[{"text":"%\\documentclass[preprintnumbers,amsmath,amssymbm,preprint]{revtex4}\n%\\documentclass[twocolumn,preprintnumbers,amsmath,amssymbm,prl]{revtex4}\n%%\\documentclass[twocolumn,preprintnumbers,amsmath,amssymbm,prd]{revtex4}\n%\\documentclass[preprintnumbers,amsmath,amssymbm,prd]{revtex4}\n%\\documentclass[prl,floats,aps,twocolumn,epsf,graphicx]{revtex4}\n\\documentclass[article,preprint,groupedaddress]{revtex4}\n\\usepackage{epsfig}\n\\usepackage{graphicx}\n\\usepackage{amssymb}\n\n\\begin{document}\n\n\\title{Universal lower bound on orbital periods around central compact objects}\n%\\title{Quantum-gravity bound on orbital periods around central compact objects}\n\\author{Shahar Hod}\n\\address{The Ruppin Academic Center, Emeq Hefer 40250, Israel}\n\\address{}\n\\address{The Hadassah Institute, Jerusalem 91010, Israel}\n\\date{\\today}\n\n\\begin{abstract}\n\n\\ \\ \\ It is proved, using the curved line element of a spherically symmetric charged object in general relativity and \nthe Schwinger discharge mechanism of quantum field theory, that the orbital periods $T_{\\infty}$ \nof test particles around central compact objects as measured by flat-space asymptotic observers \nare fundamentally bounded from below. \nThe lower bound on orbital periods \nbecomes universal (independent of the mass $M$ of the central compact object) \nin the dimensionless $ME_{\\text{c}}\\gg1$ regime, \nin which case it can be expressed in terms of the electric charge $e$ and the proper mass $m_{e}$ \nof the lightest charged particle in nature: $T_{\\infty}>{{2\\pi e\\hbar}\\over{\\sqrt{G}c^2 m^2_{e}}}$ (here \n$E_{\\text{c}}=m^2_{e}\/e\\hbar$ is the critical electric field for pair production). \nThe explicit dependence of the bound on the fundamental constants of nature $\\{G,c,\\hbar\\}$ suggests \nthat it may reflect a fundamental physical property of the elusive quantum theory of gravity. \n\\end{abstract}\n\\bigskip\n\\maketitle\n\n\\section{Introduction}\n\nThe theory of quantum gravity is notorious for its elusiveness. \nIn particular, despite the fact that the physical laws of general relativity and quantum field theory are \nwell established, it is still very difficult to reveal fundamental physical principles that are expected to remain \nvalid within the framework of the yet unknown quantum theory of gravity.\nOne such principle is the holographic entropy-area bound, whose compact \nformula $S\/A\\leq {{k_{\\text{B}}c^3}\\over{4G\\hbar}}$ contains the fundamental \nconstants of gravity ($G$), relativity ($c$), and quantum theory ($\\hbar$) \\cite{Hol1,Hol2,Hol3}.\n\nThe main goal of the present compact paper is to reveal the existence of another fundamental physical \nbound (which, admittedly, is probably far less important than the holographic entropy-area bound) \nwhose formula contains the three basic constants of nature. \nIn particular, below we shall explicitly prove that, in curved spacetimes, the orbital periods \nof test particles around central compact objects are bounded from below by a fundamental limit which is \nexpressed in terms of the basic constants of nature: $G, c$, and $\\hbar$. \n\nClosed circular motions of test particles around central compact objects provide valuable information \non the non-trivial geometries of the corresponding curved spacetimes \n(see \\cite{Bar,Chan,Shap} and references therein). \nIn particular, an empirically measured quantity which is important for the analysis of closed \ncircular motions in curved spacetimes is the orbital period $T_{\\infty}$ around the central compact object \nas measured by far away asymptotic observers. \n\nUsing a naive flat-space argument, which ignores the intriguing time-dilation\/contraction \neffect of general relativity (below we shall analyze in detail the influence of this physically important \neffect on asymptotically measured orbital periods), \nit is quite easy to prove that the orbital period of a test particle around a (possibly charged) central \ncompact object of radius $R$ should be bounded from below. \nIn particular, since the radius of a spherically symmetric object of mass $M$ and electric charge $Q$ \nthat respects the weak (positive) energy condition \\cite{HawEl} is expected to be bounded from below by \nits Schwarzschild radius \\cite{Notehun} ($R\\gtrsim M$) and also by its classical charge \nradius ($R\\geq {{Q^2}\/{2M}}$) \\cite{Got}, \nthe orbital period $T\\geq 2\\pi R$ \\cite{Notecc} around the central compact object as measured by inertial observers \nis expected to be bounded from below by the simple functional relation \\cite{Hodfast}\n\\begin{equation}\\label{Eq1}\nT\\geq T^{\\text{min}}(M,Q)=2\\pi\\cdot{\\text{max}}\\{M,Q^2\/2M\\}\\ .\n\\end{equation}\n\nIntriguingly, however, it is well established \\cite{Vio1,Vio2,Vio3,Vio4,Vio5,For} that, \ndue to quantum coherence effects, local energy densities in {\\it quantum} field theory can be negative. \nLikewise, if the matter fields inside a compact object are characterized by \na {\\it non}-minimal coupling to gravity then negative energy densities \nare not excluded even at the classical level \\cite{BekMay}. \nThe appearance of regions with negative energy densities inside a compact object may allow its radius $R$ \nto be {\\it smaller} than its classical radius, thus opening the possibility for the existence of closed \ncircular trajectories around the compact object that violate the purely classical lower bound (\\ref{Eq1}).\n\nBased on this expectation, we here raise the following physically interesting question:\nIs there a {\\it fundamental} quantum-gravity lower bound on orbital periods of test particles around central \ncompact objects? \n\nThe main goal of the present compact paper is to reveal the existence of such a bound \non the orbital periods, as measured by asymptotic observers, around spherically symmetric central compact objects, \na bound which is valid even for compact objects that may violate the classical lower bound (\\ref{Eq1}). \nIn particular, below we shall explicitly prove that \nthe Schwinger pair-production mechanism \\cite{Sch1,Sch2,Sch3}, \na purely quantum effect, sets a lower bound on the orbital periods of test particles around \ncentral compact objects in the composed Einstein-Maxwell field theory. \n\n\\section{Lower bound on orbital periods around central compact objects}\n\n\\subsection{Circular trajectories around compact objects that violate the weak energy condition}\n\nIn the present section we shall determine the shortest possible orbital period, $T^{\\text{min}}_{\\infty}(M)$, \naround a central compact object of total gravitational mass $M$ in the composed Einstein-Maxwell field theory. \n\nThe external spacetime of a spherically symmetric charged compact object of radius $R$, \ntotal mass $M$ \\cite{NoteMh}, and electric charge $Q$ \\cite{NoteQ0} is characterized by the Reissner-Nordstr\\\"om curved \nline element \\cite{Chan}\n%\\begin{eqnarray}\\label{Eq2}\n%ds^2=-\\Big[1-{{2M(r)}\\over{r}}\\Big]dt^2+\\Big[1-{{2M(r)}\\over{r}}\\Big]^{-1}dr^2\n%+r^2d\\theta^2+r^2\\sin^2\\theta d\\phi^2\\ \\ \\ \\ \\ \\text{for}\\ \\ \\ \\ \\ r\\geq R\\ ,\n%\\end{eqnarray}\n\\begin{eqnarray}\\label{Eq2}\nds^2&=&-\\Big[1-{{2M(r)}\\over{r}}\\Big]dt^2+\\Big[1-{{2M(r)}\\over{r}}\\Big]^{-1}dr^2+\\nonumber \\\\ &&\nr^2d\\theta^2+r^2\\sin^2\\theta d\\phi^2\\ \\ \\ \\ \\ \\text{for}\\ \\ \\ \\ \\ r\\geq R\\ ,\n\\end{eqnarray}\nwhere\n\\begin{equation}\\label{Eq3}\nM(r)=M-{{Q^2}\\over{2r}}\\\n\\end{equation}\nis the gravitational mass contained within a sphere of radius $r$.\n%\\begin{eqnarray}\\label{Eq2}\n%ds^2=-\\Big(1-{{2M}\\over{r}}+{{Q^2}\\over{r^2}}\\Big)dt^2+\n%\\Big(1-{{2M}\\over{r}}+{{Q^2}\\over{r^2}}\\Big)^{-1}dr^2\n%+r^2d\\theta^2+r^2\\sin^2\\theta d\\phi^2\\ \\ \\ \\ \\ \\text{for}\\ \\ \\ \\ \\ r\\geq R\\ .\n%\\end{eqnarray}\n\nOur goal is to determine the {\\it shortest} possible \norbital period $T^{\\text{min}}_{\\infty}(M)$ as measured by asymptotic observers around the central compact object. \nWe shall therefore consider test particles that move arbitrarily close to the speed of light \\cite{Notenng}, \nin which case the asymptotically measured orbital periods \ncan be determined from the curved line element (\\ref{Eq2}) with the properties \\cite{Notethet}:\n\\begin{equation}\\label{Eq4}\nds=dr=d\\theta=0\\ \\ \\ \\ \\ \\text{and}\\ \\ \\ \\ \\ \\Delta\\phi=\\pm2\\pi\\ .\n\\end{equation}\nSubstituting the relations (\\ref{Eq4}) into Eq. (\\ref{Eq2}), one obtains the functional expression \n\\begin{equation}\\label{Eq5}\nT_{\\infty}(M,Q,r)={{2\\pi r}\\over{\\sqrt{1-{{2M}\\over{r}}+{{Q^2}\\over{r^2}}}}}\\\n\\end{equation}\nfor the orbital period \naround a central (possibly charged) compact object as measured by asymptotic observers.\n\nAs discussed above, classical charged compact objects that respect the weak (positive) \nenergy condition \\cite{HawEl} must be larger than their classical charge radius [see Eq. (\\ref{Eq3})] \\cite{Got}, \n\\begin{equation}\\label{Eq6}\nR\\geq R_{\\text{c}}={{Q^2}\\over{2M}}\\ ,\n\\end{equation}\nin which case one finds the dimensionless inequality \n$g_{tt}=1-{{2M}\/{r}}+{{Q^2}\/{r^2}}\\leq1$ for external circular trajectories (with radii $r\\geq R$) \naround the central compact object. \nSubstituting this relation into Eq. (\\ref{Eq5}), one obtains the \nsimple {\\it classical} lower bound \\cite{Noterm}\n\\begin{equation}\\label{Eq7}\nT_{\\infty}(M,Q,r)\\geq 2\\pi r\\geq 2\\pi R\\geq2\\pi\\cdot{\\text{max}}\\{Q^2\/2M,M\\}\\\n\\end{equation}\non orbital periods around central compact objects. \n\nHowever, as emphasized above, negative energy densities are not always excluded \nin physics. In particular, they may appear due to a non-minimal direct coupling of matter fields \nto gravity \\cite{BekMay} and also due to quantum coherence effects in quantum field theories \\cite{Vio1,Vio2,Vio3,Vio4,Vio5,For}. \nThe possible existence of spacetime regions with negative energy densities inside a compact object \nthat violates the (classical) weak energy condition may allow its radius $R$ to be smaller than \nits classical charge radius $R_{\\text{c}}$, thus opening the possibility for the existence \nof circular trajectories, whose radii lie in the regime \n\\begin{equation}\\label{Eq8}\nr\\in[R,R_{\\text{c}})\\ ,\n\\end{equation}\nthat violate the classical lower bound (\\ref{Eq7}) for {\\it two} reasons: \n\\newline\n(1) The numerator of (\\ref{Eq5}) is smaller than $2\\pi R_{\\text{c}}$ for \ncircular trajectories in the regime $R\\leq r1$ as measured by asymptotic \nobservers. This is a general relativistic time contraction effect. \n\nBefore proceeding, it is important to emphasize that, despite the fact that the \nlocal mass (\\ref{Eq3}) contained within a compact object which is smaller than \nits classical radius is negative, we shall assume \nthat the {\\it total} ADM mass $M$ of the spacetime as measured by asymptotic observers is positive. \n\n\\subsection{A fundamental lower bound on orbital periods around central compact objects}\n\nIt is of physical interest to explore the physical and mathematical properties of closed circular motions \naround central compact objects that, due to quantum coherence effects \\cite{Vio1,Vio2,Vio3,Vio4,Vio5,For} \nor non-minimal coupling to gravity \\cite{BekMay}, may violate the classical lower bound (\\ref{Eq7}). \nIn particular, one naturally wonders whether orbital periods around central compact objects \nthat violate the weak energy condition are \nfundamentally bounded from below by the physical laws of general relativity and quantum theory? \n\nIn the present section we shall reveal the physically intriguing fact that the \nSchwinger pair-production mechanism \\cite{Sch1,Sch2,Sch3} \nsets a fundamental quantum lower bound on the orbital period (\\ref{Eq5}) around central compact objects, \na bound which is valid even for compact objects that violate the weak energy condition and can therefore \nviolate the classical lower bound (\\ref{Eq7}).% \\cite{Notenff}. \n\nIn particular, we shall now prove that, as opposed to the unbounded \nredshift (time dilation) effect which characterizes the orbital periods of test particles that circle a central black hole close \nto its horizon [$T_{\\infty}(r\\to R_{\\text{horizon}})\\to\\infty$], \nthe blueshift time contraction effect, which characterizes the orbital \nperiods of test particles that circle a central compact object with negative energy densities \\cite{Vio1,Vio2,Vio3,Vio4,Vio5,For,BekMay}, is fundamentally (quantum mechanically) bounded \nfrom above. \n\nOur goal is to determine the shortest possible orbital period $T^{\\text{min}}_{\\infty}(M)$ \naround a central compact object of a given mass $M$. \nTo this end, we first point out that, for given values of the gravitational mass $M>0$ of \nthe central compact object and the radius $r$ of the external circular trajectory, \nthe orbital period $T_{\\infty}(Q;M,r)$ as measured by asymptotic \nobservers decreases monotonically with the electric charge $Q$ of the central compact object [see \nEq. (\\ref{Eq5})]. \nThus, in order to minimize the orbital period (\\ref{Eq5}) of a test particle \naround a central object with a given total mass $M$, \none should maximize its electric charge. \nIn particular,\n\\begin{equation}\\label{Eq9}\nT^{\\text{min}}_{\\infty}(M,Q_{\\text{max}},r)=\n{{2\\pi r}\\over{\\sqrt{1-{{2M}\\over{r}}+{{Q^2_{\\text{max}}}\\over{r^2}}}}}\\ ,\n\\end{equation}\nwhere $Q_{\\text{max}}=Q_{\\text{max}}(r)$ is the maximally allowed electric charge that can be \ncontained within a sphere of radius $r$.\n\nOne naturally wonders: What physics prevents us from making the expression (\\ref{Eq9}) for the orbital period \nas small as we wish? \nOr, in other words, we ask: \nIs there a fundamental physical mechanism that bounds the electric charge $Q_{\\text{max}}(r)$ \nthat can be contained within a sphere of radius $r$? The answer is `yes'! \n\nIn particular, the Schwinger pair-production mechanism (a {\\it quantum} polarization effect) implies \nthe existence of a fundamental upper bound on the\nelectric field strength of the charged compact object \\cite{Sch1,Sch2,Sch3,Noteoom}:\n\\begin{equation}\\label{Eq10}\n{{Q}\\over{r^2}}\\leq E_{\\text{c}}\\equiv {{m^2_e}\\over{e\\hbar}}\\ ,\n\\end{equation}\nwhere $\\{e,m_e\\}$ are respectively the electric charge and the proper mass \nof the lightest charged particle in nature. \nSubstituting the upper bound (\\ref{Eq10}) into Eq. (\\ref{Eq9}), one obtains the radius-dependent \nfunctional expression\n\\begin{equation}\\label{Eq11}\nT^{\\text{min}}_{\\infty}(r;M,E_{\\text{c}})={{2\\pi r}\\over{\\sqrt{1-{{2M}\\over{r}}+E^2_{\\text{c}}r^2}}}\\\n\\end{equation}\nfor the shortest possible orbital period.\n\nInterestingly, and most importantly for our analysis, one finds from Eq. (\\ref{Eq11}) that, \nfor a given total mass $M$ of the system, \nthe orbital period is {\\it minimized} for the field-independent ($E_{\\text{c}}$-{\\it independent}) orbital radius\n\\begin{equation}\\label{Eq12}\nr^{\\text{min}}=3M\\ .\n\\end{equation}\n\nSince the electric charge is confined to the interior of the central compact object, the electric field strength \nin the exterior ($r\\geq R$) spacetime region is a monotonically decreasing function of the orbital \nradius $r$. Thus, the assumption in Eq. (\\ref{Eq11}) that the electric field along the trajectory of the test particle saturates \nthe quantum upper bound (\\ref{Eq10}) [as discussed above, for a given value $r$ of the orbital radius, \nthe larger the electric field along the trajectory of the particle, the shorter is the orbital period as measured by asymptotic observers, see Eq. (\\ref{Eq11})] corresponds to \nthe assumption that the particle moves along a circular trajectory \ninfinitesimally close to the surface of the charged compact object: \n\\begin{equation}\\label{Eq13}\nr^{\\text{min}}\\to R^+\\ .\n\\end{equation}\n\nSubstituting (\\ref{Eq12}) into Eq. (\\ref{Eq11}), one obtains the remarkably simple functional relation\n\\begin{equation}\\label{Eq14}\nT^{\\text{min}}_{\\infty}(M,E_{\\text{c}})=2\\pi\\cdot\\sqrt{{27M^2}\\over{1+27M^2E^2_{\\text{c}}}}\\\n\\end{equation}\nfor the shortest possible orbital period around a central compact object of total \ngravitational mass $M$ as measured by asymptotic observers. \n\nIt is interesting to note that, in the dimensionless regime $ME_{\\text{c}}\\ll1$, \nthe analytically derived lower bound (\\ref{Eq14}) on orbital periods around central compact objects \nyields the {\\it classical} ($\\hbar$-independent) bound\n\\begin{equation}\\label{Eq15}\nT^{\\text{min}}_{\\infty}(M,E_{\\text{c}})\\ \\to\\ 6\\sqrt{3}\\pi\\cdot M\\ \\ \\ \\ \\ \\text{for}\\ \\ \\ \\ \\ ME_{\\text{c}}\\ll1\\ .\n\\end{equation}\nOn the other hand, in the opposite dimensionless regime $ME_{\\text{c}}\\gg1$ the lower bound (\\ref{Eq14}) \nyields the purely {\\it quantum} ($\\hbar$-dependent) bound \n\\begin{equation}\\label{Eq16}\nT^{\\text{min}}_{\\infty}(M,E_{\\text{c}})\\ \\to\\ {{2\\pi}\\cdot{E^{-1}_{\\text{c}}}}\n\\ \\ \\ \\ \\ \\text{for}\\ \\ \\ \\ \\ ME_{\\text{c}}\\gg1\\ .\n\\end{equation}\nIntriguingly, the lower bound (\\ref{Eq16}) is universal in the sense that it is {\\it independent} of the \nmass $M$ of the central compact object. \n\nIt is worth stressing the fact that, in the dimensionless $ME_{\\text{c}}\\gg1$ regime, \nthe value of $T^{\\text{min}}$, as given by the analytically derived quantum \nexpression (\\ref{Eq16}), satisfies the strong inequality $T^{\\text{min}}(M,E_{\\text{c}})={{2\\pi}\/{E_{\\text{c}}}}\\ll 2\\pi\\cdot{\\text{max}}\\{Q^2\/2M,M\\}$ \\cite{NoteQM}, and it therefore violates \nthe classical bound (\\ref{Eq7}) on orbital periods. \n\n\\section{Summary and Discussion}\n\nMotivated by the fact that negative energy densities may appear in various physical situations \n(for example, due to quantum coherence effects that appear in quantum \nfield theories \\cite{Vio1,Vio2,Vio3,Vio4,Vio5,For} and also due to a possible non-minimal coupling of matter fields \nto gravity \\cite{BekMay}), in the present compact paper \nwe have raised the following physically important question:\nIs there a {\\it fundamental} lower bound on the orbital periods of test particles \naround central compact objects? \n\nIn order to address this question, we have analyzed the three-dimensional \\cite{Notethr} functional behavior \nof the orbital periods $T_{\\infty}=T_{\\infty}(M,Q,r)$ of test particles whose velocities are arbitrarily close to the \nspeed of light on the physical parameters $\\{M,Q\\}$ \nthat characterize the central compact object and on the radii $r$ of the external circular trajectories. \n\nIn particular, the main goal of the present paper was to derive a robust lower bound on orbital periods of \ntest particles around spherically symmetric central compact objects, \na bound which is valid even for compact objects that violate the weak (positive) energy condition and thus \nmay violate the classical lower bound (\\ref{Eq6}) on the radii of charged compact objects and \nthe corresponding classical lower bound (\\ref{Eq7}) on orbital periods as \nmeasured by asymptotic observers. \n\nIntriguingly, we have revealed the fact that the Schwinger pair-production mechanism \\cite{Sch1,Sch2,Sch3}, \na purely quantum effect, is responsible for the existence of a previously unknown fundamental \nlower bound on the orbital periods of test particles around central compact objects. \n\nThe main analytical results derived in this paper and their physical implications are as follows:\n\n(1) We have emphasized the fact that external circular trajectories around (possibly charged) central compact objects \nthat respect the classical weak (positive) \nenergy condition are characterized by the relation $1-{{2M}\/{r}}+{{Q^2}\/{r^2}}<1$ [see the \nlower bound (\\ref{Eq6})], in which case the orbital periods as measured by far away asymptotic observers \nare longer than the corresponding locally measured orbital times (this \nis the familiar time dilation effect in general relativity). \n\nOn the other hand, we have pointed out that charged compact objects that violate the classical positive energy \ncondition may be characterized by the presence of external circular trajectories in the regime $r\\in[R,R_{\\text{c}})$ \nwith the property $1-{{2M}\/{r}}+{{Q^2}\/{r^2}}>1$, in which case the general relativistic time contraction effect implies \nthat the asymptotically measured orbital periods are {\\it shorter} than the corresponding locally \nmeasured orbital periods. \n\nInterestingly, we have explicitly proved that, as opposed to the unbounded time dilation (redshift) \neffect, $T_{\\infty}(r\\to R_{\\text{horizon}})\\to\\infty$, which characterizes the orbital periods of test particles \nin the near-horizon region of a central black hole, the time contraction (blueshift) effect, \nwhich characterizes the orbital periods of test particles around central compact objects with negative energy densities, \nis fundamentally (quantum mechanically) bounded from above according to the analytically derived functional \nrelation (\\ref{Eq14}). \n\n(2) Using the curved line element (\\ref{Eq2}) that characterizes the exterior spacetime region of a \nspherically symmetric charged compact object in general relativity and \nthe Schwinger discharge mechanism of quantum field theory, we have explicitly proved \nthat the orbital periods $T_{\\infty}(M)$ of \ntest particles around a central compact object of total \nmass $M$ as measured by asymptotic observers are \nfundamentally bounded from below by the functional relation [see Eqs. (\\ref{Eq10}) and (\\ref{Eq14})]\n\\begin{equation}\\label{Eq17}\nT_{\\infty}(M)\\geq T^{\\text{min}}_{\\infty}(M)=\n2\\pi\\cdot\\sqrt{{27M^2}\\over{1+27M^2\\cdot({{m^2_e}\/{e\\hbar}})^2}}\\ .\n\\end{equation}\nFor a central object of total mass $M$, the minimally allowed orbital period (\\ref{Eq17}) is \nobtained for the following physical parameters of the compact object and the \ncircular trajectory: $Q=9M^2E_{\\text{c}}$ and $r\\to R=3M$ \n[see Eqs. (\\ref{Eq10}), (\\ref{Eq12}), and (\\ref{Eq13})]. \n\nIt is worth emphasizing the fact that, as opposed to the classical lower bound (\\ref{Eq7}) on orbital periods, \nthe analytically derived lower bound (\\ref{Eq17}) is valid even in the quantum regime of \nmatter fields that may violate the weak energy condition. \nIn particular, circular trajectories around central compact objects whose radii violate the \nclassical lower bound (\\ref{Eq6}) are still characterized by the fundamental \nquantum lower bound (\\ref{Eq17}) on their orbital periods. \n\n(3) The lower bound (\\ref{Eq17}) becomes universal ({\\it independent} of the mass $M$ of the central compact object) \nin the dimensionless $ME_{\\text{c}}\\gg1$ regime, in which case it can be expressed in the remarkably compact form \n(we write here the explicit dependence of $T^{\\text{min}}_{\\infty}$ on \nthe fundamental constants of nature $\\{G,c,\\hbar\\}$):\n\\begin{equation}\\label{Eq18}\nT^{\\text{min}}_{\\infty}\\to {{2\\pi\\hbar\\sqrt{k}e}\\over{\\sqrt{G}c^2 m^2_{e}}}\n\\ \\ \\ \\ \\ \\text{for}\\ \\ \\ \\ \\ ME_{\\text{c}}\\gg1\\ ,\n\\end{equation}\nwhere $\\epsilon_0=1\/4\\pi k$ is the electric constant (vacuum permittivity). \n\nThe explicit dependence of the minimally allowed orbital time $T^{\\text{min}}_{\\infty}$ \non the fundamental constants of gravity ($G$), relativity ($c$), and quantum physics ($\\hbar$) \nsuggests that it may reflect a genuine physical property of a fundamental quantum theory of gravity. \n\n(4) Interestingly, inspection of Eq. (\\ref{Eq18}) reveals the fact that, \nin order for the smallest possible orbital period $T^{\\text{min}}_{\\infty}$ to be larger than the fundamental \nscale set by the Planck time $t_{\\text{P}}=(\\hbar G\/c^5)^{1\/2}$, \none must demand the existence of a {\\it weak-gravity} bound of the form \n\\begin{equation}\\label{Eq19}\n{{Gm^2_{e}}\\over{ke^2}}\\lesssim \\alpha^{-1\/2}\\ ,\n\\end{equation}\nwhere $\\alpha$ is the dimensionless fine-structure constant.\n\n\\\n\n%\\bigskip\n\\noindent\n{\\bf ACKNOWLEDGMENTS}\n%\\bigskip\n\nThis research is supported by the Carmel Science Foundation. I would\nlike to thank Yael Oren, Arbel M. Ongo, Ayelet B. Lata, and Alona B.\nTea for helpful discussions.\n\n\\begin{thebibliography}{99}\n\n\\bibitem{Hol1} G. 't Hooft, in {\\it Salam-festschrifft}, ed. A. Aly, J. Ellis, and S. Randjbar-Daemi \n(World Scientific, Singapore 1993) [arXiv:gr-qc\/9310026].\n\n\\bibitem{Hol2} L. Susskind, J. Math. Phys. {\\bf 36}, 6377 (1995).\n\n\\bibitem{Hol3} J. D. 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B {\\bf 693}, 339 (2010) [arXiv:1009.3695].\n\n\\bibitem{NoteMh} We shall assume that the central object is macroscopic with $M\\gg\\hbar^{1\/2}$.\n\n\\bibitem{NoteQ0} We shall henceforth assume, without loss of generality, that the electric charge \nof the central compact object is characterized by the relation $Q\\geq0$. \n\n\\bibitem{Notenng} Note that the orbiting particle may use non-gravitational forces \nin order to circle the central compact object on a non-geodesic trajectory with a velocity that, in principle, \ncan be made arbitrarily close to the speed of light. \n\n\\bibitem{Notethet} One may assume, without loss of generality, the polar angular \nrelation $\\theta=\\pi\/2$ for equatorial trajectories.\n\n\\bibitem{Noterm} Here we have used the fact that a charged compact object with $Q\\leq M$ that respects \nthe weak (positive) energy condition \\cite{HawEl} is larger than its Reissner-Nordstr\\\"om \nradius $M+(M^2-Q^2)^{1\/2}\\geq M$. Charged objects that respect \nthe weak energy condition are characterized by the inequalities $R\\geq Q^2\/2M>M\/2$ in the \ncomplementary regime $Q>M$.\n\n%\\bibitem{Notenff} If you, the reader, believe that the laws of physics absolutely forbid the existence of regions with negative %energy densities, \n%then the results presented below are probably not relevant to you.\n\n\\bibitem{Noteoom} It should be noted that the expression (\\ref{Eq10}) for the \ncritical electric field should be regarded as an order-of-magnitude estimate. \n\n\\bibitem{NoteQM} Here we have used the fact that the bound (\\ref{Eq16}) is valid in the regime \n$Q\/M\\gg1$ with $ME_{\\text{c}}\\gg1$, which implies the series of strong \ninequalities $Q^2\/2M\\gg Q\\gg M\\gg 1\/E_{\\text{c}}$.\n\n\\bibitem{Notethr} We use here the term `three-dimensional' in order to emphasize the fact that the asymptotically \nmeasured orbital periods $T_{\\infty}(M,Q,r)$ depend on three physical parameters: the mass $M$ of the \ncentral compact object, its electric charge $Q$, and the radius $r$ of the external circular orbit.\n\n\\end{thebibliography}\n\n\\end{document}\n"},{"text":"\\documentclass[a4paper,11pt,fleqn]{article}\n%---------------------------------------------------------------\n%\\usepackage[color,notref,notcite]{showkeys}\n%\\usepackage{refcheck} %check unused refs if showkeys activated\n\\usepackage[obeyspaces,hyphens,spaces]{url}\n\\usepackage[dvipsnames]{xcolor}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{amsfonts}\n\\usepackage{mathtools}\n\\usepackage{euscript}\n\\usepackage{pifont} %for dingolist\n\\usepackage{theorem}\n\\usepackage{charter}\n\\usepackage{srcltx} %for forward references in xdvi\n\\usepackage{etoolbox}\n\\usepackage[scr=rsfs]{mathalfa}\n%---------------------------------------------------------------\n\\oddsidemargin -0.55cm\n\\textwidth 17cm \n\\topmargin -0.8cm\n\\headheight 0.0cm\n\\textheight 22.9cm\n\\parindent 4mm\n\\parskip 3pt\n\\tolerance 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(\\roman{enumi})}\n\\renewcommand{\\labelenumii}{\\rm (\\alph{enumii})}\n\\numberwithin{equation}{section}\n\\setlist[enumerate]{itemsep=-1pt,topsep=1pt}\n\\setlist[description]{itemsep=-1pt,topsep=1pt}\n\\setlist[itemize]{itemsep=-1pt,topsep=1pt}\n\\usepackage{soul}\n\\setstcolor{dred}\n%---------------------------------------------------------------\n%blank norm, pairing, and scalar product\n\\newcommand*\\Cdot{{\\mkern 2mu\\cdot\\mkern 2mu}}\n\\newcommand{\\rocky}{\\ensuremath{\\mbox{\\footnotesize$\\odot$}}}\n\\newcommand*{\\tran}{^{\\mkern-1.5mu\\mathsf{T}}}\n%-------------------Authors-------------------------------------\n\\usepackage{authblk}\n\\newcommand{\\email}[1]{\\href{mailto:#1}{\\nolinkurl{#1}}}\n\\renewcommand\\Affilfont{\\small}\n\n\\begin{document}\n\n\\title{\\sffamily\\huge%\nInterchange Rules for Integral Functions\\thanks{%\nContact author: P. L. Combettes.\nEmail: \\email{plc@math.ncsu.edu}.\nPhone: +1 919 515 2671.\nThe work of M. N. B\\`ui was supported by NAWI Graz and\nthe work of P. L. Combettes was supported by the National\nScience Foundation under grant DMS-1818946.\n}}\n\n\\author[1]{Minh N. B\\`ui}\n\\affil[1]{Universit\\\"at Graz\n\\authorcr\nInstitut f\\\"ur Mathematik und Wissenschaftliches Rechnen\n\\authorcr\n8010 Graz, \\\"Osterreich\n\\authorcr\n\\email{minh.bui@uni-graz.at}\\medskip\n}\n\\author[2]{Patrick L. Combettes}\n\\affil[2]{North Carolina State University\n\\authorcr\nDepartment of Mathematics\n\\authorcr\nRaleigh, NC 27695-8205, USA\n\\authorcr\n\\email{plc@math.ncsu.edu}\n}\n\n\\date{~}\n\n\\maketitle\n\n\\begin{abstract} \nWe first present an abstract principle for the interchange of\ninfimization and integration over spaces of mappings taking values\nin topological spaces. New conditions on the underlying space and\nthe integrand are then introduced to convert this principle into\nconcrete scenarios that are shown to capture those of various\nexisting interchange rules. These results are leveraged to\nimprove state-of-the-art interchange rules for evaluating Legendre\nconjugates, subdifferentials, recessions, Moreau envelopes, and\nproximity operators of integral functions by bringing the\ncorresponding operations under the integral sign. \n\\end{abstract}\n\n\\begin{keywords}\nCompliant space,\nconvex analysis,\nintegral function,\ninterchange rules,\nnormal integrand.\n\\end{keywords}\n\n\\newpage\n\n\\section{Introduction}\n\\label{sec:1}\n\nThis paper concerns the interchange of the infimization and\nintegration operations in the context of the following assumption.\n\n\\begin{assumption}\n\\label{a:1}\n\\\n\\begin{enumerate}[label={\\rm[\\Alph*]}]\n\\item\n\\label{a:1a}\n$\\XS$ is a real vector space endowed with a\nSouslin topology $\\EuScript{T}_\\XS$ and\nassociated Borel $\\sigma$-algebra $\\BE_\\XS$.\n\\item\n\\label{a:1b}\nThe mapping\n$(\\XS\\times\\XS,\\BE_\\XS\\otimes\\BE_\\XS)\\to(\\XS,\\BE_\\XS)\\colon\n(\\mathsf{x},\\mathsf{y})\\mapsto\\mathsf{x}+\\mathsf{y}$\nis measurable.\n\\item\n\\label{a:1c}\nFor every $\\lambda\\in\\RR$, the mapping\n$(\\XS,\\BE_\\XS)\\to(\\XS,\\BE_\\XS)\\colon\n\\mathsf{x}\\mapsto\\lambda\\mathsf{x}$ is measurable.\n\\item\n\\label{a:1d}\n$(\\Omega,\\FF,\\mu)$ is a $\\sigma$-finite measure space such that\n$\\mu(\\Omega)\\neq 0$, and $\\mathcal{L}(\\Omega;\\XS)$\ndenotes the vector space of measurable mappings from\n$(\\Omega,\\FF)$ to $(\\XS,\\BE_\\XS)$.\n\\item\n\\label{a:1e}\n$\\XX$ is a vector subspace of $\\mathcal{L}(\\Omega;\\XS)$.\n\\item\n\\label{a:1f}\n$\\varphi\\colon(\\Omega\\times\\XS,\\FF\\otimes\\BE_\\XS)\\to\\RXX$ is an\nintegrand in the sense that it is measurable and, for every\n$\\omega\\in\\Omega$, $\\epi\\varphi_\\omega\\neq\\emp$, where\n$\\varphi_\\omega=\\varphi(\\omega,\\Cdot)$.\n\\item\n\\label{a:1g}\nThere exists $\\overline{x}\\in\\XX$ such that\n$\\int_\\Omega\\max\\{\\varphi(\\Cdot,\\overline{x}(\\Cdot)),0\\}\nd\\mu<\\pinf$.\n\\end{enumerate}\nAs is customary, given a measurable function\n$\\varrho\\colon(\\Omega,\\FF)\\to\\RXX$, $\\int_\\Omega\\varrho d\\mu$ is\nthe usual Lebesgue integral, except when the Lebesgue integral\n$\\int_\\Omega\\max\\{\\varrho,0\\}d\\mu$ is $\\pinf$, in which case\n$\\int_\\Omega\\varrho d\\mu=\\pinf$.\n\\end{assumption}\n\nMany problems in analysis and its applications require the\nevaluation of the infimum over $\\XX$ of the function\n$f\\colon x\\mapsto\\int_\\Omega\\varphi(\\Cdot,x(\\Cdot))d\\mu$. \nA simpler task is to evaluate the function \n$\\phi\\colon\\omega\\mapsto\\inf\\varphi(\\omega,\\XS)$ and then compute \n$\\int_\\Omega\\phi d\\mu$. In general, this provides only a\nlower bound as $\\inf f(\\XX)\\geq\\int_\\Omega\\phi d\\mu$. Conditions\nunder which the two quantities are equal have been established in\n\\cite{Hiai77}, \\cite{Perk18}, and \\cite{Roc76k} under various\nhypotheses on $\\XS$, $(\\Omega,\\FF,\\mu)$, $\\XX$, and $\\varphi$.\nThe resulting infimization-integration interchange rule\nis a central tool in areas such as \nmultivariate analysis \\cite{Hiai77},\ncalculus of variations \\cite{Ioff74},\neconomics \\cite{Levi85},\nstochastic processes \\cite{Penn18},\nstochastic optimization \\cite{Penn23},\nfinance \\cite{Perk18},\nconvex analysis \\cite{Roc76k},\nvariational analysis \\cite{Rock09},\nand stochastic programming \\cite{Shap21}.\nNote that, in Assumption~\\ref{a:1}\\ref{a:1a}--\\ref{a:1c}, we do\nnot require that $(\\XS,\\EuScript{T}_\\XS)$ be a topological vector\nspace to accommodate certain applications. For instance, in\n\\cite{Perk18}, $\\XS$ is the space of c\\`adl\\`ag functions on\n$[0,1]$ and $\\EuScript{T}_\\XS$ is the Skorokhod topology. In this\ncontext, $(\\XS,\\EuScript{T}_\\XS)$ is a Polish space\n\\cite[Chapter~3]{Bill68} which is not a topological vector space\n\\cite{Pest95} but which satisfies\nAssumption~\\ref{a:1}\\ref{a:1a}--\\ref{a:1c}.\n\nOur first contribution is Theorem~\\ref{t:1} below, which provides,\nunder the umbrella of Assumption~\\ref{a:1}, a broad setting for\nthe interchange of infimization and integration.\n\n\\begin{theorem}[interchange principle]\n\\label{t:1}\nSuppose that Assumption~\\ref{a:1} and the following hold:\n\\begin{enumerate}\n\\item\n\\label{t:1i}\n$\\inf\\varphi(\\Cdot,\\XS)$ is $\\FF$-measurable.\n\\item\n\\label{t:1ii}\nThere exists a sequence $(x_n)_{n\\in\\NN}$ in\n$\\mathcal{L}(\\Omega;\\XS)$ such that the following are\nsatisfied:\n\\begin{enumerate}\n\\item\n\\label{t:1iia}\n$\\inf\\varphi(\\Cdot,\\XS)=\n\\inf_{n\\in\\NN}\\varphi(\\Cdot,x_n(\\Cdot)+\\overline{x}(\\Cdot))$\n$\\mae$\n\\item\n\\label{t:1iib}\nThere exists an increasing sequence $(\\Omega_k)_{k\\in\\NN}$\nof finite $\\mu$-measure sets in $\\FF$ such that\n$\\bigcup_{k\\in\\NN}\\Omega_k=\\Omega$ and\n\\begin{equation}\n\\label{e:99}\n\\bigcup_{n\\in\\NN}\\bigcup_{k\\in\\NN}\n\\menge{1_Ax_n}{\\FF\\ni A\\subset\\Omega_k\\,\\,\\text{and}\\,\\,\n\\overline{x_n(A)}\\,\\,\\text{is compact}}\\subset\\XX.\n\\end{equation}\n\\end{enumerate}\n\\end{enumerate}\nThen\n\\begin{equation}\n\\label{e:1}\n\\inf_{x\\in\\XX}\\int_\\Omega\\varphi\\big(\\omega,x(\\omega)\\big)\n\\mu(d\\omega)=\n\\int_\\Omega\\inf_{\\mathsf{x}\\in\\XS}\\varphi(\\omega,\\mathsf{x})\\,\n\\mu(d\\omega).\n\\end{equation}\n\\end{theorem}\n\nTheorem~\\ref{t:1} is proved in Section~\\ref{sec:3}. The second\ncontribution is the introduction of two new tools --- compliant\nspaces and an extended notion of normal integrands. This is done in\nSection~\\ref{sec:4}, where these notions are illustrated through\nvarious examples. In Section~\\ref{sec:5}, compliance and normality\nare utilized to build a pathway between the abstract interchange\nprinciple of Theorem~\\ref{t:1} and separate conditions\non $\\XX$ and $\\varphi$ that capture various application settings.\nThe main result of that section is Theorem~\\ref{t:8}, which\nencompasses in particular the interchange rules of\n\\cite{Hiai77,Perk18,Roc76k}, as well as those implicitly present\nin \\cite{Roc68a,Rock71,Vala75}. These different frameworks have so\nfar not been brought together and we improve them in several\ndirections, for instance by not requiring the completeness of\n$(\\Omega,\\FF,\\mu)$ and by relaxing the assumptions on $\\XS$. This\nleads to new concrete scenarios under which \\eqref{e:1} holds.\nOur third contribution, presented in Section~\\ref{sec:6}, concerns\nconvex-analytical operations on integral functions. By combining\nTheorem~\\ref{t:1}, compliance, and normality, we broaden\nconditions for evaluating Legendre conjugates, subdifferentials,\nrecessions, Moreau envelopes, and proximity operators of integral\nfunctions by bringing the corresponding operations under the\nintegral sign. These results improve state-of-the-art convex\ncalculus rules from\n\\cite{Livre1,Penn18,Penn23,Rock71,Roc76k,Vala75}.\n\n\\section{Notation and background}\n\\label{sec:2}\n\n\\subsection{Measure theory}\n\nWe set $\\RXX=\\left[{-}\\infty,{+}\\infty\\right]$. Let $(\\Omega,\\FF)$\nbe a measurable space and let $A$ be a subset of $\\Omega$. The\ncharacteristic function of $A$ is denoted by $1_A$ and the\ncomplement of $A$ is denoted by $\\complement A$. Now let\n$(\\XS,\\EuScript{T}_\\XS)$ be a Hausdorff topological space with\nBorel $\\sigma$-algebra $\\BE_\\XS$. We denote by\n$\\mathcal{L}(\\Omega;\\XS)$ the vector space of measurable mappings\nfrom $(\\Omega,\\FF)$ to $(\\XS,\\BE_\\XS)$. Given a measure $\\mu$ on\n$(\\Omega,\\FF)$, $\\mathcal{L}^1(\\Omega;\\RR)$ is the subset of\n$\\mathcal{L}(\\Omega;\\RR)$ of integrable functions, and\n$\\mathcal{L}^1(\\Omega;\\RXX)$ is defined likewise. Given a separable\nBanach space $(\\XS,\\|\\Cdot\\|_\\XS)$, we set\n$\\mathcal{L}^\\infty(\\Omega;\\XS)=\n\\menge{x\\in\\mathcal{L}(\\Omega;\\XS)}{\\sup\\|x(\\Omega)\\|_\\XS<\\pinf}$.\n\n\\subsection{Topological spaces}\nGiven topological spaces $(\\YS,\\EuScript{T}_\\YS)$ and\n$(\\ZS,\\EuScript{T}_\\ZS)$,\n$\\EuScript{T}_\\YS\\boxtimes\\EuScript{T}_\\ZS$ denotes the\nstandard product topology.\n\nLet $(\\XS,\\EuScript{T}_\\XS)$ be a Hausdorff topological space.\nThe Borel $\\sigma$-algebra of $(\\XS,\\EuScript{T}_\\XS)$\nis denoted by $\\BE_\\XS$. Furthermore, $(\\XS,\\EuScript{T}_\\XS)$ is:\n\\begin{itemize}\n\\item\nregular \\cite[Section~I.8.4]{Bour71} if, for every closed\nsubset $\\mathsf{C}$ of $(\\XS,\\EuScript{T}_\\XS)$\nand every $\\mathsf{x}\\in\\complement\\mathsf{C}$, there exist\n$\\mathsf{V}\\in\\EuScript{T}_\\XS$ and\n$\\mathsf{W}\\in\\EuScript{T}_\\XS$ such that\n$\\mathsf{C}\\subset\\mathsf{V}$, $\\mathsf{x}\\in\\mathsf{W}$, and\n$\\mathsf{V}\\cap\\mathsf{W}=\\emp$;\n\\item\na Polish space \\cite[Section~IX.6.1]{Bour74} if it is separable and\nthere exists a distance $\\mathsf{d}$ on $\\XS$ that induces the same\ntopology as $\\EuScript{T}_\\XS$ and such that $(\\XS,\\mathsf{d})$ is\na complete metric space;\n\\item\na Souslin space \\cite[Section~IX.6.2]{Bour74} if there exist a\nPolish space $(\\YS,\\EuScript{T}_\\YS)$ and a continuous surjective\nmapping from $(\\YS,\\EuScript{T}_\\YS)$ to $(\\XS,\\EuScript{T}_\\XS)$;\n\\item\na Lusin space \\cite[Section~IX.6.4]{Bour74} if there exists a\ntopology $\\widetilde{\\EuScript{T}_\\XS}$ on $\\XS$ such that\n$\\EuScript{T}_\\XS\\subset\\widetilde{\\EuScript{T}_\\XS}$ and\n$(\\XS,\\widetilde{\\EuScript{T}_\\XS})$ is a Polish space;\n\\item\na Fr\\'echet space \\cite[Section~II.4.1]{Bour81} if it is a locally\nconvex real topological vector space and there exists a\ntranslation-invariant distance $\\mathsf{d}$ on $\\XS$ that induces\nthe same topology as $\\EuScript{T}_\\XS$ and such that\n$(\\XS,\\mathsf{d})$ is a complete metric space.\n\\end{itemize}\nNow let $\\mathsf{f}\\colon\\XS\\to\\RXX$.\nThe epigraph of $\\mathsf{f}$ is\n\\begin{equation}\n\\epi\\mathsf{f}=\\menge{(\\mathsf{x},\\xi)\\in\\XS\\times\\RR}{\n\\mathsf{f}(\\mathsf{x})\\leq\\xi},\n\\end{equation}\n$\\mathsf{f}$ is proper if\n$\\minf\\notin\\mathsf{f}(\\XS)\\neq\\{\\pinf\\}$, and $\\mathsf{f}$ is \n$\\EuScript{T}_\\XS$-lower semicontinuous if\n$\\epi\\mathsf{f}$ is\n$\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR$-closed.\n\n\\subsection{Duality}\n\nThe dual of a real topological vector space\n$(\\XS,\\EuScript{T}_\\XS)$, that is, the vector space of continuous\nlinear functionals on $(\\XS,\\EuScript{T}_\\XS)$, is denoted by\n$(\\XS,\\EuScript{T}_\\XS)^*$.\n\nLet $\\XS$ and $\\YS$ be real vector spaces which are in\nseparating duality via a bilinear form\n$\\pair{\\Cdot}{\\Cdot}_{\\XS,\\YS}\\colon\\XS\\times\\YS\\to\\RR$, that is\n\\cite[Section~II.6.1]{Bour81},\n\\begin{equation}\n\\begin{cases}\n(\\forall\\mathsf{x}\\in\\XS)\\quad\n\\pair{\\mathsf{x}}{\\Cdot}_{\\XS,\\YS}=0\n\\quad\\Rightarrow\\quad\n\\mathsf{x}=\\mathsf{0}\n\\\\\n(\\forall\\mathsf{y}\\in\\YS)\\quad\n\\pair{\\Cdot}{\\mathsf{y}}_{\\XS,\\YS}=0\n\\quad\\Rightarrow\\quad\n\\mathsf{y}=\\mathsf{0}.\n\\end{cases}\n\\end{equation}\nIn addition, equip $\\XS$ with a locally convex topology\n$\\EuScript{T}_\\XS$ which is compatible with the pairing\n$\\pair{\\Cdot}{\\Cdot}_{\\XS,\\YS}$ in the sense that \n$(\\XS,\\EuScript{T}_\\XS)^*\n=\\{\\pair{\\Cdot}{\\mathsf{y}}_{\\XS,\\YS}\\}_{\\mathsf{y}\\in\\YS}$\nand, likewise, equip $\\YS$ with a locally convex topology\n$\\EuScript{T}_\\YS$ which is compatible with the pairing\n$\\pair{\\Cdot}{\\Cdot}_{\\XS,\\YS}$ in the sense that\n$(\\YS,\\EuScript{T}_\\YS)^*\n=\\{\\pair{\\mathsf{x}}{\\Cdot}_{\\XS,\\YS}\\}_{\\mathsf{x}\\in\\XS}$\n\\cite[Section~IV.1.1]{Bour81}.\nFollowing \\cite{More66}, the Legendre conjugate of \n$\\mathsf{f}\\colon\\XS\\to\\RXX$ is\n\\begin{equation}\n\\label{e:l0d}\n\\mathsf{f}^*\\colon\\YS\\to\\RXX\\colon\n\\mathsf{y}\\mapsto\\sup_{\\mathsf{x}\\in\\XS}\n\\big(\\pair{\\mathsf{x}}{\\mathsf{y}}_{\\XS,\\YS}-\n\\mathsf{f}(\\mathsf{x})\\big)\n\\end{equation}\nand the Legendre conjugate of $\\mathsf{g}\\colon\\YS\\to\\RXX$ is\n\\begin{equation}\n\\mathsf{g}^*\\colon\\XS\\to\\RXX\\colon\n\\mathsf{x}\\mapsto\\sup_{\\mathsf{y}\\in\\YS}\n\\big(\\pair{\\mathsf{x}}{\\mathsf{y}}_{\\XS,\\YS}-\n\\mathsf{g}(\\mathsf{y})\\big).\n\\end{equation}\nLet $\\mathsf{f}\\colon\\XS\\to\\RXX$. If $\\mathsf{f}$ is proper, its\nsubdifferential is the set-valued operator\n\\begin{equation}\n\\label{e:s14}\n\\begin{aligned}\n\\partial\\mathsf{f}\\colon\\XS&\\to 2^\\YS\\\\\n\\mathsf{x}&\\mapsto\\menge{\\mathsf{y}\\in\\YS}{\n(\\forall\\mathsf{z}\\in\\XS)\\,\\,\n\\pair{\\mathsf{z}-\\mathsf{x}}{\\mathsf{y}}_{\\XS,\\YS}\n+\\mathsf{f}(\\mathsf{x})\\leq\\mathsf{f}(\\mathsf{z})}\n=\\menge{\\mathsf{y}\\in\\YS}{\n\\mathsf{f}(\\mathsf{x})+\\mathsf{f}^*(\\mathsf{y})=\n\\pair{\\mathsf{x}}{\\mathsf{y}}_{\\XS,\\YS}}.\n\\end{aligned}\n\\end{equation}\nIn addition, $\\mathsf{f}$ is convex if $\\epi\\mathsf{f}$ is a convex\nsubset of $\\XS\\times\\RR$, and $\\Gamma_0(\\XS)$ denotes the class of\nproper lower semicontinuous convex functions from $\\XS$ to $\\RX$.\nSuppose that $\\mathsf{f}\\in\\Gamma_0(\\XS)$ and let \n$\\mathsf{z}\\in\\dom\\mathsf{f}$. The recession function of\n$\\mathsf{f}$ is the function in $\\Gamma_0(\\XS)$ defined by\n\\begin{equation}\n\\label{e:r}\n\\rec\\mathsf{f}\\colon\\XS\\to\\RX\\colon\\mathsf{x}\\mapsto\n\\lim_{0<\\alpha\\uparrow\\pinf}\n\\frac{\\mathsf{f}(\\mathsf{z}+\\alpha\\mathsf{x})\n-\\mathsf{f}(\\mathsf{z})}{\\alpha}.\n\\end{equation}\nNow suppose that, in addition, $\\XS=\\YS$ is Hilbertian\nand $\\pair{\\Cdot}{\\Cdot}_{\\XS,\\YS}$ is the scalar product of\n$\\XS$, and let $\\gamma\\in\\RPP$. The Moreau envelope of\n$\\mathsf{f}$ of index $\\gamma$ is the function in $\\Gamma_0(\\XS)$\ndefined by\n\\begin{equation}\n\\label{e:7}\n\\moyo{\\mathsf{f}}{\\gamma}\\colon\\XS\\to\\RR\\colon\n\\mathsf{x}\\mapsto\n\\min_{\\mathsf{y}\\in\\XS}\\bigg(\\mathsf{f}(\\mathsf{y})\n+\\dfrac{1}{2\\gamma}\\|\\mathsf{x}-\\mathsf{y}\\|_\\XS^2\\bigg)\n\\end{equation}\nand the proximal point of $\\mathsf{x}\\in\\XS$ relative to\n$\\gamma\\mathsf{f}$ is the unique point\n$\\prox_{\\gamma\\mathsf{f}}\\mathsf{x}\\in\\XS$ such that\n\\begin{equation}\n\\label{e:7b}\n\\moyo{\\mathsf{f}}{\\gamma}(\\mathsf{x})\n=\\mathsf{f}(\\prox_{\\gamma\\mathsf{f}}\\mathsf{x})\n+\\dfrac{1}{2\\gamma}\n\\|\\mathsf{x}-\\prox_{\\gamma\\mathsf{f}}\\mathsf{x}\\|_\\XS^2.\n\\end{equation}\nThe proximity operator $\\prox_{\\gamma\\mathsf{f}}\\colon\\XS\\to\\XS$\nthus defined can be expressed as\n\\begin{equation}\n\\label{e:8}\n\\prox_{\\gamma\\mathsf{f}}=(\\Id+\\gamma\\partial\\mathsf{f})^{-1}.\n\\end{equation}\n\n\\section{Proof of the interchange principle}\n\\label{sec:3}\n\nProving Theorem~\\ref{t:1} necessitates a few technical facts.\n\n\\begin{lemma}\n\\label{l:8}\nLet $(\\Omega,\\FF)$ be a measurable space,\nlet $n$ be a strictly positive integer,\nand let $(\\varrho_i)_{0\\leq i\\leq n}$ be a family in\n$\\mathcal{L}(\\Omega;\\RR)$. Then there exists a family\n$(B_i)_{0\\leq i\\leq n}$ in $\\FF$ such that \n\\begin{equation}\n\\label{e:bx}\n(B_i)_{0\\leq i\\leq n}\\,\\,\\text{are pairwise disjoint},\n\\quad\\bigcup_{i=0}^nB_i=\\Omega,\n\\quad\\text{and}\\quad\n\\min_{0\\leq i\\leq n}\\varrho_i=\\sum_{i=0}^n1_{B_i}\\varrho_i.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWe proceed by induction on $n$. If $n=1$, we obtain \\eqref{e:bx}\nby choosing $B_0=[\\varrho_0\\leq\\varrho_1]$ and\n$B_1=\\complement B_0$. Now assume that the claim is true for $n$,\nlet $\\varrho_{n+1}\\in\\mathcal{L}(\\Omega;\\RR)$, and set \n\\begin{equation}\n\\varrho=\\min_{0\\leq i\\leq n}\\varrho_i,\n\\quad\nD=[\\varrho\\leq\\varrho_{n+1}],\\quad\nC_{n+1}=\\complement D,\n\\quad\\text{and}\\quad\n\\big(\\forall i\\in\\{0,\\ldots,n\\}\\big)\\;\\;C_i=B_i\\cap D.\n\\end{equation}\nThen $(C_i)_{0\\leq i\\leq n+1}$ is a family of pairwise disjoint\nsets in $\\FF$. Additionally,\n\\begin{equation}\n\\bigcup_{i=0}^{n+1}C_i\n=C_{n+1}\\cup\\bigcup_{i=0}^nC_i\n=\\big(\\complement D\\big)\\cup\\bigcup_{i=0}^n(B_i\\cap D)\n=\\big(\\complement D\\big)\\cup D\n=\\Omega\n\\end{equation}\nand\n\\begin{align}\n\\min_{0\\leq i\\leq n+1}\\varrho_i\n=\\min\\{\\varrho,\\varrho_{n+1}\\}\n=1_D\\varrho+1_{\\complement D}\\varrho_{n+1}\n=1_D\\sum_{i=0}^n1_{B_i}\\varrho_i+1_{C_{n+1}}\\varrho_{n+1}\n=\\sum_{i=0}^{n+1}1_{C_i}\\varrho_i,\n\\end{align}\nwhich concludes the induction argument.\n\\end{proof}\n\n\\begin{lemma}\n\\label{l:1}\nLet $(\\Omega,\\FF,\\mu)$ be a $\\sigma$-finite measure space\nsuch that $\\mu(\\Omega)\\neq 0$ and let $\\mathcal{R}$ be a nonempty\nsubset of $\\mathcal{L}(\\Omega;\\RXX)$. Then there exists an\nelement in $\\mathcal{L}(\\Omega;\\RXX)$, denoted by\n$\\essinf\\mathcal{R}$ and unique up to a set of $\\mu$-measure zero,\nsuch that\n\\begin{equation}\n\\label{e:od3}\n\\big(\\forall\\vartheta\\in\\mathcal{L}(\\Omega;\\RXX)\\big)\n\\quad\\big[\\;(\\forall\\varrho\\in\\mathcal{R})\\;\\;\n\\vartheta\\leq\\varrho\\,\\,\\mae\\;\\big]\n\\quad\\Leftrightarrow\\quad\n\\vartheta\\leq\\essinf\\mathcal{R}\\,\\,\\mae\n\\end{equation}\nMoreover, there exists a sequence $(\\varrho_n)_{n\\in\\NN}$ in\n$\\mathcal{R}$ such that\n$\\essinf\\mathcal{R}=\\inf_{n\\in\\NN}\\varrho_n$.\n\\end{lemma}\n\\begin{proof}\nUsing Assumption~\\ref{a:1}\\ref{a:1d}, construct\n$0<\\chi\\in\\mathcal{L}^1(\\Omega;\\RR)$ such that\n$\\int_\\Omega\\chi d\\mu=1$ and define\n$\\PP\\colon\\FF\\to[0,1]\\colon A\\mapsto\\int_A\\chi d\\mu$.\nThen $(\\forall A\\in\\FF)$ $\\mu(A)=0$ $\\Leftrightarrow$ $\\PP(A)=0$.\nHence, the assertions follow from\n\\cite[Proposition~II-4-1 and its proof]{Neve70} applied in the\nprobability space $(\\Omega,\\FF,\\PP)$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{l:2}\nLet $(\\Omega,\\FF,\\mu)$ be a measure space,\nlet $(\\XS,\\EuScript{T}_\\XS)$ be a Souslin space,\nlet $z\\colon(\\Omega,\\FF)\\to(\\XS,\\BE_\\XS)$ be measurable,\nand let $E\\in\\FF$ be such that $\\mu(E)<\\pinf$. Then\nthere exists a sequence $(E_n)_{n\\in\\NN}$ in $\\FF$ such that\n\\begin{equation}\n\\big[\\;(\\forall n\\in\\NN)\\;\\;\nE_n\\subset E\\,\\,\\text{and}\\,\\,\n\\overline{z(E_n)}\\,\\,\\text{is compact}\\;\\big]\n\\quad\\text{and}\\quad\n\\mu(E)=\\mu\\bigg(\\bigcup_{n\\in\\NN}E_n\\bigg).\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWe adapt the proof of \\cite[Lemma~5]{Vala75},\nwhere $(\\XS,\\EuScript{T}_\\XS)$ is a locally convex Souslin\ntopological vector space. Define $\\nu\\colon\\BE_\\XS\\to\\RP\\colon\n\\mathsf{K}\\mapsto\\mu(E\\cap z^{-1}(\\mathsf{K}))$.\nThen $\\nu$ is a measure on $(\\XS,\\BE_\\XS)$. Hence, since\n$(\\XS,\\EuScript{T}_\\XS)$ is a Souslin space, it follows from\n\\cite[Proposition~IX.3.3]{Bour69} that\n\\begin{equation}\n\\nu(\\XS)=\\sup\\menge{\\nu(\\mathsf{K})}{\\mathsf{K}\\,\\,\n\\text{is a compact subset of}\\,\\,(\\XS,\\EuScript{T}_\\XS)}.\n\\end{equation}\nThus, for every $n\\in\\NN$, there exists a compact set\n$\\mathsf{K}_n$ such that $\\nu(\\XS)\\leq\\nu(\\mathsf{K}_n)+2^{-n}$.\nNow define a sequence $(E_n)_{n\\in\\NN}$ of $\\FF$-measurable\nsubsets of $E$ by $(\\forall n\\in\\NN)$ $E_n=E\\cap\nz^{-1}(\\mathsf{K}_n)$. On the one hand, for every $n\\in\\NN$, since\n$\\mathsf{K}_n$ is compact and $z(E_n)\\subset\\mathsf{K}_n$,\n$\\overline{z(E_n)}$ is compact. On the other hand,\n\\begin{equation}\n(\\forall n\\in\\NN)\\quad\n\\mu\\bigg(\\bigcup_{k\\in\\NN}E_k\\bigg)\n\\leq\\mu(E)\n=\\nu(\\XS)\n\\leq\\nu(\\mathsf{K}_n)+2^{-n}\n=\\mu(E_n)+2^{-n}\n\\leq\\mu\\bigg(\\bigcup_{k\\in\\NN}E_k\\bigg)+2^{-n},\n\\end{equation}\nwhich implies that $\\mu(E)=\\mu(\\bigcup_{n\\in\\NN}E_n)$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{l:3}\nSuppose that Assumption~\\ref{a:1}\\ref{a:1a}--\\ref{a:1d} hold.\nLet $\\psi\\colon(\\Omega\\times\\XS,\\FF\\otimes\\BE_\\XS)\\to\\RXX$ be\nmeasurable, let $\\mathcal{Z}$ be a nonempty at most countable\nsubset of $\\mathcal{L}(\\Omega;\\XS)$, and let\n$(\\Omega_k)_{k\\in\\NN}$ be an increasing sequence\nof finite $\\mu$-measure sets in $\\FF$ such that\n$\\bigcup_{k\\in\\NN}\\Omega_k=\\Omega$. Define\n\\begin{equation}\n\\label{e:8vc}\n\\mathcal{D}=\\bigcup_{z\\in\\mathcal{Z}}\\bigcup_{k\\in\\NN}\n\\menge{1_Az}{\\FF\\ni A\\subset\\Omega_k\\,\\,\n\\text{and}\\,\\,\\overline{z(A)}\\,\\,\\text{is compact}}\n\\end{equation}\nand\n\\begin{equation}\n\\label{e:8ki}\n\\mathcal{R}=\\menge{\\varrho\\in\\mathcal{L}^1(\\Omega;\\RR)}{\n(\\exi x\\in\\mathcal{D})\\,\\,\\psi\\big(\\Cdot,x(\\Cdot)\\big)\\leq\n\\varrho(\\Cdot)\\,\\,\\mae}.\n\\end{equation}\nSuppose that\n\\begin{equation}\n\\label{e:h3}\n\\psi(\\Cdot,\\mathsf{0})\\leq 0.\n\\end{equation}\nThen $\\mathcal{R}\\neq\\emp$ and $\\essinf\\mathcal{R}\n\\leq\\inf_{z\\in\\mathcal{Z}}\\psi(\\Cdot,z(\\Cdot))$ $\\mae$\n\\end{lemma}\n\\begin{proof}\nTake $z\\in\\mathcal{Z}$ and note that\n$(\\forall A\\in\\FF)$ $1_Az\\in\\mathcal{L}(\\Omega;\\XS)$.\nSince $\\overline{z(\\emp)}=\\emp$ is compact,\nit results from \\eqref{e:8vc} that $0=1_\\emp z\\in\\mathcal{D}$.\nHence, by \\eqref{e:h3}, $0\\in\\mathcal{R}$. Next, thanks\nto Assumption~\\ref{a:1}\\ref{a:1d}, there exists\n$\\chi\\in\\mathcal{L}^1(\\Omega;\\RR)$ such that $\\chi>0$.\nLet us set\n\\begin{equation}\n\\label{e:5h0}\n(\\forall n\\in\\NN)\\quad A_n=\\Omega_n\\cap\n\\big[\\psi\\big(\\Cdot,z(\\Cdot)\\big)\\leq n\\chi(\\Cdot)\\big].\n\\end{equation}\nLemma~\\ref{l:2} asserts that there exists a family\n$(A_{n,k})_{(n,k)\\in\\NN^2}$ in $\\FF$ such that\n\\begin{equation}\n\\label{e:24r}\n(\\forall n\\in\\NN)\\quad\n\\begin{cases}\n(\\forall k\\in\\NN)\\;\\;\nA_{n,k}\\subset A_n\\,\\,\\text{and}\\,\\,\n\\overline{z(A_{n,k})}\\,\\,\\text{is compact}\n\\\\\n\\displaystyle\n\\mu(A_n)=\\mu\\bigg(\\bigcup_{k\\in\\NN}A_{n,k}\\bigg).\n\\end{cases}\n\\end{equation}\nIn turn, by \\eqref{e:8vc} and \\eqref{e:5h0},\n\\begin{equation}\n\\label{e:y6}\n(\\forall n\\in\\NN)(\\forall k\\in\\NN)\\quad\n1_{A_{n,k}}z\\in\\mathcal{D}.\n\\end{equation}\nDefine\n\\begin{equation}\n\\label{e:c32}\n(\\forall n\\in\\NN)(\\forall k\\in\\NN)(\\forall m\\in\\NN)\\quad\n\\varrho_{n,k,m}(\\Cdot)=\n\\max\\big\\{\\psi\\big(\\Cdot,1_{A_{n,k}}(\\Cdot)z(\\Cdot)\\big),\n-m\\chi(\\Cdot)\\big\\}.\n\\end{equation}\nFix temporarily $(n,k,m)\\in\\NN^3$. We infer from\n\\eqref{e:24r}, \\eqref{e:5h0}, and \\eqref{e:h3} that\n\\begin{align}\n(\\forall\\omega\\in\\Omega)\\quad\n\\psi\\big(\\omega,1_{A_{n,k}}(\\omega)z(\\omega)\\big)\n&=\n\\begin{cases}\n\\psi\\big(\\omega,z(\\omega)\\big),\n&\\text{if}\\,\\,\\omega\\in A_{n,k};\\\\\n\\psi(\\omega,\\mathsf{0}),\n&\\text{otherwise}\n\\end{cases}\n\\nonumber\\\\\n&\\leq\n\\begin{cases}\nn\\chi(\\omega),\n&\\text{if}\\,\\,\\omega\\in A_{n,k};\\\\\n0,\n&\\text{otherwise}\n\\end{cases}\n\\nonumber\\\\\n&\\leq n\\chi(\\omega).\n\\end{align}\nTherefore, $-m\\chi\\leq\\varrho_{n,k,m}\\leq n\\chi$, which entails\nthat $\\varrho_{n,k,m}\\in\\mathcal{L}^1(\\Omega;\\RR)$.\nIn turn, we derive from \\eqref{e:c32}, \\eqref{e:y6}, and\n\\eqref{e:8ki} that $\\varrho_{n,k,m}\\in\\mathcal{R}$.\nThus, Lemma~\\ref{l:1} guarantees that there exists\n$B_{n,k,m}\\in\\FF$ such that $\\mu(B_{n,k,m})=0$ and\n\\begin{equation}\n\\label{e:uf}\n\\big(\\forall\\omega\\in\\complement B_{n,k,m}\\big)\\quad\n(\\essinf\\mathcal{R})(\\omega)\\leq\\varrho_{n,k,m}(\\omega).\n\\end{equation}\nNow set\n\\begin{equation}\n\\label{e:a34}\nA=\\bigcap_{(n,k)\\in\\NN^2}\\complement A_{n,k},\n\\quad\nB=\\bigcup_{(n,k,m)\\in\\NN^3}B_{n,k,m},\n\\quad\\text{and}\\quad\nC=\\big[\\psi\\big(\\Cdot,z(\\Cdot)\\big)<\\pinf\\big]\\cap(A\\cup B).\n\\end{equation}\nThen $\\mu(B)=0$. Furthermore, since \\eqref{e:5h0} yields\n$[\\psi(\\Cdot,z(\\Cdot))<\\pinf]=\\bigcup_{n\\in\\NN}A_n$, it follows\nfrom \\eqref{e:a34} and \\eqref{e:24r} that\n\\begin{equation}\n\\mu\\Big(\\big[\\psi\\big(\\Cdot,z(\\Cdot)\\big)<\\pinf\\big]\\cap A\\Big)\n\\leq\\sum_{n\\in\\NN}\\mu(A_n\\cap A)\n\\leq\\sum_{n\\in\\NN}\\mu\\bigg(A_n\\cap\n\\bigcap_{k\\in\\NN}\\complement A_{n,k}\\bigg)\n=0.\n\\end{equation}\nHence, using \\eqref{e:a34}, we obtain\n\\begin{equation}\n\\label{e:c0}\n\\mu(C)=0\\quad\\text{and}\\quad\n\\complement C\n=\\big[\\psi\\big(\\Cdot,z(\\Cdot)\\big)=\\pinf\\big]\\cup\n\\big(\\complement A\\cap\\complement B\\big).\n\\end{equation}\nNow suppose that $\\omega\\in\\complement A\\cap\\complement B$.\nThen it follows from \\eqref{e:a34} that there exists\n$(n,k)\\in\\NN^2$ such that $\\omega\\in A_{n,k}\\cap\\complement B$.\nTherefore, we derive from \\eqref{e:a34}, \\eqref{e:uf}, and\n\\eqref{e:c32} that\n\\begin{equation}\n\\label{e:b9}\n(\\forall m\\in\\NN)\\quad\n(\\essinf\\mathcal{R})(\\omega)\n\\leq\\varrho_{n,k,m}(\\omega)\n=\\max\\big\\{\\psi\\big(\\omega,1_{A_{n,k}}(\\omega)z(\\omega)\\big),\n-m\\chi(\\omega)\\big\\}.\n\\end{equation}\nHence, letting $m\\uparrow\\pinf$ yields \n$(\\essinf\\mathcal{R})(\\omega)\\leq\n\\psi(\\omega,1_{A_{n,k}}(\\omega)z(\\omega))\n=\\psi(\\omega,z(\\omega))$. We have thus shown that\n$\\essinf\\mathcal{R}\\leq\\psi(\\Cdot,z(\\Cdot))$ $\\mae$\nSince $\\mathcal{Z}$ is at most countable, the proof is complete.\n\\end{proof}\n\n\\bigskip\n\n\\makeatother\n\\def\\prooft{\\noindent{\\bfseries Proof of Theorem~\\ref{t:1}}.\n\\ignorespaces}\n\\def\\endprooft{\\;\\:\\vbox{\\hrule height0.6pt\\hbox{%\n\\vrule height 1.2ex% \nwidth 0.8pt\\hskip0.8ex\\vrule width 0.8pt}\\hrule height 0.6pt}}\n\\makeatletter\n\n\\begin{prooft}\nDefine\n\\begin{equation}\n\\label{e:p9}\n\\Phi\\colon\\mathcal{L}(\\Omega;\\XS)\\to\\mathcal{L}(\\Omega;\\RXX)\\colon\nx\\mapsto\\varphi\\big(\\Cdot,x(\\Cdot)\\big)\n\\end{equation}\nand note that, thanks to Assumption~\\ref{a:1}\\ref{a:1g},\n\\begin{equation}\n\\label{e:o9}\n\\int_\\Omega\\inf\\varphi(\\Cdot,\\XS)\\,d\\mu\n\\leq\\inf_{x\\in\\XX}\\int_\\Omega\\Phi(x)d\\mu\n\\leq\\int_\\Omega\\Phi(\\overline{x})d\\mu\n<\\pinf.\n\\end{equation}\nHence, the interchange rule \\eqref{e:1} holds when\n$\\inf_{x\\in\\XX}\\int_\\Omega\\Phi(x)d\\mu=\\minf$\nand we assume henceforth that\n\\begin{equation}\n\\label{e:77}\n\\inf_{x\\in\\XX}\\int_\\Omega\\Phi(x)d\\mu\\in\\RR.\n\\end{equation}\nNow define\n\\begin{equation}\n\\label{e:80}\n\\vartheta=\\max\\big\\{\\Phi(\\overline{x}),0\\big\\}\n\\end{equation}\nand\n\\begin{equation}\n\\label{e:72}\n\\psi\\colon\\Omega\\times\\XS\\to\\RXX\\colon\n(\\omega,\\mathsf{x})\\mapsto\n\\begin{cases}\n\\varphi\\big(\\omega,\\mathsf{x}+\\overline{x}(\\omega)\\big)-\n\\vartheta(\\omega),&\\text{if}\\,\\,\\vartheta(\\omega)<\\pinf;\\\\\n\\minf,&\\text{if}\\,\\,\\vartheta(\\omega)=\\pinf.\n\\end{cases}\n\\end{equation}\nThen we derive from Assumption~\\ref{a:1}\\ref{a:1g} that\n\\begin{equation}\n\\label{e:r1}\n\\vartheta\\in\\mathcal{L}^1(\\Omega;\\RXX)\n\\end{equation}\nand, therefore, that\n\\begin{equation}\n\\label{e:r2}\n\\mu\\big([\\vartheta=\\pinf]\\big)=0.\n\\end{equation}\nOn the other hand, Assumption~\\ref{a:1}\\ref{a:1b} ensures that the\nmapping $(\\Omega\\times\\XS,\\FF\\otimes\\BE_\\XS)\\to(\\XS,\\BE_\\XS)\\colon\n(\\omega,\\mathsf{x})\\mapsto\\mathsf{x}+\\overline{x}(\\omega)$ is\nmeasurable. Thus, it follows from Assumption~\\ref{a:1}\\ref{a:1f},\n\\eqref{e:r1}, and \\eqref{e:72} that\n\\begin{equation}\n\\label{e:f78}\n\\psi\\,\\,\\text{is $\\FF\\otimes\\BE_\\XS$-measurable}.\n\\end{equation}\nAt the same time, since\n\\begin{equation}\n\\label{e:45}\n\\inf_{\\mathsf{x}\\in\\XS}\\psi(\\Cdot,\\mathsf{x})\n=\\inf_{\\mathsf{x}\\in\\XS}\n\\varphi\\big(\\Cdot,\\mathsf{x}+\\overline{x}(\\Cdot)\\big)-\n\\vartheta(\\Cdot)\n=\\inf_{\\mathsf{x}\\in\\XS}\\varphi(\\Cdot,\\mathsf{x})-\\vartheta(\\Cdot)\n\\end{equation}\nand since Assumption~\\ref{a:1}\\ref{a:1f} yields\n$\\inf\\varphi(\\Cdot,\\XS)<\\pinf$, it results from \\ref{t:1i} that\n\\begin{equation}\n\\label{e:pd}\n\\inf\\psi(\\Cdot,\\XS)\\in\\mathcal{L}(\\Omega;\\RXX).\n\\end{equation}\nLet us set\n\\begin{equation}\n\\Psi\\colon\\mathcal{L}(\\Omega;\\XS)\\to\\mathcal{L}(\\Omega;\\RXX)\\colon\nx\\mapsto\\psi\\big(\\Cdot,x(\\Cdot)\\big).\n\\end{equation}\nBy \\eqref{e:72} and \\eqref{e:r2},\n\\begin{equation}\n\\label{e:jc}\n\\big(\\forall\\omega\\in\\complement[\\vartheta=\\pinf]\\big)\n(\\forall x\\in\\XX)\\quad\n\\big(\\Psi(x)\\big)(\\omega)\n=\\big(\\Phi(x+\\overline{x})\\big)(\\omega)-\\vartheta(\\omega).\n\\end{equation}\nHence, upon invoking \\eqref{e:r1}, we deduce from\nAssumption~\\ref{a:1}\\ref{a:1e}\\&\\ref{a:1g} that\n\\begin{align}\n\\label{e:15}\n\\inf_{x\\in\\XX}\\int_\\Omega\\Psi(x)d\\mu\n&=\\inf_{x\\in\\XX}\\int_\\Omega\\big(\\Phi(x+\\overline{x})-\n\\vartheta\\big)d\\mu\n\\nonumber\\\\\n&=\\inf_{x\\in\\XX}\\int_\\Omega\\Phi(x+\\overline{x})d\\mu-\n\\int_\\Omega\\vartheta d\\mu\n\\nonumber\\\\\n&=\\inf_{x\\in\\XX}\\int_\\Omega\\Phi(x)d\\mu-\\int_\\Omega\\vartheta d\\mu\n\\end{align}\nand, likewise, from \\eqref{e:45} that\n\\begin{equation}\n\\label{e:14}\n\\int_\\Omega\\inf\\psi(\\Cdot,\\XS)\\,d\\mu\n=\\int_\\Omega\\inf\\varphi(\\Cdot,\\XS)\\,d\\mu-\\int_\\Omega\\vartheta d\\mu.\n\\end{equation}\nNow set\n\\begin{equation}\n\\label{e:d2}\n\\mathcal{D}=\\bigcup_{n\\in\\NN}\\bigcup_{k\\in\\NN}\n\\menge{1_Ax_n}{\\FF\\ni A\\subset\\Omega_k\n\\,\\,\\text{and}\\,\\,\\overline{x_n(A)}\\,\\,\\text{is compact}}\n\\end{equation}\nand\n\\begin{equation}\n\\label{e:pui}\n\\mathcal{R}=\\menge{\\varrho\\in\\mathcal{L}^1(\\Omega;\\RR)}{\n(\\exi x\\in\\mathcal{D})\\,\\,\\Psi(x)\\leq\\varrho\\,\\,\\mae},\n\\end{equation}\nand note that \\ref{t:1iib} states that\n\\begin{equation}\n\\label{e:d0}\n\\mathcal{D}\\subset\\XX.\n\\end{equation}\nUsing \\eqref{e:72} and \\eqref{e:80}, we infer from\nLemma~\\ref{l:3} applied to $\\mathcal{Z}=\\{x_n\\}_{n\\in\\NN}$ that\n$\\essinf\\mathcal{R}\\leq\\inf_{n\\in\\NN}\\Psi(x_n)$ $\\mae$ In turn, we\nderive from \\eqref{e:jc}, \\ref{t:1iia}, and \\eqref{e:45}\nthat\n\\begin{equation}\n\\essinf\\mathcal{R}\n\\leq\\inf_{n\\in\\NN}\\Psi(x_n)\n=\\inf_{n\\in\\NN}\\Phi(x_n+\\overline{x})-\\vartheta\n=\\inf\\varphi(\\Cdot,\\XS)-\\vartheta\n=\\inf\\psi(\\Cdot,\\XS)\\,\\,\\mae\n\\end{equation}\nOn the other hand, \\eqref{e:pui} implies that\n$(\\forall\\varrho\\in\\mathcal{R})$\n$\\inf\\psi(\\Cdot,\\XS)\\leq\\varrho(\\Cdot)$ $\\mae$\nHence, \\eqref{e:pd} and Lemma~\\ref{l:1} guarantee that\n$\\inf\\psi(\\Cdot,\\XS)\\leq\\essinf\\mathcal{R}$ $\\mae$\nAltogether, $\\essinf\\mathcal{R}=\\inf\\psi(\\Cdot,\\XS)$ $\\mae$\nThus, we deduce from Lemma~\\ref{l:1} that there exists a sequence\n$(\\varrho_n)_{n\\in\\NN}$ in $\\mathcal{R}$ such that\n\\begin{equation}\n\\label{e:g8}\n\\inf_{n\\in\\NN}\\varrho_n(\\Cdot)=\\inf\\psi(\\Cdot,\\XS)\\,\\,\\mae\n\\end{equation}\nFor every $n\\in\\NN$, it follows from \\eqref{e:pui} and\n\\eqref{e:d2} that there exist $\\ell_n\\in\\NN$,\n$k_n\\in\\NN$, and $\\FF\\ni A_n\\subset\\Omega_{k_n}$ such that\n\\begin{equation}\n\\label{e:o4}\n\\overline{x_{\\ell_n}(A_n)}\\,\\,\\text{is compact}\n\\quad\\text{and}\\quad\n\\Psi\\big(1_{A_n}x_{\\ell_n}\\big)\\leq\\varrho_n\\,\\,\\mae\n\\end{equation}\nLet us set\n\\begin{equation}\n\\label{e:c5}\n(\\forall n\\in\\NN)\\quad\\chi_n=\\min_{0\\leq i\\leq n}\\varrho_i.\n\\end{equation}\nFix temporarily $n\\in\\NN$. Lemma~\\ref{l:8} asserts that there\nexists a family $(B_{n,i})_{0\\leq i\\leq n}$ in $\\FF$ such that\n\\begin{equation}\n\\label{e:bi6}\n(B_{n,i})_{0\\leq i\\leq n}\\,\\,\\text{are pairwise disjoint},\n\\quad\\bigcup_{i=0}^nB_{n,i}=\\Omega,\\quad\\text{and}\\quad\n\\chi_n=\\sum_{i=0}^n1_{B_{n,i}}\\varrho_i.\n\\end{equation}\nNow set\n\\begin{equation}\n\\label{e:y8}\ny_n=\\sum_{i=0}^n1_{A_i\\cap B_{n,i}}x_{\\ell_i}.\n\\end{equation}\nFor every $i\\in\\{0,\\ldots,n\\}$, since\n$A_i\\cap B_{n,i}\\subset A_i\\subset\\Omega_{k_i}$, \\eqref{e:o4}\nimplies that $\\overline{x_{\\ell_i}(A_i\\cap B_{n,i})}$ is compact\nand, therefore, \\eqref{e:d2} and \\eqref{e:d0} yield\n$1_{A_i\\cap B_{n,i}}x_{\\ell_i}\\in\\mathcal{D}\\subset\\XX$.\nConsequently, \\eqref{e:y8} and Assumption~\\ref{a:1}\\ref{a:1e}\nensure that $y_n\\in\\XX$. At the same time, we derive from\n\\eqref{e:y8}, \\eqref{e:bi6}, and \\eqref{e:o4} that\n\\begin{equation}\n\\label{e:7gf}\n\\Psi(y_n)\n=\\sum_{i=0}^n1_{B_{n,i}}\\Psi\\big(1_{A_i}x_{\\ell_i}\\big)\n\\leq\\sum_{i=0}^n1_{B_{n,i}}\\varrho_i\n=\\chi_n\\,\\,\\mae\n\\end{equation}\nTherefore, since $y_n\\in\\XX$,\n\\begin{equation}\n\\label{e:g7}\n\\inf_{x\\in\\XX}\\int_\\Omega\\Psi(x)d\\mu\n\\leq\\int_\\Omega\\Psi(y_n)d\\mu\n\\leq\\int_\\Omega\\chi_nd\\mu.\n\\end{equation}\nOn the other hand, it results from\n\\eqref{e:15}, \\eqref{e:77}, and \\eqref{e:r1} that\n$\\inf_{x\\in\\XX}\\int_\\Omega\\Psi(x)d\\mu\\in\\RR$.\nThus, since $\\chi_n\\downarrow\\inf_{i\\in\\NN}\\varrho_i(\\Cdot)=\n\\inf\\psi(\\Cdot,\\XS)$ $\\mae$\nby virtue of \\eqref{e:c5} and \\eqref{e:g8}, \\eqref{e:g7} and\nthe monotone convergence theorem\n\\cite[Theorem~2.8.2 and Corollary~2.8.6]{Boga07}\nentail that\n\\begin{equation}\n\\inf_{x\\in\\XX}\\int_\\Omega\\Psi(x)d\\mu\n\\leq\\lim\\int_\\Omega\\chi_nd\\mu\n=\\int_\\Omega\\lim\\chi_n\\,d\\mu\n=\\int_\\Omega\\inf\\psi(\\Cdot,\\XS)\\,d\\mu.\n\\end{equation}\nConsequently, since $\\int_\\Omega\\inf\\psi(\\Cdot,\\XS)\\,d\\mu\\leq\n\\inf_{x\\in\\XX}\\int_\\Omega\\Psi(x)d\\mu$, we conclude that\n\\begin{equation}\n\\inf_{x\\in\\XX}\\int_\\Omega\\Psi(x)d\\mu\n=\\int_\\Omega\\inf\\psi(\\Cdot,\\XS)\\,d\\mu.\n\\end{equation}\nIn view of \\eqref{e:15}, \\eqref{e:14}, and \\eqref{e:r1},\nthe proof is complete.\n\\end{prooft}\n\n\\begin{remark}\n\\label{r:1}\nReplacing $\\varphi$ by $-\\varphi$ in items \\ref{a:1f} and\n\\ref{a:1g} of Assumption~\\ref{a:1} and in Theorem~\\ref{t:1}\nprovides conditions under which\n\\begin{equation}\n\\sup_{x\\in\\XX}\\int_\\Omega\\varphi\\big(\\omega,x(\\omega)\\big)\n\\mu(d\\omega)=\n\\int_\\Omega\\sup_{\\mathsf{x}\\in\\XS}\\varphi(\\omega,\\mathsf{x})\\,\n\\mu(d\\omega),\n\\end{equation}\nwith the convention that, given a measurable function\n$\\varrho\\colon(\\Omega,\\FF)\\to\\RXX$, $\\int_\\Omega\\varrho d\\mu$ is\nthe usual Lebesgue integral, except when the Lebesgue integral\n$\\int_\\Omega\\min\\{\\varrho,0\\}d\\mu$ is $\\minf$, in which case\n$\\int_\\Omega\\varrho d\\mu=\\minf$.\n\\end{remark}\n\n\\begin{remark}\n\\label{r:8}\nIn Theorem~\\ref{t:1}, suppose that\n$\\inf_{x\\in\\XX}\\int_\\Omega\\varphi(\\Cdot,x(\\Cdot))\nd\\mu>\\minf$ and let $z\\in\\XX$. Then\n\\begin{equation}\n\\label{e:1x}\n\\int_\\Omega\\varphi\\big(\\omega,z(\\omega)\\big)\n\\mu(d\\omega)\n=\\min_{x\\in\\XX}\\int_\\Omega\\varphi\\big(\\omega,x(\\omega)\\big)\n\\mu(d\\omega)\n\\quad\\Leftrightarrow\\quad\n\\varphi\\big(\\Cdot,z(\\Cdot)\\big)=\\min\\varphi(\\Cdot,\\XS)\\,\\,\\mae\n\\end{equation}\n\\end{remark}\n\n\\section{Compliant spaces and normal integrands}\n\\label{sec:4}\n\nThe objective of this section is to develop tools to convert the\ninterchange principle of Theorem~\\ref{t:1} into interchange rules\nformulated in terms of explicit conditions on the ambient space\n$\\XX$ and the integrand $\\varphi$. Our framework hinges on a\nnotion of compliant spaces and a notion of normal integrands in an\nextended sense.\n\n\\subsection{Compliant spaces}\n\\label{sec:41}\n\nWe introduce the following notion of a compliant space, which\ngeneralizes and unifies the notions of decomposability employed in\nthe interchange rules of\n\\cite{Penn23,Perk18,Rock71,Roc76k,Rock09,Shap21,Vala75}.\n\n\\begin{definition}[compliance]\n\\label{d:1}\nSuppose that Assumption~\\ref{a:1}\\ref{a:1a}--\\ref{a:1e} holds. Then\n$\\XX$ is \\emph{compliant} if, for every $A\\in\\FF$ such that\n$\\mu(A)<\\pinf$ and every $z\\in\\mathcal{L}(\\Omega;\\XS)$ such\nthat $\\overline{z(A)}$ is compact, $1_Az\\in\\XX$.\n\\end{definition}\n\n\\begin{proposition}\n\\label{p:10}\nSuppose that Assumption~\\ref{a:1}\\ref{a:1a}--\\ref{a:1e} holds,\ntogether with one of the following:\n\\begin{enumerate}\n\\item\n\\label{p:10i}\n$(\\XS,\\EuScript{T}_\\XS)$ is a Souslin topological vector space and,\nfor every $A\\in\\FF$ such that $\\mu(A)<\\pinf$ and every\n$z\\in\\mathcal{L}(\\Omega;\\XS)$ such that $z(A)$ is\n$\\EuScript{T}_\\XS$-bounded (in the sense that, for every\nneighborhood $\\mathsf{V}\\in\\EuScript{T}_\\XS$\nof $\\mathsf{0}$, there exists $\\alpha\\in\\RPP$ such that\n$z(A)\\subset\\bigcap_{\\beta>\\alpha}\\beta\\mathsf{V}$\n\\cite{Rudi91}), $1_Az\\in\\XX$.\n\\item\n\\label{p:10ii}\n$\\XS$ is a separable Banach space with strong topology\n$\\EuScript{T}_\\XS$ and, for every $A\\in\\FF$ such that\n$\\mu(A)<\\pinf$ and every\n$z\\in\\mathcal{L}^\\infty(\\Omega;\\XS)$, $1_Az\\in\\XX$.\n\\item\n\\label{p:10iii}\n$\\XS$ is a separable Banach space with strong topology\n$\\EuScript{T}_\\XS$, $\\mu(\\Omega)<\\pinf$, and\n$\\mathcal{L}^\\infty(\\Omega;\\XS)\\subset\\XX$.\n\\item\n\\label{p:10iv}\n$\\XS$ is a separable Banach space with strong topology\n$\\EuScript{T}_\\XS$ and $\\XX$ is \\emph{Rockafellar-decomposable}\n\\cite{Rock71} in the sense that, for every $A\\in\\FF$ such that\n$\\mu(A)<\\pinf$, every $z\\in\\mathcal{L}^\\infty(\\Omega;\\XS)$,\nand every $x\\in\\XX$, $1_Az+1_{\\complement A}x\\in\\XX$.\n\\item\n\\label{p:10v}\n$(\\XS,\\EuScript{T}_\\XS)$ is a Souslin locally convex topological\nvector space and $\\XX$ is \\emph{Valadier-decomposable}\n\\cite{Vala75} in the sense that, for every $A\\in\\FF$ such that\n$\\mu(A)<\\pinf$, every $z\\in\\mathcal{L}(\\Omega;\\XS)$ such\nthat $\\overline{z(A)}$ is compact, and every $x\\in\\XX$,\n$1_Az+1_{\\complement A}x\\in\\XX$.\n\\item\n\\label{p:10vi}\n$\\XS$ is the standard Euclidean space $\\RR^N$\nand, for every $A\\in\\FF$ such that $\\mu(A)<\\pinf$ and every\n$z\\in\\mathcal{L}^\\infty(\\Omega;\\XS)$, $1_Az\\in\\XX$.\n\\end{enumerate}\nThen $\\XX$ is compliant.\n\\end{proposition}\n\\begin{proof}\n\\ref{p:10i}:\nLet $A\\in\\FF$ be such that $\\mu(A)<\\pinf$ and let\n$z\\in\\mathcal{L}(\\Omega;\\XS)$ be such that $\\overline{z(A)}$\nis compact. It results from \\cite[Theorem~1.15(b)]{Rudi91} that\n$z(A)$ is $\\EuScript{T}_\\XS$-bounded. Thus $1_Az\\in\\XX$.\n\n\\ref{p:10iii}$\\Rightarrow$\\ref{p:10ii}$\\Rightarrow$\\ref{p:10i}:\nClear.\n\n\\ref{p:10iv}$\\Rightarrow$\\ref{p:10ii}: Clear.\n\n\\ref{p:10v}: Clear.\n\n\\ref{p:10vi}$\\Rightarrow$\\ref{p:10ii}: Clear.\n\\end{proof}\n\n\\subsection{Normal integrands}\n\\label{sec:42}\n\nWe introduce a notion of a normal integrand which unifies and\nextends those of \\cite{Roc68a,Rock71,Roc76k,Vala75}.\n\n\\begin{definition}[normality]\n\\label{d:n}\nLet $(\\XS,\\EuScript{T}_\\XS)$ be a Souslin space, let\n$(\\Omega,\\FF)$ be a measurable space,\nlet $\\varphi\\colon(\\Omega\\times\\XS,\\FF\\otimes\\BE_\\XS)\\to\\RXX$ be\nmeasurable, and equip $\\XS\\times\\RR$ with the topology\n$\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR$.\nThen $\\varphi$ is a \\emph{normal integrand} if there\nexist sequences $(x_n)_{n\\in\\NN}$ in\n$\\mathcal{L}(\\Omega;\\XS)$ and $(\\varrho_n)_{n\\in\\NN}$ in\n$\\mathcal{L}(\\Omega;\\RR)$ such that\n\\begin{equation}\n(\\forall\\omega\\in\\Omega)\\quad\n\\big\\{\\big(x_n(\\omega),\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\NN}\n\\subset\\epi\\varphi_\\omega\n\\quad\\text{and}\\quad\n\\overline{\\epi\\varphi_\\omega}=\n\\overline{\\big\\{\\big(x_n(\\omega),\n\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\NN}}.\n\\end{equation}\n\\end{definition}\n\nThe following theorem furnishes examples of normal integrands.\n\n\\begin{theorem}\n\\label{t:3}\nLet $(\\XS,\\EuScript{T}_\\XS)$ be a Souslin space,\nlet $(\\Omega,\\FF)$ be a measurable space, and\nlet $\\varphi\\colon\\Omega\\times\\XS\\to\\RXX$ be such that\n$(\\forall\\omega\\in\\Omega)$ $\\epi\\varphi_\\omega\\neq\\emp$.\nSuppose that one of the following holds:\n\\begin{enumerate}\n\\item\n\\label{t:3i}\n$\\varphi$ is $\\FF\\otimes\\BE_\\XS$-measurable and\none of the following is satisfied:\n\\begin{enumerate}\n\\item\n\\label{t:3ia}\nThere exists a measure $\\mu$ such that\n$(\\Omega,\\FF,\\mu)$ is complete and $\\sigma$-finite.\n\\item\n\\label{t:3ib}\n$\\Omega$ is a Borel subset of $\\RR^M$ and $\\FF$ is the\nassociated Lebesgue $\\sigma$-algebra.\n\\item\n\\label{t:3ic}\nFor every $\\omega\\in\\Omega$, there exists\n$\\boldsymbol{\\mathsf{V}}_\\omega\\in\n\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR$ such that\n$\\boldsymbol{\\mathsf{V}}_\\omega\\subset\\epi\\varphi_\\omega$\nand $\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}\n=\\overline{\\epi\\varphi_\\omega}$.\n\\item\n\\label{t:3id}\nThe functions $(\\varphi_\\omega)_{\\omega\\in\\Omega}$ are upper\nsemicontinuous.\n\\end{enumerate}\n\\item\n\\label{t:3ii}\nThe functions $(\\varphi(\\Cdot,\\mathsf{x}))_{\\mathsf{x}\\in\\XS}$ are\n$\\FF$-measurable and one of the following is satisfied:\n\\begin{enumerate}\n\\item\n\\label{t:3iia}\n$(\\XS,\\EuScript{T}_\\XS)$ is metrizable and, for every\n$\\omega\\in\\Omega$, there exists $\\boldsymbol{\\mathsf{V}}_\\omega\\in\n\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR$ such that\n$\\boldsymbol{\\mathsf{V}}_\\omega\\subset\\epi\\varphi_\\omega\n=\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}$.\n\\item\n\\label{t:3iib}\n$(\\XS,\\EuScript{T}_\\XS)$ is a Fr\\'echet space and, for every\n$\\omega\\in\\Omega$, $\\varphi_\\omega\\in\\Gamma_0(\\XS)$ and\n$\\intdom\\varphi_\\omega\\neq\\emp$.\n\\item\n\\label{t:3iic}\n$(\\XS,\\EuScript{T}_\\XS)$ is the standard Euclidean line $\\RR$\nand, for every $\\omega\\in\\Omega$, $\\varphi_\\omega\\in\\Gamma_0(\\RR)$\nand $\\dom\\varphi_\\omega$ is not a singleton.\n\\end{enumerate}\n\\item\n\\label{t:3iii}\n$(\\XS,\\EuScript{T}_\\XS)$ is a regular Souslin space,\nthe functions $(\\varphi_\\omega)_{\\omega\\in\\Omega}$ are continuous,\nand the functions $(\\varphi(\\Cdot,\\mathsf{x}))_{\\mathsf{x}\\in\\XS}$\nare $\\FF$-measurable.\n\\item\n\\label{t:3iv}\nFor some separable Fr\\'echet space $(\\YS,\\EuScript{T}_\\YS)$,\n$\\XS=(\\YS,\\EuScript{T}_\\YS)^*$, $\\EuScript{T}_\\XS$ is the weak\ntopology, the functions\n$(\\varphi_\\omega)_{\\omega\\in\\Omega}$ are $\\EuScript{T}_\\XS$-lower\nsemicontinuous, and one of the following is satisfied:\n\\begin{enumerate}\n\\item\n\\label{t:3iva}\nFor every closed subset $\\boldsymbol{\\mathsf{C}}$ of\n$(\\XS\\times\\RR,\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR)$,\n$\\menge{\\omega\\in\\Omega}{\\boldsymbol{\\mathsf{C}}\n\\cap\\epi\\varphi_\\omega\\neq\\emp}\\in\\FF$.\n\\item\n\\label{t:3ivb}\n$(\\Omega,\\EuScript{T}_\\Omega)$ is a Hausdorff topological space,\n$\\FF=\\BE_\\Omega$, and\n$\\varphi$ is $\\EuScript{T}_\\Omega\\boxtimes\\EuScript{T}_\\XS$-lower\nsemicontinuous.\n\\item\n\\label{t:3ivc}\n$(\\Omega,\\EuScript{T}_\\Omega)$ is a Lusin space,\n$\\FF=\\BE_\\Omega$, and $\\varphi$ is $\\FF\\otimes\\BE_\\XS$-measurable.\n\\end{enumerate}\n\\item\n\\label{t:3v}\n$\\XS$ is a separable reflexive Banach space, $\\EuScript{T}_\\XS$ is\nthe weak topology,\n$(\\Omega,\\EuScript{T}_\\Omega)$ is a Hausdorff topological space,\n$\\FF=\\BE_\\Omega$, the functions\n$(\\varphi_\\omega)_{\\omega\\in\\Omega}$ are\n$\\EuScript{T}_\\XS$-lower semicontinuous, and one of the following\nis satisfied:\n\\begin{enumerate}\n\\item\n\\label{t:3va}\n$\\varphi$ is $\\EuScript{T}_\\Omega\\boxtimes\\EuScript{T}_\\XS$-lower\nsemicontinuous.\n\\item\n\\label{t:3vb}\n$(\\Omega,\\EuScript{T}_\\Omega)$ is a Lusin space and\n$\\varphi$ is $\\FF\\otimes\\BE_\\XS$-measurable.\n\\end{enumerate}\n\\item\n\\label{t:3vi}\n$(\\XS,\\EuScript{T}_\\XS)$ is the standard Euclidean space $\\RR^N$,\n$\\Omega$ is a Borel subset of $\\RR^M$,\n$\\FF=\\BE_\\Omega$, $\\varphi$ is $\\FF\\otimes\\BE_\\XS$-measurable, and\nthe functions $(\\varphi_\\omega)_{\\omega\\in\\Omega}$ are lower\nsemicontinuous.\n\\item\n\\label{t:3vii}\n$(\\XS,\\EuScript{T}_\\XS)$ is a Polish space, the functions\n$(\\varphi_\\omega)_{\\omega\\in\\Omega}$ are lower semicontinuous, and\none of the following is satisfied:\n\\begin{enumerate}\n\\item\n\\label{t:3viia}\nFor every $\\boldsymbol{\\mathsf{V}}\\in\n\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR$,\n$\\menge{\\omega\\in\\Omega}{\\boldsymbol{\\mathsf{V}}\\cap\n\\epi\\varphi_\\omega\\neq\\emp}\\in\\FF$.\n\\item\n\\label{t:3viib}\n$(\\XS,\\EuScript{T}_\\XS)$ is the standard Euclidean space $\\RR^N$\nand, for every closed subset $\\boldsymbol{\\mathsf{C}}$ of\n$\\XS\\times\\RR$, $\\menge{\\omega\\in\\Omega}{\n\\boldsymbol{\\mathsf{C}}\\cap\\epi\\varphi_\\omega\\neq\\emp}\\in\\FF$.\n\\end{enumerate}\n\\item\n\\label{t:3viii}\nThere exists a measurable function\n$\\mathsf{f}\\colon(\\XS,\\BE_\\XS)\\to\\RXX$ such that\n$(\\forall\\omega\\in\\Omega)$ $\\varphi_\\omega=\\mathsf{f}$.\n\\end{enumerate}\nThen $\\varphi$ is normal.\n\\end{theorem}\n\\begin{proof}\nSet $\\boldsymbol{G}=\\menge{(\\omega,\\mathsf{x},\\xi)\\in\n\\Omega\\times\\XS\\times\\RR}{\\varphi(\\omega,\\mathsf{x})\\leq\\xi}$.\nThen\n\\begin{equation}\n\\label{e:s03}\n\\boldsymbol{G}=\\menge{(\\omega,\\mathsf{x},\\xi)\\in\n\\Omega\\times\\XS\\times\\RR}{(\\mathsf{x},\\xi)\\in\\epi\\varphi_\\omega}.\n\\end{equation}\nFurther, \\cite[Lemma~6.4.2(i)]{Boga07} yields\n\\begin{equation}\n\\label{e:s04}\n\\varphi\\,\\,\\text{is $\\FF\\otimes\\BE_\\XS$-measurable}\n\\quad\\Leftrightarrow\\quad\n\\boldsymbol{G}\\in\\FF\\otimes\\BE_\\XS\\otimes\\BE_\\RR\n=\\FF\\otimes\\BE_{\\XS\\times\\RR}.\n\\end{equation}\nWe also note that $(\\XS\\times\\RR,\n\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR)$ is a Souslin space\n\\cite[Proposition~IX.6.7]{Bour74}.\n\n\\ref{t:3ia}:\nApplying \\cite[Theorem~III.22]{Cast77} to the mapping\n$\\Upsilon\\colon\\Omega\\to\n2^{\\XS\\times\\RR}\\colon\\omega\\mapsto\\epi\\varphi_\\omega$, we deduce\nfrom \\eqref{e:s03} and \\eqref{e:s04} that there exist a sequence\n$(x_n)_{n\\in\\NN}$ of mappings from $\\Omega$ to $\\XS$ and a\nsequence $(\\varrho_n)_{n\\in\\NN}$ of functions from $\\Omega$ to\n$\\RR$ such that\n\\begin{equation}\n\\label{e:10ds}\n(\\forall n\\in\\NN)\\quad\n(\\Omega,\\FF)\\to(\\XS\\times\\RR,\\BE_{\\XS\\times\\RR})\\colon\n\\omega\\mapsto\\big(x_n(\\omega),\\varrho_n(\\omega)\\big)\n\\,\\,\\text{is measurable}\n\\end{equation}\nand\n\\begin{equation}\n(\\forall\\omega\\in\\Omega)\\quad\n\\big\\{\\big(x_n(\\omega),\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\NN}\n\\subset\\Upsilon(\\omega)\n\\quad\\text{and}\\quad\n\\overline{\\Upsilon(\\omega)}=\n\\overline{\\big\\{\\big(x_n(\\omega),\n\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\NN}}.\n\\end{equation}\nMoreover, since $\\BE_{\\XS\\times\\RR}=\\BE_\\XS\\otimes\\BE_\\RR$\n\\cite[Lemma~6.4.2(i)]{Boga07},\nit follows from \\eqref{e:10ds} that, for every $n\\in\\NN$,\n$x_n\\colon(\\Omega,\\FF)\\to(\\XS,\\BE_\\XS)$ and\n$\\varrho_n\\colon(\\Omega,\\FF)\\to(\\RR,\\BE_\\RR)$\nare measurable. Altogether, $\\varphi$ is normal.\n\n\\ref{t:3ib}$\\Rightarrow$\\ref{t:3ia}:\nTake $\\mu$ to be the Lebesgue measure on $\\Omega$.\n\n\\ref{t:3ic}:\nLet $\\{(\\mathsf{x}_n,\\xi_n)\\}_{n\\in\\NN}$ be a dense set in\n$(\\XS\\times\\RR,\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR)$ and\ndefine\n\\begin{equation}\n\\label{e:rc0}\n(\\forall n\\in\\NN)\\quad\n\\Omega_n=\\big[\\varphi(\\Cdot,\\mathsf{x}_n)\\leq\\xi_n\\big].\n\\end{equation}\nOn the one hand, the $\\FF\\otimes\\BE_\\XS$-measurability of\n$\\varphi$ ensures\nthat $(\\forall n\\in\\NN)$ $\\Omega_n\\in\\FF$.\nOn the other hand, for every $\\omega\\in\\Omega$,\nsince $\\boldsymbol{\\mathsf{V}}_\\omega$ is open,\nthere exists $n\\in\\NN$ such that\n$(\\mathsf{x}_n,\\xi_n)\\in\\boldsymbol{\\mathsf{V}}_\\omega\n\\subset\\epi\\varphi_\\omega$, which yields\n$\\omega\\in\\Omega_n$ and thus\n$\\Omega=\\bigcup_{k\\in\\NN}\\Omega_k$.\nThis yields a sequence $(\\Theta_n)_{n\\in\\NN}$ of pairwise disjoint\nsets in $\\FF$ such that\n\\begin{equation}\n\\label{e:tf0}\n\\Theta_0=\\Omega_0,\\quad\n\\bigcup_{n\\in\\NN}\\Theta_n=\\Omega,\\quad\\text{and}\\quad\n(\\forall n\\in\\NN)\\;\\;\\Theta_n\\subset\\Omega_n.\n\\end{equation}\nFor every $\\omega\\in\\Omega$, there exists a unique \n$n_\\omega\\in\\NN$ such that $\\omega\\in\\Theta_{n_\\omega}$.\nNow define\n\\begin{equation}\nz\\colon\\Omega\\to\\XS\\colon\\omega\\mapsto\\mathsf{x}_{n_\\omega}\n\\quad\\text{and}\\quad\n\\vartheta\\colon\\Omega\\to\\RR\\colon\\omega\\mapsto\\xi_{n_\\omega}.\n\\end{equation}\nThen\n\\begin{equation}\n(\\forall\\mathsf{V}\\in\\EuScript{T}_\\XS)\\quad\nz^{-1}(\\mathsf{V})=\\bigcup_{\\substack{n\\in\\NN\\\\\n\\mathsf{x}_n\\in\\mathsf{V}}}\\Theta_n\\in\\FF,\n\\end{equation}\nwhich implies that $z\\in\\mathcal{L}(\\Omega;\\XS)$. Likewise,\n$\\vartheta\\in\\mathcal{L}(\\Omega;\\RR)$. Next, define\n\\begin{equation}\n\\label{e:xxi0}\n(\\forall n\\in\\NN)\\quad\nx_n\\colon\\Omega\\to\\XS\\colon\\omega\\mapsto\n\\begin{cases}\n\\mathsf{x}_n,&\\text{if}\\,\\,\\omega\\in\\Omega_n;\\\\\nz(\\omega),&\\text{if}\\,\\,\\omega\\in\\complement\\Omega_n\n\\end{cases}\n\\end{equation}\nand\n\\begin{equation}\n\\label{e:xxi1}\n(\\forall n\\in\\NN)\\quad\n\\varrho_n\\colon\\Omega\\to\\RR\\colon\\omega\\mapsto\n\\begin{cases}\n\\xi_n,&\\text{if}\\,\\,\\omega\\in\\Omega_n;\\\\\n\\vartheta(\\omega),\n&\\text{if}\\,\\,\\omega\\in\\complement\\Omega_n.\n\\end{cases}\n\\end{equation}\nThen $(x_n)_{n\\in\\NN}$ and $(\\varrho_n)_{n\\in\\NN}$ are sequences\nin $\\mathcal{L}(\\Omega;\\XS)$ and\n$\\mathcal{L}(\\Omega;\\RR)$, respectively.\nMoreover, we deduce from \\eqref{e:xxi0}, \\eqref{e:xxi1},\n\\eqref{e:rc0}, and \\eqref{e:tf0} that\n\\begin{equation}\n(\\forall\\omega\\in\\Omega)(\\forall n\\in\\NN)\\quad\n\\big(x_n(\\omega),\\varrho_n(\\omega)\\big)\\in\\epi\\varphi_\\omega.\n\\end{equation}\nOn the other hand, for every $\\omega\\in\\Omega$,\nsince $\\{(\\mathsf{x}_n,\\xi_n)\\}_{n\\in\\NN}$ is\ndense in $(\\XS\\times\\RR,\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR)$\nand since\n$\\boldsymbol{\\mathsf{V}}_\\omega$ is open, we infer from\n\\eqref{e:xxi0}, \\eqref{e:xxi1}, and \\eqref{e:rc0} that\n\\begin{equation}\n\\overline{\\big\\{\\big(x_n(\\omega),\n\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\NN}}\n=\\overline{\\big\\{(\\mathsf{x}_n,\\xi_n)\\big\\}_{n\\in\\NN}\\cap\n\\epi\\varphi_\\omega}\n\\supset\\overline{\\big\\{(\\mathsf{x}_n,\\xi_n)\\big\\}_{n\\in\\NN}\\cap\n\\boldsymbol{\\mathsf{V}}_\\omega}\n=\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}\n=\\overline{\\epi\\varphi_\\omega}.\n\\end{equation}\nConsequently, $\\varphi$ is normal.\n\n\\ref{t:3id}$\\Rightarrow$\\ref{t:3ic}:\nSet $(\\forall\\omega\\in\\Omega)$\n$\\boldsymbol{\\mathsf{V}}_\\omega=\\menge{(\\mathsf{x},\\xi)\\in\n\\XS\\times\\RR}{\\varphi(\\omega,\\mathsf{x})<\\xi}$.\nNow fix $\\omega\\in\\Omega$ and\n$(\\mathsf{x},\\xi)\\in\\epi\\varphi_\\omega$.\nSince the sequence $(\\mathsf{x},\\xi+2^{-n})_{n\\in\\NN}$\nlies in $\\boldsymbol{\\mathsf{V}}_\\omega$ and\n$(\\mathsf{x},\\xi+2^{-n})\\to(\\mathsf{x},\\xi)$, we obtain\n$(\\mathsf{x},\\xi)\\in\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}$.\nHence $\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}=\n\\overline{\\epi\\varphi_\\omega}$.\nAt the same time, the upper semicontinuity of $\\varphi_\\omega$\nguarantees that $\\boldsymbol{\\mathsf{V}}_\\omega$ is open.\n\n\\ref{t:3iia}$\\Rightarrow$\\ref{t:3ic}:\nIt suffices to show that $\\varphi$ is\n$\\FF\\otimes\\BE_\\XS$-measurable. Let\n$\\{(\\mathsf{x}_n,\\xi_n)\\}_{n\\in\\NN}$ be dense in\n$(\\XS\\times\\RR,\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR)$,\nlet $\\boldsymbol{\\mathsf{V}}\\in\n\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR$,\nand set $\\mathbb{K}=\\menge{n\\in\\NN}{\n(\\mathsf{x}_n,\\xi_n)\\in\\boldsymbol{\\mathsf{V}}}$. Then\n\\begin{equation}\n\\label{e:vd9}\n\\overline{\\{(\\mathsf{x}_n,\\xi_n)\\}_{n\\in\\mathbb{K}}}\n=\\overline{\\{(\\mathsf{x}_n,\\xi_n)\\}_{n\\in\\NN}\\cap\n\\boldsymbol{\\mathsf{V}}}\n=\\overline{\\boldsymbol{\\mathsf{V}}}.\n\\end{equation}\nSuppose that there exists $\\omega\\in\\Omega$ such that\n\\begin{equation}\n\\label{e:tx}\n\\boldsymbol{\\mathsf{V}}\\cap\\epi\\varphi_\\omega\\neq\\emp\n\\quad\\text{and}\\quad\n(\\forall n\\in\\mathbb{K})\\;\\;\n(\\mathsf{x}_n,\\xi_n)\\notin\\epi\\varphi_\\omega.\n\\end{equation}\nSince $\\boldsymbol{\\mathsf{V}}$ is open and\n$\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}=\\epi\\varphi_\\omega$,\nthere exists $(\\mathsf{y},\\eta)\n\\in\\boldsymbol{\\mathsf{V}}\\cap\\boldsymbol{\\mathsf{V}}_\\omega$.\nTherefore, we infer from \\eqref{e:vd9} that there exists a subnet\n$(\\mathsf{x}_{k(b)},\\xi_{k(b)})_{b\\in B}$ of\n$(\\mathsf{x}_n,\\xi_n)_{n\\in\\mathbb{K}}$ such that\n$(\\mathsf{x}_{k(b)},\\xi_{k(b)})\\to(\\mathsf{y},\\eta)$.\nThis and \\eqref{e:tx} force\n$(\\mathsf{y},\\eta)\n\\in\\overline{\\complement\\epi\\varphi_\\omega}\n=\\overline{\\complement\n\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}}\n=\\complement\\inte\n\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}$,\nwhich is in contradiction with the inclusion\n$(\\mathsf{y},\\eta)\\in\\boldsymbol{\\mathsf{V}}_\\omega$.\nHence, the $\\FF$-measurability of the functions\n$(\\varphi(\\Cdot,\\mathsf{x}))_{\\mathsf{x}\\in\\XS}$ yields\n\\begin{equation}\n\\menge{\\omega\\in\\Omega}{\n\\boldsymbol{\\mathsf{V}}\\cap\\epi\\varphi_\\omega\\neq\\emp}\n=\\bigcup_{n\\in\\mathbb{K}}\\menge{\\omega\\in\\Omega}{\n(\\mathsf{x}_n,\\xi_n)\\in\\epi\\varphi_\\omega}\n=\\bigcup_{n\\in\\mathbb{K}}\n\\big[\\varphi(\\Cdot,\\mathsf{x}_n)\\leq\\xi_n\\big]\n\\in\\FF.\n\\end{equation}\nTherefore, since\n$(\\XS\\times\\RR,\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR)$ is a\nseparable metrizable space\nand the sets $(\\epi\\varphi_\\omega)_{\\omega\\in\\Omega}$ are closed,\n\\cite[Theorem~3.5(i)]{Himm75} and \\eqref{e:s03} imply that\n$\\boldsymbol{G}\\in\\FF\\otimes\\BE_{\\XS\\times\\RR}$.\nConsequently, \\eqref{e:s04} asserts that $\\varphi$ is\n$\\FF\\otimes\\BE_\\XS$-measurable.\n\n\\ref{t:3iib}$\\Rightarrow$\\ref{t:3iia}:\nSet $(\\forall\\omega\\in\\Omega)$\n$\\boldsymbol{\\mathsf{V}}_\\omega=\\inte\\epi\\varphi_\\omega$.\nFor every $\\omega\\in\\Omega$, the assumption ensures that\n$\\epi\\varphi_\\omega$ is closed and convex, and that\n$\\boldsymbol{\\mathsf{V}}_\\omega\\neq\\emp$\n\\cite[Theorem~2.2.20 and Corollary~2.2.10]{Zali02}.\nThus \\cite[Theorem~1.1.2(iv)]{Zali02} yields\n$(\\forall\\omega\\in\\Omega)$\n$\\epi\\varphi_\\omega=\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}$.\n\n\\ref{t:3iic}$\\Rightarrow$\\ref{t:3iib}:\nClear.\n\n\\ref{t:3iii}:\nIt results from \\cite{Sain76} that there exists a topology\n$\\widetilde{\\EuScript{T}_\\XS}$ on $\\XS$ such that\n\\begin{equation}\n\\label{e:8dy}\n\\EuScript{T}_\\XS\\subset\\widetilde{\\EuScript{T}_\\XS}\n\\end{equation}\nand\n\\begin{equation}\n\\label{e:8dx}\n\\big(\\XS,\\widetilde{\\EuScript{T}_\\XS}\\big)\\,\\,\n\\text{is a metrizable Souslin space}.\n\\end{equation}\nSet $(\\forall\\omega\\in\\Omega)$\n$\\boldsymbol{\\mathsf{V}}_\\omega=\\menge{(\\mathsf{x},\\xi)\\in\n\\XS\\times\\RR}{\\varphi(\\omega,\\mathsf{x})<\\xi}$.\nThen, since \\eqref{e:8dy} implies that\n\\begin{equation}\n\\label{e:gf}\n(\\forall\\omega\\in\\Omega)\\quad\n\\varphi_\\omega\\,\\,\n\\text{is $\\widetilde{\\EuScript{T}_\\XS}$-continuous},\n\\end{equation}\nit follows that\n\\begin{equation}\n\\label{e:r0d}\n(\\forall\\omega\\in\\Omega)\\quad\n\\boldsymbol{\\mathsf{V}}_\\omega\\in\n\\widetilde{\\EuScript{T}_\\XS}\\boxtimes\\EuScript{T}_\\RR\n\\quad\\text{and}\\quad\n\\overline{\\boldsymbol{\\mathsf{V}}_\\omega}^{\n\\widetilde{\\EuScript{T}_\\XS}\\boxtimes\\EuScript{T}_\\RR}\n=\\overline{\\epi\\varphi_\\omega}^{\\widetilde{\\EuScript{T}_\\XS}\n\\boxtimes\\EuScript{T}_\\RR}\n=\\epi\\varphi_\\omega.\n\\end{equation}\nOn the other hand, we derive from \\eqref{e:8dx}, \\eqref{e:8dy},\nand \\cite[Corollary~2, p.~101]{Schw73} that\nthe Borel $\\sigma$-algebra of $(\\XS,\\widetilde{\\EuScript{T}_\\XS})$\nis $\\BE_\\XS$.\nAltogether, applying \\ref{t:3iia} to the metrizable Souslin space\n$(\\XS,\\widetilde{\\EuScript{T}_\\XS})$, \nwe deduce that $\\varphi$ is $\\FF\\otimes\\BE_\\XS$-measurable and\nthat there exist sequences $(x_n)_{n\\in\\NN}$ in\n$\\mathcal{L}(\\Omega;\\XS)$ and $(\\varrho_n)_{n\\in\\NN}$ in\n$\\mathcal{L}(\\Omega;\\RR)$ such that\n\\begin{equation}\n(\\forall\\omega\\in\\Omega)\\quad\n\\big\\{\\big(x_n(\\omega),\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\NN}\n\\subset\\epi\\varphi_\\omega\n\\quad\\text{and}\\quad\n\\overline{\\epi\\varphi_\\omega}^{\n\\widetilde{\\EuScript{T}_\\XS}\\boxtimes\\EuScript{T}_\\RR}=\n\\overline{\\big\\{\\big(x_n(\\omega),\n\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\NN}}^{\n\\widetilde{\\EuScript{T}_\\XS}\\boxtimes\\EuScript{T}_\\RR}.\n\\end{equation}\nHence, by \\eqref{e:8dy} and \\eqref{e:r0d},\n\\begin{align}\n\\overline{\\big\\{\\big(x_n(\\omega),\n\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\NN}}\n\\supset\\overline{\\big\\{\\big(x_n(\\omega),\n\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\NN}}^{\n\\widetilde{\\EuScript{T}_\\XS}\\boxtimes\\EuScript{T}_\\RR}\n=\\overline{\\epi\\varphi_\\omega}^{\n\\widetilde{\\EuScript{T}_\\XS}\\boxtimes\\EuScript{T}_\\RR}\n=\\epi\\varphi_\\omega.\n\\end{align}\nConsequently, $\\varphi$ is normal.\n\n\\ref{t:3iv}:\nIt follows from \\cite[Section~II.4.3]{Bour81} that\n$(\\YS\\times\\RR,\\EuScript{T}_\\YS\\boxtimes\\EuScript{T}_\\RR)$ is a\nseparable Fr\\'echet space.\nMoreover, by \\cite[Proposition~II.6.8]{Bour81}, $\\XS\\times\\RR=\n(\\YS\\times\\RR,\\EuScript{T}_\\YS\\boxtimes\\EuScript{T}_\\RR)^*$\nand the weak topology of $\\XS\\times\\RR$ is\n$\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR$. In turn, arguing as in\n\\cite[Section~IV-1.7]{Scha99}, we deduce that there exists\na covering\n$(\\boldsymbol{\\mathsf{C}}_n)_{n\\in\\NN}$ of $\\XS\\times\\RR$,\nwith respective\n$\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR$-induced topologies\n$(\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n})_{n\\in\\NN}$,\nsuch that, for every $n\\in\\NN$,\n$(\\boldsymbol{\\mathsf{C}}_n,\n\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n})$\nis a compact separable metrizable space, hence a Polish space. We\nalso introduce\n\\begin{equation}\n\\label{e:q0x}\n(\\forall n\\in\\NN)\\quad\nQ_n\\colon\\Omega\\times\\boldsymbol{\\mathsf{C}}_n\\to\\Omega\\colon\n(\\omega,\\mathsf{x},\\xi)\\mapsto\\omega.\n\\end{equation}\nNote that, for every subset $\\boldsymbol{\\mathsf{C}}$\nof $\\XS\\times\\RR$,\n\\begin{equation}\n\\label{e:rz2}\n\\menge{\\omega\\in\\Omega}{\\boldsymbol{\\mathsf{C}}\n\\cap\\epi\\varphi_\\omega\\neq\\emp}\n=\\bigcup_{n\\in\\NN}\n\\menge{\\omega\\in\\Omega}{\\boldsymbol{\\mathsf{C}}\\cap\n\\boldsymbol{\\mathsf{C}}_n\n\\cap\\epi\\varphi_\\omega\\neq\\emp}\n=\\bigcup_{n\\in\\NN}Q_n\\Big(\\boldsymbol{G}\\cap\n\\big(\\Omega\\times(\\boldsymbol{\\mathsf{C}}\\cap\n\\boldsymbol{\\mathsf{C}}_n)\\big)\\Big).\n\\end{equation}\n\n\\ref{t:3iva}:\nFor every $n\\in\\NN$, set\n\\begin{equation}\n\\Omega_n=\\menge{\\omega\\in\\Omega}{\n\\boldsymbol{\\mathsf{C}}_n\\cap\\epi\\varphi_\\omega\\neq\\emp},\n\\end{equation}\ndenote by $\\FF_n$ the trace $\\sigma$-algebra of $\\FF$ on\n$\\Omega_n$, and observe that\n\\begin{equation}\n\\label{e:0f}\n\\Omega_n\\in\\FF\\quad\\text{and}\\quad\\FF_n\\subset\\FF.\n\\end{equation}\nNow define\n\\begin{equation}\n\\label{e:kx}\n\\mathbb{K}=\\menge{n\\in\\NN}{\\Omega_n\\neq\\emp}\n\\quad\\text{and}\\quad\n(\\forall n\\in\\mathbb{K})\\;\\;\nK_n\\colon\\Omega_n\\to 2^{\\boldsymbol{\\mathsf{C}}_n}\\colon\n\\omega\\mapsto\\boldsymbol{\\mathsf{C}}_n\\cap\\epi\\varphi_\\omega.\n\\end{equation}\nThen\n\\begin{equation}\n\\label{e:ky}\n\\mathbb{K}\\neq\\emp\\quad\\text{and}\\quad\n\\bigcup_{n\\in\\mathbb{K}}\\Omega_n=\\Omega.\n\\end{equation}\nFurthermore, the\n$\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR$-closedness of\n$(\\epi\\varphi_\\omega)_{\\omega\\in\\Omega}$ guarantees that\n\\begin{equation}\n(\\forall n\\in\\mathbb{K})(\\forall\\omega\\in\\Omega)\\quad\nK_n(\\omega)\\;\\text{is\n$\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n}$-closed}.\n\\end{equation}\nOn the other hand, for every $n\\in\\mathbb{K}$ and every\nclosed subset\n$\\boldsymbol{\\mathsf{D}}$ of $(\\boldsymbol{\\mathsf{C}}_n,\n\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n})$,\nthere exists a closed subset\n$\\boldsymbol{\\mathsf{E}}$ of\n$(\\XS\\times\\RR,\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR)$\nsuch that\n$\\boldsymbol{\\mathsf{D}}=\\boldsymbol{\\mathsf{C}}_n\\cap\n\\boldsymbol{\\mathsf{E}}$ \\cite[Section~I.3.1]{Bour71}\nand therefore, since $\\boldsymbol{\\mathsf{C}}_n$ is\n$\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR$-closed,\nwe deduce from \\eqref{e:0f} that\n\\begin{equation}\n\\menge{\\omega\\in\\Omega_n}{\\boldsymbol{\\mathsf{D}}\\cap K_n(\\omega)\n\\neq\\emp}\n=\\Omega_n\\cap\\menge{\\omega\\in\\Omega}{\n\\boldsymbol{\\mathsf{C}}_n\\cap\\boldsymbol{\\mathsf{E}}\n\\cap\\epi\\varphi_\\omega\\neq\\emp}\n\\in\\FF_n.\n\\end{equation}\nHence, for every $n\\in\\mathbb{K}$, since\n$(\\boldsymbol{\\mathsf{C}}_n,\n\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n})$ is a Polish space,\nwe deduce from\n\\cite[Theorem~3.5(i), Theorem~5.1, and Theorem~5.6]{Himm75}\nthat there exist measurable mappings\n$\\boldsymbol{y}_n$ and $(\\boldsymbol{z}_{n,k})_{k\\in\\NN}$ from\n$(\\Omega_n,\\FF_n)$ to\n$(\\boldsymbol{\\mathsf{C}}_n,\\BE_{\\boldsymbol{\\mathsf{C}}_n})$\nsuch that\n\\begin{equation}\n\\label{e:tp}\n(\\forall\\omega\\in\\Omega_n)\\quad\n\\boldsymbol{y}_n(\\omega)\\in K_n(\\omega)\\quad\\text{and}\\quad\nK_n(\\omega)\n=\\overline{\\big\\{\\boldsymbol{z}_{n,k}(\\omega)\\big\\}_{\nk\\in\\NN}}^{\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n}}\n=\\boldsymbol{\\mathsf{C}}_n\\cap\n\\overline{\\big\\{\\boldsymbol{z}_{n,k}(\\omega)\\big\\}_{k\\in\\NN}}.\n\\end{equation}\nIn addition, since \\cite[Theorem~3.5(i)]{Himm75} asserts that\n\\begin{align}\n&\n(\\forall n\\in\\mathbb{K})\\quad\n\\menge{(\\omega,\\mathsf{x},\\xi)\\in\n\\Omega_n\\times\\boldsymbol{\\mathsf{C}}_n}{\n(\\mathsf{x},\\xi)\\in\\boldsymbol{\\mathsf{C}}_n\\cap\\epi\\varphi_\\omega}\n\\nonumber\\\\\n&\\hskip 26mm\n=\\menge{(\\omega,\\mathsf{x},\\xi)\\in\n\\Omega_n\\times\\boldsymbol{\\mathsf{C}}_n}{\n(\\mathsf{x},\\xi)\\in K_n(\\omega)}\n\\nonumber\\\\\n&\\hskip 26mm\n\\in\\FF_n\\otimes\\BE_{\\boldsymbol{\\mathsf{C}}_n}\n\\nonumber\\\\\n&\\hskip 26mm\n\\subset\\FF\\otimes\\BE_{\\XS\\times\\RR},\n\\end{align}\nwe get from \\eqref{e:s03} that\n\\begin{equation}\n\\boldsymbol{G}\n=\\bigcup_{n\\in\\mathbb{K}}\\menge{(\\omega,\\mathsf{x},\\xi)\\in\n\\Omega_n\\times\\boldsymbol{\\mathsf{C}}_n}{\n(\\mathsf{x},\\xi)\\in\\boldsymbol{\\mathsf{C}}_n\\cap\\epi\\varphi_\\omega}\n\\in\\FF\\otimes\\BE_{\\XS\\times\\RR}.\n\\end{equation}\nThus, in the light of \\eqref{e:s04}, $\\varphi$ is\n$\\FF\\otimes\\BE_\\XS$-measurable.\nNext, using \\eqref{e:ky}, we construct a family\n$(\\Theta_n)_{n\\in\\mathbb{K}}$ of pairwise disjoint sets in $\\FF$\nsuch that\n\\begin{equation}\n\\label{e:yf}\n\\Theta_{\\min\\mathbb{K}}=\\Omega_{\\min\\mathbb{K}},\n\\quad\n\\bigcup_{n\\in\\mathbb{K}}\\Theta_n=\\Omega,\n\\quad\\text{and}\\quad\n(\\forall n\\in\\mathbb{K})\\;\\;\\Theta_n\\subset\\Omega_n.\n\\end{equation}\nIn turn, for every $\\omega\\in\\Omega$, there exists a unique\n$\\ell_\\omega\\in\\mathbb{K}$ such that\n$\\omega\\in\\Theta_{\\ell_\\omega}$. Therefore, appealing to\n\\eqref{e:yf}, the mapping\n\\begin{equation}\n\\label{e:8uy}\n\\boldsymbol{y}\\colon\\Omega\\to\\XS\\times\\RR\\colon\n\\omega\\mapsto\\boldsymbol{y}_{\\ell_\\omega}(\\omega)\n\\end{equation}\nis well defined and, in view of \\eqref{e:tp},\n\\begin{equation}\n\\label{e:z0}\n(\\forall\\omega\\in\\Omega)\\quad\n\\boldsymbol{y}(\\omega)\n=\\boldsymbol{y}_{\\ell_\\omega}(\\omega)\n\\in K_{\\ell_\\omega}(\\omega)\n\\subset\n\\epi\\varphi_\\omega.\n\\end{equation}\nLet $\\boldsymbol{\\mathsf{V}}\\in\n\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR$.\nThen, for every $n\\in\\mathbb{K}$,\n$\\boldsymbol{\\mathsf{V}}\\cap\\boldsymbol{\\mathsf{C}}_n$ is\n$\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n}$-open\nand thus the measurability of\n$\\boldsymbol{y}_n\\colon(\\Omega_n,\\FF_n)\\to\n(\\boldsymbol{\\mathsf{C}}_n,\\BE_{\\boldsymbol{\\mathsf{C}}_n})$ and\n\\eqref{e:0f} ensure that $\\boldsymbol{y}_n^{-1}(\n\\boldsymbol{\\mathsf{V}}\\cap\\boldsymbol{\\mathsf{C}}_n)\n\\in\\FF_n\\subset\\FF$. Hence, we infer from \\eqref{e:yf},\n\\eqref{e:8uy}, and \\eqref{e:tp} that\n\\begin{align}\n\\boldsymbol{y}^{-1}(\\boldsymbol{\\mathsf{V}})\n&=\\bigcup_{n\\in\\mathbb{K}}\\menge{\\omega\\in\\Theta_n}{\n\\boldsymbol{y}(\\omega)\\in\\boldsymbol{\\mathsf{V}}}\n\\nonumber\\\\\n&=\\bigcup_{n\\in\\mathbb{K}}\\menge{\\omega\\in\\Theta_n}{\n\\boldsymbol{y}_n(\\omega)\\in\\boldsymbol{\\mathsf{C}}_n\\cap\n\\boldsymbol{\\mathsf{V}}}\n\\nonumber\\\\\n&=\\bigcup_{n\\in\\mathbb{K}}\\big(\\Theta_n\\cap\n\\boldsymbol{y}_n^{-1}(\n\\boldsymbol{\\mathsf{C}}_n\\cap\\boldsymbol{\\mathsf{V}})\\big)\n\\nonumber\\\\\n&\\in\\FF.\n\\end{align}\nThis verifies that $\\boldsymbol{y}\\colon(\\Omega,\\FF)\\to\n(\\XS\\times\\RR,\\BE_{\\XS\\times\\RR})$ is measurable.\nWe now define\n\\begin{equation}\n\\label{e:xcd}\n(\\forall n\\in\\mathbb{K})(\\forall k\\in\\NN)\\quad\n\\boldsymbol{x}_{n,k}\\colon\\Omega\\to\\XS\\times\\RR\\colon\n\\omega\\mapsto\n\\begin{cases}\n\\boldsymbol{z}_{n,k}(\\omega),&\\text{if}\\,\\,\\omega\\in\\Omega_n;\\\\\n\\boldsymbol{y}(\\omega),\n&\\text{if}\\,\\,\\omega\\in\\complement\\Omega_n.\n\\end{cases}\n\\end{equation}\nIt results from \\eqref{e:0f} that\n$(\\boldsymbol{x}_{n,k})_{n\\in\\mathbb{K},k\\in\\NN}$\nare measurable mappings from\n$(\\Omega,\\FF)$ to $(\\XS\\times\\RR,\\BE_{\\XS\\times\\RR})$.\nFurthermore, \\eqref{e:tp} and \\eqref{e:z0} give\n\\begin{equation}\n\\label{e:sd9}\n(\\forall n\\in\\mathbb{K})(\\forall k\\in\\NN)\n(\\forall\\omega\\in\\Omega)\\quad\n\\boldsymbol{x}_{n,k}(\\omega)\\in\\epi\\varphi_\\omega.\n\\end{equation}\nFix $\\omega\\in\\Omega$ and let\n$\\boldsymbol{\\mathsf{x}}\\in\\epi\\varphi_\\omega$.\nSince $\\bigcup_{n\\in\\mathbb{K}}(\\boldsymbol{\\mathsf{C}}_n\\cap\n\\epi\\varphi_\\omega)=\\epi\\varphi_\\omega$,\nthere exists $N\\in\\mathbb{K}$ such that\n$\\omega\\in\\Omega_N$ and\n$\\boldsymbol{\\mathsf{x}}\\in\\boldsymbol{\\mathsf{C}}_N\n\\cap\\epi\\varphi_\\omega=K_N(\\omega)$.\nThus, it results from \\eqref{e:tp} and \\eqref{e:xcd} that \n\\begin{equation}\n\\boldsymbol{\\mathsf{x}}\n\\in\\overline{\\big\\{\\boldsymbol{z}_{N,k}(\\omega)\\big\\}_{k\\in\\NN}}\n=\\overline{\\big\\{\\boldsymbol{x}_{N,k}(\\omega)\\big\\}_{k\\in\\NN}}\n\\subset\\overline{\\big\\{\\boldsymbol{x}_{n,k}(\\omega)\\big\\}_{\nn\\in\\mathbb{K},k\\in\\NN}}.\n\\end{equation}\nTherefore, since $\\epi\\varphi_\\omega$ is closed,\nit follows from \\eqref{e:sd9} and \\cite[Section~I.3.1]{Bour71}\nthat\n\\begin{equation}\n\\epi\\varphi_\\omega\n=\\overline{\\big\\{\\boldsymbol{x}_{n,k}(\\omega)\\big\\}_{\nn\\in\\mathbb{K},k\\in\\NN}}.\n\\end{equation}\nAt the same time, for every $n\\in\\mathbb{K}$ and every $k\\in\\NN$,\nsince $\\BE_{\\XS\\times\\RR}=\\BE_\\XS\\otimes\\BE_\\RR$\n\\cite[Lemma~6.4.2(i)]{Boga07} and since\n$\\boldsymbol{x}_{n,k}\\colon(\\Omega,\\FF)\\to\n(\\XS\\times\\RR,\\BE_{\\XS\\times\\RR})$ is measurable,\nthere exist $x_{n,k}\\in\\mathcal{L}(\\Omega;\\XS)$ and\n$\\varrho_{n,k}\\in\\mathcal{L}(\\Omega;\\RR)$ such that\n$(\\forall\\omega\\in\\Omega)$\n$\\boldsymbol{x}_{n,k}(\\omega)\n=(x_{n,k}(\\omega),\\varrho_{n,k}(\\omega))$.\nAltogether, $\\varphi$ is normal.\n\n\\ref{t:3ivb}$\\Rightarrow$\\ref{t:3iva}:\nLet $\\boldsymbol{\\mathsf{C}}$ be a nonempty closed subset of\n$(\\XS\\times\\RR,\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR)$.\nNote that the lower semicontinuity of $\\varphi$\nensures that $\\boldsymbol{G}$ is closed.\nFor every $n\\in\\NN$, since\n$\\boldsymbol{G}\\cap(\\Omega\\times(\\boldsymbol{\\mathsf{C}}\\cap\n\\boldsymbol{\\mathsf{C}}_n))$ is closed in\n$(\\Omega\\times\\boldsymbol{\\mathsf{C}}_n,\n\\EuScript{T}_\\Omega\\boxtimes\n\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n})$,\nit follows from \\eqref{e:q0x} and\n\\cite[Corollaire~I.10.5 and Th\\'eor\\`eme~I.10.1]{Bour71} that\n$Q_n(\\boldsymbol{G}\\cap(\\Omega\\times(\\boldsymbol{\\mathsf{C}}\\cap\n\\boldsymbol{\\mathsf{C}}_n)))$ is closed in\n$(\\Omega,\\EuScript{T}_\\Omega)$\nand, therefore, that it belongs to $\\BE_\\Omega=\\FF$.\nThus, by\n\\eqref{e:rz2}, $\\menge{\\omega\\in\\Omega}{\\boldsymbol{\\mathsf{C}}\n\\cap\\epi\\varphi_\\omega\\neq\\emp}\\in\\FF$.\n\n\\ref{t:3ivc}$\\Rightarrow$\\ref{t:3iva}:\nThere exists a topology $\\widetilde{\\EuScript{T}_\\Omega}$ on\n$\\Omega$ such that\n\\begin{equation}\n\\EuScript{T}_\\Omega\\subset\\widetilde{\\EuScript{T}_\\Omega}\n\\,\\,\\text{and}\\,\\,\n\\big(\\Omega,\\widetilde{\\EuScript{T}_\\Omega}\\big)\\,\\,\n\\text{is a Polish space}.\n\\end{equation}\nIn addition, by \\cite[Corollary~2, p.~101]{Schw73},\nthe Borel $\\sigma$-algebra of\n$(\\Omega,\\widetilde{\\EuScript{T}_\\Omega})$ is $\\BE_\\Omega=\\FF$.\nLet $\\boldsymbol{\\mathsf{C}}$ be a closed subset of\n$(\\XS\\times\\RR,\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR)$ and\nfix temporarily $n\\in\\NN$. Since the\n$\\FF\\otimes\\BE_\\XS$-measurability of $\\varphi$\nand \\eqref{e:s04} ensure that\n$\\boldsymbol{G}\\in\\FF\\otimes\\BE_{\\XS\\times\\RR}$, we have\n$\\boldsymbol{G}\\cap(\\Omega\\times(\\boldsymbol{\\mathsf{C}}\\cap\n\\boldsymbol{\\mathsf{C}}_n))\n=\\boldsymbol{G}\\cap(\\Omega\\times\\boldsymbol{\\mathsf{C}})\\cap\n(\\Omega\\times\\boldsymbol{\\mathsf{C}}_n)\n\\in\\BE_{\\Omega\\times\\boldsymbol{\\mathsf{C}}_n}$.\nAt the same time, for every $\\omega\\in\\Omega$,\n\\begin{align}\n&\\menge{(\\mathsf{x},\\xi)\\in\\XS\\times\\RR}{\n(\\omega,\\mathsf{x},\\xi)\\in\n\\boldsymbol{G}\\cap\\big(\\Omega\\times(\\boldsymbol{\\mathsf{C}}\\cap\n\\boldsymbol{\\mathsf{C}}_n)\\big)}\n\\nonumber\\\\\n&\\hskip 26mm\n=\\menge{(\\mathsf{x},\\xi)\\in\\XS\\times\\RR}{(\\mathsf{x},\\xi)\\in\n\\boldsymbol{\\mathsf{C}}\\cap\\boldsymbol{\\mathsf{C}}_n\n\\,\\,\\text{and}\\,\\,\n(\\mathsf{x},\\xi)\\in\\epi\\varphi_\\omega},\n\\nonumber\\\\\n&\\hskip 26mm\n=\\boldsymbol{\\mathsf{C}}\\cap\\boldsymbol{\\mathsf{C}}_n\n\\cap\\epi\\varphi_\\omega\n\\end{align}\nis a closed subset of the compact space\n$(\\boldsymbol{\\mathsf{C}}_n,\n\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n})$.\nIn turn, since\n$(\\Omega,\\widetilde{\\EuScript{T}_\\Omega})$ and\n$(\\boldsymbol{\\mathsf{C}}_n,\n\\EuScript{T}_{\\boldsymbol{\\mathsf{C}}_n})$\nare Polish spaces, \\cite[Theorem~1]{Brow73} guarantees that\n$Q_n(\\boldsymbol{G}\\cap(\\Omega\\times(\\boldsymbol{\\mathsf{C}}\\cap\n\\boldsymbol{\\mathsf{C}}_n)))\\in\\BE_\\Omega=\\FF$.\nConsequently, we infer from \\eqref{e:rz2} that\n$\\menge{\\omega\\in\\Omega}{\\boldsymbol{\\mathsf{C}}\n\\cap\\epi\\varphi_\\omega\\neq\\emp}\\in\\FF$.\n\n\\ref{t:3v}:\nLet $(\\YS,\\EuScript{T}_\\YS)$ be the strong dual of $\\XS$.\nThen $(\\YS,\\EuScript{T}_\\YS)$ is a separable reflexive Banach\nspace. Consequently, \\ref{t:3va} follows from \\ref{t:3ivb}, and\n\\ref{t:3vb} follows from \\ref{t:3ivc}.\n\n\\ref{t:3vi}$\\Rightarrow$\\ref{t:3vb}:\nLet $\\EuScript{T}_\\Omega$ be the topology on $\\Omega$ induced by\nthe standard topology on $\\RR^M$. By\n\\cite[Corollary~1, p.~102]{Schw73},\n$(\\Omega,\\EuScript{T}_\\Omega)$ is a Lusin space.\n\n\\ref{t:3viia}:\nThe lower semicontinuity of $(\\varphi_\\omega)_{\\omega\\in\\Omega}$\nensures that the sets $(\\epi\\varphi_\\omega)_{\\omega\\in\\Omega}$ are\nclosed. Hence, since $(\\XS\\times\\RR,\n\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR)$ is a Polish\nspace, \\cite[Theorem~3.5(i)]{Himm75} and \\eqref{e:s03} yield\n$\\boldsymbol{G}\\in\\FF\\otimes\\BE_{\\XS\\times\\RR}$. Therefore, by\n\\eqref{e:s04}, $\\varphi$ is $\\FF\\otimes\\BE_\\XS$-measurable.\nConsequently, we deduce the assertion from\n\\cite[Theorem~5.6]{Himm75}.\n\n\\ref{t:3viib}$\\Rightarrow$\\ref{t:3viia}:\nThis follows from \\cite[Theorem~3.2(ii)]{Himm75}.\n\n\\ref{t:3viii}:\nThe $\\BE_\\XS$-measurability of $\\mathsf{f}$ implies that\n$\\varphi$ is $\\FF\\otimes\\BE_\\XS$-measurable. At the same time,\nsince $(\\XS\\times\\RR,\n\\EuScript{T}_\\XS\\boxtimes\\EuScript{T}_\\RR)$ is a Souslin space, we\ndeduce from \\cite[Proposition~II.0]{Schw73} that there exists a\nsequence\n$\\{(\\mathsf{x}_n,\\xi_n)\\}_{n\\in\\NN}$ in $\\epi\\mathsf{f}$ such that\n$\\overline{\\{(\\mathsf{x}_n,\\xi_n)\\}_{n\\in\\NN}}\n=\\overline{\\epi\\mathsf{f}}$. Altogether, upon setting\n\\begin{equation}\n(\\forall n\\in\\NN)\\quad\nx_n\\colon\\Omega\\to\\XS\\colon\\omega\\mapsto\\mathsf{x}_n\n\\quad\\text{and}\\quad\n\\varrho_n\\colon\\Omega\\to\\RR\\colon\\omega\\mapsto\\xi_n,\n\\end{equation}\nwe conclude that $\\varphi$ is normal.\n\\end{proof}\n\n\\begin{remark}\n\\label{r:7}\nHere are a few observations about Definition~\\ref{d:n}.\n\\begin{enumerate}\n\\item\n\\label{r:7i}\nThe setting of Theorem~\\ref{t:3}\\ref{t:3viib} corresponds to\nthe definition of normality in \\cite{Roc76k}.\n\\item\n\\label{r:7ii}\nThe setting of Theorem~\\ref{t:3}\\ref{t:3ia} corresponds to\nthe definition of normality in \\cite{Vala75}, which itself\ncontains that of \\cite{Rock71}.\n\\item\n\\label{r:7iii}\nThe frameworks of \\ref{r:7i} and \\ref{r:7ii} above are distinct\nsince the former does not require that $(\\Omega,\\FF,\\mu)$ be\ncomplete. Definition~\\ref{d:n} unifies them and, as seen\nin Theorem~\\ref{t:3}, goes beyond.\n\\end{enumerate}\n\\end{remark}\n\n\\section{Interchange rules with compliant spaces and normal\nintegrands}\n\\label{sec:5}\n\nThe main result of this section is the following interchange\ntheorem, which brings together the abstract principle of\nTheorem~\\ref{t:1}, the notion of compliance of\nDefinition~\\ref{d:1}, and the notion of normality of\nDefinition~\\ref{d:n}.\n\n\\begin{theorem}\n\\label{t:8}\nSuppose that Assumption~\\ref{a:1} holds, that $\\XX$ is compliant,\nand that $\\varphi$ is normal. Then \n\\begin{equation}\n\\label{e:311}\n\\inf_{x\\in\\XX}\\int_\\Omega\\varphi\\big(\\omega,x(\\omega)\\big)\n\\mu(d\\omega)=\n\\int_\\Omega\\inf_{\\mathsf{x}\\in\\XS}\\varphi(\\omega,\\mathsf{x})\\,\n\\mu(d\\omega).\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nWe apply Theorem~\\ref{t:1}.\nBy virtue of the normality of $\\varphi$, per Definition~\\ref{d:n},\nwe choose sequences $(z_n)_{n\\in\\NN}$ in\n$\\mathcal{L}(\\Omega;\\XS)$ and $(\\vartheta_n)_{n\\in\\NN}$ in\n$\\mathcal{L}(\\Omega;\\RR)$ such that\n\\begin{equation}\n\\label{e:n0d}\n(\\forall\\omega\\in\\Omega)\\quad\n\\big\\{\\big(z_n(\\omega),\\vartheta_n(\\omega)\\big)\\big\\}_{n\\in\\NN}\n\\subset\\epi\\varphi_\\omega\n\\quad\\text{and}\\quad\n\\overline{\\epi\\varphi_\\omega}=\\overline{\\big\\{\\big(z_n(\\omega),\n\\vartheta_n(\\omega)\\big)\\big\\}_{n\\in\\NN}}.\n\\end{equation}\nOn the other hand, Assumption~\\ref{a:1}\\ref{a:1f} ensures that\n$(\\forall\\omega\\in\\Omega)$ $\\inf\\varphi(\\omega,\\XS)<\\pinf$.\nNow fix $\\omega\\in\\Omega$ and let\n$\\xi\\in\\left]\\inf\\varphi(\\omega,\\XS),\\pinf\\right[$.\nThen there exits $\\mathsf{x}\\in\\XS$ such that\n$(\\mathsf{x},\\xi)\\in\\epi\\varphi_\\omega$. Thus,\nin view of \\eqref{e:n0d}, we obtain a subnet\n$(\\vartheta_{k(b)}(\\omega))_{b\\in B}$ of\n$(\\vartheta_n(\\omega))_{n\\in\\NN}$ such that\n$\\vartheta_{k(b)}(\\omega)\\to\\xi$. On the other hand,\n\\begin{equation}\n(\\forall b\\in B)\\quad\n\\inf\\varphi(\\omega,\\XS)\n\\leq\\inf_{n\\in\\NN}\\varphi\\big(\\omega,z_n(\\omega)\\big)\n\\leq\\varphi\\big(\\omega,z_{k(b)}(\\omega)\\big)\n\\leq\\vartheta_{k(b)}(\\omega).\n\\end{equation}\nHence $\\inf\\varphi(\\omega,\\XS)\n\\leq\\inf_{n\\in\\NN}\\varphi(\\omega,z_n(\\omega))\n\\leq\\xi$. In turn,\nletting $\\xi\\downarrow\\inf\\varphi(\\omega,\\XS)$ yields\n$\\inf\\varphi(\\omega,\\XS)=\n\\inf_{n\\in\\NN}\\varphi(\\omega,z_n(\\omega))$.\nTherefore, property~\\ref{t:1iia} in Theorem~\\ref{t:1}\nis satisfied with $(\\forall n\\in\\NN)$ $x_n=z_n-\\overline{x}$.\nAt the same time, property~\\ref{t:1iib} in Theorem~\\ref{t:1}\nfollows from Assumption~\\ref{a:1}\\ref{a:1d} and the compliance of\n$\\XX$. Finally, since the functions\n$(\\varphi(\\Cdot,z_n(\\Cdot)))_{n\\in\\NN}$ are $\\FF$-measurable by\nAssumption~\\ref{a:1}\\ref{a:1f}, so is\n$\\inf_{n\\in\\NN}\\varphi(\\Cdot,z_n(\\Cdot))=\\inf\\varphi(\\Cdot,\\XS)$.\n\\end{proof}\n\nIn the remainder of this section, we construct new scenarios for\nthe validity of the interchange rule as instantiations of\nTheorem~\\ref{t:8}.\n\n\\begin{example}\n\\label{ex:1}\nLet $\\XS$ be a separable real Banach space with strong topology\n$\\EuScript{T}_\\XS$, let $(\\Omega,\\FF,\\mu)$ be a $\\sigma$-finite\nmeasure space such that $\\mu(\\Omega)\\neq 0$, let $\\XX$ be a vector\nsubspace of $\\mathcal{L}(\\Omega;\\XS)$, and let\n$\\varphi\\colon(\\Omega\\times\\XS,\\FF\\otimes\\BE_\\XS)\\to\\RXX$ be\nmeasurable. Suppose that the following are satisfied:\n\\begin{enumerate}\n\\item\n\\label{ex:1i}\nFor every $A\\in\\FF$ such that $\\mu(A)<\\pinf$ and every\n$z\\in\\mathcal{L}^\\infty(\\Omega;\\XS)$, $1_Az\\in\\XX$.\n\\item\n\\label{ex:1ii}\n$\\varphi$ is normal.\n\\item\n\\label{ex:1iii}\nThere exists $\\overline{x}\\in\\XX$ such that\n$\\int_\\Omega\\max\\{\\varphi(\\Cdot,\\overline{x}(\\Cdot)),0\\}\nd\\mu<\\pinf$.\n\\end{enumerate}\nThen the interchange rule \\eqref{e:311} holds.\n\\end{example}\n\\begin{proof}\nNote that Assumption~\\ref{a:1} is satisfied. Hence, the assertion\nfollows from Proposition~\\ref{p:10}\\ref{p:10ii} and\nTheorem~\\ref{t:8}.\n\\end{proof}\n\n\\begin{example}\n\\label{ex:2}\nSuppose that Assumption~\\ref{a:1} holds, that $(\\Omega,\\FF,\\mu)$\nis complete, and that $\\XX$ is compliant. Then the interchange\nrule \\eqref{e:311} holds.\n\\end{example}\n\\begin{proof}\nCombine Theorem~\\ref{t:3}\\ref{t:3ia} and Theorem~\\ref{t:8}.\n\\end{proof}\n\nWhen specialized to probability in separable Banach spaces,\nTheorem~\\ref{t:8} yields conditions for the interchange of\ninfimization and expectation. Here is an illustration.\n\n\\begin{example}\n\\label{ex:4}\nLet $\\XS$ be a separable real Banach space,\nlet $(\\Omega,\\FF,\\PP)$ be a probability space,\nlet $\\XX$ be a vector subspace of\n$\\mathcal{L}(\\Omega;\\XS)$ which contains\n$\\mathcal{L}^\\infty(\\Omega;\\XS)$,\nand let $\\varphi\\colon(\\Omega\\times\\XS,\\FF\\otimes\\BE_\\XS)\\to\\RXX$\nbe normal. In addition, set $\\phi=\\inf\\varphi(\\Cdot,\\XS)$ and\n$\\Phi\\colon\\mathcal{L}(\\Omega;\\XS)\n\\to\\mathcal{L}(\\Omega;\\RXX)\\colon\nx\\mapsto\\varphi(\\Cdot,x(\\Cdot))$, and suppose that there\nexists $\\overline{x}\\in\\XX$ such that\n$\\EE\\max\\{\\Phi(\\overline{x}),0\\}<\\pinf$. Then\n\\begin{equation}\n\\inf_{x\\in\\XX}\\EE\\Phi(x)=\\EE\\phi.\n\\end{equation}\n\\end{example}\n\\begin{proof}\nThis is a special case of Example~\\ref{ex:1}.\n\\end{proof}\n\n\\begin{example}\n\\label{ex:9}\nSuppose that Assumption~\\ref{a:1} holds, that $\\XX$ is compliant,\nand that the functions $(\\varphi_\\omega)_{\\omega\\in\\Omega}$ are\nupper semicontinuous. Then the interchange rule \\eqref{e:311}\nholds.\n\\end{example}\n\\begin{proof}\nWe deduce from Assumption~\\ref{a:1}\\ref{a:1f} and\nTheorem~\\ref{t:3}\\ref{t:3id} that $\\varphi$ is normal. Thus, the\nconclusion follows from Theorem~\\ref{t:8}.\n\\end{proof}\n\nAn important realization of Example~\\ref{ex:9} is the case of\nCarath\\'eodory integrands.\n\n\\begin{example}[Carath\\'eodory integrand]\n\\label{ex:5}\nLet $(\\XS,\\EuScript{T}_\\XS)$ be a Souslin topological vector\nspace, let $(\\Omega,\\FF,\\mu)$ be a $\\sigma$-finite measure space\nsuch that $\\mu(\\Omega)\\neq 0$, let $\\XX$ be a compliant vector\nsubspace of $\\mathcal{L}(\\Omega;\\XS)$, and\nlet $\\varphi\\colon\\Omega\\times\\XS\\to\\RXX$ be\na Carath\\'eodory integrand in the sense that,\nfor every $(\\omega,\\mathsf{x})\\in\\Omega\\times\\XS$,\n$\\varphi(\\omega,\\Cdot)$ is continuous with\n$\\epi\\varphi_\\omega\\neq\\emp$, and\n$\\varphi(\\Cdot,\\mathsf{x})$ is $\\FF$-measurable. Suppose that\nthere exists $\\overline{x}\\in\\XX$\nsuch that $\\int_\\Omega\\max\\{\\varphi(\\Cdot,\\overline{x}(\\Cdot)),0\\}\nd\\mu<\\pinf$. Then the interchange rule \\eqref{e:311} holds.\n\\end{example}\n\\begin{proof}\nSince $(\\XS,\\EuScript{T}_\\XS)$ is a Souslin topological vector\nspace, \\cite[Section~35F,~p.~244]{Will70} implies that\nit is a regular Souslin space. Thus, we\ndeduce from Theorem~\\ref{t:3}\\ref{t:3iii} that $\\varphi$ is\nnormal and, in particular, it is $\\FF\\otimes\\BE_\\XS$-measurable.\nHence, Assumption~\\ref{a:1} is satisfied. Consequently,\nExample~\\ref{ex:9} yields the conclusion.\n\\end{proof}\n\n\\begin{remark}\n\\label{r:3}\nHere are connections with existing work.\n\\begin{enumerate}\n\\item\n\\label{r:3i}\nExample~\\ref{ex:1} unifies and extends the classical results of\n\\cite{Hiai77,Rock71,Roc76k}:\n\\begin{itemize}\n\\item\nIt captures \\cite[Theorem~3A]{Roc76k}, where $\\XS$ is a Euclidean\nspace and $\\XX$ is assumed to be Rockafellar-decomposable\n(see Proposition~\\ref{p:10}\\ref{p:10iv} for definition).\n\\item\nIt covers the setting of \\cite{Rock71}, where\n$(\\Omega,\\FF\\,\\mu)$ is assumed to be complete\nand where \\ref{ex:1i} and \\ref{ex:1ii} in Example~\\ref{ex:1}\nare specialized to:\n\\begin{enumerate}[label={\\rm(\\roman*')}]\n\\setcounter{enumi}{1}\n\\item\n\\label{r:3i+}\n$\\XX$ is Rockafellar-decomposable.\n\\item\n\\label{r:3ii+}\nThe functions $(\\varphi_\\omega)_{\\omega\\in\\Omega}$ are lower\nsemicontinuous.\n\\end{enumerate}\nThe fact that property~\\ref{ex:1ii} in Example~\\ref{ex:1} is\nsatisfied when $(\\Omega,\\FF,\\mu)$ is complete is shown in\nTheorem~\\ref{t:3}\\ref{t:3ia}.\n\\item\nIt captures \\cite[Theorem~2.2]{Hiai77}, where\n$\\XX=\\menge{x\\in\\mathcal{L}(\\Omega;\\XS)}{\n\\int_\\Omega\\|x(\\omega)\\|_\\XS^p\\,\\mu(d\\omega)<\\pinf}$\nwith $p\\in\\left[1,\\pinf\\right[$.\n\\end{itemize}\n\\item\nAn important contribution of Theorem~\\ref{t:8} and, in particular,\nof Example~\\ref{ex:1} is that completeness of the measure space\n$(\\Omega,\\FF,\\mu)$ is not required.\n\\item\nIn the special case when $\\XS$ is a\nBanach space, an alternative framework that recovers the\ninterchange rules of \\cite{Hiai77,Rock71,Roc76k} was proposed in\n\\cite[Theorem~6.1]{Gine09}, where the right-hand side of\n\\eqref{e:1} is replaced by the integral of an abstract essential\ninfimum. However, \\cite{Gine09} does not provide new scenarios for\n\\eqref{e:1} beyond the known cases in Banach spaces.\nAn interpretation of the framework of \\cite{Gine09}\nfrom the view point of monotone relations between partially\nordered sets is proposed in \\cite{Chan22}.\n\\item\nExample~\\ref{ex:2} captures \\cite[Theorem~4]{Perk18}, where\n$\\mu(\\Omega)<\\pinf$ and $\\XX$ is Valadier-decomposable\n(see Proposition~\\ref{p:10}\\ref{p:10v} for definition).\nIt also covers the setting of \\cite{Vala75}, where $\\XS$ is a\nSouslin topological vector space and $\\XX$ is\nValadier-decomposable.\n\\item\nExample~\\ref{ex:4} contains the interchange rule of\n\\cite{Penn23,Shap21}, where $\\XS$ is the standard Euclidean space\n$\\RR^N$ and $\\XX$ is Rockafellar-decomposable.\n\\item\nExample~\\ref{ex:5} extends\n\\cite[Theorem~3A]{Roc76k}, where $\\XS$ is the standard Euclidean\nspace $\\RR^N$ and $\\XX$ is Rockafellar-decomposable.\n\\end{enumerate}\n\\end{remark}\n\n\\section{Interchanging convex-analytical operations and\nintegration}\n\\label{sec:6}\n\nWe put the interchange principle of Theorem~\\ref{t:1}, compliance,\nand normality in action to evaluate convex-analytical objects\nassociated with integral functions, namely conjugate functions,\nsubdifferential operators, recession functions, Moreau envelopes,\nand proximity operators. This analysis results in new interchange\nrules for the convex calculus of integral functions. Throughout\nthis section, we adopt the following notation.\n\n\\begin{notation}\n\\label{n:1}\nLet $(\\XS,\\EuScript{T}_\\XS)$ be a real topological vector space,\nlet $(\\Omega,\\FF,\\mu)$ be a $\\sigma$-finite measure space such\nthat $\\mu(\\Omega)\\neq 0$, let $\\XX$ be a vector subspace of\n$\\mathcal{L}(\\Omega;\\XS)$, and let\n$\\varphi\\colon(\\Omega\\times\\XS,\\FF\\otimes\\BE_\\XS)\\to\\RXX$ be an\nintegrand. Then:\n\\begin{enumerate}\n\\item\n$\\TXX$ is the vector space of equivalence classes of $\\mae$ equal\nmappings in $\\XX$.\n\\item\nThe equivalence class in $\\TXX$ of $x\\in\\XX$ is denoted by\n$\\widetilde{x}$. Conversely, an arbitrary representative in $\\XX$\nof $\\widetilde{x}\\in\\TXX$ is denoted by $x$.\n\\item\n$\\mathfrak{I}_{\\varphi,\\TXX}\\colon\\TXX\\to\\RXX\\colon\n\\widetilde{x}\\mapsto\n\\int_\\Omega\\varphi(\\omega,x(\\omega))\\mu(d\\omega)$.\n\\end{enumerate}\n\\end{notation}\n\nWe shall require the following result. Its item \\ref{l:6i} \nappears in \\cite[Lemma~4]{Vala75} in the special case when \n$(\\Omega,\\FF,\\mu)$ is complete.\n\n\\begin{lemma}\n\\label{l:6}\nLet $(\\Omega,\\FF,\\mu)$ be a $\\sigma$-finite measure space\nsuch that $\\mu(\\Omega)\\neq 0$,\nlet $(\\XS,\\EuScript{T}_\\XS)$ be a Souslin locally convex real\ntopological vector space, and let $(\\YS,\\EuScript{T}_\\YS)$ be a\nseparable locally convex real topological vector space.\nSuppose that $\\XS$ and $\\YS$ are placed in separating duality via\na bilinear form\n$\\pair{\\Cdot}{\\Cdot}_{\\XS,\\YS}\\colon\\XS\\times\\YS\\to\\RR$ with which\n$\\EuScript{T}_\\XS$ and $\\EuScript{T}_\\YS$ are compatible.\nThen the following hold:\n\\begin{enumerate}\n\\item\n\\label{l:6i}\n$\\pair{\\Cdot}{\\Cdot}_{\\XS,\\YS}\\colon(\\XS\\times\\YS,\n\\BE_\\XS\\otimes\\BE_\\YS)\\to\\RR$ is measurable.\n\\item\n\\label{l:6ii}\nLet $\\XX\\subset\\mathcal{L}(\\Omega;\\XS)$ and\n$\\YY\\subset\\mathcal{L}(\\Omega;\\YS)$ be vector subspaces\nsuch that the following are satisfied:\n\\begin{enumerate}\n\\item\n\\label{l:6iia}\n$(\\forall x\\in\\XX)(\\forall y\\in\\YY)$\n$\\int_\\Omega|\\pair{x(\\omega)}{y(\\omega)}_{\\XS,\\YS}|\n\\mu(d\\omega)<\\pinf$.\n\\item\n\\label{l:6iib}\n$\\bigcup_{\\mathsf{x}\\in\\XS}\\menge{1_A\\mathsf{x}}{A\\in\\FF\\,\\,\n\\text{and}\\,\\,\\mu(A)<\\pinf}\\subset\\XX$.\n\\item\n\\label{l:6iic}\n$\\bigcup_{\\mathsf{y}\\in\\YS}\\menge{1_A\\mathsf{y}}{A\\in\\FF\\,\\,\n\\text{and}\\,\\,\\mu(A)<\\pinf}\\subset\\YY$.\n\\end{enumerate}\nThen $\\widetilde{\\XX}$ and $\\widetilde{\\YY}$ are in\nseparating duality via the bilinear form $\\pair{\\Cdot}{\\Cdot}$\ndefined by\n\\begin{equation}\n\\label{e:62}\n(\\forall\\widetilde{x}\\in\\TXX)(\\forall\\widetilde{y}\\in\\TYY)\\quad\n\\pair{\\widetilde{x}}{\\widetilde{y}}=\n\\int_\\Omega\\Pair{x(\\omega)}{y(\\omega)}_{\\XS,\\YS}\\mu(d\\omega).\n\\end{equation}\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n\\ref{l:6i}:\nWe deduce from \\cite[Section~35F,~p.~244]{Will70} that\n$(\\XS,\\EuScript{T}_\\XS)$ is a regular Souslin space.\nOn the other hand, since $\\EuScript{T}_\\YS$ and $\\EuScript{T}_\\XS$\nare compatible with $\\pair{\\Cdot}{\\Cdot}_{\\XS,\\YS}$,\nthe functions\n$(\\pair{\\mathsf{x}}{\\Cdot}_{\\XS,\\YS})_{\\mathsf{x}\\in\\XS}$ are\n$\\BE_\\YS$-measurable and the functions\n$(\\pair{\\Cdot}{\\mathsf{y}}_{\\XS,\\YS})_{\\mathsf{y}\\in\\YS}$ are\ncontinuous. Hence, Theorem~\\ref{t:3}\\ref{t:3iii}\nimplies that $\\pair{\\Cdot}{\\Cdot}_{\\XS,\\YS}\\colon(\\XS\\times\\YS,\n\\BE_\\XS\\otimes\\BE_\\YS)\\to\\RR$ is measurable.\n\n\\ref{l:6ii}:\nNote that \\ref{l:6i} guarantees that, for every $x\\in\\XX$ and every\n$y\\in\\YY$, $\\pair{x(\\Cdot)}{y(\\Cdot)}_{\\XS,\\YS}$ is\n$\\FF$-measurable. Now let $\\{\\mathsf{y}_n\\}_{n\\in\\NN}$ be a\ndense subset of $(\\YS,\\EuScript{T}_\\YS)$ and let\n$\\widetilde{x}\\in\\TXX$ be such\nthat $(\\forall\\widetilde{y}\\in\\TYY)$\n$\\pair{\\widetilde{x}}{\\widetilde{y}}=0$.\nThen, for every $n\\in\\NN$ and every $A\\in\\FF$ such that\n$\\mu(A)<\\pinf$, since \\ref{l:6iic} ensures that\n$1_A\\mathsf{y}_n\\in\\YY$, we deduce from \\eqref{e:62} that\n$\\int_A\\pair{x(\\omega)}{\\mathsf{y}_n}_{\\XS,\\YS}\\mu(d\\omega)\n=\\int_\\Omega\\pair{x(\\omega)}{1_A(\\omega)\\mathsf{y}_n}_{\\XS,\\YS}\n\\mu(d\\omega)=0$. Therefore, since $(\\Omega,\\FF,\\mu)$ is\n$\\sigma$-finite, it follows that $(\\forall n\\in\\NN)$\n$\\pair{x(\\Cdot)}{\\mathsf{y}_n}_{\\XS,\\YS}=0$ $\\mae$\nThus $\\widetilde{x}=0$. Likewise,\n$(\\forall\\widetilde{y}\\in\\TYY)$ $\\pair{\\Cdot}{\\widetilde{y}}=0$\n$\\Rightarrow$ $\\widetilde{y}=0$, which completes the proof.\n\\end{proof}\n\nThe main result of this section is set in the following\nenvironment, which is well defined by virtue of Lemma~\\ref{l:6}.\n\n\\begin{assumption}\n\\label{a:2}\n\\\n\\begin{enumerate}[label={\\rm[\\Alph*]}]\n\\item\n\\label{a:2a}\n$(\\XS,\\EuScript{T}_\\XS)$ is a Souslin locally convex real\ntopological vector space and $(\\YS,\\EuScript{T}_\\YS)$ is a\nseparable locally convex real topological vector space.\nIn addition, $\\XS$ and $\\YS$ are placed in separating duality via\na bilinear form\n$\\pair{\\Cdot}{\\Cdot}_{\\XS,\\YS}\\colon\\XS\\times\\YS\\to\\RR$ with which\n$\\EuScript{T}_\\XS$ and $\\EuScript{T}_\\YS$ are compatible.\n\\item\n\\label{a:2b}\n$(\\Omega,\\FF,\\mu)$ is a $\\sigma$-finite measure space such that\n$\\mu(\\Omega)\\neq 0$.\n\\item\n\\label{a:2c}\n$\\XX\\subset\\mathcal{L}(\\Omega;\\XS)$ and\n$\\YY\\subset\\mathcal{L}(\\Omega;\\YS)$ are vector subspaces\nsuch that $(\\forall x\\in\\XX)(\\forall y\\in\\YY)$\n$\\int_\\Omega|\\pair{x(\\omega)}{y(\\omega)}_{\\XS,\\YS}|\n\\mu(d\\omega)<\\pinf$. In addition,\n\\begin{equation}\n\\label{e:k7}\n\\XX\\,\\,\\text{is compliant and}\\,\\,\n\\bigcup_{\\mathsf{y}\\in\\YS}\\menge{1_A\\mathsf{y}}{A\\in\\FF\\,\\,\n\\text{and}\\,\\,\\mu(A)<\\pinf}\\subset\\YY.\n\\end{equation}\n\\item\n\\label{a:2d}\n$\\widetilde{\\XX}$ and $\\widetilde{\\YY}$ are placed in separating\nduality via the bilinear form $\\pair{\\Cdot}{\\Cdot}$ defined by\n\\begin{equation}\n\\label{e:63}\n(\\forall\\widetilde{x}\\in\\TXX)\n(\\forall\\widetilde{y}\\in\\TYY)\\quad\n\\pair{\\widetilde{x}}{\\widetilde{y}}=\n\\int_\\Omega\\Pair{x(\\omega)}{y(\\omega)}_{\\XS,\\YS}\\mu(d\\omega),\n\\end{equation}\nand they are equipped with locally convex Hausdorff topologies\nwhich are compatible with $\\pair{\\Cdot}{\\Cdot}$.\n\\item\n\\label{a:2e}\n$\\varphi\\colon(\\Omega\\times\\XS,\\FF\\otimes\\BE_\\XS)\\to\\RX$ is normal\nand we write\n$\\varphi^*\\colon\\Omega\\times\\YS\\to\\RXX\\colon\n(\\omega,\\mathsf{y})\\mapsto\\varphi_\\omega^*(\\mathsf{y})$.\n\\item\n\\label{a:2f}\n$\\dom\\mathfrak{I}_{\\varphi,\\TXX}\\neq\\emp$.\n\\end{enumerate}\n\\end{assumption}\n\n\\begin{proposition}\n\\label{p:6}\nSuppose that Assumption~\\ref{a:2} holds. Then\n$\\varphi^*$ is $\\FF\\otimes\\BE_\\YS$-measurable.\n\\end{proposition}\n\\begin{proof}\nAccording to Assumption~\\ref{a:2}\\ref{a:2e} and\nDefinition~\\ref{d:n}, there exist sequences $(x_n)_{n\\in\\NN}$ in\n$\\mathcal{L}(\\Omega;\\XS)$ and $(\\varrho_n)_{n\\in\\NN}$ in\n$\\mathcal{L}(\\Omega;\\RR)$ such that\n\\begin{equation}\n\\label{e:ef}\n(\\forall\\omega\\in\\Omega)\\quad\n\\big\\{\\big(x_n(\\omega),\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\NN}\n\\subset\\epi\\varphi_\\omega\n\\quad\\text{and}\\quad\n\\overline{\\epi\\varphi_\\omega}=\\overline{\\big\\{\\big(x_n(\\omega),\n\\varrho_n(\\omega)\\big)\\big\\}_{n\\in\\NN}}.\n\\end{equation}\nSet\n\\begin{equation}\n(\\forall n\\in\\NN)\\quad\n\\psi_n\\colon\\Omega\\times\\YS\\to\\RR\\colon(\\omega,\\mathsf{y})\\mapsto\n\\pair{x_n(\\omega)}{\\mathsf{y}}_{\\XS,\\YS}-\\varrho_n(\\omega).\n\\end{equation}\nThen, for every $n\\in\\NN$, \nAssumption~\\ref{a:2}\\ref{a:2a}--\\ref{a:2c}\nand Lemma~\\ref{l:6}\\ref{l:6i} ensure that\n$\\psi_n$ is $\\FF\\otimes\\BE_\\YS$-measurable. On the other hand,\nsince the functions\n$(\\pair{\\Cdot}{\\mathsf{y}}_{\\XS,\\YS})_{\\mathsf{y}\\in\\YS}$ are\ncontinuous, we derive from Assumption~\\ref{a:2}\\ref{a:2e},\n\\eqref{e:l0d}, and \\eqref{e:ef} that\n\\begin{align}\n\\big(\\forall(\\omega,\\mathsf{y})\\in\\Omega\\times\\YS\\big)\\quad\n\\varphi^*(\\omega,\\mathsf{y})\n&=\\sup_{(\\mathsf{x},\\xi)\\in\\epi\\varphi_\\omega}\\big(\n\\pair{\\mathsf{x}}{\\mathsf{y}}_{\\XS,\\YS}-\\xi\\big)\n\\nonumber\\\\\n&=\\sup_{(\\mathsf{x},\\xi)\\in\\overline{\\epi\\varphi_\\omega}}\\big(\n\\pair{\\mathsf{x}}{\\mathsf{y}}_{\\XS,\\YS}-\\xi\\big)\n\\nonumber\\\\\n&=\\sup_{n\\in\\NN}\\big(\\pair{x_n(\\omega)}{\\mathsf{y}}_{\\XS,\\YS}\n-\\varrho_n(\\omega)\\big)\n\\nonumber\\\\\n&=\\sup_{n\\in\\NN}\\psi_n(\\omega,\\mathsf{y}).\n\\end{align}\nThus $\\varphi^*$ is $\\FF\\otimes\\BE_\\YS$-measurable.\n\\end{proof}\n\nWe first investigate the conjugate and the subdifferential\nof integral functions.\n\n\\begin{theorem}\n\\label{t:2}\nSuppose that Assumption~\\ref{a:2} holds.\nThen the following are satisfied:\n\\begin{enumerate}\n\\item\n\\label{t:2i}\n$\\mathfrak{I}_{\\varphi,\\TXX}^*=\\mathfrak{I}_{\\varphi^*,\\TYY}$.\n\\item\n\\label{t:2ii}\nSuppose that $\\mathfrak{I}_{\\varphi,\\TXX}$ is proper,\nlet $\\widetilde{x}\\in\\TXX$, and let $\\widetilde{y}\\in\\TYY$. Then\n$\\widetilde{y}\\in\n\\partial\\mathfrak{I}_{\\varphi,\\TXX}(\\widetilde{x})$\n$\\Leftrightarrow$ $y(\\omega)\\in\\partial\\varphi_\\omega(x(\\omega))$\nfor $\\mu$-almost every $\\omega\\in\\Omega$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n\\ref{t:2i}:\nIn view of Assumption~\\ref{a:2}\\ref{a:2e} and\nProposition~\\ref{p:6}, $\\mathfrak{I}_{\\varphi,\\TXX}$\nand $\\mathfrak{I}_{\\varphi^*,\\TYY}$ are well defined.\nFurther, there exist sequences $(z_n)_{n\\in\\NN}$ in\n$\\mathcal{L}(\\Omega;\\XS)$ and $(\\vartheta_n)_{n\\in\\NN}$ in\n$\\mathcal{L}(\\Omega;\\RR)$ such that\n\\begin{equation}\n\\label{e:imd}\n(\\forall\\omega\\in\\Omega)\\quad\n\\big\\{\\big(z_n(\\omega),\\vartheta_n(\\omega)\\big)\\big\\}_{n\\in\\NN}\n\\subset\\epi\\varphi_\\omega\n\\quad\\text{and}\\quad\n\\overline{\\epi\\varphi_\\omega}=\\overline{\\big\\{\\big(z_n(\\omega),\n\\vartheta_n(\\omega)\\big)\\big\\}_{n\\in\\NN}}.\n\\end{equation}\nLet $\\widetilde{y}\\in\\TYY$, define\n$\\psi\\colon\\Omega\\times\\XS\\to\\RX\\colon\n(\\omega,\\mathsf{x})\\mapsto\\varphi_\\omega(\\mathsf{x})\n-\\pair{\\mathsf{x}}{y(\\omega)}_{\\XS,\\YS}$, and note that\n$(\\forall\\omega\\in\\Omega)$ $\\epi\\psi_\\omega\\neq\\emp$.\nAssumption~\\ref{a:2}\\ref{a:2e} and\nLemma~\\ref{l:6}\\ref{l:6i} imply that\n\\begin{equation}\n\\label{e:tys}\n\\psi\\,\\,\\text{is $\\FF\\otimes\\BE_\\XS$-measurable}.\n\\end{equation}\nMoreover, using the continuity of the linear functionals\n$(\\pair{\\Cdot}{\\mathsf{y}}_{\\XS,\\YS})_{\\mathsf{y}\\in\\YS}$,\nwe derive from \\eqref{e:imd} that\n\\begin{align}\n(\\forall\\omega\\in\\Omega)\\quad\n\\inf\\psi(\\omega,\\XS)\n&=\\inf_{(\\mathsf{x},\\xi)\\in\\epi\\varphi_\\omega}\\big(\n\\xi-\\pair{\\mathsf{x}}{y(\\omega)}_{\\XS,\\YS}\\big)\n\\nonumber\\\\\n&=\\inf_{(\\mathsf{x},\\xi)\\in\\overline{\\epi\\varphi_\\omega}}\\big(\n\\xi-\\pair{\\mathsf{x}}{y(\\omega)}_{\\XS,\\YS}\\big)\n\\nonumber\\\\\n&=\\inf_{n\\in\\NN}\\big(\\vartheta_n(\\omega)\n-\\pair{z_n(\\omega)}{y(\\omega)}\\big)\n\\nonumber\\\\\n&\\geq\\inf_{n\\in\\NN}\\big(\\varphi_\\omega\\big(z_n(\\omega)\\big)\n-\\pair{z_n(\\omega)}{y(\\omega)}\\big)\n\\nonumber\\\\\n&=\\inf_{n\\in\\NN}\\psi\\big(\\omega,z_n(\\omega)\\big)\n\\nonumber\\\\\n&\\geq\\inf\\psi(\\omega,\\XS).\n\\end{align}\nHence, $(\\forall\\omega\\in\\Omega)$\n$\\inf\\psi(\\omega,\\XS)=\\inf_{n\\in\\NN}\\psi(\\omega,z_n(\\omega))$.\nCombining this with \\eqref{e:tys},\nwe infer that $\\inf\\psi(\\Cdot,\\XS)$ is $\\FF$-measurable and that\n$\\psi$ fulfills property~\\ref{t:1iia} in Theorem~\\ref{t:1} with\n$(\\forall n\\in\\NN)$ $x_n=z_n-\\overline{x}$.\nIn turn, thanks to Assumption~\\ref{a:2}\\ref{a:2b}\nand the compliance of $\\XX$, property~\\ref{t:1iib} in \nTheorem~\\ref{t:1} is fulfilled.\nThus, by invoking \\eqref{e:63} and Theorem~\\ref{t:1}, we obtain\n\\begin{align}\n\\mathfrak{I}_{\\varphi,\\TXX}^*(\\widetilde{y})\n&=\\sup_{\\widetilde{x}\\in\\TXX}\\big(\n\\pair{\\widetilde{x}}{\\widetilde{y}}-\n\\mathfrak{I}_{\\varphi,\\TXX}(\\widetilde{x})\\big)\n\\nonumber\\\\\n&=\\sup_{x\\in\\XX}\\bigg(\n\\int_\\Omega\\Pair{x(\\omega)}{y(\\omega)}_{\\XS,\\YS}\\mu(d\\omega)\n-\\int_\\Omega\\varphi\\big(\\omega,x(\\omega)\\big)\\mu(d\\omega)\n\\bigg)\n\\nonumber\\\\\n&=-\\inf_{x\\in\\XX}\\int_\\Omega\\psi\\big(\\omega,x(\\omega)\\big)\n\\mu(d\\omega)\n\\nonumber\\\\\n&=-\\int_\\Omega\\inf_{\\mathsf{x}\\in\\XS}\\psi(\\omega,\\mathsf{x})\n\\,\\mu(d\\omega)\n\\nonumber\\\\\n&=\\int_\\Omega\\varphi_\\omega^*\\big(y(\\omega)\\big)\\mu(d\\omega),\n\\end{align}\nas desired.\n\n\\ref{t:2ii}:\nSince the functions $(\\varphi_\\omega)_{\\omega\\in\\Omega}$\nare proper by Assumption~\\ref{a:2}\\ref{a:2e}, we derive from\n\\eqref{e:s14}, \\ref{t:2i}, \\eqref{e:63}, and the\nFenchel--Young inequality that\n\\begin{align}\n\\widetilde{y}\\in\n\\partial\\mathfrak{I}_{\\varphi,\\TXX}(\\widetilde{x})\n&\\Leftrightarrow\n\\mathfrak{I}_{\\varphi,\\TXX}(\\widetilde{x})+\n\\mathfrak{I}_{\\varphi^*,\\TYY}(\\widetilde{y})\n=\\pair{\\widetilde{x}}{\\widetilde{y}}\n\\nonumber\\\\\n&\\Leftrightarrow\n\\int_\\Omega\n\\varphi_\\omega\\big(x(\\omega)\\big)\\mu(d\\omega)\n+\\int_\\Omega\\varphi_\\omega^*\\big(y(\\omega)\\big)\\mu(d\\omega)\n=\\int_\\Omega\\Pair{x(\\omega)}{y(\\omega)}_{\\XS,\\YS}\\mu(d\\omega)\n\\nonumber\\\\\n&\\Leftrightarrow\n\\varphi_\\omega\\big(x(\\omega)\\big)\n+\\varphi_\\omega^*\\big(y(\\omega)\\big)\n=\\Pair{x(\\omega)}{y(\\omega)}_{\\XS,\\YS}\\,\\,\\mae\n\\nonumber\\\\\n&\\Leftrightarrow\ny(\\omega)\\in\\partial\\varphi_\\omega\\big(x(\\omega)\\big)\\,\\,\\mae,\n\\end{align}\nwhich completes the proof.\n\\end{proof}\n\nA first important consequence of Theorem~\\ref{t:2}\\ref{t:2i} is the\nfollowing.\n\n\\begin{proposition}\n\\label{p:8}\nSuppose that Assumption~\\ref{a:2} holds,\nthat $(\\YS,\\EuScript{T}_\\YS)$ is a Souslin space,\nthat $\\dom\\mathfrak{I}_{\\varphi^*,\\TYY}\\neq\\emp$,\nthat $\\YY$ is compliant, and that\n$(\\forall\\omega\\in\\Omega)$ $\\varphi_\\omega\\in\\Gamma_0(\\XS)$.\nThen the following are satisfied:\n\\begin{enumerate}\n\\item\n\\label{p:8i}\n$\\mathfrak{I}_{\\varphi,\\TXX}\\in\\Gamma_0(\\TXX)$.\n\\item\n\\label{p:8ii}\nSet $\\rec\\varphi\\colon\\Omega\\times\\XS\\to\\RX\\colon\n(\\omega,\\mathsf{x})\\mapsto(\\rec\\varphi_\\omega)(\\mathsf{x})$.\nThen $\\rec\\varphi$ is $\\FF\\otimes\\BE_\\XS$-measurable and \n$\\rec\\mathfrak{I}_{\\varphi,\\TXX}=\\mathfrak{I}_{\\rec\\varphi,\\TXX}$.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\n\\ref{p:8i}:\nLet $\\widetilde{x}\\in\\TXX$ and set\n\\begin{equation}\n\\label{e:r3}\n\\psi\\colon\\Omega\\times\\YS\\to\\RX\\colon(\\omega,\\mathsf{y})\n\\mapsto\\varphi_\\omega^*(\\mathsf{y})\n-\\pair{x(\\omega)}{\\mathsf{y}}_{\\XS,\\YS}\n\\quad\\text{and}\\quad\n\\vartheta=\\inf\\psi(\\Cdot,\\YS).\n\\end{equation}\nBy Assumption~\\ref{a:2}\\ref{a:2e},\n\\begin{equation}\n\\label{e:fxs}\n\\varphi\\big(\\Cdot,x(\\Cdot)\\big)\\,\\,\\text{is $\\FF$-measurable},\n\\end{equation}\nwhile it results from Proposition~\\ref{p:6} and\nLemma~\\ref{l:6}\\ref{l:6i} that\n\\begin{equation}\n\\label{e:yxc}\n\\psi\\,\\,\\text{is $\\FF\\otimes\\BE_\\YS$-measurable}.\n\\end{equation}\nMoreover, for every $\\omega\\in\\Omega$, since\n$\\varphi_\\omega\\in\\Gamma_0(\\XS)$, $\\varphi_\\omega^*$ is proper\nand hence $\\epi\\psi_\\omega\\neq\\emp$.\nOn the other hand, the Fenchel--Moreau biconjugation theorem\nyields\n\\begin{equation}\n\\label{e:8cp}\n(\\forall\\omega\\in\\Omega)\\quad\n\\vartheta(\\omega)\n=-\\varphi_\\omega^{**}\\big(x(\\omega)\\big)\n=-\\varphi_\\omega\\big(x(\\omega)\\big)\n\\end{equation}\nand it thus follows from \\eqref{e:fxs} that $\\vartheta$ is\n$\\FF$-measurable. Now define\n\\begin{equation}\n\\label{e:m3}\n(\\forall n\\in\\NN)\\quad\nM_n\\colon\\Omega\\to 2^\\YS\\colon\\omega\\mapsto\n\\begin{cases}\n\\menge{\\mathsf{y}\\in\\YS}{\\psi(\\omega,\\mathsf{y})\\leq-n},\n&\\text{if}\\,\\,\\vartheta(\\omega)=\\minf;\\\\\n\\menge{\\mathsf{y}\\in\\YS}{\\psi(\\omega,\\mathsf{y})\n\\leq\\vartheta(\\omega)+2^{-n}},\n&\\text{if}\\,\\,\\vartheta(\\omega)\\in\\RR.\n\\end{cases}\n\\end{equation}\nFix temporarily $n\\in\\NN$. By \\eqref{e:yxc},\n$\\menge{(\\omega,\\mathsf{y})}{\\mathsf{y}\\in\nM_n(\\omega)}\\in\\FF\\otimes\\BE_\\YS$. Hence, since\n$(\\YS,\\EuScript{T}_\\YS)$ is a Souslin space,\n\\cite[Theorem~5.7]{Himm75} guarantees that there exist\n$y_n\\in\\mathcal{L}(\\Omega;\\YS)$\nand $B_n\\in\\FF$ such that $\\mu(B_n)=0$ and\n$(\\forall\\omega\\in\\complement B_n)$ $y_n(\\omega)\\in M_n(\\omega)$.\nNow set $B=\\bigcup_{n\\in\\NN}B_n$. Then $\\mu(B)=0$ and, by virtue\nof \\eqref{e:r3} and \\eqref{e:m3},\n\\begin{equation}\n\\big(\\forall\\omega\\in\\complement B\\big)(\\forall n\\in\\NN)\\quad\n\\vartheta(\\omega)\n\\leq\\inf_{k\\in\\NN}\\psi\\big(\\omega,y_k(\\omega)\\big)\n\\leq\\psi\\big(\\omega,y_n(\\omega)\\big)\n\\leq\n\\begin{cases}\n-n,&\\text{if}\\,\\,\\vartheta(\\omega)=\\minf;\\\\\n\\vartheta(\\omega)+2^{-n},&\\text{if}\\,\\,\\vartheta(\\omega)\\in\\RR.\n\\end{cases}\n\\end{equation}\nThus, letting $n\\uparrow\\pinf$ yields\n$(\\forall\\omega\\in\\complement B)$\n$\\vartheta(\\omega)=\\inf_{n\\in\\NN}\\psi(\\omega,y_n(\\omega))$.\nConsequently, since $\\YY$ is compliant, property~\\ref{t:1ii} in\nTheorem~\\ref{t:1} is satisfied. In turn, we deduce from\n\\eqref{e:8cp}, Theorem~\\ref{t:1}, \\eqref{e:63}, and\nTheorem~\\ref{t:2}\\ref{t:2i} that\n\\begin{align}\n\\mathfrak{I}_{\\varphi,\\TXX}(\\widetilde{x})\n&=\\int_\\Omega\\varphi\\big(\\omega,x(\\omega)\\big)\\mu(d\\omega)\n\\nonumber\\\\\n&=-\\int_\\Omega\\inf_{\\mathsf{y}\\in\\YS}\\psi(\\omega,\\mathsf{y})\\,\n\\mu(d\\omega)\n\\nonumber\\\\\n&=-\\inf_{y\\in\\YY}\\int_\\Omega\\psi\\big(\\omega,y(\\omega)\\big)\n\\mu(d\\omega)\n\\nonumber\\\\\n&=\\sup_{y\\in\\YY}\\bigg(\n\\int_\\Omega\\Pair{x(\\omega)}{y(\\omega)}_{\\XS,\\YS}\\mu(d\\omega)\n-\\int_\\Omega\\varphi_\\omega^*\\big(y(\\omega)\\big)\\mu(d\\omega)\\bigg)\n\\nonumber\\\\\n&=\\sup_{\\widetilde{y}\\in\\widetilde{\\YY}}\n\\big(\\pair{\\widetilde{x}}{\\widetilde{y}}-\n\\mathfrak{I}_{\\varphi,\\TXX}^*(\\widetilde{y})\\big)\n\\nonumber\\\\\n&=\\mathfrak{I}_{\\varphi,\\TXX}^{**}(\\widetilde{x}).\n\\end{align}\nThus $\\mathfrak{I}_{\\varphi,\\TXX}=\n\\mathfrak{I}_{\\varphi,\\TXX}^{**}$ and,\nsince $\\mathfrak{I}_{\\varphi,\\TXX}$ is proper, we conclude\nthat $\\mathfrak{I}_{\\varphi,\\TXX}\\in\\Gamma_0(\\TXX)$.\n\n\\ref{p:8ii}:\nThe normality of $\\varphi$ implies that it is\n$\\FF\\otimes\\BE_\\XS$-measurable and that\nthere exists $u\\in\\mathcal{L}(\\Omega;\\XS)$ such that\n$(\\forall\\omega\\in\\Omega)$ $u(\\omega)\\in\\dom\\varphi_\\omega$.\nHence, for every $n\\in\\NN$, the function\n$(\\Omega\\times\\XS,\\FF\\otimes\\BE_\\XS)\\to\\RX\\colon\n(\\omega,\\mathsf{x})\\mapsto\n\\varphi_\\omega(u(\\omega)+n\\mathsf{x})-\\varphi_\\omega(u(\\omega))$\nis measurable. Since, by \\eqref{e:r},\n\\begin{equation}\n(\\forall\\omega\\in\\Omega)(\\forall\\mathsf{x}\\in\\XS)\\quad\n(\\rec\\varphi)(\\omega,\\mathsf{x})\n=(\\rec\\varphi_\\omega)(\\mathsf{x})\n=\\lim_{\\NN\\ni n\\uparrow\\pinf}\n\\dfrac{\\varphi_\\omega\\big(u(\\omega)+n\\mathsf{x}\\big)\n-\\varphi_\\omega\\big(u(\\omega)\\big)}{n},\n\\end{equation}\nit follows that $\\rec\\varphi$ is $\\FF\\otimes\\BE_\\XS$-measurable.\nNow let $\\widetilde{x}\\in\\TXX$ and\n$\\widetilde{z}\\in\\dom\\mathfrak{I}_{\\varphi,\\TXX}$. Then, for\n$\\mu$-almost every $\\omega\\in\\Omega$,\n$z(\\omega)\\in\\dom\\varphi_\\omega$ and it thus follows from the\nconvexity of $\\varphi_\\omega$ that the function\n$\\theta\\colon\\RPP\\to\\RX\\colon\\alpha\\mapsto\n(\\varphi_\\omega(z(\\omega)+\\alpha x(\\omega))\n-\\varphi_\\omega(z(\\omega)))\/\\alpha$ is increasing.\nThus, appealing to \\eqref{e:r} and the monotone convergence\ntheorem, we deduce from \\ref{p:8i} that\n\\begin{align}\n\\big(\\rec\\mathfrak{I}_{\\varphi,\\TXX}\\big)(\\widetilde{x})\n&=\\lim_{\\alpha\\uparrow\\pinf}\n\\frac{\\mathfrak{I}_{\\varphi,\\TXX}(\\widetilde{z}+\n\\alpha\\widetilde{x})-\\mathfrak{I}_{\\varphi,\\TXX}(\\widetilde{z})}{\n\\alpha}\n\\nonumber\\\\\n&=\\lim_{\\alpha\\uparrow\\pinf}\\int_\\Omega\\frac{\n\\varphi_\\omega\\big(z(\\omega)+\\alpha x(\\omega)\\big)\n-\\varphi_\\omega\\big(z(\\omega)\\big)}{\\alpha}\\mu(d\\omega)\n\\nonumber\\\\\n&=\\int_\\Omega\\lim_{\\alpha\\uparrow\\pinf}\\frac{\n\\varphi_\\omega\\big(z(\\omega)+\\alpha x(\\omega)\\big)\n-\\varphi_\\omega\\big(z(\\omega)\\big)}{\\alpha}\\,\\mu(d\\omega)\n\\nonumber\\\\\n&=\\int_\\Omega(\\rec\\varphi_\\omega)\\big(x(\\omega)\\big)\\mu(d\\omega),\n\\end{align}\nas claimed.\n\\end{proof}\n\nTwo key ingredients in Hilbertian convex analysis are the Moreau\nenvelope of \\eqref{e:7} and the proximity operator of \\eqref{e:8}\n\\cite{Livre1,More65}. To compute them for integral functions, we\nfirst observe that, in the case of Hilbert spaces identified with\ntheir duals, Assumption~\\ref{a:2} can be simplified as follows.\n\n\\begin{assumption}\\\n\\label{a:3}\n\\begin{enumerate}[label={\\rm[\\Alph*]}]\n\\item\n\\label{a:3a}\n$\\XS$ is a separable real Hilbert space with scalar product\n$\\scal{\\Cdot}{\\Cdot}_\\XS$, associated norm $\\|\\Cdot\\|_\\XS$,\nand strong topology $\\EuScript{T}_\\XS$.\n\\item\n\\label{a:3b}\n$(\\Omega,\\FF,\\mu)$ is a $\\sigma$-finite measure space such that\n$\\mu(\\Omega)\\neq 0$.\n\\item\n\\label{a:3c}\n$\\XX=\\menge{x\\in\\mathcal{L}(\\Omega;\\XS)}{\n\\int_\\Omega\\|x(\\omega)\\|_\\XS^2\\,\\mu(d\\omega)<\\pinf}$ and\n$\\TXX$ is the usual real Hilbert space\n$L^2(\\Omega;\\XS)$ with scalar product\n\\begin{equation}\n(\\forall\\widetilde{x}\\in\\TXX)(\\forall\\widetilde{y}\\in\\TXX)\\quad\n\\scal{\\widetilde{x}}{\\widetilde{y}}=\n\\int_\\Omega\\scal{x(\\omega)}{y(\\omega)}_\\XS\\,\\mu(d\\omega).\n\\end{equation}\n\\item\n\\label{a:3e}\n$\\varphi\\colon(\\Omega\\times\\XS,\\FF\\otimes\\BE_\\XS)\\to\\RX$ is a\nnormal integrand such that $(\\forall\\omega\\in\\Omega)$\n$\\varphi_\\omega\\in\\Gamma_0(\\XS)$.\n\\item\n\\label{a:3f}\n$\\dom\\mathfrak{I}_{\\varphi,\\TXX}\\neq\\emp$ and\n$\\dom\\mathfrak{I}_{\\varphi^*,\\TXX}\\neq\\emp$.\n\\end{enumerate}\n\\end{assumption}\n\n\\begin{proposition}\n\\label{p:11}\nSuppose that Assumption~\\ref{a:3} holds and\nlet $\\gamma\\in\\RPP$. Then the following are satisfied:\n\\begin{enumerate}\n\\item\n\\label{p:11i}\nLet $\\widetilde{x}\\in\\TXX$ and $\\widetilde{p}\\in\\TXX$. Then\n$\\widetilde{p}=\\prox_{\\gamma\n\\mathfrak{I}_{\\varphi,\\TXX}}\n\\widetilde{x}$ $\\Leftrightarrow$\n$p(\\omega)=\\prox_{\\gamma\\varphi_\\omega}(x(\\omega))$\nfor $\\mu$-almost every $\\omega\\in\\Omega$.\n\\item\n\\label{p:11ii}\nSet $\\moyo{\\varphi}{\\gamma}\\colon\\Omega\\times\\XS\\to\\RX\\colon\n(\\omega,\\mathsf{x})\\mapsto\\moyo{(\\varphi_\\omega)}{\\gamma}\n(\\mathsf{x})$.\nThen $\\moyo{\\varphi}{\\gamma}$ is normal and\n$\\moyo{\\mathfrak{I}_{\\varphi,\\TXX}}{\\gamma}\n=\\mathfrak{I}_{\\moyo{\\varphi}{\\gamma},\\TXX}$.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nSince Assumption~\\ref{a:3} is an instance of Assumption~\\ref{a:2},\nwe first infer from Proposition~\\ref{p:8}\\ref{p:8i} that \n$\\mathfrak{I}_{\\varphi,\\TXX}\\in\\Gamma_0(\\TXX)$.\n\n\\ref{p:11i}: We derive from \\eqref{e:8} and\nTheorem~\\ref{t:2}\\ref{t:2ii} that\n\\begin{align}\n\\widetilde{p}=\\prox_{\\gamma\\mathfrak{I}_{\\varphi,\n\\TXX}}\\widetilde{x}\n&\\Leftrightarrow\\widetilde{x}-\\widetilde{p}\\in\\gamma\\partial\n\\mathfrak{I}_{\\varphi,\\TXX}(\\widetilde{p})\n\\nonumber\\\\\n&\\Leftrightarrow x(\\omega)-p(\\omega)\\in\\gamma\\partial\n\\varphi_\\omega\\big(p(\\omega)\\big)\\,\\,\n\\text{for $\\mu$-almost every}\\,\\,\\omega\\in\\Omega\n\\nonumber\\\\\n&\\Leftrightarrow p(\\omega)=\\prox_{\\gamma\\varphi_\\omega}\nx(\\omega)\\,\\,\\text{for $\\mu$-almost every}\n\\,\\,\\omega\\in\\Omega.\n\\end{align}\n\n\\ref{p:11ii}:\nSince $\\BE_{\\XS\\times\\RR}=\\BE_\\XS\\otimes\\BE_\\RR$,\nit results from Assumption~\\ref{a:3}\\ref{a:3e} and\nDefinition~\\ref{d:n} that there exists a sequence\n$(\\boldsymbol{x}_n)_{n\\in\\NN}$\nin $\\mathcal{L}(\\Omega;\\XS\\times\\RR)$ such that\n\\begin{equation}\n\\label{e:usn}\n(\\forall\\omega\\in\\Omega)\\quad\n\\epi\\varphi_\\omega=\n\\overline{\\big\\{\\boldsymbol{x}_n(\\omega)\\big\\}_{n\\in\\NN}}.\n\\end{equation}\nSet $\\boldsymbol{\\mathsf{V}}=\n\\menge{(\\mathsf{x},\\xi)\\in\\XS\\times\\RR}{\n\\|\\mathsf{x}\\|_\\XS^2\/(2\\gamma)<\\xi}$.\nThen $\\boldsymbol{\\mathsf{V}}$ is open and therefore,\nfor every $\\boldsymbol{\\mathsf{C}}\\subset\\XS\\times\\RR$,\n$\\boldsymbol{\\mathsf{C}}+\\boldsymbol{\\mathsf{V}}\n=\\overline{\\boldsymbol{\\mathsf{C}}}+\\boldsymbol{\\mathsf{V}}$.\nThus, we derive from\n\\eqref{e:7} and \\eqref{e:usn} that\n\\begin{align}\n(\\forall\\omega\\in\\Omega)\\quad\n\\menge{(\\mathsf{x},\\xi)\\in\\XS\\times\\RR}{\n\\moyo{(\\varphi_\\omega)}{\\gamma}(\\mathsf{x})<\\xi}\n&=\\menge{(\\mathsf{x},\\xi)\\in\\XS\\times\\RR}{\n\\varphi_\\omega(\\mathsf{x})<\\xi}+\\boldsymbol{\\mathsf{V}}\n\\nonumber\\\\\n&=\\overline{\\menge{(\\mathsf{x},\\xi)\\in\\XS\\times\\RR}{\n\\varphi_\\omega(\\mathsf{x})<\\xi}}+\\boldsymbol{\\mathsf{V}}\n\\nonumber\\\\\n&=\\epi\\varphi_\\omega+\\boldsymbol{\\mathsf{V}}\n\\nonumber\\\\\n&=\\overline{\\big\\{\\boldsymbol{x}_n(\\omega)\\big\\}_{n\\in\\NN}}\n+\\boldsymbol{\\mathsf{V}}\n\\nonumber\\\\\n&=\\big\\{\\boldsymbol{x}_n(\\omega)\\big\\}_{n\\in\\NN}\n+\\boldsymbol{\\mathsf{V}}\n\\nonumber\\\\\n&=\\bigcup_{n\\in\\NN}\\big(\\boldsymbol{x}_n(\\omega)\n+\\boldsymbol{\\mathsf{V}}\\big).\n\\end{align}\nHence, for every $\\mathsf{x}\\in\\XS$ and every $\\xi\\in\\RR$,\nsince\n$(\\mathsf{x},\\xi)-\\boldsymbol{\\mathsf{V}}\\in\\BE_{\\XS\\times\\RR}$\nand $\\{\\boldsymbol{x}_n\\}_{n\\in\\NN}\\subset\n\\mathcal{L}(\\Omega;\\XS\\times\\RR)$, we obtain\n\\begin{equation}\n\\menge{\\omega\\in\\Omega}{\n\\moyo{(\\varphi_\\omega)}{\\gamma}(\\mathsf{x})<\\xi}\n=\\Menge{\\omega\\in\\Omega}{\n(\\mathsf{x},\\xi)\\in\\bigcup_{n\\in\\NN}\\big(\\boldsymbol{x}_n(\\omega)\n+\\boldsymbol{\\mathsf{V}}\\big)}\n=\\bigcup_{n\\in\\NN}\\boldsymbol{x}_n^{-1}\\big(\n(\\mathsf{x},\\xi)-\\boldsymbol{\\mathsf{V}}\\big)\n\\in\\FF,\n\\end{equation}\nwhich shows that $(\\moyo{\\varphi}{\\gamma})(\\Cdot,\\mathsf{x})$ is\n$\\FF$-measurable.\nHence, since $(\\XS,\\EuScript{T}_\\XS)$ is a Fr\\'echet space,\nTheorem~\\ref{t:3}\\ref{t:3iib} ensures that\n$\\moyo{\\varphi}{\\gamma}$ is normal. It remains to show that\n$\\moyo{\\mathfrak{I}_{\\varphi,\\TXX}}{\\gamma}\n=\\mathfrak{I}_{\\moyo{\\varphi}{\\gamma},\\TXX}$.\nLet $\\widetilde{x}\\in\\TXX$ and set\n$\\widetilde{p}=\\prox_{\\gamma\n\\mathfrak{I}_{\\varphi,\\TXX}}\n\\widetilde{x}$.\nThen, by \\ref{p:11i}, for \n$\\mu$-almost every $\\omega\\in\\Omega$,\n$p(\\omega)=\\prox_{\\gamma\\varphi_\\omega}(x(\\omega))$\nand, therefore, \\eqref{e:7b} yields\n$\\moyo{(\\varphi_\\omega)}{\\gamma}(x(\\omega))\n=\\varphi_\\omega(p(\\omega))\n+\\|x(\\omega)-p(\\omega)\\|_\\XS^2\/(2\\gamma)$. Hence\n\\begin{align}\n\\moyo{\\mathfrak{I}_{\\varphi,\\TXX}}{\\gamma}(\\widetilde{x})\n&=\\mathfrak{I}_{\\varphi,\\TXX}(\\widetilde{p})\n+\\frac{1}{2\\gamma}\\|\\widetilde{x}-\\widetilde{p}\\|_{\\TXX}^2\n\\nonumber\\\\\n&=\\int_\\Omega\\varphi_\\omega\\big(p(\\omega)\\big)\\mu(d\\omega)\n+\\frac{1}{2\\gamma}\\int_\\Omega\n\\|x(\\omega)-p(\\omega)\\|_\\XS^2\\mu(d\\omega)\n\\nonumber\\\\\n&=\\int_\\Omega\n\\moyo{(\\varphi_\\omega)}{\\gamma}\\big(x(\\omega)\\big)\\mu(d\\omega)\n\\nonumber\\\\\n&=\\mathfrak{I}_{\\moyo{\\varphi}{\\gamma},\\TXX}(\\widetilde{x}),\n\\end{align}\nwhich concludes the proof.\n\\end{proof}\n\n\\begin{remark}\\\n\\label{r:5}\nTheorem~\\ref{t:2}, Proposition~\\ref{p:8}, and\nProposition~\\ref{p:11} extend the state of the art on several\nfronts, in particular by removing completeness of\n$(\\Omega,\\FF,\\mu)$ when $\\XS$ is infinite-dimensional.\n\\begin{enumerate}\n\\item\nThe conclusion of Theorem~\\ref{t:2}\\ref{t:2i} first appeared in\n\\cite[Theorem~2]{Roc68a} in the special case when $\\XS$ is the\nstandard Euclidean space $\\RR^N$ and $\\XX$ is\nRockafellar-decomposable (see Proposition~\\ref{p:10}\\ref{p:10iv}\nfor definition).\n\\item\nIn view of Proposition~\\ref{p:10}\\ref{p:10iv} and\nTheorem~\\ref{t:3}\\ref{t:3ia}, Theorem~\\ref{t:2} subsumes\n\\cite[Theorem~2 and Equation~(25)]{Rock71} (see also\n\\cite[Theorem~21]{Rock74}), where $\\XS$ is a separable Banach\nspace, $\\XX$ is Rockafellar-decomposable,\nand $(\\Omega,\\FF,\\mu)$ is complete.\n\\item\nThe conclusion of Theorem~\\ref{t:2}\\ref{t:2i} appears in\n\\cite{Vala75} in the special case when $\\XX$ is\nValadier-decomposable (see Proposition~\\ref{p:10}\\ref{p:10v} for\ndefinition) and $(\\Omega,\\FF,\\mu)$ is complete.\n\\item\nProposition~\\ref{p:8}\\ref{p:8i} subsumes\n\\cite[Corollary p.~227]{Rock71}, where $\\XS$ is a separable Banach\nspace, $\\XX$ is Rockafellar-decomposable, and $(\\Omega,\\FF,\\mu)$ is\ncomplete.\n\\item\nThe conclusion of Proposition~\\ref{p:8}\\ref{p:8ii} first appeared\nin \\cite[Proposition~1]{Bism73} in the context where\n$\\XS$ is a separable reflexive Banach space,\n$\\XX$ is Rockafellar-decomposable,\nand $(\\Omega,\\FF,\\mu)$ is a complete probability space.\nAnother special case is \\cite[Theorem~2]{Penn18},\nwhere $\\XX$ is Valadier-decomposable and either $\\XS=\\RR^N$ or\n$(\\Omega,\\FF,\\mu)$ is complete.\n\\item\nProposition~\\ref{p:11}\\ref{p:11i} appears in \n\\cite[Proposition~24.13]{Livre1} in the special case when \n$(\\Omega,\\FF,\\mu)$ is complete, for every $\\omega\\in\\Omega$\n$\\varphi_\\omega=\\mathsf{f}$, and either $\\mu(\\Omega)<\\pinf$ or\n$\\mathsf{f}\\geq\\mathsf{f}(\\mathsf{0})\\geq 0$.\n\\end{enumerate}\n\\end{remark}\n\n\n\\begin{thebibliography}{99}\n\\setlength{\\itemsep}{0pt}\n\n\\bibitem{Livre1} \nH. 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Saint-Pierre,\nUne remarque sur les espaces sousliniens r\\'eguliers,\n{\\em C. R. Acad. Sci. Paris S\\'er. A},\nvol. 282, pp. 1425--1427, 1976.\n\n\\bibitem{Scha99}\nH. H. Schaefer,\n{\\em Topological Vector Spaces},\n2nd ed. Springer, New York, 1999.\n\n\\bibitem{Schw73}\nL. Schwartz,\n{\\em Radon Measures on Arbitrary Topological Spaces\nand Cylindrical Measures}.\nTata Institute of Fundamental Research, Bombay, 1973.\n\n\\bibitem{Shap21}\nA. Shapiro, D. Dentcheva, and A. Ruszczy\\'nski,\n{\\em Lectures on Stochastic Programming},\n3rd ed. SIAM, Philadelphia, PA, 2021.\n\n\\bibitem{Vala75}\nM. Valadier,\nConvex integrands on Souslin locally convex spaces,\n{\\em Pacific J. Math.},\nvol. 59, pp. 267--276, 1975.\n\n\\bibitem{Will70}\nS. Willard,\n{\\em General Topology}.\nAddison-Wesley, Reading, MA, 1970.\n\n\\bibitem{Zali02} \nC. Z\\u{a}linescu, \n{\\em Convex Analysis in General Vector Spaces.} \nWorld Scientific Publishing, River Edge, NJ, 2002.\n\n\\end{thebibliography}\n\n\\end{document}\n"},{"text":"\n\\documentclass[aps,twocolumn,superscriptaddress,prb]{revtex4-2}\n\n\\usepackage{bm,graphicx,textcomp,amssymb,amsmath,dcolumn,color}\n\n%\\usepackage[utf8]{inputenc}\n%\\usepackage[T1]{fontenc}\n% commands\n\\newcommand{\\tvec}{\\vec}\n\\newcommand{\\action}{\\mathscr{S}}\n\\newcommand{\\pathdiff}[1]{\\!\\mathscr{D}#1\\,}\n\\newcommand{\\diff}[1]{\\!\\mathrm{d}#1\\,}\n\\newcommand{\\dottau}{\\accentset{\\boldsymbol\\circ}}\n\\newcommand{\\dott}{\\accentset{\\mbox{\\large .}}}\n\\newcommand{\\ddott}{\\accentset{\\mbox{\\large ..}}}\n\\newcommand{\\dd}[2][]{\\frac{\\mathrm{d} #1}{\\mathrm{d} #2}}\n\\newcommand{\\pdd}[2][]{\\frac{\\partial #1}{\\partial #2}}\n\\newcommand{\\Tr}{\\operatorname{Tr}}\n\\newcommand{\\tr}{\\operatorname{tr}}\n\\newcommand{\\atanh}{\\operatorname{atanh}}\n\\newcommand{\\sech}{\\operatorname{sech}}\n\\newcommand{\\keld}{\\tilde{\\gamma}}\n\\renewcommand{\\Re}{\\operatorname{Re}}\n\\renewcommand{\\Im}{\\operatorname{Im}}\n\\newcommand\\numberthis{\\addtocounter{equation}{1}\\tag{\\theequation}}\n\n\\newcommand{\\ket}[1]{\\left|#1\\right>}\n\\newcommand{\\bra}[1]{\\left<#1\\right|}\n\\newcommand{\\braket}[2]{\\left<#1|#2\\right>}\n\\newcommand{\\nn}{\\nonumber\\\\}\n\\newcommand{\\ul}{\\underline}\n\\newcommand{\\f}[1]{\\mbox{\\boldmath$#1$}}\n\\newcommand{\\fk}[1]{\\mbox{\\boldmath$\\scriptstyle#1$}}\n\\newcommand{\\vau}{\\mbox{\\boldmath$v$}}\n\\newcommand{\\na}{\\mbox{\\boldmath$\\nabla$}}\n\\newcommand{\\bea}{\\begin{eqnarray}}\n\\newcommand{\\ea}{\\end{eqnarray}}\n\\newcommand{\\eea}{\\end{eqnarray}}\n\\newcommand{\\ord}{\\,{\\cal O}}\n\\newcommand{\\li}{\\,\\widehat{\\cal L}}\n%\\newcommand{\\tr}{\\,{\\rm Tr}}\n\\newcommand{\\vc}[1]{\\mathbf{#1}}\n\\newcommand{\\sumint}[1]\n{\\begin{array}{c} \\\\\n{{\\textstyle\\sum}\\hspace{-1.1em}{\\displaystyle\\int}}\\\\\n{\\scriptstyle{#1}}\n\\end{array}}\n\n\\newcommand{\\rs}[1]{{\\color{blue} #1}}\n\n\\begin{document}\n\n\\title{Doublon-holon pair creation in Mott-Hubbard systems in analogy to QED}\n\n\\author{Friedemann Queisser}\n\n\\affiliation{Helmholtz-Zentrum Dresden-Rossendorf, \nBautzner Landstra{\\ss}e 400, 01328 Dresden, Germany,}\n\n\\affiliation{Institut f\\\"ur Theoretische Physik, \nTechnische Universit\\\"at Dresden, 01062 Dresden, Germany,}\n\n\\author{Konstantin Krutitsky} \n\n\\affiliation{Fakult\\\"at f\\\"ur Physik,\nUniversit\\\"at Duisburg-Essen, Lotharstra{\\ss}e 1, Duisburg 47057, Germany,} \n\n\\author{Patrick Navez} \n\n\\affiliation{Department of Physics, Loughborough University, Loughborough LE11 3TU, United Kingdom} \n\n\\author{Ralf Sch\\\"utzhold}\n\n\\affiliation{Helmholtz-Zentrum Dresden-Rossendorf, \nBautzner Landstra{\\ss}e 400, 01328 Dresden, Germany,}\n\n\\affiliation{Institut f\\\"ur Theoretische Physik, \nTechnische Universit\\\"at Dresden, 01062 Dresden, Germany,}\n\n\\date{\\today}\n\n\\begin{abstract}\nVia the hierarchy of correlations, we study doublon-holon pair creation in \nthe Mott state of the Fermi-Hubbard model induced by a time-dependent \nelectric field.\n%\nSpecial emphasis is placed on the analogy to electron-positron pair creation from \nthe vacuum in quantum electrodynamics (QED).\n%\nWe find that the accuracy of this analogy depends on the spin structure \nof the Mott background. \n%\nFor Ising type anti-ferromagnetic order, we derive an effective Dirac equation. \n%\nA Mott state without any spin order, on the other hand, does not explicitly \ndisplay such a quasi-relativistic behavior. \n\\end{abstract}\n\n\\maketitle\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Introduction}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nNon-equilibrium dynamics in strongly interacting quantum many-body systems is a\nrich and complex field displaying many fascinating phenomena. \n%\nAs the {\\em drosophila} of strongly interacting quantum many-body systems, \nwe consider the Mott insulator phase of the Fermi-Hubbard model \\cite{Hub63,Aro22}.\n%\nAn optical laser can then serve as the external stimulus driving the system \nout of equilibrium -- leading to the creation of doublon-holon pairs, \nsee also \\cite{Oka03,Eck10,Oka05,Oka10,Avi20,Lig18}.\n\nThe intuitive similarity between the upper and lower Hubbard bands \none the one hand and the Dirac sea and the positive energy continuum in \nquantum electrodynamics (QED) on the other hand suggests analogies \nbetween doublon-holon pair creation from the Mott state and \nelectron-positron pair creation from the vacuum, see also \\cite{Sau31,Sch51,Rit85,Niki85,Sch08,Dun09,Pop01,Kohl22,Pop71,Bre70,Kim02,Nar70,Dun05,G05,Nar04}.\n%\nIn the following, we study this analogy in more detail, with special emphasis \non the space-time dependence in more than one dimension, such as the \npropagation of doublons and holons. \n%\nMore specifically, we strive for an analytic understanding without mapping the \nFermi-Hubbard Hamiltonian to an effective single site model, see also \\cite{Geor92,Eck10}. \n\nOf course, analogies between electron-positron pair creation in QED and other \nsystems at lower energies have already been discussed in previous works. \n%\nExamples include ultra-cold atoms in optical lattices \n\\cite{Pin19,Witt11,Cir10,Zhu07,Hou09,Lim08,Boa11,Gold09,Kas16,Quei12,Szp12,Szp11} as well as electrons\nin semi-conductors \\cite{Smo09,Hri93,Lin18} graphene \\cite{\nAll08,Aka16,Kat12,Novo05,Kat06,Chei06,Vand10,Been08,Fill15,Son09,Ros10,Kao10,Kum23,Schmitt23,Shy09,S84,Dor10,Gav12} and $^3$He \\cite{Scho92}. \n%\nHowever, as we shall see below, there are important differences to the \nFermi-Hubbard model considered here.\n%\nFirst, the Mott gap arises naturally through the interaction \n(see also \\cite{Mott49,Hub63}) % Bose-Hubbard\nand does not have to be introduced by hand. \n%\nSecond, the particle-hole symmetry between the upper and lower Hubbard band \n-- analogous to the $\\cal C$ symmetry in QED -- is also an intrinsic property \n(in contrast to the valence and conduction bands in semi-conductors, for example).\n%\nThird, the quantitative analogy to the Dirac equation (in 1+1 dimensions) \nand the resulting quasi-relativistic relativistic behavior does also emerge \nwithout additional fine-tuning (at least in the case of Ising type spin order, \nsee below). \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Extended Fermi-Hubbard Model}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nIn terms of the fermionic creation and annihilation operators\n$\\hat c_{\\mu s}^\\dagger$ and $\\hat c_{\\nu s}$ \nat the lattice sites $\\mu$ and $\\nu$ with spin $s\\in\\{\\uparrow,\\downarrow\\}$ \nand the associated number operators $\\hat n_{\\mu s}$, \nthe extended Fermi-Hubbard Hamiltonian reads ($\\hbar=1$) \n%\n\\bea\n\\label{Fermi-Hubbard}\n\\hat H=-\\frac1Z\\sum_{\\mu\\nu s} T_{\\mu\\nu} \\hat c_{\\mu s}^\\dagger \\hat c_{\\nu s} \n+U\\sum_\\mu\\hat n_\\mu^\\uparrow\\hat n_\\mu^\\downarrow\n+\\sum_{\\mu s}V_\\mu\\hat n_{\\mu s} \n\\,.\\;\n\\ea\n%\nHere the hopping matrix $T_{\\mu\\nu}$ equals the tunneling strength $T$ \nfor nearest neighbors $\\mu$ and $\\nu$ and is zero otherwise. \n%\nThe coordination number $Z$ counts the number of nearest neighbors $\\mu$\nfor a given lattice site $\\nu$ and is assumed to be large $Z\\gg1$.\n%\nIn order to describe the Mott insulator, the on-site repulsion $U$ is also \nsupposed to be large $U\\gg T$.\n%\nFinally, the potential $V_\\mu(t)$ represents the external electric field, \ne.g., an optical laser. \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Hierarchy of Correlations}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nTo obtain an approximate analytical solution, we consider the reduced density \nmatrices of one $\\hat\\rho_\\mu$ and two $\\hat\\rho_{\\mu\\nu}$ lattice sites etc.\n%\nNext, we split up the correlated parts via \n$\\hat\\rho_{\\mu\\nu}^{\\rm corr}=\\hat\\rho_{\\mu\\nu}-\\hat\\rho_{\\mu}\\hat\\rho_{\\nu}$ \netc. \n%\nFor large $Z\\gg1$, we may employ an expansion into powers of $1\/Z$ where we find \nthat higher-order correlators are successively suppressed \\cite{Nav10,Krut14,Queiss19,Queiss14,Nav16}. \n%\nMore precisely, the two-point correlator scales as \n$\\hat\\rho_{\\mu\\nu}^{\\rm corr}=\\ord(1\/Z)$,\nwhile the three-point correlation is suppressed as \n$\\hat\\rho_{\\mu\\nu\\lambda}^{\\rm corr}=\\ord(1\/Z^2)$ etc. \n\nVia this expansion into powers of $1\/Z$, we may find approximate \nsolutions of the evolution equations %($\\hbar=1$)\n%\n\\bea\n\\label{evolution}\ni\\partial_t \\hat\\rho_\\mu \n&=& \nF_1(\\hat\\rho_\\mu,\\hat\\rho_{\\mu\\nu}^{\\rm corr})\n\\,,\\nn\ni\\partial_t \\hat\\rho_{\\mu\\nu}^{\\rm corr} \n&=& \nF_2(\\hat\\rho_\\mu,\\hat\\rho_{\\mu\\nu}^{\\rm corr},\\hat\\rho_{\\mu\\nu\\lambda}^{\\rm corr})\n\\,.\n\\ea\n%\nUsing $\\hat\\rho_{\\mu\\nu}^{\\rm corr}=\\ord(1\/Z)$, \nthe first evolution equation can be approximated by \n$i\\partial_t \\hat\\rho_\\mu = F_1(\\hat\\rho_\\mu,0)+\\ord(1\/Z)$. \n%\nIts zeroth-order solution $\\hat\\rho_\\mu^0$ yields the mean-field background,\nwhich will be specified below. \n\nNext, the suppression $\\hat\\rho_{\\mu\\nu\\lambda}^{\\rm corr}=\\ord(1\/Z^2)$ \nallows us to approximate the second equation~\\eqref{evolution} to leading order \nin $1\/Z$ via $i\\partial_t \\hat\\rho_{\\mu\\nu}^{\\rm corr}\\approx \nF_2(\\hat\\rho_\\mu^0,\\hat\\rho_{\\mu\\nu}^{\\rm corr},0)$. \n%\nIn order to solve this leading-order equality, it is convenient to split to \nfermionc creation and annihilation operators in particle $I=1$ \nand hole $I=0$ contributions via \n%\n\\bea\n\\hat c_{\\mu s I}=\\hat c_{\\mu s}\\hat n_{\\mu\\bar s}^I=\n\\left\\{\n\\begin{array}{ccc}\n \\hat c_{\\mu s}(1-\\hat n_{\\mu\\bar s}) & {\\rm for} & I=0 \n \\\\ \n \\hat c_{\\mu s}\\hat n_{\\mu\\bar s} & {\\rm for} & I=1\n\\end{array}\n\\right.\n\\,,\n\\ea\n%\nwhere $\\bar s$ denotes the spin opposite to $s$. \n%\nIn terms of these particle and hole operators, the correlations \n(for $\\mu\\neq\\nu$) are determined by \n%\n\\bea\n\\label{correlations}\ni\\partial_t\n\\langle\\hat c^\\dagger_{\\mu s I}\\hat c_{\\nu s J}\\rangle^{\\rm corr}\n=\n\\frac1Z\\sum_{\\lambda L} T_{\\mu\\lambda}\n\\langle\\hat n_{\\mu\\bar s}^I\\rangle^0\n\\langle\\hat c^\\dagger_{\\lambda s L}\\hat c_{\\nu s J}\\rangle^{\\rm corr}\n\\nn\n-\n\\frac1Z\\sum_{\\lambda L} T_{\\nu\\lambda}\n\\langle\\hat n_{\\nu\\bar s}^J\\rangle^0\n\\langle\\hat c^\\dagger_{\\mu s I}\\hat c_{\\lambda s L}\\rangle^{\\rm corr}\n\\nn\n+\n\\left(U_\\nu^J-U_\\mu^I+V_\\nu-V_\\mu\\right) \n\\langle\\hat c^\\dagger_{\\mu s I}\\hat c_{\\nu s J}\\rangle^{\\rm corr}\n\\nn\n+\\frac{T_{\\mu\\nu}}{Z}\n\\left(\n\\langle\\hat n_{\\mu\\bar s}^I\\rangle^0\n\\langle\\hat n_{\\nu s}\\hat n_{\\nu\\bar s}^J\\rangle^0\n-\n\\langle\\hat n_{\\nu\\bar s}^J\\rangle^0\n\\langle\\hat n_{\\mu s}\\hat n_{\\mu\\bar s}^I\\rangle^0\n\\right) \n\\,,\n\\ea\n%\nwhere $\\langle\\hat X_\\mu\\rangle^0={\\rm Tr}\\{\\hat X_\\mu\\hat\\rho_\\mu^0\\}$\ndenote expectation values in the mean-field background.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\subsection{Factorization}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nThe evolution equations~\\eqref{correlations} for the correlators can be simplified \nby factorizing them via the following effective linear equations for the particle \nand hole operators\n%\n\\bea\n\\label{factorization}\n\\left(i\\partial_t-U^I_\\mu-V_\\mu\\right)\\hat c_{\\mu s I}\n=\n-\\frac1Z\\sum_{\\nu J} \nT_{\\mu\\nu} \\langle\\hat n_{\\mu\\bar s}^I\\rangle^0 \\hat c_{\\nu s J}\n\\,.\n\\ea\n%\nOf course, the hierarchy of correlations is not the only way to derive such \neffective evolution equations, similar results can be obtained by other \napproximation schemes, e.g., \\cite{Fis08}. \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Ising type spin order}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nIn order to analyze the effective equations~\\eqref{factorization}, we have to \nspecify the mean-field background $\\hat\\rho_\\mu^0$. \n%\nThe Mott insulator state corresponds to having one particle per lattice site,\nwhich leaves to determine the remaining spin degrees of freedom. \n%\nAs our first example, we consider anti-ferromagnetic spin order of the Ising type \\cite{Hir89}. \n%\nTo this end, we assume a bi-partite lattice which can be spit into two sub-lattices \n$\\cal A$ and $\\cal B$ where all neighbors $\\nu$ of a lattice site $\\mu\\in\\cal A$\nbelong to $\\cal B$ and vice versa. \n%\nThen, the zeroth-order mean-field background reads \n%\n\\bea\n\\label{mean-field-Ising}\n\\hat\\rho_\\mu^0\n=\n\\left\\{ \n\\begin{array}{ccc}\n\\ket{\\uparrow}_\\mu\\!\\bra{\\uparrow} & {\\rm for} & \\mu\\in\\cal A\n\\\\\n\\ket{\\downarrow}_\\mu\\!\\bra{\\downarrow} & {\\rm for} & \\mu\\in\\cal B\n\\end{array}\n\\right. \n\\,.\n\\ea\n%\nThis state\nminimizes the Ising type anti-ferromagnetic interaction \n$\\hat S^z_\\mu\\hat S^z_\\nu$. \n%\nNote that the Fermi-Hubbard Hamiltonian~\\eqref{Fermi-Hubbard} does indeed \ngenerate an effective anti-ferromagnetic interaction via second-order hopping \nprocesses, but it would correspond to a Heisenberg type anti-ferromagnetic \ninteraction $\\hat{\\f{S}}_\\mu\\cdot\\hat{\\f{S}}_\\nu$ \\cite{Cha78}. \n%\nAlthough the state~\\eqref{mean-field-Ising} does not describe the exact minimum \nof this interaction $\\hat{\\f{S}}_\\mu\\cdot\\hat{\\f{S}}_\\nu$, it can be regarded as \nan approximation or a simplified toy model for such an anti-ferromagnet.\n%\nAlternatively, one could imagine additional spin interactions between the \nelectrons (stemming from the full microscopic description) which are not \ncontained in the tight-binding model~\\eqref{Fermi-Hubbard} and stabilize \nthe state~\\eqref{mean-field-Ising}. \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Effective Dirac Equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nThis background~\\eqref{mean-field-Ising} supports \nhole excitations $I=0$ of spin $\\uparrow$ \nand particle excitations $I=1$ of spin $\\downarrow$ \nfor the sub-lattice $\\cal A$, and vice versa for the sub-lattice $\\cal B$. \n%\nFor the other terms, such as $\\hat c_{\\mu\\in{\\cal A},s=\\uparrow,I=1}$, \nthe right-hand side of Eq.~\\eqref{factorization} vanishes and thus they \nbecome trivial and are omitted in the following.\n\nAs a result, particle excitations in the sub-lattice $\\cal A$ are \ntunnel coupled to hole excitations of the same spin in the sub-lattice $\\cal B$\nand vice versa. \n%\nSince the spin components $s=\\uparrow$ and $s=\\downarrow$ evolve independent \nof each other, we drop the spin index in the following. \n%\nIntroducing the effective spinor in analogy to the Dirac equation \n%\n\\bea\n\\hat\\psi_\\mu\n=\n\\left(\n\\begin{array}{c}\n\\hat c_{\\mu I=1} \n\\\\\n\\hat c_{\\mu I=0} \n\\end{array}\n\\right) \n\\,,\n\\ea\n%\nthe evolution equation can be cast into the form \n%\n\\bea\n\\label{pre-Dirac}\ni\\partial_t\\hat\\psi_\\mu\n=\n\\left(\n\\begin{array}{cc} \nV_\\mu+U & 0 \n\\\\\n0 & V_\\mu \n\\end{array}\n\\right)\n\\cdot\\hat\\psi_\\mu\n-\\frac1Z\\sum_{\\nu}T_{\\mu\\nu}\n%\\left(\n%\\begin{array}{cc} \n%0 & 1 \n%\\\\\n%1 & 0\n%\\end{array}\n%\\right)\n\\sigma_x\\cdot\\hat\\psi_\\nu\n\\,,\n%\\nn\n\\ea\n%\nwhere $\\sigma_x$ is the Pauli spin matrix. \n%\nThis form is already reminiscent of the Dirac equation in 1+1 dimensions. \n%\nTo make the analogy more explicit, we first apply a simple phase \ntransformation $\\hat\\psi_\\mu\\to\\hat\\psi_\\mu\\exp\\{itU\/2\\}$ after which \nthe $U$ term reads $\\sigma_zU\/2$. \n\nSince the wavelength of an optical laser is typically much longer than all \nother relevant length scales, we may approximate it \n(in the non-relativistic regime) by a purely time-dependent electric field \n$\\f{E}(t)$ such that the potential reads $V_\\mu(t)=q\\f{r}_\\mu\\cdot\\f{E}(t)$\nwhere $\\f{r}_\\mu$ is the position vector of the lattice site $\\mu$. \n%\nThen we may use the Peierls transformation \n$\\hat\\psi_\\mu\\to\\hat\\psi_\\mu\\exp\\{i\\varphi_\\mu(t)\\}$ with \n$\\dot\\varphi_\\mu=V_\\mu$ to shift the potential $V_\\mu$ into \ntime-dependent phases of the hopping matrix \n$T_{\\mu\\nu}\\to T_{\\mu\\nu}e^{i\\varphi_\\mu(t)-i\\varphi_\\nu(t)}=T_{\\mu\\nu}(t)$.\n%\nNext, a spatial Fourier transformation simplifies Eq.~\\eqref{pre-Dirac} to \n%\n\\bea\n\\label{Dirac-k}\ni\\partial_t\\hat\\psi_{\\fk{k}}\n=\n\\left(\n\\frac{U}{2}\\,\\sigma_z\n-\nT_{\\fk{k}}(t)\\sigma_x\n\\right)\\cdot\\hat\\psi_{\\fk{k}}\n\\,,\n\\ea\n%\nwhere $T_{\\fk{k}}(t)$ denotes the Fourier transform of the hopping matrix \nincluding the time-dependent phases, which yields the usual minimal coupling \nform $T_{\\fk{k}}(t)=T_{\\fk{k}-q\\fk{A}(t)}$. \n%\nNote that the Peierls transformation is closely related to the gauge \ntransformation $A_\\mu\\to A_\\mu+\\partial_\\mu\\chi$ in electrodynamics. \n%\nUsing this gauge freedom, one can represent an electric field $\\f{E}(t)$\nvia the scalar potential $\\phi$ as $\\partial_t+iq\\phi$ in analogy to the $V_\\mu$ \nor via the vector potential $\\f{A}$ as $\\nabla-iq\\f{A}$ in analogy to the\n$T_{\\fk{k}}(t)=T_{\\fk{k}-q\\fk{A}(t)}$. \n\nIn the absence of the electric field, the dispersion relation following from \nEq.~\\eqref{Dirac-k} reads \n%\n\\bea\n\\label{dispersion-relativistic}\n\\omega_{\\fk{k}}=\\pm\\sqrt{\\frac{U^2}{4}+T_{\\fk{k}}^2}\n\\,.\n\\ea\n%\nThe positive and negative frequency solutions correspond to the upper and lower \nHubbard bands, which are separated by the Mott gap. \n%\nUnless the electric field is too strong or too fast, one expects the main \ncontributions to doublon-holon pair creation near the minimum gap, i.e., \nthe minimum of $T_{\\fk{k}}^2$, typically at $T_{\\fk{k}}=0$.\n%\nThen, a Taylor expansion $\\f{k}=\\f{k}_0+\\delta\\f{k}$\naround a zero $\\f{k}_0$ of $T_{\\fk{k}}$ yields \n%\n\\bea\n\\label{Dirac-approx}\ni\\partial_t\\hat\\psi_{\\fk{k}}\n\\approx \n\\left(\n\\frac{U}{2}\\,\\sigma_z\n-\n\\f{c}_{\\rm eff}\\cdot[\\delta\\f{k}-q\\f{A}(t)] \n\\sigma_x\n\\right)\\cdot\\hat\\psi_{\\fk{k}}\n\\,,\n\\ea\n%\nwhere $\\f{c}_{\\rm eff}=\\nabla_{\\fk{k}}T_{\\fk{k}}|_{\\fk{k}_0}$ \ndenotes the effective propagation velocity, \nin analogy to the speed of light. \n%\nNote that the validity of this approximation does not only require \n$\\delta\\f{k}$ to be small, it also assumes that $q\\f{A}(t)$ does not become \ntoo large, e.g., that we are far away from the regime of Bloch oscillations \\cite{Eck11,Niu96}. \n\nOn the other hand, even if $q\\f{A}(t)$ varies over a larger range, \nthe above approximation~\\eqref{Dirac-approx} could still provide a reasonably \ngood description for strong-field doublon-holon pair creation. \n%\nThis process can be understood as Landau-Zener tunneling occurring when an \navoided level crossing is traversed with a finite speed \n(set by $\\f{E}=-\\dot{\\f{A}}$). \n%\nSince this tunneling process mainly occurs in the vicinity of the minimum gap, \ni.e., where $T_{\\fk{k}}(t)=T_{\\fk{k}-q\\fk{A}(t)}=0$, \nit is sufficient to consider the region around $\\f{k}-q\\f{A}=\\f{k}_0$. \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Analogy to QED}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nUp to simple phase factors $\\exp\\{i\\f{k}_0\\cdot\\f{r}_\\mu\\}$, \nEq.~\\eqref{Dirac-approx} displays a qualitative analogy to the Dirac equation \nin 1+1 dimensions, where $c_{\\rm eff}$ plays the role of the speed of light \nwhile $U\/2$ corresponds to the mass $m_{\\rm eff}c_{\\rm eff}^2=U\/2$. \n%\nAs a result, we may now apply many of the results known from \nquantum electrodynamics (QED) \\cite{\nDirac1,Dirac2,Klein29,Sau32,Hei36,Zen32,Zen34,Lan32,Bun70,Niki67,Kel65}. \n%\nFor example, the effective spinor $\\hat\\psi_{\\fk{k}}$ can be expanded into \nparticle and hole contributions (first in the absence of an electric field) \n%\n\\bea\n\\hat\\psi_{\\fk{k}}\n=\nu_{\\fk{k}}\\hat d_{\\fk{k}}+v_{\\fk{k}}\\hat h_{\\fk{k}}^\\dagger \n\\,,\n\\ea\n%\nwhere the quasi-particles are usually referred to as doublons $\\hat d_{\\fk{k}}$\n(upper Hubbard band) and holons $\\hat h_{\\fk{k}}^\\dagger$\n(lower Hubbard band). \n%\nThe Mott state $\\ket{\\rm Mott}$ is then determined by \n$\\hat d_{\\fk{k}}\\ket{\\rm Mott}=\\hat h_{\\fk{k}}\\ket{\\rm Mott}=0$.\n\nIn the presence of an electric field, these operators can mix -- \nas described by the Bogoliubov transformation \n%\n\\bea\n\\hat d_{\\fk{k}}^{\\rm out}\n=\n\\alpha_{\\fk{k}}\\hat d_{\\fk{k}}^{\\rm in}\n+\n\\beta_{\\fk{k}}\\left(\\hat h_{\\fk{k}}^{\\rm in}\\right)^\\dagger \n\\,,\n\\ea\n%\nwhere the structure of the Dirac equation~\\eqref{Dirac-approx} implies the \nnormalization $|\\alpha_{\\fk{k}}|^2+|\\beta_{\\fk{k}}|^2=1$. \n%\nStarting in the Mott state \n$\\hat d_{\\fk{k}}^{\\rm in}\\ket{\\rm Mott}=\\hat h_{\\fk{k}}^{\\rm in}\\ket{\\rm Mott}=0$,\nthe $\\beta_{\\fk{k}}$ coefficient yields the amplitude for doublon-holon pair \ncreation \n$\\hat d_{\\fk{k}}^{\\rm out}\\ket{\\rm Mott}\\propto\\beta_{\\fk{k}}$. \n%\nIn analogy to QED, we may now discuss different regimes. \n%\nFor weak electric fields $\\f{E}$ oscillating near resonance $\\omega\\approx U$,\nwe find the usual lowest-order perturbative scaling \n$\\beta_{\\fk{k}}\\sim|q\\f{c}_{\\rm eff}\\cdot\\f{E}|$ \\cite{Bre34}. \n%\nHigher orders of perturbation theory then lead us in the multi-photon regime, \nfor example $n\\omega\\approx U$ with \n$\\beta_{\\fk{k}}\\sim|q\\f{c}_{\\rm eff}\\cdot\\f{E}|^n$. \n\nNote that a completely different kind of resonances such as $\\omega\\approx2U$ \ncan occur if we take higher-order correlations into account \\cite{Quei19,Quei19b}, but these \nare beyond our effective description~\\eqref{Dirac-approx}.\n\nIf the electric field becomes stronger and slower, we enter the non-perturbative\nregime of the Sauter-Schwinger effect where the pair-creation amplitude displays \nan exponential scaling \\cite{Sau31,Sch51}\n%\n\\bea\n\\beta_{\\fk{k}}\\sim\\exp\\left\\{-\\frac{\\pi U^2}{8q|\\f{c}_{\\rm eff}\\cdot\\f{E}|}\\right\\}\n\\,.\n\\ea\n%\nThe quantitative analogy to the Dirac equation even allows us to directly \ntransfer further results from QED, for example the dynamically assisted \nSauter-Schwinger effect, where pair creation by a strong and slowly varying \nelectric field is enhanced by adding a weaker and faster varying field, see,\ne.g., \\cite{Sch08}.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Unordered spin state}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nLet us compare our findings above to the case of a mean-field \nbackground without any spin order\n%\n\\bea\n\\label{mean-field-unordered}\n\\hat\\rho_\\mu^0\n=\n\\frac{\\ket{\\uparrow}_\\mu\\!\\bra{\\uparrow}+\\ket{\\downarrow}_\\mu\\!\\bra{\\downarrow}}{2}\n\\,, \n\\ea\n%\nwhich could arise for a finite temperature which is too small to excite \ndoublon-holon pairs but large enough to destroy the spin order. \n%\nAnother option could be a weak magnetic disorder potential and\/or spin \nfrustration. \n\nIn this case, we do not have to distinguish the two sub-lattices $\\cal A$ \nand $\\cal B$ and all lattice sites can support particle and hole excitations.\n%\nSince all expectation values $\\langle\\hat n_{\\mu s}^I\\rangle^0$ yield $1\/2$ \n(instead of zero or unity), the analog of Eq.~\\eqref{Dirac-k} now reads \n(after the same phase transformation)\n%\n\\bea\n\\label{matrix-unordered}\ni\\partial_t\n\\hat\\psi_{\\fk{k}}\n%\\left(\n%\\begin{array}{c}\n%\\hat f_{\\mathbf{k}s}\n%\\\\\n%\\hat e_{\\mathbf{k}s}^\\dagger \n%\\end{array}\n%\\right)\n=\n\\frac12\n\\left(\n\\begin{array}{cc}\nU-T_{\\bf k} & -T_{\\bf k}\n\\\\\n-T_{\\bf k} & -U-T_{\\bf k}\n\\end{array}\n\\right)\n\\cdot\n\\hat\\psi_{\\fk{k}}\n%\n%\\left(\n%\\begin{array}{c}\n%\\hat f_{\\mathbf{k}s}\n%\\\\\n%\\hat e_{\\mathbf{k}s}^\\dagger \n%\\end{array}\n%\\right)\n\\,.\n\\ea\n%\nThe eigenvalues of the above $2\\times2$-matrix yield the \nquasi-particle frequencies \\cite{Hub63,Herr97}\n%\n\\bea\n\\label{quasi-particle-energies}\n\\omega^\\pm_\\mathbf{k}\n=\n\\frac12\\left(-T_\\mathbf{k}\\pm\\sqrt{T_\\mathbf{k}^2+U^2}\\right)\n\\,.\n\\ea\n%\nIn view of the additional term $-T_\\mathbf{k}$ in front of the square root, \nthis dispersion relation does not display the same quasi-relativistic form \nas in Eq.~\\eqref{dispersion-relativistic}.\n%\nPutting it another way, we find that Eq.~\\eqref{matrix-unordered}\nis not formally equivalent to the Dirac equation (in 1+1 dimensions). \n\nAs a consequence, the propagation of quasi-particles in the two mean-field \nbackgrounds~\\eqref{mean-field-Ising} and \\eqref{matrix-unordered} is quite \ndifferent. \n%\nFor the Ising type order~\\eqref{mean-field-Ising}, the coherent propagation \nof a doublon or holon (without changing the background structure) \nrequires second-order hopping processes. \n%\nHence $\\omega_{\\fk{k}}$ in Eq.~\\eqref{dispersion-relativistic} \nis a quadratic function of $T_{\\fk{k}}$.\n%\nFor the unordered background~\\eqref{mean-field-unordered}, on the other hand, \ndoublons and holons can propagate coherently via first-order hopping processes. \n%\nThis is reflected in the linear contribution $-T_\\mathbf{k}$ in \nEq.~\\eqref{quasi-particle-energies}.\n\nHowever, if we do not consider quasi-particle propagation but focus on the \nprobability for creating a doublon-holon pair in a given mode $\\f{k}$,\nwe may again derive a close analogy to QED. \n%\nTo this end, we apply yet another $\\f{k}$-dependent phase transformation \n$\\hat\\psi_{\\fk{k}}\\to e^{i\\vartheta_{\\fk{k}}(t)}\\hat\\psi_{\\fk{k}}$ with \n$\\dot\\vartheta_{\\fk{k}}(t)=T_{\\fk{k}}(t)\/2$.\n%\nNote that $T_{\\fk{k}}(t)$ contains the vector potential $\\f{A}(t)$, i.e., \nthe time integral of the electric field $\\f{E}(t)$.\n%\nThus, the phase $\\vartheta_{\\fk{k}}(t)$ involves yet another time integral,\nwhich makes it a even more non-local function of time.\n%\nAfter this phase transformation, Eq.~\\eqref{matrix-unordered} becomes again \nformally equivalent to the Dirac equation in 1+1 dimensions, but now with \nthe effective speed of light being reduced by a factor of two.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Conclusions}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nWe study doublon-holon pair creation from the Mott insulator state of the \nFermi-Hubbard model induced by an external electric field $\\f{E}(t)$ which \ncould represent an optical laser, for example.\n%\nWe find that the creation and propagation dynamics of the doublons and holons \ndepends on the spin structure of the mean-field background. \n%\nFor Ising type anti-ferromagnetic order, we observe a quantitative analogy \nto QED.\n%\nMore specifically, in the vicinity of the minimum gap \n(i.e., the most relevant region for pair creation), \nthe doublons and holons are described by an effective\nDirac equation in 1+1 dimensions in the presence of an electric field \n$\\f{E}(t)$.\n\nAs a consequence of this quantitative analogy, we may employ the machinery of QED \nand apply many results regarding electron-positron pair creation to our set-up. \n%\nFor example, in the perturbative (single- or multi-photon) regime with the \nthreshold conditions $\\omega\\geq nU$ for the $n$-th order, the doublon-holon \npair creation amplitude $\\beta_{\\fk{k}}$ yields the perturbative scaling \n$\\beta_{\\fk{k}}\\sim|q\\f{c}_{\\rm eff}\\cdot\\f{E}|^n$. \n\nFor stronger and slower electric fields, we enter the non-perturbative (tunneling) \nregime in analogy to the Sauter-Schwinger effect in QED and thus recover \nthe exponential dependence already discussed earlier regarding the dielectric \nbreakdown of Mott insulator, see, e.g., \\cite{Eck10}. \n%\nNote that the quantitative analogy established above (see also Table~\\ref{table1})\nunambiguously determines the \npair-creation exponent and pre-factor without and free fitting parameters. \n%\n%This analogy is sketched in the following table: \n\nIf we consider the annihilation of doublon-holon pairs instead of their \ncreation, this analogy applies to the stimulated annihilation within an \nexternal field, but not to the spontaneous annihilation of an electron-positron\npair by emitting a pair of photons, for example. \n%\nIn order to model this process, one has to include a mechanism for dissipating\nthe energy, e.g., by coupling the Fermi-Hubbard model to an environment, see \nalso \\cite{Quei19c}.\n\nFor a mean-field background without any spin order, on the other hand, \nthe creation and propagation of doublons and holons does not display such \na quasi-relativistic behavior. \n%\nThe dispersion relation is different and the evolution equation deviates from \nthe Dirac equation. \n%\nStill, for a purely time-dependent electric field $\\f{E}(t)$ considered here,\nthe doublon-holon pair creation amplitude $\\beta_{\\fk{k}}$ for a given mode \n$\\f{k}$ can again be related to QED.\n%\nAfter a $\\f{k}$-dependent phase transformation (which is non-local in time), \nthe amplitude $\\beta_{\\fk{k}}$ is given by the same expression, just with \nthe effective speed of light $c_{\\rm eff}$ being reduced by a factor of two. \n\nIn view of the $\\f{k}$-dependence of the phase transformation, this mapping \ndoes only work for purely time-dependent electric field $\\f{E}(t)$.\n%\nFor space-time dependent electric fields $\\f{E}(t,\\f{r})$, the deviation of \nthe dispersion relation and the resulting difference in propagation become \nimportant -- which will be the subject of further studies. \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{table}[ht] \n\\begin{tabular}{|c|c|}\n\\hline \nMott insulator & QED vacuum \n\\\\\nupper Hubbard band & positive energy continuum \n\\\\ \nlower Hubbard band & Dirac sea \n\\\\\ndoublons \\& holons & electrons \\& positrons \n\\\\\nMott gap $U=2m_{\\rm eff}c_{\\rm eff}^2$ & electron mass\n\\\\\nvelocity $\\f{c}_{\\rm eff}=\\nabla_{\\fk{k}}T_{\\fk{k}}|_{\\fk{k}_0}$ \n& speed of light $c$ \n\\\\\nLandau-Zener tunneling & Sauter-Schwinger effect \n\\\\\nPeierls transformation & gauge transformation\n\\\\\n\\hline \n\\end{tabular}\n\\label{table1}\n\\caption{Sketch of the analogy. \nThe electric field $\\f{E}(t)$ and elementary charge $q$ play the same role \nin both cases.}\n\\end{table}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\acknowledgments\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n%R.S.~acknowledges valuable discussions with S.~Popescu and W.G.~Unruh. \n%\nFunded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) \n-- Project-ID 278162697-- SFB 1242. \nPN thanks support from the EU project\nSUPERGALAX (Grant agreement ID: 863313).\n\n\\begin{thebibliography}{99}\n\n%Hubbard model\n\n\n\\bibitem{Hub63}\nJ.~Hubbard,\n\\textit{Electron correlations in narrow energy bands},\nProc.~R.~Soc.~Lond.~A {\\bf 276}, 238 (1963).\n\n\n\\bibitem{Aro22}\nD.~P.~Arovas, E.~Berg, S.~A.~Kivelson, and 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quantum-relativistic phenomena in graphene}\nSol.~S.~Comm.~{\\bf 149}, 1087 (2009).\n\n\n\\bibitem{S84}\nG.~W.~Semenoff,\n\\textit{Condensed-Matter Simulation of a Three-Dimensional Anomaly},\nPhys.~Rev.~Lett.~{\\bf53}, 2449 (1984).\n\n\n%Helium-3\n\n\\bibitem{Scho92}\nN.~Schopohl and G.~E.~Volovik,\n\\textit{Schwinger Pair Production in the Orbital Dynamics of $^3$He-B}\nAnn.~Phys.~{\\bf 215}, 372 (1992)\n\n\n\n\\bibitem{Mott49}\nN.~F.~Mott,\n\\textit{The Basis of the Electron Theory of Metals, with Special Reference to the Transition Metals},\nProc.~Phys.~Soc.~A {\\bf 62} 416 (1949).\n\n\n\n\n\n\n% \\bibitem{Fis89}\n% M.~P.~A.~Fisher, P.~B.~Weichman, G.~Grinstein, and D.~S.~Fisher,\n% \\textit{Boson localization and the superfluid-insulator transition},\n% Phys.~Rev.~B {\\bf 40}, 546 (1989).\n\n\n\n\n\n\n\n\n\n\n\\bibitem{Nav10}\nP.~Navez and R.~Sch\u00fctzhold,\n\\textit{Emergence of coherence in the Mott-insulator\u2013superfluid quench of the Bose-Hubbard model},\nPhys.~Rev.~A {\\bf 82}, 063603 (2010).\n\n\n\\bibitem{Krut14}\nK.~V.~Krutitsky, P.~Navez, F.~Queisser, and R.~Sch\u00fctzhold,\n\\textit{Propagation of quantum correlations after a quench in the Mott-insulator regime of the Bose-Hubbard model}, \nEPJ Quantum Technology {\\bf 1}, 12 (2014).\n\n\n\\bibitem{Queiss19}\nF.~Queisser, and R.~Sch\u00fctzhold,\n\\textit{Environment-induced prerelaxation in the Mott-Hubbard model},\nPhys.~Rev.~B {\\bf99}, 155110 (2019).\n\n\n\\bibitem{Nav16}\nP.~Navez, F.~Queisser, and R.~Sch\u00fctzhold,\n\\textit{Large-coordination-number expansion of a lattice Bose gas at finite temperature},\nPhys.~Rev.~A {\\bf 94}, 023629 (2016).\n\n\n\\bibitem{Queiss14}\nF.~Queisser, K.~V.~Krutitsky, P.~Navez, and R.~Sch\u00fctzhold,\n\\textit{Equilibration and prethermalization in the Bose-Hubbard and Fermi-Hubbard models},\nPhys.~Rev.~A {\\bf 89}, 033616 (2014).\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibitem{Fis08}\nU.~R.~Fischer, R.~Sch\u00fctzhold, and M.~Uhlmann,\n\\textit{Bogoliubov theory 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quantum theory of the Electron. Part II},\nProc.~R.~Soc.~A {\\bf 118} 351 (1928).\n\n\\bibitem{Klein29}\nO.~Klein,\n\\textit{Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac},\nZeit.~f.~Phys.~{\\bf 53}, 157 (1929).\n\n\\bibitem{Sau32}\nF.~Sauter,\n\\textit{Zum ,,Kleinschen Paradoxon''},\nZ.~Phys.~{\\bf73}, 547 (1932).\n\n\\bibitem{Hei36}\nW.~Heisenberg and H.~Euler, \n\\textit{Folgerungen aus der Diracschen Theorie des Positrons},\nZeit.~Phys.~{\\bf 98}, 714 (1936).\n\n\n\\bibitem{Zen32}\nC.~Zener,\n\\textit{Non-adiabatic crossing of energy levels},\nProc.~Roy.~Soc.~A {\\bf 137} 696 (1932).\n\n\\bibitem{Zen34}\nC.~Zener,\n\\textit{A theory of the electrical breakdown of solid dielectrics},\nProc.~Roy.~Soc.~A {\\bf 145}, 523 (1934).\n\n\n\\bibitem{Lan32}\nL.~D.~Landau,\n\\textit{On the Theory of Transfer of Energy at Collisions II.}\nPhys.~Z.~Sow.~{\\bf 2}, 46 (1932).\n\n\n\\bibitem{Bun70}\nF.~V.~Bunkin and I.~I.~Tugov, \n\\textit{Possibility of Creating Electron-Positron Pairs in a Vacuum by the Focusing of Laser Radiation},\nSov.~Phys.~Dokl.~{\\bf 14}, 678 (1970).\n\n\n\\bibitem{Niki67}\nA.~I.~Nikishov and V.~I.~Ritus,\n\\textit{Ionization of atoms by an electromagnetic wave field},\nSov.~Phys.~JETP, {\\bf 25} 1135 (1967).\n\n\n\\bibitem{Kel65}\nL.~V.~Keldysh,\n\\textit{Ionization in the field of a strong electromagnetic wave},\nSov.~Phys.~JETP {\\bf 20}, 1307 (1965).\n\n\\bibitem{Bre34}\nG. Breit and John A. Wheeler,\n\\textit{Collision of Two Light Quanta},\nPhys.~Rev.~{\\bf 46}, 1087 (1934).\n\n\n\n\n\n\n\n\n\n\\bibitem{Quei19}\nF.~Queisser, and R.~Sch\u00fctzhold,\n\\textit{Boltzmann relaxation dynamics in the strongly interacting Fermi-Hubbard model},\nPhys.~Rev.~A {\\bf 100}, 053617 (2019).\n\n\\bibitem{Quei19b}\nF.~Queisser, S.~Schreiber, P.~Kratzer, and R.~Sch\u00fctzhold,\n\\textit{Boltzmann relaxation dynamics of strongly interacting spinless fermions on a lattice},\nPhys.~Rev.~B {\\bf 100}, 245110 (2019).\n\n\n\n\n\n\n\\bibitem{Herr97}\nT.~Herrmann and W.~Nolting,\n\\textit{Magnetism in the single-band Hubbard model},\nJ.~Magn.~Magn.~Mater {\\bf 170}, 253 (1997),\n\n\n\n\n\n\\bibitem{Quei19c}\nF.~Queisser, and R.~Sch\u00fctzhold,\n\\textit{Environment-induced prerelaxation in the Mott-Hubbard model},\nPhys.~Rev.~B {\\bf 99}, 155110 (2019).\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\end{thebibliography}\n\n\n\n\\end{document}\n"},{"text":"\\documentclass{amsart}\n\\newcommand\\hmmax{0}\n\\newcommand\\bmmax{0}\n\n\\usepackage[utf8]{inputenc}\n\\usepackage[top=1.25in, bottom=1.25in, left=1.1in, right=1.1in]{geometry, mathtools}\n\\usepackage[all]{xy}\n\\usepackage[toc,page]{appendix}\n\\usepackage{amsthm, amsmath, amssymb, amscd, latexsym, multicol, verbatim, enumerate, graphicx,xy, color}\n\\usepackage{mathrsfs}\n\\usepackage{tikz,stackrel}\n\\usepackage{mathdots}\n\\usepackage[all]{xy}\n\\usetikzlibrary{calc}\n\\usetikzlibrary{matrix,arrows,decorations.pathmorphing}\n\\usetikzlibrary{intersections}\n\\usetikzlibrary{decorations.markings}\n\\usetikzlibrary{external}\n%\\usepackage[external]{includetikz}\n\\tikzexternalize[\n mode=graphics if exists,\n prefix=Pictures\/\n ]\n\\usepackage{setspace}\n%\\linespread{1.2}\n\n\\newcommand{\\inputtikz}[1]{%\n \\tikzsetnextfilename{#1}%\n 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the clef extends below the baseline, so we raise and smash it\n \\raisebox{.412\\height}[0pt][0pt]{\\usefont{U}{musix}{m}{n}\\symbol{73}}%\n }%\n}\n\\newcommand{\\base}{\\mathbassclef}\n\n\\newcommand{\\Np}{N_{\\prin}}\n\\newcommand{\\Mp}{M_{\\prin}}\n\\newcommand{\\Kp}{K_{\\prin}}\n\\newcommand{\\Npc}{N^\\circ_{\\prin}}\n\\newcommand{\\Mpc}{M^\\circ_{\\prin}}\n\\newcommand{\\Kpc}{K^\\circ_{\\prin}}\n\n\n\n\n\\tikzexternalize[prefix=Pictures\/]\n\n\\newcommand\\tim[1]{{\\color{purple} \\sf \\Bicycle Tim: [#1]}}\n\\newcommand\\lara[1]{{\\color{cyan} \\sf $\\clubsuit$ Lara: [#1]}}\n\\newcommand\\alfredo[1]{{\\color{blue} \\sf $\\infty$ Alfredo: [#1]}}\n\\newcommand\\mandy[1]{{\\color{magenta} \\sf $\\nabla$ Mandy: [#1]}}\n\n\\title{Newton--Okounkov bodies and minimal models for cluster varieties}\n\n\n\n\\author{Lara Bossinger, Man-Wai Cheung, Timothy Magee and Alfredo N\\'ajera Ch\\'avez}\n\\date{\\today}\n\n\\address{\nInstituto de Matem\\'aticas Unidad Oaxaca, \nUniversidad Nacional Aut\\'onoma de M\\'exico,\nLe\\'on 2, altos, \nCentro Hist\\'orico,\n68000 Oaxaca,\nMexico}\n\\email{lara@im.unam.mx}\n\n\\address{\nSchool of Mathematics, Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Japan, 277-8583}\n\\email{manwai.cheung@ipmu.jp}\n\n\n\\address{Department of Mathematics, King's College London, Strand, London WC2R 2LS, UK}\n\\email{timothy.magee@kcl.ac.uk}\n\n\n\\address{\nConsejo Nacional de Ciencia y Tecnolog\u00eda - Instituto de Matem\\'aticas Unidad Oaxaca, \nUniversidad Nacional Aut\\'onoma de M\\'exico,\nLe\\'on 2, altos, \nCentro Hist\\'orico,\n68000 Oaxaca,\nMexico}\n\\email{najera@im.unam.mx}\n\n\\begin{document}\n\n\\maketitle\n\n\n\\begin{abstract}\nLet $Y$ be a (partial) minimal model of a scheme $V$ with a cluster structure (of type $\\cA$, $\\cX$ or of a quotient of $\\cA$ or a fibre of $\\cX$). Under natural assumptions, for every choice of seed we associate a Newton--Okounkov body to every divisor on $Y$ supported on $Y \\setminus V$ and show that these Newton--Okounkov bodies are positive sets in the sense of Gross, Hacking, Keel and Kontsevich \\cite{GHKK}. This construction essentially reverses the procedure in loc. cit. that generalizes the polytope construction of a toric variety to the framework of cluster varieties.\n\nIn a closely related setting, we consider cases where $Y$ is a projective variety whose universal torsor $\\UT_Y$ is a partial minimal model of a scheme with a cluster structure of type $\\cA$. If the theta functions parametrized by the integral points of the associated superpotential cone form a basis of the ring of algebraic functions on $\\UT_Y$ and the action of the torus $T_{\\text{Pic}(Y)^*}$ on $\\UT_Y$ is compatible with the cluster structure, then for every choice of seed we associate a Newton--Okounkov body to every line bundle on $Y$. We prove that any such Newton--Okounkov body is a positive set and that $Y$ is a minimal model of a quotient of a cluster $\\cA$-variety by the action of a torus. \n\nOur constructions lead to the notion of the intrinsic Newton--Okounkov body associated to a boundary divisor in a partial minimal model of a scheme with a cluster structure. This notion is intrinsic as it relies only on the geometric input, making no reference to the auxiliary data of a valuation or a choice of seed.\nThe intrinsic Newton--Okounkov body lives in a real tropical space rather than a real vector space. \nA choice of seed gives an identification of this tropical space with a vector space, and in turn of the intrinsic Newton--Okounkov body\nwith a usual Newton--Okounkov body associated to the choice of seed.\nIn particular, the Newton--Okounkov bodies associated to seeds are related to each other by tropicalized cluster transformations providing a wide class of examples of Newton-Okoukov bodies exhibiting a wall-crossing phenomenon in the sense of Escobar--Harada \\cite{EH20}.\n\nThis approach includes the partial flag varieties that arise as minimal models of cluster varieties (for example full flag varieties and Grassmannians). For the case of Grassmannians, our approach recovers, up to interesting unimodular equivalences, the Newton--Okounkov bodies constructed by Rietsch--Williams in \\cite{RW}. \n\n\n\n\\end{abstract}\n\n\\tableofcontents\n\n\\section{Introduction}\n\n\\subsection{Overview} Cluster varieties are certain schemes constructed by gluing a (possibly infinite) collection of algebraic tori using distinguished birational maps called cluster transformations.\nThese schemes were introduced in \\cite{FG_Teich,FG_cluster_ensembles} and can be studied from many different points of view. \nThey are closely related to cluster algebras and $Y$-patterns defined by Fomin and Zelevinsky in \\cite{FZ_clustersI, FZ_clustersIV}. \nIn this paper we approach them from the perspectives of birational and toric geometry, mainly following \\cite{GHK_birational,GHKK}.\nIn \\cite{GHKK}, the authors show that certain sets called \\emph{positive polytopes} can be used to produce compactifications of cluster varieties and toric degenerations of such compactifications. \nIn the trivial case where the cluster variety in question is just a torus, a positive lattice polytope is simply a usual convex lattice polytope and this construction produces the toric variety associated to such a polytope.\nOne of the main goals of this paper is to reverse this construction in a systematic way and understand this process from the view-point of Newton--Okounkov bodies. \nWe also study the wall-crossing phenomenon for Newton--Okounkov bodies arising from cluster structures.\nWe treat independently the case of the Grassmannians as, in this context, we compare the Newton--Okounkov bodies we construct with those constructed in \\cite{RW} and explore some consequences.\nMoreover, throughout the text we systematically consider not only cluster varieties but also quotients and fibres associated to them (see \\S\\ref{sec:quotients-fibres} for the precise definitions of these quotients and fibres). For simplicity, in this introduction our main focus is on cluster varieties. \nWe fix once and for all an algebraically closed field $\\Bbbk$ of characteristic zero. \nUnless otherwise stated, all the schemes we consider are over $\\Bbbk$.\n\n\\subsection{The tropical spaces} Let $\\cV $ be a cluster variety. By definition, $\\cV$ is endowed with an atlas of algebraic tori of the form \n\\[\n\\cV= \\bigcup_{\\seed} T_{L;\\seed},\n\\]\nwhere $L$ is a fixed lattice, $T_{L; \\seed} $ is a copy of the algebraic torus $T_L= \\Spec(\\Bbbk[L^*])$ associated to $L$ (so $L^*=\\text{Hom}(L, \\Z)$) and the tori in the atlas are parametrized by \\emph{seeds $\\seed$ for} $\\cV$. \nWe will exploit the fact that $\\cV$ is a log-Calabi--Yau variety. \nThis property implies that $\\cV$ is endowed with a canonical up-to-scaling volume form $\\Omega$. \nMoreover, recall that a cluster variety is of one of the types: $\\cA$ or $\\cX$.\n\nJust like in toric geometry where one can consider the dual torus $T_L^{\\vee}:=T_{L^*}$, the \\emph{dual} of $\\cV$ is a cluster variety $\\cV^{\\vee}$ whose defining atlas consists of tori of the form $T^\\vee_L$.\nIt is well known that the ring $H^{0} (T_L,\\mathcal{O}_{T_{L}})$ of algebraic functions on $T_L$ has a distinguished basis --the set of characters of $T_L$-- parametrized by $L^*$. \nFor nearly 10 years it was conjectured that this fact can be generalized for $\\cV$ using this notion of duality.\nIn order to state such a generalization, we consider the integral tropicalization of $\\cV^{\\vee}$, which we denote by $\\Trop_{\\Z}(\\cV^{\\vee})$.\nThe precise definition of $\\Trop_{\\Z}(\\cV^{\\vee})$\n can be found in \\S\\ref{ss:tropicalization}.\n For this introduction the key fact that we need is that a prime divisor $D$ on a variety birational to $\\cV^\\vee $ determines a point of $\\Trop_{\\Z}(\\cV^{\\vee})$ if $\\Omega $ has a pole along $D$.\nIn \\cite{FG_cluster_ensembles} Fock--Goncharov conjectured that $H^{0}(\\cV, \\mathcal{O}_\\cV)$ has a canonical vector space basis parametrized by $\\Trop_{\\Z}(\\cV^{\\vee})$.\nAlthough false in general, this conjecture does hold in many of the cases of wide interest.\nIn \\cite{GHKK} the authors linked this conjecture to the log Calabi--Yau mirror symmetry conjecture \\cite[Cnjecture 0.6]{GHK_logCY}, suggesting that the canonical basis proposed by Fock--Goncharov is the \\emph{theta basis}.\nAs we would like to be as close to toric geometry as possible we systematically assume that the full Fock--Goncharov conjecture holds for the cluster variety $ \\cV$ under consideration.\nSo, under under the assumption that the full Fock--Goncharov conjecture holds for $\\cV$, one may consider $\\Trop_{\\Z}(\\cV^{\\vee})$ as replacing $L^*$ and the characters of $T_L$ are replaced by the theta functions on $\\cV$.\nMoreover, the real vector space $L^*\\otimes \\R$ is replaced by the real tropicalization $ \\Trop_{\\R}(\\cV^{\\vee})$ and convex polyhedra inside $L^*\\otimes \\R$\nare replaced by positive sets in the real tropical space $\\Trop_{\\R}(\\cV^{\\vee})\\supset \\Trop_{\\Z}(\\cV^{\\vee})$ (see \\S\\ref{ss:tropicalization} and Definition \\ref{def:positive_set} for the definitions of $\\Trop_{\\R}(\\cV^{\\vee})$ and of positive set, respectively). \n\n\nBesides the trivial case where $\\cV$ is just a torus (and hence $\\cV^{\\vee}$ is just the dual torus), the tropical spaces $\\Trop_{\\Z}(\\cV^{\\vee})$ and $\\Trop_{\\R}(\\cV^{\\vee})$ do not possess a linear structure (there is no natural notion of addition in these spaces and only multiplication by positive scalars makes sense).\nHowever, in certain situations these tropical spaces do contain subsets where addition and scalar multiplication make sense, which we call \\emph{linear subsets}. \nIn any case, every choice of seed $\\seed^\\vee$ for $\\cV^{\\vee}$ gives rise to a bijection $\\mathfrak{r}_{\\seed^\\vee}:\\Trop_{\\R}(\\cV^{\\vee}) \\longrightarrow \\R^d$ that restricts to a bijection $\\Trop_{\\Z}(\\cV^{\\vee}) \\overset{\\sim}{\\longrightarrow} \\Z^d$, where $d$ is the dimension of both $\\cV$ and $\\cV^\\vee$. \nIn general, different seeds lead to different bijections. \nWhen we fix one such identification $\\mathfrak{r}_{\\seed^\\vee}$ and talk about linear subsets of $\\Z^d$ and positive subsets of $\\R^d$, what we mean is that the inverse image of such a set under $\\mathfrak{r}_{\\seed^\\vee}$ has the given property.\n\n\\subsection{Positive Newton--Okounkov bodies and minimal models} Newton--Okounkov bodies are convex closed sets in real vector spaces. Their systematic study was developed by Lazarsfeld--Musta\\c{t}\\u{a} \\cite{LM09} and Kaveh--Khovanskii \\cite{KK12} based on the work of Okounkov \\cite{Oko96,Oko03}. \nThis concept is a far reaching generalization of both the Newton polytope of a Laurent polynomial and the polytope of a polarized projective toric variety. \nIn \\cite{KK12} the authors introduced Newton--Okounkov bodies for Cartier divisors on irreducible varieties.\nIn this paper we consider Newton--Okounkov bodies associated to Weil divisors in the setting of minimal models for cluster varieties.\nMore precisely, let $D$ be a Weil divisor on a $d$-dimensional normal variety $Y$ admitting a non-zero global section, that is, the space $H^0(Y, \\mathcal{O}(D))$ is non-zero, where $\\mathcal{O}(D)$ is the coherent sheaf associated to $D$. \nThe section ring of $D$ is a graded ring \n\\[\nR(D)=\\bigoplus_{k\\in \\Z_{\\geq 0}}{R}_k(D) \n\\] \nwhose $k$-th homogeneous component is the vector space $R_k(D)=H^0(Y, \\mathcal{O}(kD)) \\subset \\Bbbk (Y)$. \nFix a non-zero element $\\tau\\in R_1(D)$, and suppose we are given a total order on $\\Z^d$ and a valuation $\\nu: \\Bbbk(Y)^* \\to \\Z^d$. \nThen the Newton--Okounkov body associated to this data is:\n\\eqn{\n\\Delta_\\nu(D,\\tau) := \\overline{\\conv\\Bigg( \\bigcup_{k\\geq 1} \\lrc{\\frac{\\nu\\lrp{f\/\\tau^k}}{k} \\mid f\\in R_k(D)\\setminus \\{0\\} } \\Bigg) }\\subseteq \\R^d.\n}\n\nGiven a cluster variety $\\cV$, our first goal is to use its cluster structure to construct Newton--Okounkov bodies associated to divisors in compactifications of $\\cV$, generalizing the construction of the polytope of a torus invariant divisors on a toric variety.\nHence, we need to establish the class of compactifications of $\\cV $, the divisors therein and the valuations we consider.\n\nWe begin discussing valuations obtained from the cluster structure. \nIn case $\\cV$ is a cluster $\\cA$-variety, this is closely related to the work of Fujita and Oya \\cite{FO20}. \nHowever, our approach includes the cases where $\\cV$ is a cluster $\\cX$-variety, a quotient of a cluster $\\cA$-variety, or a fibre of a cluster $\\cX$-variety.\nIn order to be able to use the cluster structure of $\\cV$ to construct a valuation on $ \\Bbbk(\\cV)$ certain conditions (depending on whether $\\cV$ is of type $\\cA$ or of type $\\cX$) need to be fulfilled. \nFor instances, if $\\cV$ is of type $\\cA$, a sufficient condition is that the rectangular matrix $\\widetilde{B}$ determining the cluster structure of $\\cV$ has full rank\\footnote{A weaker condition is enough to construct a valuation for a given seed, see Remark \\ref{rem:dom_order}.}; if $\\cV$ is of type $\\cX$ we need that the full Fock--Goncharov conjecture holds for $\\cX$ (as we are assuming), see \\S\\ref{sec:cluster_valuations} for more details, including the cases of quotients of $\\cA$ and fibres of $\\cX$.\nIn case the necessary conditions are satisfied then for every $\\seed$ for $\\cV$ we have a cluster valuation\n\\[\n\\nu_\\seed: \\Bbbk(\\cV) \\setminus\\{0\\} \\to (\\Z^d, <_{\\seed}).\n\\]\nThe total order $<_{\\seed}$ on $\\Z^d$ depends also on the type of $\\cV$.\nMoreover, in case $\\cV$ is of type $\\cA$ in the literature this valuation is generally denoted by $\\gv_{\\seed} $ and called a $\\gv$-\\emph{vector valuation} as it is closely related to the $\\gv$-vectors associated to cluster monomials introduced in \\cite{FZ_clustersIV}.\nIn case $\\cV$ is of type $\\cX$ the associated cluster valuation has not been systematically defined yet in the literature to the best of our knowledge. \nIn this case we also denote $\\nu_{\\seed}$ by $\\cv_{\\seed}$ and call it a $\\cv$-\\emph{vector valuation} since this valuation is closely related to the $\\cv$-vectors associated to $Y$-variables introduced in \\cite{NZ} and more generally to {\\bf c}-vectors of theta functions on $\\cX$ defined in \\cite{BFMNC}, and currently investigated in \\cite{ML23}. \nIn any case, for every seed $\\seed$ the theta basis of $H^0(\\cV, \\mathcal{O}_{\\cV})$ is adapted for the cluster valuation $\\nu_{\\seed}$.\nIn particular, if $ Y$ is a variety birational to $\\cV $ and $D$ is a divisor in $Y$, then, upon a choice of non-zero section $\\tau \\in R_1(D)$ and a seed $\\seed$, we can construct a Newton--Okounkov body $\\Delta_{\\nu_\\seed}(D,\\tau) $. \nWe are primarily interested in conditions ensuring that such a Newton--Okounkov body is a positive set.\nOn the one hand this is a condition that needs to be satisfied if one seeks to reverse Gross--Hacking--Keel--Kontsevich's construction of a compactification of a cluster variety from a positive set.\nOn the other hand, we are further interested in describing how the change of seed affects the Newton--Okounkov body and positivity plays the key role in understanding this. \n If $\\Delta_{\\nu_\\seed}(D,\\tau)$ is positive then any other $\\Delta_{\\nu_{\\seed'}}(D,\\tau)$ is obtained from $\\Delta_{\\nu_\\seed}(D,\\tau)$ by a composition of tropicalized cluster transformations. \n This will be discussed in more detail in the next subsection of the introduction.\nIn order to be able to show that $\\Delta_{\\nu_\\seed}(D,\\tau) $ is positive we restrict the class of compactifications of $\\cV$, the divisors we consider, and the sections we choose.\n\nOne can define a partial minimal model for $\\cV$\\footnote{Throughout the text we consider more generally (partial) minimal models for schemes $V$ with a cluster structure given by a birational map $\\cV\\dashrightarrow V$.} is an inclusion $\\cV \\subset Y$ such that $Y$ is normal and $\\Omega$ has a simple pole along every irreducible divisorial component of the boundary $D=Y \\setminus \\cV$, see \\cite[Remark~1.3]{GHK_birational}. It is a minimal model if $Y$ is projective over $\\Bbbk$.\nThese are the kind of (partial) compactifications of $\\cV $ we consider.\nThe main reason for this is that any prime divisor supported on $D$ determines a primitive point of $\\Trop_{\\Z}(\\cV)$. \nLet $D'$ be a divisor supported on $D$. \nWe say that $R(D')$ has a {\\it graded theta basis} if for each $k$ \nthe set of theta functions on $\\cV$ contained in $H^0(Y,\\mathcal{O}(kD'))$ forms a basis (see Definition~\\ref{def:graded_theta_basis}).\nThen we can prove the following result.\n\n\\begin{theorem*}\n(Theorem \\ref{NO_bodies_are_positive})\nLet $D'$ be a Weil divisor supported on the boundary $D$ of the minimal model $\\cV \\subset Y $ such that $R(D')$ has a graded theta basis. Let $\\tau\\in R_1(D')$ be such that $\\nu_{\\seed}(\\tau) $ belongs to a linear subset of $ \\Z^d$. Then the Newton--Okounkov body $\\Delta_{\\nu_{\\seed}}(D',\\tau)$ is a positive polytope.\n\\end{theorem*}\n\nIn Lemma~\\ref{lem:graded_theta_basis} we provide sufficient conditions ensuring that $R(D)$ has a graded theta basis. \nMoreover, the work of Mandel \\cite{Man19} provides conditions ensuring that a line bundle on a cluster $\\cX$-variety has a graded theta basis.\n\nWe further study another setting where we can use cluster structures to construct Newton--Okounkov bodies and show that they are positive polytopes:\nsuppose that $Y$ is a normal projective variety such that its Picard group is free and finitely generated.\nThe universal torsor of $Y$ is a scheme $\\UT_Y$ whose ring of algebraic functions is isomorphic to the direct sum of all the spaces of sections associated to all (isomorphism classes of) line bundles over $Y$.\nWe assume that $\\UT_Y$ is a partial minimal model of a cluster $\\cA$-variety, which we denote by $\\cA \\subset \\UT_Y$.\nFor example, we encounter this situation frequently in the study of homogeneous spaces, where moreover the ring of global functions on $\\UT_Y$ has a representation theoretic interpretation due to the Borel--Weil--Bott Theorem (Remark~\\ref{rmk:borel weil bott}).\nThis fact is commonly used when constructing Newton--Okounkov bodies in Lie theory, see e.g. \\cite{FFL15} and the references therein.\n\nLet $D_1, \\dots, D_s$ be the irreducible divisorial components of $D= \\UT_Y \\setminus \\cV$ and let $\\tf^{\\cA^{\\vee}}_{i}$ be the theta function on $\\cA^{\\vee}$ parametrized by the point in $\\Trop_{\\Z}(\\cA)$ associated to $D_i$.\nThe (theta) superpotential\\footnote{If $Y$ is Fano, then this should be considered as the superpotential used for mirror symmetry purposes.} associated to the inclusion $\\cA \\subset \\UT_Y$ is\n\\[\nW_{\\UT_Y} = \\sum_{i=1}^s \\tf^{\\cA^{\\vee}}_i.\n\\]\nThe associated superpotential cone is the subset $\\Xi_{\\UT_Y}$ of $\\Trop_{\\R}(\\cV^{\\vee})$ where the tropicalized superpotential takes non-negative values.\nGiven a choice of seed $ \\seed^\\vee$ for $\\cA^\\vee$, $\\Xi_{\\UT_Y}$ is identified with a polyhedral cone $\\Xi_{\\UT_Y, \\seed^\\vee}\\subset \\R^d$.\nAs discussed in \\cite{GHKK}, in many cases the integral points of $\\Xi_{\\UT_Y}$ parametrize the set of theta functions on $\\cA$ that extend to $\\UT_Y$. \nThis happens for example if $\\cA$ has \\emph{theta reciprocity} (see Definition \\ref{def:theta_reciprocity}), a condition that is conjectured to be true in situations more general than ours.\nEven stronger, in many of the examples arising in nature the integral points of $\\Xi_{\\UT_Y}$ parametrize a basis of $H^0(\\UT, \\mathcal{O}_{\\UT_Y})$.\nIn \\cite{GHKK}, Gross--Hacking--Keel--Kontsevich give criteria ensuring that this is satisfied.\nThese conditions hold true in many cases of interest in representation theory, as was proven in several papers including \\cite{BF, FO20, GKS_polyhedral,GKS_typeA, GKS_string, Mag20, SW18} and \\cite[\\S9]{GHKK}.\nMoreover, for special choices of seeds $\\seed^{\\vee} $, in these cases the cone $\\Xi_{\\UT_Y, \\seed^\\vee}$ agrees with known polyhedral cones such as the Gelfand--Tsetlin cone, string cones or the Knudson--Tao hive cone.\nMuch of the inspiration of this paper is due to the representation theoretic results that precede it.\nIn the case where the integral points of $\\Xi_{\\UT_Y}$ parametrize the set of theta functions on $\\cA$ that extend to $\\UT_Y$, we can restrict a {\\bf g}-vector valuation $\\gv_\\seed $ from $\\Bbbk(\\cA)$ to $H^0(\\UT_Y, \\mathcal{O}_{\\UT_Y})$. Therefore, given a line bundle $\\lb$ on $Y$ we can construct a Newton--Okounkov body $\\Delta_{\\gv_\\seed}(\\lb)$ in a similar way as before.\nIn order to show that $\\Delta_{\\gv_{\\seed}}(\\lb)$ is a positve polytope we need to consider torus actions on $ \\cA$ and fibrations of $\\cA^{\\vee}$ over a torus as we now explain. \n\nThe universal torsor $\\UT_Y$ is endowed with the action of the torus $T_{\\text{Pic}(Y)^*}$ associated to the dual of the Picard group of $Y$. \nWe first need this torus action to preserve $\\cA$ \nand that the induced action on $\\cA$ is cluster in the sense of \\thref{k_and_pic} (roughly speaking this means that the restricted action can be identified with the action induced by the choice of a sublattice of the kernel of $\\widetilde{B}$).\nIn such situations we have a cluster fibration \n\\[\nw:\\cA^{\\vee}\\to T_{\\text{Pic}(Y)}.\n\\]\nRecall that the choice of seed gives rise to the identification $\\mathfrak{r}_{\\seed^\\vee}:\\Trop_{\\R}(\\cA^{\\vee}) \\to \\R^d$. \nThe tropicalization of $w$ expressed using such an identification is a linear map $w^T:\\R^d \\to \\text{Pic}(Y)\\otimes \\R$. \nUnder the conditions above the Newton--Okounkov body $\\Delta_{\\gv_{\\seed}}(\\lb)$ can be described as a slicing of the superpotential cone. More precisely, we have the following result (see Definition \\ref{def:quotient_fibre}).\n\\begin{theorem*}\n(\\thref{thm:k_and_pic})\nAssume that the theta functions on $\\cV$ parametrized by the integral points of $\\Xi_{\\UT_Y}$ form a basis of $H^0(\\UT_Y, \\mathcal{O}_{\\UT_Y})$. \nIf the action of $ T_{\\text{Pic}(Y)^*}$ restricts to a cluster action of $ T_{\\text{Pic}(Y)^*}$ on $ \\cA$ then for any class $[\\lb]\\in \\text{Pic}(Y) $ the Newton--Okounkov body $\\Delta_{{\\bf g}_{\\seed}}(\\lb)$ can be describe as\n\\[\n\\Delta_{{\\bf g}_{\\seed}}(\\lb)=\\Trop_{\\R}(w)^{-1}([ \\lb ])\\cap \\Xi_{\\UT_Y, \\seed}.\n\\]\nIn particular, $\\Delta_{{\\bf g}_{\\seed}}(\\lb)$ is a positive subset of $\\Trop_{\\R}(\\cV^{\\vee})$ and $Y$ is a minimal model of the quotient of $\\cA$ by the action of $T_{\\text{Pic}(Y)^*}$.\n\\end{theorem*}\n\nThe case where $Y$ is the Grassmannian $\\text{Gr}_{n-k}(\\C^n)$ fits the framework above so it is possible to use the cluster $\\cA$ structure to construct Newton--Okounkov bodies associated to arbitrary line bundles over $\\text{Gr}_{n-k}(\\C^n)$. \nWe show that the Newton--Okounkov bodies we construct are unimodular to the Newton--Okounkov bodies constructed for $\\text{Gr}_{n-k}(\\C^n)$ by Rietsch and Williams in \\cite{RW} using the cluster $\\cX$ structure on Grassmannians (see Theorem~\\ref{thm: val and gv}).\nMoreover, the flow valuations of \\cite{RW} are instances of $\\bf c$-vector valuations.\n\nThis comparison result already has interesting consequences related to toric degenerations:\n\\begin{enumerate}\n \\item Given a rational polytopal Newton--Okounkov body $\\Delta$ for a (very ample) line bundle $\\lb$ over $Y$ Anderson's main result in \\cite{An13} applies and it yields a toric degeneration of $Y$ to a toric variety (whose normalization is) defined by $\\Delta$. As the semigroup algebras of the {\\bf g}-vector valuations are saturated, no normalization is necessary.\n \\item The construction of Gross--Hacking--Keel--Kontsevich in \\cite[\\S8]{GHKK} associates to a positive polytope $P$ a minimal model $\\cV\\subset Y$ and moreover, using Fomin--Zelevinsky's principal coefficients, a toric degneration of $Y$ to the toric variety defined by $P$.\n As our Newton--Okounkov bodies are positive polytopes, this construction applies in our setting.\n\\end{enumerate}\nThe identification of the Newton--Okounkov bodies constructed by Rietsch--Williams and our Newton--Okounkov bodies constructed from {\\bf g}-vectors implies the following result.\n\n\\begin{theorem*}(Theorem~\\ref{thm: val and gv} and Remark~\\ref{rmk:toric degen})\nThe toric degenerations of $\\text{Gr}_{n-k}(\\C^n)$ determined by the Newton--Okounkov polytopes constructed by Rietsch--Williams using Anderson's result coincide with the toric degenerations of $\\text{Gr}_{n-k}(\\C^n)$ given by Gross--Hacking--Keel--Kontsevich construction using principal coefficients.\n\\end{theorem*}\n\n\\subsection{The intrinsic Newton--Okounkov body} \nUnderstanding how Newton--Okounkov bodies change upon changing the valuation is an interesting problem that has attracted the attention of several authors, see for example \\cite{EH20, BMNC, FH21,CHM22,HN23}.\nSo let us return to the discussion on how the Newton--Okounkov bodies constructed above transform if we change the choice of seed.\nGiven any two seeds $\\seed $ and $\\seed'$ for $ \\cV^\\vee$ there is a piecewise linear bijection $\\Trop_{\\R}(\\mu^{\\cV^\\vee}_{\\seed,\\seed'}):\\R^d \\to \\R^d$ relating the identifications of $\\Trop_{\\R}(\\cV^{\\vee})$ with $\\R^d$.\nMore precisely, we have a commutative diagram\n\\[\n\\xymatrix{\n&\\Trop_{\\R}(\\cV^{\\vee})\n\\ar_{\\mathfrak{r}_{\\seed}}[dl] \\ar^{\\mathfrak{r}_{\\seed'}}[dr] & \\\\\n\\R^d \\ar^{\\Trop_{\\R}(\\mu^{\\cV^\\vee}_{\\seed,\\seed'})}[rr]& & \\R^d.\n}\n\\]\nEvery map $\\Trop_{\\R}(\\mu^{\\cV^\\vee}_{\\seed,\\seed'})$ restricts to a piecewise linear bijection of $\\Z^d$ and, \nby construction, the maps $\\Trop_{\\R}(\\mu^{\\cV^\\vee}_{\\seed,\\seed'})$ are composition of tropicalized cluster transformations for $\\cV^\\vee$ (see \\S\\ref{sec:intrinsic_NOB} for a more concise description).\nFor a subset $P\\subseteq \\Trop_{\\R}(\\cV^{\\vee})$ we let $P_{\\seed}=\\mathfrak{r}_{\\seed}(P)$.\nOne of the main properties behind our interest in showing that the Newton--Okounkov bodies we have constructed are positive sets is the following:\nif $P\\subseteq \\Trop_{\\R}(\\cV^{\\vee})$ is a positive set then $\\Trop_{\\R}(\\mu^{\\cV^\\vee}_{\\seed,\\seed'})(P_{\\seed})=P_{\\seed'}$ for any two seeds, $\\seed$ and $\\seed'$.\nIn particular, in this situation the entire collection of sets $\\{P_\\seed\\}_{\\seed}$ parametrized by the seed for $\\cV^{\\vee}$ may be replaced by $P$, a single intrinsic object that can be used to recover any $P_\\seed$ in the family.\n\nIn the case where a Newton--Okounkov body $\\Delta_{\\nu_{\\seed}}$ (associated to a line bundle $\\lb$ or a pair $(D',\\tau)$ as in the previous subsection) is positive, any other Newton--Okounkov body $\\Delta_{\\nu_{\\seed'}}$ associated to the same data is also positive. \nIn this situation there is a single intrinsic object $\\Delta_{\\mathrm{BL}} \\subset\\Trop_{\\R}(\\cV^{\\vee})$ representing the entire collection $\\{ \\Delta_{\\nu_{\\seed}}\\}_{\\seed}$. \nWe call $\\Delta_{\\mathrm{BL}}$ the \\emph{intrinsic Newton--Okounkov body} (associate to the data we begin with).\nThe subindex $\\mathrm{BL}$ in $\\Delta_{\\mathrm{BL}}$ stands for \\emph{broken line}, the choice of this notation goes back to \\cite{CMNcpt} where the last three authors of this paper introduce \\emph{broken line convexity}-- a notion of convexity defined in a tropical space that ensures positivity. \nBroken lines are pieces of tropical curves in $\\Trop_{\\R}(\\cV^{\\vee})$ used to define theta functions on $\\cV$ and describe their multiplication (see \\S\\ref{sec:tf_and_parametrizations}).\nStraight line segments defining convexity in a linear space are replaced by broken line segments in the tropical space to define broken line convexity.\nThe main result of \\cite{CMNcpt} is that a closed set is broken line convex if and only if it is positive.\n\nIn the situations where we are able to show that $\\Delta_{\\nu_{\\seed}}\\subset \\R^d$ is positive, it turns out that it is moreover polyhedral, a property that fails in general, see e.g. \\cite{LM09,KLM_NObodies_spherical}. \nSince $\\Delta_{\\nu_{\\seed'}}=\\Trop_{\\R}(\\mu^{\\cV^\\vee}_{\\seed,\\seed'})(\\Delta_{\\nu_{\\seed}})$ any other $\\Delta_{\\nu_{\\seed'}}$ is also polyhedral.\nThe integral points of the convex bodies we consider are naturally associated to theta functions,\nwhich suggests is the following question: does there exist a finite set of theta functions such that $\\Delta_{\\nu_{\\seed'}}$ is the convex hull of their images under $ \\nu_{\\seed'}$ for any seed $\\seed'$?\nSuch a collection of points might vary as we change seeds as exhibited in the case of the Grassmannians in an example in \\cite[\\S9]{RW} and generalized to an infinite family of examples in \\cite[Theorem~3]{bossinger2019full}.\nGiven the notion of broken line convexity, a slight reformulation of the question becomes more natural: \ndoes there exist a finite set of theta functions such that the broken line convex hull of their images under $\\nu_{\\seed'}$ is $\\Delta_{\\nu_{\\seed'}}$ for some (and hence any) seed $\\seed'$?\nIn fact, from the intrinsic Newton--Okounkov body perspective, the valuation is replaced by integral tropical points parametrizing theta functions and there is no reference to a seed at all.\nUsing this perspective, $\\Delta_{\\mathrm{BL}}$ becomes a broken line convex subset of $\\Trop_{\\R}(\\cV^{\\vee})$ whose integral points parametrize the theta basis of the first graded piece $R_1$ of the corresponding graded ring.\nIn \\thref{taut} we give sufficient conditions ensuring that $\\Delta_{\\mathrm{BL}}$ can be described as the broken line convex hull of a finite collection of points and describe this collection.\nApplying this result to the setting of Grassmannians we obtain that if $\\lb_e$ is line bundle over $\\Grass_{n-k}(\\C^n)$ obtained by pullback of $\\mathcal{O}(1)$ under the Pl\u00fccker embedding $\\Grass_{n-k}(\\C^n)\\hookrightarrow \\mathbb P^{\\binom{n}{k}-1}$ then the intrinsic Newton--Okounkov body $\\Delta_{\\mathrm{BL}}(\\lb_e)$ is the broken line convex hull of the ${\\bf g}$-vectors of the Pl\\\"ucker coordinates (Corollary~\\ref{cor:intrinsicNO grassmannian}).\n\nBroken line convexity also allows to generalize the Newton polytope of a Laurent polynomial to the the world of cluster varieties.\nIn particular, in \\S\\ref{sec:intrinsic_NOB} we introduce the \\emph{theta function analog of the Newton polytope} of $f$, for any $f\\in H^0(\\cV, \\mathcal{O}_\\cV)$.\nThe intrinsic Newton--Okounkov bodies $\\Delta_{\\mathrm{BL}}$ can be described using this notion.\nThe key idea is exploiting the bijection between the theta basis (a special case of an \\emph{adapted basis}) and integral tropical points parametrizing them.\nThis idea is explained for full rank valuations with finitely generated value semigroup in the survey \\cite{B-toric}.\nIt is therefore interesting to continue studying this new class of objects.\n\n\\subsection{Organization of the paper}\nIn \\S\\ref{sec:background} we review background material on cluster varieties their quotients and their fibres (\\S\\ref{sec:back_ghkk}), and on tropicalization (\\S\\ref{ss:tropicalization}).\nIn \\S\\ref{sec:tf_and_parametrizations} we recall the construction of cluster scattering diagrams and the theta functions on (quotients and fibres of) cluster varieties.\nIn \\S\\ref{sec:minimal_models} we elaborate on the existence of a theta basis on the ring of regular functions on a partial minimal model of (a quotient or a fibre of) a cluster variety. This section largely follows \\cite{GHKK}.\nIn \\S\\ref{sec:cluster_valuations} we recall the {\\bf g}-vector valuations for (quotients) $\\cA$-varieties. We introduce {\\bf c}-vector valuations for (fibres of) $\\cX$-varieties.\nThe main results of the paper are contained in \\S\\ref{sec:no}. The study of Newton--Okoukov bodies associated to Weil divisors on minimal models is treated in \\S\\ref{sec:NO_bodies} while the Newton--Okoukov bodies for line bundles are treated in \\S\\ref{sec:universal_torsors}.\n The intrinsic Newton--Okounkov body and the wall-crossing phenomenon for these are addressed in \\S\\ref{sec:intrinsic_NOB}.\n Finally, in \\S\\ref{sec:NO_Grass} we apply the results of the previous section to Grassmannians. \n One of the main technical conditions to be satisfied is verified in \\S\\ref{sec:Pic_property}.\n In \\S \\ref{sec:GHKK_and_RW} we prove a unimodular equivalence between the Newton--Okounkov bodies we construct and those constructed by Rietsch--Williams in \\cite{RW}. \n In \\S\\ref{sec:Grass_intrinsic} we describe the intrinsic Newton--Okounkov bodies for Grassmannians as the broken line convex hull\n of the {\\bf g}-vectors of Pl\\\"ucker coordinates (in arbitrary seeds).\n \n\n \n\\subsubsection*{Acknowledgements} \nThe authors L. Bossinger and A. N\u00e1jera Ch\u00e1vez were partially supported by PAPIIT project IA100122 dgapa UNAM 2022 and by CONACyT project CF-2023-G-106.\nM. Cheung was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.\nT. Magee was supported by EPSRC grant EP\/V002546\/1.\n\n\\section{Preliminaries}\\label{sec:background}\n\n\n\\subsection{Cluster varieties, quotients and fibres}\\label{sec:back_ghkk}\nWe briefly recall the construction of cluster varieties, their quotients and their fibres. The reader is invited to consult \\cite{GHK_birational,GHKK} for the details we shall omit in this section. \n\nUnless otherwise stated, all tensor products are taken with respect to $\\Z$. Moreover, given a lattice $L$ we denote by $L^*:= \\Hom(L,\\Z)$ its $\\Z$-dual and let $ \\langle \\cdot , \\cdot \\rangle: L\\times L^* \\to \\Z$ be the canonical pairing given by evaluation. \nWe further denote by $L_\\R:= L \\otimes \\R$ the real vector space associated to $L$. We fix an algebraically closed field $\\Bbbk$ of characteristic $0$ and let $T_L:= \\text{Spec}(\\Bbbk [L^*])$ be the algebraic torus whose character lattice is $L^*$. \n\n\\subsubsection{Cluster varieties and their dualities}\n\\label{sec:cluster_var}\n\n\nThe {\\bf fixed data} $\\Gamma$ consist of the following:\n\\begin{itemize}\n \\item a finite set $I$ of {\\bf directions} and a distinguished subset $\\Iuf \\subseteq I$ of {\\bf mutable} (or {\\bf unfrozen}) {\\bf directions}. Elements of $I \\setminus \\Iuf $ are the {\\bf frozen directions};\n \\item a lattice $N$ of rank $|I|$ together with a saturated sublattice $N_{\\text{uf}}\\subseteq N$ of rank $|I_{\\text{uf}}|$; \n \\item a skew-symmetric bilinear form $\\{ \\cdot , \\cdot \\} : N \\times N \\rightarrow \\Q$;\n \\item a finite index sublattice $N^\\circ \\subseteq N$ such that $\\{ N, \\Nuf \\cap N^{\\circ}\\}\\subset \\Z$ and $\\{ \\Nuf, N^{\\circ} \\}\\subset \\Z$;\n \\item a collection of positive integers $\\{d_i\\}_{i \\in I}$ with greatest common divisor $1$;\n \\item the dual lattices $M = \\Hom (N, \\Z)$ and $M^{\\circ}=\\Hom(N^{\\circ},\\Z)$.\n\\end{itemize}\nA ${\\bf seed}$ for $\\Gamma$ is a tuple $\\seed := ( e_i )_{i \\in I}$ such that $\\{ e_i \\}_{i\\in I}$ is a basis for $N$, $\\{e_i\\}_{i \\in \\Iuf}$ is a basis for $\\Nuf$ and $\\{d_i e_i \\}_{i \\in I } $ is a basis for $N^{\\circ}$. \nWe let $f_i := {d_i}^{-1} e_i^*$ and observe that $\\{f_i\\}_{i\\in I}$ is a basis of $M^{\\circ}$.\nFor $i,j\\in I$ we write $\\epsilon_{ij}:= \\lbrace e_{i},d_j e_{j} \\rbrace$ and define the matrix $\\epsilon=(\\epsilon_{ij})_{i,j\\in I}$. \nWhen we work with various seeds at the same time we introduce labels of the form $e_{i;\\seed}$, $f_{i;\\seed}$, $\\epsilon_{\\seed}=(\\epsilon_{ij;\\seed})$, etc. to distinguish the data associated to $\\seed $. We can {\\bf mutate} a seed $\\seed=(e_i)_{i\\in I}$ in a mutable direction $k\\in \\Iuf$ to obtain a new seed $\\mu_k(\\seed)=(e'_i)_{i\\in I}$ given by\n\\begin{equation}\n\\label{e_mutation}\ne_i':=\\begin{cases} e_i+[\\epsilon_{ik}]_+e_k & i\\neq k,\\\\\n-e_k&i=k,\n\\end{cases}\n\\end{equation}\nwhere $[x]_+:= \\text{max}(0,x)$ for $x \\in \\R$. \n\nLet $r:=|\\Iuf|$ and let $\\mathbb{T}_r$ denote the $r$-regular tree whose edges are labeled by the elements of $\\Iuf$.\nWe refer to $r$ as the {\\bf rank} and fix it one and for all.\nBy a common abuse of notation, the set of vertices of this tree is also denoted by $\\mathbb T_r$. \nWe fix once and for all a distinguished vertex $v_0\\in \\T_r$ and let $\\orT$ be the unique orientation of $ \\T_r$ such that the $r$ edges incident to $v_0$ are oriented in outgoing direction from $v_0$, and every vertex different from $v_0$ has one incoming edge and $r-1$ outgoing edges.\nWe write $v\\overset{k}{\\longrightarrow}v'\\in \\orT$ to indicate that the edge in between the vertices $v,v'$ of $\\orT$ is oriented from $v$ to $v'$ and is labeled by $k$.\n\nFix once and for all a seed $\\seed_0=(e_i\\mid i \\in I)$ and call it the {\\bf initial seed}.\nTo every vertex $v\\in \\T_r$ we attach a seed $\\seed_v$ as follows: \nwe let $\\seed_{v_0}=\\seed_0$, if $v\\overset{k}{\\longrightarrow}v'\\in \\orT$ then $\\seed_{v'}=\\mu_k(\\seed_{v})$.\nFor simplicity we write $\\seed\\in \\orT$ if $\\seed=\\seed_v$ for some $v\\in \\orT$.\n\nFor every seed $\\seed=(e_{i;\\seed}\\mid i\\in I)\\in \\orT$ we introduce the {\\bf seed tori} $\\cA_{\\seed} = T_{N^{\\circ}} $ and $ \\cX_{\\seed} = T_{M}$ which are endowed with the {\\bf cluster coordinates} $\\{A_{i;\\seed} := z^{f_{i;\\seed}}\\}_{i \\in I}$ and $\\{X_{i;\\seed} := z^{e_{i;\\seed}}\\}_{i \\in I}$, respectively. The {\\bf $\\cA$-cluster transformation} associated to $\\seed$ and $k \\in \\Iuf$ is the birational map\n$\n\\mu^{\\cA}_{k}:\\cA_{\\seed} \\dashrightarrow \\cA_{\\mu_k(\\seed)}\n$\nspecified by the pullback formula\n\\begin{equation}\n\\label{A_mut}\n(\\mu^{\\cA}_{k})^*(z^m):=z^{m} (1+z^{v_{k;\\seed}})^{-\\langle d_k e_{k;\\seed},m\\rangle} \\ \\ \\text{ for }m\\in M^{\\circ},\n\\end{equation}\nwhere $v_{k;\\seed}:=\\{e_{k;\\seed}, \\cdot \\}\\in M^{\\circ}$. Similarly, the {\\bf $\\cX$-cluster transformation} associated to ${\\vb s}$ and $k$ is the birational map\n$\n\\mu^{\\cX}_{k}:\\cX_{\\seed} \\dashrightarrow \\cX_{\\mu_k(\\seed)} \n$\nspecified by the pull-back formula\n\\begin{equation}\n\\label{X_mut}\n(\\mu^{\\cX}_{k})^*(z^n):=z^{n} (1+z^{e_{k;\\seed}})^{-[ n,e_{k;\\seed} ]}\\ \\ \\text{ for }n\\in N,\n\\end{equation}\nwhere $[\\cdot, \\cdot]:N\\times N \\to \\Q$ is the bilinear form determined by setting $[e_i,e_j]=\\lrc{e_i, d_je_j}$. \n\n\nFor seeds $\\seed, \\seed'\\in \\orT$ connected by iterated mutation in a sequence of directions $k_1, \\dots, k_s\\in \\Iuf$, we let $\\mu^{\\cA}_{\\seed, \\seed'} $ (resp. $\\mu^{\\cX}_{\\seed, \\seed'} $) be the composition of cluster transformations in the same sequence of directions and in the same order. \nA birational transformation of the form $\\mu^{\\cA}_{\\seed, \\seed'}$ (or $\\mu^{\\cX}_{\\seed, \\seed'}$) can be used to glue its domain and range by identifying the largest open subschemes where the transformation is an isomorphism. \nWe use this kind of gluing to define cluster varieties. \nMore precisely, the cluster $ \\cA$-variety associated to $\\Gamma$ and $\\seed_0$ is\n\\[\n\\cA_{\\Gamma,\\seed_0}:=\\bigcup\\limits_{\\seed\\in \\orT} \\cA_{\\seed}\/ \\left( \\text{gluing by } \\mu^{\\cA}_{\\seed', \\seed''} \\right)_{\\seed',\\seed''\\in\\orT}.\n\\]\nThe cluster $ \\cX$-variety associated to $\\Gamma$ and $\\seed_0$ is\n\\[\n\\cX_{\\Gamma,\\seed_0}:=\\bigcup\\limits_{\\seed\\in \\orT} \\cX_{\\seed}\/ \\left( \\text{gluing by } \\mu^{\\cX}_{\\seed', \\seed''} \\right)_{\\seed',\\seed''\\in\\orT}.\n\\]\n\nFrom now on an element $\\seed\\in \\orT$ will be referred to as a seed for $\\cA$ (or $\\cX$).\nIt is important to recall that declaring another $\\seed \\in \\orT$ as an initial seed gives rise to isomorphic cluster varieties.\nWe fix the pair $(\\Gamma,\\seed) $ once and for all and denote $\\cA_{\\Gamma, \\seed_0}$ (resp. $\\cX_{\\Gamma, \\seed_0}$) simply by $\\cA$ (resp. $\\cX$). \n\n\n\\subsubsection{Quotients of $\\cA$-varieties and fibres of $\\cX $-varieties}\\label{sec:quotients-fibres} \n\nLet $N^{\\perp}_{\\text{uf}}:= \\{ m\\in M \\mid \\langle n, m \\rangle=0 \\ \\forall \\ n\\in N_{\\text{uf}} \\} $.\nIn particular, $ M\/ N^{\\perp}_{\\uf}\\cong (N_{\\text{uf}})^*$. \nBy a slight abuse of notation we also write $ M^{\\circ}\/ \\Nuf^{\\perp}$. Here $\\Nuf^\\perp$ is taken in $M^\\circ$ rather than $M$, so $M^{\\circ}\/ \\Nuf^{\\perp}$ is torsion free.\nSince $\\{ N_{\\text{uf}},N \\}\\subseteq \\Z$ the following homomorphisms are well defined\n\\begin{align} \\label{eq:p12star}\n \\begin{matrix}\n p_1^*: & N_{\\uf} & \\rightarrow & M^\\circ &\\qquad \\phantom{aaaaa} \\qquad \\qquad & p_2^* : & N & \\rightarrow& M^{\\circ}\/ N^{\\perp}_{\\uf}. \\\\\n & n &\\mapsto & \\{ n, \\cdot \\}\n & \\qquad & & n &\\mapsto & \\{ n, \\cdot \\} + \\Nuf^{\\perp}\n \\end{matrix}\n\\end{align}\nThe matrix representing $p_2^*$ with respect to a seed $\\seed\\in \\orT$ is the \\emph{extended exchange matrix} $\\widetilde{B}_{\\seed}$ of \\cite{FZ_clustersIV}.\n\n\\begin{definition}\\thlabel{def:p-star}\nA {\\bf cluster ensemble lattice map} for $\\Gamma$ is a homomorphism $p^*: N \\to M^\\circ$ such that $p^*|_{N_{\\text{uf}}} = p^*_1$ and the composition $N \\overset{p^*}{\\longrightarrow} M^\\circ \\twoheadrightarrow M^{\\circ}\/ N^{\\perp}_{\\uf}$ agrees with $p_2^*$, where $ M^\\circ \\twoheadrightarrow M^{\\circ}\/ N^{\\perp}_{\\uf}$ denotes the canonical projection. Note that different choices of $p^*$ differ by a homomorphism $N\/ N_{\\uf} \\rightarrow N^{\\perp}_{\\uf} $.\n\\end{definition}\n\nIn other words, given a seed $\\vb s $, the $|I|\\times|I| $ square matrix $B_{p^*;\\vb{s}}$ associated to a cluster ensemble lattice map $p^*$ with respect to the bases $(e_i)_{i\\in I}$ and $(f_i)_{i\\in I}$ satisfies \n\\begin{equation}\\label{eq:Mp*}\nB_{p^*;\\vb{s}} - \\epsilon^{\\rm{tr}}_\\seed=\n\\lrb{\\begin{matrix}\n0 & 0 \\\\\n0 & \\ast\n\\end{matrix}},\n\\end{equation}\nwhere the $0$ entries represent the blocks $\\Iuf\\times\\Iuf$, $\\Iuf\\times (I\\setminus \\Iuf)$, and $(I\\setminus \\Iuf)\\times \\Iuf$,\nand the $\\ast$ entry indicates that the $(I\\setminus \\Iuf)\\times(I\\setminus \\Iuf)$ block has no constraints.\nEvery cluster ensemble lattice map $p^*:N\\to M^{\\circ}$ commutes with mutation. Therefore, $p^*$ gives rise to a {\\bf cluster ensemble map}\n\\[\np:\\cA \\to \\cX.\n\\]\n\nThe map $p:\\cA \\to \\cX$ yields both, torus actions on $\\cA$ and fibrations of $\\cX$ over a torus, as we explain subsequently.\nLet\n\\begin{equation}\\label{eq:define K}\nK=\\ker(p_2^*)=\\lrc{k\\in N\\mid \\{k,n\\}=0\\,\\forall\\, n\\in N_{\\rm uf}^\\circ} \\quad \\text{and} \\quad K^{\\circ}=K \\cap N^{\\circ}. \n\\end{equation}\nTo obtain an action on $\\cA$ we consider a saturated sublattice \n\\[H_{\\cA} \\subseteq K^\\circ.\n\\]\nThe inclusion $ H_{\\cA} \\hookrightarrow N^\\circ$ gives rise to an inclusion $T_{H_{\\cA}}\\hookrightarrow T_{N^{\\circ}}$\nas a subgroup.\nSince $p^*$ commutes with mutation and $H_{\\cA}\\subseteq K $ we have a non-canonical inclusion \n\\[\nT_{H_{\\cA}}\\hookrightarrow \\cA.\n\\]\nThe action of $ T_{H_{\\cA}} $ on $T_{N^\\circ}$ given by multiplication extends to a free action of $T_{H_{\\cA}} $ on $\\cA$ and gives rise to \na geometric quotient $\\cA \\to \\cA\/T_{H_{\\cA}}$.\nThe scheme $\\cA\/T_{H_{\\cA}}$ is obtained by gluing tori of the form $T_{N^{\\circ}\/H_{\\cA}}\\cong T_{N^{\\circ}}\/T_{H_{\\cA}}$; the gluing is induced by the $\\cA$-mutations used to glue the seed tori for $\\cA$.\nMore precisely, for every seed $\\seed$ for $\\cA$ we let $(\\cA\/T_{H_{\\cA}})_{\\seed}$ be a copy of the torus $ T_{N^{\\circ}\/H_{\\cA}} $. \nFor $k\\in \\Iuf$ the mutation $\\mu^{\\cA\/T_{H_\\cA}}_{k}: (\\cA\/T_{H_{\\cA}})_{\\seed} \\dashrightarrow (\\cA\/T_{H_{\\cA}})_{\\mu_k(\\seed)}$ is given by\n\\begin{equation}\n\\label{A\/T_mut}\n\\lrp{\\mu^{\\cA\/T_{H_{\\cA}}}_{k}}^*(z^m):=z^{m} (1+z^{v_{k;\\seed}})^{-\\langle d_k e_{k;\\seed},m\\rangle} \\ \\ \\text{ for }m\\in H_{\\cA}^{\\perp}.\n\\end{equation}\nLet $\\mu^{\\cA\/T_{H_{\\cA}}}_{\\seed, \\seed'}$ denote the composition of mutations determined by the path in $\\orT$ connecting $\\seed, \\seed'\\in \\orT$. Then \n\\[\n\\cA\/T_{H_{\\cA}}:=\\bigcup\\limits_{\\seed\\in \\orT} (\\cA\/T_{H_{\\cA}})_{\\seed}\/ \\left( \\text{gluing by } \\mu^{\\cA\/T_{H_{\\cA}}}_{\\seed', \\seed''} \\right)_{\\seed', \\seed'' \\in \\orT}.\n\\]\n\nTo obtain the fibration of $\\cX$ over a torus we consider a saturated sublattice \n\\[\nH_{\\cX} \\subseteq K.\n\\]\nThe inclusion $H_{\\cX} \\hookrightarrow N$ induces a surjection $T_M:= \\Spec(\\Bbbk[N]) \\to \\Spec(\\Bbbk[H_{\\cX}])=:T_{H_{\\cX}^*}$. This extends to a globally defined map \n\\begin{equation}\n\\label{eq:weight_map}\n w_{H_{\\cX}}:\\cX\\to T_{H^*_\\cX}.\n\\end{equation}\n\n\n\\begin{remark}\nThe subindex $\\cV $ in the lattice $H_{\\cV}$ stands for the cluster variety $\\cV$ for which the choice of sublattice is relevant. \nWhen there is no risk of confusion, we drop the subindex $\\cV$ from $H_{\\cV}$ (see the end of \\S\\ref{sec:FG_dual}).\n\\end{remark}\n\nWe let $\\cX_{\\phi}$ be the fibre of the map \\eqref{eq:weight_map} over a closed point $\\phi\\in T_{H^*_{\\cX}}$.\nIn this work we mainly focus on the fibre $\\cXeH$, where ${\\bf 1}_{T_{H^*_{\\cX}}}\\in T_{H^*_{\\cX}}$ is the identity element. When there is no risk of confusion on the fibration we are considering we will denote this scheme simply by $\\cXe$.\n\nThe fibre $\\cXe$ is obtained by gluing tori isomorphic to $T_{H^\\perp_{\\cX}}$ via the restrictions of the $\\cX$-mutations used to glue the seed tori for $\\cX$ (see \\cite[\\S4]{GHK_birational} for a detailed treatment of this construction).\nAs in the previous situations, we have a description of the form \n\\[\n\\cXe:=\\bigcup\\limits_{\\seed\\in \\orT} (\\cXe)_{\\seed}\/ \\left( \\text{gluing by } \\mu^{\\cXe}_{\\seed', \\seed''} \\right)_{\\seed', \\seed'' \\in \\orT},\n\\]\nwhere $(\\cXe)_{\\seed}$ is a torus isomorphic to $T_{H_{\\cX}^\\perp}$, $\\mu^{\\cXe}_{k}: (\\cXe)_{\\seed} \\dashrightarrow (\\cXe)_{\\mu_k(\\seed)}$ is given by \n\\begin{equation}\n\\label{X_phi_mut}\n\\lrp{\\mu^{\\cXe}_{k}}^*(z^{n+H_{\\cX}}):=z^{n+H_{\\cX}}(1+z^{e_{k;\\seed}+H_{\\cX}})^{-[ n,e_{k;\\seed} ]}\\ \\ \\text{ for } n+H_{\\cX} \\in N\/H_{\\cX}\n\\end{equation}\nand $\\mu^{\\cXe}_{\\seed,\\seed'}$ is defined as for the other varieties we have introduced so far.\n\n\\begin{definition}\n\\label{def:quotient_fibre}\nA variety of the form $\\cA\/T_{H_{\\cA}} $ is referred to as a {\\bf quotient of $\\cA$}. A variety of the form $\\cXe$ is referred to as a {\\bf fibre of $\\cX$}. \nA {\\bf cluster action} on $\\cA$ is the action of a torus of the form $T_{H_{\\cA}}$.\n\\end{definition}\n\nLet $T$ be an algebraic torus endowed with a set of coordinates $z_1, \\dots , z_r$ and let $\\omega_T$ be its canonical bundle. \nA {\\bf volume form} on $T$ is a nowhere vanishing form in $H^0(T, \\omega_T) $.\nThe {\\bf standard volume form} on $T$ is (any non-zero scalar multiple of) \n\\[\n\\Omega_T= \\frac{dz_1 \\wedge \\dots \\wedge dz_r}{z_1 \\cdots z_r}.\n\\]\n\n\\begin{definition}\nA {\\bf log Calabi\u2013Yau pair} $(Y, D)$ is a smooth complex projective variety $Y$ together with a reduced normal crossing divisor $D\\subset Y$ such that $K_X+D=0$. \nWe say a scheme $V$ is log Calabi\u2013Yau if there exists a log Calabi\u2013Yau pair $(Y,D)$ such that $V$ is $Y \\setminus D$ up to codimension 2.\n\\end{definition}\n\nIt follows from \\cite{Iitaka} that any log Calabi--Yau variety $V$ is endowed with a unique up to scaling holomorphic volume form (\\ie a nowhere vanishing holomorphic top form) $\\Omega_V$ which has at worst a simple pole along each component of $D$ for any such $(Y,D)$. See \\cite{GHK_birational} for further details.\n\n\nAs explained in \\cite[\\S1]{GHK_birational} both $ \\cA$ and $\\cX$ are log Calabi--Yau, the key point being that these schemes are obtained by gluing tori via birational maps that preserve the standard volume form on each seed torus (endowed with cluster coordinates). \nFor the same reason, the schemes of the form $\\cA\/T_{H_{\\cA}}$ and $\\cX_{\\phi}$ are also log Calabi--Yau.\nThe canonical volume form on $\\cA\/T_{H_{\\cA}}$ (resp. $\\cX_{\\phi}$) is induced by (resp. the restriction of) the canonical volume form of $\\cA$ (resp. $\\cX$).\n\n\n\n\\subsubsection{Principal coefficients, $\\cX$ as a quotient of $\\cAp $ and $\\cA$ as a fibre of $\\cXp$}\n\\label{sec:principal_coefficients}\nFor the fixed data $\\Gamma=\\lrp{I, \\Iuf, N,N^{\\circ}, M, M^{\\circ}, \\{ \\cdot, \\cdot \\}, \\{d_i\\}_{i\\in I} }$, we consider its principal counterpart \n\\[\n\\Gamma_{\\prin}=\\lrp{I_{\\prin}, (I_{\\prin})_{\\text{uf}}, N_{\\prin}, N_{\\prin}^{\\circ}, M_{\\prin}, M^{\\circ}_{\\prin}, \\{ \\cdot, \\cdot \\}_{\\prin}, \\{d_i\\}_{i\\in I_{\\prin}} },\n\\]\nwhere the index set $I_\\prin$ is the disjoint union of two copies of $I$, its subset $(I_\\prin)_{\\text{uf}}$ is the set $\\Iuf$ thought of as a subset of the first copy of $I$, \n\\[\n N_{\\prin} = N \\oplus M^\\circ, \\quad N_{\\prin}^{\\circ}= N^{\\circ}\\oplus M, \\quad (N_{\\prin})_{\\text{uf}}=\\Nuf \\oplus 0, \\quad M_{\\prin} = M \\oplus N^\\circ, \\quad M_{\\prin}^{\\circ}=M^{\\circ}\\oplus N.\n\\]\nFor $i \\in I_{\\prin}$ belonging to either the first or second copy of $I$, the corresponding integer in the tuple $\\{d_i \\mid i\\in I_{\\prin}\\}$ is equal to integer indexed by $i$ for $\\Gamma$, and \n\\[\n\\{(n_1,m_1),(n_2,m_2)\\}_{\\prin}= \\{n_1, n_2\\} + \\langle n_1,m_2 \\rangle - \\langle n_2,m_1 \\rangle.\n\\]\nRecall that $\\seed_0=(e_i)_{i \\in I} $ is the initial seed for $\\Gamma$. Then the initial seed for $\\Gamma_{\\prin}$ is ${\\seed_0}_{\\prin}=\\lrp{(e_i,0),(0,f_i)}_{i\\in I}$.\nSince $\\Gamma$ and $\\seed_0$ were already fixed, we denote the cluster variety $\\cA_{\\Gamma_{\\prin},{{\\seed{_0}}_{\\prin}}}$ (resp. $\\cX_{\\Gamma_{\\prin},{{\\seed{_0}}_{\\prin}}}$) simply by $\\cAp$ (resp. $\\cX_{\\prin}$). It is moreover worth pointing out that $\\cAp$ is in fact independent of the choice of initial seed $\\seed_0$ as explained in \\cite[Remark B.8]{GHKK}.\n\nIn \\cite{GHK_birational} the authors show that the scheme $\\cX$ can be described as a quotient of $\\cAp$ in the sense of Definition \\ref{def:quotient_fibre}. \nTo obtain such a description we need to choose a cluster ensemble lattice map $p^*:N \\to M^{\\circ}$ for $\\Gamma$. \nThis choice determines the cluster ensemble map \n\\begin{equation}\n\\label{eq:def_p_prin}\np_{\\prin}: \\cAp \\to \\cXp.\n\\end{equation}\nThe map $p_{\\prin}$ is induced by the cluster ensemble lattice map\n\\begin{align*}\np_{\\prin}^*:N_{\\prin} &\\to \\Mpc\\\\\n(n,m) &\\mapsto \\lrp{p^*(n)-m,n}\n\\end{align*}\nfor $\\Gamma_\\prin$.\nSet $K_{\\prin}:=\\ker(p_{\\prin,2}^*)$ and $ K_{\\prin}^\\circ:= K_{\\prin}\\cap N^\\circ_\\prin$, where $p_{\\prin,2}^*$ corresponds to the map $p_2^*$ in \\eqref{eq:p12star} for $\\Gamma_{\\prin}$.\nWe let\n\\begin{equation}\n\\label{eq:H_Aprin} \n H_{\\cAp}:= \\lrc{\\left.\\lrp{n,-(p^*)^*(n)}\\in N^\\circ_\\prin \\, \\right| \\, n \\in N^\\circ}. \n\\end{equation}\n\nIt is straightforward to verify that $H_{\\cAp}$ is a saturated sublattice of $K^\\circ_\\prin$ that is isomorphic to $N^\\circ$.\nIn particular, we have a quotient $\\cAp\/ T_{H_{\\cAp}}$ endowed with an atlas of seed tori isomorphic to $T_M $ (indeed, $T_{N^\\circ_{\\prin}}\/T_{H_{\\cAp}}\\cong T_{N^\\circ \\oplus M}\/T_{N^\\circ}\\cong T_M$). \nThere is an isomorphism \n\\begin{equation}\n \\label{eq:def_chi}\n \\chi : \\cAp\/T_{H_{\\cAp}}\\overset{\\sim}{\\longrightarrow} \\cX\n\\end{equation}\nrespecting the cluster tori of domain and range.\nThe restriction of $\\chi $ to a seed torus is a monomial map whose pullback is given by\n\\eqn{\n\\chi^*: N &\\to (H_{\\cAp})^\\perp \n\\\\\nn &\\mapsto (p^*(n),n).\n}\nThere is also a surjective map \n\\begin{equation}\n\\label{eq:def_tilde_p}\n \\tilde{p}:\\cAp \\to \\cX.\n\\end{equation}\nrespecting seed tori.\nThe restriction of $\\tilde{p} $ to a seed torus is a monomial map whose pullback is given by\n\\begin{align*}\n \\tilde{p}^*: N &\\to \\Mpc\\\\\n \\ \\ n &\\mapsto (p^*(n),n).\n\\end{align*}\nIn particular, we have $ \\tilde{p}= \\chi\\circ \\varpi$, where \n\\begin{equation}\n \\label{eq:def_varpi}\n\\varpi: \\cAp \\to \\cAp\/T_{H_{\\cAp}} \n\\end{equation}\nis the canonical projection. \n\nIt is also possible to describe $\\cA$ as a fibre of $\\cXp$. There is an injective map\n\\begin{equation}\n\\label{eq:def_xi}\n\\xi:\\cA \\to \\cXp\n\\end{equation}\nrespecting seed tori.\nThe restriction of $\\xi $ to a seed torus is a monomial map whose pullback is given by\n\\begin{align*}\n\\xi^*: N_{\\prin} &\\to M^\\circ\n\\\\\n(n,m) &\\mapsto p^*(n)-m.\n\\end{align*}\nLet\n\\begin{equation}\n \\label{eq:H_Xprin} \nH_{\\cXp}:= \\lrc{\\lrp{n,p^*(n)}\\in N_\\prin \\mid n \\in N}.\n\\end{equation}\nIt is routine to check that $H_{\\cXp}$ is a valid choice to construct a fibration of $\\cXp $ over the torus $T_{H^*_{\\cXp}}$. \nHence, we can consider the fibre $(\\cXp)_{\\bf 1}=(\\cXp)_{{\\bf 1}_{T_{H^*_{ \\cXp}}}} $ associated to this fibration. \nThere is an isomorphism \n\\begin{equation}\n\\label{eq:def_delta}\n\\delta:\n\\cA \\overset{\\sim}{\\longrightarrow} (\\cXp)_{\\bf 1} \n\\end{equation}\nrespecting seed tori.\nThe restriction of $\\delta $ to a seed torus is a monomial map whose pullback is given by\n\\begin{align*}\n\\delta^*: N_{\\prin} \/H_{\\cXp} &\\to M^\\circ\n\\\\\n(n,m) + H_{\\cXp} &\\mapsto p^*(n)-m.\n\\end{align*}\nIn particular, we have that\n\\[\n\\xi=\\iota \\circ \\delta,\n\\]\nwhere $\\iota: (\\cXp)_{\\bf 1}\\hookrightarrow \\cXp$ is the canonical inclusion.\nFor later reference we also introduce the map\n\\begin{equation}\n\\label{eq:def_rho}\n\\rho: \\cXp \\to \\cX.\n\\end{equation}\nrespecting seed tori.\nThe restriction of $\\rho $ to a seed torus is a monomial map whose pullback is given by\n\\begin{align*}\n \\rho^*: N &\\to \\Np \\\\\n \\ \\ n & \\mapsto (n,p^*(n)).\n\\end{align*}\nIn particular, $\\rho \\circ p_{\\prin}= \\tilde{p} $.\nThe maps we have considered so far fit into the following commutative diagram \\begin{equation*}\n\\xymatrix{\n(\\cXp)_{\\bf 1} \\ar@{^{(}->}^{\\ \\iota}[r] & \\cXp \\ar_{\\rho}[d] & \\cAp \\ar_{p_{\\prin}}[l] \\ar@{->>}^{\\varpi}[d] \\ar_{\\tilde{p}}[dl] \\\\\n\\cA \\ar^{\\delta}_{\\cong}[u] \\ar_{p}[r] \\ar^{\\xi}[ru] & \\cX & \\cAp\/T_{H_{\\cAp}.} \\ar_{\\cong \\ \\ }^{\\chi \\ \\ }[l]\n}\n\\end{equation*}\n\n\\begin{remark}\n \\label{rem:labels}\nThe maps introduced in this section are associated with $\\Gamma$, hence, we label the maps with the subindex $\\Gamma$ to stress the fixed data $\\Gamma$ they are associated with.\n\\end{remark}\n\n\\subsection{Tropicalization}\n\\label{ss:tropicalization}\nIn this section we discuss tropicalizations of cluster varieties. We mainly follow \\cite[\\S1]{GHK_birational}, \\cite[\\S2]{GHKK} and \\cite[\\S1.1]{FG_cluster_ensembles}. \n\nLet $T_L$ be the torus associated to a lattice $L$. A rational function $f$ on $T_L$ is called positive if it can be written as a fraction $f=f_1\/f_2$, where both $f_1$ and $f_2$ are a linear combination of characters of $T_L$ with coefficients in $\\Z_{>0} $.\nThe collection of positive rational functions on $T_L$ forms a semifield inside $ \\Bbbk(T_L)$ denoted by $Q_{\\rm sf}(L)$.\nA rational map $f:T_L\\dashrightarrow T_{L'}$ between two tori is a {\\bf positive rational map} if the pullback $f^*:\\Bbbk(T') \\to \\Bbbk(T)$ restricts to an isomorphism $f^*:Q_{\\rm sf}(L') \\to Q_{\\rm sf}(L)$.\nIf $P$ is a semifield, then the $P$ valued points of $T_L$ form the set\n\\begin{equation}\n\\label{eq:FG_tropicalization}\nT_L(P):=\\Hom_{\\rm sf}(Q_{\\rm sf} (L), P)\n\\end{equation}\nof semifield homomorphisms from $Q_{\\rm sf} (L)$ to $ P$.\nIn particular, a positive birational isomorphism $\\mu:T\\dashrightarrow T'$ induces a bijection\n\\begin{align*}\n\\mu_*: T(P) & \\to T'(P)\\\\\n h \\ & \\mapsto \\ h \\circ f^*. \n\\end{align*}\nBy a slight but common abuse of notation the sublattice of monomials of $Q_{\\rm sf}(L)$ is denoted by $L^*$. \nConsidering $P$ just as an abelian group the restriction of an element of $Q_{\\rm sf}(L)$ to $L^*$\ndetermines a canonical bijection $T_L(P) \\overset{\\sim}{\\longrightarrow} \\Hom_{\\rm groups} (L^*, P) $. \n\n\\begin{remark}\n\\label{rem:identification}\nWe systematically identify $T_L(P)$ with $L\\otimes P$ by composing the canonical bijection $T_L(P) \\overset{\\sim}{\\longrightarrow} \\Hom_{\\rm groups} (L^*, P) $ with the canonical isomorphism $\\Hom_{\\rm groups}(L^*, P) \\cong L \\otimes P$.\n\\end{remark}\n\n\nLet $ \\cV$ be a (quotient or a fibre of a) cluster variety.\nFor every $\\seed, \\seed'\\in \\orT$ the gluing map $\\mu^{\\cV}_{\\seed , \\seed'}: \\cV_\\seed\\dashrightarrow \\cV_\\seed'$ is a positive rational map.\nSo we can glue $\\cV_{\\seed}(P) $ and $ \\cV_{\\seed'} (P)$ using $(\\mu^{\\cV}_{\\seed, \\seed'})_*$ and define \n\\[\n\\cV(P):= \\coprod_{\\seed \\in \\orT} \\cV_{\\seed}(P) \/ \\left(\\text{gluing by } (\\mu^{\\cV}_{\\seed, \\seed'})_*\\right)_{\\seed, \\seed'\\in \\orT}.\n\\]\nEvery point ${\\bf a}\\in \\cV(P)$ can be represented as a tuple $(a_{\\seed})_{\\seed\\in \\orT}$ such that $(\\mu^{\\cV}_{\\seed, \\seed'})_*(a_\\seed)=(a_{\\seed'}) $ for all $\\seed,\\seed'\\in \\orT$.\nSince all of the maps $(\\mu^\\cV_{\\seed,\\seed'})_*$ are bijections, the assignment \n\\eq{\n \\mathfrak{r}_{\\seed}:\\cV(P)&\\to \\cV_{\\seed}(P)\\quad \\text{given by} \\quad {\\bf a}=(a_{\\seed})_{\\seed \\in \\orT} \\mapsto a_{\\seed}.\n}{not:tropical_space}\ndetermines an identification of $\\cV(P) $ with $ \\cV_{\\seed}(P)$. \nIf $S\\subset \\cV(P)$ we let \n\\begin{equation}\n\\label{eq:identification}\nS_{\\seed}(P):=\\mathfrak{r}_{\\seed} (S) \\subset \\cV_\\seed(P)\n\\end{equation}\nand write $S_{\\seed}$ instead of $S_{\\seed}(P)$ when the semifield $P$ is clear from the context.\n\nThe semifields we consider in this note are the integers, the rationals and the real numbers with their additive structure together with the semifield operation determined by taking the maximum (respectively, minimum).\nWe denote these semifields by $\\Z^T$, $\\Q^T$ and $\\R^T$ (respectively, $\\Z^t$, $\\Q^t$ and $\\R^t$).\nThe canonical inclusions $\\Z \\hookrightarrow \\Q \\hookrightarrow \\R$ give rise to canonical inclusions\n\\[\n\\cV(\\Z^T) \\hookrightarrow \\cV(\\Q^T) \\hookrightarrow \\cV(\\R^T) \\quad \\quad \\text{ and } \\quad \\quad \\cV(\\Z^t) \\hookrightarrow \\cV(\\Q^t) \\hookrightarrow \\cV(\\R^t).\n\\]\nFor a set $S\\subseteq \\cV(\\R^T)$ (resp. $S\\subseteq \\cV(\\R^t)$) we let $\nS(\\Z):= S\\cap \\cV(\\Z^T)$ (resp. $S(\\Z):= S\\cap \\cV(\\Z^t)$). Moreover, for $G=\\Z, \\Q$ or $\\R$, there is an isomorphism of semifields $G^T\\to G^t$ given by $x \\mapsto -x$ induces a canonical bijection\n\\begin{align} \\label{eq:imap}\n i: \\cV(G^T) \\rightarrow \\cV(G^t). \n\\end{align}\nSince $i$ amounts to a sign change (see Remark \\ref{rem:i_map} below), we think of $i$ as an involution and denote its inverse again by $i$.\n\n\n\n\\begin{remark}\\label{rmk:geometric trop}\nThe set $\\cV(\\Z^t)$ can be identified with the {\\bf geometric tropicalization} of $\\cV$, \ndefined as\n\\begin{equation*}\n \\cV^{\\trop}(\\Z) \n \\coloneqq \\{ \\text{divisorial discrete valuations } \\nu: \\Bbbk(\\cV) \\setminus \\{ 0\\} \\rightarrow \\mathbb Z \\mid \\nu (\\Omega_{\\cV}) <0 \\} \\cup \\{ 0\\}, \n\\end{equation*}\nwhere a discrete valuation is divisorial if it is given by the order of vanishing of a $\\Z_{>0}$-multiple of a prime divisor on some variety birational to $\\cV$. \n\\end{remark}\n\n\n\\begin{remark}\n\\label{rem:i_map}\nLet $G=\\Z, \\Q$ or $\\R$. Identifying $\\cV(G^T)$ with $\\cV_{\\seed}(G^T)$ via the bijection $\\mathfrak{r}_\\seed$ the map $i$ in \\eqref{eq:imap} can be thought of as the multiplication by $-1$ (\\cf Remark \\ref{rem:identification}).\n\\end{remark}\n\nA positive rational function $g$ on $\\cV $ is a rational function on $\\cV$ such that the restriction of $g$ to every seed torus $\\cV_{\\seed}$ is a positive rational function.\n\n\\begin{definition}\n\\label{def:trop_functions}\nThe {\\bf tropicalization} of a positive rational function $g: \\cV \\dashrightarrow \\Bbbk$ with respect to $\\R^T $ is the function $g^T:\\cV(\\R^T)\\to \\R$ given by \n\\begin{equation}\n\\label{eq:restriction}\n{\\bf a}\\mapsto a_{\\seed}(g),\n\\end{equation}\nwhere $ {\\bf a}=(a_{\\seed})_{\\seed \\in \\orT}$. The tropicalization of $g$ with respect to $\\R^t$ is the function $g^t:\\cV(\\R^t)\\to \\R$ defined as\n\\[\n{\\bf v} \\mapsto -v_{\\seed}(g),\n\\]\n\\end{definition}\nwhere $ {\\bf v}=(v_{\\seed})_{\\seed \\in \\orT}$. A direct computation shows that both $g^T$ and $g^t$ are well defined. Namely, one checks that for $\\seed,\\seed'\\in \\orT$\n\\[\na_{\\seed} (g)=a_{\\seed'}(g),\n\\]\nwhere in the left (resp. right) side of the equality we think of $g$ as a rational function on $\\cV_\\seed$ (resp. $\\cV_{\\seed'}$).\nMoreover, we have that\n\\begin{equation}\n\\label{eq:comparing_tropicalizations}\n g^T({\\bf a})=g^t(i({\\bf a})),\n\\end{equation}\nfor all ${\\bf a} \\in \\cV(\\R^T)$.\n\\begin{remark}\nIn order to keep notation lighter we adopt the following conventions: \n\\begin{itemize}\n\\item given a positive rational function $g\\in \\Bbbk (\\cV)=\\Bbbk(\\cV_\\seed)$ the tropicalizations of $g$ with domains $\\cV(\\R^T)$ and $\\cV_{\\seed}(\\R^T)$ are denoted by the same symbol $g^T$ for all $\\seed\\in \\orT$;\n\n\\item the restriction of $g^T$ (resp. $g^t$) to $\\cV(\\Z^T)$ (resp. $\\cV(\\Z^t)$) is also denoted by $g^T$ (resp. $g^t$);\n\\item when $P$ is one of $\\Z^T, \\Q^T$ or $\\R^T$ (resp. $\\Z^t, \\Q^t$ or $\\R^t$) the map $(\\mu^{\\cV}_{\\seed, \\seed'})_*$ is denoted by $(\\mu^{\\cV}_{\\seed, \\seed'})^T$ (resp. $(\\mu^{\\cV}_{\\seed, \\seed'})^t$).\n\\end{itemize}\n\\end{remark}\n\n\\begin{remark}\n Later we will need to systematically consider $\\cV(\\R^t)$ when $\\cV$ is a variety of the form $\\cA$ or $\\cA\/T_H$ and $ \\cV(\\R^T)$ when $\\cV$ is a variety of the form $\\cX$ or $\\cXe$.\n In particular, from \\S\\ref{sec:FG_conj} on we use the notation $\\Trop_{\\R}(\\cV)$ that takes into account the different kinds of tropicalizations that we use for different kinds of varieties, see equation \\eqref{eq:unif}. \n\\end{remark}\n\n\nFor latter use we record the following formulae associated to the mutations determined by $\\Gamma$:\n\\begin{equation}\n \\label{eq:tropical_A_mutation}\n \\lrp{\\mu^{\\cA}_{k}}^T(n)=n+[\\langle v_k,n\\rangle]_+(-d_ke_k)\n\\end{equation}\nand\n\\begin{equation}\n \\label{eq:tropical_X_mutation}\n \\lrp{\\mu^{\\cX}_{k}}^T(m)=m+[\\langle d_ke_k,m \\rangle]_+v_k.\n\\end{equation}\nIn case we tropicalize these mutations with respect to $\\R^t$ we replace $[\\ \\cdot\\ ]_+$ by $[\\ \\cdot\\ ]_-$.\n\nFinally, if we think of $T_L (\\R^T)$ (resp. $T_L(\\R^t)$) as a vector space (see Remark \\ref{rem:identification}), the tropicalization of a positive Laurent polynomial $g= \\sum_{\\ell\\in L^*}c_{\\ell} z^{\\ell} \\in Q_{\\rm sf}(L)$ with respect to $\\R^T$ (resp. $\\R^t$) is the function $g^T: T_L(\\R^T) \\to \\R$ (resp. $g^t: T_L(\\R^t) \\to \\R$) given by \n\\begin{eqnarray*}\nx &\\mapsto& - \\max \\{ \\langle \\ell , x \\rangle \\mid \\ell\\in L^* \\text{ such that } c_{\\ell} \\neq 0 \\}\\\\\n (\\text{resp. } x &\\mapsto& \\min \\{ \\langle \\ell , x \\rangle \\mid \\ell\\in L^* \\text{ such that } c_\\ell \\neq 0 \\}).\n\\end{eqnarray*}\n\n\\section{Theta functions and their labeling by tropical points}\n\\label{sec:tf_and_parametrizations}\n\n\n\\subsection{Fock--Goncharov duality}\\label{sec:FG_dual}\nFor $\\Gamma=(I, \\Iuf, N,N^{\\circ}, M, M^{\\circ}, \\{ \\cdot, \\cdot \\}, \\{d_i\\}_{i\\in I} )$ the Langlands dual fixed data\nis $\\Gamma^\\vee=(I, \\Iuf, N^\\vee, (N^\\vee)^{\\circ}, M^\\vee, (M^\\vee)^{\\circ}, \\{ \\cdot, \\cdot \\}^\\vee, \\{d^\\vee_i\\}_{i\\in I} )$, where $d:=\\text{lcm}(d_i)_{i\\in I}$,\n\\[\n N^\\vee = N^\\circ, \\quad (N^\\vee)^{\\circ}= d\\cdot N, \\quad M^\\vee = M^\\circ, \\quad (M^\\vee)^{\\circ}=d^{-1}\\cdot M, \\quad \\{\\cdot, \\cdot \\}^\\vee= d^{-1}\\{\\cdot, \\cdot \\} \\quad \\text{and} \\quad d^\\vee_i:=d\\,d_i^{-1}.\n\\]\nIf $ \\seed=(e_i)_{ i\\in I}$ is a seed for $\\Gamma$ then the Langlands dual seed is $\\seed^\\vee:=(e_i^\\vee)_{i\\in I}$, where $e_i^\\vee:=d_ie_i$. We also set $v^\\vee_i:=\\{e^\\vee_i, \\cdot \\}^\\vee$ These constructions give rise to {\\bf Langlands dual cluster varieties} which we denote as follows\n\\begin{align*}\n\t\\begin{array}{l l l l}\n\t{}^L(\\cA_{\\Gamma;\\seed_0}) := \\cA_{\\Gamma^\\vee;\\seed_0^\\vee} \\qquad \\qquad & \t \\text{and} \\qquad \\qquad & {}^L(\\cX_{\\Gamma; \\seed_0}) := \\cX_{\\Gamma^\\vee; \\seed_0^\\vee}.\n\\end{array}\n\\end{align*}\nSince $\\Gamma $ and $\\seed_0$ were already fixed, we denote ${}^L(\\cA_{\\Gamma;\\seed_0})$ (resp. ${}^L(\\cX_{\\Gamma;\\seed_0})$) simply by $\\LA$ (resp. $\\LX$).\n\n\\begin{definition}\nThe {\\bf Fock--Goncharov dual} of $\\cA$ (resp. $\\cX$) is the cluster variety $\\cA^{\\vee}$ (resp. $\\cX^{\\vee}$) given by\n\\begin{equation*}\n \\cA^{\\vee} := {}^L\\cX \\qquad \\qquad \t \\text{and} \\qquad \\qquad \\cX^{\\vee} := {}^L\\cA.\n\\end{equation*}\n\\end{definition}\n\nIn particular, we have that\n\\[\n \\cAp^\\vee = {}^L(\\cX_\\prin)=\\cX_{(\\Gamma_\\prin)^\\vee} \\qquad \\qquad \t \\cXp^{\\vee}= {}^L(\\cAp)=\\cA_{(\\Gamma_\\prin)^\\vee}.\n\\]\n\n\\begin{remark}\n \\label{rem:Lprin}\nNotice that $\\cA_{(\\Gamma_{\\prin})^\\vee}$ (resp. $\\cX_{(\\Gamma_{\\prin})^\\vee}$) is canonically isomorphic to $\\cA_{(\\Gamma^\\vee)_{\\prin}}$ (resp. $\\cX_{(\\Gamma^\\vee)_{\\prin}}$). Hence, we frequently identify these schemes without making reference to the canonical isomorphisms between them.\n\\end{remark}\n\nIt is not hard to see that the map \n\\begin{eqnarray}\\label{eq:L p}\n {(\\Lp)^*:= -d^{-1}(p^*)^*:N^\\vee \\to (M^\\vee)^{\\circ}}\n\\end{eqnarray}\nis well defined and is a cluster ensemble lattice map for the Langlands dual data $\\LGam$, where $(p^*)^*$ is the lattice map dual to $p^*$. \nIndeed, in the bases for $ N^\\vee $ and $ (M^\\vee)^{\\circ}$ determined by $\\seed^\\vee $, and in comparison with the matrix $B_{p^*;\\vb{s}}$ in \\eqref{eq:Mp*}, the matrix of $(\\Lp)^*$ is of the form\n\\begin{equation*}\nB_{(\\Lp)^*;\\seed^\\vee}= -B_{p^*;\\seed}^{\\rm{tr}}.\n\\end{equation*}\nIn particular, we have an associated dual cluster ensemble map\n\\[p^\\vee:\\cA^\\vee \\to \\cX^\\vee.\\]\n\nWe proceed to introduce the Fock--Goncharov dual for a quotient of $\\cA$. \nSo consider a cluster ensemble lattice map $p^*:N\\to M^{\\circ}$ for $\\Gamma$ and the cluster ensemble lattice map $(p^\\vee)^*:N^\\vee\\to (M^\\vee)^\\circ$ for $\\Gamma^\\vee$.\nRecall from \\eqref{eq:define K} that $K=\\ker(p_2^*)$.\nSimilarly, we set\n\n\\[\nK^\\vee=\\ker((p^\\vee)_2^*)=\\{k\\in N^\\circ\\mid \\{k,n\\}=0 \\text{ for all } n\\in d\\cdot N_{\\rm uf}\\},\n\\]\nwhere $(p^\\vee)_2^*$ is the map $p^*_2$ of \\eqref{eq:p12star} for $\\Gamma^\\vee$.\nLet $H_{\\cA}\\subseteq K^\\circ$ be a saturated sublattice and consider the quotient $\\cA\/T_{H_\\cA}$. \nRecall from \\S\\ref{sec:quotients-fibres} that $\\cA\/T_{H_\\cA}$ is obtained by gluing tori of the form $T_{N^{\\circ}\/H_{\\cA}}$. \nSince $N^{\\circ}\/H_{\\cA} $ and $H_{\\cA}^{\\perp} \\subset M^\\circ$ are dual lattices the Fock--Goncharov dual of $\\cA\/T_{H_{\\cA}}$ should be a fibre of $\\cA^\\vee$ obtained by gluing tori of the form $T_{H_{\\cA}^{\\perp}}$. In order to construct it notice that for $n$ in $\\Nuf$ we have $\\langle k,p^*(n)\\rangle = -{d}^{-1}\\{k,dn\\}=\\langle dk,-(p^\\vee)^*(n)\\rangle$. This implies that\n\\[\nK^\\circ=p^*(N_{\\rm uf})^\\perp=K^\\vee.\n\\]\nIn particular, $H_{\\cA}$ is a saturated sublattice of $K^\\vee$ as it is saturated in $K^\\circ$.\nIt is therefore possible to find $T_{H_{\\cA}^*}$ as the base of a fibration of the form \\eqref{eq:weight_map} for $\\cA^{\\vee}$ as we are allowed to set\n\\[\nH_{\\cA^{\\vee}}=H_{\\cA}\\subseteq K^\\vee.\n\\]\nSo consider the fibration\n\\[\nw_{H_{\\cA}}:\\cAm \\to T_{H_\\cA^*}.\n\\]\nNotice that the fibre $(\\cAm)_{{\\bf 1}_{T_{H_\\cA^*}}}$ is obtained gluing tori of the form $T_{H_{\\cA}^\\perp}$ as desired. \nTherefore, we define the Fock--Goncharov dual of the quotient $\\cA\/T_{H_\\cA}$ as\n\\[\n\\cAHAm := (\\cAm)_{{\\bf 1}_{T_{H_\\cA^*}}}=\\lrp{{}^L\\cX}_{{\\bf 1}_{T_{H_\\cA^*}}}.\n\\]\n\nSimilarly, let $H_{\\cX}\\subseteq K$ be a saturated sublattice and let $w_{H_{\\cX}}:\\cX \\to T_{H^*_\\cX}$ be the associated fibration. \nRecall that $\\cX_{{\\bf 1}_{T_{H^*_\\cX}}}$ is obtained by gluing tori of the form $T_{H_{\\cX}^{\\perp}}$. \nIts Fock--Goncharov dual is a quotient of $\\cXm$ glued from tori of the form $T_{(H^\\perp_\\cX)^*}$ which we construct next.\nA direct computation shows that $d\\cdot H_{\\cX}$ is a saturated sublattice of $(K^\\vee)^\\circ$. \nIn particular, we are allowed to choose\n\\[\nH_{\\cXm}= d\\cdot H_{\\cX}\\subseteq (K^\\vee)^\\circ \n\\]\nas a sublattice giving rise to a quotient $ \\LA\/T_{d\\cdot H_{\\cX}}$. This quotient is obtained by gluing tori of the form $T_{d\\cdot N}\/T_{d\\cdot H_\\cX}\\cong T_{N\/H_\\cX}\\cong T_{(H^{\\perp}_\\cX)^*}$. \nTherefore, we define the Fock--Goncharov dual of $\\cX_{{\\bf 1}_{T_{H^*_{\\cX}}}}$ as \n\\[\n\\cXeHm := \\cXm\/T_{ H_{\\cX^{\\vee}}}={}^L\\cA\/T_{d\\cdot H_\\cX}.\n\\]\n\nIn what follows, when we consider a saturated sublattice $H$ of $K^\\circ$ and write expressions such as $\\cA\/T_{H}$ or $w_{H}:\\cAm\\to T_{H^*}$ we will be implicitly assuming that we have set \n\\[\nH_{\\cA}= H = H_{\\cAm}.\n\\]\nSimilarly, when $H$ is a saturated sublattice of $K$ and we write expressions such as $w_{H}:\\cX \\to T_{H^*}$, $\\cXe$ or $\\cXem$ we will be implicitly assuming that we have set\n\\[\nH_{\\cX}= H = d^{-1}\\cdot H_{\\cXm},\n\\]\n\\[\n\\quad \\cXe = \\cXeH \\quad \\quad \\text{and} \\quad \\quad \\cXem= \\cXm \/T_{H_{\\cXm}}.\n\\]\n\n\\begin{remark}\nLet $\\cV$ be (a quotient of) $\\cA$ or (a fibre of) $\\cX$. In the skew-symmetric case Arg\\\"uz and Bousseau \\cite{AB22} showed that $\\cV$ and $\\cV^{\\vee}$ are mirror dual schemes from the point of view of \\cite{GS22}.\nA similar result is proven for the skew-symmetrizable case when $\\cV$ has dimension $2$ in \\cite{Mandy_rank2_MS} with arguments that may be generalized to arbitrary dimension.\n\\end{remark}\n\n\\subsection{Scattering diagrams and theta functions}\n\\label{sec:scat}\nTheta functions are a particular class of global function on (quotients and fibres of) cluster varieties introduced in \\cite{GHKK}.\nIn this subsection we outline their construction.\nThe main case to consider is the one of $\\cAp$ since scattering diagrams and theta functions for (quotients of) $\\cA$ and (fibres of) $\\cX$ can be constructed from this case.\n\n\\begin{remark}\n\\label{rem:full_rank_assumption}\nFrom now on, whenever we consider the variety $\\cA=\\cA_{\\Gamma,\\seed_0}$ we will assume $\\Gamma$ is of {\\bf full-rank}. \nBy definition this means that the map $p_1^*:\\Nuf \\to M^\\circ$ given by $n \\mapsto \\{ n , \\cdot \\}$ is injective. \nThere are various results of this article for $\\cA$ that are valid even if $\\Gamma$ is not of full-rank. \nHowever, various key results we shall use do need the full-rank condition (\\cf Remark \\ref{rem:all_from_cAp}). \nEven though we are imposing full-rank assumption we will frequently recall that we are assuming it to insist on the necessity of the assumption.\n\\end{remark}\n\n\\subsubsection{Theta functions on full-rank $\\cA$}\n\\label{sec:tf_A}\nThroughout this section we systematically identify $\\cA^{\\vee}_{\\seed^\\vee}(\\R^T)$ with $M^\\circ_{\\R}$, see \\S\\ref{ss:tropicalization}.\nA {\\bf wall} in $M^{\\circ}_\\R$ is a pair $(\\wall, f_{\\wall})$ where\n $\\wall \\subseteq M^{\\circ}_\\R$ is a convex rational polyhedral cone of codimension one, contained in $n^{\\perp}$ for some $n \\in N_{\\uf, \\seed}^+$, and \n $f_{\\wall} = 1+ \\sum_{k \\geq 1} c_k z^{kp^*_1(n)}$ is called a {\\bf scattering function}, where $c_k \\in \\Bbbk$. \nA {\\bf scattering diagram} $\\scat $ in $M^{\\circ}_\\R$ is a (possibly infinite) collection of walls satisfying a certain finiteness condition (see \\cite[\\S1.1]{GHKK}). \nThe {\\bf support} and the {\\bf singular locus} of $\\scat $ are defined as\n\\[\n\\Supp(\\scat):= \\bigcup_{\\wall \\in \\scat} \\wall \\ \\ \\ \\text{and} \\ \\ \\ \\Sing(\\scat):= \\bigcup_{\\wall \\in \\scat} \\partial\\wall \\ \\cup \\bigcup_{\\overset{\\wall_1,\\wall_2 \\in \\scat}{\\text{dim}(\\wall_1 \\cap \\wall_2) = |I|-2}} \\wall_1 \\cap \\wall_2.\n\\]\n\nA wall $(\\wall, f_{\\wall})$ defines a {\\bf wall-crossing automorphism} $\\mathfrak{p}_{\\wall}$ of $\\Bbbk (M)$ \ngiven in a generator $ z^m$ by $\n\\mathfrak{p}_{\\wall}(z^m)=z^m f_{\\wall}^{\\langle n_{\\wall}, m \\rangle }$,\nwhere $n_{\\wall}$ is the primitive normal vector of the wall $\\wall$ with a choice of direction going against the flow of the path $\\gamma$. \nIf we fix a scattering diagram $\\scat$ and a piecewise linear proper map $ \\gamma:[0,1]\\to M^\\circ_{\\R}\\setminus \\Sing(\\scat)$ intersecting $\\text{Supp}( \\mathfrak{D})$ transversely \nthen the {\\bf path ordered product} $ \\mathfrak{p}_{\\gamma , \\scat}$ is defined as the composition of automorphisms of the form $\\mathfrak{p}_{\\wall}$, where we consider the walls $\\wall$ that are transversely crossed by $\\gamma $. However, observe that $\\gamma$ might cross an infinite number of walls, therefore, we would be potentially composing an infinite number of automorphisms and such infinite composition is well defined. \nAgain, the reader is referred to \\cite[\\S 1.1]{GHKK} for a detailed discussion.\n\n\\begin{definition}\nA scattering diagram $\\scat$ is {\\bf consistent} if for all $\\gamma$ as above $\\mathfrak{p}_{\\gamma, \\scat}$ only depends on the endpoints of $\\gamma$. Two scattering diagrams $\\scat$ and $\\scat'$ are {\\bf equivalent} if $\\mathfrak{p}_{\\gamma, \\scat}= \\mathfrak{p}_{\\gamma, \\scat'}$ for all $\\gamma$.\n\\end{definition}\n\nTo define cluster scattering diagrams for $\\cA$ one first considers\n\\[\n\\scat_{{\\rm in}, \\seed}^{\\cA} := \\lrc{\\left.\\left( e_i^{\\perp} , 1+z^{ p_{1}^*\\left( e_i \\right)}\\right) \\right| \\ i \\in \\Iuf }.\n\\]\nA {\\bf cluster scattering diagram} for $\\cA$ is a consistent scattering diagram in $M^{\\circ}_{ \\R}$ containing $\\scat_{{\\rm in}, \\seed}^{\\cA}$. By the following theorem, cluster scattering diagrams for $\\cA$ do exist (provided $\\Gamma$ is of full-rank). \n\n\\begin{theorem} \\cite[Theorem 1.12 and 1.13]{GHKK}\n\\label{thm:consistent_scattering_diagrams}\nAssume $\\Gamma$ is of full-rank. Then for every seed $\\seed$ there is a consistent scattering diagram $\\scat_{\\seed}^{\\cA} $ such that $\\scat_{{\\rm in}, \\seed}^{\\cA} \\subset \\scat_{\\seed}^{\\cA}$.\nFurthermore $\\scat_{\\seed}^{\\cA}$ is equivalent to a scattering diagram all of whose scattering functions are of the form ${ f_{\\wall} = (1+ z^{p_{1}^*(n)})^c}$, for some $n \\in N$, and $c$ a positive integer. \n\\end{theorem}\n\n\\begin{definition} \\thlabel{def:genbroken}\nFix a cluster scattering diagram $\\scat^{\\cA}_{\\seed}$.\nLet $\\mono \\in M^{\\circ} \\setminus \\{0\\}$ and $x_0 \\in M^{\\circ}_{\\R} \\setminus \\text{Supp}(\\scat)$. \nA (generic) {\\bf broken line} for $\\scat^{\\cA}_{\\seed}$ with initial exponent $\\mono$ and endpoint $x_0$ is a piecewise linear continuous proper path $\\gamma : ( - \\infty , 0 ] \\rightarrow M^\\circ_{\\R} \\setminus \\Sing (\\scat^{\\cA}_{\\seed})$ bending only at walls, with a finite number of domains of linearity $L$ and a monomial $c_L z^{\\mono_L} \\in \\Bbbk[M^\\circ]$ for each of these domains. The path $\\gamma$ and the monomials $c_L z^{\\mono_L}$ are required to satisfy the following conditions:\n\\begin{itemize}\n\\setlength\\itemsep{0em}\n \\item $\\gamma(0) = x_0$.\n \\item If $L$ is the unique unbounded domain of linearity of $\\gamma$, then $c_L z^{\\mono_L} = z^{\\mono}$.\n \\item For $t$ in a domain of linearity $L$, $\\gamma'(t) = -\\mono_L$.\n \\item If $\\gamma$ bends at a time $t$, passing from the domain of linearity $L$ to $ L'$ then $c_{L'}z^{\\mono_{L'}}$ is a term in $\\mathfrak{p}_{{\\gamma}|_{(t-\\epsilon,t+\\epsilon)},\\scat_t} (c_L z^{m_L}) $, where ${\\scat_t = \\lrc{\\left.(\\wall, f_{\\wall}) \\in \\scat^{\\cA}_{\\seed}\\right| \\gamma (t) \\in \\wall }}$.\n\\end{itemize}\nWe refer to $\\mono_L$ as the {\\bf slope} or {\\bf exponent vector} of $\\gamma$ at $L $ and set \n\\begin{itemize}\n \\item $I(\\gamma) = \\mono$;\n \\item $\\text{Mono} (\\gamma) = c(\\gamma)z^{F(\\gamma)}$\nto be the monomial attached to the unique domain of linearity of $\\gamma$ having $x_0 $ as an endpoint. \n\\end{itemize}\n\\end{definition}\n\n\\begin{definition}\\thlabel{def:theta}\nChoose a point $x_0$ in the interior of $\\mathcal{C}_\\seed^+:=\\{m\\in M^{\\circ}_{\\R}\\mid \\langle e_i, m \\rangle \\geq 0 \\text{ for all } i \\in \\Iuf\\}$ and let $\\mono\\in \\cA^\\vee_{\\seed^\\vee}(\\Z^T)=M^\\circ$. \nThe {\\bf theta function} on $\\cA$ associated to $\\mono$ is \n\\begin{equation}\n\\label{eq:tf}\n \\tf^{\\cA}_{ \\mono}:= \\sum_{\\gamma} \\text{Mono} (\\gamma),\n\\end{equation}\nwhere the sum is over all broken lines $\\gamma$ with $I(\\gamma)=\\mono$ and $\\gamma(0)=x_0$. \nFor $\\mono = 0$ we define $\\vartheta^{\\cA}_{0} =1$. We say $\\tf^{\\cA}_{ \\mono}$ is {\\bf polynomial} if the sum in \\eqref{eq:tf} is finite.\n\\end{definition}\n\n\\begin{remark}\n\\label{rem:on_tf}\nIt is a nontrivial fact that $\\tf^{\\cA}_{\\mono}$ is independent of the point $x_0\\in \\mathcal{C}_{\\seed}^+$ we have chosen, see \\cite[\\S3]{GHKK}.\nMoreover, in general $\\tf^\\cA_{\\mono}$ can be an infinite sum and in order to think of $\\tf^{\\cA}_{\\mono} $ as a function on a space one needs to work formally an consider a degeneration of $\\cA$, see \\cite[Proposition 3.4 and \\S6]{GHKK} for the details.\nHowever, in case $\\tf^{\\cA}_{\\mono}$ is polynomial then $\\tf^{\\cA}_{\\mono}\\in H^0(\\cA,\\mathcal{O}_{\\cA})$, that is, $\\tf^{\\cA}_{\\mono}$ is an algebraic function on $\\cA$. \nThe definition of $\\tf^{\\cA}_{\\mono}$ in \\eqref{eq:tf} corresponds to the expression of such function written in the coordinates of the seed torus $\\cA_{\\seed}$.\n\\end{remark}\n\n\\subsubsection{Labeling by tropical points}\n\nRecall that we are identifying $\\cA^{\\vee}_{\\seed^\\vee}(\\R^T)$ and $ M^{\\circ}_{\\R}$. \nBy construction, a theta function on $\\cA$ is labeled by a point $\\mono \\in \\cA^\\vee_{\\seed^\\vee}(\\Z^T)= M^{\\circ}$. \nBy \\cite[Proposition 3.6]{GHKK}, this labeling upgrades to a labeling by a point in $\\cA^\\vee(\\Z^T)$. \nThe main point being that if we let $m'=(\\mu^{\\cA^\\vee}_{k})^T(m)\\in \\cA^{\\vee}_{\\mu_k(\\seed^\\vee)}(\\Z^T)$ for $k \\in \\Iuf$ then $\\tf^{\\cA}_\\mono$ and $\\tf^{\\cA}_{m'}$ correspond to the same function (see Remark \\ref{rem:on_tf}) expressed, however, in different cluster coordinates.\nThis fact is of great importance for this paper so we would like to highlight it:\n\n\\begin{center}\n \\emph{every theta function on $\\cA$ is naturally labeled by a point of $\\cA^\\vee(\\Z^T)$}.\n\\end{center}\n\nIn light of the discussion just above, from now on we label theta functions on $\\cA$ either by elements of $\\cA^{\\vee}(\\Z^T)$ or of $\\cA^{\\vee}_{\\seed^\\vee}(\\Z^T)$. \nFor sake of clarity, tropical points are denoted in bold font and as tuples. \nThat is, ${\\bf m}=(m_{\\seed^\\vee})_{\\seed^\\vee\\in \\orT}$ denotes an element of $\\cA^\\vee(\\Z^T)$ and $m_{\\seed^\\vee}=\\mathfrak{r}_{\\seed^\\vee}({\\bf m})$. \n With this notation we have the following identity\n \\[\n\\tf^{\\cA}_{\\bf m}=\\tf^{\\cA}_{m_{\\seed^\\vee}}.\n \\]\n\nEven further, we can think of $\\Supp(\\scat^{\\cA}_{\\seed})$ as a subset of $\\cA^{\\vee}_{\\seed^\\vee}(\\R^T) $. By \\cite[Theroem 1.24]{GHKK} we have that for every $k \\in \\Iuf $, $\\mu^{\\cA^\\vee}_{\\seed^\\vee,\\mu_k(\\seed^\\vee)}\\lrp{\\Supp(\\scat^{\\cA}_{\\seed})}=\\Supp(\\scat^{\\cA}_{\\mu_k(\\seed)})$ and that $\\scat^{\\cA}_{\\seed}$ and $\\scat^{\\cA}_{\\mu_k(\\seed)}$ are equivalent. Hence there is a well defined subset $\\Supp(\\scat^\\cA) \\subset \\cA^{\\vee}(\\R^T)$ such that\n\\[\n\\mathfrak{r}_{\\seed^\\vee}\\lrp{\\Supp(\\scat^\\cA) }= \\Supp(\\scat^\\cA_{\\seed})\n\\]\nfor every $\\seed\\in \\orT$. \nThe point here is that $\\Supp(\\scat^\\cA)$ is seed independent.\nSimilarly, the map $\\mu^{\\cA^\\vee}_{\\seed^\\vee,\\mu_k(\\seed^\\vee)} $ determines a bijection between the set of broken lines for $\\scat^{\\cA}_{\\seed}$ and the set of broken lines for $\\scat^{\\cA}_{\\mu_k(\\seed)}$ (see \\cite[Proposition 3.6]{GHKK}). \nIn particular, supports of broken lines make sense in $\\cA^\\vee(\\R^T)$.\n\n\n\\begin{remark}\n It is possible to upgrade $\\Supp(\\scat^\\cA)$ to a scattering diagram inside $\\cA^{\\vee}(\\R^T)$. In this generality scattering functions are described using log Gromov--Witten invariants. \n See \\cite{KY19} for details.\n\\end{remark}\n\n\\subsubsection{The middle cluster algebra}\n\nLet us recall now that broken lines also encode the multiplication of theta functions. \nThat is, given a product of arbitrary theta functions $\\tf^{\\cA}_p \\tf^{\\cA}_q$ with $p,q \\in \\cA^\\vee_{\\seed^\\vee}(\\Z^T)$,\nwe can use broken lines to express the structure constants $\\alpha\\lrp{p,q,r}$ in the expansion \n\\begin{align} \\label{eq:product}\n \\vartheta^{\\cA}_p \\vartheta^{\\cA}_q = \\sum_{r\\in \\cA^\\vee_{\\seed^\\vee}(\\Z^T)} \\alpha(p,q,r) \\vartheta^{\\cA}_r.\n\\end{align}\nWe review the construction here.\nFirst, pick a general endpoint $z$ near $r$.\nThen define (\\cite[Definition-Lemma~6.2]{GHKK})\n\\eq{\n \\alpha_z (p, q, r) := \\sum_{\\substack{\\lrp{\\gamma^{(1)}, \\gamma^{(2)}} \\\\ I(\\gamma^{(1)})= p,\\ I(\\gamma^{(2)})= q\\\\ \\gamma^{(1)}(0) = \\gamma^{(2)}(0) = z\\\\\n F(\\gamma^{(1)}) + F(\\gamma^{(2)}) = r }} c(\\gamma^{(1)})\\ c(\\gamma^{(2)}), }{eq:multibrokenline}\nwhere the sum is over all pairs of broken lines $\\lrp{\\gamma^{(1)}, \\gamma^{(2)}}$ ending at $z$ with initial slopes $I(\\gamma^{(1)}) = p$, $I(\\gamma^{(2)}) = q$ and final slopes satisfying $F(\\gamma^{(1)})+F(\\gamma^{(2)}) =r$.\nGross--Hacking--Keel--Kontsevich show that for $z$ sufficiently close to $r$, $\\alpha_z (p, q, r)$ is independent of $z$ and gives the structure constant $\\alpha (p, q, r)$ (see \\cite[Proposition~6.4]{GHKK}). \n\n\\begin{definition}\n\\thlabel{def:cmid}\nLet $\\Theta(\\cA):= \\{ {\\bf m} \\in \\cA^{\\vee}(\\Z^T) \\mid \\vartheta^{\\cA}_{{\\bf m}} \\text{ is polynomial}\\}$. The {\\bf middle cluster algebra} $\\text{mid}(\\cA)$ is the $\\Bbbk$-algebra whose underlying vector space is $\\{ \\tf^{\\cA}_{{\\bf m}} \\mid {\\bf m} \\in \\Theta(\\cA) \\}$, the multiplication of the basis elements is given by \\eqref{eq:product} and extended linearly to all $\\cmid(\\cA)$.\n\\end{definition}\n\n\\subsubsection{Theta functions on $\\cAp$}\nThe data $\\Gamma_{\\prin}$ is of full-rank. Therefore, this case is a particular case of \\S\\ref{sec:tf_A}. So we can talk about scattering diagrams, broken lines and theta functions for $\\cAp $. The following result follows from Theorem \\ref{thm:consistent_scattering_diagrams} and the definition of theta functions.\n\n\n\\begin{lemma}\nFix a seed $\\widetilde{\\seed}$ for $\\cAp$ and express theta functions on the cluster coordinates determined by $\\widetilde{\\seed}$.\nFor $(m,n)\\in \\mathfrak{r}_{\\widetilde{\\seed}}(\\Theta(\\cAp))$ we have that $\\tf^{\\cAp}_{(m,n)}=\\tf^{\\cAp}_{(m,0)}\\tf^{\\cAp}_{(0,n)}$ and $\\tf^{\\cAp}_{(0,n)}$ is the Laurent monomial on the coefficients given by $n$.\n\\end{lemma}\n\n\n\nNote that, for $(m_1,n_1),(m_2,n_2) \\in M^{\\circ}_{\\rm prin}$, in general we have that $\\tf^{\\cAp}_{(m_1+m_2,n_1+n_2)} \\neq \\tf^{\\cAp}_{(m_1,n_1)} \\tf^{\\cAp}_{(m_2,n_2)}$. \nThe above lemma holds because the decomposition is only separating the unfrozen and frozen parts (\\cf \\thref{g_is_val} below). \n\n\\begin{remark}\n\\label{rem:all_from_cAp}\nScattering diagrams for $\\cAp $ can be used to define scattering diagrams, broken lines therein and theta functions on a variety $\\cV $ of form $\\cA$ (even if $\\Gamma$ is not of full-rank), $\\cX$, $\\cA\/T_{H}$ and $\\cX_{{\\bf 1}}$. \nFurther, in each one of these cases we can define the associated middle cluster algebra $\\cmid(\\cV)$ and the set $\\Theta(\\cV)$ parametrizing its theta basis. \nIn the following subsections we explain the cases of $\\cA\/T_{H}$, $\\cX$, and $\\cX_{{\\bf 1}}$ individually. \nWe do not treat the case of $ \\cA $ for $\\Gamma$ when $\\Gamma$ is not of full-rank since the results of \\S\\ref{sec:cluster_valuations} do not apply to this case.\n\\end{remark}\n\n\n\n\n\\subsubsection{Theta functions on $\\cA\/T_{H}$}\n\\label{tf_quotient}\nSuppose that $\\Gamma $ is of full-rank (\\cf Remark \\ref{rem:all_from_cAp}). Let $H \\subset K^\\circ$ be a saturated sublattice and consider the quotient $\\cA\/T_{H}$ and the fibration $w_H: \\cAm \\to H^*$ (see the end of \\S\\ref{sec:FG_dual}).\nThe next result shows that theta functions on $\\cA$ have a well defined $T_{H}$-weight.\n\n\n\\begin{proposition}\n\\thlabel{prop:dual_fibration}\nEvery polynomial theta function on $\\cA$ is an eigenfunction with respect to the $T_{H}$-action. For every ${\\bf q}\\in \\Theta(\\cA)$ the $T_H$-weight of $\\tf^{\\cA}_{\\bf q}$ is the image of ${\\bf q} \\in \\cA^{\\vee}(\\Z^T)$ under the tropicalized map $w^{T}_{H}:\\cAm(\\Z^T) \\to H^*$. Under the isomorphism $ H^* \\cong M^\\circ\/H^\\perp$ and in the lattice identification of $ \\cA^\\vee_{\\seed^\\vee}(\\Z^T)$ of $\\cA^\\vee (\\Z^T)$ the map $w^T_{H}$ is given by\n\\begin{align*}\n w^{T}_{H} : \\ & \\cA^{\\vee}_{\\seed^\\vee}(\\Z^T) \\to M^\\circ\/H^{\\perp},\\\\\n & \\ \\ \\ \\ \\ \\ q \\longmapsto q + H^{\\perp}.\n\\end{align*}\n\\end{proposition}\nThe claims are essentially contained in the literature already\n(see for instance \\cite[Proposition 7.7]{GHKK}). \nThe differences are that we are acting by a potentially smaller torus (Gross--Hacking--Keel--Kontsevich act by $T_{K^\\circ}$ rather than $T_{H}$) and, regarding the map $w_{H}: \\cA^\\vee \\to T_{H^*}$, \nwe are including $\\Bbbk[H]$ into $\\Bbbk[N^\\vee]=\\Bbbk[N^\\circ]$ rather than including $\\Bbbk[K^\\circ]$ into $\\Bbbk[N^\\circ]$. \nFor the convenience of the reader we give a proof of the statement.\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:dual_fibration}]\nBy \\cite[Theorem~1.13]{GHKK} all scattering functions may be taken to be of the form $\\lrp{1+z^{p^*(n)}}^c$ for some $n \\in \\Nuf$ and some positive integer $c$.\\footnote{That is, the equivalence class of consistent scattering diagrams for $\\cA$ contains a representative whose scattering functions are of this form.}\nFor $q\\in \\Theta(\\cA)_{\\seed^\\vee}$ we have that $\\tf^{\\cA}_q$ is as a Laurent polynomial in $\\Bbbk[M^\\circ]$.\nAll monomial summands of $\\tf^{\\cA}_{q}$ have the form $c_m z^{q + m}$ for some $m \\in p^*(\\Nuf)$ and $c_m \\in \\Z_{>0}$.\nThe $T_{H}$-weight of this monomial is obtained from the map\n\\eqn{\nT_{H} \\to T_{\\Z}=\\Bbbk^*\\quad \\text{given by } \\quad z^{h} \\mapsto z^{\\lra{q+m , h}} \\quad \\text{for $h \\in {H}$}.\n}\nSince $H \\subset p^*\\lrp{\\Nuf}^\\perp$ we have that\n$z^{\\lra{q+m , h}} = z^{\\lra{q, h}}$. \nThat is, the $T_{H}$-weight of each monomial $z^{m'}$, $m'\\in M^\\circ$, is the character of $T_{H}$ given by $m' + H^\\perp \\in M^\\circ\/H^\\perp \\cong H^*$. \nMoreover, all monomial summands of $\\tf^{\\cA}_q$ have the $T_{H}$-weight $q + H^\\perp \\in H^*$.\nNext, the piecewise linear map $(\\mu_k^{\\cA^\\vee})^T:M^\\circ_\\seed \\to M^\\circ_{\\mu_k(\\seed)}$ sends $m$ to $m+m'$ for some $m'\\in p^*(\\Nuf)$.\nSo, the choice of torus does not affect the $T_{H}$-weight.\nTherefore, $\\tf^{\\cA}_q$ is an eigenfunction whose weight is $q + H^\\perp$.\nFurthermore, the projection \n\\eqn{M^\\circ &\\to M^\\circ\/H^\\perp\\quad \\text{given by} \\quad q \\mapsto q + H^\\perp } \ndualizes the inclusion $H \\hookrightarrow N^\\circ$.\nSo, restricting to seed tori, this is precisely the tropicalization of the map ${T_{M^\\circ} \\rightarrow T_{H^*}}$ whose pullback is the inclusion $H \\hookrightarrow N^\\circ$.\nSince $p^*$ commutes with mutation,\nwe see that the $T_{H}$-weight of $\\tf^{\\cA}_{\\bf q}$ is the image of ${\\bf q}$ under the tropicalization of \n$ w_{H}: \\cA^\\vee \\rightarrow T_{H^*}$.\n\\end{proof}\n\nEvery weight $0$ eigenfunction on $ \\cA$ induces a well defined function on $\\cA\/T_{H}$. \nSo in order to construct a scattering-diagram-like structure $\\scat^{\\cA\/T_H}$ defining theta functions on $\\cA\/T_{H}$ we consider the {\\bf weight zero slice} inside $\\cA^{\\vee}(\\R^T)$ defined as \n$(w^T_H)^{-1}(0)$. \nObserve that identifying $ \\cA^\\vee$ with $M^\\vee$ via a choice of seed, then $(w^T_H)^{-1}(0)$ corresponds to $H^{\\perp}_{\\R}$.\nWith this in mind, we define $\\supp(\\scat^{\\cA\/T_{H}})$ as\n\\[\n\\supp(\\scat^{\\cA\/T_{H}}):=\\supp (\\scat^{\\cA})\\cap (w^T_H)^{-1}(0).\n\\]\nThe scattering functions attached to the walls of $\\mathfrak{r}_{\\seed^\\vee}(\\supp(\\scat^{\\cA\/T_{H}}))$ are the same as the corresponding functions attached to the walls of $\\scat^{\\cA}_\\seed$. \nThis gives rise to a scattering diagram $\\scat^{\\cA\/T_{H}}_{\\seed}$ inside $(\\cA\/T_{H})^\\vee_{\\seed^\\vee}(\\R^T)$ for every $\\seed \\in \\orT$.\nThe broken lines for $\\scat^{\\cA\/T_{H}}_\\seed$ are the broken lines for $\\scat^{\\cA}_\\seed $ entirely contained in $\\mathfrak{r}_{\\seed^\\vee}(w^{-1}_{H}(0))$.\n\nIn order to label a theta function on $\\cA\/T_{H}$ with an element of $(\\cA\/T_{H})^{\\vee}(\\Z^T)$ it suffices to consider a bijection $\n(\\cA\/T_{H})^{\\vee}(\\R^T) \\overset{\\sim}{\\longrightarrow} (w_H^T)^{-1}(0)$.\nSuch a bijection can be obtained tropicalizing the inclusion $\\mathfrak{i}_H:(\\cA\/T_{H})^{\\vee} \\hookrightarrow \\cA^\\vee $. \nIndeed, in lattice identifications of the tropical spaces given by a seed $\\seed$, the map\n$ \\mathfrak{i}_H^T:(\\cA\/T_{H})^{\\vee}_{\\seed}(\\Z^T)\\hookrightarrow \\cA^\\vee_{\\seed}(\\Z^T)$ correspond to the inclusion $H^\\perp \\hookrightarrow M^\\vee$ and $w^{-1}_{H}(0)(\\Z) $ corresponds to $H^\\perp$. \n\n\nIn particular, we obtain (as one should have expected) that the theta functions on $\\cA\/T_{H}$ are precisely the functions on $\\cA\/T_{H}$ induced by the $T_H$-weight zero theta functions on $\\cA$. \nSo we let\n$\\Theta(\\cA\/T_H)\\subset (\\cA\/T_H)^{\\vee}(\\Z^T)$ be the preimage of $\\Theta(\\cA)\\cap (w_H^T)^{-1}(0)$ under $\\mathfrak{i}^T_H$ and define the middle cluster algebra $\\cmid(\\cA\/T_H)$ as in the case of $\\cA$ (see \\thref{def:cmid}).\nIn particular, for ${\\bf m}\\in \\Theta (\\cA\/T_H)$ the theta function $\\tf^{\\cA\/T_H}_{\\bf m}$ is the function on $\\cA\/T_H$ induced by $ \\tf^{\\cA}_{\\mathfrak{i}^T_H({\\bf m})}$. So,\n\\begin{center}\n \\emph{every theta function on $\\cA\/T_H$ is naturally labeled by a point of $(\\cA\/T_H)^\\vee(\\Z^T)$}.\n\\end{center}\n\n\n\\subsubsection{Theta functions on $\\cX$}\n\\label{sec:tf_X}\nRecall from \\S\\ref{sec:principal_coefficients} that there is an isomorphism $\\chi: \\cAp\/T_{H_{\\cAp}}\\to \\cX$, where \\[\nH_{\\cAp}=\\lrc{\\lrp{n,-(p^*)^*(n)}\\in N^\\circ_\\prin \\mid n \\in N^\\circ} \\subset K^{\\circ}_{\\prin}. \n\\]\nHence, the construction of theta functions on $\\cX$ is already covered in the previous subsection. \nHowever, there is a very subtle difference created by treating $ \\cAp\/T_{H_{\\cAp}}$ as a cluster $\\cX$-variety as opposed to a quotient of $\\cAp$: \n\\begin{center}\n\\emph{every theta function on $ \\cX$ is naturally labeled by a point of $\\cX^\\vee(\\Z^t)$ as opposed to $\\cX^\\vee(\\Z^T)$.}\n\\end{center} \nIf we would proceed as in the previous subsection we would label theta functions on $ \\cAp\/T_{H_{\\cAp}}$ by points of $(\\cAp\/T_{H_{\\cAp}})^\\vee(\\R^T)$.\nThe origin of the difference is made explicit by the following lemma.\n\n\\begin{lemma}\n\\label{lem:right_tropical_space}\nThere is a canonical bijection between $\\cX^{\\vee}(\\R^t)$ and $\\lrp{w^T_{H_{\\cAp}}}^{-1}(0)\\subset \\cAp^\\vee(\\R^T)$.\n\\end{lemma}\n\\begin{proof}\nOne can verify directly that the composition $\\xi^T_{\\Gamma^\\vee}\\circ i$ gives rise to the desired bijection, where \\[\ni:\\cX^\\vee(\\R^t) \\to \\cX^\\vee(\\R^T)\n\\]\nis the bijection discussed in \\S\\ref{ss:tropicalization} and \\[\n\\xi_{\\Gamma^\\vee}^T: \\cX^\\vee(\\R^T) \\to \\cAp^{\\vee}(\\R^T) \n\\]\nis the tropicalization of the map $\\xi_{\\Gamma^\\vee}:\\cX^\\vee=\\cA_{\\Gamma^\\vee} \\to \\cX_{(\\Gamma^\\vee)_{\\prin}}\\cong \\cX_{(\\Gamma_{\\prin})^\\vee}=\\cAp^{\\vee} $ described in \\eqref{eq:def_xi}, see Remarks \\ref{rem:labels} and \\ref{rem:Lprin}. \nHowever, for the convenience of the reader we include computations that show in a rather explicit way the necessity to consider \n$\\cX^\\vee(\\Z^t)$ as opposed to $\\cX^\\vee(\\Z^T)$. For simplicity throughout this proof we denote $w_{H_{\\cAp}}$ simply by $w$.\n\nPick a seed $\\seed=(e_i)_{i \\in I}\\in \\orT$ for $\\Gamma$ and consider the seed $\\seed^\\vee$ for $\\Gamma^\\vee$. Denote by $\\widetilde{\\seed}^\\vee$ the seed for $(\\Gamma_{\\prin})^\\vee$ obtained mutating $ \\seed_{0_{\\prin}}$ in the same sequence of directions needed to obtain $\\seed$ from $\\seed_0$. Then\n\\[\n\\lrp{(w^T)^{-1}(0)}_{\\widetilde{\\seed}^\\vee}(\\R^T)=H^\\perp_{\\cAp}=\\{(p^*(n),n)\\in M^\\circ_{\\prin,\\R} \\mid n\\in N_{\\R}\\}\\subset M^\\circ_{\\prin, \\R}=M^\\circ_{\\R}\\oplus N_{\\R}\n\\]\n(see \\eqref{eq:identification} to recall the meaning of $\\lrp{(w^T)^{-1}(0)}_{\\widetilde{\\seed}^\\vee}(\\R^T)$). We now verify that for every $k\\in \\Iuf$ there is a commutative diagram\n\\[\n\\xymatrix{\n\\lrp{(w^T)^{-1}(0)}_{\\widetilde{\\seed}^{\\vee}}(\\R^T) \\ar^{\\lrp{\\mu^{\\cAp^\\vee}_{k}}^T}[rr] \\ar_{\\pi^{\\cX^\\vee}_1}[d] & & \\lrp{(w^T)^{-1} (0)}_{\\mu_k(\\widetilde{\\seed}^{\\vee})}(\\R^T) \\ar^{\\pi^{\\cX^\\vee}_2}[d]\n\\\\\n\\cX^{\\vee}_{\\seed^{\\vee}}(\\R^t)\\ar^{\\lrp{\\mu^{\\cX^\\vee}_{k}}^t}[rr] & & \\cX^{\\vee}_{\\mu_k(\\seed^{\\vee})}(\\R^t),\n}\n\\]\nwhere the vertical maps $\\pi^{\\cX^\\vee}_1$ and $\\pi^{\\cX^\\vee}_2$ are both given by $(p^*(n),n)\\mapsto dn$ (recall that $\\cX^{\\vee}_{\\seed^{\\vee}}(\\R^t)=(N^\\vee)^\\circ_{\\R}= (d\\cdot N)_{\\R} =\\cX^{\\vee}_{\\mu_k(\\seed^{\\vee})} (\\R^t)$). By definition we have that\n\\begin{eqnarray*}\n \\lrp{\\mu^{\\cAp^\\vee}_{k}}^T(p^*(n),n)& \\overset{\\eqref{eq:tropical_X_mutation}}{=} & (p^*(n),n)+[\\langle (d e_k,0),(p^*(n),n)\\rangle]_+\\{d_ke_k, \\cdot \\}^{\\vee}_{\\prin} \\\\\n &=& (p^*(n),n)+ [p^*(n)(de_k)]_+(\\{d_ke_k, \\cdot\\}^\\vee,d_ke_k)\\\\\n &=&(p^*(n) + [\\{n, de_k\\}]_+\\{d_ke_k, \\cdot\\}^\\vee,n+ [\\{n, de_k\\}]_+d_ke_k).\n\\end{eqnarray*}\nUsing the facts that $d, d_k>0$ and \n that $d\\max(a,b)=\\max(da,db)$ and $\\max(a,b)=-\\min(-a,-b)$ for all $a,b \\in \\R$, we compute that\n\\begin{eqnarray*}\n\\pi^{\\cX^\\vee}_2\\lrp{\\lrp{\\mu^{\\cAp^\\vee}_{k}}^T(p^*(n),n)} &=& dn+ d[\\{n, de_k\\}^\\vee]_+d_ke_k\\\\\n&=& dn+ [\\{dn, de_k\\}^\\vee]_+d_ke_k\\\\\n &=& dn+[-\\{de_k,dn\\}^{\\vee}]_+d_ke_k\\\\\n &=& dn+[-\\{d_ke_k,dn\\}^{\\vee}]_+de_k\\\\\n &=& dn-[\\{d_ke_k,dn\\}^{\\vee}]_-de_k\\\\\n &=& dn-[\\langle v_k^\\vee,dn\\rangle]_-de_k\\\\\n&=& dn+[\\langle v_k^\\vee,dn\\rangle]_-(-d_k^\\vee e^\\vee_k)\\\\\n&\\overset{\\eqref{eq:tropical_A_mutation}}{=}& \\lrp{\\mu^{\\cX^\\vee}_{k}}^t (dn)\\\\\n&=& \\lrp{\\mu^{\\cX^\\vee}_{k}}^t \\lrp{\\pi^{\\cX^\\vee}_1(p^*(n),n)}.\n\\end{eqnarray*}\nThis gives the commutativity of the diagram. \nNotice moreover that $\\pi^{\\cX^\\vee}_1$ and $\\pi^{\\cX^\\vee}_2$ are canonical bijections. \nThese two facts together imply that we have a well defined bijection\n\\[\n\\pi^{\\cX^\\vee}: (w^T)^{-1}(0)(\\R^T) \\overset{\\sim}{\\longrightarrow} \\cX^{\\vee}(\\R^t).\n\\]\nThe fact that $\\xi^T_{\\Gamma^\\vee}\\circ i$ is the inverse of $\\pi^{\\cX^\\vee} $ follows from noticing that, in lattice identifications of the domain and codomian of $\\xi^T_{\\Gamma^\\vee}$ given by a choice of seed, we have that\n\\[\n\\xi^T_{\\Gamma^\\vee}(dn)=(-p^*(n),-n).\n\\]\n\\end{proof}\nWe can now define cluster scattering diagrams for $ \\cX$ using cluster scattering diagrams for $\\cAp$ and the quotient map $\\tilde{p}:\\cAp \\to \\cX$ described in \\eqref{eq:def_tilde_p} and the content of Lemma \\ref{lem:right_tropical_space}.\nWe define $\\supp(\\scat^{\\cX})$ as \n\\[\n\\supp(\\scat^{\\cX}):=\\pi^{\\cX^\\vee}\\lrp{\\supp (\\scat^{\\cAp})\\cap (w^T_H)^{-1}(0)}\\subset \\cX^\\vee(\\Z^t).\n\\]\nBy definition the support of the scattering diagram $ \\scat^{\\cX}_{\\seed}$ is $\\mathfrak{r}_{\\seed^\\vee}\\lrp{\\supp(\\scat^{\\cX})}$.\nThe scattering functions attached to the walls of $\\supp(\\scat^{\\cX}_\\seed)$ are obtained by applying $\\tilde{p}^*$ to the scattering functions of the corresponding walls of $\\scat^{\\cAp}_\\seed$. \nWe proceed in an analogous way to define broken lines for $\\scat^{\\cX}_\\seed$.\nAs in the previous cases, supports of broken lines are well defined inside $\\cX^\\vee(\\Z^t)$.\n\nThe labeling of a theta function on $\\cX$ with an element of $\\cX^{\\vee}(\\Z^t)$ is obtained using the bijection of Lemma \\ref{lem:right_tropical_space}. More precisely,\nfor ${\\bf n} \\in \\cX^\\vee(\\Z^t)$ with ${\\bf n}\\in \\Theta(\\cX)$ we have\n\\[\n\\tilde{p}^*(\\tf^\\cX_{\\bf n})=\\tf^{\\cAp}_{\\xi^T_{\\Gamma^\\vee}\\circ i({\\bf n})}.\n\\]\nExplicitly, in lattice identifications of the tropical spaces, we have that for $dn \\in \\cX^\\vee_{\\seed^\\vee}(\\Z^t) $\n\\[\n\\tilde{p}^*\\lrp{\\tf^{\\cX}_{dn}}:= \n\\tf^{\\cAp}_{(p^*(n),n)}.\n\\]\n\n\n\\begin{example}\n\\label{running_example_1}\nLet $\\epsilon\n=\n\\lrp{\\begin{matrix}\n0 & 2 \\\\\n-1 & 0\n\\end{matrix}}$\nand $d_1=1, d_2=2$. Using the above parametrization we compute\n\\[\n\\tf^{\\cX}_{2(-1,-2)}=X_1^{-1}X_2^{-2}+2X_1^{-1}X_2^{-1}+X_1^{-1}.\n\\]\nIndeed, we have that $\\xi^T_{\\Gamma^\\vee}\\circ i(2(-1,-2))=(2,-2)$ and\n\\[\n\\tf^{\\cAp}_{(2,-2),(-1,-2)}= \\lrp{\\tf^{\\cAp}_{(1,-1),(0,0)}}^2 \\tf^{\\cAp}_{(0,0),(-1,-2)} = \\lrp{\\dfrac{A_1+t_2}{A_2}}^2t_1^{-1}t_2^{-2}= \\tilde{p}^*(X_1^{-1}X_2^{-2}+2X_1^{-1}X_2^{-1}+X_1^{-1}).\n\\]\n\\end{example}\n\n\n\n\\subsubsection{Theta functions on $\\cXe$}\n\\label{tf_fibre}\nAs in the previous subsections we would like to highlight that \n\\begin{center}\n\\emph{every theta function on $ \\cXe$ is naturally labeled by a point of $(\\cXe)^\\vee(\\Z^t)$}\n\\end{center} \nas we now explain.\nThe tropical space $ (\\cXe)^{\\vee}(\\R^t) $ is the quotient of $ \\cX^{\\vee} (\\R^t)$ by the tropicalization of the action of $T_H$ on $\\cX^{\\vee}$. \nIn other words, since the variety $(\\cXe)^{\\vee}$ is a quotient of $\\cX^\\vee$, we can consider the quotient map by $\\varpi_{H}: \\cX^\\vee \\to (\\cXe)^\\vee$\nto obtain a surjection\n\\[\n\\varpi_H^t: \\cX^{\\vee}(\\R^t) \\to (\\cXe)^{\\vee} (\\R^t).\n\\]\nThen, given $\\overline{\\bf n}\\in (\\cXe)^{\\vee} (\\R^t)$ and ${\\bf n}\\in (\\varpi_H^t)^{-1}(\\overline{\\bf n})$ we define \n\\[\n\\tf^{\\cXe}_{\\overline{\\bf n}}=\\tf^{\\cX}_{\\bf n}|_{\\cXe}.\n\\]\nMore concretely, working in lattice identifications of the tropical spaces, we have that $\\cX^{\\vee}(\\R^t)_{\\seed^\\vee} = N_\\R$ and $ (\\cXe)^{\\vee}_{\\seed^\\vee}(\\R^t) {\\cong}N_\\R\/H_{\\R}$. \nThen for every $n \\in N$\n\\[\n\\tf^{\\cXe}_{d n + H}=\\tf^{\\cX}_{dn}|_{\\cXe}.\n\\]\nOne can proceed in an analogous way as in the previous cases to construct a scattering diagram like structure $\\scat^{\\cXe}_{\\seed}$ inside $(\\cXe)^\\vee_{\\seed}(\\Z^t)$. In turn we obtain a description of $ \\tf^{\\cXe}_{\\overline{\\bf n}}$ using broken lines and use these to define $\\cmid (\\cXe)$ and $\\Theta(\\cXe)$.\n\n\n\n\\subsubsection{The full Fock--Goncharov conjecture}\n\\label{sec:FG_conj}\n\n\n\nLet $\\cV$ be a scheme of the form $ \\cA$, $\\cX$, $\\cA\/T_{H}$ or $\\cX_{{\\bf 1}}$. \nThe {\\bf upper cluster algebra} of $\\cV$ is defined as \n\\[\n\\text{up}(\\cV):=H^0(\\cV,\\mathcal{O}_{\\cV}).\n\\]\nEvery polynomial theta function on $\\cV$ belongs to $\\text{up}(\\cV)$, therefore, we have a natural $\\Bbbk$-linear map $\\cmid(\\cV)\\to \\text{up}(\\cV)$.\nIf $\\cV$ is one of $\\cA$ (see Remark \\ref{rem:full_rank_assumption}) or $\\cX$ it was proved in \\cite[Theorem 7.5, Corollary 7.13, Theorem 7.16]{GHKK} that this map is in fact an injective homomorphism of algebras.\nThese cases already imply that the same is true is $\\cV$ if of the form $\\cA\/T_H$ or $\\cXe$.\n\n\\begin{remark}\n\\label{rem:integral_domain}\n If $\\cV= \\cA$, $\\cX$, $\\cA\/T_{H}$ or $\\cX_{{\\bf 1}}$ then $\\cmid(\\cV)$ is an integral domain. Indeed, $\\cmid(\\cV)$ is a subalgebra of $ \\up(\\cV)=H^0(\\cV,\\mathcal{O}_{\\cV})$ which is a domain as $\\cV$ is irreducible.\n\\end{remark}\n\n\nAs we have seen in the previous subsections theta functions on varieties of the form $ \\cA$ or $\\cA\/T_H$ are naturally labeled by the $\\Z^T$-points of its Fock--Goncharov dual, whereas theta functions on varieties of the form $ \\cX$ or $\\cXe$ are naturally labeled by the $\\Z^t$-points of its Fock--Goncharov dual. \nSince we would like to consider all these cases simultaneously we introduce the following notation. For $G= \\Z, \\Q$ or $\\R$ we set\n\n\\begin{equation}\n\\label{eq:unif}\n \\Trop_G(\\cV):=\n \\begin{cases}\n \\cV(G^t) &\\text{ if } \\cV=\\cA \\text{ or } \\cV=\\cA\/T_H\\vspace{1mm}\\\\\n \\cV(G^T) \\ & \\text{ if } \\cV=\\cX \\text{ or } \\cV=\\cXe.\n \\end{cases}\n\\end{equation}\n\nSimilarly, for a positive rational function $g: \\cV \\dashrightarrow \\Bbbk $ we let\n\\begin{equation}\n\\label{eq:unif_function}\n \\Trop_G(g):=\n \\begin{cases}\n g^t &\\text{ if } \\cV=\\cA \\text{ or } \\cV=\\cA\/T_H\\vspace{1mm}\\\\\n g^T \\ &\\text{ if } \\cV=\\cX \\text{ or } \\cV=\\cXe.\n \\end{cases}\n\\end{equation}\n\nIn particular, if we think of the seed torus $\\cV_\\seed$ as a cluster variety with only frozen directions then $\\Trop_G(\\cV_\\seed)=\\mathfrak{r}_{\\seed}(\\Trop_G(\\cV))=\\cV_{\\seed}(G^t)$, if $\\cV$ is of the form $\\cA$ or $\\cA\/T_H$ and $\\Trop_G(\\cV_\\seed)=\\mathfrak{r}_{\\seed}(\\Trop_G(\\cV))=\\cV_{\\seed}(G^T)$, if $\\cV$ is of the form $\\cX$ or $\\cXe$. For later use we also set\n\\begin{equation}\n \\label{eq:Theta_seed}\n\\Theta(\\cV)_{\\seed^\\vee}:=\\mathfrak{r}_{\\seed^\\vee}(\\Theta(\\cV))\\subset \\Trop_{\\Z}(\\cV^\\vee),\n\\end{equation}\nsee the line just below equation \\eqref{eq:identification}. \nFollowing \\cite{GHKK} we introduce the following definition.\n\\begin{definition}\n\\label{def:full_FG}\nLet $\\cV$ be a scheme of the form $ \\cA$, $\\cX$, $\\cA\/T_{H}$ or $\\cX_{{\\bf 1}}$. We say that {\\bf the full Fock--Goncharov conjecture} holds for $\\cV$ if\n\\begin{itemize}\n \\item $\\Theta(\\cV)=\\Trop_{\\Z}(\\cV^{\\vee})$, and\n \\item the natural map $\\cmid(\\cV) \\to \\text{up}(\\cV)$ is an isomorphism.\n\\end{itemize}\n\\end{definition}\n\n\n\\section{Bases of theta functions for partial minimal models}\n\\label{sec:minimal_models}\n\nIn \\cite{GHKK}, the authors obtained nearly optimal conditions ensuring that the full Fock--Goncharov conjecture holds for a cluster variety. \nHowever, they were able to prove that the ring of regular functions of a partial compactifications of a cluster varieties has a basis of theta functions under much stronger conditions. \nIn this section we outline this framework, including quotients and fibres of cluster varieties, and refer to \\cite[\\S9]{GHKK} for a detailed treatment. \nThe main class of (partial) compactifications we shall consider are the (partial) minimal models defined below.\n\n\\begin{definition}{\\cite{GHK_birational}}\n\\label{def:cv_minimal_model}\nLet $\\cV$ be a scheme of the form $\\cA, \\cX, \\cA\/T_H$ or $\\cXe$. An inclusion $\\cV \\subset Y$ as an open subscheme of a normal variety $Y$ is a {\\bf partial minimal model} of $ \\cV$ if the canonical volume form on $\\cV$ has a simple pole along every irreducible divisor of $Y$ contained in $ Y \\setminus \\cV$. It is a {\\bf minimal model} if $Y$ is, in addition, projective. We call $ Y \\setminus \\cV$ the {\\bf boundary} of $\\cV \\subset Y$.\n\\end{definition}\n\nFor example, if $\\cV$ is a cluster $\\cA$-variety with frozen variables we can let these variables vanish to obtain a partial minimal model of $\\cV$ as in \\cite[Construction B.9]{GHKK}. \nSimilarly, if we consider a torus as a cluster variety (by letting $\\Iuf = \\emptyset$) then a partial minimal model is simply a normal toric variety.\n\n\nGiven a partial minimal model $\\cV\\subset Y$, where $\\cV$ is a scheme of the form $\\cA, \\cX, \\cA\/T_H$ or $\\cXe$, we would like to describe the set of theta functions on $\\cV$ (resp. $\\cV^\\vee$) that extend to $Y$ in a similar way as the ring of algebraic functions on a normal toric variety is described in toric geometry using polyhedral fans. \nIn order to be able to do so we need that the pair $(\\cV, \\cV^\\vee)$ satisfies a technical condition --\\emph{theta reciprocity}-- that we will introduce shortly. \nFor this, we need to discuss first the \\emph{tropical pairings} associated to the pair $(\\cV,\\cV^{\\vee})$.\n \nIn order to define the tropical pairings we temporarily assume that $\\cV$ is a variety of the form $\\cA$ or $\\cA\/T_{H}$ so that $\\cV^\\vee$ is a cluster $\\cX$-variety or a fibre of a cluster $\\cX$-variety, respectively. \nIn particular, $\\Theta(\\cV)\\subset \\cV^\\vee(\\Z^T)= \\Trop_{\\Z}(\\cV^\\vee)$ and $\\Theta(\\cV^\\vee)\\subset \\cV(\\Z^t)=\\Trop_{\\Z}(\\cV)$, see \\eqref{eq:unif}. \nRecall from Remark~\\ref{rmk:geometric trop} that the set $\\cV(\\Z^t)$ (resp. $\\cV^\\vee(\\Z^t)$) is canonically identified with the geometric tropicalization\n$\\cV^\\trop(\\Z)$ (resp. $(\\cV^\\vee)^\\trop(\\Z)$). \nTherefore, we systematically think of the elements of $\\cV(\\Z^t)$ (resp. $\\cV^\\vee(\\Z^t)$) as divisorial discrete\nvaluations on $\\Bbbk(\\cV)$ (resp. $\\Bbbk(\\cV^\\vee)$).\nWe also consider the bijection $i : \\cV^\\vee(\\Z^T) \\to \\cV^\\vee(\\Z^t )$ introduced in \\S\\ref{ss:tropicalization} (see the comment bellow \\eqref{eq:imap}).\nThe {\\bf tropical pairings} associated to the pair $(\\cV,\\cV^\\vee) $ are the functions $\n \\langle \\cdot , \\cdot \\rangle : \\Theta(\\cV^{\\vee}) \\times \\Theta (\\cV) \\to \\Z $ and $ \\langle \\cdot , \\cdot \\rangle^{\\vee} : \\Theta(\\cV^{\\vee}) \\times \\Theta (\\cV) \\to \\Z$ given by\n\\[\n \\langle {\\bf v} , {\\bf b} \\rangle = {\\bf v}(\\tf^{\\cV}_{\\bf b}) \\ \\ \\ \\ \\ \\ \\ \\text{and} \\ \\ \\ \\ \\ \\ \\ \\langle {\\bf v} , {\\bf b} \\rangle^{\\vee} = i({\\bf b}) (\\tf^{\\cV^{\\vee}}_{\\bf v}),\n\\]\n\n\n\\begin{definition}\n\\label{def:theta_reciprocity}\n Let $\\cV$ be a scheme of the form $\\cA, \\cX, \\cA\/T_H$ or $\\cXe$. The pair $(\\cV,\\cV^\\vee)$ has {\\bf theta reciprocity} if $\\Theta(\\cV)=\\Trop_{\\Z}(\\cV^\\vee)$, $\\Theta(\\cV^{\\vee})=\\Trop_{\\Z}(\\cV)$, and $ \\langle {\\bf v} , {\\bf b} \\rangle = \\langle {\\bf v} , {\\bf b} \\rangle^{\\vee} $ for all $({\\bf v},{\\bf b})\\in \\Trop_{\\Z}(\\cV) \\times \\Trop_{\\Z}(\\cV^\\vee)$.\n\\end{definition}\n\\begin{remark}\n Definition \\ref{def:theta_reciprocity} shall not be considered artificial. In fact, an analogous conjecture for affine log Calabi--Yau varieties with maximal boundary is expected to hold true, see \\cite[Remark 9.11]{GHKK}.\n\\end{remark}\n\n\n\\begin{lemma}\n\\label{lem:tf_that_extend}\n Let $\\cV$ be a scheme of the form $\\cA, \\cX, \\cA\/T_H$ or $\\cXe$ and let $\\cV\\subset Y$ be a (partial) minimal model. Suppose that the pair $(\\cV,\\cV^\\vee)$ has theta reciprocity.\n Then for every seed $\\seed\\in \\orT$ the set of theta functions on $\\cV$ that extend to $Y$ can be described as the intersection of $\\Theta(\\cV^\\vee)_{\\seed^\\vee}$ (see \\eqref{eq:Theta_seed}) with a polyhedral cone of the vector space $\\Trop_{\\R}(\\cV^{\\vee}_{\\seed^\\vee})$ (see the sentence bellow equation \\eqref{eq:unif}).\n\\end{lemma}\n\\begin{proof}\nWe treat the cases $\\cV= \\cA$ or $\\cA\/T_H$ as the proof is completely analogous for the cases $\\cV= \\cX$ or $\\cXe$.\nLet $D_1, \\dots, D_s$ be the irreducible divisors of $Y$ contained in the boundary of $\\cV \\subset Y $. \nSince $Y$ is normal, to describe the theta functions on $\\cV$ that extend to $Y$ it is enough to describe the set of theta functions that extend to $D_1, \\dots , D_s$ since $Y\\setminus (\\cV \\cup D_1, \\dots , D_s)$ has co-dimension greater or equal to $2$ in $Y$.\nLet $\\ord_{D_j}$ be the discrete valuation on $ \\Bbbk(\\cV)\\setminus \\{ 0 \\}$ associated to the irreducible divisor $D_j$.\nSince $\\cV \\subset Y$ is a partial minimal model, $\\ord_{D_j}$ determines a point of $ \\cV(\\Z^t) $. Since $\\Theta(\\cV^{\\vee})= \\cV(\\Z^t)$ we have $\\ord_{D_j} \\in \\Theta (\\cV^{\\vee})$. Therefore, $\n\\tf^{\\cV^{\\vee}}_{\\ord_{D_j}}$ is a polynomial theta function and its tropicalization is the function\n\\[\n(\\tf_{\\ord_{D_j}}^{\\cV^\\vee})^t:\\cV^{\\vee}( \\Z^t)\\to \\Z\\quad \\text{given by} \\quad v \\mapsto v (\\tf^{\\cV^{\\vee}}_{\\ord_{D_j}}).\n\\]\nIn other words, $(\\tf_{\\ord_{D_j}}^{\\cV^{\\vee}})^t(v)=\\langle \\ord_{D_j}, i(v) \\rangle$. \nSince $\\Theta(\\cV)= \\cV^\\vee(\\Z^T)$ we have that $i(v)\\in \\Theta(\\cV)$ and, therefore, $\\tf^\\cV_{i(v)}$ is a polynomial theta function.\nThe assumption $ \\langle{\\bf v} , {\\bf b} \\rangle = \\langle {\\bf v} , {\\bf b} \\rangle^{\\vee} $ for all ${\\bf v}$ and ${\\bf b}$ implies that\n\\[\n(\\tf_{\\ord_{D_j}}^{\\cV^{\\vee}})^t(v)= (\\tf^{\\cV}_{i(v)})^t(\\ord_{D_j}),\n\\]\nsince\n\\[\n(\\tf_{\\ord_{D_j}}^{\\cV^{\\vee}})^t(v) =\n\\langle \\ord_{D_j}, i(v) \\rangle =\n\\langle \\ord_{D_j}, i(v)\\rangle^{\\vee} =\n\\ord_{D_j}(\\tf^{\\cV}_{i(v)}) =\n(\\tf^{\\cV}_{i(v)})^t(\\ord_{D_j}).\n\\]\nThus a theta function $\\tf^{\\cV}_{i(v)} \\in \\cmid(\\cV)$ extends to $D_j$ if and only if $0\\leq (\\tf^{\\cV^{\\vee}}_{\\ord_{D_j}})^t(v)$. \nIn particular, a theta function $\\tf^\\cV_{i(v)}$ extends to $Y$ if and only if\n\\[\ni(v)\\in \\bigcap_{i=1}^s\\{b\\in\\cV^{\\vee}_{\\seed^\\vee}(\\R^T)\\mid 0\\leq (\\tf^{\\cV^{\\vee}}_{\\ord_{D_j}})^T(b)\\}\n\\]\nsince $g^T(b)=g^t(i(b))$ for every positive function $g$ on $\\cV$, see \\eqref{eq:comparing_tropicalizations}. By definition of tropicalization, the set $\\bigcap_{i=1}^s\\{b\\in\\cV^{\\vee}_{\\seed^\\vee}(\\R^T)\\mid 0\\leq (\\tf^{\\cV^{\\vee}}_{\\ord_{D_j}})^T(b)\\}$ is a polyhedral cone of $\\cV^{\\vee}_{\\seed^\\vee}(\\R^T)=\\Trop_{\\R}(\\cV^{\\vee}_{\\seed^\\vee})$. \n\\end{proof}\n\nWe now turn to the problem of understanding when the theta functions on $\\cV$ that extend to a (partial) minimal model $\\cV \\subset Y$ form a basis of $H^0(Y, \\mathcal{O}_Y)$. \nThe following notion is central.\n\\begin{definition}\n \\label{def:respect_order}\n Let $\\cV$ be a scheme of the form $\\cA, \\cX, \\cA\/T_H$ or $\\cXe$. We say that the theta functions on $\\cV$ {\\bf respect the order of vanishing} if\n for all ${\\bf v}\\in \\cV(\\Z^t)$ and $\\displaystyle \\sum_{{\\bf q}\\in \\Theta(\\cV)} \\alpha_{\\bf q} \\tf^{\\cV}_{\\bf q}\\in \\cmid(\\cV)$ then \n\\[\n{\\bf v}\\lrp{\\sum_{{\\bf q}\\in \\Theta(\\cV)} \\alpha_{\\bf q} \\tf^{\\cV}_{\\bf q}} \\geq 0 \\ \\ \\text{ if and only if }\\ \\ {\\bf v}(\\tf_{\\bf q})\\geq 0 \\text{ for all } {\\bf q} \\text{ such that } \\alpha_{\\bf q}\\neq 0.\n\\]\n\\end{definition}\n\nNotice that in \\cite[Conjecture 9.8]{GHKK} the authors conjecture that the theta functions on $\\cAp$ respect the order of vanishing.\nThe {\\bf superpotential} associated to a partial minimal model $\\cV \\subset Y $ is the function on $\\cV^\\vee$ defined as\n\\begin{equation}\\label{eq:def superpotential}\n W_{Y}:=\\sum_{j=1}^n \\tf^{\\cV^{\\vee}}_{j},\n\\end{equation}\nwhere \n\\begin{equation}\n\\label{eq:def superpotential_summands}\n \\tf^{\\cV^{\\vee}}_{j}=\\begin{cases}\n \\tf^{\\cV^{\\vee}}_{\\ord_{D_j}} &\\text{ if } \\cV=\\cA \\text{ or } \\cV=\\cA\/T_H\\vspace{1mm}\\\\\n \\tf^{\\cV^{\\vee}}_{i(\\ord_{D_j})} \\ &\\text{ if } \\cV=\\cX \\text{ or } \\cV=\\cXe.\n \\end{cases}\n\\end{equation}\nThe {\\bf superpotential cone} associated to $W_Y$ is\n\\begin{equation}\\label{eq:def Xi}\n \\Xi_Y:= \\{ {\\bf v} \\in \\Trop_{\\R}(\\cV^\\vee) \\mid \\Trop_{\\R}(W_Y)({\\bf v})\\geq0 \\},\n\\end{equation} \nsee equation \\eqref{eq:unif_function}.\n\nWe further set $\\Xi_{Y;\\seed^\\vee}:= \\mathfrak{r}_{\\seed^\\vee}(\\Xi_Y)\\subset \\Trop_{\\R}(\\cV^{\\vee}_{\\seed^\\vee})$.\nNotice that if the theta functions on $\\cV$ respect the order of vanishing then $\\Xi_{Y;\\seed}$ is precisely the polyhedral subset of Lemma \\ref{lem:tf_that_extend}.\nThe next results follows at once from the definitions.\n\n\\begin{lemma}\n\\label{lem:basis_for_pmm}\n Let $\\cV$ be a scheme of the form $\\cA, \\cX, \\cA\/T_H$ or $\\cXe$ and let $\\cV\\subset Y$ be a (partial) minimal model. Suppose that the full Fock--Goncharov conjecture holds for $\\cV$, that the pair $(\\cV, \\cV^\\vee) $ has theta reciprocity and that the theta functions on $\\cV$ respect the order of vanishing. \n Then the set of theta functions on $\\cV$ parametrized by the points of $\\Xi_Y(\\Z)$ is a basis of $H^0(Y, \\mathcal{O}(Y))$.\n\\end{lemma}\n\n\\begin{lemma}\nSuppose there is a cluster ensemble map $p:\\cA \\to \\cX$ that is an isomorphism. Then theta functions on $\\cA$ respect the order of vanishing if and only theta functions on $\\cX$ respect the order of vanishing.\n\\end{lemma}\n\\begin{proof}\n The result follows at once from the fact that $p^*(\\tf^{\\cX}_{\\bf n})= \\tf^{\\cA}_{(p^{\\vee})^T\\circ i ({\\bf n})}$. \n\\end{proof}\n\n\nWe propose the following definition that allows to have the benefits of Lemma \\ref{lem:basis_for_pmm} without having to verify all its assumptions. \nWe apply this in \\S\\ref{sec:NO_Grass}.\n\n\\begin{definition}\n\\label{def:enough_tf}\nWe say that $ \\cV \\subset Y$ has {\\bf enough theta functions} if the full Fock--Goncharov conjecture holds for $\\cV$ and the theta functions on $\\cV$ parametrized by $\\Xi_{Y} (\\Z)$ form a basis of $H^0(Y, \\mathcal{O}_Y)$.\n\\end{definition}\n\nWe now recall an important notion introduced in \\cite[Definition 9.1]{GHKK} that can be used to verify in a combinatorial way that a partial minimal model $\\cA\\subset Y$ has enough theta functions provided $Y$ is obtained by letting the frozen variables vanish.\n\n\\begin{definition}\nWe say that a seed $\\seed=(e_i)_{ i \\in I}$ is {\\bf optimized} for a point $ {\\bf n} \\in \\cA(\\Z^t) $ if under the identification of $\\cA(\\Z^t)$ with $N^\\circ$ afforded by $\\seed$ we have that $\\{ e_k, n_{\\seed} \\}\\geq 0 $ for all $k \\in \\Iuf$.\n\\end{definition}\n\n\\begin{lemma}\n\\thlabel{lemm:enough_tf}\nAssume that $\\cA$ satisfies the full Fock--Goncharov conjecture. Let $\\cA\\subset Y$ be a partial minimal model of $\\cA$ and let $D_1, \\dots , D_s$ be the irreducible divisors of $Y$ contained in $Y\\setminus \\cA$. Assume that $p^*_2|_{N^{\\circ}}: N^{\\circ}\\to \\Nuf^*$ is surjective and that the point $\\ord_{D_j}\\in \\cA^{\\vee}(\\Z^t)$ has an optimized seed for every $1 \\leq j \\leq s$. Then the partial minimal model $\\cA \\subset Y$ has enough theta functions.\n\\end{lemma}\n\\begin{proof}\nSince $p^*_2|_{N^{\\circ}}$ is surjective we have that $\\cAp$ is isomorphic to $\\cA\\times T_M $ (see \\cite[Lemma B.7]{GHKK}). \nConsider the partial compactification $\\cAp \\subset Y \\times T_M$. Its boundary is isomorphic to $D\\times T_M$ and the irreducible components of the boundary are the divisors $\\widetilde{D}_1, \\dots, \\widetilde{D}_s$, where $\\widetilde{D}_j:=D_j \\times T_M $. \nBy hypothesis $\\ord_{D_j} $ is optimized for some seed $\\seed_j$. \nLet $\\widetilde{\\seed}_j$ be the seed for $\\Gamma_{\\prin}$ obtained mutating $ \\seed_{0_{\\prin}}$ in the same sequence of directions needed to obtain $\\seed_j$ from $\\seed_0$.\nObserve that for every $1\\leq j \\leq s$, under the identifications \n\\[\n\\cA_{\\prin,\\widetilde{\\seed}_j}(\\Z^t) = N_\\prin^{\\circ} = \\cA_{\\seed_j}(\\Z^t) \\oplus T_M(\\Z^t), \n\\]\nthe point $ \\ord_{\\widetilde{D}_j}$ of $\\cAp(\\Z^t)$ corresponds to the point $ (\\ord_{D_j},0)$ of $\\cA(\\Z^t)\\times T_M(\\Z^t)$.\n\nRecall that the index set of unfrozen indices for $\\cAp$ is $\\Iuf$. \nIn particular, for every $k \\in \\Iuf$ we have that the $k^{\\text{th}}$ element of $\\widetilde{\\seed}_{j}$ is of the form $( e_{k;j},0)$, where $e_{k;j}$ is the $k^{\\text{th}}$ element of $\\seed_j$. Then for each $1\\leq j\\leq s$ we compute\n\\begin{align*}\n\\{ (e_{k;j},0), \\ord_{\\widetilde{D}_j}\\} & = \\{ (e_{k;j},0), (\\ord_{D_j},0)\\} \\\\\n& = \\{e_{k;j}, \\ord_{D_j} \\} \\geq 0.\n\\end{align*}\n\nThis tells us that $\\ord_{\\widetilde{D}_j}$ is optimized for $\\widetilde{\\seed}_j$.\nLet $W_{Y\\times T_M}=\\sum_{j}^{s}\\tf^{\\cAp^\\vee}_{\\ord{\\widetilde{D}_j}}$ be the superpotential associated to $\\cAp \\subset Y \\times T_M$.\nBy Proposition 9.7 and Lemma 9.10 (3) of \\cite{GHKK} the integral points of $\\Xi_{Y \\times T_M}$ can be described as\n\\[\n\\Xi_{Y \\times T_M}\\cap (\\Z) = \\{ b \\in \\Theta(\\cAp) \\mid \\ord_{i(b)} (\\tf^{\\cAp^{\\vee}}_j)\\geq 0 \\text{ for all } j\\}.\n\\] \nWe define $\\cmid(Y\\times T_M)$ to be the vector subspace of $\\cmid (\\cAp)$ spanned by the theta functions parametrized by $\\Xi_{Y \\times T_M}(\\Z^T)$. \nFor the convenience of the reader we point out that in the notation of \\cite[\\S9]{GHKK} the partial compactification $Y\\times T_M$ of $\\cAp$ would be denoted by $\\overline{\\cA}_{\\text{prin}}^{S}$ and $\\Xi_{Y \\times T_M}(\\Z)$ by $\\Theta(\\overline{\\cA}_{\\text{prin}}^{S})$, where $S:=\\{ i(\\ord_{\\widetilde{D}_1}),\\dots , i(\\ord_{\\widetilde{D}_s})\\}$.\nBy \\cite[Lemma 9.10(2)]{GHKK} we have\n\\[\\cmid(Y\\times T_M)=H^0(Y\\times T_M, \\mathcal{O}_{Y\\times T_M}) \\cong H^0(Y, \\mathcal{O}_{Y})\\otimes_{\\Bbbk} H^0( T_M, \\mathcal{O}_{ T_M}).\n\\]\nIn particular, $H^0(Y\\times T_M, \\mathcal{O}_{Y\\times T_M})$ has a theta basis parametrized by $\\Theta(Y\\times T_M)$.\nThe theta function $ \\tf^{\\cA}_{\\ord_{D_j}}$ is obtained from $\\tf^{\\cAp}_{\\widetilde{D}_j}$ by specializing the coefficients to $1$. This implies that\n\\[\n\\Xi_{Y}\\cap \\Trop_{\\Z}(\\cA^{\\vee})= \\Xi_{Y \\times T_M} \\cap \\Trop_{\\Z}(\\cA^{\\vee}).\n\\]\nWe conclude that $ H^0(Y, \\mathcal{O}_Y)$ has a theta basis parametrized by the integral point of $\\Xi_{Y}$.\n\n\\end{proof}\n\n\\section{Valuations on middle cluster algebras and adapted bases}\n\\label{sec:cluster_valuations}\n In \\cite{FO20} the authors noticed that the so-called {\\bf g}-vectors associated to cluster variables can be used to construct valuations on $\\Bbbk(\\cA)$ provided $\\Gamma$ is of full-rank. In this section we study some properties of these valuations. We extend this approach for quotients of $\\cA$ and (fibres of) $\\cX$.\n\n Let $\\cV$ be a scheme of the form $\\cA, \\cX, \\cA\/T_H$ or $\\cXe$. \n Recall from \\S\\ref{sec:tf_and_parametrizations} that every theta function on $\\cV$ is labeled with a point of $\\Trop_{\\Z}(\\cV^\\vee)$, see \\eqref{eq:unif}. \n\n\\begin{definition}\n\\label{def:dom_order}\nSuppose $\\Gamma$ is of full-rank and let $ \\seed \\in \\orT$ be a seed for $\\Gamma$. \nThe {\\bf opposite dominance order} on $M^\\circ$ defined by $\\seed$ is the partial order $\\preceq_{\\seed}$ on $M^\\circ$ determined by the following condition:\n\\begin{equation}\n\\label{eq:dom_order}\nm_1 \\prec_{\\seed} m_2 \\ \\Leftrightarrow \\ m_2= m_1 + p^{\\ast}_1(n) \\text{ for some }n\\in N^+_{\\uf, \\seed}.\n\\end{equation}\n\\end{definition}\n\n\\begin{remark}\n\\label{rem:dom_order}\nIn Definition \\ref{def:dom_order}, $m_1\\preceq_{\\seed} m_2$ means that either $m_1 \\prec_\\seed m_2 $ or $m_1=m_2$. We will also adopt this notation for other orders we consider.\nThe dominance order was originally considered in \\cite[Proof of Proposition 4.3]{Labardini_et_al_CC-alg} and it is the opposite order to the one given in Definition \\ref{def:dom_order}. \nThis order was exploited by \\cite{Qin17,Qintropical} in his work on bases for cluster algebras.\nThe full-rank condition is needed so that $\\preceq_{\\seed}$ is reflexive. However, observe that for every seed $\\seed$ such that $\\text{ker}(p_1^*)\\cap N^+_{\\uf , \\seed} = \\emptyset$, equation \\eqref{eq:dom_order} still determines a partial order on $M^\\circ$ even if $\\Gamma$ is not of full-rank. \nNonetheless, whenever we talk about an (opposite) dominance order in this paper we will be tacitly assuming that $\\Gamma$ is of full-rank.\n\\end{remark}\n\nIt is straightforward to verify that $\\preceq_{\\seed}$ is {\\bf linear}. That is, $ m_1 \\preceq_\\seed m_2 $ implies that $m_1 + m \\preceq_\\seed m_2 + m$ for all $m \\in M^\\circ$.\n\n\n\\begin{definition}\n\\label{def:val}\nLet $A$ be an integral domain with a $\\Bbbk$-algebra structure, $L$ a lattice isomorphic to $\\Z^r$ and $\\leq$ a total order on $ L$. A {\\bf valuation} on $A$ with values in $L$ is a function $\\nu : A\\setminus \\{0 \\} \\to (L,<)$ such that \n\\begin{itemize}\n \\item[(1)] $\\nu(f+g) \\geq \\min\\{\\nu(f), \\nu(g)\\}$, unless $f+g=0$,\n \\item[(2)] $\\nu(fg)= \\nu(f) + \\nu(g)$,\n \\item[(3)] $\\nu(cf)=\\nu(f)$ for all $c \\in \\Bbbk^* $.\n\\end{itemize}\nFor $l \\in L$ we define the subspace\n$\nA_{\\nu \\geq l}:= \\{ x\\in A \\setminus \\{0\\} \\mid \\nu(x)\\geq l\\} \\cup \\{ 0 \\}\n$\nof $A$. The subspace $\nA_{\\nu > l}$ is defined analogously.\nWe say that $\\nu $ has {\\bf 1-dimensional leaves} if the dimension of the quotient\n\\begin{equation}\n\\label{eq:graded_piece}\nA_l:=A_{\\nu \\geq l} \\big{\/} A_{\\nu > l} \n\\end{equation}\nis either $0$ or $1$ for all $l\\in L$. A basis $B$ of $A$ is {\\bf adapted} for $ \\nu $ if for all $l\\in L$ the set $B\\cap A_{\\nu \\geq l}$ is a basis of $A_{\\nu\\geq l}$.\n\\end{definition}\n\n\\begin{lemma}\n\\thlabel{product_and_order}\n\nAssume $\\Gamma $ is of full-rank. \nLet $\\vartheta^{\\cA}_{m_1},\\vartheta^{\\cA}_{m_2}\\in \\text{mid}(\\cA)$ with $m_1,m_2\\in =\\Trop_{\\Z}(\\cA^{\\vee}_{\\seed^\\vee})=M^\\circ$. \nThen the product $\\vartheta^{\\cA}_{m_1}\\vartheta^{\\cA}_{m_2}$ expressed in the theta basis of $\\text{mid}(\\cA)$ has the following form\n\\[\n\\vartheta^{\\cA}_{m_1}\\vartheta^{\\cA}_{m_2}= \\vartheta^{\\cA}_{m_1+m_2}+ \\sum_{m_1+m_2 \\prec_{\\seed} m}c_{m}\\vartheta^{\\cA}_{m}.\n\\]\n\\end{lemma}\n\n\n\\begin{proof}\nFirst notice that for any broken line $\\gamma $ we have that\n\\begin{equation*}\nF(\\gamma)=I(\\gamma) +a_{1}p^*_1(n_{1})+ \\dots + a_{r}p^*_1(n_{r}), \n\\end{equation*}\nwhere $a_1, \\dots , a_r $ are non-negative integers and $n_1, \\dots , n_r \\in N^+_{\\uf, \\seed}$. \nThis follows from \\cite[Theorem 1.13]{GHKK} and the bending rule of broken lines (\\ie \\thref{def:genbroken}(4)). \nIn particular, we have that $a_{1}n_{1}+ \\dots + a_{r}n_{r}\\in N^+_{\\uf, \\seed} \\cup \\{ 0 \\} $. \nMoreover, \n$a_{1}p^*_1(n_{1})+ \\dots + a_{r}p^*_1(n_{r}) = 0$ if and only if $a_1=\\cdots = a_r =0$.\nTherefore, $I(\\gamma) \\preceq_{\\seed} F(\\gamma)$ and $I(\\gamma)=F(\\gamma)$ if and only if $\\gamma$ does not bend at all.\n\nThe statement we want to prove already follows from the observations made above.\nIndeed, by \\cite[Definition-Lemma 6.2]{GHKK} we know that $ \\alpha(m_1,m_2,m)\\neq 0$ if and only if there exist broken lines $\\gamma_1$ and $\\gamma_2$ such that \n$I(\\gamma_i)=m_i$ for $i \\in \\{1,2\\}$ and $F(\\gamma_1)+F(\\gamma_2)=m=\\gamma_1(0)=\\gamma_2(0)$.\nTherefore, if $ \\alpha(m_1,m_2,m)\\neq 0$ then $ m_1 + m_2=I(\\gamma_1) + I(\\gamma_2) \\preceq_{\\seed} m $. \nMoreover, the equality\n$ m_1+ m_2=m$\nholds if and only if both $\\gamma_1$ and $\\gamma_2$ do not bend at all. \nThis latter case can be realized in a unique way, therefore, $\\alpha(m_1,m_2,m_1+m_2)=1$. \n\\end{proof}\n\nFrom now on the symbol $\\leq_{\\seed} $ is used to denote a total order on $M^\\circ$ refining $\\preceq_{\\seed}$.\n\n\\begin{definition}\nLet ${\\bf m}=(m_{\\seed^\\vee})\\in \\Trop_{\\Z}(\\cA^{\\vee})$. The {\\bf g-vector of} $ \\tf^{\\cA}_{\\bf m}$ {\\bf with respect to} $\\seed $ is\n\\begin{equation}\n\\label{eq:red-g-val-A}\n{\\bf g}_{\\seed}\\left(\\tf^{\\cA}_{\\bf m}\\right):= m_{\\seed^\\vee}\n\\in \\Trop_{\\Z}(\\cA^{\\vee}_{\\seed^\\vee}).\n\\end{equation}\n\\end{definition}\n\n\\begin{definition}\n\\thlabel{g_valuation_A}\nAssume $\\Gamma$ is of full-rank and think of $M^{\\circ}$ as $\\Trop_{\\Z}(\\cA^{\\vee}_{\\seed^\\vee})$. Let $\\gv_{\\seed}:\\cmid(\\cA) \\setminus \\{ 0\\} \\to (M^{\\circ},\\leq_{\\seed})$ be the map given by \n\\begin{equation}\n\\label{eq:g_val}\n \\gv_{\\seed}(f):= \\min{}_{\\leq_{\\seed}}\\{m_1, \\dots , m_t\\},\n\\end{equation}\nwhere $f=c_1\\vartheta^{\\cA}_{m_1} + \\dots + c_t\\vartheta^{\\cA}_{m_t}$, $m_j\\in M^\\circ$ and $c_j\\not=0$ for all $j=1,\\dots,t$ is the expression of $f$ in the theta basis of $\\text{mid}(\\cA)$.\n\\end{definition}\n\n\n\n\n\\begin{lemma}\n\\thlabel{g_is_val}\nFor every seed $\\seed$ the map $\\gv_{\\seed} $ is a valuation on $\\cmid(\\cA)$ with 1-dimensional leaves and the theta basis $\\{ \\tf_{m} \\mid m\\in \\Theta (\\cA) \\}$ is adapted for $\\gv_{\\seed} $.\n\\end{lemma}\n\\begin{proof}\nThis statement follows from \\cite[Remark 2.30]{KM_Khovanskii_bases} but for the convenience of the reader we give a proof here. Items (1) and (3) of Definition~\\ref{def:val} follow directly from the definition of $\\gv_{\\seed}$. \nFor item (2) consider the expressions $f=\\sum_{i=1}^r c_i\\vartheta^{\\cA}_{m_i}$ and $g=\\sum_{j=1}^s c'_j\\vartheta^{\\cA}_{m'_j}$ where all $c_i$ and $c'_j$ are non-zero.\nThen by \\thref{product_and_order}\n\\begin{eqnarray}\\label{eq:fg in basis}\nfg=\\sum_{i,j} c_ic'_j\\left(\\vartheta^{\\cA}_{m_i+m'_j} + \\sum_{m_i+m'_j\\prec_{\\seed} m}c_{m}\\vartheta^{\\cA}_{m}\\right).\n\\end{eqnarray}\nBy definition of $\\gv_{\\seed}$ we have $m_\\mu:=\\gv_{\\seed}(f)\\prec_{\\seed} m_i$ for all $i\\in \\{1,\\dots, r\\} \\setminus \\{ \\mu \\}$ and $m'_\\nu:=\\gv_{\\seed}(g)\\prec_{\\seed} m'_j$ for all $j\\in \\{1,\\dots,s\\}\\setminus \\{\\nu\\}$. \nWe need to show that the term $\\vartheta_{m_\\mu+m'_\\nu}$ appears with non-zero coefficient in $fg$.\nAssume there exist $i\\not =\\mu$ and $j\\not=\\nu$ such that $m_\\mu+m'_\\nu=m_i+m'_j$. \nThen as $\\prec_{\\seed}$ is linear we have\n\\[\nm_\\mu +m'_\\nu \\prec_{\\seed} m_\\mu + m'_j \\prec_{\\seed} m_i + m'_j, \n\\]\na contradiction. \nHence, the term $\\vartheta_{m_\\mu+m'_\\nu}$ appears in the expression \\eqref{eq:fg in basis} of $fg$ with coefficient $c_\\mu c'_\\nu\\not =0$ and $\\gv_{\\seed}(fg)=m_\\mu+m'_\\nu=\\gv_{\\seed}(f)+\\gv_{\\seed}(g)$.\n\nThe fact that ${\\bf g}_{\\seed}$ has one dimension leaves follows directly from (\\ref{eq:g_val}). It is also clear from the definitions that for $m\\in M^{\\circ}$ the subspace $\\cmid(\\cA)_{m} $ as in (\\ref{eq:graded_piece}) is isomorphic to $\\Bbbk\\cdot \\tf^{\\cA}_{m}$ if $m \\in \\Theta(\\cA)$ and $0$-dimensional otherwise. \nIn particular, the fact that we have a bijection between the set of values of $\\gv_\\seed$ and the elements of the theta basis is equivalent to the theta basis being an adapted basis, see \\cite[Remark 2.30]{KM_Khovanskii_bases} .\n\\end{proof}\n\n\n\n\\begin{corollary}\nThe image of the valuation ${\\bf g}_{\\seed}$ is independent of the linear refinement $\\leq_{\\seed}$ of $\\preceq_{\\seed}$.\n\\end{corollary}\n\\begin{proof}\nSince the theta basis is adapted for ${\\bf g}_{\\seed}$ we have \n\\[\n{\\bf g}_{\\seed}\\lrp{\\cmid(\\cA)\\setminus \\{0\\}}= {\\bf g}_{\\seed}\\lrp{\\Theta(\\cA)}.\n\\]\nThe result follows.\n\\end{proof}\n\n\\begin{remark}\n\\thlabel{g-val-field}\nSince $\\cmid(\\cA)$ is a domain (see Remark \\ref{rem:integral_domain}) whose associated field of fractions is isomorphic to $\\Bbbk(A_i :i \\in I)$, we can extend the valuation $ {\\bf g}_{\\vb s} $ on $\\text{mid}(\\cA)$ to a valuation on $\\Bbbk(A_i :i \\in I)$ by declaring ${\\bf g}_{\\vb s} (f\/g):={\\bf g}_{\\vb s} (f)- {\\bf g}_{\\vb s} (g) $.\n\\end{remark}\n\nThe valuation ${\\bf g}_{\\seed} $ is called the {\\bf {\\bf g}-vector valuation associated to $\\seed$}. \n%Shortly we will discuss {\\bf g}-vector valuations for quotients of $\\cA$ so we will write ${\\bf g}^{\\cA}_{\\seed} $ to stress that this valuation is defined on $\\Bbbk(\\cA)$.\n\n\nWe now turn our attention to quotients of $\\cA $. We keep the assumption that $\\Gamma$ is of full-rank and consider a saturated sublattice $H=H_{\\cA}$ of $K^\\circ$. \nRecall from \\S\\ref{tf_quotient} that \n\\[\n\\Trop_{\\Z}((\\cA\/T_H)^{\\vee}_{\\seed^\\vee})= H^{\\perp}.\n\\]\nSince $ \\Theta(\\cA\/T_{H})_{\\seed^\\vee}\\subset H^ \\perp$, we can restrict restrict the total order $\\leq_{\\seed}$ on $M^{\\circ}$ to $H^{\\perp}$ to obtain a {\\bf g}-vector valuation on $\\cmid(\\cA\/T_{H})$ associated to $\\seed$ as in the previous cases:\n\\[\n{\\bf g}_{\\seed}: \\cmid(\\cA\/T_H)\\setminus\\{0\\} \\to \\Trop_{\\Z}((\\cA\/T_H)^{\\vee}_{\\seed^\\vee}).\n\\]\n\n\\begin{remark}\n\\label{rem:g_val_quotient}\n As opposed to the case of $\\cA$, in general the field of fractions of $\\cmid(\\cA\/T_H)$ might not be isomorphic to $\\Bbbk(\\cA\/T_H)$. \n This fails for example if the smallest cone in $\\Trop_{\\R}((\\cA\/T_H)^{\\vee}_{\\seed^\\vee})$ containing $\\Theta(\\cA\/T_H)_{\\seed^\\vee}$ is not full-dimensional.\n However, the field of fractions of $\\cmid(\\cA\/T_H)$ is isomorphic to $\\Bbbk(\\cA\/T_H)$ provided $\\cA\/T_H$ satisfies the full Fock--Goncharov conjecture. \n In such a case, a {\\bf g}-vector valuation on $\\cmid(\\cA\/T_H)$ can be extended to $\\Bbbk(\\cA\/T_H)$ as in \\thref{g-val-field}.\n\\end{remark}\n\nWe now treat the case of $\\cX$. So fix a cluster ensemble lattice map $p^*:N \\to M^{\\circ} $ and a seed $\\seed $. \nConsider the identifications $\\Trop_{\\Z}(\\cX^\\vee_{\\seed})= d\\cdot N$ and $\\Trop_{\\Z}(\\cA^{\\vee}_{\\prin,\\widetilde{\\seed}^\\vee}) = M^{\\circ}_{\\prin}=M^{\\circ}\\oplus N$ where $\\widetilde{\\seed}$ is the seed for $\\Gamma_\\prin$ obtained mutating $\\seed_{0_\\prin}$ in the same sequence of directions needed to obtain $\\seed$ from $\\seed_0$. \nRecall from \\S\\ref{sec:tf_X} that we have an inclusion $\\Trop_{\\Z}(\\cX^\\vee_{\\seed})\\to \\Trop_{\\Z}(\\cA^{\\vee}_{\\prin,\\widetilde{\\seed}^\\vee})$ given by $dn \\mapsto (p^*(n),n)$.\n\n\\begin{definition}\nLet ${\\bf n}=(dn_{\\seed^\\vee})\\in \\Trop_{\\Z}(\\cX^{\\vee})$. The {\\bf c-vector of} $ \\tf^{\\cX}_{\\bf n}$ with respect to $\\seed $ is\n\\begin{equation}\n\\label{eq:red-g-val-X}\n{\\bf c}_{\\seed}\\left(\\tf^{\\cX}_{\\bf n}\\right):= dn_{\\seed^\\vee}\n\\in \\Trop_{\\Z}(\\cX^{\\vee}_{\\seed^\\vee}).\n\\end{equation}\n\\end{definition}\n\n\\begin{remark}\nObserve that $\\cv_\\seed (\\tf^{\\cX}_{\\bf n})$ is an element of $d\\cdot N$. In practice we could work with the lattice $N$ as opposed to $d\\cdot N$ as they are canonically isomorphic. \nThe lattice $N$ is the set where the ${\\bf c}$-vectors (in the sense of \\cite{NZ}) live. \n\\end{remark}\n\n\\begin{definition}\nThe {\\bf divisibility order} on $ N$ determined by $\\seed$ is the partial order $\\preceq_{\\seed, \\text{div}}$ given by\n\\[\nn_1 \\preceq_{\\seed, \\text{div}} n_2 \\text{ if and only if } \nn_2- n_1 \\in N_{\\seed}^+.\n\\]\n\\end{definition}\n\n\\begin{lemma}\n\\thlabel{lem:restriction}\nThe restriction of $\\preceq_{\\widetilde{\\seed}^\\vee}$ to the $N$ component of $ M^\\circ_{\\prin}$ coincides with the divisibility order $\\prec_{\\seed,\\text{div}}$ on $N$.\n\\end{lemma}\n\\begin{proof}\nLet $p^*_{\\prin,1}:N_{\\uf, \\prin}\\to M^\\circ_\\prin$ be the given by $(n,m)\\mapsto \\{ (n,m), \\cdot \\}_{\\prin}$ (in other words, $p^*_{\\prin,1}$ corresponds to the map $p_1^*$ in \\eqref{eq:p12star} for $\\Gamma_{\\prin}$). \nIn particular, $p^*_{\\prin,1} (n,0) = (p^*_1(n), n) $.\nLet $n_1,n_2 \\in N$ be distinct elements such that $n_2-n_1 \\in N^+_\\seed$. Let $ \\widetilde{m}_i=(p_1(n_i),n_i)$ for $i = 1,2$. Then $\\widetilde{m}_2 -\\widetilde{m}_1= (p^*_1(n_2-n_1), n_2 -n_1)$. The result follows.\n\\end{proof}\n\nThe next result follows at once from \\thref{lem:restriction} and \\thref{product_and_order}.\n\n\\begin{lemma}\n\\label{lem:tf_X_pointedness}\nLet $ \\tf^{\\cX}_{dn_1},\\tf^{\\cX}_{dn_2} \\in \\cmid(\\cX)$ with $d_1n_1, d_2n_2 \\in \\Trop_{\\Z}(\\cX^\\vee_{\\seed^\\vee})=d\\cdot N$.\n Then the product $\\vartheta^{\\cX}_{dn_1}\\vartheta^{\\cX}_{dn_2}$ expressed in the theta basis of $\\cmid(\\cX)$ is of the following form\n\\[\n\\vartheta^{\\cX}_{dn_1}\\vartheta^{\\cX}_{dn_2}= \\vartheta^{\\cX}_{dn_1+dn_2}+ \\sum_{n_1+n_2 \\ \\prec_{\\seed, \\text{div}} \\ n} c_{n}\\vartheta^{\\cX}_{dn}.\n\\]\n\\end{lemma}\n\nFrom now on we let $\\leq_{\\seed,\\text{div}}$ be any total order refining $\\preceq_{\\seed, \\text{div}}$. \n\n\\begin{corollary}\\label{cor:gv on midX}\nLet ${\\bf c}_{\\seed}:\\cmid(\\cX) \\setminus \\{ 0\\} \\to (d \\cdot N,\\leq_{\\seed,\\text{div}})$ be the map defined by \n\\[\n{\\bf c }_{\\seed}(f):= \\min{}_{\\leq_{\\seed,\\text{div}}}\\{n_1, \\dots , n_t\\},\n\\]\nwhere $f=c_1\\tf^{\\cX}_{d n_1} + \\dots + c_t\\tf^{\\cX}_{d n_t}$ is the expression of $f$ in the theta basis of $\\text{mid}(\\cX)$. Then ${\\bf c }_{\\seed}$ is a valuation with 1-dimensional leaves and the theta basis for $\\cmid(\\cX) $ is adapted for ${\\bf c}_\\seed$.\n\\end{corollary}\n\n\n\nWe now let $\\cX_{\\bf 1}$ be the fibre of $\\cX$ associated to a sublattice $H:= H_{\\cX} \\subset K$. In order to define a {\\bf c}-vector valuation on $\\cmid(\\cX_{\\bf 1})$ we need that \n\\[\nH\\cap N^+_{\\seed}= \\emptyset.\n\\]\nSince, if this condition holds, $\\preceq_{\\seed, \\text{div}}$ induces a well partial order on $N\/H =\\mathcal X_{\\bf 1,\\seed}$ defined as\n\\[\nn_1 + H \\preceq_{\\seed, \\text{div}} n_2+H \\quad \\text{ if and only if } \\quad n_2 - n_1 \\in N^+_{\\seed}+ H.\n\\]\nThe rest of the construction follows from the cases already treated.\n\n\\begin{lemma}\\label{lem:cval_gval}\nSuppose $\\Gamma$ is of full-rank and let $p: \\cA \\to \\cX$ be a cluster ensemble map. Then we have a commutative diagram\n\\[\n\\xymatrix{\n\\cmid(\\cX) \\setminus \\{0\\} \\ar^{p^*}[r] \\ar_{{\\bf c}_{\\seed}}[d] & \\cmid(\\cA) \\setminus \\{0\\} \\ar^{{\\bf g}_{\\seed}}[d] \\\\\n\\Trop_{\\Z}(\\cX^{\\vee}_{\\seed^\\vee}) \\ar_{(p^\\vee)^T\\circ i} [r] & \\Trop_{\\Z}(\\cA^{\\vee}_{\\seed^\\vee}) \n}\n\\]\n\\end{lemma}\n\\begin{proof}\nIt is enough to show that for ${\\bf n} \\in \\Theta(\\cX) $ we have\n\\[\n\\gv_{\\seed}(p^*(\\tf^\\cX_{\\bf n}))=(p^\\vee)^T\\circ i({\\bf c}_{\\seed} (\\tf^\\cX_{\\bf n}))\n\\]\nLet $dn=\\mathfrak{r}_{\\seed^\\vee}({\\bf n})$. We have that\n\\[\n\\tf^{\\cX}_{dn}=z^n + \\sum_{n\\prec_{\\seed}n'}a_{n'}z^{n'}.\n\\]\nTherefore,\n\\[\np^*(\\tf^{\\cX}_{dn})=z^{p^*(n)} + \\sum_{n<_{\\seed, \\text{div}}n'}a_{n'}z^{p^*(n')}.\n\\]\nWe conclude that $\\gv_{\\seed}(p^*(\\tf^\\cX_{\\bf n}))=p^*(n)$. On the other hand we have that $ {\\bf c}_{\\seed} (\\tf^\\cX_{\\bf n})=dn$. We compute\n\\begin{align*}\n (\\Lp)^T\\circ i (dn)= ((\\Lp)^*)^*(-dn)=\\lrp{-\\frac{1}{d}(p^*)^*)}^*(-dn)=p^*(n).\n\\end{align*}\nThe claim follows.\n\\end{proof}\n\nWe would like to treat {\\bf g}-vector valuations for varieties of the form $\\cA$ and $\\cA\/T_H$ and {\\bf c}-vector valuations on $\\cX$ and $\\cXe$ in a uniform way.\nWith this in mind we introduce the following notation.\n\n\\begin{notation}\n\\thlabel{not:g-val}\nLet $\\cV$ be a cluster variety and $ \\cV^{\\vee}$ its Fock--Goncharov dual. The cluster valuation on $\\cmid(\\cV)$ associated to a seed $\\seed\\in \\orT$ is \n\\[\n\\nu_{\\seed}:\\cmid (\\cV)\\setminus\\{0\\} \\to (\\Trop_{\\Z}(\\cV^\\vee_{\\seed^\\vee}), <_{\\seed}),\n\\]\nwhere $\\Trop_{\\Z}(\\cV^\\vee_{\\seed^\\vee})$ is as in \\eqref{eq:unif} and $<_{\\seed}$ is a linear order on $\\Trop_{\\Z}(\\cV^\\vee_{\\seed^\\vee})$ refining $\\prec_\\seed$ in case $\\cV=\\cA$ or $\\cA\/T_H$ and it refines $\\prec_{\\seed,\\text{div}}$ if $\\cV=\\cX$ or $\\cXe$. \n\\end{notation}\n\n\\section{Newton--Okounkov bodies} \\label{sec:no}\n\nIn this section we provide a general approach to construct Newton--Okounkov bodies associated to certain partial minimal models of varieties with a cluster structure. \nIn particular, we treat a situation that often arises in representation theory where the universal torsor of a projective variety has a cluster structure of type $\\cA$. \nThe Newton--Okounkov bodies we construct depend on the choice of an initial seed. Hence we discuss how the bodies associated to different choices of initial seed are related and introduce the intrinsic Newton--Okounkov body which is seed independent.\n\n\\subsection{Schemes and ensembles with cluster structure}\n\n\\begin{definition}\\thlabel{def:cluster-structure}\nWe say a smooth \nscheme (over $\\Bbbk$) $V$ {\\bf can be endowed with cluster structure of type} $\\cV$ if there is a birational map $ \\Phi: \\cV \\dashrightarrow V$ which is an isomorphism outside a codimension two subscheme of the domain and range.\nIn this setting, we say that the pair $(V,\\Phi)$ is {\\bf{a scheme with cluster structure of type}} $\\cV$.\n\\end{definition}\n\n\\begin{remark}\nWe are straying slightly from \\cite{CMNcpt} in \\thref{def:cluster-structure}.\nSpecifically, we are now including $\\Phi$ as part of the data defining a scheme with cluster structure.\nSo, given two different birational maps $\\Phi_1:\\cV_1 \\dashrightarrow V$ and $\\Phi_2: \\cV_2 \\dashrightarrow V$ as in \\thref{def:cluster-structure}, we now consider $(V,\\Phi_1)$ and $(V,\\Phi_2)$ different as schemes with cluster structure (as is the case, for example, for open positroid varieties, see Remark~\\ref{rmk:open positroid}). \nNevertheless, when the map $\\Phi$ is clear from the context or we are just dealing with a single birational map $\\cV \\dashrightarrow V$, we will simply say that $V$ has a cluster structure of type $\\cV$.\n\\end{remark}\n\nLet $V=(V,\\Phi)$ be a scheme with a cluster structure of type $\\cV$. Since $V$ is normal and isomorphic to $\\cV$ up to co-dimension $2$ then $V$ and $\\cV$ have isomorphic rings of regular functions. In turn, we can talk about polynomial theta functions on $V$ which we denote by $\\tf^V_{\\bf v}$ for ${\\bf v}\\in \\Theta (\\cV)$.\nMoreover, recall that $\\cV$ is log Calabi--Yau. By \\cite[Lemma~1.4]{GHK_birational} $V$ is also log Calabi--Yau. Hence, $V$ has a canonical volume form whose pullback by $\\Phi $ coincides with the canonical volume form on $\\cV$. Moreover, a (partial) minimal model $V\\subset Y$ and its boundary can be defined as in Definition \\ref{def:cv_minimal_model}.\n\n\n\\begin{definition}{\\cite{GHK_birational}}\n An inclusion $V \\subset Y$ as an open subscheme of a normal variety $Y$ is a {\\bf partial minimal model} of $ V$ if the canonical volume form on $V$ has a simple pole along every irreducible divisor of $Y$ contained in $ Y \\setminus V$. It is a {\\bf minimal model} if $Y$ is, in addition, projective. We call $ Y \\setminus V$ the {\\bf boundary} of $V \\subset Y$.\n\\end{definition}\n\n\n\\begin{definition}\n Suppose $\\Phi:\\cV \\dashrightarrow V$ endows $V$ with a cluster structure of type $\\cV$ and that the cluster valuation $\\nu_{\\seed}$ extends to $\\Bbbk(\\cV) $. \n Then the {\\bf cluster valuation} $\\nu^{\\Phi}_{\\seed}:\\Bbbk(V)^*\\to \\Trop_{\\Z}(\\cV^\\vee)$ is given by\n \\[\n \\nu^{\\Phi}_{\\seed}(f)= \\nu_{\\seed}(\\Phi^*(f)).\n \\]\n\\end{definition}\n\n\\begin{definition}\n Suppose $\\Phi_{\\cA}:\\cA \\dashrightarrow V_1$ and $\\Phi_{\\cX}:\\cX \\dashrightarrow V_2$ endow $V_1$ (resp. $V_2$) with cluster structures of type $ \\cA$ (resp. $\\cX$). We say that $V_1 \\overset{\\tau}{\\to} V_2$ is a cluster ensemble structure if there exists a cluster ensemble map $p:\\cA \\to \\cX$ such that the following diagram commutes\n \\[\n \\xymatrix{\n V_1 \\ar^{\\tau}[r] & V_2 \\\\\n \\cA \\ar@{-->}^{\\Phi_{\\cA}}[u] \\ar_p[r] & \\cX \\ar@{-->}_{\\Phi_{\\cX}}[u].\n }\n \\]\n\\end{definition}\n\n\n\n\n\\subsection{Newton--Okounkov bodies for Weil divisors supported on the boundary}\n\\label{sec:NO_bodies}\nThroughout this section we let $\\cV$ be a scheme of the form $ \\cA$, $\\cX$, $\\cA\/T_{H}$ or $\\cX_{{\\bf 1}}$. Whenever we talk about a cluster valuation on $\\cmid(\\cV)$ we are implicitly assuming we are in a setting where such valuation exist, see \\S\\ref{sec:cluster_valuations}.\n\n\\begin{definition}{\\cite{GHKK}}\n\\label{def:positive_set}\nA closed subset $S\\subseteq \\Trop_{\\R}(\\cV^{\\vee})$ is {\\bf positive} if for any positive integers $d_1, d_2$, any $p_1\\in d_1\\cdot S(\\Z)$, $p_2\\in d_2\\cdot S(\\Z)$ and any $r \\in \\Trop_{\\Z}(\\cV^{\\vee})$ such that $\\alpha (p_1,p_2,r)\\neq 0$, we have that $r \\in (d_1 +d_2)\\cdot S(\\Z)$. \n\\end{definition}\n\n\\begin{remark}\n\\label{rem:positive_sets_in_vs}\n We can also define positive sets inside $\\Trop_{\\R}(\\cV^{\\vee})_{\\seed^\\vee}$ in exactly the same way they are defined in Definition \\ref{def:positive_set}. \n In particular we have that $S\\subset \\Trop_{\\R}(\\cV^{\\vee})$ is positive if and only if $\\mathfrak{r}_{\\seed^\\vee}(S)\\subset \\Trop_{\\R}(\\cV^\\vee_{\\seed})$ is positive.\n\\end{remark}\n\nIn \\cite[\\S8]{GHKK} the authors discuss how positive sets give rise to both, partial minimal models of cluster varieties and toric degenerations of such. In this section we study the inverse problem. \nNamely, we let $(V,\\Phi)$ be a scheme with a cluster structure of type $\\cV$ and construct Newton--Okounkov bodies associated to a partial minimal model $V \\subset Y$ (see \\S\\ref{sec:minimal_models}). \nThen we show that under suitable hypotheses these Newton--Okounkov bodies are positive sets. \nWe let $D_1, \\dots , D_s $ be the irreducible divisors of $Y$ contained in the boundary of $V\\subset Y$ and let $D:=\\bigcup_{j=1}^s D_j$.\n%\\footnote{Since $Y$ is normal a Cartier divisor $\\{(U_i,f_i)\\}$ on $Y$ is fully determined by the corresponding Weil divisor $\\sum \\text{div}(f_i)$, where $\\text{div}(f_i)$ is the principal divisor associated to $f_i$.} \n\n\nGiven a Weil divisor $D'$ on $Y$ we denote by $R(D')$ the associated {\\bf section ring}. \nRecall that $R(D')$ can be described as the $\\Z_{\\geq 0}$-graded ring whose $k^{\\mathrm{th}}$ homogeneous component is\n\\begin{equation*}\n R_k(D') := H^0(Y, \\mathcal{O}(kD'))= \\lrc{ f\\in \\Bbbk(Y)^* \\mid \\text{div}(f)+kD'\\geq 0 }\\cup \\{ 0\\},\n\\end{equation*}\nwhere $\\text{div}(f)$ is the principal divisor associated to $f$.\nEven more concretely, if $D'=c_1 D'_1 + \\cdots + c_{s'}D'_{s'}$, where $D'_1, \\dots , D'_{s'}$ are distinct prime divisors of $Y$ and $c_1, \\dots , c_{s'}$ are non-negative integers, then $ R_k(D')$ is the vector space consisting of the rational functions on $Y$ that are regular on the complement of $\\bigcup_{j=1}^{s'} D'_j$ and whose order of vanishing along every prime divisor $D'_j$ is bounded below by $-kc_j$. The multiplication of $R(D')$ is induced by the multiplication on $ \\Bbbk(Y)$. \n\n\\begin{definition}\n\\thlabel{def:NOlb}\nLet $\\nu:\\Bbbk(Y)\\setminus \\{ 0 \\} \\to L$ be a valuation, where $(L, < )$ is a linearly ordered lattice. Let $D'$ be a Weil divisor on $Y $ having a non-zero global section. For a choice of non-zero section $\\tau \\in R_1 (D')$ the associated {\\bf Newton--Okounkov body} is\n\\eqn{\n\\Delta_\\nu(D',\\tau) := \\overline{\\conv\\Bigg( \\bigcup_{k\\geq 1} \\lrc{\\frac{\\nu\\lrp{f\/\\tau^k}}{k} \\mid f\\in R_k(D')\\setminus \\{0\\} } \\Bigg) }\\subseteq L\\otimes \\R,\n}\nwhere $\\conv $ denotes the convex hull and the closure is taken with respect to the standard topology of $L\\otimes \\R$.\n\\end{definition}\n\nFrom now on we assume that $D'$ has a non-zero global section.\nWe would like to use a cluster valuation $\\cval: \\Bbbk(V)\\setminus \\{ 0\\} \\to (\\Trop_{\\Z}(\\cV^{\\vee}_{\\seed^\\vee}),<_{\\seed})$ to construct Newton--Okounkov bodies. Notice that if $\\cV$ satisfies the full Fock--Goncharov conjecture, then it is possible to do so as we can extend $\\nu_{\\seed}$ from $\\cmid(\\cV)=\\up(\\cV)$ to $\\Bbbk(\\cV) = \\Bbbk(Y)$. \nObserve, moreover, that if $D'$ is supported on $D$ (that is $D'=\\sum_{j=1}^s c_jD_j$ for some integers $c_1,\\dots , c_s$) then every graded piece $R_k(D') $ is contained in $H^{0}(V,\\mathcal{O}_V)\\cong H^{0}(\\cV,\\mathcal{O}_{\\cV})$, so elements of $R_k(D')$ can be described using the theta basis for $H^0(\\cV,\\mathcal{O}_{\\cV})$. \nMoreover, $\\ord_{D_j}\\in \\cV(\\Z^t)$, so we can define $\\tf^{\\cV}_j$ as in \\eqref{eq:def superpotential_summands}.\n\n\\begin{definition}\n\\label{def:graded_theta_basis}\nAssume $\\cV$ satisfies the full Fock--Goncharov conjecture and that $D'$ is of the form $D'=\\sum_{j=1}^s c_jD_j$. We say that $R(D')$ {\\bf has a graded theta basis} if for every integer $k\\geq 0$ the set of theta functions on $\\cV$ parametrized by the integral points of\n\\[\nP_k(D'):= \\bigcap_{j=1}^s \\lrc{b\\in \\Trop_{\\R}(\\cV^{\\vee}) \\mid \\Trop_{\\R}(\\tf^{\\cV^\\vee}_j)(b) \\geq -kc_j}\n\\] \nis a basis for $R_k(D')$.\n\\end{definition}\n\nThe reader should notice that in case $\\cV$ has theta reciprocity (see Definition \\ref{def:theta_reciprocity}), then the definition of $P_k(D')$ becomes very natural from the perspective of toric geometry, see \\S\\ref{sec:minimal_models}. We now introduce a notion that allows us to make a good choice for the section $\\tau$.\n\n\n\n \\begin{definition}\n \\label{def:linear_action}\n A subset $L\\subset \\Theta(\\cV)$ is {\\bf linear} if \n \\begin{itemize}\n \\item for any $a,b\\in L$ there exists a unique $r\\in\\Theta(\\cV)$ such that $\\alpha(a,b,r)\\neq 0$ and moreover, $r\\in L$,\n \\item for each $a\\in L$ there exists a unique $b\\in L$ such that $\\tf^{\\cV}_a \\tf^{\\cV}_b=1 $. \n \\end{itemize}\nWe further say that a linear subset $L$ {\\bf acts linearly} on $\\Theta(\\cV)$ if for any $a\\in L$ and $ b \\in \\Theta(\\cV)$ there exists a unique $r\\in \\Trop_{\\Z}(\\cV^{\\vee})$ such that $\\alpha(a,b,r)\\neq 0$. \n \\end{definition}\n \n\nFor example, if $\\cV=\\cA$ then $\\mathfrak{r}_{\\seed}^{-1}(\\Nuf^\\perp)$ is linear and acts linearly on $\\Theta(\\cV)$. If $\\cV =\\cX$\nthen $\\mathfrak{r}_{\\seed}^{-1}(\\ker(p_2^*))$ is linear and acts linearly on $\\Theta(\\cV)$.\n\n\n\\begin{theorem}\n\\label{NO_bodies_are_positive}\nLet $V\\subset Y$ be a partial minimal model. Assume the full Fock--Goncharov conjecture holds for $ \\cV$. Let $D'=\\sum_{j=1}^s c_j D_j$ be a Weil divisor on $Y$ supported on $D$ such that $R(D')$ has a graded theta basis. Let $\\tau\\in R_1(D')$ be such that $\\nu^{\\Phi}_{\\seed}(\\tau) $ belongs to a linear subset of $ \\Trop_{\\Z}(\\cV^{\\vee}) $ acting linearly on $\\Trop_{\\Z}(\\cV^{\\vee}) $. Then the Newton--Okounkov body $\\Delta_{\\nu^{\\Phi}_{\\seed}}(D',\\tau)\\subset \\Trop_{\\R}(\\cV^{\\vee}_{\\seed^\\vee})$ is a positive set.\n\\end{theorem}\n\n\\begin{proof}\nTo make notation lighter, throughout this proof we denote $\\Delta_{\\nu_{\\seed}}(D',\\tau) $ simply by $ \\Delta $, $P_k(D')_\\seed$ by $P_k$ and $\\nu^{\\Phi}_{\\seed}$ by $\\nu_{\\seed}$.\nWe work in the lattice identification $ \\Trop_{\\Z}(\\cV^{\\vee}_{\\seed^\\vee})$ of $\\Trop_{\\Z}(\\cV^{\\vee})$.\nThe linear subset of the statement corresponds to a sublattice $L \\subseteq \\Trop_{\\Z}(\\cV^{\\vee}_{\\seed^\\vee})$.\n\nConsider $d_1, d_2 \\in \\Z_{>0}$ and $p_1\\in d_1\\Delta(\\Z)$, $p_2\\in d_2\\Delta(\\Z)$. We have to show that for any $r \\in \\Trop_{\\Z}(\\cV^{\\vee}_{\\seed^\\vee})$ with $\\alpha (p_1,p_2,r)\\neq 0$ then $r \\in (d_1 +d_2)\\Delta(\\Z)$. \nFor this it is enough to show that $k\\Delta = P_k - k\\nu_{\\seed}(\\tau)$ for all $k \\in \\Z_{>0}$ as we now explain.\\footnote{In fact, it is enough that the equality holds at the level of integral points, namely, $k\\Delta(\\Z)= P_k(\\Z) - k\\nu_{\\seed}(\\tau)$. However, we are able to show the stronger condition $k\\Delta= P_k - k\\nu_{\\seed}(\\tau)$.}\nIf this is the case then for $i=1,2$, the point $p_i+d_i\\nu_\\seed(\\tau)$ belongs to $P_{d_i}(\\Z)$.\nBy hypothesis $\\tf^V_{p_i+d_i \\nu_{\\seed}(\\tau)}\\in R_{d_i}(D')$.\nIn particular, the product $\\tf^V_{p_1+d_1 \\nu_{\\seed}(\\tau)}\\tf^V_{p_2+d_2 \\nu_{\\seed}(\\tau)} $ must belong to $R_{d_1+d_2}(D')$ and this product must be expressed as a linear combination of theta functions that belong to $R_{d_1+d_2}(D')$. \nTo finish we just need to convince ourselves that \n\\[\n\\alpha(p_1+d_1\\nu_\\seed(\\tau),p_2+d_2\\nu_\\seed(\\tau), r+(d_1+d_2)\\nu_\\seed(\\tau))\\neq 0\n\\]\nas this would imply \n\\[\nr+(d_1+d_2)\\nu_\\seed(\\tau)\\in P_{d_1+d_2}(\\Z)=(d_1+d_2)\\Delta(\\Z)+ (d_1+d_2)\\nu_\\seed(\\tau) .\n\\]\nHowever, this follows at once from the fact that $\\nu_\\seed(\\tau)$ belongs to the linear subset $L$. \nIndeed, the condition $\\alpha(p_1,p_2,r)\\neq 0$ implies the existence of a pair of broken lines $\\gamma_1, \\gamma_2$ such that $I(\\gamma_i)=p_i$ and $ F(\\gamma_1)+F(\\gamma_2)=r$. \nSince $\\nu_\\seed(\\tau)\\in L$ we can construct new broken lines $\\gamma'_1$ and $\\gamma'_2$ such that $I(\\gamma'_i)=p_i+d_i\\nu_\\seed(\\tau)$ and $ F(\\gamma'_1)+F(\\gamma'_2)=r+(d_1+d_2)\\nu_\\seed(\\tau)$ by changing the direction of all the domains of linearity of $\\gamma_i$ by $d_i\\nu_\\seed(\\tau)$. \n\nWe now proceed to show that $k\\Delta= P_k-k\\nu_\\seed(\\tau) $ for all $k \\in \\Z_{>0}$. \nFirst notice that $aP_1= P_a$ for all $a\\in \\R_{\\geq 0}$ (if $g$ is a positive Laurent polynomial then $g^T(ax)=ag^T(x)$ provided $a$ is non-negative). \nSince $P_k$ is closed and convex in order to show that $k \\Delta \\subset P_k- k\\nu_\\seed(\\tau)$ it is enough to show that $\\frac{k}{k'}\\ \\nu_\\seed(f\/\\tau^{k'})=\\frac{k}{k'}\\ \\nu_\\seed(f)-k \\nu_\\seed(\\tau)$ belongs to $P_k-k\\nu_\\seed(\\tau)$ for all $k'\\geq 1$ and all $f\\in R_{k'}(D')\\setminus \\{0\\}$. This follows at once from the fact that $\\frac{k}{k'}\\nu_\\seed(f)\\in P_k$ as $\\frac{k}{k'}P_{k'}=P_k$.\nTo obtain the reverse inclusion it is enough to show that the inclusion holds at the level of rational points, namely, $P_k(\\Q)-k\\nu_\\seed(\\tau)\\subset k\\Delta(\\Q)$. \nIndeed, since $P_k$ is a finite intersection of rational hyperplanes in $\\Trop_{\\R}(\\cV^{\\vee}_{\\seed^\\vee})$ it can be described as the convex hull of its rational points. \nIf $x\\in P_k(\\Q)$ then $\\frac{x}{k}\\in \\frac{1}{k}P_k(\\Q)=P_1(\\Q)$. \nLet $d\\in \\Z_{>0}$ be such that $x':=\\frac{dx}{k} \\in \\Trop_{\\Z}(\\cV^{\\vee}_{\\seed^\\vee})$. \nIn particular, $x'\\in P_{d}(\\Z)_{\\seed}$ which gives that $d^{-1}\\nu_\\seed(\\frac{\\tf_{x'}}{\\tau^{d}})\\in \\Delta$. Finally, notice that $d^{-1}\\nu_\\seed(\\frac{\\tf_{x'}}{\\tau^{d}})=d^{-1}(\\nu_\\seed(\\tf_{x'})-d\\nu_\\seed(\\tau))=d^{-1}x'-\\nu_\\seed(\\tau)$ which implies $x-k\\nu_\\seed(\\tau) \\in k\\Delta$. \n\\end{proof}\n\nIn Theorem \\ref{NO_bodies_are_positive} the assumption that $R(D')$ has a graded theta basis might seem rather strong. We now provide a situation in which this hypothesis holds and in the next subsection we treat a more robust framework in which this condition follows directly from the equivariant nature of theta functions.\n\n\n\\begin{lemma}\n\\label{lem:graded_theta_basis}\nLet $V\\subset Y$ be a minimal model. Assume $D=\\sum_{j=1}^n D_j$ is ample with $D'=cD$ very ample for some $c\\in \\Z_{>0}$. Assume further that the image of the embedding of $Y$ into a projective space given by $D'$ is projectively normal. \nIf $\\cV$ has theta reciprocity and the theta functions on $\\cV$ respect the order of vanishing (see Definition~\\ref{def:respect_order}), then $R(D')$ has a graded theta basis.\n\\end{lemma}\n\\begin{proof}\nIt is enough to treat the case $\\cV=V$.\nConsider the affine cone $\\widetilde{Y}$ of the embedding of $Y$ into a projective space given by $D'$. We consider the canonical projection $\\widetilde{Y}\\setminus \\{ 0\\} \\overset{\\pi}{\\to } Y $ and let $ \\cV':= \\pi^{-1}(\\cV) $.\nObserve that $\\cV'\\cong \\cV \\times \\C^*$. \nWe may think of $\\cV'$ as the cluster variety obtained from $\\cV$ by adding a frozen index and extending trivially the bilinear form in the fixed data defining $\\cV$. \nIn particular, $\\text{up}(\\cV')= \\text{up}(\\cV)[x^{\\pm 1}]$, where $x$ is the coordinate for the $\\C^*$ component. Notice that the theta functions on $\\cV'$ are of the form $\\tf^{\\cV'}_{(p,h)}=\\tf^{\\cV'}_{(0,h)}\\tf^{\\cV'}_{(p,0)} =x^h\\tf^{\\cV}_p$, where $\\tf^{\\cV}_p$ is a theta function on $\\cV$ and $h \\in \\Z=\\Trop_{\\Z}(\\C^*) $.\nAn analogous description holds for the theta functions on $(\\cV')^\\vee \\cong \\cV^\\vee \\times \\C^* $. Namely, these theta functions are of the form $x^h\\tf_q^{\\cV^\\vee}$ for some $h\\in \\Z$.\nWe consider the inclusion $R(D')\\hookrightarrow \\text{up}(\\cV')$ given by sending a homogeneous element $f\\in R_k(D')$ to $x^kf$. The map is well defined since $f$ is regular on $ \\cV $. \nMoreover, if we let $\\widetilde{D}_j:= \\pi^{-1}(D_j)$ then for all $j$ we have $\\ord_{\\widetilde{D}_j}\\lrp{x^{k}}=k$ and $\\ord_{\\widetilde{D}_j}\\lrp{\\tf^{\\cV'}_{(p,0)}}=\\ord_{D_j}\\lrp{\\tf^V_p}$. \nIn particular, thinking of $\\Trop_{\\Z}(\\cV')$ as $\\Trop_{\\Z}(\\cV)\\times \\Z$ we have $\\ord_{\\widetilde{D}_k}=(\\ord_{D_k},1)$. \nSince theta functions on $\\cV$ respect the order of vanishing, the same holds for the theta functions on $\\cV'$.\n%This is the argument Tim and I came up with (we might wnat to leave it as a comment and add it if the referee asks for it): Indeed, let $f=\\sum_{i=1}^sc_ix^{a_i}\\tf^{\\cV}_{b_i}$ be such that $\\ord_{D'}(f)\\geq 0$. Without loss of generality we may assume $\\ord_{D'}(x^{a_1}\\tf^{\\cV}_{b_1}),\\dots ,\\ord_{D'}(x^{a_k}\\tf^{\\cV}_{b_k})\\leq 0$ and $\\ord_{D'}$ of the other terms are non-negative. For each $1\\leq i \\leq k$ let $\\sum_{r_i}\\alpha_{ r_i}x^{a_i}z^{b_{r_i}}$ be the terms in $x^{a_i}\\tf^{\\cV}_{b_i}$ achieving the worst pole of $(x^{a_i}\\tf^{\\cV}_{b_i})$ along $D'$. Since $\\ord_{D'}(f)\\geq 0 $ we have that $\\sum_{i=1}^kc_i\\sum_{r_i}x^{a_i}z^{b_{r_i}}=0$. Since $x$ does not appear in the monmilas of the form $z^{b_r}$ we have that $a_1=\\dots=a_k$ and that $\\sum_{i=1}^kc_i\\sum_{r_i}z^{b_{r_i}}=0$. It follows that if we write $ord_{D'}= (r, \\ord_{D})$ for $D$ some divisor at inifinity for $\\cV$ then $\\ord_{D}(\\sum_{i=1}^S c_i \\tf^{\\cV}_{b_i})\\geq 0$. Since theta functions on $ \\cV$ respect the order of vanishing we have that $\\ord_{D}(\\tf^{\\cV}_{b_1})\\geq 0$ for all $i$. In particular the poles of $x^{a_1}\\tf^{\\cV}_{b_1},\\dots , x^{a_k}\\tf^{\\cV}_{b_k}$ come from $x$ ant these poles cancel in $f$. Without loss loss of generality $x^{a_1}$ gives the worst pole of $x$ along $D'$. Consider $f_2:=x^{a_1}(\\sum_{i=1}^k x^{a_k-ai}\\tf^{\\cV}_{b_i})= \\sum_{i=1}^k c_ix^{a_i}\\tf^{\\cV}_{b_i} $. Since $\\ord_{D'}(f_2)\\geq 0$ we have that $\\sum_{i=1}^k c_ix^{a_k-ai}\\tf^{\\cV}_{b_i}=0$, but this is impossible since theta functions are linearly independent.\nThis implies that for every $a \\in \\Z$ and every $j$, $\\ord_{D_j}\\lrp{\\sum_q \\alpha_q \\tf_q^{\\cV}}\\geq a$ if and only if $\\ord_{D_j}(\\tf_q^{\\cV})\\geq a$ for all $q$ such that $\\alpha_q \\neq 0$. \nTo see this there is only one implication to be checked (the other follows from the axioms of valuations). \nSo assume $\\ord_{D_j}\\lrp{\\sum_q \\alpha_q \\tf_q^{\\cV}}\\geq a$.\nSince $\\ord_{D_j}\\lrp{\\sum_q \\alpha_q \\tf_q^{\\cV}}=\\ord_{\\widetilde{D}_j}\\lrp{\\sum_q \\alpha_q \\tf_q^{\\cV}}$ and $x^{-a}\\tf_q^{\\cV}$ is a theta function on $\\cV'$ for all $q$ we have the following\n\\begin{align*}\n \\ord_{D_j}\\lrp{\\sum_q \\alpha_q \\tf_q^{\\cV}}\\geq a & \\Longleftrightarrow \\ord_{\\widetilde{D}_j}\\lrp{\\sum_q \\alpha_q \\tf_q^{\\cV}}\\geq a \\\\\n & \\Longleftrightarrow \\ord_{\\widetilde{D}_j}\\lrp{x^{-a}\\sum_q \\alpha_q \\tf_q^{\\cV}} \\geq 0 \\\\\n & \\Longleftrightarrow \\ord_{\\widetilde{D}_j}(x^{-a} \\tf_q^{\\cV})\\geq 0 \\text{ for all } q \\text{ such that } \\alpha_q\\neq 0 \\\\\n & \\Longleftrightarrow \\ord_{\\widetilde{D}_j}( \\tf_q^{\\cV})\\geq a \\text{ for all } q \\text{ such that } \\alpha_q\\neq 0 \\\\\n & \\Longleftrightarrow \\ord_{D_j}( \\tf_q^{\\cV})\\geq a \\text{ for all } q \\text{ such that } \\alpha_q\\neq 0.\n\\end{align*}\nSince $D'$ is very ample and $Y$ is projectively normal in its embedding given by $D'$ we have that $H^0(\\widetilde{Y}, \\mathcal{O}_{\\widetilde{Y}}) \\cong R(D') \\hookrightarrow \\text{up}(\\cV')$.\nIn particular, if we express $f \\in R_k(D')$ as $f= \\sum_q \\alpha_q \\tf^{\\cV}_q$, we have that $\\ord_{D_j}\\lrp{\\tf^\\cV}\\geq -kc$ for all $j$ and all $q $ such that $\\alpha_q \\neq 0$. This means that $\\tf_q^{\\cV} \\in R_k(D')$ for all such $q$. \nIn particular, the theta functions of $\\cV$ that lie in $R_k(D')$ have to be a basis a of $R_k(D')$. By theta reciprocity, such theta functions are precisely those parametrized by $P_k(D')$.\n\\end{proof}\n\n\\begin{remark}\\label{rmk:toric degen}\nIf $R(D')$ is finitely generated and the semigroup generated by the image of $\\nu^\\cV_{\\seed}$ is of full-rank and finitely generated then there is a one parameter toric degeneration of $Y$ to the toric variety associated to $ \\Delta_{\\nu^\\cV_{\\seed}}(D',\\tau)$ \\cite{An13}\\footnote{That is, there is a scheme $\\mathcal Y$ and a flat morphism $ \\mathcal{Y}\\to \\mathbb A^1$ whose generic fibre is isomorphic to $Y$ and special fibre isomorphic to the toric variety associated to $ \\Delta_{\\nu_{\\seed}}(D',\\tau)$.}.\nAs explained in \\cite[\\S8.5]{GHKK} for cluster varieties of type $\\cA$ (regardless of the full-rank assumption) a polyhedral positive set defines a partial compactification $\\cA_{\\text{prin}} \\subset \\overline{\\cA}_{\\text{prin}}$. \nThis compactification comes with a flat morphism $\\overline{\\cA}_{\\text{prin}}\\to \\mathbb A^r$ having $\\overline{\\cA}=Y$ as fibre over ${\\bf 1}=(1, \\dots , 1)$ and whose fibre over $0$ is the toric variety associated to the positive set.\nTherefore, both constructions can be used to degenerate varieties with a cluster structure to the same toric variety. However, the variety given by the latter construction contains various intermediate fibres that lie in between $\\mathcal A=\\cV$ and a toric variety. Moreover, while Anderson's degenerations produces a $(\\Bbbk^*)$-equivariant family, for the latter degeneration this is the case if and only if $\\Gamma$ is of full-rank.\n\\end{remark}\n\n\\subsection{Newton--Okounkov bodies for line bundles via universal torsors}\n\\label{sec:universal_torsors}\n\nIn this section we consider a particularly nice geometric situation that arises often in representation theory. We let $Y$ be an irreducible normal projective scheme whose Picard group $\\text{Pic}(Y) $ is free of finite rank $\\rho \\in \\Z_{>0}$ (recall that $\\text{Pic}(Y) $ is always abelian).\nFollowing \\cite[\\S2]{Hau02} (see also \\cite[\\S3]{BH03}, \\cite[Chapter 1]{ADHL}, \\cite[\\S4]{GHK_birational} or \\cite[\\S2]{HK00}), we consider the universal torsor of $Y$ and the associated Cox ring (\\cf Remark \\ref{rem:Cox}). For the convenience of the reader we recall these concepts. We begin by considering the quasi-coherent sheaf of $\\mathcal{O}_Y$-modules\n\\[\n\\bigoplus_{[\\lb] \\in \\text{Pic}(Y)} \\lb. \n\\]\nIn essence, the universal torsor of $Y$ is obtained by applying a relative spectrum construction (also denoted by {\\bf Spec}) to this sheaf. \nHowever, the choice of the representative $\\lb $ in the class $[\\lb]$ prevents this sheaf from having a natural $\\mathcal{O}_Y$-algebra structure. \nTo address this situation one can proceed as in \\cite[\\S2]{HK00} and consider line bundles $\\lb_1, \\dots, \\lb_{\\rho} $ whose isomorphism classes form a basis of $\\text{Pic}(Y)$. For $v=(v_{1},\\dots, v_{\\rho})\\in \\Z^{\\rho}$ we let $\\lb^{v}= \\lb_1^{\\otimes v_1}\\otimes \\cdots \\otimes \\lb_{\\rho}^{\\otimes v_{\\rho}}$ and consider the quasi-coherent sheaf\n\\[\n\\bigoplus_{v \\in \\Z^{\\rho}}\\lb^{v}.\n\\]\nThis sheaf has a natural structure of a reduced $\\mathcal{O}_Y$-algebra that is locally of finite type over $\\mathcal{O}_Y$ (the component associated to the zero element of $\\text{Pic}(Y)$).\nThis means that for sufficiently small\naffine open subsets $U$ of $Y$, the space $\\bigoplus_{v \\in \\Z^{\\rho}}\\lb^{v}(U)$ is a finitely generated $\\mathcal{O}_Y(U)$-algebra.\nThe universal torsor of $Y$ is obtained by gluing the affine schemes $\\text{Spec}\\lrp{\\bigoplus_{v \\in \\Z^{\\rho}}\\lb^{v}(U)}$.\n\n\\begin{definition}\nThe {\\bf universal torsor} of $ Y$ is \n\\[\n\\UT_Y= \\textbf{Spec}\\lrp{\\bigoplus_{v \\in \\Z^{\\rho}}\\lb^{v} }.\n\\]\nThe {\\bf Cox ring} of $Y$ is\n\\[\n\\text{Cox}(Y)= H^0 (\\UT_Y,\\mathcal{O}_{\\UT_Y}).\n\\]\n\\end{definition}\n\nUniversal torsors can be used to generalize the construction of a projective variety from its affine cone as follows.\nObserve that the inclusion of $ \\mathcal{O}_Y $ as the degree $0$ part of $\\bigoplus_{v \\in \\Z^{\\rho}}\\lb^{v} $ gives rise to an affine regular map $\\UT_Y\\to Y$.\nSince $\\text{Cox}(Y)$ is $\\text{Pic}(Y)$-graded there is an action of $T_{\\text{Pic}(Y)^*}= \\text{Spec}(\\C[\\text{Pic}(Y)])$ on $\\UT_Y$. \nThis action is free and the map $\\UT_Y\\to Y$ is the associated quotient map (see \\cite[Remark 1.4]{Hau02}).\n\n\\begin{remark}\n\\label{rem:Cox}\nThe notion of a Cox ring associated to a projective variety (satisfying some technical assumptions) was first introduced in \\cite[Definition 2.6]{HK00}.\nThis notion was generalized in \\cite{BH03} for any divisorial variety with only constant globally invertible functions, in particular, for any quasi-projective variety (over very general ground fields). However, in \\cite{BH03} the term \\emph{Cox ring} was not used.\nThe importance of considering universal torsors and Cox rings in the context of cluster varieties was pointed out in \\cite[\\S4]{GHK_birational} (see also \\cite{Man19}) and satisfactorily pursued in representation theoretic contexts where Cox rings arise naturally, see for example \\cite{Mag20}.\n\\end{remark}\n\n\\begin{remark}\nFor simplicity we are assuming that $\\text{Pic}(Y)$ is free. In case it has torsion we can still construct a universal torsor which might not be unique as it depends on the choice of a \\emph{shifting family} as in \\cite[\\S3]{BH03} (see \\cite[\\S3]{Man19} for a related discussion). Generalizations of the results of this section to the torsion case shall be treated elsewhere. \n\\end{remark}\n\n\n\\begin{remark}\nIf $Y$ is smooth we can construct the Cox ring of $Y$ and the universal torsor (still assuming that $\\text{Pic(Y)}$ is torsion free) in an equivalent way. The Cox ring can be defined as $\\text{Cox}(Y)=\\bigoplus_{v\\in \\Z^{\\rho}} H^0 (Y, \\lb^v)$. If $\\text{Cox}(Y)$ is finitely generated over $\\mathcal{O}_Y$-algebra then the universal torsor $\\UT_Y $ is obtained from $\\text{Spec}(\\text{Cox}(Y))$ by removing the unstable locus of the natural $T_{\\text{Pic(Y)}^*}$-action on $\\text{Spec}(\\text{Cox}(Y))$.\n\\end{remark}\n\nFrom now on we assume $ V\\subset \\UT_Y$ is a partial minimal model where $(V,\\Phi)$ is a scheme with a cluster structure of type $\\cA$. \nIn most of the result of this section we assume that $ V\\subset \\UT_Y$ has enough theta functions.\nUnder certain conditions that we discuss next, it is possible to show that $Y$ is a minimal model for a scheme with a cluster structure given by a quotient of $\\cA$ and construct Newton--Okounkov bodies for elements of $\\text{Pic}(Y)$.\nThe key point is to relate the action of $T_{\\text{Pic}(Y)^*}$ on $\\UT_Y$ with the torus actions on $\\cA$ arising from cluster ensemble maps.\n\n\\begin{lemma}\n\\label{lem:positivity_of_q_slice}\nLet $p:\\cA \\to \\cX$ be a cluster ensemble map and $H\\subset K^{\\circ}$ be a saturated sublattice. Consider the quotient $\\cA\/T_H$ and the fibration $w_H:\\cA^\\vee \\to T_{H^*}$ (see \\S\\ref{sec:FG_dual}).\nThen the set\n\\[\n\\lrc{ \\tf^{\\cA}_{\\bf m} \\in \\cmid(\\cA) \\mid {\\bf m} \\in \\lrp{\\Trop_{\\Z}(w_H)}^{-1}(q) \\cap \\Theta(\\cA)}\n\\]\nconsists precisely of the polynomial theta functions on $\\cA$ whose $T_H$-weight is $q$. Moreover, \nfor every $q \\in H^*$ the set $\\lrp{\\Trop_{\\R}(w_H)}^{-1}(q)\\subset \\Trop_{\\R}(\\cA^{\\vee})$ is positive.\n\\end{lemma}\n\\begin{proof}\nThe first claim follows from \\thref{prop:dual_fibration}. \nSo we only need to show that $\\lrp{\\Trop_{\\R}(w_H)}^{-1}(q)$ is positive. \nIn order to show this it is convenient to work with a condition equivalent to positivity called broken line convexity, see \\S\\ref{sec:intrinsic_NOB}.\nWe work in the lattice identification $ \\Trop_{\\R}(\\cA^{\\vee}_{\\seed^\\vee})$ of $\\Trop_{\\R}(\\cA^{\\vee})$.\nWe first argue that the set $ \\lrp{\\Trop_{\\R}(w_H)}^{-1}(0)_{\\seed^\\vee}$ is positive.\nFirst notice that any linear segment $L$ of a broken line segment contained in $ \\lrp{\\Trop_{\\R}(w_H)}^{-1}(0)_{\\seed^\\vee}$ has itself tangent direction in $ \\lrp{\\Trop_{\\Z}(w_H)}^{-1}(0)_{\\seed^\\vee}$. \nLet $m\\in \\lrp{\\Trop_{\\Z}(w_H)}^{-1}(0)_{\\seed^\\vee}$ be the tangent direction of $L$. The tangent direction of the following linear segment is of form $m+cp^*(n)$ for some $n\\in N^+_{\\seed}$ and $c\\in \\Z_{\\geq 0}$.\nFor any $h\\in H^\\circ$ we have\n\\[\n\\langle m+cp^*(n),h\\rangle =\\langle m,h\\rangle + c\\{n,h\\}=0,\n\\]\nas $H^\\circ\\subset K^\\circ$. So the next tangent direction also belongs to $ \\lrp{\\Trop_{\\Z}(w_H)}^{-1}(0)_{\\seed^\\vee}$.\nWe conclude that the set $\\lrp{\\Trop_{\\R}(w_H)}^{-1}(0)_{\\seed^\\vee}$ is broken line convex and by the main result of \\cite{CMNcpt} (see Theorem \\ref{thm:mainCMN} below) the set $\\lrp{\\Trop_{\\Z}(w_H)}^{-1}(0)_{\\seed^\\vee}$ is positive.\nThis already implies that for any $ x\\in \\lrp{\\Trop_{\\R}(w_H)}^{-1}(q)_{\\seed^\\vee}$ the set $x+ \\lrp{\\Trop_{\\R}(w_H)}^{-1}(0)_{\\seed^\\vee}$ remains positive. \nIndeed, let $y, z \\in x+ \\lrp{\\Trop_{\\R}(w_H)}^{-1}(0)_{\\seed^\\vee}$. Then $y- z \\in \\lrp{\\Trop_{\\R}(w_H)}^{-1}(0)_{\\seed^\\vee} $. \nIn other words, any line segment within the set $ x+\\lrp{\\Trop_{\\R}(w_H)}^{-1}(0)_{\\seed^\\vee}$ has tangent direction in $\\lrp{\\Trop_{\\R}(w_H)}^{-1}(0)_{\\seed^\\vee}$. \nTherefore, after bending it will remain in the set $x+\\lrp{\\Trop_{\\R}(w_H)}^{-1}(0)_{\\seed^\\vee}$. \nFinally, observe that $x+\\lrp{\\Trop_{\\R}(w_H)}^{-1}(0)_{\\seed^\\vee}=\\lrp{\\Trop_{\\R}(w_H)}^{-1}(q)_{\\seed^\\vee}$. \n\\end{proof}\n\nHaving in mind Proposition~\\ref{prop:dual_fibration} and the action of the $T_{\\text{Pic}(Y)^*}$ on $\\UT_Y$ we introduce the following notion.\n\n\\begin{definition}\n\\thlabel{k_and_pic}\nThe pair $(p,H) $ has the {\\bf Picard property} with respect to $V\\subset \\UT_Y$ if\n\\begin{itemize}\n \\item $H$ and $\\text{Pic}(Y)^*$ have the same rank, and\n \\item the action of $T_{H}$ on $\\cA$ coincides with the action of $T_{\\text{Pic}(Y)^*}$ on $\\UT_Y$ restricted to the image of $\\Phi:\\cA \\dashrightarrow V $.\n\\end{itemize}\n\\end{definition}\n\nRecall the definitions of the superpotential and its associated cone of tropical points from \\eqref{eq:def superpotential} and \\eqref{eq:def Xi} in \\S\\ref{sec:minimal_models}.\nThe following result adapts the content of Proposition \\ref{prop:dual_fibration} to this framework. \n\n\n\\begin{lemma}\n\\label{lem:basis_of_tf}\nSuppose that $ V \\subset \\UT_Y$ is a partial minimal model with enough theta functions and that $(p,H)$ has the Picard property with respect to this model.\nThen for every class $[\\lb]\\in \\text{Pic}(Y)\\cong H^*$ we have that the theta functions parametrized by the integral points of the set $ \\lrp{\\Trop_{\\R}(w_H)}^{-1}\\lrp{[\\lb]}\\cap \\Xi_{\\UT_Y}$ is a basis for $H^0(Y, \\lb)$.\nIn particular, $\\Cox(Y)$ has a basis of theta functions which are $T_{\\text{Pic}(Y)^*}$-eigenfunctions. \n\\end{lemma}\n\nWe consider the section ring $R(\\lb)=\\bigoplus_{k\\geq 0} R_k(\\lb) $. The $k^{\\mathrm{th}}$ homogeneous component is defined as $R_k(\\lb )=H^0(Y, \\lb^{\\otimes k})$. The product of $R(\\lb)$ is given by the tensor product of sections. \nFix a seed $\\seed\\in \\orT$, a linear dominance order $<_{\\seed}$ on $ \\Trop_{\\Z}(\\cA^{\\vee}_{\\seed^\\vee})$ and consider the valuation\n$\n\\gv^{\\Phi}_{\\seed}:\\Bbbk(V)\\setminus \\{ 0 \\} \\to (\\Trop_{\\Z}(\\cA^{\\vee}_{\\seed^\\vee}), <_{\\seed}). \n$\nObserve that $R_k(\\lb)\\subset \\Cox(Y)$ for all $k$. Hence we can define the Newton--Okounkov body\n\\eqn{\n\\Delta_{\\gv^{\\Phi}_{\\seed}}(\\lb) := \\overline{\\conv\\Bigg( \\bigcup_{k\\geq 1}\\lrc{ \\frac{1}{k}\\gv^{\\Phi}_{\\seed} (f) \\mid f\\in R_k(\\mathcal L)\\setminus \\{0\\} } \\Bigg) }\\subseteq \\Trop_{\\Z}(\\cA^\\vee_{\\seed^\\vee})=M^{\\circ}_{\\R}.\n}\n\n\\begin{theorem}\\thlabel{thm:k_and_pic}\nSuppose that $ V \\subset \\UT_Y$ is a partial minimal model with enough theta functions and that $(p,H)$ has the Picard property with respect to this model.\nThen for any line bundle $\\lb $ on $Y$\n\\[\n\\Delta_{{\\bf g}^{\\Phi}_{\\seed}}(\\lb)=\\lrp{\\Trop_{\\R}(w_H)}^{-1}\\lrp{[\\lb]}_{\\seed^\\vee}\\cap \\Xi_{\\UT_Y, \\seed^\\vee}.\n\\]\nIn particular, $\\Delta_{{\\bf g}^{\\Phi}_{\\seed}}(\\lb)$ is a positive subset of $\\Trop_{\\R}(\\cA^{\\vee}_{\\seed^\\vee})$.\n\\end{theorem}\n\\begin{proof}\nTo make notation lighter, throughout this proof we let $S=\\lrp{\\Trop_{\\R}(w_H)}^{-1}\\lrp{[\\lb]}_{\\seed^\\vee}\\cap \\Xi_{\\UT_Y,\\seed^\\vee}$ and denote $\\gv^{\\Phi}_{\\seed}$ simply by $\\gv_{\\seed}$. Observe that $[\\lb^{\\otimes k}]=k[\\lb]$ in $\\text{Pic}(Y)$. \nTherefore, by Lemma \\ref{lem:basis_of_tf} we have that ${\\bf g}_{\\seed}(R_k(\\lb))\\subseteq \\lrp{\\Trop_{\\R}(w_H)}^{-1}\\lrp{k[\\lb]}_{\\seed^\\vee}$ for all $k\\geq 1$. \nIn particular, $\\dfrac{1}{k}{\\bf g}_{\\seed}(R_k(\\lb))\\subseteq \\lrp{\\Trop_{\\R}(w_H)}^{-1}\\lrp{[\\lb]}_{\\seed^\\vee}$ for all $k \\geq 1$.\nSince $\\lrp{\\Trop_{\\R}(w_H)}^{-1}\\lrp{[\\lb]}_{\\seed^\\vee} $ is closed in $\\Trop_{\\R}(\\cA^{\\vee}_{\\seed^\\vee})$ and convex we have that $\\Delta_{{\\bf g}_{\\seed}}(\\lb)\\subseteq \\lrp{\\Trop_{\\R}(w_H)}^{-1}\\lrp{[\\lb]}_{\\seed^\\vee}$.\nLet $ \\mathbb B_k$ be the theta basis of $R_k(\\lb)$, see \\thref{g_is_val}. Since the theta basis is adapted for ${\\bf g}_{\\seed}$ we have that ${\\bf g}_{\\seed}(R_k(\\lb))={\\bf g}_{\\seed}(\\mathbb B_k)$. \nSince $\\cA \\subseteq \\UT_Y$ has enough theta functions, every theta function $\\tf \\in \\mathbb B_k$ is a global function on $\\UT_Y$, therefore, we have that ${\\bf g}_{\\seed}(\\tf) \\in \\Xi_{\\UT_Y}$.\nSince $\\Xi_{\\UT_Y}$ is closed in $\\Trop_{\\R}(\\cA^{\\vee}_{\\seed^\\vee})$, convex and closed under positive scaling then $\\Delta_{{\\bf g}_{\\seed}}(\\lb)\\subseteq \\Xi_{\\UT_Y,\\seed^\\vee}$.\nHence, $\\Delta_{{\\bf g}_{\\seed}}(\\lb)\\subseteq S$. \nTo see the reverse inclusion we notice that the set of rational points of $S$ coincide with the set $ \\bigcup_{k\\geq 1} \\frac{1}{k} \\gv_{\\seed }(\\mathbb B_k)= \\bigcup_{k\\geq 1} \\frac{1}{k} \\gv_{\\seed }\\lrp{R_k(\\lb)}$.\nSince $S$ can be expressed as the closure of its set of rational points we have that $S\\subseteq \\Delta_{\\gv_{\\seed}}(\\lb)$.\nFinally, since $\\lrp{\\Trop_{\\R}(w_H)}^{-1}\\lrp{[\\lb]}_{\\seed^\\vee}$ and $\\Xi_{\\UT_Y,\\seed^\\vee}$ are positive sets then $S=\\Delta_{{\\bf g}_{\\seed}}(\\lb)$ is an intersection of positive sets. Hence, it is positive.\n\\end{proof}\n\n\\begin{remark}\n\\label{rem:comparing_NO_bodies}\nUnder the assumptions of \\thref{thm:k_and_pic} we have that $Y$ is a minimal model with enough theta functions for an open subscheme $V'\\subset Y$ with a cluster structure given by a birational map $ \\Phi':\\cA\/T_{H}\\dashrightarrow V'$ induced by $\\Phi$.\nTo relate the Newton--Okounkov bodies constructed in this section with those constructed in the former we let $\\lb $ be isomorphic to $\\mathcal{O}(D')$ for some Weil divisor $D'$ on $Y$ satisfying the framework of \\S\\ref{sec:NO_bodies}.\nUnder the identification $ \\Trop_{\\R}(\\cA^{\\vee}_{\\seed^\\vee}) = M^\\circ_\\R$ we realize $ \\Trop_{\\R}((\\cA\/T_H)^{\\vee}_{\\seed^\\vee})$ as the subset of $M^\\circ_\\R$ orthogonal to $H$ (see \\S\\ref{tf_quotient}). \nFor any $\\tau \\in R_1(D')$ we have $\\Delta_{\\gv^{\\Phi'}_\\seed}(D',\\tau)\\subset M_{\\R}^\\circ\\cap \\lrp{\\Trop_{\\R}(w_H)}^{-1}(0)_{\\seed^\\vee}$ and by construction\n\\[\n\\Delta_{\\gv^{\\Phi'}_\\seed}(D',\\tau) =\\Delta_{\\gv^{\\Phi}_\\seed}(\\lb)- \\gv^{\\Phi}_{\\seed}(\\tau).\n\\]\n\\end{remark}\n\n\\begin{example}\\thlabel{exp:full_flag}\nAn important class of examples is provided by the base affine spaces. \nConsider $G=SL_{n+1}(\\Bbbk)$ and $B\\subset G$ a Borel subgroup with unipotent subgroup $U\\subset B$.\nThen $G\/U$ is a universal torsor for $G\/B$. \nMoreover, $G\/U$ carries a cluster structure induced by the double Bruhat cell $G^{e,w_0}:=B_-\\cap Bw_0B$, where $B_-\\subset G$ is the Borel subgroup opposite to $B$ (i.e. $B\\cap B^-=:T$ is a maximal torus) and $w_0$ the longest element in this Weyl group $S_n$ is identified with a matrix representative in $N_G(T)\/C_G(T)$ (the normalizer of $T$ modulo the centralizer of $T$).\nThe cluster structure on $G^{e,w_0}$ was introduced by Berenstein--Fomin--Zelevinsky in \\cite{BFZ05} and it follows that (up to co-dimension 2) $G^{e,w_0}$ agrees with the corresponding $\\mathcal A$-cluster variety. \nBy \\cite[Proposition 23]{Mag15} there is an embedding $G^{e,w_0}\\hookrightarrow G\/U$ compatible with the cluster structure. \nIn particular, $G\/U$ is a partial compactification of the $\\mathcal A$-cluster variety $G^{e,w_0}$ obtained by adding the locus where frozen variables are allowed to vanish.\nMagee further proved in \\cite{Mag20} that the full Fock--Goncharov conjecture holds and a cluster ensemble map satisfying \\thref{k_and_pic} is provided in \\cite{Mag20}.\nHence, we obtain a ${\\bf g}$-vector valuation ${\\bf g}_{\\seed}$ on $H^0(G\/U,\\mathcal O_{G\/U})$ for every choice of seed $\\seed$.\n\nIn particular, \\thref{thm:k_and_pic} applies: recall that the Picard group of $G\/B$ is isomorphic to the lattice spanned by the fundamental weights $\\omega_1,\\dots,\\omega_{n}$. \nLet $\\Lambda$ denote the dominant weights, \\ie its elements are $\\lambda=a_1\\omega_1+\\dots+a_n\\omega_n$ with $a_i\\in \\mathbb Z_{\\ge 0}$ and let $\\mathcal L_\\lambda\\to G\/B$ be the associated line bundle.\nThe ring of regular functions on the quasi-affine variety $G\/U$ coincides with the Cox ring of the flag variety:\n\\[\nH^0(G\/U,\\mathcal O_{G\/U})\\cong \\bigoplus_{\\lambda \\in \\Lambda} H^0 (G\/B,\\mathcal L_\\lambda).\n\\]\nHence, we may restrict the ${\\bf g}$-vector valuations ${\\bf g}_{\\seed}$ for all seeds $\\seed$ to the section ring of any line bundle on $G\/B$.\nThe resulting Newton--Okounkov polytopes coincide with slices of the tropicalization of the superpotential corresponding to the compactification. It has been shown in \\cite{BF,GKS_typeA} that for certain choices of seeds these polytopes are unimodularly equivalent to Littelmann's string polytopes (see \\cite{Lit98,BZ01}).\n\\end{example}\n\n\\begin{example}\nGrassmannians also form a distinguished class of examples fitting this framework. We treat this class separately in \\S\\ref{sec:NO_Grass}.\n\\end{example}\n\n\n\n\n\\subsection{The intrinsic Newton--Okounkov body}\\label{sec:intrinsic_NOB} \nIn the situation of \\S\\ref{sec:NO_bodies} or \\S\\ref{sec:universal_torsors}, we can choose two seeds $\\seed, \\seed'\\orT$ to obtain two Newton--Okounkov bodies, say $\\Delta_{\\nu_\\seed}$ and $\\Delta_{\\nu_\\seed'}$ (these are associated to a line bundle $\\lb$ in case we are in a framework as in \\S\\ref{sec:universal_torsors} or to a divisor $D'$ and a section $\\tau$ in case our framework is as in \\S\\ref{sec:NO_bodies}).\nIn the same spirit as in \\cite{EH20,FH21} (see also \\cite[\\S4]{BMNC} and \\cite{HN23,CHM22}), in this section we show that if one of $\\Delta_{\\nu_{\\seed}}$ or $\\Delta_{\\nu_{\\seed'}}$ (equivalently both) is a positive set then these Newton--Okounkov bodies are related to each other by a distinguished piecewise linear transformation and, moreover, any such Newton--Okounkov body can be intrinsically described as a \\emph{broken line convex hull} (see Theorems \\ref{thm:intrinsic} and \\ref{thm:intrinsic_lb} below).\nIn order to obtain the last assertion we rely on \\cite{CMNcpt}. Along the way we introduce a theta function analog of the Newton polytope associated to a regular function on a torus.\n\nWe start by considering Newton--Okounkov bodies associated to Weil divisors as in \\S\\ref{sec:NO_bodies}. Let $\\cV$ be a scheme of the form $\\cA$, $\\cX$, $\\cA\/T_{H}$ or $\\cX_{\\bf 1}$ and $(V, \\Phi)$ a scheme with a cluster structure of type $\\cV$.\nDenote by $\\mathbb{B}_{\\tf}(\\cV)=\\{\\tf^{\\cV}_{\\bf v}\\mid {\\bf v}\\in \\Theta(\\cV)\\}$ the theta basis of $\\cmid(\\cV)$.\nWe begin by observing that a cluster valuation $\\nu_{\\seed}$ on $\\cmid(\\cV)$ can be thought of as an extension of the composition of the seed-independent map \n\\begin{eqnarray}\n \\label{eq:nu_seed_free}\n \\nu: \\mathbb{B}_{\\tf}(\\cV) &\\to& \\Trop_{\\Z}(\\cV^\\vee)\\\\\n \\nonumber\n\\tf^{\\cV}_{\\bf v} &\\mapsto &{\\bf v},\n\\end{eqnarray}\nwith the identification $\\mathfrak{r}_{\\seed^\\vee}:\\Trop_{\\Z}(\\cV^\\vee) \\to \\Trop_{\\Z}(\\cV^\\vee_{\\seed^\\vee})$. \nIf $ \\mathbb B_{\\tf}(V)$ denotes the set of polynomial theta functions on $V$ then we can define $\\nu^{\\Phi} : \\mathbb B_{\\tf}(V) \\to \\Trop_{\\Z}(\\cV^\\vee)$ analogously.\nMoreover, even though $\\Trop_{\\Z}(\\cV^{\\vee})$ may not have a linear structure, if $\\Theta (\\cV)= \\Trop_{\\Z}(\\cV^{\\vee})$ and $L\\subseteq \\Trop_{\\Z}(\\cV^{\\vee})$ is a linear subset acting linearly on $\\Trop_{\\Z}(\\cV^{\\vee})$ (see Definition \\ref{def:linear_action}) then for every $y\\in L$ we have a well defined ``subtraction\" function\n\\eqn{ (\\ \\cdot \\ )-y: \\Trop_{\\Z}(\\cV^{\\vee}) &\\to \\Trop_{\\Z}(\\cV^{\\vee})\\\\\nx &\\mapsto x-y, }\nwhere $-y $ is the unique point of $ \\Trop_{\\Z}(\\cV^{\\vee})$ such that $\\tf_y\\tf_{-y}=1$ and $x-y$ is the unique point of $\\Trop_{\\Z}(\\cV^{\\vee})$ such that $\\tf_{x}\\tf_{-y}=\\tf_{x-y}$. \n\nWe now define our notion of convexity. Recall from \\S\\ref{sec:tf_A} that we might think of supports of broken lines as seed independent objects. In light of this we consider the following.\n\n\\begin{definition}\\label{def:blc_intro} \\cite{CMNcpt}\nA closed subset $S$ of $\\Trop_{\\R}(\\cV)$ is {\\bf{broken line convex}} \nif for every pair of rational points $s_1, s_2$ in $S(\\Q)$,\nevery segment of a broken line with endpoints $s_1$ and $s_2$ is entirely contained in $S$.\n\\end{definition}\n\n\\begin{remark}\n\\label{rem:non-generic_bl}\nThe broken lines considered in Definition \\ref{def:blc_intro} include those that are \\emph{non-generic}. Namely, broken lines that are obtained as limits of the generic broken lines introduced in \\thref{def:genbroken}. See \\cite[Definition~3.3]{CMNcpt} for details. \n\\end{remark}\n\nThe main result of \\cite{CMNcpt} asserts that positivity of a set is equivalent to its broken line convexity:\n\n\\begin{theorem}\n\\thlabel{thm:mainCMN}\n\\cite[Theorem 6.1]{CMNcpt}\nLet $\\cV$ be a variety of the form $\\cA $, $\\cX$, $\\cA\/T_{H}$ or $\\cX_{\\bf 1}$. Then a closed subset $S$ of $\\Trop_{\\R}(\\cV)$ is is broken line convex if and only if it is positive.\n\\end{theorem}\n\nMorally, this means that broken line convexity in $\\Trop_{\\R}(\\cV^\\vee)$ play the same role in describing partial minimal models of $\\cV$ that usual convexity in $M_\\R$ plays in describing normal toric varieties $T_N \\subset X$.\nOne appealing feature of the broken line convexity notion is that it makes no reference to any auxiliary data-- given $\\cV$, we can talk about broken line convexity in $\\Trop_{\\R}(\\cV^{\\vee})$.\nIn contrast, the Newton--Okounkov bodies we discussed in \\S\\ref{sec:NO_bodies}\nand \\S\\ref{sec:universal_torsors}\n are convex bodies whose construction depends upon a choice of seed $\\seed$.\nMore generally, a usual Newton--Okounkov body depends not only on the geometric data of a projective variety together with a divisor but also on the auxiliary data of a choice of valuation.\nBroken line convexity makes no reference to any such auxiliary data and will lead us to an intrinsic version of a Newton--Okounkov body. \n\n\\begin{definition}\n\\label{def:bl_convex_hull}\nLet $S \\subset\\Trop_{\\R}(\\cV^{\\vee})$ be a set. The {\\bf{broken line convex hull of $S$}}, denoted by $\\bconv(S)$, is the intersection of all broken line convex sets containing $S$. \n\\end{definition}\n\n\\begin{remark}\n We can also define broken line convexity \n and broken line convex hulls inside $\\Trop_{\\R}(\\cV^{\\vee}_{\\seed^\\vee})$ in exactly the same way they are defined in Definitions \\ref{def:blc_intro} and \\ref{def:bl_convex_hull}. \n In particular, we have that $S\\subset \\Trop_{\\R}(\\cV^{\\vee})$ is broken line convex if and only if $\\mathfrak{r}_{\\seed^\\vee}(S)\\subset \\Trop_{\\R}(\\cV^\\vee_{\\seed})$ is broken line convex.\n\\end{remark}\n\nUsing this convexity notion, we describe a set analogous to the Newton polytope of a function on a torus.\n\\begin{definition}\n\tGiven a regular function ${f= \\sum_{{\\bf v} \\in \\Trop_{\\Z}(\\cV^{\\vee})}} a_{\\bf v} \\tf^{V}_{\\bf v}$ on $V$, we define the {\\bf{$\\tf$-function analogue of the Newton polytope of $f$}} to be \n\t\\eqn{ \\NewtT(f) := \\bconv\\lrc{ {\\bf v} \\in \\Trop_{\\Z}(\\cV^{\\vee}) \\mid a_{\\bf v} \\neq 0 }. }\n\\end{definition}\n\nThis leads to an intrinsic version of the Newton--Okounkov bodies we have constructed. So consider a partial minimal model $ V \\subset Y$ and let $D'$ be a divisor on $Y$ supported on the boundary of $V\\subset Y$. \n\n\\begin{definition}\nAssume that $R(D')$ has a graded theta basis (see Definition \\ref{def:graded_theta_basis}). Then the associated {\\bf{intrinsic Newton--Okounkov body}} is\n\\eqn{\n\\Delta_{\\mathrm{BL}}(D'):= \\bconv\\Bigg( \\bigcup_{k\\geq 1} \\Bigg(\\bigcup_{f \\in R_k(D')} \\frac{1}{k} \\NewtT(f) \\Bigg) \\Bigg)\\subseteq \\Trop_{\\R}(\\cV^\\vee).\n}\n\\end{definition}\n\nIn order to describe how the different realizations of intrinsic Newton--Okounkov bodies are related we record the tropicalization of the gluing map $\\mu^{\\cV^\\vee}_k:\\cV^{\\vee}_\\seed \\dashrightarrow \\cV^{\\vee}_{\\seed'}$ in terms of the fixed data $\\Gamma$ and inital seed $\\seed_0=(e_i)_{i\\in I}$ defining $\\cV$. \n\\begin{equation*}\n\\Trop_{\\R}\\lrp{\\mu^{\\cA^\\vee}_{k}}(m)=\\begin{cases} m + \\langle d_ke_k, m \\rangle v_k & \\text{if } \\langle e_k, m \\rangle \\geq 0,\\\\\nm & \\text{if } \\langle e_k, m \\rangle \\leq 0,\n\\end{cases}\n\\end{equation*}\nfor $m \\in M^{\\circ}$. \n\\[\n\\Trop_{\\R}\\lrp{\\mu^{\\cX^\\vee}_{k}}(n)=\\begin{cases} n + \\{n,d_ke_k \\} e_k & \\text{if } \\{ n,e_K \\}\\geq 0,\\\\\nn & \\text{if } \\{ n,e_K\\} \\leq 0,\n\\end{cases}\n\\]\nfor $n \\in N$. \n\\[\n\\Trop_{\\R}\\lrp{\\mu^{(\\cXe)^\\vee}_{k}}(n+H)=\\begin{cases} n + \\{n,d_ke_k \\}e_k + H & \\text{if } \\{ n, e_k \\}\\geq 0,\\\\\nn + H& \\text{if } \\{ n, e_k \\} \\leq 0,\n\\end{cases}\n\\]\nfor $n + H \\in N\/H$. \n\\[\n\\Trop_{\\R}\\lrp{\\mu^{(\\cA\/T_H)^\\vee}_{k}} = \\Trop_{\\R}\\lrp{\\mu^{\\cA^\\vee}_{k}} \\mid_{H^\\perp}.\n\\]\n\n\n\\begin{theorem}\n\\thlabel{thm:intrinsic}\nLet $(V,\\Phi)$ be a scheme with a cluster structure of type $\\cV$ and let $V \\subset Y$ be a partial minimal model. Assume that the full Fock--Goncharov conjecture holds for $\\cV$ and that there exists a theta function $\\tau \\in R_1(D')$ such that $\\nu^{\\Phi}_{\\seed}(\\tau)$ lies in a linear subset of $\\Trop_{\\Z}(\\cV^\\vee)$. If $\\Delta_{\\nu^{\\Phi}_{\\seed}}(D',\\tau)$ is positive then for every seed $\\seed \\in \\orT$ we have that $\\mathfrak{r}_{\\seed^\\vee}(\\Delta_{\\mathrm{BL}}(D')-\\nu^{\\Phi}(\\tau))= \\Delta_{\\nu^{\\Phi}_{\\seed}}(D',\\tau) $.\nIn particular, for any other seed $\\seed'\\in \\orT $ we have that\n\\[\n\\Delta_{\\nu_{\\seed'}}(D', \\tau )= \\Trop_{\\R}\\lrp{\\mu^{\\cV^\\vee}_{\\seed,\\seed'}}\\lrp{\\Delta_{\\nu_{\\seed}}(D', \\tau)}.\n\\] \n\\end{theorem}\n\\begin{proof}\nIt is enough to treat the case $V= \\cV$. We consider the broken line convex hull of \n\\[\nS=\\bigcup_{k\\geq 1}\\lrc{\\dfrac{\\nu_{\\seed}(f)}{k}-\\nu_{\\seed}(\\tau)\\mid f\\in R_k(D') \\setminus\\{0\\}} \n\\]\nin $\\Trop_{\\R}(\\cV^{\\vee}_{\\seed^\\vee})$. \nSince all line segments of $\\Trop_{\\R}(\\cV^{\\vee}_{\\seed^\\vee})$ can be thought of as a segment of a broken line and $\\Delta_{\\nu_{\\seed}}(D', \\tau)$ is closed we have that $ \\Delta_{\\nu_{\\seed}}(D', \\tau)\\subseteq \\bconv(S)$. \nBy \\thref{thm:mainCMN} $\\Delta_{\\nu_{\\seed}}(D', \\tau)$ is broken line convex. Since $S\\subset \\Delta_{\\nu_{\\seed}}(D', \\tau)$ we have the reverse inclusion. \nThe last statement follows from the fact that broken line convex sets are preserved by $\\Trop_{\\R}(\\mu^{\\cV^{\\vee}}_k)$. \n\\end{proof}\n\nThere is an analogous result for line bundles fitting the framework of \\S\\ref{sec:universal_torsors}.\n\n\\begin{definition}\n\\label{def:intrinsic_lb}\nLet $Y$ be a projective variety such that $\\text{Pic}(Y)$ is free of finite rank. Assume $(V, \\Phi)$ is a scheme with a cluster structure of type $\\cA$ and that $V \\subset \\UT_Y$ is a partial minimal model with enough theta functions. Let $(p, H)$ have the Picard property (see \\thref{k_and_pic}). The {\\bf{intrinsic Newton--Okounkov body} associated to a class $[ \\lb ]\\in \\text{Pic}(Y)\\cong H^*$} is\n\\eqn{\n\\Delta_{\\mathrm{BL}}(\\lb):= \\bconv\\Bigg( \\bigcup_{k\\geq 1} \\Bigg(\\bigcup_{f \\in R_k(\\lb)} \\frac{1}{k} \\NewtT(f) \\Bigg) \\Bigg)\\subseteq \\Trop_{\\R}(\\cA^{\\vee}).\n}\n\\end{definition}\n\nIn this case we have the following theorem whose proof is completely analogous to the proof of \\thref{thm:intrinsic}. Moreover, it uses the fact that $\\nu_{\\seed}(\\lb)$ is a positive set, as shown in \\thref{thm:k_and_pic}.\n\n\\begin{theorem}\n\\thlabel{thm:intrinsic_lb}\nKeep the assumptions of Definition \\ref{def:intrinsic_lb}.\nFor every seed $\\seed\\in \\orT$ we have that $\\Delta_{\\nu^{\\Phi}_{\\seed}}(\\lb)=\\mathfrak{r}_{\\seed^\\vee}(\\Delta_{\\mathrm{BL}}(\\lb))$. In particular, for every $\\seed' \\in \\orT $ we have that\n\\[\n\\Delta_{\\nu^\\Phi_{\\seed'}}(\\lb )= \\lrp{\\mu^{\\cV^\\vee}_{\\seed^\\vee, \\seed'^\\vee}}^T(\\Delta_{\\nu^\\Phi_{\\seed}}(\\lb)).\\] \n\\end{theorem}\n\\begin{proof}\nWe showed in \\thref{thm:k_and_pic} that $\\Delta_{\\nu_{\\seed}}(\\lb)$ is a positive set. The proof of this result is completely analogous to the proof of \\thref{thm:intrinsic}.\n\\end{proof}\n\nIn either situation (divisors or line bundles) we are of course free to compute the intrinsic Newton--Okounkov body as a usual Newton--Okounkov body in any vector space realization of $\\Trop_{\\R}(\\cV^{\\vee})$.\nHowever, the intrinsic definition has certain advantages as we now explain.\nFor simplicity, from now on we concentrate on line bundles as in \\thref{thm:intrinsic_lb}; the reader can make the appropriate changes for the case of divisors as in \\thref{thm:intrinsic}.\nIt is often the case that $\\Delta_{\\mathrm{BL}}(\\lb) = \\bconv \\Big( \\bigcup_{k=1}^\\ell \\frac{1}{k} \\nu^{\\Phi}\\lrp{R_k(\\lb)} \\Big)$ for some finite $\\ell$, meaning in these cases the infinite union reduces to finite union.\nConsider such an instance and let $\\ell_{\\seed}$ be the smallest integer such that $\\Delta_{\\nu^{\\Phi}_{\\seed}}(\\lb)=\\conv \\Big( \\bigcup_{k=1}^{\\ell_{\\seed}} \\frac{1}{k} \\nu^{\\Phi}_{\\seed}\\lrp{R_k(\\lb)} \\Big)$. \nThen the corresponding $\\ell$ for the intrinsic Newton--Okounkov body is at most $\\min_{\\seed}\\lrc{\\ell_{\\seed}}$.\nMoreover, we can give conditions indicating when $\\ell$ has been attained. \nWe will start with a condition that, after adopting a slightly different perspective on theta functions, becomes tautological.\\footnote{This perspective is essentially the {\\it{jagged path}} description of theta functions rather than the broken line description. See for example \\cite[Section~3]{GS12}.} \nWe will then adapt this condition to give a sufficient criterion that is more likely to be known for a given minimal model (and a known line bundle or Weil divisor).\n\n\\begin{proposition}\\thlabel{taut}\nLet $\\lb$ be as in \\thref{thm:intrinsic_lb}. Suppose there exists a positive integer $\\ell$ such that for all $h>\\ell$, each theta function $\\tf^V_r$ in $R_h(\\lb)$ appears as a summand (with non-zero coefficient) of some product $\\tf^V_p \\tf^V_q$, where $\\tf^V_p \\in R_i(\\lb)$ and $\\tf^V_q \\in R_j(\\lb)$ for some positive integers $i$ and $j$ with $i+j =h$. \nThen \\eqn{\n\\Delta_{\\mathrm{BL}}(\\lb) = \\bconv\\Bigg( \\bigcup_{k=1}^{\\ell} \\Bigg(\\bigcup_{f \\in R_k(\\lb)} \\frac{1}{k} \\NewtT(f) \\Bigg) \\Bigg) .\n}\n\\end{proposition}\n\n\\begin{proof}\nThis is an immediate consequence of results in \\cite{CMNcpt}. We adopt the terminology and conventions of {\\it loc. cit.} for this proof. In particular, we allow non-generic broken lines (see Remark \\ref{rem:non-generic_bl}).\n\nSince the structure constant $\\alpha(p,q,r)$ is non-zero, there exists a pair of broken lines $\\lrp{\\gamma_1,\\gamma_2}$ with $I(\\gamma_1) = p $, $I(\\gamma_2) = q $, $\\gamma_1(0)=\\gamma_2(0) = r$, and $F(\\gamma_1)+ F(\\gamma_2) = r$.\nThen the construction of \\cite[\\S4]{CMNcpt} yields a broken line segment from $\\frac{p}{i}$ to $\\frac{q}{j}$ passing through $\\frac{r}{h}$. \nAs a consequence, we have \n\\eqn{\\frac{r}{h} \\in \\bconv\\Bigg( \\bigcup_{k=1}^{\\max(i,j)} \\Bigg(\\bigcup_{f \\in R_k(\\lb)} \\frac{1}{k} \\NewtT(f) \\Bigg) \\Bigg) . }\nBy hypothesis, $R_k(\\lb)$ has a basis of theta functions for all $k$, so \n\\eqn{\n\\bconv \\Bigg(\\bigcup_{f \\in R_h(\\lb)} \\frac{1}{h} \\NewtT(f) \\Bigg) = \\bconv \\lrp{ \\frac{r}{h} \\mid \\tf^V_r \\in R_h(\\lb)} .\n}\nWe have just seen that each such $\\frac{r}{h}$ is contained in \n\\eqn{\\bconv\\Bigg( \\bigcup_{k=1}^{h-1} \\Bigg(\\bigcup_{f \\in R_h(\\lb)} \\frac{1}{h} \\NewtT(f) \\Bigg) \\Bigg), }\nso \n\\eqn{\n\\bconv \\Bigg(\\bigcup_{f \\in R_h(\\lb)} \\frac{1}{h} \\NewtT(f) \\Bigg) \\subset \\bconv\\Bigg( \\bigcup_{k=1}^{h-1} \\Bigg(\\bigcup_{f \\in R_k(\\lb)} \\frac{1}{k} \\NewtT(f) \\Bigg) \\Bigg).\n}\nAs this holds for all $h>\\ell$, we conclude that\n\\eqn{\n\\Delta_{\\mathrm{BL}}(\\lb) = \\bconv\\Bigg( \\bigcup_{k=1}^{\\ell} \\Bigg(\\bigcup_{f \\in R_k(\\lb)} \\frac{1}{k} \\NewtT(f) \\Bigg) \\Bigg) .\n}\n\\end{proof}\n\n\\begin{remark}\nIn dimension 2, Mandel \\cite{Man16} showed that the assumption in \\thref{taut} implies that $r=p+q$ in some seed. It is a very interesting problem to determine if this holds for higher dimensions. \n\\end{remark}\n\nNote that as we have (by assumption) a theta basis for $R(\\lb)$, the condition of \\thref{taut} is implied by the following condition:\n\n\\noindent\n\\begin{condition}\\thlabel{condition_section ring}\nThere exists a positve integer $\\ell$ such that for all $h>\\ell$, the natural map $R_i (\\lb) \\otimes R_j(\\lb) \\to R_h (\\lb)$ is surjective for some positive integers $i$ and $j$ with $i+j =h$.\n\\end{condition} \n\n\\begin{remark}\\label{rmk:borel weil bott}\nThe \\thref{condition_section ring} is satisfied in our main class of examples coming from representation theory: recall the setting of \\thref{exp:full_flag} where line bundles $\\mathcal L_\\lambda$ of the full flag variety $G\/B$ are indexed by dominant weights $\\lambda$.\nBy the Borel--Weil--Bott Theorem the graded pieces $R_i(\\mathcal L_\\lambda)$ of the section rings of these line bundles satisfy\n\\[\nR_i(\\mathcal L_\\lambda)\\cong V(i\\lambda)^*,\n\\]\nwhere $V(i\\lambda)$ is the irreducible $G$-representation of highest weight $i\\lambda$ and $i\\ge 0$. \nBy work of Baur \\cite{Baur_CartanComp} the tensor product $V(i\\lambda)\\otimes V(j\\lambda)$ contains among its irreducible components the unique component of maximal weight, called Cartan component, which is $V((i+j)\\lambda)$. \nHence,\n\\[\nR_i(\\mathcal L_\\lambda)\\otimes R_j(\\mathcal{L}_\\lambda)\\cong V(i\\lambda)^*\\otimes V(j\\lambda)^*\\twoheadrightarrow V((i+j)\\lambda)^*\\cong R_{i+j}(\\mathcal L_\\lambda).\n\\]\nAlthough in \\thref{exp:full_flag} we only treat the case of $SL_{n+1}(\\Bbbk)$ it is worth noticing that the Borel--Weil(--Bott) Theorem holds for semisimple Lie groups and algebraic groups over $\\Bbbk$ and Baur's result holds for irreducible representations of connected, simply-connected complex reductive groups.\nNotice further that these observations also hold for partial flag varieties, \\ie quotient $G\/P$ by parabolic subgroups $P\\subset G$ as the cohomology of an equivariant line bundles on $G\/P$ is equal to the cohomology of its pullback along the natural projection $G\/B\\twoheadrightarrow G\/P$. So the cohomology of the line bundle on $G\/P$ can be calculated using the usual Borel--Weil(--Bott) Theorem for $G\/B$, by the Leray spectral sequence.\n\\end{remark}\n\n\n\n\\section{The case of the Grassmannian}\n\\label{sec:NO_Grass}\n\n\nWe now consider in detail the case of the Grassmannians. \nThroughout this section we work over the complex numbers, fix two positive integers $k] (-1.25,9.875) -- (-.25,9.125);\n\\draw[->,blue,dashed] (7,9.25) to [out=150,in=0] (-1.25,10.125);\n\\draw[->,blue,dashed] (-.25,4.75) to [out=155,in=-90] (-1.5,9.75);\n\\node at (0,9) {$\\tiny{\\yng(1)}$};\n\\node at (1.5,9) {$\\tiny{\\yng(2)}$};\n\\node at (3,9) {$\\tiny{\\yng(3)}$};\n\\node at (5,9) {$\\tiny{\\yng(4)}$};\n\\node[blue] at (7,9) {$\\tiny{\\yng(5)}$};\n%right arrow\n\\draw[->] (.25,9) -- (1.125,9);\n\\draw[->] (1.875,9) -- (2.5,9);\n\\draw[->] (3.5,9) -- (4.375,9);\n\\draw[->] (5.625,9) -- (6.25,9);\n%down arrows\n\\draw[->] (0,8.75) -- (0,8.25);\n\\draw[->] (1.5,8.75) -- (1.5,8.25);\n\\draw[->] (3,8.75) -- (3,8.25);\n\\draw[->] (5,8.75) -- (5,8.25);\n\\draw[->,blue,dashed] (7,8.75) -- (7,8.25);\n%diagonal arrows\n\\draw[<-] (.25,8.75) -- (1.25,8.25);\n\\draw[<-] (1.875,8.75) -- (2.75,8.25);\n\\draw[<-] (3.5,8.75) -- (4.375,8.25);\n\\draw[<-] (5.625,8.75) -- (6.5,8.25);\n\n\\node at (0,7.875) {$\\tiny{\\yng(1,1)}$};\n\\node at (1.5,7.875) {$\\tiny{\\yng(2,2)}$};\n\\node at (3,7.875) {$\\tiny{\\yng(3,3)}$};\n\\node at (5,7.875) {$\\tiny{\\yng(4,4)}$};\n\\node[blue] at (7,7.875) {$\\tiny{\\yng(5,5)}$};\n\\draw[->] (.25,7.875) -- (1.125,7.875);\n\\draw[->] (1.875,7.875) -- (2.5,7.875);\n\\draw[->] (3.5,7.875) -- (4.375,7.875);\n\\draw[->] (5.625,7.875) -- (6.25,7.875);\n\n\\draw[->] (0,7.5) -- (0,7);\n\\draw[->] (1.5,7.5) -- (1.5,7);\n\\draw[->] (3,7.5) -- (3,7);\n\\draw[->] (5,7.5) -- (5,7);\n\\draw[->,blue,dashed] (7,7.5) -- (7,7);\n%diagonal arrows\n\\draw[<-] (.25,7.5) -- (1.125,7);\n\\draw[<-] (1.875,7.5) -- (2.75,7);\n\\draw[<-] (3.5,7.5) -- (4.375,7);\n\\draw[<-] (5.625,7.5) -- (6.5,7);\n\n\\node at (0,6.5) {$\\tiny{\\yng(1,1,1)}$};\n\\node at (1.5,6.5) {$\\tiny{\\yng(2,2,2)}$};\n\\node at (3,6.5) {$\\tiny{\\yng(3,3,3)}$};\n\\node at (5,6.5) {$\\tiny{\\yng(4,4,4)}$};\n\\node[blue] at (7,6.5) {$\\tiny{\\yng(5,5,5)}$};\n%right arrows\n\\draw[->] (.25,6.5) -- (1.125,6.5);\n\\draw[->] (1.875,6.5) -- (2.5,6.5);\n\\draw[->] (3.5,6.5) -- (4.375,6.5);\n\\draw[->] (5.625,6.5) -- (6.25,6.5);\n%down arrows\n\\draw[->] (0,6) -- (0,5.375);\n\\draw[->] (1.5,6) -- (1.5,5.375);\n\\draw[->] (3,6) -- (3,5.375);\n\\draw[->] (5,6) -- (5,5.375);\n\\draw[->,blue,dashed] (7,6) -- (7,5.375);\n%diagonal arrows\n\\draw[<-] (.25,6) -- (1.125,5.375);\n\\draw[<-] (1.875,6) -- (2.75,5.375);\n\\draw[<-] (3.5,6) -- (4.375,5.375);\n\\draw[<-] (5.625,6) -- (6.5,5.375);\n\n\\node[blue] at (0,4.75) {$\\tiny{\\yng(1,1,1,1)}$};\n\\node[blue] at (1.5,4.75) {$\\tiny{\\yng(2,2,2,2)}$};\n\\node[blue] at (3,4.75) {$\\tiny{\\yng(3,3,3,3)}$};\n\\node[blue] at (5,4.75) {$\\tiny{\\yng(4,4,4,4)}$};\n\\node[blue] at (7,4.75) {$\\tiny{\\yng(5,5,5,5)}$};\n\\draw[->,blue,dashed] (.25,4.75) -- (1.125,4.75);\n\\draw[->,blue,dashed] (1.875,4.75) -- (2.5,4.75);\n\\draw[->,blue,dashed] (3.5,4.75) -- (4.375,4.75);\n\\draw[->,blue,dashed] (5.625,4.75) -- (6.25,4.75);\n\\end{tikzpicture}\n\\caption{The quiver of the plabic graph $G^{\\rm rec}_{5,9}$ forming the initial seed for $\\mathcal A\\subset \\widetilde{Y}=\\widetilde{\\text{Gr}}_{4}(\\mathbb C^9)$ with the frozen arrows determining the cluster ensemble map $p^*$.}\n\\label{fig:quivGrec59}\n\\end{figure}\n\nWe start by defining a lattice map \n\\[\n\\psi:N_{\\seed_{\\Yng(1)}} \\to M_{\\seed_{\\Yng(1)}}\n\\]\nwhich is given with respect to the bases induced by $s_{\\Yng(1)}$ as follows:\nfor $i\\times j$ a mutable vertex and $a\\times b$ with either $a=n-k$ or $b=k$ a frozen vertex we define\n\\begin{eqnarray*}\n e_{i\\times j} &\\mapsto& f_{(i-1)\\times(j-1)} - f_{(i-1)\\times j} + f_{i\\times(j+1)} - f_{(i+1)\\times (j+1)} + f_{(i+1)\\times j} - f_{i\\times (j-1)} \\\\\n % e_{1\\times 1} &\\mapsto& f_{1\\times 2} - f_{2\\times 2} + f_{2\\times 1} - f_{\\varnothing}\\\\\n e_{a\\times b} &\\mapsto& f_{a\\times b} - f_{(a-1)\\times b} + f_{(a-1)\\times(b-1)} - f_{a\\times (b-1)} \\\\\n e_{\\varnothing} &\\mapsto& f_{\\varnothing} - f_{1\\times k} + f_{1\\times 1} - f_{(n-k)\\times 1}\n\\end{eqnarray*}\nwith the convention that $f_{0\\times j}=f_{i\\times 0}=0$ whenever $i,j\\not =0$ and $f_{0\\times 0}=f_{\\varnothing}$. \nWe may present the map pictorially by recording the coefficient of the basis element $e_{i\\times j}$ in the $i\\times j$'th position of the grid (with an extra position $0\\times 0$ representing the vertex $\\varnothing$). \n\\begin{eqnarray}\\label{eq:pictorial p*}\n\\begin{tikzpicture}[scale=.4]\n\\node at (-3,0){$e_{i\\times j}$};\n\\draw[dashed,opacity=.5] (-1.5,.5) -- (1.5,.5);\n\\draw[dashed,opacity=.5] (-1.5,-0.5) -- (1.5,-0.5);\n\\draw[dashed,opacity=.5] (-0.5,-1.5) -- (-0.5,1.5);\n\\draw[dashed,opacity=.5] (0.5,-1.5) -- (0.5,1.5);\n\\node[opacity=.5] at (-1,-1) {\\small $0$};\n\\node[opacity=.5] at (-1,0) {\\small$0$};\n\\node[opacity=.5] at (-1,1) {\\small$0$};\n\\node[opacity=.5] at (0,-1) {\\small$0$};\n\\node at (0,0) {\\small$1$};\n\\node[opacity=.5] at (0,1) {\\small$0$};\n\\node[opacity=.5] at (1,-1) {\\small$0$};\n\\node[opacity=.5] at (1,0) {\\small$0$};\n\\node[opacity=.5] at (1,1) {\\small$0$};\n\n\\node at (2.5,0) {$\\mapsto$};\n\n\\begin{scope}[xshift=5cm]\n\\draw[dashed,opacity=.5] (-1.5,.5) -- (1.5,.5);\n\\draw[dashed,opacity=.5] (-1.5,-0.5) -- (1.5,-0.5);\n\\draw[dashed,opacity=.5] (-0.5,-1.5) -- (-0.5,1.5);\n\\draw[dashed,opacity=.5] (0.5,-1.5) -- (0.5,1.5);\n\\node[opacity=.5] at (-1,-1) {\\small$0$};\n\\node at (-1,0) {\\small$-1$};\n\\node at (-1,1) {\\small$1$};\n\\node at (0,-1) {\\small$1$};\n\\node[opacity=.5] at (0,0) {\\small$0$};\n\\node at (0,1) {\\small$-1$};\n\\node at (1,-1) {\\small$-1$};\n\\node at (1,0) {\\small$1$};\n\\node[opacity=.5] at (1,1) {\\small$0$};\n\\end{scope}\n\n\n\\begin{scope}[xshift=15cm]\n\\node at (-3,0){$e_{a\\times b}$};\n\\draw[dashed,opacity=.5] (-1.5,.5) -- (1.5,.5);\n\\draw[dashed,opacity=.5] (-1.5,-0.5) -- (1.5,-0.5);\n\\draw[dashed,opacity=.5] (-0.5,-1.5) -- (-0.5,1.5);\n\\draw[dashed,opacity=.5] (0.5,-1.5) -- (0.5,1.5);\n\\node[opacity=.5] at (-1,-1) {\\small $0$};\n\\node[opacity=.5] at (-1,0) {\\small$0$};\n\\node[opacity=.5] at (-1,1) {\\small$0$};\n\\node[opacity=.5] at (0,-1) {\\small$0$};\n\\node at (0,0) {\\small$1$};\n\\node[opacity=.5] at (0,1) {\\small$0$};\n\\node[opacity=.5] at (1,-1) {\\small$0$};\n\\node[opacity=.5] at (1,0) {\\small$0$};\n\\node[opacity=.5] at (1,1) {\\small$0$};\n\n\\node at (2.5,0) {$\\mapsto$};\n\n\\begin{scope}[xshift=5cm]\n\\draw[dashed,opacity=.5] (-1.5,.5) -- (1.5,.5);\n\\draw[dashed,opacity=.5] (-1.5,-0.5) -- (1.5,-0.5);\n\\draw[dashed,opacity=.5] (-0.5,-1.5) -- (-0.5,1.5);\n\\draw[dashed,opacity=.5] (0.5,-1.5) -- (0.5,1.5);\n\\node[opacity=.5] at (-1,-1) {\\small$0$};\n\\node at (-1,0) {\\small$-1$};\n\\node at (-1,1) {\\small$1$};\n\\node[opacity=.5] at (0,-1) {\\small$0$};\n\\node at (0,0) {\\small$1$};\n\\node at (0,1) {\\small$-1$};\n\\node[opacity=.5] at (1,-1) {\\small$0$};\n\\node[opacity=.5] at (1,0) {\\small$0$};\n\\node[opacity=.5] at (1,1) {\\small$0$};\n\\end{scope}\n\\end{scope}\n\\end{tikzpicture}\n\\end{eqnarray}\nAll entries in the grid above \\emph{not} corresponding to vertices in the particular case considered should simply be neglected.\nA straightforward computation reveals the following\n\n\\begin{proposition}\\thlabel{prop:-p* dual}\nWe have $\\ker(\\psi)=\\langle{(1,1,\\dots,1)}\\rangle=K_{\\seed_{\\Yng(1)}}$ and $\\psi(N_{\\seed_{\\Yng(1)}})={(1,1,\\dots,1)}^\\perp$. So, the induced map $\\psi:N\/K\\to K^\\perp$ is a lattice isomorphism. \n\\end{proposition}\n\nIn fact, $\\psi$ defines a cluster ensemble lattice map (Definition~\\ref{def:p-star}), so we obtain\n\\begin{eqnarray}\\label{eq:p-map Gr}\n p:\\cA\\to\\cX, \\quad \\text{determined by }\\quad (p\\vert_{\\cA_{\\seed_{\\Yng(1)}}})^*=\\psi.\n\\end{eqnarray}\nThere is a combinatorial way to obtain the map $\\psi$ by introducing \\emph{frozen arrows} to the quiver of the initial seed to \\emph{close cycles} involving frozen vertices (see Figure~\\ref{fig:quivGrec59}).\nThese arrows are used to determine the submatrix denoted by $*$ in \\eqref{eq:Mp*}.\n\n\nAs a direct consequence of \\eqref{eq:p-map Gr} and \\thref{prop:-p* dual} we observe that the action of $T_K$ on $\\cA$ coincides with the $\\C^*$-action (of simultaneously scaling Pl\\\"ucker coordinates) on $\\UT_Y$ restricted to $\\cA$.\nIn particular:\n\\begin{corollary}\n The Picard property holds for $(p,K)$ with respect to $\\mathcal A\\hookrightarrow\\UT_Y$.\n\\end{corollary}\n\n\\subsection{Valuations and Newton--Okounkov bodies}\n\\label{sec:GHKK_and_RW}\nThis subsection is the core of our application to the Grassmannian. \nWe show in Theorem~\\ref{thm: val and gv} that certain Newton--Okounkov bodies as they appear in \\thref{thm:k_and_pic} (see also Remark~\\ref{rem:comparing_NO_bodies}) are unimodularly equivalent to Newton--Okounkov bodies of Rietsch--Williams. \nWe first introduce the combinatorics that govern Rietsch--Williams' flow valuation and the ${\\bf g}$-vector valuation in this case.\n\n\\subsubsection{The flow valuation}\nBased on Postnokiv's \\emph{boundary measurement map} for plabic networks \\cite[\\S11]{Pos06} Rietsch--Williams associate a {\\bf flow valuation} \\cite[Definition 8.1]{RW} to every plabic graph $G$ or more generally every seed $\\seed$ making use of the $\\cX$-type cluster structure on the Grassmannian\nWe denote it by \n\\[\n\\val_\\seed:\\mathbb C(Y)\\setminus \\{0\\}\\to \\mathbb Z^{(n-k)\\times k}.\n\\]\nThe valuation is defined as the multidegree of the lowest degree summand (with respect a fixed graded lexicographic order) on Laurent polynomials in $\\cX$ variables and then extended to rational functions in the natural way.\nThe lattice is of dimension $(n-k)k$ (as apposed to $(n-k)k+1$ which is the number of vertices), as the the variable corresponding to $\\varnothing$ never appears (more details below in \\S\\ref{sec:NO_Grass_equal}).\nNotice that it therefore coincides with our definition of a {\\bf c}-vector valuation for cluster $\\cX$ varieties (Corollary~\\ref{cor:gv on midX}).\nFor $G=G_{\\Yng(1)}$ we simply write $\\val_G=\\val_{\\Yng(1)}$.\nThe flow valuation with respect to the rectangles plabic graph can be computed in a particularly explicit way as Rietsch--Williams show in \\cite[\\S14]{RW}.\nWe briefly summarize some of their findings.\n\n\\begin{proposition}\\cite[Proposition 14.4 and Figure 18]{RW}\\thlabel{prop:val grec}\nFor $J\\in\\binom{[n]}{n-k}$, the valuation $\\val_{\\Yng(1)}(p_J)$ can be represented by a \\emph{GT tableau} (defined as follows, see \\cite[\\S14]{RW}) of size $(n-k)\\times k$ whose $(i\\times j)^{\\text{th}}$ entry represents the coefficient of the corresponding basis element.\nThe entries of the GT tableau are obtained as in four steps:\n\\begin{itemize}\n \\item[\\bf Step 1:] draw the Young diagram $\\mu_J$ whose south border is the path $w_J$ associated to $J$ in the $(n-k)\\times k$-rectangle; \n \\item[\\bf Step 2:] draw another copy of $w_{J}$ shifted by {\\it one step south} and {\\it one step east} (this implies that some steps of the new path $w_J^1$ lie outside of the $(n-k)\\times k$-rectangle);\n \\item[\\bf Step 3:] continue repeating Step 2 until the new copy of $w_J$ lies \\emph{entirely} outside of the $(n-k)\\times k$-rectangle;\n \\item[\\bf Step 4:] lastly, place an $i$ inside every box (that is part of the $(n-k)\\times k$-rectangle) in between the paths $w_{J}^{i-1}$ and $w_{J}^i$.\n\\end{itemize}\nAll other boxes are filled with zeros.\n\\end{proposition}\nRietsch--Williams compute the Newton--Okounkov bodies associated to this valuation. In our notation they are of form $\\Delta_{\\val_{\\Yng(1)}}(D_{n-k},p_{(n-k)\\times k})$, where $p_{(n-k)\\times k}=p_{[1,n-k]}$ is the Pl\\\"ucker coordinate (and hence section of $\\mathcal L_e$) associated to the frozen vertex $(n-k)\\times k$.\n\n\\begin{example}\\label{exp:Grec6,13}\n The procedure of \\thref{prop:val grec} is depicted in Figure~\\ref{fig:val grec} for $J=\\{3,4,7,9,11,12\\}\\subset [13]$. \n\\end{example}\n\n\\begin{figure}\n \\centering\n\\begin{tikzpicture}[scale=.4]\n\\node[left] at (-1,3) {$\\val_{\\Yng(1)}(p_J)=$};\n\\draw (0,0) -- (0,6) -- (5,6);\n\\draw[thick,magenta] (7,6) -- (5,6) -- (5,4) -- (3,4) -- (3,3) -- (2,3) -- (2,2) -- (1,2) -- (1,0) -- (0,0);\n\\node at (2,4.5) {$\\mu_J$};\n\\draw (7,6) -- (7,0) -- (1,0);\n\\draw[magenta,opacity=.4,thick] (2,0) -- (2,-1) -- (1,-1);\n\\draw[magenta,opacity=.4,thick] (2,0) -- (2,1) -- (3,1) -- (3,2) -- (4,2) -- (4,3) -- (6,3) -- (6,5) -- (7,5);\n\\draw[magenta,opacity=.4,thick] (7,5) -- (8,5);\n\\draw[magenta,opacity=.4,thick] (2,-2) -- (3,-2) -- (3,0) -- (4,0) -- (4,1) -- (5,1) -- (5,2) -- (7,2) -- (7,4) -- (9,4);% -- (10,3) -- (10,5) -- (11,5);;\n\\draw[magenta,opacity=.4,thick] (3,-3) -- (4,-3) -- (4,-1) -- (5,-1) -- (5,0) -- (6,0) -- (6,1) -- (8,1) -- (8,3) -- (10,3);\n\\draw[dashed,opacity=.5] (0,0) -- (3.5,-3.5);\n\\draw[dashed,opacity=.5] (7,6) -- (10.5,2.5);\n\\draw[opacity=.4,dashed] (5,6) -- (7,6);\n\\draw[opacity=.4,dashed] (5,5) -- (7,5);\n\\draw[opacity=.4,dashed] (5,4) -- (7,4);\n\\draw[opacity=.4,dashed] (3,3) -- (7,3);\n\\draw[opacity=.4,dashed] (2,2) -- (7,2);\n\\draw[opacity=.4,dashed] (1,1) -- (7,1);\n\\draw[opacity=.4,dashed] (6,6) -- (6,0);\n\\draw[opacity=.4,dashed] (5,5) -- (5,0);\n\\draw[opacity=.4,dashed] (4,4) -- (4,0);\n\\draw[opacity=.4,dashed] (3,3) -- (3,0);\n\\draw[opacity=.4,dashed] (2,2) -- (2,0);\n\\draw[opacity=.4,dashed] (1,1) -- (1,0); \n\\node at (1.5,.5) {1};\n\\node at (1.5,1.5) {1};\n\\node at (2.5,.5) {2};\n\\node at (2.5,1.5) {1};\n\\node at (2.5,2.5) {1};\n\\node at (3.5,.5) {2};\n\\node at (3.5,1.5) {2};\n\\node at (3.5,2.5) {1};\n\\node at (3.5,3.5) {1};\n\\node at (4.5,.5) {3};\n\\node at (4.5,1.5) {2};\n\\node at (4.5,2.5) {2};\n\\node at (4.5,3.5) {1};\n\\node at (5.5,.5) {3};\n\\node at (5.5,1.5) {3};\n\\node at (5.5,2.5) {2};\n\\node at (5.5,3.5) {1};\n\\node at (5.5,4.5) {1};\n\\node at (5.5,5.5) {1};\n\\node at (6.5,.5) {4};\n\\node at (6.5,1.5) {3};\n\\node at (6.5,2.5) {2};\n\\node at (6.5,3.5) {2};\n\\node at (6.5,4.5) {2};\n\\node at (6.5,5.5) {1};\n\n\\node at (11.5,3) {${\\longmapsto}$};\n\\node at (11.5,3.75) {\\small $-\\psi$};\n \n\\begin{scope}[xshift=14cm]\n\\draw (0,0) -- (0,6) -- (5,6) -- (5,4) -- (3,4) -- (3,3) -- (2,3) -- (2,2) -- (1,2) -- (1,0) -- (0,0);\n\\draw (5,6) -- (7,6) -- (7,0) -- (1,0);\n\\draw[opacity=.4,dashed] (0,6) -- (7,6);\n\\draw[opacity=.4,dashed] (0,5) -- (7,5);\n\\draw[opacity=.4,dashed] (0,4) -- (7,4);\n\\draw[opacity=.4,dashed] (0,3) -- (7,3);\n\\draw[opacity=.4,dashed] (0,2) -- (7,2);\n\\draw[opacity=.4,dashed] (0,1) -- (7,1);\n\\draw[opacity=.4,dashed] (6,6) -- (6,0);\n\\draw[opacity=.4,dashed] (5,6) -- (5,0);\n\\draw[opacity=.4,dashed] (4,6) -- (4,0);\n\\draw[opacity=.4,dashed] (3,6) -- (3,0);\n\\draw[opacity=.4,dashed] (2,6) -- (2,0);\n\\draw[opacity=.4,dashed] (1,6) -- (1,0); \n\\node at (.5,.5) {$1$};\n\\node[opacity=.5] at (.5,1.5) {$0$};\n\\node[opacity=.5] at (.5,3.5) {$0$};\n\\node[opacity=.5] at (.5,4.5) {$0$};\n\\node[opacity=.5] at (.5,5.5) {$0$};\n\\node at (.5,2.5) {$-1$};\n\\node at (1.5,2.5) {$1$};\n\\node at (1.5,3.5) {$-1$};\n\\node[opacity=.5] at (1.5,.5) {$0$};\n\\node[opacity=.5] at (1.5,1.5) {$0$};\n\\node[opacity=.5] at (1.5,4.5) {$0$};\n\\node[opacity=.5] at (1.5,5.5) {$0$};\n\\node at (2.5,3.5) {$1$};\n\\node[opacity=.5] at (2.5,.5) {$0$};\n\\node[opacity=.5] at (2.5,1.5) {$0$};\n\\node[opacity=.5] at (2.5,2.5) {$0$};\n\\node[opacity=.5] at (2.5,5.5) {$0$};\n\\node at (2.5,4.5) {$-1$};\n\\node[opacity=.5] at (3.5,.5) {$0$};\n\\node[opacity=.5] at (3.5,1.5) {$0$};\n\\node[opacity=.5] at (3.5,2.5) {$0$};\n\\node[opacity=.5] at (3.5,3.5) {$0$};\n\\node[opacity=.5] at (3.5,4.5) {$0$};\n\\node[opacity=.5] at (3.5,5.5) {$0$};\n\\node at (4.5,4.5) {$1$};\n\\node[opacity=.5] at (4.5,.5) {$0$};\n\\node[opacity=.5] at (4.5,1.5) {$0$};\n\\node[opacity=.5] at (4.5,2.5) {$0$};\n\\node[opacity=.5] at (4.5,3.5) {$0$};\n\\node[opacity=.5] at (4.5,5.5) {$0$};\n\\node[opacity=.5] at (5.5,.5) {$0$};\n\\node[opacity=.5] at (5.5,1.5) {$0$};\n\\node[opacity=.5] at (5.5,2.5) {$0$};\n\\node[opacity=.5] at (5.5,3.5) {$0$};\n\\node[opacity=.5] at (5.5,4.5) {$0$};\n\\node[opacity=.5] at (5.5,5.5) {$0$};\n\\node at (6.5,.5) {$-1$};\n\\node[opacity=.5] at (6.5,1.5) {$0$};\n\\node[opacity=.5] at (6.5,2.5) {$0$};\n\\node[opacity=.5] at (6.5,3.5) {$0$};\n\\node[opacity=.5] at (6.5,4.5) {$0$};\n\\node[opacity=.5] at (6.5,5.5) {$0$};\n\n\\node[right] at (7.5,3) {$=\\bar{\\bf g}_{{\\Yng(1)}}(p_{J})$};\n\\end{scope}\n\\end{tikzpicture}\n \\caption{\n On the left: the pictorial representation of the GT tableau for $J=\\{3,4,7,9,11,12\\} \\subset [13]$ (\\thref{prop:val grec}). The south steps of the path cutting out the Young diagram $\\mu_J$ correspond to indices in $J$. \n On the right, we depict its image under $-\\psi$ which coincides with the ${\\bf g}$-vector of $p_J$ up to homogenization, see \\eqref{eq:g-vectors for Grec} and \\eqref{eq:homogenized g vector op}.\n }\n \\label{fig:val grec}\n\\end{figure}\n\n\n\\subsubsection{A combinatorial description of ${\\bf g}$-vectors}\\label{sec:g-vects}\nIn this subsection we consider the cluster variety $\\cA^{\\rm op}$ whose initial quiver is obtained by opposing the initial quiver for $\\cA$. It is well known that $\\cA$ and $\\cA^{\\rm op}$ are isomorphic (in general opposing the quiver gives rise to isomorphic cluster $ \\cA$-varieties). \nWe also have a partial minimal model $\\cA^{\\op} \\hookrightarrow \\UT_Y$.\nWe write $\\seed_{\\Yng(1)}^{\\rm op}$ to denote the seed $\\seed_{\\Yng(1)}$ of equation \\eqref{eq_seed} thought of as the initial seed for $\\cA^{\\rm op}$.\nNotice that $-\\psi$ determines a cluster ensemble map $p^{\\rm op}:\\cA^{\\rm op} \\to \\cX^{\\rm op}$\nso that the Picard property holds for $(p^{\\rm op},K)$ with respect to $\\cA^{\\rm op}\\hookrightarrow \\text{UT}_Y$.\n\nIn this setting an explicit combinatorial formula to compute {\\bf g}-vectors of Pl\\\"ucker coordinates can be deduced from the categorification of the Grassmannian cluster algebra developed in \\cite{JKS16,BKM16}.\nWe learned about it from Bernhard Keller in private email communication.\nThe below formula describes ${\\bf g}$-vectors with respect to the seed $\\seed_{\\Yng(1)}^{\\rm op}$ for the cluster variety $\\cA^{\\rm op}$ which we think of as another cluster structure on $\\UT_{Y}$.\n\n\n\\begin{corollary}\\thlabel{cor:gv Grec}\n(Hook formula for {\\bf g}-vectors)\nConsider the seed $\\seed_{\\Yng(1)}^{\\rm op}$ and $J \\in \\binom{[n]}{n-k}$. We let $i_1\\times j_1,\\dots ,i_s \\times j_s $ be the rectangles corresponding to the turning points in the path $w_J$ that cuts out $\\mu_J$ inside the $(n-k)\\times k$-rectangle.\nThen \n\\begin{eqnarray}\\label{eq:g-vectors for Grec}\n{\\bf g}_{\\Yng(1)^{\\rm op}}(p_J):={\\bf g}^{\\cA^{\\rm op}}_{\\seed_{\\Yng(1)}^{\\rm op}}(p_J)= \\sum_{p=1}^{s}f_{i_{p}\\times j_{p}}-f_{i_{p}\\times j_{p+1}},\n\\end{eqnarray}\nwhere we set $f_{i_s\\times j_{s+1}}:=0$.\n\\end{corollary}\n\n\\begin{example}\nThe \nConsider $n-k=4$, $n=9$, and $J=\\{2,4,6,7\\}$. We have that $\\mu_{J}=\\Yng(4,3,2,2)$ and by \\thref{cor:gv Grec}\n\\[\n{\\bf g}_{\\Yng(1)^{\\rm op}}\\lrp{p_{\\Yngs(4,3,2,2)}}= f_{\\Yngs(4)}- f_{\\Yngs(3)}+f_{\\Yngs(3,3)}-f_{\\Yngs(2,2)}+f_{\\Yngs(2,2,2,2)}.\n\\]\n\\end{example}\n\n\\subsubsection{Equality of the Newton--Okounkov bodies}\\label{sec:NO_Grass_equal}\n The aim of this section is to identify the Newton--Okounkov bodies of flow valuations with Newton--Okounkov bodies of {\\bf g}-vector valuations for $\\cA^{\\rm op}$. \nWe use a particular cluster ensemble lattice map for the identification and work in the initial seed $\\seed^{\\rm op}_{\\Yng(1)}$ (whose quiver is opposite to the quiver depicted in Figure~\\ref{fig:quivGrec59} for $n=9,k=5$).\n\n\n\nWe think of the open positroid variety inside $Y=\\text{Gr}_{n-k}(\\C^n)$ as the quotient of the cluster variety $\\cA^{\\rm op}$ by the torus $T_{K}$.\nWe choose a section \n\\[\n\\sigma: N\/K \\to N \\quad \\text{with image} \\quad N \\cap f_{\\varnothing}^\\perp;\n\\]\nthat is, a coset $n\\mod K$ is sent to its unique representative satisfying $\\langle n,f_{\\varnothing}\\rangle=0$.\nIt is not hard to see that $\\sigma$ induces an isomorphism between the rings of rational functions $\\mathbb C(T_{M\/\\langle f_{\\varnothing}\\rangle})$ and $\\mathbb C(T_{K^\\perp})$ that commutes with cluster $\\mathcal X$ mutation.\nWe use $\\sigma$ to realize $\\Trop_{\\Z}(({\\cX_{\\bf 1}^\\vee})_{\\seed^\\vee})=\\Trop_{\\Z}((\\cA^{\\rm op}\/T_K)_{\\seed^{\\rm op}})= N\/K$ inside $ \\Trop_\\Z(\\cA^{\\rm op}_{\\seed^{\\rm op}})=N =\\Trop_\\Z(\\cX^\\vee_{\\seed^\\vee})$ for every seed.\nMoreover, the dual of $\\sigma$ induces an isomorphism of lattices\n\\[\n\\sigma^*:M\/\\langle f_{\\varnothing}\\rangle \\to K^\\perp.\n\\]\nNotice that $T_{K^\\perp}=\\pi^{-1}(\\bf 1)$ where $\\pi:T_M\\to T_{K^*}$ is the restriction of $\\cX\\to T_{K^*}$ to a cluster chart.\nAs alluded to above, we obtain an isomorphism of cluster $\\cX$-varieties\n\\[\n\\sigma^*:\\cX_{\\setminus \\varnothing}\\to \\cX_{\\bf 1},\n\\]\nwhere $\\cX_{\\setminus \\varnothing}$ is the $\\cX$-variety associated with the initial data obtained by deleting the index $\\varnothing$ upon realizing $M\/\\langle f_{\\varnothing}\\rangle$ as $\\langle f_{i\\times j}\\mid 1\\le i\\le n-k,1\\le j\\le k\\rangle \\subset M$.\nGiven a seed $\\seed$ we denote the corresponding seed of $\\cX_{\\setminus\\varnothing}$ by $\\seed_{\\setminus\\varnothing}$.\nIn particular, we have\n$\\Trop_{\\Z}((\\mathcal X^\\vee_{\\setminus\\varnothing})_{\\seed^\\vee_{\\setminus \\varnothing}})=N_{\\seed}\\cap f_{\\varnothing}^\\perp$.\nThe flow valuation is defined on ring of rational functions on the positroid variety which coincides with\n\\[\n\\mathbb C(\\cX_{\\setminus \\varnothing}) \\cong \\mathbb C(x_{i\\times j}:1\\le i\\le n-k,1\\le j\\le k).\n\\]\n\n\nThe next result follows from the preceding discussion and Corollary~\\ref{cor:gv on midX}.\n\n\\begin{proposition}\\label{prop:flow is gv for X}\nFor every choice of seed $\\seed$ the diagram commutes:\n\\[\n\\xymatrix{\n\\mathbb C(\\cX_{\\bf 1})\\setminus \\{0\\} \\ar[d]_{(\\sigma^*)^*}\\ar[r]^{\\cv_{\\seed}} & N\/K\\ar[d]^{\\sigma}\\\\\n\\mathbb C(\\cX_{\\setminus \\varnothing})\\setminus \\{0\\} \\ar[r]_{\\val_{\\seed}} & N_{\\seed}\\cap f_{\\varnothing}^\\perp.\n}\n\\]\n\\end{proposition}\nThe flow valuation is a ${\\bf c}$-vector valuation for the variety $\\cX_{\\setminus \\varnothing}$ as both are defined by picking the lowest degree exponent of a Laurent polynomial with respect to the same order. That is:\n\\[\n\\val_{\\seed}= \\cv^{\\cX_{\\setminus \\varnothing}}_{\\seed\\setminus\\varnothing}.\n\\]\nAlternatively, in light of Proposition \\ref{prop:flow is gv for X} we may think of the flow valuation as a ${\\bf c}$-vector valuation for $\\cX_{\\bf 1}$.\nOur aim now is to identify the images of $\\val_\\seed$ with \nthose of a $\\bf g$-vector valuation for $\\cA^{\\rm op}$, or more precisely a ${\\bf g}$-vector valuation for $\\cA^{\\rm op}\/T_{K}$.\nTo avoid confusion we introduce the following notation\n\\begin{eqnarray}\\label{eq:homogenized g vector op}\n\\bar {\\bf g}_{\\Yng(1)^{\\rm op}}:R\\setminus \\{0\\}\\longrightarrow K^\\perp\\cong M_{\\seed}\/\\langle f_{(n-k)\\times k}\\rangle \n\\end{eqnarray}\ndefined for a homogeneous element $h\\in R_q\\setminus \\{0\\}$ by\n\\[\nh\\longmapsto \n{\\bf g}_{\\Yng(1)^{\\rm op}}\\lrp{\\frac{h}{\np_{(n-k)\\times k}^q}},\n\\]\nwhere ${\\bf g}_{\\Yng(1)^{\\rm op}}(p_{(n-k)\\times k})=f_{(n-k)\\times k}$.\nNotice that $\\bar {\\bf g}_{\\Yng(1)^{\\rm op}}$ is the restriction of ${\\bf g}_{\\Yng(1)^{\\rm op}}: \\mathbb C(Y)\\setminus \\{0\\}\\to M$ to the section ring $R\\hookrightarrow \\mathbb C(Y)$ where the embedding is defined by $R_q\\ni h\\mapsto h\/p_{(n-k)\\times k}^q$.\n\n \n\\begin{theorem}\\thlabel{thm: val and gv}\n\\begin{samepage}\nFor every $J\\in \\binom{[n]}{n-k}$ we have\n\\begin{eqnarray}\n-\\psi(\\val_{\\Yng(1)}(p_J))=\\bar {\\bf g}_{\\Yng(1)^{\\rm op}}(p_{J}). \n\\end{eqnarray}\nIn particular, \nthe Newton--Okounkov bodies $\\Delta_{\\val_{\\Yng(1)}}(D_{n-k},p_{(n-k)\\times k})$ and $\\Delta_{\\bar{\\bf g}_{\\Yng(1)^{\\rm op}}}(D_{n-k},p_{(n-k)\\times k})$ are unimodularly equivalent with lattice isomorphism given by $-\\psi$. \n\\end{samepage}\n\\end{theorem}\n\n\\begin{proof}\nWe prove the claim in several steps. \nFirst, we need to describe $\\val_{\\Yng(1)^{\\rm op}}(p_J)$. \nFortunately, this is straightforward using \\thref{prop:val grec}.\nLet us analyze the image of the \\emph{$i$-strip}, \\emph{i.e.} the image of the elements of form $-ie_{a\\times b}$ corresponding to a box in position $a\\times b$ of the grid lying between the path $w_J^i$ and $w_J^{i-1}$. \nWe deduce\n\\begin{center}\n\t\\begin{tikzpicture}[scale=.6]\n\\draw[thick,teal] (1,3) -- (4,3) -- (4,6);\n\\draw[thick,magenta] (2,2) -- (5,2) -- (5,5);\n\\node at (4.5,5.5) {\\tiny $\\vdots$};\n\\node at (1.5,2.5) {\\tiny $\\cdots$};\n\\node at (2.5,2.5) {\\tiny $-i$};\n\\node at (3.5,2.5) {\\tiny $-i$};\n\\node at (4.5,2.5) {\\tiny $-i$};\n\\node at (4.5,3.5) {\\tiny $-i$};\n\\node at (4.5,4.5) {\\tiny $-i$};\n\\draw[dashed,opacity=.5] (1,2.5) -- (1,4.5);\n\\draw[dashed,opacity=.5] (2,1.5) -- (2,5.5);\n\\draw[dashed,opacity=.5] (3,.5) -- (3,6.5);\n\\draw[dashed,opacity=.5] (4,.5) -- (4,6.5);\n\\draw[dashed,opacity=.5] (5,.5) -- (5,5.5);\n\\draw[dashed,opacity=.5] (6,.5) -- (6,4.5);\n\\draw[dashed,opacity=.5] (1.5,5) -- (5.5,5);\n\\draw[dashed,opacity=.5] (.5,4) -- (6.5,4);\n\\draw[dashed,opacity=.5] (.5,3) -- (6.5,3);\n\\draw[dashed,opacity=.5] (1.5,2) -- (6.5,2);\n\\draw[dashed,opacity=.5] (2.5,1) -- (6.5,1);\n\\node[above right,magenta] at (5,5) {\\tiny $w_J^i$};\n\\node[above right,teal] at (4,6) {\\tiny $w_J^{i-1}$};\n\n\\node at (8,3) {$\\mapsto$};\n\n\\begin{scope}[xshift=9 cm]\n\\node at (.5,3.5) {\\tiny $\\cdots$};\n\\node at (1.5,2.5) {\\tiny $i$};\n\\node at (2.5,1.5) {\\tiny $-i$};\n\\node at (3.5,6.5) {\\tiny $\\vdots$};\n\\node at (4.5,5.5) {\\tiny $i$};\n\\node at (5.5,4.5) {\\tiny $-i$};\n\\draw[thick] (1,4) -- (3,4) -- (3,6);\n\\draw[thick,teal] (1,3) -- (4,3) -- (4,6);\n\\draw[thick,magenta] (2,2) -- (5,2) -- (5,5);\n\\draw[thick] (3,1) -- (6,1) -- (6,4);\n\\node at (1.5,3.5) {\\tiny $-i$};\n\\node at (2.5,3.5) {\\tiny $0$};\n\\node at (3.5,3.5) {\\tiny $i$};\n\\node at (3.5,4.5) {\\tiny $0$};\n\\node at (3.5,5.5) {\\tiny $-i$};\n\\node at (2.5,2.5) {\\tiny $i$};\n\\node at (3.5,2.5) {\\tiny $0$};\n\\node at (4.5,2.5) {\\tiny $-2i$};\n\\node at (4.5,3.5) {\\tiny $0$};\n\\node at (4.5,4.5) {\\tiny $i$};\n\\node at (3.5,1.5) {\\tiny $0$};\n\\node at (4.5,1.5) {\\tiny $0$};\n\\node at (5.5,1.5) {\\tiny $i$};\n\\node at (5.5,2.5) {\\tiny $0$};\n\\node at (5.5,3.5) {\\tiny $0$};\n\\draw[dashed,opacity=.5] (1,2.5) -- (1,4.5);\n\\draw[dashed,opacity=.5] (2,1.5) -- (2,5.5);\n\\draw[dashed,opacity=.5] (3,.5) -- (3,6.5);\n\\draw[dashed,opacity=.5] (4,.5) -- (4,6.5);\n\\draw[dashed,opacity=.5] (5,.5) -- (5,5.5);\n\\draw[dashed,opacity=.5] (6,.5) -- (6,4.5);\n\\draw[dashed,opacity=.5] (1.5,5) -- (5.5,5);\n\\draw[dashed,opacity=.5] (.5,4) -- (6.5,4);\n\\draw[dashed,opacity=.5] (.5,3) -- (6.5,3);\n\\draw[dashed,opacity=.5] (1.5,2) -- (6.5,2);\n\\draw[dashed,opacity=.5] (2.5,1) -- (6.5,1);\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\nNotice that unless $i=1$ all non zero entries in the picture cancel with the images of the $(i-1)$- and the $(i+1)$-strips.\nWhen $i=1$ however, the entry $i=1$ above the path $w_J^{0}=w_J$ stays.\nHence, for every corner in $w_{J}$ corresponding to a south step followed by a west step $-\\psi\\left(\\val_{\\Yng(1)}(p_J)\\right)$ has coefficient $1$ for $f_{a\\times b}$ where $a\\times b$ corresponds to the box whose south east corner coincides with this corner of $w_J$.\nThe case of a corner in $w_{J}$ corresponding to a west step followed by a south step is very similar, with the only difference that the signs change. \nIn particular,\n$-\\psi\\left(\\val_{\\Yng(1)}(p_J)\\right)$ has coefficient $-1$ for $f_{a\\times b}$ where $a\\times b$ corresponds to the box whose south east corner is adjacent to this corner of $w_J$.\n\n\nIt is left to analyze the parts of $-\\psi\\left(\\val_{\\Yng(1)}(p_J)\\right)$ corresponding to \\emph{frozen} vertices. \nThe arguments here are very similar, the only special case being the south east corner of the $(n-k)\\times k$-rectangle.\nHence, we restrict our attention to this case and omit the others.\n\nConsider the vertex in position $(n-k)\\times k$ and assume in $\\val_{\\Yng(1)}(p_J)$ the corresponding entry is $i$.\nNotice, that coefficient for the vertex $(n-k-1)\\times (k-1)$ necessarily is $i-1$.\nSo, applying $-\\psi$ we see that\n\\begin{center}\n\t\\begin{tikzpicture}[scale=.6]\n\\draw (.5,0) -- (3,0) -- (3,2.5);\n\\node at (2.5,.5) {\\tiny $-i$};\n\\node at (1.5,1.5) {\\tiny $-i+1$};\n\\node at (1.5,.5) {\\tiny $\\cdots$};\n\\node at (2.5,1.5) {\\tiny $\\vdots$};\n\n\\draw[opacity=.5,dashed] (.5,1) -- (3,1);\n\\draw[opacity=.5,dashed] (.5,2) -- (3,2);\n\\draw[opacity=.5,dashed] (1,0) -- (1,2.5);\n\\draw[opacity=.5,dashed] (2,0) -- (2,2.5);\n\n\\node at (4,1.25) {$\\mapsto$};\n\n\\begin{scope}[xshift=5cm]\n\\draw (.5,0) -- (3,0) -- (3,2.5);\n\\node at (2.5,.5) {\\tiny $-1$};\n\\node at (1.5,1.5) {\\tiny $-i$};\n\\node at (1.5,.5) {\\tiny $1$};\n\\node at (2.5,1.5) {\\tiny $1$};\n\n\\draw[opacity=.5,dashed] (.5,1) -- (3,1);\n\\draw[opacity=.5,dashed] (.5,2) -- (3,2);\n\\draw[opacity=.5,dashed] (1,0) -- (1,2.5);\n\\draw[opacity=.5,dashed] (2,0) -- (2,2.5);\n\\end{scope}\n\\end{tikzpicture} \n\\end{center}\nObserve that the entries $1$ and $-i$ cancel by similar arguments as above.\nThe only non-zero coefficient in this picture is $-1$ for $f_{(n-k)\\times k}$.\nSummarizing, we have\n\\begin{eqnarray*}\n-\\psi\\left(\\val_{\\Yng(1)}(p_J)\\right) &=& \\sum_{\\begin{smallmatrix}\n \\text{south to west}\\\\\n \\text{corners of }w_J\n\\end{smallmatrix} } f_{a'\\times b'} -f_{(n-k)\\times k}\n-\\sum_{\\begin{smallmatrix}\n \\text{west to south}\\\\\n \\text{corners of }w_J\n\\end{smallmatrix} } f_{a\\times b}\\\\\n&\\overset{\\text{Equation~\\eqref{eq:g-vectors for Grec}}}{=}& {\\bf g}_{\\Yng(1)^{\\rm op}}(p_{J}) - f_{(n-k)\\times k}=\\bar{\\bf g}_{\\Yng(1)^{\\rm op}}(p_J).\n\\end{eqnarray*}\nThis implies $-\\psi(\\Delta_{\\val_{\\Yng(1)}}(D_{n-k},p_{(n-k)\\times k})=\\Delta_{\\bar{\\bf g}_{\\Yng(1)}^{\\rm op}}(D_{n-k},p_{(n-k)\\times k})$. \n\\end{proof}\n\n\\begin{remark}\\label{rmk:val and cval}\n The attentive reader might notice that the Theorem~\\ref{thm: val and gv} and the discussion preceding it closely resemble Lemma~\\ref{lem:cval_gval}.\n However, the difference in convention choices in \\cite{RW} and the present paper yield the necessity of a non-trivial change of coordinates.\n To avoid lengthening the exposition even more we decided to give the result in a single seed but allude to the fact that Theorem~\\ref{thm: val and gv} indeed is an instance of Lemma~\\ref{lem:cval_gval} (after non-trivial changes of cluster coordinates).\n After making the appropriate change of coordinate one can show that the map $-\\psi$ may be described as the tropicalization of a cluster ensemble map. In particular, Theorem~\\ref{thm: val and gv} can be extended to all seeds.\n\\end{remark}\n\n\\subsection{The intrinsic Newton--Okounkov body for Grassmannians}\n\\label{sec:Grass_intrinsic}\nAs before, let $\\lb_e$ be the bundle over $\\text{Gr}_{n-k}(\\C^n)$ obtained by pullback of $\\mathcal{O}(1)$ under the Pl\\\"ucker embedding $\\text{Gr}_{n-k}(\\C^n)\\hookrightarrow \\mathbb{P}^{\\binom{n}{k}-1}$. Recall the definition of the intrinsic Newton--Okounkov body from Definition~\\ref{def:intrinsic_lb}.\n\n\\begin{corollary}\\label{cor:intrinsicNO grassmannian}\nConsider the partial minimal model $\\cA^{\\rm op}\\hookrightarrow \\UT_{Y} $, the minimal model $\\cA^{\\op}\/T_K \\hookrightarrow Y$ and the map ${\\bf g}:\\mathbb{B}_{\\tf}(\\cA^{\\op})\\to \\Trop_{\\Z}((\\cA^{\\op})^\\vee)$ of \\eqref{eq:nu_seed_free}. Then\n\\eqn{\n\\Delta_{\\mathrm{BL}}(\\lb_e) = \\bconv\\Bigg( \\lrc{ \\gv \\lrp{p_J} \\mid J \\in \\binom{[n]}{n-k}} \\Bigg).\n}\n\\end{corollary}\n\n\\begin{proof}\nThe Newton--Okounkov polytope for the flow valuation with respect to $\\seed_{\\Yng(1)}$ is the convex hull of the images of Pl\\\"ucker coordinates (see \\cite[\\S16.1]{RW}).\nSo by Theorem~\\ref{thm: val and gv} the same is true for the Newton--Okounkov body $\\Delta_{\\bar{\\gv}_{\\Yng(1)^{\\op}}}(D_{n-k},p_{(n-k)\\times k})$.\nBy Theorem~\\ref{NO_bodies_are_positive}, $\\Delta_{\\bar{\\gv}_{\\Yng(1)^{\\op}}}(D_{n-k},p_{(n-k)\\times k})$ is positive, hence it is broken line convex.\nTherefore, the broken line convex hull of the set $\\lrc{ \\bar{\\gv}_{\\Yng(1)^{\\op}} \\lrp{p_J} \\mid J \\in \\binom{[n]}{n-k}} $ in the lattice $\\Trop_\\R({\\cA^{\\op}\/T_{K}}^{\\vee}_{\\seed_{\\Yng(1)^{\\op}}})$ \ncoincides with its convex hull.\nWe take into account Remark~\\ref{rem:comparing_NO_bodies} to get that $ \\Delta_{\\gv_{\\Yng(1)^{\\op}}}(\\lb_e) = \\Delta_{\\bar{\\gv}_{\\Yng(1)^{\\op}}}(D_{n-k},p_{(n-k)\\times k}) + \\gv_{\\Yng(1)^{\\op}}(p_{(n-k)\\times k}) $.\nThis implies that $ \\Delta_{\\gv_{\\Yng(1)^{\\op}}}(\\lb_e) $ is the broken line convex hull of the set $\\lrc{ \\gv_{\\Yng(1)^{\\op}} \\lrp{p_J} \\mid J \\in \\binom{[n]}{n-k}} $.\nBeing a broken line convex set is independent of the choice of seed, the result follows.\n\\end{proof}\n\n\\begin{remark}\nIn the proof of Corollary~\\ref{cor:intrinsicNO grassmannian}, we implicitly use the $\\cA$ cluster structure to view the intrinsic Newton--Okounkov body $\\Delta_{\\mathrm{BL}}(\\lb_e)$ as the broken line convex hull of tropical points indexing Pl\\\"ucker coordinates.\nWe could alternatively use the $\\cX$ cluster structure as Rietsch--Williams do, and define theta functions with the corresponding $\\cX$ scattering diagram.\nBy identifying the Rietsch--Williams valuation with the $\\cv$-vector valuation (Corollary~\\ref{cor:gv on midX}) and noting that there is a cluster ensemble automorphism of the open positroid variety (see \\cite[Theorem~7.1, Corollary~5.11]{MullSp}, \\cite[Theorem~7.3, Proposition~7.4]{RW}),\nwe can apply Lemma~\\ref{lem:cval_gval} to give a completely analogous statement to Corollary~\\ref{cor:intrinsicNO grassmannian} which uses the Rietsch--Williams valuation rather than the $\\gv$-vector valuation.\nIn fact, in \\S\\ref{sec:Gr36} we present an example of an explicit computation related to the intrinsic Newton--Okounkov body $\\Delta_{\\mathrm{BL}}(\\lb_e)$ defined via the $\\Xnet$ scattering diagram.\n\\end{remark}\n\n\n\n\\subsubsection{Example}\\label{sec:Gr36}\nIn this subsection, we will give an example of the intrinsic Newton--Okounkov body for the case of $\\Grass_3\\lrp{\\C^6}$ and compare this to a Newton--Okounkov body of \\cite{RW}. \nIn particular, in \\cite[\\S9]{RW}, Rietsch--Williams discuss a non-integral vertex appearing in the Newton--Okounkov body $\\Delta_{\\val_{G}}(D_{3},p_{123})$ associated to the plabic graph $G$ of Figure~\\ref{fig:3-6}.\nWe illustrate how this non-integral vertex in the usual Newton--Okounkov body framework corresponds to a point in the interior of a broken line segment in $\\Delta_{\\mathrm{BL}}(\\lb_e)$ and thus is not a genuine vertex from the intrinsic Newton--Okounkov body perspective.\nHere, to facilitate comparison with \\cite{RW}, we will view the open positroid variety as $\\Xnet_{\\vb{1}}$ (up to codimension 2).\nSo, the scattering diagram we use to define $\\Delta_{\\mathrm{BL}}(\\lb_e)$ in the subsection will be $\\scat^{\\Xnet_{\\mathbf{1}}}_{\\text{in},\\seed_G}$ for a particular choice of initial seed $\\seed_G$. \nThe choice of seed is encoded by the plabic graph illustrated in Figure~\\ref{fig:3-6}.\n\n\\begin{figure}[ht]\n \\centering\n\t\n\\tikzexternaldisable\n\\begin{tikzpicture}[scale=.5]\n\\tikzset{->-\/.style={decoration={\n markings,\n mark=at position #1 with {\\arrow{>}}},postaction={decorate}}}\n \\tikzset{-<-\/.style={decoration={\n markings,\n mark=at position #1 with {\\arrow{<}}},postaction={decorate}}}\n \n\\draw (0,0) circle [radius=5]; \n \n\\draw[-<-=.5] (-1,2) -- (1,2);\n\\draw[->-=.5] (1,2) -- (2.25,0);\n\\draw[->-=.5] (2.25,0) -- (1,-2);\n\\draw[->-=.5] (1,-2)-- (-1,-2);\n\\draw[-<-=.5] (-1,-2) -- (-2.25,0);\n\\draw[-<-=.5] (-2.25,0)-- (-1,2);\n\\draw[-<-=.5] (-1,2) -- (-1,3.5);\n\\draw[->-=.5] (-1,3.5)-- (1,3.5);\n\\draw[->-=.5] (1,3.5) -- (1,2);\n\\draw[-<-=.5] (2.25,0) -- (3.5,-1);\n\\draw[->-=.7] (3.5,-1) -- (2.375,-3);\n\\draw[-<-=.5] (2.375,-3) -- (1,-2);\n\\draw[->-=.5] (-2.25,0) -- (-3.5,-1);\n\\draw[->-=.5] (-3.5,-1) -- (-2.375,-3);\n\\draw[-<-=.5] (-2.375,-3) -- (-1,-2);\n\\draw[->-=.5] (-1,4.9) -- (-1,3.5);\n\\draw[->-=.5] (1,4.9) -- (1,3.5);\n\\draw[->-=.5] (4.7,-1.75) -- (3.5,-1);\n\\draw[->-=.5] (2.375,-3) -- (3.35,-3.75);\n\\draw[->-=.5] (-3.5,-1) -- (-4.7,-1.75);\n\\draw[->-=.5] (-2.375,-3) -- (-3.35,-3.75);\n\n \n\\draw[fill] (1,3.5) circle [radius=.175]; \n\\draw[fill] (-1,2) circle [radius=.175]; \n\\draw[fill] (2.25,0) circle [radius=.175]; \n\\draw[fill] (2.375,-3) circle [radius=.175]; \n\\draw[fill] (-1,-2) circle [radius=.175]; \n\\draw[fill] (-3.5,-1) circle [radius=.175]; \n\n\\draw[fill, white] (-1,3.5) circle [radius=.175]; \n\\draw (-1,3.5) circle [radius=.175]; \n\\draw[fill, white] (1,2) circle [radius=.175]; \n\\draw (1,2) circle [radius=.175]; \n\\draw[fill, white] (3.5,-1) circle [radius=.175]; \n\\draw (3.5,-1) circle [radius=.175];\n\\draw[fill, white] (1,-2) circle [radius=.175]; \n\\draw (1,-2) circle [radius=.175]; \n\\draw[fill, white] (-2.25,0) circle [radius=.175]; \n\\draw (-2.25,0) circle [radius=.175]; \n\\draw[fill, white] (-2.375,-3) circle [radius=.175]; \n\\draw (-2.375,-3) circle [radius=.175]; \n\n\\node[above] at (-1,4.9) {1};\n\\node[above] at (1,4.9) {2};\n\\node[right] at (4.7,-1.75) {3};\n\\node[right] at (3.35,-3.75) {4};\n\\node[left] at (-4.7,-1.75) {6};\n\\node[left] at (-3.35,-3.75) {5};\n\n\\node at (0,0) {\\scalebox{.3}{ $\\yng(2,1)$}};\n\\node at (0,2.75) {\\scalebox{.3}{$\\yng(2)$}};\n\\node at (0,4.25) {\\scalebox{.3}{$\\yng(3)$}};\n\\node at (3,1.75) {\\scalebox{.3}{$\\yng(3,3)$}};\n\\node at (2.3,-1.5) {\\scalebox{.3}{$\\yng(3,3,2)$}};\n\\node at (3.7,-2.25) {\\scalebox{.3}{$\\yng(3,3,3)$}};\n\\node at (0,-3.5) {\\scalebox{.3}{$\\yng(2,2,2)$}};\n\\node at (-2.3,-1.5) {\\scalebox{.3}{$\\yng(1,1)$}};\n\\node at (-3.7,-2.25) {\\scalebox{.3}{$\\yng(1,1,1)$}};\n\\node at (-3,1.75) {{$\\varnothing$}};\n\n\\begin{scope}[xshift=14cm]\n\n\\def\\op{.4}\n\n\\draw[opacity=\\op, name path = boundary] (0,0) circle [radius=5]; \n \n\\node [circle, draw=black, fill=black, inner sep=0pt, minimum size=5pt, opacity=\\op] (1b) at (-1,2) {}; \n\\node [circle, draw=black, fill=black, inner sep=0pt, minimum size=5pt, opacity=\\op] (2b) at (1,3.5) {}; \n\\node [circle, draw=black, fill=black, inner sep=0pt, minimum size=5pt, opacity=\\op] (3b) at (2.25,0) {}; \n\\node [circle, draw=black, fill=black, inner sep=0pt, minimum size=5pt, opacity=\\op] (4b) at (2.375,-3) {}; \n\\node [circle, draw=black, fill=black, inner sep=0pt, minimum size=5pt, opacity=\\op] (5b) at (-1,-2) {}; \n\\node [circle, draw=black, fill=black, inner sep=0pt, minimum size=5pt, opacity=\\op] (6b) at (-3.5,-1) {}; \n\n\\node [circle, draw=black, fill=white, inner sep=0pt, minimum size=5pt, opacity=\\op] (1w) at (-1,3.5) {};\n\\node [circle, draw=black, fill=white, inner sep=0pt, minimum size=5pt, opacity=\\op] (2w) at (1,2) {};\n\\node [circle, draw=black, fill=white, inner sep=0pt, minimum size=5pt, opacity=\\op] (3w) at (3.5,-1) {};\n\\node [circle, draw=black, fill=white, inner sep=0pt, minimum size=5pt, opacity=\\op] (4w) at (1,-2) {};\n\\node [circle, draw=black, fill=white, inner sep=0pt, minimum size=5pt, opacity=\\op] (5w) at (-2.375,-3) {};\n\\node [circle, draw=black, fill=white, inner sep=0pt, minimum size=5pt, opacity=\\op] (6w) at (-2.25,0) {};\n\n\n\\draw[color=gray, opacity=\\op] (1b) -- (2w) ;\n\\draw[color=gray, opacity=\\op] (2w) -- (3b) ;\n\\draw[color=gray, opacity=\\op] (3b) -- (4w) ;\n\\draw[color=gray, opacity=\\op] (4w) -- (5b) ;\n\\draw[color=gray, opacity=\\op] (5b) -- (6w) ;\n\\draw[color=gray, opacity=\\op] (6w) -- (1b) ;\n\n\\draw[color=gray, opacity=\\op] (1b) -- (1w) ;\n\\draw[color=gray, opacity=\\op] (1w) -- (2b) ;\n\\draw[color=gray, opacity=\\op] (2b) -- (2w) ;\n\\draw[color=gray, opacity=\\op] (3b) -- (3w);\n\\draw[color=gray, opacity=\\op] (3w) -- (4b);\n\\draw[color=gray, opacity=\\op] (4b) -- (4w);\n\\draw[color=gray, opacity=\\op] (6w) -- (6b);\n\\draw[color=gray, opacity=\\op] (6b) -- (5w);\n\\draw[color=gray, opacity=\\op] (5w) -- (5b);\n\n\\path [name path = 1p] (1w) -- (-1,5);\n\\path [name intersections={of=boundary and 1p, by = 1e}];\n\\path [name path = 2p] (2w) -- (1,5);\n\\path [name intersections={of=boundary and 2p, by = 2e}];\n\\path [name path = 3p] (3b) --++ (2.5,-2);\n\\path [name intersections={of=boundary and 3p, by = 3e}];\n\\path [name path = 4p] (4b) --++ (1.25,-1);\n\\path [name intersections={of=boundary and 4p, by = 4e}];\n\\path [name path = 5p] (5w) --++ (-1.25,-1);\n\\path [name intersections={of=boundary and 5p, by = 5e}];\n\\path [name path = 6p] (6b) --++ (-1.25,-1);\n\\path [name intersections={of=boundary and 6p, by = 6e}];\n\n\n\\draw[opacity=\\op] (1w) -- (1e) node [pos=1, above,opacity=1] {$1$};\n\\draw[opacity=\\op] (2b) -- (2e) node [pos=1, above,opacity=1] {$2$};\n\\draw[opacity=\\op] (3w) -- (3e) node [pos=1, right,opacity=1] {$3$};\n\\draw[opacity=\\op] (4b) -- (4e) node [pos=1, right,opacity=1] {$4$};\n\\draw[opacity=\\op] (5w) -- (5e) node [pos=1, left,opacity=1] {$5$};\n\\draw[opacity=\\op] (6b) -- (6e) node [pos=1, left,opacity=1] {$6$};\n\n\n\n\\node (246) at (0,0) {\\footnotesize $246$};\n\\node (256) at (0,2.75) {\\footnotesize $256$};\n\\node (156) at (0,4.25) {\\footnotesize $156$};\n\\node (126) at (3,1.75) {\\footnotesize $126$};\n\\node (124) at (2.3,-1.5) {\\footnotesize $124$};\n\\node (123) at (3.8,-2.35) {\\footnotesize $123$};\n\\node (234) at (0,-4) { \\footnotesize $234$};\n\\node (346) at (-2.3,-1.5) {\\footnotesize $346$};\n\\node (345) at (-3.8,-2.35) {\\footnotesize $345$};\n\\node (456) at (-3,1.75) {\\footnotesize $456$};\n\n\\draw [->] (246) -- (456);\n\\draw [->] (246) -- (126);\n\\draw [->] (246) -- (234);\n\n\\draw [->] (256) -- (246);\n\\draw [->] (124) -- (246);\n\\draw [->] (346) -- (246);\n\n\\draw [->] (256) -- (156);\n\n\\draw [->] (2.5,-1.8) -- (3.35,-2.25) ;\n\n\\draw [->] (-2.5,-1.8) -- (-3.35,-2.25) ;\n\n\\draw [->] (456) -- (346);\n\\draw [->] (234) -- (346);\n\\draw [->] (456) -- (256);\n\\draw [->] (126) -- (256);\n\\draw [->] (126) -- (124);\n\\draw [->] (234) -- (124);\n\n\n\\end{scope}\n\n\\end{tikzpicture}\n\n\\tikzexternalenable\n\n\n \\caption{A plabic graph $G$ for $\\Grass_3\\lrp{\\C^6}$ for which $\\Delta_{\\val_{G}}(D_{3},p_{123})$ has non-integral vertex (see \\cite[\\S9]{RW}). On the left, the labels are in terms of Young diagrams. On the right, we display the quiver and label faces by Pl\\\"ucker coordinates.}\n \\label{fig:3-6}\n\\end{figure}\n\n\n\n\nRecall that the Young diagrams in Figure~\\ref{fig:3-6} label the network parameters used in flow polynomial expressions (see \\cite[Equation (6.3)]{RW}).\nThe $\\cA$-cluster determined by trips in the plabic graph $G$ consists of the Pl\\\"ucker coordinates whose indices are given in Figure~\\ref{fig:3-6} (see \\cite[Definition 3.5]{RW}).\n\n\n\nAccording to \\cite[\\S9]{RW}, a non-integral vertex in the Newton--Okounkov polytope comes from half the valuation of the flow polynomial for the element $ f = (p_{124} p_{356} - p_{123} p_{456}) \/ {p_{123}^2}$.\nThey compute $\\frac{1}{2}\\val_G(f)$ and express its entries in tabular form (see \\cite[Table~3]{RW}) as we have reproduced in Table~\\ref{table}.\nThe function $p_{123}^2\\, f$ is one of the two $\\cA$ cluster variables that are not Pl\\\"ucker coordinates, see {\\it e.g.} \\cite[Eq.(4), p.42]{Sco06}. \nIt is obtained by mutation at $\\Yng(2,1)$.\n\n\\begin{table}\n\\begin{center}\n\\captionsetup{type=table}\n\\begin{tabular}{|C|C|C|C|C|C|C|C|C|} \n\\hline\n \\vphantom{\\Yng(1,1,1,1)} \\Yng(3,3,3) & \\Yng(3,3,2) & \\Yng(2,2,2) & \\Yng(1,1,1) & \\Yng(3,3) & \\Yng(2,1) & \\Yng(1,1) & \\Yng(3) & \\Yng(2)\\\\\n\\hline\n \\vphantom{\\Yng(1,1,1)_{\\Yng(1,1,1)}}\\frac{3}{2} & \\frac{3}{2} & 1 & \\frac{1}{2} & 1 & \\frac{1}{2} &\\frac{1}{2} &\\frac{1}{2} &\\frac{1}{2} \\\\\n\\hline\n\\end{tabular} \n\\captionof{table}{The rational vertex $\\frac{1}{2}\\val_G(f)$\\label{table}}\n\\end{center}\n\\end{table}\n\n\n\nNote we can re-interpret the expression for $f$ as the expansion of a product of theta functions.\nAll Pl\\\"ucker coordinates are $\\cA$ cluster variables, and all $\\cA$ cluster monomials are theta functions.\nThen\n\\eqn{p_{124}\\, p_{356} = \np_{123}^2\\, f + p_{123}\\, p_{456},}\nand the right hand side is a sum of two theta functions. \nThis means there are only two balanced pairs of broken lines contributing to the product.\nWe will see that the pair with no bending corresponds to the summand $ p_{123}\\, p_{456}$,\nwhile the other involves a maximal bend at an initial wall.\n(Since the bend is at an initial wall, we are able to see the relevant broken line segment without constructing the consistent scattering diagram.)\n\nWe interpret the Rietsch--Williams valuation as being valued in $ \\Trop_{\\Z}((\\cX_{\\mathbf{1}})^{\\vee})$ (see Proposition~\\ref{prop:flow is gv for X}) and consider broken lines in the associated $\\cX$ cluster scattering diagram.\nThe choice of seed identifies $\\Trop_{\\Z}((\\cX_{\\mathbf{1}})^\\vee)$ with $ N^\\vee \/ {\\lra{(1,1,\\dots, 1)}}$ and we draw the scattering in $\\lrp{ N^\\vee \/ {\\lra{(1,1,\\dots, 1)}}} \\otimes \\R$.\n\n\nWe will use Figure \\ref{fig:3-6} to define the fixed data and the seed data for the cluster structure. \nThe initial scattering diagram for the $\\Xnet$ variety is \n\\[ \\mathfrak{D}^{\\cX_{}}_{\\text{in},\\seed_G}= \\lrc{ \\left( (v_{\\mu})^{\\perp}, \\ 1+ z^{e_{\\mu}}\\right) \\mid \\mu \\in \\{ \\Yng(2), \\Yng(2,1), \\Yng(3,3,2), \\Yng(1,1) \\}}.\\]\nTo get initial scattering diagram for the fibre over $\\mathbf{1}$ we take the quotient of the support $\\mathfrak{D}^{\\Xnet}_{\\text{in},\\seed_G}$ by $\\left(\\R \\cdot (1,1,\\dots, 1) \\right)$. (Observe that $(v_{\\mu})^{\\perp}$ is invariant under translations by $\\R \\cdot (1,1,\\dots, 1)$.) \n\n\\[ \\mathfrak{D}^{\\cX_{\\mathbf{1}}}_{\\text{in},\\seed_G}= \\lrc{ \\left( (v_{\\mu})^{\\perp}\/ \\left(\\R \\cdot (1,1,\\dots, 1) \\right) , \\ 1+ z^{e_{\\mu}}\\right) \\mid \\mu \\in \\{ \\Yng(2), \\Yng(2,1), \\Yng(3,3,2), \\Yng(1,1) \\}}\\]\n\nAll pertinent valuations may be found in \\cite[Table 3]{bossinger2019full}.\nWe record them here using the ordering of Table~\\ref{table}. We choose the representative whose coefficient of $e_\\varnothing$ is $0$ and do not record this entry.\n\\eqn{\\val_{G}({p}_{124}) = (1,0,0,0,0,0,0,0,0)}\n\\eqn{\\val_{G}({p}_{356}) = (2,2,2,1,2,1,1,1,1)}\n\\eqn{\\val_{G}({p}_{123}) = (0,0,0,0,0,0,0,0,0) }\n\\eqn{\\val_{G}({p}_{456}) = (3,2,2,1,2,1,1,1,1) }\nSo, we have $\\val_{G}({p}_{124}) + \\val_G( p_{356} ) = \\val_{G}( p_{123} \\, p_{456} )$.\nThe summand $p_{123} \\, p_{456} $ in the product $p_{124} \\, p_{356}$ corresponds to the straight broken line segment from $ \\val_{G}({p}_{356})$ to $\\val_G( p_{124} )$, whose midpoint is $\\frac{1}{2} \\val_{G}( p_{123} \\, p_{456} )$. \n\nThe bending wall for the other broken line segment is $\\lrp{(v_{\\Yngs(3,3,2)})^{\\perp}, 1+z^{e_{\\Yngs(3,3,2)}}}$.\nNote that $v_{\\Yngs(3,3,2)}= f_{\\Yngs(3,3,3)} + f_{\\Yngs(2,1)}-f_{\\Yngs(3,3)} - f_{\\Yngs(2,2,2)}$, and $\\frac{1}{2}\\val(f) = \\frac{1}{2}\\val(p_{123}^2\\, f)$ is perpendicular to this vector.\nSo, $\\frac{1}{2}\\val(p_{123}^2\\, f)$ lies in the support of this wall.\nWe will see that there is a broken line segment from $\\val_{G}(p_{356})$ to $\\val_{G}(p_{124})$ passing through $\\frac{1}{2}\\val(f)$ and bending maximally here, as depicted in Figure~\\ref{fig:maxbend}.\n\n\\begin{figure}[ht]\n \\centering\n \\begin{tikzpicture}\n \\draw[->] (-2,0) -- (1,0) node[anchor=west]{$\\lrp{(v_{\\Yngs(3,3,2)})^{\\perp}, 1+z^{e_{\\Yngs(3,3,2)}}}$};\n \\filldraw [gray] (0,0) circle (2pt);\n \\filldraw [purple] (-1.5,0) circle (1pt) node[anchor=south east]{$\\frac{1}{2}\\val_G(f)$};\n \\draw[purple] (-1.5,0) -- node[anchor=south east]{$ \\ell_2$} (0,1);\n \\filldraw [purple] (0,1) circle (1pt) node[anchor=south]{$\\val_G(p_{124})$};\n \\draw[purple] (-1.5,0) --node[anchor=north east]{$\\ell_1$} (0,-1);\n \\filldraw [purple] (0,-1) circle (1pt) node[anchor=north]{$\\val_G(p_{356})$};\n \\end{tikzpicture}\n \\caption{Rational point obtained from broken line bending maximally at an initial wall. In this broken line segment, $\\ell_1$ and $\\ell_2$ will take equal time, corresponding to the summand $p_{123}^2\\, f$ in the product $p_{124} \\, p_{356}$.\n\\label{fig:maxbend}} \n\\end{figure}\n\n\nRecall that the exponent vector of the decoration monomial along $\\ell_i$ is the negative of the velocity vector there. \nTraveling along $\\ell_1$, this velocity vector is positively proportional to \n\\[\\frac{1}{2}\\val_G(f) - \\val_G(p_{356}) = \n-\\left(\\frac{1}{2}, \\frac{1}{2}, 1, \\frac{1}{2}, 1, \\frac{1}{2} , \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}\\right).\\] \nWe can take a broken line with exponent vector $ v_1=(1,1,2,1,2,1,1,1,1) $ along $\\ell_1$.\nThe possible bendings after crossing the wall correspond to summands of\n\\eqn{z^{v_1}\\lrp{1+z^{e_{\\Yngs(3,3,2)}}}^{-\\lra{v_1,v_{\\Yngs(3,3,2)}}}=z^{v_1}\\lrp{1+z^{e_{\\Yngs(3,3,2)}}}^{2}.}\nThe maximal bending corresponds to the summand $z^{v_1+ 2 e_{\\Yngs(3,3,2)}} = z^{(1,3,2,1,2,1,1,1,1)}$. Let us call $v_2 := (1,3,2,1,2,1,1,1,1)$.\nThen observe that $\\frac{1}{2}\\val_G(f) - \\frac{1}{2}v_2 = \\val_G(p_{124})$. \nSo, we have a broken line segment $\\gamma$ traveling from $\\val_G(p_{356})$ to $\\frac{1}{2}\\val_G(f)$ with decoration monomial $z^{v_1}$, bending maximally and continuing to $\\val_G(p_{124})$ with decoration monomial $z^{v_2}$. Precisely $\\frac{1}{2}$ a unit of time is spent in each straight segment.\nFrom this perspective, $\\frac{1}{2}\\val_G(f)$ is not a genuine vertex; $\\frac{1}{2}\\val_G(f)$ is in the relative interior of the support of $\\gamma$, and the endpoints of $\\gamma$ are in the Newton--Okounkov body.\n\n\\footnotesize\n\\begin{thebibliography}{BFMMNC20}\n\n\\bibitem[AB22]{AB22}\nH\\\"ulya Arg\\\"uz and Pierrick Bousseau.\n\\newblock Fock-{G}oncharov dual cluster varieties and {G}ross-{S}iebert\n mirrors.\n\\newblock {\\em arXiv:2206.10584[math.AG]}, 2022.\n\n\\bibitem[ADHL15]{ADHL}\nIvan Arzhantsev, Uwe Derenthal, J\u00fcrgen Hausen, and Antonio Laface.\n\\newblock {\\em Cox rings}, volume 144 of {\\em Cambridge Studies in Advanced\n Mathematics}.\n\\newblock Cambridge University Press, Cambridge, 2015.\n\n\\bibitem[And13]{An13}\nDavid Anderson.\n\\newblock Okounkov bodies and toric degenerations.\n\\newblock {\\em Mathematische Annalen}, 356(3):1183--1202, 2013.\n\n\\bibitem[{Bau}03]{Baur_CartanComp}\nKarin {Baur}.\n\\newblock {Cartan components and decomposable tensors}.\n\\newblock {\\em {Transform. 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Algebra Geom.}, 3(2):292--336, 2019.\n\n\\bibitem[KY23]{KY19}\nSean {Keel} and Tony~Yue {Yu}.\n\\newblock The {F}robenius structure theorem for affine log-{C}alabi-{Y}au\n varities containing a torus.\n\\newblock {\\em arXiv preprint arXiv:1908.09861.v2 [math.AG], to appear in\n Annals of Mathematics}, 2023.\n\n\\bibitem[Lit98]{Lit98}\nPeter Littelmann.\n\\newblock Cones, crystals, and patterns.\n\\newblock {\\em Transformation groups}, 3(2):145--179, 1998.\n\n\\bibitem[LM09]{LM09}\nRobert Lazarsfeld and Mircea Musta\\c{t}\\u{a}.\n\\newblock Convex bodies associated to linear series.\n\\newblock {\\em Ann. Sci. \\'Ec. Norm. Sup\\'er. 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Marsh and Jeanne Scott.\n\\newblock Twists of {P}l\\\"{u}cker coordinates as dimer partition functions.\n\\newblock {\\em Comm. Math. Phys.}, 341(3):821--884, 2016.\n\n\\bibitem[MS17]{MullSp}\nGreg Muller and David~E. Speyer.\n\\newblock The twist for positroid varieties.\n\\newblock {\\em Proc. Lond. Math. Soc. (3)}, 115(5):1014--1071, 2017.\n\n\\bibitem[NZ12]{NZ}\nTomoki Nakanishi and A.~Zelevinsky.\n\\newblock On tropical dualities in cluster algebras.\n\\newblock In {\\em Algebraic groups and quantum groups}, volume 565 of {\\em\n Contemp. Math.}, pages 217--226. Amer. Math. Soc., Providence, RI, 2012.\n\n\\bibitem[Oko96]{Oko96}\nAndrei Okounkov.\n\\newblock Brunn-{M}inkowski inequality for multiplicities.\n\\newblock {\\em Invent. Math.}, 125(3):405--411, 1996.\n\n\\bibitem[Oko03]{Oko03}\nAndrei Okounkov.\n\\newblock Why would multiplicities be log-concave?\n\\newblock In {\\em The orbit method in geometry and physics ({M}arseille,\n 2000)}, volume 213 of {\\em Progr. Math.}, pages 329--347. Birkh\\\"{a}user\n Boston, Boston, MA, 2003.\n\n\\bibitem[Pos06]{Pos06}\nAlexander Postnikov.\n\\newblock Total positivity, {G}rassmannians, and networks.\n\\newblock {\\em arXiv preprint arXiv:math\/0609764 [math.CO]}, 2006.\n\n\\bibitem[Qin17]{Qin17}\nFan Qin.\n\\newblock Triangular bases in quantum cluster algebras and monoidal\n categorification conjectures.\n\\newblock {\\em Duke Math. J.}, 166(12):2337--2442, 2017.\n\n\\bibitem[Qin22]{Qintropical}\nFan Qin.\n\\newblock Bases for upper cluster algerbas and tropical points.\n\\newblock {\\em Journal of the European Mathematical Society (2022),\n DOI:10.4171\/JEMS\/1308.}, 2022.\n\n\\bibitem[RW19]{RW}\nKonstanze {Rietsch} and Lauren {Williams}.\n\\newblock Newton--{O}kounkov bodies, cluster duality, and mirror symmetry for\n {G}rassmannians.\n\\newblock {\\em Duke Math. J.}, 168(18):3437--3527, 2019.\n\n\\bibitem[Sco06]{Sco06}\nJeanne Scott.\n\\newblock Grassmannians and cluster algebras.\n\\newblock {\\em Proc. London Math. Soc. (3)}, 92(2):345--380, 2006.\n\n\\bibitem[SW20]{SW18}\nLinhui Shen and Daping Weng.\n\\newblock Cyclic sieving and cluster duality of {G}rassmannian.\n\\newblock {\\em SIGMA Symmetry Integrability Geom. Methods Appl.}, 16:067, 41\n pages, 2020.\n\n\\end{thebibliography}\n\n\n\\end{document}\n"},{"text":"\\documentclass[12pt,leqno]{amsart}\n\\usepackage{amssymb,amsthm}\n\\usepackage{amsmath,amssymb,color}\n\\oddsidemargin 0pt \\evensidemargin 0pt \\marginparwidth 1in\n\\marginparsep 0pt \\leftmargin 1.00in \\topmargin 8pt\n%\\footheight 0.25in\n%\\footskip 0.25in\n\\textheight 9in\n\\textwidth 6.5in\n\n%\\begin{document}\n%\\begin{document}\n%\n%\n\\baselineskip=18pt\n%\\theoremstyle{plain}\n%\\newtheorem{MainThm}{Theorem}\n%\\newtheorem{thm}{Theorem}[section]\n%\\newtheorem{clry}[thm]{Corollary}\n%\\newtheorem{prop}[thm]{Proposition}\n%\\newtheorem{lem}[thm]{Lemma}\n%\\newtheorem{deft}[thm]{Definition}\n%\\newtheorem{hyp}{Assumption}\n%\\newtheorem*{ThmLeU}{Theorem (J.~Lee, G.~Uhlmann)}\n\n%\\theoremstyle{definition}\n%\\newtheorem{rem}[thm]{Remark}\n%\\newtheorem*{acknow}{Acknowledgments}\n%\n%\\numberwithin{equation}{section}\n%\\renewcommand{\\baselinestretch}{1.1}\n%% definition 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\\{p_k\\}_{k\\in \\N}}\n\\newcommand{\\rbra}{\\rangle_{H^1_0(\\Omega)}}\n\\newcommand{\\lbran}{_{(H^{-1}(\\Omega))^N}{\\langle}}\n\\newcommand{\\rbran}{\\rangle_{(H^1_0(\\Omega))^N}}\n\\newcommand{\\llll}{L^{\\infty}(\\Omega\\times (0,t_1))}\n\\newcommand{\\sumkzero}{\\sum_{k=0}^{\\infty}}\n\\newcommand{\\sumkone}{\\sum_{k=1}^{\\infty}}\n%\\newcommand{\\sumk}{\\sum_{k=1}^{\\infty}}\n\\renewcommand{\\baselinestretch}{1.5}\n%\n\\renewcommand{\\div}{\\mathrm{div}\\,} %div\n\\newcommand{\\grad}{\\mathrm{grad}\\,} %grad\n\\newcommand{\\rot}{\\mathrm{rot}\\,} %rot\n\\newcommand{\\urur}{\\left( \\frac{1}{r}, \\theta\\right)}\n\n\\allowdisplaybreaks\n%% item\n\\renewcommand{\\theenumi}{\\arabic{enumi}}\n\\renewcommand{\\labelenumi}{(\\theenumi)}\n\\renewcommand{\\theenumii}{\\alph{enumii}}\n\\renewcommand{\\labelenumii}{(\\theenumii)}\n\\newcommand{\\nuA}{\\ppp_{\\nu_A}} \n\\def\\thefootnote{{}}\n\n\\title\n[]\n{\nInverse parabolic problem with initial data by a single \nmeasurement\n}\n% with\n%time order $\\alpha \\in (1,2)$.\n%}\n\n\\pagestyle{myheadings}\n\\markboth{M.~Yamamoto}\n{classes of solutions}\n\n\\author{\n$^1$ O.Y. ~Imanuvilov and $^{2,3,4}$ M.~Yamamoto}\n\n\\thanks{\n$^1$ Department of Mathematics, Colorado State University,\\\\\n101 Weber Building, Fort Collins, CO 80523-1874, USA \\\\\ne-mail: {\\tt oleg@math.colostate.edu}\n\\\\\n$^2$ Graduate School of Mathematical Sciences, The University\nof Tokyo, Komaba, Meguro, Tokyo 153-8914, Japan \\\\\n$^3$ Honorary Member of Academy of Romanian Scientists,\nIlfov, nr. 3, Bucuresti, Romania \\\\\n$^4$ Correspondence member of Accademia Peloritana dei Pericolanti,\\\\\nPalazzo Universit\\`a, Piazza S. Pugliatti 1 98122 Messina Italy \\\\\ne-mail: {\\tt myama@ms.u-tokyo.ac.jp}\n}\n\n\\date{}\n\\begin{document}\n\\maketitle\n\n\\baselineskip 18pt\n\n\\begin{abstract}{\\it \nWe consider \n%an inverse coefficient problem for \ninitial boundary value problems with the homogeneous Neumann \nboundary condition. Given an initial value, we establish\nthe uniqueness in determining a spatially varying coefficient \nof zeroth-order term by a single measurement of Dirichlet data\non an arbitrarily chosen subboundary. The uniqueness holds\nin a subdomain where the initial value is positive, \nprovided that it is sufficiently smooth which is specified \nby decay rates of the Fourier coefficients.\nThe key idea is the reduction to an inverse elliptic problem\nand relies on elliptic Carleman estimates.}\n\\\\\n{\\bf Key words.}\ninverse coefficient problem, parabolic equation, uniqueness,\ninitial boundary value problem, inverse elliptic problem,\nCarleman estimate\n\\\\\n{\\bf AMS subject classifications.}35R30, 35K15\n\\end{abstract}\n\n\\section{Introduction}\n\nLet $\\Omega$ be a bounded domain in $\\Bbb R^n$ with $C^2-$ boundary\n$\\ppp\\OOO$. \nWe set $\\nu=(\\nu_1(x),\\dots,\\nu_n(x))$ be an outward unit normal vector to \n$\\ppp\\OOO$, and\n$$\n\\ppp_k = \\frac{\\ppp}{\\ppp x_k}, \\quad 1\\le k\\le n, \\quad\n\\ppp_t = \\frac{\\ppp}{\\ppp t}.\n$$\nWe assume \n$$\na_{ij}=a_{ji} \\in C^2(\\ooo{\\OOO}) \\quad \\mbox{for all $1\\le i,j\\le n$},\n\\quad c_1, c_2 \\in C(\\ooo{\\OOO}),\n$$\nand there exists a constant $\\kappa_1>0$ such that\n$$\n\\sum_{i,j=1}^n a_{ij}(x)\\xi_i\\xi_j\n\\ge \\kappa_1\\vert \\xi\\vert^2 \\quad \\mbox{for all $x \\in \\ooo{\\OOO}$ and\n$\\xi=(\\xi_1, ... \\xi_n) \\in \\Bbb R^n$}.\n$$\nWe set \n$$\n\\ppp_{\\nu_A}v: = \\sumij a_{ij}(x)\\nu_i(x)\\ppp_jv(x), \\quad \nx\\in \\ppp\\OOO.\n$$\nMoreover let \n$$\nA(x,D)v(x) = - \\sumij \\ppp_i(a_{ij}(x)\\ppp_jv(x)) \\quad \n\\mbox{with} \\quad\n\\DDD(A):= \\{ v\\in H^2(\\OOO);\\, \\ppp_{\\nu_{A_0}} v= 0\\quad\n\\mbox{on $\\ppp\\OOO$}\\}. \\eqno{(1.1)}\n$$\n\nIn this article, we consider the folowing \n\\\\\n{\\bf Inverse problem.}\n\\\\\n{\\it\nLet \n$$\\left\\{ \\begin{array}{rl}\n& \\ppp_tu = -A(x,D)u + c_1(x)u \\quad \\mbox{in $\\OOO\\times (0,T)$}, \\\\\n& \\nuA u = 0 \\quad\\mbox{on $\\ppp\\OOO$}, \\\\\n& u(x,0) = a(x), \\quad x\\in \\OOO,\n\\end{array}\\right.\n \\eqno{(1.2)}\n$$\nand\n$$\\left\\{ \\begin{array}{rl}\n& \\ppp_tv = -A(x,D)v + c_2(x)v \\quad \\mbox{in $\\OOO\\times (0,T)$}, \\\\\n& \\nuA v = 0 \\quad\\mbox{on $\\ppp\\OOO$}, \\\\\n& v(x,0) = a(x), \\quad x\\in \\OOO.\n\\end{array}\\right.\n \\eqno{(1.3)}\n$$\nLet an initial value $a$ be suitably given and\n$\\gamma \\subset \\ppp\\OOO$ be an arbitrarily chosen non-empty connected\nrelatively open subset of $\\ppp\\OOO$.\nThen \n$$\n\\mbox{$u=v$ on $\\gamma \\times (0,T)$ implies $c_1=c_2$ in $\\OOO$?} \n$$\n}\n\nThis inverse problem has been intensively studied in the literature. \nMost general results are obtained for the case when a time of observation \n$t_0$ belongs to the open interval $(0,T)$. In this case, based on the method introduce in \nBukhgeim and Klibanov \\cite{BK}, Imanuvilov and Yamamoto \\cite{IY98} proved \nthe uniqueness and the Lipschitz stability in determination of coefficients \ncorresponding to the zeroth order term. \nRecently in Imanuvilov and Yamamoto \\cite{IY23},\nthe authors proved a conditional Lipschitz stability \nestimate as well as the uniqueness for the case $t_0=T$.\nSee also Huang, Imanuvilov and Yamamoto \\cite{HIY}.\n\nIn case an observation is taken at the initial moment $t_0=0,$ \nto the authors' best knowledge, the question of uniqueness of a solution \nof inverse problem is open in general.\nLimited to the one-dimensional case, a result in this direction was \nobtained by Suzuki \\cite{S} and Suzuki and Murayama \\cite{SM}. \nKlibanov \\cite{Kl} proved the uniqueness of determination of zeroth order \nterm coefficient in the case when $a_{ij}=\\delta_{ij}$ (the case of \nthe Laplace operator). Also it is assumed that the domain of observation \n$\\Gamma=\\partial\\Omega$.\nThe method proposed in \\cite{Kl} is based on an integral transform and \nsubsequent reduction of the original problem to the problem of determination \nof a coefficient of zeroth order term for a hyperbolic equation. \nAfter that, the method in \\cite{BK} is applied. \nIt should be mentioned that the method introduced by \nBukhgeim and Klibanov is based on Carleman estimates and the Carleman type \nestimates for hyperbolic equations are subjected to so-called non-trapping \nconditions. Therefore both assumptions made in \\cite{Kl} are critically \nimportant for the application of this method except of the one dimensional case. \nIn \\cite{IY23}, the authors extended results of \\cite{Kl} to the case of \ngeneral second order hyperbolic equation. \nThe main purpose of the current work is to remove the non-trapping assumptions\nand prove the uniqueness without any geometric constraints on the \nobservation subbondary $\\gamma$.\n\\\\\n\nHenceforth we set \n$$\n-A_1(x,D) = -A(x,D) + c_1(x), \\quad -A_2(x,D) = -A(x,D) + c_2(x)\n$$\nwith the domains $\\DDD(A_1) = \\DDD(A_2) = \\DDD(A)$.\nIt is known that the spectrum \n$\\sigma(A_k)$ of $A_k$, $k=1,2$, consists entirely of eigenvalues with \nfinite multiplicities.\n\nBy changing $\\www{u}:= e^{Mt}u$ with some constant $M$, it suffices to \nassume that there exists a constant $\\kappa_2>1$ \nsuch that $(A_1u,u)_{L^2(\\Omega)} \\ge \\kappa_2\\Vert u\\Vert^2_{L^2(\\Omega)}$ \nfor $u \\in \\DDD(A_1)$\nand $(A_2v,v)_{L^2(\\Omega)} \\ge \\kappa_2\\Vert v\\Vert^2_{L^2(\\Omega)}$\nfor $v\\in \\DDD(A_2)$.\n\nThen, setting\n$$\n\\sigma(A_1)= \\{\\la_k\\}_{k\\in \\N}, \\quad \\sigma(A_2) = \\{ \\mu_k\\}_{k\\in \\N},\n$$\nwe can number as\n$$\n1 < \\la_1 < \\la_2 < \\cdots, \\qquad 1<\\mu_1 < \\mu_2 < \\cdots.\n$$\n\nLet $P_k$ be the projection for $\\la_k$, $k \\in \\N$ which is defined by\n$$\nP_k = \\frac{1}{2\\pi\\sqrt{-1}} \\int_{\\gamma(\\la_k)}\n(z-A_1)^{-1} dz, \\quad\nQ_k = \\frac{1}{2\\pi\\sqrt{-1}} \\int_{\\gamma(\\mu_k)}\n(z-A_2)^{-1} dz,\n$$\nwhere $\\gamma(\\la_k)$ is a circle centered at $\\la_k$ with sufficiently \nsmall radius such that the disc bounded by $\\gamma(\\la_k)$ does not\ncontain any points in $\\sigma(A_1)\\setminus \\{\\la_k\\}$, and \n$\\gamma(\\mu_k)$ is a similar sufficiently small circle centered \nat $\\mu_k$. \nThen $P_k:L^2(\\OOO) \\longrightarrow L^2(\\OOO)$ is a bounded linear operator\nto a finite dimensional space \nand $P_k^2 = P_k$ and $P_kP_{\\ell} = 0$ for $k, \\ell\\in \\N$ with \n$k \\ne \\ell$. Then \n$P_kL^2(\\OOO) = \\{ b\\in \\DDD(A_1);\\, A_1b=\\la_kb\\}$, and we have\n$a = \\sum_{k=1}^{\\infty} P_ka$ in $L^2(\\OOO)$ for each \n$a \\in L^2(\\OOO)$ (e.g., Agmon \\cite{Ag}, Kato \\cite{Ka}).\nSetting $m_k:= \\mbox{dim}\\, P_kL^2(\\OOO)$, we have \n$m_k<\\infty$, and we call $m_k$ the multiplicity of $\\la_k$.\nSimilarly let $n_k$ and $Q_k$ be the multiplicity and\nthe eigenprojection for $\\mu_k$. \n\\\\\n\nMoreover we set $Q:= \\OOO \\times (0,T)$, and \n$$\nH^{2,1}(Q):= \\{ w\\in L^2(Q);\\, \nw, \\, \\ppp_iw,\\, \\ppp_i\\ppp_jw,\\, \\ppp_tw \\in L^2(Q)\n\\,\\, \\mbox{for $1\\le i,j\\le n$}\\}.\n$$\n%Let $\\Vert \\cdot\\Vert$ denote the norm in $L^2(\\OOO)$, and we \n%specify the norm in a space $X$ by $\\Vert \\cdot\\Vert_X$ if we \n%consider other space norms other than $L^2(\\OOO)$.\n\nLet \n$$\n\\Gamma=\\{x\\in \\gamma;\\, \\vert a(x)\\vert >0\\}.\n$$\n\nWe assume that \n$$\n\\Gamma\\ne \\emptyset. \\eqno{(1.4)}\n$$\nFor $a\\in C(\\ooo{\\OOO})$, we set \n$$\n\\OOO_0 := \\{ x\\in \\OOO;\\, \\vert a(x)\\vert > 0\\}. \n$$\n%Let $\\omega \\subset\\OOO_0$ be a union of open connected sets with the \n%following property: If $\\tilde \\omega$ is one of these sets there exist an \n%open subset $\\tilde \\Gamma(\\tilde \\omega)\\subset \\overline{\\tilde \\omega}\n%\\cap \\Gamma$ and \n%$$\n%\\vert a(x)\\vert>0\\quad \\mbox{on}\\quad \\tilde\\Gamma(\\tilde \\omega).\n%$$ \n\nFor $\\Gamma$, we define \n$$\n \\mbox{$\\omega: = \\{ x\\in \\OOO_0;\\,$ there exist a point $x_*\\in \\Gamma$ \nand}\n$$\n$$\n\\mbox{a smooth curve $\\ell \\in C^\\infty[0,1]$ such that \n$\\ell(\\xi) \\in \\OOO_0$ for $0<\\xi\\le 1$ and $\\ell(0)=x_*$, $\\ell(1)=x\\}$}.\n \\eqno{(1.5)}\n$$\nWe remark that the definition (1.5) implies $\\ell \\setminus \\{x_*\\}\n\\subset \\omega$.\n\nIn (1.5), replacing smooth curves by piecewise\nsmooth curves, we still have the same definition for $\\omega$.\n\nWe note that $\\omega$ is not necessarily a connected set.\nHowever, if in addition we suppose that\n$$\n\\mbox{$\\Gamma$ is a connected subset of $\\partial\\Omega$}, \\eqno{(1.6)}\n$$\nthen one can verify that $\\omega \\subset \\OOO$ is a domain, that is,\na connected open set. Indeed, choosing $x, \\www{x}\\in \\omega$ arbitrarily, \nwe will show that we can find a piecewise smooth curve $L \\subset \\omega$ \nconnecting $x$ and $\\www{x}$ as follows. \nFirst we can choose smooth curves $\\ell, \\www{\\ell}\n\\subset \\OOO_0 \\cup \\Gamma$ and points $x_*, \\www{x}_* \\in \\Gamma$\nsuch that $\\ell$ connects $x$ and $x_*$, $\\www{\\ell}$ connects $\\www{x}$ and\n$\\www{x}_*$. The definition implies that $\\ell \\setminus \\{x_*\\},\n\\www{\\ell} \\setminus \\{ \\www{x_*}\\} \\subset \\omega$.\nSince $\\vert a\\vert >0$ in $\\Gamma$, we can \nfind a smooth curve $\\www{\\gamma} \\subset \\OOO_0$ connecting $x_*$ and \n$\\www{x}_*$. Therefore, since $\\www{\\gamma} \\subset \\omega$, it follows that \n$x$ and $\\www{x}$ can be connected by a piecewise smooth curve \n$L \\subset \\omega$ composed by $\\ell, \\www{\\ell}, \\www{\\gamma}$,\nwhich means that $\\omega$ is a connected set.\nMoreover, if $x\\in \\omega$, then we see that any point $\\www{x} \\in \\OOO_0$ \nwhich is sufficiently close to $x$, can be connected to some point $\\www{x}_*\n\\in \\Gamma$ by some smooth curve in $\\OOO_0$. \nTherefore, $\\omega$ is a connected and \nopen set, that is, $\\omega$ is a domain.\n$\\blacksquare$\n\nWe can understand that $\\omega$ is the maximal set such that \nall the points of $\\omega$ is connected by a curve \nin $\\OOO_0$ to $\\Gamma$.\nBy (1.4), we note that\n$\\omega \\ne \\emptyset$. \n\\\\\n{\\bf Examples.}\n\\\\\n(i) Under condition (1.4), we have $\\omega = \\OOO$ if $\\OOO_0 = \\OOO$.\nIn general, if $\\{ x\\in \\OOO;\\, a(x) = 0\\}$ has no interior points, then \n$\\omega = \\OOO_0$. \n\\\\\n(ii) Assume that (1.4) and (1.6) hold true. Let subdomains $D_1, ..., D_m \\subset \\OOO$ satisfy \n$\\ooo{D_1}, ..., \\ooo{D_2} \\subset \\OOO$ and\n$a=0$ on $\\ooo{D_k}$ for $1\\le k \\le m$ and $\\vert a\\vert > 0$ in \n$\\OOO \\setminus \\ooo{\\bigcup_{k=1}^m D_k}$. Then \n$\\omega = \\OOO \\setminus \\ooo{\\bigcup_{k=1}^m D_k}$.\n\\\\\n(iii) Assume that (1.4) and (1.6) hold true. Let sudomains $D_1, D_2$ satisfy $\\ooo{D_1} \\subset D_2$, \n$\\ooo{D_2} \\subset \\OOO$, $a=0$ in $\\ooo{D_2 \\setminus D_1}$ and\n$\\vert a\\vert > 0$ in $D_1 \\cup (\\OOO \\setminus \\ooo{D_2})$. Then \n$\\omega = \\OOO \\setminus \\ooo{D_2}$. We note that $D_1$ is not included in \n$\\omega$ although $\\vert a\\vert > 0$ in $D_1$.\n\\\\\n\nNow we state the main uniqueness result.\n\\\\\n{\\bf Theorem 1.}\n\\\\\n{\\it\nLet $ a\\in C(\\ooo\\Omega),$ $u,v \\in H^{2,1}(Q)$ satisfy (1.2) and (1.3) \nrespectively and $\\ppp_tu, \\ppp_tv \\in H^{2,1}(Q)$ and let (1.4) hold true.\nAssume\n\\\\\n{\\bf Condition 1:} there exists a function $\\theta \\in C[1,\\infty)$ satisfying\n$$\n\\lim_{\\eta\\to\\infty} \\frac{\\theta(\\eta)}{\\eta^{\\frac{2}{3}}} = +\\infty\n$$\nand\n$$\n\\sumk e^{\\theta(\\la_k)} \\Vert P_ka\\Vert^2_{L^2(\\Omega)} < \\infty \n\\quad \\mbox{or}\n\\quad \\sumk e^{\\theta(\\mu_k)} \\Vert Q_ka\\Vert^2_{L^2(\\Omega)} < \\infty.\n \\eqno{(1.7)}\n$$\nThen, \n$$\nu=v \\quad \\mbox{on $\\gamma \\times (0,T)$}\n$$\nimplies $c_1=c_2$ on $\\ooo{\\omega}$.\n}\n\nAs is seen by the proof, without the assumption (1.7), we can prove \nat least the coincidence of the eigenvalues of $A_1$ and $A_2$ of \nnon-vanishing modes:\n\\\\\n{\\bf Corollary.}\n\\\\\n{\\it \nLet $u=v$ on $\\gamma \\times (0,T)$. Then\n$$\n\\{ \\la_k;\\, k\\in \\N, \\, P_ka \\ne 0 \\quad \\mbox{in $\\OOO$}\\}\n= \\{ \\mu_k;\\, k\\in \\N, \\, Q_ka \\ne 0 \\quad \\mbox{in $\\OOO$}\\}\n$$\nand if $P_ka \\ne 0$ in $\\OOO$ for $k\\in \\N$, then \n$$\nP_ka = Q_ka \\quad \\mbox{on $\\gamma$},\n$$\nafter suitable re-numbering of $k$.\n}\n\nThe corollary means that $u=v$ on $\\gamma \\times (0,T)$ implies \nthat there exists $N_1 \\in \\N \\cup \\{\\infty\\}$ such that we can find\nsequences $\\{i_k\\}_{1\\le k\\le N_1}, \\, \\{j_k\\}_{1\\le k\\le N_1}\n\\subset \\N$ satisfying \n$$\n\\left\\{ \\begin{array}{rl}\n& \\la_{i_k} = \\mu_{j_k}, \\quad P_{i_k}a \\ne 0, \\,\\, \nQ_{j_k}a \\ne 0 \\quad \\mbox{in $\\OOO$},\\quad\n P_{i_k}a = Q_{j_k}a = 0 \\quad \\mbox{on $\\gamma$}\n\\quad \\mbox{for $1\\le k \\le N_1$}, \\\\\n& P_ia = 0 \\quad\\mbox{in $\\OOO$ if $i\\not\\in \\{ i_k\\}_{1\\le k\\le N_1}$},\n\\quad\n Q_ja = 0 \\quad\\mbox{in $\\OOO$ if $j\\not\\in \\{ j_k\\}_{1\\le k\\le N_1}$}.\n\\end{array}\\right.\n$$\nWe remark that even in the case $N_1=\\infty$, we may have\n$\\{ i_k\\}_{1\\le k \\le N_1} \\subsetneqq \\N$.\n\\\\\n{\\bf Remark.}\nIn (1.7), consider a function $\\theta(\\eta) = \\eta^p$.\nTheorem 1 asserts the uniqueness \nif the initial value $a$ is smooth in the sense (1.7).\nWe emphasize that in (1.7), the critical exponent of $\\la_k$ should be \ngreater than $\\frac{2}{3}$. If we assume\nthe stronger condition $p=1$, that is,\n$$\n\\sumk e^{\\sigma \\la_k} \\Vert P_ka\\Vert^2_{L^2(\\Omega)} < \\infty \\quad \\mbox{and}\n\\quad \\sumk e^{\\sigma \\mu_k} \\Vert Q_ka\\Vert^2_{L^2(\\Omega)} < \\infty\n \\eqno{(1.8)}\n$$\nwith some constant $\\sigma>0$, then the uniqueness is trivial \nbecause we can extend the solutions $u(\\cdot,t)$ and $v(\\cdot,t)$\nto the time interval $(-\\delta, 0)$ with small $\\delta > 0$.\nIndeed, since $\\sumk \\vert e^{\\hhalf \\sigma \\la_k}\\vert^2 \\Vert P_ka\\Vert^2_{L^2(\\Omega)} \n< \\infty$, we can verify that $u(\\cdot,t) = \\sumk e^{-\\la_kt}P_ka$ in \n$L^2(\\OOO)$ for $t > -\\hhalf\\sigma$. Therefore we can extend \n$u(\\cdot,t)$ to $\\left(-\\frac{\\sigma}{2},\\, 0\\right)$ in $L^2(\\OOO)$ and also\nto $(-\\delta, 0)$ with sufficiently small $\\delta>0$. The extension of\n$v(\\cdot,t)$ is similarly done.\nTherefore, under (1.8), our inverse problem is reduced to the case where\nthe spatial data of $u,v$ are given at an intermediate time of the whole\ntime interval under consideration, \nwhich has been already solved in Bukhgeim and Klibanov\n\\cite{BK}, Imanuvilov and Yamamoto \\cite{IY98}, Isakov \\cite{Is}.\n\nCondition corresponding to the case $p=\\hhalf$ in (1.7) \nappears in the controllability of a parabolic equation. \nWe know that a function $a(\\cdot)$ in $\\OOO$ satisfying the condition (1.7)\nwith $\\theta(\\eta) = \\eta^{\\hhalf}$ and $c_1\\equiv 0$\nbelongs to the reachable set \n$$\n\\{ u(\\cdot,0);\\, b\\in L^2(\\OOO),\\, h \\in L^2(\\ppp\\OOO \\times (-\\tau,0)\\},\n$$\nwhere $u$ is the solution to \n$$\n\\left\\{ \\begin{array}{rl}\n& \\ppp_tu = \\Delta u \\quad \\mbox{in $\\OOO \\times (-\\tau,0)$}, \\\\\n& \\ppp_{\\nu}u = h \\quad \\mbox{on $\\ppp\\OOO \\times (-\\tau,0)$},\\\\\n& u(\\cdot,-\\tau) = b \\quad \\mbox{in $\\OOO$}\n\\end{array}\\right.\n$$\n(Theorem 2.3 in Russell \\cite{R}).\nSee also (1.9) stated below.\n\\\\\n\nThe article is composed of four sections. In Section 2, we show \nCarleman estimates for elliptic operator. In Section 3, we prove the\nuniqueness for our inverse problem first under a condition:\nthere exists a constant $\\sigma_1>0$ such that \n$$\n\\sumk e^{\\sigma_1 \\la_k^{\\hhalf}}\\Vert P_ka\\Vert^2_{L^2(\\Omega)} \n+ \\sumk e^{\\sigma_1 \\mu_k^{\\hhalf}} \\Vert Q_ka\\Vert^2_{L^2(\\Omega)} < \\infty,\n \\eqno{(1.9)}\n$$\nand next by proving that (1.7) yields (1.9), we complete\nthe proof of Theorem 1.\nIn Section 4, we prove a Carleman estimate used for deriving (1.9) from \n(1.7).\n%\n%\n\\section{Key Carleman estimate}\n\nThe proof of Theorem 1 relies essentially on the reduction of our\ninverse parabolic problem to an inverse elliptic problem.\nAfter the reduction, we prove the uniqueness by the method developed in \n\\cite{BK} or, \\cite{HIY}, \\cite{IY98}, \nand so we need a relevant Carleman estimate \nfor an elliptic equation.\nFor the statement of Carleman estimate, we introduce a weight function.\n\nWe arbitrarily fix $y \\in \\omega$. For $y$, we construct a non-empty\ndomain $\\omega_y \\subset \\OOO$ satisfying\n$$\n\\left\\{ \\begin{array}{rl}\n&\\mbox{(i)} \\,\\,y \\in \\omega_y, \\quad \\omega_y \\subset \\omega. \\\\\n&\\mbox{(ii)} \\,\\,\\mbox{$\\ppp\\omega_y$ is of $C^{\\infty}$-class.}\\\\\n&\\mbox{(iii)} \\,\\, \\mbox{$\\ppp\\omega_y \\cap \\Gamma$ has interior points \nin the topology of $\\ppp\\OOO$.} \\\\\n&\\mbox{(iv)} \\,\\, \\vert a(x)\\vert > 0 \\quad \\mbox{for all \n $x\\in \\ooo{\\omega_y}$}.\n\\end{array}\\right.\n \\eqno{(2.1)}\n$$\nIndeed, since $y\\in \\omega$, by the definition of $\\omega$, we can find\n$y_*\\in \\Gamma$ and a smooth curve $\\ell \\in C^{\\infty}[0,1]$ such that \n$\\ell(1) = y$ and $\\ell(0) = y_*$, $\\ell(\\xi) \\in \\omega$ for \n$0<\\xi \\le 1$. Then as $\\omega_y$, we can choose\na sufficiently thin neighborhood of the curve $\\{\\ell(\\xi);\\, 0<\\xi\\le 1\\}$\nwhich is included in $\\omega$. \n\nFor the proof of Theorem 1, we will show that if $y\\in\\omega$ then $y\\notin \\mbox{supp}\\, f.$ This of course implies that $f=0$ on $\\omega$.\nFirst we establish a Carleman estimate in $\\omega_y \\times (-\\tau,\\tau)$ with \na constant $\\tau > 0$.\n \nWe know that there exists a function\n$d\\in C^2(\\ooo{\\omega_y})$ such that \n$$\n\\vert \\nabla d(x)\\vert > 0 \\quad \\mbox{for $x\\in \\ooo{\\omega_y}$}, \\quad\nd(x) > 0 \\quad \\mbox{for $x\\in \\omega_y$}, \\quad\nd(x) = 0 \\quad \\mbox{for $x\\in \\ppp\\omega_y\\setminus \\Gamma$}.\n \\eqno{(2.2)}\n$$\nThe existence of such $d$ is proved for example in Imanuvilov \\cite{Im}.\nSee also Fursikov and Imanuvilov \\cite{FI}.\n\nFor a constant $\\tau>0$, we set \n$$\n{\\mathcal Q}_\\tau:= \\omega_y \\times (-\\tau, \\, \\tau),\n$$\n$\\ppp_0 := \\frac{\\ppp}{\\ppp t}$, and \n$$\n\\alpha(x,t) := e^{\\la(d(x) - \\beta t^2)}, \n\\quad (x,t)\\in {\\mathcal Q}_{\\tau} \\eqno{(2.3)}\n$$\nwith an arbitrarily chosen constant $\\beta > 0$ and sufficiently large \nfixed $\\la > 0$.\nThen\n\\\\\n{\\bf Lemma 2.1 (elliptic Carleman estimate).}\n\\\\\n{\\it \nThere exists a constant $s_0 > 0$ such that we can find a constant \n$C>0$ such that \n\\begin{align*}\n& \\int_{{\\mathcal Q}_\\tau} \\left\\{ \\frac{1}{s}\\sum_{i,j=0}^n \\vert \\ppp_i\\ppp_jw\\vert^2\n+ s\\vert \\ppp_tw\\vert^2 + s\\vert \\nabla w\\vert^2\n+ s^3\\vert w\\vert^2\\right) e^{2s\\alpha} dxdt\\\\\n\\le& C\\int_{{\\mathcal Q}_\\tau} \\vert \\ppp_t^2w - A_1w\\vert^2 e^{2s\\alpha} dxdt\n\\end{align*}\nfor all $s \\ge s_0$ and $w\\in H^2_0({\\mathcal Q}_\\tau)$.\n}\n\\\\\n\nHere we recall that $-A_1w = \\sumij \\ppp_i(a_{ij}(x)\\ppp_jw)\n+ c_1(x)w$. \nThe constants $s_0>0$ and $C>0$ can be chosen uniformly provided that \n$\\Vert c_1\\Vert_{L^{\\infty}(\\omega)} \\le M$: arbitrarily fixed constant\n$M>0$.\n\nWe note that Lemma 2.1 is a Carleman estimate for the elliptic operator \n$\\ppp_t^2 - A_1w$. Since $(\\nabla \\alpha, \\, \\ppp_t\\alpha)\n= (\\nabla d, \\, -2\\beta t) \\ne (0,0)$ on $\\ooo{{\\mathcal Q}_{\\tau}}$ by (2.2), \nthe proof of\nthe lemma relies directly on integration by parts and standard, similar for\nexample to the proof of Lemma 7.1 (p.186) in \nBellassoued and Yamamoto \\cite{BY}.\nSee also H\\\"ormander \\cite{H}, Isakov \\cite{Is}, where the estimation\nof the second-order derivatives is not included but can be be \nderived by the a priori estimate for the elliptic boundary value problem.\n\nFor the proof of Theorem 1, we further need another Carleman estimate \nin $\\OOO$ for an elliptic \nequation. We can find $\\rho\\in C^2(\\ooo{\\OOO})$ such that \n$$\n\\rho(x) > 0 \\quad \\mbox{for $x \\in\\OOO$}, \\quad\n\\vert \\nabla \\rho(x) \\vert > 0 \\quad \\mbox{for $x \\in\\ooo{\\OOO}$}, \\quad\n\\ppp_{\\nu_A}\\rho(x) \\le 0 \\quad \\mbox{for $x\\in \\ppp\\OOO\\setminus \\gamma$}.\n \\eqno{(2.4)}\n$$\nThe construction of $\\rho$ can be found in Lemma 2.3 in \\cite{IY98} for \nexample.\nMoreover, fixing a constant $\\la>0$ large, we set \n$$\n\\psi(x):= e^{\\la(\\rho(x) - 2\\Vert \\rho\\Vert_{C(\\ooo{\\OOO})})}, \\quad \nx\\in \\OOO.\n$$\nThen\n\\\\\n{\\bf Lemma 2.2.}\n\\\\\n{\\it\nThere exist constants $s_0>0$ and $C>0$ such that \n$$\n \\int_{\\OOO} (s^3\\vert g\\vert^2 + s\\vert \\nabla g\\vert^2)e^{2s\\psi(x)} dx\n\\le C\\int_{\\OOO} \\vert A_2g\\vert^2 e^{2s\\psi(x)} dx \n+ cs^3\\int_{\\gamma} (\\vert g\\vert^2 + \\vert \\nabla g\\vert^2) e^{2s\\psi} dS\n$$\nfor all $s>s_0$ and $g \\in H^2(\\OOO)$ satisfying $\\ppp_{\\nu_A}g=0$ \non $\\ppp\\OOO$.\n}\n\nWe postpone the proof of Lemma 2.2 to Section 4.\n%\n%\n\\section{Proof of Theorem 1.1}\n\nWe divide the proof into four steps.\nIn Steps 1-3, we assume the condition (1.9) to prove the \nconclusion on uniqueness in Theorem 1.1.\n\\\\\n{\\bf First Step.}\n\\\\\nWe write $u(t):= u(\\cdot,t)$ and $v(t):= v(\\cdot,t)$ for $t>0$.\nWe recall that\n$$\nu(t) = \\sumk e^{-\\la_kt}P_ka, \\quad\nv(t) = \\sumk e^{-\\mu_kt}Q_ka \\quad \\mbox{in $H^2(\\OOO)$ for $t>0$.}\n$$\n\nWe can choose subsets $\\N_1, \\M_1 \\subset \\N$ such that \n$$\n\\N_1:= \\{k \\in \\N;\\, P_ka \\not\\equiv 0 \\quad \\mbox{in $\\OOO$}\\}, \\quad\n\\M_1:= \\{k \\in \\N;\\, Q_ka \\not\\equiv 0 \\quad \\mbox{in $\\OOO$}\\}.\n \\eqno{(3.1)}\n$$\nWe note that $\\N_1 = \\N$ or $\\M_1 = \\N$ may happen.\n\nWe can renumber the sets $\\N_1$ and $\\M_1$ as \n$$\n\\N_1 = \\{1, ...., N_1\\}, \\quad \\M_1=\\{ 1, ...., M_1\\},\n$$\nwhere $N_1 = \\infty$ or $M_1 = \\infty$ may occur. \nBy $^{\\ssss}\\N_1$ we mean the cardinal number of the set $\\N_1$.\nWote that \n$$\n\\la_1< \\la_2 < \\cdots < \\la_{N_1} \\quad \\mbox{if $^{\\ssss}\\N_1 < \\infty$} \\quad\n\\la_1< \\la_2 < \\cdots \\quad \\mbox{if $^{\\ssss}\\N_1 = \\infty$}\n$$\nand\n$$\n\\mu_1< \\mu_2 < \\cdots < \\mu_{M_1} \\quad \\mbox{if $^{\\ssss}\\M_1 < \\infty$} \\quad\n\\mu_1< \\mu_2 < \\cdots \\quad \\mbox{if $^{\\ssss}\\M_1 = \\infty$}.\n$$\n\nAssuming that $u=v$ on $\\gamma \\times (0,T)$, by the time analyticity\nof $u(t)$ and $v(t)$ for $t>0$ (e.g., Pazy \\cite{Pa}), we obtain\n$$\n\\sum_{k=1}^{N_1} e^{-\\la_kt}P_ka = \\sum_{k=1}^{M_1} e^{-\\mu_kt}Q_ka \n\\quad \\mbox{on $\\gamma \\times (0,\\infty)$}. \n \\eqno{(3.2)}\n$$\nWe will prove that $\\la_1 = \\mu_1$.\nAssume that $\\la_1 < \\mu_1$. Then \n$$\nP_1a + \\sum_{k=2}^{N_1} e^{-(\\la_k-\\la_1)t}P_ka \n= \\sum_{k=1}^{M_1} e^{-(\\mu_k-\\la_1)t}Q_ka \n\\quad \\mbox{on $\\gamma \\times (0,\\infty)$}.\n$$\nSince $\\la_k - \\la_1 > 0$ for $2\\le k \\le N_1$ and \n$\\mu_k - \\la_1 > 0$ for $1\\le k \\le M_1$, letting $t\\to \\infty$,\nwe see that $P_1a=0$ on $\\Gamma$.\nTherefore\n$$\n\\left\\{ \\begin{array}{rl}\n& (A_1-\\la_1)P_1a = 0 \\quad \\mbox{in $\\OOO$}, \\\\\n& P_1a\\vert_{\\Gamma} = 0, \\quad \\nuA P_1a\\vert_{\\ppp\\OOO} = 0.\n\\end{array}\\right.\n$$\nThe unique continuation for the elliptic equation $A_1P_1a = \\la_1P_1a$, (see e.g. \\cite{H})\nyields that\n$$\nP_1a = 0 \\quad \\mbox{in $\\OOO$}.\n$$\nThis is a contradiction by $1 \\in \\N_1$.\nThus the inequality $\\la_1 < \\mu_1$ is impossible.\nSimilarly we can see that the inequality $\\la_1 > \\mu_1$ is impossible. \nTherefore $\\la_1 = \\mu_1$ follows.\n\nBy (3.2) and $\\la_1 = \\mu_1$, we have\n$$\nP_1a - Q_1a \n= -\\sum_{k=2}^{N_1} e^{-(\\la_k-\\la_1)t}P_ka \n+ \\sum_{k=2}^{M_1} e^{-(\\mu_k-\\la_1)t}Q_ka \n\\quad \\mbox{on $\\gamma \\times (0,\\infty)$}.\n$$\nHence, by $\\la_k-\\la_1 > 0$ and $\\mu_k - \\la_1 = \\mu_k - \\mu_1 > 0$ for \nall $k \\ge 2$, letting $s \\to \\infty$ we obtain $P_1a = Q_1a$ on $\\gamma$.\n\nIn view of (3.2), we obtain\n$$\n\\sum_{k=2}^{N_1} e^{-\\la_kt}P_ka \n= \\sum_{k=2}^{M_1} e^{-\\mu_kt}Q_ka \n\\quad \\mbox{on $\\gamma \\times (0,\\infty)$}. \n$$\nRepeating the same argument as much as possible, we reach \n$$\nN_1 = M_1, \\quad \\la_k = \\mu_k, \\quad\nP_ka = Q_ka \\quad \\mbox{on $\\gamma \\times (0,\\infty)$ for\n$1\\le k \\le N_1$}. \\eqno{(3.3)}\n$$\n\\\\\n{\\bf Second Step.}\n\nWe consider two initial boundary value problems for elliptic equations:\n$$\n\\left\\{ \\begin{array}{rl}\n& \\ppp_t^2w_1 - A_1w_1 = 0 \\quad \\mbox{in $\\OOO\\times (0,\\tau)$}, \\\\\n& \\nuA w_1 = 0 \\quad \\mbox{on $\\ppp\\OOO \\times (0,\\tau)$}, \\\\\n& w_1(x,0) = a(x), \\quad \\ppp_tw_1(x,0) = 0, \\quad x\\in \\OOO\n\\end{array}\\right.\n \\eqno{(3.4)}\n$$\nand\n$$\n\\left\\{ \\begin{array}{rl}\n& \\ppp_t^2w_2 - A_2w_2 = 0 \\quad \\mbox{in $\\OOO\\times (0,\\tau)$}, \\\\\n& \\nuA w_2 = 0 \\quad \\mbox{on $\\ppp\\OOO \\times (0,\\tau)$}, \\\\\n& w_2(x,0) = a(x), \\quad \\ppp_tw_2(x,0) = 0, \\quad x\\in \\OOO.\n\\end{array}\\right.\n \\eqno{(3.5)}\n$$\nSince we have the spectral representations by (1.9), we can obtain \n$$\ne^{-tA_1^{\\hhalf}}a = \\sumk e^{-\\la_k^{\\hhalf}t}P_ka, \\quad\ne^{tA_1^{\\hhalf}}a = \\sumk e^{\\la_k^{\\hhalf}t}P_ka \\quad \n\\mbox{in $L^2(\\OOO)$ for $t>0$}\n$$\nand similar representations hold for $e^{\\pm tA_2^{\\hhalf}}a$.\n \nThen by the assumption (1.9) on $a$, we see that \n$$\nw_1(t) = \\frac{1}{2}(e^{-tA_1^{\\hhalf}}a + e^{tA_1^{\\hhalf}}a), \\quad\nw_2(t) = \\frac{1}{2}(e^{-tA_2^{\\hhalf}}a + e^{tA_2^{\\hhalf}}a) \n$$\nin $H^2(\\Omega\\times (0,\\tau))$ for $t \\in (0,\\tau)$, satisfy (3.4) and (3.5)\nrespectively if $\\tau>0$ is chosen sufficiently small.\n\nIn view of (3.3) and the definition (3.1) of $\\N_1$, \nthe spectral representations imply\n\\begin{align*}\n& w_1(x,t) = \\hhalf \\sum_{k=1}^{N_1} (e^{-\\la_k^{\\hhalf}t}P_ka\n+ e^{\\la_k^{\\hhalf}t}P_ka)\n+ \\hhalf \\sum_{k\\in \\N \\setminus \\{1, ..., N_1\\}}(e^{-\\la_k^{\\hhalf}t}P_ka\n+ e^{\\la_k^{\\hhalf}t}P_ka)\\\\\n=& \\hhalf \\sum_{k=1}^{N_1} (e^{-\\la_k^{\\hhalf}t}P_ka\n+ e^{\\la_k^{\\hhalf}t}P_ka) \\quad \\mbox{in $\\OOO\\times (0,\\tau)$,}\n\\end{align*}\nand \n$$\nw_2(x,t) = \\hhalf \\sum_{k=1}^{N_1} (e^{-\\la_k^{\\hhalf}t}Q_ka\n+ e^{\\la_k^{\\hhalf}t}Q_ka) \\quad \\mbox{in $\\OOO\\times (0,\\tau)$}.\n$$\nTherefore (3.3) yields\n$$\nw_1 = w_2 \\quad \\mbox{on $\\gamma \\times (0,\\tau)$.} \\eqno{(3.6)}\n$$\n\nNow we reduce our inverse problem for the parabolic equations to the one\nfor elliptic equations for (3.4) and (3.5).\nThis is the essence of the proof.\n\\\\\n{\\bf Third Step.}\n\\\\\nBy (1.9), we can readily verify further regularity $\\ppp_tw_1, \\ppp_tw_2\n\\in H^2(0,\\tau;H^2(\\OOO))$.\nSetting $y:= w_1 - w_2$ and $R:=w_2$ in $\\OOO\\times (0,\\tau)$ \nand $f:= c_2-c_1$ in $\\OOO$, by (3.4) -(3.6) we have\n$$\n\\left\\{ \\begin{array}{rl}\n& \\ppp_t^2y - A_1y = f(x)R(x,t) \\quad \\mbox{in $\\OOO\\times (0,\\tau)$}, \\\\\n& \\nuA y = 0 \\quad \\mbox{on $\\ppp\\OOO \\times (0,\\tau)$}, \\\\\n& y=0 \\quad \\mbox{on $\\gamma \\times (0,\\tau)$}, \\\\\n& y(x,0) = \\ppp_ty(x,0) = 0, \\quad x\\in \\OOO.\n\\end{array}\\right.\n \\eqno{(3.7)}\n$$\n\nNow we will prove that for any pair $(y,f)$ solving problem (3.7) we have $y\\notin \\mbox{supp}, f.$ Since $ y$ was chosen as an arbitrary point from $\\omega$ this implies\n$$\nf=0\\quad \\mbox{in}\\quad \\omega.\n$$ The argument relies on \\cite{IY98}.\n\nWe set\n$$\n\\www{y}(x,t) = \n\\left\\{ \\begin{array}{rl}\n& y(x,t), \\quad 00$ we choose \n$\\beta > 0$ sufficiently large,\nso that \n$$\n\\Vert d\\Vert_{C(\\ooo{\\omega_y})} - \\beta \\tau^2 < 0, \\eqno{(3.10)}\n$$\nWe choose constants $\\delta_1, \\delta_2 > 0$ such that\n$$\n\\Vert d\\Vert_{C(\\ooo{\\omega_y})} - \\beta \\tau^2 < 0 < \\delta_1 < \\delta_2 \n \\quad \\mbox{and}\\quad d(y)>\\delta_2 \\eqno{(3.11)}\n$$\nand we define $\\chi \\in C^{\\infty}(\\ooo{{\\mathcal Q}_{\\tau}})$ satisfying\n$$\n\\chi(x,t) = \n\\left\\{ \\begin{array}{rl}\n& 1, \\quad d(x) - \\beta t^2 > \\delta_2, \\\\\n& 0, \\quad d(x) - \\beta t^2 < \\delta_1. \n\\end{array}\\right.\n \\eqno{(3.12)}\n$$\nIn particularl, $d(y) > \\delta_2$ implies \n$$\n\\chi(y,0)=1.\n$$\n%The parameter $\\ep>0$ will be used later for approximating $\\omega$ \n%by $\\{x\\in \\omega;\\, d(x) > \\ep\\}$.\n \nSetting $z:= \\chi \\www{z}$ on $\\ooo{{\\mathcal Q}_{\\tau}}$, we see that \n$$\nz = \\nuA z = 0 \\quad \\mbox{on $\\ppp\\omega_y \\times (-\\tau, \\tau)$}\n \\eqno{(3.13)}\n$$\nand\n$$\nz = \\ppp_t z = 0 \\quad \\mbox{on $\\omega_y \\times \\{ \\pm \\tau\\}$}.\n \\eqno{(3.14)}\n$$\nIndeed, $(x,t) \\in (\\ppp\\omega_y \\setminus \\Gamma) \\times (-\\tau, \\tau)$ \nimplies\n$d(x) - \\beta t^2 = -\\beta t^2 \\le 0 < \\delta_1$ by (2.2), and so \nthe definition (3.12) of $\\chi$ yields that \n$\\chi(x,t) = 0$ in a neighborhood of such $(x,t)$.\nFor $(x,t) \\in (\\ppp\\omega_y \\cap \\Gamma) \\times (-\\tau,\\tau)$, \nby (3.8) we see that \n$z(x,t) = \\nuA z(x,t) = 0$, which verifies (3.13).\nMoreover, on $\\omega_y\\times \\{ \\pm \\tau\\}$, by (3.11) we have \n$$\nd(x) - \\beta t^2 \\le \\Vert d\\Vert_{C(\\ooo{\\omega_y})} - \\beta\\tau^2\n< 0 < \\delta_1,\n$$\nso that $\\chi(x,t) = 0$ in a neighborhood of such $(x,t)$. \nThus (3.14) has been verified.\n$\\blacksquare$\n\nConsequently we prove that $z\\in H^2_0(Q_{\\tau})$.\nMoreover, we can readily obtain\n$$\n\\left\\{ \\begin{array}{rl}\n& \\ppp_t^2z - A_1z = \\chi(\\ppp_t\\www {R})f + R_0(x,t), \\quad \n(x,t) \\in {\\mathcal Q}_{\\tau}, \\\\\n& z = \\vert \\nabla z\\vert = 0 \\quad \\mbox{on $\\ppp {\\mathcal Q}_{\\tau}$},\n\\end{array}\\right.\n \\eqno{(3.15)}\n$$\nwhere $R_0$ is a linear combination of $\\nabla \\www{z}$, $\\ppp_t\\www{z}$,\nwhose coefficients are linear combinations of $\\nabla\\chi$ and \n$\\ppp_t\\chi$. Therefore (3.12) implies \n$$\nR_0(x,t) \\ne 0 \\quad \\mbox{only if $\\delta_1 \\le d(x) - \\beta t^2\n\\le \\delta_2$}. \\eqno{(3.16)}\n$$\n\nTherefore we can apply Lemma 2.1 to (3.15) with (3.16):\n$$\n\\int_{{\\mathcal Q}_{\\tau}} \\left( \\frac{1}{s}\\vert \\ppp_t^2z\\vert^2\n+ s\\vert \\ppp_tz\\vert^2 \\right) e^{2s\\alpha} dxdt \n \\eqno{(3.17)}\n$$\n\\begin{align*}\n\\le & C\\int_{{\\mathcal Q}_{\\tau}} \\vert \\chi(\\ppp_t\\www{R})f \\vert^2e^{2s\\alpha} dxdt\n+ C\\int_{{\\mathcal Q}_{\\tau}} \\vert R_0(x,t)\\vert^2 e^{2s\\alpha} dxdt \\\\\n\\le & C\\int_{{\\mathcal Q}_{\\tau}}\\chi^2 \\vert f\\vert^2 e^{2s\\alpha} dxdt \n+ Ce^{2se^{\\la\\delta_2}}\n\\end{align*}\nfor all large $s>0$.\n\nOn the other hand, since $\\ppp_tz(\\cdot,-\\tau) = 0$ in $\\omega_y$ by \n(3.14), we have\n\\begin{align*}\n& \\int_{\\omega_y} \\vert \\ppp_tz(x,0)\\vert^2 e^{2s\\alpha(x,0)} dx \n= \\int^0_{-\\tau} \\ppp_t\\left( \\int_{\\omega_y} \\vert \\ppp_tz(x,t)\\vert^2\ne^{2s\\alpha(x,t)} dx \\right) dt\\\\\n=& \\int^0_{-\\tau} \\int_{\\omega_y} \\{ 2(\\ppp_tz)(x,t)\\ppp_t^2z(z,t)\n+ \\vert \\ppp_tz \\vert^2 2s(\\ppp_t\\alpha))\\} e^{2s\\alpha(x,t)} dxdt.\n\\end{align*}\nSince \n$$\n\\vert (\\ppp_tz)(\\ppp_t^2z)\\vert \\le \\frac{1}{2}\n\\left( s\\vert \\ppp_tz\\vert^2 + \\frac{1}{s}\\vert \\ppp_t^2z\\vert^2\n\\right) \\quad \\mbox{in $\\mathcal{Q}_\\tau$},\n$$\nin terms of (3.17) we obtain\n$$\n \\int_{\\omega_y} \\vert \\ppp_tz(x,0)\\vert^2 e^{2s\\alpha(x,0)} dx \n\\le C\\int_{{\\mathcal Q}_{\\tau}} \\left( \\frac{1}{s} \\vert \\ppp_t^2z(x,t)\\vert^2 \n+ s\\vert \\ppp_tz\\vert^2 \\right) e^{2s\\alpha} dxdt \n \\eqno{(3.18)}\n$$\n$$\n\\le C\\int_{{\\mathcal Q}_{\\tau}}\\vert \\chi\\vert^2\\vert f\\vert^2 e^{2s\\alpha} \ndxdt + Ce^{2se^{\\la\\delta_2}}\n$$\nfor all large $s>0$.\n\nMoreover, we have\n\\begin{align*}\n& \\ppp_tz(x,0) = \\ppp_t(\\chi\\www{z})(x,0)\n= (\\ppp_t\\chi)(x,0)\\www{z}(x,0) + \\chi(x,0)\\ppp_t\\www{z}(x,0)\\\\\n=& \\chi(x,0)f(x)a(x)\n\\end{align*}\nby (3.9) and $\\www{z}(x,0) = 0$ for $x \\in \\omega_y$ in (3.8).\nTherefore, in terms of (2.1)-(iv), we obtain\n$$\n\\vert \\ppp_tz(x,0)\\vert \\ge C\\vert \\chi(x,0)f(x)\\vert, \\quad \nx\\in \\ooo{\\omega_y}.\n$$\nConsequently (3.18) implies\n$$\n\\int_{\\omega_y} \\vert\\chi(x,0)\\vert^2 \\vert f(x) \\vert^2 e^{2s\\alpha(x,0)} dx \n\\le C\\int_{\\mathcal{Q}_{\\tau}} \\vert \\chi f\\vert^2 e^{2s\\alpha} dxdt \n+ Ce^{2se^{\\la\\delta_2}} \\eqno{(3.19)} \n$$\nfor all large $s>0$.\n\nMoreover, we see\n\\begin{align*}\n&\\int_{{\\mathcal Q}_{\\tau}} \\vert \\chi f\\vert^2 e^{2s\\alpha} dxdt \n= \\int^{\\tau}_{-\\tau} \\int_{\\omega_y} \\vert\\chi(x,0)f(x)\\vert^2 e^{2s\\alpha} \ndxdt\\\\\n= & \\int_{\\omega_y} \\vert \\chi(x,0) f(x)\\vert^2 e^{2s\\alpha(x,0)} \n\\left( \\int^{\\tau}_{-\\tau} e^{2s(\\alpha(x,t) - \\alpha(x,0))} dt\\right)dx.\n\\end{align*}\nSince \n$$\n\\int^{\\tau}_{-\\tau} e^{2s(\\alpha(x,t) - \\alpha(x,0))} dt\n= \\int^{\\tau}_{-\\tau} e^{2se^{\\la d(x)}(e^{-\\la\\beta t^2} - 1)} dt\n\\le \\int^{\\tau}_{-\\tau} e^{Cs(e^{-\\la\\beta t^2} - 1)} dt\n= o(1)\n$$\nas $s \\to \\infty$ by the Lebesgue convergence theorem.\nHence, (3.19) yields\n$$\n \\int_{\\omega_y} \\vert \\chi(x,0)f(x)\\vert^2 e^{2s\\alpha(x,0)} dx\n\\le o(1)\\int_{\\omega_y} \\vert \\chi(x,0)f(x)\\vert^2 e^{2s\\alpha(x,0)} dx\n+ Ce^{2se^{\\la\\delta_2}},\n$$\nand we can absorb the first term on the right-hand side into the \nleft-hand side to reach\n$$\n\\int_{\\omega_y} \\vert \\chi(x,0)f(x)\\vert^2 e^{2s\\alpha(x,0)} dx\n\\le Ce^{2se^{\\la\\delta_2}} \\eqno{(3.20)}\n$$\nfor all large $s>0$. \n\nHenceforth we set $B(y,\\ep):= \\{ x;\\, \\vert x-y\\vert < \\ep\\}$.\nThen, we can choose $\\delta_3 > \\delta_2$ and\na sufficiently small $\\ep > 0$ such that\n$B(y,\\ep) \\subset \\omega_y$ and $d(x) \\ge \\delta_3$ for all \n$x\\in B(y,\\ep)$. This is possible, because $d(y) > \\delta_2$ in \n(3.11) and $\\omega_y$ is an open set including $y$.\n\nWe shrink the integration region of the left-hand side of (3.20)\nto $B(y,\\ep)$ and obtain\n$$\n\\int_{B(y,\\ep)} \\vert \\chi(x,0)f(x)\\vert^2 e^{2s\\alpha(x,0)} dx\n\\le Ce^{2se^{\\la\\delta_2}}\n$$\nfor all large $s>0$. \nSince $d(x) \\ge \\delta_3 > \\delta_2$ for $x\\in B(y,\\ep)$, the condition (3.12)\nyields $\\chi(x,0) = 1$ and $\\alpha(x,0) = e^{\\la d(x)}\n\\ge e^{\\la\\delta_3}$ for all $x\\in B(y,\\ep)$. Therefore,\n$$\n\\left( \\int_{B(y,\\ep)} \\vert f(x)\\vert^2 dx\\right) e^{2se^{\\la\\delta_3}}\n\\le Ce^{2se^{\\la\\delta_2}},\n$$\nthat is, $\\Vert f\\Vert^2_{L^2(B(y,\\ep))} \\le e^{-2s(e^{\\la\\delta_3}\n- e^{\\la\\delta_2})}$ for all large $s>0$. \n\nIn terms of $\\delta_3 > \\delta_2$, letting $s\\to \\infty$, \nwe see that the right-hand side\ntends to $0$, and so $f=0$ in $B(y,\\ep)$. Since $y$ is arbitrarily chosen, \nwe reach $f=c_2-c_1 = 0$ in $\\omega$.\nThus the conclusion of Theorem 1.1 is proved under condition (1.9).\n$\\blacksquare$\n%\n%\n\\\\\n{\\bf Fourth Step.}\n\\\\\nWe will complete the proof of Theorem 1.1 by demonstrating\nthat (1.7) implies (1.9).\nWithout loss of generality, we can assume\n$$\n\\sumk e^{\\theta(\\la_k)}\\Vert P_ka\\Vert^2_{L^2(\\Omega)} < \\infty.\n \\eqno{(3.21)}\n$$\nIt suffices to prove that there exists a constant $\\sigma_1>0$ such that \n$$\n\\sum_{k=1}^{\\infty} e^{\\sigma_1\\la_k^{\\hhalf}}\\Vert Q_ka\\Vert^2_{L^2(\\Omega)} < \\infty,\n \\eqno{(3.22)}\n$$\nwith the assumption that the set of $k\\in \\N$ such that \n$Q_ka\\ne 0$ in $\\OOO$ is infinite.\nFor simplicity, we can consider the case where \n$P_ka \\ne 0$ in $\\OOO$ for all $k\\in \\N$. We can argue similarly \nin the rest cases.\nThen, by Corollary which was already proved in First Step, we choose \na subset $\\M_1 \\subset \\N$ such that \n$$\n\\{ \\la_i\\}_{i\\in\\N} = \\{ \\mu_j\\}_{j\\in \\M_1}, \\quad\nQ_ja = 0 \\quad \\mbox{in $\\OOO$ for $j\\in \\N \\setminus \\M_1$}.\n$$\nNow it suffices to prove (3.22) in the case where $Q_ka\\ne 0$ in \n$\\OOO$ for all $k\\in \\N$.\n\nAfter re-numbering, we can obtain\n$$\n\\la_k = \\mu_k, \\quad P_ka = Q_ka \\quad \\mbox{on $\\gamma$ for all\n$k\\in \\N$}. \\eqno{(3.23)}\n$$\nThe trace theorem and the a priori estimate for an elliptic operator\nyields\n$$\n\\Vert P_ka\\Vert_{H^1(\\Gamma)} \\le C\\Vert P_ka\\Vert_{H^2(\\OOO)}\n\\le C(\\Vert A_1P_ka\\Vert_{L^2(\\Omega)} + \\Vert P_ka\\Vert_{L^2(\\Omega)})\n= C(\\la_k+1)\\Vert P_ka\\Vert_{L^2(\\Omega)}. \\eqno{(3.24)}\n$$\nHere and henceforth $C>0$ denotes generic constants which are\nindependent of $s>0$ and $k\\in \\N$.\n\nSince $A_2Q_k = \\la_kQ_k$, by (3.23) we apply Lemma 2.2 to have\n$$\ns^3\\int_{\\OOO} \\vert Q_ka\\vert^2 e^{2s\\psi} dx \n\\le C\\int_{\\OOO} \\la_k^2\\vert Q_ka\\vert^2 e^{2s\\psi} dx\n+ Cs^3\\int_{\\gamma} (\\vert Q_ka\\vert^2 + \\vert \\nabla (Q_ka)\\vert^2)\ne^{2s\\psi} dx \\eqno{(3.25)}\n$$\n$$\n\\le C\\int_{\\OOO} \\la_k^2\\vert Q_ka\\vert^2 e^{2s\\psi} dx\n+ Cs^3e^{2sM}\\Vert P_ka\\Vert^2_{H^1(\\gamma)}\n$$\nfor all large $s>0$. Here we set \n$M:= \\max_{x\\in \\ooo{\\Gamma}} \\psi(x)$.\n\nWe choose $s>0$ sufficiently large and set \n$s_k:= s^*\\la_k^{\\frac{2}{3}}$ for $k\\in \\N$. Then, using (3.24),\nwe obtain\n\\begin{align*}\n& ({s^*}^3\\la_k^2 - C\\la_k^2)\\int_{\\OOO} \\vert Q_ka\\vert^2\ne^{2s_k\\psi} dx\n\\le C{s^*}^3\\la_k^2e^{2s_kM}\\Vert P_ka\\Vert^2_{H^1(\\OOO)}\\\\\n\\le& C{s^*}^3\\la_k^2e^{2s_kM}(\\la_k+1)^2\\Vert P_ka\\Vert^2_{L^2(\\Omega)}.\n\\end{align*}\nSince $\\psi \\ge 0$ in $\\OOO$ and we can take $s^*>0$ sufficiently large, we see\n$$\n {s^*}^3\\la_k^2 \\Vert Q_ka\\Vert^2_{L^2(\\Omega)}\n\\le C{s^*}^3\\la_k^2\\la_k^2 e^{2s_kM}\\Vert P_ka\\Vert^2_{L^2(\\Omega)},\n$$\nthat is,\n$$\n\\Vert Q_ka\\Vert^2_{L^2(\\Omega)} \\le C\\la_k^2 e^{C_1\\la_k^{\\frac{2}{3}}}\n\\Vert P_ka\\Vert^2_{L^2(\\Omega)},\n$$\nwhere we set $C_1:= 2s^*M$. Here we note that $s^*$ and $M$, and so\nthe constant $C_1$ are independent of $k\\in \\N$.\n\nTherefore, since we can find a constant $C_2>0$ such that \n$\\eta^2 e^{C_1\\eta^{\\frac{2}{3}} + \\sigma_1\\eta^{\\hhalf}}\n\\le C_2e^{C_2\\eta^{\\frac{2}{3}}}$ for all $\\eta \\ge 0$, \nwe see\n$$\n\\sumk e^{\\sigma_1\\la_k^{\\hhalf}}\\Vert Q_ka\\Vert^2_{L^2(\\Omega)}\n\\le C\\sumk \\la_k^2e^{C_1\\la_k^{\\frac{2}{3}}+\\sigma_1\\la_k^{\\hhalf}}\n\\Vert P_ka\\Vert^2_{L^2(\\Omega)}\n\\le C_2\\sumk e^{C_2\\la_k^{\\frac{2}{3}}}\\Vert P_ka\\Vert^2_{L^2(\\Omega)}.\n$$\nMoreover, $\\lim_{k\\to\\infty} \\frac{\\theta(\\la_k)}{\\la_k^{\\frac{2}{3}}}\n= \\infty$ yields that for the constant $C_2>0$ we can choose $N\\in \\N$ \nsuch that $C_2\\la_k^{\\frac{2}{3}} \\le \\theta(\\la_k)$ for $k \\ge N$. \nConsequently, \n$$\n\\sum_{k=N}^{\\infty} e^{\\sigma_1\\la_k^{\\hhalf}}\\Vert Q_ka\\Vert^2_{L^2(\\Omega)}\n\\le C\\sum_{k=N}^{\\infty} e^{\\theta(\\la_k)}\\Vert P_ka\\Vert^2_{L^2(\\Omega)}\n< \\infty,\n$$\nand so\n$$\n\\sum_{k=1}^{\\infty} e^{\\sigma_1\\la_k^{\\hhalf}}\\Vert Q_ka\\Vert^2_{L^2(\\Omega)}\n< \\infty.\n$$\nThus (3.21) completes the proof of Theorem 1.1.\n$\\blacksquare$\n%\n%\n\\section{Appendix: Proof of Lemma 2.2}\n\nWe can prove the lemma by integration by parts similarly to \nLemma 7.1 (p.186) in Bellassoued and Yamamoto \\cite{BY}) for example, but\nhere we derive from a Carleman estimate for the parabolic equation\nby Imanuvilov \\cite{Im}.\n\nWe set $Q:= \\OOO\\times (0,T)$.\nWe choose $\\ell \\in C^{\\infty}[0,T]$ such that \n$$\n\\left\\{ \\begin{array}{rl}\n& \\ell(t) = 1 \\quad \\mbox{for $\\frac{T}{4}\\le t\\le \\frac{3}{4}T$},\\\\\n& \\ell(0) = \\ell(T) = 0,\\\\\n& \\mbox{$\\ell$ is strictly increasing on $\\left[ 0, \\, \\frac{T}{4}\\right]$\nand strictly decreasing on $\\left[ \\frac{T}{4}, \\,T\\right]$}.\n\\end{array}\\right.\n \\eqno{(4.1)}\n$$\nIn particular, $\\ell(t) \\le 1$ for $0\\le t\\le T$. Choosing $\\la>0$ \nsufficiently large, we set\n$$\n\\alpha(x,t) := \\frac{e^{\\la\\rho(x)} - e^{2\\la\\Vert \\rho\\Vert_{C(\\ooo{\\OOO})}}}\n{\\ell(t)}, \\quad\n\\va(x,t) := \\frac{e^{\\la\\rho(x)}}{\\ell(t)}, \\quad (x,t)\\in \\OOO\\times (0,T).\n$$\nThen we know\n\\\\\n{\\bf Lemma 4.1}\n\\\\\n{\\it\nThere exist constants $s_0>0$ and $C>0$ such that \n\\begin{align*}\n& \\int_Q (s\\va\\vert \\nabla U\\vert^2 + s^3\\va^3\\vert U\\vert^2)\ne^{2s\\alpha} dxdt \\\\\n\\le& C\\int_Q \\vert \\ppp_tU - A_2U\\vert^2 e^{2s\\alpha} dxdt\n+ C\\int_{\\gamma} (\\vert \\ppp_tU\\vert^2 + s\\va\\vert \\nabla U\\vert^2\n+ s^3\\va^3\\vert U\\vert^2) e^{2\\alpha} dSdt\n\\end{align*}\nfor all $s \\ge s_0$ and $U\\in H^{2,1}(Q)$ satisfying \n$\\nuA U = 0$ on $\\ppp\\OOO \\times (0,T)$.\n}\n\nThe proof is found in Chae, Imanuvilov and Kim \\cite{CIK}.\n\nWe apply Lemma 4.1 to $g(x)$ satisfying $\\nuA g = 0$ on $\\ppp\\OOO$ to\nobtain\n$$\n\\int_Q (s\\va(x,t)\\vert \\nabla g(x)\\vert^2 + s^3\\va^3(x,t)\\vert g(x)\\vert^2)\ne^{2s\\alpha(x,t)} dxdt \n \\eqno{(4.2)}\n$$\n$$\n\\le C\\int_Q \\vert A_2g\\vert^2 e^{2s\\alpha(x,t)} dxdt\n+ C\\int^T_0 \\int_{\\gamma} (s\\va(x,t)\\vert \\nabla g(x)\\vert^2\n+ s^3\\va^3\\vert g(x)\\vert^2) e^{2\\alpha} dSdt\n$$\nfor all $s \\ge s_0$.\nMoreover, in terms of (4.1) and $e^{\\la\\rho(x)} \n= e^{2\\la\\Vert \\rho\\Vert_{C(\\ooo{\\OOO})}}\\psi(x)$ for $x\\in \\OOO$,\nwe have \n$$\n\\int_Q (s\\va(x,t)\\vert \\nabla g(x)\\vert^2 + s^3\\va^3(x,t)\\vert g(x)\\vert^2)\ne^{2s\\alpha(x,t)} dxdt\n \\eqno{(4.3)}\n$$\n%\\begin{align*}\n$$\n\\ge \\int^{\\frac{3}{4}T}_{\\frac{T}{4}} \\int_{\\OOO}\n (se^{\\la\\rho(x)}\\vert \\nabla g(x)\\vert^2 \n+ s^3e^{3\\la\\rho(x)}\\vert g(x)\\vert^2) \n\\exp( 2s(e^{\\la\\rho(x)} - e^{2\\la\\Vert \\rho\\Vert_{C(\\ooo\\OOO)}}) ) dxdt\n$$\n%\\\\\n$$\n\\ge C\\frac{T}{2}\n\\int_{\\OOO} (s\\vert \\nabla g(x)\\vert^2 + s^3\\vert g(x)\\vert^2) \ne^{2s(e^{2\\la\\Vert\\rho\\Vert_{C(\\ooo\\OOO)}}\\psi(x))} dx \ne^{-2se^{2\\la\\Vert \\rho\\Vert_{C(\\ooo\\OOO)}}}\n$$\n$$\n\\ge C\\frac{T}{2}\n\\int_{\\OOO} (s\\vert \\nabla g(x)\\vert^2 + s^3\\vert g(x)\\vert^2) \ne^{2s\\psi(x)} dx \ne^{-2se^{2\\la\\Vert \\rho\\Vert_{C(\\ooo\\OOO)} }}.\n$$\n%\\end{align*}\nHere $C>0$ depends on $\\la$ but not on $s>0$.\nBy $e^{2s\\alpha(x,t)} \\le 1$ in $Q$, (4.2) and (4.3), we obtain\n$$\nC\\frac{T}{2}\\int_{\\OOO} (s\\vert \\nabla g(x)\\vert^2 + s^3\\vert g(x)\\vert^2) \ne^{2s\\psi(x)} dx e^{-2se^{2\\la\\Vert \\rho\\Vert_{C(\\ooo\\OOO)}}}\n \\eqno{(4.4)}\n$$\n$$\n\\le C\\int_Q \\vert A_2g\\vert^2 e^{2s\\alpha(x,t)} dxdt\n+ C\\int^T_0 \\int_{\\gamma} (s\\va(x,t)\\vert \\nabla g(x)\\vert^2\n+ s^3\\va^3(x,t)\\vert g(x)\\vert^2) \ne^{2s\\alpha(x,t)} dSdt.\n$$\nSince $\\sup_{(x,t)\\in \\gamma \\times (0,T)} \n\\vert (s\\va)^ke^{2s\\alpha(x,t)}\\vert \n< \\infty$ for $k=1,3$ and \n$$\ne^{2s\\psi(x)} = e^{2se^{\\la(\\rho(x) - 2\\Vert \\rho\\Vert_{C(\\ooo{\\OOO})})}}\n\\ge e^{2se^{-\\la\\Vert \\rho\\Vert_{C(\\ooo\\OOO)}}},\n$$\nwe can find a constant $C_1 = C_1(\\la) > 0$ such that \n$$\n(s\\va(x,t))^k e^{2s\\alpha(x,t)} \\le C_1e^{2s\\psi(x)}, \\quad\n(x,t) \\in \\gamma \\times (0,T).\n$$\nTherefore\n\\begin{align*}\n& \\int_{\\OOO} (s\\vert \\nabla g\\vert^2 + s^3\\vert g\\vert^2)\ne^{2s\\psi(x)} dx \\\\\n\\le &Ce^{2se^{2\\la\\Vert \\rho\\Vert_{C(\\ooo{\\OOO})}}}\n\\left( \\int_{\\OOO} \\vert A_2g\\vert^2 e^{2s\\psi} dx \n+ \\int_{\\gamma} (s\\vert \\nabla g\\vert^2 + s^3\\vert g(x)\\vert^2) \ne^{2s\\psi} dS\\right).\n\\end{align*}\nSubstituting (4.3) and (4.4) into (4.2), we complete the proof of \nLemma 2.2.\n$\\blacksquare$\n\n{\\bf Acknowledgements.}\nThe work was supported by Grant-in-Aid for Scientific Research (A) 20H00117 \nof Japan Society for the Promotion of Science.\n%\n%\n%\n\\begin{thebibliography}{99} %\n\n\\bibitem{Ag}\nS. Agmon,\n{\\it Lectures on Elliptic Boundary Value Problems},\nVan Nostrand, Princeton, 1965.\n\n\\bibitem{BK}\nA.L. Bukhgeim and M.V. Klibanov, \n{\\it Global uniqueness of a class of multidimensional inverse problems,}\nSov. Math.-Dokl. {\\bf 24} (1981) 244-247.\n\n\\bibitem{BY}\nM. Bellassoued and M. Yamamoto, {\\it Carleman Estimates and Applications to\nInverse Problems for Hyperbolic Systems}, Springer-Japan, Tokyo, 2017.\n\n\\bibitem{CIK} \nD. Chae, O. Y. Imanuvilov and S Kim, \n{\\it Exact controllability for semilinear parabolic equations with Neumann boundary \nconditions,} J. Dynam. Control Systems {\\bf 2} (1996) 449-483.\n\n\\bibitem{FI}\nA. V. Fursikov and O. Y. Imanuvilov, {\\it Controllability of Evolution \nEquations}, Lecture Notes Series vol 34, 1996, Seoul National University.\n\n\\bibitem{H}\nL. H\\\"ormander, {\\it Linear Partial Differential Operators}, \nSpringer-Verlag, Berlin, 1963.\n\n\\bibitem{HIY}\nX. Huang, O.Y. Imanuvilov and M. Yamamoto, \n{\\it Stability for inverse source problems by Carleman estimates, }\nInverse Problems {\\bf 36} (2020) 125006\n\n\\bibitem{Im}\nO.Y. Imanuvilov, {\\it Controllability of parabolic equations,} Sbornik Math. \n{\\bf 186} (1995) 879-900.\n\n\\bibitem{IY98}\nO.Y. Imanuvilov and M. Yamamoto,\n{\\it Lipschitz stability in inverse parabolic problems by the Carleman\nestimate,} Inverse Problems {\\bf 14} (1998) 1229-1245.\n\n\\bibitem{IY23}\nO.Y. Imanuvilov and M. Yamamoto,\n{\\it Inverse parabolic problems by Carleman estimates with data taken initial or \nfinal time moment of observation,} preprint \narXiv:2211.11930\n\n\\bibitem{Is}\nV. Isakov, {\\it Inverse Source Problems}, American Math. Soc., Providence,\nRI, 1990.\n\n\\bibitem{Ka}\nT. Kato, {\\it Perturbation Theory for Linear Operators},\nSpringer-Verlag, Berlin, 1980.\n\n\\bibitem{Kl}\nM.V. Klibanov, {\\it Inverse problems and Carleman estimates,} Inverse Problems \n{\\bf 8} (1992) 575-596.\n\n\\bibitem{Pa}\nA. Pazy, {\\it Semigroups of Linear Operators and Applications to \nPartial Differential Equations}, Springer-Verlag, Berlin, 1983.\n\n\\bibitem{R}\nD.L. Russell, \n{\\it A unified boundary controllability theory for \nhyperbolic and parabolic partial differential equations,}\nStudies in Appl. Math. {\\bf 52} (1973) 189-211.\n\n\\bibitem{S} T. Suzuki, \n{\\it Gelfand-Levitan's theory, deformation formulas and inverse problems, }\nJ. Fac. Sci. Univ. Tokyo sect. IA. Math., {\\bf 32} (1985) 223-271.\n\n\\bibitem{SM} \nT. Suzuki and R. Murayama, \n{\\it A Uniqueness theorem in an identification problem for coefficient of \nparabolic equation,} Proc. Japan. {\\bf 56} Ser. A. (1980) 259-263.\n\n%\\bibitem{Y09}\n%M. Yamamoto, Carleman estimates for parabolic equations and applications, \n%Inverse Problems {\\bf 25} (2009) 123013\n\n\n\\end{thebibliography}\n%\n\\end{document}\n\n\n\n\n\n\\bibitem{Im}\nO.Y. Imanuvilov, {\\it Controllability of parabolic equations, }\nSbornik Math. {\\bf 186} (1995) 879-900.\n\n\\bibitem{IY98}\nO. Y. Imanuvilov and M. Yamamoto, \n{\\it Lipschitz stability in inverse parabolic problems by the Carleman\nestimate,} Inverse Problems {\\bf 14} (1998) 1229-1245.\n\n\n\\bibitem{Is06}\nV. Isakov, {\\it Inverse Problems for Partial Differential Equations}, \nSpringer-Verlag, Berlin, 2006.\n\n\\bibitem{Ka}\nT. Kato, {\\it Perturbation Theory for Linear Operators}, Springer-Verlag,\nBerlin, 1980.\n\n\\bibitem{Kli}\nM.V. Klibanov, {\\it Inverse problems and Carleman estimates,}\nInverse Problems {\\bf 8} (1992) 575-596.\n\n\\bibitem{KT}\nM.V. Klibanov and A.A. Timonov, \n{\\it Carleman Estimates for Coefficient Inverse Problems and Numerical\nApplications}, VSP, Utrecht, 2004.\n\n\\bibitem{LRS}\nM.M. Lavrent'ev, V.G. Romanov and S.P. Shishat$\\cdot$ski{\\u\\i},\n{\\it Ill-posed Problems of Mathematical Physics and Analysis},\nAmerican Mathematical Society, Providence, Rhode Island, 1986. \n\n\\bibitem{LM}\nJ.-L. Lions and E. Magenes, {\\it Non-homogeneous Boundary Value Problems and \nApplications}, Springer-Verlag, Berlin, 1972.\n\n\n\n\\bibitem{Ag1}\nS. Agmon, {\\it On the eigenfunctions and on the eigenvalues of general\nelliptic boundary value problems,} \nCommun. Pure Appl. Math. {\\bf 15} (1962) 119-147.\n\n\\bibitem{Ag2} S. Agmon,\n{\\it Lectures on Elliptic Boundary Value Problems},\nVan Nostrand, Princeton, 1965.\n\n\n\n\n\n\n"},{"text":"\\documentclass[prd,aps,floats,showkeys,nofootinbib,notitlepage]{revtex4}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{amsbsy}\n\\usepackage{float}\n\\restylefloat{table}\n%\\usepackage{natbib}\n\\usepackage{graphicx}\n\\usepackage{epsfig}\n\\usepackage{epstopdf}\n\\usepackage{subfig}\n%\\usepackage{color}\n\\usepackage[colorlinks=true,linkcolor=red]{hyperref}\n\\usepackage{listings} \n\\usepackage{xcolor} % for setting colors\n\\usepackage{hyperref}\n\\usepackage{tabularx}\n\\usepackage{multirow}\n\\usepackage{enumerate}\n\\usepackage{hyperref}\n%\\usepackage{caption}\n\n\n%%% this force table caption on top %%%%%%%%%\n\\floatstyle{plaintop}\n\\restylefloat{table}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\newcolumntype{Y}{>{\\centering\\arraybackslash}X} % Use Y instead of X for centering text in tabularx table cell\n\n\\newcommand{\\xmark}{\\ding{55}}\n\n\n\\def\\bibsection{\\section*{\\refname}}\n\n%\\allowdisplaybreaks % for a pagebreak inside an aligned equation\n\n\\newcommand{\\matrixHorizon}[1]{\\ensuremath{\\overset{\\xrightarrow[\\hphantom{#1}]{\\text{flipping horizontally}}}{#1}}} % for putting arrows above the matrix\n\\newcommand{\\matrixVertical}[1]{\\ensuremath{\\left\\downarrow\\vphantom{#1}\\right.{#1}}} % for putting arrows beside the matrix\n%\\renewcommand*\\thelstnumber{\\arabic{lstnumber}:}\n\n\n\n\n\n\\definecolor{ForestGreen}{RGB}{34,139,34}\n\\definecolor{OliveGreen}{RGB}{72.9,72.2,42.4}\n\\lstset{ %Using different colors for different keywords in lstlisting\nframe=top,frame=bottom,\nbasicstyle=\\small\\normalfont\\sffamily, % the size of the fonts that are used for the code\nstepnumber=1, % the step between two line-numbers. If it is 1 each line will be numbered\nnumbersep=10pt, % how far the line-numbers are from the code\ntabsize=2, % tab size in blank spaces\nextendedchars=true, %\nbreaklines=true, % sets automatic line breaking\ncaptionpos=t, % sets the caption-position to top\nmathescape=true,\n%stringstyle=\\color{white}\\ttfamily, % Farbe der String\nshowspaces=false, % Leerzeichen anzeigen ?\nshowtabs=false, % Tabs anzeigen ?\nxleftmargin=17pt,\nframexleftmargin=17pt,\n%framexrightmargin=17pt,\nframexbottommargin=5pt,\nframextopmargin=5pt,\nshowstringspaces=false % Leerzeichen in Strings anzeigen ?\nkeywordstyle=\\color{blue}\\bf,\ncommentstyle=\\color{OliveGreen},\nstringstyle=\\color{red},\n%numbers=left,\n%numberstyle=\\tiny,\n%numbersep=5pt,\nnumbers=none,\nbreaklines=true,\nshowstringspaces=false,\nbasicstyle=\\footnotesize,\nemph={str},emphstyle={\\color{magenta}}}\n\n\\DeclareCaptionFormat{listing}{\\rule{\\dimexpr\\textwidth}{0.4pt}\\par\\vskip1pt#1#2#3}\n\\captionsetup[lstlisting]{format=listing,singlelinecheck=false, margin=0pt, font={sf},labelsep=space,labelfont=bf}\n\n\\begin{document}\n\t\\title{A New Algorithm to determine Adomian Polynomials for nonlinear polynomial functions}\n\t\\author{Mithun Bairagi}\n\t\\email{bairagirasulpur@gmail.com}\n\t\\affiliation{Mangal Chandi High School, Khosalpur, 722206, Patrasayer, Bankura, West Bengal, India}\n\t\n\t\n\t\\begin{abstract}\n\tWe present a new algorithm by which the Adomian polynomials can be determined for scalar-valued nonlinear polynomial functional in a Hilbert space. This algorithm calculates the Adomian polynomials without the complicated operations such as parametrization, expansion, regrouping, differentiation, etc. The algorithm involves only some matrix operations. Because of the simplicity in the mathematical operations, the new algorithm is faster and more efficient than the other algorithms previously reported in the literature. We also implement the algorithm in the MATHEMATICA code. The computing speed and efficiency of the new algorithm are compared with some other algorithms in the one-dimensional case.\n\t\n\t\\end{abstract}\n\t\\keywords{Adomian decomposition method; Adomian polynomials; Nonlinear operators; Matrix; ODE; Series solution.}\n\t\\maketitle\n\n\t\n\t\\section{introduction} \n\tThe Adomian Decomposition Method (ADM) \\cite{adm,adm1,adm2,adm3,adm4} has gained huge attention in different fields of science and engineering for solving nonlinear functional equations.\nIn practice, many nonlinear problems do not admit exact solutions, and in most cases, we have to find approximate solutions by employing numerical or analytical approximation techniques. \nThe ADM is a reliable technique for solving wide classes of nonlinear systems, including ordinary differential, partial differential, integro-differential, algebraic, differential-algebraic, non-integer-order differential, integral equations, and so on \\cite{Mavoungou,Ngarhasta,admapp,admapp1,admapp2,admapp3,admapp4}). This technique can provide an analytical approximation to the exact solutions in the series form that converge very rapidly \\cite{admconver,admconver1,admconver2}.\nThe Adomian decomposition method coupled with the Laplace transform, develops a powerful method called the Laplace Adomian decomposition method (LADM). LADM has also been used in numerous articles to find the numerical solution of fractional-order nonlinear differential equations, as can be seen in \\cite{ladm1,ladm2,ladm3,ladm4,ladm5}.\n\t\nFollowing \\cite{Mavoungou,Ngarhasta,Duan}, let us recall the basic ideas of the Adomian Decomposition Method. We consider a nonlinear ODE in order $p$ with independent variable $x$ (real and scalar) and dependent variable $u$ in the general form \\cite{Mavoungou,Ngarhasta}\n\\begin{equation}\n\t\\mathcal{F}u=g(x),\n\\end{equation}\nwhere $\\mathcal{F}$ is the nonlinear operator from a Hilbert space $H$ into $H$. In ADM, $\\mathcal{F}$ is assumed to be decomposed into\n\\begin{equation}\\label{nlode}\n\tLu+Ru+Nu=g(x),\n\\end{equation}\nwhere $L$ is the highest-order linear differential operator $L[.]=\\frac{d^p}{dx^p}[.]$ which is assumed to be invertible, $R$ is a linear differential operator containing the linear derivatives of less order than $L$, $N$ is a nonlinear operator containing all other nonlinear terms, $g(x)\\in H$ is a given analytic function. Here we should note that the choice of the operator $L$ is not generally unique \\cite{adomian95,wazwaz02,wazwaz06}. For example, in \\cite{wazwaz02}, A. Wazwaz has chosen the linear differential operator $L[.]$ as $L[.]=x^{-2}\\frac{d}{dx}\\left(x^2\\frac{d}{dx}\\right)$ for the Lane-Emden equation.\nIt is also notable that $u$ is a scalar function of real variable $x$ in Eq. \\eqref{nlode}. For a system of differential equations, $u$ will be a vector-valued function. However, in this paper, our studies are restricted to single ODE where $u$ is a scalar-valued function.\nThe principle step of the decomposition method is to suppose a series solution defined by\n\t\\begin{equation}\\label{seriesSolu}\n\t\tu=\\sum_{i=0}^{\\infty}u_i,\n\t\\end{equation}\nand then the ADM scheme corresponding to the functional equation \\eqref{nlode} converges rapidly to $u\\in H$ which is the unique solution to the functional equation \\cite{Mavoungou,Bougoffa}.\nEquation \\eqref{seriesSolu} decomposes the nonlinear term $Nu$ into an infinite series \n\t\\begin{equation}\\label{Nu}\n\t\tNu=\\sum_{i=0}^{\\infty}A_i,\n\t\\end{equation}\n\twhere $A_i$ are the so-called Adomian polynomials which depend on the solution components $u_0,u_1,\\ldots,u_i$. For a given nonlinear functional $Nu=F(u)$ ($F(u)$ is assumed to be an analytic function of variable $u$ in Hilbert space $H$), the Adomian polynomials are determined by the following definitional\n\tformula introduced by G. Adomian \\cite{adm,adm1,adm2,Duan3}:\n\t\\begin{equation}\\label{admdef}\n\t\tA_M=\\left.\\frac{1}{M!}\\frac{d^M }{d \\lambda^M}{F\\left(\\sum_{k=0}^{\\infty}u_k\\lambda^k\\right)}\\right|_{\\lambda=0},\\;\\; \\;\\;M=0,1,2,\\ldots,\n\t\\end{equation}\n\twhere the analytic parameter $\\lambda$ is simply a grouping parameter. An important property of Adomian polynomial $A_M$ is that it depends by construction only on the solution components $(u_0,u_1,\\ldots,u_M)$ and does not depend on higher-order solution components $u_k$ with $k>M$ \\cite{Wazwaz,Azreg}. Therefore, the higher-order terms for $k>M$ do not contribute in summation in Eq. \\eqref{admdef}. \n\t\n\tMain step of ADM is to determine the Adomian polynomials of the nonlinear term $Nu$.\n\tUsing the definitional formula \\eqref{admdef} it is difficult to calculate higher-order Adomian terms due to the complexity in calculations of higher-order derivatives. Later, many authors have developed several convenient algorithms for fast generation of the one-variable and the multi-variable Adomian polynomials. Adomian and Rach \\cite{AR} produced a recurrence rule that provides a systematic computational procedure to determine Adomian polynomials. Later, Rach in his paper \\cite{Rach} established simple symmetry rules (which is called Rach's rule) in Adomian and Rach's algorithms, by which Adomian\u2019s polynomials can be determined quickly to higher orders. Using the algorithm presented by Wazwaz in \\cite{Wazwaz}, we need to collect the terms from the expansion, which takes a large computational time for higher orders. Applying the algorithm in \\cite{Biazar}, we require to compute the derivative after substitution in a recurrence relation between the Adomian polynomials. Recently in \\cite{Agom}, the authors modified the formula \\eqref{admdef} to determine the Adomian polynomials for nonlinear polynomial functionals. \n\tIn \\cite{Duan,Duan1}, Duan has developed more efficient and fast recurrence algorithms for the rapid generation of the Adomian polynomials for one-variable (which is the one-dimensional case in our studies) and multi-variable cases. Duan\u2019s Corollary 1 algorithm \\cite{Duan} (called index recurrence algorithm) and Duan\u2019s Corollary 3 algorithm \\cite{Duan1} do not involve the differentiation operator in determining the reduced polynomials in one dimension. We only require the operations of addition and multiplication, which make these algorithms faster and more efficient techniques. \n\t\n\tIn this work, we have presented a new algorithm for fastest computations of Adomian polynomials for scalar-valued nonlinear polynomial functional (with index as positive integers) in a Hilbert space $H$ with the help of matrix formulations rather than recurrence processes. Our proposed algorithm does not require complex mathematical operations such as parametrization, expansion, regrouping, and differentiation. In this algorithm, the higher-order Adomian polynomials can be determined through few matrix operations, making it faster and more efficient than the other existing algorithms in the literature. We have generalized the new algorithm in two dimensions where the solution $u$ depends on two-state variables such as $t,x$. \n\t\n\t\n\tThe paper is organized as follows: In Sec. \\ref{algo} we present our algorithm to determine Adomian polynomials for nonlinear polynomial functional. In Sec. \\ref{comp}, we apply our algorithms to the polynomial functions, and the computation times are compared with some other popular algorithms previously reported in the literature. In Sec. \\ref{con}, we discuss our results and make some conclusions on our works. We list the MATHEMATICA code for the new algorithms in Listing \\ref{onedOur} for one-dimensional case and in Listing \\ref{twodOur} for two-dimensional case in Appendix: \\ref{oned}, \\ref{twod} respectively. We have also listed the MATHEMATICA code for some other algorithms which are Duan\u2019s Corollary 1 algorithm \\cite{Duan} and Duan\u2019s Corollary 3 algorithm \\cite{Duan1,Duan2} with the one-dimensional case in Listings \\ref{onedDuan}, \\ref{onedDuan1} in Appendix: \\ref{oned}. \n\n\\section{description of our proposed algorithm}\\label{algo}\n\tIn this section, we have described a new algorithm for calculating the Adomian polynomials. This algorithm is only applicable for scalar-valued nonlinear polynomial functional (with index as positive integers) in a Hilbert space $H$ for the two-dimensional case. In order to increase the calculating efficiency in this algorithm, all the mathematical operations are performed in the matrix forms. \n\t\n\tLet us now consider a nonlinear polynomial functional $F$ depends on two different functions $u$ and $v$ in $H$. \n\tThe functions $u$ and $v$ can be expanded into the following two-dimensional series\n\t\\begin{equation}\\label{uvSeries}\n\t\tu=\\sum_{i=0}^{\\infty}\\sum_{j=0}^{\\infty}u_{ij}\\;\\;\\;\\text{and}\\;\\;\\;v=\\sum_{i=0}^{\\infty}\\sum_{j=0}^{\\infty}v_{ij}\\;.\n\t\\end{equation}\n\tTo illustrate our algorithm, we take the nonlinearity $F$ in the simple form \n\t\\begin{equation}\\label{nlF}\n\t\tF=uv.\n\t\\end{equation}\n\tAnd this nonlinear function can be decomposed by a series\n\t\\begin{equation}\n\t\tF=\\sum_{i=0}^{\\infty}\\sum_{j=0}^{\\infty}A_{ij},\n\t\\end{equation}\n where $A_{ij}$ are called Adomian polynomials of the components $u_{ij},v_{ij}\\;(i=0,1,\\ldots,j=0,1,\\ldots)$.\n\tNow, we divide the algorithm into six main steps (labeled from Step-1 to Step-6), and to illustrate each step, we have used the nonlinear polynomial function \\eqref{nlF}.\n\n\t\\begin{itemize}\n\t\t\\item[]\\textbf{Step-1} (Express the functions $u$ and $v$ in the matrix forms): In this step, the functions $u$ and $v$ are expressed in the matrix forms. For computations in computer, we truncate the infinite series \\eqref{uvSeries} up to the finite terms $i=m,j=n$. We can increase the accuracy in our results by increasing the values of $m,n$ as far as possible. The functions $u,v$ in the Eq. \\eqref{uvSeries} can be expressed by $(m+1)\\times (n+1)$ matrices\n\t\t\\begin{equation}\\label{soluMatrix}\n\t\t\tU=\n\t\t\t\\begin{pmatrix}\n\t\t\t\tu_{00}& u_{01}& \\ldots& u_{0l}& \\ldots& u_{0n}\\\\ \n\t\t\t\t\\vdots&\t \\vdots& \\ldots& \\vdots& \\ldots& \\vdots\\\\\n\t\t\t\tu_{k0}& u_{k1}& \\ldots& u_{kl}& \\ldots& u_{kn}\\\\ \n\t\t\t\t\\vdots&\t \\vdots& \\ldots& \\vdots& \\ldots& \\vdots\\\\\n\t\t\t\tu_{m0}& u_{m1}& \\ldots& u_{ml}& \\ldots& u_{mn}\n\t\t\t\\end{pmatrix} \\;\\;\\text{and}\\;\\; %\\overset{r_1+r_2}{\\longrightarrow} \n\t\t\tV=\n\t\t\t\\begin{pmatrix}\n\t\t\t\tv_{00}& v_{01}& \\ldots& v_{0l}& \\ldots& v_{0n}\\\\ \n\t\t\t\t\\vdots&\t \\vdots& \\ldots& \\vdots& \\ldots& \\vdots\\\\\n\t\t\t\tv_{k0}& v_{k1}& \\ldots& v_{kl}& \\ldots& v_{kn}\\\\ \n\t\t\t\t\\vdots&\t \\vdots& \\ldots& \\vdots& \\ldots& \\vdots\\\\\n\t\t\t\tv_{m0}& v_{m1}& \\ldots& v_{ml}& \\ldots& v_{mn}\n\t\t\t\\end{pmatrix}. %\\underset{\\overset{r_1-4r_2}{\\longrightarrow}}{\\overset{r_1+r_2}{\\longrightarrow}}\n\t\t\\end{equation}\n\t\t%Then, using Eqs. \\eqref{soluMatrix} we express Eq. \\eqref{nlF} in the matrix notation.\n\t\n\t\t\\item[]\\textbf{Step-2} (Extracting the submatrices from the matrices $U$ and $V$): The Adomian polynomials corresponding to any matrix elements (let the matrix elements $u_{kl},v_{kl}$ located at row $k+1$, column $l+1$) in Eq. \\eqref{soluMatrix}, depend on the other matrix elements whose row number ($r$) and column number ($c$) are less than or equal to $k+1$ and $l+1$ respectively, but do not depend on the matrix elements located at $r>k+1$ and $c>l+1$. In order to calculate the Adomian polynomials for the elements $u_{kl}$ and $v_{kl}$ in $U$ and $V$, we extract the submatrices formed by the elements with rows $r\\leq k+1$ and columns $c\\leq l+1$ of the matrices $U$ and $V$ in Eq. \\eqref{soluMatrix}. These submatrices are given by \n\t\t\\begin{equation}\\label{subMatrix}\n\t\t\tU[{0,1,\\ldots,k;0,1,\\ldots,l}]=\n\t\t\t\\begin{pmatrix}\n\t\t\t\tu_{00}& u_{01}& \\ldots& u_{0l}\\\\ \n\t\t\t\t\\vdots&\t \\vdots& \\ldots& \\vdots\\\\\n\t\t\t\tu_{k0}& u_{k1}& \\ldots& u_{kl}\\\\ \t\t\t\t\n\t\t\t\\end{pmatrix} \\;\\;\\text{and}\\;\\; %\\overset{r_1+r_2}{\\longrightarrow} \n\t\t\tV[{0,1,\\ldots,k;0,1,\\ldots,l}]=\n\t\t\t\\begin{pmatrix}\n\t\t\t\tv_{00}& v_{01}& \\ldots& v_{0l}\\\\ \n\t\t\t\t\\vdots&\t \\vdots& \\ldots& \\vdots\\\\\n\t\t\t\tv_{k0}& v_{k1}& \\ldots& v_{kl}\t\t\t\n\t\t\t\\end{pmatrix}. %\\underset{\\overset{r_1-4r_2}{\\longrightarrow}}{\\overset{r_1+r_2}{\\longrightarrow}}\n\t\t\\end{equation}\n\t%These are the submatrices for the elements located at $(k+1)$-th row and $(l+1)$-th column of the matrices $U$ and $V$.\n\t\n\t\\item[] \\textbf{Step-3} (Flipping the submatrix): In this step, all the matrix elements of any one of the submatrices in Eq. \\eqref{subMatrix} are flipped horizontally and then vertically or vice versa. Here we perform the flipping operation on the submatrix $V[{0,1,\\ldots,k;0,1,\\ldots,l}]$. The flipping operation along horizontal axis can be shown in the following way\n\t\\begin{equation}\n\t\t\\matrixHorizon{\t\\begin{pmatrix}\n\t\t\tv_{00}& v_{01}& \\ldots& v_{0l}\\\\ \n\t\t\t\\vdots&\t \\vdots& \\ldots& \\vdots\\\\\n\t\t\tv_{k0}& v_{k1}& \\ldots& v_{kl}\t\t\t\n\t\t\\end{pmatrix}}\\longrightarrow\n\t\t\\begin{pmatrix}\n\t\tv_{0l}& v_{0l-1}& \\ldots& \tv_{00}\\\\ \n\t\t\t\\vdots&\t \\vdots& \\ldots& \\vdots\\\\\n\t\t\tv_{kl}& v_{kl-1}& \\ldots& v_{k0}\t\t\t\n\t\t\\end{pmatrix}=V[{0,1,\\ldots,k;l,l-1,\\ldots,0}].\n\t\\end{equation}\n\tThen, the flipping operation along vertical axis is performed on the above flipped submatrix, which can be shown as \n\t\\begin{equation}\n\t\t\\text{\\tiny flipping vertically}\\matrixVertical{\t\\begin{pmatrix}\n\t\t\t\tv_{0l}& v_{0l-1}& \\ldots& \tv_{00}\\\\ \n\t\t\t\t\\vdots&\t \\vdots& \\ldots& \\vdots\\\\\n\t\t\t\tv_{kl}& v_{kl-1}& \\ldots& v_{k0}\t\t\t\n\t\t\\end{pmatrix}}\\longrightarrow\n\t\t\\begin{pmatrix}\n\t\t\tv_{kl}& v_{kl-1}& \\ldots& \tv_{k0}\\\\ \n\t\t\t\\vdots&\t \\vdots& \\ldots& \\vdots\\\\\n\t\t\tv_{0l}& v_{0l-1}& \\ldots& v_{00}\t\t\t\n\t\t\\end{pmatrix}=V[{k,k-1,\\ldots,0;l,l-1,\\ldots,0}].\n\t\\end{equation}\n\n\t\\item[]\\textbf{Step-4} (Element-wise matrices multiplication): In the element-wise multiplication (also known as the Hadamard product), each element $i,j$ in the two matrices are multiplied together. We perform the element-wise multiplication between two matrices $U[{0,1,\\ldots,k;0,1,\\ldots,l}]$ and $V[{k,k-1,\\ldots,0;l,l-1,\\ldots,0}]$, given by\n\t\\begin{equation} \n\t\tU[{0,1,\\ldots,k;0,1,\\ldots,l}]\\circ V[{k,k-1,\\ldots,0;l,l-1,\\ldots,0}]=W[{0,1,\\ldots,k;0,1,\\ldots,l}]\n\t\\end{equation}\n\tand in the matrix notation the above equation can be expressed by\n\t\\begin{equation}\n\t\t\\begin{pmatrix}\n\t\t\tu_{00}& u_{01}& \\ldots& u_{0l}\\\\ \n\t\t\t\\vdots&\t \\vdots& \\ldots& \\vdots\\\\\n\t\t\tu_{k0}& u_{k1}& \\ldots& u_{kl}\\\\ \t\t\t\t\n\t\t\\end{pmatrix}\\circ \\begin{pmatrix}\n\t\t\t\t\t\t\t\tv_{kl}& v_{kl-1}& \\ldots& \tv_{k0}\\\\ \n\t\t\t\t\t\t\t\t\\vdots&\t \\vdots& \\ldots& \\vdots\\\\\n\t\t\t\t\t\t\t\tv_{0l}& v_{0l-1}& \\ldots& v_{00}\t\t\t\n\t\t\t\t\t\t\t\\end{pmatrix}\n\t\t = \t\\begin{pmatrix}\n\t\t\t \tu_{00} v_{kl}& u_{01} v_{kl-1}& \\ldots& u_{0l}v_{k0}\\\\ \n\t\t\t \t\\vdots&\t \\vdots& \\ldots& \\vdots\\\\\n\t\t\t \tu_{k0}v_{0l}& u_{k1}v_{0l-1}& \\ldots& u_{kl}v_{00}\t\t\t\n\t\t \t\\end{pmatrix}.\n\t\\end{equation}\n\tHere the symbol $\\circ$ denotes the element-wise multiplication between two matrices.\n\t\\item[]\\textbf{Step-5} (Summation over matrix elements): In this step, we take summation over all the elements of the matrix $W[{0,1,\\ldots,k;0,1,\\ldots,l}]$ and this summation is\n\t\\begin{equation}\\label{Akl}\n\t\tA_{kl}=\\sum_{i=0}^{k}\\sum_{j=0}^{l}W_{ij} = u_{00} v_{kl}+ u_{01} v_{kl-1}+\\ldots+u_{kl}v_{00}.\n\t\\end{equation}\n\tHere $A_{kl}$ is the Adomian polynomial for the two matrix elements $u_{kl},v_{kl}$. In the Adomian polynomial $A_{kl}$, notably, the sum of the first index at subscripts of the components of $u,v$ in each term in $A_{kl}$ are same. Similarly, the sum of the second index of the components of $u,v$ in each term in $A_{kl}$ are also same (here for the first index, the sum is $k$ and for the second index, the sum is $l$), which obey the important property of the Adomian polynomial given in \\cite{Wazwaz}. \n\t\n\t\\item[]\\textbf{Step-6} (Constructing Adomian matrix): Repeating the previous steps from Step-1 to Step-5, the Adomian polynomials corresponding to each matrices elements in Eq. \\eqref{soluMatrix} are determined. All the calculated Adomian polynomials are stored in a matrix and can be expressed by\n\t\\begin{equation}\\label{adomianM}\n\t\tA=\n\t\t\\begin{pmatrix}\n\t\t\tA_{00}& A_{01}& \\ldots& A_{0l}& \\ldots& A_{0n}\\\\ \n\t\t\t\\vdots&\t \\vdots& \\ldots& \\vdots& \\ldots& \\vdots\\\\\n\t\t\tA_{k0}& A_{k1}& \\ldots& A_{kl}& \\ldots& A_{kn}\\\\ \n\t\t\t\\vdots&\t \\vdots& \\ldots& \\vdots& \\ldots& \\vdots\\\\\n\t\t\tA_{m0}& A_{m1}& \\ldots& A_{ml}& \\ldots& A_{mn}\n\t\t\\end{pmatrix}.\n\t\\end{equation}\n\tWe call the matrix $A$ in \\eqref{adomianM} as Adomian matrix for the given polynomial nonlinearity \\eqref{nlF}.\n\t\n\\end{itemize} \n \nWe present the pseudo-code for the algorithms described in Step-1 to Step-6 in Listing \\ref{alg1} which compute the Adomian matrix of Eq. \\eqref{nlF}.\nHere, it is worthwhile to note how a few simple matrix operations in Step-1 to Step-6 generate the Adomian polynomials of Eq. \\eqref{nlF}. \nIt is clear from Step-1 to Step-6 that only $4(m+1)(n+1)-(m+n+2)$ number of matrix operations ($2(m+1)(n+1)-(m+n+2)$ number of flippings, $(m+1)(n+1)$ number of element-wise matrices multiplications and $(m+1)(n+1)$ number of matrix summations) are required to compute the Adomian matrix of Eq. \\eqref{nlF} with $i=m,j=n$ in Eq. \\eqref{uvSeries}. \nThis simplicity in mathematical operations enhances the computing efficiency of this algorithm.\n\n\\begin{lstlisting}[numbers=left,keywordstyle=\\color{black}\\bfseries,\tkeywords={,input, output, function, for, to, do, end, return, },mathescape=true,caption={Computation of Adomian matrix $A$ of Eq. \\eqref{nlF} in pseudo-code.}, label={alg1}]\ninput: Functions $u$ and $v$ of Eq. $\\eqref{nlF}$\noutput: Adomian matrix $A$\nfunction AdomianMatrix($u,v$)\n\tExpress $u$ in matrix form $U$: $U$ $\\gets$ Matrix($\\sum_{i=0}^{m}\\sum_{j=0}^{n}u_{ij}$)\n\tExpress $v$ in matrix form $V$: $V$ $\\gets$ Matrix($\\sum_{i=0}^{m}\\sum_{j=0}^{n}v_{ij}$)\n\tfor $k\\gets m$ to $k\\geq 0$ do\n\t\tfor $l\\gets n$ to $l\\geq 0$ do\n\t\t\t$U[{0,1,\\ldots,k;0,1,\\ldots,l}]$ $\\gets$ the submatrix of $U$ $\\text{for}$ the elements $U_{kl}$\n\t\t\t$V[{0,1,\\ldots,k;0,1,\\ldots,l}]$ $\\gets$ the submatrix of $V$ $\\text{for}$ the elements $V_{kl}$\n\t\t\t$V[{k,k-1,\\ldots,0;l,l-1,\\ldots,0}]$ $\\gets$ $V[{0,1,\\ldots,k;0,1,\\ldots,l}]$ are flipped horizontally and then vertically\n\t\t\tElement-wise multiplication: $W[{0,1,\\ldots,k;0,1,\\ldots,l}]$ $\\gets$ $U[{0,1,\\ldots,k;0,1,\\ldots,l}]\\circ V[{k,k-1,\\ldots,0;l,l-1,\\ldots,0}]$\n\t\t\t$A_{kl}$ $\\gets$ $\\sum_{i=0}^{k}\\sum_{j=0}^{l}W_{ij}$\n\t\tend for\n\tend for\n\treturn A\nend function \n\\end{lstlisting}\n\\newpage\n\\begin{lstlisting}[numbers=left,keywordstyle=\\color{black}\\bfseries,\tkeywords={,input, output, function, for, to,do, end, return, },mathescape=true,caption={Computation of Adomian matrix $A$ of Eq. \\eqref{genF} in pseudo-code.}, label={alg2}]\ninput: Functions $u^{(1)},u^{(2)},u^{(3)},\\ldots, u^{(P-2)},u^{(P-1)},u^{(P)}$ of Eq. $\\eqref{genF}$\noutput: Adomian matrix $A$\nfunction AdomianMatrix2($u^{(1)},u^{(2)},u^{(3)},\\ldots, u^{(P-2)},u^{(P-1)},u^{(P)}$)\n\tExpress $u^{(1)},u^{(2)},u^{(3)},\\ldots, u^{(P-2)},u^{(P-1)},u^{(P)}$ in matrix forms: $U^{(P)}$ $\\gets$ Matrix($\\sum_{i=0}^{m}\\sum_{j=0}^{n}u_{ij}^{(P)}$)\n\t$A$ $\\gets$ $U^{(P)}$\n\tfor $k\\gets P$ to $k\\geq 2$ do\n\t\tA $\\gets$ AdomianMatrix($U^{(k-1)}$,$A$)\t\n\tend for\n\treturn A\nend function\t \n\\end{lstlisting}\n\n\n\\subsection{$F$ in general form}\nLet us now consider the nonlinear polynomial functional $F$ in the following general form\n\\begin{equation}\\label{genF}\n\tF=u^{(1)}u^{(2)}u^{(3)}\\ldots u^{(P-2)}u^{(P-1)}u^{(P)},\n\\end{equation} \nwhere $F$ depends on $P$ number of two-dimensional functions $u^{(1)},u^{(2)},u^{(3)},\\ldots ,u^{(P)}$. For $P=2$ and $u^{(1)}=u,u^{(2)}=v$, Eq. \\eqref{genF} is reduced to Eq. \\eqref{nlF}. The algorithms presented in the Step-1 to Step-6 also work for Eq. \\eqref{genF} in the following way.\nLet $U^{(1)},U^{(2)},U^{(3)},\\ldots ,U^{(P)}$ are the matrix forms of the two-dimensional functions $u^{(1)},u^{(2)},u^{(3)},\\ldots ,u^{(P)}$ respectively. \nIn order to determine the Adomian matrix of Eq. \\eqref{genF}, at first, we will start to determine the Adomian matrix for the first two matrices $U^{(1)},U^{(2)}$ or for the last two matrices $U^{(P-1)},U^{(P)}$ using the algorithms presented in the Step-1 to Step-6.\nLet $A^{(P-1)(P)}$ is the Adomian matrix of the last two matrices $U^{(P-1)}$ and $U^{(P)}$. Next, we determine the Adomian matrix of the two matrices $A^{(P-1)(P)}$ and the previous one $U^{(P-2)}$. This process is continued up to first matrices $U^{(1)}$. After completing this process, finally, we will get the Adomian matrix of $F$ given in Eq. \\eqref{genF}. We present this process in pseudo-code in Listing \\ref{alg2} which determines the Adomian matrix of Eq. \\eqref{genF}.\n\nNow, we consider the nonlinear polynomial functional $F$ in the more general and complicated form (a sum raised to a power)\n\\begin{equation}\\label{FpN}\n\tF=\\left(u^{(1)}+u^{(2)}+u^{(3)}+\\ldots+ u^{(P-2)}+u^{(P-1)}+u^{(P)}\\right)^\\mathcal{N}\n\\end{equation}\nwhere the power index $\\mathcal{N}$ is a positive integer number. In this case, at first, we expand Eq. \\eqref{FpN} in sum of product terms. Then we can easily determine the Adomian matrix of each term of the expansion using the above algorithms for Eq. \\eqref{genF}. Finally, simply adding all the Adomian matrices of each term, we get the Adomian matrix of Eq. \\eqref{FpN}. \n\nIn a one-dimensional case, the series \\eqref{uvSeries} have only one index (say $i$). Therefore, all the matrices are one dimension, and in this case, in Step-3, we have to perform only a horizontal flipping operation. Besides this, all the algorithms described from Step-1 to Step-6 are identical in a one-dimensional case. In the following, we call the new algorithm presented by us the Adomian matrix algorithm. \n\n\\section{Software implementation and comparisons with other algorithms}\\label{comp}\nWe have implemented the algorithm described in Sec. \\ref{algo} (called Adomian matrix algorithm) into MATHEMATICA code in Listings \\ref{onedOur} (one-dimensional case), \\ref{twodOur} (two-dimensional case) of Appendix: \\ref{oned}, \\ref{twod} respectively.\nThese MATHEMATICA programs can determine one-dimensional (using Listing \\ref{onedOur}) and two-dimensional (using Listing \\ref{twodOur}) Adomian polynomials of the following polynomial functional\n\\begin{equation}\\label{FN}\n\tF=u^\\mathcal{N},\n\\end{equation}\nwhere the power index $\\mathcal{N}$ is an positive integer number that represents the order of nonlinearity. To determine the Adomian polynomials of Eq. \\eqref{FN}, we have to input the power index $\\mathcal{N}$ and the order of the Adomian matrix in the function arguments (detailed descriptions of these function arguments are given in the Appendix) of the MATHEMATICA functions, and these functions print the Adomian polynomials in the output cell of the MATHEMATICA notebook. \n\n MATHEMATICA codes for some other algorithms such as Duan\u2019s Corollary 1 algorithm \\cite{Duan}, Duan\u2019s Corollary 3 algorithm \\cite{Duan1} for one-dimensional case are also presented in Listings \\ref{onedDuan}, \\ref{onedDuan1} of Appendix: \\ref{twod}.\n The MATHEMATICA programs in Listings \\ref{onedDuan} and Listings \\ref{onedDuan1} are taken from Appendix: A.1 in \\cite{Duan} and from Appendix: A in \\cite{Duan2} respectively. Here to make the programs more faster we have modified the programs (given in \\cite{Duan}, \\cite{Duan2}) which work only with the polynomial functional \\eqref{FN} and evaluate the differentiation of \\eqref{FN} using the factorial formula $\\frac{d^iF}{du^i}=\\frac{\\mathcal{N}!}{(\\mathcal{N}-i)!}u^{\\mathcal{N}-i}$.\n \n We have compared the Adomian matrix algorithm with other algorithms by employing the MATHEMATICA programs given in Listings \\ref{onedOur}, \\ref{onedDuan}, \\ref{onedDuan1}, \\ref{twodOur} and using the polynomial functional \\eqref{FN}.\nIn Table \\ref{Tab:comp}, we have shown the comparisons between the computing speeds (measured in seconds) of the Adomian matrix algorithm (3rd column) and two different other algorithms (4th and 5th columns) for the one-dimensional case using the MATHEMATICA programs given in Listings \\ref{onedOur}, \\ref{onedDuan}, \\ref{onedDuan1} in Appendix: \\ref{oned}. We measure the computing times by MATHEMATICA\n9.0 on the laptop with Intel(R) Core(TM) i5-7200U CPU $@$ 2.50 GHz and 8 GB RAM, using the MATHEMATICA command \\verb+Timing[]+ with suppressing output (i.e., the results are retained in memory). \nTable \\ref{Tab:comp} displays that the Adomian matrix algorithm is faster and more efficient than the other two algorithms: Duan\u2019s Corollary 1 algorithm \\cite{Duan} and Duan\u2019s Corollary 3 algorithms \\cite{Duan1}.\nFor example, we observe that in calculating the first $50$ Adomian polynomials, the Adomian matrix algorithm is almost $10^4$ times faster for $\\mathcal{N}=3$ and almost $10^3$ times faster for $\\mathcal{N}=10$ in comparison to the other two algorithms. Moreover, in calculating the first $100$ Adomian polynomials, the Adomian matrix algorithm spends the time $\\sim 10^{-2}$ s, but, notably, the other two algorithms are unable to give results within an elapsed time of $600$ s.\n\nWe have also checked the computation efficiency of the Adomian matrix algorithm in the two-dimensional cases using the MATHEMATICA code in Listing \\ref{twodOur}. For example, the Adomian polynomials of Eq. \\eqref{FN} in the order of $40\\times40$ are generated within $2.6$ s for $\\mathcal{N}=3$ and within $19.5$ s for $\\mathcal{N}=10$.\n\n\\begin{table}[H]\n\t\t\\centering\n\t\t\\caption{Comparisons of computing times (unit: seconds) of the Adomian matrix algorithm with some other algorithms using different values of $\\mathcal{N}$ in \\eqref{FN} and the different number $(n)$ of Adomian polynomials in one dimension. In some table cells, $\\boldsymbol{\\times}$ symbols indicate the algorithm in the corresponding column is unable to compute Adomian polynomials after spending almost $600$ s. }\n\t\t\t% We use >{\\centering\\arraybackslash} for centering text in cell with p{}\n\t\t\\begin{tabularx}{14.2cm}{|>{\\centering\\arraybackslash}p{2.1cm}|>{\\centering\\arraybackslash}p{2.9cm}|>{\\centering\\arraybackslash}p{2.4cm}|>{\\centering\\arraybackslash}p{3cm}|Y|}\n\t\t\t\\hline\n\t\t\t Nonlinearity \\newline\n\t\t\t $(\\mathcal{N})$& Number of Adomian\\newline polynomials $(n)$& Adomian matrix\\newline algorithm& Duan\u2019s Corollary 1\\newline algorithm \\cite{Duan}& Duan\u2019s Corollary 3\\newline algorithm \\cite{Duan1} \\\\\n\t\t\t\\hline\n\t\t\t\\multirow{4}{*}{3}& 10& 0.00047& 0.0020& 0.0025\\\\\n\t\t\t\\cline{2-5}\n\t\t\t& 30& 0.002& 0.83& 0.76\\\\\n\t\t\t\\cline{2-5}\n\t\t\t& 50& 0.0047& 62& 46\\\\\n\t\t\t\\cline{2-5}\n\t\t\t& 100& 0.017& $\\boldsymbol{\\times}$& $\\boldsymbol{\\times}$\\\\\n\t\t\t\\hline\n\t\t\t\\multirow{4}{*}{5}& 10& 0.00078& 0.0026& 0.0025\\\\\n\t\t\t\\cline{2-5}\n\t\t\t& 30& 0.0039& 0.87& 0.68\\\\\n\t\t\t\\cline{2-5}\n\t\t\t& 50& 0.0092& 62.5& 46.4\\\\\n\t\t\t\\cline{2-5}\n\t\t\t& 100& 0.037& $\\boldsymbol{\\times}$& $\\boldsymbol{\\times}$\\\\\n\t\t\t\\hline\n\t\t\t\\multirow{4}{*}{10}& 10& 0.0033& 0.0037& 0.0029\\\\\n\t\t\t\\cline{2-5}\n\t\t\t& 30& 0.012& 0.96& 0.65\\\\\n\t\t\t\\cline{2-5}\n\t\t\t& 50& 0.026& 62.7& 46.7\\\\\n\t\t\t\\cline{2-5}\n\t\t\t& 100& 0.095& $\\boldsymbol{\\times}$& $\\boldsymbol{\\times}$\\\\\n\t\t\t\\hline\n\t\t\\end{tabularx}\n\t\\label{Tab:comp}\n\t\\end{table}\n\n\n\n\t\n\t\\section{Conclusion}\\label{con}\n\tWe have presented a new algorithm (called the Adomian matrix algorithm) to determine the Adomian polynomials for scalar-valued nonlinear polynomial functional (with index as positive integers) in a Hilbert space $H$. The computations in the Adomian matrix algorithm do not need complicated mathematical operations such as parametrization, expansion, regrouping, differentiation, and so on. It is clear from Step-1 to Step-6 in Sec. \\ref{algo} that the Adomian polynomials are determined entirely by some simple matrix operations. Because of the simplicity in mathematical operations, the algorithm is more efficient for the fast generation of the Adomian polynomials. We have designed two MATHEMATICA programs (one-dimensional case in Listing \\ref{onedOur} and two-dimensional case in Listing \\ref{twodOur}) based on the Adomian matrix algorithm, and compared its efficiency in computations for the one-dimensional cases with other two popular and powerful algorithms, which are Duan\u2019s Corollary 1 algorithm \\cite{Duan} and Duan\u2019s Corollary 3 algorithms \\cite{Duan1}. We have observed that the computation efficiency of the Adomian matrix algorithm is better than the other two algorithms. For example, in calculating the first $50$ Adomian polynomials in one dimension with the nonlinearity index $\\mathcal{N}=3$ in Eq. \\eqref{FN}, the Adomian matrix algorithm is almost $10^4$ times faster than the other two algorithms. \n\tFor $\\mathcal{N}=10$, we are able to find the first $100$ Adomian polynomials using this new algorithm in just $10^{-2}$ s, whereas for $\\mathcal{N}=3$ and $n=100$, the other two algorithms fail to produce any results until $600$ s have passed.\n\tTherefore, we can conclude that the Adomian matrix algorithm can be used to determine a large number of Adomian polynomials of nonlinear polynomial functionals that make the solutions more accurate. \n\t\n\t\\newpage\n\t\\appendix\n\t\n\t\\section{Mathematica programs for one-dimensional case }\\label{oned}\n\tThe following three MATHEMATICA programs can determine one-dimensional Adomian polynomials of the nonlinear function \\eqref{FN}. The function arguments N\\_ and n\\_ represent the nonlinear power index $\\mathcal{N}$ in Eq. \\eqref{FN} and the number of first Adomian polynomials, respectively.\n\t%\\subsection{Program based on the Adomian matrix algorithm}\\label{onedOur}\n\\begin{lstlisting}[language=Mathematica,keywordstyle=\\color{blue}\\bf,caption={Program based on the Adomian matrix algorithm.}, label={onedOur}]\n AdomMatAlgo1D[N_, n_] := Module[{h, j, k},\n u =.;\n mat = Table[Subscript[u, h], {h, 0, n - 1}];\n temmat = Table[Subscript[u, h], {h, 0, n - 1}];\n For[j = 1, j <= N - 1, j++,\n For[k = n, k >= 1, k--,\n mat[[k]] = Total[temmat[[;; k]]*Reverse[mat[[;; k]]]];\n ];\n ];\n mat\n ]\n\\end{lstlisting}\n\t\n\t%\\subsection{Program based on Duan's Corollary 1 algorithm \\cite{Duan}}\\label{onedDuan}\n\\begin{lstlisting}[language=Mathematica,keywordstyle=\\color{blue}\\bf,caption={Program based on the Duan's Corollary 1 algorithm \\cite{Duan}.}, label={onedDuan}]\n\tDuanIndexAlgoAdom[N_, n_] := Module[{Apoly, Zpoly, dirClt}, \n\tSubscript[Apoly, 0] = Subscript[u, 0]^N;\n\tZpoly = Table[0, {i, 1, n - 1}, {j, 1, i}];\n\tDo[Zpoly[[suInd, 1]] = Subscript[u, suInd], {suInd, 1, n - 1}];\n\tFor[i = 2, i <= n - 1, i++, \n\t For[j = 2, j <= i, j++, \n\t Zpoly[[i, j]] = Expand[Subscript[u, 1]*Zpoly[[i - 1, j - 1]]];\n\t If[Head[Zpoly[[i, j]]] === Plus, \n\t Zpoly[[i, j]] = Map[#\/Exponent[#, Subscript[u, 1]] &, Zpoly[[i, j]]], \n\t Zpoly[[i, j]] = Map[#\/Exponent[#, Subscript[u, 1]] &, Zpoly[[i, j]], {0}]]];\n\t For[j = 2, j <= Floor[i\/2], j++, \n\t Zpoly[[i, j]] = Zpoly[[i, j]] + (Zpoly[[i - j, j]] \/. \n\t Subscript[u, sub_] -> Subscript[u, sub + 1])]];\n\tdirClt = Table[Factorial[N]\/Factorial[N - j]*(Subscript[u, 0]^(N - j)), {j, 1, n - 1}];\n\tDo[Subscript[Apoly, suInd] = Take[dirClt, suInd].Zpoly[[suInd]], {suInd, 1, n - 1}];\n\tTable[Subscript[Apoly, suInd], {suInd, 0, n - 1}]]\n\\end{lstlisting}\n\t\n\t%\\subsection{Program based on Duan's Corollary 3 algorithm \\cite{Duan1,Duan2}}\\label{onedDuan1}\n\\begin{lstlisting}[language=Mathematica,keywordstyle=\\color{blue}\\bf,caption={Program based on the Duan's Corollary 3 algorithm \\cite{Duan1,Duan2}.}, label={onedDuan1}]\n DuanCoro3AlgoAdm[N_, n_] := Module[{cPoly, i, k, j, derClt}, \n Table[cPoly[i, k], {i, 1, n - 1}, {k, 1, i}];\n derClt = Table[Factorial[N]\/Factorial[N - k]*(Subscript[u, 0]^(N - k)), \n {k,1, n - 1}];\n Apoly[0] = Subscript[u, 0]^N;\n For[i = 1, i <= n - 1, i++, \n cPoly[i, 1] = Subscript[u, i];\n For[k = 2, k <= i, k++, \n cPoly[i, k] = Expand[1\/i*Sum[(j + 1)*Subscript[u, j + 1]*cPoly[i - 1 - j, k - 1], \n {j, 0,i - k}]]];\n Apoly[i] = Take[derClt, i].Table[cPoly[i, k], {k, 1, i}]];\n Table[Apoly[i], {i, 0, n - 1}]]\n\\end{lstlisting}\n\t\n\t\\section{Mathematica programs for two-dimensional case }\\label{twod}\n\t The following MATHEMATICA program can determine two-dimensional Adomian polynomials of the nonlinear function \\eqref{FN}.\n\tThe function arguments N\\_, m\\_ and n\\_ represent the nonlinear power index $\\mathcal{N}$ in Eq. \\eqref{FN}, the number of rows and number of columns in the Adomian matrix, respectively.\n\t%\\subsection{Program based on the Adomian matrix algorithm}\\label{twodOur}\n\t\n\t\\begin{lstlisting}[language=Mathematica,keywordstyle=\\color{blue}\\bf,caption={Program based on the Adomian matrix algorithm.}, label={twodOur}]\n\t AdomMatAlgo2D[N_, m_, n_] := Module[{g, h, j, k, l},\n\t u =.;\n\t mat = Table[Subscript[u, g, h], {g, 0, m - 1}, {h, 0, n - 1}];\n\t temmat = Table[Subscript[u, g, h], {g, 0, m - 1}, {h, 0, n - 1}];\n\t For[j = 1, j <= N - 1, j++,\n\t For[k = m, k >= 1, k--,\n\t For[l = n, l >= 1, l--,\n\t mat[[k, l]] = Total[temmat[[;; k, ;; l]]*Reverse[Reverse[mat[[;; k, ;; l]], 1], 2], 2];\n\t ];\n\t ];\n\t ];\n\t mat\n\t ]\n\t\\end{lstlisting}\n\n\t\n\t\n\t\\begin{thebibliography}{00}\n\t\t\\bibitem{adm} G. 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Wazwaz, A reliable algorithm for positive solutions of nonlinear boundary value problems by the multistage Adomian decomposition method, Open Engineering, 5 (2014) 59\u201374.\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\n\t\\end{thebibliography}\n\\end{document}"},{"text":"\\documentclass[12pt]{article}\n\n% FORMATTING ----------------------------------------------------\n\\setlength{\\textwidth}{6in} % Change to 1-inch margins.\n\\setlength{\\textheight}{9in}\n\n% For page width\n\\oddsidemargin=0.0in \\evensidemargin=0in \\textwidth=6.5in\n\n% For paper height\n\\topmargin=-0.5in \\textheight=9.0in\n\n\\setlength{\\evensidemargin}{0in} \\setlength{\\oddsidemargin}{0in} \\setlength{\\topmargin}{-.5in}\n\n\\usepackage{amssymb,graphicx,amsthm}\n\\usepackage{amsmath}\n\\usepackage{fancyvrb}\n\\usepackage{tikz}\n\n\n\\def \\bull{\\vrule height 1.5ex width 1.4ex depth -.1ex} %square bullet\n\\def\\A{{\\cal A}}\n\\def\\Y{{\\cal Y}}\n\\def\\I{{\\cal I}}\n\\def\\U{{\\cal U}}\n\\def\\S{{\\cal S}}\n\\def\\L{{\\cal L}}\n\\def\\B{{\\cal B}}\n\\def\\C{{\\cal C}}\n\\def\\P{{\\cal P}}\n\\def\\O{{\\cal O}}\n\\def\\PG{{\\cal PG}}\n\\def\\bt{\\beta}\n\\def\\bv{{\\bf v}}\n\\def\\bw{{\\bf w}}\n\\def\\bx{{\\bf x}}\n\\def\\by{{\\bf y}}\n\\def\\bz{{\\bf 0}}\n\\def\\bzd{{\\bf z}}\n\\def\\om{\\omega}\n\\def\\U{{\\cal U}}\n\\def\\C{{\\cal C}}\n\\def\\S{{\\cal S}}\n\\newcommand{\\LHD}{\\mathbin{< \\hbox to -.415em{}\\hbox{\\vrule\n height .51em depth .04em}\\hbox to .2em{}}}\n\\newcommand{\\leftemptybowtie}{{>\\mkern-9mu\\LHD}}\n\\def\\sq{{\\hfill\\Box}}\n\\def\\nsq{{\\hfill\\not\\!\\!\\Box}}\n\\counterwithin{figure}{section}\n\n\\newtheorem{thm}{Theorem}[section]\n\\newtheorem{lemma}[thm]{Lemma}\n\\newtheorem{cor}[thm]{Corollary}\n\\newtheorem{example}[thm]{Example}\n\\newtheorem{prop}[thm]{Proposition}\n\\newtheorem{conj}[thm]{Conjecture}\n\\renewcommand{\\baselinestretch}{1.2}\n\n\\begin{document}\n\n\\title{{\\bf On the minimum blocking semioval in $PG(2,11)$}}\n\n\\author{Jeremy M. Dover}\n\n\\maketitle\n\n\\begin{abstract}\nA blocking semioval is a set of points in a projective plane that is both a blocking set (i.e., every line meets the set, but the set contains no line) and a semioval (i.e., there is a unique tangent line at each point). The smallest size of a blocking semioval is known for all finite projective planes of order less than 11; we investigate the situation in $PG(2,11)$.\n\\end{abstract}\n\n\\section{Introduction}\nIn a projective plane $\\pi$, a {\\bf semioval} is a set of points $S$ such that there is a unique tangent line (i.e., line with one point of contact) at each point. The size of a semioval $\\S$ in $PG(2,q)$ is known to satisfy $q+1 \\le |\\S| \\le q\\sqrt{q}+1$. As defined by Kiss~\\cite{kiss}, a semioval $\\S$ is {\\em small} if it satisfies $|\\S| \\le 3q+3$. Several articles, including Kiss, et al.~\\cite{kmp} and Bartoli~\\cite{bartoli}, investigate the spectrum of sizes for small semiovals in Desarguesian planes.\n\nA set of points $S$ in $\\pi$ is called a {\\bf blocking set} if every line of $\\pi$ meets $S$ in at least one point, but $S$ contains no line. A set of points in $\\pi$ is called a {\\bf blocking semioval} if it is both a blocking set and a semioval. The smallest possible size of a blocking semioval is known for all planes of order less than 11. In the smallest cases, the author~\\cite{dover1} showed there are no blocking semiovals in $PG(2,2)$, and the smallest possible sizes in $PG(2,q)$ for $q=3,4,5,7$ are $6, 9, 11, {\\rm and}\\; 16$, respectively. In $PG(2,8)$ the author~\\cite{dover2} exhibited a blocking semioval of size 19, and showed that no smaller blocking semiovals can exist.\n\nRegarding the planes of order 9, the author~\\cite{dover2} showed that no plane of order 9 can have a blocking semioval of size 20 or smaller. Nakagawa and Suetake~\\cite{suetake3} had previously shown the existence of blocking semiovals of size 21 in the three non-Desarguesian planes of order 9, and Dover, Mellinger and Wantz~\\cite{dmw} exhibited a blocking semioval of size 21 in $PG(2,9)$.\n\nIn this article, we continue this program by studying the smallest blocking semioval in $PG(2,11)$.\n\n\\section{Some introductory analysis}\nTo begin scoping the problem, we first appeal to the two best known lower bounds for the size of blocking semiovals. For any projective plane of order $q$, the author~\\cite{dover2} shows that a blocking semioval must have at least $2q + \\sqrt{2q-\\frac{47}{4}}-\\frac12$ points. Specializing to the $q=11$ case, we calculate that a blocking semioval in $PG(2,11)$ must have at least 25 points. Since $PG(2,11)$ is Desarguesian, the lower bound of H\\'{e}ger and Tak\\'{a}ts~\\cite[Corollary 33]{hegertakats} also applies, which states that a blocking semioval in $PG(2,q)$ must have at least $\\frac94q-3$ points. However, this result provides a bound less than 25, from which we conclude that the smallest possible size for a blocking semioval in $PG(2,11)$ is 25 points.\n\nLet $\\S$ be a putative blocking semioval of size 25 in $PG(2,11)$. The author~\\cite{dover3} (Proposition 2.1 and Theorem 2.2) shows that $\\S$ cannot fully contain a line, nor can it have an $11$-secant. We then apply Theorem 3.1 from Dover~\\cite{dover1}, which states that if $\\S$ has an (11-$k$)-secant for $1\\leq k < 10$, then $\\S$ has at least $11 \\frac{3k+4}{k+2} - k$ points. Evaluating all of these bounds as $k$ varies from 1 to 10, we find that $\\S$ cannot have any 7-secants, 8-secants or 9-secants.\n\nOne possibility is that $\\S$ could have a 10-secant, in which case Theorem 4.2 in~\\cite{dover3} shows that there is only one 10-secant. On the other hand, $\\S$ could have no lines meeting it in more than 6 points. We deal with these cases separately.\n\n\\section{The 10-secant case}\nSuppose that a blocking semioval $\\S$ of size 25 in $PG(2,11)$ has a 10-secant $\\ell_{10}$. Let $Q$ and $R$ be the two points on $\\ell_{10}$ that are not in $\\S$. If $P$ is a point of $\\S$ not on $\\ell_{10}$, then the tangent line to $\\S$ through $P$ must pass through either $Q$ or $R$; let $\\S_Q$ (resp. $\\S_R$) be the set of points in $\\S$ not on $\\ell_{10}$ whose tangent passes through $Q$ (resp. $R$). If we denote the set of points in $\\S$ not on $\\ell_{10}$ as $\\S'$, we have $\\S' = \\S_Q \\dot\\cup \\S_R$, so we must have $|\\S_Q| + |\\S_R| = 15$.\n\nSince $\\S$ is a blocking set, every line through $Q$ must meet $\\S$ in at least one point. One of these is $\\ell_{10}$, but the remaining 11 lines through $Q$ must be covered by the 15 points of $\\S'$. The lines through $Q$ and points of $\\S_Q$ are tangents to $\\S$ and thus contain only a single point of $\\S_Q$, and no points of $\\S_R$; hence there are $|\\S_Q|$ of these. A line $m$ through $Q$ containing a point $X$ of $\\S_R$ cannot be a tangent to $\\S$, since the tangent to $\\S$ at $X$ meets $\\ell_{10}$ in $R$. Hence $m$ contains at least 2 points of $\\S$, but these points cannot be on $\\ell_{10}$, and by the above cannot be in $\\S_Q$, so $m$ must contain at least 2 points of $\\S_R$.\n\nSince every line through $Q$ is covered by $\\S$, we have:\n$$ 12 \\leq 1 + |\\S_Q| + \\frac12|\\S_R|$$\n\nUsing the fact that $|\\S_Q| = 15 - |\\S_R|$, we can substitute in this inequality to obtain $|\\S_Q| \\geq 7$. However, the exact same argument holds for $R$, showing $|\\S_R| \\geq 7$, from which we conclude that one of $|\\S_Q|$ and $|\\S_R|$ is 7 and the other is 8. Since our choice of labelling $Q$ and $R$ is arbitrary, we assume without loss of generality that $|\\S_Q| = 8$ and $|\\S_R| = 7$.\n\nSince $|\\S_Q|$ contains 8 points, eight lines through $Q$ are tangents to $\\S$, and one is $\\ell_{10}$, so the remaining three lines through $Q$ must meet $\\S_R$ in at least 2 points each. As $|\\S_R| = 7$, this implies that these remaining 3 lines consist of two 2-secants and a 3-secant to $\\S$. Similar analysis shows that $R$ lies on $\\ell_{10}$, seven tangents and four 2-secants to $\\S$.\n\nWith a structure this well-defined, we have a reasonable hope that a computer search can identify whether or not a blocking semioval of this form exists in $PG(2,11)$. We develop a search strategy by picking items in a potential blocking semioval in order, using Magma~\\cite{magma} at each stage to determine the number of projectively inequivalent options we have that respect previous choices.\n\nWe begin by coordinatizing $PG(2,11)$ with homogeneous coordinates such that $Q = (1,0,0)$ and $R = (0,1,0)$. The automorphism group leaving $Q$ and $R$ invariant is transitive on lines through $Q$ other than $\\ell_{10}$, thus we may assume that the 3-secant to $\\S$ through $Q$ is $[0,1,0]$.\n\n\\textbf{Pick two other lines through $Q$ to be 2-secants to $\\S$.} We use Magma to compute the group of automorphisms in $PG(2,11)$ that leave $Q$ and $R$ invariant, as well as the line $[0,1,0]$, and calculate the orbits of pairs of lines through $Q$. As a result of this calculation, we find that we may always take one of our 2-secants through $Q$ to be $[0,1,1]$, and then the other must be one of $[0,1,2],[0,1,3],[0,1,5],[0,1,7],[0,1,10]$, giving 5 possible options for this pair of lines.\n\n\\textbf{Pick three points on the 3-secant to be in $\\S_R$.} For each of the five selections of non-tangent lines through $Q$ above, we again use Magma to calculate the group of automorphisms that leave each of $Q$, $R$, and the three non-tangent lines through $Q$ fixed. This group is doubly-transitive on the points of the 3-secant through $Q$ distinct from $Q$, allowing us to assume that $(0,0,1)$ and $(1,0,1)$ are in $\\S$. However, this is the extent to which the automorphism group can help us, and we must consider each of the other 9 points on this line as candidates to be in $\\S$, leaving 9 options for this point set.\n\n\\textbf{Pick two points on $[0,1,1]$ to be in $\\S_R$.} We have already picked three points on the 3-secant to be in $\\S_R$, and the lines joining each of these points to $R$ cannot contain any other point of $\\S$. This eliminates three points of $[0,1,1]$ from consideration, as well as $Q$, so there are ${8 \\choose 2} = 28$ candidate point pairs.\n\n\\textbf{Pick two points on the other 2-secant through $Q$ to be in $\\S_R$.} As before, we may not pick points of this line that already lie on lines connecting points of $\\S_R$ and $R$, which eliminates 6 possible points; there can be no overlap since all of these lines intersect at $R$. Hence there are ${6 \\choose 2} = 15$ candidate point pairs.\n\n\\textbf{Pick the points of $\\S_Q$.} There will be four remaining lines through $R$ which do not pass through a point of $\\S_R$. Order them arbitrarily. On the first, pick two points to be in $\\S_Q$, noting that the line between either of these points and $Q$ must be a tangent. As above, this choice affects the number of candidate points for each of the remaining 2-secants through $R$. These point selections can be made in ${8 \\choose 2} {6 \\choose 2} {4 \\choose 2} = 2520$ ways.\n\nThere are $47,638,000 \\approx 2^{25.5}$ possible configurations, which is a managable number for computer search. For each of these configurations, we must check if the resulting set, when added to the ten points on $\\ell_{10}$ distinct from $Q$ and $R$, forms a blocking semioval. However, the way we've constructed the set ensures that it is a blocking set, and that every point of $\\S'$ has a unique tangent line; all that remains to check at this stage is whether there is a unique tangent line at each of the points of $\\ell_{10}$ distinct from $Q$ and $R$, or equivalently if each of the points of $\\ell_{10}$ distinct from $Q$ and $R$ lies on exactly two lines not blocked by $\\S'$, as one of those two will be $\\ell_{10}$.\n\nWe executed this search in Magma, and we found no blocking semioval of this form. Hence we conclude that there is no blocking semioval of size 25 in $PG(2,11)$ with a 10-secant.\n\n\\section{Constraint Programming}\n\\label{sat_solver}\nIn order to confirm the results of the previous section, and to set up for further searching, we have developed a constraint programming model to search for blocking semiovals, implemented using Google's OR-Tools~\\cite{ortools} Boolean satisfiability (SAT) solver, using Python. For each point $(x,y,z)$, we create a Boolean variable $p\\_x\\_y\\_z$ which is true if $(x,y,z)$ is in our blocking semioval and false otherwise. Mechanically, we also need to create a 0\/1-valued integer variable tied to the value of the Boolean (0 for false, 1 for true) to calculate line intersection sizes with the blocking semioval, but for purposes of the model description we will conflate these two variables. Similarly, for each line $[a,b,c]$ we create a Boolean variable $\\ell\\_a\\_b\\_c$ which is true if and only if the line is tangent to our blocking semioval.\n\nGiven these variable definitions, every blocking semioval corresponds to a solution of the constraint program defined with only three types of constraints:\n\\begin{eqnarray*}\n\\sum_{(x,y,z) \\in [a,b,c]} p\\_x\\_y\\_z &=& 1\\; {\\rm if} \\; \\ell\\_a\\_b\\_c\\\\\n\\sum_{(x,y,z) \\in [a,b,c]} p\\_x\\_y\\_z &>& 1\\; {\\rm if} \\; \\neg\\ell\\_a\\_b\\_c\\\\\n\\sum_{[a,b,c] \\ni (x,y,z)} \\ell\\_a\\_b\\_c &=& 1\\; {\\rm if} \\; p\\_x\\_y\\_z\\\\\n\\end{eqnarray*}\nThe first equation type asserts that if $\\ell$ is a tangent, then only one of its points lies on the blocking semioval. The second equation type asserts that all non-tangent lines meet the blocking semioval in more than one point. The final equation type asserts that for each point of the blocking semioval, only one of the lines through it is a tangent. Applying these conditions across all points and lines in the plane, we see that any solution of this constraint program is a blocking semioval, and vice versa.\n\nThe SAT problem is known to be NP-complete in general, but there are numerous algorithms and implementations which can solve many specific instances. In particular, we have modeled the specific search of the last section using constraint programming by adding the additional conditions derived to our generic model: that the total number of points must be 25, that certain points must be or not be on the blocking semioval, and that certain lines must meet the blocking semioval in 1, 2 or 3 points. By way of comparison, the Magma search conducted in the previous section took approximately 53 minutes to determine that there was no blocking semioval of the claimed form. Google's OR-Tools SAT solver reached the same conclusion in 47 seconds; excerpts of the code are given in Appendix A. Using the constraint programming approach will be particularly valuable for us in the next section.\n\n\\section{The 6-secant case}\nWe have concluded that if $\\S$ is a blocking semioval $PG(2,11)$ with 25 points, $\\S$ cannot have a 10-secant. Hence if $\\S$ exists, no line can meet $\\S$ in more than 6 points. Let $x_i$ denote the number of lines of $PG(2,11)$ meeting $\\S$ in exactly $i$ points. From~\\cite{dover2}, we have the following relations, specialized to $PG(2,11)$:\n\\begin{eqnarray}\nx_2 + x_5 + 3x_6 & = & 123 \\label{eq1}\\\\\nx_3 - 3x_5 - 8x_6 & = & -89 \\label{eq2}\\\\\nx_4 + 3x_5 + 6x_6 & = & 74 \\label{eq3}\n\\end{eqnarray}\n\nNote that Equation~\\ref{eq3}, when reduced modulo 3, shows that $x_4 \\equiv 2 \\pmod{3}$, and in particular must be at least 2. Hence $3x_5 + 6x_6 \\le 72$ by Equation~\\ref{eq3}. On the other hand Equation~\\ref{eq2} shows that $3x_5 + 8 x_6 \\ge 89$. Hence $2 x_6 \\ge 17$, from which we conclude $x_6 \\geq 9$. We proceed through with a series of small propositions that will gradually restrict the possible structure of $\\S$.\n\n\\begin{prop}\\label{prop1}\nLet $\\S$ be a blocking semioval in $PG(2,11)$ with 25 points and no 10-secant. Then no point of $\\S$ lies on more than three 6-secants to $\\S$, and at least four points of $\\S$ lie on exactly three 6-secants to $\\S$.\n\\end{prop}\n\n\\begin{proof}\nLet $\\L_6$ be the set of 6-secants to $\\S$, and define $y_i$ to be the number of points of $\\S$ lying on exactly $i$ 6-secants in $\\L_6$. First note that $y_i = 0$ for all $i \\ge 5$. Indeed if $P \\in \\S$ lies on five (or more) 6-secants, then each of these 6-secants contains 5 points of $\\S$ distinct from $P$, giving at least $25+1 = 26$ points on $\\S$.\n\nWe also note that $y_4 = 0$. If $P \\in \\S$ lies on four 6-secants, then these four 6-secants contain 20 points of $\\S$ distinct from $P$. One of the other lines through $P$ is a tangent to $\\S$ at $P$, leaving seven additional lines through $P$, each of which must contain at least one other point of $\\S$ distinct from $P$. This forces $\\S$ to have at least 28 points, again a contradiction. Hence no point of $\\S$ can lie on more than three 6-secants in $\\L_6$.\n\nCounting points of $\\S$ and flags of points of $\\S$ lying on lines of $\\L_6$, we obtain the following relations on the $y_i$:\n\\begin{eqnarray*}\ny_0 + y_1 + y_2 + y_3 & = & 25\\\\\ny_1 + 2y_2 + 3y_3 & = & 6x_6 \\ge 54\n\\end{eqnarray*}\n\nAs all of the $y_i$ are non-negative, we certainly have $50 + y_3 = 2y_0 + 2y_1 + 2y_2 + 2y_3 + y_3 \\ge y_1 + 2y_2 + 3y_3 \\ge 54$, from which we derive that $y_3 \\ge 4$.\n\\end{proof}\n\n\\begin{prop}\\label{prop2}\nLet $\\S$ be a blocking semioval in $PG(2,11)$ with 25 points and no 10-secant. Then there exists a pair of points $Q,R \\in \\S$ such that $Q$ and $R$ each lie on three 6-secants to $\\S$, and the line $\\overline{QR}$ is also a 6-secant to $\\S$.\n\\end{prop}\n\n\\begin{proof}\nLet $\\Y_3$ be the set of points in $\\S$ that lie on exactly three 6-secants. Let $P_1$, $P_2$, $P_3$, and $P_4$ be four distinct points of $\\Y_3$, which must exist by Proposition~\\ref{prop1}. If any pair of the $P_i$ lie together on a 6-secant to $\\S$, we are done, so assume otherwise. The three 6-secants through $P_1$ each contains 5 points of $\\S$ in addition to $P_1$, and we define $\\S_1$ to be this set of 15 points lying on a 6-secant through $P_1$, distinct from $P_1$. Note that by assumption none of $P_2$, $P_3$ or $P_4$ can be in $\\S_1$, as otherwise two points of $\\Y_3$ would lie together on a 6-secant.\n\nNow consider the three 6-secants through $P_2$. Again, none of these lines can contain any of $P_1$, $P_3$ or $P_4$. Moreover, each of these lines meets $\\S_1$ in at most three points, one from each of the 6-secants through $P_1$. Hence each 6-secant through $P_2$ contains at least two points distinct from the $P_i$ and $\\S_1$. But there are four $P_i$, and 15 points in $\\S_1$. Since $\\S$ only has 25 points, each 6-secant through $P_2$ must contain exactly two points distinct from the $P_i$ and $\\S_1$, and thus exactly three points of $\\S_1$. In particular, exactly nine points of $\\S_1$ lie on a 6-secant through $P_1$ and a 6-secant through $P_2$. Call these points $\\S_{12}$.\n\nThe same analysis holds for the three 6-secants through $P_3$. Hence there are exactly nine points of $\\S_1$ that lie on a 6-secant through $P_1$ and a 6-secant through $P_3$, which we denote $\\S_{13}$. Since $\\S_1$ only has 15 points, its 9-point subsets $\\S_{12}$ and $\\S_{13}$ must intersect in at least three points, and any point of that intersection is a point of $\\S$ that lies on exactly three 6-secants (it cannot lie on more, by Proposition~\\ref{prop1}) and lies on a 6-secant together with another point of $\\Y_3$. This completes the proof.\n\\end{proof}\n\n%We finish with a quick counting proposition.\n%\\begin{prop}\\label{prop3}\n%Let $\\S$ be a blocking semioval in $PG(2,11)$ with 25 points and no 10-secant. If $P \\in \\S$ lies on three 6-secants to $\\S$, then $P$ lies on one tangent, seven 2-secants, one 3-secant and %three 6-secants to $\\S$.\n%\\end{prop}\n\n%\\begin{proof}\n%Let $P$ be as in the proposition statement. Since $\\S$ is a blocking semioval, $P$ necessarily lies on a unique tangent line. The remaining 11 lines through $P$ must each contain at least two %points of $\\S$. By hypothesis, three of these are 6-secants, leaving eight lines through $P$, each of which contains at least one of the 9 remaining points of $\\S$. Hence seven of these %remaining lines must be 2-secants, and one must be a 3-secant.\n%\\end{proof}\n\nFrom Proposition~\\ref{prop2}, our blocking semioval $\\S$ must contain two points $Q$ and $R$ which each lie on three 6-secants, one of which is their common line $n$. Let $\\ell_1$ and $\\ell_2$ be the other two 6-secants through $Q$ and $m_1$ and $m_2$ be the other two 6-secants through $R$. Define $\\I$ to be the set of four points which are the intersections of $\\ell_i$ and $m_j$ for $i,j \\in \\{1,2\\}$. Figure~\\ref{fig1} provides a graphical depiction of these definitions.\n\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}\n\\draw[black,thick] (-6,2) -- (6,2) node[pos=1.05,text=black]{$n$};\n\\draw[black,thick] (-5.625,2.5) -- (1.25,-3) node[pos=1.05,text=black]{$\\ell_1$};\n\\draw[black,thick] (5.625,2.5) -- (-1.25,-3) node[pos=1.05,text=black]{$m_1$};\n\\draw[black,thick] (-5.5,2.5) -- (0.5,-3.5) node[pos=1.05,text=black]{$\\ell_2$};\n\\draw[black,thick] (5.5,2.5) -- (-0.5,-3.5) node[pos=1.05,text=black]{$m_2$};\n\\filldraw[black] (-5,2) circle (4pt) node[anchor=north east]{$Q$};\n\\filldraw[black] (5,2) circle (4pt) node[anchor=north west]{$R$};\n\\filldraw[black] (-3,2) circle (4pt);\n\\filldraw[black] (3,2) circle (4pt);\n\\filldraw[black] (-1,2) circle (4pt);\n\\filldraw[black] (1,2) circle (4pt);\n\\draw[black](0,-2.5) circle [radius=1] node[]{$\\I$};\n\\end{tikzpicture}\n\\caption{Initial structure of blocking semioval $\\S$ based on Proposition~\\ref{prop3}}\n\\label{fig1}\n\\end{figure}\n\n\\begin{prop}\\label{prop3}\nLet $\\S$ be a blocking semioval in $PG(2,11)$ with 25 points and no 10-secant. Define the points $Q$ and $R$, the lines $n$, $\\ell_1$, $\\ell_2$, $m_1$ and $m_2$, and the set $\\I$ as above. Then at least three points of $\\I$ must lie in $\\S$.\n\\end{prop}\n\n\\begin{proof}\nWe begin by counting the number of points in the subset $\\S_6$ of $\\S$ consisting of the points of $\\S$ that lie on our five noted 6-secants. There are six points on $n$, and then each of $\\ell_1$, $\\ell_2$, $m_1$ and $m_2$ contributes five additional points (not counting $Q$ and $R$), with the points of $\\I$ counted twice. Hence $\\S_6$ contains $26 - |\\I|$ points, showing immediately that $\\I$ contains at least one point of $\\S$.\n\nBut recall that $\\S$ must have at least nine 6-secants. Except for the five 6-secants defining $\\S_6$, no 6-secant can contain more than five points of $\\S_6$. So all of the additional 6-secants, of which there are at least four, must contain a point of $\\S$ that is not in $\\S_6$. However, they cannot all contain the same point off $\\S_6$, since by Proposition~\\ref{prop1} a point of $\\S$ can lie on at most three 6-secants. So there must be at least two points of $\\S$ not in $\\S_6$, forcing $\\S_6$ to contain at most 23 points, from which we conclude that at least three points of $\\I$ lie in $\\S$.\n\\end{proof}\n\nDefine $\\S^*$ to be the set of points in $\\S$ that lie on one or more of the lines $n$, $\\ell_1$, $\\ell_2$, $m_1$ and $m_2$. Notice that the configuration given in Figure~\\ref{fig1} shows five 6-secants to our putative blocking semioval $\\S$ (as well as $\\S^*$); but recall that $\\S$ must have at least nine 6-secants. As the points of $\\S^*$ are on the union of five lines, any 6-secant to $\\S$ not shown must contain at least one point not in $\\S^*$. On the other hand, $\\S^*$ contains either 22 or 23 of the points of the blocking semioval, depending on whether $\\I$ has 4 or 3 points in $\\S$, respectively. Let's address the latter case first.\n\n\\begin{prop}\\label{propI=3}\nLet $\\S$ be a blocking semioval in $PG(2,11)$ with 25 points and no 10-secant. Define the points $Q$ and $R$, the lines $n$, $\\ell_1$, $\\ell_2$, $m_1$ and $m_2$, and the set $\\I$ as above. If $\\I$ has three points in $\\S$, then there exists a point $P \\in \\S \\setminus \\S^*$ which lies on at least two 6-secants to $\\S$ such that neither of these 6-secants contains any point of $\\S \\setminus \\S^*$ except $P$.\n\\end{prop}\n\n\\begin{proof}\nSince $\\I$ has three points in $\\S$, $\\S^*$ contains 23 points of $\\S$, leaving two points of $\\S$ not in $\\S^*$, which we call $A$ and $B$. Since $\\S$ has at least nine 6-secants and only five of these contain points strictly within $\\S^*$, there must be at least four 6-secants to $\\S$ which contain at least one of $A$ or $B$. It is possible that the line containing $A$ and $B$ is a 6-secant to $\\S$, but this leaves three 6-secants to $\\S$ that contain exactly one of $A$ or $B$. The pigeonhole principle completes the proof, as one of these two points must lie on two such 6-secants.\n\\end{proof}\n\nThough a seemingly simple result, Proposition~\\ref{propI=3} is incredibly powerful from a computational perspective. In order to set up for a search, let us coordinatize $PG(2,11)$ such that the points of $\\I$ are $\\ell_2 \\cap m_2 = (1,0,0)$, $\\ell_2 \\cap m_1 = (0,1,0)$, $\\ell_1 \\cap m_2 = (0,0,1)$ and $\\ell_1 \\cap m_1 = (1,1,1)$, with $(1,1,1)$ not in $\\S$. This forces $Q=(1,1,0)$ and $R=(1,0,1)$ to also be in $\\S$. Any automorphism which fixes this configuration must leave $(1,1,1)$ and $(1,0,0)$ fixed, but it could permute $(0,1,0)$ and $(0,0,1)$, thus we do have a non-trivial automorphism group of order 2 which we can use to narrow the search space for our point $P \\in \\S$ whose existence is guaranteed by Proposition~\\ref{propI=3}. Using Magma, we find that there are 45 orbits of candidate points for $P$.\n\nOnce $P$ is chosen, we need to pick two lines through it to be 6-secants. But in order to be 6-secants, these lines must meet each of $n$, $\\ell_1$, $\\ell_2$, $m_1$ and $m_2$ in ${\\bf distinct}$ points, which must all be in $\\S$. Thus we need to pick two lines through $P$ out of 6 or 7 candidates (depending on whether or not $P$ lies on a line containing two points of $\\I$), which can be done in either 15 or 21 ways. Running out all the possibilities with Magma, we find that there are 760 possible configurations for $P$ and its two 6-secants. Combined with the 5 points we coordinatized above, we have 760 starting configurations, each of which provides 16 out of 25 points on a putative blocking semioval. This puts us in the realm of easy computation.\n\nFor each of these 760 starting configurations, we create a constraint programming model as in Section~\\ref{sat_solver}. Starting with the basic relations used to define a blocking semioval, we add the additional constraints:\n\\begin{enumerate}\n\\item the total number of points on the blocking semioval is 25;\n\\item the points $\\{(1,0,0),(0,1,0),(0,0,1),(1,1,0),(1,0,1)\\}$ are in the blocking semioval;\n\\item the point $(1,1,1)$ is not in the blocking semioval;\n\\item the point $P$ is in the blocking semioval; and\n\\item the points of intersection of the two lines chosen to be 6-secants to the blocking semioval with each of $\\ell_1$, $\\ell_2$, $m_1$, $m_2$ and $n$ are in the blocking semioval.\n\\end{enumerate}\n\nThe SAT solver runs through these cases at a rate of roughly two per second, and in each case determined that the model was infeasible, from which we can conclude that there is no blocking semioval of the form discussed here with just three points of $\\I$ in the blocking semioval. Now we follow a similar program to Proposition~\\ref{propI=3} for the case where all four points of $\\I$ lie in $\\S$.\n\n\\begin{prop}\\label{propI=4}\nLet $\\S$ be a blocking semioval in $PG(2,11)$ with 25 points and no 10-secant. Define the points $Q$ and $R$, the lines $n$, $\\ell_1$, $\\ell_2$, $m_1$ and $m_2$, and the set $\\I$ as above. If all four points of $\\I$ lie in $\\S$, then there exists a point $P \\in \\S \\setminus \\S^*$ which lies on at least two 6-secants to $\\S$ such that at least one of these 6-secants contains no point of $\\S \\setminus \\S^*$ except $P$.\n\\end{prop}\n\n\\begin{proof}\nSince all four points of $\\I$ lie in $\\S$, $\\S^*$ contains 22 points of $\\S$, leaving three points of $\\S$ not in $\\S^*$. As before $\\S$ has at least nine 6-secants and only five of these contain points strictly within $\\S^*$, so there must be at least four 6-secants to $\\S$ which contain at least one point not in $\\S^*$. Since there are only three points in $\\S \\setminus \\S^*$, at most three of our four 6-secants can contain more than one point of $\\S \\setminus \\S^*$, so there must exist a point $P \\in \\S \\setminus \\S^*$ which lies on a 6-secant that contains no other point of $\\S \\setminus \\S^*$.\n\nIf there is another 6-secant through $P$ we are done. If no other 6-secant passes through $P$, then the remaining three 6-secants must pass through either of $A$ or $B$, the two points of $\\S \\setminus \\S^*$ distinct from $P$. Again using the pigeonhole principle, at least one of these points (say $A$) must lie on at least two 6-secants, and at most one of those 6-secants can contain another point (necessarily $B$) of $\\S \\setminus \\S^*$. In this case $A$ lies on at least two 6-secants, at least one of which contains no other point of $\\S \\setminus \\S^*$, proving the claim.\n\\end{proof}\n\nWhile not as powerful as Proposition~\\ref{propI=3}, it gets the job done. We again coordinatize $PG(2,11)$ as above, but in this case all of $(1,0,0)$, $(0,1,0)$, $(0,0,1)$ and $(1,1,1)$ are in $\\S$, as well as $(1,1,0)$ and $(1,0,1)$. There is an automorphism group of order 8 that fixes $\\{(1,1,0),(1,0,1)\\}$ and $\\{(1,0,0),(0,1,0),(0,0,1),(1,1,1)\\}$ as sets, which allows us to narrow down the choice of our point $P$, whose existence is guaranteed by Proposition~\\ref{propI=4}, to one of 15 possible orbits. We then choose a line to be the 6-secant which contains no other point of $\\S \\setminus \\S^*$, which we can pick in 6 or 7 ways, as described above. However, our remaining 6-secant could be any other line through $P$ except $\\ell_{10}$, yielding 1,056 cases.\n\nFor each of these 1,056 starting configurations, we again create a constraint programming model. In addition to the basic relations used to define a blocking semioval, we add:\n\\begin{enumerate}\n\\item the total number of points on the blocking semioval is 25;\n\\item the points $\\{(1,0,0),(0,1,0),(0,0,1),(1,1,1),(1,1,0),(1,0,1)\\}$ are in the blocking semioval;\n\\item the point $P$ is in the blocking semioval;\n\\item the points of intersection of the first line chosen to be a 6-secant to the blocking semioval with each of $\\ell_1$, $\\ell_2$, $m_1$, $m_2$ and $n$ are in the blocking semioval; and\n\\item the second line chosen to be a 6-secant to the blocking semioval contains six points of the blocking semioval.\n\\end{enumerate}\n\nThe SAT solver resolves each of these case in approximately 4 seconds, and again determined that each case was infeasible. This allows us to state our main result:\n\n\\begin{thm}\nThere is no blocking semioval with exactly 25 points in $PG(2,11)$.\n\\end{thm}\n\n\\section{Conclusion}\n\nAs part of testing our constraint programming code to double check the 10-secant case, we commented out the constraint forcing the number of points in the blocking semioval to be 25, assuming the code would find a vertexless triangle or some other known blocking semioval. Instead, we found a blocking semioval in $PG(2,11)$ with 26 points, well smaller than the 29 points contained in the smallest previously known blocking semioval in that plane. The coordinates for the points of this blocking semioval are given in Table~\\ref{T2}.\n\\begin{table}[!h]\n\\centering\n\\caption{Points of a 26-point semioval in $PG(2,11)$}\n\\begin{tabular}{ c c c c c c c c c }\n$(1,1,0)$ & $(1,2,0)$ & $(1,3,0)$ & $(1,4,0)$ & $(1,5,0)$ & $(1,6,0)$ & $(1,7,0)$ & $(1,8,0)$ & $(1,9,0)$\\\\\n$(1,10,0)$ & $(0,0,1)$ & $(1,0,2)$ & $(1,0,6)$ & $(1,0,7)$ & $(1,0,8)$ & $(1,0,10)$ & $(1,1,3)$ & $(1,2,3)$\\\\\n$(1,3,5)$ & $(1,4,4)$ & $(1,5,9)$ & $(1,6,5)$ & $(1,7,1)$ & $(1,8,4)$ & $(1,9,1)$ & $(1,10,9)$\\\\\n\\end{tabular}\n\\label{T2}\n\\end{table}\n\n\nThe stabilizer $G$ of this blocking semioval has order 5, and has three fixed points $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$. It is generated by a collineation induced by the matrix:\n$$\n\\left(\\begin{matrix}\n1 & 0 & 0\\\\\n0 & 9 & 0\\\\\n0 & 0 & 4\n\\end{matrix}\\right)\n$$\n\nThe three lines of the triangle generated by the fixed points are a 10-secant ($[0,0,1]$), a tangent ($[1,0,0]$, at $(0,0,1)$), and a 6-secant which contains $(0,0,1)$ and a 5-point orbit under $G$. Using this combinatorial structure and an analogous group structure, we attempted to generalize this blocking semioval to larger planes, particularly $PG(2,19)$, using a fairly straightforward exhaustion in Magma using all possible pairs of squares in $GF(19)$ to define our automorphism group. This search was unsuccessful; however, analyzing the list of blocking semiovals in $PG(2,7)$~\\cite{ranson} shows that the unique smallest blocking semioval in that plane, of size 16, has an analogous automorphism group and combinatorial structure.\n\nTable~\\ref{T1} provides a summary of the size of the smallest blocking semiovals in small Desarguesian planes, where these values are known.\n\n\\begin{table}[!h]\n\\centering\n\\caption{Sizes of smallest known blocking semiovals}\n\\begin{tabular}{|c|c||c|c|}\n\\hline\nPlane & Minimum Size & Plane & Minimum Size\\\\ \\hline\n$PG(2,2)$ & none & $PG(2,3)$ & 6 \\\\ \\hline\n$PG(2,4)$ & 9 & $PG(2,5)$ & 11 \\\\ \\hline\n$PG(2,7)$ & 16 & $PG(2,8)$ & 19 \\\\ \\hline\n$PG(2,9)$ & 21 & $PG(2,11)$ & 26 \\\\ \\hline\n\\end{tabular}\n\\label{T1}\n\\end{table}\n\n\n\\bibliographystyle{plain}\n\n\\begin{thebibliography}{1}\n\\bibitem{bartoli}\nDaniele Bartoli.\n\\newblock On the Structure of Semiovals of Small Size.\n\\newblock {\\em J.Combin. Des.}, 22:525--536, 2014.\n\n\\bibitem{magma}\nWieb Bosma, John Cannon, and Catherine Playoust.\n\\newblock The Magma algebra system. I. The user language.\n\\newblock {\\em J. Symbolic Comput.}, 24(3--4):235--265, 1997.\n\n\\bibitem{dover3}\nJeremy M.~Dover.\n\\newblock Semiovals with large collinear subsets.\n\\newblock {\\em J. Geom.}, 69:58--67, 2000.\n\n\\bibitem{dover1}\nJeremy M.~Dover.\n\\newblock A lower bound on blocking semiovals.\n\\newblock {\\em European J. Combin.}, 21(5):571--577, 2000.\n\n\\bibitem{dover2}\nJeremy M.~Dover.\n\\newblock Some new results on small blocking semiovals.\n\\newblock {\\em Austral. J. Combin.}, 52:269--280, 2012.\n\n\\bibitem{dmw}\nJeremy M.~Dover, Keith E.~Mellinger and Kenneth L.~Wantz.\n\\newblock A minimum blocking semioval in $PG(2,9)$.\n\\newblock {\\em J. Geom.}, 107:119--123, 2016.\n\n\\bibitem{ortools}\nGoogle's OR-Tools.\n\\newblock https:\/\/developers.google.com\/optimization\/, version 9.5.2237.\n\n\\bibitem{hegertakats}\nTam\\'{a}s H\\'{e}ger and Marcella Tak\\'{a}ts.\n\\newblock Resolving sets and semi-resolving sets in finite projective planes.\n\\newblock {\\em Electron. J. Combin.}, 19(4):30, 2012.\n\n\\bibitem{kiss}\nGy. Kiss.\n\\newblock Small semiovals in $PG(2,q)$.\n\\newblock {\\em J. Geom.}, 88:110--115, 2008.\n\n\\bibitem{kmp}\nGy\u00f6rgy Kiss, Stefano Marcugini and Fernanda Pambianco.\n\\newblock On the spectrum of the sizes of semiovals in $PG(2,q)$, $q$ odd.\n\\newblock {\\em Discrete Math.} 310:3188--3193, 2010.\n\n\\bibitem{suetake3}\nNobuo~Nakagawa and Chihiro Suetake.\n\\newblock On blocking semiovals with an 8-secant in projective planes of order 9.\n\\newblock {\\em Hokkaido Math. J.}, 35(2):437--456, 2006.\n\n\\bibitem{ranson}\nB.B.~Ranson and J.M.~Dover\n\\newblock Blocking semiovals in $PG(2,7)$ and beyond.\n\\newblock {\\em European J. Combin.}, 24(2):183--193, 2003.\n\\end{thebibliography}\n\n\\section*{Appendix A: Constraint programming primitives}\nThe constraint program needed to define the incidences in $PG(2,11)$ is rather lengthy; we actually developed a Magma program to write the constraint program. Here we provide some excerpts from the constraint programs used in this paper. The code here is excerpted from a Python program using Google's OR Tools library. Note that sums with ellipses continue over all points on a line or lines through a point, and are redacted for readability considerations.\n\\begin{Verbatim}\n###\n#Variables describing a point\n###\n# Boolean is true\/false as (1,0,0) is in\/not in the blocking semioval\np_1_0_0 = model.NewBoolVar(\"point_1_0_0\")\n# Integer value of the Boolean variable\np_1_0_0_int = model.NewIntVar(0,1,\"P![1,0,0]\")\n#Constraints to tie the Boolean and integer variables together\nmodel.Add(p_1_0_0_int == 1).OnlyEnforceIf(p_1_0_0)\nmodel.Add(p_1_0_0_int == 0).OnlyEnforceIf(p_1_0_0.Not())\n\n###\n#Variables describing a line\n###\n# Boolean is true\/false as [1,0,0] is\/is not tangent to blocking semioval\nl_1_0_0 = model.NewBoolVar(\"line_1_0_0\")\n# Integer value of the Boolean variable\nl_1_0_0_int = model.NewIntVar(0,1,\"line_1_0_0_int\")\n#Constraints to tie the Boolean and integer variables together\nmodel.Add(l_1_0_0_int == 1).OnlyEnforceIf(l_1_0_0)\nmodel.Add(l_1_0_0_int == 0).OnlyEnforceIf(l_1_0_0.Not())\n\n###\n#Blocking semioval constraints\n###\n# Only one point on [1,0,0] is on the blocking semioval if it is a tangent\nmodel.Add(p_0_1_0_int+p_0_1_1_int+ ... == 1).OnlyEnforceIf(l_1_0_0)\n# More than one point on [1,0,0] is on the blocking semioval if not\nmodel.Add(p_0_1_0_int+ ... > 1).OnlyEnforceIf(l_1_0_0.Not())\n# Only one line through (1,0,0) is tangent to the blocking semioval\nmodel.Add(l_0_1_0_int+l_0_1_1_int+ ... == 1).OnlyEnforceIf(p_1_0_0)\n\n###\n#Additional constraints (not necessarily used simultaneously)\n###\n# Force [0,0,1] to be a 6-secant\nmodel.Add(p_1_0_0_int+p_1_1_0_int+p_0_1_0_int+ ... == 6)\n# Assert (1,0,0) is in the blocking semioval, while (1,1,1) is not\nmodel.AddBoolAnd([p_1_0_0,p_1_1_1.Not()])\n# Assert the blocking semioval has 25 points\nmodel.Add(p_1_0_0_int+p_0_1_0_int+p_0_0_1_int+ ... == 25)\n\n\\end{Verbatim}\n\n\n\\end{document}\n"},{"text":"\\documentclass[12pt]{amsproc}\n\n\\usepackage{fancyhdr}\n%\\pagestyle{}\n%\\fancyhead{}\\fancyfoot{}\n%\\fancyhead[RO,RE]{\\normalsize\\thepage}\n\n%\\usepackage[usanames]{color}\n\n\\author[Rodrigues]{Eliana Rodrigues}\n\\address{Department of Academic Areas, Instituto Federal de Goi\\'as, Formosa-GO 73813-816, Brazil}\n\\email{eliana.rodrigues@ifg.edu.br}\n\n\\author[de Melo]{Emerson de Melo}\n\\address{Department of Mathematics, University of Bras\\'ilia, Bras\\'ilia-DF 70910-900, Brazil}\n\\email{emerson@mat.unb.br}\n\n\\author[Ercan]{G\u00fclin Ercan}\n\\address{Department of Mathematics, Middle East Technical University,\n06800, Ankara\/Turkey}\n\\email{ercan@metu.edu.tr}\n\n\\keywords{Frobenius groups, Frobenius-like groups, Dihedral groups, Automorphisms, Nilpotent residual}\n\\subjclass{20D45}\n\n\\title{NILPOTENT RESIDUAL\\\\ OF A FINITE GROUP }\n\\chead[Eliana Rodrigues and Emerson de Melo]{Nilpotent residual of a finite group}\n \n\\newtheorem{theo}{\\sc Theorem}[section]\n\\newtheorem{lem}[theo]{\\sc Lemma}\n\\newtheorem{proposition}[theo]{\\sc Proposition}\n\\newtheorem{corollary}[theo]{\\sc Corollary}\n\\newtheorem{remark}[theo]{\\sc Remark}\n\\newtheorem{definition}[theo]{\\sc Definition}\n\n\n\\begin{document}\n%\\maketitle\n\n\\begin{abstract}\nLet $F$ be a nilpotent group\nacted on by a group $H$ via automorphisms and let the group $G$ admit the\nsemidirect product $FH$ as a group of automorphisms so that $C_G(F) = 1$. We prove that the order of $\\gamma_\\infty(G)$, the rank of $\\gamma_\\infty(G)$ are bounded in terms of the orders of $\\gamma_{\\infty}(C_G(H))$ and $H$, the rank of $\\gamma_{\\infty}(C_G(H))$ and the order of $H$, respectively in cases where either $FH$ is a Frobenius group; $FH$ is a Frobenius-like group satisfying some certain conditions; or $FH=\\langle \\alpha,\\beta\\rangle$ is a dihedral group generated by the involutions $\\alpha$ and $\\beta$ with $F =\\langle \\alpha\\beta\\rangle$ and $H =\\langle\\alpha \\rangle$. \n\n\\end{abstract}\n\n\\maketitle\n\n\n\\section{Introduction}\n\nThroughout all groups are finite. Let a group $A$ act by automorphisms on a group $G$. For any $a \\in A$, we denote by $C_G(a)$ the set $\\{x\\in G : x^a=x\\},$ and write $C_G(A)=\\bigcap_{a\\in A}C_G(a).$ In this paper we focus on a certain question related to the strong influence of the structure of such fixed point subgroups on the structure of $G$, and present some new results when the group $A$ is a Frobenius group or a Frobenius-like group or a dihedral group of automorphisms. \n\nIn what follows we denote by $A^\\#$ the set of all nontrivial elements of $A$, and we say that $A$ acts coprimely on $G$ if $(|A|,|G|)=1$. Recall that a Frobenius group $A=FH$ with kernel $F$ and complement $H$ can be characterized as a semidirect product of a normal subgroup $F$ by $H$ such that $C_F(h)=1$ for every $h \\in H^\\#$. Prompted by Mazurov's problem $17.72$ in the Kourokva Notebook \\cite{KN}, some attention was given to the situation where a Frobenius group $A=FH$ acts by automorphisms on the group $G$. In the case where the kernel $F$ acts fixed-point-freely on $G$, some results on the structure of $G$ were obtained by Khukhro, Makarenko and Shumyatsky in a series of papers \\cite{MS}, \\cite{MKS}, \\cite{K1}, \\cite{K2}, \\cite{K3}, \\cite{KM1}, \\cite{ENP}. They observed that various properties of $G$ are in a certain sense close to the corresponding properties of the fixed-point subgroup $C_G(H)$, possibly also depending on $H$. In particular, when $FH$ is metacyclic they proved that if $C_G(H)$ is nilpotent of class $c$, then the nilpotency class of $G$ is bounded in terms of $c$ and $|H|$. In addition, they constructed examples showing that the result on the nilpotency class of $G$ is no longer true in the case of non-metacyclic Frobenius groups. However, recently in \\cite{EJ2} it was proved that if $FH$ is supersolvable and $C_G(H)$ is nilpotent of class $c$, then the nilpotency class of $G$ is bounded in terms of $c$ and $|FH|$. \n\n\nLater on, as a generalization of Frobenius group the concept of a Frobenius-like group was introduced by Ercan and G\u00fclo\u011flu in \\cite{EG1}, and their action studied in a series of papers \\cite{EG2}, \\cite{EGK1},\\cite{EGK2},\\cite{EGK3},\\cite{EG4},\\cite{EG5}. A finite group $FH$ is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup $F$ with a\nnontrivial complement $H$ such that $FH\/F'$ is a Frobenius group with Frobenius kernel $F\/F'$ and complement $H$ where $F'=[F,F]$. Several results about the properties of a\nfinite group $G$ admitting a Frobenius-like group of automorphisms $FH$ aiming at restrictions on $G$ in terms of $C_G(H)$ and focusing mainly on bounds for the Fitting height and related parameters as a generalization of earlier results obtained for\nFrobenius groups of automorphisms; and also new theorems for Frobenius-like groups based on new representation-theoretic results. In these papers two special types of Frobenius-like groups have been handled. Namely, Frobenius-like groups $FH$ for which $F'$ is of prime order and is contained in $C_F(H)$; and the Frobenius-like groups $FH$ for which $C_F(H)$ and $H$ are of prime orders, which we call Type I and Type II, respectively throughout the remainder of this paper.\n\nIn \\cite{PS} Shumyatsky showed that the techniques developed in \\cite{ENP} can be \nused in the study of actions by groups that are not necessarily Frobenius. He considered a dihedral group $D=\\langle \\alpha, \\beta \\rangle$ generated by two involutions $\\alpha$ and $\\beta$ acting on a finite group $G$ in such a manner that $C_G(\\alpha \\beta)=1$. In particular, he proved that if $C_G(\\alpha)$ and $C_G(\\beta)$ are both nilpotent of class $c$, then $G$ is nilpotent and the nilpotency class of $G$ is bounded solely in terms of $c$. In \\cite{EJ1}, a similar result was obtained for other groups. It should also be noted that in \\cite{EG4} an extension of \\cite{PS} about the nilpotent length obtained by proving that the nilpotent length of a group $G$ admitting a dihedral group of automorphisms in the same manner is equal to the maximum of the nilpotent lengths of the subgroups\n$C_G(\\alpha)$ and $C_G(\\beta)$. \n\nThroughout we shall use the expression ``$(a,b,\\dots )$-bounded'' to abbreviate ``bounded from above in terms of $a, b,\\dots$ only''. Recall that the rank $\\mathbf r(G)$ of a finite group $G$ is the minimal number $r$ such that every subgroup of $G$ can be generated by at most $r$ elements. Let $\\gamma_\\infty(G)$ denote the \\textit{nilpotent residual} of the group $G$, that is the intersection of all normal subgroups of $G$ whose quotients are nilpotent. Recently, in \\cite{EAP}, de Melo, Lima and Shumyatsky considered the case where $A$ is a finite group of prime exponent $q$ and of order at least $q^3$ acting on a finite $q'$-group $G$. Assuming that $|\\gamma_\\infty(C_G(a))| \\leq m$ for any $a \\in A^\\#$, they showed that $\\gamma_\\infty(G)$ has $(m,q)$-bounded order. In addition, assuming that the rank of $\\gamma_\\infty(C_G(a))$ is at most $r$ for any $a \\in A^\\#$, they proved that the rank of $\\gamma_\\infty(G)$ is $(m,q)$-bounded. Later, in \\cite{E}, it was proved that the order of $\\gamma_\\infty(G)$ can be bounded by a number independent of the order of $A$. \n\nThe purpose of the present article is to study the residual nilpotent of finite groups admitting a Frobenius group, or a Frobenius-like group of Type I and Type II, or a dihedral group as a group of automorphisms. Namely we obtain the following results.\n\n\n\\textbf{Theorem A} \nLet $FH$ be a Frobenius, or a Frobenius-like group of Type I or Type II, with kernel $F$ and complement $H$. \nSuppose that $FH$ acts on a finite group $G$ in such a way that $C_G(F)=1$. Then\n\\begin{itemize}\n\\item[a)] $|\\gamma_\\infty(G)|$ is bounded solely in terms of $|H|$ and $|\\gamma_{\\infty}(C_G(H))|$;\n\\item[b)] the rank of $\\gamma_\\infty(G)$ is bounded in terms of $|H|$ and the rank of $\\gamma_{\\infty}(C_G(H))$.\n\\end{itemize}\n\n\n\\textbf{Theorem B} \nLet $D= \\langle\\alpha, \\beta \\rangle$ be a dihedral group generated by two involutions $\\alpha$ and $\\beta$. Suppose that $D$ acts on a finite group $G$ in such a manner that $C_G(\\alpha\\beta)=1$. Then \n\n\\begin{itemize}\n\\item[a)] $|\\gamma_\\infty(G)|$ is bounded solely in terms of $|\\gamma_{\\infty}(C_G(\\alpha))|$ and $|\\gamma_\\infty(C_G(\\beta))|$;\n\\item[b)] the rank of $\\gamma_\\infty(G)$ is bounded in terms of the rank of $\\gamma_{\\infty}(C_G(\\alpha))$ and $\\gamma_\\infty(C_G(\\beta))$. \n\\end{itemize}\n\n\n\n\n \n\nThe paper is organized as follows. In Section 2 we list some results to which we appeal frequently. Section 3 is devoted to the proofs of two key propositions which play crucial role in proving Theorem A and Theorem B whose proofs are given in Section 4.\n\n\n\n\\section{Preliminaries}\n\nIf $A$ is a group of automorphisms of $G$, we use $[G,A]$ to denote the subgroup generated by elements of the form $g^{-1}g^a$, with $g \\in G$ and $a \\in A$. Firstly, we recall some well-known facts about coprime\naction, see for example \\cite{GO}, which will be used without any further references.\n\n\\begin{lem} \\label{lema 1}\nLet $Q$ be a group of automorphisms of a finite group $G$ such that $(|G|,|Q|) = 1$. Then\n\\begin{itemize}\n\\item[(a)] $G= C_G(Q)[G,Q]$. \n\\item[(b)] $Q$ leaves some Sylow $p$-subgroup of $G$ invariant for each prime $p \\in \\pi(G)$.\n\\item[(c)] $C_{G \/N}(Q) = C_G(Q) N \/N$ for any $Q$-invariant normal subgroup $N$ of $G$.\n\\end{itemize}\n\\end{lem}\n\nWe list below some facts about the action of Frobenius and Frobenius-like groups. Throughout, a non-Frobenius Frobenius-like group is always considered under the hypothesis below.\n\n\\textbf{Hypothesis*} Let $FH$ be a non-Frobenius Frobenius-like group with kernel $F$ and complement $H$. Assume that a Sylow $2$-subgroup of $H$ is cyclic and normal, and $F$ has no extraspecial sections of order $p^{2m+1}$ such that $p^m + 1 = |H_1|$ for some subgroup $H_1 \\leq H$. \n \nIt should be noted that Hypothesis* is automatically satisfied if either $|FH|$ is odd or $|H| = 2$.\n\n\\begin{theo}\\label{theFrob}\nSuppose that a finite group $G$ admits a Frobenius group or a Frobenius-like group of automorphisms $FH$ with kernel F and complement H such that $C_G(F)=1$. Then $C_G(H)\\ne 1$ and $\\mathbf r(G)$ is bounded in terms of $\\mathbf r(C_G(H))$ and $|H|.$\n\\end{theo}\n\n\n\\begin{proposition}\\label{Fr}\nLet $FH$ be a Frobenius, or a Frobenius-like group of Type I or Type II. Suppose that $FH$ acts on a $q$-group $Q$ for some prime $q$ coprime\nto the order of $H$ in case $FH$ is not Frobenius. Let $V$ be a $kQFH$-module where $k$ is a field with characteristic not dividing $|QH|.$ Suppose further that $F$ acts fixed-point freely on the semidirect\nproduct $VQ$. Then we have $C_V(H)\\ne 0$ and $$Ker(C_Q(H) \\ \\textrm{on} \\ C_V(H)) = Ker(C_Q(H) \\ \\textrm{on} \\ V).$$ \\end{proposition}\n\n\\begin{proof} See \\cite{EGO} Proposition 2.2 when $FH$ is Frobenius; \\cite{EG2} Proposition C when $FH$ is Frobenius-like of Type I; and \\cite{EG3} Proposition 2.1 when $FH$ is Frobenius-like of Type II. It can be easily checked that \\cite{\nEGO} Proposition 2.2 is valid when $C_Q(F)=1$ without the coprimeness condition $(|Q|,|F|)=1.$\n\\end{proof}\n\nThe proof of the following theorem can be found in \\cite{PS} and in \\cite{E1}.\n\n\\begin{theo}\\label{theDih}\nLet $D= \\langle\\alpha, \\beta \\rangle$ be a dihedral group generated by two involutions $\\alpha$ and $\\beta$. Suppose that $D$ acts on a finite group $G$ in such a manner that $C_G(\\alpha\\beta)=1$. Then\n\\begin{itemize}\n\\item[(a)] $G = C_G(\\alpha)C_G(\\beta)$;\n\n\\item[(b)] the rank of $G$ is bounded in terms of the rank of $C_G(\\alpha)$ and $C_G(\\beta)$;\n\n\\end{itemize}\n\\end{theo}\n\n\n\\begin{proposition} \\label{dih}Let $D =\\langle \\alpha,\\beta\\rangle$ be a dihedral group generated by the involutions $\\alpha$ and $\\beta.$ Suppose that $D$ acts on a $q$-group $Q$ for some prime $q$ and let V be a $kQD$-module for a field $k$ of characteristic different from\n$q$ such that the group $F =\\langle \\alpha\\beta\\rangle$ acts fixed point freely on the semidirect\nproduct $VQ$. If $C_Q(\\alpha)$ acts nontrivially on $V$ then we have $C_V (\\alpha)\\ne 0$\nand $Ker(C_Q(\\alpha) \\ \\textrm{on} \\ C_V(\\alpha)) = Ker(C_Q(\\alpha) \\ \\textrm{on} \\ V)$. \n\\end{proposition}\n\\begin{proof} This is Proposition C in \\cite{EG4}.\n\\end{proof}\nThe next two results were established in \\cite[Lemma 1.6]{KS2} .\n\n\\begin{lem}\\label{sink1}\nSuppose that a group $Q$ acts by automorphisms on a group $G$. If $Q=\\langle q_1,\\ldots , q_n \\rangle$, then $[G,Q]=[G,q_1]\\cdots [G,q_n].$\n\\end{lem}\n\n\\begin{lem}\\label{sink2}\nLet $p$ be a prime, $P$ a finite $p$-group and $Q$ a $p'$-group of automorphisms of $P$. \n\\begin{itemize}\n \\item [a)] If $|[P,q]|\\leq m$ for every $q\\in Q$, then $|Q|$ and $|[P,Q]|$ are $m$-bounded.\n\\item[b)] If $r([P,q])\\leq m$ for every $q\\in Q$, then $r(Q)$ and $r([P,Q])$ are $m$-bounded.\n\\end{itemize}\n\\end{lem}\n\n\n\n\nWe also need the following fact whose proof can be found in \\cite{CP}.\n\n\\begin{lem} \\label{lema 3.1}\nLet $G$ be a finite group such that $\\gamma_\\infty(G) \\leq F(G)$. Let $P$ be a Sylow $p$-subgroup of $\\gamma_\\infty(G)$ and $H$ be a Hall $p'$-subgroup of $G$. Then $P= [P,H]$.\n\\end{lem}\n\n\n\n\n\\section{Key Propositions} We prove below a new proposition which studies the actions of Frobenius and Frobenius-like groups and forms the basis in proving Theorem A. \n\\\\\n\n\n\\begin{proposition}\\label{prop1}\nAssume that $FH$ be a Frobenius group, or a Frobenius-like group of Type I or Type II with kernel $F$ and complement $H$. Suppose that $FH$ acts on a $q$-group $Q$ for some prime $q$. Let $V$ be an irreducible $\\mathbb{F}_pQFH$-module where $\\mathbb{F}_p$ is a field with characteristic $p$ not dividing $|Q|$ such that $F$ acts fixed-point-freely on the semidirect product $VQ$. Additionaly, we assume that $q$ is coprime to $|H|$ in case where $FH$ is not Frobenius. Then $\\mathbf r([V,Q])$ is bounded in terms of $\\mathbf r([C_V(H), C_Q(H)])$ and $|H|$.\n\\end{proposition}\n\n\\begin{proof} Let $\\mathbf r([C_V(H), C_Q(H)])=s.$ We may assume that $V=[V,Q]$ and hence $C_V(Q)=0$. \nBy Clifford's Theorem, $V=V_1\\oplus \\cdots \\oplus V_t$, direct sum of \n of $Q$-homogeneous components $V_i$ , which are transitively permuted by $FH$. Set $\\Omega =\\{V_1,\\dots, V_t\\}$ and fix an $F$-orbit $\\Omega_1$ in $\\Omega$. Throughout, $W=\\Sigma_{U\\in \\Omega_1}U.$\n \n Now, we split the proof into a sequence of steps.\\\\\n\n{\\it (1) We may assume that $Q$ acts faithfully on $V$. Furthermore $Ker(C_Q(H) \\ \\textrm{on} \\ C_V(H)) = Ker(C_Q(H) \\ \\textrm{on} \\ V)=1$. }\n\n\n\\begin{proof} Suppose that $Ker(Q \\ \\rm{on} \\ V)\\neq 1$ and set $\\overline{Q} =Q\/Ker(Q \\ \\rm{on} \\ V)$. Note that since $C_Q(F)=1$, $F$ is a Carter subgroup of $QF$ and hence also a Carter subgroup of $\\overline{Q}F$ which implies that $C_{\\overline{Q}}(F)=1$. Notice that the equality $\\overline{C_Q(H)}=C_{\\overline{Q}}(H)$ holds in case $FH$ is Frobenius (see \\cite{ENP} Theorem 2.3). The same equality holds in case where $FH$ is non-Frobenius due to the coprimeness condition $(q,|H|)=1.$ Then $[C_V(H),C_Q(H)]=[C_V(H),C_{\\overline{Q}}(H)]$ and so we may assume that $Q$ acts faithfully on $V$. \nNotice that by Proposition \\ref{Fr} we have\n$$Ker(C_Q(H) \\ \\textrm{on} \\ C_V(H)) = Ker(C_Q(H) \\ \\textrm{on} \\ V)=1$$ establishing the claim. \n\\end{proof}\n\n{\\it (2) We may assume that $Q=\\langle c^F \\rangle$ for any nonidentity element $c\\in C_{Z(Q)}(H)$ of order $q$. In particular $Q$ is abelian.} \n\n\\begin{proof} We obtain that $C_{Z(Q)}(H)\\ne 1$ as $C_Q(F)=1$ by Proposition \\ref{Fr}. Let now $1\\ne c \\in C_{Z(Q)}(H)$ of order $q$ and consider $\\langle c^{FH} \\rangle=\\langle c^F \\rangle$, the minimal $FH$-invariant subgroup containing $c$. Since $V$ is an irreducible $QFH$-module on which $Q$ acts faithfully we have that $V=[V,\\langle c^F \\rangle]$. Thus we may assume that $Q=\\langle c^F \\rangle$ as claimed. \n\\end{proof}\n\n{\\it (3) $V=[V,c]\\cdot [V,c^{f_1}] \\cdots[V,c^{f_n}]$ where $n$ is a $(s,|H|)$-bounded number. Hence it suffices to bound $\\mathbf r([W,c])$. }\n\n\\begin{proof}\nNotice that the group $C_Q(H)$ embeds in the automorphism group of $[C_V(H),C_Q(H)]$ by step $\\it(1)$. Then $C_Q(H)$ has $s$-bounded rank by Lemma \\ref{sink2}. This yields by Theorem \\ref{theFrob} that $Q$ has $(s,|H|)$-bounded rank. Thus, there exist $f_1=1,\\ldots ,f_n$ in $F$ for an $(s,|H|)$-bounded number $n$ such that $Q=\\langle c^{f_1},\\ldots,c^{f_n} \\rangle$. Now $V=[V,c]\\cdot [V,c^{f_2}] \\cdots[V,c^{f_n}]=\\prod_{i=1}^n [V,c]^{f_i}$ by Lemma \\ref{sink1}. This shows that we need only to bound $\\mathbf r([V,c])$ suitably. In fact it suffices to show that $\\mathbf r([W,c])$ is suitably bounded as $V=\\Sigma_{h\\in H}W^h.$\n\\end{proof}\n\n\\textit{(4) $H_1=Stab_H(\\Omega_1)\\ne1$. Furthermore the rank of the sum of members of $\\Omega_1$ which are not centralized by $c$ and contained in a regular $H_1$-orbit, is suitably bounded.}\n\n\\begin{proof} Fix $U\\in \\Omega_1$ and set $Stab_F(U)=F_1$. Choose a transversal $T$ for $F_1$ in $F.$ Let $W=\\sum_{t\\in T}U^{t}$ where $T$ is a transversal for $F_1$ in $F$ with $1\\in T.$ Then we have $V=\\sum_{h\\in H}W^h$. Notice that $[V,c]\\ne 0$ by $\\it(1)$ which implies that $[W,c]\\neq 0$ and hence $[U^t,c]=U^t$ for some $t\\in T$. Without loss of generality we may assume that $[U,c]=U.$ \n\nSuppose that $Stab_H(\\Omega_1)=1$. Then we also have $Stab_H(U^t)=1$ for all $t\\in T$ and hence the sum $X_t=\\sum_{h\\in H}U^{th}$ is direct for all $t\\in T.$ Now, $U\\leq X_1$. It holds that $$C_{X_t}(H)=\\{ \\sum_{h\\in H}v^{h} \\ : \\ v\\in U^t\\}.$$ Then $|U|=|C_{X_1}(H)|=|[C_{X_1}(H),c]|\\leq |[C_V(H),C_Q(H)]|$ implies $\\mathbf r(U)\\leq s.$ On the other hand $V=\\bigoplus_{t\\in T}X_t$ and $$[C_V(H),c]=\\bigoplus \\{ [C_{X_t}(H),c] : t\\in T \\,\\, \\text{with }\\,\\,[U^t,c]\\ne 0\\}\\leq [C_V(H),C_Q(H)].$$ In particular, $\\{t\\in T : [U^t,c]\\ne 0\\}$ is suitably bounded whence $\\mathbf r([W,c])$ is $(s,|H|)$-bounded. Hence we may assume that $Stab_H(\\Omega_1)\\ne 1.$\n\nNotice that every element of a regular $H_1$-orbit in $\\Omega_1$ lies in a regular $H$-orbit in $\\Omega$. Let $U\\in \\Omega_1$ be contained in a regular $H_1$-orbit of $\\Omega_1.$ Let $X$ denote the sum of the members of the $H$-orbit of $U$ in $\\Omega$, that is $X=\\bigoplus_{h\\in H}U^h$. Then $C_X(H)=\\{ \\sum_{h\\in H}v^{h} \\ : \\ v\\in U\\}$. If $[U,c]\\ne 0$ then by repeating the same argument in the above paragraph we show that $\\mathbf r(U)\\leq s$ is suitably bounded. On the other hand the number, say $m$, of all $H$-orbits in $\\Omega$ containing a member $U$ such that $[U,c]\\ne 0$ is suitably bounded because $m\\leq \\mathbf r([C_V(H),c])\\leq s.$ It follows then that the rank of the sum of members of $\\Omega_1$ which are not centralized by $c$ and contained in a regular $H_1$-orbit, is suitably bounded. \n\\end{proof}\n\n{\\it (5) We may assume that $FH$ is not Frobenius.}\n\n\\begin{proof} Assume the contrary that $FH$ is Frobenius. Let $H_1=Stab_H(\\Omega_1)$ and pick $U\\in \\Omega_1$. Set $S=Stab_{FH_1}(U)$ and $F_1=F\\cap S$. Then $|F:F_1|=|\\Omega_1|=|FH_1:S|$ and so $|S:F_1|=|H_1|$. Since $(|F_1|,|H_1|)=1$, by the Schur-Zassenhaus theorem there exists a complement, say $S_1$ of $F_1$ in $S$ with $|H_1|=|S_1|$. Therefore there exists a conjugate of $U$ which is $H_1$-invariant. There is no loss in assuming that $U$ is $H_1$-invariant. On the other hand if $1\\neq h\\in H_1$ and $x\\in F$ such that $U^{xh}=U^x$, then $[h,x]\\in Stab_{F}(U)=F_1$ and so $F_1x=F_1x^h=(F_1x)^h$. This implies that $ F_1x\\cap C_F(h)$ is nonempty. Now the Frobenius action of $H$ on $F$ forces that $x\\in F_1$. This means that for each $x\\in F\\setminus F_1$ we have $Stab_{H_1}(U^x)=1$. Therefore $U$ is the unique member of $\\Omega_1$ which is $H_1$-invariant and all the $H_1$-orbits other than $\\{U\\}$ are regular. By $\\it(4)$, the rank of the sum of all members of $\\Omega_1$ other than $U$ is is suitably bounded. In particular $\\mathbf r(U)$ and hence $\\mathbf r([W,c])$ is suitably bounded in case where $[U^x,c]\\ne 0$ for some $x\\in F\\setminus F_1$. Thus we may assume that $c$ is trivial on $U^x$ for all $x\\in F\\setminus F_1$. Now we have $[W,c]=[U,c]=U.$\n\nDue to the action by scalars of the abelian group $Q$ on $U$, it holds that $[Q,F_1]\\leq C_Q(U)$. We also know that $c^x$ is trivial on $U$ for each $x\\in F\\setminus F_1$. Since $C_Q(F)=1$, there are prime divisors of $|F|$ different from $q.$ Let $F_{q'}$ denote the $q'$-Hall subgroup of $F.$ Clearly we have $C_Q(F_{q'})=1$. Let now $y=\\prod_{f\\in F_{q'}}c^f$. Then we have $$1=y=(\\prod_{f\\in F_1\\cap F_{q'} }c^f)(\\prod_{f\\in F_{q'}\\setminus F_1}c^f)\\in c^{|F_1\\cap F_{q'}|}C_Q(U).$$ As a consequence $c\\in C_Q(U)$, because $q$ is coprime to $|F_{q'}|$. This contradiction establishes the claim. \n\\end{proof}\n\n{\\it (6) We may assume that the group $FH$ is Frobenius-like of Type II.}\n\n\\begin{proof} On the contrary we assume that $FH$ is Frobenius-like of Type I. By $\\it(4),$ we have $H_1=Stab_H(\\Omega_1)\\ne 1$. Choose a transversal $T_1$ for $H_1$ in $H.$ Now $V=\\bigoplus_{h\\in T_1}W^h.$ Also we can guarantee the existence of a conjugate of $U$ which is $H_1$-invariant by means of the Schur-Zassenhaus Theorem as in $\\it(5)$. There is no loss in assuming that $U$ is $H_1$-invariant.\n\n Set now $Y=\\Sigma_{x\\in F'}U^x$ and $F_2=Stab_F(Y)$ and $F_1=Stab_F(U).$ Clearly, $F_2=F'F_1$ and $Y$ is $H_1$-invariant. Notice that for all nonidentity $h\\in H$, we have $C_F(h)\\leq F'\\leq F_2$ . Assume first that $F=F_2$. This forces that we have $V=Y$. Clearly, $Y\\ne U$, that is $F'\\not \\leq F_1$, because otherwise $Q=[Q,F]=1$ due to the scalar action of the abelian group $Q$ on $U$. So $F'\\cap F_1=1$ which implies that $|F:F_1|$ is a prime. Then $F_1\\unlhd F$ and $F'\\leq F_1$ which is impossible. Therefore $F\\ne F_2$. \n \n If $1\\neq h\\in H$ and $t\\in F$ such that ${Y}^{th}={Y}^{t}$ then $[h,t]\\in F_2$. Now, $F_2t=F_2t^h=(F_2t)^h$ and this implies the existence of an element in $F_2t\\cap C_F(h)$. Since $ C_F(h)\\leq F'\\leq F_2$ we get $t\\in F_2$. In particular, for each $t\\in F\\setminus F_2$ we have $Stab_{H}(Y^t)=1$. \n\nLet $S$ be a transversal for $F_2$ in $F$. For any $t\\in S\\setminus F_2$ set $Y_t=Y^t$ and consider $Z_t=\\Sigma_{h\\in H}{Y_t}^h$. Notice that $V=Y\\oplus \\bigoplus_{t\\in S\\setminus F_2}Z_t$. As the sum $Z_t$ is direct we have $$C_{Z_t}(H)=\\{ \\sum_{h\\in H}v^{h} \\ : \\ v\\in Y_t\\}$$ with $|C_{Z_t}(H)|=|Y_t|.$ Then $\\mathbf r([Y_t,c])=\\mathbf r([C_{Z_t}(H),c])\\leq s$ for each $t\\in S\\setminus F_2$ with $[Y_t,c]\\ne 0$.\nOn the other hand, $$ \\Sigma\\{\\mathbf r([C_{Z_t}(H),c]) : t\\in S \\,\\,\\text{with}\\,\\,[Y_t,c]\\ne 0\\}\\leq \\mathbf r([C_V (H), c])\\leq s$$ whence $|\\{t\\in S\\setminus F_2 : [Y_t,c]\\ne 0\\}|$ is suitably bounded. So the claim is established if there exists $t\\in S\\setminus F_2$ such that $[Y_t,c]\\ne 0$, since we have $V=Y\\oplus \\bigoplus_{t\\in S\\setminus F_2}Z_t$. Thus we may assume that $c$ is trivial on $\\bigoplus_{t\\in S\\setminus F_2}Z_t$ and hence $[V,c]=[Y,c].$\n\nThere are two cases now: We have either $F'\\cap F_1=1$ or $F'\\leq F_1.$ First assume that $F'\\leq F_1.$ Then we get $F_1=F_2$ because $F_2=F'F_1.$ Now $U=Y.$ Due to the action by scalars of the abelian group $Q$ on $U$, it holds that $[Q,F_1]\\leq C_Q(U)$. From this point on we can proceed as in the proof of step $\\it(5)$ and observe that $C_Q(F_{q'})=1$. Letting now $y=\\prod_{f\\in F_{q'}}c^f$, we have $$1=y=(\\prod_{f\\in F_1\\cap F_{q'} }c^f)(\\prod_{f\\in F_{q'}\\setminus F_1}c^f)\\in c^{|F_1\\cap F_{q'}|}C_Q(U).$$implying that $c\\in C_Q(U)$, because $q$ is coprime to $|F_{q'}|$.\n\nThus we have $F_1\\cap F'=1$. First assume that $H_1=H$. Then $Y$ is $H$-invariant and $F_1H$ is a Frobenius group. Note that $C_U(F_1)=1$ as $C_V(F)=1$, and hence $C_{Y}(F_1)=1$ since $F'\\leq Z(F).$ We consider now the action of $QF_1H$ on $Y$ and the fact that $\\mathbf r([C_{Y}(H),C_Q(H)])\\leq s.$ Then step $\\it(5)$, we obtain that $\\mathbf r(Y)=\\mathbf r([Y,Q])$ is $(s,|H|)$-bounded. Next assume that $H_1\\ne H.$ Choose a transversal for $H_1$ in $H$ and set $Y_1=\\Sigma_{h\\in T_1}Y^h$. Clearly this sum is direct and hence $$C_{Y_1}(H)=\\{ \\sum_{h\\in T_1}v^{h} \\ : \\ v\\in Y\\}$$ with $|[C_{Y_1}(H),c]|=|[Y,c]|.$ Then $\\mathbf r([Y,c])=\\mathbf r([C_{Y_1}(H),c])\\leq s$\nestablishing claim $\\it(6)$.\n \\end{proof}\n \n {\\it (7) The proposition follows.}\n\n\\begin{proof} From now on $FH$ is a Frobenius-like group of Type II, that is, $H$ and $C_F(H)$ are of prime orders. By step $\\it(4)$ we have $H=H_1 =Stab_H( \\Omega_1)$ since $|H|$ is a prime. Now $V=W$. We may also assume by the Schur-Zassenhaus theorem as in the previous steps that there is an $H$-invariant element, say $U$ in $\\Omega$. Let $T$ be a transversal for $F_1=Stab_F(U)$ in $F$. Then $F= \\bigcup_{t\\in T}{F_1}t$ implies $V=\\bigoplus_{t\\in T}U^t$. It should also be noted that we have $|\\{t\\in T : [U^t,c]\\ne 0\\}|$ is suitably bounded as $$[C_V(H),c]=\\bigoplus \\{ [C_{X_t}(H),c] : t\\in T \\,\\, \\text{with }\\,\\,[U^t,c]\\ne 0\\}\\leq [C_V(H),C_Q(H)]$$ where $X_t=\\bigoplus_{h\\in H}U^{th}$.\n\n\nLet $X$ be the sum of the components of all regular $H$-orbits on $\\Omega$, and let $Y$ denote the sum of all $H$-invariant elements of $\\Omega$. Then $V=X\\oplus Y.$ Suppose that ${U}^{th}={U}^t$ for $t\\in T$ and $1\\ne h\\in H$. Now $[t,h]\\in F_1$ and so the coset $F_1t$ is fixed by $H$. Since the orders of $F$ and $H$ are relatively prime we may assume that $t\\in C_F(H).$ Conversely for each $t\\in C_F(H)$, ${U}^t$ is $H$-invariant. Hence the number of components in $Y$ is $|T\\cap C_F(H)|=|C_F(H):C_{F_1}(H)|$ and so we have either $C_F(H)\\leq F_1$ or not. \n\nIf $C_F(H)\\not\\leq F_1$ then $C_{F_1}(H)=1$ whence $F_1H$ is Frobenius group acting on $U$ in such a way that $C_{U}(F_1)=1$. Then $\\mathbf r(U)$ is $(s,|H|)$-bounded by step $\\it(5)$ since $\\mathbf r([C_{U}(H),C_Q(H)])\\leq s$ holds. This forces that $\\mathbf r([V,c])$ is bounded suitably and hence the claim is established.\n\nThus we may assume that $C_F(H)\\leq F_1.$ Then $Y=U$ is the unique $H$-invariant $Q$-homogeneous component. If $[U^t,c]\\ne 0$ for some $t\\in F\\setminus F_1$ we can bound $\\mathbf r(U)$ and hence $\\mathbf r([V,c])$ suitably. Thus we may assume that $c$ is trivial on $U^t$ for each $t\\in F\\setminus F_1.$ Due to the action of the abelian group $Q$ on $U$, it holds that $[Q,F_1]\\leq C_Q(U)$. From this point on we can proceed as in the proof of step $\\it(5)$ and observe that $C_Q(F_{q'})=1$. Letting now $y=\\prod_{f\\in F_{q'}}c^f$, we have $$1=y=(\\prod_{f\\in F_1\\cap F_{q'} }c^f)(\\prod_{f\\in F_{q'}\\setminus F_1}c^f)\\in c^{|F_1\\cap F_{q'}|}C_Q(U).$$implying that $c\\in C_Q(U)$, because $q$ is coprime to $|F_{q'}|$. This final contradiction completes the proof of Proposition 3.1.\n\\end{proof}\n\\end{proof}\n\nThe next proposition studies the action of a dihedral group of automorphisms and is essential in proving Theorem B. \n\n\\begin{proposition}\\label{prop2}\nLet $D= \\langle\\alpha, \\beta \\rangle$ be a dihedral group generated by two involutions $\\alpha$ and $\\beta$. Suppose that $D$ acts on a $q$-group $Q$ for some prime $q$. Let $V$ be an irreducible $\\mathbb{F}_pQD$-module where $\\mathbb{F}_p$ is a field with characteristic $p$ not dividing $|Q|$. Suppose that $C_{VQ}(F)=1$ where $F=\\langle \\alpha\\beta\\rangle$. If $max\\{\\mathbf r([C_V(\\alpha), C_Q(\\alpha)]),\\mathbf r([C_V(\\beta), C_Q(\\beta)])\\}\\leq s$, then $\\mathbf r([V,Q])$ is $s$-bounded. \n\\end{proposition}\n\n\\begin{proof} \n\nWe set $H=\\langle \\alpha\\rangle$. So $D=FH$. By Lemma \\ref{sink1} and Theorem \\ref{theDih}, we have $[V,Q]=[V, C_Q(\\alpha)][V,C_Q(\\beta)]$. Then it is sufficient to bound the rank of $[V,C_Q(H)]$. Following the same steps as in the proof of Proposition \\ref{prop1} by replacing Proposition 2.3 by Proposition 2.4, we observe that $Q$ acts faithfully on $V$ and $Q=\\langle c^F\\rangle$ is abelian with $c\\in C_{Z(Q)}(H)$ of order $q$. Furthermore $Ker(C_Q(H) \\ \\textrm{on} \\ C_V(H)) = Ker(C_Q(H) \\ \\textrm{on} \\ V)=1$. Note that it suffices to bound $\\mathbf r([V,c])$ suitably.\n\nLet $\\Omega$ denote the set of $Q$-homogeneous components of the irreducible $QD$-module $V.$ Let $\\Omega_1$ be an $F$-orbit of $\\Omega$ and set $W=\\Sigma_{U\\in \\Omega_1}U.$ Then we have $V=W+W^{\\alpha}$. Suppose that $W^{\\alpha}\\ne W$. Then for any $U\\in \\Omega_1$ we have $Stab_H(U)=1$. Let $T$ be a tranversal for $Stab_F(U)=F_1$ in $F$ . It holds that $V=\\Sigma_{t\\in T}X_t$ where $X_t=U^t+U^{t\\alpha}.$ Now $[V,c]=\\Sigma_{t\\in T}[X_t,c]$ and $C_V(H)=\\Sigma_{t\\in T} C_{X_t}(H)$ where $C_{X_t}(H)=\\{w+w^{\\alpha} : w\\in U^t\\}$. Since $[V,c]\\ne 0$ there exists $t\\in T$ such that $[U^t,c]\\ne 0$, that is $[U^t,c]=U^t.$ Then $[C_{X_t}(H),c]=C_{X_t}(H).$ Since $\\mathbf r([C_V(H),C_Q(H)])\\leq s$ we get $\\mathbf r(U)=\\mathbf r(C_{X_t}(H))\\leq s$. Furthermore it follows that $|\\{t\\in T : [U^t,c]\\ne 0\\}|$ is $s$-bounded and as a consequence $\\mathbf r([V,c])$ is suitably bounded. Thus we may assume that $W^{\\alpha}=W$ which implies that $\\Omega_1=\\Omega$ and $H$ fixes an element, say $U$, of $\\Omega$ as desired.\n\n\n\n\n Let $U^t\\in \\Omega$ be $H$-invariant. Then $[t,\\alpha]\\in F_1.$ On the other hand $t^{-1}t^{\\alpha}=t^{-2}$ since $\\alpha$ inverts $F$. So $F_1t$ is an element of $F\/F_1$ of order at most $2$ which implies that the number of $H$-invariant elements of $\\Omega$ is at most $2$. Let now $Y$ be the sum of all $H$-invariant elements of $\\Omega$. Then $V=Y\\oplus \\bigoplus_{i=1}^m X_i$ where $X_1,\\ldots X_m$ are the sums of elements in $H$-orbits of length $2.$ Let $X_i=U_i\\oplus U_i^{\\alpha}$. Notice that if $[U_i,c]\\ne 0$ for some $i$, then we obtain $\\mathbf r(U)=\\mathbf r(U_i)\\leq s$ by a similar argument as above. On the other hand we observe that the number of $i$ for which $[U_i,c]\\ne 0$ is $s$-bounded by the the hypothesis that $\\mathbf r([C_V(H),c])\\leq s$. It follows now that $\\mathbf r([V,c])$ is suitably bounded in case where $[U_i,c]\\ne 0$ for some $i$. \n\nThus we may assume that $c$ centralizes $\\bigoplus_{i=1}^m X_i$ and that $[U,c]=U$. Due to the scalar action by scalars of the abelian group $Q$ on $U$, it holds that $[Q,F_1]\\leq C_Q(U)$. As $F_1\\unlhd FH$, we have $[Q,F_1]\\leq C_Q(V)=1$. Clearly we have $C_Q(F_{q'})=1$ where $F_{q'}$ denotes the Hall $q'$-part of $F$ whose existence is guaranteed by the fact that $C_Q(F)=1.$ Let now $y=\\prod_{f\\in F_{q'}}c^f$. Then we have $$1=y=(\\prod_{f\\in F_1\\cap F_{q'} }c^f)(\\prod_{f\\in F_{q'}\\setminus F_1}c^f)\\in c^{|F_1\\cap F_{q'}|}C_Q(U).$$ As a consequence $c\\in C_Q(U)$, because $q$ is coprime to $|F_{q'}|$. This contradiction completes the proof of Proposition \\ref{prop2}.\n\n\n\n\\end{proof}\n\n\n\n\n\\section{Proofs of theorems}\n\nFirstly, we shall give a detailed proof for Theorem A part (b). The proof of Theorem A (a) can be easily obtained by just obvious modifications of the proof of part (b).\n\nFirst, we assume that $G = PQ$ where $P$ and $Q$ are $FH$-invariant subgroups such that $P$ is a normal $p$-subgroup for a prime $p$ and $Q$ is a nilpotent $p'$-group with $|[C_P(H),C_Q(H)]|=p^s$. We shall prove that $\\mathbf r(\\gamma_{\\infty}(G))$ is $((s,|H|)$-bounded. Clearly $\\gamma_{\\infty}(G)=[P,Q]$. Consider an unrefinable $FH$-invariant normal series $$P=P_{1}>P_{2}>\\cdots>P_{k}>P_{k+1}=1.$$ Note that its factors $P_i\/P_{i+1}$ are elementary abelian. Let $V=P_{k}$. Since $C_V(Q)=1$, we have that $V=[V,Q]$. We can also assume that $Q$ acts faithfully on $V$. Proposition \\ref{prop1} yields that $\\mathbf r(V)$ is $(s, |H|)$-bounded. Set $S_i=P_i\/P_{i+1}$. If $[C_{S_i}(H),C_Q(H)]=1$, then $[S_i,Q]=1$ by Proposition \\ref{Fr}. Since $C_P(Q)=1$ we conclude that each factor $S_i$ contains a nontrivial image of an element of $[C_P(H),C_Q(H)]$. This forces that $k \\leq s$. Then we proceed by induction on $k$ to obtain that $\\mathbf r([P,Q])$ is an $(s,|H|)$-bounded number, as desired. \n\nLet $F(G)$ denote the Fitting subgroup of a group $G$. Write $F_{0}(G)=1$ and let $F_{i+1}(G)$ be the inverse image of $F(G\/F_{i}(G))$. As is well known, when $G$ is soluble, the least number $h$ such that $F_{h}(G)=G$ is called the Fitting height $h(G)$ of $G$. Let now $r$ be the rank of $\\gamma_{\\infty}(C_G(H))$. Then $C_G(H)$ has $r$-bounded Fitting height (see for example Lemma 1.4 of \\cite{KS2}) and hence $G$ has $(r,|H|)$-bounded Fitting height. \n\nWe shall proceed by induction on $h(G)$. Firstly, we consider the case where $h(G)=2$. Indeed, let $P$ be a Sylow $p$-subgroup of $\\gamma_{\\infty}(G)$ and $Q$ an $FH$-invariant Hall $p'$-subgroup of $G$. Then, by the preceeding paragraphs and Lemma \\ref{lema 3.1}, the rank of $P=[P,Q]$ is $(r,|H|)$-bounded and so the rank of $\\gamma_{\\infty} (G)$ is $(r,|H|)$-bounded. Assume next that $h(G)>2$ and let $N=F_2(G)$ be the second term of the Fitting series of $G$. It is clear that the Fitting height of $G\/\\gamma_{\\infty} (N)$ is $h-1$ and $\\gamma_{\\infty} (N)\\leq \\gamma_{\\infty}(G)$. Hence, by induction we have that $\\gamma_{\\infty}(G)\/\\gamma_{\\infty} (N)$ has $(r,|H|)$-bounded rank. As a consequence, it holds that $${\\bf r}(\\gamma_{\\infty}(G))\\leq {\\bf r}( \\gamma_{\\infty}(G)\/\\gamma_{\\infty} (N))+{\\bf r}(\\gamma_{\\infty}(N))$$ completing the proof of Theorem A(b).\n\nThe proof of Theorem B can be directly obtained as in the above argument by replacing Proposition \\ref{prop1} by Proposition \\ref{prop2}; and Proposition \\ref{Fr} by Proposition \\ref{dih}. \n\n\n\n\n\n\n\n\n\n\n\\begin{thebibliography}{00}\n\n\n\n\n\n\n\\bibitem{CP} C. Acciarri, P. Shumyatsky and A Thillaisundaram, \\textit{Conciseness of coprime commutators in finite groups}, Bull. Aust. Math. \\textbf{89} (2014), 252-258.\n\n\\bibitem{E1} E. de Melo, \\textit{Fitting Height of a Finite Group with a Metabelian Group of Automorphisms}. Communication in Algebra \\textbf{43} (2015), 4797-4808.\n\n\\bibitem{E} E. de Melo, \\textit{Nilpotent residual and Fitting subgroup of fixed points in finite groups}. J. Group Theory \\textbf{22} (2019), 1059-1068.\n\n\\bibitem{EAP} E. de Melo, A. S. Lima and P. Shumyatsky \\textit{Nilpotent residual of fixed points}. Arch. Math \\textbf{111} (2018) 13-21.\n\n\\bibitem{EJ1} E. de Melo, J. Caldeira, \\textit{On finite groups admitting automorphisms with nilpotent\ncentralizers}. J. Algebra \\textbf{493} (2018) 185-193.\n\n\\bibitem{EJ2} E. de Melo, J. Caldeira, \\textit{Supersolvable Frobenius groups with nilpotent centralizers}. J. Pure and Applied Algebra \\textbf{223} (2019) 1210-1216.\n\n\\bibitem{GO} D. Gorenstein, \\textit{Finite Groups}, Harper and Row, London, New York, 1991.\n\n\\bibitem{MS} N. Y. Makarenko and P. Shumyatsky, \\textit{Frobenius groups as groups of automorphisms}, Proc.\nAm. Math. Soc. \\textbf{138} No. 10 (2010) 3425-3436 . \n\n\\bibitem{MKS} N. Yu. Makarenko, E. I. Khukhro, and P. Shumyatsky, \\textit{Fixed points of Frobenius groups of\nautomorphisms}, Dokl. Akad. Nauk, 437, No. 1 (2011) 20-23.\n\n\n\n\\bibitem{K1} E. I. Khukhro, \\textit{The nilpotent length of a finite group admitting a Frobenius group of automorphisms with a fixedpoint-free kernel. } Algebra Logika, \\textbf{49} (2010), 819-833; English transl, Algebra Logic, \\textbf{49} (2011) 551-560.\n\n\\bibitem{K2} E. I. Khukhro, \\textit{Fitting height of a finite group with a Frobenius group of automorphisms,} J. Algebra \\textbf{366} (2012), 1-11.\n\n\\bibitem{K3} E. I. Khukhro, \\textit{Rank and order of a finite group admitting a Frobenius group of automorphisms. }Algebra Logika \\textbf{52} (2013) 99-108; English transl., Algebra Logic \\textbf{52} (2013) 72-78.\n\n\\bibitem{KM1} E.I. Khukhro EI, N.Yu Makarenko, \\textit{Finite groups and Lie rings with a metacyclic Frobenius group of automorphisms.} J. Algebra \\textbf{386} (2013) 77-104.\n\n\\bibitem{ENP} E. I. Khukhro, N. Yu Makarenko and P. Shumyatsky, \\textit{Frobenius groups of automorphisms and their fixed points}, Forum Math. \\textbf{26} (2014), 73-112.\n\n\\bibitem{KS2} E. I. Khukhro, P. Shumyatsky, \\ {\\em Finite groups with Engel sinks of bounded rank}, Glasgow Mathematical Journal \\textbf{60} (2018), 695-701.\n\n\n\\bibitem{EG1} \u0130.\u015e. G\u00fclo\u011flu, G. Ercan, \\textit{Action of a\nFrobenius-like group}, J Algebra \\textbf{ 402}, (2014) 533--543.\n\n\\bibitem{EGO} G. Ercan, \u0130.\u015e. G\u00fclo\u011flu, E. \u00d6\u011f\u00fct, \\textit{Nilpotent Length of a Finite Solvable Group with a Frobenius Group of Automorphisms,} Com. Algebra \\textbf{42} issue 11, (2014) 4751-4756.\n\n\\bibitem{EG2} G. Ercan, \u0130.\u015e. G\u00fclo\u011flu, \\textit{ Action of a\nFrobenius-like group with fixed-point-free kernel}. J Group Theory \\textbf{17}, (2014) 863-873.\n\n\\bibitem{EGK1} G. Ercan, \u0130.\u015e. G\u00fclo\u011flu, E.I. Khukhro, \\textit{Rank and Order of a Finite Group admitting a Frobenius-like Group of\nAutomorphisms}, Algebra and Logic, \\textbf{ 53} Issue 3, (2014) 258\u2013265.\n\n\\bibitem{EGK2} G. Ercan, \u0130.\u015e. G\u00fclo\u011flu, E.I.Khukhro, \\textit{ Derived length of a Frobenius-like kernel}, J Algebra \\textbf{412}, (2014) 179-188.\n\\bibitem{EG5} G. Ercan, \u0130.\u015e. G\u00fclo\u011flu, \\textit{Action of a Frobenius-like group with kernel having central derived subgroup, }International Journal of Algebra and Computation \\textbf{26} No. 6 (2016) 1257\u20131265.\n\n\\bibitem{EG3} G. Ercan, \u0130.\u015e. G\u00fclo\u011flu, \\textit{On the influence of fixed point free nilpotent automorphism groups,} Monat. Math. \\textbf{184} (2017) 531\u2013538. \n\n \\bibitem{EGK3} G. Ercan, \u0130.\u015e. G\u00fclo\u011flu, E.I. Khukhro, \\textit{Frobenius-like groups as groups of automorphisms}. Turk J Math. \\textbf{38} (2014) 965 \u2013 976.\n \n\\bibitem{EG4} G. Ercan, \u0130.\u015e. G\u00fclo\u011flu, (2017) \\textit{Finite groups admitting a dihedral group of automorphisms.} Algebra and Discrete Mathematics \\textbf{23}. Number 2, 223\u2013229.\n\n\n\\bibitem{PS} P. Shumyatsky, \\textit{The dihedral group as group of automorphisms}. J. Algebra \\textbf{375} (2013), 1-12.\n\n\n\n\n\n\n\n\n\n\\bibitem{KN} \\textit{Unsolved problems in group theory.} The Kourovka Notebook. 18th edition, Institute of Mathematics, Novosibirsk 2014.\n\n\n\n\\end{thebibliography}\n\\end{document}\n\n\n\n"},{"text":"\\documentclass[11pt,twoside]{article}\n\\usepackage{amsmath, amssymb, amsfonts, amstext, amsthm, textcomp, enumerate}\n\\usepackage[mathscr]{euscript}\n\\usepackage{float}\n%\\usepackage{bbm}\n%\\usepackage[section]{placeins}\n%\\usepackage{multirow}\n\\usepackage{booktabs}\n\\usepackage{mathtools}\n%\\usepackage[left=10mm,top=0.1in,bottom=15mm]{geometry}\n\\usepackage{graphicx}\n\\usepackage{caption}\n\\usepackage{epstopdf}\n\\usepackage{longtable}\n\\usepackage[utf8]{inputenc}\n\\usepackage{color}\n\\usepackage{hyperref}\n\\usepackage{graphicx}\n%\\usepackage{natbib}\n%\\usepackage{subfigure}\n\\usepackage{dcolumn}% Align table columns on decimal point\n\\usepackage{bm}% bold 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x}}\\def\\y{{\\sf y}}\n\\def\\f{{\\sf f}}\n\\def\\p{{\\sf p}}\n\\def\\q{{\\sf q}}\n\\def\\v{{\\sf v}}\n\\def\\0{{\\sf 0}}\n\\def\\SS{{\\sf s}}\n\\def\\so{{\\sf supp}}\n\\def\\sen{{\\rm sen}}\n\\def\\tr{{\\sf tr}}\n\\def\\proof{{\\noindent\\bf Proof.}\\hskip 0.3truecm}\n\\def\\BBox{\\kern -0.2cm\\hbox{\\vrule width 0.15cm height 0.3cm}}\n\\def\\endproof{\\hspace{.20in} \\BBox\\vspace{.20in}}\n\\def\\Circ{{\\rm Circ}}\n\\def\\AA{{\\sf A}}\n\\def\\BB{{\\sf B}}\n\\def\\B{\\mathcal{B}}\n\\def\\E{\\mathcal{E}}\n\\def\\C{\\mathcal{C}}\n\\def\\D{\\mathcal{D}}\n\\def\\Cc{{\\sf C}}\n\\def\\DD{{\\sf D}}\n\\def\\EE{{\\sf E}}\n\\def\\F{\\mathcal{F}}\n\\def\\G{\\mathcal{G}}\n\\def\\H{\\mathcal{H}}\n\\def\\I{\\mathcal{I}}\n\\def\\J{\\mathcal{J}}\n\\def\\K{\\mathcal{K}}\n\\def\\L{\\mathcal{L}}\n\\def\\M{\\mathcal{M}}\n\\def\\P{\\mathcal{P}}\n\\def\\Q{\\mathcal{Q}}\n\\def\\R{\\mathcal{R}}\n\\def\\S{\\mathcal{S}}\n\\def\\V{\\mathcal{V}}\n\n\\def\\FF{{\\sf F}}\n\\def\\GG{{\\sf G}}\n\\def\\II{{\\sf I}}\n\\def\\JJ{{\\sf J}}\n\\def\\KK{{\\sf K}}\n\\def\\kk{{\\sf k}}\n\\def\\LL{{\\sf L}}\n\\def\\MM{{\\sf M}}\n\\def\\OO{{\\sf O}}\n\\def\\PP{{\\sf P}}\n\\def\\SS{{\\sf S}}\n\\def\\Rr{{\\sf R}}\n\\def\\XX{{\\sf X}}\n\\def\\NN{\\mathbb{N}}\n\\def\\ZZ{\\mathbb{Z}}\n\\def\\QQ{\\matbb{Q}}\n\\def\\RR{\\mathbb{R}}\n\\def\\CC{\\mathbb{C}}\n%\\parskip .5mm\n%\\parindent 2cc\n\n\n\n\n\n\n\\begin{document}\n\\begin{center} \n\\begin{large} {\\bf On an Analogue of a Property of Singular $M$-matrices, for the Lyapunov and the Stein Operators}\n\\end{large}\t\n\\end{center}\n\t\t\\begin{center}\n\t\t\n\t\t\t\\it{A.M. Encinas $^1$, Samir Mondal $^2$ and K.C. Sivakumar $^2$\n\t\t\t}\\\\\n \\end{center}\n\\footnotetext[1]{Department of Mathematics, Polytechnic University of Catalunya, Barcelona, Spain (andres.marcos.encinas@upc.edu).}\n\n \\footnotetext[2]{Department of Mathematics, Indian Institute of Technology Madras,\n\t\t\tChennai 600036, India (ma19d750@smail.iitm.ac.in, kcskumar@iitm.ac.in).}\n \n\t\n\t\t\n\t\t\n\t\t%%% ----------------------------------------------------------------------\n\t\t\\begin{abstract}\n\t\t\tIn the setting of real square matrices, it is known that, if $A$ is a singular irreducible $M$-matrix, then the only nonnegative vector that belongs to the range space of $A$ is the zero vector. In this paper, we prove an analogue of this result for the Lyapunov and the Stein operators.\n\t\t\\end{abstract}\n\t\t%%% ----------------------------------------------------------------------\n\t\t\n\t\t\\vskip.25in\n\t\t\\textit{Keywords:} \n\t\t$M$-matrix, Singular irreducible $M$-matrix, Almost monotonicity, Lyapunov operator, Stein operator.\n\t\t\n\t\t%\\vskip.05in\n\t\t\\textit{AMS Subject Classifications:}\n\t\t15A48, %Positive matrices and their generalizations; cones of matrices\n\t\t15A23, %Factorization of matrices\n\t\t15A09, %Matrix inversion, generalized inverses\n\t\t15A18 %Eigenvalues, singular values, and eigenvectors\n\t\t%90C33. %Linear Complementarity\n\t\t\n\\newpage\n\t\n\\section{Introduction}\n\nThe set of all real matrices of order $n \\times n$ will be denoted by $\\mnr$. For a matrix $X \\in \\mnr$, we use $X\\geq 0$ to denote the fact that all the entries of $X$ are nonnegative. If all the entries of $X$ are positive, we denote that by $X >0$. We use the same notation for vectors. \n\n\nAs usual, for any $A\\in \\mnr$, $N(A)$ and $R(A)$ denote the null and the range spaces of $A\\in \\mnr$, respectively. The {\\it rank} of $A$ is ${\\rm rk}(A)$, the dimension of $R(A)$. In addition, $\\rho(A)$ denotes the {\\it spectral radius} of $A$; that is, the maximum of the absolute values of its eigenvalues.\n\nA matrix $A\\in\\mnr$ is said to be {\\it reducible} if there\nis a permutation matrix $P\\in\\mnr$ such that $P^{T}AP$ has\nthe form\n\\[%\n\\begin{bmatrix}\nS_{11} & S_{12}\\\\\n0 & S_{22}%\n\\end{bmatrix}\n\\]\nfor some square matrices $S_{11}$ and $S_{22}$ of order at least one. A matrix\nis {\\it irreducible} if it is not reducible. \n\nThe {\\it group inverse} of a matrix $A\\in \\mnr$ is the unique matrix $X \\in \\mnr$, if it exists, that satisfies the equations $AXA=A, XAX=X$ and $AX=XA$. If it exists, then the group inverse is denoted by $A^{\\#}$. Of course, when $A$ is nonsingular, then $A^\\#$ exists and moreover $A^\\#=A^{-1}$.\n\nA necessary and sufficient condition for the group inverse of a matrix $A$ to exist is that $A$ has index $1$; that is, $R(A^2)=R(A)$, which is equivalent to the condition $N(A^2)=N(A)$, see \\cite[Theorem 2, Section 4.4]{bg}. Another characterization is that $A^{\\#}$ exists iff $R(A)$ and $N(A)$ are complementary subspaces of $\\mathbb{R}^n$. It is easy to show that any symmetric matrix has group inverse, that a nilpotent matrix $A$ (viz., $A^n=0$), does not have group inverse, whereas the group inverse of an idempotent matrix (viz., $A^2=A$), (exists and) is itself. We refer the reader to \\cite[Chapter 4]{bg} for more details.\n\nIf ${\\cal S}^n(\\RR)$ is the subspace of symmetric matrices in $\\mnr$, for $X \\in {\\cal S}^n(\\RR),$ let us signify $X \\succeq 0$ to denote the fact that $X$ is a positive semidefinite matrix. We use $X \\succ 0,$ when $X$ is positive definite. \n\n\n\\begin{defn}\nA matrix $A\\in \\mnr$ is called a $Z$-matrix, if all the off-diagonal entries of $A$ are nonpositive. Any $Z$-matrix $A$ has the representation $A=sI-B,$ where $s \\geq 0$ and $B\\ge 0$. If $s\\ge \\rho(B)$, then $A$ is called an {\\it $M$-matrix}. \n\\end{defn}\n\nThe notion of $M$-matrix introduced by A. Ostrowski in 1935 in honor of H. Minkowski who worked with this class of matrices around 1900. $M$-matrices possess many interesting nonnegativity properties. For instance, in the representation as above, let $s > \\rho(B).$ Then, $A=s\\big(I-\\frac{B}{s}\\big)$ and considering the {\\it Neumann series} for $\\frac{B}{s}$; that is,\n%\n$$\\Big(I-\\frac{B}{s}\\Big)^{-1}=\\sum_{m=0}^{\\infty} \\Big(\\frac{B}{s}\\Big)^m \\geq 0.$$ \nwe conclude that $A$ is invertible and moreover $A^{-1}=\\frac{1}{s}\\Big(I-\\frac{B}{s}\\Big)^{-1}\\ge 0$. In such a case, we shall refer to $A$ as an {\\it invertible $M$-matrix}. Otherwise, we call $A$, a {\\it singular $M$-matrix}. \n\nHere is a characterization for invertible $M$-matrices. For a proof, we refer the reader to the book \\cite[Chapter 6]{berpl}, where fifty different characterizations are given. \n\n\\begin{thm}\\cite[Theorem 2.3, Chapter 6]{berpl}\\label{matrixcase} \\leavevmode \\\\\nLet $A$ be a $Z$-matrix. Then the following statements are equivalent:\\\\\n(a) $A$ is an invertible $M$-matrix.\\\\\n(b) There exists $x>0$ such that $Ax>0$.\\\\\n(c) For every $q >0,$ there exiss $x>0$ such that $Ax=q.$\\\\\n(d) $A$ is monotone, i.e. $Ax \\geq 0 \\Longrightarrow x\\geq 0$.\\\\ \n$A$ is inverse-nonnegative, i.e. $A$ is invertible and $A^{-1}\\ge 0$. \\\\\n(f) $A$ is a $P$-matrix, i.e. all principal minors of $A$ are positive.\\\\\n(g) $A$ is positive stable i.e. if the real part of each of its eigenvalues is positive. \\\\\nSuppose that $A$ has the representation $A=sI-B$, with $B \\geq 0$ and $s\\ge \\rho(B)$. \nThen each of the above statements is equivalent to:\\\\\n(h) $s<\\rho(B)$.\\\\\nWhen in addition, $A$ is irreducible the above statements are equivalent to:\\\\\n(e') $A$ is inverse-positive, i.e. $A$ is invertible and $A^{-1}> 0$.\n%\n\\end{thm}\n\n\nWe will have the occasion to consider versions of Theorem \\ref{matrixcase} for two special classes of operators on $S^n(\\RR)$ (Theorem \\ref{lyapunovcase} and Theorem \\ref{steincase}), in the next section.\n\nNext, let us turn our attention to the case singular $M$-matrices. A distinguished subclass of such matrices, due to their relevance in many applications, is the set of {\\it singular irreducible $M$-matrices}. For such matrices, we us recall a well known result, which will serve to motivate the contents of this article. \n\n\n\n\\begin{thm}\\label{berplthm}\\cite[Theorem 4.16, Chapter 6]{berpl},\\cite[Theorem 3]{plem}\\leavevmode \\\\\nLet $A \\in \\mnr$ be a singular irreducible $M$-matrix. Then the following hold:\\\\\n(a) $\\rm rk(A)=n-1.$\\\\\n(b) There exists a vector $x >0$ such that $Ax=0.$\\\\\n(c) $A^{\\#}$ exists and is nonnegative on $R(A)$, i.e. $x \\geq 0, x \\in R(A) \\Longrightarrow A^{\\#}x \\geq 0$.\\\\\n(d) All the principal submatrices of $A$, except $A$ itself is an invertible $M$-matrix. \\\\\n(e) $A$ is almost monotone, i.e. $Ax \\geq 0 \\Longrightarrow Ax=0.$\n\\end{thm}\n\nWe remark that in \\cite[Theorem 4.16, Chapter 6]{berpl} statement (c) was established as $A$ has \"property c\", and that the equivalence with (c) in Theorem \\ref{berplthm} was proved in \\cite[Theorem 3]{plem}. Recall that a singular $M$-matrix $A$ is said to have \"property c\", if $A=sI-B$, where $B \\geq 0, s>0$ and the matrix $\\frac{1}{s}B$ is semi-convergent. Semi-convergence of a matrix $X$ means that the matrix sequence $\\{X^k\\}$ converges. Note that for an invertible $M$-matrix, such a sequence converges to zero. One of the most prominent situations where such matrices arise concerns irreducible Markov processes, where one deals with matrices of the form $I-T$, where $T$ is an irreducible column stochastic matrix. \nWe refer the reader to \\cite{li} for additional results that characterize singular irreducible $M$-matrices. \n\n\n\\begin{defn}\nA matrix $A$ is referred to as {\\it range monotone (see, for instance \\cite{miskcs}}), if \n$$Ax \\geq 0, x \\in R(A) \\Longrightarrow x \\geq 0.$$ If $A$ is range monotone, then we say that $A$ has the {\\it range monotonicity} property.\n\\end{defn}\n\nObserve that range monotonicity of $A$ is equivalent to: $A^2x \\geq 0 \\Longrightarrow Ax\\ge 0.$ Moreover, \\cite[Theorem 3 (e)]{plem} shows that statement $(c)$ of Theorem \\ref{berplthm} is equivalent to the range monotonicity of $A$.\n\n\nLet us recall a notion that is stronger than range monotonicity. This will play a central role in this article.\n\n\\begin{defn}\nLet $A \\in \\mnr$. Then $A$ is called trivially range monotone if \n$$Ax \\geq 0, x \\in R(A) \\Longrightarrow x = 0.$$\n\\end{defn}\n\nTrivial range monotonicity of $A$ is the same as: $A^2x \\geq 0 \\Longrightarrow Ax=0.$ It is easy to verify that the matrix $A=\\begin{pmatrix}\n~~1 & -1 \\\\\n-1 & ~~1\n\\end{pmatrix}$ is trivially range monotone. \n\n\\begin{thm}\\label{trivrangemon}\nLet $A \\in \\mnr$ be a singular irreducible $M$-matrix. Then $A$ is trivially range monotone.\n\\end{thm}\n\\proof\nSuppose that $Ax \\geq 0, x \\in R(A).$ Then by the almost monotonicity of $A$ (item (e), Theorem \\ref{berplthm}), it follows that $Ax=0$. So, $x \\in N(A)\\cap R(A)$ and since the group inverse $A^{\\#}$ exists, we have $x=0.$ \n\\qed\n\nIn this work, we shall be interested in the problem of determining trivial range monotonicity for the Lyapunov and the Stein operators on the space of real symmetric matrices. We consider four classes of matrices that give rise to specific instances of these operators, for which we give affirmative\/negative answers. The main results are presented in Theorem \\ref{idem_result}, Theorem \\ref{gpinvexist} and Theorem \\ref{trangemonotone}. Pertinent counter\\-examples are presented in Section \\ref{cex}, followed by a summary table consolidating the findings. \n\n\n\n\\section{Preliminaries}\nA nonempty subset $C$ of a finite dimensional Hilbert space $H$ is said to be a {\\it proper cone} if $C+C=C, \\alpha C \\subseteq C$, for all $\\alpha \\geq 0$, $C \\cap -C =\\{0\\}$ and $C$ has a nonempty interior. Given a cone $C$, we define its {\\it dual cone} $C^*$ by \n%$$\\hbox{\\textst{C^*:=\\{ y \\in \\mathbb{R}^n: x^Ty \\geq 0, ~\\forall x \\in C\\}.}}$$\n$$ C^*:=\\{ y \\in \\mathbb{R}^n: \\langle x,y\\rangle \\geq 0, ~\\forall x \\in C\\}.$$\nIt follows that $C^*$ is indeed a proper cone. A well known example of a proper cone is the nonnegative orthant $\\mathbb{R}^n_+$, viz., the cone of nonnegative vectors. Let the space of all real symmetric matrices of order $n,$ be denoted by ${\\cal S}^n$. Then ${\\cal S}^n$ is a real Hilbert space with the trace inner product, i.e., $\\langle A,B \\rangle = tr(AB),~A,B \\in {\\cal S}^n.$ In ${\\cal S}^n,$ the set of all positive semidefinite matrices denoted by ${\\cal S}^n_+$, is a cone. Both these cones satisfy the condition of self-duality, viz., $C^*=C,$ when $C=\\mathbb{R}^n_+$ or ${\\cal S}^n_+$.\n\nNext, let us observe that if $A$ is a $Z$-matrix, then $\\langle Ae_j,e_i \\rangle = a_{ij} \\leq 0 ,\\;i\\neq j$, where $e_i$ denotes the $i$th standard basis vector of $\\mathbb{R}^n$. These basis vectors are mutually orthogonal and positive, as well. It then follows that an equivalent formulation for a matrix $A$ to be a $Z$-matrix is:\n\\begin{center}\n$x \\geq 0, \\,y \\geq 0$\\; and \\;$\\langle x,y \\rangle =0 \\;\\Longrightarrow \\; \\langle Ax,y \\rangle \\leq 0$.\n\\end{center}\nThis in turn is the same as saying (with $C=\\mathbb{R}^n_+$):\n\\begin{center}\n$x \\in C, \\,y \\in C^*$ \\; and \\;$\\langle x,y \\rangle =0 \\; \\Longrightarrow \\; \\langle Ax,y \\rangle \\leq 0$.\n\\end{center}\n\nMotivated by the reformulation above, a linear operator $T:{\\cal S}^n \\longrightarrow {\\cal S}^n$ is called a {\\it $Z$-operator} if it satisfies: \n\\begin{center}\n$X \\succeq 0, Y \\succeq 0$ \\; and \\;$\\langle X,Y \\rangle =0 \\; \\Longrightarrow \\; \\langle T(X),Y \\rangle \\leq 0$,\n\\end{center}\nwhere $U \\succeq 0$ stands for the fact that the symmetric matrix $U$ is positive semidefinite. When $-U \\succeq 0,$ we will use the notation $U \\preceq 0.$\nIn this article, we shall be concerned with two important classes of operators on ${\\cal S}^n$. These are the Lyapunov operator and the Stein operator. Let us first look at their definitions.\nGiven $A \\in \\mnr$, the {\\it Lyapunov operator} $L_A$ on ${\\cal S}^n$ is the operator\n \\[ L_A(X):=AX+XA^T, \\;X \\in {\\cal S}^n \\]\nand the {\\it Stein operator} $S_A$ is defined by\n \\[S_A(X):=X-AXA^T, \\;X \\in {\\cal S}^n.\\]\n\nLet $L$ denote either the Lyapunov or the Stein operator for a given matrix $A$. Then $L$ satisfies the following (independent of $A$):\n\\begin{center}\n$X \\succeq 0,\\, Y \\succeq 0$ \\; and \\; $\\langle X,Y \\rangle =0 \\; \\Longrightarrow \\; \\langle L(X),Y \\rangle \\leq 0.$\n\\end{center}\nThus, both these operators could be thought of as analogues of $Z$-matrices, for linear maps on ${\\cal S}^n$. \n\nAn operator $T$ on ${\\cal S}^n$ will be called a {\\it positive stable $Z$-operator} if $T$ is an invertible $Z$-operator and $T^{-1}({\\cal S}^n_+) \\subseteq {\\cal S}^n_+$. Note that this generalizes what we know for an invertible $M$-matrix, namely that it is invertible and that its inverse is nonnegative, i.e. leaves the cone $\\mathbb{R}^n_+$ invariant.\n\nThe following result is a version of (items (b) and (c) of) Theorem \\ref{matrixcase} for the Lyapunov operator \\cite{gowsong}. The notation $U \\succ 0$ denotes the fact that the symmetric matrix $U$ is positive definite. \n\n\\begin{thm} \\label{lyapunovcase} (\\cite[Theorem 5]{gowsong}) \\leavevmode \\\\\nFor $A \\in \\mnr$, the following statements are equivalent:\\\\\n(a) There exists $E \\succ 0$ such that $L_A(E) \\succ 0$.\\\\\n(b) For every $Q \\succ 0$ there exists $X \\succ 0$ such that $L_A(X)=Q$.\\\\\n(c) $A$ is positive stable.\n\\end{thm}\n\nA version for the Stein operator is stated next. The original result holds for complex matrices \\cite{gowtp}. We present only the real case. Recall that a matrix $A$ is called Schur stable if all its eigenvalues lie in the open unit disc of the complex plane.\n\n\\begin{thm} \\label{steincase} (\\cite[Theorem 11]{gowtp})\\leavevmode \\\\\nFor $A \\in \\mnr$, the following statements are equivalent:\\\\\n(a) There exists $E \\succ 0$ such that $S_A(E) \\succ0$.\\\\\n(b) For every $Q \\succ 0$ there exists $X \\succ 0$ such that $S_A(X)=Q$.\\\\\n(c) $A$ is Schur stable.\n\\end{thm}\n\nThese results may be considered as analogues of Theorem \\ref{matrixcase}. Let $L$ stand for either the Lyapunov operator or the Stein operator. Then, $(b)$ of Theorem \\ref{lyapunovcase} or Theorem \\ref{steincase} states that the range of the operator $L$ contains the open set of all (symmetric) positive definite matrices. Hence $L$ is surjective and so, it is injective, too. Thus, $L$ is invertible. By the same statement, it also follows that $L^{-1}({\\cal S}^n_+) \\subseteq {\\cal S}^n_+$. Thus, these two operators are examples of positive stable $Z$-operators. \n\n\n\\section{Main Results}\nWe are interested in identifying some matrix classes for which the Lyapunov operator and\/or the Stein operator are\/is trivially range monotone. Let us define this notion, first. Motivated by the definitions for matrices, we call a linear operator $T:{\\cal S}^n \\rightarrow {\\cal S}^n$ {\\it range monotone} if $$T(X) \\succeq 0, X \\in R(T) \\Longrightarrow X \\succeq 0$$ and refer to $T$ as {\\it trivially range monotone} if $$T(X) \\succeq 0, X \\in R(T) \\Longrightarrow X =0. $$\nAs in the case of matrices, one may observe that range monotonicity of a linear operator $T$ is equivalent to the condition: $T^2(X) \\succeq 0 \\Longrightarrow T(X) \\succeq 0$ and that trivial range monotonicity is the same as: $T^2(X) \\succeq 0 \\Longrightarrow T(X) =0.$\n\nNext, we give an example of a range monotone operator and also one which is not range monotone. \n\n\\begin{ex}\nLet $A=\n\\begin{pmatrix}\n1 & 1 \\\\\n0 & 0\n\\end{pmatrix}.$ The corresponding Lyapunov operator is given by $$L_A(X)=\n\\begin{pmatrix}\n2(x_{11}+x_{12}) & x_{12}+x_{22} \\\\\nx_{12}+x_{22} & 0\n\\end{pmatrix},$$\nwhere $X= \\begin{pmatrix}\nx_{11} & x_{12} \\\\\nx_{12} & x_{22}\n\\end{pmatrix}.$ The choice $x_{11}=x_{22}=1, x_{12}=-1$ shows that $L_A$ is not invertible. It is easily seen that if $X \\in R(L_A)$, then $x_{22}=0.$ Next, for such an $X$, if $L_A(X)$ is a positive semidefinite matrix, then $X= \\begin{pmatrix}\nx_{11} & 0 \\\\\n0 & 0\n\\end{pmatrix}$ is also positive semidefinite. Thus, $L_A$ is a range monotone operator. It is not trivially range monotone, as $X= \\begin{pmatrix}\n1 & 0 \\\\\n0 & 0\n\\end{pmatrix} \\in R(L_A) \\cap {\\cal S}^n_+.$\n\\end{ex}\n\n\\begin{ex}\\label{remstein}\nLet $A=\n\\begin{pmatrix}\n1 & 1 \\\\\n0 & 1\n\\end{pmatrix}.$ Then the associated Stein operator is given by \n$$S_A(X)=-\n\\begin{pmatrix}\n2b +c & c \\\\\nc & 0\n\\end{pmatrix},$$\ngiven $X=\n\\begin{pmatrix}\na & b \\\\\nb & c\n\\end{pmatrix}.$\nSet $X=\\begin{pmatrix}\n-1 & 0 \\\\\n~~0 & 0\n\\end{pmatrix}$ and $Y=\\begin{pmatrix}\n0 & \\frac{1}{2} \\\\\n\\frac{1}{2} & 0\n\\end{pmatrix}.$ Then $S_A(Y)=X,$ so that $X \\in R(S_A)$. Further, $S_A(X)=0$. This shows that $S_A$ is not range monotone. \n\\end{ex}\n\nIt is easy to prove that if an operator is idempotent, then it is range monotone. The only fact that is used, is that an idempotent operator acts like the identity operator on its range space. Let us record this statement. \n\n\\begin{lem}\\label{idemrangemonotone}\nLet $T:V \\rightarrow V$ be idempotent. Then $T$ is range monotone.\n\\end{lem}\n\nAs a first step, we address the question of when the Lyapunov and the Stein operators are idempotent. The relevant results are Theorem \\ref{Ly:ch} and Theorem \\ref{St:ch}. It appears that this question has not been addressed, to the best of our knowledge.\n\nLet us fix some more notation. The all ones vector in $\\RR^n$ is denoted by $e$. For any $j=1,\\ldots,n$, we define the symmetric matrix $E_{jj}=[0,\\ldots,\\stackrel{j\\atop \\downarrow}{e_j},\\ldots,0]$ and for any $1\\le ii$ and $d_j\\not=d_i$, then $ c_j=0$. Hence $ 0= c_j=d_j c_j=d_i c_j$. \n\\qed\n\n\n\nIn view of Lemma \\ref{idemrangemonotone}, Theorem \\ref{Ly:ch} and Theorem \\ref{St:ch}, we obtain the following result. We skip the proof.\n\n\\begin{thm}\\label{idem_result}\nLet $A \\in \\mnr.$ We then have the following:\\\\\n(a) If $A$ is a diagonal matrix, whose diagonal entries are either $0$ or $\\frac{1}{2}$, then the Lyapunov operator $L_A$ is range monotone.\\\\\n(c) If $A$ satisfies $A^2=\\pm A$, then the Stein operator $S_A$ is range monotone.\n\\end{thm}\n \n\n\n\n\n\n%In what follows, in the next two results, we identify a matrix class each for the Stein operator and the Lyapunov operator, relative to which they turn out to be invertible $M$-operators. Even though these two matrix classes satisfy the hypotheses of Theorem \\ref{steincase} and Theorem \\ref{lyapunovcase}, respectively, we provide independent and elementary proofs. \n\n%\\begin{thm}\\label{nilpsa}\n%Let $A$ be a nilpotent real square matrix. Then $S_A$ is an invertible $M$-operator. \n%\\end{thm}\n%\\begin{proof}\n%Let $k$ be the index of nilpotency of $A$. We have $S_A(X)=X-AXA^T$ and so if $S_A(X) \\succeq 0,$ then $X \\succeq AXA^T.$ Pre-multiplying the positive semidefinite matrix $Y:=X- AXA^T$ by $A$ and post-multiplying it by $A^T$ yields that $0 \\preceq AYA^T=AXA^T-A^2X(A^2)^T,$ so that $X \\succeq AXA^T \\succeq A^2X(A^2)^T.$ Proceeding inductively, we btain $X \\succeq A^kX(A^k)^T =0.$ Thus $$S_A(X) \\succeq0 \\Longrightarrow X \\succeq0,$$ proving that $S_A$ is invertible and that its inverse leaves the cone ${\\cal S}^n_+$ invariant. \n%\\end{proof}\n\n\n%Much like the conclusion of Theorem \\ref{nilpsa}, for the matrix of Example \\ref{unipotstein}, we show that the Lyapunov operator is an invertible $M$-operator. Once again, we include the proof, even though it follows from Theorem \\ref{lyapunovcase}. \n\n%\\begin{thm}\n%Let $A=\\begin{pmatrix} 1 & 1 \\\\ 0 & 1 %\\end{pmatrix}.$ Then the Lyapunov operator $L_A$ is an invertible $M$-operator.\n%\\end{thm}\n%\\begin{proof}\n%For the given $A,$ $$L_A(X)=\n%\\begin{pmatrix} 2(a+b) & 2b+d \\\\ 2b+d & 2d\n%\\end{pmatrix},$$ where $X=\n%\\begin{pmatrix} a & b \\\\ b & d\n%\\end{pmatrix}.$\n\n%Let $L_A(X) \\succ 0.$ Then $\\rm{trace(L_A(X))}>0,$ and $\\rm{det(L_A(x))}>0.$ Which conclude that \n%\\begin{equation}\\label{traceinq}\n% a+b+d >0 \n%\\end{equation} and \\begin{equation}\\label{detinq}\n% 4(ad-b^2)-d^2 >0. \n%\\end{equation}\n%Equation \\ref{detinq} give guarantee that either $a$ and $d$ both positive or both negative. Which implies that $$ad-b^2=\\rm {det(X)}>0.$$ Now, if we show that $a+d>0,$ then we done. If possible let $a+d <0,$ then by Equation \\ref{traceinq}, we have $b^2 >(a+d)^2.$ From this we can conclude that $4(ad-b^2)-d^2<0,$ which contradicts Equation \\ref{detinq}. So, $a+d >0.$ Hence $X \\succ 0.$ Therefore $L_A$ is an invertible $M$-operator.\n%\\end{proof}\n\nLet us briefly discuss the notion of the group inverse of a linear operator on a finite dimensional vector space $V$ \\cite{rob}. Let $T:V \\rightarrow V$ be linear. $T$ is said to have a group inverse $L:V \\rightarrow V$ if $TLT=T, LTL=L$ and $TL=LT$. \nThe group inverse of $T$ need not exist, but when it does exist, it is unique and will be denoted by $T^{\\#}.$ We make use of the following in our discussion.\n\n\\begin{thm}\\cite[Theorem 5]{rob}\\label{rob}\nLet $T:V \\rightarrow V$ be linear. Then the following are equivalent: \\\\\n(a) $T^{\\#}$ exists.\\\\\n(b) The subspaces $R(T)$ and $N(T)$, of $V$, are complementary.\\\\\n(c) $R(T^2)=R(T).$\\\\\n(d) $N(T^2)=N(T).$\n\\end{thm}\n\nLet us recall that an operator $T$, on an inner product space, is called normal if $TT^{\\ast}=T^{\\ast}T$, where $T^{\\ast}$ denotes the adjoint of $T$. It is easy to see that if $T$ is normal, then $R(T^{\\ast})=R(T)$. So, if $T$ is normal, then the subspaces $R(T)$ and $N(T)$ are complementary. Thus, we have the following consequence of Theorem \\ref{rob}.\n\n\\begin{cor}\nLet $T:V \\rightarrow V$ be linear, where $V$ is a finite dimensional inner product space. If $T$ is a normal operator, then $T^{\\#}$ exists. \n\\end{cor}\n\n\n\\begin{defn}\nAn operator $T:{\\cal S}^n \\longrightarrow {\\cal S}^n$, is called generalized $k$-potent ($k \\geq 2$) if there exists a constant $\\alpha$ such that $T^k=\\alpha T.$ \n\\end{defn}\n\n\\begin{pro}\\label{idemtripo}\nLet $T:{\\cal S}^n \\rightarrow {\\cal S}^n$ be a generalized $k$-potent. Then $T^{\\#}$ exists. In particular, any idempotent operator is group invertible.\n\\end{pro}\n\\proof\nWe have $T^k=\\alpha T$, for some $k \\geq 2.$ Let $T^2(x)=0.$ Then $0=T^k(x)=0=\\alpha T(x)=0,$ proving that $T(x)=0.$ Thus $N(T^2)=N(T).$ So, the group inverse of $T$ exists. \n\\qed\n\nIn the next result, we identify four classes of matrices for which the Lyapunov as well as the Stein operators are group invertible. We shall make use of the following observation. Let $A \\in \\mathbb{R}^{n \\times n}$. Then, $L_A^{\\ast}=L_{A^T}$ and $S_A^{\\ast}=S_{A^T}.$\n\n\n\\begin{thm}\\label{gpinvexist}\\leavevmode\\\\\nThe group inverses $L_A^{\\#}$ and $S_A^{\\#}$ exist, if $A$ satisfies any of the following conditions:\\\\\n(a) $A^2=-I$.\\\\\n(b) $A^2=I.$\\\\\n(c) $A^T=-A$.\\\\\n(d) $A^T=A.$\n\\end{thm}\n\\proof\n(a) Let $A^2=-I$. First, we consider the Lyapunov operator $L_A$. We have $$L_A(X)=AX+XA^T, X \\in {\\cal S}^n.$$ Thus, \n\\begin{eqnarray*}\nL_{A}^2(X) & = & A^2X + AXA^T + AXA^T+X(A^T)^2 \\\\\n & = & -2(X-AXA^T). \n \\end{eqnarray*}\nSo, \n\\begin{eqnarray*}\nL_A^3(X) & = & -2(L_A(X) - L_A(AXA^T)\\\\\n& = & -2(AX + XA^T - A^2XA^T - AX(A^T)^2) \\\\\n& = & -4(AX+XA^T) \\\\\n& = & -4L_A(X).\n\\end{eqnarray*} \nThus, $L_A^3=-4L_A$ and so by Proposition \\ref{idemtripo}, $L_A^\\#$ exists. \\\\\nNext, we take the case of the Stein operator. We have $$S_A(X)=X-AXA^T, X \\in {\\cal S}^n.$$ So, \n\\begin{eqnarray*}\nS_A^2(X) & = & S_A(X-AXA^T) \\\\\n& = & X-AXA^T-AXA^T+A^2X(A^T)^2\\\\\n&=& 2(X-AXA^T)\\\\\n& =& 2S_A(X).\n\\end{eqnarray*}\nSo, $S_A^2=S_A$ and again, by Proposition \\ref{idemtripo} it follows that $S_A^{\\#}$ exists. In fact, in this case, $S_A^{\\#}=2S_A$. This proves (a).\\\\\n(b) Let $A^2=I.$ Then, a calculation done as earlier, leads to the formula $$L_{A}^2(X)=2(X+AXA^T).$$ This, in turn, implies that $$L_{A}^3=4L_A.$$ The conclusion on the existence of the group inverse of $L_A$, now follows. \\\\\nFor the Stein operator, just as in the case $A^2=-I,$ it follows that $$S_A^2=2S_A$$ and so, $S_A^{\\#}$ exists.\\\\\n(c) Since $A^T=-A,$ we have $L_A(X)=AX-XA.$ Now, for $X,Y \\in {\\cal S}^n,$\n\\begin{eqnarray*}\n \\langle X, L_A(Y)\\rangle & = & \\rm tr (X(AY-YA))\\\\\n & = & \\rm tr(XAY) - \\rm tr (XYA)\\\\\n & = & -(\\rm tr (XYA) - \\rm tr (XAY))\\\\\n & = & -(\\rm tr (AXY) - \\rm tr (XAY))\\\\\n & = & -(\\rm tr((AX-XA)Y))\\\\\n & = & -\\langle L_A(X), Y\\rangle.\n\\end{eqnarray*}\nHence $L_A^*=-L_{A},$ which means that $R(L_A)=R(L_A^{\\ast})$. Thus $R(L_A)$ and $N(L_A)$ are complementary subspaces. Thus, $L_A^\\#$ exists. \n\nNext, let us consider the Stein operator. Since $A^T=-A,$ we have $S_A(X)=X+AXA.$ Now, \n\\begin{eqnarray*}\n\\langle X, S_A(Y)\\rangle & = & \\rm tr(X(Y+AYA))\\\\\n& = & \\rm tr(XY+XAYA) \\\\\n&=& \\rm tr(XY) + \\rm tr(AXAY) \\\\\n&=& \\rm tr((X+AXA)Y)\\\\\n&=& \\langle S_A(X), Y\\rangle.\n\\end{eqnarray*}\nHence $S_A^*=S_{A}.$ Once again, the subspaces $R(S_A)$ and $N(S_A)$ are complementary and so $S_A^\\#$ exists. \\\\\n(d) When $A=A^T,$ as was remarked earlier, we have $L_A^*=L_{A^T}=L_A$ and $S_A^*=S_{A^T}=S_A.$ Thus, $L_A^\\#$ and $S_A^\\#$ exist.\n\\qed\n\n\nLet $A \\in \\mnr$. We say that $A$ is {\\it nonnegative stable}, if all the eigenvalues of $A$ have a nonnegative real part. $A$ will be referred to as {\\it Shur semi-stable}, if all its eigenvalues lie in the closed unit disc of the complex plane. \n\n\\begin{rem}\\label{remlyapunov}\nSome remarks are in order. Let matrix $A$ be nilpotent (so that $A$ is nonnegative stable). Then it is easy to show that the operator $L_A$ is also nilpotent. As noted earlier, it follows that the group inverse of $L_A$ does not exist. So, $L_A$ is not trivially range monotone. Nevertheless, we identify a class of matrices $A$, which are nonnegative stable, for which the associated Lyapunov operator is trivially range monotone. Example \\ref{remstein} shows that not all Schur semi-stable matrices are trivially range monotone. However, we identify a class of Schur semi-stable matrices $A$, for which the Stein operator is trivially range monotone. Both these are presented in the next result. \n\\end{rem}\n\n\n\n\n\n%In what follows, we consider the question of trivial range monotonocity of the Lyapunov and the Stein operators corresponding to matrices that satisfy one of the conditions of Theorem \\ref{gpinvexist}. \n\n%\\begin{thm}\\label{negativeinvol}\n%Let $A^2=-I$. Then both the Lyapunov and the Stein operators are trivially range monotone.\n%\\end{thm}\n%\\begin{proof}\n%Let $L_A(X)\\succeq 0$ with $X\\in R(L_A).$ Then $0 \\preceq L_A^3(X) =-4L_A(X).$ This means that $L_A(X)=0.$ This, in turn, implies that $X=0$, since $X\\in R(L_A)$. We have used the fact that the subspaces $R(L_A)$ and $N(L_A)$ are complementary. \\\\\n%Let $X\\in R(S_A)$ so that there exists $Y \\in \\mathcal{S}^n$ with $S_{A}(Y)=X$. Also, let $S_A(X)\\succeq 0.$ Then, from the calculation as above, we have \n%\\begin{eqnarray*}\n%0 \\preceq S_A(X)=S_A^2(Y)=2S_A(Y)=2(Y-AYA^T).\n%\\end{eqnarray*}\n%Set $Z:=Y-AYA^T.$ Then, $Z \\succeq 0$. This means that $$0 \\preceq AZA^T= AYA^T-A^2Y(A^T)^2=AYA^T-Y=-Z.$$\n%Thus, $Z=0$, i.e. $S_A(X)=0,$ which in turn implies that that $X=0,$ as $X\\in R(S_A).$\n%\\end{proof}\n\n%The Lyapunov operator, is not trivially range monotone, when $A$ is an involutory matrix. Next, we show this, by a numerical example.\n\n%\\begin{ex}\\label{invlyo}\n%Let $A=\n%\\begin{pmatrix}\n%0 & 1 \\\\\n%1 & 0\n%\\end{pmatrix},$ so that for any $X=\n%\\begin{pmatrix}\n%a & b \\\\\n%b & c\n%\\end{pmatrix} \\in {\\cal S}^2,$ we have $$L_A(X)=\n%\\begin{pmatrix}\n%2b & a+c \\\\\n%a+c & 2b\n%\\end{pmatrix}.$$ Note that, if $X=\\begin{pmatrix}\n%1 & 0 \\\\\n%0 & -1\n%\\end{pmatrix},$ then $L_A(X)=0,$ proving that $L_A$ is singular. Next, take $Y=A$ and $U=\n%\\begin{pmatrix}\n%1 & 0 \\\\\n%0 & 0\n%\\end{pmatrix}.$ Then $L_A(U)=Y$ so that $Y \\in R(L_A).$ We have $L_A(Y)=2I \\succeq 0$, but $Y \\nsucceq 0.$ Thus, the Lyapunov operator is not even range monotone, if $A$ is involutory.\n%\\end{ex}\n%\\begin{ex}\n%Given that $A\\in \\mathcal{S}^n$ such that $L_A(X')=0,$ $L_A(U)=Y,$ $L_A(Y)=2I$ where $Y\\in R(L_A)$ with $Y\\nsucceq 0.$ Now consider $\\tilde{A}\n%=\\begin{pmatrix}\n%A & 0\\\\\n%0 & \\alpha\n%\\end{pmatrix}.$ Now \n%$$L_{\\tilde{A}}\\begin{pmatrix}\n%X & x^T\\\\\n%x & a\n%\\end{pmatrix}=\n%\\begin{pmatrix}\n%L_A(X) & 0 \\\\\n%0 & 2a\\alpha\n%\\end{pmatrix}.$$\n%It is clear that, if $\\tilde{X}=\n%\\begin{pmatrix}\n%X' & 0 \\\\\n%0 & 0\n%\\end{pmatrix},$ then $L_{\\tilde{A}}(\\tilde{X})=0,$ proving that $L_{\\tilde{A}}$ is singular. Next if, $\\tilde{Y}=\n%\\begin{pmatrix}\n%Y & 0 \\\\\n%0 & 0\n%\\end{pmatrix}$ and $\\tilde{U}=\n%\\begin{pmatrix}\n%U & 0 \\\\\n%0 & 0\n%\\end{pmatrix}$ then $L_{\\tilde{A}}(\\tilde{U})=\\tilde{Y}.$ So that $\\tilde{Y}\\in R(L_A).$ We have $L_{\\tilde{A}}(\\tilde{Y})=2\n%\\begin{pmatrix}\n%I & 0 \\\\\n%0 & 0\n%\\end{pmatrix}\\succeq.$ However $\\tilde{Y}\\nsucceq 0,$ proving that Lyapunov operator is not trivially range monotone. \\\\\n\n%When $A$ is involutory then we will consider $\\tilde{A}=\n%\\begin{pmatrix}\n%A & 0 \\\\\n%0 & \\pm 1\n%\\end{pmatrix}.$ And we observe that for this case also Lyapunov operator is not trivially range monotone.\n%\\end{ex}\n%However, the Stein operator, retains the property of trivial range monotonicity, when $A$ is involutory. The proof of this assertion is entirely similar to the second part of Theorem \\ref{negativeinvol} and is skipped. \n\n%\\begin{thm}\\label{invol}\n%Let $A^2=I$. Then the Stein operator is trivially range monotone.\n%\\end{thm}\n\n%Next, let $A^T=-A.$ Since all the eigenvalues of $A$ are purely imaginary, $A$ is nonnegative stable. We show the trivial range monotonicity property for the Lyapunov operator. We also show that the Stein operator does not satisfy this property.\n\n%\\begin{thm}\\label{skewlyap}\n%Let $A$ be skew-symmetric. Then the Lyapunov operator is trivially range monotone.\n%\\end{thm}\n%\\begin{proof}\n%Let $Z:=L_A(X)=AX-XA \\geq 0,$ with $X\\in R(L_A).$ Then all the eigenvalues of $Z$ are non-negative. Further, $\\rm tr (Z)=\\rm tr(AX-XA)=0$ and so, all the eigenvalues of $Z$ are zero. Since $Z$ is diagonalizable, $0=Z=L_A(X).$ Therefore $X=0,$ as $X\\in R(L_A).$ This shows the trivial range monotonicity property of $L_A$.\n%\\end{proof}\n\\begin{comment}\n\\begin{thm}\nLet $A \\in \\mathbb{R}^{2 \\times 2}$ be skew-symmetric. Then the Stein operator is trivially range monotone.\n\\end{thm}\n\\begin{proof}\nFor this case the only choice of $A$ are $$\n\\begin{pmatrix}\n~0 & \\pm 1 \\\\\n\\mp 1 & ~0\n\\end{pmatrix}.$$\n\\TR{You need to consider the case when the off-digaonals are some $\\alpha$; or else show that taking $\\alpha=1$ suffices.}\n\n \n\nThen, any symmetric matrix $X=\\begin{pmatrix}\na & b \\\\\nb & d\n\\end{pmatrix},$\nWe have \n$$ S_A(X)=\n\\begin{pmatrix}\na-d & 2b \\\\\n2b & d-a\n\\end{pmatrix}.\n$$\n$S_A$ is a singular operator, since, for the choice $X=\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & 1\n\\end{pmatrix},$ we have $S_A(X)=0.$\\\\\n\nNext, let $$ S_A(X)=\n\\begin{pmatrix}\na-d & 2b \\\\\n2b & d-a\n\\end{pmatrix}\n\\succeq 0.$$\nThen $a=d$ and so $b=0$. Further, if $X \\in R(S_A),$ then we have $a=-d$, and so $a=d=0,$ proving that $X=0$. This proves the range monotonicity of $S_A.$\n\\end{proof}\n\nIn the next example, we show that the conclusion of Theorem \\ref{skewlyap} does not hold in general, for the Stein operator.\n\\begin{ex}\\label{skewsstein}\nLet $A=\\frac{1}{\\sqrt 2}\n\\begin{pmatrix}\n~0 & ~1 & ~0 \\\\\n-1 & ~0 & ~1 \\\\\n~0 & -1 & ~0 \n\\end{pmatrix},$ so that for any symmetric matrix $X=\n\\begin{pmatrix}\na & b & c\\\\\nb & d & e \\\\\nc & e & f \n\\end{pmatrix},$ we have \n$$S_A(X)= \\frac{1}{2}\\begin{pmatrix}\n2a-d & \\ 3b-e & 2c+d \\\\\n3b-e & 2d+2c-a-f & 3e-2b \\\\\n2c+d & 3e-2b & 2f-d\n\\end{pmatrix}.$$\nIf $X= \n\\begin{pmatrix}\n-1 & ~0 & ~1\\\\\n~0 & -2 & ~0\\\\\n~1 & ~0 & -1\n\\end{pmatrix},$ then $S_A(X)=0,$ proving that $S_A$ is singular.\nNext, if $U=\n\\begin{pmatrix}\n2 & 0 & 1 \\\\\n0 & 1 & 0 \\\\\n1 & 0 & 2\n\\end{pmatrix}$ and $Y=\\frac{1}{2}\n\\begin{pmatrix}\n3 & 0 & 3 \\\\\n0 & 0 & 0 \\\\\n3 & 0 & 3\n\\end{pmatrix}$ then $S_A(U)=Y$ so that $Y \\in \\ R(S_A).$ We have \n$S_A(Y)=\\frac{1}{2}\n\\begin{pmatrix}\n3 & 0 & 3 \\\\\n0 & 0 & 0 \\\\\n3 & 0 & 3\n\\end{pmatrix}\\succeq 0$ and $Y\\succeq 0,$ proving that Stein operator is not trivially range monotone.\n\\end{ex}\n\n\\begin{ex}\nLet $A=\n\\begin{pmatrix}\n~0 & 1 & ~0 & 0 \\\\\n-1 & 0 & ~0 & 0 \\\\\n~0 & 0 & ~0 & 2 \\\\\n~0 & 0 & -2 & 0\n\\end{pmatrix},$ so that for any symmetric matrix $X=\n\\begin{pmatrix}\na & b & c & d \\\\\nb & e & f & g \\\\\nc & f & h & i \\\\\nd & g & i & j\n\\end{pmatrix},$ we have $$S_A(X)=\n\\begin{pmatrix}\na-e & 2b & c-2g & d+2f \\\\\n2b & e-a & f+2d & g-2c \\\\\nc-2g & f+2d & h-4j & 5i \\\\\nd+2f & g-2c & 5i & j-4h\n\\end{pmatrix}.$$\nIt is clear that, if $X=\n\\begin{pmatrix}\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0\n\\end{pmatrix},$ then $S_A(X)=0,$ proving that $S_A$ is singular. Next, if $Y=\n\\begin{pmatrix}\n0 & 0 & ~0 & ~0 \\\\\n0 & 0 & ~0 & ~0 \\\\\n0 & 0 & -3 & ~0 \\\\\n0 & 0 & ~0 & -3\n\\end{pmatrix}$ and $U=\n\\begin{pmatrix}\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1\n\\end{pmatrix},$ then $S_A(U)=Y$ so that $Y \\in R(S_A).$ We have $S_A(Y)=\n\\begin{pmatrix}\n0 & 0 & ~0 & ~0 \\\\\n0 & 0 & ~0 & ~0 \\\\\n0 & 0 & ~9 & ~0 \\\\\n0 & 0 & ~0 & ~9\n\\end{pmatrix} \\succeq 0.$ However, $Y \\nsucceq 0,$ proving that the Stein operator is not even range monotone.\n\\end{ex}\n\n\\begin{obs}\\label{skewsnottrmstein}\nFor $n= 5,$ consider a skew symmetric block diagonal matrix $B,$ whose one block matrix is $A,$ as in example \\ref{skewsstein} and other block matrix is \n$\\begin{pmatrix}\n 0 & -1 \\\\\n 1 & ~0\n\\end{pmatrix}$ and using the same argument as in the above example we claim that $S_B$ is not trivially range monotone. Similarly we can show that for $n\\geq 6,$ $S_A$ is not trivially range monotone when $A$ is a skew symmetric matrix.\\\\ So, for $n\\geq 3,$ $S_A$ is not trivially range monotone, when $A$ is skew symmetric matrix.\n\\end{obs}\n\nNext, we consider the situation when the matrix $A$ is symmetric. First, we show that the associated Stein operator need not be trivially range monotone. \n\n\\begin{ex}\\label{symstein}\nConsider the symmetric matrix $A=\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & 2\n\\end{pmatrix},$ so that for any $X=\n\\begin{pmatrix}\na & b \\\\\nb & c\n\\end{pmatrix}\\in \\mathcal{S}^2,$ $S_A$ is given by \n$$S_A(X)=\n\\begin{pmatrix}\n~0 & -b \\\\\n-b & -3c\n\\end{pmatrix}.$$ $S_A$ is not invertible, as $S_A(X)=0,$ for $X=\n\\begin{pmatrix}\n1 & 0\\\\\n0 & 0\n\\end{pmatrix}.$ Note that, by item $(d)$ of Proposition \\ref{gpinvexist}, $S_A^{\\#}$ exists. Next, if $Y=\n \\begin{pmatrix}\n 0 & ~0 \\\\\n 0 & -1\n \\end{pmatrix}$ then $S_A(U)=Y$, where $U=\n\\begin{pmatrix}\n0 & 0 \\\\\n0 & \\frac{1}{3}\n\\end{pmatrix},$ showing that $Y \\in R(S_A).$ Also, $Y \\nsucceq 0,$ whereas, $S_A(Y)=\n\\begin{pmatrix}\n0 & 0 \\\\\n0 & 3\n\\end{pmatrix}\\succeq 0$. \n\\end{ex}\n\\end{comment}\n\n\n%In the next theorem, we discuss about the trivial range monotonicity of $L_A$ and $S_A,$ where $A$ satisfy the condition as in Theorem \\ref{gpinvexist}.\n\nIn what follows, we consider the question of trivial range monotonicity of the Lyapunov and the Stein operators corresponding to matrices that satisfy one of the first three conditions of Theorem \\ref{gpinvexist}.\n\n\\begin{thm}\\label{trangemonotone}\n For the Lyapunov operator $L_A$ and Stein operator $S_A$ the following are true:\\\\\n $(a)$ Let $A^2=-I.$ Then $L_A$ and $S_A$ are trivially range monotone.\\\\\n $(b)$ Let $A^2=I.$ Then $S_A$ is trivially range monotone.\\\\\n $(c)$ Let $A^T=-A.$ then $L_A$ is trivially range monotone. \n \\end{thm}\n\\proof\nFirst, we observe that both the Lyapunov and the Stein operators are group invertible, under any of the three conditions on the matrix $A$, as above (in view of Theorem \\ref{gpinvexist}). Thus in all these cases, the range space and the null space (of either the Lyapunov operator or the Stein operator) are complementary subspaces of ${\\cal S}^n.$ We shall make repeated use of this fact, in our proofs.\n\n $(a)$ Let $L_A(X)\\succeq 0$ with $X\\in R(L_A).$ Then $0 \\preceq L_A^3(X) =-4L_A(X).$ This means that $L_A(X)=0.$ Thus, $X\\in R(L_A) \\cap N(L_A)=\\{0\\}$, proving that $L_A$ is trivially range monotone. \n \nNext, we show the trivial range monotonicity of the Stein operator. Let $X\\in R(S_A)$ so that there exists $Y \\in \\mathcal{S}^n$ with $S_{A}(Y)=X$. Also, let $S_A(X)\\succeq 0.$ Then, from the calculation as earlier, we have \n\\begin{eqnarray*}\n0 \\preceq S_A(X)=S_A^2(Y)=2S_A(Y)=2(Y-AYA^T).\n\\end{eqnarray*}\nSet $Z:=Y-AYA^T.$ Then, $Z \\succeq 0$. This means that $$0 \\preceq AZA^T= AYA^T-A^2Y(A^T)^2=AYA^T-Y=-Z.$$\nThus, $Z=0$, i.e. $S_A(X)=0,$ which in turn implies that that $X=0,$ as $X\\in R(S_A) \\cap N(S_A).$\n\n$(b)$ The proof of the trivial range monotonicity of the Stein operator, is entirely similar to the second part of the above result and is skipped. \n\n$(c)$ Let $Z:=L_A(X)=AX-XA \\succeq 0,$ with $X\\in R(L_A).$ Then all the eigenvalues of $Z$ are non-negative. Further, $\\rm tr (Z)=\\rm tr(AX-XA)=0$ and so, all the eigenvalues of $Z$ are zero. Since $Z$ is diagonalizable, $0=Z=L_A(X).$ Therefore $X=0,$ as $X\\in R(L_A) \\cap N(L_A).$ This shows the trivial range monotonicity property of $L_A$. \\\\\n\\qed\n\nLet matrix $A$ be such that either $A^2=I$ or $A^2=-I.$ Then $A^{-1}$ exists and shares the same property as that of $A$. If $A$ is skew-symmetric, then $A^{\\#}$ exists (since $R(A)$ and $N(A)$ are complementary) and $A^{\\#}$ is also skew-symmetric. Thus we have the following immediate consequence of Theorem \\ref{trangemonotone}.\n\n\\begin{cor}\nLet $A^2=-I.$ Then $L_{A^{-1}}$ and $S_{A^{-1}}$ are trivially range monotone. If $A^2=I,$ then $S_{A^{-1}}$ is trivially range monotone. Let $A^T=-A.$ Then $L_{A^{\\#}}$ is trivially range monotone. \n\\end{cor}\n\nThe first example below illustrates (a) of Theorem \\ref{trangemonotone}.\n\n\\begin{ex}\\label{illus_st(a)}\nLet $A=\n\\begin{pmatrix}\n~~0 & 1 \\\\\n-1 & 0\n\\end{pmatrix},$ so that $A^2=-I.$ For any $X=\n\\begin{pmatrix}\na & b \\\\\nb & c\n\\end{pmatrix} \\in {\\cal S}^2,$ we have $$S_A(X)=\n\\begin{pmatrix}\na-c & 2b \\\\\n2b & c-a\n\\end{pmatrix}.$$ If $S_A(X) \\succeq 0,$ then $a=c$ and so $b=0$. Thus, \n$X=\\begin{pmatrix}\na & 0 \\\\\n0 & a\n\\end{pmatrix}.$ If we impose the condition that $X \\in R(S_A),$ then $a= -a$, proving that $X=0.$ Thus, $S_A$ is trivially range monotone.\n\\end{ex}\n\nThe next example illustrates item (b) of Theorem \\ref{trangemonotone}.\n\n\\begin{ex}\\label{illus_st(b)}\nLet $A=\n\\begin{pmatrix}\n0 & 1 \\\\\n1 & 0\n\\end{pmatrix},$ so that $A^2=I.$ For any $X=\n\\begin{pmatrix}\na & b \\\\\nb & c\n\\end{pmatrix} \\in {\\cal S}^2,$ we have $$S_A(X)=\n\\begin{pmatrix}\na-c & 0 \\\\\n0 & c-a\n\\end{pmatrix}.$$ If $S_A(X) \\succeq 0,$ then $a=c$ and so \n$X=\\begin{pmatrix}\na & b \\\\\nb & a\n\\end{pmatrix}.$ Further, if $X \\in R(S_A),$ then $a= -a$ and $b=0$, proving that $X=0.$ Thus, $S_A$ is trivially range monotone.\n\\end{ex}\n\nItems (a) (for the Lyapunov operator) and (c) of Theorem \\ref{trangemonotone} are illustrated, next.\n\n\\begin{ex}\\label{illus_lyst}\nLet $A=\n\\begin{pmatrix}\n~~0 & 1 \\\\\n-1 & 0\n\\end{pmatrix}.$ Then $A$ is skew-symmetric and $A^2=-I.$ For any $X=\n\\begin{pmatrix}\na & b \\\\\nb & c\n\\end{pmatrix} \\in {\\cal S}^2,$ we have $$L_A(X)=\n\\begin{pmatrix}\n2b & c-a \\\\\nc-a & -2b\n\\end{pmatrix}.$$ The requirement that $L_A(X) \\succeq 0$ yields $b=0$ and so $a=c$. Thus $X=a I$. The condition that $X \\in R(L_A)$ then implies that $a=0$, so that $X=0,$ proving the trivial range monotonicity of $L_A.$ \n\\end{ex}\n\n\n\\section{Counterexamples}\\label{cex}\nThe Lyapunov operator, is not trivially range monotone, in general, when $A$ is an involutory matrix. \n\n\\begin{ex}\\label{invlyo}\nLet $A=\n\\begin{pmatrix}\n0 & 1 \\\\\n1 & 0\n\\end{pmatrix},$ so that $A^2=I$ and for any $X=\n\\begin{pmatrix}\na & b \\\\\nb & c\n\\end{pmatrix} \\in {\\cal S}^2,$ we have $$L_A(X)=\n\\begin{pmatrix}\n2b & a+c \\\\\na+c & 2b\n\\end{pmatrix}.$$ Note that, if $X=\\begin{pmatrix}\n1 & 0 \\\\\n0 & -1\n\\end{pmatrix},$ then $L_A(X)=0,$ proving that $L_A$ is singular. Next, take $Y=A$ and $U=\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & 0\n\\end{pmatrix}.$ Then $L_A(U)=Y$ so that $Y \\in R(L_A).$ We have $L_A(Y)=2I \\succeq 0$, but $Y \\nsucceq 0.$ Thus, the Lyapunov operator is not even range monotone, if $A$ is involutory. \n\\end{ex}\n\nThe conclusion above holds for matrices of higher order, too. For instance, \nif $A$ is involutory and if one sets $\\tilde{A}\n=\\begin{pmatrix}\nA & 0\\\\\n0 & \\pm 1\n\\end{pmatrix},$ then $L_{\\tilde{A}}$ not range monotone. We omit the details.\n\n\n\\begin{ex}\\label{skewssteinorder2}\nConsider the case, when $A^T=-A.$ For $n=1, S_A(X)=X.$ Here, $S_A$ is even monotone. For $n=2,$ (after normalizing) $A$ must be of the following form \n$$\\begin{pmatrix}\n~0 & \\pm 1 \\\\\n\\mp 1 & ~0\n\\end{pmatrix}.$$\nThen, for any symmetric matrix $X=\\begin{pmatrix}\na & b \\\\\nb & d\n\\end{pmatrix},$\nwe have \n$$ S_A(X)=\n\\begin{pmatrix}\na-d & 2b \\\\\n2b & d-a\n\\end{pmatrix}.\n$$\n$S_A$ is singular since $S_A(I)=0.$ Next, if $S_A(X) \\succeq 0,$, then $a=d$ and so $b=0$. Further, if $X \\in R(S_A),$ then we have $a=-d$, and so $a=d=0,$ proving that $X=0$. This proves the trivial range monotonicity of $S_A,$ for $n=2.$\n\\end{ex}\n\nIn the next two examples, we show that the Stein operator is not trivially range monotone for $n=3,4$, for a skew-symmetric matrix $A$.\n\n\\begin{ex}\\label{skewsstein1}\nLet $A=\\frac{1}{\\sqrt 2}\n\\begin{pmatrix}\n~0 & ~1 & ~0 \\\\\n-1 & ~0 & ~1 \\\\\n~0 & -1 & ~0 \n\\end{pmatrix},$ so that for any symmetric matrix $X=\n\\begin{pmatrix}\na & b & c\\\\\nb & d & e \\\\\nc & e & f \n\\end{pmatrix},$ we have \n$$S_A(X)= \\frac{1}{2}\\begin{pmatrix}\n2a-d & \\ 3b-e & 2c+d \\\\\n3b-e & 2d+2c-a-f & 3e-2b \\\\\n2c+d & 3e-2b & 2f-d\n\\end{pmatrix}.$$\n$S_A$ is singular, since $S_A(X)=0,$ for $X= \n\\begin{pmatrix}\n-1 & ~0 & ~1\\\\\n~0 & -2 & ~0\\\\\n~1 & ~0 & -1\n\\end{pmatrix}.$ Next, if $U=\n\\begin{pmatrix}\n2 & 0 & 1 \\\\\n0 & 1 & 0 \\\\\n1 & 0 & 2\n\\end{pmatrix}$ and $Y=\\frac{1}{2}\n\\begin{pmatrix}\n3 & 0 & 3 \\\\\n0 & 0 & 0 \\\\\n3 & 0 & 3\n\\end{pmatrix}$ then $S_A(U)=Y$ so that $Y \\in \\ R(S_A).$ We have \n$S_A(Y)=\\frac{1}{2}\n\\begin{pmatrix}\n3 & 0 & 3 \\\\\n0 & 0 & 0 \\\\\n3 & 0 & 3\n\\end{pmatrix}\\succeq 0$ and $Y\\succeq 0.$ Since $Y \\neq 0$, we conclude that the Stein operator is not trivially range monotone.\n\\end{ex}\n\n\n\\begin{ex}\\label{skewsstein2}\nLet $A=\n\\begin{pmatrix}\n~0 & 1 & ~0 & 0 \\\\\n-1 & 0 & ~0 & 0 \\\\\n~0 & 0 & ~0 & 2 \\\\\n~0 & 0 & -2 & 0\n\\end{pmatrix},$ so that for any symmetric matrix $X=\n\\begin{pmatrix}\na & b & c & d \\\\\nb & e & f & g \\\\\nc & f & h & i \\\\\nd & g & i & j\n\\end{pmatrix},$ we have $$S_A(X)=\n\\begin{pmatrix}\na-e & 2b & c-2g & d+2f \\\\\n2b & e-a & f+2d & g-2c \\\\\nc-2g & f+2d & h-4j & 5i \\\\\nd+2f & g-2c & 5i & j-4h\n\\end{pmatrix}.$$\nIt is clear that, if $X=\n\\begin{pmatrix}\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0\n\\end{pmatrix},$ then $S_A(X)=0,$ proving that $S_A$ is singular. Next, if $Y=\n\\begin{pmatrix}\n0 & 0 & ~0 & ~0 \\\\\n0 & 0 & ~0 & ~0 \\\\\n0 & 0 & -3 & ~0 \\\\\n0 & 0 & ~0 & -3\n\\end{pmatrix}$ and $U=\n\\begin{pmatrix}\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1\n\\end{pmatrix},$ then $S_A(U)=Y$ so that $Y \\in R(S_A).$ We have $S_A(Y)=\n\\begin{pmatrix}\n0 & 0 & ~0 & ~0 \\\\\n0 & 0 & ~0 & ~0 \\\\\n0 & 0 & ~9 & ~0 \\\\\n0 & 0 & ~0 & ~9\n\\end{pmatrix} \\succeq 0.$ However, $Y \\nsucceq 0,$ proving that the Stein operator is not even range monotone.\n\\end{ex}\n\n\\begin{ex}\\label{skewsstein3}\nFor $n = 5,$ consider the skew symmetric block diagonal matrix $B,$ whose leading principal sub-block is $A,$ as in example \\ref{skewsstein1} and whose trailing principal sub-block matrix is \n$\\begin{pmatrix}\n 0 & -1 \\\\\n 1 & ~0\n\\end{pmatrix}.$ Then, by the argument given as earlier, it may be shown that $S_B$ is not trivially range monotone. This inductive argument allows us to conclude that, {\\it in general}, $S_A$ is not trivially range monotone when $A$ is a skew symmetric matrix, of any order $n \\geq 3$. \n\\end{ex}\n\nThis finishes the discussion for skew-symmetric case. \n\nNext, let $A$ be symmetric. Example \\ref{invlyo}, shows that $L_A$ is not trivially range monotone. The following example shows that $S_A$ is not trivially range monotone.\n\n\\begin{ex}\\label{symstein}\nConsider the symmetric matrix $A=\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & 2\n\\end{pmatrix},$ so that for any $X=\n\\begin{pmatrix}\na & b \\\\\nb & c\n\\end{pmatrix}\\in \\mathcal{S}^2,$ $S_A$ is given by \n$$S_A(X)=\n\\begin{pmatrix}\n~0 & -b \\\\\n-b & -3c\n\\end{pmatrix}.$$ $S_A$ is not invertible, as $S_A(X)=0,$ for $X=\n\\begin{pmatrix}\n1 & 0\\\\\n0 & 0\n\\end{pmatrix}.$ Note that, by item $(d)$ of Proposition \\ref{gpinvexist}, $S_A^{\\#}$ exists. Next, if $Y=\n \\begin{pmatrix}\n 0 & ~0 \\\\\n 0 & -1\n \\end{pmatrix}$ then $S_A(U)=Y$, where $U=\n\\begin{pmatrix}\n0 & 0 \\\\\n0 & \\frac{1}{3}\n\\end{pmatrix},$ showing that $Y \\in R(S_A).$ Also, $Y \\nsucceq 0,$ whereas, $S_A(Y)=\n\\begin{pmatrix}\n0 & 0 \\\\\n0 & 3\n\\end{pmatrix}\\succeq 0$. \n\\end{ex}\n\nWe summarize the findings in the following table. Note that, while \"Yes\" stands for an affirmative answer, \"No\" means that counterexamples exist to show that the answers are negative, in general. \n\n\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n{\\bf Matrix classes} & \\multicolumn{2}{c|}{\\bf Trivial Range Monotonicity} \\\\\n\n\\cline{2-3} & $L_A$ & $S_A$ \\\\\n\\hline $A^2=-I$ & Yes [Theorem \\ref{trangemonotone}(a)] & Yes [Theorem \\ref{trangemonotone}(a)]\\\\\n \n\\hline $A^2=I$ & No [Example \\ref{invlyo}] & Yes [Theorem \\ref{trangemonotone}(b)]\\\\\n\n\\hline $A^T=-A$ & Yes [Theorem \\ref{trangemonotone}(c)] & Yes ($n=2$) [Example \\ref{skewssteinorder2}] \\\\\n& & No ($n\\geq 3$) [Examples \\ref{skewsstein1}, \\ref{skewsstein2}, \\ref{skewsstein3}]\\\\\n \n% \\hline &&SMD1:\\\\\n% Six monthly DC meeting& After five years from the date of&SMD2:\\\\\n% ®istration, upto maximum&SMD3:\\\\\n% &period of the programme&SMD4:\\\\\n\\hline $A^T=A$ & No [Example \\ref{invlyo}] & No [Example \\ref{symstein}]\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\n\\section{Concluding Remarks}\n\nWe have studied four matrix classes in the context of the notion of trivial range monotonicity of the associated Lyapunov and the Stein operators. The motivation, as was mentioned in the introduction, is to obtain a generalization of the trivial range monotonicity property of singular irredudcible $M$-matrices. It would be interesting to bring the matrix classes for which we could obtain affirmative results, under a more general framework. Similarly, to unify the matrix classes for which negative results for operator analogues, have been proved. It would be another interesting question to study the notion of irreducibility of the Lyapunov operator and the Stein operator. It is pertinent to point to the fact that there are different (possibly nonequivalent) ways of defining irreducibility of a \"positive\" operator. So one could propose such a notion for the Stein operator (which is the difference: $I$ minus a positive operator $X \\rightarrow AXA^T$), simply by introducing the assumption of irreducibility for the second term, which is positive (meaning, that the matrix $AXA^T$ is symmetric and positive semidefinite, whenever so is $X$). However, it is not clear how one could propose irreducibility for the Lyapunov operator and the likes of it. While we will pursue these questions in future, we note that, motivated by the considerations of this article, the notion of irreducibility has been investigated for $Z$-operators over Euclidean Jordan Algebras \\cite{gowdanew}.\n\n\\section{Acknowledgements}\nSamir Mondal acknowledges funding received from the Prime Minister's Research Fellowship (PMRF), Ministry of Education, Government of India, for carrying out this work. The first and the third authors thank ASEM-DUO for financial support enabling the former's visit to India and the latter, to Spain. The first author was also partially supported by the Spanish I+D+i program under project PID2021-122501NB-I00. The authors thank Prof. M.S. Gowda for his suggestions and comments that have helped in a clearer presentation of the material. He suggested the nomenclature \"positive stable $Z$-operators\".\n\n\\begin{thebibliography}{10}\n\n\\bibitem{bg}\nA. Ben-Israel and T. N. E. Greville, {\\it Generalized Inverses - Theory and Applications}, CMS Books in Mathematics vol. 15. Springer-Verlag, New York, 2003.\n\n\\bibitem{berpl}\nA. Berman and R.J. Plemmons, {\\it Nonnegative Matrices in the Mathematical Sciences}, Classics in Applied Mathematics, {\\bf 9} SIAM, Philadelphia, 1994.\n\n\\bibitem{gowdanew}\nM.S. Gowda, {\\it Completely mixed linear games corresponding to\n$Z$-transformations over self-dual cones}, April 2023, private communication.\n\n\\bibitem{gowtp}\nM.S. Gowda and T. 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Appl., {\\bf 22} (1968), 658-669. \n\n\\end{thebibliography}\n\n\n\n\n\n\n\n\n\\end{document}"},{"text":"\\documentclass[preprint,12pt]{elsarticle}\n%% Use the option review to obtain double line spacing\n%% \\documentclass[preprint,review,12pt]{elsarticle}\n\n\\usepackage{a4wide,amsfonts, amsmath, amscd}\n\\usepackage[psamsfonts]{amssymb}\n\n\n%\\usepackage{refcheck}\n%% Use the options 1p,twocolumn; 3p; 3p,twocolumn; 5p; or 5p,twocolumn\n%% for a journal layout:\n%% \\documentclass[final,1p,times]{elsarticle}\n%% \\documentclass[final,1p,times,twocolumn]{elsarticle}\n%% \\documentclass[final,3p,times]{elsarticle}\n%% \\documentclass[final,3p,times,twocolumn]{elsarticle}\n%% \\documentclass[final,5p,times]{elsarticle}\n%% \\documentclass[final,5p,times,twocolumn]{elsarticle}\n\n%% if you use PostScript figures in your article\n%% use the graphics package for simple commands\n%% \\usepackage{graphics}\n%% or use the graphicx package for more complicated commands\n%% \\usepackage{graphicx}\n%% or use the epsfig package if you prefer to use the old commands\n%\\usepackage{inputenc}\n\\usepackage{graphicx}\n%\\usepackage[left=3cm,right=1.5cm, top=2cm,bottom=2cm,bindingoffset=0cm]{geometry}\n\\usepackage[cp1251]{inputenc}\n%% The amssymb package provides various useful mathematical symbols\n\\usepackage{amssymb}\n%% The amsthm package provides extended theorem environments\n\\usepackage{amsthm}\n\\usepackage{cmap}\n\n%% The lineno packages adds line numbers. 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