definition \[inftyFdef\], write $$F(X):= \ho\Lim_{j \in \Delta} \oR F(O(X_j)).$$ This gives us spaces $\cP(X,n)$, $\PreSp(X,n)$ and $\Comp(X,n)$ of Poisson structures, pre-symplectic structures and compatible pairs. Explicitly, strictification theorems then imply that an element of $ \cP(X,n)$ can be represented by a cosimplicial diagram $P$ of $\hat{P}_{n+1}$-algebras equipped with a weak equivalence from $O(X)$ to the cosimplicial chain CDGA underlying $P$. \[inftyFXwell\] If $Y \to X$ is a trivial DG DM hypergroupoid, then the morphism $$F(X) \to F(Y)$$ is an equivalence for any of the constructions $F= \cP, \Comp, \PreSp$. The morphism $F(X) \to F(Y)$ exists because the maps $Y_j \to X_j$ are all étale. By Propositions \[DRobs\], \[compatobs\] and the corresponding statement for $\cP$, it suffices to prove that the morphisms $$\begin{aligned} \Omega^p_{X_j/R} &\to \Omega^p_{Y_j/R},\\ \HHom_{O(X_j)}(\CoS^p((\Omega^1_{X_j/R})_{[-n-1]}), O(X_j)) &\to \ \HHom_{O(Y_j)}(\CoS^p((\Omega^1_{Y_j/R})_{[-n-1]}), O(Y_j)) \end{aligned}$$ induce quasi-isomorphisms on the homotopy limits over $j \in \Delta$. This follows by faithfully flat descent for quasi-coherent sheaves, since $\fX :=X^{\sharp} \simeq Y^{\sharp}$, the expressions reducing to $\oR \Gamma(\fX, \oL\Omega^p_{\fX/R})$ and $\oR\HHom_{\O_{\fX}}(\CoS^p((\bL_{\fX/R})_{[-n-1]}), \O_{\fX}) $, respectively. Thus the following is well-defined: \[DMFdef\] Given a strongly quasi-compact DG DM $N$-stack $\fX$, define the spaces $\cP(\fX,n)$, $\Comp(\fX,n)$, $\Sp(\fX,n)$ to be the spaces $ \cP(X,n), \Comp(X,n), \Sp(X,n) $ for any DG DM $N$-hypergroupoid $X$ with $X^{\sharp} \simeq \fX$. By the results of §\[descentsn\], an immediate consequence of Corollary \[compatcor2\] is then: \[DMthm\] There are natural weak equivalences $$\Sp(\fX,n) \la \Comp(\fX,n)^{\nondeg}\to \cP(\fX,n)^{\nondeg}.$$ Compatible structures on derived Artin stacks {#Artinsn} ============================================= We now show how to extend the comparisons above to derived Artin $N$-stacks over a fixed chain CDGA $R \in dg_+\CAlg(\Q)$. The main difficulty is in establishing an appropriate notion for shifted Poisson structures in this setting. It is tempting just to consider DG Artin $N$-hypergroupoids equipped with Poisson structures levelwise, with weak equivalences generated by trivial DG Artin $N$-hypergroupoids. With this approach, it is not easy to compute the obstructions arising, and it seems unlikely that they will be related to the cotangent complex in the desired way. By contrast, deformations of a DG Artin hypergroupoid $X$ do recover all deformations of the associated derived Artin stack, at least on the level of objects (cf. [@stacks2 Theorem \[stacks2-deformstack\]]). The key in that case was the comparison between $\Omega^1(X/R)$ and the cotangent complex $\bL_{X/R}$ in [@stacks2 Lemma \[stacks2-deform2\]], which relies on strong boundedness properties. The shifts involved in the definition of Poisson structures preclude any analogous result for the obstruction tower associated to a semi-strict Poisson structure. Stacky CDGAs ------------ Our solution is to replace Artin $N$-hypergroupoids in CDGAs with DM hypergroupoids in a suitable category of graded-commutative algebras in double complexes, which we refer to as stacky CDGAs. The idea is to globalise one of the intermediate steps [@ddt1 Theorem \[ddt1-dequiv\]] in the proof of the equivalence between formal moduli problems and DGLAs. Given a stacky CDGA $A$, the definition of an $n$-shifted Poisson structure is fairly obvious: it is a Lie bracket of total cochain degree $-n$, or rather an $L_{\infty}$-structure in the form of a sequence $[-]_m$ of $m$-ary operations of cochain degree $1-(n+1)(m-1)$. However, the precise formulation (Definition \[bipoldef\]) is quite subtle, involving lower bounds on the cochain degrees of the operations. When working with stacky CDGAs, we write the double complexes as chain cochain complexes, enabling us to distinguish easily between derived (chain) and stacky (cochain) structures: Define a chain cochain complex $V$ over $\Q$ to be a bigraded $\Q$-vector space $V= \bigoplus_{i,j}V^i_j$, equipped with square-zero linear maps $\pd \co V^i_j \to V^{i+1}_j$ and $\delta \co V^i_j \to V^i_{j-1}$ such that $\pd\delta + \delta \pd =0$. There is an obvious tensor product operation $\ten$ on this category, and a stacky CDGA is then defined to be a chain cochain complex $A$ equipped with a commutative product $A\ten A \to A$ and unit $\Q \to A$. We regard all chain complexes as chain cochain complexes $V= V^0_{\bt}$. Given a chain CDGA $R$, a stacky CDGA over $R$ is then a morphism $R \to A$ of stacky CDGAs. We write $DGdg\CAlg(R)$ for the category of stacky CDGAs over $R$, and $DG^+dg\CAlg(R)$ for the full subcategory consisting of objects $A$ concentrated in non-negative cochain degrees. Say that a morphism $U \to V$ of chain cochain complexes is a levelwise quasi-isomorphism if $U^i \to V^i$ is a quasi-isomorphism for all $i \in \Z$. Say that a morphism of stacky CDGAs is a levelwise quasi-isomorphism if the underlying morphism of chain cochain complexes is so. \[bicdgamodel\] There is a cofibrantly generated model structure on stacky CDGAs over $R$ in which fibrations are surjections and weak equivalences are levelwise quasi-isomorphisms. We first prove the corresponding statement for chain cochain complexes over $\Q$. Let $D^i$ be the chain complex $k_{[-i]} \to k_{[1-i]}$ and $S^i:= k_{[-i]}$, so there is an obvious map $S^i \to D^i$. Write $D_i$ for the cochain complex $k^{[1-i]} \to k^{[-i]}$. We can then take the set $I$ of generating cofibrations to consist of the morphisms $$\{D_i\ten S^j \to D_i\ten D^j\}_{i,j \in \Z},$$ with the set $J$ of generating trivial cofibrations given by $$\{0 \to D_i\ten D^j\}_{i,j \in \Z}.$$ It is straightforward to verify that this satisfies the conditions of [@hovey Theorem 2.1.19]. The forgetful functor from stacky CDGAs to chain cochain complexes has a left adjoint $V \mapsto R\ten \Symm_{\Q}V$, which transfers the model structure by [@Hirschhorn Theorem 11.3.2]. As described for instance in [@ddt1 Definition \[ddt1-nabla\]], there is a denormalisation functor $D$ from non-negatively graded cochain CDGAs to cosimplicial algebras, with left adjoint $D^*$. Given a cosimplicial chain CDGA $A$, $D^*A$ is then a stacky CDGA, with $ (D^*A)^i_j=0$ for $i<0$. The functor $D$ satisfies $(DB)^m \cong \bigoplus_{i=0}^m (B^i)^{\binom{m}{i}}$, with multiplication coming from the shuffle product, so in particular the iterated codegeneracy map $(DB)^m \to B^0$ is always an $m$-nilpotent extension. As a result, the left adjoint $D^*$ factors through the functor sending $A$ to its pro-nilpotent completion over $A^0$. \[Dstarlemma\] The functor $D^*$ is a left Quillen functor from the Reedy model structure on cosimplicial chain CDGAs to the model structure of Lemma \[bicdgamodel\]. The denormalisation functor $D$ on non-negatively graded cochain complexes extends to all cochain complexes by composing with brutal truncation. The right adjoint to $D^*$ is given by applying $D$ to the cochain index of a stacky CDGA $A$. Since fibrations and weak equivalences in the Reedy model structure are levelwise surjections and levelwise quasi-isomorphisms, it follows immediately that $D$ is right Quillen, so $D^*$ is left Quillen. To any DG Artin $N$-hypergroupoid $X$ over $R$, we can then associate the stacky CDGA $D^*O(X)$. This behaves well because DG Artin hypergroupoids are Reedy fibrant, so there is no need to replace $D^*$ with an associated left-derived functor. Because $DA$ is always a nilpotent extension of $A^0$, we cannot recover $X$ from the levelwise quasi-isomorphism class of $D^*O(X)$, but we can recover $X_0$ and the formal completion of $X$ along $X_0$. Thus $D^*O(X)$ will play the same role as the formal affine derived stacks of [@CPTVV]. For any DG Artin $N$-hypergroupoid $X$, the diagram $X^{\Delta^m}$ is another DG Artin $N$-hypergroupoid resolving the same stack, and we will then consider the cosimplicial stacky CDGA $m \mapsto D^*O(X^{\Delta^m})$, which we can think of as a kind of DM $N$-hypergroupoid in stacky CDGAs. \[DstarBG\] If $Y$ is a derived affine scheme equipped with an action of a smooth affine group scheme $G$, then the nerve $X:=B[Y/G]$ is a hypergroupoid resolving the derived Artin $1$-stack $[Y/G]$. Since $B[Y/G]_i = Y \by G^i$, the simplicial derived scheme $X^{\Delta^m}$ is given by $B[Y \by G^m/G^{m+1}]$, with action $$(y,h_1, \ldots,h_m)(g_0, \ldots,g_m)= (y g_0, g_0^{-1}h_1g_1,g_1^{-1}h_2g_2, \ldots, g_{m-1}^{-1}h_mg_m).$$ Taking Reedy fibrant replacement of $X$ gives us a DG Artin hypergroupoid $X'$ and a levelwise quasi-isomorphism $X \to X'$. Since the face maps of $X$ are smooth, the map $D^*O((X')^{\Delta^{\bt}}) \to D^*O(X^{\Delta^{\bt}})$ is a levelwise quasi-isomorphism of cosimplicial stacky CDGAs. Thus Poisson structures on the stack $[Y/G]$ can be defined in terms of polyvectors on the cosimplicial stacky CDGA $D^*O(X^{\Delta^{\bt}}) $ when $Y$ is fibrant. The completion $O(\hat{X})$ of $O(X)$ over $O(X_0)$ is a cosimplicial CDGA with $O(\hat{X})^i= O(Y)\llbracket(\g^{\vee})^{\bigoplus i}\rrbracket$, for $\g$ the Lie algebra of $G$, where we write $A\llbracket M \rrbracket$ for the $(M)$-adic completion of $A\ten_R\Symm_R(M)$. Thus the stacky CDGA $D^*O(X)$ is the Chevalley–Eilenberg double complex $$O([Y/\g]):=( O(Y) \xra{\pd} O(Y)\ten \g^{\vee} \xra{\pd} O(Y)\ten \Lambda^2\g^{\vee}\xra{\pd} \ldots)$$ of $\g$ with coefficients in the chain $\g$-module $O(Y)$. The same calculation for $[Y \by G^m/G^{m+1}]$ shows that the cosimplicial stacky CDGA $D^*O(X^{\Delta^{\bt}})=D^*O((B[Y/G])^{\Delta^{\bt}})$ is given by a diagram $$\xymatrix@1{ O([Y/\g]) \ar@<1ex>[r] \ar@<-1ex>[r] & \ar@{.>}[l] O([Y \by G/\g^{\oplus 2}]) \ar[r] \ar@/^/@<0.5ex>[r] \ar@/_/@<-0.5ex>[r] & \ar@{.>}@<0.75ex>[l] \ar@{.>}@<-0.75ex>[l] O([Y \by G^2/\g^{\oplus 3}]) \ar@/^1pc/[rr] \ar@/_1pc/[rr] \ar@{}[rr]|{\cdot} \ar@{}@<1ex>[rr]|{\cdot} \ar@{}@<-1ex>[rr]|{\cdot} && {\phantom{E}}\cdots .}$$ Beware that maps $(Y',G')\to (Y,G)$ only induce levelwise quasi-isomorphisms $D^*O(B[Y'/G']) \to D^*O(B[Y/G])$ when $Y'$ is weakly equivalent to $Y$ and $\g'$ isomorphic to $\g$ — it is not enough for the Lie algebra cohomology groups to be isomorphic. ### Modules over stacky CDGAs For now, we fix a cofibrant stacky CDGA $A$ over a $\Q$-CDGA $R$. Given a chain cochain complex $V$, define the cochain complex $\hat{\Tot} V \subset \Tot^{\Pi}V$ by $$%(\hat{\Tot} V)^m := (\prod_{i \le 0} V^i_{i-m}) \oplus (\bigoplus_{i>0} V^i_{i-m}), (\hat{\Tot} V)^m := (\bigoplus_{i < 0} V^i_{i-m}) \oplus (\prod_{i\ge 0} V^i_{i-m})$$ with differential $\pd \pm \delta$. An alternative description of $\hat{\Tot} V$ is as the completion of $\Tot V$ with respect to the filtration $ \{\Tot \sigma^{\ge m }V\}_m$, where $\sigma^{\ge m}$ denotes brutal truncation in the cochain direction. In fact, we can write $$\Lim_m \LLim_n \Tot( (\sigma^{\ge - n }V)/(\sigma^{\ge m }V)) = \hat{\Tot} V = \LLim_n \Lim_m\Tot( (\sigma^{\ge -n }V)/(\sigma^{\ge m }V)).$$ The latter description also shows that there is a canonical map $(\hat{\Tot} U)\ten( \hat{\Tot} V) \to \hat{\Tot} (U \ten V)$ — the same is not true of $\Tot^{\Pi}$ in general. Write $DGdg\Mod(A)$ for the category of $A$-modules in chain cochain complexes. When $A \in DG^+dg\CAlg(\Q)$, write $DG^+dg\Mod(A) \subset DGdg\Mod(A) $ for the full subcategory of objects concentrated in non-negative cochain degrees. Given $A$-modules $M,N$ in chain cochain complexes, we define internal $\Hom$s $\cHom_A(M,N)$ by $$\cHom_A(M,N)^i_j= \Hom_{A^{\#}_{\#}}(M^{\#}_{\#},N^{\#[i]}_{\#[j]}),$$ with differentials $\pd f:= \pd_N \circ f \pm f \circ \pd_M$ and $\delta f:= \delta_N \circ f \pm f \circ \delta_M$, where $V^{\#}_{\#}$ denotes the bigraded vector space underlying a chain cochain complex $V$. We then define the $\Hom$ complex $\hat{\HHom}_A(M,N)$ by $$\hat{\HHom}_A(M,N):= \hat{\Tot} \cHom_A(M,N). %\HHom_A(M,N):= \Tot^{\Pi} \tau^{\le 0}\cHom_A(M,N)$$ Observe that there is a multiplication $\hat{\HHom}_A(M,N)\ten \hat{\HHom}_A(N,P)\to \hat{\HHom}_A(M,P)$ — the same is not true for $\Tot^{\Pi} \cHom_A(M,N)$ in general. \[denormmod\] For $A \in DG^+dg\CAlg(\Q)$, the denormalisation functor $D$ defines an equivalence between $DG^+dg\Mod(A)$, and the category $cdg\Mod(DA)$ of $DA$-modules in cosimplicial chain complexes. The inverse functor is just given by the cosimplicial conormalisation functor $N_c$, together with the Alexander–Whitney cup product. For an arbitrary cosimplicial chain CDGA $B$, the cup product on $N_cB$ need not be commutative, but because the shuffle product is left inverse to the cup product, $N_cDA\cong A$ as a chain cochain DGA. We now write $D \co DGdg\Mod(A) \to cdg\Mod(DA)$ for the composition of $D$ with the brutal truncation functor $\sigma^{\ge 0}$. We call a $DA$-module $M$ levelwise cofibrant if each $M^i$ is cofibrant in $dg\Mod(D^iA)$. \[Homrepmod\] For $A \in DG^+dg\CAlg(\Q)$, a levelwise cofibrant $DA$-module $M $ in $cdg\Mod(DA)$, and $P \in DGdg\Mod( A)$, there is a canonical quasi-isomorphism $$\oR \HHom_{DA}(M, DP) \simeq \Tot^{\Pi}\sigma^{\ge 0}\cHom_{A}(N_cM, P).$$ The equivalence of Lemma \[denormmod\] gives us an isomorphism $$\HHom_{DA}(M, DP) \cong \z^0\cHom_{A}(N_cM, P).$$ There is a model structure on $cdg\Mod(DA)$ in which cofibrations and and weak equivalences are defined levelwise in cosimplicial degrees. Similarly, there is a model structure on $DGdg\Mod(A)$ in which weak equivalences are defined levelwise in cochain degrees, and a morphism $M \to N$ is a cofibration if $M^{\#} \to N^{\#}$ has the left lifting property with respect to all surjections of $A^{\#}$-modules in graded chain complexes. These are essentially the injective diagram model structures (somewhat confusingly, the diagrams take values in the projective model structure on chain complexes). The functors $N_c \dashv D$ then form a Quillen adjunction, so to calculate $\oR \HHom_{DA}$ it suffices to take a fibrant replacement for $P$. Write $S$ for the chain cochain complex $S^{\#}_{\#} =\bigoplus_{i\ge 0} (\Q^{[-i]}_{[-i]} \oplus \Q^{[1-i]}_{[-i]})$, with all possible differentials in $S$ being the identity. Thus $S$ looks like a staircase in the second quadrant, with all columns acyclic. A fibrant replacement of $P$ is given by $\cHom_{\Q}(S,P)$, so $$\begin{aligned} \oR \HHom_{DA}(M, DP) &\simeq \HHom_{A}(M, D \cHom_{\Q}(S,P))\\ &\cong \z^0\cHom_A(N_cM, D \cHom_{\Q}(S,P))\\ &\cong \z^0\cHom_{\Q}(S, \cHom_{A}(N_cM, P))\\ &\cong \Tot^{\Pi} \sigma^{\ge 0} \cHom_{A}(N_cM, P).\end{aligned}$$ Given a stacky CDGA $A$, say that an $A$-module $M$ in chain cochain complexes is homotopy-Cartesian if the maps $$A^i\ten^{\oL}_{A^0}M^0 \to M^i$$ are quasi-isomorphisms for all $i$. Comparing DG Artin hypergroupoids and stacky CDGAs -------------------------------------------------- Given a DG Artin $N$-hypergroupoid $X$, and an $O(X)$-module $M$, we may pull $M$ back along the unit $\eta \co O(X) \to DD^*O(X)$ of the adjunction $D^*\dashv D$, and then apply Lemma \[denormmod\] to obtain a $D^*O(X)$-module $N_c\eta^*M$. As for instance in [@stacks2 Definition \[stacks2-delta\*\]], define almost cosimplicial diagrams to be functors on the subcategory $\Delta_*$ of the ordinal number category $\Delta$ containing only those morphisms $f$ with $f(0)=0$; define almost simplicial diagrams dually. Thus an almost simplicial diagram $X_*$ in $\C$ consists of objects $X_n \in \C$, with all of the operations $\pd_i, \sigma_i$ of a simplicial diagram except $\pd_0$, satisfying the usual relations. Given a simplicial (resp. cosimplicial) diagram $X$, we write $X_{\#}$ (resp. $X^{\#}$) for the underlying almost simplicial diagram (resp. almost cosimplicial) diagram. The denormalisation functor $D$ descends to a functor from graded-commutative algebras to almost cosimplicial algebras, with $D^*$ thus descending to a functor in the opposite direction. In other words, $(D^*B)^{\#}$ does not depend on $\pd^0_B$, and $\pd^0_{DA}$ is the only part of the structure on $DA$ to depend on $\pd_A$. From this, it can be seen that for any DG Artin $N$-hypergroupoid $X$, the graded-commutative algebra $\H_0D^*O(X)^{\#}$ is freely generated over $\H_0O(X)^0= \H_0D^*O(X)^0$ by a graded projective module, and that $$\H_i^*O(X)^{\#}\cong \H_0D^*O(X)^{\#}\ten_{\H_0O(X)^0}\H_iO(X)^0.$$ If $M$ is a homotopy-Cartesian $O(X)$-module, the map $$(\eta^*M)^0\ten_{D^0D^*O(X)}D^{\#}D^*O(X) \to (\eta^*M)^{\#}$$ of almost cosimplicial chain complexes is a levelwise quasi-isomorphism, from which it follows that $N_c^0(\eta^*M)\ten_{D^*O(X)^0} D^*O(X)^{\#} \to N_c(\eta^*M)^{\#}$ is a levelwise quasi-isomorphism, so $N_c(\eta^*M)^{\#}$ is also homotopy-Cartesian. ### Equivalences of hypersheaves {#replaceresn} Given a simplicial presheaf $F$ on $DG\Aff(R)$, there is an induced simplicial presheaf $FD$ on $DGdg\Alg(R)$ given by $$(FD)(A):= \ho\Lim_{i\in \Delta} F(D^iA).$$ In the case of a simplicial derived affine $X$, regarded as a functor from $dg\Alg(R)$ to simplicial sets, $XD$ is represented by the cosimplicial chain CDGA $D^*O(X^{\Delta^{\bt}})$, given in cosimplicial level $m$ by $D^*O(X^{\Delta^m})$. Applying $D$ then gives a bisimplicial derived affine $\Spec DD^*O(X^{\Delta^{\bt}})$, and it is natural to consider the diagonal simplicial object $\diag \Spec DD^*O(X^{\Delta^{\bt}})$. \[replaceprop\] For any Reedy fibrant simplicial derived affine $X$, the morphism $$X \to \diag \Spec DD^*O(X^{\Delta^{\bt}}) %\ho \LLim_{i\in \Delta} X_i \to \ho \LLim_{i,j\in \Delta\by \Delta} \Spec D^jD^*O(X^{\Delta^i}),$$ of simplicial derived affines, coming from the isomorphisms $ D^*O(X^{\Delta^m})^0 = O(X_m)$, induces a weak equivalence on the associated simplicial presheaves on $DG\Aff(R)$. We can rewrite the morphism above as $$\ho \LLim_{i\in \Delta} X_i \to \holim_{\substack{\lra \\ i,j\in \Delta\by \Delta}} \Spec D^jD^*O(X^{\Delta^i}).$$ It will therefore suffice to show that for all $j$, the maps $\eta^j \co X \to \Spec D^jD^*O(X^{\Delta^{\bt}})$ give weak equivalences of the associated simplicial presheaves. For this it is enough to show that each $\eta^j$ is a simplicial deformation retract. Now, the bisimplicial derived affine $j \mapsto X^{\Delta^j}$ admits a map from the bisimplicial object $j \mapsto X$, which we denote by $cX$. Forgetting the coface maps $\pd^0 \co \Delta^j \to \Delta^{j+1}$ in $\Delta^{\bt}$ gives us an almost cosimplicial simplicial set $\Delta^{\#}$, and hence an almost simplicial simplicial derived affine $X^{\Delta^{\#}} $. Inclusion of the $0$th vertex makes $\Delta^0$ a deformation retract of $\Delta^j$, with obvious contracting homotopy $\Delta^1 \by \Delta^j \to \Delta^j$. These homotopies combine to give a homotopy $\Delta^1 \by \Delta^{\#} \to \Delta^{\#}$ of almost cosimplicial simplicial sets. This homotopy makes $O(cX_{\#})$ a cosimplicial deformation retract of $O(X^{\Delta^{\#}})$ (as cosimplicial almost cosimplicial rings). Since $DD^*$ descends to a functor on almost cosimplicial algebras, applying $DD^*$ then makes $O(cX_{\#}) $ a cosimplicial deformation retract of $DD^*O(X^{\Delta^{\bt}})^{\#}$, so in particular the maps $\eta^j \co X \to \Spec D^jD^*O(X^{\Delta^{\bt}})$ are all simplicial deformation retracts. If $X$ is a DG Artin $N$-hypergroupoid, with associated $N$-stack $\fX$, this means that we can recover $\fX$ from the cosimplicial stacky CDGA $ D^*O(X^{\Delta^{\bt}})$, just as well as from the cosimplicial CDGA $O(X)$. Proposition \[replaceprop\] has the following immediate consequence: \[gooddescent\] For any simplicial presheaf $F$ on $DG\Aff(R)$ and any Reedy fibrant simplicial derived affine $X$, there is a canonical weak equivalence $$\ho \Lim_{j\in\Delta} \map( \Spec DD^*O(X^{\Delta^j}), F) \to \map (X, F).$$ ### Tangent and cotangent complexes {#Artintgtsn} Now consider a DG Artin $N$-hypergroupoid $X$ over $R$. Combining the adjunction $D^* \dashv D$ with Lemma \[denormmod\] and the universal property of Kähler differentials, observe that there is an isomorphism $$N_c(\Omega^1_{O(X)/R}\ten_{O(X)} DD^*O(X)) \cong \Omega^1_{D^*O(X)/R}.$$ Since $O(X)$ is Reedy cofibrant, Lemma \[Dstarlemma\] ensures that $D^*O(X)$ is cofibrant in the model structure of Lemma \[bicdgamodel\]. If we write $\cone_h$ and $\cone_v$ for cones in the chain and cochain directions respectively, then for any $D^*O(X)$-module $M$, the map $\cone_h\cone_v(M)^{[m-1]}_{[i]} \onto \cone_h(M)^{[m]}_{[i]}$ is levelwise acyclic, so cofibrancy of $D^*O(X)$ implies lifting of the respective spaces of derivations, or equivalently surjectivity of $$\cHom_{D^*O(X)}(\Omega^1_{D^*O(X)}, M)^{m-1}_{i} \onto \z^m\cHom_{D^*O(X)}(\Omega^1_{D^*O(X)}, M)_{i}.$$ In other words, the columns of $\cHom_{D^*O(X)}(\Omega^1_{D^*O(X)}, M) $ are acyclic. Because $X$ is an Artin $N$-hypergroupoid, there are in fact other restrictions on $\Omega^1_{D^*O(X)}$. The results below (or an argument adapted from [@stacks2 Lemma \[stacks2-truncate2\]]) show that for $M \in DG^+dg\Mod({D^*O(X)})$, the rows $ \cHom_{D^*O(X)}(\Omega^1_{D^*O(X)}, M)^i$ are acyclic for $i<-N$. Moreover, the argument of [@ddt1 Corollary \[ddt1-cohowelldfn\]] shows that when $M$ is concentrated in degree $(0,0)$, we have $\H_j\cHom_{D^*O(X)}(\Omega^1_{D^*O(X)}, M)^i=0$ for $i,j \ne 0$, although we will not need this. \[contractlemma\] If, for $A\in DG^+dg\CAlg(R)$, an $A$-module $M \in DG^+dg\Mod(R)$ admits an $A$-linear contracting homotopy, then the map $$%\ho\LLim_i (\Spec D^iA)^{\sharp} \to \ho\LLim_i(\Spec D^i(A \oplus M))^{\sharp} \ho\LLim_{i\in \Delta} \Spec D^iA \to \ho\LLim_{i\in \Delta} \Spec D^i(A \oplus M)$$ of simplicial presheaves on $DG^+\Aff(R)$ is a weak equivalence. A contracting homotopy is the same as a section of the canonical map $\cocone_v(M) \to M$ of $A$-modules, where the vertical cocone $\cocone_v(M)$ is given by $M \ten \cocone_v(\Q)$ for $\cocone_v(\Q)= \Q \oplus \Q^{[-1]}$ with $\pd$ the identity. On applying $D$, we then have a section $s$ of $ D\cocone_v(M) \to DM$ as $DA$-modules in cosimplicial chain complexes. The Alexander–Whitney cup product gives a canonical morphism $D\cocone_v(M) \to (DM)\ten D\cocone_v(\Q)$. Since $\Q^{\Delta^1} = \Q \oplus D\cocone_v(\Q)$, our section $s$ then gives a contracting homotopy $DM \to (DM)^{\Delta^1}$, so evaluation at $0 \in \Delta^1$ is the zero map, and evaluation at $1 \in \Delta^1$ the identity. We therefore have a homotopy $\Delta^1 \by \Spec D(A \oplus M) \to \Spec D(A \oplus M)$ of simplicial derived affine schemes (and hence of simplicial presheaves) realising $\Spec DA$ as a deformation retract of $\Spec D(A \oplus M)$. From now on, we will simply write $(\Spec DA)^{\sharp}:= \ho\LLim_{i\in \Delta} (\Spec D^iA)^{\sharp}$ for the étale hypersheafification of the simplicial derived affine $\Spec DA$. \[suspendcor\] For $A\in DG^+dg\CAlg(R)$, $M \in DG^+dg\Mod(A)$ and $m> 0$, there is a weak equivalence $$(\Spec D(A \oplus M^{[-m]}))^{\sharp}\simeq (\Spec D(A \oplus M) \by S^m)^{\sharp}\cup^{\oL}_{\Spec D(A)^{\sharp} \by S^m} (\Spec DA)^{\sharp}$$ of étale hypersheaves on $DG\Aff(R)$, where $S^m \simeq \Delta^m/\pd \Delta^m$ is the $m$-sphere. We have an exact sequence $0 \to M[-1] \to \cocone_v(M) \to M \to 0$. Since $\cocone_v(M)$ has a contracting homotopy, Lemma \[contractlemma\] shows that $(\Spec D(A \oplus \cocone_v(M))^{\sharp} \simeq (\Spec DA)^{\sharp}$. Thus $ (\Spec D(A \oplus M[-1]))^{\sharp}$ is the homotopy pushout of the nilpotent maps $$(\Spec DA)^{\sharp} \la (\Spec D(A \oplus M))^{\sharp} \to (\Spec DA)^{\sharp},$$ so is the suspension of $(\Spec D(A \oplus M))^{\sharp}$ over $ (\Spec DA)^{\sharp}$. Replacing $M$ with $M[-j]$ for $0 \le j \le m$ then gives the desired result by induction. For any stacky CDGA $A$ over $R$, the module $\Omega^1_{A/R}$ of Kähler differentials is an $A$-module in chain cochain complexes, and we define $\Omega^p_{A/R}:= \Lambda_A^p \Omega^1_{A/R}$, denoting its differentials (inherited from $A$) by $(\pd,\delta)$. There is also a de Rham differential $d \co \Omega^p_{A/R} \to \Omega^{p+1}_{A/R}$. \[tgtcor1\] For a DG Artin $N$-hypergroupoid $X$ over $R$, and any $M \in DG^+dg\Mod(D^*O(X))$, the cotangent complex $\bL^{X/R}$ satisfies $$\begin{aligned} %\oR\HHom_{\O_{\fX}}( \bL^{\fX/R}, DM) \oR\HHom_{O(X)}(\bL^{X/R}, DM) &\simeq \Tot^{\Pi}\sigma^{\ge -N}\cHom_{D^*O(X)}(\Omega^1_{D^*O(X)/R}, M)\\ &\simeq \hat{\HHom}_{D^*O(X)}(\Omega^1_{D^*O(X)/R}, M).\end{aligned}$$ The same is true for any homotopy-Cartesian module $M \in DGdg\Mod(D^*O(X))$. If $\fX= X^{\sharp}$ is the derived $N$-stack associated to $X$, then for $f \co X \to \fX$, Lemma \[suspendcor\] gives an equivalence $$\oR\HHom_{\O_{\fX}}(\oL c^* \bL^{\fX^{S^N}}, \oR f_* DM) \simeq \oR\HHom_{O(X)}(\Omega^1_{X/R}, D(M^{[-N]})),$$ for the map $c \co \fX \to \fX^{S^N}$. Since $\fX$ is an $N$-stack, there is a canonical equivalence $\bL^{\fX} \simeq \bL^{\fX^{S^N}}_{[N]}$ (cf. [@stacks2 Corollary \[stacks2-loopcot\]]), so Lemma \[Homrepmod\] gives $$\oR\HHom_{O(X)}(\bL^{X/R}, DM) \simeq \Tot^{\Pi}\sigma^{\ge -N}\cHom_{D^*O(X)}(\Omega^1_{D^*O(X)/R}, M).$$ Since $X$ is *a fortiori* an $(N+r)$-hypergroupoid for all $r\ge 0$, we may replace $\sigma^{\ge -N}$ with $\sigma^{\ge -N-r}$. Taking the filtered colimit over all $r$ then replaces $\Tot^{\Pi}\sigma^{\ge -N}\cHom$ with $\hat{\Tot}\cHom= \hat{\HHom}$, yielding the second quasi-isomorphism. Finally, observe that for $M$ homotopy-Cartesian, we must have $M^i$ acyclic for $i<0$, because $A^i=0$. Thus we can replace $M$ with $\sigma^{\ge 0}M$, and the results apply. \[tgtcor2\] For a DG Artin $N$-hypergroupoid $X$ over $R$, any $M \in DG^+dg\Mod(D^*O(X))$, and any $r \ge 0$, we have $$\oR\HHom_{O(X)}((\bL^{X/R})^{\ten r}, DM) \simeq \hat{\HHom}_{D^*O(X)}((\Omega^1_{D^*O(X)/R})^{\ten r}, M).$$ There is a natural map $\bL^{X/R} \to \Omega^1_{X/R}$ coming from Definition \[cotdef\]. Corollary \[Homrepmod\] thus ensures that we have a natural map from right to left. The spectral sequence for the cochain brutal truncation filtration for $M$ allows us to reduce to the case where $M=M^i[-i]$ is an $O(X_0)$-module, since $O(X)^0=D^*O(X)^0$. Writing $L$ for the pullback of $\bL^{X/R}$ to $X_0$, and $\bar{\Omega}$ for the pullback of $\Omega^1_{D^*O(X)/R} $ to $O(X_0)$, we therefore wish to show that the map $$\hat{\Tot} \cHom_{O(X_0)}(\bar{\Omega}^{\ten r}, M) \to \oR\HHom_{O(X_0)}(L^{\ten r}, M)$$ is a quasi-isomorphism. Proposition \[tgtcor1\] implies that $\bar{\Omega}$ is levelwise quasi-isomorphic to the brutal cotruncation $\sigma^{\le N}\bar{\Omega}:= \bar{\Omega}/\sigma^{\ge N+1}$, so we may replace $\bar{\Omega}^{\ten r} $ with a bicomplex in cochain degrees $[0, Nr]$. The resulting $\cHom$ bicomplex is bounded below, so $\hat{\Tot}$ is equivalent to $\Tot^{\Pi}$, and we need only show a quasi-isomorphism $$\oR\HHom_{O(X_0)}( (\Tot \sigma^{\le N}\bar{\Omega})^{\ten r}, M) \simeq \oR\HHom_{O(X_0)}(L^{\ten r}, M),$$ which follows because $L \simeq \Tot \sigma^{\le N}\bar{\Omega}$ by Proposition \[tgtcor1\]. (We could also take the limit over all $N$ to give $ L \simeq \Tot^{\Pi}\bar{\Omega}$.) Poisson and symplectic structures {#bipoisssn} --------------------------------- To allow for some flexibility in computations and resolutions, we will not just consider stacky CDGAs of the form $D^*O(X)$ for DG Artin $N$-hypergroupoids $X$, but will fix $A \in DG^+dg\CAlg(R)$ satisfying: 1. for any cofibrant replacement $\tilde{A}\to A$ in the model structure of Lemma \[bicdgamodel\], the morphism $\Omega^1_{\tilde{A}/R}\to \Omega^1_{A/R}$ is a levelwise quasi-isomorphism, 2. the $A^{\#}$-module $(\Omega^1_{A/R})^{\#}$ in graded chain complexes is cofibrant (i.e. it has the left lifting property with respect to all surjections of $A^{\#}$-modules in graded chain complexes), 3. there exists $N$ for which the chain complexes $(\Omega^1_{A/R}\ten_AA^0)^i $ are acyclic for all $i >N$. Observe that these conditions are satisfied by $D^*O(X)$, by the results above, but they are also satisfied by more general stacky CDGAs. The first two conditions do not require $A$ to be cofibrant in the model structure of Lemma \[bicdgamodel\]; it is enough for $A^{\#}$ to be cofibrant as a graded chain CDGA over $R$, or for $A \in DG^+dg_+\CAlg(R)$ with $A^{\#}_{\#}$ ind-smooth as a bigraded commutative algebra over $R_{\#}$. ### Polyvectors All the definitions and properties of §\[affinesn\] now carry over. In particular: \[bipoldef\] Given a stacky CDGA $A$ over $R$ as above, define the complex of $n$-shifted polyvector fields on $A$ by $$\widehat{\Pol}(A,n):= \prod_{j \ge 0} \hat{\HHom}_A(\CoS_A^j((\Omega^1_{A/R})_{[-n-1]}),A).$$ This has a filtration by complexes $$F^p\widehat{\Pol}(A,n):= \prod_{j \ge p} \hat{\HHom}_A(\CoS_A^j((\Omega^1_{A/R})_{[-n-1]}),A),$$ with $[F^i,F^j] \subset F^{i+j-1}$ and $F^i F^j \subset F^{i+j}$, where the commutative product and Schouten–Nijenhuis bracket are defined as before. We now define the space $\cP(A,n)$ of Poisson structures and its tangent space $T\cP(A,n)$ by the formulae of Definitions \[poissdef\], \[Tpoissdef\]. As in Definition \[sigmadef\], there is a canonical tangent vector $\sigma \co \cP(A,n) \to T\cP(A,n)$. As well as composition for internal $\cHom$’s, we have maps $\cHom(M_1,P_1)\ten \cHom(M_2, P_2) \to \cHom(M_1\ten M_2, P_1\ten P_2)$, and hence $$\begin{aligned} \cHom_A(\Omega_A^1, \cHom_A(M, A))^{\ten p} &\to \cHom_A((\Omega_A^1)^{\ten p}, \cHom_A(M, A)^{\ten p})\\ & \to \cHom_A((\Omega_A^1)^{\ten p}, \cHom_A((M^{\ten p}, A)).\end{aligned}$$ Substituting $M= \bigoplus_{j=2}^{r-1} \CoS_A^j((\Omega^1_{A/R})_{[-n-1]})$, taking shifts and $S_p$-coinvariants, and applying $\hat{\Tot}$, we see that an element $$\pi \in F^2\widehat{\Pol}(A,n)^{n+2}/F^r$$ thus defines a contraction morphism $$\mu(-, \pi) \co \Tot^{\Pi} \Omega_A^p \to F^p\widehat{\Pol}(A,n)^{n+2}/F^{p(r-1)}$$ of bigraded vector spaces, noting that $\Tot^{\Pi} \Omega_A^p = \hat{\Tot}\Omega_A^p $. When $r=3$ and $p=1$, we simply denote this morphism by $\pi_2^{\sharp}\co \Tot^{\Pi} \Omega_A^1 \to \hat{\HHom}_A(\Omega^1_A, A)[-n]$. \[binondegdef\] We say that a Poisson structure $\pi \in \cP(A,n)/F^p$ is non-degenerate if $\Tot^{\Pi} (\Omega_{A/R}^1\ten_AA^0)$ is a perfect complex over $A^0$ and the map $$\pi_2^{\sharp}\co \Tot^{\Pi} (\Omega_{A/R}^1\ten_AA^0) \to \hat{\HHom}_A(\Omega^1_A, A^0)[-n]$$ is a quasi-isomorphism. \[binondeglemma\] If $\pi_2 \in \cP(A,n)/F^3$ is non-degenerate, then the maps $$\mu(-, \pi_2) \co \Tot^{\Pi}(\Omega_A^p\ten_AM) \to \hat{\HHom}_A(\CoS_A^p((\Omega^1_{A/R})_{[-n-1]}),M)$$ are quasi-isomorphisms for all $M \in DG^+dg\Mod(A)$. We may proceed as in Proposition \[tgtcor2\]. By hypothesis, $ (\Omega_{A/R}^1\ten_AA^0)^i$ is acyclic for all $i>N$, so $ \cHom_A(\CoS_A^p((\Omega^1_{A/R})_{[-n-1]}),A)^i$ is acyclic for all $i<-Np$, meaning that we may replace $\hat{\HHom}$ with $\Tot^{\Pi}\cHom$. Since $M= \Lim_i (M/\sigma^{\ge i}M)$, a spectral sequence argument allows us to reduce to the case where $M \in dg\Mod(A^0)$. We may also replace $ (\Omega_{A/R}^1\ten_AA^0)$ with its brutal truncation $ \sigma^{\le N}(\Omega_{A/R}^1\ten_AA^0)$, so all complexes are bounded in the cochain direction, and then observe that non-degeneracy of $\pi$ gives us quasi-isomorphisms $$(\Tot \sigma^{\le N}(\Omega_{A/R}^1\ten_AA^0))^{\ten_{(A^0)}^p}\ten_{A^0}M \to \HHom_{A^0}( \sigma^{\le N}(\Omega_{A/R}^1\ten_AA^0)^{\ten_{(A^0)}^p}, M)[-np];$$ the required result follows on taking $S_p$-coinvariants. ### The de Rham complex {#biprespsn} \[biDRdef\] Define the de Rham complex $\DR(A/R)$ to be the product total complex of the double cochain complex $$\Tot^{\Pi} A \xra{d} \Tot^{\Pi}\Omega^1_{A/R} \xra{d} \Tot^{\Pi}\Omega^2_{A/R}\xra{d} \ldots,$$ so the total differential is $d \pm \delta \pm \pd$. We define the Hodge filtration $F$ on $\DR(A/R)$ by setting $F^p\DR(A/R) \subset \DR(A/R)$ to consist of terms $\Tot^{\Pi}\Omega^i_{A/R}$ with $i \ge p$. The definitions of shifted symplectic structures from §\[prespsn\] now carry over: \[biPreSpdef\] Define the space $\PreSp(A,n)$ of $n$-shifted pre-symplectic structures on $A/R$ by writing $\PreSp(A,n)/F^{i+2}:= \mmc(F^2 \DR(A/R)[n+1]/F^{i+2})$, and setting $ \PreSp(A,n):= \Lim_i \PreSp(A,n)/F^{i+2}$. Say that an $n$-shifted pre-symplectic structure $\omega$ is symplectic if $\Tot^{\Pi} (\Omega_{A/R}^1\ten_AA^0)$ is a perfect complex over $A^0$ and the map $$\omega_2^{\sharp}\co \hat{\HHom}_A(\Omega^1_A, A^0)[-n]\to \Tot^{\Pi} (\Omega_{A/R}^1\ten_AA^0)$$ is a quasi-isomorphism. Let $\Sp(A,n) \subset \PreSp(A,n)$ consist of the symplectic structures — this is a union of path-components. For a DG Artin $N$-hypergroupoid $X$, the space of closed $p$-forms of degree $n$ from [@PTVV] is given by $$\cA^{p,cl}_R(X^{\sharp},n)\simeq\ho\Lim_{j\in\Delta}\mmc(F^p \DR(D^*O(X^{\Delta^j})/R)^{[n+p-1]}).$$ This follows by combining Corollary \[gooddescent\] and Proposition \[tgtcor2\]. ### Compatible pairs {#Artincompat} The proof of Lemma \[mulemma\] adapts to give maps $$(\pr_2 + \mu\eps) \co \PreSp(A,n)/F^{p} \by \cP(A,n)/F^{p} \to T\cP(A,n)/F^{p}$$ over $\cP(A,n)/F^{p}$ for all $p$, compatible with each other. We may now define the space $\Comp(A, n)$ of compatible pairs as in Definition \[compdef\], with the results of §\[affinesn\] all adapting to show that the maps $\Sp(A,n) \la \Comp(A,n)^{\nondeg} \to \cP(A,n)^{\nondeg}$ are weak equivalences. Diagrams and functoriality {#Artindiagramsn} -------------------------- We now extend the constructions of §\[DMdiagramsn\] to stacky CDGAs. ### Definitions Given a small category $I$, an $I$-diagram $A$ of stacky CDGAs over $R$, and $A$-modules $M,N$ in $I$-diagrams of chain cochain complexes, we can define the cochain complex $\hat{\HHom}_A(M,N)$ to be the equaliser of the obvious diagram $$\prod_{i\in I} \hat{\HHom}_{A(i)}(M(i),N(i)) \implies \prod_{f\co i \to j \text{ in } I} \hat{\HHom}_{A(i)}(M(i),f_*N(j)).$$ All the constructions of §\[bipoisssn\] then adapt immediately; in particular, we can define $$\widehat{\Pol}(A,n):= \prod_{j \ge 0} \hat{\HHom}_A(\CoS_A^j((\Omega^1_{A/R})_{[-n-1]}),A),$$ leading to a space $\cP(A,n)$ of Poisson structures. We can now adapt the formulae of §\[DMdiagramsn\] to this setting, defining pre-symplectic structures by $$\PreSp(A,n):= \PreSp(A(0),n)= \Lim_{i\in [m]} \PreSp(A(i),n)$$ for any $[m]$-diagram $A$ of stacky CDGAs over $R$, and setting $ \Comp(A,n)$ be the homotopy vanishing locus of the obvious maps $$(\mu - \sigma) \co \PreSp(A,-1) \by \cP(A,n) \to T\cP(A,n).$$ over $\cP(A,n)$. The obstruction functors and their towers from §\[towersn\] also adapt immediately, giving the obvious analogues of the obstruction spaces defined in terms of $$\hat{\HHom}_A(\CoS^p_A(\Omega^1_{A/R}[n+1]),A), \quad \Tot^{\Pi}\Omega^p_{A(0)}.$$ ### Functors and descent {#bidescentsn} For $[m]$-diagrams in $DG^+dg\CAlg(R)$, we will consider the injective model structure, so an $[m]$-diagram $A$ is cofibrant if each $A(i)$ is cofibrant for the model structure of Lemma \[bicdgamodel\], and is fibrant if the maps $A(i) \to A(i+1)$ are all surjective. \[bicalcTlemma2\] If $D=(A\to B)$ is a fibrant cofibrant $[1]$-diagram in $DG^+dg\CAlg(R)$ which is formally étale in the sense that the map $$\{\Tot \sigma^{\le q} (\Omega_{A}^1\ten_{A}B^0)\}_q \to \{\Tot \sigma^{\le q}(\Omega_{B}^1\ten_BB^0)\}_q$$ is a pro-quasi-isomorphism, then the map $$\hat{\HHom}_D(\CoS_D^k\Omega^1_D,D) \to \hat{\HHom}_{A}(\CoS_{A}^k\Omega^1_{A},A),$$ is a quasi-isomorphism for all $k$. This follows by reasoning as in the proof of Proposition \[tgtcor2\]. Write $DG^+dg\CAlg(R)_{c, \onto}\subset DG^+dg\CAlg(R) $ for the subcategory with all cofibrant stacky CDGAs (in the model structure of Lemma \[bicdgamodel\]) over $R$ as objects, and only surjections as morphisms. For the notion of being formally étale from Lemma \[bicalcTlemma2\], we may extend the conditions of Properties \[Fproperties\] to constructions on $DG^+dg\CAlg(R)_{c, \onto}$, with quasi-isomorphisms taken levelwise. The constructions $\cP(-,n)$, $\Comp(-,n)$, $\PreSp(-,n)$, and their associated filtered and graded functors, all satisfy these properties. The first two properties follow from the right lifting property for fibrations (in the injective model structure on diagrams), and the third from Lemma \[bicalcTlemma2\]. Thus the simplicial classes $\coprod_{ A \in B_m DG^+dg\CAlg(R)_{c, \onto}} F(A)$ fit together to give a complete Segal space $\int F$ over the nerve $BDG^+dg\CAlg(R)_{c, \onto} $. Definition \[LintFdef\] then adapts to give us an $\infty$-category $\oL\int F$, and Definition \[inftyFdef\] adapts to give an $\infty$-functor $$\oR F \co \oL DG^+dg\CAlg(R)^{\et} \to \oL s\Set$$ with $(\oR F)(A) \simeq F(A)$ for all cofibrant stacky CDGAs $A$ over $R$, where $DG^+dg\CAlg(R)^{\et} \subset DG^+dg\CAlg(R)$ is the subcategory of homotopy formally étale morphisms. An immediate consequence of §\[Artincompat\] is that the canonical maps $$\begin{aligned} \oR\Comp(-,n)^{\nondeg} &\to& \oR\Sp(-,n) \\ \oR\Comp(-,n)^{\nondeg} &\to& \oR\cP(-,n)^{\nondeg} \end{aligned}$$ are weak equivalences of $\infty$-functors on the full subcategory of $\oL DG^+dg\CAlg(R)^{\et}$ consisting of objects satisfying the conditions of §\[bipoisssn\]. Corollary \[gooddescent\] and Proposition \[tgtcor2\] ensure that if a morphism $X \to Y$ of DG Artin $N$-hypergroupoids becomes an equivalence on hypersheafifying, then $D^*O(Y) \to D^*O(X)$ is formally étale in the sense of Lemma \[bicalcTlemma2\]. In particular this means that the maps $\pd^i \co D^*O(X^{\Delta^j}) \to D^*O(X^{\Delta^{j+1}})$ and $\sigma^i \co D^*O(X^{\Delta^{j+1}})\to D^*O(X^{\Delta^j})$ are formally étale. Thus $D^*O(X^{\Delta^{\bt}})$ can be thought of as a DM hypergroupoid in stacky CDGAs, and we may make the following definition: \[biinftyFXdef\] Given a DG Artin $N$-hypergroupoid $X$ over $R$ and any of the functors $F$ above, write $$F(X):= \ho\Lim_{j \in \Delta} \oR F(D^* O(X^{\Delta^j})).$$ \[biinftyFXwell\] If $Y \to X$ is a trivial DG Artin hypergroupoid, then the morphism $$F(X) \to F(Y)$$ is an equivalence for any of the constructions $F= \cP, \Comp, \PreSp$. The proof of Proposition \[inftyFXwell\] adapts, replacing Propositions \[DRobs\] and \[compatobs\] with Corollary \[gooddescent\] and Proposition \[tgtcor2\]. Thus the following is well-defined: \[PdefArtin\] Given a strongly quasi-compact DG Artin $N$-stack $\fX$, define the spaces $\cP(\fX,n)$, $\Comp(\fX,n)$, $\Sp(\fX,n)$ to be the spaces $ \cP(X,n), \Comp(X,n), \Sp(X,n) $ for any DG Artin $N$-hypergroupoid $X$ with $X^{\sharp} \simeq \fX$. \[2PBG\] If $R=\H_0R$, with $Y/R$ a smooth affine scheme equipped with an action of a Lie algebra $\g$, we start by considering $2$-shifted Poisson structures on the the Chevalley–Eilenberg complex $O([Y/\g])$ of Example \[DstarBG\] (a cochain CDGA). In this case, the DGLA $F^i\widehat{\Pol}( [Y/\g],2)[3]$ is concentrated in cochain degrees $[2i-3, \infty)$. In particular, this means that $\cP( [Y/\g],2) \cong \z^4(\gr_F^2\widehat{\Pol}( [Y/\g],2))$ is a discrete space (so all homotopy groups are trivial). Explicitly, $$\cP( [Y/\g],2) \cong \{ \pi \in (S^2\g \ten O(Y))^{\g} ~:~ [\pi, a]=0 \in \g \ten O(Y) ~\forall a \in O(Y)\}.$$ Specialising to the case $Y=\Spec R$, we have $\cP( B\g,2) \cong (S^2\g)^{\g}$, the set of quadratic Casimir elements. If $\g$ is the Lie algebra of a linear algebraic group $G$ as in Example \[DstarBG\], then for $X:=B[*/G]$, the spaces $\cP(D^*O(X^{\Delta^j}),2)$ are all discrete sets, so the space $\cP(BG,2)$ is just the equaliser of the maps $\cP( B\g,2) \implies \cP([G/\g^{\oplus 2}],2) $ coming from the vertex maps of the cosimplicial CDGA $D^*O(X^{\Delta^{\bt}})$ given by $$\xymatrix@1{ O(B\g) \ar@<1ex>[r] \ar@<-1ex>[r] & \ar@{.>}[l] O([G/\g^{\oplus 2}]) \ar[r] \ar@/^/@<0.5ex>[r] \ar@/_/@<-0.5ex>[r] & \ar@{.>}@<0.75ex>[l] \ar@{.>}@<-0.75ex>[l] O([G^2/\g^{\oplus 3}]) \ar@/^1pc/[rr] \ar@/_1pc/[rr] \ar@{}[rr]|{\cdot} \ar@{}@<1ex>[rr]|{\cdot} \ar@{}@<-1ex>[rr]|{\cdot} && O([G^3/\g^{\oplus 4}]){} \ar@/^1.2pc/[rr] \ar@/_1.2pc/[rr]\ar@{}[rr]|{\cdot} \ar@{}@<1ex>[rr]|{\cdot} \ar@{}@<-1ex>[rr]|{\cdot}& & {\phantom{E}}\cdots .}$$ 1ex These étale vertex maps are induced by applying $S^2$ to the maps $ \g \implies \g^{\oplus 2}\ten O(G) $ which when evaluated on $g \in G$ send $v \in \g$ to $(v, gvg^{-1})$ and $(gvg^{-1},v)$ respectively. The equaliser $\cP(BG,2)\subset (S^2\g)^{\g}$ is thus $$\cP(BG,2) \cong (S^2\g)^G,$$ with each element corresponding to a cosimplicial $P_3$-algebra structure on $D^*O((B[*/G])^{\Delta^{\bt}})$. Taking $G$ reductive and restricting to path components recovers an example in [@CPTVV §3.1]. \[cfCPTVV\] To relate Definition \[PdefArtin\] with the Poisson structures of [@CPTVV], first note that reindexation gives an equivalence of categories between double complexes and the “graded mixed complexes” of [@PTVV; @CPTVV] (but beware [@CPTVV Remark 1.1.2]: in the latter paper, “graded mixed complexes” do not have mixed differentials). Thus “graded mixed cdgas” are just stacky CDGAs, and to a derived stack $\fX$, [@CPTVV Definition 4.2.11] associates a sheaf $\bD_{\fX/\fX_{\dR}}$ of stacky CDGAs on the de Rham stack $\fX_{\dR}$, defining Poisson structures in terms of polyvectors on $\bD_{\fX/\fX_{\dR}}$. A comparison with our definition should then involve the observation that $\Spec DD^*O(X^{\Delta^j})$ is a model for the relative de Rham stack $(X_j/\fX)_{\dR}= (X_j)_{\dR}\by^h_{\fX_{\dR}}\fX$, possibly with $D^*O(X^{\Delta^j}) \simeq \bD_{X_j/\fX}$. Combined with the results above, an immediate consequence of §\[Artincompat\] is: \[Artinthm\] For any strongly quasi-compact DG Artin $N$-stack $\fX$ over $R$, there are natural weak equivalences $$\Sp(\fX,n) \la \Comp(\fX,n)^{\nondeg}\to \cP(\fX,n)^{\nondeg}.$$ Modulo the comparison suggested in Remark \[cfCPTVV\], an alternative proof of Theorem \[Artinthm\] is given as [@CPTVV Theorem 3.2.4]. [^1]: This work was supported by the Engineering and Physical Sciences Research Council \[grant number EP/I004130/2\].

--- abstract: 'We describe the two-dimensional Mott transition in a Hubbard-like model with nearest neighbors interactions based on a recent solution to the Zamolodchikov tetrahedron equation, which extends the notion of integrability to two-dimensional lattice systems. At the Mott transition, we find that the system is in a $d$-density wave or staggered flux phase that can be described by a double Chern Simons effective theory with symmetry $\su2 \otimes \su2$. The Mott transition is of topological nature, characterized by the emergence of vortices in antiferromagnetic arrays interacting strongly with the electric charges and an electric-magnetic duality. We also consider the effect of small doping on this theory and show that it leads to a quantum gas-liquid coexistence phase, which belongs to the Ising universality class and which is consistent with several experimental observations.' author: - 'Federico L. Bottesi' - 'Guillermo R. Zemba' title: Mott transition and integrable lattice models in two dimensions --- In spite of substantial advances in our theoretical understanding of strongly correlated electron systems, several problems still continue to provide estimulating challenges. One of the most interesting among these is the Mott transition, or metal-insulator transition driven by correlations. As early as in 1939, Mott argued that if the electron density in a metallic system was lowered enough, the Coulomb repulsion would dominate over the kinetic energy so that the system would undergo a transition to an insulating regime [@Mott]. From the experimental point of view, there exist several systems which display a Mott-type transition, such as vanadium oxide $V_2O_3$, several organic conductors, some doped semiconductors, and even underdoped high $T_C$ superconductors. Moreover, coexistence between phases of different densities has been observed in several experiments [@Algun-Exp]. From the theoretical point of view, finding solutions to even the simplest models (such as the Hubbard model) is difficult, given the failure of perturbative approaches due to the narrow differences separating the localized regime of the electrons in the insulating phase and the itinerant one in the conducting state. As of today, there exist two basic approaches for studying this transition: one is the dynamical mean field theory (DMFT) method[@Kotliar], valid in the limit of infinite dimensions (or infinite coordination number), which maps the Hubbard model onto the impurity Anderson model, with the addition of a self-consistency condition. This framework neglects spatial correlations while retaining the on-site quantum ones. The second approach consists in finding analytic expressions for physical observables in integrable models exhibiting the Mott behavior, for example, by using the Bethe Ansatz or the bosonization methods [@Lieb] [@Giamarchi] [@Shankar].However, the main restriction of these models is that they are formulated in one spatial dimension, unlike most system of experimental interest. The goal of the present article is to extend this approach to a two-dimensional (integrable) lattice model that exhibits the Mott transition and writ down an Efective fiel Theory to futher analyze the behavior af the model. Let us start by consider a system of spinless fermions on a square (two-dimensional) lattice with hamiltonian: H&=&-\_[i,]{} \[\^(i+e\_) e\^[i A\_]{} (i)+h.c.\]\ && +U\_[i,]{} (i)(i+e\_) , \[Model-Ferm-2d\] where $i$ labels the lattice sites and $e_\mu$ are unit lattice vectors, $t$ is the hopping parameter, $U$ is the (constant) Coulomb potential, $\rho(i)$ is the normal ordered charge density with respect to the half-filling ground state, $\rho(i)=:\psi^{\dagger}_i\psi_i: -1/2$ and $A_\mu $ is the abelian statistical gauge field, which after imposing Gauss’ law constraint reads A\_(i)=\_k\[(k,i)-(k,i+e\_)\]\^\_k \_k , where $\Theta(k,i)$ is the angle between the chosen direction $i$ and an arbitrary one $k$ on the lattice. Note that, for the one-dimensional case, the gauge field is irrelevant, in agreement with the fact that quantum statistics in one spatial dimension is arbitrary [@Fradkin-Book] and does not involve any physical gauge field. Using the two-dimensional Jordan-Wigner transformation[@Tsvelik][@Fradkin-Book] : && S\^+\_j=\^\_j U\_[2d]{}(j)\ && S\^-\_j=U\_[2d]{}(j)\_j\ && S\^z=\^ \_j \_j - ,where $U_{2d}(j)=\exp{[i\sum_{k\neq j}\Theta(k,j) \psi ^\dagger_k\psi_k]}$,the hamiltonian (\[Model-Ferm-2d\])becomes that of a $XXZ$ Heisenberg model H\_[XXZ]{}=\_[i,j ]{} \[ -(S\^x\_jS\^x \_j + S\^y\_i S\^y\_j)+S\^z\_jS\^z\_j \]  , \[hxxz\] where we have rescaled the terms such that $\Delta=U/t$. Following [@Jakel-Maillard] we define an [*interaction star*]{} as the set of points where the spins entering in an elementary interaction are localized, [*i.e.*]{}, the central site and their nearest-neighbors in the $XXZ$ model. The $n$-th interaction star has an energy $E_{XXZ}([\sigma]_n)$ which depends on the spin configuration in the star and on the local Boltzmann weights $W([\sigma]_n)$ . Therefore, the partition function takes the form $Z=\sum_{\sigma}\prod_n W([\sigma]_n)$, where the sum is taken over all possible configurations of the entire lattice. In two dimensional quantum systems and three dimensional statistical models the integrability is guaranteed by the existence of a set of mutual commuting layer-to-layer transfer matrices $T_{mn}(\l,\mu)$, which is tantamount to the existence of solutions of the so-called Zamolodchikov’s tetrahedron equation(TE) [@Zamolodchikov][@Bazhanov-1] : R\_[abc]{}R\_[ade]{}R\_[bdf]{}R\_[cef]{}=R\_[cef]{}R\_[bdf]{}R\_[adc]{}R\_[abc]{}  ,\[tetraeq\]where the operators $R_{ijk}$ define the mapping $ R_{ijk}: V_i \otimes V_j \otimes V_k \rightarrow V_i \otimes V_j \otimes V_k $, and $V_n$ is the spin one-half representation space , such that their matrix elements are the Boltzmann weights of the vertex $R_{ijk}=W([\sigma]_{ijk})$( the indices $i,j,k$ label the interaction star). These can be rewritten as the $LLLR-RLLL$ operator conditions, which express the associativity of the Zamolodchikov algebra: L\_[12,a]{}L\_[13,b]{}L\_[23,c]{}R\_[abc]{}=R\_[abc]{}L\_[23c]{}L\_[13,c]{}L\_[13,b]{} , \[LLLR-RLLL\]where, for example, the operator $L_{12a}$ acts on $V_1 \otimes V_2\otimes F_a $, $V_1$, $V_2$ are the auxiliary spaces and $F_a$ is the quantum space. If $F_a$ is the representation space of some algebra $\cal{A}$, it is possible to interpret the operators $L_{ij,a}$ as operator-valued matrices acting on $V_1 \otimes V_2$, and depending ‘parametrically’ on the generators of the algebra $\cal{A}$ denoted by $v_a$ and, possibility, on some $c$-numbers denoted by $s_a$: $L_{12a}=L_{12}(v_a,s_a)$ . In this case, the equation (\[LLLR-RLLL\]) can be expressed as a ‘local Yang Baxter’ equation: && L\_[12]{}([**v**]{}\_a,s\_a) L\_[13]{}([**v**]{} \_b,s\_b) L\_[23]{}([**v**]{} \_c,s\_c)=\ && L\_[23]{}([**v’**]{}\_c,s\_c) L\_[13]{}([**v’**]{}\_b,s\_b) L\_[1a]{}([**v’**]{}\_a,s\_a)  . \[Local-Yang-Baxter\]The tetrahedron equation (\[tetraeq\]) is highly non-trivial to solve, but recently a new solution to it has been found in [@Bazhanov-1]. The solution is associated to the finite-dimensional highest-weight representations of the quantum affine algebras $\uqa$, displaying the three-dimensional structure of quantum groups. It may be understood as a quantization of the spatial fluctuations of geometrical extended objects, and we shall see that in our case that these may be reinterpreted as (discrete) charge density waves. For completeness, we now briefly review the new solution (for details see [@Bazhanov-1]). The solution is inspired in the geometry of transformations applied to an hexahedron (see fig.(\[Fig1\]): there are three independent angles on each face and nine angles to fix the spatial orientation of the hexahedron. Therefore, nine independent angles are needed to specify it. Let us consider the mapping, \_[123]{}: \[\_j,\_j ,\_j\]  ,\[mapeofuncional-1\] where $\a$, $\b$ and $\gamma$ are the angles of the $j$-th face, and the primed variables refer to the opposite faces. For each quadrilateral face, say the $1'$, the relationship between opposite sides is given by $(l'_p,l'_q)^t=X({\cal A}_1) (l_p,l_q)^t$, where $X({\cal A}_1)$ is a matrix acting non trivialy on the face $1'$ , that depends on the angles on that face (${\cal A}_1=(\a_1,\b_1,\g_1$)), and which for a circular lattice reads: $$X ({\cal A}_1) = \left [ \begin{array}{ccc} {k_1} & {a^*_1} & {0} \\ {-a_1} & {k_1} &{ 0} \\ {0} & {0} & {1}\ \\ \end{array}\right]$$ where $k_1=\cos \a _1 \sin \b _1$, $a=\cos \a_1 \sin(\a _1+\b_1)$ and $a_1^*=\cos \a_1 \sin (\a_1-\b_1)$. In the general case, considering three faces , we have: (l’\_p,l’\_q,l’\_r)\^t=X\_[pq]{}([A]{}\_1)X\_[qr]{}([A]{}\_2)X\_[rs]{}([A]{}\_3)(l\_p,l\_q,l\_r)\^t .\[longitudes\]The same result (\[longitudes\]) is obtained by using the opposite faces with angles (${\cal A}'_1$,${\cal A}'_2$,${\cal A}'_3$). Therefore , it is easy to see that there exists a functional mapping given by $ X_{pq}({\cal A}_1)X_{qr}({\cal A}_2)X_{rs}({\cal A}_3)={\it R}_{123} (X_{rs}({\cal A}_3) X_{qr}({\cal A}_2)X_{pq}({\cal A}_3))$. This relation could be considered as a ’gauge symmmetry’. It can be shown that the mapping ${\it R}_{123}$ satisfies the functional tetrahedron equation ($FTE$), a ’classical version’ of the tetrahedron equation: && k’\_2a’\^\*\_1=k\_3a\^\*\_1-k\_1a\^\*\_2a\_3 k’\_2a’\_1=k\_3a\_1-k\_1a\^\*\_2a\^\*\_3\ && a’\^\*\_2=a\^\*\_2a\^\*\_3 +k\_1k\_3a\^\*\_2 a\_2’=a\_2a\_3 +k\_1k\_3a\_2\ && k’\_2a’\^\*\_3=k\_1a\^\*\_3-k\_3a\_1a\^\*\_2 k’\_2a’\_3=k\_1a\_3-k\_3a\^\*\_1a\_2  ,  \[R123\]  with $k'=\sqrt{1-a_2a^*_2}$. This map defines a canonical transformation of the Poisson algebra, which in terms of the angles reads: $\{\a_i\b_j\}=\delta_{ij}\quad \{\a_i,\a_j\}=0 \quad \{\b_i,\b_j\}=0 $. We now canonically quantize this theory, [*i.e.*]{}, by replacing the angles by Hilbert space operators and the Poisson brackets by commutators, so that $[\a,\b]=\zeta \hbar $, where $\zeta$ is a complex parameter. It can be shown that the quantum operators corresponding to $ k,a,a^*$, satisfy the commutation relation of the $q$-oscillator algebra && qa\^a-q\^[-1]{}a a\^=q-q\^[-1]{}\ &&ka\^=q a\^k ka=q\^[-1]{}ak ,\ with quantum deformation parameter $q=e^{\zeta \hbar}$ and $k^2=q(1-a^{\dagger}a)$. Upon this quantization, the map $R_{123}$ becomes a quantum operator satisfying by construction the quantum tetrahedron equation (\[tetraeq\]). It has been shown in [@Bazhanov-1] that it is possible to construct the matrix elements $\langle n'_1,n'_2,n'_3|R|n_1,n_2,n_3\rangle$ in the basis of the Fock space constructed from the $q$-oscillator algebra. The operator $R_{ijk}$ define an automorphism of the triplets of tensor products of the $q$-oscillator algebra $ O_q^{\otimes^3}\rightarrow O_q^{\otimes^3}$, and it also has the property of being non-degenerate in $F^{\otimes3}$. This fact, together with the tetrahedron equation (\[tetraeq\]), implies the validity of the standard Yang-Baxter equation (which signals the integrability in two dimensions) . In fact, tracing out in the Fock space $F_a$, the following equations are obtained: R\_[bc]{}R\_[bd]{}R\_[cd]{} &=& R\_[cd]{}R\_[bd]{}R\_[bc]{} \[Yang-Baxter\]\ L\_[Vb]{}L\_[Vc]{}R\_[bc]{} &=& R\_[bc]{}L\_[Vc]{}L\_[vb]{} \[LLR-RLL\]  .One affinization of the solution of [@Bazhanov-1] has been given in [@Bazhanov-2] as follows: consider the layer to layer transfer matrix $T_{mn}(\l,\mu)$ which are related to the $L$-operators by $T_{mn}(\l,\mu)=\prod_{i=1} ^n \prod_{j=m} ^1L_{îj}(\l_i\mu_j)$ and may be obtained from an ansatz that solves the local Yang-Baxter equation: $$L_{1,2}(u_3,\l_3) = \left [ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \l_3 k_3 & a_3^{\dagger} & 0 \\ 0 & -q^{-1}\l_3 \mu a_3 & \mu _3 k_3 & 0 \\ 0 & 0 & 0 & -q^{-1}\l_3 \mu_3\ \label{lunif} \end{array} \right]$$ where we have chosen the Fock space $F_a$ as $F_3$, for convenience. It has been shown that the solution of the affine TE (when all parameters $\l_i$ are equal to each other) has symmetry $\uqan$, where $n$ is an arbitrary integer and can be considered as the emergent coordinate of the third dimension. Equations (\[Yang-Baxter\])(\[LLR-RLL\]) can be associated either to an integrable two-dimensional statistical system or to a quantum system in $(1+1)$ dimensions. Therefore, the new solution of the Zamolodchikov equation tells us that is possible to break up the $(2+1)$-dimensional lattice system (in a consistent fashion) into $(1+1)$-dimensional ones for each row and column of the lattice or into classical statistical systems in $(2+0)$-dimensions at a fixed time. This result relates the one-dimensional Mott point ([*i.e.*]{}, the critical value of $t/U$ in the one-dimensional analogue of (\[Model-Ferm-2d\]) and its order parameter) to its two-dimensional counterpart as follows: on the one hand, in each row or column the $XXZ$ spin system with hamiltonian, H\_[XXZ]{}\^[1d]{}=-\_[i=1]{}\^[L]{}(S\^x\_iS\^x \_[i+1]{} + S\^y\_i S\^y\_[i+1]{} S\^z\_iS\^z\_[i+1]{}) + H\_[b]{} \[hxxz2\] where $H_{b} = \a(S^z_1-S^z_L)$ and $\Delta= (q+q^{-1})/2$ and $\a=(q-q^{-1})/2$, is know to posses the symmetry $U_q(sl(2))\otimes U_q(sl(2)) $ [@Sierra-Book]. Note that $q=1$ at $\Delta=1$, so that the boundary conditions are irrelevant, and we can choose periodic boundary conditions without breaking the quantum group symmetry. The system defined by (\[hxxz2\]) can be mapped to a one-dimensional nearest neighbors Hubbard (fermionic) system (using a Jordan-Wigner transformation) and may also be bosonized as a Luttinger system, which is a conformal field theory (CFT) [@cft] with central charge $c=1$ [@Affleck-lecture][@Voight]. It has been shown that this system undergoes a Mott transition at $\Delta=1$ [@Shankar], becoming an insulator. Morever, at the Mott transition (with $q=1$), the quantum group symmetry becomes the ordinary $su(2)$ symmetry (as far as the algebraic and symmetry properties are concerned, we treat $su(2)$ and $sl(2)$ as interchangeable) and the CFT will have symmetry $\su2\otimes\su2$ (because two CFTs, one for each chirality are needed) which becomes $su(2)\otimes su(2)$ at long distances. This CFT can be realized by a Wess-Zumino-Witten (WZW) model at level $k=1$, or by two chiral bosons (of opposite chirality) compactified on a circle at the self-dual radius (under $R$-duality) , where the vertex operators, known as currents $J^{\pm}=e^{i\sqrt{2}\phi}$,are well defined under a shift in the zero-mode of the fundamental bosonic field of the WZW model $\phi \rightarrow \phi+2\pi r$ ($r$ is the compactification radius). The currents have scaling dimension $(1,0)$ and together with $J^3(z)=i\de \phi$, of the same dimension, satisfy the $\su2$ algebra. The $R$-duality in these systems is well-known ([@cft]), for which an exchange of $r$ for $1/r$ leaves the spectrum unchanged but the elementary degrees of freedom are exchanged between charges and vortices. For $\Delta=1+\epsilon$, the system (\[hxxz2\]) develops an energy gap in the spectrum $E_g=4\exp(a/\epsilon)$ and exhibits a charge density wave (CDW) order parameter [@Shankar], defined by $\langle \rho(i) \rangle=1/2[1+(-1)^i P]$, where $ P=\langle\psi^\dagger_r (i)\psi_l (i)+ \psi^\dagger_l (i)\psi_r (i)\rangle = 1/\sqrt{\epsilon}\exp{(-a/\sqrt{\epsilon})}$ ($a$ is a constant, $i$ labels the lattice site and the expetaction value is taken in the ground state). The corresponding low-energy effective theory is given by the Sine-Gordon theory for the non-chiral effective bosonic field $\phi$, of Lagrangian density: L\_[SG]{}=()\^2+  . The last term can be interpreted as having origin in the Umkplap processes naturally arising in the lattice fermionic description, in which it is given by $\langle \psi_L\psi_L \psi_R\psi_R \rangle$ (where $\psi_L=\exp(-i\phi_L/\sqrt2)$ and $\psi_R=\exp(i\phi_R/\sqrt2)$, with $\phi=\phi_L+\phi_R$. As it has been pointed out in [@Nayak], for $\beta> 0$ (repulsive interactions), the ground state energy is minimized for a constant ground state density $\langle \phi \rangle =\pi/\sqrt{2}$, which in the fermionic representation corresponds to $\langle\psi^\dagger_R\psi_L\rangle = -\langle\psi^\dagger_R\psi_L\rangle =if$, where $f$ is a real function that changes sign under a $\pi/2$ rotation around any axis (with appropriate generalizations for the one-dimensional case [@Nayak]) . Therefore, this one-dimensional system exhibits a $d$-wave density order parameter, meaning that on the ground state the quantity $\langle \psi^\dagger(k)\psi(k+\pi/(a))\rangle$ breaks individually time reversal, translation invariance by one lattice site and $\pi/2$ rotation (around any axis) symmetries, while preserving the composition of any two of them. (Note that $\langle\psi^\dagger_R\psi_L\rangle = -\langle\psi^\dagger_R\psi_L\rangle$ yields $P=0$, implying a constant density profile. As consequence of the projection-like character of the solution of [@Bazhanov-1], by consitency, the system on the lattice must be in a (two-dimensional) $d$-density wave and also will be described by a CFT with $c=1$ in all phases. In orther to further account for the charge neutrality of the spin excitations and the additional lattice symmetries [@crystal], the CFT should be taken on the orbifold $S^1/Z_2$ and modded out by the lattice symmetry $D_4$. This CFT has been identified as characterizing the critical point of the six-vertex model or the four-state Potts model [@Ginsparg]. On the other hand, the lattice fermion system can be viewed in an alternative way: consider a two-layered system periodic in the ’temporal axis’, and trace out over the temporal dimension. The resulting Yang-Baxter equation has symmetry $\uqa \otimes \uqa $ corresponding to the symmetry of the six-vertex model [@Sierra-Book], whose phase diagram depends on the Boltzmann weigths at each vertex of the lattice throught the parameter $\Delta_{6v}=(a^2+b^2 -2c^2)/2ab$, where $a=\exp(-\beta E_a)$, $b=\exp(-\beta E_b)$, and $c=\exp(-\beta E_c)$ are the weights at each vertex. The transfer matrix of the six-vertex model is given by that of the $XXZ$ model through a Wick rotation. At the Mott critical point this implies that ($\Delta=-\Delta_ {6v } =1$) it is in the antiferroelectric phase.( Note that the spins are also rotated changing the sign of the spin-wave term [@Affleck-lecture]) Now we would like to write down an effective field theory (EFT) for the model on the square lattice at the Mott critical point [@Polchinski]. We first choose the effective degrees of freedom, and impose their characteristic symmetries on the theory under construction. In our case, both symmetries are given by the exact solution discussed above. As it was first pointed out by Witten, there is a close relationship between quantum groups, vertex models and Chern-Simons (CS) gauge theories: the expectation values of the Wilson loops can be calculated as statistical sums of Boltzmann weights in suitable defined vertex models, so that the mathematical structure of quantum groups encodes the topology of planar Wilson loops [@Witten-Vertex]. However, CS theories posses naturally the symmetry $\uqa$ (with $q=\exp(2\pi i/k)$ where $k$ is the CS coupling constant) [@Kogan] [@Grinseng], and not $\uqa \otimes \uqa$ . This mismatch is a consequence of the absence of parity conservation in the CS gauge theories. The simplest CS-type theories that preserve parity are the double CS gauge theories (which contain two $u(1)$ chiral gauge fields of opposite chirality, namely right and left)[@Carlo-Topics] : S\_[DCS]{}= d\^3x a\_R da\_R - d\^3x  a\_Lda\_L \[Double-CS\] where $ a_R$ ($a_L$) denotes the right (left) gauge field. This theory is known to be equivalent to the BF theory [@Carlo-Supercond] and it can also be written as a mixed CS theory. Here we are using $a \wedge da$ as a short-hand notation for the lattice version of the CS coupling $a_\mu K_{\mu,\nu} a_\nu$ with $K_{\mu,\nu}=S_{mu}\epsilon_{\mu,\a,\nu}d_\a$, $S_\mu f(x)=f(x+a\epsilon_\mu)$, $S_\mu f(x)=(f(x+a\epsilon_\mu)-f(x))/a$, (with $a$ the lattice spacing). At the Mott transition we have $k=1$, since the coupling $k$ fixes the unit of charge and the statistics of the excitations, and we find consistency with the fact that the effective degrees of freedom are density waves of the underlying electron system. Note that the $u(1)$ CS theory can be considered as the broken parity phase of the $su(2)$ CS theory, where the relation to the six-vertex model has been established [@Witten-Vertex] [@Alekseev]. We now impose periodic boundary conditions to the EFT, [*i.e.*]{}, compactify the space domain on a torus. Cutting down the torus along any cycle, induces the loosing of gauge symmetry on the cycle, so that the gauge fields become boundary dynamical degrees of freedom [@Wen][@Witten-2][@Stone] which are free chiral bosons ($c=1$ CFTs), representing charge density waves (CDW) described also by Luttinger systems . In the quantum theory obtained after quantizing these classical bosonic waves, there is a shift in the coupling parameter $k$ that is properly taken into account by the Sugawara construction: $k\rightarrow k+c_v$, where $c_v$ is the dual coxeter number of the symmetry algebra of the gauge group ( $c_v=2$ for $su(2)$). However, the identification of the Mott transition is done at the classical level, implying that the topological order remains given by the relation $ q=\exp(2\pi i /k)$, with no shift in $k$. We would now like to show that in the EFT, the emergence of a $d$-wave order considered before is natural. For this, we focus on the electric-magnetic duality between charges and vortices implicit in the EFT. Let us consider the charge current $j^\mu(x)$ degrees of freedom in the direct lattice, and corresponding vortex current $\phi^\mu(X_d) $ in the dual lattice (whose sites are in the center on each paquette of the direct lattice). We assume that these degrees of freedom can couple, and the low-energy action for their interaction is given by a mixed Chern Simons theory S\_[MCS]{}=d\^3x a da -d\^3x b db + a\_\_  where we have introduced two gauge fields $a_\mu$ and $b_\mu $ for the current and vortex degrees of freedom, respectively. The relevant definitions are $j^\mu=k \hat{K}_{\mu,\nu} a_\nu$, $\phi^\mu=k \hat{K}_{\mu,\nu}b_\nu$, $\hat{K}_{\mu,\nu}=\hat{S}_{mu}\epsilon_{\mu,\a,\nu}\hat{d}_\a$, $\hat{S}_\mu f(x)=f(x+a\epsilon_\mu)$, $\hat{d}_\mu f(x)=(f(x-a\epsilon_\mu)-f(x))/a$. In the dual lattice, the system dual to the original one is a two-dimensional $XXZ$ spin system with coupling constant $\Delta^ {-1}$. The degrees of freedom corresponding to these spin variables are vortices. For $\Delta<1$ the spins in the dual system are frozen, there are no spin waves, and the system is in the antiferomagnetic phase. Therefore, its effective action is given by a CS theory with punctures[@Trugenberger-1]: S\_[CS]{}&=&d\^3x a\^ K\_[,]{} a\_+ \^[’]{}\_p \^0  , \[CS-Vortex\] where $a_\lambda$ is a (different) abelian CS field and $\sum^{'}_p$ means that the sum is taken over all fundamental domains . Each domain has period $2a$ and contains four vortices in antiferromagnetic array. Therefore, the classical low-lying states reproduce an antiferromagnetic current pattern. At the quantum level, Gauss’ law selects the physical states from the lattice CS gauge theory with punctures. The quantum order of the ground state of this theory (staggered flux phase) breaks translation and parity symmetries by one lattice site and also time reversal invariance, but it is, however, invariant under the composition of any two symmetry transformations, satisfying the definition of the $d$-density wave order invariance ([@Nayak]). Therefore, the EFT analysis shows that the Mott transition for the system defined by (\[Model-Ferm-2d\]) is of topological ([*i.e.*]{}, Kosterlitz-Thoulouse type) nature, characterized by the emergence of CS vortices in antiferromagnetic arrays. Note that the resulting theory breaks the $Z_2$ vortex symmetry associated and, therefore, the two-dimensional chirality on each plaquette (\[Double-CS\]). Finally, we would like to discuss the behavior of the previously considered EFT away from the Mott critical point. By analogy with the CS theory of the quantum Hall effect, we could expect a ground state stable against small doping. In that case, for the simplest Laughlin inverse filling fractions $k=m$ ($m$ odd integer), the ground state is a droplet of incompressible quantum liquid [@Laughlin] (however, other phases with more exotic quantum orders, like Nematic phases are also possible in other regimes) and is stable under small perturbations away from the center of a given plateu in the conductivity. In the Mott system, we have already assumed that the dynamically generated vortices act as external statistical fields for the new electrons injected in the system by doping (this can be considered as an extension of the $R$-duality).At the self-dual point, statistical magnetic fields can be interchanged with statistical electric fields (on a torus). After imposing the lattice symmetries, the low-lying effective Hamiltonian for the injected electrons (in first quantization) is: H=\_i \[-\^2( +)+\_i(x\_i\^2- y\_i\^2)\]  ,\[effective-force\] where $\lambda _i$ can take the values $\pm \lambda$ Therefore, the electric potential changes sign in $x=\pm y$, producing domain walls between regions with different electron densities. Similar results can be obtained using the $W_4$ symmetry, which is related to the relevant perturbations of the Ashkin-Teller and six-vertex models away from the critical point [@Bottesi-Zemba][@Gaite]. One consequence of having discussed the EFT is that, [*a-posteriori*]{}, the behavior of the electrons can be more easily understood. It can be shown that the interaction term in the hamiltonian (\[Model-Ferm-2d\]) in the the continuum limit contains a chemical potential term of the form $-\mu\ \rho$, with $\mu=\Delta$ , which ensures the half-filling condition. Therefore, changing the chemical potencial by doping in $\delta \mu$ modifies the hamiltonian (in the spin representation (\[hxxz\]) by $H(\Delta)\rightarrow H(\Delta)+\delta \mu \sum_{\langle i j\rangle} S^z_i S^z_j $. At $\Delta=1$,the dynamics of the electron system is given by the double CS theory (\[Double-CS\]), whose hamiltonian can be defined as the temporal component of the stress-energy tensor $H_{cs}=T_{00}$, where $T_{\mu\nu}=\delta S_{cs}/\delta g_{\mu\nu}$ and where $g_{\mu\nu}$ is the metric tensor. However, the CS action is topological and, therefore, independent of the metric and $H_{cs}=0$ for each chiral component leading to $H(\Delta=1)=0$. This means that doping the system away from the critical point, the dynamics is controlled by an effective Ising hamiltonian. 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--- abstract: 'Deciphering the complex information contained in jets produced in collider events requires a physical organization of the jet data. We introduce two-particle correlations (2PCs) by pairing individual particles as the initial jet representation from which a probabilistic model can be built. Particle momenta, as well as particle types and vertex information are included in the correlation. A novel, two-particle correlation neural network (2PCNN) architecture is constructed by combining neural network based filters on 2PCs and a deep neural network for capturing jet kinematic information. The 2PCNN is applied to boosted boson and heavy flavor tagging, and it achieves excellent performance by comparing to models based on telescoping deconstruction. Major correlation pairs exploited in the trained models are also identified, which shed light on the physical significance of certain jet substructure.' author: - 'Kai-Feng Chen' - 'Yang-Ting Chien' bibliography: - 'ref.bib' title: 'Deep Learning Jet Substructure from Two-Particle Correlation' --- In high energy collider events, hundreds or even thousands of particles are produced, and the understanding of their high-dimensional probability distributions can be a formidable task. An emergent structure consisting of collimated particles, referred to as jets, are typically observed. The kinematic distribution and many aspects of the internal structure of jets have been the testing ground of quantum chromodynamics (QCD) in perturbative calculations and non-perturbative modeling, with remarkable success witnessed in reasonably accurate descriptions of collider data via Monte Carlo (MC) simulations and analytic calculations. However, the dynamical hadronization process which turns partonic degrees of freedom to hadronic degrees of freedom has not been fully understood and remains the holy grail of QCD. In this letter two-particle correlations (2PCs) are explored as a representation of jet information, and such organization can illuminate the physics underlying jet formation from parton evolution to hadronization. This is one step beyond processing individual particle information by constructing particle pairs as basic information elements, from which probabilistic models can be built and physical analysis can be performed. The model includes not only the particle momenta for energy flow information, but also the electric charges and vertex information which are sensitive to hadronization as well as bottom quark decays. Note that, the number of 2PCs scales quadratically with the number of particles therefore it creates a redundancy in the jet representation. Moreover, an advantage can be gained from the effectiveness of how the relevant information is contained in each 2PC pair, and how these 2PC pairs build up significant features which one can identify and define concrete observables to probe. Modern computation power has made possible the rise of machine learning techniques, and many methods have been applied successfully on classification and regression problems in particle and nuclear physics, such as jet classification [@deOliveira:2015xxd; @Baldi:2016fql; @Kasieczka:2017nvn; @Macaluso:2018tck; @Butter:2017cot; @Louppe:2017ipp; @Cheng:2017rdo; @Egan:2017ojy; @Chien:2018dfn; @Fraser:2018ieu; @Lin:2018cin; @Andreassen:2018apy; @Andreassen:2019txo; @Kasieczka:2019dbj; @Pang:2016vdc; @Pang:2019aqb; @Lim:2018toa; @Chakraborty:2019imr; @Chen:2019uar], correlation of particles [@Komiske:2018cqr; @Qu:2019gqs], anomaly detection [@Collins:2018epr; @Farina:2018fyg; @Blance:2019ibf; @Roy:2019jae], event generation [@Paganini:2017dwg; @deOliveira:2017pjk; @Paganini:2017hrr], and other tasks [@Komiske:2017ubm; @Andreassen:2019cjw]. We will tackle classic classification problems such as boosted boson and heavy flavor jet tagging, as a way to discover and highlight certain jet properties which are relevant in these tasks. Specifically, the discrimination of two-prong jets ($W$ jets and Higgs jets from the $H\rightarrow b\bar b$ decay channel) and three-prong jets (fully hadronic top jets) against light quark $q$ ($q=u,d,c,s$ quark) jets, as well as $W^+$ versus $W^-$ [@Chen:2019uar], and quark versus gluon jet discrimination, are studied. Excellent performance of 2PC-based neural network will be presented in all of the tasks. In particular, the network optimized for $W$ tagging successfully identifies the two-prong structure and isolation of $W$ jets by weighing strongly on these two features. The model behavior will be cross-checked by examining collinear and soft contributions from soft-drop [@Dasgupta:2013ihk; @Larkoski:2014wba] and collinear-drop [@Chien:2019osu] constituents and their correlations. A combination of machine learning and physics analysis methods benefits significantly from the use of a physically organized and unbiased jet representation so that one can extract the physics features the model identifies. The analysis is performed with samples generated from MC simulations using MadGraph [@Alwall:2014hca] for hard scattering processes and <span style="font-variant:small-caps;">Pythia8</span> [@Sjostrand:2007gs] for parton shower and hadronization. Jets are defined using the anti-$k_T$ algorithm [@Cacciari:2008gp] implemented in <span style="font-variant:small-caps;">FastJet</span> 3 [@Cacciari:2011ma], with $R=0.8$ for the studies of tagging high $p_T$ two or three-prong jets, and with $R=0.4$ for the studies of quark gluon discrimination. The high $p_T$ $R=0.8$ jets are generated using decays of hypothetical heavy $Z'$ bosons ($Z'\rightarrow W^+W^-, ZH, t \bar t, q \bar q$) with invariant mass fixed at 2 TeV, while the jets used in quark gluon discrimination are generated with the standard model QCD processes. For the samples generated using $Z'$ decays, jets are produced and reconstructed in the same kinematic region therefore the classification is not affected by the hard process kinematics. The truth particle information is passed through a Delphes [@deFavereau:2013fsa] fast detector simulation and converted into particle flow candidates, with track, electromagnetic calorimeter and hadronic calorimeter information. A parametric model based on CMS detector [@Chatrchyan:2008aa] at the Large Hadron Collider is introduced in the simulation. ![The schematic view of the 2PCNN model. It processes two-particle correlations as inputs and uses filters with shared weights to benchmark the importance of each 2PC pair. The top-$k$-ranked filter outputs, together with jet kinematic information, are feed into a fully connected network for decision making. []{data-label="fig:2PCNN_model"}](2PCNN_model.pdf){width="48.00000%"} Two different sets of 2PC inputs are included. A basic set contains only the energy flow information[^1], including the transverse momentum fraction $z = p_T^i/p_T({\rm jet})$, relative pseudorapidity $\Delta\eta = \eta^i - \eta({\rm jet})$, and relative azimuthal angle $\Delta\phi = \phi^i - \phi({\rm jet})$ of the jet constituents labelled by the index $i$. Here $p_T^i$, $\eta^i$ and $\phi^i$ are the transverse momentum, pseudorapidity and azimuthal angle of particle $i$, respectively, and $p_T({\rm jet})$, $\eta({\rm jet})$ and $\phi({\rm jet})$ are the corresponding quantities of the jet. A rotation in $\Delta\eta$-$\Delta\phi$ coordinate system is performed to align the principle axis of the jet constituents horizontally. The other set of inputs contains the 2PCs of charged tracks, while the vertex position and the charge of each particle are introduced in addition. Based on the 2PC inputs, we design a two-particle correlation neural network (2PCNN)[^2] to model the probability distribution of jet particles (see FIG. \[fig:2PCNN\_model\]), which is implemented using Keras [@chollet2015keras] with TensorFlow backend [@tensorflow2015-whitepaper]. Since the number of jet particles can vary, the 2PCNN layer is designed to handle inputs with variable sizes. Inspired by one of the key ideas from the convolutional neural network, the 2PCNN model implements a collection of filters[^3] with shared weights to process the input 2PC data. In the prototype model the number of filters is set to 64 to extract features from the energy flow information. The vertex and charge information is processed with a parallel 2PCNN layer containing 32 filters. Each filter processes and gives outputs to all input 2PCs. The filter outputs are then ranked according to their numerical values, and only the top-$k$ ranked 2PCs of each filter are kept as the inputs for the subsequent decision-making, fully connected network. In order to balance between performance and complexity, $k=4$ has been set; therefore the total number of output nodes is $256=64\times 4$, which is equal to the number of filters times $k$. Besides the 2PCNN layers, we use a dense network to include the jet kinematic information $p_T({\rm jet})$, $\eta({\rm jet})$ and $\phi({\rm jet})$ which is the baseline input for standard analysis. The outputs of the dense network and the 2PCNN layer are sent to another fully connected layer of 128 nodes (or 256 nodes if two 2PCNN layers are used), followed by two output nodes with softmax activation function for final decision. The model is optimized by minimizing a categorical cross-entropy loss function with the Adam optimizer [@adam]. Input samples for each task are split into three subsets: one set consisting of 80k jets is used to optimize the weights in the model, and another set of 40k jets is used to validate if the model reaches its optimal performance. The other set of 40k jets is used for an independent measure of the model performance. ![The receiver operating characteristic curves for classification of Higgs jets versus light quark jets (left), and top jets versus light quark jets (right). The solid curves show the performance of the 2PCNN model based on energy flow information. The dashed curves correspond to the 2PCNN model with additional electric charges and vertex inputs. The dotted curves give the result from the T-jet model.[]{data-label="fig:ROC"}](pub_roc_h_vs_q.pdf "fig:"){width="23.00000%"} ![The receiver operating characteristic curves for classification of Higgs jets versus light quark jets (left), and top jets versus light quark jets (right). The solid curves show the performance of the 2PCNN model based on energy flow information. The dashed curves correspond to the 2PCNN model with additional electric charges and vertex inputs. The dotted curves give the result from the T-jet model.[]{data-label="fig:ROC"}](pub_roc_t_vs_q.pdf "fig:"){width="23.00000%"} In order to benchmark the 2PCNN performance, we compare with a deep neural network model based on telescoping deconstruction of energy flow information (referred to as the T-jet model) [@Chien:2013kca; @Chien:2014hla; @Chien:2017xrb; @Chien:2018dfn]. The method systematically decomposes jet information into a fast-converging subjet series expansion $\sum_N {\rm T}_N$ which is ordered by the number of subjets $N$. These subjets are defined as the sets of particles along dominant energy flow directions within a variable subjet radius. Such organization is motivated by the infrared structure of QCD. Energetic, collinear particles are captured at lower orders, and the series gradually reaches out to soft, wide-angle particles. In this paper, the T-jet model includes jet information up to the $T_3$ order and scans energy flows with 4 values of subjet radius. The energy flow directions and subjet kinematics consist of 60 input variables. Together with the jet kinematic information, these inputs are processed by a fully connected network layer of 128 nodes followed by two output nodes. The same activation function, loss function and optimizer are adopted as in the 2PCNN model. FIG. \[fig:ROC\] shows the receiver operating characteristic (ROC) curves, plotting background rejection rate as a function of signal efficiency, for two discrimination tasks as representative examples: high $p_T$ Higgs jet versus light quark jet, as well as top jet versus light quark jet. The model performances are quantified by the area under ROC curve (AUC) and the average accuracy (ACC), which is the fraction of correctly-predicted jet samples. As summarized in TABLE \[tab:performance\], the 2PCNN and the T-jet model based on energy flow information show nearly the same performance, confirming the baseline capability of the 2PCNN model which is comparable to the state-of-the-art methods that are all capable of modeling the energy flow probability distributions very well. With the additional vertex and charge information, the 2PCNN model achieves excellent performance in all the classification tasks. The vertex information has a strong impact on tagging jets which contain one or more secondary vertices such as the high $p_T$ Higgs and top jets. The electric charges of particles are also essential for separating jets from $W^+$ and $W^-$ bosons. ---------------- ------- ------- ------- ------- ------- ------- Task ACC AUC ACC AUC ACC AUC $W$ vs quark 0.881 0.945 0.881 0.946 0.880 0.945 Higgs vs quark 0.873 0.939 0.959 0.993 0.866 0.934 top vs quark 0.900 0.962 0.929 0.978 0.900 0.963 $W^+$ vs $W^-$ 0.505 0.502 0.757 0.839 0.502 0.502 quark vs gluon 0.738 0.810 0.748 0.823 0.732 0.802 ---------------- ------- ------- ------- ------- ------- ------- : The performance of the 2PCNN and T-jet models, as quantified by the average accuracy (ACC) and the area under the receiver operating characteristic curve (AUC), for $W$, Higgs and top tagging as well as $W^+$ versus $W^-$ and quark versus gluon discrimination. The energy flow 2PCNN model has comparable performance with the T-jet model. The 2PCNN model with additional information of electric charges and vertex of charged tracks outperforms significantly the other two models in most of the tasks. The uncertainty due to finite sample size in ACC is smaller than 0.003.[]{data-label="tab:performance"} We now discuss the physics properties of the 2PCs and focus on the task of $W$ jet and light quark jet separation using the energy flow 2PCNN model, aiming to identify the key features which are useful for distinguishing the two jet samples. Many other detailed studies will be presented in a forth coming paper. Thanks to the internal ranking of 2PC pairs, the importance of the top-$k$ ranked 2PC pairs within a filter can potentially be quantified by their filter output values. These sets of outputs represent the weights on 2PCs which the 2PCNN has learned from separating the two samples and are task-dependent. Therefore intrinsic features of each jet sample can be illuminated by contrasting with different jet samples potentially having distinct features. FIG. \[fig:jet\_disp\] shows the display of a typical two-prong $W$ jet and a typical one-prong light quark jet. The jet constituents are shown as scattered circles and squares, with their sizes proportional to the particle transverse momenta. The top-one ranked 2PC pair of each active 2PCNN filter is indicated by a solid line, with the thickness representing the strength of the filter output. Two distinct signatures of the high-ranked 2PCs are identified: (1) strong internal correlations within and between the prongs, and (2) strong correlations between high $p_T$ constituents within the prongs and low $p_T$ constituents scattered at wide angle. ![Displays of a typical $W$ jet (left) and a typical light quark jet (right) in $\Delta\eta$-$\Delta\phi$ plane. The charged tracks of jet particles are shown as circles with charge signs, while the neutral clusters are shown as squares. The sizes of the circles or the squares are proportional to the $p_T$’s of jet constituents. The solid lines indicate the top-one ranked 2PCs of the filters in the energy flow 2PCNN model. The strength of filter outputs are represented by the line thickness.[]{data-label="fig:jet_disp"}](pub_disp_w_jet004.pdf "fig:"){width="23.00000%"} ![Displays of a typical $W$ jet (left) and a typical light quark jet (right) in $\Delta\eta$-$\Delta\phi$ plane. The charged tracks of jet particles are shown as circles with charge signs, while the neutral clusters are shown as squares. The sizes of the circles or the squares are proportional to the $p_T$’s of jet constituents. The solid lines indicate the top-one ranked 2PCs of the filters in the energy flow 2PCNN model. The strength of filter outputs are represented by the line thickness.[]{data-label="fig:jet_disp"}](pub_disp_q_jet051.pdf "fig:"){width="23.00000%"} {width="24.60000%"} {width="24.60000%"} {width="24.60000%"} {width="24.60000%"} Such behaviors of the high-ranked 2PC pairs are further examined by the spatial distance $\Delta R = \sqrt{(\eta^i-\eta^j)^2+(\phi^i-\phi^j)^2}$ between the $i$-th and $j$-th particles forming the 2PC, and their $p_T$ asymmetry $\mathcal{A}(p_T) = |p_T^i-p_T^j|/(p_T^i+p_T^j)$. FIG. \[fig:dr\_ptasy\] shows the comparisons of a variety of $\Delta R$ and $\mathcal{A}(p_T)$ distributions of $W$ jets and light quark jets. In order to maximize the sensitivity to the features extracted by the 2PCNN, the distributions corresponding to the top-ranked 2PCs weighed by the output values of 2PCNN filters, as an indication of their importance, are presented in the lower panels. For $W$ jets, strong features are identified at $\Delta R\approx 0$ and $\Delta R\approx 0.2\sim 2m_W/p_T({\rm jet})$, whereas for light quark jets the $\Delta R\approx 0$ feature is strong and the $\Delta R\approx 0.2$ feature is absent. This indicates the intrinsic jet property of particle collimation for both samples, and the two-prong structure of $W$ jets. The filters tend to either select the 2PCs within the same prong therefore with small $\Delta R$ values, or emphasize the correlations between the two prongs for $W$ jets and build up the $\Delta R\approx 0.2$ feature. On the other hand, a clear feature at $\mathcal{A}(p_T)\approx 1$ shows up in the filter-output weighed $\mathcal{A}(p_T)$ distributions for both samples. Such signature corresponds to highly unbalanced $p_T$’s in the 2PCs therefore one of the particle has to be soft. This shows the importance of low $p_T$ constituents which are often neglected or suppressed in many other jet tagging methods. In order to further examine the properties of 2PCs which are responsible for the learned jet features, soft-drop and collinear-drop with parameters $z_{\rm cut}=0.2$ and $\beta=0$ are used to classify jet constituents into two categories. The jet constituents surviving soft-drop are referred to as “groomed," while those surviving collinear-drop belong to the “dropped" category. Therefore the 2PCs form three distinct sets: groomed-groomed, groomed-dropped and dropped-dropped. We can see that the one and two-prong structures are dominantly determined by the groomed-groomed 2PC pairs from the two soft-drop branches. Also, there is a significant dropped-dropped contribution at medium and large $\Delta R$ values for light quark jets. On the other hand, the feature at $\mathcal{A}(p_T)\approx 1$ dominantly comes from the groomed-dropped 2PC pairs which correlate hard, collinear particles to soft, wide-angle particles, while most other 2PC pairs form a fairly flat $\mathcal{A}(p_T)$ distribution. To highlight the power and sensitivity of 2PCNN in feature extraction, we contrast with the $\Delta R$ and $\mathcal{A}(p_T)$ distributions of $W$ and light-quark jets formed with equal weight for all the 2PC pairs (upper panels of FIG. \[fig:dr\_ptasy\]). Evidence of one- or two-prong structure from the falling $\Delta R$ distribution with a “shoulder" around $\Delta R \approx 0.2$, as well as the significant soft particle contributions in the $\mathcal{A}(p_T)\approx 1$ region, are observed. Similar conclusions can be reached by decomposing the distributions into groomed-groomed, groomed-dropped and dropped-dropped components; however all the features are much more convincingly identified by the 2PCNN model as a very useful guide for physics analysis. In conclusion, we have constructed a new neural network architecture which utilizes two-particle correlations (2PCs) as a fundamental description of jets. The input for 2PCNN is dynamically determined by the number of jet constituents with no artificial reduction of input information and no particular biased ordering of jet particles. The structure of the 2PC neural network is driven by the physics needs, rather than a direct application of existing deep learning methods developed for solving problems in other subjects. We demonstrate that the 2PCNN model based on energy flow information has comparable performance with the model using variables from telescoping deconstruction, which is one of the most effective method for factorizing jet information. By including additional information from charged tracks, such as electric charges and vertex, the 2PCNN model achieves an unprecedentedly promising power for a variety of jet tagging tasks. Besides the excellent tagging performance, an important benefit of the 2PCNN model is the ranking of 2PCs which can be directly extracted from the filter outputs. Since two-particle correlations are fundamental descriptions of particles relations, this physical machine-learning method can be potentially useful in subsequent physics studies such as hadronization process and collective behaviors of quark-gluon plasma remnants in high energy collisions. The 2PCNN will shed light on physics signatures which are difficult to identify with conventional methods. Acknowledgements ================ The authors thank George Sterman for encouraging and useful conversations, as well as Cheng-Wei Chiang, Frédéric Dreyer, Sung Hak Lim, Matthew Schwartz and Jesse Thaler for helpful comments and suggestions. Y.-T. Chien was supported by the National Science Foundation grant PHY-1915093. K.-F. Chen was supported by the grant 106-2112-M-002-006 of Ministry of Science and Technology, Taiwan. [^1]: The energy flow input here includes infrared and collinear unsafe information. [^2]: The prototype 2PCNN example code and test samples are available from https://github.com/kfjack/2PCNN. [^3]: The filter consists of a fully-connected dense network with 2PCs as the input, processed with a hidden layer, and then a layer of single nodes as the output. We use the ReLU [@nair2010rectified] activation function at each layer therefore the output can only be non-negative floating-point numbers.

--- abstract: | Inverse Probability Weighting (IPW) is widely used in empirical work in economics and other disciplines. As Gaussian approximations perform poorly in the presence of “small denominators,” trimming is routinely employed as a regularization strategy. However, ad hoc trimming of the observations renders usual inference procedures invalid for the target estimand, even in large samples. In this paper, we first show that the IPW estimator can have different (Gaussian or non-Gaussian) asymptotic distributions, depending on how “close to zero” the probability weights are and on how large the trimming threshold is. As a remedy, we propose an inference procedure that is robust not only to small probability weights entering the IPW estimator but also to a wide range of trimming threshold choices, by adapting to these different asymptotic distributions. This robustness is achieved by employing resampling techniques and by correcting a non-negligible trimming bias. We also propose an easy-to-implement method for choosing the trimming threshold by minimizing an empirical analogue of the asymptotic mean squared error. In addition, we show that our inference procedure remains valid with the use of a data-driven trimming threshold. We illustrate our method by revisiting a dataset from the National Supported Work program. 1.5em Keywords: Inverse probability weighting; Trimming; Robust inference; Bias correction; Heavy tail. author: - 'Xinwei Ma[^1]' - 'Jingshen Wang[^2]' bibliography: - 'Ma-Wang-2019-RobustIPW--References.bib' title: ' Robust Inference Using Inverse Probability Weighting[^3] ' --- Introduction {#section-1:introduction} ============ Inverse Probability Weighting (IPW) is widely used in empirical work in economics and other disciplines. In practice, it is common to observe small probability weights entering the IPW estimator. This renders inference based on standard Gaussian approximations invalid, even in large samples, because these approximations rely crucially on the probability weights being well-separated from zero. In a recent study, [-@busso2014finite] investigated the finite sample performance of commonly used IPW treatment effect estimators, and documented that small probability weights can be detrimental to statistical inference. In response to this problem, observations with probability weights below a certain threshold are often excluded from subsequent statistical analysis. The exact amount of trimming, however, is usually ad hoc and will affect the performance of the IPW estimator and the corresponding confidence interval in nontrivial ways. In this paper, we show that the IPW estimator can have different (Gaussian or non-Gaussian) asymptotic distributions, depending on how “close to zero” the probability weights are and on how large the trimming threshold is. We propose an inference procedure that adapts to these different asymptotic distributions, making it robust not only to small probability weights, but also to a wide range of trimming threshold choices. This “two-way robustness” is achieved by combining subsampling with a novel bias correction technique. In addition, we propose an easy-to-implement method for choosing the trimming threshold by minimizing an empirical analogue of the asymptotic mean squared error, and show that our inference procedure remains valid with the use of a data-driven trimming threshold. To understand why standard inference procedures are not robust to small probability weights, and why their performance can be sensitive to the amount of trimming, we first study the large-sample properties of the IPW estimator $$\begin{aligned} \hat{\theta}_{n,b_n} = \frac{1}{n}\sum_{i=1}^n \frac{D_iY_i}{\hat{e}( X_i)}{\mathds{1}}_{\hat{e}(X_i)\geq b_n},\end{aligned}$$ where $D_i\in\{0,1\}$ is binary, $Y_i$ is the outcome of interest, $e( X_i)={\mathbb{P}}[D_i=1| X_i]$ is the probability weight conditional on the covariates with $\hat{e}(X_i)$ being its estimate, and $b_n$ is the trimming threshold (the untrimmed IPW estimator is a special case with $b_n=0$). The asymptotic framework we employ is general and allows, but does not require that the probability weights have a heavy tail near zero. Specifically, if the tail is relatively thin, the asymptotic distribution will be Gaussian; otherwise a slower-than-$\sqrt{n}$ convergence rate and a non-Gaussian asymptotic distribution can emerge, and they will depend on the trimming threshold $b_n$. In the latter case, $$\begin{aligned} \label{eq:trimmed IPW limiting distribution} \frac{n}{a_{n,b_n}}\Big(\hat\theta_{n,b_n} - \theta_0 - {\mathsf{B}}_{n,b_n} \Big) {\overset{\mathrm{d}}{\to}}\mathcal{L}(\gamma_0,\alpha_+(\cdot),\alpha_-(\cdot)),\end{aligned}$$ where $\theta_0$ is the parameter of interest, $a_{n,b_n}\to \infty$ is a sequence of normalizing factors, and ${\overset{\mathrm{d}}{\to}}$ denotes convergence in distribution. First, a trimming bias ${\mathsf{B}}_{n,b_n}$ emerges. This bias has order ${\mathbb{P}}[e(X)\leq b_n]$, and hence it will vanish asymptotically if the trimming threshold shrinks to zero. What matters for inference, however, is the asymptotic bias, defined as the trimming bias scaled by the convergence rate: $\frac{n}{a_{n,b_n}}\mathsf{B}_{n,b_n}$. This asymptotic bias may not vanish even in large samples, and can be detrimental to statistical inference, as it shifts the asymptotic distribution away from the target estimand. Second, the asymptotic distribution, $\mathcal{L}(\cdot)$, depends on three parameters. The first parameter $\gamma_0$ is related to tail behaviors of the probability weights near zero, and the other two parameters characterize shape and tail properties of the asymptotic distribution. In particular, $\mathcal{L}(\cdot)$ does not need to be symmetric. Third, the convergence rate, $n/a_{n,b_n}$, is usually unknown, and depends again on how “close to zero” the probability weights are and how large the trimming threshold is. As the large-sample properties of the IPW estimator are sensitive to small probability weights and to the amount of trimming, it is important to develop an inference procedure that automatically adapts to the relevant asymptotic distributions. However, the presence of additional nuisance parameters makes it challenging to base inference on estimating the asymptotic distribution. In addition, the standard nonparametric bootstrap is known to fail in our setting [@athreya1987bootstrap; @knight1989bootstrap]. We instead propose the use of subsampling [@politis1994large]. We show that subsampling provides valid approximations to the asymptotic distribution in . With self-normalization (i.e., subsampling a Studentized statistic), it also overcomes the difficulty of having a possibly unknown convergence rate. Subsampling alone does not suffice for valid inference due to the bias induced by trimming. To make our inference procedure also robust to a wide range of trimming threshold choices, we combine subsampling with a novel bias correction method based on local polynomial regressions [@fan-Gijbels_1996_Book]. To be precise, our method regresses the outcome variable on a polynomial basis of the probability weight in a region local to the origin, and estimates the trimming bias with the regression coefficients. We also address the question of how to choose the trimming threshold. One extreme possibility is fixed trimming ($b_n=b>0$). Although fixed trimming helps restore asymptotic Gaussianity by forcing the probability weights to be bounded away from zero, this practice is difficult to justify, unless one is willing to re-interpret the estimation and inference result completely. We instead propose to determine the trimming threshold by taking into account the bias and variance of the (trimmed) IPW estimator. We suggest an easy-to-implement method to choose the trimming threshold by minimizing an empirical analogue of the asymptotic mean squared error. This paper relates to a large body of literature on program evaluation and causal inference [@imbens2015causal; @abadie2018econometric; @hernan2018book]. Estimators with inverse weighting are widely used in missing data models [@robins1994estimation; @wooldridge2007inverse] and treatment effect estimation [@hirano2003efficient; @cattaneo2010efficient]. They also feature in settings such as instrumental variables [@abadie2003semiparametric], difference-in-differences [@abadie2005semiparametric], and counterfactual analysis [@dinardo1996labor]. [@khan2010irregular] show that, depending on tail behaviors of the probability weights, the variance bound of the IPW estimator can be infinite, which leads to a slower-than-$\sqrt{n}$ convergence rate. [@sasaki2018ratio] propose a trimming method and a companion sieve-based bias correction technique for conducting inference for moments of ratios, which complement our paper. [@chaudhuri2014heavy] propose a different trimming strategy based on $|DY/e( X)|$ rather than the probability weight, and an inference procedure relying on asymptotic Gaussianity. [-@crump2009dealing] also study the problem of trimming threshold selection, the difference is that their method is based on minimizing a variance term, and hence can lead to a much larger trimming threshold than what we propose. The untrimmed IPW estimator (i.e., $b_n=0$) is a special case of , and the asymptotic distribution is known as the Lévy stable distribution. Stable convergence has been established in many contexts. For example, [@vaynman2014stable] show that stable convergence may arise for the variance targeting estimator, and hence the tail trimming of [@hill2012variance] can be crucial for establishing asymptotic Gaussianity. [@khan2013Uniform] also establish a stable convergence result for the untrimmed IPW estimator. However, they do not discuss the impact of trimming or how the trimming threshold should be chosen in practice. [-@hong2018inference] consider a different setting where observations fall into finitely many strata. They demonstrate that for estimating treatment effects the effective sample size can be much smaller as a result of disproportionately many treated or control units (a.k.a. limited overlap), and relate the rate of convergence to how fast the probability weight approaches an extreme. With the IPW estimator as a special case, [@cattaneo2018kernelBased] and [-@cattaneo2018manyCovs] show how an asymptotic bias can arise in a two-step semiparametric setting as a result of overfitting the first step. [-@chernozhukov2018locally] develop robust inference procedures against underfitting bias. The trimming bias we document in this paper is both qualitatively and quantitatively different, as it will be present even when the probability weights are observed, and certainly will not disappear with model selection or machine learning methods [@athey2018approximate; @belloni2018HighDimensional; @farrell2015Robust; @farrell2018neural]. The rest of the paper is structured as follows. In Section \[section-2: large sample properties\], we state and discuss the main assumptions, and study the large-sample properties of the IPW estimator. In Section \[section-3: robust inference\], we discuss in detail our robust inference procedure, including how the bias correction and the subsampling are implemented. A data-driven method to choose the trimming threshold is also proposed. Section \[section-4:empirical\] showcases our methods with an empirical example. Section \[section:conclusion\] concludes. To conserve space, we collect auxiliary lemmas, additional results, simulation evidence, and all proofs in the online Supplementary Material. We also discuss in the Supplementary Material how our IPW framework can be generalized to provide robust inference for treatment effect estimands and parameters defined by general (nonlinear) estimating equations. Large-Sample Properties {#section-2: large sample properties} ======================= Let $(Y_i,D_i, X_i)$, $i=1, 2, \cdots, n$ be a random sample from $Y\in\mathbb{R}$, $D\in\{0,1\}$ and $ X\in\mathbb{R}^{d_{x}}$. Recall that the probability weight is defined as $e( X)= {\mathbb{P}}[D=1| X]$. Define the conditional moments of the outcome variable as $$\mu_s(e( X))={\mathbb{E}}[Y^s|e( X),D=1],\quad s>0,$$ then the parameter of interest is $\theta_0 = {\mathbb{E}}[DY/e(X)] = {\mathbb{E}}[\mu_1(e( X))]$. At this level of generality, we do not attach specific interpretations to the parameter and the random variables in our model. To facilitate understanding, one can think of $Y$ as an observed outcome variable and $D$ as an indicator of treatment status, hence the parameter is the population average of one potential outcome. As previewed in Introduction, large-sample properties of the IPW estimator $\hat{\theta}_{n,b_n}$ depend on the tail behavior of the probability weights near zero: if $e(X)$ is bounded away from zero, the IPW estimator is $\sqrt{n}$-consistent and asymptotically Gaussian; in the presence of small probability weights, however, non-Gaussian distributions can emerge. In this section, we first discuss the assumptions and formalize the notion of “probability weights being close to zero” or “having a heavy tail.” Then we characterize the asymptotic distribution of $\hat{\theta}_{n,b_n}$, and show how it is affected by the trimming threshold $b_n$. Tail Behavior {#subsection-2-1: tail behavior} ------------- For an estimator that takes the form of a sample average (or more generally can be linearized into such), distributional approximation based on the central limit theorem only requires a finite variance. The problem with inverse probability weighting with “small denominators,” however, is that the estimator may not have a finite variance. In this case, distributional convergence relies on tail features, which we formalize in the following assumption. \[assumption:tail index\] For some $\gamma_0>1$, the probability weights have a regularly varying tail with index $\gamma_0-1$ at zero: $$\begin{aligned} \lim_{t\downarrow 0} \frac{{\mathbb{P}}[e( X)\leq tx]}{{\mathbb{P}}[e( X)\leq t]} = x^{\gamma_0-1},\qquad \text{for all $x>0$}.\end{aligned}$$ Assumption \[assumption:tail index\] only imposes a local restriction on the tail behavior of the probability weights, and is common when dealing with sums of heavy-tailed random variables. It is equivalent to ${\mathbb{P}}[e( X)\leq x] = c(x)x^{\gamma_0-1}$ with $c(x)$ being a slowly varying function (see the Supplementary Material or @fellerVol2 [Chapter XVII] for a definition). A special case of Assumption \[assumption:tail index\] is “approximately polynomial tail,” which requires $\lim_{x\downarrow 0}c(x)=c>0$. To see how the tail index $\gamma_0$ features in data, we illustrate in Section \[section-4:empirical\] with estimated probability weights from an empirical example, and it is clear that the probability weights exhibit a heavy tail near zero. Later in Theorem \[thm: IPW asy distribution\], we show that $\gamma_0=2$ is the knife-edge case that separates the Gaussian and the non-Gaussian asymptotic distributions for the (untrimmed) IPW estimator. With $\gamma_0=2$, the probability weights are approximately uniformly distributed, a fact that can be used in practice as a rough guidance on the magnitude of this tail index. \[remark:subexponential tail of X\] To see how the tail behavior of the probability weights is related to that of the covariates $ X$, we consider a Logit model: $e( X) = {\exp(X^{\mathrm{T}}\pi_0)}/({1+\exp(X^{\mathrm{T}}\pi_0)})$, which implies ${\mathbb{P}}[e( X)\leq x]= {\mathbb{P}}[ X^{\mathrm{T}}\pi_0 \leq -\log(x^{-1}-1) ]$. As a result, Assumption \[assumption:tail index\] is equivalent to that, for all $x$ large enough, ${\mathbb{P}}[ X^{\mathrm{T}}\pi_0 \leq -x ]\approx e^{-(\gamma_0-1)x}$, meaning that the (left) tail of $X^{\mathrm{T}}\pi_0$ is approximately sub-exponential. [$\parallel$]{} Assumption \[assumption:tail index\] characterizes the tail behavior of the probability weights. However, it alone does not suffice for the IPW estimator to have a asymptotic distribution. The reason is that, for sums of random variables without finite variance to converge in distribution, one needs not only a restriction on the shape of the tail, but also a “tail balance condition.” For this purpose, we impose the following assumption. \[assumption:conditional distribution of true Y\] (i) For some ${\varepsilon}>0$, ${\mathbb{E}}\big[|Y|^{(\gamma_0\vee2)+{\varepsilon}}\big|e( X)=x, D=1\big]$ is uniformly bounded. (ii) There exists a probability distribution $F$, such that for all bounded and continuous $\ell(\cdot)$, ${\mathbb{E}}[\ell(Y)|e( X)=x,D=1]\to \int_{\mathbb{R}} \ell(y)F({\mathrm{d}}y)$ as $x\downarrow 0$. This assumption has two parts. The first part requires the tail of $Y$ to be thinner than that of $D/e( X)$, therefore the tail behavior of $DY/e( X)$ is largely driven by the “small denominator $e(X)$.” As our primary focus is the implication of small probability weights entering the IPW estimator rather than a heavy-tailed outcome variable, we maintain this assumption. The second part requires convergence of the conditional distribution of $Y$ given $e( X)$ and $D=1$. Together, they help characterize the tail behavior of $DY/e(X)$. \[lemma:tail of DY/E\] Under Assumption \[assumption:tail index\] and \[assumption:conditional distribution of true Y\], $$\begin{aligned} \lim_{x\to \infty}\frac{x{\mathbb{P}}[DY/e( X) > x]}{{\mathbb{P}}[e( X)<x^{-1}]} &= \frac{\gamma_0-1}{\gamma_0}\alpha_+(0),\quad \lim_{x\to \infty}\frac{x{\mathbb{P}}[DY/e( X) < -x]}{{\mathbb{P}}[e( X)<x^{-1}]} = \frac{\gamma_0-1}{\gamma_0}\alpha_-(0),\end{aligned}$$ where $\alpha_+(x) = \lim_{t\to 0}{\mathbb{E}}[|Y|^{\gamma_0}{\mathds{1}}_{Y> x}\ |e( X)=t, D=1]$ and $\alpha_-(x) = \lim_{t\to 0}{\mathbb{E}}[|Y|^{\gamma_0}{\mathds{1}}_{Y< x}\ |e( X)=t, D=1]$. Assuming the distribution of the outcome variable is nondegenerate conditional on the probability weights being small (i.e., $\alpha_+(0) + \alpha_-(0) > 0$), Lemma \[lemma:tail of DY/E\] shows that $DY/e(X)$ has regularly varying tails with index $-\gamma_0$. As a result, $\gamma_0$ determines which moment of the IPW estimator is finite: for $s<\gamma_0$, ${\mathbb{E}}[|DY/e( X)|^s]<\infty$. We compare to a common assumption made in the IPW literature, which requires the probability weights to be bounded away from zero. This assumption is sufficient but not necessary for asymptotic Gaussianity. In fact, the IPW estimator is asymptotically Gaussian as long as $\gamma_0\geq 2$. Intuitively, small denominators appear so infrequently that they will not affect the large-sample properties. For $\gamma_0\in (1,2)$, the IPW estimator no longer has a finite variance, as the distribution of $e(X)$ does not approach zero fast enough (or equivalently, the density of $e(X)$, if it exists, diverges to infinity). This scenario represents the empirical difficulty of dealing with small probability weights entering the IPW estimator, for which regular asymptotic analysis no longer applies. Thanks to Assumption \[assumption:conditional distribution of true Y\](ii), Lemma \[lemma:tail of DY/E\] also implies that $DY/e(X)$ has balanced tails: the ratio $\frac{{\mathbb{P}}[DY/e( X) > x]}{{\mathbb{P}}[|DY/e( X)| > x]}$ tends to a finite constant. It turns out that without a finite variance, the asymptotic distribution of the IPW estimator is non-Gaussian, and the asymptotic distribution depends on both the left and right tail of $DY/e(X)$. Thus, tail balancing (and Assumption \[assumption:conditional distribution of true Y\](ii)) is indispensable for developing a large-sample theory allowing for small probability weights entering the IPW estimator. Asymptotic Distribution {#subsection-2-2: large sample properties} ----------------------- The following theorem characterizes the asymptotic distribution of the IPW estimator, both with and without trimming. To make the result concise, we assume the oracle (rather than estimated) probability weights are used, making the IPW estimator a one-step procedure. We extend the theorem to estimated probability weights in the following subsection. In the Supplementary Material, we also discuss how our IPW framework can be generalized to provide robust inference for treatment effect estimands and parameters defined by general (nonlinear) estimating equations. \[thm: IPW asy distribution\] Assume Assumption \[assumption:tail index\] and \[assumption:conditional distribution of true Y\] hold with $\alpha_+(0)+\alpha_-(0)>0$, $b_n\to 0$, and let $a_n$ be defined such that $$\begin{aligned} \frac{n}{a_n^2}{\mathbb{E}}\Big[ \left|\frac{DY}{e(X)}-\theta_0\right|^2{\mathds{1}}_{|DY/e(X)|\leq a_n} \Big]\to 1.\end{aligned}$$ (i) If $\gamma_0\geq 2$, holds with $a_{n,b_n}=a_n$, and the asymptotic distribution is standard Gaussian.\ (ii.1) No trimming, light trimming and moderate trimming: if $\gamma_0< 2$ and $b_na_n\to t\in[0,\infty)$, holds with $a_{n,b_n}=a_n$, and the asymptotic distribution is infinitely divisible with characteristic function $$\begin{aligned} &\ \psi(\zeta) = \exp\left\{ \int_{\mathbb{R}}\frac{e^{i\zeta x} - 1 - i\zeta x}{x^2} M({\mathrm{d}}x) \right\},\\ &\ \qquad \qquad \text{where }M({\mathrm{d}}x) = {\mathrm{d}}x \left[\frac{2-\gamma_0}{\alpha_+(0)+\alpha_-(0)}|x|^{1-\gamma_0}\Big(\alpha_+(tx){\mathds{1}}_{x\geq 0} +\alpha_-(tx){\mathds{1}}_{x< 0} \Big)\right].\end{aligned}$$ (ii.2) Heavy trimming: if $\gamma_0< 2$ and $b_na_n\to \infty$, holds with $a_{n,b_n} = \sqrt{n{\mathbb{V}}[DY/e( X){\mathds{1}}_{e( X)\geq b_n}]}$, and the asymptotic distribution is standard Gaussian. To provide some insight, we first consider the untrimmed IPW estimator ($b_n=0$), whose large-sample properties are summarized in part (i) and (ii.1). Theorem \[thm: IPW asy distribution\] demonstrates how a non-Gaussian asymptotic distribution can emerge when the untrimmed IPW estimator does not have a finite variance ($\gamma_0<2$). The asymptotic distribution in this case is also known as the Lévy stable distribution, which has the following equivalent representation, $$\begin{aligned} \psi(\zeta) &= -|\zeta|^{\gamma_0} \frac{\Gamma(3-\gamma_0)}{\gamma_0(\gamma_0-1)} \left[-\cos\left(\frac{\gamma_0\pi}{2}\right) + i\frac{\alpha_+(0)-\alpha_-(0)}{\alpha_+(0)+\alpha_-(0)}\mathrm{sgn}(\zeta)\sin\left(\frac{\gamma_0\pi}{2}\right) \right],\end{aligned}$$ where $\Gamma(\cdot)$ is the gamma function and $\mathrm{sgn}(\cdot)$ is the sign function. From this alternative form, we deduce several properties of the asymptotic Lévy stable distribution. First, this distribution is not symmetric unless $\alpha_+(0)=\alpha_-(0)$. Second, the characteristic function has a sub-exponential tail, meaning that the limiting Lévy stable distribution has a smooth density function (although in general it does not have a closed-form expression). Finally, the above characteristic function is continuous in $\gamma_0$, in the sense that as $\gamma_0\uparrow 2$, it reduces to the standard Gaussian characteristic function. Theorem \[thm: IPW asy distribution\] also shows how the convergence rate of the untrimmed IPW estimator depends on the tail index $\gamma_0$. For $\gamma_0>2$, the IPW estimator converges at the usual parametric rate $n/a_{n,b_n}\asymp\sqrt{n}$. This extends to the $\gamma_0=2$ case, except that an additional slowly varying factor is present in the convergence rate. For $\gamma_0<2$, the convergence rate is only implicitly defined from a truncated second moment, and generally does not have an explicit formula. One can consider the special case that the probability weights have an approximately polynomial tail: ${\mathbb{P}}[e(X)\leq x]\asymp x^{\gamma_0-1}$, for which $a_{n,b_n}$ can be set to $n^{1/\gamma_0}$. As a result, the untrimmed IPW estimator will have a slower convergence rate if the probability weights have a heavier tail at zero (i.e., smaller $\gamma_0$). Fortunately, the (unknown) convergence rate is captured by self-normalization (Studentization), which we employ in our robust inference procedure. Now we discuss the impact of trimming, a strategy commonly employed in practice in response to small probability weights entering the IPW estimator. We distinguish among three cases: light trimming ($b_na_n\to 0$), moderate trimming ($b_na_n\to t\in(0,\infty)$), and heavy trimming ($b_na_n\to \infty$). For light trimming, $b_n$ shrinks to zero fast enough so that asymptotically trimming becomes negligible, and the asymptotic distribution is Lévy stable as if there were no trimming. For heavy trimming, the trimming threshold shrinks to zero slowly, hence most of the small probability weights are excluded. This leads to a Gaussian asymptotic distribution. The moderate trimming scenario lies between the two extremes. On the one hand, a nontrivial number of small probability weights are discarded, making the limit no longer the Lévy stable distribution. On the other hand, the trimming is not heavy enough to restore asymptotic Gaussianity. The asymptotic distribution in this case is quite complicated, and depends on two (infinitely dimensional) nuisance parameters, $\alpha_+(\cdot)$ and $\alpha_-(\cdot)$. For this reason, inference is quite challenging. Despite the asymptotic distribution taking on a complicated form, moderate trimming as in Theorem \[thm: IPW asy distribution\](ii.1) is highly relevant. In Section \[subsection-3-1: balancing bias and variance\], we discuss how the trimming threshold can be chosen to balance the bias and variance (i.e., to minimize the mean squared error), which corresponds to this moderate trimming scenario. In addition, unless one employs a very large trimming threshold, it is unclear how well the Gaussian approximation performs in samples of moderate size. Estimated Probability Weights {#subsection-2-3: estimated weights} ----------------------------- The probability weights are usually unknown and are estimated in a first step, which are then plugged into the IPW estimator, making it a two-step estimation problem. Estimating the probability weights in a first step can affect large-sample properties of the IPW estimator through two channels: the estimated weights enter the final estimator both through inverse weighting and through the trimming function. In this subsection, we first impose high-level assumptions and discuss the impact of employing estimated probability weights. Then we verify those high-level assumptions for Logit and Probit models, which are widely used in applied work. \[assumption:first step\] The probability weights are parametrized as $e(X,\pi)$ with $\pi\in \Pi$, and $e(\cdot)$ is continuously differentiable with respect to $\pi$. Let $e(X) = e(X,\pi_0)$ and $\hat{e}(X)=e(X,\hat{\pi}_n)$. Further, (i) $\sqrt{n}(\hat{\pi}_n - \pi_0) = \frac{1}{\sqrt{n}}\sum_{i=1}^n h(D_i,X_i) + {o_{\mathrm{p}}}(1)$, where $h(D_i,X_i)$ is mean zero and has a finite variance; and (ii) For some ${\varepsilon}>0$, ${\mathbb{E}}\left[\sup_{\pi:|\pi-\pi_0|\leq {\varepsilon}} \left|\frac{e(X_i)}{e(X_i,\pi)^2}\frac{\partial e(X_i,\pi)}{\partial \pi}\right|\right] <\infty$. \[assumption:trimming threshold\] The trimming threshold satisfies $c_n\sqrt{b_n{\mathbb{P}}[e(X_i)\leq b_n]} \to 0$, where $c_n$ is a positive sequence such that, for any ${\varepsilon}>0$, $$\begin{aligned} c_n^{-1}\max_{1\leq i\leq n}\sup_{|\pi-\pi_0|\leq {\varepsilon}/\sqrt{n}} \left| \frac{1}{e(X_i)}\frac{\partial e(X_i,\pi)}{\partial \pi} \right| = {o_{\mathrm{p}}}(1).\end{aligned}$$ Now we state the analogue of Theorem \[thm: IPW asy distribution\] but with the probability weights estimated in a first step. \[prop:IPW with estimated weights\] Assume Assumption \[assumption:tail index\]–\[assumption:trimming threshold\] hold with $\alpha_+(0)+\alpha_-(0)>0$. Let $a_n$ be defined such that $$\begin{aligned} \frac{n}{a_n^2}{\mathbb{E}}\left[ \left|\frac{DY}{e(X)}-\theta_0-A_0h(D,X)\right|^2{\mathds{1}}_{|DY/e(X)-A_0h(D,X)|\leq a_n} \right]\to 1,\end{aligned}$$ where $A_0 = {\mathbb{E}}\left[ \frac{\mu_1(e(X))}{e(X)} \left.\frac{\partial e(X,\pi)}{\partial \pi}\right|_{\pi=\pi_0} \right]$. Then the IPW estimator has the following linear representation: $$\begin{aligned} \frac{n}{a_{n,b_n}}\left(\hat\theta_{n,b_n} - \theta_0 - \mathsf{B}_{n,b_n}\right) &= \frac{1}{a_{n,b_n}}\sum_{i=1}^n \left(\frac{D_iY_i}{e(X_i)}{\mathds{1}}_{e(X_i)\geq b_n} - \theta_0 - \mathsf{B}_{n,b_n} - A_0h(D_i,X_i)\right) + {o_{\mathrm{p}}}(1),\end{aligned}$$ and the conclusions of Theorem \[thm: IPW asy distribution\] hold with estimated probability weights. To understand Proposition \[prop:IPW with estimated weights\], we consider two cases. In the first case, ${\mathbb{V}}[DY/e(X)]<\infty$, and estimating the probability weights in a first step will contribute to the asymptotic variance. The second case corresponds to ${\mathbb{V}}[DY/e(X)]=\infty$, implying that the final estimator, $\hat{\theta}_{n,b_n}$, has a slower convergence rate compared to the first-step estimated probability weights. As a result, the two definitions of the scaling factor $a_n$ (in Theorem \[thm: IPW asy distribution\] and in Proposition \[prop:IPW with estimated weights\]) are asymptotically equivalent, and the asymptotic distribution will be the same regardless of whether the probability weights are known or estimated. In addition, Proposition \[prop:IPW with estimated weights\] shows that, despite the estimated probability weights entering both the denominator and the trimming function, the second channel is asymptotically negligible under an additional assumption, which turns out to be very mild in applications. Assumption \[assumption:first step\](i) is standard. In the following remark we provide primitive conditions to justify Assumption \[assumption:first step\](ii) and Assumption \[assumption:trimming threshold\] in Logit and Probit models. \[remark:logit and probit, 1\] Assuming a Logit model for the probability weights, we show in the Supplementary Material that a sufficient condition for Assumption \[assumption:first step\](ii) is the covariates having a sub-exponential tail: ${\mathbb{E}}[e^{{\varepsilon}|X|}]<\infty$ for some (small) ${\varepsilon}>0$. This condition is fully compatible with Assumption \[assumption:tail index\], as we show in Remark \[remark:subexponential tail of X\] that for Assumption \[assumption:tail index\] to hold in a Logit model, the index $X^{\mathrm{T}}\pi_0$ needs to have a sub-exponential left tail. As for the Probit model, Assumption \[assumption:first step\](ii) is implied by a sub-Gaussian tail of the covariates: ${\mathbb{E}}[e^{{\varepsilon}|X|^2}]<\infty$ for some (small) ${\varepsilon}>0$. Again, it is possible to show that Assumption \[assumption:tail index\] implies a sub-Gaussian left tail for the index $X^{\mathrm{T}}\pi_0$. To verify Assumption \[assumption:trimming threshold\], it suffices to set $c_n = \log^2(n)$ for Logit and Probit models. Therefore, we only require the trimming threshold shrinking to zero faster than a logarithmic rate. See the Supplementary Material for details. [$\parallel$]{} Robust Inference {#section-3: robust inference} ================ In the previous section, we show that non-Gaussian asymptotic distributions may arise as a result of small probability weights entering the IPW estimator and trimming. In this section, we first study the trimming bias, and show that this bias is usually non-negligible for inference purpose. Together, these findings explain why the point estimate is sensitive to the choice of the trimming threshold, and more importantly, why conventional inference procedures based on the standard Gaussian approximation perform poorly. As a remedy, we first introduce a method to choose the trimming threshold by minimizing an empirical mean squared error, and discuss how our trimming threshold selector can be modified in a disciplined way if the researcher prefers to discard more observations. Then we propose to combine subsampling with a novel bias correction technique, where the latter employs local polynomial regression to approximate the trimming bias. Bias, Variance and Trimming Threshold Selection {#subsection-3-1: balancing bias and variance} ----------------------------------------------- If the sole purpose of trimming is to stabilize the IPW estimator, one can argue that only a fixed trimming rule, $b_n=b\in(0,1)$, should be used. Such practice, however, completely ignores the bias introduced by trimming, and forces the researcher to change the target estimand and re-interpret the estimation/inference result. Practically, the trimming threshold can be chosen by minimizing the asymptotic mean squared error. We first characterize the bias and variance of the (trimmed) IPW estimator. \[lem:bias and variance\] Assume Assumption \[assumption:tail index\] and \[assumption:conditional distribution of true Y\] hold with $\gamma_0<2$. Further, assume that $\mu_1(\cdot)$ and $\mu_2(\cdot)$ do not vanish near 0. Then the bias and variance of $\hat\theta_{n,b_n}$ are: $${\mathsf{B}}_{n,b_n} = -\mu_1(0){\mathbb{P}}\left[e( X) \leq b_n\right] (1+o(1)),\qquad {\mathsf{V}}_{n,b_n} = \mu_2(0)\frac{1}{n}{\mathbb{E}}\left[e( X)^{-1}{\mathds{1}}_{e( X)\geq b_n}\right](1+o(1)).$$ In addition, ${\mathsf{B}}_{n,b_n}^2/{\mathsf{V}}_{n,b_n}\asymp nb_n{\mathbb{P}}[e( X)\leq b_n]$. To balance the bias and variance, minimizing the leading mean squared error (MSE) with respect to the trimming threshold leads to $$\begin{aligned} b_n^\dagger\cdot{\mathbb{P}}[e( X)\leq b_n^\dagger] &= \frac{1}{2n}\frac{\mu_2(0)}{\mu_1(0)^2}.\end{aligned}$$ The MSE-optimal trimming $b_n^\dagger$ helps understand the three scenarios in Theorem \[thm: IPW asy distribution\]: light, moderate and heavy trimming. More importantly, it helps clarify whether (and when) the trimming bias features in the asymptotic distribution. (The trimming bias ${\mathsf{B}}_{n,b_n}$ vanishes as long as $b_n\to 0$. What matters for inference, however, is the asymptotic bias, which is defined as the trimming bias scaled by the convergence rate: $\frac{n}{a_{n,b_n}}\mathsf{B}_{n,b_n}$. This asymptotic bias may not be negligible even in large samples.) $b_n^\dagger$ corresponds to the moderate trimming scenario, and since it balances the leading bias and variance, the asymptotic distribution of the trimmed IPW estimator is not centered at the target estimand (i.e., it is asymptotically biased). A trimming threshold that shrinks more slowly than the optimal one corresponds to the heavy trimming scenario, where the bias dominates in the asymptotic distribution. The only scenario in which one can ignore the trimming bias for inference purposes is when light trimming is used. That is, the trimming threshold shrinks faster than $b_n^\dagger$. The following theorem shows that, under very mild regularity conditions, the MSE-optimal trimming threshold can be implemented in practice by solving a sample analogue. It also provides a disciplined method for choosing the trimming threshold if the researcher prefers to employ a heavy trimming. \[thm:est optimal b\] Assume Assumption \[assumption:tail index\] holds, and $0<\mu_2(0)/\mu_1(0)^2<\infty$. For any $s>0$, define $b_n$ and $\hat{b}_n$ as: $$\begin{aligned} b_n^s{\mathbb{P}}[e( X)\leq b_n]=\frac{1}{2n}\frac{\mu_2(0)}{\mu_1(0)^2},\qquad \hat{b}_n^s\left(\frac{1}{n}\sum_{i=1}^n {\mathds{1}}_{e( X)\leq \hat{b}_n}\right)=\frac{1}{2n}\frac{\hat{\mu}_2(0)}{\hat{\mu}_1(0)^2},\end{aligned}$$ where $\hat{\mu}_1(0)$ and $\hat{\mu}_2(0)$ are some consistent estimates of $\mu_1(0)$ and $\mu_2(0)$, respectively. Then $\hat{b}_n$ is consistent for $b_n$: $\hat{b}_n/b_n{\overset{\mathrm{p}}{\to}}1$. Therefore, for $0<s<1$, $s=1$ and $s>1$, $\hat{b}_n / b_n^\dagger$ converges in probability to $0$, $1$ and $\infty$, respectively. In addition to Assumption \[assumption:first step\], if we have for any ${\varepsilon}>0$, $$\begin{aligned} \max_{1\leq i\leq n}\sup_{|\pi-\pi_0|\leq {\varepsilon}/\sqrt{n}} \left| \frac{1}{e(X_i)}\frac{\partial e(X_i,\pi)}{\partial \pi} \right| = {o_{\mathrm{p}}}\left(\sqrt{\frac{n}{\log(n)}}\right),\end{aligned}$$ then $\hat{b}_n$ can be constructed with estimated probability weights. This theorem states that, as long as one can construct a consistent estimator for the ratio $\mu_2(0)/\mu_1(0)^2$, the optimal trimming threshold can be implemented in practice with the unknown distribution ${\mathbb{P}}[e( X)\leq x]$ replaced by the standard empirical distribution function. In addition, Theorem \[thm:est optimal b\] allows the use of estimated probability weights to construct $\hat{b}_n$. The extra condition turns out to be quite weak, and is easily satisfied if the probability weights are estimated in a Logit or Probit model. (See Remark \[remark:logit and probit, 1\], and the Supplementary Material for further discussion.) In the following, we show that distributional convergence of the IPW estimator is unaffected by the use of data-driven trimming threshold. Establishing distributional convergence with data-driven tunning parameters tends to be quite difficult in general. In our setting, however, incorporating estimated trimming threshold does not require additional (strong) assumptions, as it is possible to exploit the specific structure that the trimming threshold enters only through an indicator function. \[prop:data-driven trimming\] Assume the assumptions of Theorem \[thm:est optimal b\] hold. Then Theorem \[thm: IPW asy distribution\] and Proposition \[prop:IPW with estimated weights\] hold with data-driven trimming threshold $b_n=\hat{b}_n$. Bias Correction {#subsection-3-2:bias correction} --------------- To motivate our bias correction technique, recall that the bias is ${\mathsf{B}}_{n,b_n}=-{\mathbb{E}}[\mu_1(e( X)){\mathds{1}}_{e( X)\leq b_n}]$, where $\mu_1(\cdot)$ is the expectation of the outcome $Y$ conditional on the probability weight and $D=1$. Next, we replace the expectation by a sample average, and the unknown conditional expectation by a $p$-th order polynomial expansion, which is then estimated by local polynomial regressions [@fan-Gijbels_1996_Book]. To be precisely, one first implements a $p$-th order local polynomial regression of the outcome variable on the probability weight using the $D=1$ subsample in a region $[0,h_n]$, where $(h_n)_{n\geq 1}$ is a bandwidth sequence. The estimated bias is then constructed by replacing the unknown conditional expectation function and its derivatives by the first-step estimates. Following is the detailed algorithm, which is also illustrated in Figure \[figure:local pol\]. \[algorithm-1: loc pol bias correction\] \ **Step 1**. With the $D=1$ subsample, regress the outcome variable $Y_i$ on the (estimated) probability weight in a region $[0,h_n]$: $$\begin{aligned} \Big[ \hat{\beta}_0, \hat{\beta}_1, \cdots, \hat{\beta}_p \Big]' = \operatorname*{argmin}_{\beta_0,\beta_1,\cdots,\beta_p} \sum_{i=1}^n D_i\Big[Y_i - \sum_{j=0}^p \beta_j \hat{e}( X_i)^j\Big]^2{\mathds{1}}_{\hat{e}( X_i)\leq h_n}.\end{aligned}$$ **Step 2**. Construct the bias correction term as $$\begin{aligned} \hat{{\mathsf{B}}}_{n,b_n} = -\frac{1}{n}\sum_{i=1}^n \left(\sum_{j=0}^p \hat{\beta}_j\hat{e}( X_i)^j\right) {\mathds{1}}_{\hat{e}( X_i)\leq b_n},\end{aligned}$$ so that the bias-corrected estimator is $\hat{\theta}_{n,b_n}^{\mathsf{bc}} = \hat{\theta}_{n,b_n} - \hat{{\mathsf{B}}}_{n,b_n}$. [$\parallel$]{} By inspecting the bias-corrected estimator, our procedure can be understood as a “local regression adjustment,” since we replace the trimmed observations by its conditional expectation, which is further approximated by a local polynomial. In the local polynomial regression step, it is possible to incorporate other kernel functions: we use the uniform kernel to avoid introducing additional notation, but all the main conclusions continue to hold with other commonly employed kernel functions. As for the order of local polynomial regression, common choices are $p=1$ and $2$, which reduce the bias to a satisfactory level without introducing too much additional variation. Standard results form the local polynomial regression literature require the density of the design variable to be bounded away from zero, which is not satisfied in our context. In the $D=1$ subsample which we use for the local polynomial regression, the distribution of the probability weights quickly vanishes near the origin. (More precisely, ${\mathbb{P}}[e(X)\leq x|D=1]\prec x$ as $x\downarrow 0$, meaning that the conditional density of the probability weights (if it exists) tends to zero: $f_{e(X)|D=1}(0)=0$.) As a result, nonstandard scaling is needed to derive large-sample properties of $\hat{\mu}_1^{(j)}(0)$. See the Supplementary Material for a precise statement. \[thm:bias correction validity\] Assume Assumption \[assumption:tail index\] and \[assumption:conditional distribution of true Y\] (and in addition Assumption \[assumption:first step\] and \[assumption:trimming threshold\] with estimated probability weights) hold. Further, assume (i) $\mu_1(\cdot)$ is $p+1$ times continuously differentiable; (ii) $\mu_2(0)-\mu_1(0)^2>0$; (iii) the bandwidth sequence satisfies $nh_n^{2p+3}{\mathbb{P}}[e( X)\leq h_n] \asymp 1$; (iv) $nb_n^{2p+3}{\mathbb{P}}[e( X)\leq b_n]\to 0$. Then the bias correction is valid, and does not affect the asymptotic distribution: $\hat{\theta}_{n,b_n}^{\mathsf{bc}} - \theta_0 = (\hat{\theta}_{n,b_n} - {\mathsf{B}}_{n,b_n} - \theta_0)(1+{o_{\mathrm{p}}}(1))$. Theorem \[thm:bias correction validity\] has several important implications. First, our bias correction is valid for a wide range of trimming threshold choices, as long as the trimming threshold does not shrink to zero too slowly: $nb_n^{2p+3}{\mathbb{P}}[e( X)\leq b_n]\to 0$. However, fixed trimming $b_n=b\in(0,1)$ is ruled out. This is not surprising, since under fixed trimming the correct scaling is $\sqrt{n}$, and generally the bias cannot be estimated at this rate without additional parametric assumptions. Second, it gives a guidance on how the bandwidth for the local polynomial regression can be chosen. In practice, this is done by solving $n\hat{h}_n^{2p+3}\hat{{\mathbb{P}}}[e( X)\leq \hat{h}_n]=c$ for some $c>0$, so that the resulting bandwidth makes the (squared) bias and variance of the local polynomial regression the same order. A simple strategy is to set $c=1$. It is also possible to construct a bandwidth that minimizes the leading mean squared error of the local polynomial regression, for which $c$ has to be estimated in a pilot step (see the Supplementary Material for a characterization of the leading bias and variance). Third, it shows how trimming and bias correction together can help improve the convergence rate of the (untrimmed) IPW estimator. From Theorem \[thm: IPW asy distribution\](ii), we have $|\hat{\theta}_{n,b_n}-\theta_0-{\mathsf{B}}_{n,b_n}|={O_{\mathrm{p}}}((n/a_{n,b_n})^{-1})$, where the convergence rate $n/a_{n,b_n}$ is typically faster when a heavier trimming is employed. This, however, should not be interpreted as a real improvement, as the trimming bias can be quite large. With bias correction, it is possible to achieve a faster rate of convergence for the target estimand, since under the assumptions of Theorem \[thm:bias correction validity\], one has $|\hat{\theta}_{n,b_n}^{\mathsf{bc}}-\theta_0|={O_{\mathrm{p}}}((n/a_{n,b_n})^{-1})$, which is valid for a wide rage of trimming threshold choices. Finally, we note that when Assumption \[assumption:tail index\] is violated, bias correction may not be feasible. For example, if in some region of the covariate distribution there are lots of observations from one group but not the other, there will be a spike very close zero (or at zero) in the probability weights distribution. As a result, bias correction in either case requires extrapolating a local polynomial regression, which can be unreliable. Robust Inference {#subsection-3-3: inference} ---------------- The asymptotic distribution of the IPW estimator can be quite complicated and depend on multiple nuisance parameters which are usually difficult to estimate. We propose the use of subsampling, which is a powerful data-driven method for distributional approximation. It draws samples of size $m\ll n$ and recomputes the statistic with each subsample. Together with our bias correction technique, subsampling can be employed to conduct statistical inference and to construct confidence intervals that are valid for the target estimand. Although Theorem \[thm:bias correction validity\] states that estimating the bias does not have a first order contribution to the asymptotic distribution, it may still introduce additional variability in finite samples [@calonico2018effect]. Therefore, we recommend subsampling the bias-corrected statistic. \[algorithm-2: robust inference\] Let $\hat{\theta}_{n,b_n}^{\mathsf{bc}}$ be defined as in Algorithm \[algorithm-1: loc pol bias correction\], and $$\begin{aligned} T_{n,b_n}=\frac{\hat{\theta}_{n,b_n}^{\mathsf{bc}}-\theta_0}{S_{n,b_n}/\sqrt{n}},\qquad S_{n,b_n} = \sqrt{\frac{1}{n-1}\sum_{i=1}^n \left( \frac{D_iY_i}{\hat{e}( X_i)}{\mathds{1}}_{\hat{e}( X_i)\geq b_n} - \hat{\theta}_{n,b_n} \right)^2}.\end{aligned}$$ **Step 1**. Draw $m\ll n$ observations from the original data without replacement, denoted by $(Y_i^\star,D_i^\star,X_i^\star)$, $i=1,2,\cdots, m$.\ **Step 2**. Construct the trimmed IPW estimator and the bias correction term from the new subsample, and write the bias-corrected and self-normalized statistic as $$\begin{aligned} T_{m,b_m}^\star &= \frac{\hat{\theta}_{m,b_m}^{\star\mathsf{bc}} - \hat{\theta}_{n,b_n}^{\mathsf{bc}} }{S_{m,b_m}^\star/\sqrt{m}},\qquad S_{m,b_m}^\star = \sqrt{\frac{1}{m-1}\sum_{i=1}^m \left( \frac{D_i^\star Y_i^\star}{\hat{e}^\star( X_i^\star)}{\mathds{1}}_{\hat{e}^\star( X_i^\star)\geq b_m} - \hat{\theta}_{m,b_m}^\star \right)^2}.\end{aligned}$$ **Step 3**. Repeat Step 1 and 2, and a $(1-\alpha)\%$-confidence interval can be constructed as $$\begin{aligned} \left[ \hat{\theta}_{n,b_n}^{\mathsf{bc}} - q_{1-\frac{\alpha}{2}}(T_{m,b_m}^\star)\frac{S_{n,b_n}}{\sqrt{n}}\quad,\quad \hat{\theta}_{n,b_n}^{\mathsf{bc}} - q_{\frac{\alpha}{2}}(T_{m,b_m}^\star)\frac{S_{n,b_n}}{\sqrt{n}} \right],\end{aligned}$$ where $q_{(\cdot)}(T^\star_{m,b_m})$ denotes the quantile of the statistic $T^\star_{m,b_m}$. [$\parallel$]{} Subsampling validity typically relies on the existence of an asymptotic distribution [@politis1994large; @romano1999subsampling]. We follow this approach and justify our robust inference procedure by showing that the self-normalized statistic $T_{n,b_n}$ converges in distribution. Under $\gamma_0>2$, $S_n$ converges in probability and $T_{n,b_n}$ converges to a Gaussian distribution. Asymptotic Gaussianity of $T_{n,b_n}$ continues to hold for $\gamma_0=2$. Under $\gamma_0<2$, $T_{n,b_n}$ still converges in distribution, although the limit will depend on the trimming threshold. Establishing the asymptotic distribution in the heavy trimming case is relatively easy (Lindeberg-Feller central limit theorem). With light or moderate trimming, however, the asymptotic distribution of $T_{n,b_n}$ is quite complicated. This technical by-product generalizes [-@logan1973limit]. (To be precise, with light trimming, we obtain the same distribution as in [@logan1973limit], while the asymptotic distribution of $T_{n,b_n}$ with moderate trimming is new.) We leave the details to the Supplementary Material. \[thm:subsampling validity\] Assume the assumptions of Theorem \[thm: IPW asy distribution\] (or Proposition \[prop:IPW with estimated weights\] with estimated probability weights) and Theorem \[thm:bias correction validity\] hold, $m\to \infty$, and $m/n\to 0$. Then $\sup_{t\in\mathbb{R}}| {\mathbb{P}}[T_{n,b_n}\leq t] - {\mathbb{P}}^\star[T_{m,b_m}^\star \leq t] | {\overset{\mathrm{p}}{\to}}0$. Before closing this section, we address two practical issues when applying the robust inference procedure. First, it is desirable to have an automatic and adaptive procedure to capture the possibly unknown convergence rate $n/a_{n,b_n}$, as the convergence rate depends on the tail index $\gamma_0$. In the subsampling algorithm, this is achieved by self-normalization (Studentization). Second, one has to choose the subsample size $m$. Some suggestions have been made in the literature: [@arcones1989bootstrap] suggest to use $m=\lfloor n/\log\log(n)^{1+{\varepsilon}}\rfloor$ for some ${\varepsilon}>0$, although they consider the $m$-out-of-$n$ bootstrap. [@romano1999subsampling] propose a calibration technique. We use $m=\lfloor n/\log (n)\rfloor$ for our simulation study in the Supplementary Material, which performs reasonably well. Empirical Illustration {#section-4:empirical} ====================== In this section, we revisit a dataset from the National Supported Work (NSW) program. Our aim is not to discuss to what extent experimental estimates can be recovered by non-experimental methods. Rather, we use this dataset to showcase our robust inference procedure. The NSW program is a labor training program implemented in 1970’s by providing work experience to selected individuals. It has been analyzed in multiple studies since [@lalonde1986evaluating]. We use the same dataset employed in [@dehejia1998rcausal]. Our sample consists of the treated individuals in the NSW experimental group ($D=1$, sample size $n_1=185$), and a nonexperimental comparison group from the Panel Study of Income Dynamics (PSID, $D=0$, sample size $n_0=1,157$). The outcome variable $Y$ is the post-intervention earning measured in 1978. The covariates $X$ include information on age, education, marital status, ethnicity and earnings in 1974 and 1975. We refer interested readers to [@dehejia1998rcausal; @dehejia2002propensity] and [@smith2005does] for more details on variable definition and sample inclusion. We follow the literature and focus on the treatment effect on the treated (ATT), $$\begin{aligned} \hat{\tau}_{n,b_n}^{\mathtt{ATT}} &= \frac{1}{n_1}\sum_{i=1}^n \left[D_iY_i - \frac{\hat{e}(X_i)}{1-\hat{e}(X_i)}(1-D_i)Y_i{\mathds{1}}_{1-\hat{e}(X_i)\geq b_n}\right],\end{aligned}$$ which requires weighting observations from the comparison group by $\hat{e}(X)/(1-\hat{e}(X))$. As a result, probability weights that are close to 1 can pose a challenge to both estimation and inference. (We discuss in the Supplemental Material how our IPW framework can be generalized to provide robust inference for treatment effect estimands.) The probability weight is estimated in a Logit model with $\mathtt{age}$, $\mathtt{education}$, $\mathtt{earn1974}$, $\mathtt{earn1975}$, $\mathtt{age}^2$, $\mathtt{education}^2$, $\mathtt{earn1974}^2$, $\mathtt{earn1975}^2$, three indicators for $\mathtt{married}$, $\mathtt{black}$ and $\mathtt{hispanic}$, and an interaction term $\mathtt{black}\times \mathtt{u74}$, where $\mathtt{u74}$ is the unemployment status in 1974. Figure \[subfig:estimated p-score\] plots the distribution of the estimated probability weights, which clearly exhibits a heavy tail near 1. Since $\gamma_0=2$ roughly corresponds to uniformly distributed probability weights, the tail index in this dataset should be well below $2$, suggesting that standard inference procedures based on the Gaussian approximation may not perform well. In Figure \[figure:empirical\], we plot the bias-corrected ATT estimates (solid triangles) and the robust 95% confidence intervals (solid vertical lines) with different trimming thresholds. For comparison, we also show conventional point estimates and confidence intervals (solid dots and dashed vertical lines, based on the Gaussian approximation) using the same trimming thresholds. Without trimming, the point estimate is $\$1,451$ with a confidence interval $[-1,763,\ 2,739]$. The robust confidence interval is asymmetric around the point estimate, which is a feature also predicted by our theory. For the trimmed IPW estimator, the trimming thresholds are chosen following Theorem \[thm:est optimal b\], and the region used for local polynomial bias estimation is $[0.71, 1]$, corresponding to a bandwidth $h_n=0.29$. Under the mean squared error optimal trimming, units in the comparison group with probability weights above $0.96$ (five observations) are discarded. Compared to the untrimmed case, the robust confidence interval becomes more symmetric. In this empirical example, a noteworthy feature of our method is that both the bias-corrected point estimates and the robust confidence intervals remain quite stable for a range of trimming threshold choices, and the point estimates are very close to the experimental benchmark ($\$1,794$). This is in stark contrast to conventional confidence intervals that rely on Gaussian approximation. First, conventional confidence intervals fail to adapt to the non-Gaussian asymptotic distributions we documented in Theorem \[thm: IPW asy distribution\], and are overly optimistic/narrow. Second, by ignoring the trimming bias, they are only valid for a pseudo-true parameter implicitly defined by the trimming threshold. As a result, the researcher changes the target estimand each time a different trimming threshold is used, making conventional confidence intervals very sensitive to $b_n$. Conclusion {#section:conclusion} ========== We study the large-sample properties of the Inverse Probability Weighting (IPW) estimator. We show that, in the presence of small probability weights, this estimator may have a slower-than-$\sqrt{n}$ convergence rate and a non-Gaussian asymptotic distribution. We also study the effect of discarding observations with small probability weights, and show that such trimming not only complicates the asymptotic distribution, but also causes a non-negligible bias. We propose an inference procedure that is robust not only to small probability weights entering the IPW estimator but also to a range of trimming threshold choices. The “two-way robustness” is achieved by combining resampling with a novel local polynomial-based bias-correction technique. We also propose a method to choose the trimming threshold, and show that our inference procedure remains valid with the use of a data-driven trimming threshold. To conserve space, we report additional results and simulation evidence in the online Supplementary Material. In particular, we discuss there how our IPW framework can be generalized to provide robust inference for treatment effect estimands and parameters defined by general (nonlinear) estimating equations. [^1]: Department of Economics, University of California, San Diego. [^2]: Division of Biostatistics, University of California, Berkeley. [^3]: The authors are deeply grateful to Matias Cattaneo for the comments and suggestions that significantly improved the manuscript. The authors also thank Sebastian Calonico, Max Farrell, Yingjie Feng, Andreas Hagemann, Xuming He, Michael Jansson, Lutz Kilian, Jose Luis Montiel Olea, Kenichi Nagasawa, Rocío Titiunik, Gonzalo Vazquez-Bare, the editor, an associate editor, and two referees for their valuable feedback and thoughtful discussions.

--- abstract: 'In this paper we consider an optimal control problem governed by a time-dependent variational inequality arising in quasistatic plasticity with linear kinematic hardening. We address certain continuity properties of the forward operator, which imply the existence of an optimal control. Moreover, a discretization in time is derived and we show that every local minimizer of the continuous problem can be approximated by minimizers of modified, time-discrete problems.' address: 'Chemnitz University of Technology, Faculty of Mathematics, D–09107 Chemnitz, Germany' author: - Gerd Wachsmuth - Gerd Wachsmuth bibliography: - 'Optimal\_Control\_of\_Quasistatic\_Plasticity\_Part\_I.bib' - 'Plasticity.bib' title: - | Optimal Control of Quasistatic Plasticity with Linear Kinematic Hardening\ Part I: Existence and Discretization in Time - | Optimal Control of Quasistatic Plasticity with Linear Kinematic Hardening\ Part I: Existence and Discretization in Time --- Introduction {#sec:intro} ============ The optimization of elastoplastic systems is of significant importance for industrial deformation processes, e.g., for the control of the springback of deep-drawn metal sheets. In this paper we consider an optimal control problem for the quasistatic problem of small-strain elastoplasticity with linear kinematic hardening. The strong formulation of the forward system (in the stress based, so-called dual formulation) reads (cf. [@HanReddy1999 Chapters 2, 3]) $$\label{eq:Forward_problem_strong_form} \left. \begin{aligned} \C^{-1} \dot\bsigma - \bvarepsilon(\dot\bu) + \lambda \, (\bsigma^D + \bchi^D) & = \bnull & & \mrep{\text{in }}{\text{on }} (0,T)\times\Omega, \\ \H^{-1} \dot\bchi \phantom{{}-\bvarepsilon(\dot\bu)} + \lambda \, (\bsigma^D + \bchi^D) & = \bnull & & \mrep{\text{in }}{\text{on }} (0,T)\times\Omega, \\ \div \bsigma & = -\bf & & \mrep{\text{in }}{\text{on }} (0,T)\times\Omega, \\ \begin{aligned} \smash{\text{with complem.\ conditions}} && \; 0 \le \lambda \; \perp \; \phi(\bSigma) \\ \smash{\text{and boundary conditions}} && \;\bu \end{aligned} & \hspace*{-0.5mm} \begin{aligned} & \le 0 \\ & = \bnull \end{aligned} & & \mspace{-3mu} \begin{aligned} & \mrep{\text{in }}{\text{on }} (0,T)\times\Omega, \\ & \text{on } (0,T)\times\Gamma_D, \end{aligned} \\ \bsigma \cdot \bn & = \bg & & \text{on } (0,T)\times\Gamma_N. \end{aligned} \quad \right\}$$ The system is subject to the initial condition $(\bsigma(0),\bchi(0),\bu(0)) = \bnull$. The state variables consist of the stress $\bsigma$ and back stress $\bchi$, whose values are symmetric matrices. Both matrix functions are combined into the generalized stress $\bSigma = (\bsigma,\bchi)$. The state variables also comprise the vector-valued displacement $\bu$ and the scalar-valued plastic multiplier $\lambda$ associated with the yield condition $\phi(\bSigma) \le 0$ of von Mises type, see . The first two equations in , together with the complementarity conditions, represent the material law of quasistatic plasticity. The tensors $\C^{-1}$ and $\H^{-1}$ are the inverses of the elasticity tensor and the hardening modulus, respectively, and $\bsigma^D$ denotes the deviatoric part of $\bsigma$, see . As usual, $\bvarepsilon(\bv) = \big(\nabla \bv + (\nabla \bv)^\top\big)/2$ denotes the (linearized) strain, where $\nabla\bv$ is the gradient of $\bv$. The third equation in is the equilibrium of forces. By $\div \bsigma$ we denote the (row-wise) divergence of the matrix-valued function $\bsigma$. The boundary conditions correspond to clamping on $\Gamma_D \subset \partial\Omega$ and the prescription of boundary loads $\bg$ on the remainder $\Gamma_N = \partial\Omega \setminus \Gamma_D$, whose outer unit normal vector is denoted by $\bn$. We will see in that can be reformulated as a mixed, time-dependent and rate-independent variational inequality of the first kind. The boundary loads $\bg$ act as control variables. There would be no difficulty in admitting volume forces $\bf$ as additional control variables but for practical reasons and simplicity of the presentation, $\bf$ is assumed to be zero throughout. The optimal control problem under consideration reads $$\label{eq:upper_level_in_intro} \tag{$\mathbf{P}$} \left. \begin{aligned} \text{Minimize} \quad & \psi(\bu) + \frac{\nu}{2} \norm{\bg}_{H^1(0,T;L^2(\Gamma_N;\R^d))}^2 \\ \text{such that} \quad & (\bSigma, \bu, \lambda) \text{ solves the quasistatic plasticity problem~\eqref{eq:Forward_problem_strong_form}}, \\ \text{and} \quad & \bg \in {U_\textup{ad}}. \end{aligned} \quad\right\}$$ Here, $\psi$ is a functional favoring certain displacements $\bu$. A typical example would be to track the final deformation after unloading. The set ${U_\textup{ad}}$ realizes constraints on the control $\bg$. The assumptions on $\psi$ and ${U_\textup{ad}}$ are given in , followed by a number of examples. The aim of this paper and of the subsequent works [@Wachsmuth2011:2; @Wachsmuth2011:4] is the derivation of first order necessary optimality conditions for . Let us highlight the main contributions of this paper. 1. We show that the derivative $(\dot\bSigma,\dot\bu)$ of the solution depends continuously on the derivative $\dot\bg$ of the right-hand side, see . This continuity result of the solution operator of (the weak formulation of) is novel compared to the stability results known from the literature. 2. Since the solution map of the system is nonlinear, its weak continuity is not obvious. We prove this weak continuity in . An immediate consequence is the existence of optimal controls of , see . 3. We show the convergence of the solutions of a time-discrete variant of w.r.t. the time-step size. In particular, we prove a rate of convergence of the generalized stresses $\bSigma$ in $L^\infty(0,T)$ *without* assuming additional regularity, see . Moreover, the convergence of the plastic multiplier $\lambda$ is shown in . These findings improve the results known from the literature. 4. We study the approximation of local minima of by local minima of its time-discrete variant, see . The results of this section are interesting in their own right and they are also used in order to derive necessary optimality conditions of , see . Moreover, these approximation properties are important for a numerical realization of the optimal control problem. There are only few references concerning the optimal control of rate-independent systems, see [@Rindler2008; @Rindler2009; @KocvaraOutrata2005; @KocvaraKruzikOutrata2006; @Brokate1987]. In the case of an infinite dimensional state space (i.e. $W^{1,p}(0,T;X)$, $\dim X = \infty$), there are no contributions providing optimality conditions for rate-independent optimal control problems. [@Rindler2008; @Rindler2009] study the optimal control of rate-independent evolution processes in a general setting. The existence of an optimal control and the approximability by solutions of discretized problems is shown. The system of quasistatic plasticity in its *primal formulation* (see [@HanReddy1999 Section 7]) is contained as a special case. However, in contrast to our analysis, the boundedness of the control in $W^{1,\infty}(0,T;U)$ (cf. [@Rindler2008 Assumption (U)]) is required and no optimality conditions are proven. A finite dimensional situation (i.e. with state space $W^{1,p}(0,T; \R^n)$) is considered in [@KocvaraOutrata2005; @KocvaraKruzikOutrata2006], who consider spatially discretized problems, and [@Brokate1987], who deals with optimal control of an ODE involving a rate-independent part. The *static* (i.e. time-independent) version of the optimal control problem was considered in [@HerzogMeyerWachsmuth2009:2; @HerzogMeyerWachsmuth2010:2]. For locally optimal controls, optimality systems of B- and C-stationarity type were obtained. Let us sketch the outline of this paper. In the remainder of this section, we introduce the notation and fix the functional analytic framework. Moreover, we state the weak formulation of and give some references concerning its analysis. is devoted to the continuous optimal control problem . We give a brief introduction to evolution variational inequalities (EVIs). Restating as an EVI, we are able to prove some new continuity properties of its solution operator. Moreover, we show the weak continuity of the control-to-state map of . In we consider a discretization in time of the control problem . We show two convergence properties of the solution of the time-discrete forward problems, similar to the strong and weak continuity of the solution map of . Finally, these results are used in to prove that *every* local minimizer $\bg$ of the control problem can be approximated by local minimizers of a slightly modified time-discrete problem , see . This paper can be understood as a prerequisite for the analysis contained in [@Wachsmuth2011:2; @Wachsmuth2011:4]. Since the problems under consideration and their analysis are non-standard, the results presented here are believed to be of independent interest. We give a brief overview over [@Wachsmuth2011:2; @Wachsmuth2011:4]. We regularize the time-discrete forward problem and show the Fréchet differentiability of the associated solution map. This result requires some subtle arguments. The regularized time-discrete optimal control problems are differentiable and consequently, optimality conditions can be derived in a straightforward way. The passage to the limit in the regularization parameter $\varepsilon$ leads to an optimality system of C-stationary type for the time-discrete problem. Finally, we pass to the limit with respect to the discretization in time. This part also requires new convergence arguments. Due to the weak mode of convergence of the adjoint variables, the sign condition for the multipliers is lost in the limit and we finally obtain a system of weak stationarity for the optimal control of . Notation and assumptions ------------------------ Our notation follows [@HanReddy1999] and [@HerzogMeyerWachsmuth2009:2]. ### Function spaces {#function-spaces .unnumbered} Let $\Omega \subset \R^d$ be a bounded Lipschitz domain with boundary $\Gamma = \partial\Omega$ in dimension $d = 3$. The boundary consists of two disjoint parts $\Gamma_N$ and $\Gamma_D$. We point out that the presented analysis is not restricted to the case $d = 3$, but for reasons of physical interpretation we focus on the three dimensional case. In dimension $d = 2$, the interpretation of has to be slightly modified, depending on whether one considers the plane strain or plane stress formulation. We denote by $\S := \R^{d \times d}_\textup{sym}$ the space of symmetric $d$-by-$d$ matrices, endowed with the inner product $\bsigma \dprod \btau = \sum_{i,j=1}^d \sigma_{ij} \tau_{ij}$, and we define $$\begin{aligned} V &= H^1_D(\Omega;\R^d) = \{\bu \in H^1(\Omega;\R^d): \bu = \bnull \text{ on } \Gamma_D \}, & S &= L^2(\Omega;\S)\end{aligned}$$ as the spaces for the displacement $\bu$, stress $\bsigma$, and back stress $\bchi$, respectively. The control $\bg$ belongs to the space $$U = L^2(\Gamma_N; \R^d).$$ The control operator $E : U \to V'$, $\bg \mapsto \ell$, which maps boundary forces (i.e. controls) $\bg \in U$ to functionals (i.e. right-hand sides of the weak formulation of , see ) $\ell \in V'$ is given by $$\label{eq:def_E} \dual{\bv}{E\bg}_{V,V'} := -\int_{\Gamma_N} \bv \cdot \bg \, \d s \quad \text{for all } \bv \in V.$$ Hence, $E = -\tau_N^\star$, where $\tau_N$ is the trace operator from $V$ to $U = L^2(\Gamma_N; \R^d)$. Clearly, $E: U \to V'$ is compact. Starting with , we will omit the indices on the duality bracket $\dual{\cdot}{\cdot}$. We denote by $\dual{\cdot}{\cdot}$ the dual pairing between $V$ and its dual $V'$, or the scalar products in $S$ or $S^2$, respectively. This will simplify the notation and cause no ambiguities. For a Banach space $X$ and $p \in [1,\infty]$, we define the Bochner-Lebesgue space $$L^p(0,T;X) = \{u : [0,T] \to X, \; u \text{ is Bochner measurable and $p$-integrable}\}.$$ In the case $p = \infty$ one has to replace $p$-integrability by essential boundedness. The norm in $L^p(0,T;X)$ is given by $$\norm{u}_{L^p(0,T;X)} = \bignorm{ \norm{u(\cdot)}_X }_{L^p(0,T)}.$$ By $W^{1,p}(0,T;X)$ we denote the Bochner-Sobolev space consisting of functions $u \in L^p(0,T;X)$ which possess a weak derivative $\dot u \in L^p(0,T;X)$. Two equivalent norms on $W^{1,p}(0,T;X)$ are given by $$\label{eq:norms_in_w1p} \big( \norm{u}_{L^p(0,T;X)}^p + \norm{\dot u}_{L^p(0,T;X)}^p \big)^{1/p} \quad\text{and}\quad \big( \norm{u(0)}_X^p + \norm{\dot u}_{L^p(0,T;X)}^p \big)^{1/p},$$ where the extension to the case $p = \infty$ is clear. We use $H^1(0,T;X) = W^{1,2}(0,T;X)$. Moreover, we define the space of functions in $H^1(0,T;X)$ vanishing at $t = 0$ $$\label{eq:h01} H_{\{0\}}^1(0,T;X) = \{ u \in H^1(0,T;X) : u(0) = 0\}.$$ Details on Bochner-Lebesgue and Bochner-Sobolev spaces can be found in [@Yosida1965], [@GajewskiGroegerZacharias1974], [@DiestelUhl1977], or [@Ruzicka2004]. ### Yield function and admissible stresses {#yield-function-and-admissible-stresses .unnumbered} We restrict our discussion to the von Mises yield function. In the context of linear kinematic hardening, it reads $$\label{eq:Yield_function} \phi(\bSigma) = \big( \abs{\bsigma^D + \bchi^D}^2 - \tilde \sigma_0^2 \big) / 2$$ for $\bSigma = (\bsigma,\bchi) \in S^2$, where $\abs{\cdot}$ denotes the pointwise Frobenius norm of matrices and $$\label{eq:deviator} \bsigma^D = \bsigma - \frac{1}{d} \, (\trace \bsigma) \, \bI$$ is the deviatoric part of $\bsigma$. The yield function gives rise to the set of admissible generalized stresses $$\KK = \{ \bSigma \in S^2: \phi(\bSigma) \le 0 \quad \text{a.e.\ in } \Omega\}.$$ Let us mention that the structure of the yield function $\phi$ given in implies the *shift invariance* $$\label{eq:shift-invariance} \bSigma \in \KK \quad\Leftrightarrow\quad \bSigma + (\btau, -\btau) \in \KK \quad\text{for all }\btau \in S.$$ This property is exploited quite often in the analysis. Due to the structure of the yield function $\phi$, $\bsigma^D + \bchi^D$ appears frequently and we abbreviate it and its adjoint by $$\label{eq:DD} \DD \bSigma = \bsigma^D + \bchi^D \quad\text{and}\quad \DD^\star\bsigma = \begin{pmatrix} \bsigma^D\\\bsigma^D \end{pmatrix}$$ for matrices $\bSigma \in \S^2$ as well as for functions $\bSigma \in S^2$ and $\bSigma \in L^p(0,T;S^2)$. When considered as an operator in function space, $\DD$ maps $S^2$ and $L^p(0,T;S^2)$ continuously into $S$ and $L^p(0,T;S)$, respectively. For later reference, we also remark that $$\DD^\star \DD \bSigma = \begin{pmatrix} \bsigma^D + \bchi^D \\ \bsigma^D + \bchi^D \\ \end{pmatrix} \quad \text{and} \quad (\DD^\star \DD)^2 = 2\,\DD^\star \DD$$ holds. Due to the definition of the operator $\DD$, the constraint $\phi(\bSigma) \le 0$ can be formulated as $\norm{\DD\bSigma}_{L^\infty(\Omega;\S)} \le \tilde\sigma_0$. Hence, we obtain $$\label{eq:bSigma_in_Linfty} \bSigma \in \KK \quad\Rightarrow\quad \DD\bSigma \in L^\infty(\Omega;\S).$$ Here and in the sequel we denote linear operators, e.g. $\DD : S^2 \to S$, and the induced Nemytzki operators, e.g. $\DD : H^1(0, T; S^2 ) \to H^1(0, T; S)$ and $\DD : L^2(0, T; S^2) \to L^2(0, T; S)$, with the same symbol. This will cause no confusion, since the meaning will be clear from the context. ### Operators {#operators .unnumbered} The linear operators $A : S^2 \to S^2$ and $B : S^2 \to V'$ are defined as follows. For $\bSigma = (\bsigma,\bchi) \in S^2$ and $\bT = (\btau,\bmu) \in S^2$, let $A\bSigma$ be defined through $$\label{eq:Definition_of_a} \dual{\bT}{A\bSigma}_{S^2} = \int_\Omega \btau \dprod \C^{-1} \bsigma \, \dx + \int_\Omega \bmu \dprod \H^{-1} \bchi \, \dx.$$ The term $(1/2) \, \dual{A \bSigma}{\bSigma}_{S^2}$ corresponds to the energy associated with the stress state $\bSigma$. Here $\C^{-1}(x)$ and $\H^{-1}(x)$ are linear maps from $\S$ to $\S$ (i.e., they are fourth order tensors) which may depend on the spatial variable $x$. For $\bSigma = (\bsigma,\bchi) \in S^2$ and $\bv \in V$, let $$\label{eq:Definition_of_b} \dual{B\bSigma}{\bv}_{V',V} = - \int_\Omega \bsigma \dprod \bvarepsilon(\bv) \, \dx.$$ We recall that $\bvarepsilon(\bv) = \big(\nabla \bv + (\nabla \bv)^\top\big)/2$ denotes the (linearized) strain tensor. ### Standing assumptions {#standing-assumptions .unnumbered} Throughout the paper, we require \[asm:standing\_assumptions\] 1. The domain $\Omega \subset \R^d$, $d = 3$ is a bounded Lipschitz domain in the sense of [@Grisvard1985 Chapter 1.2]. The boundary of $\Omega$, denoted by $\Gamma$, consists of two disjoint measurable parts $\Gamma_N$ and $\Gamma_D$ such that $\Gamma = \Gamma_N \cup \Gamma_D$. While $\Gamma_N$ is a relatively open subset, $\Gamma_D$ is a relatively closed subset of $\Gamma$. Furthermore $\Gamma_D$ is assumed to have positive measure. In addition, the set $\Omega \cup \Gamma_N$ is regular in the sense of Gröger, cf. [@Groeger1989]. A characterization of regular domains for the case $d \in \{2,3\}$ can be found in [@HallerDintelmannMeyerRehbergSchiela2009 Section 5]. This class of domains covers a wide range of geometries. 2. The yield stress $\tilde \sigma_0$ is assumed to be a positive constant. It equals $\sqrt{2/3} \, \sigma_0$, where $\sigma_0$ is the uni-axial yield stress. 3. $\C^{-1}$ and $\H^{-1}$ are elements of $L^\infty(\Omega;\LL(\S,\S))$, where $\LL(\S,\S)$ denotes the space of linear operators $\S \to \S$. Both $\C^{-1}(x)$ and $\H^{-1}(x)$ are assumed to be uniformly coercive. Moreover, we assume that $\C^{-1}$ and $\H^{-1}$ are symmetric, i.e., $\btau \dprod \C^{-1}(x) \, \bsigma = \bsigma \dprod \C^{-1}(x) \, \btau$ and a similar relation for $\H^{-1}$ holds for all $\bsigma, \btau \in \S$. is not restrictive. It enables us to apply the regularity results in [@HerzogMeyerWachsmuth2009:3] pertaining to systems of nonlinear elasticity. The latter appear in the time-discrete forward problem and its regularizations. Additional regularity leads to a norm gap, which is needed to prove the differentiability of the control-to-state map. Moreover, implies that Korn’s inequality holds on $\Omega$, i.e., $$\label{eq:Korns_inequality} \norm{\bu}^2_{H^1(\Omega;\R^d)} \le c_K \, \big( \norm{\bu}^2_{L^2(\Gamma_D;\R^d)} + \norm{\bvarepsilon(\bu)}^2_S \big)$$ for all $\bu \in H^1(\Omega;\R^d)$, see e.g. . Note that entails in particular that $\norm{\bvarepsilon(\bu)}_S$ is a norm on $H^1_D(\Omega;\R^d)$ equivalent to the standard $H^1(\Omega;\R^d)$ norm. A further consequence is that $B^\star$ satisfies the inf-sup condition $$\label{eq:inf-sup} \norm{\bu}_V \le \sqrt{c_K} \; \norm{B^\star \bu}_{S^2} \quad\text{for all } \bu \in V.$$ is satisfied, e.g., for isotropic and homogeneous materials, for which $$\C^{-1}\bsigma = \frac{1}{2\,\mu} \bsigma - \frac{\lambda}{2\,\mu\,(2\,\mu+d\,\lambda)} \trace(\bsigma) \, \bI$$ with Lamé constants $\mu$ and $\lambda$, provided that $\mu > 0$ and $d \, \lambda + 2 \, \mu > 0$ hold. These constants appear only here and there is no risk of confusion with the plastic multiplier $\lambda$. A common example for the hardening modulus is given by $\H^{-1} \bchi = \bchi / k_1$ with hardening constant $k_1>0$, see [@HanReddy1999 Section 3.4]. Clearly, shows that $\dual{A\bSigma}{\bSigma}_{S^2} \ge \underline{\alpha} \, \norm{\bSigma}^2_{S^2}$ for some $\underline{\alpha} > 0$ and all $\bSigma \in \S^2$. Hence, the operator $A$ is $S^2$-elliptic. Weak formulation of and known results {#subsec:stress_based_formulation} ------------------------------------- Testing the strong formulation of the equilibrium of forces $$\div \bsigma(t) = \bnull \quad\text{in } \Omega \qquad\text{and}\qquad \bsigma(t) \cdot \bn = \bg(t) \quad\text{on }\Gamma_N$$ with $\bv \in V$ and integrating by parts, we obtain $$\dual{\bSigma(t)}{B^\star \bv}_{S^2} = -\int_\Omega \bsigma(t) \dprod \bvarepsilon(\bv) \, \d x = -\int_{\Gamma_N} \bg(t) \cdot \bv \, \d s = \dual{\ell(t)}{\bv}_{V', V}$$ for all $\bv \in V,$ or equivalently $B\bSigma(t) = \ell(t)$ in $V'$, with $\ell = E \bg$, see . In order to derive the weak formulation of the first two equations of , we fix an arbitrary test function $\bT = (\btau, \bmu) \in \KK$. Testing the first and second equation of with $\btau-\bsigma(t)$ and $\bmu-\bchi(t)$, respectively, we obtain $$\dual{A \dot\bSigma(t) + B^\star \dot\bu(t)}{\bT - \bSigma(t)}_{S^2} + \int_\Omega \lambda(t) \, \DD\bSigma(t)\dprod \big(\DD\bT - \DD\bSigma(t) \big) \, \d x = 0.$$ Due to the complementarity relation in and $\bSigma(t), \bT \in \KK$ we find $$\int_\Omega \lambda(t) \, \DD\bSigma(t)\dprod(\DD\bT - \DD\bSigma(t)) \, \d x \le \int_\Omega \lambda(t) \, \big( \abs{\DD\bSigma(t)} \, \abs{\DD\bT} - \tilde\sigma_0^2 \big) \, \d x \le 0.$$ Hence, we have $$\dual{A \dot\bSigma(t) + B^\star \dot\bu(t)}{\bT - \bSigma(t)}_{S^2} \ge 0 \quad \text{for all } \bT \in \KK.$$ We have derived the weak formulation of in the stress-based (so-called dual) form. It is represented by a time-dependent, rate-independent variational inequality (VI) of mixed type: find generalized stresses $\bSigma \in H^1(0,T;S^2)$ and displacements $\bu \in H^1(0,T;V)$ which satisfy $\bSigma(t) \in \KK$ and $$\label{eq:VI_lower_level_introduction} \tag{$\mathbf{VI}$} \begin{aligned} \dual{A\dot\bSigma(t)+B^\star\dot\bu(t)}{\bT - \bSigma(t)}_{S^2} &\ge \mrep{0}{\ell(t)} \quad \text{for all } \bT \in \KK, \\ B\bSigma(t) &= \ell(t) \quad \text{in } V', \end{aligned}$$ f.a.a. $t \in (0,T)$. Moreover, is subject to the initial condition $(\bSigma(0),\bu(0)) = (\bnull,\bnull)$. In order to guarantee the existence of a solution, we have to require $\ell(0) = \bnull$. Note that a weak formulation involving the plastic multiplier $\lambda$ is given in . The remainder of this section is devoted to known results on the analysis of . We start with the results given in [@HanReddy1999 Section 8]. We point out that the authors handle a general situation which includes the case of kinematic hardening as a special case. ### Reformulation as a sweeping process in $\bSigma$ {#reformulation-as-a-sweeping-process-in-bsigma .unnumbered} For $\ell \in V'$, we denote by $$\label{eq:KK_ell} \KK_\ell = \{ \bSigma \in \KK: B\bSigma = \ell \}$$ the subset of $\KK$ on which the constraint $B\bSigma = \ell$ is fulfilled. Testing the first equation of with $\bT \in \KK_{\ell(t)}$ results in: given $\ell \in H_{\{0\}}^1(0,T;V')$, find $\bSigma \in H_{\{0\}}^1(0,T;V')$, satisfying $\bSigma(t) \in \KK_{\ell(t)}$ and $$\label{eq:VI} \dual{A \dot\bSigma(t) }{ \bT - \bSigma(t) }_{S^2} \ge 0 \qquad \text{for all } \bT \in \KK_{\ell(t)} \text{ and almost all } t \in (0,T).$$ This is called the stress problem of plasticity. It is a time-dependent VI, where the associated convex set $\KK_{\ell(t)}$ changes in time. Such an equation was introduced in [@Moreau1977]. We mention that there are existence and uniqueness results, but no continuity results for the abstract situation considered in [@Moreau1977] seem to be available. ### Stress problem {#stress-problem .unnumbered} [@HanReddy1999 Section 8] deals with the so-called dual formulation . In [@HanReddy1999 Theorem 8.9] the existence and uniqueness of a solution $\bSigma$ of is shown together with the a-priori bound $$\norm{\bSigma}_{H^1(0, T; S^2)} \le C \, \norm{\ell}_{H^1(0, T; V')}. \label{eq:a-priori_bsigma}$$ Let us denote the solution map by $\GG^\bSigma$, i.e. $\bSigma = \GG^\bSigma(\ell)$. Additionally, [@HanReddy1999 Theorem 8.10] shows the local Hölder continuity of index $1/2$ of $\GG^\bSigma : H_{\{0\}}^1(0, T; V') \to L^\infty(0, T; S^2)$, i.e., $$\label{eq:local_1/2_hoelder_dual} \bignorm{\GG^\bSigma(\ell_1) - \GG^\bSigma(\ell_2)}_{L^\infty(0, T; S^2)}^2 \le C \, \big(\norm{\dot\ell_1}_{L^2(0, T; V')} + \norm{\dot\ell_2}_{L^2(0, T; V')}\big) \, \norm{\ell_1 - \ell_2}_{L^2(0, T; V')}.$$ We mention that this is not a typical estimate. As long as the derivatives of $\ell_i$ remain bounded in $L^2(0,T;V')$, one can control the $L^\infty$-norm of $\bSigma_1-\bSigma_2$ solely through the $L^2$-norm of the difference $\ell_1 - \ell_2$. We give a generalization of this estimate in in the context of EVIs. Finally, [@HanReddy1999 Theorem 8.12] shows that one can introduce a (not necessarily unique) multiplier associated with the equilibrium of forces $B\bSigma = \ell$, which can be interpreted as the displacement $\bu$. As a consequence, $(\bSigma, \bu)$ satisfies . Thus, given $\ell \in H_{\{0\}}^1(0,T; V')$, there exists $(\bSigma, \bu) \in H_{\{0\}}^1(0,T; S^2 \times V)$ such that $\bSigma(t) \in \KK$ and \[eq:Lower-Level\_Problem\] $$\begin{aligned} \label{eq:Lower-Level_Problem1} \dual{A\dot\bSigma(t)+B^\star\dot\bu(t)}{\bT - \bSigma(t)}_{S^2} &\ge \mrep{0}{\ell(t)} \quad \text{for all } \bT \in \KK, \\ \label{eq:Lower-Level_Problem2} B\bSigma(t) &= \ell(t) \quad \text{in } V' \end{aligned}$$ holds for almost all $t \in (0, T)$. This shows the equivalence of and . Note that is equivalent to . In the sequel we use either reference as appropriate. ### Kinematic hardening {#kinematic-hardening .unnumbered} In the case of kinematic hardening, the uniqueness of $\bu$ is obtained easily. Let us test with $\bT = \bSigma(t) + (\btau, -\btau) = (\bsigma(t)+\btau, \bchi(t)-\btau)$. Due to the shift invariance , we have $\bT \in \KK$ for all $\btau \in S$. This yields $$\dual{ A \dot\bSigma(t) + B^\star \dot\bu(t) }{ (\btau, -\btau) }_{S^2} = 0 \quad \text{for all }\btau \in S.$$ Using the definitions of $A$ and $B$, see and , respectively, we obtain $$\C^{-1} \dot\bsigma - \bvarepsilon(\dot\bu) - \H^{-1} \dot\bchi = \bnull \quad\text{almost everywhere in } (0,T) \times \Omega.$$ Integrating from $0$ to $t$ and using the initial condition $(\bSigma(0), \bu(0)) = \bnull$ yields $$\label{eq:relation_sigma_chi_u} \C^{-1} \bsigma - \bvarepsilon(\bu) - \H^{-1} \bchi = \bnull \quad\text{almost everywhere in } (0,T) \times \Omega.$$ Together with the inf-sup condition of $B^\star = (-\bvarepsilon, \bnull)$, see , this proves the uniqueness of $\bu$. Using yields the a-priori estimate $$\norm{\bSigma}_{H^1(0, T; S^2)} + \norm{\bu}_{H^1(0, T; V)} \le C \, \norm{\ell}_{H^1(0, T; V')}. \label{eq:a-priori_bsigma_bu}$$ We denote the solution mapping $\ell \mapsto \bu$ of by $\GG^\bu$. Moreover, the solution operator of mapping $\ell \to (\bSigma, \bu)$ is denoted by $\GG = (\GG^\bSigma, \GG^\bu)$. ### Primal problem {#primal-problem .unnumbered} In [@HanReddy1999 Section 7] the primal formulation of is considered. Both formulations are equivalent, see [@HanReddy1999 Theorem 8.3]. For the primal formulation and under the (additional) assumption of kinematic (or combined kinematic-isotropic) hardening, the (global) Lipschitz continuity of the solution operator from $W^{1,1}(0,T; Z')$ to $L^{\infty}(0, T; Z)$ was proven, see [@HanReddy1999 pp. 170–171]. Here, $Z$ is the appropriate function space for the analysis of the primal formulation. Due to the equivalence of the primal and the dual problem, this Lipschitz estimate carries over to the dual problem. We obtain $$\label{eq:lipschitz_continuity_W11_Linfty} \norm{\GG(\ell_1) - \GG(\ell_2)}_{L^\infty(0,T; S^2\times V)} \le C \, \norm{\ell_1 - \ell_2}_{W^{1,1}(0,T;V')}.$$ This Lipschitz estimate was actually already contained in rather unknown works of Gröger, see [@Groeger1978:1; @Groeger1978:2; @Groeger1978]. In [@Groeger1978 Section 4] the system is reformulated as an evolution equation associated with a maximal monotone operator. Then the Lipschitz estimate follows by the classical result [@Brezis1973 Lemma 3.1]. \[rem:continuity\_not\_sufficient\] In order to derive optimality conditions, the continuity results mentioned above are not sufficient. In addition, we need the continuity of $\GG : H_{\{0\}}^1(0,T;V') \to H_{\{0\}}^1(0,T;S^2 \times V)$ on two occasions. First, this continuity is needed to prove the approximability of local solutions by time-discrete minimizers in . Second, the strong convergence of $(\dot\bSigma, \dot\bu)$ in $L^2(0,T;S^2 \times V)$ is needed to pass to the limit in the optimality system, see . The required continuity of $\GG : H_{\{0\}}^1(0,T;V') \to H_{\{0\}}^1(0,T;S^2\times V)$ is shown in by building on some results of [@Krejci1996; @Krejci1998] about evolution variational inequalities (EVIs). To our knowledge, this is a new result for the analysis of . ### Existence, uniqueness and regularity of the plastic multiplier {#existence-uniqueness-and-regularity-of-the-plastic-multiplier .unnumbered} We have already seen that the generalized plastic strain $\bP = -A\bSigma - B^\star \bu$ is unique. Thus by [@HerzogMeyerWachsmuth2010:1] we obtain the existence and uniqueness of the plastic multiplier $\lambda \in L^2(0,T; L^2(\Omega))$ which can be understood as a multiplier associated with the constraint $\bSigma \in \KK$ or rather $\phi(\bSigma) \le 0$. We obtain the system \[eq:Lower-Level\_Problem\_multi\] $$\begin{aligned} A \dot\bSigma + B^\star \dot\bu + \lambda \, \DD^\star \DD \bSigma &= \mrep{\bnull}{\ell} \quad \text{in } L^2(0,T; S^2), \label{eq:Lower-Level_Problem_multi1} \\ B\bSigma &= \ell \quad \text{in } L^2(0,T;V'), \label{eq:Lower-Level_Problem_multi2}\\ 0 \le \lambda \quad \perp \quad \phi(\bSigma) & \le \mrep{0}{\ell} \quad \text{a.e.\ in } (0,T) \times \Omega. \end{aligned}$$ Endowed with the initial condition $(\bSigma(0),\bu(0)) = (\bnull,\bnull)$, this system is equivalent to and . Analysis of the continuous optimal control problem {#sec:continuous} ================================================== In this section we study the continuous (i.e. non-discretized) optimal control problem. In the first subsection, we give a brief introduction to evolution variational inequalities (EVIs) and state the continuity of the solution map of an EVI from $H^1(0,T;X)$ into itself, see . This enables us to show the continuity of the solution map of , see . is devoted to show the weak continuity of the control-to-state map of . Due to this weak continuity, we are able to conclude the existence of an optimal control in , see . Introduction to evolution variational inequalities -------------------------------------------------- In this section we give a short introduction to *evolution variational inequalities*. A comprehensive presentation of this topic can be found in [@Krejci1998]. We also mention the contributions [@KrejciLovicar1990; @Krejci1996; @BrokateKrejci1998]. For convenience of the reader who wishes to consult [@Krejci1998] in parallel, we use the notation of [@Krejci1998] in this subsection and turn back to our notation in the next subsection. Let $X$ be some Hilbert space and $Z \subset X$ be some convex, closed set. Given a function $u : [0,T] \to X$ and an initial value $x_0 \in Z$, find $x : [0,T] \to Z \subset X$ such that $x(0) = x_0$ and $$\label{eq:abstract_evi} \tag{$\mathbf{EVI}$} \scalarprod{\dot u(t) - \dot x(t)}{x(t) - \tilde x}_X \ge 0 \quad \text{for all } \tilde x \in Z.$$ There are many applications that lead to an EVI, see [@Krejci1998 page 2] for references and further comments on this topic. We only mention the strain or stress driven problem of plasticity with linear kinematic hardening. In the strain (stress) driven problem, the strain (stress) is viewed as a known quantity, whereas the stress (strain) has to be determined. We show that the dual formulation is an appropriate starting point for deriving the strain driven problem. Krejčí derived the stress and the strain driven problems in [@Krejci1998 (1.29) and (1.31)], respectively. In the strain driven problem, the strain $\bvarepsilon(\bu)$ (or, equivalently, the strain rate $(\bvarepsilon(\dot\bu),\bnull) = -B^\star \dot\bu$) is viewed as a given quantity. The variational inequality reduces to find a function $\bSigma : [0,T] \to \KK \subset S^2$, such that $$\label{eq:strain_driven_plasticity} \vdual{ \dot\bSigma(t) - \begin{pmatrix} \C \, \bvarepsilon(\dot\bu(t)) \\ 0 \end{pmatrix} }{\bT - \bSigma(t)}_A \ge 0 \quad \text{for all } \bT \in \KK,$$ where $\dual{\cdot}{\cdot}_A$ is the scalar product on $S^2$ induced by $A$, i.e., $$\label{eq:norm_A} \dual{\bSigma}{\bT}_A := \dual{A\bSigma}{\bT}_{S^2} \quad\text{for }\bSigma,\bT \in S^2.$$ Using the coercivity of $A$, which is ensured by , we find that $\dual{\cdot}{\cdot}_A$ is scalar product on $S^2$ which is equivalent to the standard scalar product. The equation is of the form in the Hilbert space $X = S^2$ equipped with the scalar product $\dual{\cdot}{\cdot}_A$ and the feasible set $Z = \KK$. \[rem:differences\_evi\_strain\_driven\] We remark that there is a fundamental difference between the dual formulation and the strain driven problem . Whereas in *both* $\bSigma$ and $\bu$ are unknown quantities, $\bvarepsilon(\bu)$ has to be *a-priori known* in . Moreover, the solution $\bSigma$ of has to fulfill . Let us denote the solution map of which maps $\bu \to \bSigma$ by $\SS$. Therefore, yields $\ell = B\bSigma = B\SS(\bu)$. Thus, is equivalent to $$\text{Find } \bu \text{ such that } B\SS(\bu) = \ell.$$ This technique is used in [@Krejci1996 Chapter III] for hyperbolic equations arising in plasticity, i.e., the author considers plasticity problems where the acceleration term $\rho \, \ddot\bu$ is included in the balance of forces . The following theorem summarizes results on given in [@Krejci1996; @Krejci1998]. \[thm:solution\_of\_evi\] Let $u \in W^{1,1}(0, T; X)$ and $x_0 \in Z$ be given. Then there exists a unique solution $S(x_0, u) := x \in W^{1,1}(0, T; X)$ of . Moreover, $S : Z \times W^{1,1}(0, T; X) \to L^\infty(0, T; X)$ is globally Lipschitz continuous and $S : Z \times W^{1,p}(0, T; X) \to W^{1,p}(0, T; X)$ is continuous for all $p < \infty$. The operator $S$ is called the *stop operator*, whereas $P(x_0,u) = u - S(x_0,u)$ is called the *play operator*. Let us show a result which generalizes the local Hölder estimate . In the context of EVIs this is a new result. It shows that the play operator $P$ possesses an additional smoothing property which is not shared by the stop operator $S$. Another such property is that $P$ maps $C(0,T;X)$ (continuous functions mapping $[0,T] \to X$) to $CBV(0,T;X)$ (continuous functions of bounded variation mapping $[0,T] \to X$), see [@Krejci1998 Theorem 3.11]. Again, this does not hold for the stop operator $S$. \[thm:local\_1/2\_hoelder\_evi\] Let $p \in [1, \infty]$ be given and denote by $q$ its dual exponent. Let $u_1, u_2 \in W^{1,q}(0,T; X)$ and initial values $x_1^0, x_2^0 \in Z$ be given. Let $\xi_i = P(x_i^0, u_i)$, $i = 1,2$. Then the estimate $$\begin{aligned} \norm{\xi_1-\xi_2}_{L^\infty(0,T; X)}^2 &\le 2 \, \big( \norm{\dot u_1}_{L^q(0,T; X)} + \norm{\dot u_2}_{L^q(0,T; X)} \big) \norm{ u_1 - u_2 }_{L^p(0,T; X)} \notag \\ & \notag \qquad + \norm{\xi_1(0) - \xi_2(0)}_X^2 \end{aligned}$$ holds, where $\xi_i(0) = u_i(0) - x_i^0$, $i = 1,2$. First we recall [@Krejci1998 (3.16ii)], i.e., $\scalarprod{\dot\xi_1(t)}{\dot x_1(t)}_X = 0$ a.e. in $(0,T)$, where $x_1 = S(x_1^0,u_1)$. Using $x_1 = u_1 - \xi_1$, this implies $\norm{\dot\xi_1(t)}_X \le \norm{\dot u_1(t)}_X$ a.e. in $(0,T)$. Analogously, we obtain $\norm{\dot\xi_2(t)}_X \le \norm{\dot u_2(t)}_X$ a.e. in $(0,T)$. Using $\tilde x = u_2(t) - \xi_2(t) = x_2(t) \in Z$ as a test function in shows $$\scalarprod{\dot\xi_1(t)}{u_1(t) - u_2(t) + \xi_2(t) - \xi_1(t)}_X \ge 0 \quad\text{a.e.\ in }(0,T).$$ Similarly, we obtain $$\scalarprod{\dot\xi_2(t)}{u_2(t) - u_1(t) + \xi_1(t) - \xi_2(t)}_X \ge 0 \quad\text{a.e.\ in }(0,T).$$ Adding these inequalities, we have $$\scalarprod{\dot\xi_1(t) - \dot\xi_2(t)}{\xi_1(t) - \xi_2(t)}_X \le \scalarprod{\dot\xi_1(t) - \dot\xi_2(t)}{u_1(t) - u_2(t)}_X \quad\text{a.e.\ in }(0,T).$$ This shows $$\begin{aligned} \frac12 \, \frac{\d}{\d t} \, \norm{\xi_1 - \xi_2}_X^2 & = \scalarprod{\dot\xi_1(t) - \dot\xi_2(t)}{\xi_1(t) - \xi_2(t)}_X \\ & \le \scalarprod{\dot\xi_1(t) - \dot\xi_2(t)}{u_1(t) - u_2(t)}_X \\ & \le (\norm{\dot\xi_1(t)}_X + \norm{\dot\xi_2(t)}_X) \, \norm{u_1(t) - u_2(t)}_X \\ & \le (\norm{\dot u_1(t)}_X + \norm{\dot u_2(t)}_X) \, \norm{u_1(t) - u_2(t)}_X \quad\text{a.e.\ in }(0,T). \end{aligned}$$ Integrating from $0$ to $t$ yields $$\begin{aligned} & \frac12 \, \norm{\xi_1(t) - \xi_2 (t)}_X^2 - \frac12 \, \norm{\xi_1(0) - \xi_2(0)}_X^2 \\ & \qquad \le \int_0^t (\norm{\dot u_1(s)}_X + \norm{\dot u_2(s)}_X) \, \norm{u_1(s) - u_2(s)}_X \, \d s \\ & \qquad \le (\norm{\dot u_1}_{L^q(0,T; X)} + \norm{\dot u_2}_{L^q(0,T; X)}) \, \norm{u_1 - u_2}_{L^p(0,T; X)}. \end{aligned}$$ Taking the supremum $t \in (0,T)$ yields the claim. Quasistatic plasticity as an EVI {#sec:evolution_vi} -------------------------------- In this section we give an equivalent reformulation of which fits into the framework of [@Krejci1998]. Once this reformulation is established, the continuity of $\GG : H_{\{0\}}^1(0, T; V') \to H_{\{0\}}^1(0, T; S^2 \times V)$ is a consequence of . As mentioned in , the continuity of $\GG$ is used in several places. In order to reformulate , we need some preparatory work. Due to the inf-sup condition we obtain that for all $\ell \in V'$ there is a $\bsigma_\ell \in S$, such that $$\label{eq:bsigma(ell)} (\bsigma_\ell,\bnull) \in (\ker B)^\perp \quad\text{and}\quad B(\bsigma_\ell,\bnull) = \ell,$$ see [@Brezzi1974]. Moreover, this mapping $\ell \mapsto \bsigma_\ell$ is linear and continuous. For $\ell \in H_{\{0\}}^1(0,T;V')$, we define $\bsigma_\ell(t) := \bsigma_{\ell(t)}$. Hence, $\bsigma_{(\cdot)}$ maps $H_{\{0\}}^1(0,T;V') \to H_{\{0\}}^1(0,T;S)$ continuously. For convenience, we also introduce the notation $$\label{eq:bSigma(ell)} \bSigma_\ell = (\bsigma_\ell, -\bsigma_\ell).$$ Let us mention that this is a useful tool to construct test functions $\bT$ for and . Indeed, for arbitrary $\bSigma \in \KK$ and $\ell \in V'$, let us define the test function $\bT = \bSigma + \bSigma_{\ell - B\bSigma}$. The shift invariance implies $\bT \in \KK$ and $B\bT = \ell$ is ensured by the definition of $\bSigma_\ell$. Hence we obtain $\bT \in \KK_\ell$, i.e., it is an admissible test function for and . Let $\ell \in H_{\{0\}}^1(0,T;V')$ be arbitrary. Using we are able to reformulate as an EVI. As stated in , $\bSigma \in \KK$ holds if and only if $\bSigma + (\btau, -\btau) \in \KK$ holds for all $\btau \in S$. Since $\bSigma_\ell = (\bsigma_\ell, -\bsigma_\ell)$ by definition, $\bSigma \in \KK$ if and only if $\bSigma - \bSigma_\ell \in \KK$. This gives rise to the decomposition $\bSigma = \bSigma_0 + \bSigma_\ell$. There are two immediate consequences: $\bSigma \in \KK$ is equivalent to $\bSigma_0 \in \KK$ and $B\bSigma = \ell$ is equivalent to $\bSigma_0 \in \ker B$. Therefore, $\bSigma_0 \in \KK \cap \ker B$. We define $\KK_B := \KK \cap \ker B$. Let $\bT \in \KK_B$ be arbitrary. Testing or with $\bT + \bSigma_{\ell}(t) \in \KK$ yields an equivalent reformulation: find $\bSigma_0 \in \KK_B$ such that $$\dual{ A( \dot\bSigma_0 + \dot\bSigma_\ell) }{ \bT - \bSigma_0} \ge 0 \quad \text{for all } \bT \in \KK_B. \label{eq:eq:Lower-Level_Problem_evolution_vi}$$ Hence, is an EVI in the Hilbert space $S^2$ equipped with $\dual{\cdot}{\cdot}_A$. yields the continuity of the mapping $\bSigma_\ell \mapsto \bSigma_0$ from $H_{\{0\}}^1(0, T; S^2)$ to $H_{\{0\}}^1(0, T; S^2)$. Since $\ell \mapsto \bSigma_\ell$ is continuous from $H_{\{0\}}^1(0,T;V') \to H_{\{0\}}^1(0,T;S^2)$, we obtain the continuity of the mapping $\ell \mapsto \bSigma$ from $H_{\{0\}}^1(0,T; V')$ to $H_{\{0\}}^1(0,T; S^2)$. The uniqueness and continuous dependence of $\bu$ on $\ell$ follows from and the inf-sup condition of $B^\star$, see . Moreover, also shows the Lipschitz continuity of $\GG: \ell \mapsto (\bSigma, \bu)$, $W^{1,1}(0,T; V') \to L^\infty(0,T; S^2 \times V)$, see . Summarizing, we have found \[cor:continuity\_GG\_in\_H1\] The solution operator $\GG$ of is continuous from $H_{\{0\}}^1(0,T;V')$ to $H_{\{0\}}^1(0,T;S^2\times V)$ and globally Lipschitz continuous to $L^\infty(0,T;S^2 \times V)$. The technique presented in this subsection is not restricted to kinematic hardening only, the arguments remain valid for all hardening models involving some kinematic part, see also [@Groeger1978]. To be precise, the technique is applicable whenever the hardening variable can be split into a kinematic part $\bchi$ and some other part $\boldeta$, such that the yield function $\phi$ can be written as $\phi(\bSigma) = \tilde\phi( \bsigma+\bchi, \boldeta )$, where $\bSigma = (\bsigma, \bchi, \boldeta)$. It is interesting to note that this is exactly the case for which [@HanReddy1999] were able to prove Lipschitz continuity of the forward operator (in primal formulation) from $W^{1,1}(0,T; Z')$ to $L^\infty(0,T; Z)$, see the discussion in . Weak continuity of the forward operator {#subsec:weak_continuity} --------------------------------------- In this section we show the weak continuity of the control-to-state map $\GG \circ E$ of the optimal control problem . Here, $\GG$ is the solution map of and $E$ is the control operator, see . Since the solution operator $\GG \circ E$ is nonlinear, the proof of its weak continuity is non-trivial. The weak continuity of $\GG \circ E$ from $H_{\{0\}}^1(0, T; U)$ to $H_{\{0\}}^1(0, T; S^2 \times V)$ is essential for proving the existence of an optimal control in . A main ingredient to prove the weak continuity is the compactness of the control operator $E$ from $U = L^2(\Gamma_N; \R^d)$ to $V' = (H^1_D(\Omega; \R^d))'$, see the proof of . Due to this compactness, Aubin’s lemma, see, e.g., [@Simon1986 Equation (6.5)], implies that $H^1(0,T; U)$ embeds compactly into $L^2(0,T; V')$. Let us mention that the weak continuity of $\GG \circ E$ is a non-trivial result: although $E : U \to V'$ is compact, the operator $E : H^1(0, T; U) \to H^1(0, T; V')$ (applied in a pointwise sense) is not compact. Hence the weak continuity of $\GG \circ E$ does not simply follow from the compactness of $E$ nor the continuity of $\GG$. First, we need a preliminary result. It can be interpreted as an upper semicontinuity result for the left-hand side of . Let $\{(\bSigma_k, \bu_k)\} \subset H_{\{0\}}^1(0, T; S^2 \times V)$ and $(\bSigma, \bu) \in H^1(0, T; S^2 \times V)$ be given. Moreover, we assume that there are $\bg_k, \bg \in H^1(0,T; U)$ such that $B\bSigma_k = E \bg_k$ and $B\bSigma = E \bg$ hold. If $$\begin{aligned} (\bSigma_k, \bu_k) &\weakly (\bSigma, \bu) \text{ in } H^1(0, T; S^2 \times V) \quad \text{and}\\ \bg_k &\weakly \mrep{\bg}{(\bSigma, \bu)} \text{ in } H^1(0, T; U), \end{aligned}$$ then for all $\bT \in L^2(0, T; S^2)$ we have $$\limsup_{k \to \infty} \int_0^T \dual{A \dot\bSigma_k + B^\star \dot\bu_k }{ \bT - \bSigma_k } \, \d t \le \int_0^T \dual{A \dot{\bSigma} + B^\star \dot{\bu} }{ \bT - \bSigma } \, \d t.$$ \[lem:limsup\] Due to the weak convergence of $(\bSigma_k, \bu_k)$ in $H^1(0,T; S^2 \times V)$ we have $$\int_0^T \dual{A \dot\bSigma_k + B^\star \dot\bu_k }{ \bT } \, \d t \to \int_0^T \dual{A \dot{\bSigma} + B^\star \dot{\bu} }{ \bT } \, \d t.$$ Since $U$ embeds compactly into $V'$, Aubin’s lemma yields that $H^1(0, T; U)$ embeds compactly into $L^2(0, T; V')$ and hence $\bg_k \weakly \bg$ in $H^1(0, T; U)$ implies $E \bg_k \to E \bg$ in $L^2(0, T; V')$. This shows $$\begin{aligned} \int_0^T \dual{B^\star \dot\bu_k }{ \bSigma_k } \, \d t &= \int_0^T \dual{\dot\bu_k }{ B \bSigma_k } \, \d t = \int_0^T \dual{\dot\bu_k }{ E \bg_k } \, \d t \\ &\to \int_0^T \dual{\dot{\bu} }{ E \bg } \, \d t = \int_0^T \dual{\dot{\bu} }{ B\bSigma } \, \d t = \int_0^T \dual{B^\star \dot{\bu} }{ \bSigma } \, \d t. \end{aligned}$$ To address the remaining term, we use integration by parts and obtain $$\int_0^T \dual{ A\dot\bSigma_k }{ \bSigma_k } \, \d t = \frac12 \big( \dual{A\bSigma_k(T)}{\bSigma_k(T)} - \dual{A\bSigma_k(0)}{\bSigma_k(0)} \big).$$ The functionals $\bSigma \mapsto \dual{A\bSigma(t)}{\bSigma(t)}$ are continuous (w.r.t. the $H^1(0,T;S^2)$-norm) and convex, hence weakly lower semicontinuous. Due to $\bSigma_k(0) = \bnull$, this already implies $\dual{A\bSigma(0)}{\bSigma(0)} = 0$. Now the weak lower semicontinuity of $\bSigma \mapsto \dual{A\bSigma(T)}{\bSigma(T)}$ yields $$\liminf_{k\to\infty} \dual{A\bSigma_k(T)}{\bSigma_k(T)} \ge \dual{A\bSigma(T)}{\bSigma(T)}.$$ Therefore, $$\liminf_{k\to\infty} \int_0^T \dual{ A\dot\bSigma_k }{ \bSigma_k } \, \d t \ge \frac12 \dual{A\bSigma(T)}{\bSigma(T)} = \int_0^T \dual{ A\dot{\bSigma} }{ \bSigma } \, \d t$$ holds. By combining the results above, we obtain the assertion $$\limsup_{k\to\infty} \int_0^T \dual{A \dot\bSigma_k + B^\star \dot\bu_k }{ \bT - \bSigma_k } \, \d t \le \int_0^T \dual{A \dot{\bSigma} + B^\star \dot{\bu} }{ \bT - \bSigma } \, \d t.$$ The previous lemma enables us to prove The operator $\GG \circ E$ is weakly continuous from $H_{\{0\}}^1(0, T; U)$ to $H_{\{0\}}^1(0, T; S^2 \times V)$. \[thm:weak\_continuity\_solution\_operator\] Let $\{ \bg_k \} \subset H_{\{0\}}^1(0, T; U)$ be a sequence converging weakly towards $\bg \in H_{\{0\}}^1(0, T; U)$. The solution of with right-hand side $E\bg$, $E\bg_k$ is denoted by $(\bSigma, \bu)$, $(\bSigma_k, \bu_k)$, respectively, i.e., $(\bSigma_k, \bu_k) = \GG( E \bg_k)$ and $(\bSigma, \bu) = \GG( E \bg )$. Due to the boundedness of $E$ and $\GG$, see , there exists a subsequence (for simplicity denoted by the same symbol) $\{ ( \bSigma_k, \bu_k ) \}$ which converges weakly towards some $(\tilde\bSigma, \tilde\bu)$ in $H^1(0,T;S^2 \times V)$. We shall prove $(\tilde\bSigma, \tilde\bu) = (\bSigma, \bu)$, therefore the whole sequence converges weakly and this simplification of notation is justified. To this end, we show that $(\tilde\bSigma, \tilde\bu)$ is a solution to with right-hand side $E\bg$. First we address the admissibility of $\tilde\bSigma$ in : the set $\{ \bT \in H^1(0, T; S^2): \bT(t) \in \KK \text{ f.a.a.\ } t \in (0,T)\}$ is convex and closed, hence weakly closed. Since all $\bSigma_k$ belong to this set, $\tilde\bSigma$ belongs to it as well. Therefore $\tilde\bSigma(t) \in \KK$ holds for almost all $t \in (0,T)$. Since $B$ is linear and bounded, it is weakly continuous and hence $B\bSigma_k \weakly B\tilde\bSigma$ in $H^1(0, T; V')$. Due to $B\bSigma_k = E\bg_k \weakly E\bg$ in $H^1(0,T; V')$, we have $B\tilde\bSigma = E\bg$ in $H^1(0, T; V')$, this shows . Let $\bT \in L^2(0, T; S^2)$ with $\bT(t) \in \KK$ f.a.a. $t \in (0,T)$ be given. This implies $$\dual{A \dot\bSigma_k(t) + B^\star \dot\bu_k(t) }{ \bT(t) - \bSigma_k(t) } \ge 0 \quad\text{f.a.a.\ } t \in [0,T],$$ see . Integrating w.r.t. $t$ yields $$\int_0^T \dual{A \dot\bSigma_k + B^\star \dot\bu_k }{ \bT - \bSigma_k } \, \d t \ge 0.$$ By applying , we obtain $$\int_0^T \dual{A \dot{\tilde\bSigma} + B^\star \dot{\tilde\bu} }{ \bT - \tilde\bSigma } \, \d t \ge \limsup_{k\to\infty} \int_0^T \dual{A \dot\bSigma_k + B^\star \dot\bu_k }{ \bT - \bSigma_k } \, \d t \ge 0.$$ Since $\bT \in L^2(0, T; S^2)$ with $\bT(t) \in \KK$ f.a.a. $t \in (0,T)$ was arbitrary, $$\label{eq:in_the_proof_of_some_thm} \dual{A \dot{\tilde\bSigma}(t) + B^\star \dot{\tilde\bu}(t) }{ \bT - \tilde\bSigma(t) } \ge 0 \quad\text{for all } \bT \in \KK$$ holds for every common Lebesgue point $t$ of the functions $$\begin{aligned} t &\mapsto \dual{A \dot{\tilde\bSigma}(t) + B^\star \dot{\tilde\bu}(t) }{ \tilde\bSigma(t) } \in L^1(0,T; \R) \\ \text{and} \quad t &\mapsto \phantom{\langle{}} A \dot{\tilde\bSigma}(t) + B^\star \dot{\tilde\bu}(t) \in L^2(0,T; S^2). \end{aligned}$$ Lebesgue’s differentiation theorem, see [@Yosida1965 Theorem V.5.2], yields that almost all $t \in (0,T)$ are Lebesgue points of these functions. This implies that holds for almost all $t \in (0,T)$. Hence $(\tilde\bSigma, \tilde\bu)$ solves . Since the solution is unique we obtain $(\tilde\bSigma, \tilde\bu) = (\bSigma, \bu)$. This implies that the weak limit is indeed independent of the chosen subsequence. Hence, the whole sequence converges weakly, i.e. $(\bSigma_k, \bu_k) \weakly (\bSigma, \bu)$ in $H^1(0,T;S^2 \times V)$. That is, $\GG(E\bg_k) = (\bSigma_k, \bu_k) \weakly (\bSigma, \bu) = \GG(E\bg)$ in $H^1(0,T;S^2 \times V)$. Since $\{\bg_k\}$ and $\bg$ were arbitrary, $\GG \circ E$ is weakly continuous. Existence of optimal controls {#subsec:continuous_ulp} ----------------------------- In this section we prove the existence of an optimal control. The key tool in the proof of is the weak continuity of the forward operator proven in . For convenience, we repeat the definition of the optimal control problem under consideration $$\label{eq:continuous_ulp} \tag{$\mathbf{P}$} \left. \begin{aligned} \text{Minimize}\quad & F(\bu,\bg) = \psi( \bu ) + \frac{\nu}{2} \norm{\bg}_{H^1(0,T;U)}^2 \\ \text{such that}\quad & (\bSigma, \bu) = \GG(E\bg) \\ \text{and}\quad & \bg \in {U_\textup{ad}}. \end{aligned} \quad\right\}$$ The admissible set ${U_\textup{ad}}$ is a convex closed subset of $H_{\{0\}}^1(0,T;U)$. Here, $\norm{\cdot}_{H^1(0,T;U)}$ can be any norm such that $H^1(0,T;U)$ is a Hilbert space, see . Let us fix the assumptions on the objective $\psi$ and on the set of admissible controls ${U_\textup{ad}}$. \[asm:psi\_lsc\] 1. The function $\psi : H^1(0,T; V) \to \R$ is weakly lower semicontinuous, continuous and bounded from below. 2. The cost parameter $\nu$ is a positive, real number. 3. The admissible set ${U_\textup{ad}}$ is nonempty, convex and closed in $H_{\{0\}}^1(0,T;U)$. Let us give some examples which satisfy these conditions. Since the domain of definition of $\psi$ is $H^1(0,T;V)$ we can track the displacements, the strains or even their time derivatives (or point evaluations). As examples for $\psi$ we mention $$\begin{aligned} \psi^1(\bu) &= \frac12\,\norm{\bu - \bu_d}_{L^2(0,T; L^2(\Omega; \R^d))}^2, & \psi^2(\bu) &= \frac12 \, \norm{\bu(T) - \bu_{T,d}}_{L^2(\Omega; \R^d)}^2, \\ \psi^3(\bu) &= \frac12 \, \norm{\bvarepsilon(\bu(T)) - \bvarepsilon_{T,d}}_{L^2(\Omega; \S)}^2.\end{aligned}$$ Here, $\bu_d \in L^2(0,T;L^2(\Omega;\R^d))$, $\bu_{T,d} \in L^2(\Omega; \R^d)$ and $\bvarepsilon_{T,d} \in L^2(\Omega; \S)$ are desired displacements and strains, respectively. Let us give some examples for the admissible set ${U_\textup{ad}}$. The sets $$\begin{aligned} {U_\textup{ad}}^1 &= \{ \bg \in H^1(0,T;U) : \bg(0) = \bnull, \norm{\bg}_{L^2(\Gamma_N; \R^d)} \le \rho \text{ a.e.\ in } (0,T) \} \\ {U_\textup{ad}}^2 &= \{ \bg \in H^1(0,T;U) : \bg(0) = \bg(T) = \bnull \}\end{aligned}$$ satisfy for every $\rho \ge 0$. Of special interest is the combination of $\psi^2$ and ${U_\textup{ad}}^2$. Since we consider a quasistatic process, the condition $\bg(T) = \bnull$ implies that at time $t = T$ the solid body is unloaded. Due to the plastic behaviour, the remaining (and lasting) deformation $\bu(T)$ is typically non-zero. Due to the choice of $\psi^2$, the displacement is controlled towards the desired deformation $\bu_{T,d}$. Thus, choosing $\psi^2$ and ${U_\textup{ad}}^2$ allows the control of the springback of the solid body. This is of great interest in applications, e.g. deep-drawing of metal sheets. Using standard arguments, see, e.g. [@Troeltzsch2010:1 Theorem 4.15], the existence of a global minimizer of is a straightforward consequence of the weak continuity of $\GG\circ E$ proven in . There exists a global minimizer of . \[thm:existence\_continuous\] Time discretization {#sec:time_discretization} =================== In this section we study a discretization in time of the optimal control problem . The time-discrete version of the forward system is introduced in . The strong convergence of the time-discrete solutions $(\bSigma^\tau,\bu^\tau)$ is shown in . In particular, we prove a new rate of convergence of $(\bSigma^\tau,\bu^\tau)$ in $L^\infty(0,T;S^2 \times V)$ without assuming additional regularity of $\bSigma$, see . Moreover, we show the convergence of $(\bSigma^\tau,\bu^\tau)$ in $H^1(0,T;S^2\times V)$ (by using an idea of [@Krejci1998]) and the convergence of $\lambda^\tau$ in $L^2(0,T;L^2(\Omega))$, see and \[thm:strong\_convergence\_lambda\], respectively. These strong convergence results are new for the analysis of the quasistatic problem and both are essential for passing to the limit in the optimality system in . In we prove the weak convergence of $(\bSigma^\tau,\bu^\tau)$ in $H^1(0,T;S^2 \times V)$ assuming the weak convergence of the controls $\bg^\tau$ in $H^1(0,T;U)$. This result is necessary in order to show in that *every* local minimizer of can be approximated by local minimizers of (slightly modified) time-discrete problems , see . This approximability of local minimizers is essential for the derivation of *necessary* optimality conditions which are satisfied for *every* local minimizer of in . ### Notation {#notation .unnumbered} Throughout the paper we partition the time horizon $[0,T]$ into $N$ intervals, each of constant length $\tau = T/N$ for simplicity. We use a superscript $\tau$, i.e. $(\cdot)^\tau$, to indicate variables and operators associated with the discretization in time. For a time-discrete variable $f^\tau \in X^N$, where $X$ is some Banach space, the components of $f^\tau$ are denoted by $f\taui$, $i = 1,\ldots,N$. When necessary, we may refer to $f^\tau_0$ with a pre-defined value (interpreted as an initial condition), mostly $f^\tau_0 = 0$. If $f^\tau \in X^N$ is the time-discretization of a variable $f \in H_{\{0\}}^1(0, T; X)$, we identify $f^\tau$ with its *linear* interpolation, i.e., for $t \in [(i-1)\,\tau, i\,\tau]$ we define $$\label{eq:linear_interpolation} f^\tau(t) = \frac{t - (i-1)\,\tau}{\tau} \, f\taui + \frac{i\,\tau - t}{\tau} \, f\tauim,$$ with $f^\tau_0 = 0$. Therefore, $X^N$ is identified with a subspace of $H_{\{0\}}^1(0,T; X)$. We make use of this identification for the variables $\bSigma$, $\bu$ and $\bg$. For later reference, we remark $$ \dot f^\tau(t) = \frac{1}{\tau} \, (f\taui - f\tauim) \quad\text{and}\quad \label{eq:linear_interpolation_vs_constant} f\taui = f^\tau(t) - ( t - i \, \tau ) \, \dot f^\tau(t)$$ for almost all $t \in ((i-1)\,\tau, i\,\tau)$ and $i = 1,\ldots,N$. On the other hand, if $f^\tau \in X^N$ is the discretization in time of a variable belonging to $L^2(0, T; X)$, we identify $f^\tau$ with a piecewise *constant* function in $L^2(0,T;X)$, i.e., for $t \in [(i-1)\,\tau, i\,\tau)$ we define $$\label{eq:constant_interpolation} f^\tau(t) = f\taui.$$ This is used for the plastic multiplier $\lambda$. Introduction of a discretization in time of the forward problem {#subsec:introduction_time_discretization} --------------------------------------------------------------- In this section we introduce a discretization in time of the forward problem . Replacing the time derivatives by backward differences, we obtain the discretized problem: given $\ell^\tau \in (V')^N$, find $(\bSigma^\tau, \bu^\tau) \in (S^2 \times V)^N$ such that $\bSigma\taui \in \KK$ and \[eq:Lower-Level\_Problem\_semidiscretized\] $$\begin{aligned} \dual{ A (\bSigma\taui- \bSigma\tauim) + B^\star (\bu\taui-\bu\tauim) }{ \bT - \bSigma\taui } &\ge \mrep{0}{\ell\taui} \quad \text{for all }\bT \in \KK, \label{eq:Lower-Level_Problem_semidiscretized1} \\ B\bSigma\taui &= \ell\taui \quad \text{in } V', \label{eq:Lower-Level_Problem_semidiscretized2} \end{aligned}$$ holds for all $i \in \{1,\ldots,N\}$, where $(\bSigma^\tau_0,\bu^\tau_0) = \bnull$. We denote the solution operator which maps $\ell^\tau \to (\bSigma^\tau, \bu^\tau)$ by $\GG^\tau$. Let us mention that, for fixed $i$, can be interpreted as the solution of a *static* plasticity problem. Several properties of this time discretization are proven in [@HanReddy1999 Proof of Theorem 8.12, page 196]. We recall the results which are important for the following analysis. In order to show an analog to for the time-discrete problem, we test with $\bT = \bSigma\taui + (\btau, -\btau)$, where $\btau \in S$ is arbitrary. Using $(\bSigma^\tau_0, \bu^\tau_0) = \bnull$ we obtain $$\label{eq:relation_sigma_chi_u_tau} \C^{-1} \bsigma^\tau - \bvarepsilon(\bu^\tau) - \H^{-1} \bchi^\tau = \bnull \quad\text{a.e.\ in } (0,T) \times \Omega.$$ In [@HanReddy1999 Lemma 8.8 and (8.37)], we find the a-priori estimate $$\label{eq:a-priori_time_bsigma} \norm{\bSigma^\tau}_{H^1(0,T; S^2)} \le C \, \norm{\ell^\tau}_{H^1(0,T; V')}.$$ Together with the inf-sup condition of $B^\star$ and , we obtain $$\label{eq:a-priori_time} \norm{\bSigma^\tau}_{H^1(0,T; S^2)} + \norm{\bu^\tau}_{H^1(0,T; V)} \le C \, \norm{\ell^\tau}_{H^1(0,T; V')}.$$ Due to this a-priori estimate, [@HanReddy1999] are able to prove the weak convergence of $(\bSigma^\tau,\bu^\tau) = \GG^\tau(\ell^\tau)$ towards the solution $(\bSigma,\bu) = \GG(\ell)$ of in $H^1(0,T;S^2\times V)$ if the data $\ell\taui$ in is chosen as the point evaluation of $\ell$. From an optimization point of view, this result is too weak for two reasons. First, the convergence is not w.r.t. the strong topology in $H^1(0,T;S^2\times V)$. Second, the convergence is proven for a restricted choice of $\ell^\tau$. We therefore strengthen the results of [@HanReddy1999] in . As for the continuous problem, the time-discrete problem can be formulated as a complementarity system. The results in prove the existence of $\lambda^\tau \in L^2(\Omega)^N$ such that the complementarity system \[eq:Lower-Level\_Problem\_semidiscretized\_multi\] $$\begin{aligned} A (\bSigma\taui-\bSigma\tauim) + B^\star (\bu\taui-\bu\tauim) + \tau \, \lambda\taui \, \DD^\star \DD \bSigma\taui &= \mrep{\bnull}{\ell\taui} \quad \text{in } S^2, \label{eq:Lower-Level_Problem_semidiscretized_multi1} \\ B\bSigma\taui &= \ell\taui \quad \text{in } V', \label{eq:Lower-Level_Problem_semidiscretized_multi2}\\ 0 \le \lambda\taui \quad \perp \quad \phi(\bSigma\taui) & \le \mrep{0}{\ell\taui} \quad \text{a.e.\ in } \Omega \label{eq:Lower-Level_Problem_semidiscretized_multi3} \end{aligned}$$ is satisfied. Here, we scaled the multiplier $\lambda^\tau$ appropriately. Similarly to we obtain a representation formula for $\lambda^\tau$, $$\label{eq:-h_chi=lambda_DD_bsigma_discrete} -\C^{-1} (\bsigma\taui - \bsigma\tauim) + \bvarepsilon( \bu\taui - \bu\tauim ) = -\H^{-1} (\bchi\taui - \bchi\tauim) = \tau \, \lambda\taui \, \DD\bSigma\taui.$$ \[rem:same\_time\_discretization\_as\_krejci\] The same time discretization approach is used in [@Krejci1996; @Krejci1998] to prove the existence of a solution of . Indeed, by applying the time discretization [@Krejci1998 (3.6)] to the reduced formulation , we obtain: find $\bSigma\taui \in \KK_{\ell\taui}$, such that $$ \dual{ A (\bSigma\taui- \bSigma\tauim) }{ \bT - \bSigma\taui } \ge 0 \quad \text{for all }\bT \in \KK_{\ell\taui}.$$ As discussed for the continuous problem , we can introduce a Lagrange multiplier associated with $B\bSigma\taui = \ell\taui$ such that $(\bSigma^\tau, \bu^\tau)$ satisfy . In the following two subsections we prove that the discretization in time converges in a strong and a weak sense, similar to the continuity properties of the forward operator $\GG$ proven in and \[subsec:weak\_continuity\]. Under the assumption that $\ell^\tau$ converges strongly to $\ell$ in the space $H^1(0,T;V')$ or that $\bg^\tau$ converges weakly in the space $H^1(0,T;U)$, the corresponding solutions $(\bSigma^\tau, \bu^\tau)$ are shown to converge strongly (or weakly) to $(\bSigma, \bu)$ in $H^1(0,T;S^2\times V)$, respectively. Both statements will be needed in to show the approximability of local solutions of mentioned in the introduction of this section. Strong convergence {#subsec:strong_convergence_forward} ------------------ The strong convergence of the discretization in time of an EVI can be found in [@Krejci1996 Proposition I.3.11]. He proves the convergence in $W^{1,1}(0,T;X)$ of the time-discrete solutions. The data of the time-discretized problems are taken as the point evaluations of the data of the continuous problem. We use these ideas to prove the convergence of $(\bSigma^\tau, \bu^\tau)$ in $H^1(0,T; S^2 \times V)$ whenever the right-hand sides $\ell^\tau$ converge in $H^1(0,T;V')$ as $\tau \searrow 0$. Moreover, we show the convergence of $\lambda^\tau$ in $L^2(0,T;L^2(\Omega))$ as $\tau \searrow 0$. This is a new result for the analysis of and a crucial ingredient for proving the optimality system in . Let us consider a sequence of time steps $\{\tau_k\}$ and a sequence of loads $\ell^{\tau_k} \in (V')^{N_k}$, with $N_k \, \tau_k = T$, such that the linear interpolants $\ell^{\tau_k} \in H_{\{0\}}^1(0,T; V')$ converge in $H^1(0,T; V')$ towards some $\ell \in H_{\{0\}}^1(0,T; V')$. We denote by $(\bSigma^{\tau_k}, \bu^{\tau_k}) \in (S^2 \times V)^{N_k}$ the solutions of the time-discrete problem , as well as their piecewise linear interpolants. The aim of this section is to prove $(\bSigma^{\tau_k}, \bu^{\tau_k}) \to (\bSigma, \bu)$ in $H^1(0,T; S^2 \times V)$, where $(\bSigma, \bu)$ is the solution to the continuous problem . For simplicity of notation, we omit the index $k$ and we refer to “$\ell^{\tau_k} \to \ell$ if $k \to \infty$” simply as “$\ell^\tau \to \ell$ as $\tau \searrow 0$”. The proof of relies on the convergence argument which is tailored to the analysis of EVIs. Its prerequisite is the convergence of $\bSigma^\tau$ in $L^\infty(0,T; S^2)$. This is addressed in a preliminary step and required in the proof of . \[lem:sigma\_tau\_to\_sigma\_in\_L\_infty\] Let $\ell^\tau$ be a bounded sequence in $H^1_{\{0\}}(0,T; V')$ such that $\ell^\tau \to \ell$ in $W^{1,1}(0,T; V')$ as $\tau \searrow 0$. Then $$\label{eq:sigma_tau_to_sigma_in_L_infty} \norm{\bSigma - \bSigma^\tau}_{L^\infty(0,T; S^2)}^2 \le C \, \big(\norm{\dot\ell - \dot\ell^\tau}_{L^1(0,T; V')}^2 + \tau \big).$$ In particular, $\bSigma^\tau \to \bSigma$ in $L^\infty(0,T;S^2)$ as $\tau \searrow 0$. We have $$\label{eq:time_derivative_of_norm} \frac12 \frac{\d}{\d t} \norm{\bSigma(t) - \bSigma^\tau(t)}_A^2 = \dual{A\dot\bSigma(t) - A\dot\bSigma^\tau(t)}{\bSigma(t) - \bSigma^\tau(t)} \quad\text{f.a.a.\ } t \in (0,T).$$ Let us fix some $t \in ((i-1)\,\tau, i\,\tau)$, $i \in \{1, \ldots, N\}$. Since $\bSigma^\tau(t)$ is a convex combination of $\bSigma\taui$ and $\bSigma\tauim$, see , and $\bSigma\tauim, \bSigma\taui \in \KK$, we obtain $\bSigma^\tau(t) \in \KK$. Hence, $\bT = \bSigma^\tau(t) + \bSigma_{\ell-\ell^\tau}(t) \in \KK$. Using $B\bT = \ell(t)$, implies $$\label{eq:continuous_estimate} \bigdual{ A \dot\bSigma(t)}{\bSigma^\tau(t)-\bSigma(t) + \bSigma_{\ell-\ell^\tau}(t) } \ge 0.$$ Testing with $\bT = \bSigma(t) + \bSigma_{\ell\taui - \ell(t)}$ yields $$\bigdual{ A (\bSigma\taui-\bSigma\tauim)}{\bSigma(t) - \bSigma\taui + \bSigma_{\ell\taui - \ell(t)} } \ge 0.$$ Using we obtain $$\bigdual{ A \dot\bSigma^\tau(t)}{\bSigma(t) - \bSigma^\tau(t) - (i\,\tau-t) \, \dot\bSigma^\tau + \bSigma_{\ell^\tau(t) - \ell(t) + (i\,\tau-t)\,\dot\ell^\tau(t)} } \ge 0.$$ Together with this yields $$\begin{aligned} \dual{A\dot\bSigma(t) - A\dot\bSigma^\tau(t)}{\bSigma(t) - \bSigma^\tau(t)} &\le \bigdual{A\dot\bSigma(t) - A\dot\bSigma^\tau(t)}{ \bSigma_{\ell-\ell^\tau}(t) } \\ &\qquad + (i\,\tau-t) \, \bigdual{ A \dot\bSigma^\tau(t)}{-\dot\bSigma^\tau(t) + \bSigma_{\dot\ell^\tau}(t) }. \end{aligned}$$ Integrating over $t$ and using , together with the boundedness of $\dot\bSigma^\tau$ and $\bSigma_{\dot\ell^\tau}$ in $L^2(0,T; S^2)$, see , yields $$\begin{aligned} \frac12 \norm{\bSigma(t) - \bSigma^\tau(t)}_A^2 \le \int_0^t \bigdual{A\dot\bSigma(s) - A\dot\bSigma^\tau(s)}{ \bSigma_{\ell-\ell^\tau}(s) } \, \d s + C \, \tau. \end{aligned}$$ Integrating by parts and using $\bSigma(0) = \bSigma^\tau(0) = \bnull$ gives $$\begin{aligned} \frac12 \norm{\bSigma(t) - \bSigma^\tau(t)}_A^2 &\le \bigdual{A\bSigma(t) - A\bSigma^\tau(t)}{\bSigma_{\ell-\ell^\tau}(t)} \\ &\qquad- \int_0^t \bigdual{A\bSigma(s) - A\bSigma^\tau(s)}{ \bSigma_{\dot\ell-\dot\ell^\tau}(s) } \, \d s + C \, \tau\\ & \le C \, \big(\norm{\bSigma(t) - \bSigma^\tau(t)}_{S^2} \, \norm{\ell(t) - \ell^\tau(t)}_{V'} \\ & \qquad + \norm{\bSigma - \bSigma^\tau}_{L^\infty(0,T; S^2)} \, \norm{\dot\ell - \dot\ell^\tau}_{L^1(0,T; V')} + \tau \big). \end{aligned}$$ Taking the supremum over $t \in (0,T)$ on both sides shows $$\begin{aligned} \norm{\bSigma - \bSigma^\tau}_{L^\infty(0,T; S^2)}^2 & \le C \, \big(\norm{\bSigma - \bSigma^\tau}_{L^\infty(0,T; S^2)} \, \norm{\ell - \ell^\tau}_{L^\infty(0,T;V')} \\ & \qquad + \norm{\bSigma - \bSigma^\tau}_{L^\infty(0,T; S^2)} \, \norm{\dot\ell - \dot\ell^\tau}_{L^1(0,T; V')} + \tau \big). \end{aligned}$$ Finally, Young’s inequality and $W^{1,1}(0,T; V') \embeds L^\infty(0,T; V')$ yields $$\norm{\bSigma - \bSigma^\tau}_{L^\infty(0,T; S^2)}^2 \le C \, \big(\norm{\dot\ell - \dot\ell^\tau}_{L^1(0,T; V')}^2 + \tau \big).$$ Let us comment on the case when $\ell^\tau$ is the point evaluation of $\ell$. The estimate shows the order of convergence of $\tau^{1/2}$ w.r.t. the $L^\infty$-norm provided that $\dot\ell \in W^{1,1}(0,T;V')$. This rate of convergence is a new result for the time-discretization of . In [@HanReddy1999 Theorem 13.1] the estimate $$\norm{\bSigma - \bSigma^\tau}_{L^\infty(0,T;S^2)} \le c \, \tau \, \norm{\bSigma}_{W^{2,1}(0,T;S^2)}$$ is shown under the assumption $\bSigma \in W^{2,1}(0,T;S^2)$. However, this assumption is unlikely to hold for the solution of an EVI. There are many examples where the solution of an EVI only belongs to $W^{1,\infty}(0,T;X)$, even if $X$ is one-dimensional and the input is smooth, see e.g. [@KrejciLovicar1990 Examples 1, 2]. Hence, $\bSigma \in W^{2,1}(0,T;S^2)$ cannot be considered the generic case. \[thm:strong\_convergence\] Let us assume $\ell^\tau \to \ell$ in $H_{\{0\}}^1(0,T; V')$ as $\tau \searrow 0$. Then $(\bSigma^\tau, \bu^\tau) \to (\bSigma, \bu)$ in $H_{\{0\}}^1(0,T; S^2 \times V)$ as $\tau \searrow 0$. We follow the idea of [@Krejci1998 Proof of Theorem 3.6]. We will apply with the setting $$\label{eq:setting_for_cor:convergence_in_W^1,p} \begin{aligned} X &= S^2_A, & u_n &= \bSigma_{\ell^\tau} - 2 \, \bSigma^\tau, & u_0 &= \bSigma_\ell - 2 \, \bSigma, \\ p &= 2, & g_n &= \norm{\bSigma_{\dot\ell^\tau}(\cdot)}_A, & g_0 &= \norm{\bSigma_{\dot\ell}(\cdot)}_A. \end{aligned}$$ to show the convergence of $\bSigma^\tau$. Here, $S^2_A$ denotes the Hilbert space $S^2$ equipped with the inner product induced by $A$. To apply , we have to verify the prerequisites and . By definition of the discrete problem , we have for a fixed $i \in \{ 1, 2, \ldots, N \}$ $$\dual{ A( \bSigma_i^\tau - \bSigma_{i-1}^\tau) }{ \bT - \bSigma_i^\tau} \ge 0 \quad\text{for all } \bT \in \KK,\; B \bT = \ell\taui,$$ which is the time-discrete analog to . We choose $\bT = \bSigma_{i-1}^\tau + \bSigma_{\ell\taui - \ell\tauim}$, see for the definition of $\bSigma_\ell$. By definition of $\bSigma_\ell$ we have $B \bT = \ell_i^\tau$ and the shift invariance implies $\bT \in \KK$. This shows $$\Bigdual{ A( \bSigma_i^\tau - \bSigma_{i-1}^\tau) }{ \bSigma_{i-1}^\tau + \bSigma_{\ell\taui - \ell\tauim} - \bSigma_i^\tau} \ge 0.$$ Therefore dividing by $\tau^2$ and using the notation of linear interpolants, see , implies $$\Bigdual{ A \dot{\bSigma}^\tau(t) }{ \bSigma_{\dot\ell^\tau}(t) - \dot{\bSigma}^\tau(t)} \ge 0 \quad\text{f.a.a.\ } t \in (0,T).$$ This shows $$\label{eq:thm:strong_convergence_1} \norm{\bSigma_{\dot\ell^\tau}(t) - 2 \, \dot{\bSigma}^\tau(t)}_A \le \norm{ \bSigma_{\dot\ell^\tau}(t)}_A \quad\text{f.a.a.\ } t \in (0,T),$$ where $\norm{\cdot}_A$ is the norm on $S^2$ induced by $A$, see . To derive a similar formula for the continuous solution, we test with $\bT = \bSigma(t+h) + \bSigma_{\ell(t)-\ell(t+h)}$ for $h >0$ and obtain $$\Bigdual{A\dot\bSigma(t)}{ \bSigma_{\ell(t)-\ell(t+h)} + \bSigma(t+h)-\bSigma(t) } \ge 0.$$ Passing to the limit $h \searrow 0$, using [@Krejci1998 Theorem 8.14], yields $$\Bigdual{A\dot\bSigma(t)}{ \bSigma_{\dot\ell}(t) - \dot\bSigma(t) } \le 0\quad\text{f.a.a.\ } t \in (0,T).$$ Analogously, we obtain by $h \nearrow 0$ $$\Bigdual{A\dot\bSigma(t)}{ \bSigma_{\dot\ell}(t) - \dot\bSigma(t) } \ge 0\quad\text{f.a.a.\ } t \in (0,T).$$ Hence $$\label{eq:thm:strong_convergence_2} \norm{\bSigma_{\dot\ell}(t) - 2\,\dot{\bSigma}(t)}_A = \norm{\bSigma_{\dot\ell}(t)}_A\quad\text{f.a.a.\ } t \in (0,T).$$ In order to apply with the setting we check the prerequisites. - ensures $\bSigma^\tau \to \bSigma$ in $L^\infty(0,T; S^2_A)$ and the assumption $\ell^\tau \to \ell$ in $H^1(0,T;V')$ implies $\bSigma_{\ell^\tau} \to \bSigma_\ell$ in $L^\infty(0,T;S^2)$. - By the linearity of $\ell \mapsto \bSigma_\ell$, the assumption $\ell^\tau \to \ell$ in $H^1(0,T; V')$ implies $\norm{\bSigma_{\dot\ell^\tau}(\cdot)}_A \to \norm{\bSigma_{\dot\ell}(\cdot)}_A$ in $L^2(0,T; \R)$. - This was shown in . - This was shown in . Therefore, yields $$\norm{\bSigma_{\ell^\tau} - 2 \, \bSigma^\tau - (\bSigma_\ell - 2 \, \bSigma)}_{H^1(0,T;S^2_A)} \to 0.$$ By the assumption $\ell^\tau \to \ell$ in $H^1(0,T;V')$ we infer $$\norm{\bSigma^\tau - \bSigma}_{H^1(0,T; S^2_A)} \to 0 \quad\text{as } \tau \searrow 0.$$ Using , and the inf-sup condition of $B$, we obtain $\bu^\tau \to \bu$ in $H^1(0,T; V)$ as $\tau \searrow 0$. Using the representation formula for $\lambda^\tau$ we are able to prove the strong convergence of $\lambda^\tau$ in $L^2(0,T;L^2(\Omega))$ towards $\lambda$. We remark that this is a novel result for the analysis of . Let us mention that this strong convergence is crucial for the analysis in [@Wachsmuth2011:4]. Without this strong convergence, we could neither pass to the limit in the complementarity relation $\mu^\tau \, \lambda^\tau = 0$ nor in the adjoint equation, see . Hence, this convergence is essential to prove the necessity of the system of weakly stationary type in [@Wachsmuth2011:4]. \[thm:strong\_convergence\_lambda\] Let $\ell^\tau \to \ell$ in $H_{\{0\}}^1(0,T; V')$ as $\tau \searrow 0$. Then $\lambda^\tau \to \lambda$ in $L^2(0,T; L^2(\Omega))$ as $\tau \searrow 0$. *Step (1):* We show that $\lambda^\tau \to \lambda$ in $L^1(0,T;L^1(\Omega))$. Using $\lambda\taui \, \abs{\DD\bSigma\taui}^2 = \tilde\sigma_0^2 \, \lambda\taui$ by and , we obtain $$\lambda\taui = \frac{1}{\tilde\sigma_0^2} \, \lambda\taui \, \DD\bSigma\taui \dprod \DD\bSigma\taui = \frac{1}{\tau \, \tilde\sigma_0^2} (-\H^{-1} (\bchi\taui-\bchi\tauim ) \dprod \DD\bSigma\taui).$$ Using the notion of piecewise linear interpolations $\bchi^\tau$, $\bSigma^\tau$ and the piecewise constant interpolation $\lambda^\tau$, we obtain for $t \in ((i-1)\,\tau, i\,\tau)$, see , $$\lambda^\tau(t) = \frac{-1}{\tilde\sigma_0^2} \Big(\H^{-1} \dot\bchi^\tau(t) \dprod \big[\DD\bSigma^\tau(t) - (t -i\,\tau)\,\DD\dot\bSigma^\tau(t)\big]\Big).$$ Using $\lambda = -\H^{-1} \dot\bchi \dprod \DD\bSigma / \tilde\sigma_0^2$, by , this implies $$\begin{aligned} \tilde\sigma_0^2 \, \abs{\lambda^\tau(t) - \lambda(t)} &\le \Bigabs{ \H^{-1} \dot\bchi^\tau(t) \dprod \DD\bSigma^\tau(t) - \H^{-1}\dot\bchi(t) \dprod \DD\bSigma(t)} \\ &\qquad + (t -i\,\tau)\, \Bigabs{ \H^{-1}\dot\bchi^\tau(t)\dprod\DD\dot\bSigma^\tau(t) }\\ &\le \Bigabs{ \H^{-1} \dot\bchi^\tau(t) \dprod \DD\bSigma^\tau(t) - \H^{-1}\dot\bchi(t) \dprod \DD\bSigma(t)} \\ &\qquad + \tau \, \Bigabs{ \H^{-1}\dot\bchi^\tau(t)\dprod\DD\dot\bSigma^\tau(t) }. \end{aligned}$$ Hence, $$\begin{aligned} \norm{\lambda^\tau - \lambda}_{L^1(0,T;L^1(\Omega))} &\le c \, \Big( \norm{\dot\bchi^\tau} \, \norm{\DD\bSigma^\tau - \DD\bSigma} + \norm{\DD\bSigma} \, \norm{\dot\bchi^\tau - \dot\bchi}\\ &\qquad + \tau \, \norm{\dot\bchi^\tau} \, \norm{\DD\dot\bSigma^\tau} \Big), \end{aligned}$$ where all norms on the right-hand side are those of $L^2(0,T;S)$. Using $\bSigma^\tau \to \bSigma$ in $H^1(0,T; S^2)$ implies $\lambda^\tau \to \lambda$ in $L^1(0,T; L^1(\Omega))$. *Step (2):* We show that $\norm{\lambda^\tau}_{L^2(0,T;L^2(\Omega))} \to \norm{\lambda}_{L^2(0,T;L^2(\Omega))}$. Using that $\lambda^\tau$ is the piecewise constant interpolation of $(\lambda\taui)_{i=1}^N$, we obtain $$\norm{\lambda^\tau}_{L^2(0,T;L^2(\Omega))}^2 = \tau \sum_{i=1}^{N} \int_\Omega (\lambda\taui)^2 \, \d x.$$ Using $\lambda\taui \, \abs{\DD\bSigma\taui} = \tilde\sigma_0 \, \lambda\taui$ by , we obtain $$\begin{aligned} \norm{\lambda^\tau}_{L^2(0,T;L^2(\Omega))}^2 & = \frac{\tau}{\tilde\sigma_0^2} \sum_{i=1}^{N} \int_\Omega \abs{\lambda\taui \, \DD\bSigma\taui}^2 \, \d x \\ & = \frac{\tau}{\tilde\sigma_0^2} \sum_{i=1}^{N} \int_\Omega \Bigabs{- \H^{-1}\frac{\bchi\taui-\bchi\tauim}{\tau}}^2 \, \d x && \text{(by \eqref{eq:-h_chi=lambda_DD_bsigma_discrete})} \\ & = \frac{1}{\tilde\sigma_0^2}\norm{\H^{-1}\dot\bchi^\tau}_{L^2(0,T;S)}^2. \end{aligned}$$ An analogous calculation shows $\norm{\lambda}_{L^2(0,T; L^2(\Omega))} = \frac{1}{\tilde\sigma_0}\norm{\H^{-1}\dot\bchi}_{L^2(0,T;S)}$. Hence, by $\bSigma^\tau \to \bSigma$ in $H^1(0,T; S^2)$, see , we infer the convergence of norms $\norm{\lambda^\tau}_{L^2(0,T; L^2(\Omega))} \to \norm{\lambda}_{L^2(0,T; L^2(\Omega))}$. *Step (3):* We show that $\lambda^\tau \to \lambda$ in $L^2(0,T;L^2(\Omega))$. By Step (2) we know that for every subsequence of $\tau$, there is a subsequence $\tau_k$ such that $\lambda^{\tau_k} \weakly \tilde\lambda$ in $L^2(0,T;L^2(\Omega))$ for some $\tilde \lambda$. Step (1) implies $\tilde\lambda = \lambda$. Hence, the whole sequence converges weakly. In view of the convergence of norms, $\lambda^\tau$ converges strongly to $\lambda$ in $L^2(0,T;L^2(\Omega))$ as $\tau\searrow0$. Weak convergence {#sec:weak_convergence_of_time_discretization} ---------------- In this section we show the weak convergence of the time-discrete states $(\bSigma^\tau, \bu^\tau) = \GG^\tau(E \bg^\tau)$ in $H_{\{0\}}^1(0,T;S^2 \times V)$, under the assumption that the controls $\bg^\tau$ converges weakly towards $\bg$ in $H_{\{0\}}^1(0,T;U)$. This result is essential for proving the approximability results in . Similarly to the weak continuity of the forward operator proven in , we need the weak convergence of the right-hand sides $E \bg^\tau$ with respect to a stronger norm in space, see the discussion in the beginning of . \[thm:weak\_convergence\_of\_time\_discretization\] Let $\bg^\tau \in U^n$ such that the linear interpolants converge weakly, i.e., there is $\bg \in H_{\{0\}}^1(0,T; U)$ such that $\bg^\tau \weakly \bg$ in $H_{\{0\}}^1(0, T; U)$ as $\tau \searrow 0$. Then the solutions $(\bSigma^\tau, \bu^\tau)$ of the time-discrete problem with right-hand side $\ell^\tau = E\bg^\tau$ converge weakly in $H_{\{0\}}^1(0, T; S^2 \times V)$ towards the solution $(\bSigma, \bu)$ of with right-hand side $\ell = E\bg$ as $\tau \searrow 0$. Due to the a-priori bound , there exists a weakly convergent subsequence of $(\bSigma^\tau, \bu^\tau)$, denoted by the same symbol, i.e. $(\bSigma^\tau, \bu^\tau) \weakly (\widetilde\bSigma, \widetilde\bu)$ in $H^1(0,T;S^2 \times V)$ as $\tau \searrow 0$. As in the proof of , we shall show $(\widetilde\bSigma, \widetilde\bu) = (\bSigma, \bu)$. Hence the whole sequence converges weakly and this simplification of notation is justified. Let $\bT \in L^2(0, T; S^2)$ with $\bT \in \KK$ a.e. in $(0,T)$ be arbitrary. Let $i \in \{0, \ldots, N-1\}$ and $\kappa \in (0,1)$ be given. Set $t = (i + \kappa ) \, \tau$. Testing with $\bT(t)$ yields $$\dual{A \dot\bSigma^\tau(t) + B^\star \dot\bu^\tau(t)}{\bT(t) - \bSigma^\tau( (i+1) \, \tau)} \ge 0.$$ Using $ \bSigma^\tau( (i+1) \, \tau) = \bSigma^\tau(t) + ( 1 - \kappa ) \, \tau \, \dot\bSigma^\tau(t) $ by , and integrating over $\kappa \in (0,1)$, i.e. $t \in (i\,\tau, (i+1)\,\tau)$ yields $$\begin{aligned} 0 &\le \int_{i\,\tau}^{(i+1)\,\tau} \dual{A \dot\bSigma^\tau(t) + B^\star \dot\bu^\tau(t)}{\bT(t) - \bSigma^\tau( (i+1) \, \tau)} \, \d t \\ &= \int_{i\,\tau}^{(i+1)\,\tau} \dual{A \dot\bSigma^\tau + B^\star \dot\bu^\tau}{\bT - \bSigma^\tau} \, \d t - \frac{\tau}{2} \int_{i\,\tau}^{(i+1)\,\tau} \dual{A \dot\bSigma^\tau + B^\star \dot\bu^\tau}{\dot\bSigma^\tau} \, \d t. \end{aligned}$$ Hence, summing over $i = 0,\ldots, N-1$ implies $$\label{eq:in_thm:weak_convergence_of_time_discretization} 0 \le \int_{0}^{T} \dual{A \dot\bSigma^\tau + B^\star \dot\bu^\tau}{\bT - \bSigma^\tau} \, \d t - \frac{\tau}{2} \int_{0}^{T} \dual{A \dot\bSigma^\tau + B^\star \dot\bu^\tau}{\dot\bSigma^\tau} \, \d t.$$ Due to the boundedness of $(\bSigma^\tau, \bu^\tau)$ in $H^1(0,T; S^2\times V)$, the second addend goes to zero as $\tau \searrow 0$. For the first addend we can proceed as in the proof of : the application of yields $$\limsup_{\tau \searrow 0} \int_{0}^{T} \dual{A \dot\bSigma^\tau + B^\star \dot\bu^\tau}{\bT - \bSigma^\tau} \, \d t \le \int_{0}^{T} \dual{A \dot{\widetilde\bSigma} + B^\star \dot{\widetilde\bu} }{\bT - \widetilde\bSigma } \, \d t.$$ Together with this implies $$0 \le \int_{0}^{T} \dual{A \dot{\widetilde\bSigma} + B^\star \dot{\widetilde\bu} }{\bT - \widetilde\bSigma } \, \d t \quad\text{for all } \bT \in L^2(0,T; S^2) \text{, } \bT \in \KK \text{ a.e.\ in } (0,T).$$ Therefore, $(\widetilde\bSigma, \widetilde\bu)$ is the solution of . This implies $(\widetilde\bSigma, \widetilde\bu) = (\bSigma, \bu)$. The time-discrete optimal control problem {#subsec:approx_by_time_discrete} ----------------------------------------- In this section we consider a time discretization of . The time-discrete controls belong to the space $U^N$, which is identified with a subspace of $H_{\{0\}}^1(0,T;U)$ via the piecewise linear interpolation . The discretization of the admissible set is given by $\Uadtau = {U_\textup{ad}}\cap U^N$, where $\tau = T/N$. Now, the time-discrete optimal control problem reads $$\label{eq:time-discrete_ulp} \tag{$\mathbf{P}^\tau$} \left. \begin{aligned} \text{Minimize}\quad & F(\bu^\tau,\bg^\tau) = \psi( \bu^\tau ) + \frac{\nu}{2} \norm{\bg^\tau}_{H^1(0,T;U)}^2 \\ \text{such that}\quad & (\bSigma^\tau, \bu^\tau) = \GG^\tau( E \bg^\tau) \\ \text{and}\quad & \bg^\tau \in \Uadtau. \end{aligned} \quad \right\}$$ For the analysis of we need the additional assumption that all $\bg \in {U_\textup{ad}}$ can be approximated by time-discrete controls $\bg^\tau \in \Uadtau$. \[asm:approximability\_of\_Uad\] In addition to we suppose that for all $\bg \in {U_\textup{ad}}$, there exists $\bg^\tau \in \Uadtau$, such that $\bg^\tau \to \bg$ in $H^1(0,T;U)$ as $\tau \searrow 0$. We remark that this condition is satisfied for both examples of ${U_\textup{ad}}$ given after . The aim of this section is to answer the question whether optimal controls of can be approximated by optimal controls of . Similar results for a regularization of an optimal control problem of static plasticity have been proved in [@HerzogMeyerWachsmuth2009:2 Section 3.2]. These results are very common for the approximation of minimizers of optimal control problems and the technique of proof is applicable to various problems, see also [@CasasTroeltzsch2002:1; @Barbu1981:1]. To be precise, we prove that - one global optimum of can be approximated by global solutions of , see , - every strict local optimum of can be approximated by local solutions of , see , - every local optimum of can be approximated by local solutions of a perturbed problem , see . These results are very important for the analysis in [@Wachsmuth2011:4]. Without these results at hand, one cannot show the necessity of the optimality conditions therein. Although both of $\bu^\tau$ and $\bg^\tau$ are optimization variables in , for simplicity we refer solely to $\bg^\tau$ being (locally or globally) optimal. This is justified in view of the continuous solution map $\GG^\tau$ of . First we check that there exists a minimizer of the time-discrete optimal control problem . \[lem:p\_tau\_has\_minimizer\] There exists a global minimizer of . In we show in a more general framework that the solution operator $\GG^\tau$ of is Lipschitz continuous from $(V')^N$ to $(S^2\times V)^N$. Since the control operator $E$ from $U^N$ into $(V')^N$ is compact, $\GG^\tau$ is compact from $U^N$ to $(S^2\times V)^N$. Now we proceed similarly as in the proof of . Now, we are going to prove the approximation properties. \[thm:continuous\_approximation\_with\_global\_solutions\] Suppose that is fulfilled. Let $\{\tau\}$ be a sequence tending to $0$ and let $\bg^\tau$ denote a global solution to . 1. Then there exists an accumulation point $\bg$ of $\{\bg^\tau\}$ in $H^1(0,T;U)$ and 2. every weak accumulation point of $\{\bg^\tau\}$ in $H^1(0,T;U)$ is in fact a strong accumulation point in $H^1(0,T;U)$ and a global solution of . By , we can choose $\bg_0 \in {U_\textup{ad}}$. By , there exists a sequence $\bg^\tau_0 \in \Uadtau$ which converges towards $\bg_0$ in $H_{\{0\}}^1(0,T;U)$. Hence, by the corresponding displacements $\bu^\tau_0$ converge towards $\bu_0$ in $H_{\{0\}}^1(0,T; V)$. Together with the continuity of $\psi$, this implies the convergence of $F(\bu^\tau_0, \bg^\tau_0)$. Since $\bg^\tau$ is a global optimum of , we have $F(\bu^\tau, \bg^\tau) \le F(\bu^\tau_0, \bg^\tau_0)$. Hence, $\{\bg^\tau\}$ is bounded in $H^1(0,T;U)$. This implies the existence of a weakly convergent subsequence in this space. Therefore, assertion (1) follows by assertion (2). To prove assertion (2), let $\{\bg^\tau\}$ converge weakly towards $\bg$ in $H^1(0,T;U)$ as $\tau \searrow 0$. We denote by $(\bSigma^\tau, \bu^\tau) = \GG^\tau(E\bg^\tau)$ the (time-discrete) solution to and by $(\bSigma, \bu) = \GG(E\bg)$ the solution to . Due to the weak convergence proven in , we have $\bu^\tau \weakly \bu$ in $H^1(0, T; V)$. Let $\tilde\bg \in {U_\textup{ad}}$ with corresponding displacement $\tilde\bu$ be arbitrary. By , there is $\tilde\bg^\tau \in \Uadtau$, such that $\tilde\bg^\tau \to \tilde\bg$ in $H^1(0, T; L^2(\Gamma_N; \R^d))$. We denote the corresponding displacements by $\tilde\bu^\tau$. By we infer $\tilde\bu^\tau \to \tilde\bu$. We have $$\begin{aligned} F( \bu, \bg ) & \le \liminf F( \bu^\tau, \bg^\tau ) && \text{by weak lower semicontinuity of $F$} \\ & \le \liminf F( \tilde\bu^\tau, \tilde\bg^\tau ) && \text{by global optimality of $(\bu^\tau, \bg^\tau)$} \\ & = F( \tilde\bu, \tilde\bg ). && \text{by convergence of $(\tilde\bu^\tau, \tilde\bg^\tau)$} \end{aligned}$$ This shows that $\bg$ is a global optimal solution. Inserting $\tilde\bg = \bg$ yields the convergence of norms and hence the strong convergence of $\bg^\tau$ in $H^1(0,T; U)$. \[thm:continuous\_approximation\_with\_local\_solutions\] Suppose is fulfilled. Let $\bg$ be a strict local optimum of w.r.t. the topology of $H^1(0,T;U)$. Then, for every sequence $\{\tau\}$ tending to $0$, there is a sequence $\{\bg^\tau\}$ of local solutions to , such that $\bg^\tau \to \bg$ strongly in $H^1(0,T;U)$ as $\tau \searrow 0$. *Step (1):* Let $\varepsilon > 0$, such that $\bg$ is the unique global optimum in the closed ball $B_\varepsilon(\bg)$ with radius $\varepsilon$ centered at $\bg$. We define ${\hat U_\textup{ad}}= {U_\textup{ad}}\cap B_\varepsilon(\bg)$. *Step (2):* We check that is fulfilled for ${\hat U_\textup{ad}}$. Clearly we have $\bg \in {\hat U_\textup{ad}}$. Let $\tilde\bg \in {\hat U_\textup{ad}}$ be arbitrary. We construct a sequence $\{\tilde\bg^\tau\} \subset {\hat U_\textup{ad}}$, $\tilde\bg^\tau \in U^N$, such that $\tilde\bg^\tau \to \tilde\bg$. Since ${U_\textup{ad}}$ satisfies , there are sequences of time-discrete functions $(\bg^\tau), (\hat\bg^\tau) \subset {U_\textup{ad}}$, converging towards $\bg$ and $\tilde\bg$, respectively. If $\norm{\tilde\bg - \bg}_{H^1(0,T;U)} < \varepsilon$, the convergence $\hat\bg^\tau \to \tilde\bg$ implies $\hat\bg^\tau \in B_\varepsilon(\bg)$, and hence $\hat\bg^\tau \in {\hat U_\textup{ad}}\cap \Uadtau$ for sufficiently small $\tau$. Otherwise, if $\norm{\tilde\bg-\bg}_{H^1(0,T;U)} = \varepsilon$, we obtain $\lim \norm{\bg - \hat\bg^\tau}_{H^1(0,T;U)} = \varepsilon$. Therefore, there exists $\tau_0$, such that for all $\tau \le \tau_0$, we have $\norm{\bg - \bg^\tau}_{H^1(0,T;U)} \le \varepsilon/2$ and $\norm{\bg - \hat\bg^\tau}_{H^1(0,T;U)} > \varepsilon/2$. Now, we construct a convex combination $\tilde\bg^\tau$ of $\bg^\tau$ and $\hat\bg^\tau$ which belongs to $B_\varepsilon(\bg)$ and approximates $\tilde\bg$. We define the sequence $\tilde\bg^\tau$ by $$\tilde\bg^\tau = (1-\eta^\tau) \, \hat\bg^\tau + \eta^\tau \, \bg^\tau, \quad\text{with } \eta^\tau = \max\left(0, \frac{\norm{\bg - \hat\bg^\tau}_{H^1(0,T;U)} - \varepsilon}{\norm{\bg - \hat\bg^\tau}_{H^1(0,T;U)} - \varepsilon/2} \right) \in [0,1].$$ In case $\norm{\bg - \hat\bg^\tau}_{H^1(0,T;U)} > \varepsilon$, we find (all norms are those of $H^1(0,T;U)$) $$\begin{aligned} \norm{\bg - \tilde\bg^\tau} & \le (1 - \eta^\tau) \, \norm{\bg - \hat\bg^\tau} + \eta^\tau \, \norm{\bg - \bg^\tau} \\ & = \frac{1}{\norm{\bg - \hat\bg^\tau} - \varepsilon/2} \, \Bigh(){ \frac\varepsilon2 \, \norm{\bg - \hat\bg^\tau} + (\norm{\bg - \hat\bg^\tau} - \varepsilon) \, \norm{\bg - \bg^\tau} } \\ & \le \frac{1}{\norm{\bg - \hat\bg^\tau} - \varepsilon/2} \, \Bigh(){ \frac\varepsilon2 \, \norm{\bg - \hat\bg^\tau} + (\norm{\bg - \hat\bg^\tau} - \varepsilon) \, \frac\varepsilon2 } \\ & = \varepsilon. \end{aligned}$$ This shows $\norm{\bg - \tilde\bg^\tau}_{H^1(0,T;U)} \le \varepsilon$. Moreover, $\eta^\tau \in [0,1]$ implies $\tilde\bg^\tau \in {U_\textup{ad}}$. Therefore, $\tilde\bg^\tau \in {\hat U_\textup{ad}}$. Further, $\lim \norm{\bg - \hat\bg^\tau}_{H^1(0,T;U)} = \varepsilon$ implies $\eta^\tau \to 0$ and hence $\tilde\bg^\tau \to \tilde\bg$. This shows that ${\hat U_\textup{ad}}$ satisfies . *Step (3):* We define the auxiliary problem $$\label{eq:ulp_aux} \tag{$\mathbf{P}_{\bg,\varepsilon}$} \left. \begin{aligned} \text{Minimize}\quad & F(\bu,\bg) = \psi( \bu ) + \frac{\nu}{2} \norm{\bg}_{H^1(0,T;U)}^2 \\ \text{such that}\quad & (\bSigma,\bu) = \GG(E \bg) \\ \text{and}\quad & \bg \in {U_\textup{ad}}\cap B_\varepsilon(\bg). \end{aligned} \quad \right\}$$ Since $\bg$ is the unique minimum in ${\hat U_\textup{ad}}= {U_\textup{ad}}\cap B_\varepsilon(\bg)$, it is the unique global minimizer of . Invoking yields the existence of a sequence $\bg^\tau$ of global solutions of the associated time-discrete problem $\textup{($\mathbf{P}_\varepsilon^\tau$)}$, such that $\bg^\tau \to \bg$ in $H^1(0,T;U)$. *Step (4):* It remains to check that $\bg^\tau$ are local solutions of . The convergence $\bg^\tau \to \bg$ in $H^1(0,T;U)$ implies that there is a $\tau_0$ such that $\norm{\bg - \bg^\tau}_{H^1(0,T;U)} \le \varepsilon/2$ holds for all $\tau \le \tau_0$. Let $\tau \le \tau_0$ and $\tilde\bg^\tau \in B_{\varepsilon/2}(\bg^\tau) \cap \Uadtau$ be arbitrary. By the triangle inequality we infer $\tilde\bg^\tau \in B_\varepsilon(\bg)$. This implies $\tilde\bg^\tau \in {\hat U_\textup{ad}}\cap \Uadtau$. The global optimality of $\bg^\tau$ implies $F(\GG^{\tau, \bu}(\bg^\tau), \bg^\tau) \le F(\GG^{\tau, \bu}(\tilde\bg^\tau), \tilde\bg^\tau)$. Hence, $\bg^\tau$ is a local optimum of in the neighborhood $B_{\varepsilon/2}(\bg^\tau)$. Finally, we address the approximability of a local minimum, which is not assumed to be strict. Let $\bg$ be a local optimum of w.r.t. the topology of $H^1(0,T;U)$. We define the modified problem, see also [@CasasTroeltzsch2002:1; @Barbu1981:1], $$\label{eq:time-discrete_ulp_mod} \tag{$\mathbf{P}_{\bg}$} \left. \begin{aligned} \text{Minimize}\quad & F_\bg(\bu,\tilde\bg) = \psi( \bu ) + \frac{\nu}{2} \norm{\tilde\bg}_{H^1(0,T;U)}^2 + \frac12 \norm{\tilde\bg - \bg}_{H^1(0,T;U)}^2 \\ \text{such that}\quad & (\bSigma, \bu) = \GG(E\tilde\bg) \\ \text{and}\quad & \tilde\bg \in {U_\textup{ad}}\end{aligned} \right\}$$ Clearly, $\bg$ becomes a *strict* local optimum of . Analogously to we define the time-discrete approximation . \[thm:continuous\_approximation\_with\_local\_solutions\_mod\] Suppose is fulfilled. Let $\bg$ be a local optimum of w.r.t. the topology of $H^1(0,T;U)$. Then, for every sequence $\{\tau\}$ tending to $0$, there is a sequence $(\bg^\tau)$ of local solutions to , such that $\bg^\tau \to \bg$ strongly in $H^1(0,T;U)$ as $\tau \searrow 0$. Since the additional term $\norm{\tilde\bg - \bg}_{H^1(0,T;U)}^2$ is weakly lower semicontinuous, this follows analogously to . Due to this theorem, we are able to derive *necessary* optimality conditions for by passing to the limit with the optimality conditions of , see . A convergence result in Bochner-Sobolev spaces {#appendix:bochner} ============================================== For the proof of we need the following result. \[cor:convergence\_in\_W\^1,p\] Let $X$ be a Hilbert space and $p \in [1,\infty)$ be given. Let $\{u_n\} \subset W^{1,p}(0, T; X)$, $\{g_n\} \subset L^p(0, T; \R)$ be given sequences for $n \in \N \cup \{0\}$ such that \[eq:condition\_of\_cor:convergence\_in\_W\^1,p\] $$\begin{aligned} u_n &\mrep{{}\to u_0}{{}\le g_n(t)} \text{ in } L^\infty(0, T; X) \label{eq:condition_of_cor:convergence_in_W^1,p_1} \\ g_n &\mrep{{}\to g_0}{{}\le g_n(t)} \text{ in } L^p(0, T; \R), \label{eq:condition_of_cor:convergence_in_W^1,p_2} \\ \norm{ \dot u_n(t) }_X &\le g_n(t) \text{ a.e.\ in } (0,T), \text{ for all } n \in \N, \label{eq:condition_of_cor:convergence_in_W^1,p_3} \\ \norm{ \dot u_0(t) }_X &= \mrep{g_0(t)}{g_n(t)} \text{ a.e.\ in } (0,T). \label{eq:condition_of_cor:convergence_in_W^1,p_4} \end{aligned}$$ Then $u_n \to u_0$ in $W^{1,p}(0, T; X)$. Acknowledgment {#acknowledgment .unnumbered} -------------- The author would like to express his gratitude to Roland Herzog and Christian Meyer for helpful discussions on the topic of this paper and to Dorothee Knees for pointing out reference [@Krejci1998]. This work was supported by a DFG grant within the [Priority Program SPP 1253](http://www.am.uni-erlangen.de/home/spp1253/wiki) (*Optimization with Partial Differential Equations*), which is gratefully acknowledged.

--- abstract: 'We study orthogonal polynomials associated with a continued fraction due to Hirschhorn. Hirschhorn’s continued fraction contains as special cases the famous Rogers–Ramanujan continued fraction and two of Ramanujan’s generalizations. The orthogonality measure of the set of polynomials obtained has an absolutely continuous component. We find generating functions, asymptotic formulas, orthogonality relations, and the Stieltjes transform of the measure. Using standard generating function techniques, we show how to obtain formulas for the convergents of Ramanujan’s continued fractions, including a formula that Ramanujan recorded himself as Entry 16 in Chapter 16 of his second notebook.' address: - | Fakultät für Mathematik, Universität Wien\ Oskar-Morgenstern-Platz 1, 1090 Wien, Austria. - 'Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA ' author: - 'Gaurav Bhatnagar\*' - 'Mourad E. H. Ismail' title: Orthogonal polynomials associated with a continued fraction of Hirschhorn --- [^1] Introduction ============ The connection of continued fractions with orthogonal polynomials is well known. Indeed, orthogonal polynomials made an appearance in the context of continued fractions as early as 1894, in the work of Stieltjes [@TJS1894; @TJS1895]. Our objective in this paper is to study orthogonal polynomials associated to a continued fraction due to Hirschhorn [@MDH1974] (also considered by Bhargava and Adiga [@BA1984]). This continued fraction is $$\label{mikecf} \frac{1}{1-b+a}{\genfrac{}{}{0pt}{}{}{+}}\frac{b+\lambda q}{1-b+aq}{\genfrac{}{}{0pt}{}{}{+}}\frac{b+\lambda q^2}{1-b+aq^2}{\genfrac{}{}{0pt}{}{}{+}}\frac{b+\lambda q^3}{1-b+aq^3}{\genfrac{}{}{0pt}{}{}{+}}{\genfrac{}{}{0pt}{}{}{\cdots}},$$ where we have changed a few symbols in order to fit the notation used by Andrews and Berndt [@AB2005] in their edited version of Ramanujan’s Lost Notebook. Hirschhorn’s continued fraction contains three of Ramanujan’s famous continued fractions. For example, when one of $a$ or $b$ is $0$, it reduces to continued fractions in the Lost Notebook (see [@AB2005] and [@Berndt1991-RN3 Ch. 16]); when both $a$ and $b$ are $0$ it is the Rogers–Ramanujan continued fraction. Some of the orthogonal polynomials arising from these special cases have been studied before. A set of orthogonal polynomials corresponding to the $b=0$ case have been studied previously by Al-Salam and Ismail [@AI1984], and we study one more. The orthogonality measure of the polynomials associated to Hirschhorn’s continued fraction studied here has an absolutely continuous component, as opposed to the discrete measure in [@AI1984]. The techniques we use were developed by Askey and Ismail in their memoir [@AI1984]. These authors study classical orthogonal polynomials using techniques involving recurrence relations, generating functions and asymptotic methods. They also used a theorem of Nevai [@PN1979]. In addition, we apply a moment method developed by Stanton and Ismail [@IS2002]. In the context of Ramanujan’s continued fractions, these ideas have been applied previously by Al–Salam and Ismail [@Al-I1983] and Ismail and Stanton [@IS2006]. The contents of this paper are as follows. In Section \[sec:mikecf1\] we provide some background information from the theory of orthogonal polynomials. In Section \[sec:Mike-Nevai\], we translate Hirschhorn’s continued fraction to a form suitable for our study. Further, we apply Nevai’s theorem to compute a formula for the absolutely continuous component of the measure of the orthogonal polynomials associated with Hirschhorn’s continued fraction. In Section \[sec:stieltjes\], we obtain another expression for this by inverting the associated Stieltjes transform. In Section \[sec:moments\], we provide another solution of the recurrence relation consisting of functions that are moments over a discrete measure. In Section \[sec:specialcases\], we consider a continued fraction of Ramanujan obtained by taking $b=0$ in Hirschhorn’s continued fraction, where we obtain a discrete orthogonality measure. Finally, in Section \[sec:convergents\], we show how to obtain formulas for the convergents of a continued fraction. Such a formula was given by Ramanujan himself, who gave a formula for the convergents of $$\frac{1}{1}{\genfrac{}{}{0pt}{}{}{+}}\frac{\lambda q}{1}{\genfrac{}{}{0pt}{}{}{+}}\frac{\lambda q^2}{1}{\genfrac{}{}{0pt}{}{}{+}}\frac{\lambda q^3}{1}{\genfrac{}{}{0pt}{}{}{+}}{\genfrac{}{}{0pt}{}{}{\cdots}},$$ the Rogers–Ramanujan continued fraction. His formula for its convergents appears as Entry 16 in Chapter 16 of Ramanujan’s second notebook (see Berndt [@Berndt1991-RN3]). To state Ramanujan’s formula, we require some notation. We need the $q$-rising factorial ${{\left({q}; q\right)_{n}}}$, which is defined to be $1$ when $n=0$; and $${{\left({q}; q\right)_{n}}} = (1-q)(1-q^2)\cdots (1-q^{n}),$$ for $n$ a positive integer. Next we have the $q$-binomial coefficient, defined as $${\genfrac{[}{]}{0pt}{}}{n}{k}_q = \frac{{{\left({q}; q\right)_{n}}}}{{{\left({q}; q\right)_{k}}}{{\left({q}; q\right)_{n-k}}}}$$ where $n\geq k$ are nonnegative integers. When $n<k$ we take ${\genfrac{[}{]}{0pt}{}}{n}{k}_q=0.$ Ramanujan’s formula is as follows. $$\frac{N_n}{D_n} = \frac{1}{1}{\genfrac{}{}{0pt}{}{}{+}}\frac{\lambda q}{1}{\genfrac{}{}{0pt}{}{}{+}}\frac{\lambda q^2}{1}{\genfrac{}{}{0pt}{}{}{+}}\frac{\lambda q^3}{1} {\genfrac{}{}{0pt}{}{}{+}}{\genfrac{}{}{0pt}{}{}{\cdots}}{\genfrac{}{}{0pt}{}{}{+}}\frac{\lambda q^n}{1} ,\label{entry16}$$ where $$\label{num-entry16} N_n = \sum_{k\geq 0} q^{k^2+k}\lambda^k {\genfrac{[}{]}{0pt}{}}{n-k}{k}_{q}$$ and $$\label{den-entry16} D_n = \sum_{k\geq 0} q^{k^2}\lambda^k {\genfrac{[}{]}{0pt}{}}{n-k+1}{k}_{q} . $$ The sums $N_n$ and $D_n$ are finite sums. For example, the summand of $N_n$ is $0$ when the index $k$ is such that $n-k<k$. Unlike the work of Ramanujan, there will be no mystery about how such formulas are discovered. Background: From continued fractions to orthogonal polynomials {#sec:mikecf1} ============================================================== If there is a continued fraction, then there is a three-term recurrence relation. And if the recurrence relation is of the right type, it defines a set of orthogonal polynomials. Such a recurrence relation is central to the study of the associated orthogonal polynomials and examining it directs our study. The objective of this section is to collate this background information from the theory of orthogonal polynomials. We have used Chihara [@Chihara1978] and the second author’s book [@MI2009]. For introductory material on these topics we recommend Andrews, Askey and Roy [@AAR1999]. The right type of continued fraction is called the $J$-fraction, which is of the form $$\label{jfrac} \frac{A_0}{A_0x+B_0}{\genfrac{}{}{0pt}{}{}{-}}\frac{C_1}{A_1x+B_1}{\genfrac{}{}{0pt}{}{}{-}}\frac{C_2}{A_2x+B_2}{\genfrac{}{}{0pt}{}{}{-}}{\genfrac{}{}{0pt}{}{}{\cdots}}.$$ The $k$th convergent of the $J$-fraction is given by $$\frac{N_k(x)}{D_k(x)} := \frac{A_0}{A_0x+B_0}{\genfrac{}{}{0pt}{}{}{-}}\frac{C_1}{A_1x+B_1}{\genfrac{}{}{0pt}{}{}{-}}{\genfrac{}{}{0pt}{}{}{\cdots}}{\genfrac{}{}{0pt}{}{}{-}}\frac{C_{k-1}}{A_{k-1}x+B_{k-1}}.$$ The following proposition shows how to compute the convergents of a continued fraction. \[cf-conv\] Assume that $A_kC_{k+1}\neq 0$, $k=0, 1, \dots.$ Then the polynomials $N_k(x)$ and $D_k(x)$ are solutions of the recurrence relation $$\label{three-term} y_{k+1}(x) = (A_kx+B_k) y_k(x) -C_ky_{k-1}(x), \text{ for } k>0,$$ with the initial values $$D_0(x)=1, D_1(x)=A_0x+B_0, N_0(x)=0, N_1(x)=A_0.$$ Instead of , we consider a three term recurrence equation of the form $$\label{three-term2} x y_{k}(x) = y_{k+1}(x) + \alpha_k y_k(x) +\beta_k y_{k-1}(x), \text{ for } k>0,$$ where $\alpha_k$ is real for $k\geq 0$ and $\beta_k>0$ for $k>0$. This three-term recurrence can be obtained from by mildly re-scaling the functions involved. Let the polynomials $\left\{ P_k(x)\right\}$ satisfy , with the initial values $$P_0(x)=1 \text{ and }P_1(x)=x-\alpha_0.$$ We will also have occasion to consider the polynomials $\left\{ P_k^*(x)\right\}$, satisfying with the initial conditions $P_0^*(x)=0$ and $P_1^*(x)=1$. The $P_k^*(x)$ correspond to the numerator and $P_k(x)$ to the denominator of the associated $J$-fraction. Note that both $P_k^*(x)$ and $P_k(x)$ are monic polynomials, of degree $k-1$ and $k$, respectively. The next proposition shows that the $P_k(x)$ are orthogonal with respect to a measure $\mu$. \[spectral\] Given a sequence $\{P_n(x)\}$ as above, there is a positive measure $\mu$ such that $$\int_a^b P_n(x)P_m(x) d\mu(x) = \beta_1\beta_2\dots\beta_n \cdot \delta_{mn}.$$ Some pertinent facts {#some-pertinent-facts .unnumbered} -------------------- - If $\{\alpha_k\}$ and $\{\beta_k\}$ are bounded, then support of $\mu$ is bounded, and $[a,b]$ is a finite interval. In addition, the measure $\mu$ is unique. - The interval $[a,b]$ is called the ‘true interval of orthogonality’, and is a subset of the convex hull of $\operatorname{supp}(\mu)$. All the zeros of the set of polynomials $\{P_k(x)\}$ lie here; indeed, it is the smallest such interval. - The measure $\mu$ could possibly have both discrete and an absolutely continuous component. The orthogonality relation is then of the form $$\int_a^b P_n(x)P_m(x)\mu^{\prime}(x)dx + \sum_j P_n(x_j)P_m(x_j) w(x_j) = h_n\delta_{mn},$$ where $x_j$ are the points where $\mu$ has mass $w(x_j)$, and $h_n>0$. - (Blumenthal’s Theorem [@Chihara1978 Th. IV-3.5, p. 117] (rephrased)) If $\alpha_k\to \alpha$ and $\beta_k\to 0$, then the measure of the orthogonal polynomials defined by is purely discrete. However, if $\alpha_k\to\alpha$ and $\beta_k\to \beta>0$, then $\mu$ has an absolutely continuous component. Next, we have a proposition that shows the connection between the continued fraction and the Stieltjes transform of the measure. \[prop:stieltjes\] Assume that the true interval of orthogonality $[a,b]$ is bounded. Then $$\displaystyle \lim_{k\to\infty} \frac{P_k^*(x)}{P_k(x)} =\int_a^b \frac{d\mu(t)}{x-t},$$ uniformly for $x\not\in \operatorname{supp}(\mu)$. From here, we can use Stieltjes’ inversion formula (see [@MI2009 Eq. (1.2.9)]) to obtain a formula for $d\mu$. Let $$X(x)= \int_a^b \frac{d\mu(t)}{x-t},\text{ where } x\not\in \operatorname{supp}(\mu).$$ Then $$\mu(x_2)-\mu(x_1) = \lim_{\epsilon\to 0^+} \int_{x_1}^{x_2} \frac{X(x-i\epsilon)-X(x+i\epsilon)}{2\pi i} dx.$$ So $\mu^\prime$ exists at $x$, and we have [@MI2009 Eq.  (1.2.10)]: $$\label{stieltjes-inverse} \mu^{\prime}(x)= \frac{X(x-i0^+)-X(x+i0^+)}{2\pi i}.$$ To summarize, each $J$-fraction is associated with a three-term recurrence relation. The solutions of a (possibly scaled) three-term recurrence relation, under certain conditions, are orthogonal polynomials. Both the numerator and denominator of the continued fraction satisfy the recurrence relation, with differing initial conditions. The limit of their ratio, that is the value of the continued fraction, gives a formula for the orthogonality measure of the denominator polynomials. Hirschhorn’s Continued Fraction: computing the measure {#sec:Mike-Nevai} ====================================================== In this section, we begin our study of the orthogonal polynomials associated with Hirschhorn’s continued fraction. On examining the associated three-term recurrence relation, we find that the associated denominator polynomials have an orthogonality measure with an absolutely continuous component. Our goal in this section is to compute a formula for this, using a very useful theorem of Nevai [@PN1979]. Nevai’s theorem requires finding the asymptotic expression for the denominator polynomials, for which we will use Darboux’s method. We need some notation. The [*$q$-rising factorial*]{} ${{\left({a}; q\right)_{n}}}$ is defined as $${{\left({a}; q\right)_{n}}} := \begin{cases} 1 & \text{ for } n=0\cr (1-aq)(1-aq^2)\cdots (1-aq^{n-1}) & \text{ for } n=1, 2, \dots. \end{cases}$$ In addition $${{\left({a}; q\right)_{\infty}}} := \prod_{k=0}^\infty (1-aq^{k}) \text{ for } |q|<1.$$ We use the short-hand notation $$\begin{aligned} {{\left({a_1, a_2,\dots, a_r}; q\right)_{k}}} &:= {{\left({a_1}; q\right)_{k}}} {{\left({a_2}; q\right)_{k}}}\cdots {{\left({a_r}; q\right)_{k}}}.\end{aligned}$$ We now begin our study of Hirschhorn’s continued fraction by considering the more general $J$-fraction $$H(x):= \frac{1-b}{x(1-b)+a}{\genfrac{}{}{0pt}{}{}{+}}\frac{b+\lambda q}{x(1-b)+aq}{\genfrac{}{}{0pt}{}{}{+}}\frac{b+\lambda q^2}{x(1-b)+aq^2}{\genfrac{}{}{0pt}{}{}{+}}{\genfrac{}{}{0pt}{}{}{\cdots}}\label{mikecf-jfrac}$$ Note that is $H(1)/(1-b)$. On comparing with the form of the $J$-fraction in we find that $$A_k= (1-b), B_k=aq^k \text{ for } k=0, 1, 2, \dots \text{ and }C_k = -(b+\lambda q^k) \text{ for } k=1, 2, 3, \dots.$$ The corresponding three term recurrence relation is $$\label{mikecf-3term} y_{k+1}(x) = (x(1-b) + aq^k) y_k(x) + (b+\lambda q^{k}) y_{k-1}(x), \text{ for } k > 0.$$ By Proposition \[cf-conv\], the numerator and denominator polynomials (denoted by $N_n(x)$ and $D_n(x)$) satisfy and the initial values $$D_0(x)=1, D_1(x)=x(1-b)+a, N_0(x)=0, N_1(x)= 1-b.$$ On writing in the form , we note that $\beta_k=-(b+\lambda q^k)\to -b$, so if $b<0$ the measure has an absolutely continuous component. We use a theorem of Nevai [@PN1979 Th. 40, p. 143] (see [@MI2009 Th. 11.2.2, p. 294]) to find the absolutely continuous component of the measure. \[Nevai\] Assume that the set of orthogonal polynomials $\{P_k(x)\}$ are as in Proposition \[spectral\]. If $$\label{ineq:nevai} \sum_{k=1}^\infty \left( \left| \sqrt{\beta_k}-\frac{1}{2}\right| +|\alpha_k|\right) <\infty,$$ then $\mu$ has an absolutely continuous component $\mu^\prime$ supported on $[-1,1]$. Further, if $\mu$ has a discrete part, then it will lie outside $(-1,1)$. In addition, the limiting relation $$\label{eq:nevai} \limsup_{k\to\infty} \left( \frac{P_k(x)\sqrt{1-x^2}}{\sqrt{\beta_1\beta_2\cdots\beta_{k}}} - \sqrt{\frac{2\sqrt{1-x^2}}{\pi\mu^{\prime}(x)}} \sin\left((k+1)\vartheta -\phi(\vartheta)\right) \right) =0$$ holds, with $x=\cos\vartheta \in (-1,1)$. Here $\phi(\vartheta)$ does not depend on $k$. The interval $[-1,1]$ need not be the true interval of orthogonality. We first modify the recurrence relation so that the hypothesis of Proposition \[Nevai\] is satisfied. Let $$P_k(x):=\frac{y_k(\gamma x)}{\gamma^k(1-b)^k},$$ where $\gamma$ will be determined shortly. Next, divide by $\gamma^{k+1}(1-b)^{k+1}$ to see that $P_k(x)$ satisfies the recurrence $$xP_{k}(x)=P_{k+1}(x)+\frac{aq^k}{\gamma(1-b)} P_k(x)-\frac{b+\lambda q^k}{\gamma^2(1-b)^2}P_{k-1}(x).$$ We now choose $$\gamma^2 = -\frac{4b}{(1-b)^2}$$ to find that the recurrence reduces to $$\label{mike-p-3term2} xP_{k}(x)=P_{k+1}(x)+cq^k P_k(x)+\frac1{4} \left(1+\lambda q^k/b\right) P_{k-1}(x),$$ where $c=a/2{\sqrt{-b}}$. We consider the polynomials defined by with the initial conditions $P_0(x)=1$ and $P_1(x)=x-c$. A short calculation shows that the conditions for Proposition \[Nevai\] are satisfied. Here $\alpha_k=cq^k$, $\beta_k=\left(1+\lambda q^k/b\right)/4$. Assume that $0<|q|<1$. Then for $k$ large enough, we can see using the mean value theorem that $$\begin{aligned} \left| \sqrt{\beta_k}-\frac{1}{2} \right| +|\alpha_k | & = \left| \frac{1}{2}\left(1+\lambda q^k/b\right)^{1/2}-\frac{1}{2} \right| +|cq^k | \cr &\leq C \left| \frac{\lambda q^k}{4b} \right| +|cq^k | $$ for some constant $C$. Thus $$\sum_{k=1}^\infty \left( \left| \sqrt{\beta_k}-\frac{1}{2} \right| +|\alpha_k |\right) <\infty.$$ The choice of $\gamma$ is now transparent. The idea is to compare the asymptotic expression for $P_k(x)$ with to determine the formula for $\mu^\prime(x)$. For this purpose we will use Darboux’s method, which can be stated as follows. \[darboux\] Let $f(z)$ and $g(z)$ be analytic in the disk $\{ z: |z|<r\}$ and assume that $$f(z)=\sum_{k=0}^{\infty} f_kz^k, \text{ } g(z)=\sum_{k=0}^{\infty} g_kz^k, \text{ } |z|<r.$$ If $f-g$ is continuous on the closed disk $\{ z: |z|\leq r\}$ then $$f_k = g_k +o\left(r^{-k}\right).$$ With these preliminaries, we now proceed with our first result, the orthogonality relation for $P_k(x)$. Let $q$ be real satisfying $0<|q|<1$, $c\in \mathbb{R} $, and $1+\lambda q^k/b >0$. Let $P_k(x)$ be a set of polynomials defined by satisfying the initial conditions $P_0(x)=1$ and $P_1(x)=x-c$. Then we have the orthogonality relation: $$\int P_n(x)P_m(x)d\mu = \frac{1}{4^n}{{\left({-\lambda q/b}; q\right)_{n}}} \delta_{mn},$$ where $\mu$ has an absolutely continuous component, and $$\begin{aligned} \label{mu-prime} \mu^{\prime}(x) &= \frac{2}{\pi} \frac{{{\left({-\lambda q/b}; q\right)_{\infty}}}}{|R|^2\sqrt{1-x^2} } \text{ for } x\in (-1,1), \\ \intertext{with} R&=\frac{-1}{i\sin\vartheta} \sum_{m=0}^{\infty} \frac{{{\left({- \lambda qe^{i\vartheta} /2bc}; q\right)_{m}}} }{{{\left({q, qe^{2i\vartheta} }; q\right)_{m}}}} (-2c)^m e^{im\vartheta} q^{m\choose 2} ,\end{aligned}$$ and $x=\cos\vartheta$. Further, if $\mu$ has a discrete part, it will lie outside $(-1,1)$.   1. We can take $x=\cos\vartheta$ and write the part of the integral where $\mu$ has an absolutely continuously component as follows: $$\frac{2{{\left({-\lambda q/b}; q\right)_{\infty}}} }{\pi} \int_{-1}^1 \frac{P_n(x)P_m(x)}{\sqrt{1-x^2} |R|^2}dx= \frac{2{{\left({-\lambda q/b}; q\right)_{\infty}}} }{\pi} \int_{0}^\pi \frac{P_n(\cos\vartheta)P_m(\cos\vartheta)}{|R|^2}d\vartheta . $$ 2. The denominator polynomials we considered in Section \[sec:mikecf1\] are related to $P_k(x)$ as follows: $$P_k(x)=\frac{D_k(\gamma x)}{\gamma^k(1-b)^k}.$$ We have already seen that the hypothesis for Nevai’s theorem are satisfied. To use Darboux’s method to find the formula for $P_k(x)$, we require its generating function. Let $P(t)$ denote the generating function of $P_k(x)$, that is, $$P(t):= \sum_{k=0}^\infty P_k(x)t^k.$$ Multiply by $t^{k+1}$ and sum over $k\geq 0$ to find that $$P(t) = \frac{1}{1-xt+t^2/4} - \frac{ct(1+\lambda tq /4bc)}{1-xt+t^2/4} P(tq).$$ We change the variable by taking $$x=\frac{e^{i\vartheta}+e^{-i\vartheta}}{2} \text{ } (= \cos \vartheta)$$ so $$\begin{aligned} 1-xt+t^2/4 &= (1-e^{i\vartheta}t/2)(1-e^{-i\vartheta}t/2)\cr &= (1-\alpha t)(1-\beta t) .\end{aligned}$$ Using $\alpha$ and $\beta$ we can write the $q$-difference equation for $P(t)$ in the form $$\begin{aligned} P(t) &= \frac{1}{{{\left({\alpha t, \beta t }; q\right)_{1}}} } - \frac{ct(1+\lambda tq/4bc)} {{{\left({\alpha t, \beta t }; q\right)_{1}}} } P(tq) \cr &= \sum_{k=0}^{\infty} \frac{{{\left({-\lambda tq /4bc}; q\right)_{k}}} }{{{\left({\alpha t, \beta t }; q\right)_{k+1}}}} (-ct)^{k}q^{k\choose 2},$$ by iteration. So we obtain $$\label{mikecf2-gf} P(t) = \sum_{k=0}^{\infty} \frac{{{\left({-\lambda tq /4bc}; q\right)_{k}}} }{{{\left({e^{i\vartheta} t/2, e^{-i\vartheta} t/2 }; q\right)_{k+1}}}} (-ct)^{k}q^{k\choose 2}.$$ Next we use Darboux’s method to find an asymptotic expression for $P_k(x)$, where $x=\cos\vartheta$. The terms in the denominator are $$(1-e^{i\vartheta }t/2)(1-e^{i\vartheta}tq/2)\cdots (1-e^{-i\vartheta}t/2)(1-e^{i\vartheta}tq/2)\cdots.$$ The poles are at $$t=2e^{-i\vartheta}, 2e^{-i\vartheta}/q, 2e^{-i\vartheta}/q^2,\dots; \text{ and } t=2e^{i\vartheta}, 2e^{i\vartheta}/q, 2e^{i\vartheta}/q^2\dots.$$ Since $0<|q|<1 $, the poles nearest to $t=0$ are at $t=2e^{-i\vartheta}$ and $t=2e^{i\vartheta}$. We consider $$\label{mikecf2-Q} Q(t):= \sum_{m=0}^{\infty} \frac{{{\left({-2 \lambda qe^{i\vartheta} /2bc}; q\right)_{m}}} } {{{\left({q, qe^{2i\vartheta} }; q\right)_{m}}}} \frac{(-2ce^{i\vartheta})^{m}q^{m\choose 2} }{1-e^{2i\vartheta}}\frac{1}{1-\frac{t}{2}e^{-i\vartheta}}$$ and observe that $P(t)-Q(t)$ has a removable singularity at $t=2e^{i\vartheta}.$ Similarly, the we consider the conjugate $$\label{milecf2-Q-conj} \overline{Q}(t)= \sum_{m=0}^{\infty} \frac{{{\left({- \lambda qe^{-i\vartheta} /2bc}; q\right)_{m}}} }{{{\left({q, qe^{-2i\vartheta} }; q\right)_{m}}}} \frac{(-2ce^{-i\vartheta})^{m}q^{m\choose 2} }{1-e^{-2i\vartheta}}\frac{1}{1-\frac{t}{2}e^{i\vartheta}}$$ and note that $P(t)-\overline{Q}(t)$ has a removable singularity at $t=2e^{-i\vartheta}.$ Thus we see that $$P(t)-Q(t)-\overline{Q}(t)$$ is continuous in $|t|\leq 2$. Thus Darboux’s method can be used to find the formula for $P_k(x)$. Writing $$Q(t)=\sum_{k=0}^\infty Q_k t^k$$ we see that $$P_k(x) =Q_k+\overline{Q}_k + {o}(2^{-k}).$$ The geometric series implies that $$Q_k = \frac{e^{-i(k+1)\vartheta}}{2^{k+1} (-i)\sin\vartheta} \sum_{m=0}^{\infty} \frac{{{\left({- \lambda qe^{i\vartheta} /2bc}; q\right)_{m}}} }{{{\left({q, qe^{2i\vartheta} }; q\right)_{m}}}} (-2c)^m e^{im\vartheta} q^{m\choose 2}$$ and $$\overline{Q}_k = \frac{e^{i(k+1)\vartheta}}{2^{k+1} i\sin\vartheta} \sum_{m=0}^{\infty} \frac{{{\left({- \lambda qe^{-i\vartheta} /2bc}; q\right)_{m}}} }{{{\left({q, qe^{-2i\vartheta} }; q\right)_{m}}}} (-2c)^m e^{-im\vartheta} q^{m\choose 2} .$$ We denote by $R$ the part of $Q_k$ that is independent of $k$. That is, let $$R:=\frac{-1}{i\sin\vartheta} \sum_{m=0}^{\infty} \frac{{{\left({- \lambda qe^{i\vartheta} /2bc}; q\right)_{m}}} }{{{\left({q, qe^{2i\vartheta} }; q\right)_{m}}}} (-2c)^m e^{im\vartheta} q^{m\choose 2}$$ and write it in a form $$R=|R|e^{i\phi}.$$ It is clear that $\phi$ is independent of $k$ (though it depends on $\vartheta$). Now using this notation we have the asymptotic formula for $P_k(x)$: $$\begin{aligned} P_k(x) &\sim Q_k+ \overline{Q}_k =\frac{|R|}{2^{k}} \sin\left((k+1)\vartheta-\phi+\frac{\pi}{2}\right). \label{formula-pk} $$ Now that we know the asymptotic formula for $P_k(x)$ we can compare with and obtain the expression for the measure $\mu^\prime$. To do so, we informally write as $$\frac{P_k(x)\sqrt{\mu^\prime(x)}}{\sqrt{\beta_1\beta_2\cdots \beta_{k}}} \sim \sqrt{\frac{2}{\pi}} \frac{\sin\left( (k+1)\vartheta -\phi(\vartheta)\right)}{(1-x^2)^{1/4}} .$$ Note that $$\beta_1\beta_2\cdots\beta_k=\frac{1}{4^k}{{\left({-\lambda q/b}; q\right)_{k}}}.$$ Comparing with the above, we find that $$\begin{aligned} \mu^\prime (x) = \frac{2{{\left({-\lambda q/b}; q\right)_{\infty}}}}{\pi \sqrt{1-x^2} |R|^2} \label{mu-prime}\end{aligned}$$ where $x=\cos\vartheta$. In this manner, we have obtained an expression for $\mu^{\prime}$ from Nevai’s theorem. This completes the proof. We can take the special cases $\lambda=0$ and $a=0$ in and obtain analogous results for the corresponding special cases of Hirschhorn’s continued fractions. The special case $a=0$ is a continued fraction considered by Ramanujan in his Lost Notebook, see [@AB2005 Entry 6.3.1(iii)]. However, if we take $b=0$, $\mu$ does not have an absolutely convergent component. We consider this case in Section \[sec:specialcases\]. The Stieltjes Transform {#sec:stieltjes} ======================= We now recall Proposition \[prop:stieltjes\] which says that the continued fraction is given by the Steiltjes transform of the measure $\mu$. In this section, we provide the evaluation of the continued fraction. In addition, we invert the Stieltjes transform using , and obtain an alternate expression for $\mu^\prime$. Recall the notation $P_k^*(x)$ for the polynomials satisfying with initial conditions $P^*_0(x)=0$ and $P^*_1(x)=1$. The polynomials $P_k(x)$ satisfy the same recurrence with the initial conditions $P_0(x)=1$ and $P_1(x)=x-c$. We need to compute $$\label{cf-mike-transformed} X(x)= \lim_{k\to\infty} \frac{P_k^*(x)}{P_k(x)}.$$ Again we will appeal to Darboux’s theorem. However, this time the computation of the formula is slightly different. Note that since $x=\cos\vartheta$, $$e^{\pm i\vartheta} = x\pm \sqrt{x^2-1}.$$ We choose a branch of $\sqrt{x^2-1}$ in such a way that $$\sqrt{x^2-1} \sim x, \text{ as } x\to\infty,$$ so that $\left| e^{-i\vartheta}\right|<\left|e^{i\vartheta}\right|$ in the upper half plane, and $\left|e^{i\vartheta}\right|<\left|e^{-i\vartheta}\right|$ in the lower half plane. We use the notation $\rho_1=e^{-i\vartheta}$ and $\rho_2=e^{i\vartheta}$. \[th:cf-values\] Let $X(x)$ be the continued fraction in . Let $\rho_1$ and $\rho_2$ be as above. Let $F$ and $G$ be defined as follows: $$\begin{aligned} F(\rho) &= \sum_{m=0}^{\infty} \frac{{{\left({- \lambda q\rho /2bc}; q\right)_{m}}} }{{{\left({q, q\rho^2 }; q\right)_{m}}}} (-2c\rho)^{m}q^{\binom{m+1}{2}}, \\ \intertext{and} G(\rho) &= \sum_{m=0}^{\infty} \frac{{{\left({- \lambda q\rho /2bc}; q\right)_{m}}} }{{{\left({q, q\rho^2 }; q\right)_{m}}}} (-2c\rho)^{m}q^{\binom{m}{2}} .\end{aligned}$$ Then $X(x)$ converges for all complex numbers $x\not\in (-1,1)$, except possibly a finite set of points, and is given by $$X(x)= 2\rho\frac{F(\rho)}{G(\rho)},$$ where $\rho$ is given by: $$\rho = \begin{cases} \rho_1, & \text{if } {\operatorname{Im}}(x)>0 , \text{ or } x > 1 \text{ ($x$ real)} \cr \rho_2, & \text{if } {\operatorname{Im}}(x)<0, \text{ or } x < -1 \text{ ($x$ real)} \cr 1, & \text{if } x = 1, \cr -1, & \text{if } x = -1. \end{cases}$$ We first compute asymptotic formulas for $P_k^*(x)$ and $P_k(x)$ in the upper half plane. It is not difficult to see that the generating function of $P_k^{*}(x)$ is given by $$\label{mikecf2-gf-num} P^*(t) = \frac{t}{(1-\rho_1 t/2) (1-\rho_2 t/2)} \sum_{k=0}^{\infty} \frac{{{\left({-\lambda tq /4bc}; q\right)_{k}}} }{{{\left({\rho_1 qt/2, \rho_2 qt/2 }; q\right)_{k}}}} (-ct)^{k}q^{\binom{k+1}2}.$$ In the upper half-plane, the singularity nearest the origin is at $t=2\rho_1$. Let $Q^*(t)$ be the series $${Q^*}(t)= \frac{2\rho_1}{(1-\rho_1^2) (1-\rho_2 t/2)} \sum_{m=0}^{\infty} \frac{{{\left({- \lambda q\rho_1 /2bc}; q\right)_{m}}} }{{{\left({q, q\rho_1^2 }; q\right)_{m}}}} (-2c\rho_1)^{m}q^{\binom{m+1}{2}}.$$ Then $P^*(t)-Q^*(t)$ has a removable singularity at $t=2\rho_1$. By Darboux’s method, we have $$P_k^*(x) \sim \frac{2\rho_1 \rho_2^k}{2^k(1-\rho_1^2) } \sum_{m=0}^{\infty} \frac{{{\left({- \lambda q\rho_1 /2bc}; q\right)_{m}}} }{{{\left({q, q\rho_1^2 }; q\right)_{m}}}} (-2c\rho_1)^{m}q^{\binom{m+1}{2}} = \frac{2\rho_1 \rho_2^k}{2^k(1-\rho_1^2) } F(\rho_1) .$$ Similarly, considering the generating function of $P_k(x)$ in the upper half plane, we find that $$P_k (x) \sim \frac{\rho_2^k}{2^k(1-\rho_1^2) } \sum_{m=0}^{\infty} \frac{{{\left({- \lambda q\rho_1 /2bc}; q\right)_{m}}} }{{{\left({q, q\rho_1^2 }; q\right)_{m}}}} (-2c\rho_1)^{m}q^{\binom{m}{2}} = \frac{\rho_2^k}{2^k(1-\rho_1^2) } G(\rho_1) .$$ Thus in the upper half-plane, we find that $$X(x)=\lim_{k\to\infty} \frac{P_k^*(x)}{P_k(x)} =2\rho_1\frac{F(\rho_1)}{G(\rho_1)}.$$ The same calculation works when $x$ is real, and $x>1$. In the lower half-plane, since $t=2\rho_2$ is the singularity nearest to the origin, a similar calculation yields $$X(x)=\lim_{k\to\infty} \frac{P_k^*(x)}{P_k(x)} =2\rho_2\frac{F(\rho_2)}{G(\rho_2)}.$$ This is also valid for real values of $x$ such that $x<-1$. For $x=1$, we find that the generating function for $P_k^*(1)$ is given by $$P^*(t) = \frac{t}{(1- t/2)^2} \sum_{k=0}^{\infty} \frac{{{\left({-\lambda tq /4bc}; q\right)_{k}}} }{{{\left({qt/2}; q\right)_{k}}}^2} (-ct)^{k}q^{\binom{k+1}2}.$$ The singularity nearest the origin is at $t=2$. The dominating term of the comparison function is given by $${Q^*}(t)= \frac{2}{(1-t/2)^2} \sum_{m=0}^{\infty} \frac{{{\left({- \lambda q /2bc}; q\right)_{m}}} }{{{\left({q, q }; q\right)_{m}}}} (-2c)^{m}q^{\binom{m+1}{2}}.$$ (There is another term of the form $A/(1-t/2)$, but that does not have any contribution to $P_k^*(1)$). Darboux’s method yields $$P_k^*(1) \sim \frac{2(k+1)}{2^k } \sum_{m=0}^{\infty} \frac{{{\left({- \lambda q /2bc}; q\right)_{m}}} }{{{\left({q, q }; q\right)_{m}}}} (-2c)^{m}q^{\binom{m+1}{2}} = \frac{2 (k+1)}{2^k } F(1) .$$ Similarly, we find that $$P_k(1)\sim \frac{(k+1)}{2^k } G(1),$$ and so $$X(1) = 2\frac{F(1)}{G(1)},$$ as required. The computation at $x=-1$ is similar. The remarks at the end of the section explain why there may be a finite set of points outside of $(-1,1)$ where $X(x)$ does not converge. On inverting the Stieltjes Transform using , we have another formula for the absolutely continuous component of the orthogonality measure. Let $\mu^{\prime}$ be given by and let $F$ and $G$ be as in Theorem \[th:cf-values\]. Then, for $x\in (-1,1)$, we have $$\mu^{\prime} = \frac{1}{\pi i} \left( \rho_2\frac{F(\rho_2)}{G(\rho_2)}-\rho_1\frac{F(\rho_1)}{G(\rho_1)} \right) .$$ From Theorem \[th:cf-values\], it follows that in the upper half-plane, $$X(x+i0^+)=\lim_{k\to\infty} \frac{P_k^*(x)}{P_k(x)} =2\rho_1\frac{F(\rho_1)}{G(\rho_1)},$$ where now $x$ is a real number in $(-1,1)$. Similarly, we have $$X(x-i0^+)=\lim_{k\to\infty} \frac{P_k^*(x)}{P_k(x)} =2\rho_2\frac{F(\rho_2)}{G(\rho_2)}.$$ The theorem now follows from . Before closing this section, we make a few remarks concerning the discrete part of the measure $\mu$. Recall that Nevai’s theorem says that the discrete part of the measure will lie outside $(-1,1)$. Let $X(x)=F/G $ represent the continued fraction, with $F$ and $G$ entire functions (as above). Assume that $x_0$ is an isolated mass point of weight $m_0$. Then $X(x)$ is of the form $$\begin{aligned} X(x) &=\frac{F}{G} =\int \frac{d\mu(t)}{x-t}\cr &= \int_{-1}^1 \frac{\mu^{\prime }d t}{x-t} + \frac{m_0}{x-x_0} + \text{terms from other isolated mass points} .\end{aligned}$$ Thus $X(x)$ has a simple pole at $x_0$ with residue equal to $m_0$. Since the measure is positive, the residue $m_0$ is positive. 1. This implies that the mass points of the discrete part of the measure occur at the poles of the continued fraction $X(x)$ outside $(-1,1)$. Since the measure is bounded, we can only have a finite number of such points. 2. We can show that the zeros of $F(x)$ interlace with the zeros of $G(x)$. The poles of $X(x)$ occur at the zeros of $G$. If the pole is at $x=\rho$, we have $$m=\frac{F(\rho)}{G^{\prime}(\rho)}>0.$$ Thus, $F$ and $G^{\prime}$ have the same sign. Now at two successive zeros of $G(x)$, the sign of $G^{\prime}(x)$ will be different. And thus the sign of $F(x)$ changes at two successive zeros of $G(x)$. This implies that $F$ has a zero between two successive zeros of $G$. Unfortunately, we are unable to compute the zeros of $G$ from our formulas, and thus cannot say much more about the discrete part of the measure. Solutions of the recurrence that are moments {#sec:moments} ============================================ Recall the definition of the $q$-integral: $$\int_{a}^b f(t) d_q t:= b(1-q) \sum_{n=0}^\infty q^n f(bq^n) - a(1-q) \sum_{n=0}^\infty q^n f(aq^n).$$ In this section we find a solution $p_k(x)$ of of the form $$\label{pk-def} p_k(x) = \int_{t_1}^{t_2} t^k f(t) d_q t,$$ following a technique developed by Ismail and Stanton in [@IS1997; @IS1998; @IS2002]. We will use the integration by parts formula $$\label{qbyparts} \int_a^b f(t) g(qt) d_q t = \frac{1}{q} \int_a^b g(t) f(t/q) d_q t +\frac{1-q}{q} \big( ag(a)f(a/q) - bg(b)f(b/q)\big).$$ This formula follows from the definition of the $q$-integral. We will require the notation of [*basic hypergeometric series*]{} (or $_r\phi_s$ series). This series is of the form $$_{r}\phi_s \left[\begin{matrix} a_1,a_2,\dots,a_r \\ b_1,b_2,\dots,b_s\end{matrix} ; q, z \right] := \sum_{k=0}^{\infty} \frac{{{\left({a_1,a_2,\dots, a_r}; q\right)_{k}}}}{{{\left({q, b_1,b_2,\dots, b_s}; q\right)_{k}}}} \left( (-1)^kq^{\binom k2}\right)^{1+s-r} z^k.$$ When $r=s+1$, the series converges for $|z|<1$. See Gasper and Rahman [@GR90] for further convergence conditions for these series. Let $|\lambda q/b|<1$. With $x=\cos\vartheta$, we define $p_k(x)$ as the $q$-integral $$\begin{gathered} p_k(x) \label{final-pk-qint} := \frac {4 (-i\sin\vartheta)} {(1-q)} \frac {{{\left({ 2c e^{i\vartheta}, 2c e^{-i\vartheta} }; q\right)_{\infty}}}} {{{\left({q, e^{2i\vartheta}, e^{-2i\vartheta} }; q\right)_{\infty}}}}\cr \times \int_{\frac{1}{2}e^{-i\vartheta}}^{\frac{1}{2}e^{i\vartheta}} t^k \frac{{{\left({2q e^{i\vartheta} t,2q e^{-i\vartheta} t, -\lambda q/4bct}; q\right)_{\infty}}}} {{{\left({ 4c t, q/4c t}; q\right)_{\infty}}}} d_q t.\end{gathered}$$ Then $p_k(x)$ satisfies the recurrence relation . Further, let $|\lambda q/2bc|<1$. Then, for ${\operatorname{Im}}(x) \ge 0$, we have $$\begin{aligned} \label{final-pka} p_k(x) &= \frac{e^{ik\vartheta}{{\left({2ce^{-i\vartheta}}; q\right)_{k}}} {{\left({-\frac{\lambda q}{2bc }e^{-i\vartheta}}; q\right)_{\infty}}}}{2^{k} {{\left({ \frac{q}{2c}e^{-i\vartheta}}; q\right)_{\infty}}}} \ \! _{2}\phi_1 \left[ \begin{matrix} -{b}q^{-k}/{\lambda}, 0 \\ q^{1-k} e^{i\vartheta}/2c \end{matrix} ;q, -\frac{\lambda q}{2bc}e^{-i\vartheta} \right]\\ \intertext{ and, for ${\operatorname{Im}}(x) \le 0 $, we have} p_k(x) &= \frac{e^{-ik\vartheta}{{\left({2ce^{i\vartheta}}; q\right)_{k}}} {{\left({-\frac{\lambda q}{2bc }e^{i\vartheta}}; q\right)_{\infty}}}}{2^{k} {{\left({ \frac{q}{2c}e^{i\vartheta}}; q\right)_{\infty}}}} \ \! _{2}\phi_1 \left[ \begin{matrix} -{b}q^{-k}/{\lambda}, 0 \\ q^{1-k} e^{-i\vartheta}/2c \end{matrix} ;q, -\frac{\lambda q}{2bc}e^{i\vartheta} \right]. \label{final-pkb}\end{aligned}$$   1. The ratio $p_k(x)/p_0(x),$ is a solution of with value $1$ at $k=0$. 2. When $b=-\lambda$, the $_2\phi_1$ in (and ) terminates, and we find that $p_0(x)=1$ and $p_1(x)$ is a polynomial of degree $1$. Indeed, we see that $p_1(x)=x-c$, so the initial conditions will match those satisfied by the denominator polynomials $P_k(x)$ (with $b=-\lambda$) considered in Section \[sec:Mike-Nevai\]. In that case, our calculations are a special case of the calculations in Ismail and Stanton [@IS1997] in their proof of Theorem 2.1(B). For now, we call our solution $g_k(x)$ and assume it satisfies . We will show how one can guess $f(t)$, and the limits $t_1$ and $t_2$. From the recurrence relation , we must have $$\begin{aligned} \label{qint1} x\int_{t_1}^{t_2} t^k f(t) d_q t & = \int_{t_1}^{t_2} t^{k+1} f(t) d_q t + c\int_{t_1}^{t_2} (qt)^{k} f(t) d_q t \cr & \hspace{1in} + \frac{1}{4}\int_{t_1}^{t_2} t^{k-1} f(t) d_q t +\frac{\lambda q}{4b}\int_{t_1}^{t_2} (qt)^{k-1} f(t) d_q t\cr &= \int_{t_1}^{t_2} t^{k} \left(t f(t) + f(t)/4t\right) d_q t +\cr &\hspace{1in} \int_{t_1}^{t_2} t^{k} \left(c f(t/q)/q +\lambda f(t/q)/4bt\right) d_q t, \end{aligned}$$ where we use and assume that $$f(t_1/q) =0 = f(t_2/q)$$ in the last step. Now will be satisfied if $$\label{f-feqn} f(t)\big(x-t-{1}/{4t}\big) = f(t/q) \big(c/q+\lambda/4bt\big),$$ or $$\begin{aligned} f(t) &= \frac{-b (1-\alpha t)(1-\beta t)}{\lambda(1+4bct/\lambda )} f(tq),\end{aligned}$$ where $\alpha$ and $\beta$ are such that $$1-4qxt +4q^2t^2 =(1-\alpha t)(1-\beta t).$$ For convenience we change the variable by taking $$x=\cos\vartheta = \frac{e^{i\vartheta} + e^{-i\vartheta}}{2}$$ so $$\alpha = 2qe^{i\vartheta} \text{ and } \beta = 2qe^{-i\vartheta}.$$ Now if we find a function $h(t)$ such that $$\label{h-feqn} h(t)=\frac{-b}{\lambda}h(tq),$$ then we can write $f$ as $$f(t)=\frac{{{\left({\alpha t,\beta t}; q\right)_{\infty}}}}{{{\left({-4bct/\lambda}; q\right)_{\infty}}}} h(t).$$ To find an $h(t)$ which satisfies , we turn to the elliptic theta factorials, defined for $z\neq 0$ and $|q|<1$ as follows: $${\theta\!\left({z} ;q \right) }:={{\left({z, q/z}; q\right)_{\infty}}}.$$ Note the [*quasiperiodicity*]{} property $$\frac{{\theta\!\left({z} ;q \right) }}{{\theta\!\left({zq} ;q \right) }}=-z.$$ This suggests that we can take $h(t)$ of the form $$\begin{aligned} h(t)=\frac{{\theta\!\left({At} ;q \right) }}{{\theta\!\left({Bt} ;q \right) }}\cr \intertext{so that} \frac{h(t)}{h(tq)} = \frac{A}{B}.\end{aligned}$$ We postpone the selection of $A$ and $B$ until later, but assume that $$\frac{A}{B}= \frac{-b}{\lambda},$$ so that is satisfied. Thus, with $A$ and $B$ as above, we find a solution $f(t)$ of given by $$\label{f-final} f(t) = \frac{{{\left({2q e^{i\vartheta} t,2q e^{-i\vartheta} t, At, q/At}; q\right)_{\infty}}}}{{{\left({-4bct/\lambda, B t, q/B t}; q\right)_{\infty}}}} .$$ It remains to find $t_1$ and $t_2$. If we take $$t_1= \frac{1}{2}e^{-i\vartheta}, t_2= \frac{1}{2}e^{i\vartheta}$$ we will find that $$f(t_1/q)=0=f(t_2/q).$$ In this manner, we obtain an expression for a solution of the recurrence relation in the form : $$\label{gk-qint} g_k(x)= \int_{\frac{1}{2}e^{-i\vartheta}}^{\frac{1}{2}e^{i\vartheta}} t^k \frac{{{\left({2q e^{i\vartheta} t,2q e^{-i\vartheta} t, At, q/At}; q\right)_{\infty}}}}{{{\left({-4bct/\lambda, B t, q/B t}; q\right)_{\infty}}}} d_q t.$$ We will specify $A$ and $B$ shortly. Using the definition of the $q$-integral, and some elementary algebraic manipulations, we obtain another expression for $g_k(x)$: $$\begin{aligned} g_k(x) &= (1-q) \frac{e^{i(k+1)\vartheta}}{2^{k+1}} \frac {{{\left({q, q e^{2i\vartheta}, \frac{A}{2}e^{i\vartheta}, \frac{2q}{A}e^{-i\vartheta}}; q\right)_{\infty}}}} {{{\left({ -\frac{2bc}{\lambda}e^{i\vartheta}, \frac{B}{2}e^{i\vartheta}, \frac{2q}{B}e^{-i\vartheta}}; q\right)_{\infty}}}}\cr &\hspace{20pt} \times \sum_{n=0}^\infty \frac {{{\left({-\frac{2bc}{\lambda}e^{i\vartheta} }; q\right)_{n}}}} {{{\left({q, q e^{2i\vartheta}}; q\right)_{n}}}} \left(\frac{-\lambda q^{k+1}}{b}\right) ^n\cr & \hspace{50 pt} - \text{ (same term with $\vartheta\mapsto -\vartheta$)}. $$ Now using the $_r\phi_s$ notation, and collecting common terms, we can write this as $$\begin{aligned} \label{pk-sum} g_k(x) &= (1-q) \frac{e^{i(k+1)\vartheta}}{2^{k+1}} \frac {{{\left({q, q e^{2i\vartheta}, \frac{A}{2}e^{i\vartheta}, \frac{2q}{A}e^{-i\vartheta}}; q\right)_{\infty}}}} {{{\left({ -\frac{2bc}{\lambda}e^{i\vartheta}, \frac{B}{2}e^{i\vartheta}, \frac{2q}{B}e^{-i\vartheta}}; q\right)_{\infty}}}}\cr &\hspace{20pt} \times \Bigg( \ \! _{2}\phi_1 \left[ \begin{matrix} -\frac{2bc}{\lambda}e^{i\vartheta}, 0 \\ q e^{2i\vartheta} \end{matrix} ;q, -\frac{\lambda q^{k+1}}{b} \right] \cr &\hspace{20pt} - e^{-2i(k+1)\vartheta} \frac {{{\left({q e^{-2i\vartheta}, -\frac{2bc}{\lambda}e^{i\vartheta}, \frac{A}{2}e^{-i\vartheta}, \frac{2q}{A}e^{i\vartheta}, \frac{B}{2}e^{i\vartheta}, \frac{2q}{B}e^{-i\vartheta}}; q\right)_{\infty}}}} {{{\left({q e^{2i\vartheta}, -\frac{2bc}{\lambda}e^{-i\vartheta}, \frac{A}{2}e^{i\vartheta}, \frac{2q}{A}e^{-i\vartheta}, \frac{B}{2}e^{-i\vartheta}, \frac{2q}{B}e^{i\vartheta}}; q\right)_{\infty}}}} \cr &\hspace{30pt} \cdot \ \! _{2}\phi_1 \left[ \begin{matrix} -\frac{2bc}{\lambda}e^{-i\vartheta}, 0 \\ q e^{-2i\vartheta} \end{matrix} ;q, -\frac{\lambda q^{k+1}}{b} \right] \Bigg).\end{aligned}$$ Next, we wish to examine whether the term in the brackets can be simplified by using a transformation formula. Indeed, on scanning the list of transformations in Gasper and Rahman, one finds in [@GR90 Eq. (III.31)] a promising candidate. We take $a\mapsto -2bce^{i\vartheta}/\lambda$, $b\to 0$, $c\mapsto qe^{2i\vartheta}$ and $z\mapsto -\lambda q^{k+1}/b$ in this transformation formula to obtain: $$\begin{aligned} \label{III.31-special} \ \! _{2}\phi_1 & \left[ \begin{matrix} -\frac{2bc}{\lambda}e^{i\vartheta}, 0 \\ q e^{2i\vartheta} \end{matrix} ;q, -\frac{\lambda q^{k+1}}{b} \right] \cr & - e^{-2i(k+1)\vartheta} \frac {{{\left({q e^{-2i\vartheta}, -\frac{\lambda q}{2bc}e^{i\vartheta}, 2ce^{i\vartheta}, \frac{q}{2c}e^{-i\vartheta} }; q\right)_{\infty}}}} {{{\left({q e^{2i\vartheta}, -\frac{\lambda q}{2bc}e^{-i\vartheta}, 2ce^{-i\vartheta}, \frac{q}{2c}e^{i\vartheta} }; q\right)_{\infty}}}} \ \! _{2}\phi_1 \left[ \begin{matrix} -\frac{2bc}{\lambda}e^{-i\vartheta}, 0 \\ q e^{-2i\vartheta} \end{matrix} ;q, -\frac{\lambda q^{k+1}}{b} \right] \cr &\hspace{30pt} = \frac {{{\left({e^{-2i\vartheta} }; q\right)_{\infty}}}} {{{\left({ -\frac{\lambda q}{2bc}e^{-i\vartheta}, 2cq^k e^{-i\vartheta}}; q\right)_{\infty}}}} \ \! _{1}\phi_1 \left[ \begin{matrix} -{\lambda q}e^{i\vartheta}/{2bc} \\ q^{1-k}e^{i\vartheta} /{2c} \end{matrix} ;q, \frac{q^{1-k}e^{-i\vartheta}}{2c} \right].\end{aligned}$$ Now, comparing and the two terms inside the bracket in , we see that we should choose $B=4c$ and thus, since $A/B=-b/\lambda$, we must choose $A=-4bc/\lambda$. Next, we obtain . First we assume that ${\operatorname{Im}}(x)\ge 0$ or ${\operatorname{Im}}(\vartheta) \le 0$, so that $|e^{-i\vartheta}|\le 1$. Applying , we find that reduces to $$\begin{aligned} \label{pk-sum2} g_k(x) &= (1-q) \frac{e^{i(k+1)\vartheta}}{2^{k+1}} \frac {{{\left({q, q e^{2i\vartheta}, e^{-2i\vartheta}}; q\right)_{\infty}}}} {{{\left({2ce^{i\vartheta}, \frac{q}{2c}e^{-i\vartheta}, 2cq^ke^{-i\vartheta}}; q\right)_{\infty}}}}\cr &\hspace{20pt} \times \ \! _{1}\phi_1 \left[ \begin{matrix} -{\lambda q}e^{i\vartheta}/{2bc} \\ q^{1-k} e^{i\vartheta}/2c \end{matrix} ;q, \frac{q^{1-k}e^{-i\vartheta}}{2c} \right] .\end{aligned}$$ We can rewrite the $_1\phi_1$ on the right hand side using a special case of the transformation formula as a $_2\phi_1$ sum. The transformation we use is [@GR90 Eq. (III.4)]: $$\label{III.4} \ \! _{2}\phi_1 \left[ \begin{matrix} a, b \\ c \end{matrix} ;q,z \right] = \frac {{{\left({az}; q\right)_{\infty}}}} {{{\left({z}; q\right)_{\infty}}}} \ \! _{2}\phi_2 \left[ \begin{matrix} a, c/b \\ c, az \end{matrix} ;q, bz \right] .$$ We use the $a\mapsto 0$, $b\mapsto -bq^{-k}/\lambda$, $c\mapsto q^{1-k} e^{i\vartheta}/2c$, $z\mapsto -\lambda qe^{-i\vartheta}/2bc$ case of and some elementary computations to write our solution of as follows. $$\begin{aligned} g_k(x) &= (1-q) \frac{e^{ik\vartheta}{{\left({2ce^{-i\vartheta}}; q\right)_{k}}}}{2^{k+2} (-i\sin\vartheta)} \frac {{{\left({q, e^{2i\vartheta}, e^{-2i\vartheta}, -\frac{\lambda q}{2bc }e^{-i\vartheta}}; q\right)_{\infty}}}} {{{\left({ 2c e^{i\vartheta}, 2c e^{-i\vartheta}, \frac{q}{2c}e^{-i\vartheta}}; q\right)_{\infty}}}}\cr &\hspace{20pt} \times \ \! _{2}\phi_1 \left[ \begin{matrix} -{b}q^{-k}/{\lambda}, 0 \\ q^{1-k} e^{i\vartheta}/2c \end{matrix} ;q, -\frac{\lambda q}{2bc}e^{-i\vartheta} \right].\end{aligned}$$ Finally, we divide through by some of the factors that do not depend on $k$, and obtain the solution $p_k(x)$ given in . Dividing by these same factors, and inserting the values of $A$ and $B$, we obtain the $q$-integral representation . To obtain , we consider the case ${\operatorname{Im}}(x) \le 0$, replace $\vartheta$ by $-\vartheta$, and apply the transformations as above. Alternatively, we use a Heine transformation [@GR90 Eq. (III.2)]. This completes the proof. The special case when $b=0$ {#sec:specialcases} =========================== In this section we consider the special case $b=0$ of . Observe that other special cases which lead to Ramanujan’s continued fractions (when $a=0$ or $\lambda =0$) can be treated as special cases of our work earlier in this paper. But when $b=0$, Blumenthal’s theorem tells us that the measure has no absolutely continuous component, and is purely discrete. Thus this case has to be considered separately. When $b=0$, the continued fraction is $$R(x)= \frac{1}{x+a}{\genfrac{}{}{0pt}{}{}{+}}\frac{\lambda q}{x+aq}{\genfrac{}{}{0pt}{}{}{+}}\frac{\lambda q^2}{x+aq^2}{\genfrac{}{}{0pt}{}{}{+}}{\genfrac{}{}{0pt}{}{}{\cdots}}. \label{ram-jfrac}$$ When $x=1$, it reduces to Ramanujan’s continued fraction, given by the $b=0$ case of . The corresponding three-term recurrence relation is $$\label{ram-3term} y_{k+1}(x) = (x+ aq^k) y_k(x) + \lambda q^{k} y_{k-1}(x), \text{ for } k > 0.$$ By Proposition \[cf-conv\], the numerator and denominator polynomials (denoted by $Q^*_k(x)$ and $Q_k(x)$, respectively) satisfy and the initial values $$Q_0(x)=1, Q_1(x)=x+a; \; Q^*_0(x)=0, Q^*_1(x)= 1.$$ We require $0<|q|<1$ (with $q$ real), $a\in\mathbb{R}$, and $\lambda < 0$ to apply Proposition \[spectral\]. Previously, Al–Salam and Ismail [@Al-I1983] had considered a very similar recurrence relation $$\label{ai-3term} U_{k+1} = x(1+ aq^k) U_k - \lambda q^{k-1} U_{k-1}, \text{ for } k > 0,$$ with $U_0=1$, $U_1=x(1+a)$. We denote the generating function of $Q_n(x)$ by $Q(t)$ and of $Q^*_n(x)$ by $Q^*(t)$. The generating functions are as follows. $$\begin{gathered} Q(t) = \sum_{k=0}^{\infty} \frac{{{\left({-\lambda tq /a}; q\right)_{k}}} }{{{\left({x t }; q\right)_{k+1}}}} (at)^{k}q^{k\choose 2}, \cr \intertext{and} Q^*(t) = t\sum_{k=0}^{\infty} \frac{{{\left({-\lambda qt /a}; q\right)_{k}}} }{{{\left({xt }; q\right)_{k+1}}}} (at)^{k}q^{\binom{k}{2}+k}. $$ To obtain explicit expressions of the numerator and denominator polynomials, we need to extract the coefficient of powers of $t$. We need the $q$-binomial theorem in the form [@GR90 Ex. 1.2(vi)] $$\label{gr90-ex-1.2vi} {{\left({at}; q\right)_{k}}}=\sum_{j\geq 0} {\genfrac{[}{]}{0pt}{}}{k}{j}_q (-1)^j q^{j\choose 2} (at)^j.$$ In addition, we require the following special case of the $q$-binomial theorem (cf. [@GR90 Eq. 1.3.2]) valid for $|at|<1$: $$\label{q-bin-special-1} \frac{1}{{{\left({at}; q\right)_{k+1}}}} =\sum_{m=0}^{\infty} {\genfrac{[}{]}{0pt}{}}{m+k}{k} (at)^m.$$ Using these, we find that $Q(t)$ can be written as $$Q(t)=\sum_{j, k, m\geq 0} {\genfrac{[}{]}{0pt}{}}{k}{j}_{q} {\genfrac{[}{]}{0pt}{}}{k+m}{k}_{q} a^{k-j} x^m \lambda^j q^{{k\choose 2}+{j\choose 2}+j}t^{j+k+m}.$$ From here, we take the coefficient of $t^n$ to obtain an expression for $Q_n(x)$. We see that $$\begin{aligned} \label{ram-den} Q_n(x) &= \sum_{j, k \geq 0} {\genfrac{[}{]}{0pt}{}}{k}{j}_{q} {\genfrac{[}{]}{0pt}{}}{n-j}{k}_{q} a^{k-j} x^{n-j-k} \lambda^j q^{{k\choose 2}+{j\choose 2}+j} \cr &= \sum_{j\geq 0} {\genfrac{[}{]}{0pt}{}}{n-j}{j}_q \frac{{{\left({-a/x}; q\right)_{n-j}}}} {{{\left({-a/x}; q\right)_{j}}}} \lambda^j x^{n-2j} q^{j^2},\end{aligned}$$ where we obtain the last equality by summing the inner sum using . Note that the first of these sums expresses $Q_n(x)$ as a polynomial in $x$ of degree $n$, since the indices satisfy $k+j\leq n$. Similarly, $Q_n^*(x)$ can be written as $$\begin{aligned} \label{ram-n} Q_n^*(x) &= \sum_{j\geq 0} {\genfrac{[}{]}{0pt}{}}{n-j-1}{j}_q \frac{{{\left({-a/x}; q\right)_{n-j}}}} {{{\left({-a/x}; q\right)_{j+1}}}} \lambda^j x^{n-2j-1} q^{j^2+j}.\end{aligned}$$ From Proposition \[spectral\], we have the following orthogonality relation. Suppose $q$ is real with $0<|q|<1$, $a\in \mathbb{R}$, and $\lambda<0$. Let $Q_n(x)$ be given by . Then we have the orthogonality relation $$\int_{-\infty}^{\infty} Q_n(x) Q_m(x) d\mu = (-\lambda)^n q^{\binom{n+1}{2}}\delta_{mn},$$ where $\mu$ is a purely discrete positive measure. Next we find asymptotic formulas for the denominator and numerator polynomials, from the formulas for $Q_n(x)$ and $Q_n^*(x)$ above. We find that, for a fixed $x$, as $n\to\infty$, $$\begin{gathered} Q_n(x) \sim x^n {{\left({-a/x}; q\right)_{\infty}}} \ \! _{0}\phi_1 \left[ \begin{matrix} - \\ -a/x \end{matrix} ;q, \frac{\lambda q}{x^2} \right]\cr \intertext{and} Q_n^*(x) \sim x^{n-1} {{\left({-aq/x}; q\right)_{\infty}}} \ \! _{0}\phi_1 \left[ \begin{matrix} - \\ -aq/x \end{matrix} ;q, \frac{\lambda q^2}{x^2} \right].\end{gathered}$$ Thus, the Stieltjes transform of $\mu$ is given by $$\int_{-\infty}^{\infty} \frac{d\mu(t)}{x-t} = \frac{1}{(x+a)} \frac{ _{0}\phi_1 \left[ \begin{matrix} - \\ -aq/x \end{matrix} ;q, \displaystyle \frac{\lambda q^2}{x^2} \right] }{ _{0}\phi_1 \left[ \begin{matrix} - \\ -a/x \end{matrix} ;q, \displaystyle \frac{\lambda q}{x^2} \right] },$$ for $x\not\in \operatorname{supp}{\mu}$. Formulas for the convergents {#sec:convergents} ============================ In this section, we show how to obtain formulas for the convergents analogous to Ramanujan’s Entry 16, which was highlighted in the introduction. We derive a formula given by Hirschhorn [@MDH1974], and then take special cases corresponding to two of Ramanujan’s continued fractions. We have recast Hirschhorn’s original approach in terms of Proposition \[cf-conv\] in order to make it transparent how such formulas can be found. For some further examples, see Bowman, Mc Laughlin and Wyshinski  [@BMW2006]. We will require the notation of the [*$q$-multinomial coefficients*]{}, defined as $${\genfrac{[}{]}{0pt}{}}{n}{k_1,k_2,\dots, k_r}_q = \frac{{{\left({q}; q\right)_{n}}}} {{{\left({q}; q\right)_{k_1}}}{{\left({q}; q\right)_{k_2}}}\cdots {{\left({q}; q\right)_{k_r}}}{{\left({q}; q\right)_{n-(k_1+k_2+\cdots+k_r)}}}}$$ where $n, k_1, k_2, \dots, k_r$ are positive integers and $n\geq k_1+k_2+\cdots + k_r$. When $n< k_1+k_2+\cdots +k_r$, we take the $q$-multinomial coefficient to be $0.$ When $r=1$, then these reduce to the $q$-binomial coefficients. We first consider . Denote by $Y(t)$, $D(t)$ and $N(t)$ the generating functions of $y_k(x)$, $D_k(x)$ and $N_k(x)$ respectively. Multiply by $t^{k+1}$ and sum over $k\geq 0$ to find that $$(1-x(1-b)t-bt^2)Y(t) = y_0+ty_1 -xt(1-b)y_0-aty_0 +at(1+\lambda qt/a)Y(tq),$$ where we have used $y_0=y_0(x)$ and $y_1=y_1(x)$ to denote the initial values of $y_k(x)$. Thus, the generating function of $D_n(x)$ satisfies the $q$-difference equation $$D(t) = \frac{1}{1-x(1-b)t-bt^2} + \frac{at(1+\lambda tq /a)}{1-x(1-b)t-bt^2} D(tq).$$ Let $\alpha$ and $\beta$ be such that $$\label{alpha-beta-mikecf} 1-(1-b)xt-bt^2 = (1-\alpha t) (1-\beta t).$$ Using $\alpha$ and $\beta$ we can write the $q$-difference equation for $D(t)$ in a form that it can be iterated easily. As before, we obtain the generating function $$D(t) = \sum_{k=0}^{\infty} \frac{{{\left({-\lambda tq /a}; q\right)_{k}}} }{{{\left({\alpha t, \beta t }; q\right)_{k+1}}}} (at)^{k}q^{k\choose 2}.\label{mikecf-denom-gf}$$ Similarly, we obtain the generating function of the numerators $$N(t)=t(1-b)\sum_{k=0}^{\infty} \frac{{{\left({-\lambda tq /a}; q\right)_{k}}} }{{{\left({\alpha t, \beta t }; q\right)_{k+1}}}} (at)^{k}q^{{k\choose 2}+k}.$$ Notice that the $x$ is hidden implicitly in $\alpha$ and $\beta$. To obtain explicit formulas for the convergents, we need to find expressions for $N_n(x)$ and $D_n(x)$ when $x=1$. Note that when $x=1$ in , then $\alpha =1$ and $\beta = -b$. We use and to find that $D(t)$ with $\alpha =1$, $\beta=-b$ becomes $$D(t)=\sum_{j, k,l, m\geq 0} {\genfrac{[}{]}{0pt}{}}{k}{j}_{q} {\genfrac{[}{]}{0pt}{}}{k+l}{k}_{q} {\genfrac{[}{]}{0pt}{}}{k+m}{k}_{q} a^{k-j}(-b)^l \lambda^jq^{{k\choose 2}+{j\choose 2}+j}t^{j+k+l+m}.$$ We now take the coefficient of $t^n$ (so restrict the sum to $n=j+k+l+m$) to find that $$\begin{aligned} D_n(1)&=\sum_{j, k,l \geq 0} {\genfrac{[}{]}{0pt}{}}{k}{j}_{q} {\genfrac{[}{]}{0pt}{}}{k+l}{k}_{q} {\genfrac{[}{]}{0pt}{}}{n-j-l}{k}_{q} a^{k-j}(-b)^l \lambda^jq^{{k\choose 2}+{j\choose 2}+j} \cr &= \sum_{j, k,l \geq 0} {\genfrac{[}{]}{0pt}{}}{k+l}{j,l}_{q} {\genfrac{[}{]}{0pt}{}}{n-j-l}{k}_{q} a^{k-j}(-b)^l \lambda^jq^{{k\choose 2}+{j\choose 2}+j} .\end{aligned}$$ Similarly, we find that $$N_n(1) =(1-b) \sum_{j, k,l \geq 0} {\genfrac{[}{]}{0pt}{}}{k+l}{j,l}_{q} {\genfrac{[}{]}{0pt}{}}{n-j-l-1}{k}_{q} a^{k-j}(-b)^l \lambda^jq^{{k\choose 2}+k+{j\choose 2}+j}.$$ We divide $N_{n+1}(1)$ by $(1-b)D_{n+1}(1)$ to obtain Hirschhorn’s formula [@MDH1974]: $$\frac{N_{n+1}(1)}{(1-b)D_{n+1}(1)} = \frac{1}{1-b+a}{\genfrac{}{}{0pt}{}{}{+}}\frac{b+\lambda q}{1-b+aq}{\genfrac{}{}{0pt}{}{}{+}}{\genfrac{}{}{0pt}{}{}{\cdots}}{\genfrac{}{}{0pt}{}{}{+}}\frac{b+\lambda q^n}{1-b+aq^n}. \label{mikecf-jfrac-conv}$$ Taking $n\to\infty$ and invoking the two summations and we obtain Hirschhorn’s formula for his infinite continued fraction as a ratio of two sums, under the condition $|b|<1$. From we can take special cases $b=0$, $a=0$ or both to obtain results related to Ramanujan’s continued fractions. The first special case we consider is from the lost notebook [@AB2005 Entry 6.3.1(iii)] $$\frac{1}{1-b}{\genfrac{}{}{0pt}{}{}{+}}\frac{b+\lambda q}{1-b}{\genfrac{}{}{0pt}{}{}{+}}\frac{b+\lambda q^2}{1-b}{\genfrac{}{}{0pt}{}{}{+}}\frac{b+\lambda q^3}{1-b}{\genfrac{}{}{0pt}{}{}{+}}{\genfrac{}{}{0pt}{}{}{\cdots}}.$$ This is obtained by taking $a=0$ in . Here is our formula for the convergents of . We have, $$\label{g-cfrac3-n} \frac{N^\prime_n}{D^\prime_n} = \frac{1}{1-b}{\genfrac{}{}{0pt}{}{}{+}}\frac{b+\lambda q}{1-b}{\genfrac{}{}{0pt}{}{}{+}}\frac{ b+\lambda q^2}{1-b}{\genfrac{}{}{0pt}{}{}{+}}\frac{b+\lambda q^3}{1-b} {\genfrac{}{}{0pt}{}{}{+}}{\genfrac{}{}{0pt}{}{}{\cdots}}{\genfrac{}{}{0pt}{}{}{+}}\frac{b+\lambda q^n}{1-b} ,$$ where the numerator and denominator polynomials of the $(n+1)$th convergent are given by: $$N^\prime_n = \sum_{k, j\geq 0} q^{k^2+k}\lambda^k {\genfrac{[}{]}{0pt}{}}{k+j}{k}_{q} {\genfrac{[}{]}{0pt}{}}{n-k-j}{k}_{q} (-b)^j$$ and $$D^\prime_n = \sum_{k, j\geq 0} q^{k^2}\lambda^k {\genfrac{[}{]}{0pt}{}}{k+j}{k}_{q} {\genfrac{[}{]}{0pt}{}}{n-k-j+1}{k}_{q} (-b)^j . $$ When $b=0$, this immediately reduces to , Ramanujan’s Entry 16. Upon taking $n\to\infty$, we obtain Ramanujan’s continued fraction evaluation, given in Andrews and Berndt [@AB2005 Entry 6.2.1(iii)]. We define $$\begin{aligned} g(b,\lambda)&:=\sum_{k=0}^{\infty}\frac{ \lambda^k q^{k^2}}{{{\left({q}; q\right)_{k}}}{{\left({-bq}; q\right)_{k}}}}. $$ Then for $|b|<1$, $$\frac{g(b,\lambda q)}{g(b,\lambda)} =\frac{1}{1-b}{\genfrac{}{}{0pt}{}{}{+}}\frac{b+\lambda q}{1-b}{\genfrac{}{}{0pt}{}{}{+}}\frac{b+\lambda q^2}{1-b}{\genfrac{}{}{0pt}{}{}{+}}\frac{b+\lambda q^3}{1-b}{\genfrac{}{}{0pt}{}{}{+}}{\genfrac{}{}{0pt}{}{}{\cdots}}.\label{g-cfrac3}$$ The condition $|b|<1$ appears quite naturally as a requirement for the sum to be convergent. To see this, consider the limit $$\begin{aligned} \lim_{n\to\infty} N^\prime_n &= \lim_{n\to\infty} \sum_{k, j\geq 0} q^{k^2+k}\lambda^k \frac{{{\left({q}; q\right)_{k+j}}} {{\left({q}; q\right)_{n-k-j}}}} {{{\left({q}; q\right)_{k}}}{{\left({q}; q\right)_{j}}} {{\left({q}; q\right)_{n-2k-j}}} {{\left({q}; q\right)_{k}}}} (-b)^j\cr &= \sum_{k\geq 0} \frac{q^{k^2+k}\lambda^k} { {{\left({q}; q\right)_{k}}}} \sum_{j\geq 0} \frac{{{\left({q}; q\right)_{k+j}}} } {{{\left({q}; q\right)_{j}}}{{\left({q}; q\right)_{k}}} } (-b)^j\cr &= \sum_{k\geq 0} \frac{q^{k^2+k}\lambda^k} { {{\left({q}; q\right)_{k}}}{{\left({-b}; q\right)_{k+1}}}} ,\end{aligned}$$ upon invoking , assuming $|b|<1$. This shows that $$\lim_{n\to\infty} N^\prime_n = \frac{g(b,\lambda q)}{1+b}.$$ Similarly, we can see that $$\lim_{n\to\infty} D^\prime_n = \frac{g(b,\lambda )}{1+b},$$ and this completes a proof of . Next we take $b=0$ in . Ramanujan found the continued fraction (see Entry 15 of [@Berndt1991-RN3 ch. 16 ] or [@AB2005 Entry 6.3.1(ii)]) $$\begin{aligned} \frac{g(a,\lambda )}{g(a,\lambda q)} &=1+\frac{\lambda q}{1+aq}{\genfrac{}{}{0pt}{}{}{+}}\frac{\lambda q^2}{1+aq^2}{\genfrac{}{}{0pt}{}{}{+}}\frac{\lambda q^3}{1+aq^3}{\genfrac{}{}{0pt}{}{}{+}}{\genfrac{}{}{0pt}{}{}{\cdots}}. $$ A formula for the convergents of Ramanujan’s Entry 15 is as follows. Let $$\begin{gathered} \widehat{N}_n = \sum_{j\geq 0} q^{j^2}\lambda^j {\genfrac{[}{]}{0pt}{}}{n+1-j}{j}_{q} \frac{{{\left({-aq}; q\right)_{n-j}}}}{{{\left({-a}; q\right)_{j}}}} \cr \intertext{and} \widehat{D}_n = \sum_{j\geq 0} q^{j^2+j}\lambda^j {\genfrac{[}{]}{0pt}{}}{n-j}{j}_{q} \frac{{{\left({-aq}; q\right)_{n-j}}}}{{{\left({-aq}; q\right)_{j}}}}.\end{gathered}$$ Then, for $n=1, 2, 3, \dots$, we have $$(1+a)\frac{\widehat{N}_n}{\widehat{D}_n} = 1+a + \frac{\lambda q}{1+aq}{\genfrac{}{}{0pt}{}{}{+}}\frac{\lambda q^2}{1+aq^2}{\genfrac{}{}{0pt}{}{}{+}}\frac{\lambda q^3}{1+aq^3} {\genfrac{}{}{0pt}{}{}{+}}{\genfrac{}{}{0pt}{}{}{\cdots}}{\genfrac{}{}{0pt}{}{}{+}}\frac{\lambda q^n}{1+aq^n} .\label{entry16-gen1-a}$$ To obtain , we take $x=1$ in and and observe that $$\begin{gathered} Q_{n+1}(1)=(1+a)\widehat{N}_n\cr \intertext{and} Q_{n+1}^*(1)=\widehat{D}_n.\end{gathered}$$ When $a=0$, reduces to Ramanujan’s Entry 16 given in . Formula is implicit in Al-Salam and Ismail’s study [@Al-I1983] of the orthogonal polynomials associated with Rogers–Ramanujan continued fraction. Bhatnagar and Hirschhorn [@BH2016] wrote it in this form and gave an elementary proof following Euler’s approach given in [@GB2014]. Formulas and are generalizations of Ramanujan’s Entry 16, corresponding to two extensions of the Rogers–Ramanujan continued fraction given by Ramanujan in the Lost Notebook, recorded as Entry 6.3.1(ii) and (iii), respectively in [@AB2005]. As we have seen, such formulas can be discovered quite easily using generating functions. Acknowledgments {#acknowledgments .unnumbered} --------------- This work was done at the sidelines of many workshops, conferences and summer schools organized by the members of the Orthogonal Polynomials and Special Functions (OPSF) group of SIAM. We thank the organizers of the following: OPSF summer school, (July 2016), University of Maryland; the international conference on special functions: theory, computation and applications, (June 2018), Liu Bie Ju center for mathematical sciences, City University of Hong Kong, Hong Kong; and, summer research institute on $q$-series, (July-Aug 2018), Chern Institute of Mathematics, Nankai University, Tianjin, PR China. [10]{} W. A. Al-Salam and M. E. H. Ismail. Orthogonal polynomials associated with the [R]{}ogers-[R]{}amanujan continued fraction. , 104(2):269–283, 1983. G. E. Andrews, R. Askey, and R. Roy. , volume 71 of [*Encyclopedia of Mathematics and its Applications*]{}. Cambridge University Press, Cambridge, 1999. G. E. Andrews and B. C. Berndt. . Springer, New York, 2005. R. Askey and M. Ismail. Recurrence relations, continued fractions, and orthogonal polynomials. , 49(300):iv+108, 1984. B. C. Berndt. . Springer-Verlag, New York, 1991. S. Bhargava and C. Adiga. 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With two chapters by Walter Van Assche, With a foreword by Richard A. Askey, Reprint of the 2005 original. M. E. H. Ismail and D. Stanton. Classical orthogonal polynomials as moments. , 49(3):520–542, 1997. M. E. H. Ismail and D. Stanton. More orthogonal polynomials as moments. In [*Mathematical essays in honor of [G]{}ian-[C]{}arlo [R]{}ota ([C]{}ambridge, [MA]{}, 1996)*]{}, volume 161 of [*Progr. Math.*]{}, pages 377–396. Birkhäuser Boston, Boston, MA, 1998. M. E. H. Ismail and D. Stanton. -integral and moment representations for [$q$]{}-orthogonal polynomials. , 54(4):709–735, 2002. M. E. H. Ismail and D. Stanton. Ramanujan continued fractions via orthogonal polynomials. , 203(1):170–193, 2006. P. G. Nevai. Orthogonal polynomials. , 18(213):v+185, 1979. T. J. Stieltjes. Recherches sur les fractions continues. , 4(3):J76–J122, 1995. Reprint of Ann. Fac. Sci. Toulouse [[**[8]{}**]{}]{} (1894), J76–J122. T. J. Stieltjes. Recherches sur les fractions continues. , 4(4):A5–A47, 1995. Reprint of Ann. Fac. Sci. Toulouse [[**[9]{}**]{}]{} (1895), A5–A47. [^1]: \*Research supported by grants of the Austrian Science Fund (FWF): START grant Y463 and FWF grant F50-N15.

=1 Introduction ============ Let $(M,g)$ be an oriented Riemannian manifold, let ${\operatorname{vol}}_g$ denote its volume form and let $f$ be a smooth function on $M$. The triple $\big(M,g, e^{-f} {\operatorname{vol}}_g\big)$ is called a smooth metric measure space. Based on considerations from diffusion processes, Bakry–Émery [@BakryEmeryDiffusion] introduced the tensor $$\begin{gathered} {\operatorname{Ric}}_f = {\operatorname{Ric}}+ {\operatorname{Hess}}f\end{gathered}$$ as a weighted Ricci curvature for a geometric measure space. In fact, this tensor appeared earlier in work of Lichnerowicz [@LichnerowiczBETensor]. Volume comparison theorems for smooth metric measure spaces with ${\operatorname{Ric}}_f$ bounded from below have been established by Qian [@QianEstimatesWeightedVolume], Lott [@LottGeometryBETensor], Bakry–Qian [@BakryQianVolumeComparison] and Wei–Wylie [@WeiWylieComparosionGeoBE]. In this note we study the Bochner technique on smooth metric measure spaces. The distortion of the volume element introduces a diffusion term to the Bochner formula $$\begin{gathered} \Delta_f \omega = ( d d^{*}_f + d^{*}_f d) \omega= \nabla^{*}_f \nabla \omega + {\operatorname{Ric}}(\omega) - ( {\operatorname{Hess}}f ) \omega,\end{gathered}$$ where ${\operatorname{Ric}}$ is the Bochner operator on $p$-forms. Lott [@LottGeometryBETensor] proved that if ${\operatorname{Ric}}_f \geq 0$, then all $\Delta_f$-harmonic $1$-forms are parallel and, for compact manifolds, $H^1(M;{\mathbb{R}})$ is isomorphic to the space of all parallel $1$-forms $\omega$ which satisfy $\big\langle \nabla e^{-f}, \omega \big\rangle = 0$. Moreover, if ${\operatorname{Ric}}_f > 0$, then all $\Delta_f$-harmonic $1$-forms vanish. We introduce new weighted curvature conditions that imply rigidity and vanishing results for $\Delta_f$-harmonic $p$-forms for $p \geq 1$. We can restrict to $p$-forms $\omega$ for $1 \leq p \leq \big\lfloor \frac{n}{2}\big\rfloor$ since $\omega$ is parallel if and only if $\ast \omega$ is parallel, where $\ast$ denotes the Hodge star. By convention, we will refer to the eigenvalues of the curvature operator simply as the eigenvalues of the associated curvature tensor. Let $\big(M^n,g, e^{-f} {\operatorname{vol}}_g\big)$ be a smooth metric measure space. For $1 \leq p < \frac{n}{2}$ set $$\begin{gathered} h = \frac{1}{n-2p} {\operatorname{Hess}}f - \frac{\Delta f}{2(n-p)(n-2p)} g.\end{gathered}$$ Let $\omega$ be a $\Delta_f$-harmonic $p$-form with $|\omega| \in L^2\big(M, e^{-f} {\operatorname{vol}}_g\big)$ for $1 \leq p < \frac{n}{2}$. Let $\lambda_1 \leq \dots \leq \lambda_{\genfrac(){0pt}{2}{n}{2}}$ denote the eigenvalues of the weighted curvature tensor ${\operatorname{Rm}}+ h \owedge g$. If $\lambda_1 + \dots + \lambda_{n-p} \geq 0$, then $\omega$ is parallel. If in addition $M$ is compact, then $H^{p}(M)= \big\lbrace \omega \in \Omega^p(M) \, \vert \, \nabla \omega =0 \ \text{and} \ i_{\nabla f} \omega = 0 \big\rbrace$. If $\lambda_1 + \dots + \lambda_{n-p} > 0$, then $\omega$ vanishes. If in addition $M$ is compact, then the Betti numbers $b_p(M)$ and $b_{n-p}(M)$ vanish for $1 \leq p < \frac{n}{2}$. For $p=1$ the Ricci curvature of the modified curvature tensor is the Bakry–Émery Ricci tensor, and the assumption in the Theorem implies that it is nonnegative. In this sense the Theorem is a generalization of Lott’s [@LottGeometryBETensor] results for $1$-forms. A stronger curvature assumption also allows control in the middle dimension $p = \frac{n}{2}$. Recall that a curvature tensor is $l$-nonnegative (positive) if the sum of its lowest $l$ eigenvalues is nonnegative (positive). Let $\big(M^n,g, e^{-f} {\operatorname{vol}}_g\big)$ be a smooth metric measure space. Let $\mu_1 \leq \dots \leq \mu_n$ denote the eigenvalues of ${\operatorname{Hess}}f$ and let $1 \leq p \leq \big\lfloor \frac{n}{2}\big\rfloor$. Let $\omega$ be a $\Delta_f$-harmonic $p$-form with $|\omega| \in L^2\big(M, e^{-f} {\operatorname{vol}}_g\big)$. If the weighted curvature tensor $$\begin{gathered} {\operatorname{Rm}}+ \frac{\sum\limits_{i=1}^p \mu_i}{2p(n-p)} g \owedge g\end{gathered}$$ is $(n-p)$-nonnegative, then $\omega$ is parallel. If it is $(n-p)$-positive, then $\omega$ vanishes. In particular, if $M$ is compact, then $H^{p}(M)= \big\lbrace \omega \in \Omega^p(M) \, \vert \, \nabla \omega =0 \ \text{and} \ i_{\nabla f} \omega = 0 \big\rbrace$ and in case the weighted curvature tensor is $(n-p)$-positive, the Betti numbers $b_p(M)$ and $b_{n-p}(M)$ vanish. The notation in this paper builds up on the presentation in [@PetersenRiemGeom Chapter 9] and [@PetersenWinkBochner]. Preliminaries ============= Algebraic curvature tensors --------------------------- For an $n$-dimensional Euclidean vector space $(V,g)$ let $\mathcal{T}^{(0,k)}(V)$ denote the vector space of $(0,k)$-tensors and $\operatorname{Sym}^2(V)$ the vector space of symmetric $(0,2)$-tensors on $V$. Let $\mathcal{C}(V)$ denote the vector space of $(0,4)$-tensors with $T(X,Y,Z,W) = - T(Y,X,Z,W) = T(Z,W,X,Y)$. If $T$ also satisfies the algebraic Bianchi identity, then $T$ is called algebraic curvature tensor, $T \in \mathcal{C}_B(V)$. The Kulkarni–Nomizu product of $S_1, S_2 \in \operatorname{Sym}^2(V)$ is given by $$\begin{gathered} (S_1 \owedge S_2)(X,Y,Z,W) = S_1(X,Z)S_2(Y,W)-S_1(X,W)S_2(Y,Z) \\ \hphantom{(S_1 \owedge S_2)(X,Y,Z,W) =}{} +S_1(Y,W)S_2(X,Z)-S_1(Y,Z)S_2(X,W).\end{gathered}$$ With this convention the algebraic curvature tensor $I= \frac{1}{2} g \owedge g$ corresponds to the curvature tensor of the unit sphere. Recall that the decomposition of $\mathcal{C}(V)$ into $O(n)$-irreducible components is given by $$\begin{gathered} \mathcal{C}(V) = \langle I \rangle \oplus \langle \mathring{{\operatorname{Ric}}} \rangle \oplus \langle W \rangle \oplus \Lambda^4 V,\end{gathered}$$ where $\langle \mathring{{\operatorname{Ric}}} \rangle = S_0^2(V) \owedge g$ is the subspace of algebraic curvature tensors of trace-free Ricci type, $S_0^2(V)= \big\lbrace h \in \operatorname{Sym}^2(V) \, \vert \, \tr(h)= 0 \big\rbrace$, and $\langle W \rangle$ denotes the subspace of Weyl tensors. Explicitly, every algebraic curvature tensor decomposes as $$\begin{gathered} {\operatorname{Rm}}= \frac{{\operatorname{scal}}}{2(n-1)n} g \owedge g + \frac{1}{n-2} \mathring{{\operatorname{Ric}}} \owedge g + W.\end{gathered}$$ Lichnerowicz Laplacians on smooth metric measure spaces ------------------------------------------------------- Let $(M,g,f)$ be a smooth metric measure space. The formal adjoints of the exterior and covariant derivative with respect to the measure $e^{-f} {\operatorname{vol}}_g$ are given by $$\begin{gathered} d^{*}_f = d^{*} + i_{\nabla f} \qquad \text{and} \qquad \nabla^{*}_f = \nabla^{*} + i_{\nabla f}.\end{gathered}$$ More generally, for a vector field $U$ on $M$, we will consider $$\begin{gathered} d^{*}_U = d^{*} + i_{U} \qquad \text{and} \qquad \nabla^{*}_U = \nabla^{*} + i_{U}.\end{gathered}$$ The associated generalized Lichnerowicz Laplacian on $(0,k)$-tensors is given by $$\begin{gathered} \Delta_U T = \nabla^{*}_U \nabla T + {\operatorname{Ric}}(T) - (\nabla U) T,\end{gathered}$$ where the curvature term is given by $$\begin{gathered} {\operatorname{Ric}}(T)(X_1, \dots, X_k) = \sum_{i=1}^k \sum_{j=1}^n (R(X_i,e_j)T) (X_1, \dots, e_j, \dots, X_k).\end{gathered}$$ A tensor $T$ is called [*$U$-harmonic*]{} if $\Delta_U T =0$. To emphasize that the curvature term is calculated with respect to the curvature tensor ${\operatorname{Rm}}$, we will also write ${\operatorname{Ric}}_{{\operatorname{Rm}}}(T)$ for ${\operatorname{Ric}}(T)$. Recall that for an endomorphism $L$ of $V$ and a $(0,k)$-tensor $T$ we have $$\begin{gathered} (LT)(X_1, \dots, X_k) = - \sum_{i=1}^k T(X_1, \dots,L(X_i), \dots, X_k).\end{gathered}$$ In particular, the Ricci identity implies that the definition of the curvature term in the Lichnerowicz Laplacian naturally carries over to algebraic curvature tensors. \[BochnerFormulasExample\] Let $(M,g)$ be a Riemannian manifold and $U$ a vector field on $M$. For a $(0,k)$-tensor $T$ on $M$ set ${\operatorname{Ric}}_U(T)={\operatorname{Ric}}(T) - (\nabla U)T$. 1. Every $p$-form satisfies $$\begin{gathered} ( d d^{*}_U + d^{*}_U d ) \omega = \nabla^{*}_U \nabla \omega + {\operatorname{Ric}}_U(\omega).\end{gathered}$$ 2. Every symmetric $(0,2)$-tensor satisfies $$\begin{gathered} ( \nabla_X \nabla^{*}_U T )(X) + \big( \nabla^{*}_U d^{\nabla} T \big) (X,X) = ( \nabla^{*}_U \nabla T )(X,X)+ \frac{1}{2} ( {\operatorname{Ric}}_U T ) (X,X),\end{gathered}$$ where $d^{\nabla}T(Z,X,Y)= (\nabla_X T )(Y,Z) - (\nabla_Y T )(X,Z)$. \(a) The case $U=0$ recovers the well-known Bochner formula. The generalized Hodge Laplacian satisfies $$\begin{gathered} d d^{*}_U + d^{*}_U d = d d^{*} + d^{*} d + d i_U + i_U d = \Delta + L_U.\end{gathered}$$ In addition to the classical Lichnerowicz Laplacian we have on the right hand side $$\begin{gathered} \nabla_U - ( \nabla U ) = L_U\end{gathered}$$ and thus all diffusion terms balance out. \(b) As in (a), it suffices to consider all terms that depend on $U$ and show that $$\begin{gathered} ( \nabla_X i_U h ) (X) + \big( i_U d^{\nabla} h \big) (X,X) = ( \nabla_U h ) (X,X) - \frac{1}{2} ( (\nabla U) h)(X,X).\end{gathered}$$ This is a straightforward calculation $$\begin{gathered} ( \nabla_X i_U h ) (X) + \big( i_U d^{\nabla} h \big) (X,X) \\ \qquad{} = ( \nabla_X h ) (U,X) + h ( \nabla_X U, X ) + ( \nabla_U h ) (X,X) - ( \nabla_ X h ) ( U, X ) \\ \qquad{} = ( \nabla_U h ) (X,X) + h ( \nabla_X U, X ) \\ \qquad{} = ( \nabla_U h ) (X,X) - \frac{1}{2} ( ( \nabla U ) h ) (X,X).\tag*{\qed}\end{gathered}$$ The curvature tensor ${\operatorname{Rm}}$ of a Riemannian manifold satisfies $$\begin{gathered} \nabla^{*}_U \nabla {\operatorname{Rm}}+ \frac{1}{2} {\operatorname{Ric}}_U( {\operatorname{Rm}}) = \frac{1}{2} ( \nabla_X \nabla_U^{*} {\operatorname{Rm}})(Y,Z,W) - \frac{1}{2} ( \nabla_Y \nabla_U^{*} {\operatorname{Rm}}) (X,Z,W) \\ \hphantom{\nabla^{*}_U \nabla {\operatorname{Rm}}+ \frac{1}{2} {\operatorname{Ric}}_U( {\operatorname{Rm}}) =}{} + \frac{1}{2} ( \nabla_Z \nabla_U^{*} {\operatorname{Rm}})(W, X,Y) - \frac{1}{2} ( \nabla_W \nabla_U^{*} {\operatorname{Rm}}) (Z,X,Y).\end{gathered}$$ A straightforward computation based on the second Bianchi identity shows that all terms that involve $U$ cancel. The Bochner technique with diffusion relies on the following basic observations. Firstly, the maximum principle implies: \[BochnerTechniqueWithDiffusion\] Let $(M,g)$ be a Riemannian manifold, $U$ a vector field on $M$. Let $T$ be a tensor such that $$\begin{gathered} g ( \nabla_U^{*} \nabla T, T) \leq 0.\end{gathered}$$ If $| T |$ has a maximum, then $T$ is parallel. \[IsomorphismDeRhamColomology\] Note that a $p$-form $\omega$ satisfies $(dd^{*}_U + d^{*}_Ud ) \omega = 0$ if and only if $d \omega =0$ and $d^{*}_U \omega = 0$. As in [@LottGeometryBETensor], if $M$ is compact and oriented, standard elliptic theory implies that $$\begin{gathered} H^p(M) = \big\lbrace \omega \in \Omega^p(M) \, \vert \, d \omega =0 \ \text{and} \ d^{*}_U \omega = 0 \big\rbrace.\end{gathered}$$ Suppose that ${\operatorname{Ric}}_U \geq 0$ on $p$-forms. It follows that a $p$-form $\omega$ is $U$-harmonic if and only if $\omega$ is parallel and $i_U \omega = 0$. Thus, $$\begin{gathered} H^{p}(M)= \big\lbrace \omega \in \Omega^p(M) \, \vert \, \nabla \omega =0 \ \text{and} \ i_{U} \omega = 0 \big\rbrace.\end{gathered}$$ If $U=\nabla f$, then we can use integration to conclude: \[BochnerTechniqueSMMS\] Let $(M,g,f)$ be a smooth metric measure space with $\int_M e^{-f} {\operatorname{vol}}_g < \infty$. If $T$ is a $(0,k)$-tensor with $|T| \in L^{2}\big(M, e^{-f} {\operatorname{vol}}_g\big)$ and $$\begin{gathered} g ( \nabla_f^{*} \nabla T, T) \leq 0,\end{gathered}$$ then $T$ is parallel. Weighted Lichnerowicz Laplacians ================================ The idea of this section is to define a weighted curvature tensor $\widetilde{{\operatorname{Rm}}}$ so that for a given symmetric tensor $S$ the curvature term of the Lichnerowicz Laplacian satisfies $$\begin{gathered} g( {\operatorname{Ric}}_{{\operatorname{Rm}}}(T) - (S)T, T ) = g\big( {\operatorname{Ric}}_{\widetilde{{\operatorname{Rm}}}}(T),T\big).\end{gathered}$$ This will be achieved by adding a weight to the Ricci tensor of ${\operatorname{Rm}}$, leaving the Weyl curvature unchanged. The specific weight will depend on the irreducible components of the tensors of type $T$, e.g., it is different for forms and symmetric tensors. Let $T$ be a $(0,k)$-tensor. For $\tau_{ij} \in S_k$ let $T \circ \tau_{ij}$ denote the transposition of the $i$-th and $j$-th entries of $T$ and for $h \in \Sym^2(V)$ let $c_{ij}(h \otimes T)$ denote the contraction of $h$ with the $i$-th and $j$-th entries of $T$. \[GeneralFormulaBochnerOnHWedgeG\] For $h \in \Sym^2(V)$ let $H \colon V \to V$ denote the associated symmetric operator. If $T \in \mathcal{T}^{(0,k)}(V)$, then $$\begin{gathered} {\operatorname{Ric}}_{h \owedge g}(T) (X_1, \dots, X_k) = 2 \sum_{i \neq j} ( T \circ \tau_{ij}) (X_1, \dots, H(X_i), \dots, X_k) \\ \hphantom{{\operatorname{Ric}}_{h \owedge g}(T) (X_1, \dots, X_k)=}{} - \sum_{i \neq j} g(X_i, X_j) c_{ij}(h \otimes T)( X_1, \dots, \widehat{X}_i, \dots, \widehat{X}_j, \dots, X_k ) \\ \hphantom{{\operatorname{Ric}}_{h \owedge g}(T) (X_1, \dots, X_k)=}{} - \sum_{i \neq j} h(X_i, X_j) c_{ij}(g \otimes T)( X_1, \dots, \widehat{X}_i, \dots, \widehat{X}_j, \dots, X_k ) \\ \hphantom{{\operatorname{Ric}}_{h \owedge g}(T) (X_1, \dots, X_k)=}{} -(n-2) (HT)(X_1, \dots, X_k) + k \cdot \tr(h) T( X_1, \dots, X_k).\end{gathered}$$ The algebraic curvature tensor $R = h \owedge g$ satisfies $$\begin{gathered} R(X,Y,Z,W) = g(H(X),Z)g(Y,W)-g(Y,Z)g(H(X),W) \\ \hphantom{R(X,Y,Z,W) =}{} +g(X,Z)g(H(Y),W)-g(H(Y),Z)g(X,W)\end{gathered}$$ and hence $$\begin{gathered} R(X,Y)Z = ( H(X) \wedge Y + X \wedge H(Y) ) Z\end{gathered}$$ is the corresponding $(1,3)$-tensor. It follows that $$\begin{gathered} {\operatorname{Ric}}_{h \owedge g}(T)(X_1, \dots, X_k) = \sum_{i=1}^k \sum_{a=1}^n ( R(X_i, e_a)T ) ( X_1, \dots, e_a, \dots, X_k) \\ \hphantom{{\operatorname{Ric}}_{h \owedge g}(T)(X_1, \dots, X_k)}{} = \sum_{i=1}^k \sum_{a=1}^n (( H(X_i) \wedge e_a )T)( X_1, \dots, e_a, \dots, X_k) \\ \hphantom{{\operatorname{Ric}}_{h \owedge g}(T)(X_1, \dots, X_k)=}{} + \sum_{i=1}^k \sum_{a=1}^n (( X_i \wedge H(e_a) )T)( X_1, \dots, e_a, \dots, X_k).\end{gathered}$$ It is straightforward to calculate $$\begin{gathered} \sum_{i=1}^k \sum_{a=1}^n ((X_i \wedge H(e_a)) T)(X_1, \dots, e_a, \dots, X_k) \\ \qquad{} = \sum_{i \neq j} \sum_{a=1}^n T( X_1, \dots, (H(e_a) \wedge X_i) X_j, \dots, e_a, \dots, X_k) \\ \qquad\quad{} + \sum_{i=1}^k \sum_{a=1}^n T(X_1, \dots, (H(e_a) \wedge X_i)e_a, \dots, X_k) \\ \qquad {}= \sum_{i \neq j} \sum_{a=1}^n T( X_1, \dots, g(H(e_a), X_j) X_i - g(X_i, X_j)H(e_a), \dots, e_a, \dots, X_k) \\ \qquad\quad{}+ \sum_{i=1}^k \sum_{a=1}^n T(X_1, \dots, g(H(e_a),e_a) X_i - g(e_a, X_i) H(e_a), \dots, X_k) \\ \qquad {}= \sum_{i \neq j} \sum_{a=1}^n T( X_1, \dots, g(e_a, H(X_j)) X_i, \dots, e_a, \dots, X_k) \\ \qquad\quad{} - \sum_{i \neq j} \sum_{a=1}^n g(X_i, X_j) T( X_1, \dots, H(e_a), \dots, e_a, \dots, X_k) \\ \qquad\quad{}+ \sum_{i=1}^k \sum_{a=1}^n h(e_a,e_a) T(X_1, \dots, X_k) - \sum_{i=1}^k \sum_{a=1}^n T(X_1, \dots, H ( g(e_a, X_i)e_a ), \dots, X_k) \\ \qquad {}= \sum_{i \neq j} T ( X_1, \dots, X_i, \dots, H(X_j), \dots, X_k) \text{ [here } X_i \text{ is in the j-th position]} \\ \qquad\quad{}- \sum_{i \neq j} \sum_{a,b=1}^n g(X_i, X_j) h(e_a, e_b) T( X_1, \dots, e_b, \dots, e_a, \dots, X_k) + k \cdot \tr( h ) T(X_1, \dots, X_k) \\ \qquad\quad{} - \sum_{i=1}^k T(X_1, \dots, H(X_i), \dots, X_k) \\ \qquad {}= \sum_{i \neq j} (T \circ \tau_{ij}) ( X_1, \dots, H(X_j), \dots, X_i, \dots, X_k) \text{ [here } H(X_j) \text{ is in the j-th position]} \\ \qquad\quad{} - \sum_{i \neq j} g(X_i, X_j) c_{ij} (h \otimes T)( X_1, \dots, \widehat{X_i}, \dots, \widehat{X_j}, \dots, X_k) \\ \qquad\quad{} + k \cdot \tr( h ) T(X_1, \dots, X_k) + (HT)(X_1, \dots, X_k).\end{gathered}$$ Similarly one computes $$\begin{gathered} \sum_{i=1}^k \sum_{a=1}^n ((H(X_i) \wedge e_a) T)(X_1, \dots, e_a, \dots, X_k) \\ \qquad{} = \sum_{i \neq j} (T \circ \tau_{ij}) ( X_1, \dots, X_j, \dots, H(X_i), \dots, X_k) \text{ [here } X_j \text{ is in the j-th position]} \\ \qquad\quad{} -\! \sum_{i \neq j} h(X_i, X_j) c_{ij} (g \otimes T)\big(X_1, {\dots}, \widehat{X_i}, {\dots}, \widehat{X_j}, {\dots}, X_k\big) - (n-1) (HT) (X_1, {\dots}, X_k) .\end{gathered}$$ Adding up both terms yields ${\operatorname{Ric}}_{h \owedge g}(T)$ as claimed. \[WeightedCurvatureTerm\] Let $(V,g)$ be an $n$-dimensional Euclidean vector space and $h \in \Sym^2(V)$. The following hold: 1. Every $T \in \operatorname{Sym}^2(V)$ satisfies $$\begin{gathered} {\operatorname{Ric}}_{h \owedge g}(T) = - n HT - 2 \langle T,h \rangle g-2 \tr(T)h + 2 \tr(h) T , \\ g({\operatorname{Ric}}_{h \owedge g}(T),T) = - n g(HT,T) -4 \tr(T) \langle T, h \rangle + 2 \tr(h) |T|^2.\end{gathered}$$ 2. Every $p$-form $\omega$ satisfies $$\begin{gathered} {\operatorname{Ric}}_{h \owedge g}(\omega) = - (n-2p) H \omega + p \tr(h) \omega, \\ g({\operatorname{Ric}}_{h \owedge g}(\omega), \omega) = - (n-2p) g(H \omega, \omega) + p \tr(h) |\omega|^2.\end{gathered}$$ 3. Every algebraic $(0,4)$-curvature tensor ${\operatorname{Rm}}$ satisfies $$\begin{gathered} {\operatorname{Ric}}_{h \owedge g}({\operatorname{Rm}}) = -2 ( h \owedge {\operatorname{Ric}}) -2 g \owedge ( c_{24} ( h \otimes {\operatorname{Rm}}) ) -(n-2) H {\operatorname{Rm}}+ 4 \tr(h) {\operatorname{Rm}}.\end{gathered}$$ \(a) Due to the symmetry of $T$ it follows that $$\begin{gathered} {\operatorname{Ric}}_{h \owedge g}(T)(X_1,X_2) = 2 \lbrace T(H(X_1), X_2) + T(X_1, H(X_2) \rbrace \\ \hphantom{{\operatorname{Ric}}_{h \owedge g}(T)(X_1,X_2) =}{} - 2 \lbrace g(X_1, X_2) \langle h, T \rangle + h(X_1, X_2) \tr(T) \rbrace \\ \hphantom{{\operatorname{Ric}}_{h \owedge g}(T)(X_1,X_2) =}{} - (n-2) (HT)(X_1, X_2) + 2 \tr(h) T(X_1, X_2).\end{gathered}$$ \(b) Since $\omega \circ \tau_{ij} = - \omega$ for every transposition $\tau_{ij}$ it follows that $$\begin{aligned} \sum_{i \neq j} (\omega \circ \tau_{ij})(X_1, \dots, H(X_i), \dots, X_p)& = - \sum_{i \neq j} \omega(X_1, \dots, H(X_i), \dots, X_p) \\ & = - (p-1) \sum_{i=1}^p \omega(X_1, \dots, H(X_i), \dots, X_p) \\ & = (p-1) (H \omega)(X_1, \dots, X_p)\end{aligned}$$ and furthermore $c_{ij}(g \otimes \omega) = c_{ij}(h \otimes \omega)=0$ for all $i \neq j$. This implies the claim. \(c) The symmetries of the curvature tensor imply that $$\begin{gathered} \sum_{i \neq j} ( {\operatorname{Rm}}\circ \tau_{ij} )(X_1, \dots, H(X_i), \dots, X_4) \\ \!\!\qquad{}= (H {\operatorname{Rm}})(X_1, X_2, X_3, X_4) + (H {\operatorname{Rm}})( X_2, X_3, X_1, X_4) + (H {\operatorname{Rm}})( X_3, X_1, X_2, X_4)=0\end{gathered}$$ due to the first Bianchi identity. Computing with respect to an orthonormal eigenbasis of $H$ it follows that $$\begin{gathered} ( g( \cdot, \cdot ) c_{12}(h \otimes {\operatorname{Rm}})) (X, Y, Z, W) = 0, \\ ( g( \cdot, \cdot ) c_{13}(h \otimes {\operatorname{Rm}})) (X, Y, Z, W) = \sum_{a,b=1}^n g(X,Z) {\operatorname{Rm}}(g(H(e_a),e_b) e_b, Y, e_a, W) \\ \hphantom{( g( \cdot, \cdot ) c_{13}(h \otimes {\operatorname{Rm}})) (X, Y, Z, W)}{} = \sum_{a=1}^n g(X,Z) {\operatorname{Rm}}(H(e_a), Y, e_a, W) \\ \hphantom{( g( \cdot, \cdot ) c_{13}(h \otimes {\operatorname{Rm}})) (X, Y, Z, W)}{} = \sum_{a=1}^n g(Z,X) {\operatorname{Rm}}(e_a, Y, H(e_a), W) \\ \hphantom{( g( \cdot, \cdot ) c_{13}(h \otimes {\operatorname{Rm}})) (X, Y, Z, W)}{} = ( g( \cdot, \cdot ) c_{31}(h \otimes {\operatorname{Rm}})) (X, Y, Z, W).\end{gathered}$$ This implies $$\begin{gathered} \sum_{i \neq j} ( g( \cdot, \cdot ) c_{ij}(h \otimes {\operatorname{Rm}})) (X, Y, Z, W) \\ \qquad{}= 2 \sum_{i=1}^n \lbrace g(X,Z) {\operatorname{Rm}}(H(e_i), Y, e_i, W) + g(X,W) {\operatorname{Rm}}(H(e_i), Y, Z, e_i) \\ \qquad\quad{} + g(Y,Z) {\operatorname{Rm}}(X, H(e_i), e_i, W ) + g(Y,W) {\operatorname{Rm}}(X, H(e_i), Z, e_i) \rbrace \\ \qquad {}= 2 \sum_{i=1}^n \lbrace g(X,Z) {\operatorname{Rm}}(Y, H(e_i), W, e_i) - g(X,W) {\operatorname{Rm}}(Y, H(e_i), Z, e_i) \\ \qquad\quad{}- g(Y,Z) {\operatorname{Rm}}( X, H(e_i), W, e_i) + g(Y,W) {\operatorname{Rm}}(X, H(e_i), Z, e_i) \rbrace \\ \qquad {}= 2 \left( g \owedge \left[ \sum_{i=1}^n {\operatorname{Rm}}( \cdot, H(e_i), \cdot, e_i) \right] \right) (X, Y, Z,W) \\ \qquad {} = 2 \left( g \owedge c_{24} ( h \otimes {\operatorname{Rm}}) \right) (X, Y, Z,W).\end{gathered}$$ Similarly it follows that $$\begin{gathered} \sum_{i \neq j} ( h( \cdot, \cdot ) c_{ij}(g \otimes {\operatorname{Rm}})) = 2 \left( h \owedge c_{24} ( g \otimes {\operatorname{Rm}}) \right) = 2 \left( h \owedge {\operatorname{Ric}}\right).\end{gathered}$$ This completes the proof. For a Weyl tensor $W$ and $h$ a symmetric $(0,2)$-tensor it is not hard to check that ${\operatorname{Ric}}_{h \owedge g}(W)$ satisfies $$\begin{gathered} g( {\operatorname{Ric}}_{h \owedge g}(W),W) = - (n-2) g( HW, W) + 4 \tr(h) |W|^2, \\ g\big( {\operatorname{Ric}}_{h \owedge g}(W), g \owedge \mathring{{\operatorname{Ric}}}\big) = - 8(n-2) \langle c_{24}(h \otimes W), {\operatorname{Ric}}\rangle = - 8(n-2) \big\langle c_{24}(\mathring{h} \otimes W), \mathring{{\operatorname{Ric}}} \big\rangle, \\ g( {\operatorname{Ric}}_{h \owedge g}(W), g \owedge g) = 0.\end{gathered}$$ It is worth noting that there are trace-free symmetric $(0,2)$-tensors $h_1$, $h_2$ such that the curvature tensor $h_1 \owedge h_2$ is Weyl. The main Theorem follows as in Proposition \[BochnerWithDiffusionForForms\] below by using Lemma \[BochnerTechniqueSMMS\] instead of Lemma \[BochnerTechniqueWithDiffusion\]. The description of the de Rham cohomology groups follows from Remark \[IsomorphismDeRhamColomology\]. \[BochnerWithDiffusionForForms\] Let $(M,g)$ be a Riemannian manifold and let $U$ be a vector field on $M$. Set $S= \nabla U$ and for $1 \leq p < \frac{n}{2}$ set $$\begin{gathered} H = \frac{1}{n-2p} S - \frac{1}{2(n-p)(n-2p)} \tr(S) I,\end{gathered}$$ where $I \colon TM \to TM$ denotes the identity operator. Suppose that the eigenvalues $\lambda_1 \leq \dots \leq \lambda_{\genfrac(){0pt}{2}{n}{2}}$ of the weighted curvature tensor ${\operatorname{Rm}}+ h \owedge g$ satisfy $$\begin{gathered} \lambda_1 + \dots + \lambda_{n-p} \geq 0\end{gathered}$$ and let $\omega$ be a $U$-harmonic $p$-form for $1 \leq p < \frac{n}{2}$. If $| \omega |$ achieves a maximum, then $\omega$ is parallel. If in addition the inequality is strict, then $\omega$ vanishes. Proposition \[WeightedCurvatureTerm\] (b) and $- I \omega = p \omega$ imply that $$\begin{aligned} g( {\operatorname{Ric}}_{h \owedge g} \omega, \omega ) & = - (n-2p) g( H \omega, \omega) + p \tr(h) | \omega|^2 = - g( ( (n-2p)H+ \tr(h) I ) \omega, \omega ) \\ & = - g \left( \left( S- \frac{\tr(S)}{2(n-p)} I + \frac{\tr(S)}{2(n-p)} I \right) \omega, \omega \right) = - g( S \omega, \omega ).\end{aligned}$$ Thus the Bochner formula takes the form $$\begin{gathered} \Delta_U \omega = \nabla^{*}_U \nabla \omega + {\operatorname{Ric}}(\omega) - (\nabla U) \omega = \nabla^{*}_U \nabla \omega + {\operatorname{Ric}}_{{\operatorname{Rm}}+ h \owedge g}(\omega).\end{gathered}$$ The argument in [@PetersenWinkBochner proof of Theorem A] shows that ${\operatorname{Ric}}_{{\operatorname{Rm}}+h \owedge g}( \omega ) \geq 0$. Lemma \[BochnerTechniqueWithDiffusion\] implies the claim. If the inequality is strict, then the same argument shows that ${\operatorname{Ric}}_{{\operatorname{Rm}}+h \owedge g}( \omega ) > 0$ unless $\omega =0$. The above approach only works for $p = \frac{n}{2}$ if $S$ is a multiple of the identity. However, we have Let $(M,g)$ be an $n$-dimensional Riemannian manifold and let $U$ be a vector field on $M$. Set $S= \nabla U$ and fix $1 \leq p \leq \big\lfloor \frac{n}{2}\big\rfloor$. Let $\mu_1 \leq \dots \leq \mu_n$ denote the eigenvalues of $S$. Suppose that the weighted curvature tensor $$\begin{gathered} {\operatorname{Rm}}+ \frac{\sum\limits_{i=1}^p \mu_i}{2p(n-p)} g \owedge g\end{gathered}$$ is $(n-p)$-nonnegative. If $\omega$ is a $U$-harmonic $p$-form $\omega$ such that $|\omega|$ has a maximum, then $\omega$ is parallel. If in addition the weighted curvature tensor is $(n-p)$-positive, then $\omega$ vanishes. Calculating with respect to an orthonormal eigenbasis for $S$ it follows that $$\begin{gathered} - g( (S \omega), \omega) = - \sum_{i_1 < \dots < i_p} (S \omega )_{i_1 \dots i_p} \omega_{i_1 \dots i_p} = \sum_{i_1 < \dots < i_p} \left( \sum_{j=1}^p \mu_{i_j} \right) ( \omega_{i_1 \dots i_p } )^2 \geq \left( \sum_{i=1}^p \mu_{i} \right) | \omega |^2.\end{gathered}$$ Let $\lbrace \lambda_{\alpha} \rbrace$ denote the eigenvalues of (the curvature operator associated to) ${\operatorname{Rm}}$ and let $\lbrace \Xi_{\alpha} \rbrace$ be an orthonormal eigenbasis. It follows from [@PetersenWinkBochner Proposition 1.6] that $$\begin{aligned} g( {\operatorname{Ric}}_{{\operatorname{Rm}}}( \omega), \omega ) - g( S \omega, \omega) \geq \sum_{\alpha} \lambda_{\alpha} | \Xi_{\alpha} \omega |^2 + \left( \sum_{i=1}^p \mu_i \!\right) | \omega |^2 = \sum_{\alpha} \!\left(\! \lambda_{\alpha} + \frac{\sum\limits_{i=1}^p \mu_i}{p(n-p)}\! \right) | \Xi_{\alpha} \omega |^2.\end{aligned}$$ The proof can now be completed as in Proposition \[BochnerWithDiffusionForForms\]. This principle can also be applied to $(0,2)$-tensors. Let $T \in \Sym^2(V)$ with $\tr(T)=0$, let $S= \nabla U$ and set $$\begin{gathered} H=\frac{S}{n} - \frac{\tr(S)}{2n^2} I.\end{gathered}$$ Let $\lambda_1 \leq \dots \leq \lambda_{\genfrac(){0pt}{2}{n}{2}}$ denote the eigenvalues of the weighted curvature tensor ${\operatorname{Rm}}+ h \owedge g$ and suppose that $$\begin{gathered} \lambda_{1} + \dots + \lambda_{\floor{\frac{n}{2}}} \geq 0.\end{gathered}$$ If $T$ is $U$-harmonic and $|T|$ has a maximum, then $T$ is parallel. If in addition the inequality is strict, then $T$ vanishes. Proposition \[WeightedCurvatureTerm\](a) implies that $$\begin{aligned} g ( {\operatorname{Ric}}_{h \owedge g}(T), T ) & = - n g \left( \left(H + \frac{\tr(h)}{n} I \right)T ,T \right) \\ & = - n g \left( \left( \frac{S}{n} - \frac{\tr(S)}{2n^2} I + \frac{\tr(S)}{2n^2} I \right) T, T \right) = - g( ST,T ).\end{aligned}$$ It follows from Proposition \[BochnerFormulasExample\](b) that $$\begin{gathered} \left( \nabla_X \nabla^{*}_U T \right)(X) + \left( \nabla^{*}_U d^{\nabla} T \right) (X,X) = \left( \nabla^{*}_U \nabla T \right)(X,X)+ \frac{1}{2} \left( {\operatorname{Ric}}_{{\operatorname{Rm}}+h \owedge g} T \right) (X,X).\end{gathered}$$ As in [@PetersenWinkBochner Lemma 2.1 and Proposition 2.9] we conclude that ${\operatorname{Ric}}_{{\operatorname{Rm}}+ h \owedge g}(T) \geq 0$. When the inequality is strict, the argument shows moreover ${\operatorname{Ric}}_{{\operatorname{Rm}}+ h \owedge g}(T) > 0$ unless $T = 0$. This uses again that $T$ is trace-less. An application of Lemma \[BochnerTechniqueSMMS\] as before implies the claim. Acknowledgements {#acknowledgements .unnumbered} ---------------- We would like to thank the referees for useful comments. [99]{} Bakry D., Émery M., Diffusions hypercontractives, in Séminaire de probabilités, [XIX]{}, 1983/84, *Lecture Notes in Math.*, Vol. 1123, [Springer](https://doi.org/10.1007/BFb0075847), Berlin, 1985, 177–206. Bakry D., Qian Z., Volume comparison theorems without [J]{}acobi fields, in Current Trends in Potential Theory, *Theta Ser. Adv. Math.*, Vol. 4, Theta, Bucharest, 2005, 115–122. Lichnerowicz A., Variétés riemanniennes à tenseur [C]{} non négatif, *C. R. Acad. Sci. Paris Sér. A-B* **271** (1970), A650–A653. Lott J., Some geometric properties of the [B]{}akry–Émery–[R]{}icci tensor, [*Comment. Math. Helv.*](https://doi.org/10.1007/s00014-003-0775-8) **78** (2003), 865–883, [arXiv:math.DG/0211065](https://arxiv.org/abs/math.DG/0211065). Petersen P., Riemannian geometry, 3rd ed., *Graduate Texts in Mathematics*, Vol. 171, [Springer](https://doi.org/10.1007/978-3-319-26654-1), Cham, 2016. Petersen P., Wink M., New curvature conditions for the [B]{}ochner technique, [arXiv:1908.09958v3](https://arxiv.org/abs/1908.09958v3). Qian Z., Estimates for weighted volumes and applications, [*Quart. J. Math. Oxford*](https://doi.org/10.1093/qmath/48.2.235) **48** (1997), 235–242. Wei G., Wylie W., Comparison geometry for the [B]{}akry–Émery [R]{}icci tensor, [*J. Differential Geom.*](https://doi.org/10.4310/jdg/1261495336) **83** (2009), 377–405, [arXiv:0706.1120](https://arxiv.org/abs/0706.1120).

--- abstract: 'The number of shortest factorizations into reflections for a Singer cycle in $GL_n({{\mathbb{F}}}_q)$ is shown to be $(q^n-1)^{n-1}$. Formulas counting factorizations of any length, and counting those with reflections of fixed conjugacy classes are also given. The method is a standard character-theory technique, requiring the compilation of irreducible character values for Singer cycles, semisimple reflections, and transvections. The results suggest several open problems and questions, which are discussed at the end.' address: | School of Mathematics\ University of Minnesota\ Minneapolis, MN 55455, USA author: - 'J.B. Lewis' - 'V. Reiner' - 'D. Stanton' title: Reflection factorizations of Singer cycles --- [^1] Introduction and main result ============================ This paper was motivated by two classic results on the number $t(n,\ell)$ of ordered factorizations $(t_1,\ldots,t_\ell)$ of an $n$-cycle $c=t_1 t_2 \cdots t_\ell$ in the symmetric group ${\mathfrak{S}}_n$, where each $t_i$ is a transposition. .1in For $n \geq 1$, one has $ \label{Denes-theorem} t(n,n-1)=n^{n-2}. $ For $n \geq 1$, more generally $t(n,\ell)$ has ordinary generating function $$\label{Jackson-ordinary-gf-product} \sum_{\ell \geq 0} t(n,\ell) x^\ell = n^{n-2} x^{n-1} \prod_{k=0}^{n-1} \left( 1 - x n\left(\frac{n-1}{2}-k\right) \right)^{-1}$$ and explicit formulas $$\begin{aligned} \label{Jackson-difference-formula} t(n,\ell) =\frac{n^\ell}{n!} \sum_{k=0}^{n-1}(-1)^k \binom{n-1}{k} \left( \frac{n-1}{2}-k \right)^\ell =\frac{(-n)^\ell}{n!}(-1)^{n-1} \left[ \Delta^{n-1}(x^\ell)\right]_{x=\frac{1-n}{2}}.\end{aligned}$$ Here the difference operator $\Delta(f)(x):=f(x+1)-f(x)$ satisfies $ \Delta^n(f)(x):=\sum_{k=0}^n (-1)^{n-k} \binom{n}{k} f(x+k). $ Our goals are $q$-analogues, replacing the symmetric group ${\mathfrak{S}}_n$ with the [*general linear group*]{} $GL_n({{\mathbb{F}}}_q)$, replacing transpositions with [*reflections*]{}, and replacing an $n$-cycle with a [*Singer cycle*]{} $c$: the image of a generator for the cyclic group ${{\mathbb{F}}}_{q^n}^\times \cong {{\mathbb{Z}}}/(q^n-1){{\mathbb{Z}}}$ under any embedding $ {{\mathbb{F}}}_{q^n}^\times \hookrightarrow GL_{{{\mathbb{F}}}_q}({{\mathbb{F}}}_{q^n}) \cong GL_n({{\mathbb{F}}}_q) $ that comes from a choice of ${{\mathbb{F}}}_q$-vector space isomorphism ${{\mathbb{F}}}_{q^n} \cong {{\mathbb{F}}}_q^n$. The analogy between Singer cycles in $GL_n({{\mathbb{F}}}_q)$ and $n$-cycles in ${\mathfrak{S}}_n$ is reasonably well-established [@StantonWebbR §7], [@StantonWhiteR §§8-9]. Fixing such a Singer cycle $c$, denote by $t_q(n,\ell)$ the number of ordered factorizations $(t_1,\ldots,t_\ell)$ of $c=t_1 t_2 \cdots t_\ell$ in which each $t_i$ is a [*reflection*]{} in $GL_n({{\mathbb{F}}}_q)$, that is, the fixed space $({{\mathbb{F}}}_q^n)^{t_i}$ is a hyperplane in ${{\mathbb{F}}}_q^n$. \[q-Denes-theorem\] For $n \geq 2$, one has $ t_q(n,n)=(q^n-1)^{n-1}. $ \[q-factorization-theorem\] For $n \geq 2$, more generally $t_q(n,\ell)$ has ordinary generating function $$\label{q-Jackson-ordinary-gf-product} \sum_{\ell \geq 0} t_q(n,\ell) x^\ell = (q^n-1)^{n-1} x^n \cdot \left(1+x[n]_q \right)^{-1} \prod_{k=0}^{n-1} \left(1+x[n]_q(1+q^k-q^{k+1})\right)^{-1}$$ and explicit formulas $$\begin{aligned} \label{q-Jackson-sum-formula} \qquad t_q(n,\ell) &= \frac{(-[n]_q)^\ell}{q^{\binom{n}{2}}(q;q)_n} \left( (-1)^{n-1} (q;q)_{n-1}+ \sum_{k=0}^{n-1}(-1)^{k+n} q^{\binom{k+1}{2}} {\left[\begin{matrix} n-1 \\ k \end{matrix} \right]_{q}} (1+q^{n-k-1}-q^{n-k})^\ell \right) \\ \label{q-Jackson-difference-formula} &= (1-q)^{-1}\frac{(-[n]_q)^\ell}{[n]!_q} \left[ \Delta_q^{n-1} \biggl(\frac{1}{x}-\frac{(1+x(1-q))^\ell}{x}\biggr)\right]_{x=1}\\ \label{tot-num-formula} &= [n]_q^{\ell-1}\sum_{i=0}^{\ell-n} (-1)^i (q-1)^{\ell-i-1} \binom{\ell}{i} {\left[\begin{matrix} \ell-i-1 \\ n-1 \end{matrix} \right]_{q}}.\end{aligned}$$ The $q$-analogues used above and elsewhere in the paper are defined as follows: $$\begin{aligned} {\left[\begin{matrix} n \\ k \end{matrix} \right]_{q}}&:=\frac{[n]!_q}{[k]!_q [n-k]!_q}, \,\, \text{ where } \, [n]!_q:=[1]_q [2]_q \cdots [n]_q \,\, \text{ and } \, [n]_q:=1+q+q^2+\cdots+q^{n-1}, \\ (x;q)_n&:=(1-x)(1-xq)(1-xq^2)\cdots(1-xq^{n-1}), \,\, \textrm{ and }\\ \Delta_q(f)(x)&:=\frac{f(x)-f(qx)}{x-qx}, \text{ so that } \end{aligned}$$ $$\label{q-diff-iterate} \Delta_q^n(f)(x) =\frac{1}{q^{\binom{n}{2}} x^n(1-q)^n} \sum_{k=0}^n (-1)^{n-k} q^{\binom{k}{2}} {\left[\begin{matrix} n \\ k \end{matrix} \right]_{q}}f(q^{n-k} x).$$ The equivalence of the three formulas , , for $t_q(n,\ell)$ is explained in Proposition \[q-Jackson-sum-equals-tot-num-lemma\] below. In fact, we will prove the following refinement of Theorem \[q-factorization-theorem\] for $q>2,$ having no counterpart for ${\mathfrak{S}}_n$. Transpositions are all conjugate within ${\mathfrak{S}}_n$, but the conjugacy class of a reflection $t$ in $GL_n({{\mathbb{F}}}_q)$ for $q>2$ varies with its determinant $\det(t)$ in ${{\mathbb{F}}}_q^\times$. When $\det(t)=1$, the reflection $t$ is called a [*transvection*]{} [@Lang XIII §9], while $\det(t) \neq 1$ means that $t$ is a [*semisimple reflection*]{}. One can associate to an ordered factorization $(t_1,\ldots,t_\ell)$ of $c=t_1 t_2 \cdots t_\ell$ the sequence of determinants $(\det(t_1),\ldots,\det(t_\ell))$ in ${{\mathbb{F}}}_q^\ell$, having product $\det(c)$. \[fixed-det-sequence-theorem\] Let $q>2.$ Fix a Singer cycle $c$ in $GL_n({{\mathbb{F}}}_q)$ and a sequence $\alpha=(\alpha_i)_{i=1}^\ell$ in $({{\mathbb{F}}}_q^\times)^\ell$ with $\prod_{i=1}^\ell \alpha_i = \det(c)$. Let $m$ be the number of values $i$ such that $\alpha_i = 1$. Then one has $m \le \ell-1,$ and the number of ordered reflection factorizations $c=t_1 \cdots t_\ell$ with $\det(t_i)=\alpha_i$ depends only upon $\ell$ and $m$. This quantity $t_q(n,\ell,m)$ is given by these formulas: $$\begin{aligned} \label{det-sequence-difference-formula} t_q(n,\ell,m) &= [n]_q^{\ell - 1} \sum_{i = 0}^{\min(m,\ell-n)} (-1)^i \binom{m}{i} {\left[\begin{matrix} \ell - i - 1 \\ n - 1 \end{matrix} \right]_{q}} \\ \label{q-diff-nlm} &= \frac{[n]_q^\ell}{[n]!_q} \left[ \Delta_q^{n-1}\bigl( (x-1)^mx^{\ell-m-1}\bigr) \right]_{x=1}.\end{aligned}$$ In particular, setting $\ell=n$ in , the number of shortest such factorizations is $$t_q(n,n,m)= [n]_q^{n - 1},$$ which depends neither on the sequence $\alpha=(\det(t_i))_{i=1}^\ell$ nor on the number of transvections $m$. The equivalence of the formulas and for $t_q(n,\ell,m)$ is also explained in Proposition \[q-Jackson-sum-equals-tot-num-lemma\] below. Theorems \[q-factorization-theorem\] and \[fixed-det-sequence-theorem\] are proven via a standard character-theoretic approach. This approach is reviewed quickly in Section \[character-approach-section\], followed by an outline of ordinary character theory for $GL_n({{\mathbb{F}}}_q)$ in Section \[GL-character-theory\]. Section \[character-values-section\] either reviews or derives the needed explicit character values for four kinds of conjugacy classes: the identity element, Singer cycles, semisimple reflections, and transvections. Then Section \[main-result-proof-section\] assembles these calculations into the proofs of Theorems \[q-factorization-theorem\] and \[fixed-det-sequence-theorem\]. Section \[questions-remarks\] closes with some further remarks and questions. Although Theorem \[fixed-det-sequence-theorem\] is stated for $q>2$, something interesting also occurs for $q=2$. All reflections in $GL_n({{\mathbb{F}}}_2)$ are transvections, thus one always has $m=\ell$ for $q=2$. Furthermore, one can see that , give the same answer when both $q=2$ and $m=\ell$. This reflects a striking dichotomy in our proofs: for $q > 2$ the only contributions to the computation come from irreducible characters of $GL_n({{\mathbb{F}}}_q)$ arising as constituents of parabolic inductions of characters of $GL_1({{\mathbb{F}}}_q)$, while for $q =2$ the cuspidal characters for $GL_s({{\mathbb{F}}}_q)$ with $s \geq 2$ play a role, miraculously giving the same polynomial $t_q(n,\ell)$ in $q$ evaluated at $q=2$. Can one derive the formulas or via [*inclusion-exclusion*]{} more directly? \[q-Denes-proof-question\] Can one derive Theorem \[q-Denes-theorem\] [*bijectively*]{}, or by an [*overcount*]{} in the spirit of Dénes [@Denes], that counts factorizations of all conjugates of a Singer cycle, and then divides by the conjugacy class size? The character theory approach to factorizations {#character-approach-section} =============================================== We recall the classical approach to factorization counts, which goes back to Frobenius [@Frobenius]. Given a finite group $G$, let ${\operatorname{Irr}}(G)$ be the set of its irreducible ordinary (finite-dimensional, complex) representations $V$. For each $V$ in ${\operatorname{Irr}}(G)$, denote by $\deg(V)$ the [*degree*]{} $\dim_{{\mathbb{C}}}V$, and let $ \chi_{V}(g)={\operatorname{Tr}}(g: V \rightarrow V) $ be its [*character value*]{} at $g$, along with ${\widetilde{\chi}}_{V}(g):=\frac{\chi_V(g)}{\deg(V)}$ the [*normalized character value*]{}. Both functions $\chi_{V}(-)$ and ${\widetilde{\chi}}_{V}(-)$ on $G$ extend by ${{\mathbb{C}}}$-linearity to functionals on the [*group algebra*]{} ${{\mathbb{C}}}G$. \[conj-count-prop\] Let $G$ be a finite group, and $A_1,\ldots,A_\ell \subset G$ unions of conjugacy classes in $G$. Then for $g$ in $G$, the number of ordered factorizations $(t_1,\ldots,t_\ell)$ with $g=t_1 \cdots t_\ell$ and $t_i$ in $A_i$ for $i=1,2,\ldots,\ell$ is $$\label{Chapuy-Stump-varying-class-answer} \frac{1}{|G|} \sum_{V \in {\operatorname{Irr}}(G)} \deg(V) \cdot \chi_V(g^{-1}) \cdot {\widetilde{\chi}}_{V}(z_1) \cdots {\widetilde{\chi}}_{V}(z_\ell).$$ where $z_i:=\sum_{t \in A_i} t$ in ${{\mathbb{C}}}G$. This lemma was a main tool used by Jackson [@Jackson-older §2], as well as by Chapuy and Stump [@ChapuyStump §4] in their solution of the analogous question in well-generated complex reflection groups. The proof follows from a straightforward computation in the group algebra ${{\mathbb{C}}}G$ coupled with the isomorphism of $G$-representations $ {{\mathbb{C}}}G \cong \bigoplus_{V \in {\operatorname{Irr}}(G)} V^{\oplus \deg(V)} $; it may be found for example in [@LandoZvonkin Thm. 1.1.12], [@LuxPahlings Thm. 2.5.9]. Review of ordinary characters of $GL_n({{\mathbb{F}}}_q)$ {#GL-character-theory} ========================================================= The ordinary character theory of $GL_n:=GL_n({{\mathbb{F}}}_q)$ was worked out by Green [@Green], and has been reworked many times. Aside from Green’s paper, some useful references for us in what follows will be Macdonald [@Macdonald Chaps. III, IV], and Zelevinsky [@Zelevinsky §11]. Parabolic or Harish-Chandra induction ------------------------------------- The key notion is that of [*parabolic*]{} or [*Harish-Chandra induction*]{}: given an integer composition $\alpha=(\alpha_1,\ldots,\alpha_m)$ of $n$, so that $\alpha_i > 0$ and $|\alpha|:=\sum_i \alpha_i=n$, and class functions $f_i$ on $GL_{\alpha_i}$ for $i=1,2,\ldots,m$, one produces a class function $f_1 {{*}}f_2 {{*}}\cdots {{*}}f_m$ on $GL_n$ defined as follows. Regard the $m$-tuple $(f_1,\ldots,f_m)$ as a class function on the block upper-triangular parabolic subgroup $P_\alpha$ inside $GL_n$, whose typical element is $$\label{parabolic-format} p= \begin{bmatrix} A_{1,1} & * & \cdots & *\\ 0 & A_{2,2}& \cdots& *\\ \vdots& \vdots & \ddots& \vdots\\ 0 & 0 &\cdots &A_{m,m} \end{bmatrix}$$ with $A_{i,i}$ an invertible $\alpha_i \times \alpha_i$ matrix, via $(f_1,\ldots,f_m)(p)=\prod_{i=1}^m f_i(A_{i,i})$. Then apply (ordinary) induction of characters from $P_\alpha$ up to $GL_n$. In other words, for an element $g$ in $GL_n$ one has $$\label{parabolic-induction-formula} (f_1 {{*}}f_2 {{*}}\cdots {{*}}f_m)(g):= \frac{1}{|P_\alpha|} \sum_{\substack{h \in G:\\ hgh^{-1} \in P_\alpha}} f_1(A_{1,1}) \cdots f_m(A_{m,m}) \qquad \text{ if }hgh^{-1}\text{ looks as in }\eqref{parabolic-format}.$$ Identify representations $U$ up to equivalence with their characters $\chi_U$. The parabolic induction product $(f,g) \longmapsto f{{*}}g$ gives rise to a graded, associative product on the graded ${{\mathbb{C}}}$-vector space $${\operatorname{Cl}}(GL_*) = \bigoplus_{n \geq 0} {\operatorname{Cl}}(GL_n)$$ which is the direct sum of class functions on all of the general linear groups, with ${\operatorname{Cl}}(GL_0)={{\mathbb{C}}}$ by convention. Parametrizing the $GL_n$-irreducibles {#irreducible-parametrization-section} ------------------------------------- A $GL_n$-irreducible $U$ is called [*cuspidal*]{} if $\chi_U$ does not occur as a constituent in any induced character $f_1 {{*}}f_2$ for compositions $n=\alpha_1+\alpha_2$ with $\alpha_1,\alpha_2 >0$. Denote by ${{\operatorname{Cusp}}}_n$ the set of all such cuspidal irreducibles $U$ for $GL_n$, and say that [*weight ${\operatorname{wt}}(U)=n$*]{}. Let ${\operatorname{Par}}_n$ denote the partitions $\lambda$ of $n$ (that is, $|\lambda|:=\sum_i \lambda_i=n$), and define $$\begin{aligned} {\operatorname{Par}}&:=\bigsqcup_{ n \geq 0} {\operatorname{Par}}_n,\\ {{\operatorname{Cusp}}}&:=\bigsqcup_{ n \geq 1} {{\operatorname{Cusp}}}_n\end{aligned}$$ the sets of [*all*]{} partitions, and [*all*]{} cuspidal representations for all groups $GL_n$. Then the $GL_n$-irreducible characters can be indexed as $ {\operatorname{Irr}}(GL_n)=\{ \chi^{{{\underline{\lambda}}}} \} $ where ${{\underline{\lambda}}}$ runs through the set of all functions $$\begin{array}{rcl} {{\operatorname{Cusp}}}&\overset{{{\underline{\lambda}}}}{\longrightarrow}& {\operatorname{Par}}\\ U & \longmapsto & \lambda(U) \end{array}$$ having the property that $$\label{irreducible-weight-condition} \sum_{U \in {{\operatorname{Cusp}}}} {\operatorname{wt}}(U) \, |\lambda(U)|=n.$$ Although ${{\operatorname{Cusp}}}$ is infinite, this condition implies that ${{\underline{\lambda}}}$ can only take on finitely many non-$\varnothing$ values $\lambda(U_1),\ldots,\lambda(U_m)$, and in this case $$\label{general-irreducibles-are-induced} \chi^{{{\underline{\lambda}}}} = \chi^{U_1,\lambda(U_1)} {{*}}\cdots {{*}}\chi^{U_m,\lambda(U_m)}$$ where each $\chi^{U,\lambda}$ is what Green [@Green §7] called a [*primary irreducible character*]{}. In particular, a cuspidal character $U$ in ${{\operatorname{Cusp}}}_n$ is the same as the primary irreducible $\chi^{U,(1)}$. Jacobi-Trudi formulas {#Jacobi-Trudi-section} --------------------- We recall from symmetric function theory the [*Jacobi-Trudi*]{} and [*dual Jacobi-Trudi formulas*]{} [@Macdonald I (3.4),(3.5)] formulas. For a partition $\lambda=(\lambda_1 \geq \cdots \geq \lambda_\ell)$ with largest part $m:=\lambda_1$, these formulas express a [*Schur function*]{} $s_\lambda$ either as an integer sum of products of [*complete homogeneous*]{} symmetric functions $h_n=s_{(n)}$, or of [*elementary*]{} symmetric functions $e_n=s_{(1^n)}$: $$\begin{array}{rcccl} s_\lambda &=&\det( h_{\lambda_i-i+j} ) &=&\displaystyle \sum_{w \in {\mathfrak{S}}_\ell} {\operatorname{sgn}}(w) h_{\lambda_1-1+w(1)} \cdots h_{\lambda_\ell-\ell + w(\ell)}, \\ s_\lambda &=&\det( e_{\lambda'_i-i+j} ) &=&\displaystyle \sum_{w \in {\mathfrak{S}}_{m}} {\operatorname{sgn}}(w) e_{\lambda'_1-1+w(1)} \cdots e_{\lambda'_m-m + w(m)} . \end{array}$$ Here $\lambda'$ is the usual [*conjugate*]{} or [*transpose*]{} partition to $\lambda$. Also $h_0=e_0=1$ and $h_n=e_n=0$ if $n < 0$. The special case of primary irreducible $GL_n$-characters $\chi^{U,(n)}, \chi^{U,(1^n)}$ corresponding to the single row partitions $(n)$ and single column partitions $(1^n)$ are called [*generalized trivial*]{} and [*generalized Steinberg characters*]{}, respectively, by Silberger and Zink [@SilbergerZink]. One has analogous formulas expressing any primary irreducible character $\chi^{U,\lambda}$ virtually in terms of parabolic induction products of such characters: $$\begin{aligned} \label{primary-irreducible-Jacobi-Trudi} \chi^{U,\lambda}&= \sum_{w \in {\mathfrak{S}}_\ell} {\operatorname{sgn}}(w) \chi^{U,(\lambda_1-1+w(1))} {{*}}\cdots {{*}}\chi^{U,(\lambda_\ell-\ell + w(\ell))} \\ \label{primary-irreducible-dual-Jacobi-Trudi} \chi^{U,\lambda}&= \sum_{w \in {\mathfrak{S}}_{m}} {\operatorname{sgn}}(w) \chi^{U,(1^{\lambda'_1-1+w(1)})} {{*}}\cdots {{*}}\chi^{ U, (1^{\lambda'_m-m + w(m)} ) } \end{aligned}$$ where $\chi^{U,(n)}, \chi^{U,(1^n)}$ are both the zero character if $n < 0$, and the trivial character ${\mathbf{1}}_{GL_0}$ if $n=0$. The cuspidal characters: indexing and notation ---------------------------------------------- The set ${{\operatorname{Cusp}}}_n$ of cuspidal characters for $GL_n({{\mathbb{F}}}_q)$ has the same cardinality $\frac{1}{n} \sum_{d | n} \mu(n/d) q^d$ as the set of irreducible polynomials in ${{\mathbb{F}}}_q[x]$ of degree $n$, or the set of [*primitive necklaces*]{} of $n$ beads having $q$ possible colors (= free orbits under $n$-fold cyclic rotation of words in $\{0,1,\ldots,q-1\}^n$). There are at least two ways one sees ${{\operatorname{Cusp}}}_n$ indexed in the literature. - Green [@Green §7] indexes ${{\operatorname{Cusp}}}_n$ via free orbits $ [\beta]=\{\beta,\beta^q,\cdots,\beta^{q^{n-1}}\} $ for the action of the [*Frobenius map*]{} $\beta \overset{F}{\longmapsto} \beta^q$ on the multiplicative group ${{\mathbb{F}}}_{q^n}^\times$; he calls such free orbits [*$n$-simplices*]{}. In his notation, if $U$ lies in ${{\operatorname{Cusp}}}_s$ and is indexed by the orbit $[\beta]$ within ${{\mathbb{F}}}_{q^s}$, then the primary $GL_n$-irreducible character $\chi^{U,\lambda}$ for a partition $\lambda$ of $\frac{n}{s}$ is (up to a sign) what he denotes $I^\beta_s[\lambda]$. The special case $I^\beta_s[(m)]$ he also denotes $I^\beta_s[m]$. Thus the cuspidal $U$ itself is (up to sign) denoted $I^\beta_s[1]$, and he also uses the alternate terminology $J_s(\beta):=I^\beta_s[1]$; see [@Green p. 433]. - Later authors index ${{\operatorname{Cusp}}}_n$ via free orbits $ [\varphi]=\{\varphi,\varphi \circ F,\cdots, \varphi \circ F^{n-1}\} $ for the Frobenius action on the [*dual group ${\operatorname{Hom}}({{\mathbb{F}}}^\times_{q^n},{{\mathbb{C}}}^\times)$*]{}. Say that $U$ in ${{\operatorname{Cusp}}}_n$ is [*associated*]{} to the orbit $[\varphi]$ in this indexing. When $n=1$, one simply has ${{\operatorname{Cusp}}}_1={\operatorname{Hom}}({{\mathbb{F}}}^\times_{q},{{\mathbb{C}}}^\times)$. In other words, the Frobenius orbits $[\varphi]=\{\varphi\}$ are singletons, and if $U$ is associated to this orbit then $U=\varphi$ considering both as homomorphisms $$GL_1({{\mathbb{F}}}_q) = {{\mathbb{F}}}_q^\times \overset{U=\varphi}{\longrightarrow} {{\mathbb{C}}}^\times.$$ Although we will not need Green’s full description of the characters $\chi^{U,(m)}$ and $\chi^{U,\lambda}$, we will use (in the proof of Lemma \[normalized-characters-on-semisimple-reflections\] below) the following consequence of his discussion surrounding [@Green Lemma 7.2]. \[primaries-in-terms-of-cuspidals\] For $U$ in ${{\operatorname{Cusp}}}_s$, every $\chi^{U,(m)}$, and hence also every primary irreducible character $\chi^{U,\lambda}$, is in the ${{\mathbb{Q}}}$-span of characters of the form $ \chi_{U_1} {{*}}\cdots {{*}}\chi_{U_t} $ where $U_i$ is in ${{\operatorname{Cusp}}}_{n_i}$, with $s$ dividing $n_i$ for each $i$. Some explicit character values {#character-values-section} ============================== We will eventually wish to apply Proposition \[conj-count-prop\] with $g$ being a Singer cycle, and with the central elements $z_i$ being sums over classes of reflections with fixed determinants. For this one requires explicit character values on four kinds of conjugacy classes of elements in $GL_n({{\mathbb{F}}}_q)$: - the identity, giving the character degrees, - the Singer cycles, - the semisimple reflections, and - the transvections. We review known formulas for most of these, and derive others that we will need, in the next four subsections. It simplifies matters that the character value $\chi^{{{\underline{\lambda}}}}(c^{-1})$ vanishes for a Singer cycle $c$ unless $\chi^{{{\underline{\lambda}}}}=\chi^{U,\lambda}$ is a primary irreducible character and the partition $\lambda$ of $\frac{n}{s}$ takes a very special form; see Proposition \[Singer-cycle-character-values\] below. (This may be compared with, for example, Chapuy and Stump [@ChapuyStump p. 9 and Lemma 5.5].) The [*hook-shaped partitions*]{} of $n$ are $ \lambda={{\left( n-k, 1^{k} \right)}} $ for $k=0,\ldots,n-1$. Thus we only compute [*primary*]{} irreducible character values, sometimes only those of the form $\chi^{U,{{\left( \frac{n}{s}-k, 1^{k} \right)}}}$. Character values at the identity: the character degrees ------------------------------------------------------- Green computed the degrees of the primary irreducible characters $\chi^{U,\lambda}$ as a product formula involving familiar quantities associated to partitions. For a partition $\lambda$, recall [@Macdonald (1.5)] the quantity $ n(\lambda):=\sum_{i \geq 1} (i-1)\lambda_i. $ For a cell $a$ in row $i$ and column $j$ of the Ferrers diagram of $\lambda$ recall the [*hooklength*]{} [@Macdonald Example I.1] $$h(a):=h_\lambda(a):=\lambda_i+\lambda'_j-(i+j)+1.$$ \[dim-formula\] The primary irreducible $GL_n$-character $\chi^{U,\lambda}$ for a cuspidal character $U$ of $GL_s({{\mathbb{F}}}_q)$ and a partition $\lambda$ of $\frac{n}{s}$ has degree $$\deg( \chi^{U,\lambda} ) =(-1)^{n-\frac{n}{s}} (q;q)_n \frac{q^{s \cdot n(\lambda)} }{ \prod_{a \in \lambda} (1-q^{s \cdot h(a)})} =(-1)^{n-\frac{n}{s}} (q;q)_n s_\lambda(1,q^s,q^{2s},\ldots).$$ Here $s_\lambda(1,q,q^2,\ldots)$ is the [*principal specialization*]{} $x_i=q^{i-1}$ of the Schur function $s_\lambda=s_\lambda(x_1,x_2,\ldots)$. Observe that this formula depends only on $\lambda$ and $s$, and not on the choice of $U\in {{\operatorname{Cusp}}}_s$. Two special cases of this formula will be useful in the sequel. - The case of hook-shapes: $$\label{hook-degree-formula} \deg( \chi^{U, {{\left( \frac{n}{s}-k, 1^{k} \right)}}} )= (-1)^{n-\frac{n}{s}} q^{s\binom{k+1}{2}} \frac{(q;q)_n}{(q^s;q^s)_{\frac{n}{s}}} {\left[\begin{matrix} \frac{n}{s}-1 \\ k \end{matrix} \right]_{q^s}}.$$ - When $s=1$ and $U={\mathbf{1}}$ is the trivial character of $GL_1({{\mathbb{F}}}_q)$, the degree is given by the usual [*$q$-hook formula*]{} [@Stanley §7.21] $$\label{q-hook-formula} \deg( \chi^{{\mathbf{1}},\lambda} ) =f^\lambda(q):=(q;q)_n \frac{q^{n(\lambda)}}{ \prod_{a \in \lambda} (1-q^{h(a)})} =(q;q)_n s_\lambda(1,q,q^2,\ldots) =\sum_Q q^{{\operatorname{maj}}(Q)}$$ where the last sum is over all standard Young tableaux $Q$ of shape $\lambda$, and ${\operatorname{maj}}(Q)$ is the sum of the entries $i$ in $Q$ for which $i+1$ lies in a lower row of $Q$. (Such characters are called *unipotent characters*.) Character values on Singer cycles and regular elliptic elements --------------------------------------------------------------- Recall from the Introduction that a [*Singer cycle*]{} in $GL_n({{\mathbb{F}}}_q)$ is the image of a generator for the cyclic group ${{\mathbb{F}}}_{q^n}^\times \cong {{\mathbb{Z}}}/(q^n-1){{\mathbb{Z}}}$ under any embedding $ {{\mathbb{F}}}_{q^n}^\times \hookrightarrow GL_{{{\mathbb{F}}}_q}({{\mathbb{F}}}_{q^n}) \cong GL_n({{\mathbb{F}}}_q) $ that comes from a choice of ${{\mathbb{F}}}_q$-vector space isomorphism ${{\mathbb{F}}}_{q^n} \cong {{\mathbb{F}}}_q^n$. (Such an embedded subgroup ${{\mathbb{F}}}_{q^n}^\times$ is sometimes called a [*Coxeter torus*]{} or an [*anisotropic maximal torus*]{}.) Many irreducible $GL_n$-character values $\chi^{{{\underline{\lambda}}}}(c^{-1})$ vanish not only on Singer cycles, but even for a larger class of elements that we introduce in the following proposition. \[regular-elliptic-definition-proposition\] The following are equivalent for $g$ in $GL_n({{\mathbb{F}}}_q)$. 1. No conjugates $hgh^{-1}$ of $g$ lie in a proper parabolic subgroup $P_\alpha \subsetneq GL_n$. 2. There are no nonzero proper $g$-stable ${{\mathbb{F}}}_q$-subspaces inside ${{\mathbb{F}}}_q^n$. 3. The characteristic polynomial $\det(xI_n-g)$ is irreducible in ${{\mathbb{F}}}_q[x]$. 4. The element $g$ is the image of some $\beta$ in ${{\mathbb{F}}}_{q^n}^\times$ satisfying ${{\mathbb{F}}}_q(\beta)={{\mathbb{F}}}_{q^n}$ (that is, a [*primitive element*]{} for ${{\mathbb{F}}}_{q^n}$) under one of the embeddings $ {{\mathbb{F}}}_{q^n}^\times \hookrightarrow GL_{{{\mathbb{F}}}_q}({{\mathbb{F}}}_{q^n}) \cong GL_n({{\mathbb{F}}}_q). $ The elements in $GL_n({{\mathbb{F}}}_q)$ satisfying these properties are called the [**regular elliptic elements**]{}. [(i) is equivalent to (ii).]{} A proper ${{\mathbb{F}}}_q$-subspace $U$, say with $\dim_{{{\mathbb{F}}}_q} U = d < n$, is $g$-stable if and only any $h$ in $GL_n({{\mathbb{F}}}_q)$ sending $U$ to the span of the first $d$ standard basis vectors in ${{\mathbb{F}}}_q^n$ has the property that $h g h^{-1}$ lies in a proper parabolic subgroup $P_{\alpha}$ with $\alpha_1=d$. .1in [(ii) implies (iii).]{} Argue the contrapositive: if $\det(xI_n-g)$ had a nonzero proper irreducible factor $f(x)$, then $\ker(f(g): V \rightarrow V)$ would be a nonzero proper $g$-stable subspace. .1in [(iii) implies (iv).]{} If $f(x):=\det(xI_n-g)$ is irreducible in ${{\mathbb{F}}}_q[x]$, then $f(x)$ is also the minimal polynomial of $g$. Thus $g$ has rational canonical form over ${{\mathbb{F}}}_q$ equal to the companion matrix for $f(x)$. This is the same as the rational canonical form for the image under one of the above embeddings of any $\beta$ in ${{\mathbb{F}}}_{q^n}^\times$ having minimal polynomial $f(x)$, so that ${{\mathbb{F}}}_q(\beta) \cong {{\mathbb{F}}}_{q^n}$. Hence $g$ is conjugate to the image of such an element $\beta$ embedded in $GL_n({{\mathbb{F}}}_q)$, and then equal to such an element, after conjugating the embedding. .1in [(iv) implies (ii).]{} Assume that $g$ is the image of such an element $\beta$ in ${{\mathbb{F}}}_{q^n}^\times$ satisfying ${{\mathbb{F}}}_q(\beta)={{\mathbb{F}}}_{q^n}$. Then a $g$-stable ${{\mathbb{F}}}_q$-subspace $W$ of ${{\mathbb{F}}}_q^n$ would correspond to a subset of $W \subset {{\mathbb{F}}}_{q^n}$ stable under multiplication by ${{\mathbb{F}}}_q$ and by $\beta$, so also stable under multiplication by ${{\mathbb{F}}}_q(\beta)={{\mathbb{F}}}_{q^n}$. This could only be $W=\{0\}$ or $W={{\mathbb{F}}}_{q^n}$. Part (iv) of Proposition \[regular-elliptic-definition-proposition\] shows that Singer cycles $c$ in $G$ are always regular elliptic, since they correspond to elements $\gamma$ for which ${{\mathbb{F}}}_{q^n}^\times = \langle \gamma \rangle$, that is, to [*primitive roots*]{} in ${{\mathbb{F}}}_{q^n}$. Recall that associated to the extension ${{\mathbb{F}}}_{q} \subset {{\mathbb{F}}}_{q^n}$ is the [*norm map*]{} $$\begin{array}{rcl} {{\mathbb{F}}}_{q^n} &\overset{N_{{{\mathbb{F}}}_{q^n}/{{\mathbb{F}}}_{q}}}{\longrightarrow} & {{\mathbb{F}}}_{q}\\ \beta & \longmapsto & \beta \cdot \beta^q \cdot \beta^{q^2} \cdots \beta^{q^{n-1}}. \end{array}$$ The well-known surjectivity of norm maps for finite fields [@Lang VII Exer. 28] is equivalent to the following. \[norm-map-preserves-Singer\] If ${{\mathbb{F}}}_{q^n}^\times=\langle \gamma \rangle$, then ${{\mathbb{F}}}_{q}^\times=\langle N(\gamma) \rangle$. \[Singer-cycle-character-values\] Let $g$ be a regular elliptic element in $GL_n({{\mathbb{F}}}_q)$ associated to $\beta \in {{\mathbb{F}}}_{q^n}$, as in Proposition \[regular-elliptic-definition-proposition\](iv). 1. The irreducible character $\chi^{{{\underline{\lambda}}}}(g)$ vanishes unless $\chi^{{{\underline{\lambda}}}}$ is a primary irreducible character $\chi^{U,\lambda}$, for some $s$ dividing $n$ and some cuspidal character $U$ in ${{\operatorname{Cusp}}}_s$ and partition $\lambda$ in ${\operatorname{Par}}_{\frac{n}{s}}$. 2. Furthermore, $\chi^{U,\lambda}(g)=0$ except for hook-shaped partitions $\lambda={{\left( \frac{n}{s}-k, 1^{k} \right)}}$. 3. More explicitly, if $U$ in ${{\operatorname{Cusp}}}_s$ is associated to $[\varphi]$ with $\varphi$ in ${\operatorname{Hom}}({{\mathbb{F}}}_{q^s}^\times,{{\mathbb{C}}}^\times)$, then $$\begin{aligned} \chi^{U,{{\left( \frac{n}{s}-k, 1^{k} \right)}}}(g) &=(-1)^{k} \chi^{U,(\frac{n}{s})}(g)\\ &=(-1)^{\frac{n}{s}-k-1} \chi^{U,(1^{\frac{n}{s}})}(g) \\ &=(-1)^{n-\frac{n}{s}-k} \sum_{j=0}^{s-1} \varphi\left( N_{{{\mathbb{F}}}_{q^n}/{{\mathbb{F}}}_{q^s}}(\beta^{q^j}) \right). \end{aligned}$$ 4. If in addition $g$ is a Singer cycle then $$\sum_{U} \chi^{U,{{\left( \frac{n}{s}-k, 1^{k} \right)}}}(g) = \begin{cases} (-1)^{n-\frac{n}{s}-k} \mu(s) & \text{ if } q = 2,\\ 0 & \text{ if } q \neq 2. \end{cases}$$ where the sum is over all $U$ in ${{\operatorname{Cusp}}}_s$, and $\mu(s)$ is the usual number-theoretic Möbius function of $s$. The key point is Proposition \[regular-elliptic-definition-proposition\](i), showing that regular elliptic elements $g$ are the elements whose conjugates $hgh^{-1}$ lie in no proper parabolic subgroup $P_\alpha$. Hence the parabolic induction formula shows that any properly induced character $f_1 {{*}}\cdots {{*}}f_m$ will vanish on a regular elliptic element $g$. Assertion (i) follows immediately, as shows non-primary irreducibles are properly induced. Assertion (ii) also follows, as a non-hook partition $\lambda=(\lambda_1 \geq \lambda_2 \geq \cdots)$ has $\lambda_2 \geq 2$, so that in the Jacobi-Trudi-style formula for $\chi^{U,\lambda}$, each term $$\chi^{U,(\lambda_1-1+w(1))} {{*}}\chi^{U,(\lambda_2-2+w(2))} {{*}}\cdots {{*}}\chi^{U,(\lambda_\ell-\ell+w(\ell))}$$ begins with two nontrivial product factors, so it is properly induced, and vanishes on regular elliptic $g$. The first two equalities asserted in (iii) follow from similar analysis of terms in , for $\chi^{U,\lambda}$ when $\lambda={{\left( \frac{n}{s}-k, 1^{k} \right)}}$. These formulas have $2^{k+1}$ and $2^{\frac{n}{s}-k}$ nonvanishing terms, respectively, of the form $$\begin{array}{rcl} &(-1)^{k-m} & \chi^{U,(\alpha_1)} {{*}}\chi^{U,(\alpha_2)} {{*}}\cdots {{*}}\chi^{U,(\alpha_m)} \\ &(-1)^{\frac{n}{s}-k-m}& \chi^{U,(1^{\beta_1})} {{*}}\chi^{U,(1^{\beta_2})} {{*}}\cdots {{*}}\chi^{U,(1^{\beta_m})} \end{array}$$ corresponding to compositions $(\alpha_1,\ldots,\alpha_m)$ and $(\beta_1,\ldots,\beta_m)$ of $\frac{n}{s}$ with $\alpha_1 \geq k+1$ and $\beta_1 \geq \frac{n}{s}-k$, respectively. All such terms vanish on regular elliptic $g$, being properly induced, except the $m=1$ terms: $$\begin{aligned} \chi^{U,{{\left( \frac{n}{s}-k, 1^{k} \right)}}}(g) &=(-1)^{k-1} \chi^{U,(\frac{n}{s})}(g)\\ &=(-1)^{\frac{n}{s}-k-1} \chi^{U,(1^{\frac{n}{s}})}(g). \end{aligned}$$ The last equality in (iii) comes from a result of Silberger and Zink [@SilbergerZink Theorem 6.1], which they deduced by combining various formulas from Green [@Green]. For assertion (iv), say that the regular elliptic element $g$ corresponds to an element $\beta$ in ${{\mathbb{F}}}_{q^n}$ under the embedding ${{\mathbb{F}}}_{q^n}^\times \hookrightarrow GL_n({{\mathbb{F}}}_q)$, and let $\gamma:=N_{{{\mathbb{F}}}_{q^n}/{{\mathbb{F}}}_{q^s}}(\beta)$ be its norm in ${{\mathbb{F}}}_{q^s}^\times$. Assertion (iii) and the multiplicative property of the norm map $N_{{{\mathbb{F}}}_{q^n}/{{\mathbb{F}}}_{q^s}}$ imply $$\label{norm-orbit-Singer-sum} \sum_{U} \chi^{U,{{\left( \frac{n}{s}-k, 1^{k} \right)}}}(g) =(-1)^{n-\frac{n}{s}-k} \sum_U \sum_{j=0}^{s-1} \varphi\left( \gamma^{q^j} \right) \\ =(-1)^{n-\frac{n}{s}-k} \sum_U \sum_{\gamma'} \varphi(\gamma')$$ where the inner sum runs over all $\gamma'$ lying in the Frobenius orbit of $\gamma$ within ${{\mathbb{F}}}_{q^s}$. When one further assumes that $g$ is a Singer cycle, then Proposition \[norm-map-preserves-Singer\] implies ${{\mathbb{F}}}_{q^s}^\times = \langle \gamma \rangle$, so that a homomorphism $\varphi: {{\mathbb{F}}}_{q^s}^\times \rightarrow {{\mathbb{C}}}^\times$ is completely determined by its value $z:=\varphi(\gamma)$ in ${{\mathbb{C}}}^\times$. Furthermore, $\varphi$ will have a free Frobenius orbit if and only if the powers $\{z, z^q, z^{q^2},\cdots,z^{q^{s-1}}\}$ are [*distinct*]{} roots of unity. Thus one can rewrite the rightmost summation $\sum_U \sum_{\gamma'} \varphi(\gamma')$ in above as the sum over all $z$ in ${{\mathbb{C}}}^\times$ for which $z^{q^s-1}=1$ but $z^{q^t-1} \neq 1$ for any proper divisor $t$ of $s$. Number-theoretic Möbius inversion shows this is $ \sum_{t | s} \mu\left( \frac{s}{t} \right) f(t) $ where $$f(t):=\sum_{\substack{z \in {{\mathbb{C}}}^\times:\\ z^{q^t-1}=1}} z =\begin{cases} 1 & \text{ if } q = 2, t=1,\\ 0 & \text{ if } q \neq 2 \text{ or }t \neq 1. \end{cases}$$ Hence $$\label{moebius-function-equality} \sum_U \sum_{\gamma'} \varphi(\gamma')= \begin{cases} \mu(s) & \text{ if } q = 2,\\ 0 & \text{ if } q \neq 2. \end{cases} \qedhere$$ Character values on semisimple reflections {#semisimple-reflection-character-section} ------------------------------------------ Recall that a semisimple reflection $t$ in $GL_n({{\mathbb{F}}}_q)$ has conjugacy class determined by its non-unit eigenvalue $\det(t)$, lying in ${{\mathbb{F}}}_q^\times \setminus \{1\}$. Recall also the notion of the [*content*]{} $c(a):=j-i$ of a cell $a$ lying in row $i$ and column $j$ of the Ferrers diagram for a partition $\lambda$. \[normalized-characters-on-semisimple-reflections\] Let $t$ be a semisimple reflection in $GL_n({{\mathbb{F}}}_q)$. 1. Primary irreducible characters $\chi^{U,\lambda}$ vanish on $t$ unless ${\operatorname{wt}}(U)=1$, that is, unless $U$ is in ${{\operatorname{Cusp}}}_1$. 2. For $U$ in ${{\operatorname{Cusp}}}_1$, so $ {{\mathbb{F}}}_q^\times \overset{U}{\rightarrow} {{\mathbb{C}}}^\times, $ and $\lambda$ in ${\operatorname{Par}}_n$, the normalized character ${\widetilde{\chi}}^{U,\lambda}$ has value on $t$ $${\widetilde{\chi}}^{U,\lambda}(t) = U(\det(t)) \cdot \frac{\sum_{a \in \lambda} q^{c(a)}}{[n]_q}.$$ 3. In particular, for $U$ in ${{\operatorname{Cusp}}}_1$ and hook shapes $\lambda=(n-k,1^k)$, this simplifies to $${\widetilde{\chi}}^{U,(n-k,1^k)}(t)=U(\det(t)) \cdot q^{-k}.$$ For assertion (i), we start with the fact proven by Green [@Green §5 Example (ii), p. 430] that cuspidal characters for $GL_n$ vanish on [*non-primary*]{} conjugacy classes, that is, those for which the characteristic polynomial is divisible by at least two distinct irreducible polynomials in ${{\mathbb{F}}}_q[x]$. This implies cuspidal characters for $GL_n$ with $n \geq 2$ vanish on semisimple reflections $t$, since such $t$ are non-primary: $\det(xI-t)$ is divisible by both $x-1$ and $x-\alpha$ where $\alpha=\det(t) \neq 1$. Next, the parabolic induction formula shows that any character of the form $\chi_{U_1} {{*}}\cdots {{*}}\chi_{U_\ell}$ in which each $U_i$ is a $GL_{n_i}$-cuspidal with $n_i \geq 2$ will also vanish on all semisimple reflections $t$: whenever $hth^{-1}$ lies in the parabolic $P_{(n_1,\ldots,n_\ell)}$ and has diagonal blocks $(g_1,\ldots,g_\ell)$, one of the $g_{i_0}$ is also a semisimple reflection, so that $\chi_{U_{i_0}}(g_{i_0})=0$ by the above discussion. Lastly, Lemma \[primaries-in-terms-of-cuspidals\] shows that every primary irreducible $\chi^{U,\lambda}$ with ${\operatorname{wt}}(U) \geq 2$ will vanish on every semisimple reflection: $\chi^{U,\lambda}$ is in the ${{\mathbb{Q}}}$-span of characters $\chi_{U_1} {{*}}\cdots {{*}}\chi_{U_\ell}$ with each $U_i$ a $GL_{n_i}$-cuspidal in which ${\operatorname{wt}}(U)$ divides $n_i$, so that $n_i \geq 2$. Assertion (iii) is an easy calculation using assertion (ii), so it only remains to prove (ii). We first claim that one can reduce to the case where character $U$ in ${{\operatorname{Cusp}}}_1$ is the trivial character ${{\mathbb{F}}}_q^\times \overset{U={\mathbf{1}}}{\longrightarrow} {{\mathbb{C}}}^\times$. This is because one has $ \chi^{U,(n)} = U = U \otimes \chi^{{\mathbf{1}},(n)} $ and hence using one has $$\label{weight-one-cuspidal-tensor-product} \chi^{U,\lambda} = U \otimes \chi^{{\mathbf{1}},\lambda} \qquad\text{ for }\lambda\text{ in }{\operatorname{Par}}_n\text{ when }U\text{ lies in }{{\operatorname{Cusp}}}_1.$$ Thus without loss of generality, $U={\mathbf{1}}$, and we wish to show $$\label{semisimple-refn-character-value-at-unipotent} {\widetilde{\chi}}^{{\mathbf{1}},\lambda}(t) = \frac{1}{[n]_q} \sum_{a \in \lambda} q^{c(a)}.$$ \[semisimple-reflection-character-skews\] A semisimple reflection $t$ has $ \chi^{{\mathbf{1}},\lambda}(t) = \Psi( s_\lambda ) $ where $\Psi$ is the linear map on the symmetric functions $\Lambda={{\mathbb{Q}}}[p_1,p_2,\ldots]$ expressed in terms of power sums that sends $ f(x_1,x_2,\ldots) \mapsto (q;q)_{n-1} \frac{\partial f}{\partial p_1}(1,q,q^2,\ldots). $ By linearity and , it suffices to check for compositions $\alpha=(\alpha_1,\ldots,\alpha_m)$ of $n$ that $ \chi^{{\mathbf{1}},\alpha}:=\chi^{{\mathbf{1}},(\alpha_1)} {{*}}\cdots {{*}}\chi^{{\mathbf{1}},(\alpha_m)} $ has $\chi^{{\mathbf{1}},\alpha}(t) = \Psi (h_\alpha)$ where $h_\alpha = h_{\alpha_1} \cdots h_{\alpha_m}$. The character $\chi^{{\mathbf{1}},\alpha}$ is just the usual induced character ${\operatorname{Ind}}_{P_\alpha}^{GL_n} {\mathbf{1}}_{P_\alpha}$, so the permutation character on the set of [*$\alpha$-flags of subspaces*]{} $$\{0\} \subset V_{\alpha_1} \subset V_{\alpha_1+\alpha_2} \subset \cdots \subset V_{\alpha_1+\alpha_2+\cdots+\alpha_{m-1}} \subset {{\mathbb{F}}}_q^n,$$ which are counted by the $q$-multinomial coefficient $${\left[\begin{matrix} n \\ \alpha \end{matrix} \right]_{q}}:={\left[\begin{matrix} n \\ \alpha_1,\ldots,\alpha_m \end{matrix} \right]_{q}} = \frac{[n]!_q}{[\alpha_1]!_q \cdots [\alpha_m]!_q} =(q;q)_n h_\alpha(1,q,q^2,\ldots).$$ Thus $\chi^{{\mathbf{1}},\alpha}(t)$ counts the number of such flags stabilized by the semisimple reflection $t$. To count these let $H$ and $L$ denote, respectively, the fixed hyperplane for $t$ and the line which is the $\det(t)$-eigenspace for $t$. Then one can classify the $\alpha$-flags stabilized by $t$ according to the smallest index $i$ for which $L \subset V_{\alpha_1+\cdots+\alpha_i}$. Fixing this index $i$, such flags must have their first $i-1$ subspaces $V_{\alpha_1}, V_{\alpha_1+\alpha_2},\ldots,V_{\alpha_1+\cdots+\alpha_{i-1}}$ lying inside $H$, and their remaining subspaces from $V_{\alpha_1+\cdots+\alpha_i}$ onward containing $L$. From this description it is not hard to see that the quotient map ${{\mathbb{F}}}_q^n \twoheadrightarrow {{\mathbb{F}}}_q^n/L$ is a bijection between such $t$-stable $\alpha$-flags and the $(\alpha-e_i)$-flags in ${{\mathbb{F}}}_q^n/L \cong {{\mathbb{F}}}_q^{n-1}$, where $ \alpha-e_i:=(\alpha_1,\ldots,\alpha_{i-1}, \alpha_i-1,\alpha_{i+1},\ldots,\alpha_{m}). $ Consequently $$\begin{aligned} \chi^{{\mathbf{1}},\alpha}(t) = \sum_{i=1}^m {\left[\begin{matrix} n-1 \\ \alpha-e_i \end{matrix} \right]_{q}} =(q;q)_{n-1} \sum_{i=1}^m h_{\alpha-e_i}(1,q,q^2,\ldots) = (q;q)_{n-1} \frac{\partial h_{\alpha}}{\partial p_1}(1,q,q^2,\ldots) =\Psi(h_\alpha) \end{aligned}$$ using the fact [@Macdonald Example I.5.3] that $\frac{\partial h_n}{\partial p_1}=h_{n-1}$, and hence $\frac{\partial h_{\alpha}}{\partial p_1} = \sum_{i=1}^m h_{\alpha-e_i}$ via the Leibniz rule. Resuming the proof of , since [@Macdonald Example I.5.3] shows $ \partial s_\lambda/\partial p_1 = \sum_{\mu \subset \lambda:\\ |\mu|=|\lambda|-1} s_\mu, $ one concludes from Lemma \[semisimple-reflection-character-skews\] and that $${\widetilde{\chi}}^{{\mathbf{1}}, \lambda}(t) = \frac{\chi^{{\mathbf{1}}, \lambda}(t)}{\deg(\chi^{{\mathbf{1}},\lambda})} = \sum_{\substack{\mu \subset \lambda:\\ |\mu|=|\lambda|-1}} \frac{(q;q)_{n-1} s_\mu(1,q,q^2,\ldots)}{(q;q)_n s_\lambda(1,q,q^2,\ldots)} \\ = \sum_{\substack{\mu \subset \lambda:\\ |\mu|=|\lambda|-1}} \frac{ f^{\mu}(q) }{ f^\lambda(q) }$$ where $f^\lambda(q)$ is the $q$-hook formula from . Thus the desired equation becomes the assertion $$\label{q-hook-walk-formula} \sum_{\substack{\mu \subset \lambda: \\ |\mu|=|\lambda|-1}} \frac{ f^{\mu}(q) }{ f^\lambda(q) } = \frac{1}{[n]_q}\sum_{a \in \lambda} q^{c(a)}$$ which follows from either of two results in the literature: is equivalent[^2], after sending $q \mapsto q^{-1}$, to a result of Kerov [@Kerov Theorem 1 and Eqn. (2.2)], and is also the $t=q^{-1}$ specialization of a result of Garsia and Haiman [@GarsiaHaiman (I.15), Theorem 2.3]. Character values on transvections {#transvection-character-section} --------------------------------- The $GL_n$-irreducible character values on transvections appear in probabilistic work of M. Hildebrand [@Hildebrand]. For primary irreducible characters, his result is equivalent[^3] to the following. \[Hildebrand’s-calculation\] For $U$ in ${{\operatorname{Cusp}}}_s$ with $\lambda$ in ${\operatorname{Par}}_{\frac{n}{s}}$, a transvection $t$ in $GL_n({{\mathbb{F}}}_q)$ has $${\widetilde{\chi}}^{U,\lambda}(t)= \begin{cases} \displaystyle \frac{1}{1 - q^{n-1}} \left( 1 - q^{n-1} \sum_{\substack{\mu \subset \lambda: \\ |\mu|=|\lambda|-1}} \frac{ f^{\mu}(q) }{ f^\lambda(q) } \right) & \text{ if }s=1,\\ \displaystyle \frac{1}{1 - q^{n-1}} & \text{ if } s \geq 2. \end{cases}$$ One can rephrase the $s=1$ case similarly to Lemma \[normalized-characters-on-semisimple-reflections\](ii). \[normalized-characters-on-transvections\] For $U$ in ${{\operatorname{Cusp}}}_1$ with $\lambda$ in ${\operatorname{Par}}_{n}$, a transvection $t$ in $GL_n({{\mathbb{F}}}_q)$ has $${\widetilde{\chi}}^{U,\lambda}(t) = \frac{ \displaystyle 1 - q^{n-1} \left( \frac{ \sum_{a \in \lambda} q^{c(a)} }{[n]_q} \right) } {1 - q^{n-1}}.$$ In particular, for $U$ in ${{\operatorname{Cusp}}}_1$ and $0 \leq k \leq n-1$, one has $${\widetilde{\chi}}^{U,{{\left( n-k, 1^{k} \right)}}}(t)= \frac{1 - q^{n-k-1}}{1 - q^{n-1}}.$$ The first assertion follows from Theorem \[Hildebrand’s-calculation\] using , and the second from the calculation $$\sum_{a \in {{\left( n-k, 1^{k} \right)}}} q^{c(a)} = q^{-k} + q^{-k+1} + \cdots + q^{n-k-1} = q^{-k} [n]_q. \qedhere$$ Proofs of Theorems \[q-factorization-theorem\] and \[fixed-det-sequence-theorem\]. {#main-result-proof-section} ================================================================================== In proving the main results Theorems \[q-factorization-theorem\] and \[fixed-det-sequence-theorem\], it is convenient to know the equivalences between the various formulas that they assert. After checking this in Proposition \[q-Jackson-sum-equals-tot-num-lemma\], we assemble the normalized character values on reflection conjugacy class sums, in the form needed to apply . This is then used to prove Theorem \[fixed-det-sequence-theorem\] for $q>2$, from which we derive Theorem \[q-factorization-theorem\] for $q>2$. Lastly we prove Theorem \[q-factorization-theorem\] for $q=2$. Equivalences of the formulas ---------------------------- We will frequently use the easy calculation $$\label{q-difference-on-powers} \left[\Delta_q^{N} \left( x^A \right) \right]_{x=1} = \frac{(q^{A-N+1};q)_N}{(1-q)^N}$$ which can be obtained by iterating $\Delta_q$, or via and the [*$q$-binomial theorem*]{} [@GasperRahman p. 25, Exer. 1.2(vi)] $$\label{q-binomial-theorem} (x;q)_N=\sum_{k=0}^N (-x)^k q^{\binom{k}{2}}{\left[\begin{matrix} N \\ k \end{matrix} \right]_{q}}.$$ The following assertion was promised in the Introduction. \[q-Jackson-sum-equals-tot-num-lemma\] As polynomials in $q$, - the three expressions , , for $t_q(n,\ell)$ asserted in Theorem \[q-factorization-theorem\] all agree, and - the two expressions , for $t_q(n,\ell,m)$ asserted in Theorem \[fixed-det-sequence-theorem\] agree if $m\le \ell-1.$ [Assertion (i).]{} Starting with $$t_q(n,\ell) = (1-q)^{-1}\frac{(-[n]_q)^\ell}{[n]!_q} \left[ \Delta_q^{n-1} \biggl(\frac{1}{x}-\frac{(1+x(1-q))^\ell}{x}\biggr)\right]_{x=1},$$ linearity of the operator $g(x) \longmapsto \left[\Delta_q^{n-1} g(x) \right]_{x=1}$ lets one expand in two different ways its subexpression $$\label{q-Jackson-difference-formula-subexpression} \left[ \Delta_q^{n-1} \biggl(\frac{1}{x}-f(x) \biggr)\right]_{x=1} \qquad \text{ where }f(x):=\frac{\left(1+x(1-q)\right)^\ell}{x}.$$ The first way will yield , by expanding as $\left[\Delta_q^{n-1} \left( \frac{1}{x} \right) \right]_{x=1}- \left[\Delta_q^{n-1} f(x) \right]_{x=1}. $ Note that $$\left[\Delta_q^{n-1} \left( \frac{1}{x} \right) \right]_{x=1} = \frac{(q^{1-n};q)_{n-1}}{(1-q)^{n-1}} = \frac{(-1)^{n-1}}{q^{\binom{n}{2}} (1-q)^{n-1}} (q;q)_{n-1}$$ via , which accounts for the $(-1)^{n-1}(q;q)_{n-1}$ term inside the large parentheses of . Meanwhile, applying to $\left[\Delta_q^{n-1} f(x) \right]_{x=1}$ and noting that $ f(q^{n-1-k})=q^{1-n}q^{k}(1+q^{n-k-1}-q^{n-k})^\ell, $ one obtains a summation that accounts for the remaining terms inside the large parentheses of . This shows the equivalence of , . The second way will yield , by expanding $f(x)=\sum_{i=0}^{\ell} \binom{\ell}{i}(1-q)^{\ell-i} x^{\ell-i-1}$, and noting that the $i=\ell$ term cancels with the $\frac{1}{x}$ appearing inside . Therefore becomes $$\begin{aligned} t_q(n,\ell)=& (-[n]_q)^\ell \frac{(1-q)^{n-1}}{(q;q)_n} \sum_{i=0}^{\ell-1} -\binom{\ell}{i}(1-q)^{\ell-i} \left[ \Delta_q^{n-1}\left( x^{\ell-i-1} \right) \right]_{x=1}\\ =& [n]_q^{\ell-1} \sum_{i=0}^{\ell-n} (-1)^i(q-1)^{\ell-i-1}\binom{\ell}{i}{\left[\begin{matrix} \ell-i-1 \\ n-1 \end{matrix} \right]_{q}} , \end{aligned}$$ using . The summands with $\ell-n+1\le i\le \ell-1$ vanish, showing the equivalence of , . .1in [Assertion (ii).]{} Starting with , $$t_q(n,\ell,m) = \frac{[n]_q^\ell}{[n]!_q} \left[ \Delta_q^{n-1}\bigl( (x-1)^m x^{\ell-m-1}\bigr) \right]_{x=1},$$ expand the $(x-1)^m$ factor via the binomial theorem. Using , this expression for $t_q(n,\ell,m)$ becomes $$t_q(n,\ell,m) = [n]_q^{\ell-1} \sum_{i = 0}^m (-1)^i \, \binom{m}{i} \, {\left[\begin{matrix} \ell-i - 1 \\ n - 1 \end{matrix} \right]_{q}} .$$ As $i\le m\le \ell-1$, one has $\ell-i-1\ge 0$ and the sum is actually over $0 \le i \le \ell-n$, agreeing with . The normalized characters on reflection conjugacy class sums ------------------------------------------------------------ \[reflection-conjugacy-class-sum-definition\] For $\alpha$ in ${{\mathbb{F}}}_q^\times$, let $z_\alpha:=\sum_{t:\det(t)=\alpha} t$ in ${{\mathbb{C}}}GL_n$ be the sum of reflections of determinant $\alpha$. \[normalized-character-on-reflection-class\] For $U$ in ${{\operatorname{Cusp}}}_s$, and $k$ in the range $0 \leq k \leq \frac{n}{s}$, and any $\alpha$ in ${{\mathbb{F}}}_q^\times \setminus \{ 1 \}$, one has $$\begin{aligned} \label{semisimple-reflection-class-value} {\widetilde{\chi}}^{U,{{\left( \frac{n}{s}-k, 1^{k} \right)}}}(z_\alpha) &=& [n]_q \left. \begin{cases} q^{n-k-1} U(\alpha) &\text{ if }s=1 \\ 0 &\text{ if }s \geq 2. \end{cases} \right\},\\ \label{transvection-class-value} {\widetilde{\chi}}^{U,{{\left( \frac{n}{s}-k, 1^{k} \right)}}}(z_1) &=& [n]_q \left. \begin{cases} q^{n-k-1}-1 &\text{ if }s=1, \\ -1 &\text{ if }s \geq 2. \end{cases} \right\}.\end{aligned}$$ First we count the reflections $t$ in $GL_n({{\mathbb{F}}}_q)$. There are $[n]_q=1+q+q^2+\cdots+q^{n-1}$ choices for the hyperplane $H$ fixed by $t$. To count the reflections fixing $H$, without loss of generality one can conjugate $t$ and assume that $H$ is the hyperplane spanned by the first $n$ standard basis vectors $e_1,\ldots,e_{n-1}$. If $t$ is a semisimple reflection then its conjugacy class is determined by its determinant, lying in ${{\mathbb{F}}}_q^\times \setminus \{1\}$. Having fixed $\alpha:=\det(t)$, there will be $q^{n-1}$ such reflections that fix $e_1,\ldots,e_{n-1}$: each is determined by sending $e_n$ to $\alpha e_n + \sum_{i=1}^{n-1} c_i e_i$ for some $(c_1,\ldots,c_{n - 1})$ in ${{\mathbb{F}}}_q^{n-1}$. Hence follows from Lemma \[normalized-characters-on-semisimple-reflections\]. The nonsemisimple reflections $t$ are the transvections, forming a single conjugacy class, with $\det(t)=1$. There will be $q^{n-1}-1$ transvections that fix $e_1,\ldots,e_{n-1}$: each is determined by sending $e_n$ to $e_n + \sum_{i=1}^{n-1} c_i e_i$ for some $(c_1, \ldots, c_{n-1})$ in ${{\mathbb{F}}}_q^{n-1} \setminus \{ \mathbf{0} \}$. Hence follows from Theorem \[Hildebrand’s-calculation\] and Corollary \[normalized-characters-on-transvections\]. Proof of Theorem \[fixed-det-sequence-theorem\] for $q>2$. ---------------------------------------------------------- For a Singer cycle $c$ in $GL_n({{\mathbb{F}}}_q)$, and $\alpha=(\alpha_1,\ldots,\alpha_\ell)$ in $({{\mathbb{F}}}_q^\times)^{\ell}$ with $\prod_{i=1}^\ell\alpha_i=\det(c)$, Proposition \[conj-count-prop\] counts the reflection factorizations $c=t_1 t_2 \cdots t_\ell$ with $\det(t_i)=\alpha_i$ as $$\label{factorizations with fixed alpha} \frac{1}{|GL_n|} \sum_{\substack{(s,U):\\s | n\\U \in {{\operatorname{Cusp}}}_s}} \sum_{k=0}^{\frac{n}{s}-1} \deg(\chi^{U,{{\left( \frac{n}{s}-k, 1^{k} \right)}}}) \cdot \chi^{U,{{\left( \frac{n}{s}-k, 1^{k} \right)}}} (c^{-1}) \cdot \prod_{i=1}^\ell {\widetilde{\chi}}^{U,{{\left( \frac{n}{s}-k, 1^{k} \right)}}}(z_{\alpha_i}).$$ There are several simplifications in this formula. Firstly, note that the outermost sum over pairs $(s,U)$ reduces to the pairs with $s=1$: since $\det(c)$ is a primitive root in ${{\mathbb{F}}}_q^\times$ by Proposition \[norm-map-preserves-Singer\] and $q>2$, one knows that $\det(c) \neq 1$, so that at least one of the $\alpha_i$ is not $1$. Thus its factor ${\widetilde{\chi}}^{U,{{\left( \frac{n}{s}-k, 1^{k} \right)}}}(z_{\alpha_i})$ in the last product will vanish if $s \geq 2$ by . Secondly, when $s=1$ then Corollary \[normalized-character-on-reflection-class\] evaluates the product in as $$\label{varying-class-product-with-det} \prod_{i=1}^\ell {\widetilde{\chi}}^{U,{{\left( \frac{n}{s}-k, 1^{k} \right)}}}(z_{\alpha_i}) = [n]_q^\ell \,\, (q^{n-k-1}-1)^m \,\, q^{(n-k-1)(\ell-m)} \,\, U(\det(c))$$ if exactly $m$ of the $\alpha_i$ are equal to $1$, that is, if the number of transvections in the factorization is $m$. This justifies calling it $t_q(n,\ell,m)$ where $m\le \ell-1$. Thirdly, for $s=1$ Proposition \[Singer-cycle-character-values\](iii) shows[^4] that $ \chi^{U,{{\left( n-k, 1^{k} \right)}}} (c^{-1}) = (-1)^k U(\det(c^{-1})) $, so there will be cancellation of the factor $U(\det(c))$ occurring in within each summand of . Thus plugging in the degree formula from the $s=1$ case of , one obtains the following formula for , which we denote by $t_q(n,\ell,m)$, emphasizing its dependence only on $\ell$ and $m$, not on the sequence $\alpha$: $$t_q(n,\ell,m) = \frac{(q - 1)[n]_q^\ell}{|GL_n|} \sum_{k=0}^{n-1} q^{\binom{k + 1}{2}}{\left[\begin{matrix} n-1 \\ k \end{matrix} \right]_{q}} \, (-1)^k \, (q^{n-k-1}-1)^m \, q^{(n-k-1)(\ell-m)}.$$ This expression may be rewritten using the $q$-difference operator $\Delta_q$ and as $$t_q(n,\ell,m) = \frac{[n]_q^\ell}{|GL_n|} q^{\binom{n}{2}}(q-1)^n \left[ \Delta_q^{n-1}\bigl( (x-1)^mx^{\ell-m-1}\bigr) \right]_{x=1}. $$ Since $|GL_n|=q^{\binom{n}{2}} (-1)^n (q;q)_n$, this last expression is the same as . Hence by Proposition \[q-Jackson-sum-equals-tot-num-lemma\], this completes the proof of Theorem \[fixed-det-sequence-theorem\] for $q > 2$. that the final sum has all summands $0$ except for the $i = 0$ summand, whence in this case we have $[n]_q^{n - 1}$ reflections. In particular, for $\ell = n$ the number of factorizations is completely independent of the tuple $\alpha$ of determinants (provided the product of the entries of $\alpha$ actually is $\det c$). Proof of Theorem \[q-factorization-theorem\] when $q>2$. -------------------------------------------------------- We will use Theorem \[fixed-det-sequence-theorem\] for $q>2$ to derive for $q>2$. First note that one can choose a sequence of determinants $\alpha=(\alpha_1,\ldots,\alpha_\ell)$ in ${{\mathbb{F}}}_q^\times$ that has $\prod_{i=1}^\ell \alpha_i =\det(c)$ and has exactly $m$ of the $\alpha_i=1$ in a two-step process: first choose $m$ positions out of $\ell$ to have $\alpha_i=1$, then choose the remaining sequence in $\left( {{\mathbb{F}}}_q^\times\setminus\{1\} \right)^{\ell-m}$ with product equal to $\det(c)$. Simple counting shows that in a finite group $K$, the number of sequences in $(K \setminus \{1\})^N$ whose product is some fixed nonidentity element[^5] of $K$ is $$\label{factoring-non-identity-elements-count} \frac{(|K|-1)^{N}-(-1)^{N}}{|K|}.$$ Applying this to $K={{\mathbb{F}}}_q^\times$ with $N=\ell-m$ gives $$t_q(n,\ell) =\sum_{m=0}^{\ell} t_q(n,\ell,m) \binom{\ell}{m} \frac{(q-2)^{\ell-m}-(-1)^{\ell-m}}{q-1}. $$ Thus from one has $$\begin{aligned} t_q(n,\ell) =& \frac{(q-1)[n]_q^\ell}{|GL_n|}q^{\binom{n}{2}}(q-1)^{n-1} \left[ \Delta_q^{n-1} \left(\sum_{m=0}^{\ell} \binom{\ell}{m} (x-1)^{m}x^{\ell-m-1} \frac{(q-2)^{\ell-m}-(-1)^{\ell-m}}{q-1}\right) \right]_{x=1}\\ =& \frac{(-[n]_q)^\ell}{|GL_n|}q^{\binom{n}{2}}(q-1)^{n-1} \left[ \Delta_q^{n-1} \biggl(\frac{(1+x(1-q))^\ell}{x}-\frac{1}{x}\biggr) \right]_{x=1}\\ =& (1-q)^{-1}\frac{(-[n]_q)^\ell}{[n]!_q} \left[ \Delta_q^{n-1} \biggl(\frac{1}{x}-\frac{(1+x(1-q))^\ell}{x}\biggr)\right]_{x=1}\end{aligned}$$ which is . Hence by Proposition \[q-Jackson-sum-equals-tot-num-lemma\], this completes the proof of Theorem \[q-factorization-theorem\] when $q>2$. Proof of Theorem \[q-factorization-theorem\] when $q=2$. -------------------------------------------------------- Here all reflections are transvections and gives us $$\begin{aligned} t_q(n,\ell) &=\frac{1}{|GL_n|} \sum_{\chi^{{\underline{\lambda}}}\in {\operatorname{Irr}}(GL_n)} \deg(\chi^{{{\underline{\lambda}}}}) \cdot \chi^{{{\underline{\lambda}}}}(c^{-1}) \cdot {\widetilde{\chi}}^{{{\underline{\lambda}}}}(z_1)^\ell \\ &=\frac{1}{|GL_n|} \sum_{\substack{(s,U):\\s | n\\U \in {{\operatorname{Cusp}}}_s}} \underbrace{ \sum_{k=0}^{\frac{n}{s}-1} \deg(\chi^{U,{{\left( \frac{n}{s}-k, 1^{k} \right)}}}) \cdot \chi^{U,{{\left( \frac{n}{s}-k, 1^{k} \right)}}} (c^{-1}) \cdot {\widetilde{\chi}}^{U,{{\left( \frac{n}{s}-k, 1^{k} \right)}}}(z_1)^\ell }_{\text{Call this } f(s,U)} \end{aligned}$$ using the vanishing of $\chi^{{{\underline{\lambda}}}}(c^{-1})$ from Proposition \[Singer-cycle-character-values\](i,ii). We separate the computation into $s=1$ and $s \geq 2$, and first compute $\sum_{U \in {{\operatorname{Cusp}}}_1} f(s,U)$. As $q=2$ there is only one $U$ in ${{\operatorname{Cusp}}}_1$, namely $U={\mathbf{1}}$, and hence $$\begin{aligned} \sum_{U \in {{\operatorname{Cusp}}}_1} f(s,U)= f(1,{\mathbf{1}}) &=\sum_{k=0}^{n-1} \deg(\chi^{{\mathbf{1}},(n-k,1^k)}) \cdot \chi^{{\mathbf{1}},(n-k,1^k)} (c^{-1}) \cdot {\widetilde{\chi}}^{{\mathbf{1}},(n-k,1^k)}(z)^\ell \\ &=\sum_{k=0}^{n-1} q^{\binom{k+1}{2}} {\left[\begin{matrix} n-1 \\ k \end{matrix} \right]_{q}} \cdot (-1)^k \cdot [n]^\ell_q(q^{n-k}-q^{n-k-1}-1)^\ell \end{aligned}$$ using the degree formula at $s=1$, the fact that $ \chi^{({\mathbf{1}},n-k,1^k)} (c^{-1}) = (-1)^k \chi^{{\mathbf{1}}, (n)} (c^{-1}) = (-1)^k $ from Proposition \[Singer-cycle-character-values\](iii), and the value ${\widetilde{\chi}}^{{\mathbf{1}},(n-k,1^k)}(z_1) = [n]_q(q^{n-k}-q^{n-k-1}-1)$ from . For $s \geq 2$, we compute $$\begin{aligned} \sum_{\substack{(s,U):\\s |n, s \geq 2 \\\ U \in {{\operatorname{Cusp}}}_s}} f(s,U) &= \sum_{\substack{(s,U):\\s |n, s \geq 2 \\\ U \in {{\operatorname{Cusp}}}_s}} \sum_{k=0}^{ \frac{n}{s}-1 } \deg(\chi^{U,{{\left( \frac{n}{s}-k, 1^{k} \right)}}}) \cdot \chi^{U,{{\left( \frac{n}{s}-k, 1^{k} \right)}}} (c^{-1}) \cdot {\widetilde{\chi}}^{U,{{\left( \frac{n}{s}-k, 1^{k} \right)}}}(z_1)^\ell \\ &=\sum_{\substack{s |n\\ s \geq 2} } \sum_{k=0}^{ \frac{n}{s}-1 } \frac{(-1)^{n-\frac{n}{s}} q^{s\binom{k+1}{2}} (q;q)_n}{(q^s;q^s)_{\frac{n}{s}}} {\left[\begin{matrix} \frac{n}{s}-1 \\ k \end{matrix} \right]_{q^s}} \cdot \left( \sum_{U \in {{\operatorname{Cusp}}}_s} \chi^{U,{{\left( \frac{n}{s}-k, 1^{k} \right)}}}(c^{-1}) \right) \cdot (-[n]_q)^\ell \end{aligned}$$ again via , Proposition \[Singer-cycle-character-values\](iii), and . The parenthesized sum is $(-1)^{n-\frac{n}{s}-k}\mu(s)$ by Proposition \[Singer-cycle-character-values\](iii, iv), so $$\begin{aligned} \sum_{\substack{(s,U):\\s |n, s \geq 2 \\\ U \in {{\operatorname{Cusp}}}_s}} f(s,U) &=(-[n]_q)^\ell (q;q)_n \sum_{\substack{s |n\\ s \geq 2} } \frac{1}{(q^s;q^s)_{\frac{n}{s}}} \left( \sum_{k=0}^{ \frac{n}{s}-1 } (-1)^{k} q^{s\binom{k+1}{2}} {\left[\begin{matrix} \frac{n}{s}-1 \\ k \end{matrix} \right]_{q^s}} \right) \mu(s)\\ &= (-[n]_q)^\ell (q;q)_n \sum_{\substack{s |n\\ s \geq 2} } \frac{\mu(s)}{(q^s;q^s)_{\frac{n}{s}}} (q^s;q^s)_{\frac{n}{s}-1} \\ &= (-[n]_q)^\ell (q;q)_{n-1} \sum_{\substack{s |n\\ s \geq 2} } \mu(s) \\ &= - (-[n]_q)^\ell (q;q)_{n-1},\end{aligned}$$ where the second equality used the $q$-binomial theorem . Thus one has for $q=2$ that $$\label{final-answer-for-q=2} t_q(n,\ell) =\frac{1}{|GL_n|} \left( - (-[n]_q)^\ell (q;q)_{n-1} +\sum_{k=0}^{n-1} q^{\binom{k+1}{2}} {\left[\begin{matrix} n-1 \\ k \end{matrix} \right]_{q}} \cdot (-1)^k \cdot [n]^\ell_q(q^{n-k}-q^{n-k-1}-1)^\ell \right).$$ Since $|GL_n|=(-1)^n q^{\binom{n}{2}} (q;q)_n$, one finds that agrees with the expression $$t_q(n,\ell)=\frac{(-[n]_q)^\ell}{q^{\binom{n}{2}}(q;q)_n} \left( (-1)^{n-1} (q;q)_{n-1}+ \sum_{k=0}^{n-1}(-1)^{k+n} q^{\binom{k+1}{2}} {\left[\begin{matrix} n-1 \\ k \end{matrix} \right]_{q}} (1+q^{n-k-1}-q^{n-k})^\ell \right)$$ after redistributing the $[n]_q^\ell$ and powers of $-1$. This completes the proof of Theorem \[q-factorization-theorem\] for $q=2$. Further remarks and questions {#questions-remarks} ============================= Product formula versus partial fraction expansions -------------------------------------------------- The equivalence between , , and between , are explained as follows. One checks the partial fraction expansion of is $$\sum_{\ell \geq 0} t(n,\ell) x^\ell \,\, = \,\, \dfrac{n^{n-2} x^{n-1}} {\prod_{k=0}^{n-1} \left( 1 - x n\left(\frac{n-1}{2}-k\right) \right) } = \frac{1}{n!} \sum_{k=0}^{n-1} \frac{(-1)^k \binom{n-1}{k}} {1-xn\left(\frac{n-1}{2}-k\right)}$$ and comparing coefficients of $x^\ell$ gives the first equality in . Similarly, one checks that the partial fraction expansion of the right side of is $$\label{q-partial-fraction-expansion} \begin{aligned} & (q^n-1)^{n-1} \cdot \dfrac{ x^n } {\left(1+x[n]_q \right) \prod_{k=0}^{n-1} \left(1+x[n]_q(1+q^k-q^{k+1})\right)} \\ &=\dfrac{(-1)^n}{q^{\binom{n}{2}} (q^n-1)} \left( \frac{1}{1+x[n]_q} + \sum_{k=0}^{n-1} \dfrac{(-1)^{k+1} q^{\binom{k+1}{2}}}{(q;q)_k (q;q)_{n-1-k}} \cdot \dfrac{1}{1+x[n]_q(1+q^{n-k-1}-q^{n-k})} \right). \end{aligned}$$ Comparing coefficients of $x^\ell$ in gives . This proves . More observations about $t_q(n,\ell,m)$ --------------------------------------- From and one can derive $q=1$ limits $$\begin{array}{rll} t_1(n,\ell)&:= \lim_{ q\to 1} \frac{t_q(n,\ell)}{(1-q)^{n-1}} &= (-n)^{\ell-1}\binom{\ell}{n}\\ t_1(n,\ell,m)&:=\lim_{q\to 1} t_q(n,\ell,m)&=n^{\ell-1}\binom{\ell-m-1}{\ell-n}. \end{array}$$ We do not know an interpretation for these limits. Exponential generating function ------------------------------- The classical count $t(n,\ell)$ of factorizations of an $n$-cycle into $\ell$ transpositions has both an elegant ordinary generating function and [*exponential*]{} generating function $$\label{Jackson-exponential-gf} \sum_{\ell \geq 0} t(n,\ell) \frac{u^\ell}{\ell!} =\frac{1}{n!} \left( e^{u\frac{n}{2}} - e^{-u\frac{n}{2}} \right)^{n-1}.$$ This was generalized by Chapuy and Stump [@ChapuyStump] to [*well-generated*]{} finite complex reflection group $W$ as follows; we refer to their paper for the background on such groups. If $W$ acts irreducibly on ${{\mathbb{C}}}^n$, with a total of ${\operatorname{N^{\operatorname{ref}}}}$ reflections and ${\operatorname{N^{\operatorname{hyp}}}}$ reflecting hyperplanes, then for any Coxeter element $c$, the number $a_\ell$ of ordered factorizations $c=t_1 \cdots t_\ell$ into reflections satisfies $$\label{well-generated-Coxeter-element-exp-gf} \begin{aligned} \sum_{\ell \geq 0} a_{\ell} \frac{u^\ell}{\ell!} &=\frac{1}{|W|} \left( e^{u\frac{{\operatorname{N^{\operatorname{ref}}}}}{n}} - e^{-u\frac{{\operatorname{N^{\operatorname{hyp}}}}}{n}} \right)^{n} \\ &=\frac{1}{|W|} e^{-u {\operatorname{N^{\operatorname{hyp}}}}} \left( e^{u\frac{{\operatorname{N^{\operatorname{ref}}}}+{\operatorname{N^{\operatorname{hyp}}}}}{n}} - 1 \right)^{n} \\ &=\frac{1}{|W|} e^{-u {\operatorname{N^{\operatorname{hyp}}}}} \left[ \Delta^n \left( e^{ux\frac{{\operatorname{N^{\operatorname{ref}}}}+{\operatorname{N^{\operatorname{hyp}}}}}{n}} \right) \right]_{x=0} \end{aligned}$$ where the last equality uses the fact that the difference operator $\Delta$ satisfies $\left[ \Delta^n(e^{ax})\right]_{x=0}=(e^{a}-1)^n$. One can derive an exponential generating function analogous to for the number $t_q(n,\ell)$ of Singer cycle factorizations in $W=GL_n({{\mathbb{F}}}_q),$ $$\label{tq-expgf} \sum_{\ell \geq 0} t_q(n,\ell) \frac{u^\ell}{\ell!} =\frac{(q-1)^{n-1} q^{\binom{n}{2}}}{|W|} \,\, e^{-u {\operatorname{N^{\operatorname{hyp}}}}} \left[ \Delta_q^{n-1} \left( \frac{1}{x} \left( e^{ux\frac{{\operatorname{N^{\operatorname{ref}}}}+{\operatorname{N^{\operatorname{hyp}}}}}{q^{n-1}}} - 1 \right) \right) \right]_{x=1},$$ where ${\operatorname{N^{\operatorname{hyp}}}}, {\operatorname{N^{\operatorname{ref}}}}$ denote the number of reflecting hyperplanes and reflections in $W=GL_n({{\mathbb{F}}}_q)$, that is, $$\begin{aligned} {\operatorname{N^{\operatorname{hyp}}}}&=[n]_q,\\ {\operatorname{N^{\operatorname{ref}}}}&=[n]_q(q^n-q^{n-1}-1). \end{aligned}$$ To prove , use to find $$\begin{aligned} \sum_{\ell \geq 0} t_q(n,\ell) \frac{u^\ell}{\ell!} =&\frac{(1-q)^{n-1}}{(q;q)_n} \left[ \Delta_q^{n-1}\biggl( \frac{1}{x}\bigl( e^{-u[n]_q}-e^{-u[n]_q(1+x(1-q))}\bigr)\biggr) \right]_{x=1} \\ =& \frac{(-1)^n (1-q)^{n-1} q^{\binom{n}{2}}}{|W|} e^{-u[n]_q} \left[ \Delta_q^{n-1}\biggl( \frac{1}{x}\bigl( 1-e^{u x [n]_q (q-1))}\bigr)\biggr) \right]_{x=1}. \end{aligned}$$ Noting that $[n]_q={\operatorname{N^{\operatorname{hyp}}}}$, and $[n]_q(q-1)=q^n-1=({\operatorname{N^{\operatorname{hyp}}}}+{\operatorname{N^{\operatorname{ref}}}})/q^{n-1}$, and distributing some negative signs, gives . Hurwitz orbits {#Hurwitz-orbit-conjectures-section} -------------- In a different direction, one can consider the [*Hurwitz action*]{} of the [*braid group on $\ell$ strands*]{} acting on length $\ell$ ordered factorizations $c=t_1 t_2 \cdots t_\ell$. Here the braid group generator $\sigma_i$ acts on ordered factorizations as follows: $$\begin{array}{rccll} (t_1,\ldots,t_{i-1},& t_i,&t_{i+1},& t_{i+2},\ldots,t_\ell) &\overset{\sigma_i}{\longmapsto} \\ (t_1,\ldots,t_{i-1},& t_{i+1},& t_{i+1}^{-1} t_i t_{i+1},& t_{i+2},\ldots,t_\ell). \end{array}$$ For well-generated complex reflection groups $W$ of rank $n$ and taking $\ell=n$, Bessis showed [@Bessis-Kpi1 Prop. 7.5] that the set of all shortest ordered factorizations $(t_1,\ldots,t_n)$ of a Coxeter element $c=t_1 t_2 \cdots t_n$ forms a single transitive orbit for this Hurwitz action. One obvious obstruction to an analogous transitivity assertion for $c$ a Singer cycle in $GL_n({{\mathbb{F}}}_q)$ and factorizations $c=t_1 t_2 \cdots t_\ell$ is that the unordered $\ell$-element multiset $\{\det(t_i)\}_{i=1}^\ell$ of ${{\mathbb{F}}}_q^\times$ is constant on a Hurwitz orbit, but (when $q \neq 2$) can vary between different factorizations, even when $\ell=n$. Nevertheless, we make the following conjecture. \[Hurwitz-orbit-conjecture\] Any two factorizations $c=t_1 t_2 \cdots t_\ell$ with the same multiset $\{\det(t_i)\}_{i=1}^\ell$ lie in the same Hurwitz orbit. In particular, there is only one Hurwitz orbit of factorizations when $q=2$ for any $\ell$. We report here some partial evidence for Conjecture \[Hurwitz-orbit-conjecture\]. - It is true when $n=\ell=2$; here is a proof. Fix a Singer cycle $c$ in $GL_2({{\mathbb{F}}}_q)$ and $\alpha_1, \alpha_2$ in ${{\mathbb{F}}}_q^\times$ having $\det(c)= \alpha_1 \alpha_2$. Theorem \[fixed-det-sequence-theorem\] in the case $\ell = n = 2$ tells us that there will be exactly $[2]_q=q + 1$ factorizations $c = t_1 \cdot t_2$ of $c$ as a product of two reflections with $(\det(t_1),\det(t_2))=(\alpha_1,\alpha_2)$, and similarly $q + 1$ for which $(\det(t_1),\det(t_2))=(\alpha_2,\alpha_1)$. This gives a total of either $q+1$ or $2(q+1)$ factorizations with this multiset of determinants, depending upon whether or not $\alpha_1 = \alpha_2$. Now note that applying the Hurwitz action $\sigma_1$ twice sends $$(t_1, t_2) \overset{\sigma_1}{\longmapsto} (t_2 \,\, , \,\, t_2^{-1} t_1 t_2) \overset{\sigma_1}{\longmapsto} (t_2^{-1} t_1 t_2, \,\, \underbrace{t_2^{-1} t_1^{-1} t_2 t_1 t_2}_{=c^{-1} t_2 c}),$$ yielding a factorization with the same determinant sequence, but whose second factor changes from $t_2$ to $c^{-1} t_2 c$. This moves the reflecting hyperplane (line) $H$ for $t_2$ to the line $c^{-1}H$ for $c^{-1} t_2 c$. Since ${{\mathbb{F}}}_{q^2}^\times=\langle c \rangle$, one knows that the powers of $c$ act transitively on the lines in ${{\mathbb{F}}}_{q^2} \cong {{\mathbb{F}}}_q^2$, and hence there will be at least $q+1$ different second factors $\{ c^{-i} t_2 c^i \}$ achieved in the Hurwitz orbit. This shows that the Hurwitz orbit contains [*at least*]{} $q+1$ or $2(q+1)$ different factorizations, depending upon whether or not $\alpha_1 = \alpha_2$, so it exhausts the factorizations that achieve this multiset of determinants. This completes the proof. - Conjecture \[Hurwitz-orbit-conjecture\] has also been checked - for $q=2$ when $n=\ell \leq 5$ and $n=3,\ell=4$, - for $q=3$ when $n=2$ and $\ell \leq 4$, and also when $n=\ell=3$, - for $q=5$ when $n=2$ and $\ell \leq 3$. One might hope to prove Conjecture \[Hurwitz-orbit-conjecture\] similarly to the uniform proof for transitivity of the Hurwitz action on short reflection factorizations of Coxeter elements in real reflection groups, given in earlier work of Bessis [@Bessis Prop. 1.6.1]. His proof is via induction on the rank, and relies crucially on proving these facts: - The elements $w \leq c$ in the [*absolute order*]{}, that is, the elements which appear as partial products $w=t_1 t_2 \cdots t_i$ in shortest factorizations $c=t_1 t_2 \cdots t_n$, are all themselves [*parabolic Coxeter elements*]{}, that is, Coxeter elements for conjugates of standard parabolic subgroups of $W$. - All such parabolic Coxeter elements share the property that the Hurwitz action is transitive on their shortest factorizations into reflections. One encounters difficulties in trying to prove this analogously, when one examines the interval $[e,c]$ of elements lying below a Singer cycle $c$ in $GL_n({{\mathbb{F}}}_q)$: - It is no longer true that the elements $g$ in $[e,c]$ all have a transitive Hurwitz action on their own short factorizations. For example in $GL_4({{\mathbb{F}}}_2)$, the unipotent element $u$ equal to a single Jordan block of size $4$ appears as a partial product on the way to factoring a Singer cycle, but its $64$ short factorizations $u=t_1 t_2 t_3$ into reflections break up into two Hurwitz orbits, of sizes $16$ and $48$. - It also seems nontrivial to characterize intrinsically the elements in $[e,c]$ for a fixed Singer cycle $c$. For example, the elements $g$ which are [*$c$-noncrossing*]{} in the following sense appear[^6] to be always among them: arranging the elements ${{\mathbb{F}}}_{q^n}^\times=\{1,c,c^2,\ldots,c^{q^n-2}\}$ clockwise circularly, $g$ permutes them (after embedding them via ${{\mathbb{F}}}_{q^n} \cong {{\mathbb{F}}}_q^n$) in cycles that are each oriented clockwise, and these oriented arcs do not cross each other. However, starting already with $GL_4({{\mathbb{F}}}_2)$ and $GL_3({{\mathbb{F}}}_3)$, there are [*other*]{} element below the Singer cycle besides these $c$-noncrossings. $q$-Noncrossings? {#q-noncrossings-section} ----------------- The poset of elements $[e,c]$ lying below a Singer cycle $c$ in the absolute order on $GL_n({{\mathbb{F}}}_q)$ would seem like a reasonable candidate for a $q$-analogue of the usual poset of [*noncrossing partitions*]{} of $\{1,2,\ldots,n\}$; see [@Armstrong]. However, $[e,c]$ does not seem to be so well-behaved in $GL_n({{\mathbb{F}}}_q)$, although a few things were proven about it by Jia Huang in [@Huang]. For instance, he showed that the absolute length of an element $g$ in $GL_n({{\mathbb{F}}}_q)$, that is, the minimum length of a factorization into reflections, coincides with the codimension of the fixed space $({{\mathbb{F}}}_q^n)^g$. Hence the poset $[e,c]$ is ranked in a similar fashion to the noncrossing partitions of real reflection groups, and has an order- and rank-preserving map $$\begin{array}{rcl} [e,c] &\overset{\pi}{\longrightarrow}&L({{\mathbb{F}}}_q^n) \\ g &\longmapsto & ({{\mathbb{F}}}_q^n)^g \end{array}$$ to the lattice $L({{\mathbb{F}}}_q^n)$ of subspaces of ${{\mathbb{F}}}_q^n$. Because conjugation by $c$ acts transitively on lines and hyperplanes, this map is surjective for $n \leq 3$; empirically, it seems to be surjective in general. The poset $[e,c]$ also has a [*Kreweras complementation*]{} anti-automorphism $w \mapsto w^{-1}c$. However, Huang noted that the rank sizes of $[e,c]$ do not seem so suggestive. E.g., for $[e,c]$ in $GL_4({{\mathbb{F}}}_2)$ they are $(1,60,240,60,1)$, and preclude $\pi$ being an $N$-to-one map for some integer $N$, since $L({{\mathbb{F}}}_2^4)$ has rank sizes $(1,15,35,15,1)$ and $35$ does not divide $240$. Are the [*$c$-noncrossing elements*]{} mentioned in Section \[Hurwitz-orbit-conjectures-section\] a better-behaved subposet of $[e,c]$? Regular elliptic elements versus Singer cycles {#regular-elliptic-elements-remark} ---------------------------------------------- Empirical evidence supports the following hypothesis regarding the [*regular elliptic elements*]{} of $GL_n({{\mathbb{F}}}_q)$ that appeared in Proposition \[regular-elliptic-definition-proposition\]. \[reg-ell-conj\] The number of ordered reflection factorizations $g=t_1 t_2 \cdots t_\ell$ is the same for all regular elliptic elements $g$ in $GL_n({{\mathbb{F}}}_q)$, namely the quantity $t_q(n,\ell)$ that appears in Theorem \[q-factorization-theorem\]. Conjecture \[reg-ell-conj\] has been verified for $n=2$ and $n=3$ using explicit character values [@Steinberg]. In the case $\det g\neq 1$, only minor modifications are required in our arguments to prove Conjecture \[reg-ell-conj\]. The spot in our proof that breaks down for regular elliptic elements with $\det g= 1$ is the identity . For example, when $s=n=4$ and $q=2$, if one chooses $\beta$ in ${{\mathbb{F}}}_{2^4}^\times$ with $\beta^5=1$ (so still one has ${{\mathbb{F}}}_{2^4}={{\mathbb{F}}}_2(\beta)$, but ${{\mathbb{F}}}_{2^4}^\times \neq \langle \beta \rangle$), then there are three homomorphisms $\varphi$ with free Frobenius orbits and $ \sum_{\phi} \left( \varphi(\beta) + \varphi(\beta^2) + \varphi(\beta^4) + \varphi(\beta^8) \right)=-3 \quad (\neq 0 = \mu(4)). $ Nevertheless, in this $GL_4({{\mathbb{F}}}_2)$ example it appeared from [GAP]{} [@GAP] computations that such regular elliptic $g$ with $g^5=1$ had the same number of factorizations into $\ell$ reflections for all $\ell$ as did a Singer cycle in $GL_4({{\mathbb{F}}}_2)$. On the other hand, in considering transitivity of Hurwitz actions, we [*did*]{} see a difference in behavior for regular elliptic elements versus Singer cycles: in $GL_4({{\mathbb{F}}}_2)$, there are $3375=(2^4-1)^{4-1}$ short reflection factorizations $t_1 t_2 t_3 t_4$ both for the the Singer cycles (the elements whose characteristic polynomials are $x^4+x^3+1$ or $x^4+x+1$) and for the non-Singer cycle regular elliptic elements (the elements whose characteristic polynomials are $x^4+x^3+x^2+x+1$). However, for the Singer cycles, these factorizations form one Hurwitz orbit, while for the non-Singer cycle regular elliptic elements they form four Hurwitz orbits. The approach of Hausel, Letellier, and Rodriguez-Villegas --------------------------------------------------------- The number of factorizations $g=t_1 t_2 \cdots t_\ell$ where $t_1,\ldots,t_\ell,g$ come from specified $GL_n({{\mathbb{F}}}_q)$ conjugacy classes $C_1,\ldots,C_\ell,C_{\ell+1}$ appears in work of Hausel, Letellier, and Rodriguez-Villegas [@HauselLetellierRodriguez] and more recently Letellier [@Letellier]. They interpret it in terms of the topology of objects called [*character varieties*]{} under certain [*genericity conditions*]{} [@Letellier Definition 3.1] on the conjugacy classes. One can check that these conditions are satisfied in the case of interest to us, that is, when $C_{\ell+1}$ is a conjugacy class of Singer cycles and the $C_1,\ldots,C_\ell$ are all conjugacy classes of reflections. Assuming these genericity conditions, [@Letellier Theorem 4.14] gives an expression for the number of such factorizations in terms of a specialization ${{\mathbb{H}}}_\omega(q^{-\frac{1}{2}},q^{\frac{1}{2}})$ of a rational function ${{\mathbb{H}}}_\omega(z,w)$ defined in [@HauselLetellierRodriguez §1.1] via [*Macdonald symmetric functions*]{}. In principle, this expression should recover Theorem \[fixed-det-sequence-theorem\] as a very special case. However, in practice, the calculation of ${{\mathbb{H}}}_\omega(z,w)$ is sufficiently intricate that we have not verified it. Jucys-Murphy approach? ---------------------- The formulas for character values on semisimple reflections and transvections in Lemma \[normalized-characters-on-semisimple-reflections\](ii) and Corollary \[normalized-characters-on-transvections\] are remarkably simple compared to the machinery used in their proofs. Can they be developed using a $q$-analogue of the Okounkov-Vershik approach [@CST; @VershikOkounkov] to the ordinary character theory of ${\mathfrak{S}}_n$, using the commuting family of [*Jucys-Murphy elements*]{} [@Jucys; @Murphy], a multiplicity-free branching rule, a Gelfand-Zetlin basis, etc.? Such a theory might even allow one to prove $q$-analogues for more general generating function results, such as one finds in Jackson [@Jackson]. A feature of the ${\mathfrak{S}}_n$ theory (see Chapuy and Stump [@ChapuyStump §5], Jucys [@Jucys §4]) is that any symmetric function $f(x_1,\ldots,x_n)$ when evaluated on the Jucys-Murphy elements $J_1,\ldots,J_n$ acts as a scalar in each ${\mathfrak{S}}_n$-irreducible $V^\lambda$, and this scalar is $f(c(a_1),\ldots,c(a_n))$ where $c(a_i)$ are the contents of the cells of $\lambda$. Taking $f=\sum_{i=1}^n x_i$ gives a quick calculation of the irreducible characters evaluated on $\sum_{i=1}^n J_i$, the sum of all transpositions. Lemma \[normalized-characters-on-semisimple-reflections\](ii) and Corollary \[normalized-characters-on-transvections\] seem suggestive of a $q$-analogue for this assertion. It is at least clear how one might define relevant Jucys-Murphy elements. For $1 \leq m < n$ embed $GL_m \subset GL_n$ as the subgroup fixing $e_{m+1},\ldots,e_n$. Then for each $\alpha \in {{\mathbb{F}}}_q^\times$, let $J_m^{\alpha}:=\sum_t t$ be the sum inside the group algebra ${{\mathbb{C}}}GL_n$ over this subset of reflections: $$\label{fine-Jucys-Murphy-summation-set} \{ \text{reflections }t \in GL_{m}: \det(t)=\alpha\text{ and }t \not\in GL_{m-1}\}.$$ The elements $\{ J_m^{\alpha} \}$ for $m=1,2,\cdots,n$ and $\alpha $ in ${{\mathbb{F}}}_q^\times$ pairwise commute. Note that $J_n^{\alpha}$ commutes with any $g$ in $GL_{n-1}$, or equivalently, $gJ_n^{\alpha}g^{-1}=J_n^{\alpha}$, since conjugation by $g$ induces a permutation of the set in . This shows that $J_n^{\alpha}, J_m^{\beta}$ commute when $n \neq m$, since if one assumes $m<n$, then every term of $J_m^\beta$ lies in $GL_{n-1}$. To see that $[J_n^{\alpha}, J_n^{\beta}]=0$, note that our conjugacy sums $z_\alpha=:z_{n,\alpha}$ from Definition \[reflection-conjugacy-class-sum-definition\] lie the center of ${{\mathbb{C}}}GL_n$ and can be expressed as $ z_{n,\alpha}=\sum_{i=1}^n J_i^{\alpha}. $ Therefore $$0=[z_{n,\alpha},J_n^\beta] =\left[ \sum_{i=1}^n J_i^{\alpha}, J_n^{\beta} \right] =[J_n^{\alpha}, J_n^{\beta}] + \left[ \sum_{i<n} J_i^{\alpha}, J_n^{\beta} \right] = [J_n^{\alpha}, J_n^{\beta}]$$ using bilinearity of commutators, and the commutativity of $J_i^{\alpha}, J_n^{\beta}$ for $i <n$. Acknowledgements {#acknowledgements .unnumbered} ================ The authors thank A. Ram and P. Diaconis for pointing them to this work of Hildebrand [@Hildebrand] used in Section \[transvection-character-section\]. They also thank A. Henderson and E. 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Okounkov, A new approach to representation theory of symmetric groups. II. [*J. Math. Sci. (N. Y.)*]{} [**131**]{} (2005), no. 2, 5471–5494. A.V. Zelevinsky, Representations of finite classical groups: a Hopf algebra approach, [*Lecture Notes in Mathematics*]{} [**869**]{}. Springer-Verlag, Berlin-New York, 1981. [^1]: Work partially supported by NSF grants DMS-1148634 and DMS-1001933. [^2]: In checking this equivalence, it is useful to bear in mind that $f^{\lambda^t}(q) = q^{\binom{n}{2}} f^{\lambda}(q^{-1})$, along with the fact that if $\mu \subset \lambda$ with $|\mu|=|\lambda|-1$ and the unique cell of $\lambda/\mu$ lies in row $i$ and column $j$, then $n(\lambda)-n(\mu)=i-1$ and $n(\lambda^t)-n(\mu^t)=j-1$. 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--- abstract: 'After a seminal paper by Shekeey (2016), a connection between maximum $h$-scattered ${{\mathbb F}_{q}}$-subspaces of $V(r,q^n)$ and maximum rank distance (MRD) codes has been established in the extremal cases $h=1$ and $h=r-1$. In this paper, we propose a connection for any $h\in\{1,\ldots,r-1\}$, extending and unifying all the previously known ones. As a consequence, we obtain examples of non-square MRD codes which are not equivalent to generalized Gabidulin or twisted Gabidulin codes. Up to equivalence, we classify MRD codes having the same parameters as the ones in our connection. Also, we determine the weight distribution of codes related to the geometric counterpart of maximum $h$-scattered subspaces.' author: - 'Giovanni Zini and Ferdinando Zullo[^1]' title: Scattered subspaces and related codes --- *Dedicated to the memory of Elisa Montanucci.\ We unite us to her family’s pain.* 51E20, 94B27, 15A04 rank metric code; scattered subspace; linear code; linear set Introduction ============ An ${{\mathbb F}_{q}}$-subspace $U$ of an $r$-dimensional ${\mathbb{F}_{q^n}}$-vector space $V$ is said to be *$h$-scattered* if $U$ spans $V$ over ${\mathbb{F}_{q^n}}$ and, for any $h$-dimensional ${\mathbb{F}_{q^n}}$-subspace $H$ of $V$, $U$ meets $H$ in an ${{\mathbb F}_{q}}$-subspace of dimension at most $h$. This family of subspaces was introduced in [@CsMPZ] as a generalization of $1$-scattered subspaces, which are simply known as scattered subspaces and were originally presented in [@BL2000]. Since then, the theory of scattered subspaces has constantly increased its importance, mainly because of their applications to several algebraic and geometric objects, such as finite semifields, blocking sets, two-intersection sets; see [@Lavrauw; @LVdV2015; @Polverino]. After the seminal paper [@Sheekey] by Sheekey, the interest towards scattered subspaces was also boosted by their connections with the theory of rank metric codes, whose relevance in communication theory relies on its applications to random linear network coding and cryptography. A $h$-scattered ${{\mathbb F}_{q}}$-subspace of highest dimension in $V(r,q^n)$ is called *maximum $h$-scattered*; its dimension is upper bounded by $\frac{rn}{h+1}$. This bound is known to be achieved in the following cases: $h=1$, $h=r-1$, $(h+1)\mid r$ or $h=n-3$; see Section \[sec:h-scatt\]. When $h=1$ or $h=r-1$, maximum $h$-scattered subspaces are strongly related to rank metric codes having the greatest correcting and detecting capabilities for fixed dimension and ambient space, that is, to maximum rank distance (MRD) codes. This has been shown in [@Sheekey; @CSMPZ2016; @PZ] for $h=1$ and in [@Lunardon2017; @ShVdV] for $h=r-1$, while no relation was known for $1<h<r-1$. In this paper we establish a connection between ${{\mathbb F}_{q}}$-subspaces of $V$ and rank metric codes. We start by generalizing the construction of rank-metric codes $\mathcal{C}_U$ provided in [@CSMPZ2016] and defined by an ${{\mathbb F}_{q}}$-subspace $U$ of $V$. We detect those $U$’s such that $\mathcal{C}_U$ is MRD; among these are the maximum $1$- and $(r-1)$-scattered subspaces. Actually, the code $\mathcal{C}_U$ is MRD exactly when $U$ is the dual of a $h$-scattered subspace of dimension $\frac{rn}{h+1}$, for some $1\leq h\leq r-1$. Therefore, our connection extends and unifies the ones in [@Sheekey; @CSMPZ2016; @PZ; @ShVdV; @Lunardon2017]. To this aim, we exhibit two characterizations of $h$-scattered subspaces of dimension $\frac{rn}{h+1}$, which are of independent interest. Moreover we prove that, up to equivalence, the MRD codes of type $\mathcal{C}_U$ are exactly the ${{\mathbb F}_{q}}$-linear MRD codes with parameters $(\frac{rn}{h+1},n,q;n-h)$ and maximum right idealiser. An essential though difficult task is to decide whether or not two rank metric codes with the same parameters are equivalent (especially when they correspond to non-square matrices). A remarkable aspect of the MRD codes that we construct is that we are able to determine one of their idealisers; this allows to prove that some of them are not equivalent to punctured generalized Gabidulin codes nor to punctured generalized twisted Gabidulin codes. The geometric counterparts of $h$-scattered subspaces of dimension $\frac{rn}{h+1}$ are called *$h$-scattered linear sets* of rank $\frac{rn}{h+1}$. They are known to have at most $h+1$ intersection numbers with respect to the hyperplanes, and hence are of interest in coding theory when regarded as projective systems. The intersection numbers w.r.t. the hyperplanes of $h$-scattered linear sets of rank $\frac{rn}{h+1}$ have been determined in [@BL2000] for $h=1$, in [@NZ] for $h=2$, and in [@ShVdV] for $h=r-1$. We determine them for any $1\leq h\leq r-1$, by using the connection between MRD codes and $h$-scattered subspaces of dimension $\frac{rn}{h+1}$ presented in Section \[sec:subMRD\]. As a byproduct, we compute the weight distribution of the arising codes; this answers a question posed by Randrianarisoa [@Ra]. The paper is organized as follows. Section \[sec:pre\] contains preliminary results on $h$-scattered subspaces (Section \[sec:h-scatt\]), dualities of subspaces, both ordinary and Delsarte (Section \[sec:dualities\]), linear codes, equipped with the Hamming distance or with the rank metric (Section \[sec:codes\]). In Section \[sec:subMRD\] we describe the connection between ${{\mathbb F}_{q}}$-subspaces and rank metric codes, characterizing those codes which are MRD, and showing that ${{\mathbb F}_{q}}$-linear MRD $(\frac{rn}{h+1},n,q;n-h)$-codes with maximum right idealiser are exactly the codes of type $\mathcal{C}_U$, up to equivalence. This connection is shown to extend and unify the previously known ones in Section \[sec:uni\]. Section \[sec:twocharact\] completes the connection between $h$-scattered subspaces of dimension $\frac{rn}{h+1}$ and MRD codes, by means of two characterizations which are proved through the ordinary and Delsarte dualities. Section \[sec:noGab\] provides families of MRD codes which are not equivalent to punctured generalized (twisted) Gabidulin codes. Section \[sec:h+1weights\] computes the weight distribution of the linear codes arising from $h$-scattered linear sets of rank $\frac{rn}{h+1}$, seen as projective systems. Finally, in Section \[sec:open\], we resume our results and state some open questions. Preliminaries {#sec:pre} ============= Scattered $\mathbb{F}_q$-subspaces with respect to $\mathbb{F}_{q^n}$-subspaces {#sec:h-scatt} ------------------------------------------------------------------------------- Let $V=V(m,q)$ denote an $m$-dimensional ${{\mathbb F}}_q$-vector space. A $t$-spread of $V$ is a set ${{\mathcal S}}$ of $t$-dimensional ${{\mathbb F}}_q$-subspaces such that each vector of $V^*=V\setminus \{{\bf 0}\}$ is contained in exactly one element of ${{\mathcal S}}$. As shown by Segre in [@Segre], a $t$-spread of $V$ exists if and only if $t$ divides $m$. Let $V$ be an $r$-dimensional ${{\mathbb F}}_{q^n}$-vector space and let ${{\mathcal S}}$ be an $n$-spread of $V$. An ${{\mathbb F}}_q$-subspace $U$ of $V$ is called *scattered* w.r.t. ${{\mathcal S}}$ if $U$ meets every element of ${{\mathcal S}}$ in an ${{\mathbb F}}_q$-subspace of dimension at most one; see [@BL2000]. If we consider $V$ as an $rn$-dimensional ${{\mathbb F}}_q$-vector space, then it is well-known that the one-dimensional ${{\mathbb F}}_{q^n}$-subspaces of $V$, viewed as $n$-dimensional ${{\mathbb F}}_q$-subspaces, form an $n$-spread of $V$. This spread is called the *Desarguesian spread*. In this paper scattered will always mean scattered w.r.t. the Desarguesian spread. Blokhuis and Lavrauw [@BL2000] showed that the dimension of such subspaces is bounded by $rn/2$. After a series of papers it is now known that when $rn$ is even there always exist scattered subspaces of dimension $rn/2$; they are called *maximum scattered* [@BBL2000; @BGMP2015; @BL2000; @CSMPZ2016]. In [@CsMPZ], the authors introduced a special family of scattered subspaces, named $h$-scattered subspaces. Let $V$ be an $r$-dimensional ${{\mathbb F}}_{q^n}$-vector space and $h\leq r-1$ be a positive integer. An ${{\mathbb F}}_q$-subspace $U$ of $V$ is called $h$-*scattered* (or scattered w.r.t. the $h$-dimensional ${\mathbb{F}_{q^n}}$-subspaces) if ${\langle}U {\rangle}_{{{\mathbb F}}_{q^n}}=V$ and each $h$-dimensional ${{\mathbb F}}_{q^n}$-subspace of $V$ meets $U$ in an ${{\mathbb F}}_q$-subspace of dimension at most $h$. The $1$-scattered subspaces are the scattered subspaces generating $V$ over ${{\mathbb F}}_{q^n}$. The same definition applied to $h=r$ describes the $n$-dimensional ${{\mathbb F}}_q$-subspaces of $V$ defining canonical subgeometries of ${\mathrm{PG}}(V,{{\mathbb F}}_{q^n})$. If $h=r-1$ and $\dim_{{{\mathbb F}}_q}(U)=n$, then $U$ is $h$-scattered exactly when $U$ defines a scattered ${{\mathbb F}}_q$-linear set with respect to the hyperplanes, introduced in [@ShVdV Definition 14]; see also [@Lunardon2017]. Theorem \[th:bound\] bounds the dimension of a $h$-scattered subspace. \[th:bound\][[@CsMPZ Theorem 2.3]]{} If $U$ is a $h$-scattered ${{\mathbb F}}_q$-subspace of dimension $k$ in $V=V(r,q^n)$, then one of the following holds: - $k=r$ and $U$ defines a subgeometry ${\mathrm{PG}}(r-1,q)$ of ${\mathrm{PG}}(V,{{\mathbb F}}_{q^n})$; - $k\leq\frac{rn}{h+1}$. A $h$-scattered ${{\mathbb F}}_q$-subspace of highest possible dimension is said to be a [*maximum $h$-scattered*]{} ${{\mathbb F}}_q$-subspace. Theorem \[th:inter\] bounds the dimension of the intersection between a $h$-scattered subspace of dimension $\frac{rn}{h+1}$ and an ${\mathbb{F}_{q^n}}$-subspace of codimension $1$. [[@CsMPZ Theorem 2.8]]{} \[th:inter\] If $U$ is an $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}}_q$-subspace of a vector space $V=V(r,q^n)$, then for any $(r-1)$-dimensional ${{\mathbb F}}_{q^n}$-subspace $H$ of $V$ we have $$\frac{rn}{h+1}-n\leq \dim_{{{\mathbb F}}_q}(U \cap H) \leq \frac{rn}{h+1}-n+h.$$ Constructions of $h$-scattered ${{\mathbb F}}_q$-subspaces have been given in [@CsMPZ] and also in [@NPZZ]. A generalization of $h$-scattered subspaces has been recently introduced in [@BCsMT]. Two dualities for $\mathbb{F}_q$-subspaces {#sec:dualities} ------------------------------------------ In this paper we need both ordinary and Delsarte dualities. ### Ordinary duality {#sec:classicalduality} Let $\sigma \colon V\times V \rightarrow \mathbb{F}_{q^n}$ be a non-degenerate reflexive sesquilinear form over $V=V(r,q^n)$ and define $\sigma' \colon V \times V \rightarrow \mathbb{F}_q, \, (\mathbf{u},\mathbf{v})\mapsto \mathrm{Tr}_{q^n/q}(\sigma(\mathbf{u},\mathbf{v}))$. Once we regard $V$ as an $rn$-dimensional ${{\mathbb F}}_q$-vector space, $\sigma^\prime$ turns out to be a non-degenerate reflexive sesquilinear form over $V=V(rn,q)$. Let $\perp$ and $\perp'$ be the orthogonal complement maps defined by $\sigma$ and $\sigma'$ on the lattices of the ${{\mathbb F}}_{q^n}$-subspaces and the ${{\mathbb F}}_q$-subspaces of $V$, respectively. The following properties hold (see [@Polverino Section 2] for the details). - $\dim_{{{\mathbb F}}_{q^n}}(W)+\dim_{{{\mathbb F}}_{q^n}}(W^\perp)=r$, for every ${{\mathbb F}}_{q^n}$-subspace $W$ of $V$. - $\dim_{{{\mathbb F}}_{q}}(U)+\dim_{{{\mathbb F}}_{q}}(U^{\perp'})=nr$, for every ${{\mathbb F}}_{q}$-subspace $U$ of $V$. - $W^\perp=W^{\perp'}$, for every ${{\mathbb F}}_{q^n}$-subspace $W$ of $V$. - Let $W$ and $U$ be an ${{\mathbb F}}_{q^n}$-subspace and an ${{\mathbb F}}_q$-subspace of $V$ of dimension $s$ and $t$, repsectively. Then $$\label{eq:dualweight} \dim_{{{\mathbb F}}_q}(U^{\perp'}\cap W^{\perp'})-\dim_{{{\mathbb F}}_q}(U\cap W)=rn-\dim_{{{\mathbb F}}_q}(U)-sn.$$ - Let $\sigma$, $\sigma_1$ be non-degenerate reflexive sesquilinear forms over $V$ and define $\sigma^\prime$, $\sigma_1^\prime$, $\perp$, $\perp_1$, $\perp'$ and $\perp_1'$ as above. Then there exists an invertible ${{\mathbb F}}_{q^n}$-linear map $f$ such that $f(U^{\perp'})=U^{\perp_1'}$, i.e. $U^{\perp'}$ and $U^{\perp_1'}$ are $\mathrm{GL}(V)$-equivalent. When $U$ is an ${{\mathbb F}}_q$-subspace of $V$, we denote by $U^{\perp_O}$ one of the ${{\mathbb F}}_q$-subspaces $U^{\perp'}$, where $\perp'$ is defined by the restriction to ${{\mathbb F}}_q$ of any non-degenerate reflexive sesquilinear form over $V$, as defined at the beginning of this section. ### Delsarte duality {#sec:Delsarteduality} Let $U$ be a $k$-dimensional ${{\mathbb F}}_q$-subspace of a vector space $V=V(r,q^n)$, with $k>r$. By [@LuPo2004 Theorems 1, 2] (see also [@LuPoPo2002 Theorem 1]), there is an embedding of $V$ in $\operatorname{\mathbb{V}}=V(k,q^n)$ with $\operatorname{\mathbb{V}}=V \oplus \Gamma$ for some $(k-r)$-dimensional ${{\mathbb F}}_{q^n}$-subspace $\Gamma$ such that $U={\langle}W,\Gamma{\rangle}_{{{\mathbb F}}_{q}}\cap V$, where $W$ is a $k$-dimensional ${{\mathbb F}}_q$-subspace of $\operatorname{\mathbb{V}}$ satisfying $\langle W\rangle_{{{\mathbb F}}_{q^n}}=\operatorname{\mathbb{V}}$ and $W\cap \Gamma=\{{\bf 0}\}$. Then $ \varphi:V\to\operatorname{\mathbb{V}}/\Gamma$, $\mathbf{v}\mapsto \mathbf{v}+\Gamma$, is an ${\mathbb{F}_{q^n}}$-isomorphism such that $\varphi(U)=W+\Gamma$. Following [@CsMPZ Section 3], let $\beta'\colon W\times W\rightarrow{{\mathbb F}}_{q}$ be a non-degenerate reflexive sesquilinear form on $W$. Then $\beta'$ can be extended to a non-degenerate reflexive sesquilinear form $\beta\colon \operatorname{\mathbb{V}}\times\operatorname{\mathbb{V}}\rightarrow{{\mathbb F}}_{q^n}$. Let $\perp$ and $\perp'$ be the orthogonal complement maps defined by $\beta$ and $\beta'$ on the lattices of ${{\mathbb F}}_{q^n}$-subspaces of $\operatorname{\mathbb{V}}$ and of ${{\mathbb F}}_q$-subspaces of $W$, respectively. For an ${{\mathbb F}}_q$-subspace $S$ of $W$ the ${{\mathbb F}}_{q^n}$-subspace ${\langle}S {\rangle}_{{{\mathbb F}}_{q^n}}$ of $\operatorname{\mathbb{V}}$ will be denoted by $S^*$. In this case, $(S^*)^{\perp}=(S^{\perp'})^*$. \[deffff\] Let $U$ be a $k$-dimensional ${{\mathbb F}}_q$-subspace of $V=V(r,q^n)$ such that $k>r$ and $\dim_{{{\mathbb F}}_q}(M\cap U)<k-1$ for every $(r-1)$-dimensional ${\mathbb{F}_{q^n}}$-subspace $M$ of $V$. Then the $k$-dimensional ${{\mathbb F}}_q$-subspace $W+\Gamma^{\perp}$ of the quotient space $\operatorname{\mathbb{V}}/\Gamma^{\perp}$ will be denoted by $ U^{\perp_{D}}$ and will be called the *Delsarte dual* of $U$ (w.r.t. $\perp$). The Delsarte duality preserves the property of being scattered w.r.t. ${\mathbb{F}_{q^n}}$-subspaces, in the following sense. [@CsMPZ Theorem 3.3] \[thm:dual\] Let $U$ be a $k$-dimensional $h$-scattered ${{\mathbb F}}_q$-subspace of a vector space $V=V(r,q^n)$ with $n\geq h+3$. Then $U^{\perp_{D}}$ is an $\frac{rn}{h+1}$-dimensional $(n-h-2)$-scattered ${{\mathbb F}_{q}}$-subspace of $\operatorname{\mathbb{V}}/\Gamma^\perp=V(k-r,q^n)$. Proposition \[prop:property\] points out some properties of the Delsarte duality. \[prop:property\] Let $U$, $W$, $V$, $\Gamma$, $\operatorname{\mathbb{V}}$, $\perp$ and $\perp_D$ be defined as above. The following properties hold: - $(U^{\perp_D})^{\perp_D}=W+\Gamma=\varphi^{-1}(U)$; - under the assumption $n\geq h+3$, $U$ is an $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}_{q}}$-subspace of $V$ if and only if $U^{\perp_D}$ is an $\frac{rn}{h+1}$-dimensional $(n-h-2)$-scattered ${{\mathbb F}_{q}}$-subspace of $\operatorname{\mathbb{V}}/\Gamma^{\perp}$. The first property easily follows from the definition of Delsarte duality. Together with Theorem \[thm:dual\] applied to $U^{\perp_D}$, this yields the second property. Generalities on codes {#sec:codes} --------------------- In this section we recall some properties of codes that will be used in the paper. In Section \[sec:Hamming\] we consider ${{\mathbb F}_{q}}$-linear codes with respect to the Hamming metric in ${{\mathbb F}}_{q}^N$, while in Section \[sec:rank\] we consider ${{\mathbb F}_{q}}$-linear codes with respect to the rank metric in ${{\mathbb F}}_{q}^{m\times n}$. ### Projective systems and linear codes {#sec:Hamming} Let $\mathcal{C}\subseteq{{\mathbb F}}_q^N$ be an ${{\mathbb F}_{q}}$-linear code of length $N$, dimension $k$ and minimum distance $d$ over the alphabet ${{\mathbb F}_{q}}$; we denote by $[N,k,d]_q$ the parameters of $\mathcal{C}$. A generator matrix of $\mathcal{C}$ is a matrix $G\in{{\mathbb F}}_q^{k\times N}$ whose rows form a basis of $\mathcal{C}$. The weight of a codeword $\mathbf{c}\in\mathcal{C}$ is the number of nonzero components of $\mathbf{c}$, and $A_i^H$ will denote the number of codewords of weight $i$ in $\mathcal{C}$. The $N$-tuple $(A_0^H=1,A_1^H,\ldots,A_N^H)$ is called the weight distribution of $\mathcal{C}$, and the polynomial $\sum_{i=0}^{N}A_i^H z^i$ is the weight enumerator of $\mathcal{C}$. A projective $[N,k,d]_q$-system is a point subset $\mathcal{P}$ of $\Omega={\mathrm{PG}}(k-1,q)$ of size $N$, not contained in any hyperplane of $\Omega$, such that $$d= N - \max\{|\mathcal{P}\cap\mathcal{H}|\colon \mathcal{H}\,\mbox{ is a hyperlane of }\,\Omega\}.$$ The matrix $G\in{{\mathbb F}}_q^{k\times N}$ whose columns are the coordinates of the points of a projective $[N,k,d]_q$-system $\mathcal{P}$ is the generator matrix of a linear code with parameters $[N,k,d]_q$. Different choices of the coordinates yield linear codes which are equivalent by means of a diagonal matrix; we denote one of them by $\mathcal{C}_{\mathcal{P}}$. \[prop:projsyst\] Let $\mathcal{P}$ be a projective $[N,k,d]_q$-system of $\Omega$ and $\mathcal{C}_{\mathcal{P}}$ be a corresponding linear $[N,k,d]_q$-code. Then the weights of $\mathcal{C}_{\mathcal{P}}$ are the values $N-i$, where $i=|\mathcal{P}\cap\mathcal{H}|$ and $\mathcal{H}$ runs over the hyperplanes of $\Omega$. The number $A_i^H$ of codewords of $\mathcal{C}_{\mathcal{P}}$ with weight $i$ is equal to the number of hyperplanes $\mathcal{H}$ of $\Omega$ such that $|\mathcal{P}\cap\mathcal{H}|=i$. ### Rank metric codes {#sec:rank} Rank metric codes were introduced by Delsarte [@Delsarte] in 1978 and they have been intensively investigated in recent years because of their applications; we refer to [@sheekey_newest_preprint] for a survey on this topic. The set ${{\mathbb F}_{q}}^{m\times n}$ of $m \times n$ matrices over ${{\mathbb F}_{q}}$ may be endowed with a metric, called *rank metric*, defined by $$d(A,B) = \mathrm{rk}\,(A-B).$$ A subset $\operatorname{\mathcal{C}}\subseteq {{\mathbb F}_{q}}^{m\times n}$ equipped with the rank metric is called a *rank metric code* (shortly, an *RM code*). The minimum distance of $\operatorname{\mathcal{C}}$ is defined as $$d = \min\{ d(A,B) \colon A,B \in \operatorname{\mathcal{C}},\,\, A\neq B \}.$$ Denote the parameters of an RM code $\operatorname{\mathcal{C}}\subseteq{{\mathbb F}_{q}}^{m\times n}$ with minimum distance $d$ by $(m,n,q;d)$. We are interested in ${{\mathbb F}_{q}}$-*linear* RM codes, i.e. ${{\mathbb F}_{q}}$-subspaces of ${{\mathbb F}_{q}}^{m\times n}$. Delsarte showed in [@Delsarte] that the parameters of these codes must obey a Singleton-like bound. \[th:Singleton\] If $\operatorname{\mathcal{C}}$ is an RM code of ${{\mathbb F}}_q^{m\times n}$ with minimum distance $d$, then $$|\operatorname{\mathcal{C}}| \leq q^{\max\{m,n\}(\min\{m,n\}-d+1)}.$$ When equality holds, we call $\operatorname{\mathcal{C}}$ a *maximum rank distance* (*MRD* for short) code. Examples of MRD codes are resumed in [@PZ; @sheekey_newest_preprint], see also the paper [@SheekeyLondon]. For an RM code ${{\mathcal C}}\subseteq {{\mathbb F}}_{q}^{m \times n}$, the *adjoint code* of $\operatorname{\mathcal{C}}$ is $$\operatorname{\mathcal{C}}^\top =\{C^t \colon C \in \operatorname{\mathcal{C}}\},$$ where $C^t$ is the transpose matrix of $C$. Define the symmetric bilinear form $\langle\cdot,\cdot\rangle$ on ${{\mathbb F}}_q^{m \times n}$ by $$\langle M,N \rangle= \mathrm{Tr}(MN^t).$$ The *Delsarte dual code* of an ${{\mathbb F}}_q$-linear RM code $\operatorname{\mathcal{C}}\subseteq {{\mathbb F}}_{q}^{m \times n}$ is $$\operatorname{\mathcal{C}}^\perp = \{ N \in {{\mathbb F}}_q^{m\times n} \colon \langle M,N \rangle=0 \; \text{for each} \; M \in \operatorname{\mathcal{C}}\}.$$ \[rk:dualMRD\] If $\mathcal{C}\subseteq {{\mathbb F}}_{q}^{m \times n}$ is an MRD code with minimum distance $d$, then $\operatorname{\mathcal{C}}^\top$ and $\mathcal{C^\perp}$ are MRD codes with minimum distances $d$ and $\min\{m,n\}-d+2$, respectively; see [@Delsarte; @Ravagnani]. Given an RM code $\mathcal{C}$ in $\mathbb{F}_{q}^{m\times n}$ and an integer $i \in \mathbb{N}$, define $A_i=|\{M \in \mathcal{C} \colon \mathrm{rk}(M)=i\}|$. The *rank distribution* of $\mathcal{C}$ is the vector $(A_i)_{i \in \mathbb{N}}$. MacWilliams identities for RM codes are stated in Theorem \[th:MacWilliams\] and were first obtained by Delsarte in [@Delsarte] using the machinery of association schemes; see also [@Ravagnani] for a different approach. Recall that the $q$-binomial coefficient of two integers $s$ and $t$ is $${s \brack t}_q=\left\{ \begin{array}{lll} 0 & \text{if}\,\, s<0,\,\,\text{or}\,\,t<0,\,\, \text{or}\,\, t>s,\\ 1 & \text{if}\,\, t=0\,\, \text{and}\,\, s\geq 0,\\ \displaystyle\prod_{i=1}^t \frac{q^{s-i+1}-1}{q^i-1} & \text{otherwise}. \end{array} \right.$$ ([@Delsarte Theorem 3.3],[@Ravagnani Theorem 31])\[th:MacWilliams\] Let $\mathcal{C}$ be an RM code in $\mathbb{F}_q^{m\times n}$. Let $(A_i)_{i\in \mathbb{N}}$ and $(B_j)_{j\in \mathbb{N}}$ be the rank distribution of $\mathcal{C}$ and $\mathcal{C}^\perp$, respectively. For any integer $\nu \in \{ 0,\ldots,m \}$ we have $$\sum_{i=0}^{m-\nu} A_i {m-i \brack \nu}_q = \frac{|\mathcal{C}|}{q^{n\nu}} \sum_{j=0}^\nu B_j {m-j \brack \nu -j}_q.$$ As a consequence, Delsarte in [@Delsarte] and later Gabidulin in [@Gabidulin] determined precisely the weight distribution of MRD codes. \[th:weightdistribution\] Let $\mathcal{C}$ be an MRD code in $\mathbb{F}_q^{m\times n}$ with minimum distance $d$. Let $m'=\min\{m,n\}$ and $n'=\max\{m,n\}$. Then $$A_{d+\ell}={m'\brack d+\ell}_q \sum_{t=0}^\ell (-1)^{t-\ell}{\ell+d \brack \ell-t}_q q^{\binom{\ell-t}{2}}(q^{n'(t+1)}-1)$$ for any $\ell \in \{0,1,\ldots,n'-d\}$. In particular, Lemma \[lemma:weight\] holds. ([@LTZ2 Lemma 2.1],\[lemma:weight\][@Ravagnani Lemma 52])\[lemma:complete weight\] Let $\mathcal{C}$ be an MRD code in $\mathbb{F}_q^{m\times n}$ with minimum distance $d$. Let $m'=\min\{m,n\}$ and $n'=\max\{m,n\}$. Assume that the null matrix $O$ is in $\mathcal{C}$. Then, for any $0 \leq \ell \leq m'-d$, we have $A_{d+\ell}>0$, i.e. there exists at least one matrix $C \in \mathcal{C}$ such that $\mathrm{rk} (C) = d + \ell$. Theorem \[th:dualrelations\] follows from the MacWilliam identities. \[th:dualrelations\]([@Ravagnani Proof of Corollary 44]) Let $\mathcal{C}$ be an MRD code in $\mathbb{F}_q^{m\times n}$ with minimum distance $d$. Let $m'=\min\{m,n\}$ and $n'=\max\{m,n\}$. Then for any $\nu \in \{0,\ldots,m'-d \}$ we have $$\label{eq:identities} {m' \brack \nu}_q+\sum_{i=d}^{m'-\nu} A_i {m'-i \brack \nu}_q=\frac{|\mathcal{C}|}{q^{n'\nu}} {m' \brack \nu}_q.$$ By Remark \[rk:dualMRD\], the minimum distance of $\mathcal{C}^\perp$ is $m'-d+2$. Thus, Theorem \[th:MacWilliams\] proves the claim. Two RM codes $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}'$ in $\mathbb{F}_q^{m\times n}$ are *equivalent* if and only if there exist $X \in \mathrm{GL}(m,q)$, $Y \in \mathrm{GL}(n,q)$, $Z \in {{\mathbb F}}_q^{m\times n}$ and a field automorphism $\sigma$ of ${{\mathbb F}}_q$ such that $$\operatorname{\mathcal{C}}'=\{XC^\sigma Y + Z \colon C \in \operatorname{\mathcal{C}}\}.$$ The *left* and *right idealisers* $L(\operatorname{\mathcal{C}})$ and $R(\operatorname{\mathcal{C}})$ of an RM code $\mathcal{C}\subseteq{{\mathbb F}}_{q}^{m\times n}$ are defined as $$L(\operatorname{\mathcal{C}})=\{ Y \in {{\mathbb F}}_q^{m \times m} \colon YC\in \operatorname{\mathcal{C}}\hspace{0.1cm} \text{for all}\hspace{0.1cm} C \in \operatorname{\mathcal{C}}\},$$ $$R(\operatorname{\mathcal{C}})=\{ Z \in {{\mathbb F}}_q^{n \times n} \colon CZ\in \operatorname{\mathcal{C}}\hspace{0.1cm} \text{for all}\hspace{0.1cm} C \in \operatorname{\mathcal{C}}\}.$$ The notion of idealisers have been introduced by Liebhold and Nebe in [@LN2016 Definition 3.1]; they are invariant under equivalences of rank metric codes. Further invariants have been introduced in [@GiuZ; @NPH2]. In [@LTZ2], idealisers have been studied in details and the following result has been proved. \[th:propertiesideal\] Let $\mathcal{C}$ and $\mathcal{C}^\prime$ be ${{\mathbb F}_{q}}$-linear RM codes of ${{\mathbb F}_{q}}^{m\times n}$. - If $\mathcal{C}$ and $\mathcal{C}^\prime$ are equivalent, then their left and right idealisers are isomorphic as ${{\mathbb F}_{q}}$-algebras ([@LTZ2 Proposition 4.1]). - $L(\operatorname{\mathcal{C}}^\top)=R(\operatorname{\mathcal{C}})^\top$ and $R(\operatorname{\mathcal{C}}^\top)=L(\operatorname{\mathcal{C}})^\top$ ([@LTZ2 Proposition 4.2]). - Let $\mathcal{C}$ have minimum distance $d>1$. If $m \leq n$, then $L(\operatorname{\mathcal{C}})$ is a finite field with $|L(\operatorname{\mathcal{C}})|\leq q^m$. If $m \geq n$, then $R(\operatorname{\mathcal{C}})$ is a finite field with $|R(\operatorname{\mathcal{C}})|\leq q^n$. In particular, when $m=n$, $L(\operatorname{\mathcal{C}})$ and $R(\operatorname{\mathcal{C}})$ are both finite fields ([@LTZ2 Theorem 5.4 and Corollary 5.6]). Let $\mathcal{C}$ be an RM code in ${{\mathbb F}_{q}}^{n\times n}$, and $A\in{{\mathbb F}_{q}}^{m\times n}$ be a matrix of rank $m\leq n$. The RM code $A\mathcal{C}=\{AM\colon M\in\mathcal{C}\}\subseteq{{\mathbb F}_{q}}^{m\times n}$ is a *punctured code* obtained by *puncturing $\mathcal{C}$ with $A$*. \[th:punct\]([@BR Corollary 35], [@CsS Theorem 3.2]) Let $\operatorname{\mathcal{C}}$ be an MRD code with parameters $(n,n,q;d)$, $A \in {{\mathbb F}}_q^{m\times n}$ be a matrix of rank $m$, and $n-d\leq m\leq n$. Then the punctured code $A\operatorname{\mathcal{C}}$ is an MRD code with parameters $(m,n,q;d+m-n)$ and $(A\operatorname{\mathcal{C}})^\top$ is an MRD code with parameters $(n,m,q;d+m-n)$. In the literature equivalent representations of RM codes are used, other than the matrix representation that has been described above, and some of them will be used in this paper. In particular, we see the elements of an ${{\mathbb F}_{q}}$-linear RM code $\mathcal{C}$ with parameters $(m,n,q;d)$ as: - matrices of ${{\mathbb F}}_q^{m\times n}$ having rank at least $d$; - ${{\mathbb F}_{q}}$-linear maps $V\to W$ where $V=V(n,q)$ and $W=V(m,q)$, having usual map rank at least $d$; - when $m=n$, elements of the ${{\mathbb F}_{q}}$-algebra $\mathcal{L}_{n,q}$ of $q$-polynomials over ${\mathbb{F}_{q^n}}$ modulo $x^{q^n}-x$, having rank at least $d$ as an ${{\mathbb F}_{q}}$-linear map ${\mathbb{F}_{q^n}}\to{\mathbb{F}_{q^n}}$. Connection between ${{\mathbb F}_{q}}$-vector spaces and rank metric codes {#sec:subMRD} ========================================================================== In this section, an ${{\mathbb F}_{q}}$-linear RM code with parameters $(m,n,q;d)$ is regarded as a set of ${{\mathbb F}_{q}}$-linear maps $W_1=V(n,q)\to W_2=V(m,q)$. The following notation will be used. - $\omega_{\alpha}:{\mathbb{F}_{q^n}}\to{\mathbb{F}_{q^n}}$, $x\mapsto\alpha x$, for any $\alpha\in{\mathbb{F}_{q^n}}$. - $\mathcal{F}_n=\{\omega_{\alpha}\colon\alpha\in{\mathbb{F}_{q^n}}\}$, which is a field isomorphic to ${\mathbb{F}_{q^n}}$. - $\mathcal{F}_{n,q}=\{\omega_{\alpha}\colon \alpha\in{{\mathbb F}_{q}}\}$, which is a subfield of $\mathcal{F}_n$ isomorphic to ${{\mathbb F}_{q}}$. - $\tau_{\mathbf{v}}:{\mathbb{F}_{q^n}}\to W_1$, $\lambda\mapsto \lambda \mathbf{v}$, for any $\mathbf{v}\in W_1$. We define a family of ${{\mathbb F}_{q}}$-linear RM codes associated with an ${{\mathbb F}_{q}}$-vector space $U$. Let $n,r,k$ be positive integers with $k<rn$, $U$ be a $k$-dimensional ${{\mathbb F}_{q}}$-subspace of an $r$-dimensional ${\mathbb{F}_{q^n}}$-vector space $V$, $W$ be an $(rn-k)$-dimensional ${{\mathbb F}_{q}}$-vector space, and $G:V\to W$ be an ${{\mathbb F}_{q}}$-linear map with kernel $U$. For any $\mathbf{v} \in V$ define the ${{\mathbb F}_{q}}$-linear map $\Gamma_{\mathbf{v}}=G\circ\tau_{\mathbf{v}}$. \[th:construction\] Let $V=V(r,q^n)$ and $W=V(rn-k,q)$. Let $U=V(k,q)$ be an ${{\mathbb F}_{q}}$-subspace of $V$, and $G:V\to W$ be an ${{\mathbb F}_{q}}$-linear map with $\ker(G)=U$. Define $$\iota=\max\{\dim_{{{\mathbb F}_{q}}}(U\cap\langle\mathbf{v}\rangle_{{\mathbb{F}_{q^n}}}) \colon \mathbf{v}\in V^* \}.$$ If $\iota<n$, then the pair $(U,G)$ defines an ${{\mathbb F}_{q}}$-linear RM code $$\label{eq:rd} \mathcal{C}_{U,G}=\left\{ \Gamma_{\mathbf{v}}=G\circ \tau_{\mathbf{v}} \colon \mathbf{v} \in V \right\}$$ of dimension $rn$ with parameters $(rn-k,n,q;n-\iota)$, whose right idealiser contains $\mathcal{F}_n$. For any $\mathbf{v},\mathbf{w}\in V$ and $\alpha\in{{\mathbb F}_{q}}$ we have $\Gamma_{\mathbf{v}}+\Gamma_{\mathbf{w}}=\Gamma_{\mathbf{v}+\mathbf{w}}$ and $\alpha\,\Gamma_{\mathbf{v}}=\Gamma_{\alpha\mathbf{v}}$, and hence $\mathcal{C}_{U,G}$ is an ${{\mathbb F}_{q}}$-vector space. For any $\mathbf{v}\in V$, let $R_{\mathbf{v}}=\{ \lambda \in {\mathbb{F}_{q^n}}\colon \lambda \mathbf{v} \in U \}$. Clearly $\ker (\Gamma_{\mathbf{v}})=R_{\mathbf{v}}$ and, when $\mathbf{v}\ne\mathbf{0}$, $\dim_{{{\mathbb F}_{q}}}(R_{\mathbf{v}})=\dim_{{{\mathbb F}_{q}}}(U\cap\langle \mathbf{v}\rangle_{{\mathbb{F}_{q^n}}})$. Then $\dim_{{{\mathbb F}_{q}}}(\ker(\Gamma_{\mathbf{v}}))\leq\iota$ and there exists $\mathbf{u}\in V^*$ such that $\dim_{{{\mathbb F}_{q}}}(\ker(\Gamma_{\mathbf{u}}))=\iota$, so that the minimum distance of $\mathcal{C}_{U,G}$ is $n-\iota$. For any $\mathbf{v},\mathbf{w}\in V$, we have $\Gamma_{\mathbf{v}}=\Gamma_{\mathbf{w}}$ if and only if $\mathbf{v}=\mathbf{w}$. In fact, if $\Gamma_{\mathbf{v}}=\Gamma_{\mathbf{w}}$, then $G(\lambda(\mathbf{v}-\mathbf{w}))=\mathbf{0}$ for every $\lambda\in{\mathbb{F}_{q^n}}$, whence $\dim_{{{\mathbb F}_{q}}}(U\cap\langle\mathbf{v}-\mathbf{w}\rangle_{{\mathbb{F}_{q^n}}})=n>\iota$ and hence $\mathbf{v}=\mathbf{w}$. Therefore, $\dim_{{{\mathbb F}_{q}}}(\mathcal{C}_{U,G})=rn$. Finally, for any $\alpha\in{\mathbb{F}_{q^n}}$ and $\mathbf{v}\in V$ we have $\Gamma_{\mathbf{v}}\circ \omega_{\alpha}=\Gamma_{\alpha\mathbf{v}}$. Then $R(\mathcal{C}_{U,G})$ contains $\mathcal{F}_n$. We now characterize the codes $\mathcal{C}_{U,G}$ which are MRD. \[th:MRDiff\] Let $V=V(r,q^n)$ and $W=V(rn-k,q)$. Let $U=V(k,q)$ be an ${{\mathbb F}_{q}}$-subspace of $V$, $G:V\to W$ be an ${{\mathbb F}_{q}}$-linear map with $\ker(G)=U$, $\iota=\max\{\dim_{{{\mathbb F}_{q}}}(U\cap\langle\mathbf{v}\rangle_{{\mathbb{F}_{q^n}}}) \colon \mathbf{v}\in V^* \}$ with $\iota<n$, and $\mathcal{C}_{U,G}=\left\{ \Gamma_{\mathbf{v}} \colon \mathbf{v} \in V \right\}$. Then $\mathcal{C}_{U,G}$ is an ${{\mathbb F}_{q}}$-linear MRD code if and only if $$(\iota+1)\mid rn\quad\textrm{and}\quad k=\frac{\iota rn}{\iota+1}\leq(r-1)n.$$ In this case, - the parameters of $\mathcal{C}_{U,G}$ are $\left(\,\frac{rn}{\iota+1}\,,\,n\,,\,q\,;\,n-\iota\,\right)$; - the right idealiser of $\mathcal{C}_{U,G}$ is $\mathcal{F}_n$; - the weight distribution of $\mathcal{C}_{U,G}$ is $$A_{n-s}={n \brack s}_q \sum_{j=0}^{\iota-s} (-1)^{j}{n-s \brack j}_q q^{\binom{j}{2}}\left(q^\frac{rn(\iota-s-j+1)}{\iota+1}-1\right),$$ for $s \in \{0,1,\ldots,\iota\}$. If $k>(r-1)n$, then for every $\mathbf{v}\in V^*$ we have $\dim_{{{\mathbb F}_{q}}}(U\cap\langle\mathbf{v}\rangle_{{\mathbb{F}_{q^n}}})\geq1$ and hence $\dim_{{{\mathbb F}_{q}}}(\ker(\Gamma_{\mathbf{v}}))=\dim_{{{\mathbb F}_{q}}}(U\cap\langle\mathbf{v}\rangle_{{\mathbb{F}_{q^n}}})\geq1$, so that $\mathcal{C}_{U,G}$ has no elements of rank $n$. Thus, by Lemma \[lemma:weight\], $\mathcal{C}_{U,G}$ is not an MRD code. Suppose $k\leq (r-1)n$. Then $rn-k\geq n$ and the Singleton-like bound of Theorem \[th:Singleton\] reads $$rn\leq(rn-k)(n-(n-\iota)+1).$$ Therefore, $\mathcal{C}_{U,G}$ is an MRD code if and only if $\iota+1$ divides $rn$ and $k=\frac{\iota rn}{\iota+1}$. In this case, the parameters of $\mathcal{C}_{U,G}$ are provided by Theorem \[th:construction\] and the weight distribution of $\mathcal{C}_{U,G}$ follows from Theorem \[th:weightdistribution\]. Also, the right idealiser of $\mathcal{C}_{U,G}$ contains $\mathcal{F}_n$ by Theorem \[th:construction\], and hence is equal to $\mathcal{F}_n$ by Theorem \[th:propertiesideal\]. Different choices of the map $G$ yield equivalent codes, i.e. $\mathcal{C}_{U,G}$ is uniquely determined by $U$, up to equivalence. Let $V=V(r,q^n)$ and $W=V(rn-k,q)$. Let $U=V(k,q)$ be an ${{\mathbb F}_{q}}$-subspace of $V$, $G:V\to W$ and $\overline{G}:V\to W$ be two ${{\mathbb F}_{q}}$-linear maps with $\ker(G)=\ker(\overline{G})=U$. Then the codes $\mathcal{C}_{U,G}$ and $\mathcal{C}_{U,\overline{G}}$ are equivalent. Let $B_U \cup\{\mathbf{w}_1,\ldots,\mathbf{w}_{rn-k}\}$ be an ${{\mathbb F}_{q}}$-basis of $V$ such that $B_U$ is an ${{\mathbb F}_{q}}$-basis of $U$. Clearly, $G(\mathbf{w}_1),\ldots,G(\mathbf{w}_{rn-k})$ are ${{\mathbb F}_{q}}$-linearly independent, as well as $\overline{G}(\mathbf{w}_1),\ldots,\overline{G}(\mathbf{w}_{rn-k})$. Then there exists an invertible ${{\mathbb F}_{q}}$-linear map $L:V\to V$ such that $L(U)=U$ and $L(G({{\mathbf w}}_i))=\overline{G}({{\mathbf w}}_i)$ for every $i=1,\ldots,rn-k$, i.e. $L\circ G=\overline{G}$. Therefore, by choosing $R={\rm Id}_{{\mathbb{F}_{q^n}}}$ and $\sigma={\rm Id}_{{\rm Aut}({{\mathbb F}_{q}})}$, we have $$L\circ\mathcal{C}_{U,G}^{\sigma}\circ R = \left\{L\circ(G\circ\tau_{\mathbf{v}})\colon \mathbf{v} \in V\right\} =\mathcal{C}_{U,\overline{G}}.$$ The claim is proved. We recall the following conjugacy property of Singer cycles of ${\rm GL}(n,q)$. \[rem:singer\] The cyclic subgroups of $\mathrm{GL}(n,q)$ of order $q^n-1$ are called Singer cycles; it is well-known that any two Singer cycles $S_1=\langle g_1\rangle$ and $S_2=\langle g_2\rangle$ are conjugate in $\mathrm{GL}(n,q)$. In fact, let $p_{g_1}(x)$ be the minimal polynomial of $g_1$ over $\mathbb{F}_q$, and $\gamma$ be a primitive element of ${\mathbb{F}_{q^n}}$ with minimal polynomial $p_{g_1}(x)$ over ${{\mathbb F}_{q}}$. The set $\overline{S}_1=S_1\cup\{\mathbf{0}\}$ is an ${{\mathbb F}_{q}}$-subalgebra of ${{\mathbb F}_{q}}^{n\times n}$, isomorphic to ${\mathbb{F}_{q^n}}$ by the ${{\mathbb F}_{q}}$-linear map $\varphi$ mapping $(1,g_1,\ldots,g_1^{n-1})$ to $(1,\gamma,\ldots,\gamma^{n-1})$. Also, $\overline{S}_1$ is a field of order $q^n$ and $\varphi$ is a field ${{\mathbb F}_{q}}$-isomorphism. The same holds for $\overline{S}_2=S_2\cup\{\mathbf{0}\}$, so that there exists a field ${{\mathbb F}_{q}}$-isomorphism $\psi:\overline{S}_1\to\overline{S}_2$. Therefore, there exists $\hat{\psi}\in\mathrm{GL}(n,q)$ which conjugates $S_1$ to $S_2$. See also [@Huppert pag. 187] and [@Hiss Section 1.2.5 and Example 1.12]. Also the converse of Theorem \[th:MRDiff\] holds, in the sense that any MRD code as in the claim of that theorem is equivalent to $\mathcal{C}_{U,G}$ for some $U$ as in the assumption of Theorem \[th:MRDiff\]. \[th:MRDconverse\] Let $\mathcal{C}$ be an ${{\mathbb F}_{q}}$-linear MRD code with parameters $(t,n,q;n-\iota)$ such that $t\geq n$ and $|R(\mathcal{C})|=q^n$, contained in ${\rm Hom}({\mathbb{F}_{q^n}},W)$ with $W=V(t,q)$. Let $r=\dim_{R(\mathcal{C})}(\mathcal{C})$. Then the following holds. - $\iota+1$ divides $rn$ and $t=\frac{rn}{\iota+1}$. - $\mathcal{C}$ is equivalent to an ${{\mathbb F}_{q}}$-linear MRD code $\mathcal{C}^\prime$ such that $R(\mathcal{C}^\prime)=\mathcal{F}_n$. - The set $$U=\{f\in\mathcal{C}^\prime\colon f(1)=0\}\subseteq\mathcal{C}^\prime$$ is a $\frac{\iota rn}{\iota+1}$-dimensional $\mathcal{F}_{n,q}$-subspace of $\mathcal{C}^\prime$, and satisfies [^2] $$\label{eq:iota} \max\left\{\dim_{\mathcal{F}_{n,q}}\left(U\cap\langle f\rangle_{\mathcal{F}_n}\right)\colon f\in\mathcal{C}^\prime\right\}=\iota.$$ - $\mathcal{C}^\prime$ is equal to $\mathcal{C}_{U,G}$, where $G:\mathcal{C}^\prime\to W$, $f\mapsto f(1)$. Since $|\mathcal{C}|=q^{rn}$ and $t\geq n$, the Singleton-like bound of Theorem \[th:Singleton\] reads $rn\leq t(n-(n-\iota)+1)$. As $\mathcal{C}$ is MRD, this implies that $\iota+1$ divides $t$, and $t=\frac{rn}{\iota+1}$. Since $R(\mathcal{C})\setminus\{\mathbf{0}\}$ and $\mathcal{F}_n\setminus\{\omega_0\}$ are Singer cycles of $\mathrm{GL}(n,q)$, there exists by Remark \[rem:singer\] an invertible $\mathbb{F}_q$-linear map $H\colon \mathbb{F}_{q^n}\rightarrow\mathbb{F}_{q^n}$ such that $R(\mathcal{C})= H\circ\mathcal{F}_n\circ H^{-1}$. Thus, $\mathcal{C}^\prime = \mathcal{C}\circ H$. Clearly, $U$ is an $\mathcal{F}_{n,q}$-subspace of $\mathcal{C}^\prime$. For every $i\in\{1,\ldots,\iota\}$, we determine the size of $U_i=\{ f \in U \colon \dim_{{{\mathbb F}}_q}(\ker f)=i \}$. Let $g \in \mathcal{C}'$ be such that $\dim_{{{\mathbb F}}_q}(\ker g)=i$. As $\dim_{{{\mathbb F}}_q}(\ker g)>0$, there exists $\alpha \in {{\mathbb F}}_{q^n}^*$ such that $g(\alpha)=0$, that is $g\circ \omega_\alpha (1)=0$. As $\operatorname{\mathcal{C}}'$ is a right vector space over $\mathcal{F}_n$, it follows that $g\circ \omega_\alpha \in \operatorname{\mathcal{C}}'$ and, in particular, $g\circ \omega_\alpha \in U_i$. This implies that $$\{ f \circ \omega_\alpha \colon f \in U_i,\,\,\alpha \in {{\mathbb F}}_{q^n}^* \}$$ coincides with the set of all the elements in $\operatorname{\mathcal{C}}'$ of rank $n-i$. Also, for any $f \in U_i$ and $\alpha\in{\mathbb{F}_{q^n}}$, we have $f \circ \omega_\alpha \in U_i$ if and only if $\alpha \in \ker f$. Thus, $$A_{n-i}= \frac{|U_i| (q^n-1)}{q^i-1}.$$ By Lemma \[lemma:complete weight\] $A_{n-\iota}\ne0$, and follows. Furthermore, $$|U|=1+ A_{n-1}\frac{q-1}{q^n-1}+\ldots+A_{n-\iota}\frac{q^\iota -1}{q-1},$$ i.e. $$(q^n-1)(|U|-1)= A_{n-1}(q-1)+\ldots+A_{n-\iota}(q^\iota -1).$$ By Theorem \[th:dualrelations\] applied to $\mathcal{C}'$ with $\nu=1$, we get $$A_{n-1}(q-1)+\ldots+A_{n-\iota}(q^\iota -1)=(q^n-1)(q^{\frac{\iota rn}{\iota+1}}-1),$$ whence $\dim_{\mathcal{F}_{n,q}}(U)=\frac{\iota rn}{\iota +1}$. Finally, choosing $G:\mathcal{C}^\prime\to W$, $f\mapsto f(1)$ and recalling that $\tau_{f}\colon {{\mathbb F}}_{q^n}\rightarrow \operatorname{\mathcal{C}}'$, $\alpha \mapsto f\circ \omega_{\alpha}$ for any $f \in \operatorname{\mathcal{C}}^\prime$, we obtain $\operatorname{\mathcal{C}}'=\operatorname{\mathcal{C}}_{U,G}$. Theorems \[th:MRDiff\] and \[th:MRDconverse\] provide a correspondence between: - ${{\mathbb F}_{q}}$-subspaces $U=V(\frac{\iota rn}{\iota+1},q)$ of $V=V(r,q^n)$ such that $\iota=\max\{\dim_{{{\mathbb F}_{q}}}(U\cap\langle\mathbf{v}\rangle_{{\mathbb{F}_{q^n}}})\colon \mathbf{v}\in V^*\}$; and - ${{\mathbb F}_{q}}$-linear MRD codes $\mathcal{C}$ with parameters $\left(\frac{rn}{\iota+1},n,q;n-\iota\right)$ and right idealiser isomorphic to ${\mathbb{F}_{q^n}}$. When $W={{\mathbb F}}_{q^{{nr}/{(\iota+1)}}}$ and $R(\mathcal{C})=\mathcal{F}_n$, Theorem \[th:MRDconverse\] reads as follows. Let $\iota,r,n$ be positive integers such that $\iota<n$, $\,\iota<r$ and $(\iota+1)\mid rn$. Let $f_1,\ldots,f_r\colon {\mathbb{F}_{q^n}}\rightarrow {{\mathbb F}}_{q^{{nr}/{(\iota+1)}}}$ be $\mathcal{F}_n$-linearly independent (on the right) ${{\mathbb F}_{q}}$-linear maps. Then the RM code $$\operatorname{\mathcal{C}}_{f_1,\ldots,f_r}=\{f_1\circ \omega_{\alpha_1}+\ldots+f_r\circ \omega_{\alpha_r} \colon \alpha_1,\ldots,\alpha_r \in {\mathbb{F}_{q^n}}\}$$ is an MRD code if and only if $$\dim_{{{\mathbb F}_{q}}} (\ker(f_1\circ \omega_{\alpha_1}+\ldots+f_r\circ \omega_{\alpha_r})) \leq \iota$$ for every $\alpha_1,\ldots,\alpha_r \in {\mathbb{F}_{q^n}}$. In this case, $\mathcal{C}_{f_1,\ldots,f_r}$ has parameters $\left(\frac{rn}{\iota+1},n,q;n-\iota\right)$ and $R(\operatorname{\mathcal{C}}_{f_1,\ldots,f_r})=\mathcal{F}_n$. Also, the $\mathcal{F}_{n,q}$-subspace $U_{f_1,\ldots,f_r}$ of $C_{f_1,\ldots,f_r}$ given by $$U_{f_1\ldots,f_r}=\{ f_1\circ \omega_{\alpha_1}+\ldots+f_r\circ \omega_{\alpha_r} \in \operatorname{\mathcal{C}}_{f_1,\ldots,f_r} \colon f_1(\alpha_1)+\ldots+f_r(\alpha_r)=0 \}$$ has dimension $\frac{\iota rn}{\iota +1}$, and $\iota=\max\{\dim_{{{\mathbb F}_{q}}}(U_{f_1,\ldots,f_r}\cap\langle\mathbf{v}\rangle_{{\mathbb{F}_{q^n}}})\colon \mathbf{v}\in \operatorname{\mathcal{C}}_{f_1,\ldots,f_r}^*\}$. Let $\iota,n,r$ be positive integers such that $\iota<n$ and $(\iota+1)\mid r$. Define $t=r/(\iota+1)$. The code $$\mathcal{C}=\left\{x\in{{\mathbb F}}_{q^{nt}}\mapsto a_0 x+a_1 x^q+\ldots+a_{\iota}x^{q^\iota}\in{{\mathbb F}}_{q^{nt}}\;\colon\; a_0,\ldots,a_{\iota}\in{{\mathbb F}}_{q^{nt}}\right\}$$ is an MRD code with parameters $(nt,nt,q;nt-\iota)$, known as Gabidulin code; see Section \[sec:noGab\] below. Consider the code $$\mathcal{C}|_{{\mathbb{F}_{q^n}}}=\left\{f|_{{\mathbb{F}_{q^n}}}:{\mathbb{F}_{q^n}}\to{{\mathbb F}}_{q^{nt}}\colon f\in\mathcal{C}\right\}.$$ By Theorem \[th:punct\], $\mathcal{C}|_{{\mathbb{F}_{q^n}}}$ is an MRD code with parameters $(nt,n,q;n-\iota)$. Also, $R(\mathcal{C}|_{{\mathbb{F}_{q^n}}})=\mathcal{F}_n$ and an $\mathcal{F}_n$-basis of $\mathcal{C}|_{{\mathbb{F}_{q^n}}}$ (seen as a right vector space) is $$\left\{ f_{j,i} \colon x\in{\mathbb{F}_{q^n}}\mapsto \xi^{i}x^{q^j}\in{{\mathbb F}}_{q^{nt}}\;\mid\; 0\leq i\leq t-1,\,0\leq j\leq \iota \right\},$$ where $\{1,\xi,\ldots,\xi^{t-1}\}$ is an ${\mathbb{F}_{q^n}}$-basis of ${{\mathbb F}}_{q^{nt}}$. Moreover, the set of the elements $f\in\mathcal{C}|_{{\mathbb{F}_{q^n}}}$ vanishing at $1$ is equal to $$U=\left\{x\in{\mathbb{F}_{q^n}}\mapsto -(a_1+\ldots+a_{\iota}) x+a_1 x^q+\ldots+a_{\iota}x^{q^\iota}\in{{\mathbb F}}_{q^{nt}}\;\colon\; a_1,\ldots,a_{\iota}\in{{\mathbb F}}_{q^{nt}} \right\},$$ and $\iota=\max\left\{\dim_{\mathcal{F}_{n,q}}\left(U\cap\langle f\rangle_{\mathcal{F}_n}\right)\colon f\in\mathcal{C}|_{{\mathbb{F}_{q^n}}}^{\,*}\right\}$. Let $$B=\left(f_{j,i}\,\colon\, j=0,\ldots,\iota,\;i=0,\ldots,t-1\right).$$ The coordinates of a vector in $U$ with respect to $B$ are $$\left( -\sum_{k=1}^{\iota}a_{k,0}\;,\ldots,-\sum_{k=1}^{\iota}a_{k,0}\;,\;a_{1,0}^{q^{n-1}},\ldots,a_{1,t-1}^{q^{n-1}}\;,\;\ldots\ldots,\;a_{\iota,0}^{q^{n-\iota}},\ldots,a_{\iota,t-1}^{q^{n-\iota}} \right),$$ where $a_{k,i}\in {\mathbb{F}_{q^n}}$ are such that $a_k=\sum_{i=0}^{t-1}a_{k,i}\xi^{i}$. Denote by $\overline{U}$ the set of the coordinates of the vectors in $U$. Let $\sigma^{\prime}:\mathbb{F}_{q^n}^{t(\iota+1)}\times \mathbb{F}_{q^n}^{t(\iota+1)}\to\mathbb{F}_q$, $(\mathbf{u},\mathbf{v})\mapsto\mathrm{Tr}_{q^n/q}(\langle\mathbf{u},\mathbf{v}\rangle)$, where $\langle\cdot,\cdot\rangle$ is the standard inner product. Then the vectors of $\overline{U}^{\perp^\prime}$ are $$(y_0,\ldots,y_{t-1},y_0^{q^{n-1}},\ldots,y_{t-1}^{q^{n-1}},\ldots \ldots, y_0^{q^{n-\iota}},\ldots,y_{t-1}^{q^{n-\iota}}),$$ where $y_0,\ldots,y_{t-1}\in\mathbb{F}_{q^n}$. Note that $\overline{U}^{\perp^\prime}$ is the direct sum of $t$ copies of $$\{ (z,z^{q},\ldots,z^{q^\iota})\colon z\in\mathbb{F}_{q^n} \}$$ which is a $\iota$-scattered ${{\mathbb F}_{q}}$-subspace of ${\mathbb{F}_{q^n}}^{\iota+1}$. Therefore, when $\iota+1$ divides $r$, the restriction to ${\mathbb{F}_{q^n}}$ of a Gabidulin code is associated with the direct sum of $t$ copies of a $\iota$-scattered ${{\mathbb F}_{q}}$-subspace of ${\mathbb{F}_{q^n}}^{\iota+1}$ (which is a $\iota$-scattered ${{\mathbb F}_{q}}$-subspace of ${\mathbb{F}_{q^n}}^r$); see Section \[sec:twocharact\]. Previously known connections {#sec:uni} ============================ In this section, we show that the connection between ${{\mathbb F}_{q}}$-vector spaces and ${{\mathbb F}_{q}}$-linear MRD codes established in Section \[sec:subMRD\] generalizes those presented in [@Sheekey; @ShVdV; @Lunardon2017; @CSMPZ2016]. Sheekey’s connection {#sec:sheekey} -------------------- The first connection was pointed out by Sheekey in its seminal paper [@Sheekey]. Let $U$ be an ${{\mathbb F}_{q}}$-subspace of ${\mathbb{F}_{q^n}}\times{\mathbb{F}_{q^n}}$, so that $$U=U_{f_1,f_2}=\{(f_1(x),f_2(x))\colon x\in{\mathbb{F}_{q^n}}\}$$ for some $f_1(x),f_2(x)$ in $\mathcal{L}_{n,q}$. Consider the ${{\mathbb F}_{q}}$-linear RM code $$\mathcal{S}_{f_1,f_2}=\{a_1 f_1(x)+a_2 f_2(x)\colon a_1,a_2\in{\mathbb{F}_{q^n}}\}\subset\mathcal{L}_{n,q},$$ whose left idealiser is isomorphic to ${\mathbb{F}_{q^n}}$. Then $U_{f_1,f_2}$ is a maximum scattered ${{\mathbb F}_{q}}$-subspace of ${\mathbb{F}_{q^n}}\times{\mathbb{F}_{q^n}}$ if and only if $\mathcal{S}_{f_1,f_2}$ is an MRD code with parameters $(n,n,q;n-1)$; see [@Sheekey Section 5]. A generalization to maximum $(r-1)$-scattered ${{\mathbb F}_{q}}$-subspaces of ${\mathbb{F}_{q^n}}^r$ {#sec:r-1} ----------------------------------------------------------------------------------------------------- Sheekey’s connection was extended by Sheekey and Van de Voorde in [@ShVdV] as follows; see also [@Lunardon2017]. Let $U$ be an ${{\mathbb F}_{q}}$-subspace of ${\mathbb{F}_{q^n}}^r$, so that $$U=U_{f_1,\ldots,f_r}=\{(f_1(x),\ldots,f_r(x))\colon x\in{\mathbb{F}_{q^n}}\}$$ for some $f_1(x),\ldots,f_r(x)\in\mathcal{L}_{n,q}$, and consider the ${{\mathbb F}_{q}}$-linear RM code $$\label{eq:Cf1fr} \mathcal{S}_{f_1,\ldots,f_r}=\{a_1 f_1(x)+\cdots+a_r f_r(x)\colon a_1,\ldots,a_r\in{\mathbb{F}_{q^n}}\},$$ whose left idealiser is isomorphic to ${\mathbb{F}_{q^n}}$. Then $U_{f_1,\ldots,f_r}$ is a maximum $(r-1)$-scattered ${{\mathbb F}_{q}}$-subspace of ${\mathbb{F}_{q^n}}^r$ if and only if $\mathcal{S}_{f_1,\ldots,f_r}$ is an MRD code with parameters $(n,n,q;n-r+1)$; see [@ShVdV Corollary 5.7]. Clearly, when $r=2$ this connection coincides with the one of Section \[sec:sheekey\]. A generalization to maximum scattered ${{\mathbb F}_{q}}$-subspaces {#sec:JACO} ------------------------------------------------------------------- Sheekey’s connection was extended by Csajbók, Marino, Polverino and the last author in [@CSMPZ2016] by considering maximum scattered ${{\mathbb F}_{q}}$-subspaces of $V=V(r,q^n)$ for any $r\geq2$ with $rn$ even; see [@CSMPZ2016 Theorem 3.2]. Let $U=V(\frac{rn}{2},q)$ be an ${{\mathbb F}_{q}}$-subspace of $V$, $W=V(\frac{rn}{2},q)$, $G:V\to W$ be an ${{\mathbb F}_{q}}$-linear map with $\ker(G)=U$, and $\iota=\max\{\dim_{{{\mathbb F}_{q}}}(U\cap\langle\mathbf{v}\rangle_{{\mathbb{F}_{q^n}}})\colon \mathbf{v}\in V^*\}$ with $\iota<n$. Then $\mathcal{C}_{U,G}=\{\Gamma_{\mathbf{v}}\colon \mathbf{v}\in V\}$ is an ${{\mathbb F}_{q}}$-linear RM code of dimension $rn$ with parameters $(\frac{rn}{2},n,q;n-\iota)$. Moreover, $U$ is a maximum scattered ${{\mathbb F}_{q}}$-subspace of $V$ if and only if $\mathcal{C}_{U,G}$ is an MRD code. In this case, the right idealiser of $\mathcal{C}_{U,G}$ is isomorphic to ${\mathbb{F}_{q^n}}$. Conversely, in [@PZ] the authors prove that any ${{\mathbb F}_{q}}$-linear MRD code with parameters $(\frac{rn}{2},n,q;n-1)$ and right idealiser isomorphic to ${\mathbb{F}_{q^n}}$ is equivalent to an MRD code $\mathcal{C}^\prime$ containing a maximum scattered ${{\mathbb F}_{q}}$-subspace $U$ such that $\mathcal{C}^\prime=\mathcal{C}_{U,G}$ with $G:\mathcal{C}^\prime\to W$, $f\mapsto f(1)$; see [@PZ Theorem 4.7]. This family contains the adjoint codes of the codes $\mathcal{S}_{f_1,f_2}$ presented in Section \[sec:sheekey\]; see [@CSMPZ2016 Example 3.5]. A unified connection -------------------- When $\iota=1$ and $U$ is a maximum scattered ${{\mathbb F}_{q}}$-subspace of $V=V(r,q^n)$, the connection established in Theorems \[th:construction\] and \[th:MRDiff\] coincides with the one of Section \[sec:JACO\], and hence generalizes the one of Section \[sec:sheekey\]. Also, Theorem \[th:MRDconverse\] extends the result of [@PZ]. When $\iota=r-1$ and $U$ is a maximum $(r-1)$-scattered ${{\mathbb F}_{q}}$-subspace of $V=V(r,q^n)$, our connection contains the adjoint codes of the MRD codes provided in Section \[sec:r-1\]. Indeed, let $\mathcal{C}$ be as in Equation with parameters $(n,n,q;n-r+1)$ and left idealiser isomorphic to ${\mathbb{F}_{q^n}}$. By Theorem \[th:propertiesideal\], the adjoint code $\mathcal{C}^\top$ is MRD with parameters $(n,n,q;n-r+1)$ and right idealiser isomorphic to ${\mathbb{F}_{q^n}}$. Thus, by Theorem \[th:MRDconverse\], $\mathcal{C}^\top$ is equivalent to $\mathcal{C}_{U,G}$ for some $U$ and $G$. Two characterizations of $h$-scattered subspaces {#sec:twocharact} ================================================ The ${{\mathbb F}_{q}}$-subspaces $U$ of $V=V(r,q^n)$ defining an MRD code with parameters $(\frac{rn}{h+1},n,q;n-h)$ and right idealiser isomorphic to ${\mathbb{F}_{q^n}}$ are exactly those of dimension $\frac{hrn}{h+1}$ such that $h=\max\{\dim_{{{\mathbb F}_{q}}}(U\cap\langle\mathbf{v}\rangle_{{\mathbb{F}_{q^n}}})\colon\mathbf{v}\in V^*\}$. Examples of such $U$’s are provided by the ordinary duals of $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}_{q}}$-subspaces of $V$, for which several constructions are known; see [@CsMPZ; @NPZZ]. We prove that, whenever $n\geq h+3$, such $U$’s are exactly the ordinary duals of $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}_{q}}$-subspaces of $V$. To this aim we provide two characterizations of these objects, namely Corollaries \[cor:caratterizzazione\] and \[cor:caratterizzazione2\], by means of ordinary and Delsarte dualities. \[th:condhyper\] Let $r,n,h,k$ be positive integers such that $n\geq h+3$ and $k>r$. Let $U$ be a $k$-dimensional ${{\mathbb F}_{q}}$-subspace of $V=V(r,q^n)$ such that $$\label{eq:intersection} \dim_{{{\mathbb F}_{q}}}(H\cap U)\leq k-n+h$$ for every $(r-1)$-dimensional ${\mathbb{F}_{q^n}}$-subspace $H$ of $V$. Let $\Gamma,\operatorname{\mathbb{V}},\perp,\perp_D$ be as in Section \[sec:Delsarteduality\]. Then $U^{\perp_D}$ is an $(n-h-2)$-scattered subspace of $\operatorname{\mathbb{V}}/\Gamma^\perp$. As noted in Section \[sec:Delsarteduality\], there exist $\operatorname{\mathbb{V}}=V(k,q^n)$, an ${\mathbb{F}_{q^n}}$-subspace $\Gamma=V(k-r,q^n)$ of $\operatorname{\mathbb{V}}$, and an ${{\mathbb F}_{q}}$-subspace $W=V(k,q)$ of $\operatorname{\mathbb{V}}$ such that $\operatorname{\mathbb{V}}=V\oplus\Gamma$, $\langle W\rangle_{{\mathbb{F}_{q^n}}}=\operatorname{\mathbb{V}}$, $W\cap\Gamma=\{\mathbf{0}\}$, and $U=\langle W,\Gamma\rangle_{{{\mathbb F}_{q}}}\cap V$. Let $\perp^\prime$ and $\perp$ be the orthogonal complement maps which act respectively on the ${{\mathbb F}_{q}}$-subspaces of $W$ and on the ${\mathbb{F}_{q^n}}$-subspaces of $\operatorname{\mathbb{V}}$, which are defined by non-degenerate reflexive sesquilinear forms $\beta^\prime :W\times W\to{{\mathbb F}_{q}}$ and $\beta:\operatorname{\mathbb{V}}\times\operatorname{\mathbb{V}}\to{\mathbb{F}_{q^n}}$ respectively, such that $\beta$ coincides with $\beta^\prime$ on $W\times W$. Since $n\geq h+3$, $k>r$ and holds, the Delsarte duality can be applied to $U$. Then $U^{\perp_D}=W+\Gamma^{\perp}$ is a $k$-dimensional ${{\mathbb F}_{q}}$-subspace of $\operatorname{\mathbb{V}}/\Gamma^{\perp}$. Suppose that there exists an $(n-h-2)$-dimensional subspace $M$ of $\operatorname{\mathbb{V}}/\Gamma^\perp$ such that $\dim_{{{\mathbb F}_{q}}}(M\cap U^{\perp_D})\geq n-h-1$. Write $M=N+\Gamma^\perp$, where $N$ is an $(n-h-2+r)$-dimensional subspace of $\operatorname{\mathbb{V}}$ satisfying $\Gamma^\perp \subseteq N$, so that $$\dim_{{{\mathbb F}_{q}}}(N\cap W)=\dim_{{{\mathbb F}_{q}}}(M\cap U^{\perp_D})\geq n-h-1.$$ Let $S$ be an $(n-h-1)$-dimensional ${{\mathbb F}_{q}}$-subspace of $N\cap W$. As $S\subseteq W$, we have $\dim_{{\mathbb{F}_{q^n}}}(S^*)=\dim_{{{\mathbb F}_{q}}}(S)$; see [@Lun99 Lemma 1]. Since $N$ contains both $S$ and $\Gamma^\perp$, we have $N^\perp\subseteq (S^*)^\perp \cap \Gamma$, whence $$\dim_{{\mathbb{F}_{q^n}}}((S^*)^\perp \cap \Gamma)\geq \dim_{{\mathbb{F}_{q^n}}}(N^\perp) = k-(n-h-2+r).$$ This implies that $\langle(S^*)^\perp,\Gamma\rangle_{{\mathbb{F}_{q^n}}}$ is contained in an ${\mathbb{F}_{q^n}}$-subspace $T$ of $\operatorname{\mathbb{V}}$ of dimension $k-1$. Let $\hat{T}=T\cap V$. As $T$ contains $\Gamma$, we have $\dim_{{\mathbb{F}_{q^n}}}(\hat{T})=r-1$. Using $U=\langle W,\Gamma\rangle_{{{\mathbb F}_{q}}}\cap V$ and $T\cap\langle W,\Gamma\rangle_{{{\mathbb F}_{q}}}=\langle\Gamma,T\cap W\rangle_{{{\mathbb F}_{q}}}$, we obtain $$\dim_{{{\mathbb F}_{q}}}(\hat{T}\cap U)=\dim_{{{\mathbb F}_{q}}}(T\cap W).$$ As $S^{\perp^\prime}= W\cap (S^*)^\perp\subseteq W\cap T$ and $\dim_{{{\mathbb F}_{q}}}(S^{\perp^{\prime}})=k-(n-h-1)$, we obtain $$\dim_{{{\mathbb F}_{q}}}(\hat{T}\cap U)\geq k-n+h+1,$$ a contradiction to . Therefore $U^{\perp_D}$ is an $(n-h-2)$-scattered ${{\mathbb F}_{q}}$-subspace of $\operatorname{\mathbb{V}}/\Gamma^\perp$. Note that Theorem \[th:condhyper\] can also be obtained as a consequence of [@BCsMT Theorem 3.5]. By Theorem \[th:condhyper\], the following characterization is obtained. \[cor:caratterizzazione\] Let $r,n,h$ be positive integers such that $h+1$ divides $rn$ and $n\geq h+3$. Let $U$ be an $\frac{rn}{h+1}$-dimensional ${{\mathbb F}_{q}}$-subspace of $V=V(r,q^n)$. Then $U$ is an $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}_{q}}$-subspace of $V$ if and only if $$\label{eq:intermax} \dim_{{{\mathbb F}_{q}}}(H\cap U)\leq\frac{rn}{h+1}-n+h$$ for every $(r-1)$-dimensional ${\mathbb{F}_{q^n}}$-subspace $H$ of $V$. Assume that holds. By Theorem \[th:condhyper\], $U^{\perp_D}$ is a $\frac{rn}{h+1}$-dimensional $(n-h-2)$-scattered ${{\mathbb F}_{q}}$-subspace in $\operatorname{\mathbb{V}}/\Gamma^\perp$. By Proposition \[prop:property\], $U$ is a $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}_{q}}$-subspace of $V$. The converse follows from Theorem \[th:inter\]. \[rem:nonallargarti\] If $n>2$, $k<\frac{rn}{h+1}$ and $h=1$, then there exist $k$-dimensional $1$-scattered ${{\mathbb F}_{q}}$-subspaces of $V=V(r,q^n)$ such that does not hold. Therefore, Corollary \[cor:caratterizzazione\] cannot be extended to all $h$-scattered subspaces which are not $\frac{rn}{h+1}$-dimensional $h$-scattered subspaces. Indeed, let $U^\prime$ be a scattered $k^\prime$-dimensional ${{\mathbb F}_{q}}$-subspace such that $H=\langle U^\prime\rangle_{{\mathbb{F}_{q^n}}}$ is a $(r-1)$-dimensional ${\mathbb{F}_{q^n}}$-subspace of $V$, and $\mathbf{v}\in V\setminus H$. Then $U=U^\prime\oplus\langle\mathbf{v}\rangle_{{{\mathbb F}_{q}}}$ is a $1$-scattered ${{\mathbb F}_{q}}$-subspace of $V$ of dimension $k=k^\prime+1$ such that $\dim_{{{\mathbb F}_{q}}}(U\cap H)=k-1>k-n+1$, as $n>2$. By using Corollary \[cor:caratterizzazione\], a further characterization of $\frac{rn}{h+1}$-dimensional $h$-scattered subspaces is proved. \[cor:caratterizzazione2\] Let $r,n,h$ be positive integers such that $h+1$ divides $rn$ and $n\geq h+3$. Let $U$ be an $\frac{rn}{h+1}$-dimensional ${{\mathbb F}_{q}}$-subspace of $V=V(r,q^n)$. Then $U$ is an $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}_{q}}$-subspace of $V$ if and only if $U^{\perp_O}$ satisfies $$\label{eq:intermaxpoint} \dim_{{{\mathbb F}_{q}}}(\langle \mathbf{v}\rangle_{{{\mathbb F}}_{q^n}}\cap U^{\perp_O})\leq h$$ for every $\mathbf{v}\in V\setminus\{\mathbf{0}\}$. If $U$ is an $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}}_q$-subspace of $V$, then the assertion follows from Theorem \[th:inter\] and Equation . Conversely, $\dim_{{{\mathbb F}_{q}}}(U)=\frac{rn}{h+1}$ implies $\dim_{{{\mathbb F}}_q}(U^{\perp_O})=\frac{hrn}{h+1}$. Together with the assumption and Equation , this yields $$\dim_{{{\mathbb F}_{q}}}(H\cap U)\leq \frac{rn}{h+1}-n+h$$ for every $(r-1)$-dimensional ${{\mathbb F}}_{q^n}$-subspace $H$ of $V$. The claim now follows from Corollary \[cor:caratterizzazione\]. Theorems \[th:MRDiff\] and \[th:MRDconverse\], together with Corollary \[cor:caratterizzazione2\], provide a correspondence between the following objects, under the assumption $n\geq h+3$: - $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}_{q}}$-subspaces of $V(r,q^n)$; and - ${{\mathbb F}_{q}}$-linear MRD codes with parameters $\left(\frac{rn}{h+1},n,q;n-h\right)$ and right idealiser isomorphic to ${\mathbb{F}_{q^n}}$. MRD codes inequivalent to generalized (twisted) Gabidulin codes {#sec:noGab} =============================================================== In this section we prove that the family of RM codes described in Section \[sec:subMRD\] contains MRD codes which are not equivalent to punctured generalized Gabidulin codes nor to punctured generalized twisted Gabidulin codes. Let $N,k,s$ be positive integers with $k<N$ and $\gcd(s,N)=1$. The *generalized Gabidulin code* $\mathcal{G}_{k,s}$ is defined as $$\mathcal{G}_{k,s}=\left\{ x\in{{\mathbb F}}_{q^N}\mapsto a_0 x+ a_1 x^{q^s}+\ldots+a_{k-1}x^{q^{s(k-1)}}\in{{\mathbb F}}_{q^N}\,\colon\, a_0,\ldots,a_{k-1}\in{{\mathbb F}}_{q^N} \right\}$$ and is an ${{\mathbb F}_{q}}$-linear MRD code with parameters $(N,N,q;N-k+1)$. The codes $\mathcal{G}_{k,s}$ were first introduced in [@Delsarte; @Gabidulin] for $s=1$ and generalized in [@kshevetskiy_new_2005]. Let $0\leq c<N$ and $\eta\in{{\mathbb F}}_{q^N}$ be such that $\eta^{(q^N-1)/(q-1)}\ne(-1)^{Nk}$. The *generalized twisted Gabidulin code* $\mathcal{H}_{k,s}(\eta,c)$ is defined as $$\mathcal{H}_{k,s}(\eta,c)=\left\{ x\in{{\mathbb F}}_{q^N}\mapsto a_0 x+ a_1 x^{q^s}+\ldots+a_{k-1}x^{q^{s(k-1)}}+a_0^{q^c}\eta x^{q^{sk}}\in{{\mathbb F}}_{q^N}\,\colon\, a_i\in{{\mathbb F}}_{q^N} \right\}$$ and is an ${{\mathbb F}_{q}}$-linear MRD code with parameters $(N,N,q; N-k+1)$. The codes $\mathcal{H}_{k,s}(\eta,c)$ were first introduced in [@Sheekey] and investigated in [@LTZ]. As a consequence of [@TZ Theorem 3.8], the left idealisers of punctured generalized (twisted) Gabidulin codes satisfy the following property. \[lemma:TZ\] Let $g:{{\mathbb F}}_{q^N}\to{{\mathbb F}}_{q^M}$ be an ${{\mathbb F}_{q}}$-linear map of rank $M\leq N$, and consider the punctured code $\mathcal{C}$, where either $\mathcal{C}=g\circ\mathcal{G}_{k,s}$ or $\mathcal{C}=g\circ\mathcal{H}_{k,s}(\eta,c)$. If $M>k+1$ and $(M,k)\ne(4,2)$, then $|L(\mathcal{C})|=q^\ell$ where $\ell$ divides $N$. In Theorem \[th:new\] we investigate the equivalence issue between the codes $\mathcal{C}$ as in having parameters $(M,N,q;d)$ with $M\geq N$, and punctured generalized (twisted) Gabidulin codes. As the punctured $(M,N,q;d)$-codes $\mathcal{D}$ arising from Lemma \[lemma:TZ\] satisfy $M\leq N$, we need to consider the adjoint code $\mathcal{D}^{\top}$ of $\mathcal{D}$, having parameters $(N,M,q;d)$. In this sense, whenever $\mathcal{C}$ and $\mathcal{D}^\top$ are not equivalent, we will say that $\mathcal{C}$ is not equivalent to a punctured generalized (twisted) Gabidulin code. We show in Theorem \[th:new\] that the condition $(h+1)\nmid r$ is sufficient for the MRD codes of Section \[sec:subMRD\] to be inequivalent to punctured generalized (twisted) Gabidulin codes. Afterwards, we provide examples. \[th:new\] Let $r,n,h$ be positive integers such that $(h+1)$ divides $rn$, $n\geq h+3$, and $(n,h)\ne(4,1)$. Let $\mathcal{C}=\mathcal{C}_{U,G}$ be the MRD code with parameters $\left(\frac{rn}{h+1},n,q;n-h\right)$ defined in Theorem \[th:MRDiff\]. If $(h+1)$ does not divide $r$, then $\mathcal{C}$ is not equivalent to any punctured generalized Gabidulin code nor to any punctured generalized twisted Gabidulin code. Suppose that $\mathcal{C}$ is equivalent to $\mathcal{D}$, where $\mathcal{D}$ is either $(g\circ \mathcal{G}_{k,s})^\top$ or $(g\circ\mathcal{H}_{k,s}(\eta,c))^\top$. Then $k=h+1$, $N=\frac{rn}{h+1}$ and $M=n$. Since $n>h+2$ and $(n,h)\ne(4,1)$, we can apply Lemma \[lemma:TZ\] and Theorem \[th:propertiesideal\] to get $|R(\mathcal{D})|=q^\ell$, where $\ell$ divides $\frac{rn}{h+1}$. As $\mathcal{C}$ and $\mathcal{D}$ are equivalent, by Theorem \[th:propertiesideal\] we have $|R(\mathcal{C})|=|R(\mathcal{D})|=q^\ell$. Then $n=\ell$ and hence $n\mid\frac{rn}{h+1}$, a contradiction to $(h+1)\nmid r$. Let $n\geq 6$ be even and $r^\prime\geq 3$ be odd. By [@CsMPZ Theorem 3.6], there exist $\frac{rn}{2}$-dimensional $(n-3)$-scattered ${{\mathbb F}_{q}}$-subspaces $U^\prime$ of $V=V\left(\frac{r^\prime(n-2)}2,q^n \right)$. Let $h=n-3$ and $r=\frac{r^\prime(n-2)}2$, and consider $U=U^{\prime\perp_O} \subseteq V$. Note that $(h+1)\nmid r$. Choose $G$ as in Theorem \[th:construction\]. By Theorems \[th:MRDiff\] and \[th:new\], the MRD code $\operatorname{\mathcal{C}}_{U,G}$ with parameters $\left(\frac{r^\prime n}{2},n,q;3\right)$ is not equivalent to any punctured generalized Gabidulin code nor to any punctured generalized twisted Gabidulin code. Let $n\geq6$ be even and $r\geq3$ be odd. Examples 3.11, 3.12 and 3.13 in [@CSMPZ2016] provide MRD codes with parameters $\left(\frac{rn}{2},n,q;n-1\right)$ which are not equivalent to any punctured generalized Gabidulin code nor to any punctured generalized twisted Gabidulin code. $h$-scattered linear sets: intersection with hyperplanes and codes with $h+1$ weights {#sec:h+1weights} ===================================================================================== Let $V=V(r,q^n)$. A point set $L$ of $\Omega={\mathrm{PG}}(V,{{\mathbb F}}_{q^n})\allowbreak={\mathrm{PG}}(r-1,q^n)$ is an *${{\mathbb F}}_q$-linear set* of $\Omega$ of rank $k$ if it is defined by the non-zero vectors of a $k$-dimensional ${{\mathbb F}}_q$-subspace $U$ of $V$, i.e. $$L=L_U:=\{{\langle}{\bf u} {\rangle}_{\mathbb{F}_{q^n}} \colon {\bf u}\in U^* \}\}.$$ We denote the rank of $L_U$ by $\mathrm{rk}(L_U)$. Let $\mathcal{S}={\mathrm{PG}}(S,{{\mathbb F}}_{q^n})$ be a subspace of $\Omega$ and $L_U$ be an ${{\mathbb F}}_q$-linear set of $\Omega$. Then $\mathcal{S} \cap L_U=L_{S\cap U}$. If $\dim_{{{\mathbb F}}_q} (S\cap U)=i$, i.e. if $\mathcal{S} \cap L_U=L_{S\cap U}$ has rank $i$, we say that $\mathcal{S}$ has *weight* $i$ in $L_U$, and we write $w_{L_U}(\mathcal{S})=i$. Note that $0 \leq w_{L_U}(\mathcal{S}) \leq \min\{{\rm rk}(L_U),n(\dim(\mathcal{S})+1)\}$. In particular, a point $P$ belongs to an ${{\mathbb F}}_q$-linear set $L_U$ if and only if $w_{L_U}(P)\geq 1$. If $U$ is a (maximum) scattered ${{\mathbb F}}_q$-subspace of $V$, then we say that $L_U$ is (maximum) scattered. In this case, $$|L_U| = \theta_{k-1}=\frac{q^k-1}{q-1},$$ where $k$ is the rank of $L_U$; equivalently, all of its points have weight one. If $U$ is a (maximum) $h$-scattered ${{\mathbb F}}_q$-subspace of $V$, $L_U$ is said to be a (maximum) $h$-scattered ${{\mathbb F}}_q$-linear set in $\Omega={\mathrm{PG}}(V,{{\mathbb F}}_{q^n})$. Therefore, an ${{\mathbb F}}_q$-linear set $L_U$ of $\Omega$ is $h$-scattered if - $\langle L_U \rangle= \Omega$; - for every $(h-1)$-subspace $\mathcal{S}$ of $\Omega$, we have $$w_{L_{U}}(\mathcal{S}) \leq h.$$ When $h=r-1$ and $\dim_{{{\mathbb F}}_q}(U)=n$, we obtain the scattered linear sets with respect to the hyperplanes introduced in [@Lunardon2017] and in [@ShVdV]. By Theorem \[th:inter\], if $L_U$ is a $h$-scattered ${{\mathbb F}}_q$-linear set of rank $\frac{rn}{h+1}$ in $\Omega$, then for every hyperplane $\mathcal{H}$ of $\Omega$ we have $$\frac{rn}{h+1}-n\leq w_{L_U}(\mathcal{H})\leq \frac{rn}{h+1}-n+h.$$ The following question arises: for any $j \in \{\frac{rn}{h+1}-n,\ldots,\frac{rn}{h+1}-n+h\}$, how many hyperplanes of $\Omega$ have weight $j$ in $L_U$? The answer is known for $h=1$, $h=2$ and $h=r-1$; see [@BL2000; @NZ; @ShVdV]. Theorem \[th:intersectionhyper\] gives a complete answer for any admissible values of $h$, $r$ and $n$. \[th:intersectionhyper\] Let $L_U$ be a $h$-scattered ${{\mathbb F}}_q$-linear set of rank $\frac{rn}{h+1}$ in $\Omega={\mathrm{PG}}(r-1,q^n)$. For every $i\in\left\{0,\ldots,h\right\}$, the number of hyperplanes of weight $\frac{rn}{h+1}-n+i$ in $L_U$ is $$\label{eq:ti} t_i=\frac{1}{q^n-1}{n \brack i}_q \sum_{j=0}^{h-i} (-1)^{j}{n-i \brack j}_q q^{\binom{j}{2}}\left(q^\frac{rn(h-i-j+1)}{h+1}-1\right).$$ In particular, $t_i>0$ for every $i\in\left\{0,\ldots,h\right\}$. Let $\sigma,\sigma^\prime,\perp,\perp^\prime$ be defined as in Section \[sec:classicalduality\], and $\mathcal{H}={\mathrm{PG}}(H,{{\mathbb F}}_{q^n})$ be a hyperplane of $\Omega$ with weight $\frac{rn}{h+1}-n+i$ in $L_U$. By Equation , $$\dim_{{{\mathbb F}_{q}}}(U^{\perp^\prime}\cap H^\perp)=\dim_{{{\mathbb F}_{q}}}(U\cap H) +rn-\frac{rn}{h+1}-(r-1)n = i\leq h.$$ Thus, the ${{\mathbb F}}_q$-linear set $L_{U^{\perp^\prime}}$ of $\Omega$ has rank $\frac{hrn}{h+1}$, the weight in $L_{U^{\perp^\prime}}$ of a point of $\Omega$ is at most $h$, and the number of points of $\Omega$ with weight $i$ in $L_{U^{\perp^\prime}}$ equals the number $t_i$ of hyperplanes with weight $\frac{rn}{h+1}-n+i$ in $L_U$, for every $i\in \{0,\ldots,h\}$. Let $W=V(\frac{rn}{h+1},q)$ and $G\colon V \rightarrow W$ be an ${{\mathbb F}}_q$-linear map with $\ker(G)=U$. By Theorem \[th:MRDiff\], the code $\mathcal{C}_{U^{\perp^\prime},G}=\{\Gamma_{\mathbf{v}} \colon \mathbf{v}\in V\}$ is an MRD code. Note that, for every $i \in \{0,\ldots,h\}$, $t_i$ is equal to the number of maps in $\operatorname{\mathcal{C}}_{U^{\perp^\prime},G}$ having rank $n-i$, divided by $q^n-1$. In fact, if $P=\langle \mathbf{v}\rangle_{{{\mathbb F}}_{q^n}}$ is a point of weight $i$ in $L_{U^{\perp^\prime}}$, then $\Gamma_{\lambda\mathbf{v}}$ has rank $n-i$ for every $\lambda\in{{\mathbb F}}_{q^n}^*$; conversely, if $\mathbf{v}\in V^*$ is such that $\Gamma_{\mathbf{v}}$ has rank $n-i$, then $\langle\mathbf{v}\rangle_{{{\mathbb F}}_{q^n}}$ has weight $i$ in $L_{U^{\perp^\prime}}$. Thus, $t_i=A_{n-i}/(q^n-1)$. By Theorem \[th:MRDiff\], Equation follows. As $\mathcal{C}_{U^{\perp^\prime},G}$ is an MRD code, $t_i>0$ by Lemma \[lemma:complete weight\]. Under the assumptions of Theorem \[th:intersectionhyper\], the property of being scattered determines completely the intersection numbers w.r.t. the hyperplanes. \[cor:intersec\] Let $L_U$ be a $h$-scattered ${{\mathbb F}_{q}}$-linear set of rank $\frac{rn}{h+1}$ in $\Omega={\mathrm{PG}}(r-1,q^n)$. For every hyperplane $\mathcal{H}$ of $\Omega$, we have $$|\mathcal{H}\cap L_U| \in \left\{ \theta_{\frac{rn}{h+1}-n-1},\ldots,\theta_{\frac{rn}{h+1}-n+h-1} \right\}.$$ For every $i\in\{0,\ldots,h\}$, the number of hyperplanes $\mathcal{H}$ of $\Omega$ satisfying $|\mathcal{H}\cap L_U|=\theta_{\frac{rn}{h+1}-n+i-1}$ is $t_i$, as in Equation . We now consider $h$-scattered ${{\mathbb F}_{q}}$-linear sets $L_U$ of rank $\frac{rn}{h+1}$ as projective systems in $\Omega$, and the related linear codes (with the Hamming metric). By means of Theorem \[th:intersectionhyper\] we determine the weight distribution and the weight enumerator. Let $L_U$ be a $h$-scattered ${{\mathbb F}_{q}}$-linear set of rank $\frac{rn}{h+1}$ in $\Omega={\mathrm{PG}}(r-1,q^n)$, and $\mathcal{C}_{L_U}$ be the corresponding linear code over ${{\mathbb F}}_{q^n}$, having length $N=\theta_{\frac{rn}{h+1}-1}$ and dimension $k=r$. Then $\mathcal{C}_{L_U}$ has minimum distance $d=\theta_{\frac{rn}{h+1}-1}-\theta_{\frac{rn}{h+1}-n+h-1}$ and exactly $h+1$ weights, namely $w_i=\theta_{\frac{rn}{h+1}-1}-\theta_{\frac{rn}{h+1}-n+i-1}$ with $i=0,\ldots,h$. The weight enumerator of $\mathcal{C}_{L_U}$ is $$1+\sum_{i=0}^{h}A_{w_i}^H z^{w_i},$$ where $A_{w_i}^H=t_i$, as in Equation . From ${\rm rk}(L_U)=\frac{rn}{h+1}$ follows the length $N=\theta_{\frac{rn}{h+1}-1}$, and from $\langle L_U\rangle = \Omega$ follows the dimension $k=r$. By Corollary \[cor:intersec\] and Proposition \[prop:projsyst\], the minimum distance of $\mathcal{C}_{L_U}$ is $d=\theta_{\frac{rn}{h+1}-1}-\theta_{\frac{rn}{h+1}-n+h-1}$. Also, the weight distribution and the weight enumerator of $\mathcal{C}_{L_U}$ are as in the claim. Randrianarisoa in [@Ra] introduced the concept of $[N,k,d]$ $q$-system over ${\mathbb{F}_{q^n}}$ as an $N$-dimensional ${{\mathbb F}_{q}}$-subspace $U$ of ${\mathbb{F}_{q^n}}^k$ such that $\langle U\rangle_{{\mathbb{F}_{q^n}}}={\mathbb{F}_{q^n}}^k$ and $N-d=\max\left\{\dim_{{{\mathbb F}_{q}}}(U\cap H)\colon H=V(k-1,q^n)\subset {\mathbb{F}_{q^n}}^k\right\}$. For any positive integers $r,n,h$ such that $(h+1)\mid rn$ and $n\geq h+3$, Corollary \[cor:caratterizzazione\] implies that the $[\frac{rn}{h+1},r,n-h]$ $q$-systems over ${\mathbb{F}_{q^n}}$ are exactly the $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}_{q}}$-subspaces of ${\mathbb{F}_{q^n}}^r$. In this case, the code $\mathcal{C}$ considered in [@Ra Section 3] has a generator matrix $G$ whose columns form an ${{\mathbb F}_{q}}$-basis of $U$. The code $\mathcal{C}$ turns out to be obtained by $\mathcal{C}_{L_U}$ by deleting all but $\frac{rn}{h+1}$ positions (corresponding to an ${{\mathbb F}}_q$-basis of $U$). Together with [@Ra Theorem 2], this answers the question posed in [@Ra Section 8] about the correspondence between $h$-scattered linear sets and RM codes of this type. Conclusions and open questions {#sec:open} ============================== Several connections between between MRD codes and scattered ${{\mathbb F}_{q}}$-subspaces (linear sets) have been introduced in the literature. In this paper we propose a unified approach which generalizes all of these connections. To this aim, we give useful characterizations of $\frac{rn}{h+1}$-dimensional $h$-scattered subspaces. This allows to use the known constructions of $\frac{rn}{h+1}$-dimensional $h$-scattered subspaces in order to define MRD codes, and conversely. The family we construct is very large and contains some ”new” MRD codes, in the sense that they cannot be obtained by puncturing generalized (twisted) Gabidulin codes; this property is in general quite difficult to establish. We conclude the paper by determining the intersection numbers of $h$-scattered linear sets of rank $\frac{rn}{h+1}$ w.r.t. the hyperplanes and the weight distribution of the code obtained by regarding the linear set as a projective system. Several remarkable problems remain open; we list some of them. - The main open problem about $\frac{rn}{h+1}$-dimensional $h$-scattered ${{\mathbb F}_{q}}$-subspaces of $V(r,q^n)$ is their existence for every admissible values of $r$, $n$ and $h$. This would imply the existence of possibly new MRD codes. Conversely, constructions of MRD codes with parameters $(\frac{rn}{h+1},n,q;n-h)$ and right idealiser isomorphic to ${\mathbb{F}_{q^n}}$, when $(h+1)\nmid r$ and $h< n-3$, give new examples of $\frac{rn}{h+1}$-dimensional $h$-scattered subspaces. - Corollary \[cor:caratterizzazione2\] characterizes $\frac{rn}{h+1}$-dimensional $h$-scattered subspaces whenever $n\geq h+3$. 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New semifields and new MRD codes from skew polynomial rings, [*J. Lond. Math. Soc. (2)*]{} [**101(1)**]{} (2020), 432–456. MRD codes: constructions and connections, *Combinatorics and finite fields: Difference sets, polynomials, pseudorandomness and applications*, Radon Series on Computational and Applied Mathematics [**23**]{}, K.-U. Schmidt and A. Winterhof (eds.), De Gruyter (2019). Rank-metric codes, linear sets and their duality, [*Des. Codes Cryptogr.*]{} (2019), <https://doi.org/10.1007/s10623-019-00703-z>. Nuclei and automorphism groups of generalized twisted Gabidulin codes, *Linear Algebra Appl.* [**575**]{} (2019), 1–26. Giovanni Zini and Ferdinando Zullo\ Dipartimento di Matematica e Fisica,\ Università degli Studi della Campania “Luigi Vanvitelli”,\ I–81100 Caserta, Italy\ [[*{giovanni.zini,ferdinando.zullo}@unicampania.it*]{}]{} [^1]: The first author is funded by the project ”Attrazione e Mobilità dei Ricercatori” Italian PON Programme (PON-AIM 2018 num. AIM1878214-2). The research was supported by the project ”VALERE: VAnviteLli pEr la RicErca" of the University of Campania ”Luigi Vanvitelli”, and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM). [^2]: Recall that $\langle f\rangle_{\mathcal{F}_n}=\{f\circ\omega_{\alpha}\colon\alpha\in{\mathbb{F}_{q^n}}\}$.

--- abstract: 'The high-temperature expansion of the spin-spin correlation function of the two-dimensional classical XY (planar rotator) model on the square lattice is extended by three terms, from order 21 through order 24, and analyzed to improve the estimates of the critical parameters.' author: - 'P. Butera[@pb]' - 'M. Pernici[@mp]' title: ' High-temperature expansions through order 24 for the two-dimensional classical XY model on the square lattice' --- Tests of increasing accuracy [@Kenna] of the BKT theory [@BKT] of the two-dimensional XY model critical behavior have been made possible by the steady improvements of the computers performances and the progress in the numerical approximation algorithms. However, the critical parameters of this model have not yet been determined with a precision comparable to that reached for the usual power-law critical phenomena, due to the complicated and peculiar nature of the critical singularities. Therefore any effort at improving the accuracy of the available numerical methods by stretching them towards their (present) limits should be welcome. After extending the high-temperature(HT) expansions of the model in successive steps [@BC] from order $\beta^{ 10}$ to $\beta^{ 21}$, we present here a further extension by three orders for the expansions of the spin-spin correlation on the square lattice and perform a first brief analysis of our data for the susceptibility and the second-moment correlation-length. More results and further extensions both for the square and the triangular lattice[@bct] will be presented elsewhere. Our study strengthens the support of the main results of the BKT theory already coming from the analysis of shorter series and suggests a closer agreement with recent high-precision simulation studies[@Kenna; @Has] of the model. The Hamiltonian $$H\{ v \} = - 2{J} \sum_{nn} \vec v({\vec r}) \cdot \vec v({\vec r}') \label{hamilt}$$ with $\vec v({\vec r})$ a two-component unit vector at the site ${\vec r}$ of a square lattice, describes a system of $XY$ spins with nearest-neighbor interactions. Computing the spin-spin correlation function, $$C(\vec 0, \vec x;\beta)= <s(\vec 0) \cdot s(\vec x)>, \label{corfun}$$ (for all values of $\vec x$ for which the HT expansion coefficients are non-trivial within the maximum order reached), as series expansion in the variable $\beta= J/kT$, enables us to evaluate the expansions of the $l$-th order spherical moments of the correlation function: $$m^{(l)}(\beta) = \sum_{\vec x }|\vec x|^l <s(\vec 0) \cdot s(\vec x)> \label{sfermom}$$ and in particular the reduced ferromagnetic susceptibility $\chi(\beta)=m^{(0)}(\beta)$. In terms of $m^{(2)}(\beta)$ and $\chi(\beta )$ we can form the second-moment correlation length: $$\xi^2(\beta) = m^{(2)}(\beta)/4\chi(\beta). \label{corleng}$$ Our results for the nearest-neighbor correlation function (or energy $E$ per link) are: $$\begin{aligned} E &=& \beta + \frac{3}{2}\beta^{3} + \frac{1}{3}\beta^{5} - \frac{31}{48}\beta^{7} - \frac{731}{120}\beta^{9} - \frac{29239}{1440}\beta^{11} - \frac{265427}{5040}\beta^{13} - \frac{75180487}{645120}\beta^{15} \nonumber \\ &-& \frac{6506950039}{26127360}\beta^{17}- \frac{1102473407093}{2612736000}\beta^{19} - \frac{6986191770643}{14370048000}\beta^{21} \nonumber \\ &+& \frac{1657033646428733}{4138573824000}\beta^{23}+ O(\beta^{25}) \nonumber \\\end{aligned}$$ For the susceptibility we have: $$\begin{aligned} \chi &=& 1 + 4\beta + 12\beta^{2} + 34\beta^{3} + 88\beta^{4} + \frac{658}{3}\beta^{5} + 529\beta^{6} + \frac{14933}{12}\beta^{7} + \frac{5737}{2}\beta^{8} + \frac{389393}{60}\beta^{9} \nonumber \\ &+& \frac{2608499}{180}\beta^{10} + \frac{3834323}{120}\beta^{11} + \frac{1254799}{18}\beta^{12} + \frac{84375807}{560}\beta^{13} + \frac{6511729891}{20160}\beta^{14} \nonumber \\ &+& \frac{66498259799}{96768}\beta^{15}+ \frac{1054178743699}{725760}\beta^{16} + \frac{39863505993331}{13063680}\beta^{17} \nonumber \\ &+& \frac{19830277603399}{3110400}\beta^{18}+ \frac{8656980509809027}{653184000}\beta^{19}+ \frac{2985467351081077}{108864000}\beta^{20} \nonumber \\ &+& \frac{811927408684296587}{14370048000}\beta^{21}+ \frac {399888050180302157} {3448811520} \beta^{22}+ \frac {245277792666205990697} {1034643456000} \beta^{23} \nonumber \\ &+& \frac {83292382577873288741}{172440576000}\beta^{24}+ O(\beta^{25}) \nonumber \\ \end{aligned}$$ For the second moment of the correlation function we have: $$\begin{aligned} m_2 &=& 4\beta + 32\beta^{2} + 162\beta^{3} + 672\beta^{4} + \frac{7378}{3}\beta^{5} +\frac{24772}{3}\beta^{6} + \frac{312149}{12}\beta^{7} + 77996\beta^{8} \nonumber \\ &+& \frac{13484753} {60}\beta^{9}+ \frac{28201211}{45} \beta^{10}+ \frac{611969977}{360} \beta^{11}+ \frac{202640986}{45} \beta^{12}+ \frac{58900571047}{5040}\beta^{13} \nonumber \\ &+& \frac{3336209179}{112}\beta^{14}+ \frac{1721567587879}{23040 }\beta^{15}+ \frac{16763079262169}{90720}\beta^{16}+ \frac{5893118865913171}{13063680}\beta^{17} \nonumber \\ &+& \frac{17775777329026559}{16329600}\beta^{18}+ \frac{1697692411053976387}{653184000}\beta^{19}+ \frac{41816028466101527}{6804000}\beta^{20} \nonumber \\ &+& \frac{206973837048951639371}{14370048000}\beta^{21}+ \frac{721617681295019782781}{21555072000}\beta^{22} \nonumber \\ &+& \frac{79897272060888843617033}{1034643456000}\beta^{23}+ \frac{2287397511857949924319}{12933043200}\beta^{24}+ O(\beta^{25})\end{aligned}$$ The coefficients of order less than 22 were already tabulated in Refs.[@BC], but for completeness we report all known terms. As implied by eq.(\[hamilt\]), the normalization of these series reduces to that of our earlier papers[@BC] by the change $\beta \rightarrow \beta/2$. Let us now list briefly the main predictions[@BKT] of the BKT renormalization-group analysis to which the HT series should be confronted in order to extract the critical parameters. As $\beta \rightarrow \beta_{c}$, the correlation length $\xi^2(\beta) = m^{(2)}(\beta)/4\chi(\beta)$ is expected to diverge with the characteristic singularity $$\xi^2(\beta) \propto \xi^2_{as}(\beta)= exp(b/\tau^\sigma)[1+O(\tau)] \label{xias}$$ where $\tau=1-\beta/\beta_{c}$. The exponent $\sigma$ takes the universal value $\sigma=1/2$, whereas $b$ is a nonuniversal positive constant. At the critical inverse temperature $\beta=\beta_{c}$, the asymptotic behavior of the two-spin correlation function as $|\vec x |= r \rightarrow \infty$ is expected[@Amit] to be $$<s(\vec 0) \cdot s(\vec x)> \propto \frac {({\rm ln}r)^{2\theta}} {r^{\eta}} [1+O(\frac{{\rm ln}{\rm ln}r} {{\rm ln}r})] \label{coras}$$ Universal values $\eta= 1/4$ and $\theta = 1/16$ are predicted also for the correlation exponents. A simple non-rigorous argument based on eqs. (\[xias\]) and (\[coras\]) suggests that, for $l>\eta-2$, the spherical correlation moment $m^{(l)}(\beta)$ diverges as $\tau \rightarrow 0^+$ with the singularity $$m^{(l)}(\beta) \propto \tau^{-\theta}\xi^{2-\eta+l}_{as}(\beta)[1+O(\tau^{1/2}{\rm ln}\tau)] \label{momas}$$ This argument was challenged[@Bal] by a recent renormalization group analysis implying that the logarithmic factor in eq.(\[coras\]) gives rise to a less singular correction in the correlation moments, taking, for example in the case of the susceptibility, the form $$m^{(0)}(\beta) \propto \xi^{2-\eta}_{as}(\beta)[1+cQ] \label{momasb}$$ where $Q= \frac {\pi^2} {2({\rm ln}(\xi)+u)^2} +O({\rm ln}(\xi)^{-5})$ and $u$ is a non universal parameter. By eqs.(\[xias\]) and (\[momas\]), the ratios $r_n(m^{(l)})= a^{(l)}_n/a^{(l)}_{n+1}$ of the successive HT expansion coefficients of the correlation moment $m^{(l)}(\beta)$, for large $n$ should behave[@BC] as $$r_n(m^{(l)})= \beta_{c} + C_l/(n+1)^{\zeta} +O(1/n) \label{ratas}$$ with $\zeta=1/(1+\sigma)$, to be contrasted with the value $\zeta=1$ which is found for the usual power-law critical singularities. To begin with, let us assume that $\sigma=1/2$ as expected, so that $\zeta=2/3$. Fig.\[figlograpp\] gives a suggestive visual test of the asymptotic behavior of some ratio sequences $r_n(m^{(l)})$ by comparing them with eq.(\[ratas\]). The four lowest continuous curves interpolating the data points are obtained by separate three-parameter fits of the ratio sequences $r_n(\chi)$, $r_n(m^{(1/2)})$, $r_n(m^{(1)})$ and $r_n(m^{(2)})$ to the asymptotic form $a +b/(n+1)^{2/3} +c/(n+1)$ of eq.(\[ratas\]). In the same figure, the two upper sets of points are obtained by extrapolating the alternate-ratio sequence for the susceptibility, first in terms of $1/(n+1)^{2/3}$ and then in terms of $1/(n+1)$. The values of $\beta_c$ indicated by the fits of the ratio sequences, range between 0.5592 and 0.5611. A more accurate analysis can be based on the simple remark that, near the critical point, by eq.(\[xias\]) and (\[momas\]) (or eq.(\[momasb\])), one has ${\rm ln}(\chi) =c_1/\tau^{\sigma}+ c_2+..$. Therefore, if $\sigma=1/2$, the relative strength of the $1/\sqrt{\tau}$ and $1/\tau$ singularities in the function $L(a,\beta)=( a+ {\rm ln}(\chi))^{2}$ is determined by the value of the constant $a$. If we choose $a \approx 1.19$, the function $L(a,\beta)$ is approximately dominated by a simple pole and we can expect that the differential approximants (DAs)[@Gutt] will be able to determine with higher accuracy not only the position, but also the exponent of the critical singularity. Using inhomogeneous second-order DAs of $L(a,\beta)$, we can locate the critical singularity at $\beta_c= 0.5598(10)$. By analysing in the same way the series data truncated to order 21 which were previously available , we would get the estimate $\beta_c=0.5588(15)$. A consistent estimate $\beta_c=0.558(2)$ had been obtained in earlier independent[@BC; @pisa] studies of the same series using Padé approximants or first-order DAs. Older studies[@BC] of slightly shorter series also indicated values of $\beta_c$ in the same range, but with notably larger uncertainty. Thus our new series results indicate a stabilization and a sizable reduction of the spread for the $\beta_c$ estimates. Our uncertainty estimates are generally taken as the width of the distribution of the values of $\beta_c$ in the appropriate class of DAs. Fig.\[figbetacda2\] shows the singularity distribution (open histogram) of the set of quasi-diagonal DAs which yield our new estimate. These are chosen as the approximants $[k,l,m;n]$ with $17 < k+l+m+n< 22$. Moreover, we have taken $|k-l|,|l-m|,|k-m|< 3$ with $ k,l,m > 3$ and $1<n<7$. The class of DAs can be varied with no significant variation of the final estimates, for example by further restricting the extent of off-diagonality, or by varying the minimal degree of the polynomial coefficients in the DAs. No limitations have been imposed on the exponents of the singular terms or on the background terms in the DAs in order to avoid biasing the $\beta_c$ estimates. Should we require that the exponent of the most singular term in the approximants differs from -1, for example, by less than $20\%$, we would obtain $\beta_c=0.5602(5)$, well within the uncertainty of our previous unrestricted estimate. The vertical dashed line in Fig.\[figbetacda2\] shows the value $\beta_c=0.55995$ suggested by the simulation of Ref.[@Has]. Although no explicit indication of an uncertainty comes with this estimate, an upper bound to its error might be guessed from the statement[@Has] that the simulation can exclude values larger or equal than $\beta_c= 0.56045$ for the inverse critical temperature. Biasing with $\beta_c= 0.5598(10)$ the set previously specified of second-order DAs of $L(a,\beta)$, leads to the exponent estimate $\sigma=0.50(1)$. Fig.\[figbetacda2\] also shows the distribution of the exponent estimates (hatched histogram) from this biased set. The uncertainty we have reported for $\sigma$ accounts not only for the width of its distribution shown in Fig.\[figbetacda2\], but also for the variation of its central value as the bias value of $\beta_c$ is varied in the uncertainty interval of the critical inverse temperature. Essentially the same value of $\sigma$ would be obtained from the analysis of a series truncated to order 21. While, as one should expect, the DA estimate of $\beta_c$ is rather insensitive to the choice of $a$, the estimate of the exponent $\sigma$ and the width of its distribution are fairly improved by our choice of $a$. Taking for example $a=0$, we would find $\sigma= 0.53(4)$, which shows how the convergence of the exponent estimates is slowed down by the more complicated singularity structure of $L(0,\beta)$. Similar values of $\sigma$ were found in previous studies of shorter series. Probably for the same reason, also the central values of the $\eta$ estimates obtained from the usual indicators are still slightly larger than expected. For example, by studying the function $ H(\beta)={\rm ln}(1+m^{(2)}/\chi^2)/\rm ln(\chi)$ (or analogous functions of different moments), we can infer $\eta=0.260(10)$. The function[@Bal] $D(\beta)= \ln(\chi) - (2-\eta)\rm ln(\xi)$ and its first derivative are also interesting indicators of the value of $\eta$. Taking $\eta=1/4$, Padé approximants and DAs do not detect any singular behavior of $D(\beta)$ or of its derivative as $\beta \rightarrow \beta_c$, thus confirming the complete cancellation of the leading singularity in $D(\beta)$. Moreover, this behavior seems to exclude the form eq.(\[momas\]) of the corrections which implies the presence of weak subleading singularities, while it is compatible with eq.(\[momasb\]). In conclusion, our analysis suggests that, in spite of their diversity, the HT extended series approach and the latest most extensive simulation are competitive and lead to consistent numerical estimates of the highest accuracy so far possible. Acknowledgements ================ We thank Prof. Ralph Kenna for a useful correspondence. This work was partially supported by the italian Ministry of University and Research. Electronic address: [email protected] Electronic address: [email protected] R.  Kenna, cond-mat/0512356; Condens. Matter Phys. [**9**]{}, 283 (2006). V. 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Balog, M.  Niedermaier, F.  Niedermaier, A.  Patrascioiu, E.  Seiler, and P.  Weisz, Nucl. Phys. B [**618**]{}, 315 (2001). A. J. Guttmann, in [*Phase Transitions and Critical Phenomena*]{}, edited by C. Domb and J. Lebowitz (Academic, New York 1989) , Vol. 13. M. Campostrini, A. Pelissetto, P.Rossi and E. Vicari, Phys. Rev. B [**54**]{}, 7301 (1996). ![ \[figlograpp\] Ratios of the successive HT-expansion coefficients vs. $1/(n+1)^{2/3}$: for the susceptibility $\chi$ (open circles), for $m^{(1/2)}$ (rhombs), for $m^{(1)} $ (squares) and for $m^{(2)}$ (triangles). The four lowest continuous curves are obtained by separate three-parameter fits of each ratio sequence to its leading asymptotic behavior eq.(\[ratas\]). The data points represented by crossed circles are obtained by extrapolating the sequence of the susceptibility alternate ratios with respect to $1/n^{2/3}$, and the continuous line interpolating them is the result of a two-parameter fit of the last few points to the expected asymptotic form $a+b/n$. The small black circles are obtained by a further extrapolation of the latter quantities with respect to $1/n$. The continuous line interpolating the black circles is drawn only as a guide to the eye. The horizontal broken line indicates the critical value $\beta_c=0.55995$ suggested by the simulation of Ref.[@Has]](Fig1_xy_darker.eps){width="3.37"} ![\[figbetacda2\] Distribution of singularities for a class of second-order inhomogeneous DAs of $L(1.19,\beta)=(1.19+{\rm ln} \chi)^2$ versus their position on the $\beta$ axis(open histogram). The central value of the open histogram is $\beta_c=0.5598(10)$. The bin width is 0.0007. The vertical dashed line shows the critical value $\beta_c=0.55995$ indicated by the simulation of Ref.[@Has] for which one can guess an uncertainty at least twice smaller than ours. The hatched histogram represents the distribution of the exponent $\sigma$ obtained from DAs of $L(1.19,\beta)$ biased with $\beta_c= 0.5598$, vs. their position on the $\sigma$ axis. The central value of the hatched histogram is $\sigma=0.500(1)$ and the bin width is 0.0015. The variation of the central value of $\sigma$ as $\beta_c$ varies in its uncertainty interval is $0.01$. This value can be taken as a more reliable estimate of the uncertainty of $\sigma$.](Fig2_xy_darker.eps){width="3.37"}

--- abstract: 'The proton radius puzzle questions the self-consistency of theory and experiment in light muonic and electronic bound systems. Here, we summarize the current status of virtual particle models as well as Lorentz-violating models that have been proposed in order to explain the discrepancy. Highly charged one-electron ions and muonic bound systems have been used as probes of the strongest electromagnetic fields achievable in the laboratory. The average electric field seen by a muon orbiting a proton is comparable to hydrogenlike Uranium and, notably, larger than the electric field in the most advanced strong-laser facilities. Effective interactions due to virtual annihilation inside the proton (lepton pairs) and process-dependent corrections (nonresonant effects) are discussed as possible explanations of the proton size puzzle. The need for more experimental data on related transitions is emphasized.' author: - 'U. D. Jentschura' title: 'Muonic bound systems, virtual particles and proton radius' --- Introduction ============ Recent muonic hydrogen experiments [@PoEtAl2010; @AnEtAl2013] have resulted in the most severe discrepancy of the predictions of quantum electrodynamics with experiment recorded over the last few decades. In short, both (electronic, i.e., atomic) hydrogen experiments (for an overview see Ref. [@JeKoLBMoTa2005]) as well as recent scattering experiments lead to a proton charge radius of about $\left< r_p \right> \approx 0.88 \, {\rm fm}$, while the muonic hydrogen experiments [@PoEtAl2010; @AnEtAl2013] favor a proton charge radius of about $\left< r_p \right> \approx 0.84 \, {\rm fm}$. Few-body bound electronic and muonic systems belong to the most intensely studied fundamental physical entities; a combination of atomic physics and field-theoretic techniques is canonically employed [@BeSa1957; @BrMo1978; @BeKlSh1997; @SoEtAl1998; @MoPlSo1998; @SoEtAl2001; @EiGrSh2001; @EiGrSh2007; @Je2011aop1; @Je2011aop2; @AnEtAl2013aop]. Here, we aim to discuss conceivable explanations for the discrepancy and highlight a few aspects that set the muonic systems apart from any other bound states which have been studied spectroscopically so far. To this end, in Sec. \[sec2\], we briefly summarize the status of virtual particle models discussed in the literature and supplement previous approaches with a discussion of the role of axion terms that might be significant in the strong magnetic fields used in the muonic hydrogen experiments. In Sec. \[sec3\], we show that muonic hydrogen (as well as muonic hydrogenlike ions with low nuclear charge number $Z$) constitute some of the most sensitive probes of high-field physics to date; concomitant speculations about novel phenomena in the strong fields inside the proton are discussed. Finally, a possible role of process-dependent corrections in experiments is mentioned in Sec. \[sec4\]. Conclusions are reserved for Sec. \[sec5\]. We use SI mksA units unless indicated otherwise. Virtual Particles and Muonic Hydrogen {#sec2} ===================================== From the point of view of quantum field theory, the most straightforward explanation for the proton radius puzzle in muonic hydrogen would involve a “subversive” virtual particle that modifies the muon-proton interaction at distances commensurate with the Bohr radius of muonic hydrogen, $$\label{amu} a_\mu = \frac{\hbar}{\alpha_{\rm QED} \, m_r \, c} = 2.84708 \times 10^{-13} \, {\rm m} \,,$$ where $\alpha_{\rm QED}$ is the fine-structure constant and $m_r = m_\mu \, m_p/(m_\mu + m_p)$ is the reduced mass. The distance regime of $a_\mu \approx 300 \, {\rm fm}$ is intermediate between the Bohr radius of (ordinary) hydrogen and the proton radius. In consequence, the possible role of millicharged particles, which modify the Coulomb force law in this distance regime, has been analyzed in Ref. [@Je2011aop2]. These particles could conceivably modify the photon propagator at energy scales $\hbar c/a_\mu$ via vacuum-polarization insertions into the photon line. Supplementing this analysis, in Ref. [@Je2011aop1], conceivable hidden (massive) photons are analyzed. Particles with scalar and pseudo-scalar couplings are the subject of Ref. [@CaRi2012]. A model that explicitly breaks electron-muon universality, introducing a coupling of the right-handed muonic fermion sector to a $U(1)$ gauge boson, is investigated in Ref. [@BaMKPo2011]. One should notice, though, that the explicit breaking of the universality according to Eq. (7) of Ref. [@BaMKPo2011] appears as somewhat artificial. The reduction in the muonic helium nuclear radius by $\Delta r^2_{\rm He} = -0.06 \, {\rm fm}^2$ as predicted by the model proposed in Ref. [@BaMKPo2011] has the opposite sign as compared to the results of the experiments [@CaEtAl1969; @BrEtAl1972], that were carried out about four decades ago and observe a roughly 4% [*lower*]{} cross section for muons scattering off of protons as opposed to electrons being scattered off the same target. Likewise, in a recent paper on Lorentz-violating terms in effective Dirac equations [@GoKoVa2014], the authors assume an explicit breaking of electron-muon universality (see Sec. IIC3 of Ref. [@GoKoVa2014] where the authors explicitly state that they assume only muon-sector Lorentz violation, so that effects arise in H$_\mu$ spectroscopy but are absent in H spectroscopy and electron elastic scattering). Viewed with skepticism, this assumption appears to be a little artificial because it would modify the effective Dirac equation for muons as compared to that of electrons. In general, Lorentz-violating parameters may break rotational invariance, and thus have an effect on the $S$–$P$ transitions measured in [@PoEtAl2010; @AnEtAl2013] \[see the derivation in Eqs. (21)–(25) of Ref. [@GoKoVa2014]\]. In the virtual particle models from Refs. [@Je2011aop2; @CaRi2012; @BaMKPo2011], it has been found necessary to fine-tune the coupling constants in order to avoid conflicts with muon and electron $g-2$ measurements, which otherwise provide constraints on the size of the new physics terms due to their relatively good agreement with experiment (for a discussion, see Ref. [@Je2011aop2]). Furthermore, attempts to reconcile the difference based on higher moments of the proton charge distribution (its higher-order shape, see Ref. [@dR2010]) face difficulty when confronted with scattering experiments that set relatively tight constraints on the higher-order corrections to the proton’s shape. One class of models that has not been explored hitherto concerns electrodynamics with axion-like particles (ALPs, see Refs. [@PeQu1977; @GiJaRi2006prl; @AhGiJaRi2007; @AhGiJaReRi2008]). In the experiments [@PoEtAl2010; @AnEtAl2013], strong magnetic fields on the order of about $5 \, {\rm T}$ are used to collimate the muon beam. Axion terms could potentially influence the results of the spectroscopic measurements. We start from the Lagrangian [@Si1983; @Gi2008; @EhEtAl2009; @EhEtAl2010; @PDG2012] for a pseudoscalar ($0^-$) axion-like particle (temporarily setting $\hbar = c = \epsilon_0 = 1$) $$\begin{aligned} \label{calL} {\mathcal L} =& \; -\frac14 \, F^{\mu\nu} \, F_{\mu\nu} - \frac{g}{4} \, \phi \, \widetilde F^{\mu\nu} \, F_{\mu\nu} \nonumber\\[0.133ex] & \; + \frac{1}{2} \partial_\mu \phi \, \partial^\mu \phi - \frac{1}{2} m_\phi^2 \, \phi \nonumber\\[0.133ex] =& \; \frac12 \, (\vec E^2 - \vec B^2) + g \, \phi \, \vec E \cdot \vec B \nonumber\\[0.133ex] & \; + \frac{1}{2} \partial_\mu \phi \, \partial^\mu \phi - \frac{1}{2} m_\phi^2 \, \phi \,.\end{aligned}$$ Here, according to Ref. [@PDG2012], the axion’s two-photon coupling constant reads as $$g \equiv G_{A\gamma\gamma} = \frac{\alpha_{\rm QED}}{2 \pi f_A} \, \left( \frac{E}{N} - \frac23 \, \frac{4 + z}{1+z} \right)$$ ($\phi$ is the axion field, $m_\phi$ is the axion mass, $m_\phi \, f_A \approx m_\pi f_\pi$ where $f_\pi$ is the pion decay constant and $m_\pi$ the pion mass, while $z = m_u/m_d$ is the quark mass ratio). Grand unified models [@DiFiSr1981; @Zh1980; @Ki1979; @ShVaZa1980; @ChGeNi1995] assign rational fractions to the ratio $E/N$ of the electromagnetic to the color anomaly of the axial current associated with the axion. Possible values are $E/N = 8/3$ (see Refs. [@DiFiSr1981; @Zh1980]) or zero [@Ki1979; @ShVaZa1980]. In Eq. , the electromagnetic field strength tensor $F_{\mu\mu}$ and its dual $\widetilde F_{\mu\nu}$ have their usual meaning. It is interesting to consider the leading correction to the Coulomb potential in strong magnetic fields, on the order of $5 \, {\rm T}$, due to the axion-photon conversion amplitude inherent to the Lagrangian  (see Figs. \[fig1\] and \[fig2\]). We shall first assume that the vacuum expectation value of the axion field vanishes [@VCdP2013; @VC2014] and consider the tree-level correction to the Coulomb potential given in Fig. \[fig2\]. We match the scattering amplitude according to Chap. 83 of Ref. [@BeLiPi1982vol4] (see also [@CaLe1986]) and calculate the potential, generated by the axion-like particle, due to the diagram in Fig. \[fig2\]. The pseudoscalar ALP potential is given as $$\begin{aligned} V_{{\rm ALP} \, 0^-}(\vec k) =& \; (\vec k \cdot \vec B)^2 \; \frac{4 \pi Z \alpha \, g^2}{\vec k^{\,4} \, (\vec k^2 + m_\phi^2)} = (\vec k \cdot \vec B)^2 \; f(\vec k) \,, \nonumber\\[0.133ex] f(\vec k) =& \; \frac{4 \pi Z \alpha \, g^2}{\vec k^{\,4} \, (\vec k^2 + m_\phi^2)} \,.\end{aligned}$$ In coordinate space, we therefore have $$\begin{aligned} V_{{\rm ALP} \, 0^-}(\vec r) =& \; - \left( \vec B \cdot \vec \nabla \right)^2 f(\vec r) \,, \nonumber\\[0.133ex] f(\vec r) =& \; 4 \pi Z\alpha g^2 \, \left( \frac{{\mathrm{e}}^{-m_\phi \, r} - 1}{4 \pi \, m_\phi^4 \, r} - \frac{r}{8 \, \pi \, m_\phi^2} \right) \,.\end{aligned}$$ With $f(\vec r) = f(r)$, we have the second derivative as $$\left( \vec B \cdot \vec \nabla \right)^2 f(r) = \left( \frac{\vec B^{\,2}}{r} - \frac{(\vec B \cdot \vec r)^2}{r^3} \right) f'(r) + \frac{(\vec B \cdot \vec r)^2}{r^2} f''(r) .$$ Differentiating and expanding for small $m_\phi$, one obtains $$\begin{aligned} V_{{\rm ALP} \, 0^-}(\vec r) =& \; Z\alpha g^2 \, \left( \frac{\vec B^{\,2}}{3 m_\phi} - \frac{ \vec B^2 \, \vec r^{\,2} + (\vec B \cdot \vec r)^2}{8 \, r} \right) \nonumber\\[0.133ex] \sim & \; -\frac{Z\alpha g^2}{8 \, r} \, \left( \vec B^{\,2} \, \vec r^{\,2} + (\vec B \cdot \vec r)^2 \right)\end{aligned}$$ where we subtract the constant shift. This effective potential is independent of the ALP mass $m_\phi$ provided $m_\phi$ is much smaller than other mass scales in the problems, such as $m_e$ and $m_\mu$ (see also Fig. \[fig3\]). The $1S$ expectation value is $$\begin{aligned} \delta E = & \; \left< 1S \left| -\frac{Z\alpha g^2}{8 \, r} \, \left( \vec B^{\,2} \, \vec r^{\,2} + (\vec B \cdot \vec r)^2 \right) \right| 1S \right> \nonumber\\[0.133ex] =& \; - \frac{g^2 \, \vec B^{\,2}}{4 m_r} = - \epsilon_0 \, (\hbar \, c)^3 \, \frac{g^2 \, \vec B^{\,2}}{4 m_r} \,,\end{aligned}$$ where $m_r$ is the reduced mass of the bound system and SI mksA units are restored in the last step. Otherwise, according to Table 5 of Ref. [@EhEtAl2009], we have $$g < 4.9 \times 10^{-7} \, {\rm GeV} \,, \qquad m_\phi \lesssim 0.5 \, {\rm meV} \,.$$ For the parameters $| \vec B | = 5 \, {\rm T}$ and $g = 5 \times 10^{-7} {\rm GeV}^{-1}$, we obtain $$\delta E_H = -1.67 \times 10^{-31} \, {\rm eV} \,, \qquad \delta E_{\mu H} = -6.28 \times 10^{-34} \, {\rm eV} \,.$$ The smallness of these results excludes ALPs as possible explanations for the proton radius puzzle. A possible scenario with a nonvanishing vacuum expectation value of the axion field (see also Refs. [@DuvB2009; @BaCMDe2014]) is studied in Appendix \[appa\]. ![\[fig1\] ALP-photon conversion in a strong magnetic field according to the interaction term in the Lagrangian given in Eq. . The large encircled cross denotes the interaction with an external magnetic field.](fig1){width="0.5\linewidth"} ![\[fig2\] The leading (tree-level) correction to the Coulomb potential due to the ALP-photon interaction is given by the tree-level diagram shown. The upper fermion line corresponds to an electron $e$ (ordinary hydrogen) or a muon $\mu$ (muonic hydrogen).](fig2){width="0.6\linewidth"} ![\[fig3\] (Color.) Plot of the average field strength $E \equiv \langle E \rangle$ \[see Eq. \] experienced by a bound electron or muon in a one-muon ion (red line, $1 \leq Z \leq 5$), and for hydrogenlike (electronic) ions in the range $1 \leq Z \leq 92$. For comparison, the average field strength in a laser field of intensity $10^{24} \, {\rm W} \, {\rm cm}^{-2}$ is given [@YaEtAl2008]. The Schwinger critical field strength is denoted by $E_{\rm cr}$.](fig3){width="0.91\linewidth"} Strong–Field Electrodynamics {#sec3} ============================ Muonic bound systems have been used as probes of the strongest electromagnetic fields since the 1970s (see Ref. [@BrMo1978]), but progress was eventually hindered due to electron screening [@BeEtAl1986]. Typically, transitions in high-$Z$ muonic ions involve highly excited, non-$S$ states [@DeKeSaHe1980; @RuEtAl1984; @OfEtAl1991], where the average field experienced by the orbiting electron is reduced due to the higher principle quantum number. In view of the current muonic hydrogen discrepancy, it is useful to recall just how strong these fields are, especially in very simple bound systems, where shielding electrons are absent [@KaHi2012; @UmJo2014]. The conceivable presence of novel phenomena in the very strong electromagnetic fields within highly charged ions has been mentioned as a significant motivation for the study of these systems [@IoEtAl1988; @MoPlSo1998; @SoEtAl1998; @GuEtAl2005]. According to Eq. (2) of Ref. [@SoEtAl1998] and the more comprehensive discussion of Ref. [@IoEtAl1988], a conceivable nonlinear correction term (contact interaction) has been mentioned for high-field quantum electrodynamics. In view of this situation, it is indicated to compare the field strengths in highly charged (electronic) ions to those reached for low excited states in muonic hydrogen and low-$Z$ muonic ions. A measure for the strongest electromagnetic fields that can be described by perturbative electrodynamics is the Schwinger critical field strength [@Sa1931a; @Sa1931b; @HeEu1936; @Sc1951] $$\label{Ecr} E_{\rm cr} = 1.32 \times 10^{18} \, \frac{{\rm V}}{{\rm m}} \,.$$ The electric field around the proton reaches the Schwinger critical field already at a distance $0.116 \, a_\mu$ Bohr radii of the muonic hydrogen system, where $a_\mu$ is given in Eq. . Let us consider bound one-muon ions in the region of low nuclear charge numbers $1 \leq Z \leq 5$. The probability of finding a $1S$ muon inside the region of super-critical field strength, in one-muon ions of nuclear charge number $1 \leq Z \leq 5$, is evaluated as follows, $$\begin{aligned} p_{\rm cr}(Z=1) =& \; 0.17 \, \% \,, \\[0.133ex] p_{\rm cr}(Z=2) =& \; 1.18 \, \% \,, \\[0.133ex] p_{\rm cr}(Z=3) =& \; 3.36 \, \% \,, \\[0.133ex] p_{\rm cr}(Z=4) =& \; 6.73 \, \% \,, \\[0.133ex] p_{\rm cr}(Z=5) =& \; 11.2 \, \% \,.\end{aligned}$$ The field scales as $1/r^2$ for small distances. In Fig. \[fig3\], to supplement a corresponding investigation in Fig. 2 of Ref. [@SoEtAl1998], we investigate the electric field strength experienced by a bound muon in a muonic hydrogenlike system (only one orbiting particle) in the region of low nuclear charge number. We start from the ground-state expectation value of the electric-field operator, which is obtained as the gradient of the Coulomb potential. Within the nonrelativistic approximation (in SI mksA units), the result reads \[std\] $$\begin{aligned} \label{stda} & \langle E \rangle = \left< 1S \left| \left( - \frac{\partial}{\partial r} \frac{Z |e|}{4 \pi \epsilon_0 \, r} \right) \right| 1S \right> = 2 \, Z^3 \, \frac{m_r^2}{m_e^2} \, {\mathcal E}_0 \,, \\[0.133ex] \label{stdb} & {\mathcal E}_0 = \frac{e \, \alpha_{\rm QED}^2 \, m_e^2 \, c^2}{4 \pi \, \epsilon_0 \, \hbar^2} = 5.14 \times 10^{11} \, \frac{{\rm V}}{{\rm m}} \,.\end{aligned}$$ Here, $m_r$ is the reduced mass of the atomic system, $m_e$ is the electron mass, and ${\mathcal E}_0 $ denotes the “standard” atomic field strength observed at one Bohr radius in a standard hydrogen atom (it is equal to the atomic unit of the electric field strength). The prefactor $2$ in Eq.  is a consequence of our taking the quantum mechanical expectation value as opposed to evaluating the classical expression at the (shifted) Bohr radius. For ultra-relativistic systems, Eq.  is replaced by the expectation value of the fully relativistic Dirac–Coulomb wave function [@SwDr1991a]; the relativistic correction factor amounts to the replacement $$\langle E \rangle \mapsto \frac{\langle E \rangle}{2 - \sqrt{1 - (Z\alpha_{\rm QED})^2} - 2 \, (Z\alpha_{\rm QED})^2} \,,$$ which does not change the order-of-magnitude of the result. The decisive factor in Eq.  is the prefactor $Z^3 \, (m_r/m_e)^2$, which is responsible for an enhancement of the field strength by six orders of magnitude in the range $1 \leq Z \leq 92$ for the electronic system, but also for a considerable enhancement in muonic systems, where $$\left( \frac{m_r}{m_e} \right)^2 \to \left( \frac{m_\mu \, m_p}{(m_\mu + m_p) \, m_e} \right)^2 \approx 3.45 \times 10^4 \,.$$ For a one-muon ion, the average electric field strengths at $Z = 4$ and $Z = 5$ surpass the average electric field strength in hydrogenlike Uranium (see Fig. \[fig3\]). Furthermore, the average field strength experienced by a bound $1S$ electron in one-muon ions with $Z=4$ and $Z=5$ is given as $$\begin{aligned} \langle E \rangle_{\mu,Z=4} =& \; 1.72 \, E_{\rm cr} \,, \\[0.133ex] \langle E \rangle_{\mu,Z=5} =& \; 3.36 \, E_{\rm cr} \,,\end{aligned}$$ thus surpassing (in terms of quantum mechanical average) the Schwinger critical field strength. The HERCULES laser [@YaEtAl2008] (still) sets the standard for the highest achievable laser intensities to date, with a peak intensity of about $2 \times 10^{22} \, {\rm W} \, {\rm cm}^{-2}$. In the future, such facilities are supposed to reach intensities in the range $10^{23}$—$10^{24} \, {\rm W} \, {\rm cm}^{-2}$. An intensity of $10^{24} \, {\rm W} \, {\rm cm}^{-2}$ corresponds to an electric field strength of $$E_L = 2.74 \times 10^{15} \, \frac{{\rm V}}{{\rm m}} \,,$$ which is surpassed in the muonic system ($1 \leq Z \leq 5$) as well as medium-$Z$ and high-$Z$ bound quantum electrodynamic (QED) systems (with $Z \geq 14$, see Fig. \[fig3\]). It is thus evident that bound muonic systems offer a competing alternative to the exploration of the strong-field QED regime, complementary to strong laser systems [@Ke2001]. One might argue that the time average of the oscillating laser fields is zero, as much as the spatial (vector) average of the electric field (vector), taken over the spherically symmetric $S$ wave function, vanishes. However, the exploration of the strong-field domain of electrodynamics is not precluded by the oscillating or spherically symmetric nature of the fields. One easily estimates that the (fluctuating) electric fields inside the proton, given the fact that the three valence quarks cannot be further apart than $0.8\,{\rm fm}$, are of order $E_p \sim 10^{21} \, \frac{{\rm V}}{{\rm m}}$ and thus exceed the Schwinger critical field strength $E_{\rm cr}$ of about $E_{\rm cr} = 1.32 \times 10^{18} \, \frac{{\rm V}}{{\rm m}}$ by three orders of magnitude. Conceivable corrections to the muonic hydrogen spectrum due to the high field strengths have been discussed in Refs. [@Je2013pra; @PaMe2014; @Mi2015]. Just to avoid a misunderstanding, we should clarify that the recently discussed hypothesis of nonperturbative lepton pairs inside the proton [@Je2013pra; @PaMe2014; @Mi2015] certainly does not imply the production of such pairs from the vacuum inside the nucleus; the vacuum is known to “spark” only if the critical field strength is maintained over a sufficiently large space-time interval which is absent in muonic hydrogen. The hypothesis discussed in Refs. [@Je2013pra; @PaMe2014; @Mi2015] merely implies that the highly nonperturbative nature of strong interactions (quantum chromodynamics) inside the proton, which involves electrically charged constituent as well as sea quarks, might lead to effective lepton-proton interactions which have so far been overlooked in theoretical treatments (see Refs. [@Je2013pra; @PaMe2014; @Mi2015] and Appendix \[appb\]). Finally, a remark on the relationship of the light muonic systems and the strong electric fields to the “classical” strong-field systems (highly charged ions) is in order. In these latter systems, the (initially positive-energy) $1S$ level can be shown to approach the negative continuum, effectively “sparking” the vacuum [@ZePo1971; @RaMuGr1974]. A single proton of course is unable to create such an effect, but the proximity of the bound muon to the proton (nucleus) generates the extreme fields and the corresponding quantum mechanical expectation values that contribute to the interest in muonic bound systems. Non–Resonant Effects and Transition Frequencies {#sec4} =============================================== Discrepancies of Lamb shift experiments and theory have been explored for a long time. For example, a rather well-known accurate Lamb shift experiment in helium [@vWKwDr1991] had long been in disagreement with theory (the discrepancy was resolved in Refs. [@vWHoDr2000; @JeDr2004]). A measurement of the $^4$He nuclear radius using muonic helium ions is currently in progress [@PoPriv2014]. In many cases, nuclear radius determinations using electronic and muonic bound systems complement each other [@Je2011aop2]. One may add that additional experiments on electronic helium ions (as opposed to muonic helium ions) would be able to shed additional light on the “generalized” proton radius puzzle, or “nuclear size effect puzzle”, because they would enable us to compare the “electronically measured” radius of $^4$He with the “muonically measured” radius; a corresponding experimental setup was recently proposed [@HeEtAl2009]. In particular, it would be rather interesting to compare the “anisotropy method” used in Refs. [@vWKwDr1991; @vWHoDr2000] with other spectroscopic techniques. Historical developments encourage us to search for additional conceivable explanations of the proton radius puzzle in systematic effects that may not have been fully appreciated in even the most carefully planned experiments. One such set of corrections is given by so-called off-resonant corrections to frequency measurements. In Ref. [@CaRaSt1982], it was stressed that an accurate understanding of the line shape of quantum transitions to neighboring levels can lead to surprising phenomena such as prevention of fluorescence; for precision experiments, this finding highlights the necessity of including a good line-shape model. Because the non-resonant corrections to the line shape involve mixed products of dipole operators connecting the resonant and off-resonant levels, these effects are also referred to as “cross-damping” terms in quantum optics [@FiSw2002; @FiSw2004] \[see also Eq. (9) of Ref. [@JeMo2002]\]. In Sec. III of Ref. [@JeMo2002] \[see the text after Eq. (15) [*ibid.*]{}\], the authors investigate off-resonant effects in differential as opposed to angular-averaged cross sections. Quantum interference effects can be excluded as an explanation of the proton radius discrepancy in muonic systems [@AmEtAl2015], mainly because the proton radius discrepancy, converted to frequency units, is much larger than the natural linewidth of the transitions in the muonic systems. However, the situation is different for atomic hydrogen, where spectral lines have to be split to much higher relative accuracy. In order to gauge possible concomitant systematic shifts of the accurately measured frequencies, especially those involving highly excited states of (atomic) hydrogen and deuterium, improved measurements of hydrogen $2S$–$nP$ lines are currently being pursued [@UdPriv2014], while an improved measurement of the “classic” $2S$–$2P_{1/2}$ Lamb shift is also planned [@HePriv2014]. Both of these experiments have the potential of clarifying the “electronic hydrogen” side of the proton radius puzzle. In order to understand the importance of the nonresonant terms, and see if they can potentially contribute to the explanation of the proton radius puzzle, let us recall that a typical nonresonant energy shift due to neighboring levels, still displaced by an energy shift $\Delta E_n$ commensurate with a change in the principal quantum number, is [@JeMo2002; @Lo1952] $$\label{low} \delta E = \frac{(\hbar \, \Gamma)^2}{\Delta E_n} \sim \alpha_{\rm QED}^8 \, m_e \, c^2 \,,$$ where $\Gamma$ is the decay width of the reference state and the term after the “$\sim$” sign is a parametric estimate according to the $Z\alpha_{\rm QED}$-expansion [@BeSa1957]. The shift , which according to Low [@Lo1952] defines the ultimate limit to which energy levels can be resolved in spectroscopic experiments, is too small to explain the proton radius puzzle (we have $\hbar \, \Gamma \sim \alpha_{\rm QED}^5 \, m_e \, c^2$, while $\Delta E_n \sim \alpha_{\rm QED}^2 \, m_e \, c^2$ for a transition with a change in the principal quantum number). By contrast, in differential cross sections, the shift due to neighboring levels removed only by the fine-structure is proportional to [@JeMo2002] $$\label{high} \delta E = \frac{(\hbar \, \Gamma)^2}{\Delta E_{\rm fs}} \sim \alpha_{\rm QED}^6 \, m_e \, c^2 \,,$$ where $\Delta E_{\rm fs}\sim \alpha_{\rm QED}^4 \, m_e \, c^2$ is of the order of a typical fine-structure interval. According to Eqs. (9) and (12) of Ref. [@JeMo2002], there is an additional prefactor $1/2$ to consider for the shift of the center of the half-maximum values of the resonance curve, while this prefactor is $1/4$ for the Lorentzian maximum itself. The presence of this additional prefactor has no effect on the phenomenological significance of the estimates to be discussed in the following. The shift given in Eq.  is of sufficient magnitude to explain the muonic hydrogen discrepancy. Let us perform some order-of-magnitude estimate to explore the possibility of explaining the proton radius puzzle on the basis of non-resonant corrections. The reduced electron Compton wavelength is $\lambdabar_C = \hbar/(m_e \, c) = 386.159 \, {\rm fm}$. The ratio of $\lambdabar_C$ to the proton radius, which we assume to be given by $r_p \approx 0.88 \, {\rm fm}$, is given as $$\xi = \frac{r_p}{\lambdabar_C} = 2.27 \times 10^{-3} \,.$$ According to Eq. (51) of Ref. [@MoTaNe2008] (see also Table 10 of Ref. [@EiGrSh2001]), the leading-order finite-size effect for the $2S$ state is (non-recoil limit), $$E_{\rm FS} = \frac{1}{12} \, (Z\alpha)^4 \, m_e c^2 \, \xi^2 \approx 150 \, {\rm kHz} .$$ We defined the “proton puzzle prefactor” $\chi_{\rm PP}$ as $$\chi_{\rm PP} = \frac{0.88^2 - 0.84^2}{0.88^2} = 0.089 \,,$$ leading to a “proton puzzle energy shift” $E_{\rm PP}$ for the $2S$ state of $$E_{\rm PP} = \chi_{\rm PP} \, E_{\rm FS} \approx 13 \, {\rm kHz} .$$ We aim to investigate the possible presence of significant off-resonant corrections to the $2S$–$4P_{1/2}$ and $2S$–$4P_{3/2}$ frequencies [@BeHiBo1995], as well as $2S$–$8D_{3/2}$ and $2S$–$8D_{5/2}$ frequencies [@BeEtAl1997], and $2S$–$12D$ transitions [@ScEtAl1999]. To this end, we first recall that the fine-structure energy difference, for $P$ and $D$ states in hydrogen, is \[chiF\] $$\begin{aligned} {{\mathcal F}}_{nP} = & \; E_{nP_{3/2}} - E_{nP_{1/2}} = \chi_{{{\mathcal F}}P} \, \frac{(Z\alpha)^4 \, m_e \, c^2}{n^3} \,, \\[0.133ex] {{\mathcal F}}_{nD} = & \; E_{nD_{5/2}} - E_{nD_{3/2}} = \chi_{{{\mathcal F}}D} \, \frac{(Z \alpha)^4 \, m_e \, c^2}{n^3} \,, \\[0.133ex] \chi_{{{\mathcal F}}P} =& \; \frac14 \,, \qquad \qquad \chi_{{{\mathcal F}}D} = \frac{1}{12} \,.\end{aligned}$$ According to p. 266 of Ref. [@BeSa1957], the one-photon decay width of $nP$ and $nD$ states can be estimated as (independent of the total angular momentum) \[chiGamma\] $$\begin{aligned} \Gamma_{nP} \approx & \; \chi_{\Gamma P} \, \frac{\alpha \, (Z\alpha)^4 m_e c^2}{\hbar n^3} \,, \qquad \chi_{\Gamma P} = 0.311 \,, \\[0.133ex] \Gamma_{nD} \approx & \; \chi_{\Gamma D} \, \frac{\alpha \, (Z\alpha)^4 m_e c^2}{\hbar n^3} \,, \qquad \chi_{\Gamma D} = 0.107 \,.\end{aligned}$$ We now focus on a potential nonresonant correction to the transition frequencies, due to neighboring fine-structure levels. This choice is motivated in part by a remark in the text in the right-hand column of the second page of Ref. [@BeEtAl1997], where it is confirmed that neighboring hyperfine structure levels are taken into account in the line-shape model used in Ref. [@BeEtAl1997] (but those levels displaced by the fine structure apparently are not taken into account). According to Ref. [@JeMo2002], in an angle-differential cross section, the off-resonant shift due to neighboring fine-structure levels can be estimated as follows. For a $2S$–$nP$ transition, $$E_{\rm OR} = \frac{(\hbar \Gamma)^2_{n}}{{{\mathcal F}}_{n}} = \frac{\chi_{\Gamma}^2}{\chi_{{{\mathcal F}}}} \, \frac{\alpha^2 \, (Z\alpha)^4 m_e c^2}{n^3} \,,$$ where one has to replace the prefactors as $\chi_{\Gamma} \to \chi_{\Gamma P,D}$ and $\chi_{{{\mathcal F}}} \to \chi_{{{\mathcal F}}P,D}$, respectively, according to the estimates given in Eqs.  and . It is interesting to investigate the ratio of the proton size puzzle energy shift to the natural linewidth as a measure of how precisely the line has to be split in order to resolve the proton size puzzle. It is given as ($2S$-$nP$ transitions), $$R_P = \frac{E_{\rm PP}}{\hbar \Gamma_{nP}} = \frac{n^3 \, \chi_{\rm PP} \, \xi^2}{12 \, \alpha \, \chi_{\Gamma P}} = 1.68 \times 10^{-5} \, n^3 \,.$$ Example values for $2S$-$nP$ are $R_P(n=4) = 0.0011$, $R_P(n=8) = 0.008$, and $R_P(n=12) = 0.029$. So, in order to resolve the proton size puzzle based on the $2S$-$4P$ transition, one has to understand the line width to better than 1 part in 1000. The work in [@BeHiBo1995] reaches an accuracy close to this limit: The experimental accuracy for the $2S$-$4P$ transitions is on the order of $\sim 12 \, {\rm kHz}$, to be compared to a natural line width of $\sim 13 \, {\rm MHz}$. The ratio $R_P$ becomes significantly more favorable for transitions to higher excited $P$ states. The corresponding estimate for $2S$-$nD$ transitions is $$R_D = \frac{E_{\rm PP}}{\hbar \Gamma_{nD}} = \frac{n^3 \, \chi_{\rm PP} \, \xi^2}{12 \, \alpha \, \chi_{\Gamma D}} = 4.86 \times 10^{-5} \, n^3 \,.$$ For $2S$–$nD$ transitions with $n=4,8,12$, we have $R_D(n=4) = 0.0031$, $R_D(n=8) = 0.025$, and $R_D(n=12) = 0.084$. It means that in order to resolve the proton size puzzle based on the $2S$-$12D$ transition [@ScEtAl1999], one has to understand the line width only to (roughly) 1 part in 12. Another interesting quantity is the ratio of the off-resonant terms to the natural linewidth. It measures how accurately the natural line width has to be split in order to see the off-resonant effects. For $2S$–$nP$ transitions and $2S$–$nD$ transitions, it is given by $$\begin{aligned} S_P =& \; \frac{E_{\rm OR}}{\hbar \Gamma_{nP}} = \frac{\alpha \, \chi_{\Gamma P}}{\chi_{{{\mathcal F}}D}} \approx \frac{1}{110} \,, \\[0.133ex] S_D =& \; \frac{E_{\rm OR}}{\hbar \Gamma_{nD}} = \frac{\alpha \, \chi_{\Gamma D}}{\chi_{{{\mathcal F}}D}} \approx \frac{1}{106} \,,\end{aligned}$$ independent of $n$. It is also very important to compare the “proton size puzzle energy shift” to the off-resonant shift. It is given by ($2S$–$nP$ transitions) $$T_P = \frac{E_{\rm PP}}{E_{\rm OR}} = \frac{R_P}{S_P} = \frac{n^3 \, \chi_{\rm PP} \, \chi_{{{\mathcal F}}P} \, \xi^2 }{12 \, \alpha^2 \, \chi_{\Gamma P} } = 1.85 \times 10^{-3} \, n^3 \,.$$ For the $2S$–$4P$ transition, one has $T_P(n=4) = 0.118$, implying that the off-resonant, cross-damping shift due to neighboring fine-structure levels is roughly ten times larger than the energy shift corresponding to the proton size puzzle for the $2S$ level. We conclude that, unless one uses an appropriate $4 \pi$ detector to eliminate the nonresonant terms, one has to understand the line shape of the $2S$–$4P$ transition extremely well in order to resolve proton radius puzzle based on this transition. From a complementary viewpoint, the line shape of the $2S$–$4P$ transition could be an an excellent tool for studying the nonresonant cross-damping terms. For $2S$-$nD$ transitions, we have $$T_D = \frac{E_{\rm PP}}{E_{\rm OR}} = \frac{R_D}{S_D} = \frac{n^3 \, \chi_{\rm PP} \, \chi_{{{\mathcal F}}D} \, \xi^2 }{12 \, \alpha^2 \, \chi_{\Gamma D} } = 5.16 \times 10^{-3} \, n^3 \,.$$ Examples are $T_D(n=8) = 2.64$, and $T_D(n=12) = 8.92$. For the $2S$–$8D$ transitions and $2S$–$12D$ transitions studied in Refs. [@BeEtAl1997] and [@ScEtAl1999], respectively, this means that the estimated ratio of the proton size puzzle energy shift to the off-resonant contribution is larger than unity. One could thus tentatively conclude that the inclusion of any conceivable nonresonant corrections is not likely to shift the experimental results reported in Refs. [@BeEtAl1997] and [@ScEtAl1999] on a level commensurate with the proton radius puzzle energy shift. In summary, our estimates would suggest that $2S$–$nD$ transitions to highly excited $D$ states provide for the most favorable “signal-to-noise” ratio $E_{\rm PP}/E_{\rm OR}$ \[ratio of proton size puzzle energy shift to the off-resonant energy shift, with $T_D(n=12)=8.92$\]. In view of $R_D(n=12) = 0.084$, the proton puzzle energy shift enters at about $1/12$ of the natural line width [@ScEtAl1999] for $n=12$. Because $S_D \approx 1/106$, the off-resonant terms are suppressed by about two orders of magnitude in relation to the natural linewidth, which is smaller than the proton radius puzzle energy shift by roughly another order of magnitude. An inspection of Fig. 1 of Ref. [@BeEtAl2013] (see also Fig. 1 of Ref. [@BeEtAl2013jpconf]) would indicate that the $2S$–$8D$ and $2S$–$12D$ transitions are consistent with a proton radius, derived from hydrogen experiments, which is significantly larger than the muonic hydrogen result. A least-squares analysis of all accurately measured hydrogen transitions yields the proton radius $r_p = 0.8802(80) \, {\rm fm}$ (see Table XLV of Ref. [@MoTaNe2012]). For comparison, we have exclusively taken the data from the $2S$–$8D$ and $2S$–$12D$ transitions reported in Refs. [@BeEtAl1997; @ScEtAl1999], together with the latest $1S$–$2S$ result [@MaEtAl2013prl], and current theory as summarized in Refs. [@MoTaNe2008; @MoTaNe2012], and calculated a naive statistical average of the proton radii derived from $2S$–$8D$ and $2S$–$12D$ transitions (disregarding the covariances among the data which otherwise leads to a much more accurate value for the proton radius [@JeKoLBMoTa2005]). With this approach, the result from $2S$–$8D$ and $2S$–$12D$ transitions alone is $r_p = 0.873(17) \, {\rm fm}$, still larger than the muonic hydrogen value by $2\sigma$. While the reconsideration of cross-damping terms for hydrogen transitions would be very helpful in clarifying a conceivable contribution to the solution of the proton size puzzle, our estimates suggest that it would be very surprising if the proton size puzzle were to find a full explanation based on the cross-damping terms alone. The off-resonant terms seem to be most effectively suppressed in transitions to highly excited $D$ states. Conclusions {#sec5} =========== In this paper, we explored the remaining options for the explanation of the persistent proton radius discrepancy [@PoEtAl2010; @AnEtAl2013]. Specifically, in Sec. \[sec2\], we supplemented previous attempts to find an explanation for the proton radius puzzle based on “subversive” virtual particles; all of these appear to require fine-tuning of the coupling constants and no compelling set of quantum numbers has as yet been found for the virtual particle which could potentially explain the discrepancy of theory and experiment in (muonic) hydrogen within the limits set by other precision experiments such as the electron and muon $g$ factors. Virtual particle explanations appear to be disfavored at the current stage, and other models depend on rather drastic hypotheses such as symmetry breaking terms that affect only the muon sector of the Standard Model (but not electrons or positrons). Here, we supplemented the discussions of virtual particles by a calculation of the effective potential describing the leading correction to the Coulomb interaction due to axion–photon conversion in the (strong) magnetic fields used in the muonic hydrogen experiments [@PoEtAl2010; @AnEtAl2013]. In Sec. \[sec3\], we continued to explore the typical electric fields in a low-$Z$ bound muonic system. These fields are seen to be commensurate with, or even exceed the Schwinger critical field strength. Because of the lack of electron screening, the one-muon ions can be interpreted as the most sensitive probes of high-field physics to date. The hypothesis of nonperturbative lepton pairs inside the proton and their conceivable influence on electron-proton and muon-proton interactions (see Refs. [@Je2013pra; @PaMe2014; @Mi2015]) is based on the interplay of nonperturbative quantum chromodynamics with quantum electrodynamics (see Appendix \[appb\]). A breakdown of perturbative quantum electrodynamics is not necessary for the existence of the conjectured effect [@Mi2015]. Muon-proton scattering experiments will be an important cornerstone in the further clarification of the electron-muon universality in lepton-proton interactions (MUSE collaboration, see Ref. [@KoEtAl2014]). Finally, in Sec. \[sec4\], the role of nonresonant line shifts in differential as opposed to total cross sections was mentioned. Two ongoing experimental efforts [@UdPriv2014; @HePriv2014] share the aim of analyzing the process-dependent line shifts [@Lo1952] further. Transitions to highly excited $D$ states ($2S$–$nD$ transitions) in hydrogen are identified in terms of favorable parameters for the suppression of nonresonant correction terms (cross-damping terms), which otherwise could account for hitherto unexplored systematic effects in atomic hydrogen experiments. For the muonic hydrogen experiments, in contrast to the experiments on ordinary hydrogen, it is not necessary to “split” the resonance line in order to make the proton radius puzzle manifest; the discrepancy is much larger than the width of the resonance line itself (see Fig. 5 of Ref. [@PoEtAl2010]). The binding field strengths in muonic ions exceed those achievable in current and projected high-power laser systems. The benefit of the low-$Z$ muonic ions produced in the high-intensity muon beams at the Paul–Scherrer–Institute (PSI) lies in the “clean” environment provided by the one-muon ions, where all other bound electrons have been stripped and the interaction of the muon and the nucleus can be investigated spectroscopically to high accuracy. From a theoretical point of view, it appears to be hard to shed any further light on the proton radius puzzle without significant further stimulation from additional experimental spectroscopic or scattering data. Helpful conversation with I. Nándori and M. M. Bush are gratefully acknowledged. The author is grateful to the Mainz Institute for Theoretical Physics (MITP) for its hospitality and its partial support during the completion of this work. This research was supported by the National Science Foundation (Grants PHY–1068547 and PHY–1403973). Heisenberg–Euler Lagrangian and Variational Calculus {#appa} ==================================================== In many cases, the leading perturbation to the Coulomb potential due to a new interaction can be obtained by variational calculus; we illustrate the procedure here, on the basis of the Wichmann–Kroll correction to the Coulomb potential. The Maxwell Lagrangian with the Heisenberg–Euler Lagrangian reads as (we switch to natural units with $\hbar = c = \epsilon_0 = 1$) $$\begin{aligned} \label{LeffSOURCE} {{\mathcal L}}=& \; \tfrac{1}{2} \, \left( \vec{E}^2 - \vec{B}^2 \right) \nonumber\\[0.133ex] & \; + \frac{2 \, \alpha_{\rm QED}^2}{45 m^4} \left( \vec{E}^2 - \vec{B}^2 \right)^2 + \frac{14 \, \alpha_{\rm QED}^2}{45 m^4} \left( \vec{E} \cdot \vec{B} \right)^2 \,.\end{aligned}$$ If $\vec E$ is given by the gradient of a Coulomb field and the magnetic field vanishes ($\vec B = \vec 0$), then ${{\mathcal L}}$ is redefined to the expression $${{\mathcal L}}= \frac{1}{2} \, \left( \vec{\nabla} \Phi \right)^2 + \frac{2 \, \alpha_{\rm QED}^2}{45 m^4} \, \left( \vec{\nabla} \Phi \right)^4 - \rho \, \Phi \,,$$ where we add the source term. In view of the relations $$\frac{\partial {{\mathcal L}}}{\partial \vec\nabla \Phi} = \vec{\nabla} \Phi + \frac{8 \, \alpha_{\rm QED}^2}{45 m^4} \, \, \vec\nabla \Phi \, \left( \vec\nabla \Phi \right)^2 \,, \qquad \frac{ \partial {{\mathcal L}}}{\partial \Phi} = -\rho \,,$$ the variational equation $\vec\nabla \cdot \frac{\partial {{\mathcal L}}}{\partial \vec\nabla \Phi} = \frac{ \partial {{\mathcal L}}}{\partial \Phi}$ becomes $$\vec{\nabla}^2 \Phi + \frac{8 \, \alpha_{\rm QED}^2}{45 m^4}\, \left( \vec\nabla^2 \Phi \, \left( \vec\nabla \Phi \right)^2 + \vec\nabla \Phi \cdot \vec\nabla \left( \vec\nabla \Phi \right)^2 \right) = -\rho \,,$$ which can be reformulated as $$\vec{\nabla}^2 \Phi + \frac{8 \, \alpha_{\rm QED}^2}{45 m^4} \, \left( \partial_r + \frac{2}{r} \right) \, \left( \partial_r \Phi \right)^3 = -\rho \,.$$ where we assume that $\Phi$ is radially symmetric. We set $\Phi = \Phi_C + \Xi$ where $\Phi_C$ is the Coulomb potential and $\Xi$ is a quantum correction. The charge density of the nucleus and the Coulomb potential are given by $$\rho(\vec r) = Z \, |e| \, \delta^{(3)}(\vec r) \,, \qquad \Phi_C(\vec r) = \frac{Z \, |e|}{4 \pi \, r} \,,$$ where $\vec\nabla^2 \, \Phi_C(\vec r) = -\rho(\vec r)$, so that, to first order in $\Xi$, $$\label{plus_xi} \left( \partial_r^2 + \frac{2}{r} \, \partial_r \right) \Xi + \frac{8 \, \alpha_{\rm QED}^2}{45 m^4} \, \left( \partial_r + \frac{2}{r} \right) \left( \partial_r \Phi_C \right)^3 = 0 \,.$$ It is straightforward to observe that Eq.  is solved by a potential proportional to $r^{-5}$, $$\label{asymp} \Phi = \Phi_C + \Xi = \frac{Z \, |e|}{4 \pi \, r} \, \left( 1 - \frac{2}{225} \, \frac{\alpha_{\rm QED}}{\pi} \, \frac{(Z\alpha_{\rm QED})^2}{(m \, r)^4} \right) \,,$$ This is equal to the long-distance tail of the Wichmann-Kroll potential [@WiKr1954; @WiKr1956], which is relevant to a distance range $r \sim a_0$, where $a_0$ is the Bohr radius; we here confirm the result given in Appendix III of Ref. [@WiKr1956]. A few remarks are in order. Matrix elements of a term of order $(\alpha_{\rm QED}/\pi) \, (Z\alpha_{\rm QED})^3/(m^4 \, r^5)$ \[see Eq. \] generate an energy shift proportional to $\alpha_{\rm QED} \, (Z\alpha_{\rm QED})^8 \, m$ in hydrogenlike systems. By contrast, the leading term in the Wichmann–Kroll potential is otherwise proportional to a Dirac-$\delta$ function and generates an energy shift of the order of $\alpha_{\rm QED} \, (Z\alpha_{\rm QED})^6 \, m$. The latter term is given by the high-energy (short-distance) regime not covered by our variational ansatz. Namely, the atomic nucleus and the Coulomb potential and its derivative, the Coulomb field, vary considerably on the length scale of an electron Compton wavelength, which exceeds the “operational parameters” of the Heisenberg–Euler Lagrangian, so the result  cannot be used for distances closer than an electron Compton wavelength, i.e., it fails in the immediate vicinity of the nucleus. One might wonder why the functional form of the long-distance tail ($1/r^5$ for the Wichmann–Kroll potential) is different from the corresponding term for the Uehling potential, which decays exponentially at large distances (see [@Je2011aop1] and references therein). The answer to this question is that the Wichmann–Kroll potential, which is generated by Feynman diagrams with at least four electromagnetic interaction terms inside the loop, can be related to the Heisenberg–Euler effective Lagrangian, which is valid for the long-distance tail of the potential, while the corresponding term, for the Uehling potential (with only two electromagnetic interaction terms inside the loop) would otherwise generate a term proportional to $\vec E^2$ that is absorbed in the $Z_3$ renormalization of the electromagnetic charge [@ItZu1980]. Hence, the tail of the Uehling potential decays exponentially, akin to a Yukawa potential, with a range of the potential being proportional to the electron Compton wavelength (Sec. 2.4 of Ref. [@Je2011aop1]). After these intermediate considerations, we may proceed to apply our variational ansatz to a calculation of interest in the context of the subject matter of the current investigation. Namely, for a nonvanishing vacuum expectation value $\left< \phi \right> \neq 0$ of the axion-like particle as a dark matter candidate, the Lagrangian [@DuvB2009; @BaCMDe2014] $$\label{LA} {{\mathcal L}}_A = \frac12 \, (\vec E^2 - \vec B^2) + g \, \left< \phi \right> \, \vec E \cdot \vec B$$ is exact up to possible QED or axion loop corrections; in contrast to the Heisenberg-Euler Lagrangian, it is not the result of “integrating out” the fermionic degrees of freedom which limits the “operational parameters” of the Lagrangian . Hence, we are not at risk of exceeding the “operational parameters of the variational ansatz” when we use the axion background Lagrangian  to calculate a possible correction to the Coulomb potential due to dark matter physics. If $\vec E = -\vec\nabla \Phi$ is generated by a (possibly distorted) Coulomb field and $\vec B$ is the (possibly inhomogeneous) external magnetic field, then the Lagrange density ${{\mathcal L}}_A$ is redefined to read (adding the source term $\rho \, \Phi$) $${{\mathcal L}}_A = \frac{1}{2} \, \left( \vec{\nabla} \Phi \right)^2 - g \, \left< \phi \right> \, \vec B \cdot \vec\nabla \Phi - \rho \, \Phi \,.$$ The variational equation $$\label{variat} \vec\nabla \cdot \frac{\partial {{\mathcal L}}_A}{\partial \vec\nabla \Phi} = \frac{ \partial {{\mathcal L}}_A}{\partial \Phi}$$ becomes $$\vec{\nabla}^2 \Phi - g \, \left< \phi \right> \, \vec\nabla \cdot \vec B = -\rho \,.$$ In the absence of magnetic monopoles, the leading correction to the Coulomb potential mediated by the axion vacuum expectation value thus vanishes, even for very strong and inhomogeneous external magnetic fields $\vec B$. One more remark is in order. The direct coupling of the fermion to the axion [@PDG2012] is of the derivative form ${{\mathcal L}}_{Aff} = (C_f/(2 f_A)) \, \overline\psi_f \, \gamma^\mu \, \gamma_5 \, \psi_f \, \partial_\mu \phi$, where $C_f$ is a model-dependent constant. The Yukawa coupling is $g_{Aff} = C_f m_f/f_A$ and the “fine-structure constant” is $g^2_{Aff}/(4 \pi)$; energy loss arguments from the SN1987A supernova typically give bounds in the range $g^2_{Aff}/(4 \pi) \sim 10^{-21}$ (see Refs. [@GrMaPe1989; @Ra1990prep]). This implies that a single axion exchange, or an axion interaction insertion (e.g., into the fermion line of a vacuum polarization diagram) suffers from a suppression factor on the order of $g^2_{Aff}/(4 \pi) \sim 10^{-21}$ and is thus suppressed with respect to the corresponding photon exchange diagram (coupling parameter $\alpha_{\rm QED}$) by roughly 18 orders of magnitude. The fermion-axion coupling thus is too small to explain the proton radius puzzle. Axion-mediated effects as well as weak interactions [@Ei2012] can thus also be excluded as possible explanations for the proton radius puzzle. Strong Fields in the Proton,\ Interplay of QED and QCD {#appb} ============================= The presence of a very small fraction of light sea fermions, conceivably due to a nonperturbative mechanism, inside the proton, was recently mentioned in Refs. [@Je2013pra; @PaMe2014]. One might counter-argue that the QED running coupling constant, at distances commensurate with the proton radius, still is small against unity, and that this precludes a nonperturbative mechanism leading to sea fermions inside the proton. In Sec. III of Ref. [@Je2013pra], it is argued that the highly nonperturbative quantum chromodynamic (QCD) nature of a hypothetical electrically neutral proton receives a correction due to the electroweak interactions, as they are switched back on, and that, due to the highly nonlinear nonperturbative nature of QCD, this reshaping can be much larger than the electromagnetic perturbation itself. Alternatively, one might argue that the fundamentally nonperturbative nature of the QCD interaction inside the proton might leave room for effects that cannot be described by dispersion relations and perturbation theory alone. Namely, due to the nonperturbative nature of QCD, the three valence quarks of the proton are supplemented, at any given time, by a large number of virtual sea quarks that emerge from the vacuum due to quantum corrections to the gluon exchange [@GaLa2009]. The sea quarks, as much as the valence quarks, are electrically charged, off of their mass shell, and may exchange photons. The propagator of these photons, in turn, receives a correction due to vacuum polarization; hence, at any given time, the proton wave function has a nonvanishing electron-positron content due to the light fermionic vacuum bubbles. This is a persistent phenomenon because the quarks inside the proton are always highly virtual (off mass shell) in view of their strong (mutual) interactions [@Mi2015]. In Ref. [@Mi2015], the lepton pair content has recently been estimated based on a (perturbative) calculation of the electron-positron vacuum polarization insertion into the radiative correction to a constituent quark’s vector and axial vector current matrix elements. According to Ref. [@Je2013pra], the virtual annihilation channel in positronium, $$\label{B1} \delta H = \frac{\pi \alpha_{\rm QED}}{2 m_e^2} \, \left( 3 + \vec \sigma_+ \cdot \vec \sigma_- \right) \, \delta^3(r) \,,$$ corresponds to an effective Hamiltonian for electron-proton interactions of the form $$\label{B2} H_{\rm ann} = \epsilon_p \, \frac{3 \pi \alpha_{\rm QED}}{2 m_e^2} \, \delta^3(r) \,,$$ where $\epsilon_p$ measures the electron-positron pair content inside the proton and a value of $\epsilon_p = 2.1 \times 10^{-7}$ is found to be sufficient to explain the proton radius puzzle. Near Eq. (22) of Ref. [@Mi2015], it is argued that instead of Eq. , one should rather consider the Hamiltonian $$\label{B3} \delta H = \frac{\pi \alpha_{\rm QED}}{2 m_q^2} \, \left( 3 + \vec \sigma_+ \cdot \vec \sigma_- \right) \, \delta^3(r) \,,$$ where $m_q$ is a quark mass. According to Ref. [@Mi2015], an appropriate choice is $m_q \approx 600 \, m_e$ (constituent value of one third of the mass of a proton). Comparing Eqs. —, one is led to the identification $\epsilon_p \sim m_e^2/m_q^2 \approx 2.8 \times 10^{-6}$ which is “too large” to explain the proton radius puzzle. An estimate of the lepton pair content of the proton, based on electron-positron vacuum polarization insertions into the radiative correction to a constituent quark’s vector and axial vector current, likewise leads to estimates for $\epsilon_p$ that are much larger than the discrepancy. According to Eqs. (13) and (21) of Ref. [@Mi2015], an estimate based on matrix elements of the current leads to values in the range $$\epsilon_p \sim 10 \, \left( \frac{\alpha_{\rm QED}}{\pi} \right)^2 \sim 10^{-5} \gg 10^{-7} \,.$$ Conversely, if one starts from Eq.  instead of , arguing that the effective mass in the virtual annihilation diagram should be the quark mass, and [*additionally*]{} invokes the suppression factor $\epsilon_p$ \[see the text following Eq. (22) of Ref. [@Mi2015]\], then the resulting effect in muonic hydrogen becomes negligible on the level of the proton radius discrepancy. Guidance for the exploration of the conjectured sea lepton effect in future experiments is given by the discussion surrounding Eq. (23) of Ref. [@Mi2015], where the functional dependence on the charge and mass numbers of the nucleus is discussed. Nuclear structure corrections (nuclear polarizability corrections) are usually taken into account with the use of dispersion relations [@CaGoVa2014]. This is certainly a valid approach for genuine excitations of the valence quarks into excited states. However, the light sea fermions are generated by a QED correction to a nonperturbative process, namely, a correction to the nonperturbative QCD interaction inside the proton; the latter gives rise to the ubiquitous sea quarks. Dispersion relations (Cutkosky rules) are available for the treatment of the genuine excitations of the proton into its own excited states, but it is unclear if the use of dispersion relations could capture the effect of the sea fermions. Because the sea quark interaction is nonperturbative and the light fermion vacuum bubbles are inserted into the photon exchange among the (nonperturbative) sea quarks, one does not know where to cut the nonperturbative diagram and the dispersion relation is not available. 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--- abstract: | We describe a novel approach for adapting an existing software model checker to perform precise runtime verification. The software under test is allowed to communicate with the wider environment (including the file system and network). The modifications to the model checker are small and self-contained, making this a viable strategy for re-using existing model checking tools in a new context. Additionally, from the data that is gathered during a single execution in the runtime verification mode, we automatically re-construct a description of the execution environment which can then be used in the standard, full-blown model checker. This additional verification step can further improve coverage, especially in the case of parallel programs, without introducing substantial overhead into the process of runtime verification. author: - Katarína Kejstová - Petr Ročkai - Jiří Barnat bibliography: - 'common.bib' title: 'From Model Checking to Runtime Verification and Back[^1]' --- Introduction ============ While model checking is a powerful technique for software verification, it also has certain limitations and deficiencies. Many of those limitations are related to the fact that a model checker must, by design, fully isolate the program from any outside effects. Therefore, for verification purposes, the program under test is placed into an artificial environment, which gives non-deterministic (but fully reproducible) responses to the program. The existence of this model environment immediately requires trade-offs to be made. If the environment model is too coarse, errors may be missed, or spurious errors may be introduced. Creating a detailed model is, however, more costly, and the result is not guaranteed to exactly match the behaviour of the actual environment either. Moreover, a detailed model may be too rigid: programs are often executed in conditions that have not been fully anticipated, and a certain amount of coarseness in the model of the environment can highlight such unwarranted assumptions. Many of those challenges are, however, not unique to model checking. In the context of automated testing, the test environment plays a prominent role, and a large body of work deals with related problems. Unfortunately, adapting the methods used in automated testing to the context of model checking is far from straightforward. Making existing test-based setups easier to use with model checking tools is a core contribution of this paper. Both manual and automated testing are established, core techniques which play an important role in virtually every software development project. In a certain sense, then, testing provides an excellent opportunity to integrate rigorous tools into the software development process. A number of verification tools specifically tailored for this mode of operation have seen great success in the software development community, for instance the `memcheck` tool from the `valgrind` suite. We show that it is possible to tap into this potential also with a traditionally-designed software model checker: we hope that this will help put powerful verification technology into the hands of software developers in a natural and seamless fashion. The second important contribution of this paper, then, is an approach to build a runtime verification tool out of an existing software model checker. Our main motivating application is extending our existing software model checker,  [@barnat13:divine], with a runtime verification mode. In its latest version, has been split into a number of well-defined, reusable components [@rockai18:divm] and this presented an opportunity to explore the contexts in which the new components could be used. Based on this motivation, our primary goal is to bring traditional (software) model checking and runtime verification closer together. As outlined above, there are two sides to this coin. One is to make model checking fit better into existing software development practice, the second is to derive powerful runtime verification tools from existing model checkers. To ensure that the proposed approach is viable in practice, we have built a prototype implementation, which allowed us to execute simple C and C++ programs in the resulting runtime verifier. The rest of the paper is organised as follows: Section \[sec:related\] describes prior art and related work, while Section \[sec:prelim\] lays out our assumptions about the model checker and its host environment. Section \[sec:passthrough\] describes adapting a model checker to also work as a runtime verifier and Section \[sec:replay\] focuses on how to make use of data gathered by the runtime verifier in the context of model checking. Section \[sec:implementation\] describes our prototype implementation based on (including evaluation) and finally, Section \[sec:conclusion\] summarises and concludes the paper. Related Work {#sec:related} ============ There are two basic approaches to runtime verification [@havelund04:efficien]: online (real time) monitoring, where the program is annotated and, during execution, reports its actions to a monitor. In an offline mode, the trace is simply collected for later analysis. Clearly, an online-capable tool can also work in offline mode, but the reverse is not always true. An extension of the online approach allows the program to be monitored also in production, and property violations can invoke a recovery procedure in the program [@meredith12:mop]. Our work, in principle, leads to an online verifier, albeit with comparatively high execution overhead, which makes it, in most cases, unsuitable for executing code in production environments. Depending on the model checker used, it can, however, report violations to the program and invoke recovery procedures and may therefore be employed this way in certain special cases. Since our approach leads to a runtime verification tool, this can be compared to other such existing tools. With the exception of `valgrind` [@nethercote07:valgrin], most tools in this category focus on Java programs. For instance, Java PathExplorer [@havelund04:overview.runtim] executes annotated Java byte code, along with a monitor which can check various properties, including past-time LTL. Other Java-based tools include JavaMOP [@jin12:javamop] with focus on continuous monitoring and error recovery and Java-MaC [@kim04:java.mac] with focus on high-level, formal property specification. Our *replay mode* (described in Section \[sec:replay\]) is also related to the approach described in [@havelund00:using.runtim], where data collected at runtime is used to guide the model checker, with the aim of reducing the size of the state space. In our case, the primary motivation is to use the model checker for verifying more complex properties (including LTL) and to improve coverage of runtime verification. Preliminaries {#sec:prelim} ============= There are a few assumptions that we need to make about the mode of operation of the model checker. First, the model checker must be able to restrict the exploration to a single execution of the program, and it must support explicitly-valued operations. The simplest case is when the model checker in question is based on an explicit-state approach (we will deal with symbolic and/or abstract values in Section \[sec:abstract\]). If all values are represented explicitly in the model checker, exploration of a single execution is, in a sense, equivalent to simply running the program under test. Of course, since this is a model checker, the execution is subject to strict error checking. Abstract and Symbolic Values {#sec:abstract} ---------------------------- The limitation to exploring only a single execution is, basically, a limitation on *control flow*, not on the representation of variables. The root cause for the requirement of exploring only one control flow path is that we need to insert actions into the process of model checking that will have consequences in the outside world, consequences which cannot be undone or replayed. Therefore, it is not viable to restore prior states and explore different paths through the control flow graph, which is what normally happens in a model checker. It is, however, permissible to represent data in an abstract or symbolic form, which essentially means the resulting runtime verifier will also act as a symbolic executor. In this case, an additional requirement is that the values that reach the outside world are all concrete (the abstract representation used in the model checker would not be understood by the host operating system or the wider environment). Luckily, most tools with support for symbolic values already possess this capability, since it is useful in a number of other contexts. Environments in Model Checking {#sec:env} ------------------------------ A model checker needs a complete description of a system, that is, including any environment effects. This environment typically takes the form of code in the same language as the program itself, in our case C or C++. For small programs or program fragments, it is often sufficient to write a custom environment from scratch. This is analogous to how unit tests are written: effects from outside of the program are captured by the programmer and included as part of the test. When dealing with larger programs or subsystems, however, the environment becomes a lot more complicated. When the program refers to an undefined function, the model checker will often provide a fallback implementation that gives completely undetermined results. This fallback, typically, does not produce any side effects. Such fallback functions constitute a form of synthetic model environment. However, this can be overly coarse: such model environment will admit many behaviours that are not actually possible in the real one, and vice versa, lasting side effects of a program action (for instance a change in file content) may not be captured at all. Those infidelities can introduce both false positives and false negatives. For this reason, it is often important to provide a more realistic environment. A typical model checker (as opposed to a runtime verifier) cannot make use of a real operating system nor of testing-tailored, controlled environment built out of standard components (physical or virtual machines, commodity operating systems, network equipment and so on). A possible compromise is to implement an operating system which is designed to run inside a model checker, as a stand-in for the real OS. This operating system can then be re-used many times when constructing environments for model checking purposes. Moreover, this operating system is, to a certain degree, independent of the particular model checker in use. Like with standard operating systems, a substantial part of the code base can be re-used when porting the OS (that is, the host model checker is akin to a processor architecture or a hardware platform in standard operating systems). Many programs of interest are designed to run on POSIX-like operating systems, and therefore, POSIX interfaces, along with the interfaces mandated by ISO C and C++ are a good candidate for implementation. This has the additional benefit that large parts of all these specifications are implemented in open source C and/or C++ code, and again, large parts of this code are easily ported to new kernels. Together, this means that a prefabricated environment with POSIX-like semantics is both useful for verifying many programs and relatively simple to create. In the context of a model checker, the kernel of the operating system can be linked directly to the program, as if it were a library. In this approach, the model checker in question does not need any special support for loading kernel-like objects or even for privilege separation. System Calls {#sec:syscall} ------------ In this section, we will consider how traditional operating systems, particularly in the POSIX family, define and implement system calls. A traditional operating system consists of many different parts, but in our context, the most important are the kernel and the user-space libraries which implement the operating system API (the most important of these libraries is, on a typical Unix system, `libc`). From the point of view of a user program, the `libc` API *is* the interface of the operating system. However, many functions which are mandated as part of this interface cannot be entirely implemented in the user space: they work with resources that the user-space code is unable to directly access. Examples of such functions would be `read` or `write`: consider a `read` from a file on a local file system. If the implementation was done in the user space, it would need direct access to the hardware, for instance the PCI bus, in order to talk to the hard drive which contains the requisite blocks of data which represent the file system. This is, quite clearly, undesirable, since granting such access to the user program would make access control and resource multiplexing impossible. For these reasons, it is standard practice to implement parts of this functionality in separate, system-level code with a restricted interface, which makes access control and resource sharing possible. In operating system designs with monolithic kernels, this restricted interface consists of what is commonly known as system calls.[^2] A system call is, then, a mechanism which allows the user-space code to request that the system-level software (the kernel) executes certain actions on behalf of the program (subject to appropriate permission and consistency checks). The actual implementation of syscall invocation is platform-specific, but it always involves a switch from user (non-privileged) mode into kernel mode (privileged mode, *supervisor* mode or *ring 0* on x86-style processors). On POSIX-like systems, `libc` commonly provides a generic `syscall` function (it first appeared in `3BSD`). This function allows the application to issue syscalls based on their number, passing arguments via an ellipsis (i.e. by taking advantage of variadic arguments in the C calling convention). In particular, this means that given a description of a system call (its number and the number and types of its arguments), it is possible to automatically construct an appropriate invocation of the `syscall` function. Overview of Proposed Extensions {#sec:overview} ------------------------------- Under the proposed extensions, we have a model checker which can operate in two modes: *run* and *verify*. In the *run* mode, a single execution of the program is explored, in the standard execution order. We expect that all behaviour checking (enforcement of memory safety, assertion checks, etc.) is still performed in this mode. The *verify* mode, on the other hand, uses the standard model checking algorithm of the given tool. ![A scheme of components involved in our proposed approach.[]{data-label="fig:scheme"}](passthrough-scheme) The system under test (the input to this model checker), then, consists of the user program itself, along with the environment, the latter of which contains a stand-in operating system. The situation is illustrated in Figure \[fig:scheme\]. The operating system has 3 different modes: 1. a *virtual* mode, in which all interaction with the real world is simply simulated – for example, a virtual file system is maintained in-memory and is therefore part of the state of the system under test; this OS mode can be used with both *run* and *verify* modes of the model checker 2. a *passthrough* mode, which uses the `vm_syscall` model checker extension to execute system calls in the host operating system and stores a trace of all the syscalls it executed for future reference; this OS mode can only be used in the *run* mode of the model checker 3. a *replay* mode, which reads the system call trace recorded in the *passthrough* mode, but does not interact with the host operating system; this OS mode can be again used in both the *run* and *verify* mode of the model checker Syscall Passthrough {#sec:passthrough} =================== In order to turn a model checker into a runtime verifier, we propose a mechanism which we call *syscall passthrough*, where the virtual, stand-in operating system (see Section \[sec:env\]) gains the ability to execute syscalls in the host operating system (see also Section \[sec:syscall\]). Of course, this is generally *unsafe*, and only makes sense if the model checker can explore a single run of the program and do so *in order*. Thanks to the architecture of system calls in POSIX-like kernels, we only need a single new primitive function to be implemented in the model checker (we will call this new primitive function `vm_syscall` from now on; first, we need to avoid confusion with the POSIX function `syscall`, second, the model checker acts as a virtual machine in this context). The sole purpose of the function is to construct and execute, in the context of the host operating system, an appropriate call to the host `syscall` function (the interface of which is explained in more detail in Section \[sec:syscall\]). We would certainly like to avoid any system-specific knowledge in the implementation of `vm_syscall` – instead, any system-specific code should reside in the stand-in OS, which is much easier to modify than the model checker proper. To this end, the arguments to our `vm_syscall` primitive contain metadata describing the arguments `syscall` expects, in addition to the data itself. That is, `vm_syscall` needs to know whether a particular argument is an input or an output argument, its size, and if it is a buffer, the size of that buffer. The exact encoding of these metadata will be described in Section \[sec:passthrough-mc\], along with more detailed rationale for this approach. Finally, most of the implementation work is done in the context of the (stand-in) operating system (this is described in more detail in Section \[sec:passthrough-os\]). This is good news, because most of the code in the operating system, including all of the code related to syscall passthrough, is in principle portable between model checkers. Model Checker Extension {#sec:passthrough-mc} ----------------------- The model checker, on the other hand, only needs to provide one additional primitive. As already mentioned, we call this primitive `vm_syscall`, and it should be available as a variadic C function to the system under test. This is similar to other built-in functions often provided by model checkers, like `malloc` or a non-deterministic choice operator. While in the program under test, invocations of such built-ins look just like ordinary C function calls, they are handled differently in the model checker and often cause special behaviour that is not otherwise available to a C program. We would like this extension to be as platform-neutral as possible, while maintaining simplicity. Of course not all platforms provide the `syscall` primitive described in Section \[sec:syscall\], and on these platforms, the extension will be a little more complicated. Namely, when porting to a platform of this type, we need to provide our own implementation of `syscall`, which is easy to do when the system calls are available as C functions, even if tedious. In this case, we can simply assign numbers to system calls and construct a single `switch` statement which, based on a number, calls the appropriate C function. Therefore, we can rely on the `syscall` system-level primitive without substantial loss of generality or portability. The next question to ask is whether a different extension would serve our purpose better – in particular, there is the obvious choice of exposing each syscall separately as a model checker primitive. There are two arguments against this approach. First, it is desirable that the syscall-related machinery is all in one place and not duplicated in both the stand-in operating system and in the model checker. However, in the *virtual* and *replay* modes, this machinery must be part of the stand-in operating system, which suggests that this should be also the case in the *passthrough* mode. Second, the number of system calls is quite large (typically a few hundred functions) and the functions are system-dependent. When the code that is specific to the host operating system resides in the stand-in operating system, it can be ported once and multiple model checkers can benefit. Of course, the stand-in operating system needs to be ported to the model checker in question, but this offers many additional advantages (particularly the virtual mode). Now if we decide that a single universal primitive becomes part of the model checker, we still need to decide the syntax and the semantics of this extension. Since different system calls take different arguments with varying meaning, the primitive itself will clearly need to be variadic. Since one of the main reasons for choosing a single-primitive interface was platform neutrality, the primitive itself should not possess special knowledge about individual syscalls. First of all, it does not know the bit widths of individual arguments (on most systems, some arguments can be 32 bit – for instance file descriptors – and other 64 bit – object sizes, pointers, etc.). This information is crucial to correctly set up the call to `syscall` (the variadic arguments must line up). Moreover, some pointer-type arguments represent variable-sized *input* data (the buffer argument to `write`, for example) and others represent *output* data (the buffer argument to `read`). In both cases, the size of the memory allocated for the variable-sized argument must be known to `vm_syscall`, so that this memory can be correctly copied between the model checker and the system under test. ![An example invocation of `vm_syscall` performing a `read` passthrough.[]{data-label="fig:syscall"}](passthrough-syscall) For these reasons, the arguments to `vm_syscall` also contain metadata: for each real argument that ought to be passed on to `syscall`, 2 or 3 arguments are passed to `vm_syscall`. The first one is always type information: whether the following argument is a scalar (32b or 64b integer) or a pointer, whether it is an input or an output. If the value is a scalar input, the second argument is the value itself, if it is a scalar output, the following argument is a pointer to an appropriate-sized piece of memory. If the value is a pointer, the size of the pointed-to object comes second and the pointer itself comes third. An example invocation of `vm_syscall` is shown in Figure \[fig:syscall\]. The information passed to `vm_syscall` this way is sufficient to both construct a valid call to `syscall` and to copy inputs from the system under test to the host system and pass back the outputs. Operating System Extension {#sec:passthrough-os} -------------------------- The `vm_syscall` interface described above is a good low-level interface to pass syscalls through to the host operating system, but it is very different from the usual POSIX way to invoke them, and it is not very intuitive or user-friendly either. It is also an unsafe interface, because wrong metadata passed to `vm_syscall` can crash the model checker, or corrupt its memory. The proper POSIX interface is to provide a separate C function for each syscall, essentially a thin wrapper that just passes the arguments along. Calling these dedicated wrappers is more convenient, and since they are standard C functions, their use can be type-checked by the compiler. In the *virtual* mode of the operating system, those wrappers cause the execution to divert into the kernel. We can therefore re-use the entire `libc` without modifications, and implement syscall passthrough at the kernel level, where we have more control over the code. In our OS design, the kernel implements each system call as a single C++ method of a certain class (a *component*). Which exact components are activated is decided at boot time, and it is permissible that a given system call is implemented in multiple components. Since the components are arranged in a stack, the topmost component with an implementation of a given system call “wins”. In this system, implementing a passthrough mode is simply a question of implementing a suitable passthrough component and setting it up. When `libc` invokes a system call, the control is redirected into the kernel as usual, and the passthrough component can construct an appropriate invocation of `vm_syscall`. This construction requires the knowledge of a particular system call. Those are, luckily, more or less standardised by POSIX and the basic set is therefore reasonably portable. Moreover, we already need all of this knowledge in the implementation of the virtual mode, and hence most of the code related to the details of argument passing can be shared. As mentioned earlier, this means that the relevant `libc` code and the syscall mechanism it uses internally is identical in all the different modes of operation. The passthrough mode is, therefore, implemented entirely in the kernel of the stand-in operating system. Tracing the Syscalls {#sec:tracing} -------------------- The architecture of syscall passthrough makes it easy to capture argument values and results of every invoked syscall, in addition to actually passing it on to the host operating system. Namely, the implementation knows exactly which arguments are inputs and which are outputs and knows the exact size of any buffer or any other argument passed as a pointer (both input and output). This allows the implementation to store all this data in a file (appending new records as they happen). This file can then be directly loaded for use in the *replay mode* of the stand-in operating system. Syscall Replay {#sec:replay} ============== In a model checker, all aspects of program execution are fully repeatable. This property is carried over into the *virtual* operating mode (as described in this paper), but not into the *passthrough* mode. System calls in the host operating system are, in general, not repeatable: files appear and disappear and change content, network resources come and go and so on, often independently of the execution of the program of interest. What the passthrough mode can do, however, is recording the answers from the host operating system (see Section \[sec:tracing\]). When we wish to repeat the same execution of the program (recall that everything apart from the values coming from `vm_syscall` is under the full control of the model checker), we do not need to actually pass on the syscalls to the host operating system: instead, we can read off the outputs from a trace. This is achieved by simply replacing all invocations of `vm_syscall` by a different mechanism, which we will call `replay_syscall`. This new function looks at the trace, ensures that the syscall invoked by the program matches the one that comes next in the trace and then simply plays back the effects observable in the program. Since the program is otherwise isolated by the model checker, those effects are limited to the changes the syscall caused in its output parameters and the value of `errno`. The appropriate return value is likewise obtained from the trace. Motivation ---------- There are two important applications of the replay mode. First, if the model checker in question provides interactive tools to work with the state space, we can use those tools to look at real executions of the program, and in particular, we can easily step backwards in time. That is, if we have an interactive simulator (like, for example, presented in [@rockai17:simulat.llvm.bitcod]), we can derive a reversible debugger essentially for free by recording an execution in the passthrough mode and then exploring the corresponding path through the state space in the *replay* mode. Second, if the behaviour of the program depends on circumstances other than the effects and return values of system calls, it is often the case that multiple different executions of the program will result in an identical sequence of system calls. As an example, if the program contains multiple threads, one of which issues syscalls and others only participate in computation and synchronisation, the exact thread interleaving will only have a limited effect on the order and arguments of system calls, if any. The model checker is free to explore all such interleavings, as long as they produce the same syscall trace. That this is a practical ability is easily demonstrated. A common problem is that a given program, when executed in a controlled environment, sometimes executes correctly and other times incorrectly. In this case, by a controlled environment we mean that files and network resources did not change, and that the behaviour of the program does not depend on the value of the real-time clock. Therefore, we can reasonably expect the syscall trace to be identical (at least up to the point where the unexpected behaviour is encountered). If this is the case, the model checker will be able to reliably detect the problem based on a single syscall trace, regardless of whether the problem did or did not appear while running in the passthrough mode. Constructing the State Space ---------------------------- As explained above, we can use the replay mode to explore behaviours of the program that result in an identical syscall trace, but are not, computation-wise, identical to the original passthrough execution. In this case, it is important that the model checker explores only executions with this property. A primitive which is commonly available in model checkers and which can serve this purpose is typically known as `assume`[^3]. The effect of this primitive is to instruct the model checker to abandon any executions where the condition of the `assume` does not hold. Therefore, our `replay_syscall`, whenever it detects a mismatch between the syscall issued by the program and the one that is next in the trace, it can simply issue `assume( false )`. The execution is abandoned and the model checker is forced to explore only those runs that match the external behaviour of the original. Causality-Induced Partial Order ------------------------------- The requirement that the traces exactly match up is often unnecessarily constraining. For instance, it is quite obvious that the order of two read operations (with no intervening write operations) can be flipped without affecting the outcome of either of the two reads. In this sense, such two reads are not actually ordered in the trace. This means that the trace does not need to be ordered linearly – the two reads are, instead, incomparable in the causal ordering. In general, it is impossible to find the exact causal relationships between syscalls, especially from the trace alone – a write to a file may or may not have caused certain bytes to appear on the `stdin` of the program. We can, however, construct an approximation of the correct partial order, and we can do so safely: the constructed ordering will always respect causality, but it may order certain actions unnecessarily strictly. We say that two actions $a$ and $b$ (system call invocations) *commute* if the outcome of both is the same, regardless of their relative ordering (both $a$ and $b$ have the same individual effect, whether they are executed as $a, b$ or as $b, a$). Given a sequence of system calls that respects the causal relationships, swapping two adjacent entries which commute will lead to a new sequence with the same property. We can obtain an approximate partial order by constructing all such sequences and declaring that $a < b$ iff this is the case in all of the generated sequences. Prototype Implementation {#sec:implementation} ======================== We have implemented the approach described in this paper, using the 4 software model checker as a base. In particular, we rely on the component in , which is a verification-focused virtual machine based on the intermediate representation (more details in Section \[sec:llvm\]). The architecture of 4, as a model checker, is illustrated in Figure \[fig:d4\]. First, we have extended with the `vm_syscall` primitive (cf. Section \[sec:passthrough\]). Taking advantage of this extension, we have implemented the requisite support code in , as described in Section \[sec:passthrough-os\]. is a pre-existing stand-in operating system component which originally supported only the *virtual* mode of operation. As part of the work presented in this paper, we implemented both a passthrough and a replay mode in . ![The architecture of 4. The shaded part is, from a model checking point of view, the system under test. However, and most of the libraries are shipped as part of .[]{data-label="fig:d4"}](passthrough-d4) In the rest of this section, we will describe the underpinnings of 4 in more detail. The first important observation is that, since is based on interpreting bitcode, it can use a standard compiler front-end to compile C and C++ programs into the bitcode form, which can then be directly verified. We will also discuss the limitations of the current implementation and demonstrate its viability using a few examples. Bitcode {#sec:llvm} -------- bitcode (or intermediate representation) [@llvm16:llvm.languag] is an assembly-like language primarily aimed at optimisation and analysis. The idea is that -based analysis and optimisation code can be shared by many different compilers: a compiler front end builds simple IR corresponding to its input and delegates all further optimisation and native code generation to a common back end. This architecture is quite common in other compilers: as an example, GCC contains a number of different front ends that share infrastructure and code generation. The major innovation of is that the language on which all the common middle and back end code operates is exposed and available to 3rd-party tools. It is also quite well-documented and provides stand-alone tools to work with both bitcode and textual form of this intermediate representation. From a language viewpoint, IR is in partial SSA form (single static assignment) with explicit basic blocks. Each basic block is made up of instructions, the last of which is a *terminator*. The terminator instruction encodes relationships between basic blocks, which form an explicit control flow graph. An example of a terminator instruction would be a conditional or an unconditional branch or a `ret`. Such instructions either transfer control to another basic block of the same function or stop execution of the function altogether. Besides explicit control flow, also strives to make much of the data flow explicit, taking advantage of partial SSA for this reason. It is, in general, impossible to convert entire programs to a full SSA form; however, especially within a single function, it is possible to convert a significant portion of code. The SSA-form values are called *registers* in and only a few instructions can “lift” values from memory into registers and put them back again (most importantly `load` and `store`, respectively, plus a handful of atomic memory access instructions). Runtime Verification with {#runtime-verification-with-llvm} -------------------------- While bitcode is primarily designed to be transformed and compiled to native code, it can be, in principle, executed directly. Of course, this is less convenient than working with native code, but since the bitcode is appreciably more abstract than typical processor-level code, it is more amenable to model checking. The situation can be improved by providing tools which can work with hybrid object files, which contain both native code and the corresponding bitcode. This way, the same binary can be both executed natively and analysed by -based tools. Extensions for Verification {#sec:extensions} ---------------------------- Unfortunately, bitcode alone is not sufficiently expressive to describe real programs: most importantly, it is not possible to encode interaction with the operating system into instructions. When is used as an intermediate step in a compiler, the lowest level of the user side of the system call mechanism is usually provided as an external, platform-specific function with a standard C calling convention. This function is usually implemented in the platform’s assembly language. The system call interface, in turn, serves as a gateway between the program and the operating system, unlocking OS-specific functionality to the program. An important point is that the gateway function itself cannot be implemented in portable . To tackle these problems, a small set of primitives was proposed in [@rockai18:divm] (henceforth, we will refer to this enriched language as ). With these primitives, it is possible to implement a small, isolated operating system in the language alone. already provides such an operating system, called – the core OS is about 2500 lines of C++, with additional 5000 lines of code providing *virtual* POSIX-compatible file system and socket interfaces. Our implementation of the ideas outlined in Section \[sec:passthrough-os\] can, therefore, re-use a substantial part of the existing code of . Source Code ----------- The implementation consists of two parts. The model checker extension is about 200 lines of C++, some of which is quite straightforward. The extension is more complex: the passthrough component is about 1400 lines, while the replay component is less than 600. All the relevant source code, including the entire 4 model checker, can be obtained online[^4]. Limitations {#sec:limitations} ----------- There are two main limitations in our current implementation. The first is caused by a simplistic implementation of the *run* mode of our model checker (see Section \[sec:overview\]). The main drawback of such a simple implementation is that syscalls that block may cause the entire model checker to deadlock. Specifically, this could happen in cases where one program thread is waiting for an action performed by another program thread. Since there is only a single model checker thread executing everything, if it becomes blocked, no program threads can make any progress. There are two possible counter-measures: one is to convert all system calls to non-blocking when corresponding `vm_syscall` invocations are constructed, another is to create multiple threads in the model checker, perhaps even a new thread for each system call. Only the latter approach requires additional modifications to the model checker, but both require modifications to the stand-in operating system. The second limitation stems from the fact that our current `libc` implementation only covers a subset of POSIX. For instance, the `gethostbyname` interface (that is, the component of `libc` known as a resolver) is not available. This omission unfortunately prevents many interesting programs from working at the moment. However, this is not a serious limitation in principle, since the resolver component from an existing `libc` can be ported. Many of the networking-related interfaces are already present and work (in particular, TCP/IP client functionality has been tested, cf. Section \[sec:evaluation\]). Finally, a combination of both those limitations means that the `fork` system call, which would create a new process, is not available. In addition to problems with blocking calls, there are a few attributes that are allocated to each process, and those attributes can be observed by certain system calls. For example, one such attribute is the `pid` (process identifier), obtainable with a `getpid` system call, another is the working directory of the process, available through `getcwd`. Again, there are multiple ways to resolve this problem, some of which require modifications in the model checker. Evaluation {#sec:evaluation} ---------- Mainly due to the limitations outlined in Section \[sec:limitations\], it is not yet possible to use our prototype with many complete, real-world programs. The domain in which has been mainly used so far are either small, self-contained programs and unit tests for algorithms and data structures. Both sequential and parallel programs can be verified. The source distribution of includes about 600 test cases for the model checker, many of which also use POSIX interfaces, leveraging the existing *virtual* mode of . As a first part of our evaluation, we took all those test cases and executed them in the new *passthrough* mode, that is, in a mode when acts as a runtime verifier. A total of 595 tests passed without any problems, 3 timed out due to use of blocking system calls and 9 timed out due to presence of infinite loops. Of course, since runtime verification is not exhaustive, not all errors present in the 595 tests were uncovered in this mode. The second part of our evaluation was to write small programs that specifically test the *passthrough* and the *replay* mode: - `pipe`, which creates a named pipe and two threads, one writer and one reader and checks that data is transmitted through the pipe - `rw` which simply creates, writes to and reads from files - `rw-par` in which one thread writes data to a file and another reads and checks that data - `network`, a very simple HTTP client which opens a TCP/IP connection to a fixed IP address, performs an HTTP request and prints the result We tested these programs in both the *passthrough* mode and in the *replay* mode. While very simple, they clearly demonstrate that the approach works. The source code of those test programs is also available online[^5]. Clearly, our verifier incurs appreciable overhead, since it interprets the program, instead of executing it directly. Quantitative assessment of the runtime and memory overhead is subject to future work (more complex test cases are required). Conclusions and Future Work {#sec:conclusion} =========================== We have described an approach which allows us to take advantage of an existing software model checking tool in the context of runtime verification. On one hand, this approach makes model checking more useful by making it usable with real environments while retaining many of its advantages over testing. On the other hand, it makes existing model checking tools useful in cases when runtime verification is the favoured approach. The approach is lightweight, since the modification to the model checker is small and self-contained. The other component required in our approach, the stand-in operating system, is also reasonably portable between model checkers. The overall effort associated with our approach is small, compared to implementing two dedicated tools (a model checker and a runtime verifier). In the future, we plan to remove the limitations described in Section \[sec:limitations\] and offer a production-ready implementation of both a passthrough and a replay mode in 4. Since the results of the preliminary evaluation are highly encouraging, we firmly believe that a runtime verification mode based on the ideas laid out in this paper will be fully integrated into a future release of . [^1]: This work has been partially supported by the Czech Science Foundation grant No. 15-08772S and by Red Hat, Inc. [^2]: In microkernel and other design schools, syscalls in the traditional sense only exist as an abstraction, and are implemented through some form of inter-process communication. [^3]: The `assume` primitive is a counterpart to `assert` and has a similar interface. It is customary that a single boolean value is given as a parameter to the `assume` statement (function call), representing the assumed condition. [^4]: <https://divine.fi.muni.cz/2017/passthrough/> [^5]: <https://divine.fi.muni.cz/2017/passthrough/>

--- abstract: 'This is a progress report on the ongoing project dealing with ensemble asteroseismology of B-type stars in young open clusters. The project is aimed at searches for B-type pulsating stars in open clusters, determination of atmospheric parameters for some members and seismic modeling of B-type pulsators. Some results for NGC457, IC1805, IC4996, NGC6910 and $\alpha$ Per open clusters are presented. For the last cluster, BRITE data for five members were used.' author: - Dawid Moździerski - Andrzej Pigulski bibliography: - 'prospects.bib' title: Prospects for the ensemble asteroseismology in young open clusters --- Introduction ============ Stellar clusters are known as good ‘laboratories’ for studying member stars. By fitting isochrones, one can obtain distance, age and sometimes get information on metallicity of a cluster. Our project is focused on finding open clusters rich in pulsating B-type stars and subsequent seismic modeling of their members by means of ensemble asteroseismology (hereafter EnsA). EnsA takes advantage of the common parameters of the members of a cluster (e.g. age and metallicity) to put additional constraints on seismic models of member stars. It is applicable to open clusters rich in massive pulsating stars. The most promising are $\beta$ Cep stars for which many interesting results were already obtained by means of seismic modeling [e.g. @Aerts2003; @Pam2004; @Dup2004; @Dasz2010]. One of the best candidates for application of EnsA is NGC6910 [@Kol2004], for which pulsation parameters were obtained as a result of the international observational campaign [@Pig2008; @Sae2010]. In the era of nano-satellites, new possibilities of seismic studies of stars belonging to bright, nearby star clusters occured. An example is $\alpha$ Per open cluster. ![Variable stars in the color-magnitude diagram of IC4996.[]{data-label="VarCMD"}](fig1-VarCMD){width="9cm"} New observations and results ============================ The results of the observations of NGC457 and preliminary results of the search for variable stars in IC1805 were published by [@Moz2014] and [@Moz2012], respectively. Recently, we performed also search for variable stars in another young open cluster, IC4996. Observations of IC4996 were made between 2007 and 2014 in Bia[ł]{}ków Observatory (Poland) during 50 observing nights. The observations were carried out with a 60-cm reflecting telescope with the attached CCD camera covering $13^\prime\times12^\prime$ field of view. About 7500 CCD frames were acquired through the $B$, $V$, $R$, $I_{\rm C}$ and narrow-band H$\alpha$ filters. We detected 81 variable stars (Fig. \[VarCMD\]), of which 71 are new. One new $\beta$ Cep star was found. In total, in three open clusters (NGC457, IC1805, and IC4996) we have detected 231 variables. Among the most interesting are: large population of SPB stars (21) in NGC457, some of them showing frequences above 3.5 $\rm d^{-1}$, and many monoperiodic variables in IC4996 and IC1805 grouping in the lower parts of their color-magnitude diagrams. These are probably pre-main sequence stars. Small number of $\beta$ Cep stars in NGC457, IC4996 and IC1805 does not give, or gives marginal chance for a successful application of EnsA. Nonetheless, our study resulted in finding mentioned above groups of SPB and monoperiodic stars which in the future might be useful for better understanding of the incidence of variability in young open clusters at main sequence and pre-main sequence stages of evolution. The best candidate for the application of EnsA is NGC6910. The first results look promising. Using echelle spectra obtained with Apache Point Observatory (APO) ARC 3.5-m telescope and Nordic Optical Telescope (NOT), we have derived atmospheric parameters of three $\beta$ Cep stars from NGC6910, NGC6910-14, -16 and -18, using NLTE BSTAR2006 grid [@Lanz2007] of atmosperic models. Then, effective temperatures and surface gravities of these stars were used to place them in the theoretical H-R diagram and for mode identification. In calculations, we used Warsaw - New Jersey evolutionary code adopting OPAL opacities, solar mixture as determined by @Asp2009 and no overshooting from the convective core. Mode identification based on $B$, $V$, $I_{\rm C}$ time-series photometry was performed with the methods developed by [@Dasz2003] and [@Dasz2005]. We identified degree of the mode with frequency $f=5.252056$ $\rm d^{-1}$ detected in NGC6910-14 as $l$ = 4. In view of the relatively high projected rotational velocity of the star ($V_{\rm rot} = 149$ km/s), the possiblity that it is rotationally coupled $l=$ 2 mode, cannot be excluded. We also identified degrees of four modes with the highest photometric amplitudes in two other $\beta$ Cephei stars, NGC6910-16 ($f_1=5.202740$ $\rm d^{-1}$ and $f_2=4.174670$ $\rm d^{-1}$, both as $l=$ 2) and NGC6910-18 ($f_1=6.154885$ $\rm d^{-1}$ as $l$ = 0 and $f_2=6.388421$ $\rm d^{-1}$ as $l$ = 2). The full results of the application of EnsA to NGC6910 will be published elsewhere. Unfortunately, it turned out that the up-to-date BRITE observations of five B-type members of $\alpha$ Per cluster revealed only one pulsating star, HD22192. It can be classified as an SPB star with frequency groupings. Therefore, the cluster does not seem to be an object suitable for the application of EnsA.

--- abstract: 'We present some forms of uncertainty principle which involve in a new way localization operators, the concept of $\varepsilon$-concentration and the standard deviation of $L^2$ functions. We show how our results improve the classical Donoho-Stark estimate in two different aspects: a better general lower bound and a lower bound in dependence on the signal itself.' author: - | [Paolo Boggiatto, Evanthia Carypis and Alessandro Oliaro[^1]]{}\ [*Department of Mathematics, University of Torino*]{}\ [*Via Carlo Alberto, 10, I-10123 Torino (TO), Italy*]{} title: '**Two Aspects of the Donoho-Stark Uncertainty Principle**' --- Introduction ============ An uncertainty principle (UP) is an inequality expressing limitations on the simultaneous concentration of a function, or distribution, and its Fourier transform. More in general UPs can express limitations on the concentration of any time-frequency representation of a signal. According to the meaning given to the term “concentration” many different formulations are possible and, starting from the classical works of Heisenberg, a vast literature is today available on these topics, see e.g. [@BogCarOli2012; @BogFerGal; @BonDemJam; @cow; @Deg13; @DonSta; @FolSit97; @Gro2003]. In this paper we are concerned with the Donoho-Stark form of the UP of which we present an improvement in the form of a new general bound for the constant which is involved in the estimate, and a new type of estimation of the same constant in dependence on the signal. The Donoho-Stark UP relies on the concept of *$\varepsilon$-concentration* of a function on a measurable set $U\subseteq {\mathbb{R}^{d}}$. We start by recalling this definition followed by the statement of the classical theorem. \[epscon\] Given $\varepsilon\geqslant 0$, a function $f\in L^{2}({\mathbb{R}^{d}})$ is *$\varepsilon$-concentrated* on a measurable set $U\subseteq {\mathbb{R}^{d}}$ if $$\left(\int_{{\mathbb{R}^{d}}\backslash U}{|f(x)|^{2}dx}\right)^{1/2}\leqslant\varepsilon\|f\|_{2}.$$ \[DS\] (Donoho-Stark) Suppose that $f\in L^{2}({\mathbb{R}^{d}})$, $f\neq 0$, is $\varepsilon_{T}$-concentrated on $T\subseteq{\mathbb{R}^{d}}$, and $\widehat{f}$ is $\varepsilon_{\Omega}$-concentrated on $\Omega\subseteq{\mathbb{R}^{d}}$, with $T,\Omega$ measurable sets in ${\mathbb{R}^{d}}$ and $\varepsilon_T,\varepsilon_\Omega\geqslant 0$, $\varepsilon_T+\varepsilon_\Omega < 1$. Then $$\label{DSineq} |T||\Omega|\geqslant(1-\varepsilon_T - \varepsilon_{\Omega})^{2}.$$ [*(We use the convention $\widehat f(\omega)=\int_{{\mathbb{R}^{d}}}e^{-2\pi i x \omega} f(x)\, dx$.)* ]{} Many variations and related information about this result can be found e.g. in [@DonSta] and [@Gro01-1]. Our investigations of the Donoho-Stark UP are based on two well-known fundamental results, namely the $L^p$ boundedness result of Lieb for the Gabor transform, recalled in Theorem \[Lieb\], and the local UP of Price, which we recall in Theorem \[PriceUP\]. We present now in more details how the two aspects that we have mentioned are considered in the paper. The first aspect we deal with is an improvement of the estimation of the Donoho-Stark constant which appears on the right-hand side of inequality . More precisely in section 2 we prove a new estimate, Lemma \[LemmaLocOp\], for the norm of localization operators: $$\label{locop} f \longrightarrow L_{\phi,\psi}^{a}f = \int_{\mathbb{R}^{d}}a(x,\omega) V_{\phi}f(x,\omega)\, \mu_\omega\tau_x\psi \, dx d\omega$$ acting on $L^2({\mathbb{R}^{d}})$, with symbol $a\in L^q(\R2d)$, for $q\in[1,\infty]$, and “window” functions $\phi,\psi\in L^2({\mathbb{R}^{d}})$. Here, for $\phi,\psi\in L^2(\mathbb{R}^d)$, $$V_\phi f (x,\omega) = \int e^{-2\pi it\omega} f(t) \overline{\phi(t-x)}\,dt$$ is the [*Gabor transform*]{} of $f\in L^2(\mathbb{R}^d)$, whereas $\mu_\omega\tau_x\psi(t)= e^{2\pi i \omega t}\psi(t-x)$ are [*time-frequency shifts*]{} of $\psi(t)$. See e.g. [@BogOliWon-2006; @Coh95-1; @FerGal10; @Foll89; @Gro01-1] for references on this topic as well as for extensions to more general functional settings. Although not new in its functional framework as boundedness result, as far as we know, the norm estimate of Lemma \[LemmaLocOp\] does not appear before in literature. Our proof relies on Lieb’s estimate of the $L^p$ norm of the Gabor transform (Theorem \[Lieb\]). We then use Lemma \[LemmaLocOp\] and some facts from the theory of pseudo-differential operators, in particular from Weyl calculus, to obtain our main result of this section, Theorem \[opL\], which is an uncertainty principle for [*concentration operators*]{}, i.e. localization operators with characteristic functions as symbols. The reason of the interest in these operators lies in the fact that the Donoho-Stark hypothesis of $\varepsilon$-concentration can be interpreted in terms of the action of concentration operators. This is used in section 3 to compare our results with the classical Donoho-Stark UP. It turns out that, as a limit case for window functions tending to the Dirac delta in the space of tempered distributions, this UP can be reobtained with a considerable improvement of the constant appearing in the estimate, see Proposition \[ourDS\]. In section 4 we consider the second aspect of the Donoho-Stark UP starting from Price’s local UP, Theorem \[PriceUP\] (cf, [@Pri87], see also [@BogCarOli14]). Qualitatively this theorem asserts that a highly concentrated signal $f$ cannot have Fourier transform which is too concentrated on any measurable set $\Omega$, the upper bound of the local concentration of $\widehat f$ being given in terms of the Lebesgue measure of $\Omega$ itself and the standard deviation of $f$. We show that the concept of $\varepsilon$-concentration of $f$ and $\widehat f$ respectively on sets $T$ and $\Omega$ can be used in this local context to get a version of the Donoho-Stark UP with an estimation whose constant depends on the signal $f$, Theorem \[DSfdepending\]. Actually, as pointed out in Remark \[strongerresult\], the proof of Theorem \[DSfdepending\] shows more than inequalities of Donoho-Stark type, as we get independent lower bounds for the measures of the sets $T$ and $\Omega$, whereas the Donoho-Stark estimate is a lower bound only for the product of the two measures. Our final result in section 4, Theorem \[mixed\], concerns a “mixed” lower bound for the support of a signal and the standard deviation of its Fourier transform. An uncertainty principle for localization operators =================================================== Localization operators of type have been widely studied in literature (see e.g. [@BogOliWon-2006; @Gro01-1; @BenHeiWal92; @Won02]). As first result of this section, Lemma \[LemmaLocOp\], we obtain a new estimation for their $L^2$-boundedness constant by means of the classical Lieb’s $L^p$-boundedness result for the Gabor transform which we recall here for completeness (see e.g. [@Lieb; @Gro01-1]). \[Lieb\](Lieb) If $f,g\in L^2({\mathbb{R}^{d}})$ and $2\leqslant p \leqslant \infty$, then $$\|V_gf\|_p\leqslant \left(\frac{2}{p}\right)^{\frac{d}{p}}\|f\|_2\|g\|_2 \hskip.5cm \hbox{(with $(\frac{2}{p})^\frac{d}{p}=1$ for $p=+\infty$)}.$$ \[LemmaLocOp\] Let $\phi, \psi \in L^{2}({\mathbb{R}^{d}})$, $q\in [1,\infty]$ and consider the quantization (see ): $$L_{\phi,\psi}: a\in L^{q}(\mathbb{R}^{2d})\rightarrow L_{\phi,\psi}^{a}(\mathbb{R}^{2d})\in B(L^2(\mathbb{R}^{2d})).$$ Then the following estimation holds $$\|L_{\phi,\psi}^{a}\|_{B(L^{2})}\leqslant \left(\frac{1}{q'}\right)^{d/q'}\|\phi\|_{2}\|\psi\|_{2}\|a\|_{q},$$ with $\frac{1}{q}+\frac{1}{q'}=1$ and setting $(\frac{1}{q'})^\frac{1}{q'}=1$ for $q=1$. We indicate by $(\cdot,\cdot)$ the inner product in $L^2$ spaces. Recall (cf. for example [@BogOliWon-2006]) that for every $f,g,\phi,\psi\in L^2(\mathbb{R}^d)$ and $a\in L^q(\mathbb{R}^{2d})$ we have $(L^a_{\phi,\psi}f,g) = (a,V_\psi g\overline{V_\phi f})$. Then, in view of Hölder’s inequality and Lieb’s UP, for $f,g \in L^{2}({\mathbb{R}^{d}})$ we have: $$\begin{split} &|(L_{\phi,\psi}^{a}f,g)| = |(a, V_{\psi}g\overline{V_{\phi}f})|\nonumber\\ &\leqslant \|a\|_{q}\|V_{\psi}g\overline{V_{\phi}f}\|_{q'}\nonumber\\ &\leqslant\|a\|_{q}\|V_{\psi}g\|_{q'k}\|V_{\phi}f\|_{q'k'}\quad\qquad \left(\text{for}\, k\in[1,\infty],\,\frac{1}{k} + \frac{1}{k'}=1\right)\nonumber\\ &\leqslant\|a\|_{q}\left(\frac{2}{q'k}\right)^{d/q'k}\|\psi\|_{2}\|g\|_{2}\left(\frac{2}{q'k'}\right)^{d/q'k'}\|\phi\|_{2}\|f\|_{2} \nonumber\\ &=\|a\|_{q}\left(\frac{2}{q'}\right)^{\left(\frac{1}{k}+\frac{1}{k'}\right)\frac{d}{q'}}\left(\left(\frac{1}{k}\right)^{1/k}\left(\frac{1}{k'}\right)^{1/k'}\right)^{d/q'}\|\psi\|_{2}\|\phi\|_{2}\|f\|_{2}\|g\|_{2}\nonumber\\ &=\|a\|_{q}\left(\frac{2}{q'}\right)^{d/q'}\alpha_{k}^{d/q'}\|\psi\|_{2}\|\phi\|_{2}\|f\|_{2}\|g\|_{2},\nonumber\\ \end{split}$$ where we have set $$\alpha_k =\left(\frac{1}{k}\right)^{1/k}\left(\frac{1}{k'}\right)^{1/k'},$$ and we have applied Theorem \[Lieb\], supposing $k\in[1,\infty]$ such that $q'k\geqslant 2$ and $q'k'\geqslant 2$. It follows that $$\|L_{\phi,\psi}^{a}\|_{B(L^{2})}\leqslant C\|a\|_{q},$$ with $$C = \left(\frac{2}{q'}\right)^{d/q'}\|\psi\|_{2}\|\|\phi\|_{2}\left(\min\{\alpha_{k}: k\in[1,\infty] \,\text{such that}\, q'k\geqslant 2,q'k'\geqslant 2\}\right)^{d/q'}.$$ Let us now study the function $$\alpha_{k} = \left(\frac{1}{k}\right)^{1/k}\left(\frac{1}{k'}\right)^{1/k'} = f\left(\frac{1}{k}\right).$$ Setting $x= \frac{1}{k}$, we have $f(x)= x^{x}(1-x)^{(1-x)}$, which, for $x\in [0,1]$ and setting $\left(\frac{1}{\infty}\right)^{\frac{1}{\infty}} = 1$, has an absolute minimum in $x = 1/2$. It follows that $f(1/k)$ on the interval $k\in[1,\infty]$ has an absolute minimum in $k = 2$. Given $q\in [1,\infty]$, we search now the minimum of the values $\alpha_{k}$ with $k$ satisfying the conditions: $$\label{conditions} \left\{\begin{array}{ll} 1\leqslant k \leqslant \infty &\\ q'k\geqslant 2 &\\ q'k'\geqslant 2& \end{array}\right.$$ For $k\in[1,\infty]$, the condition $q'k\geqslant 2$ yields $k\geqslant \frac{2q-2}{q}$; whereas the condition $q'k'\geqslant 2$ yields $k\leqslant \frac{2q-2}{q-2}$ for $q > 2$, and $k\geqslant \frac{2q-2}{q-2}$, for $q < 2$ (observe that $q'k'\geqslant 2$ is satisfied for every $k\in [1,\infty]$ when $q=2$). Elementary considerations lead then to the conclusion that, for every $q\in[1,\infty]$, the value $k=2$ satisfies and gives the absolute minimum for $\alpha_k$. As $$\alpha_{2} = \left(\frac{1}{2}\right)^{1/2}\left(\frac{1}{2}\right)^{1/2} = \frac{1}{2},$$ we have $$\begin{split} \|L_{\phi, \psi}^{a}\|_{B(L^{2})}&\leqslant\left(\frac{2}{q'}\right)^{d/q'}2^{-d/q'}\|\phi\|_{2}\|\psi\|_{2}\|a\|_{q}\nonumber\\ &=\left(\frac{1}{q'}\right)^{d/q'}\|\phi\|_{2}\|\psi\|_{2}\|a\|_{q} \end{split}$$ as desired. We shall use the previous result to obtain an uncertainty principle involving localization operators in the special case where the symbol is the characteristic function of a set, expressing therefore [*concentration*]{} of energy on this set when applied to signals in $L^2({\mathbb{R}^{d}})$. In this case they are also known as [*concentration operators*]{}. For the proof we shall need some tools from the pseudo-differential theory which we now recall in the $L^2$ functional framework, for more general settings and reference see e.g. [@BogDedOli2010], [@Hor90], [@Tof04-1], [@Tof04-2], [@Won14]. The Wigner transform is the sesquilinear bounded map from $L^2({\mathbb{R}^{d}})\times L^2({\mathbb{R}^{d}})$ to $L^2(\R2d)$ defined by $$(f,g) \longrightarrow {\mathop{\rm Wig}}(f,g)(x,\omega)=\int_{{\mathbb{R}^{d}}}e^{-2\pi i t \omega}f(x+t/2)\overline{g(x-t/2)}\,dt.$$ For short we shall write ${\mathop{\rm Wig}}(f)$ instead of ${\mathop{\rm Wig}}(f,f)$. In connection with the Wigner transform, Weyl pseudo-differential operators are defined by the formula $$\label{WeylWig} (W^af,g)_{L^2({\mathbb{R}^{d}})}=(a,{\mathop{\rm Wig}}(g,f))_{L^2(\R2d)},$$ for $f,g\in L^2(\mathbb{R}^d)$, $a\in L^2(\mathbb{R}^{2d})$. More explicitly they are maps of the type $$f\in L^2({\mathbb{R}^{d}})\longrightarrow W^a f(x)= \int_{\R2d} e^{2\pi i (x-y)\omega} a\left(\frac{x+y}{2},\omega\right) f(y)\, dy\, d\omega\in L^2({\mathbb{R}^{d}}).$$ The fundamental connection between Weyl and localization operators is expressed by the formula which yields localization operators in terms of Weyl operators: $$\label{locWeyl} L^a_{\phi,\psi}=W^b, \ \ \ \ \ \ \ {\rm with} \ \ b=a*{\mathop{\rm Wig}}(\widetilde\psi,\widetilde\phi),$$ with $\psi,\phi \in L^2(\mathbb R^d)$ and where, for a generic function $u(x)$, we use the notation $\widetilde u(x)=u(-x)$. Of particular importance for our purpose will be the fact that Weyl operators with symbols $a(x,\omega)$ depending only on $x$, or only on $\omega$, are multiplication operators, or Fourier multiplier respectively. More precisely we have $$\label{multop-Fmultop} \begin{array}{ll} a(x,\omega)=a(x) \ \ \ \Longrightarrow \ \ \ W^af(x)=a(x)f(x) \\ a(x,\omega)=a(\omega) \ \ \ \Longrightarrow \ \ \ W^af(x)= \mathcal F^{-1}[a(\omega) \widehat f(\omega)](x). \end{array}$$ Let us now fix some notations. Let $T\subseteq \mathbb{R}_{x}^{d}$, $\Omega\subseteq\mathbb{R}_{\omega}^{d}$ be measurable sets, and write for shortness $\chi_T=\chi_{T\times\mathbb{R}^d}$ and $\chi_\Omega=\chi_{\mathbb{R}^d\times\Omega}$, in such a way that $\chi_T=\chi_T(x)$ and $\chi_\Omega=\chi_\Omega(\omega)$. Moreover, for $j = 1, 2$, $\lambda_{j}>0$, we set $\phi_{j}(x) = e^{-\pi\lambda_{j}x^{2}}$ and $\Phi_{j}(x) = c_{j}\phi_{j}(x)$, where $\Phi_{j}$ are normalized in $L^{2}({\mathbb{R}^{d}})$, i.e. $c_j = (2\lambda_j)^{d/4}$. Furthermore let $$\label{op1} L_{1}f = L_{\Phi_{1}}^{\chi_{T}}f = \int_{\mathbb{R}^{2d}}{\chi_{T}(x)V_{\Phi_{1}}f(x,\omega)\mu_{\omega}\tau_{x}\Phi_{1}dxd\omega}$$ and $$\label{op2} L_{2}f = L_{\Phi_{2}}^{\chi_{\Omega}}f = \int_{\mathbb{R}^{2d}}{\chi_{\Omega}(\omega)V_{\Phi_{2}}f(x,\omega)\mu_{\omega}\tau_{x}\Phi_{2}dxd\omega}$$ be the two localization operators with symbols $\chi_T, \chi_\Omega$ and windows $\Phi_1, \Phi_2$ respectively. We can state now the main result of this section which is an UP involving the $\varepsilon$-concentration of these two localization operators. \[opL\] With the previous assumptions on $T$, $\Omega$, $L_1$, $L_2$, suppose that $\varepsilon_{T},\varepsilon_{\Omega}>0$, $\varepsilon_T+\varepsilon_\Omega \leqslant 1$, and that $f\in L^{2}(\mathbb{R}^{d})$ is such that $$\label{Hp} \|L_{1}f\|_{2}^{2}\geqslant(1-\varepsilon_{T}^{2})\|f\|_{2}^{2} \quad \text{and}\quad\|L_{2}f\|_{2}^{2}\geqslant(1-\varepsilon_{\Omega}^{2})\|f\|_{2}^{2}.$$ Then $$\label{improv} |T||\Omega|\geqslant \sup_{r\in [1,\infty)} (1-\varepsilon_{T}-\varepsilon_{\Omega})^{r} \left(\frac{r}{r-1}\right)^{2d(r-1)}.$$ Writing the operators $L_{j}$, $j=1,2$, defined in and as Weyl operators we have: $$L_{1}f = W^{F_1}f, \quad \text{with}\quad F_1(x,\omega) = \left(\chi_{T}(x)\otimes 1_{\omega}\right)\ast {\mathop{\rm Wig}}{}{}(\Phi_{1})(x,\omega)$$ $$L_{2}f = W^{F_2}f, \quad \text{with}\quad F_2(x,\omega) = \left(1_x\otimes \chi_{\Omega}(\omega)\right)\ast {\mathop{\rm Wig}}{}{}(\Phi_{2})(x,\omega).$$ An explicit calculation yields: $$Wig(\Phi_{j})(x,\omega) = c_{j}^{2}\left(\frac{2}{\lambda_{j}}\right)^{d/2}e^{-2\pi\lambda_{j}x^2}e^{-\pi\frac{2}{\lambda_{j}}\omega^{2}}, \quad j = 1,2,$$ therefore we have $$\begin{split} F_1(x,\omega) &= c_{1}^{2}\left(\int{\left(\frac{2}{\lambda_{1}}\right)^{d/2}e^{-2\pi \lambda_{1}t^{2}}\chi_{T}(x-t)dt}\right)\left(\int{e^{-\pi \frac{2}{\lambda_{1}}s^{2}}ds}\right)\nonumber\\ &=c_{1}^{2}\int{\chi_{T}(x-t)e^{-2\pi\lambda_{1}t^{2}}dt}\nonumber\\ &=c_{1}^{2}\left(\chi_{T}\ast e^{-2\pi \lambda_{1}(\cdot)^2}\right)(x)\nonumber \end{split}$$ which shows that $F_1$ depends only on the time variable $x$. In a similar way we can prove that $F_2$ depends just on $\omega$. Precisely we have: $$\begin{split} F_2(x,\omega) &= c_{2}^{2}\left(\int{\left(\frac{2}{\lambda_{2}}\right)^{d/2}e^{-2\pi \lambda_{2}t^{2}}dt}\right)\left(\int{\chi_{\Omega}(\omega-s)e^{-\pi \frac{2}{\lambda_{2}}s^{2}}ds}\right)\nonumber\\ &=c_{2}^{2}\left(\frac{2}{\lambda_{2}}\right)^{d/2}(2\lambda_{2})^{-d/2}\left(\chi_{\Omega}\ast e^{-\pi\frac{2}{\lambda_{2}}(\cdot)^{2}}\right)(\omega)\nonumber\\ &=c_{2}^{2}\lambda_{2}^{-d}\left(\chi_{\Omega}\ast e^{-\pi \frac{2}{\lambda_{2}}(\cdot)^2}\right)(\omega).\nonumber \end{split}$$ It follows that $$L_{1}f = W^{F_1}f = F_1f,$$ i.e. $L_1$ is the multiplication operator by the function $F_1$ and $$L_{2}f = W^{F_2}f = \mathcal{F}^{-1}F_2\mathcal{F}f,$$ i.e. $L_2$ is the Fourier multiplier with symbol $F_2$. Now, for $j = 1, 2$, we compute $$\begin{split} \|f\|_{2}^{2} &= \|(f - L_{j}f) + L_{j}f\|_{2}^{2}\\ &=((f-L_{j}f) + L_{j}f,(f - L_{j}f) + L_{j}f)\\ &=\|f- L_{j}f\|_{2}^{2} + \|L_{j}f\|_{2}^{2} + (f - L_{j}f,L_{j}f) + (L_{j}f, f-L_{j}f) \quad \label{ripre} \end{split}$$ Next we show that $(f - L_{j}f,L_j f)\geqslant 0$ if $\Phi_{j}$ are normalized in $L^{2}$. For $j = 1$ we have $$\begin{split} (f - L_{1}f, L_{1}f) &= (f,L_{1}f) - (L_{1}f,L_{1}f)\nonumber\\ &=\int{f\overline{F_1}\overline{f}}-\int{F_1f\overline{F_1}\overline{f}}\nonumber\\ &=\int{(1-F_1)\overline{F_1}|f|^{2}}\geqslant 0, \end{split}$$ as $F_1$ is real, non negative, and $\|F_1\|_{\infty}\leqslant 1$; actually $$\begin{split} \|F_1\|_{\infty} &= c_{1}^{2}\|\chi_{T}\ast e^{-2\pi \lambda_{1}t^{2}}\|_{\infty}\nonumber\\ &\leqslant c_{1}^{2}\|\chi_{T}\|_{\infty}\|e^{-2\pi \lambda_{1}t^{2}}\|_{1} \nonumber\\ &= c_{1}^{2}(2\lambda_{1})^{-d/2}\nonumber\\ &=1, \end{split}$$ recalling that $c_{1} = (2\lambda_{1})^{d/4} = \|\phi_1\|_{2}^{-1}$. Analogously, if $j = 2$ we have $$\begin{split} (f - L_{2}f, L_{2}f) &= (f,L_{2}f) - (L_{2}f,L_{2}f)\nonumber\\ &=\left(f,\mathcal{F}^{-1}F_2\mathcal{F}f\right)-\left(\mathcal{F}^{-1}F_2\mathcal{F}f,\mathcal{F}^{-1}F_2\mathcal{F}f\right)\nonumber\\ &=(\widehat{f},F_2\widehat{f}) - (F_2\widehat{f},F_2\widehat{f}) \nonumber\\ &=\int{\widehat{f}\overline{F_2\widehat{f}}} - \int{F_2\widehat{f}\overline{F_2\widehat{f}}}\nonumber\\ &=\int{(1-F_2)\overline{F_2}|\widehat{f}|^{2}}\geqslant 0, \end{split}$$ as $F_2$ is real, non negative, and $\|F_2\|_{\infty}\leqslant 1$, the last inequality following from $$\begin{split} \|F_2\|_{\infty} &= c_{2}^{2}\lambda_{2}^{-d}\|\chi_{\Omega}\ast e^{-\pi \frac{2}{\lambda_{2}}s^{2}}\|_{\infty}\nonumber\\ &\leqslant c_{2}^{2}\lambda_{2}^{-d}\|\chi_{\Omega}\|_{\infty}\|e^{-\pi \frac{2}{\lambda_{2}}s^{2}}\|_{1} \nonumber\\ &= c_{2}^{2}\lambda_{2}^{-d}\left(\frac{2}{\lambda_{2}}\right)^{-d/2}\nonumber\\ &=1, \end{split}$$ as $c_{2} = (2\lambda_{2})^{d/4} = \|\phi_2\|_{2}^{-1}$. Now, from , since $(f-L_{j}f, L_{j}f)\geqslant 0$, it follows $$\|f\|_2^{2} = \|f-L_{j}f\|_2^{2}+\|L_{j}f\|_2^{2} + 2(f-L_{j}f, L_{j}f)$$ and hence $$\label{estLj} \|f-L_{j}f\|_2^{2} \leqslant \|f\|_2^{2}-\|L_{j}f\|_2^{2}.$$ From the hypothesis and we obtain $$\left\{ \begin{array}{ll} \|f-L_{1}f\|_2^{2}\leqslant \varepsilon_{T}^{2}\|f\|_2^{2}&,\\ \|f-L_{2}f\|_2^{2}\leqslant \varepsilon_{\Omega}^{2}\|f\|_2^{2}&. \end{array} \right.$$ Considering the composition of $L_{1}$ and $L_{2}$ we have $$\begin{split} \|f-L_{2}L_{1}f\|_{2}&\leqslant\|f-L_{2}f\|_{2}+\|L_{2}f-L_{2}L_{1}f\|_{2}\nonumber\\ &\leqslant\varepsilon_{\Omega}\|f\|_{2} + \|L_{2}\|\|f-L_{1}f\|_{2}\nonumber\\ &\leqslant\varepsilon_{\Omega}\|f\|_{2}+ 1\cdot\varepsilon_{T}\|f\|_{2}\nonumber\\ &= (\varepsilon_{\Omega}+ \varepsilon_{T})\|f\|_{2}, \end{split}$$ where Lemma \[LemmaLocOp\] has been used with $q = \infty$ in the estimation of the operator norm $\|L_2\|_{B(L^{2})}\leqslant\|\Phi_2\|_{2}^{2}\|\chi_{\Omega}\|_{\infty} = 1$. Then $$\begin{split} \|L_{1}L_{2}f\|_2&\geqslant\|f\|_{2} - \|f-L_{2}L_{1}f\|_{2}\nonumber\\ &\geqslant\|f\|_{2} - (\varepsilon_{\Omega}+ \varepsilon_{T})\|f\|_{2}\nonumber\\ &=(1-\varepsilon_{T}-\varepsilon_{\Omega})\|f\|_{2}, \end{split}$$ and, from this, it follows that for every $r\in [1,\infty)$ $$\begin{split} 1-\varepsilon_{\Omega} - \varepsilon_{T}&\leqslant\frac{\|L_{1}L_{2}f\|_{2}}{\|f\|_{2}}\nonumber\\ &\leqslant\|L_{1}L_{2}\|\nonumber\\ &\leqslant\|L_{1}\|\|L_{2}\|\nonumber\\ &\leqslant\|\chi_{T}\|_{r}\|\chi_{\Omega}\|_{r}\left(\frac{1}{r'}\right)^{2d/r'}\|\Phi_{1}\|_{2}^{2}\|\Phi_{2}\|_{2}^{2}\nonumber\\ &= \left(\int_{T}{dt}\right)^{1/r}\left(\int_{\Omega}{ds}\right)^{1/r}\left(\frac{1}{r'}\right)^{2d/r'}, \end{split}$$ where we have used again Lemma \[LemmaLocOp\] with $q = r<+\infty$ in order to have norms involving the measures of the sets $T$ and $\Omega$. Hence, we finally have that $$|T||\Omega|\geqslant \sup_{r\in[1,\infty)} (1-\varepsilon_{T}-\varepsilon_{\Omega})^{r}(r')^{\frac{r}{r'}2d}.$$ which proves the thesis. From we have in particular that - For $r\rightarrow 1^+$, then $|T||\Omega| \geqslant 1-\varepsilon_T-\varepsilon_\Omega$; - For $r= 2$, then $|T||\Omega| \geqslant (1-\varepsilon_T-\varepsilon_\Omega)^{2}4^{d}$; - One can prove that for any fixed value of the parameter $1-\varepsilon_T - \varepsilon_\Omega\in[0,1)$ the supremum over $r\in[1,\infty)$ in the right-hand side of is actually a maximum. For this maximum no explicit expression is available but a study of the function $f(r) = (1-\varepsilon_T - \varepsilon_\Omega)^{r}(r')^{\frac{r}{r'}2d}$ can yield an approximation in dependence on $1-\varepsilon_T - \varepsilon_\Omega$ which improves estimates (1) and (2). - Remark that in the case where the inequalities of the hypothesis are strict, the same proof yields a strict estimate in the thesis . We remark that whereas the case $\varepsilon_T=\varepsilon_\Omega=0$ in the classical Donoho-Stark UP yields $|T||\Omega|\geqslant 1$ (which is a trivial assertion since in this case either $|T|=+\infty$ or $|\Omega|=+\infty$, cf. [@Ben85]), the case $\varepsilon_T=\varepsilon_\Omega=0$ in Theorem \[opL\] is not trivial and actually yields the following result. Let $T$, $\Omega$, $L_1$, $L_2$ be as in Theorem \[opL\] and suppose that there exists $f\in L^{2}(\mathbb{R}^{d})$ such that $$\|L_{1}f\|_{2}^{2}=\|f\|_{2}^{2} \quad \text{and}\quad\|L_{2}f\|_{2}^{2}=\|f\|_{2}^{2}$$ then $|T||\Omega|\geqslant e^{2d}$. Observe that for $\varepsilon_T=\varepsilon_\Omega=0$ the hypotheses of Theorem \[opL\] become $\Vert L_1 f\Vert_2 = \Vert L_2 f\Vert_2 = \Vert f\Vert_2$, since from Lemma \[LemmaLocOp\] we have $\Vert L_1\Vert_{B(L^2)} = \Vert L_2\Vert_{B(L^2)} = 1$. Then the assertion is proved by taking $\varepsilon_T=\varepsilon_\Omega=0$ in Theorem \[opL\] and remarking that $\sup_{r\in [1,\infty)} \left(\frac{r}{r-1}\right)^{r-1}=\lim_{r\to+\infty} \left(\frac{r}{r-1}\right)^{r-1}=e$. Another consequence of Theorem \[opL\] is an UP involving the marginal distributions of the spectrogram. We recall that the [*spectrogram*]{} is the time-frequency representation given by ${\mathop{\rm Sp}}_{\psi}(f,g)(x,\omega)=V_\psi f(x,\omega) \overline{V_\psi g}(x,\omega)$, defined in terms of a Gabor transform with window $\psi\in L^2({\mathbb{R}^{d}})$. It is an important and widely used tool in signal analysis as well as in connection with the theory of pseudo-differential operators, see e.g. [@BogOliWon-2006], [@BonDemJam], [@Coh89], [@CohLou04], [@FerGal10], [@Wil00]. We denote its marginal distributions with ${\mathop{\rm Sp}}_{\psi}^{(1)}(f,g)(x)=\int_{{\mathbb{R}^{d}}}{\mathop{\rm Sp}}_{\psi}(f,g)(x,\omega)\, d\omega$ and ${\mathop{\rm Sp}}_{\psi}^{(2)}(f,g)(\omega)=\int_{{\mathbb{R}^{d}}}{\mathop{\rm Sp}}_{\psi}(f,g)(x,\omega)\, dx$. Suppose that $f,g$ are functions in $L^2({\mathbb{R}^{d}})$ for which $\|f\|_2=\|g\|_2=1$ and $$\begin{array}{ll} \left\vert\int_{T}{\mathop{\rm Sp}}_{\Phi_1}^{(1)}(f,g)(x)\, dx\right\vert \geqslant \sqrt{1-\varepsilon_T^2} & \hbox{\rm ;} \\ \left\vert\int_{\Omega}{\mathop{\rm Sp}}_{\Phi_2}^{(2)}(f,g)(\omega)\, d\omega\right\vert\geqslant \sqrt{1-\varepsilon_\Omega^2} & \hbox{.} \end{array}$$ Then $$|T||\Omega|\geqslant \sup_{r\in [1,\infty)} (1-\varepsilon_{T}-\varepsilon_{\Omega})^{r} \left(\frac{r}{r-1}\right)^{2d(r-1)}.$$ Using the fundamental connection between localization operators and spectrogram $(L^a_\psi f,g)_{L^2({\mathbb{R}^{d}})}=(a,{\mathop{\rm Sp}}_{\psi}(g,f))_{L^2(\R2d)}$, which is a consequence of and , we can rewrite the hypothesis $\|L_{1}f\|_{2}^{2}\geqslant(1-\varepsilon_{T}^{2})\|f\|_{2}^{2}$ of Theorem \[opL\] as $$\label{1} \begin{array}{ll} \sqrt{1-\varepsilon_T^2} & \leqslant \sup_{\|g\|=1} |(L_1 f,g)|=\sup_{\|g\|=1} |(\chi_T,{\mathop{\rm Sp}}_{\Phi_1}(g,f))|= \\ & \sup_{\|g\|=1}\left|\int_{T\times{\mathbb{R}^{d}}} \overline{{\mathop{\rm Sp}}_{\Phi_1}(g,f)}\, dx d\omega\right|= \sup_{\|g\|=1}\left|\int_T {\mathop{\rm Sp}}_{\Phi_1}^{(1)}(f,g)\, dx\right| \end{array}$$ In analogous way the hypothesis $\|L_{2}f\|_{2}^{2}\geqslant(1-\varepsilon_{\Omega}^{2})\|f\|_{2}^{2}$ reads $$\label{2} \sqrt{1-\varepsilon_{\Omega}^2} \leqslant \sup_{\|g\|=1}\left|\int_{\Omega} {\rm Sp}_{\Phi_2}^{(2)}(f,g)\, d\omega\right|.$$ In particular and are satisfied in our hypothesis and therefore the thesis follows from Theorem \[opL\]. Comparison with Donoho-Stark ============================ This section is dedicated to the classical version of the Donoho-Stark theorem. We use the results of the previous section to prove a substantial improvement in constant $(1-\varepsilon_T-\varepsilon_\Omega)^2$ appearing on the right-hand side of estimate . Our result is the following. \[ourDS\] Let $f\in L^2({\mathbb{R}^{d}})$, $T,\Omega\subset {\mathbb{R}^{d}}$, $\varepsilon_\Omega, \varepsilon_T>0$ satisfy the hypotheses of Theorem \[DS\] (Donoho-Stark), then $$\label{sup} |T||\Omega|\geqslant \sup_{r\in [1,\infty)} (1-\varepsilon_{T}-\varepsilon_{\Omega})^{r} \left(\frac{r}{r-1}\right)^{2d(r-1)},$$ and in particular $$\label{improvedDS} |T||\Omega| \geqslant (1-\varepsilon_T - \varepsilon_{\Omega})^2 4^d$$ Let $Pf = \chi_T f$ and $Qf= \mathcal{F}^{-1}\chi_\Omega \mathcal{F}f$, then the hypotheses of the Donoho-Stark UP (Thm. \[DS\]) can be rewritten as $$\|Pf\|_{2}^{2}\geqslant(1-\varepsilon_{T}^{2})\|f\|_2^2\quad \text{and}\quad \|Qf\|_{2}^{2}\geqslant(1-\varepsilon_{\Omega}^{2})\|f\|_{2}^{2}.$$ From the condition $\varepsilon_T+\varepsilon_\Omega<1$ we can choose $\nu_T>\varepsilon_T$, $\nu_\Omega>\varepsilon_\Omega$, also satisfying $\nu_T+\nu_\Omega<1$. For $\nu_T$, $\nu_\Omega$ the strict inequalities hold: $$\label{>} \|Pf\|_{2}^{2}>(1-\nu_T^{2})\|f\|_2^2\quad \text{and}\quad \|Qf\|_{2}^{2}>(1-\nu_\Omega^{2})\|f\|_{2}^{2}.$$ Let us consider the operators $L_1$ and $L_2$ as defined in and respectively. We recall that $L_j=W^{F_j}$, $j=1,2$, as Weyl operators with $F_1 =c_1^2(\chi_T\ast e^{-\pi 2\lambda_1(\cdot)^2})^{}(x)$ and $F_2 =c_2^2 \lambda_2^{-d}(\chi_\Omega\ast e^{-\pi \frac{2}{\lambda_2}(\cdot)^2})^{}(\omega)$. Setting now $\varphi_{\lambda}(x) = \lambda^{d/2}e^{-\pi\lambda x^2}$, we have $ F_1 = \chi_T\ast \varphi_{2\lambda_1}$ and $F_2 = \chi_\Omega \ast\varphi_{\frac{2}{\lambda_2}}$. Notice that $\|\varphi_{2\lambda_1}\|_1 = \|\varphi_{\frac{2}{\lambda_2}}\|_{1} =1$ and that $\varphi_{2\lambda_1}\rightarrow\delta$ for $\lambda_1 \rightarrow+\infty$, and $\varphi_{\frac{2}{\lambda_2}}\rightarrow\delta$ for $\lambda_2 \rightarrow 0^+$ in $\mathcal{S'}({\mathbb{R}^{d}})$, so that $\{\varphi_{2\lambda_1}\}_{\lambda_1 \in\mathbb{R}}$ and $\{\varphi_{\frac{2}{\lambda_2}}\}_{\lambda_2 \in\mathbb{R}}$ are approximate identities. We prove now that if a function $f$ is suitably regular then: - $\left\|(\chi_T\ast\varphi_{2\lambda_1})f - \chi_T f\right\|_{2}\rightarrow 0$,  i.e. $L_1 f\rightarrow Pf$ in $L^{2}({\mathbb{R}^{d}})$,  as $\lambda_1\rightarrow +\infty$. - $\|\mathcal{F}^{-1}[(\chi_\Omega\ast\varphi_{\frac{2}{\lambda_2}})\widehat{f}] - \mathcal{F}^{-1}[\chi_\Omega \widehat{f}]\|_{2}\rightarrow 0$, i.e. $L_2 f\rightarrow Qf$ in $L^{2}({\mathbb{R}^{d}})$, as $\lambda_2\rightarrow 0^+$. Let us consider $(a)$: $$\begin{aligned} \left\|(\chi_T\ast\varphi_{2\lambda_1})f - \chi_T f\right\|_{2} &= \left\|\left(\chi_T\ast\varphi_{2\lambda_1} - \chi_T\right) f\right\|_{2}\nonumber\\ &\leqslant\|\chi_T\ast\varphi_{2\lambda_1} - \chi_T\|_{2p'}\|f\|_{2p}\nonumber\end{aligned}$$ for all $p\in[1,\infty]$. From the properties of approximate identities the first norm in the last line goes to $0$ as $\lambda_1 \rightarrow\infty$, if $p'<\infty$ and the second term is constant if $f\in L^{2p}({\mathbb{R}^{d}})$. Therefore $(a)$ is valid for all $f$ for which there exists $p>1$ such that $f\in L^{2p}({\mathbb{R}^{d}})$. In particular, this is true for all functions in $\mathcal S({\mathbb{R}^{d}})$. In a similar way (b) can be proven, indeed we have: $$\begin{aligned} \left\|\mathcal{F}^{-1}\left[(\chi_{\Omega}\ast\varphi_{2/\lambda_2})\widehat{f}\right] - \mathcal{F}^{-1}\left[\chi_{\Omega}\widehat{f}\right]\right\|_{2}& = \left\|\left((\chi_{\Omega}\ast\varphi_{2/\lambda_2}) - \chi_{\Omega}\right)\widehat{f}\right\|_{2}\nonumber\\ &\leqslant\|\chi_\Omega \ast \varphi_{2/ \lambda_2} - \chi_\Omega\|_{2p'}\|\widehat{f}\|_{2p}\nonumber\end{aligned}$$ where for $p'<\infty$ the first norm in the last line goes to $0$ as $\lambda_2\rightarrow 0^+$, and the second is constant if for instance $f\in \mathcal S({\mathbb{R}^{d}})$. Suppose now that the function $f\in L^2({\mathbb{R}^{d}})$ satisfies the Donoho-Stark hypotheses and let $f_n\in \mathcal S({\mathbb{R}^{d}})$ be such that $f_n \rightarrow f$ in $L^{2}({\mathbb{R}^{d}})$. Then $Pf_n\rightarrow Pf$ in $L^{2}({\mathbb{R}^{d}})$ and, therefore, $\frac{\|Pf_n\|_{2}}{\|f_n\|_2}\rightarrow\frac{\|Pf\|_2}{\|f\|_2}$. From the first inequality in , $(1-\nu_{T}^{2})^{1/2}<\frac{\|Pf\|_2}{\|f\|_2}$, and therefore there exists $n_1$ such that for all $n>n_1$ we have $$\label{confr1} (1-\nu_{T}^{2})^{1/2}<\frac{\|Pf_n\|_2}{\|f_n\|_2}.$$ On the other hand $Qf_n\rightarrow Qf$ in $L^{2}({\mathbb{R}^{d}})$ and hence $\frac{\|Qf_n\|_{2}}{\|f_n\|_2}\rightarrow\frac{\|Qf\|_2}{\|f\|_2}$. Similarly, by the second inequality in , $(1-\nu_{\Omega}^{2})^{1/2}<\frac{\|Qf\|_2}{\|f\|_2}$, and it follows that there exists $n_2$ such that $\forall n>n_2$ $$\label{confr2} (1-\nu_{\Omega}^{2})^{1/2}<\frac{\|Qf_n\|_2}{\|f_n\|_2}.$$ For $n>\max\{n_1,n_2\}$ both and hold, i.e. the hypotheses of Donoho-Stark hold therefore on $f_n$. As $f_n \in \mathcal S({\mathbb{R}^{d}})$, it follows that $$L_1 f_n\rightarrow Pf_n, \quad \text{as} \quad \lambda_1\rightarrow +\infty$$ and $$L_2 f_n\rightarrow Qf_n \quad\text{as}\quad \lambda_2\rightarrow 0^+$$ in $L^2(\mathbb{R}^d)$. Then for $\lambda_1$ sufficiently large and $\lambda_2$ sufficiently small, from and we have: $$(1-\nu_{T}^{2})^{1/2}<\frac{\|L_{1}f_n\|_2}{\|f_n\|_2}\quad\text{and}\quad(1-\nu_{\Omega}^{2})^{1/2}<\frac{\|L_{2}f_n\|_2}{\|f_n\|_2},$$ i.e. $f_n$ satisfies the hypotheses of Theorem \[opL\] and the thesis follows from with $\nu_T$, $\nu_\Omega$ in place of $\varepsilon_T$, $\varepsilon_\Omega$ respectively, i.e. $$\label{nu} |T||\Omega|\geqslant \sup_{r\in [1,\infty)} (1-\nu_{T}-\nu_{\Omega})^{r} \left(\frac{r}{r-1}\right)^{2d(r-1)}.$$ Finally the thesis follows taking in the supremum over all $\nu_T>\varepsilon_T$ and $\nu_\Omega>\varepsilon_\Omega$. Donoho-Stark Uncertainty principle and local uncertainty principle ================================================================== The Donoho-Stark UP states that there are restrictions to the behavior of a function and its Fourier transform, from a local viewpoint. There are other results in this direction in the literature, see e.g. [@CohLou04]; here we want to consider the local UP of Price, cf. [@Pri87], and investigate some consequences as well as the connections between these two UPs. More precisely, under the hypotheses of Donoho and Stark we prove a different estimate of $\vert T\vert \vert\Omega\vert$, with a constant depending on the function $f$. Moreover, we obtain a new UP involving the measure of the support of a function and the standard deviation of its Fourier transform. We start by recalling the result of Price. \[PriceUP\] Let $\Omega\subset\mathbb{R}^d$ be a measurable set and $\alpha>d/2$. Then for every $f\in L^2(\mathbb{R}^d)$ we have $$\label{IneqLocUP} \int_\Omega \vert\widehat{f}(\omega)\vert^2\,d\omega < K_1 \vert \Omega\vert\Vert f\Vert_2^{2-d/\alpha} \Vert \vert t\vert^\alpha f\Vert_2^{d/\alpha},$$ where $K_1$ is a constant depending on $d$ and $\alpha$, given by $$\label{K1} \begin{split} K_1 &= K_1(d,\alpha) \\ &= \frac{\pi^{d/2}}{\alpha} \left( \Gamma\left(\frac{d}{2}\right) \right)^{-1} \Gamma\left(\frac{d}{2\alpha}\right) \Gamma\left(1-\frac{d}{2\alpha}\right) \left(\frac{2\alpha}{d}-1\right)^{\frac{d}{2\alpha}} \left( 1-\frac{d}{2\alpha}\right)^{-1} \end{split}$$ and $\Gamma$ is the Gamma function defined as $\Gamma(x)=\int_0^{+\infty} t^{x-1} e^{-t}\,dt$. Moreover, the constant $K_1$ is optimal, and equality in is never attained. At first, we observe that Theorem \[PriceUP\], stated in $L^2$ spaces in [@Pri87], can be easily generalized; in fact, it is proved in [@Pri87 Corollary 2.2] that for every $q\in (1,\infty]$, $\alpha>\frac{d}{q^\prime}$, and $f\in L^q(\mathbb{R}^d)$, we have $$\label{GeneralizedLemma} \Vert\widehat{f}\Vert_\infty\leq \tilde{K}\Vert f\Vert_q^{1-d/\alpha q^\prime} \Vert \vert t\vert^\alpha f\Vert_q^{d/\alpha q^\prime},$$ where $$\label{KTilde} \tilde{K}=\left[ \frac{2\pi^{d/2}}{\Gamma(d/2)}\frac{1}{\alpha q} B\left(\frac{d}{\alpha q},\frac{1}{q-1}-\frac{d}{\alpha q}\right)\right]^{\frac{q-1}{q}} \left( \frac{\alpha q^\prime}{d}-1\right)^{d/qq^\prime\alpha} \left(1-\frac{d}{\alpha q^\prime}\right)^{-1/q}$$ and $B(\cdot,\cdot)$ is the Beta function, given by $B(x,y)=\int_0^1 t^{x-1}(1-t)^{y-1}\,dt$. Then, by the same proof of [@Pri87 Theorem 1.1] we get the following result. \[GeneralizedPriceUP\] Let $\Omega\subset\mathbb{R}^d$ be a measurable set, $q\in (1,\infty]$ and $\alpha>d/q^\prime$. Then for every $f\in L^q(\mathbb{R}^d)$ we have $$\label{GeneralizedIneqLocUP} \int_\Omega \vert\widehat{f}(\omega)\vert^2\,d\omega \leq K(d,\alpha,q) \vert \Omega\vert\Vert f\Vert_q^{2-2d/\alpha q^\prime} \Vert \vert t\vert^\alpha f\Vert_q^{2d/\alpha q^\prime},$$ where $K(d,\alpha,q)=\tilde{K}^2$, and $\tilde{K}$ is given by . We only have to prove the statement when the right-hand side of is finite; in this case we have $f\in L^1(\mathbb{R}^d)$, and so $\widehat{f}$ is a continuous bounded function; we then have $$\int_\Omega \vert\widehat{f}(\omega)\vert^2\,d\omega\leq \vert\Omega\vert \Vert \widehat{f}\Vert_\infty^2,$$ and the conclusion is an application of . We can formulate our new version of the Donoho-Stark UP with constant depending on the signal $f$. \[DSfdepending\] Let $\Omega$ and $T$ be two measurable subsets of $\mathbb{R}^d$, $q_j\in (1,\infty]$, $\alpha_j>d/q^\prime_j$, $j=1,2$ and $f\in L^1(\mathbb{R}^d)$ such that $\widehat{f}\in L^1(\mathbb{R}^d)$, $f\neq 0$. Suppose that $f$ is $\varepsilon_T$-concentrated on $T$, and $\widehat{f}$ is $\varepsilon_\Omega$-concentrated on $\Omega$, with $0\leqslant\varepsilon_T,\varepsilon_\Omega\leqslant 1$ and $\varepsilon_T+\varepsilon_\Omega\leqslant 1$. Then $$\label{DSestimate} \vert T\vert\vert\Omega\vert\geq C_f (1-\varepsilon_T-\varepsilon_\Omega)^2,$$ where $C_f$ is the supremum over $\overline{t},\overline{\omega}\in\mathbb{R}^d$, $q_j\in (1,\infty]$ and $\alpha_j>d/q_j^\prime$, $j=1,2$, of the quantities $$\frac{\Vert f\Vert_2^4 \Vert \widehat{f}\Vert_{q_1}^{2d/(\alpha_1 q_1^\prime)} \Vert f\Vert_{q_2}^{2d/(\alpha_2 q_2^\prime)}}{K(d,\alpha_1,q_1) K(d,\alpha_2,q_2) \Vert f\Vert_{q_2}^2 \Vert \widehat{f}\Vert_{q_1}^2 \Vert \vert t-\overline{t}\vert^{\alpha_2} f\Vert_{q_2}^{2d/(\alpha_2 q_2^\prime)} \Vert \vert \omega-\overline{\omega}\vert^{\alpha_1} \widehat{f}\Vert_{q_1}^{2d/(\alpha_1 q_1^\prime)}}, $$ and $K(d,\alpha_j,q_j)$, $j=1,2$, are the ones appearing in . We can limit our attention to $f$ such that $C_f>0$, otherwise the result is trivial. The hypothesis $f,\widehat{f}\in L^1(\mathbb{R}^d)$ implies that $f,\widehat{f}\in L^\infty(\mathbb{R}^d)$, and so $f,\widehat{f}\in L^q(\mathbb{R}^d)$ for every $q\in [1,\infty]$. Now, writing for a translation by $\overline{t}$ of $f$, we have that the left-hand side does not change, since the Fourier transform turns translations into modulations. Moreover, in the right-hand side the only term that is affected by the translation is the last norm, and so we get the following more general estimate: $$\label{Price1} \int_\Omega \vert\widehat{f}(\omega)\vert^2\,d\omega \leq K(d,\alpha,q) \vert \Omega\vert\Vert f\Vert_q^{2-2d/\alpha q^\prime} \Vert \vert t-\overline{t}\vert^\alpha f\Vert_q^{2d/\alpha q^\prime}.$$ By interchanging the roles of $f$ and $\widehat{f}$ in , we get $$\label{Price2} \int_T \vert f(t)\vert^2\,dt \leq K(d,\alpha,q) \vert T\vert\Vert \widehat{f}\Vert_q^{2-2d/\alpha q^\prime} \Vert \vert \omega-\overline{\omega}\vert^\alpha \widehat{f}\Vert_q^{2d/\alpha q^\prime}.$$ Observe now that, by the definition of $\varepsilon_T$-concentration of $f$ on $T$, we have $$\label{concf} \int_T \vert f(t)\vert^2\,dt = \Vert f\Vert_2^2-\int_{\mathbb{R}^d\setminus T}\vert f(t)\vert^2\,dt\geqslant (1-\varepsilon_T^2)\Vert f\Vert_2^2,$$ and analogously the hypothesis that $\widehat{f}$ is $\varepsilon_\Omega$-concentrated on $\Omega$ can be rewritten as $$\label{conchatf} \int_\Omega \vert \widehat{f}(\omega)\vert^2\,d\omega \geqslant (1-\varepsilon_\Omega^2)\Vert f\Vert_2^2.$$ Combining with (with $\alpha_1$ and $q_1$ instead of $\alpha$ and $q$, respectively) and with (with $\alpha_2$ and $q_2$ instead of $\alpha$ and $q$, respectively), we obtain $$\label{separateT} \vert T\vert \geq (1-\varepsilon_T^2)\frac{\Vert f\Vert_2^2 \Vert \widehat{f}\Vert_{q_1}^{2d/(\alpha_1 q_1^\prime)}}{K(d,\alpha_1,q_1) \Vert \widehat{f}\Vert_{q_1}^2 \Vert \vert \omega-\overline{\omega}\vert^{\alpha_1} \widehat{f}\Vert_{q_1}^{2d/(\alpha_1 q_1^\prime)}},$$ $$\label{separateomega} \vert \Omega\vert \geq (1-\varepsilon_\Omega^2)\frac{\Vert f\Vert_2^2 \Vert f\Vert_{q_2}^{2d/(\alpha_2 q_2^\prime)}}{K(d,\alpha_2,q_2) \Vert f\Vert_{q_2}^2 \Vert \vert t-\overline{t}\vert^{\alpha_2} f\Vert_{q_2}^{2d/(\alpha_2 q_2^\prime)}},$$ Then, multiplying these last inequalities we get $$\label{strongerestimate} \begin{split} \vert &T\vert\vert\Omega\vert\geq (1-\varepsilon_T^2)(1-\varepsilon_\Omega^2) \\ &\cdot\frac{\Vert f\Vert_2^4 \Vert \widehat{f}\Vert_{q_1}^{2d/(\alpha_1 q_1^\prime)} \Vert f\Vert_{q_2}^{2d/(\alpha_2 q_2^\prime)}}{K(d,\alpha_1,q_1) K(d,\alpha_2,q_2) \Vert f\Vert_{q_2}^2 \Vert \widehat{f}\Vert_{q_1}^2 \Vert \vert t-\overline{t}\vert^{\alpha_2} f\Vert_{q_2}^{2d/(\alpha_2 q_2^\prime)} \Vert \vert \omega-\overline{\omega}\vert^{\alpha_1} \widehat{f}\Vert_{q_1}^{2d/(\alpha_1 q_1^\prime)}}. \end{split}$$ Observe that, since $0\leqslant\varepsilon_T,\varepsilon_\Omega\leqslant 1$ and $\varepsilon_T+\varepsilon_\Omega\leqslant 1$, we have $(1-\varepsilon_T^2)(1-\varepsilon_\Omega^2)\geqslant (1-\varepsilon_T-\varepsilon_\Omega)^2$. Then the conclusion follows from , by taking the supremum over $\overline{t}$, $\overline{\omega}$, $\alpha_1$, $\alpha_2$, $q_1$ and $q_2$ in the right-hand side. We compare now the result of Theorem \[DSfdepending\] with the classical formulation of the Donoho-Stark UP. \[strongerresult\] We observe at first that the statement of Theorem \[DSfdepending\] is given in a parallel way to the classical one, but in the proof we have proved a stronger result. In fact, the local UP by Price gives estimates separately on the amount of energy of $f$ and $\widehat{f}$ in $T$ and $\Omega$, respectively. So, under the hypotheses of Theorem \[DSfdepending\] we have deduced and , that contain lower bounds for the measures of $T$ and $\Omega$ separately. This gives more information than a lower bound of the product between them. In order to make a further comparison between and Donoho-Stark UP, we observe that in we have a constant depending on the function $f$. A very natural question is if for some $f$ the result of Theorem \[DSfdepending\] is stronger than Proposition \[ourDS\] (and then stronger than the classical Donoho-Stark UP), or even stronger than any other estimate of the kind of with a constant that does not depend on $f$ (for example, this would happen if we find a sequence $f_n$ for which the corresponding $C_{f_n}$ tends to infinity). This seems a not trivial question, and is postponed to a further study; however, since Theorem \[DSfdepending\] is based on the local estimates of Price, and such estimates are optimal, our feeling is that Theorem \[DSfdepending\] can give better estimates for some functions $f$. As particular case of Theorem \[DSfdepending\] if $d=1$, $q_1=q_2=2$, $\alpha_1=\alpha_2=1$ and $f$ satisfies the hypotheses of Theorem \[DSfdepending\] we have the inequality $$\label{Delta} \Delta f \Delta \widehat f \geqslant \frac{(1-\varepsilon_T^2)(1-\varepsilon_\Omega^2)}{4\pi^2|T||\Omega|}\|f\|_2^2$$ with $\Delta f = \left(\int |t|^2 |f(t)|^2\, dt \right)^{1/2}$ and analogous definition for $\widehat f$. This can be viewed as an $\varepsilon$-concentration version of the classical Heisenberg UP which states that $$\label{HUP} \Delta f \Delta \widehat f \geqslant \frac{\|f\|_2^2}{4\pi}$$ for every $f\in L^2(\mathbb{R})$. Inequality is clearly of any interest only for the cases where the bound on its right-hand side exceeds the one of the classical Heisenberg UP . This happens if $|T||\Omega|\leqslant \frac{1}{\pi}(1-\varepsilon_T^2)(1-\varepsilon_\Omega^2)$; On the other hand, from the improved lower bound of the Donoho-Stark UP, we know that $4(1-\varepsilon_T-\varepsilon_\Omega)^2\leqslant |T| |\Omega|$. An elementary calculation shows that actually there exists values $\varepsilon_T,\varepsilon_\Omega$ and sets $T,\Omega$ compatible with both conditions. We end this section by presenting another consequence of the local UP of Price, in the form of a “mixed” UP that relates the measure of the support of a function with the concentration of its Fourier transform. \[mixed\] Let $f\in L^2(\mathbb{R}^d)$, $\alpha>d/2$, and $\overline{t},\overline{\omega}\in\mathbb{R}^d$. We have $$\label{suppvar1} \left\vert\supp f\right\vert \Vert\vert\omega-\overline{\omega}\vert^\alpha\widehat{f}\Vert_2^{d/\alpha}>\frac{1}{K}\Vert f\Vert_2^{d/\alpha},$$ and $$\label{suppvar2} \big\vert\supp \hat{f}\big\vert \Vert\vert t-\overline{t}\vert^\alpha f\Vert_2^{d/\alpha}>\frac{1}{K}\Vert f\Vert_2^{d/\alpha},$$ where $K=K(d,\alpha,2)$. The estimates are not trivial only for functions $f$ such that one between $f$ and $\widehat{f}$ has support with finite measure. Suppose that $\left\vert\supp f\right\vert$ is finite. By , with $q=2$ and $T=\supp f$, we obtain $$\Vert f\Vert_2^2 = \int_{\supp f}\vert f(t)\vert^2\,dt<K\left\vert\supp f\right\vert \Vert f\Vert_2^{2-d/\alpha}\Vert\vert\omega-\overline{\omega}\vert^\alpha\widehat{f}\Vert_2^{d/\alpha},$$ that is . The inequality can be proved in the same way, by using with $\Omega=\supp\hat{f}$ and $q=2$. [20]{} M. Benedicks. On Fourier transforms of functions supported on sets of finite Lebesgue measure, (1), 180–183, 1985. 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--- abstract: 'The two-pion correlation function can be defined as a ratio of either the measured momentum distributions or the normalized momentum space probabilities. We show that the first alternative avoids certain ambiguities since then the normalization of the two-pion correlator contains important information on the multiplicity distribution of the event ensemble which is lost in the second alternative. We illustrate this explicitly for specific classes of event ensembles.' address: 'Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany' author: - 'Q.H. Zhang, P. Scotto and U. Heinz' title: 'Multi-boson effects and the normalization of the two-pion correlation function' --- PACS numbers: 25.75.-q, 25.75.Gz, 25.70.Pq. Introduction {#sec1} ============ Two-particle Bose-Einstein (BE) interferometry (also known as Hanbury Brown-Twiss (HBT) intensity interferometry) as a method for obtaining information on the space-time geometry and dynamics of high energy collisions has recently received intensive theoretical and experimental attention. Detailed investigations revealed that high-quality two-particle correlation data constrain not only the geometric size of the particle-emitting source but also its dynamical state at particle freeze-out [@Zajc86; @Lorstad89; @BGJ; @APW; @Pratt95; @He96]. Two different definitions of two-pion correlation function are employed in the literature [@Zajc86; @Lorstad89; @BGJ; @APW; @Pratt95; @He96; @GKW79; @Zajc84; @Mark; @UA1; @Padula; @Alex93; @CDL; @MV97]. The first starts from the measured invariant $i$-pion inclusive distribution $$\label{1} N_i({\bbox{p}_1, \bbox{p}_2,\dots, \bbox{p}_i}) = E_{\bbox{p}_1}\cdots E_{\bbox{p}_i} {1\over \sigma} \frac{d^{3i}\sigma}{d^3p_1 d^3p_2 \cdots d^3p_i} \, ,$$ which is normalized via $$\begin{aligned} \label{2} && \int {d^3p_1\over E_{\bbox{p}_1}}\cdots {d^3p_i\over E_{\bbox{p}_i}} N_i(\bbox{p}_1,\dots,\bbox{p}_i) \nonumber\\ &&\qquad\qquad\qquad = \langle n(n-1)\cdots(n-i+1) \rangle \end{aligned}$$ to the $i$th order factorial moment of the pion multiplicity distribution, and defines the two-particle correlator as $$\label{I} C^I({\bbox{p}_1,\bbox{p}_2}) = \frac{N_2(\bbox{p}_1,\bbox{p}_2)} {N_1({\bbox{p}_1})N_1({\bbox{p}_2})}\, .$$ The second definition instead employs the normalized $i$-pion production probability $$P_{i}({\bbox{p}_1,\dots,\bbox{p}_i}) =\frac{N_i({\bbox{p}_1,\dots,\bbox{p}_i})} {\langle n(n-1)\cdots(n-i+1)\rangle}$$ and defines $$\label{II} C^{II}(\bbox{p}_1,\bbox{p}_2) = \frac{P_2({\bbox{p}_1},{\bbox{p}_2})} {P_1({\bbox{p}_1})P_1({\bbox{p}_2})}\, .$$ It follows that $$\label{18} C^{II}(\bbox{p}_1,\bbox{p}_2) = \frac{\langle n \rangle^2}{\langle n(n-1)\rangle}\, C^{I}(\bbox{p}_1,\bbox{p}_2)\,.$$ Recently, Miśkowiec and Voloshin [@MV97] argued that the first definition is preferable because it is based directly on measured quantities and it is consistent with the often used theoretical expression $$\label{corr} C(\bbox{q,K}) = 1 + {\left|\int d^4x\, S(x,\bbox{K}) \, e^{iq\cdot x} \right|^2 \over \int d^4x\, S(x,\bbox{p}_1)\, \int d^4y\, S(y,\bbox{p}_2)}\, .$$ Here $\bbox{K}=(\bbox{p}_1+\bbox{p}_2)/2$, $\bbox{q}=\bbox{p}_1-\bbox{p}_2$, $q^0=E_{\bbox{p}_1}-E_{\bbox{p}_2}$, and $S(x,\bbox{K})$ is the emission function of the source. In this paper we will stress that the first definition has the additional advantage that it provides information not only about [*the shape of the correlator*]{}, but also through its normalization about [*the pion multiplicity distribution*]{} which is lost in the second definition. We will show that it can be exploited to search for multi-pion symmetrization effects and may thus be a useful ingredient in the HBT analysis of 2-pion correlation functions. A simple example {#sec2} ================ To illustrate the importance of the normalization of the 2-pion correlator let us start with a simple example in which we consider the following multi-pion states: $$\begin{aligned} \label{7} |\phi\rangle_m &=& A_m \exp [ (\hat B^\dagger)^{m} ] |0\rangle \, , \label{xx1}\\ \hat B^\dagger &=& \int d^3p\, j(\bbox{p})\, a^\dagger(\bbox{p}). \end{aligned}$$ The states (\[7\]) are normalizable for $m\leq 2$; the normalization $A_m$ ensures $_m\langle\phi|\phi\rangle_m=1$. For $m=1$, $|\phi\rangle_1$ is a standard coherent state [@GKW79] with $A_1= \exp(-n_0/2)$ and $n_0= \int d^3p \, |j(\bbox{p})|^2$. Then the pion multiplicity distribution is of Poisson form, $$\label{9} P(n)= \frac{n_0^n}{n!}\exp(-n_0),$$ and the two-pion correlators (\[I\]) and (\[II\]) are given by $$C^{I}({\bbox{p}_1,\bbox{p}_2})= C^{II}({\bbox{p}_1,\bbox{p}_2})= 1\, .$$ For $m=2$ we have $A_2 = (1-4 n_0^2)^{\frac{1}{4}}$, and the pion multiplicity distribution is given by $$P(n) = \left\{ \begin{array}{ccl} 0 &,&\ n=2k+1~{\rm odd},\\ (1-4n_0^2)^{\frac{1}{2}}\frac{(2k)!}{(k!)^2}n_0^{2k} &,&\ n=2k~{\rm even}. \end{array} \right.$$ Calculating the correlation functions we find $$C^{I}({\bbox{p}_1,\bbox{p}_2}) = 2+\frac{1}{4n_0^2}, \quad C^{II}({\bbox{p}_1,\bbox{p}_2})=1.$$ One observes that now $C^{I}$ is different from $C^{II}$ due to the fact that the pion multiplicity distribution is no longer of Poisson form. However, although $|\phi\rangle_2$ is clearly not a coherent state, $C^{II}$ is again equal to $1$. The use of the second definition (\[II\]) thus does not allow to distinguish between the states $|\phi_1\rangle$ and $|\phi_2 \rangle$, whereas the first definition (\[I\]) clearly does. One may, of course, argue that the only difference between $C^{I}$ and $C^{II}$ is the normalization factor which can be obtained independently by measuring the pion multiplicity distribution. Our point is that important information on the pion multiplicity distribution can also be extracted directly from the properly normalized correlation function, and that this opportunity should not be given away by working with probabilities rather than directly with the measured cross sections. Multi-boson effects on the correlator and its normalization {#sec3} =========================================================== We will now consider a more physical model and show again that the use of the second definition leads to a loss of interesting information about the source. It is well known that in relativistic heavy-ion collisions the pion multiplicity is so large that it may be necessary to take multi-pion BE correlations into account [@APW; @Pratt95; @Zajc; @Pratt93; @CGZ; @ZC1; @CZ1; @Zhang; @Zhang2; @SBA97; @Pol1; @Pol2; @Urs]. In the following we will use a specific class of density matrices for multi-pion systems [@Pratt93; @CGZ; @ZC1] to study multi-pion BE correlation effects on the two-pion correlation function, thereby generalizing the conclusions of Ref.[@MV97]. For this class of ensembles it was shown in [@Pratt93; @CGZ; @ZC1] that, after including multi-pion correlation effects, the two-pion and single-pion inclusive distributions can, in the notation of [@Pratt93; @CGZ; @ZC1], be written in the following simple form [@ZZZ]: $$\begin{aligned} \label{12} N_1(\bbox{p}) &=& E_{\bbox{p}} H(\bbox{p,p})\, , \\ \label{13} N_2({\bbox{p}_1,\bbox{p}_2}) &=& E_{\bbox{p}_1} E_{\bbox{p}_2} \bigl[ H(\bbox{p}_1,\bbox{p}_1) H(\bbox{p}_2,\bbox{p}_1) \nonumber\\ &&\qquad\quad + H(\bbox{p}_1,\bbox{p}_2) H(\bbox{p}_2,\bbox{p}_1) \bigr]\, , \\ \label{14} H({\bbox{p}_1,\bbox{p}_2}) &=& \sum_{i=1}^{\infty} G_i({\bbox{p}_1,\bbox{p}_2}) \, . \end{aligned}$$ The $G_i(\bbox{p,q})$ are defined as $$\label{15} G_i(\bbox{p},\bbox{q}) = \int \rho(\bbox{p},\bbox{p}_1) d^3p_1 \rho(\bbox{p}_1,\bbox{p}_2) \cdots d^3p_{i-1} \rho(\bbox{p}_{i-1},\bbox{q}) ,$$ where $\rho(\bbox{p}_i,\bbox{p}_j)$ is a Fourier transform of the source emission function $g(x,\bbox{K})$: $$\label{16} \rho(\bbox{p}_i,\bbox{p}_j) = \int d^4x\, g\left(x,\bbox{K}_{ij}\right) \, e^{iq_{ij}\cdot x}\, .$$ Here $\bbox{K}_{ij}{=}(\bbox{p}_i+\bbox{p}_j)/2$ and $\bbox{q}_{ij}{=}\bbox{p}_i-\bbox{p}_j$, $q^0_{ij}{=}E_{\bbox{p}_i}-E_{\bbox{p}_j}$. Inserting the expressions (\[12\],\[13\]) into Eq. (\[I\]) one obtains $$\label{17} C^{I}(\bbox{p}_1,\bbox{p}_2) = 1 + \frac{H(\bbox{p}_1,\bbox{p}_2) H(\bbox{p}_2,\bbox{p}_1)} {H(\bbox{p}_1,\bbox{p}_1) H(\bbox{p}_2,\bbox{p}_2)}\,.$$ This correlator goes to 1 as $\bbox{q}\to\infty$ and to 2 as $\bbox{q}\to 0$. (Final state interactions are neglected here.) Thus even dramatic multi-boson effects as discussed below do not affect the intercept of the correlator $C^I$ — although they change the multiplicity distribution towards Bose-Einstein form they do not lead to genuine phase coherence. Explicit expressions for the pion multiplicity distribution and its first two moments $\langle n\rangle$, $\langle n(n-1)\rangle$ for the model studied here can be found in [@Pratt93; @CGZ; @ZC1; @CZ1]. Since $H(\bbox{p}_1,\bbox{p}_2)=H^*(\bbox{p}_2,\bbox{p}_1)$, the second term in Eq. (\[17\]) is always positive, ensuring that $C-1$ is positive definite. The normalization conditions (\[2\]) therefore imply that for the class of systems studied here and in [@Pratt93; @CGZ; @ZC1; @CZ1] one has always $\langle n(n-1)\rangle > \langle n\rangle^2$ (see Fig. 3 below). Obviously, Eq. (\[17\]) can therefore not apply to systems with multiplicity distributions $P(n)$ which give $\langle n(n-1)\rangle \leq \langle n\rangle^2$ (e.g. for systems with fixed event multiplicity [@Pratt93; @Zhang; @Urs]). The structure of (\[17\]) permits to introduce, in analogy to (\[16\]), a modified source distribution $S(x,\bbox{K})$ via $$\label{20} H(\bbox{p}_1,\bbox{p}_2) = \int d^4x\, S(x, \bbox{K})\, e^{iq\cdot x}$$ such that the correlator (\[17\]) can be written in the form (\[corr\]). $S(x,\bbox{K})$ is related to the original source distribution $g(x,\bbox{K})$ via Eqs. (\[14\])-(\[16\]). It includes all higher order multiparticle BE symmetrization effects. When interpreting measured single particle spectra and two-particle correlations one must keep in mind that the extracted information on the source [*corresponds to the effective source distribution $S(x,\bbox{p})$ rather than to the emission function $g(x,\bbox{p})$*]{}. The following example shows that these two functions can differ considerably; but we will also see that an important clue as to how much they differ will be provided by the normalization of the correlator. As shown in [@Pratt93; @CGZ; @ZC1; @CZ1] the recursion relations for the functions $G_i$ in (\[15\]) can be solved analytically for the class of model ensembles studied here if the following source distribution $g(x,\bbox{p})$ is assumed: $$\begin{aligned} \label{21} g(\bbox{r},t,\bbox{p}) &=& n_0 \left( {1\over 2\pi R^2} \right)^{3/2} \exp\left(-\frac{\bbox{r}^2}{2R^2}\right) \nonumber\\ && \times \left( \frac{1}{2\pi \Delta^2} \right)^{3/2} \exp\left(-\frac{\bbox{p}^2}{2 \Delta^2}\right)\, \delta(t)\, . \end{aligned}$$ $g(\bbox{r},t,\bbox{p})$ is the [*Wigner*]{} density of the source in the absence of multi-pion symmetrization effects. It is obtained in [@ZC1; @CZ1] by folding the Wigner densities of individual Gaussian wavepackets with a classical phase-space distribution $\rho_{\rm class}$ for their centers (Eq. (19) in [@CH94]; see also [@ZWSH]). The parameters $R,\Delta$ in (\[21\]) are thus combinations of the wavepacket width $\sigma$ with the spatial and momentum space widths $R_{\rm class}$ and $\Delta_{\rm class}$ of $\rho_{\rm class}$ (see Eqs. (20,21) in [@Weal]). While the width parameters $R_{\rm class}, \Delta_{\rm class}$ of the classical distribution $\rho_{\rm class}$ are unconstrained, the widths $R,\Delta$ of the Wigner density $g(\bbox{r},t,\bbox{p})$ which result from the folding procedure always satisfy the quantum mechanical uncertainty relation $R\Delta \geq \hbar/2$. The [*input*]{} multiplicity distribution is taken to be Poissonian as in (\[9\]); its mean value $n_0$ can be interpreted as the mean pion multiplicity in the absence of Bose-Einstein correlations [@Pratt93; @CGZ]. By inspection of Eqs. (\[14\])-(\[16\]) one easily convinces oneself that the instantaneous character of (\[21\]) carries over to the effective source distribution $S(x,\bbox{p})$. Using the analytical expressions from Refs. [@Pratt93; @CGZ; @ZC1; @CZ1] we compute $$N_1(\bbox{p}) = E_{\bbox{p}}\,H(\bbox{p},\bbox{p}) = E_{\bbox{p}} \int d^4x\, S(x,\bbox{p})$$ as well as the normalized single-pion probability distribution in momentum space $$P_1^{\langle n \rangle}(\bbox{p}) = \frac{E_{\bbox{p}}}{\langle n \rangle} \int d^4x\, S(x,\bbox{p}) = \frac{E_{\bbox{p}}}{\langle n \rangle} \sum_{i=1}^{\infty} G_i(\bbox{p,p})\, .$$ The mean pion multiplicity $\langle n \rangle$ is given by $$\langle n \rangle = \int d^3p\, d^4x\, S(x,p) = \sum_{i=1}^{\infty} \int d^3p\, G_i(\bbox{p,p})\, .$$ For the model (\[21\]) $P_1^{\langle n \rangle}(p)$ is a function of $p=|\bbox{p}|$ only. It is shown in Fig. 1 for different [*observed*]{} average pion multiplicities $\langle n \rangle$. Next to the value $\langle n \rangle$ we also give the average pion phase-space density of the system, $$\label{d} d = {\langle n \rangle \over (2 R\Delta)^3} \, .$$ One sees that as $d$ increases the pions concentrate in momentum space at low momenta. This reflects their bosonic nature: pions like to be in the same state. The instantaneous nature of the (effective) emission functions $g(x,\bbox{p})$ and $S(x,\bbox{p})$ (see (\[21\])) allows for inversion of the Fourier transform (\[20\]): writing $S(x,\bbox{p})= S(\bbox{r},\bbox{p})\,\delta(t)$ we have $$\label{25} S(\bbox{r,K}) = \int {d^3q \over (2\pi)^3}\, H\left(\bbox{K}+{\textstyle{\bbox{q} \over 2}}, \bbox{K}-{\textstyle{\bbox{q} \over 2}}\right) e^{i\bbox{q}\cdot \bbox{r}}\,.$$ We define the normalized source distribution in coordinate space $P^{\langle n\rangle}(\bbox{r})$ via $$\begin{aligned} \label{26} && P^{\langle n \rangle}(\bbox{r}) = {\int d^3K\, S(\bbox{r,K}) \over \int d^3K\, d^3r\, S(\bbox{r,K})} \nonumber\\ && \qquad = {1\over \langle n \rangle} \int {d^3K\, d^3q\, \over (2\pi)^3}\, H\left(\bbox{K}+{\textstyle{\bbox{q} \over 2}}, \bbox{K}-{\textstyle{\bbox{q} \over 2}}\right)\, e^{i\bbox{q}\cdot \bbox{r}} \, . \end{aligned}$$ Due to the spherical symmetry of (\[21\]) it is a function of $r=|\bbox{r}|$ only. The function $H$ in (\[25\],\[26\]) is known analytically [@Pratt93; @CGZ; @ZC1; @CZ1] to be a simple Gaussian in $\bbox{q}$, rendering the Fourier transform trivial. The resulting $P^{\langle n \rangle}(r)$ is shown in Fig. 2 for different pion phase-space densities. One sees that with increasing phase-space density the multi-pion BE correlations also lead to a concentration of the pions in coordinate space. The fact that multi-pion BE correlations lead to a reduction of the HBT radius has been observed previously [@Zajc; @CGZ; @Urs]. The radius extracted from HBT interferometry reflects the typical length scale of $P^{\langle n \rangle}(r)$; it is always smaller than the input geometric radius $R$ of the source and depends on the mean pion multiplicity per event. The high density limit {#sec4} ====================== Taking the limit of a highly condensed Bose gas, $d = {\langle n \rangle \over (2 R\Delta)^3} \rightarrow \infty$ [@Comm1], the multiplicity distribution and 1- and 2-particle spectra can be determined analytically [@ZC1]: $$\begin{aligned} \label{27} P(n) &=& \frac{\langle n \rangle^n}{(\langle n \rangle +1)^{n+1}}\, , \\ \label{28} N_1(\bbox{p}) &=& E_{\bbox{p}} {\langle n\rangle \over (2\pi\Delta_{\rm eff}^2)^{3/2}}\, \exp\left(-\frac{\bbox{p}^2}{2\Delta_{\rm eff}^2}\right)\,, \\ \label{29} N_2(\bbox{p}_1,\bbox{p}_2) &=& 2\, N_1(\bbox{p}_1)\, N_1(\bbox{p}_2)\, , \\ \label{30} \Delta_{\rm eff}^2 &=& {\Delta \over 2R} \leq \Delta^2\, . \end{aligned}$$ In this limit the correlation functions are $$C^{I}(\bbox{p}_1,\bbox{p}_2) = 2, \qquad C^{II}(\bbox{p}_1,\bbox{p}_2) = 1.$$ Multi-pion BE correlations change the original Poisson distribution into the Bose-Einstein multiplicity distribution (\[27\]). Correspondingly, ${\langle n(n-1) \rangle \over \langle n\rangle^2}$ changes from 1 to 2. In (\[18\]) this change exactly compensates the fact that the correlator $C^I$ no longer decays as a function of $\bbox{q}{=}\bbox{p}_1{-}\bbox{p}_2$, and from the resulting $C^{II}\equiv 1$ one might thus be misled to conclude (incorrectly) that the source exhibits phase coherence. In Fig. 3 we show the ratio ${\langle n\rangle^2 \over \langle n(n-1)\rangle}$ as a function of the average pion phase-space density. In the lower diagram we plot it as a function of the ratio of input parameters $n_0/(2R\Delta)^3$, in the upper diagram we use as a measure of the phase-space density the analogous ratio formed with the [*measured*]{} average multiplicity $\langle n \rangle$. For low phase space densities and large systems ($2R\Delta \gg 1$) ${\langle n\rangle^2 \over \langle n(n-1)\rangle}$ is close to 1; BE symmetrization effects on the observed multiplicity distribution are then negligible, and it becomes equal to the Poissonian input distribution with $\langle n(n-1)\rangle = \langle n\rangle^2 = n_0^2$. For smaller systems ($2R\Delta \simeq 1$) the observed multiplicity distribution becomes non-Poissonian, with ${\langle n\rangle^2 \over \langle n(n-1)\rangle} < 1$, even in the limit of vanishing multiplicity, $d = \langle n \rangle /(2R\Delta)^3 \to 0$: for the model discussed here one finds analytically $$\label{lim} \lim_{n_0\to 0} {\langle n(n-1) \rangle \over \langle n\rangle^2} = 1 + {1\over (2R\Delta)^3} \leq 2\, .$$ For large phase-space densities $d{>}1$, the ratio ${\langle n\rangle^2 \over \langle n(n-1)\rangle}$ decreases, eventually approaching for $d\to \infty$ the value 0.5 which reflects Bose-Einstein statistics. The critical phase-space density for the transition from Poisson statistics with $\langle n(n-1)\rangle = \langle n\rangle^2$ to Bose-Einstein statistics with $\langle n(n-1)\rangle = 2 \langle n\rangle^2$ depends on the total phase-space volume $(2R\Delta)^3$, but for $2R\Delta \gg 1$ it occurs near $d\simeq 0.3$. For $d\gg 1$, multi-boson symmetrization effects become dominant. The Bose condensation limit is reached at a finite critical value for the mean input multiplicity $n_0$: $$\label{nc} n_0 \to n_c = \left( R\Delta + {\textstyle{1\over 2}}\right)^3 \geq 1\, .$$ As $n_0$ approaches the critical value (which for large systems $2R\Delta \gg 1$ corresponds to $n_c/(2R\Delta)^3 \approx {1\over 8}$), the observed mean multiplicity $\langle n\rangle$ and phase-space density $d$ (as well as the total energy) go to infinity [@Pratt93]. In this sense the limit $n_0\to n_c$ here is analogous to the limit $\mu_\pi \to m_\pi$ in a thermalized pion gas of infinite volume in the grand canonical formalism. Normalized correlation functions from experiment {#sec5} ================================================ A direct experimental determination of the phase-space density $d= \langle n \rangle/(2R\Delta)^3$ in high energy collisions is not easy. The sources created in such collisions feature strong collective expansion [@Pratt95; @He96], and therefore only small fractions of the total collision region (so-called regions of homogeneity) contribute effectively to the two-particle correlator [@He96]. This means that in the parametrization (\[21\]) we should use $R^2 = R_{\rm hom}^2 + 1/(4\Delta^2)$ where $R_{\rm hom}$ is the (pair momentum dependent) homogeneity radius which, in the absence of strong multi-pion effects, is equal to the HBT radius parameter $R_{\rm HBT}$. Without more detailed model studies it is then, however, unclear what fraction of the total observed multiplicity $\langle n \rangle$ comes from a single such homogeneity region. On the other hand Fig. 3 suggests that, within our model class of event ensembles, the ratio ${\langle n\rangle^2 \over \langle n(n-1)\rangle}$ is a useful indicator for the average phase-space density $d$ in the source and thereby also for the expected multi-pion symmetrization effects on the 1- and 2-particle spectra which need to be taken into account in an extraction of the source size from HBT measurements. In the experiment one usually fits the two-particle correlator with the functional form $$C^{\rm exp}(\bbox{p}_1,\bbox{p}_2) = C^{\rm exp}(\bbox{q,K}) = {\cal N} \left( 1{+}\lambda |f(\bbox{q,K})|^2\right) ,$$ where the function $f$ vanishes as $\bbox{q}\to \infty$. Obviously, ${\cal N}$ depends on the chosen definition of the correlation function: For the definition (\[I\]) the normalization is always ${\cal N}^{I}=1$ (see Eq. (\[17\])), while for the definition (\[II\]) it is ${\cal N}^{II} = {\langle n\rangle^2 \over \langle n(n-1)\rangle}$ (see Eq. (\[18\])). But in both cases ${\cal N}$ is well-defined and thus should not be treated as a free fit parameter. Therefore we now discuss shortly an algorithm for the experimental construction of the two-particle correlator which is guaranteed [@Comm2] to give the correct value for ${\cal N}$, [*without relying on an actual measurement of the multiplicity distribution*]{}. We write the single-pion inclusive distribution as $$N_1(\bbox{p}) = \frac{E_{\bbox{p}}}{N_{\rm ev}} \sum_{i=1}^{N_{\rm ev}} \nu_i(\bbox{p})\, .$$ $N_{\rm ev}$ is the total number of collision events, and $\nu_i(\bbox{p})$ is the number of pions with momentum $\bbox{p}$ in collision $i$. The two-particle distribution can be expressed as $$\label{35} N_2(\bbox{p}_1,\bbox{p}_2) = {E_{\bbox{p}_1}\, E_{\bbox{p}_2} \over N_{\rm ev}} \sum_{i=1}^{N_{\rm ev}} \tilde\nu_{i,i}(\bbox{p}_1,\bbox{p}_2)\, ,$$ where $\tilde\nu_{i,i}(\bbox{p}_1,\bbox{p}_2)$ is the number of pion pairs with momenta $(\bbox{p}_1,\bbox{p}_2)$ in collision event $i$, and the double index $i$ indicates that both particles are from the same event. These definitions satisfy the normalization conditions (\[2\]). While $N_2(\bbox{p}_1,\bbox{p}_2)$ is constructed by selecting pion pairs from the same events, the denominator $N_1(\bbox{p}_1)N_1(\bbox{p}_2)$ can be generated by combining pion pairs from different events [@Kop74; @Zajc84; @Mark; @UA1; @Alex93; @CDL; @MV97]. The proper prescription is $$\label{36} N_1(\bbox{p}_1) N_1(\bbox{p}_2) = {E_{\bbox{p}_1} \, E_{\bbox{p}_2} \over N_{\rm ev}(N_{\rm ev}-1)} \sum_{i,j=1 \atop i\ne j}^{N_{\rm ev}} \tilde\nu_{i,j}(\bbox{p}_1, \bbox{p}_2)$$ where $\tilde\nu_{i,j}(\bbox{p}_1, \bbox{p}_2) = \nu_i(\bbox{p}_1) \nu_j(\bbox{p}_2)$. One easily checks that $$\int \frac{d^3p_1}{E_{\bbox{p}_1}} \frac{d^3p_2}{E_{\bbox{p}_2}} N_1(\bbox{p}_1) N_1(\bbox{p}_2) = \langle n \rangle^2\, .$$ The ratio of (\[35\]) and (\[36\]) thus gives the properly normalized correlator $C^{I}$. The above equations are true for unbiased events. Trigger biases and limited experimental acceptances can induce residual correlations in the event-mixed “background” (\[36\]) which must be corrected for separately (see [@Zajc84] for an extensive discussion). For large $N_{\rm ev}$ the evaluation of (\[36\]) is very time consuming; it also leads to a statistically unnecessarily accurate result for the denominator in (\[I\]). In practice one can live with fewer event pairs for event mixing, by replacing in (\[36\]) the number $N_{\rm ev}$ by a much smaller number $N'_{\rm ev}$. As long as $N'_{\rm ev}(N'_{\rm ev}-1) \gg N_{\rm ev}$ one can still ensure that the contribution of the denominator to the statistical error of the final correlation function is negligible [@Comm2]. The correlator $C^{II}$ differs from $C^{I}$ only by the different normalization. It can be constructed by taking the ratio of the following two expressions [@Zajc84]: $$\begin{aligned} \label{38} P_2(\bbox{p}_1,\bbox{p}_2) &=& {E_{\bbox{p}_1} \, E_{\bbox{p}_2} \over N^c_{\rm pairs}} \sum_{i=1}^{N_{\rm ev}} \tilde\nu_{i,i}(\bbox{p}_1,\bbox{p}_2) \, , \\ \label{39} P_1(\bbox{p}_1) P_1(\bbox{p}_2) &=& {E_{\bbox{p}_1} \, E_{\bbox{p}_2} \over N^u_{\rm pairs}} \sum_{i,j=1 \atop i\ne j}^{N_{\rm ev}} \tilde\nu_{i,j}(\bbox{p}_1, \bbox{p}_2) \,. \end{aligned}$$ $N^c_{\rm pairs}$ and $N^u_{\rm pairs}$ are the total numbers of “correlated” and “uncorrelated” pion pairs, respectively: $$\begin{aligned} N^c_{\rm pairs} &=& N_{\rm ev} \cdot \langle n(n-1)\rangle \,, \\ N^u_{\rm pairs} &=& N_{\rm ev}(N_{\rm ev}-1) \cdot \langle n \rangle^2 \,. \end{aligned}$$ Conclusions {#sec6} =========== We have shown that in principle the normalization of the two-particle Bose-Einstein correlation function contains valuable information on the the multiplicity distribution of the event ensemble. Both theoretically and experimentally the absolute normalization of the correlation function should thus be controlled as well as possible. We presented a variant of a previously suggested experimental algorithm [@MV97] for the construction of the correlator which guarantees correctly normalized correlators. Within a specific model class of event ensembles which recently received extensive theoretical attention we showed that in systems with large pion phase-space densities multi-pion symmetrization effects can lead to interesting measurable effects on the normalization of the correlator. We suggest a careful study of this normalization as an alternate method for searching for strong multi-pion symmetrization effects in high-multiplicity hadronic and heavy-ion collisions. The authors thank D. Miśkowiec, U. Wiedemann, C. Slotta, and T. Csörgő for helpful discussions. Q.H.Z. gratefully acknowledges support by the Alexander von Humboldt Foundation. 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Wiedemann [*et al.*]{}, Phys. Rev. C [**56**]{}, R614 (1997). The expressions for $G_i(\bbox{p},\bbox{q})$ in Refs. [@Pratt93; @CGZ] and [@ZC1; @CZ1] look at first sight quite different, but they become identical after suitable rearrangements [@ZC1]. Since $2R\Delta \geq 1$, the limit indicated in the text can only be achieved by letting $\langle n \rangle \to \infty$ or $n_0 \to n_c$ (see Eq. (\[nc\])). Combining this result with Eqs. (\[17\]) and (\[18\]) gives a correlator of the form $C^{II}(\bbox{q,K}) = {\cal N} (1 + \vert f(\bbox{q,K})\vert^2)$ with ${\cal N} = 1/[1 + (2R\Delta)^{-3}] \approx 1 - {1\over (2R\Delta)^3}$. A similar result was also obtained in [@Urs] in the limit of small phase-space densities, although with quite different assumptions about the event ensemble. In particular, in [@Urs] the events were assumed to have a fixed pion multiplicity $n$. For this reason the factor ${\cal N}$ in [@Urs] cannot be associated with the pion multiplicity distribution $P(n)$; still, the leading dependence of the normalization of the correlator on multi-pion effects enters through the same factor involving the phase-space volume $(2R\Delta)^3$ of the source. This algorithm agrees with the recent suggestion by Miśkowiec and Voloshin [@MV97] up to a minor detail: these authors suggested to use for the denominator in the correlation function mixed pairs constructed from $N_{\rm mix}(N_{\rm mix}-1)$ event pairs, $N_{\rm mix}$ being chosen such that $N_{\rm mix}(N_{\rm mix}-1) \equiv N_{\rm ev}$ where $N_{\rm ev}$ is the number of events from which the correlated pairs for the numerator are extracted. With this choice the normalizing prefactors in the numerator and denominator of the correlation cancel automatically. Our algorithm uses a larger number of event pairs for event mixing and rescales the denominator accordingly; this ensures that the statistical error of the correlation function is dominated by the “signal” in the numerator, and it also agrees with current experimental practice (D. Miśkowiec, private communication). G.I. Kopylov, Phys. Lett. B [**50**]{}, 472 (1974).

--- abstract: 'We prove that the abundance conjecture holds on a variety $X$ with mild singularities if $X$ has many reflexive differential forms with coefficients in pluricanonical bundles, assuming the Minimal Model Program in lower dimensions. This implies, for instance, that under this condition, hermitian semipositive canonical divisors are almost always semiample, and that uniruled klt pairs in many circumstances have good models. When the numerical dimension of $K_X$ is $1$, our results hold unconditionally in every dimension. We also treat a related problem on the semiampleness of nef line bundles on Calabi-Yau varieties.' address: - 'Fachrichtung Mathematik, Campus, Gebäude E2.4, Universität des Saarlandes, 66123 Saarbrücken, Germany' - 'Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany' author: - Vladimir Lazić - Thomas Peternell bibliography: - 'biblio.bib' title: | Abundance for varieties\ with many differential forms --- [^1] Introduction ============ In this paper we prove that the abundance conjecture holds on a variety $X$ with mild singularities if $X$ has many reflexive differential forms with coefficients in pluricanonical bundles, assuming the Minimal Model Program in lower dimensions. The abundance conjecture and the existence of good models are the main open problems in the Minimal Model Program in complex algebraic geometry. A main step towards abundance is the so called nonvanishing conjecture. Various theorems presented in this paper are the first results on nonvanishing in dimensions $\geq4$ when the numerical dimension of $X$ is not $0$ or $\dim X$. As a by-product, we obtain a new proof of (the most difficult part of) nonvanishing in dimension $3$. Recall that given a $\mathbb Q$-factorial projective variety $X$ with terminal singularities (terminal variety for short), the Minimal Model Program (MMP) predicts that either $X$ is uniruled and has a birational model which admits a Mori fibration, or $X$ has a birational model $X'$ with terminal singularities such that $K_{X'}$ is nef; the variety $X'$ is called a *minimal model* of $X$. The *abundance conjecture* then says that $K_{X'}$ is semiample, i.e. some multiple $mK_{X'}$ is basepoint free; the variety $X'$ is then a *good model* of $X$. For various important reasons it is necessary to study a more general situation of klt pairs $(X,\Delta).$ We recall briefly the previous progress towards the resolution of these conjectures, concentrating mainly on the case of klt singularities. Everything is classically known for surfaces. For terminal threefolds, minimal models were constructed in [@Mor88; @Sho85] and abundance was proved in [@Miy87; @Miy88a; @Miy88b; @Kaw92; @Kol92]. The corresponding generalisations to threefold klt pairs were established in [@Sho92] and in [@KMM94]. Minimal models for canonical fourfolds exist by [@BCHM; @Fuj05], and abundance for canonical fourfolds is known when $\kappa(X,K_X)>0$ by [@Kaw85]. In arbitrary dimension, the only unconditional results are the existence of minimal models for klt pairs of log general type proved in [@HM10; @BCHM] and in [@CL12a; @CL13] by different methods, the abundance for klt pairs of log general type [@Sho85; @Kaw85b], and the abundance for varieties with numerical dimension $0$, see [@Nak04]. The non-log-general type case is notoriously difficult, and there are only several reduction steps known. The running assumption is that the Minimal Model Program holds in lower dimensions; this is of course completely natural, since the completion of the programme should be done by induction on the dimension. With this assumption in mind, minimal models for klt pairs $(X,\Delta)$ with $\kappa(X,K_X+\Delta)\geq0$ exist by [@Bir11], and good models exist for klt pairs $(X,\Delta)$ with $\kappa(X,K_X+\Delta)\geq1$ by [@Lai11]. The existence of good models for arbitrary klt pairs was reduced to the existence of good models for klt pairs $(X,\Delta)$ with $X$ terminal and $K_X$ pseudoeffective in [@DL15]. By a result of Lai, which we recall below in Theorem \[thm:lai\], the existence of good models for klt pairs is reduced to proving the existence of good models of pairs $(X,\Delta)$ with $\kappa(X,K_X+\Delta)\leq0$. We exploit this fact crucially in this paper. There are two faces of the problem of existence of good models. - The first is *nonvanishing*: showing that if $(X,\Delta)$ is a klt pair with $K_X+\Delta$ pseudoeffective, then $\kappa(X,K_X+\Delta)\geq0$. - The second is *semiampleness*: showing that if $(X,\Delta)$ is a klt pair with $\kappa(X,K_X+\Delta)\geq0$, then any minimal model of $(X,\Delta)$ is good. By [@DHP13], nonvanishing is reduced to proving $\kappa(X,K_X)\geq0$ for a smooth variety $X$ with pseudoeffective canonical class. The approach to semiampleness until now has been to find a suitable extension result for pluricanonical forms as in [@DHP13; @GM14], which usually requires working with singularities on reducible spaces. On the other hand, the nonvanishing in dimensions greater than $3$ has remained completely mysterious. The proof by Miyaoka and Kawamata in dimension $3$ resists straightforward generalisation to higher dimensions because of explicit use of surface and $3$-fold geometry. In this work we take a very different approach. The main idea is that the growth of global sections of the sheaves $\Omega_X^{[q]}\otimes\OO_X(mK_X)$ should correspond to the growth of sections of $\OO_X(mK_X)$, where $\Omega_X^{[q]}=(\bigwedge^q \Omega^1_X)^{**}$ is the sheaf of reflexive $q$-differentials on a terminal variety $X$. We use recent advances on the structure of sheaves of $q$-differentials together with the Minimal Model Program for a carefully chosen class of pairs to establish this link; the details of the main ideas of the proof are at the end of this section. A hope that such a link should exist was present already in [@DPS01 §2.7]. The situation here is somewhat similar to the semiampleness conjecture for nef line bundles $\mathcal L$ on varieties $X$ of Calabi-Yau type,\[page\] and we developed our main ideas while thinking about this related problem, see [@LOP16]. In that context, it seems that the first consideration of the sheaves $\Omega_X^q\otimes\mathcal L$ appears in [@Wi94], and parts of our proofs here are inspired by some arguments there. Sheaves of this form also appear in [@Ver10] in an approach towards the hyperkähler SYZ conjecture. We now discuss our results towards nonvanishing and semiampleness. Below, $\nu(X,D)$ denotes the numerical dimension of a $\Q$-Cartier $\Q$-divisor on a normal projective variety $X$, see Definition \[dfn:kappa\]. Nonvanishing {#nonvanishing .unnumbered} ------------ The following is our first main result. \[thm:A\] Let $X$ be a terminal projective variety of dimension $n$ with $K_X$ pseudoeffective. Assume either 1. the existence of good models for klt pairs in dimensions at most $n-1$, or 2. that $K_X$ is nef and $\nu(X,K_X)=1$. Assume that there exist a resolution $\pi\colon Y\to X$ of $X$ and a positive integer $q$ such that $$h^0(Y,\Omega^q_Y \otimes \OO_Y(m\pi^*K_X))>0$$ for infinitely many $m$ such that $mK_X$ is Cartier. Then $\kappa(X,K_X)\geq0$. Using the sheaf $\Omega_X^{[q]}$ of reflexive differentials and the results of [@GKKP11], the assumption in Theorem \[thm:A\] can be rephrased by saying that $$h^0(X,\Omega_X^{[q]} \otimes \mathcal O_X(mK_X)) > 0$$ for infinitely many $m$. We actually prove a stronger statement involving any effective tensor representation of $\Omega_Y^q$, see Theorems \[thm:nonvanishingForms\] and \[thm:nonvanishingFormsnu1\] below. As mentioned above, the assumptions on the MMP in lower dimensions are natural and expected for any result towards abundance. Using the main results of [@DPS01] and [@GM14], our application of Theorem \[thm:A\] is the following, which shows in particular, that hermitian semipositive canonical divisors are almost always semiample, assuming the MMP in lower dimensions. It follows from Corollaries \[cor:nv\], \[cor:semipositive\] and \[thm:nu1\]. \[thm:B\] Let $X$ be a terminal projective variety of dimension $n$ with $K_X$ pseudoeffective and $\chi(X,\OO_X)\neq0$. 1. Assume the existence of good models for klt pairs in dimensions at most $n-1$. If $K_X$ has a singular metric with algebraic singularities and semipositive curvature current, then $\kappa(X,K_X) \geq0$. Moreover, if $K_X$ is hermitian semipositive, then $K_X$ is semiample. 2. Assume that $K_X$ is nef and $\nu(X,K_X)=1$. Then $\kappa(X,K_X) \geq0$. Semiampleness {#semiampleness .unnumbered} ------------- Let $(X,\Delta)$ be a klt pair such that $K_X+\Delta$ is pseudoeffective. By passing to a terminalisation, in order to prove the existence of a good model for $(X,\Delta)$ we may assume that the pair is terminal, and we distinguish two cases. If $K_X$ is not pseudoeffective, then by [@BDPP] the variety $X$ is uniruled, and by modifying $X$ via a generically finite map, by [@DL15] we may assume that $\kappa(X,K_X)\geq0$. If $K_X$ is pseudoeffective, then assuming nonvanishing, we may also assume that $\kappa(X,K_X)\geq0$. Therefore, when considering the semiampleness problem, then – assuming nonvanishing – we may assume that the pair $(X,\Delta)$ under consideration is terminal and that $\kappa(X,K_X)\geq0$. In this context, the following is our second main result. \[thm:C\] Let $(X,\Delta)$ be a projective $\Q$-factorial terminal pair of dimension $n$. Assume either 1. the existence of good models for klt pairs in dimensions at most $n-1$, and that $\kappa(X,K_X)\geq0$, or 2. that $\Delta=0$, $K_X$ is nef and $\nu(X,K_X)=1$. Assume that there exist a resolution $\pi\colon Y\to X$ of $X$ and a positive integer $q$ such that $$h^0\big(Y,\Omega_Y^q(\log\lceil\pi^{-1}_*\Delta\rceil)\otimes \OO_Y(m\pi^*(K_X+\Delta))\big)>\binom{n}{q} \quad\text{for some }m.$$ Then $(X,\Delta)$ has a good model. As above, our results are stronger and apply to any effective tensor representation of the sheaf of logarithmic differentials, see Theorems \[thm:abundanceForms\] and \[thm:nonvanishingFormsnu2\] below. Theorem \[thm:C\] generalises [@Taj14 Theorem 1.5]. The following applications of Theorem \[thm:C\] prove semiampleness for a large class of varieties. Theorem \[thm:D\] follows from Corollaries \[cor:chi\] and \[cor:xx\]. \[thm:D\] Let $(X,\Delta)$ be a $\Q$-factorial terminal pair of dimension $n$ such that $|\chi(X,\OO_X)|>2^{n-1}$. Assume either 1. the existence of good models for klt pairs in dimensions at most $n-1$, and that $\kappa(X,K_X)\geq0$, or 2. that $\Delta=0$, $K_X$ is nef and $\nu(X,K_X)=1$. Then $(X,\Delta)$ has a good model. When the underlying variety of a klt pair is uniruled, by using the main result of [@DL15] we can say much more: that it in many circumstances has a good model, assuming the MMP in lower dimensions. \[thm:E\] Assume the existence of good models for klt pairs in dimensions at most $n-1$. Let $(X,\Delta)$ be a klt pair of dimension $n$ such that $X$ is uniruled and $K_X+\Delta$ is pseudoeffective. If $\vert\chi(X,\OO_X)\vert> 2^{n-1}$, then $(X,\Delta)$ has a good model. Theorem \[thm:E\] follows from Theorem \[thm:uniruled\] below. Nef bundles on Calabi-Yau varieties {#nef-bundles-on-calabi-yau-varieties .unnumbered} ----------------------------------- As mentioned on page , the problem of deciding whether the canonical class on a minimal variety is semiample is intimately related to the problem of deciding whether a nef line bundle on a variety with trivial canonical class is semiample. Using very similar ideas, one can find analogues of Theorems \[thm:A\] and \[thm:C\] in this second context; this is done in Section \[sec:CY\]. This method was already crucial in [@LOP16], and we expect it to be of similar use in the future. Immediate consequences of this approach are contained in the following result, which is Theorem \[cor:nef\]. \[thm:G\] Assume the existence of good models for klt pairs in dimensions at most $n-1$. Let $X$ be a projective klt variety of dimension $n$ such that $K_X\sim_\Q0$, and let $\mathcal L$ be a nef line bundle on $X$. 1. Assume that $\mathcal L$ has a singular hermitian metric with semipositive curvature current and with algebraic singularities. If $\chi(X,\OO_X)\neq0$, then $\kappa(X,\mathcal L)\geq0$. 2. If $\mathcal L$ is hermitian semipositive and if $\chi(X,\OO_X)\neq0$, then $\mathcal L$ is semiample. Note that when $X$ is a hyperkähler manifold and $\mathcal L$ is a hermitian semipositive line bundle on $X$, then automatically $\chi(X,\OO_X)\neq0$, and it is known unconditionally that $\kappa(X,\mathcal L)\geq0$ by [@Ver10]. We obtain in Section \[sec:CY\] also more precise information when $X$ is a Calabi-Yau manifold of even dimension. Sketch of the proof {#sketch-of-the-proof .unnumbered} ------------------- We now outline the main steps of the proof of Theorem \[thm:A\]; the proof of Theorem \[thm:C\] is similar. Assume for simplicity that $X$ is a smooth minimal variety, and that there exists a positive integer $q$ such that for infinitely many $m$ we have $$h^0(X,\Omega^q_X \otimes \OO_X(mK_X))>0.$$ By using the main results of [@CP11; @CP15], we first show that there exists a pseudoeffective divisor $F$ and divisors $N_m\geq0$ for infinitely many $m$ with $$N_m\sim mK_X-F.$$ By the basepoint free theorem and by the pseudoeffectivity of $F$ we may assume that none of $N_m$ are big, and a short additional argument allows to deal with the case where $\kappa(X,N_m)=0$ for infinitely many $m$. Then the results of [@Lai11; @Kaw91] allow us to run a Minimal Model Program with scaling for a carefully chosen pair $(X,\varepsilon N_k)$, so that $K_X$ is nef at every step of the programme. In other words, replacing $X$ by the obtained minimal model, we may additionally assume that all $K_X+\varepsilon N_m$ are semiample for $m$ sufficiently large. By using the pseudoeffectivity of $F$ again, we show the existence of a fibration $f\colon X\to Y$ to a lower-dimensional variety $Y$ and of a big $\Q$-divisor $D$ on $Y$ such that $K_X\sim_\Q f^*D$. This produces the desired contradiction. Outline of the paper {#outline-of-the-paper .unnumbered} -------------------- In Section \[sec:prelim\] we collect definitions and results which are used later in the paper. Most of the material should be well known, however we decided to give proofs when a good reference could not be found. The whole of Section \[sec:MMPtwist\] is dedicated to the MMP argument hinted at above. The subtlety of the proof consists in finding the right Minimal Model Program to run, in order to preserve all the good properties of the canonical class and to find a birational model of the initial variety, on which the canonical class is the pullback of a $\Q$-divisor from a lower-dimensional base. In Section \[sec:thmA\] we prove a more general version of Theorem \[thm:A\]. Apart from the results from Section \[sec:MMPtwist\], the main input is the stability of the cotangent bundle [@CP11; @CP15], which generalise previous results of [@Miy87a]. Using the main result of [@DPS01], we also derive a part of Theorem \[thm:B\]. Section \[sec:thmC\] is devoted to the proofs of Theorem \[thm:C\] and a part of Theorem \[thm:D\]. The proofs are similar to those in Section \[sec:thmA\], although they are somewhat more involved. Under the additional assumption that the numerical dimension of the (log) canonical class is $1$, we can deduce several results unconditionally, and this occupies Section \[sec:nd1\]. In this special case, one can avoid the MMP techniques from Section \[sec:MMPtwist\]. Previous unconditional results were only known when the numerical dimension is $0$ or maximal, see [@BCHM; @Nak04; @Dru11]. In Section \[sec:uniruled\], we consider uniruled varieties and prove Theorem \[thm:E\]. Following [@DL15], we reduce the existence of good models for uniruled pairs to the case of effective canonical class, so that we can apply the results from Section \[sec:thmC\]. Finally, we treat the related problem of the semiampleness of nef line bundles on varieties of Calabi-Yau type in Section \[sec:CY\]. The techniques are similar to those of the previous sections. We finish by proving Theorem \[thm:G\]. Preliminaries {#sec:prelim} ============= In this paper we work over $\C$, and all varieties are normal and projective. We start with the following easy result which will be used in the proof of Theorem \[thm:MMPtwist\]. \[relation\] Let $X$ be a variety and $D$ a reduced Weil divisor on $X$. Let $\{N_m\}_{m\in \N}$ a sequence of effective Weil divisors on $X$ such that $\Supp N_m\subseteq\Supp D$ for every $m$. Then there exist positive integers $k\neq \ell$ such that $N_k\geq N_\ell$. Setting $D=\sum_{i=1}^n D_i$, where $D_i$ are prime divisors, the proof is by induction on $n$. We may assume that there does not exist an integer $k\geq2$ such that $N_k\geq N_1$. Then for each $k\geq2$, we have $\mult_{D_i}N_k<\mult_{D_i}N_1$ for some $i$. Hence, by passing to a subsequence and by relabelling, we may assume that $\mult_{D_1}N_k$ is constant for all $k\geq2$, and set $$N_k'=N_k-(\mult_{D_1}N_k)D_1.$$ Then $\Supp N_k'\subseteq\Supp(D-D_1)$, and the conclusion follows. We often use without explicit mention that if $f\colon X\dashrightarrow Y$ is a birational map between $\Q$-factorial varieties which is either a morphism or an isomorphism in codimension $1$, and if $D$ is a big, respectively pseudoeffective divisor on $X$, then $f_*D$ is a big, respectively pseudoeffective divisor on $Y$. We need the following easy consequences of the Hodge index theorem and of the existence of the Iitaka fibration. \[lem:hodge\] Let $X$ be a smooth projective surface, and let $L$ and $M$ be divisors on $X$ such that $$L^2=M^2=L\cdot M=0.$$ If $L$ and $M$ are not numerically trivial, then $L$ and $M$ are numerically proportional. Let $H$ be an ample divisor on $X$. By the Hodge index theorem we have $\lambda=L\cdot H\neq0$ and $\mu=M\cdot H\neq 0$, and set $D=\lambda M-\mu L$. Then $D^2=D\cdot H=0$, hence $D\equiv 0$ again by the Hodge index theorem. \[lem:iitaka\] Let $X$ be a normal projective variety and let $L$ be a $\Q$-divisor on $X$ with $\kappa(X,L)\geq0$. Then there exist a resolution $\pi\colon Y\to X$ and a fibration $f\colon Y\to Z$: $$\xymatrix{ Y \ar[d]^{\pi} \ar[r]^{f} & Z \\ X & }$$ such that $\dim Z=\kappa(X,L)$, and for every $\pi$-exceptional $\Q$-divisor $E\geq0$ on $Y$ and for a very general fibre $F$ of $f$ we have $$\kappa\big(F,(\pi^*L+E)|_F\big)=0.$$ By passing to multiples, we may assume that $L$ and $E$ are Cartier. The result is clear when $\kappa(X,L)=0$, hence we may assume that $\kappa(X,L)\geq1$. The lemma follows essentially from the proof of [@Laz04 Theorem 2.1.33], and we use the notation from that proof. We may assume that $X_\infty$ is smooth and that $X_\infty=X_{(p)}=X_{(q)}$. By possibly blowing up $X_{(m)}$ more, we may assume that all birational maps $\xi_m\colon X_{(m)}\dashrightarrow X_\infty$ are morphisms. $$\xymatrix{ X_{(m)} \ar[r]^{\xi_m} & X_\infty \ar[r]^{\phi_\infty} \ar[d]^{u_\infty} & Y_\infty \\ & X & }$$ Then it is clear that for every $m$ we have $$\big|\xi_m^*\big(m(u_\infty^*L+E)\big)\big|=|M_m|+F_m+mE.$$ Since all the maps in the proof of [@Laz04 Theorem 2.1.33] are defined via multiples of $M_m$, it follows that the morphism $\phi_{\infty}\colon X_\infty\to Y_\infty$ is also a model for the Iitaka fibration of $u_\infty^*L+E$, and in particular, for a very general fibre $F$ of $u_\infty$ we have $\kappa\big(F,(u_\infty^*L+E)|_F\big)=0$. Then we set $\pi:=u_\infty$, $Y:=X_\infty$, $f:=\phi_\infty$ and $Z:=Y_\infty$. Good models ----------- A *pair* $(X,\Delta)$ consists of a normal variety $X$ and a Weil $\Q$-divisor $\Delta\geq0$ such that the divisor $K_X+\Delta$ is $\Q$-Cartier. Such a pair is *log smooth* if $X$ is smooth and if the support of $\Delta$ has simple normal crossings. The standard reference for the foundational definitions and results on the singularities of pairs and the Minimal Model Program is [@KM98], and we use these freely in this paper. We recall the definition of log terminal and good models. Let $X$ and $Y$ be $\Q$-factorial varieties, and let $D$ be a $\Q$-divisor on $X$. A birational contraction $f\colon X\dashrightarrow Y$ is a *log terminal model for $D$* if $f_*D$ is nef, and if there exists a resolution $(p,q)\colon W\to X\times Y$ of the map $f$ such that $p^*D=q^*f_*D+E$, where $E\geq0$ is a $q$-exceptional $\Q$-divisor which contains the proper transform of every $f$-exceptional divisor in its support. If additionally $f_*D$ is semiample, the map $f$ is a *good model* for $D$. Here we recall additionally that flips for klt pairs exist by [@BCHM]. We also use the MMP with scaling of an ample divisor as in [@BCHM §3.10]. In this context, the following result says, among other things, that if one knows that a good model for a klt pair $(X,\Delta)$ exists, then one knows that there exists also an MMP which leads to a good model; this will be crucial in the proofs in Section \[sec:MMPtwist\]. \[thm:lai\] Assume the existence of good models for klt pairs in dimensions at most $n-1$. Let $(X,\Delta)$ be a $\Q$-factorial projective klt pair such that $\kappa(X,K_X+\Delta)\geq1$. Then $(X,\Delta)$ has a good model, and every $(K_X+\Delta)$-MMP with scaling of an ample divisor terminates with a good model of $(X,\Delta)$. If $K_X+\Delta$ is additionally nef, then it is semiample. The result follows by combining [@Lai11 Propositions 2.4 and 2.5, and Theorem 4.4]. Note that [@Lai11 Theorem 4.4] is stated for a terminal variety $X$, but the proof generalises to the context of klt pairs by replacing [@Lai11 Lemma 2.2] with [@HX13 Lemma 2.10]. Numerical Kodaira dimension {#subsec:numdim} --------------------------- If $L$ is a pseudoeffective $\R$-Cartier $\R$-divisor on a projective variety $X$, we denote by $P_\sigma(L)$ and $N_\sigma(L)$ the $\R$-divisors forming the Nakayama-Zariski decomposition of $L$, see [@Nak04 Chapter III] for the definition and the basic properties. Further, we recall the definition of the numerical Kodaira dimension [@Nak04; @Kaw85]. \[dfn:kappa\] Let $X$ be a smooth projective variety and let $D$ be a pseudoeffective $\Q$-divisor on $X$. If we denote $$\sigma(D,A)=\sup\big\{k\in\N\mid \liminf_{m\rightarrow\infty}h^0(X, \mathcal O_X(\lfloor ( mD\rfloor+A))/m^k >0\big\}$$ for a Cartier divisor $A$ on $X$, then the [*numerical dimension*]{} of $D$ is $$\nu(X,D)=\sup\{\sigma(D,A)\mid A\textrm{ is ample}\}.$$ Note that this coincides with various other definitions of the numerical dimension by [@Leh13]. If $X$ is a projective variety and if $D$ is a pseudoeffective $\Q$-Cartier $\Q$-divisor on $X$, then we set $\nu(X,D)=\nu(Y,f^*D)$ for any birational morphism $f\colon Y\to X$ from a smooth projective variety $Y$. When the divisor $D$ is nef, then alternatively, $$\nu(X,D)=\sup\{k\in\N\mid D^k\not\equiv0\}.$$ We use often without explicit mention that the functions $\kappa$ and $\nu$ behave well under proper pullbacks: in other words, if $D$ is a $\Q$-divisor on a $\Q$-factorial variety $X$, and if $f\colon Y\to X$ is a proper surjective morphism, then $$\kappa(X,D)=\kappa(Y,f^*D)\quad\text{and}\quad\nu(X,D)=\nu(Y,f^*D).$$ If $f$ is birational and $E$ is an effective $f$-exceptional divisor on $Y$, then $$\kappa(X,D)=\kappa(Y,f^*D+E)\quad\text{and}\quad\nu(X,D)=\nu(Y,f^*D+E).$$ For proofs, see [@Nak04 Lemma II.3.11, Proposition V.2.7(4)], [@GL13 Lemma 2.16] and [@Leh13 Theorem 6.7]. We also need the following result [@GL13 Theorem 4.3]. \[lem:Kappa=KappaSigma\] Let $(X,\Delta)$ be a klt pair. Then $(X,\Delta)$ has a good model if and only if $\kappa(X,K_X+\Delta)=\nu(X,K_X+\Delta)$. The proof of the following simple lemma is analogous to that of [@DL15 Lemma 3.1]. \[lem:pushforward\] Let $\pi\colon X\dashrightarrow Y$ be a birational contraction between projective varieties. Let $D$ be a $\Q$-Cartier $\Q$-divisor on $X$ such that $\pi_*D$ is $\Q$-Cartier. Then $\kappa(Y,\pi_*D)\geq\kappa(X,D)$ and $\nu(Y,\pi_*D)\geq\nu(X,D)$. By passing to multiples, we may assume that both $D$ and $D'=\pi_*D$ are Cartier. Let $(p,q)\colon W\to X\times Y$ be a resolution of the map $\pi$. Write $$p^*D=p_*^{-1}D+E_p^+-E_p^-\quad\text{and}\quad q^*D'=q_*^{-1}D'+E_q^+-E_q^-,$$ where $E_p^+,E_p^-\geq0$ is $p$-exceptional and without common components and $E_q^+,E_q^-\geq0$ is $q$-exceptional and without common components. Then there are $q$-exceptional divisors $E^+,E^-\geq0$ without common components such that $E^--E^+=p_*^{-1}D-q_*^{-1}D'$, and since $\pi$ is a contraction, $E_p^+$ and $E_p^-$ are $q$-exceptional. Therefore, $$\begin{aligned} \kappa(Y,D')&=\kappa(W,q^*D'+E_p^+ +E_q^-+E^-)=\kappa(W,p_*^{-1}D+E_p^+ +E_q^++E^+)\\ & \geq\kappa(W,p^*D+E_q^+ +E^+)\geq\kappa(W,p^*D)=\kappa(X,D),\end{aligned}$$ which was to be proved. The second inequality is analogous. Torsion free and reflexive sheaves ---------------------------------- A coherent sheaf $\mathcal F$ on an algebraic variety $X$ is *reflexive* if the natural homomorphism from $\mathcal F$ to its double dual $\mathcal F^{**}$ is an isomorphism. In particular, a reflexive sheaf $\mathcal F$ is torsion free. If $X$ is locally factorial, then a reflexive sheaf of rank $1$ is a line bundle [@Har80 Proposition 1.9]. If $X$ is smooth, then a reflexive sheaf is locally free away from a codimension $3$ subset of $X$ [@Har80 Corollary 1.4], and a torsion free sheaf is locally free away from a codimension $2$ subset of $X$ [@OSS80 Corollary on p. 148]. Let $\mathcal F$ be a coherent sheaf which is a subsheaf of a locally free sheaf $\mathcal E$. The *saturation* of $\mathcal F$ in $\mathcal E$ is the largest sheaf $\mathcal F'\subseteq\mathcal E$ such that $\mathcal F\subseteq\mathcal F'$, the ranks of $\mathcal F$ and $\mathcal F'$ are the same, and the quotient $\mathcal E/\mathcal F'$ is torsion free. The saturation $\mathcal F'$ always exists and is a reflexive sheaf, see [@OSS80 Lemma 1.1.16]. If $\mathcal F$ is a torsion free coherent sheaf of rank $r$ on a smooth variety $X$, then the *determinant* of $\mathcal F$ is by definition $\det\mathcal F=\big(\bigwedge^r\mathcal F\big)^{**}$. If $\mathcal F$ is reflexive, then this definition coincides with that in [@Har80 p. 129]. Let $\mathcal E$ be a coherent sheaf on an algebraic variety $X$. Each nontrivial global section $\sigma\in H^0(X,\mathcal E)$ gives a nontrivial morphism $f_\sigma\colon\OO_X\to\mathcal E$. We say that a sheaf $\mathcal F\subseteq\mathcal E$ is *the subsheaf of $\mathcal E$ generated by global sections of $\mathcal E$* if it is the image of the morphism $\bigoplus_{\sigma\in H^0(X,\mathcal E)}f_\sigma$. The next proposition seems to be folklore, but we include the proof for the benefit of the reader. \[pro:wedge\] Let $X$ be a smooth variety and let $\mathcal F$ be a globally generated torsion free sheaf on $X$ of rank $r$. If $h^0(X,\mathcal F) \geq r+1$, then $$h^0(X, \det \mathcal F) \geq 2.$$ Arguing by contradiction, assume that $h^0(X, \det \mathcal F) \leq 1$. Let $X^\circ$ be the largest open subset where $\mathcal F$ is locally free. Then $\codim_X(X\setminus X^\circ)\geq2$, which implies that $h^0(X^\circ, \det \mathcal F|_{X^\circ}) \leq 1$ by [@Har80 Proposition 1.6], and that the restriction map $$\label{eq:restriction} H^0(X,\mathcal F) \to H^0(X^{\circ} ,\mathcal F \vert_{X^\circ})$$ is injective. Let $\eta$ be the generic point of $X$. Our assumption and imply that there are $\omega_1,\dots,\omega_r\in H^0(X^\circ,\mathcal F|_{X^\circ})$ such that $(\omega_1)_\eta,\dots,(\omega_r)_\eta$ are linearly independent in $\OO_{X,\eta}$. Therefore, $\omega_1\wedge\ldots\wedge\omega_r$ defines a non-zero global section of $\det\mathcal F|_{X^\circ}$, thus $h^0(X^\circ,\det\mathcal F|_{X^\circ})=1$. Since $\mathcal F$ is generated by global sections, so is $\det \mathcal F \vert_{X^\circ}$, hence $$\det \mathcal F \vert _{X^\circ} \simeq \mathcal O_{X^{\circ}}.$$ But then $\omega_1 \wedge \ldots \wedge \omega_r\in H^0(X^\circ,\det\mathcal F|_{X^\circ})\simeq\C$ is constant on $X^\circ$, and so the sections $\omega_1,\dots,\omega_r$ are linearly independent at every point of $X^\circ$. Therefore, the induced map $\OO_{X^\circ}^{\oplus r}\to\mathcal F|_{X^\circ}$ is injective, hence surjective as the rank of $\mathcal F|_{X^\circ}$ is $r$. In particular, $h^0(X^{\circ} ,\mathcal F \vert_{X^\circ})=r$, which contradicts our assumption and . The following result is essential for the analysis of subsheaves of sheaves of log differentials. For $\Delta=0$, this is [@CP11 Theorem 0.1], and in general, this follows from [@CP15 Theorem 1.2], combined with [@BDPP Theorem 0.2]. \[thm:CP11\] Let $(X,\Delta)$ be a log smooth projective pair, where $\Delta$ is a reduced divisor. Let $\Omega_X^1(\log\Delta)^{\otimes m}\to\mathcal Q$ be a torsion free coherent quotient for some $m\geq1$. If $K_X+\Delta$ is pseudoeffective, then $c_1(\mathcal Q)$ is pseudoeffective. Metrics on line bundles {#subsec:metric} ----------------------- Let $X$ be a normal projective variety and let $D$ be a $\Q$-Cartier divisor. Following [@DPS01] and [@Dem01], we say that $D$, or $\mathcal O_X(D)$, has a metric with *analytic singularities* and semipositive curvature current, if there exists a positive integer $m$ such that $mD$ is Cartier and if there exists a resolution of singularities $\pi\colon Y \to X$ such that the line bundle $\pi^*\OO_X(mD)$ has a singular metric $h$ whose curvature current is semipositive (as a current), and the local plurisubharmonic weights $\varphi$ of $h$ are of the form $$\varphi = \sum \lambda_j \log \vert g_j \vert + O(1),$$ where $\lambda_j$ are positive real numbers, $O(1)$ is a bounded term, and the divisors $D_j$ defined locally by $g_j$ form a simple normal crossing divisor on $Y$. We then have $$\textstyle\mathcal I(h^{\otimes m})=\OO_Y\big(-\sum\lfloor m\lambda_j\rfloor D_j\big)\quad\text{for every positive integer }m,$$ where $\mathcal I(h^{\otimes m})$ is the multiplier ideal associated to $h^{\otimes m}$. If all $\lambda_j$ are rational, then $h$ has *algebraic singularities*. Further, $\mathcal O_X(D)$ is *hermitian semipositive* if $\pi^*\big(\OO_X(mD)\big)$ has a smooth hermitian metric $h$ whose curvature $\Theta_h(D)$ is semipositive. We mostly use these notions when $D = K_X$. The following result [@DPS01 Theorem 0.1] is a generalisation of the hard Lefschetz theorem. \[thm:DPS\] Let $X$ be a compact Kähler manifold of dimension $n$ with a Kähler form $\omega$. Let $\mathcal L$ be a pseudoeffective line bundle on $X$ with a singular hermitian metric $h$ such that $\Theta_h(\mathcal L) \geq 0$. Then for every nonnegative integer $q$ the morphism $$\xymatrix{ H^0\big(X,\Omega^{n-q}_X\otimes\mathcal L\otimes\mathcal I(h)\big) \ar[r]^{\omega^q\wedge\bullet} & H^q\big(X, \Omega^n_X\otimes \mathcal L\otimes\mathcal I(h)\big) }$$ is surjective. We also need the following result [@LOP16 Lemma 3.6]. \[lem:33\] Let $X$ be a projective manifold and let $L$ be a pseudoeffective Cartier divisor on $X$. Let $h$ be a singular hermitian metric on $\OO_X(L)$ with semipositive curvature current and multiplier ideal sheaf $\mathcal I(h)$. Let $D$ be an effective Cartier divisor such that $\mathcal I(h)\subseteq\OO_X(-D)$. Then $L-D$ is pseudoeffective. MMP with a twist {#sec:MMPtwist} ================ In the Minimal Model Program, starting from a klt pair $(X,\Delta)$ with $K_X+\Delta$ pseudoeffective, one wants to produce a good model for $(X,\Delta)$. It is often difficult even to start the construction due to lack of sections of $K_X+\Delta$. However, if we are in a favourable situation that arbitrarily good approximations of $K_X+\Delta$ have (many) sections, then one can conclude the same for $K_X+\Delta$ itself. That is the content of this section. We start with the case when $K_X+\Delta$ is nef. It turns out that in this case one can control the growth of sections of $K_X+\Delta$ precisely, and similar techniques will be used in Section \[sec:CY\]. \[thm:MMPtwist\] Assume the existence of good models for klt pairs in dimensions at most $n-1$. Let $(X,\Delta)$ be a $\Q$-factorial projective klt pair of dimension $n$ such that $K_X+\Delta$ is nef. Assume that there exist a pseudoeffective $\Q$-divisor $F$ on $X$ and an infinite subset $\mathcal S\subseteq\N$ such that $$\label{eq:rel2a} N_m+F\sim_\Q m(K_X+\Delta)$$ for all $m\in\mathcal S$, where $N_m\geq0$ are integral Weil divisors. Then $$\kappa(X,K_X+\Delta)=\max\{\kappa(X,N_m)\mid m\in\mathcal S\}\geq0.$$ *Step 1.* Note first that implies $$\label{eq:rel2b} N_p-N_q\sim_\Q (p-q)(K_X+\Delta)\quad\text{for all }p,q\in\mathcal S.$$ We claim that for every $m\in\mathcal S$ and every rational $\varepsilon>0$ we have $$\label{eq:kodaira} \kappa(X,K_X+\Delta+\varepsilon N_m)\geq\kappa(X,N_m).$$ Indeed, pick $i_m\in\mathcal S$ very large such that $\varepsilon_m<\varepsilon$, where $\varepsilon_m=\frac{1}{i_m-m}$. Then by we have $$\begin{aligned} \label{eq:33a} K_X+\Delta+\varepsilon N_m& =\varepsilon_m\big((i_m-m)(K_X+\Delta)+N_m\big)+(\varepsilon-\varepsilon_m)N_m\\ &\sim_\Q \varepsilon_m N_{i_m}+(\varepsilon-\varepsilon_m)N_m,\notag\end{aligned}$$ which proves . There are now three cases to consider. *Step 2.* First assume that $K_X+\Delta+\varepsilon N_p$ is big for some $p\in\mathcal S$ and some rational number $\varepsilon>0$. Then implies that the divisor $$(1+\varepsilon p)(K_X+\Delta) \sim_\Q K_X+\Delta+\varepsilon N_p+\varepsilon F$$ is big, and the result is clear since then $N_m$ is big for $m\gg0$ by . *Step 3.* Now assume that $\kappa(X,K_X+\Delta+\varepsilon N_p)=0$ for some $p\in\mathcal S$ and some rational number $\varepsilon>0$. Fix $q\in\mathcal S$ such that $q-p>1/\varepsilon$. Then as in we have $$\textstyle K_X+\Delta+\varepsilon N_p\sim_\Q \frac{1}{q-p} N_q+\big(\varepsilon-\frac{1}{q-p}\big)N_p,$$ hence $\kappa(X,N_q)\leq\kappa(X,K_X+\Delta+\varepsilon N_p)=0$, and therefore $\kappa(X,N_q)=0$. Let $r\in\mathcal S$ be such that $r>q$. Then by we have $$K_X+\Delta\sim_\Q\frac{1}{q-p}(N_q-N_p)\quad\text{and}\quad K_X+\Delta\sim_\Q \frac{1}{r-p}(N_r-N_p),$$ so that $$(r-p)N_q\sim_\Q(q-p)N_r+(r-q)N_p\geq0.$$ Since $\kappa(X,N_q)=0$, this implies $$(r-p)N_q=(q-p)N_r+(r-q)N_p,$$ and hence $\Supp N_r\subseteq\Supp N_q$ and $\kappa(X,N_r)=0$. Therefore, for $r>q$, all divisors $N_r$ are supported on a reduced Weil divisor. By Lemma \[relation\], there are positive integers $k\neq\ell$ larger than $q$ in $\mathcal S$ such that $N_k\leq N_\ell$, and hence by , $$(\ell-k)(K_X+\Delta)\sim_\Q N_\ell-N_k\geq0,$$ hence $\kappa(X,K_X+\Delta)\geq 0$. Moreover, since then $\kappa(X,K_X+\Delta)\leq\kappa(X,K_X+\Delta+\varepsilon N_p)=0$, we have $$\kappa(X,K_X+\Delta)=0.$$ If $m$ is an element of $\mathcal S$ with $m\geq q$, then $\kappa(X,N_m)=0$ by above, and if $m<q$, then $0=\kappa(X,N_q)\geq\kappa(X,N_m)$ by , which yields the result. *Step 4.* Finally, assume that $$\label{eq:kodaira1} 0<\kappa(X,K_X+\Delta+\varepsilon N_p)<n \quad\text{for every }p\in\mathcal S\text{ and every }\varepsilon>0.$$ It suffices to show that $$\label{eq:rel222} \kappa(X,K_X+\Delta)\geq\kappa(X,N_m)\quad\text{for all large }m\in \mathcal S.$$ Indeed, then $\kappa(X,K_X+\Delta)\geq0$, hence gives $\kappa(X,K_X+\Delta)\leq\kappa(X,N_m)$ for all large $m\in \mathcal S$ and $\kappa(X,N_q)\leq\kappa(X,N_p)$ for $p,q\in\mathcal S$ with $q<p$, which implies the theorem. Fix a positive integer $t$ such that $t(K_X+\Delta)$ is Cartier. Fix integers $\ell>k$ in $\mathcal S$ and fix a rational number $0<\varepsilon\ll1$ such that: 1. the pair $(X,\Delta+\varepsilon N_k)$ is klt, and 2. $\varepsilon(\ell-k)>2nt$. Denote $\delta=\frac{\varepsilon}{\varepsilon(\ell-k)+1}$ and fix an ample divisor $A$ on $X$. We run the MMP with scaling of $A$ for the klt pair $(X,\Delta+\delta N_k)$. By we have $$\label{eq:twoMMP} \textstyle K_X+\Delta+\varepsilon N_\ell\sim_\Q\big(\varepsilon(\ell-k)+1\big)\big(K_X+\Delta+\delta N_k\big),$$ hence every step in this MMP is a $(K_X+\Delta+\varepsilon N_\ell)$-negative map. Since we are assuming the existence of good models for klt pairs in lower dimensions, by Theorem \[thm:lai\] our MMP with scaling of $A$ terminates with a good model for $(X,\Delta+\delta N_k)$. We claim that this MMP is $(K_X+\Delta)$-trivial, and hence the proper transform of $t(K_X+\Delta)$ at every step of this MMP is a nef Cartier divisor by [@KM98 Theorem 3.7(4)]. Indeed, it is enough to show the claim for the first step of the MMP, as the rest is analogous. Let $c_R\colon X\to Z$ be the contraction of a $(K_X+\Delta+\delta N_k)$-negative extremal ray $R$ in this MMP. Since by we have $$\label{eq:rel2c} K_X+\Delta+\varepsilon N_\ell\sim_\Q K_X+\Delta+\varepsilon N_k+\varepsilon(\ell-k)(K_X+\Delta),$$ and as $K_X+\Delta$ is nef, the equation implies that $R$ is also $(K_X+\Delta+\varepsilon N_k)$-negative. By the boundedness of extremal rays [@Kaw91 Theorem 1], there exists a rational curve $C$ contracted by $c_R$ such that $$(K_X+\Delta+\varepsilon N_k)\cdot C\geq {-}2n.$$ If $c_R$ were not $(K_X+\Delta)$-trivial, then $t(K_X+\Delta)\cdot C\geq1$ as $t(K_X+\Delta)$ is Cartier. But then and the condition (b) above yield $$(K_X+\Delta+\varepsilon N_\ell)\cdot C=(K_X+\Delta+\varepsilon N_k)\cdot C+\varepsilon(\ell-k)(K_X+\Delta)\cdot C>0,$$ a contradiction which proves the claim, i.e. the MMP is $(K_X + \Delta)$-trivial. *Step 5.* In particular, the numerical Kodaira dimension and the Kodaira dimension of $K_X+\Delta$ are preserved in the MMP, see [@KM98 Theorem 3.7(4)] and §\[subsec:numdim\]. Hence, $K_X+\Delta$ is not big by . Furthermore, the proper transform of $F$ is pseudoeffective, and the Kodaira dimension of the proper transform of each $N_m$ for $m\in\mathcal S$ does not decrease by Lemma \[lem:pushforward\]. Therefore, by replacing $X$ by the resulting minimal model, we may assume that $K_X+\Delta+\delta N_k$ is semiample, and hence the divisor $K_X+\Delta+\varepsilon N_\ell$ is also semiample by . Note also that $\kappa(X,K_X+\Delta+\varepsilon N_m)>0$ for all $m\in\mathcal S$ by Lemma \[lem:pushforward\]. Fix $m\in\mathcal S$ such that $m>\ell$. Then the divisor $$K_X+\Delta+\varepsilon N_m\sim_\Q K_X+\Delta+\varepsilon N_\ell+\varepsilon(m-\ell)(K_X+\Delta)$$ is nef. Notice that $K_X+\Delta+\varepsilon N_m$ is not big, since otherwise $K_X + \Delta$ would be big as in Step 2. Therefore, we have $0<\kappa(X,K_X+\Delta+\varepsilon N_m)<n$. By we have $$\textstyle K_X+\Delta+\varepsilon N_m\sim_\Q\big(\varepsilon(m-k)+1\big)\big(K_X+\Delta+\frac{\varepsilon}{\varepsilon(m-k)+1} N_k\big),$$ and the pair $(X,\Delta+\frac{\varepsilon}{\varepsilon(m-k)+1} N_k)$ is klt. Since we are assuming the existence of good models for klt pairs in lower dimensions, by Theorem \[thm:lai\] the divisor $K_X+\Delta+\varepsilon N_m$ is semiample. Let $\varphi_\ell\colon X\to S_\ell$ and $\varphi_m\colon X\to S_m$ be the Iitaka fibrations associated to $K_X+\Delta+\varepsilon N_\ell$ and $K_X+\Delta+\varepsilon N_m$, respectively. Then there exist ample $\Q$-divisors $A_\ell$ on $S_\ell$ and $A_m$ on $S_m$ such that $$K_X+\Delta+\varepsilon N_\ell\sim_\Q\varphi_\ell^*A_\ell\quad\text{and}\quad K_X+\Delta+\varepsilon N_m\sim_\Q\varphi_m^*A_m.$$ If $\xi$ is a curve on $X$ contracted by $\varphi_m$, then $$0=(K_X+\Delta+\varepsilon N_m)\cdot \xi=(K_X+\Delta+\varepsilon N_\ell)\cdot \xi+\varepsilon(m-\ell)(K_X+\Delta)\cdot \xi,$$ hence $(K_X+\Delta+\varepsilon N_\ell)\cdot \xi=(K_X+\Delta)\cdot \xi=0$. In particular, $\xi$ is contracted by $\varphi_\ell$, which implies that there exists a morphism $\psi\colon S_m\to S_\ell$ such that $\varphi_\ell=\psi\circ\varphi_m$. $$\xymatrix{ & X \ar[ld]_{\varphi_m} \ar[dr]^{\varphi_\ell} & \\ S_m \ar[rr]^{\psi} & & S_\ell }$$ Therefore, denoting $B=\frac{1}{\varepsilon(m-\ell)} (A_m-\psi^*A_\ell)$, we have $$K_X+\Delta\sim_\Q \frac{1}{\varepsilon(m-\ell)}\big((K_X+\Delta+\varepsilon N_m)-(K_X+\Delta+\varepsilon N_\ell)\big)\sim_\Q\varphi_m^*B.$$ Denoting $$\textstyle B_0=\big(m+\frac{1}{\varepsilon}\big)B-\frac{1}{\varepsilon}A_m,$$ it is easy to check from that $$F\sim_\Q\varphi_m^*B_0,$$ and hence $B_0$ is pseudoeffective. Therefore the divisor $A_m+B_0$ is big on $S_m$, and $$(1+\varepsilon m)(K_X+\Delta)\sim_\Q K_X+\Delta+\varepsilon N_m+\varepsilon F\sim_\Q\varphi_m^*(A_m+B_0).$$ By , this yields $$\begin{aligned} \label{eq:equality0} \kappa(X,K_X+\Delta)&=\kappa(S_m,A_m+B_0)=\dim S_m\\ &=\kappa(X,K_X+\Delta+\varepsilon N_m)\geq\kappa(X,N_m),\notag\end{aligned}$$ and note that this holds for all $m\in\mathcal S$ sufficiently large. This proves and finishes the proof of the theorem. Now we treat the general case when $K_X+\Delta$ is pseudoeffective. We start with the following result which is implicit already in [@DHP13; @DL15]. It says that if a klt pair $(X,\Delta)$ has a fibration to a lower dimensional variety which is not a point, then often the existence of good models holds, assuming the Minimal Model Program in lower dimensions. \[pro:contraction\] Assume the existence of good models for klt pairs in dimensions at most $n-1$. Let $(X,\Delta)$ be a $\Q$-factorial projective klt pair such that $K_X+\Delta$ is pseudoeffective. If there exists a fibration $X\to Z$ to a normal projective variety $Z$ such that $\dim Z\geq 1$ and $K_X+\Delta$ is not big over $Z$, then $(X,\Delta)$ has a good model. The divisor $K_X+\Delta$ is effective over $Z$ by induction on the dimension and by [@BCHM Lemma 3.2.1]. By [@DL15 Theorem 2.5] and by [@Fuj11 Theorem 1.1] there exists a good model $(X,\Delta)\dashrightarrow (X_{\min},\Delta_{\min})$ of $(X,\Delta)$ over $Z$; alternatively, this follows from [@HX13 Lemma 2.12]. Let $\varphi\colon X_{\min}\to X_\can$ be the corresponding fibration to the canonical model of $(X,\Delta)$ over $Z$. $$\xymatrix{ X \ar[dr] \ar@{-->}[r] & X_{\min} \ar[d] \ar[r]^{\varphi} & X_\can \ar[dl]\\ & Z & }$$ Since $K_X+\Delta$ is not big over $Z$, we have $\dim X_\can<\dim X$. By [@Amb05a Theorem 0.2], there exists a divisor $\Delta_\can\geq0$ on $X_\can$ such that the pair $(X_\can,\Delta_\can)$ is klt and $$K_{X_{\min}}+\Delta_{\min}\sim_\Q\varphi^*(K_{X_\can}+\Delta_\can).$$ Since we assume the existence of good models for klt pairs in dimensions at most $n-1$, by Lemma \[lem:Kappa=KappaSigma\] we have $$\kappa(X_\can,K_{X_\can}+\Delta_\can)=\nu(X_\can,K_{X_\can}+\Delta_\can),$$ and the result follows by the discussion in §\[subsec:numdim\] and by another application of Lemma \[lem:Kappa=KappaSigma\]. \[thm:MMPtwist3\] Assume the existence of good models for klt pairs in dimensions at most $n-1$. Let $(X,\Delta)$ be a $\Q$-factorial projective klt pair of dimension $n$ such that $K_X+\Delta$ is pseudoeffective. Assume that there exist a pseudoeffective $\Q$-divisor $F$ on $X$ and an infinite subset $\mathcal S\subseteq\N$ such that $$\label{eq:rel2a3} N_m+F\sim_\Q m(K_X+\Delta)$$ for all $m\in\mathcal S$, where $N_m\geq0$ are integral Weil divisors. Then $$\kappa(X,K_X+\Delta)=\max\{\kappa(X,N_m)\mid m\in\mathcal S\}\geq0.$$ We first observe that Steps 1–3 of the proof of Theorem \[thm:MMPtwist\] work also in the case when $K_X+\Delta$ is pseudoeffective and not only nef. The relation implies $$\label{eq:rel2b3} N_p-N_q\sim_\Q (p-q)(K_X+\Delta)\quad\text{for all }p,q\in\mathcal S,$$ and as in the proof of Theorem \[thm:MMPtwist\], for every $m\in\mathcal S$ and every rational $\varepsilon>0$ we have $$\kappa(X,K_X+\Delta+\varepsilon N_m)\geq\kappa(X,N_m).$$ Again as in that proof, we may assume that $$\label{eq:kodaira13} 0<\kappa(X,K_X+\Delta+\varepsilon N_p)<n \quad\text{for every }p\in\mathcal S\text{ and every }\varepsilon>0.$$ We first show that $$\label{eq:claim} \kappa(X,K_X+\Delta)\geq0.$$ Fix $p\in\mathcal S$ and denote $L=K_X+\Delta+N_p$. By Lemma \[lem:iitaka\] there exists a resolution $\pi\colon Y\to X$ and a morphism $f\colon Y\to Z$: $$\xymatrix{ Y \ar[d]_{\pi} \ar[r]^{f} & Z\\ X & }$$ such that $\dim Z=\kappa(X,L)\in\{1,\dots,n-1\}$, and for a very general fibre $F$ of $f$ and for every $\pi$-exceptional $\Q$-divisor $G$ on $Y$ we have $$\label{eq:exceptional} \kappa\big(F,(\pi^*L+G)|_F\big)=0.$$ There exist $\Q$-divisors $\Gamma,E\geq0$ without common components such that $$K_Y+\Gamma\sim_\Q\pi^*(K_X+\Delta)+E,$$ and it is enough to show that $\kappa(Y,K_Y+\Gamma)\geq0$. By we have $$\label{eq:Gamma} \kappa\big(F,(K_Y+\Gamma+\pi^*N_p)|_F\big)=\kappa\big(F,(\pi^*L+E)|_F\big)=0,$$ and since $\kappa\big(F,(K_Y+\Gamma)|_F)\geq0$ by induction on the dimension, the equation implies $$\kappa\big(F,(K_Y+\Gamma)|_F\big)=0.$$ But then $\kappa(Y,K_Y+\Gamma)\geq0$ by Proposition \[pro:contraction\], which proves . Note that then by we have $\kappa(X,K_X+\Delta)\leq\kappa(X,N_m)$ for all large $m\in \mathcal S$ and $\kappa(X,N_q)\leq\kappa(X,N_p)$ for $p,q\in\mathcal S$ with $q<p$, hence it suffices to show that $\kappa(X,K_X+\Delta)\geq\kappa(X,N_m)$ for all large $m\in \mathcal S$. Now, by [@DL15 Theorem 2.5] there exists a log terminal model $$\varphi\colon (X,\Delta)\dashrightarrow (X_{\min},\Delta_{\min})$$ of $(X,\Delta)$, and for all $m\in\mathcal S$ we have $$\varphi_*N_m+\varphi_*F\sim_\Q m(K_{X_{\min}}+\Delta_{\min}).$$ Then Theorem \[thm:MMPtwist\] implies $$\kappa(X_{\min},K_{X_{\min}}+\Delta_{\min})=\max\{\kappa(X_{\min},\varphi_*N_m)\mid m\in\mathcal S\},$$ and the result follows from Lemma \[lem:pushforward\]. Finally, the following is an immediate corollary of the proof of Theorem \[thm:MMPtwist3\]. \[thm:MMPtwist2\] Assume the existence of good models for klt pairs in dimensions at most $n-1$. Let $(X,\Delta)$ be a $\Q$-factorial projective klt pair of dimension $n$ such that $K_X+\Delta$ is pseudoeffective. Assume that there exists a pseudoeffective $\Q$-divisor $F$ on $X$ such that $\kappa\big(X,m(K_X+\Delta)-F\big)\geq1$ for infinitely many $m$. Then $K_X+\Delta$ has a good model. By assumption, the set $\mathcal S=\{m\in\N\mid \kappa\big(X,m(K_X+\Delta)-F\big)\geq1\}$ has infinitely many elements, and hence for every $m\in\mathcal S$, there exists a $\Q$-divisor $N_m\geq0$ such that $$N_m+F\sim_\Q m(K_X+\Delta).$$ Then as in the proof of Theorem \[thm:MMPtwist3\], we obtain $$\kappa(X,K_X+\Delta)\geq\max\{\kappa(X,N_m)\mid m\in\mathcal S\}\geq1;$$ note that the condition that the divisors $N_m$ are *integral* was only used in Step 3 of the proof of Theorem \[thm:MMPtwist\], and the situation in this step does not happen in our context by the equation . We conclude by Theorem \[thm:lai\]. Nonvanishing {#sec:thmA} ============ In this section, we prove Theorem \[thm:A\]. Note that by [@DHP13 Theorem 8.8], assuming the existence of good models for a klt pairs in dimensions at most $n-1$, the nonvanishing for *klt pairs* in dimension $n$ reduces to nonvanishing for *terminal varieties* in dimension $n$. Therefore, one does not gain any generality when one considers nonvanishing for pairs. The situation is, however, different when one considers semiampleness. The following result implies Theorem \[thm:A\], since any tensor representation of a vector bundle can be embedded as a submodule in its high tensor power, see [@Bou98 Chapter III, §6.3 and §7.4]. \[thm:nonvanishingForms\] Assume the existence of good models for klt pairs in dimensions at most $n-1$. Let $X$ be a $\Q$-factorial projective terminal variety of dimension $n$ with $K_X$ pseudoeffective. Assume that $\kappa(X,K_X) = - \infty$. Let $\pi\colon Y\to X$ be a resolution of $X$. Then for every $p\geq1$ we have $$H^0\big(Y,(\Omega^1_Y)^{\otimes p} \otimes \OO_Y(m\pi^*K_X)\big)=0\quad\text{for all $m\neq0$ sufficiently divisible}.$$ We first note that we have $K_X \not \equiv 0$, since otherwise $\kappa (X,K_X) = 0$ by [@Kaw85b Theorem 8.2]. Arguing by contradiction, assume that there exists $p\geq1$ and an infinite set $\mathcal T\subseteq \Z$ such that $$H^0\big(Y,(\Omega^1_Y)^{\otimes p} \otimes \OO_Y(m\pi^*K_X)\big) \neq 0$$ for all $m\in\mathcal T$. Denote $\mathcal E=(\Omega^1_Y)^{\otimes p} $ and $Z=\PS(\mathcal E)$ with the projection $f\colon Z\to Y$. First note that $$H^0\big(Y,\mathcal E\otimes \OO_Y(m\pi^*K_X)\big)\simeq H^0\big(Z,\OO_Z(1)\otimes f^*\OO_Y(m\pi^*K_X)\big).$$ Since $K_X$ is pseudoeffective and not numerically trivial, the line bundle $\OO_Z(1)\otimes f^*\OO_Y(m\pi^*K_X)$ is not pseudoeffective for $m\ll0$, hence there are only finitely many negative integers in $\mathcal{T}$. Therefore, we may assume that $\mathcal T\subseteq\N$. Every nontrivial section of $H^0\big(Y,\mathcal E \otimes \OO_Y(m\pi^*K_X)\big)$ gives an inclusion $\OO_Y(-m\pi^*K_X)\to \mathcal E$, and let $\mathcal F \subseteq\mathcal E$ be the image of the map $$\textstyle\bigoplus_{m\in \mathcal T}\OO_Y(-m\pi^*K_X)\to \mathcal E.$$ Then $\mathcal F$ is quasi-coherent by [@Har77 Proposition II.5.7], and therefore a torsion free coherent sheaf as it is a subsheaf of the torsion free coherent sheaf $\mathcal E$. Let $r$ be the rank of $\mathcal F$. We may assume that there exist infinitely many $r$-tuples $(m_1,\dots,m_r)$ such that the image of the map $$\label{eq:inclusion} \OO_Y(-m_1\pi^*K_X)\oplus\cdots\oplus\OO_Y(-m_r\pi^*K_X)\to\mathcal F$$ has rank $r$: indeed, if this is not the case, we replace $\mathcal T$ by a suitable infinite subset, and the rank of $\mathcal F$ is smaller than $r$. Taking determinants in yields inclusions $$\label{eq:inclusion2} \OO_Y\big({-}(m_1+\dots+m_r)\pi^*K_X\big) \to \det\mathcal F\subseteq \bigwedge^r\mathcal E.$$ There is a Cartier divisor $F_Y$ such that $\OO_Y(-F_Y)$ is the saturation of $\det\mathcal F$ in $\bigwedge^r\mathcal E$. Then by there exists an infinite set $\mathcal S\subseteq \N$ such that $$\label{eq:infmany} H^0(Y,m\pi^*K_X-F_Y) \ne 0\quad\text{for all }m\in\mathcal S.$$ Consider the exact sequence $$0 \to \OO_Y(-F_Y) \to \bigwedge^r\mathcal E\to \mathcal Q \to 0.$$ Since $\OO_Y(-F_Y)$ is saturated in $\bigwedge^r\mathcal E$, the sheaf $\mathcal Q$ is torsion free, and hence $\tilde F_Y=c_1(\mathcal Q)$ is pseudoeffective by Theorem \[thm:CP11\]. From the above exact sequence, there exists a positive integer $\ell$ such that $\ell K_Y\sim\tilde F_Y-F_Y$. From , for every $m\in\mathcal S$ we obtain an effective divisor $\tilde N_{m+\ell}$ such that $\tilde N_{m+\ell}\sim m\pi^*K_X-F_Y$, and hence $$\label{eq:relation} \tilde N_{m+\ell}+\tilde F_Y\sim m\pi^*K_X+\ell K_Y.$$ Denote $N_{m+\ell}=\pi_*\tilde N_{m+\ell}$ and $F=\pi_*\tilde F_Y$; note that $N_{m+\ell}$ is effective and that $F$ is pseudoeffective. Pushing forward the relation to $X$, we get $$\label{eq:rel2} N_{m+\ell}+F\sim_\Q (m+\ell)K_X.$$ Now Theorem \[thm:MMPtwist3\] gives a contradiction. The same proof also shows the following variant. \[thm:nonvanishingForms2\] Assume the existence of good models for klt pairs in dimensions at most $n-1$. Let $X$ be a $\Q$-factorial projective terminal variety of dimension $n$ with $K_X$ pseudoeffective. Assume that $\kappa(X,K_X) = - \infty$. Let $\pi\colon Y\to X$ be a resolution of $X$, and let $E_1,\dots,E_\ell$ be all $\pi$-exceptional prime divisors on $Y$. Then for all integers $\lambda _i$, for all $m\neq0$ sufficiently divisible and for all $q\geq0$ we have $$\textstyle H^0\big(Y,\Omega^q_Y(\log\sum E_i) \otimes \OO_Y(m(\pi^*K_X+\sum\lambda_i E_i))\big)=0.$$ In particular, $$\textstyle H^0\big(Y,\Omega^q_Y(\log\sum E_i) \otimes \OO_Y(mK_Y)\big)=0$$ for $m$ sufficiently divisible and $q\geq0$. Now we are ready to state the first corollary of Theorem \[thm:nonvanishingForms\]. The details are taken from [@DPS01 Theorem 2.14], but we include the proof for the benefit of the reader. \[cor:nv\] Assume the existence of good models for klt pairs in dimensions at most $n-1$. Let $X$ be a projective terminal variety of dimension $n$ with $K_X$ pseudoeffective. Suppose that $K_X$ has a metric with algebraic singularities and semipositive curvature current. If $\chi(X,\OO_X)\neq0$, then $\kappa(X,K_X) \geq 0$. Let $\rho\colon X'\to X$ be a $\Q$-factorialisation of $X$, see [@Kol13 Corollary 1.37]. Then $\rho$ is an isomorphism in codimension $1$, hence $K_{X'}=\rho^*K_X$ and $X'$ is terminal. By replacing $X$ by $X'$, we may thus assume that $X$ is $\Q$-factorial. Choose a resolution $\pi\colon Y\to X$ such that for some positive integer $\ell$ the divisors $\ell K_X$ and $\ell K_Y$ are Cartier, and there exists a metric $h$ with algebraic singularities on $\pi^*\OO_X(\ell K_X)$ as in §\[subsec:metric\]. Then the local plurisubharmonic weights $\varphi$ of $h$ are of the form $$\varphi = \sum_{j=1}^r \lambda_j \log \vert g_j \vert + O(1),$$ where $\lambda_j$ are positive rational numbers and the divisors $D_j$ defined locally by $g_j$ form a simple normal crossing divisor on $Y$. We have $$\label{eq:metric1} \textstyle\mathcal I(h^{\otimes m})=\OO_Y\big(-\sum_{j=1}^r\lfloor m\lambda_j\rfloor D_j\big).$$ Assume that $\kappa(X,K_X)=-\infty$. Then by Theorem \[thm:nonvanishingForms\], for all $p\geq 0$ and for all $m$ sufficiently divisible we have $$H^0\big(Y,\Omega^p_Y \otimes \pi^*\OO_X(m\ell K_X)\big)=0,$$ and thus $$H^0\big(Y,\Omega^p_Y\otimes \pi^*\OO_X(m\ell K_X)\otimes\mathcal I(h^{\otimes m})\big) = 0.$$ Theorem \[thm:DPS\] implies that for all $p\geq 0$ and for all $m>0$ sufficiently divisible, $$H^p\big(Y,\OO_Y(K_Y+m\ell\pi^* K_X)\otimes\mathcal I(h^{\otimes m})\big) = 0,$$ which together with and Serre duality yields $$\label{eq:euler} \textstyle\chi\big(Y,\OO_Y\big(\sum_{j=1}^r\lfloor m\lambda_j\rfloor D_j-m\ell\pi^* K_X\big)\big) = 0$$ for all $m>0$ sufficiently divisible. There are integers $p_j$ and $q_j\neq0$ such that $\lambda_j=p_j/q_j$, and denote $q=\prod q_j$ and $D=\sum \frac{qp_j}{q_j} D_j-q\ell\pi^* K_X$. Then implies $$\chi(Y,\OO_Y(mD)) = 0\quad \text{for $m>0$ sufficiently divisible}.$$ But then $\chi(Y,\OO_Y(mD)) = 0$ for all integers $m$, and hence $\chi(Y,\OO_Y) = 0$. Since $X$ has rational singularities, this implies $\chi(X,\OO_X) = 0$, a contradiction which finishes the proof. When $K_X$ is hermitian semipositive, the conclusion is much stronger. \[cor:semipositive\] Assume the existence of good models for klt pairs in dimensions at most $n-1$. Let $X$ be a projective terminal variety of dimension $n$. Assume that there is a positive integer $m$ and a desingularization $\pi\colon Y \to X$ such that $\pi^*\mathcal O_X(\ell K_X)$ has a singular metric $h$ with semipositive curvature current and vanishing Lelong number. If the numerical polynomial $$P(m) = \chi\big(X,\mathcal O_X(m\ell K_X)\big)$$ is not identically zero, then $K_X$ is semiample. In particular, the result holds when $K_X$ is hermitian semipositive. We follow closely the proof of Corollary \[cor:nv\]. We may assume that $X$ is $\Q$-factorial. Arguing by contradiction, assume that $\kappa(X,K_X) = - \infty$. There exists a resolution $\pi\colon Y\to X$ such that for some positive integer $\ell$ the divisors $\ell K_X$ and $\ell K_Y$ are Cartier, and there exists a smooth hermitian semipositive metric $h$ on $\pi^*\OO_X(\ell K_X)$. Note that $$\mathcal I(h^{\otimes m}) = \mathcal O_Y \quad\text{for all positive integers $m$.}$$ Then by Theorem \[thm:nonvanishingForms\], for all $p\geq 0$ and for all $m>0$ sufficiently divisible we have $$H^0\big(Y,\Omega^p_Y \otimes \pi^*\OO_X(m\ell K_X)\big)=0,$$ which together with Theorem \[thm:DPS\] and Serre duality implies $$H^{n-p}\big(Y,\pi^*\OO_X(-m\ell K_X)\big) = 0,$$ hence $\chi\big(Y,\pi^*\OO_X(m\ell K_X)\big) = 0$ for all $m$. Since $X$ has rational singularities, we deduce $$\chi\big(X,\OO_X(m\ell K_X)\big) = 0\quad\text{for all $m$},$$ a contradiction. Therefore, $\kappa (X,K_X) \geq 0$, and hence $K_X$ is semiample by [@GM14 Theorem 5.1]. The proof of Corollary \[cor:semipositive\] actually gives more: if $K_X$ is not semiample, then for all $q$ and all $m$ sufficiently divisible we have $$H^q(X,\mathcal O_X(-m K_X)) = 0.$$ Semiampleness {#sec:thmC} ============= In this section we prove Theorem \[thm:C\]. \[thm:abundanceForms\] Assume the existence of good models for klt pairs in dimensions at most $n-1$. Let $(X,\Delta)$ be a $\Q$-factorial projective terminal pair of dimension $n$ such that $\kappa(X,K_X)\geq0$. Let $\pi\colon Y\to X$ be a sufficiently high resolution of $X$, fix a tensor representation $\mathcal E$ of $\Omega_Y^1(\log\lceil\pi^{-1}_*\Delta\rceil)$, and let $q$ be the rank of $\mathcal E$. If $K_X+\Delta$ does not have a good model, then for all $m$ such that $m(K_X+\Delta)$ is Cartier, we have $$h^0\big(Y,\mathcal E\otimes \OO_Y(m\pi^*(K_X+\Delta))\big)\leq q.$$ *Step 1.* Since $\kappa(X,K_X)\geq0$, there exists a $\Q$-divisor $D\geq0$ such that $K_X\sim_\Q D$. For a rational number $0<\varepsilon\ll1$, denote $\Delta'=(1+\varepsilon)\Delta+\varepsilon D$ and $D'=(1+\varepsilon)(D+\Delta)$, so that $\lfloor\Delta'\rfloor=0$, $D'\geq0$, $\Supp\Delta'=\Supp D'$, the pair $(X,\Delta')$ is terminal, and we have $$(1+\varepsilon)(K_X+\Delta)\sim_\Q K_X+\Delta'\sim_\Q D'.$$ Let $\pi\colon Y\to X$ be a log resolution of $(X,\Delta')$, and write $$K_Y+\Gamma=\pi^*(K_X+\Delta')+E\sim_\Q \pi^*D'+E,$$ where $\Gamma$ and $E$ are effective and have no common components. Then the pair $(Y,\Gamma)$ does not have a good model by [@HX13 Lemma 2.10]. Note that $\Supp\Gamma\subseteq\Supp(\pi^*D'+E)$ since $(X,\Delta')$ is terminal, and let $\Gamma_Y=(\pi^*D'+E)_{\textrm{red}}$. Denoting $D_Y=\pi^*D'+E+\Gamma_Y-\Gamma\geq0$, we have $$\label{eq:support} K_Y+\Gamma_Y\sim_\Q D_Y\quad\text{and}\quad \Supp \Gamma_Y=\Supp D_Y=\Supp(\pi^*D'+E).$$ Then the pair $(Y,\Gamma_Y)$ is dlt, and $\kappa(Y,K_Y+\Gamma_Y)=\kappa(Y,K_Y+\Gamma)$ and $\nu(Y,K_Y+\Gamma_Y)=\nu(Y,K_Y+\Gamma)$ by [@DL15 Lemma 2.9], hence $$\kappa(Y,K_Y+\Gamma_Y)\neq\nu(Y,K_Y+\Gamma_Y)$$ by Lemma \[lem:Kappa=KappaSigma\]. Let $\mathcal E'$ be a tensor representation of $\Omega_Y^1(\log\Gamma_Y)$ corresponding to $\mathcal E$. Observing that $\lceil\pi^*\Delta\rceil\leq\Gamma_Y$, we have an inclusion $$\mathcal E\otimes \OO_Y(m\pi^*(K_X+\Delta))\to \mathcal E' \otimes \OO_Y(m(K_Y+\Gamma_Y)),$$ hence it suffices to show $$h^0\big(Y,\mathcal E' \otimes \OO_Y(m(K_Y+\Gamma_Y))\big)\leq q$$ for all sufficiently divisible $m$. *Step 2.* Arguing by contradiction, assume that there exist some $m_0 \geq 0$ sufficiently divisible such that $$h^0\big(Y,\mathcal E' \otimes \OO_Y(m_0(K_Y+\Gamma_Y))\big)\geq q+1.$$ Let $\mathcal F$ be the subsheaf of $\mathcal E' \otimes \OO_Y(m_0(K_Y+\Gamma_Y))$ generated by its global sections, and let $r$ be the rank of $\mathcal F$. Then $$\det\mathcal F\subseteq\bigwedge^r\mathcal E' \otimes \OO_Y(rm_0(K_Y+\Gamma_Y)),$$ and there exists a Cartier divisor $N$ such that $\OO_Y(N)$ is the saturation of $\det\mathcal F$ in $\bigwedge^r\mathcal E'\otimes \OO_Y(rm_0(K_Y+\Gamma_Y))$. By Proposition \[pro:wedge\] we have $$\label{eq:infmany2} h^0(Y,N)\geq2,$$ and denote $$\label{eq:18} {-}F_Y=N-rm_0(K_Y+\Gamma_Y).$$ Then we have the exact sequence $$0 \to \OO_Y(-F_Y) \to \bigwedge^r\mathcal E' \to \mathcal Q \to 0.$$ Since $\OO_Y(-F_Y)$ is saturated in $\bigwedge^r\mathcal E'$, the sheaf $\mathcal Q$ is torsion free, and hence $F=c_1(\mathcal Q)$ is pseudoeffective by Theorem \[thm:CP11\]. From the above exact sequence, there exists a positive integer $\ell$ such that $\ell (K_Y+\Gamma_Y)\sim F-F_Y$. Let $d$ be the smallest positive integer such that $H^0(Y,d(K_Y+\Gamma_Y))\neq0$. Denote $\mathcal S=\{rm_0+id\mid i\in\N\}\subseteq\Z$ and $$N_{m+\ell}=N+(m-rm_0)(K_Y+\Gamma_Y)\quad\text{for }m\in\mathcal S.$$ From , for every $m\in\mathcal S$ we have $\kappa(Y,N_{m+\ell})\geq1$, and gives $$N_{m+\ell}+F\sim (m+\ell) (K_Y+\Gamma_Y).$$ For a rational number $0<\delta\ll1$, denote $\Gamma_Y'=\Gamma_Y-\varepsilon D_Y$, and note that $\lfloor\Gamma_Y'\rfloor=0$ by . Therefore, the pair $(Y,\Gamma_Y')$ is klt by [@KM98 Proposition 2.41]. We have $(1-\delta)(K_Y+\Gamma_Y)\sim_\Q K_Y+\Gamma_Y'$ by , and therefore $$\textstyle\frac{1}{1-\delta} N_{m+\ell}+\frac{1}{1-\delta} F\sim_\Q (m+\ell)(K_Y+\Gamma_Y').$$ Now Theorem \[thm:MMPtwist2\] gives a contradiction. \[cor:chi\] Assume the existence of good models for klt pairs in dimensions at most $n-1$. Let $(X,\Delta)$ be a $\Q$-factorial projective terminal pair of dimension $n$ such that $\kappa(X,K_X)\geq0$. If $K_X+\Delta$ does not have a good model, then $$h^q(X,\OO_X)\leq \binom{n}{q}\quad\text{for all }q.$$ In particular, $\vert\chi(X,\OO_X)\vert\leq 2^{n-1}$. Let $\pi\colon Y\to X$ be a sufficiently high resolution of $X$. Applying Theorem \[thm:abundanceForms\] with $m = 0$ we obtain $$h^0\big(Y,\Omega^q_Y) \leq \binom{n}{q}\quad\text{for all }q,$$ hence the first statement follows from Hodge symmetry, since $X$ has rational singularities. Numerical dimension 1 {#sec:nd1} ===================== In this section we show that some of the previous results hold unconditionally in every dimension, if one assumes that the numerical dimension of the canonical class is $1$. The following is the key technical observation. \[thm:nu1a\] Let $X$ be a projective $\Q$-factorial variety of dimension $n$, and let $L$ be a nef divisor on $X$ such that $\nu(X,L)=1$. Assume that there exist a pseudoeffective $\Q$-divisor $F$ and a non-zero $\Q$-divisor $D \geq 0$ on $X$ such that $$D+F\sim_\Q L.$$ Then there exists a $\Q$-divisor $E\geq0$ such that $$L\equiv E\quad\text{and}\quad \kappa(X,E)\geq\kappa(X,D).$$ Let $f\colon Y\to X$ be a resolution of $X$, and denote $L'=f^*L$, $D'=f^*D$ and $F'=f^*F$, so that $D'+F'\sim_\Q L'$. Let $P=P_\sigma(F')$ and $N=N_\sigma(F')\geq0$, so that we have the Nakayama-Zariski decomposition $$F' = P + N.$$ Assume first that $P \not\equiv 0$. Let $S$ be a surface in $Y$ cut out by $n-2$ general hyperplane sections. Then $P|_S$ is nef by [@Nak04 Remark III.2.8 and paragraph after Corollary V.1.5], and in particular $$\label{eq:restrictionNef} (P|_S)^2\geq0.$$ On the other hand, since $\nu(Y,L')=1$, we have $$0=(L'|_S)^2=L'|_S\cdot P|_S+L'|_S\cdot N|_S+L'|_S\cdot D'|_S,$$ hence $$L'|_S\cdot P|_S=L'|_S\cdot N|_S=L'|_S\cdot D'|_S=0.$$ Now the Hodge index theorem implies $(P|_S)^2\leq0$, and hence $(P|_S)^2=0$ by . Then Lemma \[lem:hodge\] yields $P|_S\equiv\lambda L'|_S$ for some real number $\lambda > 0$, and hence $P\equiv \lambda L'$ by the Lefschetz hyperplane section theorem. Note that $D' \neq 0$ implies $\lambda < 1$. Therefore, setting $$E'=\frac{1}{1-\lambda}(N+D')-\varepsilon D'\geq0$$ for a rational number $0<\varepsilon\ll1$, we obtain $$L'-\varepsilon D'\equiv E'.$$ Let $E_1,\dots,E_r$ be the components of $E'$ and let $\pi\colon \Div_\R(Y)\to N^1(Y)_\R$ be the standard projection. Then $\pi^{-1}\big(\pi(L'-\varepsilon D')\big)\cap\sum\R_+E_i$ is a rational affine subspace of $\sum\R E_i\subseteq\Div(Y)_\R$ which contains $E'$, hence there exists a rational point $$0\leq E''\in \pi^{-1}\big(\pi(L'-\varepsilon D')\big)\cap\sum\R_+E_i.$$ Setting $E=f_*(E''+\varepsilon D')$, we have $L\equiv E$ and $E\geq\varepsilon D$, which proves the result in the case $P\not\equiv0$. If $P\equiv 0$, denote $E'=N+(1-\varepsilon) D'\geq0$ for a rational number $0<\varepsilon\ll1$, so that $L'-\varepsilon D'\equiv E'$. We conclude as above. \[cor:nu1\] Let $(X,\Delta)$ be a $\Q$-factorial projective klt pair such that $K_X+\Delta$ is nef and $\nu(X,K_X+\Delta)=1$. Assume that there exist a pseudoeffective $\Q$-divisor $F$ and a non-zero $\Q$-divisor $D \geq 0$ on $X$ such that $K_X+\Delta\sim_\Q D+F$. Then $\kappa(X,K_X+\Delta)\geq\kappa(X,D)$. By Theorem \[thm:nu1a\] applied to $L=K_X+\Delta$, there exists an effective $\Q$-divisor $E$ on $X$ such that $K_X+\Delta\equiv E$ and $\kappa(X,E)\geq\kappa(X,D)$. By [@CKP12 Theorem 0.1] we have $\kappa(X,K_X+\Delta)\geq\kappa(X,E)$, and the result follows. \[thm:nonvanishingFormsnu1\] Let $X$ be a minimal $\Q$-factorial projective terminal variety of dimension $n$. Assume that $\kappa(X,K_X) = - \infty$ and $\nu(X,K_X)=1$. Let $\pi\colon Y\to X$ be a resolution of $X$. Then for all $m \neq 0$ sufficiently divisible and for all $p$ we have $$H^0\big(Y,(\Omega^1_Y)^{\otimes p} \otimes \OO_Y(m\pi^*K_X)\big)=0.$$ The proof is the same as the proof of Theorem \[thm:nonvanishingForms\], by invoking Corollary \[cor:nu1\] instead of Theorem \[thm:MMPtwist3\]. \[cor:nd1\] Let $X$ be a minimal $\Q$-factorial projective terminal variety of dimension $n$ such that $\nu(X,K_X) = 1$. Assume that $K_X$ has a singular metric with algebraic singularities and semipositive curvature current. If $\chi(X,\OO_X)\neq0$, then $\kappa(X,K_X)\geq0$. In particular, the result holds if $K_X$ is hermitian semipositive. The proof is the same as that of Corollary \[cor:nv\], by invoking Theorem \[thm:nonvanishingFormsnu1\] instead of Theorem \[thm:nonvanishingForms\]. \[thm:multiplier2\] Let $X$ be a $\Q$-factorial projective variety and let $L$ be a nef divisor on $X$ with $\nu(X,L)=1$. Assume that there exists a resolution $\pi\colon Y\to X$ and a singular metric $h$ on $\pi^*\OO_X(L)$ with semipositive curvature current such that the multiplier ideal sheaf $\mathcal I(h)$ is different from $\OO_Y$. Then there exists a $\Q$-divisor $D\geq0$ such that $$L\equiv D\quad\text{and}\quad \kappa(X,D)\geq0.$$ Let $V \subseteq Y$ be the subspace defined by $\mathcal I(h)$, and let $y$ be a closed point in $V$ with ideal sheaf $\mathcal I_y$ in $y$. Let $\mu\colon \hat Y \to Y$ be the blow-up of $Y$ at $y$ and let $E = \pi^{-1}(y) $ be the exceptional divisor. Let $\hat h$ be the induced metric on $(\pi\circ\mu)^*\OO_X(L)$. By [@Dem01 Proposition 14.3], we have $$\mathcal I(\hat h) \subseteq \mu^{-1}\mathcal I(h)\cdot\OO_{\hat Y}\subseteq\mu^{-1}\mathcal I_y\cdot\OO_{\hat Y}=\OO_{\hat Y}(-E).$$ By Lemma \[lem:33\], the divisor $(\pi\circ\mu)^*L - E$ is pseudoeffective. Then by Lemma \[thm:nu1a\] there exists a $\Q$-divisor $\hat D\geq0$ on $\hat Y$ such that $(\pi\circ\mu)^*L\equiv\hat D$, and we set $D=(\pi\circ\mu)_*\hat D$. \[cor:multiplier\] Let $(X,\Delta)$ be a $\Q$-factorial projective klt pair such that $K_X+\Delta$ is nef and $\nu(X,K_X+\Delta)=1$. Assume that there exist a resolution $\pi\colon Y\to X$, a positive integer $m$ such that $m(K_X+\Delta)$ is Cartier, and a singular metric $h$ on $\pi^*\OO_X\big(m(K_X+\Delta)\big)$ with semipositive curvature current, such that the multiplier ideal sheaf $\mathcal I(h)$ is different from $\OO_Y$. Then $\kappa(X,K_X+\Delta)\geq0$. By Theorem \[thm:multiplier2\] applied to $L=m(K_X+\Delta)$, there exists an effective $\Q$-divisor $D$ on $X$ such that $K_X+\Delta\equiv D$ and $\kappa(X,D)\geq0$. By [@CKP12 Theorem 0.1] we have $\kappa(X,K_X+\Delta)\geq\kappa(X,D)$, and the result follows. The following is the main result of this section. \[thm:nu1\] Let $X$ be a minimal $\Q$-factorial projective terminal variety such that $\nu(X,K_X)=1$. If $\chi(X,\OO_X)\neq0$, then $\kappa(X,K_X)\geq0$. If there exist a resolution $\pi\colon Y\to X$, a positive integer $m$ such that $mK_X$ is Cartier, and a singular metric $h$ on $\pi^*\OO_X(mK_X)$ with semipositive curvature current, such that the multiplier ideal sheaf $\mathcal I(h)$ is different from $\OO_Y$, then the result follows from Corollary \[cor:multiplier\]. Otherwise, pick a resolution $\pi\colon Y\to X$, a positive integer $m$ such that $mK_X$ is Cartier, and a singular metric $h$ on $\pi^*\OO_X(mK_X)$ with semipositive curvature current. Then the result follows from Corollary \[cor:nd1\]; note that since $\mathcal I(h^{\otimes\ell})=\OO_Y$ for all $\ell$, here the hypothesis in Corollary \[cor:nd1\] that $h$ has algebraic singularities is not necessary. \[remark:dim3\] Theorem \[thm:nu1\] gives a new proof of the hardest part of nonvanishing for minimal terminal threefolds. Indeed, if $X$ is a minimal terminal threefold, we only need to check the cases $\nu(X,K_X)\in\{1,2\}$. If the irregularity $q(X)$ is positive, then the nonvanishing follows from known cases of Iitaka’s conjecture $C_{n,m}$ applied to the Albanese map, see for instance [@MP97 pp. 73-74]. The case $\nu(X,K_X)=2$ is a relatively quick application of Miyaoka’s inequality for Chern classes and the Kawamata-Viehweg vanishing, see [@MP97 pp. 83-84]. The remaining case $\nu(X,K_X)=1$ is the most difficult. Since we may assume that $q(X) = 0,$ we are reduced to the case that $\chi(X,\OO_X)>0$. Then the nonvanishing follows from Theorem \[thm:nu1\]. \[thm:nonvanishingFormsnu2\] Let $X$ be a minimal $\Q$-factorial projective terminal pair of dimension $n$ such that $\nu(X,K_X)=1$. Let $\pi\colon Y\to X$ be a resolution of $X$, fix a tensor representation $\mathcal E$ of $\Omega_Y^1$, and let $q$ be the rank of $\mathcal E$. If $K_X$ is not semiample, then for all $m$ such that $mK_X$ is Cartier we have $$h^0\big(Y,\mathcal E\otimes \OO_Y(m\pi^*K_X)\big)\leq q.$$ We follow the proof of Theorem \[thm:abundanceForms\] closely. Arguing by contradiction, assume that there exist some $m_0$ such that $m_0K_X$ is Cartier and $$h^0\big(Y,\mathcal E \otimes \OO_Y(m_0\pi^*K_X)\big)\geq q+1.$$ Let $\mathcal F$ be the subsheaf of $\mathcal E \otimes \OO_Y(m_0\pi^*K_X)$ generated by its global sections, and let $r$ be the rank of $\mathcal F$. Then $$\det\mathcal F\subseteq\bigwedge^r\mathcal E \otimes \OO_Y(rm_0\pi^*K_X),$$ and there exists a Cartier divisor $N$ such that $\OO_Y(N)$ is the saturation of $\det\mathcal F$ in $\bigwedge^r\mathcal E \otimes \OO_Y(rm_0\pi^*K_X)$. By Proposition \[pro:wedge\] we have $$\label{eq:infmany2a} h^0(Y,N)\geq2,$$ and denote $$\label{eq:18a} {-}F_Y=N-rm_0\pi^*K_X.$$ Then we have the exact sequence $$0 \to \OO_Y(-F_Y) \to \bigwedge^r\mathcal E \to \mathcal Q \to 0.$$ Since $\OO_Y(-F_Y)$ is saturated in $\bigwedge^r\mathcal E$, the sheaf $\mathcal Q$ is torsion free, and hence $F=c_1(\mathcal Q)$ is pseudoeffective by Theorem \[thm:CP11\]. From the above exact sequence, there exists a positive integer $\ell$ such that $\ell K_Y\sim F-F_Y$, and gives $$N+F\sim \ell K_Y+rm_0\pi^*K_X.$$ Pushing forward this relation by $\pi$, we obtain $$\pi_*N+\pi_*F\sim_\Q (rm_0+\ell)K_X.$$ Since $\pi_*F$ is pseudoeffective and $\kappa(X,\pi_*N)\geq1$ by Lemma \[lem:pushforward\], Corollary \[cor:nu1\] implies that $\kappa(X,K_X)\geq1$, and hence $\kappa(X,K_X)=\nu(X,K_X)=1$. Now Lemma \[lem:Kappa=KappaSigma\] gives a contradiction. \[cor:xx\] Let $X$ be a minimal $\Q$-factorial projective terminal variety of dimension $n$ such that $\nu(X,K_X)=1$. 1. If $\vert\chi(X,\OO_X)\vert > 2^{n-1}$, then $K_X$ is semiample. 2. If $\kappa(X,K_X)\geq0$, $\pi_1(X)$ is infinite and $\chi(X,\OO_X)\neq 0$, then $K_X$ is semiample. For (a), let $\pi\colon Y\to X$ be a resolution. If $\vert\chi(X,\OO_X)\vert =\vert\chi(Y,\OO_Y)\vert> 2^{n-1}$, then there exists $q$ such that $$h^0\big(Y,\Omega^q_Y) =h^q(Y,\OO_Y) > \binom{n}{q}.$$ The result follows by Theorem \[thm:nonvanishingFormsnu2\]. For (b), by Lemma \[lem:Kappa=KappaSigma\] it suffices to show that $\kappa(X,K_X)=\nu(X,K_X)$. Arguing by contradiction, assume that $\kappa(X,K_X)=0$. Let $\pi\colon Y\to X$ be a resolution of $X$. Note that $\vert\chi(X,\OO_X)\vert=\vert\chi(X,\OO_X)\vert\neq 0$ since $X$ has rational singularities, and that $\pi_1(Y)$ is infinite by [@Tak03 Theorem 1.1]. In order to derive contradiction, by [@Cam95 Corollary 5.3] it suffices to show that for any $q$ and for any coherent subsheaf $\mathcal F\subseteq\Omega_Y^q$ we have $\kappa(Y,\det\mathcal F)\leq0$. To this end, we follow the proof of Theorem \[thm:nonvanishingFormsnu2\]. Fix $q$, and assume that there exists a coherent subsheaf $\mathcal F\subseteq\Omega_Y^q$ of rank $r$ such that $\kappa(Y,\det\mathcal F)>0$. Then $$\det\mathcal F\subseteq\bigwedge^r\Omega_Y^q,$$ and there exists a Cartier divisor $N\geq0$ such that $\OO_Y(N)$ is the saturation of $\det\mathcal F$ in $\bigwedge^r\Omega_Y^q$. Then we have the exact sequence $$0 \to \OO_Y(N) \to \bigwedge^r\Omega_Y^q \to \mathcal Q \to 0,$$ where the sheaf $\mathcal Q$ is torsion free, and hence $F=c_1(\mathcal Q)$ is pseudoeffective by Theorem \[thm:CP11\]. From the above exact sequence, there exists a positive integer $\ell$ such that $N+F\sim \ell K_Y$. Pushing forward this relation by $\pi$, we obtain $$\pi_*N+\pi_*F\sim_\Q \ell K_X,$$ hence Corollary \[cor:nu1\] and Lemma \[lem:pushforward\] imply $$0<\kappa(Y,\det\mathcal F)\leq\kappa(Y,N)\leq\kappa(X,\pi_*N)\leq\kappa(X,K_X)=0,$$ a contradiction which finishes the proof. We also notice: Let $X$ be a minimal $\Q$-factorial projective klt variety of dimension $n$ such that $\nu(X,K_X) = 1$. If for some $q>0$ there exists $$s \in H^0\Big(X, \Omega^{[q]}_X\Big)$$ which vanishes along some divisor, then $\kappa (X,K_X) \geq 0.$ Suppose that $s$ vanishes along a divisor $D\geq0$, and let $\pi\colon Y\to X$ be a resolution. Then by [@GKKP11 Theorem 4.3], the pullback $s_Y\in H^0(Y,\Omega_Y^q)$ of $s$ exists, and vanishes along the divisor $D_Y=\pi^{-1}_*D$. There is a Cartier divisor $D'\geq D_Y$ such that $\OO_Y(D')$ is the saturation of $\OO_Y(D_Y)$ in $\Omega^q_Y$, hence we have an exact sequence $$0 \to \OO_Y(D') \to \Omega^q_Y \to \mathcal Q \to 0,$$ where $\mathcal Q$ is torsion free. The divisor $c_1(\mathcal Q)$ is pseudoeffective by Theorem \[thm:CP11\], and there exists a positive integer $\ell$ such that $$\ell K_Y \sim_\Q D' + c_1(\mathcal Q).$$ By Theorem \[thm:nu1a\], this implies $\kappa(X,K_X)\geq\kappa(Y,K_Y)\geq\kappa(Y,D')\geq0$, which finishes the proof. Uniruled varieties {#sec:uniruled} ================== When the underlying variety of a klt pair $(X,\Delta)$ is uniruled, which – as explained in the introduction – is equivalent to the canonical class not being pseudoeffective on a terminalisation of $X$, we can show that the pair in many circumstances has a good model. The following is Theorem \[thm:E\]. \[thm:uniruled\] Assume the existence of good models for klt pairs in dimensions at most $n-1$. Let $(X,\Delta)$ be a klt pair of dimension $n$ such that $X$ is uniruled and $K_X+\Delta$ is pseudoeffective. If $(X,\Delta)$ does not have a good model, then $\vert\chi(X,\OO_X)\vert\leq 2^{n-1}$. We follow closely the proof of [@DL15 Theorem 1.3]. First of all, by passing to a resolution and by [@DL15 Lemma 2.9], we may assume that the pair $(X,\Delta)$ is log smooth, the divisor $\Delta$ is reduced, and there exists a $\Q$-divisor $D$ such that $K_X+\Delta\sim_\Q D$ and $\Supp\Delta=\Supp D$. Then the proof of [@DL15 Theorem 3.5] shows that there are proper maps $$T\stackrel{\mu}{\lto} W\stackrel{g}{\lto} X'\stackrel{\pi}{\lto} X,$$ where $\pi$ and $\mu$ are finite and $g$ is birational, and a $\Q$-divisor $\Delta_T$ on $T$ such that $(T,\Delta_T)$ is a log smooth klt pair with $|K_T|\neq\emptyset$ and $$\kappa(T,K_T+\Delta_T)=\kappa(X,K_X+\Delta)\quad\text{and}\quad\nu(T,K_T+\Delta_T)=\nu(X,K_X+\Delta).$$ Therefore, the pair $(T,\Delta_T)$ does not have a good model by Lemma \[lem:Kappa=KappaSigma\], hence $$\label{eq:uniruled} h^q(T,\OO_T)\leq\binom{n}{q}\quad\text{for all }q$$ by Corollary \[cor:chi\]. Since $\OO_W$ is a direct summand of $\mu_*\OO_T$ and since $\OO_X$ is a direct summand of $\pi_*\OO_{X'}$ by [@KM98 Proposition 5.7], by the Leray spectral sequence we have for each $q$ $$\label{eq:uniruled1} h^q(T,\OO_T)\geq h^q(W,\OO_W)\quad\text{and}\quad h^q(X',\OO_{X'})\geq h^q(X,\OO_X).$$ We claim that $X'$ is a klt variety. The claim immediately implies the theorem: indeed, since then $X'$ has rational singularities, we have $h^q(W,\OO_W)=h^q(X',\OO_{X'})$, which together with and gives $$h^q(X,\OO_X)\leq\binom{n}{q}\quad\text{for all }q,$$ and hence $\chi(X,\OO_X)\leq2^{n-1}$. To show the claim, by the proof of [@DL15 Theorem 3.5] there exists a $\Q$-divisor $\Delta'$ on $X'$ such that $K_{X'}+\Delta'=\pi^*(K_X+\Delta)$ and $\Supp\Delta'=\Supp\pi^*\Delta$. Then for a rational number $0<\varepsilon\ll1$, denoting $\Delta''=\Delta'-\varepsilon\pi^*\Delta$, we have $\Delta''\geq0$ and $$K_{X'}+\Delta''=\pi^*(K_X+(1-\varepsilon)\Delta).$$ Since the pair $(X,(1-\varepsilon)\Delta)$ is klt, so is the pair $(X',\Delta'')$ by [@KM98 Proposition 5.20], hence so is $X'$, which finishes the proof. Nonvanishing on Calabi-Yau varieties {#sec:CY} ==================================== As mentioned in the introduction, similar techniques as those used above can be applied in the context of nef line bundles on varieties of Calabi-Yau type. In particular, Theorem \[thm:FormsCY0\] generalises [@LOP16 Proposition 3.4] and [@Wi94 3.1] to any dimension. An immediate corollary is Theorem \[thm:G\]. \[thm:FormsCY0\] Assume the existence of good models for klt pairs in dimensions at most $n-1$. Let $X$ be a $\Q$-factorial projective klt variety of dimension $n$ such that $K_X\sim_\Q0$, and let $L$ be a nef divisor on $X$ such that $\kappa(X,L) = - \infty$. Let $\pi\colon Y\to X$ be a resolution of $X$. Then for every $p\geq1$ we have $$H^0(Y,(\Omega^1_Y)^{\otimes p} \otimes \OO_Y(m\pi^*L))=0\quad\text{for all $m\neq0$ sufficiently divisible}.$$ We follow closely the proof of Theorem \[thm:nonvanishingForms\]. Arguing by contradiction, assume that there exists $p\geq1$ and an infinite set $\mathcal T\subseteq \Z$ such that $$H^0(Y,(\Omega^1_Y)^{\otimes p} \otimes \OO_Y(m\pi^*L)) \neq 0$$ for all $m\in\mathcal T$. We may assume that $\mathcal T\subseteq\N$: indeed, if $L\not\equiv0$, then we can achieve it as in the proof of Theorem \[thm:nonvanishingForms\], and otherwise, we possibly replace $L$ by ${-}L$. Denote $\mathcal E=(\Omega^1_Y)^{\otimes p}$. Every nontrivial section of $H^0(Y,\mathcal E \otimes \OO_Y(m\pi^*L))$ gives an inclusion $\OO_Y(-m\pi^*L)\to \mathcal E$. Then analogously as in the proof of Theorem \[thm:nonvanishingForms\], there exist a Cartier divisor $F_Y$, an integer $1\leq r\leq n$ and an infinite set $\mathcal S\subseteq \N$ such that $\OO_Y(-F_Y)$ is a saturated subsheaf of $\bigwedge^r\mathcal E$ and $$\label{eq:infmany1} H^0(Y,m\pi^*L-F_Y) \ne 0\quad\text{for all }m\in\mathcal S.$$ Consider the exact sequence $$0 \to \OO_Y(-F_Y) \to \bigwedge^r\mathcal E\to \mathcal Q \to 0.$$ Since $\OO_Y(-F_Y)$ is saturated in $\bigwedge^r\mathcal E$, the sheaf $\mathcal Q$ is torsion free, and hence $\tilde F_Y=c_1(\mathcal Q)$ is pseudoeffective by Theorem \[thm:CP11\]. From the above exact sequence, there exists a positive integer $\ell$ such that $\ell K_Y\sim\tilde F_Y-F_Y$. From , for every $m\in\mathcal S$ we obtain a divisor $\tilde N_m\geq0$ such that $\tilde N_m\sim m\pi^*L-F_Y$, and hence $$\label{eq:relation1} \tilde N_m+\tilde F_Y\sim m\pi^*L+\ell K_Y.$$ Denote $N_m=\pi_*\tilde N_m$ and $F=\pi_*\tilde F_Y$; note that $N_m$ is effective and $F$ is pseudoeffective. Pushing forward the relation to $X$, we get $$N_m+F\sim_\Q mL.$$ Now Theorem \[thm:MMPtwistCY\] gives a contradiction. \[thm:MMPtwistCY\] Assume the existence of good models for klt pairs in dimensions at most $n-1$. Let $X$ be a $\Q$-factorial projective klt variety of dimension $n$ such that $K_X\sim_\Q0$, and let $L$ be a nef divisor on $X$. Assume that there exist a pseudoeffective $\Q$-divisor $F$ on $X$ and an infinite subset $\mathcal S\subseteq\N$ such that $$\label{eq:rel2aCY} N_m+F\sim_\Q mL,$$ for all $m\in\mathcal S$, where $N_m\geq0$ are integral Weil divisors. Then $$\kappa(X,L)=\max\{\kappa(X,N_m)\mid m\in\mathcal S\}\geq0.$$ The proof is very similar to that of Theorem \[thm:MMPtwist\]. *Step 1.* Note first that implies $$\label{eq:rel2bCY} N_p-N_q\sim_\Q (p-q)L\quad\text{for all }p,q\in\mathcal S.$$ There are three cases to consider. First assume that $N_p$ is big for some $p\in\mathcal S$. Then implies that $L$ is big, and the result is clear. *Step 2.* Now assume that $\kappa(X,N_p)=\kappa(X,N_q)=0$ for some distinct $p,q\in\mathcal S$. Let $r\in\mathcal S$ be such that $r>q$. Then by we have $$L\sim_\Q\frac{1}{q-p}(N_q-N_p)\quad\text{and}\quad L\sim_\Q \frac{1}{r-p}(N_r-N_p),$$ so that $$(r-p)N_q\sim_\Q(q-p)N_r+(r-q)N_p\geq0.$$ Since $\kappa(X,N_q)=0$, we have $(r-p)N_q=(q-p)N_r+(r-q)N_p$, and hence $\Supp N_r\subseteq\Supp N_q$ and $\kappa(X,N_r)=0$. Therefore, for $r>q$, all divisors $N_r$ are supported on a reduced Weil divisor. By Lemma \[relation\], there are positive integers $k\neq\ell$ larger than $q$ in $\mathcal S$ such that $N_k\leq N_\ell$, and hence by , $$(\ell-k)L\sim_\Q N_\ell-N_k\geq0,$$ which shows that $\kappa(X,L)\geq0$. Moreover, since then $\kappa(X,L)\leq\kappa(X,N_q)=0$ by , we have $$\kappa(X,L)=0.$$ If $m$ is an element of $\mathcal S$ with $m\geq q$, then $\kappa(X,N_m)=0$ by above, and if $m<q$, then $0=\kappa(X,N_q)\geq\kappa(X,N_m)$ by , which yields the result. *Step 3.* Finally, by replacing $\mathcal S$ by its infinite subset, we may assume that $$\label{eq:kodaira1CY} 0<\kappa(X,N_p)<n \quad\text{for every }p\in\mathcal S.$$ Fix integers $\ell>k$ in $\mathcal S$ and fix $0<\varepsilon,\delta\ll1$ such that: 1. the pair $(X,\varepsilon N_k)$ is klt, 2. $\varepsilon(\ell-k)>2n$, and 3. the pair $(X,\delta N_\ell)$ is klt. Fix an ample divisor $A$ on $X$, and we run the MMP with scaling of $A$ for the klt pair $(X,\delta N_\ell)$. Since we are assuming the existence of good models for klt pairs in lower dimensions, by Theorem \[thm:lai\] our MMP with scaling of $A$ terminates with a good model for $(X,\delta N_\ell)$. We claim that this MMP is $L$-trivial, and hence the proper transform of $L$ at every step of this MMP is a nef Cartier divisor. Indeed, it is enough to show the claim for the first step of the MMP, as the rest is analogous. Let $c_R\colon X\to Z$ be the contraction of a $(K_X+\delta N_\ell)$-negative (hence $N_\ell$-negative) extremal ray $R$ in this MMP. Since by we have $$N_\ell\sim_\Q N_k+(\ell-k)L$$ and as $L$ is nef, the ray $R$ is also $N_k$-negative. By the boundedness of extremal rays [@Kaw91 Theorem 1], there exists a rational curve $C$ contracted by $c_R$ such that $\varepsilon N_k\cdot C=(K_X+\varepsilon N_k)\cdot C\geq {-}2n$. If $c_R$ were not $L$-trivial, then $L\cdot C\geq1$ as $L$ is Cartier. But then the condition (b) above yields $$\varepsilon N_\ell\cdot C=\varepsilon N_k\cdot C+\varepsilon(\ell-k)L\cdot C>0,$$ a contradiction which proves the claim, i.e. the MMP is $L$-trivial. *Step 4.* In particular, the numerical Kodaira dimension and the Kodaira dimension of $L$ are preserved in the MMP, see [@KM98 Theorem 3.7(4)] and §\[subsec:numdim\]. Hence, $L$ is not big by and . Furthermore, the proper transform of $F$ is pseudoeffective. Therefore, by replacing $X$ by the resulting minimal model, we may assume that $N_\ell$ is semiample. Note also that $\kappa(X,N_m)>0$ for all $m\in\mathcal S$ by Lemma \[lem:pushforward\]. Fix $m\in\mathcal S$ such that $m>\ell$. Then the divisor $$N_m\sim_\Q N_\ell+(m-\ell)L$$ is nef. Notice that $N_m$ is not big, since otherwise $L$ would be big as in Step 1. Therefore, we have $0<\kappa(X,N_m)<n$, and pick $0<\eta\ll1$ so that the pair $(X,\eta N_m)$ is klt. Since we are assuming the existence of good models for klt pairs in lower dimensions, by Theorem \[thm:lai\] the divisor $\eta N_m$ is semiample. Let $\varphi_\ell\colon X\to S_\ell$ and $\varphi_m\colon X\to S_m$ be the Iitaka fibrations associated to $N_\ell$ and $N_m$, respectively. Then there exist ample $\Q$-divisors $A_\ell$ on $S_\ell$ and $A_m$ on $S_m$ such that $$N_\ell\sim_\Q\varphi_\ell^*A_\ell\quad\text{and}\quad N_m\sim_\Q\varphi_m^*A_m.$$ If $\xi$ is a curve on $X$ contracted by $\varphi_m$, then by we have $$0=N_m\cdot \xi=N_\ell\cdot \xi+(m-\ell)L\cdot \xi,$$ hence $N_\ell\cdot \xi=L\cdot \xi=0$. In particular, $\xi$ is contracted by $\varphi_\ell$, which implies that there exists a morphism $\psi\colon S_m\to S_\ell$ such that $\varphi_\ell=\psi\circ\varphi_m$. Therefore, denoting $B=\frac{1}{m-\ell} (A_m-\psi^*A_\ell)$, we have $$L\sim_\Q \frac{1}{m-\ell}(N_m-N_\ell)\sim_\Q\varphi_m^*B.$$ Denoting $B_0=mB-A_m$, it is easy to check from that $$F\sim_\Q\varphi_m^*B_0,$$ and hence $B_0$ is pseudoeffective. Therefore the divisor $A_m+B_0$ is big on $S_m$, and $$mL\sim_\Q N_m+F\sim_\Q\varphi_m^*(A_m+B_0).$$ This yields $$\label{eq:equality} \kappa(X,L)=\kappa(S_m,A_m+B_0)=\dim S_m=\kappa(X,N_m),$$ and note that this holds for $m\in\mathcal S$ sufficiently large. In particular, $\kappa(X,L)\geq0$, and then $\kappa(X,L)\geq\kappa(X,N_p)$ for all $p\in\mathcal S$ by and , which finishes the proof. \[thm:FormsCY\] Assume the existence of good models for klt pairs in dimensions at most $n-1$. Let $X$ be a $\Q$-factorial projective klt variety of dimension $n$ such that $K_X\sim_\Q0$, and let $L$ be a nef divisor on $X$ which is not semiample. Let $\pi\colon Y\to X$ be a resolution of $X$. Fix a tensor representation $\mathcal E$ of $\Omega_Y^1$, and let $q$ be the rank of $\mathcal E$. Then for all $m \neq 0$ sufficiently divisible we have $$h^0\big(Y,\mathcal E\otimes \OO_Y(m\pi^*L)\big)\leq q.$$ We follow closely the proof of Theorem \[thm:abundanceForms\]. If $\kappa(X,L)=-\infty$, then the result follows from Theorem \[thm:FormsCY0\]. Therefore, in the remainder of the proof we assume that $\kappa(X,L)\geq0$. Arguing by contradiction, assume that there exist some $m_0$ and some $p$ and $q$ such that $$h^0\big(Y,\mathcal E\otimes \OO_Y(m_0\pi^*L)\big)\geq q+1.$$ Then analogously as in the proof of Theorem \[thm:abundanceForms\], there exist a Cartier divisor $N$ and an integer $1\leq r\leq n$ such that $\OO_Y(N)$ is a saturated subsheaf of $\bigwedge^r\mathcal E\otimes\OO_Y(rm_0\pi^*L)$ and $$\label{eq:infmany2CY} h^0(Y,N)\geq2.$$ Denote $$\label{eq:18CY} -F_Y=N-rm_0\pi^*L.$$ Then we have the exact sequence $$0 \to \OO_Y(-F_Y) \to \bigwedge^r\mathcal E \to \mathcal Q \to 0.$$ Since $\OO_Y(-F_Y)$ is saturated in $\bigwedge^r\mathcal E$, the sheaf $\mathcal Q$ is torsion free, and hence $\tilde F_Y=c_1(\mathcal Q)$ is pseudoeffective by Theorem \[thm:CP11\]. From the above exact sequence, there exists a positive integer $\ell$ such that $\ell K_Y\sim\tilde F_Y-F_Y$. Let $d$ be the smallest positive integer such that $H^0(Y,d\pi^*L)\neq0$. Denote $\mathcal S=\{rm_0+id\mid i\in\N\}\subseteq\Z$ and $\tilde N_m=N+(m-rm_0)\pi^*L$ for $m\in\mathcal S$. From , for every $m\in\mathcal S$ we have $\kappa(Y,\tilde N_m)\geq1$, and gives $$\label{eq:relation2CY} \tilde N_m+\tilde F_Y\sim m\pi^*L+\ell K_Y.$$ Denote $N_m=\pi_*\tilde N_m$ and $F=\pi_*\tilde F_Y$. Then $F$ is pseudoeffective, and by Lemma \[lem:pushforward\] we have $\kappa(X,N_m)\geq\kappa(Y,\tilde N_m)\geq1$. Pushing forward the relation to $X$, we get $$N_m+F\sim_\Q mL.$$ As in Steps 1, 3 and 4 of the proof of Theorem \[thm:MMPtwistCY\], we conclude that $$\kappa(X,L)=\max\{\kappa(X,N_m)\mid m\in\mathcal S\}\geq1.$$ Pick a rational number $0<\varepsilon\ll1$ such that the pair $(X,\varepsilon L)$ is klt. Then $\varepsilon L\sim_\Q K_X+\varepsilon L$ is semiample by Theorem \[thm:lai\], a contradiction which finishes the proof. Let $X$ be a projective canonical variety of dimension $n$ such that $K_X\sim_\Q0$. Then we have $$h^q(X,\OO_X) = h^0\big(X,\Omega^{[q]}_X\big) \leq \binom{n}{q}\quad\text{for all }q,$$ and in particular, $|\chi(X,\mathcal O_X)| \leq 2^{n-1}$. Indeed, assume that there exists $q$ such that $h^q(X,\OO_X) > \binom{n}{q}$, and note that $h^q(X,\OO_X) = h^0\big(X,\Omega^{[q]}_X\big)$ by [@GKP11 Proposition 6.9]. Then by Proposition \[pro:wedge\] there is a positive integer $N$ and a line bundle $\mathcal L \subseteq \big(\bigwedge^N\Omega^{[q]}_X\big)^{**}$ with $h^0(X,\mathcal L) \geq 2$. Let $C\subseteq X$ be a curve obtained as complete intersection of $n-1$ high multiples of a very ample divisor. By Miyaoka’s generic semipositivity [@Miy87; @Miy87a], the sheaf $\big(\bigwedge^N\Omega^{[q]}_X\big)^{**}|_C$ is nef, and hence semistable with respect to any polarisation since $\det(\bigwedge^N\Omega^{[q]}_X\big)^{**}=\OO_X$. Therefore, $\big(\bigwedge^N\Omega^{[q]}_X\big)^{**}$ is semistable with respect to any polarisation by the theorem of Mehta-Ramanathan [@MR82; @Fle84]. However, the slope of $\mathcal L$ with respect to any ample polarisation is positive, a contradiction. In the more general setting when $X$ has klt singularities, the techniques from Sections \[sec:thmC\] and \[sec:uniruled\] show that $|\chi(X,\mathcal O_X)| \leq 2^{n-1}$ assuming the Minimal Model Program in dimensions at most $n-1$. \[cor:nef\] Assume the existence of good models for klt pairs in dimensions at most $n-1$. Let $X$ be a projective klt variety of dimension $n$ such that $K_X\sim_\Q0$, and let $\mathcal L$ be a nef line bundle on $X$. 1. Assume that $\mathcal L$ has a singular hermitian metric with semipositive curvature current and with algebraic singularities. If $\chi(X,\OO_X)\neq0$, then $\kappa(X,\mathcal L)\geq0$. 2. If $\mathcal L$ is hermitian semipositive and if $\chi(X,\OO_X)\neq0$, then $\mathcal L$ is semiample. The proof is similar to that of Corollaries \[cor:nv\], \[cor:semipositive\] and \[cor:chi\], so we will be quick on the details. By passing to a $\Q$-factorialisation and replacing $\mathcal L$ by its pullback, we may assume that $X$ is $\Q$-factorial. For (i), choose a resolution $\pi\colon Y\to X$ as in §\[subsec:metric\], and let $h$ denote the induced metric on $\pi^*\mathcal L$. Then the local plurisubharmonic weights $\phi$ of $h$ are of the form $$\phi = \sum_{j=1}^r \lambda_j \log \vert g_j \vert + O(1),$$ where $\lambda_j$ are positive rational numbers and the divisors $D_j$ defined locally by $g_j$ form a simple normal crossing divisor on $Y$. We have $$\label{eq:metric11} \textstyle\mathcal I(h^{\otimes m})=\OO_Y\big(-\sum_{j=1}^r\lfloor m\lambda_j\rfloor D_j\big).$$ Assume that $\kappa(X,\mathcal L)=-\infty$. Then by Theorem \[thm:FormsCY0\], for all $p\geq 0$ and for all $m$ sufficiently divisible we have $$H^0(Y,\Omega^p_Y \otimes \pi^*\mathcal L^{\otimes m})=0,$$ and thus $$H^0(Y,\Omega^p_Y\otimes \pi^*\mathcal L^{\otimes m}\otimes\mathcal I(h^{\otimes m})) = 0.$$ Theorem \[thm:DPS\] implies that for all $p\geq 0$ and for all $m$ sufficiently divisible, $$H^p(Y,\OO_Y(K_Y)\otimes \pi^*\mathcal L^{\otimes m}\otimes\mathcal I(h^{\otimes m})) = 0,$$ which together with and Serre duality yields $$\textstyle\chi\big(Y,\OO_Y\big(\sum_{j=1}^r\lfloor m\lambda_j\rfloor D_j\big)\otimes \pi^*\mathcal L^{\otimes {-}m}\big) = 0$$ for all $m$ sufficiently divisible. As in the proof of Corollary \[cor:nv\], this then implies $\chi(X,\OO_X) = 0$, which proves (i). Now assume that $\mathcal L$ is hermitian semipositive and that $\chi(X,\OO_X)\neq0$. By (i) we have $\kappa(X,\mathcal L)\geq0$, and choose a Cartier divisor $L\geq0$ such that $\mathcal L\simeq\OO_X(L)$. Pick a rational number $0<\varepsilon\ll1$ such that the pair $(X,\varepsilon L)$ is klt. Then [@GM14 Theorem 5.1] implies that $\varepsilon L\sim_\Q K_X+\varepsilon L$ is semiample, which proves (ii). Let $X$ be a projective klt variety of dimension $n$ such that $K_X\sim_\Q0$. If we drop the assumption $\chi(X,\mathcal O_X) \ne 0$, then Corollary \[cor:nef\] fails in general: for instance, $X$ could be a torus. However, one might expect that there is a line bundle $\mathcal L'$ numerically equivalent to $\mathcal L$ such that the conclusion remains true. Furthermore, note that if $K_X \sim 0$ and if $n $ is odd, then we always have $\chi(X,\mathcal O_X) = 0$ by [@GKP11 Corollary 6.11]. Recall that a Calabi-Yau manifold of dimension $n$ is a simply connected projective manifold $X$ such that $K_X\sim 0$ and $h^q(X,\OO_X)=0$ for all $q=1,\dots,n-1$. By the Beauville-Bogomolov decomposition theorem, Calabi-Yau manifolds are the building blocks for all manifolds with $K_X\equiv0$, together with hyperkähler manifolds and abelian varieties. \[thm:CYnu1\] Let $X$ be a projective manifold with $K_X \sim_{\mathbb Q} 0$ and $\pi_1(X) $ finite. Let $\mathcal L$ be a nef line bundle on $X$ with $\nu(X,\mathcal L)=1$. Let $\eta\colon \tilde X \to X$ be the universal cover and assume that the Beauville-Bogomolov decomposition is of the form $$\tilde X \simeq \prod X_j,$$ where all irreducible components $X_j$ are even-dimensional. Then $\kappa(X,\mathcal L)\geq0$. Replacing $X$ by $\tilde X$, we may assume that $X$ is simply connected. Since all $X_j$ are even-dimensional Calabi-Yau or hyperkähler manifolds, we have $\chi(X_j,\OO_{X_j}) > 0$ for all $j$, hence $\chi(X,\OO_X) > 0$. Since $h^1(X,\OO_X)=0$, numerical and linear equivalence of divisors on $X$ coincide. If there exist a resolution $\pi\colon Y\to X$, a positive integer $m$, and a singular metric $h$ on $\pi^*\mathcal L^{\otimes m}$ with semipositive curvature current, such that the multiplier ideal sheaf $\mathcal I(h)$ is different from $\OO_Y$, then the result follows from Corollary \[cor:multiplier\]. Otherwise, pick a resolution $\pi\colon Y\to X$ and a singular metric $h$ on $\pi^*\mathcal L$ with semipositive curvature current. Then the result follows from the proofs of Corollary \[cor:nef\](i) and Theorem \[thm:FormsCY0\], by invoking Theorem \[thm:nu1a\] instead of Theorem \[thm:MMPtwistCY\] in the proof of Theorem \[thm:FormsCY0\]. Note that since $\mathcal I(h^{\otimes\ell})=\OO_Y$ for all $\ell$, the hypothesis in Corollary \[cor:nef\](i) that $h$ has algebraic singularities is not necessary. Note that if in the theorem above $X_j$ is a hyperkähler manifold of dimension $\geq4$ for some $j$, then $\eta^*\mathcal L|_{X_j}\simeq\OO_{X_j}$ by [@Mat99 Lemma 1]. We also have the following generalization of Theorem \[thm:CYnu1\]. Let $X$ be a projective manifold with $K_X \sim_{\mathbb Q} 0$ and let $\mathcal L$ be a nef line bundle on $X$ with $\nu(X,\mathcal L)=1$. Let $\eta\colon \tilde X \to X$ be a finite étale cover such that the Beauville-Bogomolov decomposition is of the form $$\tilde X \simeq T\times \prod X_j,$$ where the $X_j$ are even-dimensional Calabi-Yau manifolds or hyperkähler manifolds, and $T$ is an abelian variety. Then there exists a line bundle $\mathcal L'$ numerically equivalent to $\mathcal L$ such that $\kappa (X,\mathcal L') \geq 0$. Denote $Y = \prod X_j$, and let $p_1\colon Y \times T \to Y$ and $p_2\colon Y \times T \to T$ be the projections. Since $Y$ is simply connected, by [@Har77 Exercise III.12.6] there exist line bundles $\mathcal M$ and $\mathcal N$ on $Y$ and $T$, respectively, such that $$\mathcal L \simeq p_1^*\mathcal M \otimes p_2^*\mathcal N.$$ By restricting this relation to fibres of $p_1$ and $p_2$, we obtain that $\mathcal M$ and $\mathcal N$ are nef. As $T$ is an abelian variety, there exists a semiample line bundle $\mathcal N'$ on $T$ numerically equivalent to $\mathcal N$, and since $\nu(Y,\mathcal M) \leq 1$, we have $\kappa(Y,\mathcal M) \geq 0$ by Theorem \[thm:CYnu1\]. Therefore, there exists a divisor $D$ on $\tilde X$ with $\kappa(\tilde X,D)\geq0$ such that $\eta^*\mathcal L\equiv\OO_{\tilde X}(D)$, hence $\mathcal L^{\otimes\deg\eta}\equiv\OO_X(\eta_*D)$. This finishes the proof. Following [@GKP11 §8], we say that a canonical variety $X$ of dimension $n$ and with $K_X\sim0$ is a Calabi-Yau variety if for every quasi-étale cover $\tilde X\to X$ we have $H^0(\tilde X,\Omega_{\tilde X}^{[q]}) = 0$ for $q=1,\dots,n -1$; and that it is a singular irreducible symplectic variety if there is a non-degenerate reflexive holomorphic $2$-form $\omega$ on $X$ such that for every quasi-étale cover $f\colon \tilde X\to X$, every reflexive holomorphic form on $\tilde X$ is of the form $c f^*\omega^{[p]}$ with a constant $c$. These, together with abelian varieties, are conjecturally the building blocks of singular varieties with trivial canonical class. Then we have: \[sing\] Let $X$ be a normal projective klt variety such that $K_X \sim_\Q 0$. Suppose that there exists a quasi-étale cover $\eta\colon \tilde X \to X$, such that $\tilde X$ is either a Calabi-Yau variety of even dimension or a singular irreducible symplectic variety. Let $\mathcal L$ be a nef line bundle on $X$ with $\nu(X,\mathcal L)=1$. Then $\kappa(X,\mathcal L)\geq0$. By [@GKP11 Proposition 6.9] we have $$H^0(\tilde X,\Omega^{[q]}_{\tilde X}) \simeq H^q({\tilde X},\OO_{\tilde X}).$$ Therefore $\chi(\tilde X,\OO_{\tilde X}) > 0$, and we conclude as in Theorem \[thm:CYnu1\] that $\kappa(X,\mathcal L)=\kappa(\tilde X,\eta^*\mathcal L)\geq0$. We also note: \[cor:nef1\] Let $X$ be a projective klt variety of dimension $4$ such that $K_X\sim_\Q0$, and let $\mathcal L$ be a nef line bundle on $X$ with $\nu(X,\mathcal L)=1$. Let $\eta\colon \tilde X \to X$ be the canonical cover and assume that $h^1(\tilde X,\OO_{\tilde X}) = 0$. Then $\kappa (X,\mathcal L) \geq 0$. If $\mathcal L$ is hermitian semipositive, then it is semiample. By [@GKP11 Proposition 6.9 and Corollary 6.11], we have $$h^3(\tilde X,\mathcal O_{\tilde X}) = h^1(\tilde X, \mathcal O_{\tilde X}) = 0.$$ This implies $\kappa(X,\mathcal L)\geq0$ as in the proof of Theorem \[sing\]. The last claim follows as in the proof of Corollary \[cor:nef\](ii). Several results of this section can be stated appropriately for klt pairs $(X,\Delta)$ such that $K_X+\Delta\sim_\Q0$; they are often called *varieties of Calabi-Yau type*. The proofs of those generalisations are straightforward adaptations of the proofs presented above. [^1]: Lazić was supported by the DFG-Emmy-Noether-Nachwuchsgruppe “Gute Strukturen in der höherdimensionalen birationalen Geometrie". Peternell was supported by the DFG grant “Zur Positivität in der komplexen Geometrie". We would like to thank D. Huybrechts, B. Taji and L. Tasin for useful conversations related to this work.2010 *Mathematics Subject Classification*: 14E30, 14F10.*Keywords*: abundance conjecture, Minimal Model Program, differential forms, Calabi-Yau varieties.

--- abstract: 'We report a detailed and comparative study of the single crystal CeCoInGa$_3$ in both experiment and theory. Resistivity measurements reveal the typical behavior of Kondo lattice with the onset temperature of coherence, $T^*\approx 50\,$K. The magnetic specific heat can be well fitted using a spin-fluctuation model at low temperatures, yielding a large Sommerfeld coefficient, $\gamma\approx172\,$mJ/mol K$^2$ at 6 K, suggesting that this is a heavy-fermion compound with a pronounced coherence effect. The magnetic susceptibility exhibits a broad field-independent peak at $T_{\chi}$ and shows an obvious anisotropy within the $bc$ plane, reflecting the anisotropy of the coherence effect at high temperatures. These are compared with strongly correlated calculations combining first-principles band structure calculations and dynamical mean-field theory. Our results confirm the onset of coherence at about 50 K and reveal a similar anisotropy in the hybridization gap, pointing to a close connection between the hybridization strength of the low-temperature Fermi-liquid state and the high-temperature coherence effect.' author: - Le Wang - Yuanji Xu - Meng Yang - Qianqian Wang - Cuixiang Wang - Shanshan Miao - Youting Song - Youguo Shi - 'Yi-feng Yang' title: 'Anisotropic hybridization in a new Kondo lattice compound CeCoInGa$_3$' --- INTRODUCTION ============ Kondo lattice physics is governed by two competing tendencies towards either coherent heavy-electron state [@Kondo1964; @Doniach1977; @Stewart1984-1; @Coleman2001; @Yang2016] or long-range magnetic orders of localized $f$-moments [@Ruderman1954; @Kasuya1956; @Yosida1957; @Yang2008a]. As a result, it involves a cascade of experimentally well-defined temperature scales, such as the coherence temperature $T^*$, the spin fluctuation temperature $T_{\rm SF}$, and the Fermi liquid temperature $T_{\rm FL}$. Among them, $T^*$ marks the onset of heavy-electron coherence produced by collective hybridization and sets the upper boundary of an intermediate regime with coexisting heavy electrons and unhybridized local $f$-moments [@Yang2012]. The transition from fully localized $f$-moments at high temperatures to coherent heavy electrons at low temperatures is at the heart of Kondo lattice physics [@Lonzarich2017]. Consequently, one expects anomalous properties in the intermediate state in all measured quantities, accompanying the emergence of heavy electrons. For example, the susceptibility shows deviation (or even a peak) from its high temperature Curie-Weiss behavior below $T^*$ and the specific heat exhibits logarithmic divergence before it saturates while entering a heavy Fermi liquid ground state. Experimentally, these anomalies provide a unified identification of the coherence taking place below $T^*$ [@Yang2008a]. However, some also regarded the peak structure in the susceptibility as a way to determine the crystal field scheme. These different opinions reflect the difficulty and confusions in our basic understanding of heavy fermion physics. Moreover, what has been less studied in previous literatures is the anisotropy of the coherence effect [@Yang2008b; @Ohishi2009] and how this is correlated with the underlying structure of collective hybridization in the Fermi liquid state below $T_{\rm FL}$. Exploration of special anisotropic or even nodal structure of hybridization is becoming a new frontier for novel anomalous and exotic phenomena in heavy fermion research [@Dzero2010; @Ramires2012; @Chen2018].  (a) Picture of the CeCoInGa$_3$ single crystal of the size of about 0.5 mm $\times$ 0.3 mm $\times$ 2.5 mm. (b) The orthorhombic unit cell of CeCoInGa$_3$ (space group Cmcm, No. 63). (c) The representation of multiple unit cells. The zone circled by the dashed line is enlarged with the lattice planes indexed as in (a).](fig1){width="48.00000%"} Here we report the successful synthesis and comparative study of the coherence and hybridization effect in a new Kondo lattice compound CeCoInGa$_3$. Unlike its sister families, Ce-115 such as CeCoIn$_5$ and CeRhIn$_5$ [@Petrovic2001; @Tayama2002; @Park2006; @Kenzelmann2008] and Ce-113 such as CeCuGa$_3$ and CeRhGe$_3$ [@Hillier2012; @Joshi2012; @Wang2018], which show interesting cuprate-like layered structure or noncentrosymmetric structure, respectively, and have thus been widely studied, the Ce-1113 or Ce-114 family has less been investigated, possibly due to the difficulty of its synthesis. In particular, to the best of our knowledge, CeCoGa$_4$ was never studied again ever since its discovery [@Routsi1992] where only the magnetic susceptibility was reported in polycrystals to show paramagnetic behavior down to 4.6 K. The 114 family has a quite different structure from 113 and 115, which may be viewed as stacked spin chains of Ce-ions located inside Ga$_5$ pyramids. It is therefore intriguing to see what physics might be discovered in this family. CeCoInGa$_3$ was obtained by doping In atoms into the mother compound CeCoGa$_4$ using the flux method [@Canfield1992]. The In atoms occupy the 4a-site of Ga atoms and the crystal structure remains orthorhombic with the space group Cmcm. In doing so, the lattice is expanded and the system is driven further towards a potential quantum critical point, where one may hope to find different quantum critical behavior or even superconductivity. We therefore performed a systematic measurement of CeCoInGa$_3$ and our results confirm that it is a standard Kondo lattice compound. The resistivity exhibits a progressive crossover from an insulating-like state due to incoherent Kondo scattering with logarithmic temperature dependence above $T^*\approx 50\,$K to a Fermi liquid state with $T^2$-dependence below about 6 K. The specific heat exhibits a logarithmic increase due to heavy fermion formation below $T^*$ and contains a $T^3\ln T$ contribution at intermediate temperatures, indicating a possible contribution from spin fluctuations with $T_{\rm SF} \approx 9\,$K. A broad hump is observed in the susceptibility whose position $T_\chi$ varies with the direction of the magnetic field and reflects the anisotropy of the coherence effect. To understand these, we carried out comparative studies using strongly correlated band calculations combining fully consistently the density functional theory and the state-of-the-art dynamical mean-field theory (DFT+DMFT). Our results for CeCoInGa$_3$ produce the correct coherence temperature $T^*$ and reveal an anisotropy of the hybridization gap, which is consistent with the anisotropy in the coherence effect at high temperatures. This suggests that the peak and its anisotropy in the susceptibility may be related to the onset of coherence and its underlying anisotropy of hybridization, which may be further traced back to the formation of Ce-Co-Ce zigzag chains along the $c$-axis rather than the Ce-chains along the shortest $a$-axis. Unfortunately, we do not find superconductivity down to 2 K in this compound, which indicates that further chemical tuning may be needed for future investigations. ----------------------------- -------------------------------------- empirical formula CeCoInGa$_3$ formula weight 523.036 g/mol temperature 273(2) K wavelength Mo $K_\alpha$ (0.71073 Å) crystal system orthorhombic space group $Cmcm~(63)$ unit cell dimensions $a=4.2315(4)$Å $b=16.0755(18)$Å $c=6.5974(6)$Å cell volume 448.78(8) Å$^3$ $Z$ 4 density, calculated 7.741 g/cm$^3$ $h \ k \ l$ range $-5 \le h \le 5$ $-11 \le k \le 20$ $-8 \le l \le 7$ 2$\theta_{max}$ 56.39 linear absorption coeff. 36.134 mm$^{-1}$ absorption correction multi-scan no. of reflections 1184 $T_{min}/T_{max}$ 0.004/0.030 $R_{int}$ 0.0529 no. independent reflections 338 no. observed reflections 337 \[$F_o > 4\sigma (F_o)$\] $F$(000) 908 $R$ values 5.29 % ($R_1[F_o > 4\sigma (F_o)]$) 13.36 % (w$R_2$) weighting scheme $w=1/[\sigma^2(F_o^2) + (0.0672P)^2$ $+ 17.0963P]$, where $P = (F_o^2 + 2F_c^2)/3$ diff. Fourier residues \[-2.969,3.437\] e/Å$^3$ refinement software SHELXL-2014/7 ----------------------------- -------------------------------------- : \[table1\]Crystallographic data of CeCoInGa$_3$. EXPERIMENTAL DETAILS ==================== Single crystals of CeCoInGa$_3$ were grown using the In-Ga eutectic as flux in alumina crucible sealed in a fully evacuated quartz tube. The crucible was heated to 1100 $^\circ$C for 10 hours and then cooled slowly to 630 $^\circ$C where the flux was spun off by a centrifuge. Rectangle-like single crystals were yielded with the volume of about 0.5 mm $\times$ 0.3 mm $\times$ 2.5 mm as shown in Figure \[fig1\]. Elemental analysis was conducted via energy dispersive X-ray (EDX) spectroscopy using a Hitachi S-4800 scanning electron microscope at an accelerating voltage of 15 kV with an accumulation time of 90 s. Single crystal X-ray diffraction was carried out on Bruker D8 Venture diffractometer at 273(2) K using Mo K$\alpha$ radiation ($\lambda=0.71073$ Å). The crystal structure was refined by full-matrix least-squares fitting on $F^2$ using the SHELXL-2014/7 program. A well-crystallized sample was picked out for the measurements. The magnetic susceptibility ($\chi$) was performed in a Quantum Design Magnetic Property Measurement System (MPMS) from 2 K to 300 K under various applied magnetic fields up to 50 kOe in field-cooling (FC) and zero-field-cooling (ZFC) modes. The electrical resistivity ($\rho$) and the specific heat ($C_p$) were measured between 2 K and 300 K in a Physical Property Measurement System (PPMS) using a standard $dc$ four-probe technique and a thermal-relaxation method, respectively. Site WP$^a$ x y z $U_{eq}$ OP$^b$ ------ -------- --------- ------------- ------------ ------------ -------- Ce 4c 0.00000 0.37989(7) 0.25000 0.0160(5) 1 Co 4c 0.00000 0.72432(17) 0.25000 0.0163(7) 1 In 4a 0.00000 0.00000 0.00000 0.0239(5) 1 Ga1 4c 0.00000 0.57916(16) 0.25000 0.0202(6) 1 Ga2 8f 0.00000 0.19182(1) 0.05409(7) 0.00173(5) 1 : \[table2\] Atomic coordinates and equivalent isotropic thermal parameters of CeCoInGa$_3$. RESULTS AND DISCUSSION ====================== The refined results are listed in Tables \[table1\] and  \[table2\], indicating a stoichiometric composition with the orthorhombic YNiAl$_4$-type structure (space group Cmcm, No. 63) and the lattice parameters $a=4.2315(4)$ Å, $b=16.0755(18)$ Å and $c=6.5974(6)$ Å. All the crystallographic sites are fully occupied by a unique sort of atoms. The larger In atoms of CeCoInGa$_3$ replace the 4a-site of Ga atoms in CeCoGa$_4$ without changing the crystal structure. It, however, enlarges the inter-plane distance and makes the lattice plane (0 2 1) to be the easy cleavage plane. The Ce atoms locate at 4c site and are each surrounded by five Ga atoms, forming CeGa$_5$ polyhedra that are straightly packed with sharing edges along the $a$-axis. The Co atoms locate between the Ga-cages, forming a layer-like structure. As shown in Fig. \[fig1\](c), the bright crystal surfaces are indexed as (0 2 1), (0 1 0) and (0 2 -1) by single crystal X-ray diffraction, consistent with the enlarged micro-structure.  Electrical resistivity of single crystal CeCoInGa$_3$ and LaCoInGa$_3$ with $j \parallel a$, $H = 1$ T and $H \perp a$. The inset shows the temperature dependence of the magnetic resistivity $\rho_m$ after subtracting the resistivity of the isostructural compound LaCoInGa$_3$. A Kondo-type scattering ($\rho_m \sim -\ln T$) was found above the coherence temperature, $T^*\approx50\,$K, as marked by the dashed line. The low-temperature resistivity data can be fitted (dash-dotted line) by the Fermi liquid model, $\rho_m \sim T^2$, giving the Fermi liquid temperature, $T_{\rm FL}\approx 6\,$K. ](fig2){width="48.00000%"} Figure \[fig2\] presents the temperature dependence of the $a$-axis resistivity $\rho$ of both CeCoInGa$_3$ and LaCoInGa$_3$ single crystals. The residual resistivity ratio, ${\rm RRR}=\rho(300\,{\rm K})/\rho(2\,{\rm K})$, is 3.1 for CeCoInGa$_3$ and 8.3 for LaCoInGa$_3$. A magnetic field of 1 T perpendicular to the $a$-axis was applied to suppress the superconductivity of the In flux. The magnetoresistance of CeCoInGa$_3$ appears to be very small up to 9 T and is not shown here. The magnetic resistivity $\rho_m(T)$ was obtained by subtracting the nonmagnetic contribution estimated from LaCoInGa$_3$. As shown in the inset of Fig. \[fig2\], it follows a logarithmic temperature dependence above $T^*\approx 50\,$K, indicating a major contribution from Kondo scattering by localized $f$-moments at high temperatures. A larger $T^* \approx 120\,$K has been observed in CeNiAl$_4$ with the same crystal structure [@Mizushima1991], as the lattice is expanded by In and Ga atoms in CeCoInGa$_3$. The coherence peak around $T^*$ in the magnetic resistivity marks the onset of localized-to-itinerant transition. Below 6 K, we find $\rho_m=\rho_0+AT^2$ with a residual resistivity $\rho_0=11.97~\mu\Omega$ cm and a resistivity coefficient $A=0.0826~\mu\Omega$ cm/K$^2$. This defines the Landau-Fermi liquid regime with a Fermi-liquid temperature, $T_{FL}\approx 6\,$K, roughly one tenth of $T^*$ [@Kaga1988-1; @Kaga1988-2]. We conclude that CeCoInGa$_3$ is a typical Kondo lattice material with a Fermi liquid ground state. The specific heat data of CeCoInGa$_3$ and LaCoInGa$_3$ in zero field are compared in Figure \[fig3\], showing no obvious phase transition down to 2 K in both compounds. The magnetic specific heat $C_m$ can be obtained in a similar way by subtracting the lattice contribution estimated from the nonmagnetic LaCoInGa$_3$. As is seen in the inset of Fig. \[fig3\], $C_m/T$ shows a logarithmic divergence with temperature below $T^*$, marking the emergence of heavy electrons accompanying the onset of coherence in the magnetic resistivity [@Yang2008a; @Yang2016]. Interestingly, at lower temperatures, the magnetic specific heat becomes saturated and obeys the formula [@Trainor1975; @Stewart1984-2], $C_m=\gamma T+DT^3\ln(T_{\rm SF}/T)$, where $T_{\rm SF}$ corresponds to the spin-fluctuation temperature. Our best fit yields the residual specific heat $\gamma=0.172\,$J/mol K$^2$, $D=1.92 \times 10^{-4}\,$J/mol K$^4$, and $T_{\rm SF}\approx 9\,$K. The large $\gamma$ implies a heavy quasi-particle effective mass $m^*$ in the Fermi-liquid state. We can calculate the Kadowaki-Woods ratio, $A/\gamma^2\approx 0.28\times10^{-5}$ $\mu\Omega$ cm (mol K mJ$^{-1}$)$^2$, which is comparable with that of other heavy-fermion compounds [@Kadowaki1986]. A rough comparison with the prediction of the spin-1/2 Kondo model suggests a Kondo temperature of about 41 K [@Desgranges1982], roughly consistent with the magnitude of the coherence temperature, but a quantitative fit is impossible. This indicates that in the real system CeCoInGa$_3$ there may be features additionally to the Kondo physics that are not captured by the simple model.  Temperature dependence of the zero-field specific heat coefficient $C_p/T$ for CeCoInGa$_3$ and LaCoInGa$_3$. The inset shows the magnetic contribution $C_m/T$ of CeCoInGa$_3$ after subtracting the lattice contribution estimated from LaCoInGa$_3$. The low-temperature data can be fitted (solid line) using the spin-fluctuation model, $C_m/T=\gamma+DT^2\ln(T_{\rm SF}/T)$ with $T_{\rm SF} \approx 9$ K. The dashed line indicates the logarithmic divergence of the specific heat due to incoherent Kondo scattering above $T^*$. Interestingly, the two lines intersect roughly at $T_{\chi} \approx 20$ K, where a peak is seen in the magnetic susceptibility as shown in Fig. \[fig4\].](fig3){width="48.00000%"} Figure \[fig4\] plots the ZFC data of the magnetic susceptibility $\chi$ and the inverse susceptibility $\chi^{-1}$ for $H=0.1$ T and 1 T along the $a$-axis. The $M$-$H$ curve is almost linear up to at least 7 T. A Curie-Weiss fit (dashed line) above 150 K using $\chi(T)=C/(T-\theta_p)$ yields an effective magnetic moment, $\mu_{eff}=2.64~\mu_{\rm B}$, close to the theoretical value of 2.54 $\mu_{\rm B}$ of free Ce$^{3+}$ ion, and a negative Curie temperature, $\theta_p=-19.8$ K. These indicate that the Ce $f$-electrons are well located at high temperatures with an antiferromagnetic exchange coupling. A similar analysis for LaCoInGa$_3$ (inset) using a modified Curie-Weiss formula, $\chi(T)=\chi_0+C/(T-\theta_p)$, yields a diamagnetic background susceptibility $\chi_0=-9.85 \times 10^{5}\,$emu/mol and a Weiss temperature $\theta_p=-1.35$ K, possibly contributed by the Co $3d$-electrons. Thus the Co ions are essentially nonmagnetic. For CeCoInGa$_3$, the violation of the Curie-Weiss behavior below 150 K might be first due to crystal field effects. However, below $T^*$, the development of a broad peak should be attributed to the coherence effect as observed in CeAl$_3$ and URu$_2$Si$_2$ [@Yang2012]. In the two-fluid model, it has been argued that increasing hybridization could induce a more rapid delocalization of the localized $f$-moments [@Yang2012]. Therefore, the directional dependence of $T_\chi$ potentially reflects the anisotropy of the high temperature coherence effect.  The ZFC susceptibility of CeCoInGa$_3$ with the magnetic field $H=0.1$ and 1 T along the $a$-axis. A best Curie-Weiss fit (dashed line) yields an effective magnetic moment, $\mu_{eff}=2.64\,\mu_B$, and the Weiss temperature, $\theta_p=-19.8$ K. The inset shows the susceptibility of LaCoInGa$_3$ with magnetic field $H = 1$ T along the $a$-axis, revealing diamagnetic behavior at high temperatures. The solid line is a modified Curie-Weiss fit (see text) with $\chi_0=-9.85 \times 10^{-5}$ emu/mol and $\theta_p=-1.35$ K, showing that the Co ions are essentially nonmagnetic.](fig4){width="48.00000%"} Figure \[fig5\](a) plots the ZFC susceptibilities for field in parallel with or perpendicular to the (0 1 0), (0 2 1) or (0 2 -1) planes. As shown in Fig. \[fig1\](a), the single crystal of CeCoInGa$_3$ has several facets in one-to-one correspondence with its micro-structure. It is relatively easy to apply the field along these directions. Fig. \[fig5\](b) plots the polar diagram of $T_\chi$ with the data periodically extrapolated to 360$^\circ$. The angle, $\theta$, is set to zero for $H\parallel c$. We see an angular variation associated with the crystal symmetry. Interestingly, as plotted in Fig. \[fig5\](c), there exists an anti-correlation between $T_\chi$ and the residual susceptibility $\chi_0$. For $H \parallel c$, $T_\chi$ is large and $\chi_0$ is small; while for $H \parallel b$, $T_\chi$ is small and $\chi_0$ is large. In the literature, the susceptibility peak has often been attributed to the crystal field effect. We will show that it is potentially correlated with the strength of collective hybridization. Thus the hybridization is stronger along the $c$-axis. In between, the results may be roughly understood by $\chi(\theta)=\chi_c\cos^2\theta +\chi_b\sin^2\theta$.  (a) The ZFC susceptibility of CeCoInGa$_3$ with magnetic field $H$ perpendicular to the $a$-axis. The angle, $\theta$, is set to zero for $H\parallel c$, as illustrated in the inset. The peak, $T_{\chi}$, marked by the arrows, evolves as the field changes the direction within the $bc$-plane. (b) Angular dependence of $T_{\chi}$ and the residual susceptibility $\chi_0$. The solid circle and square represent the experimental data and the hollow ones are from periodic extrapolation. The upper panel gives the polar diagram of $T_\chi$ in correspondence with the crystal structure. The lower panel compares the angular dependence of $\chi_0$ and $T_{\chi}$.](fig5){width="48.00000%"} Numerical calculations ====================== To gain further insight into above results, we carried out fully consistent DFT+DMFT calculations [@Kotliar2006; @Haule2010; @Held2008]. This method has been successfully applied to CeIrIn$_5$ [@Shim2007] and some other materials [@Shorikov2015; @Yang2007]. However, comparative studies of the hybridization structure on realistic heavy fermion materials are still very few, due to the difficulty in treating the 14 spin and orbital degrees of freedom of the strongly correlated 4$f$-electrons and the extremely low temperature of coherence. For the DFT part, we have adopted the full-potential linearized augmented-plane-wave method as implemented in the WIEN2K package [@Blaha2001; @Perdew1996]. Figure \[fig6\](a) compares the Ce-4$f$ density of states at 200 K and 1 K using one-crossing approximation as the impurity solver for DMFT [@Pruschke1989]. The Coulomb interaction was set to 6 eV, an approximate value typically used for Ce 4$f$-orbitals [@Shim2007; @Shorikov2015].The broad peaks at -3 eV and 4 eV correspond to the Hubbard bands. At 1 K, a sharp resonance is seen to develop near the Fermi energy, manifesting the emergence of heavy quasiparticles. This is clearly seen in Fig. \[fig6\](b), where the quasiparticle peak grows rapidly with lowering temperature. Correspondingly, the imaginary part of the self-energy, $|{\rm Im}\Sigma(\omega=0)|$, decreases rapidly, producing a broad maximum at about $T^*\approx50\,$K in its temperature derivative. This is an indication of a crossover in the magnetic scattering rate of the 4$f$-electrons at $T^*$. Above $T^*$, the quasi-particle density of states drops rapidly to zero, marking the loss of heavy electron coherence at higher temperatures.  (a) Comparison of the Ce 4$f$ density of states (DOS) at 200 K and 1 K calculated using DFT+DMFT. (b) Temperature evolution of the height of quasi-particle peak, the imaginary part of the 4$f$ self-energy at the Fermi energy ($\omega=0$) and its temperature derivative. The results show a maximum at about 50 K and a rapid increase of the DOS at lower temperatures. (c) and (d) compare the momentum-resolved spectral functions at 200 K and 1 K.](fig6){width="48.00000%"} Figures \[fig6\](c) and \[fig6\](d) compare the momentum-resolved spectral function along high symmetry path at 200 K and 1 K. Near the Fermi energy, we see the emergence of evident flat hybridization bands at 1 K, which are not present at 200 K. The hybridization strength may be estimated by fitting each band using $ E_k^\pm=\frac12[ (\epsilon_k+\epsilon_f)\pm \sqrt{(\epsilon_k-\epsilon_f)^2+\Delta^{2}}]$, where $E_k^\pm$ are the two hybridization bands, $\epsilon_k$ is the dispersion of the corresponding conduction band from high temperatures (200 K), $\epsilon_f\approx 0$ is the renormalized $f$-electron energy level, and $\Delta$ corresponds to the direct gap and represents the strength of the hybridization. We obtain $\Delta\approx 22$ meV for the band along the $\Gamma$-Z path ($k_z$), 40 meV along U-X ($k_z$), and 16 meV along U-Z ($k_y$) and $\Gamma$-Y ($k_x$) paths. Thus the hybridization is stronger along the $c$-axis and weaker along the $a$ and $b$-axes. The origin of such anisotropy might be traced back to the hybridization pathway of the Ce-$f$ electrons. Although one might naively think that the Ce-ions are surrounded by Ga pyramids and connect to form spin chains along the $a$-axis with shortest Ce-Ce distance given by the lattice constant $a$, the Ga-ions seem to play largely the role of a support of the crystal structure, as is the role of B in YbB$_6$ [@Zhou2015], and the hybridization mainly takes place between the Ce-4$f$ and Co-3$d$ bands. The Ce and Co-ions form a zigzag chain along the $c$-axis, favoring the largest hybridization along this direction, while for other two directions, the Ce-Co-Ce bonds are out of plane or have a longer distance, causing their relatively smaller strengths of hybridization. This anisotropy is in good correspondence with the angular variation of $T_{\chi}$, confirming a correlation between the high-temperature coherence effect and the low-temperature hybridization strength. We would like to further remark that while DFT could sometimes yield useful information for understanding the Fermi surface topology of heavy fermion compounds, it alone cannot describe the development of the $f$-electron coherence with lowering temperature and therefore is incapable of quantitative or even qualitative comparison with many experiments. Moreover, in CeCoInGa$_3$ and many other cases, the Ce-4$f$ bands are predicted in DFT to exhibit a large dispersion near the Fermi energy due to the lack of Kondo renormalization, while the conduction bands are all pushed away. This makes it impossible to derive any information on the hybridization structure between the $f$ and conduction bands. It is only with DFT+DMFT that the $f$-electrons are well treated and strongly renormalized to give rise to flat bands near the Fermi energy, allowing for an unambiguous identification of their hybridization with conduction electrons. CONCLUSIONS =========== We have successfully synthesized high-quality single crystals of CeCoInGa$_3$ and LaCoInGa$_3$ by a flux method. In contrast to it sister Ce-113 and Ce-115 families, the Ce-1113 or Ce-114 family is less well studied. Our systematic investigation of its resistivity, specific heat, and susceptibility provides a unified picture of CeCoInGa$_3$ as a typical paramagnetic Kondo lattice material with logarithmic temperature-dependent specific heat coefficient at low temperatures before the system enters a Fermi liquid state. We identify three important temperature scales in this compound: the coherence temperature $T^*\approx 50\,$K, the spin-fluctuation temperature $T_{\rm SF} \approx 9\,$K and the Fermi liquid temperature $T_{\rm FL}\approx 6\,$K. A broad hump is observed below $T^*$ in the magnetic susceptibility and shows strong anisotropy, reflecting the directional dependence of heavy-electron coherence. We performed comparative numerical studies. Strongly correlated calculations based on DFT+DMFT confirms the onset of heavy-electron coherence below 50 K, and reveals a similar anisotropy in the hybridization strength, suggesting a close connection with the anisotropy of the coherence effect at high temperatures. We note that replacing Ga by In expands the lattice and drives the system towards a potential quantum critical point where superconductivity may emerge. Although this was not observed in CeCoInGa$_3$, we expect that further chemical tuning will push the system closer to the quantum critical point. A systematic investigation of its peculiar quantum criticality and potential superconductivity, in comparison with the 113 and 115 family, might improve our understanding of heavy fermion physics in association with the crystal structures and hybridization anisotropy. ACKNOWLEDGMENTS =============== This work was supported by the National Key R&D Program of China (Grant No. 2017YFA0302901, No. 2017YFA0303103, No. 2016YFA0300604), the National Natural Science Foundation of China (NSFC Grant No. 11522435, No. 11474330, No. 11774399, No. 11774401), the State Key Development Program for Basic Research of China (Grant No. 2015CB921300), the Chinese Academy of Sciences (CAS) (Grant No. XDB07020100, No. XDB07020200, No. QYZDB-SSW-SLH043), the National Youth Top-notch Talent Support Program of China, and the Youth Innovation Promotion Association of CAS. L.W. and Y.X. contributed equally to this work. [99]{} J. Kondo, Prog. Theor. Phys. **32**, 37 (1964). S. Doniach, Physica B **91**, 231 (1977). G. R. Stewart, Rev. Mod. Phys. **56**, 755 (1984). P. Coleman, C. Pépin, Q. Si, and R. Ramazashvili, J. Phys.: Condens. Matter [**13**]{}, R723-R738 (2001). Y.-F. Yang, Rep. Prog. Phys. **79**, 074501 (2016). M. A. Ruderman and C. Kittel, Phys. 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--- abstract: 'The [Fréchet distance]{} is a metric to compare two curves, which is based on monotonous matchings between these curves. We call a matching that results in the [Fréchet distance]{} a [Fréchet matching]{}. There are often many different [Fréchet matching]{}s and not all of these capture the similarity between the curves well. We propose to restrict the set of [Fréchet matching]{}s to “natural” matchings and to this end introduce *locally correct* [Fréchet matching]{}s. We prove that at least one such matching exists for two polygonal curves and give an $O(N^3 \log N)$ algorithm to compute it, where $N$ is the total number of edges in both curves. We also present an $O(N^2)$ algorithm to compute a locally correct discrete [Fréchet matching]{}.' author: - Kevin Buchin - Maike Buchin - Wouter Meulemans - Bettina Speckmann bibliography: - 'references.bib' title: 'Locally Correct Fréchet Matchings[^1]' --- Introduction {#sec:introduction} ============ Many problems ask for the comparison of two curves. Consequently, several distance measures have been proposed for the similarity of two curves $P$ and $Q$, for example, the Hausdorff and the [Fréchet distance]{}. Such a distance measure simply returns a number indicating the (dis)similarity. However, the Hausdorff and the [Fréchet distance]{} are both based on matchings of the points on the curves. The distance returned is the maximum distance between any two matched points. The [Fréchet distance]{} uses *monotonous matchings* (and limits of these): if point $p$ on $P$ and $q$ on $Q$ are matched, then any point on $P$ after $p$ must be matched to $q$ or a point on $Q$ after $q$. The *[Fréchet distance]{}* is the maximal distance between two matched points minimized over all monotonous matchings of the curves. Restricting to monotonous matchings of only the vertices results in the *discrete [Fréchet distance]{}*. We call a matching resulting in the (discrete) [Fréchet distance]{} a *(discrete) [Fréchet matching]{}*. See Section \[sec:prelims\] for more details. There are often many different [Fréchet matching]{}s for two curves. However, as the [Fréchet distance]{} is determined only by the maximal distance, not all of these matchings capture the similarity between the curves well (see Fig. \[fig:badmatch\]). There are applications that directly use a matching, for example, to map a GPS track to a street network [@Wenk2006] or to morph between the curves [@Efrat2002]. In such situations a “good” matching is important. We believe that many applications of the (discrete) [Fréchet distance]{}, such as protein alignment [@Wylie2012] and detecting patterns in movement data [@bbgll-dcpcs-11], would profit from good [Fréchet matching]{}s. ![Two Fréchet matchings for curves $P$ and $Q$.[]{data-label="fig:badmatch"}](badmatch) ![Two Fréchet matchings. Right: the result of speed limits is not locally correct.[]{data-label="fig:speedcounter"}](speedcounter) #### Results We restrict the set of [Fréchet matching]{}s to “natural” matchings by introducing *locally correct* [Fréchet matching]{}s: matchings that for any two matched subcurves are again a [Fréchet matching]{} on these subcurves. In Section \[sec:lcm\] we prove that there exists such a locally correct [Fréchet matching]{} for any two polygonal curves. Based on this proof we describe in Section \[sec:algorithm\] an $O(N^3 \log N)$ algorithm to compute such a matching, where $N$ is the total number of edges in both curves. We consider the discrete [Fréchet distance]{} in Section \[sec:discrete\] and give an $O(N^2)$ algorithm to compute locally correct matchings under this metric. #### Related work The first algorithm to compute the [Fréchet distance]{} was given by Alt and Godau [@Alt1995]. They also consider a non-monotone [Fréchet distance]{} and their algorithm for this variant results in a locally correct non-monotone matching (see Remark 3.5 in [@hr-fdre-12]). Eiter and Mannila gave the first algorithm to compute the discrete [Fréchet distance]{} [@em-cdfd-94]. Since then, the [Fréchet distance]{} has received significant attention. Here we focus on approaches that restrict the allowed matchings. Efrat [*et al.*]{} [@Efrat2002] introduced Fréchet-like metrics, the geodesic width and link width, to restrict to matchings suitable for curve morphing. Their method is suitable only for non-intersecting polylines. Moreover, geodesic width and link width do not resolve the problem illustrated in Fig. \[fig:badmatch\]: both matchings also have minimal geodesic width and minimal link width. Maheshwari [*et al.*]{} [@Maheshwari2011] studied a restriction by “speed limits”, which may exclude all [Fréchet matching]{}s and may cause undesirable effects near “outliers” (see Fig. \[fig:speedcounter\]). Buchin [*et al.*]{} [@Buchin2010] describe a framework for restricting [Fréchet matching]{}s, which they illustrate by restricting slope and path length. The former corresponds to speed limits. We briefly discuss the latter at the end of Section \[sec:algorithm\]. Preliminaries {#sec:prelims} ============= [**Curves.**]{} Let $P$ be a polygonal curve with $m$ edges, defined by vertices $p_0, \ldots, p_m$. We treat a curve as a continuous map $P : [0,m] \rightarrow \mathbb{R}^d$. In this map, $P(i)$ equals $p_i$ for integer $i$. Furthermore, $P(i+\lambda)$ is a parameterization of the $(i+1)$st edge, that is, $P(i+\lambda) = (1-\lambda) \cdot p_i + \lambda \cdot p_{i+1}$, for integer $i$ and $0 < \lambda < 1$. As a reparametrization $\sigma : [0,1] \rightarrow [0,m]$ of a curve $P$, we allow any continuous, non-decreasing function such that $\sigma(0) = 0$ and $\sigma(1) = m$. We denote by $P_\sigma(t)$ the actual location according to reparametrization $\sigma$: $P_\sigma(t) = P(\sigma(t))$. By $P_\sigma[a,b]$ we denote the subcurve of $P$ in between $P_\sigma(a)$ and $P_\sigma(b)$. In the following we are always given two polygonal curves $P$ and $Q$, where $Q$ is defined by its vertices $q_0, \ldots, q_n$ and is reparametrized by $\theta : [0,1] \rightarrow [0,n]$. The reparametrized curve is denoted by $Q_\theta$. #### [Fréchet matching]{}s We are given two polygonal curves $P$ and $Q$ with $m$ and $n$ edges. A (monotonous) *matching* $\mu$ between $P$ and $Q$ is a pair of reparametrizations $(\sigma,\theta)$, such that $P_\sigma(t)$ matches to $Q_\theta(t)$. The Euclidean distance between two matched points is denoted by $d_\mu(t) = |P_\sigma(t) - Q_\theta(t)|$. The maximum distance over a range is denoted by $d_\mu[a,b] = \max_{a \leq t \leq b} d_\mu(t)$. The *[Fréchet distance]{}* between two curves is defined as $\delta_\text{F}(P,Q) = \inf_{\mu} d_\mu[0,1]$. A *[Fréchet matching]{}* is a matching $\mu$ that realizes the [Fréchet distance]{}: $d_\mu[0,1] = \delta_\text{F}(P,Q)$ holds. #### Free space diagrams Alt and Godau [@Alt1995] describe an algorithm to compute the [Fréchet distance]{} based on the decision variant (that is, solving $\delta_\text{F}(P,Q) \leq \varepsilon$ for some given $\varepsilon$). Their algorithm uses a *free space diagram*, a two-dimensional diagram on the range $[0, m] \times [0, n]$. Every point $(x,y)$ in this diagram is either “free” (white) or not (indicating whether $|P(x) - Q(y)| \leq \varepsilon$). The diagram has $m$ columns and $n$ rows; every cell $(c,r)$ ($1 \leq c \leq m$ and $1 \leq r \leq n$) corresponds to the edges $p_{c-1}p_{c}$ and $q_{r-1}q_{r}$. To compute the [Fréchet distance]{}, one finds the smallest $\varepsilon$ such that there exists an x- and y-monotone path from point $(0,0)$ to $(m, n)$ in free space. ![Three event types. (A) Endpoints come within range of each other. (B) Passage opens on cell boundary. (C) Passage opens in row (or column). We scale every row and column in the diagram to correspond to the (relative) length of the actual edge of the curve instead of using unit squares for cells.[]{data-label="fig:events"}](events) For this, only certain *critical values* for the distance have to be checked. Imagine continuously increasing the distance $\varepsilon$ starting at $\varepsilon = 0$. At so-called *critical events*, which are illustrated in Fig. \[fig:events\], passages open in the free space. The critical values are the distances corresponding to these events. Locally correct [Fréchet matching]{}s {#sec:lcm} ===================================== We introduce *locally correct* [Fréchet matching]{}s, for which the matching between any two matched subcurves is a [Fréchet matching]{}. \[def:locallycorrect\] Given two polygonal curves $P$ and $Q$, a matching $\mu = (\sigma,\theta)$ is *locally correct* if for all $a,b$ with $0 \leq a \leq b \leq 1$ $$d_\mu[a,b] = \delta_\text{F}(P_\sigma[a,b], Q_\theta[a,b]).$$ Note that not every [Fréchet matching]{} is locally correct. See for example Fig. \[fig:speedcounter\]. The question arises whether a locally correct matching always exists and if so, how to compute it. We resolve the first question in the following theorem. \[thm:main\] For any two polygonal curves $P$ and $Q$, there exists a locally correct [Fréchet matching]{}. #### Existence We prove Theorem \[thm:main\] by induction on the number of edges in the curves. First, we present the lemmata for the two base cases: one of the two curves is a point, and both curves are line segments. In the following, $n$ and $m$ again denote the number of edges of $P$ and $Q$, respectively. \[lem:pointcurve\] For two polygonal curves $P$ and $Q$ with $m = 0$, a locally correct matching is $(\sigma, \theta)$, where $\sigma(t) = 0$ and $\theta(t) = t \cdot n$. Since $m = 0$, $P$ is just a single point, $p_0$. The [Fréchet distance]{} between a point and a curve is the maximal distance between the point and any point on the curve: $\delta_\text{F}(p_0, Q_\theta[a,b]) = d_\mu[a,b]$. This implies that the matching $\mu$ is locally correct. \[lem:linesegments\] For two polygonal curves $P$ and $Q$ with $m = n = 1$, a locally correct matching is $(\sigma, \theta)$, where $\sigma(t) = \theta(t) = t$. The free space diagram of $P$ and $Q$ is a single cell and thus the free space is a convex area for any value of $\varepsilon$. Since $\mu = (\sigma, \theta)$ is linear, we have that $d_\mu[a,b] = \max\left\{ d_\mu(a), d_\mu(b) \right\}$: if there would be a $t$ with $a < t < b$ such that $d_\mu(t) > \max\left\{ d_\mu(a), d_\mu(b) \right\}$, then the free space at $\varepsilon = \max\left\{ d_\mu(a), d_\mu(b) \right\}$ would not be convex. Since $d_\mu[a,b] = \max\left\{ d_\mu(a), d_\mu(b) \right\} \leq \delta_\text{F}(P_\sigma[a,b],Q_\theta[a,b]) \leq d_\mu[a,b]$, we conclude that $\mu$ is locally correct. ![(a) Curves with the free space diagram for $\varepsilon = \delta_\text{F}(P,Q)$ and the realizing event. (b) The event splits each curve into two subcurves. The hatched areas indicate parts that disappear after the split.[]{data-label="fig:split"}](split) For induction, we split the two curves based on events (see Fig. \[fig:split\]). Since each split must reduce the problem size, we ignore any events on the left or bottom boundary of cell $(1,1)$ or on the right or top boundary of cell $(m,n)$. This excludes both events of type A. A free space diagram is *connected* at value $\varepsilon$, if a monotonous path exists from the boundary of cell $(1,1)$ to the boundary of cell $(m,n)$. A *realizing event* is a critical event at the minimal value $\varepsilon$ such that the corresponding free space diagram is connected. Let $\mathcal{E}$ denote the set of concurrent realizing events for two curves. A *realizing set* $E_\text{r}$ is a subset of $\mathcal{E}$ such that the free space admits a monotonous path from cell $(1,1)$ to cell $(m,n)$ without using an event in $\mathcal{E} \backslash E_\text{r}$. Note that a realizing set cannot be empty. When $\mathcal{E}$ contains more than one realizing event, some may be “insignificant”: they are never required to actually make a path in the free space diagram. A realizing set is *minimal* if it does not contain a strict subset that is a realizing set. Such a minimal realizing set contains only “significant” events. \[lem:realizingset\] For two polygonal curves $P$ and $Q$ with $m > 1$ and $n \geq 1$, there exists a minimal realizing set. Let $\mathcal{E}$ denote the non-empty set of concurrent events at the minimal critical value. By definition, the empty set cannot be a realizing set and $\mathcal{E}$ is a realizing set. Hence, $\mathcal{E}$ contains a minimal realizing set. The following lemma directly implies that a locally correct [Fréchet matching]{} always exists. Informally, it states that curves have a locally correct matching that is “closer” (except in cell $(1,1)$ or $(m,n)$) than the distance of their realizing set. Further, this matching is linear inside every cell. In the remainder, we use realizing set to indicate a minimal realizing set, unless indicated otherwise. \[lem:specmatching\] If the free space diagram of two polygonal curves $P$ and $Q$ is connected at value $\varepsilon$, then there exists a locally correct [Fréchet matching]{} $\mu = (\sigma,\theta)$ such that $d_\mu(t) \leq \varepsilon$ for all $t$ with $\sigma(t) \geq 1$ or $ \theta(t) \geq 1$, and $\sigma(t) \leq m-1$ or $\theta(t) \leq n -1$. Furthermore, $\mu$ is linear in every cell. We prove this by induction on $m + n$. The base cases ($m = 0$, $n = 0$, and $m = n = 1$) follow from Lemma \[lem:pointcurve\] and Lemma \[lem:linesegments\]. For induction, we assume that $m \geq 1$, $n \geq 1$, and $m + n > 2$. By Lemma \[lem:realizingset\], a realizing set $E_\text{r}$ exists for $P$ and $Q$, say at value $\varepsilon_\text{r}$. The set contains realizing events $e_1, \ldots, e_k$ ($k \geq 1$), numbered in lexicographic order. By definition, $\varepsilon_\text{r} \leq \varepsilon$ holds. Suppose that $E_\text{r}$ splits curve $P$ into $P_1, \ldots, P_{k+1}$ and curve $Q$ into $Q_1, \ldots, Q_{k+1}$, where $P_i$ has $m_i$ edges, $Q_i$ has $n_i$ edges. By definition of a realizing event, none of the events in $E_\text{r}$ occur on the right or top boundary of cell $(m,n)$. Hence, for any $i$ ($1 \leq i \leq k+1$), it holds that $m_i \leq m$, $n_i \leq n$, and $m_i < m$ or $n_i < n$. Since a path exists in the free space diagram at $\varepsilon_\text{r}$ through all events in $E_\text{r}$, the induction hypothesis implies that, for any $i$ ($1 \leq i \leq k+1$), a locally correct matching $\mu_i = (\sigma_i,\theta_i)$ exists for $P_i$ and $Q_i$ such that $\mu_i$ is linear in every cell and $d_{\mu_i}(t) \leq \varepsilon_\text{r}$ for all $t$ with $\sigma_i(t) \geq 1$ or $\theta_i(t) \geq 1$, and $\sigma_i(t) \leq m_i-1$ or $\theta_i(t) \leq n_i -1$. Combining these matchings with the events in $E_\text{r}$ yields a matching $\mu = (\sigma,\theta)$ for $(P,Q)$. As we argue below, this matching is locally correct and satisfies the additional properties. The matching of an event corresponds to a single point (type B) or a horizontal or vertical line (type C). By induction, $\mu_i$ is linear in every cell. Since all events occur on cell boundaries, the cells of the matchings and events are disjoint. Therefore, the matching $\mu$ is also linear inside every cell. For $i < k+1$, $d_{\mu_i}$ is at most $\varepsilon_\text{r}$ at the point where $\mu_i$ enters cell $(m_i,n_i)$ in the free space diagram of $P_i$ and $Q_i$. We also know that $d_{\mu_i}$ equals $\varepsilon_\text{r}$ at the top right corner of cell $(m_i,n_i)$. Since $\mu_i$ is linear inside the cell, $d_{\mu_i}(t) \leq \varepsilon_\text{r}$ also holds for $t$ with $\sigma_i(t) > m_i-1$ and $\theta_i(t) > n_i -1$. Analogously, for $i > 0$, $d_{\mu_i}(t)$ is at most $\varepsilon_\text{r}$ for $t$ with $\sigma_i(t) < 1$ and $\theta_i(t) < 1$. Hence, $d_\mu(t) \leq \varepsilon_\text{r} \leq \varepsilon$ holds for $t$ with $\sigma(t) \geq 1$ or $\theta(t) \geq 1$, and $\sigma(t) \leq m-1$ or $\theta(t) \leq n -1$. To show that $\mu$ is locally correct, suppose for contradiction that values $a,b$ exist such that $\delta_\text{F}(P_\sigma[a,b], Q_\theta[a,b]) < d_\mu[a,b]$. If $a,b$ are in between two consecutive events, we know that the submatching corresponds to one of the matchings $\mu_i$. Since these are locally correct, $\delta_\text{F}(P_\sigma[a,b], Q_\theta[a,b]) = d_\mu[a,b]$ must hold. Hence, suppose that $a$ and $b$ are separated by at least one event of $E_\text{r}$. There are two possibilities: either $d_\mu[a,b] = \varepsilon_\text{r}$ or $d_\mu[a,b] > \varepsilon_\text{r}$. $d_\mu[a,b] < \varepsilon_\text{r}$ cannot hold, since $d_\mu[a,b]$ includes a realizing event. First, assume $d_\mu[a,b] = \varepsilon_\text{r}$ holds. If $\delta_\text{F}(P_\sigma[a,b], Q_\theta[a,b]) < \varepsilon_\text{r}$ holds, then a matching exists that does not use the events between $a$ and $b$ and has a lower maximum. Hence, the free space connects point $(\sigma(a),\theta(a))$ with point $(\sigma(b),\theta(b))$ at a lower value than $\varepsilon_\text{r}$. This implies that all events between $a$ and $b$ can be omitted, contradicting that $E_\text{r}$ is a minimal realizing set. Now, assume $d_\mu[a,b] > \varepsilon_\text{r}$. Let $t'$ denote the highest $t$ for which $\sigma(t) \leq 1$ and $\theta(t) \leq 1$ holds, that is, the point at which the matching exits cell $(1,1)$. Similarly, let $t''$ denote the lowest $t$ for which $\sigma(t) \geq m-1$ and $\theta(t) \geq n-1$ holds. Since $d_\mu(t) \leq \varepsilon_\text{r}$ holds for any $t' \leq t \leq t''$, $d_\mu(t) > \varepsilon_\text{r}$ can hold only for $t < t'$ or $t > t''$. Suppose that $d_\mu(a) > \varepsilon_\text{r}$ holds. Then $a < t'$ holds and $\mu$ is linear between $a$ and $t'$. Therefore, $d_\mu(a) > d_\mu(t)$ holds for any $t$ with $a < t < t'$. Analogously, if $d_\mu(b) > \varepsilon_\text{r}$ holds, then $d_\mu(b) > d_\mu(t)$ holds for any $t$ with $t'' < t < b$ . Hence, $d_\mu[a,b] = \max \left\{ d_\mu(a), d_\mu(b) \right\}$ must hold. This maximum is a lower bound on the [Fréchet distance]{}, contradicting the assumption that $d_\mu[a,b]$ is larger than the [Fréchet distance]{}. Matching $\mu$ is therefore locally correct. Algorithm for locally correct [Fréchet matching]{}s {#sec:algorithm} =================================================== The existence proof directly results in a recursive algorithm, which is given by Algorithm \[alg:findmatching\]. Fig. \[fig:badmatch\] (left), Fig. \[fig:speedcounter\] (left), Fig. \[fig:examplematching\], Fig. \[fig:vertexdependency\], and Fig. \[fig:lengthcounter\] (left) illustrate matchings computed with our algorithm. This section is devoted to proving the following theorem. Algorithm \[alg:findmatching\] computes a locally correct [Fréchet matching]{} of two polygonal curves $P$ and $Q$ with $m$ and $n$ edges in $O((m+n) m n \log (mn))$ time. $P$ and $Q$ are curves with $m$ and $n$ edges A locally correct [Fréchet matching]{} for $P$ and $Q$ $(\sigma, \theta)$ where $\sigma(t) = t \cdot m$, $\theta(t) = t \cdot n$ $(\sigma, \theta)$ where $\sigma(t) = \theta(t) = t$ Find event $e_\text{r}$ of a minimal realizing set \[algline:find\_er\] Split $P$ into $P_1$ and $P_2$ according to $e_\text{r}$ Split $Q$ into $Q_1$ and $Q_2$ according to $e_\text{r}$ $\mu_1 \rightarrow \texttt{ComputeLCFM}(P_1, Q_1)$ $\mu_2 \rightarrow \texttt{ComputeLCFM}(P_2, Q_2)$ concatenation of $\mu_1$, $e_\text{r}$, and $\mu_2$ Using the notation of Alt and Godau [@Alt1995], $L^F_{i,j}$ denotes the interval of free space on the left boundary of cell $(i,j)$; $L^R_{i,j}$ denotes the subset of $L^F_{i,j}$ that is reachable from point $(0,0)$ of the free space diagram with a monotonous path in the free space. Analogously, $B^F_{i,j}$ and $B^R_{i,j}$ are defined for the bottom boundary. With a slight modification to the decision algorithm, we can compute the minimal value of $\varepsilon$ such that a path is available from cell $(1,1)$ to cell $(m,n)$. This requires only two changes: $B^R_{1,2}$ should be initialized with $B^F_{1,2}$ and $L^R_{2,1}$ with $L^F_{2,1}$; the answer should be “yes” if and only if $B^R_{m,n}$ or $L^R_{m,n}$ is non-empty. ![Locally correct matching produced by Algorithm \[alg:findmatching\]. Free space diagram drawn at $\varepsilon = \delta_\text{F}(P,Q)$.[]{data-label="fig:examplematching"}](examplematching) #### Realizing set By computing the [Fréchet distance]{} using the modified Alt and Godau algorithm, we obtain an ordered, potentially non-minimal realizing set $\mathcal{E} = \{ e_1, \ldots, e_l \}$. The algorithm must find an event that is contained in a realizing set. Let $E_{k}$ denote the first $k$ events of $\mathcal{E}$. For now we assume that the events in $\mathcal{E}$ end at different cell boundaries. We use a binary search on $\mathcal{E}$ to find the $r$ such that $E_r$ contains a realizing set, but $E_{r-1}$ does not. This implies that event $e_r$ is contained in a realizing set and can be used to split the curves. Note that $r$ is unique due to monotonicity. For correctness, the order of events in $\mathcal{E}$ must be consistent in different iterations, for example, by using a lexicographic order. Set $E_r$ contains only realizing sets that use $e_r$. Hence, $E_{r-1}$ contains a realizing set to connect cell $(1,1)$ to $e_r$ and $e_r$ to cell $(m,n)$. Thus any event found in subsequent iterations is part of $E_{r-1}$ and of a realizing set with $e_r$. To determine whether some $E_{k}$ contains a realizing set, we check whether cells $(1,1)$ and $(m,n)$ are connected without “using” the events of $\mathcal{E} \backslash E_{k}$. To do this efficiently, we further modify the Alt and Godau algorithm. We require only a method to prevent events in $\mathcal{E} \backslash E_{k}$ from being used. After $L^R_{i,j}$ is computed, we check whether the event $e$ (if any) that ends at the left boundary of cell $(i,j)$ is part of $\mathcal{E} \backslash E_{k}$ and necessary to obtain $L^R_{i,j}$. If this is the case, we replace $L^R_{i,j}$ with an empty interval. Event $e$ is necessary if and only if $L^R_{i,j}$ is a singleton. To obtain an algorithm that is numerically more stable, we introduce entry points. The *entry point* of the left boundary of cell $(i,j)$ is the maximal $i' < i$ such that $B^R_{i',j}$ is non-empty. These values are easily computed during the decision algorithm. Assume the passage corresponding to event $e$ starts on the left boundary of cell $(i_\text{s}, j)$. Event $e$ is necessary to obtain $L_{i,j}^R$ if and only if $i' < i_\text{s}$. Therefore, we use the entry point instead of checking whether $L^R_{i,j}$ is a singleton. This process is analogous for horizontal boundaries of cells. Earlier we assumed that each event in $\mathcal{E}$ ends at a different cell boundary. If events end at the same boundary, then these occur in the same row (or column) and it suffices to consider only the event that starts at the rightmost column (or highest row). This justifies the assumption and ensures that $\mathcal{E}$ contains $O(m n)$ events. Thus computing $e_\text{r}$ (Algorithm \[alg:findmatching\], line \[algline:find\_er\]) takes $O(m n \log(m n))$ time, which is equal to the time needed to compute the [Fréchet distance]{}. Each recursion step splits the problem into two smaller problems, and the recursion ends when $mn \leq 1$. This results in an additional factor $m+n$. Thus the overall running time is $O((m+n) m n \log(m n))$. ![Different sampling may result in different matchings.[]{data-label="fig:vertexdependency"}](vertexdependency) #### Sampling and further restrictions Two curves may still have many locally correct [Fréchet matching]{}s: the algorithm computes just one of these. However, introducing extra vertices may alter the result, even if these vertices do not modify the shape (see Fig. \[fig:vertexdependency\]). This implies that the algorithm depends not only on the shape of the curves, but also on the sampling. Increasing the sampling further and further seems to result in a matching that decreases the matched distance as much as possible within a cell. However, since cells are rectangles, there is a slight preference for taking longer diagonal paths. Based on this idea, we are currently investigating “locally optimal” [Fréchet matching]{}s. The idea is to restrict to the locally correct [Fréchet matching]{} that decreases the matched distance as quickly as possible. We also considered restricting to the “shortest” locally correct [Fréchet matching]{}, where “short” refers to the length of the path in the free space diagram. However, Fig. \[fig:lengthcounter\] shows that such a restriction does not necessarily improve the quality of the matching. ![Two locally correct Fréchet matchings for $P$ and $Q$. Right: shortest matching.[]{data-label="fig:lengthcounter"}](lengthcounter) Locally correct discrete [Fréchet matching]{}s {#sec:discrete} ============================================== Here we study the discrete variant of [Fréchet matching]{}s. For the discrete [Fréchet distance]{}, only the vertices of curves are matched. The discrete [Fréchet distance]{} can be computed in $O(m \cdot n)$ time via dynamic programming [@em-cdfd-94]. Here, we show how to also compute a locally correct discrete [Fréchet matching]{} in $O(m \cdot n)$ time. #### Grids Since we are interested only in matching vertices of the curves, we can convert the problem to a grid problem. Suppose we have two curves $P$ and $Q$ with $m$ and $n$ edges respectively. These convert into a grid $G$ of non-negative values with $m+1$ columns and $n+1$ rows. Every column corresponds to a vertex of $P$, every row to a vertex of $Q$. Any node of the grid $G[i,j]$ corresponds to the pair of vertices $(p_i, q_j)$. Its value is the distance between the vertices: $G[i,j] = |p_i - q_j|$. Analogous to free space diagrams, we assume that $G[0,0]$ is the bottomleft node and $G[m,n]$ the topright node. #### Matchings A monotonous path $\pi$ is a sequence of grid nodes $\pi(1), \ldots, \pi(k)$ such that every node $\pi(i)$ ($1 < i \leq k$) is the above, right, or above/right diagonal neighbor of $\pi(i-1)$. In the remainder of this section a path refers to a monotonous path unless indicated otherwise. A monotonous discrete matching of the curves corresponds to a path $\pi$ such that $\pi(1) = G[0,0]$ and $\pi(k) = G[m,n]$. We call a path $\pi$ locally correct if for all $1 \leq t_1 \leq t_2 \leq k$, $\max_{t_1 \leq t \leq t_2} \pi(t) = \min_{\pi'} \max_{1 \leq t \leq k'} \pi'(t)$, where $\pi'$ ranges over all paths starting at $\pi'(1) = \pi(t_1)$ and ending at $\pi'(k') = \pi(t_2)$. $P$ and $Q$ are curves with $m$ and $n$ edges A locally correct discrete [Fréchet matching]{} for $P$ and $Q$ Construct grid $G$ for $P$ and $Q$ Let $T$ be a tree consisting only of the root $G[0,0]$ Add $G[i,0]$ to $T$ Add $G[0,j]$ to $T$ $\texttt{AddToTree}(T, G, i, j)$ path in $T$ between $G[0,0]$ and $G[m,n]$ #### Algorithm The algorithm needs to compute a locally correct path between $G[0,0]$ and $G[m,n]$ in a grid $G$ of non-negative values. To this end, the algorithm incrementally constructs a tree $T$ on the grid such that each path in $T$ is locally correct. The algorithm is summarized by Algorithm \[alg:finddiscrete\]. We define a *growth node* as a node of $T$ that has a neighbor in the grid that is not yet part of $T$: a new branch may sprout from such a node. The growth nodes form a sequence of horizontally or vertically neighboring nodes. A *living node* is a node of $T$ that is not a growth node but is an ancestor of a growth node. A *dead node* is a node of $T$ that is neither a living nor a growth node, that is, it has no descendant that is a growth node. Every pair of nodes in this tree has a *nearest common ancestor* (NCA). When we have to decide what parent to use for a new node in the tree, we look at the maximum value on the path in the tree between the parents and their NCA (excluding the value of the latter). A *face* of the tree is the area enclosed by the segment between two horizontally or vertically neighboring growth nodes (without one being the parent of another) and the paths to their NCA. The unique *sink* of a face is the node of the grid that is in the lowest column and row of all nodes on the face. Fig. \[fig:sinksandshortcuts\] (a-b) shows some examples of faces and their sinks. ![(a) Face of tree (gray area) with its unique sink (solid dot). A dashed line represents a dead path. (b) Two adjacent faces with some shortcuts indicated. (c) Tree with 3 faces. Solid dots indicate growth nodes with a growth node as parent. These nodes are incident to at most one face. All shortcuts of these nodes are indicated.[]{data-label="fig:sinksandshortcuts"}](discreteSinksAndShortcuts) #### Shortcuts To avoid repeatedly walking along the tree to compute maxima, we maintain up to two *shortcuts* from every node in the tree. The segment between the node and its parent is incident to up to two faces of the tree. The node maintains shortcuts to the sink of these faces, associating the maximum value encountered on the path between the node and the sink (excluding the value of the sink). Fig. \[fig:sinksandshortcuts\] (b) illustrates some shortcuts. With these shortcuts, it is possible to determine the maximum up to the NCA of two (potentially diagonally) neighboring growth nodes in constant time. Note that a node $g$ of the tree that has a growth node as parent is incident to at most one face (see Fig. \[fig:sinksandshortcuts\] (c)). We need the “other” shortcut only when the parent of $g$ has a living parent. Therefore, the value of this shortcut can be obtained in constant time by using the shortcut of the parent. When the parent of $g$ is no longer a growth node, then $g$ obtains its own shortcut. #### Extending the tree Algorithm \[alg:extend\] summarizes the steps required to extend the tree $T$ with a new node. Node $G[i,j]$ has three *candidate parents*, $G[i-1, j]$, $G[i-1, j-1]$, and $ G[i, j-1]$. Each pair of these candidates has an NCA. For the actual parent of $G[i,j]$, we select the candidate $c$ such that for any other candidate $c'$, the maximum value from $c$ to their NCA is at most the maximum value from $c'$ to their NCA—both excluding the NCA itself. We must be consistent when breaking ties between candidate parents. To this end, we use the preference order of $G[i-1, j] \succ G[i-1, j-1] \succ G[i, j-1]$. Since paths in the tree cannot cross, this order is consistent between two paths at different stages of the algorithm. Note that a preference order that prefers $G[i-1,j-1]$ over both other candidates or vice versa results in an incorrect algorithm. $G$ is a grid of non-negative values; any path in tree $T$ is locally correct node $G[i,j]$ is added to $T$ and any path in $T$ is locally correct $parent(G[i, j]) \leftarrow$ candidate parent with lowest maximum value to NCA Remove the dead path ending at $G[i-1, j-1]$ and extend shortcuts Make shortcuts for $G[i-1, j]$, $G[i, j-1]$, and $G[i, j]$ where necessary ![(a) Each sink has up to four sets of shortcuts. (b-d) Removing a dead path (dashed) extends at most one set of shortcuts.[]{data-label="fig:fourshortcutsets"}](discreteFourShortcutSetsAtSink) When a dead path is removed from the tree, adjacent faces merge and a sink may change. Hence, shortcuts have to be extended to point toward the new sink. Fig. \[fig:fourshortcutsets\] illustrates the incoming shortcuts at a sink and the effect of removing a dead path on the incoming shortcuts. Note that the algorithm does not need to remove dead paths that end in the highest row or rightmost column. Finally, $G[i-1, j]$, $G[i, j-1]$, and $G[i, j]$ receive shortcuts where necessary. $G[i-1, j]$ or $G[i, j-1]$ needs a shortcut only if its parent is $G[i-1, j-1]$. $G[i, j]$ needs two shortcuts if $G[i-1,j-1]$ is its parent, only one shortcut otherwise. #### Correctness To prove correctness of Algorithm \[alg:finddiscrete\], we require a stronger version of local correctness. A path $\pi$ is *strongly locally correct* if for all paths $\pi'$ with the same endpoints $\max_{1 < t \leq k} \pi(t) \leq \max_{1 < t' \leq k'} \pi'(t')$ holds. Note that the first node is excluded from the maximum. Since $\max_{1 < t \leq k} \pi(t) \leq \max_{1 < t' \leq k'} \pi'(t')$ and $\pi(1) = \pi'(1)$ imply $\max_{1 \leq t \leq k} \pi(t) \leq \max_{1 \leq t' \leq k'} \pi'(t')$, a strongly locally correct path is also locally correct. Lemma \[lem:discretecorrect\] implies the correctness of Algorithm \[alg:finddiscrete\]. \[lem:discretecorrect\] Algorithm \[alg:finddiscrete\] maintains the following invariant: any path in $T$ is strongly locally correct. To prove this lemma, we strengthen the invariant. **Invariant.** We are given a tree $T$ such that every path in $T$ is strongly locally correct. In constructing $T$, any ties were broken using the preference order. **Initialization.** Tree $T$ is initialized such that it contains two types of paths: either between grid nodes in the first column or in the first row. In both cases there is only one path between the endpoints of the path. Therefore, this path must be strongly locally correct. Since every node has only one candidate parent, $T$ adheres to the preference order. **Maintenance.** The algorithm extends $T$ to $T'$ by including node $g = G[i,j]$. This is done by connecting $g$ to one of its candidate parents ($G[i-1,j]$, $G[i-1,j-1]$, or $G[i,j-1]$), the one that has the lowest maximum value along its path to the NCA. We must now prove that any path in $T'$ is strongly locally correct. From the induction hypothesis, we conclude that only paths that end at $g$ could falsify this statement. Suppose that such an invalidating path exists in $T'$, ending at $g$. This path must use one of the candidate parents of $g$ as its before-last node. We distinguish three cases on how this path is situated compared to $T'$. The last case, however, needs two subcases to deal with candidate parents that have the same maximum value on the path to their NCA. The four cases are illustrated in Fig. \[fig:nonLCcasesAppendix\]. For each case, we consider the path $\pi_\textrm{i}$ between the first vertex and the parent of $g$ in the invalidating path (i.e. one of the three candidate parents). Note that $\pi_\textrm{i}$ need not be disjoint of the paths in $T'$. Slightly abusing notation, we also use a path $\pi'$ to denote its maximum value, excluding the first node, i.e. $\max_{1 < t \leq k'} \pi'(t)$. We now show that for each case, the existence of the invalidating path contradicts the invariant on $T$. ![The four cases for the proof of Lemma \[lem:discretecorrect\].[]{data-label="fig:nonLCcasesAppendix"}](discreteNonLCcases) *Case (a).* Path $\pi_\mathrm{i}$ ends at the parent of $g$ in $T'$. Path $\pi$ is the path in $T'$ between the first and last vertex of $\pi_\mathrm{i}$. Since $(\pi_\mathrm{i}, g)$ is the invalidating path, we know that $\max\{ \pi_\mathrm{i}, g \} < \max\{ \pi, g \}$ holds. This implies that $\pi_\mathrm{i} < \pi$ holds. In particular, this means that $\pi$, a path in $T$, is not strongly locally correct: a contradiction. *Case (b).* Path $\pi_\mathrm{i}$ ends at a non-selected candidate parent of $g$. Path $\pi_2$ ends at the parent of $g$ in $T'$ and path $\pi_3$ ends at the last vertex of $\pi_\mathrm{i}$. Let $nca$ denote the NCA of the endpoints of $\pi_2$ and $\pi_3$. The first vertex of $\pi_\mathrm{i}$ is $nca$ or one of its ancestors. Both path $\pi_2$ and $\pi_3$ start at $nca$. Let $\pi_1$ be the path from $\pi_\mathrm{i}(1)$ to $nca$. Since the endpoint of $\pi_2$ was chosen as parent over the endpoint of $\pi_3$, we know that $\pi_2 \leq \pi_3$ holds. Furthermore, since $(\pi_\mathrm{i}, g)$ is the invalidating path, we know that $\max\{ \pi_\mathrm{i}, g \} < \max\{ \pi_1, \pi_2, g \}$ holds. These two inequalities imply $\max\{ \pi_\mathrm{i}, g \} < \max\{ \pi_1, \pi_3, g \}$ holds. This in turn implies that $\pi_\mathrm{i} < \max\{ \pi_1, \pi_3 \}$ must hold. Since $(\pi_1, \pi_3)$ is a path in $T$ and the inequality implies that it is not strongly locally correct, we again have a contradiction. *Case (c).* Path $\pi_\mathrm{i}$ ends at a non-selected candidate parent of $g$. Path $\pi_2$ ends at the parent of $g$ in $T'$ and path $\pi_3$ ends at the last vertex of $\pi_\mathrm{i}$. Let $nca$ denote the NCA of the endpoints of $\pi_2$ and $\pi_3$. The first vertex of $\pi_\mathrm{i}$ is a descendant of $nca$. Path $\pi_2$ starts at $\pi_\mathrm{i}(1)$ and $\pi_3$ starts at $nca$. Let $\pi_1$ be the path from $nca$ to $\pi_\mathrm{i}(1)$. In this case, we must explicitly consider the possibility of two paths having equal values. Hence, we distinguish two subcases. *Case (c-1).* In the first subcase, we assume that the endpoint of $\pi_2$ was chosen as parent since its maximum value is strictly lower: $\max\{ \pi_1, \pi_2 \} < \pi_3$ holds. Since $(\pi_\mathrm{i}, g)$ is the invalidating path, we know that $\max\{ \pi_\mathrm{i}, g \} < \max\{ \pi_2, g \}$ holds. Since $\pi_2 \leq \max\{ \pi_1, \pi_2 \}$ always holds, we obtain that $\max\{ \pi_\mathrm{i}, g \} < \max\{ \pi_3, g \}$ must hold. This in turn implies that $\pi_\mathrm{i} < \pi_3$ holds. Similarly, since $\pi_1 \leq \max\{ \pi_1, \pi_2 \}$, we know that $\pi_1 < \pi_3$ must hold. Combining these last two inequalities yields $\max\{ \pi_1, \pi_\mathrm{i} \} < \pi_3$. Since $\pi_3$ is a path in $T$ and the inequality implies that it is not strongly locally correct, we again have a contradiction. (Note that with $\max\{ \pi_1, \pi_2 \} \leq \pi_3$, we can at best derive $\max\{ \pi_1, \pi_\mathrm{i} \} \leq \pi_3$ which is not strong enough to contradict the invariant on $T$.) *Case (c-2).* In the second subcase, we assume that the endpoint of $\pi_2$ was chosen as parent based on the preference order: the maximum values are equal, thus $\max\{ \pi_1, \pi_2 \} = \pi_3$ holds. We now subdivide $\pi_\mathrm{i}$ into two parts, $\pi_\mathrm{ia}$ and $\pi_\mathrm{ib}$. $\pi_\mathrm{ia}$ runs from the first vertex of $\pi_\mathrm{i}$ up to the first vertex along $\pi_\mathrm{i}$ that is an ancestor in $T'$ of candidate parent of $g$ that is used by $\pi_\mathrm{i}$. At the same point, we also split path $\pi_3$ into $\pi_\mathrm{3a}$ and $\pi_\mathrm{3b}$. We now obtain two more cases, $\pi_\mathrm{3a} < \max\{ \pi_1, \pi_\mathrm{ia}\}$ and $\pi_\mathrm{3a} \geq \max\{ \pi_1, \pi_\mathrm{ia}\}$. In the former case, we obtain that $\max\{\pi_\mathrm{3a}, \pi_\mathrm{ib}\} < \max\{ \pi_1, \pi_2\}$ holds and thus $(\pi_\mathrm{3a}, \pi_\mathrm{ib}, g)$ is also an invalidating path. Since this path starts at the NCA of $\pi_2$ and $\pi_3$, this is already covered by case (b). In the latter case, we have that either path $\pi_\mathrm{3a}$—which is in $T$—is not strongly locally correct (contradicting the induction hypothesis) or there is equality between the two paths $(\pi_1, \pi_\mathrm{ia})$ and $\pi_\mathrm{3a}$. In case of equality, we observe that $(\pi_1, \pi_\mathrm{ia})$ and $\pi_\mathrm{3a}$ arrive at their endpoint in the same order as $\pi_2$ and $\pi_3$ arrive at $g$. Thus $T$ does not adhere to the preference order to break ties. This contradicts the invariant. #### Execution time When a dead path $\pi_\mathrm{d}$ is removed, we may need to extend a list of incoming shortcuts at $\pi_\mathrm{d}(1)$, the node that remains in $T$. Let $k$ denote the number of nodes in $\pi_\mathrm{d}$. The lemma below relates the number of extended shortcuts to the size of $\pi_\mathrm{d}$. The main observation is that the path requiring extensions starts at $\pi_\mathrm{d}(1)$ and ends at either $G[i-1,j]$ or $G[i,j-1]$, since $G[i,j]$ has not yet received any shortcuts. \[lem:deadpath\] A dead path $\pi_\mathrm{d}$ with $k$ nodes can result in at most $2 \cdot k - 1$ extensions. ![(a) A dead path $\pi_\mathrm{d}$ and the corresponding path requiring extensions $\pi_\mathrm{e}$.(b) Endpoint of $\pi_\mathrm{e}$ has outdegree 1. None of its descendant has a shortcut to $\pi_\mathrm{e}(1)$.[]{data-label="fig:DeadPath"}](discreteDeadPath) Since $\pi_\mathrm{d}$ is a path with $k$ nodes, it spans at most $k$ columns and $k$ rows. When a dead path is removed, its endpoint is $G[i-1,j-1]$. Let $\pi_\mathrm{e}$ denote the path of $T$ that requires extensions. We know that both paths start at the same node: $\pi_\mathrm{d}(1) = \pi_\mathrm{e}(1)$. The endpoint of $\pi_\mathrm{e}$ is at either $G[i-1,j]$ or $G[i,j-1]$, since $G[i,j]$ has not yet received shortcuts when the dead path is removed. Also, note that if the endpoint of $\pi_\mathrm{e}$ is not the parent of $G[i,j]$ and has outdegree higher than 0, then it is a growth node and its descendants are also growth nodes. Hence, these descendants have parent that is a growth node and thus have shortcuts that need to be extended. Fig. \[fig:DeadPath\] illustrates these situations. Hence, we know that $\pi_\mathrm{e}$ spans either $k + 1$ columns and $k$ rows or vice versa. Therefore, the maximum number of nodes in $\pi_\mathrm{e}$ is $2 \cdot k$, since it must be monotonous. Since $\pi_\mathrm{e}(1)$ does not have a shortcut to itself, there are at most $2 \cdot k - 1$ incoming shortcuts from $\pi_\mathrm{e}$ at $\pi_\mathrm{d}(1)$. Hence, we can charge every extension to one of the $k-1$ dead nodes (all but $\pi_\mathrm{d}(1)$). A node gets at most 3 charges, since it is a (non-first) node of a dead path at most once. Because an extension can be done in constant time, the execution time of the algorithm is $O(m n)$. Note that shortcuts that originate from a living node with outdegree 1 could be removed instead of extended. We summarize the findings of this section in the following theorem. Algorithm \[alg:finddiscrete\] computes a locally correct discrete [Fréchet matching]{} of two polygonal curves $P$ and $Q$ with $m$ and $n$ edges in $O(m n)$ time. Conclusion {#sec:conclusion} ========== We set out to find “good” matchings between two curves. To this end we introduced the local correctness criterion for [Fréchet matching]{}s. We have proven that there always exists at least one locally correct [Fréchet matching]{} between any two polygonal curves. This proof resulted in an $O(N^3 \log N)$ algorithm, where $N$ is the total number of edges in the two curves. Furthermore, we considered computing a locally correct matching using the discrete [Fréchet distance]{}. By maintaining a tree with shortcuts to encode locally correct partial matchings, we have shown how to compute such a matching in $O(N^2)$ time. #### Future work Computing a locally correct discrete [Fréchet matching]{} takes $O(N^2)$ time, just like the dynamic program to compute only the discrete [Fréchet distance]{}. However, computing a locally correct continuous [Fréchet matching]{} takes $O(N^3 \log N)$ time, a linear factor more than computing the [Fréchet distance]{}. An interesting question is whether this gap in computation can be reduced as well. Furthermore, it would be interesting to investigate the benefit of local correctness for other matching-based similarity measures, such as the geodesic width [@Efrat2002]. [^1]: M. Buchin is supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 612.001.106. W. Meulemans and B. Speckmann are supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 639.022.707. A preliminary version of this paper will appear in Proc. 20th European Symposium on Algorithms (ESA 2012).

--- abstract: | In a previous work ([@Eb]), the author proposed a method employing contiguity relations to derive hypergeometric series in closed form. In [@Eb], this method was used to derive Gauss’s hypergeometric series $_2F_1$ possessing closed forms. Here, we consider the application of this method to Appell’s hypergeometetric series $F_1$ and derive several $F_1$ possessing closed forms. Moreover, analyzing these $F_1$, we obtain values of $_2F_1$ with no free parameters. Some of these results provide new examples of algebraic values of $_2F_1$. Key Words and Phrases: Gauss’s hypergeometric series, algebraic value, Appell’s hypergeometric series, hypergeometric identity. 2010 Mathematics Subject Classification Numbers: Primary 33C05 Secondary 13P15, 33C65, 33F10. author: - Akihito Ebisu title: 'Special values of Gauss’s hypergeometric series derived from Appell’s series $F_1$ with closed forms' --- Introduction ============ The study of special values of Gauss’s hypergeometric series $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{2}F_{1}\biggl(\genfrac..{0pt}{}{a,b}{c};x\biggr) \endgroup }:= \sum _{n=0} ^{\infty}\frac{(a,n)(b,n)}{(c,n)(1,n)}x^n, \label{hgs}\end{gathered}$$ where $({{\alpha}},n):={{\alpha}}({{\alpha}}+1)\cdots ({{\alpha}}+n-1)$, has a long history. The oldest and most well-known formula is $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{2}F_{1}\biggl(\genfrac..{0pt}{}{a,b}{c};1\biggr) \endgroup }= \frac{{{\Gamma}}(c){{\Gamma}}(c-a-b)}{{{\Gamma}}(c-a){{\Gamma}}(c-b)}, \label{gauss}\end{gathered}$$ where $\Re (c-a-b)>0$. The formula (\[gauss\]), due to Gauss, is called “Gauss’s summation formula”. Since this formula was derived by Gauss, many other identities for Gauss’s hypergeometric series have been obtained by many other people. For example, we have $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{2}F_{1}\biggl(\genfrac..{0pt}{}{2a,2b}{a+b+1/2};\frac{1}{2}\biggr) \endgroup }= \frac{\sqrt{\pi}\, {{\Gamma}}(a+b+1/2)}{{{\Gamma}}(a+1/2){{\Gamma}}(b+1/2)} \label{Kummer}\end{gathered}$$ (see formula (50) in Section 2.8 of [@Erd1]), $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{2}F_{1}\biggl(\genfrac..{0pt}{}{1/2,-a}{2a+5/2};\frac{1}{4}\biggr) \endgroup }= \frac{1}{3\cdot2^{2a}} \frac{\sqrt{\pi}\, {{\Gamma}}(2a+5/2)}{{{\Gamma}}(a+3/2)^2} \label{Gosper}\end{gathered}$$ (see formula (1/4.2) in [@Gos] and formula (1,3,3-1)(xvi) in [@Eb]). There are many other known identities for $_2F_1$ similar to the above, containing one or more free parameters. These identities can be found by demonstrating that the corresponding hypergeometric series possess closed forms. For instance, the formula (\[Gosper\]) is derived by showing that $F(a):={}_2F_{1}(1/2, -a; 2a+5/2; 1/4)$ has a closed form, that is, that $F(a)$ satisfies the closed-form relation $$\begin{gathered} \frac{F(a+1)}{F(a)}=\frac{(2a+5/2,2)}{2^2(a+3/2)^2}. \label{Gosper2}\end{gathered}$$ The relation (\[Gosper2\]) can be obtained by employing Gosper’s algorithm, the W-Z method, Zeilberger’s algorithm (see [@Ko] and [@PWZ]), and the method of contiguity relations, which was recently introduced in [@Eb]. Thus, using these methods, we are able to find numerous identities for $_2F_1$ with one or more free parameters. Most of the identities that can be derived with these methods are listed in [@Eb]. There are also many known identities for $_2F_1$ with no free parameters. The following are some examples: $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{2}F_{1}\biggl(\genfrac..{0pt}{}{1/4,1/2}{3/4};\frac{80}{81}\biggr) \endgroup }=\frac{9}{5} \label{jz2}\end{gathered}$$ (see formula (1.6) in [@JZ2]), $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{2}F_{1}\biggl(\genfrac..{0pt}{}{1/12,5/12}{1/2};\frac{1323}{1331}\biggr) \endgroup }= \frac{3\sqrt[4]{11}}{4} \label{bw}\end{gathered}$$ (see Theorem 3 in [@BW]). Unfortunately, it seems that such identities can not be obtained by direct application of the above methods. Indeed, if we could find (\[jz2\]) directly with one of the above methods, then there must exist $p, q, r\in {{\Bbb Q}}$ satisfying $$\begin{gathered} \frac{F(a+1)}{F(a)}\in {{\Bbb Q}}(a),\end{gathered}$$ where $$\begin{gathered} F(a):={ \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{2}F_{1}\biggl(\genfrac..{0pt}{}{pa+1/4, qa+1/2}{ra+3/4};\frac{80}{81}\biggr) \endgroup }.\end{gathered}$$ However, no such parameters $p,q,r$ have yet been identified. The same holds for other identities, including (\[bw\]). To obtain these formulae, other methods have been used, including methods employing modular forms and elliptic integrals (see [@Ar], [@BG], [@BW], [@JZ1], [@JZ2], [@JZ3]). As another approach, if we could find Appell’s hypergeometric series $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{}F_{1}\biggl(\genfrac..{0pt}{}{{{\alpha}}; \beta _1, \beta_2}{{{\gamma}}};x,y\biggr) \endgroup } :=\sum _{m=0} ^{\infty}\sum _{n=0} ^{\infty} \frac{({{\alpha}}, m+n)(\beta _1,m)(\beta _2,n)}{({{\gamma}},m+n)(1,m)(1,n)}x^my^n\end{gathered}$$ in closed forms, then it may be possible to obtain identities for $_2F_1$ with no free parameters by analyzing the corresponding closed-form relations for $F_1$. Below, we consider two examples. As the First example, we consider the following (see Example 1 in Section 3.1 for details). Using the method of contiguity relations, which is effective for deriving hypergeometric series in closed form, we find the closed-form relation $$\begin{gathered} \frac{F(a+1)}{F(a)}=\frac{3^{8}}{2^2\, 5^5}\, \frac{(2a+1/2, 2)}{(a+1/2)^2}, \label{ex1closed1}\end{gathered}$$ where $$\begin{gathered} F(a):= { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{}F_{1}\biggl(\genfrac..{0pt}{}{2a; a+1/2,4a-1}{2a+1/2};\frac{1}{81}, \frac{1}{6}\biggr) \endgroup }.\end{gathered}$$ From (\[ex1closed1\]), we obtain $$\begin{gathered} \frac{F(a+n)}{F(a)}=\frac{3^{8n}}{2^{2n}\, 5^{5n}}\, \frac{(2a+1/2, 2n)}{(a+1/2,n)^2}, \label{ex1closed2}\end{gathered}$$ and $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{}F_{1}\biggl(\genfrac..{0pt}{}{2a; a+1/2,4a-1}{2a+1/2};\frac{1}{81}, \frac{1}{6}\biggr) \endgroup } =F(a)= \frac{3^{8a}}{2^{2a}5^{5a}}\, \frac{\sqrt{\pi}\,{{\Gamma}}(2a+1/2)}{{{\Gamma}}(a+1/2)^2}. \label{ex1gamma}\end{gathered}$$ Then, from the relation $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{}F_{1}\biggl(\genfrac..{0pt}{}{{{\alpha}}; \beta _1, 0}{{{\gamma}}};x,y\biggr) \endgroup } = { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{2}F_{1}\biggl(\genfrac..{0pt}{}{{{\alpha}}; \beta _1}{{{\gamma}}};x\biggr) \endgroup }, \label{reduct}\end{gathered}$$ which follows from the definition of Appell’s hypergeometric series $F_1$, we have $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{2}F_{1}\biggl(\genfrac..{0pt}{}{1/2,3/4}{1};\frac {1}{81}\biggr) \endgroup }= \frac{9}{100} \frac{\sqrt{2}\, 5^{3/4}\,{{\Gamma}}(1/4)^2}{\pi ^{3/2}}, \label{jz3}\end{gathered}$$ by substituting $a=1/4$ into (\[ex1gamma\]). As the second example, we consider the following (see Example 2 in Section 3.2 for details). From the method of contiguity relations, we are also able to derive the closed-form relation $$\begin{gathered} \frac{F(a+1)}{F(a)}=\frac{3^4}{5^4} \label{ex2closed1}\end{gathered}$$ and, from this, $$\begin{gathered} {F(a)}=\frac{5^{4n}}{3^{4n}}F(a+n), \label{ex2closed2}\end{gathered}$$ where $$\begin{gathered} F(a):= { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{}F_{1}\biggl(\genfrac..{0pt}{}{a;2a,1-4a}{a+1/2};\frac{80}{81}, \frac{16}{15}\biggr) \endgroup }. \label{ex2}\end{gathered}$$ It is known that, in general, $F_1({{\alpha}}; \beta _1, \beta _2; {{\gamma}}; x, y)$ is absolutely convergent when $|x|<1$ and $|y|<1$, and divergent when $|x|>1$ or $|y|>1$ (for example, see Section 9.1 in [@Ba]). Therefore, $F(a)$ is not meaningful for a parameter $a$ with unrestricted value. However, $F(a)$ is meaningful if we restrict $a$ to values satisfying $a=1/4+n$, where $n$ is any non-negative integer. Thus, the relation (\[ex2closed2\]) with $a=1/4$ is meaningful. Then, investigating the asymptotic behavior of the right-hand side of this relation by taking the limit $n\rightarrow +\infty$, we are able to deduce (\[jz2\]). In this way, analyzing closed-form relations for $F_1$, we can derive values of $_2F_1$ with no free parameters. In this article, making use of the method of contiguity relations, we derive several $F_1$ in closed forms. These are listed in Table \[closed\_form\]. Moreover, in the cases that these $F_1$ are convergent, we evaluate them with (\[ex1gamma\]). These hypergeometric identities for $F_1$ are tabulated in Table \[identities\]. In addition, analyzing the closed-form relations for $F_1$ listed in Table \[closed\_form\], we derive values of $_2F_1$ with no free parameters. These values are listed in Table \[value1\]. In Tables \[value2\] and \[value3\], we present several complicated identities obtained by applying algebraic transformations to the identities listed in Table \[value1\]. As seen from these, we are able to obtain several new identities using our approach. In particular, (B$''$.3), (C$''$.1), (C$''$.4) and (C$'''$.1) provide new examples of algebraic values of Gauss’s hypergeometric series. However, it seems that we are not able to derive the beautiful formula (\[bw\]) obtained by Beukers and Wolfart with our approach. In [@Si], Siegel posed the problem of determining the nature of the following set: $$\begin{gathered} E(a,b,c):=\{x\in \bar{{{\Bbb Q}}};\ {}_2F_1(a,b;c;x)\in \bar{{{\Bbb Q}}} \}.\end{gathered}$$ This set is a so-called “exceptional set”. Since Wolfart’s celebrated work [@Wo] investigating $E(a,b,c)$, much progress has been made in the study of exceptional sets. In those studies, Gauss’s hypergeometric series $_2F_1(a,b;c;x)$ corresponding to the arithmetic triangular group and satisfying the relations $$\begin{gathered} c<1,\ 0<a<c,\ 0<b<c,\ |1-c|+|a-b|+|c-a-b|<1,\\ \frac{1}{|1-c|}, \frac{1}{|a-b|}, \frac{1}{|c-a-b|}\in {{\Bbb Z}}_{>0}\end{gathered}$$ is a focus of investigation. The formulae [(B$''$.3), (C$''$.1), (C$''$.4)]{} and [(C$'''$.1)]{} provide new examples of elements of these exceptional sets. The formulae [(B$''$.3), (C$''$.1), (C$''$.4)]{} and [(C$'''$.1)]{} arise from Gauss’s hypergeometric equations corresponding to the $(3,6,6)$, $(2,5,10)$, $(2,5,10)$ and $(2,3,10)$ triangular groups, respectively. The method of contiguity relations ================================== In this section, we review the method of contiguity relations, which was introduced in [@Eb]. Using this method, we obtain the examples of Appell’s hypergeometric series $F_1$ possessing closed forms listed in Table \[closed\_form\]. First, for simplicity, we write the parameters $({{\alpha}}; \beta _1, \beta _2; {{\gamma}})$ as ${\bm {{\alpha}}}$. Then, Appell’s hypergeometric series $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{}F_{1}\biggl(\genfrac..{0pt}{}{{{\alpha}}; \beta _1, \beta _2}{{{\gamma}}};x,y\biggr) \endgroup }\end{gathered}$$ is expressed as $F_1({\bm {{\alpha}}}; x,y)$. We also define $$\begin{gathered} {\bm e_{10}}:=(1;1,0;1),\ {\bm e_{01}}:=(1;0,1;1),\ {\bm k}:=(k; l _1, l _2; m)\in {{\Bbb Z}}^4.\end{gathered}$$ Now, we review the method of contiguity relations. It is known that for a given quadruple of integers ${\bm k} \in{{\Bbb Z}}^4$, there exists a unique triple of rational functions $(Q_{10}, Q_{01}, Q_{00}) \in ({{\Bbb Q}}({{\alpha}}; \beta _1, \beta _2; {{\gamma}}, x, y))^3$ satisfying $$\begin{aligned} \begin{split} F_1({\bm {{\alpha}}}+{\bm k};x,y) =Q_{10}\,F_1({\bm {{\alpha}}}+{\bm e_{10}};x,y) +Q_{01}\,F_1({\bm {{\alpha}}}+{\bm e_{01}};x,y) +Q_{00}\,F_1({\bm {{\alpha}}};x,y). \end{split} \label{four_term}\end{aligned}$$ The relation (\[four\_term\]) is called the “contiguity relation” for $F_1$ (or the “four-term relation” for $F_1$). Note that it is possible to compute $Q_{10}$, $Q_{01}$ and $Q_{00}$ exactly by using the method introduced in [@Ta1] (see also [@Ta2]). Next, we define $$\begin{gathered} Q_{ij} ^{(n)}:=Q_{ij} |_{{\bm {{\alpha}}} \rightarrow {\bm {{\alpha}}}+n{\bm k}},\end{gathered}$$ where each $Q_{ij} ^{(n)}$ is a rational expression in $n$ whose coefficients belong to ${{\Bbb Z}}[{{\alpha}},$ $\beta _1$,$ \beta _2$, ${{\gamma}}$, $x$, $y]$. Now, let the sextuple $({{\alpha}}; \beta _1, \beta _2; {{\gamma}}, x, y)$ satisfy $$\begin{gathered} \begin{cases} Q_{10}^{(n)}\equiv 0,\\ Q_{01}^{(n)}\equiv 0\quad \label{case} \end{cases} {\text {for every integer $n$}}.\end{gathered}$$ Such a sextuple can be found by solving the polynomial system in $({{\alpha}}, \beta _1, \beta _2, {{\gamma}}, x, y)$ arising from (\[case\]). Then, from (\[four\_term\]), we find that the relation $$\begin{aligned} F_1({\bm {{\alpha}}}+(n+1){\bm k};x,y) =Q_{00} ^{(n)}\,F_1({\bm {{\alpha}}}+n{\bm k};x,y) \label{two_term_f1}\end{aligned}$$ holds for such a sextuple. The relation (\[two\_term\_f1\]) implies that $F(n):=F_1({\bm {{\alpha}}}+n{\bm k};x,y)$ has a closed form. Thus, we are able to find $F_1$ in closed form. The above method is called the “method of contiguity relations”. As an example, we consider the case ${\bm k}=(2,1,4,2)$. Applying the method of contiguity relations to this case, we find that $$\begin{gathered} ({{\alpha}}, \beta _1, \beta _2, {{\gamma}}, x, y) =\left(2a, a+\frac{1}{2}, 4a-1, 2a+\frac{1}{2}, \frac{1}{81}, \frac{1}{6}\right)\end{gathered}$$ satisfies (\[case\]). For this sextuple, $Q _{00}$ becomes $$\begin{gathered} \frac{3^{8}}{2^2\, 5^5}\, \frac{(2a+1/2, 2)}{(a+1/2)^2}.\end{gathered}$$ Hence, we have the following relation: $$\begin{aligned} \dfrac{F_1({\bm {{\alpha}}}+(n+1){\bm k};x,y)}{F_1({\bm {{\alpha}}}+n{\bm k};x,y)} &= \dfrac{ { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{}F_{1}\biggl(\genfrac..{0pt}{}{2a+2n+2; a+n+3/2,4a+4n+3}{2a+2n+5/2};\frac{1}{81},\frac{1}{6}\biggr) \endgroup } } { { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{}F_{1}\biggl(\genfrac..{0pt}{}{2a+2n; a+n+1/2,4a+4n-1}{2a+2n+1/2};\frac{1}{81},\frac{1}{6}\biggr) \endgroup } } \\ &= \frac{3^{8}}{2^2\, 5^5}\, \frac{(2a+2n+1/2, 2)}{(a+n+1/2)^2}=Q _{00} ^{(n)}.\end{aligned}$$ Thus, with $$\begin{gathered} F(a):= { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{}F_{1}\biggl(\genfrac..{0pt}{}{2a; a+1/2,4a-1}{2a+1/2};\frac{1}{81}, \frac{1}{6}\biggr) \endgroup },\end{gathered}$$ we see that $F(a)$ has a closed form, and $F(a)$ satisfies the closed-form relation $$\begin{gathered} \frac{F(a+1)}{F(a)}= \frac{3^{8}}{2^2\, 5^5}\, \frac{(2a+1/2, 2)}{(a+1/2)^2}.\end{gathered}$$ As seen in the above example, if $\bm k \in {{\Bbb Z}}^4$ is given, then we can obtain $F_1$ in closed form by using the method of contiguity relations. Such examples are tabulated in Table \[closed\_form\]. Special values of $_2F_1$ derived from $F_1$ with closed forms ============================================================== In the previous section, we presented examples of $F_1$ possessing closed forms. These are listed in Table \[closed\_form\]. In this section, from among these $F_1$, we evaluate those that are convergent. The derived hypergeometric identities for $F_1$ are listed in Table \[identities\]. In addition, analyzing the closed-form relations for $F_1$ listed in Table \[closed\_form\], we obtain values of $_2F_1$ with no free parameters. These values are presented in Table \[value1\]. Hypergeometric identities for $F_1$ obtained from Table \[closed\_form\] ------------------------------------------------------------------------ In this subsection, we derive hypergeometric identities from the closed-form relations given in Table \[closed\_form\]. As seen in Table \[closed\_form\], setting ${\bm k}=(2,1,4,2)$, we find that $$\begin{gathered} F(a):= { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{}F_{1}\biggl(\genfrac..{0pt}{}{2a;a+1/2,4a-1}{2a+1/2};\frac{1}{81}, \frac{1}{6}\biggr) \endgroup }\end{gathered}$$ possesses a closed form, and it satisfies the closed-form relation $$\begin{gathered} \frac{F(a+1)}{F(a)}= \frac{3^{8}}{2^2\, 5^5}\, \frac{(2a+1/2, 2)}{(a+1/2)^2}. \label{gamma_ex}\end{gathered}$$ This implies $$\begin{gathered} \frac{F(a+n)}{F(a)}= \frac{3^{8n}}{2^{2n}\, 5^{5n}}\, \frac{(2a+1/2, 2n)}{(a+1/2,n)^2}. \label{gamma_ex1}\end{gathered}$$ Then, noting that $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{}F_{1}\biggl(\genfrac..{0pt}{}{0; \beta _1, \beta _2}{{{\gamma}}};x,y\biggr) \endgroup }=1,\end{gathered}$$ and, substituting $a=0$ into (\[gamma\_ex1\]), we obtain $$\begin{gathered} F(n)=G(n) \label{gamma_ex2}\end{gathered}$$ for any integer $n$, where $$\begin{gathered} G(n) := \frac{3^{8n}}{2^{2n}\, 5^{5n}}\, \frac{(1/2, 2n)}{(1/2,n)^2} = \frac{3^{8n}}{2^{2n}\, 5^{5n}}\, \frac{{{\Gamma}}(1/2){{\Gamma}}(2n+1/2)}{{{\Gamma}}(n+1/2)^2}.\end{gathered}$$ Now, we show that the identity (\[gamma\_ex2\]) holds for any complex number $a$, except for $a=-1/4, -3/4, -5/4, \ldots$. For this purpose, we use the following lemma proved by Carlson (see Section 5.3 in [@Ba]). (Carlson’s theorem) We assume that $f(a)$ and $g(a)$ are regular and of the form $O(e^{k|a|})$, where $k<\pi$, for $\Re (a) \geq 0$, and that $f(a)=g(a)$ for $a=0,1,2,\ldots$. Then, we have $f(a)=g(a)$ on $\{a\mid \,\Re (a) \geq 0\}$. Obviously, both sides of (\[gamma\_ex2\]) are regular for $\Re (a) \geq 0$. Also, we have the following: $$\begin{aligned} &|F(a)|\\ &\leq \sum _{m,n=0} ^{\infty} \frac{|2a||2a+1|\cdots |2a+m+n-1|(|a|+1/2,m)(4|a|+1,n)} {|2a+1/2||2a+3/2|\cdots|2a-1/2+m+n|(1,m)(1,n)} \left(\frac{1}{81}\right)^m \left(\frac{1}{6}\right)^n\\ &\leq \sum _{m,n=0} ^{\infty} \frac{(|a|+1/2,m)(4|a|+1,n)}{(1,m)(1,n)} \left(\frac{1}{81}\right)^m \left(\frac{1}{6}\right)^n\\ &= \sum _{m=0} ^{\infty} \frac{(|a|+1/2,m)}{(1,m)} \left(\frac{1}{81}\right)^m \sum _{n=0} ^{\infty} \frac{(4|a|+1,n)}{(1,n)} \left(\frac{1}{6}\right)^n\\ &= \left(1-\frac{1}{81}\right)^{-|a|-1/2} \left(1-\frac{1}{6}\right)^{-4|a|-1}\\ &= O\left(\exp\left(|a|\log\left(\frac{3^8}{5^5}\right)\right)\right).\end{aligned}$$ Note that we can also compute the asymptotic behavior of $F(a)$ exactly by making use of Laplace’s method (also, see Section 3 in [@Iw]). In any case, we see that $|F(a)|$ is of the form $O(e^{|a|})$, whereas it is easily demonstrated using Stirling’s formula that $|G(a)|$ is of the form $O(e^{|a|})$. Of course, we know that from (\[gamma\_ex2\]), $$\begin{gathered} F(a)=G(a) \label{gamma_ex3}\end{gathered}$$ holds for any non-negative integer $a$. Therefore, it follows from Carlson’s theorem that the identity (\[gamma\_ex3\]) is valid for $\Re(a) \geq 0$. Also, by analytic continuation, we find that (\[gamma\_ex3\]) holds for any complex number $a$, except for $a=-1/4, -3/4, -5/4, \ldots$. Thus, we derive (\[ex1gamma\]), which appears as (A$'$.1) in Table \[identities\]. In this way, it is possible to obtain hypergeometric identities for $F_1$ from the closed-form relations in Table \[closed\_form\]. These are listed in Table \[identities\]. Special values of $_2F_1$ derived from closed-form relations for $F_1$ ---------------------------------------------------------------------- In this subsection, we derive identities for $_2F_1$ with no free parameters, using the closed-form relations listed in Table \[closed\_form\] and the hypergeometric identities for $F_1$ listed in Table \[identities\]. For example, let us substitute $a=1/4$ into (A$'$.1). Then, recalling the reduction formula (\[reduct\]), we obtain (\[jz3\]). This appears as formula (A$''$.1) in Table \[value1\]. In a similar way, reducing the hypergeometric identities for $F_1$ appearing in Table \[identities\] to identities for $_2F_1$, we derive (A$''$.2), (B$''$.1), (B$''$.4), (C$''$.1), (C$''$.2), (C$''$.3), (D$''$.1), (D$''$.3) and (E$''$.1). We now derive the remaining identities that can be obtained from the relations and identities given in Tables \[closed\_form\] and \[identities\]. From Table \[closed\_form\], choosing ${\bm k}=(1,2,-4,1)$, we find that $$\begin{gathered} F(a):= { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{}F_{1}\biggl(\genfrac..{0pt}{}{a;2a,1-4a}{a+1/2};\frac{80}{81}, \frac{16}{15}\biggr) \endgroup }\end{gathered}$$ has a closed form, and it satisfies the following closed-form relation: $$\begin{gathered} \frac{F(a+1)}{F(a)}=\frac{3^4}{5^4}. \label{ex2_closed1}\end{gathered}$$ This implies that $$\begin{gathered} {F(a)}=\frac{5^{4n}}{3^{4n}}{F(a+n)} \label{ex2_closed2}\end{gathered}$$ holds for any non-negative integer $n$. Although (\[ex2\_closed2\]) (and (\[ex2\_closed1\])) is valid by virtue of analytic continuation, $F(a)$ and $F(a+n)$, which are regarded as infinite double series expressions, are meaningless. For this reason, we carry out a reduction of each of these to a finite sum of a single series expression that is meaningful. This is done, for example, by substituting $a=1/4$ into (\[ex2\_closed2\]). Then, it becomes $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{2}F_{1}\biggl(\genfrac..{0pt}{}{1/4, 1/2}{3/4};\frac{80}{81}\biggr) \endgroup } = \frac{5^{4n}}{3^{4n}} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{}F_{1}\biggl(\genfrac..{0pt}{}{1/4+n; 1/2+2n, -4n}{3/4+n};\frac{80}{81}, \frac{16}{15}\biggr) \endgroup }. \label{ex2_eq}\end{gathered}$$ Next, we evaluate the left-hand side of (\[ex2\_eq\]) by determining the asymptotic behavior of the right-hand side of (\[ex2\_eq\]) in the limit $n\rightarrow +\infty$. Because $F_1({{\alpha}}; \beta _1, \beta _2; {{\gamma}}; x,y)$ has the integral representation $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{}F_{1}\biggl(\genfrac..{0pt}{}{{{\alpha}}; \beta _1, \beta _2}{{{\gamma}}};x,y\biggr) \endgroup } =\frac{{{\Gamma}}({{\gamma}})}{{{\Gamma}}({{\alpha}}){{\Gamma}}({{\gamma}}-{{\alpha}})} \int _{0} ^{1} t^{{{\alpha}}-1}(1-t)^{{{\gamma}}-{{\alpha}}-1}(1-xt)^{-\beta _1}(1-yt)^{-\beta _2}dt,\end{gathered}$$ the right-hand side of (\[ex2\_eq\]) can be expressed as $A\cdot B$, where $$\begin{aligned} A:=\frac{5^{4n}}{3^{4n}} \frac{{{\Gamma}}(3/4+n)}{{{\Gamma}}(1/4+n){{\Gamma}}(1/2)},\ B:=\int _{0} ^{1} g(t)e ^{nh(t)}dt,\end{aligned}$$ and, here, $$\begin{gathered} g(t):=t^{-3/4}(1-t)^{-1/2}\left(1-\frac{80}{81}t\right)^{-1/2},\ h(t):=\log\left( t\left(1-\frac{80}{81}t\right)^{-2}\left(1-\frac{16}{15}t\right)^4 \right).\end{gathered}$$ The function $h(t)$ is plotted in Figure \[fig1\]. ![Graph of $h(t)$.[]{data-label="fig1"}](graph201606.eps){width="80mm"} From the relation $$\begin{gathered} h'(t)={\frac {15(256\,{t}^{2}-352\,t+81)}{t \left( 15-16\,t \right) \left( 81-80\,t \right) }},\end{gathered}$$ we find that $h'(t)$ becomes zero at $$\begin{gathered} t_0:=\frac{11}{16}-\frac{\sqrt{10}}{8}.\end{gathered}$$ We also obtain the relations $$\begin{gathered} h(t_0)=h(1)=\log\left(\frac{3^4}{5^4}\right).\end{gathered}$$ From Figure \[fig1\], it can be seen that the major contribution to the value of $B$ arises from the neighborhoods of the points $t=t_0, 1$. So, using Laplace’s method (see Section 2.4 in [@Erd2]), we find that $B$ takes the form $$\begin{gathered} B\sim \frac{9}{5}\left(\frac{3^4}{5^4}\right)^{n}\sqrt{\frac{\pi}{n}} \label{ex2_asymb}\end{gathered}$$ in the limit $n \rightarrow +\infty$. We can also compute the asymptotic behavior of $A$ using Stirling’s formula; we find that $$\begin{gathered} A\sim \left(\frac{5^4}{3^4}\right)^{n}\sqrt{\frac{n}{\pi}} \label{ex2_asyma}\end{gathered}$$ in the limit $n \rightarrow +\infty$. The formulae (\[ex2\_asymb\]) and (\[ex2\_asyma\]) yield $$\begin{gathered} \lim _{n\rightarrow +\infty}A\cdot B=\frac{9}{5}.\end{gathered}$$ Thus, we obtain (\[jz2\]). This appears as (A$''$.3) in Table \[value1\]. Similarly, some of the remaining identities can be obtained. However, among the relations given in Table \[closed\_form\], there are some cases in which it it is difficult to obtain values of $_2F_1$ by direct application of Laplace’s method. For such cases, we must use connection formulae for $_2F_1$. For example, it is apparently difficult to compute the asymptotic behavior of (B.3) with $a=1/3+n$ in the limit $n\rightarrow +\infty$, by direct use of Laplace’s method. For this reason, we use a connection formula for $_2F_1$ to obtain (B$''$.3) in Table \[value1\]. As seen in the beginning of this subsection, we already know (B$''$.1) and (B$''$.4). Also, the following is a known connection formula for $_2F_1$: $$\begin{gathered} u_1=\frac{{{\Gamma}}(c){{\Gamma}}(c-a-b)}{{{\Gamma}}(c-a){{\Gamma}}(c-b)}u_2 +\frac{{{\Gamma}}(c){{\Gamma}}(a+b-c)}{{{\Gamma}}(a){{\Gamma}}(b)}u_6, \label{connection}\end{gathered}$$ where $$\begin{aligned} u_1&:={ \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{2}F_{1}\biggl(\genfrac..{0pt}{}{a,b}{c};x\biggr) \endgroup },\\ u_2&:={ \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{2}F_{1}\biggl(\genfrac..{0pt}{}{a,b}{a+b+1-c};1-x\biggr) \endgroup },\\ u_6&:=(1-x)^{c-a-b}{ \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{2}F_{1}\biggl(\genfrac..{0pt}{}{c-a,c-b}{c+1-a-b};1-x\biggr) \endgroup }\end{aligned}$$ (see formula (33) in Section 2.9 in [@Erd1]). Then, substituting $(a,b,c,x)=(1/3$, $2/3,$ 7/6$, 5/32)$ into (\[connection\]), we deduce (B$''$.3). In this way, all the remaining identities are derived. All of the identities we have been able to derive are presented in Table \[value1\]. There, “type” refers to the type of Schwarz triangle for the Schwarz map of the corresponding hypergeometric equation. Explicitly, for a given $_2F_1(a,b;c;x)$, “type” is given by $(1/|1-c|, 1/|c-a-b|, 1/|a-b|)$. Algebraic transformations of identities in Table \[value1\] =========================================================== In this section, applying algebraic transformations to the identities in Table \[value1\], we have some complicated identities for $_2F_1$. These identities are tabulated in Tables \[value2\] and \[value3\]. As an example, here we derive (A$'''$.1) in Table \[value2\]. As seen in Table \[value1\], (A$''$.1) is known. Then, applying Goursat’s algebraic transformation $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{2}F_{1}\biggl(\genfrac..{0pt}{}{a,b}{2b};x\biggr) \endgroup } = (1-x)^{b-a}\left(1-\frac{x}{2}\right)^{a-2b} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{2}F_{1}\biggl(\genfrac..{0pt}{}{b-a/2,b+1/2-a/2}{b+1/2};\left(\frac{x}{2-x}\right)^2\biggr) \endgroup }\end{gathered}$$ (see formula (45) in [@Gour]) with $(a,b)=(3/4,1/2)$ to the left-hand side of (A$''$.1), (A$'''$.1) is easily deduced. In a similar way, we derive the following: - (A$'''$.2) from (A$''$.2) using Goursat’s algebraic transformation (51) in [@Gour], - (A$'''$.3) from (A$''$.3) using Goursat’s algebraic transformation (51) in [@Gour], - (B$'''$.1) from (B$''$.1) using Goursat’s algebraic transformation (41) in [@Gour], - (B$'''$.2) from (B$''$.2) using Goursat’s algebraic transformation (38) in [@Gour], - (B$'''$.3) using a connection formula for $_2F_1$ and (B$'''$.1) and (B$'''$.2), - (B$'''$.4) using a connection formula for $_2F_1$ and (B$'''$.1) and (B$'''$.2), - (C$'''$.1) from (C$''$.1) using Goursat’s algebraic transformation (121) in [@Gour], - (C$'''$.2) from (C$''$.2) using Goursat’s algebraic transformation (121) in [@Gour], - (C$'''$.3) using a connection formula for $_2F_1$ and (C$'''$.1) and (C$'''$.2), - (C$'''$.4) using a connection formula for $_2F_1$ and (C$'''$.1) and (C$'''$.2), - (D$'''$.1) from (D$''$.1) using Goursat’s algebraic transformation (50) in [@Gour], - (D$'''$.2) from (D$''$.2) using Goursat’s algebraic transformation (45) in [@Gour], - (D$'''$.3) from (D$''$.3) using Goursat’s algebraic transformation (45) in [@Gour], - (E$'''$.1) from (E$''$.1) using Goursat’s algebraic transformation (45) in [@Gour]. The identities in Table \[value3\] are derived as follows: - (B$''''$.1) from (B$'''$.1) using Goursat’s algebraic transformation (50) in [@Gour], - (B$''''$.2) from (B$'''$.2) using Goursat’s algebraic transformation (50) in [@Gour], - (B$''''$.3) from (B$'''$.3) using Goursat’s algebraic transformation (45) in [@Gour], - (B$''''$.4) from (B$'''$.4) using Goursat’s algebraic transformation (45) in [@Gour]. Concluding remarks ================== Applying the method of contiguity relations to Appell’s hypergeometric series $F_1$, we have obtained several identities for $_2F_1$ with no free parameters. In addition to the identities derived above, there are a number of cases in which the same method allows us to conjecture values of $_2F_1$ with no free parameters, as we now discuss. For example, for ${\bm k}=(-2, 3, 0, 1)$, we have the closed-form relation $$\begin{gathered} \frac{F(a+1)}{F(a)}= -\frac{5}{3^3}\frac{(a+1/2)(a+3/2)}{(a+5/6)(a+7/6)}, \label{conj1}\end{gathered}$$ where $$\begin{gathered} F(a):={ \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{}F_{1}\biggl(\genfrac..{0pt}{}{-2a; 3a+1. 1/2}{a+3/2};\frac{16}{25}, \frac{4}{5}\biggr) \endgroup }.\end{gathered}$$ From this relation, we can conjecture $$\begin{aligned} \begin{split} F(a)&={ \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{}F_{1}\biggl(\genfrac..{0pt}{}{-2a; 3a+1. 1/2}{a+3/2};\frac{16}{25}, \frac{4}{5}\biggr) \endgroup }\\ &=\left(\frac{5}{3^3}\right)^{a} \frac{\cos (\pi a)\, {{\Gamma}}(5/6){{\Gamma}}(7/6){{\Gamma}}(a+1/2){{\Gamma}}(a+3/2)} {{{\Gamma}}(1/2){{\Gamma}}(3/2){{\Gamma}}(a+5/6){{\Gamma}}(a+7/6)}\\ &=\frac{2}{3}\left(\frac{5}{3^3}\right)^{a} \frac{\cos(\pi a)\,{{\Gamma}}(a+1/2){{\Gamma}}(a+3/2)}{{{\Gamma}}(a+5/6){{\Gamma}}(a+7/6)}. \end{split} \label{conj2}\end{aligned}$$ Unfortunately, however, it seems difficult to compute the asymptotic behavior of $F(a)$ in the limit $|a|\rightarrow +\infty$ from a direct application of Laplace’s method. For this reason, we have not been able to prove (\[conj2\]). However, with numerical calculations, we have obtained results consistent with this identity. If indeed this identity does hold, then we have $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{2}F_{1}\biggl(\genfrac..{0pt}{}{1/2, 2/3}{7/6};\frac{4}{5}\biggr) \endgroup } = \frac{1}{40}\frac{\sqrt{3}\, 5^{2/3}\, {{\Gamma}}(1/3)^6}{\pi^3} \label{conj3}\end{gathered}$$ by substituting $a=-1/3$ into (\[conj2\]). Moreover, using the connection formula (\[connection\]), (E$''$.1) and (\[conj3\]), we obtain $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{2}F_{1}\biggl(\genfrac..{0pt}{}{1/3, 1/2}{5/6};\frac{4}{5}\biggr) \endgroup }=\frac{3}{\sqrt{5}}. \label{conj4}\end{gathered}$$ Although this relation follows from a conjecture, if indeed it does hold, it provides a new example of an algebraic value of $_2F_1$. In addition, by applying algebraic transformations to (\[conj3\]) and (\[conj4\]), we can similarly conjecture the relations $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{2}F_{1}\biggl(\genfrac..{0pt}{}{1/3,5/6}{7/6};\frac{80}{81}\biggr) \endgroup }=\frac{3}{40}\, \frac{3^{5/6}\, {{\Gamma}}(1/3)^6}{\pi ^3} \label{conj5}\end{gathered}$$ and $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{2}F_{1}\biggl(\genfrac..{0pt}{}{1/6,2/3}{5/6};\frac{80}{81}\biggr) \endgroup }=\frac{3}{5}\, 3^{2/3}\, \sqrt[6]{5}, \label{conj6}\end{gathered}$$ respectively. We can also obtain non-trivial algebraic values of $F_1$ from our approach. For instance, substituting $a=-2/3$ into (B$'$.4), we have $$\begin{gathered} { \begingroup \begingroup\lccode`~=`, \lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000 {}_{}F_{1}\biggl(\genfrac..{0pt}{}{1/2; 2/3, 1/2}{11/6};\frac{27}{32}, \frac{5}{6}\biggr) \endgroup } = \frac{2}{3}\,\sqrt{2}\sqrt{3}.\end{gathered}$$ We thus conclude that for the purpose of finding algebraic values of Gauss’s hypergeometric series $_2F_1$ and Appell’s hypergeometric series $F_1$, it is worthwhile studying $F_1$ possessing closed forms. Acknowledgement {#acknowledgement .unnumbered} =============== The author would like to thank Professor Katsunori Iwasaki for many valuable comments. The author is also grateful to Professor Frits Beukers for his encouragement. This work is supported by a Grant-in-Aid for JSPS Fellows, JSPS No. 15J00201. [KR]{} N.Archinard, [*Exceptional sets of hypergeometric series*]{}, J. Number Theory [**101**]{}(2003), no. [**2**]{}, 244–269. 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Stanton*]{}, XEROX Palo Alto Research Center, 21 December 1977. É.Goursat, [*Sur l’équation différentielle linéaire, qui admet pour intégrale la série hypergéométrique*]{}, Ann. Sci. École Norm. Sup. ([**[2]{}**]{}) [**[10]{}**]{}(1881), 3–142. I.Gessel and D.Stanton, [*Strange evaluations of hypergeometric series*]{}, SIAM J. Math. Anal., [**[13]{}**]{}(1982), no. [**[2]{}**]{}, 295–308. K.Iwasaki, [*On Some Hypergeometric Summations*]{}, e-Print arXiv:1408.5658. G.S.Joyce, I.J.Zucker, [*Special values of the hypergeometric series*]{}, Math. Proc. Cambridge Philos. Soc. [**109**]{}(1991), no. [**2**]{}, 257–261. G.S.Joyce, I.J.Zucker, [*Special values of the hypergeometric series II*]{}, Math. Proc. Cambridge Philos. Soc. [**131**]{}(2001), no. [**2**]{}, 309–319. G.S.Joyce, I.J.Zucker, [*Special values of the hypergeometric series III*]{}, Math. Proc. Cambridge Philos. Soc. [**133**]{}(2002), no. [**2**]{}, 213–222. P.W.Karlsson, [*On two hypergeometric summation formulas conjectured by Gosper*]{}, Simon Stevin [**[60]{}**]{} (1986), no. 4, 329–337. T.Kimura, [*Hypergeometric functions of two variables*]{}, Seminar Note in Math. Univ. of Tokyo, 1973. W.Koepf, [*Hypergeometric summation —An algorithmic approach to summation and special function identities, Second edition* ]{}, Universitext, Springer(2014). M.Petkovšek, H.Wilf and D.Zeilberger, [*A=B*]{}, A.K.Peters, Wellesley(1996). C.L. Siegel, [*[Ü]{}eber einige Anwendungen diophantischer Approximationen*]{}, in: Ges. Werke, Bd 1, Springer, Berlin, Heidelberg, New York, 1966, 209–266. N.Takayama, [*Gröbner basis and the problem of contiguous relations*]{}, Japan J. Appl. Math. [**6**]{}(1989), no. [**1**]{}, 147–160. N.Takayama, [*Gröbner basis for rings of differential operators and applications*]{}, in [*Gröbner bases: Statistics and software systems*]{}, Springer(2013), 279–344. R.Vidunas, [*Expressions for values of the gamma function*]{}, Kyushu J. Math. [**59**]{}(2005), no. [**2**]{}, 267–283. J.Wolfart, [*Werte hypergeometrischer Funktionen*]{}, Invent. Math. [**92**]{}(1988), no. [**1**]{}, 187–216. Akihito Ebisu\ Department of Mathematics\ Hokkaido University\ Kita 10, Nishi 8, Kita-ku, Sapporo, 060-0810\ Japan\ [email protected]

--- abstract: 'The unsupervised text clustering is one of the major tasks in natural language processing (NLP) and remains a difficult and complex problem. Conventional generally treat this task using separated steps, including text representation learning and clustering the representations. As an improvement, neural methods have also been introduced for continuous representation learning to address the sparsity problem. However, the multi-step process still deviates from the unified optimization target. Especially the second step of cluster is generally performed with conventional methods such as k-Means. We propose a pure neural framework for text clustering in an end-to-end manner. It jointly learns the text representation and the clustering model. Our model works well when the context can be obtained, which is nearly always the case in the field of NLP. We have our method on two widely used benchmarks: IMDB movie reviews for sentiment classification and $20$-Newsgroup for topic categorization. Despite its simplicity, experiments show the model outperforms previous clustering methods by a large margin. Furthermore, the model is also verified on English wiki dataset as a large corpus.' author: - | Jie Zhou\ Pattern Recognition Center,\ WeChat AI,Tencent Inc.\ `[email protected]  `\ Xingyi Cheng\ \ \ `[email protected]`\ Jinchao Zhang\ Pattern Recognition Center\ WeChat AI,Tencent Inc.\ `  [email protected]` bibliography: - 'Main.bib' title: 'An end-to-end Neural Network Framework for Text Clustering' ---

--- abstract: 'We study Cantor Staircases in physics that have the Farey-Brocot arrangement for the $\frac QP$ rational heights of stability intervals $I(\frac QP)$, and such that the length of $I(\frac QP)$ is a convex function of $\frac 1P$. Circle map staircases and the magnetization function fall in this category. We show that the fractal sets $\Omega$ underlying these staircases are connected with key sets in Number Theory via their $(\alpha, f(\alpha))$ multifractal decomposition spectra. It follows that such sets $\Omega$ are self similar when the usual (Euclidean) measure is replaced by the hyperbolic measure induced by the Farey-Brocot partition.' author: - | Grynberg, S.[^1] and Piacquadio, M.\ Departamento de Matemática\ Facultad de Ingeniería\ Universidad de Buenos Aires\ Paseo Colón 850, (1063)\ Buenos Aires, Argentina. title: 'Self-similarity of Farey Staircases' --- Introduction ============ A Cantor staircase is an increasing continuous function from $[a,b]$ to $[0,1]$, $y=g(x)$, with zero derivative almost everywhere, constant on the so-called intervals of resonance or stability $\Delta x_k, k\in\Bbb N$. The complement in $[a,b]$ of $\bigcup_{k\in\Bbb N}\Delta x_k$ is a totally disconnected Cantordust set $\Omega$. Cantor staircases are frequently observed in empirical physics, and their universal properties are of great interest. These staircases are naturally associated with a Cantordust set $\Omega$. Such an $\Omega$ reflects the particular physical problem under study. Such Cantordusts can be studied with the tools provided by Number Theory and the multidimensional $(\alpha, f(\alpha))$ decomposition of a fractal set $\Omega$. The Ising Model and the Circle Map ---------------------------------- 1. [**The Ising model.**]{} Bruinsma and Bak \[1983\] studied the one dimensional Ising model with convex long-range antiferromagnetic interaction. Only “up” spins interact, their interaction being given by a convex function depending on a parameter $a>1$, $a$ is the strength of the interaction. Let $H$ be the applied magnetic field and $q$ the proportion of up-spins. At the critical temperature $T=0$ the phase diagram $q=g(-H)$ exhibits a Cantor staircase. With $\Delta H$ we will denote the intervals of resonance or stability of the staircase $q=g(-H)$. [*Par abus de langage*]{} $\Delta H$ will be the corresponding stairstep as well as its length. $\Delta H(\frac QP)$ means: the stairstep of rational height $\frac QP$ in the staircase. Bruinsma and Bak state that $\Delta H(\frac QP)$ depends only on $P$, and that $$\label{fases} \gamma\left(\Delta H\left(\frac QP\right)\right)^{\frac 1{a+1}}\cong\frac 1P$$ where $\gamma$ is a constant depending only on $a$, the interaction strength. 2. [**The Circle Map.**]{} The simple sine circle map $$\theta_{n+1}=\theta_n+\omega+\frac{\sin(2\pi\theta_n)}{2\pi}$$ is one of the simplest models describing systems with two competing frequencies –e.g. the forced pendulum. Here $\theta$ is the angle formed by the vertical and the pendulum; $n$ is the discretized time variable; $\omega$ represents the frequency of the system in the absence of the nonlinear term given by the sine function. Let $W$ be the winding number corresponding to the average $$\lim\limits_{n\to\infty}\frac{\theta_n}{n}.$$ The graph of the function $W=g(\omega)$ is a well known Cantor staircase. With $\Delta\omega$ we denote its intervals of resonance, as well as the corresponding stairsteps and their length. Universal Properties of these two Cantor Staircases --------------------------------------------------- 1. [**Farey-Brocot.**]{} Let $y=g(x)$ be any of these two Cantor staircases. Let $\Delta x$ and $\Delta x'$ be two intervals of resonance. Let us further suppose that each $\Delta x''$ in the gap between $\Delta x$ and $\Delta x'$ has size smaller than those of both $\Delta x$ and $\Delta x'$. Let $\frac QP$ be $g(x)$ when $x\in\Delta x$, and let $\frac{Q'}{P'}$ be $g(x')$ when $x'\in\Delta x'$. Then, if $\Delta x''$ is the largest interval in the gap, and if $x''\in\Delta x''$, one has $g(x'')=\frac{Q''}{P''}=\frac{Q+Q'}{P+P'}$. 2. [**Hausdorff dimension.**]{} Let us recall that the Cantordust $\Omega$ naturally associated with a Cantor staircase is the complement –in the domain $[a,b]$ of $y=g(x)$– of the union of the intervals of resonance. For each such $\Omega$ associated with the staircases quoted above, we have $d_H(\Omega)\in (0,1)$, where $d_H$ is the Hausdorff dimension... i.e. $\Omega$ is, strictly speaking, a fractal set. For the Ising Model case, Bruinsma and Bak estimated $d_H(\Omega)$, and for the Circle Map the result $d_H(\Omega)=0.87...$ is a known universal number. The Tools Provided by Number Theory and the Multifractal Spectrum $(\alpha, f(\alpha))$ --------------------------------------------------------------------------------------- 1. [**Number Theory.**]{} The problem of approximating irrational numbers by rational ones is a key subject in Number Theory. Let $i\in(0,1)$ be an irrational number, $Q$ and $P$ in $\Bbb N$. Both in Number Theory and its applications, the approximation of $i$ by different rationals $\frac QP$ is given by the study of the distance $\left|i-\frac QP\right|$. Farey-Brocot sequences and continued fractions provide the tools to study the evolution (behaviour, dynamics) of this distance. 1. [**Farey-Brocot $(F-B)$ sequences.**]{} Farey-Brocot sequences $(F-B)_n$ with $n\in\Bbb N$ are defined thus: $(F-B)_1=\left\{\frac01,\frac11\right\}$, $(F-B)_2=\left\{\frac01,\frac{0+1}{1+1},\frac11\right\}=\left\{\frac01,\frac12,\frac11\right\}$, and, once $(F-B)_n$ is defined, we will define $(F-B)_{n+1}$ by interpolating as follows: we take each consecutive pair $\frac QP$ and $\frac{Q'}{P'}$ in $(F-B)_n$, $\frac QP<\frac{Q'}{P'}$, and we interpolate $\frac{Q+Q'}{P+P'}$ between $\frac QP$ and $\frac{Q'}{P'}$, adding the fraction $\frac{Q+Q'}{P+P'}$ to $(F-B)_n$ in order to make $(F-B)_{n+1}$. The first $(F-B)'s$ are: $$(F-B)_2=\left\{\frac01,\frac12,\frac11\right\}$$ $$(F-B)_3=\left\{\frac01,\frac13,\frac12,\frac23,\frac11\right\}$$ $$(F-B)_4=\left\{\frac01,\frac14,\frac13,\frac25,\frac12,\frac35,\frac23,\frac34,\frac11\right\}$$ ...and so on. 2. [**Continued Fractions.**]{} Any irrational number $i\in (0,1)$ can be written uniquely as $$i={1\over\displaystyle a_1+{1\over\displaystyle a_2+{1\over\displaystyle a_3+{_{~~\displaystyle\ddots}}}}}$$ $$=[a_1,a_2, a_3,...],\;a_n\in\Bbb N.$$ This infinite continued fraction $i$ when cut off at $n$, i.e. $$[a_1,a_2,..., a_n],$$ is a rational number $\frac{Q_n}{P_n}$ that well approximates $i$ as $n\to\infty$, which means that $$\left|i-\frac{Q_n}{P_n}\right|<\frac{1}{P_n^2},$$ $n\in\Bbb N$. 3. [**The relationship between continued fractions and $(F-B)$ sequences.**]{} Let $i=[a_1,a_2,..., a_n,...]$. $\frac{Q_1}{P_1}$ is in the $a_1$-th $(F-B)$ sequence, and in general, $\frac{Q_n}{P_n}$ is found in an $(F-B)$ sequence $a_n$ steps ahead of the $(F-B)$ sequence in which $\frac{Q_{n-1}}{P_{n-1}}$ appears. 2. [**The multifractal or multidimensional $(\alpha, f(\alpha))$ of a fractal set $\Omega$.**]{} Multifractal decomposition is a useful tool \[Halsey et al., 1986\], first conceived by physicists, to study a fractal $\Omega$ with a somewhat irregular geometric configuration, such as the Cantordust underlying our $y=g(x)$ staircases. Let us consider the ternary set $K\subset [0,1]$ of Cantor. $K$ is an example of a geometrically very regular fractal set. K is obtained applying contractive transformations $T_1$ and $T_2$ to $[0,1]=I$; $T_1(I)=[0,\frac13]$ and $T_2(I)=[\frac23,1]$. Successive iterations of $T_1$ and $T_2$ produce $K$. Words of $k$ letters $T_i\; (i=1,2)$, yield $2^k$ segments in $I$, which constitute the $k^{th}$ approximation to $K$. The contractors associated with $T_1$ and $T_2$ are both $\frac13$; i.e. $K$ is a $\frac13-\frac13$ fractal. We provide $K$ with a probability measure $p$ such that $p(T_1(I)\cap K)=p_1=\frac12$ and $p(T_2(I)\cap K)=p_2=\frac12$. All $2^k$ segments $I^k$ in the $k^{th}$ partition fulfill $p(I^k\cap K)=\frac{1}{2^k}$. Let $\left|I^k\right|$ be the length of segments $I^k$. The equation for the $\alpha$-index of concentration of $I^k$: $p(I^k\cap K)=\left|I^k\right|^{\alpha}$ will yield $\alpha(I^k)=\frac{\log 2}{\log 3}$, a number independent of $k$. Therefore any point $x\in K$, via its sequence of nested intervals $I^k=I^k(x), k\in\Bbb N$, will inherit a concentration $\alpha(x)=\frac{\log 2}{\log 3}$. Let us consider a more irregular Cantordust $\Omega$, say, $\Omega$ is a $\frac13-\frac14$ fractal with $p_1=p_2=\frac12$. Intervals $I^k$ in the $k$-partition do not necessarily share the same length, though we still have $p(I^k\cap \Omega)=\frac{1}{2^k}$. The concentration $\alpha$, therefore, varies from segment to segment... hence, from point in $\Omega$ to point in $\Omega$. Let $\Omega_{\alpha}$ be the set of all $x$ in $\Omega$ sharing the same concentration $\alpha$. The multifractal or multidimensional spectrum $f(\alpha)$ is, by definition, $d_H(\Omega_{\alpha})$. The Cantordusts $\Omega$ underlying the Cantor staircases quoted above are much more irregular than any $\frac1m-\frac1n$ fractal. On Continued Fractions and Jarník Classes ----------------------------------------- 1. Throughout this paper we will need a number of properties about continued fractions. Let $i=[a_1,a_2, ...,a_n,...]$, $a_n\in\Bbb N$, be an irrational number. Let us recall that $[a_1, a_2, ..., a_n]=$ $\frac{Q_n}{P_n}$ well approximates $i$ as $n\to \infty$; $\left|i-\frac{Q_n}{P_n}\right|<\frac{1}{P_n^2}$, $n\in\Bbb N$. The denominator $P_n$ tends to $\infty$ because we have $P_{n+1}=a_{n+1}P_n+P_{n-1}$; $P_{-1}=0$; $P_0=1$. The polynomial $P_n=P_n(a_1, a_2,...,a_n)$ has $F_n$ monomials, where $F_n$ is the $n^{th}$ Fibonacci number, $F_n=\frac{1}{\sqrt5}(\phi^n-\phi^{-n})\simeq c\phi^n$, $\phi$ the golden mean. Therefore, we have that $P_n\geq F_n$. 2. What we should recall about Jarník classes. The set of irrationals $i$ for which $\left|i-\frac{Q}{P}\right|<\frac{1}{P^{\beta}}$ (for infinite values of $P\in\Bbb N$) is called the $J_{\beta}$ class of Jarník, here $\beta\geq 2$ \[Falconer, 1990\]. Clearly $J_{\beta_1}\supset J_{\beta_2}$ when $\beta_1<\beta_2$. Results by Dirichlet and Jarník \[Falconer, 1990\] show that $d_H(J_{\beta})=\frac{2}{\beta}$, $\beta\geq 2$. Cantor Staircases and the Ising Model ===================================== Let $y=g(x)$ be the staircase describing the magnetization process, $x=-H$ and $y=q$ the proportion of “up” spins as described in section 1. In a previous paper \[Piacquadio and Grynberg, 1998\] we studied the size of intervals of stability $\Delta H$. We studied the size of steps $\Delta H$ near points of irrational height $i$ in the staircase. What follows is a brief sketch of the contents of that paper. Each step $\Delta H$ has rational height $\frac QP$; we studied the size of steps $\Delta H(\frac QP)$ when $\frac QP$ well approximates a certain irrational $i$. Notice that Eq.(\[fases\]) already gives the value of the length of $\Delta H(\frac QP)$. If $\frac{Q_n}{P_n}=[a_1, a_2,...,a_n]$, $i=[a_1, a_2,...,a_n,...]$, we will deal with $\Delta H(\frac{Q_n}{P_n})$. In what follows, and for short, with the same symbol $\Delta H$ we will refer to the interval of resonance in the $-H$ axis, to the corresponding stairstep in the graph of $y=g(-H)$, and to the length of either. We trust that the context will avoid confusion. Given a certain irrational $i$ there exists a unique point $A_i$ of height $i$ in the Cantor staircase and, given a certain small $\varepsilon>0$, there is an infinity of stairsteps $\Delta H$ at no-bigger-than-$\varepsilon$ distance of the point $A_i$. Let $\Delta H_{i,\varepsilon}$ be the largest of them. Clearly, as $\varepsilon$ goes to zero, so does $\Delta H_{i,\varepsilon}$. For a fixed value of $i$, we are interested in computing $\Delta H_{i,\varepsilon}$ as a diminishing function of $\varepsilon$. Now, such a stairstep $\Delta H_{i,\varepsilon}$ has rational height in the staircase. We show that 1. for the sake of our computations, $\varepsilon=\varepsilon_n=\frac{1}{P_nP_{n+1}}$ is an appropriate choice of an $\varepsilon$ going to zero. 2. for such an $\varepsilon_n=\varepsilon$ the stairstep $\Delta H_{i,\varepsilon}$ has rational height $\frac{Q_n}{P_n}$, that is, we have $\Delta H_{i,\varepsilon}=\Delta H(\frac{Q_n}{P_n})$ in the notation of Bruinsma and Bak. In order to study the behaviour of $\Delta H_{i,\varepsilon}$ as a function of $\varepsilon$, we introduced in \[Piacquadio and Grynberg, 1998\] the notion of “type”: Let $k\in\Bbb N$, $k\geq 2$. If $i$ is such that $\frac{1}{P_n}$ fulfills $$\label{tipounosobrek} \lim\limits_{\varepsilon_n\to 0}\frac{\frac{1}{P_n}}{\varepsilon_n^{\frac 1k +\delta}}=\infty\;\; \mbox{and}\;\; \lim\limits_{\varepsilon_n\to 0}\frac{\frac{1}{P_n}}{\varepsilon_n^{\frac 1k -\delta}}=0$$ for an arbitrary small $\delta>0$, we say that $\frac{1}{P_n}$ goes to zero strictly like $\varepsilon_n^{\frac 1k}$, we call $i$ “type $\frac 1k$”, we place every such $i$ in a pigeonhole $G_k$, $k\in\Bbb N$, $k\geq 2$, and we write $\frac{1}{P_n}$ as $\kappa_n\varepsilon_n^{\frac 1k}$, $\kappa_n$ bounded or not. We show that, if $i\in G_k$, we have $\Delta H_{i,\varepsilon}$ determined by $\sqrt{\varepsilon}^{d_H(J_k)}$; that is, we have $\frac{1}{P_n}$ going to zero strictly as a power, the base of which is $\sqrt{\varepsilon}=\sqrt{\varepsilon_n}$ (i.e. the smallest possible diminishing function), and the exponent is $d_H(J_k)$. Essentially, the size of steps $\Delta H_{i,\varepsilon}$ depends on the Jarník classes to which $i$ belongs. The shortest intervals $\Delta H_{i,\varepsilon_n}$ correspond to $i=[1,...,1,...]=\phi^{-1}=\phi-1$, where $\phi= \frac{1+\sqrt5}{2}$, the golden mean. In that case $\Delta H_{i,\varepsilon_n}=\Delta H_{\phi,\varepsilon_n}$ is given by $\sqrt{\varepsilon_n}$ multiplied by a coefficient $\kappa\cong 1.27$. Already for $i=[2,...,2,...]=s$, the silver mean, we have $\Delta H_{s,\varepsilon_n}$ given by $\sqrt{\varepsilon_n}$ multiplied by a coefficient strictly larger than $1.27$. Irrationals $\phi$ and $s$ are, of course, in $G_2$. So are $[1,2,...,n,...]$ and $[1^2,2^2,...,n^2,...]$. If the growth of $a_n$ is sufficiently accelerated, we have that $i=[a_1,a_2,...,a_n,...]$ may be in $G_3$, or in $G_4$,... Since we have $d_H(G_k)\leq\frac 2k=d_H(J_k)$, $k\geq 2$, $k\in\Bbb N$, the dimension of $G_k$ diminishes as $k$ grows. We had “disjointed” the $J_k$ classes ($J_2\supset J_3\supset\dots\supset J_k\supset J_{k+1}\supset\dots$) in disjoint rings $R_k=J_k-J_{k+1}$ and we had promised to show that $G_k\subset R_k$, $\forall k\in\Bbb N$, $k\geq 2$. When we exhaust all rings $R_k$ we are left with irrationals $i\in J_{\infty}$ –i.e. the class of irrationals with ultrarapid growth of $a_n$. Such class has zero Hausdorff dimension, and is contained in every $J_k$, $k\geq 2$. The condition $\lim\limits_{n\to\infty}\frac{\ln P_{n+1}}{\ln P_n}=k-1$ ensured $i\in G_k$. The condition $\lim\limits_{n\to\infty}\frac{\ln P_{n+1}}{\ln P_n}=\infty$ ensured $i\in J_{\infty}$. The Shells $G_{\beta}$ ====================== In fact, $R_k=J_k-J_{k+1}$, $k\in\Bbb N$, is a very “thick” ring, i.e. the case $k\in\Bbb N$ is very far from being a general one. In this section we will prove [**Theorem.**]{} Let $\beta\in\Bbb R$, $\beta\geq 2$. Let $G_{\beta}$ be defined by Eq. (\[tipounosobrek\]) with $\beta\in\Bbb R$ in place of $k\in\Bbb N$. Then $$G_{\beta}=\bigcap\limits_{\beta-2\geq\theta>0}J_{\beta-\theta}- \bigcup\limits_{\delta>0}J_{\beta+\delta},$$ and we have $\Delta H_{i,\varepsilon}$ given by $\sqrt{\varepsilon}^{d_H(G_{\beta})}=\sqrt{\varepsilon}^{d_H(J_{\beta})}$. We will use two known properties of continued fractions: [**Property 1.**]{} Let $i$ be an irrational number, $i\in (0,1)$, and let $\frac rs \in\Bbb Q$. If $$\label{Hardy} \left|i-\frac rs\right|<\frac{1}{2s^2},$$ then $\frac rs=\frac{Q_n}{P_n}$, for some $n\in\Bbb N$. [**Property 2.**]{} Let $i$ be an irrational number, and $\frac{Q_n}{P_n}$ be a $n$-approximant to $i$. We then have $$\label{app1} \frac{1}{P_n(P_n+P_{n+1})}<\left|i-\frac{Q_n}{P_n}\right|<\frac{1}{P_nP_{n+1}}.$$ We need three Claims: [**Claim 1.**]{} Let $i\in (0,1)$ be an irrational number, let $\theta\in (0,1)$, let $\frac rs \in\Bbb Q$, and $\beta\geq 2$. Then we have that –except for a finite number of rationals $\frac rs$– $$\left|i-\frac rs\right|<\frac{1}{s^{\beta+\theta}}$$ implies: $\frac rs$ is a $\frac{Q_n}{P_n}$ for some $n\in\Bbb N$. [**Claim 2.**]{} If $\limsup\limits_n\frac{\ln P_{n+1}}{\ln P_n}=\beta-1$, $\beta\geq 2$, and $P_n$ as above, then for any $\delta>0$ there exists $n_{\delta}\in\Bbb N$ such that $$\label{claim2} P_n(P_n+P_{n+1})<P_n^{\beta+\delta}$$ $\forall n\geq n_{\delta}$. [**Claim 3.**]{} If $\limsup\limits_n\frac{\ln P_{n+1}}{\ln P_n}=\beta-1$, $\beta\geq 2$, then for any $\theta\in (0,1)$ there exists a sequence of naturals numbers $n_j=n_j(\theta)$ such that $$\label{claim3} P_{n_j}^{\beta-\theta}<P_{n_j}P_{n_j+1}.$$ Let us prove the Theorem now. 1. Let us suppose $\limsup\limits_n\frac{\ln P_{n+1}}{\ln P_n}=\beta-1$. 1. We will prove that $$i\notin \bigcup\limits_{\delta>0}J_{\beta+\delta}.$$ Combining Eqs.(\[claim2\]) and (\[app1\]) we have $$\left|i-\frac{Q_n}{P_n}\right|>\frac{1}{P_n^{\beta+\delta}},$$ $n\geq n_{\delta}$ for the $\delta$ and the $n_{\delta}$ in [**Claim 2**]{}. This inequality, together with [**Claim 1**]{}, imply that the set $$\left\{\frac rs\in\Bbb Q: \;\left|i-\frac rs\right|<\frac 1{s^{\beta+\delta}}\right\}$$ has finite cardinality, which implies $i\notin J_{\beta+\delta}$, for any $\delta>0$, which is (a1). 2. We will prove: $$i\in\bigcap\limits_{\beta-2\geq\theta>0}J_{\beta-\theta}.$$ Combining Eq.(\[claim3\]) and the right part of Eq.(\[app1\]) we have that, if $i\in (0,1)$, there exists a sequence $n_j=n_j(\theta)$ of natural numbers such that $$\left|i-\frac{Q_{n_j}}{P_{n_j}}\right|<\frac{1}{P_{n_j}^{\beta-\theta}},$$ which implies $i\in J_{\beta-\theta}$ which, in turn, implies (a2). 2. Let us suppose $$i\in\bigcap\limits_{\beta-2\geq\theta>0}J_{\beta-\theta}- \bigcup\limits_{\delta>0}J_{\beta+\delta}.$$ We will prove that $\limsup\limits_n\frac{\ln P_{n+1}}{\ln P_n}=\beta-1$. Since $i\notin\bigcup\limits_{\delta>0}J_{\beta+\delta}$, then for each $\delta>0$ the inequality $\left|i-\frac{Q_n}{P_n}\right|<\frac 1{P_n^{\beta+\delta}}$ has finite solutions. So $\exists n(\delta)\in\Bbb N$ such that $$\left|i-\frac{Q_n}{P_n}\right|\geq\frac 1{P_n^{\beta+\delta}} \;\forall\;n\geq n(\delta).$$ Also, the right side of Eq.(\[app1\]) implies $P_nP_{n+1}<P_n^{\beta+\delta}$, that is $P_{n+1}<P_n^{\beta+\delta-1}$. Therefore $\forall n\geq n(\delta)$ we have $\frac{\ln P_{n+1}}{\ln P_n}<\beta+\delta-1,$ hence $$\limsup\limits_{n\to\infty}\frac{\ln P_{n+1}}{\ln P_n}\leq \beta+\delta-1.$$ This inequality holds $\forall\delta>0$, therefore $$\limsup\limits_{n\to\infty}\frac{\ln P_{n+1}}{\ln P_n}\leq \beta-1.$$ Now, if we had $$\limsup\limits_{n\to\infty}\frac{\ln P_{n+1}}{\ln P_n}< \beta-1,$$ then, there would exist $\theta_0>0$ such that $$\limsup\limits_{n\to\infty}\frac{\ln P_{n+1}}{\ln P_n}= \beta-\theta_0-1,$$ and from part (a) above we would have: $$i\notin\bigcup\limits_{\delta>0}J_{\beta-\theta_0+\delta},$$ from which $i\notin J_{\beta-\theta}$ for any $\theta>0$ small enough, which is a contradiction with $$i\in\bigcap\limits_{0<\theta\leq \beta-2}J_{\beta-\theta},$$ a part of our hypothesis. It remains to prove the three Claims. [**Proof of Claim 1.**]{} Let $\frac rs$ be a rational number fulfilling $$\left|i-\frac rs\right|<\frac{1}{s^{\beta+\theta}},$$ and such that $s\geq [2^\frac 1{\theta}]+1$. This inequality, in turn, implies $s^{\beta+\theta}>2s^{\beta}$, and since $\beta\geq 2$, $s\in\Bbb N$, we have $s^{\beta}\geq s^2$. Therefore $s^{\beta+\theta}>2s^2.$ From this and $$\left|i-\frac rs\right|<\frac{1}{s^{\beta+\theta}}$$ we have $$\left|i-\frac rs\right|<\frac 1{2s^2}.$$ From this and [**Property 1**]{} we have $\frac rs=\frac{Q_n}{P_n}$ for some $n\in\Bbb N$, q.e.d. Before proving [**Claim 2**]{} we need an [**Observation.**]{} 1), 2) and 3) below are equivalent statements. 1. $\limsup\limits_{n\to\infty}\frac{\ln P_{n+1}}{\ln P_n}=\beta-1$ 2. 1. $\forall\theta>0,\;\exists n(\theta)\in\Bbb N\;\mbox{such that}\;\frac{\ln P_{n+1}}{\ln {P_n}}<\beta-1+\theta\;\forall\;n\geq n(\theta).$ 2. $\forall\theta>0,\;\exists\left\{n_j(\theta)\right\}$, an infinite sequence of natural numbers, such that $$\beta-1-\theta<\frac{ln P_{n_j(\theta)+1}}{\ln P_{n_j(\theta)}}.$$ 3. 1. $\forall\theta>0,\;\exists n(\theta)\in\Bbb N\;\mbox{such that}\; P_{n+1}<P_n^{\beta-1+\theta}\;\forall\;n\geq n(\theta).$ 2. $\forall\theta>0,\;\exists\left\{n_j(\theta)\right\}$, an infinite sequence of natural numbers, such that $$P_{n_j(\theta)}^{\beta-1-\theta}< P_{n_j(\theta)+1}.$$ [**Proof of Claim 2.**]{} Let $\theta$ and $\delta$ be positive numbers, $\theta<\delta$. Our hypothesis implies that $\exists n(\theta)\in\Bbb N$ such that $$P_{n+1}<P_n^{\beta-1+\theta}$$ $\forall n\geq n(\theta)$. From this we have $$\label{aux2} P_n(P_n+P_{n+1})<P_n^2+P_n^{\beta+\theta}=P_n^{\beta+\theta}\left(P_n^{2-\beta-\theta}+1\right),$$ and since $$\frac{P_n^{\beta+\theta}\left(P_n^{2-\beta-\theta}+1\right)}{P_n^{\beta+\delta}}=P_n^{\theta-\delta}\left(P_n^{2-\beta-\theta}+1\right)$$ and $$\lim\limits_{n\to\infty}P_n^{\theta-\delta}\left(P_n^{2-\beta-\theta}+1\right)= 0$$ we have that $\exists n_0\in\Bbb N$ such that, $\forall n>n_0$ $$\label{aux5} P_n^{\beta+\theta}\left(P_n^{2-\beta-\theta}+1\right)<P_n^{\beta+\delta}$$ holds. Now, if $n>n(\delta)=\max \{n_0, n(\theta) \}$, Eqs. (\[aux2\]) and (\[aux5\]) hold simultaneously, from which $$P_n(P_n+P_{n+1})<P_n^{\beta+\delta}\;\forall n\geq n(\delta),\;\;\mbox{q.e.d.}$$ [**Claim 3**]{} is an obvious consequence of statement 3)(b) in the Observation above. [**Theorem 2**]{} With the notation in Theorem 1 we have 1. $\bigcap\limits_{0<\theta\leq \beta-2}J_{\beta-\theta}\neq J_{\beta} $ 2. $\bigcup\limits_{\delta>0}J_{\beta+\delta}\neq J_{\beta}$ [**Proof.**]{} With bits and pieces in the proof of Theorem 1, a proof of (a) and (b) can be put together, a technical exercise we leave to the reader. The Multifractal Spectrum of $\Omega$ ===================================== The Cantordust Set ------------------ The fractal set $\Omega$ underlying the Cantor staircase $q=g(-H)$ studied by Bruinsma and Bak can be constructed in a way analogous to the one used to construct the Cantordust sets in section 1.3. Let $\Delta H(\frac{Q}{P})$ be the resonance interval corresponding to the rational number $\frac{Q}{P}$, that is the step $\Delta H(\frac{Q}{P})$ in the staircase of height $\frac{Q}{P}$. We subtract from the real line $\Bbb R$ intervals $\Delta H(\frac{0}{1})$ and $\Delta H(\frac{1}{1})$, obtaining a closed and bounded interval $I^0$; this will be the equivalent of the initial interval $I=[0,1]$ in the two examples in section 1.3. Next, we subtract interval $\Delta H(\frac{1}{2})$ from $I^0$, thereby obtaining two compact intervals $I^1_1$ and $I^1_2$. Intervals $I^1$ constitute the first approximation to $\Omega$. Next, we subtract $\Delta H(\frac{1}{3})$ and $\Delta H(\frac{2}{3})$ from intervals $I^1$ obtaining four compact intervals $I^2_i,\;i=1,...,4$ which constitute the second approximation to $\Omega$. Proceeding in this way, in the $k$-th step we subtract all intervals $\Delta H(\frac{Q}{P}),\;\frac{Q}{P}$ in the $(F-B)_k$ sequence, thereby obtaining $2^k$ intervals $I^k_i,\;i=1,...,2^k$; which are the $k$-th approximation to $\Omega$. The Measure of Probability on $\Omega$ -------------------------------------- The probability $p$ induced in our Cantordust $\Omega$ will be like in section 1.3., given by $p_1=p(I^1_1\cap \Omega)=\frac12$, $p_2=p(I^1_2\cap \Omega)=\frac12$. All segments $I^k$ will be equiprobable, i.e. $p(I^k_i\cap \Omega)=\frac{1}{2^k}$ for $i=1,...,2^k$. Notice that $g(I^1_1\cap\Omega)=[0,\frac{1}{2}]$ and $g(I^1_2\cap\Omega)=[\frac{1}{2},1]$, whereas $g(I^2_1\cap\Omega)=[0,\frac{1}{3}]$, $g(I^2_2\cap\Omega)=[\frac{1}{3},\frac{1}{2}]$, $g(I^2_3\cap\Omega)=[\frac{1}{2},\frac{2}{3}]$, and $g(I^2_4\cap\Omega)=[\frac{2}{3},1]$. Therefore, the equiprobability of the $I^k$ approximating $\Omega$ is equivalent to the equiprobability of the $2^k$ segments in the $(F-B)_k$ partition. Such equiprobability on the $(F-B)_k$ segments induces a probability measure in the unit segment –the hyperbolic measure– a measure that we will study below. $\alpha_{\max}$ of $\Omega$ --------------------------- Let us go back to section 1.3. Let $x\in \Omega$. Let $I^k=I^k(x),\; k\in\Bbb N$, be the sequence of nested intervals in successive $k$-approximations to $\Omega$ to which $x$ belongs. Let us recall that the equation for the $\alpha$-index of $I^k$ is $p(I^k\cap \Omega)=\left|I^k\right|^{\alpha}$. This sequence $\alpha=\alpha(I^k)=\alpha_k,\;k\in\Bbb N$, when convergent, defines $\alpha(x)$. Notice that $p(I^k\cap \Omega)=\frac{1}{2^k}$, therefore $\alpha_k$ depends strictly on the size $\left|I^k\right|=\left|I^k(x)\right|$. Hence, if we are interested in points $x\in\Omega$ with $\alpha(x)=\alpha_{\max}$ we have to select points for which intervals $I^k(x)$ are the longest in the $k$-approximation to $\Omega,\;k\in\Bbb N$. Let us consider the point in the staircase of height $g(x)=i$, an irrational value. The steps near it were called $\Delta H_{i,\varepsilon}$ and we will call them $\Delta H(i)$ for short. We have $\Delta H(i)=\Delta H(g(x))$. We constructed $I^k(x)$ by subtracting intervals $\Delta H(i)=\Delta H(g(x))$. Therefore [**long**]{} $I^k(x)$ correspond to [**small**]{} $\Delta H(g(x))$, and viceversa. We are interested now in [**small**]{} $\Delta H(i)$. Small $\Delta H(i)$ correspond to steps near points $i\in G_2$. Among them, the smallest correspond to points $i=[a_1,...,a_n,1,...,1,...]$ and the smallest of them all correspond to $$i=[1,1,...,1,...]=\phi^{-1}=\phi-1,$$ $\phi$ being the golden mean, a result that agrees with classical ones. Notice that in the vertical $q$-axis provided with the $(F-B)$ arrangement, the [**short**]{} segments in each $(F-B)_k$ partition are precisely the ones that cover the point $i=\phi^{-1}=\phi-1$. Therefore, when $I=[0,1]$ is provided with the Hyperbolic measure of probability, we have that points in $I$ with $\alpha_{\min}$ correspond to points $x$ in $\Omega$, $i=g(x)$, with $\alpha_{\max}$. $\alpha_{\min}$ of $\Omega$ --------------------------- Following the outlines in the preceding section, points $x$ with $\alpha_{\min}$ in $\Omega$ are those with the shortest $\left|I^k(x)\right|$, which in turn correspond to the longest $\Delta H(g(x))=\Delta H(i)$. The longest $\Delta H(i)$ correspond to $i\in G_{\infty}=J_{\infty}$ \[Piacquadio and Grynberg, 1998\]. [**Liouville Numbers.**]{} Liouville constructed irrational numbers $i$ for which, for each $k\in\Bbb N$, there exists rationals $\frac QP$ such that $$\left|i-\frac QP\right|<\frac{1}{P^k},\;\;P\geq 2.$$ Let $a_1$ be arbitrary. Choose $a_2>P_1^1(a_1)$ and, given $P_{n-1}(a_1,...,a_{n-1}),$ we choose $a_n>P_{n-1}^{n-1}(a_1,...,a_{n-1})$. Such growth of the $a_n$ guarantees (see properties of $P_n$ in Sec.1.4 a)) that $i=[a_1,a_2,...,a_n,...]$ belongs to every $J_k$, hence, to $J_{\infty}=G_{\infty}$. Notice that these $a_n$ have an ultrarapid growth, that $a_{n+1}>>a_n\;\forall\;n\in\Bbb N$. Therefore, the part of the continued fraction expansion of $i$ that starts with $a_n$, i.e. $a_n+\frac{1}{a_{n+1}+\frac{1}{\ddots}}$ is almost indistinguishable from $a_n$. Let us recall that a real number is a rational number precisely when its continued fraction has a finite number of such $a_n$. Therefore, the elements of $G_{\infty}$ are called time and again “quasi-rationals” in the literature. Therefore we will study rational numbers in the vertical axis, because they have properties analogous to those of “quasi-rational” numbers, i.e. we will study quasi-rationals in $G_\infty$ via rational numbers. Let $\frac{Q}{P}$ and $\frac{Q'}{P'}$ be two rationals adjacent in a certain $(F-B)_k$. In order to avoid overlapping we will consider segments in an $(F-B)_k$ partition as closed on the left and open on the right: $[\frac{Q}{P},\frac{Q'}{P'})$ will be our $(F-B)_k$ segment covering point $\frac{Q}{P}$. The only segment in $(F-B)_{k+1}$ covering $\frac{Q}{P}$ is $[\frac{Q}{P},\frac{Q+Q'}{P+P'})$; the one in $(F-B)_{k+2}$ is $[\frac{Q}{P},\frac{2Q+Q'}{2P+P'})$, and in general, in $(F-B)_{k+n}$ it will be $[\frac{Q}{P},\frac{nQ+Q'}{nP+P'})$. The length of such general segment is $\frac{1}{P(nP+P')}\cong\frac{1}{P^2}\frac{1}{n}$, a value that diminishes like the harmonic sequence. So, while the harmonic sequence $\frac{1}{n}$ is responsible for the longest segments, the sequence $\frac{1}{\phi^{2n}}$ is responsible for the shortest ones. The Value $\alpha\in(\alpha_{\min},\alpha_{\max})$ and the Type $G_{\beta}$ --------------------------------------------------------------------------- Let us recall that $\Omega_{\alpha}\subset\Omega$ is the set of all points $x\in\Omega$ that share the same $\alpha$-concentration, i.e. $\Omega_{\alpha}=\left\{x\in\Omega:\;\alpha(x)=\alpha\right\}$. Now, as we remarked in Secs. 4.3 and 4.4, $\alpha(x)$ is large (small) if the nested $I^k(x)$, $k\in\Bbb N$ (from successive $k$-partitions of $\Omega$), are large (small). We also remarked (same Secs.) that the size of $I^k(x)$ was related to the size of $\Delta H(i)=\Delta H(g(x))$: the larger $\Delta H(g(x))$, the smaller $I^k(x)$ –hence $\alpha(x)$– will be. But the size of steps $\Delta H(i)=\Delta H(g(x))$ with height $\cong i$ in the staircase, depends strongly on the $G_{\beta}$ to which $i=g(x)$ belongs. We are saying that if $i_1\in G_{\beta}$ and $i_2\in G_{\beta}$ for the same $\beta$, then $x_1=g^{-1}(i_1)$ and $x_2=g^{-1}(i_2)$ should belong to the same $\Omega_{\alpha}$, for some $\alpha=\alpha(\beta)$. In other words: given $\beta\geq 2$ there should be $\alpha=\alpha(\beta)$ such that $g^{-1}(G_{\beta})=\Omega_{\alpha(\beta)}$. While we cannot yet categorically affirm the validity of this equality, the remarks above show that the $G_{\beta}$ and the $\Omega_{\alpha}$ are closely linked. The Spectrum $(\alpha,f(\alpha))$ of $\Omega$ --------------------------------------------- Let us consider $\alpha$ growing from $\alpha_{\min}$ to $\alpha_{\max}$, and let us for a moment accept that $g(\Omega_{\alpha})=g(\Omega_{\alpha(\beta)})=G_{\beta}$. Then we have $G_{\beta}$ changing from $G_{\infty}$ to $G_2$ as $\alpha$ grows. Notice that $f(\alpha)=d_H(\Omega_{\alpha})=d_H(\Omega_{\alpha(\beta)})$ would be directly related –via the [**increasing**]{} function $g$– to $d_H(G_{\beta})$, which strictly grows when $\beta$ changes from $\infty$ to $2$, going from $d_H(G_{\infty})=0$ to $d_H(G_{2})=1$. The function $g$ linking $\Omega_{\alpha(\beta)}$ and $G_{\beta}$, and the uneven (and hitherto poorly understood) distribution of intervals $I^k$ in a $k$-partition of $\Omega$, are two strong factors which hinder us from linking $f(\alpha)=d_H(\Omega_{\alpha(\beta)})$ with $d_H(G_{\beta})$ directly through, say, so simple a way as an equality. Still, we know: 1. For $\beta=\infty$ we have $\alpha(\beta)=\alpha_{\min}$. 2. For $\beta=2$ we have $\alpha(\beta)=\alpha_{\max}$. 3. $f(\alpha_{\min})=0$. 4. $f(\alpha)$ is increasing for the greater part of the interval $[\alpha_{\min},\alpha_{\max}]$. 5. Let $\max\limits_{\alpha}f(\alpha)=f(\alpha^{\max})$. Then $\alpha_{\max}$ is very near $\alpha^{\max}$. 6. The $\Omega_{\alpha}$ are strongly linked to the types $G_{\beta}$. 7. The sets $G_{\infty}$ and $G_{2}$, related to $\alpha_{\min}$ and $\alpha_{\max}$ in $\Omega$, are related to $\alpha_{\max}$ and $\alpha_{\min}$ in $I=[0,1]$ endowed with the hyperbolic measure of probability induced by Farey-Brocot. The Spectrum $(\alpha, f(\alpha))$ of the Fractal Set $\Omega$ underlying the Circle Map Staircase ================================================================================================== Conclusions 1) to 7) in the last section show a strong connection between the magnetization function $q=g(-H)$ and leading problems in Number Theory –viz the good approximation of irrational numbers studied with Jarník classes $J_{\beta}$ and their refinements $G_{\beta},\;\beta\geq 2$. This particular connection between magnetization and Number Theory is seen only when analyzing the multifractal spectrum of the fractal set $\Omega$ underlying the magnetization Cantor staircase. The conclusions (1) to 7)) are based, as we have seen, on two premises about the staircase $q=g(-H)$: the $(F-B)$ arrangement of the stairsteps $\Delta H$ in the staircase, and the formula given by Eq. (\[fases\]). Let us now consider the Cantor staircase $W=g(\omega)$ associated with the circle map: we land in Dynamical Systems, where connections with Number Theory are old and well explored. The stairsteps $\Delta\omega$ do satisfy the $(F-B)$ arrangement \[Cvitanovic et al., 1985\], as we remarked above. On the other hand, Fig. 1 shows that Eq. (\[fases\]) is valid with $\Delta\omega(\frac QP)$ instead of $\Delta H(\frac QP)$ –at least when we take averages over the $\Delta\omega(\frac QP)$ with the same $P$. We can, therefore, extend conclusions 1) to 7) for the case of the circle map staircase $W=g(\omega)$. Now, for the fractal set $\Omega$ underlying staircase $W=g(\omega)$ there is a spectrum $(\alpha, f(\alpha))$ associated with it \[Halsey et al., 1986\]. A natural question arises: are conclusions 1) to 7) verifiable for this $f(\alpha)$? Conclusions 1) and 2) can be checked from statements as early as “...the most extremal behaviours of this staircase are found around the golden mean sequence of dressed winding numbers... and at the harmonic sequence $\frac{1}{Q}\to 0$. The most rarified region of the staircase is located around the golden mean” “...the $\frac{1}{Q}$ series... determines the most concentrated portion of the staircase...” in \[Halsey et al.,1986\]. Conclusions 3) and 5) follow from observing the corresponding graph $(\alpha, f(\alpha))$ in Fig. 12, in the same reference. Conclusion 4) follows observing the same figure: $f(\alpha)$ is increasing for an interval $\Delta\alpha\subset[\alpha_{\min},\alpha_{\max}]$, where the length $|\Delta\alpha|$ is some $98\%$ of the length $|[\alpha_{\min},\alpha_{\max}]|$. Conclusion 7) holds just as it does for $q=g(-H)$. Conclusion 6) remains a qualitative one –and quantitatively conjectural. If we ponder on the fact that the Circle Map is universal in character, i.e. with small change of details, it describes a variety of phenomena, and if we recall that the time variables $W$ and $\omega$ in the function $W=g(\omega)$ have no connection with $q$ and the magnetic field $H$, then we can safely conclude that Number Theory links with Cantor staircases in physics in a way even more universal than the one indicated in the already explored linking with Circle Maps and Dynamical Systems. The Hyperbolic Metric ===================== Let $\Bbb H=\left\{z\in\Bbb C:\;Im(z)>0\right\}$ be the upper half plane. The geodesics are circumferences orthogonal to the real axis, and that includes semilines orthogonal to the real axis. The congruences are transformations $$z\to\frac{az+b}{cz+d},\;a,b,c,d\; \mbox{in}\; \Bbb Z,\;ad-bc=1$$ The congruences have a group structure denoted by ${\cal U}$ in the literature. We can consider ${\cal U}$ as a multiplicative group of $2\times 2$ matrices $\left(\begin{array}{cc} a&b\\ c&d\\ \end{array}\right)$ with integer entries and unit determinant. Associated with ${\cal U}$ there is a fundamental region $R\subset\Bbb H$ such that if $z\in Int R$ then $uz\in Ext R$ for every $u\in {\cal U}$, $u\neq Identity$; and given $z\in Ext R$ there exists $u\in {\cal U}$ such that $uz\in Int R$. Points on the boundary of $R$ are transformed, by some elements of ${\cal U}$, into other points on the same boundary. All regions $uR$, $u\in {\cal U}$, have disjoint interiors. The union of all $uR$, $u\in {\cal U}$, covers $\Bbb H$. Such $R$ is called a fundamental tile. Any $uR$ is another fundamental tile. Any two such tiles are congruent by means of an element in ${\cal U}$. Looking at the tiling we notice that tiles near the real axis are much smaller than other tiles. Yet, if we take off our Euclidean eyeglasses, put on a pair of Hyperbolic spectacles, and look again at the tiling, we will see all tiles equal to one another very much like, say, squares of the same size: for there exists a unique metric –the hyperbolic one– measuring with which the sizes of all tiles are the same. Matrices $P=\left(\begin{array}{cc} 1&1\\ 0&1\\ \end{array}\right)$ and $Q=\left(\begin{array}{cc} 0&1\\ -1&0\\ \end{array}\right)$ generate ${\cal U}$, $R$ being the fundamental tile situated symmetrically above the origen. We take fundamental tile $R$ and perform on it the cut-and-paste surgery indicated in \[Series, 1985; Grynberg and Piaquadio, 1995\] obtaining another fundamental tile $T$ for ${\cal U}$. The generators of ${\cal U}$ are now matrices $P$ and $A=QP^{-1}Q=\left(\begin{array}{cc} 1&0\\ 1&1\\ \end{array}\right)$, that is, given $u\in {\cal U}$ there exists a finite word in letters $A$ and $P$ such that $u=A^{a_1}P^{a_2}A^{a_3}P^{a_4}\dots\;a_i\in\Bbb N$. We denote tile $uT=A^{a_1}P^{a_2}A^{a_3}P^{a_4}\dots T$ by the finite word $A^{a_1}P^{a_2}A^{a_3}P^{a_4}\dots$ Now, $T$ is a “rhombus”, and so is tile $A$. Two opposite vertices of rhombus $A$ are points $0$ and $1$ of the real axis: we associate $A$ with segment $[0,1]$. Next, we will consider tiles associated with words starting with letter $A$, and such that all $a_i\in\Bbb N$. Two-letter words like that are $AA$ and $AP$. $AA$ is the rhombus associated with segment $[0,\frac 12]$: it is the only tile with two opposite vertices leaning on $0$ and $\frac12$ on the real line. $AP$ is likewise associated with $[\frac12,1]$. Tiles $AA$ and $AP$ are Euclideanly smaller than $A$ and are closer to $\Bbb R$ than is $A$. Three-letter words $A^3,A^2P,APA$ and $AP^2$, associated with intervals $[0,\frac13],[\frac13,\frac12],[\frac12,\frac23]$ and $[\frac23,1]$, respectively, are Euclideanly smaller than two-letter tiles, and are even closer to $\Bbb R$. Letter $A$, therefore, is associated with the “left”, and $P$ to the “right”, in a way we trust is obvious. The $2^k$ words with $k$-letters are associated with the $(F-B)_k$ sequence. An infinite word $A^{a_1}P^{a_2}A^{a_3}P^{a_4}\dots$, therefore, is associated with an irrational number $i$ in the unit segment; moreover $$A^{a_1}P^{a_2}A^{a_3}P^{a_4}\dots=[a_1,a_2,a_3,a_4,...]=\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\frac{1}{\ddots}}}}=i.$$ With $C$ indicating $A$ or $P$, according to the case, we have that tile $$A^{a_1}P^{a_2}A^{a_3}P^{a_4}\dots C^{a_n}$$ is a $2\times 2$ matrix with unit determinant $$\left(\begin{array}{cc} Q_{n-1}&Q_n\\ P_{n-1}&P_n\\ \end{array}\right)$$ where $\frac{Q_k}{P_k}$ is $[a_1,a_2,...,a_k]$. Tiles $A^{k+1},...,AP^k$ corresponding to the $2^k$ words starting with $A$ followed by $k$ letters $A$ and $P$, are hyperbolically equimeasurable, and they are associated with the $2^k$ segments in $(F-B)_k$, for all $k\in\Bbb N$. Then we say that $[0,1]$ inherits from $\Bbb H$ a measure $\mu$ that renders these $2^k$ segments equimeasurable, the $\mu$ measure of each segment being $\frac{1}{2^k}$. [*Par abus de langage*]{} we will refer to this measure $\mu$ in $[0,1]$ indistinctly as the hyperbolic measure or the $(F-B)$ measure. Hyperbolic Self-similarity of the Cantor Staircase ================================================== Let $y=g(x)$ be any Cantor staircase that fulfills both Eq. (\[fases\]) for the intervals of resonance and the $(F-B)$ arrangement for the vertical axis. The Circle map and the magnetization curve both fulfill this condition. It has been claimed \[Bruinsma and Bak, 1983; Bak, 1986\] that the graph of $y=g(x)$ is “self-similar”: the whole staircase looks like a small section of it. In this section we study the character of this self-similarity: we will focus on a section of the staircase, say, the section between two small intervals of resonance $I$ and $I'$. [*Par abus de langage*]{}, with $I$ and $I'$ we will denote the corresponding stairsteps as well. For clarity we will choose $I$ and $I'$ with heights $\frac{Q}{P}$ and $\frac{Q'}{P'}$ in the staircase, where $\frac{Q}{P}$ and $\frac{Q'}{P'}$ are two rational numbers adjacent in a $(F-B)_k$ partition, for a certain $k\in\Bbb N$. Let us consider the whole staircase, situated between steps $I(\frac{0}{1})$ and $I(\frac{1}{1})$, see Sec. 2. In the staircase there is a specific way in which the height of intervals of resonance is distributed according to the Euclidean size of the latter. Such relationship between height and size is precisely what we described as the $(F-B)$ arrangement of the vertical axis of the staircase. Let us consider now the staircase between $I=I(\frac{Q}{P})$ and $I'=I'(\frac{Q'}{P'})$. The segment $[\frac{Q}{P}, \frac{Q'}{P'}]$ is obtained from $[\frac01,\frac11]$ by means of a hyperbolic rigid movement. Such hyperbolic movement is given by a $(k+1)$-letter word in letters $A$ and $P$, starting with $A$. Therefore the distribution of heights of steps in the staircase between $I$ and $I'$ is hyperbolically equivalent to that of the whole staircase. Let ${\cal T}={\cal T}(b_1,b_2,...,b_m)=A^{b_1}P^{b_2}\dots C^{b_m}$, $b_1+b_2+\dots+b_m=k+1$ be the corresponding $(k+1)$-letter word effecting the hyperbolic transformation, and let $i=[a_1,a_2,...,a_n,...]$ be the irrationals in $[0,1]$. Then the irrationals in $[\frac{Q}{P}, \frac{Q'}{P'}]$ are written $[b_1,...,b_m;a_1,a_2,...,a_n,...]$. Let us focus in this general expression of an irrational in $[\frac{Q}{P}, \frac{Q'}{P'}]$: the first $m$-numbers $b_1,...,b_m$ give the hyperbolic transformation $${\cal T}=A^{b_1}P^{b_2}\dots C^{b_m},$$ the numbers that follow, $a_1,a_2,...,a_n,...$ give all the irrationals in $[0,1]$. Reading from left to right, this notation $[b_1,...,b_m;a_1,a_2,...,a_n,...]$ given by continued fractions for an irrational in $[\frac{Q}{P}, \frac{Q'}{P'}]$ yields this number in a natural way as ${\cal T}=A^{b_1}P^{b_2}\dots C^{b_m}$ applied to the irrationals $[a_1,a_2,...,a_n,...]$ of $[0,1]$. We have seen, then, that heights of stairsteps vis-$\grave{a}$-vis their size is, hyperbolically, the same for steps between $\frac{Q}{P}$ and $\frac{Q'}{P'}$ and for stairsteps between $\frac01$ and $\frac11$. Next, we have to compare sizes of steps between $\frac{Q}{P}$ and $\frac{Q'}{P'}$ with the size of the corresponding steps between $\frac01$ and $\frac11$. The Change of Scale for the Stairsteps -------------------------------------- Let ${\cal T}$ be the hyperbolic transformation just described: $${\cal T}[a_1,a_2,...,a_n,...]=[b_1,b_2,...,b_m;a_1,a_2,...,a_n,...],$$ ${\cal T}:[0,1]\to [\frac{Q}{P},\frac{Q'}{P'}].$ Intervals of stability with height $[a_1]$,$[a_1,a_2]$,...,$[a_1,a_2,...,a_n]$,... get nearer and nearer the point in the staircase of height $[a_1,a_2,...,a_n,...]$. The heights of the associated steps in the staircase between $I(\frac{Q}{P})$ and $I'(\frac{Q'}{P'})$ are $$[b_1,b_2,...,b_m;a_1],\;[b_1,b_2,...,b_m;a_1,a_2],...,\;[b_1,b_2,...,b_m;a_1,a_2,...,a_n],...$$ Let us recall that $[a_1,...,a_n]=\frac{Q_n}{P_n}$ is a good approximant of irrational $[a_1,...,a_n,...]$. $P_n$ is a polynomial $P_n(a_1,...,a_n)$. Let us also recall that the size of $I(\frac{Q}{P})$ is given by $\frac{1}{P}$. Therefore, lengths of steps $[a_1,a_2,...,a_n]$ and the associated steps $[b_1,b_2,...,b_m;a_1,a_2,...,a_n]$ are, respectively, given by $$\frac{1}{P_n(a_1,...,a_n)}$$ and $$\frac{1}{P_{m+n}(b_1,b_2,...,b_m;a_1,...,a_n)}.$$ How does the size of these two intervals compare? The corresponding scale factor $\lambda$, if it exists, would be $$\lambda= \frac{\frac{1}{P_{m+n}(b_1,b_2,...,b_m;a_1,...,a_n)}}{\frac{1}{P_n(a_1,...,a_n)}}=\frac{P_n(a_1,...,a_n)}{P_{m+n}(b_1,b_2,...,b_m;a_1,...,a_n)}=$$ $$\frac{P_n(a_1,...,a_n)}{P_m(b_1,...,b_m)P_n(a_1,a_2,...,a_n)+P_{m-1}(b_1,...,b_{m-1})P_{n-1}(a_2,...,a_n)}.$$ In order to see this the reader is invited to decompose, say, $$P_1(b_1),\;P_2(b_1,b_2),\;P_3(b_1,b_2,a_1),\;P_4(b_1,b_2,a_1,a_2)...$$ and verify that the decomposition of these $P's$ in terms of $P(b's\;\mbox{only:}\;b_1, b_2)$ and $P(a's\;\mbox{only: as many}\; a's\; \mbox{as you can handle})$ is $$P_4(b_1,b_2,a_1,a_2)=P_2(b_1,b_2)P_2(a_1,a_2)+P_1(b_1)P_1(a_2).$$ With [**many more lines!**]{} for the general expresion we can see that $$P_{m+n}(b_1,b_2,...,b_m,a_1,a_2,...,a_n)=$$ $$P_m(b_1,b_2,...,b_m)P_n(a_1,a_2,...,a_n)+ P_{m-1}(b_1,b_2,...,b_{m-1})P_{n-1}(a_2,...,a_n)$$ holds. Therefore, $$\lambda=\frac{P_n(a_1,...,a_n)}{P_m(b_1,...,b_m)P_n(a_1,a_2,...,a_n)+P_{m-1}(b_1,...,b_{m-1})P_{n-1}(a_2,...,a_n)}=$$ $$\frac{P_n(a_1,...,a_n)}{P_m(b_1,...,b_m)P_n(a_1,...,a_n)\left(1+\theta_m\right)}=$$ $$\frac{1}{P_m(b_1,...,b_m)\left(1+\theta_m\right)}=\left(\frac{1}{1+\theta_m}\right)\frac{1}{P_m(b_1,...,b_m)},$$ where $$\theta_m=\frac{P_{m-1}(b_1,...,b_{m-1})}{P_m(b_1,...,b_m)}\frac{P_{n-1}(a_2,...,a_n)}{P_n(a_1,a_2,...,a_n)}.$$ Since $$P_{m-1}(b_1,...,b_{m-1})<P_m(b_1,...,b_m)$$ and $$P_{n-1}(a_2,...,a_n)<P_n(a_1,a_2,...,a_n)$$ then $\theta\in (0,1)$ and $\frac{1}{1+\theta}\in (\frac12,1)$, hence $$\frac12\frac{1}{P_m(b_1,...,b_m)}<\lambda<\frac{1}{P_m(b_1,...,b_m)},$$ i.e. $\lambda$ is of the order of $\frac{1}{P_m(b_1,...,b_m)}$. Moreover, remarks in Sec. 1.4 on the growth of polynomials $P_n$ and the number of monomials in $P_n$ imply that the quotient of two successive $P_n$ and $P_{n-1}$ is no smaller than $\phi$, save for some pathological (and enumerable) cases. Hence our $\theta$ fluctuates between $0$ and $\frac{1}{\phi^2}$ and $$0,723...\frac{1}{P_m(b_1,...,b_m)}=\frac{1}{1+\frac{1}{\phi^2}}\frac{1}{P_m(b_1,...,b_m)}<\lambda<\frac{1}{P_m(b_1,...,b_m)}.$$ [**A comment.**]{} Let us have a look at $\theta$: we have in $\theta$ the product of two quotients of consecutive polynomials $P$. Let us consider two such consecutive polynomials $P_{n-1},\;P_n$ in variables $j_1,j_2,...,j_n,...,\;j_i\in\Bbb N$. Roughly speaking, these variables, being natural, can either grow or not. If they grow, we have $\frac{P_{n-1}}{P_n}\to 0$. If they don’t grow, the most extreme case is $1,1,...,1,...,$ where $\frac{P_n}{P_{n-1}}\cong\phi$. That is why $\frac{P_{n-1}}{P_n}$ fluctuates between $0$ and $\frac{1}{\phi}$. Hence our bounds on $\theta$. We have, then, that the Cantor staircase is hyperbolically self-similar in the vertical axis, whereas the size of horizontal stairsteps have an Euclidean self-similar structure: sizes of steps of height between $\frac{0}{1}$ and $\frac{1}{1}$ change to sizes of steps of height between $\frac{Q}{P}$ and $\frac{Q'}{P'}$, with a contractor $$\lambda\cong\frac{1}{P_m(b_1,...,b_m)},$$ where $b_1,...,b_m$ defines the hyperbolic transformation ${\cal T}:[0,1]\to [\frac{Q}{P},\frac{Q'}{P'}].$ There are two modes of self-similarity involved here: one hyperbolic and one Euclidean. Vertical segments $[\frac{Q}{P},\frac{Q'}{P'}]$ have length $$\frac{1}{PP'}=\frac{1}{P_m(b_1,...,b_m)P_{m-1}(b_1,...,b_{m-1})},$$ whereas the size of horizontal segments between $\frac{Q}{P}$ and $\frac{Q'}{P'}$ decrease with a factor $\lambda\cong\frac{1}{P_m(b_1,...,b_m)}$: hence vertical sizes and horizontal sizes decrease with diferent scale factors. Nevertheless, there is a particular sense in which the size of horizontal segments (stairsteps) also decreases according to a hyperbolic law: we explained above that vertical segment $[\frac{Q}{P},\frac{Q'}{P'}]$ is $[\frac{Q_n}{P_n},\frac{Q_{n-1}}{P_{n-1}}]$. Now, we also stated that rationals $\frac{Q_n}{P_n}$ well approximated a certain irrational $i$. We also stated that the size of stairsteps $I(\frac{Q_n}{P_n})$ near $i$ behaves according to the type $G_\beta$ to which $i$ belongs. By “near $i$” we mean \[Piacquadio and Grynberg, 1998\] steps situated exactly between $\frac{Q_n}{P_n}$ and $\frac{Q_{n-1}}{P_{n-1}}$. If we notice that the distribution of types $G_\beta$ in $[\frac{Q_n}{P_n},\frac{Q_{n-1}}{P_{n-1}}]$ is identical to the distribution of types in $[\frac01,\frac11]$ we conclude that there is a certain relationship of hyperbolic self-similarity between sizes of stairsteps between $\frac{Q_n}{P_n}\;\&\;\frac{Q_{n-1}}{P_{n-1}}$ and sizes of stairsteps between $\frac01\;\&\;\frac11$: there is a subtle underlying law, $(F-B)$ based –hence hyperbolic in nature– that rules the way in which the size of stairsteps decreases. Self-similarity of the underlying Set $\Omega$ ============================================== In the last section we saw that vertical sizes in the staircase decrease hyperbolically and horizontal sizes Euclideanly. We also saw that there was a particular aspect in which we could study the shrinking of intervals of resonance according to a hyperbolic law as well. Such ubiquitous hyperbolic changes point to a question that arises in a natural way: is the underlying set $\Omega$ hyperbolically self-similar? We will consider the set $\Omega$ underlying the circle-map. We have two intervals of resonance $I^1_1$, $I^1_2$ associated with $[\frac01,\frac12]$ and $[\frac12,\frac11]$, respectively, [**via the staircase**]{} (see Sec. 4.2). We have four intervals $I^2_i,\;i=1,...,4$ associated (via the staircase) respectively with segments $[\frac01,\frac13]$, $[\frac13,\frac12]$, $[\frac12,\frac23]$ and $[\frac23,\frac11]$ of the $(F-B)_2$ partition. In general we have $I^k_i,\;i=1,...,2^k$ associated with the $2^k$ segments in the $(F-B)_k$ partition. Segments in the $(F-B)_k$ partition decrease, when $k$ grows, in a hyperbolically self-similar way, and we want to compare sizes of segments $I^k_i$ with sizes of segments in the corresponding $(F-B)_k$ partition. Figs. $2,3,4$ and $5$ show, for $k=3,4,5$ and $6$, sizes of $I^k_i$ in the horizontal axis plotted against the corresponding $(F-B)_k$ sizes in the vertical axis. The fact that each comparative figure shows a straight line indicates that $(F-B)_k$ segments and $I^k$ segments have sizes proportional to one another, the constant of proportionality $m_k$ being the slope of said straight line. It remains to show the law governing the change of these slopes $m_k$ when $k$ grows. A look at Fig. $6$ suggest that $m_k$ grows [**linearly**]{} with $k$. So $\Omega$ is hyperbolically self-similar. If we took [**another**]{} circle-map, would the corresponding $\Omega$ be hyperbolically self-similar as well? Would the $\Omega$ underlying $q=g(-H)$ be hyperbolically self-similar? We conjecture that the answer to both questions is “yes”: provided that the staircase fulfills equation (\[fases\]) and has the $(F-B)$ arrangement in the vertical axis, we conjecture that the underlying $\Omega$ should be hyperbolically self-similar. Conclusions =========== The leading subject in Number Theory of approximating irrational numbers by rational ones can be tackled precisely when real numbers are expressed as continued fraction expansions. There is a partition of $I=[0,1]$ naturally associated with this expansion: the $(F-B)$ partition,... in much the same way as the decimal expansion of real numbers is naturally associated with the decimal partition of $I$ in ten segments of equal length, each of the latter in another ten... and so on. This $(F-B)$ partition is, in turn, naturally associated with the hyperbolic measure in $\Bbb H$. Now, in order to tackle the problem of approximating an irrational $i=[a_1,...,a_n,...]$ by rationals $[a_1,...,a_n]=\frac{Q_n}{P_n}$, Jarník classified irrationals in $J_\beta$ classes, $\beta\geq 2$, according to the corresponding degree of approximation –i.e. to the speed of convergence of $\frac{Q_n}{P_n}$ to $i$. In order to study Cantor staircases in physics –forced pendulum, magnetization, etc.– showing the $(F-B)$ arrangement for intervals $I(\frac{Q}{P})$, a natural connection with Number Theory appears, precisely due to the ubiquitous presence of the $(F-B)$ partition. But when closely examining the behaviour of these staircases, we were forced to considerably refine the $J_\beta$ nested classes into the $G_\beta$ disjoint ones. [**We are saying that problems in empirical physics produced a refinement of key tools in Number Theory.**]{} The properties of these $G_\beta$, $\beta\geq 2$, allowed us to extract theoretical and practical information about the multifractal spectrum of such cantordusts $\Omega$ underlying Cantor staircases in physics, and about the nature of the self-similarity of said stircases. [**References**]{} Bak, P. \[1986\] “The Devil’s staircase,” [*Phys. Today*]{}, December 1986, 38-45. Bruinsma, R. and Bak, P. \[1983\] “Self-similarity and fractal dimension of the devil’s staircase in the one-dimensional Ising model,” [*Phys. Rev.*]{} [**B27**]{}(9), 5924-5925. Cvitanovic, P., Jensen, M., Kadanoff, L. and Procaccia, I. \[1985\] “Renormalization, unstable manifolds, and the fractal structure of mode locking,” [*Phys. Rev. Lett.*]{} [**55**]{}(4), 343-346. Falconer, K. \[1990\] [*Fractal Geometry*]{} (John Wiley and Sons, Chichester-New York). Grynberg, S. and Piacquadio, M. \[1995\] “Hyperbolic geometry and multifractal spectra. Part. II,” [*Trabajos de Matemática 252, Instituto Argentino de Matemática, CONICET.*]{} Halsey, T., Jensen, M., Kadanoff, L., Procaccia, I. and Shraiman, B. \[1986\] “Fractal measures and their singularities: The characterization of strange sets,” [*Nucl. Phys. B, Proc. Suppl. 2*]{}, 513-516. Piacquadio, M. and Grynberg, S. \[1998\] “Cantor staircases in physics and diophantine approximations,” [*International Journal of Bifurcation and Chaos*]{} [**8**]{}(6), 1095-1106. Series, C. \[1985\] “The modular surface and continued fractions,” [*Journal of the London Mathematical Society*]{} [**31**]{}(2), 69-80. [**Figure Captions**]{} {width="120.00000%"} ![Horizontal variable $X$ is the size of $I^3_i$ for $i=1,2,3,4$. Variable $Y$ is the size of the corresponding $(F-B)_3$ segment. The [*other*]{} four $(i=5,6,7,8)$ out of a total of $2^3$ points coincide exactly with the four points shown in the figure.](d4.eps){width="120.00000%"} {width="120.00000%"} {width="120.00000%"} {width="120.00000%"} {width="120.00000%"} [^1]: e-mail: [email protected]

--- author: - 'H. Lehmann' - 'A. Tkachenko' - 'T. Semaan' - 'J. Gutiérrez-Soto' - 'B. Smalley' - 'M. Briquet' - 'D. Shulyak' - 'V. Tsymbal' - 'P. de Cat' date: 'Received ; accepted' title: 'Spectral analysis of Kepler SPB and $\beta$Cep candidate stars[^1]' --- Introduction ============ The Kepler satellite delivers light curves of unique accuracy and time coverage, providing unprecedented data for the asteroseismic analysis. The identification of non-radial pulsation modes and the asteroseismic modeling, on the other hand, require a classification of the observed stars in terms of , , , and metallicity. These basic stellar parameters can only be obtained from the colors of the stars which are not measured by the satellite. That is why ground-based multi-color and spectroscopic observations of the Kepler target stars are urgently needed. We describe a semi-automatic method of spectrum analysis based on high-resolution spectra of stars in the Kepler satellite field of view that have been proposed by the Working Groups 3 and 6 of the Kepler Asteroseismic Science Consortium (KASC) to be candidates for SPB and $\beta$Cep pulsators. The object selection was mainly based on the data given in Kepler Input Catalogue (KIC). The spectral types given for these stars in the SIMBAD database are based on only a few, older measurements and in most cases no luminosity class is given. For the KIC data, on the other hand, an uncertainty of the surface gravity of $\pm$0.5 dex is stated in the catalogue, much too high for an accurate classification in terms of non-radial pulsators, and there are hints that the temperature values given in the KIC show larger deviations for the hotter stars (see Molenda-Żakowicz et al. [@Molenda]). For two of the selected stars we found no classification in the SIMBAD database and for two no classification in the KIC. The aim of this work is to provide fundamental stellar parameters like effective temperature , surface gravity , metallicity $\epsilon$, and projected rotation velocity  for an asteroseismic modeling of the stars, to compare our results of spectral analysis with the KIC data and to estimate the expected type of variability of the different target stars. For the analysis, we used stellar atmosphere models and synthetic spectra computed under the LTE assumption. The advantage of the applied programs (see Sect.3) is that they allow to use different metallicity and individual abundances of He and metals and that they are running fast in parallel mode on a cluster PC installed at TLS. In the investigated spectral region, the spectra of the B-type stars are dominated by the lines. Auer & Mihalas ([@Auer]) state that we have to expect deviations in the equivalent widths of the He lines due to the effects of departure from local thermodynamic equilibrium (LTE) for stars in the 15000 to 27500 K temperature range in the order of 10% for 4026Å$<$$\lambda$$<$5047Å and of up to 30% for the 5876Å line. There is a controversial discussion about these results, however (see Sect.6.1). For that reason, to check for the reliability of our results of the spectral analysis obtained in the LTE regime, we repeated the analysis for the four hottest stars of our sample using programs with non-local thermodynamic equilibrium (NLTE) capability (Sect.4) but had to assume constant, solar metallicity and He abundance. One further test is described in Sect.5, where we derive the effective temperatures of the stars from their available photometry and try to explain the deviations of the given in the KIC from our spectroscopically determined values. The results obtained from the different methods are discussed in Sect.6. Observations and spectrum reduction =================================== Spectra of 16 bright (V$<$10.4) suspected SPB and $\beta$ Cep stars selected from the KIC have been taken with the Coude-Echelle spectrograph attached to the 2-m telescope at the Thüringer Landessternwarte (TLS) Tautenburg. The spectra have a resolution of 32000 and cover the wavelength range from 470 to 740 nm. Table\[objects\] gives KIC number, common designation, suspected type of variability, $V$-magnitude, number of observed spectra, and the signal-to-noise of the averaged spectra for all observed stars. Spectra have been reduced using standard ESO-MIDAS packages. The reduction included filtering of cosmic rays, bias and stray light subtraction, flat fielding, optimum extraction of the Echelle orders, wavelength calibration using a Th-Ar lamp, normalization to the local continuum, and merging of the orders. Small shifts of the instrumental zero-point have been corrected by an additional wavelength calibration using a large number of telluric O$_2$-lines. Finally, the difference in the radial velocities (RVs) between all spectra of the same star have been determined from cross-correlation and the spectra have been rebinned according to theses differences and added to build the mean, averaged spectrum. 1.5mm designation type $N$ S/N ---------- ------------------- ----------------- ------ ----- ----- 3240411 GSC 03135$-$00115 $\beta$Cep 10.2 2 67 3756031 GSC 03135$-$00619 $\beta$Cep 10.0 2 80 5130305 HD  226700 SPB 10.2 2 74 5217845 HD  226628 SPB 9.3 2 103 5479821 HD  226795 SPB 9.9 1 65 7599132 HD  180757 SPB 9.3 1 75 8177087 HD  186428 SPB, $\beta$Cep 8.1 1 138 8389948 HD  189159 SPB 9.1 1 81 8451410 HD  188459 SPB 9.1 2 104 8459899 HD  190254 $\beta$Cep 8.7 2 127 8583770 HD  189177 SPB 10.1 2 74 8766405 HD  187035 SPB, $\beta$Cep 8.8 1 103 10960750 BD+482781 SPB, $\beta$Cep 9.7 1 59 11973705 HD  234999 SPB 9.1 2 116 12207099 BD+502787 $\beta$Cep 10.3 2 69 12258330 HD  234893 SPB, $\beta$Cep 9.3 2 105 : List of observed stars. \[objects\] LTE based analysis ================== The method ---------- Due to the large  of many of the observed stars it is impossible to find a sufficiently large number of unblended spectral lines for a spectral analysis based on the comparison of the equivalent widths of the lines of single elements. Thus we decided to analyse the spectra by computing synthetic spectra for a wider spectral region, including H$_{\beta}$ and a larger number of metal lines. We used the range from 472 to 588 nm (lower end of the covered wavelength range up to the wavelength where stronger telluric lines occur). We used the LLmodels program (Shulyak et al. [@Shulyak]) in its most recent parallel version to compute the atmosphere models and the SynthV program (Tsymbal 1996) in a parallelized version written by A.T. to compute the synthetic spectra. The LLmodels code is a 1-D stellar model atmosphere code for early and intermediate type stars assuming LTE which is intended for as accurate a treatment as possible of the line opacity using a direct method for the line blanketing calculation. This line-by-line method is free of any approximations so that it fully describes the dependence of the line absorption coefficient on frequencies and depths in a model atmosphere, it does not require pre-calculated opacity tables. The code is based on modified ATLAS9 subroutines (Kurucz [@KuruczA]) and the continuum opacity sources and partition functions of iron-peak elements from ATLAS12 (Kurucz [@KuruczB]) are used. Like the SynthV program, it can handle individual elemental abundances. Actually, the main limitation with respect to hot stars is that both programs assume LTE. The line tables have been taken from the VALD data base (Kupka et al. [@Kupka]). They are adjusted by the mentioned programs according to the different spectral types. For the comparison with the synthetic spectra, the observed spectra have been rebinned in wavelength according to their RVs obtained from the cross-correlation with the computed spectra and averaged in the case of several observations per star. We computed the synthetic spectra on a grid in $T_{\rm eff}$, $\log{g}$, $v\sin{i}$, and metallicity $\epsilon$, based on a pre-calculated library of atmosphere models. The models have been computed by scaling the solar metal abundances according to the different metallicities from $-$0.8 to +0.8 dex, assuming constant, solar He abundance and a micro-turbulent velocity of 2. At this point, we derived the value of the metallicity and its error that we will give later in Table\[ResPar\]. In a second step, we fixed the previously derived parameters to its optimum values and took the abundance table corresponding to the determined metallicity as the starting point to readjust the abundances of He and all metals for which we found measurable contributions in the observed spectra. Here, we iterated the individual abundances together with . \ In the final step, we readjusted the values of , , and  based on the abundances determined in step two and added the micro-turbulent velocity $\xi_t$ as a free parameter. We were not able to compute all the atmosphere models for the full parameter space including the individual abundances of He and metal lines and different $\xi_t$, however. We used the atmosphere models computed for the fixed, optimum metallicity determined in step 1 but computed the synthetic spectra with the SynthV program based on the individual abundances determined in step 2. A comparison of the derived metallicities and metal abundances showed that the derived values are compatible in most cases. Only for the three stars where we found larger deviations of the He abundance (the He weak star KIC5479821 and the He strong stars KIC8177087 and KIC12258330), we had to calculate and use atmosphere models based on exactly the determined He abundances. The final values of the parameters and its errors have been taken from step 1 for the metallicity, from step 2 for the individual abundances (here, we did no error calculation but assume the error derived for the metallicity to be the typical error), and from the last step for all the other parameters. The applied method of grid search allows to find the global minimum of $\chi^2$ and for a realistic estimation of the errors of the parameters as we will show in the next section. Testing the method ------------------ The method has been tested on a spectrum of Vega taken with the same instrument and resolution, with the aim to check for the reliability of the obtained values and for the influence of the different parameters on the accuracy of the results. Fig.\[VegaFig\] shows the $\chi^2$-distributions obtained from the grid search. Each panel contains all $\chi^2$ values up to a certain value obtained from all combinations of the different parameters versus one of the parameters. The dashed lines indicate the 1$\sigma$ confidence level obtained from the $\chi^2$-statistics assuming that for a large number of degrees of freedom, the $\chi^2$-distribution approaches a Gaussian one. The continuous curves show the polynomial fit to the smallest $\chi^2$-values. The three crosses in each panel show the optimum value and the $\pm 1\sigma$ error limits of the corresponding parameter. Table\[VegaRes\] lists the resulting values. Table\[VegaComp\] compares the results with values from the literature. Our values of $T_{\rm eff}$ and $\log{g}$ agree well with those from previous investigations. The metallicity and micro-turbulent velocity have been used by us as free parameters as well. The obtained values confirm those assumed by the other authors and the obtained value of  is identical with that measured by Hill et al. ([@Hill]). 1.2mm -------------------------------------------- ------------------------------------------- ---------------------------------- $T_{\rm eff} = 9540^{+140}_{-100}$ K $\xi_t = 2.41^{+0.37}_{-0.35}$ kms$^{-1}$ $\log(g) = 3.92^{+0.07}_{-0.05}$ $v\sin(i) = 21.9^{+1.1}_{-1.1}$ kms$^{-1}$ $\epsilon = -0.58^{+0.10}_{-0.08}$ dex -------------------------------------------- ------------------------------------------- ---------------------------------- : Vega: Comparison of the results. \[VegaRes\] 2.6mm ---- ---------- ------ ------ --------- --------- ------ model $\xi_t$ K 1) ATLAS6 9400 3.95    0.0 2.0 2) MARCS 9650 3.90    0.0 3.0 3) ATLAS6 9500 3.90    0.0 2.0 4) ATLAS6 9500 3.90 $-$0.5 2.0 5) ATLAS12 9550 3.95 $-$0.5 2.0 6) ATLAS9 9506 4.00 $-$0.6 1.1 21.9 7) LLmodels 9540 3.92 $-$0.58 2.4 21.9 ---- ---------- ------ ------ --------- --------- ------ : Vega: Comparison of the results. \ 1) Kurucz (1979), 2) Dreiling & Bell (1980), 3) Lane & Lester (1984), 4) Gigas (1986), 5) Castelli & Kurucz (1994), 6) Hill et al. (2004), 7) our result \[VegaComp\] ------------------------------------------------------------------------------------ $\epsilon$ $\xi_t$ $\log(g)$ $T_{\rm eff}$ $v\sin(i)$ ------------ --------- -------------------------------- --------------- ------------ fixed 86 **67 & **64 & 98\ 100 & fixed & 100 & 100 & 100\ **56 & 81 & fixed & **39 & 77\ **56 & 78 & **50 & fixed & 95\ 100 & 97 & 100 & 98 & fixed\ ************ ------------------------------------------------------------------------------------ : Reduction in the errors of the parameters in % by fixing one of them. \[VegaErr\] 1.97mm ----------------- ------- ------------------------- ------------------------------------- ------------------- ------------------------ ----------- --------------------- ----------- ----------------------- ---------------------------------- ----- ---------- ------------ $\frac{\d 1\sigma}{\d T_{\rm eff}}$ $\log{g}^{\rm K}$ $\log{g}$ $1\sigma$ $\xi_t$ $1\sigma$ $v\sin{i}$ $\frac{\d 1\sigma}{\d v\sin{i}}$ % kms$^{-1}$ % CDS KIC new 3240411   $20\,980_{-840}^{+880}$ 4.1 $4.01_{-0.11}^{+0.12}$ 0.11 $4.8_{-4.7}^{+2.9}$ 3.8 $42.6_{-4.9}^{+5.1}$ 11 – – B2V 3756031   11177 $15\,980_{-300}^{+310}$ 1.9 4.24 $3.75_{-0.06}^{+0.06}$ 0.06 $0.5_{-0.5}^{+2.3}$ 2.3 $30.8_{-3.1}^{+3.8}$ 11 – B8.5V B5V-IV 5130305   9533 $10\,670_{-200}^{+180}$ 1.8 4.14 $3.86_{-0.07}^{+0.07}$ 0.07 $1.4_{-1.0}^{+0.7}$ 0.8 $155_{-13}^{+13}$ 8.4 B9 A0V B9V-IV 5217845$^{1)}$ 8813 $11\,790_{-260}^{+240}$ 2.1 3.70 $3.41_{-0.08}^{+0.10}$ 0.09 $2.1_{-0.9}^{+0.8}$ 0.8 $237_{-16}^{+16}$ 6.8 B8 A3IV B8.5III 5479821   10850 $14\,810_{-290}^{+350}$ 2.2 4.19 $3.97_{-0.07}^{+0.09}$ 0.08 $0.1_{-0.1}^{+1.3}$ 1.3 $ 85_{-8}^{+8}$ 9.4 B8 B9V B5.5V 7599132   10251 $11\,090_{-140}^{+100}$ 1.1 3.62 $4.08_{-0.06}^{+0.06}$ 0.06 $1.6_{-0.6}^{+0.5}$ 0.6 $ 63_{-4}^{+5}$ 7.1 B9 B9.5IV B8.5V 8177087   9645 $13\,330_{-170}^{+220}$ 1.5 4.10 $3.42_{-0.06}^{+0.06}$ 0.06 $1.3_{-0.7}^{+0.5}$ 0.6 $ 22.2_{-1.7}^{+1.5}$ 7.2 B9 A0V B7III 8389948   8712 $10\,240_{-220}^{+340}$ 2.7 3.61 $3.86_{-0.10}^{+0.12}$ 0.11 $0.8_{-0.8}^{+0.9}$ 0.9 $ 142_{-11}^{+12}$ 8.1 B9V A3IV B9.5V-IV 8451410$^{2)}$ 8186 $8\,490_{-100}^{+100}$ 1.2 3.81 $3.51_{-0.05}^{+0.07}$ 0.06 $3.8_{-0.4}^{+0.4}$ 0.4 $39.8_{-1.4}^{+1.4}$ 3.5 B9V A5IV A3.5IV-III 8459899$^{3)}$ 9231 $15\,760_{-210}^{+240}$ 1.4 4.22 $3.81_{-0.05}^{+0.05}$ 0.05 $1.4_{-1.4}^{+1.6}$ 1.5 $53_{-4}^{+4}$ 7.5 B8 A1V B4.5IV 8583770   7659 $ 9\,690_{-170}^{+230}$ 2.1 3.47 $3.39_{-0.05}^{+0.08}$ 0.07 $1.3_{-0.8}^{+0.6}$ 0.7 $ 102_{-7}^{+9}$ 7.8 B9 A7IV-III A0.5IV-III 8766405$^{3)}$ 10828 $12\,930_{-220}^{+210}$ 1.7 3.67 $3.16_{-0.08}^{+0.08}$ 0.08 $0.0^{+1.2}$ 1.2 $240_{-12}^{+12}$ 5.0 B8 B9IV B7III 10960750   $19\,960_{-880}^{+880}$ 4.4 $3.91_{-0.11}^{+0.11}$ 0.11 $0.0^{+3.1}$ 3.1 $253_{-15}^{+15}$ 5.9 B8 – B2.5V 11973705$^{4)}$ 7404 (4.04) $103_{-10}^{+10}$ 9.7 B9 A9V-IV B8.5V-IV 12207099$^{3)}$ 10711 4.07 $43_{-3}^{+5}$ 9.3 A0 B9V B9III–II 12258330   13224 $14\,700_{-200}^{+200}$ 1.4 4.86 $3.85_{-0.04}^{+0.04}$ 0.04 $0.0^{+0.8}$ 0.8 $130_{-8}^{+8}$ 6.2 B9V B7V B5.5V-IV ----------------- ------- ------------------------- ------------------------------------- ------------------- ------------------------ ----------- --------------------- ----------- ----------------------- ---------------------------------- ----- ---------- ------------ \ $^{1)}$binary, $^{2)}$suspected SB2, RV var., $^{3)}$suspected SB2, $^{4)}$SB2 \[ResPar\] ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- $\epsilon$ He C N O Mg Si S Ca Fe ----------------- ---------- ------------------- ------------------------------------------------------------------------------------------------------------------------------ --------- ----------- ----------- --------- --------- ----------- ----------- --------- 8451410   8490 $+0.10$$\pm$0.10 – – – $+$0.03 $+$0.11 $+$0.03 $-$0.05 $-$0.47 $+$0.34 8583770   9690 $+0.18$$\pm$0.09 0.89 – – $+$0.18 $+$0.31 $+$0.13 $-$0.20 $+$0.23 $+$0.24 8389948   10240 $+0.10$$\pm$0.12 1.24 – – $+$0.23 $+$0.26 $+$0.28 $\pm$0.00 $\pm$0.00 $+$0.14 5130305   10670 $-0.07$$\pm$0.11 1.13 – – $+$0.03 $-$0.04 $+$0.03 $-$0.05 $-$0.37 $+$0.09 12207099$^{1)}$ $<$11000 ($>$0.8) 1.40 – – $\pm$0.00 $+$0.56 $-$0.47 $-$0.10 $-$0.62 $+$0.49 7599132   11090 $+0.06$$\pm$0.10 0.89 – – $+$0.23 $+$0.16 $+$0.13 $\pm$0.00 $-$0.07 $+$0.09 11973705$^{1)}$ 11150   (0.00$\pm$0.12) 0.77 – – $+$0.33 $+$0.21 $-$0.57 $-$0.40 $+$0.43 $+$0.04 5217845   11790 $-0.06$$\pm$0.10 1.08 $+$0.15 $\pm$0.00 $\pm$0.00 $+$0.31 $+$0.03 $+$0.05 $+$0.38 $-$0.11 8177087   13330 $-0.11$$\pm$0.11 **1.70 &$+$0.20 &$-$0.10 &$+$0.23 &$+$0.06 &$-$0.12 &$-$0.15 &$-$0.17 &$-$0.11\ 8766405   & 12930 &$-$**0.41$\pm$0.12&1.11 & – &$+$**1.10 &$+$**1.11&$-$0.09 &$-$**0.37 &$-$**0.30 & – &$-$**0.56\ 12258330   & 14700 &$-$**0.30$\pm$0.16&**2.10&$\pm$0.00 &$\pm$0.00 &$-$0.07 &$-$0.14 &$-$0.03 &$-$0.30 & – &$-$0.16\ 5479821   & 14810 &($-0.11$$\pm$0.15) &**0.46&**$+$0.95&**$+$1.56&$\pm$0.00 &$-$**0.64&$+$0.13 &$-$**0.40 & – &$+$0.29\ 8459899   & 15760 &$-$**0.45$\pm$0.11&1.47 &$+$0.05 &$\pm$0.00 &$+$**0.38&$-$**0.54&**$-$0.47 &$-$**0.35 & – &$-$**0.46\ 3756031   & 15980 &$-$**0.57$\pm$0.08&1.49 &$-$0.20 &$-$0.09 &$+$**0.38&$-$**0.44&**$-$0.67&**$-$0.60& – &$-$**0.36\ 10960750   & 19960 &$-0.04$$\pm$0.16 &1.00 &$\pm$0.00 &$\pm$0.00 &$\pm$0.00 &$\pm$0.00 &$\pm$0.00 &$\pm$0.00 & – &$\pm$0.00\ 3240411   & 20980 &$-$**0.30$\pm$0.14&1.40 &$-$**0.30 &$+$0.06 &$-$0.12 &$-$0.19 &**$-$0.92&**$-$0.55& – &$-$**0.46\ & && Na & Sc & Ti & Cr & Mn & Y & Ba &&\ 8451410   & 8490 &&**$-$0.58&**$-$2.01&$-$0.11&$+$0.40&$+$0.25 &**$+$0.93&**$+$1.57&&\ 8583770   & 9690 && – &$\pm$0.00 &$+$0.24&$+$0.30&**$+$0.50&**$+$1.13& – &&\ 8389948   & 10240 && – &$\pm$0.00 &$-$0.26&$-$0.20& – & – & – &&\ 5130305   & 10670 && – & – &$-$0.01&$-$0.05& – & – & – &&\ 12207099$^{1)}$&$<$11000 && – & – &$-$0.16&$+$2.30&$\pm$0.00 & – & – &&\ 7599132   & 11090 && – & – &$+$0.09&$+$0.10& – & – & – &&\ ************************************************************************** ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- \ $^{1)}$formal solution, suspected SB2 star \[ResAbund\] Table\[VegaErr\] gives the fraction of the computed error of each parameter in the case of fixing one of the parameters to the error computed when all parameters are considered to be free. The values indicating the largest effects are set in bold face. It shows that there is a strong correlation between , , and metallicity. This can be easily understood because the line strengths are determined by  and $\epsilon$ and the shape of the Balmer line is strongly influenced by  and $\epsilon$.  and $\xi_t$, on the other hand, are much less influenced by the other parameters and fixing them to their optimum values in the error calculation has no effect on the derived errors of the other parameters. The result shows that we have to vary at least , , and metallicity together to get reliable error estimations for each single parameter. Results ------- Table\[ResPar\] lists the fundamental parameters obtained for the 16 target stars.  and  taken from the KIC are given for comparison. The 1$\sigma$-errors of  and  are given in per cent of the determined values, all other errors are absolute values. The spectral types are compared between those given in the SIMBAD database at CDS, those derived from the  and  listed in the KIC, and from our values of  and . For the derivation we used an interpolation based on the tables by Schmidt-Kaler ([@Schmidt]) and by de Jager & Nieuwenhuijzen ([@Jager]). The accuracy in $T_{\rm eff}$ is about 2% (mean value of 1.8%) for most of the stars, only for the two hottest stars we obtain twice this value. Looking for any correlations between the errors of measurement and the absolute values of determined parameters, we found only one, namely for the micro-turbulent velocity in dependence on  (Fig.\[Xi\_err\]). It seems that our method can determine $\xi_t$ with an accuracy of about 0.6for the cooler stars but that the error raises up to almost 4for the hottest stars. It means that for stars hotter than about 15000K it is not possible to determine their micro-turbulent velocity and that for those stars the determination of the other parameters is practically independent of the value of $\xi_t$. There is also a slight correlation between the relative error of  and  that we attribute to the fact that the accuracy of  is lowered by the inclusion of H$_{\beta}$ into the measurement and that for lower  the Balmer lines show narrower profiles. Due to the inclusion of H$_{\beta}$, the mean accuracy of our  determination is of only 8%. The mean error in  is of 0.07. We refer to the previous section where it was shown that all the errors get larger (and more reliable) by determining them from the complete grid in all parameters as we did here. Table\[ResAbund\] lists the resulting metallicities and individual abundances. The upper part gives the abundances of elements that were found in the spectra of most of the stars, the lower part those that have been additionally found for the cooler stars. Larger deviations from the solar values are highlighted in bold face. The solar values refer to the solar chemical composition given by Grevesse et al. ([@Grevesse]). In the following, we give remarks on peculiarities found in some of the stars. A general comparison of our results with the KIC data will be given in Sect.6.2. [*KIC3240411 and 10960750:*]{} Not any data about these two stars from previous investigations could be found. According to our measurement, they are the hottest stars of our sample. KIC3240411 has low metallicity, in agreement with a distinct depletion in Fe, whereas KIC10960750 has solar abundance. [*KIC3756031:*]{} This star shows the lowest metallicity of our sample. All metals where we found contributions in its spectrum are depleted, whereas He is enhanced. The profiles of some of the metal lines are strongly asymmetric. This may indicate pulsation (we will show that the star falls into the SPB instability region) or, because of the chemical peculiarity of the star, rotational modulation due to spots (as found, e.g., by Briquet et al. ([@Briquet]) for four B-type stars showing inhomogeneous surface abundance distributions). [*KIC5217845:*]{} The star has been identified from the Kepler satellite light curve as eclipsing binary of uncertain type with a period of 1.67838 days ([@2010arXiv1006.2815P]). No RV or line profile variations could be detected from our two spectra, however. [*KIC5479821:*]{} The star is He-weak by a factor of 0.46 compared to the solar value. It shows a strong enhancement of C and N and a depletion of Mg and S. All final parameter values have been calculated from atmosphere models based on the derived individual abundances. The given metallicity, derived from a previous step, has no physical meaning. [*KIC8177087:*]{} This is the sharpest-lined star of our sample with =22. Whereas He is strongly enhanced, all metal abundances are close to the solar values. [*KIC8451410:*]{} This is a suspected SB2 star. We observe a RV shift between our two spectra and cannot fit the shifted and co-added spectrum as perfect as in the other cases. The observed strong Ba and Y overabundance and the strong depletion in Sc may be related to the presence of a second component in the observed spectrum. The determined low temperature of about 8500K agrees within the errors of measurement with that given in the KIC. [*KIC8459899:*]{} We suspect, from the slightly larger O-C residuals of the spectrum fit, that this star may be a SB2 star as well but that the obtained parameters are still reliable. It has low metallicity, the derived value agrees well with the depleted metal abundances, except for O which is enhanced. [*KIC8583770:*]{} This is a double star (WDS 19570+4441) with a 3-magnitude fainter companion at a separation of 0.9. According to our analysis, it is Si-strong, also Mg is enhanced. [*KIC8766405:*]{} The star has low metallicity, the derived value agrees well with the depleted metal abundances, except for N and O which are enhanced. [*KIC11973705:*]{} This is certainly a SB2 star, the typical spectrum of a cooler star of 8000 to 9000K can be seen in the O-C residuals. All values given here result from a formal solution for the hotter component that shows the stronger lines. Due to the unknown flux ratio between the two components, the derived values, except for , are not reliable. [*KIC12207099:*]{} For this sharp-lined star we have got no satisfactory solution. Maybe that it is a Chromium star (we obtain a Cr abundance of 2.30 dex above the solar one) and vertical stratification of the elemental abundances has to be included into the calculations, or it is a SB2 star as well. [*KIC12258330:*]{} The star is He-strong, it shows an enhancement in He abundance by a factor of more than two compared to the solar value. The metal abundances are close to the solar values, the derived metallicity is too low (see the remark for KIC5479821). NLTE based analysis =================== Our aim in this attempt was not to investigate the NLTE effects in detail but to check for the order of the deviations in the results of the LTE calculations that may arise from neglecting these effects. We used the GIRFIT program (Frémat et al. [@Fremat06]) to determine the fundamental parameters , , and . This program adjusts synthetic spectra interpolated on a grid of stellar fluxes to the observed spectra using the least squares method. The temperature structure of the atmospheres was computed as in Castelli & Kurucz ([@Castelli03]) using the ATLAS9 computer code (Kurucz[@KuruczA]). Non-LTE level populations were then computed for each of the atoms listed in Table\[table:atoms\] using the TLUSTY program (Hubeny & Lanz [@Hubeny]) and keeping the temperature and the density distributions obtained with ATLAS9 fixed. For the spectral region considered, we used the specific intensity grids computed by Frémat (private communication) for  and ranging from 15000 K to 27000 K and from 3.0 to 4.5, respectively. For $<$15000K we used LTE calculation. We assumed solar metallicity and a micro-turbulence of 2 for all the grids. atom ion levels ----------- ----- ---------------------------------- Hydrogen 8 levels + 1 superlevel 1 level Helium 24 levels 20 levels 1 level Carbon 53 levels, all individual levels 12 levels 9 levels + 4 superlevels 1 level Nitrogen 13 levels 35 levels + 14 superlevels 11 levels 1 level Oxygen 14 levels + 8 superlevels 36 levels + 12 superlevels 9 levels 1 level Magnesium 21 levels + 4 superlevels 1 level : Atomic models used for the treatment of NLTE. \[table:atoms\] Kepler-ID  () ----------------- ----------------- ----------------- ---------------- KIC3240411  20200$\pm$1000 4.0$\pm$0.2     45$\pm$15 (20980$\pm$860) (4.0$\pm$0.1)   (43$\pm$5) KIC10960750  18100$\pm$1000 3.7$\pm$0.2   250$\pm$20 (19960$\pm$880) (3.9$\pm$0.1)   (253$\pm$15) KIC3756031  16100$\pm$800 3.8$\pm$0.2     40$\pm$15 (15980$\pm$300) (3.75$\pm$0.06)   (31$\pm$4) KIC12258330  15400$\pm$800 4.3$\pm$0.2   125$\pm$15 KIC12258330$^*$  14700$\pm$800 4.1$\pm$0.2   125$\pm$15 (14700$\pm$200) (3.85$\pm$0.04) (130$\pm$8) : Fundamental parameters derived from the NLTE model. LTE based values taken from Table\[ResPar\] are given in parentheses. \ $^*$ based on H$_{\beta}$ only \[table:results\] Our investigation of NLTE effects in the four hottest stars of our sample (not regarding the suspected SB2 candidates) is based on the 4720-5050Å spectral region. For the hotter stars, we used the two helium lines at 4921 and 5016 Å that are present in this region to measure . Then we determined the other parameters,  and , by fitting the spectrum in the whole domain between 4720 and 5050Å. For the cooler stars of the sample, the determination of  is based on the metallic and the He I lines. The derived  values of these stars are consistent with those obtained for the $H_{\beta}$ lines. The coolest star, KIC12258330, was analysed using LTE calculations to check for the influence of the usage of different wavelength ranges and the differences in the applied programs. Table \[table:results\] lists the results. The given errors are computed according to the error determination method introduced by Martayan et al. ([@Martayan]) that is based on the computation of theoretical spectra where a Poisson distributed noise has been added. For KIC12258330, we did not find a satisfying solution that fits both the Balmer and the He lines and give two solutions here. For comparison, we also list the parameter values obtained in Sect.3 and visualize the differences in the results in Fig.\[CompTlgg\]. For two of the stars, among them the hottest star KIC3240411, the values of both  and agree within the errors of measurement. For the other two stars, surprisingly for KIC12258330 where we assumed LTE, we observe large differences between the values obtained from the two approaches. These results will be discussed in detail in Sect.6.1. Stellar temperatures from spectral energy distributions ======================================================= The method ---------- Stellar effective temperatures can be determined from the spectral energy distributions (SEDs). For our target stars, these were constructed from photometry taken from the literature. 2MASS ([@2006AJ....131.1163S]), Tycho B and V magnitudes (), USNO-B1R magnitudes ([@2003AJ....125..984M]), and TASSI magnitudes ([@2006PASP..118.1666D]), supplemented with CMC14 $r'$ magnitudes () and TD-1 ultraviolet flux measurements ([@1979BICDS..17...78C]) where available. The SED can be significantly affected by interstellar reddening. We have determined the reddening from interstellar NaD lines present in our spectra. For resolved multi-component interstellar NaD lines, the equivalent widths of the individual components were measured using multi-Gaussian fits. The total  in these cases is the sum of the reddening per component, since interstellar reddening is additive ().  was determined using the relation given by these authors. Several of stars have $UBV$ photometry which allows us to determine  using the Q-method ([@1973IAUS...54..231H]). For these stars, there is good agreement with the extinction obtained from the NaD lines. The SEDs were de-reddened using the analytical extinction fits of [@1979MNRAS.187P..73S] for the ultraviolet and [@1983MNRAS.203..301H] for the optical and infrared. The stellar $T_{\rm eff}$ values were determined by fitting solar-composition (Kurucz [@KuruczA]) model fluxes to the de-reddened SEDs. The model fluxes were convolved with photometric filter response functions. A weighted Levenberg-Marquardt, non-linear least-squares fitting procedure was used to find the solution that minimizes the difference between the observed and model fluxes. Since  is poorly constrained by our SEDs, we fixed =4.0 for all the fits. 1.7mm ---------- ------------------- ------ ------------------- ----------------- SED Notes NaD QM 3240411 0.07$\pm$0.01 22280$\pm$1320 CMC14 $r'$ 3756031 0.12$\pm$0.01$^*$ 18470$\pm$   970 CMC14 $r'$ 5130305 0.09$\pm$0.01 11590$\pm$   470 CMC14 $r'$ 5217845 0.25$\pm$0.03 18780$\pm$2250 CMC14 $r'$ 5479821 0.24$\pm$0.03 25280$\pm$3390 7599132 0.02$\pm$0.01 10300$\pm$   130 CMC14 $r'$, TD1 8177087 0.12$\pm$0.01 0.09 13120$\pm$   200 TD1 8389948 0.20$\pm$0.02 0.19 12270$\pm$   550 CMC14 $r'$, TD1 8451410 0.04$\pm$0.01   8560$\pm$   120 CMC14 $r'$ 8459899 0.13$\pm$0.01$^*$ 0.16 14780$\pm$   310 TD1 8583770 0.38$\pm$0.06 16290$\pm$3440 CMC14 $r'$ 8766405 0.10$\pm$0.01$^*$ 0.12 15460$\pm$   750 10960750 0.06$\pm$0.01 0.06 20530$\pm$   980 CMC14 $r'$, TD1 11973705 0.02$\pm$0.01$^*$   7920$\pm$   100 TD1 12207099 0.03$\pm$0.01 12160$\pm$   520 12258330 0.04$\pm$0.01 0.08 15820$\pm$   370 TD1 ---------- ------------------- ------ ------------------- ----------------- :  determined from the NaD lines and from the Q-method, and   obtained from SED-fitting. \[Teff-SED\] $^*$ indicates multi-component interstellar NaD lines Results ------- The results are given in Table \[Teff-SED\]. The uncertainty in $T_{\rm eff}$ includes the formal least-squares error and that from the uncertainty in  added in quadrature. The differences between the photometric, spectroscopic, and the KIC values are shown in Fig.\[Comp3\]. KIC10960750 has $uvby\beta$ photometry. Using the [uvbybeta]{} and [tefflogg]{} codes of [@1985CommULO.....78M], we obtain =0.05, =19320$\pm$800 K, and =3.60$\pm$0.07, which is in good agreement with what we determined from spectroscopy. For three more stars, the  derived from SED fitting agree with those from the spectroscopic analysis within the errors of measurement. For 12 of the 16 targets, the photometric   is close to the spectroscopic  confirming the spectroscopically obtained values. In only one case, for KIC11973705, the photometric  is in favour of the KIC value. This is the star that we identified as a SB2 star where we see the lines of a secondary component in its spectrum. KIC5217845, 5479821 and 8583770 suggest that the interstellar lines comprise of un-resolved multiple components leading to an overestimation of the interstellar reddening and to much too high . One of them, KIC8583770, is a double star (WDS 19570+4441) with a 3-magnitude fainter companion at a separation of 0.9. Discussion ========== The influence of NLTE effects ----------------------------- Four stars have been analysed by two different methods using a) slightly different input physics, b) LTE or NLTE treatment for three of the stars, and c) different wavelength ranges, short in NLTE, wider in LTE. For two of the stars, KIC3240411 and KIC3766081, the results agree well within the errors of measurement. For the other two stars, KIC10960750 and KIC12258330, we find significant deviations. The fact that we find a good agreement for the hottest star and a distinct deviation for the coolest one that was analysed using LTE in both attempts, is surprising and raises questions about the origin of the observed deviations, i.e. about the order of the influence of the differences in the two methods that we labeled a) to c) in before. For KIC12258330, we can clearly show that the difference in the results comes from the facts that it is a helium-strong star and that our second approach assumed solar metallicity and He abundance. Fig.\[CompHe\] shows the quality of different fits, all calculated in LTE. Model 1 is the original LTE solution with the He abundance enhanced by a factor of 2.1 against the solar one and =14700K, =3.85. Models 2 and 3 are based on =15400 K, =4.30 with solar He abundance in model 2 and enhanced He abundance in model 3. It can be seen that varying the He abundance effects not only the strength of the He lines but also the shape of the wings of  and so the derived . Compared to solution 1), the reduced $\chi^2$ of solution 2) is 1.5 times higher and that of solution 3) 3.2 times higher. We assume that the parameter values derived in Sect.3 are the correct ones and that KIC12258330 is helium-strong. A closer investigation of the application to KIC10960750 shows that the differences in  and mainly come from the usage of different wavelength regions. If we shorten the region for the LTE calculations to that used in NLTE we end up with =(18940$\pm$840)K and =3.78$\pm$0.08 which comes much closer to the results obtained from the NLTE calculations. Here, we have to solve the question if the difference in the results comes from the fact that one more stronger He line at 5876Å was included in the wider spectral range used in the LTE approximation which falsifies the LTE results due to additional NLTE effects, or if the difference simply comes from the fact that a wider spectral range gives more accurate results which favours the LTE results. As already mentioned in the introduction, Auer & Mihalas ([@Auer]) state that the deviations in the equivalent widths of the He lines due to the effects of departure from NLTE increases for B-type stars with wavelength and can reach 30% for the 5876Å line and more for redder lines. According to their calculations, NLTE effects are negligible only for the He lines in the blue (up to 4471Å) but produce deeper line cores for the He lines at longer wavelengths whereas the line wings remain essentially unaffected. Hubeny & Lanz ([@Hube2]) state as well that the core of strong lines and lines from minor ions will be most affected by departures from LTE which implies that the abundance of some species might be overestimated from LTE predictions. Also the surface gravities derived from the Balmer line wings tend to be overestimated. Mitskevich & Tsymbal ([@Mitsk]), on the other hand, computed model atmospheres of B-stars in LTE and NLTE and found no remarkable differences. In particular the temperature inversion in the upper layers of the atmospheres, intrinsic to the models of Auer & Mihalas, is absent in their NLTE models and the departure coefficients for the first levels of H and He in the upper layers are lower by three orders of magnitude. The authors believe that the reason for the discrepancy in the results is the absence of agreement between radiation field and population levels in the program applied by Auer & Mihalas. And there is a second point. The TLUSTY program (Hubeny & Lanz [@Hubeny]) provides NLTE fully line-blanketed model atmospheres, whereas Auer & Mihalas did not take the line blanketing into account. In a more recent article, Nieva & Przybilla ([@Nieva]) investigate the NLTE effects in OB stars. Using ATLAS9 for computing the stellar atmospheres in LTE and the DETAIL and SURFACE programs to include the NLTE level populations and to calculate the synthetic spectra, respectively, they compute the Balmer and lines over a wide spectral range and compare the results with those of pure LTE calculations. In the result, they obtained narrower profiles of the Balmer lines in LTE compared to NLT for stars hotter than 30000K, leading to an overestimation of their surface gravities. The calculations done for one cooler star of 20000K which is in the range of the hottest stars of our sample, but for  of 3.0, did not show such effect but differences in the line cores, increasing from H$_{\delta}$ to . The same they observed for the cooler star for the lines. Many of them experience significant NLTE-strengthening, in particular in the red, but without following a strong rule. So the strengthening of the 5876Å line is less then that of the 4922Å and 6678Å lines. From the LTE calculations, we obtain solar metal and He abundance for KIC10960750. Unfortunately, our actually available grid of NLTE synthetic spectra does not comprise the wavelength region of the 5876Å line and so we cannot reproduce the LTE calculations one by one. Since the quality of the fit of the observed spectrum obtained from the LTE and NLTE calculations is the same we can not directly decide if the difference in the parameters comes from NLTE effects of the He 5876Å line as discussed above. But since we do not observe any deviation in the results for the hottest star of our sample, we believe that NLTE effects are only second order effects and cannot give rise to the deviations observed for KIC10960750. Comparison with the KIC data ---------------------------- Comparing our $T_{\rm eff}$ and $\log{g}$ with the values given for 12 of the stars in the KIC (for two of the 16 stars of our sample there are no entries, and we excluded KIC11973705 and 12207099), we see systematic differences. We obtain higher temperatures in general, the difference increases with increasing temperature of the stars. Already Molenda-Żakowicz et al. ([@Molenda]) found that the $T_{\rm eff}$ given in the KIC is too low for stars hotter than about 7000K, by up to 4000K for the hottest stars. The trend in the temperature difference derived from our values is shown in Fig.\[Teff\_KIC\] (the error in the difference also includes the 200K error typical for the KIC data). We have drawn the curve of a second order polynomial resulting from a fit that observes the boundary condition that the difference should be zero for =7000K. Our analysis seems to confirm the finding by Molenda-Żakowicz et al. Comparing our values derived for  with those given in the KIC, we find that they are systematically lower (Fig.\[logg\_KIC\]). Counting for the large errors that mainly result from the $\pm$0.5dex error of the KIC data, we cannot say that this difference is significant, however. The same holds true for the Fe/H ratios given in the KIC, the corresponding error of $\pm$0.5dex prevents us from any comparison with our derived values. There are two extreme cases: KIC3756031, where our derived temperature is 16000K whereas the KIC gives about 11200K, and KIC12258330, where we obtain a $\log{g}$ of 3.85 which is by 1 dex lower than given in the KIC. A closer investigation shows, however, that the KIC values are very unlikely. Fig.\[K01\] shows, in its upper panel, the fit of a part of the spectrum of KIC3756031 assuming our value of $T_{\rm eff}$. The lower panel gives the same for $T_{\rm eff}$ taken from the KIC. Neither H$_{\beta}$ nor the He lines are fitted well. The best fit of H$_{\beta}$ at this temperature is obtained assuming a $\log{g}$ below 3.0, but in this case the He lines can also not be reproduced. The use of LTE in our program cannot balance a temperature difference of 5000K and the 11200 K given in the KIC cannot be true for this star. Fig.\[K02\] shows, in its upper panel, the fit of a part of the spectrum of KIC122583330 assuming our values of  and . The lower panel gives the same for  and   taken from the KIC. The resulting deviation in the shape of H$_{\beta}$ cannot be explained in terms of a wrong continuum normalization in the H$_{\beta}$ range. The $\log{g}$ given for this star in the KIC is by about 1 dex too large.  from spectral energy distributions ----------------------------------- Our method of deriving  from the EWs of the NaD lines may overestimate  in the cases where we observe non-resolved interstellar contributions to the NaD line profiles. This is one reason, besides the poor photometric data for some of the stars, why this method is of lower accuracy compared to the spectroscopic analysis. The facts that the  derived from the NaD lines and from the Q-method for the stars where $UBV$ or, in one case, $uvby\beta$ photometry was available are in a good agreement and that all derived  are in favour of our spectroscopic values confirm that the  given in the KIC must be too low, however. Thus, the results from the SED-fitting based on the photometric data reveal the reason why the  given in the KIC deviate from our findings. For most of the hotter stars, the interstellar reddening was not properly taken into account leading to an underestimation of the stellar temperatures. It also explains why the difference in the derived temperatures between the KIC and our spectroscopic analysis rises with increasing temperature of the stars. The hotter the stars, the more luminous and the farther they are and the more the ignored reddening plays a role. Positions of the stars compared to SPB and $\beta$Cep instability strips ------------------------------------------------------------------------ position spectral type $N$ ------------------------------ ---------- --------------- ----- 3240411 B2   V a \[-1.2ex\][$\beta$Cep/SPB]{} 10960750 B2.5V m 3756031 B5   IV-V b 5479821 B5.5V e \[-1.2ex\][SPB]{} 8459899 B4.5IV j 12258330 B5.5IV-V p 5130305 B9   IV-V c 5217845 B8.5III d 7599132 B8.5V f \[-1.2ex\][possibly SPB]{} 8177087 B7   III g 8389948 B9.5IV-V h 8451410 A3.5IV-III i \[-1.2ex\][too cool]{} 8583770 A0.5IV-III k too evolved 8766405 B7   III l uncertain 11973705 B8.5VI-V n (SB2 stars) 12207099 B9   II-III o \[pulsators\] : Positions of the stars with respect to the instability regions. \[positions\] In the result of our analysis, we can directly place the stars into a –  diagram to compare their positions with the known instability domains of main-sequence B-type pulsators. Fig.\[logTeff\_logg\] shows the resulting plots where the boundaries of the theoretical $\beta$Cep (the hottest region in Fig.\[logTeff\_logg\]) and SPB instability strips have been taken from Miglio et al. ([@miglio]). A core convective overshooting parameter of 0.2 pressure scale heights was used in the stellar models since asteroseismic modeling results of $\beta$Cep targets have given evidence for the occurrence of core overshooting of that order (e.g. Aerts et al. [@aerts]). It is well-known that the choice of the metal mixture, opacities and metallicity also has a large influence on the extent of the instability regions. Here, we illustrate these domains for the OP (upper panel) and the OPAL (lower panel) opacity tables, as well as for two values of the metal mass fraction $Z$=0.01 (continuous boundaries of the instability regions) and $Z$=0.02 (dashed boundaries). However, we only adopt the metal mixture by Asplund et al. ([@asplund]) corrected with the Ne abundance determined by Cunha et al. ([@cunha]). Another choice of metal mixture (e.g., that by Grevesse & Noels [@grevesse_noels]) leads to narrower instability domains, as shown in Miglio et al. ([@miglio]). The stars in Fig.\[logTeff\_logg\] are marked by the letters given in Table\[positions\]. The SB2 star KIC11973705 and the other presumed SB2 star KIC12207099 are marked by two asterisks, as their determined values are uncertain and no error bars can be given. It can be seen that 4 stars (b, e, j and p) fall into the middle of the SPB region. The two hottest stars (a and m) can be of SPB and/or $\beta$Cep nature. Both low order p- and g-modes, and high-order g-modes can be expected for these possibly ’hybrid’ pulsators. Six of the stars (c, d, f, g, h and n) lie on or close to the boundaries of the SPB instability strips so that they possibly exhibit high-order g-mode pulsations. The remaining 4 stars lie outside the instability regions, two of them (k and i) are too cool to be main-sequence SPB stars and the two other ones (l and o) have too low . Table\[positions\] lists the potential pulsators together with their spectral types as derived in Sect.3.3. For the two suspected SB2 stars we do not want to make a classification in terms of pulsators because their determined spectral types may be completely wrong. Conclusions =========== We tried to determine the fundamental parameters of B-type stars from the combined analysis of  and the neighbouring metal lines in high-resolution spectra. Our results obtained for the test star Vega show that we can reproduce the values of , , , metallicity and micro-turbulent velocity known from the literature and that our method works well at least in the 10000K range. The application of our programs to stars hotter than 15000K sets limitations in the accuracy of the results due to the used LTE approximation. The independent analysis of the four hottest stars of our sample by NLTE-based programs showed that the derived parameters agree within the errors of measurements for two of the stars, among them the hottest star. The deviations obtained for the other two stars can be explained by other limitations of the applied methods than the use of LTE and thus we believe that our results are valid within the derived errors of measurement. In particular, the use of LTE cannot explain the large deviations in $T_{\rm eff}$ and $\log{g}$ following for some of the stars from the KIC data, however, as we showed on two examples. From our results, there is strong evidence that the KIC systematically underestimates the temperatures of hotter stars, the difference increases with increasing . This finding confirms the results by Molenda-Żakowicz et al. ([@Molenda]) which observed the same tendency for stars hotter than about 7000K. The calculation of  using SED-fitting based on the available photometric data revealed the reason why the  listed in the KIC are too low and why the difference is largest for the hottest stars: the stellar temperatures have been underestimated because the interstellar reddening was not properly taken into account. Eight stars of our sample show larger abundance anomalies. Five of them have reduced metallicity, two are He-strong, one is He-weak, and one is Si-strong. According to our measurements, two of the 16 investigated stars fall into the overlapping range of the $\beta$Cep and SPB instability regions and could show, as so-called hybrid pulsators, both low-order p- and g-modes and high-order g modes. These are the two hottest stars in our sample, KIC3240411 and KIC10960750. Four stars fall into the SPB instability region, and five more are located close to the borders of this region. The two coolest stars, KIC8451410, and KIC8583770, lie between the SPB instability region and the blue edge of the classical instability strip. Two of the stars, KIC11973705 and KIC12207099, could not be classified because of their SB2 nature and one star, KIC8766405, is too evolved to show $\beta$Cep or SPB-type pulsations. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France, the Vienna Atomic Line Database (VALD), and of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. T.S. is deeply indebted to Dr. Yves Fr[é]{}mat for providing the NLTE specific intensity grids for this study. 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--- abstract: | In this paper the kinematical correlations from the [*phase conjugated optics (* ]{}equivalently with [*crossing*]{} [*symmetric spontaneous parametric down conversion (SPDC) phenomena)*]{} in the nonlinear crystals are used for the description of a new kind of optical[* *]{}device called SPDC-[*quantum mirrors.* ]{}Then,[* s*]{}ome important laws of the[* plane*]{} [*SPDC-quantum mirrors*]{} combined with usual mirrors or lens are proved only by using geometric optics concepts. In particular, these results allow us to obtain a new interpretation of the recent experiments on the [*two-photon geometric optics*]{}. PACS:  42. 50. Tv ; 42. 50. Ar ; 42. 50. Kb ; 03. 65. Bz.  author: - 'M. L. D. Ion and D. B. Ion' title: '[**PLANE SPDC-QUANTUM MIRROR**]{}' --- [**Introduction**]{}[* *]{} =========================== The [*spontaneous parametric down conversion*]{} (SPDC) is a nonlinear optical process \[1\] in which a laser pump (p) beam incident on a nonlinear crystal leads to the emission of a correlated pair of photons called signal (s) and idler (i). If the [*S-matrix crossing symmetry \[2\]* ]{}of the electromagnetic interaction in the [*spontaneous parametric down conversion*]{} (SPDC) crystals is taken into account, then the existence of the [*direct SPDC process*]{} $$\label{1} p\rightarrow s+i$$ will imply the existence of the following [*crossing symmetric processes \[3\]*]{} $$\label{2} p+\stackrel{\_}{s}\rightarrow i$$ $$\label{3} p+\stackrel{\_}{i}\rightarrow s$$ as real processes which can be described by the same[* transition amplitude.*]{} Here, by $\stackrel{\_}{s}$ and $\stackrel{\_}{i}$ we denoted the [*time reversed*]{} [*photons (*]{}or antiphotons in sense introduced in Ref. \[4\][*)*]{} relative to the original photons $s$ and $i$, respectively. In fact the SPDC effects (1)-(3) can be identified as being directly connected with the $\chi ^{(2)}$-[*second-order nonlinear effects*]{} called in general [*three wave mixing* ]{}(see[* *]{}Ref.\[5\][*).* ]{}So,[* *]{}the process (1) is just the [*inverse of second-harmonic generation,* ]{}while, the effects (2)-(3) can be interpreted just as emission of [*optical phase conjugated replicas*]{} in the presence of pump laser via [*three wave mixing.* ]{} In this paper a new kind of geometric optics called [*quantum SPDC-geometric optics*]{} is systematically[* *]{}developed by using [*kinematical correlations of the pump, signal and idler photons*]{} from the SPDC processes. Here we discuss only the plane quantum mirror. Other kind of the SPDC-quantum mirrors, such as spherical SPDC quantum mirrors, parabolic quantum mirrors, etc., will be discussed in a future paper.[* *]{} [**Quantum kinematical correlations**]{}[* *]{} =============================================== In the SPDC processes (1) the energy and momentum of photons are conserved: $$\label{4} \omega _p=\omega _s+\omega _i,\smallskip\ {\bf k_p}={\bf k_s}+{\bf k_i}$$ Moreover, if the crossing SPDC-processes (2)-(3) are interpreted just as emission of [*optical phase conjugated replicas* ]{}in the presence of input pump laser then Eqs. (4) can be identified as being the[* phase matching*]{} [*conditions*]{} in the three wave mixing (see again Ref. \[5\]). Indeed, this scheme exploits the second order optical nonlinearity in a crystal lacking inversion symmetry. In such crystals, the presence of the input pump (${\bf % E_p}$) and of the signal (${\bf E_s}$[**)**]{} fields[* *]{} induces in the medium a [*nonlinear optical polarization* ]{}(see Eqs. (26)-(27) in Pepper and Yariv Ref.\[5\]) which is: $P_i^{NL}=\chi _{ijk}^{(2)}E_{pj}(\omega _p)E_{sk}^{*}(\omega _s)\exp \{i[(\omega _p-\omega _s)t-({\bf k_p-k_s)\cdot r% }]\}+c.c.,$ where $\chi _{ijk}^{(2)}$ is the susceptibility of rank two tensor components of the crystal. Consequently, such polarization, acting as a [*source*]{} in the [*wave equation*]{} will radiate a [*new wave*]{} $% {\bf E_i}$ at frequency $\omega _i=$ $\omega _p-\omega _i,$ with an amplitude proportional to ${\bf E_i^{*}}(\omega _i),$ i.e., to the[*complex*]{} [*conjugate*]{} of the spatial amplitude of the low-frequency probe wave at $\omega _s.$ Then, it is easy to show that a necessary condition for a [*phase-coherent*]{} cumulative buildup of [*conjugate-field*]{} [*radiation*]{} at $\omega _i=\omega _p-\omega _s$ is that the wave vector [**k$% _i$**]{} at this new frequency must be equal to[** k$_i$**]{}$={\bf k}_p-{\bf k}% _s,$ i.e., we have the phase matching conditions (4). Hence, the [*optical*]{} [*phase conjugation by three-wave mixing* ]{} help[* *]{} us to obtain a complete proof of the existence of the crossing reactions (2)-(3) as real processes which take place in the nonlinear crystals when the [*energy-momentum* ]{}(or [*phase*]{} [*matching) conditions*]{} (4) are fulfilled.[* *]{} Now, it is important to introduce the [*momentum projections*]{}, parallel and orthogonal to the pump momentum, and to write the momentum conservation law from (4) as follows $$\label{5} k_p=k_s\cos \theta _{ps}+k_i\cos \theta _{pi}$$ $$\label{6} k_s\sin \theta _{ps}=k_i\sin \theta _{pi}$$ where the angles $\theta _{pj,}$ $j=s,i,$are the angles (in crystal) between momenta of the [*pump*]{} (p)$\equiv $[*($\omega _p,$*]{}${\bf k}$[**$% _p,e_p,\mu _p)$**]{}, [*signal*]{} (s)$\equiv ($[*$\omega _s,$*]{}${\bf k}$[**$% _s,$**]{}${\bf e}$[**$_s,\mu _s)$**]{} and [*idler* ]{}(i)$\equiv $([*$\omega _i,$*]{}${\bf k}$[**$_i,e_i,\mu _i)$**]{} [*photons.*]{} By[** e$_j$**]{} and $\mu _j{\bf ,\ }j\equiv p,s,i,$[** **]{}we denoted the photon polarizations and photon helicities, respectively. Now, let $\beta _{ps},$and $\beta _{pi}$ be the corresponding exit angles of the signal and idler photons from crystal. Then from (6) in conjunction with Snellius law, we have $$\label{7} \sin \beta _{ps}=n_s\sin \theta _{ps},\smallskip\ \sin \beta _{pi}=n_i\sin \theta _{pi}$$ $$\label{8} \omega _s\sin \beta _{ps}=\omega _i\sin \beta _{pi}$$ **Quantum mirrors via SPDC phenomena** ======================================= (D.1) [**Quantum Mirror** ]{}(QM). By definition a [*quantum mirror (QM) is a combination of standard devices*]{} (e.g., usual lenses, usual mirrors, lasers, etc.) with a nonlinear crystal [*by which one involves the use of a variety of quantum phenomena to exactly transform ${\bf \ }$not only the direction of propagation of a light beam but also their polarization characteristics.*]{} (D.2) [**SPDC**]{}-[**Quantum Mirror**]{} (SPDC-QM). A [*quantum mirrors*]{} is called SPDC-QM if is based on the quantum SPDC phenomena (1)-(3) in order to transform [*signal photons*]{} characterized by [*($\omega _s,{\bf % k_s,e_s,\mu _s)}$*]{}${\bf \ }$into [*idler photons*]{} with [*($\omega _p-\omega _s,{\bf k_p-}{\bf k_s,e_s^{*},-\mu _s)\equiv }$($\omega _i,{\bf % k_i,e_i,\mu _i)}$*]{}. Now, since the crossing symmetric SPDC effects (2)-(3) can be interpreted just as emission of [*optical phase conjugated replicas*]{} in the presence of pump laser via [*three wave mixing,* ]{}the high quality of the SPDC-QM will be given by the following peculiar characteristics: (i)[* Coherence*]{}:The SPDC-QM [*preserves high coherence*]{} between s-photons and i-photons; (ii) [*Distortion undoing:*]{} The SPDC-QM [*corrects all the aberrations*]{} which occur in signal or idler beam path; (iii) [*Amplification:* ]{}A SPDC-QM [*amplifies the conjugated wave*]{} if some conditions are fulfilled.   [*3.1. Plane SPDC-quantum mirrors.*]{} The quantum mirrors can be [*plane quantum mirrors* ]{}(P-QM) (see Fig.1), [*spherical quantum mirrors (S-QM), hyperbolic quantum mirror (H-QM), parabolic quantum mirrors (PB-QM), etc., *]{}according with the character of incoming laser wave fronts ( [*plane waves,*]{} [*spherical waves, etc.).* ]{}Here we discuss only the[* plane SPDC-quantum mirror.*]{} Other kind of the [*SPDC-quantum mirrors*]{}, such as [*spherical SPDC quantum mirrors, parabolic quantum mirrors*]{}, etc., will be discussed in a future paper.[* *]{} In order to avoid many complications, in the following we will work only in the [*thin crystal approximation*]{}. Moreover, we do not consider here the so called optical aberrations. (L.1)[* *]{}Law of [*thin plane SPDC-quantum mirror*]{}: Let BBO be a SPDCcrystal illuminated uniform by a high quality laser pump. Let Z$_s$ and Z$_i $ be the distances shown in Fig.1 ( from the [*object point*]{} P to crystal (point A) and from crystal (point A) to [*image point*]{} I$.$ Then, the system behaves as a [*plane mirror*]{} but satisfying the following important laws: $$\label{9} \frac{Z_i}{Z_s}=\frac{\omega _i}{\omega _s}=\frac{\sin \beta _{ps} }{\sin \beta _{pi}}=\frac{n_s\sin \theta _{ps}}{n_i\sin \theta _{pi}},\smallskip\ M=% \frac{\omega _sZ_i}{\omega _iZ_s}=1$$ where M is the[* linear magnification* ]{}of[* *]{}the plane SPDC-quantum mirror. [*3.2. Plane SPDC-QM combined with thin lens.* ]{}The basic optical geometric configurations of a plane SPDC-QM combined with thin lens is presented in Figs. 2a and 2b. The system in this case behaves as in usual geometric optics but with some modifications in the non degenerate case introduced by the presence of the [*plane SPDC-quantum mirror*]{}. The remarkable law in this case is as follows. (L.2) [*Law*]{} of the[* thin lens combined with a plane SPDC-QM: The distances S ( lens-object), S’(lens-crystal-image plane), D$_{CI}$ (crystal-image plane) and f (focal distance of lens), satisfy the following thin lens equation $$\label{10} \frac 1S+\frac 1{S^{\prime }+(\frac{\omega _s}{\omega _i}-1)\:% D_{CI}}=\frac 1f$$ The SPDC-QM system in this case has the magnification M given by*]{} $$\label{11} M=\frac{S^{\prime }+(\frac{\omega _s}{\omega _i}-1)\:D_{CI}}S=M_0+(% \frac{\omega _s}{\omega _i}-1)\frac{D_{CI}}S$$ In degenerate case $(\omega _s=\omega _i=\omega _p/2)$ we obtain the usual [*Gauss law for thin lens*]{} with the magnification $M_0=S^{\prime }/S$. [*Proof:*]{} The proof of the predictions (10)-(11) can be obtained by using the basic geometric optical configuration presented in Fig. 2a. Hence, the image of the object P in the thin lens placed between the crystal and object is located according to the Gauss law $$\label{12} \frac 1S+\frac 1{S_1}=\frac 1f$$ where $S_1$ is the distance from lens to image I$_1.$ Now the final image I of the image I$_1$ in the plane SPDC-QM is located according to the law (9). Consequently, if d is the lens-crystal distance then we have $$\label{13} S_1=S^{^{\prime }}+(Z_s-Z_i)=S^{^{\prime }}+(\frac{\omega _s}{\omega _i}% -1)D_{CI}$$ since S$_1=d+Z_s,$ S’=d+Z$_i,$and D$_{CI}$ is the crystal-image distance. A proof a the magnification factor can be obtained on the basis of geometric optical configuration from Fig. 2b. Hence, the magnification factor is $$\label{14} M=\frac{y_I}{y_O}=\frac{y_I}{y_I^{\prime }}\cdot \frac{y_I^{\prime }}{y_O}=% \frac{y_I^{\prime }}{y_O}$$ since the plane SPDC-QM has the magnification $\frac{y_I}{y_I^{\prime }}=1.$ Obviously, from $\Delta PP^{\prime }V\sim \Delta I_1I_1^{\prime }V,$we get y$% _I^{\prime }$/y$_O=S_1/S$ and then with (13) we obtain the magnification (11). [*(L.3)*]{} [*Law of thin lens + plane SPDC-QM with the null crystal-lens distance*]{} $$\label{15} \frac 1S+\frac 1{\frac{\omega _s}{\omega _i}S^{\prime }}=\frac 1f \:,% \smallskip\ M=\frac{\omega _s}{\omega _i}\frac{S^{\prime }}S$$ [*Proof:* ]{}Here we note that (L.4) is the particular case of (L.3) with d=0 for which we get S$_1=Z_s,$and S’=Z$_i.$ Then from (9) and (12) we obtain (15). [*3.3. Thin lens combined with plane SPDC-QM and classical mirror.*]{} [*(L.4)* ]{} [*Law of thin lens + plane SPDC-QM +classical mirror (*]{} see the basic geometric optical configuration presented in Fig. 3). The distances S ( lens-object), S$_1^{^{\prime }}$(lens-crystal-first image plane I$_1$), S$_2^{^{\prime }}$(lens-crystal-second image plane I$_2$), D$% _{CI_1}$ (crystal-first image plane), D$_{CI_2}$ (crystal-second image plane) and f (focal distance of lens), must satisfy the following law[*  $$\label{16} \frac 1S+\frac 1{S_1^{\prime }+(\frac{\omega _s}{\omega _i}-1) \:% D_{CI_1}}=\frac 1f$$* ]{}and the magnification M$_1$ given by $$\label{17} M_1=\frac{S_1^{\prime }+(\frac{\omega _s}{\omega _i}-1)\:D_{CI_1}}S$$ and[*  $$\label{18} \frac 1{S+2D_{OM}}+\frac 1{S_2^{\prime }+(\frac{\omega _s}{\omega _i}-1)% \:D_{CI_2}}=\frac 1f$$* ]{} the magnification M$_2$ given by $$\label{19} M_2=\frac{S_2^{\prime }+(\frac{\omega _s}{\omega _i}-1)\:D_{CI_2}}S$$ where D$_{OM}$ is the distance from object to the classical mirror M (see Fig. 3). The [*proof of*]{} [*(L.4)*]{} is similar to that of[* (L.3)*]{} and here will be omitted. **Experimental tests for the geometric SPDC-quantum optics** ============================================================ For an experimental test of [*the Gauss like law of the thin lens combined with a plane SPDC-QM* ]{}we propose an experiment based on a detailed setup presented in Fig. 4 and in the optical geometric configuration shown in Fig. 2b. Then, we predict that the image I of the object P (illuminated by a high quality signal laser SL with s[*($\omega _s,{\bf k_s,e_s,\mu _s))}$*]{} will be observed in the idler beam, i[*($\omega _i,{\bf % k_i,e_i,\mu _i)\equiv }$*]{}i($\omega _p$-$\omega _s$,[**k$_p$-**]{}[**k$_s$,e$% _s^{*}$,-$\mu _s$**]{}), when distances lens-object (S), lens-crystal-image plane (S’), crystal-image plane (D$_{CI})$ and focal distance f of lens, satisfy [*thin lens+QM law (10).* ]{}Moreover[*,* ]{}if [*thin lens+QM law (10)*]{} is satisfied, the image I of that object P can be observed even when instead of the signal source SL we put a detector D$_s.$ This last statement is clearly confirmed recently, in the degenerate case $\omega _s=\omega _i=\omega _p/2,$ by a remarkable [*two-photon imaging experiment*]{} \[8\]. Indeed, in these recent experiments, inspired by the papers of Klyshko et al (see refs. quoted in \[9\]), was demonstrated some unusual [*two-photon effects*]{}, which looks very strange from classical point of view. So, in these experiments, an argon ion laser is used to pump a nonlinear BBO crystal ($\beta -BaB_2O_4)$ to produce pairs of [*orthogonally polarized photons*]{} (see Fig. 1 in ref. \[8\] for detailed experimental setup). After the separation of the [*signal*]{} and[* idler* ]{}beams, an aperture (mask) placed in front of one of the detectors (D$_s$) is illuminated by the [*signal beam* ]{}through a convex lens. The surprising result consists from the fact that an image of this aperture is observed in coincidence counting rate by scanning the other detector (D$_i$) in the transverse plane of the idler beam, even though both detectors single counting rates remain constants. For understanding the physics involved in their experiment they presented an ”equivalent ” scheme ( in Fig. 3 in ref. \[8\]) of the experimental setup. By comparison of their ”scheme” with our optical configuration from Fig. 2b we can identify that the observed validity of the [*two-photon*]{} [*Gaussian thin-lens equation*]{} $$\label{20} \frac 1f=\frac 1S+\frac 1{S^{\prime }}$$ as well as of the[* linear magnification*]{} $$\label{21} M_0=\frac{S^{\prime }}S=2$$ can be just explained by our results on the two-photon geometric law (10)-(11) [*of the thin lens combined with a plane SPDC-QM* ]{}for the degenerate case $\omega _s=\omega _i=\omega _p/2.$ Therefore, the general tests of the predictions (10)-(11) using a setup described in Fig. 4, are of great importance not only in measurements in presence of the signal laser LS (with and without coincidences between LS and idler detector D$_i),$ but also in the measurements in which instead of the laser LS we put the a signal detector D$_s$ in coincidence with D$_i.$   **Conclusions** =============== In this paper the class of the [*SPDC-phenomena*]{} (1) is enriched by the introducing the [*crossing symmetric*]{} [*SPDC-processes* ]{}(2)-(3) satisfying the same energy-momentum conservation law (4). Consequently, the kinematical correlations (4)-(8) in conjunction with the Snellius relations (7) allow us to introduce a new kind of optical devices called [*quantum mirrors.* ]{}Then, some laws of the [*quantum mirrors,* ]{}such as:[* *]{}law (9) of [*thin plane SPDC-quantum mirror,* ]{}the [*law*]{} (10)-(11) [*of the thin lens combined with a plane SPDC-QM,* ]{}as well as,[* the laws (16)-(19),*]{} [* *]{}are proved. These results are natural steps towards a [*new geometric optics* ]{}which can be constructed for the kinematical correlated SPDC-photons. In particular, the results obtained here are found in a very good agreement with the recent results \[8\] on [*two-photon imaging experiment*]{}. Moreover, we recall that [* *]{}all the results obtained in the [*two-photon ghost interference-diffraction*]{} experiment \[6\] was recently explained by using the concept of [*quantum mirrors (*]{}see Ref. \[3\]). Finally, we note that all these results can be extended to the case of the [*spherical quantum mirrors.* ]{}Such results, which are found in excellent agreement to the recent experimental results \[7\] on [*two-photon geometric optics*]{}, will be presented in a future publication.   (This paper was published in Romanian Journal of Physics, Vol.45, P. 15, Bucharest 2000) [99]{} A. Yariv,[* Quantum Electronics,* ]{}Wiley, New York, 1989[*.*]{} See e. g., A. D. Martin and T. D. Spearman[*, Elementary Particle Theory*]{}, Nord Holland Publishing Co., Amsterdam, 1970. D. B. Ion and P. Constantin, [*A New Interpretation of two-photon entangled*]{} [*Experiments*]{}, [*NIPNE-1996 Scientific Report*]{}, National Institute for Physics and Nuclear Engineering Horia Hulubei, Bucharest, Romania, p. 139; D. B. Ion, P. Constantin and M. L. D. Ion, Rom. J. Phys. [**43** ]{}(1998) 3. M. W. Evans, in [*Modern Nonlinear Optics*]{}, [*Vol. 2*]{}, (M. W. Evans and S. Kielich (Eds)), John Wiley&Sons, Inc.,1993, pp.249. For a review see for example: D. M. Pepper and A. Yariv, in R. A. Fischer (Ed.), [*Optical Phase Conjugation,*]{} Academic Press, Inc., 1983, p 23; See also: H. Jagannath et al., [*Modern Nonlinear Optics, Vol 1,*]{} (M. Evans and S .Kielich (Eds)) John Wiley&Sons, Inc.,1993, pp.1. D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, Phys. Rev. Lett. [**74** ]{}(1995) 3600. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, Phys. Rev. [**A 52 **]{}(1995) R3429. T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, Phys. Rev. [**A 53 **]{}(1996) 2804.  A. V. Belinski and D. N. Klyshko, Sov. JETP [**78** ]{}(1994) 259.         Fig. 1: : The basic optical configuration of a[* plane SPDC-quantum mirror.*]{}   Fig. 2a: : The basic optical configuration for usual lens combined with a [*plane SPDC-quantum mirror.*]{}   Fig. 2b: : The basic optical configuration for a proof of magnification factor for a usual lens combined with a [*plane*]{} [*SPDC-quantum mirror*]{}.  Fig. 3: : The basic optical configuration for usual lens combined with a [*plane*]{} [*SPDC-quantum mirror* ]{}and with[* *]{}a[* classical mirror*]{}.  Fig. 4: : The scheme of the experimental setup for a test of the geometric optics of correlated photons. The QM indicates the SPDC- [*quantum mirror*]{}, PBS is a polarization beam splitter, SL is a signal laser, P is an object, L a convergent lens, D$_{i}$ is an idler detector and CC is the coincidence circuit.

--- abstract: 'We report the results of a first experimental search for lepton number violation by four units in the neutrinoless quadruple-$\beta$ decay of $^{150}$Nd using a total exposure of $0.19$ kg$\cdot$y recorded with the NEMO-3 detector at the Modane Underground Laboratory (LSM). We find no evidence of this decay and set lower limits on the half-life in the range $T_{1/2}>(1.1\text{--}3.2)\times10^{21}$ y at the $90\%$ CL, depending on the model used for the kinematic distributions of the emitted electrons.' author: - 'R. Arnold' - 'C. Augier' - 'A.S. Barabash' - 'A. Basharina-Freshville' - 'S. Blondel' - 'S. Blot' - 'M. Bongrand' - 'D. Boursette' - 'V. Brudanin' - 'J. Busto' - 'A.J. Caffrey' - 'S. Calvez' - 'M. Cascella' - 'C. Cerna' - 'J.P. Cesar' - 'A. Chapon' - 'E. Chauveau' - 'A. Chopra' - 'L. Dawson' - 'D. Duchesneau' - 'D. Durand' - 'V. Egorov' - 'G. Eurin' - 'J.J. Evans' - 'L. Fajt' - 'D. Filosofov' - 'R. Flack' - 'X. Garrido' - 'H. Gómez' - 'B. Guillon' - 'P. Guzowski' - 'R. Hodák' - 'A. Huber' - 'P. Hubert' - 'C. Hugon' - 'S. Jullian' - 'A. Klimenko' - 'O. Kochetov' - 'S.I. Konovalov' - 'V. Kovalenko' - 'D. Lalanne' - 'K. Lang' - 'Y. Lemière' - 'T. Le Noblet' - 'Z. Liptak' - 'X. R. Liu' - 'P. Loaiza' - 'G. Lutter' - 'M. Macko' - 'C. Macolino' - 'F. Mamedov' - 'C. Marquet' - 'F. Mauger' - 'B. Morgan' - 'J. Mott' - 'I. Nemchenok' - 'M. Nomachi' - 'F. Nova' - 'F. Nowacki' - 'H. Ohsumi' - 'C. Patrick' - 'R.B. Pahlka' - 'F. Perrot' - 'F. Piquemal' - 'P. Povinec' - 'P. Přidal' - 'Y.A. Ramachers' - 'A. Remoto' - 'J.L. Reyss' - 'C.L. Riddle' - 'E. Rukhadze' - 'R. Saakyan' - 'R. Salazar' - 'X. Sarazin' - 'Yu. Shitov' - 'L. Simard' - 'F. Šimkovic' - 'A. Smetana' - 'K. Smolek' - 'A. Smolnikov' - 'S. Söldner-Rembold' - 'B. Soulé' - 'D. Štef[á]{}nik' - 'I. Štekl' - 'J. Suhonen' - 'C.S. Sutton' - 'G. Szklarz' - 'J. Thomas' - 'V. Timkin' - 'S. Torre' - 'Vl.I. Tretyak' - 'V.I. Tretyak' - 'V.I. Umatov' - 'I. Vanushin' - 'C. Vilela' - 'V. Vorobel' - 'D. Waters' - 'F. Xie' - 'A. Žukauskas' title: 'Search for neutrinoless quadruple-$\beta$ decay of $^{150}$Nd with the NEMO-3 detector' --- In the standard model (SM) of particle physics, leptons are assigned a lepton number of $+1$ and anti-leptons are assigned $-1$. All experimental observations thus far are consistent with the assumption that the total lepton number $L$ is conserved in particle interactions [@pdg]. However, since this is not due to a fundamental symmetry, there is no reason to assume that $L$ is generally conserved in theories beyond the SM. Lepton-number violating processes could be directly linked to the possible Majorana nature of neutrinos. If Majorana mass terms are added to the SM Lagrangian, processes appear that violate $L$ by two units ($\Delta L=2$) [@valle]. Searches for $\Delta L=2$ processes such as neutrinoless double-$\beta$ ($0\nu2\beta$) decay have therefore been the focus of many experiments [@double-beta-searches]. In this letter, we present a first search for processes with $\Delta L=4$, which are allowed even if neutrinos are Dirac fermions and $\Delta L=2$ processes are forbidden [@quad-theory]. Models with $\Delta L=4$ have some power in explaining naturally small Dirac masses of neutrinos [@dirac-mass] and could mediate leptogenesis [@leptogenesis]. The models have also been linked with dark matter candidates [@DM] and with *CP* violation in the lepton sector [@CP]. Processes with $\Delta L=4$ could also be probed at the Large Hadron Collider (LHC), for example in the pair production and decay of triplet-Higgs states to four identical charged leptons [@LHC]. An experimental signature of some models with $\Delta L=4$ would be the neutrinoless quadruple-$\beta$ ($0\nu 4\beta$) decay of a nucleus, $(A,Z) \to (A,Z+4) + 4 e^-$, where four electrons are emitted with a total kinetic energy equal to the energy $Q_{4\beta}$ of the nuclear transition. The $0\nu4\beta$ half-life is expected to depend strongly on the unknown mass scale $\Lambda_\textrm{NP}$ of the new $\Delta L=4$ phenomena [@quad-theory]. The search for $0\nu 4\beta$ decay is experimentally challenging, since only three long-lived isotopes can undergo this decay, $^{136}$Xe ($Q_{4\beta}=0.079$ MeV [@qvals]), $^{96}$Zr ($Q_{4\beta}=0.642$ MeV), and $^{150}$Nd, which has the highest $Q_{4\beta}$ value of $2.084$ MeV. The NEMO-3 detector contained two of these isotopes, $^{96}$Zr and $^{150}$Nd. The $^{96}$Zr decay has too low a $Q_{4\beta}$ value to be detected with high enough efficiency in NEMO-3 since low-energy electrons would be absorbed in the source. It could instead be studied using geochemical methods [@zr96-geochem]. The value of $Q_{4\beta}$ of the decay $^{150}$Nd$\to^{150}$Gd, however, is sufficiently large for four electrons to be observable in the NEMO-3 detector. We search for $0\nu 4\beta$ decay by exploiting the unique ability of the NEMO-3 experiment to reconstruct the kinematics of each final-state electron. In the absence of a more complete theoretical treatment of the kinematics of the decay [@heeck], we test four models of the electron energy distributions, labeled uniform, symmetric, semi-symmetric, and anti-symmetric. This choice is designed to cover a wide range of models and used to demonstrate that the final result is largely model-independent. The uniform model has all four electron kinetic energies $T_i$ distributed uniformly on the simplex $T_1+T_2+T_3+T_4=Q_{4\beta}$ with each kinetic energy $T_i>0$. The decay rates $\mathrm{d}N$ for the other three models are distributed according to the differential phase space given by $$\begin{aligned} \lefteqn{ \frac{\mathrm{d^4}N}{\prod_{i=1}^4\mathrm{d}T_i} \propto } \\ & & A_m \delta\left(Q_{4\beta}-\sum_{i=1}^4 T_i\right)\cdot \prod_{i=1}^4 (T_i+m_e)p_i F(T_i, Z) , \nonumber \label{eqn:kinem}\end{aligned}$$ which is an extension of the $0\nu2\beta$-decay phase space [@supernemo]. Here, $i$ labels the electrons, $m_e$ the electron mass, $p_i = \sqrt{T_i(T_i+2m_e)}$, and $A_m$ is a model-dependent factor. The Fermi function $F(T, Z) \propto p^{2s-2} e^{\pi u} \left| \Gamma(s+iu)\right|^2$ describes the Coulomb attraction between the electrons and the daughter nucleus with atomic number $Z$. In this function, $s=\sqrt{1-(\alpha Z)^{2}}$, $u=\alpha Z (T+m_e) / p$, $\Gamma$ is the gamma function, and $\alpha$ is the fine structure constant. The three different phase space distributions differ by the factors $A_m$ that depend on the energy asymmetry of electron pairs. For the symmetric distribution $A_m=\mathcal{S}\{1\times1\}$, the semi-symmetric distribution has $A_m=\mathcal{S}\{1\times(T_k-T_l)^2\}$, and for the anti-symmetric distribution $A_m=\mathcal{S}\{(T_i-T_j)^2\times(T_k-T_l)^2\}$, where $\mathcal{S}\{\cdots\}$ is a sum over symmetric interchange of labels $i,j,k,l$ of the four electrons. In all models, each electron angular distribution is generated isotropically. ![Normalized distribution of the individual electron kinetic energies $T$ in each $0\nu 4\beta$ decay for the four kinematic models.[]{data-label="fig:true_energies"}](plots/kine_spec.eps){width="1.1\columnwidth"} Since electrons produced in the NEMO-3 source foil must have a minimum energy of $\approx 250$ keV to fall into the acceptance, the efficiency is smaller for models producing more low-energy electrons. We show the electron kinetic energy distributions for the four kinematic models in Fig. \[fig:true\_energies\]. We perform the search with the NEMO-3 detector on data collected between $2003$ and $2011$ using $36.6$ g of enriched $^{150}$Nd source, with a live time of $5.25$ y. The detector is optimized to search for $0\nu 2\beta$ decays by reconstructing the full decay topology. It is cylindrical in shape, with the cylinder axis oriented vertically, a height of $3$ m and a diameter of $5$ m, and is divided into $20$ sectors of equal size. Thin foils with a thickness of $40\text{--}60$ mg/cm$^2$ contain $7$ different isotopes. The Nd foil has a height of $2.34$ m and a width of $6.5$ cm. The foils are located between two concentric tracking chambers composed of $6180$ drift cells operating in Geiger mode. Surrounding the tracking chambers on all sides are calorimeter walls composed of $1940$ scintillator blocks coupled to low-activity photomultipliers that provide timing and energy measurements. The calorimeter energy resolution is $(14.1\text{--}17.7)\%$ (FWHM) at an electron energy of $1$ MeV. A vertically oriented magnetic field of $\approx 25$ G allows discrimination between electrons and positrons. Detailed descriptions of the experiment and data sets are given in Refs. [@detector; @mo]. In Ref. [@nd150-2beta], we describe a measurement of the two-neutrino double-$\beta$ ($2\nu2\beta$) decay of $^{150}$Nd, and provide details of the background model and measured activities that are used in this analysis. The backgrounds are categorized as internal (within the source foil, including contamination of $^{208}$Tl and $^{214}$Bi), external to the foil (electrons and photons produced in or outside of the detector components), radon diffusion that can deposit background isotopes on the surface of the detector components, and also internal contamination in the source foils neighboring the Nd foil, which can have a falsely reconstructed vertex in the Nd foil. Internal conversions, M[ø]{}ller and Compton scattering are sources of additional electrons in single-$\beta$ or double-$\beta$ decays that can mimic four-electron final states. The largest contribution to the background is $2\nu2\beta$ decay of $^{150}$Nd to the ground state (g.s.) of $^{150}$Sm with a half-life of $T_{1/2}=9.34\times10^{18}$ y [@nd150-2beta]. An additional background source not considered in Ref. [@nd150-2beta] is the double-$\beta$ decay of $^{150}$Nd to the $0^+_1$ excited state of $^{150}$Sm [@nd150-new-excited], for which we use a half-life of $T_{1/2}=1.33\times10^{20}$ y [@nd150-excited] in the simulation. The selection requires candidate decays that produce three or four tracks originating in the foil. If there are three tracks, all three must be matched to calorimeter hits, which is the signature of a reconstructed electron candidate, while the fourth $\beta$ electron is assumed to be absorbed in the foil ($3e$ topology). We further distinguish two topologies in the four-track final state, where either all four tracks are associated with calorimeter hits ($4e$ topology) or one of the tracks has no calorimeter hit ($3e1t$ topology). [ 1= ]{} [ 1= ]{} An additional set of selections is applied to all topologies to ensure events are well reconstructed and to reject instrumental backgrounds. Decay vertices in regions of high activity in the foil corresponding to localized contaminations from $^{234m}$Pa and $^{207}$Bi (hot spots) are rejected. The locations of these hot spots have been determined in Ref. [@nd150-2beta]. Events where more than one electron track is associated with the same calorimeter hit are removed. The energy of each associated calorimeter hit must be $>150$ keV. Events in the $4e$ topology with one associated calorimeter hit below $150$ keV are treated as $3e1t$ candidates. The vertical component of the distance between the intersection points of the tracks with the foil must be $<8$ cm. We apply no requirement in the horizontal direction, since the foil has a width of $6.5$ cm. For each event, the track lengths, calorimeter hit times and energies, along with their uncertainties, are used to construct two $\chi^2$ values assuming all tracks originate in the foil (internal hypothesis) or one track originates outside the foil and scatters in the foil producing secondary tracks (external). The probabilities of the internal hypothesis must be $>0.1\%$ and of the external hypothesis $<4\%$. Finally, events with unassociated calorimeter hits with energies $>150$ keV in time with the electron candidates are rejected, since this would indicate that photons were emitted in the decay. Topology Symmetric Uniform Semi-symm. Anti-symm. ---------- ----------- --------- ------------ ------------ 4e 0.20 0.13 0.04 0.01 3e 3.55 3.11 2.39 1.67 3e1t 0.86 0.64 0.30 0.13 Total 4.61 3.88 2.73 1.81 : Signal efficiencies (in $\%$) of the four kinematic models for the three topologies.[]{data-label="tab:effs"} For the $3e1t$ topology only, we require that there are no delayed hits with times up to $700$ $\mu$s near the decay vertex or the track end points, caused by an $\alpha$ decay of the $^{214}$Po daughter of $^{214}$Bi $\beta$ decays [@nd150-2beta]. These decays can occur on the surface of the tracker wires with the $\beta$ electron scattering in the foil producing secondaries. The $\beta$ electron in this type of decay would have no associated calorimeter hit. To validate the background model, the selection is applied to the foils containing the isotopes $^{100}$Mo and $^{82}$Se, which are expected to contain no $0\nu4\beta$ signal. The energy-sum distributions for the $3e$ topology, which have higher statistics, are shown in Fig. \[fig:sidebands\]. We observe no events in the $4e$ topology in the $^{82}$Se foil, where $0.05\pm0.01$ are expected. We observe two $4e$-candidates in the $^{100}$Mo foil, with an expectation of $2.3\pm 0.5$ events, of which $2.0\pm0.4$ are due to $2\nu2\beta$ decays followed by double M[ø]{}ller scattering. A display of one of these two data events is shown in Fig. \[fig:evdisp\]. ![Display of a decay with four reconstructed electrons in NEMO-3 data, originating in the $^{100}$Mo source foil, in the horizontal plane.[]{data-label="fig:evdisp"}](plots/evdisp.eps){width="0.7\linewidth"} The total efficiencies for signal decays are shown in Tab. \[tab:effs\] and range from $1.81\%$ to $4.61\%$ depending on kinematic model. The expected background yields are given in Tab. \[tab:backgrounds\] for the energy range $1.2 \le \Sigma E \le 2.0$ MeV, where $\Sigma E$ is the electron energy sum, obtained by summing over the calorimeter hits for all reconstructed electrons. All activities and systematic uncertainties, except for the $2\nu2\beta$ $0^+_1$ process, are taken from Ref. [@nd150-2beta]. Origin 4e $[\times10^{-2}]$ 3e 3e1t $[\times10^{-2}]$ ----------------------------------- ---------------------- --------------- ------------------------ $^{150}$Nd $2\nu2\beta$ (g.s.) $2.08\pm0.57$ $9.43\pm0.84$ $8.98\pm0.92$ $^{150}$Nd $2\nu2\beta$ ($0^+_1$) $0.85\pm0.36$ $2.39\pm0.63$ $3.98\pm1.07$ $^{208}$Tl internal $0.74\pm0.15$ $1.28\pm0.21$ $5.37\pm1.21$ $^{214}$Bi internal $0.19\pm0.07$ $0.74\pm0.18$ $1.08\pm0.30$ Other internals $0.82\pm0.11$ $1.01\pm0.51$ Neighboring foils $1.61\pm0.45$ $1.95\pm1.91$ Radon $0.43\pm0.15$ Externals $0.12\pm0.09$ $6.50\pm4.12$ Total $3.86\pm0.74$ $16.8\pm1.7$ $28.9\pm5.4$ : Expected number of background events for an exposure of $36.6$ g$\times5.25$ y in the $^{150}$Nd source foil in the range $1.2 \le \Sigma E \le 2.0$ MeV for the three topologies, with their total systematic uncertainties.[]{data-label="tab:backgrounds"} The distributions of the electron energy-sum for events originating from the Nd foil are shown in Fig. \[fig:ene\_specs\]. The energies of the signal distributions are lower than $Q_{4\beta}=2.084$ MeV due to electron energy losses in the source foil. In addition, only three of the electrons have an associated calorimeter energy measurement for the $3e1t$ candidate events. The distributions show that there are no large differences between the shapes for the different kinematic models. We observe no candidate events in the $4e$ and $3e1t$ topologies, with expected background rates of $0.04\pm0.01$ and $0.29 \pm 0.05$ events, respectively. There is also no significant excess of data in the $3e$ topology, with 22 observed events in the range $1.2 \le \Sigma E \le 2.0$ MeV, compared to $16.8 \pm 1.7$ expected background events. Source 4e 3e 3e1t --------------------------------------------- ------------- ------------- ------------- Reconstruction efficiency ($\epsilon_{2e}$) $\pm5.5\%$ $\pm5.5\%$ $\pm5.5\%$ Reconstruction efficiency ($\epsilon_{3e}$) $\pm 8.5\%$ $\pm 8.5\%$ $\pm 8.5\%$ Energy scale $\pm12.1\%$ $\pm4.4\%$ $\pm8.5\%$ Angular distribution $\pm5.7\%$ $\pm1.9\%$ $\pm4.5\%$ : Systematic uncertainties on the signal normalization for the three topologies.[]{data-label="tab:sigsyst"} We consider several sources of systematic uncertainty. The systematic uncertainties on the background model given in Tab. \[tab:backgrounds\] are the same as used in Ref. [@nd150-2beta], apart from the $25\%$ uncertainty on the half-life of the $2\nu 2\beta$ $0^+_1$ excited state decay [@nd150-excited]. The uncertainties of the signal efficiency are given in Tab. \[tab:sigsyst\]. The uncertainty on the reconstruction efficiency is determined using the $^{100}$Mo data. It is broken down into two independent components, one based on a two-electron efficiency ($\epsilon_{2e}$) uncertainty of $5.5\%$ and the second on a three-electron ($\epsilon_{3e}$) uncertainty of $8.5\%$. The first value is obtained by comparing the independently measured activity of a $^{207}$Bi calibration source with the in-situ measurements. This uncertainty can only be determined for decays with a maximum of two electrons in the final state. The three-electron uncertainty ($\epsilon_{3e}$) of $8.5\%$ is obtained by comparing the normalization of the $3e$ selection in the simulation and data for the $^{100}$Mo foils. This additional uncertainty is taken to be correlated between signal and background, and assumed to be the same size in the $4e$ and $3e1t$ topologies. A variation of $2\%$ on the energy scale for all electrons is applied in the simulation to cover uncertainties on the $Q_{4\beta}$ value, the calorimetric energy reconstruction of $0.2\%$ [@mo], and uncertainties on the simulated energy loss in the foil. The uncertainty due to the assumption of an isotropic angular distribution of the electrons from the $0\nu 4\beta$ decay is derived from the variation of the reconstruction efficiency as a function of the generated angles between electron pairs. [ 1= ]{} [ 1= ]{} [ 1= ]{} For the $4e$ and $3e1t$ topologies, where no candidate events are observed, we set limits using a single bin for each topology, as for a counting experiment. For the $3e$ topology, we use the binned distribution of Fig. \[fig:ene\_specs\](b). Limits at the $90\%$ CL are calculated using the modified-frequentist [*CL*]{}$_s$ method [@cls], which includes the systematic uncertainties with Gaussian priors. The observed and expected half-life limits $T_{1/2}^{0\nu 4\beta}$ are shown in Tab. \[tab:limits\]. We obtain the best sensitivity in the $3e$ topology, due to the much higher signal efficiency compared to the $4e$ topology. The combined lower limit at the $90\%$ CL on the $0\nu 4\beta$ half-life is $3.2\times10^{21}$ y, with a sensitivity, given by the median expected limit, of $3.7\times10^{21}$ y, assuming a symmetric energy distribution. The combined limits lie in the range $(1.1\mbox{--}3.2)\times10^{21}$ y for the different models. This result represents the first search for neutrinoless quadruple-$\beta$ decay in any isotope, and the first search for lepton-number violation by 4 units. ---------- ----- ----- ----- ----- ----- ----- ------ ------ obs exp obs exp obs exp obs exp 4e 0.5 0.3 0.3 0.2 0.1 0.1 0.03 0.02 3e 1.6 2.4 1.5 2.1 1.2 1.7 0.9 1.2 3e1t 2.0 1.9 1.5 1.4 0.7 0.6 0.3 0.3 Combined 3.2 3.7 2.6 3.0 1.7 2.0 1.1 1.3 ---------- ----- ----- ----- ----- ----- ----- ------ ------ : Observed and median expected lower limits at the $90\%$ CL on the $0\nu4\beta$ half-life (in units of $10^{21}$ y) for the four signal models. Systematic uncertainties are taken into account with Gaussian priors.[]{data-label="tab:limits"} To improve on this limit in the future using the NEMO-3 technique would not only require more exposure, but also an optimization of the foil density and thickness which causes the main loss of efficiency for low-energy electrons and increases background from M[ø]{}ller scattering. Even with reduced isotope mass, a thinner foil should increase sensitivity. Since our search strategy is largely model-independent, this first limit on $0\nu 4\beta$ decay can provide valuable constraints on new-physics models. The authors of Ref. [@quad-theory] estimate for their particular model the ratio $R$ of the $0\nu4\beta$ half-life to the $2\nu2\beta$ half-life to be $R\approx 10^{46} (\Lambda_\textrm{NP}/\textrm{TeV})^4$. For $T_{1/2}^{0\nu 4\beta} > 1.1\times10^{21}~$years, this translates to a limit of $R> 120$. For $0\nu 4\beta$ processes to be observable at the current experimental sensitivities, significant enhancement factors are therefore required. This result thus motivates further theoretical and experimental studies of $\Delta L=4$ processes in nuclear decays and at colliders. The authors would like to thank the staff of the Modane Underground Laboratory for their technical assistance in operating the detector. We thank Werner Rodejohann for useful discussions. We acknowledge support by the funding agencies of the Czech Republic, the National Center for Scientific Research/National Institute of Nuclear and Particle Physics (France), the Russian Foundation for Basic Research (Russia), the Slovak Research and Development Agency, the Science and Technology Facilities Council and the Royal Society (United Kingdom), and the National Science Foundation (United States). -4mm [99]{} For a review of experimental limits on lepton-number violation, see C. Patrignani *et al.* (Particle Data Group), Chin. Phys. C [**40**]{}, 100001 (2016). J. Schechter and J. W. F. Valle, Phys. Rev. D [**25**]{}, 2951 (1982). R. Arnold *et al.* (NEMO Collab.), Phys. Rev. D [**89**]{}, 111101(R) (2014); J. B. Albert *et al.* (EXO-200 Collab.), Nature [**510**]{}, 229 (2014); K. Alfonso *et al.*, (CUORE Collab.) Phys. Rev. Lett. [**115**]{}, 102502 (2015); R. Arnold *et al.* (NEMO Collab.), Phys. Rev. D [**93**]{}, 112008 (2016); A. Gando *et al.*, (KamLAND-Zen Collab.) 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--- author: - Yi Xie bibliography: - 'gravity.bib' date: 'Received   month day; accepted    month day' title: 'Testing Lorentz violation with binary pulsars: constraints on standard model extension$^*$ ' --- Introduction {#sect:intro} ============ Unification of general relativity (GR) and quantum mechanics is a grand challenge in the fundamental physics. Some candidates of a self-consistent quantum theory of gravity emerge from tiny violations of Lorentz symmetry [@Kostelecky2005; @Mattingly2005]. To describe observable effects of the violations, effective field theories could be a theoretical framework for tests. The standard model extension (SME) is one of those effective theories. It includes the Lagrange densities for GR and the standard model for particle physics and allows possible breaking of Lorentz symmetry [@Bailey2006]. The SME parameters $\bar{s}^{\mu\nu}$ control the leading signals of Lorentz violation in the gravitational experiments in the case of the pure-gravity sector of the minimal SME. By analyzing archival lunar laser ranging data, @Battat2007 constrain these dimensionless parameters at the range from $10^{-11}$ to $10^{-6}$, which means no evidence for Lorentz violation at the same level. However, tighter constraints on $\bar{s}^{\mu\nu}$ would be hard to obtain in the solar system because the gravitational field is weak there. Thus, for this purpose, binary pulsars provide a good opportunity. Because of their stronger gravitational fields, for example the relativistic periastron advance in the double pulsars could exceed the corresponding value for Mercury by a factor of $\sim 10^5$, these systems are taken as an ideal and clean test-bed for testing GR, alternative relativistic theories of gravity and modified gravity, such as the works by @Bell1996, @Damour1996, @Kramer2006, @Deng2009 and @Deng2011. Motivated by this advantage of binary pulsars, we will try to test Lorentz invariance under the SME framework with five binary pulsars: PSR J0737-3039, PSR B1534+12, PSR J1756-2251, PSR B1913+16 and PSR B2127+11C. In Sec. \[sec:model\], the orbital dynamics of double pulsars in the SME will be briefed. Observational data will be used to constrain the SME parameters in Sec. \[sec:obs\]. The conclusions and discussions will be presented in Sec. \[sec:con\]. Orbital dynamics of double pulsars in SME {#sec:model} ========================================= When the pure-gravity sector of the minimal SME is considered, it will cause secular evolutions of the orbits of double pulsars. Since timing observations of double pulsars could obtain its value very precisely, the periastron advance plays a much more important role in constraining $\bar{s}^{\mu\nu}$ and, with widely used notations in celestial mechanics, it reads [@Bailey2006] $$\begin{aligned} \label{domegadtsme} \bigg<\frac{d\omega}{dt}\bigg>\bigg|_{\mathrm{SME}} & = & -\frac{n}{\tan i (1-e^2)^{1/2}}\bigg[\frac{\varepsilon}{e^2}\bar{s}_{kP}\sin\omega+\frac{(e^2-\varepsilon)}{e^2}\bar{s}_{kQ}\cos\omega-\frac{\delta m}{M}\frac{2na\varepsilon}{e}\bar{s}_k\cos\omega\bigg]\nonumber\\ & & -n\bigg[\frac{(e^2-2\varepsilon)}{2e^4}(\bar{s}_{PP}-\bar{s}_{QQ})+\frac{\delta m}{M}\frac{2na(e^2-\varepsilon)}{e^3(1-e^2)^{1/2}}\bar{s}_Q\bigg],\end{aligned}$$ where $M = m_1+m_2$, $\delta m = m_2-m_1$ ($m_2>m_1$) and $\varepsilon = 1-(1-e^2)^{1/2}$. In this expression, the coefficients $\bar{s}_{\cdot}$ and $\bar{s}_{\cdot\cdot}$ for Lorentz violation with subscripts $P$, $Q$ and $k$ are projections of $\bar{s}^{\mu\nu}$ along the unit vectors $\bm{P}$, $\bm{Q}$ and $\bm{k}$. The unit vector $\bm{k}$ is perpendicular to the orbital plane of the binary pulsars, $\bm{P}$ points from the focus to the periastron, and $\bm{Q}=\bm{k}\times\bm{P}$. By definitions [@Bailey2006], $\bar{s}_k\equiv \bar{s}^{0j}k^j$, $\bar{s}_Q \equiv \bar{s}^{0j}Q^j$, $\bar{s}_{kP}\equiv \bar{s}^{ij}k^iP^j$, $\bar{s}_{kQ}\equiv \bar{s}^{ij}k^iQ^j$, $\bar{s}_{PP}\equiv \bar{s}^{ij}P^iP^j$ and $\bar{s}_{QQ}\equiv \bar{s}^{ij}Q^iQ^j$. However, according to Eq. (\[domegadtsme\]) , it is easy to see that the measurement of $\dot{\omega}$ is sensitive to a combination of $\bar{s}^{\mu\nu}$ instead of its individual components. @Bailey2006 define the combination as $$\begin{aligned} \label{} \bar{s}_{\omega} & \equiv & \bar{s}_{kP}\sin\omega+(1-e^2)^{1/2}\bar{s}_{kQ}\cos\omega-\frac{\delta m}{M}2nae\bar{s}_k\cos\omega\nonumber\\ & & +\tan i\frac{(1-e^2)^{1/2}(e^2-2\varepsilon)}{2e^2\varepsilon}(\bar{s}_{PP}-\bar{s}_{QQ})+\frac{m}{M}2na\tan i \frac{(e^2-\varepsilon)}{e\varepsilon}\bar{s}_Q,\end{aligned}$$ and crudely estimate its value at the level of $10^{-11}$. Together with the contribution from GR, the total secular periastron advance of a double pulsars system is $$\begin{aligned} \label{dodttot} \dot{\omega} & = & 3\bigg(\frac{P_b}{2\pi}\bigg)^{-5/3}\bigg(\frac{GM}{c^3}\bigg)^{2/3}(1-e^2)^{-1} -\frac{n\varepsilon}{\tan i (1-e^2)^{1/2}e^2}\bar{s}_{\omega}\nonumber\\ & = & 3\bigg(\frac{P_b}{2\pi}\bigg)^{-5/3}T_{\sun}^{2/3}\bigg(\frac{M}{M_{\sun}}\bigg)^{2/3}(1-e^2)^{-1} -\frac{2\pi\varepsilon s}{P_b (1-e^2)^{1/2}e^2 (1-s^2)^{1/2}}\bar{s}_{\omega},\end{aligned}$$ where $T_{\sun}\equiv GM_{\sun}/c^3=4.925490947$ $\mu$s and $$\label{eqns} s=x\bigg(\frac{P_b}{2\pi}\bigg)^{-2/3}T_{\sun}^{-1/3}M^{2/3}m_2^{-1}.$$ The quantity $x$ in Eq. (\[eqns\]) is the projected semi-major axis, which is usually given by the timing observations, while, in some cases, $s$ could be measured directly so that there is no necessity to evaluate it from this equation. In this work, Eq. (\[dodttot\]) will be taken to find the constraints on $\bar{s}_{\omega}$ with timing measurements of double pulsars. Observational constraints {#sec:obs} ========================= Long-term timing observations can determine the geometrical and physical parameters of binary pulsars very well. Among them, PSR J0737-3039 [@Kramer2006], PSR B1534+12 [@Staris2002], PSR J1756-2251 [@Faulkner2005], PSR B1913+16 [@Weisberg2010] and PSR B2127+11C [@Jacoby2006] are good samples for gravitational tests. Some of their timing parameters are listed in the Table \[Tab:timingpm\]. In terms of the estimated uncertainties given in parentheses after $\dot{\omega}$, the data pool is divided into two groups: Group I, all the double pulsars are taken; and Group II, including PSR B1913+16, PSR B1534+12 and PSR B2127+11C, which have the smallest uncertainties. By weighted least square method, the parameter $\bar{s}_{\omega}$ is estimated (see Table \[Tab:somega\]). The estimation made by Group I is $\bar{s}_{\omega}=(-1.24\pm0.54)\times 10^{-10}$ and Group II gives $\bar{s}_{\omega}=(-1.42\pm0.75)\times 10^{-10}$. For comparison, @Bailey2006 propose the attainable experimental sensitivity of $\bar{s}_{\omega}$ is $10^{-11}$, which is 10 times less than the results we obtain. [lllllll]{} PSR & $P_b$(d) & $M$ ($M_{\sun}$) & $e$ & $s$ & $\dot{\omega}$ ($\degr$ yr$^{-1}$) & Reference\ J0737-3039 & 0.10225156248 & 2.58708 & 0.0877775 & 0.99974 & 16.89947(68) & @Kramer2006\ B1534+12 & 0.420737299122 & 2.678428 & 0.2736775 & 0.975 & 1.755789(9) & @Staris2002\ J1756-2251 & 0.319633898 & 2.574 & 0.180567 & 0.961${}^a$ & 2.585(2) & @Faulkner2005\ B1913+16 & 0.322997448911 & 2.828378 & 0.6171334 & 0.733650${}^a$ & 4.226598(5) & @Weisberg2010\ B2127+11C & 0.33528204828 & 2.71279 & 0.681395 & 0.76762${}^a$ & 4.4644(1) & @Jacoby2006\ ${}^a$Derived value according to Eq. (\[eqns\]). [cccc]{} & Group I & Group II & Predicted sensitivity\ & & & [@Bailey2006]\ $\bar{s}_{\omega}$ & $(-1.24\pm0.54)\times 10^{-10}$ & $(-1.42\pm0.75)\times 10^{-10}$ & $10^{-11}$\ Conclusions and Discussion {#sec:con} ========================== In this work, we test Lorentz violation with five binary pulsars under the framework of standard model extension. It finds that $\bar{s}_{\omega}$, which is a dimensionless combination of SME parameters, is at the order of $10^{-10}$, whether all five systems are taken or top three systems with the smallest estimated uncertainties of periastron advances are used. This value, one order of magnitude greater than the estimation by @Bailey2006, implies no evidence for the break of Lorentz invariance at $10^{-10}$ level. Nevertheless, as mentioned by @Bailey2006, the secular evolution of the eccentricity of the double pulsars should be included in the analysis. Its contribution is [@Bailey2006] $$\label{} \bigg<\frac{de}{dt}\bigg> = \frac{1}{e^3}n(1-e^2)^{1/2}(e^2-2\varepsilon)\bar{s}_e,$$ where $$\label{} \bar{s}_e = \bar{s}_{PQ}-\frac{\delta m}{M}\frac{2nae\varepsilon}{e^2-2\varepsilon}\bar{s}_P.$$ $\bar{s}_e$ is a combination of coefficients in $\bar{s}^{\mu\nu}$ and sensitive to observations. However, there is lacking of timing observations on double pulsars running for a long enough time so that rare observations could show the secular change of $e$. Even though a few numbers could be derived from data, the uncertainties of them are quite larger than those of periastron advances. Timing observations usually could set the upper bounds only, such as $|\dot{e}|<1.9\times10^{-14}$ s${}^{-1}$ for PSR B1913+16 [@Taylor1989] and $|\dot{e}|<3\times10^{-15}$ s${}^{-1}$ for PSR B1534+12 [@Staris2002]. Hence, we suppose that, at least in current stage, the constraints made by $\dot{e}$ might be looser and the resulting upper bound is $|\bar{s}_e|<3\times 10^{-10}$. Although it is consistent with the values of $\bar{s}_{\omega}$ we obtain, the exact value of $\bar{s}_e$ remains unknown. Therefore, unless timing observations could provide much more definitive results about $\dot{e}$, the secular changes of eccentricity would not impose a tight constraint on $\bar{s}^{\mu\nu}$ or combinations of $\bar{s}^{\mu\nu}$. Another issue for future work is to constrain the components of $\bar{s}^{\mu\nu}$ directly with double pulsars. However, the choice of reference frame affects the values of these components so that a certain reference frame must be specified first and the projections of $\bar{s}^{\mu\nu}$ will be along its standard unit basis vectors. For example, for comparing the constraints due to double pulsars and lunar laser ranging, $\bar{s}^{\mu\nu}$ has to be projected along the same triad of vectors. It means the unit vectors $\bm{P}$, $\bm{Q}$ and $\bm{k}$ (see Sec.\[sec:model\]) have to be decomposed in terms of these vectors, which requires the geometrical information of the orbit of the double pulsars, such as the orbital elements $\Omega$ and $\omega$. Unfortunately, timing observations are not sensitive to those two elements. This makes the components of $\bar{s}^{\mu\nu}$ hard to access directly for now and demonstrates the advantages and availability of $\bar{s}_{\omega}$. This work is funded by the National Natural Science Foundation of China (NSFC) under Nos. 10973009 and 11103010, the Fundamental Research Program of Jiangsu Province of China under No. BK2011553, the Research Fund for the Doctoral Program of Higher Education of China under No. 20110091120003 and the Fundamental Research Funds for the Central Universities under No. 1107020116.

[**Adaptive Approximation of Functions with Discontinuities**]{}\  \ Licia Lenarduzzi and Robert Schaback\  \ Version of Nov. 09, 2015 [**Abstract**]{}: One of the basic principles of Approximation Theory is that the quality of approximations increase with the smoothness of the function to be approximated. Functions that are smooth in certain subdomains will have good approximations in those subdomains, and these [*sub-approximations*]{} can possibly be calculated efficiently in parallel, as long as the subdomains do not overlap. This paper proposes a class of algorithms that first calculate sub-approximations on non-overlapping subdomains, then extend the subdomains as much as possible and finally produce a global solution on the given domain by letting the subdomains fill the whole domain. Consequently, there will be no Gibbs phenomenon along the boundaries of the subdomains. Throughout, the algorithm works for fixed scattered input data of the function itself, not on spectral data, and it does not resample. Key words: Kernels, classification, localized approximation, adaptivity, scattered data AMS classification: 65D05, 62H30, 68T05 Introduction ============ Assume that a large set $\{(\bx_i,f_i),i=1,\ldots,N\}$ of data is given, where the points $\bx_i$ are scattered in ${\ensuremath{\mathbb{R}}}^d$ and form a set $X$. We want to find a function $u$ that recovers the data on a domain $\Omega$ containing the points, i.e. $$\begin{array}{rcl} u&:& \Omega\to {\ensuremath{\mathbb{R}}},\\ u(\bx_i)&\approx& f_i,\;i=1,\ldots,N. \end{array}$$ We are particularly interested in situations where the data have smooth interpolants in certain non-overlapping subdomains $\Omega_j$, but not globally. The reason may be that there are discontinuities in the function itself or its derivatives. Thus a major goal is to identify subdomains $\Omega_j\subseteq\Omega,\;1\leq j\leq J$ and smooth functions $u_j,\;1\leq j\leq J$ such that $$\begin{array}{rcl} u_j&:& \Omega_j\to {\ensuremath{\mathbb{R}}},\\ u_j(\bx_i)&\approx& f_i \fa \bx_i\in X\cap \Omega_j. \end{array}$$ The solution to the problem is piecewise defined as $$u(\bx):=u_j(\bx)\fa \bx\in \Omega_j,\;1\leq j\leq J.$$ Our motivation is the well-known fact that errors and convergence rates in Approximation Theory always improve with increasing smoothness. Thus on each subdomain we expect to get rather small errors, much smaller than if the problem would have been treated globally, where the non-smoothness is a serious limiting effect. From the viewpoint of Machine Learning this is a mixture of classification and regression. The domain points have to be classified in such a way that on each class there is a good regression model. The given training data are used for both classification and regression, but in this case the classification is dependent on the regression, and the regression is dependent on the classification. Furthermore, there is a serious amount of geometry hidden behind the problem. The subdomains should be connected, their interiors should be disjoint, and the union of their closures should fill the domain completely. This is why a black-box machine learning approach is not pursued here. Instead, Geometry and Approximation Theory play a dominant part. For the same reason, we avoid to calculate edges or fault lines first, followed by local approximations later. The approximation properties should determine the domains and their boundaries, not the other way round. In particular, [*localized approximation*]{} will combine Geometry and Approximation Theory and provide a central tool, together with [*adaptivity*]{}. The basic idea is that in the interior of each subdomain, far away from its boundary, there should be a good and simple approximation to the data at each data point from the data of its neighbors. An Adaptive Algorithm ===================== Localized approximation will be used as the first phase of an [*adaptive algorithm*]{}, constructing disjoint localized subsets of the data that allow good and simple [*local approximations*]{}. Thus this “localization” phase produces a subset $X^g \subseteq X$ of “good” data points that is the union of disjoint sets $X_1^g,\ldots,X_J^g$ consisting of data points that allow good approximations $u_j^g\in U,\;1\leq j\leq J$ using only the data points in $X_j^g$. In some sense, this is a rough classification already, but only of data points. The goal of the second phase is to reduce the number of unclassified points by enlarging the sets of classified points. It is tacitly assumed that the final number of subdomains is already obtained by the number $J$ of classes of “good” points after the first phase. The “blow–up” of the sets $X_j^g$ should maintain [*locality*]{} by adding neighboring data points first, and adding them only if the local approximation $u_j^g$ does not lose too much quality after adding that point and changing the approximation. The second phase usually leaves a small number of “unsure” points that could not be clearly classified by blowing up the classified sets. While the blow-up phase focuses on each single set $X_j^g$ in turn and tries to extend it by looking at all “unsure” points for good extension candidates, the third phase works the other way round. It focuses on each single “unsure” point $\bx_i$ in turn and looks at all sets $X_j^g$ and the local approximations $u_j$ on these, and assigns the point $\bx_j$ to one of the sets $X_j^g$ so that $u_j(\bx_i)$ is closest to $f(\bx_i)$. It is a “final assignment” phase that should classify all data points and it should produce the final sets $X^f_j\supseteq X^g_j$ of data points. The sets $X^f_j$ should be disjoint and their union should be $X$. After phase 3, each local approximation $u_j^f\in U$ is based on the points in $X_j^f$ only, but there still are no well-defined subdomains $\Omega_j\supseteq X_j^f$ as domains of $u_j^f$. Thus the determination of subdomain boundaries from a classification of data points could be the task of a fourth phase. It could, for instance, be handled by any machine learning program that uses the classification as training data and classifies each given point $\bx$ accordingly. But this paper does not implement a fourth phase, being satisfied if each approximation $u_j^f$ is good on each set $X_j^f$, and much better than any global approximation $u^*\in U$ to all data. Implementation ============== The above description of a three-phase algorithm allows a large variation of different implementations that compete for efficiency and accuracy. We shall describe a basic implementation together with certain minor variants, and provide numerical examples demonstrating that the overall strategy works fine. We work on the unit square of ${\ensuremath{\mathbb{R}}}^2$ for simplicity and take a trial space $U$ spanned by translates of a fixed positive definite radial kernel $K$. In our examples, $K$ may be a Gaussian or an inverse multiquadric. For details on kernels, readers are referred to standard texts , for example. When working on finite subsets of data points, we shall only use the translates with respect to this subset. Since the kernel $K$ is fixed, also the Hilbert space $H$ is fixed in which the kernel is reproducing, and we can evaluate the norm $\|.\|_K$ of trial functions cheaply and exactly. To implement locality, we assume that we have a computationally cheap method that allows to calculate for each $\bx\in{\ensuremath{\mathbb{R}}}^2$ its $n$ nearest neighbors from $X$. This can, for instance, be done via a range query after an initialization of a kd-tree data structure . Phase 1: Localization --------------------- This is carried out by a first step picking all data points with good localized approximation properties, followed by a second step splitting the set $X^{g}$ of good points into $J$ disjoint sets $X_j^{g}$. ### Good Data Points We assume that the global fill distance $$h(X, \Omega):=\sup_{\boy\in\Omega} \min_{\bx_k\in X}\|\boy-\bx_k\|_2$$ of the full set of data points with respect to the full domain $\Omega$ is roughly the same as the local fill distances $h(X_j^{f},\Omega_j)$ of the final splitting. The basic idea is to loop over all $N$ data points of $X$ and to calculate for each data point $\bx_i,\;1\leq i\leq N$ a number $\sigma_i$ that is a reliable indicator for the quality of [*localized approximation*]{}. Using a threshold $\sigma$, this allows to determine the set $X^{g}\subseteq X$ of “good” data points, without splitting it into subsets. There are many ways to do this. The implementation of this paper fixes a number $n$ of neighbors and loops over all $N$ data points to calculate for each data point $\bx_i,\;1\leq i\leq N$ 1. the set $N_i$ of their $n$ nearest neighbors from $X$, 2. the kernel-based interpolant $s_i$ of the data $(\bx_k, f(x_k))$ for all $n$ neighboring data points $\bx_k\in N_i$, 3. the norm $\sigma_i:=\|s_i\|_K$. This loop can be executed with roughly ${\cal O}(Nn^3)$ complexity and ${\cal O}(N+n^3)$ storage, and with easy parallelization, if necessary at all. A similar indicator would be the error obtained when predicting $f(\bx_i)$ from the values at the $n$ neighboring points. Practical experience shows that the numbers $\sigma_i$ are good indicators of locality, because adding outliers to a good interpolant usually increases the error norm dramatically. Many of the $\sigma_i$ can be expected to be small, and thus the threshold $$\sigma_i< 2 M_\sigma$$ will be used to determine “good” points within the next splitting step, see Section \[SecP1S\], where $M_\sigma$ is the median of all $\sigma_i$. This is illustrated for a data set by Figure \[loglogsigma\]: it represents, in base loglog scale, the sorted $\{{\bf \sigma}_i\}$ and the constant line relevant to the value of the threshold. \ ### Splitting The set $X^{g}$ of points with good localization must now be split into $J$ disjoint subsets of points that are close to each other. We assume that the inner boundaries of the subdomains are everywhere clearly determined by large values of ${\bf \sigma}_i$. The implementation of this paper accomplishes the splitting by a variation of Kruskal’s algorithm for calculating minimal spanning trees in graphs. The Kruskal algorithm sorts the edges by increasing weight and starts with an output graph that has no edges and no vertices. When running, it keeps a number of disconnected graphs as the output graph. It gradually adds edges with increasing weight that either connect two previously disconnected graphs or add an edge to an existing component or define a new connected component by that single edge. In the current implementation the edges that connect each $x_i$ with its $n-1$ nearest neighbors are collected in an edge list. The edge list is sorted by increasing length of the edges and then, by $n\mid X \mid$ comparisons, many repetitions of edges are removed, and these are all repetitions if any two different edges have different length. Then the thresholding of the $\{{\bf \sigma}_i\}$ by $${\bf \sigma}_i< 2 M_{\bf \sigma}$$ is executed, and it is known which points are good and which are bad. All edges with one or two bad end points are removed from the edge list with cost $n\mid X\mid$. After the spanning tree algorithm is run, the list of the points of each tree is intersected with itself to avoid eventual repetitions that are left. At the end, each connected component is associated to its tree in exactly one way. In rare cases, the splitting step may return only one tree, but these cases are detected and repaired easily. Phase 2: Blow-up ---------------- This is also an adaptive iterative process. It reduces the set $ \displaystyle{ X^u:=X\setminus \cup_{j=1}^J X_j^g} $ of “unsure” data points gradually, moving points from $X^u$ to one of the nearest sets $X_j^g$. In order to deal with easy cases first, the points $\bx_i$ in $X^u$ are sorted by their locality quality such that points with better localization come first. We also assume that for each point $\bx_i\in X^u$ we know its distance to all sets $X_j^g$, and we shall update this distance during the algorithm, when the sets $X^u$ and $X_j^g$ change. We also use the distances to the sets $X_j^{g,0}$ that are the output of the localization phase and serve as a start-up for the sets $X_j^g$. In an outer loop we run over all points $\bx_i\in X^u$ with decreasing quality of local approximation. In our implementation, this means increasing values of $\sigma_i$. The inner loop runs over the $m$ sets $X_j^g$ to which $\bx_i$ has shortest distance. In most cases, and in particular in ${\ensuremath{\mathbb{R}}}^2$, it will suffice to take $m=2$. The basic idea is to find the nearby set $X_j^g$ of “good” points for which the addition of $\bx_i$ does least damage to the local approximation quality. Our implementation of the inner loop over $m$ neighboring sets $X_j^g$ works as follows. In $X_j^{g,0}$, the point $\boy_j$ with shortest distance to $\bx_i$ is picked, and its $n$ nearest neighbors in $X_j^{g,0}$ are taken, forming a set $Y_j^g$. On this set, the data interpolant $s_j^g$ is calculated, and then the number $\sigma_j^g:=\|s_j^g\|_K$ measures the local approximation quality near the point $\boy_j$ if only “good” points are used. Then the “unsure” point $\bx_i$ is taken into account by forming a set $Y_j^u$ of points consisting of $\bx_i$ and the up to $n-1$ nearest neigbors to $\bx_i$ from $X_j^g$. On this set, the data interpolant $s_j^u$ is calculated, and the number $\sigma_j^u:=\|s_j^u\|_K$ measures the local approximation quality if the “unsure” point $\bx_i$ is added to $X_j^g$. The inner loop ends by maintaining the minimum of quotients $\sigma_j^u/\sigma_j^g$ over all nearby sets $X_j^g$ checked by the loop. These quotients are used to indicate how much the local approximation quality would degrade if $\bx_i$ would be added to $X_j^g$. Note that this strategy maintains locality by focusing on “good” nearest neighbors of either $\bx_i$ or $\boy_j$. By using the fixed sets $X_j^{g,0}$ instead of the growing sets $X_j^g$, the algorithm does not rely heavily on the newly added points. An illustration is attached to Example $1$ in the next section; there the point $\boy_1$ and the sets $X_1^g$ and $Y_1^u$ associated to a point $\bx_i$ will be shown. After the inner loop, if the closest set to ${\bf x}_i$ among all sets $X_k^g$ is $X_j^g$ and $\sigma_j^u/\sigma_j^g$ is less than $\sigma_k^u/\sigma_k^g$ for $k\neq j$, then ${\bf x}_i$ is moved from $X^u$ to $X_j^g$. If the closest set to ${\bf x}_i$ is $X_j^g$ but if it is not true that $\sigma_j^u/\sigma_j^g$ is less than $\sigma_k^u/\sigma_k^g$ for $k\neq j$, then ${\bf x}_i$ remains “unsure”. The “unsure” points are those that seriously degrade the local approximation quality of all nearby sets of “good” points. Phase $3$: Final Assignment --------------------------- The assignment of a point ${\bf x}_i\in X^u$ to a set $X_j^g$ is done on the basis of how well the function value $f({\bf x}_i)$ is predicted by $u_j({\bf x}_i)$. We loop over all points $\bx_i\in X^u$ and first determine two sets $X_j^g$ and $X_k^g$ to which ${\bf x}_i$ has shortest distance. This is done in order to make sure that $\bx_i$ is not assigned to a far-away $X_j^g$. We then could add ${\bf x}_i$ to $X_j^g$ if $|f({\bf x}_i)-u_j({\bf x}_i)|\leq |f({\bf x}_i)-u_k({\bf x}_i)|$, otherwise to $X_k^g$, but in case that we have more than one unsure point, we want to make sure that under all unsure points, $\bx_i$ fits better into $X_j^g$ than into $X_k^g$. Therefore we calculate $$\begin{array}{rcl} d_j(\bx_i)&:=& |f({\bf x}_i)-u_j({\bf x}_i)|\\ \mu_j&:=&\displaystyle{\min_{\bx_i\in X^u} d_j(\bx_i) }\\ M_j&:=&\displaystyle{\max_{\bx_i\in X^u} d_j(\bx_i) }\\ D_j(\bx_i)&:=&\displaystyle{\frac{d_j(\bx_i)-\mu_j}{M_j-\mu_j} } \end{array}$$ for all $j$ and $i$ beforehand, and assign ${\bf x}_i$ to $X_j^g$ if $D_j(\bx_i)\leq D_k(\bx_i)$, otherwise to $X_k^g$. Examples ======== Some test functions are considered now, each of which is smooth on $J=2$ subdomains of $\Omega$. The algorithm constructs $X_1^g$ and $X_2^g$ with $X_1^g\cup X_2^g=X$. Concerning the error of approximation of $u$, we separate what happens away from the boundaries of $\Omega_j$ from what happens globally on $[0,1]^2$. This is due to the fact that standard domain boundaries, even without any domain splittings, let the approximation quality decrease near the boundaries. To be more precise, let $\Omega_{safe}$ be the union of the circles of radius $$q:=\displaystyle{ \min_{1\leq i<j\leq N}\|\bx_i-\bx_j\|_2},$$ the separation distance of the data sites, centered at those points of $X_j^g,\; j=1,2$ such that the centered circles of radius $2 {q}$ do not contain points of $X_k^g$ with $k\neq j$. We then evaluate $$L_\infty^{safe}(u):=\| u-f\|_{\infty,\Omega_{safe}\cap [0,1]^2}$$ and $$L_\infty(u):=\| u-f\|_{\infty,[0,1]^2}.$$ The chosen kernel for calculating the local kernel-based interpolants is the inverse multiquadric kernel $\phi(r)=(1+2 r^2/{\delta^2})^{-1/2}$ with parameter $\delta=0.35$. In all cases, $N=900$ data locations are mildly scattered on a domain $\Omega$ that extends $[0,1]^2$ a little, with $q=0.04$. We shall restrict to $[0,1]^2$ to evaluate the subapproximant, calculated by the basis in the Newton form. Such a basis is much more stable than the standard basis, see . The error is computed on a grid with step $0.01$. [**Example 1**]{}. The function $$f_1(x,y):= \log(\mid x-(0.2 \sin(2\pi y)+0.5)\mid+0.5),$$ has a derivative discontinuity across the curve $x=0.2 \sin(2\pi y)+0.5$. We get $$L_\infty^{safe}(u)=1.6\cdot 10^{-5},\;\; L_\infty(u)=6.0\cdot 10^{-2}.$$ For comparison, the errors of the global interpolant are $$L_\infty^{safe}(u^{\star})=1.1\cdot 10^{-1},\;\; L_\infty(u^{\star})=1.1\cdot 10^{-1}.$$ The classification turns out to be correct. $890$ out of $900$ data points are correctly classified as output of phase $3.2$, and then phase $3.3$ completes the classification. Figure \[f1new\] shows the points of $X_1^f$ as dotted and those of $X_2^f$ as crossed. The points both dotted and circled of $X_1^f$, respectively the points both crossed and circled of $X_2^f$, are the result of the splitting (Section \[SecP1S\]), while the points dotted only, respectively crossed only, are those added by the blow-up phase (Section \[SecP1BlU\]). The points squared are the result of the final assignment phase (Section \[SecP1FV\]). The true splitting line is traced too. The convention of the marker types will be used in the next examples as well. The function $u$ is defined as $u_1^f$ where the subdomain $\Omega_1$ is determined and as $u_2^f$ on $\Omega_2$. The actual error $L_\infty(u)=6.0\cdot 10^{-2}$ is not much affected if we omit Phase 3 and and ignore the remaining 10 “unsure” points after the blow-up phase. A similar effect is observed for the other examples to follow. A zoomed area of $\Omega$ is considered in Figure \[f4zoom\]. The details are related to an iteration of the blow-up phase, where the “unsure” point ${\bf x}_i$ (both squared and starred) is currently examined. Points of ${X}_2^{g,0}$ are shown as crosses. At the current iteration, the dots are points inserted in ${X}_1^{g}$ up to now, those belonging to ${X}_1^{g,0}$ bold dotted, while the points inserted in ${X}_2^g$ up to now are omitted in this illustration. The points squared are those of ${ Y}_1^u$, while the points as diamonds are those of ${ Y}_1^g$. The point ${\bf y}_1$ is both written as diamond and star. [**Example 2**]{}. The function $$\RSlabel{fun1modif} f_2(x,y):=\left\{\begin{array}{ll} f_1(x,y) \quad\quad\quad {\rm if}\quad x<=0.2\, \sin(2\pi y)+0.5 \\ f_1(x,y)+0.01 \quad {\rm if}\quad x>0.2\, \sin(2\pi y)+0.5 \end{array} \right.$$ has a discontinuity across the curve $x=0.2\,\sin(2\pi y) + 0.5$. We get $$L_\infty^{safe}(u)=1.6\cdot 10^{-5},\;L_\infty(u)=6.0\cdot 10^{-2}.$$ For comparison, the errors of the global interpolant are $$L_\infty^{safe}(u^\star)=1.3\cdot 10^{-1} ,\;L_\infty(u^\star)=1.3\cdot 10^{-1}.$$ The classification turns out to be correct. $888$ out of $900$ data points are classified correctly as output of phase $3.2$, and phase $3.3$ completes the classification for the remaining 12 points. It might be that $u_j^f$ is more accurate on the safe zone, and also globally. [**Example 3**]{}. The function $$\RSlabel{fun3} f_3(x,y):= \arctan(10^3 (\sqrt{(x+0.05)^2+(y+0.05)^2}-0.7))$$ is regular but has a steep gradient. Our algorithm yields $$L_\infty^{safe}(u)=9.0\cdot 10^{-2} \hbox{ and } L_\infty(u)= 2.67\cdot 10^0,$$ while for the global interpolant we get $$L_\infty^{safe}(u^\star)=2.31\cdot 10^{0} ,\;L_\infty(u^\star)=3.26\cdot 10^{0}.$$ Figure \[sigm1\] shows the points of $X_1^f$ as dotted and those of $X_2^f$ as crossed; $X_1^f$ and $X_2^f$ stay at the opposite sides of the mid range line $f(x,y)=0$ . [**Example 4**]{}. The function $$\RSlabel{fun4} f_4(x,y):= ((x-0.5)^2+(y-0.5)^2)^{0.35}+0.05*(x-0.5)^0_+$$ has a jump on the line $x=0.5$ and a derivative singularity on it at $(0.5, 0.5)$. It has rather a steep gradient too. One data point close to the singularity is not classified correctly. We get $$L_\infty^{safe}(u)=9.9\cdot 10^{-4} \hbox{ and } L_\infty(u)=7.3\cdot 10^{-2},$$ while the global interpolant $u^{\star}$ has $$L_\infty^{safe}(u^{\star})=1.5\cdot 10^{-2} \hbox{ and } L_\infty(u^{\star})= 8.5\cdot 10^{-2}.$$ Figure \[punta1\] shows the points of $X^f_1$ as dotted and those of $X^g_2$ as crossed. All examples show that the transition from a global to a properly segmented problem decreases the achievable error considerably. But the computational cost is serious, and it might be more efficient to implement a multiscale strategy that works on coarse data first, does the splitting of the domain coarsely, and then refines the solution on more data, without recalculating everything on the finer data. \ \ \ \ [10]{} J.L. Bentley. Multidimensional binary search trees used for associative searching. , 18:509–517, 1975. M.D. Buhmann. . Cambridge University Press, 2003. N. Cristianini and J. Shawe-Taylor. . Cambridge University Press, Cambridge, 2000. G. Fasshauer and M. McCourt. , volume 19 of [*Interdisciplinary Mathematical Sciences*]{}. World Scientific, Singapore, 2015. J.B. Kruskal. On the shortest spanning subtree of a graph and the traveling salesman problem. , 7:48–50, 1956. St. Müller and R. Schaback. A [N]{}ewton basis for kernel spaces. , 161:645–655, 2009. doi:10.1016/j.jat.2008.10.014. R. Schaback and H. Wendland. Kernel techniques: from machine learning to meshless methods. , 15:543–639, 2006. B. Schölkopf and A.J. Smola. . MIT Press, Cambridge, 2002. J. Shawe-Taylor and N. Cristianini. . Cambridge University Press, 2004. H. Wendland. . Cambridge University Press, 2005.

--- author: - 'S. Forti' - 'S. Link' - 'A. Stöhr' - 'Y. R. Niu' - 'A. A. Zakharov' - 'C. Coletti' - 'U. Starke' bibliography: - './Biblio.bib' title: | Supplementary information for:\ Semiconductor to metal transition in two-dimensional gold and its van der Waals heterostack with graphene --- Statistical analysis of the 2D ML Au semiconducting character {#statanal} ============================================================= In order to present a robust and statistically solid set of measurements for proving the actual semiconductor character of the single-layer 2D gold, we carried out a systematic analysis of all our ARPES measurements, done on several gold-intercalated graphene samples, at different photon energies and at different synchrotron facilities. Such an extensive analysis allows us to present strong and consistent evidence of the semiconducting behavior of the 2D gold layer. The result of such analysis is reported in Fig. \[FigS1\]. In panel (a), we show an exemplary ARPES measurement cutting through both the K points of graphene and 2D gold (cf. green line in the inset). In panel (b), we report the energy distribution curves (EDCs) through the gray dashed lines at graphene’s pi bands and at the [[$\overline{\textrm{K}}_{\textup{Au}}$]{}]{} point. The two profiles, i.e. the Fermi edge and the position of the gold valence band maximum (VBM), are fitted with a sigmoidal curve and the fit result is plotted as two vertical dashed lines on the graph. The same procedure was applied for every sample measured and the overall result is reported in the lower part of the figure, where the shaded region represents the error associated with the extracted value. ![(a) ARPES cut along the green line in the inset, showing both the [[$\overline{\textrm{K}}_{\textup{Gr}}$]{}]{} and [[$\overline{\textrm{K}}_{\textup{Au}}$]{}]{} points. The horizontal lines account for the energy level as extracted by the sigmoidal fit of the profiles in panel (b). (b) Profiles taken along the vertical grey-dashed lines in panel (a). (c) distribution of the energy difference between the gold VBM and the Fermi level for different samples.[]{data-label="FigS1"}](./FigS1.pdf){width="75.00000%"} The $x$ axis of panel (c) shows the names of the samples measured, the photon energy and the facility at which they have been measured. The mean value extracted from this statistics is 67 meV, i.e. the energy distance from the 2D-Au VBM and the Fermi level. Two-dimensionality of Au states {#Gold2D} =============================== For completeness of what is reported in the main text, in Fig. \[FigS2\](a) we show the TB bands calculated in the NNN approximation on a 2D triangular lattice. In panel (a), on the right side we show the density of states (DOS) derived from the TB model. The 2D-Au exhibits a Van Hove singularity at its [[$\overline{\textrm{M}}$]{}]{} point, which is located just about 400 meV from the Fermi level. This means that this system has an instability point which is at an energy reachable by electrical gating. To prove the two-dimensionality of the gold bands, we show high-resolution ARPES data, collected with different photon energies to measure the dispersion of the $k$-vector along the $z$ direction in reciprocal space. ![(a) NNN-TB bands of gold (blue-dashed) and graphene (black-dashed). In the right panel, the DOS of gold (blue), graphene (black) and the sum of the two (red) is shown in the same energy range. (b) ARPES spectrum acquired at 75 eV along the gold [[$\overline{\textrm{K}}$]{}[$\overline{\Gamma}$]{}[$\overline{\textrm{K'}}$]{}]{} direction. (c) $k_z$ dispersion obtained by extracting the spectral weight of spectra similar to the one in panel (a) at 1.7 eV (see dashed line), recorded at photon energies from 30 to 100 eV with 5 eV of energy step.[]{data-label="FigS2"}](./FigS2.jpg){width="95.00000%"} In panel (b) we show an exemplary of ARPES spectrum acquired along the Au [[$\overline{\textrm{K}}$]{}[$\overline{\Gamma}$]{}[$\overline{\textrm{K'}}$]{}]{} direction and centered in [[$\overline{\Gamma}$]{}]{} with photon energy 75 eV. The spectral weight visible in the vicinity of [[$\overline{\Gamma}$]{}]{} is due to gold replica bands as it will be further explained in sec. \[replicas\]. The spectral weight at binding energy 1.7 eV, as traced by the dashed line in the panel, was extracted for every energy from 30 to 100 eV with energy step 5 eV and plotted as shown in panel (c), following the relation that binds the $k_z$ vector to the kinetik energy and the emission angle: $k_{z} = \sqrt{\frac{2m}{\hbar}}\sqrt{E_{kin}\cos^2(\theta)+V_0}$, where $V_0$ is the *inner potential* and $\theta$ the photoemission angle. We chose $V_0=14.5$ eV, according to Refs. [@ZhouAnnPhys2006; @OhtaPRL2007] and considering the difference in the valence band minimum between gold and graphene as extracted from the TB calculations. The absence of dispersion in the direction perpendicular to the surface of the $k$-vector indicates the actual two-dimensionality of the interfacial gold layer. The spectral weight visible towards the Fermi level in [[$\overline{\Gamma}$]{}]{} is ascribed to gold replica bands, due to electron diffraction at the surface. The scattering vector corresponds to the reciprocal lattice vector of graphene, hence rotated by 30 degrees with respect to the gold BZ alignment (see also sec. \[replicas\]). The electronic properties of graphene over a single layer of gold {#manybody} ================================================================= The effects of gold intercalation over the decoupled monolayer graphene on SiC(0001) have been already discussed in the past literature[@GierzPRB2010; @WalterPRB2011]. Nevertheless, in this section we point out what are the effects of a very careful preparation on the properties of the intercalated graphene.\ The very careful preparation of the samples via the intercalation of precisely one Au monolayer with respect to the SiC(0001) atomic areal density, translates in a very clean ARPES signal and allows for the extraction of band parameters from the data. Fig. \[FigS3\](a) shows the ARPES spectrum of the Dirac cone acquired along [[$\overline{\Gamma}$]{}[$\overline{\textrm{K}}_{\textup{Gr}}$]{}]{} at 40 eV. The measured bands are accompained by the fitting of the positions of the maxima of the energy dispersion curves, shown as green and red circles superimposed to the raw data. Two distinct dispersing states are fitted, as it is apparent from the figure. This effect has been reported for the first time in graphene by Bostwick and coworkers [@BostwickScience2010]. What is visible below the Dirac point is the energy dispersion of the *plasmaron* quasiparticle[@LundqvistPKM1969], i.e. a photohole that is coupled to a plasmon with similar group velocity. According to Ref. [@WalterPRB2011], by measuring the energy and momentum separation of the hole and plasmaron bands, one can estimate the effective dielectric constant of the investigated sample. We determined the energy and momentum spread of the diamond formed by the crossing of the hole and the plasmaron dispersions (cf. Fig. \[FigS4\](a)). This is done in Fig. \[FigS3\](b) and (c), where the energy (b) and momentum (c) distribution curves of the Dirac cone are fitted with two Voigt functions, colored in gray. The two band traces are about 253 meV (b) and 0.02 [Å]{}$^{-1}$ (c) apart from each other and the Dirac energy of the system is defined as E$_0$ and located (685$\pm$5) meV below the Fermi level (cf. Fig. \[FigS4\]). To give an estimation of the effective coupling constant and effective dielectric constant, we compare our results with the work of Ref. . However, we point out that the normalization of the diamond width in momentum space with respect to the Fermi momentum $k_F$ is somewhat inconsistent since for such high doping level, the Dirac cone is strongly warped (see Fig. \[FigS4\](b) and (c)). We therefore rely on the separation of the cones in energy. In this way, the $\alpha_{ee}$, or graphene fine constant is found to be $\alpha_{ee}=0.30\pm0.02$, which is translated into an effective dielectric constant $\epsilon_{eff}=7\pm1$, meaning a substrate dielectric constant of $\epsilon_s=2*\epsilon_{eff}-1=13\pm2$. Such a small effective dielectric constant enhances the *e-e* coupling and makes collective phenomena such as plasmarons observable. In their paper[@WalterPRB2011], Walter and coworkers reported a dielectric constant value about five times higher. This is simply because they started from a $p$-type Au intercalated graphene, which is induced by twice the amount of gold at the interface (cf. Fig. 3(c)), and counter $n$-doped it with K atoms in order to set the Fermi level of the system well above the Dirac point and thereby observe the plasmaron band. ![[**Observed plasmarons in graphene.**]{} (a) ARPES spectrum at 40 eV of the Dirac cone along the $k_x$ direction of a Au-intercalated epitaxial graphene, $n$-phase. The empty circles are fits of the peak position of the energy distribution curves. (b) and (c) line profiles of the plasmaron diamond, from which to extract the value of dielectric constant. See also Fig. \[FigS4\].[]{data-label="FigS3"}](./FigS3.jpg){width="95.00000%"} For completeness, we also extract the band parameters from the measured ARPES. To do this, we analyze the ARPES data of the graphene Dirac cone measured at 40 eV, as displayed in Fig. \[FigS4\]. In panel (a) we show the ARPES spectrum of the $n$-doped Dirac cone as in Fig. 1 of the main text, with the addition of dashed lines to distinguish between the hole (green) and plasmaron (red) dispersions. Panel (b) displays the Fermi surface, whereas panel (c) shows the amplitude of the Fermi vector as a function of the angle as centered in [[$\overline{\textrm{K}}$]{}]{}, highlighting the threefold symmetry of the Fermi surface and the dark corridor[@ShirleyPRB1995], where the intensity vanishes. This supports the argumentation promulgated in the main text, that considers unreliable the extraction of the dielectric constant value by measuring the distance in $k$-space between the hole and plasmaron dispersion, normalized for the Fermi vector. Such a normalization procedure is clearly direction-dependent, instead of the energy difference between the two dispersions, which is not, and it is therefore more reliable. ![(a) ARPES spectrum of the graphene’s Dirac cone measured perpendicular to the [[$\overline{\Gamma}$]{}[$\overline{\textrm{K}}_{\textup{Gr}}$]{}]{} direction at 40 eV. Photohole and plasmaron dispersion are superimposed to guide the eye. (b) Fermi surface of graphene. (c) Angular dependence of the Fermi momentum. (d) FWHM and band slope of the Dirac cone extracted along the [[$\overline{\Gamma}$]{}[$\overline{\textrm{K}}_{\textup{Gr}}$]{}]{} direction at 40 eV. (e) Imaginary part of the self energy and quasiparticle lifetime extracted from the FWHM (see text). (f) Scanning tunneling spectroscopy of Gr/SC Au/SiC(0001) acquired at 78 K. $E_0$ is the Dirac energy, as defined in the main text.[]{data-label="FigS4"}](./FigS4.pdf){width="95.00000%"} We extracted the band parameters by fitting the momentum dispersion curves (MDCs) with Lorentzian lineshapes. Fig. \[FigS4\](d) shows the full width at half maximum (FWHM) of the bands and their slope $\partial E/\partial k$, in the energy range displayed in panel (a). Panel (e) shows the imaginary component of the self energy and the photohole lifetime, which depend on the FWHM through the relation: $\Sigma^{''}=(\delta k/2)(\hbar v_F)$ and $\tau=1/(v_F \delta k)$, where $\delta k$ is the FWHM and $v_F$ the Fermi velocity. The self energy is a particularly sensitive quantity for measuring the interactions between quasiparticles. Indeed, a strong peak is visible in proximity of the Dirac point, corresponding to the photohole relaxation through the emission of a plasmon. The other two less pronounced peaks are ascribed to the interaction between graphene and gold bands, although in the dispersion no actual gap is observed. The first low-energy kink in the spectrum is due instead to quasiparticle relaxation via phonon emission. Fig. \[FigS4\](f) displays the scanning tunneling spectroscopy (STS) spectrum of the semiconducting Au phase intercalated in between graphene and SiC(0001), acquired at 78 K. The Dirac point is observed at around -625 meV, in line with the ARPES measurements. Interfacial atomic ordering derived from periodicity of ARPES replica bands {#replicas} =========================================================================== In this section we show the final state effects caused by the diffraction of photoemitted electrons as they exit the material.\ In Fig. \[FigS5\](a) a portion of the $k$-space is shown, recorded with ARPES at 40 eV centered in [[$\overline{\Gamma}$]{}]{}. At this binding energy, no intensity should be observable. However, some spectral weight clearly appears. Four Dirac cone profiles are distinguishable and correspond to graphene replica bands. Those replicas are induced by a reciprocal lattice vector with module and orientation corresponding to the SiC(0001) surface vector. They are therefore generated by the gold atoms arranged on the SiC (1$\times$1), as shown in panel (c). ![(a) ARPES constant energy surface close to the Fermi level measured about [[$\overline{\Gamma}$]{}]{} at 40 eV. (b) Model for the SC gold replica bands, imposing the graphene reciprocal lattice vector as generator. (c) Model for the graphene replica bands using a SC Au reciprocal lattice vector as generator.[]{data-label="FigS5"}](./FigS5.pdf){width="95.00000%"} The intensity around [[$\overline{\Gamma}$]{}]{} is due to gold replica bands. In this case the generator vector is the reciprocal lattice vector of graphene. In panel (b) we show an ARPES spectrum recorded at 75 eV along the gold [[$\overline{\textrm{K}}$]{}[$\overline{\Gamma}$]{}[$\overline{\textrm{K'}}$]{}]{} direction with superimposed the theoretical gold SC bands derived from TB (in yellow) and displaced by a graphene reciprocal lattice vector (other colors). The matching is quite remarkable, confirming the proposed model and also justifying the residual intensity visible in the $k_z$ vs. $E$ plot in Fig. \[FigS2\](c). Details about HRXPS data fitting {#SecXPS} ================================ In this section we report the details about the fitting for the HRXPS data, as shown in Fig. 2 of the main text.\ The Au 4f spectrum of the 2D SC phase can be well fitted with a single Voigt doublet lineshape (cf. Fig. 2(a)). The Si 2p spectrum instead, needs a double Voigt doublet function (cf. Fig. 2(b)). For the Au 4f, spin splitting was set to 3.68 eV and the branching ratio was 0.75. Standard constraints were set for the Si 2p fit, such as spin splitting of 0.63 eV and branching ratio of 0.5 for the single doublets. Within the respective doublets, both components were set to the same Lorentzian and Gaussian widths. The resulting fitting parameters can be found in Tab. \[Tab1\]. ---------------------------------- ----------------------- -------------------------- -------------------------- ----------- E$_{\textup{B}}$ (eV) $\omega_\textup{L}$ (eV) $\omega_\textup{G}$ (eV) ratio (%) Au 4f$_{\sfrac{7}{2}}$ 84.35 0.15 0.20 Si 2p$_{\sfrac{3}{2}}$, bulk SiC 100.78 0.12 0.32 68 Si 2p$_{\sfrac{3}{2}}$, Au-Si 100.31 0.12 0.13 32 ---------------------------------- ----------------------- -------------------------- -------------------------- ----------- : Fit parameters of the Si 2p$_{\textup{\sfrac{3}{2}}}$ and the Au 4f$_{\sfrac{7}{2}}$ components of the n-phase Au intercalated ZLG. The binding energies of the respective doublet partners are given by the spin splitting mentioned in the text, which was set as a constrain for the fit.[]{data-label="Tab1"} The small Lorentzian and Gaussian widths for the Au 4f levels is indicative for the presence of only one chemical species of Au. The 4f$_{\sfrac{7}{2}}$ component’s binding energy of 84.35 eV is shifted by about 350 meV towards higher binding energy compared to metallic Au. This means firstly, that one can exclude the presence of Au clusters on the sample surface, like observed earlier [@GierzPRB2010]. Also, all Au atoms within the interface have the same chemical environment. Judging from the two very distinct chemical components in the Si 2p spectrum, it is reasonable to assume that these Au atoms are chemically bound to the topmost Si of the SiC(0001) surface. All components of the Si 2p spectrum show a Lorentzian broadening of 0.12 eV, which can be viewed standard for this core level. The Gaussian width is 0.32 eV in the 100.78 eV binding energy doublet and 0.13 eV in the 100.31 eV doublet (both binding energies denote to the 2p$_{ \sfrac{3}{2}}$ components). A Gaussian broadening of 0.32 eV is generally observed for bulk SiC even if the experimental resolution is much better. However, the Gaussian width of Si 2p spectra can be much smaller in other materials, as it is the case for instance in the (7$\times$7)-reconstruction of the Si(111) surface [@PaggelPRB1994]. This peculiar broadness within the SiC system is not understood up until now, yet it can be seen as characteristic for bulk SiC. It is therefore reasonable to assign the narrower component to Si, which is bound to Au on the surface of the SiC. Au adsorbed on the (0001) surface of 4H-SiC have been investigated by Stoltz and coworkers[@StoltzJPCM2007]. They report a Si 2p component distribution, which is similar to our case, but with slightly different binding energies. There, they also assign the higher binding energy component to Si in the bulk of the SiC and the lower binding energy component to Si, which is bound to Au on the surface. The differences in binding energy can be caused by the different SiC polytype used here (6H-SiC) and by them (4H-SiC). Also the additional graphene layer on top in our case has influence on the band alignment and accompanied surface band bending in the SiC [@RisteinPRL2012; @Huefner], which alters the observed binding energies.\ It should be noted, that one can compare the intensity ratio of the signal produced by the two chemical Si species to other intercalation systems based on graphene on SiC(0001). The intensity ratio from Si in the bulk SiC and Si bound to Au is 68:32. In the case of H intercalation, the intensity ratio of Si in the bulk SiC to Si bound to H is 63:37 and 67:33 for photon energies 140 eV and 330 eV, respectively [@RiedlDissertation]. In this case, every Si atom on the surface of the SiC is bound to one H. Keeping in mind that the photon energy in the measurements presented here is 210 eV, it is reasonable to assume that also in the Au case here all Si atoms on the surface are bound to Au.\ ![(a) $\mu$XPS spectrum of C 1s recorded on Gr/SC 2D Au/SiC(0001) at 330 eV. (b) Photoemission cut-off spectra recorded on Gr/Au/SiC(0001) regions where the graphene was $n$ and $p$ doped.[]{data-label="FigS6"}](./FigS6.pdf){width="95.00000%"} The decoupling of the graphene is well illustrated by the C 1s core level peak in Fig. \[FigS6\](a). The SiC peak binding energy is measured at about 283 eV, as observed in other intercalated systems[@RiedlPRL2009; @FortiPRB2011; @Forti2DMat2016]. The graphene peak is here fitted with a single Doniach-[Ŝ]{}unji[ć]{} (DS) centered at 285 eV with a gaussian width of 0.45 eV, which takes into account the different chemical environments present inside the supercell[@Forti2DMat2016; @PreobrajenskiPRB2008]. The binding energy of the sp$^2$ carbon peak is increased by about 600 meV from the nominal peak position of 284.4 eV. Such a shift is mostly due to the $n$-type doping of about $n\sim0.035$ electrons per graphene unit cell[@Schroeder2DMater2016]. In turn, we observe this doping being accompanied by a decrease of work function. Being a 2D material, whenever the bands are rigidly moved up or down with respect to the Fermi level, a corresponding change in the work function must occur. This is actually observed and measured through the cut-off energy of the photoemitted electrons, displayed in Fig. \[FigS6\](b). We look at the low-kinetic energy part of the photoemission spectrum until we reach the edge where no electron is extracted from the material. In this way we can easily measure the difference in work function between regions which are $n$ and $p$ doped (corresponding to semiconducting and metallic gold phases, respectively). We measure an energy difference of (690$\pm$5) meV. Considering that the $p$-phase is almost neutral (cf. Fig. 3 in the main text), such a measure shows very well how tightly related the charge transfer and the variation of work function are in 2D materials.\ STM measurements {#STMsec} ================ In this section, we show scanning tunneling microscopy (STM) measurements of the gold-intercalated graphene in its SC and M phase. Fig. \[FigS7\](a) shows a sample’s region where the transition between the SC and M gold region is visible. The SC (M) region is shown zoomed-in in panel b (e), whereas the FFT-filtered and 2D-FFT of the zoomed-in region are shown in panel c and d (f and g), respectively. The superstructure on the M region is so large and strong that it can be seen even on the large-scale image. The corrugation of the M region pattern is indeed of the order of 200 pm, as visible also in the line profile of panel (h) (cf. also Fig. \[FigS8\]), while on the SC phase is about 25 pm. ![(a) STM topography of a SC/M transition region. Au-intercalated graphene recorded with 400 pA of constant tunnel current and -250 mV of tip voltage. Panels (b,c,d) and (e,f,g) show a zoomed-in portion of the scanned area corresponding to the frames on panel (a), the FFT filtered image of the same region and the 2D-FFT of the image of the SC and M regions, respectively. Panel (h) shows the black line profile across the transition region between the SC and the M regions in panel (a). (i) simple sketch of the system.[]{data-label="FigS7"}](./FigS7.pdf){width="95.00000%"} As visible from panel (e) and Fig. \[FigS8\], the superstructure stemming from the M region has a periodicity of (7$\sqrt{3}\times$7$\sqrt{3}$)R30 graphene unit cells over (10$\times$10) gold, as also confirmed by $\mu$LEED measurements (see discussion in the main text). The height difference of about 3.5 [Å]{} observed between the two regions supports a model where only two gold layers are intercalated in the M phase. ![[**M-phase superstructure measure with STM.**]{} (a) Raw topographical data. In the top-right inset, the line profile over the red line is shown. (b) 2D-FFT filtered image with indicated the lattice vectors of the superperiodicity and of graphene, the length of which has been increased by a factor 5.[]{data-label="FigS8"}](./FigS8.pdf){width="95.00000%"} ![(a) 2D-FFT of SC Au phase, retrieved from Fig. 2(d). Graphene and gold reciprocal lattice spots are indicated by a black and red hexagon, respectively. The spots stemming from the (13$\times$13) periodicity are circled in blue. (b) Theoretical reciprocal lattice grid of the (13$\times$13), representing the kinematic LEED pattern in the first quadrant. Highlighted are the spots visible in panel (a). (c) portion of Fig. 2(c) from the main text with enhanced contrast, to highlight the high order diffraction spots on the (13$\times$13) grid.[]{data-label="FigS9"}](./FigS9.pdf){width="95.00000%"} The 2D fast Fourier transformed (FFT) image FFT pattern in Fig. \[FigS9\](a) clearly shows the graphene and gold reciprocal lattice spots, indicated with a black and red hexagon, respectively. Other spots are also visible and they belong to the (13$\times$13) grid, as explained in panel (b), where the top-left quadrant of the theoretical LEED pattern up to the graphene’s first diffraction order is shown. The blue circle in panel (a) encloses three spots, two of which are visible along every direction. The third and innermost spot is less visible in the FFT. As illustrated in panel (b), those spots are the (7,-1)/13, (6,1)/13 and 5/13 reflexes of the (13$\times$13) pattern. The same spots are measured as well by $\mu$LEED, as we prove by showing in panel (c) a portion of Fig. 2(c) with enhanced contrast. We briefly point out that the (13$\times$13) is a rather natural periodicity for the graphene on SiC system and it is often observed, also for other intercalated systems[@Forti2DMat2016]. In Ref.[@PremlalAPL2009] , for example, they have observed several different superperiodicities induced by the intercalated gold. Even the (13$\times$13), but in that case, the gold was aligned with the graphene and it had a different lattice constant. A configuration very similar to what has been observed for the copper-intercalated graphene[@Forti2DMat2016; @KazumaAPL2014]. References {#references .unnumbered} ==========

[to]{} [ **Standard Model with hidden scale invariance and light dilaton**]{} [**Archil Kobakhidze and Shelley Liang\ **]{} [ *ARC Centre of Excellence for Particle Physics at the Terascale,\ School of Physics, The University of Sydney, NSW 2006, Australia,\ E-mails: archil.kobakhidze, [email protected]\ *]{} [**Abstract**]{} We consider the minimal Standard Model as an effective low-energy description of an unspecified fundamental theory with spontaneously broken conformal symmetry. The effective theory exhibits classical scale invariance which manifest itself through the dilaton field. The mass of the dilaton is generated via the quantum scale anomaly at two-loop level and is proportional to the techically stable hierarchy between the electroweak scale and a high energy scale given by a dilaton vacuum expectation value. We find that a generic prediction of this class of models is the existence of a very light dilaton with mass between $\sim 0.01$ $\mu$eV to $\sim 100$ MeV, depending on the hierarchy of scales. Searches for such a light scalar particle may reveal a fundamental role of conformal invariance in nature. Introduction ============ The discovery of the Higgs boson completes the Standard Model (SM) and confirms of the basic picture of mass generation through the spontaneous electroweak symmetry breaking. At the same time, the quadratic sensitivity of the Higgs mass under the quantum correction from ultraviolet physics and the related mass hierarchy problem remains a mystery. The measured Higgs boson mass, $m_{h}\simeq125$ GeV, can hardly be accommodated in the most popular minimal supersymmetric extension of the SM, which for a long time has been assumed as a prototype model for the solution of the hierarchy problem. As an alternative to supersymmetry, scale invariance has been advocated as a potential solution to the hierarchy problem for quite some time now [@Wetterich:1983bi; @Bardeen:1995kv; @Meissner:2006zh; @Foot:2013hna; @Aoki:2012xs; @Kobakhidze:2014afa]. Conformal invariance and supersymmetry are believed to be symmetries of fundamental string theory. Conformal invariance is typically assumed to be broken at Planck/string scale, while supersymmetry survives all the way down the electroweak scale. As a logical possibility, one may also consider scenario where supersymmetry is broken at high energy scales, while conformal invariance is maintaining down to lower energy scales. In this paper we consider a low-energy effective description of such a scenario without trying to specify the ultraviolet completion. Spontaneously broken conformal invariance manifests in the effective theory through the dilaton field[^1]. More specifically we consider the minimal Standard Model with hidden scale invariance and demonstrate that technically natural hierarchy between the electroweak scale given by the Higgs vacuum expectation value (VEV) $v_{ew}\approx 246$ GeV and high energy scale $\Lambda$ defined by the dilaton VEV, $\epsilon =v_{ew}/\Lambda$, can be maintained in the effective theory. A rather generic prediction of our theory is the existence of light dilaton, which develops its mass via the dimensional transmutation mechanism due to the quantum scale anomaly. Assuming the vacuum energy is tuned to be (nearly) zero, as required by observations, the dilaton mass is generated at two-loop level in perturbation theory. It is also suppressed as $\propto \epsilon$ and ranges between $\sim 0.01$ $\mu$eV to $\sim 100$ MeV. The model ========= There is little doubt that SM is an effective low energy description of some more fundamental theory, which incorporates dark matter and neutrino masses and perhaps also addresses other theoretical problems, such as strong CP problem and flavour problem as well as provides a framework for a consistent quantum description of gravity. As an effective theory SM contains an additional mass parameter, the ultraviolet cut-off $\Lambda$, which is not just a mathematical tool to regulate divergent amplitudes, but is a physical parameter that encapsulates physics (massive fields and high momenta modes of light fields) which we are agnostic of. The Higgs potential defined at this ultraviolet scale reads: $$V(\Phi^{\dagger}\Phi)=V_0(\Lambda)+\lambda(\Lambda)\left[\Phi^{\dagger}\Phi - v_{ew}^2(\Lambda)\right]^2 +..., \label{1}$$ where $\Phi$ is the electroweak doublet Higgs field, $V_0$ is the field-independent constant (bare cosmological constant parameter) and the ellipsis denote all possible dimension $> 4$ (irrelevant), gauge invariant operators, $\left(\Phi^{\dagger}\Phi\right)^n$, $n=3,4...$. Other bare parameters include dimensionless couplings $\lambda(\Lambda)$ and a mass dimension parameter $v_{ew}(\Lambda)$, the bare Higgs expectation value. In principle, this potential with infinite number of nonrenormalisable operators and $\Lambda$-dependent parameters must fully encode the physics beyond SM. In practice, however, the parameters are measured in low-energy experiments, which are not particularly sensitive to irrelevant operators. The truncated theory contains finite number of parameters and is reliable only in the low-energy domain. Now, if one computes quantum correction $\delta m_{\Phi}^2$ to the Higgs mass parameter $m_{\Phi}^2\equiv 2\lambda v^2_{ew}$ one finds that it is $\propto \Lambda^2$. Taking this computation as a guiding estimate, one comes to the conclusion that a light Higgs $(m_{\Phi}^2+\delta m_{\Phi}^2)/\Lambda^2 << 1$, necessarily implies fine-tuning between the tree-level parameter $m_{\Phi}^2(\Lambda)$ and the quantum correction to it, $\delta m_{\Phi}^2$. However, this naive conclusion is not necessarily correct if the theory exhibits additional symmetries such as softly broken supersymmetry or classical scale invariance (for more discussion see Ref. [@Kobakhidze:2014afa]). Assume now that a fundamental theory maintains spontaneously broken scale invariance, such that all mass parameters (including gravitational constant) have the common origin. To make this symmetry manifest in our effective theory, we promote all mass parameters to a dynamical field $\chi$, the dilaton, as follows[^2]: $$\Lambda \to \Lambda \frac{\chi}{f_{\chi}},~~v_{ew}^2(\Lambda) \to \frac{v_{ew}^2(\chi)}{f_{\chi}^2}\chi^2\equiv \frac{\xi (\chi)}{2}\chi^2,~~V_0(\Lambda)\to \frac{V_0(\chi)}{f_{\chi}^4}\chi^4\equiv \frac{\rho(\chi)}{4}\chi^4~, \label{2}$$ where $f_{\chi}$ is the dilaton decay constant (in analogy with the pion decay constant in the effective chiral theory), which we assume to be equal to $\Lambda$ in what follows. Then, Eq. (\[1\]) turns into the Higgs-dilaton potential, $$V(\Phi^{\dagger}\Phi, \chi)=\lambda(\chi)\left[\Phi^{\dagger}\Phi -\frac{\xi(\chi)}{2}\chi^2 \right]^2 +\frac{\rho(\chi)}{4}\chi^4~. \label{3}$$ This potential is manifestly scale invariant up to the quantum scale anomaly, which is engraved in $\chi$-dependence of dimensionless couplings[^3]. Indeed, the Taylor expansion around an arbitrary fixed scale $\mu$ reads: $$\lambda^{(i)}(\chi)=\lambda^{(i)}(\mu)+\beta_{\lambda^{(i)}}(\mu)\ln\left(\chi/\mu\right)+\beta'_{\lambda^{(i)}}(\mu)\ln^2\left(\chi/\mu\right)+..., \label{4}$$ where $\lambda^{(i)} \equiv (\lambda, \xi, \rho)$ and $$\beta_{\lambda^{(i)}}(\mu)=\left. \frac{\partial \lambda^{(i)}}{\partial \ln\chi}\right |_{\chi=\mu}~, \label{5}$$ is the renormalisation group (RG) $\beta$-functions for respective couplings $\lambda^{(i)}$ defined at a scale $\mu$, while $\beta'_{\lambda^{(i)}}(\mu)=\left. \frac{\partial^2 \lambda^{(i)}}{\partial (\ln\chi )^2}\right |_{\chi=\mu}$, etc. Note that while the lowest order contribution in $\beta$-functions is one-loop, i.e. $\sim {\cal O}(\hbar)$, $n$-th derivative of $\beta$ is higher $nth$ order in the perturbative loop expansion, $\sim {\cal O}(\hbar^n)$. In order to analyse minima of the potential (\[3\]) it is convenient to set an arbitrary renormalisation scale $\mu$ to be equal to the dilaton VEV, $\langle \chi \rangle \equiv v_{\chi}$. We also need to satisfy phenomenologically important constraint that the vacuum energy density is (nearly) zero $V(v_{ew}, v_(\chi))=0$ as it is required by astrophysical observations. The later constraint is nothing but a fine-tuning of the cosmological constant, which in scale invariant theories results in a certain relation between dimensioneless couplings [@Foot:2010et; @Foot:2011et]. For our model we find: $$\begin{aligned} V(v_{ew}, v_{\chi})=0 \Longrightarrow \rho(v_{\chi})=0~. \label{6}\end{aligned}$$ This relation, together with the extremum condition $\left. \frac{dV}{d\chi}\right |_{\Phi=\langle\Phi\rangle, \chi=\langle\chi\rangle}=0$, actually implies: $$\begin{aligned} \frac{\beta_{\rho}(v_{\chi})}{4}+\rho(v_{\chi})=0 \Longrightarrow \beta_{\rho}(v_{\chi})=0~. \label{7}\end{aligned}$$ One of the above equations (\[6\], \[7\]) can be used to define the dilaton VEV (dimensional transmutation) and another represents tuning of the cosmological constant. The second extremum condition $\left. \frac{dV}{d\Phi}\right |_{\Phi=\langle\Phi\rangle, \chi=\langle\chi\rangle}=0$ sets the hierarchy of VEVs: $$\begin{aligned} \xi(v_{\chi})\equiv \epsilon^2 = \frac{v^2_{ew}}{v^2_{\chi}}~. \label{8}\end{aligned}$$ This is a good place to remark on the the stability of the hierarchy of scales. Unlike the Higgs self-coupling $\lambda$, both Higgs-dilatton and self-dilaton couplings, $\lambda \xi$ and $\lambda \xi^2 +\rho$, respectively, exhibit trivial infrared fixed points, i.e., $\xi=\rho=0$, and hence they do not change much under the RG running, if taken to be small at some renormalisation scale. This implies that the ratio of VEVs in Eq. (\[8\]) can be hierarchically small in the sense of technical naturalness [@Wetterich:1983bi; @Foot:2013hna; @Kobakhidze:2014afa], that is, no radiative corrections can change the hierarchy $\epsilon$ appreciably as it is defined through the small coupling $\xi$ according to Eq. (\[8\]). We also note that in the classical (or exact conformal) limit where $\beta_{\lambda^{(i)}}= 0$, VEVs and consequently the hierarchy are undetermined and thus can be arbitrary, as expected. The light dilaton ================= Next we compute the scalar mass spectrum. The 2-by-2 mass squared matrix of the neutral Higgs scalar and the dilaton fields is given by $$\begin{aligned} \mathbf{M}^2(v_{\chi})= v_{ew}^2 \left( \begin{tabular}{rr} $2\lambda(v_{\chi})$ & $-\frac{\lambda(v_{\chi})}{\epsilon}\left(\beta_{\xi}(v_{\chi}) + 2\epsilon^2 \right)$ \\ $-\frac{\lambda(v_{\chi})}{\epsilon}\left(\beta_{\xi}(v_{\chi}) + 2\epsilon^2 \right)$ & $\frac{\lambda(v_{\chi})}{2\epsilon^2}\left(\beta_{\xi}(v_{\chi}) + 2\epsilon^2 \right)^2+\frac{\beta'_{\rho}(v_{\chi})}{4\epsilon^2}$ \\ \end{tabular} \right) \label{11}\end{aligned}$$ We immediately notice that in the limit $\beta'_{\rho}(v_{\chi})\to 0$ the above matrix becomes degenerate and hence the dilaton running mass tends to zero at the scale $v_{\chi}$[^4]. Thus the dilaton mass in our model emerges at ${\cal O}(\hbar ^2)$ in the perturbative loop expansion. This is in accord with the earlier observation in [@Foot:2011et] that cancellation of the scalar vacuum energy implies that the dilaton mass is generated at 2-loop level. More specifically, we find for the scalar running masses, $$\begin{aligned} m_h^2\simeq2\lambda (v_{\chi}) v^2_{ew}~,~~ m_{\chi}^2\simeq\frac{\beta'_{\rho}(v_{\chi})}{4\epsilon^2}v^2_{ew}~, \label{12}\end{aligned}$$ where, to a good accuracy, we can express $\beta'_{\rho}(v_{\chi})$ through the Higgs self-coupling beta-function as: $$\begin{aligned} \beta'_{\rho}(v_{\chi})= \epsilon^4\beta'_{\lambda}(v_{\chi})+2\epsilon^2\beta'_{\lambda_{h\chi}}(v_{\chi})+\beta'_{\lambda_{\chi}}(v_{\chi})-\frac{2}{\lambda(v_{\chi})}\left(\epsilon^2\beta_{\lambda}(v_{\chi})+\beta_{\lambda_{h\chi}}(v_{\chi})\right)^2~, \label{13}\end{aligned}$$ where the relevant beta-functions are given in appendix. We observe that the Higgs boson mass in Eq. (\[12\]) is essentially the same as in SM, while dilaton mass is suppressed by the hierarchy parameter, $m_{\chi}\propto \frac{\epsilon}{16\pi^2} v_{ew}$. We also find that Higgs-dilaton mixing is small and is also controlled by the hierarchy parameter: $$\begin{aligned} \tan2\alpha \approx -\epsilon~, \label{14}\end{aligned}$$ ,e.g. $\alpha \lesssim 0.01$ for $v_{\chi}\gtrsim 10$ TeV. Thus, our model predicts a very light dilaton with very small mixing with the Higgs boson. ![RG evolution of the dilaton mass square for various cut-off scales: $10^4,~10^{10},~10^{16}$ and $M_P= 1.2\cdot 10^{19}$ GeV. The dotted, solid and dashed lines corresponds to top quark mass $m_t= 171, 173$ and 174 GeV, respectively.[]{data-label="mass"}](m_e4 "fig:"){width="0.5\linewidth"} ![RG evolution of the dilaton mass square for various cut-off scales: $10^4,~10^{10},~10^{16}$ and $M_P= 1.2\cdot 10^{19}$ GeV. The dotted, solid and dashed lines corresponds to top quark mass $m_t= 171, 173$ and 174 GeV, respectively.[]{data-label="mass"}](m_e10 "fig:"){width="0.5\linewidth"} ![RG evolution of the dilaton mass square for various cut-off scales: $10^4,~10^{10},~10^{16}$ and $M_P= 1.2\cdot 10^{19}$ GeV. The dotted, solid and dashed lines corresponds to top quark mass $m_t= 171, 173$ and 174 GeV, respectively.[]{data-label="mass"}](m_e16 "fig:"){width="0.5\linewidth"} ![RG evolution of the dilaton mass square for various cut-off scales: $10^4,~10^{10},~10^{16}$ and $M_P= 1.2\cdot 10^{19}$ GeV. The dotted, solid and dashed lines corresponds to top quark mass $m_t= 171, 173$ and 174 GeV, respectively.[]{data-label="mass"}](m_mp "fig:"){width="0.5\linewidth"} The sign of the running masses in Eq. (\[12\]) at the cut-off scale (recall $v_{\chi}=\Lambda$) is essentially defined by the largest scalar coupling $\lambda (\Lambda)$. The RG evolution of this coupling (and thus its $\Lambda$-dependence) in our model is very similar to the one in SM: $\lambda (\mu)$ becomes negative at a scale $\mu_I\sim 10^{8}$ GeV, signalling instability of the effective potential[^5]. We find that the dilaton mass square $m^2_{\chi}(\Lambda)$ is positive for negative $\lambda(\Lambda)$, i.e. for $\mu_I\lesssim\Lambda \lesssim 10^{17}$ GeV, and negative when $\lambda(\Lambda)>0$. However, evaluating this running mass down to the infrared region, we find that it is always positive and well approximated by the formulae: $m_{\chi}\propto \frac{\epsilon}{16\pi^2} v_{ew}$. Hence depending on cut-off $\Lambda \in \left[10^4~ {\rm GeV},~~10^{19}~ {\rm GeV} \right]$, we predict light dilaton with mass in the range from $\sim 0.01$ $\mu$eV to $\sim 100$ MeV. The results of these calculations are presented on Figure \[mass\]. The couplings of the dilaton with the SM particles are defined through the mixing with the Higgs boson and scale anomaly. Since the dilaton is a pseudo-Goldstone boson of spontaneously broken anomalous scale invariance, its couplings are suppressed by powers of $1/v_{\chi}$. For large $v_{\chi}$, the dilaton is a very light state which feebly interacts with SM particles and can, in principle, play the role of dark matter. We will study phenomenology of the light dilaton in details elsewhere. Conclusion ========== In this paper we have presented a very simple extension of the effective SM with hidden scale invariance. The scale invariance manifests at low energies through the dilaton field. All mass scales in the model (including Wilsonian cut-off of the effective theory) are generated through the dilaton VEV $v_{\chi}$ through the quantum effect of dimensional transmutation. We have argued that $v_{ew}/v_{\chi}\ll 1$ is technically natural. In addition, assuming cancellation of the scalar vacuum energy, the dilaton mass is generated at 2-loop level, $\sim {\cal O}(\hbar^2)$, in the perturbative loop expansion and is proportional to the hierarchy of scales $\epsilon=v_{ew}/v_{\chi}$. Therefore, light dilaton with mass between $\sim 0.01$ $\mu$eV to $\sim 100$ MeV, depending on $\epsilon$, is a generic prediction of our model. Searches for such a light scalar particle may reveal a fundamental role of conformal invariance in nature. Besides the phenomenological studies of the light dilaton, which we delegate to future work, an interesting extension of our present work would be construction of a model which also addresses other outstanding problems of SM, such as neutrino masses and the strong CP problem. Inclusion of gravity in the current framework and study of early universe models of electroweak phase transition [@new] or inflationary scenarios along the lines of Ref. [@Barrie:2016rnv] would also be interesting. #### Acknowledgement. The work was supported in part by the Australian Research Council. Beta functions ============== For reader’s convenience here we include the relevant one-loop beta-functions, $\beta_{C}=\frac{dC}{d\ln(\mu)}$, used in our calculations: $$\begin{aligned} \beta_{g_{Y}} & = & \frac{g_{Y}^{3}}{16\pi^{2}}\frac{41}{6},\quad\beta_{g_{2}}=\frac{g_{2}^{3}}{16\pi^{2}}\left(-\frac{19}{6}\right),\quad\beta_{g_{3}}=\frac{g_{3}^{3}}{16\pi^{2}}\left(-7\right)\\ \beta_{y_{t}} & = & \frac{y_{t}}{16\pi^{2}}\left(-\frac{9}{4}g_{2}^{2}-8g_{3}^{2}-\frac{17}{12}g_{Y}^{2}+\frac{9}{2}y_{t}^{2}\right)\\ \beta_{\lambda} & = & \frac{1}{16\pi^2}\left[\lambda\left(-9g_{2}^{2}-3g_{Y}^{2}+24\lambda+12y_{t}^{2}\right)+\frac{3}{4}g_{2}^{2}g_{Y}^{2}+\frac{9}{8}g_{2}^{4}+\frac{3}{8}g_{Y}^{4}-6y_{t}^{4}+2\lambda_{h\chi}^{2}\right]\\ \beta_{\lambda_{\chi}} & = & \frac{1}{16\pi^{2}}\left(18\lambda_{\chi}+8\lambda_{h\chi}^{2}\right)\\ \beta_{\lambda_{h\chi}} & = & \frac{\lambda_{h\chi}}{16\pi^2}\left(-\frac{9}{2}g_{2}^{2}-\frac{3}{2}g_{Y}^{2}+12\lambda+6\lambda_{\chi}+8\lambda_{h\chi}+6y_{t}^{2}\right)\\ \beta_{m_{\chi}^{2}} & = & \frac{v^{2}}{16\pi^{2}}8\lambda\lambda_{h\chi}\end{aligned}$$ The RG equations were solved using the values of couplings at $\mu=m_t$ and relations between couplings at $\mu=v_{\chi}$ steaming from minimization conditions as described in the main text: $$\begin{aligned} \lambda(m_{t}) & = & \frac{1}{2}\frac{m_{h}^{2}}{v_{ew}^{2}}\approx 0.129\\ g_{Y}(m_{t}) & = & 0.35761+0.00011\left(\frac{m_{t}}{\text{GeV}}-173.10\right)\\ g_{2}(m_{t}) & = & 0.64822+0.00004\left(\frac{m_{t}}{\text{GeV}}-173.10\right)\\ g_{3}(m_{t}) & = & 1.1666-0.00046\left(\frac{m_{t}}{\text{GeV}}-173.10\right)+\frac{0.00314\left(\alpha_{3}(m_{Z})-0.1184\right)}{0.0007}\\ y_{t}(m_{t}) & = & 0.93558+0.0055\left(\frac{m_{t}}{\text{GeV}}-173.10\right)-\frac{0.00042(\alpha_{3}(m_{Z})-0.1184)}{0.0007}\\ \alpha_{3}(m_{Z}) & = & 0.1185\pm0.0006,\end{aligned}$$ where $m_{t}$ is the pole mass of top quark and the Higgs mass is taken to be $m_{h}=125.09$ GeV. 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[^1]: Despite 5 broken generators, spontaneously broken conformal invariance in 4D results in a single (pseudo)Goldstone boson, the dilaton. [^2]: See also earlier work [@Buchmuller:1990pz] for a similar inclusion of a dilaton within the dimensionally regularised SM. In a theory with gravity, the conformal scalar may play the role of the dilaton [@Kobakhidze:2015jya]. [^3]: In this we differ substantially from the so-called quantum scale-invariant SM [@Shaposhnikov:2008xi; @Ghilencea:2016dsl]. In their approach SM is extrapolated to an arbitrary high energy scale and regularized by invoking dilaton-dependent renormalization scale, $\mu=\mu(\chi)$ [^4]: Note however, very small mass is still expected due to the RG running in the infrared. Obviously, the dilaton would be strictly massless in the full conformal limit, as it is a true Goldstone boson in this limit. [^5]: It is known that the electroweak vacuum in SM is a metastable state (see the most recent analysis in Refs [@Bednyakov:2015sca; @Espinosa:2015qea]) and is consistent with observations, unless the rate of inflation is large [@Kobakhidze:2013tn; @Espinosa:2015qea]. The potential instability due to the fast inflation, however, must be re-analysed in our model, since dilaton is expected to play a significant role.

--- author: - 'Maciej Konacki, Guillermo Torres, Saurabh Jha & Dimitar D. Sasselov' title: '**A new transiting extrasolar giant planet**' --- 23.8cm-0.5cm+1.5cm The advent of high-precision Doppler and timing techniques in the past decade has brought a rich bounty of giant planets[@May:95::; @Mar:98::; @Schneider:02::] as well as smaller, terrestrial-mass pulsar planets[@Wol:92::]. To date over one hundred extrasolar giant planets have been found by different groups using precise radial velocity measurements[@Schneider:02::]. Photometric observations of transiting planets, when combined with radial velocities, yield entirely new diagnostics: the planet size and mean density[@Char:00::; @Hen:00::]. Transits supply the orbital inclination and a precise mass for the planet, and they additionally enable a number of follow-up studies [@Jha:00::; @Brown:01::; @Charbonneau:02::; @Brown:02::]. Hence, a large number of transit searches are already underway or under development[@Horne:03::]. However, photometry alone cannot distinguish whether the occulting object is a gas giant planet ($\sim$1–13 Jupiter masses), a brown dwarf ($\sim$13–80 Jupiter masses) or a very late type dwarf star, because such objects have nearly constant radius over a range from $\sim$0.001 to 0.1 Solar masses. This critical parameter, the mass of the companion, can be determined from the amplitude of the radial velocity variation induced in the star. One of the most successful searches to date is the Optical Gravitational Lensing Experiment (OGLE), which uncovered 59 transiting candidates in three fields in the direction of the Galactic centre (OGLE-III)[@U1:02::; @U2:02::], with estimated sizes for the possible companions of $\sim$1–4 Jupiter radii. The large number of relatively faint ($V =$ 14–18 mag) candidates to study led to our strategy of a preliminary spectroscopic reconnaissance to detect and reject large-amplitude (high-mass) companions, followed by more precise observations of the very best candidates that remained. Of the 59 OGLE candidates, 20 were unsuitable: one is a duplicate entry, 4 have no ephemeris (only one transit was recorded), 8 show obvious signs in the light curve of a secondary eclipse and/or out-of-eclipse variations (clear indications of a stellar companion), and 7 were considered too faint to follow up. We undertook low-resolution spectroscopy of the other 39 candidates in late June and mid-July 2002 on the Tillinghast 1.5-m telescope at the F. L. Whipple Observatory (Arizona) and the 6.5-m Magellan I Baade telescope at Las Campanas Observatory (Chile). These spectra were used to eliminate stellar binaries, which can produce shallow, planet-like eclipses due to blending with light from another star, grazing geometry, or the combination of a large (early-type) primary and a small stellar secondary, but are betrayed by large, easily-detected velocity variations (tens of ). We found 25 of the 39 candidates to be stellar binaries, and 8 to be of early spectral type. Only 6 solar-type candidates remained with no detected variations at the few  level (G. Torres et al., in prep.). Subsequently, we used the high resolution echelle spectrograph (HIRES)[@Vog:94::] on the Keck I 10-m telescope at the W. M. Keck Observatory (Hawaii) on the nights of July 24-27, 2002, to obtain spectra of 5 of these candidates and measure more precise velocities. OGLE-TR-3 turned out to be the result of grazing eclipses and blending (with even a hint of a secondary eclipse present in the light curve), and the data for OGLE-TR-33, OGLE-TR-10, and OGLE-TR-58 are as yet inconclusive and require further measurements. OGLE-TR-33 exhibits a complex spectral line profile behaviour and could also be a blend. OGLE-TR-10 shows insignificant velocity variation, which is consistent with a sub-Jovian mass planetary companion; OGLE-TR-58 is still inconclusive because of the uncertain ephemeris (M. Konacki et al., in prep.). Only OGLE-TR-56 showed clear low-amplitude velocity changes consistent with its 1.21190-day photometric variation[@U2:02::], revealing the planetary nature of the companion. With only one bona-fide planet (or at most 3) among the $39+8$ objects examined spectroscopically or ruled out on the basis of their light curves, the yield of planets in this particular photometric search has turned out to be very low: at least 94% (possibly up to 98%) of the candidates are “false positives". This is likely to be due in part to the crowded field towards the Galactic centre, which increases the incidence of blends. We report here our results for OGLE-TR-56 ($I \simeq 15.3$ mag). Radial velocities were obtained using exposures of a Th-Ar lamp before and after the stellar exposure for wavelength calibration, and standard cross-correlation against a carefully-matched synthetic template spectrum (see Table 1). Our nightly-averaged measurements rule out a constant velocity at the 99.3% confidence level, and are much better represented by a Keplerian model of an orbiting planet (Figure 1a and 1b). Note that the period and phase of the solid curve are entirely fixed by the transit photometry, as the ephemeris is constrained extremely well by the 12 transits detected so far (A. Udalski, private communication). The only remaining free parameters are the amplitude of the orbital motion (the key to establishing the mass of the companion) and the centre-of-mass velocity, both of which can be accurately determined from our velocity measurements. The properties of the planet and the star are summarised in Table 2, and Figure 1c shows a phased light curve of the transit together with our fitted model. We performed numerous tests to place limits on any systematic errors in our radial velocities and to examine other possible causes for the variation. These are crucial to assess the reality of our detection. On each night we observed two “standards" (HD 209458 and HD 179949) which harbor close-in planets with known orbits[@Mazeh:00::; @Tinney:01::]. We derived radial velocities using the same Th-Ar method as for OGLE-TR-56, and also using the I$_2$ gas absorption cell to achieve higher accuracy than is possible for our faint OGLE candidates. In Figure 2 we show that the measured velocity difference between our two standards (HD 209458 minus HD 179949) is similar for the Th-Ar and I$_2$ techniques, and more importantly, that both are consistent with the expected velocity change. This indicates that we are able to detect real variations at a level similar to those we see in OGLE-TR-56. We can rule out the possibility that OGLE-TR-56 is a giant star eclipsed by a smaller main-sequence star, both from a test based on the star’s density inferred directly from the transit light curve[@Seager:02::], and from the very short orbital period. We also examined the spectra for sky/solar spectrum contamination from scattered moonlight; a very small contribution from this source was removed using TODCOR, a two-dimensional cross-correlation technique[@Zucker:94::]. The separation between the sky lines and the stellar lines is large enough ($\sim$30 ) that the effect on our derived velocities is very small. Blending of the light with other stars is the most serious concern[@Queloz:01::; @San:02::] in a crowded field such as toward the Galactic centre. We have examined the profiles of the stellar spectral lines for asymmetries and any phase-dependent variations that can result from blending. Very little asymmetry is present, and no correlation with phase is observed. In addition, we performed numerical simulations to fit the observed light curve assuming OGLE-TR-56 is blended with a fainter eclipsing binary. Extensive tests show that with a photometric precision similar to the OGLE data ($\sigma \simeq$ 0.003–0.015 mag), almost any transit-like light curve can be reproduced as a blend, and only with spectroscopy can these cases be recognized. For each trial simulation, the relative brightness and velocity amplitude of the primary in the eclipsing binary can be predicted. Although a good fit to the photometry of OGLE-TR-56 can indeed be obtained for a model with a single star blended with a fainter system comprising a G star eclipsed by a late M star, the G star would be bright enough that it would introduce strong line asymmetries (which are not seen), or would be detected directly by the presence of a second set of lines in the spectrum. Careful inspection using TODCOR[@Zucker:94::] rules this out as well. Therefore, based on the data available, a blend scenario seems extremely unlikely. This is the faintest ($V \simeq 16.6$ mag) and most distant ($\sim1500$ pc) star around which a planet with a known orbit has been discovered. The planet is quite similar to the only other extrasolar giant planet with a known radius, HD 209458b, except for having an orbit which is almost two times smaller. Thus its substellar hemisphere can heat up to $\sim$1900 K. However, this is still insufficient to cause appreciable planet evaporation (with thermal r.m.s. velocity for hydrogen of $\sim$7  compared to a surface escape velocity of $\sim$50 ). The tidal Roche lobe radius of OGLE-TR-56b at its distance from the star is $\sim$2 planet radii. The planet’s orbit is most likely circularised ($e=0.0$) and its rotation tidally locked, but the star’s rotation is not synchronised ($v \sin i \simeq$ 3 ). Thus the system appears to have adequate long-term stability. Interestingly, OGLE-TR-56b is the first planet found in an orbit much shorter than the current cutoff of close-in giant planets at 3–4 day periods ($\sim$0.04 AU)[@Schneider:02::]. This might indicate a different mechanism for halting migration in a protoplanetary disk. For example, OGLE-TR-56b may be representative of a very small population of objects — the so-called class II planets, which have lost some of their mass through Roche lobe overflow[@Trilling:98::] but survived in close proximity to the star; a detailed theoretical study of OGLE-TR-56b will be presented elsewhere (D. Sasselov, in prep.). These observations clearly show that transit searches provide a useful tool in adding to the amazing diversity of extrasolar planets being discovered. [10]{} , D., [Brown]{}, T. M., [Latham]{}, D. W. & [Mayor]{}, M. [Detection of Planetary Transits Across a Sun-like Star]{}. , L45–L48 (2000). , G. W., [Marcy]{}, G. W., [Butler]{}, R. P. & [Vogt]{}, S. S. 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[A planetary system around the millisecond pulsar PSR1257 + 12]{}. , 145–147 (1992). , S. *et al.* [Multicolor Observations of a Planetary Transit of HD 209458]{}. , L45–L48 (2000). , T. M., [Charbonneau]{}, D., [Gilliland]{}, R. L., [Noyes]{}, R. W. & [Burrows]{}, A. [Hubble Space Telescope Time-Series Photometry of the Transiting Planet of HD 209458]{}. , 699–709 (2001). , D., [Brown]{}, T. M., [Noyes]{}, R. W. & [Gilliland]{}, R. L. [Detection of an Extrasolar Planet Atmosphere]{}. , 377–384 (2002). , T. M., [Libbrecht]{}, K. G. & [Charbonneau]{}, D. [A Search for CO Absorption in the Transmission Spectrum of HD 209458b]{}. , 826–832 (2002). , K. Status and Prospects of Planetary Transit Searches: Hot Jupiters Galore, to appear in *Proc. Scientific Frontiers of Research on Extrasolar Planets* (eds. Deming, D. & Seager, S.), in the press (ASP, San Francisco, 2003). , S. S. *et al.* in [*Proc. SPIE Instrumentation in Astronomy VIII*]{} (eds Crawford, D. L. & Craine, E. R.) 362–375 (SPIE, Bellingham, WA, 1994). , C. G. *et al.* [First Results from the Anglo-Australian Planet Search: A Brown Dwarf Candidate and a 51 Peg-like Planet]{}. , 507–511 (2001). , S. & [Mallén-Ornelas]{}, G. On the Unique Solution of Planet and Star Parameters from an Extrasolar Planet Transit Light Curve. *Astrophys. J.* (in press); preprint http://arXiv.org/astro-ph/0206228 (2002). , S. & [Mazeh]{}, T. [Study of spectroscopic binaries with TODCOR. I: A new two-dimensional correlation algorithm to derive the radial velocities of the two components]{}. , 806–810 (1994). , D. *et al.* [No planet for HD 166435]{}. , 279–287 (2001). , N. C. *et al.* [The CORALIE survey for southern extra-solar planets. IX. A 1.3-day period brown dwarf disguised as a planet]{}. , 215–229 (2002). , D. E. *et al.* [Orbital Evolution and Migration of Giant Planets: Modeling Extrasolar Planets]{}. , 428–439 (1998). , T. MAKEE Keck Observatory HIRES Data Reduction Software, web page http://spider.ipac.caltech.edu/staff/tab/makee/ (2001). , A. M. & [Sasselov]{}, D. D. [HD 209458: Physical Parameters of the Parent Star and the Transiting Planet]{}. , 451–458 (2002). Acknowledgements {#acknowledgements .unnumbered} ================ We wish to thank A. Udalski and the OGLE team for generous contributions to this project. We are very grateful to S. Kulkarni for invaluable support, to R. Noyes and D. Latham for helpful comments, to T. Barlow for assistance with the spectroscopic reductions using MAKEE[@Barlow:01::], and to K. Stanek for his continuous encouragement. The data presented herein were obtained at the W. M. Keck Observatory (operated by Caltech, Univ. of California, and NASA), which was made possible by the generous financial support of the W. M. Keck Foundation. M.K. gratefully acknowledges the support of NASA through the Michelson Fellowship programme. G.T. acknowledges support from NASA’s Kepler Mission. S.J.thanks the Miller Institute for Basic Research in Science at UC Berkeley for support via a research fellowship. Correspondence and requests for materials should be addressed to M.K. (e-mail: [email protected]) ------------ ---------- ------- Date RV Error MJD 52480.4239 $-$49.26 0.20 52481.4011 $-$49.44 0.08 52481.4178 $-$49.24 0.09 52483.3984 $-$49.60 0.06 52483.4152 $-$49.78 0.11 ------------ ---------- ------- : [**OGLE-TR-56 radial velocities.**]{} The velocities (reduced to the solar system barycentre) and formal errors are given for each of our individual spectra of OGLE-TR-56. The data indicate a significant variation; a flat line fit gives $\chi^2 \simeq 20$ with 4 degrees of freedom (0.06% false alarm probability), which is considerably worse than a fit to a Keplerian orbit model with a fixed ephemeris ($\chi^2 \simeq 5$ with 3 degrees of freedom; 17% probability). Having shown, as a check, that the velocities from separate exposures on the same night (originally intended for cosmic ray removal) are not significantly different, we have adopted the nightly averages for subsequent use. A similarly high significance is found for the conclusion that the average velocities are not well fit by a flat line (99.3% confidence level). Parameter Value --------------------------------------- ------------------------------- Velocity amplitude $0.167 \pm 0.027$ Centre-of-mass velocity $-49.49 \pm 0.02$ Orbital period $1.21190 \pm 0.00001$ days Reference transit epoch (MJD) $52072.185 \pm 0.003$ Star mass $1.04 \pm 0.05$ M$_{\odot}$ Star radius $1.10 \pm 0.10$ R$_{\odot}$ Limb darkening coefficient ($I$ band) $0.56 \pm 0.06$ Orbital inclination $86 \pm 2$ deg Planet distance from star 0.0225 AU Planet mass $0.9 \pm 0.3$ M$_{\rm Jup}$ Planet radius $1.30 \pm 0.15$ R$_{\rm Jup}$ Planet density $0.5 \pm 0.3$ g cm$^{-3}$ : [**Derived stellar and planetary parameters.**]{} The physical properties of the star were derived by modelling the high-resolution spectra with numerical model atmospheres. We find that OGLE-TR-56 is very similar to the Sun, with a temperature of $T_{\rm eff} \sim 5900$ K. The star’s mass and radius were computed from our stellar evolution tracks[@Cody:02::]. Combining the stellar parameters with the OGLE-III $I$-band photometry yields the planetary radius and orbital inclination. The uncertainties shown for the orbital elements are formal errors; the errors for the planet mass and radius additionally reflect our conservative estimate of systematic uncertainties.

--- abstract: 'We introduce the [*diffusion $K$-means*]{} clustering method on Riemannian submanifolds, which maximizes the within-cluster connectedness based on the diffusion distance. The diffusion $K$-means constructs a random walk on the similarity graph with vertices as data points randomly sampled on the manifolds and edges as similarities given by a kernel that captures the local geometry of manifolds. Thus the diffusion $K$-means is a multi-scale clustering tool that is suitable for data with non-linear and non-Euclidean geometric features in mixed dimensions. Given the number of clusters, we propose a polynomial-time convex relaxation algorithm via the semidefinite programming (SDP) to solve the diffusion $K$-means. In addition, we also propose a nuclear norm (i.e., trace norm) regularized SDP that is adaptive to the number of clusters. In both cases, we show that exact recovery of the SDPs for diffusion $K$-means can be achieved under suitable between-cluster separability and within-cluster connectedness of the submanifolds, which together quantify the hardness of the manifold clustering problem. We further propose the [*localized diffusion $K$-means*]{} by using the local adaptive bandwidth estimated from the nearest neighbors. We show that exact recovery of the localized diffusion $K$-means is fully adaptive to the local probability density and geometric structures of the underlying submanifolds.' address: - 'Department of StatisticsUniversity of Illinois at Urbana-ChampaignS. Wright Street, Champaign, IL 61820: <[email protected]>: <http://publish.illinois.edu/xiaohuichen/> ' - 'Department of StatisticsUniversity of Illinois at Urbana-ChampaignS. Wright Street, Champaign, IL 61820: <[email protected]>: <https://sites.google.com/site/yunyangstat/> ' author: - Xiaohui Chen - Yun Yang bibliography: - 'clustering\_sdp.bib' date: 'First arXiv version: March 11, 2019. This version: ' title: 'Diffusion $K$-means clustering on manifolds: provable exact recovery via semidefinite relaxations' --- [^1] Introduction {#sec:introduction} ============ This article studies the clustering problem of partitioning $n$ data points to $K$ disjoint (smooth) Riemannian submanifolds with $1 {\leqslant}K {\leqslant}n$. Problem formulation ------------------- Let ${\mathcal{D}}_{k},k=1,\dots,K$ be compact and connected Riemannian manifolds of dimension $q_{k}$. Suppose that ${\mathcal{D}}_{k}$ can be embedded as a [*submanifold*]{} of an ambient Euclidean space ${\mathbb{R}}^{p}$ equipped with the Euclidean metric $\|\cdot\|$ (i.e., there is an immersion $\varphi_{k}: {\mathcal{D}}_{k} \to {\mathbb{R}}^{p}$ such that the differential ${\mathrm{d}}\varphi_{x}$ is injective for all $x \in {\mathcal{D}}_{k}$ and $\varphi_{k}$ is a homeomorphism onto $\varphi_{k}({\mathcal{D}}_{k}) \subset {\mathbb{R}}^{p}$; cf. [@doCarmo1992_RG]). In our clustering setting, we work with [*disjoint*]{} submanifolds ${\mathcal{D}}_{1},\dots,{\mathcal{D}}_{K}$ in ${\mathbb{R}}^{p}$ and denote $S = \bigsqcup_{k=1}^{K} {\mathcal{D}}_{k}$ as their disjoint union. Each smooth submanifold ${\mathcal{D}}_{k}$ is endowed with the Riemannian metric $\rho_{k}$ induced from $\|\cdot\|$, and we denote ${\mathscr{D}}_{k}$ as the Borel $\sigma$-algebra on ${\mathcal{D}}_{k}$ (i.e., the $\sigma$-algebra generated by the open balls in ${\mathcal{D}}_{k}$ with respect to $\rho_{k}$). Let $X_{1}^{n} := \{X_{1},\dots,X_{n}\}$ be a sequence of independent random variables taking values in $S$. Suppose that there exists a clustering structure $G_{1}^{*},\dots,G_{K}^{*}$ (i.e., a partition on $[n] := \{1,\dots,n\}$ satisfying $\bigsqcup_{k=1}^{K} G_{k}^{*} = [n]$) such that each of the $n$ data points belongs to one of the $K$ clusters: if $i \in G_{k}^{*}$, then $X_{i} \sim \mu_{k}$ for some probability distribution $\mu_{k}$ supported on ${\mathcal{D}}_{k}$. Given the observations $X_{1}^{n}$, the task of this paper is to develop computationally tractable algorithms with strong theoretical guarantees for recovering the true clustering structure $G_{1}^{*},\dots,G_{K}^{*}$. ![Comparison of the $K$-means and the SDP relaxed diffusion $K$-means clustering methods on a synthetic data sampled from three clusters with one disk and two annuli.[]{data-label="fig:kmeans_demo"}](kmeans_demo.pdf "fig:")\ ![Comparison of the $K$-means and the SDP relaxed diffusion $K$-means clustering methods on a synthetic data sampled from three clusters with one disk and two annuli.[]{data-label="fig:kmeans_demo"}](diffusion_kmeans_demo.pdf "fig:") Classical clustering methods such as $K$-means [@MacQueen1967_kmeans] and mixture models [@FraleyRaftery2002_JASA] assume that data points from each cluster are sampled in the neighborhood (with the same dimension) of a [*centroid*]{}, where ${\mathcal{D}}_{k}$ contains only one point in ${\mathbb{R}}^{p}$. Such methods are effective for partitioning data with ellipsoidal contours, which implicitly implies that the similarity (or affinity) criteria of centroid-based clustering methods target on some notions of “compactness". In modern applications such as image processing and computer vision [@ShiMalik2000_IEEEPAMI; @SouvenirPless2005_ICCV; @ElhmifarVidal2011_NIPS], structured data with geometric features are commonly seen as clusters without necessarily being close together and having the same dimension. Figure \[fig:kmeans\_demo\] is an illustration for such observation on a synthetic data sampled from a noisy version of three clusters with one disk and two annuli. In this example, it is visually clear to distinguish the three clusters, however the $K$-means method fails to correctly cluster the data points. There are two main reasons for the failure of $K$-means. First, the north pole and south pole in the outer annulus have the largest Euclidean distance among all data points, even though they belong to the same cluster. Second, the annuli and the disk live in different dimensions. In particular, the annulus is a one-dimensional circle in ${\mathbb{R}}^{2}$ that is locally isometric to the real line and the disk has dimension two. Thus these geometric concerns motivate us to seek a more natural and flexible notion of closeness for clustering analysis. In this paper, we shall focus on the clustering criterion based on the [*connectedness*]{}, which is suitable for simultaneously addressing the two issues. First, connectedness is a graph property that does not rely on the physical distance: two vertices are connected if there is a path joining them. This extends the closeness from the local neighborhood to the global sense. Second, connectivity is a viable notion for clustering components of mixed dimensions, as long as all clusters live in the same ambient space where the graph connectivity weights can be computed. In the population version, a clustering component can be viewed as a smooth submanifold, embedded in ${\mathbb{R}}^{p}$. In Riemannian geometry, a Riemannian submanifold ${\mathcal{M}}$ in ${\mathbb{R}}^{p}$ is said to be [*connected*]{} if for any $x, y \in {\mathcal{M}}$, there is a parameterized regular curve joining $x$ and $y$. Thus an appealing notion of ${\mathcal{M}}$ for being a cluster is that ${\mathcal{M}}$ is a compact and connected component in ${\mathbb{R}}^{p}$. In our setting, a clustering model is the union of $K$ disjoint submanifolds ${\mathcal{D}}_{1},\dots,{\mathcal{D}}_{K}$, and each ${\mathcal{D}}_{k}$ is equipped with a probability distribution $\mu_{k}$. Thus for the underlying true clustering model, there are $K$ connected graphs that do not overlap. In the sample version, data points are randomly generated from $\{({\mathcal{D}}_{k}, \mu_{k})\}_{i=1}^{k}$ with a clustering structure $\{G_{k}^{*}\}_{k=1}^{K}$. Typically, weighted graphs computed from the observed data points are fully connected (e.g., based on the Gaussian kernel). Thus a fundamental challenge of clustering analysis is to recover the true clustering structure from a noisy and fully connected weighted graph on the data. Our contributions ----------------- In this paper, we propose a new clustering method, termed as the [*diffusion $K$-means*]{}, for manifold clustering. The diffusion $K$-means contains two key ingredients. First, it constructs a random walk (i.e., a Markov chain) on the weighted random graph with data points $X_{1}^{n}$ as vertices and edge weights computed from a kernel representing the similarity of data in a local neighborhood. By running the Markov chain forward in time, the local geometry (specified by the kernel bandwidth) will be integrated at multiple time scales to reveal global (topological) structures such as the connectedness properties of the graph. In the limiting case as the sample size tends to infinity and the bandwidth tends to zero, the random walk becomes a diffusion process over the the manifold. By looking at the spectral decomposition of this limiting diffusion process, the evaluations of the eigenfunctions at vertices $X_{1}^{n}$ can be viewed as a continuous embedding of the data, called [*diffusion map*]{}, into a higher-dimensional Euclidean space. Second, once the diffusion map is obtained, we can compute the diffusion distance [@coifman2006diffusion] and the $K$-means algorithm (with the Euclidean metric) can be naturally extended with the diffusion affinity as the similarity measure. Since the diffusion distance/affinity captures the connectedness among vertices on the weighted random graph, the diffusion $K$-means aims to maximize the within-cluster connectedness, which can be recast as an [*assignment*]{} problem via a 0-1 integer program. Because 0-1 integer programming problems with a non-linear objective function is generally $\mathsf{NP}$-hard, solving the diffusion $K$-means is computationally intractable, i.e., polynomial-time algorithms with exact solutions only exists in special cases. This motivates us to consider semidefinite programming (SDP) relaxations. We propose two versions of SDP relaxations of the diffusion $K$-means. The first one requires the knowledge of the number of clusters, and it can be viewed as an extension from Peng and Wei’s SDP relaxation [@PengWei2007_SIAMJOPTIM] for the $K$-means (as well as Chen and Yang’s SDP relaxation [@ChenYang2018] for the generalized $K$-means for non-Euclidean data in an inner product space) to the manifold clustering setting with diffusion distances. Figure \[fig:kmeans\_demo\] (on the right) shows that the SDP relaxed diffusion $K$-means can correctly identify the three clusters in the previous example. The second SDP relaxation does not require the number of clusters as an input. The idea is to drop the constraint on the trace of the clustering membership matrix (which involves number of clusters $K$), and to add a penalization term on the diffusion $K$-means objective function. Thus it can be seen as a nuclear norm [*regularized*]{} version of the SDP for diffusion $K$-means that is adaptive to the number of clusters. For both SDP relaxations of the diffusion $K$-means, we show that exact recovery can be achieved when the underlying submanifolds are well separated and subsamples within each submanifold are well connected. Since the diffusion $K$-means and its regularized version have only one (non-adaptive) bandwidth parameter to control the local geometry, they may fail for clustering problems with unbalanced sizes, mixed dimensions, and different densities. In such situations, a random walk on the vertices sampled from regions of low density mixes slower than that from regions of high density. This motivates us to consider a variant of diffusion $K$-means, termed as the [*localized diffusion $K$-means*]{}, by using data-dependent local bandwidth. We adopt the self-tuning procedure from [@zelnik2005self] where local adaptive bandwidth is estimated from the nearest neighbors and we show that the localized diffusion $K$-means is adaptive to the local geometry and the local sampling density for the purpose of exact recovery of the true clustering structure. To summarize, our contributions are listed as below. 1. We introduce the diffusion $K$-means clustering method for manifold clustering, which integrates the nonlinear embedding via the diffusion maps and the $K$-means clustering. 2. We propose two versions of the SDP relaxations of the diffusion $K$-means: one requires to know the number of clusters, and the other one does not require such knowledge as an input (and thus it is adaptive to the unknown number of clusters). 3. We derive the exact recovery property of the SDP relaxed diffusion $K$-means in terms of two hardness parameters of the clustering problem: one reflects the separation of the submanifolds, and the other one quantifies the degree of connectedness of the submanifolds. 4. We combine the local scaling procedures with the diffusion $K$-means and its regularized version, and derive their adaptivity when the clustering problems have unbalanced sizes, mixed dimensions, and different densities. Related work ------------ There is a large collection of clustering methods and algorithms in literature, which can be broadly classified into two categories: hierarchical clustering and partition-based clustering. Hierarchical clustering recursively divides data points into groups in either a top-down or bottom-up way. Such algorithms are greedy and they often get stuck into local optimal solutions. Partition-based clustering methods such as $K$-means clustering [@MacQueen1967_kmeans] and spectral clustering [@vanLuxburg2007_spectralclustering] directly assign each data point with a group membership. Perhaps one of the most widely used clustering methods is the $K$-means method, due to the existence of algorithms with linear sample complexity (such as Lloyd’s algorithm [@Lloyd1982_TIT]). However, the $K$-means clustering converges locally to a stationary point that depends on the initial partition. Recent theoretical studies in [@LuZhou2016] show that, given a proper initialization (such as spectral clustering), Lloyd’s algorithm for optimizing the $K$-means objective function can consistently recover the clustering structures. Exact and approximate recovery of various convex relaxations for the $K$-means and mixture models are studied in literature [@PengWei2007_SIAMJOPTIM; @LiLiLingStohmerWei2017; @FeiChen2018; @Royer2017_NIPS; @BuneaGiraudRoyerVerzelen2016]. To the best of our knowledge, existing theoretical guarantees developed for the convex relaxed $K$-means clustering assumes that the clusters are sampled in a neighborhood of a centroid. Thus results derived for $K$-means in literature cannot be directly compared with our results. On the other hand, spectral clustering methods [@ShiMalik2000_IEEEPAMI; @NgJordanWeiss2001_NIPS] take the similarity matrix as the input and solve the clustering problem by applying $K$-means to top eigenvectors of the graph Laplacian matrix or its normalized versions [@Chung1996_SpectralGraphTheory]. In essence, spectral clustering contains two steps: (i) the Laplacian eigenmaps embed data into feature spaces, and (ii) $K$-means on top eigenvectors serves as a rounding procedure to obtain the true clustering structure [@vanLuxburg2007_spectralclustering]. Conventional intuition for spectral clustering is that the embedding step (i) often “magnifies" the cluster structure from the dataset to the feature space such that it can be revealed by a relatively simple algorithm (such as $K$-means) in step (ii). However, theoretical guarantees (such as exact recovery) for the spectral clustering is rather vague in literature, partially due to its two-step nature which complicates its theoretical analysis. For instance, [@vonLuxburgBelkinBousquet2008_AoS] study the convergence of spectral properties of random graph Laplacian matrices constructed from sample points and they establish the consistency of the spectral clustering in terms of eigenvectors. However, they do not address the problem of the exact recovery property of the clustering structure. Similar results along this direction can be found in [@Rosasco2010_JMLR; @schiebinger2015; @TrillosHoffmannHosseini2019]. [@LingStrohmer2019] propose similar SDP relaxations for the spectral clustering as in the present paper with the diffusion distance metric replaced with the graph Laplacian. Specifically, it is shown in [@LingStrohmer2019] that those SDP relaxations can exactly recover the true clustering structure under a spectral proximity condition. Such condition is deterministic and difficult to check for general data generation models. (A particular checkable random model is the stochastic ball model [@LingStrohmer2019].) During the preparation of this work, we notice a recent work [@maggioni2018learning] which proposes a similar idea of applying the diffusion distance as the similarity metric for clustering based on fast search and find of density peaks clustering (FSFDPS) [@rodriguez2014clustering]. To prove exact recovery, [@maggioni2018learning] requires strong deterministic assumptions on the Markov transition matrix associated with the diffusion process that could be difficult to check under their stochastic clustering model. Literature on theoretical guarantees for manifold clustering is rather scarce, with a few exceptions [@Arias-Castro2011_IEEETIT; @LittleMaggioniMurphy2017]. Near-optimal recovery of some emblematic clustering methods based on pairwise distances of data is derived under a condition that the [*minimal*]{} signal separation strength over all pairs of submanifolds is larger than a threshold. Compared with our diffusion $K$-means with local scaling, results established in [@Arias-Castro2011_IEEETIT] are non-adaptive to the local density and (geometric) structures of the submanifolds (cf. Theorem \[thm:main\_adaptive\_h\] and \[thm:main\_adaptive\] ahead). [@LittleMaggioniMurphy2017] derive recovery guarantees for manifold clusters using a data-dependent metric called the longest-leg path distance (LLPD) that adapts to the geometry of data, where the data points are drawn from a mixture of uniform distributions on disjoint low-dimensional geometric objects. Notation -------- For a matrix $A\in{\mathbb{R}}^{n\times n}$ and index subsets $G,G'\subset [n]$, we use notation $A_{GG'}$ to denote the submatrix of $A$ with rows being selected by $G$ and columns by $G'$, and $\mbox{diag}(A)$ the $n$-dimensional vector composed of all diagonal entries of $A$. Let $\|A\|_{\infty} =\max_{1 {\leqslant}i,j {\leqslant}n} |A_{ij}|$ and $\|A\|_1=\sum_{i,j=1}^{n} |A_{ij}|$ denote the $\ell_\infty$ and the $\ell_1$ norm of the vectorization $\mbox{vec}(A)$ of matrix $A$. Let ${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert A \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast}$ and ${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert A \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\mbox{\scriptsize op}}$ denote the nuclear norm and the operator norm of matrix $A$. We shall use $c, c', c_{1}, c_{2},\dots,C,C', C_{1},C_{2},\dots$ to denote positive and finite (non-random) constants whose values may depend on the submanifolds $\{{\mathcal{D}}_{k}\}_{k=1}^{K}$ and the probability distributions $\{\mu_{k}\}_{k=1}^{K}$ supported on $\{{\mathcal{D}}_{k}\}_{k=1}^{K}$ and whose values may vary from place to place. The rest of the paper is organized as follows. In Section \[sec:prelims\], we discuss some related background on diffusion distances and nonlinear embeddings in Euclidean spaces, as well as the Laplace-Beltrami operator for the heat diffusion process on Riemannian manifolds. In Section \[sec:diffusion\_Kmeans\], we introduce the diffusion $K$-means and its SDP relaxations. Regularized and localized diffusion $K$-means clustering methods are also proposed in this section. In Section \[sec:main\_results\], we present our main results on the exact recovery property of the SDP relaxed diffusion $K$-means. Simulation studies are presented in Section \[sec:simulations\]. Proofs are relegated to Section \[sec:proofs\]. Preliminaries {#sec:prelims} ============= Let $S \subset {\mathbb{R}}^{p}$ and $\mu$ be a probability distribution on $S$. Let $S_n=\{X_1,X_2,\ldots,X_n\}$ be $n$ i.i.d. random variables in $S$ sampled from $\mu$, and $\mu_n =n^{-1} \sum_{i=1}^n \delta_{X_i}$ the empirical distribution. In Section \[sec:prelims\], we discuss the Euclidean embedding via the diffusion distance in the general setting. Thus in this section, we do not assume $S$ to be a disjoint union of $K$ submanifolds in ${\mathbb{R}}^{p}$ and the sample $S_{n}$ does not necessarily have a clustering structure. Euclidean embedding and diffusion distances {#subsec:Euclidean_embedding_diffusion_dist} ------------------------------------------- Let $\kappa : S \times S \to {\mathbb{R}}$ be a positive semidefinite kernel that satisfies: symmetry: $\kappa(x, y) = \kappa(y, x)$, positivity preserving: $\kappa(x, y) {\geqslant}0$. A kernel is a similarity measure between points of $S$. A widely used example is the Gaussian kernel: $$\label{eqn:gaussian_kernel} \kappa(x, y) = \exp \left( -{ \|x-y\|^{2} \over 2 h^{2}} \right),$$ where $h > 0$ is the bandwidth parameter that captures the local similarity of points in $S$. Given a kernel $\kappa$ with property (i) and (ii), we can define a reversible Markov chain on $S$ via the normalized graph Laplacian constructed as follows. Specifically, for any $x \in S$, let $$d(x) = \int_{S} \kappa (x, y)\, {\mathrm{d}}\mu(y)$$ be the degree function of the graph on $S$. For simplicity, we assume $d(x)>0$ for all $x\in S$. Define $$\label{eqn:transition_kernel} p(x, y) = {\kappa(x, y) \over d(x)},$$ which satisfies the positivity preserving property (ii) and the conservation property $$\int_{S} p(x, y) \, {\mathrm{d}}\mu(y) = 1.$$ Thus $p(x, y)$ can be viewed as the one-step transition probability of a (stationary) Markov chain on $S$ from $x$ to $y$. We shall write this Markov chain (i.e., random walk) as ${\mathcal{W}}= (S, \mu, p)$, where $p(\cdot, \cdot)$ is called the [*transition kernel*]{} of ${\mathcal{W}}$. Equivalently, we can describe ${\mathcal{W}}$ by the bounded linear operator $P : L^{2}({\mathrm{d}}\mu) \to L^{2}({\mathrm{d}}\mu)$ defined as $$Pf(x) = \int_{S} p(x, y) f(y) \, {\mathrm{d}}\mu(y).$$ Here $L^{2}({\mathrm{d}}\mu) := L^{2}(S, {\mathrm{d}}\mu)$ is the class of squared integrable functions on $S$ with respect to $\mu$. In literature, $P$ is often called the [*diffusion operator*]{} for the following reason. If we denote $p_{t}(x, y)$ as the $t$-step transition probability of the Markov chain ${\mathcal{W}}$ from $x$ to $y$ in $S$, then $$P_{t}f(x) = \int_{S} p_{t}(x, y) f(y) \, {\mathrm{d}}\mu(y),\qquad t=1,2,\ldots,$$ form a semi-group of bounded linear operators on $L^{2}({\mathrm{d}}\mu)$. Let $\Pi$ be a stationary distribution of the Markov chain ${\mathcal{W}}$ over $S$. Then $\Pi$ is absolutely continuous with respect to $\mu$, and the probability density function $\pi$ of $\Pi$ with respect to $\mu$ is given by the Radon-Nikodym derivative $$\label{eqn:stationary_dist} \pi(x) = \frac{{\mathrm{d}}\Pi}{{\mathrm{d}}\mu} (x) = { d(x) \over \int_{S} d(y)\, {\mathrm{d}}\mu(y)}.$$ Since $\Pi$ is the stationary measure of the Markov chain ${\mathcal{W}}$ with transition $P$, we have $$\Pi(P_{t}f) = \Pi(f)$$ for all bounded measurable functions $f$, where $\Pi(f) := \int_{S} f(x) \, {\mathrm{d}}\Pi(x)$. Note that, since the kernel $\kappa$ is symmetric, ${\mathcal{W}}$ is reversible and satisfies the detailed balance condition: $$\pi(x)\, p(x, y) = \pi(y) \,p(y, x),\quad\forall x,y\in S.$$ \[lem:spectral\_decomposition\_Markov\_chain\] Let $$R(x,y) = {\kappa(x,y) \over \sqrt{\pi(x)} \, \sqrt{\pi(y)}},\quad\forall x,y\in S.$$ If $$\label{eqn:kernel_integrability_condition} \int_{S} \int_{S} R(x,y)^{2} \, {\mathrm{d}}\mu(x) \, {\mathrm{d}}\mu(y) < \infty,$$ then the following statements hold. 1. There exists a sequence of nonnegative eigenvalues $\lambda_{0} {\geqslant}\lambda_{1} {\geqslant}\cdots {\geqslant}0$ such that $$R(x,y) = \sum_{j=0}^{\infty} \lambda_{j} \phi_{j}(x) \phi_{j}(y),$$ where $\{\phi_{j}\}_{j=0}^{\infty}$ is the set of associated eigenfunctions to $\{\lambda_{j}\}_{j=0}^{\infty}$, and $\{\phi_{j}\}_{j=0}^{\infty}$ forms an orthonormal basis of $L^{2}({\mathrm{d}}\mu)$. 2. The transition probability $p(x,y)$ admits the following decomposition $$p(x,y) = \sum_{j=0}^{\infty} \lambda_{j} \psi_{j}(x) \varphi_{j}(y),$$ where $\psi_{j}(x) = \phi_{j}(x) / \sqrt{\pi(x)}$ and $\varphi_{j}(x) = \phi_{j}(x) \sqrt{\pi(x)}$. 3. The diffusion operator $P$ satisfies $$P\, \psi_{j} = \lambda_{j} \,\psi_{j}, \quad j = 0, 1, \dots.$$ In addition, $\lambda_0=1$ and $\psi_0\equiv 1$. The proof of Lemma \[lem:spectral\_decomposition\_Markov\_chain\] is given in Appendix \[app:A\], and our argument is similar to Lemma 12.2 in [@levin2017markov] in the finite-dimensional setting. If ${\mathcal{W}}$ is irreducible (i.e., the graph on $S$ is connected in that for all $x,y \in S$, there is some $t>0$ such that $p_{t}(x,y) > 0$), then the stationary distribution $\pi$ is unique. Thus if we run this Markov chain ${\mathcal{W}}$ forward in time, then the local geometry (captured by the kernel $\kappa$ which is parameterized by the bandwidth $h$) will be integrated to reveal global structures of $S$ at multiple (time) scales. In particular, we can define a class $\{D_{t}\}_{t \in {\mathbb{N}}_{+}}$ of [*diffusion distances*]{} [@coifman2006diffusion] on $S$ by $$D_{t}(x, y) := \|\, p_{t}(x, \cdot) - p_{t}(y, \cdot)\,\|_{L^{2}({\mathrm{d}}\mu/\pi)} = \left\{ \int_{S} [\, p_{t}(x, z) - p_{t}(y, z)]^{2}\, {{\mathrm{d}}\mu(z) \over \pi(z)} \right\}^{1/2}.$$ Roughly speaking, for each $t\in {\mathbb{N}}_{+}$ and $x,y\in S$, the diffusion distance $D_{t}(x,y)$ quantifies the the total number of paths with length $t$ connecting $x$ and $y$ (see Figure \[fig:diffussion\_dist\]), thereby reflecting the local connectivity at the time scale $t$. ![Illustration of the diffusion distance between two red dots as the total number of paths connecting them. The region on the left panel (the Cheeger dumbbell) is “less" connected than the region on the right as there are fewer paths in the former due to the narrow bottleneck in the middle. In particular, the second smallest eigenvalue associated with the Laplace-Beltrami operator (or the Cheeger isoperimetric constant) of the left region is smaller that of the right.[]{data-label="fig:diffussion_dist"}](examples.pdf "fig:") ![Illustration of the diffusion distance between two red dots as the total number of paths connecting them. The region on the left panel (the Cheeger dumbbell) is “less" connected than the region on the right as there are fewer paths in the former due to the narrow bottleneck in the middle. In particular, the second smallest eigenvalue associated with the Laplace-Beltrami operator (or the Cheeger isoperimetric constant) of the left region is smaller that of the right.[]{data-label="fig:diffussion_dist"}](examples_2.pdf "fig:") \[lem:spectral\_representation\_diffusion\_distances\] If the Markov chain ${\mathcal{W}}= (S, \mu, p)$ is irreducible, then we have $$D_{t}^2(x, y) = \sum_{j=0}^{\infty} \lambda_{j}^{2t} \, [\psi_{j}(x) - \psi_{j}(y)]^{2}$$ for all $t\in{\mathbb{N}}_{+}$ and $x,y\in S$. The proof of Lemma \[lem:spectral\_representation\_diffusion\_distances\] is given in Appendix \[app:A\]. For an irreducible Markov chain, the spectral gap is strictly positive (i.e., $|\lambda_{j}| < 1$ for all $j > 0$). Based on the spectral decomposition in Lemma \[lem:spectral\_representation\_diffusion\_distances\] and noting that $\psi_{0} \equiv 1$, we see that the diffusion distance can be written as $$\begin{aligned} \label{Eqn:Spectral_rep} D_{t}(x, y) = \left\{ \sum_{j=1}^{\infty} \lambda_{j}^{2t} \, [\psi_{j}(x) - \psi_{j}(y)]^{2} \right\}^{1/2}.\end{aligned}$$ In this case, the diffusion distance $D_{t}(x, y)$ decays to zero as $t$ increases, provided that $x$ and $y$ belong to a connected component of the graph on $S$. In particular, the decay rate of the spectrum quantifies the connectivity of points in the graph on $S$. Given a positive integer $q \in {\mathbb{N}}_{+}$, the diffusion maps $\{\Psi_{t}\}_{t \in {\mathbb{N}}}$ are defined as $$\Psi_{t}^{(q)}(x) = (\lambda_{1}^{t} \psi_{1}(x), \dots, \lambda_{q}^{t} \psi_{q}(x))^{T},$$ where the $\ell$-th component $\Psi_{t\ell}^{(q)}(x)$ is the $\ell$-th diffusion coordinate in ${\mathbb{R}}^{q}$. Thus we obtain an embedding of $(S, \mu)$ into the Euclidean space ${\mathbb{R}}^{q}$ in the limiting sense that $$D_{t}(x, y) = \lim_{q \to \infty} \|\Psi_{t}^{(q)}(x) - \Psi_{t}^{(q)}(y)\|_{2}.$$ Empirical diffusion embedding {#subsec:empirical_diffusion_embedding} ----------------------------- Recall that $S_n=\{X_1,X_2,\ldots,X_n\}$ are $n$ i.i.d. random variables in $S$ sampled from $\mu$, and $\mu_n =n^{-1} \sum_{i=1}^n \delta_{X_i}$ is the empirical distribution. Given $S_n$, we can consider finite sample approximations $\{D_{n,t}\}_{t\in{\mathbb{N}}_{+}}$ to the underlying population level quantities $\{D_{t}\}_{t\in{\mathbb{N}}_{+}}$. More precisely, consider a weighted graph with nodes corresponding to the elements in $S_n$, where the weight between a pair $(X_i,X_j)$ of nodes is $\kappa(X_i,X_j)$, for $i,j\in [n]$. Define the (rescaled) empirical degree function $d_n:\, S_n\to {\mathbb{R}}_{+}$ by $$\begin{aligned} d_n(x) = n\,\int_{S_n} \kappa(x, y)\, {\mathrm{d}}\mu_n (y) =\sum_{i=1}^n \kappa(x, X_i),\quad \forall x\in S_n,\end{aligned}$$ where we added an extra $n$-factor so that $d_n(X_i)$ is also the degree of node $X_i$ in the weighted graph. Let $D_n$ denote the $n$-by-$n$ diagonal matrix whose $i$-th diagonal entry is $d_n(X_i)$. Consider the (empirical) random walk ${\mathcal{W}}_n=(S_n, \mu_n, P_n)$ over $S_n$ with transition probability $$\begin{aligned} P_n (x, y) = \frac{\kappa(x,y)}{d_n(x)}\quad\forall x,y\in S_n.\end{aligned}$$ The (empirical) stationary distribution $\pi_n$ of the random walk ${\mathcal{W}}_n$ over $S_n$ becomes $$\begin{aligned} \pi_n(x) = \frac{d_n(x)}{\sum_{i=1}^{n} d_n(X_i)} \quad \forall x\in S_n.\end{aligned}$$ For any vector $v\in{\mathbb{R}}^n_{+}$, let $L^2(v)=\{u=(u_1,\ldots,u_n)\in{\mathbb{R}}^n:\, \|u\|_{L^2(v)} = \sum_{i=1}^n v_i\, u_i^2\}$ denote a weighted $L^2$ space over $S_n$. We define the [*empirical diffusion distances*]{} $\{D_{n,t}\}_{t\in {\mathbb{N}}_{+}}$ as $$\begin{aligned} D_{n,t}(x,y) =\|P^t_n(x,\cdot) - P^t_n(y,\cdot)\|_{L^2(\mbox{diag}(D_n^{-1}))} = \left\{ \sum_{i=1}^n [\, P_{n}^t(x, X_i) - P^t_n(y, X_i)]^{2}\, {1 \over d_n(X_i)} \right\}^{1/2},\end{aligned}$$ for all $x,y\in S_n$ and $t\in{\mathbb{N}}_{+}$. Roughly speaking, $\sqrt{n^{-1}\sum_{i=1}^{n} d_n(X_i)}\,D_{n,t}$ provides an empirical estimate to $D_t$. Similar to the spectral representation  for $D_t$, we also have the following spectral representation of $D_{n,t}$ (see Appendix \[app:B\]), $$\begin{aligned} D_{n,t}(x, y)& = \left\{ \sum_{j=0}^{n-1} \lambda_{n, j}^{2t} \, [\psi_{n,j}(x) - \psi_{n,j}(y)]^{2} \right\}^{1/2}, \quad\forall t\in{\mathbb{N}}_{+} \mbox{ and }x,y\in S_n,\end{aligned}$$ where $1=\lambda_{n,0}{\geqslant}\lambda_{n,1} {\geqslant}\cdots {\geqslant}\lambda_{n,n-1} {\geqslant}0$ are the nonnegative eigenvalues (due to the positive semidefiniteness of the kernel $\kappa$) of the transition probability operator $P_n$, which can be identified with a matrix in ${\mathbb{R}}^{n\times n}$ with $[P_{n}]_{ij} = P_n(X_i,X_j)$ as its $(i,j)$-th element, and $\psi_{n,0},\psi_{n,1},\ldots,\psi_{n,n-1}:\,S_n \to{\mathbb{R}}$ are the associated eigen-functions on $S_n$ with unit $L^2(\mbox{diag}(D_n))$ norm, which can be identified with vectors in ${\mathbb{R}}^n$ with $[\psi_{n,j}]_i = \psi_{n,j}(X_i)$ as the $i$-th element of $\psi_{n,j}$ for $i\in [n]$ and $j =0,1\ldots,n-1$. The empirical diffusion distance $D_{n,t}(X_i, X_j)$ between two nodes $X_i$ and $X_j$ is also the Euclidean distance between their embeddings $\Psi_{n,t}(X_i)$ and $\Psi_{n,t}(X_j)$ via the empirical diffusion map $$\begin{aligned} \Psi_{n,t}:\, S_n \to {\mathbb{R}}^{n},\quad x\mapsto \big(\lambda_{n,1}^t \psi_{n,1}(x),\ldots,\lambda_{n,n}^t\psi_{n,n}(x)\big)^T.\end{aligned}$$ The Laplace-Beltrami operator on Riemannian manifolds {#subsec:Laplace-Beltrami_operator} ----------------------------------------------------- The Laplace-Beltrami operator on Riemannian manifolds is a generalization of the Laplace operator on Euclidean spaces. Let $f : {\mathcal{M}}\to {\mathbb{R}}$ be an (infinitely) differentiable function with continuous derivatives on a $q$-dimensional compact and smooth Riemannian manifold and $\nabla_{{\mathcal{M}}} f$ be the gradient vector field on ${\mathcal{M}}$ (i.e., $\nabla_{{\mathcal{M}}} f(x)$ is the deepest direction of ascent for $f$ at the point $x \in {\mathcal{M}}$). The Laplace-Beltrami operator $\Delta_{{\mathcal{M}}}$ is defined as the divergence of the gradient vector $$\Delta_{{\mathcal{M}}} f = -\mbox{div}(\nabla_{{\mathcal{M}}} f),$$ where the $\mbox{div}$ operator is relative to the volume form $\mbox{Vol}_{{\mathcal{M}}}$ of ${\mathcal{M}}$. Here we adopt the convention with the minus sign of the divergence such that $\Delta_{{\mathcal{M}}}$ is a positive-definite operator. With integration-by-parts, we have for any two differentiable functions $f$ and $g$, $$\int_{{\mathcal{M}}} g(x) \Delta_{{\mathcal{M}}} f \,{\mathrm{d}}\mbox{Vol}_{{\mathcal{M}}}(x) = \int_{{\mathcal{M}}} \langle \nabla_{{\mathcal{M}}} g(x), \nabla_{{\mathcal{M}}} f(x) \rangle \,{\mathrm{d}}\mbox{Vol}_{{\mathcal{M}}}(x),$$ where the inner product is taken in the $q$-dimensional tangent space of ${\mathcal{M}}$ (at the point $x$). In a Euclidean space (i.e., ${\mathcal{M}}= {\mathbb{R}}^{q}$), the Laplace-Beltrami operator is the usual Laplace operator $$\Delta f = -\sum_{j=1}^{q}{\partial^{2}f \over \partial x_{j}^{2}}.$$ On a general $q$-dimensional Riemannian manifold ${\mathcal{M}}$, the Laplace-Beltrami operator in a local coordinate system $(e^{1},\dots,e^{q})$ with a metric tensor ${\mathbf{G}}= (g_{ij})_{i,j=1}^{q}$ is given by $$\Delta_{{\mathcal{M}}} f = -{1 \over \sqrt{\det({\mathbf{G}})}} \sum_{j=1}^{q} {\partial \over \partial e^{j}} \left( \sqrt{\det({\mathbf{G}})} \sum_{i=1}^{q} g^{ij} {\partial f \over \partial e^{i}} \right),$$ where $g^{ij}$ are the entries of ${\mathbf{G}}^{-1}$. In the special case ${\mathcal{M}}= {\mathbb{R}}^{q}$, ${\mathbf{G}}$ is the $q \times q$ identity matrix. Note that $\Delta_{{\mathcal{M}}}$ is a self-adjoint positive-definite compact operator, its spectrum contains a sequence of nonnegative eigenvalues $0 {\leqslant}\lambda_{0} {\leqslant}\lambda_{1} {\leqslant}\cdots$. If in addition ${\mathcal{M}}$ is connected, then the second smallest eigenvalue $\lambda_{1} > 0$. As we will show, $\lambda_{1}$ depends on the connectivity of the manifold (Figure \[fig:diffussion\_dist\]), thus characterizing the limiting mixing time of the empirical random walk ${\mathcal{W}}_n$ over the $S_n$ as $n\to \infty$ and $h\to 0^{+}$, when $S_n$ is sampled from the manifold ${\mathcal{M}}$. Diffusion $K$-means {#sec:diffusion_Kmeans} =================== Recall that in our clustering model, $S_n=\{X_{1},X_2,\ldots,X_n\}$ is a sample of independent random variables taking values in $S$, where $S$ is the union of $K$ disjoint Riemannian submanifolds ${\mathcal{D}}_{1},\dots,{\mathcal{D}}_{K}$ embedded in the ambient space ${\mathbb{R}}^p$. The clustering problem is to divide these $n$ data points into $K$ clusters, so that points in the same cluster belongs to the same connected component in $S$, based on certain similarity measures between the points. In particular, the (classical) $K$-means clustering method minimizes the total intra-cluster squared Euclidean distances in ${\mathbb{R}}^p$ $$\min_{G_{1},\dots,G_{K}} \sum_{k=1}^{K} {1 \over |G_{k}|} \sum_{i,j \in G_{k}} \|X_{i}-X_{j}\|^{2}$$ over all possible partitions on $[n]$, where $|G_{k}|$ is the cardinality of $G_{k}$. Dropping the sum of squared norms $\sum_{i=1}^{n} \|X_{i}\|^{2}$, we see that the $K$-means clustering is equivalent to the maximization of the total within-cluster covariances $$\max_{G_{1},\dots,G_{K}} \sum_{k=1}^{K} {1 \over |G_{k}|} \sum_{i,j \in G_{k}} a_{ij}, \quad\mbox{with }a_{ij} = X_{i}^{T} X_{j}.$$ Here, $a_{ij}=X_{i}^{T} X_{j}$ can be viewed as a similarity measure specified by the Euclidean space inner product $\langle X_{i}, X_{j}\rangle_{{\mathbb{R}}^p}$. In general, we can replace the Euclidean inner product with any other inner product over $S_n$ [@ChenYang2018]. For manifold clustering, we replace it with the inner product induced from the empirical diffusion distance, that is, $$\begin{aligned} \langle x,\, y\rangle_{D_{n,t}} = \langle \Psi_{n,t}(x), \,\Psi_{n,t}(y) \rangle_{{\mathbb{R}}^{n}} =\sum_{j=1}^{n} \lambda_{n,j}^{2t} \,\psi_{n,j}(x) \, \psi_{n,j}(y),\quad \forall x,y\in S_n.\end{aligned}$$ Henceforth, we will refer to $\langle \cdot,\, \cdot\rangle_{D_{n,t}}$ as the [*diffusion affinity*]{}. Interestingly, we can obtain this diffusion affinity value without explicitly conducting eigen-decomposition (spectral decomposition) to the transition probability matrix $P_n=D_n^{-1} K_n$ (or the symmetrized matrix $D_n^{-1/2} K_n D_n^{-1/2}$), where recall that $D_n=\mbox{diag}\big(d_n(X_1),\ldots,d_n(X_n)\big)\in{\mathbb{R}}^n$ is the degree diagonal matrix, and $K_n=\big[\kappa(X_i,X_j)\big]_{n\times n}\in{\mathbb{R}}^{n\times n}$ is the empirical kernel matrix. In fact, we may use the following relation that links the empirical diffusion affinity with entries in matrix $P_n$ raising to power $2t$ (see Appendix \[app:B\] for details), $$\begin{aligned} \langle x,\, y\rangle_{D_{n,t}} = \sum_{j=1}^{n} \lambda_{n,j}^{2t} \,\psi_{n,j}(x) \, \psi_{n,j}(y)=[P_n^{2t}D_n^{-1}](x,y).\end{aligned}$$ This motivates a $K$-means clustering method via diffusion distances, referred to as the *diffusion $K$-means* as $$\begin{aligned} \label{Eqn:Diffussion_K_Means} \max_{G_{1},\dots,G_{K}} \sum_{k=1}^{K} {1 \over |G_{k}|} \sum_{i,j \in G_{k}} [P_n^{2t}D_n^{-1}]_{ij},\end{aligned}$$ for the tuning parameter $t$ interpreted as the number of steps in the empirical random walk ${\mathcal{W}}_n$. Note that here the affinity matrix $P_n^{2t}D_n^{-1}= D_n^{-1/2} (D_n^{-1/2} K_n D_n^{-1/2})^{2t} D_n^{-1/2} \in{\mathbb{R}}^{n\times n}$ is symmetric. In light of the connections between the diffusion distance and the random walk ${\mathcal{W}}_n$ over $S_n$ in Section \[subsec:empirical\_diffusion\_embedding\], the diffusion $K$-means attempts to maximize the total within-cluster connectedness. \[rem:intution\_DKM\] In Section \[subsec:Euclidean\_embedding\_diffusion\_dist\], we see that, on a connected submanifold ${\mathcal{D}}_{k}$, the (population) diffusion process  converges to the stationary distribution : $$p_{t}(x,y) \to \pi(y) = {\int_{{\mathcal{D}}_{k}} \kappa(x,y) \, {\mathrm{d}}\mu(x) \over \iint_{{\mathcal{D}}_{k} \times {\mathcal{D}}_{k}} \kappa(x,z) \, {\mathrm{d}}\mu(x)\, {\mathrm{d}}\mu(z)} \quad \text{as } t \to \infty.$$ In fact, since the kernel $\kappa$ is positive semidefinite, this convergence holds at a geometric rate governed by the spectral gap of the Laplace-Beltrami operator on ${\mathcal{D}}_{k}$ (cf.  in the proof of Lemma \[Lem:T\_2\]). Thus the empirical version of the diffusion (i.e., the Markov chain on the random graph generated by $X_{1}^{n}$ and $\kappa$) obeys $${P^{t}_{n}(X_{i}, X_{j}) \over \sum_{\ell \in G_{k}^{*}} \kappa(X_{\ell}, X_{j})} \approx {1 \over \sum_{\ell,\ell' \in G_{k}^{*}} \kappa(X_{\ell}, X_{\ell'})}$$ for any two data points $X_{i}, X_{j} \in {\mathcal{D}}_{k}$. On the other hand, if the separation between the submanifolds is large enough and $t$ is not so large, then the probability to diffuse from one cluster to another one is small (cf. Lemma \[lem:between\_cluster\_random\_walk\]). Thus we expect that $${P^{t}_{n}(X_{i}, X_{j}) \over \sum_{\ell \in G_{k}^{*}} \kappa(X_{\ell}, X_{j})} \approx 0$$ for any two data points $X_{i} \in {\mathcal{D}}_{k}$ and $X_{j} \in {\mathcal{D}}_{m}$ such that $k \neq m$. This means that the within-cluster entries of the empirical diffusion affinity are larger than the between-cluster entries. In particular, for suitably large $t \in {\mathbb{N}}_{+}$, the empirical diffusion affinity matrix $A_{n} := A_{n,t} = P_{n}^{2t} K_{n}^{-1}$ tends to become close to a block-diagonal matrix $$\begin{aligned} \label{Eqn:approx_form_A_n} A_n \approx \begin{pmatrix} \displaystyle \frac{1}{N_1} \mathbf{1}_{G_1^\ast} \mathbf{1}^T_{G_1^\ast} & 0 & \cdots & 0\\ 0 & \displaystyle \frac{1}{N_2} \mathbf{1}_{G_2^\ast} \mathbf{1}^T_{G_2^\ast} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & \cdots & 0 & \displaystyle \frac{1}{N_K} \mathbf{1}_{G_K^\ast} \mathbf{1}^T_{G_K^\ast} \end{pmatrix},\end{aligned}$$ where $N_{k} = \sum_{\ell,\ell' \in G_{k}^{*}} \kappa(X_{\ell}, X_{\ell'})$. Since each diagonal block of $A_{n}$ tends to be a constant matrix, if we run the Markov chain for a suitably long time, then this block-diagonal structure in the limit precisely conveys the true clustering structure in that $i, j \in G_{k}^{*}$ if and only if $\lim_{t \to \infty} [A_{n,t}]_{i,j} = N_{k}^{-1} > 0$. The trade-off regime of $t$ (cf.  in Theorem \[thm:main\]) is determined by the non-asymptotic bounds on the convergence of the empirical diffusion maps to its population version (cf. Lemma \[lem:within\_cluster\_random\_walk\] and \[lem:between\_cluster\_random\_walk\]), as well as the submanifolds separation. Note that, for every partition $G_{1},\dots,G_{K}$, there is a one-to-one $n \times K$ [*assignment matrix*]{} $H = (h_{ik}) \in \{0,1\}^{n \times K}$ such that $h_{ij} = 1$ if $i \in G_{k}$ and $h_{ij} = 0$ if $i \notin G_{k}$. Thus the diffusion $K$-means clustering problem can be recast as a 0-1 integer program: $$\label{eqn:kernel_Kmeans_integer_program} \max \left\{ \langle P_n^{2t}D_n^{-1}, H B H^{T} \rangle : H \in \{0,1\}^{n \times K}, H {\mathbf{1}}_{K} = {\mathbf{1}}_{n} \right\},$$ where ${\mathbf{1}}_{n}$ denotes the $n \times 1$ vector of all ones and $B = {\text{diag}}(n_{1}^{-1},\dots,n_{K}^{-1})$, where $n_{k}=|G_{k}|$ for $k=1,\ldots,K$ is the size of the $k$-th cluster. The diffusion $K$-means clustering problem (\[eqn:kernel\_Kmeans\_integer\_program\]) is often computationally intractable, namely, polynomial-time algorithms with exact solutions only exist in special cases [@SongSmolaGrettonBorgwardt2007_ICML]. For instances, the (classical) $K$-means clustering is an $\mathsf{NP}$-hard integer programming problem with a non-linear objective function [@PengWei2007_SIAMJOPTIM]. Exact and approximate recovery of various SDP relaxations for the $K$-means [@PengWei2007_SIAMJOPTIM; @LiLiLingStohmerWei2017; @FeiChen2018; @Royer2017_NIPS; @GiraudVerzelen2018] are studied in literature. However, it remains a challenging task to provide statistical guarantees for clustering methods that can capture non-linear features of data taking values on manifolds. Semidefinite programming relaxations ------------------------------------ We consider the SDP relaxations for the diffusion $K$-means clustering. Note that every partition $G_{1},\dots,G_{K}$ of $[n]$ can be represented by a partition function $\sigma : [n] \to [K]$ via $G_{k}=\sigma^{-1}(k), k=1,\dots,n$. If we change the variable $Z = H B H^{T}$ in the 0-1 integer program formulation (\[eqn:kernel\_Kmeans\_integer\_program\]) of the diffusion $K$-means, then $Z$ satisfies the following properties: $$\label{eqn:constraints_clustering_generic_integer_program} Z^{T} = Z, \quad Z \succeq 0, \quad \operatorname{tr}(Z) = \sum_{k=1}^{K} n_{k} b_{kk}, \quad (Z {\mathbf{1}}_{n})_{i} = \sum_{k=1}^{K} n_{k} b_{\sigma(i)k}, \; i=1,\dots,n.$$ For the diffusion $K$-means $B = {\text{diag}}(n_{1}^{-1},\dots,n_{K}^{-1})$, the last constraint in (\[eqn:constraints\_clustering\_generic\_integer\_program\]) reduces to $Z {\mathbf{1}}_{n} = {\mathbf{1}}_{n}$, which does not depend on the partition function $\sigma$. Thus we can relax the diffusion $K$-means clustering to the SDP problem: $$\label{eqn:clustering_Kmeans_sdp} \begin{gathered} \hat{Z} = \operatorname{argmax}\left\{ \langle A, Z \rangle : Z \in {\mathscr{C}}_K \right\} \\ \qquad \mbox{with } {\mathscr{C}}_K = \{Z \in {\mathbb{R}}^{n \times n} : Z^{T} = Z, Z \succeq 0, \operatorname{tr}(Z) = K, Z {\mathbf{1}}_{n} = {\mathbf{1}}_{n}, Z {\geqslant}0 \}, \end{gathered}$$ where $Z \succeq 0$ means that $Z$ is positive semidefinite and $Z {\geqslant}0$ means that all entries of $Z$ are non-negative, and matrix $A=[a_{ij}]:=A_{n}=P_n^{2t}D_n^{-1}\in{\mathbb{R}}^{n\times n}$. We shall use $\hat{Z}$ to estimate the true “membership matrix" $Z^{*}$, where $$\label{eqn:Kmeans_true_membership_matrix} Z_{ij}^{*} = \left\{ \begin{array}{cc} 1/n_{k} & \text{if } i, j \in G_{k}^{*} \\ 0 & \text{otherwise} \\ \end{array} \right. .$$ Note that $Z^{*}$ is a block diagonal matrix (up to a permutation) of rank $K$. If $X_{1},\dots,X_{n} \in {\mathbb{H}}$ (i.e., ${\mathbb{S}}= {\mathbb{H}}$) for some Hilbert space ${\mathbb{H}}$ and $a_{ij} = \langle X_{i}, X_{j} \rangle_{{\mathbb{H}}}$ is the inner product between $X_i$ and $X_j$, then (\[eqn:clustering\_Kmeans\_sdp\]) is the SDP for kernel $K$-means proposed in [@ChenYang2018]. In particular, [@PengWei2007_SIAMJOPTIM] consider the special case for the (Euclidean) $K$-means, where ${\mathbb{H}}= {\mathbb{R}}^{p}$ and $a_{ij} = X_{i}^{T} X_{j}$. Observe that the SDP relaxation (\[eqn:clustering\_Kmeans\_sdp\]) does not require the knowledge of the cluster sizes other than the number of clusters $K$. Thus it can handle the general case for unequal cluster sizes. Regularized diffusion $K$-means {#subsec:regularized_diffusion_Kmeans} ------------------------------- In practice, the number $K$ of clusters is rarely known. Note that the SDP problem  depends on $K$ only through the constraint $\operatorname{tr}(Z) = K$. Therefore we propose a [*regularized diffusion $K$-means*]{} estimator by dropping the constraint on the trace and penalizing $\operatorname{tr}(Z)$ as follows, $$\label{eqn:clustering_Kmeans_sdp_unknown_K} \begin{gathered} \tilde{Z} := \tilde{Z}_{\lambda} = \operatorname{argmax}\left\{ \langle A, Z \rangle - n\,\lambda \operatorname{tr}(Z) : Z \in {\mathscr{C}}\right\} \\ \mbox{with } {\mathscr{C}}= \{Z \in {\mathbb{R}}^{n \times n} : Z^{T} = Z, Z \succeq 0,\, Z {\mathbf{1}}_{n} = {\mathbf{1}}_{n}, Z {\geqslant}0 \}, \end{gathered}$$ where $\lambda>0$ is the regularization parameter. Recall that ${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert Z \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast}$ denotes the nuclear norm of a matrix $Z$ (i.e., ${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert Z \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast}$ is the sum of the singular values of $Z$). Since ${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert Z \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast} = \operatorname{tr}(Z)$ for $Z \in {\mathscr{C}}$, it is interesting to note that  is the same as the nuclear norm penalized diffusion $K$-means $$\label{eqn:clustering_Kmeans_sdp_unknown_K_nuclear-norm_form} \begin{gathered} \tilde{Z} = \operatorname{argmax}\left\{ \langle A, Z \rangle - n\,\lambda {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert Z \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast} : Z \in {\mathscr{C}}\right\}. \end{gathered}$$ Recall that the true membership matrix $Z^{*}$ has a block diagonal structure with rank $K$, the nuclear norm penalized diffusion $K$-means  can be thought as an $\ell^{1}$ norm convex relaxation of the $\mathsf{NP}$-hard rank minimization problem (i.e., minimizing the number of non-zero eigenvalues). Thus the parameter $K$ in the clustering problem plays a similar role as the sparsity (or low-rankness) parameter in the matrix completion context [@CandesRecht2009_FoCM]. Hence the SDP problem  can be viewed as a (further) convex relaxation of the infeasible SDP problem  when $K$ is unknown. Note that similar regularizations have been considered in [@BuneaGiraudRoyerVerzelen2016] for the $G$-latent clustering models and in [@YanSarkarCheng2018_AISTATS] for stochastic block models. It remains a question to choose the value of $\lambda$. Larger values of $\lambda$ will lead to solutions containing less number of clusters (with larger sizes). In particular, when matrix $A$ is positive-definite, the following Lemma \[lem:feasibility\_SDP\_lambda\_infinity\] shows that the solution reduces to a rank one matrix that assigns all points into a giant cluster when $\lambda$ is large enough, and becomes the identity matrix that assigns $n$ points into $n$ distinct clusters when $\lambda$ is small enough. In addition, the trace $\operatorname{tr}(\tilde Z_\lambda)$ of the solution is nonincreasing in $\lambda>0$. \[lem:feasibility\_SDP\_lambda\_infinity\] Suppose $A$ is a positive definite matrix, and let $\lambda_{\max}(A)$ and $\lambda_{\min}(A)$ denote its respective largest and smallest eigenvalues. (1) If $n \lambda > \lambda_{\max}(A)$, then $Z^{\diamond} = n^{-1} J_{n}$, where $J_{n}$ is the $n \times n$ matrix of all ones, is the unique solution of the SDP . (2) If $n \lambda < \lambda_{\min}(A)$, then $Z^{\dagger} = I_n$, the $n \times n$ identity matrix, is the unique solution of the SDP . (3) If $\tilde Z_{\lambda_1}$ and $\tilde Z_{\lambda_2}$ are two solutions of the SDP  with the regularization parameter taking values $\lambda_1$ and $\lambda_2$, respectively. If $\lambda_1 < \lambda_2$, then $\operatorname{tr}(\tilde Z_{\lambda_1}){\geqslant}\operatorname{tr}(\tilde Z_{\lambda_2})$. Furthermore, if at least one of the two SDPs has a unique solution, then $\operatorname{tr}(\tilde Z_{\lambda_1})> \operatorname{tr}(\tilde Z_{\lambda_2})$. According to the interpretation of the SDP , the trace $\operatorname{tr}(Z)$ of the solution can be viewed as the fitted number of clusters. Consequently, Lemma \[lem:feasibility\_SDP\_lambda\_infinity\] implies that smaller values of $\lambda$ will result in more clusters (with smaller sizes) in $\tilde{Z}_\lambda$. In practice, we need to properly select the tuning parameter $\lambda$. We propose the following decision rule for this purpose. For each $\lambda$, we run the SDP problem  and extract the value of $\operatorname{tr}(\tilde{Z}_\lambda)$. Then we plot the solution path $\operatorname{tr}(\tilde{Z}_\lambda)$ versus $\log(\lambda)$ and pick the values of $\lambda$ which spend the longest time (on the logarithmic scale) with a flat value of $\operatorname{tr}(\tilde{Z}_\lambda)$. Here we recommend using the logarithmic scale since values of $\lambda$ with non-trivial solutions $\operatorname{tr}(\tilde Z_\lambda)$ tends to be close to zero. Algorithm \[alg1\] below summarizes this decision rule. \[alg1\] Set an increasing sequence $\{\lambda_j\}_{j=1}^J$ of candidate values for $\lambda$, for example, a geometric sequence in the interval $[n^{-1}\lambda_{\min}(A),\,n^{-1}\lambda_{\max}(A)]$. Set an upper bound $K_{\max}$ of $K$ and a tolerance level $\varepsilon\in (0,1/2)$.\ Figure \[fig:diffusion\_kmeans\_lambda\_demo\] shows the empirical result on the three clusters (one disk and two annuli) example in Section \[sec:introduction\]. According to Lemma \[lem:feasibility\_SDP\_lambda\_infinity\], the estimated number of clusters, proxied by $\operatorname{tr}(\tilde{Z}_\lambda)$, is a non-increasing function of $\lambda$. In particular, the trace $\operatorname{tr}(\tilde Z_\lambda)$ in the solution path in the upper left panel of Figure \[fig:diffusion\_kmeans\_lambda\_demo\] stabilizes around $2$ and $3$, indicating that both $2$ and $3$ are candidate values for the number of clusters. In particular, the interval of $\lambda$ (on the logarithmic scale) corresponding to value $3$ is much larger than that to value $2$, indicating that $3$ is more likely to be the true number of clusters (cf. the (correct) case in Figure \[fig:diffusion\_kmeans\_lambda\_demo\]). In Section \[sec:main\_results\], we will use our theory to explain this stabilization phenomenon, which partially justifies our $\lambda$ selection rule. In addition, as we can see from the rest three panels in Figure \[fig:diffusion\_kmeans\_lambda\_demo\], by gradually increasing $\lambda$, the adaptive diffusion $K$-means method produces a hierarchical clustering structure. Unlike the top-down or bottom-up clustering procedures which are based on certain greedy rule and can incur inconsistency, the hierarchical clustering structure produced by our approach is consistent — it does not depend on the order of partitioning or merging due to the uniqueness of the global solution from the convex optimization via the SDP. Similar observations can be drawn on another example shown in Figure \[fig:diffusion\_kmeans\_lambda\_demo\_DGP2\] containing a uniform sample on three rectangles (see DGP 2 in our simulation studies Section \[sec:simulations\] for details). Further, it is interesting to observe in Figure \[fig:diffusion\_kmeans\_lambda\_demo\] that the regularized diffusion $K$-means tuned with two clusters yields a merge between the outer annulus and the disk, which gives the largest total diffusion affinity in the objective function  among the three possible combinations of the true clusters. Since diffusion affinity decays exponentially fast to zero in the squared Euclidean distance (for the Gaussian kernel), the diffusion affinity matrix $A = P_{n}^{2t} D_{n}^{-1}$ tends to have a block diagonal structure, as weights between points belonging to different clusters are exponentially small (cf.  and Lemma \[Lemma:total\_degree\]). Thus running SDP for examples with relatively well separated clusters, such as the one in Figure \[fig:diffusion\_kmeans\_lambda\_demo\], tends to merge two clusters with largest within-cluster diffusion affinities that is irrespective of the between-cluster Euclidean distances. This may lead to a visually less appealing merge as in the Euclidean distance case (cf. the (under) case in Figure \[fig:diffusion\_kmeans\_lambda\_demo\]). On the other hand, the regularized diffusion $K$-means is able to produce more reasonable partition in splitting the clusters (cf. the bottom-left panel in Figure \[fig:diffusion\_kmeans\_lambda\_demo\]). In particular, if the regularization parameter $\lambda$ is chosen such that the corresponding number $\hat{K}$ of clusters in the SDP solution is greater than $K$, then this will cause a split in one of the true clustering structures that minimizes the between-cluster diffusion affinities after the splitting. Moreover, in our simulation studies (setup DGP 3 in Section \[sec:simulations\]), we observe that the SDP relaxed regularized diffusion $K$-means performs much better in harder cases than the spectral clustering methods when the true clusters are not well separated. ![Plot of the estimated trace norms along the path of tuning parameter $\lambda$ (on the log-scale) in the SDP relaxed regularized diffusion $K$-means clustering method  on the synthetic data in Figure \[fig:kmeans\_demo\]. The clustered data are shown for estimated number of clusters 2 (under), 3 (correct), 4 (over).[]{data-label="fig:diffusion_kmeans_lambda_demo"}](diffusion_kmeans_lambda_demo.pdf "fig:") ![Plot of the estimated trace norms along the path of tuning parameter $\lambda$ (on the log-scale) in the SDP relaxed regularized diffusion $K$-means clustering method  on the synthetic data in Figure \[fig:kmeans\_demo\]. The clustered data are shown for estimated number of clusters 2 (under), 3 (correct), 4 (over).[]{data-label="fig:diffusion_kmeans_lambda_demo"}](diffusion_kmeans_adaptive_correct_demo "fig:")\ ![Plot of the estimated trace norms along the path of tuning parameter $\lambda$ (on the log-scale) in the SDP relaxed regularized diffusion $K$-means clustering method  on the synthetic data in Figure \[fig:kmeans\_demo\]. The clustered data are shown for estimated number of clusters 2 (under), 3 (correct), 4 (over).[]{data-label="fig:diffusion_kmeans_lambda_demo"}](diffusion_kmeans_adaptive_over_demo "fig:") ![Plot of the estimated trace norms along the path of tuning parameter $\lambda$ (on the log-scale) in the SDP relaxed regularized diffusion $K$-means clustering method  on the synthetic data in Figure \[fig:kmeans\_demo\]. The clustered data are shown for estimated number of clusters 2 (under), 3 (correct), 4 (over).[]{data-label="fig:diffusion_kmeans_lambda_demo"}](diffusion_kmeans_adaptive_under_demo "fig:") ![Plot of the estimated trace norms along the path of tuning parameter $\lambda$ (on the log-scale) in the SDP relaxed regularized diffusion $K$-means clustering method  for DGP=2. The clustered data are shown for estimated number of clusters 2 (under), 3 (correct), 4 (over).[]{data-label="fig:diffusion_kmeans_lambda_demo_DGP2"}](diffusion_kmeans_lambda_demo_DGP2.pdf "fig:") ![Plot of the estimated trace norms along the path of tuning parameter $\lambda$ (on the log-scale) in the SDP relaxed regularized diffusion $K$-means clustering method  for DGP=2. The clustered data are shown for estimated number of clusters 2 (under), 3 (correct), 4 (over).[]{data-label="fig:diffusion_kmeans_lambda_demo_DGP2"}](diffusion_kmeans_adaptive_correct_demo_DGP2 "fig:")\ ![Plot of the estimated trace norms along the path of tuning parameter $\lambda$ (on the log-scale) in the SDP relaxed regularized diffusion $K$-means clustering method  for DGP=2. The clustered data are shown for estimated number of clusters 2 (under), 3 (correct), 4 (over).[]{data-label="fig:diffusion_kmeans_lambda_demo_DGP2"}](diffusion_kmeans_adaptive_over_demo_DGP2 "fig:") ![Plot of the estimated trace norms along the path of tuning parameter $\lambda$ (on the log-scale) in the SDP relaxed regularized diffusion $K$-means clustering method  for DGP=2. The clustered data are shown for estimated number of clusters 2 (under), 3 (correct), 4 (over).[]{data-label="fig:diffusion_kmeans_lambda_demo_DGP2"}](diffusion_kmeans_adaptive_under_demo_DGP2 "fig:") ![Plots of the SDP diffusion $K$-means clustering method without (left) and with (right) local scaling for the data generation mechanism DGP=3 in the simulation studies Section \[sec:simulations\].[]{data-label="fig:diffusion_kmeans_local_scaling_demo"}](diffusion_kmeans_no_local_scaling_demo.pdf "fig:") ![Plots of the SDP diffusion $K$-means clustering method without (left) and with (right) local scaling for the data generation mechanism DGP=3 in the simulation studies Section \[sec:simulations\].[]{data-label="fig:diffusion_kmeans_local_scaling_demo"}](diffusion_kmeans_local_scaling_demo.pdf "fig:") Localized diffusion $K$-means {#subsec:localized_diffusion_Kmeans} ----------------------------- For clustering problems with different sizes, dimensions and densities, the diffusion $K$-means may have limitations since only one bandwidth parameter $h$ is used to control the local geometry on the domain. More precisely, according to our theory (for example, Theorem \[thm:main\] below), the optimal choice of the bandwidth parameter for a $q$-dimensional submanifold as one connected component in our clustering model is $h\asymp (\log \tilde{n}/\tilde{n})^{1/q}$, where $\tilde n$ corresponds to the sample size within this cluster and thus depends on the local cluster size or density level. Figure \[fig:diffusion\_kmeans\_local\_scaling\_demo\] demonstrates such an example for a mixture of three bivariate Gaussians, which consist of one larger Gaussian component with low density and two smaller Gaussian components with high density. Empirically, the diffusion $K$-means fails on this example (even after tuning) for the reason that the larger and smaller clusters have very different local densities. This motivates us to consider a variant of diffusion $K$-means, termed as *localized diffusion $K$-means*, by using local adaptive bandwidth $h_i=h(X_i)$ for each $X_i$, $i\in[n]$. In particular, we adopt the self-tuning procedure from [@zelnik2005self] by setting $h_i$ to be $\|X_i-X_i^{(k_0)}\|$, where $X_i^{(k_0)}$ denotes the $k_0$-th nearest neighbor to $X_i$, and replacing $K_n$ with $K_n^\dagger=[K^\dagger(X_i,X_j)]_{n\times n}$ (and accordingly replacing $D_n$ with $D_n^\dagger$ corresponding to the diagonal degree matrix associated with $K^\dagger_n$) given by $$\begin{aligned} K^\dagger_n (X_i,X_j) = \exp\Big(-\frac{\|X_i-X_j\|^2}{2h_ih_j}\Big),\end{aligned}$$ in the SDP . Note that $K^\dagger_n$ is generally no longer a positive semidefinite matrix. Intuitively for $i \in G_{k}^{*}$, the local scaling $h_{i}$ automatically adapts to the local density $p_{k}(X_{i})$ about $X_{i}$, the cluster size $n_k$ and the dimension $q_{k}$ for the $k$-th Riemannian submanifold. Specifically, for each cluster $k=1,\ldots,K$, the $n_k$-by-$n_k$ submatrix $[K_n^\dagger]_{G^\ast_kG^\ast_k}$ resembles the a Gaussian kernel matrix with a homogeneous bandwidth $h_k \asymp (\log n_k/n_k)^{1/q_k}$ that adapts to the local geometry in ${\mathcal{D}}_k$. For points $X_i$ and $X_j$ belonging to distinct clusters that are properly separated, $K^\dagger_n (X_i,X_j)$ tends to be close to zero and is less affected by the choice of $h_i$ and $h_j$. Heuristically, $h_{i}$ is larger for lower density regions where the degree function of $X_{i}$ is smaller so that the random walk can speed up mixing at such lower density regions. Overall, such a locally adaptive choice of bandwidth improves the mixing time of the random walk within each cluster, while leaves the between cluster jumping probabilities remaining small. As a consequence, the pairwise diffusion affinity matrix $A$ in our SDP formulation  tends to exhibit a clearer block form reflecting the clustering structure. To compute $h_{i}$, we only need to specify $k_{0}$ to replace the (non-adaptive) bandwidth parameter $h$ whose value depend on the unknown cluster sizes $n_{k}$, dimensions $q_{k}$ of submanifolds ${\mathcal{D}}_{k}$, and the underlying probability density functions $p_{k}$ on ${\mathcal{D}}_{k}$. In contrast, the simple choice $k_{0} = \lfloor C \log{n} \rfloor$ guarantees that the local scaling $h_{i}$ adapts to the local density (cf. Theorem \[thm:main\_adaptive\_h\] in Section \[sec:main\_results\]). Main results {#sec:main_results} ============ In this section, we assume each ${\mathcal{D}}_k$ is a compact connected $q_k$-dimensional Riemannian submanifold embedded in ${\mathbb{R}}^p$ with bounded diameter, absolute sectional curvature value, and injectivity radius. Throughout the rest of the paper, we assume that each $\mu_k$ has a Lipschitz density function $p_k$ with respect to the Riemannian volume measure on ${\mathcal{D}}_k$, such that $$\begin{aligned} \label{Eqn:density_condition} c{\leqslant}p_k(x) {\leqslant}\frac{1}{c}\quad \mbox{for all }x\in {\mathcal{D}}_k,\end{aligned}$$ for some constant $c>0$. Exact recovery of diffusion $K$-means ------------------------------------- Let $\delta = \min_{1{\leqslant}k\neq k'{\leqslant}K} \|{\mathcal{D}}_k-{\mathcal{D}}_{k'}\|$, where $\|A-B\|=\inf_{x\in A, y\in B} \|x-y\|$ denote the (Euclidean) distance between two disjoint sets $A$ and $B$. Recall that the size of the true cluster $n_{k} =|G_{k}^{*}|$ for $k=1,\ldots,K$ and let $\underline{n} = \min_{1 \le k \le K} n_{k}$ denote the minimal cluster size. \[thm:main\] Let $c_1,c_2, c_3,c_4,C_1,C_2$ be some positive constants only depending on $S=\bigsqcup_{k=1}^K {\mathcal{D}}_k$, and $c$ some constant that depends on $\max_{1 \le k \le K}q_k$. If $$\label{eqn:bandwidth_condition_nonadaptive} c_1 (\log n_k/ n_k)^{1/q_k}\,(\log n_k)^{1(q_k=2)/4}{\leqslant}h {\leqslant}c_2 \quad \mbox{for each } k\in[K]$$ for some sufficiently large constant $c_{1}$ and sufficiently small constant $c_{2}$, then we can achieve exact recovery, that is $\hat Z= Z^\ast$, with probability at least $1- c_3K\,\underline{n}^{-c_4}$ as long as $$\begin{aligned} \label{eqn:exact_recovery_condition_SDP_diffusion_Kmeans_LB_eigenval} C_1\,nt\, \exp\Big\{-\frac{\delta^2}{2h^2}\Big\} +C_1\,\exp\Big\{-c\, \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\}\, h^2 t\Big\} < \frac{C_2}{n\, \max_{1 {\leqslant}k {\leqslant}K} \{n_k h^{q_k}\}},\end{aligned}$$ where $\lambda_1({\mathcal{D}}_k)>0$ denotes the second smallest eigenvalue of the Laplace-Beltrami operator (with the minus sign) on ${\mathcal{D}}_k$. A proof of this theorem is provided in Section \[Sec:proof\_thm\_main\]. A crucial step in the proof is bounding from below the absolute spectral gap of the transition matrix associated with the restricted random walk $\mathcal W_n$ onto each submanifold $\mathcal D_k$ for $k=1,\ldots,K$. We do so by applying a comparison theorem of Markov chains to connect this spectral gap with the eigensystem of the Laplace-Beltrami operator over the submanifold ${\mathcal{D}}_k$ that captures the diffusion geometry. In particular, our proof borrows existing results [@burago2014graph; @trillos2018error] on error estimates of using the spectrum of a random geometric graph to approximate the eigensystem of the Laplace-Beltrami operator in the numerical analysis literature. \[rem:comments\_condition\_main\_thm\] We begin with comments on the second term on the left hand side of . We point out that this is essentially a mixing condition on random walks over the $K$ submanifolds. In particular, it is due to a relation between the mixing times of the heat diffusion process on each submanifold and its discretized random walk ${\mathcal{W}}_n$ over vertices sampled from the submanifold. For simplicity, we illustrate this relation for $K = 1$ and ${\mathcal{D}}_{1} = {\mathbb{S}}^{1}$, where ${\mathbb{S}}^{1}$ is the unit circle in ${\mathbb{R}}^{2}$. As a one-dimensional compact smooth manifold (without boundary), ${\mathbb{S}}^{1}$ can be parametrized by the angle $\theta \in [-\pi, \pi)$. Under this parametrization, the density function $u(\tau,\theta)$ of the heat diffusion process on ${\mathbb{S}}^{1}$, as a function of time $\tau$ and location $\theta$, is determined by the corresponding heat equation, $$\label{eqn:heat_eq} {\partial \over \partial \tau}u(\tau,\theta) + \Delta u(\tau,\theta) = 0, \quad (\tau, \theta) \in (0,\infty) \times {\mathbb{S}}^{1},$$ where $\Delta$ is the Laplace-Beltrami operator on ${\mathbb{S}}^{1}$. Under the same parametrization, the Laplace operator $\Delta = -{{\mathrm{d}}^{2} \over {\mathrm{d}}\theta^{2}}$ on ${\mathbb{S}}^{1}$ (with the minus sign) admits the following eigen-decomposition $$\Delta e^{\iota n \theta} = n^{2} e^{\iota n \theta}, \quad n=0,\pm1,\pm2,\dots,$$ where $\iota = \sqrt{-1}$. That is, $(\lambda_{n}({\mathbb{S}}^{1}), e_{n}) := (n^{2}, e^{\iota n \theta})$ is an eigen-pair of $\Delta$, which implies that $\Delta$ is a positive semidefinite and unbounded operator on $L^{2}({\mathbb{S}}^{1})$ functions. Now we can solve the heat equation  by expanding it with respect to this orthonormal basis. More precisely, for any $f \in L^{2}({\mathbb{S}}^{1})$, the Fourier transform of $f$ is given by $$f(\theta) = \sum_{n=-\infty}^{\infty} a_{n} e_{n} := \sum_{n=-\infty}^{\infty} \langle f, e_{n} \rangle e_{n},$$ where $\langle f, g \rangle = (2\pi)^{-1} \int_{{\mathbb{S}}^{1}} f(\theta) \overline{g(\theta)} \, {\mathrm{d}}\theta$ is the standard inner product on ${\mathbb{S}}^{1}$. Then the solution to heat equation with the initial distribution $u(0,\theta) = f(\theta)$ is given by $$u(\tau,\theta) = \sum_{n=-\infty}^{\infty} a_{n} e^{-n^{2}\tau} e_{n}.$$ So if ${\mathbb{S}}^{1}$ is insulated, then as $\tau \to \infty$ the heat flow has a constant equilibrium state with the value equal to the average of the initial heat distribution, namely $\lim_{\tau\to\infty} u(\tau,\theta) = (2\pi)^{-1} \int_{{\mathbb{S}}^{1}} f(\theta)\,{\mathrm{d}}\theta$. In particular, the second smallest eigenvalue $\lambda_{1}({\mathbb{S}}^{1})=1$ characterizes the mixing rate of the heat diffusion process. Now we consider the “inverse Fourier transform" by expressing the solution $u$ in terms of Green’s function, also called the heat kernel, on ${\mathbb{S}}^{1}$ as $$\label{eqn:heat_kernel_unit_circle} K_{{\mathbb{S}}^{1}}(\tau, \theta, \varphi) = \sum_{n=-\infty}^{\infty} e^{-n^{2}\tau} {\tilde{e}}^{\iota n (\theta-\varphi)},$$ where $ {\tilde{e}}^{\iota n \theta} = e_{n} / \sqrt{2\pi}$ is the rescaled orthonormal basis of $L^{2}({\mathbb{S}}^{1})$. Then we obtain that $$u(\tau,\theta) = {\mathcal{K}}_{{\mathbb{S}}^{1}}^{\tau} f (\theta), \mbox{ where } {\mathcal{K}}_{{\mathbb{S}}^{1}}^{\tau} f (\theta) := \int_{{\mathbb{S}}^{1}} K_{{\mathbb{S}}^{1}}(\tau, \theta, \varphi) f(\varphi) \,{\mathrm{d}}\varphi$$ defines an (integral) heat diffusion operator on ${\mathbb{S}}^{1}$. Then the Laplace operator on ${\mathbb{S}}^{1}$ can be seen as the generator of the heat diffusion process: $$\Delta f(\theta) = - \left.{\partial \over \partial \tau}u(\tau,\theta) \right|_{\tau=0} = - \left.{\partial \over \partial \tau}{\mathcal{K}}_{{\mathbb{S}}^{1}}^{\tau}f(\theta) \right|_{\tau=0} = \lim_{\tau \to 0^{+}} {f(x) - {\mathcal{K}}_{{\mathbb{S}}^{1}}^{\tau} f(x) \over \tau}.$$ Similarly, the normalized graph Laplacian $L_n = I_n - P_n = I_n- D_n^{-1} K_n$ corresponding to the random walk ${\mathcal{W}}_n$ over a random sample of $S_n$ on ${\mathbb{S}}^{1}$ can also be seen as a discrete generator of ${\mathcal{W}}_n$. Recall that in the heat diffusion process, the second smallest eigenvalue $\lambda_{1}({\mathbb{S}}^{1})$ of its generator, i.e., the Laplace-Beltrami operator, characterizes the mixing rate of the heat diffusion process. Similarly, in the random walk ${\mathcal{W}}_n$ (as a discretization of the heat process), the second smallest eigenvalue $\lambda_{j}(L_n)$ of its discrete generator, i.e., the normalized graph Laplacian operator $L_n$, characterizes its mixing rate. From Lemma \[lem:eigenval\_convergence\_normalized\_graph\_Laplacian\], the spectrum of these two operators are related in the sense that for each $j=1,2\dots,$ with probability at least $1-c_{1} n^{-c_{2}}$, $$\lambda_{j}({\mathbb{S}}^{1}) \asymp h^{2} \lambda_{j}(L_n),$$ where $\lambda_{j}({\mathbb{S}}^{1})$ is the $j$-th eigenvalue of $\Delta$ and $\lambda_{j}(L_n)$ is the $j$-th eigenvalue of the normalized graph Laplacian. This means that we must change the time clock unit of the random walk on the graph by multiplying a factor of $h^{2}$ to approximate its underlying heat diffusion process. Thus the term $h^{2}t$ in the second term of is the right time scale $\tau$ for running the heat diffusion process on the manifold, and we need $h^{2}t \to \infty$ for the heat diffusion process converges to an equilibrium distribution. On finite data, this means that the random walk converges to its stationary distribution over the points $S_n$ sampled from the submanifold. Using this correspondence, the second term on the left hand side of is a mixing condition on random walks over the $K$ submanifolds. The first term on the left hand side of can be seen as a separation requirement of the $K$ disjoint submanifolds ${\mathcal{D}}_{1},\dots,{\mathcal{D}}_{K}$. In particular, if $t = n^{\epsilon}$ for some $\epsilon > 0$ (i.e., we run the random walk in polynomial times/steps), then the minimal separation should obey $$\label{eqn:lower_bound_Delta} \delta \gtrsim h \sqrt{\log n}$$ in order to achieve the exact recovery for the manifold clustering problem. Combining the two terms of , we see that steps of the random walk must be properly balanced: we would like the random walk on the similarity graph to sufficiently mix within each cluster (second term of ), while it does not overly mix to merge the true clusters (first term of ). This reflects the [*multi-scale*]{} property of the diffusion $K$-means. \[rem:thresholding\] In view of the approximation property  of the empirical diffusion affinity matrix $A_{n} = P_{n}^{2t} K_{n}^{-1}$ to a block-diagonal matrix in Remark \[rem:intution\_DKM\], one can show that a simple thresholding of the matrix $A_{n}$ also yields the exact recovery for a properly chosen threshold. Indeed, by the triangle inequality, we have for any $i,j \in G_k^\ast$ in the same cluster, $$[A_n]_{ij} {\geqslant}N_{k}^{-1} -\max_{1 {\leqslant}k {\leqslant}K} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty.$$ Choosing a threshold value $\gamma$ such that $$\label{eqn:thresholding_value} \max_{1 {\leqslant}k\neq m {\leqslant}K} \|[A_n]_{G_k^\ast G_m^\ast}\|_\infty < \gamma < \min_{1{\leqslant}k{\leqslant}K}\Big\{\frac{1}{N_k}\Big\} - \max_{1 {\leqslant}k {\leqslant}K} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty,$$ one can completely separate the block-diagonal entries from the off-diagonal ones, thus achieving exact recovery. This argument leads to the following lemma. \[lem:thresholding\_master\_bound\] If $$\label{eqn:thresholding_exact_recover_master_condition} \max_{1 {\leqslant}k\neq m {\leqslant}K} \|[A_n]_{G_k^\ast G_m^\ast}\|_\infty + \max_{1 {\leqslant}k {\leqslant}K} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty < \min_{1{\leqslant}k{\leqslant}K}\Big\{\frac{1}{N_k}\Big\},$$ then thresholded estimator on the empirical diffusion affinity matrix $A_{n}$ with the threshold value satisfying  yields exact recovery. Note that condition  in Lemma \[lem:thresholding\_master\_bound\] is slightly weaker than the master condition  of the SDP relaxed diffusion $K$-means in Lemma \[lem:DKM\_SDP\_master\_bound\] (up to a factor of $K^{-1}$ for balanced clusters, say). On the other hand, thresholding has a tuning parameter $\gamma$ and it is unclear how to develop a principled procedure to choose the threshold value satisfying . On the contrary, our SDP relaxed diffusion $K$-means only requires the knowledge of the number of cluster $K$ and it is tuning-free in that sense. It is worthy to note that the second smallest eigenvalue of the Laplace-Beltrami operator in condition of Theorem \[thm:main\] can be regarded as characterizations of the connectedness of the submanifolds, where the latter can be formally quantified by the Cheeger isoperimetric constant defined as follows. Let ${\mathcal{M}}$ be a $q$-dimensional compact Riemannian manifold. Let $\mbox{Vol}({\mathcal{A}})$ denote the volume of a $q$-dimensional submanifold ${\mathcal{A}}\subset {\mathcal{M}}$ and $\mbox{Area}({\mathcal{E}})$ denote the $(q-1)$-dimensional area of a submanifold ${\mathcal{E}}$. The [*Cheeger isoperimetric constant*]{} of ${\mathcal{M}}$ is defined to be $${\mathfrak{h}}({\mathcal{M}}) = \inf_{{\mathcal{E}}} \left\{ \frac{\mbox{Area}({\mathcal{E}})}{\min(\mbox{Vol}({\mathcal{M}}_{1}), \mbox{Vol}({\mathcal{M}}_{2}))} \right\},$$ where the infimum of the normalized manifold cut (in the curly brackets) is taken over all smooth $(q-1)$-dimensional submanifolds ${\mathcal{E}}$ of ${\mathcal{M}}$ that cut ${\mathcal{M}}$ into two disjoint submanifolds ${\mathcal{M}}_{1}$ and ${\mathcal{M}}_{2}$ such that ${\mathcal{M}}= {\mathcal{M}}_{1} \bigsqcup {\mathcal{M}}_{2}$. In words, ${\mathfrak{h}}({\mathcal{M}})$ quantifies the minimal area of a hypersurface that bisects ${\mathcal{M}}$ into two disjoint pieces (cf. Figure \[fig:diffussion\_dist\]). Smaller values of ${\mathfrak{h}}({\mathcal{M}})$ mean that ${\mathcal{M}}$ is less connected – in particular, ${\mathfrak{h}}({\mathcal{M}}) = 0$ implies that there are two disconnected components in ${\mathcal{M}}$. The Cheeger isoperimetric constant may also be analogously defined for a graph and its value (i.e., the conductance of the graph) is closely related to the normalized graph cut problem. Suppose we have an i.i.d. sample $X_{1},\dots,X_{n}$ drawn from the uniform distribution on ${\mathcal{M}}$ and ${\mathcal{G}}_{n}$ is the neighborhood random graph with an edge between $X_{i}$ and $X_{j}$ if $\|X_{i}-X_{j}\| \le h$. It is shown in [@Arias-CastroPelletierPudlo2012_AAP] that the normalized graph cut (after a suitable normalization) converges to the normalized manifold cut, yielding an asymptotic upper bound on the conductance of ${\mathcal{G}}_{n}$ based on ${\mathfrak{h}}({\mathcal{M}})$. See also [@Trillos:2016_JMLR] for improved results. \[cor:main\] Under the setting of Theorem \[thm:main\], if $$\begin{aligned} \label{eqn:exact_recovery_condition_SDP_diffusion_Kmeans_Cheeger} C_1\,nt\, \exp\Big\{-\frac{\delta^2}{2h^2}\Big\} +C_1\,\exp\Big\{-c\, \underline{{\mathfrak{h}}}^{2}\, h^2 t\Big\} < \frac{C_2}{n\,\max_{1 {\leqslant}k {\leqslant}K} \{n_kh^{q_k}\}},\end{aligned}$$ where $\underline{{\mathfrak{h}}} = \min_{1 {\leqslant}k {\leqslant}K} {\mathfrak{h}}({\mathcal{D}}_k)$, then $\hat Z= Z^\ast$ with probability at least $1- c_3K\,\underline{n}^{-c_4}$. Since $\delta$ reflects the separation of the submanifolds of $S = \bigsqcup_{k=1}^{K} {\mathcal{D}}_{k}$ and ${\mathfrak{h}}({\mathcal{D}}_k)$ reflects the degree of connectedness of the submanifold ${\mathcal{D}}_{k}$, the (overall) hardness of the manifold clustering problem is determined by $(\delta, \underline{{\mathfrak{h}}})$. In particular, if $t = n^{\epsilon}$ for some $\epsilon > 0$, then we require that $$\underline{{\mathfrak{h}}} \gtrsim {1 \over h} \sqrt{\log n \over n^{\epsilon}},$$ in addition to . Our results in the rest subsections can also be stated via this geometric quantity of the Cheeger isoperimetric constant. \[rem:kernel\_choice\] In Theorem \[thm:main\] and Corollary \[cor:main\], the kernel $k$ is assumed to be the Gaussian kernel in . However, these exact recovery results do not rely on the particular choice of the Gaussian kernel. Specifically, Theorem \[thm:main\] and Corollary \[cor:main\] still hold, as long as the kernel is isotropic and satisfies the exponential decay in the squared Euclidean distance. For the heat kernel on ${\mathbb{R}}$ (i.e., Green’s function associated with the heat equation on ${\mathbb{R}}$) $$\label{eqn:heat_kernel_real_line} H(\tau,x,y) = (4\pi \tau)^{-1/2} e^{-{(x-y)^{2} \over 4\tau}}, \quad x,y \in {\mathbb{R}},$$ it can be viewed as an approximation to the short time dynamics of the heat kernel  on ${\mathbb{S}}^{1}$ in Remark \[rem:comments\_condition\_main\_thm\] as $\tau \to 0^{+}$ (cf. Chapter 1 in [@Rosenberg1997]). Hence, we can approximate the short time behavior of the heat flow on the compact manifold ${\mathbb{S}}^{1}$ by that of the non-compact manifold ${\mathbb{R}}$, where the latter is governed by the Gaussian heat kernel on ${\mathbb{R}}$. Setting $h^{2} = 2\tau$ in and noticing that the normalization $(4\pi \tau)^{-1/2}$ does not affect the results in Theorem \[thm:main\] and Corollary \[cor:main\] since the SDP solution in is invariant under scaling. Thus the bandwidth parameter in the Gaussian kernel in has the time scale interpretation in terms of the heat flow dynamics, in addition to capturing the local neighborhood geometry of the submanifolds. Exact recovery of the regularized and localized diffusion $K$-means ------------------------------------------------------------------- In this subsection, we extend the exact recovery results to the two variants of the diffusion $K$-means. First, we consider the regularized diffusion $K$-means  that does not require knowledge of the true number of clusters $K$. \[thm:main\_adaptive\_lambda\] Suppose all conditions in Theorem \[thm:main\] are true. In addition, if the regularization parameter satisfies $$\begin{aligned} \label{Eqn:lambda_condition} C_1\,nt\, \exp\Big\{-\frac{\delta^2}{2h^2}\Big\} &+C_1\,\exp\Big\{-c\, \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\}\, h^2 t\Big\} < \lambda {\leqslant}\frac{C_2}{n\max_{1 {\leqslant}k {\leqslant}K}\{n_kh^{q_k}\}},\end{aligned}$$ then we can achieve exact recovery for $\tilde{Z}$ from the regularized diffusion $K$-means with probability at least $1- c_3K\,\underline{n}^{-c_4}$. Condition , as a sufficient condition for the exact recovery, provides some justification of our $\lambda$ selection Algorithm \[alg1\], in particular, the reason of why using the logarithmic scale. More precisely, we observe from Lemma \[lem:feasibility\_SDP\_lambda\_infinity\] that an upper bound for $\lambda$ to produce non-trivial clustering is $n^{-1}$ times the largest eigenvalue of the affinity matrix $A$ in the SDP, which is of order $n^{-1}\max_{k}\{n_kh^{q_k}\}^{-1} =\mathcal O((n\log n)^{-1})$ (from the approximating form  and Lemma \[Lemma:total\_degree\] in the proof of Theorem \[thm:main\]). As a consequence, in the original scale, the interval length of those $\lambda$ that underestimates $K$ is of order $(n\log n)^{-1}$, which is comparable to the range of $\lambda$ corresponding to exact recovery (correct $K$) implied by  as $(n\log n)^{-1}$. On the other hand, the range of $\lambda$ corresponding to exact recovery will dominate if we instead consider the logarithmic scale. Precisely, on the logarithmic scale, the interval length for underestimating $K$ is of order $\log n$, while the range of $\log \lambda$ implied by  becomes $\log n - C'\,\big(\log n - \min\big\{\delta^2/(2h^2), \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\}\, h^2 t\big\}\big)$, which is of order $\Omega(n^{\iota})$ for some constant $C'>0$ and $\iota>0$ as long as both $\delta^2/h^2$ and $h^2t$ are bounded below by $n^{\iota}$. (The latter requirement can be easily satisfied, for example, in our simulations the choice of $t=n^{1.2}$ is good enough to produce robust results.) In particular, this suggests that the interval length of $\log \lambda$ for exact recovery is proportional to $\delta^2/h^2$, which can be viewed as a signal-to-noise ratio characteristic. Now let us turn to the localized diffusion $K$-means that locally selects the node-wise bandwidth adapting to the local geometric structure. \[thm:main\_adaptive\_h\] Let $\delta_{kk'}=\|{\mathcal{D}}_k-{\mathcal{D}}_{k'}\|$. If the number of neighbor parameter $k_0$ satisfies $k_0=\lfloor C\log n\rfloor$ for some constant $C>0$, and $$\delta_{kk'} {\geqslant}C'\, \max\{(\log n/ n_k)^{1/q_k},(\log n/ n_{k'})^{1/q_{k'}}\}$$ for each distinct $k,k'\in[K]$, then with probability at least $1- c_3K\,\underline{n}^{-c_4}$, the followings are true. (1) For each $i\in G^\ast_k$, its local bandwidth parameter $h_i$ satisfies $$\begin{aligned} \label{Eqn:local_h_nounds} c_1 (\log n/ n_k)^{1/q_k}{\leqslant}h_i {\leqslant}c_2 (\log n/ n_k)^{1/q_k}.\end{aligned}$$ (2) We can achieve exact recovery for $\tilde{Z}$ from the localized diffusion $K$-means as long as $$\begin{aligned} &C_1\,nt\, \exp\Big\{-c\,\Big(\min_{k,k'\in[K]}\frac{\delta_{kk'}}{\max\{(\log n/ n_k)^{1/q_k},(\log n/ n_{k'})^{1/q_{k'}}\}}\Big)^2\Big\}\notag \\ &\qquad\qquad\qquad+C_1\,\exp\Big\{-c\, \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\, (\log n/ n_k)^{2/q_k}\}\, t\Big\} < \frac{C_2}{n\log n}.\label{eqn:local_condition}\end{aligned}$$ \[rem:localization\] The first result  Part (1) in Theorem \[thm:main\_adaptive\_h\] shows that our localized selection scheme via nearest neighbors truly leads to bandwidth adaptation to the unknown submanifold dimension $q_k$ and the unknown true cluster size $n_k$, by only sacrificing a $\log n$ term (from $\log n_k$ to $\log n$ as compared with the optimal bandwidth choice $(\log n_k/ n_k)^{1/q_k}$ from Theorem \[thm:main\]). The second result Part (2) in Theorem \[thm:main\_adaptive\_h\] indicates the advantages of using the localized node-wise bandwidth, by comparing the condition   with those in Theorem \[thm:main\]. In particular, in order for the lower bound condition on the global bandwidth $h$ in Theorem \[thm:main\] to hold, the smallest $h$ would be $\max_{k} h_k$, where $h_k=(\log n_k/n_k)^{q_k}$ denotes the optimal bandwidth in the $k$-th cluster ${\mathcal{D}}_k$. Note that this lower bound on $h$ is uniformly larger than the magnitudes of localized bandwidth provided in . As a consequence, this large $h$ would require the same separation condition as $\delta_{kk'} {\geqslant}\max_{k}h_k$ for each pair $({\mathcal{D}}_k,{\mathcal{D}}_{k'})$ of distinct clusters. In comparison, the new sufficient condition  for exact recovery only needs a cluster-dependent separation condition as $\delta_{kk'} {\geqslant}\max\{h_k,h_{k'}\}$, which can be substantially weaker than $\delta_{kk'} {\geqslant}\max_{k}h_k$ if clusters are highly unbalanced with unequal sizes and mixed dimensions. Finally, we can further combine the regularized diffusion $K$-means with local adaptive bandwidths into the *localized and regularized diffusion $K$-means*. The following result is an immediate consequence by combining the proofs of Theorem \[thm:main\_adaptive\_lambda\] and Theorem \[thm:main\_adaptive\_h\], and thus its proof is omitted. \[thm:main\_adaptive\] Suppose all conditions in Theorem \[thm:main\] and Theorem \[thm:main\_adaptive\_h\] are true. In addition, if the regularization parameter satisfies $$\begin{aligned} &C_1\,nt\, \exp\Big\{-c\,\Big(\min_{k,k'\in[K]}\frac{\delta_{kk'}}{\max\{(\log n/ n_k)^{1/q_k},(\log n/ n_{k'})^{1/q_{k'}}\}}\Big)^2\Big\} \notag\\ &\qquad\qquad\qquad+C_1\,\exp\Big\{-c\, \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\, (\log n/ n_k)^{2/q_k}\}\, t\Big\}< \lambda {\leqslant}\frac{C_2}{n\log n},\label{Eqn:local_lambda_condition}\end{aligned}$$ then we can achieve exact recovery for $\tilde{Z}$ from the localized and regularized diffusion $K$-means with probability at least $1- c_3K\,\underline{n}^{-c_4}$. \[rem:bandwidth\_adaptivity\] It is interesting to note the signal separation and random walk mixing of the localized diffusion $K$-means and its regularized version are adaptive to local probability density and local geometric structures of the Riemannian submanifolds. In [@Arias-Castro2011_IEEETIT], nearly-optimal exact recovery of a collection of clustering methods based on pairwise distances of data is derived under a condition that the [*minimal*]{} signal separation strength over all pairs of submanifolds is larger than a threshold (even for their local scaling version, cf. Proposition 3 therein). Thus results established in [@Arias-Castro2011_IEEETIT] are non-adaptive to the local density and (geometric) structures of the submanifolds. In addition, the $\max_k\{n_k h_k^{q_k}\}$ on the right hand side of now reduces to $\log(n)$ as in . This means that the localized diffusion $K$-means tends to increase the signal-to-noise ratio (cf. Remark \[rem:localization\]) as well as the upper bound on $\lambda$ for exact recovery, thereby widening the interval length of $\log(\lambda)$ corresponding to the true clustering structure and improves the performance of the $\lambda$ selection Algorithm \[alg1\]. Simulations {#sec:simulations} =========== In this section, we assess the empirical performance of the diffusion $K$-means on some simulation examples. We generate $n=768$ data points from the following three data generation mechanisms (DGPs). The clustering structure contains three disjoint submanifolds: - ${\mathcal{D}}_{1} = \mbox{unit disk}$ and $n/4$ data points are uniformly sampled on ${\mathcal{D}}_{1}$, - ${\mathcal{D}}_{2} = \mbox{annulus with radius } 2.5$ and $n/4$ data points are uniformly sampled on ${\mathcal{D}}_{2}$, - ${\mathcal{D}}_{3} = \mbox{annulus with radius } 4$ and $n/2$ data points are uniformly sampled on ${\mathcal{D}}_{3}$, where all ${\mathcal{D}}_{1},{\mathcal{D}}_{2},{\mathcal{D}}_{3}$ are centered at the origin $(0,0)$. The clustering structure contains three disjoint rectangles: - ${\mathcal{D}}_{1} = \{(-15,-8), (-15,8), (-8,8), (8,8)\}$, - ${\mathcal{D}}_{2} = \{(10,3), (10,8), (15,3), (15,8)\}$, - ${\mathcal{D}}_{3} = \{(10,-8), (10,-3), (15,-8), (15,-3)\}$, where data points are uniformly distributed on ${\mathcal{D}}_{1} \bigsqcup {\mathcal{D}}_{2} \bigsqcup {\mathcal{D}}_{3}$. The clustering structure is a mixture of three bivariate Gaussians: $$\alpha_{1} N(\mu_{1}, \sigma_{1}^{2} {\text{Id}}_{2}) + \alpha_{2} N(\mu_{2}, \sigma_{2}^{2} {\text{Id}}_{2}) + \alpha_{3} N(\mu_{3}, \sigma_{3}^{2} {\text{Id}}_{2}),$$ where $(\alpha_{1}, \alpha_{2}, \alpha_{3}) = (1/3, 1/3, 1/3)$, $\mu_{1} = (-6,0), \mu_{2} = (0,0), \mu_{3} = (2.5,0)$, $\sigma_{1} = 2$, and $\sigma_{2} = \sigma_{3} = 0.5$. Our simulation setups are similar to [@zelnik2005self; @NadlerGalun2006_NIPS]. Note that the sampling density in DGP 1 and 2 is uniform on the disjoint submanifolds, the hardness of the problems is mainly determined by the geometry, and we thus expect the diffusion $K$-means and its localized version can both succeed in these two cases. In addition, since DGP 1 contains two annuli that are less connected than the rectangles and ellipsoids, we expect that, for the localized diffusion $K$-means (with self-tuned bandwidths), more random walk steps are needed for DGP 1 to correctly identify the clusters than those for DGP 2 and DGP 3. In our simulation studies, we use $t = n^{2}$ for DPG 1 and $t = n^{1.2}$ for both DGP 2 and DGP 3 (all with local scaling). Further, DGP 3 has a mixture of Gaussian densities, the local scaling is expected to improve the performance of the diffusion $K$-means. In fact, we have observed in Figure \[fig:diffusion\_kmeans\_local\_scaling\_demo\] that the diffusion $K$-means without local scaling does not work for DGP 2. It is also known that spectral clustering methods fail on such setup [@NadlerGalun2006_NIPS]. Thus we do not report results on DGP 2 without local scaling for all competing methods since it does not provide meaningful comparisons with other setups. For the SDP relaxed diffusion $K$-means clustering methods, we report the $\ell^{1}$ estimation error for estimating the true clustering membership $Z^{*}$ and the (normalized) Hamming distance error for classifying the clustering labels. In each setup, our results are reported on 1,000 simulations. For brevity, DKM stands for the diffusion $K$-means, RDKM for the (nuclear norm) regularized diffusion $K$-means, LDKM for the localized diffusion $K$-means, and LRDKM for the localized and regularized diffusion $K$-means. In the cases of no local scaling, the steps of random walks is fixed as $t = n^{1.2}$ in all setups. In the cases of local scaling, the nearest neighborhood size is chosen as $\lfloor \log{n} \rfloor$ for DPG 1 and DGP 3, and as $\lfloor 0.5 \log{n} \rfloor$ for DGP 2. For the comparison purpose, we also include three spectral clustering methods: the unnormalized spectral clustering (SC-UN), the random walk normalized spectral clustering (SC-RWN) [@ShiMalik2000_IEEEPAMI], a symmetrically normalized spectral clustering (SC-NJW) proposed in [@NgJordanWeiss2001_NIPS]. For each spectral clustering method, we also consider their localized versions (LSC-UN, LSC-RWN, LSC-NJW) by replacing the kernel matrix $K_{n}$ with $K^{\dagger}_{n}$. We can draw several observations from the simulation studies. First, the estimation error agrees well with our exact recovery theory for SDP relaxed DKM and LDKM, given the number of clusters (cf. Table \[tab:estimation\_errors\]). Second, all methods works relatively better for DGP 1 and DGP 2 since the separation signal strength is stronger than DGP 3 (cf. Table \[tab:classification\_errors\]). Third, the RDKM and LRDKM perform well in selecting the true number of clusters (cf. Table \[tab:percentages\_correctly\_estimated\_noc\]). We also modify DGP 3 to make the problem harder. We consider the mixture of three Gaussian with parameters $(\alpha_{1}, \alpha_{2}, \alpha_{3}) = (1/4, 1/4, 1/2)$, $\mu_{1} = (-6,0), \mu_{2} = (0,0), \mu_{3} = (1.45,0)$, $\sigma_{1} = 2$, and $\sigma_{2} = \sigma_{3} = 0.5$. This setup is denoted as DGP 3’. For DGP 3’, LDKM has much smaller classification errors than all spectral methods with local scaling (i.e., LSC-UN, LSC-RWN, LSC-NJW); see last column of Table \[tab:classification\_errors\]. ------ ------------------------- ------------------------- ----------- DGP=1 DGP=2 DGP=3 DKM $4.7642 \times 10^{-6}$ $3.3258 \times 10^{-4}$ [**–**]{} LDKM $5.2835 \times 10^{-5}$ 0.0049 0.0451 ------ ------------------------- ------------------------- ----------- : $\ell^{1}$ estimation errors of the SDP solutions of various diffusion $K$-means clustering methods.[]{data-label="tab:estimation_errors"} --------- ------- ------------------------- ----------- ----------- DGP=1 DGP=2 DGP=3 DGP=3’ DKM 0 $1.3021 \times 10^{-4}$ [**–**]{} [**–**]{} SC-UN 0 0 [**–**]{} [**–**]{} SC-RWN 0 0 [**–**]{} [**–**]{} SC-NJW 0 0 [**–**]{} [**–**]{} LDKM 0 0.0018 0.0086 0.0594 LSC-UN 0 $4.7917 \times 10^{-4}$ 0.0098 0.0801 LSC-RWN 0 0.0016 0.0105 0.0802 LSC-NJW 0 0.0399 0.0084 0.0884 --------- ------- ------------------------- ----------- ----------- : Classification errors of various diffusion $K$-means and spectral clustering methods.[]{data-label="tab:classification_errors"} Method DGP=1 DGP=2 DGP=3 -------- -------- -------- ----------- RDKM 94.30% 95.10% [**–**]{} LRDKM 99.20% 83.10% 97.70% : Percentages of correctly estimated number of clusters by the regularized diffusion $K$-means and its local scaling version.[]{data-label="tab:percentages_correctly_estimated_noc"} Proofs {#sec:proofs} ====== Recall that $n_k=|G_k^\ast|$ is the size of $k$-th true cluster index set $G_k^\ast$, and let $N_k = \sum_{i,j\in G_k^\ast} \kappa(X_i,\,X_j)$ denote the total within-weight in $G_k^\ast$. For any subset $G\subset [n]$, we use $\mathbf{1}_{G}$ to denote the all-one vector whose size equal to the size of $G$. Proof of Theorem \[thm:main\] {#Sec:proof_thm_main} ----------------------------- For simplicity of notation, we use $A:=A_n$ to denote the empirical diffusion affinity matrix $P_n^{2t}D_n^{-1}$ in the proof, and recall $$\begin{gathered} \hat{Z} = \operatorname{argmax}\left\{ \langle A, Z \rangle : Z \in {\mathscr{C}}_K \right\} \\ \qquad \mbox{with } {\mathscr{C}}_K = \{Z \in {\mathbb{R}}^{n \times n} : Z^{T} = Z, Z \succeq 0, \operatorname{tr}(Z) = K, Z {\mathbf{1}}_{n} = {\mathbf{1}}_{n}, Z {\geqslant}0 \}, \end{gathered}$$ At a high level, our strategy is to show that for suitably large $t\in{\mathbb{N}}_{+}$, the matrix $A_n$ tends to become close to a block-diagonal matrix, where each diagonal block tends to be a constant matrix (cf. equation ). Based on this approximation, we expect the global optimum $\hat Z$ to share a similar block-diagonal structure, thereby recovers the true membership matrix $Z^\ast$ in  which takes the form of $$\begin{aligned} Z^\ast = \begin{pmatrix} \displaystyle \frac{1}{n_1} \mathbf{1}_{G_1^\ast} \mathbf{1}^T_{G_1^\ast} & 0 & \cdots & 0\\ 0 & \displaystyle \frac{1}{n_2} \mathbf{1}_{G_2^\ast} \mathbf{1}^T_{G_2^\ast} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & \cdots & 0 & \displaystyle \frac{1}{n_K} \mathbf{1}_{G_K^\ast} \mathbf{1}^T_{G_K^\ast} \end{pmatrix}.\end{aligned}$$ To put this intuition in a technical form, since $Z^\ast$ defined in  is also a feasible solution belonging to the convex set ${\mathscr{C}}_K$, we have by the optimality of $\hat Z$ that $$\begin{aligned} \label{Eqn:basic_ineq} 0{\leqslant}\langle A_n,\, \hat Z - Z^\ast\rangle =\sum_{1{\leqslant}k\neq m{\leqslant}K} \big\langle\, [A_n]_{G_k^\ast G_m^\ast},\, [\hat Z - Z^\ast]_{G_k^\ast G_m^\ast}\,\big\rangle + \sum_{k=1}^K \big\langle\, [A_n]_{G_k^\ast G_k^\ast},\, [\hat Z - Z^\ast]_{G_k^\ast G_k^\ast}\,\big\rangle.\end{aligned}$$ We analyze the two sums separately as follows. [**The first sum:**]{} By noticing that $Z^\ast_{G_k^\ast G_m^\ast}$ is a zero matrix for each pair $k\neq m \in [K]$, we have the following bound $$\begin{aligned} &\sum_{1{\leqslant}k\neq m{\leqslant}K} \big\langle\, [A_n]_{G_k^\ast G_m^\ast},\, [\hat Z - Z^\ast]_{G_k^\ast G_m^\ast}\,\big\rangle \notag \\ = &\,\sum_{1{\leqslant}k\neq m{\leqslant}K} \big\langle\, [A_n]_{G_k^\ast G_m^\ast},\, \hat Z_{G_k^\ast G_m^\ast}\,\big\rangle {\leqslant}\max_{1{\leqslant}k\neq m{\leqslant}K} \|[A_n]_{G_k^\ast G_m^\ast}\|_\infty \, \sum_{1{\leqslant}k\neq m{\leqslant}K} \|\hat Z_{G_k^\ast G_m^\ast}\|_1,\label{eqn:k_neq_m}\end{aligned}$$ where the leading factor $\max_{k\neq m} \|[A_n]_{G_k^\ast G_m^\ast}\|_\infty$ is expected to be small due to the approximating structure . [**The second sum:**]{} Since we expect the $k$th block $[A_n]_{G_k^\ast G_k^\ast}$ of $A_n$ in the diagonal to be close to $N_k^{-1} \mathbf{1}_{G_k^\ast} \mathbf{1}^T_{G_k^\ast}$, we can subtract and add the same term to decompose it into $$\begin{aligned} &\sum_{k=1}^K \big\langle\, [A_n]_{G_k^\ast G_k^\ast},\, [\hat Z - Z^\ast]_{G_k^\ast G_k^\ast}\,\big\rangle\\ = &\, \sum_{k=1}^K \big\langle\, [A_n]_{G_k^\ast G_k^\ast} - N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T,\, [\hat Z - Z^\ast]_{G_k^\ast G_k^\ast}\,\big\rangle + \sum_{k=1}^K N_k^{-1} \,\big\langle\, \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T,\, [\hat Z - Z^\ast]_{G_k^\ast G_k^\ast}\,\big\rangle.\end{aligned}$$ The first term on the right hand side can be bounded by applying Hölder’s inequality, $$\begin{aligned} \sum_{k=1}^K& \big\langle\, [A_n]_{G_k^\ast G_k^\ast} - N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T,\, [\hat Z - Z^\ast]_{G_k^\ast G_k^\ast}\,\big\rangle \\ &\,{\leqslant}\max_{1 {\leqslant}k {\leqslant}K} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} 1_{G_k^\ast}1_{G_k^\ast}^T\,\big\|_\infty \, \sum_{k=1}^K \big\|\, [\hat Z -Z^\ast]_{G_k^\ast G_k^\ast}\big\|_1,\end{aligned}$$ where again the leading factor $\max_{k} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty $ is expected to be small due to the approximating structure . Now consider the second term. By the definition of ${\mathscr{C}}_K$, the sums of entries in each row of $\hat Z$ and $Z^\ast$ are equal (to one), we have for fixed $k\in[K]$ and each $i\in G^\ast_k$, $$\begin{aligned} \sum_{j\in G_k^\ast} [\hat Z - Z^\ast]_{ij} + \sum_{j\not\in G_k^\ast} [\hat Z - Z^\ast]_{ij}=0.\end{aligned}$$ Since $Z^\ast_{ij}=0$ for each pair $(i,j)$ with $i\in G_k^\ast$ and $j\notin G_k^\ast$, and $\hat Z$ has nonnegative entries, we can sum up the preceding display over all $i\in G_k^\ast$ to obtain $$\begin{aligned} \big\langle\, \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T,\, [\hat Z - Z^\ast]_{G_k^\ast G_k^\ast}\,\big\rangle = - \sum_{m:\,m\neq k} \|\hat Z_{G_k^\ast G_m^\ast}\|_1,\quad \forall k\in[K].\end{aligned}$$ Putting pieces together, we obtain $$\label{Eqn:k_eq_m} \begin{aligned} &\sum_{k=1}^K \big\langle\, [A_n]_{G_k^\ast G_k^\ast},\, [\hat Z - Z^\ast]_{G_k^\ast G_k^\ast}\,\big\rangle \\ {\leqslant}&\, \max_{1 {\leqslant}k {\leqslant}K} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty \, \sum_{k=1}^K \big\|\, [\hat Z -Z^\ast]_{G_k^\ast G_k^\ast}\big\|_1- \sum_{1{\leqslant}k\neq m{\leqslant}K} \frac{1}{N_k}\|\hat Z_{G_k^\ast G_m^\ast}\|_1. \end{aligned}$$ Now by combining inequalities , and , we can reach the following inequality $$\label{Eqn:key_ineq} \begin{aligned} &\sum_{1{\leqslant}k\neq m{\leqslant}K} \frac{1}{N_k}\|\hat Z_{G_k^\ast G_m^\ast}\|_1 {\leqslant}\max_{1 {\leqslant}k\neq m {\leqslant}K} \|[A_n]_{G_k^\ast G_m^\ast}\|_\infty \, \sum_{1{\leqslant}k\neq m{\leqslant}K} \|\hat Z_{G_k^\ast G_m^\ast}\|_1\\ &\qquad \qquad\qquad +\max_{1 {\leqslant}k {\leqslant}K} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty \, \sum_{k=1}^K \big\|\, [\hat Z -Z^\ast]_{G_k^\ast G_k^\ast}\big\|_1. \end{aligned}$$ Since $\hat{Z} \in {\mathscr{C}}_{K} \subset {\mathscr{C}}$, according to inequalities - in Lemma \[lem:some\_ineq\_feasible\_set\] in Appendix C, $$\begin{aligned} \label{eqn:DKM_SDP_another_core_inequality} \sum_{1{\leqslant}k\neq m{\leqslant}K} \frac{1}{n_k}\|\hat Z_{G_k^\ast G_m^\ast}\|_1 {\geqslant}\frac{1}{n}\,\big\|\hat Z - Z^\ast\big\|_1.\end{aligned}$$ The last two displays  and  imply the exact recovery $\hat Z=Z^\ast$ as long as $$\label{eqn:DKM_SDP_exact_recover_master_condition} \max_{1 {\leqslant}k\neq m {\leqslant}K} \|[A_n]_{G_k^\ast G_m^\ast}\|_\infty + \max_{1 {\leqslant}k {\leqslant}K} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty < \frac{1}{n}\,\min_{1{\leqslant}k{\leqslant}K}\Big\{\frac{n_k}{N_k}\Big\}.$$ \[lem:DKM\_SDP\_master\_bound\] If  holds, then we can achieve exact recovery $\hat{Z} = Z^{*}$. To further proceed, we will make use of following two lemmas to provide high probability bounds for the empirical diffusion affinity entries deviating from their expectations. Proofs of Lemma \[lem:within\_cluster\_random\_walk\] and \[lem:between\_cluster\_random\_walk\] are deferred to the following subsections. \[lem:within\_cluster\_random\_walk\] Let $\kappa = \max_{1{\leqslant}k\neq k'{\leqslant}K}\sup_{x\in {\mathcal{D}}_k,\,x'\in {\mathcal{D}}_{k'}} \kappa(x,\,x')$ and $\tau = \inf_{x,y\in S:\, \|x-y\| {\leqslant}h} \kappa(x,y)$. If $c_1 (\log n_k/ n_k)^{1/q_k}\,(\log n_k)^{1(q_k=2)/4}{\leqslant}h$, then it holds with probability at least $1-c_2\,n_k^{-c_3}$ that $$\begin{aligned} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty {\leqslant}C_0\, t\,(n_k\, h^{q_k})^{-2} \,n\,\kappa+ C_0\, (n_k\,h^{q_k})^{-1}\, e^{-2t\,\gamma(P_{n,k})},\end{aligned}$$ where the spectral gap $\gamma(P_{n,k})$, defined as one minus the second largest eigenvalue of $P_{n,k}$, satisfies $$\begin{aligned} \label{Eqn:Spectral_gap_lower} \gamma(P_{n,k}) {\geqslant}\bigg(1- C\,\Big(c_1^{-1}+(\sqrt{\lambda_1({\mathcal{D}}_k)} + 1)\,h+ h^2\Big)\bigg)\,\tau \, C_k \, \lambda_1({\mathcal{D}}_k)\,h^2,\end{aligned}$$ for some constants $C_0,C>0$ that only depends on ${\mathcal{D}}_k$ and $p_k$ and $C_k$ only depends on $q_k$. Here $\lambda_1({\mathcal{D}}_k)>0$ is the second smallest eigenvalue of the Laplace-Beltrami operator on ${\mathcal{D}}_k$ (cf. Section \[subsec:Laplace-Beltrami\_operator\]). \[lem:between\_cluster\_random\_walk\] Let $\kappa = \max_{1{\leqslant}k\neq k'{\leqslant}K}\sup_{x\in {\mathcal{D}}_k,\,x'\in {\mathcal{D}}_{k'}} \kappa(x,\,x')$. Suppose conditions  and  in Theorem \[thm:main\] are satisfied. Then it holds with probability at least $1 - c_2 K\, \underline{n}^{-c_3}$ that $$\begin{aligned} \|[A_n]_{G_k^\ast G_m^\ast}\|_\infty {\leqslant}C\, t \, (n_k h^{q_k}\,n_m h^{q_m})^{-1}\,n\,\kappa, \quad\forall k\neq m \in[K],\end{aligned}$$ where recall $\underline n = \min_{k\in [K]} n_k$ and $C$ is a constant only depending on $\{{\mathcal{D}}_k, \mu_k\}_{k = 1}^{K}$. Recall that $\kappa(x,y) = \exp\{-\|x-y\|^2/(2h^2)\}$ is the Gaussian kernel. Consequently, we may choose $\kappa = \exp\{-\delta^2/(2h^2)\}$ and $\tau = e^{-1/2}$ in these two lemmas. By Weyl’s law for the growth of eigenvalues of the Laplace-Beltrami operator (cf. Remark 6 in [@trillos2018error]), we have $\lambda_{j}({\mathcal{D}}_{k}) \sim j^{2/q_{k}}$. Thus $\lambda_1({\mathcal{D}}_k)$ is bounded. Since the bandwidth parameter satisfies $c_1 (\log n_k/ n_k)^{1/q_k}\,(\log n_k)^{1(q_k=2)/4}{\leqslant}h {\leqslant}c_2$ for some sufficient large constant $c_1$ and sufficiently small constant $c_2$ for all $k \in[K]$, inequality  in Lemma \[lem:within\_cluster\_random\_walk\] becomes $$\begin{aligned} \gamma(P_{n,k}) {\geqslant}C_k \, \lambda_1({\mathcal{D}}_k)\,h^2,\quad\forall k\in[K],\end{aligned}$$ for some constant $C_k$ only depending on $q_k$. Therefore, the following two bounds hold with probability at least $1 - c_2 K\, \underline{n}^{-c_3}$, $$\begin{aligned} \max_{1 {\leqslant}k {\leqslant}K} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty &{\leqslant}C\, nt\,\exp\Big\{-\frac{\delta^2}{2h^2}\Big\}+ C\, \exp\Big\{-c\, \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\}\, h^2 t\Big\},\\ \max_{1{\leqslant}k\neq m{\leqslant}K}\|[A_n]_{G_k^\ast G_m^\ast}\|_\infty & {\leqslant}C\,nt\,\exp\Big\{-\frac{\delta^2}{2h^2}\Big\},\end{aligned}$$ for some constant $C$ only depending on $\{{\mathcal{D}}_k, \mu_k\}_{k = 1}^{K}$ and $c$ only on $\{q_k\}_{k=1}^K$. The last two inequalities combining with Lemma \[lem:DKM\_SDP\_master\_bound\] imply the exact recovery $\hat Z=Z^\ast$ as long as $$\begin{aligned} C\,nt\, \exp\Big\{-\frac{\delta^2}{2h^2}\Big\} +C\,\exp\Big\{-c\, \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\} \, h^2 t\Big\} < \frac{1}{n}\,\min_{1{\leqslant}k{\leqslant}K}\Big\{\frac{n_k}{N_k}\Big\}. \end{aligned}$$ Finally, the claimed result follows from the preceding display and the following lemma of a high probability bound on $N_k$, whose proof is postponed to Section \[Sec:Proof\_total\_degree\]. \[Lemma:total\_degree\] Suppose the density $p_{k}$ satisfies  and the bandwidth $h {\leqslant}c$ for some constant $c>0$. Then it holds with probability at least $1-c_2\,n_k^{-c_3}$ that $$\begin{aligned} \bigg| \frac{N_k}{n_k} - (\sqrt{2\pi}\, h)^{q_k} \,n_k \beta_k\bigg| {\leqslant}C \left( n_{k} h^{q_{k}+1} + \sqrt{n_kh^{q_k}\log n_k} \right), \end{aligned}$$ where $\beta_k = \mathbb E_{X\sim p_k}[p_k(X)]$ and $C>0$ is a constant depends only on $p_{k}, {\mathcal{D}}_{k}$, and $c$. Proof of Corollary \[cor:main\] ------------------------------- Corollary \[cor:main\] follows from Theorem \[thm:main\] and the Cheeger inequality [@Cheeger1970]: $$\lambda_{1}({\mathcal{D}}_{k}) {\geqslant}\frac{{\mathfrak{h}}({\mathcal{D}}_{k})^{2}}{4}.$$ Proof of Theorem \[thm:main\_adaptive\_lambda\] {#Sec:proof_thm_main_adaptive} ----------------------------------------------- Similar to the proof of Theorem \[thm:main\] in Section \[Sec:proof\_thm\_main\], we can use the optimality of $\tilde Z$ and the feasibility of $Z^\ast$ for the SDP program  to obtain the following basic inequality, $$\begin{aligned} 0&{\leqslant}\langle A_n,\, \tilde Z - Z^\ast\rangle + n \lambda\big(\operatorname{tr}(Z^\ast) - \operatorname{tr}(\tilde Z)\big) \notag \\ &=\sum_{1{\leqslant}k\neq m{\leqslant}K} \big\langle\, [A_n]_{G_k^\ast G_m^\ast},\, [\tilde Z - Z^\ast]_{G_k^\ast G_m^\ast}\,\big\rangle + \sum_{k=1}^K \big\langle\, [A_n]_{G_k^\ast G_k^\ast},\, [\tilde Z - Z^\ast]_{G_k^\ast G_k^\ast}\,\big\rangle \notag\\ &\qquad\qquad\qquad+ n\,\lambda\,\big(\operatorname{tr}(Z^\ast) - \operatorname{tr}(\tilde Z)\big).\label{Eqn:basic_inequality_adaptive}\end{aligned}$$ Since the only place where we used the constraint $\operatorname{tr}(Z) =K$ in Section \[Sec:proof\_thm\_main\] is Lemma \[lem:some\_ineq\_feasible\_set\] in Appendix C, the analysis of the first two sums in  still apply, leading to $$\label{Eqn:final_bound_adaptive} \begin{aligned} \sum_{1{\leqslant}k\neq m{\leqslant}K} \frac{1}{N_K}\|\tilde Z_{G_k^\ast G_m^\ast}\|_1 {\leqslant}&\, C\,\bigg( nt\, \exp\Big\{-\frac{\delta^2}{2h^2}\Big\} +\exp\Big\{-c\, \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\}\, h^2 t\Big\}\bigg)\, \big\|\tilde Z - Z^\ast\big\|_1\\ &\,\qquad + n\,\lambda\,\big(\operatorname{tr}(Z^\ast) - \operatorname{tr}(\tilde Z)\big), \end{aligned}$$ which holds with probability at least $1-c_3K\,\underline{n}^{-c_3}$. Now we apply Lemma \[lem:some\_ineq\_feasible\_set\_adaptive\] in Appendix C to obtain $$\begin{aligned} n\, \lambda\, \big(\operatorname{tr}(Z^\ast) - \operatorname{tr}(\tilde Z)\big) {\leqslant}4n\,\lambda\sum_{1{\leqslant}k\neq m{\leqslant}K} \frac{1}{n_k}\|\tilde Z_{G_k^\ast G_m^\ast}\|_1-\lambda\,\big\|\tilde Z - Z^\ast\big\|_1.\end{aligned}$$ Combining this inequality with , we obtain $$\begin{aligned} \bigg(\lambda - C\,\Big( nt\, \exp\Big\{-\frac{\delta^2}{2h^2}\Big\} &+\exp\Big\{-c\, \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\}\, h^2 t\Big\}\Big)\bigg)\, \big\|\tilde Z - Z^\ast\big\|_1\\ &\qquad\qquad+\,\bigg(1-4n\lambda \max_{1{\leqslant}k{\leqslant}K}\Big\{\frac {N_k}{n_k}\Big\}\bigg)\,\sum_{1{\leqslant}k\neq m{\leqslant}K} \frac{1}{N_k}\|\tilde Z_{G_k^\ast G_m^\ast}\|_1 {\leqslant}0.\end{aligned}$$ This implies the exact recovering $\tilde Z=Z^\ast$ provided that $$\begin{aligned} C_1\, \Big( nt\, \exp\Big\{-\frac{\delta^2}{2h^2}\Big\} &+\exp\Big\{-c\, \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\}\, h^2 t\Big\}\Big) < \lambda {\leqslant}\frac{C_2}{n}\,\min_{1{\leqslant}k{\leqslant}K}\Big\{\frac{n_k}{N_k}\Big\}.\end{aligned}$$ Finally, the claimed result follows by combining the above with Lemma \[Lemma:total\_degree\]. Proof of Lemma \[lem:within\_cluster\_random\_walk\] ---------------------------------------------------- We consider a fixed $k\in[K]$ throughout this proof. Recall that $P_n=D_n^{-1} K_n$ defines the random walk ${\mathcal{W}}_n$ over $S_n=\{X_1,X_2,\ldots,X_n\}$, and $A_n = P_n^{2t} D_n^{-1}$. Now consider a new random walk ${\mathcal{W}}_{n,k}$ over the $k$th cluster $G_k^\ast$ defined in the following way. For simplicity of notation, we may rearrange the nodes order so that $G_k^\ast = \{1,2,\ldots,n_k\}$. Then the transition probability matrix $P_{n,k}\in{\mathbb{R}}^{n_k\times n_k}$ of ${\mathcal{W}}_{n,k}$ is defined as $$\begin{aligned} [P_{n,k}]_{ij} =\frac{\kappa(X_i,\,X_j)}{d_{n,k}(X_i)},\quad \forall i,j\in G_k^\ast,\quad \mbox{where } d_{n,k}(x) = \sum_{j\in G_k^\ast} \kappa(x,\,X_j)\end{aligned}$$ is the induced degree function within cluster $G_k^\ast$. Similar to the diagonal degree matrix $D_n$, we denote by $D_{n,k}\in{\mathbb{R}}^{n_k\times n_k}$ the diagonal matrix whose $i$th diagonal entry is $d_{n,k}(X_i)$ for $i\in[n_k]$. Note that $N_k=\sum_{i\in G_k^\ast} d_{n,k}(X_i)$ the total degrees within $G_k^\ast$ so that $N_k {\geqslant}n_k \min_{i\in G^\ast_k} d_{n,k}(X_i)$. It is easy to see that the limiting distribution of ${\mathcal{W}}_{n,k}$ is $\pi_{n,k} = N_k^{-1} \mbox{diag}(D_{n,k})\in{\mathbb{R}}^{n_k}$. Under the separation condition on $\delta$ in the lemma, the probability of moving out from $G_k^\ast$ is exponentially small, suggesting that we may approximate the sub-matrix $[P_n^{2t}]_{G_k^\ast G_k^\ast} $ of $P_n^{2t}$ with $P_{n,k}^{2t}$. We will formalize this statement in the rest of the proof. First, we apply the triangle inequality to obtain $$\begin{aligned} &\big\|\,[A_n]_{G_k^\ast G_k^\ast} - N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty \notag\\ & {\leqslant}\big\| \,[P_n^{2t}]_{G_k^\ast G_k^\ast} [D_{n}]_{G_k^\ast G_k^\ast}^{-1} - P_{n,k}^{2t} D_{n,k}^{-1}\big\|_\infty + \big\| \,P_{n,k}^{2t} D_{n,k}^{-1} - N_k^{-1} \mathbf{1}_{G_k^\ast} \mathbf{1}_{G_k^\ast}^T \big\|_\infty =: T_1 + T_2,\label{Eqn:within_random_walk}\end{aligned}$$ where $T_1$ captures the difference between $[P_{n}^{2t}]_{G_k^\ast G_k^\ast}$ and $P_{n,k}^{2t}$, and $T_2$ characterizes the convergence of the Markov chain ${\mathcal{W}}_{n,k}$ to its limiting distribution $\pi_{n,k}$ after $2t$ steps. Recall that $\kappa = \max_{1{\leqslant}k\neq k'{\leqslant}K}\sup_{x\in {\mathcal{D}}_k,\,x'\in {\mathcal{D}}_{k'}} \kappa(x,\,x')$ is the minimal between-cluster affinity. The first term $T_1$ and the second term $T_2$ can be bounded via two lemmas below. \[Lem:T\_1\] If $n \kappa {\leqslant}\min_{i \in G_{k}^{*}} d_{n,k}(X_{i})$, then $$\begin{aligned} T_1 {\leqslant}(2t+1)\,n\,\kappa\,\max_{i\in[n]}d_{n,k}^{-2}(X_i). \end{aligned}$$ \[Lem:T\_2\] Let $\tau = \inf_{x,y\in S:\, \|x-y\| {\leqslant}h} \kappa(x,y)$. For each $i\in G_k^\ast$, let $d_{k}^\dagger(X_{i})$ denote total number of points in $\{X_j\}_{j\in G_k^\ast}$ inside the $d$-ball centered at $X_i$ with radius $h$. If $c_1 (\log n_k/ n_k)^{1/q_k}\,(\log n_k)^{1(q_k=2)/4}{\leqslant}h$, then it holds with probability at least $1-c_2\,n_k^{-c_3}$ that $$\begin{aligned} T_2 {\leqslant}e^{-2t\,\gamma(P_{n,k})} \,\max_{i\in G_k^\ast} d_{n,k}^{-1}(X_i),\quad\forall t=1,2,\ldots,\end{aligned}$$ where $$\begin{aligned} \label{Eqn:spectral_gap_bound} \gamma(P_{n,k}) {\geqslant}\bigg(1- C\,\Big(c_1^{-1}+(\sqrt{\lambda_1({\mathcal{D}}_k)} + 1)\,h+ h^2\Big)\bigg) \,\bigg[\min_{i\in G_k^\ast} \frac{d_{k}^\dagger(X_{i})}{d_{n,k}(X_i)}\bigg]\,\tau\, C_k \, \lambda_1({\mathcal{D}}_k)\,h^2,\end{aligned}$$ where recall that $\lambda_1({\mathcal{D}}_k)>0$ denotes the second smallest eigenvalue of the Laplace-Beltrami operator on ${\mathcal{D}}_k$. Proofs of these two lemmas are provided in Sections \[Sec:Proof\_T\_1\] and \[Sec:Proof\_T\_2\]. Combining upper bounds for the two terms in inequality  together, we can reach $$\begin{aligned} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty {\leqslant}(2t+1)\,n\,\kappa\, \max_{i\in[n]}d_{n,k}^{-2}(X_i) + e^{-2t\,\gamma(P_{n,k})}\,\max_{i\in[n]}d_{n,k}^{-1}(X_i),\end{aligned}$$ where the spectral gap $\gamma(P_{n,k})$ satisfies  . It remains to prove some high probability bounds for $d_{n,k}(X_i)$ and $d_{k}^\dagger(X_i)$, which are summarized in the following lemma. \[lem:node\_degree\] Suppose the density $p_{k}$ satisfies  and the bandwidth $h {\leqslant}c$ for some constant $c>0$. Then it holds with probability at least $1-c_2\,n_k^{-c_3}$ that $$\begin{aligned} \max_{i\in G_k^\ast} \bigg|\,\frac{d_{n,k}(X_i)}{n_k\,(\sqrt{2\pi} \,h)^{q_k}} - p_k(X_i) \bigg| &{\leqslant}C\bigg( h + \sqrt{\frac{\log n_k}{n_k h^{q_{k}}}}\,\bigg), \quad \mbox{and} \\ \max_{i\in G_k^\ast} \bigg|\,\frac{d_{k}^\dagger(X_i)}{n_k\,\nu_{q_k}\,h^{q_k}} - p_k(X_i) \bigg| &{\leqslant}C\bigg( h^2 + \sqrt{\frac{\log n_k}{n_k h^{q_{k}}}}\,\bigg),\end{aligned}$$ where $\nu_{q_k}$ denotes the volume of an unit ball in ${\mathbb{R}}^{q_k}$, and $C>0$ is some constant depends only on $p_{k}, {\mathcal{D}}_{k}$, and $c$. A proof of this lemma is deferred to Section \[Sec:Proof\_lem:node\_degree\]. Finally, by combining this lemma with the last display, and applying the uniform boundedness condition  on $p_k$, we obtain $$\begin{aligned} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty {\leqslant}C_0\,n\,t\,\kappa\, (n_k\,h^{q_k})^{-2}+ C_0\, (n_k\,h^{q_k})^{-1}\, e^{-2t\,\gamma(P_{n,k})}\end{aligned}$$ with probability at least $1-c_{2} \, n_{k}^{-c_{3}}$, where the spectral gap $\gamma(P_{n,k})$ satisfies $$\begin{aligned} \gamma(P_{n,k}) {\geqslant}\bigg(1- C\,\Big(c_1^{-1}+(\sqrt{\lambda_1(D_k)} + 1)\,h+ h^2\Big)\bigg)\,\tau \, C_1 \, \lambda_1(D_k)\,h^2,\end{aligned}$$ for some constants $C_1,C_2,C>0$ that only depends on ${\mathcal{D}}_k$ and $p_k$. Proof of Lemma \[lem:between\_cluster\_random\_walk\] ----------------------------------------------------- For each indices $i$ and $j$ that belong to two difference clusters $G_k^\ast$ and $G_m^\ast$ with $k \neq m$, we have $$\begin{aligned} [A_n]_{ij}= [P_{n}^{2t}]_{ij} \cdot d_n(X_j)^{-1}.\end{aligned}$$ Let ${\mathcal{W}}_n=\{Y_t:\,t{\geqslant}0\}$, with $Y_t$ denote the state of the Markov chain ${\mathcal{W}}_n$ at time $t$. Define $T_k(i)=\min\big\{t \in {\mathbb{N}}_{+}:\, Y_t \not\in G_k^\ast, \, Y_{0} = i\big\}$ denote the first exit time from $G_k^\ast$ of ${\mathcal{W}}_n$ starting from $Y_0=i$. Then it is easy to see that $$\begin{aligned} [P_{n}^{2t}]_{ij} = {\mathds{P}}(Y_0=i,\,Y_{2t} = j) {\leqslant}{\mathds{P}}(T_k(i) {\leqslant}2t) = 1 - {\mathds{P}}(T_k(i) > 2t).\end{aligned}$$ Since for each $i\in G_k^\ast$, the one-step transition probability of moving out from $G_k^\ast$ is bounded by $n\,\kappa\,\max_{i\in G_k^\ast} d_{n}^{-1}(X_i)$, we have $$\begin{aligned} {\mathds{P}}(T_k(i) > 2t) {\geqslant}(1-n\,\kappa\,\max_{i\in G_k^\ast} d_{n}^{-1}(X_i))^{2t} {\geqslant}1-2n\,t \,\kappa\,\max_{i\in G_k^\ast} d_{n,k}^{-1}(X_i), \end{aligned}$$ provided that $n\,\kappa {\leqslant}\min_{i\in G_k^\ast} d_{n}(X_i)$. Therefore, for each $i\in G_k^\ast$ and $j\in G_m^\ast$ with $k\neq m$, we have $$\begin{aligned} \big|[A_n]_{ij}\big| {\leqslant}2n\,t \,\kappa\,\max_{i'\in G_k^\ast} d_{n,k}^{-1}(X_{i'}) \,\max_{j'\in G_m^\ast} d_{n,m}^{-1}(X_{j'}).\end{aligned}$$ By Lemma \[lem:node\_degree\] and , we have that with probability at least $1-c_{2}n^{-c_{3}}$, $C_{1} n_{k} h^{q_{k}} {\leqslant}d_{n,k}(X_{i}) {\leqslant}C_{2} n_{k} h^{q_{k}}$ for some constants $C_1$ and $C_{2}$. Note that condition  in Theorem \[thm:main\] yields that $n \kappa = n e^{-\delta^{2} / (2h^{2})} {\leqslant}C n_{k} h^{q_{k}}$. Then the claimed bound is implied by the above combined with the union bound. Proof of Lemma \[Lem:T\_1\] {#Sec:Proof_T_1} --------------------------- By adding and subtracting the same term we obtain $$\begin{aligned} T_1 {\leqslant}&\, \big\| \big([P_n^{2t}]_{G_k^\ast G_k^\ast} - P_{n,k}^{2t} \big)\,D_{n,k}^{-1}\big\|_\infty +\big\| \,[P_n^{2t}]_{G_k^\ast G_k^\ast}\,\big( [D_{n}]_{G_k^\ast G_k^\ast}^{-1} - D_{n,k}^{-1}\big)\big\|_\infty \notag \\ {\leqslant}&\, \big\| \,[P_n^{2t}]_{G_k^\ast G_k^\ast} - P_{n,k}^{2t} \big\|_\infty \,\|D_{n,k}^{-1}\|_\infty+ \big\| [D_{n}]_{G_k^\ast G_k^\ast}^{-1} - D_{n,k}^{-1}\big\|_\infty \notag \\ = &\, \big\| \,[P_n^{2t}]_{G_k^\ast G_k^\ast} - P_{n,k}^{2t} \big\|_\infty \,\max_{i\in G_k^\ast} d_{n,k}^{-1}(X_i)+\max_{i\in G_k^\ast} |d_n^{-1}(X_i)-d_{n,k}^{-1}(X_i)| \notag\\ {\leqslant}&\, \big\| \,[P_n^{2t}]_{G_k^\ast G_k^\ast} - P_{n,k}^{2t} \big\|_\infty \,\max_{i\in G_k^\ast}d_{n,k}^{-1}(X_i) +\max_{i\in G_k^\ast} |d_n(X_i)-d_{n,k}(X_i)|\,\max_{i\in G_k^\ast}d_{n,k}^{-2}(X_i),\label{Eqn:T_1_bound}\end{aligned}$$ where the second inequality is due to the fact that each row sum of $[P_n^{2t}]_{G_k^\ast G_k^\ast}$ is at most one. Now we consider the first term in . Recall that ${\mathcal{W}}_n=\{Y_t:\,t{\geqslant}0\}$, where $Y_t$ is the state of the Markov chain ${\mathcal{W}}_n$ at time $t$. Note that for each $i,j\in G_k^\ast$, we have $$\begin{aligned} [P_{n,k}]_{ij} =\frac{\kappa(X_i, X_j)}{\sum_{j\in G_k^\ast} \kappa(X_i,X_j)}= \frac{{\mathds{P}}(Y_{2t}=j\,|\, Y_{2t-1}=i)}{{\mathds{P}}(Y_{2t}\in G_k^\ast\,|\, Y_{2t-1}=i)} = {\mathds{P}}(Y_{2t}=j\, |\,Y_{2t-1}=i,\,Y_{2t}\in G_k^\ast).\end{aligned}$$ As a consequence, we have by the law of total probability and the Markov property of ${\mathcal{W}}_n$ that for each $i,j\in G_k^\ast$ and $s\in{\mathbb{N}}_{+}$, $$\begin{aligned} [P_n^{s}]_{ij} &= {\mathds{P}}(Y_{s} = j \,|\, Y_0 =i) \\ &=\sum_{\ell\in G_k^\ast} {\mathds{P}}(Y_{s} = j \,|\, Y_{s-1} =\ell,\, Y_0=i, \,Y_{s}\in G_k^\ast) \\ & \qquad \ \ \ \ \cdot {\mathds{P}}(Y_{s}\in G_k^\ast \,|\, Y_{s-1} =\ell,\, Y_0=i)\cdot {\mathds{P}}(Y_{s-1} = \ell\,|\, Y_0=i) \\ & \ \ \ \ + \sum_{\ell\not\in G_k^\ast} {\mathds{P}}(Y_{s} = j\,|\, Y_{s-1} =\ell,\, Y_0=i) \cdot {\mathds{P}}(Y_{s-1} =\ell\,|\, Y_0=i)\\ &= \sum_{\ell\in G_k^\ast} [P_n^{s-1}]_{i\ell} \cdot [P_{n,k}]_{\ell j}\cdot {\mathds{P}}(Y_{s}\in G_k^\ast \,|\, Y_{s-1} =\ell)\\ & \ \ \ \ + \sum_{\ell\not\in G_k^\ast} {\mathds{P}}(Y_{s} = j\,|\, Y_{s-1} =\ell) \cdot {\mathds{P}}(Y_{s-1} =\ell\,|\, Y_0=i).\end{aligned}$$ For each pair $(j,\ell)$ belonging to different clusters, noting that $d_{n,k}(X_i) {\leqslant}d_n(X_i)$, we have $$\begin{aligned} {\mathds{P}}(Y_{s} = j\,|\, Y_{s-1} =\ell) = [P_n]_{\ell j} =\frac{\kappa(X_\ell, X_j)}{d_n(X_\ell)} {\leqslant}\kappa\, \max_{i\in[n]} d_n^{-1}(X_i) {\leqslant}\max_{i\in[n]} d_{n,k}^{-1}(X_i),\end{aligned}$$ which implies that for each $\ell \in G_k^\ast$, $$\begin{aligned} 0{\leqslant}1- {\mathds{P}}(Y_{s}\in G_k^\ast \,|\, Y_{s-1} =\ell) = {\mathds{P}}(Y_{s} \not\in G_{k}^{\ast} \,|\, Y_{s-1}=\ell) {\leqslant}n \,\kappa\, \max_{i\in[n]} d_{n,k}^{-1}(X_i) {\leqslant}1.\end{aligned}$$ Combining the last three displays, we obtain for each $i,j\in G_k^\ast$ and $s\in{\mathbb{N}}^{+}$, $$\begin{aligned} \big(1- n \,\kappa\, \max_{i\in[n]} d_{n,k}^{-1}(X_i)\big)\,\sum_{\ell\in G_k^\ast} [P_n^{s-1}]_{i\ell} & \cdot [P_{n,k}]_{\ell j} {\leqslant}[P_n^{s}]_{ij} \\ {\leqslant}& \sum_{\ell\in G_k^\ast} [P_n^{s-1}]_{i\ell} \cdot [P_{n,k}]_{\ell j} + n \,\kappa\, \max_{i\in[n]} d_{n,k}^{-1}(X_i),\end{aligned}$$ which can be further simplified into $$\begin{aligned} \big(1- n \,\kappa\, \max_{i\in[n]} d_{n,k}^{-1}(X_i)\big)\,\big[ [P_n^{s-1}]_{G_k^\ast G_k^\ast} P_{n,k}\big]_{ij} {\leqslant}[P_n^{s}]_{ij}{\leqslant}\big[ [P_n^{s-1}]_{G_k^\ast G_k^\ast} P_{n,k}\big]_{ij} + n \,\kappa\, \max_{i\in[n]} d_{n,k}^{-1}(X_i).\end{aligned}$$ Now we can recursively apply this two-sided inequality to get $$\begin{aligned} \big(1- n \,\kappa\, \max_{i\in[n]} d_{n,k}^{-1}(X_i)\big)^s\,\big[ P_{n,k}^s]_{ij} {\leqslant}[P_n^{s}]_{ij}{\leqslant}\big[ P_{n,k}^s ]_{ij} + n\, s \,\kappa\, \max_{i\in[n]} d_{n,k}^{-1}(X_i),\quad\forall i,j\in G_k^\ast.\end{aligned}$$ By taking $s=2t$ and applying the inequality $(1-x)^s{\geqslant}1-xs$ for $s\in{\mathbb{N}}_{+}$ and $x\in[0,1]$, the above can be further reduced into $$\begin{aligned} \label{Eqn:T_1_first_term} \big\| \,[P_n^{2t}]_{G_k^\ast G_k^\ast} - P_{n,k}^{2t} \big\|_\infty {\leqslant}2n\, t \,\kappa\, \max_{i\in[n]} d_{n,k}^{-1}(X_i).\end{aligned}$$ Then we get $$\big\| \,[P_n^{2t}]_{G_k^\ast G_k^\ast} - P_{n,k}^{2t} \big\|_\infty \,\max_{i\in G_k^\ast}d_{n,k}^{-1}(X_i) {\leqslant}2n\, t \,\kappa\, \max_{i\in[n]} d_{n,k}^{-2}(X_i).$$ which is an upper bound to the first term in inequality  for $T_1$. The second term in inequality  can be bounded as $$\begin{aligned} \label{Eqn:T_1_second_term} & \max_{i\in G_k^\ast} |d_n(X_i)-d_{n,k}(X_i)| =\max_{i\in G_k^\ast} \sum_{j\not\in G_k^\ast} \kappa(X_i,X_j) {\leqslant}n \kappa.\end{aligned}$$ Finally, by combining , and , we obtain $$\begin{aligned} T_1 {\leqslant}(2t+1) \, n\,\kappa\,\max_{i\in[n]}d_n^{-2}(X_i).\end{aligned}$$ Proof of Lemma \[Lem:T\_2\] {#Sec:Proof_T_2} --------------------------- Recall that $\gamma(P_{n,k}) = 1-\lambda_1(P_{n,k})$ denote spectral gap of the transition matrix $P_{n,k}$, where $\lambda_1(P_{n,k})$ denotes the second largest eigenvalue of $P_{n,k}\in {\mathbb{R}}^{n_k\times n_k}$ (due to similar arguments as in Appendix B, $P_{n,k}$ has $n_k$ real eigenvalues with $1$ as the largest one). In addition, since the kernel function $k$ is positive semidefinite, all eigenvalues of $P_{n,k}$ are nonnegative, meaning that $\gamma(P_{n,k})$ is equal to the absolute spectral gap $1-\max\{\lambda_1(P_{n,k}),\, \lambda_{n_k-1}(P_{n,k})\}$ where $\lambda_{n_k-1}(P_{n,k})$ is the $n_k$th (smallest) eigenvalue of $P_{n,k}$. Therefore, according to the relationship between the mixing time of a Markov chain and its absolute spectral gap (see, for example, equation (12.11) in [@levin2017markov]), we have for each $i,j\in G_k^\ast$, $$\label{eqn:mixing_T2} \bigg|\frac{[P_{n,k}^{2t}]_{ij}}{[\pi_{n,k}]_j} - 1\bigg| {\leqslant}\frac{e^{-2t\,\gamma(P_{n,k})}}{\min_{\ell \in G_k^\ast} [\pi_{n,k}]_\ell},\quad\forall t=1,2,\ldots,$$ where $\pi_{n,k} =\big(d_{n,k}(X_1)/ N_k,\ldots, d_{n,k}(X_{n_k})/ N_k\big)^T\in{\mathbb{R}}^{n_k}$ is the limiting distribution of induced Markov chain ${\mathcal{W}}_{n,k}$ over $G_k^\ast$ with transition probability matrix $P_{n,k}$. This leads to a bound on $T_2$ as $$\begin{aligned} T_2 & = \big\| \,P_{n,k}^{2t} D_{n,k}^{-1} - N_k^{-1} \mathbf{1}_{G_k^\ast} \mathbf{1}_{G_k^\ast}^T \big\|_\infty= \max_{i,j\in G_k^\ast}\frac{1}{N_k}\, \bigg|\frac{[P_{n,k}^{2t}]_{ij}}{[\pi_{n,k}]_j} - 1\bigg| \\ &{\leqslant}\frac{1}{N_k} \, \frac{e^{-2t\,\gamma(P_{n,k})}}{\min_{\ell \in G_k^\ast} [\pi_{n,k}]_\ell}=e^{-2t\,\gamma(P_{n,k})} \,\max_{i\in G_k^\ast} d_{n,k}^{-1}(X_i),\quad\forall t=1,2,\ldots.\end{aligned}$$ Therefore, it remains to provide a lower bound on the spectral gap $\gamma(P_{n,k}) = 1-\lambda_1(P_{n,k})$. We do so by applying a comparison theorem of Markov chains (Lemma 13.22 in [@levin2017markov]), where we compare the spectral gap of $P_{n,k}$ with that of a standard random walk on a random geometric graph over $\{X_i\}_{i\in G_k^\ast}$, where each pair of nodes are connected if and only if they are at most $h$ far away from each other. The spectrum of the normalized graph Laplacian of the latter is known to behave like the eigensystem of the Laplace-Beltrami operator over the submanifold corresponding to the $k$-th connected subset ${\mathcal{D}}_k$. In particular, we will use existing results [@burago2014graph; @trillos2018error] on error estimates by using the spectrum of a random geometric graph to approximate the eigensystem of the Laplace-Beltrami operator in the numerical analysis literature. Let us first formally define a random geometric graph over $\{X_i\}_{i\in G_k^\ast}$ as i.i.d. samples from the compact connected $q_k$-dimensional Riemannian submanifold ${\mathcal{D}}_k$ in ${\mathbb{R}}^{p}$ with bounded diameter, absolute sectional curvature value, and injective radius. Recall that $\mu_k$ is a probability measure on ${\mathcal{D}}_k$ that has a Lipschitz density $p_k$ with respect to the Riemannian volume measure on ${\mathcal{D}}_k$ satisfying . $\{X_i\}_{i\in G_k^\ast}$ can be viewed as a sequence of i.i.d. samples from $\mu_k$, and without loss of generality, we may assume $G_k^\ast=\{1,2,\ldots,n_{k}\}$. Consider the random geometric graph ${\mathcal{G}}_k^{\dagger}=(V_k, E_k)$, with $V_k=\{X_i\}_{i\in G_k^\ast}$ being its set of vertices and $E_k$ set of edges, constructed by putting an edge between $X_i$ and $X_j$ (write $i\sim j$ and call $X_i$ to be a neighbor of $X_j$) if and only if $\|X_i-X_j\| {\leqslant}h$. We define the natural random walk ${\mathcal{W}}_{k}^\dagger$ as a reversible Markov chain on $V_k$ that moves to neighbors of the current state with equal probabilities. In other words, the transition probability matrix ${\mathcal{P}}_k^{\dagger}\in {\mathbb{R}}^{n_k\times n_k}$ satisfies $$\begin{aligned} [{\mathcal{P}}_k^{\dagger}]_{ij} = \begin{cases} \displaystyle (d^{\dagger}_{k,i})^{-1}, & \quad\mbox{if } j\sim i\\ \displaystyle 0, &\quad\mbox{otherwise}, \end{cases}\end{aligned}$$ where $d^\dagger_{k,i} := d^\dagger_{k}(X_{i}) = \sum_{j=1}^{n_k} 1(j\sim i)$ denotes the degree of vertex $X_i$. It is easy to see that $\pi^\dagger_k = (d^\dagger_{k,1}/d^\dagger_k,d^\dagger_{k,2}/d^\dagger_k,\ldots,d^\dagger_{k,n_k}/d^\dagger_k)^T$, where $d^\dagger_k=\sum_{i=1}^{n_k}d^{\dagger}_{k,i}$ denotes the total degree, is the stationary distribution of this random walk. Let $1= \lambda_0( {\mathcal{P}}_k^{\dagger}){\geqslant}\lambda_1({\mathcal{P}}_k^{\dagger}){\geqslant}\ldots{\geqslant}\lambda_{n-1}({\mathcal{P}}_k^{\dagger}){\geqslant}-1$ denote the eigenvalues of matrix ${\mathcal{P}}_k^{\dagger}$, and $\gamma({\mathcal{P}}_k^{\dagger})=1-\lambda_1({\mathcal{P}}_k^{\dagger})$ denote its spectral gap. Let $L_{{\mathcal{G}}_k^\dagger}=D_k^\dagger - A_k^\dagger\in{\mathbb{R}}^{n_k\times n_k}$ denote the graph Laplacian matrix associated with graph ${\mathcal{G}}_k^\dagger=(V_k,E_k)$, where $D_k^\dagger\in{\mathbb{R}}^{n_k\times n_k}$ is a diagonal matrix with $[D^\dagger_k]_{ii} = d_{k,i}^\dagger$, and $A_k^\dagger\in {\mathbb{R}}^{n_k\times n_k}$ is the adjacency matrix with $[A_k^\dagger]_{ij}=1(i\sim j)$ for all distinct pair $(i,j)\in [n_k]^2$. Define the normalized Laplacian of ${\mathcal{G}}_k^\dagger$ as $L^N_{{\mathcal{G}}_k^\dagger}=(D_k^\dagger)^{-1/2}L_{{\mathcal{G}}_k^\dagger} (D_k^\dagger)^{-1/2}= I - (D_k^\dagger)^{-1/2} A_k^\dagger (D_k^\dagger)^{-1/2}$, and denote its ordered eigenvalues by $0 {\leqslant}\lambda_0(L^N_{{\mathcal{G}}_k^\dagger}) {\leqslant}\lambda_1(L^N_{{\mathcal{G}}_k^\dagger}){\leqslant}\cdots{\leqslant}\lambda_{n_k-1}(L^N_{{\mathcal{G}}_k^\dagger})$. Since $(D_k^\dagger)^{-1/2} A_k^\dagger (D_k^\dagger)^{-1/2}= (D_k^\dagger)^{1/2} {\mathcal{P}}_k^\dagger (D_k^\dagger)^{-1/2}$ is a similarity transformation of ${\mathcal{P}}_k^\dagger$, they share the same eigenvalues. Therefore, we have the relation $\lambda_j(L^N_{{\mathcal{G}}_k^\dagger}) = 1-\lambda_j({\mathcal{P}}_k^\dagger)$ for all $j=0,1,\ldots,n_k-1$. In particular, by taking $j=1$, we can relate the spectral gap of ${\mathcal{P}}_k^\dagger$ with the second smallest eigenvalue of the normalized Laplacian matrix $L^N_{{\mathcal{G}}_k^\dagger}$ as $\gamma({\mathcal{P}}_k^\dagger) = \lambda_1(L^N_{{\mathcal{G}}_k^\dagger})$. It is known that the eigenvalues of the normalized Laplacian $L^N_{{\mathcal{G}}_k^\dagger}$ of the geometric random graph ${\mathcal{G}}_k^{\dagger}$ approaches (up to a scaling factor) the eigenvalues of the Laplace-Beltrami operator on ${\mathcal{D}}_k$. More concretely, let $L^2({\mathcal{D}}_k, {\mathrm{d}}\mu_k)$ be the space of all square integrable functions on ${\mathcal{D}}_k$, and $\Delta_{{\mathcal{D}}_k}$ denote the Laplace-Beltrami operator on ${\mathcal{D}}_k$ (cf. Section \[subsec:Laplace-Beltrami\_operator\]). Let $0=\lambda_0({\mathcal{D}}_k){\leqslant}\lambda_1({\mathcal{D}}_k){\leqslant}\cdots$ denote the sequence of nonnegative eigenvalues of $\Delta_{{\mathcal{D}}_k}$. The connectedness of ${\mathcal{D}}_k$ implies that its second smallest eigenvalue $\lambda_1({\mathcal{D}}_k)$ is strictly positive. We will invoke Corollary 2 of [@trillos2018error], which generalizes Theorem 1 of [@burago2014graph] from the uniform density to any Lipschitz continuous density satisfying , for relating the spectrum of the Laplace-Beltrami operator $\Delta_{{\mathcal{D}}_k}$ on ${\mathcal{D}}_k$ with the spectrum of the discrete normalized graph Laplacian $L^N_{{\mathcal{G}}_k^\dagger}$, as summarized in the following. \[lem:eigenval\_convergence\_normalized\_graph\_Laplacian\] Let $\nu_{q_k}$ to denote the volume of an unit ball in ${\mathbb{R}}^{q_k}$. For each $j=0,1,\ldots$, suppose the radius $h$ and the $\varepsilon_{n,k}$ to be defined below satisfy $(\sqrt{\lambda_j({\mathcal{D}}_k)} + 1)\,h +\varepsilon_{n,k}/h {\leqslant}c_1$. Then it holds with probability at least $1-c_2\,n_k^{-c_3}$ $$\begin{aligned} \bigg| \frac{2(q_k+2)}{\nu_{q_k} h^2}\cdot\frac{\lambda_j(L^N_{{\mathcal{G}}_k^\dagger})}{\lambda_j({\mathcal{D}}_k)} - 1 \bigg| {\leqslant}C\, \bigg(\frac{\varepsilon_{n,k}}{h}+(\sqrt{\lambda_j({\mathcal{D}}_k)} + 1)\,h+ h^2\bigg),\end{aligned}$$ where constants $c_1,c_2, c_3,C>0$ depend only on the submanifold ${\mathcal{D}}_k$ and the density $p_k$, and $$\begin{aligned} \varepsilon_{n,k} = \begin{cases} \displaystyle\frac{(\log n_k)^{3/4}}{n_k^{1/2}} , & \quad\mbox{if } q_k=2\\[2ex] \displaystyle \Big(\frac{\log n_k}{n_k}\Big)^{1/q_k}, & \quad\mbox{otherwise} \end{cases}.\end{aligned}$$ In particular, this lemma (taking $j=1$) implies a lower bound on the spectral gap of ${\mathcal{P}}_k^\dagger$ as $$\begin{aligned} \label{Eqn:RGG_spectral_lower_bound} \gamma({\mathcal{P}}_k^\dagger) {\geqslant}\bigg(1- C\,\Big(c_1^{-1}+(\sqrt{\lambda_1({\mathcal{D}}_k)} + 1)\,h+ h^2\Big)\bigg) C_k \, \lambda_1({\mathcal{D}}_k)\,h^2\end{aligned}$$ as long as $h {\geqslant}c_1 \varepsilon_{n,k}$, where constant $C_k$ only depends on $q_k$. Next, we will apply the following comparison theorem to relate the spectral gaps of Markov chains ${\mathcal{W}}_{n,k}$ and ${\mathcal{W}}_{k}^\dagger$. \[lem:comparison\_thm\] Let $P$ and $P'$ be transition matrices of two reversible Markov chains on the same state space $\Omega$, whose stationary distributions are denoted by $\pi$ and $\pi'$, respectively. Let ${\mathcal{E}}(f)$ and ${\mathcal{E}}'(f)$ denote the Dirichlet forms associated to the pairs $(P,\pi)$ and $(P',\pi')$, where $$\begin{aligned} \label{Eqn:Dirichlet_form} {\mathcal{E}}(f) =\frac{1}{2} \sum_{x,y\in\Omega} [\,f(x) - f(y)]^2\,\pi(x)\, P(x,y),\quad \forall f\in L^2(\Omega),\end{aligned}$$ and ${\mathcal{E}}'(f)$ can be similarly defined. If there exists some constant $B>0$ such that ${\mathcal{E}}'(f) {\leqslant}B \,{\mathcal{E}}(f)$ for all $f$, then $$\begin{aligned} \gamma(P') {\leqslant}\bigg[\max_{x\in\Omega} \frac{\pi(x)}{\pi'(x)}\bigg] \, B\,\gamma(P),\end{aligned}$$ where $\gamma(P)$ and $\gamma(P')$ denote the spectral gaps associated with $P$ and $P'$, respectively. We will apply this comparison theorem with $P_{n,k}\to P, {\mathcal{P}}_{k}^\dagger \to P'$, and $\Omega = G_{k}^{*}$. Let us find the constant $B$ such that ${\mathcal{E}}'(f){\leqslant}B\, {\mathcal{E}}(f)$ for all $f$, where in our setting, $$\begin{aligned} {\mathcal{E}}(f) &=\frac{1}{2 N_k} \sum_{1{\leqslant}i,j{\leqslant}n_k} (\,f_i - f_j)^2\,\kappa(X_i,X_j),\quad\mbox{and}\\ {\mathcal{E}}'(f) &=\frac{1}{2 d_k^\dagger} \sum_{(i,j):\, \|X_i-X_j\| {\leqslant}h} (\,f_i - f_j)^2,\quad \forall f=(f_1,\ldots,f_{n_k})^T\in{\mathbb{R}}^{n_k}, \end{aligned}$$ According to the definition of $\tau$ as $\inf_{x,y\in S:\, \|x-y\| {\leqslant}h} \kappa(x,y)$, we can simply choose $B = (N_k/d_k^\dagger)\, \tau^{-1}$. In addition, we have the bound $$\begin{aligned} \max_{i\in G_k^\ast} \frac{[\pi_{n,k}]_i}{[\pi_k^\dagger]_i} = \frac{d_k^\dagger}{\tilde N_k}\, \max_{i\in G_k^\ast} \frac{d_{n,k}(X_i)}{d_{k,i}^\dagger}.\end{aligned}$$ Therefore, we can apply Lemma \[lem:comparison\_thm\] to get $$\begin{aligned} \gamma(P_{n,k}) {\geqslant}\tau \bigg[\min_{i\in G_k^\ast} \frac{d_{k,i}^\dagger}{d_{n,k}(X_i)}\bigg]\, \gamma({\mathcal{P}}_{k}^\dagger),\end{aligned}$$ which combined with inequality  leads to $$\begin{aligned} \gamma(P_{n,k}) {\geqslant}\bigg(1- C\,\Big(c_1^{-1}+(\sqrt{\lambda_1({\mathcal{D}}_k)} + 1)\,h+ h^2\Big)\bigg) \,\bigg[\min_{i\in G_k^\ast} \frac{d_{k,i}^\dagger}{d_{n,k}(X_i)}\bigg]\,\tau\, C_k \, \lambda_1({\mathcal{D}}_k)\,h^2.\end{aligned}$$ Proof of Lemma \[lem:node\_degree\] {#Sec:Proof_lem:node_degree} ----------------------------------- Recall that $\kappa(x,y)=\exp\{-\|x-y\|^2/(2h^2)\}$ is the Gaussian kernel with bandwidth parameter $h>0$. For $x \in S$, define $d_{n,k}(x) = \sum_{j \in G_{k}^{*}} \kappa(x, X_{j})$ as the induced degree function of $x$ within cluster $G_{k}^{*}$. Then for each $i\in G_k^\ast$ we have $d_{n,k}(X_{i}) = 1+\sum_{j\in G_k^\ast,\,j\neq i} \kappa(X_i,X_j) =: 1+\tilde{d}_{n,k}(X_{i})$. Denote $\alpha_{k}(x) = \operatorname{\mathds{E}}_{X \sim p_{k}} \kappa(x,\, X)$ and $v_{k}(x) = {\text{Var}}_{X \sim p_{k}}[\kappa(x,\, X)]$. Applying Lemma \[lem:expectation\_variance\_bound\] with ${\mathcal{M}}={\mathcal{D}}_k$ and $f(x)=p_k(x)$, we have $v_{k}(x) {\leqslant}\alpha_{k}(x) {\leqslant}C h^{q_{k}}$ for all $x \in {\mathcal{D}}_{k}$. Then for any fixed $x \in {\mathcal{D}}_{k}$, using the bound in  and the boundedness of $\kappa$, we may apply Bernstein inequality (cf. Lemma 2.2.9 in [@vandervaartwellner1996]) to obtain that for all $u>0$, $$\begin{aligned} {\mathds{P}}\left( \Big|d_{n,k}(x) - n_k \alpha_{k}(x) \Big| {\geqslant}u \right) {\leqslant}2 \exp\left( -{u^{2} \over 2 C n_{k} \alpha_{k}(x) + {2 \over 3} u} \right). \end{aligned}$$ Choosing $u = t n_{k} \alpha_{k}(x)$ for $t \in (0, C]$, we have $$\label{Eqn:degree_con} {\mathds{P}}\left( \Big|d_{n,k}(x) - n_k \alpha_{k}(x) \Big| {\geqslant}t n_{k} \alpha_{k}(x) \right) {\leqslant}2 \exp \left( -{t^{2} n_{k} \alpha_{k}(x) \over 2 C + {2 \over 3} t} \right) \le 2 \exp \left( -C n_{k} \alpha_{k}(x) t^{2} \right).$$ Now choosing $t = c_{1} \sqrt{\log(n_{k})/(\alpha_{k}(x) n_{k})}$ for some large enough constant $c_{1} > 0$, we get $${\mathds{P}}\left( \Big|d_{n,k}(x) - n_k \alpha_{k}(x) \Big| {\geqslant}c_{1} \sqrt{\alpha_{k}(x) n_{k} \log{n_{k}}} \right) {\leqslant}2 n_{k}^{-C c_{1}^{2}},$$ provided that $\log(n_{k}) {\leqslant}C \alpha_{k}(x) n_{k} {\leqslant}C n_{k} h^{q_{k}}$ in view of the uniform bounds  and . But this is ensured by our bandwidth assumption . Thus for any fixed $x \in {\mathcal{D}}_{k}$, we have with probability at least $1 - c_{2} n_{k}^{-c_{3}}$, $$\left| {d_{n,k}(x) \over n_{k}} - \alpha_{k}(x) \right| {\leqslant}c_{1} \sqrt{\alpha_{k}(x) \log{n_{k}} \over n_{k}}.$$ Then it follows that with probability at least $1 - c_{2} n_{k}^{-c_{3}}$, $$\left| {d_{n,k}(x) \over n_{k} (\sqrt{2\pi} h)^{q_{k}}} - p_{k}(x) \right| {\leqslant}{C \over (\sqrt{2\pi})^{q_{k}}} h + {c_{1} \over (\sqrt{2\pi} h)^{q_{k}}} \sqrt{\alpha_{k}(x) \log{n_{k}} \over n_{k}} {\leqslant}C \left( h + \sqrt{\log{n_{k}} \over n_{k} h^{q_{k}}} \right).$$ This implies that the rescaled degree function $n_k^{-1}\,\big(\sqrt{2\pi} h\big)^{-q_k}\,d_{n,k}(x)$ provides a good estimate of the density $p_k(x)$ at $x$. Since $X_{i} \in {\mathcal{D}}_{k}$ are i.i.d. for $i \in G_{k}^{*}$, we have with probability at least $1 - c_{2} n_{k}^{-c_{3}}$, $$\left| {\tilde{d}_{n,k}(X_{i}) \over (n_{k}-1) (\sqrt{2\pi} h)^{q_{k}}} - p_{k}(X_{i}) \right| {\leqslant}C \left( h + \sqrt{\log{n_{k}} \over n_{k} h^{q_{k}}} \right).$$ Then union bound implies that $$\max_{i \in G_{k}^{*}} \left| {d_{n,k}(X_{i}) \over n_{k} (\sqrt{2\pi} h)^{q_{k}}} - p_{k}(X_{i}) \right| {\leqslant}C \left( h + \sqrt{\log{n_{k}} \over n_{k} h^{q_{k}}} + {1 \over n_{k} h^{q_{k}}} \right) {\leqslant}C \left( h + \sqrt{\log{n_{k}} \over n_{k} h^{q_{k}}} \right)$$ with probability at least $1-c_{2}n_{k}^{-c_{3}}$ (by choosing the constant $c_{1}$ large enough). The second part about the concentration of $d_{k}^\dagger(X_{i})$ can be analogously proved by applying the Chernoff bound for sum of i.i.d. Bernoulli random variables. Let $$\begin{aligned} d_{k}^\dagger(x) = \sum_{j \in G_{k}^{*}} 1(\|x-X_j\|{\leqslant}h)\end{aligned}$$ so that $d_{k}^\dagger(X_{i}) = 1+\tilde{d}_{k}^\dagger(X_{i})$ where $\tilde{d}_{k}^\dagger(X_{i}) = \sum_{j \in G_{k}^{*}, j\neq i} 1(\|X_{i}-X_j\|{\leqslant}h)$. Note that Section 2.2 of [@burago2014graph] provides a uniform estimate of the expectation ${\mathbb{E}}_{X\sim p_k} [1(\|x-X\|{\leqslant}h)]$ in terms of the density $p(x)$ as $$\begin{aligned} \label{Eqn:Expected-degree} \sup_{x \in {\mathcal{D}}_{k}} \Big|{\mathbb{E}}_{X\sim p_k} [1(\|x-X\|{\leqslant}h)] - \nu_{q_k} \, h^{q_k}\,p_k(x) \Big| {\leqslant}C\, h^{q_k+2},\end{aligned}$$ where recall that $\nu_{q_k}$ denotes the volume of unit ball in ${\mathbb{R}}^{q_k}$. The rest of the proof follows a similar line as the proof of the first part, and we omit the details. \[lem:expectation\_variance\_bound\] Let ${\mathcal{M}}$ be a $q$-dimensional compact submanifold in ${\mathbb{R}}^p$ with bounded absolute sectional curvature and injective radius, and $\mbox{Vol}_{{\mathcal{M}}}$ denote its volume form. Let $f$ be a Lipschitz probability density function on ${\mathcal{M}}$ such that $c {\leqslant}f(x) {\leqslant}c^{-1}$ for some constant $c > 0$. Let $\alpha(x) = \operatorname{\mathds{E}}_{X \sim f} [\kappa(x,\, X)]$ and $v(x) = {\text{Var}}_{X \sim f}[\kappa(x,\, X)]$. Then we have $$\label{Eqn:expectation_bound} \sup_{x\in {\mathcal{M}}} \Big| \alpha(x) - \big(\sqrt{2\pi} h\big)^{q} \,f(x)\Big| {\leqslant}C \, h^{q+1}$$ and $v(x) {\leqslant}C \alpha(x)$ for all $x \in {\mathcal{M}}$, where the constant $C$ only depends on $f$, ${\mathcal{M}}$, and $c$. Consequently, we have $\sup_{x \in {\mathcal{M}}} \alpha(x) {\leqslant}C h^{q}$ and $\sup_{x \in {\mathcal{M}}} v(x) {\leqslant}C h^{q}$. Note that for each $x\in {\mathcal{M}}$ and any Lipschitz probability density function $f$ on ${\mathcal{M}}$, the expectation ${\mathbb{E}}[\kappa(x,\, X)^2]$ takes the form $$\begin{aligned} {\mathbb{E}}_{X \sim f} [\kappa(x,\, X)] = \int_{{\mathcal{M}}} \exp\{-\|x-y\|^2/(2h^2)\} \, f(y)\, {\mathrm{d}}\mbox{Vol}_{{\mathcal{M}}} (y).\end{aligned}$$ Then  follows from Lemma \[Lem:convolution\_bound\]. Similarly, we can bounded the variance of $\kappa(x,\, X)$ via $\mbox{Var}_{X \sim f} [\kappa(x,\, X)] {\leqslant}{\mathbb{E}}_{X \sim f} [\kappa(x,\, X)^2]$, where $$\begin{aligned} {\mathbb{E}}_{X \sim f} [\kappa(x,\, X)^2]= \int_{{\mathcal{M}}} \exp\{-\|x-y\|^2/(h^2)\} \, f(y)\, {\mathrm{d}}\mbox{Vol}_{{\mathcal{M}}} (y).\end{aligned}$$ By a second application of Lemma \[Lem:convolution\_bound\] with $h/\sqrt{2} \to h$ to obtain $$\begin{aligned} \Big| {\mathbb{E}}_{X \sim f} [\kappa(x,\, X)^2] - \big(\sqrt{\pi} h\big)^{q} \,f(x)\Big| {\leqslant}C' \, h^{q+1}.\end{aligned}$$ This together with the uniform boundedness condition on $f$ and inequality  imply an upper bound to the variance by the expectation, $$\begin{aligned} \label{Eqn:variance_bound} \mbox{Var}_{X \sim f} [\kappa(x,\, X)] {\leqslant}C \,h^{q}{\leqslant}C \,{\mathbb{E}}_{X \sim f} [\kappa(x,\, X)],\quad \mbox{for some $C>0$}.\end{aligned}$$ The bounds $\sup_{x \in {\mathcal{M}}} \alpha(x) {\leqslant}C h^{q}$ and $\sup_{x \in {\mathcal{M}}} v(x) {\leqslant}C h^{q}$ follow from the fact that $h {\leqslant}c$. \[Lem:convolution\_bound\] Let ${\mathcal{M}}$ be a $q$-dimensional compact submanifold in ${\mathbb{R}}^p$ with bounded absolute sectional curvature and injective radius, and $\mbox{Vol}_{{\mathcal{M}}}$ denote its volume form. Then there exists some constant $h_0>0$ only depending on ${\mathcal{M}}$, such that for all $h\in(0, h_0]$ and any Lipschitz function $f$ on ${\mathcal{M}}$, we have $$\begin{aligned} \bigg| \frac{1}{\big(\sqrt{2\pi} h\big)^q}\,\int_{{\mathcal{M}}} \exp\{-\|x-y\|^2/(2h^2)\} \, f(y)\, {\mathrm{d}}\mbox{Vol}_{{\mathcal{M}}} (y) - f(x)\bigg| {\leqslant}C_{f}\,h,\end{aligned}$$ where constant $C_f$ only depends on $f$ and ${\mathcal{M}}$. This lemma follows from Lemma 5.2 on pages 895–898 in [@yang2016bayesian]. Proof of Lemma \[Lemma:total\_degree\] {#Sec:Proof_total_degree} -------------------------------------- Recall that $N_k=\sum_{i,j\in G^\ast_k} \kappa(X_i,X_j)$ is the total within-weight in $G^\ast_k$. According to Lemma \[lem:node\_degree\] (note that $N_k=\sum_{i\in G_k^\ast} d_{n,k}(X_i)$ using the notation therein), it holds with probability at least $1-c_2 n_k^{-c_3}$ that $$\begin{aligned} \bigg| \frac{N_k}{n_k} - (\sqrt{2\pi}\, h)^{q_k}\, \sum_{i\in G_k^\ast} p_k(X_i)\bigg| {\leqslant}C\, n_{k} h^{q_k}\,\bigg(h + \sqrt{\frac{\log n_k}{n_kh^{q_{k}}}} \, \bigg).\end{aligned}$$ According to the sandwiched bound on the density function $p_k$, $\{p_k(X_i):\,i\in G_k^\ast\}$ are independent and bounded random variables. Therefore, we may apply Hoeffding’s inequality to obtain that $$\begin{aligned} \bigg|\sum_{i\in G_k^\ast} p_k(X_i) - n_k \,\beta_k\bigg| {\leqslant}C\,\sqrt{n_k\log n_k}\end{aligned}$$ holds with probability at least $1-c_2 n_k^{-c_3}$, where $\beta_k=\mathbb E[p_k(X_i)]$. Combining the two preceding inequalities, we obtain that $$\begin{aligned} \bigg| \frac{N_k}{n_k} - (\sqrt{2\pi}\, h)^{q_k} \,n_k \beta_k\bigg| {\leqslant}C \left( n_{k} h^{q_{k}+1} + \sqrt{n_kh^{q_k}\log n_k} \right),\end{aligned}$$ which completes the proof. Proof of Theorem \[thm:main\_adaptive\_h\] {#Sec:Proof_thm:main_adaptive_h} ------------------------------------------ #### *Proof of Part (1):* Consider $X_i$, where $i\in G_k^\ast$ for some $k\in[K]$. Fix $h_U=c_U(\log n/n_k)^{1/q_k}$ and $h_L=c_L(\log n/n_k)^{1/q_k}$ for two sufficiently large constant $c_U$ and $c_L$ with $c_U=2c_L$. We use notation $N(X_i,h)$ to denote the number of points from $\{X_i\}_{i=1}^n$ that is within $h$ distance from $X_i$. Recall that $d_k^\dagger(X_i) := d_{k,h}^\dagger(X_i)$ in Lemma \[lem:node\_degree\] is the number of points from $\{X_i\}_{i\in G_k^\ast}$ that is within $h$ distance to $X_i$ (we will choose $h=h_L$ and $h=h_U$ later). Here we put an subscript $h$ in $d_{k,h}^\dagger(X_i)$ to indicate the dependence of $d_k^\dagger$ on $h$. The condition on $\delta_{kk'}$ implies that any point outside the $k$th cluster $D_k$ has distance at least $C'(\log n/n_k)^{1/q_k} {\geqslant}\max\{h_L,h_U\}$. Therefore all points that are within $h_L(h_U)$ distance to $X_i$ must belong to $\mathcal D_k$, implying $N(X_i,h) = d_{k,h}^\dagger(X_i)$. Consequently, from the proof of Lemma \[lem:node\_degree\] (take $h=h_U$ and $h=h_L$ respectively), we have (a concentration inequality as  plus the expectation bound ), $$\begin{aligned} &\mathbb P \big( N(x,h_L) {\leqslant}c_1 (1-t)\, n_k h_L^{q_k} \big) {\leqslant}\exp\{- c_1' n_k h_L^{q_k} t^2\},\\ &\mathbb P \big( N(x,h_U) {\geqslant}c_2 (1+t)\, n_k h_U^{q_k} \big) {\leqslant}\exp\{- c_2' n_k h_U^{q_k} t^2\},\quad t>0, x \in {\mathcal{D}}_{k}, \end{aligned}$$ where constants $(c_1,c_1')$ and $(c_2,c_2')$ only depend on the constant $c$ in the two-sided bound on density $p_k(\cdot)$ on the $k$th region $\mathcal D_k$. Let $c_U$ be sufficiently large such that $c_2 c_U^{q_k} = C$, where $C$ is the constant appearing in the neighborhood parameter $k_0=\lfloor C\log n\rfloor$. By taking $t$ such that $(1-t) c_1 c_L^{q_k} = C/2$ in the first inequality of the preceding display, and $t=1$ in the second inequality, we obtain that $$\begin{aligned} &\mathbb P \big( N(X_{i},h_L) {\leqslant}k_0/2 \big) {\leqslant}\exp\{- c_1'' \,C \log n\},\\ &\mathbb P \big( N(X_{i},h_U) {\geqslant}2k_0 \big) {\leqslant}\exp\{- c_2''\, C\log n\},\end{aligned}$$ where $c_1''$ and $c_2''$ are two constants independent of $C$. For large enough $C$, we can make these two probabilities smaller than $1/n^3$. Since the event $\{N(X_i,h_L) {\leqslant}k_0/2\}\cap\{N(X_i,h_U) {\geqslant}2k_0\}$ implies $h_i \in[h_L,h_U]$ for each $i\in[n]$, a union bound argument over $i=1,2,\ldots,n$ leads to the claimed two sided bound for $h_i$ (with probability at least $1-n^{-1}$). #### *Proof of Part (2):* Using the two sided bound in the proof of Part (1), the proof follows same steps in the proof of Theorem \[thm:main\], with the only exception in proving a counterpart of Lemma \[Lem:T\_2\] for bounding the spectral gap $\gamma(P_{n,k})$ of chain $P_{n,k}$ on $\{X_i\}_{i\in G_k^\ast}$ in cluster $G_k^\ast$ for $k\in [K]$, since $P_{n,k}$ now has different bandwidth parameter $h_i$ at each observed point $X_i$ (that is, a bandwidth parameter inhomogeneous chain). More precisely, it remains to show that with high probability, it holds that $$\begin{aligned} \gamma(P_{n,k}){\geqslant}C' \lambda_1(\mathcal D_k) \,(\log n /n_k)^{2/q_k}, \quad \mbox{for some constant $C'>0$.}\end{aligned}$$ Here, we assume without loss of generality that the absolute spectral gap of $P_{n,k}$ is dominated by one minus its second largest eigenvalue. Otherwise, we can always consider the lazy random walk by replacing $P_n$ with $P_n/2 +I_n/2$ in the diffusion $K$-mean SDP, whose absolute spectral gap is $\gamma(P_{n,k})/2$. Our proof strategy is again based on the Markov chains comparison theorem (Lemma \[lem:comparison\_thm\]) by comparing this bandwidth parameter inhomogeneous chain with a bandwidth parameter homogeneous chain with $h_i \equiv h_L$, for each $i\in G_k^\ast$, where $h_L$ is the lower bound of $h_i$ in the proof of Part (1). In particular, a lower bound on the spectral gap of the latter is already derived in Lemma \[Lem:T\_2\] as of order $\lambda_1(\mathcal D_k) h_L^2$. Fix the cluster index $k\in [K]$, and without loss of generality assume $G_k^\ast=\{1,2,\ldots,n_k\}$. To avoid confusion of notation, we put an superscript “IH" indicate the bandwidth parameter inhomogeneous chain, and “H" to denote the bandwidth parameter homogeneous chain with $h_i \equiv h_L$. For example, $P_{n,k}^{IH}$ and $\mathcal E^{IH}$ denote the transition probability matrix and the Dirichlet form  (defined in Lemma \[lem:comparison\_thm\]), respectively, associated with the bandwidth parameter inhomogeneous chain. Due to the fact that $h_i {\geqslant}h_L$ for all $i\in G^\ast_k$, we immediately have $$\begin{aligned} 2 N_k^H \mathcal E^{H}(f) &= \sum_{1{\leqslant}i,j{\leqslant}n_k} (f_i-f_j)^2 \kappa^{H}(X_i,X_j) \\ &{\leqslant}\sum_{1{\leqslant}i,j{\leqslant}n_k} (f_i-f_j)^2 \kappa^{IH}(X_i,X_j) = 2 N_k^{IH}\mathcal E^{IH}(f), \quad\mbox{for all $f\in\mathbb R^{n_k}$,}\end{aligned}$$ where $k^{H}(X_i,X_j) = \exp\{-\|X_i-X_j\|^2/(2h_L^2)\}$ and $\kappa^{IH}(X_i,X_j) = \exp\{-\|X_i-X_j\|^2/(2h_ih_j)\}$, and $(N_k^H,N_k^{IH})$ are the respective total degrees within $G_k^\ast$. Moreover, recall that the stationary distributions of $P_{n,k}^{IH}$ and $P_{n,k}^{H}$ are $\pi_{n,k}^{IH} = d_{n,k}^{IH}(X_i) / N_k^{IH}$ and $\pi_{n,k}^{H} = d_{n,k}^{H}(X_i) / N_k^{H}$, for $i\in G_k^\ast$, respectively, where $d_{n,k}^{IH}(X_i) =\sum_{j\in G_k^\ast} \kappa^{IH}(X_i,X_j)$, $d_{n,k}^{H}(X_i) =\sum_{j\in G_k^\ast} \kappa^{H}(X_i,X_j)$ are the node degrees, and $N_k^{IH}= \sum_{i\in G_k^\ast} d_{n,k}^{IH}(X_i)$, $N_k^{H}= \sum_{i\in G_k^\ast} d_{n,k}^{H}(X_i)$ are the total degrees. Now we can apply Lemma \[lem:comparison\_thm\] with $B= N_k^{IH}/ N_k^{H}$ and Lemma \[Lem:T\_2\] (to the homogeneous chain $P_{n,k}^H$) to obtain $$\begin{aligned} \gamma(P_{n,k}^{IH}) {\geqslant}\min_{i\in G_k^\ast} \bigg[\frac{d_{n,k}^{H}(X_i)}{d_{n,k}^{IH}(X_i)} \bigg]\, \gamma(P_{n,k}^{H}) \quad\mbox{and}\quad \gamma(P_{n,k}^{H}){\geqslant}C \lambda_1(\mathcal D_k) \,h_L^2,\end{aligned}$$ for some constant $C>0$. Similar to the proof of Lemma \[lem:node\_degree\], we can apply the concentration inequality to the nodes degree with bandwidth $h=h_U$ and $h=h_L$ to obtain that with probability at least $1-c_2\, n^{-c_3}$, $$\begin{aligned} \max_{i\in G_k^\ast} \bigg|\,\frac{\sum_{j\in G_k^\ast}\exp\{-\|X_i-X_j\|^2/(2h_L^2)\} }{n_k(\sqrt{2\pi} h_L)^{q_k}} - p_k(X_i) \bigg| {\leqslant}C'\bigg( h_L + \sqrt{\frac{\log n_k}{n_k h_{L}^{q_{k}}}}\,\bigg),\\ \max_{i\in G_k^\ast} \bigg|\,\frac{\sum_{j\in G_k^\ast}\exp\{-\|X_i-X_j\|^2/(2h_U^2)\} }{n_k(\sqrt{2\pi} h_U)^{q_k}} - p_k(X_i) \bigg| {\leqslant}C'\bigg( h_U+ \sqrt{\frac{\log n_k}{n_k h_{U}^{q_{k}}}}\,\bigg), \end{aligned}$$ for all $i\in G_k^\ast$. Combining this with the sandwiched bound for $h_i$ in Part (1), we obtain $$\begin{aligned} c_1\, n_k h_L^{q_k} &{\leqslant}d_{n,k}^{H}(X_i)=\sum_{j\in G_k^\ast}\exp\{-\|X_i-X_j\|^2/(2h_L^2)\} \\ &{\leqslant}d_{n,k}^{IH}(X_i) = \sum_{j\in G_k^\ast} \exp\{-\|X_i-X_j\|^2/(2h_ih_j)\} \\ &{\leqslant}\sum_{j\in G_k^\ast} \exp\{-\|X_i-X_j\|^2/(2h_U^2)\} {\leqslant}c_1'\, n_k h_U^{q_k},\ \ \ \mbox{for all $i\in G_k^\ast$.}\end{aligned}$$ Putting all pieces together, we obtain that it holds with probability at least $1-c_2\, n^{-c_3}$ that $$\begin{aligned} \gamma(P_{n,k}^{IH}){\geqslant}C' \lambda_1(\mathcal D_k) \,(\log n /n_k)^{2/q_k}, \quad \mbox{for some constant $C'>0$.}\end{aligned}$$ Proof of Lemma \[lem:feasibility\_SDP\_lambda\_infinity\] --------------------------------------------------------- *Proof of Part (1):* Since $n\lambda >\lambda_{\max}(A)$, the matrix $n\lambda I_n - A$ is positive definite. For any $Z \in {\mathscr{C}}$, from the constraint $Z {\mathbf{1}}_{n} = {\mathbf{1}}_{n}$, we see that $(1, n^{-1/2} {\mathbf{1}}_{n})$ is an eigen-pair of $Z$. In addition, since $Z \succeq 0$, all eigenvalues $\lambda_{1},\dots,\lambda_{n}$ of $Z$ are non-negative. Let $U_1=n^{-1/2}{\mathbf{1}}_{n}, U_2,\ldots, U_n$ denote the corresponding eigenvectors of $Z$. Thus the objective function $$\begin{aligned} &\langle A ,Z\rangle - n\lambda \operatorname{tr}(Z) = - \langle n\lambda I_n - A, Z\rangle \\ &= -\frac{1}{n}\,{\mathbf{1}}_{n}^T(n\lambda I_n - A){\mathbf{1}}_{n} - \sum_{i=2}^n \lambda_i\, U_i^T(n\lambda I_n - A)U_i {\leqslant}-\frac{1}{n}{\mathbf{1}}_{n}^T(n\lambda I_n - A){\mathbf{1}}_{n},\end{aligned}$$ where the equality holds if and only if $\lambda_2=\cdots=\lambda_n=0$. Note that $Z^{\diamond} \in {\mathscr{C}}$ is a feasible solution for  and $Z^{\diamond}$ has a non-zero eigenvalue equal to $1$ and $(n-1)$ zero eigenvalues. Therefore, $Z^{\diamond}$ is the unique solution of the SDP . *Proof of Part (2):* For any $Z\in {\mathscr{C}}$, since $Z$ is a symmetric matrix satisfying $Z {\geqslant}0$ and $Z {\mathbf{1}}_{n} = {\mathbf{1}}_{n}$, $Z$ is a stochastic matrix and all its eigenvalues have absolute values less than or equal to one. Moreover, from the positive semi-definiteness of $Z$, all eigenvalues of $Z$ lie in the $[0,1]$ interval. Now since $n\lambda <\lambda_{\min}(A)$, the matrix $A- n\lambda I_n$ is positive definite. From matrix Hölder’s inequality, the objective function satisfies $$\begin{aligned} &\langle A ,Z\rangle - n\lambda \operatorname{tr}(Z) = \langle A- n\lambda I_n, Z\rangle \\ &{\leqslant}{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert A- n\lambda I_n \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast}\, {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert Z \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\mbox{\scriptsize op}} = {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert A- n\lambda I_n \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast},\end{aligned}$$ where the equality holds if and only if all eigenvalues of $Z$ equal to one (since $A- n\lambda I_n$ is strictly positive definite). Note that $Z^{\dagger} = I_{n} \in {\mathscr{C}}$ is a feasible solution for . Therefore, $Z^{\dagger}$ is the unique solution of the SDP . *Proof of Part (3):* By the optimality and feasibility of the solutions $\tilde Z_{\lambda_1}$ and $\tilde Z_{\lambda_2}$, we have $$\label{Eqn:optimality} \begin{aligned} \langle A ,\tilde Z_{\lambda_2}\rangle - \lambda_1 \operatorname{tr}(\tilde Z_{\lambda_2}) &{\leqslant}\langle A ,\tilde Z_{\lambda_1}\rangle - \lambda_1 \operatorname{tr}(\tilde Z_{\lambda_1}),\ \ \mbox{and}\\ \langle A ,\tilde Z_{\lambda_1}\rangle - \lambda_2 \operatorname{tr}(\tilde Z_{\lambda_1}) &{\leqslant}\langle A ,\tilde Z_{\lambda_2}\rangle - \lambda_2 \operatorname{tr}(\tilde Z_{\lambda_2}). \end{aligned}$$ Adding these two inequalities together yields $$\begin{aligned} (\lambda_1-\lambda_2) \big( \operatorname{tr}(\tilde Z_{\lambda_1}) - \operatorname{tr}(\tilde Z_{\lambda_2})\big) {\leqslant}0,\end{aligned}$$ which implies $\operatorname{tr}(\tilde Z_{\lambda_1}) {\geqslant}\operatorname{tr}(\tilde Z_{\lambda_2})$ when $\lambda_1> \lambda_2$. Moreover, if at least one of the SDPs has a unique solution, then at least one of the two inequalities in  is strict, implying $$(\lambda_1-\lambda_2) \big( \operatorname{tr}(\tilde Z_{\lambda_1}) - \operatorname{tr}(\tilde Z_{\lambda_2})\big) < 0,$$ and $\operatorname{tr}(\tilde Z_{\lambda_1}) > \operatorname{tr}(\tilde Z_{\lambda_2})$. Proofs on spectral decompositions {#app:A} ================================= Since $k$ is symmetric and positive semidefinite, so is $R$. Thus the corresponding operator $R$ is self-adjoint in $L^2({\mathrm{d}}\mu)$ and is also compact if holds. Then $R$ has a discrete set of nonnegative eigenvalues $\lambda_0{\geqslant}\lambda_1{\geqslant}\cdots {\geqslant}0$, and has the following eigen-decomposition $$\begin{aligned} R(x,y) = \sum_{j=0}^\infty \lambda_j\, \phi_j(x)\,\phi_j(y),\quad\forall x,y\in S,\end{aligned}$$ where $\{\phi_j\}_{j=0}^\infty$ is an orthonormal basis of $L^2({\mathrm{d}}\mu)$. Note that $$\begin{aligned} R(x,y) = \sqrt{\frac{\pi(x)}{\pi(y)}}\,p(x,y), \quad\forall x,y\in S.\end{aligned}$$ This implies a decomposition of the transition probability $p(x,y)$ as $$\begin{aligned} p(x,y) = \sum_{j=0}^\infty \lambda_j\, \psi_j(x)\,\varphi_j(y),\quad\forall x,y\in S,\end{aligned}$$ where $\psi_j(x) = \phi_j(x)/\sqrt{\pi(x)}$ and $\varphi_j(x) = \phi_j(x)\sqrt{\pi(x)}$. In particular, for each $j=0,1,\ldots$, $$\begin{aligned} P\psi_j(x) &= \sum_{l=0}^\infty \lambda_l \psi_l(x)\, \int_S \varphi_l(y)\, \psi_j(y)\,{\mathrm{d}}\mu(y)=\sum_{l=0}^\infty \lambda_l \psi_l(x)\,\delta_{lj} = \lambda_j \psi_j(x),\quad\forall x\in S,\end{aligned}$$ implying that $\{\psi_j\}_{j=0}^\infty$ are the corresponding (right) eigenfunctions of $P$, with unit $L^2(\pi {\mathrm{d}}\mu)$ norm, associated with the same eigenvalues $\lambda_0{\geqslant}\lambda_1{\geqslant}\cdots {\geqslant}0$. Since $P$ is the transition operator of a Markov chain, $\lambda_0=1$ and $\psi_0\equiv 1$. For $t\in{\mathbb{N}}_{+}$ and $x,y\in S$, let $p_{t}(x,y)$ be the $t$-step transition probability from $x$ to $y$. By Lemma \[lem:spectral\_decomposition\_Markov\_chain\], we have $$\begin{aligned} p_t(x,y) = \sum_{j=0}^\infty \lambda_j^t\, \psi_j(x)\,\varphi_j(y) \end{aligned}$$ and $\{\varphi_j\}_{j=0}^\infty$ forms an orthonormal basis of $L^2({\mathrm{d}}\mu/\pi)$. Consequently, by viewing $\lambda_j^t\, \psi_j(x)$ as the coefficient associated with $\varphi_j$ in the orthogonal expansion of function $p_t(x,\cdot)$, we have $$\begin{aligned} D_{t}^2(x, y) =\|\, p_{t}(x, \cdot) - p_{t}(y, \cdot)\,\|^2_{L^{2}({\mathrm{d}}\mu/\pi)}= \sum_{j=0}^{\infty} \lambda_{j}^{2t} \, [\psi_{j}(x) - \psi_{j}(y)]^{2}. \end{aligned}$$ Empirical diffusion affinity {#app:B} ============================ Similar to the derivations in Appendix \[app:A\], if we define matrix $R_n\in{\mathbb{R}}^{n\times n}$ with $$\begin{aligned} [R_n]_{ij}=\frac{\kappa(X_i,X_j)}{\sqrt{d_n(X_i)}\sqrt{d_n(X_j)}},\end{aligned}$$ then $\{\lambda_{n,j}\}_{j=0}^{n-1}$ are also the eigenvalues of $R_n$. Let $\phi_{n,j}\in{\mathbb{R}}^n$ denote the unit Euclidean norm eigenvector associated with $\lambda_{n,j}$. Then the empirical probability transition matrix $P_n$ has the decomposition $$\begin{aligned} P_{n}^t = \sum_{j=0}^{n-1} \lambda_{n,j}^t\, \psi_{n,j}\,\varphi_{n,j}^T,\end{aligned}$$ where $\psi_{n,j} = D_n^{-1/2}\,\phi_{n,j}\in{\mathbb{R}}^n$ and $\varphi_{n,j} = D_n^{1/2}\phi_{n,j}\in{\mathbb{R}}^n$, so that $\psi_{n,j}=D_n^{-1} \varphi_{n,j}$ for each $j\in\{0,1,\dots,n-1\}$. In particular, $\psi_{n,j}$ has unit $L^2(\mbox{diag}(D_n))$ norm, and $\varphi_{n,j}$ has unit $L^2(\mbox{diag}(D_n^{-1}))$ norm, for each $j\in\{0,1,\dots,n-1\}$. In addition, we have the following relation between the diffusion affinity and $P_n^{2t}$, $$\begin{aligned} \langle X_i,\, X_j\rangle_{D_{n,t}} = \sum_{l=0}^{n-1} \lambda_{n,l}^{2t} \,[\psi_{n,l}]_i \, [\psi_{n,l}]_j =\sum_{l=0}^{n-1} \lambda_{n,l}^{2t} \, [\psi_{n,l}]_i \, [\varphi_{n,l}]_j d_n^{-1}(X_j) =[P_n^{2t}D_n^{-1}]_{ij}.\end{aligned}$$ Technical proofs ================ In this appendix, we collect some technical lemmas used in the proofs of our main results. \[lem:some\_ineq\_feasible\_set\] Let $Z^{*}$ be defined in (\[eqn:Kmeans\_true\_membership\_matrix\]). Then for any $Z \in {\mathscr{C}}$ defined in , we have $$\begin{aligned} \label{eqn:ineq_1_feasible_set} \|Z^{*} - Z^{*} Z Z^{*}\|_{1} = \|Z^{*} - Z^{*} Z\|_{1} =&\, 2\sum_{k=1}^K \sum_{m\neq k} \|Z_{G_{k}^{*} G_{m}^{*}}\|_{1}.\end{aligned}$$ In addition, if $Z$ also satisfies $\operatorname{tr}(Z)=\operatorname{tr}(Z^\ast)$, or $Z\in {\mathscr{C}}_K$, where ${\mathscr{C}}_K$ is defined in , then $$\begin{aligned} \label{eqn:ineq_2_feasible_set} {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert (I-Z^{*}) Z (I-Z^{*}) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast} = &\, \sum_{k=1}^K \sum_{m\neq k} \frac{1}{n_k}\, \|Z_{G_{k}^{*} G_{m}^{*}}\|_{1} {\leqslant}{\|Z^{*} - Z^{*} Z\|_{1} \over 2 \underline{n}}, \\ \label{eqn:ineq_3_feasible_set} \|Z^{*} - Z^{*} Z\|_{1} {\leqslant}\|Z^{*} - Z\|_{1} {\leqslant}& \, n\,{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert (I-Z^{*}) Z (I-Z^{*}) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast} {\leqslant}{2 n \over \underline{n}} \|Z^{*} - Z^{*} Z\|_{1}.\end{aligned}$$ Inequalities  and  follows from Lemma 1 in [@GiraudVerzelen2018]. Inequality  is due to inequality (57) in [@BuneaGiraudRoyerVerzelen2016]. \[lem:some\_ineq\_feasible\_set\_adaptive\] Let $Z^{*}$ be defined in (\[eqn:Kmeans\_true\_membership\_matrix\]). Then for any $Z \in {\mathscr{C}}$ defined in (\[eqn:clustering\_Kmeans\_sdp\_unknown\_K\]), we have $$\begin{aligned} {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert (I-Z^{*}) Z (I-Z^{*}) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast}{\leqslant}&\, \sum_{k=1}^K \sum_{m\neq k} \frac{1}{n_k}\, \|Z_{G_{k}^{*} G_{m}^{*}}\|_{1} + \operatorname{tr}(Z) - \operatorname{tr}(Z^\ast),\\ \|Z^{*} - Z\|_{1} {\leqslant}& \,4n\,\sum_{k=1}^K \sum_{m\neq k} \frac{1}{n_k}\, \|Z_{G_{k}^{*} G_{m}^{*}}\|_{1}+ n \big(\operatorname{tr}(Z) - \operatorname{tr}(Z^\ast)\big),\end{aligned}$$ and  holds for $Z \in {\mathscr{C}}$. The first inequality follows from inequality (57) in [@BuneaGiraudRoyerVerzelen2016]. The second one follows from the first,  , and the following decomposition, $$\begin{aligned} Z - Z^\ast = (I-Z^{*}) Z (I-Z^{*}) + (Z^\ast Z - Z^\ast) + (Z Z^\ast - Z^\ast) + (Z^\ast - Z^\ast ZZ^\ast),\end{aligned}$$ with inequality $\|U\|_1 {\leqslant}n {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert U \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\mbox{\scriptsize op}} {\leqslant}n {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert U \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast}$ for any matrix $U\in {\mathbb{R}}^{n\times n}$. [^1]: X. Chen’s research is supported in part by NSF DMS-1404891, NSF CAREER Award DMS-1752614, and UIUC Research Board Awards (RB17092, RB18099). Y. Yang’s research is supported in part by NSF DMS-1810831. This work is completed in part with the high-performance computing resource provided by the Illinois Campus Cluster Program at UIUC. Authors are listed in alphabetical order

--- abstract: 'We discuss the statistical properties of a recently introduced unbiased stochastic approximation to the score equations for maximum likelihood calculation for Gaussian processes. Under certain conditions, including bounded condition number of the covariance matrix, the approach achieves $O(n)$ storage and nearly $O(n)$ computational effort per optimization step, where $n$ is the number of data sites. Here, we prove that if the condition number of the covariance matrix is bounded, then the approximate score equations are nearly optimal in a well-defined sense. Therefore, not only is the approximation efficient to compute, but it also has comparable statistical properties to the exact maximum likelihood estimates. We discuss a modification of the stochastic approximation in which design elements of the stochastic terms mimic patterns from a $2^n$ factorial design. We prove these designs are always at least as good as the unstructured design, and we demonstrate through simulation that they can produce a substantial improvement over random designs. Our findings are validated by numerical experiments on simulated data sets of up to 1 million observations. We apply the approach to fit a space–time model to over 80,000 observations of total column ozone contained in the latitude band $40^\circ$–$50^\circ$N during April 2012.' address: - | M. L. Stein\ Department of Statistics\ University of Chicago\ Chicago, Illinois 60637\ USA\ - | J. Chen\ M. Anitescu\ Mathematics and Computer\ Science Division\ Argonne National Laboratory\ Argonne, Illinois 60439\ USA\ \ author: - - - title: Stochastic approximation of score functions for Gaussian processes --- , Introduction {#sec1} ============ Gaussian process models are widely used in spatial statistics and machine learning. In most applications, the covariance structure of the process is at least partially unknown and must be estimated from the available data. Likelihood-based methods, including Bayesian methods, are natural choices for carrying out the inferences on the unknown covariance structure. For large data sets, however, calculating the likelihood function exactly may be difficult or impossible in many cases. Assuming we are willing to specify the covariance structure up to some parameter $\theta\in\Theta\subset{{\mathbb R}}^p$, the generic problem we are faced with is computing the loglikelihood for $Z\sim N(0,K(\theta))$ for some random vector $Z\in{{\mathbb R}}^n$ and $K$ an $n\times n$ positive definite matrix indexed by the unknown $\theta$. In many applications, there would be a mean vector that also depends on unknown parameters, but since unknown mean parameters generally cause fewer computational difficulties, for simplicity we will assume the mean is known to be 0 throughout this work. For the application to ozone data in Section \[sec6\], we avoid modeling the mean by removing the monthly mean for each pixel. The simulations in Section \[sec5\] all first preprocess the data by taking a discrete Laplacian, which filters out any mean function that is linear in the coordinates, so that the results in those sections would be unchanged for such mean functions. The loglikelihood is then, up to an additive constant, given by $${{\mathcal}L}(\theta) = -\tfrac{1}{2}Z'K(\theta)^{-1}Z - \tfrac{1}{2}\log\det\bigl\{K(\theta)\bigr\}.$$ If $K$ has no exploitable structure, the standard direct way of calculating ${{\mathcal}L}(\theta)$ is to compute the Cholesky decompositon of $K(\theta)$, which then allows $Z'K(\theta)^{-1}Z$ and $\log\det\{K(\theta)\}$ to be computed quickly. However, the Cholesky decomposition generally requires $O(n^2)$ storage and $O(n^3)$ computations, either of which can be prohibitive for sufficiently large $n$. Therefore, it is worthwhile to develop methods that do not require the calculation of the Cholesky decomposition or other matrix decompositions of $K$. If our goal is just to find the maximum likelihood estimate (MLE) and the corresponding Fisher information matrix, we may be able to avoid the computation of the log determinants by considering the score equations, which are obtained by setting the gradient of the loglikelihood equal to 0. Specifically, defining $K_i = \frac{\partial}{\partial\theta_i}K(\theta)$, the score equations for $\theta$ are given by (suppressing the dependence of $K$ on $\theta$) $$\label{score} \tfrac{1}{2}Z'K^{-1}K_iK^{-1}Z -\tfrac{1}{2} \operatorname{tr}\bigl(K^{-1}K_i\bigr) = 0$$ for $i=1,\ldots,p$. If these equations have a unique solution for $\theta\in\Theta$, this solution will generally be the MLE. Iterative methods often provide an efficient (in terms of both storage and computation) way of computing solves in $K$ (expressions of the form $K^{-1}x$ for vectors $x$) and are based on being able to multiply arbitrary vectors by $K$ rapidly. In particular, assuming the elements of $K$ can be calculated as needed, iterative methods require only $O(n)$ storage, unlike matrix decompositions such as the Cholesky, which generally require $O(n^2)$ storage. In terms of computations, two factors drive the speed of iterative methods: the speed of matrix–vector multiplications and the number of iterations. Exact matrix–vector multiplication generally requires $O(n^2)$ operations, but if the data form a partial grid, then it can be done in $O(n\log n)$ operations using circulant embedding and the fast Fourier transform. For irregular observations, fast multipole approximations can be used \[@anitescu2012mfa\]. The number of iterations required is related to the condition number of $K$ (the ratio of the largest to smallest singular value), so that preconditioning \[@chenbook\] is often essential; see @steinchenanitescufiltering for some circumstances under which one can prove that preconditioning works well. Computing the first term in (\[score\]) requires only one solve in $K$, but the trace term requires $n$ solves (one for each column of $K_i$) for $i=1,\ldots,p$, which may be prohibitive in some circumstances. Recently, @anitescu2012mfa analyzed and demonstrated a stochastic approximation of the trace term based on the Hutchinson trace estimator \[@hutchinson\]. To define it, let $U_1,\ldots,U_N$ be i.i.d. random vectors in ${{\mathbb R}}^n$ with i.i.d. symmetric Bernoulli components, that is, taking on values 1 and $-1$ each with probability $\frac{1}{2}$. Define a set of estimating equations for $\theta$ by $$\label{ascore} g_i(\theta,N) = \frac{1}{2}Z'K^{-1}K_iK^{-1}Z -\frac{1}{2N}\sum_{j=1}^N U_j'K^{-1}K_iU_j = 0$$ for $i=1,\ldots,p$. Throughout this work, $E_\theta$ means to take expectations over $Z\sim N(0,K(\theta))$ and over the $U_j$’s as well. Since $E_\theta(U_1'K^{-1}K_iU_1) = \operatorname{tr}(K^{-1}K_i)$, $E_\theta g_i(\theta,N) = 0$ and (\[ascore\]) provides a set of unbiased estimating equations for $\theta$. Therefore, we may hope that a solution to (\[ascore\]) will provide a good approximation to the MLE. The unbiasedness of the estimating equations (\[ascore\]) requires only that the components of the $U_j$’s have mean 0 and variance 1; but, subject to this constraint, @hutchinson shows that, assuming the components of the $U_j$’s are independent, taking them to be symmetric Bernoulli minimizes the variance of $U_1'MU_1$ for any $n\times n$ matrix $M$. The Hutchinson trace estimator has also been used to approximate the GCV (generalized cross-validation) statistic in nonparametric regression \[@girard [@wahba]\]. In particular, @girard shows that $N$ does not need to be large to obtain a randomized GCV that yields results nearly identical to those obtained using exact GCV. Suppose for now that it is possible to take $N$ much smaller than $n$ and obtain an estimate of $\theta$ that is nearly as efficient statistically as the exact MLE. From here on, assume that any solves in $K$ will be done using iterative methods. In this case, the computational effort to computing (\[score\]) or (\[ascore\]) is roughly linear in the number of solves required (although see Section \[sec4\] for methods that make $N$ solves for a common matrix $K$ somewhat less than $N$ times the effort of one solve), so that (\[ascore\]) is much easier to compute than (\[score\]) when $N/n$ is small. An attractive feature of the approximation (\[ascore\]) is that if at any point one wants to obtain a better approximation to the score function, it suffices to consider additional $U_j$’s in (\[ascore\]). However, how exactly to do this if using the dependent sampling scheme for the $U_j$’s in Section \[sec4\] is not so obvious. Since this stochastic approach provides only an approximation to the MLE, one must compare it with other possible approximations to the MLE. Many such approaches exist, including spectral methods, low-rank approximations, covariance tapering and those based on some form of composite likelihood. All these methods involve computing the likelihood itself and not just its gradient, and thus all share this advantage over solving (\[ascore\]). Note that one can use randomized algorithms to approximate $\log\det K$ and thus approximate the loglikelihood directly \[@zhangY\]. However, this approximation requires first taking a power series expansion of $K$ and then applying the randomization trick to each term in the truncated power series; the examples presented by @zhangY show that the approach does not generally provide a good approximation to the loglikelihood. Since the accuracy of the power series approximation to $\log\det K$ depends on the condition number of $K$, some of the filtering ideas described by @steinchenanitescufiltering and used to good effect in Section \[sec4\] here could perhaps be of value for approximating $\log\det K$, but we do not explore that possibility. See @aune for some recent developments on stochastic approximation of log determinants of positive definite matrices. Let us consider the four approaches of spectral methods, low-rank approximations, covariance tapering and composite likelihood in turn. Spectral approximations to the likelihood can be fast and accurate for gridded data \[@whittle [@guyon; @dahlhaus]\], although even for gridded data they may require some prefiltering to work well \[@stein1995\]. In addition, the approximations tend to work less well as the number of dimensions increase \[@dahlhaus\] and thus may be problematic for space–time data, especially if the number of spatial dimensions is three. Spectral approximations have been proposed for ungridded data \[@fuentes\], but they do not work as well as they do for gridded data from either a statistical or computational perspective, especially if large subsets of observations do not form a regular grid. Furthermore, in contrast to the approach we propose here, there appears to be no easy way of improving the approximations by doing further calculations, nor is it clear how to assess the loss of efficiency by using spectral approximations without a large extra computational burden. Low-rank approximations, in which the covariance matrix is approximated by a low-rank matrix plus a diagonal matrix, can greatly reduce the burden of memory and computation relative to the exact likelihood \[@cressiejohannesson [@eidsvik]\]. However, for the kinds of applications we have in mind, in which the diagonal component of the covariance matrix does not dominate the small-scale variation of the process, these low-rank approximations tend to work poorly and are not a viable option \[@steinkorea\]. Covariance tapering replaces the covariance matrix of interest by a sparse covariance matrix with similar local behavior \[@furrer\]. There is theoretical support for this approach \[@kaufman [@wangloh]\], but the tapered covariance matrix must be very sparse to help a great deal with calculating the log determinant of the covariance matrix, in which case @steintaper finds that composite likelihood approaches will often be preferable. There is scope for combining covariance tapering with the approach presented here in that sparse matrices lead to efficient matrix–vector multiplication, which is also essential for our implementation of computing (\[ascore\]) based on iterative methods to do the matrix solves. @sang show that covariance tapering and low-rank approximations can also sometimes be profitably combined to approximate likelihoods. We consider methods based on composite likelihoods to be the main competitor to solving (\[ascore\]). The approximate loglikelihoods described by @vecchia [@steinchiwelty; @caragea] can all be written in the following form: for some sequence of pairs of matrices $(A_j,B_j)$, $j=1,\ldots,q$, all with $n$ columns, at most $n$ rows and full rank, $$\label{composite} \sum_{j=1}^q \log f_{j,\theta}(A_j Z\mid B_j Z),$$ where $f_{j,\theta}$ is the conditional Gaussian density of $A_j Z$ given $B_j Z$. As proposed by @vecchia and @steinchiwelty, the rank of $B_j$ will generally be larger than that of $A_j$, in which case the main computation in obtaining (\[composite\]) is finding Cholesky decompositions of the covariance matrices of $B_1Z,\ldots, B_q Z$. For example, @vecchia just lets $A_j Z$ be the $j$th component of $Z$ and $B_j Z$ some subset of $Z_1,\ldots,Z_{j-1}$. If $m$ is the largest of these subsets, then the storage requirements for this computation are $O(m^2)$ rather than $O(n^2)$. Comparable to increasing the number of $U_j$’s in the randomized algorithm used here, this approach can be updated to obtain a better approximation of the likelihood by increasing the size of the subset of $Z_1,\ldots,Z_{j-1}$ to condition on when computing the conditional density of $Z_j$. However, for this approach to be efficient from the perspective of flops, one needs to store the Cholesky decompositions of the covariance matrices of $B_1Z,\ldots,B_q Z$, which would greatly increase the memory requirements of the algorithm. For dealing with truly massive data sets, our long-term plan is to combine the randomized approach studied here with a composite likelihood by using the randomized algorithms to compute the gradient of (\[composite\]), thus making it possible to consider $A_j$’s and $B_j$’s of larger rank than would be feasible if one had to do exact calculations. Section \[sec2\] provides a bound on the efficiency of the estimating equations based on the approximate likelihood relative to the Fisher information matrix. The bound is in terms of the condition number of the true covariance matrix of the observations and shows that if the covariance matrix is well conditioned, $N$ does not need to be very large to obtain nearly optimal estimating equations. Section \[sec3\] shows how one can get improved estimating equations by choosing the $U_j$’s in (\[ascore\]) based on a design related to $2^n$ factorial designs. Section \[sec4\] describes details of the algorithms, including methods for solving the approximate score equations and the role of preconditioning. Section \[sec5\] provides results of numerical experiments on simulated data. These results show that the basic method can work well for moderate values of $N$, even sometimes when the condition numbers of the covariance matrices do not stay bounded as the number of observations increases. Furthermore, the algorithm with the $U_j$’s chosen as in Section \[sec3\] can lead to substantially more accurate approximations for a given $N$. A large-scale numerical experiment shows that for observations on a partially occluded grid, the algorithm scales nearly linearly in the sample size. Section \[sec6\] applies the methods to OMI (Ozone Monitoring Instrument) Level 3 (gridded) total column ozone measurements for April 2012 in the latitude band $40^\circ$–$50^\circ$N. ![Demeaned ozone data (Dobson units) plotted using a heat color map. Missing data is colored white.[]{data-label="figozone"}](627f01.eps) The data are given on a $1^\circ\times1^\circ$ grid, so if the data were complete, there would be a total of $360\times10\times30 = 108\mbox{,}000$ observations. However, as Figure \[figozone\] shows, there are missing observations, mostly due to a lack of overlap in data from different orbits taken by OMI, but also due to nearly a full day of missing data on April 29–30, so that there are 84,942 observations. By acting as if all observations are taken at noon local time and assuming the process is stationary in longitude and time, the covariance matrix for the observations can be embedded in a block circulant matrix, greatly reducing the computational effort needed for multiplying the covariance matrix by a vector. Using (\[ascore\]) and a factorized sparse inverse preconditioner \[@kolo\], we are able to compute an accurate approximation to the MLE for a simple model that captures some of the main features in the OMI data, including the obvious movement of ozone from day to day visible in Figure \[figozone\] that coincides with the prevailing westerly winds in this latitude band. Variance of stochastic approximation of the score function {#sec2} ========================================================== This section gives a bound relating the covariance matrices of the approximate and exact score functions. Let us first introduce some general notation for unbiased estimating equations. Suppose $\theta$ has $p$ components and $g(\theta)=(g_1(\theta ),\ldots, g_p(\theta))'=0$ is a set of unbiased estimating equations for $\theta$ so that $E_\theta g(\theta)=0$ for all $\theta$. Write $\dot{g}(\theta)$ for the $p\times p$ matrix whose $ij$th element is $\frac{\partial}{\partial\theta_i}g_j(\theta)$ and $\operatorname{cov}_\theta \{g(\theta)\}$ for the covariance matrix of $g(\theta)$. The Godambe information matrix \[@varin\], $$\mathcal{E}\bigl\{g(\theta)\bigr\}= E_\theta\bigl\{\dot{g}(\theta)\bigr \} \bigl[\operatorname{cov}_\theta\bigl\{g(\theta)\bigr\} \bigr]^{-1} E_\theta\bigl\{\dot{g}(\theta)\bigr\}$$ is a natural measure of the informativeness of the estimating equations \[@Heyde, Definition 2.1\]. For positive semidefinite matrices $A$ and $B$, write $A\succeq B$ if $A-B$ is positive semidefinite. For unbiased estimating equations $g(\theta)=0$ and $h(\theta)=0$, then we can say $g$ dominates $h$ if $\mathcal{E}\{g(\theta)\} \succeq\mathcal{E}\{h(\theta)\}$. Under sufficient regularity conditions on the model and the estimating equations, the score equations are the optimal estimating equations \[@Bhapkar\]. Specifically, for the score equations, the Godambe information matrix equals the Fisher information matrix, ${{\mathcal}I}(\theta)$, so this optimality condition means ${{\mathcal}I}(\theta)\succeq\mathcal{E}\{g(\theta)\}$ for all unbiased estimating equations $g(\theta)=0$. Writing $M_{ij}$ for the $ij$th element of the matrix $M$, for the score equations in (\[score\]), ${{\mathcal}I}_{ij}(\theta) = \frac{1}{2}\operatorname{tr}(K^{-1}K_iK^{-1}K_j)$ \[@stein-book, page 179\]. For the approximate score equations (\[ascore\]), it is not difficult to show that $E_\theta\dot{g}(\theta,N)=-{{\mathcal}I}(\theta)$. Furthermore, writing $W^i$ for $K^{-1}K_i$ and defining the matrix ${{\mathcal}J}(\theta)$ by ${{\mathcal}J}_{ij}(\theta) = \operatorname{cov}(U_1'W^iU_1,U_1'W^jU_1)$, we have $$\label{B} \operatorname{cov}_\theta\bigl\{g(\theta,N)\bigr\} = {{\mathcal}I}(\theta) + \frac{1}{4N}{{\mathcal}J}(\theta),$$ so that $\mathcal{E}\{g(\theta,N)\}={{\mathcal}I}(\theta) \{{{\mathcal}I}(\theta) +\frac{1}{4N}{{\mathcal}J}(\theta) \}^{-1}{{\mathcal}I}(\theta)$, which, as $N\to\infty$, tends to ${{\mathcal}I}(\theta)$. In fact, as also demonstrated empirically by @anitescu2012mfa, one may often not need $N$ to be that large to get estimating equations that are nearly as efficient as the exact score equations. Writing $U_{1j}$ for the $j$th component of $U_1$, we have $$\begin{aligned} \label{Jij} {{\mathcal}J}_{ij}(\theta) & = & \sum_{k,\ell,p,q=1}^n \operatorname{cov}\bigl(W_{k\ell}^i U_{1k}U_{1\ell}, W_{pq}^j U_{1p}U_{1q}\bigr) \nonumber \\ & = & \sum_{k\ne\ell}\bigl\{\operatorname{cov} \bigl(W_{k\ell}^i U_{1k}U_{1\ell },W_{k\ell}^j U_{1k}U_{1\ell}\bigr)+\operatorname{cov}\bigl(W_{k\ell}^i U_{1k}U_{1\ell},W_{\ell k}^j U_{1k}U_{1\ell}\bigr)\bigr\} \nonumber\\[-8pt]\\[-8pt] & = & \sum_{k\ne\ell}\bigl(W_{k\ell}^i W_{k\ell}^j + W_{k\ell}^i W_{\ell k}^j\bigr) \nonumber \\ & = & \operatorname{tr}\bigl(W^i W^j\bigr) + \operatorname{tr} \bigl\{W^i \bigl(W^j\bigr)' \bigr\} - 2\sum _{k=1}^n W_{kk}^iW_{kk}^j.\nonumber\end{aligned}$$ As noted by @hutchinson, the terms with $k=\ell$ drop out in the second step because $U_{1j}^2=1$ with probability 1. When $K(\theta)$ is diagonal for all $\theta$, then $N=1$ gives the exact score equations, although in this case computing $\operatorname{tr}(K^{-1}K_i)$ directly would be trivial. Writing $\kappa(\cdot)$ for the condition number of a matrix, we can bound$\operatorname{cov}_\theta\{g(\theta,N)\}$ in terms of ${{\mathcal}I}(\theta)$ and $\kappa(K)$. The proof of the following result is given in the . \[tmain\] $$\label{Bbound} \operatorname{cov}_\theta\bigl\{g(\theta,N)\bigr\} \preceq{{\mathcal}I}(\theta) \biggl\{ 1 + \frac{(\kappa(K)+1)^2}{4N\kappa(K)} \biggr\}.$$ It follows from (\[Bbound\]) that $$\mathcal{E}\bigl\{g(\theta,N)\bigr\} \succeq\biggl\{ 1+ \frac{(\kappa (K)+1)^2}{4N\kappa(K)} \biggr\}^{-1}{{\mathcal}I}(\theta).$$ In practice, if $\frac{(\kappa(K)+1)^2}{4N\kappa(K)} < 0.01$, so that the loss of information in using (\[ascore\]) rather than (\[score\]) was at most 1%, we would generally be satisfied with using the approximate score equations and a loss of information of even 10% or larger might be acceptable when one has a massive amount of data. For example, if $\kappa(K)=5$, a bound of 0.01 is obtained with $N=180$ and a bound of 0.1 with $N=18$. It is possible to obtain unbiased estimating equations similar to (\[ascore\]) whose statistical efficiency does not depend on $\kappa(K)$. Specifically, if we write $\operatorname{tr}(K^{-1}K_i)$ as $\operatorname{tr}((G')^{-1}K_iG^{-1})$, where $G$ is any matrix satisfying $G'G=K$, we then have that $$\label{symscore} h_i(\theta,N) = \frac{1}{2}Z'K^{-1}K_iK^{-1}Z -\frac{1}{2N}\sum_{j=1}^N U_j'\bigl(G'\bigr)^{-1}K_iG^{-1}U_j = 0$$ for $i=1,\ldots,p$ are also unbiased estimating equations for $\theta$. In this case, $\operatorname{cov}_\theta\{h(\theta,N)\}\preceq ( 1+\frac{1}{N} ){{\mathcal}I}(\theta)$, whose proof is similar to that of Theorem \[tmain\] but exploits the symmetry of $(G')^{-1}K_iG^{-1}$. This bound is less than or equal to the bound in (\[Bbound\]) on $\operatorname{cov}_\theta\{g(\theta,N)\}$. Whether it is preferable to use (\[symscore\]) rather than (\[ascore\]) depends on a number of factors, including the sharpness of the bound in (\[Bbound\]) and how much more work it takes to compute $G^{-1}U_j$ than to compute $K^{-1}U_j$. An example of how the action of such a matrix square root can be approximated efficiently using only $O(n)$ storage is presented by @chen2011computing. Dependent designs {#sec3} ================= \[secdependent\] Choosing the $U_j$’s independently is simple and convenient, but one can reduce the variation in the stochastic approximation by using a more sophisticated design for the $U_j$’s; this section describes such a design. Suppose that $n=Nm$ for some nonnegative integer $m$ and that $\beta_1,\ldots,\beta_N$ are fixed vectors of length $N$ with all entries $\pm1$ for which $\frac{1}{N}\sum_{j=1}^N \beta_j\beta'_j = I$. For example, if $N=2^q$ for a positive integer $q$, then the $\beta_j$’s can be chosen to be the design matrix for a saturated model of a $2^q$ factorial design in which the levels of the factors are set at $\pm1$ \[@BHH, Chapter 5\]. In addition, assume that $X_1,\ldots,X_m$ are random diagonal matrices of size $N$ and $Y_{jk}$, $j=1,\ldots,N; k=1,\ldots,m$ are random variables such that all the diagonal elements of the $X_j$’s and all the $Y_{jk}$’s are i.i.d. symmetric Bernoulli random variables. Then define $$\label{Uj} U_j = \pmatrix{ Y_{j1}X_1 \cr \vdots \cr Y_{jm}X_m} \beta_j.$$ One can easily show that for any $Nm\times Nm$ matrix $M$, $E (\frac{1}{N}\sum_{j=1}^N U_j'MU_j ) = \operatorname{tr}(M)$. Thus, we can use this definition of the $U_j$’s in (\[ascore\]), and the resulting estimating equations are still unbiased. This design is closely related to a class of designs introduced by @avron, who propose selecting the $U_j$’s as follows. Suppose $H$ is a Hadamard matrix, that is, an $n\times n$ orthogonal matrix with elements $\pm1$. @avron actually consider $H$ a multiple of a unitary matrix, but the special case $H$ Hadamard makes their proposal most similar to ours. Then, using simple random sampling (with replacement), they choose $N$ columns from this matrix and multiply this $n\times N$ matrix by an $n\times n$ diagonal matrix with diagonal entries made up of independent symmetric Bernoulli random variables. The columns of this resulting matrix are the $U_j$’s. We are also multiplying a subset of the columns of a Hadamard matrix by a random diagonal matrix, but we do not select the columns by simple random sampling from some arbitrary Hadamard matrix. The extra structure we impose yields beneficial results in terms of the variance of the randomized trace approximation, as the following calculations show. Partitioning $M$ into an $m\times m$ array of $N\times N$ matrices with $k\ell$th block $M^b_{k\ell}$, we obtain the following: $$\label{UjMUj} \frac{1}{N}\sum_{j=1}^N U_j'MU_j = \frac{1}{N} \sum _{k,\ell=1}^m\sum_{j=1}^N Y_{jk}Y_{j\ell} \beta_j'X_k M^b_{k\ell} X_\ell\beta_j.$$ Using $Y_{jk}^2=1$ and $X_k^2 = I$, we have $$\begin{aligned} \frac{1}{N}\sum_{j=1}^N Y_{jk}^2 \beta_j'X_k M^b_{kk} X_k \beta_j & = & \frac{1}{N}\operatorname{tr}\Biggl(X_k M^b_{kk} X_k \sum_{j=1}^N \beta_j\beta_j' \Biggr) \\ & = & \operatorname{tr}\bigl(M^b_{kk}X_k^2 \bigr) \\ & = & \operatorname{tr}\bigl(M^b_{kk}\bigr),\end{aligned}$$ which is not random. Thus, if $M$ is block diagonal (i.e., $M^b_{k\ell}$ is a matrix of zeroes for all $k\ne\ell$), (\[UjMUj\]) yields $\operatorname{tr}(M)$ without error. This result is an extension of the result that independent $U_j$’s give $\operatorname{tr}(M)$ exactly for diagonal $M$. Furthermore, it turns out that, at least in terms of the variance of $\frac{1}{N}\sum_{j=1}^N U_j'MU_j$, for the elements of $M$ off the block diagonal, we do exactly the same as we do when the $U_j$’s are independent. Write $B(\theta)$ for $\operatorname{cov}\{g(\theta,N)\}$ with $g(\theta,N)$ defined as in (\[ascore\]) with independent $U_j$’s. Define $g^d(\theta,N)=0$ for the unbiased estimating equations defined by (\[ascore\]) with dependent $U_j$’s defined by (\[Uj\]) and $B^d(\theta)$ to be the covariance matrix of $g^d(\theta,N)$. Take $T(N,n)$ to be the set of pairs of positive integers $(k,\ell)$ with $1\le\ell<k \le n$ for which $\lfloor k/N\rfloor= \lfloor\ell /N\rfloor$. We have the following result, whose proof is given in the . \[tdependent\] For any vector $v=(v_1,\ldots,v_p)'$, $$\label{improve} v'B(\theta)v - v'B^d(\theta)v = \frac{2}{N} \sum_{(k,\ell)\in T(N,n)} \Biggl\{ \sum _{i=1}^p v_i \bigl( W_{k\ell}^i+W_{\ell k}^i \bigr) \Biggr\}^2.$$ Thus, $B(\theta) \succeq B^d(\theta)$. Since $E_\theta\dot{g} (\theta,N) = E_\theta\dot{g}^d(\theta,N)=-\mathcal{I}(\theta)$, it follows that $\mathcal{E}\{g^d(\theta,N)\}\succeq\mathcal{E}\{g(\theta,N)\}$. How much of an improvement will result from using dependent $U_j$’s depends on the size of the $W_{k\ell}^i$’s within each block. For spatial data, one would typically group spatially contiguous observations within blocks. How to block for space–time data is less clear. The results here focus on the variance of the randomized trace approximation. @avron obtain bounds on the probability that the approximation error is less than some quantity and note that these results sometimes give rankings for various randomized trace approximations different from those obtained by comparing variances. Computational aspects {#sec4} ===================== Finding $\theta$ that solves the estimating equations (\[ascore\]) requires a nonlinear equation solver in addition to computing linear solves in $K$. The nonlinear solver starts at an initial guess $\theta^0$ and iteratively updates it to approach a (hopefully unique) zero of (\[ascore\]). In each iteration, at $\theta^i$, the nonlinear solver typically requires an evaluation of $g(\theta^i,N)$ in order to find the next iterate $\theta^{i+1}$. In turn, the evaluation of $g$ requires employing a linear solver to compute the set of vectors $K^{-1}Z$ and $K^{-1}U_j$, $j=1,\ldots,N$. The Fisher information matrix $\mathcal{I}(\theta)$ and the matrix $\mathcal{J}(\theta)$ contain terms involving matrix traces and diagonals. Write ${\operatorname{diag}}(\cdot)$ for a column vector containing the diagonal elements of a matrix and $\circ$ for the Hadamard (elementwise) product of matrices. For any real matrix $A$, $${\operatorname{tr}}(A)=E_U\bigl(U'AU\bigr) \quad\mbox{and}\quad {\operatorname{diag}}(A)=E_U(U\circ AU),$$ where the expectation $E_U$ is taken over $U$, a random vector with i.i.d. symmetric Bernoulli components. One can unbiasedly estimate $\mathcal{I}(\theta)$ and $\mathcal{J}(\theta)$ by $$\label{Iij} {\widehat}{\mathcal{I}}_{ij}(\theta)=\frac{1}{2N_2}\sum _{k=1}^{N_2}U_k'W^iW^jU_k$$ and $$\begin{aligned} \label{Jij2} {\widehat}{\mathcal{J}}_{ij}(\theta) & = & \frac{1}{N_2}\sum _{k=1}^{N_2}U_k'W^iW^jU_k +\frac{1}{N_2}\sum_{k=1}^{N_2}U_k'W^i \bigl(W^j\bigr)'U_k \nonumber\\[-8pt]\\[-8pt] &&{} -2\sum_{\ell=1}^n \Biggl[ \frac{1}{N_2}\sum_{k=1}^{N_2} \bigl(U_k\circ W^iU_k\bigr) \Biggr]_{\ell} \Biggl[\frac{1}{N_2}\sum_{k=1}^{N_2} \bigl(U_k\circ W^jU_k\bigr) \Biggr]_{\ell}. \nonumber\end{aligned}$$ Note that here the set of vectors $U_k$ need not be the same as that in (\[ascore\]) and that $N_2$ may not be the same as $N$, the number of $U_j$’s used to compute the estimate of $\theta$. Evaluating ${\widehat}{\mathcal{I}}(\theta)$ and ${\widehat}{\mathcal {J}}(\theta)$ requires linear solves since $W^iU_k=K^{-1}(K_iU_k)$ and $(W^i)'U_k=K_i(K^{-1}U_k)$. Note that one can also unbiasedly estimate $\mathcal{J}_{ij}(\theta)$ as the sample covariance of $U'_kW^iU_k$ and $U'_kW^jW_k$ for $k=1,\ldots,N$, but (\[Jij2\]) directly exploits properties of symmetric Bernoulli variables (e.g., $U^2_{1j}=1$). Further study would be needed to see when each approach is preferred. Linear solver {#sec4.1} ------------- We consider an iterative solver for solving a set of linear equations $Ax=b$ for a symmetric positive definite matrix $A\in{{\mathbb R}}^{n\times n}$, given a right-hand vector $b$. Since the matrix $A$ (in our case the covariance matrix) is symmetric positive definite, the conjugate gradient algorithm is naturally used. Let $x^i$ be the current approximate solution, and let $r^i=b-Ax^i$ be the residual. The algorithm finds a search direction $q^i$ and a step size $\alpha^i$ to update the approximate solution, that is, $x^{i+1}=x^i+\alpha^iq^i$, such that the search directions $q^i,\ldots,q^0$ are mutually $A$-conjugate \[i.e., $(q^i)'Aq^j = 0$ for $i\ne j$\] and the new residual $r^{i+1}$ is orthogonal to all the previous ones, $r^i,\ldots,r^0$. One can show that the search direction is a linear combination of the current residual and the past search direction, yielding the following recurrence formulas: $$\begin{aligned} x^{i+1}&=&x^i+\alpha^iq^i, \\ r^{i+1}&=&r^i-\alpha^iAq^i, \\ q^{i+1}&=&r^{i+1}+\beta^iq^i,\end{aligned}$$ where $\alpha^i= \langle r^i,r^i \rangle/ \langle Aq^i,q^i \rangle$ and $\beta^i= \langle r^{i+1},r^{i+1} \rangle/ \langle r^i,r^i \rangle$, and $ \langle\cdot,\cdot\rangle$ denotes the vector inner product. Letting $x^*$ be the exact solution, that is, $Ax^*=b$, then $x^i$ enjoys a linear convergence to $x^*$: $$\label{eqncgconverge} \bigl\|x^i-x^*\bigr\|_A\le2 \biggl( \frac{\sqrt{\kappa(A)}-1}{\sqrt{\kappa (A)}+1} \biggr)^{i}\bigl\|x^0-x^*\bigr\|_A,$$ where $\|\cdot\|_A= \langle A\cdot,\cdot\rangle^{{1/2}}$ is the $A$-norm of a vector. Asymptotically, the time cost of one iteration is upper bounded by that of multiplying $A$ by $q^i$, which typically dominates other vector operations when $A$ is not sparse. Properties of the covariance matrix can be exploited to efficiently compute the matrix–vector products. For example, when the observations are on a lattice (regular grid), one can use the fast Fourier transform (FFT), which takes time $O(n\log n)$ \[@toeplitzbook\]. Even when the grid is partial (with occluded observations), this idea can still be applied. On the other hand, for nongridded observations, exact multiplication generally requires $O(n^2)$ operations. However, one can use a combination of direct summations for close-by points and multipole expansions of the covariance kernel for faraway points to compute the matrix–vector products in $O(n\log n)$, even $O(n)$, time \[@treecode [@fmm]\]. In the case of Matérn-type Gaussian processes and in the context of solving the stochastic approximation (\[ascore\]), such fast multipole approximations were presented by @anitescu2012mfa. Note that the total computational cost of the solver is the cost of each iteration times the number of iterations, the latter being usually much less than $n$. The number of iterations to achieve a desired accuracy depends on how fast $x^i$ approaches $x^*$, which, from (\[eqncgconverge\]), is in turn affected by the condition number $\kappa$ of $A$. Two techniques can be used to improve convergence. One is to perform preconditioning in order to reduce $\kappa$; this technique will be discussed in the next section. The other is to adopt a block version of the conjugate gradient algorithm. This technique is useful for solving the linear system for the same matrix with multiple right-hand sides. Specifically, denote by $AX=B$ the linear system one wants to solve, where $B$ is a matrix with $s$ columns, and the same for the unknown $X$. Conventionally, matrices such as $B$ are called *block vectors*, honoring the fact that the columns of $B$ are handled simultaneously. The block conjugate gradient algorithm is similar to the single-vector version except that the iterates $x^i$, $r^i$ and $q^i$ now become block iterates $X^i$, $R^i$ and $Q^i$ and the coefficients $\alpha^i$ and $\beta^i$ become $s\times s$ matrices. The detailed algorithm is not shown here; interested readers are referred to @olearyblockcg. If $X^*$ is the exact solution, then $X^i$ approaches $X^*$ at least as fast as linearly: $$\label{eqnbcgconverge} \bigl\|\bigl(X^i\bigr)_j-\bigl(X^* \bigr)_j\bigr\|_A\le C_j \biggl(\frac{\sqrt{\kappa_s(A)}-1}{\sqrt{\kappa_s(A)}+1} \biggr)^{i},\qquad j=1,\ldots,s,$$ where $(X^i)_j$ and $(X^*)_j$ are the $j$th column of $X^i$ and $X^*$, respectively; $C_j$ is some constant dependent on $j$ but not $i$; and $\kappa_s(A)$ is the ratio between $\lambda_n(A)$ and $\lambda_s(A)$ with the eigenvalues $\lambda_k$ sorted increasingly. Comparing (\[eqncgconverge\]) with (\[eqnbcgconverge\]), we see that the modified condition number $\kappa_s$ is less than $\kappa$, which means that the block version of the conjugate gradient algorithm has a faster convergence than the standard version does. In practice, since there are many right-hand sides (i.e., the vectors $Z$, $U_j$’s and $K_iU_k$’s), we always use the block version. Preconditioning/filtering {#sec4.2} ------------------------- Preconditioning is a technique for reducing the condition number of the matrix. Here, the benefit of preconditioning is twofold: it encourages the rapid convergence of an iterative linear solver and, if the effective condition number is small, it strongly bounds the uncertainty in using the estimating equations (\[ascore\]) instead of the exact score equations (\[score\]) for estimating parameters (see Theorem \[tmain\]). In numerical linear algebra, preconditioning refers to applying a matrix $M$, which approximates the inverse of $A$ in some sense, to both sides of the linear system of equations. In the simple case of left preconditioning, this amounts to solving $MAx=Mb$ for $MA$ better conditioned than $A$. With certain algebraic manipulations, the matrix $M$ enters into the conjugate gradient algorithm in the form of multiplication with vectors. For the detailed algorithm, see @saadbookiterativemethod. This technique does not explicitly compute the matrix $MA$, but it requires that the matrix–vector multiplications with $M$ can be efficiently carried out.=-1 For covariance matrices, certain filtering operations are known to reduce the condition number, and some can even achieve an optimal preconditioning in the sense that the condition number is bounded by a constant independent of the size of the matrix \[@steinchenanitescufiltering\]. Note that these filtering operations may or may not preserve the rank/size of the matrix. When the rank is reduced, then some loss of statistical information results when filtering, although similar filtering is also likely needed to apply spectral methods for strongly correlated spatial data on a grid \[@stein1995\]. Therefore, we consider applying the same filter to all the vectors and matrices in the estimating equations, in which case (\[ascore\]) becomes the stochastic approximation to the score equations of the *filtered* process. Evaluating the filtered version of $g(\theta,N)$ becomes easier because the linear solves with the filtered covariance matrix converge faster. Nonlinear solver {#sec4.3} ---------------- \[secnonlinearsolver\] The choice of the nonlinear solver is problem dependent. The purpose of solving the score equations (\[score\]) or the estimating equations (\[ascore\]) is to maximize the loglikelihood function $\mathcal{L}(\theta)$. Therefore, investigation into the shape of the loglikelihood surface helps identify an appropriate solver. In Section \[sec5\], we consider the power law generalized covariance model ($\alpha>0$): $$\label{GC1} G(x;\theta)= \cases{\Gamma(-\alpha/2) r^{\alpha}, &\quad if $ \alpha/2\notin{{\mathbb N}}$, \cr (-1)^{1+\alpha/2}r^{\alpha}\log r, &\quad if $\alpha/2\in {{\mathbb N}}$,}$$ where $x=[x_1,\ldots,x_d]\in{{\mathbb R}}^d$ denotes coordinates, $\theta$ is the set of parameters containing $\alpha>0$, $\ell=[\ell_1,\ldots,\ell_d]\in{{\mathbb R}}^d$, and $r$ is the elliptical radius $$\label{GC2} r=\sqrt{\frac{x_1^2}{\ell_1^2}+\cdots+\frac{x_d^2}{\ell_d^2}}.$$ Allowing a different scaling in different directions may be appropriate when, for example, variations in a vertical direction may be different from those in a horizontal direction. The function $G$ is conditionally positive definite; therefore, only the covariances of authorized linear combinations of the process are defined \[@geostatisticsbook, Section 4.3\]. In fact, $G$ is $p$-conditionally positive definite if and only if $2p+2>\alpha$ \[see @geostatisticsbook, Section 4.5\], so that applying the discrete Laplace filter (which gives second-order differences) $\tau$ times to the observations yields a set of authorized linear combinations when $\tau\ge\frac {1}{2}\alpha$. @steinchenanitescufiltering show that if $\alpha=4\tau-d$, then the covariance matrix has a bounded condition number independent of the matrix size. Consider the grid $\{\delta\mathbf{j}\}$ for some fixed spacing $\delta$ and $\mathbf{j}$ a vector whose components take integer values between $0$ and $m$. Applying the filter $\tau$ times, we obtain the covariance matrix $$K_{\mathbf{i}\mathbf{j}}={\operatorname{cov}}\bigl\{\Delta^{\tau}Z(\delta\mathbf{i}),\Delta ^{\tau}Z(\delta\mathbf{j})\bigr\},$$ where $\Delta$ denotes the discrete Laplace operator $$\Delta Z(\delta\mathbf{j})=\sum_{p=1}^d\bigl \{Z(\delta\mathbf{j}-\delta\mathbf{e}_p)-2Z(\delta\mathbf{j})+Z(\delta\mathbf{j}+ \delta\mathbf{e}_p)\bigr\}$$ with $\mathbf{e}_p$ meaning the unit vector along the $p$th coordinate. If $\tau=\operatorname{round}((\alpha+d)/4)$, the resulting $K$ is both positive definite and reasonably well conditioned. Figure \[figloglik\] shows a sample loglikelihood surface for $d=1$ based on an observation vector $Z$ simulated from a 1D partial regular grid spanning the range $[0,100]$, using parameters $\alpha=1.5$ and $\ell=10$. (A similar 2D grid is shown later in Figure \[figgrid\].) The peak of the surface is denoted by the solid white dot, which is not far away from the truth $\theta=(1.5,10)$. The white dashed curve (profile of the surface) indicates the maximum loglikelihoods $\mathcal{L}$ given $\alpha$. The curve is also projected on the $\alpha-\mathcal{L}$ plane and the $\alpha-\ell$ plane. One sees that the loglikelihood value has small variation (ranges from $48$ to $58$) along this curve compared with the rest of the surface, whereas, for example, varying just the parameter $\ell$ changes the loglikelihood substantially. ![A sample loglikelihood surface for the power law generalized covariance kernel, with profile curve and peak plotted.[]{data-label="figloglik"}](627f02.eps) A Newton-type nonlinear solver starts at some initial point $\theta^0$ and tries to approach the optimal point (one that solves the score equations).[^1] Let the current point be $\theta^i$. The solver finds a direction $q^i$ and a step size $\alpha^i$ in some way to move the point to $\theta^{i+1}=\theta^i+\alpha^iq^i$, so that the value of $\mathcal{L}$ is increased. Typically, the search direction $q^i$ is the inverse of the Jacobian multiplied by $\theta ^i$, that is, $q^i=\dot{g}(\theta^i,N)^{-1}\theta^i$. This way, $\theta^{i+1}$ is closer to a solution of the score equations. Figure \[figloglik\] shows a loglikelihood surface when $d=1$. The solver starts somewhere on the surface and quickly climbs to a point along the profile curve. However, this point might be far away from the peak. It turns out that along this curve a Newton-type solver is usually unable to find a direction with an appropriate step size to numerically increase $\mathcal{L}$, in part because of the narrow ridge indicated in the figure. The variation of $\mathcal{L}$ along the normal direction of the curve is much larger than that along the tangent direction. Thus, the iterate $\theta^i$ is trapped and cannot advance to the peak. In such a case, even though the estimated maximized likelihood could be fairly close to the true maximum, the estimated parameters could be quite distant from the MLE of $(\alpha,\ell)$. To successfully solve the estimating equations, we consider each component of $\ell$ an implicit function of $\alpha$. Denote by $$\label{ascore2} g_i(\alpha,\ell_1,\ldots, \ell_d)=0,\qquad i=1,\ldots,d+1,\vadjust{\goodbreak}$$ the estimating equations, ignoring the fixed variable $N$. The implicit function theorem indicates that a set of functions $\ell_1(\alpha),\ldots,\ell_d(\alpha)$ exists around an isolated zero of (\[ascore2\]) in a neighborhood where (\[ascore2\]) is continuously differentiable, such that $$g_i\bigl(\alpha,\ell_1(\alpha),\ldots, \ell_d(\alpha)\bigr)=0\qquad \mbox{for } i=2,\ldots,d+1.$$ Therefore, we need only to solve the equation $$\label{ascore3} g_1\bigl(\alpha,\ell_1(\alpha),\ldots, \ell_d(\alpha)\bigr)=0$$ with a single variable $\alpha$. Numerically, a much more robust method than a Newton-type method exists for finding a root of a one-variable function. We use the standard method of Forsythe, Malcolm and Moler \[([-@fzero]), see the Fortran code \] for solving (\[ascore3\]). This method in turn requires the evaluation of the left-hand side of (\[ascore3\]). Then, the $\ell_i$’s are evaluated by solving $g_2,\ldots,g_{d+1}=0$ fixing $\alpha$, whereby a Newton-type algorithm is empirically proven to be an efficient method. Experiments {#sec5} =========== \[secexp\] In this section we show a few experimental results based on a partially occluded regular grid. The rationale for using such a partial grid is to illustrate a setting where spectral techniques do not work so well but efficient matrix–vector multiplications are available. A partially occluded grid can occur, for example, when observations of some surface characteristics are taken by a satellite-based instrument and it is not possible to obtain observations over regions with sufficiently dense cloud cover. The ozone example in Section \[sec6\] provides another example in which data on a partial grid occurs. This section considers a grid with physical range $[0,100]\times[0,100]$ and a hole in a disc shape of radius $10$ centered at $(40,60)$. An illustration of the grid, with size $32\times32$, is shown in Figure \[figgrid\]. The matrix–vector multiplication is performed by first doing the multiplication using the full grid via circulant embedding and FFT, followed by removing the entries corresponding to the hole of the grid. Recall that the covariance model is defined in Section \[secnonlinearsolver\], along with the explanation of the filtering step. ![A $32\times32$ grid with a region of missing observations in a disc shape. Internal grid points are grouped to work with the dependent design in Section \[secdependent\].[]{data-label="figgrid"}](627f03.eps) When working with dependent samples, it is advantageous to group nearby grid points such that the resulting blocks have a plump shape and that there are as many blocks with size exactly $N$ as possible. For an occluded grid, this is a nontrivial task. Here we use a simple heuristic to effectively group the points. We divide the grid into horizontal stripes of width $\lfloor\sqrt{N}\rfloor$ (in case $\lfloor\sqrt{N}\rfloor$ does not divide the grid size along the vertical direction, some stripes have a width $\lfloor\sqrt{N}\rfloor+1$). The stripes are ordered from bottom to top, and the grid points inside the odd-numbered stripes are ordered lexicographically in their coordinates, that is, $(x,y)$. In order to obtain as many contiguous blocks as possible, the grid points inside the even-numbered stripes are ordered lexicographically according to $(-x,y)$. This ordering gives a zigzag flow of the points starting from the bottom-left corner of the grid. Every $N$ points are grouped in a block. The coloring of the grid points in Figure \[figgrid\] shows an example of the grouping. Note that because of filtering, observations on either an external or internal boundary are not part of any block. Choice of $N$ {#sec5.1} ------------- One of the most important factors that affect the efficacy of approximating the score equations is the value $N$. Theorem \[tmain\] indicates that $N$ should increase at least like $\kappa(K)$ in order to guarantee the additional uncertainty introduced by approximating the score equations be comparable with that caused by the randomness of the sample $Z$. In the ideal case, when the condition number of the matrix (possibly with filtering) is bounded independent of the matrix size $n$, then even taking $N=1$ is sufficient to obtain estimates with the same rate of convergence as the exact score equations. When $\kappa$ grows with $n$, however, a better guideline for selecting $N$ is to consider the growth of $\mathcal{I}^{-1}\mathcal{J}$. Figure \[figpowerbounds\] plots the condition number of $K$ and the spectral norm of $\mathcal{I}^{-1}\mathcal{J}$ for varying sizes of the matrix and preconditioning using the Laplacian filter. Although performing a Laplacian filtering will yield provably bounded condition numbers only for the case $\alpha=2$, one sees that the filtering is also effective for the cases $\alpha=1$ and $1.5$. Moreover, the norm of $\mathcal {I}^{-1}\mathcal{J}$ is significantly smaller than $\kappa$ when $n$ is large and, in fact, it does not seem to grow with $n$. This result indicates the bound in Theorem 1 is sometimes far too conservative and that using a fixed $N$ can be effective even when $\kappa$ grows with $n$. ![Growth of $\kappa$ compared with that of $\|\mathcal {I}^{-1}\mathcal{J}\|$, for power law kernel in 2D. Left: $\alpha=1$; right: $\alpha=1.5$.[]{data-label="figpowerbounds"}](627f04.eps) Of course, the norm of $\mathcal{I}^{-1}\mathcal{J}$ is not always bounded. In Figure \[figmaternbounds\] we show two examples using the Matérn covariance kernel with smoothness parameter $\nu=1$ and 1.5 (essentially $\alpha=2$ and 3). Without filtering, both $\kappa(K)$ and $\|\mathcal{I}^{-1}\mathcal{J}\|$ grow with $n$, although the plots show that the growth of the latter is significantly slower than that of the former. ![Growth of $\kappa$ compared with that of $\|\mathcal {I}^{-1}\mathcal{J}\|$, for Matérn kernel in 1D, without filtering. Left: $\nu=1$; right: $\nu=1.5$.[]{data-label="figmaternbounds"}](627f05.eps) If the occluded observations are more scattered, then the fast matrix–vector multiplication based on circulant embedding still works fine. However, if the occluded pixels are randomly located and the fraction of occluded pixels is substantial, then using a filtered data set only including Laplacians centered at those observations whose four nearest neighbors are also available might lead to an unacceptable loss of information. In this case, one might instead use a preconditioner based on a sparse approximation to the inverse Cholesky decomposition as described in Section \[sec6\]. A 32x32 grid example {#sec5.2} -------------------- Here, we show the details of solving the estimating equations (\[ascore\]) using a $32\times32$ grid as an example. Setting the truth $\alpha =1.5$ and $\ell=(7,10)$ \[i.e., $\theta=(1.5, 7, 10)$\], consider exact and approximate maximum likelihood estimation based on the data obtained by applying the Laplacian filter once to the observations. Writing $\mathcal{G}$ for $\mathcal{E}\{g(\theta,N)\}$, one way to evaluate the approximate MLEs is to compute the ratios of the square roots of the diagonal elements of $\mathcal{G}^{-1}$ to the square roots of the diagonal elements of $\mathcal{I}^{-1}$. We know these ratios must be at least 1, and that the closer they are to 1, the more nearly optimal the resulting estimating equations based on the approximate score function are. For $N=64$ and independent sampling, we get 1.0156, 1.0125 and 1.0135 for the three ratios, all of which are very close to 1. Since one generally cannot calculate $\mathcal{G}^{-1}$ exactly, it is also worthwhile to compare a stochastic approximation of the diagonal values of $\mathcal{G}^{-1}$ to their exact values. When this approximation was done once for $N=64$ and by using $N_2=100$ in (\[Iij\]) and (\[Jij2\]), the three ratios obtained were 0.9821, 0.9817 and 0.9833, which are all close to 1. ![Effects of $N$ (1, 2, 4, 8, 16, 32, 64). In each plot, the curve with the plus sign corresponds to the independent design, whereas that with the circle sign corresponds to the dependent design. The horizontal axis represents $N$. In plots , and , the vertical axis represents the mean squared differences between the approximate and exact MLEs divided by the mean squared errors for the exact MLEs, for the components $\alpha$, $\ell_1$ and $\ell_2$, respectively. In plot , the vertical axis represents the mean squared difference between the loglikelihood values at the exact and approximate MLEs.[]{data-label="figN"}](627f06.eps) Figure \[figN\] shows the performance of the resulting estimates (to be compared with the exact MLEs obtained by solving the standard score equations). For $N=1$, $2$, $4$, $8$, $16$, $32$ and $64$, we simulated 100 realizations of the process on the $32\times32$ occluded grid, applied the discrete Laplacian once, and then computed exact MLEs and approximations using both independent and dependent (as described in the beginning of Section \[secexp\]) sampling. When $N=1$, the independent and dependent sampling schemes are identical, so only results for independent sampling are given. Figure \[figN\] plots, for each component of $\theta$, the mean squared differences between the approximate and exact MLEs divided by the mean squared errors for the exact MLEs. As expected, these ratios decrease with $N$, particularly for dependent sampling. Indeed, dependent sampling is much more efficient than independent sampling for larger $N$; for example, the results in Figure \[figN\] show that dependent sampling with $N=32$ yields better estimates for all three parameters than does independent sampling with $N=64$. Large-scale experiments {#sec5.3} ----------------------- We experimented with larger grids (in the same physical range). We show the results in Table \[tablargescale\] and Figure \[figlargescale\] for $N=64$. When the matrix becomes large, we are unable to compute $\mathcal{I}$ and $\mathcal{G}$ exactly. Based on the preceding experiment, it seems reasonable to use $N_2=100$ in approximating $\mathcal{I}$ and $\mathcal{G}$. Therefore, the elements of $\mathcal{I}$ and $\mathcal{G}$ in Table \[tablargescale\] were computed only approximately. [@lcd[2.4]{}d[2.4]{}d[2.4]{}cc@]{} **Grid size** & & & & & &\ ${\widehat}{\theta}^N$ & 1.5355 & 1.5084 & 1.4919 & 1.4975 & 1.5011 & 1.5012\ & 6.8507 & 6.9974 & 7.1221 & 7.0663 & 6.9841 & 6.9677\ & 9.2923 & 10.062 & 10.091 & 10.063 & 9.9818 & 9.9600\ \[6pt\] $\sqrt{(\mathcal{I}^{-1})_{ii}}$ & 0.0882 & 0.0406 & 0.0196 & 0.0096 & 0.0048 & 0.0024\ & 0.5406 & 0.3673 & 0.2371 & 0.1464 & 0.0877 & 0.0512\ & 0.8515 & 0.5674 & 0.3605 & 0.2202 & 0.1309 & 0.0760\ \[6pt\] $\frac{\sqrt{(\mathcal{G}^{-1})_{ii}}}{\sqrt{(\mathcal{I}^{-1})_{ii}}}$ & 1.0077 & 1.0077 & 1.0077 & 1.0077 & 1.0077 & 1.0077\ & 1.0062 & 1.0070 & 1.0073 & 1.0074 & 1.0075 & 1.0076\ & 1.0064 & 1.0071 & 1.0073 & 1.0075 & 1.0075 & 1.0076\ ![Running time for increasingly dense grids. The dashed curve fits the recorded times with a function of the form of $n\log n$ times a constant.[]{data-label="figlargescale"}](627f07.eps) One sees that as the grid becomes larger (denser), the variance of the estimates decreases as expected. The matrices $\mathcal{I}^{-1}$ and $\mathcal{G}^{-1}$ are comparable in all cases and, in fact, the ratios stay roughly the same across different sizes of the data. The experiments were run for data size up to around one million, and the scaling of the running time versus data size is favorable. The results show a strong agreement of the recorded times with the scaling $O(n\log n)$. Application {#sec6} =========== Ozone in the stratosphere blocks ultraviolet radiation from the sun and is thus essential to all land-based life on Earth. Satellite-based instruments run by NASA have been measuring total column ozone in the atmosphere daily on a near global scale since 1978 (although with a significant gap in 1994–1996) and the present instrument is the OMI. Here, we consider Level 3 gridded data for the month April 2012 in the latitude band $40^\circ$–$50^\circ$N \[Aura OMI Ozone Level-3 Global Gridded ($1.0\times1.0$ deg) Data Product-OMTO3d (V003)\]. Because total column ozone shows persistent patterns of variation with location, we demeaned the data by, for each pixel, subtracting off the mean of the available observations during April 2012. Figure \[figozone\] displays the resulting demeaned data. There are potentially $360\times10 = 3600$ observations on each day in this latitude strip. However, Figure \[figozone\] shows 14 or 15 strips of missing observations each day, which is due to a lack of overlap in OMI observations between orbits in this latitude band (the orbital frequency of the satellite is approximately 14.6 orbits per day). Furthermore, there is nearly a full day of missing observations toward the end of the record. For the 30-day period, a complete record would have 108,000 observations, of which 84,942 are available. The local time of the Level 2 data on which the Level 3 data are based is generally near noon due to the sun-synchronous orbit of the satellite, but there is some variation in local time of Level 2 data because OMI simultaneously measures ozone over a swath of roughly 3000 km, so that the actual local times of the Level 2 data vary up to about 50 minutes from local noon in the latitude band we are considering. Nevertheless, @fang showed that, for Level 3 total column ozone levels (as measured by a predecessor instrument to the OMI), as long as one stays away from the equator, little distortion is caused by assuming all observations are taken at exactly local noon and we will make this assumption here. As a consequence, within a given day, time (absolute as opposed to local) and longitude are completely confounded, which makes distinguishing longitudinal and temporal dependencies difficult. Indeed, if one analyzed the data a day at a time, there would be essentially no information for distinguishing longitude from time, but by considering multiple days in a single analysis, it is possible to distinguish their influences on the dependence structure. Fitting various Matérn models to subsets of the data within a day, we found that the local spatial variation in the data is described quite well by the Whittle model (the Matérn model with smoothness parameter 1) without a nugget effect. Results in @stein07 suggest some evidence for spatial anisotropy in total column ozone at midlatitudes, but the anisotropy is not severe in the band $40^\circ$–$50^\circ$N and we will ignore it here. The most striking feature displayed in Figure \[figozone\] is the obvious westerly flow of ozone across days. Based on these considerations, we propose the following simple model for the demeaned data $Z(\mathbf{x},t)$. Denoting by $r$ the radius of the Earth, $\varphi$ the latitude, $\psi $ the longitude, and $t$ the time, we assume $Z$ is a 0 mean Gaussian process with covariance function (parameterized by $\theta_0$, $\theta_1$, $\theta_2$ and $v$): $${\operatorname{cov}}\bigl\{Z(\mathbf{x}_1, t_1), Z(\mathbf{x}_2, t_2)\bigr\}= \theta_0{\mathrm{M}}_1 \biggl(\sqrt {\frac{T^2}{\theta_1^2}+\frac {S^2}{\theta_2^2}} \biggr),$$ where $T=t_1-t_2$ is the temporal difference, $S=\|\mathbf{x}(r,\varphi_1, \psi_1-vt_1)-\mathbf{x}(r,\varphi_2,\allowbreak \psi _2-vt_2)\|$ is the (adjusted for drift) spatial difference and $\mathbf{x}(r,\varphi, \psi)$ maps a spherical coordinate to ${{\mathbb R}}^3$. Here, ${\mathrm{M}}_{\nu}$ is the Matérn correlation function $$\label{spacetime} {\mathrm{M}}_{\nu}(x)=\frac{(\sqrt{2\nu}x)^{\nu}{\mathrm{K}}_{\nu}(\sqrt{2\nu }x)}{2^{\nu-1}\Gamma(\nu)}$$ with ${\mathrm{K}}_{\nu}$ the modified Bessel function of the second kind of order $\nu$. We used the following unit system: $\varphi$ and $\psi$ are in degrees, $t$ is in days, and $r\equiv1$. In contravention of standard notation, we take longitude to increase as one heads westward in order to make longitude increase with time within a day. Although the use of Euclidean distance in $S$ might be viewed as problematic \[@gneiting\], it is not clear that great circle distances are any more appropriate in the present circumstance in which there is strong zonal flow. The model (\[spacetime\]) has the virtues of simplicity and of validity: it defines a valid covariance function on the sphere${}\times{}$time whenever $\theta_0,\theta_1$ and $\theta_2$ are positive. A more complex model would clearly be needed if one wanted to consider the process on the entire globe rather than in a narrow latitude band. Because the covariance matrix $K(\theta_0,\theta_1,\theta_2,v)$ can be written as $\theta_0M(\theta_1,\break\theta_2,v)$, where the entries of $M$ are generated by the Matérn function, the estimating equations (\[ascore\]) give ${\widehat}\theta_0=Z'M({\widehat}{\theta}_1,{\widehat}{\theta}_2,{\widehat}{v})^{-1}Z/n$ as the MLE of $\theta_0$ given values for the other parameters. Therefore, we only need to solve (\[ascore\]) with respect to $\theta_1$, $\theta_2$ and $v$. Initial values for the parameters were obtained by applying a simplified fitting procedure to a subset of the data. We first fit the model using observations from one latitude at a time. Since there are about 8500 observations per latitude band, it is possible, although challenging, to compute the exact MLEs for the observations within a single band using the Cholesky decomposition. However, we chose to solve (\[ascore\]) with the number $N$ of i.i.d. symmetric Bernoulli vectors $U_j$ fixed at 64. A first order finite difference filtering \[@steinchenanitescufiltering\] was observed to be the most effective in encouraging the convergence of the linear solve. Differences across gaps in the data record were included, so the resulting sizes of the filtered data sets were just one less than the number of observations available in each longitude. Under our model, the covariance matrix of the observations within a latitude can be embedded in a circulant matrix of dimension 21,600, greatly speeding up the necessary matrix–vector multiplications. Table \[tabozoneonelat\] summarizes the resulting estimates and the Fisher information for each latitude band. The estimates are consistent across latitudes and do not show any obvious trends with latitude except perhaps at the two most northerly latitudes. The estimates of $v$ are all near $-7.5^\circ$, which qualitatively matches the westerly flow seen in Figure \[figozone\]. The differences between $\sqrt{(\mathcal{G}^{-1})_{ii}}/\sqrt{(\mathcal {I}^{-1})_{ii}}$ and $1$ were all less than $0.01$, indicating that the choice of $N$ is sufficient. [@lccd[2.3]{}ccccc@]{} & & & & &\ &&&&&\ **Latitude** & $\bolds{{\widehat}{\theta}_0^N}$ [$\bolds{(\times}$**10**$\bolds{^3)}$]{} & $\bolds{{\widehat}{\theta}_1^N}$ & & $\bolds{{\widehat}{v}^N}$ & [$\bolds{(\times}$**10**$\bolds{^3)}$]{} & & &\ $40.5^\circ$N & 1.076 & 2.110 & 11.466 & $-6.991$ & 0.106 & 0.127 & 0.586 & 0.244\ $41.5^\circ$N & 1.182 & 2.172 & 11.857 & $-6.983$ & 0.123 & 0.136 & 0.634 & 0.251\ $42.5^\circ$N & 1.320 & 2.219 & 12.437 & $-7.118$ & 0.144 & 0.145 & 0.698 & 0.266\ $43.5^\circ$N & 1.370 & 2.107 & 12.104 & $-7.369$ & 0.145 & 0.136 & 0.660 & 0.285\ $44.5^\circ$N & 1.412 & 2.059 & 11.845 & $-7.368$ & 0.145 & 0.130 & 0.628 & 0.294\ $45.5^\circ$N & 1.416 & 2.010 & 11.814 & $-7.649$ & 0.147 & 0.128 & 0.632 & 0.313\ $46.5^\circ$N & 1.526 & 2.075 & 12.254 & $-8.045$ & 0.166 & 0.138 & 0.686 & 0.320\ $47.5^\circ$N & 1.511 & 2.074 & 11.939 & $-7.877$ & 0.161 & 0.135 & 0.654 & 0.319\ $48.5^\circ$N & 1.325 & 1.887 & 10.134 & $-7.368$ & 0.128 & 0.114 & 0.505 & 0.303\ $49.5^\circ$N & 1.246 & 1.846 & 9.743 & $-7.120$ & 0.117 & 0.110 & 0.473 & 0.305\ The following is an instance of the asymptotic correlation matrix, obtained by normalizing each entry of $\mathcal{I}^{-1}$ (at $49.5^\circ$N) with respect to the diagonal: $$\left[ \matrix{\hphantom{-}1.0000 & \hphantom{-}0.8830 & \hphantom{-}0.9858 & -0.0080 \cr \hphantom{-}0.8830 & \hphantom{-}1.0000 & \hphantom{-}0.8767 & -0.0067 \cr \hphantom{-}0.9858 & \hphantom{-}0.8767 & \hphantom{-}1.0000 & -0.0238 \cr -0.0080 & -0.0067 & -0.0238 & \hphantom{-}1.0000 } \right].$$ We see that ${\widehat}\theta_0,{\widehat}\theta_1$ and ${\widehat}\theta_2$ are all strongly correlated. The high correlation of the estimated range parameters ${\widehat}\theta_1$ and ${\widehat}\theta_2$ with the estimated scale ${\widehat}\theta_0$ is not unexpected considering the general difficulty of distinguishing scale and range parameters for strongly correlated spatial data \[@zhang\]. The strong correlation of the two range parameters is presumably due to the near confounding of time and longitude for these data. Next, we used the data at all latitudes and progressively increased the number of days. In this setting, the covariance matrix of the observations can be embedded in a block circulant matrix with blocks of size $10\times10$ corresponding to the 10 latitudes. Therefore, multiplication of the covariance matrix times a vector can be accomplished with a discrete Fourier transform for each pair of latitudes, or ${10 \choose2} = 55$ discrete Fourier transforms. Because we are using the Whittle covariance function as the basis of our model, we had hoped filtering the data using the Laplacian would be an effective preconditioner. Indeed, it does well at speeding the convergence of the linear solves, but it unfortunately appears to lose most of the information in the data for distinguishing spatial from temporal influences, and thus is unsuitable for these data. Instead, we used a banded approximate inverse Cholesky factorization \[@kolo, (2.5), (2.6)\] to precondition the linear solve. Specifically, we ordered the observations by time and then, since observations at the same longitude and day are simultaneous, by latitude south to north. We then obtained an approximate inverse by subtracting off the conditional mean of each observation given the previous 20 observations, so the approximate Cholesky factor has bandwidth 21. We tried values besides 20 for the number of previous observations on which to condition, but 20 seemed to offer about the best combination of fast computing and effective preconditioning. The number $N$ of i.i.d. symmetric Bernoulli vectors $U_j$ was increased to 128, in order that the differences between $\sqrt{(\mathcal{G}^{-1})_{ii}}/ \sqrt{(\mathcal{I}^{-1})_{ii}}$ and $1$ were around $0.1$. The results are summarized in Table \[tabozonealllat\]. One sees that the estimates are reasonably consistent with those shown in Table \[tabozoneonelat\]. Nevertheless, there are some minor discrepancies such as estimates of $v$ that are modestly larger (in magnitude) than found in Table \[tabozonealllat\], suggesting that taking account of correlations across latitudes changes what we think about the advection of ozone from day to day. [@lcccccccc@]{} & & & & &\ & & & & &\ **Days** & $\bolds{{\widehat}{\theta}_0^N}$ [$\bolds{(\times}$**10**$\bolds{^3)}$]{} & $\bolds{{\widehat}{\theta}_1^N}$ & $\bolds{{\widehat}{\theta }_2^N}$ & $\bolds{{\widehat}{v}^N}$ & [$\bolds{(\times}$**10**$\bolds{^3)}$]{} & & &\ \ 1–3 & $1.594$ & $2.411$ & $12.159$ & $-8.275$ & $0.362$ & $0.334$ & $1.398$ & $0.512$\ 1–10 & $1.301$ & $1.719$ & $11.199$ & $-8.368$ & $0.146$ & $0.121$ & $0.639$ & $0.407$\ 1–20 & $1.138$ & $1.774$ & $10.912$ & $-9.038$ & $0.090$ & $0.085$ & $0.436$ & $0.252$\ 1–30 & $1.265$ & $1.918$ & $11.554$ & $-8.201$ & $0.089$ & $0.081$ & $0.414$ & $0.198$\ \[6pt\]\ 1–30 & $1.260$ & $1.907$ & $11.531$ & $-8.211$ & $0.088$ & $0.079$ & $0.406$ & $0.200$\ Note that the approximate inverse Cholesky decomposition, although not as computationally efficient as applying the discrete Laplacian, is a full rank transformation and thus does not throw out any statistical information. The method does require ordering the observations, which is convenient in the present case in which there are at most 10 observations per time point. Nevertheless, we believe this approach may be attractive more generally, especially for data that are not on a grid. We also estimated the parameters using the dependent sampling scheme described in Section \[sec3\] with $N=128$ and obtained estimates given in the last row of Table \[tabozonealllat\]. It is not as easy to estimate $B^d$ as defined in Theorem \[tdependent\] as it is to estimate $B$ with independent $U_j$’s. We have carried out limited numerical calculations by repeatedly calculating $g^d({\widehat}{\theta},N)$ for ${\widehat}{\theta}$ fixed at the estimates for dependent samples of size $N=128$ and have found that the advantages of using the dependent sampling are negligible in this case. We suspect that the reason the gains are not as great as those shown in Figure \[figN\] is due to the substantial correlations of observations that are at similar locations a day apart. Discussion {#sec7} ========== We have demonstrated how derivatives of the loglikelihood function for a Gaussian process model can be accurately and efficiently calculated in situations for which direct calculation of the loglikelihood itself would be much more difficult. Being able to calculate these derivatives enables us to find solutions to the score equations and to verify that these solutions are at least local maximizers of the likelihood. However, if the score equations had multiple solutions, then, assuming all the solutions could be found, it might not be so easy to determine which was the global maximizer. Furthermore, it is not straightforward to obtain likelihood ratio statistics when only derivatives of the loglikelihood are available. Perhaps a more critical drawback of having only derivatives of the loglikelihood occurs when using a Bayesian approach to parameter estimation. The likelihood needs to be known only up to a multiplicative constant, so, in principle, knowing the gradient of the loglikelihood throughout the parameter space is sufficient for calculating the posterior distribution. However, it is not so clear how one might calculate an approximate posterior based on just gradient and perhaps Hessian values of the loglikelihood at some discrete set of parameter values. It is even less clear how one could implement an MCMC scheme based on just derivatives of the loglikelihood. Despite this substantial drawback, we consider the development of likelihood methods for fitting Gaussian process models that are nearly $O(n)$ in time and, perhaps more importantly, $O(n)$ in memory, to be essential for expanding the scope of application of these models. Calling our approach nearly $O(n)$ in time admittedly glosses over a number of substantial challenges. First, we need to have an effective preconditioner for the covariance matrix $K$. This allows us to treat $N$, the number of random vectors in the stochastic trace estimator, as a fixed quantity as $n$ increases and still obtain estimates that are nearly as efficient as full maximum likelihood. The availability of an effective preconditioner also means that the number of iterations of the iterative solve can remain bounded as $n$ increases. We have found that $N=100$ is often sufficient and that the number of iterations needed for the iterative solver to converge to a tight tolerance can be several dozen, so writing $O(n)$ can hide a factor of several thousand. Second, we are assuming that matrix–vector multiplications can be done in nearly $O(n)$ time. This is clearly achievable when the number of nonzero entries in $K$ is $O(n)$ or when observations form a partial grid and a stationary model is assumed so that circulant embedding applies. For dense, unstructured matrices, fast multipole methods can achieve this rate, but the method is only approximate and the overhead in the computations is substantial so that $n$ may need to be very large for the method to be faster than direct multiplication. However, even when using exact multiplication, which requires $O(n^2)$ time, despite the need for $N$ iterative solves, our approach may still be faster than computing the Cholesky decomposition, which requires $O(n^3)$ computations. Furthermore, even when $K$ is dense and unstructured, the iterative algorithm is $O(n)$ in memory, assuming that elements of $K$ can be calculated as needed, whereas the Cholesky decomposition requires $O(n^2)$ memory. Thus, for example, for $n$ in the range 10,000–100,000, even if $K$ has no exploitable structure, our approach to approximate maximum likelihood estimation may be much easier to implement on the current generation of desktop computers than an approach that requires calculating the Cholesky decomposition of $K$. The fact that the condition number of $K$ affects both the statistical efficiency of the stochastic trace approximation and the number of iterations needed by the iterative solver indicates the importance of having good preconditioners to make our approach effective. We have suggested a few possible preconditioners, but it is clear that we have only scratched the surface of this problem. Statistical problems often yield covariance matrices with special structures that do not correspond to standard problems arising in numerical analysis. For example, the ozone data in Section \[sec6\] has a partial confounding of time with longitude that made Laplacian filtering ineffective as a preconditioner. Further development of preconditioners, especially for unstructured covariance matrices, will be essential to making our approach broadly effective. \[app\] Appendix: Proofs {#appendix-proofs .unnumbered} ================ [Proof of Theorem \[tmain\]]{} Since $K$ is positive definite, it can be written in the form $S\Lambda S'$ with $S$ orthogonal and $\Lambda$ diagonal with elements $\lambda_1\ge\cdots\ge\lambda_n>0$. Then $Q^i:= S' K_i S$ is symmetric, $$\label{firstterm}\quad \operatorname{tr}\bigl(W^i W^j\bigr) = \operatorname{tr} \bigl(S'K^{-1}SS'K_iSS'K^{-1}SS'K_jS \bigr) = \operatorname{tr}\bigl(\Lambda^{-1}Q^i\Lambda^{-1}Q^j \bigr)$$ and, similarly, $$\label{secondterm} \operatorname{tr} \bigl\{W^i \bigl(W^j\bigr)' \bigr\} = \operatorname{tr}\bigl(\Lambda^{-1}Q^iQ^j \Lambda^{-1}\bigr).$$ For real $v_1,\ldots,v_p$, $$\label{lastterm} \sum_{i,j=1}^p v_iv_j \sum_{k=1}^n W_{kk}^iW_{kk}^j = \sum_{k=1}^n \Biggl\{\sum _{i=1}^p v_iW_{kk}^i \Biggr\}^2 \ge0.$$ Furthermore, by (\[firstterm\]), $$\label{firstquad} \sum_{i,j=1}^p v_iv_j \operatorname{tr}\bigl(W^i W^j\bigr) = \sum _{k,\ell=1}^n \frac{1}{\lambda_k\lambda_\ell} \Biggl\{\sum _{i=1}^p v_iQ^i_{k,\ell} \Biggr\}^2$$ and, by (\[secondterm\]), $$\label{secondquad} \sum_{i,j=1}^p v_iv_j \operatorname{tr} \bigl\{W^i \bigl(W^j\bigr)' \bigr \} = \sum_{k,\ell=1}^n \frac{1}{\lambda_k^2} \Biggl \{\sum_{i=1}^p v_iQ^i_{k,\ell} \Biggr\}^2.$$ Write $\gamma_{k\ell}$ for $\sum_{i=1}^p v_iQ^i_{k,\ell}$ and note that $\gamma_{k\ell}=\gamma_{\ell k}$. Consider finding an upper bound to $$\frac{\sum_{i,j=1}^p v_iv_j\operatorname{tr} \{W^i (W^j)' \}} { \sum_{i,j=1}^p v_iv_j\operatorname{tr}(W^i W^j)} = \frac{\sum_{k=1}^n {\gamma_{kk}^2}/{\lambda_k^2} + \sum_{k>\ell} \gamma_{k\ell}^2 ({1}/{\lambda_k^2} + {1}/{\lambda _\ell^2} )} { \sum_{k=1}^n {\gamma_{kk}^2}/{\lambda_k^2} + \sum_{k>\ell} {2\gamma_{k\ell}^2}/{\lambda_k\lambda_\ell}}.$$ Think of maximizing this ratio as a function of the $\gamma_{k\ell}^2$’s for fixed $\lambda_k$’s. We then have a ratio of two positively weighted sums of the same positive scalars (the $\gamma_{k\ell}^2$’s for $k\ge\ell$), so this ratio will be maximized if the only positive $\gamma_{k\ell}^2$ values correspond to cases for which the ratio of the weights, here $$\label{ratioweights} \frac{{1}/{\lambda_k^2}+{1}/{\lambda_\ell^2}}{ {2}/({\lambda_k \lambda_\ell})} = \frac{1+ ({\lambda_k}/{\lambda_\ell } )^2} { {2\lambda_k}/{\lambda_\ell}}$$ is maximized. Since we are considering only $k\ge\ell$, $\frac{\lambda_k}{\lambda _\ell} \ge1$ and $\frac{1+x^2}{2x}$ is increasing on $[1,\infty)$, so (\[ratioweights\]) is maximized when $k=n$ and $\ell=1$, yielding $$\frac{\sum_{i,j=1}^p v_iv_j\operatorname{tr} \{W^i (W^j)' \}} { \sum_{i,j=1}^p v_iv_j\operatorname{tr}(W^i W^j)} \le\frac{\kappa(K)^2 +1}{2\kappa(K)}.$$ The theorem follows by putting this result together with (\[B\]), (\[Jij\]) and (\[lastterm\]). [Proof of Theorem \[tdependent\]]{} Define $\beta_{ia}$ to be the $a$th element of $\beta_i$ and $X_{\ell a}$ the $a$th diagonal element of $X_\ell$. Then note that for $k\ne\ell$ and $k'\ne\ell'$ and $a,b\in\{1,\ldots,N\}$, $$\begin{aligned} & & (U_{i,(k-1)N+a}U_{i,(\ell-1)N+b}, U_{j,(k'-1)N+a'}U_{j,(\ell'-1)N+b'}) \\ & &\qquad = (\beta_{ia}\beta_{ib}Y_{ik}X_{k a}Y_{i\ell}X_{\ell b}, \beta_{ja'}\beta_{jb'}Y_{jk'}X_{k' a'}Y_{j\ell'}X_{\ell'b'})\end{aligned}$$ have the same joint distribution as for independent $U_j$’s. Specifically, the two components are independent symmetric Bernoulli random variables unless $i=j, a=a', b=b'$ and $k=k'\ne\ell=\ell'$ or $i=j,a=b',b=a'$ and $k=\ell'\ne\ell=k'$, in which case they are the same symmetric Bernoulli random variable. Straightforward calculations yield (\[improve\]). Acknowledgments {#acknowledgments .unnumbered} =============== The data used in this effort were acquired as part of the activities of NASAs Science Mission Directorate, and are archived and distributed by the Goddard Earth Sciences (GES) Data and Information Services Center (DISC). [44]{} , (). . . , (). . . (). . . (). . . (). . . , (). , ed. , . (). . . (). . . , . (). . . , . , (). . . (). , ed. , . (). . . (). . . , , (). . . (). . . , (). . , . (). . . , (). . . (). . . (). . . . (). . . (). . . (). . , . (). . . , (). . . (). . . (). . . (). , ed. , . (). . . (). . . (). . , . (). . . (). . . (). . , (). . . , (). . . , (). . . (). . . (). . . (). . . (). . . (). . In . , . , , , , , (). . . [^1]: To facilitate understanding, we explain here the process for solving the score equations (\[score\]). Conceptually it is similar to that for solving the estimating equations (\[ascore\]).

--- abstract: 'We consider an optimization problem for spatial power distribution generated by an array of transmitting elements. Using ultrasound hyperthermia cancer treatment as a motivating example, the signal design problem consists of optimizing the power distribution across the tumor and healthy tissue regions, respectively. The models used in the optimization problem are, however, invariably subject to errors. deposition as well as inefficient treatment. To combat such unknown model errors, we formulate a robust signal design framework that can take the uncertainty into account using a worst-case approach. This leads to a semi-infinite programming (SIP) robust design problem which we reformulate as a tractable convex problem, potentially has a wider range of applications.' author: - 'Nafiseh Shariati, Dave Zachariah, Johan Karlsson, Mats Bengtsson' bibliography: - 'bibliokthNafis.bib' title: Robust Optimal Power Distribution for Hyperthermia Cancer Treatment --- Introduction {#sec:introduction} ============ Local hyperthermia is a noninvasive technique for cancer treatment in which targeted body tissue is exposed to high temperatures to damage cancer cells while leaving surrounding tissue unharmed. This technique is used both to kill off cancer cells in tumors and as a means to enhance other treatments such as radiotherapy and chemotherapy. Hyperthermia has the potential to treat many types of cancer, including sarcoma, melanoma, and cancers of the head and neck, brain, lung, esophagus, breast, bladder, rectum, liver, appendix, cervix, etc. . Hyperthermia treatment planning involves modeling patient-specific tissue, using medical imaging techniques such as microwave, ultrasound, magnetic resonance or computed tomography, and calculating the spatial distribution of power deposited in the tissue to heat it [@13:; @Paulides]. There exist two major techniques to concentrate the power in a well-defined tumor region: electromagnetic and ultrasound, each with its own limitations. The drawback of electromagnetic microwaves is poor penetration in biological tissue, while for ultrasound the short acoustic wavelength renders the focal spot very small. Using signal design methods, however, one can improve the spatial power deposition generated by an array of acoustic transducers. Specifically, standard phased array techniques do not make use of combining a diversity of signals transmitted at each transducer. When this diversity exploited it is possible to dramatically improve the power distribution in the tumor tissue, thus improving the effectiveness of the method and reducing treatment time . Given a set of spatial coordinates that describe the tumor region and the healthy tissue, respectively, the transmitted waveforms can be designed to optimize the spatial power distribution while subject to certain design constraints. One critical limitation, however, is the assumption of an ideal wave propagation model from the transducers to a given point in the tissue. Specifically, model mismatches may arise from hardware imperfections, tissue inhomogeneities, inaccurately specified propagation velocities, etc. Thus the actual power distribution may differ substantially from the ideal one designed by an assumed model. This results in suboptimal clinical outcome due to loss of power in the tumor region and safety issues due to the possible damage of healthy tissue. These considerations motivate developing robust design schemes that take such unknown errors into account. In this paper we derive a robust optimization method that only assumes the unknown model errors to be bounded. The power is then optimized with respect to ‘worst-case’ model errors. By using a worst-case model, we provide an optimal signal design scheme that takes into account all possible, bounded model errors. Such a conservative approach is warranted in signal design for medical applications due to safety and health considerations. Our method further generalizes the approach in [@08:Guo] by obviating the need to specify a fictitious tumor center point. The framework developed here has potential use in wider signal design applications where the resulting transmit power distributions are subject to model inaccuracies. More specifically, the design problem formulated in this paper and the proposed robust scheme can be exploited to robustify the spatial power distribution for applications that an array equipped with multiple elements is used to emit waveforms in order to deliver power to an area of interest in a controlled manner. The core of this study is built upon exploiting waveform diversity which has been introduced in multiple-input multiple-output (MIMO) radar literature [@Stoica2007a], and later has been applied for local hyperthermia cancer treatment improvement in [@08:Guo]. In the MIMO radar field, robustness studies have been carried out in different applications under varying design parameter uncertainties, cf., [@07:Yang; @11:Grossi]. Recently, in [@14:Shariati2014c], we have studied the robustification of the waveform diversity methodology for MIMO radar applications. It should be highlighted that in this paper a more generic problem formulation has been studied with respect to those of [@14:Shariati2014c], where a new application area is considered to illustrate the performance of our proposed robust design. In the array processing literature, beamforming under array model errors has also spawned extensive work, cf., [@04:Li; @08:Yan; @08:Kim; @12:Khabbazibasmenj]. For hyperthermia therapy, the need for robust solutions when optimizing for phase and amplitude of conventional phased array, has been investigated in [@deGreef2010a] considering perfusion uncertainties, and in [@deGreef2011a] considering dielectric uncertainties. The authors emphasize on the role of uncertainty in such designs (hyperthermia planning) since it influences the calculation of power distribution, and correspondingly temperature distribution. The paper is organized as follows: In Section \[sec:system model\], we describe the system model and the relevant variables. In Section \[sec:problem formulation\], the signal design problem is presented. First, we consider the state-of-the-art method based on ‘waveform diversity’ [@08:Guo; @Stoica2007a; @JLi2009a], then we generalize the design problem by introducing a deterministic and bounded set of possible model errors which results in an infinite number of constraints. Importantly we show that this seemingly intractable problem can be equivalently formulated as a tractable convex optimization problem. In Section \[sec:numerical\_results\], we evaluate the design scheme. We evaluate the performance of our proposed robust power distribution scheme specifically for local hyperthermia breast cancer treatment. This example application is motivated by the alarming statistics pointing to breast cancer as one of the leading causes of death among women worldwide [@CancerReport2014UK; @CancerReport2014US; @CancerReport2013French][^1]. The case of no model mismatch is investigated first, and then the robust design scheme is applied where its power distribution in the worst-case model is evaluated and compared to the nonrobust formulation. *Notation:* Boldface (lower case) is used for column vectors, $\mathbf{x}$, and (upper case) for matrices, $\mathbf{X}$. $ \| \a \|_{\mathbf{W}} \triangleq \sqrt{\a^H {\mathbf{W}} \a}$ where ${\mathbf{W}} \succ {\mathbf{0}}$. ${\mathbf{x}}^T$ and ${\mathbf{x}}^H$ denote transpose and Hermitian transpose. ${\mathbf{R}} \succeq {\mathbf{0}}$ signifies positive semidefinite matrix and ${\mathbf{R}}^{1/2}$ a matrix square-root, e.g., Hermitian. The set of complex numbers is denoted by $\mathcal{C}$. *Abbreviations:* Semi-infinite programming (SIP); multiple-input multiple-output (MIMO); Semidefinite program (SDP); linear matrix inequality (LMI). system model {#sec:system model} ============ We consider an array of $M$ acoustic transducers to heat target points. These transducers are located at known positions $\th_m$, for $m = 1,2,\dots,M$, around the tissue at risk, cf., [@08:Guo; @14:Shariati2014c]. We parameterize an arbitrary point in 3D space using Cartesian coordinates $\mathbf{r} = [x \: y \: z]^T$. Let $x_m(n)$ denote the baseband representation of narrowband discrete-time signal transmitted at the $m$th transducer, at sample $n = 1, \dots, N$. Then the baseband signal received at a generic location $\r$ equals the superposition of signals from all $M$ transducers, i.e., $$\label{eq:baseband_recieved} \begin{aligned} y(\r,n) &= \sum_{m=1}^M a_m(\r) x_m(n) , \hspace{.25cm} n = 1, \ldots, N& \\ &= \a^H(\r) \x(n) , \hspace{.25cm} n = 1, \ldots, N,& \end{aligned}$$ where the $m$th signal is attenuated by a factor $a_m({\mathbf{r}})$ which depends on the properties of the transducers, the carrier wave and the tissue. This factor is modeled as $$\label{eq:a_m(r)} a_m(\r) = \frac{e^{-j2\pi f_c \tau_m(\r)}}{\|\th_m - \r\|^{\frac{1}{2}}},$$ where $f_c$ is the carrier frequency, and $\tau_m(\r) = \frac{\|\th_m - \r\|}{c}$ is the required time for any signal to arrive at location $\r$ where $c$ is the sound speed inside the tissue. Note that the root-squared term in the denominator in represents the distance dependent propagation attenuation of the acoustic waveforms. In , the narrowband signals are represented in vector form $\mathbf{x}(n) = [x_1(n) \: \ldots \: x_m(n) \: \ldots \: x_M(n)]^T \in \mathcal{C}^{M \times 1}$ and $\a({\mathbf{r}}) \triangleq [a_1(\r) \: \ldots \: a_m(\r) \: \ldots a_M(\r)]^T \in \mathcal{C}^{M \times 1}$ is the array steering vector as a function of $\r$. At a generic location $\r$ in the tissue, the power of the transmitted signal, i.e., *the transmit beampattern*, is given by $$\label{eq:transmit_beampattren} p(\r) = \mathbb{E} \{ | y(\r,n) |^2 \} = \a^H(\r)\R \a(\r),$$ where $$\R \triangleq \mathbb{E} \{ \x(n) \x^H(n)\}$$ is the $M \times M$ covariance matrix of the signal $\x(n)$. As equation suggests, the transmit beampattern is dependent on the waveform covariance matrix $\R$ and the array steering vector $\a(\r)$. In the following we analyze how one can form and control the beampattern by optimizing the covariance matrix $\R$, so as to heat up the tumor region of the tissue while keeping the power deposition in the healthy tissue minimal. In this work, we consider schemes which allow for the lowest possible power leakage to the healthy area. Once an optimal covariance matrix $\R$ has been determined, the waveform signal $\x(n)$ can be synthesized accordingly. One simple approach is $\x(n) = \R^{1/2} \mathbf{w}(n)$, where $\mathbf{w}(n)$ is a sequence of independent random vectors with mean zero and covariance matrix $\mathbf{I}$. For detailed discussion see [@Stoica2007b; @08:Fuhrmann][@12:He ch. 14]. A significant challenge to this approach, however, is that the *true* steering vector $\a(\r)$ in does not exactly match the model in for a host of reasons: array calibration imperfections, variations in transducing elements, tissue inhomogeneities, inaccurately specified propagation velocity, etc. We will therefore consider the aforementioned design problem subject to model uncertainties in the array steering vector at any given point $\r$. We refer to this approach as robust waveform diversity. problem formulation {#sec:problem formulation} =================== The waveform-diversity-based technique [@04:Fuhrmann; @Stoica2007a; @Stoica2007b; @08:Guo; @08:Fuhrmann; @14:Shariati2014c] have been used for designing beampatterns subject to practical constraints. In general, we aim to control and shape the spatial power distribution at a set of target points while simultaneously minimizing power leakage in the remaining area. By exploiting a combination of different waveforms in , the degrees of freedom increase for optimizing the beampattern under constraints. After reviewing the standard waveform diversity approach, we focus on the practical scenario where the assumed array steering vector model is subject to perturbations. In the subsequent section, the proposed robust technique is evaluated by numerical simulations, comparing the performance with and without robustified solution under perturbed steering vectors. Waveform Diversity based Ultrasound System ------------------------------------------ In the MIMO radar literature, sidelobe minimization is a beampattern design problem that has been addressed by using the waveform diversity methodology, cf., [@04:Fuhrmann; @Stoica2007a; @Stoica2007b; @08:Fuhrmann]. This design problem can be thought of as an optimization problem where the probing waveforms covariance matrix $\R$ is the optimization variable to be chosen under positive semi-definiteness assumption and with a constraint on the total power. The waveform-diversity-based scheme for ultrasound system has been introduced and explained in detail in [@08:Guo] based on the transmit beampattern design technique for MIMO radar systems [@04:Fuhrmann; @Stoica2007a]. In the following we consider the practical power constraint where all array elements have the same power. Therefore, the covariance matrix $\R$ belongs to the following set $\mathcal{R}$: $$\mathcal{R} \triangleq \{\mathbf{R} \hspace{.1cm}|\hspace{.1cm} \mathbf{R}\succeq {\mathbf{0}} , R_{mm} = \frac{\gamma}{M}, m=1,2,...,M \},$$ where $\gamma$ is the total transmitted power and $R_{mm}$ is the $m$th diagonal element of $\R$ corresponding to the power emitted by $m$th transducer. The healthy tissue and the tumor regions are represented by two sets of discrete control points $\r$: $$\begin{split} \Omega_S &= \{ \r_1,\r_2,\ldots,\r_{N_S}\} \\ \Omega_T &= \{ \r_1,\r_2,\ldots,\r_{N_T} \}, \end{split}$$ where $N_S$ and $N_T$ denote the number of points in the healthy tissue region and the tumor regions, respectively. Without loss of generality, let $\r_0$ be a representative point which is taken to be the center of the tumor region $\Omega_T$. The objectives for this optimization problem can be summarized as follows: Design the waveform covariance matrix ${\mathbf{R}}$ so as to - maximize the gap between the power at the tumor center $\r_0$ and the power at the control points $\r$ in the healthy tissue region $\Omega_S$; - while guaranteeing a certain power level for control points $\r$ in the tumor region $\Omega_T$. Mathematically, this problem is formulated as (see [@08:Guo]) $$\label{eq:sidelobe min} \begin{aligned} \underset{\R,t}{\textrm{max}} \hspace{.5cm} &t& \\ \textrm{s.t.} \hspace{.5cm} &\a^H(\r_0) \R \a(\r_0) - \a^H(\r) \R \a(\r) \geq t, \forall \r \in \Omega_S &\\ & \a^H(\r) \R \a(\r) \geq (1 - \delta) \a^H(\r_0) \R \a(\r_0), \forall \r \in \Omega_T& \\ & \a^H(\r) \R \a(\r) \leq (1 + \delta) \a^H(\r_0) \R \a(\r_0), \forall \r \in \Omega_T&\\ &\R \in \mathcal{R}& \end{aligned}$$ where $t$ denotes the gap between the power at $\r_0$ and the power at the control points $\r$ in the healthy region $\Omega_S$. The parameter $\delta$ is introduced here to control the required certain power level at the control points in the tumor region. For instance, if we set $\delta = 0.1$, then we aim for having power at the tumor region $\Omega_T$ to be within 10% of $p(\r_0)$, i.e., the power at the tumor center. This is an SDP problem which can be solved efficiently in polynomial time using any SDP solver, e.g., `CVX` [@cvx2013; @gb08]. Robust Waveform Diversity based Ultrasound System ------------------------------------------------- The convex optimization problem and consequently its optimal solution, i.e., the optimal covariance matrix ${\mathbf{R}}$, are functions of the steering vectors $\a(\r)$. In practice, however, the assumed steering vector model used to optimize ${\mathbf{R}}$ is inaccurate. Hence using *nominal* steering vectors $\ahat(\r)$ based on an ideal model, in lieu of the unknown *true* steering vectors $\a(\r)$ in , may result in undesired beampatterns with low power at the tumor region and damaging power deposition in the healthy tissue region. Such health considerations in medical applications motivate an approach that is robust with respect to the worst-case model uncertainties. In order to formulate the robust design problem mathematically, we parameterize the steering vector uncertainties as follows. Let the true steering vector for the transducer array be $\mathbf{a(\r)} = \mathbf{\hat{a}(\r)} + \mathbf{\tilde{a}(\r)}$ where $\mathbf{\tilde{a}(\r)}$ is an unknown perturbation from the nominal steering vector. The deterministic perturbation at any generic point $\r$ belongs to uncertainty set $\mathcal{E}_{\r}$ that is bounded $$\begin{split} \mathcal{E}_{\r} \triangleq \{ \mathbf{\tilde{a}}(\r) \hspace{.2cm} | \hspace{.2cm} \| \mathbf{\tilde{a}}(\r) \|_{\mathbf{W}}^2 \leq \epsilon_{\r} \}, \end{split}$$ where $\mathbf{W}$ is an $M \times M$ diagonal weight matrix with positive elements. The weight matrix $\mathbf{W}$ can be derived based on the type of uncertainty. Using $\mathbf{W}$, the set $\mathcal{E}_{\r}$ indicates an ellipsoidal region. The bound $\epsilon_{\r}$ for the set can be a constant or a function of $\r$, i.e., $\epsilon_{\r} = f(\r)$. This set enables parameterization of element-wise uncertainties in the nominal steering vector $\mathbf{\hat{a}(\r)}$ at each $\r$. Besides this consideration, we generalize the problem formulation further by setting a uniform bound (power level) $P$ across the tumor region $\Omega_T$ as an optimization variable to which the power of all the control points in the healthy region $\Omega_S$ are compared. This is in contrast to and the robust formulation in [@14:Shariati2014c], where the power levels of all the healthy grid points $\Omega_S$ are compared with the power of only a single reference point at fictitious tumor center $\r_0$. There is no need to limit our problem to a single point as a reference power level. Rather, the desired tightness of the power level across $\Omega_T$ is specified by the parameter $0 \leq \delta < 1$. This generalization also improves the efficiency when it comes to solving the robust design problem. With these considerations, the robust beampattern design problem can be formulated as$$\label{eq:robust beampattern problem} \begin{aligned} \underset{\R,t,P}{\textrm{max}} \hspace{.3cm} &t \hspace{.3cm} \textrm{subject to}&\\ &\hspace{-1cm} P - \left(\ahat(\r) \!\!+\!\! \atil(\r)\right)^H \R \left(\ahat(\r) \!\!+\!\! \atil(\r)\right) \geq t, \forall \atil(\r) \in \mathcal{E}_{\r}, \r \in \Omega_S&\\ &\hspace{-1.3cm}\left(\ahat(\r) \!\!+\!\! \atil(\r)\right)^H \R \left(\ahat(\r) \!\!+\!\! \atil(\r)\right) \geq (1-\delta)P, \forall \atil(\r) \in \mathcal{E}_{\r}, \r \in \Omega_T&\\ &\hspace{-1.3cm}\left(\ahat(\r) \!\!+\!\! \atil(\r)\right)^H \R \left(\ahat(\r) \!\!+\!\! \atil(\r)\right) \leq (1+\delta)P, \forall \atil(\r) \in \mathcal{E}_{\r}, \r \in \Omega_T&\\ &\hspace{-1cm}\R \in \mathcal{R},& \end{aligned}$$ where $t$ is the gap between the desired power level set across $\Omega_T$ and power deposition in the healthy tissue $\Omega_S$, similar to . Note that we take into account every possible perturbation $\atil(\r) \in \mathcal{E}_{\r}$. In contrast to the optimization problem which is a tractable convex problem, the robust problem is an SIP problem. For a given $\R$ in , there are infinite number of constraints in terms of $\atil(\r)$ to satisfy which makes the problem non-trivial. However, in the following theorem, extending the approach in [@14:Shariati2014c], we reformulate the robust power deposition problem as a convex SDP problem whose solution is the optimally robust covariance matrix. \[theo:robustSDP\] The robust power deposition for an M-element transducer array with the probing signal covariance matrix $\R \in \mathcal{R}$ and the perturbation vector $\atil(\r) \in \mathcal{E}_{\r}$, i.e., the solution of , is given as a solution to the following SDP problem $$\label{eq:robustSDP} \begin{aligned} &\underset{\R,t,P,\beta_i,\beta_{j,1},\beta_{j,2}}{\textrm{max}} \hspace{.3cm} t \hspace{.5cm} \textrm{subject to}& \\ &\Omega_S\!\!:\!\!\left[ \begin{array}{cc} \beta_i \mathbf{W} - \R & -\R \ahat(\r_i) \\ -\ahat(\r_i)^H \R & P - t - \ahat(\r_i)^H \R \ahat(\r_i) - \beta_i \epsilon_{\r_i} \end{array} \right] \succeq {\mathbf{0}}, &\\ &\Omega_T\!\!:\!\!\left[ \begin{array}{cc} \beta_{j,1} \mathbf{W} + \R & \R \ahat(\r_j) \\ \ahat(\r_j)^H \R & \ahat(\r_j)^H \R \ahat(\r_j) \!\!-\!\! (1-\delta)P \!\!-\!\! \beta_{j,1} \epsilon_{\r_j} \end{array} \right] \succeq {\mathbf{0}}, &\\ &\Omega_T\!\!:\!\!\left[ \begin{array}{cc} \beta_{j,2} \mathbf{W} - \R & -\R \ahat(\r_j) \\ -\ahat(\r_j)^H \R & (1+\delta)P \!\!-\!\! \ahat(\r_j)^H \R \ahat(\r_j) \!\!-\!\! \beta_{j,2} \epsilon_{\r_j} \end{array} \right] \succeq {\mathbf{0}}, &\\ &\R \in \mathcal{R},\beta_i,\beta_{j,1},\beta_{j,2} \geq 0, i=1,\ldots,N_S, j=1,\ldots,N_T.& \end{aligned}$$ *Proof:* See Appendix \[app A\]. Observe that the notations $\Omega_S$ and $\Omega_T$ indicate that the corresponding linear matrix inequalities (LMIs) should be satisfied for the points $\r_i \in \Omega_S$ and $\r_j \in \Omega_T$, respectively. Note that the robust SDP problem in this paper, which is stated in Theorem \[theo:robustSDP\], can be solved more efficiently than the SDP problem in [@14:Shariati2014c] since the matrices $\R$ and $\mathbf{W}$ in the current formulation have half of the size of the matrices involved in the latter problem. This occurs due to the generalization of the robust problem by using the uniform power level as a benchmark. Note that other robust problems with similar objectives can also be addressed using the above approach which are outlined in the following subsection. Alternative robust formulations {#subsec:alternative} ------------------------------- Similar robust problems to that of can be formulated in many different ways. For example, by restricting the power level outside the tumor in a weighted fashion. $$\begin{aligned} \min_{t,R}&& t \hspace{.3cm} \mbox{subject to}\\ && \hspace{-1.3cm}(\ahat(\r)\!\!+\!\!\atil (\r))^H R (\ahat(\r) \!\!+\!\! \atil (\r)) \le t w(\r), \forall \atil (\r)\in \mathcal{E}_\r, \r\in \Omega_{\rm S}\\ &&\hspace{-1.3cm} (\ahat (\r)\!\!+\!\!\atil (\r))^H R (\ahat(\r) \!\!+\!\! \atil (\r)) \ge (1-\delta)P, \forall \atil (\r)\in \mathcal{E}_\r, \r\in \Omega_{\rm T}\\ && \hspace{-1.3cm}(\ahat (\r)\!\!+\!\!\atil (\r))^H R (\ahat(\r) \!\!+\!\! \atil (\r)) \le (1+\delta)P, \forall \atil (\r)\in \mathcal{E}_\r, \r\in \Omega_{\rm T}\\ &&\hspace{-1.3cm}R\in \mathcal{R}\end{aligned}$$ where $P,\delta$ are fixed and $w(\r)$ is a weighting function constructed, e.g., so that the energy bound close to the tumor is less restrictive. One could also construct problems that minimize the sum of the energy in the non-tumor area where $t(\r)$ denotes the energy at $\r$: $$\begin{aligned} \min_{t(\r),R}&& \sum_{\r\in\Omega_{\rm S}}t(\r) \hspace{.3cm} \mbox{subject to} \\ && \hspace{-1.3cm} (\ahat(\r) \!\!+\!\! \atil (\r))^H R (\ahat(\r) \!\!+\!\! \atil (\r)) \le t(\r), \forall \atil (\r)\in \mathcal{E}_\r, \r\in \Omega_{\rm S}\\ && \hspace{-1.3cm} (\ahat(\r) \!\!+\!\! \atil (\r))^H R (\ahat(\r) \!\!+\!\! \atil (\r)) \ge (1-\delta)P, \forall \atil (\r)\in \mathcal{E}_\r, \r\in \Omega_{\rm T}\\ && \hspace{-1.3cm} (\ahat(\r) \!\!+\!\! \atil (\r))^H R (\ahat(\r) \!\!+\!\! \atil (\r)) \le (1+\delta)P, \forall \atil (\r)\in \mathcal{E}_\r, \r\in \Omega_{\rm T}\\ &&\hspace{-1.3cm} R\in \mathcal{R}.\end{aligned}$$ Both of the alternative formulations described above can be addressed following the steps derived in Appendix \[app A\] by using $\mathcal{S}$-lemma since we are still dealing with quadratic constraints. In the next section, we illustrate the reference performance of a nominal scenario where the steering vectors are perfectly known. Then we observe how much power can leak to the healthy tissue and cause damages when subject to uncertain steering vectors. Finally, we evaluate the proposed robust scheme in terms of improving the power deposition along our stated design goals. Numerical Results {#sec:numerical_results} ================= To illustrate the performance of the proposed robust scheme, we consider a 2D model of the organ at risk. Here, similar to [@08:Guo], we focus on the ultrasonic hyperthermia treatment for breast cancer where a 10-cm-diameter semi-circle is assumed to model breast tissues with a 16-mm-diameter tumor embedded inside. The tumor center is located at $\r_0 = [0 \hspace{.1cm} 34]^T$ mm. Fig. \[fig:2D\_model\] shows this schematic model. We consider a curvilinear array with $M=51$ acoustic transducers and half wavelength element spacing. Acoustic waveforms used to excite the array have the carrier frequency of $500$ kHz. The acoustic wave speed for the breast tissue is considered $1500$ m/s. ![A schematic 2-D breast model with an $16$-mm embedded tumor at $(0,34)$ as a reference geometry. A curvilinear ultrasonic array with $51$ transducers is located near to the organ at risk. The ultrasonic array is used for hyperthermia treatment.[]{data-label="fig:2D_model"}](schematic_2D_model_col "fig:"){width="\columnwidth" height="5.5cm"}\ To characterize (discretize) the healthy tissue region $\Omega_S$ and the tumor region $\Omega_T$, two grid sets with the spacing $4$mm are considered. For optimization, a rectangular surface of the dimension $ 64 \times 42$ in mm is assumed symmetric around the tumor to model the healthy region $\Omega_S$, while the grid points belonging to the circular tumor region are excluded from this surface and they model $\Omega_T$. Overall, $174$ and $13$ number of control points are considered to characterize $\Omega_S$ and $\Omega_T$ in order to optimize the array beampattern. The total transmitted power is constrained to $\gamma = 1$. For simplicity, the uncertainty set $\mathcal{E}_{\r}$ is modeled with ${\mathbf{W}} = {\mathbf{I}}_M$ with $\epsilon_{\r} \equiv \epsilon$ for all $\r$ where $\epsilon = 0.25$. Furthermore, the tightness of the desired power level in the across tumor region, $\delta$, is set to $0.7$. Note that for the small values of $\delta$ and/or large values of $\epsilon$, the problem may turn infeasible. In general, the feasibility of the problem depends on the value of the tightness bound $\delta$ relative to the size of the existing uncertainty in the system, i.e., the volume of the uncertainty set $\epsilon$, and the number of grid points $N_S$ and $N_T$ used to control the beampattern at the area of interest. When $\delta$ is too small, the desired power level across $\Omega_T$ is close to uniform and there may not exist enough degrees of freedom for the design problem to have a solution. For reference, the optimal covariance matrix when no uncertainty is taken into account, $\mathbf{R}_{nr}$, is obtained by solving problem using only nominal steering vectors $\ahat(\r)$, i.e., $\atil(\r) \equiv {\mathbf{0}}$. The optimal robust covariance matrix, denoted $\R^\star$, is obtained by solving where $\atil(\r) \in \mathcal{E}_{\r}$. For performance evaluation, we consider the power deposition in the tissue under the worst-case perturbations of the steering vectors. This scenario provides a lower bound to the achievable performances of all steering vector perturbations $\atil(\r)$ which belong to the deterministic uncertainty set $\mathcal{E}_{\r}$. In other words, for the points $\r$ in the healthy region $\Omega_S$, the worst-case performance is rendered by the steering vectors which provide the highest power, whereas for the points $\r$ in the tumor region $\Omega_T$, those steering vectors which attain the lowest power are the ones which contribute in the worst-case performance. They are collectively referred to as the *worst steering vectors*. Therefore, for a given $\R$, either $\R_{nr}$ or $\R^\star$, the worst steering vectors for the control points $\r$ in $\Omega_S$ and $\Omega_T$, are obtained by maximizing and minimizing the transmit beampattern , respectively. Observe that finding the worst steering vectors for the points in the tumor region $\Omega_T$ equals solving the following convex minimization problem at each $\r \in \Omega_T$, i.e., $$\underset{\|\atil(\r)\|^2 \leq \epsilon}{\textrm{min}} \hspace{.2cm} (\ahat(\r) + \atil(\r))^H \R (\ahat(\r) + \atil(\r))$$ using `CVX` [@cvx2013; @gb08]. Whereas, for finding the worst steering vectors for the points in the healthy region $\Omega_S$, we obtain a local optimum for the following non-convex maximization problem at each $\r \in \Omega_S$, i.e., $$\underset{\|\atil(\r)\|^2 \leq \epsilon}{\textrm{max}} \hspace{.2cm} (\ahat(\r) + \atil(\r))^H \R (\ahat(\r) + \atil(\r)),$$ using semidefinite relaxation techniques from [@Beck06strongduality]. We evaluate the designed beampatterns plotting the spatial power distribution in decibel scale, i.e., $20 \log_{10} (p(\r))$. Two different scenarios are considered, namely, *nominal* and *perturbed*, to evaluate the proposed robust power distribution scheme for the ultrasonic array. In the first scenario, nominal, we assume that the array steering vectors are precisely modeled, i.e., $\atil(\r) = \mathbf{0}$. In Fig. \[fig:nominal\], the beampattern generated by the array is plotted for the nominal scenario. This figure represents how power is spatially distributed over the organ at risk in an idealistic situation. Here, the covariance matrix of the waveforms is optimized under the assumption that the steering vectors are accurately modeled by , and the performance is evaluated using exactly the same steering vectors without any perturbations. The power is noticeably concentrated in the tumor region and importantly the power in the healthy tissue is several decibels lower. ![Power distribution (transmit beampattern in dB) for the nominal scenario, i.e., using $\R_{nr}$ and $\atil(\r) \equiv 0$.[]{data-label="fig:nominal"}](nominal_05_07lines "fig:"){width="\columnwidth" height="5.5cm"}\ In the second scenario, perturbed, the idealistic assumptions are relaxed and model uncertainties and imperfections are taken into account. The second scenario represents the case where the true steering vectors are perturbed versions of the nominal steering vectors $\ahat(\r)$, i.e., the true steering vector equals $\ahat(\r) + \atil(\r)$ where $\atil(\r) \in \mathcal{E}_{\r}$. The perturbation vectors $\atil(\r)$ are unknown but deterministically bounded. In the following we illustrate the worst-case performance, i.e., using the worst steering vectors to calculate the power distribution at each point. We start by illustrating the beampattern for the non-robust covariance matrix $\R_{nr}$ under the worst steering vectors. Fig. \[fig:NonRobustWorst\] shows how steering vector errors can degrade the array performance. Notice that in the worst-case, there is a substantial power leakage that occurs in the healthy tissue surrounding the tumor compared to Fig. \[fig:nominal\]. While, in Fig. \[fig:RobustWorst\], the robust optimal covariance matrix $\R^\star$, i.e., the solution to , is used to calculate the power for the worst steering vectors. Comparing Fig. \[fig:NonRobustWorst\] and Fig. \[fig:RobustWorst\], we see that by taking model uncertainties into account it is possible to obtain a noticeable increase in power in the tumor region for the worst case, and importantly, dramatic reductions of power deposited in the healthy tissue. ![Power distribution (transmit beampattern in dB) for the perturbed scenario, i.e., using $\R_{nr}$ and $\atil(\r) \in \mathcal{E}_{\r}$.[]{data-label="fig:NonRobustWorst"}](worst_nr_05_07lines "fig:"){width="\columnwidth" height="5.5cm"}\ ![Power distribution (transmit beampattern in dB) for the perturbed scenario, i.e., using $\R^\star$ and $\atil(\r) \in \mathcal{E}_{\r}$.[]{data-label="fig:RobustWorst"}](worst_r_05_07lines "fig:"){width="\columnwidth" height="5.5cm"}\ To finalize the numerical analysis, we provide a quantitative description for the performance of our proposed scheme summarized in Table \[table:power\]. It shows the average power calculated in dB received at the tumor region $\Omega_T$ and at the healthy region $\Omega_S$. Scenarios $\Omega_T$ $\Omega_S$ ----------------------- ------------ ------------ Nominal, $\R_{0}$ $-16.54$ $-29.78$ Perturbed, $\R_{0}$ $-36.40$ $-11.69$ Perturbed, $\R^\star$ $-27.17$ $-17.43$ : Average power for different regions[]{data-label="table:power"} Conclusion ========== The robust transmit signal design for optimizing spatial power distribution of an multi-antenna array is investigated. A robustness analysis is carried out to combat against inevitable uncertainty in model parameters which results in performance degradation. Such degradation occurs in practice quite often due to relying on imperfect prior and designs based upon them. Particularly, in this paper, the transmit signal design is based on exploiting the waveform diversity property but where errors in the array steering vector are taken into account. These errors are modeled as belonging to a deterministic set defined by a weighted norm. Then, the resulting robust signal covariance optimization problem with infinite number of constraints is translated to a convex problem which can be solved efficiently, by using the $\mathcal{S}$-procedure. Designs that are robust with respect to the worst case are particularly vital in biomedical applications due to health risks and possible damage. Herein we have focused on local hyperthermia therapy as one of the cancer treatments to be used either individually or along with other treatments such as radio/chemotherapy. Specifically, we consider hyperthermia treatment of breast cancer motivated by the fact that breast cancer is a major global health concern. The proposed robust signal design scheme aims to reduce unwanted power leakage into the healthy tissue surrounding the tumor while guaranteeing certain power level in the tumor region itself. We should emphasize on the fact that the robust design problem formulation and the analysis carried out herein yielding to the robust waveforms are general enough to be exploited whenever spatial power distribution is a concern to be addressed in real world scenarios dealing with uncertainties, e.g., for radar applications. Numerical examples representing different scenarios are given to illustrate the performance of the proposed scheme for hyperthermia therapy. We have observed significant power leakage into the healthy tissue that can occur if the design is based on uncertain model parameters. Importantly, we have shown how such damaging power deposition can be avoided using the proposed robust design for optimal spatial power distribution. Acknowledgement =============== The authors would like to acknowledge Prof. Jian Li for providing an implementation of examples from [@08:Guo]. \[sec:appendix\] Proof of Theorem \[theo:robustSDP\] {#app A} ----------------------------------- We start the proof by first stating the $\mathcal{S}$-Procedure lemma which helps us to turn the optimization problem with infinitely many quadratic constraints into a convex problem with finite number of LMIs. \[lem 6\]($\mathcal{S}$-Procedure [@Beck2009a Lemma 4.1]): Let $f_k(\x): \mathbb{C}^n \rightarrow \mathbb{R}$, $k = 0,1$, be defined as $f_k(\x) = \x^H\mathbf{A}_k \x + 2 \textrm{Re} \{\mathbf{b}_k^H \x \} + c_k$, where $\mathbf{A}_k = \mathbf{A}_k^H \in \mathbb{C}^{n \times n}, \mathbf{b}_k \in \mathbb{C}^n$, and $c_k \in \mathbb{R}$. Then, the statement (implication) $f_0(\x) \geq 0$ for all $\x \in \mathbb{C}^n$ such that $f_1(\x) \geq 0$ holds if and only if there exists $\beta \geq 0$ such that[^2] $$\left[ \begin{array}{cc} \mathbf{A}_0 & \mathbf{b}_0 \\ \mathbf{b}_0^H & c_0 \end{array}\right] - \beta \left[ \begin{array}{cc} \mathbf{A}_1 & \mathbf{b}_1 \\ \mathbf{b}_1^H & c_1 \end{array}\right] \succeq 0,$$ if there exists a point $\mathbf{\hat{x}}$ with $f_1(\mathbf{\hat{x}}) > 0$. The constraints in the optimization problem can be rewritten as the following functions of $\atil(\r)$ for $\r \in \Omega_S$ and $\r \in \Omega_T$. For notation simplicity we only specify the set from which the control points are drawn, and we also drop $\r$. $$\Omega_S: \begin{cases} f_0 = -\atil^H \R \atil - 2 \textrm{Re}(\ahat^H \R \atil) - \ahat^H \R \ahat - t + P \geq 0 \\ f_1 = - \atil^H \mathbf{W} \atil + \epsilon_{\r} \geq 0 \end{cases}$$ $$\Omega_T: \begin{cases} f_0 = \atil^H \R \atil + 2 \textrm{Re}(\ahat^H \R \atil) + \ahat^H \R \ahat - (1-\delta)P \geq 0 \\ f_1 = - \atil^H \mathbf{W} \atil + \epsilon_{\r} \geq 0 \end{cases}$$ $$\Omega_T: \begin{cases} f_0 = -\atil^H \R \atil - 2 \textrm{Re}(\ahat^H \R \atil) - \ahat^H \R \ahat + (1+\delta)P \geq 0 \\ f_1 = - \atil^H \mathbf{W} \atil + \epsilon_{\r} \geq 0 \end{cases}$$ Now, according to the $\mathcal{S}$-Procedure lemma, each pair of the quadratic constraints above is replaced with an LMI for each grid points in the pre-defined sets. In other words, all these quadratic constraints are satisfied simultaneously if we find $\beta_i$ for $i=1,\ldots,N_S$, $\beta_{j,1}$ and $\beta_{j,2}$ for $j=1,\ldots,N_T$ for which the mentioned LMIs in Theorem \[theo:robustSDP\] holds. Thus, the problem boils down to the SDP problem with $2N_T + N_S$ LMIs of the size $(M+1) \times (M+1)$ as the constraints. $\Box$ [^1]: Breast cancer is the most common cancer in the UK [@CancerReport2014UK]. The risk of being diagnosed with breast cancer is $1$ in $8$ for women in the UK and US [@CancerReport2014UK; @CancerReport2014US]. Breast cancer is also stated to be a leading cause of cancer death in the less developed countries [@CancerReport2013French]. [^2]: Note that $\mathcal{S}$-Procedure is lossless in complex space for the case of at most two constraints [@01:Jonsson].

--- abstract: | The importance of a research article is routinely measured by counting how many times it has been cited. However, treating all citations with equal weight ignores the wide variety of functions that citations perform. We want to automatically identify the subset of references in a bibliography that have a central academic influence on the citing paper. For this purpose, we examine the effectiveness of a variety of features for determining the academic influence of a citation. By asking authors to identify the key references in their own work, we created a dataset in which citations were labeled according to their academic influence. Using automatic feature selection with supervised machine learning, we found a model for predicting academic influence that achieves good performance on this dataset using only four features. The best features, among those we evaluated, were features based on the number of times a reference is mentioned in the body of a citing paper. The performance of these features inspired us to design an *influence-primed* (the hip-index). Unlike the conventional , it weights citations by how many times a reference is mentioned. According to our experiments, the hip-index is a better indicator of researcher performance than the conventional . author: - | **[Xiaodan Zhu and Peter Turney]{}**\ *National Research Council Canada, Ottawa, ON K1A 0R6, Canada.*\ *Email: [email protected]*\ **[Daniel Lemire]{}**\ *TELUQ, Université du Québec, Montreal, QC H2S 3L5, Canada.*\ **[André Vellino]{}**\ *School of Information Studies, University of Ottawa, Ottawa, ON K1N 6N5, Canada.* bibliography: - 'affinity.bib' title: | Measuring academic influence:\ Not all citations are equal --- =1 Introduction {#introduction .unnumbered} ============ One of the functions of citation analysis is to determine the impact of an author’s work on a research community. A first approximation for measuring this impact is to count the number of times an author is cited. Various other measures, such as the  [@Hirsch:2005], the  [@Egghe:2006] and the  [@schreiber2008share], refine this basic measure using functions based on the distribution of citations [@bornmann2008there]. Yet other measures of author impact are based on methods for scoring articles with weights and thresholds that depend on the journals in which they were published and the number of times the article was cited [@marchant2009score]. However, all these indexes and score-based rankings treat each citation as having equal significance. It has long been recognized that not all citations are created equal and hence they should not be counted equally. estimated that authors read only 20% of the works they cite. This estimate was based on a detailed analysis of the frequency of replication of distinctive errors in citations, such as incorrect page numbers or volume numbers. When an error in a citation is replicated many times, it seems likely that the citers have copied the citation without actually reading the cited paper. As an illustration, reported that a commonly cited paper by Gerard Salton does not actually exist. An incorrect citation was accidentally created by mixing the citations for two separate papers. This incorrect citation has since been cited by more than 300 papers. If the citers had tried to read the paper before citing it, they would have discovered that the paper does not exist. Like , we are concerned about the side-effects of counting insignificant references: . Indeed, based on an analysis of hundreds of references, found that a third of the references were redundant and 40% were perfunctory. In an independent study, found that the majority (62.7%) of the references could not be attributed a specific function whereas the fraction of references that provided an essential component for the citing paper (definition, tool, starting point) was 18.9%. The aim of our work is to determine the most effective features for identifying references that have high *academic influence* on the citing paper. An *influential* reference is one that inspired a new idea, method, experiment, or research problem that is a core contribution of the citing paper. We use the terms *influence* and *influential* to indicate the degree of academic influence of a single citation. In contrast, used the term *citation influence* to refer to the academic influence of a journal. Many attempts have been made to automatically identify which citations are most influential. Readers can often tell quickly whether a citation is shallow from the text itself, which has prompted several efforts to categorize citations by the linguistic context of their occurrence; that is, by the words near the citation in the body of the citing paper [@Teufel:2006; @Hanney:2005; @Mercer:2004; @Pham:2003]. In contrast to approaches based solely on linguistic context, our method uses machine learning to evaluate a number of citation features. We examine features based on linguistic context as well as other features, such as - the location of the citation in the text, - the semantic similarities between the titles of the cited papers and the content of the citing paper, - the frequency with which the articles are cited in the literature, - and the number of times a given reference is cited in the body of the paper. We test the effectiveness of the features by applying machine learning to the problem of identifying the influential references. One of our most important contributions is to identify a set of four features that are particularly useful to determine influence. For example, the two best features are the number of times a reference appears and the similarity between the title of the cited paper and the core sections of the citing paper. A secondary purpose of citation measures is to predict the future performance of authors, such as whether they will win a Nobel Prize [@Garfield:1968; @Gingras:2010]. The importance of researchers is reflected in the amount of influence they have on the research of their colleagues. Citation frequency is a measure of this influence, but a better measure would take into account *how* a researcher is cited, rather than giving all citations equal weight. As a test of our method’s ability to determine whether a cited paper substantially influenced the citing paper, we attempt to identify which researchers in computational linguistics are Fellows of the Association for Computational Linguistics (ACL), based solely on their publication records and citations. For each ACL Fellow, we compare their conventional (unweighted) with an computed from citations weighted by our measure of academic influence. We get a better average precision measure with weighted citations. Defining academic influence {#defining-academic-influence .unnumbered} =========================== What does it mean to say that one reference had more academic influence on a given citing paper than another reference? If our aim is to distinguish references according to their degrees of academic influence, then we must be precise about the meaning of academic influence. As researchers, we know that some papers have influenced the course of our research more than others, but how can we pin down this intuition? A paper written by an evolutionary biologist is likely to have been influenced by Darwin’s *On the Origin of Species*, but we are more interested in the *proximate* influences on the paper. A good research paper contributes a new idea to the literature. What prior work was the proximate cause, the impetus for that new idea? We believe that this question is best answered by the authors of the citing paper, because they are in the best position to decide which of their references should be labeled *influential* and which should be labeled *non-influential*. It could be said that we avoid the problem of precisely defining influence; instead we give a kind of *operational definition*: A cited paper is influential for a given citing paper if the authors of the citing paper say that it is influential. We acknowledge that authors may be wrong about whether a paper was influential. Two types of error are possible: Authors may say a reference is *influential* when it is actually *non-influential* or authors may say a reference is *non-influential* when it is actually *influential*. Both types seem plausible to us. In the first case, the authors might feel obliged to say that a paper is *influential*, because the paper is very popular, very respected, or very well written. In the second case, a paper might have greatly influenced the authors at a subconscious level, but they might mistakenly say it is *non-influential*, or they might not want to admit that there was any influence, due to professional jealousy. Nevertheless, although the authors might be wrong, we know of no better, more reliable way of determining which references were influential. Therefore we base our experiments on author-labeled data. also rely on author-labeled data. They collected a data set of twenty-two papers labeled by their authors. Each reference was labeled on a Likert scale and they experimented with unsupervised prediction of citation influence. Unlike us, their purpose was to model topical inheritance via citations. Motivation {#motivation .unnumbered} ========== Suppose that we have a model for predicting the label (*influential* or *non-influential*) of a paper–reference pair, consisting of a given citing paper and a given citation within that citing paper. We label pairs rather than references alone, because a reference that is influential for one citing paper is not necessarily influential for another citing paper. Such a model would have many potential applications. Wherever citation counts play an important role, the model could be applied to filter or weight the citations. Some potential applications follow. Summarizing: Given a paper with a long list of references, the model could identify the most influential references and list them. For those who are familiar with the field of the given paper, this list would rapidly convey the topic and nature of the paper. For those who are new to the field, this list would suggest further reading material. Citations have generally proven useful for summarization [@Qazvinian:2008:SPS:1599081.1599168; @Qazvinian:2010:INC:1858681.1858738; @Abu-Jbara:2011:CCS:2002472.2002536; @nanba1999towards; @Taheriyan:2011:SCR:2023568.2023579; @Kaplan:2009:AEC:1699750.1699764]. Improved measures of an author’s impact: Indexes such as the  [@Hirsch:2005],  [@Egghe:2006], and  [@schreiber2008share] could be made less sensitive to noise by filtering citation counts with a model of influence. Beyond reducing sensitivity to noise, a model of influence could also put more weight on original contributions. It is known that survey papers and methodology papers tend to be more highly cited than research contributions in general [@ioannidis2006concentration]. went as far as to recommend focusing on reviews: . However, survey and methodology papers seem less likely to us to be labeled as influential by authors. Filtering citation counts by a model of influence might decrease the impact of survey and methodology papers, putting more weight on innovative research. Improved journal impact factors: As with measures of author impact, measures of journal impact [@bollen2009principal] could be made less sensitive to noise by filtering citation counts with a model of influence. Improved measures of research organization impact: Citations counts are also used to evaluate research organizations. As with journal impact and author impact, performance measures that are based on citation counts may benefit from filtering by a model of influence. Meme tracking: Historians of science are interested in tracking the spread of ideas (memes) [@Haque:2011:PSC:1998076.1998081; @Leskovec:2009:MDN:1557019.1557077]. Citations are a noisy way to track how ideas spread, because a reference may be cited for many reasons other than being the source of an influential idea [@bornmann2008citation]. Filtering by a model of influence may result in a better analysis of the spread of an idea. Research network analysis: Scientists belong to networks of people who collaborate with each other or influence each other’s work. Filtering citations with a model of influence may make it easier to identify these networks automatically. Improved hyperlink analysis: In many ways, hypertext links in web pages are analogous to citation links in research papers. A good model of citation influence could suggest a model of hypertext link importance. This could improve measures of the importance of web pages, such as PageRank [@Qi:2007:MSD:1244408.1244418]. Improved recommender systems: Researchers often need help identifying relevant work that they should read. Filtering out less relevant citations might help paper recommender systems [@MEET:MEET14504701330; @springerlink:10.1007/978-3-642-23535-1_35]. Related work {#related-work .unnumbered} ============ The idea that the mere counting of citations is dubious is not new [@chubin1975content]: The field of *citation context analysis* has a long history dating back to the early days of citation indexing. There is a wide variety of reasons for a researcher to cite a source and many ways of categorizing them. For instance, identified fifteen such reasons, including giving credit for related work, correcting a work, and criticizing previous work. For articles in the field of high energy physics, distinguished four major classes of polar opposite pairs, conceptual–operational, organic–perfunctory, evolutionary–juxtapositional, and confirmative–negational. They found that the fraction of negational references, i.e., citations indicating that the cited source is wrong, is not negligible (14%). presented one of the first automatic citation indexing systems (CiteSeer). It could parse citations and use them to compute similarities between documents. might have implemented the first automated classification systems for citations. They used over 200 manually selected rules to classify citations in one of 35 categories. Machine learning methods for automatic classification can be applied to the text of a citing document. distinguish categories of citations that can be identified via linguistic cues in the text. They are able to classify citations into one of four categories (weak, positive, contrast, neutral) with an average of 68%. For a classification in three categories (weakness, positive, neutral), they get an average of 71%. Their classifier relies on 892 manually selected cue phrases, such as whether the citation is a self-citation, the location of the citation in the text, and manually acquired verb clusters. annotated a corpus of 43 open-access full-text biomedical articles. They built classifiers using Support Vector Machine (SVM) and Multinomial Naïve Bayes (MNB) models using the open-source Java library Weka. They report an average of 76.5%. They used unigrams (individual words) and bigrams (two consecutive words) as features. They ranked their features using mutual information. They found the SVM models were generally superior to the MNB models. Our own methodology differs from and in at least one significant way: We asked the authors of the citing papers themselves to identify the influential references whereas they used independent annotations. We believe that it is difficult for an independent annotator to classify citations. This concern was raised by : . Nevertheless, report moderately good inter-annotator agreement. To address concerns about the consistency of the , ranking and scoring have been proposed as alternative measures [@waltman2012inconsistency]. Like the , these measures are based on citation counts, and they too could benefit from filtering or weighting citations with a model of influence. There are other good reasons, beside assessing researchers, to make distinctions between different types of citations. The need also arises from the desire by publishers to provide scholarly research with semantic annotations. Thus CiTO, the Citation Typing Ontology [@Peroni:2012; @shotton2010cito], provides a rich machine-readable RDF metadata ontology for the characterization of bibliographic citations. From among the almost ninety semantic relations for citations identified in CiTO (for example, *agrees with*, *obtains background from*, *supports*, and *uses conclusions from*), it is natural to generalize at least two broad categories of citations, ones that acknowledge a fundamental intellectual legacy (such as *critiques*, *extends*, and *disputes*) and ones that are incidental (such as *cites for information*, *obtains support from*, and *cites as related*). Several authors have proposed weighting citations based on factors such as the prestige of the citing journal [@ding2011applying; @yan2010weighted; @ding2011popular; @gonzalez2010new]. Others have proposed weighting citations by the mean number of references of the citing journal [@zitt2008modifying]. proposed giving more weights to citations from more prestigious authors. He also argues that the citation of a paper published in a less prestigious venue should be considered more significant: . propose to use frequency to assess the importance of a citation. That is, if a reference was cited 10 times in the citing paper it gets a weight of 10. They show that by weighting citations by the in-paper frequency, review articles lose part of their advantage over original contributions when counting citations. They state that greater credit is reverted to the discoverers. They also show that closely related references are cited more often in the body of a citing paper than less related references (on average, 3.35 times versus 1.88 times). They define *closely related references* as papers having at least ten references in common with the given paper. studied hedging as a means to classify citations. used finite-state machines for classification of citations. Features for supervised learning {#sec:features .unnumbered} ================================ We are concerned with a binary classification problem: Given a research paper, classify its references as either *influential* or *non-influential*. The task is to create a model that takes a pair, consisting of a given author’s paper (the citer) and a reference in the given paper (the cited paper), as input, and generate a label (influential or non-influential) as output. Our goal is to create a model that can predict the labels assigned by the authors in our gold-standard dataset. Our approach to this task is to use supervised machine learning. For supervised machine learning, we must generate feature vectors that represent a variety of properties of each paper–reference pair. Given a training set of manually labeled paper–reference feature vectors, a learning algorithm can create a model for predicting the label of a paper–reference pair in the testing set. We use a standard supervised machine learning algorithm in our experiments (a support vector machine). The main contribution of this paper is that we evaluate a wide range of different features for representing the paper–reference pairs. Finding good features is the key to successful prediction. In our experiments, we consider five general classes of features: 1. Count-based features 2. Similarity-based features 3. Context-based features 4. Position-based features 5. Miscellaneous features Not all of these features are useful. However, many of these features are intuitively attractive, and we can only find out if (and the extent to which) they are useful through experiments. We describe these features in the following subsections. Count-based features {#count-based-features .unnumbered} -------------------- The count-based features are based on the intuition that a reference that is frequently mentioned in the body of a citing paper is more likely to be influential than a reference that is only mentioned once. We created five different count-based features: 1. Count-based features 1. countsInPaper\_whole 2. countsInPaper\_secNum 3. countsInPaper\_related 4. countsInPaper\_intro 5. countsInPaper\_core We count the occurrences of each reference in the entire citing paper ($counts\allowbreak{}InPaper\allowbreak{}\_whole$), in the introduction ($countsInPaper\_intro$), in the related work ($countsInPaper\_related$), and in the core sections ($countsInPaper\_core$), where *core sections* include all the other sections, excluding those already mentioned and excluding the acknowledgment, conclusion, and future-work sections. We added a feature ($countsInPaper\_secNum$) to indicate the number of different sections in which a reference appears. This feature is based on the intuition that a reference that appears in several different sections is more significant than a reference that appears in only one section (even if it may have a high frequency within that one section). Similarity-based features {#similarity-based-features .unnumbered} ------------------------- It seems natural to suppose that the influence of a cited paper on a given citing paper (the citer) is proportional to the overlap in the semantic content of the cited paper and the citer. That is, if there is a high degree of semantic similarity between the text of the citer and the text of the cited paper, then it seems likely that the cited paper had a significant influence on the citer. Accordingly, we explored a variety of features that attempt to capture semantic similarity. We assume that the text of the citer is given, but we do not assume that we have access to the text of the cited paper. The cited paper might not be readily available online, due to subscription charges or the age of the paper. A benefit of our approach is that all these features can be implemented efficiently: All features can often be computed from a single document. Even when the titles of references are not included by the journal, it may still be easier to locate the missing titles than the full text. Since (by choice) we do not have access to the full text of the cited paper, we use the title of the cited paper as a surrogate for the full text. The first five similarity-based features compare the title of the cited paper to various parts of the citing paper: 1. Similarity-based features 1. sim\_titleTitle 2. sim\_titleCore 3. sim\_titleIntro 4. sim\_titleConcl 5. sim\_titleAbstr We calculate the similarities between the title of a reference and the title ($sim\_titleTitle$), the abstract ($sim\_titleAbstr$), the introduction ($sim\_titleIntro$), the conclusion ($sim\_titleConcl$), and the core sections ($sim\_titleCore$) of the citing paper. These features should be able to capture the semantic similarity between the citer and a reference; in most cases, a title, abstract, and conclusion section are good summaries of the given citing paper. *Core sections* here refer to the same sections as in the count-based features. More specifically, we calculated cosine similarity scores. A piece of text (e.g., a title or abstract) is first represented as a vector in the word space, where each dimension is a word (a word type; not a word token). The values in the vector are the word frequencies of each word appearing in this piece of text. stemming algorithm was used to stem the text (remove suffixes) and we kept stop words (function words), since removing them did not improve the performance of our models during their development. Readers can refer to for further discussions of vector space models of semantic similarity. In a given citing paper (the citer), when a reference is mentioned in the body of the citer, the text that appears near the mention is called the *citation context*. Like the title of the cited paper, the citation context provides information about the cited paper; hence we can use the citation context as a surrogate for the full text of the cited paper, in the same way that we used the title as a surrogate. The next four similarity-based features compare the citation context to various parts of the citing paper: 1. Similarity-based features 1. sim\_contextTitle 2. sim\_contextIntro 3. sim\_contextConcl 4. sim\_contextAbstr For each reference, we calculate the similarities between the citation contexts and the title ($sim\_contextTitle$), abstract ($sim\_contextAbstr$), introduction ($sim\_contextIntro$), and conclusion ($sim\_contextConcl$) of the citing paper. When a reference appears multiple times in the citer, we take the average of the similarities over all its contexts. As with title similarity (features 2.1 to 2.5), we use cosine similarity, after the text was preprocessed with stemmer. During development, we experimented with different window sizes, ranging from two words around a citation to several sentences around it. We found that using the entire sentence in which the citation appears gave the best results. In contrast, found that contexts larger than one sentence were better for indexing purposes: further work might be needed to identify the optimal window. Context-based features {#context-based-features .unnumbered} ---------------------- The citation context of a reference could indicate the academic influence of the reference in other ways, beyond its value as as a surrogate for the full text of the cited paper (as in the above features 2.6 to 2.9). For example, if a citation $X$ appears in the context “the work of $X$ inspired us”, then $X$ seems likely to be influential for the given citing paper. For these features, we define the citation context to be a window of ten words around a citation (five words on each side). If a reference appears multiple times in the citing paper, we calculate its average score. The first three context-based features are based on the relation between the citation and the citation context: 1. Context-based features 1. contextMeta\_authorMentioned 2. contextMeta\_appearAlone 3. contextMeta\_appearFirst The first feature ($contextMeta\_authorMentioned$) indicates whether the authors of a reference are explicitly mentioned in the citation context; for example, “the work of Smith et al. \[4\]” mentions the authors (Smith et al.) in the citation context of the reference (\[4\]). The second feature ($contextMeta\_appearAlone$) indicates whether a citation is mentioned by itself (e.g., “\[4\]”) or together with other citations (e.g., “\[3,4,5\]”). When a citation is mentioned with other citations, the third feature indicates whether it is mentioned first (e.g., “\[4\]” is first in “\[4,5,6\]”). These three features may be biased by the different citation format requirements of various journals, but we leave it to the supervised learning system to decide whether the features are useful. A feature may be useful for prediction even when it has some bias. (However, we will see later that these features were not particularly effective in our experiments.) The next twelve context-based features are based on the meaning of the words in the citation context: 1. Context-based features 1. contextLex\_relevant 2. contextLex\_recent 3. contextLex\_extreme 4. contextLex\_comparative 5. contextLexOsg\_wnPotency 6. contextLexOsg\_wnEvaluative 7. contextLexOsg\_wnActivity 8. contextLexOsg\_giPotency 9. contextLexOsg\_giEvaluative 10. contextLexOsg\_giActivity 11. contextLexEmo\_emo 12. contextLexEmo\_polarity We manually created four relatively short lists of words that we designed to detect whether the citation context suggests that the cited paper is especially relevant to the citer ($contextLex\_relevant$), whether the citation context signals that the cited paper is new ($contextLex\_recent$), whether the citation context implies that the cited paper is extreme in some way ($contextLex\_extreme$), and whether the citation context makes some kind of comparison with the cited paper ($contextLex\_comparative$). The names of these features convey the kinds of words in the short lists. In Table \[table:shortlists\], we give the full lists for $contextLex\_relevant$ and $contextLex\_recent$ and a few terms for the other two features (as they both contain over 100 words). --------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- $contextLex\_relevant$ relevant, relevantly, related, relatedly, similar, similarly, likewise, pertinent, applicable, appropriate, useful, pivotal, influential, influenced, comparable, original, originally, innovative, suggested, interesting, inspiring, inspired $contextLex\_recent$ recent, recently, up-to-date, latest, later, late, latest, subsequent, subsequently, previous, previously, initial, initially, continuing, continued, sudden, current, currently, future, unexpected, upcoming, expected, ongoing, imminent, anticipated, unprecedented, proposed, startling, preliminary, ensuing, repeated, reported, new, old, early, earlier, earliest, existing, further, update, renewed, revised, improved, extended $contextLex\_extreme$ greatly, intensely, acutely, almighty, awfully, drastically, exceedingly, exceptionally, excessively, … $contextLex\_comparative$ easy, easier, easiest, strong, stronger, strongest, vague, vaguer, vaguest, weak, weaker, weakest, … --------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- : Manually created lists of words to classify the citation context \[table:shortlists\] We also created features based on semantic differential categories ($contextLexOsg$). discovered that three main factors accounted for most of the variation in the connotative meaning of adjectives. The three factors were *evaluative* (good–bad), *potency* (strong–weak), and *activity* (active–passive). The General Inquirer lexicon [@stone1966general] represents these three factors using six labels, *Positiv* and *Negativ* for the two ends of the *evaluative* continuum, *Strong* and *Weak* for the two ends of the *potency* continuum, and *Active* and *Passive* for the two ends of the *activity* continuum.[^1] The feature $contextLexOsg\_giEvaluative$ is the number of words in the citation context that are labeled *Positiv* in the General Inquirer lexicon, $context\allowbreak{}LexOsg\allowbreak{}\_giPotency$ is the number of words labeled *Strong*, and $context\allowbreak{}LexOsg\allowbreak{}\_giActivity$ is the number of words labeled *Active*. The intuition behind these features is that a citation is more likely to be influential if positive, strong, active words occur in the citation context. The General Inquirer lexicon has labels for 11,788 words. Using the algorithm of , we automatically extended the labels to cover 114,271 words. The additional words are from the WordNet lexicon. The WordNet features ($contextLexOsg\_wn$) are similar to the corresponding General Inquirer features ($contextLexOsg\_gi$), except they include these additional words.[^2] Since the citation context is a window of ten words around the citation, the values of these features range from zero to ten. If a reference is cited multiple times in the body of the citing paper, we calculate its average value. For increased precision, we only considered the words in the citation context that have an adjective or adverb sense in WordNet. We used an emotion lexicon [@Mohammad:2010] to check whether the citation context includes words that convey sentiment ($context\allowbreak{}LexEmo\allowbreak{}\_polarity$) or emotion ($context\allowbreak{}LexEmo\allowbreak{}\_emo$). The lexicon contains human annotation of emotion associations for about 14,200 words. The annotations in the lexicon indicate whether a word is positive or negative (known as *sentiment*, *polarity*, or *semantic orientation*), and whether it is associated with eight basic emotions (joy, sadness, anger, fear, surprise, anticipation, trust, and disgust). The feature $contextLexEmo\_polarity$ is the number of words in the citation context that are labeled either positive or negative. The feature $context\allowbreak{}LexEmo\allowbreak{}\_emo$ is the number of words that are labeled with any of the eight basic emotions. The idea behind these features is that any kind of sentiment or emotion in the words in the citation context might indicate that the citation is influential, even if the sentiment or emotion is negative. As with the other $contextLex$ features, the values of $contextLexEmo$ range from zero to ten. Multiple occurrences of a citation are averaged. Position-based features {#position-based-features .unnumbered} ----------------------- The location of a citation in the body of a citing paper might be predictive of whether the cited paper was influential. Intuitively, the earlier the citation appears in the text, the more important it seems to us. The first two types of position-based features are based on the location of a citation in a sentence: 1. Position-based features 1. posInSent\_begin 2. posInSent\_end These are binary features indicating whether a citation appears at the beginning ($posInSent\_begin$) or the end ($posInSent\_end$) of the sentence. If a reference appears more than once in the citing paper, we calculated the percentages; for example, if a reference is cited three times in the paper and two of the three appear at the beginning of the sentences, the $posInSent\_begin$ feature takes the value of 0.667. The next four position-based features are based on the location of a citation in the entire citing paper: 1. Position-based features 1. posInPaper\_stdVar 2. posInPaper\_mean 3. posInPaper\_last 4. posInPaper\_first We measured the positions of the sentences that cite a given reference, including the mean ($posInPaper\_mean$), standard variance ($posInPaper\_stdVar$), first ($posInPaper\_first$), and last position ($posInPaper\_last$) of these sentences. These features are normalized against the total length (the total number of sentences) of the citing paper; thus the position ranges from 0 (the beginning of the citing paper) to 1 (the end of the citing paper). More sophisticated location-based features are possible but not considered. For example, references appearing in a methodology section might be more influential than those appearing solely in the related work section. Miscellaneous features {#miscellaneous-features .unnumbered} ---------------------- The next three features do not fit into the previous four classes and they have little in common with each other. We arbitrarily put them together as *miscellaneous* features: 1. Miscellaneous features 1. aux\_citeCount 2. aux\_selfCite 3. aux\_yearDiff The citation count a paper has received (in the general literature; not the number of occurrences within a specific paper) is widely used as a metric for estimating the academic contribution of a paper, which in turn is an essential building block in calculating other metrics (e.g., ) for evaluating the academic contribution of a researcher, organization, or journal. We are interested in understanding its usefulness in deciding academic influence (in a specific paper). That is, when cited in a given paper, is a more highly cited paper more likely to have academic influence on the citer? To explore this question, we collected the raw citation counts of each reference in Google Scholar ($aux\_citeCount$). In accordance with convention, self-citation refers to the phenomenon where a citer and a reference share at least one common author. We are interested in knowing whether a self-citation would have a positive or negative correlation with academic influence. To study this, we manually annotated self-citation among the references and used it as a binary feature ($aux\_selfCite$). Are older papers, if cited, more likely to be academically influential? We incorporated the publication year of a reference as a feature ($aux\_yearDiff$). We calculated the difference in publication dates between a reference and the citer by subtracting the former from the latter, which resulted in a non-negative integer feature. Contextual normalization {#contextual-normalization .unnumbered} ------------------------ Many of the above features are sensitive to the length of the citing paper. For example, the number of occurrences of each reference in the entire citing paper ($countsInPaper\_whole$) tends to range over larger values in a long paper than in a short paper. For predicting whether a reference is influential, it is useful to normalize the raw feature values for a given paper–reference pair by considering the range of values in the given citing paper. This is a form of *contextual normalization* [@Turney:1993; @Turney:1996], where the citing paper is the context of a feature. We normalize all our features so that their values are in the range $[0,1]$. This kind of normalization is standard practice in data mining, as it improves the accuracy of most supervised learning algorithms [@witten2011data]. For example, consider the feature $countsInPaper\_whole$, where we count how many times a given reference is cited in the whole text. Suppose we find that, in a given citing paper, one reference is cited ten times, but all other references are cited only once. We would then give a score of 1 to the most cited reference, and a score of $1/10$ to the other references. That is, the most often cited references in any given citing paper always get a score of 1. We formalize the normalization as follows. Our feature set contains both binary and real-valued features that take non-negative values. Binary features do not require normalization: Their values are 0 and 1.[^3] Other features are normalized to $[0,1]$. Let $\langle p_i, r_{ij} \rangle$ be a paper–reference pair, where $p_i$ is the citing paper and $r_{ij}$ is the reference in $p_i$. Let $f_k$ be the feature in our feature set and let $v(p_i,r_{ij},f_k)$ be the value of the feature $f_k$ in the paper–reference pair $\langle p_i, r_{ij} \rangle$. Suppose that $p_i$ contains $n$ distinct references, $\langle p_i, r_{i1} \rangle, \dots, \langle p_i, r_{in} \rangle$, resulting in $n$ values for $f_k$, $v(p_i,r_{i1},f_k), \dots, v(p_i,r_{in},f_k)$. Let $\max (p_i,r_{i*},f_k)$ be the maximum of the $n$ values, $v(p_i,r_{i1},f_k), \dots, v(p_i,r_{in},f_k)$. We normalize each $v(p_i,r_{ij},f_k)$ to range from zero to one, using the formula $v(p_i,r_{ij},f_k) / \allowbreak{}\max (p_i,r_{i*},f_k)$. If $\max (p_i,r_{i*},f_k)$ is zero, then we normalize $v(p_i,r_{ij},f_k)$ to zero. Experiments with features {#experiments-with-features .unnumbered} ========================= Using a labeled dataset, we first identify the features that are most correlated with academic influence. We then combine some of these features to achieve a good classification score. Gold-standard dataset {#gold-standard-dataset .unnumbered} --------------------- We believe that the authors of a paper are in the best position to determine whether a given reference had a strong influence on their research. In a blog posting, we invited authors to help us create a *gold-standard dataset* of labeled references.[^4] The authors were directed to fill in an online form.[^5] The instructions on the form were as follows: > We believe that most papers are based on 1, 2, 3 or 4 essential references. By an essential reference, we mean a reference that was highly influential or inspirational for the core ideas in your paper; that is, a reference that inspired or strongly influenced your new algorithm, your experimental design, or your choice of a research problem. Other references merely support the work. > > We believe that authors are the best experts to assess which references are essential. We are interested in automatically finding these references. To know how well we are doing, we need your help: please give us the title of a few of your papers and list for each paper the references that you feel are most essential, those without which the work would not have been possible. Forty different researchers filled out our online form (see Table \[table:vol\]). About half of them are from the USA and Canada. Three quarters of them are in computer science. This gold-standard dataset provides us with a benchmark for supervised machine learning.[^6] The authors gave us the titles of their papers and they indicated which references in each paper were influential for them. From the titles, we obtained PDF copies of their papers and converted them to plain text. We then extracted the references from the text and labeled them as influential or non-influential. In total, the authors contributed 100 of their papers. OpenNLP was used to detect sentence boundaries and conduct tokenization.[^7] We then used ParsCit to parse the papers [@Councill:2008]. ParsCit is an open-source package for parsing references and document structure in scientific papers. We first ran the papers through ParsCit and then used a few hand-coded regular expressions to capture citation occurrences in paper bodies that were not detected by ParsCit. The contents of the papers were then further annotated. First, the section names were standardized to twelve predefined labels: *title*, *author*, *abstract*, *introduction*, *related*, *main*, *conclusion*, *future*, *acknowledgment*, *reference*, *appendix*, and *date* (the year of publication of the given paper). The default label for a section was *main* (the core or main body of the paper). For example, *previous work* and *related work* would both be standardized to *related*. Second, the bibliographic items were manually corrected and meta-data about them (e.g., the Google citation counts) was included. The citations of these items in the main body of the paper were also manually corrected. For example, if references were cited as “\[7-10\]”, we modified the citation to , so as to explicitly include references \[8\] and \[9\]. stemmer (mentioned in the preceding section) was only applied as a preprocessing step when generating feature vectors; the stemmer was not applied during the corpus annotation step described here. The basic units in our study are paper–reference pairs, not papers. The 100 papers yield 3143 paper–reference pairs (that is, 3143 data points; 3143 feature vectors). In the main bodies of the 100 papers, there are 5394 occurrences of the references, so each paper contains an average of around 31 references (in the bibliographies) and 54 citations (in the main text). The dataset contains 322 (10.3% of 3143) *influential* references (strictly speaking, 322 influential paper–reference pairs). That is, their authors identified an average of 3.2 influential references per research paper. Correlation between labels and features {#correlation-between-labels-and-features .unnumbered} --------------------------------------- We seek to determine which features are better able to predict the academic influence of a reference. The Pearson correlation coefficients between the various features and the gold influence labels are a simple indication of how useful a feature might be. We show the coefficients in Figure \[fig:correlation\]. {width="84.90000%"} First, consider the correlation coefficients for the count-based features: 1. Count-based features 1. countsInPaper\_whole 2. countsInPaper\_secNum 3. countsInPaper\_related 4. countsInPaper\_intro 5. countsInPaper\_core Figure \[fig:correlation\] shows that the most correlated individual features to academic influence are in-paper count features ($counts\allowbreak{}InPaper\allowbreak{}\_whole$ and $counts\allowbreak{}InPaper\allowbreak{}\_secNum$). This is a convenient result, because one of the best features, $counts\allowbreak{}InPaper\allowbreak{}\_whole$ (the number of times a reference is cited in a paper), is also one of the easiest to compute from a technical point of view. Moreover, it suggests simple but potentially effective schemes for modifying the standard citation count (i.e., the number of papers that cite a given paper in the general literature): For each paper $X$ that cites a paper $Y$, increment the citation count for $Y$ - only if $Y$ was cited more than once in the body of $X$, - only if $Y$ is cited more often in the body of $X$ than most of the other references in $X$, or - only if $Y$ is cited in more than one section of $X$. Intuitively, these modified citation counts may also be more robust than the standard citation count, considering that an author seems unlikely to cite a paper more than once when the paper is included in the references because it is *de rigueur* in the field. Next, consider the similarity-based features. These are features that measure the semantic similarity between a citer and a reference. In general, we found such features well correlated with academic influence. The first group of similarity-based features compares the similarities between the title of a cited paper and the title, introduction, conclusion, and abstract of the citer: 1. Similarity-based features 1. sim\_titleTitle 2. sim\_titleCore 3. sim\_titleIntro 4. sim\_titleConcl 5. sim\_titleAbstr As shown in Figure \[fig:correlation\], the correlation coefficients of the features (e.g., $sim\allowbreak{}\_title\allowbreak{}Abstr$) rank right after those of the two in-paper count features. As we will show soon, these features (e.g., $sim\_titleCore$) can work synergetically with count-based features for predicting academic influence. The second group of similarity-based features use citation context instead of the title of a cited paper: 1. Similarity-based features 1. sim\_contextTitle 2. sim\_contextIntro 3. sim\_contextConcl 4. sim\_contextAbstr These features compare the similarities between citation contexts and the title, abstract, and conclusion of the citing paper. We found that the context–abstract ($sim\_contextAbstr$) similarity feature is the one in this group that is most correlated with academic influence, followed by context–conclusion ($sim\_contextIntro$), context–title($sim\_contextTitle$), and context–introduction ($sim\_contextConcl$). We turn to the context-based features. First, we focus on the features that consider the relation between the citation and the citation context: 1. Context-based features 1. contextMeta\_authorMentioned 2. contextMeta\_appearAlone 3. contextMeta\_appearFirst In this group, $contextMeta\_authorMentioned$ has the highest correlation coefficient. This feature indicates whether the names of the authors appear in the citation context (e.g., “Smith et al. \[4\]”). The second group of context-based features are based on the meaning of the words in the citation context: 1. Context-based features 1. contextLex\_relevant 2. contextLex\_recent 3. contextLex\_extreme 4. contextLex\_comparative 5. contextLexOsg\_wnPotency 6. contextLexOsg\_wnEvaluative 7. contextLexOsg\_wnActivity 8. contextLexOsg\_giPotency 9. contextLexOsg\_giEvaluative 10. contextLexOsg\_giActivity 11. contextLexEmo\_emo 12. contextLexEmo\_polarity We used several different types of lexicons to capture different aspects of semantics in the citation contexts, including sentiment, emotion, and  semantic differential categories. As we mentioned in the preceding section, $contextLexEmo\_polarity$ is the number of words in the citation context that are labeled either positive or negative and $contextLexEmo\_emo$ is the number of words that are labeled with any of the eight basic emotions. Our hope was that any kind of sentiment or emotion in the words in the citation context might indicate that the citation is influential, even if the sentiment or emotion is negative, but Figure \[fig:correlation\] shows that neither feature has a high correlation with the gold labels. This suggests to us that it might be better to split these features into more specific features for each of the possible categories. To test this idea, we split $contextLexEmo\_polarity$ into two features, one for positive polarity and one for negative polarity, and we split $contextLexEmo\_emo$ into eight features, one for each of the eight basic emotions. Figure \[fig:emo\_correlation\] shows the correlation coefficients for each of these more specific features. ![Detailed Pearson correlation coefficients between emotional and sentimental features and the gold labels.[]{data-label="fig:emo_correlation"}](emo_correlation.pdf) We see in Figure \[fig:emo\_correlation\] that positive polarity has a higher correlation than negative polarity. Among the eight basic emotions, surprise has the highest correlation. These results are intuitively reasonable. However, none of the correlations is greater than 0.06. It seems that none of these features are likely to be of much use for predicting influence. Let us consider the position-based features. First, we examine the position of a citation in a sentence: 1. Position-based features 1. posInSent\_begin 2. posInSent\_end Our results suggest that references located at the beginning of a sentence might be more influential. The second group of position-based features calculates the locations of citations in the body of the paper: 1. Position-based features 1. posInPaper\_stdVar 2. posInPaper\_mean 3. posInPaper\_last 4. posInPaper\_first The best paper-position-based feature is the standard variance of a reference’s positions ($posInPaper\_stdVar$). Note that this feature is likely to overlap with the in-paper-counts features to some degree: A larger in-paper-counts number could correspond to a higher position variance, so these features may not have additive benefit when used together. Finally, we examine the miscellaneous features: 1. Miscellaneous features 1. aux\_citeCount 2. aux\_selfCite 3. aux\_yearDiff Figure \[fig:correlation\] shows that the correlation coefficient between citation counts and the influence labels is positive. This confirms the previous finding that highly cited papers are more likely to be cited in a meaningful manner [@Bornmann2008]. However, we find that the correlation is moderate: It is smaller than that of half of the features we tested. Hence, while highly cited papers may have more academic influence, the citation count is not an ideal indicator of influence. This result is consistent with the fact that papers are often cited for reasons other than academic influence. When a paper is highly cited, the authors of the citing paper may feel obliged to cite it, yet might not take the time to read it. From Figure \[fig:correlation\], we see a small positive correlation between self-citation and the gold influence labels. We will see later that $aux\_selfCite$ is useful as a corrective factor in the final model. Are older papers more likely to be influential? In Figure \[fig:correlation\], the $aux\_yearDiff$ feature has a small positive correlation with the influence. We discretized the feature over the ranges 0, 1, …, 10, 11–20, 21–30, $31+$. The corresponding Pearson coefficients are given in Figure \[fig:yearDiff\]. The figure shows that, when the cited paper is one or four to seven years older than the citing paper, there is a positive correlation with academic influence. More recent papers (0, 2 and 3 years) and older papers ($\geq 8$ years) are poorly or negatively correlated with academic influence. This result is consistent with our interest (mentioned in the section on defining academic influence) in the *proximate* influences on a citing paper. Results of predicting academic influence {#results-of-predicting-academic-influence .unnumbered} ---------------------------------------- We used the LIBSVM support vector machine (SVM) package [@Chang:2011] as our supervised learning algorithm.[^8] We chose a second-degree polynomial as our kernel function.[^9] The was applied to evaluate the performance of the learned model. The is defined as the harmonic mean of precision $P$ and recall $R$: $F = 2PR / (P+R)$. The precision of a model is the conditional probability that a paper–reference pair is *influential* (according to the gold-standard author-generated label), given that the model guesses that it is *influential*. The recall of a model is the conditional probability that the model guesses that a paper–reference pair is *influential*, given that it actually is *influential* (according to the gold-standard). Another metric is *accuracy*, defined as the ratio of properly classified paper–reference pairs (as either *influential* or not) over the total number of classified pairs. However, the classes in our data are imbalanced (10.3% in the *influential* class and 89.7% in the *non-influential* class). This makes accuracy inappropriate as a performance measure for our task, because we could achieve an accuracy of 89.7% by the trivial strategy of always guessing the *non-influential* class. The is a better performance measure for imbalanced classes, because it rewards a model that has a balance of precision and recall. Always guessing *non-influential* yields an of zero (we take division by zero to be zero). Always guessing *influential* yields an of 18.7% ($2 \cdot 0.103 \cdot 1 / (0.103 + 1) = 0.187$). Unlike accuracy, the penalizes trivial models. The SVM algorithm is designed to optimize accuracy, whereas we want to optimize the . Since an SVM does not directly optimize the and our data are not balanced, we used a simple down-sampling method to handle this. In each fold of cross validation, we randomly down-sampled the negative instances (non-influential references) in the training data to make their number equal to that of the positive ones (influential references). Table \[tab:bestSys\] shows the scores of different models under ten-fold cross-validation. We included two baselines. The first baseline randomly labels a reference with a probability equal to the distribution of the labels in the training data. The second predicts the academic influence of references based on their Google citation counts ($aux\_citeCount$). Note that the macro-averaged F-measure is not necessarily between the averaged precision and recall. For example, for model (3) in the table, 0.35 is not between 0.36 and 0.41. Starting with model (3) in the table, we added features greedily: In each round, the feature resulting in the maximum improvement of the was added. That is, model (3) is the best model that uses only one single feature, and the best performing model (with four features) is model (6). [clSlSS]{} Model & Features & & [Precision]{} & [Recall]{}\ (1) & random & 0.10 & & 0.10 & 0.10\ (2) & aux\_citeCount & 0.12 & & 0.12 & 0.13\ (3) & countsInPaper\_whole & 0.35 & \* & 0.36 & 0.41\ (4) & (3) + sim\_titleCore & 0.39 & \*& 0.40 & 0.44\ (5) & (4) + countsInPaper\_secNum & 0.41 & \*& 0.42 & 0.46\ (6) & (5) + aux\_selfCite & 0.42 & \*& 0.43 & 0.48\ In Table \[tab:bestSys\], all of the models marked with a dagger sign (${\dag}$) are statistically significantly better than model (3). The models marked with an asterisk (\*) are statistically significantly better than the two baselines. We use a one-tailed paired t-test with a 99% significance level. The first feature chosen for the model by greedy feature selection is $counts\allowbreak{}InPaper\allowbreak{}\_whole$, the feature with the highest correlation in Figure \[fig:correlation\]. The best model achieves an of about 42% (see Table \[tab:bestSys\]). The model uses only four features, two of which are count-based ($countsInPaper$) and one semantics-based ($sim\_titleCore$). Adding more features to model (6) did not result in further improvement. Using all features presented in Figure \[fig:correlation\] results in an of 37%, which is significantly better than model (1) and (2) ($p<0.01$), insignificantly better than the best single-feature model (3) ($p>0.05$), and worse than model (6) ($p<0.01$). This observation supports the hypothesis that feature selection is useful for this task. In general, feature selection removes useless or detrimental features, which often leads to better performance (e.g., higher ) and greater efficiency. We find it interesting that a semantic feature ($sim\_titleCore$, the similarity between the title of the cited paper and the core sections of the citing paper) is the second feature chosen. It seems that this feature complements the count-based feature; it covers some papers that are influential but have lower counts. The improvement of model (4) over model (3) is about 4% in terms of , which is statistically significant at a level of $p<0.01$. Using another count-based feature, $counts\allowbreak{}InPaper\allowbreak{}\_secNum$, additionally improves the performance. Although $aux\_selfCite$ by itself has a small correlation with influence (see Figure \[fig:correlation\]), it seems to be useful when combined with the other three features. Note that the we used here is the macro-averaged [@Lewis:1991]. That is, we calculated the for each paper individually and then computed the arithmetic average over all the obtained. For each reference in a given paper, we used the SVM model to estimate the probability that the reference is labeled [*influential*]{}. In the training data, the average paper contained three influential references. Therefore the model guesses that the top three references in the given paper, with the highest estimated probabilities, are [*influential*]{}, and the remaining references, with lower probabilities, are [*non-influential*]{}. It is difficult to describe an SVM model intuitively. Moreover, given an SVM model, it is not straightforward to describe the importance of a feature. To further assess the importance of the four features in model (6), we have also applied logistic regression to our data [@long1997regression]. Logistic regression assumes that the probability distribution of some binary random variable $Y$ is of the form $$P(Y=1|\vec{X})= \frac{1}{1+e^{-(\beta_0+\sum_i \beta_i X_i)}}$$ where $\vec{X}=(X_1, X_2, \ldots )$ are feature values and $\beta_0, \beta_1, \beta_2, \ldots$ are weight vectors. Given $N$ training instances, we can solve for the weights $\vec{W} = (\beta_0, \beta_1, \ldots)$ by maximizing the likelihood function $$l(\vec{W}) = \prod\nolimits_{i=1}^N P(Y=1|X)^{t_i}(1-P(Y=1|X))^{1-t_i}$$ where $t_i \in \left\{0,1\right\}$ is the binary gold label of the $i^{th}$ training instance. Once we have solved for the weights, we can classify instances by using a threshold $\omega$. That is, given an instance with feature values $\vec{X}$, we predict $Y=1$ if $P(Y=1|\vec{X})>\omega$ and we predict $Y=0$ otherwise. For our application, we set the threshold so that the relative number of influential citations is the same as in the training set. With logistic regression, the magnitude (absolute value) of the weights, $\beta_i$, indicates the importance of the corresponding feature in the model. We used the *mnrfit* command in Matlab to conduct logistic regression on our data. The weights assigned to the features $countsInPaper\_whole$, $sim\_titleCore$, $countsInPaper\_secNum$, and $aux\_selfCite$ are 2.7228, 1.2683, 1.1763, and -0.0923. Their absolute values correspond to the order they are selected by SVM in Table \[tab:bestSys\]: The most important feature is $countsInPaper\_whole$ and the least important is $aux\_selfCite$. Note that the weight for $aux\_selfCite$ is smallest, which corresponds to the observation that self-citations are less likely to be influential. We have also used logistic regression to classify the references, using the same experimental setup as we used for the SVM. The logistic regression performance is slightly below that of the SVM ($\approx 0.37$ vs. $0.41$).[^10] When only one feature is used with SVM (as in model (3)), the classification task can be regarded as setting a threshold on the feature, to separate the influential references from the rest. In Figure \[fig:threshClr\], we vary such a threshold to provide a full view of the of the two most relevant features shown in Figure \[fig:correlation\]. Different thresholds here resulted in different percentages of references being predicted as *influential*, corresponding to the x-axis in the figure. Note that our thresholds are in the range \[0, 1\], since we have normalized the count values (and all other features) to this range (as we discussed earlier). ![Detailed curves for *countsInPaper\_secNum* and *countsInPaper\_whole*.[]{data-label="fig:threshClr"}](fByThreshClr.pdf){width="70.00000%"} Figure \[fig:threshClr\] also includes the curve of random guesses (*random* in the figure). This curve serves as a minimum baseline for comparison with the other features. Model (1) in Table \[tab:bestSys\] corresponds to the point on the random curve such that the percentage of references predicted as *influential* equals the size of the *influential* class ( 322/3143 = 10.3%). In Figure \[fig:threshClr\], the peak of our best feature, $countsInPaper\_whole$, is 0.37 (when the value on the x-axis is 13%). Model (3) attained an of 0.35 in Table \[tab:bestSys\], slightly below the value of 0.37 in Figure \[fig:threshClr\]. Model (3) was trained and tested with ten-fold cross validation, whereas the of 0.37 is based on using the whole dataset as training data, with no independent testing data; thus the small gap in the (0.35 versus 0.37) indicates that SVM is performing well. Experiments with in-paper citation counts {#experiments-with-in-paper-citation-counts .unnumbered} ========================================= In the preceding section, we made a number of observations. A significant one is that the in-paper citation counts (how many times a reference is cited in a paper) are the most predictive features for academic influence. The following experiments are designed to further validate our results. Since the in-paper counts convey influence information, which is ignored in the conventional counting of citations, we wondered whether incorporating in-paper citation numbers into global citation counting would result in different rankings of papers and authors. In contrast to the conventional citation counting, we refer to the methods that take into consideration the in-paper counts as *influence-primed citation counts*. We conducted two types of experiments. First, we explored the correlation between the rankings of papers and authors with and without influence-primed citation counts. Second, we tackled the task of identifying ACL Fellows. Influence-primed citation counts {#influence-primed-citation-counts .unnumbered} -------------------------------- A classical citation network is a graph in which the nodes (vertices) correspond to papers and there is a directed link (directed edge) from one paper to another if the first paper cites the second paper. The network is usually acyclic (it has no loops), due to time: An earlier paper rarely cites a later paper (with some exceptions, due to overlap in the gestation periods of publications). A slightly more sophisticated citation network could have weights or labels associated with the edges in the graph. For example, a directed edge from citing paper $X$ to cited paper $Y$ might be labeled as *$Y$ provides evidence that supports claims in $X$*, or the directed edge might be weighted with a number that indicates how influential $Y$ is to $X$. There are various ways that in-paper citation counts can be used to modify classical citation networks. We have experimented with using in-paper citation counts for *filtering* edges in the graph and for *weighting* edges in the graph. A simple filtering method is to drop the edge from citing paper $X$ to cited paper $Y$ when the in-paper citation count for $Y$ in $X$ is below a threshold. Equivalently, when building a citation network, only add an edge when the in-paper citation count is greater than or equal to a threshold. We have also tried filtering with a combination of two thresholds, $T_1$ and $T_2$. For a given paper–reference pair (i.e., a given citing–citer pair in the citation network), when building a network, we add an edge from citing paper $X$ to cited paper $Y$ based on the in-paper citation count of the reference $Y$ (i.e., the number of times $Y$ is mentioned anywhere in the body of $X$) and the rank of the reference $Y$ relative to the other references in $X$ (i.e., the rank in a list, sorted in descending order of in-paper citation counts). An edge is added only if the in-paper citation count is at least $T_1$ and the rank is less than $T_2$ (lower rank is better, because the list is sorted in descending order). An alternative to filtering is weighting edges. The edge between a reference and a citer can be weighted by the in-paper citation counts. Given a citing paper $X$ and a cited paper $Y$, suppose that $Y$ is mentioned $c$ times in the body of $X$. There are many functions that we might apply to convert the in-paper citation count $c$ to a weight. Any linear or polynomial function might be useful. In the following experiments, we use the square of the in-paper count to weight an edge: We weight the edge from $X$ to $Y$ with $c^2$. Squaring $c$ gives more weight to higher values of $c$. The *conventional citation count* for a paper is calculated from a citation network. It is the number of edges in the graph that are directed into the vertex that represents the given paper. That is, the conventional citation count for a paper is the number of papers that cite the given paper. We weight citations according to the square of the number of times the reference is mentioned in the text. For example, being cited once in a paper that mentions the reference once counts for one whereas being cited once in a paper that mentions the reference twice counts for four. Weights are added up: Being cited four times by four papers that mention the reference once counts the same as being cited once by a paper that mentions the reference two times. The *influence-primed citation count* is like the conventional citation count, except it weights each edge by $c^2$, instead of 1. We define *influence-primed citation count* formally as follows: Given two papers, $p_i$ and $p_j$, let $c(p_i,p_j)$ be the number of times paper $p_i$ mentions paper $p_j$ in the body of its text, excluding the reference section. If the paper $p_i$ cites paper $p_j$, then $c(p_i,p_j) > 0$; otherwise $c(p_i,p_j) = 0$. Let $L$ (the literature) be the set of all papers in the given citation network. The *influence-primed citation count* of paper $p_j$, $\textrm{cip}(p_j)$, is $$\sum_{p_i \in L} c(p_i,p_j)^2.$$ The function name, $\textrm{cip}(\cdot)$, stands for *citations, influence-primed*. The *conventional* for an author is the largest number $h$ such that at least $h$ of the author’s papers are cited by at least $h$ other papers. Each citation of a paper has a weight of 1. The *influence-primed* for an author is like the conventional , except it weights each edge by $c^2$, instead of 1. The *influence-primed* for an author is the largest number $h$ such that at least $h$ of the author’s papers have an *influence-primed citation count* of at least $h$. For example, if an author has four papers, each one cited only once, but each time they are cited, they are mentioned twice, then the influence-primed is 4. In contrast, with the conventional , the same author would receive an of 1. We define *influence-primed* formally as follows: An author, $a_i$, with a set of papers $O(a_i)$ (the œuvre) has an *influence-primed* , $\textrm{hip}(a_i)$, of $h$ if $h$ is the largest value such that $$| \{p_j \in O(a_i) | \textrm{cip}(p_j)\geq h\} | \geq h.$$ The function name, $\textrm{hip}(\cdot)$, stands for *, influence-primed*. We refer to this as the hip-index. ACL Anthology Network {#acl-anthology-network .unnumbered} --------------------- For this experiment, we use the AAN (ACL Anthology Network) dataset [@radev2009acl]. AAN is a citation network constructed from the papers published in Association for Computational Linguistics (ACL) venues (conferences, workshops, and journals since 1965), approximately 20,000 papers. The AAN citation network is a closed graph; edges from or to the papers published outside ACL venues are not included. In effect, by using the AAN citation network, we can measure the impact of researchers and papers on the ACL community. This restriction might be desirable when we try to identify the recipients of honours granted by ACL. Table \[tab:aan\] shows the basic statistics for the dataset.[^11] Statistics Values --------------------------- -------- Number of venues 341 Number of papers 18,290 Number of authors 14,799 Number of paper citations 84,237 : Statistics of the AAN dataset.[]{data-label="tab:aan"} To obtain in-paper counts, we used regular expressions to locate citations. Our regular expressions are good at both precision and recall according to our manual examination, but they still make a few errors. For example, the regular expressions have trouble with citations that span two lines of text and with multiple papers written by the same author in the same year. Another problem is automatically distinguishing the main body text from the reference section of a paper. The regular expressions may wrongly increment the in-paper count by matching citations in the reference section. Numerical citations (e.g., “\[1\]” or “\[1,2,3\]”) are more difficult to process than textual citations (e.g., “Smith et al. (1998)”). We used a random number generator to select a sample of 100 papers from AAN and then we manually determined their citation types. In this random sample, 7% used numerical citations. Since numerical citations are relatively rare in the AAN dataset, we simply ignored them. We did not normalize the in-paper citation count $c$ for these experiments with the AAN dataset. In this section, the in-paper citation count $c$ is a non-negative integer value. The reason for this is that the AAN network is a closed graph: All citations to and from papers outside of the AAN dataset are ignored in the AAN citation network. The maximum value that we used for contextual normalization in the preceding section, $\max (p_i,r_{i*},f_k)$, could be distorted by the ignored citations. For example, a paper that mainly cites the AAN papers could be normalized very differently from one that mainly cites non-AAN papers. The maximum value may be highly sensitive to whether a citing paper is influenced by cited work that is outside of the AAN network. The in-paper citation count $c$ that we use in this section is essentially a raw (unnormalized) variation of $countsInPaper\_whole$. We did not use $counts\allowbreak{}InPaper\allowbreak{}\_secNum$, because it might be more sensitive to noise introduced by the process of automatically detecting section boundaries. (We manually detected sections in the preceding experiments, but this manual process does not scale up from 100 papers to 20,000 papers.) Figure \[fig:threshClr\] suggests that $countsInPaper\_whole$ performs better than $countsInPaper\_secNum$, and it is easier to compute. Conventional versus influence-primed counting {#conventional-versus-influence-primed-counting .unnumbered} --------------------------------------------- A natural question is whether there is any difference between conventional citation counts and influence-primed citation counts. In particular, do the two approaches yield different rankings of the papers? Table \[tab:paperRank\] shows the Spearman correlation coefficients between the AAN papers. We grouped the papers according to their ranks in the conventional counting. For each group, we calculated the Spearman correlation coefficient between the conventional counts and the influence-primed counts. [rS]{} Papers & [Correlations]{}\ 1–100 & 0.67\ 101–200 & 0.12\ 201–300 & 0.11\ 301–400 & -0.04\ 401–500 & 0.07\ 501–600 & 0.05\ 601–700 & -0.13\ 701–800 & 0.30\ 801–900 & 0.22\ 901–1000 & 0.06\ For example, papers 1–100 are the top 100 most highly cited papers, according to conventional citation counts, where each edge directed into a given paper increments that paper’s count by one. For these 100 papers, we have a vector of 100 conventional citation counts. We also calculate a vector of 100 influence-primed citation counts. The Spearman correlation between these two vectors is 0.67.[^12] For the top 100 most highly cited papers, conventional citations counts and influence-primed citation counts have a high correlation. As we move down the list, the correlation drops. The two counts agree on the most highly ranked papers, but they disagree on the less cited papers. Weighting makes a difference. Table \[tab:authorRank\] shows the Spearman correlation coefficients for the AAN authors, under these two different counting methods. The of the authors were calculated and were used to rank the authors. For each group of authors, we calculate the Spearman correlation between the and the hip-indexes. Comparing Tables \[tab:paperRank\] and \[tab:authorRank\], we see that the authors’ correlations steadily decline as we go down the rows of Table \[tab:authorRank\], but the papers’ correlations fluctuate with no clear trend as we go down the rows of Table \[tab:paperRank\]. This indicates that conventional citation counts and influence-primed citation counts are not related by a simple linear transformation. That is, influence-primed counting is different from conventional counting in a non-trivial way. [rS]{} Authors & [Correlations]{}\ 1–100 & 0.74\ 101–200 & 0.49\ 201–300 & 0.21\ 301–400 & 0.40\ 401–500 & 0.22\ 501–600 & 0.10\ 601–700 & 0.01\ 701–800 & 0.03\ 801–900 & -0.04\ 901–1000 & 0.03\ Identifying ACL Fellows {#identifying-acl-fellows .unnumbered} ----------------------- The Association for Computational Linguistics has seventeen fellows.[^13] We might assume that these seventeen fellows can be identified by selecting the authors having the best scores. Indeed, found that the was superior at predicting the future performance of a researcher than the number of citations, the number of papers, and mean citations per paper [@lehmann2006measures]. Table \[tab:fellow\] shows the precision of the conventional and influence-primed hip-index at identifying ACL Fellows. For a given value of $N$, we sort all authors in AAN in descending order of their and then count the number of ACL Fellows among the top $N$ authors in the sorted list. We also sort all authors in AAN in descending order of the hip-indexes and then count the number of ACL Fellows among the top $N$. Precision is the number of ACL Fellows in the top $N$ divided by $N$. Table \[tab:fellow\] shows the precision as $N$ ranges from 1 to 17: we stop at 17 because there are 17 ACL Fellows in total. For example, the second row in the body of the table, with $N = 2$, shows that zero of the top two ranked authors are ACL Fellows (precision 0%), but one of the top two hip-index ranked authors is an ACL Fellow (precision 50%). For $N = 3$, finds zero ACL Fellows but hip-index finds two Fellows. [SSSS]{} &\ [$N$]{} & [$h$-index]{} & & [hip-index]{}\ 1 & 0 & & 0\ 2 & 0 & \* & 50\ 3 & 0 & \* & 67\ 4 & 25 & \* & 50\ 5 & 20 & \* & 40\ 6 & 33 & & 33\ 7 & 29 & & 29\ 8 & 25 & & 25\ 9 & 22 & \* & 33\ 10 & 30 & \* & 40\ 11 & 36 & \* & 45\ 12 & 42 & & 42\ 13 & 38 & & 38\ 14 & 36 & & 36\ 15 & 33 & & 33\ 16 & 31 & & 31\ 17 & 29 & & 29\ & 10 & \* & 14\ The asterisks (\*) in the table mark where the two methods have different results. The table shows that the influence-primed model identified the ACL fellows with a better precision for seven values of $N$. When we limit $N$ to less than or equal to seventeen, hip-index is never worse but often better than . As $N$ grows, the differences between the two indexes become negligible. At $N=17$, both indexes identify 5 out 17 ACL Fellows. The last row of Table \[tab:fellow\] shows the *average precision* measure (AveP), which is commonly used to evaluate search engines [@Buckley:2000]. The formula for AveP is $$\textrm{AveP}(n_c) = \frac{\sum_{k=1}^{n_c}(P(k) \times \textrm{rel}(k))}{n_r} ,$$ where $n_c$ is the point at which we cut off the list of search results for a search engine (in our case, $n_c = 17$, where we cut off the ranked lists of authors), $n_r$ is the total number of relevant documents (in our case, the total number of ACL Fellows, $n_r = 17$), $P(k)$ is the precision at each observation point (in our case, $k$ ranges from 1 to 17), and $\textrm{rel}(k)$ is an indicator function that equals 1 if the document is relevant to the given query and 0 otherwise (in our case, $\textrm{rel}(k)$ is 1 if the author is an ACL Fellow and 0 otherwise). From the table, we can see that the AveP score of the influence-primed model is 14% whereas that of the conventional model is 10%. This better average precision measure is encouraging evidence that weighting the citations by our measures of influence could improve the identification of the best scientists. Future work and limitations {#future-work-and-limitations .unnumbered} =========================== Further work is needed to validate these results over more extensive and different datasets. We rely on authors to annotate their own papers. However, we did not assess the reliability of authors at identifying the key references. For papers with several authors, we could ask more than one author to provide annotation. Thus we could quantify the inter-annotator agreement. We could also ask the same authors, after a long delay, to annotate their own papers again. Furthermore, we would find it interesting to compare the performance of a machine learning approach with human-level performance. For this purpose, we could recruit independent annotators having specific degrees of expertise. We assumed that the full text of the citing paper was available. Yet some citation indexing databases have a more limited access to the content due to copyright restrictions or technical limitations. It should not be difficult to extend these databases so that they have the necessary information to identify influential references or to count the number of times a paper mentions another. For example, this information could be provided by the copyright owner without giving access to the full text. Intentionally, we limited our feature set so that we did not have to recover full text of the cited work (only the title). However, many other features are possible if we access the full text of both the citer and cited paper: measure similarity by the overlap in the reference section. We could also add other features such as the prestige of the cited venue or the prestige of the cited authors [@zitt2008modifying]. We also did not take into account the relationships between authors. Maybe authors who are similar or related are more likely to influence each other. In related work, proposed to measure the number of citers (authors who cite) rather than the number of citations. Moreover, identifying the genuinely significant citations might be viewed as an adversarial problem. Indeed, some authors and editors attempt to game (i.e., manipulate or exploit) citations counts. In a survey, found that 20% of all authors were coerced into citing some references by an editor, after their manuscript had undergone normal peer review. In fact, the majority of authors reported that they were willing to add superfluous citations if it is an implied requirement by an editor. If we could determine that many non-influential references in some journals are citing some specific journals, this could indicate unethical behavior. Our approach for identifying celebrated scientists is simple. State-of-the-art approaches such as that of can achieve better precision and recall. We expect that they would benefit from an identification of the influential citations. In some circumstances counting multiple occurrences of a citation might require coreference resolution [@Soon:2001; @Athar:2012:DIC:2391171.2391176]. By convention, some authors and editors only ever cite a reference once but mention it several times either with a nominal, or pronominal reference. Mazloumian [@mazloumian2012predicting] found that a useful predictor of future performance was the annual number of citations at the time of citation. Maybe the annual number of influential citations could be a superior predictor. A finer view of the problem is to rank or rate the references by their degrees of academic influence, which however could bring further complexity that we are avoiding in the present work; e.g., comparing two less influential or uninfluential references could be a harder task even for human annotators, and such annotation may be difficult to interpret. Nevertheless, a weighted citation measure based on the number of occurrences of a citation could significantly alter other evaluation metrics that depend on simple citation counts, such as Impact Factor [@Garfield:2006], Eigenfactor, and Article Influence [@bergstrom:2007]. Even though such a refinement would not address the *statistical* charges leveled against citation-based evaluation metrics [@Adler:2009], it would at least address to some degree the need to distinguish between citations that acknowledge an *intellectual debt* and *de-rigeur* citations. We conjecture that some types of research papers that tend to be highly cited, such as review or methodological articles, are less likely to be perceived as influential. found that weighting citations by the in-paper frequency reduced the importance of reviews. We should further investigate this issue to verify whether original contributions are significantly more likely to be perceived as influential. Conclusions {#conclusions .unnumbered} =========== One of our main results is that counting the number of times a paper is cited ($countsInPaper\_whole$) is one of the best predictors of how influential a reference is. This confirms an earlier result by who stated, “Citation frequency of individual articles in other papers more fairly measures their scientific contribution than mere presence in reference lists.” Alternatively, we can count the number of sections in which a paper is cited ($countsInPaper\_secNum$). We believe that in assessing the influence of a research paper or researcher, weighting the citations by these features (e.g., $countsInPaper\_whole$) would provide more robust results. It should also be used when tracking follow-up work or recommending research papers. We have also shown that we could combine the in-paper citation counts ($countsInPaper$) and the semantic relatedness between a reference and the citing paper, to derive a superior classifier. Though self-citations are only slightly correlated with academic influence, a classifier can derive some benefits when combining it with other features. Acknowledgments {#acknowledgments .unnumbered} =============== We are grateful to the volunteers who identified key citations in their own work. Daniel Lemire acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC) with grant number 26143. We thank M. Couture, V. Larivière and the anonymous reviewers for their helpful comments. [^1]: See <http://www.wjh.harvard.edu/~inquirer/> to obtain a copy of the General Inquirer lexicon. [^2]: See <http://wordnet.princeton.edu/> to download the WordNet lexicon. [^3]: This normalization leaves binary values unchanged, so it makes no difference whether it is applied to them. [^4]: See <http://tinyurl.com/counting-citations>. [^5]: See <http://tinyurl.com/influential-references>. [^6]: The dataset is freely available online at <http://lemire.me/citationdata/>. [^7]: See <http://opennlp.apache.org/> to download OpenNLP. [^8]: LIBSVM is available for download at <http://www.csie.ntu.edu.tw/~cjlin/libsvm/>. [^9]: The LIBSVM parameters we used are “-s 0 -d 2 -t 1 -r 1”. [^10]: With logistic regression, the precision and recall are 0.38 and 0.42. [^11]: The AAN corpus is available at <http://clair.eecs.umich.edu/aan/index.php>. [^12]: Spearman correlation is specifically intended for comparing ranked lists, whereas Pearson correlation is more appropriate when numerical values are more important than ranks. [^13]: ACL Fellows are listed at <http://aclweb.org/aclwiki/index.php?title=ACL_Fellows>. When we performed our experiments, there were only seventeen fellows, but there are more now.

--- abstract: 'Starting with a subclass of the four-dimensional spaces possessing two commuting Killing vectors and a non-trivial Killing tensor, we fully integrate Einstein’s vacuum equation with a cosmological constant. Although most of the solutions happen to be already known, we have found a solution that, as far as we could search for, has not been attained before. We also characterize the geometric properties of this new solution, it is a Kundt spacetime of Petrov type II possessing a null Killing vector field and an isometry algebra that is three-dimensional and abelian. In particular, such solution becomes a $pp$-wave spacetime when the cosmological constant is set to zero.' author: - Gabriel Luz Almeida and Carlos Batista title: A Class of Integrable Metrics II --- Introduction ============ Due to the nonlinearity of Einstein’s equation, it is virtually impossible to integrate it analytically without imposing restrictions over the initial ansatz. The most common way of doing so is by the imposition of symmetries. For instance, Schwarzschild solution has been found assuming that the spacetime has spherical symmetry, namely it has three Killing vectors whose Lie algebra is $\mathfrak{so}(3)$. Likewise, Kerr solution has been obtained relying on the existence of two commuting Killing vectors [@Kerr], i.e. the spacetime was assumed to be stationary and axisymmetric. It is important to keep in mind that the hypothesis of two commuting Killing vectors is not over-restrictive from the physical point of view, since the rigidity theorem states that the equilibrium state of an astronomical object should be stationary and axisymmetric [@HawkingRigidity; @Chrusciel:1996bj]. Besides the symmetries of the spacetime, which are generated by Killing vectors, one can also impose symmetries on the geodesic motion, which are generated by Killing tensors and Killing-Yano tensors [@Carter-KleinG; @Santillan]. Since the metric is always a Killing tensor, the existence of an extra Killing tensor along with two independent Killing vectors leads to four first integrals for the geodesic motion, which enables full integrability. Nevertheless, one might wonder whether it is plausible to assume the existence of a Killing tensor in physical spacetimes. The known examples tell us that the answer is yes. For instance, four-dimensional Kerr metric and, more generally, Kerr-NUT-(A)dS spacetimes in arbitrary dimension [@KerrNutAds], are all endowed with enough Killing tensors to allow the integrability of the geodesic motion [@Kubiz; @Krtous]. Thus, some of the most physically important exact solutions for Einstein’s vacuum equation are endowed with Killing tensors. In addition to being related to the integrability of the geodesic motion, these Killing tensors are also related to the integrability of field equations in such spacetimes, like scalar fields [@Frol-KG], electromagnetic fields [@KrtousMaxwell; @Teukolsky], and spin 1/2 fields [@OotaDirac]. Probably, the existence of these objects might also be related to the integrability of Einstein’s equation itself [@Yasui], as hinted by the successful integration of gravitational perturbations through the use of Killing tensors [@OotaGrav]. Moreover, these Killing and Killing-Yano tensors can play an important role in supersymmetric theories [@KY-SUSY; @Cariglia]. With these motivations in mind, in the present article we will search for solutions of Einstein’s vacuum equation with a cosmological constant within the class of spacetimes possessing a Killing tensor and two commuting Killing vectors. The general form of the spaces with such symmetry properties has been found by Benenti and Francaviglia in Ref. [@BenentiFrancaviglia] and is given by: $$\begin{aligned} g^{ab}\partial_{a}\partial_{b} =\frac{1}{S_{x}+S_{y}}\,& \Big[ \,G_{x}^{ij}\,\partial_{\sigma_i}\partial_{\sigma_j}\,+\,G_{y}^{ij}\,\partial_{\sigma_i}\partial_{\sigma_j} \nonumber\\ & \;\;\;\quad + \Delta_{x}\, \partial_{x}^{2}+\Delta_{y}\, \partial_{y}^{2}\, \Big] \, , \label{BFmetric}\end{aligned}$$ where functions with subscript $x$ are arbitrary functions of $x$, while those with subscript $y$ are arbitrary functions of $y$. For instance, $\Delta_{x} = \Delta_{x}(x)$. The indices $i,j$ run through $\{1,2\}$ and label the cyclic coordinates $\sigma_1$ and $\sigma_2$. Note that we can assume that $G_{x}^{ij} = G_{x}^{ji}$ and $G_{y}^{ij} = G_{y}^{ji}$, due to the symmetry of the metric. The rank two Killing tensor associated to this metric is given by $$\begin{aligned} \boldsymbol{K}\,=\,\frac{1}{S_{x}+S_{y}}\,\Big[& \,S_{x}\,G_{y}^{ij}\, \partial_{\sigma_i}\partial_{\sigma_j} + S_{x}\,\Delta_{y} \,\partial_{y}^{2} \nonumber\\ & - S_{y}\,G_{x}^{ij}\,\partial_{\sigma_i}\partial_{\sigma_j} - S_{y} \,\Delta_{x} \, \partial_{x}^{2} \, \Big] \,. \label{KillingT1}\end{aligned}$$ In recent previous works we have already exploited the integrability of Einstein’s equation of some spaces within the class of metrics (\[BFmetric\]). In Ref. [@AnabalonBatista], one of us (C.B.) along with A. Anabalón investigated the subcase in which the determinants of the matrices $G_x^{ij}$ and $G_y^{ij}$ are both zero. It has been found that Einstein’s vacuum equation with a cosmological constant is fully integrable for such a subcase, with Kerr-NUT-(a)dS being a particular solution. Latter, the present authors also considered the subcase of vanishing determinant for $G_x^{ij}$ and $G_y^{ij}$ but, instead of vacuum, a gauge field of arbitrary gauge algebra have been considered as a source for the gravitational field [@GabrielBatista]. In particular, new exact solutions have been attained in Ref. [@GabrielBatista]. Now, the idea is to explore another subcase of the class of spaces (\[BFmetric\]). Namely, the one in which one of the matrices $G_{x}^{ij}$ or $G_{y}^{ij}$ vanishes identically. For definiteness, we shall assume $G_x^{ij} = 0$. In this case it is immediate to notice that the line element is given by $$\label{metric1} ds^2 = (S_x + S_y) \left[ H_y^{ij}\,d\sigma_i d\sigma_j + \frac{dx^2}{\Delta_x} + \frac{dy^2}{\Delta_y} \right] \,,$$ where $H_y^{ij}$ are arbitrary functions of $y$. Note that in the general case, when $G_{x}^{ij}$ and $G_{y}^{ij}$ are both nonzero, the line element would have the same algebraic structure above, but the components $H^{ij}$ would be convoluted combinations of functions of $x$ and functions of $y$. As we shall see in the sequel, Einstein’s vacuum equation for the class of spaces described by (\[metric1\]) is integrable. It will be shown that although most of the solutions found within this class are already known, we arrive at a particular solution that, as far as the authors know, has not been attained before. The outline of the article is the following. At the next section we start the integration of Einstein’s equation and conclude that the calculations should be split in three different cases depending on the constancy of the functions $S_x$ and $S_y$. The case in which both functions are constant is tackled in subsection \[SubSecA\], which yields flat spaces as the only solutions. Then, the case in which $S_y$ is constant while $S_x$ is non-constant is considered in subsection \[SubSecB\], with the only solutions being spaces of constant curvature. Finally, the case in which just $S_x$ is constant is considered in subsection \[SubSecC\]. During the integration process of the latter case we conclude that there is a special subcase that must be considered separately. In \[SubSecC1\] we treat the general case and arrive at a generalization of Kasner spacetime, while the special subcase is tackled in subsection \[SubSecC2\] and leads to a solution that, as far as the authors know, has not been described in the literature yet. Then, in Sec. \[Sec.NewSOL\] we investigate the geometrical features of the new solution. We show that this solution is a Kundt spacetime of Petrov type II possessing a null Killing vector field and that it reduces to a $pp$-wave spacetime when the cosmological constant vanishes. Its isometry algebra is three-dimensional and abelian, so that it is Bianchi type I, but, differently from the most known solutions of this type, the line element cannot be diagonalized using the cyclic coordinates associated to the Killing vectors. The regularity of the new solution and its asymptotic form are also investigated. The conclusions and perspectives are presented at Sec. \[Sec.Conc\]. At appendix \[AppendixA\] we show that at the asymptotic limit, the new solution goes to a Kasner spacetime. Integrating Einstein’s Equation {#Sec.Integration} =============================== The goal of this work is to integrate Einstein’s field equation in vacuum with a cosmological constant $\Lambda$. That is, we want to find the most general solution of the equation $$\label{Einsteinseq} R_{ab}=\Lambda g_{ab},$$ for line elements of the form (\[metric1\]), where $R_{ab}$ stands for the Ricci tensor. Nevertheless, before doing so, it is useful to replace the three arbitrary functions $H_y^{11}$, $H_y^{22}$ and $H_y^{12}=H_y^{21}$ appearing in (\[metric1\]) by the three functions $P_y$, $Q_y$ and $\Omega_y$ defined in a way that the line element assumes the following form: $$\begin{aligned} \label{metric2} ds^2=& \,S \,\Big( -\frac{1}{\Omega_y}d\sigma_1^2+ \frac{Q_y^2-P_y^2}{\Omega_y}d\sigma_2^2 \nonumber\\ & \quad \;\;+ \frac{2P_y}{\Omega_y} d\sigma_1 d\sigma_2 +\frac{dx^2}{\Delta_x}+\frac{dy^2}{\Delta_y} \,\Big),\end{aligned}$$ where $S=S_x+S_y$. This represents no loss of generality. Now, an immediate integration of the component $R_{\sigma_1}^{{\phantom}{\sigma_1}\sigma_2}=0$ of Einstein’s equation for the function $\Delta_y$ provides $$\label{Dy} \Delta_y=\frac{c_1 \,Q_y^2 \,\Omega_y^2}{(S_x + S_y)^2(P_y')^2} \,,$$ where $c_1$ is an arbitrary integration constant and the prime denotes a derivative with respect to the variable on which a function depends. Although Eq. (\[Dy\]) is correct when $S_x$ is a constant function, such equation cannot be used when $S_x$ is non-constant, otherwise $\Delta_y$ would also depend on $x$. Thus, the case in which $S_x$ is a non-constant function of $x$ must be handled with special care[^1]. Doing so, we find that the equation $R_{\sigma_1}^{{\phantom}{\sigma_1}\sigma_2}=0$ yields the following constraints: $$\label{Dy2} \Delta_y=\frac{c_1 \,Q_y^2 \,\Omega_y^2}{(P_y')^2} \;\; \textrm{ and } \;\; S'_y = 0 \,. \;\;\; (\textrm{when } S'_x\neq 0)$$ In order to attain both of the expressions and , we have considered that $P_y'\neq 0$. The special case in which $P_y$ is constant will be considered latter. Now, assuming either or to hold, and then integrating $R_{\sigma_2}^{{\phantom}{\sigma_2}\sigma_1}=0$, we find that in both cases $Q_y$ must be given by $$\label{Qy} Q_y=\sqrt{(P_y-a_1)(P_y-a_2)} \,,$$ with $a_1$ and $a_2$ being arbitrary integration constants. Also, irrespective of assuming the latter expressions for $\Delta_y$ and $Q_y$, the integration of the component $R_x{}^y=0$ leads to the following constraint: $$S_x'\,S_y'=0 \,.$$ Thus, we face three possible cases to be followed depending on whether the functions $S_x(x)$ and $S_y(y)$ are constant or not. Namely, (A) the functions $S_x$ and $S_y$ are both constant, (B) $S_x$ is non-constant and $S_y$ constant, and (C) $S_x$ is constant and $S_y$ non-constant. Particularly, note that in cases (A) and (C), $\Delta_y$ is given by Eq. , while in the case (B) we must use Eq. . In the following section, each of these three cases will be treated separately. As we shall see, the cases (A) and (B) do not provide any particularly interesting solutions, while the case (C) will lead to solutions with richer physics: a generalization of Kasner solution, that is already available in the literature, and a new solution of Petrov type II possessing a null Killing vector field and whose isometry algebra is three-dimensional and abelian. Subcase $S_x'=0$ and $S_y'=0$ {#SubSecA} ----------------------------- In this subsection we investigate the simplest of the three possible cases regarding the constancy of the functions $S_x$ and $S_y$, namely we shall consider that they are both constant. This gives rise to a constant conformal factor $S$ which can be easily incorporated into the coordinates by a scaling transformation, so that we can set $$S= S_x + S_y = 1 \,.$$ Then, assuming $S=1$, along with Eqs. (\[Dy\]) and (\[Qy\]) for $\Delta_y$ and $Q_y$, it follows that $R_{x}^{{\phantom}{x}x}$ is automatically zero, so that the equation $R_{x}^{{\phantom}{x}x} = \Lambda \delta^x_x$ states that the cosmological constant must vanish, $\Lambda = 0$. Then, integrating $R_{\sigma_1}^{{\phantom}{\sigma_1}\sigma_1}=\Lambda=0$, we find $$\label{Omegay1} \Omega_y=c_2\,(P_y-a_1)^{d} \, (P_y-a_2)^{1-d} \,,$$ where $c_2$ and $d$ are arbitrary integration constants. In order to attain (\[Omegay1\]) it was necessary to assume $a_1 \neq a_2$. Indeed, the special case $a_1=a_2$ would lead to a different expression for $\Omega_y$, but let us put this particular case aside and deal with it at the end of this subsection. With the latter expressions for $S$, $\Delta_y$, $Q_y$, and $\Omega_y$ at hand, the equation $R_{y}^{{\phantom}{y}y}=\Lambda=0$ leads to the constraint $d = 0$. Actually, another possibility for solving $R_{y}^{{\phantom}{y}y}=0$ is $d=1$, but this is equivalent to $d=0$ when we interchange the arbitrary constants $a_1$ and $a_2$, so that we just need to consider $d=0$. Thus, $\Omega_y$ should be given by: $$\Omega_y=c_2\, (P_y-a_2) \,.$$ With this expression for $\Omega_y$ along with the latter expressions for $S$, $Q_y$ and $\Delta_y$, it follows that the Riemann tensor is identically zero. Thus, the solution is the flat space. In particular, the Ricci tensor vanishes, so that we must have $\Lambda= 0$. In the latter integration, we have excluded two possibilities, namely the case $a_1=a_2$ and the case in which $P_y$ is a constant function. Nevertheless, integrating these cases separately we have checked that, in both circumstances, the solution can only be attained for $\Lambda=0$ and that, likewise, these solutions turn out to be flat spaces. Thus, summing up, the case considered in this subsection, namely $S_x'=0$ and $S_y'=0$, do not lead to any interesting solution. More precisely, all solutions in such subcase are flat. Subcase $S_x'\neq0$ and $S_y'=0$ {#SubSecB} -------------------------------- Now, let us integrate Einstein’s vacuum equation for the subcase $S_x'\neq0$ and $S_y'=0$. Since the functions $S_x$ and $S_y$ appear in the metric only through the combination $S_x + S_y$, it follows that the constant value of $S_y$ can be absorbed into $S_x$. Thus, without loss of generality, we can set $$S_y = 0 \,.$$ Assuming that $P_y'\neq 0 $, it follows that $\Delta_y$ and $Q_y$ should be given by Eqs. (\[Dy2\]) and (\[Qy\]), respectively. With these at hand, it follows that integration of the component $R_{\sigma_1}^{{\phantom}{\sigma_1}\sigma_1}- R_{y}^{{\phantom}{y}y}=0$ of Einstein’s equation yields $$\label{SSy} \Omega_y = c_2 + c_3 P_y,$$ with $c_2$ and $c_3$ being integration constants. Also, using the equation $R_{x}^{{\phantom}{x}x}=\Lambda$, we obtain $$\label{D1} \Delta_x=\frac{ c_4\, S_x^2 - 4 \Lambda \, S_x^3 }{ 3(S_x')^2 },$$ where $c_4$ is another integration constant. Finally, imposing $R_{\sigma_1}^{{\phantom}{\sigma_1}\sigma_1}= \Lambda$ we arrive at the following constraint on the integration parameters: $$\label{c2c4} c_4 = -3 c_1(c_2+a_1 c_3)(c_2+a_2 c_3)\,.$$ Then, once assumed that $c_4$ is given by Eq. (\[c2c4\]), it follows that Einstein’s vacuum equation $R_{a}^{{\phantom}{a}b}=\Lambda\delta_{a}^{{\phantom}{a}b}$ are fully obeyed. Nevertheless, one can check that this final solution has vanishing Weyl tensor, so that the Riemann tensor obeys $$R_{abcd}=\frac{1}{3}\Lambda \left(g_{ac}g_{bd}-g_{ad}g_{bc}\right).$$ Thus, the solution that we have found have constant curvature, i.e. they are de Sitter and anti de Sitter spacetimes when Lorentzian signature is assumed and $\Lambda\neq 0$, while it is the flat space for vanishing cosmological constant. A possibility that has not been considered yet for the present subcase ($S_x'\neq0$ and $S_y'=0$) is $P_y'=0$, in which case Eqs. (\[Dy2\]) and (\[Qy\]) are not valid. However, integrating Einstein’s equation for $S_y=0$ and $P_y'=0$ we also eventually find that the solution is a maximally symmetric space. Thus, all solutions of the subcase $S_x'\neq0$ and $S_y'=0$ turn out to be the “non-interesting” spaces of constant curvature. Subcase $S_x'=0$ and $S_y'\neq0$ {#SubSecC} -------------------------------- Finally, let us consider the subcase $S_x'=0$ and $S_y'\neq 0$, in which case we can, without loss of generality, absorb the constant value of $S_x$ into $S_y$ and set $$\label{Sx3} S_x = 0 \,.$$ Moreover, we can easily redefine the coordinate $x$ ($dx\rightarrow d\tilde{x} = dx/\sqrt{\Delta_x}$) in order to eliminate the function $\Delta_x$. Doing so, and dropping the tilde over the new coordinate, we find that this is equivalent to setting $$\label{Dx3} \Delta_x = 1 \,.$$ In particular, note that due to Eqs. (\[Sx3\]) and (\[Dx3\]) the metric is independent of the coordinate $x$. Thus, besides the Killing vector fields $\partial_{\sigma_1}$ and $\partial_{\sigma_2}$, $\partial_{x}$ does also generate a symmetry. These three independent Killing vector fields commute with each other and, therefore, yields an abelian three-dimensional algebra. According to Bianchi’s classification of three-dimensional Lie Algebras, this isometry algebra is of Bianchi type I [@LBianchi]. Moreover, it is worth noting that the Killing tensor (\[KillingT1\]) is trivial in this subcase. Indeed, with the choices (\[Sx3\]) and (\[Dx3\]) we get ${\boldsymbol}{K}=-\partial_x^2$, so that the first integral associated to ${\boldsymbol}{K}$ for the geodesic motion is just the square of the one associated to the Killing vector $\partial_{x}$ [@Santillan]. Postponing the analysis of the special case in which $P_y$ is constant, we can assume expressions (\[Dy\]) and (\[Qy\]) to hold. Doing so, and using and , it follows from the integration of $R_{\sigma_1}^{{\phantom}{\sigma_1}\sigma_1}- R_{x}^{{\phantom}{x}x}=0$ that $\Omega_y$ must be given by $$\label{Omegay3} \Omega_y=c_2(P_y-a_1)^{d}(P_y-a_2)^{1-d},$$ where $c_2$ and $d$ are arbitrary integration constants. Then, assuming to hold, it follows from the integration of $R_{\sigma_1}^{{\phantom}{\sigma_1}\sigma_1}- R_{y}^{{\phantom}{y}y}=0$ that $$\label{S2} S_y=\left[b_1\left(\frac{P_y-a_1}{P_y-a_2}\right)^{d_+} + b_2\left(\frac{P_y-a_1}{P_y-a_2}\right)^{d_-}\right]^{-2/3}.$$ In the above expression, while $b_1$ and $b_2$ are, for the time being, arbitrary integration constants, $d_{\pm}$ are not arbitrary, rather they are given in terms of $d$ by: $$\label{epm} d_{\pm} =\frac{1}{2}\left[1 - 2\,d \pm \sqrt{d(d-1)+1}\right] \,.$$ Finally, integrating $R_x^{{\phantom}{x}x}=\Lambda$, we find that $b_1$ and $b_2$ must be constrained by the following relation: $$\label{b1b2} b_1 b_2 = \frac{3\Lambda}{c_1 c_2^2(a_1-a_2)^2[d(d-1)+1]} \,.$$ In order for the latter expression to be meaningful we need to have $a_1\neq a_2$. The special case $a_1=a_2$ will be considered latter. With the above prescriptions, namely Eqs. (\[Dy\]), (\[Qy\]), and (\[Sx3\])-(\[b1b2\]), we have that Einstein’s vacuum equation is fully obeyed. Notice that in this solution the function $P_y$, apart from being non-constant, has not been constrained. This freedom on the choice of $P_y$ is expected from the fact that in the metric we could, for instance, have set $\Delta_y=1$ by means of a redefinition of the coordinate $y$. Thus, we have started with more degrees of freedom than necessary. The important point is that different choices of $P_y$ can be understood as different choices of the coordinate $y$ and, therefore, represent the same physical space. ### Turning the metric into a diagonal form {#SubSecC1} Now, let us try to identify the solution just found. Integrating the Killing equation, we can check that this solution admits no other independent generators of symmetries besides the commuting Killing vector fields $\partial_{\sigma_1}$, $\partial_{\sigma_2}$, and $\partial_{x}$. Thus, this solution is, indeed, a Bianchi Type I space. A well-known class of spacetimes that are Bianchi type I are the so-called *Bianchi type I cosmological spacetimes*, which have the diagonal form $$\label{bianchiI} ds^2=-d\tau^2+(A^1_{\tau})^2dz_1^2+ (A^2_{\tau})^2 dz_2^2+ (A^3_{\tau})^2 dz_3^2 \,,$$ where $A^1_{\tau}$, $A^2_{\tau}$ and $A^3_{\tau}$ are arbitrary functions of $\tau$. These spacetimes are used by cosmologists to incorporate anisotropy at the space-like hyper-surfaces $\tau=constant$, providing a generalization of the FRW cosmological model [@Jacobs]. The Killing vectors $\partial_{z_1}$, $\partial_{z_2}$ and $\partial_{z_3}$ generate a three-dimensional abelian isometry group, so that the isometry algebra is of Bianchi type I. This isometry group acts transitively on the three-dimensional hyper-surfaces given by $\tau = constant$. The diagonal form of this line element indicates that the coordinate vector fields are orthogonal to family hyper-surfaces. In particular, the Killing vectors $\partial_{z_1}$, $\partial_{z_2}$, and $\partial_{z_3}$ are hyper-surface orthogonal. For instance, $\partial_{z_1}$ is orthogonal to the hyper-surfaces $z_1=constant$. Coming back to our Bianchi type I solution found in the present subsection, one can see that while $\partial_x$ is a hyper-surface orthogonal Killing vector, the existence of the term $d\sigma_1 d\sigma_2$ in the line element (\[metric2\]) indicates that the Killing vector fields $\partial_{\sigma_1}$ and $\partial_{\sigma_2}$ are not orthogonal to families of hyper-surfaces. Indeed, we can check that $$(\partial_{\sigma_1})_{[a}\nabla_b(\partial_{\sigma_1})_{c]}\neq0 \quad \text{and} \quad (\partial_{\sigma_2})_{[a}\nabla_b(\partial_{\sigma_2})_{c]}\neq0 \,.$$ Thus, let us try to find two independent Killing vector fields that are orthogonal to families of hyper-surfaces to replace $\partial_{\sigma_1}$ and $\partial_{\sigma_2}$. Defining the Killing vector field $${\boldsymbol}{k} = \alpha\,\partial_{\sigma_1} + \partial_{\sigma_2}$$ and imposing the condition $k_{[a}\nabla_bk_{b]}=0$, one can find that as long as $Q_y=\sqrt{(P-a_1)(P-a_2)}$, irrespective of form of the other functions appearing in the line element (\[metric2\]), we end up with two possible values for the constant parameter $\alpha$: either $\alpha=a_1$ or $\alpha=a_2$. Thus, whenever $a_1\neq a_2$ we can exchange the independent Killing vector fields $\partial_{\sigma_1}$ and $\partial_{\sigma_2}$ by $$\label{k1k2} {\boldsymbol}{k_1} = a_1\partial_{\sigma_1}+\partial_{\sigma_2} \quad \text{and} \quad {\boldsymbol}{k_2} = a_2\partial_{\sigma_1}+\partial_{\sigma_1}\,,$$ which are also independent if $a_1\neq a_2$. The important point is that ${\boldsymbol}{k_1}$ and ${\boldsymbol}{k_2}$ are hyper-surface orthogonal, differently from $\partial_{\sigma_1}$ and $\partial_{\sigma_2}$. Since ${\boldsymbol}{k_1}$ and ${\boldsymbol}{k_2}$ commute with each other, we can associate to them coordinates $\phi_1$ and $\phi_2$ such that ${\boldsymbol}{k_1} = \partial_{\phi_1}$ and ${\boldsymbol}{k_2} = \partial_{\phi_2}$. Indeed, $\phi_1$ and $\phi_2$ are defined by $$\sigma_1 = a_1 \phi_1 + a_2 \phi_2 \quad \text{and} \quad \sigma_2= \phi_1 +\phi_2 \,.$$ In terms of these coordinates, the line element (\[metric2\]) takes the form below: $$\begin{aligned} ds^2=& \frac{S_y}{\Delta_y}dy^2 -\frac{(a_2-a_1)(P_y-a_1)S_y}{\Omega_y}d\phi_1^2 \nonumber \\ &+\frac{(a_2-a_1)(P_y-a_2)S_y}{\Omega_y}d\phi_2^2 + S_ydx^2. \label{metric4}\end{aligned}$$ This diagonal line element can be easily put in the general form (\[bianchiI\]) by redefining the coordinate $y$. The fact that the investigated solution could be diagonalized using three cyclic coordinates, $\phi_1$, $\phi_2$ and $x$, could be anticipated from the fact that if we take a general Killing vector field, ${\boldsymbol}{\eta} = \lambda_1 \partial_{\sigma_1} + \lambda_2 \partial_{\sigma_2} + \lambda_3 \partial_{x} $ and compute its squared norm, we will conclude that if $a_1\neq a_2$ then $\eta^a\eta_a = 0$ only if $\lambda_1=\lambda_2=\lambda_3 = 0$. Thus, the hyper-surfaces $y=constant$, spanned by the Killing vector fields, have metrics that are either positive-definite or negative-definite. In this circumstance, there is a result on the literature stating that the metric can be diagonalized. Indeed, in [@Jacobs] it is shown that it is always possible to diagonalize a metric of the form $ds^2=-dt^2+\gamma_{ij}dx^i dx^j$, where $\gamma_{ij}$ is a positive/negative-definite three-dimensional metric, whenever the Einstein’s vacuum equation with cosmological constant is imposed. Nevertheless, for the case in which $a_1=a_2$ we can have a non-zero light-like Killing vector, so that the diagonalization cannot be attained using cyclic coordinates. Remember that the non-constant function $P_y$ has not been constrained, which was a consequence of the freedom in the choice of the coordinate $y$, as argued above. Thus, without any loss of generality, we can set $$\label{Py} P_y = \frac{a_2 F_y -a_1}{F_y -1} \,,$$ with the function $F_y$ being defined by $$F_y=\left[\sqrt{\frac{b_2}{b_1}}\tan\left(\frac{\sqrt{3\Lambda}y}{2}\right)\right]^{2/\sqrt{d(d-1)+1}}.$$ This choice in the coordinate $y$ was made so that the component $g_{yy}$ of the metric became equal to the unit. Then, assuming (\[Py\]) to hold and replacing the cyclic coordinates $\phi_1$, $\phi_2$ and $x$ by their rescaled versions defined by $$\begin{aligned} x_1 &= \sqrt{ \frac{(3\Lambda)^{p_1}(a_1-a_2) b_2^{p_1-2/3}}{2^{2p_1} \,c_2\, b_1^{p_1}}} \,\, \phi_1 \,,\\ x_2 &= \sqrt{\frac{(3\Lambda)^{p_2} (a_2-a_1) b_2^{p_2-2/3}}{2^{2p_2} \,c_2\, b_1^{p_2}}} \,\, \phi_2 \,, \\ x_3 &= \sqrt{\frac{(3\Lambda)^{p_3} b_2^{p_3-2/3} }{ 2^{2p_3} \ b_1^{p_3}}} \,\, x \,,\end{aligned}$$ with the constant parameters $p_i$ given by $$\begin{aligned} p_1&=\frac{2-d}{3\sqrt{d(d-1)+1}} + \frac{1}{3}\,,\\ p_2 &=-\,\frac{d+1}{3\sqrt{d(d-1)+1}} + \frac{1}{3} ,\\ p_3&=\frac{2d-1}{3\sqrt{d(d-1)+1}} + \frac{1}{3} ,\end{aligned}$$ it follows that the line element (\[metric4\]) becomes $$\label{KasnerM} ds^2=dy^2+L_y^{2/3}\Bigg[ \sum_{i=1}^{3}e^{2(p_i-\frac{1}{3})N_y}(dx_i)^2\Bigg]\,,$$ where $$L_y =\frac{\sin(\sqrt{3\Lambda}y)}{\sqrt{3\Lambda}} \, ,\;\; N_y =\textrm{Log}\left[\frac{2 \,\tan\left( \sqrt{3\Lambda}y/2\right)}{\sqrt{3\Lambda}} \right].$$ Note that the parameters $p_i$ obey the following constraint. $$\sum_{i=1}^{3} p_i= 1\,, \; \textrm{ and } \; \sum_{i=1}^{3} p_i^2 = 1.$$ The solution (\[KasnerM\]) is a generalization of the Kasner metric for the case in which the cosmological constant is different from zero. This particular solution is already available in the literature, see chapter 13 of Ref. [@Stephani]. In the limit $\Lambda\rightarrow 0$ the solution becomes $$ds^2=dy^2+y^{2p_1}dx_1^2+ y^{2p_2}dx_2^2+y^{2p_3}dx_3^2,$$ which is Kasner Metric [@Stephani; @EKasner]. Such a solution is used in cosmology to model an anisotropic vacuum universe [@Jacobs]. In order to obtain the latter solution we have avoided two special cases, namely we have assumed that $P_y$ is non-constant, so that (\[Dy\]) hold, and have assumed $a_1\neq a_2$. Thus, for completeness, we should also tackle these cases. First, considering $P_y$ constant and following steps analogous to the ones adopted above, one can check that solutions can be attained but all these solutions are either equivalent to (\[KasnerM\]) or one of its subcases. So, the special case in which $P_y$ is constant does not lead to new solutions. Differently, the special case $a_1=a_2$ will yield a new solution that is not available in the literature. In the case $a_1=a_2$, Eqs. (\[S2\]) and (\[b1b2\]) are not valid so that the calculations should be done separately, which we shall do in the next subsection. Note that in this special case the Killing vectors (\[k1k2\]) are not independent from each other, so that the diagonal form above cannot be attained, as hinted by the fact that the coordinates $\phi_1$ and $\phi_2$ are proportional to each other when $a_1=a_2$. ### The special case $a_1=a_2$ {#SubSecC2} The special case $a_1=a_2$ will be considered in the present section. It turns out that this will be the most interesting case, since, as far as the authors know, the obtained solution has not been described in the literature yet. In the sequel, we will assume $$\label{SxDxDy} S_x=0 \;,\;\; \Delta_x=1 \;,\; \textrm{and } \; \Delta_y=\frac{c_1Q_y^2\Omega_y^2}{S_y^2(P_y')^2}\,,$$ as assumed for the general case, whereas the function $Q_y$ reduces to $$\label{delta2Q} Q_y = P_y-a_1 \,,$$ since now $a_1=a_2$. Then, from the integration of the equation $R_{\sigma_1}^{{\phantom}{\sigma_1}\sigma_1}- R_{x}^{{\phantom}{x}x}=0$, we obtain $$\label{Sk} \Omega_y=c_2\,Q_y\,e^{-\tilde{d}/Q_y},$$ where $c_2$ and $\tilde{d}$ are arbitrary integration constants. Using this result for integrating the equation $R_{\sigma_1}^{{\phantom}{\sigma_1}\sigma_1}- R_{y}^{{\phantom}{y}y}=0$, we find that $$\label{S2k} S_y=\left[ b_1 e^{3\tilde{d}/(2Q_y)} + b_2\, e^{\tilde{d}/(2Q_y)} \right]^{-2/3},$$ with $b_1$ and $b_2$ being arbitrary integration constants. Finally, solving $R_{\sigma_1}^{{\phantom}{\sigma_1}\sigma_1}=\Lambda$, we conclude that the constants $b_1$ and $b_2$ must be related to $\Lambda$ as follows: $$\label{b1b2k} b_1 b_2=\frac{3\Lambda}{ c_1 \,c_2^2 \, \tilde{d}^2}.$$ This concludes the integration, as it can be checked that the remaining components of Einstein’s equations are obeyed. Thus, we have completely integrated Einstein’s equations for the particular case in which $a_1=a_2$, the general solution being given by the line element (\[metric2\]) with its functions given by (\[SxDxDy\])-(\[b1b2k\]). An interesting fact is that this solution for the case $a_1=a_2$ can be obtained from the case $a_1\neq a_2$ by defining $$d = \frac{\tilde{d}}{a_1 - a_2}$$ and then taking the singular limit $a_2\rightarrow a_1$ in the expressions (\[Omegay3\]), (\[S2\]), and (\[b1b2\]). Now, let us try to put the solution just found in a neater form. First, let us make use of the degree of freedom on the choice of $P_y$ to set $$P_y = y \,.$$ As explained before, this amounts to no loss of generality. Then, we shall perform the coordinate transformation $(\sigma_1, \sigma_2, x, y)\rightarrow (t,\phi,\theta,r)$, where the new coordinates are defined by $$\begin{aligned} \sigma_1 &= - \frac{\sqrt{c_2} \, b_1^{1/3}}{2\sqrt{\tilde{d}} } \left[ (\tilde{d}+ a_1\,\tilde{c}) \,t - a_1 \phi\right] \\ \sigma_2 &= \frac{\sqrt{c_2} \, b_1^{1/3}}{2 a_1 \sqrt{\tilde{d}} } \left[ (\tilde{d}- a_1\,\tilde{c}) \,t + a_1 \phi\right]\\ x&= b_1^{1/3} \, e^{(a_1\tilde{c}-\tilde{d})/(2a_1)} \,\, \theta \,,\\ y &= \frac{a_1^2\,(r + \tilde{c} )}{a_1 \, r + a_1\tilde{c}-\tilde{d}} \, ,\end{aligned}$$ with the constant $\tilde{c}$ standing for $$\tilde{c} = \frac{\tilde{d}}{a_1}-\log(c_1\,c_2^2\,b_1^2\, \tilde{d}^2 /3)\,.$$ In terms of these new coordinates the line element is given by $$\label{metric23} ds^2=\frac{e^{-r} dr^2}{3(1+\Lambda e^{-r})^2}+\frac{e^{-r}d\theta^2-dt(r\, dt+d\phi)}{(1+ \Lambda e^{-r} )^{2/3}}\,.$$ Notice that we were able to get rid of all of the integration constants, so that this solution depends just on the cosmological constant, which is an external parameter. In these coordinates the metric is Lorentzian, although the signature could be easily changed by means of Wick rotations. Analyzing The New Solution {#Sec.NewSOL} =========================== In this section we shall analyze the geometrical properties of the line element (\[metric23\]) aiming the identification of the spacetime. As we will argue in the sequel, such analysis hints that the metric given in Eq. (\[metric23\]) might be a new exact solution for Einstein’s vacuum equation. In order to arrive at this conclusion, we have tried to characterize this line element as much as possible and then looked for known solutions with the same geometrical features. The bottom line is that as far as the authors were able to investigate, the solution (\[metric23\]) has not been defined in the literature yet. First, let us point out that the special case of vanishing cosmological constant of the solution (\[metric23\]) is already described in the literature. Indeed, when $\Lambda=0$ it follows that $\partial_\phi$ is a covariantly constant null vector field, so that the line element represents a $pp$-wave spacetime [@Stephani; @EhlersKundt]. The $pp$-wave spacetimes are Petrov type $N$ and all their curvature scalars vanish identically (VSI spacetimes), for more in this class of spaces see Ref. [@VPravdaEtAl]. Thus, it remains to analyze the general case $\Lambda\neq0$. A good starting point is to investigate the isometry group of the solution (\[metric23\]). A complete integration of the Killing equation yields that the isometry group is three-dimensional and abelian, with the trivial Killing vectors $\partial_t$, $\partial_\theta$, and $\partial_\phi$ being a basis for the isometry Lie algebra. So, the isometry algebra is of Bianchi type I. Forming a general linear combination of these Killing vectors, we can see that the only ones that are orthogonal to families of hyper-surfaces are $\partial_\theta$ and $\partial_\phi$. Moreover, note that the Killing vector field $\partial_\phi$ is null. In particular, the existence of a null Killing vector implies that the line element cannot be put in a diagonal form using cyclic coordinates, differently from the previous case $a_1\neq a_2$, see the discussion on the paragraph below Eq. (\[metric4\]). Besides studying the isometry group, another geometric way to characterize the solution is analysing its Petrov type. In order to do so, we need to use a so-called null tetrad frame $\{{\boldsymbol}{\ell},{\boldsymbol}{n},{\boldsymbol}{m},{\boldsymbol}{\bar{m}}\}$, in which the vector fields ${\boldsymbol}{\ell}$ and ${\boldsymbol}{n}$ are real, while ${\boldsymbol}{m}$ and ${\boldsymbol}{\bar{m}}$ are complex and conjugated to each other. The only non-vanishing inner products in such a frame are $\ell^a n_{a} = -1$ and $m^a \bar{m}_{a} = 1$. Using one of the null tetrad frames below, i.e. choosing either the $+$ frame or the $-$ frame, $$\begin{aligned} {\boldsymbol}{\ell} &= \partial_\phi \,, \\ {\boldsymbol}{n}_{\pm} &= \pm\frac{e^r\sqrt{2}}{\sqrt{\Lambda}} (1+ \Lambda e^{-r})^{2/3}(3+ \Lambda e^{-r})^{1/2} \, \partial_\theta \ + 2 (1+ \Lambda e^{-r})^{2/3} \partial_t + \frac{1}{ \Lambda } (1+ \Lambda e^{-r})^{2/3}[3 e^r + \Lambda(1-2r) ] \, \partial_\phi \,, \\ {\boldsymbol}{m}_{\pm} &= \frac{\sqrt{3}e^{r/2}}{\sqrt{2\,}} (1+ \Lambda e^{-r}) \, \partial_r \ + i\,\frac{e^{r/2}}{\sqrt{2}} (1+ \Lambda e^{-r})^{1/3} \partial_\theta \pm i\, \frac{e^{r/2}}{\sqrt{\Lambda}} (3+ \Lambda e^{-r})^{1/2} (1+ \Lambda e^{-r})^{1/3} \, \partial_\phi \,, \\ {\boldsymbol}{\bar{m}}_{\pm} &= \frac{\sqrt{3}e^{r/2}}{\sqrt{2\,}} (1+ \Lambda e^{-r}) \, \partial_r \ - i\,\frac{e^{r/2}}{\sqrt{2}} (1+ \Lambda e^{-r})^{1/3} \partial_\theta \mp i\, \frac{e^{r/2}}{\sqrt{\Lambda}} (3+ \Lambda e^{-r})^{1/2} (1+ \Lambda e^{-r})^{1/3} \, \partial_\phi \,,\end{aligned}$$ it follows that the only Weyl scalars different from zero are, respectively, $$\begin{aligned} \Psi_2 &= \frac{\Lambda}{6} (1+ \Lambda e^{-r}) \, ,\, \textrm{ and} \\ \Psi_3 &=\mp \,i \sqrt{\frac{\Lambda e^r}{4 }} (1+ \Lambda e^{-r})^{4/3} (3+ \Lambda e^{-r})^{1/2}\,.\end{aligned}$$ The fact that $\Psi_0$, $\Psi_1$, and $\Psi_4$ all vanish in these frames means that ${\boldsymbol}{\ell}=\partial_\phi$ is a repeated principal null direction of the Weyl tensor, while ${\boldsymbol}{n}_{\pm}$ are non-degenerated principal null directions. Moreover, this implies that the Weyl tensor is of Petrov type II. For some review on the Petrov classification, see Ref. [@Bat-Book-art2]. Another important geometric characterization of this spacetime is that the null vector field $\partial_\phi$ is geodesic, shear-free, twist-free, and expansion-free. This means that the above solution is contained in the Kundt class of spacetimes. For a recent review on this class of spacetimes see [@ColeyPapadopoulos]. All the above features of the solution (\[metric23\]) have been extensively used in order to try to find it in the literature. In particular, a thorough search has been performed by the authors on the books [@Stephani; @GrifPodol-Book]. In fact, the closest that we could get from finding such a solution in the literature was in chapter 31 of Stephani et. al. book [@Stephani], where they exhibit the Kundt’s class of spacetimes. In particular, for solutions of Petrov type II with non-zero cosmological constant, the authors of [@Stephani] refer to two papers, [@Garcia] and [@Khlebnikov], where special solutions in such a class of spacetimes are found. However, our solution (\[metric23\]) could not be found there, inasmuch as their solutions contain strictly nonzero electromagnetic fields. In light of this, it seems to the authors of the present paper that the spacetime described by the metric (\[metric23\]) has not been presented in the literature so far, being a new solution of Einstein’s field equations with cosmological constant. Actually, the analysis of the existing literature revealed that there are few known exact vacuum solutions of Petrov type II. In contrast, solutions of Petrov type D are much more abundant. For instance, W. Kinnersley has been able to fully integrate Einstein’s vacuum equation with vanishing cosmological constant for the entire class of type D spacetimes [@typeD], yielding a plethora of solutions, a particular example being Kerr metric. Concerning the regularity of the line element (\[metric23\]), it seems that it is regular at all range of the coordinate $r$ except for the point $r=-\infty$ and when the denominator $(1+\Lambda e^{-r})$ vanishes. Computing some curvature scalars we have found the following pattern: $$\begin{aligned} & R^{a_1b_1}_{{\phantom}{a_1b_1}a_2b_2} R^{a_2b_2}_{{\phantom}{a_1b_1}a_3b_3} \cdots R^{a_nb_n}_{{\phantom}{a_nb_n}a_1b_1} = \nonumber\\ & \quad 4 \sum_{j=0}^{n}\,\binom{n}{j}\, \frac{e^{-jr} }{3^j} \Lambda^{n+j} + \frac{2}{3^n} (-2 \Lambda^2 e^{-r})^n\,, \label{Scalars2}\end{aligned}$$ where $R_{abcd}$ stands for the Riemann tensor. Note that all these scalars are finite for $r\neq - \infty$. On the other hand, in the limit $r\rightarrow- \infty$ these scalars diverge exponentially like $e^{n|r|}$. Thus, the point $r=-\infty$ is a singularity of the spacetime, while other points are regular. Likewise, computing the curvature scalar $$\label{DR2} \nabla^{a}R^{bcde}\nabla_{a}R_{bcde} = \frac{20}{3} \Lambda^4 (1+\Lambda e^{-r})^2 e^{-r}\,,$$ we check that there is a divergence just at $r=-\infty$. Note that when the cosmological constant is negative the denominator $(1+\Lambda e^{-r})$ can vanish, which could indicate the existence of a real singularity at $r= \log(-\Lambda)$, inasmuch as the line element (\[metric23\]) blows up. However, the fact that the curvature scalars (\[Scalars2\]) and (\[DR2\]) are perfectly regular at $r= \log(-\Lambda)$ reveals that this is not the case. In other words, the divergence of the metric components at $r= \log(-\Lambda)$, when $\Lambda < 0$, is just a coordinate singularity. The asymptotic limit $r\rightarrow \infty$ has a particularly simple structure concerning the curvature scalars. While Eq. (\[DR2\]) reveals that the square of the derivative of the curvature tensor goes to zero in this limit, the powers of the Riemann tensor given in Eq. (\[Scalars2\]) goes to $4\Lambda^n$ when $r\rightarrow \infty$. Such a simple structure reminds of spaces of constant curvature like (anti-)de Sitter, $(a)dS_4$, which is a four-dimensional Lorentzian space of constant curvature, and (anti-)Nariai, $(a)N_4$, which is a solution of Einstein’s equation that is the direct product of two spaces of constant curvature. However, although these two spacetimes have covariantly constant Riemann tensors, so that $\nabla^{a}R^{bcde}\nabla_{a}R_{bcde}=0$, in agreement with the behaviour of Eq. (\[DR2\]) in the limit $r\rightarrow \infty$, the powers of the Riemann tensor differ from the ones of our spacetime. Instead of $4\Lambda^n$, which is obtained from Eq. (\[Scalars2\]) in the limit $r\rightarrow \infty$, for these solutions we have $$R^{a_1b_1}_{{\phantom}{a_1b_1}a_2b_2} \cdots R^{a_nb_n}_{{\phantom}{a_nb_n}a_1b_1} = \left\{ \begin{array}{ll} (a)dS_4:\;\; 6 (2\Lambda/3)^n \,, \\ \quad \\ (a)N_4:\;\; 2 (2\Lambda)^n\,. \end{array} \right.$$ Thus, we can state that the new solution is neither asymptotically $(a)dS_4$ nor asymptotically $(a)N_4$. In order to investigate the asymptotic limit of our solution, we shall focus on the block related to $dt$ and $d\phi$ in the line element (\[metric23\]), namely let us consider $$ds^2_{t\phi} \equiv - dt(\,r \,dt + d\phi) \,.$$ Then, performing the coordinate transformation $(t,\phi) \rightarrow (\tilde{t},\tilde{\phi})$, where $$\tilde{t} = r\,t \,, \;\; \textrm{and} \;\; \tilde{\phi} = r^{-1}\,\phi \,,$$ it follows that $ds^2_{t\phi}$ becomes: $$ds^2_{t\phi} = -d\tilde{t} \, d\tilde{\phi} - \frac{1}{r}\left[ d\tilde{t}^2 + \tilde{\phi}\, d\tilde{t} dr - \tilde{t}\, d\tilde{\phi}dr \right] + O\left( r^{-2} \right),$$ where $O\left(r^{-2}\right)$ denotes terms that fall off as $r^{-2}$, or faster, when $r\rightarrow\infty$. Thus, in terms of the coordinates $(\tilde{t},\tilde{\phi})$, the asymptotic limit of the block $ds^2_{t\phi}$ becomes $$ds^2_{t\phi}|_{r\rightarrow\infty} \simeq -d\tilde{t} \, d\tilde{\phi} \,.$$ Hence, we can say that in the asymptotic limit the solution (\[metric23\]) converges to $$\label{metric23-Limit_1} ds^2|_{r\rightarrow\infty} \simeq \frac{ e^{-r} dr^2}{3(1+ \Lambda e^{-r})^2}+ \frac{e^{-r}d\theta^2-d\tilde{t}\,d\tilde{\phi} }{(1+ \Lambda e^{-r} )^{2/3}}\,.$$ This limit spacetime turn out to be a particular member of the generalized Kasner class of solutions, as demonstrated in App. \[AppendixA\]. More precisely, the solution (\[metric23-Limit\_1\]) corresponds to the choice $(p_1,p_2,p_3)=(2/3,2/3,-1/3)$ of the generalized Kasner metric (\[KasnerM\]). As shown in App. \[AppendixA\], this limit spacetime is of Petrov type D and possess a four-dimensional isometry algebra. Curiously, one can check that the curvature scalars of the line element (\[metric23-Limit\_1\]) are exactly the same as the ones of the solution (\[metric23\]), namely Eqs. (\[Scalars2\]) and (\[DR2\]) are also valid for the solution (\[metric23-Limit\_1\]). This, however, do not imply that these two spacetimes are the same. Indeed, it is well-known that two geometries can have the same curvature scalars and still be different from each other [@Coley:2009eb; @Olver]. A famous example is given by $pp$-wave spacetimes, which, in spite of having all curvature scalars equal to zero, are not flat. Thus, here we have obtained another example of two different spacetimes with the same curvature scalars. Conclusions and Perspectives {#Sec.Conc} ============================ In this paper we have completely integrated Einstein’s vacuum equation with a cosmological constant for a subclass of the most general four-dimensional metric containing two commuting Killing vector fields and a non-trivial Killing tensor of rank two. As we have seen, most of the solutions then found have already been described in the literature. Among them, we have obtained flat space, spaces of constant curvature, and a generalization of the Kasner metric to the case of non-zero cosmological constant. Nevertheless, we have also obtained a solution that, as far as we know, have never been described in the literature before, see Eq. (\[metric23\]). In order to arrive at this conclusion some features of this solution were investigated, such as its isometry group, its Petrov type, and the optical scalars related to the null Killing vector field of this solution. More precisely, we have obtained that the isometry algebra of this solution is three-dimensional and abelian, which means that it is Bianchi type I, its Weyl tensor is of Petrov type II, and the solution is contained in the Kundt class of spacetimes. Then, we searched in the literature pre-existing vacuum solutions having the same features, but no match occurred. Finally, we have proved that in the asymptotic limit $r\rightarrow\infty$ this solution approaches a member of the class of generalized Kasner spacetimes which have the same curvature scalars. We hope that this new solution, along with the characterization given in this paper could give rise to applications within the framework of gravitation, cosmology and beyond. The analysis of the physical properties of the solution (\[metric23\]) can give a hint on the range of its applicability. Therefore, in a future work we intend to investigate the physics of such exact solution . C. B. would like to thank Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for the partial financial support through the research productivity fellowship. Likewise, C. B. thanks Universidade Federal de Pernambuco for the funding through Qualis A project. G. L. A. thanks CNPq for the financial support. Another Solution Through a Singular Limit {#AppendixA} ========================================= In this appendix we shall investigate some geometric properties of the spacetime (\[metric23-Limit\_1\]), which is the asymptotic limit of the new solution (\[metric23\]) when $r\rightarrow\infty$. In particular, we will prove that its Weyl tensor is type D according to the Petrov classification and that its isometry group is four-dimensional. Let us start proving that the spacetime (\[metric23-Limit\_1\]) can be obtained from the new solution (\[metric23\]) by means of a singular coordinate transformation. Replacing the coordinates $t$ and $\phi$ in the line element (\[metric23\]) by $\tilde{t}$ and $\tilde{\phi}$ defined as $$\label{t phi LimitMetric} \tilde{t} = \lambda^{-1/2}\,t \;\, \textrm{ and } \,\; \tilde{\phi} = \lambda^{1/2}\,\phi \,,$$ where $\lambda$ is a positive constant parameter, we are led to $$ds^2=\frac{ e^{-r} dr^2}{3(1+\Lambda e^{-r})^2}+ \frac{e^{-r}d\theta^2-d\tilde{t}(\,\lambda \,r\, d\tilde{t}+d\tilde{\phi})}{(1+ \Lambda e^{-r} )^{2/3}}\,.$$ Note that although the limit $\lambda\rightarrow0$ is forbidden at the level of the coordinates, since $\tilde{t}$ and $\tilde{\phi}$ become ill-defined, the line element obtained in this limit is perfectly regular and is given by $$\label{metric23-Limit} ds^2=\frac{ e^{-r} dr^2}{3(1+ \Lambda e^{-r})^2}+ \frac{e^{-r}d\theta^2-d\tilde{t}\,d\tilde{\phi} }{(1+ \Lambda e^{-r} )^{2/3}}\,.$$ Despite the line element (\[metric23-Limit\]) being obtained from our solution through a coordinate transformation, the metric (\[metric23-Limit\]) can represent a completely different spacetime, since the coordinate transformation is singular at $\lambda=0$. For instance, another example of singular coordinate transformations that yield a different space is provided by Nariai spacetime, which can be obtained from the degenerated Schwarzschild-dS solution[^2] by means of a singular coordinate transformation [@BatistaNariai]. Now, let us investigate some properties of the solution (\[metric23-Limit\]). Note that besides being invariant under translations in the coordinates $\theta$, $\tilde{t}$ and $\tilde{\phi}$, the line element (\[metric23-Limit\]) is also invariant under the boost transformation $$r\rightarrow r \,,\;\; \theta \rightarrow \theta \,,\;\; \tilde{t}\rightarrow a\,\tilde{t} \,,\;\; \tilde{\phi}\rightarrow \frac{1}{a}\,\tilde{\phi} \,,$$ with $a$ being an arbitrary constant parameter. This is an extra symmetry, whose generator is the Killing vector field $${\boldsymbol}{\tilde{k}} = \tilde{t}\, \partial_{\tilde{t}} - \tilde{\phi}\, \partial_{\tilde{\phi}} \,.$$ One can check that this is the only extra independent killing vector of the solution (\[metric23-Limit\]) besides the obvious ones $\partial_\theta$, $\partial_{\tilde{t}}$, and $\partial_{\tilde{\phi}}$, so that the isometry group is four-dimensional and nonabelian. In particular, this implies that, in spite of having the same curvature scalars, the solutions (\[metric23\]) and (\[metric23-Limit\]) represent different spacetimes, since they have different isometry groups. In order to obtain the Petrov classification of the solution (\[metric23-Limit\]), let us introduce the following null tetrad $$\begin{aligned} {\boldsymbol}{\ell} &= \partial_{\tilde{\phi}} \,,\\ {\boldsymbol}{n} &= 2(1+ \Lambda e^{-r})^{2/3}\partial_{\tilde{t}} \,,\\ {\boldsymbol}{m} &= \sqrt{\frac{3 e^{r}}{2}}\,(1+ \Lambda e^{-r})\partial_{r} + i \, \sqrt{\frac{e^{r}}{2}}\,(1+ \Lambda e^{-r})^{1/3}\partial_{\theta}\,,\\ {\boldsymbol}{\bar{m}} &= \sqrt{\frac{3 e^{r}}{2}}\,(1+ \Lambda e^{-r})\partial_{r} - i \, \sqrt{\frac{e^{r}}{2}}\,(1+ \Lambda e^{-r})^{1/3}\partial_{\theta}\,.\end{aligned}$$ Then, computing the Weyl scalars with such tetrad, we find that $\Psi_0$, $\Psi_1$, $\Psi_3$, and $\Psi_4$ vanish, while, for $\Lambda\neq 0$, $\Psi_2$ is different from zero and given by $$\Psi_2 = \frac{\Lambda}{6} (1+ \Lambda e^{-r})\,.$$ This means that ${\boldsymbol}{\ell}$ and ${\boldsymbol}{n}$ are both repeated principal null directions and that the solution (\[metric23-Limit\]) is of Petrov type D if $\Lambda\neq 0$, which differs from the Petrov classification of the line element (\[metric23\]). Note also that in the case $\Lambda=0$ the latter tetrad is well-defined, so that we can use it to compute the Weyl scalars. Doing so, we see that, when $\Lambda$ is zero, $\Psi_2$ vanishes along with the $\Psi_0$, $\Psi_1$, $\Psi_3$, and $\Psi_4$. Since all Weyl scalars vanish it follows that the Weyl tensor is identically zero, which along with the fact that the metric (\[metric23-Limit\]) is Ricci-flat when $\Lambda=0$ means that the spacetime is flat. Thus, the case $\Lambda=0$ of the line element (\[metric23-Limit\]) is just Minkowski spacetime. Continuing the characterization of the limit solution (\[metric23-Limit\]), we can also verify that the null Killing vector fields $\partial_{\tilde{t}}$ and $\partial_{\tilde{\phi}}$ are geodesic, shear-free, twist-free, and expansion-free, so that the line element (\[metric23-Limit\]) is contained in the Kundt class. Concerning the regularity of the solution and its asymptotic limit when $r\rightarrow \infty$, all the comments made for the solution (\[metric23\]) remains valid for the limit solution (\[metric23-Limit\]), since these spaces have exactly the same curvature scalars. Finally, we can check that the solution (\[metric23-Limit\]) is, actually, a member of the generalized Kasner solutions that we have obtained in Eq. (\[KasnerM\]). Indeed, performing the coordinate transformation $(r,\theta,\tilde{t},\tilde{\phi})\rightarrow (y,x_1,x_2,x_3)$, where $$\begin{aligned} r&= 2 \log\Big[\sqrt{\Lambda}\tan\Big(\frac{\sqrt{3\Lambda}y}{2}\Big)\Big]\\ \tilde{t}&= \Big(\frac{4}{3\Lambda}\Big)^{1/3}(i x_1+x_2)\\ \tilde{\phi}&= \Big(\frac{4}{3\Lambda}\Big)^{1/3}(i x_1-x_2)\\ \theta&= \sqrt{\Lambda}\Big(\frac{4}{3\Lambda}\Big)^{-1/6}x_3\\\end{aligned}$$ we can see that the line element (\[metric23-Limit\]) takes the form (\[KasnerM\]) with the choice $(p_1,p_2,p_3)=(2/3,2/3,-1/3)$. 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--- abstract: 'Following the pioneering Okubo scrutiny of gauge simple groups for the quantum chromodynamics we show the constraints coming from the wondrous predictive leptoquark-bilepton flavordynamics connecting the number of color charges to solution of the flavor question and to an electric charge quantization unconstrained from the Dirac, Majorana or Dirac–Majorana character of massive neutrino.' address: 'Departamento de Física, Universidade Federal do Paraná 81531-990 Curitiba PR, Brazil' author: - 'A. Doff and F. Pisano' title: On the color simple group from chiral electroweak gauge groups --- Nobody has put to the test the interplay between the Fermi-TeV-kTeV [@ktevFnal] and the end of space-time Planck scales but everybody believes that there is more physics beyond the standard QCD and QFD. Since the standard model [@Glashetc] of nuclear and electromagnetic interactions is very well confirmed [@pdgwww; @nu] up to the TeV scale the possibilities for chiral gauge semisimple group extensions have not been exhausted yet. The most general chiral gauge semisimple group, expanding the number of color charges ($n_c$) and the weak isospin group ($m$) is $$G_{n_c m_L 1_N} \equiv {\rm SU}(n_c)_c\otimes {\rm SU}(m)_L\otimes {\rm U(1)}_N$$ where the minimal extension is the $G_{331}$ gauge symmetry [@pp92; @fra92]. Although the data accumulated on the scaling violations in deep inelastic scattering experiments are consistent with SU(3) gauge structure of strong interactions, extensions of the color sector [@Okubo77] in which quarks transform under the fundamental representations of SU($n_c$)$_c$, $n_c = 4,5$ were considered in the context of the electroweak standard model preserving the consistency at low energies [@Footetal]. However, for the first time too many fundamental questions of physics are answered and within the minimal semisimple gauge group extension of the standard model. Although the weak isospin group is minimally enlarged to SU(3)$_L$ preserving the color and Abelian group factors there is a first unusual capacity in answering fundamental questions only by exploring the minimal enlargement of the electroweak gauge group. Consider, for instance, the following: 1. Each generation is anomalous and the anomalies cancel when the number of leptonic generations is divisible by the number of colors [@pp92; @fra92; @fp96; @DP2000]. There is a relation between the strong and electroweak sectors of the model which does not exist in the standard model with the solution for the flavor question; 2. The electroweak mixing angle, $\theta_{\rm W}$, is limited from above with an upper bound determined by their Landau pole [@Ng94]; 3. The neutrino and the charged leptons masses are constrained in the cubic seesaw relation $$m_{\nu_\ell}\propto\frac{m^3_\ell}{M^2_W}, \quad \ell = e,\mu,\tau$$ with outcomes for the solar neutrino problem and hot dark matter [@fr94]; 4. The Yukawa couplings have a Peccei–Quinn [@PQ] symmetry which can be extended to all sectors of the Lagrangian with an invisible axion solving the strong-CP problem [@Pal95]; 5. Spontaneous CP violation in the electroweak sector [@Epele95]. There are several natural sources of explicit and spontaneous CP violations [@vicente]; 6. The quark mass hierarchy [@austr]; 7. Although the leptoquark-bilepton models do not conserve each generation lepton number $L_\ell$, the neutrinoless double beta decay is forbidden because of the conservation of the quantum number ${\cal F}\equiv L + B$, where $B$ is the barion number and $L=\sum_\ell L_\ell$ is the total lepton number. If this global symmetry is explicitly violated in the Higgs potential, there are contributions to the decay which depend less on the neutrino mass than they do in too many extensions of the standard model [@pt93]. The double beta decay with Majoron emission is possible as well [@fpshe1]; 8. There is an electric charge quantization without any constraint on the Dirac, Majorana or Dirac–Majorana [@Esposito] character of the massive neutral fermions [@doff]. Representation contents are determined by embedding the electric charge operator $$\frac{{\cal Q}}{|e|} = (\Lambda_3 + \xi\Lambda_8 + \zeta\Lambda_{15}) + N \label{opcar}$$ in the neutral generators $\Lambda_{3,8,15} = \lambda^{\rm SU(4)}_{3,8,15}/2$ of the largest weak isospin group SU(4) extension and $N$ is the new U(1)$_N$ charge equivalent to the electric charge average of the fermions contained in each flavor multiplet. If we consider the lightest leptons as the fermions which determine the approximate symmetry, and also independent flavor generations, then SU(4)$\times$U(1) is the largest non-symmetric gauge group of the electroweak sector. There is no room for the chiral semisimple group SU(5)$\times$U(1) if lepton electric charges are only $0$, $\pm 1$. The weak hypercharge of the $G_{321}$ standard model is $$\frac{Y}{2} = (\xi\Lambda_8 + \zeta\Lambda_{15})+N \label{anaa}$$ and in the minimal $G_{331}$ leptoquark-bilepton model, $\xi=-\sqrt 3$, $\zeta = 0$, are contained 17 gauge vector fields, $$\begin{aligned} {\rm SU}(3)_c & : & g^i_\mu\sim ({\bf 8},{\bf 1},N=0); \quad i=1,2,...,8; \nonumber \\ {\rm SU}(3)_L & : & W^j_\mu\sim ({\bf 1},{\bf 8},0); \quad j=1,2,...,8; \\ {\rm U}(1)_N & : & B_\mu\sim ({\bf 1},{\bf 1},0), \nonumber\end{aligned}$$ nine lepton fields connected through charge conjugation of the charged fields in three triplets, $$L_\ell\sim ({\bf 1},{\bf 3},0), \quad \ell = e, \mu, \tau; \label{nonoh}$$ three families of quarks, $$\begin{aligned} Q_{1L} & \sim & ({\bf 3},{\bf 3},+2/3) \nonumber \\ u_R & \sim & ({\bf 3},{\bf 1},+2/3) \nonumber \\ d_R & \sim & ({\bf 3},{\bf 1},-1/3) \nonumber \\ J_{1R} & \sim & ({\bf 3},{\bf 1},+5/3)\end{aligned}$$ for the first family, and $$\begin{aligned} Q_{\alpha L} & \sim & ({\bf 3},{\bf\bar 3},-1/3) \nonumber \\ c_{\alpha R} & \sim & ({\bf 3},{\bf 1},+2/3) \nonumber \\ s_{\alpha R} & \sim & ({\bf 3},{\bf 1},-1/3) \nonumber \\ J_{\alpha R} & \sim & ({\bf 3},{\bf 1},-4/3)\end{aligned}$$ where $\alpha = 2,3$ labels the second and third families. Taking into account three color charges we have an amount of 54 quark fields. The $J_1$ and $J_\alpha$ leptoquark fermions are color-triplet particles with electric charge $\pm\frac{5}{3}$ and $\mp\frac{4}{3}$ which carry baryon number and lepton number, $B_{J_{1,\alpha}}=+\frac{1}{3}$, and $L_{J_\alpha}=-L_{J_1}=+2$. All masses are generated with four multiplets of scalar fields $$\begin{aligned} \eta & \sim & ({\bf 1}, {\bf 3}, 0) \nonumber \\ \rho & \sim & ({\bf 1}, {\bf 3},+1) \nonumber \\ \chi & \sim & ({\bf 1}, {\bf 3},-1) \nonumber \\ S_{ij} & \sim & ({\bf 1},{\bf\bar 6}_S,0)\end{aligned}$$ and in the symmetric phase of the theory they are parametrized by 30 real scalar fields. Such unavoidable scalarland is the most desirable field sector to have an experimental comprovation, since in theories with spontaneous symmetry breaking of the gauge symmetry it is essential but the unique field of the standard model which does not present evidences is the Higgs scalar boson. The total number of massless fields in the $G_{331}$ model is 110 and there are not spin-$\frac{3}{2}$ Rarita–Schwinger fields. Our main purpose is to select the possible color gauge simple groups from the leptoquark-bilepton flavordynamics. Let us remark that in a theory whith the SU($n_c$)$_c$ gauge simple group the ’t Hooft [@thft] limit $n_c\rightarrow \infty$ and the Maldacena [@Mald] conjecture provide the evidence of the gauge to string theories limit. In the four-dimensional super Yang–Mills type IIB string theory arises the color confinement and a mass gap within the 5-brane of the eleven-dimensional M-theory [@Witten98]. The $n_c=3$ standard QCD is an asymptotically free theory including its non-perturbative confinement property. The perturbative strong coupling constant is $$\alpha_{\rm s}(q) = \frac{g^2_{{\rm SU}(n_c)_c}(q)}{4\pi} = 4\pi \left [\beta_0 %%%%%%\left (\frac{11}{3} n_c - \frac{2}{3} n_f\right ) \ln\left (\frac{q}{\Lambda_{\rm QCD}}\right )^2\right ]^{-1} \label{strcoupl}$$ with $$\beta_0 = \frac{11}{3} n_c - \frac{2}{3} n_f \label{eexxuno}$$ and the fundamental scale $\Lambda_{\rm QCD}\simeq 250$ MeV $\simeq 10^{-3} (\sqrt 2 G_{\rm F})^{-\frac{1}{2}} \simeq 246 \times 10^{-3}$ GeV $= 10^{-3}\Lambda_{\rm QFD}$ where quarks form the hadrons as a direct effect of the color confinement and $n_f$ is the number of quark flavors. The $\Lambda_{\rm QFD}$ is the Fermi scale of electroweak spontaneous symmetry breaking $G_{321}\rightarrow$ SU(3)$_c\times$U(1)$_{\rm em}$. There are two limits, $$\lim_{n_c\rightarrow\infty}\alpha_{\rm s} (q) = 0, \quad \lim_{q\rightarrow\infty}\alpha_{\rm s} (q) = 0. \label{limote}$$ At high energy, $q^2/\Lambda^2_{\rm QCD}\gg 1$, the strong coupling constant is small and the QCD is described by the perturbation theory. In ’t Hooft original expansion the number of flavors is kept fixed when $n_c\rightarrow\infty$. The SU($n_c$)$_c$ exact symmetry is realized in the hidden way. In the weak coupling limit $$a^2\Lambda^2_{\rm QCD} = \exp\left \{-\frac{1}{\beta_0}\left (\frac{4\pi}{ g_{{\rm SU}(n_c)_c}(a)}\right )^2 \right \} \label{acplqcd}$$ where $a$ could be the spacing scale of a lattice gauge theory of strong couplings the Maldacena conjecture provides a special evidence that a string theory comes out from a gauge theory [@DiVe]. The $G_{331}$ theory has two anomalies containing the color gauge group. Characterizing each triangle anomaly by three generators associated to the gauge group they are \[SU(3)$_c$\]$^3$ and \[SU(3)$_c$\]$^2$\[U(1)$_N$\]. The pure cubic color anomaly cancel since the QCD has a vector-like fermion representation content so there is independent anomaly cancellation in each color triplet and the associated antitriplets of quarks. Setting the notation for the standard quark chiral flavors $$\begin{aligned} N_{u_R} = N_{c_R} = N_{t_R} & \equiv & N_{U_R}\,,\\ \label{xaxii} N_{d_R} = N_{s_R} = N_{b_R} & \equiv & N_{D_R}\,, \label{xqk}\end{aligned}$$ and $$\begin{aligned} N_{Q_{2L}} = N_{Q_{3L}} & \equiv & N_{Q_{\alpha L}}, \\ \label{nqua} N_{J_{2R}} = N_{J_{3R}} & \equiv & N_{J_{\alpha R}} \label{nqiu}\end{aligned}$$ also for the leptoquark flavors, the Tr(\[[SU(3)]{}$_c$\]$^2$\[U(1)$_N$\]$)=0$ constraint is $$3(N_{Q_{1L}} + 2N_{Q_{\alpha L}}) - 3(N_{U_R} + N_{D_R}) - N_{J_{1R}} - 2 N_{J_{\alpha R}} = 0 \label{ampa}$$ and since the $\sum_{L_\ell}N_{L_\ell}$ term vanishes coincides with the mixed gravitational-gauge anomaly constraint Tr(\[[graviton]{}\]$^2$\[U(1)$_N$\])$=0$. Also the $N_{Q_{1L}} + 2N_{Q_{\alpha L}}$ term vanishes in the minimal and extended leptoquark-bilepton models [@pp92; @fra92; @sufour]. Being ${\rm N}_\ell$ and ${\rm N}_{\rm q}$ the number of lepton and quark generations let us consider the SU($n_c$)$_c$ possibilities for $n_c\geq 3$ where the ${\rm N}_\ell = {\rm N}_{\rm q} = {\rm N}_{\rm generations}$ coincidence is evaded. Denoting as $n_{\bf m}$ and $n_{{\bar{\bf m}}}$ the number of quark generation multiplets transforming as ${\bf m}$ and ${{\bar{\bf m}}}$ in the fundamental representation under the SU($m$)$_L$ flavor group factor we have the universality breaking condition in the lepton sector $${\rm N}_\ell = |n_c\,(n_{\bf m} - n_{{\bar{\bf m}}})|, \label{gng}$$ and $${\rm N}_{\rm q} = n_{\bf m} + n_{\bar{\bf m}} \label{unaltr}$$ for the number of quark flavor generations. The condition in Eq. (\[gng\]) involves the following possibilities: (1) $n_{\bf m} > n_{\bar{\bf m}}$, when the leptons must transform as ${\bar{\bf m}}$; (2) $n_{\bf m} < n_{\bar{\bf m}}$ when the lepton multiplets are attributed to the ${\bf m}$ representation; (3) $n_{\bf m} = n_{\bar{\bf m}}$. For $n_{\bf m} > n_{\bar{\bf m}}$ and in the case of even $n_c$, $n_c = 2k$, $k\geq 2$ we have the ratio $$\frac{n_{\bf m}}{n_{\bar{\bf m}}} = \frac{2k + 1}{2k - 1} \label{ratgen}$$ but for odd $n_c = 2k + 1$ the ratio is $$\frac{n_{\bf m}}{n_{\bar{\bf m}}} = 1+\frac{1}{k}. \label{impa}$$ The SU(5)$_c$ group consistent with the standard flavordynamics [@Footetal] satisfies the ${\rm N}_\ell = {\rm N}_{\rm q}$ condition if $n_{\bf m}/n_{\bar{\bf m}} = 3/2$ for $k=2$ with five generations in a universal representation content. 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--- abstract: 'Bilayer graphene provides a unique platform to explore the rich physics in quantum Hall effect. The unusual combination of spin, valley and orbital degeneracy leads to interesting symmetry broken states with electric and magnetic field. Conventional transport measurements like resistance measurements have been performed to probe the different ordered states in bilayer graphene. However, not much work has been done to directly map the energetics of those states in bilayer graphene. Here, we have carried out the magneto capacitance measurements with electric and magnetic field in a hexagonal boron nitride encapsulated dual gated bilayer graphene device. At zero magnetic field, using the quantum capacitance technique we measure the gap around the charge neutrality point as a function of perpendicular electric field and the obtained value of the gap matches well with the theory. In presence of perpendicular magnetic field, we observe Landau level crossing in our magneto-capacitance measurements with electric field. The gap closing and reopening of the lowest Landau level with electric and magnetic field shows the transition from one ordered state to another one. Further more we observe the collapsing of the Landau levels near the band edge at higher electric field ($\bar D > 0.5$ V/nm), which was predicted theoretically. The complete energetics of the Landau levels of bilayer graphene with electric and magnetic field in our experiment paves the way to unravel the nature of ground states of the system.' author: - Manabendra Kuiri - Anindya Das bibliography: - 'ref\_blgAUG.bib' title: Energetics of complex phase diagram in a tunable bilayer graphene probed by quantum capacitance --- Introduction ============ Bilayer graphene (BLG) provides a unique two-dimensional system in condensed matter physics, where the low energy spectrum is gapless touching at K and K’ points and an external electric field opens up a tunable gap at the valley points[@PhysRevLett.96.086805; @ohta2006controlling]. In clean samples the $e-e$ interactions lead to gap opening even without an external electric field[@PhysRevLett.104.156803; @PhysRevLett.108.076602] and interesting phases like quantum spin Hall, anomalous quantum Hall[@PhysRevB.82.115124], layer antiferromagnet[@PhysRevB.87.195413], and nematic[@PhysRevB.81.041401] states were suggested to be the possible ground state at the neutrality point[@PhysRevLett.108.186804]. Bilayer graphene is even more interesting in presence of magnetic field due to the additional orbital degeneracy of the lowest Landau level (LL) together with spin and valley degeneracy, resulting in complex quantum Hall states (QHS)[@feldman2009broken; @maher2014tunable]. The coupling of electric and magnetic field leads to transitions between different spin, valley and orbital ordering leading to unique interaction driven symmetry broken states[@PhysRevLett.120.047701; @velasco2014transport; @hunt2017direct; @maher2013evidence; @kou2014electron; @lee2014chemical; @PhysRevB.87.161402; @velasco2012transport; @PhysRevB.85.115408; @PhysRevLett.107.016803; @velasco2014competing]. Thus BLG provides an excellent platform to probe the phase transitions between different ordered states[@PhysRevB.85.235460; @leroy2014emergent; @PhysRevB.86.075450; @PhysRevLett.109.046803].\ There has been extensive studies to find the nature of ordered states in BLG, both theoretically[@PhysRevB.86.075450; @PhysRevLett.109.046803; @PhysRevB.86.195435] and experimentally[@weitz2010broken; @PhysRevLett.107.016803; @lee2014chemical; @hunt2017direct; @zibrov2017tunable]. The model employed in Refs. [@PhysRevLett.109.046803; @lee2014chemical; @leroy2014emergent] shows that at finite magnetic field ($B$), the LLs are spin splitted and the orbital and valley degeneracies are lifted by the application of electric field. However, model employed in Refs.[@hunt2017direct; @zibrov2017tunable] showed that at finite $B$ both the spin and orbital degeneracies are lifted, and the application of electric field results in lifting the valley degeneracy only. However, there is no common consensus about the order of the ground state of these symmetry broken states[@maher2013evidence; @kou2014electron; @hunt2017direct].\ Recent transport measurements in dual gated geometry have observed the crossing of LLs leading to the closing of gap which is attributed to the phase transition between different type of ordered states[@weitz2010broken; @PhysRevLett.107.016803]. Although transport measurements can provide an indication of the gap size, but the true energetics of these states cannot be estimated by conventional transport measurements. Therefore, thermodynamic measurement is desirable to directly probe the electronic properties as well as the energetics of the these states[@PhysRevLett.105.256806]. The proper knowledge of the energetics of these LL crossing points together with the variation of LL energy by external electric and magnetic fields provide key insights to the nature of ground state, which has been employed to probe the magnetization of quantum hall states[@de2000resistance] and many body enhanced susceptibility[@PhysRevLett.90.056805] in two dimensional electron gas (2DEG).\ In order to obtain the energetics in BLG with electric and magnetic field, we employ magneto capacitance studies in a hexagonal boron nitride (hBN) encapsulated dual gated BLG device. At zero magnetic field, using our quantum capacitance measurement we measure the gap around the charge neutrality point as a function of perpendicular electric field($\bar D$), where the obtained value of the gap matches well with the previously reported values[@zhang2009direct]. In presence of perpendicular magnetic field, we observe LL crossing in our magneto-capacitance measurements with $\bar D$. The gap closing and reopening of the lowest LL with $\bar D$ and $B$ shows the transition from one ordered state to another one. The values of critical electric field ($\bar D_c$) required to close the gap as a function of magnetic field matches well with the earlier reports[@weitz2010broken; @PhysRevLett.107.016803]. We further obtain the energetics of the LLs as a function of $\bar D$ and $B$, where the renormalization of LL spectrum at higher electric field ($\bar D > 0.5$ V/nm) is clearly visible. It has been shown theoretically that at higher electric fields the LLs collapses at the band edge due to LL coupling and hybridization [@PhysRevLett.96.086805; @PhysRevB.73.245426; @PhysRevB.87.075417], which has not been observed experimentally prior to this report. {width="80.00000%"} experimental details ==================== Dual gated bilayer graphene device was fabricated using van der Waals assembly, following the procedure developed by Wang *et.al.* [@wang2013one]. Briefly bilayer graphene (BLG) was first mechanically exfoliated onto a piranha cleaned Si/SiO$_2$ substrate from bulk single crystal of natural graphite. On another clean substrate hBN was mechanically exfoliated and potential thin hBN was looked for using optical microscope. Using dark field microscope imaging hBN flake with uniform smooth surface and free of bubbles was chosen. hBN, BLG and hBN were picked up sequentially one on top of another and the complete stack (hBN-BLG-hBN) was deposited onto a $n++$ doped Si/SiO$_2$ substrate with 285 nm oxide. The stack was then annealed at 200$^\circ$C in vacuum to get a uniform surface free of bubbles. The electrical contacts were fabricated using electron beam lithography followed by etching the hBN-BLG-hBN stack, and one-dimensional contact was established by thermally evaporating Cr/Au (5nm/70nm)[@wang2013one]. Another step of lithography and thermal deposition was carried out to define the topgate electrode (see supplemental material; SM-Sec.I for details). The optical image of the final device is shown in Fig. 1a. The schematic of the device and the measurement scheme are shown in Fig. 1b. The top hBN thickness $\sim$ 11 nm and bottom hBN thickness $\sim$ 15 nm were measured using atomic force microscope (see SM-Sec.II). The thickness of top hBN was found independently using period of oscillation of the capacitance minima in magnetic field [@yu2013interaction]. The excellent dielectric properties of hBN serves the purpose of using thin gate dielectric for measuring detectable change in total capacitance (C$_t$). All the measurements were carried out in a $^3$He refrigerator with a base temperature $T\sim $ 240 mK.\ For the capacitance measurements we have used the measurement scheme described in our earlier works[@bppaper; @PhysRevB.98.035418] using a home built differential current amplifier with a gain of $10^7$. The capacitance has been measured between the topgate electrode and BLG with a small ac excitation voltage of $\sim$ 10-15 mV at a frequency of $\sim$ 5 kHz with a resolution of $\sim 0.5~fF$. All wires were shielded to reduce the parasitic capacitance. In a parallel plate capacitor made of a normal bulk metal and a two dimensional material like graphene, adding a charge requires electrostatic energy, but also kinetic energy due to the change in chemical potential, thereby contributing to the total capacitance[@luryi1988quantum]. The total measured differential capacitance in such a system is given by $$C_t= \left(\frac{1}{C_g} + \frac{1}{C_q} \right)^{-1} +C_p \label{eqn:eq1}$$ where, $C_g$ is the geometric capacitance, $C_q=Se^2\frac{dn}{d\mu}$ is the quantum capacitance; $e$ is the electronic charge; $S$ is the area under the topgate electrode; $\frac{dn}{d\mu}$ is the thermodynamic compressibility, $C_p$ is the parasitic capacitance arising due to the wirings plus the stray capacitances. In BLG, the application of electric field between the layers results in breaking the inversion symmetry, which in turn opens up a band gap [@zhang2009direct] at the charge neutrality point. Dual gated geometry allows us to independently control electronic density ($n$) and electric displacement field ($\bar D$) under the topgated region. The net transverse electric field in a dual gated device is given by $\bar D=[C_{bg}(V_{bg}-V_{bg}^0)-C_{tg}(V_{tg}-V_{tg}^0)]/2\epsilon_0$ and the total carrier density is given by $n=[C_{bg}(V_{bg}-V_{bg}^0)+C_{tg}(V_{tg}-V_{tg}^0)]/e$ ; $\epsilon_0$ is the vacuum permittivity, $e$ is the electronic charge, $C_{bg}(C_{tg})$ is the capacitance per unit area of the backgate(topgate) region and $V_{bg}^0,V_{tg}^0$ are the charge neutrality points. {width="100.00000%"} Capacitance Data at B=0T ======================== Fig. 1c shows the colorplot of the measured total capacitance, $C_t$ as a function of backgate voltage ($V_{bg}$) and topgate voltage ($V_{tg}$) at B = 0T. The data was taken by sweeping the topgate voltage for different values of backgate voltages. Tuning of topgate and backgate changes both the total carrier density ($n$) and the band gap ($\Delta_g$). The diagonal white dashed marked in Fig. 1c shows the direction of $n$ and solid black line shows the direction of $\bar D$. For $\bar D\sim 0$, C$_t$ exhibits a minimum at zero density, signifying the hyperbolic nature of band structure for ungapped bilayer graphene[@PhysRevB.82.041412]. As $|\bar D|$ increases the capacitance minima decreases revealing the formation of gap in the energy spectrum in bilayer graphene[@PhysRevB.85.235458]. The diagonal line in Fig. 1c corresponds to the charge neutrality point under the topgated region. Along the diagonal line the capacitance minima decreases signifying the electric field induced band gap opening. The charge neutrality points ($V_{bg}^0,V_{tg}^0$) are located at 0.3V, -8.5V. From the slope of the diagonal line we can effectively estimate the ratio of the capacitive coupling between the top and bottom gates $C_{tg}/C_{bg} \sim 27$ ($C_{tg}\Delta V_{tg}=C_{bg}\Delta V_{bg}$ along the diagonal line in Fig. 1c, $d_{bg}\sim$ 300 nm, $\epsilon_{hBN}=\epsilon_{SiO_2}\sim~3.9$, yields $d_{tg}\sim$ 10.75 nm, which matches well with the value of $d_{tg}\sim$ 11 nm obtained using AFM, see SM-Sec.II). Fig. 1d shows the cut lines of $C_t$ as a function of $V_{tg}$ for several value of $V_{bg}$. The geometric capacitance $C_g\sim 66~fF$ is marked with dashed black line. Noting the area of our device $S\sim 21~\mu m^2$, the effective geometric capacitance was $C_g\sim 66~fF$. The parasitic capacitance was estimated by comparing the experimental capacitance data at $\bar D=0$ with the theoretical one (Eq.\[eqn:eq1\]), where only adjusting parameter was C$_p$ (see SM-Sec.III; the density of states for ungapped bilayer graphene with effective mass $m_*=0.03m_e$ was calculated from Ref[@PhysRevLett.96.086805]). The parasitic capacitance C$_p$ in our device is $\sim$ 152 fF. This value of C$_p$ is subtracted from all the data presented in this paper.\ {width="100.00000%"} In order to get a better insight to the experimental data we need to extract the quantum capacitance ($C_q$) as a function of Fermi energy ($E_F$) from the experimentally measured $C_t$ as a function of backgate and topgate voltages . The Fermi energy and band gap are independently controlled by changing $V_{bg}$ and $V_{tg}$. Thus the quantum capacitance should be extracted along the constant $\bar{D}$ lines as a function of Fermi energy. We have followed a similar approach as described in Ref [@kanayama2015gap] (see SM-Sec.IV for details). The Fermi energy of bilayer graphene is given by the charge conservation relation $E_F=e\int_{0}^{V_{tg}}\left(1-\frac{C_t}{C_g}\right)dV_{tg}$ [@droscher2010quantum]. Fig. 2a shows the colorplot of total capacitance ($C_t$) as a function of Fermi energy and electric field. It can be seen that the band gap opens with the increment of $\bar D$. The maximum $\bar D$ we could reach was 0.8 V/nm with a band gap opening $\Delta_g \sim$ 80 meV in agreement with previously reported values[@zhang2009direct]. The extracted quantum capacitance ($C_q^{-1}=C_t^{-1}-C_g^{-1}$) for several values of $\bar D$ is shown in Fig. 2b. It can be seen that with the increment of $\bar D$, $C_q$ decreases signifying the increase of band gap. We have observed asymmetry in the $C_q$ for the electron and hole side, which has also been previously observed by other groups[@PhysRevB.85.235458; @kanayama2015gap]. The 1/$\sqrt{E}$ van hove singularity is also observed at the band edge as predicted[@mccann2013electronic]. The extracted $\Delta_g$ as a function of $\bar D$ has been shown in Fig. 2c. The measured band gap values matches well with the theoretical band gap calculated using tight binding model[@PhysRevB.75.155115]. Magneto-capacitance data ======================== The competing magnetic and electric field leads to various interesting phases in the LL spectrum of BLG. To visualize the energetics of the LLs as a function of $\bar D$ and B, we present our magneto capacitance data. For an ungapped pristine BLG, in absence of any interactions, the LL energies in a perpendicular magnetic field is given by $E_N=\pm\hbar\omega_c \sqrt{N(N-1)}$, where $\omega_c=eB/m^*$ is the cyclotron frequency, and $N=0,\pm1,\pm2...$ are the orbital index. For $N=0,1; ~E_N=0$. Thus, the zeroth energy LL is eight-fold degenerate, whereas all other landau levels ($N\geq2$) are four fold degenerate (two spin and two valley)[@PhysRevLett.96.086805].\ ![(Color Online) (a) Critical electric field ($\bar D_c$) as a function of $B$. (b) The LL energies for $\nu=0$ state as a function of $B$ for $\bar D=0$. []{data-label="fig:fig4"}](fig5.pdf){width="35.00000%"} {width="80.00000%"} Fig. 3a shows the experimental LL fan diagram for $\bar D=0$. Here, $C_t$ was measured by sweeping $V_{tg}$,$V_{bg}$ synchronously keeping the $\bar D=0$ and changing only the carrier density. The dips in the capacitance data corresponds to the LL gap. The gap around the zeroth LL start appearing for $B>5T$. The LL corresponding to $N=\pm2, \pm3,\pm4$ can be seen in Fig. 3a. The geometric capacitance C$_g$ was determined independently from the fact that spacing $\Delta V_{tg}$ between the adjacent capacitance minima in Fig. 3a is given by the amount of charge required to fill each Landau level[@yu2013interaction] ($C_g\Delta V_g = \frac{4Se^2B}{h}$, where $\Delta V_g\sim 0.48V$) (see SM-Sec.V) yielding an effective $C_g\sim65.5~fF$ which matches quite well as extracted from the colorplot of Fig. 1c and AFM imaging (see SM-Sec.II). The conversion of $x-$ axis in Fig. 3a, which is a combination of topgate voltage and backgate voltage, to Fermi energy is shown in SM-Sec.VI. Fig. 3b shows the result of such a conversion where we plot the extracted $C_q$ as a function Fermi energy for different values of magnetic field. The solid lines are generated using single particle LL energies for ungapped BLG ($E_N=\pm\hbar\omega_c \sqrt{N(N-1)}$, with effective mass $m_*=0.03m_e$). It can be seen that upto $B<6T$, the extracted LL spectrum matches quite well with the theory. However, for $B>6T$ we observe noticeable mismatch between the experimental and the theoretical values (10%-20%), which has also been addressed in previous studies, employing magneto-capacitance measurements[@yu2013interaction; @yu2014hierarchy; @PhysRevB.98.035418]. This mismatch has been attributed to the inaccurate conversion in determining $E_F$ at higher magnetic field as the bulk becomes more insulating and increasingly isolated from electrical contacts leading to excess deep in the $C_t$ versus gate voltage curve.\ We now show the LL spectrum as a function of electric and magnetic field. Figure. 3c shows the measured $C_t$ as a function of $V_{bg}$ and $V_{tg}$ for $B=10T$. In Fig. 3d we have shown the extracted quantum capacitance as a function of $E_F$ and $\bar D$ for B=10T. The parallel lines are the different LLs which evolves with $\bar D$. The most striking feature is the evolution of the zeroth energy LL with $\bar D$. In Fig. 3f we have shown the zoomed part of the Fig. 3d (white dashed box). The emergence of the $\nu=0$ insulating state can be seen for $\bar D=0$. With the increment of $\bar D$, we see the evolution of the $\nu=0$ insulating state. For small values of $\bar D$, $\nu=0$ state remains gapped, with increase in $\bar D$, the gap decreases monotonically, and then for a critical value $\bar D_c=0.08$mV/nm the gap closes, further increase in $\bar D$ the gap again re-opens and remains gapped for high $\bar D$ (maximum $\bar D$ for our device was $\bar D\sim $1V/nm). This electric field induced gap closing and re opening is a signature of phase transition[@leroy2014emergent]. In Fig. 3e we show the evolution of the $\nu=0$ state with $\bar D$ and $B$. Here, the topgate and backgate were swept synchronously to maintain zero carrier density and vary only $\bar D$ as described earlier. The $V$ shaped yellow structure in Fig. 3e separate out two insulating states (blue regions inside and outside of the $V$), which is in consistent with earlier reports. Fig. 4a shows the plot of critical electric field, $\bar D_c$ as a function of B. The $\bar D_c$, which determines the transition point, can be written as a linear function of magnetic field as $\bar D_c=\bar D_{off}+\alpha B$, where $\bar D_{off}$ is the offset electric field and $\alpha$ is the slope. For our case $\bar D_{off}=18$ and $\alpha=7~mV/nm~\times B[T]$ matches well with the theoretically predicted values[@PhysRevB.83.115455] and experimentally observed values for $\bar D_c$ reported using resistance measurements[@PhysRevLett.107.016803], where the $\nu=0$ QHS undergoes a phase transition between the spin polarized phase and the layer polarized phase in the ($B-\bar D$) plane. Further more the $\nu=0$ gap at $\bar D = 0$ as a function of B is shown in Fig. 4b, where the gap increase linearly with $B$, with a slope of $3~meV/T$, in agreement with previous reports[@PhysRevLett.105.256806], which suggests the ground state is spin polarized ($\bar D = 0$) and rules out the possibility that the ground state is valley polarized[@kou2014electron]. Landau Levels with high D ========================= Theoretical work employing tight binding calculations have shown that the existence of interlayer bias between the layers ($U$) will have compelling effect on the LL spectrum of BLG[@PhysRevB.76.115419]. In this section we will discuss about the evolution of LL spectrum with high interlayer bias. Figure 5a shows the LL energies as a function of $\bar D$ for B=6T. One striking feature is the reduction of the energy separation between the LLs as the band gap increases, specially between the LLs near the band edge. For $\bar D > 0.5~ V/nm$ we see the LLs near the band edge merge with each other. Fig. 5b shows colorplot of LL spectrum (as $C_q$) as a function of $E_F$ for $\bar D=0.8~V/nm$ ($\Delta_g\sim 80~meV$). One can clearly see the differences between the LL spectrum at $\bar D=0$ (Fig. 3b) and $\bar D=0.8~V/nm$ (Fig. 5b). At $\bar D=0$ the LLs are clearly visible at B = 2T where as at $\bar D=0.8~V/nm$ LLs can be hardly seen even at B = 8T. It can be also seen from the Fig. 5b that the LLs are broadened and the broadening is higher for lower LLs near the band edge. In Fig. 5c, we also show the evolution of the gap for $\nu=2$ state as a function of $\bar D$ for B = 10T. One can notice that for a fixed magnetic field the LLs gap decreases almost linearly with increasing $\bar D$. It has been shown theoretically in Ref[@PhysRevB.84.075451] that the LL spectrum in presence of B and $\bar D$ has the following energy eigenvalues for $n>0$ $$\begin{split} E^{\pm}_{n,s_1,-}&=\bigg(n+\frac{1}{2}\bigg)\beta\widetilde\Delta\mp \beta U\\ &+ s_1 \sqrt{ \Big[ (2n + 1)\beta U \mp \dfrac{\beta \widetilde\Delta}{2} - U \Big]^2 \!\! + n (n + 1)\,\beta^2 \gamma_1^2\,} \,. \end{split} \label{eqn:E_low_energy}$$ where, $\gamma_1=0.4~eV$, $\widetilde\Delta=59~meV$, $\beta =\frac{\omega_0^2}{\gamma_1^2}, \omega_0=\sqrt{2}\frac{\hbar v_0}{l_B}$; $l_B$ is the magnetic length, and $v_0 = \sqrt{3} \gamma_0 a_0/\hbar\approx 1.0\times10^8\,{\rm cm}/{\rm s}$ is the Fermi velocity. Fig. 5d shows the calculated LL energies as a function of energy gap (U) for B=6T. The solid and the dashed lines correspond to $K$ and $K'$ valleys. The LLs start to merge for $U>50~meV$ which matches well with the experimentally observed values as can be seen in Fig. 5a. We do not observe the splitting of the K and K’ valleys due to the large broadening of our device ($\delta E_F\sim 20$ meV). Instead we observe the broadening of the LLs with increasing $\bar D$.\ conclusion ========== In summary, we have mapped the complete energetics of the Landau level spectrum in a bilayer graphene with magnetic and electric field. We model a possible ground state based on our observations. We have also demonstrated the smearing of the LLs at high broken inversion symmetry in agreement with theoretical predictions.

--- abstract: | *Bregman divergences* are important distance measures that are used extensively in data-driven applications such as computer vision, text mining, and speech processing, and are a key focus of interest in machine learning. Answering *nearest neighbor* (NN) queries under these measures is very important in these applications and has been the subject of extensive study, but is problematic because these distance measures lack metric properties like symmetry and the triangle inequality. In this paper, we present the first provably *approximate nearest-neighbor* (ANN) algorithms. These process queries in $O(\log n)$ time for Bregman divergences in fixed dimensional spaces. We also obtain $\text{poly}\log n$ bounds for a more abstract class of distance measures (containing Bregman divergences) which satisfy certain structural properties . Both of these bounds apply to both the regular asymmetric Bregman divergences as well as their symmetrized versions. To do so, we develop two geometric properties vital to our analysis: a *reverse triangle inequality* (RTI) and a relaxed triangle inequality called *$\mu$-defectiveness* where $\mu$ is a domain-dependent parameter. Bregman divergences satisfy the RTI but *not* $\mu$-defectiveness. However, we show that the square root of a Bregman divergence does satisfy $\mu$-defectiveness. This allows us to then utilize both properties in an efficient search data structure that follows the general two-stage paradigm of a ring-tree decomposition followed by a quad tree search used in previous near-neighbor algorithms for Euclidean space and spaces of bounded doubling dimension. Our first algorithm resolves a query for a $d$-dimensional $(1+{\varepsilon})$-ANN in $O \left((\frac{\log n}{{\varepsilon}})^{O(d)} \right)$ time and $O \left(n \log^{d-1} n \right)$ space and holds for generic $\mu$-defective distance measures satisfying a RTI. Our second algorithm is more specific in analysis to the Bregman divergences and uses a further structural constant, the maximum ratio of second derivatives over each dimension of our domain ($c_0$). This allows us to locate a $(1+{\varepsilon})$-ANN in $O(\log n)$ time and $O(n)$ space, where there is a further $(c_0)^d$ factor in the big-Oh for the query time. author: - | Amirali Abdullah\ University of Utah - | John Moeller\ University of Utah - | Suresh Venkatasubramanian\ University of Utah bibliography: - 'nn.bib' title: 'Approximate Bregman near neighbors in sublinear time: Beyond the triangle inequality' --- Introduction {#Introduction} ============ intro Related Work {#sec:related} ============ related Definitions {#sec:defn} =========== In this paper we study the approximate nearest neighbor problem for distance functions $D$: Given a point set $P$, a query point $q$, and an error parameter ${\varepsilon}$, find a point $\text{nn}_q \in P$ such that $D(\text{nn}_q,q) \leq (1 + {\varepsilon})\min_{p \in P} D(p,q)$. We start by defining general properties that we will require of our distance measures. In what follows, we will assume that the distance measure $D$ is *reflexive*: $D(x,y) = 0$ iff $x = y$. \[monotonedefn\] Let $M\subset \reals$, $D:M\times M\to\reals$ be a distance function, and let $a,b,c\in M$ where $a<b<c$. If the following are true for any such choice of $a,b,$ and $c$: that $0 \leq D(a,b) < D(a,c)$, that $0 \leq D(b,c) < D(a,c)$, and that $D(x,y) = 0$ iff $x =y$, then we say that $D$ is *monotonic*. For a general distance function $D : M \times M \to \reals$, where $M \subset \reals^d$, we say that $D$ is monotonic if it is monotonic when restricted to any subset of $M$ parallel to a coordinate axis. Let $M$ be a subset of $\reals$. We say that a monotone distance measure $D : M \times M \to \reals$ satisfies a *reverse triangle inequality* or RTI if for any three elements $a \le b \le c \in M$, $ D(a,b) + D(b,c) \le D(a,c) $ \[musimdefn\] Let $D$ be a symmetric monotone distance measure satisfying the reverse triangle inequality. We say that $D$ is *$\mu$-defective* with respect to domain $M$ if for all $a,b,q \in M$, $$|D(a,q) - D(b,q)| < \mu D(a,b)$$ For an asymmetric distance measure $D$, we define left and right sided $\mu$-defectiveness respectively as $$|D(q,a) - D(q,b)| < \mu D(a,b)$$ $$|D(a,q) - D(b,q)| < \mu D(b,a)$$ Note that by interchanging $a$ and $b$ and using the symmetry of the modulus sign, we can also rewrite left and right sided $\mu$-defectiveness respectively as $|D(q,a) - D(q,b)| < \mu D(b,a)$ and $|D(a,q) - D(b,q)| < \mu D(a,b)$. #### Two technical notes. The distance functions under consideration are typically defined over $\reals^d$. We will assume in this paper that the distance $D$ is *decomposable*: roughly, that $D((x_1, \ldots, x_d), (y_1, \ldots, y_d))$ can be written as $g( \sum_i f(x_i, y_i))$, where $g$ and $f$ are monotone. This captures all the Bregman divergences that are typically used (with the exception of the Mahalanobis distance). We will also need to compute the diameter of an axis parallel box of side length $\ell$. Our results hold as long as the diameter of such a box is $O(\ell d^{O(1)} )$: note that this captures standard distances like those induced by norms, as well as decomposable Bregman divergences. In what follows, we will mostly make use of the *square root* of a Bregman divergence, for which the diameter of a box is $\ell (\mu+1) d^{\frac{1}{2}}$ or $\ell d^{\frac{1}{2}}$ , and so without loss of generality we will use this in our bounds. #### Bregman Divergences. Let $\phi: M\subset \reals^d \to \reals$ be a *strictly convex* function that is differentiable in the relative interior of $M$. The *Bregman divergence* ${\ensuremath{D_\phi}}$ is defined as $${\ensuremath{D_\phi}}(x,y) = \phi(x) - \phi(y) - \langle \nabla \phi(y), x-y\rangle$$ In general, ${\ensuremath{D_\phi}}$ is asymmetric. A *symmetrized* Bregman divergence can be defined by averaging: $${\ensuremath{D_{s\phi}}}(x,y) = \frac{1}{2}({\ensuremath{D_\phi}}(x,y) + {\ensuremath{D_\phi}}(y,x)) = \frac{1}{2}\langle x - y, \nabla \phi(x) - \nabla \phi(y) \rangle$$ An important subclass of Bregman divergences are the *decomposable* Bregman divergences. Suppose $\phi$ has domain $M = \prod_{i=1}^d M_i $ and can be written as $\phi(x) = \sum_{i=1}^d \phi_i(x_i)$, where $\phi_i :M_i \subset \reals \to \reals$ is also strictly convex and differentiable in relint($S_i$). Then $ {\ensuremath{D_\phi}}(x,y) = \sum_{i=1}^d D_{\phi_i}(x_i, y_i)$ is a *decomposable* Bregman divergence. Most commonly used Bregman divergences are decomposable:  [@cayton-thesis Chapter 3] illustrates some of the commonly used ones, including the Euclidean distance, the KL-divergence, and the Itakura-Saito distance . In this paper we will hence limit ourselves to considering decomposable distance measures. We note that due to the primal-dual relationship of ${\ensuremath{D_\phi}}(a,b)$ and $D_{\phi^*}(b^*, a^*)$, for our results on the asymmetric Bregman divergence we need only consider right-sided $\mu$-defective distance measures. Properties of Bregman Divergences {#sec:prop-sqrtd_s-phi} ================================= The previous section defined key properties that we desire of a distance function $D$. The Bregman divergences (or modifications thereof) satisfy the following properties, as can be shown by direct computation. \[lefttr\] Any one-dimensional Bregman divergence is monotonic. \[cover\] Any one-dimensional Bregman divergence satisfies the reverse triangle inequality. Let $a \leq b \leq c$ be three points in the domain of ${\ensuremath{D_\phi}}$. Then it holds that: $${\ensuremath{D_\phi}}(a,b) + {\ensuremath{D_\phi}}(b,c) \leq {\ensuremath{D_\phi}}(a,c)$$ $${\ensuremath{D_\phi}}(c,b) + {\ensuremath{D_\phi}}(b,a) \leq {\ensuremath{D_\phi}}(c,a)$$ We prove the first case, the second follows almost identically. $$\begin{aligned} {\ensuremath{D_\phi}}(a,b) + {\ensuremath{D_\phi}}(b,c) &= \phi(a) - \phi(b) - \phi'(b)(a-b) + \phi(b) - \phi(c) - \phi'(c)(b-c) \\ &= \phi(a) - \phi(c) - \phi'(b)(a-b) - \phi'(c)(b-c)\end{aligned}$$ But since $\phi''(x) \geq 0$ for all $x \in \mathbb{R}$, by property of Bregman divergences, we have that $\phi'(b) \leq \phi'(c)$. This allows us to make the substitution. $$\begin{aligned} {\ensuremath{D_\phi}}(a,b) + {\ensuremath{D_\phi}}(b,c) &= \phi(a) - \phi(c) - \phi'(b)(a-b) - \phi'(c)(b-c) \\ & \leq \phi(a) - \phi(c) - \phi'(c)(a-b) - \phi'(c)(b-c) \\ &= \phi(a) - \phi(c) - \phi'(c)(a-c) \\ &= {\ensuremath{D_\phi}}(a,c)\end{aligned}$$ Note that this lemma can be extended similarly by induction to any series of $n$ points between $a$ and $c$. Further, using the relationship between ${\ensuremath{D_\phi}}(a,b)$ and the “dual” distance $D_{\phi^*}(b^*, a^*)$, we can show that the reverse triangle inequality holds going “left” as well: ${\ensuremath{D_\phi}}(c,b) + {\ensuremath{D_\phi}}(b,a) \leq {\ensuremath{D_\phi}}(c,a)$. These two separate reverse triangle inequalities together yield the result for ${\ensuremath{D_{s\phi}}}$. We also get a similar result for $\sqrt{{\ensuremath{D_{s\phi}}}}$ by algebraic manipulations. \[Aklreverse\] $\sqrt{{\ensuremath{D_{s\phi}}}}$ satisfies the reverse triangle inequality. Fix $a \le x \le b$, and assume that the reverse triangle inequality does not hold: $$\begin{aligned} \sqrt{{\ensuremath{D_{s\phi}}}(a,x)} + \sqrt{{\ensuremath{D_{s\phi}}}(x,b)} &> \sqrt{{\ensuremath{D_{s\phi}}}(a,b)} \\ \sqrt{(x-a) (\phi'(x) - \phi'(a))} + \sqrt{(b-x) (\phi'(b) - \phi'(x))} &> \sqrt{(b-a) (\phi'(b) - \phi'(a))} \end{aligned}$$ Squaring both sides, we get: $$\begin{aligned} (x-a)( \phi'(x)-\phi'(a)) + (b-x) (\phi'(b) - \phi'(x)) \hspace{1.5in}& \\ + 2 \sqrt{(x-a)(b-x)(\phi'(x) - \phi'(a))(\phi'(b) - \phi'(x))} &> (b-a)( \phi'(b) - \phi'(a)) \\ (b-x)(\phi'(x) - \phi'(a)) + (x-a)(\phi'(b) - \phi'(x)) \hspace{1.5in}& \\ - 2 \sqrt{(x-a)(b-x)(\phi'(x) - \phi'(a))(\phi'(b) - \phi'(x))} &< 0 \\ \left( \sqrt{(b-x)(\phi'(x) - \phi'(a))} - \sqrt{(x-a)(\phi'(b) - \phi'(x))} \right)^2 &< 0\end{aligned}$$ which is a contradiction, since the LHS is a perfect square. While the Bregman divergences satisfy both monotonicity and the reverse triangle inequality, they are not $\mu$-defective with respect to *any* domain! An easy example of this is $\ell_2^2$, which is also a Bregman divergence. A surprising fact however is that $\sqrt{{\ensuremath{D_{s\phi}}}}$ and $\sqrt{{\ensuremath{D_\phi}}}$ do satisfy $\mu$-defectiveness (with $\mu$ depending on the bounded size of our domain). While we were unable to show precise bounds for $\mu$ in terms of the domain, the values are small. For example, for the symmetrized KL-divergence on the simplex where each coordinate is bounded between $0.1$ and $0.9$, $\mu$ is $1.22$. If each coordinate is between $0.01$ and $0.99$,then $\mu$ is $2.42$. \[Arootmu\] Given any interval $I=[x_1 x_2]$ on the real line, there exists a finite $\mu$ such that $\sqrt{{\ensuremath{D_{s\phi}}}}$ is $\mu$-defective with respect to $I$. Consider three points $a,b,q \in I$. Due to symmetry of the cases and conditions, there are three cases to consider: $a<q<b$, $a < b <q$ and $q < b < a$. **Case 1:** : Here $a < q < b$. The following is trivially true by the monotonicity of $\sqrt{{\ensuremath{D_{s\phi}}}}$. $$\left| \sqrt{{\ensuremath{D_{s\phi}}}(q,a)} - \sqrt{{\ensuremath{D_{s\phi}}}(q,b)} \right| < \sqrt{{\ensuremath{D_{s\phi}}}(a,b)}$$ **Cases 2 and 3:** : For the remaining symmetric cases, $a < b< q$ and $q<b<a$, note that since $\sqrt{{\ensuremath{D_{s\phi}}}(q,a)} - \sqrt{{\ensuremath{D_{s\phi}}}(q,b)}$ and $\sqrt{{\ensuremath{D_{s\phi}}}(a,b)}$ are both bounded, continuous functions on a compact domain (the interval $[x_1 x_2]$), we need only show that the following limit exists: $$\label{mu-def-limit} \lim_{a \to b} \frac{\left| \sqrt{{\ensuremath{D_{s\phi}}}(q,a)} - \sqrt{{\ensuremath{D_{s\phi}}}(q,b)} \right|}{\sqrt{{\ensuremath{D_{s\phi}}}(a,b)}}$$ First consider $a<b<q$, and we assume $\lim_{b \to a}$ We will use the following substitutions repeatedly in our derivation: $b = a +h$, $\lim_{h \to 0} \phi(a+h) = \lim_{h\to 0}\left( \phi(a) + h\phi'(a) \right)$, and $\lim_{h \to 0} \sqrt{1+h} = \lim_{h\to 0}(1 + h/2)$. For ease of computation, we replace $\phi'$ by $\psi$, to be restored at the last step. $$\label{mucomp} \lim_{a \to b} \frac{\sqrt{{\ensuremath{D_{s\phi}}}(a,q)} - \sqrt{{\ensuremath{D_{s\phi}}}(b,q)}}{\sqrt{{\ensuremath{D_{s\phi}}}(a,b)}} = \frac{\lim_{a \to b} \left(\sqrt{(q-a)(\psi(q) - \psi(a)) } - \sqrt{(q-b)(\psi(q) - \psi(b)) } \right)} {\lim_{a \to b}\sqrt{(b-a)(\psi(b) - \psi(a)) }}$$ Computing the denominator: $$\begin{aligned} \lim_{b \to a}\sqrt{(b-a) (\psi(b) - \psi(a)) } &=\lim_{h \to 0} \sqrt{(a+h-a) (\psi(a+h) - \psi(a) } \\ &=\lim_{h \to 0} \sqrt{h ( \psi(a) + h \psi'(a) - \psi(a)) } \\ &=\lim_{h \to 0} \sqrt{h (h \psi'(a))} =\lim_{h \to 0} h \sqrt{\psi'(a)} \end{aligned}$$ We now address the numerator: $$\begin{aligned} \lim_{b \to a}&\sqrt{(q-a)(\psi(q) - \psi(a))} - \sqrt{(q-b) (\psi(q) - \psi(b)) } \\ &= \lim_{h \to 0} \sqrt{(q-a) (\psi(q) - \psi(a))} - \sqrt{(q-a-h) (\psi(q) - \psi(a) - h \psi'(a)) } \\ &= \lim_{h \to 0} \sqrt{(q-a) (\psi(q) - \psi(a))} - \sqrt{(q-a)\left( 1-\frac{h}{q-a} \right) (\psi(q) - \psi(a)) \left( 1 - h \frac{\psi'(a)}{\psi(q) - \psi(a)} \right)} \\ &= \lim_{h \to 0} \sqrt{(\psi(q) - \psi(a))(q-a)} \left(1 - \sqrt{ 1 - \frac{h}{q-a}} \sqrt{ 1 - h \frac{\psi'(a)}{\psi(q) - \psi(a)} } \right) \\ &= \lim_{h \to 0} \sqrt{(\psi(q) - \psi(a))(q-a)} \left(1 - \left( 1- \frac{h}{2(q-a)} \right) \left(1 - h \frac{\psi'(a)}{2(\psi(q) - \psi(a))} \right) \right) \\ &= \lim_{h \to 0} \sqrt{(\psi(q) - \psi(a))(q-a)} \left( \frac{h}{2(q-a)} + h \frac{\psi'(a)}{2(\psi(q) - \psi(a))} - \frac{h^2}{4(q-a)(\psi(q) - \psi(a))} \right) \end{aligned}$$ Dropping higher order terms of $h$, the above reduces to: $$\lim_{h \to 0} h \sqrt{(\psi(q) - \psi(a))(q-a)} \left( \frac{1}{2 (q-a)} + \frac{\psi'(a)}{2 (\psi(q) - \psi(a))} \right)$$ Now combine numerator and denominator back in equation \[mucomp\]. $$\begin{aligned} \lim_{b \to a} \frac{\sqrt{{\ensuremath{D_{s\phi}}}(a,q)} - \sqrt{{\ensuremath{D_{s\phi}}}(b,q)}}{\sqrt{{\ensuremath{D_{s\phi}}}(a,b)}} &= \frac{\lim_{h \to 0} h \sqrt{(\psi(q) - \psi(a))(q-a)} \left( \frac{1}{2 (q-a)} + \frac{\psi'(a)}{2 (\psi(q) - \psi(a))} \right)} {\lim_{h \to 0} h \sqrt{\psi'(a)}} \\ &= \sqrt{\frac{(\psi(q) - \psi(a))(q-a)}{\psi'(a)}} \left( \frac{1}{2 (q-a)} + \frac{\psi'(a)}{2 (\psi(q) - \psi(a))} \right) \\ &= \frac{1}{2} \left( \sqrt{\frac{\psi(q) - \psi(a)}{\psi'(a)(q-a)}} + \sqrt{\frac{\psi'(a)(q-a)}{\psi(q) - \psi(a)}} \right)\end{aligned}$$ Substituting back $\phi'(x)$ for $\psi(x)$, we see that limit \[mu-def-limit\] exists, provided $\phi$ is strictly convex: $$\frac{1}{2} \left( \sqrt{\frac{\phi'(q) - \phi'(a)}{\phi''(a)(q-a)}} + \sqrt{\frac{\phi''(a)(q-a)}{\phi'(q) - \phi'(a)}} \right)$$ The analysis follows symmetrically for case 3, where $q < b < a$. We note that the result for $\sqrt{{\ensuremath{D_\phi}}}$ is proven by establishing the following relationship between ${\ensuremath{D_\phi}}(a,b)$ and ${\ensuremath{D_\phi}}(b,a)$ over a bounded interval $I \subset \reals$, and with some further computation. \[firstTosecond\] Given a Bregman divergence ${\ensuremath{D_\phi}}$ and a bounded interval $I \subset \reals$, $\sqrt{{\ensuremath{D_\phi}}(a,b)}/ \sqrt{{\ensuremath{D_\phi}}(b,a)}$ is bounded by a constant $c_0$ $\forall a,b \in I$ where $c_0$ depends on the choice of divergence and interval. By continuity and compactness, over a finite interval $I$ we have that $ c_ 0 = \max_x \phi_i'' (x)/ \min_y \phi_i''(y) $ is bounded. Now by using the Lagrange form of $\sqrt{{\ensuremath{D_\phi}}(a,b)}$, we get that $\sqrt{{\ensuremath{D_\phi}}(a,b)}/ \sqrt{{\ensuremath{D_\phi}}(b,a)} < \sqrt{c_0}$ \[Arootmubreg\] Given any interval $I=[x_1 x_2]$ on the real line, there exists a finite $\mu$ such that $\sqrt{{\ensuremath{D_\phi}}}$ is right-sided $\mu$-defective with respect to $I$ Consider any three points $a,b,q \in I$. We will prove that there exists finite $\mu$ such that: $$\left| \sqrt{{\ensuremath{D_\phi}}(a,q)} - \sqrt{{\ensuremath{D_\phi}}(b,q)} \right| < \mu \sqrt{{\ensuremath{D_\phi}}(b,a)}$$ Here there are now six cases to consider: $a<q<b$, $b<q<a$ , $a < b <q$, $b < a <q$ , $q < b < a$, and $q<a<b$ . **Case 1 and 2:** : Here $a < q < b$. By monotonicity we have that: $$\left| \sqrt{{\ensuremath{D_\phi}}(a,q)} - \sqrt{{\ensuremath{D_\phi}}(b,q)} \right| < \sqrt{{\ensuremath{D_\phi}}(a,b)} + \sqrt{{\ensuremath{D_\phi}}(b,a)}$$ But by lemma \[firstTosecond\], we have that $\sqrt{{\ensuremath{D_\phi}}(a,b)} < c \sqrt{{\ensuremath{D_\phi}}(b,a)}$ for some constant $c$ defined over $I$. This implies that $ \left| \sqrt{{\ensuremath{D_\phi}}(a,q)} - \sqrt{{\ensuremath{D_\phi}}(b,q)} \right| / \sqrt{{\ensuremath{D_\phi}}(b,a)} < c + 1$, i.e, it is bounded over $I$. A similar analysis works for Case 2 where $b<q<a$. **Cases 3 and 4:** : For these two cases, $a < b< q$ and $b < a < q$, note that since $\sqrt{{\ensuremath{D_\phi}}(q,a)} - \sqrt{{\ensuremath{D_\phi}}(q,b)}$ and $\sqrt{{\ensuremath{D_\phi}}(b,a)}$ are both bounded, continuous functions on a compact domain (the interval $[x_1 x_2]$), we need only show that the following limit exists: $$\label{mubreg-def-limit} \lim_{a \to b} \frac{\left| \sqrt{{\ensuremath{D_\phi}}(a,q)} - \sqrt{{\ensuremath{D_\phi}}(b,q)} \right|}{\sqrt{{\ensuremath{D_\phi}}(b,a)}}$$ First consider $a<b<q$, and we assume $\lim_{b \to a}$. We will use the following substitutions repeatedly in our derivation: $b = a +h$, $\lim_{h \to 0} \phi(a+h) = \lim_{h\to 0}(\phi(a) + h\phi'(a))$, $\lim_{h \to 0} \phi(b) = \phi(a+h) = \lim_{h\to 0}(\phi(a) + h \psi(a) + \frac{h^2 \psi'(a)}{2})$ and $\lim_{h \to 0} \sqrt{1+h} = \lim_{h\to 0}(1 + h/2)$. For ease of computation, we replace $\phi'$ by $\psi$, to be restored at the last step. $$\label{mucom} \lim_{a \to b} \frac{\sqrt{{\ensuremath{D_\phi}}(a,q)} - \sqrt{{\ensuremath{D_\phi}}(b,q)}}{\sqrt{{\ensuremath{D_\phi}}(b,a)}} = \lim_{a \to b} \frac{ \sqrt{\phi(a) - \phi(q) - \psi(q)(a-q) } - \sqrt{\phi(b) - \phi(q) - \psi(q)(b-q)}} {\sqrt{ \phi(b) - \phi(a) - \psi(a)(b-a)}}$$ Computing the denominator: $$\begin{aligned} \lim_{a \to b}\sqrt{ \phi(b) - \phi(a) - \psi(a)(b-a)} &=\lim_{h \to 0} \sqrt{\phi(a) + h \psi(a) + \frac{h^2 \psi'(a)}{2} - \phi(a) - h \psi(a) } \\ &=\lim_{h \to 0} \sqrt{\frac{h^2 \psi'(a)}{2}} \\ &=\lim_{h \to 0} h \sqrt{\frac{\psi'(a)}{2}} \end{aligned}$$ We now address the numerator: $$\begin{aligned} &\lim_{a \to b} \left(\sqrt{\phi(a) - \phi(q) - \psi(q)(a-q) } - \sqrt{\phi(b) - \phi(q) - \psi(q)(b-q)} \right) \\ &= \lim_{h \to 0} \sqrt{\phi(a) - \phi(q) - \psi(q)(a-q) } - \sqrt{\phi(a) - \phi(q) - \psi(q)(a -q) + h(\psi(a) - \psi(q))} \\ &= \lim_{h \to 0} \sqrt{{\ensuremath{D_\phi}}(a,q)} - \sqrt{{\ensuremath{D_\phi}}(a,q)(1 + \frac{h(\psi(a) - \psi(q))}{{\ensuremath{D_\phi}}(a,q)})} \\ &= \lim_{h \to 0} \sqrt{{\ensuremath{D_\phi}}(a,q)} \left(1 - \sqrt{ 1 - \frac{h(\psi(q) - \psi(a))}{{\ensuremath{D_\phi}}(a,q)}} \right) \\ &= \lim_{h \to 0} \sqrt{{\ensuremath{D_\phi}}(a,q)} \left(1 - \left(1 - \frac{h(\psi(q) - \psi(a))}{2 {\ensuremath{D_\phi}}(a,q)}\right) \right) \\ &= \lim_{h \to 0} \frac{h \left(\psi(q) - \psi(a) \right)}{2 \sqrt{{\ensuremath{D_\phi}}(a,q)}} \end{aligned}$$ Now combine numerator and denominator back in equation \[mucom\], and note that ${\ensuremath{D_\phi}}(a,q) = \frac{1}{2}(\psi'(x))(q-a)^2$, for some $x \in [ab]$. $$\begin{aligned} \lim_{a \to b} \frac{\sqrt{{\ensuremath{D_\phi}}(a,q)} - \sqrt{{\ensuremath{D_\phi}}(b,q)}}{\sqrt{{\ensuremath{D_\phi}}(b,a)}} &= \frac{\lim_{h \to 0} \frac{h(\psi(q) - \psi(a))}{2 \sqrt{{\ensuremath{D_\phi}}(a,q)}} } {\lim_{h \to 0} h \sqrt{\frac{\psi'(a)}{2}} } \\ &= \frac{(\psi(q) - \psi(a))}{q - a}\frac{\sqrt{\psi'(a)}}{\sqrt{ \psi'(x)}} \end{aligned}$$ Substituting back $\phi'(x)$ for $\psi(x)$, we see that limit \[mubreg-def-limit\] exists, provided $\phi$ is strictly convex: $$\frac{(\phi'(q) - \phi'(a))}{q - a}\frac{\sqrt{\phi''(a)}}{\sqrt{ \phi''(x)}}$$ The analysis follows symmetrically for case 4, by noting that $\lim_{a \to b} \frac{\sqrt{{\ensuremath{D_\phi}}(a,b)}}{\sqrt{{\ensuremath{D_\phi}}(b,a)}} = 1$ and that $\sqrt{{\ensuremath{D_\phi}}(a,q)} - \sqrt{{\ensuremath{D_\phi}}(b,q)} = - (\sqrt{{\ensuremath{D_\phi}}(b,q)} - \sqrt{{\ensuremath{D_\phi}}(a,q)})$, i.e we may suitably interchange $a$ and $b$. **Cases 5 and 6:** : Here $q<a<b$ or $q<b<a$. Looking more carefully at the analysis for cases 3 and 4, note that the ordering $q<a<b$ vs $a<b<q$ does not affect the magnitude of the expression for limit \[mubreg-def-limit\], only the sign. Hence we can use the same analysis to prove $\mu$-defectiveness for cases 5 and 6. Given any interval $I=[x_1 x_2]$ on the real line, there exists a finite $\mu$ such that $\sqrt{{\ensuremath{D_\phi}}}$ is left-sided $\mu$-defective with respect to $I$ Follows from similar computation. We extend our results to $d$ dimensions naturally now by showing that if $M$ is a domain such that $\sqrt{{\ensuremath{D_{s\phi}}}}$ and $\sqrt{{\ensuremath{D_\phi}}}$ are $\mu$-defective with respect to the projection of $M$ onto each coordinate axis, then $\sqrt{{\ensuremath{D_{s\phi}}}}$ and $\sqrt{{\ensuremath{D_\phi}}}$ are $\mu$-defective with respect to all of $M$. \[AallDmusim\] Consider three points, $a = (a_1 , \ldots, a_i, \ldots, a_d)$, $b = (b_1 , \ldots, b_i, \ldots, b_d)$, $q = (q_1 , \ldots, q_i, \ldots, q_d)$ such that $ | \sqrt{{\ensuremath{D_{s\phi}}}(a_i, q_i)} - \sqrt{{\ensuremath{D_{s\phi}}}(b_i,q_i)}| < \mu \sqrt{{\ensuremath{D_{s\phi}}}(a_i, b_i)}, \forall 1 \leq i \leq d$. Then $$\left| \sqrt{{\ensuremath{D_{s\phi}}}(a, q)} - \sqrt{{\ensuremath{D_{s\phi}}}(b,q)} \right| < \mu \sqrt{{\ensuremath{D_{s\phi}}}(a, b)}$$ Similarly, if $| \sqrt{{\ensuremath{D_\phi}}(a_i, q_i)} - \sqrt{{\ensuremath{D_\phi}}(b_i,q_i)}| < \mu \sqrt{{\ensuremath{D_\phi}}(a_i, b_i)}, \forall 1 \leq i \leq d$. Then $$\left| \sqrt{{\ensuremath{D_\phi}}(a, q)} - \sqrt{{\ensuremath{D_\phi}}(b,q)} \right| < \mu \sqrt{{\ensuremath{D_\phi}}(b,a)}$$ $$\begin{aligned} \left| \sqrt{{\ensuremath{D_{s\phi}}}(a, q)} - \sqrt{{\ensuremath{D_{s\phi}}}(b,q)} \right| &< \mu \sqrt{{\ensuremath{D_{s\phi}}}(a, b)} \\ {\ensuremath{D_{s\phi}}}(a,q) + {\ensuremath{D_{s\phi}}}(b,q) - 2 \sqrt{{\ensuremath{D_{s\phi}}}(a,q) {\ensuremath{D_{s\phi}}}(b,q)} &< \mu^2 {\ensuremath{D_{s\phi}}}(a,b) \\ \sum_{i=1}^{d} \left({\ensuremath{D_{s\phi}}}(a_i,q_i) + {\ensuremath{D_{s\phi}}}(b_i, q_i)\right) - 2 \sqrt{{\ensuremath{D_{s\phi}}}(a,q) {\ensuremath{D_{s\phi}}}(b,q)} &< \mu^2 \sum_{i=1}^{d} {\ensuremath{D_{s\phi}}}(a_i,b_i) \\ \sum_{i=1}^{d} \left({\ensuremath{D_{s\phi}}}(a_i, q_i) + {\ensuremath{D_{s\phi}}}(b_i,q_i) - \mu^2 {\ensuremath{D_{s\phi}}}(a_i,b_i)\right) &< 2 \sqrt{{\ensuremath{D_{s\phi}}}(a,q){\ensuremath{D_{s\phi}}}(b,q)}\end{aligned}$$ The last inequality is what we need to prove for $\mu$-defectiveness with respect to $a,b,q$. By assumption we already have $\mu$-defectiveness w.r.t each $a_i,b_i, q_i$, for every $1 \leq i \leq d$: $$\begin{aligned} {\ensuremath{D_{s\phi}}}(a_i,q_i) + {\ensuremath{D_{s\phi}}}(b_i,q_i) - \mu^2 {\ensuremath{D_{s\phi}}}(a_i,b_i) &< 2 \sqrt{{\ensuremath{D_{s\phi}}}(a_i,q_i) {\ensuremath{D_{s\phi}}}(b_i,q_i)} \\ \sum_{i=1}^{d} \left( {\ensuremath{D_{s\phi}}}(a_i,q_i) + {\ensuremath{D_{s\phi}}}(b_i,q_i) - \mu^2 {\ensuremath{D_{s\phi}}}(a_i,b_i) \right) &< 2 \sum_{i=1}^{d} \sqrt{{\ensuremath{D_{s\phi}}}(a_i,q_i) {\ensuremath{D_{s\phi}}}(b_i,q_i)} \end{aligned}$$ So to complete our proof we need only show: $$\label{sec:prop-sqrts-skl} \sum_{i=1}^{d} \sqrt{{\ensuremath{D_{s\phi}}}(a_i,q_i)} \sqrt{{\ensuremath{D_{s\phi}}}(b_i,q_i)} \leq \sqrt{{\ensuremath{D_{s\phi}}}(a,q)}\sqrt{{\ensuremath{D_{s\phi}}}(b,q)}$$ But notice the following: $$\begin{aligned} \sqrt{{\ensuremath{D_{s\phi}}}(a,q)} &= \left( \sum_{i=1}^{d}{\ensuremath{D_{s\phi}}}(a_i,q_i) \right)^{\frac{1}{2}} = \left( \sum_{i=1}^{d} \left( \sqrt{{\ensuremath{D_{s\phi}}}(a_i,q_i)} \right)^2 \right)^{\frac{1}{2}} \\ \sqrt{{\ensuremath{D_{s\phi}}}(b,q)} &= \left( \sum_{i=1}^{d}{\ensuremath{D_{s\phi}}}(b_i,q_i) \right)^{\frac{1}{2}} = \left( \sum_{i=1}^{d} \left( \sqrt{{\ensuremath{D_{s\phi}}}(b_i,q_i)} \right)^2 \right)^{\frac{1}{2}}\end{aligned}$$ So inequality \[sec:prop-sqrts-skl\] is simply a form of the Cauchy-Schwarz inequality, which states that for two vectors $u$ and $v$ in $\reals^d$, that $\left|\left< u, v \right>\right| \leq \|u\| \|v\|$, or that $$\left| \sum_{i=1}^d u_iv_i \right| \leq \left( \sum_{i=1}^du_i^2 \right)^{\frac{1}{2}} \left( \sum_{i=1}^dv_i^2 \right)^{\frac{1}{2}}$$ The second part of the proposition can be derived by an essentially identical argument. Packing and Covering Bounds {#covering} =========================== The aforementioned key properties (monotonicity, the reverse triangle inequality, decomposability, and $\mu$-defectiveness) can be used to prove packing and covering bounds for a distance measure $D$. We now present some of these bounds. \[1dintersect\] Consider a monotone distance measure $D$ satisfying the reverse triangle inequality, an interval $[ab]$ such that $D(a,b) = s$ and a collection of disjoint inprovetervals intersecting $[ab]$, where $I = \{[x x'] \mid [x x'] , D(x, x') \geq \ell\}$. Then $|I| \leq \frac{s}{\ell}+2$. Let $I'$ be the intervals of $I$ that are totally contained in $[ab]$. The combined length of all intervals in $I'$ is at most $|I'|\ell$, but by the reverse triangle inequality, their total length cannot exceed $s$, so $|I'| \leq \frac{s}{\ell}$. There can be only two members of $I$ not in $I'$, so $|I| \leq \frac{s}{\ell} + 2$. A simple greedy approach yields a constructive version of this lemma. \[1dcover\] Given any two points, $a \leq b$ on the line s.t $D(a,b) = s$, we can construct a packing of $[ab]$ by $r \le \frac{1}{{\varepsilon}}$ intervals $[x_i x_{i+1}]$, $1 \leq i \leq r$ such that $D(a, x_0) = D(x_i, x_{i+1}) = {\varepsilon}s$, $\forall i$ and $D(x_r , b) \leq {\varepsilon}s$. Here $D$ is a monotone distance measure satisfying the reverse triangle inequality. Recall here that ${\ensuremath{D_\phi}}$, ${\ensuremath{D_{s\phi}}}$ and $\sqrt{{\ensuremath{D_{s\phi}}}}$ satisfy the conditions of Lemma \[1dintersect\] and corollary \[1dcover\] as they satisfy an RTI and are decomposable. However, since $\sqrt{{\ensuremath{D_\phi}}}$ may not satisfy the reverse triangle inequality, we instead prove a weaker packing bound on $\sqrt{{\ensuremath{D_\phi}}}$ by using ${\ensuremath{D_\phi}}$. \[1dsqrtbregint\] Given distance measure $\sqrt{{\ensuremath{D_\phi}}}$ and an interval $[ab]$ such that $\sqrt{{\ensuremath{D_\phi}}}(a,b) = s$ and a collection of disjoint intervals intersecting $[ab]$ where $I = \{[x x'] \mid [x x'] , \sqrt{{\ensuremath{D_\phi}}}(x, x') \geq \ell \}$. Then $|I| \leq \frac{s^2}{\ell^2}+2$. Such a set of intervals can be explicitly constructed. We note that here ${\ensuremath{D_\phi}}(a,b) = s^2$, and $I = \{[x x'] \mid [x x'] , {\ensuremath{D_\phi}}(x, x') \geq \ell^2 \}$. The result then follows trivially from lemma \[1dintersect\], since ${\ensuremath{D_\phi}}$ satisfies the conditions of lemma \[1dintersect\]. The above bounds can be generalized to higher dimensions to provide packing bounds for balls and cubes (which we define below) with respect to a monotone, decomposable distance measure. \[cube\] Given a collection of $d$ intervals $a_i, b_i$ , s.t $D(a_i, b_i) = s$ where $1 \leq i \leq d$, the *cube* in $d$ dimensions is defined as $\prod_{i=i}^{d} [a_i b_i]$ and is said to have side length $s$. \[cubeCover\] Given a $d$ dimensional cube $B_1$ of side length $s$ under distance measures ${\ensuremath{D_\phi}}$, ${\ensuremath{D_{s\phi}}}$ and $\sqrt{{\ensuremath{D_{s\phi}}}}$, we can cover it with at most ${\varepsilon}^d$ cubes of side length *exactly* ${\varepsilon}s$. In the case of $\sqrt{{\ensuremath{D_\phi}}}$, we can cover it with at most ${\varepsilon}^{2d}$ cubes of side length ${\varepsilon}s$. Note that ${\ensuremath{D_{s\phi}}}$, ${\ensuremath{D_\phi}}$ , $\sqrt{{\ensuremath{D_{s\phi}}}}$ satisfy conditions of corollary \[1dcover\]. Hence we can construct a gridding of at most $\frac{1}{{\varepsilon}}$ points in each dimension spaced ${\varepsilon}s$ apart. We then take a product over all $d$ dimensions, and the lemma follows trivially. For $\sqrt{{\ensuremath{D_\phi}}}$, we refer to the RTI for and follow the same procedure, gridding by at most $\frac{1}{{\varepsilon}^2}$ points in each dimension, spaced ${\varepsilon}s$ apart. \[ballcover\] Consider a ball $B$ of radius $s$ and center $C$ with respect to a distance measure $D$. Then in the case of ${\ensuremath{D_{s\phi}}}$ and ${\ensuremath{D_\phi}}$ it can be covered with $\frac{2^d}{{\varepsilon}^d}$ balls of radius $d {\varepsilon}s$. In the case of $\sqrt{{\ensuremath{D_{s\phi}}}}$, $B$ can be covered with $\frac{2^d}{{\varepsilon}^d}$ balls of radius $\sqrt{d} {\varepsilon}s$. And for $\sqrt{{\ensuremath{D_\phi}}}$ , $B$ can be covered by $\frac{2^d}{{\varepsilon}^{2d}}$ balls of radius $\sqrt{d} {\varepsilon}s$. We divide the ball into $2^d$ orthants around the center $c$. Each orthant can be covered by a cube of size $s$. We now consider each case separately. For ${\ensuremath{D_{s\phi}}}$, ${\ensuremath{D_\phi}}$ and $\sqrt{{\ensuremath{D_{s\phi}}}}$ , by lemma \[cubeCover\] each such cube can be broken down into $\frac{1}{{\varepsilon}^d}$ sub-cubes of side length ${\varepsilon}s$. For $\sqrt{{\ensuremath{D_\phi}}}$, we can break down each cube into $\frac{1}{{\varepsilon}^{2d}}$ sub-cubes of side length ${\varepsilon}s$. For ${\ensuremath{D_{s\phi}}}$ we can trivially cover each sub-cube by a ball of radius $d {\varepsilon}s$ placed at any corner. Similarly, for $\sqrt{{\ensuremath{D_{s\phi}}}}$, we can cover each sub-cube by a ball of radius $\sqrt{d} {\varepsilon}{s}$ placed at any corner. (This latter result follows by considering the sub-cube of side length ${\varepsilon}s$ under $\sqrt{{\ensuremath{D_{s\phi}}}}$ as one of side length ${\varepsilon}^2 s^2$ under ${\ensuremath{D_{s\phi}}}$ and placing a ball of radius $d {\varepsilon}^2 s^2$ under ${\ensuremath{D_{s\phi}}}$ on any corner). We now consider the cases of ${\ensuremath{D_\phi}}$ and $\sqrt{{\ensuremath{D_\phi}}}$. For each orthant, we construct the gridding by Lemmas \[1dcover\] and \[1dsqrtbregint\] in each dimension for ${\ensuremath{D_\phi}}$ and $\sqrt{{\ensuremath{D_\phi}}}$ respectively. This gives us $d$ sets of points $X_i$, $1 \leq i \leq d$, where $X_i$ lies on the $i$-th axis passing through the center of the ball $C$. For each $X_i$, we have an ordering (by construction) of points $C, x_{i1}, x_{i2}, \ldots$, s.t $D(x_i, x_{i+1}) = {\varepsilon}s$. Clearly every subcube is induced by the product of $d$ pairs of points of the form $\{x_{i (m_i-1)}, x_{i m_i}\}$ where $1 \leq i \leq d$ and $m_i$ is some positive integer. Now to each subcube assign the lowest corner $L_c$, defined as the product of the points $x_{i (m_i -1)}$, $1 \leq i \leq d$. The Bregman ball of radius $d {\varepsilon}s$ with center $L_c$ will cover this subcube for the case ${\ensuremath{D_\phi}}$, and the Bregman ball of radius $\sqrt{d} {\varepsilon}s$ with center $L_c$ will cover this subcube for the case $\sqrt{{\ensuremath{D_\phi}}}$. Note that this argument will also extend to covering the cells of a quadtree produced by recursive decomposition, by a ball of required size placed on appropriate ”lowest" corner. Since there are $\frac{1}{{\varepsilon}^{2d}}$ and $\frac{1}{{\varepsilon}^d}$ sub-cubes to each orthant for $\sqrt{{\ensuremath{D_\phi}}}$ and ${\ensuremath{D_\phi}}$ respectively, the lemma now follows by covering each subcube with a Bregman ball of the required radius. Computing a rough approximation {#sec:ringsec} =============================== Armed with our packing and covering bounds, we now describe how to compute a $O(\log n)$ rough approximate nearest-neighbor on our point set $P$, which we will use in the next section to find the $(1+{\varepsilon})$-approximate nearest neighbor. The technique we use is based on ring separators. Ring separators are a fairly old concept in geometry, notable appearances of which include the landmark paper by Indyk and Motwani [@indykmotwani]. Our approach here is heavily influenced by Har-Peled and Mendel [@peledmendel], and by Krauthgamer and Lee [@blackbox], and our presentation is along the template of the textbook by Har-Peled [@snotes Chapter 11]. We note here that the constant of $d^{d/2}$ which appears in our final bounds for storage and query time is specific to $\sqrt{{\ensuremath{D_{s\phi}}}}$. However, an argument on the same lines will yield a constant of $d^{O(d)}$ for any generic $\mu$-defective, symmetric RTI-satisfying decomposable distance measure such that the diameter of a cube of side length $1$ is bounded by $d^{O(1)}$. Let $B(m,r)$ denote the ball of radius $r$ centered at $m$, and let $B'(m,r)$ denote the complement (or exterior) of $B(m,r)$. A *ring* $R$ is the difference of two concentric balls: $R = B(m, r_2) \setminus B(m, r_1), r_2 \ge r_1$. We will often refer to the larger ball $B(m, r_2)$ as $B_{\text{out}}$ and the smaller ball as $B_{\text{in}}$. We use $P_{\text{out}}(R)$ to denote the set $P \cap B_{\text{out}}$, and similarly use $P_{\text{in}}(R)$ as $P \cap B_{\text{in}}$, where we may drop the reference to $R$ when the context is obvious. A *$t$-ring separator* $R_{P,c}$ on a point set $P$ is a ring such that $\frac{n}{c} < |P_{\text{in}}| < (1 - \frac{1}{c})n $, $\frac{n}{c} < |P_{\text{out}}| < (1 - \frac{1}{c})n$, $r_2 \geq (1 + t) r_1$ and $B_{\text{out}} \setminus B_{\text{in}}$ is empty. A $t$-ring tree is a binary tree obtained by repeated dispartition of our point set $P$ using a $t$-ring separator. Note that later on in this section, we will abuse this notation slightly by using ring-separators where the annulus is not actually empty, but we will bound the added space complexity and tree depth introduced Finally, denote the minimum sized ball containing at least $\frac{n}{c}$ points of $P$ by $B_{\text{opt},c}$; its radius is denoted by $r_{\text{opt},c}$. We demonstrate that for any point set $P$, a ring separator exists and secondly, it can always be computed efficiently. Applying this “separator” recursively on our point structure yields a ring-tree structure for searching our point set. Before we proceed further, we need to establish some properties of disks under a $\mu$-defective distance. Lemma \[circle\] is immediate from the definition of $\mu$-defectiveness, Lemma \[randommufraction\] is similar to one obtained by Har-Peled and Mazumdar [@smallestdisk] and the idea of repeating points in both children of a ring-separator derives from a result by Har-Peled and Mendel [@peledmendel]. \[circle\] Let $D$ be a $\mu$-defective distance, and let $B(m,r)$ be a ball with respect to $D$. Then for any two points $x,y \in B(m,r)$, $D(x,y) < (\mu+1) r$. Follows from the definition of $\mu$-defectiveness. $$\begin{aligned} D(x,y) - D(m,y) &< \mu D(m,x) \\ D(x,y) < \mu r + D(m,y) &= (\mu + 1) r \end{aligned}$$ \[randommufraction\] Given a constant $1 \leq c \leq n$ , we can compute in $O(nc)$ randomized time a $\mu+1$ approximation to the smallest radius ball containing $\frac{n}{c}$ points. As described by Har Peled-Mazumdar ([@smallestdisk]) we let $S$ be a random sample from $P$, generated by choosing every point of $P$ with probability $\frac{c}{n}$. Next, compute for every $p \in S$, the smallest disk centered at $p$ containing $c$ points. By median selection, this can be done in $O(n)$ time and since $E(|S|) = c$, this gives us the expected running time of $O(nc)$. Now, let $r'$ be the minimum radius computed. Note that by lemma \[circle\], if $|S \cap B_{\text{opt},c} | > 0$ then we have that $r' \leq (\mu+1) r_{opt}$. But since $B_{\text{opt},c}$ contains $\frac{n}{c}$ points, we can upper bound the probability of failure as the probability that we do not select any of the $\frac{n}{c}$ points in $B_{\text{opt}}$ in our sample. Hence: $$\begin{aligned} Pr(|S \cap B_{\text{opt},c}| > 0) = 1 - (1 - \frac{c}{n})^{\frac{n}{c}} \geq 1 - \frac{1}{e}\end{aligned}$$ Note that one can obtain a similar approximation deterministically by brute force search, but this would incur a prohibitive $O(n^2)$ running time. We can now use Lemma \[randommufraction\] to construct our ring-separator. \[improvedRing\] For arbitrary $t$ s.t $1 < t < n$, we can construct a $\frac{1}{t}$-ring separator $R_{P,c}$ in $O(n)$ expected time on a point set $P$ by repeating points. Using Lemma \[randommufraction\], we compute a ball $S=B(m,r_1)$ (where $m \in P$) containing $\frac{n}{c}$ points such that $r_1 \leq (\mu+1)r_{\text{opt},c}$ where $c$ is a constant to be set. Consider the ball $\bar{S} = B(m,2r_1)$. We shall argue that there must be $\frac{n}{c}$ points of $P$ in $\bar{S}'$ , for careful choices of $c$. As described in Lemma \[ballcover\], $\bar{S}$ can be covered by $2^d$ hypercubes of side length $2 r_1$, the union of which we shall refer to as $H$. Set $L = (\mu+1)\sqrt{d}$. Imagine a partition of $H$ into a grid, where each cell is of side-length $\frac{r_1}{L}$ and hence of diameter at most $\Delta(\frac{r_1}{L}, d) = \frac{r_1}{\mu+1} \leq r_{\text{opt},c}$. A ball of radius $r_{\text{opt},c}$ on any corner of a cell will contain the entire cell, and so it will contain at most $\frac{n}{c}$ points, by the definition of $r_{\text{opt},c}$. By Lemma \[cubeCover\] the grid on $H$ has at most $2^d(2r_1/\frac{r_1}{L})^d = (4 (\mu+1) \sqrt{d})^d$ cells. Set $c = 2(4 (\mu+1) \sqrt{d})^d$. Then we have that $\bar{S} \subset H$ contains at most $\frac{n}{c} (4 (\mu+1) \sqrt{d})^d = \frac{n}{2}$ points. Since the inner ball $S$ contains at least $\frac{n}{c}$ points, and the outer ball $\bar{S}$ contains at most $\frac{n}{2}$ points, hence the annulus $\bar{S} \setminus S$ contains at most $\frac{n}{2} - \frac{n}{c}$ points. Now, divide $\bar{S} \setminus S$ into $t$ rings of equal width, and by the pigeonhole principle at least one of these rings must contain at most $ O(\frac{n}{t})$ points of $P$. Now let the inner ball corresponding to this ring be $B_{\text{in}}$ and the outer ball be $B_{\text{out}}$ and add these points to *both* children. Even for $t = 1$, each child contains at most $\frac{n}{2} + (\frac{n}{2} - \frac{n}{c}) = (1 - \frac{1}{c})n$ points. Also, the thickness of the ring is bounded by $\frac{2 r_1 - r_1}{t}/2r_1 = \frac{1}{2t}$, i.e it is a $O(\frac{1}{t})$ ring separator. Finally, we can check in $O(n)$ time if the randomized process of Lemma \[randommufraction\] succeeded simply by verifying the number of points in the inner and outer ring. \[ringsep\] Given any point set $P$, we can construct a $O(\frac{1}{\log n})$ ring-separator tree $T$ of depth $O(d^{\frac{d}{2}} (\mu+1)^d \log n)$. Repeatedly partition $P$ by lemma \[randommufraction\] into $P^{v}_{\text{in}}$ and $P^{v}_{\text{out}}$ where $\textbf{v}$ is the parent node. Store only the single point $\text{rep}_v = m \in P$ in node $\textbf{v}$, the center of the ball $B(m,r_1)$. We continue this partitioning until we have nodes with only a single point contained in them. Since each child contains at least $\frac{n}{c}$ points (by proof of Lemma \[improvedRing\]), each subset reduces by a factor of at least $1 - \frac{1}{c}$ at each step, and hence the depth of the tree is logarithmic. We calculate the depth more exactly, noting that in Lemma \[improvedRing\], $c = O(d^{\frac{d}{2}} (\mu+1)^d)$. Hence the depth $x$ can be bounded as: $$\begin{aligned} n (1 - \frac{1}{c})^x &= 1 \\ (1- \frac{1}{c})^x &= \frac{1}{n} \\ x & = \frac{ \ln \frac{1}{n}}{\ln (1 - \frac{1}{c})} = \frac{-1}{\ln (1 - \frac{1}{c})} \ln n \\ x &\leq c \ln n = O \left( d^{\frac{d}{2}} (\mu+1)^d \log n \right)\end{aligned}$$ Finally, we verify that the storage space require is not excessive. \[ringstorage\] To construct a $O(\frac{1}{\log n})$ ring-separator tree requires $O(n)$ storage and $O(d^{\frac{d}{2}} (\mu+1)^d n \log n)$ time. By Lemma \[ringsep\] the depth bounds still hold upon repeating points. For storage, we have to bound the total number of points in our data structure after repetition, let us say $P_R$. Since each node corresponds to a splitting of $P_R$,there may be only $O(P_R)$ nodes and total storage. Note in the proof of Lemma \[improvedRing\], for a node containing $x$ points, at most an additional $\frac{x}{\log n}$ may be duplicated in the two children. To bound this over each level of our tree, we sum across each node to obtain that the number of points $T_i$ at the $i$-th level, as: $$T_i = T_{i-1} \left( 1 + \frac{1}{\log T_{i-1}} \right)$$ Note also by Lemma \[ringsep\], the tree depth is $O(\log n)$ or bounded by $k \log n$ where $k$ is a constant. Hence we only need to bound the storage at the level $i = O(\log n)$. We solve the recurrence, noting that $T_0 = n$ and $T_i > n$ for all $i$ and hence $T_i < T_{i-1}(1 + \frac{1}{\log n})$. Thus the recurrence works out to: $$\begin{aligned} T_i &< n \left( 1+ \frac{1}{\log n} \right)^{O(\log n)} < n \left( \left( 1+ \frac{1}{\log n} \right)^{\log n} \right)^k < n(e^k).\end{aligned}$$ Where the main algebraic step is that $(1+ \frac{1}{x})^x < e$. This proves that the number of points, and hence our storage complexity is $O(n)$. Multiplying the depth by $O(n)$ for computing the smallest ball across nodes on each level, gives us the time complexity of $O(n \log n)$. We note that other tradeoffs are available for different values of approximation quality ($t$) and construction time / query time. #### Algorithm and Quality Analysis Let $\text{best}_q$ be the best candidate for nearest neighbor to $q$ found so far and $D_{\text{near}} = D(\text{best}_q, q)$. Let $\text{nn}_q$ be the exact nearest neighbor to $q$ from point set $P$ and $D_{\text{exact}} = D(\text{nn}_q,q)$ be the exact nearest neighbor distance. Finally, let $\textbf{curr}$ be the tree node currently being examined by our algorithm, and $\text{rep}_{\text{curr}}$ be a representative point $p \in P$ of $\textbf{curr}$. By convention $r_v$ represents the radius of the *inner* ball associated with a node $\textbf{v}$, and within each node $\textbf{v}$ we store $\text{rep}_v = m_v$, which is the center of $B_{\text{in}} (m_v , r_v)$. The node associated with the inner ball $B_{\text{in}}$ is denoted by $\mathbf{v_{\text{in}}}$ and the node associated with $B_{\text{out}}$ is denoted by $\mathbf{v_{\text{out}}}$. \[ringsearch\] Given a $t$-ring tree $T$ for a point set with respect to a $\mu$-defective distance $D$, where $t \leq \frac{1}{\log n}$ and query point $q$ we can find a $O(\mu + \frac{2 \mu^2 }{t})$ nearest neighbor to $q$ in $O( (\mu+1)^d d^{\frac{d}{2}} \log n)$ time. Our search algorithm is a binary tree search. Whenever we reach node $\textbf{v}$, if $D(\text{rep}_v,q) < D_{\text{near}}$ set $\text{best}_q =\text{rep}_v$ and $D_{\text{near}} = D(\text{rep}_v,q)$ as our current nearest neighbor and nearest neighbor distance respectively. Our branching criterion is that if $D (\text{rep}_v, q) < (1 + \frac{t}{2}) r_v$, we continue search in $\mathbf{v_{\text{in}}}$, else we continue the search in $\mathbf{v_{\text{out}}}$. Since the depth of the tree is $O(\log n)$ by Lemma \[ringsep\], this process will take $O(\log n)$ time. Turning now to quality, let $\textbf{w}$ be the first node such that $\text{nn}_q \in \mathbf{w_{\text{in}}}$ but we searched in $\mathbf{w_{\text{out}}}$, or vice-versa. After examining $\text{rep}_w$, $D_{\text{near}} \leq D(\text{rep}_w, q)$ and $D_{\text{near}}$ can only decrease at each step. An upper bound on $D(q, \text{rep}_w)/D(q,\text{nn}_q)$ yields a bound on the quality of the approximate nearest neighbor produced. In the first case, suppose $\text{nn}_q \in \mathbf{w_{\text{in}}}$, but we searched in $\mathbf{w_{\text{out}}}$. Then $D(\text{rep}_w, q) > \left( 1 + \frac{t}{2} \right) r_w$ and $D(\text{rep}_w, \text{nn}_q) < r_w$. Now $\mu$-defectiveness implies that $\mu D(q,\text{nn}_q) > D (\text{rep}_w, q) - D(\text{rep}_w, \text{nn}_q)$, so we have $D(q,\text{nn}_q) > \frac{t}{2 \mu} r_w$. And for the upper bound on $D(\text{rep}_w,q)/D(q,\text{nn}_q)$, we again apply $\mu$-defectiveness to conclude that $D(\text{rep}_w,q) - D(q, \text{nn}_q) < \mu D(\text{nn}_q, \text{rep}_w)$, which yields $\frac{D(\text{rep}_w,q)}{D(q,\text{nn}_q)} < 1 + \mu \frac{r_w}{D(q, \text{nn}_q)} < 1 + \mu \frac{r_w}{\frac{t}{2 \mu}r_w} = 1 + 2 \frac{\mu^2}{t}$. We now consider the other case. Suppose $\text{nn}_q \in \mathbf{w_{\text{out}}}$ and we search in $\mathbf{w_{\text{in}}}$ instead. By construction we must have $D(\text{rep}_w, q) < \left( 1 + \frac{t}{2} \right) r_w$ and $D(\text{rep}_w, \text{nn}_q) > ( 1 + t) r_w$. Again, $\mu$-defectiveness yields $D(q,\text{nn}_q) > \frac{t}{2 \mu} r_w$. Now we can simply take the ratios of the two: $\frac{D(\text{rep}_w,q)}{D(q, \text{nn}_q)} < \frac{(1 + \frac{t}{2}) r_w}{\frac{t}{2 \mu} r_w } = \mu + \frac{2 \mu}{t}$. Taking an upper bound of the approximation provided by each case, the ring tree provides us a $\mu + 2 \frac{\mu^2}{t}$ approximation. The space/running time bound follows from Lemma \[ringstorage\], and noting that taking a thinner ring ($t \leq \frac{1}{\log n}$) in the proof there only decrease the depth of the tree due to lesser duplication of points. Setting $t = \frac{1}{\log n}$, we can find a $O(\mu + 2 \mu^2 \log n)$ approximate nearest neighbor to a query point $q$ in $O(d^{\frac{d}{2}} (\mu+1)^d \log(n))$ time, using a $O(\frac{1}{\log n})$ ring separator tree constructed in $O(d^{\frac{d}{2}} (\mu+1)^d n \log(n))$ expected time. By Lemma \[ringsep\], Lemma \[improvedRing\], Lemma \[ringstorage\] and Lemma \[ringsearch\]. Note that we are slightly abusing notation in Lemma \[improvedRing\], in that the separating ring obtained there is not empty of points of $P$ as originally stipulated. However remember that if $\text{nn}_q$ is in the ring, then $\text{nn}_q$ repeats in *both* children and cannot fall off the search path. Hence we can “pretend” the ring is empty as in our analysis in Lemma \[ringsearch\]. Overall algorithm {#sec:finalized-algorithm} ================= We give now our overall algorithm for obtaining a $1 + {\varepsilon}$ nearest neighbor in $O\left( \frac{1}{{\varepsilon}^d}\log^{2d} n \right)$ query time. Preprocessing ------------- We first construct an improved ring-tree $R$ on our point set $P$ in $O(n \log n)$ time as described in Lemma \[ringstorage\], with ring thickness $O(\frac{1}{\log n})$. We then compute an efficient orthogonal range reporting data structure on $P$ in $O(n \log ^{d-1} n)$ time, such as that described in [@rangesearching] by Afshani [*et al*]{}. We note the main result we need: \[rangesearch\] We can compute a data structure from $P$ with $O(n \log ^{d-1} n)$ storage (and same construction time), such that given an arbitrary axis parallel box $B$ we can determine in $O(\log^{d} n)$ query time a point $p \in P \cap B$ if $|P \cap B| > 0$ Query handling -------------- Given a query point $q$, we use $R$ to obtain a point $q_{\text{rough}}$ in $O(\log n)$ time such that $D_{\text{rough}} = D(q, q_{\text{rough}}) \leq (1 + \mu^2 \log n) D(q, \text{nn}_q)$. Given $q_{\text{rough}}$, we can use Lemma \[ballcover\] to find a family $F$ of $2^d$ cubes of side length exactly $D_{\text{rough}}$ such that they cover $B(q, D_{\text{rough}})$. We use our range reporting structure to find a point $p \in P$ for all non-empty cubes in $F$ in a total of $2^d \log^d n$ time. These points act as representatives of the cubes for what follows. Note that $\text{nn}_q$ must necessarily be in one of these cubes, and hence there must be a ($1+{\varepsilon}$)-nearest neighbor $q_{\text{approx}} \in P$ in some $G \in F$. To locate this $q_{\text{approx}}$, we construct a quadtree [@snotes Chapter 11] [@skipquadtrees] for repeated bisection and search on each $G \in F$. Algorithm \[algo\] describes the overall procedure. We call the collection of all cells produced during the procedure a *quadtree*. We borrow the presentation in Har-Peled’s book [@snotes] with the important qualifier that we construct our quadtree at runtime. The terminology here is as introduced earlier in section \[sec:ringsec\]. Instantiate a queue $Q$ containing all cells from $F$ along with their representatives and enqueue **root** $\log n$ Let $D_{\text{near}} =D(\text{rep}_{\text{root}}, q)$, $\text{best}_q = \text{rep}_{\mathbf{root}}$ Pull off the head of the queue and place it in $\mathbf{curr}$. Let $\text{best}_q = \text{rep}_{\text{curr}}$, $D_{\text{near}} = D(\text{best}_q, q)$ Bisect $\textbf{curr}$ according to procedure of Lemma \[bisectionProcedure\]; place the result in $\{G_i\}$. As described in \[bisectionProcedure\], check if $G_i$ is non-empty by passing it to our range reporting structure, which will also return us some $p \in P$ if $G_i$ is not empty. Also check if $G_i$ may contain a point closer than $(1- \frac{{\varepsilon}}{2}) D_{\text{near}}$ to $q$. (This may be done in $O(d)$ time for each cell, given the coordinates of the corners.) Let $\text{rep}_{G_i} = p$ Enqueue $G_i$ $Q$ is empty Return $\text{best}_q$ \[algo\] \[correctness\] Algorithm \[algo\] will always return a $(1+{\varepsilon})$-approximate nearest neighbor. Let $\text{best}_q$ be the point returned by the algorithm at the end of execution. By the method of the algorithm, for all points $p$ for which the distance is directly evaluated, we have that $ D(\text{best}_q, q) < D(p,q) $. The terminology here is as in section \[sec:ringsec\]. We look at points $p$ which are *not* evaluated during the running of the algorithm, i.e. we did not expand their containing cells. But by the criterion of the algorithm for not expanding a cell, it must be that $D(\text{best}_q, q) (1 - \frac{{\varepsilon}}{2}) < D(p,q)$. For ${\varepsilon}<1$, this means that $(1+ {\varepsilon})D(p,q) > D(\text{best}_q,q)$ for any $p \in P$, including $\text{nn}_q$. So $\text{best}_q$ is indeed a $1 + {\varepsilon}$ approximate nearest neighbor. We must analyze the time complexity of a single iteration of our algorithm, namely the complexity of a subdivision of a cube $G$ and determining which of the $2^d$ subcells of $G$ are non-empty. \[bisectionProcedure\] Let $G$ be a cube with maximum side length $s$ and $G_i$ its subcells produced by bisecting along each side of $G$. For all non-empty subcubes $G_i$ of $G$, we can find $p_i \in P \cap G_i$ in $O(2^d \log^{d} n)$ total time complexity, and the maximum side length of any $G_i$ is at most $\frac{s}{2}$. Note that $G$ is defined as a product of $d$ intervals. For each interval, we can find an approximate bisecting point in $O(1)$ time and by the RTI each subinterval is of length at most $\frac{s}{2}$. This leads to an $O(d)$ cost to find a bisection point for all intervals, which define $O(2^d)$ subcubes or children. We pass each subcube of $G$ to our range reporting structure. By lemma \[rangesearch\], this takes $O(\log^{d} n)$ time to check emptiness or return a point $p_i \in P$ contained in the child, if non-empty. Since there are $O(2^d)$ non-empty children of $G$, this implies a cost of $2^d(\log^{d} n)$ time incurred. Checking each of the non-empty subcubes $G_i$ to see if it may contain a point closer than $(1 - \frac{{\varepsilon}}{2}) D_{\text{near}}$ to $q$ takes a further $O(d)$ time per cell or $O(d 2^d)$ time. We now bound the number of cells that will be added to our search queue. We do so indirectly, by placing a lower bound on the maximum side length of all such cells. \[depthCube\] Algorithm \[algo\] will not add the children of node $\textbf{C}$ to our search queue if the maximum side length of $\textbf{C}$ is less than $\frac{{\varepsilon}D(q, \text{nn}_q)}{2 \mu \sqrt{d}}$. Let $\Delta(\textbf{C})$ represent the diameter of cell $\textbf{C}$. By construction, we can expand $\textbf{C}$ only if some subcell of $\textbf{C}$ contains a point $p$ such that $D(p , q) \leq (1 - \frac{{\varepsilon}}{2}) D_{\text{near}}$. Note that since $\textbf{C}$ is examined, we have $D_{\text{near}} \leq D(\text{rep}_C , q)$. Now assuming we expand $\textbf{C}$, then we must have: $$\mu \Delta(\textbf{C}) > D(\text{rep}_C,q) - D(p, q) \geq D_{\text{near}} - (1 - \frac{{\varepsilon}}{2})D_{\text{near}} = \frac{{\varepsilon}}{2} D_{\text{near}}$$ So ${\varepsilon}/(2 \mu) D_{\text{near}} < \Delta(\textbf{C})$. First note $D (\text{rep}_C, q) < D_{\text{near}} $. Also, by definition, $D(q, \text{nn}_q) < D_{\text{near}}$. And $\Delta(\textbf{C}) < \sqrt{d} s$ where $s$ is the maximum side length of $\textbf{C}$. Making the appropriate substitutions yields us our required bound. Given the bound on quadtree depth (Lemma \[depthCube\]), and using the fact that at most $2^{xd}$ nodes are expanded at level $x$, we have: \[timeFinal\] Given a cube $G$ of side length $D_{\text{rough}}$, we can compute a $(1 + {\varepsilon})$-nearest neighbor to $q$ in $O\left( \frac{1}{{\varepsilon}^d} 2^d \mu^d d^{\frac{d}{2}} \left( \frac{D_{\text{rough}}}{D(q, \text{nn}_q)} \right)^d \log^d n \right )$ time. Consider a quadtree search from $q$ on a cube $G$ of side length $D_{\text{rough}}$. By lemma \[depthCube\], our algorithm will not expand cells with all side lengths smaller than $\frac{ {\varepsilon}D(q, \text{nn}_q)}{2 \mu \sqrt{d}}$. But since the side length reduces by at least half in each dimension upon each split, all side lengths are less than this value after $x =\log \left( D_{\text{rough}}/\frac{ {\varepsilon}D(q, \text{nn}_q)}{2 \mu \sqrt{d}} \right)$ repeated bisections of our root cube. Noting that $O(\log^d n)$ time is spent at each node by lemma \[bisectionProcedure\], and that at the $x$-th level the number of nodes expanded is $2^{xd}$, we get a final time complexity bound of $O\left( \frac{1}{{\varepsilon}^d} 2^d \mu^d d^{\frac{d}{2}} \left( \frac{D_{\text{rough}}}{D(q, \text{nn}_q)} \right)^d \log^d n \right)$. Substituting $D_{\text{rough}} = \mu^2 \log n D(q, \text{nn}_q) $ in Lemma \[timeFinal\] gives us a bound of $O\left(2^d \frac{1}{{\varepsilon}^d} \mu^{3d} d^{\frac{d}{2}} \log^{2d} n\right)$. This time is per cube of $F$ that covers $B(q,D_{\text{rough}})$. Noting that there are $2^d$ such cubes gives us a final time complexity of $O\left(2^{2d} \frac{1}{{\varepsilon}^d} \mu^{3d} d^{\frac{d}{2}} \log^{2d} n \right)$. For the space complexity of our run-time queue, observe that the number of nodes in our queue increases only if a node has more than one non-empty child, i.e, there is a split of our $n$ points. Since our point set may only split $n$ times, this gives us a bound of $O(n)$ on the space complexity of our queue. Logarithmic bounds, with further assumptions. {#sec:condition} ============================================= For a given ${\ensuremath{D_{s\phi}}}$, let $c_0 = \max_{ i\in [1..d]} \sqrt{\frac{ \max_x \phi_i'' (x)} { \min_y \phi_i''(y)}}$. We conjecture that $c_0 = \Theta(\mu)$ although we cannot prove it. In particular, we show that if we assume a bounded $c_0$ (in addition to $\mu$), we can obtain a $1 + {\varepsilon}$ nearest neighbor in time $O(\log n + (\frac{1}{{\varepsilon}})^d)$ time for $\sqrt{{\ensuremath{D_{s\phi}}}}$. We do so by constructing a *Euclidean* quadtree $T$ on our set in preproccessing and using $c_0$ and $\mu$ to express the bounds obtained in terms of $\sqrt{{\ensuremath{D_{s\phi}}}}$. We will refer to the Euclidean distance $l_2$ as $D_e$ and note first the following key relation between $\sqrt{{\ensuremath{D_{s\phi}}}}$ and $D_e$. \[EucBregBisect\] Suppose we are given a interval $I = [x_1 x_2] \subset \reals$ s.t. $x_1 < x_2$, $D_e(x_1, x_2) = r_e$, and $\sqrt{{\ensuremath{D_{s\phi}}}(x_1,x_2)} = r_{\phi}$. Suppose we divide $I$ into $m$ subintervals of equal length with endpoints $x_1 = a_0, a_1,\ldots a_{m-1}, a_m = x_2$, where $a_i < a_{i+1}$ and $D_e (a_i, a_{i+1}) = r_e /m$, $\forall i\in[0..m-1]$. Then $\frac{r_{\phi} }{c_0 m } \leq \sqrt{{\ensuremath{D_{s\phi}}}(a_i, a_{i+1})} \leq \frac{c_0 r_{\phi}}{m}$. We can relate $\sqrt{{\ensuremath{D_{s\phi}}}}$ to $D_e$ via the Taylor expansion of $\sqrt{{\ensuremath{D_{s\phi}}}}$: $\sqrt{{\ensuremath{D_{s\phi}}}(a,b)} = \sqrt{ \phi''(\bar{x}) } D_e(a,b)$ for some $\bar{x} \in [a b]$. Combining this with $c_0$ yields $$\frac{ \min_{i} \sqrt{{\ensuremath{D_{s\phi}}}(a_i , a_{i+1})}}{\sqrt{{\ensuremath{D_{s\phi}}}(x_1,x_2) } } \geq \frac{D_e(a_i, a_{i+1})}{c_0 D_e(x_1, x_2)} = \frac{1}{c_0 m}$$ and $$\frac{ \max_{i} \sqrt{{\ensuremath{D_{s\phi}}}(a_i , a_{i+1})}}{\sqrt{{\ensuremath{D_{s\phi}}}(x_1,x_2) } } \leq c_0 \frac{D_e(a_i, a_{i+1})}{D_e(x_1, x_2)} = \frac{c_0}{m}.$$ \[repbis\] If we recursively bisect an interval $I = [x_1 x_2] \subset \reals$ s.t. $D_e(x_1, x_2) = r_e$ and $\sqrt{{\ensuremath{D_{s\phi}}}(x_1,x_2)} = r_{\phi}$ into $2^i$ equal subintervals (under $D_e$), then $\frac{r_{\phi} }{c_0 2^i } \leq \sqrt{{\ensuremath{D_{s\phi}}}(a_k, a_{k+1}) }\leq \frac{c_0 r_{\phi}}{2^i}$ for any of the subintervals $[a_k a_{k+1}]$ so obtained. Hence after $\log \frac{c_0 r_{\phi}}{ x}$ subdivisions, all intervals will be of length at most $x$ under $\sqrt{{\ensuremath{D_{s\phi}}}}$. Also, given a cube of initial side length $r_{\phi}$, after $\log \frac{c_0 r_{\phi} }{ x}$ repeated bisections (under $D_e$) the diameter will be at most $\sqrt{d} x$ under $\sqrt{{\ensuremath{D_{s\phi}}}}$. We find the smallest enclosing Bregman cube of side length $s$ that bounds our point set, and then construct our compressed Euclidean quadtree in preprocessing. Corollary \[repbis\] gives us that for cells formed at the $i$-th level of decomposition, the side length under $\sqrt{{\ensuremath{D_{s\phi}}}}$ is between $\frac{s}{c_0 2^i}$ and $\frac{c_0 s}{ 2^i}$. Refer to these cells formed at the $i$-th level as $L_i$. \[packingTheTree\] Given a ball $B$ of radius $r$ under $\sqrt{{\ensuremath{D_{s\phi}}}}$, let $i = \log \frac{ s}{c_0 r}$. Then $|L_i \cap B| \leq O(2^d)$ and the side length of each cell in $L_i$ is between $r$ and ${c_0}^2 r$ under $\sqrt{{\ensuremath{D_{s\phi}}}}$. We can also explicitly retrieve the quadtree cells corresponding to $|L_i \cap B|$ in $O(2^d \log n)$ time. Note that for cells in $L_i$, we have side lengths under $\sqrt{{\ensuremath{D_{s\phi}}}}$ between $\frac{s}{c_0 2^i}$ and $\frac{c_0 s}{2^i}$ by Corollary \[repbis\]. Substituting $i = \log \frac{s}{c_0 r}$, these cells have side length between $r$ and ${c_0}^2 r$ under $\sqrt{{\ensuremath{D_{s\phi}}}}$. By the reverse triangle inequality and a similar argument to Lemma \[ballcover\], we get our required bound for $|L_i \cap B|$. In preconstruction of our quadtree $T$ we maintain for each dimension the corresponding interval quadtree $T_k$, $\forall k\in [1..d]$. Observe this incurs at most $O(n)$ storage, with $d$ in the big-Oh. For retrieving the actual cells $|L_i \cap B|$, we first find the $O(1)$ intervals from level $i$ in each $T_k$ that may intersect $B$. Taking a product of these, we get $O(2^d)$ cells which are a superset of the canonical cells $L_i \subset T$. Each cell may be looked up in $O(\log n)$ time from the compressed quadtree [@snotes] so our overall retrieval time is $O (2^d \log n)$. Given query point $q$, we first obtain in $O(\log n)$ time with our ring-tree a rough $O(n)$ ANN $q_{\text{rough}}$ s.t.\ $D_{\text{rough}} = \sqrt{{\ensuremath{D_{s\phi}}}(q, q_{\text{rough}})} = \mu^2 n \sqrt{{\ensuremath{D_{s\phi}}}(q, \text{nn}_q)}$. Note that we can actually obtain a $O(\log n)$-ANN instead, using the results of Section \[ringsep\]. But a coarser approximation of $O(n)$-ANN suffices here for our bound. The tree depth (and implicitly the storage and running time) is still bounded by the $O(d^{\frac{d}{2}} (\mu+1)^d \log n)$ of Lemma \[ringsep\], since in using thinner rings we have less point duplication and the same proportional reduction in number of points in each node at each level. Now Lemma \[packingTheTree\], we have $O(2^d)$ quadtree cells intersecting $B (q, \sqrt{{\ensuremath{D_{s\phi}}}(q, q_{\text{rough}})} )$. Let us call this collection of cells $Q$. We then carry out a quadtree search on each element of $Q$. Note that we expand only cells which may contain a point nearer to query point $q$ than the current best candidate. We bound the depth of our search using $\mu$-defectiveness similar to Lemma \[depthCube\]: \[modDepth\] We will not expand cells of diameter less than\ $\frac{{\varepsilon}\sqrt{{\ensuremath{D_{s\phi}}}(q, \text{nn}_q)}}{2 \mu}$ or cells whose side-lengths w.r.t. $\sqrt{{\ensuremath{D_{s\phi}}}}$ are less than $\frac{{\varepsilon}\sqrt{{\ensuremath{D_{s\phi}}}(q, \text{nn}_q)}}{2 \mu \sqrt{d}}$. For what follows, refer to our *spread* as $\beta = \frac{D_{\text{rough}}}{\sqrt{{\ensuremath{D_{s\phi}}}(q,\text{nn}_q)}}$. \[treeDepth\] We will only expand our tree to a depth of\ $k= \log (2 {c_0}^3 \mu \beta \sqrt{d}/{\varepsilon})$. Using Lemma \[modDepth\] and Corollary \[repbis\], each cell of $Q$ will be expanded only to a depth of $k= \log \left( c_0 {c_0}^2 D_{\text{rough}} / \frac{{\varepsilon}\sqrt{{\ensuremath{D_{s\phi}}}(q, \text{nn}_q)}}{2 \mu \sqrt{d}} \right)$. This gives us a depth of $\log (2 {c_0}^3 \mu \beta \sqrt{d}/{\varepsilon})$. \[breathnum\] The number of cells examined at the $i$-th level is $n_i < 2^d \left( \mu^d d^{\frac{d}{2}} c_0^{4d}+ (\frac{2^i c_0}{\beta})^d \right)$. Recalling that the cells of $Q$ start with side length at most $c_0^2 D_{\text{rough}}$, at the $i$-th level the diameter of cells is at most $\frac{c_0^3 \sqrt{d} D_{\text{rough}}} {2^i}$ under $\sqrt{{\ensuremath{D_{s\phi}}}}$, by Corollary \[repbis\]. Hence by $\mu$-defectiveness, there must be some point examined by our algorithm at distance at most $D_{\text{best}} =\sqrt{{\ensuremath{D_{s\phi}}}(q,\text{nn}_q)} + \frac{\mu c_0^3 \sqrt{d} D_{\text{rough}}}{2^i}$. Note that our algorithm will only expand cells within this distance of $q$. The side-length of a cell $\textbf{C}$ at this level is at least $\Delta(\textbf{C}) = \frac{ D_{\text{rough}}}{c_0 2^i}$. Applying the packing bounds from Lemma \[cubeCover\], and the fact that $(a+b)^d < 2^d(a^d + b^d)$, the number of cells expanded is at most $$n_i = \left( \frac{D_{\text{best}}}{\Delta(\textbf{C})} \right)^d < 2^d \left( \mu^d d^{\frac{d}{2}} c_0^{4d} + \left( \frac{c_0 2^i}{\beta} \right)^d \right). \qedhere$$ Finally we add the $n_i$ to get the total number of nodes explored: $$\sum_i n_i = O \left( 2^d \mu^d d^{\frac{d}{2}} c_0^{4d} \log (2 {c_0}^3 \mu \beta \sqrt{d}/{\varepsilon}) + 2^{2d} c_0^{4d} \mu^d d^{\frac{d}{2}}/{\varepsilon}^d \right).$$ Recalling that $\beta =\frac{D_{\text{rough}}}{\sqrt{{\ensuremath{D_{s\phi}}}(q,\text{nn}_q)}} = \mu^2 n$, substituting back and ignoring lower order terms, the time complexity is $$O \left(2^d \mu^d d^{\frac{d}{2}} c_0^{4d} \log n + 2^{2d} c_0^{4d} \mu^d d^{\frac{d}{2}}/{\varepsilon}^d \right).$$ Accounting for the $2^d$ cells in $Q$ that we need to search, this adds a further $2^d$ multiplicative factor. This time complexity of this quadtree phase(number of cells explored) of our algorithm dominates the time complexity of the ring-tree search phase of our algorithm, and hence is our overall time complexity for finding a $(1+ {\varepsilon})$ ANN to $q$. For space and pre-construction time, we note that compressed Euclidean quadtrees can be built in $O(n \log n)$ time and require $O(n)$ space [@snotes], which matches our bound for the ring-tree construction phase of our algorithm requiring $O(n \log n)$ time and $O(n)$ space. The General Case: Asymmetric Divergences {#sec:generalizations} ======================================== Without loss of generality we will focus on the *right-sided* nearest neighbor: given a point set $P$, query point $q$ and $\epsilon \ge 0$, find $x \in P$ that approximates $\min_{p \in P} D(p, q)$ to within a factor of $(1+\epsilon)$. Since a Bregman divergence is not in general $\mu$-defective, we will consider instead $\sqrt{{\ensuremath{D_\phi}}}$: by monotonicity and with an appropriate choice of $\epsilon$, the result will carry over to ${\ensuremath{D_\phi}}$. We list three issues that have to be resolved to complete the algorithm. Firstly, because of asymmetry, we cannot bound the diameter of a quadtree cell $\textbf{C}$ of side length $s$ by $s\sqrt{d}$. However, as the proof of Lemma \[ballcover\] shows, we can choose a *canonical corner* of a cell such that a (directed) ball of radius $s\sqrt{d}$ centered at that corner covers the cell. By $\mu$-defectiveness, we can now conclude that the diameter of $\mathbf{C}$ is at most $(\mu+1)s\sqrt{d}$ (note that this incurs an extra factor of $\mu+1$ in all expressions). Secondly, since while $\sqrt{{\ensuremath{D_\phi}}}$ satisfies $\mu$-defectiveness (unlike ${\ensuremath{D_\phi}}$) the opposite is true for the reverse triangle inequality, which is satisfied by ${\ensuremath{D_\phi}}$ but not $\sqrt{{\ensuremath{D_\phi}}}$. This requires the use of a weaker packing bound based on Lemma \[1dsqrtbregint\], introducing dependence in $1/\epsilon^2$ instead of $1/\epsilon$. And thirdly, the lack of symmetry means we have to be careful of the use of directionality when proviing our bounds. Note that for this section, when we speak of assymetric $\mu$-defective distance measure $D$, we are referring to $\sqrt{{\ensuremath{D_\phi}}}$. With some small adjustments, similar bounds can be obtained for more generic asymmetric, monotone, decomposable and $\mu$-defective distance measures satisfying packing bounds. The left-sided asymmetric nearest neighbor can be determined analogously. Finally, given a bounded domain $D$, we have that $\sqrt{{\ensuremath{D_\phi}}}$ is left-sided $\mu$-defective for some $\mu_L$ and right sided $\mu$-defective for some $\mu_R$ (see Lemma \[Arootmubreg\] for detailed proof). For what follows, let $\mu = \max(\mu_L, \mu_R)$ and describe $D$ as simply $\mu$-defective. Most of the proofs here mirror their counterparts in Sections \[sec:ringsec\] and \[sec:finalized-algorithm\]. Asymmetric ring-trees {#subsec:ringextension} --------------------- Since we focus on *right*-near-neighbors, all balls and ring separators referred to will use *left-balls* i.e balls $B(m,r) = \{x \mid D(m,x) < r\}$. As in Section \[sec:ringsec\], we will design a ring-separator algorithm and use that to build a ring-separator tree. \[leftcircle\] Let $D$ be a $\mu$-defective distance, and let $B(m,r)$ be a left-ball with respect to $D$. Then for any two points $x,y \in B(m,r)$, $D(x,y) < (\mu+1) r$. Follows from the definition of right sided $\mu$-defectiveness. $$\begin{aligned} D(x,y) - D(m,y) &< \mu D(m,x) \\ D(x,y) < \mu r + D(m,y) &= (\mu + 1) r \end{aligned}$$ As in Lemma \[randommufraction\] we can construct (in $O(nc)$ expected time) a $(\mu+1)$-approximate left-ball enclosing $\frac{n}{c}$ points. This in turn yields a ring-separator construction, and from it a ring tree with an extra $(\mu+1)^d d^{\frac{d}{2}}$ factor in depth as compared to symmetric ring-trees ,due to the weaker packing bounds for $\sqrt{{\ensuremath{D_\phi}}}$. We note that the asymptotic bounds for ring-tree storage and construction time follow from purely combinatorial arguments and hence are unchanged for $\sqrt{{\ensuremath{D_\phi}}}$. Once we have the ring- tree, we can use it as before to identify a rough near-neighbor for a query $q$; once again, exploiting $\mu$-defectiveness gives us the desired approximation guarantee for the result. \[randommufractionassym\] Given any constant $1 \leq c \leq n$, we can compute in $O(nc)$ randomized time a left-ball $B(m,r')$ such that $r' \leq (\mu + 1) r_{\text{opt},c}$ and $B(m,r') \cap P \geq \frac{n}{c}$. Follows identically to the proof of Lemma \[randommufraction\]. \[ringassym\] There exists a constant $c$ (which depends only on $d$ and $\mu$), such that for any $d$-dimensional point set $P$ and any $\mu$-defective distance $D$, we can find a $O(\frac{1}{\log n})$ left-ring separator $R_{P,c}$ in $O(n)$ expected time. First, using our randomized construction, we compute a ball $S=B(m,r_1)$ (where $m \in P$) containing $\frac{n}{c}$ points such that $r_1 \leq (\mu+1)r_{\text{opt},c}$, where $c$ is a constant to be set. Consider the ball $\bar{S} = B(m,2r_1)$. As described in Lemma \[ballcover\], $\bar{S}$ can be covered by $2^d$ hypercubes of side length $2 r_1$, the union of which we shall refer to as $H$. Set $L = (\mu+1)\sqrt{d}$. Imagine a partition of $H$ into a grid, where each cell is of side-length $\frac{r_1}{L}$. Each cell in this grid can be covered by a ball of radius $\Delta(\frac{r_1}{L}, d) = \frac{r_1}{\mu+1} \leq r_{\text{opt},c}$ centered on it’s lowest corner. This implies any cell will contain at most $\frac{n}{c}$ points, by the definition of $r_{\text{opt},c}$. By Lemma \[cubeCover\] the grid on $H$ has at most $2^d(2r_1/\frac{r_1}{L})^{2d} = (4 (\mu+1) \sqrt{d})^{2d}$ cells. Each cell may contain at most $\frac{n}{c}$ points. In particular, set $c = 2(4 (\mu+1) \sqrt{d})^{2d}$. Then we have that $H$ may contain at most $\frac{n}{c} (4 (\mu+1) \sqrt{d})^{2d} = \frac{n}{2}$ points, or since $\bar{S} \subset H$, $\bar{S}$ contains at most $\frac{n}{2}$ points and $\bar{S}'$ contains at least $\frac{n}{2}$ points. The rest of the proof goes through as in Lemma \[improvedRing\] We proceed now to the construction of our ring-tree using the basic ring-separator structure of Lemma \[ringassym\]. \[assymringsep\] Given any point set $P$, we can construct a $O(\frac{1}{\log n})$ left ring-separator tree $T$ of depth $O(d^d (\mu+1)^{2d} \log n)$. Repeatedly partition $P$ by Lemma \[ringassym\] into $P^{v}_{\text{in}}$ and $P^{v}_{\text{out}}$ where $\textbf{v}$ is the parent node. Store only the single point $\text{rep}_v = m \in P$ in node $\textbf{v}$, the center of the ball $B(m,r_1)$. We continue this partitioning until we have nodes with only a single point contained in them. Since each child contains at least $\frac{n}{c}$ points, each subset reduces by a factor of at least $1 - \frac{1}{c}$ at each step, and hence the depth of the tree is logarithmic. We calculate the depth more exactly, noting that in Lemma \[ringassym\], $c = O(d^d (\mu+1)^{2d})$. Hence the depth $x$ can be bounded as: $$\begin{aligned} n (1 - \frac{1}{c})^x &= 1 \\ (1- \frac{1}{c})^x &= \frac{1}{n} \\ x & = \frac{ \ln \frac{1}{n}}{\ln (1 - \frac{1}{c})} = \frac{-1}{\ln (1 - \frac{1}{c})} \ln n \\ x &\leq c \ln n = O \left( d^d (\mu+1)^{2d} \log n \right)\end{aligned}$$ Note that Lemma \[assymringsep\] also serves to bound the query time of our data structure. We need only now bound the approximation quality. The derivation is similar to Lemma \[ringsearch\], but with some care about directionality. \[ringsearchassym\] Given a $t$-ring tree $T$ for a point set with respect to a right-sided $\mu$-defective distance $D$, where $t \leq \frac{1}{\log n}$, and query point $q$ we can find a $O(\mu + \frac{2 \mu^2 }{t}$ nearest neighbor to query point $q$ in $O( (\mu+1)^{2d} d^d \log n)$ time. Our search algorithm is a binary tree search. Whenever we reach node $\textbf{v}$, if $D(\text{rep}_v,q) < D_{\text{near}}$ set $\text{best}_q =\text{rep}_v$ and $D_{\text{near}} = D(\text{rep}_v,q)$ as our current nearest neighbor and nearest neighbor distance respectively. Our branching criterion is that if $D (\text{rep}_v, q) < (1 + \frac{t}{2}) r_v$, we continue search in $\mathbf{v_{\text{in}}}$, else we continue the search in $\mathbf{v_{\text{out}}}$. Since the depth of the tree is $O(\log n)$ by Lemma \[assymringsep\], this process will take $O(\log n)$ time. Let $\textbf{w}$ be the first node such that $\text{nn}_q \in \mathbf{w_{\text{in}}}$ but we searched in $\mathbf{w_{\text{out}}}$, or vice-versa. The analysis goes by cases. In the first case as seen in figure \[case1\], suppose $\text{nn}_q \in \mathbf{w_{\text{in}}}$, but we searched in $\mathbf{w_{\text{out}}}$. Then $$\begin{aligned} D(\text{rep}_w, q) &> \left( 1 + \frac{t}{2} \right) r_w \\ D(\text{rep}_w, \text{nn}_q) &< r_w.\end{aligned}$$ Now left-sided $\mu$-defectiveness implies a lower bound on the value of $D(\text{nn}_q,q)$: $$\begin{aligned} \mu D(\text{nn}_q, q) &> D (\text{rep}_w, q) - D(\text{rep}_w, \text{nn}_q) \\ \mu D(\text{nn}_q,q) &> \left( 1+ \frac{t}{2} \right) r_w - r_w \\ D(\text{nn}_q,q) &> \frac{t}{2 \mu} r_w,\end{aligned}$$ And for the upper bound on $D(\text{rep}_w,q)/D(\text{nn}_q,q)$. First by right-sided $\mu$-defectiveness: $$\begin{aligned} D(\text{rep}_w,q) - D(\text{nn}_q,q) &< \mu D( \text{rep}_w, \text{nn}_q) \\ D(\text{rep}_w,q) &< D(\text{nn}_q,q) + \mu r_w \\ \frac {D(\text{rep}_w,q)}{D(\text{nn}_q,q)} &< 1 + \mu \frac{r_w}{D( \text{nn}_q,q)} \\ \frac{D(\text{rep}_w,q)}{D(\text{nn}_q,q)} &< 1 + \mu \frac{r_w}{\frac{t}{2 \mu}r_w} \\ \frac{D(\text{rep}_w,q)}{D(\text{nn}_q,q)} &< 1 + \mu \frac{2 \mu}{t} \\ \frac{D(\text{rep}_w,q)}{D(\text{nn}_q,q)} &< 1 + 2 \frac{\mu^2}{t}\end{aligned}$$ We now consider the other case. Suppose $\text{nn}_q \in \mathbf{w_{\text{out}}}$ and we search in $\mathbf{w_{\text{in}}}$ instead. The analysis is almost identical. By construction we must have: $$\begin{aligned} D(\text{rep}_w, q) &< \left( 1 + \frac{t}{2} \right) r_w \\D(\text{rep}_w, \text{nn}_q) &> ( 1 + t) r_w \end{aligned}$$ Again, left-sided $\mu$-defectiveness yields: $$\begin{aligned} D(\text{nn}_q,q) &> \frac{t}{2 \mu} r_w\end{aligned}$$ We can simply take the ratios of the two: $$\begin{aligned} \frac{D(\text{rep}_w,q)}{D( \text{nn}_q,q)} &< \frac{(1 + \frac{t}{2}) r_w}{\frac{t}{2 \mu} r_w } = \mu + \frac{2 \mu}{t}\end{aligned}$$ Taking an upper bound of the approximation quality provided by each case, we get that the ring separator provides us a $\mu + 2 \frac{\mu^2}{t}$ rough approximation. Substitute $t \leq \frac{1}{\log n}$ and the time bound follows from the bound of the depth of the tree in Lemma \[assymringsep\]. We can find a $O(\mu + 2 \mu^2 \log n)$ nearest neighbor to qury point $q$ in $O( (\mu+1)^{2d} d^d \log n)$ time using a $O(\frac{1}{\log n}$ ring-tree constructed in $O(d^d (\mu+1)^{2d} n \log(n))$ expected time. Set $t = \frac{1}{\log n}$, using Lemma \[assymringsep\]. The construction time for the ring tree follows by combining Lemmas \[assymringsep\] and \[ringassym\]. Asymmetric quadtree decomposition {#subsec:quadextension} --------------------------------- As in Section \[sec:finalized-algorithm\], we use the approximate near-neighbor returned by the ring-separator-tree query to progressively expand cells, using a subdivide-and-search procedure similar to Algorithm \[algo\]. A key difference is the procedure used to bisect a cell. \[bisectionasymmetric\] Let $G$ be a cube with maximum side length $s$ and $G_i$ its subcells produced by partitioning each side of $G$ into two equal intervals. For all non-empty subcubes $G_i$ of $G$, we can find $p_i \in P \cap G_i$ in $O(2^d \log^{d} n)$ total time complexity, and the maximum side length of any $G_i$ is at most $\frac{s}{\sqrt{2}}$. Note that $G$ is defined as a product of $d$ intervals. For each interval, we can find an approximate bisecting point in $O(1)$ time. Here the bisection point $x$ of interval $[a b]$ is such that $\sqrt{{\ensuremath{D_\phi}}(a,x)} = \sqrt{{\ensuremath{D_\phi}}(x,b)}$. By resorting to the RTI for ${\ensuremath{D_\phi}}$, we get that ${\ensuremath{D_\phi}}(a,x) + {\ensuremath{D_\phi}}(x,b) < s^2$ and hence ${\ensuremath{D_\phi}}(a,x) = {\ensuremath{D_\phi}}(x,b) < \frac{s^2}{2}$ which implies $\sqrt{{\ensuremath{D_\phi}}(a,x)} = \sqrt{{\ensuremath{D_\phi}}(x,b)} < \frac{s}{\sqrt{2}}$. The rest of our proof follows as in Lemma \[bisectionProcedure\] We now bound the number of cells that will be added to our search queue. We do so indirectly, by placing a lower bound on the maximum side length of all such cells, and note that for the asymmetric case we get an additional factor of $\frac{1}{\mu+1}$. \[depthAsymmetricCube\] Algorithm \[algo\] will not add the children of node $\textbf{C}$ to our search queue if the maximum side length of $\textbf{C}$ is less than $\frac{{\varepsilon}D({nn}_q,q)}{2 \mu (\mu+1) \sqrt{d}}$. Let $\Delta(\textbf{C})$ represent the maximum distance between any two points of cell $\textbf{C}$. By construction, we can expand $\textbf{C}$ only if some subcell of $\textbf{C}$ contains a point $p$ such that $D(p , q) \leq (1 - \frac{{\varepsilon}}{2}) D_{\text{near}}$. Note that since $\textbf{C}$ is examined, we have $D_{\text{near}} \leq D(\text{rep}_C , q)$. Now assuming we expand $\textbf{C}$, then we must have: $$\begin{aligned} D(\text{rep}_C,q) - D(p, q) &< \mu \Delta(\textbf{C}) \\ D_{\text{near}} - (1 - \frac{{\varepsilon}}{2})D_{\text{near}} &< \mu \Delta(\textbf{C}) \\ \frac{{\varepsilon}}{2} D_{\text{near}} &< \mu \Delta(\textbf{C}) \\ \frac{{\varepsilon}}{2 \mu} D_{\text{near}} &< \Delta(\textbf{C})\end{aligned}$$ Note that we substitute $D (\text{rep}_C, q) < D_{\text{near}} $ and that by the definition of $D_{\text{near}}$ as our candidate nearest neighbor distance, $D( \text{nn}_q,q) < D_{\text{near}}$. Our main modification from the symmetric case is that here $\Delta(\textbf{C}) < (\mu+1)\sqrt{d} s$, where $s$ is the maximum side length of $\textbf{C}$, as opposed to $\sqrt{d}s$. Since cell $\textbf{C}$ may be covered by a left-ball of radius $\sqrt{d} s$ placed at a suitably chosen corner (as explained in Lemma \[ballcover\]), lemma \[leftcircle\] gives the required bound on $\Delta(\textbf{C})$ The main difference between this lemma and Lemma \[depthCube\] is the extra factor of $\mu+1$ that we incur (as discussed) because of asymmetry. We only need do a little more work to obtain our final buonds: \[timeFinalAsymmetric\] Given a cube $G$ of side length $D_{\text{rough}}$, and letting $x= \frac{1}{{\varepsilon}^d} 2^d \mu^d (\mu+1)^d d^{\frac{d}{2}} \left( \frac{D_{\text{rough}}}{D(\text{nn}_q,q)} \right)^d $ we can compute a $(1 + {\varepsilon})$- right sided nearest neighbor to $q$ in $O(x^2 \log^d n)$ time. Consider a quadtree search from $q$ on a cube $G$ of side length $D_{\text{rough}}$. By lemma \[depthAsymmetricCube\], our algorithm will not expand cells with all side lengths smaller than ${\varepsilon}D( \text{nn}_q,q) / 2 \mu (\mu+1) \sqrt{d}$. But since the side length reduces by at least a factor of $\sqrt{2}$ in each dimension upon each split, all side lengths are less than this value after $k = \log_{\sqrt{2}} \left( 2D_{\text{rough}} \mu (\mu+1) \sqrt{d}/ {\varepsilon}D(\text{nn}_q,q) \right)$ repeated bisections of our root cube. Observe now that $O(\log^d n)$ time is spent at each node by Lemma \[bisectionasymmetric\] , that at the $k$-th level the number of nodes expanded is $2^{kd}$, and that $\log_{\sqrt{2}} n = (\log_{2} n )^2$. We then get a final time complexity bound of $O\left( (1/{\varepsilon}^{2d}) 2^{2d} \mu^{2d} (\mu+1)^{2d} d^d \left( D_{\text{rough}}/D( \text{nn}_q, q) \right)^{2d} \log^d n \right)$. Substituting $D_{\text{rough}} = \mu^2 \log (n) D( \text{nn}_q,q) $ in Lemma \[timeFinalAsymmetric\] gives us a bound of $O\left(2^{2d} \frac{1}{{\varepsilon}^{2d}} \mu^{6d} (\mu+1)^{2d} d^d \log^{3d} n\right)$. This time is per cube of $F$ that covers right-ball $B(q,D_{\text{rough}})$. Noting that there are $2^d$ such cubes gives us a final time complexity of $O\left(2^{3d} \frac{1}{{\varepsilon}^{2d}} \mu^{6d} (\mu+1)^{2d} d^d \log^{3d} n \right)$. The space bound follows as in Section \[sec:finalized-algorithm\]. #### Logarithmic bounds for Asymmetric Bregman divergences {#subsec:assymcondition} We now extend our logarithmic bounds from Section \[sec:condition\] to asymmetric Bregman divergence $\sqrt{{\ensuremath{D_\phi}}}$. First note that the following Lemma goes through by identical argument to Lemma \[EucBregBisect\]. \[EucABregBisect\] Suppose we are given an interval $I = [x_1 x_2] \subset \reals$ s.t. $x_1 < x_2$, $D_e(x_1, x_2) = r_e$ and $\sqrt{D_{\phi}(x_1,x_2)} = r_{\phi}$. Suppose we divide $I$ into $m$ subintervals of equal length with endpoints $x_1 = a_0 < a_1 < \ldots < a_{m-1} < a_m = x_2$ where $D_e (a_i, a_{i+1}) = r_e /m$, for all $i\in [0..m-1]$. Then $\frac{r_{\phi} }{c_0 m } \leq \sqrt{D_{ \phi} (a_i, a_{i+1})} \leq \frac{c_0 r_{\phi}}{m}$. \[repabis\] If we recursively bisect an interval $I = [x_1 x_2] \subset \reals$ s.t. $D_e(x_1, x_2) = r_e$ and $\sqrt{{\ensuremath{D_\phi}}(x_1,x_2)} = r_{\phi}$ into $2^i$ equal subintervals (under $D_e$), then $\frac{r_{\phi} }{c_0 2^i } \leq \sqrt{{\ensuremath{D_\phi}}(a_k, a_{k+1}) }\leq \frac{c_0 r_{\phi}}{2^i}$ for any of the subintervals $[a_k a_{k+1}]$ so obtained. Hence after $i = \lceil \log \frac{c_0 r_{\phi}}{ x} \rceil$ subdivisions, all intervals will be of length at most $x$ under $\sqrt{{\ensuremath{D_\phi}}}$. We now construct a compressed Euclidean quad tree as before, modifying the Section \[sec:condition\] analysis slightly to account for the weaker packing bounds for $\sqrt{{\ensuremath{D_\phi}}}$ and the extra $\mu+1$ factor on the diameter of a cell. Given an asymmetric decomposable Bregman divergence $D_\phi$ that is $\mu$-defective over a domain with associated $c_0$ as in Section \[sec:condition\], we can compute a $(1+\epsilon)$-approximate right-near-neighbor in time $O \left((\mu+1)^d d^{\frac{d}{2}} \log n + ( \frac{2{c_0}^4 ( \mu+1) \mu^3 \sqrt{d}}{{\varepsilon}})^d \right)$. We note our first new Lemma, a slightly modified packing bound due to $\sqrt{{\ensuremath{D_\phi}}}$ not having a direct RTI. \[cpack\] Given an interval $[x_1 x_2] \subset \reals$ s.t. $\sqrt{{\ensuremath{D_\phi}}(x_1,x_2)} = r > 0$, and intervals with endpoints $a_0 < a_1 < \ldots < a_{m-1} < a_m$, s.t. for all $i\in[0..m-1]$, $\sqrt{{\ensuremath{D_\phi}}(a_i, a_{i+1})} \geq l$, at most $O(\frac{c_0 r}{l})$ such intervals intersect $[x_1 x_2]$. By the Lagrange form, $$\frac{l}{r} < \frac{\sqrt{{\ensuremath{D_\phi}}(a_i, a_{i+1})}}{\sqrt{{\ensuremath{D_\phi}}(x1,x2)}} < c_0 \frac{D_e(a_i,a_{i+1})}{D_e(x_1,x_2)},$$ or we can say that $\frac{D_e(a_i,a_{i+1})}{D_e(x1,x2)} > \frac{l}{r c_0}$. The RTI for $D_e$ then gives us the required result. \[cpackd\] Given a ball $B$ of radius $r$ under $\sqrt{{\ensuremath{D_\phi}}}$, there can be at most $c_0^d (\frac{r}{l})^d$ disjoint cubes that can intersect $B$ where each cube has side length at least $l$ under $\sqrt{{\ensuremath{D_\phi}}}$. As before, we find the smallest enclosing Bregman cube of side length $s$ that encloses our point set, and then construct a compressed Euclidean quad-tree in pre-processing. Let $L_i$ denote the cells at the $i$-th level. \[apackingTheTree\] Given a ball $B$ of radius $r$ under $\sqrt{D_{ \phi}}$, let $i = \log \frac{ s}{c_0 r}$. Then $|L_i \cap B| \leq O({c_0}^d)$ and the side length of each cell in $L_i$ is between $r$ and ${c_0}^2 r$ under $\sqrt{{\ensuremath{D_{s\phi}}}}$. We can also explicitly retrieve the quadtree cells corresponding to $|L_i \cap B|$ in $O({c_0}^d \log n)$ time. Note that for cells in $L_i$, we have side lengths between $\frac{s}{c_0 2^i}$ and $\frac{c_0 s}{2^i}$ by Corollary \[repabis\]. Substituting $i = \log \frac{s}{c_0 r}$, these cells have side length between $r$ and ${c_0}^2 r$ under $\sqrt{{\ensuremath{D_{s\phi}}}}$. Now, we look in each dimension at the number of disjoint intervals of length at least $r$ that can intersect $B$. By Lemma \[cpack\], this is at most $c_0$. The rest of the proof follows as in Lemma \[packingTheTree\]. We first obtain in $O(\log n)$ time with our asymmetric ring-tree an $O(n)$ ANN $q_{\text{rough}}$ to query point $q$, such that $D_{\text{rough}} = \sqrt{{\ensuremath{D_\phi}}( q_{\text{rough}},q)} = O \left(\mu^2 n \sqrt{{\ensuremath{D_\phi}}( \text{nn}_q,q)} \right)$. We then use Lemma \[apackingTheTree\] to get $O({c_0}^d)$ cells of our quadtree that intersect right ball $B \left(q, \sqrt{{\ensuremath{D_\phi}}( q_{\text{rough}},q)} \right)$. Let us call this collection of cells as $Q$. We then carry out a quadtree search on each element of $Q$. Note that we expand only cells which may contain a point nearer to query point $q$ than the current best candidate. We bound the depth of our search using $\mu$-defectiveness similar to Lemma \[treeDepth\]. \[modADepth\] We need only expand cells of diameter greater than $\frac{{\varepsilon}\sqrt{{\ensuremath{D_\phi}}( \text{nn}_q,q)}}{2 \mu}$ By $\mu$-defectiveness, similar to Lemma \[depthCube\]. \[asidelen\] We will not expand cells where the length of each side is less than $x =\frac{{\varepsilon}\sqrt{ {\ensuremath{D_\phi}}( \text{nn}_q,q)}}{2 \mu (\mu+1) \sqrt{d}}$ Note that a quadtree cell $\textbf{C}$ whose side length is less than $x$ can be covered by a ball of radius $\sqrt{d} x$ under $\sqrt{{\ensuremath{D_\phi}}}$ with appropriately chosen corner as center of ball, as explained in proof of Lemma \[ballcover\]. Now by Lemma \[leftcircle\], $\sqrt{{\ensuremath{D_\phi}}(a,b)} \leq (\mu+1) \sqrt{d} x$, $\forall a,b \in \textbf{C}$. Substituting for $x$ from Lemma \[modADepth\], the diameter of $\textbf{C}$ is at most $\frac{{\varepsilon}\sqrt{{\ensuremath{D_\phi}}(\text{nn}_q,q)}}{2 \mu}$. Let the spread be $\beta = \frac{D_{\text{rough}}}{\sqrt{{\ensuremath{D_{s\phi}}}(\text{nn}_q, q)}} = O(\mu^2 n)$. \[AfinalTreeDepth\] We will only expand our tree to a depth of $k = \log(2 c_0^3 \mu (\mu+1) \beta \sqrt{d} / {\varepsilon})$. Note first that $D_{\text{rough}}= O \left(\beta \sqrt{{\ensuremath{D_\phi}}( \text{nn}_q,q)} \right)$. Then by Lemma \[apackingTheTree\], each of the cells of our corresponding quadtree is of side length at most $c_0^2 D_{\text{rough}}$. Using \[asidelen\] to upper bound the maximum side length of any quadtree cell expanded, and \[repabis\] to bound number of bisections needed to achieve this gives us out bound. \[atreebreadth\] The number of cells expanded at the $i$-th level is $n_i < 2^d(\mu^d d^{\frac{d}{2}} c_0^{5d} + (\frac{c_0^2 2^i}{\beta})^d)$. Recalling that the cells of $Q$ start with side length at most $c_0^2 D_{\text{rough}}$, at the $i$-th level the side length of a cell $\textbf{C}$ is at most $\frac{c_0^3 D_{\text{rough}}}{2^i}$ under $\sqrt{{\ensuremath{D_\phi}}}$ by Corollary \[repabis\]. And using Lemma \[leftcircle\], $\Delta{\textbf{C}} < \sqrt{d}(\mu+1) \frac{c_0^3 D_{\text{rough}}}{2^i}$. Hence by $\mu$-defectiveness there must be a point at distance at most $D_{\text{best}} = \sqrt{{\ensuremath{D_\phi}}(\text{nn}_q,q)} + \frac{\mu (\mu+1) c_0^3 \sqrt{d} D_{\text{rough}}}{2^i}$. The side length of a cell $C$ at this level is at least $\frac{D_{\text{rough}}}{c_0 2^i}$, so the number of cells expanded is at most $n_i = c_0^d(\frac{D_{\text{best}}}{\Delta{\textbf{c}}})^d = c_0^d( \mu (\mu+1) \sqrt{d} c_0^4 + \frac{c_0 2^i}{\beta})^d$, by Corollary \[cpackd\]. Using the fact that $(a+b)^d < 2^d (a^d + b^d)$, we get $n_i < 2^d \left(\mu^d (\mu+1)^d d^{\frac{d}{2}} c_0^{5d} + (\frac{c_0^2 2^i}{\beta})^d \right)$. Simply summing up all $i$, the total number of nodes explored is $$O(2^d \mu^d (\mu+1)^d c_0^{5d} \log(2 c_0^3 \mu \beta \sqrt{d} /{\varepsilon}) + 2^{2d} c_0^{5d} \mu^d (\mu+1)^d d^{\frac{d}{2}}/{\varepsilon}^d)$$, or $$O \left(2^d \mu^d (\mu+1)^d c_0^{5d} \log n + 2^{2d} c_0^{5d} \mu^d (\mu+1)^d d^{\frac{d}{2}}/{\varepsilon}^d \right)$$, after substituting back for $\beta$ and ignoring smaller terms. Recalling that there are $c_0^d$ cells in $Q$ adds a further $c_0^d$ multiplicative factor. This time complexity of this quadtree phase(number of cells explored) of our algorithm dominates the time complexity of the ring-tree search phase of our algorithm, and hence is our overall time complexity for finding a $(1+ {\varepsilon})$ ANN to $q$. For space and pre-construction time, we note that compressed Euclidean quadtrees can be built in $O(n \log n)$ time and require $O(n)$ space [@snotes], which matches our bound for the ring-tree construction phase of our algorithm requiring $O(n \log n)$ time and $O(n)$ space. Numerical arguments for bisection {#numerical} ================================= In our algorithms, we are required to *bisect* a given interval with respect to the distance measure $D$, as well as construct points that lie a fixed distance away from a given point. We note that in both these operations, we do not need exact answers: a constant factor approximation suffices to preserve all asymptotic bounds. In particular, our algorithms assume two procedures: 1. [Given interval $[ab] \subset \reals$, find $\bar{x} \in [ab]$ such that $(1 - \alpha)\sqrt{{\ensuremath{D_{s\phi}}}(a,\bar{x})} < \sqrt{{\ensuremath{D_{s\phi}}}(\bar{x}, b)} < (1 + \alpha) \sqrt{{\ensuremath{D_{s\phi}}}(a,\bar{x})}$]{}to a lesser degree by 2. [ Given $q \in \reals$ and distance $r$, find $\bar{x}$ s.t $|\sqrt{{\ensuremath{D_{s\phi}}}(q,\bar{x})} - r| < \alpha r$]{} For a given $\sqrt{{\ensuremath{D_{s\phi}}}} : \reals \to \reals$ and precision parameter $0 < \alpha < 1$, we describe a procedure that yields an $0 <\alpha < 1$ approximation in $O(\log c_0 + \log \mu + \log \frac{1}{\alpha})$ steps for both problems, where $c_0$ implicitly depends on the domain of convex function $\phi$: $$c_0 =\sqrt{\max_{ 1 \leq i \leq d} \left(\max_x \phi_i'' (x)/ \min_y \phi_i''(y) \right)}$$ Note that this implies linear convergence. While more involved numerical methods such as Newton’s method may yield better results, our approximation algorithm serve as proof-of-concept that the numerical precision is not problematic. A careful adjustment of our NN-analysis now gives a $O\left( \left(\log \mu + \log c_0 + \log \frac{1}{\alpha} \right )2^{2d} (1+ \alpha)^d \frac{1}{{\varepsilon}^d} \mu^{3d} d^{\frac{d}{2}} \log^{2d} n \right)$ time complexity to compute a $(1 + {\varepsilon})$-ANN to query point $q$. We now describe some useful properties of ${\ensuremath{D_{s\phi}}}$. \[ratio\] Consider $\sqrt{{\ensuremath{D_{s\phi}}}} : \reals \to \reals$ such that $c_0 = \sqrt{\max_x \phi''(x) / \min_y \phi''(y)}$. Then for any two intervals $[x_1 x_2] ,[x_3 x_4] \subset \reals$ , $$\frac{1}{c_0} \frac{|x_1 - x_2| }{ |x_3 - x_4|} < \frac{\sqrt{{\ensuremath{D_{s\phi}}}(x_1, x_2)} }{ \sqrt{{\ensuremath{D_{s\phi}}}(x_3,x_4)}} < c_0 \frac{ |x_1 - x_2| }{ |x_3 - x_4|}$$ The lemma follows by the definition of $c_0$ and by direct computation from the Lagrange form of $\sqrt{{\ensuremath{D_{s\phi}}}(a,b)}$, i.e, $\sqrt{{\ensuremath{D_{s\phi}}}(a,b)} = \sqrt{\phi''(\bar{x}_{ab})} |b -a|$, for some $\bar {x}_{ab} \in [ab]$. \[approxnum\] Given a point $q \in \reals$, distance $r \in \reals$, precision parameter $0 <\alpha < 1$ and a $\mu$-defective $\sqrt{{\ensuremath{D_{s\phi}}}} : \reals \to \reals$, we can locate a point $x_i$ such that $|\sqrt{{\ensuremath{D_{s\phi}}}(q,x_i)} - r| < \alpha r$ in $O(\log \frac{1}{\alpha} + \log \mu + \log c_0)$ time. Let $x$ be the point such that $\sqrt{{\ensuremath{D_{s\phi}}}(q,x)} = r$. We outline an iterative process, \[finalalgo\], with $i$-th iterate $x_i$ that converges to $x$. Let $x_0 > q$ be such that $\frac{\sqrt{\phi''(q)}}{c_0} (x_0 - q) = r$ Let $\text{step} = (x_0 - q)/2$ $x_{i+1} = x_i + \text{step}$ $ x_{i+1} = x_i- \text{step}$ $\text{step} = \text{step}/2$ Return $\bar{x} = x_i$ \[finalalgo\] First note that $\frac{\sqrt{ \phi''(q)}}{c_0} \leq \sqrt{\min_y \phi''(y)}$ and $\frac{\sqrt{\phi''(q)}}{ c_0} \geq \frac{\max_z \sqrt{\phi''(z)}} {c_0^2}$. It immediately follows that $r \leq\sqrt{{\ensuremath{D_{s\phi}}}(q,x_0)} \leq c_0^2 r$. By construction, $|x_i - x| \leq |x_0 - q|/2^i$. Hence by Lemma \[ratio\], $\sqrt{{\ensuremath{D_{s\phi}}}(x_i,x)} < \frac {c_0 ^3 r}{2^i}$. We now use $\mu$-defectiveness to upper bound our error $|\sqrt{{\ensuremath{D_{s\phi}}}(q,x_i)} - \sqrt{{\ensuremath{D_{s\phi}}}(q,x)}|$ at the $i$-th iteration: $$\left|\sqrt{{\ensuremath{D_{s\phi}}}(q,x_i)} - \sqrt{{\ensuremath{D_{s\phi}}}(q,x)} \right | < \frac{\mu c_0^3 r}{2^i}$$ Choosing $i$ such that $(\mu c_0^3)/2^i \leq \alpha$ implies that $i \leq \log \frac{1}{\alpha} + \log \mu + 3 \log c_0$. An almost identical procedure can locate an approximate bisection point of interval $[ab]$ in $O(\log \mu + \log c_0 + \log \frac{1}{\alpha})$ time, and similar techniques can be applied for $\sqrt{{\ensuremath{D_\phi}}}$. We omit the details here. Further work ============ A major open question is whether bounds independent of $\mu$-defectiveness can be obtained for the complexity of ANN-search under Bregman divergences. As we have seen, traditional grid based methods rely heavily on the triangle inequality and packing bounds, and there are technical difficulties in adapting other method such as cone decompositions [@chanNN] or approximate Voronoi diagrams [@plebs]. We expect that we will need to exploit geometry of Bregman divergences more substantially. Acknowledgements ================ We thank Sariel Har-Peled and anonymous reviewers for helpful comments.

--- abstract: 'It is now known that, apart from black holes, some naked singularities can also cast shadows which provide their possible observable signatures. We examine the relevant question here as to how to distinguish then these entities from each other, in terms of further physical signatures. We point out that black holes always admit timelike bound orbits having positive perihelion precession. Also, while a naked singularity with a photon sphere can cast a shadow, it could also admit positive perihelion precession for such orbits, thereby mimicking a black hole. This indicates that compact objects with photon spheres (shadows) always admit positive perihelion precession of timelike bound orbits around them. On the other hand, a naked singularity without a photon sphere could admit both positive and negative perihelion precession but need not have a shadow. In this paper, we construct a spacetime configuration which has a central naked singularity but no photon sphere, and it can give both shadow and a negative perihelion precession. Our results imply that, whereas the presence of a shadow and a positive perihelion precession implies either a black hole or a naked singularity, the presence of a shadow and a negative perihelion precession simultaneously would imply a naked singularity only. We discuss our results in the context of stellar motions (motion of the ‘S’ stars) around the Sgr-A\* galactic center.' author: - Dipanjan Dey - Rajibul Shaikh - 'Pankaj S. Joshi' title: Perihelion Precession and Shadows near Blackholes and Naked Singularities --- Introduction ============ Recent observation of the shadow of M87 galactic center by the Event Horizon Telescope (EHT) collaboration [@Akiyama:2019fyp], triggers a lot of attention to understand the nature and dynamics of the object at the galactic center [@Shaikh:2019hbm; @Gralla:2019xty; @Abdikamalov:2019ztb; @Yan:2019etp; @Vagnozzi:2019apd; @Gyulchev:2019tvk; @Shaikh:2019fpu]. There are a lot of literature where timelike, lightlike geodesics around the black hole and naked singularity are investigated [@Dey:2013yga; @Dey+15; @levin1; @Glampedakis:2002ya; @Chu:2017gvt; @Dokuchaev:2015zxe; @Borka:2012tj; @Martinez:2019nor; @Fujita:2009bp; @Wang:2019rvq; @Suzuki:1997by; @Zhang:2018eau; @Pugliese:2013zma; @Farina:1993xw; @Dasgupta:2012zf; @Shoom:2015slu; @Eva; @Eva1; @Eva2; @tsirulev; @Bambhaniya:2019pbr; @Joshi:2019rdo]. Generally, shadow is considered to be formed due to the existence of a photon sphere outside the event horizon of a black hole. However, in [@Shaikh:2018lcc], it is shown that a naked singularity spacetime known as Joshi-Malafarina-Narayan (JMN) spacetime [@JMN11] can cast similar type of shadow which is expected to be seen in a black hole spacetime. In [@Shaikh:2018lcc], it is shown that only the first type of JMN spacetime (JMN1) can cast shadow with a specific range of parameter’s value. JMN1 spacetime is a spherically symmetric, naked singularity spacetime which can be formed as an end state of gravitational collapse in a large comoving time [@JMN11]. JMN1 spacetime can be written as, $$ds^2=-(1-\chi)\left(\frac{r}{r_b}\right)^{\frac{\chi}{1-\chi}}dt^2+\frac{dr^2}{1-\chi}+r^2d\Omega^2\,\, , \label{JMNspt}$$ where $\chi$ is a constant parameter which can have values from zero to one and at $r=r_b$ this spacetime can be matched with external Schwarzschild spacetime. Throughout the paper, we consider Newton’s gravitational constant $G_N=1$ and light velocity $C=1$. In [@Shaikh:2018lcc], it is shown that for $\chi>\frac23$, JMN1 spacetime casts similar shadow as what can be seen in Schwarzschild spacetime. For both the Schwarzschild and JMN1 spacetimes, the central shadow depends upon the total Schwarzschild mass ($M_{TOT}$). In [@Shaikh:2018lcc], the theoretical results are not compared with any observational results. The main goal of that paper is to show theoretically how a naked singularity can cast similar shadow that a black hole can cast. The EHT collaboration will possibly release the picture of the central supermassive object (Sgr-A\*) of the Milky way in this year. On the other hand, GRAVITY, SINFONI collaborations are continuously observing the stellar motion around Sgr-A\* [@M87; @Eisenhauer:2005cv; @center1]. There are many ‘S’ stars (e.g. S02, S102, S38, etc.) which are orbiting around the Sgr-A\*. Among them some stars have perihelion points very close ($\sim 0.006~ Parsec$) to the central object of our Milky way. Their orbital behaviour can revel very important information about the spacetime structure around the Sgr-A\*. In [@Bambh], it is shown that in a naked singularity spacetime, the perihelion precession of the bound timelike orbits can be negative, which is never possible in a Schwarzschild black hole spacetime. We always have a positive perihelion precession in this Schwarzschild black hole case. In [@Dey:2019fpv], the future trajectory of S02 star is predicted considering both positive and negative precession. The negative perihelion precession occurs when a massive particle travels less than $2\pi$ angular distance in between two successive perihelion points, whereas, for positive precession, the particle has to travel greater than $2\pi$ distance in between the two successive perihelion points. For JMN1 spacetime, negative and positive precessions occur when $\chi<\frac13$ and $\chi>\frac13$ respectively [@Bambh]. Also, as shown in [@Shaikh:2018lcc], the JMN1 naked singularity, when matched to an exterior Schwarzschild geometry, cast shadow for $\chi>\frac23$. Therefore, the presence of both a shadow and a positive perihelion precession can mean either a Schwarzschild black hole, or a JMN1 naked singularity which has $\chi>\frac23$ and which is matched to an exterior Schwarzschild geometry. In both these cases, the bound orbits with positive perihelion precession lie in the Schwarzschild spacetime. However, if a shadow and bound orbits with negative perihelion precession are observed simultaneously, then it cannot be explained using these two scenarios as we need either a Schwazschild black hole or a JMN1 naked singularity with $\chi>\frac23$ for the shadow and only a JMN1 naked singularity with $\chi<\frac13$ for the negative precession. In this paper, we present a spacetime configuration which can explain this latter case where we have both shadow and negative perihelion precession simultaneously. We consider a spacetime configuration where an interior JMN1$_{int}$ with $\chi_{int}>\frac23$ is matched to a second JMN1$_{ext}$ with $\chi_{ext}<\frac23$ at some radius $r=r_{b1}$ (say) and then the second JMN1$_{ext}$ is matched to an exterior Schwarzschild geometry at some greater radius $r=r_{b2}$ (say), where $r_{b2}>r_{b1}$. As we show below, such spacetime configuration allows shadow because of JMN1$_{int}$ with $\chi_{int}>\frac23$ and timelike bound orbits with negative and positive perihelion precession, respectively, for $\chi_{ext}<\frac13$ and $\frac13<\chi_{ext}<\frac23$. In the next section (\[structure\]), we begin with a discussion of our spacetime structure. In that section, using Israel junction conditions, we show how the different $\chi$s’ values can create a thin shell of matter at the matching radius. In section (\[shadow\]), we show how a thin matter shell can create a shadow, though the photon sphere does not exist in the proposed spacetime structure, and we discuss the distinguishable properties of a black hole shadow and a shadow cast by a thin shell of matter. In that section, we also briefly review the work done in [@Bambh],[@Dey:2019fpv] and show that bound timelike orbits with positive and negative precession are possible in the JMN1$_{ext}$. In this paper, we do not compare our theoretical results with any observations. We mainly emphasize on the fact that the proposed spacetime structure with a central naked singularity allows bound timelike geodesics and a shadow of the central object which can be formed due to the presence of a thin matter shell. In section (\[conclusion\]), we discuss our results and the outcomes implied. Perihelion Precession and Shadow ================================ In [@Bambh], we derive the following orbit equations for JMN1 spacetime (eq. \[JMNspt\]), $$\frac{d^2u}{d\phi^2} + (1 - \chi) u - \frac{\gamma^2}{2h^2}\frac{\chi}{(1- \chi)}\left(\frac{1}{u}\right)\left(\frac{1}{u~r_{b}}\right)^\frac{-\chi}{(1- \chi)}=0\,\, , \label{eqJMNorbit}$$ where $u=\frac{1}{r}$ which is a function of azimuthal distance $\phi$. This equation can be solved numerically. In [@Bambh], we solve it numerically and also present an approximate analytic solution. The approximate analytic solution of the above orbit equation can be derived by considering small value of eccentricity and it can be written as, $$\tilde{u}=\frac{1}{p}\left[1+e\cos(m\phi)+O(e^2)\right]\,\, , \label{orbitsch1}$$ where $\tilde{u}=ur_b$, $p$, $m$ are positive real numbers and $e$ is the eccentricity of the orbit. Using the above solution and the differential eq. (\[eqJMNorbit\]), one can show that for JMN spacetime the parameter $m$ can be written as, $$m=\sqrt{2-3\chi}\,\, , \label{mJMN}$$ which shows that for $\chi<\frac13$ and $\chi>\frac13$ we get negative $(m>1)$ and positive $(m<1)$ perihelion precession respectively. However, when we use the same approximate solution for Schwarzschild spacetime, it can be shown ([@Bambh]) that only positive precession of timelike bound orbit is allowed in this spacetime. In [@Shaikh:2018lcc], it is shown that, when the JMN1 spacetime with $\chi>\frac23$ is matched with an external Schwarzschild spacetime, then in the external Schwarzschild spacetime, there exist a photon sphere which casts a shadow. However, for $\chi<\frac23$, no photon sphere exist and therefore, there will be no shadow. Therefore, note that, when there is a photon sphere which cast a shadow, the perihelion precession is positive always. On the other hand, when there is no photon sphere, the perihelion precession can be both positive and negative. This same results exist in Janis-Newman-Winicour (JNW) naked singularity spacetime also. This spacetime is a mass-less scalar field solution of Einstein equation and it can be written as, $$ds^2_{JNW} = -\left(1-\frac{b}{r}\right)^n dt^2 + \frac{dr^2}{\left(1-\frac{b}{r}\right)^{n}} + r^2\left(1-\frac{b}{r}\right)^{1-n}d\Omega^2\,\, , \label{JNWmetric}$$ where $b=2\sqrt{M^2+q^2}$ and $n=\frac{2M}{q}$. The parameters $q$ and $M$ represent charge of the scalar field and the ADM mass respectively. From the expression of $b$ and $n$ one can show that $0<n<1$. Using eq. (\[orbitsch1\]) in eq. (58) of [@Bambh], one can show $$m=\sqrt{Qp-2R-\frac{3S}{p}}\label{mJNW}\,\, ,$$ where, $$p_{\pm}=\frac{R\pm\sqrt{R^2+4QS}}{2Q}$$ $$Q = \left[\frac{b^2\gamma^2(1-n)}{h^2} - \frac{b^2(2-n)}{2h^2}\right]\,\, ,$$ $$R = \left[\frac{b^2\gamma^2(1-n)(1-2n)}{h^2} - \frac{b^2(2-n)(1-n)}{2h^2} - 1\right]\,\, ,$$ $$S = \left[\frac{3}{2} - \frac{b^2(2-n)(1-n)n}{4h^2} + \frac{b^2\gamma^2(1-2n)(1-n)n}{h^2}\right]\,\, ,$$ where $\gamma$ and $h$ are the conserved energy and angular momentum per unit rest mass (see [@Bambh] for details). In fig. (\[mvsn\]), it is shown that stable bound orbit with negative precession ($m>1$) is only possible when $n<\frac12$. On the other hand, positive precession ($m<1$) of timelike bound orbit is possible for both $n<\frac12$ and $n>\frac12$. In [@Shaikh:2019hbm], it is shown that for $n<\frac12$, JNW spacetime cannot cast a shadow. JNW can cast shadow only for $\frac12<n<1$. Therefore, from the above results, we can conclude that, when both the JMN1 and JNW naked singularities have photon spheres and hence cast shadows, they admit only positive perihelion precession of bound orbits. On the other hand, when they do not have any photon sphere, they admit both negative and positive precession but no shadows. Then the question arises is, can both shadows and negative precession exist simultaneously? In the next section, using an internal JMN1$_{int}$ and an external JMN1$_{ext}$, we construct a spacetime configuration which has a central naked singularity but no photon sphere, and it can give both shadow and a negative perihelion precession. The Spacetime Structure {#structure} ======================= In [@Dey:2019fja], we discussed how a galactic halo like structure can form due the gravitational collapse of General Collapsing Metric (GCM), where we model the GCM as the spacetime of collapsing baryonic matter and dark matter. In general relativity, a spherically symmetric general collapsing metric can be written as, $$ds^2_{\text{GCM}} = - e^{2\nu(r,t)} dt^2 + {R'^2\over G(r,t)}dr^2 + R^2(r,t) d\Omega^2\,\, ,$$ where $r,t$ are the comoving radial and temporal coordinates respectively and $R(r,t)$ is the physical radius. In the above equation $G(r,t)$ and $\nu(r,t)$ are the functions of comoving radius and comoving time, where $G(r,t)$ can have positive values only. In [@Dey:2019fja], we considered the above metric to be seeded by baryonic matter and dark matter which are collapsing together quasistatically at the initial stage of gravitational collapse. As the cooling time of baryonic matter is less than its dynamic time, baryonic matter cools down and accumulates at the central region of halo. In [@Dey:2019fja], this situation is described by an internal and an external GCM spacetimes. It can be shown that a collapsing matter cloud can reach to an equilibrium state after a large enough comoving time if there exists a non-zero pressure[@JMN11]. Similarly, the spacetime structure made with an internal GCM and an external GCM can also reach to an equilibrium state in asymptotic time. In [@Dey:2019fja], it is shown that the above mentioned spacetime structure, in a large comoving time, can transform into a static, spherically symmetric spacetime structure which can be described by an internal JMN1 (JMN1$_{int}$)and an external JMN1 (JMN1$_{ext}$) spacetime, where JMN1$_{ext}$ is matched with an external Schwarzschild spacetime. This static spacetime structure can be described as, $$\begin{aligned} ds^2_{\text{int}} &=& -(1-\chi_{ext})\left(\frac{\tilde{r}_{b1}}{\tilde{r}_{b2}}\right)^{\frac{\chi_{ext}}{1-\chi_{ext}}}\left(\frac{r}{r_{b1}}\right)^{\frac{\chi_{int}}{1-\chi_{int}}}dt^2 \nonumber + \frac{dr^2}{1-\chi_{int}} + r^2 d\Omega^2, \\ ds^2_{\text{ext}} &=& -(1-\chi_{ext})\left(\frac{\tilde{r}}{\tilde{r}_{b2}}\right)^{\frac{\chi_{ext}}{1-\chi_{ext}}}dt^2 \! + \! \frac{d\tilde{r}^2}{1-\chi_{ext}} \! + \! \tilde{r}^2 d\Omega^2. \nonumber\\ ds^2_{\text{Schw}}&=&-\left(1 - \frac{\chi_{ext} \tilde{r}_{b2}}{\tilde{r}}\right)dt^2 \! + \! \frac{d\tilde{r}^2}{\left(1 - \frac{\chi_{ext} \tilde{r}_{b2}}{\tilde{r}}\right)} \! + \! \tilde{r}^2d\Omega^2\,\, , \label{spctimestruc}\end{aligned}$$ where JMN1$_{int}$ and JMN1$_{ext}$ are matched at a timelike hypersurface $r-r_{b1}=0$ and JMN1$_{ext}$ is matched with external Schwarzschild metric at a timelike hypersurface $r-r_{b2}=0$. For the smooth matching of JMN1$_{int}$ and JMN1$_{ext}$, we need to match the induced metrics ($h_{ab}$) and the extrinsic curvatures ($K_{ab}$) at the matching hypersurface $r-r_{b1}=0$. It can be shown that, in order to match the induced metrics and extrinsic curvatures for JMN1$_{int}$ and JMN1$_{ext}$, we need $\tilde{r}=r$ and $\chi_{int}=\chi_{ext}$. Since the spacetime configuration we are considering has a mismatch in $\chi$ of internal and external asymptotic spacetimes (JMN1$_{int}$ and JMN1$_{ext}$), then there is a spherically symmetric thin matter shell at the timelike hypersurface $r-r_{b1}=0$. The external JMN1$_{ext}$, however, is smoothely matched to the external Schwarzschild spacetime [@Shaikh:2018lcc]. In the next section, we discuss shadows and prehelion precession in the above-mentioned spacetime. Shadow cast by the proposed spacetime configuration {#shadow} ==================================================== A spherically symmetric static spacetimes can be written as, $$ds^2=-A(r)dt^2+B(r)dr^2+r^2d\Omega^2\,\, , \label{sptm}$$ where $A(r)$ and $B(r)$ are the positive valued functions of $r$. For light like geodesics in these spacetimes, we can write, $$A(r)B(r)\left(\frac{dr}{d\lambda}\right)^2+V_{eff}=e^2\,\, , \label{sphersptm}$$ where $\lambda$ is the affine parameter and $V_{eff}$ is effective potential of lightlike geodesics which can be written as, $V_{eff}=\frac{A(r)}{B(r)}l^2$, where $e$ and $l$ are the conserved energy and angular momentum of photon. In the above equation, we use $k_{\mu}k^{\mu}=0$, where $k^{\mu}$ is the nulllike four velocity. From the effective potential of photon, one can get the information about the turning points and stable and unstable circular orbits of light like geodesics. When the effective potential has a maximum point at $r_{ph}$, where $V_{eff}(r_{ph})=e^2$, $V^{\prime}_{eff}(r_{ph})=0$ and $V^{\prime\prime}_{eff}(r_{ph})<0$, we can say that at $r=r_{ph}$, lightlike unstable, circular geodesics are possible. This timelike spherical surface of radius $r_{ph}$ is known as photon sphere. In a spherically symmetric spacetime, a photon sphere exists when above mentioned conditions of $V_{eff}$ are fulfilled. Turning points ($r_{tp}$) of lightlike geodesics can be found from $V_{eff}(r_{tp})=e^2$. From $V_{eff}(r_{tp})=e^2$, one can write $r_{tp}=b\sqrt{A(r)}$, where $b=\frac{l}{e}$ which is known as impact parameter. One can verify that photon sphere in JMN1 spacetime is not possible, as the effective potential of null geodesic cannot fulfil the previously mentioned conditions. On the other hand, in Schwarzschild spacetime photon sphere exists. In fig. (\[Schpoten\]), the effective potential for light like geodesics in Schwarzschild spacetime is shown, where the effective potential has a maximum point. Therefore, in Schwarzschild spacetime photon sphere exists. However, in [@Shaikh:2018lcc], it is shown that a spacetime structure, which is internally JMN1 spacetime and externally Schwarzschild spacetime, can have a photon sphere in the external Schwarzschild spacetime when $\chi>\frac23$. Therefore, the spacetime structure can cast same type of shadow what a solely Schwarzschild spacetime can cast. \ The spacetime structure which is mentioned in eq. (\[spctimestruc\]) does not allow any photon sphere inside the JMN1$_{int}$ and JMN1$_{ext}$ spacetimes. As it was discussed before, at the junction of two JMN1 spacetimes a thin matter shell can exist due to the mismatch of $\chi_{int}$ and $\chi_{ext}$. Now, if one consider $\chi_{int}>\frac23$ and $\chi_{ext}<\frac23$ then the spacetime structure described in eq. (\[spctimestruc\]) have a cusp like potential as shown in fig. (\[JMNpoten\]). When the effective potential has a cusp like nature at a point $r=r_{csp}$, where $V_{eff}(r_{csp})=e^2$ and $V^{\prime\prime}_{eff}(r_{csp})\rightarrow -\infty$, light like geodesic cannot have unstable circular orbits at $r=r_{csp}$. Therefore, due to the existence of cusp like potential, the lightlike geodesics can show some distinguishable properties which cannot be seen when photon sphere exists. At the cusp point ($r_{csp}$), as the $V^{\prime}_{eff}(r)$ is non-zero, there will be no photon sphere at this point. Therefore, relativistic Einstein rings does not form due to the cusp point of the effective potential. However, incoming light like geodesics can have innermost turning point at $r=r_{csp}$. Therefore, there exist a critical impact parameter, $b_{csp}=\frac{r_{csp}}{\sqrt{A(r_{csp})}}$, corresponding to $r_{csp}$. Ingoing photons with an impact parameter greater than $b_{csp}$ must be deflected away by the cusp potential and those with impact parameter less than $b_{csp}$ would fall into the central singularity. Therefore, in such a case, we have a shadow of radius $b_{csp}$. Next, following [@Shaikh:2018lcc], we consider an optically thin, radiating, radially infalling, accreting matter around the central singularity and produce the intensity map of the image. Beside the ingoing photons with impact parameter $b<b_{csp}$ getting absorbed by the central singularity, the outdoing photons which are emitted from the region $r<r_{csp}$ and have $b<b_{csp}$ can escape and are highly redshifted. Consequently, there should be a sudden drop in the observed intensity ($I_{obs}(X,Y)$) in the region $0\leq b\leq b_{csp}$, where $(X,Y)$ is the point in observer sky and impact parameter $b = \sqrt{X^2+Y^2}$. The observed intensity can be written as [@Bambi:2013nla] (see also [@Shaikh:2018lcc]), $$I_{obs}(X,Y)=-\int_{\gamma}\frac{g^3k_t}{r^2 k^r}dr\,\, , \label{Iobs}$$ where the integration is done along the photon trajectory ($\gamma$) and $g$, $k^t$ and $k^r$ are the redshift factor, temporal part and radial part of null four velocity respectively. The redshift factor $g$ can be written as [@Shaikh:2018lcc], $$g=\frac{1}{\frac{1}{A(r)}\pm\sqrt{\left(\frac{1}{A(r)}-1\right)\left(\frac{1}{A(r)}-\frac{b^2}{r^2}\right)}}\,\, ,$$ where we consider photon trajectory inside the spherically symmetric spacetime mentioned in eq. (\[sptm\]). Here we consider only a simple model where a optically thin radiating matter radially freely falling towards the center with an emissivity proportional to $r^{-2}$ and the emitted radiation is monochromatic [@Bambi:2013nla]. We can use eq. (\[Iobs\]) to derive the intensity map with respect to the stationary asymptotic observer. The intensity variation and the intensity map of the images for the Schwarzschild black hole and the proposed spacetime structure shown in figs. (\[schinten\]), (\[schshadow\]), (\[JMNinten\]) and (\[JMNshadow\]). In Schwarzschild black hole spacetime, it is the shadow of the photon sphere which can be observed by asymptotic observer. The photon sphere exists in Schwarzschild spacetime at $r_{ph}=3M_{TOT}$ and therefore, the radius of the shadow in the observer sky will be, $$b_{ph}=\frac{r_{ph}}{\sqrt{\left(1-\frac{2M_{TOT}}{r_{ph}}\right)}}=3\sqrt3 M_{TOT}\,\, . \label{shadowSCH}$$ In fig. (\[JMNSCHshadow\]), we consider the total mass $M_{TOT}=0.5$ for the Schwarzschild black hole. Therefore, the shadow radius will be $b_{ph}=2.59$, which can be seen in fig. (\[schshadow\]). On the other hand, in the spacetime structure (eq. (\[spctimestruc\])), it is the shadow of thin matter shell. The thin matter shell exists at a timelike hypersurface $r-r_{b1}=0$. Therefore, the shadow radius in the observer sky will be, $$b_{csp}=\frac{r_{b1}}{\sqrt{1-\chi_{ext}}\left(\frac{r_{b1}}{r_{b2}}\right)^{\frac{\chi_{ext}}{2(1-\chi_{ext})}}}\,\, , \label{shadowStruc}$$ where we can see that the radius of the shadow not only depends upon the radius ($r_{b1}$) of thin matter shell, but also it depends upon the $\chi_{ext}$ of JMN1$_{ext}$ and the radius $r_{b2}$ where the JMN1$_{ext}$ is matched with the external Schwarzschild spacetime. Therefore, for a fix value of the radius ($r_{b1}$) of the thin matter shell, radius of the shadow can vary for different values of $r_{b2}$ and $\chi_{ext}$. On the other hand, in Schwarzschild black hole spacetime, shadow radius depends upon the Schwarzschild mass or the radius of the photon sphere only (eq. (\[shadowSCH\])). Therefore, shadow of the photon sphere in Schwarzschild spacetime only carries the information of Schwarzschild mass, on the other hand shadow of a thin matter shell carries the information of the radius of the shell ($r_{b1}$), $\chi_{ext}$ and the radius ($r_{b2}$) of the outer edge of the external JMN1 spacetime. So, we can say that the radius of the shadow of thin matter shell carries the information of the whole structure of spacetime. Using the eq. (\[spctimestruc\]), one can verify that the $\chi_{ext}$ and $r_{b2}$ together fix the total Schwarzschild mass of the spacetime structure, $M_{TOT}=\frac{\chi_{ext}r_{b2}}{2}$. For the spacetime structure, in fig. (\[JMNSCHshadow\]), we consider the total mass ($M_{TOT}$) to be unity and we consider the mass ($M_{in}$) enclosed by the thin matter shell to be half of the total mass ($M_{TOT}$) of the entire spacetime structure. Therefore, $r_{b2}=\frac{2}{\chi_{ext}}$ and $r_{b1}=\frac{1}{\chi_{int}}$. In fig. (\[JMNSCHshadow\]), we take the $\chi_{int}=0.75$ and $\chi_{ext}=0.1$ for JMN1$_{int}$ and JMN1$_{ext}$ respectively, then the radius of the thin matter shell will be $r_{b1}=1.333$ and the radius of the outer edge of the JMN1$_{ext}$ will be $r_{b2}=20$. The shadow radius in observer sky will be $b_{csp}=\frac{1.054~r_{b1}}{\left(\frac{r_{b1}}{r_{b2}}\right)^{0.055}}=1.63$, which is $1.22$ times of the radius ($r_{b1}$) of thin matter shell. The intensity variation and the shadow of the thin matter shell for the above mentioned values of $\chi_{int}$, $\chi_{ext}$, $r_{b1}$ and $r_{b2}$, are shown in fig. (\[JMNinten\]), (\[JMNshadow\]) respectively. As we have mentioned, the spacetime structure given in eq. (\[spctimestruc\]) may be used to model the galactic halo structure, where $r_{b2}$ will be the hallo radius. In [@Dey:2019fja], it is shown how this type of spacetime structure can be formed in the cosmological scenario. If one models the dark matter halo by the spacetime structure, then one has to investigate stellar motion around the galactic center. As we previously discussed, the motion of different ‘S’ stars around the Milky-way galactic center Sgr-A\* is being observed continuously by different collaborations(e.g. GRAVITY, SINFONI, etc.). As these stars are very close to the Sgr-A\*, their orbit’s precession due to the spacetime curvature may be observed in the near future. In [@Bambh], we investigated the perihelion precession of timelike orbits in different naked singularity spacetimes and compared the results with the perihelion precession of timelike orbits in Schwarzschild spacetime. We find out that the perihelion precession in naked singularity spacetimes can be opposite to the direction of motion of particles. We showed that this opposite or negative precession cannot be possible in Schwarzschild spacetime. Therefore, any evidence of negative precession can rule out the existence of vacuum blackhole spacetime around the center of the Milky-way. As we have mentioned previously, in [@Bambh], it is shown that for JMN1 naked singularity spacetime, negative precession and positive precession of timelike bound orbits are possible when $\chi<\frac13$ and $\chi>\frac13$ respectively. In fig. (\[JMNSCH\]), we show that both the positive and negative precession of bound timelike orbits are possible in the external JMN1 spacetime. On the other hand, we also show the shadow cast by spherically symmetric thin matter shell which exists at the matching radius of the internal and external JMN1 spacetimes. The timelike orbit equation in external JMN1 spacetime can be written in the following form, $$\frac{d^2u}{d\phi^2} + (1 - \chi_{ext}) u - \frac{\gamma^2}{2h^2}\frac{\chi_{ext}}{(1- \chi_{ext})}\left(\frac{1}{u}\right)\left(\frac{1}{u~r_{b2}}\right)^\frac{-\chi_{ext}}{(1- \chi_{ext})}=0\,\, .$$ Using the above orbit equation, in fig. (\[JMNSCH\]), we show the precession of stable timelike bound orbits in external JMN1 spacetime. In fig. (\[JMNorbitshadow1\]), we show the possibility of negative precession of the timelike bound orbits and the shadow of the central thin shell of matter by considering $\chi_{int}=0.75$, $\chi_{ext}=0.3$, $r_{b2}=1000$ and $r_{b1}=\frac{1}{\chi_{int}}=1.333$. From eq. (\[shadowStruc\]), we get the shadow radius $b_{csp}=6.58$. As we know, for positive precession we need $\chi_{ext}>\frac13$. Therefore, in fig. (\[JMNorbitshadow2\]), to get the positive precession, we consider $\chi_{ext}=0.36$. For this case, the shadow radius becomes $b_{csp}=10.7$. As we have mentioned, thin shell shadow size depends upon various parameters’ values of the entire spacetime structure. Therefore, in figs. (\[JMNorbitshadow1\],\[JMNorbitshadow2\], \[JMNshadow\]), the shadow radius of the thin shell of matter can be changed by changing different parameters’ values of the proposed spacetime structure. Conclusion ========== In this paper, we have investigated the shadow of a spherically symmetric thin matter shell and we compared that with the shadow cast by a Schwarzschild black hole. We also show that stable bound timelike orbits with positive and negative precession can be incorporated considering the spacetime structure mentioned in eq. (\[spctimestruc\]). To get the shadow of the thin matter shell we need $\chi_{int}>\frac23$ and $\chi_{ext}<\frac23$. On the other hand, for positive and negative precession, we need $\chi_{ext}>\frac13$ and $\chi_{ext}<\frac13$ respectively. Few important properties of a shadow of the thin matter shell are coming out from our discussion, which are as below. - Negative precession with a central shadow is forbidden in JMN1 and JNW naked singularity spacetimes. We show that, in the proposed spacetime structure (eq. (\[spctimestruc\])) constructed using the JMN1$_{int}$ and JMN1$_{ext}$ naked singularity spacetimes, both shadow and negative precession can simultaneously exist. On the other hand, shadow with positive precession also can exist in the proposed spacetime configuration. Therefore, any observational result of negative precession of ‘S’ stars with a central shadow can abandon the possibility of existence of a central black hole. In this case the proposed naked singularity spacetime configuration can be one of the candidates to explain the observed phenomena. On the other hand, both the black hole or naked singularity can exist if positive precession of ‘S’ stars with central shadow is observed. One can form same type of spacetime configuration (eq. (\[spctimestruc\])) with JNW naked singularity spacetime which can allow negative precession of timelike bound orbits with a central shadow. - Shadow of the thin matter shell can be formed due to the curvature around the matter shell. We do not consider any chemical absorption of light by the matter present at the thin matter shell. - In black hole spacetimes, the radius of the shadow only depends upon the mass, angular momentum and charge of the central object. On the other hand, the radius of the shadow of a thin matter shell carries the information of the detailed structure of internal and external spacetime (eq. (\[shadowStruc\])). If we model the dark matter halo structure by the structure described in eq. (\[spctimestruc\]), then the radius of the central shadow of the thin matter shell should be correlated with the radius of dark matter halo ($r_{b2}$) and the total dark matter halo mass ($\frac{\chi_{ext}r_{b2}}{2}$). This correlation in eq. (\[shadowStruc\]), can be verified when we have sufficient number of data of the central shadow radius for different galaxies. As was previously mentioned, in this paper, we consider Newton’s gravitational constant and light velocity as unity. Therefore, in this paper, we do not attempt to fit the theoretical prediction with any observational data. Fig. (\[JMNSCH\]) shows only the possibility of simultaneous existence of negative precession (or positive precession) and central shadow. In that figure, we show the astrophysical importance of proposed spacetime configuration in the context of bound timelike orbits of ‘S’ stars around the galactic center, and the shadow of the galactic center. If one wants to fit the theoretical results with ‘S’ stars’ data, one needs to consider actual values of Newton’s gravitational constant and light velocity. A detailed phenomenological discussion of ‘S’ stars orbits and central shadow will be reported elsewhere. [99]{} K. 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--- abstract: 'This topical review describes the methodology of continuum variational and diffusion quantum Monte Carlo calculations. These stochastic methods are based on many-body wave functions and are capable of achieving very high accuracy. The algorithms are intrinsically parallel and well-suited to petascale computers, and the computational cost scales as a polynomial of the number of particles. A guide to the systems and topics which have been investigated using these methods is given. The bulk of the article is devoted to an overview of the basic quantum Monte Carlo methods, the forms and optimisation of wave functions, performing calculations within periodic boundary conditions, using pseudopotentials, excited-state calculations, sources of calculational inaccuracy, and calculating energy differences and forces.' address: 'Theory of Condensed Matter Group, Cavendish Laboratory, Cambridge CB3 0HE, UK' author: - 'R J Needs, M D Towler, N D Drummond and P López Ríos' title: Continuum variational and diffusion quantum Monte Carlo calculations --- Introduction \[sec:introduction\] ================================= The variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC) methods are stochastic approaches for evaluating quantum mechanical expectation values with many-body Hamiltonians and wave functions [@foulkes_2001]. VMC and DMC methods are used for both continuum and lattice systems, but here we describe their application only to continuum systems. The main attraction of these methods is that the computational cost scales as some reasonable power (normally from the second to fourth power) of the number of particles [@note_exp_scaling]. This scaling makes it possible to deal with hundreds or even thousands of particles, allowing applications to condensed matter. Continuum quantum Monte Carlo (QMC) methods, such as VMC and DMC, occupy a special place in the hierarchy of computational approaches for modelling materials. QMC computations are expensive, which limits their applicability at present, but they are the most accurate methods known for computing the energies of large assemblies of interacting quantum particles. There are many problems for which the high accuracy achievable with QMC is necessary to give a faithful description of the underlying science. Most of our work is concerned with correlated electron systems, but these methods can be applied to any combination of fermion and boson particles with any inter-particle potentials and external fields *etc.* Being based on many-body wave functions, these are zero-temperature methods, and for finite temperatures one must use other approaches such as those based on density matrices. Both the VMC and DMC methods are variational, so that the calculated energy is above the true ground state energy. The computational costs of VMC and DMC calculations scale similarly with the number of particles studied, but the prefactor is larger for the more accurate DMC method. QMC algorithms are intrinsically parallel and are ideal candidates for taking advantage of the petascale computers (10$^{15}$ flops) which are becoming available now and the exascale computers (10$^{18}$ flops) which will be available one day. DMC has been applied to a wide variety of continuum systems. A partial list of topics investigated within DMC and some references to milestone papers are given below. - Three-dimensional electron gas [@ceperley_1980; @moroni_1995; @zong_2002; @drummond_2004]. - Two-dimensional electron gas [@tanatar_1989; @moroni_1992; @attaccalite_2002; @drummond_2008_2d]. - The equation of state and other properties of liquid $^3$He [@lee_1981; @holzmann_2006]. - Structure of nuclei [@carlson_2007]. - Pairing in ultra-cold atomic gases [@carlson_2003; @astrakharchik_2004; @carlson_2008]. - Reconstruction of a crystalline surface [@healy_2001] and molecules on surfaces [@filippi_2002; @kim_2006]. - Quantum dots [@ghosal_2006]. - Band structures of insulators [@mitas_1994; @williamson_1998; @towler_2000]. - Transition metal oxide chemistry [@towler_2003; @wagner_2003; @wagner_2007a]. - Optical band gaps of nanocrystals [@williamson_2002; @drummond_2005_dia]. - Defects in semiconductors [@leung_1999; @hood_2003; @alfe_2005a]. - Solid state structural phase transitions [@alfe_2005b]. - Equations of state of solids [@natoli_1993; @delaney_2006; @maezono_2007a; @pozzo_2008]. - Binding of molecules and their excitation energies [@manten_2001; @grossman_2002; @aspuru-guzik_2004; @gurtubay_2006; @gurtubay_2007]. - Studies of exchange-correlation [@hood_1997; @hood_1998; @nekovee_2001; @nekovee_2003]. The same basic QMC algorithm can be used for each of the applications mentioned above with only minor modifications. The complexity and sophistication of the computer codes arises not from the algorithm itself, which is in fact quite simple, but from the diversity of the Hamiltonians and many-body wave functions which are involved. A number of computer codes are currently available for performing continuum QMC calculations of the type described here [@qmc_wiki]. We have developed the <span style="font-variant:small-caps;">casino</span> code [@casino], which can deal with systems of different dimensionalities, various interactions including the Coulomb potential, external fields, mixtures of particles of different types and different types of many-body wave function. The VMC and DMC methods are described in section \[sec:qmc\] and the types of many-body wave function we use are described in section \[sec:psi\_trial\]. The optimisation of parameters in wave functions using stochastic methods which are both subtle and unique to the field is described in section \[sec:optimise\_psi\_trial\]. QMC calculations within periodic boundary conditions are described in section \[sec:pbc\], the use of pseudopotentials in QMC calculations is discussed in section \[sec:pseudopots\] and excited-state DMC calculations are briefly described in section \[sec:excited\_states\]. The scaling of the QMC methods with system size is discussed in section \[sec:scaling\]. Sources of errors in the DMC method and practical methods for handling errors in QMC results are described in section \[subsec:errors\]. In section \[sec:other expectation values\] we describe how to evaluate other expectation values apart from the energy. Section \[sec:energy differences and energy derivatives\] deals with the calculation of energy differences and energy derivatives in the VMC and DMC methods, and we make our final remarks in section \[sec:conclusions\]. Quantum Monte Carlo methods {#sec:qmc} =========================== The VMC method is conceptually very simple. The energy is calculated as the expectation value of the Hamiltonian with an approximate many-body trial wave function.  In the more sophisticated DMC method the estimate of the ground state energy is improved by performing a process described by the evolution of the wave function in imaginary time. Throughout this article we will consider only systems with spin-independent Hamiltonians and collinear spins. We will also restrict the discussion to systems with time-reversal symmetry, for which the wave function may be chosen to be real. It is, however, straightforward to generalise the VMC algorithm to work with complex wave functions, and only a little more complicated to generalise the DMC algorithm to work with them [@ortiz_1993]. The VMC method {#subsec:vmc} -------------- The variational theorem of quantum mechanics states that, for a real, proper [@footnote:proper] trial wave function $\Psi_{\rm T}$, the variational energy, $$\begin{aligned} \label{eq:variational_energy} E_{\rm V} = \frac{\int \Psi_{\rm T}({\bf R}) \hat{H} \Psi_{\rm T}({\bf R})\, d{\bf R}}{\int \Psi_{\rm T}^2({\bf R}) \, d{\bf R}} \;,\end{aligned}$$ is an upper bound on the exact ground state energy $E_0$, *i.e.*, $E_{\rm V} \geq E_0$. In equation (\[eq:variational\_energy\]), $\hat{H}$ is the many-body Hamiltonian and ${\bf R}$ denotes a $3N$-dimensional vector of particle coordinates. As discussed in section \[subsec:Slater-Jastrow wave functions\], the spin variables in equation (\[eq:variational\_energy\]) are implicitly summed over. To facilitate the stochastic evaluation, $E_{\rm V}$ is written as $$\begin{aligned} \label{eq:variational_energy_2} E_{\rm V} = \int p({\bf R}) E_{\rm L}({\bf R}) \, d{{\bf R}} \;,\end{aligned}$$ where the probability distribution $p$ is $$\begin{aligned} \label{eq:distribution} p({\bf R}) = \frac{\Psi_{\rm T}^2({\bf R})}{\int \Psi_{\rm T}^2({\bf R}^{\prime}) \, d{\bf R}^{\prime}} \;,\end{aligned}$$ and the local energy, $$\begin{aligned} \label{eq:local_energy} E_{\rm L}({\bf R}) = \Psi_{\rm T}^{-1} \hat{H} \Psi_{\rm T} \;.\end{aligned}$$ is straightforward to evaluate at any ${\bf R}$. In VMC the Metropolis algorithm [@metropolis_1953] is used to sample the probability distribution $p({\bf R})$. Let the electron configuration at a particular step be ${\bf R}^\prime$. A new configuration ${\bf R}$ is drawn from the probability density $T({\bf R} \leftarrow {\bf R}^\prime)$, and the move is accepted with probability $$A({\bf R}\leftarrow{\bf R}^\prime)= \min \left\{ 1,\frac{T({\bf R}^\prime \leftarrow {\bf R})\Psi_{\rm T}^2({\bf R})}{T({\bf R} \leftarrow {\bf R}^\prime)\Psi_{\rm T}^2({\bf R}^\prime)} \right\}.$$ It can easily be verified that this algorithm satisfies the *detailed balance* condition $$\Psi_{\rm T}^2({\bf R})T({\bf R}^\prime \leftarrow {\bf R}) A({\bf R}^\prime \leftarrow {\bf R}) = \Psi_{\rm T}^2({\bf R}^\prime)T({\bf R} \leftarrow {\bf R}^\prime) A({\bf R} \leftarrow {\bf R}^\prime).$$ Hence $p({\bf R})$ is the equilibrium configuration distribution of this Markov process and, so long as the transition probability is ergodic (i.e., it is possible to reach any point in configuration space in a finite number of moves), it can be shown that the process will converge to this equilibrium distribution. Once equilibrium has been reached, the configurations are distributed as $p({\bf R})$, but successive configurations along the random walk are in general correlated. The variational energy is estimated as $$\begin{aligned} \label{eq:variational_energy_3} E_{\rm V} \simeq \frac{1}{M} \sum_{i=1}^M E_{\rm L}({\bf R}_i),\end{aligned}$$ where $M$ configurations ${\bf R}_i$ have been generated after equilibration. The serial correlation of the configurations and therefore local energies $E_{\rm L}({\bf R}_i)$ complicates the calculation of the statistical error on the energy estimate: see section \[subsec:statistical errors\]. Other expectation values may be evaluated in a similar manner to the energy. Equation (\[eq:variational\_energy\_2\]) is an importance sampling transformation of equation (\[eq:variational\_energy\]). Equation (\[eq:variational\_energy\_2\]) exhibits the zero variance property: as the trial wave function approaches an exact eigenfunction ($\Psi_{\rm T} \rightarrow \phi_i$), the local energy approaches the corresponding eigenenergy, $E_i$, everywhere in configuration space. As $\Psi_{\rm T}$ is improved, $E_{\rm L}$ becomes a smoother function of ${\bf R}$ and the number of sampling points, $M$, required to achieve an accurate estimate of $E_{\rm V}$ is reduced. VMC is a simple and elegant method. There are no restrictions on the form of trial wave function which can be used and it does not suffer from a fermion sign problem. However, even if the underlying physics is well understood it is often difficult to prepare trial wave functions of equivalent accuracy for two different systems, and therefore the VMC estimate of the energy difference between them will be biased. We use the VMC method mostly to optimise parameters in trial wave functions (see section \[sec:optimise\_psi\_trial\]) and our main calculations are performed with the more sophisticated DMC method, which is described in the next section. The DMC method {#subsec:dmc} -------------- In DMC the operator $\exp(-t \hat{H})$ is used to project out the ground state from the initial state. This can be viewed as solving the imaginary-time Schrödinger equation, which for electrons is $$\begin{aligned} \label{eq:imaginary_time_se} -\frac{\partial}{\partial t} \Phi({\bf R},t) = \left(\hat{H} - E_{\rm T} \right) \Phi({\bf R},t) = \left(-\frac{1}{2} \nabla^2_{\bf R}+ V({\bf R}) - E_{\rm T} \right) \Phi({\bf R},t) \;,\end{aligned}$$ where $t$ is a real variable measuring the progress in imaginary time, $V$ is the potential energy (assumed to be local for the time being), and $E_{\rm T}$ is an arbitrary energy offset known as the reference energy. Throughout this article we use Hartree atomic units where $m_e = \hbar = |e| = 4\pi \epsilon_0 = 1$, where $m_e$ is the mass of the electron and $e$ is its charge. Equation (\[eq:imaginary\_time\_se\]) can be solved formally by expanding $\Phi({\bf R},t)$ in the eigenstates $\phi_i$ of the Hamiltonian, $$\begin{aligned} \label{eq:expansion} \Phi({\bf R},t) = \sum_i c_i(t) \phi_i({\bf R}) \;,\end{aligned}$$ which leads to $$\begin{aligned} \label{eq:expansion_2} \Phi({\bf R},t) = \sum_i \exp[-(E_i-E_{\rm T})t] \, c_i(0) \phi_i({\bf R}) \;.\end{aligned}$$ For long times one finds $$\begin{aligned} \label{eq:expansion_3} \Phi({\bf R},t\rightarrow \infty) \simeq \exp[-(E_0-E_{\rm T})t] \, c_0(0) \phi_0({\bf R}) \;,\end{aligned}$$ which is proportional to the ground state wave function, $\phi_0$. The Hamiltonian is the sum of kinetic and potential terms: $\hat{H} = -(1/2) \nabla^2_{\bf R}+ V({\bf R})$. Suppose for a moment that we can interpret the initial state, $\sum_i c_i(0)\phi_i$, as a probability distribution. If we neglect the potential term then the imaginary-time Schrödinger equation (\[eq:imaginary\_time\_se\]) reduces to a diffusion equation in the configuration space. If, on the other hand, we neglect the kinetic term, (\[eq:imaginary\_time\_se\]) reduces to a rate equation. It should not be surprising that a short time slice of the imaginary-time evolution can be simulated by taking a population of configurations $\{{\bf R}_i\}$ and subjecting them to random hops to simulate the diffusion process, and “birth” and “death” of configurations to simulate the rate process. By “birth” and “death” we mean replicating some configurations and deleting others at the appropriate rates, a process which is often referred to as “branching”. Unfortunately the wave function cannot in general be interpreted as a probability distribution. A wave function for two or more identical fermions must have positive and negative regions, as should an excited state of any system. One can construct algorithms which are formally exact using two distributions of configurations with positive and negative weights [@kalos_2005], but they are inefficient and the scaling of the computational cost with system size is unclear. The fixed-node approximation [@anderson_1975; @anderson_1976] provides a way to evade the sign problem. (In a 3D system, the nodal surface is the $(3N-1)$-dimensional surface on which the wave function is zero and across which it changes sign.) The fixed-node approximation is equivalent to placing an infinite repulsive potential barrier on the nodal surface of the trial wave function which is sufficiently strong to force the wave function to be zero on the nodal surface. In effect we solve the Schrödinger equation exactly within each pocket enclosed by the nodal surface, subject to the boundary condition that the wave function is zero on the nodal surface. The infinite repulsive potential barrier has no effect if the trial nodal surface is placed correctly but, if it is not, the energy is always raised. It follows that the DMC energy is always less than or equal to the VMC energy with the same trial wave function, and always greater than or equal to the exact ground-state energy. The fixed-node DMC algorithm described above is extremely inefficient and a vastly superior algorithm can be obtained by introducing an importance sampling transformation [@grimm_1971; @kalos_1974]. Consider the mixed distribution, $$\begin{aligned} \label{eq:f} f({\bf R},t) = \Psi_{\rm T}({\bf R}) \Phi({\bf R},t) \;,\end{aligned}$$ which has the same sign everywhere if and only if the nodal surface of $\Phi({\bf R},t)$ equals that of $\Psi_{\rm T}({\bf R})$. Substituting in equation (\[eq:imaginary\_time\_se\]) for $\Phi$ we obtain $$\begin{aligned} \label{eq:importance_sampled_imaginary_time_se} -\frac{\partial f}{\partial t} = -\frac{1}{2} \nabla_{\bf R}^2 f + \nabla_{\bf R} \cdot [{\bf v}f] + [E_{\rm L}-E_{\rm T}]f \;,\end{aligned}$$ where the $3N$-dimensional drift velocity is defined as $$\begin{aligned} \label{eq:drift_velocity} {\bf v}({\bf R}) = \Psi_{\rm T}^{-1}({\bf R}) \nabla_{\bf R} \Psi_{\rm T}({\bf R}) \;.\end{aligned}$$ The three terms on the right-hand side of equation (\[eq:importance\_sampled\_imaginary\_time\_se\]) correspond to diffusion, drift and branching processes, respectively. The importance sampling transformation has several consequences. First, the density of configurations is increased where $|\Psi_{\rm T}|$ is large, so that the more important parts of the wave function are sampled more often. Second, the rate of branching is now controlled by the local energy which is normally a much smoother function than the potential energy. This is particularly important for the Coulomb interaction, which diverges when particles are coincident. The importance sampling transformation, together with an algorithm that imposes $f({\bf R},t) \ge 0$, ensures that $\Psi_{\rm T}$ and $\Phi({\bf R},t)$ have the same nodal surfaces, as can be seen in equation (\[eq:f\]). The importance sampling transformation also reduces the statistical error bar on the estimate of the energy and leads to a zero variance property analogous to that in VMC. The importance-sampled imaginary-time Schrödinger equation may be written in integral form: $$\begin{aligned} \label{eq:time evolution of f} f({\bf R},t) = \int G({\bf R} \leftarrow {\bf R}^{\prime}, t-t^{\prime}) f({\bf R}^{\prime},t^{\prime}) \, d{\bf R}^{\prime} \;,\end{aligned}$$ where the Green’s function $G({\bf R} \leftarrow {\bf R}^{\prime}, t-t^{\prime})$ is a solution of equation (\[eq:importance\_sampled\_imaginary\_time\_se\]) satisfying the initial condition $G({\bf R} \leftarrow {\bf R}^{\prime}, 0)=\delta({\bf R} - {\bf R}^{\prime})$. The exact Green’s function can be sampled using the Green’s function Monte Carlo (GFMC) algorithm developed by Kalos and coworkers [@kalos_1962; @kalos_1967; @kalos_1974; @ceperley_1986b; @schmidt_1987]. Let us interpret $f({\bf R},t)$ as the probability distribution of a discrete population of $P$ configurations with positive weights: $$\label{eq:f_discrete} f({\bf R},t) = \left< \sum_{p=1}^P w_p(t) \, \delta[{\bf R}-{\bf R}_p(t)] \right>,$$ where the $p$th configuration at time $t$ has position ${\bf R}_p(t)$ in configuration space and weight $w_p(t)$, and the angled brackets denote an ensemble average. Using equation (\[eq:time evolution of f\]), the evolution of $f({\bf R},t)$ to time $t+\tau$ yields $$\begin{aligned} \label{eq:f_prime} f({\bf R},t+\tau) & = & \left< \sum_{p=1}^P w_p(t) \, G[{\bf R} \leftarrow {\bf R}_p(t),\tau] \right> \nonumber \\ & = & \left< \sum_{p=1}^P w_p(t+\tau) \, \delta[{\bf R}-{\bf R}_p(t+\tau)] \right>. \end{aligned}$$ The dynamics of the configurations and their weights is governed by the Green’s function. The GFMC algorithm is computationally expensive, but considerably faster calculations can be made using an approximate Green’s functions which becomes exact in the limit of infinitely small time steps. Within the short-time approximation $$\begin{aligned} \label{eq:short time G} G({\bf R} \leftarrow {\bf R}^{\prime}, \tau) \simeq G_{\rm st}({\bf R} \leftarrow {\bf R}^{\prime}, \tau) = G_{\rm D}({\bf R} \leftarrow {\bf R}^{\prime}, \tau)G_{\rm B}({\bf R} \leftarrow {\bf R}^{\prime}, \tau) \;,\end{aligned}$$ where $$\begin{aligned} \label{eq:short time GD} G_{\rm D}({\bf R} \leftarrow {\bf R}^{\prime}, \tau) = \frac{1}{(2\pi\tau)^{3N/2}} \exp \left(- \frac{\left[ {\bf R}-{\bf R}^{\prime} - \tau {\bf v}({\bf R}^\prime) \right]^2}{2\tau} \right)\end{aligned}$$ is the drift-diffusion Green’s function and $$\begin{aligned} \label{eq:short time GB} G_{\rm B}({\bf R} \leftarrow {\bf R}^{\prime}, \tau) = \exp \left( -\frac{\tau}{2} \left[E_{\rm L}({\bf R}) + E_{\rm L}({\bf R}^{\prime}) - 2E_{\rm T} \right] \right)\end{aligned}$$ is the branching factor. The process described by $G_{\rm D}({\bf R}\leftarrow {\bf R}^\prime,\tau)$ is simulated by making each configuration ${\bf R}^\prime$ in the population drift through a distance $\tau{\bf v}({\bf R}^\prime)$, then diffuse by a random distance drawn from a Gaussian distribution of variance $\tau$. Each configuration is then copied or deleted in such a fashion that, on average, $G_{\rm B}({\bf R}\leftarrow {\bf R}^\prime,\tau)$ configurations continue from the new position ${\bf R}$. When using the short time approximation, configurations occasionally attempt to cross the nodal surface but such moves may simply be rejected. The short time approximation leads to a dependence of DMC results on the time step. It is important to investigate the size of the time step dependence, and it is common practice to extrapolate the energy to zero time step: see figure \[fig:time\_step\_errors\]. It turns out that $G_{\rm st}$ does not precisely satisfy the detailed-balance condition, but it is standard practice to reinstate detailed balance by incorporating an accept-reject step. The importance-sampled fixed-node fermion DMC algorithm was first used by Ceperley and Alder in their ground-breaking study of the homogeneous electron gas (HEG) [@ceperley_1980]. It can be seen that the reference energy $E_{\rm T}$ appears in the branching factor of equation (\[eq:short time GB\]). By adjusting the reference energy during the simulation we may keep the total population close to a target value, preventing the population from either increasing exponentially or dying out. An example of the behaviour of the total population and the reference energy can be seen in figure \[fig:graphit\_silane\] [@foulkes_2001]. Another important aspect of practical implementations is that the particles are normally moved one at a time in both VMC and DMC algorithms. The trial wave function can usually be evaluated more rapidly when a single particle has been moved than if all particles have been moved, and a longer time step can be employed for an equivalent time-step error. The correlation length of the local energy is shorter for single-particle moves and overall the efficiency is considerably increased [@lopez-rios_2006]. The initial configurations are normally taken from a VMC calculation and equilibrated within DMC for a period of imaginary time. The importance-sampled DMC algorithm generates configurations asymptotically distributed according to $f({\bf R})= \Psi_{\rm T}({\bf R})\phi_0({\bf R})$, where $\phi_0$ is the ground state of the Schrödinger equation subject to the fixed-node boundary condition. Noting that $\hat{H} \phi_0 = E_0 \phi_0$ everywhere (except on the nodal surface where $\phi_0=0$) the fixed-node DMC energy can be evaluated using the formula $$\begin{aligned} \label{eq:diffusion_energy} E_{\rm D} \equiv E_0 = \frac{\langle \phi_0 | \hat{H} | \Psi_{\rm T} \rangle}{\langle \phi_0 | \Psi_{\rm T} \rangle} & = & \frac{\int f({\bf R}) E_{\rm L}({\bf R}) \, d{\bf R}}{\int f({\bf R}) \, d{\bf R}} \\ & \simeq & \frac{1}{M} \sum_{i=1}^M E_{\rm L}({\bf R}_i) \;.\end{aligned}$$ Some example DMC data are shown in figure \[fig:graphit\_silane\]. ![DMC data for a silane (SiH$_4$) molecule, with the ions represented by pseudopotentials. The upper panel shows the fluctuations in the population of configurations arising from the branching process used to simulate equation (\[eq:short time GB\]). The reference energy, $E_{\rm T}$, is altered during the run to control the population. Specifically, the reference energy is set to return the population to the target population (128,000 configurations) on the same time-scale as the autocorrelation period of the energy data [@foulkes_2001]. The total energy is shown in the lower panel as a function of the move number. The black line shows the instantaneous value of the local energy averaged over the current population of configurations, the red line is the reference energy $E_{\rm T}$ and the green line is the best estimate of the DMC energy as the simulation progresses. The configurations at move number zero are from the output of a VMC simulation, and the energy decays rapidly from its initial VMC value of about -6.250 a.u. and reaches a plateau with a DMC energy of about -6.305 a.u. The data up to move 1000 are deemed to form the equilibration phase, and are discarded. []{data-label="fig:graphit_silane"}](graphit_silane){width="100.00000%"} Trial wave functions {#sec:psi_trial} ==================== Trial wave functions are of central importance in VMC and DMC calculations because they introduce importance sampling and control both the statistical efficiency and accuracy obtained. The accuracy of a DMC calculation depends on the nodal surface of the trial wave function via the fixed-node approximation, while in VMC the accuracy depends on the entire trial wave function. VMC energies are therefore more sensitive to the quality of the trial wave function than DMC energies. Slater-Jastrow wave functions {#subsec:Slater-Jastrow wave functions} ----------------------------- QMC calculations require a compact trial wave function which can be evaluated rapidly. Most studies of electronic systems have used the Slater-Jastrow form, in which a pair of up- and down-spin determinants is multiplied by a Jastrow correlation factor, $$\label{eq:slater-jastrow} \Psi_{\rm SJ}({\bf R})\! \! = e^{J({\bf R})} \det{\left[ \psi_n({\bf r}_i^{\uparrow})\right]} \det{\left[ \psi_n({\bf r}_j^{\downarrow})\right]} \;,$$ where $e^{J}$ is the Jastrow factor and $\det{\left[ \psi_n({\bf r}_i^{\uparrow}) \right]}$ is a determinant of single-particle orbitals for the up-spin electrons. The quality of the single-particle orbitals is very important, and they are often obtained from density functional theory (DFT) or Hartree-Fock (HF) calculations. Note that the spin variables themselves do not appear in equation (\[eq:slater-jastrow\]). Formally the sum over spin variables in the expectation values in equations (\[eq:variational\_energy\]) and (\[eq:diffusion\_energy\]) has already been performed and the single determinant with spin variables is replaced by two determinants of up- and down-spin orbitals whose arguments are the up- and down-spin electron coordinates ${\bf R}_{\uparrow}$ and ${\bf R}_{\downarrow}$, respectively. This is explained in more detail in reference [@foulkes_2001]. The Jastrow factor is taken to be symmetric under the interchange of identical particles and its positivity means that it does not alter the nodal surface of the trial wave function. The Jastrow factor introduces correlation by making the wave function depend explicitly on the particle separations. The optimal Jastrow factor is normally small when particles with repulsive interactions (for example, two electrons) are close to one another and large when particles with attractive interactions (for example, an electron and a positron) are close to one another. The Jastrow factor can also be used to ensure that the trial wave function obeys the Kato cusp conditions [@kato_1957], which leads to smoother behaviour in the local energy $E_{\rm L}({\bf R})$. When two particles interacting via the Coulomb potential approach one another, the potential energy diverges, and therefore the exact wave function $\Psi$ must have a cusp so that the local kinetic energy $-(1/2) \Psi^{-1}\nabla^2 \Psi$ supplies an equal and opposite divergence. It seems very reasonable to enforce the cusp conditions on trial wave functions because they are obeyed by the exact wave function. Imposition of the cusp conditions is in fact very important in both VMC and DMC calculations because divergences in the local energy lead to poor statistical behaviour and even instabilities in DMC calculations due to divergences in the branching factor. Figure \[fig:1\] shows the local energies generated during two VMC runs for a silane molecule in which the Si$^{4+}$ and H$^{+}$ ions are described by smooth pseudopotentials. In figure \[fig:1\](a) the trial wave function consists of a product of up- and down-spin Slater determinants of molecular orbitals. The Kato cusp conditions for electron-electron coalescences are therefore not satisfied and the local energy shows very large positive spikes when two electrons are close together. Figure \[fig:1\](b) shows the effect of adding a Jastrow factor which satisfies the electron-electron cusp conditions. The large positive spikes in the local energy are removed and the mean energy is lowered. Some small spikes remain, and the frequency and size of the positive and negative spikes are roughly equal. These spikes arise from electrons approaching the nodes of the trial wave function, where the local kinetic energy diverges positively on one side of the node and negatively on the other side. The basic Jastrow factor that we use for systems of electrons and ions contains the sum of homogeneous, isotropic electron-electron terms $u$, isotropic electron-nucleus terms $\chi$ centred on the nuclei and isotropic electron-electron-nucleus terms $f$, also centred on the nuclei [@ndd_newjas]. We use a Jastrow factor of the form $\exp [J({\bf R})]$, where $$\begin{aligned} \label{eq:basic_J} J(\{{\bf r}_i\},\{{\bf r}_I\}) = \sum_{i>j}^{N} u(r_{ij}) + \sum_{I=1}^{N_{\rm ions}} \sum_{i=1}^N \chi_I(r_{iI}) + \sum_{I=1}^{N_{\rm ions}} \sum_{i>j}^{N} f_I(r_{iI},r_{jI},r_{ij}) \;,\end{aligned}$$ $N$ is the number of electrons, $N_{\rm ions}$ is the number of ions, ${\bf r}_{ij} = {\bf r}_{i} - {\bf r}_{j}$, ${\bf r}_{iI} = {\bf r}_{i} - {\bf r}_{I}$, ${\bf r}_i$ is the position of electron $i$ and ${\bf r}_I$ is the position of nucleus $I$. The functions $u$, $\chi$ and $f$ are represented by power expansions with optimisable coefficients. Different coefficients are used for terms involving different spins. Note that, even if the determinant part of the Slater-Jastrow wave function is an eigenfunction of the spin operator $\hat{S}^2$, the use of different coefficients for parallel-spin and antiparallel-spin pairs of electrons generally leads to a trial wave function that is not an eigenfunction of $\hat{S}^2$. When using periodic boundary conditions, we often add a plane-wave term in the electron-electron separations, $p({\bf r}_{ij})$, which describes similar sorts of correlation to the $u$ term. The $u(r_{ij})$ term, however, is cut off at a distance less than or equal to the Wigner-Seitz radius of the simulation cell, and the $p$ term adds variational freedom in the corners of the simulation cell. Occasionally we add a plane-wave expansion in electron position, $q({\bf r}_{i})$, and also occasionally add three-body electron-electron-electron terms. ![Local energy of a silane (SiH$_4$) molecule from a VMC calculation (a) using a Slater-determinant trial wave function and (b) including a Jastrow factor.[]{data-label="fig:1"}](vmc){width=".7\textwidth"} We have recently developed a more general form of Jastrow factor [@general_jastrow] which allows the inclusion of higher order terms than those of equation (\[eq:basic\_J\]), such as terms involving the distances between four or more particles. An example of the application of such a Jastrow factor to the H${}_2$ molecule is shown in figure \[fig:h2jastrow\]. The molecular orbital was calculated within Hartree-Fock theory and VMC calculations were performed including Jastrow factors of increasing complexity. The Jastrow factor of equation (\[eq:basic\_J\]) includes electron-nucleus (e-N *etc.*), e-e and e-e-N terms, but the additional reductions in energy from including the e-N-N and e-e-N-N terms are clearly visible in figure \[fig:h2jastrow\]. ![The difference between the VMC energy and the exact ground state energy against the variance of the VMC local energies on logarithmic scales for H${}_2$ at a bond length of 1.397453 a.u. obtained using Jastrow factors of increasing complexity. “HF” indicates a wave function consisting of a molecular orbital obtained from a Hartree-Fock calculation and “e-e-N” denotes a term in the Jastrow factor involving the three distances between two electrons and one proton, *etc*. []{data-label="fig:h2jastrow"}](h2jastrow){width=".7\textwidth"} Pairing wave functions {#subsec:Pairing wave functions} ---------------------- Slater-Jastrow wave functions are not appropriate for all systems. For example, the strongly attractive interaction between electrons and holes within an effective-mass theory leads to the formation of excitons, which are not well described by a Slater-Jastrow wave function. A more appropriate wave function [@depalo_2002] is formed from the antisymmetrised product of identical electron-hole pairing functions $\psi$, multiplied by a Jastrow factor, $$\label{eq:singlet-pairing} \Psi_{\rm SP}({\bf R}) = e^{J({\bf R})} \det{\left[\psi({\bf r}_i^{\uparrow},{\bf r}_j^{\downarrow})\right]} \;.$$ It is also possible to include additional orbitals for unpaired particles within this wave function. Multi-determinant wave functions {#subsec:Multi-determinant wave functions} -------------------------------- Multi-determinant expansions have been used with considerable success over many decades within the quantum chemistry community. The trial wave function can be written as $$\label{eq:multi-slater-jastrow} \Psi_{\rm MD}({\bf R})\! \! = e^{J({\bf R})} \sum_n c_n \det{\left[ \psi_n({\bf r}_i^{\uparrow})\right]} \det{\left[ \psi_n({\bf r}_j^{\downarrow})\right]} \;,$$ where the $c_n$ are coefficients. This method provides a systematic approach to improving the trial wave function, and there have been numerous applications of multi-determinant trial wave functions in QMC calculations for small molecules [@filippi_1996; @schautz_2004; @harkless_2006]. Such trial wave functions can capture near-degeneracy effects (also known as *static correlation*). Multi-determinant wave functions are not in general suitable for large systems because the number of determinants required to retrieve a given fraction of the correlation energy increases exponentially with system size. An exception to this occurs if only a small region of the system requires a multi-determinant description. An example of a DMC calculation of this type is the study of the electronic states formed by the strongly interacting dangling bonds at a neutral vacancy in diamond by Hood *et al.* [@hood_2003]. Backflow wave functions {#subsec:Backflow wave functions} ----------------------- Additional correlation effects can be incorporated in the trial wave function using backflow transformations [@feynman_1954; @feynman_1956]. Consider a solid ball falling through a classical liquid. The incompressible liquid is pushed out of the way and it fills in behind the ball to form a characteristic flow pattern. One can imagine that similar correlations occur as a quantum particle moves through a quantum fluid, as shown in figure \[fig:bfplot\]. Much of this correlation can be captured in a Jastrow factor which, however, preserves the nodal surface of the wave function. The backflow motion gives an additional contribution which leaves its imprint on the nodes. Quantum backflow was discussed by Feynman and coworkers [@feynman_1954; @feynman_1956] for excitations in $^{4}$He and the effective mass of a $^{3}$He impurity in liquid $^{4}$He. Backflow wave functions have been used successfully in QMC studies of liquid He [@schmidt_1981; @holzmann_2006], the electron gas [@kwon_1993; @kwon_1998; @zong_2002], hydrogen systems [@delaney_2006], and various inhomogeneous systems [@lopez-rios_2006; @drummond_2006b; @brown_2007]. The backflow wave functions we use [@lopez-rios_2006] can be written as $$\label{eq:backflow} \Psi_{\rm BF}({\bf R}) = e^{J({\bf R})} \det{\left[\psi_i({\bf r}_i^{\uparrow}+{\bm \xi}_i({\bf R}))\right]} \det{\left[\psi_i({\bf r}_j^{\downarrow}+{\bm \xi}_j({\bf R}))\right]} \;.$$ For a system of $N$ electrons and $N_{\rm ion}$ classical ions we write the backflow displacement for electron $i$ in the form $$\label{eq:backflow displacement} {\bm \xi}_i = \sum_{j\neq i}^N \eta_{ij} {\bf r}_{ij} + \sum_I^{N_{\rm ion}} \mu_{iI} {\bf r}_{iI} + \sum_{j\neq i}^N \sum_I^{N_{\rm ion}} {\big (} \Phi_i^{jI} {\bf r}_{ij} + \Theta_i^{jI} {\bf r}_{iI} {\big )} \;.$$ In this expression $\eta_{ij}=\eta(r_{ij})$ is a function of electron-electron separation, $\mu_{iI}=\mu(r_{iI})$ is a function of electron-ion separation, and $\Phi_i^{jI}=\Phi(r_{iI},r_{jI},r_{ij})$ and $\Theta_i^{jI}=\Theta(r_{iI},r_{jI},r_{ij})$. We parameterise the functions $\eta$, $\mu$, $\Phi$ and $\Theta$ using power expansions with optimisable coefficients [@lopez-rios_2006]. ![Effect of the motion of an electron (black, with the arrow showing the direction of motion) on the backflow-transformed coordinates of three opposite-spin electrons (red, green and blue). Circles with the same colour intensity correspond to the same instant in the motion. []{data-label="fig:bfplot"}](bfplot){width=".7\textwidth"} Other wave functions {#subsec:Other wave functions} -------------------- The wave function types of equations (\[eq:slater-jastrow\]), (\[eq:singlet-pairing\]), (\[eq:multi-slater-jastrow\]), and (\[eq:backflow\]) can be combined in various ways within the <span style="font-variant:small-caps;">casino</span> code [@casino] so that, for example, it is possible to use Slater-Jastrow-pairing-backflow wave functions, *etc.* Of course the range of possible wave functions could be extended by, for example, including Pfaffian wave functions [@bajdich_2006; @bajdich_2008], *etc.* Optimisation of trial wave functions {#sec:optimise_psi_trial} ==================================== Optimising trial wave functions is a very important part of QMC calculations which can consume large amounts of human and computing resources. With modern stochastic methods it is possible to optimise hundreds or even thousands of parameters in the wave function. The parameters which can be optimised include those in the Jastrow factor, the coefficients of determinants in a multi-determinant wave function, the parameters in the backflow functions and the parameters in single-particle and pairing orbitals. The trial wave function used in a DMC calculation should ideally be optimised within DMC, but reliable and efficient methods to achieve this are still under development [@luchow_2007; @reboredo_2009]. Minimisation of the DMC energy has been performed “by hand” for small numbers of parameters [@drummond_2004; @drummond_2008_2d]. Wave function optimisation within <span style="font-variant:small-caps;">casino</span> is performed by minimising the VMC energy or its variance. Optimising wave functions by minimising the variance of the energy is an old idea dating back to the 1930s. The first application within Monte Carlo methods may have been by Conroy [@conroy_1964], but the method was popularised within QMC by the work of Umrigar and coworkers [@umrigar_1988]. It is now generally believed that it is better to minimise the VMC energy than its variance, but it has proved more difficult to develop robust and efficient algorithms for this purpose. Since the trial wave function forms used cannot generally represent energy eigenstates exactly, except in trivial cases, the minima in the energy and variance do not coincide. Energy minimisation should therefore produce lower VMC energies, and although it does not necessarily follow that it produces lower DMC energies, experience indicates that, more often than not, it does. Variance minimisation {#subsec:variance minimisation} --------------------- The variance of the VMC energy is $$\label{eq:sigma^2_vmc} \sigma^2({\bm \alpha}) = \frac{\int [\Psi_{\rm T}^{{\bm \alpha}}({\bf R})]^2 [E_{\rm L}^{{\bm \alpha}}({\bf R}) - E_{\rm V}^{{\bm \alpha}}]^2 \, d{\bf R}}{\int [\Psi_{\rm T}^{{\bm \alpha}}({\bf R})]^2 \, d{\bf R}} \;,$$ where ${{\bm \alpha}}$ denotes the set of variable parameters. The minimum possible value of $\sigma^2({{\bm \alpha}})$ is zero, which is obtained if and only if $\Psi_{\rm T}^{{\bm \alpha}}$ is an exact eigenstate of $\hat{H}$. In practice the trial wave function forms used are incapable of representing the exact eigenstates. Nevertheless, the minimum value of $\sigma^2({{\bm \alpha}})$ is still expected to correspond to a reasonable set of wave function parameters. Minimisation of $\sigma^2({{\bm \alpha}})$ is carried out via a correlated sampling approach in which a set of configurations distributed according to $[\Psi_{\rm T}^{{\bm \alpha}_0}]^2$ is generated, where ${{\bm \alpha}_0}$ is an initial set of parameter values [@dewing_2002]. $\sigma^2({{\bm \alpha}})$ is then evaluated as $$\label{eq:sigma^2_vmc correlated sampling} \sigma^2({{\bm \alpha}}) = \frac{\int [\Psi_{\rm T}^{{\bm \alpha}_0}]^2 \; w_{{\bm \alpha}_0}^{{\bm \alpha}} \; [E_{\rm L}^{{\bm \alpha}} - E_{\rm V}^{{\bm \alpha}}]^2 \, d{\bf R}}{\int [\Psi_{\rm T}^{{\bm \alpha}_0}]^2 \; w_{{\bm \alpha}_0}^{{\bm \alpha}} \, d{\bf R}} \;,$$ where the integrals contain weights, $w_{{\bm \alpha}_0}^{{\bm \alpha}}$, given by $$\label{eq:W} w_{{\bm \alpha}_0}^{{\bm \alpha}}({\bf R}) = \frac{[\Psi_{\rm T}^{{\bm \alpha}}]^2} {[\Psi_{\rm T}^{{\bm \alpha}_0}]^2} \;,$$ and $E_{\rm V}$ is evaluated using $$\label{eq:vmc energy correlated sampling} E_{\rm V} = \frac{ \int [\Psi_{\rm T}^{{\bm \alpha}_0}]^2 \; w_{{\bm \alpha}_0}^{{\bm \alpha}} \; E_{\rm L}^{{\bm \alpha}} \, d{\bf R} } {\int [\Psi_{\rm T}^{{\bm \alpha}_0}]^2 \; w_{{\bm \alpha}_0}^{{\bm \alpha}} \, d{\bf R}} \;.$$ After generating the initial set of configurations, the optimisation proceeds using standard techniques to locate the new parameter values which minimise $\sigma^2({{\bm \alpha}})$. With perfect sampling $\sigma^2({{\bm \alpha}})$ is independent of the initial parameter values ${{\bm \alpha}_0}$. For real (finite) sampling, however, one runs into problems because the values of $w_{{\bm \alpha}_0}^{{\bm \alpha}}$ for different configurations can vary by many orders of magnitude if ${{\bm \alpha}}$ and ${{\bm \alpha}_0}$ differ substantially. During the minimisation procedure a few configurations (often only one) acquire very large weights and the estimate of the variance is reduced almost to zero by a poor set of parameter values. This optimisation scheme is therefore often unstable, and in practice modified versions of it are used. The above scheme can be made much more stable by altering the weights $w_{{\bm \alpha}_0}^{{\bm \alpha}}$. A robust procedure is to set all the weights $w_{{\bm \alpha}_0}^{{\bm \alpha}}$ in equation (\[eq:sigma\^2\_vmc correlated sampling\]) to unity, which is reasonable because the minimum value of $\sigma^2({{\bm \alpha}}) = 0$ is still obtained only if $E_{\rm L}({\bf R})$ is a constant independent of ${\bf R}$, which holds only for eigenstates of the Hamiltonian. We call this the “unreweighted variance” minimisation method. The procedure is cycled until the parameters converge to their optimal values (within the statistical noise). For a number of model systems it was found that the trial wave functions generated by unreweighted variance minimisation iterated to self-consistency have a lower variational energy than wave functions optimised by reweighted variance minimisation [@drummond_2005_min]. If the Jastrow factor of equation (\[eq:basic\_J\]) can be written in the form $$\label{eq:linear Jastrow factor} J({\bf R}) = \sum_n \alpha_n f_n({\bf R}) \;,$$ then it is possible to simplify the calculation of the variance of the VMC energy [@moroni_1995b; @drummond_2005_min]. It can be shown that the unreweighted variance is a quartic function of the linear parameters $\alpha_n$ [@drummond_2005_min]. This has two advantages: ($i$) the unreweighted variance can be evaluated extremely rapidly at a cost which depends only on the number of parameters and is independent of the number of particles; and ($ii$) the unreweighted variance along a line in parameter space is a quartic polynomial. This is useful because it allows the exact global minimum of the unreweighted variance along the line to be computed analytically by solving the cubic equation obtained by setting the derivative equal to zero. The unreweighted variance minimisation method works well for optimising Jastrow factors, but it often performs poorly when parameters which alter the nodal surface of $\Psi_{\rm T}$ are optimised. The problem is that the local energy $E_{\rm L}$ generally diverges for a configuration on the nodal surface. As the parameter values are changed during a minimisation cycle the nodal surface can move through a configuration, resulting in a very large (positive or negative) value of $E_{\rm L}$, which adversely affects the optimisation. Such an effect would not occur when using the weights $w_{{\bm \alpha}_0}^{{\bm \alpha}}$ because they go to zero on the nodal surface. We have developed two schemes which solve this problem. In the first scheme we limit the weights by replacing them with ${\rm min}(w_{{\bm \alpha}_0}^{{\bm \alpha}},W)$, so that the weight goes to zero on the nodal surface but can never become larger than a chosen value $W$. In the second scheme we use a weight which goes smoothly to zero as $E_{\rm L}$ deviates from an estimate of the energy. Unreweighted variance minimisation belongs to a wider class of wave-function optimisation methods which are based on minimising a measure of the spread of the set of local energies. Another measure of spread that we have used with considerable success for wave-function optimisation is the mean absolute deviation of the local energies of a set of configurations from the median energy, $$\label{eq:madmin} {\cal M} = \frac{\int [\Psi_{\rm T}^{{\bm \alpha}_0}({\bf R})]^2 |E_{\rm L}^{{\bm \alpha}}({\bf R}) - E_{\rm m}^{{\bm \alpha}}| \, d{\bf R}}{\int [\Psi_{\rm T}^{{\bm \alpha}_0}({\bf R})]^2 \, d{\bf R}} \;.$$ In this expression, $E_{\rm m}^{{\bm \alpha}}$ is the median value of the local energies evaluated with the parameter values ${{\bm \alpha}}$. This is useful for optimising parameters that affect the nodal surface, because outlying local energies are less significant. Energy minimisation {#subsec:energy minimisation} ------------------- A well-known method for finding approximations to the eigenstates of a Hamiltonian is to express the wave function as a linear combination of basis states $g_i$, $$\label{eq:linear parameters} \Psi_{\rm T}({\bf R}) = \sum_{i=1}^p \beta_i \, g_i({\bf R}) \;,$$ calculate the matrix elements $H_{ij} = \langle g_i| \hat{H}| g_j \rangle$ and $S_{ij} = \langle g_i| g_j \rangle$, and solve the two-sided eigenproblem $\sum_j H_{ij}\beta_j = E \sum_j S_{ij}\beta_j$ by standard diagonalisation techniques. One can also do this in QMC [@riley_2003], although the statistical noise in the matrix elements leads to slow convergence with respect to the number of configurations used to evaluate the integrals. Nightingale and Melik-Alaverdian [@nightingale_emin] reformulated the diagonalisation procedure as a least-squares fit rather than integral evaluation, which leads to much faster convergence with the number of configurations. Let us assume that the set $\{g_i\}$ spans an invariant subspace of $\hat{H}$, which means that the result of acting $\hat{H}$ on any member of the set $\{g_i\}$ can be expressed as a linear combination of the $\{g_i\}$, *i.e.*, $$\label{eq:invariant subspace} \hat{H} g_i({\bf R}) = \sum_{i=1}^p {\cal{E}}_{ij} g_j({\bf R}) \;\;\;\;\; \forall \; i \;.$$ The eigenstates and associated eigenvalues of $\hat{H}$ can then be obtained by diagonalising the matrix ${\cal{E}}_{ij}$. Within a Monte Carlo approach we could evaluate the $g_i({\bf R})$ and $\hat{H} g_i({\bf R})$ for $p$ uncorrelated configurations generated by a VMC calculation and solve the resulting set of linear equations for the ${\cal{E}}_{ij}$. For problems of interest, however, the assumption that the set $\{g_i\}$ span an invariant subspace of $\hat{H}$ does not hold and there exists no set of ${\cal{E}}_{ij}$ which solves equation (\[eq:invariant subspace\]). If we took $p$ configurations and solved the set of $p$ linear equations, the values of ${\cal{E}}_{ij}$ would depend on which configurations had been chosen. To overcome this problem, a number of configurations $M \gg p$ is sampled to obtain an overdetermined set of equations which can be solved in a least-squares sense using singular value decomposition. In fact Nightingale and Melik-Alaverdian recommended that equation (\[eq:invariant subspace\]) be divided by $\Psi_{\rm T}({\bf R})$ so that in the limit of perfect sampling the scheme corresponds precisely to standard diagonalisation. The method of Nightingale and Melik-Alaverdian works very well for linear variational parameters as in equation (\[eq:linear parameters\]). The natural generalisation to parameters which appear non-linearly in $\Psi_{\rm T}$ is to consider the basis of the initial trial wave function ($g_0 = \Psi_{\rm T}$) and its derivatives with respect to the variable parameters, $$\label{eq:non-linear parameters} g_i = \left. \frac{\partial \Psi_{\rm T}}{\partial \beta_i} \right|_{\beta_i^0} \;.$$ In its simplest form this algorithm turns out to be highly unstable because the first-order approximation in equation (\[eq:non-linear parameters\]) is often inadequate. Umrigar and coworkers [@umrigar_emin; @toulouse_emin] showed how this method can be stabilised. The details of the stabilisation procedures are quite involved and we refer the reader to the original papers [@umrigar_emin; @toulouse_emin] for the details. The stabilised algorithm works well and is quite robust. The VMC energies given by this method are usually lower than those obtained from any of the variance-based algorithms described in section \[subsec:variance minimisation\], although the difference is often small. QMC calculations within periodic boundary conditions {#sec:pbc} ==================================================== QMC calculations for extended systems may be performed using cluster models or periodic boundary conditions, just as in other techniques. Periodic boundary conditions are preferred because they give smaller finite size effects. One can also use the standard supercell approach for systems which lack three-dimensional periodicity where a cell containing, for example, a point defect and a small part of the host crystal, are repeated periodically throughout space. Just as in other electronic structure methods, one must ensure that the supercell is large enough for the interactions between defects in different supercells to be small. When using standard single-particle-like theories within periodic boundary conditions such as density functional theory, the charge density and potentials are taken to have the periodicity of a chosen unit cell or supercell. The single particle orbitals can then be chosen to obey Bloch’s theorem and the results for the infinite system are obtained by summing quantities obtained from the different Bloch wave vectors within the first Brillouin zone. This procedure can also be applied within HF calculations, although the Coulomb interaction couples the Bloch wave vectors in pairs. The situation with the many-particle wave functions described in section \[sec:psi\_trial\] is somewhat different. Although the many-particle wave function satisfies Bloch theorems [@kpoints_1; @kpoints_2], it is not possible to perform a many-particle calculation using a set of ${\bf k}$-points; one has to perform it at a single ${\bf k}$-point. A single ${\bf k}$-point normally gives a poor representation of the infinite-system result, so that one either chooses a larger non-primitive simulation cell, or averages over the results of QMC calculations at a set of different ${\bf k}$-points [@lin_2001], or both. Many-body techniques such as QMC also suffer from finite size errors arising from long-ranged interactions, most notably the Coulomb interaction. Coulomb interactions are normally included within periodic boundary conditions calculations using the Ewald interaction. Long-ranged interactions induce long-ranged exchange-correlation interactions, and if the simulation cell is not large enough these effects are described incorrectly. Such effects are absent in local DFT calculations because the interaction energy is written in terms of the electronic charge density, but HF calculations show very strong effects of this kind and various ways to accelerate the convergence have been developed. The finite size effects arising from the long-ranged interaction can be divided into potential and kinetic energy contributions [@fin_chiesa; @ndd_fin]. The potential energy component can be removed from the calculations by replacing the Ewald interaction by the so-called model periodic Coulomb (MPC) interaction [@fraser_1996; @finsize; @finlong]. Recent work has added substantially to our understanding of finite size effects, and theoretical expressions have been derived for them [@fin_chiesa; @ndd_fin], but at the moment it seems that they cannot entirely replace extrapolation procedures. Kwee *et al.* [@kwee_2008] have developed an alternative approach for estimating finite size errors in QMC calculations. DMC results for the three-dimensional HEG are used to obtain a system-size-dependent local density approximation (LDA) functional. The correction to the total energy is given by the difference between the DFT energies for the finite-sized and infinite systems. This approach appears promising, although it does rely on the LDA giving a reasonable description of the system. Pseudopotentials in QMC calculations {#sec:pseudopots} ==================================== The computational cost of a DMC calculation increases with the atomic number $Z$ of the atoms as roughly $Z^{5.5}$ [@ceperley_1986; @ma_2005] which makes calculations with $Z>10$ extremely expensive. This problem can be solved by using pseudopotentials to represent the effect of the atomic core on the valence electrons. The use of non-local pseudopotentials within VMC is quite straightforward [@fahy_prl; @fahy_prb], but DMC poses an additional problem because the use of a non-local potential is incompatible with the fixed-node boundary condition. To circumvent this difficulty an additional approximation is made. In the “locality approximation” [@mitas_1991] the non-local part of the pseudopotential $\hat{V}_{\rm nl}$ is taken to act on the trial wave function rather than the DMC wave function, *i.e.*, $\hat{V}_{\rm nl}$ is replaced by $\Psi_{\rm T}^{-1} \hat{V}_{\rm nl} \Psi_{\rm T}$. The leading-order error term in the locality approximation is proportional to $(\Psi_{\rm T} - \phi_0)^2$ [@mitas_1991], where $\phi_0$ is the exact fixed-node ground state wave function, although it can be of either sign, so that the variational property of the algorithm is lost. Casula *et al.* [@casula_2005; @casula_2006] have introduced a fully variational “semi-localisation” scheme for dealing with non-local pseudopotentials within DMC, which also shows superior numerical stability to the locality approximation. Currently it is not possible to generate pseudopotentials entirely within a QMC framework, and therefore they are obtained from other sources. There is evidence that HF theory provides better pseudopotentials than DFT for use within QMC calculations [@greeff_1998], and we have developed smooth relativistic HF pseudopotentials for H to Ba and Lu to Hg, which are suitable for use in QMC calculations [@trail_2005_1; @trail_2005_2; @casino_page]. Another set of pseudopotentials for use in QMC calculations has been developed by Burkatzki *et al.* [@burkatzki07]. In the few cases where reliable tests have been performed [@trail_2008; @santra_2008], the pseudopotentials of references [@trail_2005_1; @trail_2005_2; @casino_page] and those of [@burkatzki07] have produced almost identical results, although those of references [@trail_2005_1; @trail_2005_2; @casino_page] are a little more efficient as they have smaller core radii. DMC calculations for excited states {#sec:excited_states} =================================== The fixed-node DMC algorithm is useful for studying excited states because it gives the exact excited-state energy if the nodal surface of the trial wave function matches that of the exact excited state and it gives an approximation to the excited-state energy if a trial wave function with an approximate nodal surface is used. This can be proved as follows. The local energy calculated with the exact excited-state wave function is equal to the exact excited-state energy throughout configuration space, and, by definition, the wave function is zero at the nodal surface and nowhere else. Hence within each nodal pocket the exact excited-state wave function is the ground-state solution of the Schrödinger equation subject to the boundary condition of being zero on the pocket boundary. Therefore the ground-state pocket eigenvalues are all equal to the exact excited-state energy, and the fixed-node DMC algorithm indeed gives the exact excited-state energy. An important difference from the ground state case is that the existence of a variational principle for excited state energies cannot in general be guaranteed, and it depends on the symmetry of the trial wave function [@foulkes_1999]. In practice DMC works quite well for excited states [@williamson_1998; @towler_2000; @porter_2001a; @porter_2001b; @williamson_2002; @drummond_2005_dia; @bande_2006]. Ceperley and Bernu [@ceperley_1988] have devised a method which combines DMC and the variational principle to calculate the eigenvalues of several different excited states simultaneously. However, this method suffers from stability problems in large systems. Scaling of computational effort with system size {#sec:scaling} ================================================ Over the accessible range of system sizes, the computational cost of a single configuration move in a VMC or DMC calculation is usually determined by the time taken to evaluate each of the ${\cal O}(N)$ orbitals in the Slater part of the wave function at each of the $N$ electron positions [@foulkes_2001]. If the delocalised orbitals are expanded in localised basis functions then the time taken to move a configuration scales as ${\cal O}(N^2)$. However, the number of configuration moves required to achieve a given error bar on the total energy grows as ${\cal O}(N)$, because the variance of the energy is proportional to the system size. Hence the time taken to evaluate the total energy to within a given statistical error bar scales as ${\cal O}(N^3)$. (Note that the time taken to evaluate the Slater determinants during the run scales as ${\cal O}(N^4)$, but with a small prefactor. In fact, for the DMC method the scaling with system size is ultimately exponential due to correlations within the configuration population [@note_exp_scaling].) The scaling of the QMC methods can be improved by using localised orbitals, so that the number of nonzero orbitals to be evaluated at each electron position is independent of the system size [@williamson_2001; @alfe_2004]. In this case the CPU time required to achieve a given error bar on the total energy scales as ${\cal O}(N^2)$ over the relevant range of system sizes. To maximise the localisation of the orbitals, the orthogonality constraint can be dropped, for it is irrelevant in QMC. However, it is not possible to “cheat” on the size of the orbital localisation regions in QMC, because this would compromise the high accuracy of the method. (The use of localised orbitals enables the use of sparse linear algebra to compute the Slater determinants, improving the scaling of this part of the algorithm by a factor of $N$ as well.) In calculations of the energy per particle of a periodic crystal the number of moves required to achieve a given error per particle falls off as ${\cal O}(N^{-1})$. Hence the CPU time required to achieve a given error bar on the energy per particle increases as ${\cal O}(N)$ in the standard algorithm and is roughly independent of the system size when localised orbitals are used. Sources of error and statistical analysis {#subsec:errors} ========================================= Sources of error in DMC calculations ------------------------------------ The potential sources of errors in DMC calculations may be summarised as follows. - Statistical errors. The standard error in the mean is proportional to $1/\sqrt{M}$, where $M$ is the number of particles moves. It therefore costs a factor of 100 in computer time to reduce the statistical error bars by a factor of 10. On the other hand, a random error is much better than a systematic one as its size can normally be reliably estimated. - Fixed-node error. This is the central approximation of the DMC technique, and is normally the limiting factor in the accuracy of the results. - Time-step bias. The short time approximation leads to a bias in the $f$ distribution and hence in expectation values. This bias is often significant and can be of either sign, but it can be largely removed by performing calculations for different time steps and extrapolating to zero time step or by simply choosing a small enough time step. An example of time-step extrapolation is shown in figure \[fig:time\_step\_errors\]. - Population control bias. The $f$ distribution is represented by a finite population of configurations which fluctuates due to branching. The population may be controlled in various ways, but this introduces a population control bias which is positive and falls off as the reciprocal of the population. In practice the population control bias is normally so small that it is difficult to detect [@umrigar_1993; @drummond_2004]. - Finite size errors within periodic boundary conditions calculations. It is important to correct for finite size effects carefully, as mentioned in section \[sec:pbc\]. - The pseudopotential approximation inevitably introduces errors. In DMC there is an additional error arising from the localisation [@mitas_1991] or semi-localisation [@casula_2006] of the non-local pseudopotential operator. The localisation error appears to be quite small in the cases for which it has been tested [@drummond_2006b]. ![DMC energy against time step for a 64-electron ferromagnetic 2D hexagonal Wigner crystal at density parameter $r_s=50$ a.u. with a Slater-Jastrow wave function. The solid line is a linear fit to the data. []{data-label="fig:time_step_errors"}](ferro_rs50_N064_exp0002_E_v_dt){width=".7\textwidth"} Practical methods for handling statistical errors in QMC results {#subsec:statistical errors} ---------------------------------------------------------------- Two main practical problems are encountered when dealing with errors in the QMC data: the data are serially correlated and the underlying probability distributions are non-Gaussian. The probability distribution of the local energies has $|E-E_0|^{-4}$ tails, where $E_0$ is a constant. These tails arise from singularities in the local energy such as the divergence at the nodal surface [@trail_2005_1; @trail_2005_2], as shown in figure \[fig:local\_energy\]. In consequence, although the mean energy and its variance are well defined, the variance of the variance is infinity. For other quantities the problem may be even more severe; for example, the probability distributions for the Pulay terms in the forces described in section \[subsec:forces\] decay as $|F-F_0|^{-5/2}$, so that the variance of the force is infinity [@badinski_2009]. Reasonably robust estimates of the errors can still be made, although it has to be accepted that they are not as well founded as for Gaussian statistics. ![Variation in the local energy $E_{\rm L}$ of a silane (SiH$_4$) molecule as an electron moves through the nodal surface at $x=0$. The local energy diverges as $1/x$.[]{data-label="fig:local_energy"}](nodal_divergence){width=".7\textwidth"} The data produced by VMC and DMC calculations are correlated from one step to the next. The problem is very important in DMC because short time steps are used to reduce the effect of the approximation in the Green’s function. The simulation effectively produces only one independent data point per correlation time, so that the estimate of the statistical error obtained on the assumption that the data points are independent is too small. We use the “blocking method” to obtain an estimate of the error. In this approach adjacent data points are averaged to form block averages [@Flyvbjerg_1989]. This procedure is carried out recursively so that after each blocking transformation the number of data points is reduced by one half. An example of blocking is shown in figure \[fig:reblock\]. The computed value of the standard error $\Delta_k$ increases with the number of blocking transformations $k$ until a limiting value is reached when the block length starts to exceed the correlation time. The standard error in the mean is estimated by the value of $\Delta$ on the plateau. Because the sizes of the error bars on QMC expectation values are themselves approximate estimates, apparent outliers in QMC data can be more common than one might expect on the basis of Gaussian statistics. ![Blocking analysis of data for an (all-electron) lithium atom. The blocking analysis indicates that the true standard error in the mean is about $\Delta = 2.6 \times 10^{-5}$ a.u., which is reached at about blocking transformation $k=10$, while the raw value is $\Delta_0 = 7.0 \times 10^{-6}$ a.u.[]{data-label="fig:reblock"}](reblock){width=".7\textwidth"} Evaluating other expectation values {#sec:other expectation values} =================================== As mentioned in section \[sec:introduction\], VMC and DMC can be used to calculate expectation values of many time-independent operators, not just the Hamiltonian. Typical quantities of interest are particle densities, pair correlation functions and one- and two-body density matrices, all of which can be evaluated using the <span style="font-variant:small-caps;">casino</span> code. It is not possible to obtain unbiased expectation values directly from the DMC distribution, $f({\bf R})$, for operators which do not commute with the Hamiltonian (which includes all of the quantities mentioned in the previous sentence). Unbiased (within the fixed-node approximation) estimates can be obtained as pure expectation values, $$\begin{aligned} \label{eq:pure expectation value} \langle \hat{A} \rangle & = & \frac{\int \phi_0({\bf R}) \hat{A} \phi_0({\bf R}) \, d{\bf R}}{\int \phi_0^2({\bf R}) \, d{\bf R}} \;.\end{aligned}$$ Pure expectation values can be obtained using a variety of methods: the approximate (but often very accurate) extrapolation technique [@ceperley_1986b], the future walking technique [@liu_1974; @barnett_1991], which is formally exact but statistically poorly behaved, and the reptation QMC technique of Baroni and Moroni [@baroni_1999], which is formally exact and well behaved, but quite expensive. The extrapolation technique can be used for any operator, but the future walking and reptation techniques are limited to spatially local multiplicative operators. Here we shall illustrate the use of the extrapolation technique [@ceperley_1986b] to calculate the charge density of a Wigner crystal. The pure estimate of the charge density $\rho$ is approximated as $$\begin{aligned} \label{eq:extrapolation} \rho_{\rm ext} \simeq 2\rho_{\rm DMC}-\rho_{\rm VMC}.\end{aligned}$$ The errors in both the VMC and DMC charge densities $\rho_{\rm VMC}$ and $\rho_{\rm DMC}$ are linear in the error in the trial wave function, but the error in the extrapolated estimate $\rho_{\rm ext}$ is quadratic in the error in the wave function. ![Charge density of a triangular antiferromagnetic Wigner crystal at density parameter $r_s=30$ a.u., plotted along a line between a pair of nearest-neighbour lattice sites. Two different wave functions are used: wave function 1 was optimised by variance minimisation, while wave function 2 was optimised by energy minimisation. The inset shows the extrapolation with wave function 1 at the minimum in greater detail.[]{data-label="fig:crys_cden"}](wigner_cden_rs30){width=".7\textwidth"} At low densities the HEG freezes into a Wigner crystal to minimise the electrostatic repulsion between electrons. The charge density of a 2D Wigner crystal [@drummond_2008_2d; @drummond_2009_2d] close to the crystallisation density is shown in figure \[fig:crys\_cden\]. VMC, DMC and extrapolated results are shown for two different trial wave functions. It can be seen that the dependence of the extrapolated estimate on the trial wave function is much smaller than for the raw VMC and DMC estimates, so we may have more confidence in the extrapolated estimate of the charge density. Energy differences and energy derivatives {#sec:energy differences and energy derivatives} ========================================= In electronic structure theory one is almost always interested in the differences in energy between systems. All electronic structure methods for complex systems rely for their accuracy on the cancellation of errors in energy differences. In DMC this helps with all the sources of error mentioned in section \[subsec:errors\] except the statistical errors. Fixed-node errors tend to cancel because the DMC energy is an upper bound, but even though DMC often retrieves 95% or more of the correlation energy, non-cancellation of nodal errors is the most important source of error in DMC results. Energy differences in QMC {#subsec:energy differences} ------------------------- Correlated sampling methods allow the computation of the energy difference between two similar systems with a smaller statistical error than those obtained for the individual energies [@dewing_2002]. Correlated sampling is relatively straightforward in VMC, and a version of it is described in section \[subsec:variance minimisation\] in the context of optimising wave functions by variance minimisation. Energy derivatives (forces) in QMC {#subsec:forces} ---------------------------------- Atomic forces are useful for relaxing the structures of molecules and solids, calculating their vibrational properties, and for performing molecular dynamics (MD) simulations. It has proved difficult to develop accurate and efficient methods for calculating atomic forces within QMC, although considerable progress has been made in recent years. Difficulties have arisen in obtaining accurate expressions for DMC forces which can readily be evaluated and in the statistical properties of the expressions, which are not as advantageous as those for the energy. According to the Hellmann-Feynman theorem (HFT), the derivative of the energy with respect to a parameter $\lambda$ in the Hamiltonian is $$\begin{aligned} \label{eq:HFT force} E^{\prime} & = & \frac{\int \Psi \, \hat{H}^{\prime} \, \Psi \, d{\bf R}}{\int \Psi \, \Psi \, d{\bf R}} \;,\end{aligned}$$ where the primes denote derivatives with respect to $\lambda$. This expression is valid when $\Psi$ is an exact eigenstate of $\hat{H}$. Unfortunately the HFT is not normally applicable within QMC because the wave functions are approximate. Exact expressions for the VMC and DMC forces must therefore contain additional Pulay terms which depend on $\Psi_{\rm T}^{\prime}$. To define the force properly it is therefore necessary to define and evaluate $\Psi_{\rm T}^{\prime}$. The DMC algorithm solves for the ground state of the fixed-node Hamiltonian exactly and therefore the HFT holds. Unfortunately the fixed-node Hamiltonian is different from the physical Hamiltonian because it contains an additional infinite potential barrier on the nodal surface of $\Psi_{\rm T}$ which forces the DMC wave function $\phi_0$ to go to zero. As $\lambda$ varies, the nodal surface, and hence the infinite potential barrier, moves, giving a contribution to $\hat{H}^{\prime}$ [@huang_2000; @schautz_2000; @badinski_2008a] which depends on $\Psi_{\rm T}$ and $\Psi_{\rm T}^{\prime}$ and is classified as a Pulay term. The Pulay terms arising from the derivative of the mixed estimate of the energy of equation (\[eq:diffusion\_energy\]) contain $\phi_0^{\prime}$, the derivative of the DMC wave function. This quantity cannot readily be evaluated, and the approximation $$\begin{aligned} \label{eq:reynolds} \frac{\phi_0^{\prime}}{\phi_0} & \simeq & \frac{\Psi_{\rm T}^{\prime}}{\Psi_{\rm T}}\end{aligned}$$ has normally been used [@reynolds_1986b; @assaraf_1999; @casalegno_2003; @assaraf_2003; @lee_2005; @badinski_2007; @badinski_2008a; @badinski_2008b; @badinski_2008c]. However, it leads to errors of first order in $(\Psi_{\rm T}-\phi_0)$ and $(\Psi_{\rm T}^{\prime}-\phi_0^{\prime})$; therefore its accuracy depends sensitively on the quality of $\Psi_{\rm T}$ and $\Psi_{\rm T}^{\prime}$, and in practice this approximation is often inadequate. The pure DMC energy, $$\begin{aligned} \label{eq:pure dmc energy} E_{\rm D} = \frac{\int \phi_0 \hat{H} \phi_0 \, d{\bf R}}{\int \phi_0 \phi_0 \, d{\bf R}} \;,\end{aligned}$$ is equal to the mixed DMC energy. Forces may also be calculated within pure DMC, and although this is more expensive it brings significant advantages. The derivative $E_{\rm D}^{\prime}$ contains the derivative of the DMC wave function, $\phi_0^{\prime}$. However, Badinski *et al.* [@badinski_2008a] showed that $\phi_0^{\prime}$ can be eliminated from the pure DMC expression, giving the exact result $$\begin{aligned} \label{eq:derivative of pure dmc energy 1} E_{\rm D}^{\prime} & = & \frac{\int \phi_0 \phi_0 \, \phi_0^{-1} \hat{H}^{\prime} \phi_0 \, d{\bf R}}{\int \phi_0 \phi_0 \, d{\bf R}} - \frac{1}{2} \frac{\int \phi_0 \phi_0 \, \Psi_{\rm T}^{-2} |\nabla_{\bf R} \Psi_{\rm T}| \Psi_{\rm T}^{\prime} \, d{\bf S}}{\int \phi_0 \phi_0 \, d{\bf R}} \;,\end{aligned}$$ where $d{\bf S}$ denotes an element of the nodal surface. Unfortunately it is not straightforward to evaluate integrals over the nodal surface. The nodal surface integral can be converted into a volume integral in which $\phi_0^{\prime}$ does not appear using an approximation with an error of order $(\Psi_{\rm T}-\phi_0)^2$, giving $$\begin{aligned} \label{eq:derivative of pure dmc energy 2} E_{\rm D}^{\prime} & = & \frac{\int \phi_0 \phi_0 \, \left[\phi_0^{-1} \hat{H}^{\prime} \phi_0 + \Psi_{\rm T}^{-1} \left(\hat{H}-E_{\rm D}\right) \Psi_{\rm T}^{\prime} \right] \, d{\bf R}}{\int \phi_0 \phi_0 \, d{\bf R}} + \\ && \frac{\int \Psi_{\rm T} \Psi_{\rm T} \, \left(E_{\rm L}-E_{\rm D}\right) \Psi_{\rm T}^{-1} \Psi_{\rm T}^{\prime} \, d{\bf R}}{\int \Psi_{\rm T} \Psi_{\rm T} \, d{\bf R}} + {{\cal{O}}[(\Psi_{\rm T}-\phi_0)^2}] \;.\end{aligned}$$ This expression is readily calculable if one generates configurations distributed according to the pure ($\phi_0^2$) and variational ($\Psi_{\rm T}^2$) distributions. The approximation is in the Pulay terms, which are smaller in pure than in mixed DMC and, in addition, the approximation in equation (\[eq:derivative of pure dmc energy 2\]) is second order compared with the first-order error in equation (\[eq:reynolds\]). Equation (\[eq:derivative of pure dmc energy 2\]) satisfies the zero variance condition; if $\Psi_{\rm T}$ and $\Psi_{\rm T}^{\prime}$ are exact the variance of the force obtained from equation (\[eq:derivative of pure dmc energy 2\]) is zero. Equation (\[eq:derivative of pure dmc energy 2\]) has been used to obtain very accurate forces in small molecules [@badinski_2008c; @badinski_2009]. The calculation of accurate DMC forces is still in its infancy, but it does appear that equation (\[eq:derivative of pure dmc energy 2\]) offers a very promising way forward. Conclusions {#sec:conclusions} =========== QMC methods provide a framework for computing the properties of correlated quantum systems to high accuracy within polynomial time [@note_exp_scaling], facilitating applications to large systems. They can be applied to fermions and bosons with arbitrary inter-particle potentials and external fields. These intrinsically parallel methods are ideal for utilising current and next-generation massively parallel computers. Their accuracy, generality and wide applicability suggest that they will play an important role in improving our understanding of the behaviour of large assemblies of quantum particles. It is believed [@troyer_2005] that a complete solution to the fermion sign problem may be impossible, and any exact fermion method may be exponentially slow on a classical computer. Accurate quantum chemistry techniques such as the “gold standard” coupled cluster with single and double excitations and perturbative triples \[CCSD(T)\] have been applied with considerable success to correlated electron problems but, although they are also polynomial time algorithms, their cost increases much more rapidly with system size than for QMC methods. DFT methods have proved extremely useful in describing correlated electron systems, but there are many examples where the accuracy of current density functionals has proved wanting. It is important to remember that trial wave functions for QMC calculations could be improved by developing new wave function forms and better optimisation methods, whereas improving approximate DFT methods requires the development of better density functionals, which seems likely to be a much harder problem. These considerations motivate the development of approximate QMC methods such as those described in this review. Although the basics of the DMC algorithm used by Ceperley and Alder in 1980 [@ceperley_1980] have remained unchanged, enormous progress has been made in using more complex trial wave functions and in optimising the many parameters in them. There is every reason to believe that the current high rate of progress will continue for many years to come. Although these QMC methods will remain approximate, it is clear that they can deliver highly accurate results provided the trial wave functions are accurate enough. Development of sophisticated computer packages [@qmc_wiki] such as the <span style="font-variant:small-caps;">casino</span> code [@casino; @casino_page] should help to promote these methods. Acknowledgements {#sec:acknowledgments} ================ We would like to thank all of our collaborators who have contributed so much to our QMC project. Much of this work has been supported by the Engineering and Physical Sciences Research Council (EPSRC) of the UK. NDD acknowledges support from the Leverhulme Trust and Jesus College, Cambridge, and MDT acknowledges support from the Royal Society. Computing resources were provided by the Cambridge High Performance Computing Service. References {#references .unnumbered} ========== [999]{} Foulkes W M C, Mitas L, Needs R J and Rajagopal G 2001 *Rev. Mod. Phys.* **73** 33 The scaling of DMC is expected to become exponential in very large systems because of increasing correlation in the data, see Nemec N 2009 *Diffusion Monte Carlo computational cost scales exponentially for large systems* arXiv:0906.0501. VMC does not suffer from this problem. 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