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TITLE: Is the book wrong about this corollary?
QUESTION [2 upvotes]: My book states
The group $\prod_{i = 1\dots n} \mathbb{Z}_{m_i}$ is cyclic and isomorphic to $\mathbb{Z}_{m_1m_2\dots m_n} \iff$ the numbers $m_i$ for $i = 1\dots n$ are such that the gcd of any two of them is $1$.
So if I take $\mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_{10}$, then gcd(2,3) = 1, so $\mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_{10} \equiv \mathbb{Z}_6 \times \mathbb{Z}_{10}$, but gcd(6,10) $\neq 1$, so $\mathbb{Z}_6 \times \mathbb{Z}_{10}$ is not isomorphic to $\mathbb{Z}_{60}$, so this is wrong?
REPLY [2 votes]: I think this wording is extremely unclear. "If any two members of a set S satisfy condition P, then Q" parses (in my mind) as "If there exists a pair of elements in S that satisfy P, then Q". But the authors intend it to mean "If all pairs of elements in S satisfy P, then Q." I think that is the essence of your question. | 27,350 |
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Inflammation is the latest buzzword, which has seen a rise in the demand for anti-inflammatory foods, supplements and even life-style practices! Unlike some health trends, we can actually all benefit from the inclusion of more anti-inflammatory foods in our diet. Anti-inflammatory foods have a powerful presence in the body and offer a host of nutrients, minerals and anti-oxidant compounds to help us glow from within. Here we are going to explore how to beat inflammation through diet in the most delicious ways possible.
Inflammation Defined
Inflammation is the body’s natural response to injury (inflammatory trigger) and involves the production of pro-inflammatory chemicals to fight off the inflammatory trigger. If this response occurs day in day out due to a poor diet, it can leave us with tissue damage, accumulation of free radicals and a weakened defence system. Inflammation has been identified as an underlying driver of many chronic health conditions such as type 2 diabetes, gastrointestinal disease and even mood disorders.
Fighting Inflammation with Food
A diet rich in anti-inflammatory foods is hands down the best protection against inflammation and associated poor health.
The most cited pro-inflammatory foods are:
- High-glycaemic index carbohydrates (refined white bread, pasta, cereal)
- Processed and refined sugar found in cakes, biscuits, soft drinks
- Processed meats and high-intake of fatty red meat
- Vegetable oil, canola oil and deep-fried foods
- Dairy and gluten (if you are sensitive/intolerant to such foods)
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- Salmon, small white fish (whiting or snapper)
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- Green leafy vegetables
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You will see anti-inflammatory foods have a few things in common:
- Low-glycaemic index and high-fibre carbohydrates (slow-releasing complex carbohydrates for blood sugar regulation)
- Source of monounsaturated and polyunsaturated fatty acids
- Rich in vitamins, minerals and anti-oxidants (eg Vitamin C, Vitamin E, Carotene, Resveratrol)
If you are serious about targeting inflammation at its core, here are a few key players to pop on your plate:
Curcumin (turmeric)
Curcumin is the active compound found in the orange spice, turmeric. Curcumin is famous for its anti-inflammatory abilities and has been studied for its positive effects in inflammatory bowel disease, arthritis and even diabetes. Curcumin is best absorbed alongside a source of fat, try freshly grated turmeric in a home-made curry paste with coconut oil as the base or add to a smoothie alongside coconut or almond milk.
Omega-3 Fatty acids
Omega-3 fatty acids, which are found in fatty fish, chia seeds and walnuts exert an anti-inflammatory effect in the body whereas omega-6 is more pro-inflammatory. Omega-3 and omega-6 are both needed in the diet because humans do not have the enzymes required to make these fats. However, omega-3 and omega-6 compete for absorption and because omega-3 has a greater anti-inflammatory effect its important to eat a diet higher in omega-3 fatty acids and limit omega-6. The easiest way to do this is to by-pass heavily processed foods with added oils and favour whole foods.
Resveratrol
Resveratrol is associated with red wine and is in fact a polyphenol (antioxidant) found in the skin of grapes and blueberries. Resveratrol has both an anti-inflammatory and anti-oxidant effect on the body and has shown promise in various disease states such as arthritis, gastrointestinal disease and cardiovascular disease. Adding a handful of blueberries to your breakfast or a sprinkling of freeze-dried berry powder to smoothies is a delicious way to include resveratrol in your diet.
If you are looking to beat inflammation through diet, the key is to crowd out the inflammatory foods with anti-inflammatory foods to flip the balance towards greater health. And I promise, the taste won’t disappoint!
Try my Sweet Blueberry Marinated Salmon for a powerful combo of omega-3 fatty acids and Resveratrol!. | 38,558 |
\begin{document}
\address{Laboratoire Jean Leray, Universit\'e de Nantes,\\ 2, rue de la Houssinire \\ Nantes 44300, France\\
email:\,\tt{[email protected]}}
\title{On algebraic structures of the Hochschild complex}
\maketitle
\abstract{ We review various algebraic structures on the Hochschild homology and cohomology of a differential graded algebra $A$ under a weak Poincar\'e duality hypothesis. This includes a BV-algebra structure on $HH^*(A,A^\vee)$ or $HH^*(A,A)$, which in the latter case is an extension of the natural Gerstenhaber structure on $HH^*(A,A)$.
In sections 6 and 7 we construct similar structures for open Frobenius DG-algebras. In particular we prove that the Hochschild homology and cohomology of an open Frobenius algebra is a BV-algebra. In other words we prove that Hochschild chains complex is homotopical BV and coBV algebra. In Section 7 we present an action of the Sullivan diagrams on the Hochschild (co)chain complex of an open Frobenius DG-algebra. This recovers Tradler-Zeinalian \cite{TZ} result for closed Froebenius algebras using the isomorphism $C^*(A ,A) \simeq C^*(A,A^\vee)$.} Our description of the action can be easily and without much of modification extended to the homotopy Frobenius algebras.
\tableofcontents
\section{Introduction}
In this article we study the algebraic structures of Hochschild homology and cohomology of differential graded associative algebras over a field $\kk$ in four settings: Calabi-Yau algebras, derived Poincar\'e duality algebras, open Frobenius algebras and closed Frobenius algebras. For instance we prove the existence of a Batalin-Vilkovisky (BV) algebra structure on the Hochschild cohomology $HH^*(A,A)$ in the first two cases, and on the Hochschild cohomology $HH^*(A,A^\vee)$ in the last case. Let us explain the main motivation of the results presented in this chapter.
One knows from Chen \cite{Chen} and Jones \cite{Jones} work that the homology of $LM=C^\infty(S^1,M)$, the free loop space of a simply connected manifold $M$, can be computed by
\begin{equation}\label{eqgiso}
H_*(LM)\simeq HH^*(A,A^\vee),
\end{equation}
where $A=C^*(M)$ is the singular cochain algebra of $M$. Jones also proved an equivariant verion $H^{S^1}(LM)\simeq HC^*(A)$. Starting with Chas-Sullivan's work many different algebraic structures of $H_*(LM)$ have been discovered. This includes a BV-algebra structure on $H_*(LM)$ \cite{CS1}; and an action of Sullivan chord diagrams on $H_*(LM)$ which in particular implies that $H_*(LM)$ is an open Frobenius algebra. In order to find an algebraic model of these structures using the Hochschild complex and the isomorphism above, one has to equip the cochains algebra $A$ with further structures.
In order to find the Chas-Sullivan BV structure on $HH^*(A,A^\vee)$, one should take into account the Poincar\'e duality for $M$. Over a field $\kk$, we have a quasi-isomorphism $ A \rightarrow C_*(M) \simeq A^\vee $ given by capping with the fundamental class of $M$. Therefore one can use the result of Section 5 to find a BV-algebra structure on $HH^*(A,A^\vee)\simeq HH^*(A,A)$. Said more explicitly, $H_*(LM)$ is isomorphic to $HH^*(A,A)$ as a BV-algebra where the underlying Gerstenhaber structure of the BV structure on $HH^*(A,A)$ is the standard one (see Theorem \ref{Gersten}). This last statement which is true over a field, is a result to which many authors have contributed: \cite{CJ, FT2,Tradler,Mer}. The statement is not proved as yet for integer coefficients.
As we will see in Section \ref{secmoore}, an alternative way to find an algebraic model for the BV-structure of $H_*(LM)$ is via the Burghelea-Fiedorowicz-Goodwillie isomorphism $H_*(LM)\simeq HH_*(C_*(\Omega M), C_*(\Omega M))$, where $\Omega M$ is the based loop space of $M$. This approach has the advantage of working for all closed manifolds and it does not require $M$ to be simply connected. Moreover there is not much of a restriction on the coefficients \cite{Malm}.
Now we turn our attention to the action of the Sullivan chord diagrams and the open Frobenius algebra structure of $H_*(LM)$ (see \cite{CG}). For that one has to assume that the cochain algebra has some additional structures. The results of Section 7 show that in order to have an action of Sullivan chord diagram on
on $HH^*(A,A^\vee)$ and $HH_*(A,A)$ we have to start with an open Frobenius algebra structure on $A$. As far as we know, such structure is not known on $C^*(M)$ but only on the differential forms $\Omega^*(M)$ (see \cite{Wils}). Therefore the isomorphism (\ref{eqgiso}) is an isomorphism of algebras over the PROP of Sullivan chord diagrams if we work with real coefficients (see also \cite{CTZ}).
Here is a brief description of the organization of the chapter. In Section 2 we introduce the Hochschild homology and cohomology of a differential graded algebra and various classical operations such as cup and cap product. We also give the definition of Gerstenhaber and BV-algebras. In particular we give an explicit description of the Gerstenhaber algebra structure one the Hochschild cohomology of $A$ with coefficient in $A$, $HH^*(A,A)$. In Section 3 we explain how we can see Hochschild (co)homology as a derived functor. The section includes a quick review of model categories which can be skipped by the reader.
In Sections 4 and 5 we work with algebras which verify a sort of derived Poincar\'e duality rather than being equipped with an inner product. In these two sections, we introduce a BV structure on $HH^*(A,A)$, whose underlying Gerstenhaber structure is the standard one (see Section 2).
In Section 6, we show that the Hochschild homology $HH_*(A,A)$ and cohomology $HH^*(A,A^\vee)$ of an open Frobenius algebra $A$, are BV-algebras. Note that we don't find a BV-algebra structure on $HH^*(A,A)$ although it is naturally a Gerstenhaber algebra (see Section 2).
In Section 7, we aim at constructing an action of the Sullivan chord diagrams on the Hochschild chains of an open Frobenius algebra $A$, which can be extended to an action of the homology of the moduli space of curves. In particular there is a BV and coBV structure on $HH_*(A,A)$ and on the dual theory $HH^*(A,A^\vee)$. Our construction is based on \cite{TZ} for closed Frobenius algebras. This formulation is very much suitable for an extension to the moduli as it is given in \cite{C} and \cite{CTZ}. An open Frobenius algebra with a counit can be naturally equipped with a symmetric inner product. If the inner product induces a quasi-isomorphism $A\simeq A^\vee$ (of $A$-bimodules), then we obtain a BV structure on $HH^*(A,A)$ whose underlying Gerstenhaber structure is the standard one \cite{Gers}. Finally in Section 8 we will show how these BV structures induce a graded Lie algebra, and even better a gravity algebra structure on the cyclic cohomology of $A$.
\bigskip
\noindent\textbf{Acknowledgment:} I would like to thank Janko Latschev and Alexandru Oancea for encouraging me to write this chapter. I am indebted to Aur\'elien Djament, Alexandre Quesney and Friedrich Wagemann for reading the first draft and suggesting a few corrections. I am also grateful to Nathalie Wahl for many helpful communications.
\section{Hochschild Complex} \label{section-CH}
Throughout this paper $\kk$ is a field. Let $A=\kk\oplus \bar{A}$ be an augmented unital differential $\kk$-algebra with $\deg d_A=+1$, $\bar{A}=A/\kk$ or $\bar{A}$ is the kernel of the augmentation $\epsilon: A \rightarrow \kk$.
A differential graded $(A,d)$-module, or $A$-module for short, is a $\kk$-complex $(M,d)$ together with an (left) $A$-module structure
$\cdot: A\times M\rightarrow M$ such that $ d_M(am)=d_A(a)m+(-1)^{|a|}a d_M(m)$. The multiplication map is of degree zro \emph{i.e.} $\deg(am)=\deg a+\deg m $. In particular, the identity above implies that the differential of $M$ has to be of degree $1$.
Similarly for a $(M,d_M)$ a graded differential $(A,d)-bimodule$, we have
$$
d_M(amb)=d_A(a)mb-(-1)^{|a|}ad_M (m)b+(-1)^{|a|+|m|}amd_Ab,
$$
or equivalently, $M$ is a $(A^e:=A\otimes A^{op}, d_A\otimes 1 +1\otimes d_A)$ DG-module where $A^{op}$ is the algebra whose
underlying graded vector space is $A$ with the opposite multiplication of $A$, \emph{i.e.} $a\overset{op}{\cdot}b=(-1)^{|a|.|b|}b\cdot a$.
From now on $Mod(A)$ denotes the category of (left or right) (differential) $A$-modules and $Mod(A^e)$ denotes the category of differential $A$-bimodules. All modules considered in this article are differential modules. We will also drop the indices from the differential when there is no possibility of confusion.
We recall that the \emph{two-sided bar construction} (\cite{CarEilen,MacLane})is given by
$B(A,A,A):= A\otimes T(s\bar{A})\otimes A$ equipped with the differential $d=d_{0}+d_{1}$ where
\begin{equation}
\begin{split}
d_{1}(a[a_{1}, \cdots , a_{n}] b)=(-1)^{|a|}aa_{1}[a_{2}, \cdots , a_{n}] b+\\ \sum_{i=1}^{n-1}
(-1)^{\epsilon_{i}}a[a_{1} , \cdots , a_{i}a_{i+1}, \dots , a_{n}] b\\ -(-1)^{\epsilon_{n}}a[a_{1}, \cdots , a_{n-1}]
a_{n}b
\end{split}
\end{equation}
and $d_{0}$ is the internal differential for the tensor product complex $A\otimes T(s\bar{A})\otimes A$. Here $\epsilon_{i}=|a_{1}| +\cdots
|a_{i}|-i$. The degree on $B(A,A,A)$ is defined by $\deg (a[a_{1}, \cdots , a_{n}]b)=\sum_{i=1}^n |s(a_i)|=|a|+|b|+ \sum_{i=1}^n |a_i|-n$, therefore $\deg (d_0+d_1)=+1$. We recall that $sA$ stands for the suspension of $A$, \emph{i.e.} the shift in degree by $-1$.
We equip $A$ and $A\otimes_\kk A$, or $A\otimes A$ for short, with the \emph{outer} $A$-bimodule structure that is $a(b_1\otimes b_2)c=(ab_1)\otimes (b_2c)$.
Similarly $B(A,A,A)$ is equipped with the outer $A$-bimodule structure. This is a free resolution of $A$ as an $A$-bimodule which allows us to define \emph{Hochschild chains and cochains} of $A$ with
coefficients in $M$. Then \emph{(normalized) Hochschild chain complex} with coefficients in $M$ is
\begin{equation}
C_*(A,M):= M \otimes_{A^e} B(A,A,A) =M \otimes T(s \bar{A})
\end{equation}
and comes equipped with a degree +1 differential $ D=d_{0}+d_{1}$. We recall that $TV=\oplus_{n\geq 0} V^{\otimes n}$ denotes the tensor algebra of a $\kk$-module $V$.
The internal differential is given by
\begin{equation}
\begin{split}
d_{0}(m [a_{1},\cdots , a_{n}])&= \sum_{i=1}^{n-1} (-1)^{\epsilon_{i}}m [a_{1},\cdots,d_A a_{i},\dots a_{n}]\\
&-(-1)^{\epsilon_{n}}d_M m [a_{1},\cdots, a_{n}],
\end{split}
\end{equation}
and the external differential is
\begin{equation}
\begin{split}
d_{1}(m [a_{1}, \cdots , a_{n}] )&= m a_{1} [a_{2}, \cdots , a_{n}]+\\ &\sum_{i=1}^{n-1} (-1)^{\epsilon_{i}} m [a_{1},
\cdots a_{i}a_{i+1} \cdots \,a_{n}]\\ &-(-1)^{\epsilon_{n}} a_{n}m [a_{1}, \cdots , a_{n-1}],
\end{split}
\end{equation}
with $\epsilon_{i}=|a_{1}| +\cdots |a_{i}|-i $. Note that the degree of $m [a_{1}, \cdots , a_{n}]$ is $\sum_{i=1}^{n}|a_i|-n+|m|$.
When $M=A$, by definition $(C_*(A), D=d_0+d_1):=(C_*(A,A),D=d_0+d_1)$ is the \emph{Hochschild chain complex of} $A$ and $HH_*(A,A):=\ker
D/\im D$ is the Hochschild homology of $A$.
Similarly we define the $M$-valued \emph{Hochschild cochain} of $A$ to be the dual complex $$ C^*(A,M):=\Hom_{A^e} (B(A,A,A),
M)=\Hom_{\kk}(T(s\bar{A}),M). $$ For a homogenous cochain $f\in C^n(A,M)$, the degree $|f|$ is defined to be the degree of the linear
map $f: (s\bar{A})^{\otimes n}\rightarrow M$. In the case of Hochschild cochains, the external differential of $f\in \Hom (s\bar{A}^{\otimes n},
M)$ is
\begin{equation}
\begin{split}
d_{1}(f)(a_{1}, \cdots , a_{n})&=-(-1)^{(|a_1|+1)|f|}a_{1}f(a_{2}, \cdots , a_{n})+\\ &-\sum_{i=2}^{n} (-1)^{\epsilon_i} f(a_{1}, \cdots ,
a_{i-1}a_{i}, \cdots a_{n})+(-1)^{\epsilon_n} f( a_{1}, \cdots , a_{n-1})a_n,
\end{split}
\end{equation}
where $\epsilon_i=|f|+|a_1|+\cdots +|a_{i-1}|-i+1$. The internal differential of $f\in C^*(A,M)$ is
\begin{equation}
\begin{split}
d_{0}f(a_{1}, \cdots , a_{n})&=d_Mf(a_{1}, \cdots , a_{n})-\sum_{i=1}^n (-1)^{\epsilon_i} f(a_{1}, \cdots d_A a_i \cdots , a_{n}).
\end{split}
\end{equation}
\subsubsection*{Gerstenhaber bracket and cup product:}
When $M=A$, for $x\in C^m(A,A)$ and $y\in C^n(A,A)$ one defines the \emph{cup product} $x\cup y\in C^{m+n}(A,A)$ and the \emph{Gerstenhaber
bracket} $[x,y] \in C^{m+n-1}(A,A)$ by
\begin{equation}
(x\cup y)(a_{1},\cdots , a_{m+n}):=(-1)^{|y|(\sum_{i\leq m} |a_{i}|+1)}x(a_{1},\cdots , a_{m})y(a_{n+1}, \cdots , a_{m+n}),
\end{equation}
and
\begin{equation}
[x,y]:= x\circ y - (-1)^{(|x|+1)(|y|+1)}y\circ x,
\end{equation}
where
$$
(x\circ_j y)( a_{1},\cdots ,a_{m+n-1})=(-1)^{(|y|+1)\sum_{i\leq j} (|a_{i}|+1)} x(a_{1}, \cdots , a_{j}, y(a_{j+1}, \cdots , a_{j+m}),\cdots ).
$$
and
\begin{equation}
x\circ y=\sum_j x\circ_j y
\end{equation}
Note that this is not an associative product. It turns out that the operations $\cup$ and $[-,-]$ are chain maps, hence they define two well-defined operations on $HH^*(A,A)$. Moreover,
\begin{theorem}\label{Gersten}(Gerstenhaber \cite{Gers}) $(HH^*(A,A),\cup,[-,-])$ is a \textit{Gerstenhaber algebra} that is:
\begin{enumerate}
\item $\cup$ is an associative and graded commutative product,
\item $[x,y\cup z] = [x,y]\cup z + (-1)^{(|x|-1)|y|}y\cup[x,z] $ (Leibniz rule),
\item $[x,y] = (-1)^{(|x|-1)(|y|-1)} [y,x] $,
\item $[[x,y],z] = [x,[y,z]] -(-1)^{(|x|-1)(|y|-1)}[y,[x,z]] $ (Jacobi identity).
\end{enumerate}
The homotopy for the commutativity of the cup product $x\cup y$ is given by $x\circ y$.
\end{theorem}
In this article we show that under some kind of Poincar\'e duality condition this Gerstanhaber structure is part of a BV structure.
\begin{definition}\label{BV-def}(Batalin-Vilkovisky algebra) A BV-algebra is a Gerstenhaber algebra $(A^*,\cdot,[-,-])$ with a degree one operator
$\Delta: A^*\rightarrow A^{*+1}$ whose deviation from being a derivation for the product $\cdot$ is the bracket $[-,-]$, \textit{i.e.}
$$ [a,b]:=(-1)^{|a|}\Delta(ab)-(-1)^{|a|}\Delta(a)b-a \Delta(b), $$
and $\Delta^2=0$.
\end{definition}
It follows from $\Delta^2=0$ that $\Delta$ is a derivation for the bracket.
In fact the Leibniz identity for $[-,-]$ is equivalent to the 7-term relation \cite{Getz}
\begin{equation}\label{7term}
\begin{split}
\Delta(abc)&= \Delta (ab)c +(-1)^{|a|} a\Delta(bc)+ (-1)^{(|a|-1)|b|}b\Delta (ac)\\
&-\Delta (a) bc - (-1)^{|a|}a \Delta(b) c-(-1)^{|a|+|b|}ab\Delta c.
\end{split}
\end{equation}
Definition \ref{BV-def} is equivalent to the following one:
\begin{definition}
A BV-algebra is a graded commutative associative algebra $(A^*,\cdot)$ equipped with a degree one operator $\Delta: A^*\rightarrow A^{*+1}$ which satisfies the 7-term relation (\ref{7term}) and $\Delta^2=0$. It follows from the 7-term relation that $[a,b]:=(-1)^{|a|}\Delta(ab)-(-1)^{|a|}\Delta(a)b-a \Delta(b)$ is a Gerstenhaber bracket for the graded commutative associative algebra $(A^*,\cdot)$.
\end{definition}
As we said before the Leibniz identity is equivalent to the 7-term identity and the Jacobi identity follows from $\Delta^2=0$ and the 7-term identity. We refer the reader interested in the homotopic aspects of BV-algebras to \cite{ColVal}.
For $M=A^\vee:=\Hom_\kk(A,\kk)$, by definition $(C^*(A), D=d_0+d_1):=(C^*(A,A^\vee),d_0+d_1)$ is the \emph{Hochschild cochain complex of}
$A$ and $HH^*(A):=\ker D/ \im D$ is the \emph{Hochschild cohomology} of $A$. It is clear that $C^*(A)$ and $\Hom_k(C_*(A),k)$ are isomorphic as $k$-complexes, therefore the Hochschild cohomology $A$ is the dual theory of the
Hochschild homology of $A$. The Hochschild homology and cohomology of an algebra have an extra feature and that is Connes operator $B$ (\cite{connes}). On the chains we have
\begin{equation}\label{B1}
B(a_0 [a_1, a_2 \cdots, a_n ])=\sum_{i=1}^{n+1}(-1)^{\epsilon_i} 1[a_{i+1} \cdots a_n,a_0,\cdots,a_i]
\end{equation}
and on the dual theory $C^*(A)=\Hom_\kk (T(s\bar{A}),A^\vee)=\Hom (A \otimes T(s\bar{A}), \kk)$ is given by
$$ (B^\vee \phi)(a_0 [a_1, a_2 \cdots, a_n ])=(-1)^{|\phi|}\sum_{i=1}^{n+1}(-1)^{\epsilon_i} \phi (1[a_{i+1} \cdots a_n,a_0,\cdots,a_i] ) $$
where $\phi \in C^{n+1}(A)=\Hom(A\otimes (s\bar{A})^{\otimes n+1} , \kk)$ and $\epsilon_i=(|a_0|+\dots |a_{i-1}|-i)(|a_i|+\dots |a_{n}|-n+i-1)$. In other words $$B^\vee (\phi)= (-1)^{|\phi|} \phi \circ B.$$
Note that $
\deg(B)=-1 \text{ and } \deg B^\vee=+1.
$
\subsubsection*{Warning:} The degree $k$ of a cycle $x\in HH_k(A,M)$, is not given by the number terms in a tensor product but by the total degree.
\begin{remark} In this article we use normalized Hochschild chains and cochains. It turns out that they are quasi-isomorphic to the
non-normalized Hochschild chains and cochains. The proof is the same as the one on page 46 of \cite{Loday} for the algebras. One only
has to modify the proof to the case of simplicial objects in the category of differential graded algebras. The proof of the Lemma 1.6.6 of \cite{Loday} works in this setting since the degeneracy maps commute with the internal differential of a simplicial differential graded algebra.
\end{remark}
\subsubsection*{Chain and cochain pairings and noncommutative calculus} Here we borrow some definitions and facts from noncommutative calculus
\cite{CTS}. Roughly said, one should think of $HH^*(A,A)$ and $HH_*(A,A)$ respectively as multi-vector fields and differential forms, and of $B$ as the
de Rham differential.
\subsubsection*{1. Contraction or cap product:} The pairing between $a_0 [a_1,\cdots, a_n] \in C_n(A,A)$ and $f\in C^k(A,A)$, $n\geq k$ is given by
\begin{equation}\label{contr-1t}
i_f(a_0 [a_1,\cdots, a_n])=(-1)^{|f|(\sum_{i=1}^k (|a_i|+1) )}a_0 f(a_1,\cdots, a_k)[ a_{k+1},\cdots, a_n]\in C_{n-k}(A,A).
\end{equation}
It is a chain map $C^*(A,A)\otimes C_*(A,A)\rightarrow C_*(A,A)$ and it induces a pairing at cohomology and homology level.
\subsubsection*{2. Lie derivative:} The next operation is the infinitesimal Lie algebra action of $HH^*(A,A)$ on $HH_*(A,A)$ and is given by Cartan's formula
\begin{equation}
L_{f}=[B,i_f].
\end{equation}
Note that the Gerstenhaber bracket on $HH^*(A,A)$ becomes a (graded) Lie bracket after a shift of degrees by one.
This explains also the sign convention below. The triple $i_f$, $L_f$ and $ B$ form a calculus \cite{CTS} that is,
\begin{eqnarray}
L_f&=&[i_f,B]\\
i_{[f,g]}&=&[L_f,i_g]\\
i_{f\cup g}&=& i_f\circ i_g\\
L_{[f,g]}&=&[L_f,L_g]\\
L_{fg}&=&[L_f,i_g]
\end{eqnarray}
As $HH^*(A,A)$ acts on $HH_*(A)=HH_*(A,A)$ by contraction, it also acts on the dual theory \textit{i.e.} $HH^*(A)= HH^*(A,A^\vee)$. More explicitly, $i_f (\phi) \in
C^{n}(A,A^\vee)$ is given by
\begin{equation}\label{contr-2}
i_f(\phi)(a_0[a_1,\cdots, a_n]):=(-1)^{|f|(|\phi|+ \sum_{i=0}^k (|a_i|+1) )}\phi (a_0 [f(a_1,\cdots, a_k), a_{k+1},\cdots, a_n])
\end{equation}
where $\phi\in C^{n-k}(A,A^\vee)$ and $f\in C^k (A,A)$, in other words
$$
i_f(\phi) :=(-1)^{|f||\phi|} \phi\circ i_f.
$$
\section{Derived category of DGA and derived functors}
Now we try to present the Hochschild (co)homology in a more conceptual way \emph{i.e.} as a derived functor on the category of $A$-bimodules. We must first introduce an appropriate class of objects which can approximate all $A$-bimodules. This is done properly using the concept of model category introduced by Daniel Quillen \cite{Qui}. It is also he right language for constructing homological invariants of homotopic categories. It will naturally lead us to the construction of derived categories as well.
\subsection{A quick review of model categories and derived functors}
The classical references for this subject are Hovey's book \cite{Hov} and the Dwyer-Spalinsky manuscript \cite{DwSpa}.
The reader who gets to know the notion of model category for the first time, should not worry about the word ``closed'' which now has only a historical bearing. From now on we drop the word ``closed'' from closed model category.
\begin{definition} Let $\c$ be a category with three classes of morphisms $\cC$ (cofibrations), $\cF$(fibrations) and $\cW$ (weak equivalences) such that:\\
\noindent (MC1) $\c$ is closed under finite limits and colimits.\\
\noindent(MC2) Let $f,g\in Mor(\c)$ such that $fg$ is defined. If any two among $f,g$ and $fg$ are in $\cW$, then the third one is in $\cW$.\\
\noindent (MC3) Let $f$ be a retract of $g$. If $g\in \cC$ (resp. $\cF$ or $\cW)$, then $f\in \cC$ (resp. $\cF$ or $\cW)$.
\noindent(MC4) For a commutative diagram, as below, with $i \in \cC$ and $p\in \cF$, the morphism $f$ making the diagram commutative exists if
\begin{numlist}
\item $i\in \cW$ (left lifting property (LLP) of fibrations $f\in\cF$ with respect to acyclic cofibrations $i\in \cW\cap \cC$).
\item $p\in \cW$ (right lifting property (RLP) of cofibrations $i\in\cC$ with respect to acyclic fibrations $p\in \cW\cap \cF$).
\end{numlist}
\begin{equation}
\xymatrix{A\ar[d]_-i\ar[r] & X\ar[d]^-p\\ B\ar@{-->}[ru]^-f\ar[r] & Y}
\end{equation}
The reader should have noticed that we call the elements of $ \cW\cap \cC$ (resp. $\cW\cap \cF$) acyclic cofibrations (resp. fibrations).
\noindent(MC5) Any morphism $f:A\rightarrow B$ can be written as one of the following:
\begin{numlist}
\item $f=pi$ where $p\in \cF$ and $i\in \cC\cap \cW $;
\item $f=pi$ where $p\in \cF\cap \cW $ and $i\in \cC $.
\end{numlist}
\end{definition}
In fact in a model category the lifting properties characterize the fibrations and cofibrations:
\begin{proposition}
In a model category:
\begin{romlist}
\item The cofibrations are the morphisms which have the RLP with respect to
acyclic fibrations.
\item The acyclic cofibrations are the morphisms which have the RLP with respect
to fibrations.
\item The fibrations are the morphisms which have the LLP with respect to acyclic
cofibrations.
\item The acyclic fibrations in C are the maps which have the LLP with respect
to cofibrations.
\end{romlist}
\end{proposition}
It follows from (MC1) that a model category $\c$ has an initial object $\emptyset$ and a terminal object $\ast$. An object $A\in Obj(\c)$ is called \emph{cofibrant} if the morphism $\emptyset
\rightarrow A$ is a cofibration and is said to be \emph{fibrant} if the morphism
$A\rightarrow \ast$ is a fibration.
\noindent \textbf{Example 1:}\label{ex-CHR} For any unital associative ring $ R$, let $\ch(R)$ be the category of non-negatively graded chain complexes of left $R$-modules.
The following three classes of morphisms endow $\ch(R)$ with a model category structure:
\begin{numlist}
\item Weak equivalences $\cW$ are the quasi-isomorphims \text{i.e.} maps of $R$-complexes $f=\{f_k\}_{k\geq 0}:\{M_k\}_{k\in \Z}\rightarrow \{N_k\}_{k\geq 0}$ inducing an isomorphism $f_*: H_*(M)\rightarrow H_*(N)$ in homology.
\item Fibrations $\cF$: $f$ is a fibration if it is (componentwise) surjective \textit{i.e} for all $k\geq 1$, $f_k: M_k\rightarrow N_k$ is surjective.
\item Cofibrations $\cC$: $f=\{f_k\}$ is a cofibration if for all $k\geq 0$, $f_k: M_k\rightarrow N_k$ is injective with a projective $R$-module as its cokernel.
Here projective is the standard notion \text{i.e.} \emph{a direct summand of free $R$-module}.
\end{numlist}
\noindent \textbf{Example 2:}
The category \textbf{Top} of topological spaces can be given the structure
of a model category by defining a map $f : X \rightarrow Y$ to be
\begin{romlist}
\item a weak equivalence if $f$ is a homotopy equivalence;
\item a cofibration if $f$ is a Hurewicz cofibration;
\item a fibration if $f$ is a Hurewicz fibration.
\end{romlist}
Let $A$ be a closed subspace of a topological space $B$. We say that the inclusion $i:A\hookrightarrow B$ is a \emph{Hurewicz cofibration} if it has the homotopy extension property that is for all maps $f:B\rightarrow X$, any homotopy $F:A\times [0,1]\rightarrow X$ of $f|_A$ can be extended to a homotopy of $f:B\rightarrow X$.
\begin{equation*}
\xymatrix{B\cup (A\times [0,1]) \ar[r]^-{f\cup F} \ar[d]_-{ id\times {0}\cup (i\times id)} & X \\ B \times[0,1]\ar@{-->}[ur]& }
\end{equation*}
A \emph{Hurewicz fibration} is a continuous map $E\rightarrow B$ which has the homotopy lifting property with respect to all continuous maps $X\rightarrow B$, where $X\in$ \textbf{Top}.
\noindent \textbf{Example 3:} The category \textbf{Top} of topological spaces can be given the
structure of a model category by defining $f : X \rightarrow Y$ to be
\begin{romlist}
\item a weak equivalence when it is a weak homotopy equivalence.
\item a cofibration if it is a retract of a map $X \rightarrow Y'$ in which $Y'$ is obtained
from $X$ by attaching cells,
\item a fibration if it is a Serre fibration.
\end{romlist}
We recall that a \emph{Serre fibration} is a continuous map $E\rightarrow B$ which has the homotopy lifting property with respect to all continuous maps $X\rightarrow B$ where $X$ is a CW-complex (or equivalently cubes).
\bigskip
\subsubsection*{Cylinder, path objects and homotopy relation.} After setting up the general framework, we define the notion of homotopy. A \emph{cylinder object} for $A\in obj(\c)$ is an object $A\wedge I \in obj (\c)$ with a \emph{weak equivalence}
$\sim: A\wedge I \rightarrow A$ which factors the natural map $id_A\sqcup id_A : A\coprod A \rightarrow A$:
$$
id_A\sqcup id_A:A \coprod A \overset{i}{\rightarrow} A\wedge I \overset{\sim}{\rightarrow} A
$$
Here $A\coprod A\in obj(\c) $ is the colimit, for which one has two structural maps $in_{0},in_{1}:A\to A\coprod A $. Let $i_{0}= i\circ in_{0}$ and $i_{1}= i\circ in_{1}$. A cylinder object $A\wedge I$ is said to be \emph{good} if $A \coprod A \rightarrow A\wedge I$ is a cofibration. By (MC5), every $A \in obj(\c)$ has a good cylinder object.
\begin{definition}
Two maps $f,g: A\rightarrow B $ are said to be \emph{left homotopic} $ f\overset{l}{\sim} g$ if there is a cylinder object $A\wedge I$ and $H: A\wedge I\rightarrow B$ such that $f=H\circ i_{0}$ and $g=H\circ i_{1}$. A left homotopy is said to be \emph{good} if the cylinder object $A\wedge I$ is good. It turns out that every left homotopy relation can be realized by a good cylinder object. In addition one can prove that if $B$ is a fibrant object, then a left homotopy for $f$ and $g$ can be refined into a \emph{very good one} \emph{i.e} $A\wedge I\rightarrow A$ is a fibration.
\end{definition}
It is easy to prove the following:
\begin{lemma}\label{equi-rel}
If $A$ is cofibrant, then left homotopy $\overset{l}{\sim}$ is an equivalence relation on $\Hom_{\c}(A,B)$.
\end{lemma}
Similary, we introduce the notion of path objects which will allow us to define right homotopy relation. A \emph{path object} for $A\in obj(\c)$ is an object $A^{I}\in obj(\c)$ with a weak equivalence $A\overset{\sim}{\rightarrow} A^I$ and a morphism $p:A^{I}\rightarrow A\times A$
which factors the diagonal map
$$
(id_{A},id_{A}): A\overset{\sim}{\rightarrow} A^{I}\overset{p}{\rightarrow}A\times A
$$
Let $pr_{0}, pr_{1}: A\times A \rightarrow A$ be the structural projections. Define $p_{i}=pr_{i}\circ p$. A path object $A^ I$ is said to be \emph{good} if $A^{I}\rightarrow A\times A$ is a fibration. By (MC5) every $A \in obj(\c)$ has a good path object.
\begin{definition} Two maps $f,g: A\rightarrow B $ are said to be \emph{right homotopic} $ f\overset{r}{\sim} g$ if there is a path object $B^ I$ and $H: A\rightarrow B^I$ such that $f=p_{0}\circ H$ and $g=p_{1}\circ H$. A right homotopy is said to be good if the path object $P^I$ is good. It turns out that every right homotopy relation can be refined into a good one. In addition one can prove that if $B$ is a cofibrant object then a right homotopy for $f$ and $g$ can be refined into a \emph{very good one} \emph{i.e} $B\rightarrow B^I$ is a cofibration.
\end{definition}
\begin{lemma}\label{equi-rel-r}
If $B$ is fibrant, then right homotopy $\overset{r}{\sim}$ is an equivalence relation on $\Hom_{\c}(A,B)$
\end{lemma}
One naturally asks whether being right and left homotopic are related. The following result answers this question.
\begin{lemma}\label{r-l} Let $f,g: A\rightarrow B$ be two morphisms in a model category $\c$.
\begin{numlist}
\item If $A$ is cofibrant then $f\overset{l}{\sim}g$ implies $f\overset{r}{\sim}g$
\item If $B$ is fibrant then $f\overset{r}{\sim}g$ implies $f\overset{l}{\sim}g$.
\end{numlist}
\end{lemma}
\subsubsection*{Cofibrant and Fibrant replacement and homotopy category.} By applying (MC5) to the canonical morphism
$\emptyset \rightarrow A$, there is a cofibrant object (not unique) $QA$ and an \emph{acyclic fibration} $p:QA\overset{\sim}{\rightarrow} A $
such that $\emptyset \rightarrow QA\overset{p}{\rightarrow} A$. If $A$ is cofibrant we can choose $QA=A$.
\begin{lemma}\label{lift}
Given a morphism $f:A\rightarrow B$ in $\c$, there is a morphism $\tilde{f}:QA\rightarrow QB$ such that the following diagram commutes:
\begin{equation}
\xymatrix{QA\ar[d]^-{p_A} \ar[r]^-{\tilde{f}} & QA \ar[d]^-{p_B}\\ A \ar[r]^-f & B }
\end{equation}
The morphism $\tilde{f}$ depends on $f$ up to left and right homotopy, and is a weak equivalence if and only $f$ is. Moreover, if $B$ is fibrant then
the right or left homotopy class of $\tilde{f}$ depends only on the left homotopy class of $f$.
\end{lemma}
Similarly one can introduce a fibrant replacement by applying (MC5) to the terminal morphism $A\rightarrow \ast$ and obtain a fibrant object $RA$ with an \emph{acyclic cofibration} $i_A:A\rightarrow RA$.
\begin{lemma}\label{desc}
Given a morphism $f:A\rightarrow B$ in $\c$, there is a morphism $\tilde{f}:RA\rightarrow RB$ such that the following diagram commutes:
\begin{equation}
\xymatrix{A\ar[d]^-{i_A} \ar[r]^-{f} & B \ar[d]^-{i_B}\\ RA \ar[r]^-{\tilde{f}} & RB }
\end{equation}
The morphism $\tilde{f}$ depends on $f$ up to left and right homotopy, and is a weak equivalence if and only $f$ is. Moreover, if $A$ is cofibrant then right or left homotopy class of $\tilde{f}$ depends only on the right homotopy class of $f$.
\end{lemma}
\begin{remark}\label{remk-fib-cofib}
For a cofibrant object $A$, $RA$ is also cofibrant because the trivial morphism $(\emptyset \rightarrow RA)=(\emptyset \rightarrow A \overset{i_A}{\rightarrow}RA)$ can be written as the composition of two cofibrations, therefore is a cofibration.
In particular, for any object $A$, $RQA$ is fibrant and cofibrant. Similarly, $QRA$ is a fibrant and cofibrant object.
\end{remark}
Putting the last three lemmas together, one can make the following definition:
\begin{lemma}\label{w-c-f}Suppose that $f : A \rightarrow X $ is a map in $\c$ between objects $A$ and $X$
which are both fibrant and cofibrant. Then $f$ is a weak equivalence if and only if f
has a homotopy inverse, \emph{i.e.}, if and only if there exists a map $g : X \rightarrow A$ such that
the composites $gf$ and $fg$ are homotopic to the respective identity maps.
\end{lemma}
\begin{definition} The \emph{homotopy category} $\Ho(\c)$ of a model category $\c$ has the same objects as $\c$ and the morphism set $\Hom_{\Ho(\c)}(A,B)$ consists of the (right or left) homotopy classes of the morphism $\Hom_{\c}(RQA,RQB)$. Note that since $RQA$ and $RQB$ are fibrant and cofibrant, the left and right homotopy relations are the same. There is a natural functor
$H_{\c}: \c\rightarrow \Ho(\c)$ which is the identity on the objects and sends a morphism $f:A\rightarrow B$ to the homotopy class of the morphism
obtained in $\Hom_{\c}(RQ A,RQ B)$ by applying consecutively Lemma \ref{lift} and Lemma \ref{desc}.
\end{definition}
\subsubsection*{Localization functor.} Here we give a brief conceptual description of the homotopy category of a model category. This description relies only on the class of weak equivalences and suggests that the weak equivalences encode most of the homotopic properties of the category. Let $W$ be a subset of the morphisms in a category $\c$. A functor $F:\c\rightarrow \mathbf{D}$ is said to be a \emph{localization} of $\c$ with respect to $W$ if the elements of $W$ are sent to isomorphisms and if $F$ is universal for this property \emph{i.e.} if $G:\c \rightarrow \mathbf{D'}$ is another localizing functor then $G$ factors through $F$ via a functor $G':\mathbf{D}\rightarrow \mathbf{D'}$ for which $G'F=G$. It follows from Lemma \ref{w-c-f} and a little work that:
\begin{theorem} For a model category $\c$, the natural functor $H_{\c}: \c\rightarrow \Ho(\c)$ is a localization of $\c$ with respect to the weak equivalences.
\end{theorem}
\subsubsection*{Derived and total derived functors:}
In this section we introduce the notions of \emph{left derived} $LF$ and \emph{right derived} $RF$ of a functor $F:\c\rightarrow \d$ of model categories. In particular, we spell out the necessary conditions for the existence of $LF$ and $RF$ which provide us a factorization of $F$ via the homotopy categories. All functors considered here are covariant, however see Remark \ref{rem-op}.
\begin{definition} For a functor $F:\c\rightarrow \d$ on a model category $\c$, we consider all pairs $(G,s)$ where $G:\Ho(\c)\rightarrow\d$ is a functor and $s:G H_{\c}\rightarrow F$ is a natural transformation. The \emph{left derived} functor of $F$ is such a pair $(LF,t)$ which is universal from left \emph{i.e.} for another such pair $(G,s)$ there is a unique natural transformation $t': G\rightarrow LF$ such that $t(t'H_C):GH_{\c}\rightarrow F$ is $s$.
Similarly one can define the right derived functor $RF: \Ho(\c) \rightarrow D$ which provides a factorization of $F$ and satisfies the usual universal property from the right. A right derived functor for $F$ is a pair $(RF,t)$ where $RF:\Ho(\c)\rightarrow \d$ and $t$ is a natural transformation $t: F \rightarrow RFH_{\c} $ such that for any such pair $(G,s)$ there is a unique natural transformation $t':RF\rightarrow G$ such that $t'H_C t:F \rightarrow GH_{\c}$ is $s$.
The reader can easily check that the derived functors of $F$ are unique up to canonical equivalence.
\end{definition}
The following result tells us when do derived functors exist.
\begin{proposition}\label{prop-derived}
\begin{numlist}
\item Suppose that $F:\c\rightarrow \d$ is a functor from a model categories $\c$ to a category $d$, which sends acyclic cofibration between cofibrant objects to isomorphims. Then $(LF,t)$ the left derived functor of $F$ exists. Moreover for any cofibrant object $X$ the map $t_x:LF(X) \rightarrow F(X)$ is an isomorphism.
\item Suppose that $F:\c\rightarrow \d$ is a functor between two model categories which sends acyclic fibrations between fibrant objects to isomorphisms. Then $(RF,t)$ the right derived functor of $F$ exists. Moreover for all fibrant object $X$ the map $t_X: RF(X) \rightarrow F(X) $ is an isomorphism.
\end{numlist}
\end{proposition}
\begin{definition} Let $F:\c\rightarrow \d$ be a functor between two model categories. The \emph{total left derived functor} $\L F:\Ho(\c)\rightarrow \Ho(\d)$ is a left derived functor of $H_{\d} F: \c\rightarrow \Ho(\d)$. Similarly one defines the total right derived functor $\R F:\c\rightarrow \d$ to be the right derived functor of $H_{\d} F: \c\rightarrow \Ho(\d)$.
\end{definition}
\begin{remark}\label{rem-op}
Till now we have defined and discussed the derived functor for covariant functors. We can defined the derived functors for contravariant functors as well, for that we have to only work with the opposite category of the source of the functor. A morphism $A\rightarrow B$ in the opposite category is a cofibration (resp. fibration, weak equivalence) if the corresponding morphism $B\rightarrow A$ is a fibration (resp. cofibration, weak equivalence).
\end{remark}
We finish this section with an example.
\medskip
\noindent \textbf{Example 4:} Consider the model category $\ch(R)$ of Example 1 in Section 3 and let $M$ be a fixed $R$-module. One defines the functor $F_M:\ch(R)\rightarrow \ch(\Z)$ given by $F_M(N_*)=M\otimes_R N_*$ where $N_*\in \ch(R)$ is a complex of $R$-modules. Let us check that $F=H_{\ch(R)}F_M:\ch(R)\rightarrow \ch(\Z)$ satisfies the conditions of Proposition \ref{prop-derived}.
Note that in $\ch(R)$ every object is fibrant and a complex $A_*$ is cofibrant if for all $k$, $A_k$ is a projective $R$-module. We have to show that
an acyclic cofibration $f: A_*\rightarrow B_*$ between cofibrant objects $A$ and $B$ is sent by $F$ to an isomorphism. So for all $k$, we have a short exact sequence $0\rightarrow A_*\rightarrow B_*\rightarrow B_*/A_*\rightarrow 0$ where for all $k$, $B_k/A_k$ is also projective.
Since $f$ is a quasi-isomorphism the homology long exact sequence of this short exact sequence tells us that the complex $B_*/A_*$ is acyclic.
The lemma below shows that $ B_*/A_*$ is in fact a projective complex. Therefore we have $B_*\simeq A_*\oplus B_*/A_*$. So $F_M(B_*)\simeq F_M(A_*)\oplus F_M(B_*/A_*)\simeq F_M(A_*)\oplus \bigoplus_n F_M(D(Z_{n-1}(B_*/A_*),n))$. Here $Z_*(X_*):= \ker (d: X_*\to X_{*+1})$ stands for the graded module of the cycles in a given complex $X_*$, and the complex $D(X,n)_*$ is defined as follows: To any $R$-module $X$ and a positive integer $n$, one can associate a complex $\{D(X,n)_k\}_{k\geq 0}$,
\begin{equation*}
D(X,n)_k=\begin{cases} 0, \text{ if } k\neq n,n-1,\\
X, \text{ if } k=n,n-1,
\end{cases}
\end{equation*}
where the only nontrivial differential is the identity map.
It is a direct check that each $F_M(D(Z_{n-1}(B_*/A_*),n))$ is acyclic
therefore $H_{\ch(\Z)}(F_(B))$ is isomorphic to $H_{\ch(\Z)}(F_M(A))$ in the homotopy category $\Ho(\ch(\Z))$.
\begin{lemma} Let $\{C_k\}_{k\geq 0}$ be an acyclic complex where each $C_k$ is projective $R$-module. Then $\{C_k\}_{k\geq 0}$ is a projective complex \text{i.e.} any level-wise surjective chain complex map $D_*\rightarrow E_*$ can be lifted via any chain complex map $C_*\rightarrow E_*$.
\end{lemma}
\begin{proof}
It is easy to check that if $X$ is a projective $R$-module then $ D_n(X)$ is a projective complex. Let $C_*^{(m)}$ be the complex
\begin{equation*}
C_k^{(m)}=\begin{cases} C_k, \text{ if } k\geq m\\
Z_k(C), \text{ if } k=m-1\\
0\text{ otherwise }
\end{cases}
\end{equation*}
Here, $Z_k(C)$ denotes the space of cycles in $C_k$, and $B_k(C)$ is the space of boundary elements in $C_k$. The acyclicity condition implies that $C_*^{(m)}/C_*^{(m+1)}\simeq D(Z_{m-1}(C),m)$. Note that $Z_0(C)=C_0$ is a projective $R$-module and $C_*=C^{(1)}=C^{(2)}\oplus D_1(Z_0(C))$.
Now $D_1(Z_0(C))$ is a projective complex and $C^{(2)}$ also satisfies the assumption of the lemma and vanishes in degree zero. Therefore by applying the same argument one sees that $C^{(2)}=C^{(3)}\oplus D(Z_1(C),2)$. Continuing this process one obtains $C_*=D(Z_0(C),1)\oplus D(Z_1(C),2)\cdots \oplus D(Z_{k-1},k)\oplus\cdots $ where each factor is a projective complex, thus proving the statement.
\end{proof}
We finish this example by computing the left derived functor. For any $R$-module $N$ let $K(N,0)$ be the chain complex concentrated in degree zero where there is a copy of $N$. Since every object is fibrant, a fibrant-cofibrant replacement of $K(N,0)$ is simply a cofibrant replacement. A cofibrant replacement $P_*$ of $K(N,0)$ is exactly a projective resolution (in the usual sense) of $N$ in the category of $R$-modules. In the homotopy category of $\ch(R)$, $K(N,0)$ and $P$ are isomorphic because by definition
$
\Hom_{Ho(\ch(R))}(K(N,0),P)
$
consists of the homotopy classes of $\Hom_{\ch(R)}(RQK(N,0),RQP_*)=\Hom_{\ch(R)}(P_*,P_*)$ which contains the identity map. Therefore $\L F(K(N,0))\simeq \L F (P_*)$ and $ \L F (P_*)$ by Proposition \ref{prop-derived} and the definition of total derived functor is isomorphic to $H_{\ch(R)}F(P_*)=M\otimes_R P_*$.
In particular,
$$
H_* (\L F(K(N,0))= \Tor_*^R(N,M),
$$
where $\Tor^R_*$ is the usual $\Tor_R$ in homological algebra. We usually denote the derived functor $\L F(N)=N\otimes_R^L M$.
Similarly one can prove that the contravariant functor $N_*\mapsto \Hom_R(N_*, M)$ has a total right derived functor, denoted by $\RHom_R (N_*,M)$ and
$$
H^*(\RHom_R(K(N,0),M))\simeq \Ext^*_R(N,M),
$$
is just the usual $\Ext$ functor (see Remark \ref{rem-op}).
\bigskip
\subsection{Hinich's theorem and Derived category of DG module}
The purpose of this section is to introduce a model category and derived functors of DG-modules over a fixed differential graded $\kk$-algebra . From now on we assume that $\kk$ is a field. The main result is essentially due to Hinich \cite{Hinich} who introduced a model category structure for algebras over a vast class of operads.
Let $C(\kk)$ be the category of (unbounded) complexes over $\kk$. For $d\in \Z$ let $M_d\in C(\kk)$ be the complex
$$
\cdots \rightarrow 0\rightarrow \kk=\kk\rightarrow 0\rightarrow 0\cdots
$$
concentrated in degree $d$ and $d+1$.
\begin{theorem}(V. Hinich)\label{hinich} Let $\c$ be a category which admits finite limits and arbitrary colimits and is endowed with
two right and left adjoint functors $(\#,F)$
\begin{equation}
\#: C\rightleftarrows C(\kk):F
\end{equation}
such that for all $A \in obj(A)$ the canonical map $A\rightarrow A\coprod F(M_d)$ induces a quasi-isomorphism $A^\#
\rightarrow (A
\coprod F(M_d))^{\#}$. Then there is a model category structure on $\c$ where the three distinct classes of morphisms are:
\begin{numlist}
\item Weak equivalences $\cW$: $f\in Mor(\c)$ is in $\cW$ iff $f^\#$ is quasi-isomorphism.
\item Fibrations $\cF$: $f\in Mor(\c)$ is in $\cF$ if $f^\#$ is (component-wise) surjective.
\item Cofibrations $\cC$: $f\in Mor(\c)$ is a cofibration if it satisfies the LLP property with respect to all acyclic fibrations $\cW\cap \cF$.
\end{numlist}
\end{theorem}
As an application of Hinich's theorem, one obtains a model category structure on the category $Mod(A)$ of
(left) differential graded modules over a differential graded algebra $A$. Here $\#$ is the forgetful functor and $F$ is given by tensoring $F(M)= A\otimes_\kk M$.
\begin{corollary}
The category $Mod(A)$ of DG $A$-modules is endowed with a model category structure where
\begin{romlist}
\item weak equivalences are the quasi-isomorphisms.
\item fibrations are level-wise surjections. Therefore all objects are fibrant.
\item cofibrations are the maps have the left lifting property with respect to all acyclic fibrations.
\end{romlist}
\end{corollary}
In what follows we give a description of cofibrations and cofibrant objects. An excellent reference for this part is \cite{FT4}.
\begin{definition} An $A$-module $P$ is called a \emph{semi-free} extension of $M$ if $P$ is a union of an increasing family of $A$-modules
$M=P(-1)\subset P(0) \subset\cdots $ where each $P(k)/P(k-1)$ is a free $A$-modules generated by cycles. In particular $P$ is said to be a \emph{semi-free} $A$-module if it is a semi-free extension of the $0$.
A \emph{semi-free resolution} of an $A$-module morphism $f:M\rightarrow N $ is a semi-free extension $P$ of $M$
with a quasi-isomorphism $P\rightarrow N$ which extends $f$.
In particular a \emph{semi-free} resolution of an $A$-module $M$ is a semi-free resolution of the trivial map $0\rightarrow M$.
\end{definition}
The notion of semi-free modules can be traced back to \cite{GugMay}, and \cite{Drinfeld} is another nice reference for the subject. A $\kk$-complex $(M,d)$ is called semi-free, if it is semi-free as a differential $\kk$-module. Here $\kk$ is equipped with the trivial differential. In the case of a field $\kk$, every positively graded $\kk$-complex is semi-free. It is clear from the definition that a finitely generated semi-free $A$-module is obtained through a finite sequence of extensions of some free $A$-modules of the form $A[n]$, $n\in\Z$. Here is $A[n]$ is $A$ after in shift in degree by $-n$.
\begin{lemma}\label{lemm-filt}Let $M$ be an $A$-module with a filtration $F_0\subset F_1\subset F_2\cdots$ such that $F_0$ and all $F_{i+1}/F_i$ are semifree $A$-modules. Then $M$ is semifree.
\end{lemma}
\begin{proof}
Since $F_k/F_{k-1}$ is semifree, it has a filtration $ \cdots P^k_l\subset P^k_{l+1}\cdots $ such that $P^k_l/ P^k_{l+1}$ is generated as an $(A,d)$-module by cycles. So one can write $F_k/F_{k-1}=\oplus_l (A\otimes Z'_k(l))$ where $Z'_k(l)$ are free (graded) $\kk$-modules such that $d(Z_k(l))\subset \oplus_{j\leq l} Z_k(j)$. Therefore there are free $\kk$-modules $Z_k(l)$ such that $$F_k=F_{k-1}\bigoplus_{l\geq 0} Z'_k(l)$$ and
$$d(Z_k(l))\subset F_{k-1}\bigoplus_{j<l} A\otimes Z_k(j).$$
In particular $M$ is the free $\kk$-module generated by the union of all basis elements $\{z_\alpha\}$ of $Z_k(l)$'s. Now consider the filtration $P_0\subset P_1 \cdots $ of free $\kk$-modules constructed inductively as follows: $P_0$ is generates as $\kk$-module by the $z_\alpha$' which are cycles, \emph{i.e.} $dz_\alpha=0$. Then $P_k$ is generated by those $z_\alpha$'s such that $dz_\alpha \in A\cdot P_{k-1}$. This is clearly a semifree resolution if we prove that $M=\cup_k P_k$. For that, we show by induction on degree that for all $\alpha$, $z_\alpha$ belongs to some $P_k$. Suppose that $z_\alpha \in Z_k(l)$, then $dz_\alpha \in \oplus A. Z_i(j)$ where $i<k$ or $i=k$ and $j<l$. By induction hypothesis all $z_\beta$'s in the sum $dz_\alpha$ are in some $P_{m_\beta}$. Therefore $z_\alpha\in P_{m}$ where $m= \max_\beta m_\beta$ and this finishes the proof.
\end{proof}
\begin{remark}\label{rmk-mod}
If we had not assumed that $\kk$ is a field but only a commutative ring then we could still put a model category on $Mod(A)$. This is a special case of the Schwede-Shipely theorem (Theorem 4.1 \cite{SchShip}). More details are provided in pages 503-504 of \cite{SchShip}.
\end{remark}
\begin{proposition} In the model category of $A$-modules, a maps $f:M\rightarrow N$ is a cofibration if and only if it is a retract of a semi-free extension $M\hookrightarrow P$. In particular an $A$-module $M$ is cofibrant iff it is a retract of a semi-free $A$-module, in other words it is a direct summand of a semi-free $A$-module.
\end{proposition}
Here is a list of properties of semi-free modules which allow us to define the derived functor by means of semi-free resolutions.
\begin{proposition}\label{prop-ess}
\begin{romlist}
\item Any morphism $f: M\rightarrow N$ of $A$-modules has a semi-free resolution. In particular every $A$-module has a semi-free resolution.
\item If $P$ is a semi-free $A$-module then $\Hom_A(P,-)$ preserves the quasi-isomorphisms.
\item Let $P$ and $Q$ be semi-free $A$-modules and $f:P\rightarrow Q$ a quasi-isomorphism. Then
$$
g\otimes f :M\otimes _A P\rightarrow N\otimes_A Q
$$
is a quasi-isomorphism if $g:M\rightarrow N$ is a quasi-isomorphism.
\item Let $P$ and $Q$ be semi-free $A$-modules and $f:P\rightarrow Q$ a quasi-isomorphism. Then
$$
\Hom_R(g, f) :\Hom_A(Q,M)\rightarrow \Hom_A(P,N)
$$
is a quasi-isomorphism if $g:M\rightarrow N$ is a quasi-isomorphism.
\end{romlist}
\end{proposition}
The second statement in the proposition above implies that a quasi-isomorphism $f:M\rightarrow N$ between semi-free $A$-modules is a homotopy equivalence \emph{i.e} there is a map $f':N\rightarrow M$ such that $ ff'-id_N= [d_N,h']$ and $f'f-id_M=[d_M,h]$ for some $h:M\rightarrow N$ and $h':N\rightarrow M$. In fact part (iii) and (iv) follow easily from this observation.
The properties listed above imply that the functors $-\otimes _AM $ and $\Hom_A(-,M)$ preserves enough weak equivalences, ensuring that the derived functors $\otimes^L_A $ and $\RHom_A(-,M)$ exist for all $A$-modules $M$.
Since we are interested in Hochschild and cyclic (co) homology, we switch to the category of DG $A$-bimodules. This category is the same as the category of DG $A^e$-modules. Therefore one can endow $A$-bimodules with a model category structure and define the derived functors $-\otimes_{A^e} ^L M$ and $\RHom_{A^e} (-,M)$ by mean of fibrant-cofibrant replacements.
More precisely, for two $A$-bimodule $M$ and $N$ we have
$$
\Tor_*^{A^{e}}(M,N)= H_*(P\otimes_{A^e} N)
$$
and
$$
\Ext^*_{A^{e}}(M,N)= H^*(\Hom_{A^{e}}(P, N))
$$
where $P$ is cofibrant replacement for $M$.
By Proposition \ref{prop-ess} every $A^e$-module has a semi-free resolution for which there is an explicit construction using the
two-sided bar construction. For right and left $A$-modules $P$ and $M$, let
\begin{equation}
B(P,A,M)=\bigoplus_{k\geq 0} P\otimes (s\bar{A})^{\otimes k} \otimes M
\end{equation}
equipped with the differential:
if $k=0$,
\begin{equation*}
D(p[\quad] m)= dp [\quad] n+(-1)^{|p|} p[\quad] dm
\end{equation*}
if $ k>0$
\begin{equation*}
\begin{split}
&D( p[a_{1},\cdots , a_{k}]m)=d_0( p[a_{1},\cdots , a_{k}]m)+ d_1( p[a_{1},\cdots , a_{k}]m))
\\&= dp[a_{1},\cdots , a_{k}]m+\sum_{i=1}^{k} (-1)^{\epsilon_{i}}p [a_{1},\cdots,d a_{i},\dots a_{k}]m+(-1)^{\epsilon_{k+1}}p [a_{1},\cdots, a_{k}]dm\\
& +(-1)^{|p|} pa_{1}[a_{2},\cdots , a_{k}]m +\sum_{i=2}^{k} (-1)^{\epsilon_{i}}p [a_{1},\cdots,a_{i-1}a_{i},\dots a_{k}]m + (-1)^{\epsilon_k}p[a_{1},\cdots ,a_{k-1} ]a_{k}m
\end{split}
\end{equation*}
where
$$\epsilon_{i}=|p|+ |a_{1}| +\cdots |a_{i-1}|-i-1 $$
Let $P=A$ and $\epsilon_M:B(A,A,M)\rightarrow M$ be defined by
\begin{equation}
\epsilon_M(a [a_{1}, \cdots , a_{k}])m)=\begin{cases}0 \text{ if } k\geq 1\\ am \text{ if } k=0
\end{cases}
\end{equation}
It is clear that $\epsilon_M$ is a map of left $A$-modules if $M$.
\begin{lemma} In the category of left $A$-modules, $\epsilon_M:B(A,A,M)\rightarrow M$ is a semi-free resolution.
\end{lemma}
\begin{proof}
Let us first prove that this is a resolution. Let $h:B(A,A,M)\rightarrow B(A,A,M)$ be defined
\begin{equation}
h(a[a_1,a_2,\cdots a_k]m)= \begin{cases}[a,a_1,\cdots a_k]m, \text{ if } k\geq 1,\\
[a]m,\text{ if } k=0.
\end{cases}
\end{equation}
On can easily check that for $ [D,h]= id$ on $\ker \epsilon_M$, which implies $H_*(\ker (\epsilon_M))=0$. Since $\epsilon_M$ is surjective, $\epsilon_M$ is a quasi-isomorphism.
Now we prove that $B(A,A,M)$ is a semifree $A$-module. Let
$F_k= \bigoplus_{i\leq k} A\otimes T(s\bar{A})^{\otimes i} \otimes M$. Since $d_1(F_{k+1})\subset F_k$, then $F_{k+1}/F_k$ as a differential graded $A$-module is isomorphic to $ (A\otimes (sA)^{\otimes k}\otimes M, d_0)=(A,d)\otimes_\kk ( (sA)^{\otimes k},d)\otimes (M,d)$. The later is a semifree $(A,d)$-module since $( (sA)^{\otimes k},d)\otimes_\kk(M,d)$ is a semifree $\kk$-module via the filtration
$$
0 \hookrightarrow \ker (d\otimes 1+1\otimes d) \hookrightarrow ( (sA)^{\otimes k},d)\otimes_\kk(M,d).
$$
Therefore $B(A,A,M)$ is semi-free by Lemma \ref{lemm-filt}.
\end{proof}
\begin{corollary}\label{cor-reso} The map $\epsilon_A: B(A,A):=B(A,A,\kk)\rightarrow \kk$ given by
\begin{equation*}
\epsilon_k(a[a_1,a_2\cdots a_n])=\begin{cases} \epsilon(a) \text{ if } n=0\\ 0 \text{ otherwise}
\end{cases}
\end{equation*}
is a resolution. Here $\epsilon:A\rightarrow \kk$ is the augmentation of $A$. In other words $ B(A,A)$ is acyclic.
\end{corollary}
\begin{proof} In the previous lemma, let $M=\kk$ be the differential $A$-module with trivial differential and the module structure
$a.k:= \epsilon(a)k$
\end{proof}
\begin{lemma} In the category $Mod(A^e)$, $\epsilon_A:B(A,A,A)\rightarrow A$ is a semifree resolution.
\end{lemma}
\begin{proof}
The proof is similar to the previous lemma. First of all, it is obvious that this is a map of $A^e$-modules. Let
$F_k= \bigoplus_{i\leq k} A\otimes T(s\bar{A})^{\otimes i} \otimes A$. Then $F_{k+1}/F_k$ as a differential graded $A$-module is isomorphic to $ (A\otimes (sA)^{\otimes k}\otimes A, d_0)=(A,d)\otimes_\kk ( (sA)^{\otimes k},d)\otimes (A,d)$. The later is semi-free as $A^e$-module since $( (sA)^{\otimes k},d)$ is a semi-free $\kk$-module via the filtration $\ker d \hookrightarrow (sA)^{\otimes k}$.
\end{proof}
Since the two-sided bar construction $B(A,A,A)$ provides us with a semi-free resolution of $A$ we have that
$$
HH_*(A,M)=H_* (B(A,A,A)\otimes_{A^e}M)= \Tor_*^{A^{e}}(A,M)
$$
and
$$
HH^*(A,M)=H^* (Hom_{A^e}(B(A,A,A), M))= \Ext^*_{A^{e}}(A,M).
$$
In some special situations, for instance that of Calabi-Yau algebras, one can choose smaller resolutions to compute Hochschild homology or cohomology.
The following will be useful.
\begin{lemma}\label{lem-util} If $H^*(A)$ is finite dimensional then for all finitely generated semi-free $A$-bimodules $P$ and $Q$, $H^*(P)$, $H^*(Q)$ and $H^*(\Hom_{A^e}(P,Q))$ are also finite dimensional.
\end{lemma}
\begin{proof}
Since $A$ has finite dimensional cohomology, we see that $H^*(A\otimes A^{op})$ is finite dimensional. Similarly $P$ (or $Q$) has finite cohomological dimension since it is obtained via a finite sequence of extensions of free bimodules of the form $(A\otimes A^{op})[n]$. Note that since $\Hom_{A^{e}}(A\otimes A^{op},A\otimes A^{op})\simeq A\otimes A^{op},A\otimes A^{op}$ is a free $A$ bimodule of finite cohomological dimension, therefore $\Hom_{A^e}(P,Q)$ is obtained through a finite sequence of extensions of shifted free $A$-bimodules, proving that it has finite cohomological dimension.
\end{proof}
\section{Calabi-Yau DG algebras}
Throughout this section $(A,d)$ is a differential graded algebra, and by an $A$-bimodule we mean a differential graded $(A,d)$-bimodule.
In this section we essentially explain how an isomorphism
\begin{equation}\label{vdb}HH^*(A,A)\simeq HH_*(A,A)\end{equation}
(of $HH^*(A,A)$-modules) gives rise to a BV structure on $HH^*(A,A)$ extending its canonical Gerstenhaber structure. For Calabi-Yau DG algebra one does have such an isomorphism (\ref{vdb}) and this is a special case of a more general statement due to Van den Bergh \cite{VdBergh}. The main idea is due to V. Ginzburg \cite{Ginz} who proved that for a Calabi-Yau algebra $A$, $HH^*(A,A)$ is a BV-algebra. However he works with ordinary algebras rather than DG algebras. But here we have adapted his result to the case of Calabi-Yau DG algebras. For this purpose one has to work in the correct derived category of $A$-bimodules, and this is the derived category of perfect $A$-bimodules as it is formulated below. All this can be extended to the case of $A_\infty$ but for simplicity we refrain from doing so.
\subsection{Calabi-Yau algebras}
We first give the definition of Calabi-Yau algebra which were introduced by Ginzburg in \cite{Ginz} for algebras with no differentialº and then generalized by Kontsevich-Soibelman \cite{KS} to the differential graded algebras.
\begin{definition} (Kontsevich-Soibelman \cite{KS})
\begin{numlist}
\item An $A$-bimodule is \emph{perfect} if it is quasi-isomorphic to a direct summand of a finitely generated semifree $A$-bimodules.
\item $A$ is said to be \emph{homologically smooth} if it is perfect as an $A$-bimodule.
\end{numlist}
\end{definition}
\begin{remark} In \cite{KS}, the definition of perfectness uses the notion of \emph{extension} \cite{Keller} and it is essentially the same as ours.
\end{remark}
We define \emph{DG-projective $A$-modules} to be the direct summands of semifree $A$-modules. As a consequence, an $A$-bimodule is perfect iff it is quasi-isomorphic to a finitely generated DG-projective $A$-bimodule. We call the latter a finitely generated \emph{DG-projective $A$-module resolution}. This is analoguous to having a bounded projective resolution in the case of ordinary modules (without differential). By Proposition \ref{prop-ess}, DG-projectives have all the nice homotopy theoretic properties that one expects.
The content of the next lemma is that $A^!:=\RHom_{A^{e}}(A,A^e)$, called the \emph{derived dual} of $A$, is also a perfect $A$-bimodule. The $A$-bimodule structure of $A^!$ is induced by the right action of $A^e$ on itself. Recall that for an $A$-bimodule $M$, $\RHom_{A^{e}}(-,M)$ is the right derived functor of $\Hom_{A^{e}}(-,M)$ \textit{i.e.} for an $A$-bimodule $N$, $\RHom_{A^{e}}(N,M)$ is the complex $\Hom _{A^{e}}(P,M)$ where $P$ is a DG-projective $A$-bimodule quasi-isomophic to $N$. In general, $M^!=\RHom_{A^{e}}^*(M,A^{e})$ is different from the usual dual $$M^\vee=\Hom_{A^{e}} (M,A^{e}).$$
\begin{lemma}\label{dual-perfect}\cite{KS}
If A is homologically smooth then $A^!$ is a perfect $A$-bimodule.
\end{lemma}
\begin{proof} Let $P=P_i \twoheadrightarrow A$ be a finitely generated DG-projective resolution. Note that $A^!= \RHom_{A^e}(A, A^e)$ is quasi-isomorphic to the complex $\Hom_{A^e}(P_i, A^e)$. Each $P_i$ being a direct summand of a semi-free module $Q_i$. Since $Q_i$ is obtained through a finite sequence of extensions of a free $A^e$-modules,
$\Hom(Q_i, A^e)$ is also a semi-free module. Clearly $\Hom_{A^e}(P_i, A^e)$ is a direct summand of the semi-free module $\Hom(Q_i, A^e)$, therefore DG-projective
This proves the lemma.
\end{proof}
We say that a DG algebra $A$ is \emph{compact} if the cohomology $H^*(A)$ is finite dimensional.
\begin{lemma} \label{bad-lemma} A compact homologically smooth DG algebra $A$ has finite dimensional Hochschild cohomology $HH^*(A,A)$.
\end{lemma}
\begin{proof}By assumption $A$ has finite dimensional cohomology and so does $A^e=A\otimes A^{op}$. Now let $P\twoheadrightarrow A$ be a finitely generated DG-projective resolution of $A$-bimodules. We have a quasi-isomorphism of complexes $C^*(A,A)\simeq \RHom_{A^{e}} (A,A)\simeq Hom_{A^{e}} (P,P)$, which by Lemma \ref{lem-util} has finite dimensional cohomology.
\end{proof}
\begin{definition}(Ginzburg, Kontsevich-Soibelman \cite{Ginz},\cite{KS})
A $d$-dimensional \emph{Calabi-Yau differential graded algebra} is a homologically smooth DG-algebra endowed with an $A$-bimodule quasi-isomorphism
\begin{equation}\label{cond1}
\psi:A\overset{\simeq}{\longrightarrow} A^{!}[d]
\end{equation}
such that
\begin{equation}\label{cond2}
\psi ^!=\psi [d].
\end{equation}
\end{definition}
The main reason to call such algebras Calabi-Yau is that a tilting generator $\mathcal{E}\in D^b(Coh (X))$ of the bounded derived category of coherent sheaves on a smooth algebraic variety $X$ is a Calabi-Yau algebra iff $X$ is a Calabi-Yau (see \cite{Ginz} Proposition 3.3.1 for more details).
There are many other examples provided by representation theory. For instance most of the three dimensional Calabi-Yau algebras are obtained as a quotient of the free associative algebras $F=\mathbb{C}\langle x_1,\cdots x_n\rangle$ on $n$ generators. An element $\Phi$ of $F_{cyc}:=F/[F,F]$ is called a cyclic potential. One can defines the partial derivatives $\frac{\partial}{\partial x_i}: F_{cyc} \rightarrow F$ in this setting. Many of 3-dimensional Calabi-Yau algebras are obtained as a quotient $ \mathcal{U}(F, \phi)= F/\{ \frac{\partial \Phi}{\partial x_i=0}_{i=0,\cdots n}\} $. For instance for $\Phi(x,y,z)= xyz-yzx$, we obtain $\mathcal{U}(F, \phi)=\mathbb{C}[x,y,z]$, the polynomial algebra in $3$ variables. The details of this discussion is irrelevant to the context of this chapter which is the algebraic models of free loop spaces. We, therefore, refer the reader to \cite{Ginz} for further details.
Here $A^!=\RHom_{A^{e}}(A,A^{e})$ is called the \emph{dualizing the bi-module}, which is also a $A$-bimodule using outer multiplication. Condition (\ref{cond1}) amounts to the following.
\begin{proposition}\label{ext-tor}(Van den Bergh Isomorphism \cite{VdBergh}) Let $A$ be a Calabi-Yau DG algebra of dimension $d$. Then for all $A$-bimodules $A$ we have
\begin{equation} \label{iso-VdB}
HH_{d-*}(A, N) \simeq HH^*(A,N).
\end{equation}
\end{proposition}
\begin{proof} We compute
\begin{equation}
\begin{split}
HH^*(A,N)&\simeq \Ext^*_{A^e}(A,N)\simeq H^*(\RHom_{A^e}(A,N)) \simeq H^*(\RHom_{A^e}(A,A^e)\otimes^L_{A^e} N)) \\ & \simeq H^* (A^!\otimes^L_{A^e}N) \simeq H_*(A[-d]\otimes_{A^e}^L N)\simeq \Tor_*^{A^e}(A,N)\simeq HH_{d-*}(A,N) \\
\end{split}
\end{equation}
\end{proof}
Note that the choice of the $A$-bimodule isomorphism $\psi$ is important and it is characterized by the image of the unit $\pi=\psi (1_A)\in A^!$. By definition, $\pi$ is the \emph{volume} of the Calabi-Yau algebra $A$. For a Calabi-Yau algebra $A$ with a volume $\pi$ and $N=A$, we obtain an isomorphism
\begin{equation}
D=D_\pi: HH_{d-*}(A, A) \rightarrow HH^*(A,A).
\end{equation}
\noindent One can use $D$ to transfer the Connes operator $B$ from $ HH_{*}(A, A)$ to $HH^*(A,A)$,
$$
\Delta=\Delta_{\pi}:= D\circ B \circ D^{-1}
$$
In the following lemmas $A$ is a Calabi-Yau algebra with a fixed volume $\pi$ and the associated operator $\Delta$ .
\begin{lemma}\label{lem0} $f \in HH^*(A,A)$ and $a\in HH_*(A,A)$
$$
D(i_fa)=f\cup D a.
$$
\end{lemma}
\begin{proof}
To prove the lemma we use the derived description of Hochschild (co)homology, cap and cup product. Let $(P,d)$ be a projective resolution of $A$. Then $HH_*(A,A)$ is computed by the complex $(P\otimes_{A^{e}} P, d)$, and similarly $HH^*(A,A)$ is computed by the complex $(\End (P), \ad(d)= [-,d])$. Here $\End (P)=\oplus_{r\in \Z} \Hom_{A^{e}}(P, P[r])$. Then the cap product corresponds to the natural pairing
\begin{equation}
ev: (P\otimes_{ A^e} P ) \otimes_\kk \End (P) \rightarrow P\otimes_{A^e} P
\end{equation}
given by $ev: (p_1\otimes p_2)\otimes f\mapsto p_1\otimes f(p_2)$. The quasi-isomorphism $\psi: A\rightarrow A^![d]$ yields a morphism $\phi: P\rightarrow P^\vee$. Let us explain this in detail.
Using the natural identification $P^\vee \otimes _{A^e}P = \End(P)$, we have a commutative diagram
\begin{equation}\label{diagcom1}
\xymatrix{ \End (P) \otimes_\kk (P\otimes_{ A^e} P ) \ar[rr]^-{ev}\ar[d]^{id \otimes(\phi\otimes id)}&& P\otimes_{A^e} P \ar[d]^-{\phi\otimes\id}\\
\End (P) \otimes_\kk \End (P)\ar[rr]^-{\text{composition}} & & \End (P)=P^\vee \otimes _{A^e}P}
\end{equation}
The evaluation map is defined by $ev (\psi\boxtimes (x\otimes y)):= x\otimes \psi (y)$. After passing to (co)homology, $\phi\otimes\id$ becomes $D$, the composition induces the cup product and $ev$ is the contraction (cap product), hence
$$
D(i_f a)= f\cup Da
$$
which proves the lemma.
\end{proof}
\begin{lemma}\label{lem1} For $f, g\in HH^*(A,A)$ and $a\in HH_*(A,A)$ we have
\begin{equation*}
\begin{split}
[f,g]\cdot a&= (-1)^{|f|} B((f\cup g)a)-f\cdot B(g\cdot a)+ (-1)^{(|f|+1)(|g|+1)}g\cdot B(f\cdot a)\\& +(-1)^{|g|} (f\cup g)\cdot B(a).
\end{split}
\end{equation*}
\end{lemma}
\begin{proof} We compute
\begin{equation}\label{BV1}
\begin{split}
i_{[f,g]}&=L_f i_g - i_gL_f \\
&= i_fBi_g-Bi_fi_g -i_gBi_f-i_gi_fB\\
&=i_fBi_g-i_gBi_f-Bi_{f\cup g}-i_gi_fB
\end{split}
\end{equation}
\end{proof}
\begin{corollary}\label{cor1}
For all $f,g\in HH^*(A,A)$ and $a\in HH_*(A,A)$, we have
\begin{equation*}
\begin{split}
[f,g]\cup D(a)&=(-1)^{|f|} \Delta(f\cup g\cup D a) -f\cup \Delta (g\cup D(a))\\& +(-1)^{(|f|+1)(|g|+1)}g\cup \Delta (f\cup D(a))- f \cup g \cup DB(a).
\end{split}
\end{equation*}
\end{corollary}
\begin{proof}
After taking $D$ of the identity in the previous lemma, we obtain
\begin{equation*}
\begin{split}
D(i_{[f,g]}a)& =(-1)^{|f|} D(Bi_{f\cup g}a)-D(i_fBi_g(a))+(-1)^{(|f|+1)(|g|+1)}D(i_gBi_f(a)) \\ &+(-1)^{|g|}D(i_fi_gB(a))
\end{split}
\end{equation*}
By Lemma \ref{lem0} and $DB= \Delta D $, this reads
\begin{equation*}
\begin{split}
[f,g]\cup D(a)&=(-1)^{|f|} \Delta D(i_{f\cup g}a) -f\cup \Delta D (i_g(a))+(-1)^{(|f|+1)(|g|+1)}g\cup \Delta D(i_f(a))\\&- g\cup f \cup DB(a).
\end{split}
\end{equation*}
Once again by Lemma \ref{lem0} we get
\begin{equation*}
\begin{split}
[f,g]\cup D(a)&=(-1)^{|f|} \Delta(f\cup g\cup D a) -f\cup \Delta (g\cup D(a))\\&+(-1)^{(|f|+1)(|g|+1)}g\cup \Delta (f\cup D(a))+(-1)^{|g|} f\cup g \cup DB(a).
\end{split}
\end{equation*}
\end{proof}
\begin{theorem} \label{main-theo1} For a Calabi-Yau algebra $A$ with a volume $\pi\in A^!$, $(HH^*(A,A), \cup, \Delta)$ is a BV-algebra \emph{i.e.}
\begin{equation}
[f,g]=(-1)^{|f|}\Delta(f\cup g)-(-1)^{|f|}\Delta(f)\cup g-f\cup \Delta(g).
\end{equation}
\end{theorem}
\begin{proof}
In the statement of the previous lemma choose $a\in HH_d(A,A)$ such that $D(a)=1\in HH^0(A,A)$. The identity (\ref{BV-cond}) will follow since $B(a)=0$ for obvious degree reason.
\end{proof}
\subsection{Chains of Moore based loop space}\label{secmoore}
Let us finish this section with some interesting examples of DG Calabi-Yau algebras. We will also dicuss chains of the Moore based loop space. This example plays an important role in symplectic geometry where it appears as the generator of a particular type of Fukaya category called the wrapped Fukaya category (see \cite{abou} for more details). One can then compute the Hochschild homology of wrapped Fukaya categories using Burghelea-Fiedorowicz-Goodwillie theorem. This theorem implies that the Hochschild homology of the Fukaya category of a closed oriented manifold is isomorphic to the homology of the free loop space of the manifold.
We start with a more elementary example \emph{i.e.} the Poincar\'e duality groups.
Among the examples, we have the fundamental group of closed oriented aspherical manifolds. Closed oriented irreducible 3-manifolds are aspherical, therefore they provide us an interesting large class of examples.
\begin{proposition} \label{pro-PDG}Let $G$ be finitely generated oriented Poincar\'e duality group of dimension $d$. Then
$\kk[G]$ is a Calabi-Yau algebra of dimension $d$, therefore $HH^*(\kk[G],\kk[G])$ is a BV algebra
\end{proposition}
\begin{proof}
First note that $\kk[G]$ is only an ordinary algebra without grading and differential. The hypothesis that
$G$ is a finitely generated oriented Poincar\'e duality group of dimension $d$ means that $\kk$ has a bounded finite projective resolution $P=\{P_d\rightarrow \cdots P_1\rightarrow P_0\} \twoheadrightarrow \kk$ as a left $\kk[G]$-module, and $H^d(G,\kk[G])\simeq \kk$ as a $\kk[G]$. Here $\kk$ is equipped with the trivial action and $\kk[G]$ acts on $H^d(G,\kk[G])$ from left via the coefficient module. In particular, we
\begin{equation}
\Ext_{\kk[G]}^i (\kk,\kk[G]) \simeq \left\{
\begin{array}{ll}
\kk , & i=d\\
0 , & \hbox{otherwise.}
\end{array}
\right.
\end{equation}
In other words the resolution $P$ has the property that $P^\vee:= \Hom_{\kk[G]} (P,\kk[G])$, after a shift in degree by $d$, is also a resolution of $\kk$ as a right $\kk[G]$-module (See \cite{Brown} for more details).
Note that using the map $g\to (g,g^{-1})$, we can turn $\kk[G]^e$ into a right $\kk[G]$-module. More precisely $(g_1\otimes g_2)g:=g_1g\otimes g^{-1}g_2$. The tensor product $\kk[G]^e\otimes_{\kk[G]} \kk$ is isomorphic to $\kk[G]$ as a left $\kk[G]^e$-module. The isomorphism is given by $(g_1\otimes g_2)\boxtimes 1\mapsto g_1g_2$. Similarly $\kk[G]^e$ can be considered as a left $\kk[G]$-module using the action $g(g_1\otimes g_2):=gg_1\otimes g_2g^{-1}$ and once again
$\kk\otimes_{\kk[G]} \kk[G]^e\simeq \kk[G]$ as a right $\kk[G]^e$-module using the isomorphism $1 \boxtimes (g_1\otimes g_2)\mapsto g_2g_1$.
It is now clear that $\kk[G]^e \otimes_{\kk[G]}P_d\rightarrow \cdots \kk[G]^e \otimes_{\kk[G]}P_1 \rightarrow \kk[G]^e \otimes_{\kk[G]}P_0 \twoheadrightarrow \kk[G]\otimes_{\kk[G]} \kk \simeq \kk[G]$ is a projective resolution of $\kk[G]$ as a $\kk[G]$-bimodule, proving that $\kk[G]$ is homologically smooth. Similarly $\Hom_{\kk[G]} (P,A)\otimes _{\kk[G]}\kk[G]^e$ is a projective resolution of $\kk[G]$ as $\kk[G]$-bimodule. Therefore we have a homotopy equivalence
$$
\phi: \kk[G]^e\otimes_{\kk[G]}P\to\Hom_{\kk[G]} (P,A)\otimes _{\kk[G]}\kk[G]^e .
$$
Now if we take $Q= \kk[G]^e\otimes_{\kk[G]}P$ as a DG-projective resolution of $\kk[G]$ as $\kk[G]$-bimodule, then
$ \kk[G]^!= \Hom_{\kk[G]^e}( \kk[G]^e\otimes_{\kk[G]}P, \kk[G]^e ) \simeq \Hom_{\kk[G]}( P,\kk[G]^e)$ where the right $\kk[G]$-module structure of $\kk[G]^e$ was described above. Since the $P$ is $\kk[G]$-projective, the natural map $ \Hom_{\kk[G]}( P,\kk[G])\otimes _{\kk[G]} \kk[G]^e \overset{\simeq}{\longrightarrow }\Hom_{\kk[G]}( P,\kk[G]^e) $ is an isomorphism. Therefore $\phi$ is nothing but an equivalence $\kk[G]\overset{\phi}{\simeq} \kk[G]^!$ in the derived category of the $\kk[G]$-bimodules. It remains to prove that after $\phi^!\simeq \phi[d]$ in the derived category of $\kk[G]$-bimodules. We have
\begin{equation}
\begin{split}
\phi^!=\phi^\vee: \Hom_{\kk[G]^e}( \Hom_{\kk[G]}(P,\kk[G])\otimes_{\kk[G]} \kk[G]^e, \kk[G]^e)\rightarrow &\Hom_{\kk[G]^e}(\kk[G]^e\otimes_{\kk[G]}P,\kk[G]^e)\\&\simeq \kk[G]^!.
\end{split}
\end{equation}
On the other hand we have the natural inclusion map
\begin{equation}
\begin{split}
i: \kk[G]^e\otimes_{\kk[G]}P\to &\Hom_{\kk[G]^e}(\Hom_{\kk[G]^e}(\kk[G]^e\otimes_{\kk[G]}P, \kk[G]^e))\\ &\simeq \Hom_{\kk[G]^e}( \Hom_{\kk[G]}(P,\kk[G])\otimes_{\kk[G]} \kk[G]^e, \kk[G]^e)
\end{split}
\end{equation}
and one can easily check that $\phi^!\circ i=\phi$ after a shift in degree by $d$. This proves that $\phi^! \simeq \phi[d] $ in the derived category.
\end{proof}
\begin{remark}
In the case of $G= \pi_1(M)$ the fundamental group of an aspherical manifold $M$, Vaintrob \cite{Vaint} has proved that the BV structure on $HH^{*+d}(\kk[G],\kk[G]) \simeq HH_*(\kk[G],\kk[G])$ corresponds to the Chas-Sullivan BV structure on $H_*(LM,\kk) $.
\end{remark}
Let $(X,\ast)$ be a finite CW complex with a basepoint and Poincar\'e duality. The Moore loop space of $X$, $\Omega X=\{\gamma:[0,s]| \gamma(0)=\gamma(s)=*, s\in \R^{>0}\}$ is equipped with the standard concatenation which is strictly associative. Therefore the cubic chains $C_*(\Omega X)$ can be made into a strictly associative algebra using the Eilenberg-Zilber map and the concatenation.
In \cite{Ginz} there is a sketch of the proof that $C_*(\Omega X)$ is homologically smooth. Here we prove more using a totally different method.
\begin{proposition} For a Poincar\'e duality finite CW-complex $X$, $C_*(\Omega X)$ is Calabi-Yau DG algebra.
\end{proposition}
\begin{proof} The main idea of the proof is essentially taken from \cite{FT4}. Let $A=C_*(\Omega X)$ be the cubic singular chains complex of the Moore loop space. By composing the Eilenberg-Zilber and contcatenation maps $C_*( \Omega X) \otimes C_*(\Omega X) \overset{EZ}{\longrightarrow} C_*(\Omega X\times \Omega X)
\overset{concaten.}{\longrightarrow} C_*(\Omega X)$ one can define an associative product on $A$ . The product is often called the Pontryagin product.
One could switch to the simplicial singular chain complex of the standard base loop space $\{\gamma:[0,1]| \gamma(0)=\gamma(1)=*\}$
but then one has to work with $A_\infty$-algebras and $A_\infty$-bimodules and their derived category. All these work nicely \cite{KS,Malm} and the reader may wish to write down the details in this setting.
Note that $A$ has some additional structures. First of all, the composition of Alexander-Withney and the diagonal maps $C_*(\Omega X)\overset{diagonal}{\longrightarrow} C_*(\Omega X\times \Omega X)
\overset{A-W}{\longrightarrow} C_*(\Omega X)\otimes C_*(\Omega X)$ provides $A$ with a coassociative coproduct, which together with the Pontryagin product make $C_*(\Omega X)$ into a bialgebra.
One can consider the inverse map on $\Omega X$ which makes the bialgebra $C_*(\Omega X)$ into a differential graded Hopf algebra up to homotopy. In order to get a strict differential graded Hopf algebra, one finds a topological group $G$ which is homotopy equivalent to $\Omega X$ (see \cite{Kan,HessTonks}). This can be done, and one can even find a simplicial topological group homotopy equivalent to $\Omega X$. Therefore from now on, we assume that $\Omega X=G $ is a topological group and $C_*(\Omega X) $ is a differential graded Hopf algebra $(A, \cdot, \delta, S)$ with the coproduct $\delta$ and antipode map $S$.
First we prove that $A$ has a finitely generated semifree resolution as an $A$-bimodule. The proof which is essentially taken from \cite{FT4} (Proposition 5.3) relies on the cellular structure of $X$.
Consider the path space $E=\{\gamma:[0,s]\rightarrow X| \gamma(s)=\ast\}$. Using the concatenation of paths and loops, one can define an action of $\Omega X$ on $E$, and thus $C_*(E)$ becomes a $C_*(\Omega X)$-module. This action translates to an action of $A$ on $E$ which is from now on an $A$-module.
Let $ G=\Omega X \rightarrow E\rightarrow X$ be the path space fibration of $X$. We will construct a finitely generated semifree resolution of $C_*(E)$ as an $A$-module which, since $E$ is contractible, provides us with a finitely generated semifree resolution of $\kk\simeq C_*(E)$. Now by tensoring this resolution with $A^e$ over $A$ we obtain a finitely generated semifree resolution of $A$ as $A$-bimodule. Here the $A$-module structure of $A^e$ is defined via the composite $Ad_0:= (A\otimes S) \delta : A\to A^e)$, similar to the case of the Poincar\'e duality groups (see the proof Proposition \ref{pro-PDG}).
The semifree resolution of $ C_*(E)$ is constructed as follows. Let $X_1\subset X_2 \subset \cdots \subset X_m$ be the skeleta of $X$ and $D_n=\coprod D_\alpha^n$ the disjoint union of $n$-cells and $\Sigma_n=\coprod S_\alpha^{n-1}$ .
Let $V_n=H_*(X_n,X_{n-1})$ be the free $\kk$-module on the basis $v_\alpha^n$. Using the cellular structure of $X$ we construct an $A=C_*(G)$-linear quasi-isomorphism $\phi: (V\otimes A)\rightarrow C_*(E)$ where
$V=\oplus_n V_n$, inductively from the restrictions $\phi_n=\phi | \oplus_{i\leq n} V_i \otimes A\rightarrow C_*(E_n)$. The induction step $n-1$ to $n$ goes as follows: Let $f: (D_n,\Sigma_n )\rightarrow (X_n,X_{n-1})$ be the characteristic map. Since the (homotopy) $G$-fibration $E$ can be trivialized over $D^n$, one has a homotopy equivalence of pairs
$$
\Phi: (D_n,\Sigma_n)\times G\rightarrow (E_n,E_{n-1}),
$$
where $E_i= \pi^{-1}(X_i)$. We have a commutative diagram
\begin{equation}
\xymatrix{ & C_*(E_n)\ar[d]_q \\ C_*(D_n,\Sigma_n)\otimes C_*(G) \ar[d]\ar[r]^-{ \Phi_*\circ EZ} & C_*(E_n,E_{n-1})\ar[d]_-{\pi_*} \\
C_*(D_n,\Sigma_n) \ar[r]_-{f_*} & C_*(X_n,X_{n-1}) }
\end{equation}
whose horizontal arrows are quasi-isomorphisms. Here $q$ is the standard projection map and $\pi$ is the fibration map. Since $q$ is surjective there is an element $w^n_\alpha \in C_*(E_n)$ such that
$q_*(w_n^\alpha)= \Phi_*\circ EZ(v^\alpha_n\otimes 1)$. Since $ v^\alpha_n\otimes 1$ is a cycle we have that $dw_n^\alpha \in C_*(E_{n-1})$. Because we have assumed that
$m_{n-1}$ is a quasi-isomorphism, there is a cycle $z^\alpha_{n-1}\in \oplus_{i\leq n-1} V_i \otimes A$ such that $\phi_{n-1}(z^\alpha_{n-1})=dw_ n^\alpha$. First we extend the differential by
$d(v_\alpha\otimes 1)=z_\alpha$. We extends $\phi_{n-1}$ to $\phi_n$ by defining
$\phi_n(v^\alpha_n\otimes 1 )= w_\alpha$. The fact that $\phi_n $ is an quasi-isomorphism follows from an inductive argument and 5 Lemma and the fact that on the quotient $\phi_n: V_n\otimes C_*(G)\rightarrow C_*(E_n, E_{n-1})$ is a quasi-isomorphism.
Next, we prove that $A\simeq A^!$ in the derived category of $A^e$-bimodules which is a translation of Poincar\'e duality. Let $Ad_0:A\rightarrow A^e$ be defined by $Ad_0= (id\otimes S) \delta$. For an $A$-bimodule $M$ let $Ad_0^*(M)$ be the $A$-module
whose $A$-module structure is induced using pull-back by $Ad_0$. By applying the result of F\'elix-Halperin-Thomas on describing the chains of the base space of a $G$-fiberation $G \rightarrow EG\rightarrow BG\simeq X$, we get a quasi-isomorphism
$$
C_*(X)\simeq B(\kk,A,\kk),
$$
as coalgebras. Note that $ B(\kk,A,\kk)\simeq B(\kk, A,A)\otimes_A B(A,A, \kk)$. The Poincar\'e duality for $X$ implies that there is a cycle $z_1\in C_*(X)$ such that capping with $z_1$
\begin{equation}\label{poin}
-\cap z_1: C^*(X)\rightarrow C_{*-d}(X),
\end{equation}
is a quasi-isomorphism. The class $z_1$ corresponds to a cycle $z\in B(\kk, A,A)\otimes_A B(A,A, \kk)$ and the quasi-isomorphism (\ref{poin}) corresponds to the quasi-isomorphism
\begin{equation}
ev_{z,P}: \Hom_k ( B(\kk, A,A) , P) \rightarrow B(A,A, \kk)\otimes P.
\end{equation}
given by $ ev_{z}(f)= \sum f(z_i)z'_i $, where $f\in \Hom_k ( B(\kk, A,A) , \kk) $ and $z=\sum z_i\otimes z'_i$.
Let $E=Ad_0^*(A^e)$. Then we have the quasi-isomorphisms of $A^e$-modules,
\begin{equation}
\begin{split}
A&\simeq B(A,A,A)\simeq B(Ad^*(A^e),A,\kk)\simeq E \otimes_A B(A,A,\kk),
\end{split}
\end{equation}
where $A^e=A\otimes A $ acts on the latter from the left and on the factor $E$. On the other hand
\begin{equation}
\begin{split}
A^!&\simeq \Hom_{A^e}(B(A,A,A),A^e)\simeq \Hom_{A^e} (B(\kk,A,Ad^*(A^e)), A^e)\\ &\simeq \Hom_{A} (B(\kk,A,A),\Hom_{A^e}(Ad_0^*(A^e), A^e) )\\
& \simeq \Hom_{A} (B(\kk,A,A),Ad_0^*(A^e)).
\end{split}
\end{equation}
Therefore $ev_{z,E}$ is a quasi-isomorphism of $A^e$-modules from $A^!$ and $A[-d]$.
\end{proof}
\begin{corollary} For a closed oriented manifold $M$, $HH^*(C^*(\Omega M), C^*(\Omega M))$ is a BV-algebra.
\end{corollary}
\begin{proof}
Note that in the proof of Theorem \ref{main-theo1} we don't use the second part of the Calabi-Yau condition. We only use the derived equivalence $A\simeq A^!$.
\end{proof}
\begin{remark}
Recently E. Malm \cite{Malm} has proved that the Burghelea-Fiedorowicz-Goodwillie isomorphism (\cite{BurFied,Good})
$$HH^*( C_*(\Omega M),C_*(\Omega M) )\simeq HH_*(C_*(\Omega M),C_*(\Omega M))\overset{\text{{\tiny Burghelea-Fiedorowicz-Goodwillie}}}{\simeq} H_*(LM).$$
is an isomorphism of BV-algebras where $H_*(LM)$ is equipped with the Chas-Sullivan \cite{CS1} BV-structure.
\end{remark}
\section{Derived Poincar\'e duality algebras}
In this section we essentially show how an isomorphism
\begin{equation*}HH^*(A,A)\simeq HH^*(A,A^\vee)\end{equation*} of $HH^*(A,A)$-modules gives rise to a BV structure on $HH^*(A,A)$ whose underlying Gerstenhaber structure is the canonical one. The next lemma follows from Lemma \ref{lem1}.
\begin{lemma}\label{lem2} For $a, b\in HH^*(A,A)$ and $\phi\in HH^*(A,A^{\vee})$ we have
\begin{equation}\label{lem2-eq}
[f,g]\cdot \phi= (-1)^{|f|} B^{\vee}((f\cup g)\phi)-f \cdot B^{\vee}(g \cdot \phi)+ (-1)^{(|f|+1)(|g|+1)}g\cdot B^\vee(f \cdot \phi) +(-1)^{|g|} (f\cup g) \cdot B^{\vee}(\phi).
\end{equation}
\end{lemma}
\begin{proof}
To prove the identity, one evaluates the cochains in $C^* (A,A^\vee)=\Hom_{\kk}(T(s\bar{A}),A^\vee)\simeq \Hom_{\kk}(A\otimes T(s\bar{A}),\kk)$ on both sides on a chain $ x=a_0[a_1,\cdots, a_n] \in A\otimes T(s\bar{A})$. By (\ref{contr-2}) and Lemma \ref{lem2}, we have:
\begin{equation*}
\begin{split}
&([f,g]\cdot \phi)(x)=(i_ {[f,g]}\phi)(x)=(-1)^{|[f,g]|\cdot |\phi|}\phi(i_{[f,g]}(x))\\
&=(-1)^{|[f,g]|\cdot |\phi|} \phi((-1)^{|f|}B((f\cup g)\cdot x)-f\cdot B(g \cdot x) + (-1)^{(|f|+1)(|g|+1)}g \cdot B(f \cdot x)\\&+(-1)^{|g|} (f\cup g) \cdot B(x))=(-1)^{|[f,g]| \cdot |\phi|+|f|+ |\phi|}B^\vee(\phi)(i_{f\cup g}x)-(-1)^{|[f,g]| \cdot |\phi|} \phi(i_f B(i_gx))\\
&(-1)^{|[f,g]| \cdot |\phi|+(|f|+1)(|g|+1)} \phi(i_g B(i_fx)) + (-1)^{|[f,g]|.|\phi|+|g|} \phi(i_{f\cup g}B(x))\\
&=(-1)^{|[f,g]| \cdot |\phi|+|f|+|\phi| +(|\phi|+1)|f\cup g|}i_{f\cup g}(B^\vee(\phi))(x)\\&-(-1)^{|[f,g]||\phi|+|g|(|\phi|+|f|+1)+|f|+|\phi|+|f|\cdot |\phi|} i_g( B^\vee(i_f\phi))(x)\\
&(-1)^{|[f,g]|\cdot |\phi|+(|f|+1)(|g|+1)+|f|(|\phi|+|g|+1)+|g|+|\phi|+ |g|\cdot |\phi| } i_f( B^\vee(i_g\phi))(x)\\& +(-1)^{|[f,g]|\cdot |\phi|+|g|+(|\phi|+1)|f\cup g|+|\phi|} i_{f\cup g} (B^\vee \phi)(x)\\&=(-1)^{|g|}i_{f\cup g}(B^\vee(\phi))(x)+(-1)^{(|f|+1)(|g|+1)}i_gB^\vee(i_f \phi)(x)-i_fB^\vee(i_g \phi)(x)\\& +(-1)^{|f|} B^{\vee}((f\cup g)\cdot \phi)(x).
\end{split}
\end{equation*}
This proves the statement.
\end{proof}
Now let us suppose that we have an equivalence $A \simeq A^\vee[d]$ in the derived category of $A$-bimodules.
This property provides us with an isomorphism $D: HH^*(A,A^\vee)\rightarrow HH^{*+d}(A,A)$ which allows us to transfer the Connes operator on $HH^*(A,A^\vee)$ to $HH^*(A,A)$,
$$
\Delta := D \circ B^\vee \circ D^{-1}.
$$
\begin{lemma} Let $A$ be a DGA algebra with an equivalence $A\simeq {A^\vee}[d]$ in the derived category of $A$-bimodules. Then the induced isomorphism $D: HH^*(A,A^\vee)\rightarrow HH^{*+d}(A,A)$ is an isomorphism of $HH^*(A,A)$-modules, \textit{i.e.} for all $f\in HH^*(A,A)$ and $\phi\in HH^*(A,A^\vee)$ we have
\begin{equation}
D(i_f(\phi))= f\cup D(\phi)
\end{equation}
\end{lemma}
\begin{proof}
The proof is identical to the proof of Lemma \ref{lem0}. One uses a
resolution by semi-free modules in the category of $A$-bimodules and adapts diagram (\ref{diagcom1}) to the case of $C^*(A,A^\vee)$, the dual theory of $C_*(A,A)$.
\end{proof}
\begin{corollary}\label{coropda} Let $A$ be a DG algebra with an equivalence $A\simeq {A^\vee}[d]$ in the derived category of $A$-bimodules. Then for $f,g\in HH^*(A,A)$ and $\phi \in HH^*(A,A^\vee)$ we have
\begin{equation}
\begin{split}
[f,g]\cup D(\phi)&= (-1)^{|f|} \Delta((f\cup g)\cdot D\phi)-f\cdot \Delta (g\cdot D\phi)+ (-1)^{(|f|+1)(|g|+1)}g\cdot \Delta(f\cdot D\phi)\\ & +(-1)^{|g|} (f\cup g)\cdot DB^\vee(\phi).
\end{split}
\end{equation}
where $D:HH^*(A,A^\vee)\rightarrow HH^*(A,A)$ is the isomorphism induced by the derived equivalence.
\end{corollary}
\begin{proof}
This is a consequence of Lemma \ref{lem2}, the proof being similar to that of Corollary \ref{cor1}.
\end{proof}
\begin{definition}\label{PDPDA}
Let $A$ be a differential graded algebra such that $A$ is equivalent to $A^\vee[d]$ in the derived category of $A$-bimodules.
This means that there is a quasi-isomorphism of $A$-bimodules $\psi: P\rightarrow A^\vee[d]$ where $P$ is a semi-free resolution of $A$.
Then $\psi$ is a cocycle in $\Hom_{A^{e}}(P,A^\vee)$ for which $-\cap [\psi]: HH^*(A,A)\rightarrow HH^*(A,A^\vee)$ is an isomorphism. Under this assumption, $A$ is said to be a \emph{derived Poincar\'e duality} algebra (DPD for short) of dimension $d \in \Z$ if $B^\vee([\psi])=0$.
\end{definition}
\begin{remark}For a cocycle $\psi \in C^d(A,A^\vee)$, it is rather easy to check when $-\cap [\psi]: HH^*(A,A)\rightarrow HH^{*+d}(A,A^\vee)$ is an isomorphism\footnote{Intuitively, one should think of $HH^*(A,A)$ as the homology of the free loop space of some space, which includes a copy of the homology of the underlying space by the inclusion of constant loops. This condition means that one has to check that the restriction of the cap product to the constants loops corresponds to the Poincar\'e duality of the underlying manifold.}: one only has to check that $-\cap [\phi]: H^*(A)\hookrightarrow HH^*(A,A)\overset{\cap [\phi]}{\hookrightarrow} H^*(A^\vee) $ is an isomorphism (See \cite{Me}, Proposition 11).
\end{remark}
Two immediate consequences of the previous lemma are the following theorems.
\begin{theorem}
For a DPD algebra $A$, $(HH^*(A,A), \cup, \Delta)$ is a BV-algebra i.e.
\begin{equation}\label{BV-cond}
[f,g]=(-1)^{|f|}\Delta(f\cup g)-(-1)^{|f|}\Delta(f)\cup g-f\cup \Delta(g).
\end{equation}
\end{theorem}
\begin{proof}
Suppose that the derived equivalence $A^\vee[d] \simeq A$ is realized by a quasi-isomorphism $\psi: P\rightarrow A^\vee$,
where $\epsilon: P\rightarrow A$ is a semi-free resolution of $A$.
\begin{equation}
\xymatrix{P\ar[r]^\psi & A^\vee \\ P\ar[r]^{id} & P\ar[u]^\psi \ar[d]^\epsilon\\ P\ar[r]^\epsilon & A}
\end{equation}
One can then use $\Hom_{A^{e}}(P,A^\vee)$ to compute $HH^*(A,A^\vee)$, and similarly \linebreak $\Hom_{A^{e}}(P,A)$ or $\Hom_{A^{e}}(P,P)$ to compute the cohomology $HH^*(A,A)$. Let $ D= (-\cap [\psi])^{-1}: HH^*(A,A^\vee)\rightarrow HH^*(A,A)$
be the isomorphism induced by the derived equivalence.
Then the cohomology class represented by $id\in \Hom_{A^{e}}(P,P)$ corresponds to $1\in HH^*(A,A)$ using $\epsilon_* : \Hom_{A^{e}}(P,P) \rightarrow \Hom_{A^{e}}(P,A) $, and to $\psi$ by the map $$\psi_*:\Hom_{A^{e}}(P,P) \rightarrow \Hom_{A^{e}}(P,A^\vee).$$
Therefore, $D([\psi])= 1\in HH^*(A,A)$ where $D=\epsilon_*\circ \psi_*^{-1}$. Now take $\phi=[\psi]$ in the statement of Corollary \ref{coropda}.
\end{proof}
A similar theorem can be proved under a slightly different assumption.
\begin{theorem}(Menichi \cite{Me}) Let $A$ be a differential graded algebra equipped with a quasi-isomorphism $m: A\rightarrow A^\vee [d]$. Then $HH^*(A,A)$ has a BV algebra structure extending its natural Gerstenhaber algebra structure. The BV operator is $\Delta= DB^\vee D^{-1} $ where
$D: HH^*(A,A^\vee)\rightarrow HH^*(A,A)$ is the isomorphism induced by $m$.
\end{theorem}
\begin{proof}
The proof is very similar to that of the previous theorem. For simplicity we take the two-sided bar resolution $ \epsilon: B(A,A,A) \rightarrow A $, where $ \epsilon: A\otimes \kk \otimes A\subset B(A,A,A) \rightarrow A$ is given by the multiplication of $A$. Then $\psi= m\circ \epsilon: B(A,A,A)\rightarrow A^\vee$ is a quasi-isomorphism.
Let $[\psi] \in HH^*(A,A^\vee)$ be the class represented by $\psi$, and $D=(-\cap [\psi ])^{-1}: HH^*(A,A^\vee)\rightarrow HH^*(A,A)$ be the inverse of the isomorphism induced by $\psi$. Similarly to the proof of the previous theorem, we have $D([\psi])=1\in HH^*(A,A)$. It only remains to prove that $DB^\vee([\psi])=0$. For that we compute $B^\vee([\psi])=[B^\vee (\psi)]$. Note thatwe have $\kk$-module isomorphism $C^*(A,A^\vee)=\Hom_{A^{e}}(B(A,A,A), A^\vee)\simeq \Hom(A\otimes T(s\bar{A}), \kk)= (A\otimes T(s\bar{A}))^\vee$. The image of the Connes operator $B:A\otimes T(s\bar{A})\rightarrow A\otimes T(s\bar{A})$ is included in $A\otimes T(s\bar{A})^+$. Since $\epsilon_{|A \otimes T(s\bar{A})^+\otimes A}=0$, we have $m\circ B=\psi\circ\epsilon \circ B=0$.
\end{proof}
\section{Open Frobenius algebras}
\label{section2}
In this section we study the algebraic structure of the Hochschild cohomology of an open Frobenius algebra. More precisely, we introduce a BV algebra structure on the Hochschild cohomology and homology of such algebras. In fact this structure is a consequence of the action of the Sullivan chord diagrams on the Hochschild (co)homology of an open Frobenius algebra as we'll explain in the following section. But in this section we will present explicitly all the operations and homotopies related to the BV structures. The main theorems of this section hint that there should be a bi-BV structure of the Hochschild homology or cohomology of Frobenius algebras but we won't get into that. Let us just speculate that there should be a sort of Drinfeld compatibility for the Gerstenhaber bracket and cobracket \textit{i.e.} the cobracket is a Chevalley-Eilenberg cocycle with respect to the bracket. The results of this section are due to T. Tradler, M. Zeinalian and the author \cite{ATZ2}.
\medskip
All over this section, like the rest of the chapter, the signs are determined by Koszul's rule, we won't give them explicitly.
Readers interested in a more detailed sign discussion are referred to \cite{ChenGan} and \cite{TZ}, and they will be treated in \cite{ATZ2} as well.
\begin{definition}(DG open Frobenius algebra). A \textit{differential graded open Frobenius $\kk$-algebra} of degree $m$ is a triple $(A,\cdot,\delta)$ such that:
\begin{enumerate}
\item $(A,\cdot)$ is a unital differential graded associative algebra whose product has degree zero,
\item $(A, \delta)$ is differential graded cocommutative coassociative coalgebra whose coproduct has degree $m$,
\item $\delta : A\rightarrow A\otimes A$ is a right and left $A$-module map which using (simplified) Sweedler' notation reads
$$
\sum_{(x.y)} (x.y)'\otimes (xy)''=\sum_{(y)}x.y'\otimes y''= \sum_{(x)}(-1)^{m|x|} x'\otimes x''.y
$$
\end{enumerate}
Here we have simplified Sweedler's notation for the coproduct $ \delta x=\sum_{i} x'_i\otimes x''_i$, to $ \delta x=\sum_{(x)} x'\otimes x''$ where $(x)$ should be thought of as the index set for $i$'s.
\end{definition}
We recall that an ordinary (DG) Frobenius algebra, sometimes called \emph{closed Frobenius (DG) algebra}, is
a \emph{finite dimensional} unital associative commutative differential graded algebra equipped with an \emph{nondegenerate inner product} $\langle -,- \rangle$ which is \emph{invariant i.e}
$$
\langle xy,z \rangle=\langle x,yz\rangle.
$$
In particular the inner product allows us to identify $A$ with its dual $A^\vee$ (as $A$-modules, and even $A$-bimodules ) and define a coproduct on $A$ by
$$
A\simeq A^\vee \overset{\text{dual of product}}{\longrightarrow }(A \otimes A)^\vee \simeq A^\vee\otimes A^\vee\simeq A\otimes A.
$$
The coproduct is cocommutative and coassociative and satisfies condition (3) of the definition above, in other words a closed Frobenius algebra is also an open Frobenius algebra. Moreover, $\epsilon: A\rightarrow \kk$ defined by $\eta(x)=\langle x| 1 \rangle$ is a counit (trace). So we have the following identity:
\begin{equation}
\sum_{(x)}\eta(x')x''= x.
\end{equation}
As a consequence we have the following identities
\begin{equation}\label{iden-counit}
\sum_{(x)}\eta(x'y)x''=\sum_{(x)}(-1)^{|x'|m}\eta(x''y)x'= xy.
\end{equation}
\bigskip
\noindent \textbf{Exercise: }Prove that an open Frobenius algebra with a counit is a closed Frobenius algeba, and in particular it is finite dimensional. \textbf{Hint:} To prove that it has finite dimension, prove that for all $x$, we have $x=\sum_{(1)} 1'\langle x|1''\rangle$ where $\delta 1=\sum_{(1)} 1'\otimes1''$ therefore $A\subset Span_{(1)} \{ 1'\}$ hence finite dimensional. This explains why the homology of the free space $H_*(LM)$ (and its algebra models) is generally not a closed Frobenius.
There are plenty of examples of closed Frobenius algebras, for instance the cohomology and homology of a closed oriented manifold. This is a consequence of Poincar\'e duality. Over the rationals it is possible to lift this Frobenius algebra structure to the cochains level.
By a result of Lambrechts and Stanley \cite{LamStan}, there is a connected finite dimensional commutative DG algebra $A$ which is quasi-isomorphic to $C^*(M)$ the cochains of a given $n$-dimensional manifold $M$ and is equipped with a bimodule isomorphism $A\rightarrow A^\vee $ inducing the Poincar\'e duality $H^*(M)\rightarrow H_{*-n}(M)$. For more interesting examples of open Frobenius algebras see Section 6.
As we know $HH^*(A,A^\vee)$ is already equipped with a BV operator namely the Connes operator $B^\vee $, so we just need a product on $HH^*(A,A^\vee)$ or equivalently a coproduct on the Hochshild chains. This is given by
\begin{equation} \label{eq-cup-frob}
\theta (a_0[a_1,\cdots,a_n])=\sum_{(a_0), 1\leq i\leq n} \pm (a_0' [a_1,\cdots,a_{i-1},a_i ]) \bigotimes (a_0'' [a_{i+1}, \cdots,a_{n}])
\end{equation}
Then we can define the cup product of $f,g \in C^*(A,A^\vee)= \Hom (A \otimes T(s\bar{A}), \kk)$ by
$$
(f \ast g)(x):= \mu(f\otimes g )\theta(x)
$$
where $\mu: \kk\otimes \kk \rightarrow \kk$ is the multiplication. More explicitly
$$(f \ast g)(a_0 [a_1,\cdots,a_n])= \sum_{(a_0), 1\leq i< n} \pm f(a'_0 [a_1,\cdots,a_{i-1}, a_i]) g(a''_0 [a_{i+1}, \cdots,a_{n}]).$$
In the case of a close Froebnius algbera this product corresponds to the standard cup product on $HH^*(A,A)$ using the isomorphism
\begin{equation*}
HH^*(A,A)\simeq HH^*(A,A^\vee)
\end{equation*}
induced by the inner product on $A$. More explicitly we identify $A$ with $A^\vee$
using the map $ a\mapsto (a^\vee(x):=\langle a,x\rangle )$. Therefore to a cocycle
$f \in C^*(A,A)$, $f:A^{\otimes n}\to A$, corresponds a cocycle $\tilde{f}\in C^*(A)= C^*(A,A^\vee)$ given
by $\tilde{f} \in \Hom (A^{\otimes n }, A^\vee)\simeq \Hom (A^{\otimes (n+1)},\kk)$,
\begin{equation*}
\tilde{f}(a_0,a_1,\cdots a_n):=\langle f(a_1,\cdots a_n),a_0\rangle.
\end{equation*}
The inverse of this isomorphism is given by
\begin{equation*}
f(a_1,\cdots a_n):=\sum_{(1)} \tilde {f}(1',a_1,\cdots a_n)1'' .
\end{equation*}
For two cocyles $f: A^{\otimes p} \to A$ and $g: A^{\otimes q} \to A$ in $C^*(A,A)$ we have
\begin{equation*}
\begin{split}
\widetilde{f \cup g}(a_0, a_1,\cdots a_{p+q})&= \langle ( f \cup g)( a_1,\cdots a_{p+q}), a_0\rangle = \langle f(a_1,\cdots a_{p})g(a_{p+1},\cdots a_{p+q}),a_0\rangle\\ &=\langle f(a_1,\cdots a_{p}),g(a_{p+1},\cdots a_{p+q})a_0\rangle= \sum_{(a_0)} \langle f(a_1,\cdots a_{p}),\langle g(a_{p+1},\cdots a_{p+q}),a''_0 \rangle a'_0\rangle\\ &= \sum_{(a_0)} \langle f(a_1,\cdots a_{p}),\langle g(a_{p+1},\cdots a_{p+q}),a''_0 \rangle a'_0\rangle\\&=\sum_{(a_0)} \langle f(a_1,\cdots a_{p}),a'_0\rangle \langle g(a_{p+1},\cdots a_{p+q}),a''_0 \rangle=
(\tilde{f} \ast \tilde{g})(a_0[a_1,\cdots a_{p+q}])\\
\end{split}
\end{equation*}
\begin{remark}
By a theorem of F\'elix-Thomas \cite{FTBV}, this cup product on $HH^*(A,A^\vee)$ provides an algebraic model for the Chas-Sullivan product on $H_*(LM)$ the homology of the free loop space of closed oriented manifold $M$. Here one must over a field of characteristic zero and for $A$ one can take the closed (commutative) Frobenius algebra provided by Lambreschts-Stanley result \cite{LamStan} on the existence of an algebraic model with Poincar\'e duality for the cochains of a closed oriented maniflold.
\end{remark}
\begin{theorem} For an open Frobenius algebra $A$, $(HH^*(A,A^\vee), \cup, B^\vee)$ is a BV-algebra.
\end{theorem}
\begin{proof}
To prove the theorem we show that $(C_*(A,A), \theta, B)$ is a homotopy coBV coalgebra.
It is a direct check that $\theta$ is co-associative. Just like the computation above, we transfer the homotopy for commutative (in the case of closed Frobenius algebra) of the cup product as given in Theorem \ref{Gersten} to
$C^*(A,A^\vee)$ and then dualize it. It turns out that the obtained formula only depends on the product and coproduct, so it makes also sense for open Frobenius algebras.
The homotopy for co-commutativity is given by
\begin{equation}\label{homotpy-cocom}
\begin{split}
h(a_0 [a_1, \cdots ,a_n])&:=\sum_{(1), 0\leq i<j\leq n+1} ( a_0 [ a_1 ,\cdots ,a_i, 1'',a_{j},\cdots a_{n}] )\bigotimes (1' [ a_{i+1},\cdots,a_{j-1}] ).
\end{split}
\end{equation}
where for $j=n+1$ and $i=0$ the correspondings terms are respectively
$$
(a_0 [ a_1 ,\cdots ,a_i, 1''] )\bigotimes (1' [ a_{i+1},\cdots,a_{n}]).
$$
and
$$
(a_0 [ 1'',a_{j},\cdots a_{n}] )\bigotimes (1' [ a_{1},\cdots, a_{j-1}]).
$$
It is a direct check that $ hd-(d\otimes 1+ 1\otimes d)h=\theta -\tau \circ \theta$ where $\tau: C_*(A) \otimes C_*(A)\rightarrow C_*(A) \otimes C_*(A)$ is given by $\tau (\alpha_1\otimes \alpha_2)=\pm \alpha_2\otimes \alpha_1$.
To prove that the 7-term (coBV) relation holds, we use the Chas-Sullivan \cite{CS1} idea (see also \cite{Tradler}) in the case of the free loop space adapted to the combinatorial (simplicial) situation. First we identify the Gerstenhaber co-bracket explicitly. Let
$$
S:= h-\tau\circ h
$$
Once proven $S$ is, up to homotopy, the deviation of $B$ from being a coderivation for $\theta$, the $7$-term homotopy coBV relation is equivalent to the homotopy co-Leibniz identity for $S$.
\subsubsection*{Co-Leibniz identity:} The idea of the proof is identical to Lemma 4.6 \cite{CS1}. We prove that up to some homotopy we have
\begin{equation}\label{coleib}
(\theta \otimes id) S=(id\otimes \tau)(S\otimes id)\theta+ (id \otimes S) \theta
\end{equation}
It is a direct check that
$$
(id\otimes \tau)(h\otimes id)\theta + (id \otimes h)\theta=(\theta \otimes id)h,
$$
so to prove (\ref{coleib}) we should prove that up to some homotopy
\begin{equation}
(id\otimes \tau)(\tau h\otimes id)\theta + (id\otimes \tau h)\theta=(\theta \otimes id)\tau h.
\end{equation}
The homotopy is given by $H: C^*(A)\rightarrow (C^*(A))^{\otimes 3} $
\begin{equation*}
\begin{split}
H(a_0 [a_1, \cdots ,a_n])=\sum_{0\leq l< i\leq j<k} \sum_{(1),(1)} ( 1''[a_l,\cdots a_{i-1}]) & \bigotimes (1'' [a_{j},\cdots a_{k-1}])\\ &\bigotimes a_0[a_{1},\dots , a_{l},1',a_i\cdots a_{j-1},1',a_k,\cdots, a_n].
\end{split}
\end{equation*}
Note that in the sum above the sequence $a_i\cdots a_{j-1}$ can be empty. The identity
$$(d\otimes id \otimes id+id\otimes d \otimes id+id\otimes id \otimes d)H-Hd=(id\otimes \tau)(\tau h\otimes id)\theta + (id\otimes \tau h)\theta-(\theta \otimes id)\tau h $$
can be checked directly.
\subsubsection*{Compatibility of $B$ and $S$:} The final step is to prove that $S= \theta B \pm (B \otimes id \pm id\otimes B) \theta$ up to homotopy. To that end we prove that $h$ is homotopic to $(\theta B)_2-(B\otimes id)\theta$ and similarly $\tau h \simeq (\theta B)_1-(id \otimes B)\theta $ where $\theta B= (\theta B)_1 +(\theta B)_2$, with
\begin{equation*}
\begin{split}
(\theta B)_1(a_0 [a_1,\cdots ,a_n])= \sum_{0\leq i\leq j\leq n} \sum_{(1)}( 1' [a_i,\cdots , a_{j}])\bigotimes (1'' [a_{j+1},\cdots ,a_n, a_0 \cdots , a_{i-1}]).
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
(\theta B)_2(a_0 [a_1,\cdots , a_n])= \sum_{0<j<i\leq n} \sum_{(1)} (1' [a_{i},\cdots ,a_n ,a_0, a_1 \cdots , a_{j}])\bigotimes (1'' [a_{j+1},\cdots , a_{i-1}]).
\end{split}
\end{equation*}
It can be easily checked that
\begin{equation*}
\begin{split}
H(a_0 [a_1,\cdots , a_n])= \sum_{0 \leq k< j\leq i\leq n} \sum_{(a_i)} (&1[ a_{i},\cdots a_n, a_0,a_1,\cdots,a_{k},1',a_{j},\cdots a_{i-1}] )\\ &\bigotimes (1'' [ a_{k+1},\cdots,a_{j-1}] ),
\end{split}
\end{equation*}
is a homotopy between $h$ and $(\theta B)_2-(B \otimes id)\theta$. In the formulae describing $H$, the sequence $a_j,\cdots, a_{i-1}$ can be empty.
While computing $dH$ we encounter, the terms corresponding to $k=0$ is exactly
\begin{equation*}
\begin{split}
(B\otimes 1)\theta(a_0 [a_1,\cdots ,a_n])=\sum (1 [a_{j},\cdots, a_i,a'_0,a_1,\cdots, a_{j-1}] ) \bigotimes (a''_0 [a_{i+1}, \cdots , a_n]).
\end{split}
\end{equation*}
Similarly one proves that $\tau h\simeq (\theta B)_1-(id \otimes B)\theta $.
\end{proof}
There is a dual statement as follows.
\begin{theorem}\label{thm-BV_hom} The shifted Hochschild homology $HH_*(A)[1-m]$ of a degree $m$ open Frobenius algebra $A$ is a BV algebra whose BV-operator is the Connes operator and the product is given by
\begin{equation}
\begin{split}
x\cdot y & = \sum_{(a_0b_0)}\pm (a_0b_0)'[ a_1 , \dots , a_m , (a_0b_0)'' , b_1 , \dots , b_n]\\
& = \sum_{a_0)}\pm a'_0[ a_1 , \dots , a_m , a_0''b_0 , b_1 , \dots , b_n]\\
& = \sum_{b_0}\pm a_0b_0'[ a_1 , \dots , a_m , b_0'' , b_1 , \dots , b_n]\\
\end{split}
\end{equation}
for $x=a_0 [a_1,\cdots ,a_m]$ and $y= b_0 [b_1, \cdots ,b_n] \in C_*(A,A)$.
\end{theorem}
Note that the identities above hold because $A$ is an open Frobenius algebra. The product defined above is strictly associative, but commutative only up to homotopy, and the homotopy being given by
\begin{equation}
\begin{split}
H_{1} (x, y)=& \sum_{(a_0b_0)} \pm
1[ a_{1},\cdots , a_{n},(a_0b_0)', b_1,\cdots , b_m , (a_0b_0)'']
\\ & + \sum_{i=1}^n \sum_{(a_0b_0)} \pm
1[ a_{i+1},\cdots , a_{n} ,(a_0b_0)', b_1 , \cdots, b_m , (a_0b_0)'' , a_1,\cdots ,a_i].
\end{split}
\end{equation}
To prove that the 7-term relation holds, we adapt once again Chas-Sullivan's \cite{CS1} idea to a simplicial situation. First we identify the Gerstenhaber bracket directly. Let
\begin{equation}
\begin{split}
x\circ y:= \sum _{i= 0 }^m \sum_{(a_0)} b_0[ b_1 , \cdots b_{i} , a_{0}' , a_{1}, \cdots a_{n}, a_{0}'', b_{i+1}, \cdots b_{m}],
\end{split}
\end{equation}
and then define $\{x,y\}:=x\circ y\pm y\circ x$. Next we prove that the bracket $\{-,-\}$ is homotopic to the deviation of the BV operator from being a derivation. For that we decompose the Connes operator, our BV operator, in two pieces:
\begin{equation*}
\begin{split}
B_{1} (x,y):= \sum_{j=1}^{m} \sum_{(a_0b_0)}\pm 1[b_{j+1}, \dots b_{m}, (a_0b_0)', a_1, \dots, a_n , (a_0b_0)'' , b_1 , \dots , b_j],
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
B_{2} (x,y):= \sum_{j=1}^{m} \sum_{(a_0b_0)}\pm 1[a_{j+1},\dots a_{n}, (a_0b_0)', b_1, \dots, b_m , (a_0b_0)'' , a_1 , \dots , a_j],
\end{split}
\end{equation*}
so that $B=B_1+B_2$. Then $x\circ y $ is homotopic $ B_1(x,y)-x.By$. In fact the homotopy is given by
\begin{equation*}
\begin{split}
H_{2}(x,y) &= \sum_{0\leq j\leq i\leq m} \sum_{(a_0)}1 [b_{j+1}, \cdots , b_{i} , (a_{0})' , a_{1}, \cdots a_{n}, (a_{0})'' , b_{i+1},\cdots , b_m, b_{0}, \cdots b_{j}].
\end{split}
\end{equation*}
Similarly for $y\circ x$ and $B_2(x\cdot y) -Bx \cdot y$. Therefore we have proved that on $HH_*(A,A)$ the following identity holds:
$$
\{x,y\}=B(x\cdot y)- Bx\cdot y\pm x\cdot By.
$$
Now proving the 7-term relation is equivalent to prove the Leibniz rule for the bracket and the product, \textit{i.e.}
$$
\{x,y\cdot z\}= \{x,y\}\cdot z\pm y\cdot \{x,z\}.
$$
It is a direct check that $x\circ (y \cdot z)=(x\circ y)\cdot z+ y\cdot(x\circ z)$. On the other hand $(y\cdot z) \circ x$ is homotopic to
$(y\circ x) \cdot z - y\cdot (z\circ x) $ using the homotopy
\begin{eqnarray*}
H_{3}(x,y,z) &=&\sum a_0 [a_1, \dots , a_i, b_0', b_1 , \dots , b_n,b_0'', a_{i+1}\dots , a_j , c_0', c_1, \dots, c_m, c_0'', a_{j+1}, \dots ,a_p].
\end{eqnarray*}
Here $z=c_0[c_1, \dots , c_p]$. This proves that the Leibniz rule holds up to homotopy.
\begin{remark} In \cite{ChenGan} Chen and Gan prove that for an open Frobenius algebra $A$, the Hochschild homology of $A$ seen as a \emph{coalgebra}, is a BV algebra. They also prove that the \emph{reduced} Hochschild homology is a BV and coBV algebra. It is necessary to take the reduced Hochschild homology in order to get the coBV structure.
\end{remark}
\section{Towards the action of the moduli space of curves: Sullivan chord diagrams}\label{chord-section}
In this section we extend the operations introduced in the previous section to an action of Sullivan chord diagrams on the Hochschild chains $C_*(A,A)$ [and cochains $C^*(A,A^\vee)$] and on the Hochschild cochains $C^*(A,A^\vee)$ of an open Frobenius algebra. Our formulation implies to \cite{TZ} and can be extended to homotopy Frobenius algebras without much of modification. Because the inner product induces an isomorphism $A\simeq A^\vee$ of $A$-bimodules then all structures can be transferred from $HH^*(A,A^\vee)$ to $HH^*(A,A)$ and this recovers the main result of \cite{TZ}. Since we are describing the action on the Hochschild chains there is difference in our terminology with that of \cite{TZ}. Here the incoming cycles of Sullivan chord diagram correspond to outgoing cycles of the same diagram in \cite{TZ} and \cite{CG}.
This action is part of the action of the homology of the moduli space on the Hochschild chains of a closed Frobenius algebra. We refer the reader to \cite{CTZ} for a description of the action of of the moduli space of curves on the Hochschild chains of Hermitian Calabi-Yau spaces. More recently there has been some progress has been annonced by N. Wahl and C. Westerland \cite{WahWest} who claimed to have extended these results for integral coefficients and strong homotopy Frobenius algebras .
A \emph{Sullivan chord diagram} \cite{CS2,CG} of type $(g,m,n)$ is a graph which is a union of $m$ labeled disjoints oriented circles, called \emph{output circles} or \emph{outgoing boundaries}, and some \emph{disjointly immersed} trees whose endpoints land on the outcoming circles. The trees are called \emph{chords}, which have length zero. We assume that each vertex is at least trivalent, therefore there is no vertex on a circle which is not an end of a tree. The graphs don't need to be connected.
The cyclic ordering basically tells us how to draw the graph on the plane. The cyclic ordering should be such that the $m$ ouput circles are among the boundary components, which is best visualized by thickening the graph to a surface whose genus is $g$. The cyclic orderings also allow us to identify the remaining $n$ labeled \emph{input circles} or \emph{intgoing boundaries}. Therefore this surface has $n+m$ labeled boundary components. We also assume that each incoming circle has a marked point, called \emph{input marked point}, and similarly each outgoing boundary has a marked point, called \emph{outgoing marked point}. Like\cite {WahWest} one may think of the the input marked point as a leaf, connecting a degree vertex to the corresponding input cycle, but we don't.
We don't consider marked point as vertices of the graphs but as some \emph{special points} on the graph.
The marked points and the endpoints of the chords may correspond. Because of the cyclic ordering at each vertex, there is a well-defined cyclic ordering on the special points attached to a tree (chord).
\begin{figure}[htb]
\centering
\includegraphics[scale=0.6]{Irma-graph-Sulli.eps}
\caption{}
\label{graph}
\end{figure}
Figure \ref{graph} displays a chord diagram with 5 incoming cycles and 3 outgoing cycles. There is an obvious composition rule for two Sullivan chord diagrams if the number of output circles of the first graph equals the number of input circles of the second one. Of course the labeling matters and marked points get identified. This composition rule makes the space of Sullivan chord diagrams into a PROP (see \cite{TZ} or \cite{CG} for more details). Here we don't give the definition of a PROP and we refer the interested reader to \cite{May} and \cite{MSS} for more detail.
The combinatorial degree of a diagram of type $(g,m,n)$ is the number of connected components obtained after removing all special points. Let $CS_k(g,m,n)$ denote the space of $(g,m,n)$-diagrams of degree $k$. For instance the combinatorial degree of the diagram in Figure \ref{diag3} is one which corresponds to the BV operator. One makes $\{CS_k(g,m,n)\}_{k\geq 0}$ into a complex using a boundary map which is defined by collapsing an edge (arc) on input circles and considering the induced cyclic ordering. In what follows we describe the action of chord diagrams on chains in $C_*(A,A)$ whose degree is exactly the combinatorial degree of the given diagram. In other words we construct a chain map $(CS_k(g,m,n)\rightarrow (\Hom(C_*(A,A)^{\otimes m},C_*(A,A)^{\otimes n}), D:=[d_{Hoch},-])$. Moreover this action is compatible with the composition rule of the diagrams. Said formally, $ C_*(A,A)$ is a differential graded algebra over the differential operad $\{CS_k(g,m,n)\}_{k\geq 0}$. We won't deal with this last statement.
\subsubsection*{The equivalence relation for graphs and essentially trivalent graphs:} Two graphs are considered equivalent if one is obtained from the other using on of the following moves:
\begin{itemize}
\item sliding, one each time, a vertex on the chord over edges of the chord.
\item sliding an input marked point over the chord tree.
\end{itemize}
By doing so one can easily see that each Sullivan chord diagram is equivalent to a Sullivan chord diagram whose each vertex is trivalent or has an input marked point, and no input marked point coincides is on a chord end point.
\subsubsection*{The action of the diagrams:}
Let $\gamma$ be a chord diagram with $m$ input circles and $n$ output circles. We assume that in $\gamma$ all vertices are trivalent and no input marked point coincides with a chord end point, otherwise we will replace with an equivalent trivalent graph as explained above.
The aim is to associate to $\gamma$ a chain map $(C_*(A,A))^{\otimes m} \rightarrow (C_*(A,A))^{\otimes n}$. Let $x_i=a_0^i [a_1^i|\cdots| a_{k_i}^i]$, $1\leq i\leq m$, be $m$ Hochschild chains.
\bigskip
\noindent Step 1) Write down $a_0^i, a_1^i,\cdots, a_{k_i}^i$ on the $i$th input circle by putting first $a_0^i$ on the input marked point and then the rest following the orientation of the cycle, on those parts of the $i$th input cycle which are not part of the chord tree (at this stage we don't use the output marked point).
We consider all the possible ways of placing $a_1^i,\cdots, a_{k_i}^i$ on the $i$th cycle following rules specified above.
\noindent Step 2) At an output marked point which is not a chord end point or an input marked point we place a $1$, otherwise we move to the next step.
\bigskip
\noindent Step 3) On the end points of a chord tree with $r$ end points and no input marked point, we place following orientation of the plane $1',1''\cdots 1^{(r)} $ where
$$
(\delta\otimes id ^{(r-2)})\otimes \cdots \otimes (\delta\otimes id) \delta (1) =\sum_{(1)} 1'\otimes 1''\otimes \dots 1^{(r)} \in A^{\otimes r},
$$
\bigskip
\noindent Step 4) On the end points of a chord tree with $r$ endpoint which has $s$ input marked points on its vertices we do as follows: We organize the chord tree as a rooted tree whose roots are input marked point. Now the tree defines a a well-define (because of the Frobenius relations) operation $A^{\otimes s}\to A^{\otimes r}$ defined using the product and coproduct of $A$. Now by applying this operation on the element placed on the input marked points (the roots of the tree) we obtain a sum $\sum_i x_i^1\otimes\cdots \otimes x_i^r$. We decorate the end points of the chord tree by $x_i^r, \cdots , x_i^r$ following the orientation of the plane.
\bigskip
\noindent Step 5) For each output circle, starting from its output marked point and following its orientation, read off all the elements on the outgoing cycle \emph{i.e.} $a_j^i$ left or created after the previous steps, possibly $1$ and the $a^{(k)}$, and note them as an element of $C_*(A,A)$. Since the output circles are labeled we therefore obtain a well-defined element of $ (C_*(A,A))^{\otimes n}$.
Take the sum over all possible sum appeared in steps 3 and 4. The result is an element of $C_*(A,A)^{\otimes n}$.
\bigskip
\noindent We clarify this procedure with some examples. The BV operator clearly corresponds to the diagram in Figure \ref{diag3}.
\begin{figure}[htb]
\centering
\includegraphics[scale=0.5]{Figurenew7.eps}
\caption{BV operator}
\label{diag3}
\end{figure}
The coproduct (\ref{eq-cup-frob})
\begin{equation}
\theta (a_0[a_1,\cdots,a_n])=\sum_{(a_0), 1\leq i\leq n} \pm (a_0' [a_1,\cdots,a_{i-1},a_i ]) \bigotimes (a_0'' [a_{i+1}, \cdots,a_{n}]),
\end{equation}
corresponds to the diagram in Figure \ref{Figurenew3}. The dual of $\theta$ induces a product on $HH^*(A,A^\vee)$ which under isomorphism $HH^*(A,A^\vee) \simeq HH^*(A,A)$ corresponds to the cup product on $HH^*(A,A)$ (see Section \ref{section2}). On should think of the latter as the the algebraic model of the Chas-Sullivan \cite{CS1} product on $H_*(LM)$.
The homotopy $h$ for co-commutativity of the $\theta$ as defined in (\ref{eq-cup-frob}) corresponds to the diragram in Figure \ref{Figurenew4}.
\begin{figure}[htb]
\centering
\includegraphics[scale=0.6]{Figurenew3.eps}
\caption{String topology coproduct on $HH_*(A,A)$ [the dual of the cup product on cohomology $HH^*(A,A^\vee)\simeq HH^*(A,A)$] }
\label{Figurenew3}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[scale=0.7]{Figurenew4.eps}
\caption{The homotopy for cocommutativity of $\theta$}
\label{Figurenew4}
\end{figure}
The degree zero coproduct as defined in Cohen-Godin on $H_*(LM)$ is the dual of the following product on $HH_*(A,A)$:
\begin{equation}
(a_0[a_1,\cdots,a_n]) \ast (b_0[b_1,\cdots,b_m])=\begin{cases} 0 \text{ if } n\geq 1 \\ a'a''b_0[b_1,\cdots,b_m] \text{ otherwise }
\end{cases}
\end{equation}
The product $\ast$ corresponds to the diagram in Figure \ref{Figurenew1} which is equivalent to the essentially trivalent graph in Figure \ref{Figurenew2}.
\begin{figure}[htb]
\centering
\includegraphics[scale=0.6]{Figurenew1.eps}
\caption{The dual of Cohen-Godin coproduct}
\label{Figurenew1}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[scale=0.5]{Figurenew2.eps}
\caption{Essentially trivalent graph corresponding to Cohen-Godin coproduct }
\label{Figurenew2}
\end{figure}
The degree 1 string topology coproduct on $H_*(LM)$ which corresponds to the diagram in Figure \ref{Figurenew5} is
\begin{equation*}
\begin{split}
x\bullet y&=\sum \pm (a_0b_0)' [a_1|\cdots | a_n|(a_0b_0)''|b_1|\cdots b_m],
\end{split}
\end{equation*}
\begin{figure}[htb]
\centering
\includegraphics[scale=0.5]{Figurenew5.EPS}
\caption{The dual of the Chas-Sullivan-Turaev coproduct}
\label{Figurenew5}
\end{figure}
This is exactly the product introduced in the statement of Theorem \ref{thm-BV_hom}. This diagram is equivalent to the essentially trivalent graph in Figure \ref{Figurenew6}.
\begin{figure}[htb]
\centering
\includegraphics[scale=0.5]{Figurenew6.EPS}
\caption{The trivalent graph of the Chas-Sullivan-Turaev coproduct}
\label{Figurenew6}
\end{figure}
The dual of the equivariant version (on the cyclic homology) of this product corresponds to the Chas-Sullivan\cite{chassulli2}/Turaev \cite{Turaev} coproduct on $H^{S^1} (LM)$.
\begin{remark}
A similar coproduct was also studied by Chen-Gan \cite{ChenGan} for co-Hochschild homology.
\end{remark}
Now it remains to deal with differentials. This is quite easy to check since collapsing the arcs on the input circles corresponds to the components of the Hochschild differential. The only nontrivial part concerns collapsing the arcs attached to the special points and this follows from the hypothesis that $A$ is an open Frobenius algebra with a counit. This shows that to cycles in $( \{CS_k(g,m,n)\} _{k\geq 0},\partial)$, the action associates a chain map and the homotopies between operations correspond to the action of the boundaries of corresponding chains in $( \{CS_k(g,m,n)\}_{k\geq 0},\partial)$. We refer the reader to \cite{TZ} for more details. Now one can explain all the homotopies in the previous section using this language.
The main the result of this section can be formulated as follows:
\begin{theorem} For an open Frobenius (DG) algebra $A$, $C_*(A,A)$ the Hochschild chain complex of $A$ is an algebra over the PROP of Sullivan chord diagrams. Similarly the Hochschild cochain complex $C^*(A,A)$ of $A$ is an algebra over the PROP of Sullivan chord diagrams.
\end{theorem}
In particular the action of Sullivan chord diagrams implies (see \cite{CG} for more details):
\begin{corollary}For an open Frobenius (DG) algebra $A$, $HH_*(A,A)$ and $HH^*(A,A^\vee)$ are open Frobenius algebras.
\end{corollary}
Note that we had already identified the product and coproduct of this open Frobenius algebra structure.
\begin{remark}
As we saw above, the product and coproduct on $HH_*(A,A)$ require only an open Frobenius algebra structure on $A$. The results of this section do not prove that the product and coproduct are compatible so that $HH_*(A,A)$ is an open Frobenius algebra. The reason is that our proof of the compatibility identities uses the action of some chord diagrams whose actions are defined only if $A$ is a closed Frobenius algebra. Still it could be true that $HH_*(A,A)$ is an open Frobenius algebra if $A$ is only an open Frobenius algebra, but this needs a direct proof.
\end{remark}
\section{Cyclic cohomology}
In this section we briefly describe some of the structures of the cyclic homology and negative cyclic homology, which are induced by those of the Hochschild cohomology via Connes' long exact sequence. We recall that the cyclic and negative cyclic chain complexes of a DG- algebra $A$ are
\begin{equation*}
\begin{split}
CC_*(A)&= (C_*(A,A)[[u,u^{-1}]/u\kk[[u]], d+ uB),\\
CC_*^-(A)&= (C_*(A,A)[[u]], d+ uB).\\
\end{split}
\end{equation*}
Here is $u$ is formal variable of degree $2$, $d=d_{Hoch}$ and $B$ is the Connes operator, whereas $u\kk[[u]]$ stands for the ideal generated by $u$ in $C_*(A,A)[[u,u^{-1}]$. Here $\kk[[u,u^{-1}]$ stands for the Laurent series in $u$.
The cyclic and negative cyclic cochain complexes are defined to be:
\begin{equation*}
\begin{split}
CC^*(A)&= (C^*(A,A^\vee)[v], d^\vee+ vB^\vee),\\
CC^*_-(A)&= (C^*(A,A^\vee)[[v,v^{-1}]/v\kk[[v]], d^\vee+ vB^\vee).\\
\end{split}
\end{equation*}
The cyclic homology of $A$ is denoted $CH_*(A)$ and is the homology of the complex $CC_*(A) $. The negative cyclic homology $HC^-_*(A)$ is the homology of $CC_*^-(A)$. Here $v$ is a formal variable of degree $-2$.
\begin{lemma} Let $(A^*,\cdot,\Delta)$ be a BV-algebra and $L^*$ a graded vector space with a long exact sequence
\begin{equation}\label{Connes}
\xymatrix{\cdots \ar[r]& L^{k+2} \ar[r] & L^{k} \ar[r]^-m & A^{k+1} \ar[r]^-e & L^{k+1}\ar[r] & L^{k-1} \ar[r]^-m& A^k \ar[r]^-e &\cdots}
\end{equation}
such that $\Delta= m\circ e$. Then
$$
\{a,b\}:=(-1)^{|a|} e(ma\cdot mb)
$$
defines a graded Lie bracket on the graded vector space $L^*$. Moreover $m$ sends the Lie bracket to the opposite of the Gerstenhaber bracket i.e.
$$
m\{a,b\}=-[ma,mb].
$$
\end{lemma}
\begin{proof}
We have,
\begin{equation}\begin{split}
&\{a,\{b,c\}\}=(-1)^{|a|+|b|} e (ma \cdot \Delta(mb \cdot mc) )\\
&\{\{a,b \},c\}= (-1)^{|b|} e (\Delta(ma \cdot mb) \cdot mc) )\\
&\{b,\{a,c\}\}=(-1)^{|b|+|a|} e( mb \cdot \Delta(ma \cdot mc)) .
\end{split}
\end{equation}
Then
\begin{equation}\begin{split}
& \{a,\{b,c\}\}- \{\{a,b \},c\}-(-1)^{|a|.|b|} \{b,\{a,c\}\} \\=&(-1)^{|a|+|b|} e [ma \cdot \Delta(mb \cdot mc) +(-1)^{|a|+1}\Delta(ma \cdot mb) \cdot mc+(-1)^{|a||b|+1}mb \cdot \Delta(ma \cdot mc)]\\=&
(-1)^{|b|+1}e [\Delta(ma \cdot mb) \cdot mc +(-1)^{|a|+1} ma \cdot \Delta(mb \cdot mc) +(-1)^{|a|(|b|+1)}mb \cdot \Delta(ma \cdot mc)].
\end{split}
\end{equation}
By replacing $a$, $b$, and $c$ in the 7-term relation (\ref{7term}) respectively by $ma$, $mb$ and $mc$, we see that the last line in the above identity is equal to $ (-1)^{|b|+1}e \Delta (ma.mb.mc)=-1)^{|b|+1}e m e (ma.mb.mc)=0$ because of the exactness of the long exact sequence.
Therefore $ \{a,\{b,c\}\}- \{\{a,b \},c\}-(-1)^{|a|.|b|} \{b,\{a,c\}\}=0$, proving the Jacobi identity.
As for the second statement,
\begin{equation}\begin{split}
m\{a,b\}&=(-1)^{|a|} me(ma\cdot mb)= (-1)^{|a|} \Delta (ma\cdot mb)\\&=(-1)^{|a|} ( (-1)^{|a|+1}[ma,mb] - \Delta (ma)\cdot mb+ (-1)^{|a|+1}ma\cdot \Delta (mb)]=-[ma,mb].
\end{split}
\end{equation}
\end{proof}
Using this lemma and Connes' exact sequence for the cyclic cohomology (or homology),
\begin{equation}\label{Connes2}
\xymatrix{\cdots HC^{k+2}(A) \ar[r] & HC^{k}(A) \ar[r]^-b & HH^{k+1}(A,A^*) \ar[r]^-e & HC^{k+1}(A)\ar[r] &\cdots}
\end{equation}
we have:
\begin{corollary}
The cyclic cohomology and negative cyclic cohomology of an algebra whose Hochschild cohomology is a BV algebra, has a natural graded Lie algebra structure given by
$$
\{x,y\}:= e(m(x)\cup m(y)).
$$
\end{corollary}
In fact one can prove something slightly better.
\begin{definition}
A \emph{gravity} algebra is a graded vector space $L^* $ equipped with maps
$$
\{\cdot,\cdots, \cdot\}: L^{\otimes k}\rightarrow L
$$
satisfying the following identities:
\begin{equation}
\begin{split}
\sum_{i,j}(-1)^{\epsilon_{i,j}}\{\{x_i,x_j\},x_1 \cdots ,\hat{x}_i,\cdots, &\hat{x}_j,\cdots,x_k,y_1,\cdots,y_l\}\\&=\begin{cases}
0, \text{ if } l=0\\
\{ \{ x_1 ,\cdots ,\cdots,x_k\},y_1,\cdots,y_l \}, \text{ if } l>0
\end{cases}
\end{split}
\end{equation}
\end{definition}
It is quite easy to prove that
\begin{proposition} The cyclic and negative cyclic cohomology of an algebra whose Hochschild cohomology is a BV algebra, is naturally a gravity algebra where the brackets are given by
\end{proposition}
$$
\{x_1,\cdots,x_k\}:= e(m(x_1)\cup \cdots \cup m(x_k)).
$$
The proof is a consequence of the following identity for BV algebras:
\begin{equation}
\begin{split}
\Delta( x_1\cdots x_n)=\sum \pm \Delta(x_ix_j ) \cdot x_1\cdots \hat{x}_i \cdots \hat{x}_j \cdots x_n
\end{split}
\end{equation}
This is a generalized form of the 7-term identity which is rather easy to prove. We refer the reader to \cite{West} for a more operadic approach on the gravity algebra structure.
\medskip
The Lie bracket on cyclic homology is known in the literature under the name \emph{string bracket}. For surfaces it was discovered by W. Goldman \cite{Gold} who studied the symplectic structure of the representation variety of fundamental groups of surfaces, or equivalently the moduli space of flat connections. His motivation lied in the dynamics of Teichm\"uller theory and Hamiltonian vector fields of Thurston earthquakes. It was then generalized by Chas-Sullivan using a purely topological construction to manifolds of all dimension. A geometrical description of the string bracket is given in \cite{AZ} (and
\cite{ATZ}) which generalizes Goldman's computation for surfaces using Chen iterated integrals to arbitrary even dimensions.
\bibliography{Bib-IRMA-Hoch}{}
\bibliographystyle{amsalpha}
\end{document} | 19,566 |
This past weekend Rachel my girl visited for the weekend and we had a great time spending time together. She got to spend some time with my roommates. We went to a couple parks and watched Wings of Desire, a German film about guardian angels. Quite good.
This past week(Monday through Friday), my roommate JT and I participated in a puzzle competition based at the University of Illinois campus.
It was quite fun but very tiring and I get burned out at the end after I finished solving a very hard cryptographic cipher.
We finished the last puzzle 4th, which got us some free pizza, but really that was just because the organizers had extra pizza they wouldn't be able to finish. Yes, we met the people who made the puzzles, and got to talk with them a bit. You see, the last puzzle required us to go to the computer science building and find them, in order to get the final clue from them.
But we spent a very long time on those puzzles all week, and now we must get back to work :-)
Sunday, October 09, 2005 | 145,439 |
Thesaurus
sacerdotal
Synonyms and Antonyms of sacerdotal
of, relating to, or characteristic of the clergy <sacerdotal garments such as a cassock and miter> Synonyms clerkly, ministerial, pastoral, priestly, clericalRelated sacerdotal
Dictionary: Definition of "sacerdotal" Spanish Central: Translation of "sacerdotal"
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What made you want to look up sacerdotal? Please tell us where you read or heard it (including the quote, if possible). | 1,057 |
\section{Proof Sketches}\label{appendix_proofs}
\subsection{Proof of Corollary~\ref{th:itksm_exact}} \label{proof_itksm_exact}
The proof is analogue to the one of Theorem~\ref{th:itksm}. We only need to take into account that without noise we have $C_r=1$ and that in all estimates the constant $B+1$ can be replaced by $B$, since for noise free signals $y_n=\dico x_{c_n, p_n,\sigma_n}$ we have $\|y_n\|_2 \leq B$. Further since the coefficients are strongly $S$-sparse, thresholding using the generating dictionary $\dico$ will always (almost surely) recover the generating support with a margin $u_s \geq (\Delta_S - 2\mu S) c_n(1)$, that is $\min_{k\in I_n} |\ip{\atom_k}{y_n}| \geq \max_{k\notin I_n} |\ip{\atom_k}{y_n}| + u_s$, compare \cite{sc14b}. Therefore the event that thresholding using $\pdico$ fails or that the empirical signs differ from the generating ones is contained in
\begin{align}\label{eventFsdef}
\mathcal F^s_n&:= \Big\{y_n: \exists k \mbox{ s.t. } \omega_k\Big|\sum_j \sigma_n(j) c_n\big(p_n(j)\big) \ip{\atom_j}{ z_k}\Big| \geq \frac{u_s -\frac{\eps^2c_n(S)}{2}}{2} \Big\}
\end{align}
and we get
\begin{align}
\left\| \bar \patom_k - \frac{\gamma_{1,S}}{K}\atom_k\right\|_2 &\leq \frac{2 \sqrt{B}}{N} \sharp \{ n: y_n \in \mathcal F^s_n\} + \left\|\frac{1}{N} \sum_n y_n \, \sigma_n(k) \, \chi(I_n,k) - \frac{\gamma_{1,S}}{K}\atom_k\right\|_2,
\end{align}
which can be estimated as before.
\subsection{Proof of Theorem~\ref{th:itkrm}} \label{proof_itkrm}
As already mentioned we use the same two step procedure and ideas as in the proof of Theorem~\eqref{th:itksm}. \\
\noindent \emph{Step 1:} We first check how often thresholding with $\pdico$ fails. Assuming thresholding recovers the generating support we show that the difference of the residuals using
$\dico$ or $\pdico$ concentrates around its expectation, which is small. Finally we show that the sum of residuals using $\dico$ converges to a scaled version of $\atom_k$.
To make the ideas precise we define the thresholding residual based on $\pdico$
\begin{align} \label{defRt}
R^t(\pdico, y_n, k) := \big[y_n - P(\pdico_{I_{\pdico,n}^t}) y_n + P(\patom_k) y_n\big] \cdot \signop(\ip{\patom_k}{y_n}) \cdot \chi(I_{\pdico,n}^t, k)
\end{align}
and the oracle residual based on the generating support $I_n=p_n^{-1}(\Sset)$, the generating signs $\sigma_n$ and $\pdico$.
\begin{align}\label{defRo}
R^o(\pdico, y_n, k) := \big[y_n - P(\pdico_{I_n}) y_n + P(\patom_k) y_n\big] \cdot \sigma_n(k) \cdot \chi(I_n,k).
\end{align}
We can now write,
\begin{align}
\bar \patom_k &= \frac{1}{N} \sum_n \left[R^t(\pdico, y_n, k) - R^o(\pdico, y_n, k)\right] + \frac{1}{N} \sum_n \left[R^o(\pdico, y_n, k) - R^o(\dico, y_n, k)\right]+\frac{1}{N} \sum_n R^o(\dico, y_n, k) \notag \\
&= \frac{1}{N} \sum_n \left[R^t(\pdico, y_n, k) - R^o(\pdico, y_n, k)\right] + \frac{1}{N} \sum_n \left[R^o(\pdico, y_n, k) - R^o(\dico, y_n, k)\right] \notag\\
&\hspace{2cm}+\frac{1}{N} \sum_n \big[y_n - P(\dico_{I_n}) y_n\big] \cdot \sigma_n(k) \cdot \chi(I_n,k) + \left( \frac{1}{N} \sum_n \ip{y_n}{\atom_k} \cdot \sigma_n(k) \cdot \chi(I_n,k)\right) \atom_k .
\end{align}
Abbreviating $s_k=\frac{1}{N} \sum_n \ip{y_n}{\atom_k} \cdot \sigma_n(k) \cdot \chi(I_n,k)$ we get
\begin{align}
\| \bar \patom_k - s_k \atom_k\|_2 \leq \frac{1}{N} \Big\| \sum_n &\left[R^t(\pdico, y_n, k) - R^o(\pdico, y_n, k)\right] \Big\|_2 \notag\\
&\hspace{2cm}+ \frac{1}{N} \Big\| \sum_n \left[R^o(\pdico, y_n, k) - R^o(\dico, y_n, k)\right] \Big\|_2\notag\\
&\hspace{5cm}+\frac{1}{N} \Big\| \sum_n \big[y_n - P(\dico_{I_n}) y_n\big] \cdot \sigma_n(k) \cdot \chi(I_n,k) \Big\|_2.\label{itkrmsplit}
\end{align}
We first estimate the norm of the first sum using the fact that the operator $\I_d- P(\pdico_{I_n}) + P(\patom_k)$ is an orthogonal projection and that $\|y_n\|_2\leq\sqrt{B+1}$,
\begin{align}
\frac{1}{N} \Big\| \sum_n &\left[R^t(\pdico, y_n, k) - R^o(\pdico, y_n, k)\right] \Big\|_2 \leq \frac{2\sqrt{B+1}}{N} \cdot \sharp \{n : R^t(\pdico, y_n, k) \neq R^o(\pdico,y_n, k)\}.
\end{align}
Next note that on the draw of $y_n$ the event that the thresholding residual using $\pdico$ is different from the oracle residual using $\pdico$, $\{y_n : R^t(\pdico, y_n, k) \neq R^o(\pdico,y_n, k) \}$ for any $k$ is again contained in the events $\mathcal E_n \cup \mathcal F_n$ as defined in \eqref{eventEdef}/\eqref{eventFdef},
\begin{align}
\{y_n : R^t(\pdico, y_n, k) \neq R^o(\pdico,y_n, k) \} \subseteq \{y_n : I_{\pdico,n}^t \neq I_n\} \cup \{y_n : \signop(\pdico_{I_n}^\star y_n) \neq \sigma_n(I_n)\} \subseteq \mathcal E_n \cup \mathcal F_n.
\end{align}
Substituting the corresponding bounds into \eqref{itkrmsplit} we get,
\begin{align}
\| \bar \patom_k - s_k \atom_k\|_2 &\leq \frac{2\sqrt{B+1}}{N} \cdot \sharp \{ n: y_n \in \mathcal E_n\} +\frac{2\sqrt{B+1}}{N}\cdot \sharp \{ n: y_n \in \mathcal F_n\}\notag\\
&+ \frac{1}{N} \Big\| \sum_n \left[R^o(\pdico, y_n, k) - R^o(\dico, y_n, k)\right] \Big\|_2 +\frac{1}{N} \Big\| \sum_n \big[y_n - P(\dico_{I_n}) y_n\big] \cdot \sigma_n(k) \cdot \chi(I_n,k) \Big\|_2 .
\end{align}
For the first two terms on the right hand side we use the same estimates as in the proof of Theorem~\ref{th:itksm}. To estimate the remaining two terms on the right hand side as well as $s_k$ we use the corresponding lemmata in the appendix. From Lemma \ref{lemma3b} we know that
\begin{align}
\P\left( \left| \frac{1}{N}\sum_n \chi(I_n,k) \sigma_n(k) \ip{y_n}{\atom_k}\right| \leq (1-t_0) \frac{C_r \gamma_{1,S}}{K} \right)\leq\exp\left(- \frac{ N t_0^2C_r^2 \gamma_{1,S}^2}{2K(1+ \frac{SB}{K}+S\nsigma^2 + t_0 C_r \gamma_{1,S}\sqrt{B+1}/3)}\right).
\end{align}
From Lemma~\ref{lemma4} we get that if $ S\leq \min \{\frac{K}{98B}, \frac{1}{98\nsigma^2}\}$, $\eps \leq \frac{1}{32\sqrt{S}}$ and $\eps_\delta \leq \frac{1}{24(B+1)}$ then
\begin{align}
&\P\left(\frac{1}{N} \left\| \sum_n \left[R^o(\pdico, y_n, k)-R^o(\dico, y_n, k) \right]\right\|_2 \geq \frac{ C_r \gamma_{1,S}}{K}(0.381\eps + t_3)\right)\notag\\
&\hspace{4cm}\leq \exp\left(- \frac{ t_3 C^2_r \gamma^2_{1,S} N}{40K\max\{S,B+1\}} \min\left\{\frac{t_3}{\eps^2 + \eps_\delta \left(1-\gamma_{2,S}+ d \nsigma^2\right)/160}, \frac{5}{3}\right\} +\frac{1}{4}\right).
\end{align}
Finally from Lemma~\ref{lemma3a} we know that for $0\leq t_4 \leq 1-\gamma_{2,S} + d\nsigma^2$, we have
\begin{align}
&\P\left( \left\| \frac{1}{N}\sum_n \big[y_n - P(\dico_{I_n})y_n \big]\cdot \sigma_n(k) \cdot \chi(I_n,k) \right\|_2 \geq \frac{C_r\gamma_{1,S}}{ K}\, t_4 \right)\notag\\
&\hspace{5cm}\leq \exp\left(- \frac{t_4^2C^2_r\gamma_{1,S}^2 N}{8 K \max\{S,B+1\}\left(1-\gamma_{2,S} + d\nsigma^2\right)}+\frac{1}{4}\right).
\end{align}
Thus with high probability we have
\begin{align}
\left\| \bar \patom_k - s_k \atom_k\right\|_2 &\leq \frac{C_r\gamma_{1,S}}{K} \left(\eps_{\mu,\nsigma} + t_1 + \tau \eps + t_2 + 0.381 \eps + t_3 + t_4\right) \quad \mbox{and} \quad s_k \geq (1-t_0) \frac{C_r \gamma_{1,S}}{K}.
\end{align}
To be more precise, if we choose a target precision $\epstarget \geq 8\eps_{\mu,\nsigma}$ and set $t_1=\epstarget/24$, $t_2=t_3= \max\{\epstarget, \eps\}/24$, $\tau=1/24$, $t_4=\epstarget/8$ and $t_0=1/50$ we get
\begin{align}
\max_k \left\| \bar \patom_k - \frac{C_r\gamma_{1,S}}{K}\atom_k\right\|_2 &\leq 0.8\cdot \frac{C_r\gamma_{1,S}}{K} \max\{\epstarget,\eps\} \quad \mbox{and} \quad \min_k s_k \geq 0.98 \cdot \frac{C_r \gamma_{1,S}}{K}.
\end{align}
except with probability
\begin{align}
\exp\left( \frac{-C_r\gamma_{1,S} N\epstarget}{336\, K\sqrt{B+1} }\right) + \exp\left( \frac{-C_r\gamma_{1,S} N \max\{\epstarget, \eps\} }{144\,K\sqrt{B+1} }\right)+K\exp\left(\frac{-C^2_r\gamma_{1,S}^2N}{K(5103 +34\,C_r\gamma_{1,S} \sqrt{B+1})}\right) \notag\\
+ 2K\exp\left(\frac{-C^2_r\gamma_{1,S}^2N\epstarget^2}{512K \max\{S,B+1\} \left(1-\gamma_{2,S} + d\nsigma^2\right)}\right) +
2K\exp\left(\frac{-C^2_r\gamma_{1,S}^2N\max\{\epstarget, \eps\}^2}{576K\max\{S,B+1\}\left(\eps +1 - \gamma_{2,S} + d\nsigma^2\right)}\right). \notag
\end{align}
Note that in case the target precision $\epstarget$ is larger than $\eps_\delta$, as happens for instance as soon as $\beta_S\leq \frac{1}{7\sqrt{S}}$ and therefore $\eps_{\mu, \nsigma}\geq \eps_\delta $, the last term in the sum above reduces to
\begin{align}
2K\exp\left(\frac{-C^2_r\gamma_{1,S}^2N\max\{\epstarget, \eps\}}{576 K \max\{S,B+1\}\left(2-\gamma_{2,S} + d\nsigma^2\right)}\right).
\end{align}
Lemma~\ref{lemma_rescale} then again implies that
\begin{align}
d(\bar \pdico, \dico)=\max_k \left\| \frac{\bar \patom_k}{\|\bar \patom_k\|_2} - \atom_k\right\|_2 \leq 0.92 \max\{\epstarget,\eps\}.
\end{align}
\noindent \emph{Step 2:} The second step is analogue to the one in the proof of Theorem~\ref{th:itksm}.
\subsection{Proof of Theorem~\ref{th:itkrm_exact}}\label{proof_itkrm_exact}
We follow the proof of Theorem~\ref{th:itkrm} but take into account that
in case of exactly $S$-sparse, noiseless signals the bound \eqref{itkrmsplit} reduces to
\begin{align}
\| \bar \patom_k - s_k \atom_k\|_2 &\leq \frac{2\sqrt{B}}{N}\cdot \sharp \{ n: y_n \in \mathcal F^s_n\}+ \frac{1}{N} \Big\| \sum_n \left[R^o(\pdico, y_n, k) - R^o(\dico, y_n, k)\right] \Big\|_2.
\end{align}
Since the relative gap $\Delta_S > 2 \mu S$ we get $\delta_S \leq \mu S \leq \frac{1}{2}$ and by Lemma~\ref{lemma1b}
\begin{align}
&\P\left(\sharp \{ n: y_n \in \mathcal F^s_n\} \geq \frac{\gamma_{1,S}N}{ 2K\sqrt{B}}\cdot(\tau \eps + t_2) \right)\leq \exp\left( \frac{-t_2^2\gamma_{1,S} N}{2K\sqrt{B}\,(2\tau \eps + t_2) }\right),
\end{align}
whenever
\begin{align}
\eps \leq \frac{\Delta_S - 2\mu S}{\sqrt{12}\left(\frac{1}{4} +\sqrt{\log\left(\frac{23K^2 \sqrt{B}}{(\Delta_S - 2\mu S) \gamma_{1,S}\tau}\right)}\right)}.
\end{align}
Further by Lemma~\ref{lemma4}
\begin{align}
&\P\left(\frac{1}{N} \left\| \sum_n \left[R^o(\pdico, y_n, k)-R^o(\dico, y_n, k) \right]\right\|_2 \geq \frac{ \gamma_{1,S}}{K}(1 \eps + t_3)\right) \leq \exp\left(- \frac{ t_3 \gamma^2_{1,S} N}{32\eps K\max\{S,B\}} \min\left\{\frac{t_3}{\eps}, 1\right\} +\frac{1}{4}\right),\notag
\end{align}
and again by \ref{lemma3b}
\begin{align}
\P\left( \left| \frac{1}{N}\sum_n \chi(I_n,k) \sigma_n(k) \ip{y_n}{\atom_k}\right| \leq (1-t_0) \frac{C_r \gamma_{1,S}}{K} \right)\leq\exp\left(- \frac{ N t_0^2 \gamma_{1,S}^2}{2K(1+ \mu^2(S-1) + t_0 \gamma_{1,S}\sqrt{B}/3)}\right).
\end{align}
Thus with high probability we have
\begin{align}
\left\| \bar \patom_k - s_k \atom_k\right\|_2 &\leq \frac{\gamma_{1,S}}{K} \left(\tau \eps + t_2 + 0.611 \eps + t_3 \right) \quad \mbox{and} \quad s_k \geq (1-t_0) \frac{\gamma_{1,S}}{K}.
\end{align}
The final result follows as before from setting $t_0=1/50$, $\tau = 1/24$, $t_2=\max\{\tilde \eps,\eps\}/24$ and $t_3=2t_2$. | 210,296 |
Something.
So imagine my delight when upon my arrival in LA, my dad offered to be my date to the CineConcerts: The Godfather Live!
For those who haven’t hear about these yet, CineConcerts are amazing live events created by composer/conductor Justin Freer, where you go watch a classic movie and the score is played by a live orchestra.
With most of today’s movies having soundtracks rather than scores and those that are scored are shown with a prerecorded score, the experience of seeing a movie with live music is essentially all but lost. Justin Freer’s goal with these concerts is to not only preserve live music as a part of the cinematic experience (you know, like back before the talkies when films were accompanies by a live organ), but to highlight the art of scoring a film.
In the age of the soundtrack, the musical score has fallen out of vogue in favor of pop songs that are more recognizable to audiences and therefore more marketable. This really is a shame when you think of all the original scores that really completed, rounded out, and “made” the movies they accompanied: Jaws, Gone With the Wind, The Last of the Mohicans.
Aside from just being a novel experience, The Godfather Live simply made me more aware of the score and what a masterpiece it is in and of itself, let alone when combined with the greatest movie ever.
The score is just a perfectly crafted as the film and together they are a lesson in “less is more” and the art of reservation.
If you are either a film buff, a music buff, or simply a human who loves cool stuff, see if you can get yourself to a CineConcert. It’s such a unique experience and you’ll be sure to see, or rather hear, your favorite movies in a whole new light. | 308,335 |
Everywhere's a mousse mousse...
What I'm referring to is the Maybelline Dream Mousse line! I don't tend to buy makeup from the same line but looking through my makeup drawer, I found three of these Maybelline Dream Mousse stuff.
I have the Dream Mousse Blush in Rose Petal, Bronzer in Sun Glow, and Concealer in Beige.
The first thing that I love love love about these products is the texture!! DUH right! I love the mousse feel - so soft on your skin and I think it just blends so easily - especially the blush. As you guys can see, the blush and bronzer have been well loved - definitely a dip.
- I first got the blush last summer and I have really enjoyed it. I don't like that it has shimmer in it nor do I like that the staying power is not so great but I love using it as a light tint on my cheeks. It is such a natural glow. People might not think that I'm wearing blush - the only thing that would give it away is the shimmer! I also love using this as a base for my blush. Really fun to blend out because, again, the texture makes it feel like AIR!!
- Second, I swapped the bronzer in April (a bit before summer started) and I really enjoy it. It gave me like a golden glow so it worked so great as a highlighter! I think you girls that are trying to achieve that bronzy glow - beachy look. This is such a great highlighter for you. Again, best staying power is when a powder is used to set but not necessary. This is very shimmery and sheer though so it is not something you apply all over your cheek or your face.
- Third, I just wanted a concealer to match my skin tone and I decided to pick this up. I wanted a creamy texture for my face, unlike my other Maybelline Mineral Power Concealer which is more of a liquid and it is too light for my face. Let me tell you, this baby has been used to death this past few weeks. I don't know what happened but I have been breaking out like crazy and I had so many pimples and nastiness to conceal. I think it has a decent staying power when set with a powder and I use Maybelline's Dream Matte powder.
I have not tried the foundation or the new Maybelline Smooth Mousse - let me know if you have and if you like it or not!
Hopefully this helps you guys decide what to get at the drugstore. The only other line that I tend to reach for a lot is the Revlon Colorstay line.
So what about you guys??? Is there a line that you guys really reach for at the drugstore... or anywhere really???
--------------
Also, I just wanted to let you guys know about another giveaway!!! PoorCollegeStudent has a beauty blog called Maquilagem Masstige and I really enjoy reading her blog. She has fantastic skin care and haul posts!! Please check out her giveaway - the prizes are very exciting.
--------------
Besides all that madness - how is everyone? I really wanted to update but got really busy this entire weekend!!! It is insane... I had time on Friday I guess but man, I was just a busy bee...
Also, school is starting for me. I'm doing a full week of Pharmacy school orientation right now and Monday will be the first day of school. I am apologizing in advanced right now that I probably won't update as often as I did in July. I tend to not update that often (not as often as I want) anyways, but still, SORRY!!!
K, have a great day.
i must admit, i sang those first few lines to the tune of old mcdonald... hahaha! so cute! i like mousse-y texture too! its like a dream to blend in right? my sister uses the blusher, it really nice! :)
Yes, I love the Maybelline Dream products, but I'm with you on the lack of staying power...but I still love them! I am in LOVE with the Dream Matte Mousse...it's my HG as of this summer. I want to try the new Dream Smooth Mousse now though, of course!
I have the Dream Matte Mousse foundation, and it is amazing! The texture is awesome and it has good coverage. I highly recommend, and I hoping to put a review up on my blog in a few days!
I still haven't tried this line! *blushes* Nice collection, girlie. =) Good luck in school!
I have tried the blush before but never the concealer or the bronzer. thanks for the review!
My friend gave me the mousse blush, but I didn't think it was my color so I gave it away.& I like ur post title, very catchy haha ^.^
Hi, hope it's OK to contact you here. We would love to include your blog on our giveaway search engine: Giveaway Scout (). Have a look and if interested, use our online form to add your blog ( ). thanks, Josh | 398,035 |
Donald Glover Is Spider-Man At Last (In Disney XD’s Animated Spider-Man Series)
"My name is Miles Morales, and I'm Spider-Man." With those words, Donald Glover takes his place among the ranks of official on-screen Spider-Men.
It's been known for a while now that Miles Morales, the Ultimate Universe version of Spider-Man, would make his screen debut in an upcoming episode of the Disney XD series Ultimate Spider-Man: Web Warriors, but only now do we know that the part will be voiced by actor and Spidey fan Donald Glover. It's a brilliant casting decision that we choose to interpret as the first step towards more Miles in other media, rather than an end in its own right. | 99,808 |
\begin{document}
\title{ \bf THE TOPOLOGICAL STRUCTURE\\
OF SCALING LIMITS\\
OF LARGE PLANAR MAPS}
\author{
Jean-Fran\c cois {\sc Le Gall}\footnote{DMA-ENS, 45 rue d'Ulm, 75005 Paris, France --- e-mail:
[email protected]\ , fax: (33) 1 44 32 20 80}{}\\ {\small Ecole normale sup\'erieure de Paris}}
\date{\small\today}
\maketitle
\begin{abstract}
We discuss scaling limits of large bipartite planar maps.
If $p\geq 2$ is a fixed integer, we consider, for every integer $n\geq 2$, a random planar
map $M_n$ which is uniformly distributed over the set of all rooted $2p$-angulations
with $n$ faces. Then, at least along a suitable subsequence, the
metric space consisting of the set of vertices of $M_n$, equipped with the graph distance rescaled by the
factor
$n^{-1/4}$, converges in distribution as $n\to\infty$ towards a limiting random compact
metric space, in the sense of the Gromov-Hausdorff distance. We prove that the topology of the limiting
space is uniquely determined independently of $p$ and of the subsequence, and that
this space can be obtained as the quotient of the
Continuum Random Tree for an equivalence relation which is defined from Brownian labels attached to
the vertices. We also verify that the Hausdorff dimension of the
limit is almost surely equal to $4$.
\end{abstract}
\section{Introduction}
The main purpose of the present work is to investigate
continuous limits of rescaled planar maps. We concentrate
on bipartite planar maps, which are known to be in
one-to-one correspondence with certain labeled trees
called mobiles (Bouttier, Di Francesco, Guitter \cite{BDG}).
In view of the
correspondence between maps and mobiles, it seems plausible that scaling limits of large
bipartite planar maps
can be described in terms of continuous random trees. This
idea already appeared in the pioneering work of Chassaing
and Schaeffer \cite{CS}, and was then developed by
Marckert and Mokkadem \cite{MaMo}, who
defined and studied the so-called Brownian map. It was argued in \cite{MaMo} that
the Brownian map is in some weak sense the limit of rescaled uniformly
distributed random quadrangulations of the plane (see also Marckert and Miermont
\cite{MaMi} for recent work along the same lines). The point of view
of the present paper is however different from the one in \cite{MaMo}
or in \cite{MaMi}. For every given planar map $M$, we equip the
set ${\bf m}$ of its vertices with the graph distance, and our aim is to study
the resulting compact metric space when the number of faces
of the map tends to infinity. Assuming that the
map $M$ is chosen uniformly over the set of all rooted $2p$-angulations with
$n$ faces, we discuss the convergence in distribution when $n$ tends to infinity of the
associated random metric spaces, rescaled with the factor $n^{-1/4}$,
in the sense of the Gromov-Hausdorff distance between compact metric spaces
(see e.g. Chapter 7 of \cite{BBI}, and subsection 2.3 below, for the definition of the Gromov-Hausdorff
distance). This is in contrast with \cite{MaMo}, which does not consider the limiting
behavior of distances between two points other than the root vertex.
Before we describe our main results in a more precise way, we need to
set some definitions. Recall that a planar map is a proper embedding, without edge
crossings, of a connected graph in the two-dimensional sphere. Loops and multiple edges
are a priori allowed. The faces of the map are the connected components of the complement
of the union of edges. A planar map is rooted
if it has a distinguished oriented edge called the root edge,
whose origin is called the root vertex. The set of vertices will always be equipped
with the graph distance: If $a$ and $a'$ are two vertices, $d_{gr}(a,a')$ is the
minimal number of edges on a path from $a$ to $a'$. Two rooted planar maps
are said to be equivalent if the second one is the image of the first one
under an orientation-preserving
homeomorphism of the sphere, which also preserves the root edges.
From now on we deal only with equivalence classes of rooted planar maps.
Given an integer $p\geq 2$, a
$2p$-angulation is a planar map where each face has degree $2p$, that is $2p$ adjacent
edges (one should count edge sides, so that if an edge lies entirely inside a face
it is counted twice). We denote by $\m^p_n$ the set of all
rooted $2p$-angulations with $n$ faces.
Let us now discuss the continuous trees that will arise in scaling limits
of planar maps.
We write $\t_\eg$ for the continuum random tree or CRT,
which was introduced and studied by Aldous \cite{Al1}, \cite{Al3}. The CRT
can be viewed as a random variable taking
values in the space of all rooted compact real trees (see e.g. \cite{survey}, or subsection 2.3
below). It turns out that the CRT
is the limit in distribution of several (suitably rescaled) classes of discrete trees when the
number of edges tends to infinity. For instance, it is relatively easy to show that if $\tau_n$
is distributed uniformly over the set of all plane trees with $n$ edges, then the vertex set
of $\tau_n$, viewed as a metric space for the graph distance rescaled by the factor $(2n)^{-1/2}$, will
converge in distribution to the CRT as $n\to\infty$, in the sense of the Gromov-Hausdorff distance. Our
notation
$\t_\eg$ reflects the fact that the CRT can be defined as the real tree coded by a normalized Brownian
excursion $\eg=(\eg_t)_{0\leq t\leq 1}$. This coding, which plays a major role in the
present work, is recalled in subsection 2.3 below. In addition to the usual genealogical order
of the tree, the CRT $\t_\eg$ inherits a lexicographical order from the coding, in a way analogous
to the ordering of (discrete) plane trees from the left to the right. We write $d_\eg$
for the distance on the tree $\t_\eg$ and $\rho$ for the root of $\t_\eg$.
We can assign Brownian labels to the vertices of the CRT. This means that given $\t_\eg$,
we consider a centered Gaussian process $(Z_a)_{a\in\t_\eg}$, such that $Z_\rho=0$ and the variance of
$Z_a-Z_b$ is equal to $d_\eg(a,b)$ for every
$a,b\in\t_\eg$. The pair $(\t_\eg,(Z_a)_{a\in\t_\eg})$ is the probabilistic
object that allows us to describe the continuous limit of random planar maps. We use
the Brownian labels to define a mapping $D^\circ$ from $\t_\eg\times \t_\eg$
into $\R_+$, via the formula
$$D^\circ(a,b)=Z_a+Z_b - 2\inf_{c\in[a,b]} Z_c$$
where $[a,b]$ denotes the ``lexicographical'' interval between $a$ and $b$. The preceding
definition is a little informal, since there are two lexicographical intervals between $a$
and $b$, corresponding to the two possible ways of going from $a$ to $b$ around the tree.
It should be understood that we take the lexicographical interval that minimizes
the value of $D^\circ(a,b)$ as defined above (see Section 3 below for a more
rigorous presentation). The intuition behind the definition of $D^\circ$ comes from the
discrete picture where each (bipartite) planar map is coded by a labeled tree, in such a way
that vertices of the map other than the root are in one-to-one correspondence with
vertices of the tree (\cite{BDG}, see subsection 2.1 below). From the properties of this coding, and more
precisely from the way edges of the map are reconstructed from the labels in the tree, one sees that any
two vertices
$a$ and $b$
that satisfy a discrete version of the relation $D^\circ(a,b)=0$ will be connected by
an edge of the map. See subsection 2.1 for more details.
The function $D^\circ$ does not satisfy the triangle inequality, but we may set
$$D^*(a,b)=\inf\left\{\sum_{i=1}^q D^\circ(a_{i-1},a_i)\right\}$$
where the infimum is over all choices of the integer $q\geq 1$
and of the finite sequence $a_0,a_1,\ldots,a_q$
in $\t_{{\bf e}}$ such that $a_0=a$ and $a_q=b$. We then define an equivalence relation
on $\t_{\eg}$ by setting $a\approx b$ if and only if $D^*(a,b)=0$. Although this is not
obvious, it turns out that the latter condition is equivalent to $D^\circ(a,b)=0$, outside a
set of probability zero.
Moreover one can check that equivalence classes for $\approx$ contain $1$, $2$ or at
most $3$ points, almost surely.
The quotient space $\t_\eg\,/\!\approx$ equipped with the metric $D^*$ is compact.
Let us now come to our main results. For every integer $n\geq 2$, let
$M_n$ be a random rooted $2p$-angulation uniformly distributed over $\m^p_n$. Denote
by ${\bf m}_n$ the set of vertices of $M_n$ and by $d_n$ the
graph distance on ${\bf m}_n$. We view $({\bf m}_n,d_n)$ as a random
variable taking values in the space of isometry classes of compact metric spaces.
Recall that the latter space equipped with the Gromov-Hausdorff distance is a Polish space,
as a simple consequence of Gromov's compactness theorem (\cite{BBI}, Theorem 7.4.15).
It can be checked that the sequence of the laws of
$({\bf m}_n,n^{-1/4}d_n)$ is tight, and so, at least along a subsequence, we
may assume that $({\bf m}_n,n^{-1/4}d_n)$ converges in distribution towards
a certain random compact metric space. The Skorokhod representation
theorem even allows us to get
an almost sure convergence, at the cost of replacing each map $M_n$
by another random map with same distribution. The principal contribution of the present
work is to identify the limiting compact metric space up to homeomorphism.
Precisely, our main result (Theorem \ref{main}) can be stated as follows.
From any
sequence of integers converging to $+\infty$, we can extract a subsequence and for every $n$
belonging to this subsequence
we can construct a random $2p$-angulation $M_n$ that is uniformly distributed over $\m^p_n$, in such a way
that we have the almost sure
convergence
\be
\label{mainintro}
\left({\bf m}_n,\Big(\frac{9}{4p(p-1)}\Big)^{1/4}\,n^{-1/4} d_n\right)
\build{\la}_{n\to\infty}^{} (\t_{\eg}\,/\!\approx,D)
\ee
in the sense of the Gromov-Hausdorff distance. Here $D$ is a (random) metric on the quotient space
$\t_{\eg}\,/\!\approx$, such that $D(a,b)\leq D^*(a,b)$ for every $a,b$. The random metric $D$ may
a priori depend on the choice of the subsequence and on the value of $p$. However,
since $\t_\eg\,/\!\approx$ equipped with the metric $D^*$ is compact and $D\leq D^*$, a standard argument
shows that the metric spaces $(\t_{\eg}\,/\!\approx,D)$ and $(\t_{\eg}\,/\!\approx,D^*)$
are homeomorphic, so that the topological structure of the limit in (\ref{mainintro})
is uniquely determined. In the companion paper \cite{LGP}, we prove that
$(\t_{\eg}\,/\!\approx,D)$, or equivalently $(\t_{\eg}\,/\!\approx,D^*)$, is a.s.
homeomorphic to the sphere $S^2$. We conjecture that $D=D^*$, and then the
convergence (\ref{mainintro}) would not require the use of a subsequence, and the limit would
not depend on $p$ (the constant $(9/(4p(p-1))^{1/4}$ in (\ref{mainintro})
is relevant mainly because we expect the limit to be independent of $p$). Although we are not able to
prove this, we can derive enough information about the limiting metric space in (\ref{mainintro}) to prove
that its Hausdorff dimension is equal to $4$ almost surely (Theorem \ref{Hausdim}).
Let us briefly comment on the proof of our main result.
The compactness argument that we use to get the existence of a limit in (\ref{mainintro})
along a suitable subsequence also shows that
this limit can be written as a quotient of the CRT $\t_\eg$ corresponding to a certain random
pseudo-metric $D$. The point is then to check that $a\approx b$ holds if and
only if $D(a,b)=0$. In other words, the points of the CRT that we
need to identify in order to get the limit in (\ref{mainintro})
are given by the equivalence relation
$\approx$, which is defined in terms of $D^*$ or of $D^\circ$. Once we know
that $D\leq D^*$, it is obvious that
$a\approx b$ implies $D(a,b)=0$. The hard core of the proof is thus to check the reverse
implication. The above-mentioned interpretation of the condition $D^\circ(a,b)=0$
in the discrete setting makes
it clear that any two points satisfying this condition must be identified. However,
other pairs of points could conceivably have been identified.
Roughly speaking, the proof that this is not the
case proceeds as follows. Given $a$ and $b$ in $\t_\eg$,
we can construct corresponding vertices $a_n$ and $b_n$ in $M_n$ such that
the sequence $(a_n)$ converges to $a$ and the sequence $(b_n)$
converges to $b$, in some suitable sense. The condition $D(a,b)=0$
entails that $d_n(a_n,b_n)=o(n^{1/4})$ as $n\to \infty$. We can then use this estimate
together with some combinatorial considerations and certain delicate properties
of the ``Brownian tree'' $(\t_\eg,(Z_a)_{a\in\t_\eg})$, in order to conclude
that we must have $D^\circ(a,b)=0$.
Let us discuss previous work related to the subject of the present article.
Planar maps were first studied by Tutte \cite{Tu} in connection with his work
on the four colors theorem. Because of their relations with Feynman diagrams,
planar maps soon attracted the attention of specialists of theoretical physics. The
pioneering papers \cite{tH} and \cite{BIPZ} related enumeration problems for planar
maps with asymptotics of matrix integrals. The interest for random planar
maps in theoretical physics grew significantly when these combinatorial
objects were interpreted as models of random surfaces, especially in
the setting of the theory of quantum gravity (see in particular \cite{Da}
and the book \cite{ADJ}). On the other hand, the idea of coding
planar maps with simpler combinatorial objects such as labeled trees
appeared in Cori and Vauquelin \cite{CV} and was much developed in Schaeffer's thesis
\cite{Sc}. In the present work, we use a version of the bijections between maps and trees
that was obtained in the recent paper of Bouttier, Di Francesco and Guitter \cite{BDG}.
See Bouttier's thesis \cite{Bo} and the references therein for applications of these bijections to
the statistical physics of random surfaces.
Other applications in the spirit of
the present work can be found in the recent papers \cite{CS}, \cite{MaMi} and \cite{MaMo}
that were mentioned earlier. Note in particular that the random metric space
$(\t_\eg\,/\!\approx,D^*)$ that is discussed above is essentially equivalent
to the Brownian map of \cite{MaMo}, although the presentation there is
different. See also \cite{An}, \cite{AS}, \cite{CD} and \cite{Kr} for various results
about random infinite planar triangulations and quadrangulations and their asymptotic
properties.
The paper is organized as follows. Section 2 gives a number of preliminaries
concerning bijections between maps and trees, the coding of real trees and
the construction of the Brownian tree $(\t_\eg,(Z_a)_{a\in\t_\eg})$. We also
state three important lemmas about the Brownian tree. Section 3 contains our main results.
The presentation is
slightly different (although equivalent) from the one that is given above,
because we prefer to argue with the tree $\t_\eg$ re-rooted at the vertex
with the minimal label, and the labels $Z_a$ shifted accordingly so that the label
of the root is still zero. Indeed, it is the genealogical structure of this
re-rooted tree that plays a major role in our approach. Section 4 is devoted to the main step
of our arguments, that is the proof that $D(a,b)=0$ implies $D^\circ(a,b)=0$.
Section 5 gives the proof of three technical lemmas that were stated in
Section 2. The proofs of these lemmas depend on some rather intricate properties
of Brownian trees, which we found convenient to derive using the path-valued
process called the Brownian snake \cite{Zu}. In order to make most of the paper
accessible to the reader who is unfamiliar with the Brownian snake, we have preferred
to postpone these proofs to Section 5. At last, Section 6 contains the calculation
of the Hausdorff dimension of the limiting metric space.
As a final remark, it is very plausible that our results
can be extended to the more general setting of Boltzmann distributions
on bipartite maps, which is considered in \cite{MaMi}
and in \cite{We}. We have chosen to concentrate on the particular
case of uniform $2p$-angulations for the sake of simplicity and
to keep the present work to a reasonable size.
\medskip
\noindent{\bf Acknowledgments}. I am indebted to Gr\'egory Miermont for a number of very
stimulating discussions. I also thank Fr\'ed\'eric Paulin for several useful
conversations and helpful comments, and Oded Schramm for his remarks on a preliminary version of this work.
\section{Preliminaries}
\subsection{Planar maps and the Bouttier-Di Francesco-Guitter bijection}
Recall that we have fixed an integer $p\geq 2$ and that $\m^p_n$ denotes the set of all
rooted $2p$-angulations with $n$ faces. We start this section with a precise
description of the Bouttier-Di Francesco-Guitter bijection between
$\m^p_n$ and the set of all $p$-mobiles with $n$ black vertices.
We use the standard formalism for plane trees as found in \cite{Neveu}
for instance. Let
$${\cal U}=\bigcup_{n=0}^\infty \N^n $$
where $\N=\{1,2,\ldots\}$ and by convention $\N^0=\{\varnothing\}$.
The generation of $u=(u_1,\ldots,u_n)\in\N^n$ is $|u|=n$. If
$u=(u_1,\ldots u_m)$ and
$v=(v_1,\ldots, v_n)$ belong to $\cal U$, $uv=(u_1,\ldots u_m,v_1,\ldots ,v_n)$
denotes the concatenation of $u$ and $v$. In particular $u\varnothing=\varnothing u=u$.
If $v$ is of the form $v=uj$ for $u\in{\cal U}$ and $j\in\N$,
we say that $u$ is the {\it father} of $v$, or that $v$ is a {\it child} of $u$.
More generally, if $v$ is of the form $v=uw$ for $u,w\in{\cal U}$, we say that
$u$ is an {\it ancestor} of $v$, or that $v$ is a {\it descendant} of $u$.
A plane tree $\tau$ is a finite subset of
$\cal U$ such that:
\begin{description}
\item{(i)} $\varnothing\in \tau$.
\item{(ii)} If $v\in \tau$ and $v\ne\varnothing$, the father of
$u$ belongs to $\tau$.
\item{(iii)} For every $u\in\tau$, there exists an integer $k_u(\tau)\geq 0$
such that $uj\in\tau$ if and only if $1\leq j\leq k_u(\tau)$.
\end{description}
A $p$-tree is a plane tree $\tau$ that satisfies the
following additional property:
\begin{description}
\item{(iv)} For every $u\in\tau$ such that $|u|$ is odd, $k_u(\tau)=p-1$.
\end{description}
If $\tau$
is a $p$-tree, vertices $u$ of $\tau$ such that $|u|$ is even are called
white vertices, and vertices of $u$ such that $|u|$ is odd are called
black vertices. We denote by $\tau^\circ$ the set of all white vertices
of $\tau$ and by $\tau^\bullet$ the set of all black vertices.
See the left side of Fig.1 for an example of a $3$-tree.
\begin{center}
\begin{picture}(450,140)
\linethickness{1.5pt}
\put(160,0){\circle{12}}
\put(146,-14){$\varnothing$}
\put(155,3){\thinlines\line(-2,1){56}}
\put(165,3){\thinlines\line(2,1){56}}
\put(220,30){\circle*{12}}
\put(228,17){$2$}
\put(245,55){\circle{12}}
\put(223,33){\line(1,1){18}}
\put(195,55){\circle{12}}
\put(180,42){$21$}
\put(217,33){\line(-1,1){18}}
\put(252,42){$22$}
\put(100,30){\circle*{12}}
\put(90,17){$1$}
\put(125,55){\circle{12}}
\put(132,42){$12$}
\put(103,33){\line(1,1){18}}
\put(75,55){\circle{12}}
\put(61,42){$11$}
\put(97,33){\line(-1,1){18}}
\put(75,85){\circle*{12}}
\put(55,72){$111$}
\put(75,61){\thinlines\line(0,1){20}}
\put(100,110){\circle{12}}
\put(78,88){\line(1,1){18}}
\put(50,110){\circle{12}}
\put(23,97){$1111$}
\put(72,88){\line(-1,1){18}}
\put(106,97){$1112$}
\thinlines \put(300,0){\vector(1,0){140}}
\put(300,0){\circle*{3}}
\thinlines \put(300,0){\vector(0,1){130}}
\put(314,56){\circle*{3}}
\thicklines \put(300,0){\line(1,4){14}}
\thicklines \put(314,56){\line(1,4){14}}
\thicklines \put(328,112){\line(1,0){14}}
\put(328,112){\circle*{3}}
\thicklines \put(342,112){\line(1,-4){14}}
\put(342,112){\circle*{3}}
\thicklines \put(356,56){\line(1,0){14}}
\put(356,56){\circle*{3}}
\thicklines \put(370,56){\line(1,-4){14}}
\put(370,56){\circle*{3}}
\thicklines \put(384,0){\line(1,4){14}}
\put(384,0){\circle*{3}}
\thicklines \put(398,56){\line(1,0){14}}
\put(398,56){\circle*{3}}
\thicklines \put(412,56){\line(1,-4){14}}
\put(412,56){\circle*{3}}
\put(426,0){\circle*{3}}
\put(440,-10){$i$}
\put(282,125){$C^{\tau^\circ}_i$}
\put(422,-10){$pn$}
\put(314,0){\line(0,1){4}}
\put(312,-10){$1$}
\put(293,52){$1$}
\put(300,55){\line(1,0){4}}
\end{picture}
\vspace{8mm}
Figure 1. A $3$-tree $\tau$ and the associated
contour function $C^{\tau^\circ}$ of $\tau^\circ$.
\end{center}
A (rooted) $p$-mobile is a pair $\theta=(\tau,(\ell_u)_{u\in\tau^\circ})$
that consists of a $p$-tree $\tau$ and a collection of integer labels
attached to the white vertices of
$\tau$, such that the following properties hold:
\begin{description}
\item{(a)} $\ell_\varnothing=1$ and $\ell_u\geq 1$ for each $u\in\tau^\circ$.
\item{(b)} Let $u\in \tau^\bullet$, let $u_{(0)}$ be the father of $u$ and let
$u_{(j)}=uj$ for every $1\leq j\leq p-1$. Then for every $j\in\{0,1,\ldots,p-1\}$,
$\ell_{u_{(j+1)}}\geq \ell_{u_{(j)}}-1$, where by convention $u_{(p)}=u_{(0)}$.
\end{description}
\begin{center}
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\vspace{4mm}
Figure 2. A $3$-mobile $\theta$ with $5$ black vertices
and the associated spatial contour function.
\end{center}
The left side of Fig.2 gives an example of a $p$-mobile with $p=3$. The numbers appearing
inside the circles representing white vertices are the labels assigned to
these vertices. Condition (b) above means that if one lists the white vertices
adjacent to a given black
vertex in clockwise order, the labels of these vertices can decrease by at most one
at each step.
We will now describe the Bouttier-Di Francesco-Guitter bijection between
$\m^p_n$ and the set of all $p$-mobiles with $n$ black vertices.
This bijection can be found in Section 2 of \cite{BDG} in the more general
setting of bipartite planar maps. Also \cite{BDG} deals with pointed
planar maps rather than with rooted planar maps. It is however easy to verify that
the results described below are simple consequences of \cite{BDG}.
Let $\tau$ be a $p$-tree with $n$ black vertices and let $k=\#\tau -1=pn$. The
search-depth sequence of $\tau$ is the sequence $u_0,u_1,\ldots,u_{2k}$ of vertices
of $\tau$ which is obtained by induction as follows. First $u_0=\varnothing$, and
then for every $i\in\{0,\ldots,2k-1\}$, $u_{i+1}$ is either the first child of
$u_i$ that has not yet appeared in the sequence $u_0,\ldots,u_i$, or the father
of $u_i$ if all children of $u_i$ already appear in the sequence $u_0,\ldots,u_i$.
It is easy to verify that $u_{2k}=\varnothing$ and that all vertices of $\tau$
appear in the sequence $u_0,u_1,\ldots,u_{2k}$ (of course some of them
appear more than once).
It is immediate to see that vertices $u_i$ are white when $i$ is even
and black when $i$ is odd.
The search-depth sequence of $\tau^\circ$ is by definition the sequence
$v_0,\ldots,v_k$ defined by $v_i=u_{2i}$ for every $i\in\{0,1,\ldots,k\}$.
Now let $(\tau,(\ell_u)_{u\in\tau^\circ})$ be a $p$-mobile
with $n$ black vertices. Denote by $v_0,v_1,\ldots,v_{pn}$ the search-depth
sequence of $\tau^\circ$. Suppose that the tree $\tau_n$ is drawn
in the plane as pictured on Fig.3 and add an extra vertex $\partial$.
We associate with $(\tau,(\ell_u)_{u\in\tau^\circ})$ a $2p$-angulation $M$
with $n$ faces,
whose set of vertices is
$$\tau^\circ \cup\{\partial\}$$
and whose edges are obtained by the following device: For every
$i\in\{0,1,\ldots,pn-1\}$,
\begin{description}
\item{$\bullet$} if $\ell_{v_i}=1$, draw an edge between $v_i$ and $\partial$\ ;
\item{$\bullet$} if $\ell_{v_i}\geq 2$, draw an edge between $v_i$ and the first
vertex in the sequence $v_{i+1},\ldots,v_{pn}$ whose label is
$\ell_{v_i}-1$ (this vertex will be called a {\it successor} of $v_i$ -- note that a given
vertex $v$ can appear several times in the search-depth sequence and so
may have several different successors).
\end{description}
Notice that $\ell_{v_{pn}}=\ell_\varnothing=1$ and that condition (b)
in the definition of a $p$-tree
entails that
$\ell_{v_{i+1}}\geq \ell_{v_i}-1$ for every $i\in\{0,1,\ldots,pn-1\}$.
This ensures that whenever $\ell_{v_i}\geq 2$ there is at least one
vertex among $v_{i+1},v_{i+2},\ldots,v_{pn}$ with label $\ell_{v_i}-1$.
The construction can be made in such a way
that edges do not intersect: See
Section 2 of \cite{BDG}. The resulting planar
graph $M$ is a $2p$-angulation, which is rooted at the oriented edge
between $\partial$ and $v_0=\varnothing$, corresponding
to $i=0$ in the previous construction. Each black
vertex of
$\tau$ is associated with a face of the map $M$. Furthermore the graph
distance in $M$ between the root vertex $\partial$ and another vertex
$u\in\tau^\circ$ is equal to $\ell_u$. See Fig.3 for the $6$-angulation
associated with the $3$-mobile of Fig.2.
It follows from \cite{BDG} that the preceding construction yields
a bijection between the set $\T^p_n$ of all $p$-mobiles with $n$
black vertices and the set $\m^p_n$.
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\vspace{6mm}
Figure 3. The Bouttier-Di Francesco-Guitter bijection: A rooted $3$-mobile with $5$ black vertices
and the associated
$6$-angulation with $5$ faces
\end{center}
\subsection{Genealogical structure of maps}
Let $\theta=(\tau,(\ell_u)_{u\in\tau^\circ})$ be a $p$-mobile
with $n$ black vertices. The set
$\tau^\circ$ of white vertices can also be viewed as a graph, by declaring that
there is an edge between $u$ and $v$ if and only if $u$ is the grandfather
of $v$ (that is, there exist $j$ and $k\in\N$ such that $v=ujk$) or conversely
$v$ is the grandfather of $u$. Obviously $\tau^\circ$ is a tree in
the graph-theoretic sense. If $u,v\in\tau^\circ$, we then denote by
$\llbracket u,v\rrbracket$ the set of points of $\tau^\circ$ that lie on
the unique shortest path from $u$ to $v$ in $\tau^\circ$. As
usual,
$\rrbracket u,v\llbracket=
\llbracket u,v\rrbracket\backslash\{u,v\}$. We also denote by $u\wedge v$
the ``most recent common ancestor'' of $u$ and $v$ in $\tau^\circ$, which may be
defined by $\llbracket \varnothing,u\wedge v\rrbracket=
\llbracket \varnothing,u\rrbracket\cap \llbracket \varnothing, v\rrbracket$.
Notice that $u\wedge v$ is not necessarily the most recent ancestor of
$u$ and $v$ in the tree $\tau$.
We denote
by $\prec$ the genealogical relation on $\tau^\circ$: $u\prec v$ if and only
if $u$ is an ancestor of $v$ (in the tree $\tau$). We use $u\leq v$ for the
lexicographical order on $\tau^\circ$. As usual $u<v$ if and
only if $u\leq v$ and $u\ne v$. It will also be
convenient to introduce a ``reverse'' lexicographical order denoted by $\ll$. This
is the total order on $\tau^\circ$ defined as follows. If neither of the
relations
$u\prec v$ and $v\prec u$ holds, then $u\ll v$ if and only if $u\leq v$. On
the other hand, if $u\prec v$, then $v\ll u$ (although $u \leq v$).
Let $v_0,v_1,\ldots,v_{pn}$ be the search-depth sequence of $\tau^\circ$, as defined in
the preceding subsection. If $x,y\in\tau^\circ$, the condition $x\leq y$
implies that the first occurence of $x$ in the sequence $v_0,\ldots,v_{pn}$
occurs before the first occurence of $y$, and conversely the condition $x\ll y$
implies that the last occurence of $x$ occurs before the last occurence of $y$.
The {\it contour function} of $\tau^\circ$ is the discrete sequence
$C^{\tau^\circ}_0,C^{\tau^\circ}_1,\ldots,C^{\tau^\circ}_{pn}$ defined by
$$C^{\tau^\circ}_i=\frac{1}{2}\,|v_i|\ ,\hbox{ for every }0\leq i\leq pn.$$
See Fig.1 for an example with $p=n=3$.
It is easy to verify that the contour function determines $\tau^\circ$, which
in turn determines the $p$-tree $\tau$ uniquely. We will also
use the {\it spatial contour function} of $\theta=(\tau,(\ell_u)_{u\in\tau^\circ})$, which
is the discrete sequence $(V^\theta_0,V^\theta_1,\ldots,V^\theta_{pn})$ defined by
$$V^\theta_i=\ell_{v_i}\ ,\hbox{ for every }0\leq i\leq pn.$$
From property (b) of the labels and the definition of the search-depth sequence, it
is clear that $V^\theta_{i+1}\geq V^\theta_i-1$ for every $0\leq i\leq pn-1$
(cf Fig.2). This fact will
be used many times below.
The pair $(C^{\tau^\circ},V^\theta)$ determines $\theta$ uniquely. For our
purposes it will sometimes be convenient to view
$C^{\tau^\circ}$ or $V^\theta$ as functions of the continuous parameter
$t\in[0,pn]$, simply by interpolating linearly on the intervals
$[i-1,i]$, $1\leq i\leq n$ (as it is suggested by Figs 1 and 2).
Let $[pn]$ stand for the set $\{0,1,\ldots,pn\}$. Define an equivalence
relation $\sim$ on $[pn]$ by setting $i\sim j$
if and only if $v_i=v_j$. The quotient space $[pn]/\sim$
is then obviously identified with $\tau^\circ$. This
identification plays an important role throughout this work.
If $i\leq j$, the
relation $i\sim j$ implies
$$\inf_{i\leq k\leq j} C^{\tau^\circ}_k=C^{\tau^\circ}_i=C^{\tau^\circ}_j.$$
The converse is not true (except if $p=2$) but
the conditions $j>i+1$, $C^{\tau^\circ}_i=C^{\tau^\circ}_j$ and
$$C^{\tau^\circ}_k>C^{\tau^\circ}_i\ ,\hbox{ for every }
k\in]i,j[\cap \Z$$
imply that $i\sim j$. Similarly, if $i<j$, the condition $v_i\prec v_j$ implies
$$\inf_{i\leq k\leq j} C^{\tau^\circ}_k=C^{\tau^\circ}_i.$$
The converse is not true, but the condition
$$\inf_{i< k\leq j} C^{\tau^\circ}_k>C^{\tau^\circ}_i$$
forces $v_i\prec v_j$.
Let $u,v\in\tau^\circ$ with $u\prec v$, and let $w\in\rrbracket u,v\llbracket$.
The set
$$\tau^\circ_{(v,w)}:=\{x\in\tau^\circ: x\wedge v=w\hbox{ and }x\leq v\}$$
is called the subtree from the left side of $\llbracket u,v\rrbracket$
with root $w$. Similarly, the set
$$\widetilde\tau^\circ_{(v,w)}
:=\{x\in\tau^\circ: x\wedge v=w\hbox{ and }v\ll x\}$$
is called the subtree from the right side of $\llbracket u,v\rrbracket$
with root $w$. Let $j\in[pn]$ be such that $v_j=v$, and
set
\ba
&&k=\inf\{i\in\{0,1,\ldots,j\}:v_i=w\},\\
&&k'=\sup\{i\in\{0,1,\ldots,j\}:v_i=w\}.
\ea
Then $\tau^\circ_{(v,w)}$ exactly consists of the vertices $v_i$
for $k\leq i\leq k'$: We will say that
$[k,k']\cap\Z$ is the interval coding $\tau^\circ_{(v,w)}$. Similar remarks apply to
$\widetilde\tau^\circ_{(v,w)}$.
Recall from the preceding subsection that the $p$-mobile
$(\tau,(\ell_u)_{u\in\tau^\circ})$ corresponds to
a $2p$-angulation $M$ via the Bouttier-Di Francesco-Guitter bijection.
Through this correspondence, vertices of $M$ (with the exception of the root vertex $\partial$)
are identified with elements of $\tau^\circ$. From now on, we systematically do
this identification.
Let $d_M$ stand for the graph distance on the set of vertices of $M$. A geodesic path in $M$ is
a discrete path $\gamma=(\gamma(i),0\leq i\leq k)$ in $M$ such that
$d_M(\gamma(i),\gamma(j))=|i-j|$ for every $i,j\in\{0,\ldots,k\}$.
The following lemma plays an important role in our proofs.
\begin{lemma}
\label{combi}
Let $\gamma=(\gamma(i),0\leq i\leq k)$ be a geodesic path in $M$ which does not
visit the root vertex $\partial$. Let
$u=\gamma(0)$ be the starting point of the path $\gamma$ and let $y=\gamma(k)$
be its final point.
Let $\tau_1$ be a subtree from the left
side of $\llbracket \varnothing,y\rrbracket$ (respectively
from the right
side of $\llbracket \varnothing,y\rrbracket$) with root $w\in\rrbracket
\varnothing,y\llbracket$. Let $v=\gamma(1)$ be the point following $u$ on
the path $\gamma$, and assume that:
\begin{description}
\item{\rm (i)} $v\in\tau_1$ and $v\ne w$.
\item{\rm (ii)} $u\leq w$ (resp. $w\ll u$).
\item{\rm (iii)} For every $i\in\{1,\ldots,k\}$, one has $w\leq \gamma(i)$
(resp.
$\gamma(i)\ll w$).
\end{description}
Then, for any point $x$ of $\tau_1\backslash\{w\}$ such that
\be
\label{combitech1}
\ell_z>\sup_{0\leq i\leq k} \ell_{\gamma(i)},\hbox{ for every }z\in\llbracket w,x
\rrbracket
\ee
one has
$$d_M(x,y)\leq d_M(u,y)+\ell_x-\inf_{0\leq i\leq k} \ell_{\gamma(i)}.$$
\end{lemma}
\proof We only treat the case when $\tau_1$ is a subtree from the left
side of $\llbracket \varnothing,y\rrbracket$. We fix a point
$x\in\tau_1\backslash\{w\}$ such that (\ref{combitech1}) holds. Denote by $b$ the
first point on the geodesic $\gamma$ such that $x\leq b$. This makes sense
because $x\leq y$ by the definition of subtrees. Also $b\ne u$ because $u\leq w$
and $w<x$. So we can also introduce the point $a$ preceding $b$
on the geodesic $\gamma$.
Let us first assume that $b\ne v$, or equivalently $a\ne u$. Then
$w\leq a$ by (iii), and $a\leq x$, which forces $a\in \tau_1$.
On the other hand, assumption (\ref{combitech1}) guarantees that
$a\notin\llbracket w,x
\rrbracket$. Since $a\leq x\leq b$, it follows that $a$ cannot be an
ancestor of $b$. Any occurence of $a$ in the search-depth sequence of
$\tau^\circ$ thus happens before the first occurence of $b$ in this sequence.
Now notice that $a$ and $b$ are connected by an edge of the
map $M$, and recall the construction of these edges at the end
of the preceding subsection. It follows that $\ell_z\geq \ell_a$ for every vertex
$z$ such that $a\ll z <b$, whereas $\ell_b=\ell_a-1$.
Note that $a\ll x<b$ (the case $a\prec x$ is excluded since $a\in\tau_1$
and $a\notin\llbracket w,x
\rrbracket$, and $x=b$ is impossible by (\ref{combitech1})), so that the previous
sentence applies to $z=x$. Set $q=\ell_x-\ell_b\geq 1$. We let $i_0$ be the first
index
such that
$v_{i_0}=x$, and observe that $x\leq v_i$ for every $i\geq i_0$. We then define $i_1,\ldots,i_q$ by
setting
$$i_j=\inf\{i\geq i_0:\ell_{v_i}=\ell_x-j\}\hbox{ for every }1\leq j\leq q.$$
By the preceding considerations, we have $\ell_z\geq \ell_a=\ell_x-q+1$
for every $z$ such that $x\leq z<b$. It follows that $v_{i_q}=b$. On the other hand, $v_{i_0}=x$
and $d_M(v_{j},v_{j+1})=1$ for every $0\leq j\leq q-1$, by the construction of
edges in
$M$. We thus get
$$d_M(x,b)\leq q=\ell_x-\ell_b.$$
Finally,
$$d_M(x,y)\leq d_M(b,y)+d_M(x,b)\leq d_M(u,y)+\ell_x-\ell_b,$$
which gives the desired bound.
In the case when $a=u$ and $b=v$, the argument is almost the same. Note that
$u<w$ ($u=w$ is excluded by (\ref{combitech1})) and $x<b=v$ as previously.
The existence of an edge between $u$ and $v$ warrants that $\ell_v=\ell_u-1$
and that $\ell_z\geq \ell_u$ for every vertex $z$ of $\tau_1$
such that $z<v$. In the same way as before, we get $d_M(x,v)\leq \ell_x-\ell_v$
which leads to the desired bound. \cq
\subsection{Real trees}
We will now discuss the continuous trees that are scaling limits of
our discrete plane trees. We start with a basic definition.
\begin{definition}
A metric space $(\t,d)$ is a real tree if the following two
properties hold for every $a,b\in \t$.
\begin{description}
\item{\rm(i)} There is a unique
isometric map
$f_{a,b}$ from $[0,d(a,b)]$ into $\t$ such
that $f_{a,b}(0)=a$ and $f_{a,b}(
d(a,b))=b$.
\item{\rm(ii)} If $q$ is a continuous injective map from $[0,1]$ into
$\t$, such that $q(0)=a$ and $q(1)=b$, we have
$$q([0,1])=f_{a,b}([0,d(a,b)]).$$
\end{description}
\noindent A rooted real tree is a real tree $(\t,d)$
with a distinguished vertex $\rho=\rho(\t)$ called the root.
\end{definition}
In what follows, real trees will always be rooted and compact, even if this
is not mentioned explicitly.
Let us consider a rooted real tree $(\t,d)$.
The range of the mapping $f_{a,b}$ in (i) is denoted by
$\llbracket a,b\rrbracket$ (this is the line segment between $a$
and $b$ in the tree), and we also use the obvious notation
$\rrbracket a,b\llbracket$.
In particular, for every $a\in \t$, $\llbracket \rho,a\rrbracket$ is the path
going from the root to $a$, which we will interpret as the ancestral
line of vertex $a$. More precisely we can define a partial order on the
tree, called the genealogical order, by setting $a\prec b$
if and only if $a\in\llbracket \rho,b
\rrbracket$.
If $a,b\in\t$, there is a unique $c\in\t$ such that
$\llbracket \rho,a
\rrbracket\cap \llbracket \rho,b
\rrbracket=\llbracket \rho,c
\rrbracket$. We write $c=a\wedge b$ and call $c$ the most recent
common ancestor to $a$ and $b$. The multiplicity of a vertex $a\in\t$ is the number of
connected components of $\t\backslash\{a\}$. In particular, $a$ is
called a leaf if it has multiplicity one.
In a way similar to the discrete case, real trees can be coded by
``contour functions''. If $E$ and $F$ are two topological spaces, we write $C(E,F)$ for the space
of all continuous functions from $E$ into $F$.
Let $\sigma>0$ and let
$g\in C([0,\sigma],[0,\infty[)$ be such that $g(0)=g(\sigma)=0$. To
avoid trivialities, we will also assume that $g$ is not identically zero.
For every $s,t\in[0,\sigma]$, we set
$$m_g(s,t)=\inf_{r\in[s\wedge t,s\vee t]}g(r),$$
and
$$d_g(s,t)=g(s)+g(t)-2m_g(s,t).$$
It is easy to verify that $d_g$ is a pseudo-metric
on $[0,\sigma]$. As usual, we introduce the equivalence
relation
$s\simeq_g t$ if and only if $d_g(s,t)=0$ (or equivalently if and only if $g(s)=g(t)=m_g(s,t)$).
The function $d_g$ induces a distance on the quotient space $\t_g
:=[0,\sigma]\,/\!\simeq_g$, and we keep the
notation $d_g$ for this distance. We denote by
$p_g:[0,\sigma]\longrightarrow
\t_g$ the canonical projection. Clearly $p_g$ is continuous (when
$[0,\sigma]$ is equipped with the Euclidean metric and $\t_g$ with the
metric $d_g$), and therefore $\t_g=p_g([0,\sigma])$ is a compact metric space.
By Theorem 2.1 of \cite{DuLG},
the metric space $(\t_g,d_g)$ is a real tree.
We will always view $(\t_g,d_g)$ as a rooted real tree with root $\rho=p_g(0)=p_g(\sigma)$.
Then, if $s,t\in[0,\sigma]$, the
property $p_g(s)\prec p_g(t)$ holds if and only if $g(s)=m_g(s,t)$.
Let us recall the definition of the Gromov-Hausdorff distance. Let
$(E_1,d_1)$ and $(E_2,d_2)$ be two compact metric spaces.
The Gromov-Hausdorff distance between $(E_1,d_1)$
and $(E_2,d_2)$ is
$$d_{GH}(E_1,E_2)=\inf\Big(d_{Haus}(\varphi_1(E_1),\varphi_2(E_2))\Big),$$
where the infimum is over all isometric embeddings $\varphi_1:E_1\la E$ and
$\varphi_2:E_2\la E$ of $E_1$ and $E_2$ into the same metric
space $(E,d)$, and $d_{Haus}$ stands for the usual Hausdorff distance
between compact subsets of $E$.
Then Lemma 2.3 of \cite{DuLG} shows that $\t_g$ depends continuously on $g$,
in the sense that
$$d_{GH}(\t_g,\t_g')\leq 2 \|g-g'\|$$
where $\|g-g'\|$ is the supremum norm of $g-g'$.
In addition to the genealogical order $\prec$, the tree
$\t_g$ inherits a lexicographical order from the coding
through the function $g$. Precisely if $a,b\in\t_g$ we
write $a\leq b$ if and only if $s\leq t$, where
$s$, respectively $t$, is the smallest representative
of $a$, resp. of $b$, in $[0,\sigma]$.
We can also introduce a ``reverse'' lexicographical order $\ll$, by replacing smallest by greatest
in the previous sentence. If neither
of the relations $a\prec b$ or $b\prec a$ holds, we have $a\ll b$
if and only if $a\leq b$. On the other hand, if $a\prec b$, we have $b\ll a$.
Let $a,b\in \t_g$. If $a\leq b$, or if $a\ll b$, we define the lexicographical interval
$[a,b]$ as the image
under the projection
$p_g$ of the minimal interval $[s,t]$ such that $s\leq t$, $p_g(s)=a$ and $p_g(t)=b$. If
neither of the relations $a\leq b$ or $a\ll b$ holds, then there is no such interval
and we take $[a,b]=\varnothing$.
If $[a,b]$ is nonempty,
then $\llbracket a, b\rrbracket\subset
[a,b]$. Furthermore if $a\prec b$, then both $[a,b]$ and
$[b,a]$ are nonempty, and $[a,b]\cap [b,a]=\llbracket a,b\rrbracket$.
Let $a,b\in \t_g$ with $a\prec b$, and let $c\in\rrbracket a,b\llbracket$.
Suppose that the set
$$\t^1=\{u\in\t_g: u\wedge b =c\hbox{ and }u\leq b\}$$
is not the singleton $\{c\}$.
Then the set $\t^1$ is called a
subtree from the left side of $\llbracket a,b\rrbracket$ with root $c$
(it is straightforward to verify that $\t^1$ is itself a real tree).
Moreover, if $s=\inf p_g^{-1}(a)$ and $t=\inf p_g^{-1}(b)$, there is
a unique subinterval $[\alpha,\beta]$ of $]s,t[$ such that
$\t^1=p_g([\alpha,\beta])$, $p_g(\alpha)=p_g(\beta)=c$ and
$$\alpha=\sup\{r\in[s,t]: g(r)<g(\alpha)\}\ ,\
\beta=\sup\{r\in[s,t]: g(r)\leq g(\alpha)\}.$$
We say that $[\alpha,\beta]$ is the coding interval of $\t^1$. In a
similar way we can define subtrees from the right side of $\llbracket
a,b\rrbracket$: $\t^2$ is such a subtree if there exists
$c'\in\rrbracket a,b\llbracket$ such that
$$\t^2=\{u\in\t_g: u\wedge b =c'\hbox{ and }b\ll u\}$$
and $\t^2\ne\{c'\}$.
\subsection{Brownian trees and conditioned Brownian trees}
We first explain how we can assign Brownian labels to
the vertices of the real tree $(\t_g,d_g)$ defined in
the previous subsection. To this end, we consider the centered real-valued
Gaussian process $(\Gamma_t)_{t\in[0,\sigma]}$ with covariance function
\begin{equation}
\label{covariance}
{\rm cov}(\Gamma_s,\Gamma_t)=m_g(s,t)
\end{equation}
for every $s,t\in[0,\sigma]$ (it is a simple exercise
to check that $m_g(s,t)$ is a covariance function). Note that $\Gamma_0=\Gamma_\sigma=0$ and that the form
of the covariance gives
$E[(\Gamma_s-\Gamma_t)^2]=d_g(s,t)$. Suppose that $g$ is H\"older
continuous with some exponent $\delta>0$, which will always
hold in what follows. Then an application of the
classical Kolmogorov lemma shows that the process
$(\Gamma_t)_{t\in[0,\sigma]}$ has a continuous modification, and from
now on we consider only this modification. We write ${\bf Q}_g$
for the distribution of $(\Gamma_t)_{t\in[0,\sigma]}$, which is
a probability measure on the space $C([0,\sigma],\R)$.
From the formula $E[(\Gamma_s-\Gamma_t)^2]=d_g(s,t)$ and
a continuity argument, we immediately get that a.s. for every
$s,t\in[0,\sigma]$ such that $s\simeq_g t$, we have $\Gamma_s=\Gamma_t$.
Therefore we may also view $\Gamma$ as a Gaussian process
indexed by the tree $\t_g$. Indeed, it is natural to interpret
$(\Gamma_a,a\in \t_g)$ as Brownian motion indexed by
$\t_g$ and started from $0$ at the root of $\t_g$.
Note that formula (\ref{covariance}) may be rewritten in the form
$${\rm cov}(\Gamma_a,\Gamma_b)=d_g(\rho,a\wedge b)$$
for every $a,b\in\t_g$.
We now randomize the coding function $g$.
Let $\eg=(\eg_t)_{t\in[0,1]}$ be the normalized Brownian excursion, and take
$g=\eg$ and $\sigma=1$
in the previous discussion.
The random real tree $(\t_\eg,d_\eg)$ coded by $\eg$ is the so-called CRT, or
Continuum Random Tree. Using the fact that local minima of Brownian motion
are distinct, one easily checks that points of $\t_\eg$ can have multiplicity at most $3$.
We then consider
the real-valued process $(Z_t)_{t\in[0,1]}$ such that conditionally given $\eg$,
$(Z_t)_{t\in[0,1]}$ has distribution ${\bf Q}_\eg$. As explained
above, we can also view $(Z_t)_{t\in[0,1]}$ as parametrized
by the tree $\t_\eg$, and then interpret $(Z_a)_{a\in\t_\eg}$
as Brownian motion indexed by $\t_\eg$. This interpretation creates some
technical difficulties since $\t_\eg$ is now a random index set -- to
circumvent these difficulties it is often more convenient to view
$Z$ as indexed by $[0,1]$, keeping in mind that $Z_t$ only
depends on the equivalence class of $t$ in $\t_\eg$.
In view of our applications it is important
to consider the pair $(\eg,Z)$
conditioned on the event
$$Z_t\geq 0\ \hbox{ for every }t\in[0,1].$$
Here some justification is needed for the conditioning,
since the latter event has probability zero. The paper \cite{LGW}
describes several limit procedures that allow one to make sense
of the previous conditioning. These procedures all lead to
the same limiting pair $(\ov\eg,\ov Z)$ which can be described as
follows from the original pair $(\eg,Z)$. Set
$$\un Z=\inf_{t\in[0,1]} Z_t$$
and let $s_*$ be the (almost surely) unique time in $[0,1]$ such
that $Z_{s_*}=\un Z$. The fact that $\un Z$ is attained at a unique
time (\cite{LGW} Proposition 2.5) entails that the vertex $p_\eg(s_*)$ is a
leaf of the tree $\t_\eg$. For
every $s,t\in[0,1]$, set $s\oplus t=s+t$ if $s+t\leq 1$ and $s\oplus t=s+t-1$ if $s+t>1$. Then, for every
$t\in[0,1]$,
\begin{description}
\item{$\bullet$}
$\displaystyle{\ov\eg_t=\eg_{s_*}+\eg_{s_*\oplus t}-2\,m_{\eg}(s_*,
{s_*\oplus t})}$;
\item{$\bullet$} $\ov Z_t=Z_{s_*\oplus t} -Z_{s_*}$.
\end{description}
The formula for $\ov Z$ makes it obvious that $\ov Z_t\geq 0$
for every $t\geq 0$, in agreement with the above-mentioned conditioning. The function
$\ov\eg$ is continuous on $[0,1]$ and such that $\ov\eg(0)=\ov\eg(1)=0$.
Hence the tree $\t_{\ov\eg}$ is well defined, and this tree is
isometrically identified with the tree $\t_\eg$ re-rooted at the
(minimizing) vertex $p_\eg(s_*)$: See Lemma 2.2 in \cite{DuLG}.
Moreover we have $s\simeq_{\ov \eg} t$ if and only if $s_*\oplus s\simeq_{\eg}s_*\oplus t$
and so $\ov Z_t$ only depends on the equivalence class of $t$
in the tree $\t_{\ov\eg}$. Therefore we may and will sometimes
view $\ov Z$ as indexed by vertices of the tree $\t_{\ov\eg}$.
By a well-known property of the Brownian excursion, the law of pair $(\eg_t,Z_t)_{t\in[0,1]}$
is invariant under time reversal, meaning that $(\eg_t,Z_t)_{t\in[0,1]}$
has the same distribution as $(\eg_{1-t},Z_{1-t})_{t\in[0,1]}$. A similar
time-reversal invariance property then holds for the pair
$(\ov\eg_t,\ov Z_t)_{t\in[0,1]}$.
In what follows we use the notation $\rho$ for the root of $\t_\eg$
and $\ov \rho$ for the root of $\t_{\ov\eg}$.
We now state three important lemmas which are key ingredients
of the proofs of our main results.
\begin{lemma}
\label{increasepoint}
We say that $s\in[0,1[$ is an increase point of the pair
$(\eg,Z)$, respectively of the pair $(\ov\eg,\ov Z)$, if there
exists $\varepsilon>0$ such that $\eg_t\geq \eg_s$ and
$Z_t\geq Z_s$, resp. $\ov\eg_t\geq \ov\eg_s$ and
$\ov Z_t\geq \ov Z_s$, for every $t\in[s,(s+\varepsilon)\wedge 1]$.
Then a.s. there is no increase point of
$(\eg,Z)$, and $s=0$ is the only increase point of $(\ov\eg,\ov Z)$.
\end{lemma}
Before stating
the next lemma we need to introduce some additional notation. The
uniform measure $\lambda$ on $\t_\eg$, resp. on $\t_{\ov\eg}$, is the
image of Lebesgue measure on $[0,1]$ under the canonical projection
$p_{\eg}$, resp. $p_{\ov\eg}$. There is no ambiguity in using the same
notation $\lambda$ for both cases, since it really corresponds to the same
measure when $\t_{\ov\eg}$ is identified to $\t_\eg$ up to re-rooting.
We also let $\cal I$ and $\ov{\cal I}$ be the random measures on $\R$
defined by
$$\langle {\cal I},f\rangle =\int_{\t_\eg} \lambda(da)\,f(Z_a)=\int_0^1 dt\,f(Z_t)
\ ,\ \langle \ov{\cal I},f\rangle =\int_{\t_{\ov\eg}} \lambda(da)\,f(\ov Z_a)=\int_0^1
dt\,f(\ov Z_t).$$
The random measure $\cal I$ is sometimes called (one-dimensional)
ISE. Notice that $\ov{\cal I}$ is supported on $[0,\infty[$ and is just the image of
$\cal I$ under the shift $x\la x-\un Z$.
\begin{lemma}
\label{estimateISE}
For every $\alpha>0$,
$$\lim_{\varepsilon\to 0} \varepsilon^{-2} P(\ov{\cal I}([0,\varepsilon])\geq
\alpha\varepsilon^2)=0.$$
\end{lemma}
Our last lemma is concerned with values of
$\ov Z$ over subtrees of $\t_{\ov\eg}$. Roughly speaking it asserts
that, for a given $\beta>0$ and a subtree $\t^1$ with root $c$, if both
$\ov Z_c>\beta$ and the minimum of the values
of $\ov Z$ over $\t^1$ is strictly less than $\beta$, then
the mass (for the uniform measure $\lambda$) of those vertices $x$ of
$\t_{\ov\eg}$ with label $\ov Z_x\in[\beta,\beta+\varepsilon]$,
and such that the label of any ancestor of $x$ in $\t^1$
is greater than $\beta$, will be of
order at least
$\varepsilon^2$. The precise statement is as follows.
\begin{lemma}
\label{occuptree}
Almost surely, for every $\mu>0$, for every $a\in \t_{\ov\eg}$
and every subtree $\t^1$ from $\llbracket \ov \rho,a\rrbracket$
with root $c\in \rrbracket \ov \rho,a\llbracket$,
the condition
$$\inf_{b\in \t^1} \ov Z_b< \ov Z_{c}-\mu$$
implies that
$$\liminf_{\varepsilon\to 0} \varepsilon^{-2}\,\lambda\Big(\Big\{ x\in\t^1: \ov Z_x\leq
\ov Z_c-\mu+\varepsilon
\hbox{ and }\ov Z_y\geq \ov Z_c-\mu+\frac{\varepsilon}{8}
\hbox{ for every }y\in\llbracket c,x\rrbracket\Big\}\Big)
>0.$$
\end{lemma}
Although Lemma \ref{occuptree} is stated in terms of the pair
$(\ov\eg,\ov Z)$, in view of our applications, the proof will show
that this lemma reduces to a similar statement for the pair $(\eg,Z)$.
The proof of the preceding three lemmas depends on some properties of
the path-valued process called the Brownian snake, and recalling these
properties at the present stage would take us too far from our
main concern. For this
reason, we prefer to postpone the proofs to Section 5.
\subsection{Invariance principles}
In this subsection, we recall the basic invariance principles that
relate the discrete labeled trees of subsection 2.1 to the Brownian trees
of subsection 2.4. Recall that the integer $p\geq 2$ is fixed.
Let $\theta_n=(\tau_n,(\ell^n_u)_{u\in\tau^\circ_n})$ be uniformly distributed
over the set $\T^p_n$ of all $p$-mobiles with $n$ black vertices.
We denote by $C^n=(C^n_t)_{0\leq t\leq pn}$ the contour function of
$\tau_n^\circ$ and by
$V^n=(V^n_t)_{0\leq t\leq pn}$ the spatial contour function of $\theta_n$
(it
is convenient to view $C^n$ and $V^n$ as continuous functions of
$t\in[0,pn]$, as explained in subsection 2.2).
Recall that the pair $(C^n,V^n)$ determines $\theta_n$.
\begin{theorem}
\label{invar1}
We have
\be
\label{basicinvar}
\left({1\over 2}\sqrt{\frac{p}{p-1}}\,n^{-1/2}\,C^n_{pnt},
\Big(\frac{9}{4p(p-1)}\Big)^{1/4}\,n^{-1/4} V^n_{pnt}\right)_{0\leq t\leq 1}
\build{\la}_{n\to\infty}^{\rm(d)} (\ov{\bf e}_t,\ov Z_t)_{0\leq t\leq 1}.
\ee
in the sense of weak convergence of the laws in the space of probability
measures on $C([0,1],\R^2)$.
\end{theorem}
The case $p=2$ of Theorem \ref{invar1} is a special case of Theorem 2.1
in \cite{LGinvar}, which is itself a
conditional version of invariance principles relating
discrete snakes to the Brownian snake \cite{JM}. See the discussion in Section 8 of \cite{LGinvar}.
Similar results were obtained before
by Chassaing and Schaeffer \cite{CS}. In the general case, Theorem \ref{invar1}
is a consequence of Theorem 3.3 in \cite{We}, and is also
closely related to Theorem 11 in \cite{MaMi}.
Although Theorem \ref{invar1} will be our main tool, we will also need
another asymptotic result, which does not easily follow from Theorem \ref{invar1}
but fortunately can be deduced from the results in \cite{MaMi}. Let
$M_n$ be the random element of ${\cal M}^p_n$ that corresponds to
$\theta_n$ via the Bouttier-Di Francesco-Guitter bijection. Obviously $M_n$
is uniformly distributed over ${\cal M}^p_n$.
Conditionally on $M_n$, let us choose
a vertex $Y_n$ of $M_n$ uniformly at random.
The pair $(M_n,Y_n)$ is then uniformly
distributed over the set of all rooted and pointed $2p$-angulations
with $n$ faces. Theorem 3 (iii) of \cite{MaMi} gives
precise information
about the profile of distances to the point $Y_n$ in the
map $M_n$ (to be precise, \cite{MaMi} imposes a special constraint on the orientation
of the root edge depending on the distinguished point in the map, but since every rooted and pointed map with
this constraint corresponds exactly to two unconstrained
rooted and pointed maps, the results of \cite{MaMi} immediately carry over to
our setting). In our special
situation, we can restate this result as follows. We write $d_{n}$ for the graph distance on the set ${\bf
m}_n$ of vertices of
$M_n$, and for every $R>0$ and $x\in {\bf m}_n$ we denote by $B_n(x,R)$
the closed ball with radius $R$ centered at $x$ in the metric space
$({\bf m}_n,d_n)$.
\begin{proposition}
\label{pointedmap}
For every $\alpha,\beta> 0$,
$$P\Big[\frac{1}{(p-1)n} \#B_n(Y_n,\alpha n^{1/4})
\geq \beta \Big]
\build{\la}_{n\to\infty}^{} P\Big[\ov{\cal I}\Big(
\Big[0,\Big(\frac{9}{4p(p-1)}\Big)^{1/4}\alpha\Big]\Big)\geq \beta\Big].
$$
\end{proposition}
Since $Y_n$ is uniformly distributed over ${\bf m}_n$ and $\#({\bf m}_n)=(p-1)n+2$, the convergence of
the proposition can be restated as follows. For every $\alpha,\beta>0$,
\begin{equation}
\label{pointed}
E\Big[\frac{1}{(p-1)n}\#\{y\in{\bf m}_n: \#B_n(y,\alpha n^{1/4})
\geq \beta\,(p-1)n\} \Big]
\build{\la}_{n\to\infty}^{} P\Big[\ov{\cal I}\Big(
\Big[0,\Big(\frac{9}{4p(p-1)}\Big)^{1/4}\alpha\Big]\Big)\geq \beta\Big].
\end{equation}
\section{Main results}
Recall the notation introduced in the previous section. In particular,
$M_n$ is a random rooted $2p$-angulation which is uniformly distributed over the set
$\m^p_n$, $\mm_n$ denotes the
set of vertices of $M_n$, and $\theta_n=(\tau_n,(\ell^n_u)_{u\in\tau^\circ_n})$ is the
random mobile corresponding to $M_n$ via the Bouttier-Di Francesco-Guitter bijection.
We constantly use the identification
$$\mm_n=\tau^\circ_n \cup\{\partial_n\}$$
where $\partial_n$ is the root vertex of $M_n$. The graph distance on $\mm_n$
is denoted by $d_n$. In particular, if $a,b\in\tau^\circ_n$, $d_n(a,b)$ denotes the
graph distance between $a$ and $b$ viewed as vertices in
the map $M_n$.
As in subsection 2.5, $C^n$ and $V^n$ are respectively the contour function
of the tree $\tau^\circ_n$ and the spatial
contour function of $\theta_n$.
Following subsection 2.2, the equivalence relation $\sim_n$ on $[pn]=\{0,1,\ldots,pn\}$
is defined by declaring that $i \sim_n j$ if and only if
the $i$-th vertex in the search-depth sequence of $\tau^\circ_n$
is the same as the $j$-th vertex in the same sequence. Recall that this implies
$$C^n_i=C^n_j=\inf_{i\wedge j\leq k\leq i\vee j} C^n_k .$$
The quotient set $[pn]\,/\!\sim_n$ is then canonically identified
with $\tau^\circ_n$ and thus with the set of vertices of
$M_n$ other than the root $\partial_n$. If $a\in \tau^\circ_n$ and
$i\in [pn]$, we will abuse notation by writing $a\sim_n i$
if $i$ is a representative of $a$ viewed as an element of
$[pn]\,/\!\sim_n$ (similar abuses of notation will occur
for other equivalence relations). With this notation,
if $a\sim_n i$, we have $d_n(\partial_n,a)=\ell^n_a=V^n_i$, by the properties
the Bouttier-Di Francesco-Guitter bijection. If $i,j\in [pn]$ and $a,b\in \tau^\circ_n$
are such that $a\sim_n i$ and $b\sim_n j$, we will also
write $d_n(i,j)=d_n(a,b)$.
For every $i,j\in[pn]$, we put
$$d_n^\circ(i,j)=
V^n_i+V^n_j -2\inf_{i\wedge i\leq k\leq i\vee j} V^n_k +2.$$
\begin{lemma}
\label{simple-bound}
For every $i,j\in [pn]$,
$$d_n(i,j)\leq d_n^\circ(i,j).
$$
\end{lemma}
\proof Fix $i\in [pn]$ and let $a\in \tau^\circ_n$ be such that
$a\sim_n i$. Let $q=V^n_i=d_n(\partial_n,a)$. We set $i_q=i$
and for every $k\in\{1,\ldots,q-1\}$,
$$i_k=\inf\{\ell \geq i:V^n_\ell = k\}.$$
From the construction of edges in the Bouttier-Di Francesco-Guitter bijection,
it is immediate to see that $d_n(i_k,i_{k-1})=1$
for every $2\leq k\leq q$.
We also fix $j\in[pn]$ and let $b\in \tau^\circ_n$ be such that
$b\sim_n j$, and we set $r=V^n_j=d_n(\partial_n,b)$. We define
similarly the sequence $j_r=j,j_{r-1},\ldots,j_1$. Then:
\begin{description}
\item{$\bullet$} Either $\inf_{i\wedge i\leq k\leq i\vee j} V^n_k=1$
and the bound of the lemma is just the triangle inequality
$d_n(i,j)=d_n(a,b)\leq d_n(\partial_n,a)+d_n(\partial_n,b)$.
\item{$\bullet$} Or $\inf_{i\wedge i\leq k\leq i\vee j} V^n_k=\ell\geq 2$,
and we have $i_{\ell-1}=j_{\ell-1}$. The bound of the lemma
follows by writing:
$$d_n(i,j)\leq d_n(i_{\ell-1},i_q)+d_n(j_{\ell-1},j_r)\leq q+r-2\ell +2.$$
\end{description}
\par\cq
We extend the definition of
$d_n(i,j)$ and $d_n^\circ(i,j)$ to noninteger values
of $i$ and $j$ by linear interpolation. If $s,t\in[0,pn]$,
we set
\begin{eqnarray*}
d_n(s,t)&=&(s-\lfloor s\rfloor)(t-\lfloor
t\rfloor)d_n(\lceil s\rceil,
\lceil t\rceil) +(s-\lfloor s\rfloor)(\lceil t\rceil -t)d_n(\lceil
s\rceil,
\lfloor t\rfloor)\cr
&+& (\lceil s\rceil-s)(t-\lfloor
t\rfloor)d_n(\lfloor s\rfloor,
\lceil t\rceil)
+(\lceil s \rceil -s)(\lceil t\rceil -t)d_n(\lfloor
s\rfloor,
\lfloor t\rfloor),
\end{eqnarray*}
with the notation $\lfloor s\rfloor=\sup\{k\in\Z:k\leq s\}$
and $\lceil s\rceil=\inf\{k\in\Z:k>s\}$. We
define $d_n^\circ(s,t)$ in a similar way. Obviously the bound
$d_n(s,t)\leq d_n^\circ(s,t)
$ remains valid for reals $s,t\in[0,pn]$. Furthermore, the triangle inequality
$d_n(s,u)\leq d_n(s,t)+d_n(t,u)$ also holds for every
$s,t,u\in [0,pn]$.
As a straightforward consequence of (\ref{basicinvar}) and the definition
of $d^\circ_n(s,t)$, we have
\be
\label{basic-dist}
\left(
\Big(\frac{9}{4p(p-1)}\Big)^{1/4}\,n^{-1/4}
d^\circ_n(pns,pnt)\right)_{0\leq s\leq 1, 0\leq t\leq 1}
\build{\la}_{n\to\infty}^{\rm(d)} (D^\circ(s,t))_{0\leq
s\leq 1, 0\leq t\leq 1}
\ee
where
$$D^\circ(s,t)
=\ov Z_s+ \ov Z_t -2 \inf_{s\wedge t\leq r \leq s\vee t} \ov Z_r$$
and the limit holds in the sense of weak convergence
in the space of probability measures on $C([0,1]^2,\R)$.
\begin{proposition}
\label{tightness}
The sequence of the laws of the processes
$$\left(n^{-1/4}\,d_n(pns,pnt)\right)_{0\leq s\leq 1,0\leq t\leq 1}$$
is tight in the space of probability measures on
$C([0,1]^2,\R)$. Let $\bf C$ be the space of isometry classes of
compact metric spaces, which is equipped with the Gromov-Hausdorff measure.
The sequence of the laws of the metric spaces $(\mm_n,n^{-1/4}d_n)$
is tight in the space of probability measures on $\bf C$.
\end{proposition}
\proof First observe that, for every $s,t,s',t'\in[0,1]$,
\begin{eqnarray}
\label{tight0}
|n^{-1/4}\,d_n(pns,pnt)-n^{-1/4}\,d_n(pns',pnt')|
&\leq&n^{-1/4}(d_n(pns,pns')+d_n(pnt,pnt'))\nonumber\\
&\leq&n^{-1/4}(d^\circ_n(pns,pns')+d^\circ_n(pnt,pnt')).
\end{eqnarray}
From the convergence (\ref{basic-dist}), we have for every
$\delta,\varepsilon>0$,
\begin{equation}
\label{tight1}
\limsup_{n\to \infty} P\left(\sup_{|s-s'|\leq \delta}
n^{-1/4}\,d^\circ_n(pns,pns') \geq \varepsilon\right)
\leq P\left(\sup_{|s-s'|\leq \delta} D^\circ(s,s')\geq
\Big(\frac{4p(p-1)}{9}\Big)^{1/4}\,\varepsilon\right).
\end{equation}
Let $\eta>0$ and for every $k\geq 1$ set $\varepsilon_k=2^{-k}$.
We apply (\ref{tight1}) with $\varepsilon=\varepsilon_k$ and note that
we can then choose $\delta_k>0$ sufficiently small so that the
right-hand side of (\ref{tight1}) is strictly less than $2^{-k}\eta$.
Therefore, there exists an integer $n_k$ such that,
for every $n\geq n_k$,
\be
\label{tight2}
P\left(\sup_{|s-s'|\leq \delta_k}
n^{-1/4}\,d^\circ_n(pns,pns') \geq \varepsilon_k\right)
\leq 2^{-k}\eta.
\ee
By choosing $\delta_k$ even smaller if necessary, we may assume that
(\ref{tight2}) holds for every $n\geq 1$. It follows that,
for every $n\geq 1$,
\be
\label{tight3}
P\left(\bigcap_{k\geq 1}\left\{\sup_{|s-s'|\leq \delta_k}
n^{-1/4}\,d^\circ_n(pns,pns') \leq \varepsilon_k\right\}\right)
\geq 1-\eta.
\ee
Let $K$ denote the set of all functions $\omega\in C([0,1]^2,\R)$
such that $\omega(0,0)=0$ and, for every $k\geq 1$,
$$\sup\{|\omega(s,t)-\omega(s',t')|:|s-s'|\leq \delta_k,|t-t'|\leq
\delta_k\}\leq 2\,\varepsilon_k.$$
Then $K$ is a compact subset of $C([0,1]^2,\R)$. By (\ref{tight0})
and (\ref{tight3}), the probability that the random function
$(s,t)\la n^{-1/4}d_n(pns,pnt)$ belongs to $K$ is bounded
below by $1-\eta$, for every $n\geq 1$. Since $\eta$
was arbitrary, this completes the proof of the first assertion.
The second assertion is an easy consequence of the first one and
the Gromov compactness
criterion (Theorem 7.4.15 in \cite{BBI}). We omit details, since this
result is not really needed in what follows.
\cq
\medskip
From (\ref{basicinvar}) and Proposition \ref{tightness}, there exists a
strictly increasing sequence
$(n_k)_{k\geq 1}$ such that along this sequence we have
the joint convergence in distribution
\begin{eqnarray}
\label{basic}
&&\hspace{-6mm}\left({1\over 2}\sqrt{\frac{p}{p-1}}\,n^{-1/2}\,C^n_{pnt},
\Big(\frac{9}{4p(p-1)}\Big)^{1/4}\,n^{-1/4} V^n_{pnt},
\Big(\frac{9}{4p(p-1)}\Big)^{1/4}\,n^{-1/4}\,d_n(pns,pnt)\!\right)
_{0\leq s\leq 1, 0\leq
t\leq 1}\nonumber\\
\noalign{\smallskip}
&&\qquad\build{\la}_{n\to\infty}^{} \left(\ov{\bf e}_t,\ov Z_t
,D(s,t)\right)_{0\leq s\leq 1,0\leq t\leq
1}.
\end{eqnarray}
Here the limiting triple $\left(\ov{\bf e}_t,\ov Z_t
,D(s,t)\right)$ is defined on a suitable probability space, the pair
$(\ov{\bf e},\ov Z)$ obviously has the same distribution as before, and
$D$ is a continuous process indexed by $[0,1]^2$ and taking values
in $\R_+$. In the remaining part of this work, we restrict our attention
to values of $n$ belonging to the sequence $(n_k)_{k\geq 1}$.
In particular, when we pass to the limit as $n\to \infty$, this
always means along the sequence $(n_k)_{k\geq 1}$.
Thanks to the Skorokhod representation theorem, we may and will assume
that the convergence (\ref{basic}) holds almost surely, in the
sense of uniform convergence over $[0,1]^2$.
Strictly speaking, we should replace for every $n\geq 1$ the
random mobile $\theta_n$ (respectively the random map $M_n$)
with another random mobile $\wt\theta_n$
(resp. another random map $\wt M_n$)
having the same distribution, but we do not keep
track of this replacement in the notation.
The next proposition records some properties of the
random function $D(s,t)$. We write $\simeq$ instead
of $\simeq_{\ov\eg}$
for the equivalence relation defining the tree
$\t_{\ov{\bf e}}\;$: $\t_{\ov{\bf e}}=[0,1]\,/\!\simeq$
as was explained in subsection 2.3.
\begin{proposition}
\label{prop-distance} The following properties hold almost surely.
\par\noindent{\rm(i)} For every $s,t,u\in[0,1]$,
\ba
&&D(s,s)=0\\
&&D(s,t)=D(t,s)
\ea
and
$$D(s,u)\leq D(s,t) + D(t,u).$$
{\rm (ii)} For every $s,t\in [0,1]$,
$$D(s,t)\leq D^\circ(s,t).$$
{\rm(iii)} For every $s,t\in [0,1]$, the property
$s \simeq t$ implies $D(s,t)=0$.
\par\noindent{\rm (iv)} For every $s\in[0,1]$, $D(0,s)=\ov Z_s$.
\end{proposition}
\proof Except for the first one, the properties in (i) are immediate from
the analogous properties for $d_n$ and the (almost sure) convergence
(\ref{basic}). Similarly, (ii) follows from Lemma \ref{simple-bound} and
the convergence (\ref{basic-dist}), which holds a.s. along the
sequence $(n_k)_{k\geq 1}$ if (\ref{basic}) also holds a.s.
along this sequence. The first
property in (i) then readily follows from (ii).
Let us prove (iii). Let $s,t\in[0,1]$ with $s<t$. If
$s\simeq t$, we have
$$\ov{\bf e}_s=\ov{\bf e}_t=\inf_{s\leq r\leq t} \ov{\bf e}_r.$$
Suppose first that $\ov{\bf e}_r> \ov{\bf e}_s$ for every
$r\in]s,t[$. From the uniform convergence of the function
${1\over 2}\sqrt{{p}/{(p-1)}}\,n^{-1/2}\,C^n_{pnt}$ towards
$\ov{\bf e}_t$, an elementary argument yields the existence of
two sequences $(i_n)$ and $(j_n)$ of integers in $[pn]$ such that:
\begin{description}
\item{$\bullet$} $\displaystyle\frac{i_n}{pn}\la s$ and $
\displaystyle\frac{j_n}{pn}\la
t$ as $n\to \infty$.
\item{$\bullet$} For $n$ sufficiently large, $j_n\geq i_n+ 2$
and $C^n_{i_n}=C^n_{j_n}<{\displaystyle\inf_{i_n<k<j_n} C^n_k}$.
\end{description}
As we already noticed in subsection 2.2, the last property ensures that $i_n\sim_n j_n$ and thus
$d_n(i_n,j_n)=0$. By passing to the limit $n\to\infty$, we get
$D(s,t)=0$.
If $\ov{\bf e}_r= \ov{\bf e}_s$ for some
$r\in]s,t[$, then $r$ is necessarily unique, because otherwise
the tree $\t_{\ov{\bf e}}$, which is isometric to $\t_\eg$, would have a point with multiplicity
strictly greater than $3$. By the preceding argument,
$D(s,r)=D(r,t)=0$ and thus $D(s,t)=0$ by the triangle inequality
in (i).
Let us finally prove (iv). Let $s\in[0,1]$ and let $(i_n)$ be
a sequence of integers such that $i_n/(pn) \la s$ as $n\to\infty$.
From the properties of the
Bouttier-Di Francesco-Guitter bijection, we know that
$d_n(0,i_n)=V^n_{i_n}$. On the other hand,
(\ref{basic}) ensures that $(9/(4p(p-1)))^{1/4}n^{-1/4}d_n(0,i_n)$
converges to $D(0,s)$, and that $(9/(4p(p-1)))^{1/4}n^{-1/4}V^n_{i_n}$
converges to $\ov Z_s$. The desired result follows.
\cq
\smallskip
We define an equivalence relation
$\approx$ on $[0,1]$ by setting
$$s\approx t\quad\hbox{if and only if}\quad D(s,t)=0.$$
Clearly, $D$ induces a metric, which we still denote by $D$, on the
quotient set $[0,1]\,/\!\approx$. The bound $D\leq D^\circ$ ensures that the canonical projection
from $[0,1]$ onto $[0,1]\,/\!
\approx$ is continuous when
$[0,1]\,/\!
\approx$ is equipped with the metric $D$. In particular the metric space
$([0,1]\,/\!\approx, D)$ is compact.
For our purposes, it will be convenient to view this metric space as
a quotient of the real tree $\t_{\ov{\bf e}}$.
By property (iii) of the previous proposition, we may define
$D(a,b)$ for $a,b\in \t_{\ov{\bf e}}=[0,1]\,/\!\simeq$ simply by setting
$D(a,b)=D(s,t)$ where $s$, resp. $t$, is any representative of $a$, resp.
$b$, in $[0,1]$. The equivalence relation $\approx$
then makes sense on $\t_{\ov{\bf e}}$, and the quotient space
$(\t_{\ov{\bf e}}\,/\!
\approx,D)$ is obviously isometric to $([0,1]\,/\!\approx, D)$.
As a consequence of Proposition \ref{prop-distance} (iv)
and the triangle inequality,
for every $a,b\in\t_{\ov{\bf e}}$, the condition $D(a,b)=0$
implies $\ov Z_a=\ov Z_b$.
Before stating the main result, we need to introduce some
additional notation. For every $a,b\in \t_{\ov{\bf e}}$, we set
$$D^\circ(a,b)=\inf\{D^\circ(s,t):s,t\in[0,1], a\simeq s, b\simeq t\}.$$
Suppose that neither of the relations $a\prec b$
and $b\prec a$ holds, and assume for definiteness
that $a<b$. Then the infimum in the definition of $D^\circ(a,b)$
is attained when $[s,t]$ is the minimal subinterval of $[0,1]$ such
that $a\simeq s$ and $b\simeq t$, and it follows that
$$D^\circ(a,b)=\ov Z_a +\ov Z_b - 2 \inf_{c\in [a,b]} \ov Z_c$$
where $[a,b]$ is the lexicographical interval between $a$
and $b$ in $\t_{\ov {\bf e}}$, as defined in subsection 2.3. On the other hand, if
$a\prec b$, then the preceding formula does not necessarily hold: We
have instead
$$D^\circ(a,b)=\ov Z_a +\ov Z_b - 2 \sup\left(\inf_{c\in [a,b]} \ov Z_c,
\inf_{c\in [b,a]} \ov Z_c\right).$$
The function $D^\circ(a,b)$, $a,b\in\t_{\ov{\bf e}}$ needs not satisfy the
triangle inequality. For this reason, we set
for every $a,b\in\t_{\ov{\bf e}}$,
$$D^*(a,b)=\inf\left\{\sum_{i=1}^q D^\circ(a_{i-1},a_i)\right\}$$
where the infimum is over all choices of the integer $q\geq 1$
and of the finite sequence $a_0,a_1,\ldots,a_q$
in $\t_{\ov{\bf e}}$ such that $a_0=a$ and $a_q=b$.
Since $D\leq D^\circ$, and $D$ satisfies the triangle inequality, it
is clear that we have
$$0\leq D(a,b)\leq D^*(a,b)\leq D^\circ (a,b)$$
for every $a,b\in\t_{\ov{\bf e}}$.
We can now state our main result. Recall that we are restricting
our attention to values of $n$ belonging to the sequence
$(n_k)_{k\geq 1}$, and that we assume that the convergence
(\ref{basic}) holds a.s. along this sequence.
\begin{theorem}
\label{main}
We have almost surely
$$\left({\bf m}_n,\Big(\frac{9}{4p(p-1)}\Big)^{1/4}\,n^{-1/4} d_n\right)
\build{\la}_{n\to\infty}^{} (\t_{\ov{\bf e}}\,/\!\approx,D)$$
in the sense of the Gromov-Hausdorff distance on compact
metric spaces. In addition, a.s. for every $a,b\in\t_{\ov{\bf e}}$,
the relation $a\approx b$ holds if and only if one of the
following equivalent properties holds:
\begin{description}
\item{\rm(i)} $D(a,b)=0$.
\item{\rm(ii)} $D^*(a,b)=0$.
\item{\rm(iii)} $D^\circ(a,b)=0$.
\end{description}
\end{theorem}
\noindent{\bf Remarks.} (a)
Although the process $D$ may depend on the sequence $(n_k)_{k\geq 1}$,
the equivalence relation $\approx$ does not, since it can be defined
by either (ii) or (iii) in Theorem \ref{main}.
As was already observed in the introduction, this guarantees that the
limiting compact metric space $(\t_{\ov{\bf e}}\,/\!\!\approx,D)$
is homeomorphic to $(\t_{\ov{\bf e}}\,/\!\approx,D^*)$, and thus that
its topology does not depend on the choice of the
sequence $(n_k)_{k\geq 1}$ (nor on the value of $p$). Still it is tempting
to conjecture that $D(a,b)=D^*(a,b)$, for every $a,b\in\t_{\ov\eg}$.
If this conjecture is correct, the convergence (\ref{basic}), or
that of Theorem \ref{main}, does not require the use of
a subsequence.
\noindent(b) It is not hard to prove that equivalence classes in $\t_{\ov\eg}$ for the
equivalence relation $\approx$ can contain only $1$, $2$ or $3$ points. For
every fixed $s\in[0,1]$,
it is easy to verify that the equivalence
class of $p_{\ov\eg}(s)$ is a singleton a.s. Furthermore,
one can check that a.s. for every rational numbers $r,s,t,u$
such that $0\leq r<s<t<u\leq 1$ one has
$$\inf_{r\leq x\leq s} \ov Z_x \ne \inf_{t\leq x\leq u} \ov Z_x.$$
(The easiest way to derive this property is to use the Brownian snake approach
that is presented below in Section 5.) It follows that an equivalence class
cannot contain more than $3$ points. Conversely, if
we are given two rationals $0<r<s<1$, there exists an a.s. unique $y\in]r,s[$
such that
$$\ov Z_y=\inf_{r\leq x\leq s} \ov Z_x,$$
and the vertex of $\t_{\ov\eg}$ corresponding to $y$ is a leaf of $\t_{\ov\eg}$.
Set $t_1=\sup\{u\leq r:\ov Z_u=\ov Z_y\}$ and $t_2=\inf\{u\geq s:\ov Z_u=\ov Z_y\}$.
Then $t_1\approx y\approx t_2$, and $t_1,y$ and $t_2$
correspond to different vertices of the tree $\t_{\ov{\bf e}}$. To summarize, the equivalence class of a
typical vertex $a\in\t_{\ov\eg}$ is a singleton, but there is a continuum of equivalence
classes consisting of pairs, and there are countably many equivalence classes
containing three elements. These properties are not used below. They will be derived
in greater detail
in the subsequent paper \cite{LGP} where they play an important role.
\medskip
\noindent{\bf Proof of Theorem \ref{main} (first part):} The main difficulty in the proof of Theorem
\ref{main} comes from the implication (i)$\Rightarrow$(iii). Notice that the other
implications (iii)$\Rightarrow$(ii)$\Rightarrow$(i) are trivial. The
implication (i)$\Rightarrow$(iii) is established in the next section.
We now prove the first assertion of
Theorem \ref{main}.
Recall that the metric spaces
$(\t_{\ov{\bf e}}\,/\!\approx,D)$ and $([0,1]\,/\!\approx,D)$
are isometric. For every integer $n$, consider the equivalence
relation $\approx_n$ defined on $[0,1]$ by setting
$$s\approx_n t\hbox{\quad if and only if\quad} d_n(\lfloor pns\rfloor,
\lfloor pnt\rfloor)=0.$$
Clearly, the quotient space $E_n:=[0,1]\,/\!\!\approx_n$
equipped with the metric $\delta_n(s,t)=d_n(\lfloor pns\rfloor,
\lfloor pnt\rfloor)$ is isometric to $(\tau_n^\circ,d_n)$
or equivalently to $({\bf m}_n\backslash\{\partial_n\},d_n)$.
Since
$$d_{GH}(({\bf m}_n\backslash\{\partial_n\},n^{-1/4}d_n),({\bf
m}_n,n^{-1/4}d_n))
\build{\la}_{n\to\infty}^{} 0$$
the first part of Theorem \ref{main} reduces to checking that
we have a.s.
\be
\label{convDGH}
d_{GH}\left(\Big(E_n,
\Big(\frac{9}{4p(p-1)}\Big)^{1/4}\,n^{-1/4}\delta_n\Big),(E_\infty,D)\right)
\build{\la}_{n\to\infty}^{} 0
\ee
where $E_\infty:=[0,1]\,/\!\approx$.
To this end, we construct a correspondence between the
metric spaces $E_n$ and $E_\infty$ by setting
$${\mathcal C}_n=\{(a,b)\in E_n\times E_\infty:
\hbox{there exists }t\in[0,1]\hbox{ such that } a\approx_n t
\hbox{ and }b\approx t\}.$$
In order to bound the distortion of this correspondence,
consider two pairs $(a,b),(a',b')\in{\mathcal C}_n$. By definition,
there exist $s,t\in[0,1]$
such that $a\approx_n s,b\approx s$ and $a'\approx_n t,
b'\approx t$. Then we have
\ba
&&\delta_n(a,a')=d_n(\lfloor pns\rfloor,\lfloor pnt\rfloor)\\
&&D(b,b')=D(s,t).
\ea
Thus, when $E_n$ is equipped with the distance
$({9}/{(4p(p-1))})^{1/4}\,n^{-1/4}\delta_n$, and $E_\infty$ with the
distance $D$, the distortion of ${\mathcal C}_n$ is
\ba
&&\sup_{(a,b),(a',b')\in{\mathcal C}_n}
\Big|\Big(\frac{9}{4p(p-1)}\Big)^{1/4}\,n^{-1/4}\delta_n(a,a')
-D(b,b')\Big|\\
&&\qquad\leq \sup_{s,t\in[0,1]}
\Big|\Big(\frac{9}{4p(p-1)}\Big)^{1/4}\,n^{-1/4}
d_n(\lfloor pns\rfloor,\lfloor pnt\rfloor)-
D(s,t)\Big|,
\ea
which tends to $0$ a.s. by (\ref{basic}). The first assertion of Theorem \ref{main} now follows
from the known result connecting the Gromov-Hausdorff
distance between two compact metric spaces with the infimum
of the distortion of correspondences between these two spaces (Theorem 7.3.25
in \cite{BBI}).
\cq
\medskip
Before proceeding to the second part of the proof of Theorem \ref{main}, let
us state and prove a closely related result.
\begin{proposition}
\label{finite-margi}
Let $k\geq 1$ be an integer. For every $n\geq 1$, let $Y^n_1,\ldots,Y^n_k$ be
$k$ random variables which conditionally given $M_n$ are independent and
uniformly distributed over ${\bf m}_n$. Also, given the triple $(\ov\eg,\ov Z,D)$, let
$Y^\infty_1,\ldots,Y^\infty_k$ be random variables
with values in $\t_{\ov\eg}$ which are independent and distributed according to $\lambda$.
Then,
$$\left(\Big(\frac{9}{4p(p-1)}\Big)^{1/4}\,n^{-1/4}
\,d_n(Y^n_i,Y^n_j)\right)_{1\leq i\leq k,1\leq j\leq k}
\build{\la}_{n\to\infty}^{\rm (d)}
\left(D(Y^\infty_i,Y^\infty_j)\right)_{1\leq i\leq k,1\leq j\leq k}.$$
\end{proposition}
\noindent{\bf Remarks.} (a) Informally, Proposition \ref{finite-margi} means that the convergence in
Theorem \ref{main} can be reinforced in the sense of convergence of measured metric spaces, provided ${\bf
m}_n$ is equipped with the uniform probability measure and $\t_{\ov{\bf e}}\,/\!\approx$
is equipped with the image of $\lambda$ under the canonical projection. We could give other
versions of this reinforcement: See Chapter $3\frac{1}{2}$ of the book \cite{Gro}
for various notions of convergence of measured metric spaces. Here we content ourselves
with the preceding proposition, which will be useful in Section 6 below.
\noindent (b) The reader may be puzzled by our
assumption on $Y^\infty_1,\ldots,Y^\infty_k$, since we seem
to be dealing with random variables taking values in a {\it random} state space. It is however
a straightforward matter to give a mathematically rigorous (although less intuitive)
version of the statement of the proposition.
\medskip
\proof Recall that ${\bf m}_n=\tau^\circ_n \cup\{\partial_n\}$. We may
and will assume that $Y^n_1,\ldots,Y^n_k$ are
uniformly distributed over $\tau^\circ_n$ rather than over ${\bf m}_n$.
Then let $U_1,\ldots,U_k$ be $k$ independent random variables which are uniformly
distributed over $[0,1]$ and independent of all other random quantities we have
considered until now. We may then take $Y^\infty_i=p_{\ov\eg}(U_i)$
for every $1\leq i\leq k$. Also, for every $1\leq i\leq k$, we let
$\wt Y^n_i$ be the equivalent class of $\lfloor pnU_i \rfloor$ in the quotient set
$[pn]\,/\!\sim_n=\tau^\circ_n$. The (almost sure) convergence (\ref{basic}) implies that
$$\left(\Big(\frac{9}{4p(p-1)}\Big)^{1/4}\,n^{-1/4}
\,d_n(\wt Y^n_i,\wt Y^n_j)\right)_{1\leq i\leq k,1\leq j\leq k}
\build{\la}_{n\to\infty}^{\rm (a.s.)}
\left(D(Y^\infty_i,Y^\infty_j)\right)_{1\leq i\leq k,1\leq j\leq k}.$$
This does not immediately give us the desired result, because the
variables $\wt Y^n_i$ are not uniformly distributed over $\tau^\circ_n$.
Still we will see that in a sense they are close enough to variables that have
the desired uniform distribution. To this end, for every $n$ and every $1\leq i\leq k$, set
$$k^n_i=\lceil ((p-1)n+2)U_i\rceil$$
and let $Y^n_i$ be the $k^n_i$-th element in the sequence of vertices of $\tau_n^\circ$
listed in lexicographical order. Clearly, the variables $Y^n_i$ have the properties
stated in the proposition. To complete the proof, it is therefore enough to
check that, for every $1\leq i\leq k$,
\be
\label{margitech1}
n^{-1/4}d_n(Y^n_i,\wt Y^n_i)\build{\la}_{n\to\infty}^{\rm a.s.} 0.
\ee
Note that a.s. for every $t\in]0,1]$, the number of distinct vertices of $\tau^\circ_n$
that appear in the search-depth sequence before rank $\lfloor pnt\rfloor$ behaves
as $(p-1)nt$ when $n\to\infty$. To see this, observe
that in the evolution of the contour function
of $\tau^\circ_n$ each step which is not downwards corresponds in the
search-depth sequence to a vertex of $\tau^\circ_n$ that has not been visited
before, and then use (\ref{basic}) to see that the number of downward steps
before time $\lfloor pnt\rfloor$ behaves like $nt$ when $n\to\infty$ (indeed the difference between
the numbers of upward and downward steps is $O(n^{1/2})$ as $n\to\infty$).
From the preceding remarks, we get that a.s. for every $s,t\in[0,1]$
such that $s<U_i<t$, if $n$ is large enough, the vertex $Y^n_i$
is visited by the search-depth sequence during the time interval $[\lfloor pns\rfloor,\lfloor pnt\rfloor]$.
Thus, for $n$ sufficiently large,
$$n^{-1/4}d_n(Y^n_i,\wt Y^n_i)\leq n^{-1/4}\sup_{\lfloor pns\rfloor\leq j\leq \lfloor pnt\rfloor}
d_n(j,\lfloor pnU_i\rfloor).$$
The right-hand side can be made arbitrarily small when $n$ is large by choosing $s$ and $t$
sufficiently close to $U_i$. This completes the proof of
(\ref{margitech1}) and of Proposition \ref{finite-margi}. \cq
\section{The key step}
This section is devoted to the second part of the proof of Theorem \ref{main}, that is
to the proof of the implication
(i)$\Rightarrow$(iii) in this theorem. We start with a lemma.
\begin{lemma}
\label{linesegment}
Almost surely, for every $a,b\in\t_{\ov{\bf e}}$, the condition
$D(a,b)=0$ implies $\ov Z_c\geq \ov Z_a=\ov Z_b$
for every $c\in\llbracket a,b\rrbracket$.
\end{lemma}
\proof We already noticed that the condition $D(a,b)=0$ forces
$\ov Z_a=\ov Z_b$. We can immediately exclude the case $a=\ov\rho$
since this would imply $\ov Z_b=\ov Z_a=0$
and $b=\ov \rho=a$. Then we can assume without loss of
generality that $a<b$. We argue by contradiction, assuming that
there exists $c\in\rrbracket a,b \llbracket$ such that
$\ov Z_c< \ov Z_a$. For definiteness, we assume that
$c\in \llbracket a\wedge b,a\llbracket$. The symmetric case
$c\in \llbracket a\wedge b,b\llbracket$ is treated in a
similar manner.
Let $s<t$ be such that $a\simeq s$ and $b\simeq t$.
We can then find $r\in]s,t[$ such that $c\simeq r$. Choose
$i_n,j_n,k_n\in[pn]$,
with $i_n\leq k_n\leq j_n$, such that $i_n/(pn)\la s$,
$j_n/(pn)\la t$ and $k_n/(pn)\la r$.
Denote by $a_n,b_n,c_n$ the vertices in $\tau^\circ_n$
corresponding respectively to $i_n,j_n,k_n$. Since
$c\in \llbracket a\wedge b,a\llbracket$, a simple argument using the
convergence of the first components in (\ref{basic}),
and the remarks of the beginning of subsection 2.2,
shows that $k_n$ can be chosen in such a way that
$c_n\in\rrbracket a_n\wedge b_n,a_n\llbracket$ for every
$n$ sufficiently large. Denote by $\tau^\circ_n(c_n)$
the set of all descendants of $c_n$ in $\tau^\circ_n$. Then
$a_n\in\tau^\circ_n(c_n)$ but $b_n\notin\tau^\circ_n(c_n)$.
By (\ref{basic}) and our assumption $D(a,b)=0$
we know that $d_n(a_n,b_n)=o(n^{1/4})$ as $n\to\infty$. Let
$\gamma_n=(\gamma_n(i),0\leq i\leq d_n(a_n,b_n))$ be a geodesic path
from $a_n$ to $b_n$ in the map $M_n$. When $n$ is large,
the path $\gamma_n$ must lie entirely in $\tau^\circ_n$,
because if $\partial_n$ belongs to this path the
equality $d_n(a_n,b_n)=d_n(a_n,\partial_n)+ d_n(\partial_n,b_n)=V^n_{i_n}+V^n_{j_n}$
yields a contradiction with the property $d_n(a_n,b_n)=o(n^{1/4})$.
Denote by $g_n$
the last point on the geodesic $\gamma_n$ that belongs
to $\tau^\circ_n(c_n)$. Since $g_n$ is a point of the
geodesic $\gamma_n$ and $d_n(a_n,b_n)=o(n^{1/4})$, we have
$$\ell^n_{g_n}=d_n(\partial_n,g_n)=d_n(\partial_n,a_n)+o(n^{1/4})
=n^{1/4} \ov Z_a +o(n^{1/4})$$
as $n\to\infty$. On the other hand, since $k_n/(pn)\la r$
and $c\simeq r$,
$$\ell^n_{c_n}=V^n_{k_n}=n^{1/4}\ov Z_c +o(n^{1/4})$$
as $n\to\infty$. Hence, for $n$ large we
must have $\ell^n_{g_n}>\ell^n_{c_n}$.
Using the way edges of the map $M_n$ are reconstructed from the
mobile $\theta_n$, we now see that any edge starting from $g_n$
in $M_n$ connects $g_n$ with another point of $\tau^\circ_n(c_n)$.
Indeed, any successor of the vertex $g_n$ must clearly lie
in $\tau^\circ_n(c_n)$ because in the search-depth sequence
of $\tau^\circ_n$, a vertex with label $\ell^n_{g_n}-1$ will be
visited after the last visit of $g_n$ before coming back to $c_n$ and exiting the tree $\tau^\circ_n(c_n)$.
Similarly, $g_n$ cannot be a successor of a
vertex $h\notin\tau^\circ_n(c_n)$: If this were the case we would have
$\ell^n_h-1=\ell^n_{g_n}>\ell^n_{c_n}$, and the search-depth sequence
of $\tau^\circ_n$ would visit a vertex with label $\ell^n_h-1$ after visiting $h$
before entering the set $\tau^\circ_n(c_n)$. Finally, the fact that
$g_n$ is not connected to any point outside $\tau^\circ_n(c_n)$
gives a contradiction with our choice of $g_n$. \cq
\begin{proposition}
\label{ancestorline}
Almost surely, for every pair $(a,b)$ in $\tree$ such that
$a$ is an ancestor of $b$ and $a\ne b$, we have $D(a,b)>0$.
\end{proposition}
\proof We argue by contradiction, assuming that there exists
a pair $(a,b)$ in $\tree$ such that
$a$ is an ancestor of $b$, $a\ne b$ and $D(a,b)=0$.
Notice that the case $a=\ov\rho$ is excluded since we already
know from Proposition \ref{prop-distance} (iv) that $D(\ov\rho,b)=\ov Z_b>0$
for every $b\ne
\ov\rho$. So we assume that $a\ne \ov\rho$.
Recall that we have automatically $\ov Z_a=\ov Z_b$.
Let $s,t\in[0,1]$ be such that
$a\simeq s$ and $b\simeq t$. Since $a$ is an ancestor of $b$
we can choose $s$ and $t$ such that $s<t$ and $\ov{\bf e}_r>\ov{\bf e}_s$
for every $r\in]s,t]$.
Since
$D(s,t)=D(a,b)=0$,
(\ref{basic}) gives
$$n^{-1/4}d_n(pn s,pnt)\build{\la}_{n\to\infty}^{} 0.$$
So, for every $n$, we can find $i^\circ_n,j_n\in[pn]$ such that
$i^\circ_n\leq j_n$,
$|i^\circ_n-pns|\leq 1$, $|j_n-pnt|\leq 1$ and
$$n^{-1/4}d_n(i^\circ_n,j_n)\build{\la}_{n\to\infty}^{} 0.$$
Let $i_n=\sup\{k\in[i^\circ_n,j_n]\cap\Z: C^n_k=C^n_{i^\circ_n}\}$. By
(\ref{basic}) and the condition $\ov{\bf e}_r>\ov{\bf e}_s$
for every $r\in]s,t]$, we must have $n^{-1}(i_n-i^\circ_n)\la 0$
as $n\to \infty$. As a consequence, $n^{-1/4}d_n(i^\circ_n,i_n)\la 0$.
Let $a_n$ and $b_n$ be the vertices in $\tau^\circ_n$ such that
$a_n\sim_n i_n$ and $b_n\sim_n j_n$. Then provided $n$ is
sufficiently large, the remarks of subsection 2.2 show that $a_n$ is an ancestor of $b_n$. Moreover
we have
\be
\label{distzero}
n^{-1/4}d_n(a_n,b_n)=n^{-1/4}d_n(i_n,j_n)\build{\la}_{n\to\infty}^{} 0.
\ee
By Lemma \ref{linesegment}, we also know that $\ov Z_c\geq \ov Z_a$
for every $c\in\llbracket a,b\rrbracket$. Recall that the conditioned
tree $(\tree, \ov Z)$ is obtained by re-rooting the unconditioned tree
$(\t_{\bf e},Z)$ at the vertex corresponding to the minimal spatial
position, and that along a given line segment of $\t_{\bf e}$,
$Z$ evolves like linear Brownian motion. Since local minima
of linear Brownian motion are distinct, a simple argument then
shows that the equality $\ov Z_c=\ov Z_a$ can hold for at most
one value of $c\in\rrbracket a,b\llbracket$. Hence, we can find
$\eta>0$ such that the properties $c\in\rrbracket a,b\llbracket$ and
$d_{\ov{\bf e}}(a,c)<\eta$ imply $\ov Z_c>\ov Z_a$.
Since $\ov{\bf e}_r>\ov{\bf e}_s$
for every $r\in]s,t]$, Lemma \ref{increasepoint}
implies that for every $\varepsilon>0$,
$$\inf_{r\in[s,s+\varepsilon]}\ov Z_r<\ov Z_s.$$
It follows that there exists one (in fact infinitely many) subtree
$\t^1$ from the left side of $\llbracket a,b\rrbracket$, with
root $\rho^1\in\rrbracket a,b \llbracket$, such that
$d_{\ov{\bf e}}(a,\rho^1)<\eta$ and
$$\inf_{c\in\t^1} \ov Z_c <\ov Z_a.$$
We denote by $[\alpha,\beta]$ the interval coding $\t^1$: The elements
of $\t^1$ are exactly the equivalence classes
in $\tree$ of the reals in $[\alpha,\beta]$, and in particular
$\rho^1\simeq \alpha\simeq\beta$.
In a similar way, using a time reversal argument, we can construct
a subtree $\t^2$ from the right side of $\llbracket a,b\rrbracket$, with
root $\rho^2\in\rrbracket a,b \llbracket$, such that
$d_{\ov{\bf e}}(a,\rho^2)<\eta$ and
$$\inf_{c\in\t^2} \ov Z_c <\ov Z_a.$$
We can always choose $\t^1$ and $\t^2$ in such a way
that $\rho^1\prec \rho^2$. From our choice of $\eta$, we
have then
$$\inf_{c\in\llbracket \rho^1,\rho^2\rrbracket}\ov Z_c > \ov Z_a.$$
We now exploit the convergence (\ref{basic}) to get similar
properties for the discrete trees $(\tau^\circ_n,\ell^n)$. We can
find a positive number $\kappa$ such that the following holds for
$n$ sufficiently large. There exists a subtree $\tau^1_n$
from the left side of $\llbracket a_n,b_n\rrbracket$ with
root $\rho^1_n\in \rrbracket a_n,b_n\llbracket$ such that
\be
\label{altech1}
\inf_{x\in\tau^1_n} \ell^n_x \leq \ell^n_{a_n} -\kappa n^{1/4}.
\ee
The subtree $\tau^1_n$ is coded by an interval $[\alpha_n,\beta_n]\cap\Z$
(via the identification
$\tau^\circ_n=[pn]/\sim_n$) such that $\alpha_n/(pn)\la \alpha$
and $\beta_n/(pn)\la \beta$.
Similarly, there exists a subtree $\tau^2_n$
from the right side of $\llbracket a_n,b_n\rrbracket$ with
root $\rho^2_n\in \rrbracket a_n,b_n\llbracket$ such that
\be
\label{altech2}
\inf_{x\in\tau^2_n} \ell^n_x \leq \ell^n_{a_n} -\kappa n^{1/4}.
\ee
Furthermore, $\rho^1_n\prec \rho^2_n$ and
\be
\label{altech3}
\inf_{x\in\llbracket \rho^1_n,\rho^2_n\rrbracket} \ell^n_x \geq \ell^n_{a_n}
+\kappa n^{1/4}.
\ee
Let $\gamma_n=(\gamma_n(i),0\leq i\leq d_n(a_n,b_n))$ be a geodesic path
from $a_n$ to $b_n$ in $M_n$. As in the proof of
Lemma \ref{linesegment}, we know that the path $\gamma_n$
lies in $\tau^\circ_n$ when $n$ is large. Denote by $u_n$ the last point on the geodesic
$\gamma_n$
that does not belong to the set
$$\{x\in\tau^\circ_n: \rho^1_n\leq x\ll \rho^2_n\}.$$
This definition makes sense because $a_n<\rho^1_n$. Also $u_n\ne b_n$ since
$\rho^1_n\leq b_n\ll \rho^2_n$. Denote by $v_n$ the point following $u_n$ on the
geodesic
$\gamma_n$.
Since $n^{-1/4}d_n(a_n,b_n)\la 0$ as $n\to\infty$, we know that
$$n^{-1/4}\sup_{0\leq i\leq d_n(a_n,b_n)} d_n(a_n,\gamma_n(i))
\build{\la}_{n\to\infty}^{} 0,$$
and therefore
\be
\label{altech4}
n^{-1/4}\sup_{0\leq i\leq d_n(a_n,b_n)} |\ell^n_{a_n} - \ell^n_{\gamma_n(i)} |
\build{\la}_{n\to\infty}^{} 0.
\ee
The preceding properties imply that $v_n\in \tau^1_n\cup\tau^2_n$
for $n$ sufficiently large. Indeed, we have
$\rho^1_n\leq v_n\ll \rho^2_n$ by construction and we also know that
$\ell^n_{v_n}>\ell^n_{a_n}-\kappa n^{1/4}$ if $n$ is large, by (\ref{altech4}).
Suppose that
$v_n\notin \tau^1_n\cup\tau^2_n$. Then, by
(\ref{altech1}) and (\ref{altech2}),
$v_n$ can
be connected to a point $y$ that does not belong to $\{x\in\tau^\circ_n:
\rho^1_n\leq x\ll
\rho^2_n\}$ only if $y\in
\llbracket \rho^1_n,\rho^2_n\rrbracket$.
Thus we get that $u_n\in
\llbracket \rho^1_n,\rho^2_n\rrbracket$, but this is impossible by
(\ref{altech3}) and (\ref{altech4}), if $n$ is large enough.
So, for $n$ sufficiently large, we have either
$v_n\in \tau^1_n$ or
$v_n\in \tau^2_n$.
One of these two cases has to occur infinitely often.
For definiteness, we assume that
the property $v_n\in \tau^1_n$ occurs infinitely often
and from now on until the final part of the proof we restrict our attention to integers $n$ such that
this property holds.
Then the following properties hold for
$n$ large:
\begin{description}
\item{\rm(i)}
$v_n\in \tau^1_n$ and $v_n\ne \rho^1_n$.
\item{\rm(ii)} $u_n\leq \rho^1_n$.
\item{\rm (iii)} Every point $w$ that comes after $v_n$ on the
geodesic $\gamma_n$ satisfies $\rho^1_n\leq w$.
\end{description}
The property $v_n\ne\rho^1_n$ is clear from (\ref{altech4}) and
(\ref{altech3}). To get (ii), recall that by construction
we have either $u_n\leq \rho^1_n$ or $\rho^2_n\ll u_n$ (or both together).
Suppose that $\rho^2_n\ll u_n$.
If $n$ is large, the fact
that $u_n$ is connected with a point of
$\tau^1_n\backslash\{\rho^1_n\}$ and the property (\ref{altech2})
then imply that $u_n\in\llbracket \varnothing,\rho^2_n\rrbracket$.
However $u_n\in \llbracket \rho^1_n,\rho^2_n\rrbracket$ is excluded
by (\ref{altech3}) and (\ref{altech4}), and thus we get
$u_n\in\llbracket \varnothing,\rho^1_n\rrbracket$, so that
in particular $u_n\leq \rho^1_n$. Finally, (iii) is clear from the definition
of $u_n$.
Thanks to (i)--(iii), we can apply Lemma \ref{combi}, and we
get that if $n$ is large enough, for every point
$y$ of $\tau^1_n\backslash\{\rho^1_n\}$ such that
\be
\label{claim0}
\ell^n_x>
\sup_{0\leq i\leq d_n(a_n,b_n)} \ell^n_{\gamma_n(i)}\;
\hbox{ for every }
x\in\llbracket\rho^1_n,y\rrbracket
\ee
we have
\be
\label{claim1}
d_n(y,b_n)\leq d_n(u_n,b_n)+\ell^n_y -
\inf_{0\leq i\leq d_n(a_n,b_n)} \ell^n_{\gamma_n(i)}.
\ee
For every $\varepsilon>0$, denote by $\u^\varepsilon_n$
the set of all vertices $y\in\tau^1_n$ such that:
\begin{description}
\item{$\bullet$} $\ell^n_y\leq \ell^n_{a_n}+ {3\varepsilon\over 2} n^{1/4}$;
\item{$\bullet$} $\ell^n_x\geq \ell^n_{a_n}+{\varepsilon\over 16} n^{1/4}$,
for every $x\in\llbracket \rho^1_n,y\rrbracket$.
\end{description}
Recall that $[\alpha,\beta]$ is the interval coding $\t^1$
and that $s\simeq a$.
We denote by $\u^\varepsilon_\infty$ the set of all
$r\in[\alpha,\beta]$ such that
\begin{description}
\item{$\bullet$} $\ov Z_r< \ov Z_s+
(\frac{9}{4p(p-1)})^{1/4}\,\varepsilon$;
\item{$\bullet$} $\ov Z_{r'}>\ov
Z_s+(\frac{9}{4p(p-1)})^{1/4}\,\frac{\varepsilon}{8}$, for every
$r'\in\llbracket \rho^1,r\rrbracket$.
\end{description}
(When writing $\llbracket \rho^1,r\rrbracket$ we slightly abuse notation by identifying $r$
with the corresponding vertex in $\t_{\ov{\bf e}}$.) Notice that
$\u^\varepsilon_\infty$ is open.
Moreover, let
$]u,v[$ be a connected component of $\u^\varepsilon_\infty$,
and let $[u',v']$ be a compact subinterval of $]u,v[$. We claim that
for every $n$ sufficiently large, we must have
\be
\label{discreconti}
[pnu',pnv']\cap\Z\subset \u^\varepsilon_n
\ee
in the sense that every vertex $y$ of $\tau^\circ_n$
such that $y\sim_n k$ for some $k\in [pnu',pnv']\cap\Z$
belongs to $\u^\varepsilon_n$. To see this, first note that
the property $[pnu',pnv']\cap\Z\subset\tau^1_n$ holds for $n$
sufficiently large because $]u,v[\subset[\alpha,\beta]$. Then suppose
that for every $n$ belonging to a subsequence converging to $\infty$
we can find a vertex $y_n\in\tau^1_n$ such that
$y_n\sim_n k_n$ for some $k_n\in [pnu',pnv']\cap\Z$
and at least one of the two conditions
\begin{description}
\item{(a)} $\ell^n_{y_n}\leq \ell^n_{a_n}+ {3\varepsilon\over 2} n^{1/4}$,
\item{(b)} $\ell^n_{x}\geq \ell^n_{a_n}+{\varepsilon\over 16} n^{1/4}$,
for every $x\in\llbracket \rho^1_n,y_n\rrbracket$,
\end{description}
does not hold. By compactness we can assume
that $k_n/(pn)\la r\in[u',v']$.
If condition (a) fails for
infinitely many values of $n$, (\ref{basic}) gives
$$\ov Z_r\geq \ov Z_s+
\Big(\frac{9}{4p(p-1)}\Big)^{1/4}\frac{3\varepsilon}{2},$$
which contradicts the fact that $[u',v']\subset \u^\varepsilon_\infty$.
If (b) fails for infinitely values of $n$, then for these values
of $n$ we can find $\ov k_n\in[\alpha_n,k_n]\cap \Z$
such that
$$C^n_{\ov k_n}=\inf_{\ov k_n\leq k\leq k_n} C^n_k$$
and
$$V^n_{\ov k_n}< \ell^n_{a_n}+{\varepsilon\over 16} n^{1/4}.$$
Again by compactness, we can assume that $\ov k_n/(pn)\la \ov r\in[\alpha,r]$.
We have then
$$\ov{\bf e}_{\ov r}=\inf_{\ov r\leq r'\leq r} \ov{\bf e}_{r'}$$
so that $\ov r\in \llbracket \rho^1,r\rrbracket$, and
$$\ov Z_{\ov r}\leq \ov Z_s+\Big(\frac{9}{4p(p-1)}\Big)^{1/4}
\frac{\varepsilon}{16}$$
thus contradicting the fact that $[u',v']\subset \u^\varepsilon_\infty$.
This completes the proof of our claim (\ref{discreconti}).
If $I$ is a finite union of closed
subintervals of $[0,1]$ the number of vertices of
$\tau^\circ_n=[pn]/\sim_n$ for which the first representative in $[pn]$ belongs
to $pnI$ behaves like $(p-1)n|I|$ as $n\to\infty$, where
$|I|$ denotes the Lebesgue measure of $I$. When $I$ is of the type $[0,t]$, this
was observed in the proof of Proposition \ref{finite-margi}, and the general case
follows by a simple argument. Thus (\ref{discreconti}) implies that
\be
\label{altech5}
\liminf_{n\to\infty} \frac{1}{(p-1)n}\,\#\u^\varepsilon_n
\geq \lambda(\u^\varepsilon_\infty).
\ee
We can now use (\ref{distzero}), (\ref{altech4}) and (\ref{claim1})
to see that for $n$ sufficiently large, for every $y\in\u^\varepsilon_n
\backslash\{\rho^1_n\}$,
we have
$$d_n(y,b_n)\leq 2\varepsilon\,n^{1/4}.$$
(Notice that condition (\ref{claim0}) is satisfied for every
$y\in\u^\varepsilon_n$ when $n$ is large enough.) Hence, for every
$y,y'\in\u^\varepsilon_n$ we have also
$$d_n(y,y')\leq 4\varepsilon\,n^{1/4}.$$
Recall that $B_n(y,R)$ denotes the closed ball with radius $R$ centered
at $y$ in the metric space $({\bf m}_n,d_n)$. We have thus
$\# B_n(y,4\varepsilon n^{1/4})\geq \#\u^\varepsilon_n$
for every $y\in \u^\varepsilon_n$.
Let $(\varepsilon_k)$ be any fixed sequence
monotonically decreasing to $0$. By Lemma \ref{occuptree},
we can find $\delta_0>0$ and an integer $k_0$ such that for
every $k\geq k_0$,
$$\lambda(\u^{\varepsilon_k}_\infty) \geq 2\delta_0\,\varepsilon_k^2.$$
From (\ref{altech5}) we then see that for every $k\geq k_0$,
if $n$ is sufficiently large, we have
$$\#\u^{\varepsilon_k}_n\geq \delta_0\,\varepsilon_k^2\,(p-1)n.$$
By preceding remarks, this entails that for every $k\geq k_0$,
if $n$ is sufficiently large,
\be
\label{altech6}
\#\{y\in {\bf m}_n:\# B_n(y,4\varepsilon_kn^{1/4})\geq
\delta_0\varepsilon_k^2(p-1)n \} \geq \delta_0\varepsilon_k^2 (p-1)n.
\ee
Since we restricted our attention to integers $n$ such that $v_n\in\tau^1_n$,
the bound (\ref{altech6}) only holds for those integers. However, a symmetric
argument shows that (\ref{altech6}) also holds for all (sufficiently large) integers $n$
such that $v_n\in\tau^2_n$, possibly with different values of $\delta_0$ and $k_0$.
Thus by changing $\delta_0$ and $k_0$ if necessary, we can assume that
(\ref{altech6}) holds for all sufficiently large integers $n$.
On the other hand, (\ref{pointed}) shows that for every $\delta>0$ and every $k$,
\ba
&&E\Big[\frac{1}{(p-1)n}\#\{y\in {\bf m}_n:\# B_n(y,4\varepsilon_kn^{1/4})\geq
\delta\varepsilon_k^2(p-1)n \} \Big]\\
&&\qquad\build{\la}_{n\to\infty}^{} P\Big[\ov{\cal I}\Big(\Big[0,
4\Big(\frac{9}{4p(p-1)}\Big)^{1/4}
\varepsilon_k\Big]\Big)\geq \delta\varepsilon_k^2\Big].
\ea
By Lemma \ref{estimateISE}, we have for every $\delta>0$,
$$P\Big[\ov{\cal I}\Big(\Big[0,
4\Big(\frac{9}{4p(p-1)}\Big)^{1/4}
\varepsilon_k\Big]\Big)\geq \delta\varepsilon_k^2\Big]
=o(\varepsilon_k^2)$$
as $k\to\infty$. Hence, Fatou's lemma gives
$$E\Big[\liminf_{n\to\infty}
\frac{1}{n}\#\{y\in {\bf m}_n:\# B_n(y,4\varepsilon_kn^{1/4})\geq
\delta\varepsilon_k^2(p-1)n \}\Big]=o(\varepsilon_k^2)$$
as $k\to\infty$.
Another application of Fatou's lemma yields that
$$E\Big[\liminf_{k\to\infty}\Big(
\liminf_{n\to\infty}
\frac{1}{\varepsilon_k^2n}\#\{y\in {\bf m}_n:\#
B_n(y,4\varepsilon_kn^{1/4})\geq
\delta\varepsilon_k^2(p-1)n \}\Big)\Big]=0.$$
By applying the above to a sequence of values of $\delta$ decreasing to $0$, we obtain that a.s. for every
$\delta>0$,
$$\liminf_{k\to\infty}\Big(
\liminf_{n\to\infty}
\frac{1}{\varepsilon_k^2n}\#\{y\in {\bf m}_n:\#
B_n(y,4\varepsilon_kn^{1/4})\geq
\delta\varepsilon_k^2(p-1)n \}\Big)=0.$$
This contradicts (\ref{altech6}), thus completing the proof of
Proposition \ref{ancestorline}. \cq
\begin{proposition}
\label{notancestor}
Almost surely, for every pair $(a,b)$ in $\tree$ such that
$a$ is not an ancestor of $b$ and $b$ is not an ancestor
of $a$, the condition
$D(a,b)=0$ implies $D^\circ(a,b)=0$.
\end{proposition}
\proof The proof is similar to that of Proposition
\ref{ancestorline} but the fact that we already know the
property stated in this proposition makes the argument
a little simpler. We again argue by contradiction,
assuming that there exists a pair $(a,b)$ satisfying the
condition of the proposition, such that $D(a,b)=0$
and $D^\circ(a,b)>0$. Without loss of generality we may
and will assume that $a<b$. Recall that we have
automatically $\ov Z_a=\ov Z_b$.
Let $[s,t]$ be the smallest subinterval of $[0,1]$
such that $a\simeq s$ and $b\simeq t$. As in the proof of Proposition
\ref{ancestorline}, we can find $a_n,b_n\in\tau^\circ_n$
in such a way that there exist $i_n,j_n\in [pn]$ with
$i_n\leq j_n$, $a_n\sim_n i_n$, $b_n\sim_n j_n$ and
$i_n/(pn)\la s$, $j_n/(pn)\la t$ as $n\to\infty$. We have then
$$n^{-1/4}d_n(a_n,b_n)\build{\la}_{n\to\infty}^{} D(a,b)=0.$$
From Lemma \ref{linesegment}, we also know that
$\ov Z_c\geq \ov Z_a$ for every $c\in\llbracket a,b \rrbracket$.
Recall that we assumed
$$\ov Z_a + \ov Z_b -2 \inf_{c\in[a,b]} \ov Z_c =D^\circ(a,b) >0.$$
It follows that
$$\inf_{c\in[a,b]\backslash\llbracket a,b \rrbracket} \ov Z_c
<\ov Z_a=\ov Z_b.$$
Since the minimum of $\ov {\bf e}$ over $[s,t]$ is attained at a unique
time corresponding to the vertex $a\wedge b$ (otherwise the tree $\t_{\ov{\bf e}}$ would have a point
with multiplicity strictly greater than $3$), we have
$$[a,b]\backslash\llbracket a,b \rrbracket
=([a,a\wedge b]\backslash\llbracket a,a\wedge b \rrbracket)
\cup ([a\wedge b,b]\backslash\llbracket a\wedge b,b\rrbracket).$$
Thus at least one of the following two conditions holds:
\be
\label{nota1}
\inf_{c\in[a,a\wedge b]\backslash\llbracket a,a\wedge b \rrbracket} \ov
Z_c <\ov Z_a,
\ee
or
\be
\label{nota2}
\inf_{c\in[a\wedge b,b]\backslash\llbracket a\wedge b,b\rrbracket} \ov
Z_c <\ov Z_a.
\ee
For definiteness, we assume that (\ref{nota2}) holds. The symmetric case
where (\ref{nota1}) holds is treated in a similar manner.
Under (\ref{nota2}), there exists a subtree $\t^1$ from the left
side of $\llbracket a\wedge b,b\rrbracket$, with root
$\rho^1\in \rrbracket a\wedge b,b\llbracket$, such that
$$\inf_{c\in\t^1} \ov Z_c <\ov Z_a.$$
We let $[\alpha,\beta]$ be the interval coding $\t^1$.
As in the proof
of Proposition \ref{ancestorline}, we can find a positive number
$\kappa$ such that the following holds for $n$ sufficiently large. There exists a
subtree $\tau^1_n$ of $\tau^\circ_n$, from the left side of
$\llbracket a_n\wedge b_n,b_n\rrbracket$, with root
$\rho^1_n\in \rrbracket a_n\wedge b_n,b_n\llbracket$
and such that
\be
\label{nota3}
\inf_{x\in \tau^1_n} \ell^n_{x} \leq \ell^n_{a_n} - \kappa n^{1/4}.
\ee
Furthermore, $\tau^1_n$ is coded by an interval $[\alpha_n,\beta_n]
\cap \Z$, with $i_n\leq \alpha_n\leq \beta_n\leq j_n$, and
$\alpha_n/(pn)\la \alpha$, $\beta_n/(pn)\la \beta$
as $n\to\infty$.
Let $\gamma_n=(\gamma_n(i),0\leq i\leq d_n(a_n,b_n))$ be a geodesic path
from $a_n$ to $b_n$. As previously, we know that $\gamma_n$ lies entirely in
$\tau^\circ_n$ when $n$ is large. Furthermore, as in the proof of Proposition
\ref{ancestorline}, we have
\be
\label{notatech1}
n^{-1/4}\sup_{0\leq i\leq d_n(a_n,b_n)} | \ell^n_{a_n} - \ell^n_{\gamma_n(i)} |
\build{\la}_{n\to\infty}^{} 0.
\ee
We first
claim that for
$n$ sufficiently large the path $\gamma_n$ does not intersect $\llbracket
\varnothing,\rho^1_n
\rrbracket$. Indeed, suppose that $\gamma_n$ intersects
$\llbracket \varnothing,\rho^1_n
\rrbracket$ for infinitely many values of $n$, and for such values
write $g_n$ for one of the intersection points. Let
$k_n\in[pn]$ be such that $g_n\sim_n k_n$, and let
$r$ be any accumulation point of $k_n/(pn)$ in $[0,1]$. If
$c\in\tree$ is such that $c\simeq r$, the property
$d_n(g_n,b_n)\leq d_n(a_n,b_n)=o(n^{1/4})$ ensures that
$D(c,b)=0$. However, the fact that $g_n\in \llbracket \varnothing,\rho^1_n
\rrbracket$ easily implies that $c\in\llbracket \ov\rho,\rho^1\rrbracket$.
Hence we have both $D(c,b)=0$ and $c\in\llbracket \ov\rho,b\rrbracket$
with $c\ne b$. By Proposition \ref{ancestorline} this cannot occur.
Now let $u_n$ be the last point on the geodesic $\gamma_n$
that belongs to $\{x\in\tau^\circ_n:x<\rho^1_n\}$. This makes
sense since $a_n$ belongs to the latter set. Also $u_n\ne b_n$
since $\rho^1_n\leq b_n$. Let $v_n$ be the point following $u_n$
on the geodesic $\gamma_n$. We claim that $v_n\in\tau^1_n$
if $n$ is sufficiently large. Indeed, the property (\ref{nota3})
warrants that a vertex $y$ belonging to the set
$$\{x\in\tau^\circ_n: \rho^1_n\leq x\}\backslash \tau^1_n$$
and such that $\ell^n_y > \ell^n_{a_n} -\kappa n^{1/4}$ cannot be connected
to $u_n$, except possibly if
$u_n\in\llbracket \varnothing,\rho^1_n\rrbracket$. However we just saw that
this case does not occur for $n$ sufficiently large. By applying
the preceding considerations to $y=v_n$, using (\ref{notatech1}), we get our claim.
Then the following properties hold for
$n$ large:
\begin{description}
\item{\rm(i)}
$v_n\in \tau^1_n$ and $v_n\ne \rho^1_n$.
\item{\rm(ii)} $u_n\leq \rho^1_n$.
\item{\rm (iii)} Every point $w$ that comes after $v_n$ on the
geodesic $\gamma_n$ satisfies $\rho^1_n\leq w$.
\end{description}
From Lemma \ref{combi},
we
get that if $n$ is large enough, for every point
$y$ of $\tau^1_n\backslash\{\rho^1_n\}$ such that
$$\ell^n_x>
\sup_{0\leq i\leq d_n(a_n,b_n)} \ell^n_{\gamma_n(i)}\;
\hbox{ for every }
x\in\llbracket\rho^1_n,y\rrbracket
$$
we have
$$d_n(y,b_n)\leq d_n(u_n,b_n)+\ell^n_y -
\inf_{0\leq i\leq d_n(a_n,b_n)} \ell^n_{\gamma_n(i)}.
$$
The end of the argument is now entirely similar to the end
of the proof of Proposition \ref{ancestorline}: We use
(\ref{basic}), Lemma \ref{estimateISE} and Lemma \ref{occuptree} to show
that the preceding properties lead to a contradiction. This
completes the proof of Proposition \ref{notancestor}. \cq
The implication (i)$\Rightarrow$(iii) in Theorem \ref{main}
is a consequence of Propositions \ref{ancestorline}
and \ref{notancestor}. This completes the proof
of Theorem \ref{main}.
\section{Proof of the technical estimates}
In this section, we prove the three lemmas that were stated
at the end of subsection 2.4. We first need to recall some
basic properties of the Brownian snake. More information
can be found in the monograph \cite{Zu}.
The (one-dimensional) Brownian snake is a Markov process taking values
in the space $\W$ of finite paths in $\R$. Here a finite path is simply
a continuous mapping $\w:[0,\zeta]\la \R$, where
$\zeta=\zeta_{(\w)}$ is a nonnegative real number called the
lifetime of $\w$. The set $\W$ is a Polish space when equipped with the
distance
$$d(\w,\w')=|\zeta_{(\w)}-\zeta_{(\w')}|+\sup_{t\geq 0}|\w(t\wedge
\zeta_{(\w)})-\w'(t\wedge\zeta_{(\w')})|.$$
The endpoint (or tip) of the path $\w$ is denoted by $\wh \w=\w(\zeta_{(\w)})$.
Let
$\Omega:=C(\R_+,\W)$ be the space of all continuous functions from
$\R_+$ into $\W$, which is equipped with the topology
of uniform convergence on every compact subset of $\R_+$.
The canonical process on $\Omega$ is then denoted by
$W_s(\omega)=\omega(s)$ for $\omega\in\Omega\;,$
and we write $\zeta_s=\zeta_{(W_s)}$ for the lifetime of $W_s$.
Let $\w\in\W$. The law of the Brownian snake started from $\w$
is the probability measure $\P_\w$ on $\Omega$ which can be
characterized as follows. First, the process $(\zeta_s)_{s\geq 0}$
is under $\P_\w$ a reflected Brownian motion in $[0,\infty[$ started
from $\zeta_{(\w)}$. Secondly, the conditional distribution
of $(W_s)_{s\geq 0}$ knowing $(\zeta_{s})_{s\geq 0}$, which is
denoted by $\Theta^\zeta_\w$, is characterized by the
following properties:
\begin{description}
\item{(i)} $W_0=\w$, $\Theta^\zeta_\w$ a.s.
\item{(ii)} The process $(W_s)_{s\geq 0}$ is time-inhomogeneous
Markov under $\Theta^\zeta_\w$. Moreover, if $0\leq s\leq
s'$,
\begin{description}
\item{$\bullet$} $W_{s'}(t)=W_{s}(t)$ for every $t\leq m(s,s'):=
\inf_{[s,s']}\zeta_r$, \ $\Theta^\zeta_\w$ a.s.
\item{$\bullet$} Under
$\Theta^\zeta_\w$, $(W_{s'}(m(s,s')+t)-W_{s'}(m(s,s')))_{0\leq t\leq
\zeta_{s'}- m(s,s')}$ is independent of $W_s$ and distributed as a
one-dimensional Brownian motion started at $0$.
\end{description}
\end{description}
Informally, the value $W_s$ of the Brownian snake at time $s$
is a random path with a random lifetime $\zeta_s$ evolving like
reflecting Brownian motion in $[0,\infty[$. When $\zeta_s$ decreases,
the path is erased from its tip, and when $\zeta_s$ increases, the path
is extended by adding ``little pieces'' of Brownian paths at its tip.
We denote by $n(de)$ the It\^o measure of positive Brownian excursions,
which is a $\sigma$-finite measure on the space $C(\R_+,\R_+)$,
and we write
$$\sigma(e)=\inf\{s> 0:e(s)=0\}$$
for the duration of excursion $e$.
For $s>0$, $n_{(s)}$ will denote the conditioned measure $n(\cdot\mid\sigma=s)$. In particular
$n_{(1)}(de)$ is the law of the normalized excursion $\eg$, or more
precisely of $(\eg_{t\wedge 1})_{t\geq 0}$.
Our normalization of the excursion measure is fixed by the relation
\be
\label{decomItomeas}
n=\int_0^\infty {ds\over 2\sqrt{2\pi s^3}}\;n_{(s)},
\ee
and we have then $n(\sup_{s\geq 0}e(s)>\varepsilon)=(2\varepsilon)^{-1}$ for every $\varepsilon>0$.
If $x\in\R$, the excursion measure
$\N_x$ of the Brownian snake from $x$ is given by
$$\N_x=\int_{C(\R_+,\R_+)} n(de)\;\Theta^e_{\ov x}$$
where $\ov x$ denotes the trivial element of $\W$ with lifetime $0$
and initial point $x$. With a slight abuse of notation
we also write $\sigma(\omega)=\inf\{s>0:\zeta_s(\omega)=0\}$ for
$\omega\in\Omega$. We can then consider the conditioned measures
$$\N_x^{(s)}=\N_x(\cdot\mid \sigma=s)=
\int_{C(\R_+,\R_+)} n_{(s)}(de)\;\Theta^e_{\ov x}\;.$$
We can now relate the Brownian snake to the Brownian trees of
subsection 2.4: We may define the pair $(\eg,Z)$ under the probability measure
$\N^{(1)}_0$ by taking $\eg_s=\zeta_s$ and $Z_s=\wh W_s$, for every $0\leq s\leq 1$.
Furthermore the path $(W_s(t),0\leq t\leq \zeta_s)$ is then interpreted in terms
of the labels attached to the ancestors of $p_\eg(s)$: If $a=p_\eg(s)$ is a vertex of the
tree
$\t_\eg$, and
$c\in\llbracket \rho,a\rrbracket$ is the ancestor of $a$ at generation $t=d_\eg(\rho,c)$,
we have $Z_c=W_s(t)$. These identifications follow very easily from the
properties of the Brownian snake.
For future reference, we state a crude
bound on the increments of the process $(\wh W_s)_{s\geq 0}$ under $\N_0$. In a way analogous
to subsection 2.3 we set, for every $s,t\geq 0$,
$$d_\zeta(s,t)=\zeta_s+\zeta_t-2\inf_{s\wedge t\leq r\leq s\vee t} \zeta_r\;.$$
\begin{lemma}
\label{crudebound}
Let $b\in]0,1/2[$. Then $\N_0(d\omega)$ a.e. there exists $\varepsilon_0(\omega)>0$ such that
for every $s,t\geq 0$ with $d_\zeta(s,t)\leq \varepsilon_0$, one has
$$|\wh W_s-\wh W_t|\leq (d_\zeta(s,t))^b.$$
\end{lemma}
\proof Conditionally on $(\zeta_r)_{r\geq 0}$, the process $(\wh W_r)_{r\geq 0}$ is
Gaussian with mean $0$ and such that $E[(\wh W_s-\wh W_t)^2]=d_\zeta(s,t)$
for every $s,t\geq 0$. The bound of the lemma then follows
from standard chaining arguments. We leave details to the reader. \cq
\smallskip
\noindent{\bf Proof of Lemma \ref{estimateISE}:} Set
$$\un W=\inf_{s\geq 0} \wh W_s,$$
and, for every $\varepsilon>0$,
$${\cal J}(\varepsilon)=\int_0^\sigma ds\,{\bf 1}{\{\wh W_s-\un W\leq \varepsilon\}}.$$
Thanks to the remarks preceding Lemma \ref{crudebound}, the quantity $\ov{\cal I}([0,\varepsilon])$
in Lemma \ref{estimateISE} has the same distribution as ${\cal J}(\varepsilon)$
under $\N^{(1)}_0$. Therefore, the
statement of Lemma \ref{estimateISE} reduces to checking that
$$\N^{(1)}_0({\cal J}(\varepsilon)\geq \alpha \varepsilon^2)=o(\varepsilon^2)$$
as $\varepsilon\to 0$. From (\ref{decomItomeas}) and simple scaling arguments, it is enough to verify
that
\be
\label{estmISEtech}
\N_0({\cal J}(\varepsilon)\geq \alpha \varepsilon^2\,,\,\sigma>1/2)=o(\varepsilon^2)
\ee
as $\varepsilon\to 0$. For every $\delta>0$, we have
${\cal J}(\varepsilon)={\cal J}_\delta(\varepsilon)+{\cal J}'_\delta(\varepsilon)$,
where
$${\cal J}_\delta(\varepsilon)=\int_0^\sigma ds\,{\bf 1}{\{\wh W_s-\un W\leq \varepsilon,
\zeta_s<\delta\}}\ ,\
{\cal J}'_\delta(\varepsilon)=\int_0^\sigma ds\,{\bf 1}{\{\wh W_s-\un W\leq \varepsilon,
\zeta_s\geq\delta\}}.
$$
Let us fix
$\beta>0$. By Lemma 3.2 in \cite{LGW}, we can choose $\delta>0$ small enough so that,
for every $\varepsilon\in]0,1[$,
$$\N_0\Big({\bf 1}_{\{\sigma >1/2\}}\,{\cal J}_\delta(\varepsilon)\Big)\leq \beta\,\varepsilon^4.$$
On the other hand, Lemma 3.3 in \cite{LGW} yields the existence of a constant $K_\delta$
such that, for every $\varepsilon\in]0,1[$,
$$\N_0\Big(({\cal J}'_\delta(\varepsilon))^2\Big)\leq K_\delta\,\varepsilon^8.$$
Then,
\ba
\N_0({\cal J}(\varepsilon)\geq \alpha \varepsilon^2\,,\,\sigma>1/2)&\leq&
\N_0({\cal J}_\delta(\varepsilon)\geq \frac{\alpha}{2}\,\varepsilon^2\,,\,\sigma>1/2)
+\N_0({\cal J}'_\delta(\varepsilon)\geq \frac{\alpha}{2}\,\varepsilon^2)\\
&\leq& \frac{2}{\alpha \varepsilon^2}
\N_0\Big({\bf 1}_{\{\sigma >1/2\}}\,{\cal J}_\delta(\varepsilon)\Big)
+\frac{4}{\alpha^2\varepsilon^4}\,\N_0\Big(({\cal J}'_\delta(\varepsilon))^2\Big)\\
&\leq& \frac{2\beta}{\alpha}\,\varepsilon^2+\frac{4 K_\delta}{\alpha^2}\,\varepsilon^4.
\ea
It follows that
$$\limsup_{\varepsilon\to 0} \varepsilon^{-2}\N_0({\cal J}(\varepsilon)\geq \alpha
\varepsilon^2\,,\,\sigma>1/2)\leq \frac{2\beta}{\alpha}.$$
Since $\beta$ was arbitrary, this completes the proof of (\ref{estmISEtech}) and of the lemma. \cq
\medskip
\noindent{\bf Proof of Lemma \ref{increasepoint}:}
We first explain why it is enough to prove the statement concerning
the pair $(\eg,Z)$. This follows from a re-rooting argument. Recall
the notation of subsection 2.4. For every fixed $s\in[0,1[$, set
\begin{description}
\item{$\bullet$}
$\displaystyle{\eg^{[s]}_t=\eg_{s}+\eg_{s\oplus t}-2\,m_{\eg}(s,
{s\oplus t})}$;
\item{$\bullet$} $Z^{[s]}_t=Z_{s\oplus t} -Z_{s}$,
\end{description}
for every $t\in[0,1]$. By construction, $(\ov\eg,\ov Z)=(\eg^{[s_*]},Z^{[s_*]})$.
Also, $(\eg^{[s]},Z^{[s]})\build{=}_{}^{\rm(d)} (\eg,Z)$ for every fixed
$s\in[0,1[$: See Proposition 4.9 in \cite{MaMo} or
Theorem 2.3 in \cite{LGW}. Hence, if $U$ is uniformly distributed
over $[0,1[$ and independent of $(\eg,Z)$, we have also
$(\eg^{[U]},Z^{[U]})\build{=}_{}^{\rm(d)} (\eg,Z)$.
Suppose there exists an increase point $r\in]0,1[$ of the pair
$(\ov\eg,\ov Z)=(\eg^{[s_*]},Z^{[s_*]})$. Then for every
$s$ sufficiently close to $s_*$, $r+s_*-s$ will be an increase point
of the pair $(\eg^{[s]},Z^{[s]})$ (this can be verified by
direct inspection of the formulas defining the pair
$(\eg^{[s]},Z^{[s]})$, keeping in mind that $s_*$ corresponds to
a leaf of the tree $\t_{\eg}$, so that immediately after
or immediately before $s_*$, $\eg_s$ takes values strictly less
than $\eg_{s_*}$). In particular, the pair $(\eg^{[U]},Z^{[U]})$
will have an increase point with positive probability, which contradicts the
first assertion of the lemma.
Let us now prove the statement concerning
the pair $(\eg,Z)$. In terms of the Brownian snake, we need
to check that $\N^{(1)}_0$ a.s. the pair $(\zeta_s,\wh W_s)_{0\leq s\leq 1}$
has no increase point. By a simple scaling argument, it is enough to
verify that the same property holds for the pair $(\zeta_s,\wh W_s)_{0\leq s\leq \sigma}$
under the excursion measure $\N_0$ (obviously time $1$ is now replaced by $\sigma$
in the definition of an increase point). To this end, we will use the
following lemma.
\begin{lemma}
\label{increaselemma}
Let $\delta>0$. Let $\w\in{\cal W}$ with $\w(0)=0$ and $\zeta_{(\w)}=a>0$, and
let $\varepsilon\in]0,a]$. Consider the
stopping times
\ba
&&T=\inf\{s\geq 0:\zeta_s=a+\delta\},\\
&&T'=\inf\{s\geq 0:\zeta_s=a-\varepsilon\}.
\ea
On the event $\{T<T'\}$, we also define
$$L=\sup\{s\leq T:\zeta_s=a\}.$$
Then there exists a constant $C_\delta$, which only depends on $\delta$, such that,
for every $\eta\in]0,1]$,
$$\P_\w(T<T'\hbox{ and }\wh W_s>\wh W_L-\eta\hbox{ for every }s\in[L,T])\leq
C_\delta\,\varepsilon\,\eta^3.$$
\end{lemma}
\noindent{\bf Remark.} The exponent $3$ in $\eta^3$ is sharp and related to the
fact that the bound of the lemma is a ``one-sided'' estimate. This should be
compared with the exponent $4$ that appears in similar two-sided estimates
derived in \cite{LGW}.
\proof Under $\P_\w$, $(\zeta_s)_{s\geq 0}$ is distributed as a reflected linear
Brownian motion started from $a$. In particular,
$$\P_\w(T<T')=\frac{\varepsilon}{\varepsilon+\delta}.$$
Moreover, from standard connections between linear Brownian motion and
the three-dimen\-sional Bessel process, we know that under the conditional
probability $\P_\w(\cdot\mid T<T')$, the shifted process
$$Y_s:=\zeta_{(L+s)\wedge T}-a\ ,\ s\geq 0$$
is distributed as a three-dimensional Bessel process started from $0$ and stopped
when it first hits $\delta$. At this point, it is convenient to introduce the
future infimum process of $Y$,
$$J_s:=\inf_{r\geq s} Y_r\ ,\ s\geq 0$$
and the excursions of $Y-J$ away from $0$: Let $]\alpha_i,\beta_i[$, $i\in I$, be the connected
components of the open set $\{s\geq 0:Y_s>J_s\}$, and for every $i\in I$ set
$$e_i(s)=Y_{(\alpha_i+s)\wedge \beta_i}-Y_{\alpha_i}.$$
Then the point measure
$$\sum_{i\in I} \delta_{(Y_{\alpha_i},e_i)}(dt\,de)$$
is Poisson with intensity
$$2 \,{\bf 1}_{[0,\delta]}(t)\,dt\;n\Big(de \cap\Big\{\sup_{s\geq 0}e(s)<\delta-t\Big\}\Big).$$
The last property follows from standard facts of excursion theory. See e.g. Lemma 1
in \cite{AW} for a detailed derivation.
We can then combine the preceding excursion decomposition of the paths of $Y$
with the spatial displacements of the Brownian snake, in a way similar
to the proof of Lemma V.5 in \cite{Zu}. Let $H=\sup_{s\geq 0}\zeta_s$
denote the maximum of the lifetime process. It follows that
\begin{eqnarray}
\label{increasetech1}
&&\P_\w(T<T'\hbox{ and }\wh W_s>\wh W_L-\eta\hbox{ for every }s\in[L,T])\nonumber\\
&&\quad = \frac{\varepsilon}{\varepsilon+\delta}\,
E_0\Big[{\bf 1}_{\{\xi[0,\delta]\subset]-\eta,\infty[\}}\,\exp\Big(-2\int_0^\delta dt
\,\N_{\xi_t}(H<\delta-t,\un W\leq -\eta)\Big)\Big]
\end{eqnarray}
where $(\xi_t)_{t\geq 0}$ is a linear Brownian motion started from $x$ under the
probability measure $P_x$, and we use the notation $\xi[0,\delta]$ for the range of
$\xi$ over the time interval $[0,\delta]$.
From this point, the argument is very similar to the end of the proof of Proposition 4.2 in \cite{LGW},
to which we refer the reader for more details. For every
$x>0$, we set
$$f(x)=\N_0(\un W>-x\mid H=1)$$
and
$$G(x)=4\int_0^x u(1-f(u))\,du.$$
Note that $G(+\infty)=6$ (see Section 4 in \cite{LGW}).
By conditioning with respect to $H$ and then using a scaling argument, we get
$$\int_0^\delta dt\,\N_{\xi_t}(H<\delta-t,\un W\leq -\eta)
=\int_0^\delta dt\int_0^{\delta-t} {du\over 2 u^2}\,(1-f(\frac{\xi_t+\eta}{\sqrt{u}})).$$
Hence, the right-hand side of (\ref{increasetech1}) can be written as
\ba
&&\frac{\varepsilon}{\varepsilon+\delta}\,
E_0\Big[{\bf 1}_{\{\xi[0,\delta]\subset]-\eta,\infty[\}}\,\exp\Big(-\int_0^\delta dt
\int_0^{\delta-t} {du\over u^2}\,(1-f(\frac{\xi_t+\eta}{\sqrt{u}}))\Big)\Big]\\
&&=\frac{\varepsilon}{\varepsilon+\delta}\,
E_\eta\Big[{\bf 1}_{\{\xi[0,\delta]\subset]0,\infty[\}}\,\exp\Big(-\int_0^\delta dt
\int_0^{\delta-t} {du\over u^2}\,(1-f(\frac{\xi_t}{\sqrt{u}}))\Big)\Big].
\ea
From the definition of $G$, the property $G(+\infty)=6$ and a change of variables, we have
$$\int_0^{\delta-t} {du\over u^2}\,(1-f(\frac{\xi_t}{\sqrt{u}}))
=(\xi_t)^{-2} \Big(3-\frac{1}{2}G(\frac{\xi_t}{\sqrt{\delta-t}})\Big).$$
Hence we get
\begin{eqnarray}
\label{increasetech2}
&&\P_\w(T<T'\hbox{ and }\wh W_s>\wh W_L-\eta\hbox{ for every }s\in[L,T])\nonumber\\
&&\quad =\frac{\varepsilon}{\varepsilon+\delta}\,
E_\eta\Big[{\bf 1}_{\{\xi[0,\delta]\subset]0,\infty[\}}\,\exp\Big(-3\int_0^\delta \frac{dt}{\xi_t^2}
+\frac{1}{2}\int_0^\delta \frac{dt}{\xi_t^2}\,
G(\frac{\xi_t}{\sqrt{\delta-t}})\Big)\Big].
\end{eqnarray}
Proposition 2.6 of \cite{LGW}, which reformulates absolute continuity relations between
Bessel processes due to Yor, implies that the right-hand side of (\ref{increasetech2})
is equal to
$$\frac{\varepsilon}{\varepsilon+\delta}\,\eta^3\,E^{(7)}_\eta\Big[
(R_\delta)^{-3}\,\exp\Big(\frac{1}{2}\int_0^\delta \frac{dt}{R_t^2}\,G(\frac{R_t}{\sqrt{\delta-t}})\Big)
\Big],$$
where $(R_t)_{t\geq 0}$ is a Bessel process of dimension $7$ started from $\eta$
under the probability measure $P^{(7)}_\eta$. Finally, we can argue as in the end
of the proof of Proposition 4.2 in \cite{LGW} to verify the existence of a
constant $C'_\delta$ such that, for every $\eta>0$,
$$E^{(7)}_\eta\Big[
(R_\delta)^{-3}\,\exp\Big(\frac{1}{2}\int_0^\delta \frac{dt}{R_t^2}\,G(\frac{R_t}{\sqrt{\delta-t}})\Big)
\Big]
\leq C'_\delta.$$
Lemma \ref{increaselemma} follows with $C_\delta=\delta^{-1}C'_\delta$. \cq
\smallskip
We come back to the proof of Lemma \ref{increasepoint}. We fix $\delta\in]0,1[$. For every
$\varepsilon\in]0,1[$, we introduce the sequence of stopping times defined
inductively by
$$T^\varepsilon_0=0\ ,\
T^\varepsilon_{i+1}=\inf\{s>T^\varepsilon_i:|\zeta_s-\zeta_{T^\varepsilon_i}|=\varepsilon\},$$
with the usual convention $\inf\varnothing=\infty$. For every index $i$ such that $T^\varepsilon_i<\infty$,
we also set
\ba
&&S^\varepsilon_i=\inf\{s>T^\varepsilon_i:\zeta_s=\zeta_{T^\varepsilon_i}+\delta\},\\
&&\wt T^\varepsilon_i=\inf\{s>T^\varepsilon_i:\zeta_s=\zeta_{T^\varepsilon_i}-\varepsilon\}.
\ea
On the event $\{T^\varepsilon_i=\infty\}$ simply set $S^\varepsilon_i=\wt T^\varepsilon_i=\infty$.
Finally, on the event $\{T^\varepsilon_i<\infty\}\cap \{S^\varepsilon_i<\infty\}$, we put
$$L^\varepsilon_i=\sup\{s<S^\varepsilon_i:\zeta_s=\zeta_{T^\varepsilon_i}\}.$$
Fix $A>0$, and let ${\cal A}_{\varepsilon,\eta}$ be the event that there exists
$i\geq 1$ such that $T^\varepsilon_i<\infty$, $\zeta_{T^\varepsilon_i}\in]0,A]$, $S^\varepsilon_i<\wt
T^\varepsilon_i<\infty$ and
$$\wh W_s> \wh W_{L^\varepsilon_i}-\eta$$
for every $s\in[L^\varepsilon_i,S^\varepsilon_i]$.
From Lemma \ref{increaselemma} and the strong Markov property for
the Brownian snake, we have
\ba
\N_0({\cal A}_{\varepsilon,\eta})
\!\!\!&\leq&\!\!\!\N_0\Big(\!\sum_{i=1}^\infty {\bf
1}{\{T^\varepsilon_i<\infty,\zeta_{T^\varepsilon_i}\in]0,A]\}}
\,{\bf 1}{\{S^\varepsilon_i<\wt
T^\varepsilon_i<\infty\}}\,{\bf 1}{\{\wh W_s>\wh W_{L^\varepsilon_i}-\eta\,,\,\forall s
\in[L^\varepsilon_i,S^\varepsilon_i]\}}\!\Big)\\
\!\!\!&\leq&\!\!\! C_\delta \varepsilon\,\eta^3\,\N_0\Big(
\sum_{i=1}^\infty {\bf 1}{\{T^\varepsilon_i<\infty,\zeta_{T^\varepsilon_i}\in]0,A]\}}\Big).
\ea
Standard properties of linear Brownian motion give
$$\N_0\Big(\sum_{i=1}^\infty {\bf 1}{\{T^\varepsilon_i<\infty,\zeta_{T^\varepsilon_i}\in]0,A]\}}\Big)
=\frac{1}{\varepsilon}\,\lfloor\frac{A}{\varepsilon}\rfloor.$$
Therefore we have obtained the bound
$$\N_0({\cal A}_{\varepsilon,\eta})\leq C_\delta A\,\varepsilon^{-1}\eta^3.$$
We apply this estimate with $\varepsilon=\varepsilon_p=2^{-p}$, for every integer $p\geq 1$,
and $\eta=(\varepsilon_p)^b$, where $b\in]\frac{1}{3},\frac{1}{2}[$. It follows that
$\N_0$ a.e. for all $p$ sufficiently large the event ${\cal A}_{\varepsilon_p,(\varepsilon_p)^b}$
does not occur.
To complete the argument, notice that it is enough to prove that there cannot exist $r>0$ such that
$\inf\{u\geq r:\zeta_u=\zeta_r+2\delta\}<\infty$ and
$$\zeta_s\geq \zeta_r\hbox{ and }\wh W_s\geq \wh W_r\;,\ \hbox{for every } s\in[r,\inf\{u\geq
r:\zeta_u=\zeta_r+2\delta\}].$$
We argue by contradiction and suppose that there is such a value of $r$. Let $i\geq 1$
be such that $r\in]T^{\varepsilon_p}_{i-1},T^{\varepsilon_p}_i]$. If $p$ has been taken
large enough, we have $S^{\varepsilon_p}_i<\wt T^{\varepsilon_p}_i\wedge
\inf\{u\geq r:\zeta_u=\zeta_r+2\delta\}$, and
for every $s\in[T^{\varepsilon_p}_i,S^{\varepsilon_p}_i]$,
$$\wh W_s\geq \wh W_r > W_{L^{\varepsilon_p}_i}-(\varepsilon_p)^b\;,$$
where the last inequality follows from Lemma \ref{crudebound} since
$d_\zeta(r,L^{\varepsilon_p}_i)< \varepsilon_p$. We thus get a contradiction with the
fact that ${\cal A}_{\varepsilon_p,(\varepsilon_p)^b}$ does not occur when $p$
is large. This contradiction completes the proof. \cq
\medskip
\noindent{\bf Proof of Lemma \ref{occuptree}:} We first observe that it
is enough to prove the statement of Lemma \ref{occuptree} when the pair $(\t_{\ov\eg},\ov Z)$ is replaced
by $(\t_\eg,Z)$, and of course $\ov\rho$ is also replaced by the root $\rho$
of $\t_\eg$. This follows from a re-rooting argument analogous to the one we used at
the beginning of the proof of Lemma \ref{increasepoint}. Let us only sketch
the argument. We assume that the property of Lemma \ref{occuptree} has been derived
when the pair $(\t_{\ov\eg},\ov Z)$ is replaced
by $(\t_\eg,Z)$. Suppose that the
conclusion of this lemma fails for some subtree of $\t_{\ov\eg}$. Then it
will also fail for some subtree of the re-rooted tree $\t_{\eg^{[s]}}$, provided
that $s$ is sufficiently close to $s_*$. Hence with positive probability it will
fail for some subtree of $\t_{\eg^{[U]}}$, where $U$ is uniformly distributed
over $[0,1]$. Since we saw that $(\eg^{[U]},Z^{[U]})\build{=}_{}^{\rm(d)} (\eg,Z)$,
this leads to a contradiction.
Then, we notice that by a symmetry
argument we need only consider subtrees of $\t_\eg$ from the right side of $\llbracket\rho,a\rrbracket$.
Furthermore, as we already observed, the pair
$(\eg_s,Z_s)_{0\leq s\leq 1}$ has the same distribution as $(\zeta_s,\wh W_s)_{0\leq s\leq 1}$ under
$\N^{(1)}_0$. By scaling, it is then enough to prove that the analogue of Lemma \ref{occuptree} holds for
the pair $(\zeta_s,\wh W_s)_{0\leq s\leq \sigma}$ under $\N_0$. We can thus reformulate
the desired property in the following way. Let us fix $s>0$, and argue on the event $\{s<\sigma\}$.
Denote by $a=p_\zeta(s)$ the vertex corresponding to $s$ in the tree $\t_\zeta$.
The subtrees of $\t_\zeta$ from the right side of $\llbracket \rho,a\rrbracket$
exactly correspond to the excursions of the shifted process $(\zeta_{s+r})_{r\geq 0}$
above its past minimum process. More precisely, set
$$\zeta^{(s)}_r=\zeta_{s+r}\ ,\ \check\zeta^{(s)}_r=\inf_{0\leq u\leq r}\zeta^{(s)}_u$$
for every $r\geq 0$. Denote by $]\alpha_i,\beta_i[$, $i\in I$, the connected components
of the open set $\{r\geq 0:\zeta^{(s)}_r>\check\zeta^{(s)}_r\}$. Then for each
$i\in I$, the set $\t^1_i:=p_\zeta([s+\alpha_i,s+\beta_i])$ is a subtree of $\t_\zeta$
from the right side of $\llbracket \rho,a\rrbracket$ with root
$p_\zeta(s+\alpha_i)=p_\zeta(s+\beta_i)$, and conversely all subtrees
from the right side of $\llbracket \rho,a\rrbracket$ are obtained in this way.
Recall the interpretation of the path $(W_{s+r}(t),0\leq t\leq \zeta_{s+r})$ as giving the
labels of the ancestors of the vertex $p_\zeta(s+r)$ in the tree $\t_\zeta$.
In order to get the statement of Lemma \ref{occuptree}, it is enough to
prove the following claim.
\smallskip
\noindent{\bf Claim}. {\it $\N_0$ a.e. on the event $\{s<\sigma\}$, for every $\mu>0$ and every
$i\in I$ such that
\be
\label{occuptech0}
\inf_{\alpha_i\leq r\leq \beta_i} \wh W_{s+r} < \wh W_{s+\alpha_i} -\mu
\ee
we have
\be
\label{occuptech1}
\liminf_{\varepsilon\to 0}
\varepsilon^{-2}\!\! \int_{\alpha_i}^{\beta_i} dr\,
{\bf 1}{\{\wh W_{s+r}\leq \wh W_{s+\alpha_i}-\mu+\varepsilon\}}
{\bf 1}{\{W_{s+r}(t)\geq \wh W_{s+\alpha_i}-\mu+\frac{\varepsilon}{8},
\forall t\in[\zeta_{s+\alpha_i},\zeta_{s+r}]\}}>0.
\ee}
Note that the preceding claim is concerned with subtrees from the right side of one particular vertex
$a=p_\zeta(s)$, whereas the statement of Lemma \ref{occuptree} holds simultaneously for all
choices of the vertex $a$. However, assuming that the claim is proved, it immediately follows that
the desired property holds for all subtrees from the left side of $\llbracket \rho, p_\zeta(s)\rrbracket$,
for all rational numbers $s>0$, outside a single set of zero $\N_0$-measure. Since a subtree
from the right side of $\llbracket \rho,p_\zeta(s)\rrbracket$ is also a subtree from the right side of
$\llbracket \rho,p_\zeta(s')\rrbracket$ as soon as $s'$ is close enough to $s$, we then get the desired
result simultaneously for all choices of $a=p_\zeta(s)$.
\smallskip
Let us now discuss the proof of the claim. Recall that $s>0$ is fixed and that we
argue on the event $\{s<\sigma\}$. For every $i\in I$ and every $r\geq 0$ set
\ba
&&\zeta^i_r=\zeta_{(s+\alpha_i+r)\wedge (s+\beta_i)}-\zeta_{s+\alpha_i}\\
&&W^i_r(t)=W_{(s+\alpha_i+r)\wedge (s+\beta_i)}(\zeta_{s+ \alpha_i}+t)-\wh W_{s+\alpha_i}\ ,\
\hbox{ for every }t\in[0,\zeta^i_r]
\ea
and view $W^i_r$ as a finite path with lifetime $\zeta^i_r$, so that $W^i=(W^i_r)_{r\geq 0}$
is a random element of $\Omega=C(\R_+,{\cal W})$. Also set
$\sigma_i=\beta_i-\alpha_i$, which corresponds to the duration of
the ``excursion'' $\zeta^i$. By combining the Markov property at time
$s$ with Lemma V.5 in \cite{Zu}, we get that under the probability
measure $\N_0(\cdot\mid s<\sigma)$ and conditionally on $W_s$, the point measure
$$\sum_{i\in I} \delta_{W^i}(d\omega)$$
is Poisson on $\Omega$ with intensity
$2\,\zeta_s\,\N_{0}(d\omega)$.
Now observe that condition (\ref{occuptech0}) reduces to
$$\inf_{r\geq 0} \wh W^i_r<-\mu$$
and that the integral in (\ref{occuptech1}) is equal to
$$\int_{0}^{\sigma_i} dr\,
{\bf 1}{\{\wh W^i_{r}\leq -\mu+\varepsilon\}}\,
{\bf 1}{\{W^i_{r}(t)\geq -\mu+\frac{\varepsilon}{8}\;,\;
\forall t\in[0,\zeta^i_{r}]\}}.$$
Thanks to these observations and to our previous description of the conditional
distribution of the point measure $\sum_{i\in I} \delta_{W^i}$, we see that our claim
follows from the next lemma.
\begin{lemma}
\label{occuplast}
$\N_0$ a.e. for every $\mu\in]0,-\un W[$, we have
$$\liminf_{\varepsilon\to 0}
\varepsilon^{-2} \int_{0}^{\sigma} dr\,
{\bf 1}{\{\wh W_{r}\leq -\mu+\varepsilon\}}\;
{\bf 1}{\{W_{r}(t)\geq -\mu+\frac{\varepsilon}{8}\;,\;
\forall t\in[0,\zeta_{r}]\}}>0.$$
\end{lemma}
\noindent{\bf Proof of Lemma \ref{occuplast}.} We fix an integer $N\geq 2$. Without
loss of generality, we may and will
restrict our attention to values $\mu\in[N^{-1},N]$.
We also consider another integer $n\geq N$. If $j$ is
the integer such that $(j-1)2^{-n-3}< \mu \leq j2^{-n-3}$, and
if $2^{-n-1}\leq \varepsilon\leq 2^{-n}$,
we have the following simple
inequalities:
\ba
&&{\bf 1}{\{\wh W_{r}\leq -\mu+\varepsilon\}}\;
{\bf 1}{\{W_{r}(t)\geq -\mu+\frac{\varepsilon}{8}\;,\;
\forall t\in[0,\zeta_{r}]\}}\\
&&\qquad\geq {\bf 1}{\{\wh W_{r}\leq -\mu+2^{-n-1}\}}\;
{\bf 1}{\{W_{r}(t)\geq -\mu+2^{-n-3}\;,\;
\forall t\in[0,\zeta_{r}]\}}\\
\noalign{\smallskip}
&&\qquad\geq {\bf 1}{\{\wh W_{r}\leq -j2^{-n-3}+2^{-n-1}\}}\;
{\bf 1}{\{W_{r}(t)\geq -j2^{-n-3}+2^{-n-2}\;,\;
\forall t\in[0,\zeta_{r}]\}}.
\ea
So, for every integer $j$ such that $N^{-1}2^{n+3}\leq j \leq N 2^{n+3}$, we set
$$U_{n,j} =\int_0^\sigma dr\,{\bf 1}{\{\wh W_{r}\leq -j2^{-n-3}+2^{-n-1}\}}\;
{\bf 1}{\{W_{r}(t)\geq -j2^{-n-3}+2^{-n-2}\;,\;
\forall t\in[0,\zeta_{r}]\}}.$$
For every $r>0$, denote by $L^r$ the total mass of the exit measure
of the Brownian snake from
the open set $]-r,\infty[$ (see e.g. Chapter 6 of \cite{Zu} for the definition
and main properties of exit measures). Note that $\{\un W<-r\}=\{L^r>0\}$,
$\N_0$ a.e. Put
$$r_{n,j}:=-j2^{-n-3}+2^{-n-1}\leq -N^{-1}/2<0$$
to simplify notation. By the special Markov property (cf Section 2.4 in \cite{LGW}),
conditionally on $\{L^{r_{n,j}}=\ell\}$, the variable $U_{n,j}$ is distributed as
$$\int \n(d\omega)\,X_n(\omega)$$
where $\n$ is a Poisson point measure with intensity $\ell \N_0$, and
$$X_n=\int_0^\sigma dr\,{\bf 1}{\{\wh W_{r}\leq 0\}}\;
{\bf 1}{\{W_{r}(t)\geq -2^{-n-2}\;,\;
\forall t\in[0,\zeta_{r}]\}}.$$
From scaling properties of $\N_0$,
$$\int \n(d\omega)\,X_n(\omega)\build{=}_{}^{\rm(d)} 2^{-4n}\int \n_n(d\omega)\,X_0(\omega),$$
where $\n_n$ is a Poisson point measure with intensity $\ell 2^{2n}\N_0$. Note that the
quantity
$$2^{-2n}\int \n_n(d\omega)\,X_0(\omega)$$
is the mean of $2^{2n}$ independent nonnegative random variables distributed as
$\int \n(d\omega)\,X_0(\omega)$.
We can then use standard large deviations estimates for sums of i.i.d.
random variables to derive the following. If $\eta>0$ is fixed, we can find
two positive constants $\nu$ and $\kappa$ such that, for every $n$ large enough,
for every integer $j\geq N^{-1}2^{n+3}$,
$$\N_0(\{2^{2n}U_{n,j}\leq \nu\}\cap\{L^{r_{n,j}}\geq \eta\})\leq \exp(-\kappa
2^{2n})\,\N_0(L^{r_{n,j}}\geq \eta)\leq c_0\,\exp(-\kappa
2^{2n}),$$
where $c_0=\N_0(\un W\leq -N^{-1}/2)$ is a positive constant. In the last
inequality we use the fact that $\N_0(L^{r_{n,j}}\geq \eta)\leq \N_0(L^{r_{n,j}}>0)
= \N_0(\un W\leq r_{n,j})$. We can sum the preceding estimate over values
of $j\in [N^{-1}2^{n+3},N 2^{n+3}]$, and then use the Borel-Cantelli lemma to
get that $\N_0$ a.e. for all $n$ sufficiently large and all $j\in [N^{-1}2^{n+3},N 2^{n+3}]$ we have either
$L^{r_{n,j}}<\eta$ or $U_{n,j} > \nu \,2^{-2n}$.
Now recall the elementary inequalities of the beginning of the proof. It follows that
$\N_0$ a.e., for all $\mu\in[N^{-1},N]$ we have either
\begin{equation}
\label{lasttech1}
\inf_{r\in [-\mu,-(2N)^{-1}]\cap \Q} L^r \leq \eta
\ee
or, for $\varepsilon$ small enough,
\be
\label{lasttech2}
\int_{0}^{\sigma} dr\,
{\bf 1}{\{\wh W_{r}\leq -\mu+\varepsilon\}}\;
{\bf 1}{\{W_{r}(t)\geq -\mu+\frac{\varepsilon}{8}\;,\;
\forall t\in[0,\zeta_{r}]\}}\geq \nu \varepsilon^2.
\ee
A simple application of the special Markov property shows that under the
probability measure $\N_0(\cdot\mid \un W\leq (2N) ^{-1})$ the process
$(L^{-(2N) ^{-1}-a})_{a\geq 0}$ is a continuous-state branching process, hence
a Feller Markov process which is absorbed at the origin. Thus, for every $a>0$, we have
\be
\label{lasttech3}\inf_{r\in [-(2N)^{-1}-a,-(2N)^{-1}]\cap \Q} L^r >0,\qquad
\N_0\hbox{ a.e. on the event }\{\un W< -(2N)^{-1}-a\}.
\ee
We now take
$\eta=\eta_k=2^{-k}$, for every integer $k\geq 1$ (then $\nu=\nu_k$ also depends on $k$). If
$$\mu_k=\inf\{a\in[(2N)^{-1},\infty[\cap \Q: L^{-a}\leq \eta_k\}$$
condition (\ref{lasttech1}) fails for all $\mu\in[N^{-1},N\wedge \mu_k[$, and
so (\ref{lasttech2}) must hold for the same values of $\mu$. Since (\ref{lasttech3}) shows that
$\mu_k\uparrow -\un W$ as $k\uparrow\infty$, $\N_0$ a.e. on $\{\un W <-(2N) ^{-1}\}$, this completes
the proof of Lemma \ref{occuplast} and Lemma \ref{occuptree}. \cq
\section{Hausdorff dimension}
In this section we compute the Hausdorff dimension of the limiting metric space
appearing in Theorem \ref{main}. Although the metric $D$ is not known explicitly,
it turns out that we have enough information to determine this
Hausdorff dimension.
\begin{theorem}
\label{Hausdim}
We have a.s.
$${\rm dim}(\t_{\ov\eg}\,/\!\approx,D)=4.$$
\end{theorem}
\proof We first derive the upper bound ${\rm dim}(\t_{\ov\eg}\,/\!\approx,D)\leq 4$.
Recall that the process $(Z_t)_{t\in[0,1]}$ is Gaussian conditionally given $(\eg_t)_{t\geq 0}$,
and that the conditional second moment of $Z_t-Z_s$ is $m_\eg(s,t)$. Also recall that the
function $t\la \eg_t$ is a.s. H\"older continuous with exponent $\frac{1}{2}-\varepsilon$, for any
$\varepsilon>0$. From this fact and an application of the classical Kolmogorov lemma, we
get that the mapping $t\la Z_t$ is a.s. H\"older continuous with exponent $\frac{1}{4}-\varepsilon$, for
any
$\varepsilon\in]0,\frac{1}{4}[$. Clearly the same holds if $Z$ is replaced by $\ov Z$. Hence, if
$\varepsilon\in]0,\frac{1}{4}[$ is fixed, there exists a (random) constant $C_1$ such that, for every
$s,t\in[0,1]$,
$$|\ov Z_s-\ov Z_t|\leq C_1\,|s-t|^{\frac{1}{4}-\varepsilon}.$$
It immediately follows that, for every $s,t\in[0,1]$,
$$D^\circ(s,t)\leq 2C_1\,|s-t|^{\frac{1}{4}-\varepsilon}.$$
Since $D\leq D^\circ$, we see that the canonical projection from $[0,1]$
onto $[0,1]/\approx$ (equipped with the metric $D$) is H\"older continuous with exponent
$\frac{1}{4}-\varepsilon$. It follows that ${\rm dim}([0,1]\,/\!\approx,D)\leq
(\frac{1}{4}-\varepsilon)^{-1}$ and since $\varepsilon$ was arbitrary, we
get ${\rm dim}(\t_{\ov\eg}\,/\!\approx,D)={\rm dim}([0,1]\,/\!\approx,D)\leq 4$.
The proof of the corresponding lower bound requires the following lemma. Recall that
$\lambda$ denotes the uniform probability measure on $\t_{\ov\eg}$ (cf subsection 2.4). For every
$a\in\t_{\ov\eg}$ and every $\varepsilon>0$, we set
$B_D(a,\varepsilon)=\{b\in \t_{\ov\eg}:D(a,b)<\varepsilon\} $.
\begin{lemma}
\label{Hauslem}
There exists a constant $C$ such that, for every $r\in]0,1]$,
$$E\Big[\int_{\t_{\ov\eg}} \lambda(da)\,\lambda(B_D(a,r))\Big]\leq C\,r^4.$$
\end{lemma}
Assume that the result of the lemma holds, and fix $\varepsilon\in]0,1]$. From the bound
of the lemma, we get that, for every integer $k\geq 1$,
$$E[\lambda(\{a\in\t_{\ov\eg} : \lambda(B_D(a,2^{-k}))\geq 2^{-k(4-\varepsilon)}\})]
\leq C\,2^{-k\varepsilon}.$$
By summing this estimate over $k$, we obtain
$$\limsup_{k\to\infty} \frac{\lambda(B_D(a,2^{-k}))}{2^{-k(4-\varepsilon)}}\leq 1\ ,\quad
\lambda(da)\hbox{ a.e.,\ a.s.}$$
By standard density theorems for Hausdorff measures, this implies that
${\rm dim}(\t_{\ov\eg}\,/\!\approx,D)\geq 4-\varepsilon$, a.s., which completes the
proof of Theorem \ref{Hausdim}. It only remains to prove Lemma \ref{Hauslem}. \cq
\smallskip
\noindent{\bf Proof of Lemma \ref{Hauslem}:} We rely on the case $k=2$
of Proposition \ref{finite-margi}.
With the notation of this proposition, we have
\ba
E\Big[\int_{\t_{\ov\eg}} \lambda(da)\,\lambda(B_D(a,r))\Big]
&=&E\Big[\int_{\t_{\ov\eg}\times\t_{\ov\eg}} \lambda(da)\lambda(db)\,
{\bf 1}_{\{D(a,b)<r\}}\Big]\\
&=&P[D(Y^\infty_1,Y^\infty_2)<r]\\
&\leq&\liminf_{n\to \infty} P\Big[d_n(Y^n_1,Y^n_2)<(4p(p-1)/9)^{1/4} n^{1/4}r
\Big].
\ea
On the other hand, it follows from Proposition \ref{pointedmap} that
\ba
P\Big[d_n(Y^n_1,Y^n_2)<(4p(p-1)/9)^{1/4} n^{1/4}r
\Big]\!\!\!\!&=&\!\!\!\!E\Big[\frac{1}{(p-1)n+2}\,\#B_n(Y^1_n,(4p(p-1)/9)^{1/4}n^{1/4}r)\Big]\\
&\build{\la}_{n\to\infty}^{}&E[\ov{\cal I}([0,r])].
\ea
Therefore we have obtained the bound
$$E\Big[\int_{\t_{\ov\eg}} \lambda(da)\,\lambda(B_D(a,r))\Big]\leq E[\ov{\cal
I}([0,r])].$$
Recall the notation of the proof of Lemma \ref{estimateISE} in Section 5. We know
that
$\ov{\cal I}([0,r])$ has the same distribution as ${\cal J}(r)$ under
$\N^{(1)}_0$. Furthermore the estimates recalled in the proof of
Lemma \ref{estimateISE} imply that, for every $r\in]0,1]$,
$$\N_0\Big({\bf 1}_{\{\sigma>1/2\}}{\cal J}(r)\Big)\leq C'\,r^4$$
for a certain constant $C'$. A simple scaling argument then gives, with another constant $C$,
$$E[\ov{\cal
I}([0,r])]=\N^{(1)}_0({\cal J}(r))\leq C\,r^4.$$
This completes the proof of Lemma \ref{Hauslem}. \cq | 131,743 |
A cannabis farm worth up to £390,000 was busted by police who raided flats above a Stockton pub.
In total, 469 cannabis plants were found in multiple rooms at the property on the corner of Yarm Street and Yarm Lane in central Stockton on Saturday morning.
Police vans were parked outside The Jokers pub for most of the day on Saturday, while electrical engineers had to dig up the road to disconnect a dangerous electricity supply.
Officers believe the drugs had a street value of between £130,000 and £390,000.
No arrests have yet been made, and inquiries are ongoing.
T/Sergeant Mitch Baldwin, from Cleveland Police, was a big success and will prevent further crime in our area.”
A spokesperson for Northern Powergrid confirmed they were also tasked to the scene: "We were called at around 10am to Yarm Street in Stockton by Cleveland Police, and carried out a safe disconnection at a property.
"We were at the scene until around 3.15pm."
Stockton Neighbourhood Police Team officers had posted a video inside the property to social media on Saturday morning.
In it, a long corridor was filled with large black plastic bags, and three rooms off the hallway are covered with sheeting.
Inside, curtained off rooms are filled with what looks like cannabis plants.
Police Dog Elsa also helped officers who were searching for suspects potentially still at the scene.
The team said it had responded to a tip-off from the public.
"Officers responded to reports from the community and located the farm in upstairs flats," it said.
"The condition of the electricity supply was particularly dangerous and posed a risk to neighbouring properties."
Anyone with further information should contact police on 101 or Crimestoppers anonymously on 0800 555 111.
Teesside Live has attempted to contact The Jokers about the drama. | 4,195 |
TITLE: Prove that $C_n < 4n^2$ for all n greater than or equal to 1
QUESTION [1 upvotes]: $C_1 = 0$, $C_n = C_{\lfloor n/2\rfloor} + n^2$ for all $n \ge 1$
Prove that $C_n < 4n^2$ for all $n \ge 1$.
I don't know how to even approach this. I remember something about inductive proofs...but i really don't understand that, could you please explain that to me?
REPLY [1 votes]: Why not unroll the recursion to get a precise answer?
Let $$n = \sum_{k=0}^{\lfloor \log_2 n \rfloor} d_k 2^k$$
be the binary representation of $n.$
This yields the exact formula
$$C_n = \sum_{k=0}^{\lfloor \log_2 n \rfloor - 1}
\left(\sum_{j=k}^{\lfloor \log_2 n \rfloor} d_j 2^{j-k}\right)^2$$
where $n\ge 2$ and zero otherwise.
Clearly this is maximized when all digits are equal to one, giving
$$C_n\le \sum_{k=0}^{\lfloor \log_2 n \rfloor - 1}
\left(\sum_{j=k}^{\lfloor \log_2 n \rfloor} 2^{j-k}\right)^2
= \frac{16}{3} 2^{2 \lfloor \log_2 n \rfloor}
- 8 \times 2^{\lfloor \log_2 n \rfloor} + \lfloor \log_2 n \rfloor
+ \frac{8}{3}. $$
For a string of ones we have $n=2^j-1$ for some $j$ with $j-1 = \lfloor \log_2 n \rfloor.$ This gives
$$2^{\lfloor \log_2 n \rfloor} = \frac{1}{2} (n+1)
\quad\text{or}\quad
\lfloor \log_2 n \rfloor = \log_2(n+1) - 1.$$
Hence we have the following bound on the leading term
$$ \frac{16}{3} 2^{2 \lfloor \log_2 n \rfloor}
= \frac{16}{3} 2^{2 \log_2(n+1) - 2}
= \frac{4}{3} (n+1)^2.$$
Furthermore recalling the condition on $n$ we have
$$- 8 \times 2^{\lfloor \log_2 n \rfloor} = -8 \times \frac{1}{2} (n+1).$$
Putting these pieces together we find that
$$C_n\le \frac{4}{3}n^2-\frac{4}{3}n + \lfloor \log_2 n \rfloor
< \frac{4}{3}n^2$$
because we certainly have $4/3 \times n > \log_2 n.$
To conclude, let us briefly consider the lower bound, which occurs when there is a one followed by a string of zeros, giving
$$C_n\ge \sum_{k=0}^{\lfloor \log_2 n \rfloor - 1}
\left(2^{\lfloor \log_2 n \rfloor - k}\right)^2
= \frac{4}{3} 2^{2\lfloor \log_2 n \rfloor} -\frac{4}{3}.$$
In this scenario we have $\lfloor \log_2 n \rfloor = \log_2 n,$ so that the lower bound gives $$C_n\ge \frac{4}{3} n^2 -\frac{4}{3}.$$
Note that the bounds from the two bit patterns are no longer directly comparable because the substitutions for $\lfloor \log_2 n \rfloor$ are different. What we may say, however, is that $$C_n \sim \frac{4}{3} n^2.$$
The coefficient on the leading term of the bounds in $\lfloor \log_2 n \rfloor$ fluctuates between $4/3$ and $16/3$ on each interval where $\lfloor \log_2 n \rfloor$ is constant. However when we reach the end of this interval $\lfloor \log_2 n \rfloor$ is off by almost one from the true value, reducing $16/3$ to $4/3$ when the bounds are expressed in $n.$
This link points to a series of similar calculations. | 15,639 |
TITLE: How to simplify $\frac{\partial^m}{\partial y_i^m}\mathrm{div }(A\nabla u({\bf x}({\bf y})))$
QUESTION [6 upvotes]: Let $\bf{x}\in\mathbb{R}^n$ (interesting in $n\in\{2,3\}$) and let $A=A_{n\times n}=\mathrm{diag}(a_1({\bf x}),\dots,a_n({\bf x}))$, that is
$$A_{2\times2}=\begin{bmatrix}
a_1(\bf{x})&0\\0&a_2(\bf{x})
\end{bmatrix}
\quad \text{or}\quad
A_{3\times3}=\begin{bmatrix}
a_1(\bf{x})&0&0\\0&a_2(\bf{x})&0\\
0&0& a_3(\bf{x})
\end{bmatrix} $$
Assume that $a_i\ne a_j, \forall 1\le i,j\le n$.
Let $\bf y$ be some curvilinear coordinates, e.g. elliptic, spherical, normal-tangential.
I'm interesting to differentiate $$\mathrm{L}u({\bf x}) =\{\mathrm{div }(A\nabla u)\}({\bf x})$$ with respect to $y_i$ , that is to express
$$\frac{\partial^m}{\partial y_i^m} \mathrm{L}u({\bf x}),\quad \text{interesting in no more then }0\le m\le4$$
explicitly but compact (by $m=0$ I mean just change of coordinates).
The expression, that I got so far (using scale factors\metrics), become too long and ugly. Say it is become pretty much unmanageable when I need to program it, which make it vulnerable for bugs and may affect numerical stability as well.
Example in 2D
Let $a_1=a, a_2=b$ one translates $$ \mathrm{L} u = \frac{\partial}{\partial x} \left( a u_x \right) + \frac{\partial}{\partial y} \left( b u_y \right) =
a_x u_x + a u_{xx} + b_y u_y + b u_{yy}
$$
to some (s,t) coordinates to get
$$
\alpha u_s + \beta u_t +\gamma u_{st}
+\delta u_{tt}
+\sigma u_{ss}
$$
and differentiate it, for an instant with respect to $s$ it to get
$$\begin{align}
& \alpha_s u_s +\alpha u_{ss} + \beta_s u_t + \beta u_{ts}
+\gamma_s u_{st}+\gamma u_{sst}
+\delta_s u_{tt}+\delta u_{tts}
+\sigma_s u_{ss}+\sigma u_{sss}\\
=& \alpha_s u_s + \beta_s u_t+
(\alpha +\sigma_s) u_{ss} +\delta_s u_{tt}+ (\beta +\gamma_s) u_{st}
+\gamma u_{sst}+\delta u_{tts}+\sigma u_{sss}
\end{align}$$
Any help will be appreciated.
REPLY [1 votes]: I represent my solution in some steps. As it may become too lengthy, I leave the proof of some parts to you. Also, in what follows, $u$ is a scalar, ${\bf{v}}$, ${\bf{a}}$, ${\bf{b}}$ are vectors, ${\bf{A}}$, ${\bf{B}}$ are second order tensors (or Matrices). Furthermore, you should notice the following definitions
$$\begin{array}{l}
{\bf{A}}:{\bf{B}} = {A_{ij}}{B_{ij}}\\
\nabla .{\bf{A}} = \frac{{\partial {A_{ij}}}}{{\partial {x_i}}}{{\bf{e}}_j}
\end{array}$$
where summation is implied over repeated indices. These definitions can be found in any standard tensor calculus book. In fact, you should be familiar with $:$, the scalar product of two second order tensors and with $\otimes $, the tensor product, and you can just derive the above formulas by direct computation.
Step 1. The following formula is valid
$$\begin{array}{l}
\nabla .\left( {{\bf{A}}.{\bf{v}}} \right) = \left( {\nabla .{\bf{A}}} \right).{\bf{v}} + {\bf{A}}:\nabla {\bf{v}}\\\end{array}\tag{1}$$
In this formula and what follows, you can put ${\bf{v}} = \nabla u$ to get what you want.
Step 2. Consider these formulas
$$\begin{array}{l}
{\nabla ^{\left( n \right)}}\left( {{\bf{a}}.{\bf{b}}} \right) = \sum\limits_{k = 0}^{n - 1} {\left( {\begin{array}{*{20}{c}}
{n - 1}\\
k
\end{array}} \right)\left( {{\nabla ^{\left( {n - k} \right)}}{\bf{a}}.{\nabla ^{\left( k \right)}}{\bf{b}} + {\nabla ^{\left( {n - k} \right)}}{\bf{b}}.{\nabla ^{\left( k \right)}}{\bf{a}}} \right)} ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,n \ge 1\\
{\nabla ^{\left( n \right)}}\left( {{\bf{A}}:{\bf{B}}} \right) = \sum\limits_{k = 0}^{n - 1} {\left( {\begin{array}{*{20}{c}}
{n - 1}\\
k
\end{array}} \right)\left( {{\nabla ^{\left( {n - k} \right)}}{\bf{A}}:{\nabla ^{\left( k \right)}}{\bf{B}} + {\nabla ^{\left( {n - k} \right)}}{\bf{B}}:{\nabla ^{\left( k \right)}}{\bf{A}}} \right)} ,\,\,\,\,\,\,\,\,n \ge 1\\
{\nabla ^{\left( n \right)}} = \underbrace {\nabla \nabla ...\nabla }_n
\end{array}\tag{2}$$
I proved theses formulas by induction. These are some generalization of the well known formula
$${\left( {fg} \right)^{(n)}} = \sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}{c}}
n\\
k
\end{array}} \right){f^{(n - k)}}{g^{(k)}}}$$
Step 3. Use $(1)$ and $(2)$ to obtian
$$\begin{array}{l}
{\nabla ^{\left( n \right)}}\left[ {\nabla .\left( {{\bf{A}}.{\bf{v}}} \right)} \right] = {\nabla ^{\left( n \right)}}\left[ {\left( {\nabla .{\bf{A}}} \right).{\bf{v}}} \right] + {\nabla ^{\left( n \right)}}\left[ {{\bf{A}}:\nabla {\bf{v}}} \right]\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \sum\limits_{k = 0}^{n - 1} {\left( {\begin{array}{*{20}{c}}
{n - 1}\\
k
\end{array}} \right)\left( {{\nabla ^{\left( {n - k} \right)}}\left( {\nabla .{\bf{A}}} \right).{\nabla ^{\left( k \right)}}{\bf{v}} + {\nabla ^{\left( {n - k} \right)}}{\bf{v}}.{\nabla ^{\left( k \right)}}\left( {\nabla .{\bf{A}}} \right)} \right)} \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \sum\limits_{k = 0}^{n - 1} {\left( {\begin{array}{*{20}{c}}
{n - 1}\\
k
\end{array}} \right)\left( {{\nabla ^{\left( {n - k} \right)}}{\bf{A}}:{\nabla ^{\left( k \right)}}\nabla {\bf{v}} + {\nabla ^{\left( {n - k} \right)}}\nabla {\bf{v}}:{\nabla ^{\left( k \right)}}{\bf{A}}} \right)}
\end{array}\tag{3}$$
and some clear simplifications will lead to
$${\nabla ^{\left( n \right)}}\left[ {\nabla .\left( {{\bf{A}}.{\bf{v}}} \right)} \right] = \sum\limits_{k = 0}^{n - 1} {\left( {\begin{array}{*{20}{c}}
{n - 1}\\
k
\end{array}} \right)\left( \begin{array}{l}
{\nabla ^{\left( {n - k} \right)}}\left( {\nabla .{\bf{A}}} \right).{\nabla ^{\left( k \right)}}{\bf{v}} + {\nabla ^{\left( {n - k} \right)}}{\bf{v}}.{\nabla ^{\left( k \right)}}\left( {\nabla .{\bf{A}}} \right)\\
+ {\nabla ^{\left( {n - k} \right)}}{\bf{A}}:{\nabla ^{\left( {k + 1} \right)}}{\bf{v}} + {\nabla ^{\left( {n - k + 1} \right)}}{\bf{v}}:{\nabla ^{\left( k \right)}}{\bf{A}}
\end{array} \right)}\tag{4}$$
Step 4. The last step is to apply the chain-rule. In all above formulas it was assumed that $\nabla = {\nabla _{\bf{x}}}$. It can be proved that
$$\begin{array}{*{20}{l}}
{{\nabla _{\bf{y}}}{\bf{v}} = {\nabla _{\bf{y}}}{\bf{x}}.{\nabla _{\bf{x}}}{\bf{v}}}\\
{{\nabla _{\bf{y}}}{\bf{A}} = {\nabla _{\bf{y}}}{\bf{x}}.{\nabla _{\bf{x}}}{\bf{A}}}
\end{array}\tag{5}$$
However, I didn't have time to go further. But you should derive some formulas based on $(5)$ to obtain something like
$$\nabla _{\bf{y}}^{(n)}\left[ {{\nabla _{\bf{y}}}.\left( {{\bf{A}}.{\bf{v}}} \right)} \right] = \left( {..?..} \right)\nabla _{\bf{x}}^{(n)}\left[ {{\nabla _{\bf{x}}}.\left( {{\bf{A}}.{\bf{v}}} \right)} \right]\tag{6}$$
and then combine $(6)$ with $(4)$ to derive the final result. But this won't be easy! For further help on this last step see this link. | 203,283 |
The coronavirus outbreak has seen small business owners and employees across the country make the transition to working from home. For the first few days, this change may have been exciting, confusing or a bit of a novelty for many. That said, it’s an important change to daily life that the whole country is making with one goal in mind: to protect the NHS and save lives.
Realising that working from home could last longer than planned, or become the new ‘normal’, people are looking to upgrade their home-office environments and make their work stations comfortable. So, we’ve rounded up some helpful tips that you could use to improve your posture and take a weight off your shoulders, literally. But, first, let’s explore why posture is important and how you can take better care of yourself.
Perfect posture doesn’t exist. That said, there are some general rules and guidelines that can help you work towards maintaining a good posture. For seated posture, this mainly means sitting in a way that doesn’t either put too much pressure on your joints or engage your muscles all of the time.
Good posture means that you’re taking care of your health by minimising the damage that can be caused to your body by sitting in the same space in an uncomfortable way, for probably much longer than you know that you should.
There’s a psychological element to posture, too. If you’re able to fully get ‘into the zone’ and concentrate on your work while forgetting about your surroundings, then you’re likely to be more productive and creative. This is much harder to achieve when you’re in pain or uncomfortable- and aware of it.
For the reasons above, here are our five posture tips that we hope will help you feel comfortable throughout your working day.
It may be difficult to purchase a brand new office chair with the surge in demand brought about by coronavirus. That said, the chair you sit on day-in, day-out has a huge impact on your posture. So, try to look for a chair with ergonomic qualities where possible- meaning that it’s designed to give you the support you need when seated, and that it’s adjustable.
Things to look out for include lumbar support (normally a small cushion that supports your lower back) and a cushioned seat. Ideally your chair can be raised or lowered so that you can adjust yourself to sit comfortably at a desk.
For a business owner, upgrading office chairs and equipment for your team can be a nice way of showing your staff that you’re looking out for them. And, who doesn’t want a more motivated team?
Loosely speaking, it’s healthy to shape your body into some right angles while you’re sitting at your desk. Having your calves at 90° to your thighs, your thighs at 90° to your body, and your forearms at 90° to your biceps, while typing, is desirable.
You can adjust your desk height to make sure that your hands are flat while you’re typing, which should help, too. Also, try to keep your eyes level with the top of your screen. This could help you avoid prolonged neck-tilt which can cause a lot of pain.
The NHS has a fantastic guide on this, and it’s worth noting that you shouldn’t try too hard to maintain an exact posture. Relaxing is a part of proper posture, so if you’re always tensing your muscles constantly to keep a certain seated position, you may be doing it wrong.
One of the more popular posture exercises for office workers is simply taking a break. Getting up and moving around for five minutes roughly once per hour could see you feeling looser and more relaxed.
As an employer, you can make sure that your employees don’t feel bad about doing this when in the office, too, by reassuring them that it’s okay to take breaks. If you’re struggling to think of a reason to take a break despite knowing that you probably should, then perhaps you could try making a coffee run, or ‘breaking out’ to work in a different environment for a while.
Have you heard of the 20-20-20 rule? The idea is simple; for every twenty minutes you spend looking at a screen, look twenty feet away, for twenty seconds.
How come? When you’re staring at a screen for long periods of time, you may often forget to blink. This could leave your eyes dehydrated, which can lead to eye-strain. The 20-20-20 rule is a great way of helping to avoid this, and standing up while doing it can even count as a mini-break to give your posture that extra little bit of care and attention.
If you find that you’ll struggle to remember to take a quick break that often, using a productivity app could be a great solution. By automating that reminder, you could find yourself building a habit that will protect your eyes from harm in the long run.
Just as with any muscle group in your body- making repeated actions over long periods of time without properly stretching can cause tension- which makes your muscles prone to damage. This is as true for someone working out at the gym as it is for someone sitting at home, typing on a laptop.
A tip for how to improve your posture, then, is to learn some fundamental stretches that can release tension in your hands and forearms. Adarsh Williams’ video is a great visual guide to how this can be done.
Bending your fingers back gently and curling your wrists are great examples of exercises to improve your posture. If you do these alongside taking regular breaks, you may be less likely to run into health complications such as repetitive strain injury or arthritis in the long run.
Finally, it’s worth mentioning that self-care is incredibly important. Share this knowledge with your colleagues and friends, and maybe you’ll help make their day-to-day lives more comfortable in a way they could not have predicted. We wish you all the best throughout the current crisis- keep washing your hands, stay safe and look after yourself. | 409,124 |
TITLE: On a formula of the norm of an element of a finite extension of a field
QUESTION [7 upvotes]: Theorem
Let $F$ be a field.
Let $K$ be a finite extension of $F$.
Let $[K : F]_i$ be the inseparable degree of $K/F$.
Let $\bar{K}$ be an algebraic closure of $K$.
Let $S$ be the set of $F$-embeddings of $K$ into $\bar{K}$.
Let $\alpha \in K$.
Then $N_{K/F}(\alpha) = (\prod_{\sigma \in S}\sigma(\alpha))^{[K : F]_i}$
This is proved in Theorem 60 in page 39 of the lecture note written by Pete L. Clark.
I don't understand the proof.
Would any one please enlighten me?
EDIT
Why down votes??
Asking for help to understand a proof should be frowned upon?
EDIT
I understand the prerequisites for the proof, i.e. the content of section 6 and Corollary 58.
EDIT
Related question 1,
Related question 2
EDIT
Let $f(X)$ be the characteristic polynomial of $\alpha$.
Let $g(X)$ be the minimal polynomial of $\alpha$.
By Corollary 58, $f(X)$ = $g(X)^{[K:F(\alpha)]}$.
The set {$\sigma(\alpha); \sigma ∈ S$} is the set of the roots of g(X).
However, it is not clear to me that the equation $N_{K/F}(\alpha) = (\prod_{\sigma \in S}\sigma(\alpha))^{[K : F]_i}$ follows immediately.
Would anyone please explain why the equation follows other than just saying it is straightforward?
EDIT
It's amazing that some people regard questioning a proof as an attack to the author's credibilty. Everybody makes a mistake. Even Grothendieck made a non-trivial mistake(see Misconceptions About $K_{X}$ by Kleiman).
I think this attitude is harmful to healthy development of mathematics.
I'm not claiming that the proof is wrong, though.
EDIT
To anyone who thinks the proof is straightforward, please explain to me in detail.
I am not as smart as you.
EDIT
It's surprising that no one explained the proof so far.
I believe every step of any proof can be reduced to (really) trivial statements.
If you think it's straightforward, please reduce it to more trivial statements that anybody who has basic knowledge of abstract algebra can understand.
REPLY [3 votes]: Well the theorem is a bit too hastily stated, for example if you happen to pick $\alpha \in F$, then $f(t) = (t-\alpha)^n$ and has only one distinct root.
The correct statement is :
Let $K/F$ be a field extension of degree $n < \infty$ and separable degree $m$. Let $\overline{K}$ be an algebraic closure of $K$. Let $\alpha \in K$ and let $f(t)$ be the characteristic polynomial of $\alpha \bullet \in End_F(K)$. Let $\sigma_1 \ldots \sigma_m$ be the distinct $F$-algebra embeddings of $K$ into $\overline{K}$. Then the factorisation of $f(t)$ on $\overline{K}$ is $f(t) = \prod_{i=1}^m (t- \sigma_i(\alpha))^{n/m}$.
First, let $K_0 = F(\alpha)$, $n_0 = [K_0 : F]$, $m_0$ be the separable degree of $[K_0 : F]$, $\tau_1 \ldots \tau_{m_0}$ be the embeddings of $K_0$ into $\overline{K}$, and $f_0(t)$ be the minimal polynomial of $\alpha$ over $F$.
Then, by corollary 58, $f(t) = f_0(t)^{[K : K_0]} = f_0(t)^{n/n_0}$.
According to the proof of theorem 38, each $\tau_i$ is the restriction to $K_0$ of $m/m_0$ many embeddings $\sigma_j$, therefore
$\prod_{i=1}^m (t- \sigma_i(\alpha))^{n/m} = \prod_{i=1}^{m_0} (t- \tau_i(\alpha))^{n/m_0} $, so we only need to prove that $f_0(t) = \prod_{i=1}^{m_0} (t- \tau_i(\alpha))^{n_0/m_0}$.
From now, suppose we are in the case $K = K(\alpha)$. Let $K_s$ be the separable closure of $F$ in $K$, so by Proposition 50, $K/K_s$ is purely inseparable and $K_s/F$ is separable. By corollary 47, there is an integer $a\ge 0$ such that $K_s = K(\alpha^{p^a})$.
Let $g(t)$ be the minimal polynomial of $\alpha^{p^a}$ over $F$.
Then $f(t) = g(t^{p^a})$, because $g(\alpha^{p^a}) = 0$ and they have the same degree $n = p^a m$.
Meanwhile, $(t - \sigma_i(\alpha))^{p^a} = (t^{p^a} - \sigma_i(\alpha^{p^a}))$ so we only need to prove that $g(t) = \prod_{i=1}^m (t - \sigma_i(\alpha^{p^a}))$.
So we only need to prove the theorem when $\alpha$ is separable.
But then this is clear : the $\sigma_i(\alpha)$ are roots of $f(t)$ because $\sigma_i$ are embeddings of $F$-algebras, and every factor in the product is distinct, so the product has to divide $f(t)$, and both polynomials have the same degree, so they must be equal. | 169,518 |
Models for our students are needed for our training classes:
– Permanent Makeup Class models for microblading, permanent eyeliner and permanent lip procedures, color correction and/or removal.
– Paramedical Class models for scar relaxation and camouflage, areola and nipple repig. | 315,580 |
The Lonely Cache Challenge is a type of Challenge Cache.
Description[]
The Lonely Cache Challenge involves finding a lonely cache that has been unvisited in over a year. The Wisconsin Geocaching Association host a Lonely Cache Game using Lonely Caches located in Wisconsin.
Rules[]
Usually, there is only a single rule. A cache must be found that has not been found in over a year. Other variations may include how many lonely caches you must find or some kind of point system based on how long a cache went unfound, requiring a certain number of points to claim the challenge. | 359,405 |
Paul McLane is editor in chief.
We’ve been telling you about some of the many LPFM stations that will be coming on the air in the United States. Here’s one that’s cut from my kind of cloth.
“Mississippi non-profit Classic Book Radio is starting a low-power FM radio station which will broadcast readings from classic works of literature,” states the press release.
It said Classic Book Radio was established to broadcast classic audio books from the archive at LibriVox.org. It has a construction permit for 95.5 MHz in Columbus, with call sign WMFH(LP).
The station is trying to raise $15,000 via “crowd funding” on the Indigogo platform.
Founder Christopher Howard, who has audio and amateur radio experience, is quoted in the release saying the station wants to promote literacy. “We feel this can be an important tool for introducing people to the large body of quality English literature they might not otherwise experience, including timeless works from Aesop to Melville and from Shakespeare to Mark Twain.”
There are no royalties to pay for the text, because the material is all in the public domain, he indicated. “Our programming content is normal people sitting down to read to you.” Further, Howard writes on the station’s fundraising page: “We have a place to operate. And we have a radio tower, 60 feet tall, which I have used for my amateur radio hobby. We have reliable electricity and Internet. We have a PC running the Rivendell radio library/scheduling/playback software and a steadily growing body of downloaded content.” But he is seeking to raise money for studio gear, EAS system, transmitter and streaming hosting service.
The Indiegogo campaign runs til late March. Christopher Howard is seen in a promotional video below.
I wish him well; this kind of endeavor seems to me to represent the spirit of what LPFM is for. (Tell us about your own LPFM plans during this historic expansion of the FM service. Email [email protected].) | 405,623 |
12, 2010
Nanny's Tired
It's Friday. It's raining out. N had an epic meltdown a few hours ago involving broccoli throwing, and I'm conserving all my energy for an event I'm attending (hint: it involves 30 varieties of gourmet chili for me to eat). And since brevity, my dears, is the soul of wit, I'm sharing but two things.
Please find above in all of its glory the song, "I Eat Doughnuts," by artist Sibley, which I tracked down after enjoying so much in Stella McCartney's ad for her Kids line, as posted earlier this week (Pour les Enfants).
Updates to come on what N has to say about the song, and "what it make her booty do."
And lastly, because Fridays are supposed to be en Francais, I thought I would share my favorite French term of endearment:
mon petit chou chou - "my little cabbage"
As in, "Awwww, je t'adore, mon petit chou chou!"
Je l'adore.
Voila! C'est bien. Bonne Vendredi.<<
I don't like donuts and I do like shoes, and yet somehow, I also like this song.
Okay, I guess I do like donuts if they're from Voodoo Donuts or Sweetpea Donuts, but those aren't regular donuts, they're like circles of yum. If you're ever in Portland don't get your donuts nowhere else (I'm for real, Voodoo Donuts is nationally known for its awesome donuts and Sweetpea makes VEGAN donuts that somehow still manage to be delicious).
I could not have said that first sentence more perfectly.
I'm just going to hire you as a tour guide next time I go to Portland. K? Delicious vegan doughnuts intrigue me. Also: the name "Sweetpea Donuts." | 399,754 |
Favorite Posts
Here's a list of a few of my favorite posts over the years:
A Complete List of Bookwords!
Book Blogging Explained
Reading: Luxury or Necessity?
Author Fun Facts: Markus Zusak and John Green
A Few of My Favorite Things
A Comment or Two on Comments
A Dream BEA Panel
The Trouble with Author Signings
True or False: 90% of Everything is Crap
Attempting to List Genres
Robin Hood: Fact or Fiction
1990's: My Decade of Lost Reading
Lurkers! Delurk.... please?
Music Munday Guest Post
Love Month: Favorite TV Couples
Let's Talk About Kissing
Five Stars: Real or Not Real
Why Fantasy?
The Color of Villains
Discussing Books... With Family
Edgy Books vs. Gentle Books
How to Start a Book Club
Why I Love Robin Hood Guest Post
My Kindle: Six Months Later
Getting to Know Markus Zusak
A Photo Essay of My Literary European Trip
Life Story: Music Memories
Five Stars: Real or Not Real?
A Day in the Life: Monday, March 23, 2015
The Scoop on Dr. Who
The Worth of Celebrity
Speaking Like Our Favorite Books
Life Story: Movie Memories
Guest Post at Book Bloggers International: Swooning!
Top Ten Bookish Problems | 309,465 |
Our group of allies is increasing! Artists raise their voices against the total destruction of Balkan rivers – they want to see them protected, not dammed.
“Drinking water is one of the most important resources for preserving life as we know it and is the bloodstream of our planet Earth. By endangering this vital – but at the same time sensitive – system, we are committing the slow suicide of human civilization.” says photographer Mario Maduna as he contributes to #ArtistsForBalkan Rivers
Mario Maduna is one of the ten winners at the Sony World Photography Awards 2021, which awards the best single photographs from 2020. He is also a professional diver and has been involved in many environmental actions around the Balkans.
Find his full album in this post | 213,241 |
TITLE: Describe the elements in the following set
QUESTION [0 upvotes]: $x \notin \bigcap\limits_{i\in I}A_{i}\diagdown \bigcup\limits_{j\in J}B_{j} $
This was a problem on exam and I think the correct answer should be
$x$ such that $x \notin A_i$ for some $i \in I$ or $x \in B_j$ for some $j \in J$
The only answer close to this on the test(multiple choice) was $x \notin A_i$ for some $i \in I$ and $x \in B_j$ for some $j \in J$
Other answers were:
$x \notin A_i$ for some $i \in I$ and $x \in B_j$ for all $j \in J$
$x \notin A_i$ for all $i \in I$ and $x \in B_j$ for some $j \in J$
$x \notin A_i$ for all $i \in I$ and $x \in B_j$ for all $j \in J$
Which one is correct(if any!) also is there a different way to express the first statement using and?
REPLY [1 votes]: The test is in error and you are right. This can occur if $x \not \in \cap A_i$, and it can occur if $x \in \cup B_j$. But it isn't required that both be true. Just one or the other. SO it should be "OR"; not "AND".
===
If $x \not \in \cap_{i\in I}A_i \setminus \cup_{j\in J}B_j$
The $x$ is not in all $A_i$ unless $x$ is in some $B_j$.
If that isn't clear then consider.
$\cap_{i\in I}A_i$ is just the elements in all the $A_i$. What's not in that set are the elements in some but not all $A_i$ and the elements that aren't in any $A_i$.
$ \cap_{i\in I}A_i \setminus \cup_{j\in J}B_j$ are just the elements in all the $A_i$ unless they also are in $\cup_{j\in J}B_j$. Any element in any or some of the $B_j$ will be excluded. So $ \cap_{i\in I}A_i \setminus \cup_{j\in J}B_j$ are the elements in all the $A_i$ but not in any of $B_j$.
So the way $x$ might not be in $ \cap_{i\in I}A_i \setminus \cup_{j\in J}B_j$ is if it isn't in all the $A_i$ (it might be in some of them but not all) or if it is in all the $A_i$ it can avoid being in $ \cap_{i\in I}A_i \setminus \cup_{j\in J}B_j$ if it is in any or some of the $B_j$.
In other words the answer you picked:
$x$ is either not in all $A_i$ which means there are some (maybe all) $A_i$ that don't contain $x$ OR $x$ is in one or more of the $B_j$.
The test was in error.
It is not required that either $x$ is not in all $A_i$ (it could by in all $A_i$ but also in at least one $B_j$) or that it must be in any $B_j$ (it could be in none of the $B_j$ but not be in some of the $A_i$). It is only required that one or the other be true. It's not required that both be true. | 108,639 |
This worn over the shoulder. | 227,748 |
TITLE: Help with the definition of a bilinear form $\omega$
QUESTION [1 upvotes]: According to this for $V$ a $2n$ (real) dimensional space any bilinear form $\omega: V \times V \to \mathbb{R}$ induces a linear map $\tilde{\omega}: V \to V^*$ via
$$ \tilde{\omega}(v) := \omega(v, \bullet) $$
where, from what I understand $v \in V$ but then what is this $\bullet \,\,$? Can you give an example in the context of normal differential forms maybe? Or, if this $\omega$ is the symplectic form, say for simplicity in $\mathbb{R}^2$, $\omega = dx \wedge dy$, then what is $\tilde{\omega}$? Also is it correct to say that $\omega(x,y)=dx \wedge dy$?
REPLY [0 votes]: We have that $dx(v)\wedge dy:V\to\Bbb R$ is just the linear map $u\mapsto dx\wedge dy(v,u)$ | 127,816 |
Keep in touch with Zazzle custom postcards! Add your favorite image to a blank postcard or say “hi” with a pre-existing design. Save paper and mail a note without wasting envelopes!
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Comment Wall | 41,013 |
Call for Papers 2nd Annual Institute of African Studies Undergraduate Research Conference
15 October 2016
Ottawa, Canada
Theme: Africa after Africa Rising: Politics, Development, Youth and Innovation in an Era of Globalization
Calling all undergraduate students with a passion for African Studies! The Institute of African Studies (IAS) at Carleton University (carleton.ca/africanstudies) is hosting its second international undergraduate research conference, organized by the Institute of African Studies Students Association (IASSA). The goal of this one-day interdisciplinary conference is to provide a platform for emerging researchers in the field of African Studies. It is understood that this may be a student’s first time completing a scholarly paper, therefore this conference has been timed to allow students the summer to work on their research papers and present at Carleton University in mid-October.
This year’s theme,.
Papers which must remain within the scope of African Studies, may address the following necessarily inexhaustive subtopics:
- Globalization, Popular and Youth Cultures
- Population, Displacement, Migration and Land
- Gender, Sexuality and Intersectionality
- Critiques of Imperialism, Colonialism and the Slave Trade
- Natural Resources and International Development
- Race, Identity, and Diaspora
- Afropolitanism
- Crime, Security, and Political Leadership
- Environment and Conservation
We encourage students coming from the following disciplines: Anthropology, Sociology, Psychology, Law, Global and International Relations, Policy and Political Science, Art History, Cultural Studies, Musicology, Religion, Film Studies, Business, Media and Communication Studies, Technology, Literature and Critical Studies. Students interested in participating are invited to submit abstracts by July 1st, 2016 to the organizing committee at: [email protected] Abstracts of between 300-500 words in length should include the following:
- Name, institution, field of study, address, email and phone number.
- The title of your presentation.
- A thesis statement, two or more research questions, and a brief description of the argument the paper seeks to make.
Research Papers are due September 1st, 2016 and must be a minimum of 3000 words.
For participating students, this conference is an excellent opportunity for improving academic portfolios, especially for those interested in future graduate studies or a career that requires writing reports or policy papers for government agencies, NGOs, etc. There will be prizes for the best papers at the conference, including being awarded publication opportunity in Nokoko, the open-access academic journal of Carleton University’s Institute of African Studies.
The IASSA plans to make this a memorable experience for students by incorporating distinguished guest-speakers, cultural components, and an environment that fosters the kind of support and encouragement that undergraduate students treasure. More information on these activities as well as about accommodation, possible travel subsidies, possible tour of Ottawa (Canada’s capital city), and so forth will be made available in due course.
For more information, please email the organizing committee at: [email protected]
Applicants will receive confirmation of acceptance by July 20th, 2016 | 147,907 |
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Save musicians, entertainers, speakers, and services, and they’ll appear here.
Ms. Twisty Pants
- Rockford, IL
- Balloon Twister
- Travels up to 120 miles
- Contact for rates
Overview
Hello there! My name is (Tracy Smith) Twisted Tracy aka Ms. Twisty Pants, and I am a balloon artist and face painter. I have been performing since April 2010 in Illinois and 2011 in California. My balloon art can be seen on the cartoon network show Incredible Crew, episode name Super annoying guy on a plane. (video attached!) I have also performed for Big Red Entertainment run by Bill Parks, who has been in several television shows, commercials and sitcoms. I am available for parties and special occasions of any kind, and my rates are the most reasonable you will find. You can also visit my website,. Please contact me to schedule your special event today!
2 Reviews
Children's Party Entertainment, Clown, Face Painter, Balloon Twister
Her balloons were beautiful , unique and creative!! She was kind and fun with the children!! She squeezed in our event- even though she had a very busy day and she was wonderful about everything!!
Children's Party Entertainment, Balloon Twister
Response from Ms. Twisty Pants:
Thank you so much for the 5 star rating! I'm so glad you were happy with the services I provided!
Booking Info
Price Range: Contact for rates
Gig Length: 60 - 999 minutes
Languages: English
Insurance: $1,000,000 - Liability
Website: Visit website
What to Expect
You can expect smiling faces, fun for kids of all ages, professional and courteous attitude from the performer, games with balloons available upon request, kid friendly jokes, and clean family entertainment.
About
I am very professional at all events, and you can count on good clean family fun for kids of all ages. I tell lots of kid friendly jokes while creating the requested balloon art, games with balloons such as balloon limbo are available upon request, and I can provide face painting from cheek art to full face masks. I have been performing since April 2010 and I love it! My passion for creating quality entertainment is evident at every performance.
Additional Booking Notes
I am fully self contained and bring all of my own equipment. If you are requesting face painting, I will need 2 chairs and a table if possible. Please let me know if I need to bring my own.
Set List
I provide balloon creations of all kinds and face paint designs by request. If I haven't done it before, I will try my best to fulfill every request!
Team
Influences & Inspiration
Tim Adamz of Adamz Entertainment, Unpoppables, Stacie Chase of Party Savers.
Setup Requirements
table and 2 chairs for face painting
Social | 234,215 |
Americans Support Full Due-Process Rights for Terrorism Suspects: Majorities Oppose Rendition of Suspects to Countries that Practice Torture Most Believe Abu-Ghraib-type Abuses Still Occurring
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Date2006-01-17
Author
Program on International Policy Attitudes (PIPA)
MetadataShow full item record
Abstract. | 386,678 |
China crisis? Posted on 30 Jan 2014 Professor Mark Exworthy, Health Services Management Centre David Nicholson famously claimed that the recent NHS reforms could be seen from space. If he visits China, he will need to revise his analogy. My participation (on behalf of HSMC) in a health care `trade mission’1 to China (led by Ken Clarke MP) in January 2014 brought home the scale of change and enormity of the challenges facing China and its health care system.. Until about 30 years ago, 20% of China health care costs were out-of-pocket expenses, with the rest met by the state. Then, health care reforms made hospitals take care of their own finances, as commercial entities. However, such incentives led to over 50% of hospital income currently being derived from drug sales and unnecessary medical procedures. The Chinese government is now seeking to devolve away from the centre, to reduce bureaucracy and shift from (direct) provision to regulation. So far, the agenda seems remarkably similar to English health care reform. Indeed, many Chinese policy-makers and practitioners have a high admiration for the NHS, seeking to learn from it and even emulate it. Yet, enormous challenges remain for the health-care system. These include: Scale of change: In the past 10 years, over 100 million people have migrated to cities and even more will move in the next 10 years. Meeting the health needs of such a population is becoming increasingly difficult. For example, just 30 minutes from Beijing (admittedly by the very fast Harmony train) is Tianjin, a city of over 12 million, with 304 “hospitals” and over 2000 “village health rooms.” The pace of change is stretching existing capacity and capabilities here, as elsewhere. From secondary to primary care: Hospitals dominate the health care landscape. In China, 90% of health care contacts are in secondary care, unlike the NHS where 90% of contacts are in primary care. Although the policy rhetoric is to move away from existing funding steams at prospective payment models (similar to PbR), the shift to primary care remains problematic. For example, minor surgery in primary care has largely disappeared. Moreover, the public has expectations of seeing a `specialist’ in secondary care. The continued dominance of secondary care and the weak infrastructure in primary care might augur higher costs and reduced access in the long term. This could exacerbate health inequalities. Current investment in primary care facilities will remedy this situation somewhat but the scale of the challenge may hamper such ambitions. Health spending: China spends 5.2% of its GDP on health care (compared to 9.3% in UK)2. In the past decade, there has been a growth in social insurance coverage. In some places, it is currently over 90% whereas it was only 15% ten years ago. Packages of essential care are still provided by the state. Despite such social insurance, patients still make significant co-payments of 40-60% of the costs of their care. In such circumstances, it is understandable that the public seek to save as much of their income as they can, not least for cases of catastrophic health care costs3. The effect across the country is an economy unbalanced towards investment and not consumption. Corruption remains a significant challenge as, in some sectors, it is endemic. Some strides are being made to address this. The GSK case of bribery has become a notable illustration of the state’s commitment to tackling this issue4. Corruption has also, some claim, had an antagonistic effect on doctor-patient relations (given the role of the former in prescribing drugs). Social determinants of health: Throughout the week of the visit, `smog’ pollution was ever-present. Although we visited only cities (where pollution might be expected to be at its worst), train travel between them revealed persistent `grey’ skies. Combined with on-going urbanisation, reliance on coal fired power stations, and growing car usage, deleterious health effects will be substantial. Rising life expectancy (2.4 years increase in the last decade, for example) may not be sustainable. Moreover, policy-makers are especially concerned with models of care for the growing elderly population and those with associated conditions such as dementia. A final word is merited about purpose of the `trade mission.’ it was interesting to note the mix of 50 or so delegates from the UK – from IT companies, the NHS and universities. While IT companies had a particular focus on digital health, the NHS (mostly specialist Trusts) was exploring the development of education, research and commercial links which may ultimately create a new income stream for the NHS. For universities, there was interest in advancing existing research and educational programmes. The University of Birmingham has a major collaboration with Guangzhou. HSMC has won British Academy funding to conduct a learning network on health care reform in China (see HSMC latest Newsletter). To that end, my visit helped HSMC pursue yet more international links and in doing so, revealed familiar challenges of health care reform in a completely different context. Healthcare UK: World Bank, 2011 China Daily, 3 Sept 2013: ; Guardian, 23 Oct 2013:. | 375,936 |
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The full list of official partners to the British Academy Television Awards.
The Academy is extremely grateful to the following companies who supply products and services to our annual Television Awards. We are proud to have such a prestigious range of companies represented as our Official Partners and are deeply indebted to them all.
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The official list of nominee gift providers for the British Academy Television Awards.
Stay up-to-date with the latest BAFTA news, events and online content. | 288,358 |
8Cancun, MX 77500
Nightly Rates: ( $130.00 - $479.00 )
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A Unique family concept on Cancuns best rated beach featuring the largest 1, 2 and 3 bedroom suites with kitchen and terraces, fully furnished studios, walking distance from shopping, dining, convention center, night life, nearby golf, a full service resort, great amenities, intimate tranquil setting, perfect location, the comforts of home, the ambiance of a resort. Book your ambiance suite now!
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Cancel by 5pm hotel time, 2 days prior to arrival to avoid a penalty of one night's room and tax.12/21/04 - 01/02/05 - Cancellations must be recieved 14 days prior to day of arrival in order to get refunded.
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The hotel is located in the heart of Cancun's dining, disco, convention center & shopping area. Five miles away from downtown, 12 miles from Cancun's International Airport. | 251,396 |
Filed under: cpd23 | Tags: reflection
I love attending events, and go to as many as I can. As I’m based in London there are more events going on than I have the time to attend, many of which are free or low-cost (and yes, I realise I’m extremely lucky). .
My response to this thing can be summed up in 3 words: yes, maybe, no.
Just a quick post for this Thing as my thoughts on social media can be found scattered throughout my blog archives. However, here are a few brief thoughts about social media (if only to justify actually posting this!)….
Although).
This ought to be a short post – I didn’t do a graduate traineeship, neither am I chartered, although I do have two Masters degrees, of which one is the coveted MA LIS! | 9,995 |
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RS3 Christmas Rares
Santa's elves have come to Burthorpe bearing sacks of presents for everyone in RuneScape. For the next two weeks, they'll be dishing out parcels of Christmas joy - which can include brand-new rares.
From 3rd December 00:00 GMT until 16th December 23:59 GMT, you can collect enchanted snow through skilling, combat, chatting to Santa's head elf in Burthorpe once per day, and through Treasure Hunter.
Rare Gifts
Every 1000 snow gathered can be made into an enchanted snowball, which can be handed to the head elf.
The next day, you'll be able to collect a number of Christmas presents from the head elf - the same as the number of snowballs you handed in, up to 50. These contain prizes, including a limited number of rare items across all those awarded:
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They're also the only way that members can get the glowing cracker. When pulled with another player, this gives 1000 Bonus XP in a random skill to both. Its owner also gets a prize, which could be the rarest of all Christmas gifts: the holly wreath.
Of the new rares on offer, the holly wreath, Christmas scythe and rubber turkeys are tradeable, so get hold of them during the two week period if you can!
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During the first week (3rd December 00:00 GMT - 9th December 23:59 GMT) you'll get double the snow you usually would from killing NPCs. On the second, (10th December 00:00 GMT - 16th December 23:59 GMT you'll get double snow from non-combat skilling activities.
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All regular accounts get at least one Treasure Hunter Key per day, and RuneScape members get two - as well as access to the golden cracker! Go for Premier Club and you can get up to three Keys per day. You can earn more through gameplay – just visit the wiki to find out how, and to learn more about Treasure Hunter itself.
Head to our billing page to buy more Keys, redeem Bonds in-game, or simply click ‘Buy Keys’ in Treasure Hunter itself. | 66,483 |
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Take it away, Jonathan Coulton:
I’ll be having a grand day out in lovely PDX today, but feel free to join the virtual party by leaving a comment here (or on Facebook, or by e-mail, or via passenger pigeon) which answers this question:
But where are the clowns?
Did you try looking in the tiny Volkswagen?
I'll have to ask my ten-inch pianist where he parked it. | 62,684 |
TITLE: Why $\sum_{k=1}^n \frac{1}{2k+1}$ is not an integer?
QUESTION [5 upvotes]: Let $S=\sum_{k=1}^n \frac{1}{2k+1}$, how can we prove with elementary math reasoning that $S$ is not an integer?
Can somebody help?
REPLY [6 votes]: Hint: Recall the (elementary) proof that $\sum_{i=1}^n \frac{1}{i}$ is not an integer (for $n>1$):
Let $2^k$ be the largest power of 2 that is smaller than or equal to $n$. Consider making (smallest) common denominator of the form $2^k L$, where $L$ is odd. The numerator of every term will be even except for the term of $\frac{1}{2^k}$, which contributes an odd term $L$. Hence, the numerator is odd and the denominator is even. This cannot be an integer.
Hint: For finite sum of odd reciprocals, show that when we make a (smallest) common denominator, the numerator is not a multiple of $3$.
Solution: Let $3^k$ be the largest power of 3 that is smaller than or equal to $n$. Consider making common denominator of the form $3^k L$, where $L$ is not a multiple of 3. The numerator of every term will be a multiple of 3, except for terms of the form $\frac{ 1}{ a3^k}$ for some integer $a$.
Use the fact that $a=2$ does not appear in the sequence since it is even. Also, by the definition of $3^k$, no other higher multiple can appear. Hence, there is only 1 term which contributes a non-multiple of 3. Thus, the numerator is not a multiple of 3.
REPLY [1 votes]: Let $p=2k'+1$ be the largest odd prime number less than or equal to $2n+1$. Now consider the sum:
$$\frac13 + \frac15 + \frac17 + \cdots + \frac{1}{2k'-1} + \frac{1}{p} + \frac{1}{2k'+3} + \cdots + \frac{1}{2n+1}.$$
We can find a common denominator for this fraction by taking the product:
$$3\cdot5\cdots (2k'-1) \cdot p \cdot (2k' +3) \cdots (2n+1) = M.$$
In the numerator we have a sum of several terms:
$$\frac{M}{3} + \frac{M}{5} + \cdots + \frac{M}{p} + \cdots + \frac{M}{(2n+1)}.$$
Each term here is an integer.
I claim that all but $\frac{M}{p}$ is divisible by $p$. This would mean the numerator is not divisible by $p$ and is congruent to $\frac{M}{p}$ modulo $p$. Since $p$ appears in the denominator, this sum cannot be an integer.
Suppose that $\frac{M}{p}$ was indeed divisible by $p$. This means that we must have had another multiple of $p$ appear in one of our denominators. The next valid term would be $3p$. However, Bertrand's postulate tells us that for any natural number $n$ we have a prime between $n$ and $2n$. This tells us that there would have been a new largest prime between $p$ and $3p$. Which is a contradiction.
REPLY [0 votes]: You can write
$$S=\frac{\sum_j\prod_{k\neq j}(2k+1)}{\prod (2k+1)}$$
Now the largest prime factor less or equal than $2n+1$ divides the denominator and all but one of the summands in the numerator, hence not the sum. | 169,833 |
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TITLE: Connection between cyclic group and exponential function
QUESTION [12 upvotes]: I have been thinking about this for a while, but now got to the point where I got stuck. I don't know if it might be considered as a research level question, but I would be very happy if somebody knew of any direction on how to move on:
Define the functions $t_k(x) = \sum_{n=0}^{\infty}{\frac{x^{3n+k}}{(3n+k)!}}$ for $k=0,1,2$. The functions then satisfy:
$
\begin{pmatrix}\exp(x) \\ \exp(\omega x) \\ \exp(\omega^2 x)\end{pmatrix} =
\begin{pmatrix}1 & 1 & 1\\1 & \omega & \omega^2 \\1 & \omega^2 & \omega\end{pmatrix} \cdot \begin{pmatrix}t_0(x) \\ t_1(x) \\ t_2(x)\end{pmatrix}$
where $\omega = \exp(2\pi i / 3)$.
As might be known, the matrix in the equation is the character table of the cyclic group $C_3$ and also a Vandermonde matrix. Using this last matrix equation one can prove the following addition theorem:
$\begin{pmatrix}t_0(x+y) \\ t_1(x+y) \\ t_2(x+y)\end{pmatrix} = \begin{pmatrix}t_0(x) & t_2(x) & t_1(x) \\ t_1(x) & t_0(x) & t_2(x)\\ t_2(x) & t_1(x) & t_0(x)\end{pmatrix} \cdot \begin{pmatrix}t_0(y)\\t_1(y) \\ t_2(y)\end{pmatrix}$
As might be known the matrix in the last equation is a circulant matrix, which is also the group matrix $(x_{gh^{-1}})_{g,h\in G}$ as defined by Dedekind.
Now it is clear how to do this construction for every cyclic group $C_n$, which we just did for the cyclic group $C_3$, namely define the functions $f_g(x)$, $g\in C_n$using the character table of $C_n$.
If one takes the determinant of the last matrix, one can show (using the theory of circular matrix) that this is equal to one:
$t_0(x)^3+t_1(x) ^3+t_2(x)^3-3t_0(x)t_1(x)t_2(x) = 1$ for all $x$.
I am able to prove that the determinant is equal to $1$ for every cyclic group. Notice also that the determinant is the group determinant of $C_3$ as defined by Frobenius and Dedekind .
(1) Is it true, that for a general cyclic group the functions defined fullfill the "addition theorem" which is given by the Dedekind group matrix?
(2) How does one proceed with arbitrary finite groups (for example the Klein four group and the symmetric group on three elements)?
(3) Does somebody know of any other context, where the specific functions $t_k$ $k=0,1,2$ appear?
REPLY [1 votes]: First, let me apologize for posting this second answer. But it gives a definitive answer; at least I think so.
The problem is to find functions $t_g(x)$ for $g\in G$ (a finite group) and $x\in{\mathbb R}$ (or ${\mathbb C}$), such that
$$\sum_{h\in G}t_{gh^{-1}}(x)t_h(y)=t_g(x+y).$$
What seems reasonable is to impose the restriction that each $t_\bullet(x)$ be a class function (it depends only upon the conjugacy class of $g$).
As noted by stackExchangeUser, by Fourier transform, this amounts to finding functions $\widehat{t_x}(\rho)$ satisfying
$$\widehat{t_x}(\rho)\widehat{t_y}(\rho)=\widehat{t_{x+y}}(\rho),$$
where now $\rho$ runs over complex representations of $G$. Actually, it is enough to verify the identity above for irreducible representations.
What remained un-noticed is that the latter identity can be rewritten
$$M_\rho(x+y)=M_\rho(x)M_\rho(y),$$
where $M_\rho(x)$ is another way to write $\widehat{t_x}(\rho)$. Now, we see that $M_\rho$ is nothing but an exponential:
$$M_\rho(x)=\exp(xA_\rho),\qquad A_\rho:=M_\rho'(0)=\sum_{g\in G}t_g'(0)\rho(g).$$
This shows how to find the most general solution $T=(t_g)_{g\in G}$ of the problem: choose any class function $a:G\to{\mathbb C}$. Consider the regular representation $R$, acting over ${\mathbb C}[G]=:V$. Form $A:=\sum_{g\in G}a_gR(g)\in{\rm End}(V)$. Then define $M(x)=\exp(xA)$. Because the image $R(G)$ is closed under composition, $M(x)$ belongs to the linear subspace of ${\rm End}(V)$ spanned by $R(G)$. It even belongs to the subspace spanned by the class-sums $R_s:=\sum_{h\in c}R(h)$ ($c$ are conjugacy classes). Therefore $M(x)$ decomposes as
$$M(x)=\sum_{\rm classes}m_s(x)R_s=\sum_{g\in G}t_g(x)R(g),$$
where $t_\bullet(x)$ are class functions.
The identity $M(x+y)=M(x)M(y)$ writes $\sum_{g\in G}b_g(x)R(g)=0$ where $b_g(x):=t_g(x+y)-\sum_{h\in G}t_{gh^{-1}}(x)t_h(y)$. The latter identity implies classically $b\equiv0$, thus the functions $t_g$ solve the problem.
About the determinant. The matrix of $M(x)$ in the canonical basis of ${\mathbb C}[G]$ is the ``circulant''
$${\rm Circ}(T)=(t_{gh^{-1}})_{g,h\in G}.$$
We have
$$\det{\rm Circ}(T)=\det M(x)=\exp(x{\rm Tr}A)=\exp(|G|xa_e).$$
In particular $\det{\rm Circ}(T)\equiv1$ whenever $a_0=0$, which happens in the example given in OP's question.
Edit. As it has been known after Frobenius, $\det{\rm Circ}(T)$ factorizes as
$$\prod_{\chi{\rm irrep}} \Delta_\chi^{{\rm deg}\chi},$$
where $\Delta_\chi$ is a polynomial in $T$, homogeneous of degre ${\rm deg}\chi$. Each factor is the determinant of the restriction of ${\rm Circ}(T)$ on the $\chi$-component of $V$. Again this restriction is the exponential of $x\sum_Ga_gR(g)_\chi$, the linear combination of the corresponding restrictions of $R$. Thus
$$\Delta_\chi^{{\rm deg}\chi}=\det\left(x\sum_Ga_gR(g)_\chi\right)=\exp x\sum_Ga_g({\rm deg}\chi)\chi(g).$$
This yields
$$\Delta_\chi=\exp(x|G|\langle a,\chi\rangle).$$ | 51,340 |
Firstly, I would like to apologise to all my lovely blogger friends for not checking your beautiful blogs much lately. My tablet is no longer functioning properly and I need to buy a new one but have not managed to get around to that yet. Meanwhile I have to wait until I can get on our main computer which is situated in hubby's domain. Hubby was ill for a week which also preoccupied me and whenever I had some free time I chose to do some sewing therapy. I am a hands on Grandma once a week and twice on alternate weeks and I also have had to deal with my own health issues. So I just hope you haven't given up on me.
Despite not having the right blue thread for my munchkin's project I utilised a DMC variegated thread that I had. Unfortunately, I now find myself running out of this particular thread so I have had to stitch more of the design in the red trying to conserve what is left of the blue. I just hope that the blue variegated is still available to purchase as I think it is quite old. Otherwise I shall just pick out a dark blue that I have. Fingers crossed.
I have finished appliqueing one pumpkin.
Another book has been added to my ever growing pile of books I have read. By 'pile', I do mean 'pile'. I literally have 2 piles of books sitting on the floor in front of my overflowing book shelf. "Birthright" has been written by another Australian author. We are very lucky having so much talent here. Every time I search for a new book I come across another Aussie author.
It is Saturday, (despite what the date says on here), and I have been baking my Clever Gutz bread. This is the 3rd loaf I have made and I quite like it. It is nice to have with soup or for breakfast. Thankfully, I have never been a big bread eater which has proven to be a good thing for me as I need to restrict my diet quite a lot due to gut problems. This is a bread I am "allowed" to eat.
NBN appears to be working well at the moment , however , I do have an issue with not being able to have the regular phone line. Like a lot of things these days "progress" seems to actually mean that we are going backwards. We have quite a few power surges here and the phone cut out on me while I was in the middle of a conversation with someone. Some people do not have reception for mobile phones in their area and if the power goes out they cannot use the regular phone as it is all connected to power now. Thus endeth my whinge for the week.
May your week ahead be a beautiful one filled with an abundance of all good things.
Angel Blessings. | 124,784 |
A]
Le Ultime News
Sei appassionato di cucina? Ecco qualche consiglio per te
Links | 389,608 |
TITLE: Determine whether the following groups are isomorphic.
QUESTION [0 upvotes]: Let $G=\lbrace \frac{p}{q}:p,q\in\mathbb{N} \rbrace$, and let $G'=\lbrace \frac{p}{q}:p,q\in\mathbb{N},p$ and $q$ both are odd $ \rbrace$. It is clear that both $G$ and $G'$ are groups under common multiplication.
Now the question : Is $G$ and $G'$ isomorphic as groups ?
I clearly don't have any idea how to prove/disprove it. Please give me some hints.
REPLY [2 votes]: Hints:
Each positive rational number $x$ has a unique factorization as
$$x = 2^{e_2}\cdot 3^{e_3} \cdot 5^{e_5}\cdot 7^{e_7} \cdots$$
where $e_2,e_3,e_5,e_7, ...\;$are integers.
Define $f:G \to G'$ by
$$
f(2^{e_2}\cdot 3^{e_3} \cdot 5^{e_5}\cdot 7^{e_7} \cdots)
=
3^{e_2}\cdot 5^{e_3} \cdot 7^{e_5}\cdot 11^{e_7} \cdots
$$
Verify that $f$ is an isomorphism.
REPLY [1 votes]: Consider the map $r:\mathbb{N}\rightarrow\mathbb{N}$ that sends $1$ to $1$, $p_{i}$ to $p_{i+1}$ for all $i\geq 1$ ($p_{i}$ denoting the $i$th prime, so that in particular, $p_{1}=2$), and for $n\in\mathbb{N},$ $n=p_{j_{1}}^{k_{1}}\cdots p_{j_{\ell}}^{k_{\ell}},$ $r(n):=p_{j_{1}+1}^{k_{1}}\cdots p_{j_{\ell}+1}^{k_{\ell}}.$ If we define $R:G\rightarrow G'$ by $R(p/q)=r(p)/r(q),$ this clearly maps into $G',$ and by our definition of $r$ and $R$, $$R\left(\frac{p}{q}\cdot\frac{p'}{q'}\right)=R\left(\frac{pp'}{qq'}\right)=\frac{r(pp')}{r(qq')}=\frac{r(p)r(p')}{r(q)r(q')}=R\left(\frac{p}{q}\right)R\left(\frac{p'}{q'}\right),$$ so this is a group homomorphism (you'll need to check that it's well-defined, since $p/q=p'/q'$ is possible for $p\neq p',q\neq q',$ but this is easy). Moreover, it is injective, since $R(p/q)=1$ implies $r(p)=r(q),$ which implies $p=q.$ To show that it is surjective, suppose $p$ and $q$ are odd natural numbers. Then their prime expansions do not have any powers of $2$, so if $p=p_{j_{1}}^{k_{1}}\cdots p_{j_{\ell}}^{k_{\ell}},$ then all $j_{i}\geq 2$, and therefore $r^{-1}(p)=p_{j_{1}-1}^{k_{1}}\cdots p_{j_{\ell}-1}^{k_{\ell}}$ is well-defined, and similarly for $r^{-1}(q).$ Since $R(r^{-1}(p)/r^{-1}(q))=p/q$, $R$ is surjective, and the obvious definition of $R^{-1}(p/q)$, $p/q\in G$, is $r^{-1}(p)/r^{-1}(q).$ A bijective group homomorphism is an isomorphism, so we are done. | 163,355 |
TITLE: Problem about proving fermat's little theorem
QUESTION [1 upvotes]: We know that, there is an important step to prove Fermat's little theorem, two side times $(n- 1)! \cdot a^{n-1} = (a\cdot1)\cdot(a\cdot2)\cdot...\cdot(a\cdot(n-1)) \equiv (n-1)! \mod(n) $
Example:
If $\gcd(a,b)=1, x = x_1\mod (n), y = y_1 \mod (n), z = z_1 \mod (n)$, then ${a\cdot x,a\cdot y,a\cdot z} \mod n = {x,y,z}$, only the arrangement different, what is the step to prove that?
REPLY [2 votes]: Note first that in Fermat's Theorem, the number you call $n$ is prime. So we change the name and call it $p$.
We want to show that if $a$ is not divisible by $p$, then the numbers $a\cdot 1, a\cdot 2,\dots, a\cdot (p-1)$ are all incongruent modulo $p$.
Suppose to the contrary that $ax\equiv ay\pmod{p}$, where $1\le x\lt y \le p-1$. Then $p$ divides $ay-ax$, that is, $p$ divides $a(y-x)$.
Since $p$ does not divide $a$, it must divide $y-x$. This is impossible, since $1\le y-x\lt p$.
Recall now that any number not divisible by $p$ is congruent modulo $p$ to one of $1,2,\dots,p-1$. Our numbers $a\cdot 1,a\cdot 2,\dots, a\cdot (n-1)$ are congruent to different numbers in the interval from $1$ to $p-1$. But there are $p-1$ numbers in our collection $a\cdot 1,a\cdot 2,\dots, a\cdot (n-1)$. So they must be congruent, in some order, to all the numbers from $1$ to $p-1$.
REPLY [2 votes]: It is easy to see that if $p$ is prime and $a$ is not a multiple of $p$ then the numbers
$$a,\,2a,\,\ldots,\,(p-1)a\tag{*}$$
are not multiples of $p$. Also, no two of these numbers are congruent modulo $p$: if say $ia\equiv ja\pmod p$ with $1\le i<j\le p-1$, then
$$(j-i)a\equiv0\pmod p$$
with
$$1\le j-i\le p-1\ ,$$
which is impossible. So the $p-1$ numbers (*) are all different and non-zero modulo $p$, so they must, in some order, be congruent to $1,2,\ldots,p-1$. Hence
$$(a)(2a)\cdots((p-1)a)\equiv(1)(2)\cdots(p-1)\pmod p\ .$$
I think this is the step you are asking for. | 108,928 |
TITLE: Ways to choose $k$ items out $n$ without overlap in the chosen sets
QUESTION [1 upvotes]: This is part of a much larger and harder problem I am solving. I feel like it's a somewhat easy combinatorics problem, but that is not my field, and I can't find a solution online.
So I have a set of $n$ distinct items. Say, $\{0,1,2,3\}$ (so $n=4$), or the positions of characters on this string: "$1111$". We know $n$ to be a power of two, and $k$ to be even.
I want to know how many ways there are to choose $k$ items from the set, without any overlap.
With $\{0,1,2,3\}$ and $k=2$, if I choose $0,1,$ then I can only choose $2,3$. If I choose $0,2,$ then I can only choose $1,3.$ Etc. With those parameters, we can see there are $3$ ways to choose sets of two without overlap (aabb, abab, abba; or their complements (with a the chosen items and b the non-chosen)).
With $\{0,1,2,3,4,5,6,7\}$ and $k=2$, if I choose $0,1,$ then I can choose any two out of $\{2,3,...,7\}$.
I tried to puzzle this out. I thought I had $\binom42$ ways of choosing the first two items, and then I had the remaining $\binom{4-2}{2}$ choices, but I'm not sure how to combine those facts, since $\binom{4-2}{2}$ is 1, and multiplying, subtracting, or dividing it with $\binom42$ doesn't give the right answer.
And clearly $\binom{4-2}{2}/2 = 3$ seems to be right, but I don't know how to explain the "$/2$", and if it generalizes to other values of $k$ or $n$.
Thanks for any help you can provide.
REPLY [1 votes]: I'm going to interpret your problem as "how many ways can I choose $k$ things from $n$, and then choose $k$ more things from the remaining $n-k$ things?" In that case, let's think of the $n$ objects as being mapped to three different letters, $a,b,c$. Then $a$ will denote the objects in the first set, $b$ will denote the objects in the second set, and $c$ will denote objects not chosen. In the final answer, which set was chosen first/second will not matter, but we introduce this notion to make things easier to count.
This is the same as counting the number of "words" with $n$ letters that have $k$ $a$'s, $k$ $b$'s and $n-2k$ $c$'s, where the position of the letter gives the value of the entry corresponding to that letter. For example, $n=8$ and $k=2$:
$aacbccbc$ corresponds to $\{0,1\}$ for the first set, $\{4,7\}$ for the second, and the rest not chosen.
Standard combinatorics tells us that there are $${n}\choose{k, k, n-2k}$$ ways to do this. However, because you don't care which set was chosen first and which set was chosen second, the words $aacbccbc$ and $bbcaccac$ are indistinguishable to us. Since there's exactly $2$ ways for this to happen each time, you divide by $2$ again to get $$\frac{1}{2}\binom{n}{k, k, n-2k}$$ different ways. Hope this helps! | 104,723 |
Espacio Danza is a Centre for the Development, Diffusion and Investigation of Contemporary Dance in Bolivia.
One of the main objectives of Espacio Danza is to strengthen dance art in Bolivia. For this we are trying to develop an exchange platform which may benefit professional dancers and choreographers with the possibility to confront their work with notable international artists.
In March of 2007 we had the honor to work with Mr. Gerardo Agudo, dancer, choreographer and pedagogue, Director of Nang Teatro, Argentina.
María José Rivera
Founder of Espacio Danza. in performance arts; primarily dance, as a visual art.
Since 1988 she has been in the main theatres within Bolivia and Mexico, and had individual presentations en the United States. She has participated in various festivals in Bolivia and Mexico. As an independent artist she has participated in Festival ANDANZA 06 and 07, at the XII Internacional Dance Festival “Habana Vieja, Ciudad en Movimiento” in Cuba and recently has received the danceWEB scholarship to attend ImpulsTanz 08 in Wien, Austria.
She has collaborated with Universidad Mayor de San Andrés and Universidad Católica Boliviana as a teacher and choreographer, and continuously works with the Ballet Municipal de El Alto and Summa Artis, and complementary teaches at the Sub Alcaldía de Mallasa.
Amancaya Rivera
Initiated her dance studies in classical ballet. Complementary has taken many workshops in Bolivia and the United States.
She has performed with Summa Artis and Ballet Municipal de El Alto, where she has also teached.
On 2005 has worked with Frederik Theatre in Mexico City.
At the present she is pursuing her dance degree in France. | 169,528 |
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Call with China on green cities
NWO, through the Merian Fund, and the Chinese Academy of Sciences (CAS) have launched a joint call for proposals on Green Cities. This topic is of great interest and importance to both China and The Netherlands due to the increasingly urban nature of their populations. The deadline for applications is 21 January 2020... | 145,004 |
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In this section we estimate the computational complexity of the
dense-block and the compressed-block algorithms presented in
Section \ref{sec:inversion}, under a number of assumptions on the
rank structure of blocks of $\cA[A]$.
We first define a framework to estimate work and storage for
algorithms defined on a binary tree $\cA[T]$, and use it to analyze
all types of algorithms described in this paper.
We consider both non-translation invariant (\textbf{NTI}), and
translation-invariant kernels (\textbf{TI}) for which significant
performance gains can be obtained.
\subsection{Complexity of algorithms on binary tries}
\label{sec:framework}
All our algorithms compute and store matrix blocks associated with boxes
organized into a binary tree $\cA[T]$. To produce complexity estimates
for a given accuracy $\acc$, we introduce bounds for work $W_{\ell}(n_{\ell},\acc)$ and storage $M_{\ell}(n_{\ell},\acc) $ at each level $\ell$ of the tree, where $n_{\ell}=2^{-\ell}N$ is the maximum number of points in a box at this level (we assume that the work per box on a given level has small variance).
\begin{lemma}
\label{lemma:complexity}
Let $n_{\ell}=2^{-\ell}N$, and $d = \log_2(N/n_{max})$, and exponents
$p,q \ge 0$. Then if $W_{\ell}(n_{\ell},\acc)$ has the form $C_{\acc} n_{\ell}^p \log_2^q(n_{\ell})$,
the total work has complexity
\beqn
\textbf{NTI: }\sum_{\ell = 0}^{d} {2^{\ell} W_{\ell}(n_{\ell},\acc)} =
\left\{ \begin{array}{lr}
O(N) & : 0 \le p<1 \\
O(N \log_2^{q+1}N) & : p = 1 \\
O(N^p \log_2^{q}N) & : p > 1
\end{array} \right.
\eeqn
\beqn
\textbf{TI: } \sum_{\ell = 0}^{d} {W_{\ell}(n_{\ell},\acc)} = O(N^p \log^{q}N)
\eeqn
\end{lemma}
For NTI algorithms, on each level $\ell$ we obtain an estimate by adding the bound for work per box for the $2^{\ell}$ boxes. The polynomial growth in $W_{\ell}$ or $M_{\ell}$ is compensated by the fact that the number of boxes decreases exponentially going up the tree.
If the rate of growth of $W_{\ell}$ is slower than linear ($p<1$),
the overall complexity is linear. If $W_{\ell}$ grows linearly, we
accumulate a $\log N$ factor going up the tree.
If the growth of $W_{\ell}$ is superlinear ($p>1$), the work
performed on the top boxes dominates, and we obtain the same
complexity as in $W_{\ell}$ for the overall algorithm.
In the TI case, work/storage on the top boxes dominates the
calculation, since only one set of matrices needs to be computed and
stored per level. Hence, the interpretation is simpler: the
single-box bound for complexity at the top levels reflects the overall complexity.
\subsection{Assumptions on matrix structure}
\label{sec:assumptions}
Complexity estimates for the work per box require assumptions on
matrix structure. Let us restate some assumptions already
made in Sections \ref{sec:background} and \ref{sec:inversion}:
\begin{enumerate}
\item \label{asm:skeleton} \textbf{Skeleton size scaling:}
The maximum size of skeleton sets for boxes at level $\ell$
grows as $O(n_{\ell}^{1/2})$. This determines the size of
blocks within the HSS structure, and can be proved for
non-oscillatory PDE kernels in 2D.
\item \label{asm:eqdensity} \textbf{Localization:} Equivalent
densities may be used to represent long range interactions to
within any specified accuracy, cf.~Section \ref{sec:hss}.
\item \label{asm:skeleton-struct} \textbf{Skeleton structure:}
The skeleton set for any box may be chosen from within a thin
layer of points close to the boundary of the box, cf.~Section \ref{sec:skeleton_construction}.
\item \label{asm:compressed-blocks} \textbf{Compressed block structure:}
Experimental evidence and physical intuition from scattering problems
allows us to assume that the blocks of $F$ and $E$ discussed in
Section~\ref{sec:inversion} have one-dimensional HSS or low-rank
structure, with logarithmic rank growth ( $O(\log(n_{\ell}))$ ).
\end{enumerate}
\noindent
These assumptions arise naturally in the context of
solving integral equations with non-oscillatory PDE kernels in 2D. All
assumptions excluding the last one are relevant for both dense and
compressed block algorithms. The last one is needed only for the
compressed-block algorithms.
We note assumption \ref{asm:skeleton-struct} implies \ref{asm:skeleton}: being able to pick skeletons from a thin boundary layer determines how their sizes scale. We mention them separately to distinguish their roles on the design and complexity analysis of our algorithms: while \ref{asm:skeleton} mainly impacts block sizes on the outer HSS structure, \ref{asm:skeleton-struct} is much more specific and refers to a priori knowledge of skeleton set structure which we exploit extensively in the compressed-block algorithm.
\subsection{Estimates}
\label{sec:estimates}
We analyze work and storage for the algorithms of Section~\ref{sec:inversion}. Since they all use the same
set of fast operations (see Section \ref{sec:fast_arithmetic}), we can make unifying observations:
\subsubsection*{Work}
\begin{enumerate}
\item Assumptions \ref{asm:skeleton} and \ref{asm:skeleton-struct}
imply that our fast subroutines perform operations with HSS blocks of
size $O(n_{\ell}^{1/2})$.
Further, Assumption~\ref{asm:compressed-blocks} states that these
behave like HSS matrices representing boundary integral operators,
for which all one-dimensional HSS operations are known to be linear
in matrix size. Thus, all \textbf{HSS1D} operations are $O(n_{\ell}^{1/2})$, including matrix application.
\item As indicated in Remark~\ref{rem:matrix-matrix}, for an HSS
matrix of size $k \times k$, products of HSS and low-rank matrices
require $O(kq)$ work. Assumption~\ref{asm:compressed-blocks} implies
all such products are $O(n_{\ell}^{1/2} \log_2(n_{\ell}))$. Low-rank
matrix matrix-vector multiplication has the same complexity.
\item Finally, both \textbf{LR\_to\_HSS1D} and \textbf{Rand\_ID}
involve interpolative decompositions of a matrix of size
$O(n_{\ell}^{1/2}) \times O(\log_2(n_{\ell}))$, and, therefore have complexity $O(n_{\ell}^{1/2} \log^2_2(n_{\ell}))$. Products between low-rank matrices are also of this complexity.
\end{enumerate}
\subsubsection*{Storage}
\begin{enumerate}
\item Again, since all HSS blocks behave as operators acting on one-dimensional
box boundaries, storage is linear with respect to the number of nodes along the
boundary of the box: $O(n_{\ell}^{1/2})$.
\item A low-rank matrix of size $m \times n$ and rank $q$ occupies
$O((m+n)q)$ space in storage. By Assumption~\ref{asm:compressed-blocks}, storage of low rank blocks ($\ttA[L],\ttA[R]$ and off-diagonal blocks of F) is $O(n_{\ell}^{1/2} \log_2(n_{\ell}))$.
\end{enumerate}
Algorithms~\ref{alg:inter-lowrank}, \ref{alg:build-inv}, and \ref{alg:builde},
require only the operations listed above. We observe that all algorithms contain
at least one $O(n_{\ell}^{1/2} \log^2_2(n_{\ell}))$ operation. In terms of storage,
\textbf{INTER\_LOWRANK} and \textbf{BUILD\_Finv} store both HSS and low-rank blocks
($O(n_{\ell}^{1/2} \log_2(n_{\ell}))$), and \textbf{BUILD\_E} one HSS block ($O(n_{\ell}^{1/2})$).
Hence, the \emph{compressed-block} interpolation operator build and inverse compression
algorithms perform $W^{CB}_{\ell} =
O(n_{\ell}^{1/2} \log^2_2(n_{\ell}))$ work per box and require $M^{CB}_{\ell} = O(n_{\ell}^{1/2} \log_2(n_{\ell}))$ storage for each set of matrices computed at level $\ell$. As a contrast, their \emph{dense-block} counterparts have $W^{DB}_{\ell} = O(n_{\ell}^{3/2})$ and $M_{\ell}^{DB} = O(n_{\ell})$.
We note that a more detailed complexity analysis may be performed to
obtain constants for each subroutine, given the necessary experimental
data about our kernel for a given accuracy $\acc$. The specific
dependance of these constants on accuracy is briefly discussed and
tested in Section~\ref{sec:vary-accuracy}.
We summarize the complexity estimates in the following proposition
\begin{proposition}
\label{prop:complexity}
{\textbf{Dense-Block Algorithms}} Let $\cA[A]$ be an $N \times N$ system matrix such that assumptions 1-3 hold.
Then, the dense-block tree build and inverse compression algorithms perform $O(N^{3/2})$ work. For NTI kernels, storage requirements and matrix apply are both $O(N \log N)$. For TI kernels, storage is $O(N)$, and matrix apply is $O(N \log N)$.
\textbf{NTI Compressed-Block Algorithms} Let $\cA[A]$ be an $N \times N$, non translation invariant system matrix such
that assumptions 1-4 hold. Then compressed-block tree build, inverse compression and HSS apply all perform $O(N)$ work,
and require $O(N)$ storage.
\textbf{TI Compressed-Block Algorithms} Let $\cA[A]$ be an $N \times N$,
translation invariant system matrix such that assumptions 1-4 hold.
Then compressed-block tree build, inverse compression and HSS apply
all perform $O(N)$ work, and require $O(N)$ storage. In fact, inverse compression work and storage are sublinear:
$O(N^{1/2} \log^2_2 N)$ and $O(N^{1/2} \log_2 N)$, respectively.
\end{proposition}
The limitations of dense-block algorithms now become clear. With notation as in Lemma \ref{lemma:complexity},
dense-block algebra corresponds to $(p,q) = (3/2,0)$ for work and $(p,q) = (1,0)$ for storage, which precludes
overall linear complexity. For the compressed block algorithms, on the other hand, we have
$(p,q)=(1/2,2)$ for work and $(p,q)=(1/2,1)$ for storage, which does yield linear complexity.
\begin{remark}
As we have previously observed, the compressed-block algorithm may be generalized to system matrices with other rank growth and behavior. More specifically, we observe that a sufficient condition to maintain optimal complexity could be that the outer HSS structure ranks grow as $O(n_{\ell}^p)$, for $p<1$, and that the most expensive operations such as the randomized interpolative decomposition also remain being sublinear. This would require all low-rank blocks to be of rank $q_{\ell} \sim n_{\ell}^{min\{1/3,(p-1)/2p\}}$.
\end{remark}
\begin{remark}
In all our practical implementations of the compressed-block algorithms, we perform dense computations for blocks up to a fixed threshhold skeleton size $k_{\rm cut}$, after which we switch to the fast routines. This may then be tuned as a parameter to further speed up these algorithms, and a straight-forward computation shows it does not alter their computational complexity.
\end{remark} | 115,628 |
Editorial Board
Alfio Ferlito
Editor-in-Chief
Faculty of Medicine (2003-2009), Professor and Chairman, ENT Clinic
Udine University Faculty of Medicine (1997-2013), Italy
Abdallah BADOU
Professor of Immunology, Faculty of Medicine and Pharmacy-Casablanca
University Hassan II Casablanca, Morocco
Ana Isabel Rocha Faustino
Lusophone University of Humanities and Technologies, Portugal
-
Arnab Ghosh
Postdoctoral Fellow, Cancer Research Unit
Missouri, USA
Stoycho Dimitrov Stoev
Faculty of Veterinary Medicine
Trakia University, Bulgaria
Arun Kanakkanthara
Devision of Oncolcogy, Mayo Clinic
USA
Antoni Llueca
Multidisciplinary Unit for Abdomino-Pelvic Oncology Surgery (MUAPOS), Department of Medicine
University General Hospital of Castellon, Castellon, Spain | 319,524 |
\begin{document}
\def\smfbyname{}
\begin{abstract}
In this paper, we will give an upper bound of the number of auxiliary hypersurfaces in the determinant method, which reformulates a work of Per Salberger by Arakelov geometry. One of the key constants will be determined by the pseudo-effective threshold of certain line bundles.
\end{abstract}
\begin{altabstract}
Dans cet article, on donnera une majoration du nombre de hypersurfaces auxiliaires dans la m\'ethode de d\'eterminant, qui reformule un travail de Per Salberger par la g\'eom\'etrie d'Arakelov. Une des constantes cl\'ees sera d\'etermin\'ee par le seuill de pseudo-effectivit\'e de certains fibr\'es en droites.
\end{altabstract}
\maketitle
\tableofcontents
\section{Introduction}
Let $K$ be a number field, and $X\hookrightarrow\mathbb P^n_K$ be a projective variety. Let $\xi\in X(K)$, and $H_K(\xi)$ be a height of $\xi$ with respect to the above closed immersion, for example, the classic Weil height. A height function of rational points $H_K(\ndot)$ often evaluates the arithmetic complexity of rational points. Let $B\in\mathbb R$, and
\[S(X;B)=\{\xi\in X(K)|\;H_K(\xi)\leqslant B\}\]
be the set of rational points of bounded heights with respect to the above closed immersion. Usually, a good height function has the so-called Northcott's property, which means that the cardinality $N(X;B)=\#S(X;B)$ is finite when $B$ is fixed. In this case, the map $N(X;\cdot):\;\mathbb R\rightarrow \mathbb N$ is a function which gives a description of the density of rational points in $X$.
It is a central subject to understand different kinds of properties of the function $N(X;B)$ with the variable $B\in\mathbb R$ for different kinds of $X$. For this target, lots of methods have been involved in. In this article, we will concern on the uniform upper bound of $N(X;B)$. The word "uniform" means that we want to obtain a good upper bound of $N(X;B)$ for all projective varieties with fixed degree and dimension, or maybe for those varieties satisfying certain common conditions.
\subsection{Determinant method}
In this article, we will focus on the so-called determinant method proposed in \cite{Heath-Brown} to study the density of rational points in arithmetic varieties.
\subsubsection{Basic ideas and the developments}
Tranditionally, we consider a projective variety $X\hookrightarrow\mathbb P^n_{\Q}$ over $\Q$ for simplicity, since the operations over arbitrary number fields sometimes bring us extra technical troubles. In \cite{Bombieri_Pila} (see also \cite{Pila95}), Bombieri and Pila proposed a method of determinant argument to study this kind of problems. The monomials of a certain degree evaluated on a family of rational points in $S(X;B)$ having the same reduction modulo some prime numbers form a matrix whose determinant is zero by a local estimate. By Siegal's Lemma, we can assure the existence of the hypersurfaces in $\mathbb P^n_{\Q}$ with bounded degree which contain all rational points in the family, but do not contain the generic point of $X$. If we can control the number of these auxiliary hypersurfaces and their maximal degree, it will play a significant role in controlling the upper bound of $N(X;B)$.
In \cite{Heath-Brown}, Heath-Brown generalized the method of \cite{Bombieri_Pila} to the higher dimensional case. His idea is to focus on a sub-set of $S(X;B)$ whose reductions modulo a prime number are a same regular point, and he proved that this sub-set can be covered by a bounded degree hypersurface which do not contain the generic point of $X$. Then he counted the number of regular points over finite fields, and control the regular reductions. In \cite{Broberg04}, Broberg generalized it to the case over arbitrary number fields.
In \cite{Heath-Brown}, Heath-Brown also proposed a so-called \textit{the dimension growth conjecture}. Let $\dim(X)=d$. It is said that for all $d\geqslant2$ and $\delta\geqslant2$, we have $N(X;B)\ll_{d,\delta,\epsilon}B^{d+\epsilon}$ for all $\epsilon>0$. He proved this conjecture for some special cases. Later, Browning, Heath-Brown and Salberger had some contributions on this subject, see \cite{Browning_Heath06I,Browning_Heath06II,Bro_HeathB_Salb,Salberger07,Salberger_preprint2013} for the refinements of the determinant method and the proofs under certain conditions.
In \cite{Walsh_2015}, Walsh refined the so-called global determinant method in firstly proposed by Salberger in \cite{Salberger_preprint2013}. He applied the global determinant to a series of hypersurfaces, whose equations of definition satisfy certain conditions, and then he obtained a better estimate. In \cite{Cluckers2019}, Castryck, Cluckers, Dittmann and Nguyen refined \cite{Walsh_2015} on giving an explicit dependence on $\delta$, and obtained a better estimate of $N(X;B)$.
In \cite{Salberger2015}, Salberger considered the case of cubic hypersurfaces. In this case, we have a better estimate on a key invariant, so for this case, a better result than that in \cite{Bro_HeathB_Salb,Salberger_preprint2013} was obtained. Actually, this is the motivation of this article to explore the refinement of that invariant by the pseudo-effective thresholds of line bundles.
In order to study the density of rational points of a higher dimensional variety, it is also important to understand its lower dimensional sub-varieties of particular degrees, see \cite[Appendix]{Heath-Brown} of J.-L. Colliot-Th\'el\`ene, \cite[Appendix]{Browning_Heath06II} of J. M. Starr, and \cite[\S5, \S7]{Salberger07} for instance. Since it has no direct relation with the determinant method itself, we do not plan to study this issue in this article.
\subsubsection{Reformulation by Arakelov geometry}
In \cite{Chen1,Chen2}, H. Chen reformulated the works of Salberger \cite{Salberger07} by the slope method in Arakelov geometry. By this formulation, we replace the matrix of monomials by the evaluation map which sends a global section of a particular line bundle to a family of rational points. By the slope inequalities, we can control the height of the evaluation map in the slope method, which replaces the role of Siegal's lemma in controlling heights.
There are two advantages by the approach of Arakelov geometry. First, we can work over arbitrary number fields, while the classic formulation often matches well over $\Q$ only. Second, it is easier to obtain explicit estimates, since usually the constants obtained by the slope method are given explicitly.
But in this article, because of certain obstructions in the positivity of line bundles, we are not able to give effective estimates for all invariants.
\subsection{Application of the pseudo-effective threshold}
In a mini-course in the summer school "Arakelov Geometry and Diophantine applications" at Institut Fourier in 2017, and a mini-course in the thematic activity "Reinventing rational points" at Institut Henri Poincar\'e in 2019, Salberger gave lectures on the application of the pseudo-effective thresholds of certain line bundles on projective varieties to estimate the number of auxiliary hypersurfaces in the determinant method. In \cite{Salberger2015}, he has applied this idea to study the density of rational points in the complement of the union of all lines of cubic surfaces in $\mathbb P^3$.
In this article, we will reformulate the above works of Salberger by Arakelov geometry following the strategy of \cite{Chen1, Chen2}, where we will consider the case of general projective varieties.
\subsubsection{Pseudo-effective threshold}
Let $X\hookrightarrow\mathbb P^n_K$ be a projective variety over the number field $K$ of degree $\delta$ and dimension $d$, $\pi:\;\widetilde{X}\rightarrow X$ be the blow-up at the non-singular rational point $\eta$, $E$ is the exceptional divisor of this blow-up, $H$ be a Cartier divisor on $X$ given by a hyperplane section on $\mathbb P^n_K$, and $D,m\in\mathbb N$. We consider the sum
\begin{equation}\label{R(E)_introduction}
R(\eta,D)=\sum_{m=1}^\infty \dim_K H^0\left(\widetilde{X},D\pi^*H-mE\right),
\end{equation}
which plays a significant role in the refinement of determinant method in this article. Next, we denote
\begin{equation}\label{I_X_introduction}
I_X(H,\eta)=\int_0^\infty \vol (\pi^*H-\lambda E)d\lambda,
\end{equation}
where $\vol(\ndot)$ is the usual volume function of $\mathbb R$-divisors. To the author's knowledge, the invariant $I_X(H,\eta)$ is first introduced by Per Salberger in 2006 at a talk in Mathematical Science Research Institute (MSRI), Berkeley, USA. In \cite[\S4]{McKinnonRoth_2015}, D. Mckinnon and M. Roth also introduced this invariant for the research of Diophantine approximations over higher dimensional projective varieties, which is a generalization of Roth's theorem.
In Theorem \ref{R(E) by GIT}, we will prove the estimate
\begin{equation}\label{R(E)_introduction2}
R(\eta,D)=\frac{I_X(H,\eta)}{d!}D^{d+1}+O_{d,\delta}(D^d).
\end{equation}
By this fact, we can refine some former results on the determinant method.
\subsubsection{An improved upper bound of the number of auxiliary hypersurfaces}
Let $\mathscr X\hookrightarrow\mathbb P^n_{\O_K}$ be the Zariski closure of $X\hookrightarrow\mathbb P^n_K$, and $\p$ be a maximal ideal of $\O_K$ whose residue field is $\f_\p$. Let $\xi\in\mathscr X(\f_\p)$, and we denote by $S(X;B,\xi)$ the sub-set of $S(X;B)$ the reduction modulo $\p$ of whose Zariski closures in $\mathscr X$ is $\xi$. We can prove that the invariant $I_X(H,\eta)$ only depends on its reduction class if its reduction is regular. By Lemma \ref{I_X(H,xi) same}, if for the family of maximal ideals $\p_1,\ldots,\p_r$ of $\O_K$, the point $\xi_j$ is regular in $\mathscr X$ for all $j=1,\ldots,r$ and $\bigcap\limits_{j=1}^rS(X;B,\xi_j)\neq\emptyset$, then all $I_X(H,\xi_j)$ are equal, noted by $I_X(H,\xi_J)$ for simplicity. Then we have the result below, and Salberger has proved the case of $K=\Q$ announced in the talks and lectures mentioned above.
\begin{theo}[Theorem \ref{semi-global determinant method mid}]\label{semi-golbal determinant method introduction}
We keep all the above notations. Let $\p_1,\ldots,\p_r$ be a family of maximal ideals of $\O_K$, $N(\p_j)=\#\left(\O_K/\p_j\right)$, and $\epsilon>0$. Suppose that the point $\xi_j\in\mathscr X(\f_{\p_j})$ is regular in $\mathscr X$ for all $j=1,\ldots,r$. If the inequality
\[\sum_{j=1}^r\log N(\p_j)\gg_{K,n,\delta,\epsilon}\frac{\delta}{I_X(H,\xi_J)}\log B\]
is verified, then there exists a hypersurface of degree $O_{d,\delta,\epsilon}(1)$, which covers $\bigcap\limits_{j=1}^rS(X;B,\xi_j)$ but do not contain the generic point of $X$.
\end{theo}
The implicit constant in Theorem \ref{semi-golbal determinant method introduction} will be given explicitly in Theorem \ref{semi-global determinant method mid}. By this result, let $\epsilon>0$ and
\[I_X(H)=\inf\limits_{\begin{subarray}{c} \eta\in S(X;B)\\ \eta\hbox{ regular}\end{subarray}}I_X(H,\eta).\]
Then we have the following estimate of the number of auxiliary hypersurfaces, which is also proved by Salberger for the case of $K=\Q$ announced in the talks and lectures mentioned above without the application of Arakelov geometry.
\begin{theo}[Theorem \ref{number of hypersurfaces}]\label{number of auxiliary hypersurface_introduction}
With all the notations above. There exists an explicit constant $C_4(\epsilon, \delta, n,d, K)$ such that $S(X;B)$ is covered by no more than
\begin{equation}\label{number of hypersurfaces_introduction}
C_4(\epsilon, \delta, n,d, K)B^{\frac{(1+\epsilon)d\delta}{I_X(H)}}
\end{equation}
hypersurfaces of degree $O_{n,\delta,\epsilon}(1)$ which do not contain the generic point of $X$.
\end{theo}
By \cite[Corollary 4.2]{McKinnonRoth_2015}, for every regular closed point $\eta$ in $X$, we have
\[I_X(H,\eta)\geqslant\frac{d\delta^{1+\frac{1}{d}}}{(d+1)}.\]
So the upper bound of auxiliary hypersurfaces given in \eqref{number of hypersurfaces_introduction} can be considered as a refinement of some former results (\cite{Heath-Brown, Salberger07, Chen2}, for example). If we focus on some particular varieties $X$ with clearer information on $I_X(H,\eta)$ defined at \eqref{I_X_introduction}, we may obtain a better estimate on the number of auxiliary hypersurfaces, see \cite{Salberger2015} for such an example, where the case of cubic hypersurfaces in $\mathbb P^3$ is considered.
\subsubsection{Non-effective estimate}
In the above argument, we have
\[\dim_KH^0\left(\widetilde{X},D\pi^*H-mE\right)=\frac{D^d}{d!}\vol\left(\pi^*H-\frac{m}{D}E\right)+O_{d,\delta}(D^{d-1}).\]
However, up to the author's knowledge, we cannot obtain an effective estimate above. Thus we are only able to make sure that the maximal degree of auxiliary hypersurfaces can depend only on $n$, $\delta$ and $\epsilon$, but we cannot get an explicit bound up to the author's ability until now.
\subsection{Organization of the article}
This article is organized as follows. In \S2, we will recall some useful preliminary knowledge to this program and propose the basic setting, where we follow the approach of \cite{Chen1,Chen2}. In \S3, we will give a bound involving the invariant $R(\eta,D)$ defined at \eqref{R(E)_introduction} and both geometric and arithmetic Hilbert-Samuel functions of arithmetic varieties, which is a generalization of \cite[Lemma 16.9]{Salberger2015}. In \S4, we will prove the finiteness of the sum \eqref{R(E)_introduction} and the asymptotic estimate \eqref{R(E)_introduction2}. In \S5, we will prove Theorem \ref{semi-golbal determinant method introduction}, and give the upper bound \eqref{number of hypersurfaces_introduction} in Theorem \ref{number of auxiliary hypersurface_introduction} by applying it.
\subsection*{Acknowledgement}
This work is inspired by a mini-course of Prof. Per Salberger in the summer school "Arakelov Geometry and Diophantine applications" at Institut Fourier in 2017, and a mini-course in the thematic activity "Reinventing rational points" at Institut Henri Poincar\'e in 2019. I would like to thank Prof. Salberger for introducing me his brilliant work \cite{Salberger_preprint2013} and some useful personal notes, and also for lots of useful private discussion with me. At the same time, I would like to thank Prof. Yuji Odaka for some useful suggestions on the pseudo-effective thresholds. Chunhui Liu was supported by JSPS KAKENHI Grant Number JP17F17730, and is supported by JSPS grant (S) 16H06335 now.
\section{Preliminaries and the basic setting}
In this section, we will provide the preliminary knowledge in order to formulate the determinant method by Arakelov geometry, where we follow the strategy of H. Chen in \cite{Chen1,Chen2}. We will also present some other useful preliminary knowledge.
\subsection{Classic height function of rational points}
Let $K$ be a number field, and $\O_K$ be its ring of integers. We denote by $M_{K,f}$ the set of finite places of $K$, and by $M_{K,\infty}$ the set of infinite places of $K$. In addition, we denote by $M_K=M_{K,f}\sqcup M_{K,\infty}$ the set of places of $K$. For every $v\in M_{K,f}$, if $\Q_v$ is the $p$-adic field, we define the absolute value $|x|_v=\left|N_{K_v/\Q_v}(x)\right|_p^\frac{1}{[K_v:\Q_v]}$, where $|\ndot|_p$ is the $p$-adic absolute value. For every $v\in M_{K,\infty}$, we define $|x|_v=\left|N_{K_v/\Q_v}(x)\right|^\frac{1}{[K_v:\Q_v]}$, where $|\ndot|$ is the usual absolute values over $\mathbb R$ or $\mathbb C$.
For every $a\in K^\times$, we have the \textit{product formula} (cf. \cite[Chap. III, Proposition 1.3]{Neukirch})
\begin{equation}\label{product formula}
\prod_{v\in M_K}|a|_v^{[K_v:\Q_v]}=1.
\end{equation}
Let $\xi=[\xi_0:\cdots:\xi_n]\in\mathbb P^n_K(K)$. We define the \textit{absolute height} of $\xi$ in $\mathbb P^n_K$ as
\begin{equation}\label{classic absolute height}
H_K(\xi)=\prod_{v\in M_K}\max_{0\leqslant i\leqslant n}\left\{|\xi_i|_v\right\}^{[K_v:\Q_v]}.
\end{equation}
Next, we define the \textit{logarithmic height} of $\xi$ as
\begin{equation}\label{log height}
h(\xi)=\frac{1}{[K:\Q]}\log H_K(\xi),
\end{equation}
which is independent of the choice of $K$ (cf. \cite[Lemma B.2.1]{Hindry}).
Suppose $X$ is a closed integral sub-scheme of $\mathbb P^n_K$ of degree $\deg(X)=\delta$ and dimension $\dim(X)=d$, and $\phi:X\hookrightarrow\mathbb P^n_K$ is the projective embedding. For $\xi\in X(K)$, we define $H_K(\xi)=H_K\left(\phi\left(\xi\right)\right)$ for simplicity, and usually we omit the closed immersion $\phi$ if there is no confusion. Next, we define
\[S(X;B)=\{\xi\in X(K)|H_K(\xi)\leqslant B\},\hbox{ and } N(X;B)=\#S(X;B).\]
By the Northcott's property (cf. \cite[Theorem B.2.3]{Hindry}), the cardinality $N(X;B)$ is finite for a fixed real number $B\geqslant1$.
The objective of counting rational points of bounded height is to understand the function $N(X;B)$ with some particular projective varieties $X$ and real numbers $B\geqslant1$.
\subsection{Multiplicity of points in a scheme}\label{local algebra}
In this part, we will define the multiplicity of closed points in schemes induced by the local Hilbert-Samuel function. This notion will be useful in the determinant method.
Let $X$ be a Noetherian scheme of pure dimension $d$, which means all its irreducible components have the same dimension. Let $\xi$ be a closed point of $X$, $\sm_{X,\xi}$ be the maximal ideal of the local ring $\O_{X,\xi}$, and $\kappa(\xi)$ be its residue field. We define
\begin{equation}\label{local hilbert of a closed point}
H_\xi(s)=\dim_{\kappa(\xi)}\left(\sm_{X,\xi}^s/\sm_{X,\xi}^{s+1}\right)
\end{equation}
as the \textit{local Hilbert-Samuel function} of $X$ at the closed point $\xi$ with the variable $s\in\mathbb N$, where we define $\sm_{X,\xi}^0=\O_{X,\xi}$ for simplicity. For this function, when $d\geqslant2$, we have the polynomial asymptotic extension
\[H_\xi(s)=\frac{\mu_\xi(X)}{(d-1)!}s^{d-1}+O(s^{d-2}),\]
where we define the positive integer $\mu_\xi(X)$ as the \textit{multiplicity} of point $\xi$ in $X$. If $d=1$, then $\O_{X,\xi}$ is a local Artinian ring. The multiplicity $\mu_\xi(X)$ is then defined as the length of the local ring $\O_{X,\xi}$ as a $\O_{X,\xi}$-module.
If $\O_{X,\xi}$ is a regular local ring, we say that $\xi$ is \textit{regular} in $X$. In this case we have $\mu_\xi(X)=1$. Else we say that $\xi$ is \textit{singular} in $X$. We refer the reader for \cite[Exercise 2.4]{Roberts98} in Page 41 for an example of non-regular local ring of multiplicity $1$. In addition, if $X$ is pure dimensional and has no embedded component, then from the fact that $\xi$ is singular in $X$ by the above definition, we deduce $\mu_\xi(X)\geqslant2$ (cf. \cite[(40.6)]{Nagata62}).
We denote by $X^{\mathrm{reg}}$ the regular locus of $X$, and by $X^{\mathrm{sing}}$ the singular locus of $X$. By the semi-continuity of the multiplicity function, the singular locus $X^{\mathrm{sing}}$ is a closed sub-set of $X$. If $X$ is reduced and pure dimensional, the set $X^{\mathrm{reg}}$ is open dense in $X$ (cf. \cite[Corollary 8.16, Chap. II]{GTM52}).
\subsection{Normed vector bundles}
The normed vector bundle is one of the main research objects in Arakelov geometry. Let $K$ be a number field and $\O_K$ be its ring of integers. A \textit{normed vector bundle} on $\spec\O_K$ is a pair $\E=\left(E,\left(\|\ndot\|_v\right)_{v\in M_{K,\infty}}\right)$, where:
\begin{itemize}
\item $E$ is a projective $\O_K$-module of finite rank;
\item $\left(\|\ndot\|_v\right)_{v\in M_{K,\infty}}$ is a family of norms, where $\|\ndot\|_v$ is a norm over $E\otimes_{\O_K,v}\C$ which is invariant under the action of $\gal(\C/K_v)$.
\end{itemize}
If the norms $\left(\|\ndot\|_v\right)_{v\in M_{K,\infty}}$ are Hermitian for all $v\in M_{K,\infty}$, we call $\E$ a \textit{Hermitian vector bundle} over $\spec\O_K$. In particular, if $\rg_{\O_K}(E)=1$, we say that $\E$ is a \textit{Hermitian line bundle} since all Archimedean norms are Hermitian in this case.
Suppose that $F$ is a sub-$\O_K$-module of $E$. We say that $F$ is a \textit{saturated} sub-$\O_K$-module of $E$ if $E/F$ is a torsion-free $\O_K$-module.
Let $\E=\left(E,\left(\|\ndot\|_{E,v}\right)_{v\in M_{K,\infty}}\right)$ and $\F=\left(F,\left(\|\ndot\|_{F,v}\right)_{v\in M_{K,\infty}}\right)$ be two Hermitian vector bundles on $\spec\O_K$. If $F$ is a saturated sub-$\O_K$-module of $E$ and $\|\ndot\|_{F,v}$ is the restriction of $\|\ndot\|_{E,v}$ over $F\otimes_{\O_K,v}\C$ for every $v\in M_{K,\infty}$, we say that $\F$ is a \textit{sub-Hermitian vector bundle} of $\E$ on $\spec\O_K$.
We say that $\G=\left(G,\left(\|\ndot\|_{G,v}\right)_{v\in M_{K,\infty}}\right)$ is a \textit{quotient Hermitian vector bundle} of $\E$ on $\spec\O_K$, if for every $v\in M_{K,\infty}$, the module $G$ is a projective quotient $\O_K$-module of $E$ and $\|\ndot\|_{G,v}$ is the induced quotient space norm of $\|\ndot\|_{E,v}$.
For simplicity, we denote by $E_K=E\otimes_{\O_K}K$ in the remainder part of this article.
\subsection{Arakelov invariants}
We will introduce some useful invariants in Arakelov geometry in this part.
\subsubsection{Arakelov degree}
Let $\E$ be a Hermitian vector bundle on $\spec\O_K$, and $\{s_1,\ldots,s_r\}$ be a $K$-basis of the vector space $E_K$. The \textit{Arakelov degree} of $\E$ is defined as
\begin{eqnarray*}
\adeg(\E)&=&-\sum_{v\in M_{K}}[K_v:\Q_v]\log\left\|s_1\wedge\cdots\wedge s_r\right\|_v\\
&=&\log\left(\#\left(E/\O_Ks_1+\cdots+\O_Ks_r\right)\right)-\frac{1}{2}\sum_{v\in M_{K,\infty}}\log\det\left(\langle s_i,s_j\rangle_{v,1\leqslant i,j\leqslant r}\right),
\end{eqnarray*}
where $\left\|s_1\wedge\cdots\wedge s_r\right\|_v$ follows the definition in \cite[2.1.9]{Chen10b} for all $v\in M_{K,\infty}$, and $\langle s_i,s_j\rangle_{v,1\leqslant i,j\leqslant r}$ is the Gram matrix of the basis $\{s_1,\ldots,s_r\}$ with respect to $v\in M_{K,\infty}$. For those $v\in M_{K,f}$, we take the norms defined by models.
We refer the readers to \cite[2.4.1]{Gillet-Soule91} for a proof of the equivalence of the above two definitions. The Arakelov degree is independent of the choice of the basis $\{s_1,\ldots,s_r\}$ by the product formula \eqref{product formula}. In addition, we define
\[\adeg_n(\E)=\frac{1}{[K:\Q]}\adeg(\E)\]
as the \textit{normalized Arakelov degree} of $\E$, which is independent of the choice of the base field $K$.
\subsubsection{Slope}
Let $\E$ be a non-zero Hermitian vector bundle on $\spec\O_K$, and $\rg(E)$ be the rank of $E$. The \textit{slope} of $\E$ is defined as
\[\wmu(\E):=\frac{1}{\rg(E)}\adeg_n(\E).\]
In addition, we denote by $\wmu_{\max}(\E)$ the maximal slope of all its non-zero Hermitian sub-bundles, and by $\wmu_{\min}(\E)$ the minimal slope of all its non-zero Hermitian quotients bundles of $\E$.
\subsubsection{Height of linear maps}
Let $\E$ and $\F$ be two non-zero Hermitian vector bundles on $\spec\O_K$, and $\phi:\; E_K\rightarrow F_K$ be a non-zero homomorphism of $K$-vector spaces. The \textit{height} of $\phi$ is defined as \[h(\phi)=\frac{1}{[K:\Q]}\sum_{v\in M_K}\log\|\phi\|_v,\]
where $\|\phi\|_v$ is the operator norm of $K_v$-linear map $\phi_v:E\otimes_KK_v\rightarrow F\otimes_KK_v$ induced by the above linear homomorphism with respect to every $v\in M_K$.
We refer the readers to \cite[Appendix A]{BostBour96} for some equalities and inequalities on Arakelov degrees and corresponding heights of homomorphisms.
\subsection{Arithmetic Hilbert-Samuel function}\label{basic setting}
Let $\overline{\mathcal E}$ be a Hermitian vector bundle of rank $n+1$ on $\spec\O_K$, and $\mathbb P(\sE)$ be the projective space which represents the functor from the category of commutative $\O_K$-algebras to the category of sets mapping all $\O_K$-algebra $A$ to the set of projective quotient $A$-module of $\sE\otimes_{\O_K}A$ of rank $1$.
Let $\O_{\mathbb P (\sE)}(1)$ (or by $\O(1)$ if there is no confusion) be the universal bundle, and $\O_{\mathbb P (\sE)}(D)$ (or by $\O(D)$) be the line bundle $\O_{\mathbb P (\sE)}(1)^{\otimes D}$ for simplicity. The Hermitian metrics on $\sE$ induce by quotient of Hermitian metrics (i.e. Fubini-Study metrics) on $\O_{\mathbb P(\sE)}(1)$ which define a Hermitian line bundle $\overline{\O_{\mathbb P(\sE)}(1)}$ on $\mathbb P(\sE)$.
For every $D\in\mathbb N^+$, let
\begin{equation}\label{definition of E_D}
E_D=H^0\left(\mathbb P(\sE),\O_{\mathbb P(\sE)}(D)\right),
\end{equation} and let $r(n,D)$ be its rank over $\O_K$. In fact, we have
\begin{equation}\label{def of r(n,D)}
r(n,D)={n+D\choose D}.
\end{equation}
For each $v\in M_{K,\infty}$, we denote by $\|\ndot\|_{v,\sup}$ the norm over $E_{D,v}=E_D\otimes_{\O_K,v}\C$ such that
\begin{equation}\label{definition of sup norm}
\forall s\in E_{D,v},\;\|s\|_{v,\sup}=\sup_{x\in\mathbb P(\sE_K)_v(\C)}\|s(x)\|_{v,\mathrm{FS}},
\end{equation}
where $\|\ndot\|_{v,\mathrm{FS}}$ is the corresponding Fubini-Study norm.
Next, we will introduce the \textit{metric of John}, see \cite{Thompson96} for a systematic introduction to this notion. In general, for a given symmetric convex body $C$, there exists the unique ellipsoid $J(C)$, called \textit{ellipsoid of John}, contained in $C$ whose volume is maximal.
For the $\O_K$-module $E_D$ and any place $v\in M_{K,\infty}$, we take the ellipsoid of John of its unit closed ball defined via the norm$\|\ndot\|_{v,\sup}$, and this ellipsoid induces a Hermitian norm, noted by $\|\ndot\|_{v,\mathrm{John}}$. For every section $s\in E_{D}$, the inequality
\begin{equation}\label{john norm}
\|s\|_{v,\sup}\leqslant\|s\|_{v,\mathrm{John}}\leqslant\sqrt{r(n,D)}\|s\|_{v,\sup}
\end{equation}
is verified by \cite[Theorem 3.3.6]{Thompson96}. In fact, these constants do not depend on the choice of the symmetric convex body.
Let $A$ be a ring, and $E$ be an $A$-module. We denote by $\sym^D_{A}(E)$ the symmetric product of degree $D$ of the $A$-module $E$, or by $\sym^D(E)$ if there is no confusion on the base ring.
If we consider the above $E_D$ defined in \eqref{definition of E_D} as a $\O_K$-module, we have the isomorphism of $\O_K$-modules $E_D\cong \sym^D(\mathcal{E})$. Then for every place $v\in M_{K,\infty}$, the Hermitian norm $\|\ndot\|_v$ over $\mathcal{E}_{v,\C}$ induces a Hermitian norm $\|\ndot\|_{v,\mathrm{sym}}$ by the symmetric product. More precisely, this norm is the quotient norm induced by the quotient morphism
\[\sE^{\otimes D}\rightarrow\sym^D(\sE),\]
where the vector bundle $\overline{\sE}^{\otimes D}$ is equipped with the norms of tensor product of $\overline{\sE}$ on $\spec\O_K$ (see \cite[D\'efinition 2.10]{Gaudron08} for the definition). We say that this norm is the \textit{symmetric norm} over $\sym^D(\sE)$. For any place $v\in M_{K,\infty}$, the norms $\|\ndot\|_{v,\mathrm{John}}$ and $\|\ndot\|_{v,\mathrm{sym}}$ are invariant under the action of the unitary group $U(\sE_{v,\C},\|\ndot\|_v)$ of order $n+1$. Then they are proportional and the ratio is independent of the choice of $v\in M_{K,\infty}$ (see \cite[Lemma 4.3.6]{BGS94} for a proof). We denote by $R_0(n,D)$ the constant such that, for every section $0\neq s\in E_{D,v}$, the equality
\begin{equation}\label{symmetric norm vs John norm}
\log\|s\|_{v,\mathrm{John}}=\log\|s\|_{v,\mathrm{sym}}+R_0(n,D).
\end{equation}
is verified.
\begin{defi}\label{definition of E_D with norm}
Let $E_D$ be the $\O_K$-module defined at \eqref{definition of E_D}. For every place $v\in M_{K,\infty}$, we denote by $\E_D$ the Hermitian vector bundle on $\spec\O_K$ which $E_D$ is equipped with the norm of John $\|\ndot\|_{v,\mathrm{John}}$ induced by the norms $\|\ndot\|_{v,\sup}$ defined in \eqref{definition of sup norm}. Similarly, we denote by $\E_{D,\mathrm{sym}}$ the Hermitian vector bundle on $\spec\O_K$ where $E_D$ is equipped with the norms $\|\ndot\|_{v,\mathrm{sym}}$ introduced above.
\end{defi}
With all the notations in Definition \ref{definition of E_D with norm}, we have the following result.
\begin{prop}[\cite{Chen1}, Proposition 2.7]\label{symmetric norm vs John norm, constant}
With all the notations in Definition \ref{definition of E_D with norm}, we have
\[\wmu_{\min}(\E_D)=\wmu_{\min}(\E_{D,\mathrm{sym}})-R_0(n,D).\]
In the above equality, the constant $R_0(n,D)$ defined in the equality \eqref{symmetric norm vs John norm} satisfies the inequality
\begin{equation*}
0\leqslant R_0(n,D)\leqslant\log\sqrt{r(n,D)},
\end{equation*}
where the constant $r(n,D)=\rg(E_D)$ follows the definition in the equality \eqref{def of r(n,D)}.
\end{prop}
Let $X$ be a pure dimensional closed sub-scheme of $\mathbb{P}(\mathcal{E}_K)$, and $\mathscr{X}$ be the Zariski closure of $X$ in $\mathbb{P}(\mathcal{E})$. We denote by
\begin{equation}\label{evaluation map}
\eta_{X,D}:\;E_{D,K}=H^0\left(\mathbb{P}(\mathcal{E}_K),\O(D)\right)\rightarrow H^0\left(X,\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\right)
\end{equation}
the \textit{evaluation map} over $X$ induced by the closed immersion of $X$ into $\mathbb P(\sE_K)$. In addition, we denote by $F_D$ the largest saturated sub-$\O_K$-module of $H^0\left(\mathscr{X},\O_{\mathbb P(\sE)}(1)|_\mathscr{X}^{\otimes D}\right)$ such that $F_{D,K}=\im(\eta_{X,D})$. When the integer $D$ is large enough, the homomorphism $\eta_{X,D}$ is surjective, which means $F_D=H^0(\mathscr{X},\O_{\mathbb P(\sE)}(1)|_\mathscr{X}^{\otimes D})$ (cf. \cite[Chap. III, Theomrem 5.2 (b)]{GTM52}). We refer the readers to \cite[Lemma 2.1]{Poonen2004} for an explicit lower bound of such a positive integer $D$.
The $\O_K$-module $F_D$ is equipped with the quotient metrics (from $\E_D$) such that $F_D$ is a Hermitian vector bundle on $\spec \O_K$, noted by $\F_D$ this Hermitian vector bundle. Moreover, in the remainder part of this article, we denote by $r_1(D)$ the rank of the $\O_K$-module $F_D$.
\begin{defi}\label{arithmetic hilbert function}
We denote by $\F_D$ the Hermitian vector bundle on $\spec\O_K$ defined above from the map \eqref{evaluation map}. We say that the function which maps the positive integer $D$ to $\wmu(\F_D)$ is the \textit{arithmetic Hilbert-Samuel function} of $X$ with respect to the Hermitian line bundle $\overline{\O_{\mathbb P(\sE)}(1)}$.
\end{defi}
\begin{rema}\label{definition of arakelov height}
With all the notations in Definition \ref{arithmetic hilbert function}. Let
\begin{equation}\label{definition of arakelov height}
h_{\overline{\O_{\mathbb P(\sE)}(1)}}(X)=\adeg_n\left(\widehat{c}_1\left(\overline{\O_{\mathbb P(\sE)}(1)}\right)^{d+1}\cdot\left[\mathscr X\right]\right).
\end{equation}
In fact, \eqref{definition of arakelov height} is a height of $X$ be defined by the arithmetic intersection theory (cf. \cite[Definition 2.5]{Faltings91}). By \cite[Th\'eor\`eme A]{Randriam06}, we have
\[h_{\overline{\O_{\mathbb P(\sE)}(1)}}(X)=\lim_{D\rightarrow+\infty}\frac{\adeg_n(\F_D)}{D^{d+1}/(d+1)!}.\]
\end{rema}
By \cite[Corollary 2.9]{Chen1}, we have the following trivial lower bound of $\wmu(\F_D)$, which is
\begin{equation}\label{trivial lower bound of F_D}
\wmu(\F_D)\geqslant-\frac{1}{2}D\log(n+1).
\end{equation}
\subsection{Height functions given by Arakelov theory}
We will give a definition of the height of rational points by Arakelov geometry, which provides the possibility of the reformulation of counting rational points problem by Arakelov geometry. Let $\overline {\mathcal E}$ be a Hermitian vector bundle of rank $n+1$ on $\spec\O_K$, $P\in \mathbb P(\mathcal E_K)(K)$, and $\mathcal P\in\mathbb P(\mathcal E)(\O_K)$ be its Zariski closure in $\mathbb P(\mathcal E)$. Let $\overline {\O_{\mathbb P(\mathcal E)}(1)}$ be the universal bundle equipped with the corresponding Fubini-Study metric at each $v\in M_{K,\infty}$, then $\mathcal P^*\overline {\O_{\mathbb P(\mathcal E)}(1)}$ is a Hermitian line bundle on $\spec\O_K$. We define the \textit{height} of the rational point $P$ as
\begin{equation}\label{arakelov height}
h_{\overline {\O_{\mathbb P(\mathcal E)}(1)}}(P)=\adeg_n\left(\mathcal P^*\overline {\O_{\mathbb P(\mathcal E)}(1)}\right).
\end{equation}
Actually, \eqref{arakelov height} is the same as the definition \eqref{definition of arakelov height} where we choose $X$ to be a rational point in $\mathbb P(\sE_K)$ considered as its closed integral sub-scheme.
\begin{rema}\label{Comparing heights}
We keep all the above notations in this part. Now we choose $\overline{\sE}=\left(\O_K^{\oplus(n+1)},\left(\|\ndot\|_v\right)_{v\in M_{K,\infty}}\right)$, where for every $v\in M_{K,\infty}$, $\|\ndot\|_v$ is the $\ell^2$-norm mapping $(t_0,\ldots,t_n)$ to $\sqrt{|v(t_0)|^2+\cdots+|v(t_n)|^2}$. We suppose that $P$ has a $K$-rational projective coordinate $[x_0:\cdots:x_n]$, then we have (cf. \cite[Proposition 9.10]{Moriwaki-book})
\begin{eqnarray*}
h_{\overline {\O_{\mathbb P(\mathcal E)}(1)}}(P)&=&\sum\limits_{v\in M_{K,f}}\frac{[K_v:\Q_v]}{[K:\Q]}\log \left(\max\limits_{1\leqslant i\leqslant n}|x_i|_v\right)\\
& &\;\;+\frac{1}{2}\sum\limits_{v\in M_{K,\infty}}\frac{[K_v:\Q_v]}{[K:\Q]}\log\left(\sum\limits_{j=0}^n|v(x_j)|^2\right).
\end{eqnarray*}
In addition, let the logarithmic height $h(\ndot)$ be as defined in \eqref{log height}. Then by some elementary calculation, the inequality
\[\left|h(P)-h_{\overline {\O_{\mathbb P(\mathcal E)}(1)}}(P)\right|\leqslant\frac{1}{2}\log(n+1)\]
is verified uniformly for all $P\in\mathbb P(\sE_K)$ when we choose $\overline{\sE}$ as above.
\end{rema}
\subsection{Further notations on counting rational points problem}
Let $\psi:X\hookrightarrow\mathbb P(\sE_K)$ be a closed immersion of $X$ in $\mathbb P(\sE_K)$, and $P\in X(K)$. We denote the height of $P$ by $h_{\overline {\O_{\mathbb P(\mathcal E)}(1)}}(\psi(P))$ at \eqref{arakelov height}. We will just use the notations $h_{\overline {\O_{\mathbb P(\mathcal E)}(1)}}(P)$, $h_{\overline {\O(1)}}(P)$ or $h(P)$ if there is no confusion of the morphism $\psi$ and the Hermitian line bundle $\overline{\O_{\mathbb P(\mathcal E)}(1)}$. This height also satisfies the Northcott's property for arbitrary Hermitian vector bundle $\overline{\sE}$ (cf. \cite[Theorem 5.3]{Yuan_ICCM2010}), so it can be used in the counting rational points problem. Actually, the line bundle $\O_{\mathbb P(\sE_K)}(1)$ can be replaced by arbitrary ample line bundle for the correctness of the Northcott's property.
In the rest part of this article, unless specially mentioning, we will use the height function defined at \eqref{arakelov height}, and we will use the notation $h(\ndot)$ to denote this height function. The classic height defined at \eqref{classic absolute height} and \eqref{log height} will not be essentially used any longer.
\section{A refinement of the determinant method}
In this section, we will generalize a result in the determinant method.
\subsection{Estimates of norms}\label{estimates of norms}
In this part, we will estimate the norms of some local homomorphisms, which can be viewed as a generalization of parts of \cite[\S 3]{Chen2}. The original idea comes from \cite[\S 16.2]{Salberger2015}. This estimate is more precise than that in \cite[Lemma 2.4]{Salberger07} and \cite[Proposition 3.4]{Chen2}, but will be more implicit because of some technical obstructions.
Firstly, we refer a useful auxiliary result in \cite{Chen2}, which will be useful in the approach of Arakelov geometry. Before this, we recall an useful notion. Let $(k,|\ndot|)$ be a non-Archimedean field, and $(V,\|\ndot\|)$ be normed vector space over $(k,|\ndot|)$. We say that $(V,\|\ndot\|)$ is \textit{ultranormed} if for all $x,y\in U$, we always have $\|x+y\|\leqslant \max\left\{\|x\|,\|y\|\right\}$.
\begin{lemm}[\cite{Chen2}, Lemma 3.3]\label{untrametric operator norm}
Let $k$ be a field equipped with a non-archimedean absolute value $|\ndot|$, $U$ and $V$ be two $k$-linear ultranormed spaces of finite rank and $\phi:U\rightarrow V$ be a $k$-linear homomorphism. Let $m=\dim_k(U)$. For any integer $1\leqslant i\leqslant m$, let
\[\lambda_i=\inf_{\begin{subarray}{c}W\subset U\\ \codim_U(W)=i-1\end{subarray}}\|\phi|_W\|.\]
If $i>m$, let $\lambda_i=0$. Then for any integer $r>0$, we have
\begin{equation}
\left\|\wedge^r\phi\right\|\leqslant\prod_{i=1}^r\lambda_i.
\end{equation}
\end{lemm}
In the rest part of this section, unless specially mentioned, we denote by $K$ a number field, and by $\O_K$ its ring of integers. We fix a Hermitian vector bundle $\overline{\sE}$ of rank $n+1$ on $\spec\O_K$, a closed integral sub-scheme $X$ of $\mathbb P(\sE_K)$, and the Zariski closure $\mathscr X$ of $X$ in $\mathbb P(\sE)$. We denote by $\mathscr X_K\rightarrow \spec K$ the generic fiber of $\mathscr X\rightarrow \spec\O_K$, which is essentially $X\rightarrow\spec K$ in the above case.
Let $\p$ be a maximal ideal of $\O_K$, $\f_\p$ be the residue field of $\O_K$ at $\p$. Let $\xi$ be an $\f_\p$-point of $\mathscr X$, and $k\in\mathbb N^+$. We suppose that $\{f_i\}_{1\leqslant i\leqslant k}$ is a family of local homomorphisms of $\O_{K,\p}$-algebras from $\O_{\mathscr X,\xi}$ to $\O_{K,\p}$. Let $\mathfrak a$ be the kernel of $f_1$, then we have $\O_{\mathscr X,\xi}/\mathfrak a\cong\O_{K,\p}$, which shows that $\mathfrak a$ is a prime ideal. Furthermore, since $\O_{\mathscr X,\xi}$ is a local ring with the maximal ideal $\sm_\xi$, we have $\sm_\xi\supseteq\mathfrak a$. Moreover, for $f_1$ is a local homomorphism, we have the fact $\mathfrak a+\p\O_{\mathscr X,\xi}=\sm_\xi$.
In addition, we suppose that the point $\xi$ is regular in $\mathscr X$, which means $\O_{\mathscr X,\xi}$ is a regular local ring. In this case, the ideal $\mathfrak a$ is generated by $\dim\left(\O_{\mathscr X,\xi}\right)-1$ regular parameters (cf. \cite[Proposition 4.10]{LNM146}). Since these elements form a regular sequence on $\O_{\mathscr X,\xi}$ (cf. \cite[Chap. III, Proposition 6]{SerreLocAlg}), we have $\sym^m(\mathfrak a/\mathfrak a^2)\cong\mathfrak a^m/\mathfrak a^{m+1}$ as free $\O_{K,\p}$-modules for all $m\geqslant0$ due to \cite[Chap. IV, \S2, Corollary 2.4]{FultonLang1985}, where we define $\mathfrak a^0=\O_{\mathscr X,\xi}$ for convenience.
Let $S=\O_{\mathscr X,\xi}\smallsetminus\mathfrak a$, and we denote by $R_{\mathscr X,\xi}=S^{-1}\left(\O_{\mathscr X,\xi}\right)$ the localization of $\O_{\mathscr X,\xi}$ at the prime ideal $\mathfrak a$. We denote by $m_\xi$ the maximal ideal of the ring $R_{\mathscr X,\xi}$, and then we have $m_\xi=\mathfrak a R_{\mathscr X,\xi}$ by the definition of this localization.
Let $u\in S$ and $r\in\mathfrak a^m$ for every $m\geqslant0$. If we have $ur\in \mathfrak a^{m+1}$, since $(u+\mathfrak a)(r+\mathfrak a^{m+1})=\mathfrak a^{m+1}$ is verified, then we have $r\in \mathfrak a^{m+1}$. Therefore, we obtain
\begin{equation}\label{m cap a}
m_\xi^{m+1}\cap\mathfrak a^m=\left(\mathfrak a^{m+1}\cdot R_{\mathscr X,\xi}\right)\cap\mathfrak a^m=\mathfrak a^{m+1}
\end{equation}
for all $m\geqslant0$.
Let $E$ be a free sub-$\O_{K,\p}$-module of finite type of $\O_{\mathscr X,\xi}$ and let
\begin{equation}\label{f_i}
f=(f_i|_E)_{1\leqslant i\leqslant k}:\;E\rightarrow\O_{K,\p}^k
\end{equation}
be an $\O_{K,\p}$-linear homomorphism. As $f_1$ is a homomorphism of $\O_{K,\p}$-algebras, it is surjective.
We consider $\left(E\cap\mathfrak a^j\right)/\left(E\cap \mathfrak a^{j+1}\right)$ and $\left(E\cap m_\xi^j\right)/\left(E\cap m_\xi^{j+1}\right)$ as two free $\O_{K,\p}$-modules, where we consider $E$ as a sub-$\O_{K,\p}$-module of $R_{\mathscr X,\xi}$ when necessary. Then we have the isomorphism
\begin{eqnarray}\label{isom of E cap m}
& &\left(E\cap\mathfrak a^j\right)/\left(E\cap \mathfrak a^{j+1}\right)\cong\left(E\cap\mathfrak a^j\right)/\left((E\cap \mathfrak a^{j})\cap (E\cap m_\xi^{j+1})\right)\\
&\cong&\left((E\cap\mathfrak a^j)+(E\cap m_\xi^{j+1})\right)/\left(E\cap m_\xi^{j+1}\right)\cong\left(E\cap m_\xi^j\right)/\left(E\cap m_\xi^{j+1}\right)\nonumber
\end{eqnarray}
by \eqref{m cap a}, where we use the fact $\mathfrak a^jR_{\mathscr X,\xi}+m_\xi^{j+1}=m_\xi^j$ in $R_{\mathscr X,\xi}$.
We suppose that the reductions of all the above local homomorphisms $f_1,\ldots,f_k$ modulo $\p$ are same, which means all the composed homomorphisms $\O_{\mathscr X,\xi}\xrightarrow{f_i} \O_{K,\p}\rightarrow\f_\p$ are same for every $i=1,\ldots,k$, where the last arrow is the canonical reduction morphism modulo $\p$. Let $N(\p)=\#\f_\p$. In this case, the restriction of $f$ on $E\cap \mathfrak a^j$ has its norm smaller than $N(\p)^{-j}$. In fact, for any $1\leqslant i\leqslant k$, we have $f_i(\mathfrak a)\subset\p\O_{K,\p}$ and hence we have $f_i(\mathfrak a^j)\subset\p^j\O_{K,\p}$.
From the above construction, we have the following result, which is a reformulation of \cite[Lemma 16.9]{Salberger2015}.
\begin{prop}\label{upper bound of operator norm}
Let $\p$ be a maximal ideal of $\O_K$, and $\xi\in\mathscr X(\f_\p)$ be a non-singular point. Suppose that $\{f_i\}_{1\leqslant i\leqslant k}$ is a family of local $\O_{K,\p}$-linear homomorphisms from $\O_{\mathscr X,\xi}$ to $\O_{K,\p}$ whose reductions module $\p$ are same. Let $E$ be a free sub-$\O_{K,\p}$-module of finite type of $\O_{\mathscr X,\xi}$ and $f=(f_i|_E)_{1\leqslant i\leqslant k}$ be as defined at \eqref{f_i}, $N(\p)=\#(\O_K/\p)$. We consider $E$ as a sub-$\O_{K,\p}$-module of $R_{\mathscr X,\xi}$, and let
\begin{equation}\label{R(E)}
\mathcal R_\xi(E)=\sum_{k=1}^\infty\dim_{K}\left(E\cap m_\xi^k\right)_{K}.
\end{equation}
Then if $r=\dim_K(E_K)$, we have
\[\log\|\wedge^rf_K\|\leqslant -\mathcal R_\xi(E)\log N(\p).\]
\end{prop}
\begin{proof}
By the above notations and argument, we have the filtration
\[\mathcal F:\; E\supset E\cap \mathfrak a\supset\cdots\supset E\cap \mathfrak a^j\supset E\cap \mathfrak a^{j+1}\supset\cdots\]
of $E$, whose $j$-th subquotient $\left(E\cap\mathfrak a^j\right)/\left(E\cap \mathfrak a^{j+1}\right)$ is a free $\O_{K,\p}$-module. The restriction of $f$ on $E\cap \mathfrak a^j$ has norm smaller than $N(\p)^{-j}$. Meanwhile, let $\{q_\xi(m)\}_{m=1}^\infty$ be the series of non-negative integers where the integer $m$ appears exactly
\[\dim_K\left(E\cap m_\xi^m\right)_K-\dim_K\left(E\cap m_\xi^{m+1}\right)_K\]
times. Then by the isomorphism \eqref{isom of E cap m}, the free $\O_{K,\p}$-modules $\left(E\cap\mathfrak a^j\right)/\left(E\cap\mathfrak a^{j+1}\right)$ and $\left(E\cap m_\xi^j\right)/\left(E\cap m_\xi^{j+1}\right)$ have the same rank for all $j\geqslant0$. Thus we have
\begin{equation}\label{q without wedge}
\inf_{\begin{subarray}{c}W\subset E_K\\ \codim_{E_K}(W)=j-1\end{subarray}}\|f_K|_W\|\leqslant N(\p)^{-q_\xi(j)}.
\end{equation}
Since the above filtration $\mathcal F$ is of finite length, then by some elementary calculation, we obtain the equality
\[\sum\limits_{m=1}^\infty q_\xi(m)=\sum\limits_{m=1}^\infty\dim_K\left(E\cap m_\xi^m\right)_K.\]
Finally by applying Lemma \ref{untrametric operator norm} to \eqref{q without wedge}, we obtain the result.
\end{proof}
\subsection{Existence of auxiliary hypersurfaces}
In this part, we will reformulate the determinant method by the slope method. Different from \cite[Theorem 3.2]{Salberger07} and \cite[Theorem 16.12]{Salberger2015}, our estimate will depend on the term $\mathcal R_\xi(E)$ for some special modules $E$ defined at \eqref{R(E)}. We will give an estimate of $\mathcal R_\xi(E)$ in \S4 for our application such that we can control the number of auxiliary hypersurfaces by this result. The strategy is similar to that of \cite[Theorem 3.1]{Chen2}.
The following slope equality is useful in this reformulation, which is obtained by the slope equalities and inequalities.
\begin{prop}[\cite{Chen1}, Proposition 2.2]\label{slope of evaluation map}
Let $\overline E$ be a Hermitian vector bundle of rank $r>0$ on $\spec\O_K$, and $\{\overline L_i\}_{i\in I}$ be a family of Hermitian line bundles on $\spec\O_K$. If $\phi:\; E_K\rightarrow\bigoplus\limits_{i\in I}L_{i,K}$ is an injective homomorphism of $K$-vector spaces, then there exists a subset $I_0$ of $I$ whose cardinality is $r$ such that the following equality
\[\wmu(\E)=\frac{1}{r}\left(\sum_{i\in I_0}\wmu(\overline L_i)+h\left(\wedge^r(\pr_{I_0}\circ\phi)\right)\right)\]
is verified, where $\pr_{I_0}:\;\bigoplus\limits_{i\in I}L_{i,K}\rightarrow\bigoplus\limits_{i\in I_0}L_{i,K}$ is the canonical projection.
\end{prop}
\
The following result is a refined determinant method, which is a direct generalization of \cite[Theorem 3.1]{Chen2} by involving the term $\mathcal R_\xi(E)$ defined at \eqref{R(E)}.
Before providing the statement of this generalized determinant, we will introduce the operation below. Let $\overline{\sE}$ be a Hermitian vector bundle of rank $n+1$ on $\spec\O_K$, $X$ be a closed integral sub-scheme of $\mathbb P(\sE_K)$, and $\mathscr X$ be the Zariski closure of $X$ in $\mathbb P(\sE)$. We choose a $P\in X(K)$, and let $\mathcal P\in\mathscr X(\O_K)$ be the Zariski closure of $P$ in $\mathscr X$. If we say that the reduction of $P$ modulo a maximal ideal $\p$ of $\O_K$ is $\xi\in\mathscr X(\f_\p)$, we mean that we consider the reduction of $\mathcal P$ modulo $\p$, whose image is $\xi$. We will use this representation multiple times in this article below.
\begin{theo}\label{semi-global determinant method}
Let $\{\p_j\}_{j\in J}$ be a finite family of maximal ideals of $\O_K$, and $\{P_i\}_{i\in I}$ be a family of rational points of $\mathscr X_K$ such that, for any $i\in I$ and any $j\in J$, the reduction of $P_i$ modulo $\p_j$ coincides with the same non-singular point $\xi_j\in\mathscr X(\f_{\p_j})$. Let $\F_D$ be as defined at Definition \ref{arithmetic hilbert function}, $\mathcal R_{\xi_j}(F_D)$ be defined as \eqref{R(E)}, $r_1(D)=\rg(F_D)$, $N(\p_j)=\#(\O_K/\p_j)$, and the height function $h(\ndot)$ of rational points follows the definition at \eqref{arakelov height} by Arakelov theory. If the inequality
\begin{equation}
\sup_{i\in I}h(P_i)<\frac{\wmu(\overline{F}_D)}{D}-\frac{\log r_1(D)}{2D}+\frac{1}{[K:\Q]}\sum_{j\in J}\frac{\mathcal R_{\xi_j}(F_{D})}{Dr_1(D)}\log N(\p_j)
\end{equation}
is verified for a positive integer $D$, then there exists a section $s\in E_{D,K}$ (see \eqref{definition of E_D} for its definition), which contains $\{P_i\}_{i\in I}$ but does not contain the generic point of $\mathscr X_K$. In other words, $\{P_i\}_{i\in I}$ can be covered by a hypersurfaces in $\mathbb P(\sE_K)$ of degree $D$ which does not contain the generic point of $\mathscr X_K$.
\end{theo}
\begin{proof}
We suppose the section predicted by this theorem does not exist. Then the evaluation map
\[f:\;F_{D,K}\rightarrow \bigoplus\limits_{i\in I}P_i^*\O_{\mathbb P(\sE_K)}(D)\]
is injective. We can replace $I$ by one of its sub-sets such that the above homomorphism $f$ is an isomorphism.
For every $v\in M_{K,\infty}$, we have
\[\frac{1}{r_1(D)}\log\|\wedge^{r_1(D)}f\|_v\leqslant\log\|f\|_v\leqslant\log\sqrt{r_1(D)},\]
where the first inequality comes from Hadamard's inequality, and the second one is due to the definition of metrics of John introduced at \S\ref{basic setting}.
For every $v\in M_{K,f}$, let $\p$ be the maximal ideal of $\O_K$ corresponding to the place $v$. By definition, the isomorphism $f$ is induced by a homomorphism $\O_K$-modules
\[F_D\rightarrow\bigoplus_{i\in I}\mathcal P_i^*\O_{\mathbb P(\sE)}(D),\]
where $\mathcal P_i$ is the $\O_K$-point of $\mathscr X$ extending $P_i$. Hence for any maximal ideal $\p$, we have $\log\|\wedge^{r_1(D)}f\|_\p\leqslant0$.
We fix a $j\in J$. For each $i\in I$, the $\O_K$-point $\mathcal P_i$ defines a local homomorphism from $\O_{\mathscr X,\xi_j}$ to $\O_{K,\p_j}$ which is $\O_{K,\p_j}$-linear. By taking a local trivialization of $\O_{\mathbb P(\sE)}(D)$ at $\xi_j$, we identify $F_D$ as a sub-$\O_{K,\p_j}$-module of $\O_{\mathscr X,\xi_j}$. Then by Proposition \ref{upper bound of operator norm}, we have
\[\log\|\wedge^{r_1(D)}f\|_{\p_j}\leqslant-\mathcal R_{\xi_j}(F_D)\log N(\p_j).\]
From the above two upper bounds of the operator norms, combined with Proposition \ref{slope of evaluation map}, we obtain
\[\frac{\wmu(\F_D)}{D}\leqslant\sup_{i\in I}h(P_i)+\frac{1}{2D}\log r_1(D)-\frac{1}{[K:\Q]}\sum_{j\in J}\frac{\mathcal R_{\xi_j}(F_D)}{Dr_1(D)}\log N(\p_j),\]
which leads to a contradiction. That is the end of the proof.
\end{proof}
\section{Estimates of $\mathcal R_{\xi_j}(F_D)$}
In order to apply Theorem \ref{semi-global determinant method}, more information about the term $\mathcal R_{\xi_j}(F_D)$ need to be gathered. The aim of this section is to give an asymptotic estimate of $\mathcal R_{\xi_j}(F_D)$. These estimates are proved by Per Salberger in his unpublished notes.
\subsection{Finiteness of $\mathcal R_{\xi_j}(F_D)$}
Formally, the sum in $\mathcal R_{\xi_j}(F_D)$ defined at \eqref{R(E)} is infinite. But since the filtration $\mathcal F$ introduced in the proof of Proposition \ref{upper bound of operator norm} is finite, then $\mathcal R_{\xi_j}(F_D)$ is essentially a finite sum. Then when the positive integer $m$ is large enough in $F_D\cap m_{\xi_j}^m$, it will be a zero module, so essentially it is a finite sum.
The following result is a generalization of \cite[Lemma 16.10]{Salberger2015}, at which the case of cubic hypersurfaces in $\mathbb P^3$ was considered only, but their proofs are quite similar.
\begin{prop}\label{R(E)-kernel}
We keep all notations and conditions in Theorem \ref{semi-global determinant method}. Let $\eta_j\in X(K)$ be a rational point which specializes to $\xi_j$ with respect to the operation in Theorem \ref{semi-global determinant method}, $m_{\xi_j}$ be the maximal ideal of $R_{\mathscr X,\xi_j}$ defined at \S\ref{estimates of norms}, and $\sn_{\eta_j}$ be the maximal ideal of $\O_{X}$ at the point $\eta_j$. Then for every $m\in\mathbb N^+$ and $j\in J$ in Theorem \ref{semi-global determinant method}, we have
\begin{eqnarray*}
\dim_K\left(F_{D}\cap m_{\xi_j}^m\right)_K&=&\dim_K\ker\left(F_{D,K}\rightarrow H^0\left(X,\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\otimes\O_{X}/\sn_{\eta_j}^m\right)\right)\\
&\geqslant&\max\left\{0,r_1(D)-{d+m-1\choose m-1}\right\},
\end{eqnarray*}
where we identify $F_D$ as a sub-$\O_{K,\p_j}$-module of $\O_{\mathscr X,\xi_j}$ for $j\in J$ above.
\end{prop}
\begin{proof}
Let $s_1,\ldots,s_{r_1(D)}\in F_D$ which generate $F_D$. Let $T_0,\ldots,T_n$ be the homogeneous coordinate of $\mathscr X\hookrightarrow\mathbb P(\sE)$. Without loss of generality, we suppose that $T_0(\xi_j)\neq0$ with respect to the canonical morphism. Let $r_i=s_i/T_0^D$ for all $i=1,\ldots,r_1(D)$. and $W_D\subset R_{\mathscr X,\xi_j}$ be the vector space over $K$ generated by the images of $r_1,\ldots,r_{r_1(D)}$ in $R_{\mathscr X,\xi_j}$, which is also of dimension $r_1(D)$. Thus for each $s\in F_D$, its image in $H^0\left(X,\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\otimes\O_{X}/\sn_{\eta_j}^m\right)$ is zero if and only if $s/T_0^D\in\ker\left(W_D\rightarrow W_D/m_{\xi_j}^m\right)$ considered as an element in $R_{\mathscr X,\xi_j}$, which means it holds if and only if $s/T_0^D\in W_D\cap m_{\xi_j}^m$. Thus there exists an isomorphism of $K$-vector spaces from $F_{D,K}$ to $W_D$, which maps $\ker\left(F_{D,K}\rightarrow H^0\left(X,\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\otimes\O_{X}/\sn_{\eta_j}^m\right)\right)$ onto $W_D\cap m_{\xi_j}^m$, and then we obtain the first equality in the assertion.
By the fact that $\dim(X)=d$ and the point $\xi_j$ is regular in $\mathscr X$, then the point $\eta_j$ is also regular in $X$, and the ring $R_{\mathscr X,\xi_j}$ is a regular local ring of Krull dimension $d$. By these facts, we have $\dim_{K}\left(R_{\mathscr X,\xi_j}/m_{\xi_j}^m\right)={d+m-1\choose m-1}$ for all $m\in\mathbb N^+$. Furthermore, we have $\dim_K\left(W_D/\left(W_D\cap m_{\xi_j}^m\right)\right)\leqslant\dim_K\left(R_{\mathscr X,\eta_j}/m_{\xi_j}^m\right)$. Hence we have
\begin{eqnarray*}
\dim_K\left(F_D\cap m_{\xi_j}^m\right)_K&=&\dim_K\left(W_D\right)-\dim_K\left(W_D/\left(W_D\cap m_{\xi_j}^m\right)\right)\\
&\geqslant& r_1(D)-{d+m-1\choose m-1},
\end{eqnarray*}
which completes the proof.
\end{proof}
\subsubsection{A naive lower bound of $\mathcal R_{\xi_j}(F_D)$}
In order to apply Theorem \ref{semi-global determinant method}, we need a lower bound of $\mathcal R_{\xi_j}(F_D)$ defined at \eqref{R(E)}. By the definition of $\mathcal R_{\xi_j}(F_D)$, we have
\[\mathcal R_{\xi_j}(F_D)\geqslant\sum_{m=1}^\infty\max\left\{0,r_1(D)-{d+m-1\choose m-1}\right\}\]
from Proposition \ref{R(E)-kernel} directly, where $j\in J$ follows the assertions of Theorem \ref{semi-global determinant method}. When $m$ is large enough, we have
\[r_1(D)-{d+m-1\choose m-1}\leqslant0,\]
so essentially the above sum is finite. Moreover, by an estimate of Sombra in \cite{Sombra97}, the inequality
\begin{equation}\label{lower bound of geometric Hilbert-Samuel function}
r_1(D)\geqslant{D+d+1\choose d+1}-{D-\delta+d+1\choose d+1}
\end{equation}
is verified uniformly for all $D\geqslant1$, and the equality holds when $X$ is a hypersurface of degree $\delta$ in $\mathbb P^n_K$. So it is possible to obtain an effective lower bound of $\mathcal R_{\xi_j}(F_D)$.
In order to provide such a lower bound from Proposition \ref{R(E)-kernel}, the inequality
\[\frac{(N-k+1)^k}{k!}\leqslant{N\choose k}\leqslant \frac{\left(N-(k-1)/2\right)^k}{k!}\]
will be useful, which is verified for all $N\geqslant k\geqslant1$. Then we have the following result.
\begin{prop}\label{explicit lower bound of R(E)}
Let $\mathcal R_{\xi_j}(F_D)$ be the same as in Theorem \ref{semi-global determinant method}, then we have
\[\mathcal R_{\xi_j}(F_D)\geqslant\frac{d\delta^{1+\frac{1}{d}}}{(d+1)!}D^{d+1}+B_2(d,\delta)D^d,\]
where $B_2(d,\delta)$ is an explicit constant depending on $d$ and $\delta$.
\end{prop}
\begin{proof}
By \eqref{lower bound of geometric Hilbert-Samuel function}, we have
\[r_1(D)\geqslant\sum_{i=1}^\delta{D-\delta+d+i\choose d}\geqslant\frac{\delta(D-\delta+2)^d}{d!}.\]
Meanwhile, we also have
\[{d+m-1\choose m-1}\leqslant\frac{(m-(d+1)/2)^d}{d!}.\]
Then from the relation
\[\frac{\delta(D-\delta+2)^d}{d!}\leqslant\frac{(m-(d+1)/2)^d}{d!},\]
we deduce
\[m\leqslant \sqrt[d]{\delta}(D-\delta+2)+\frac{d+1}{2}.\]
We denote $B(D,d,\delta)=\left[\sqrt[d]{\delta}(D-\delta+2)+\frac{d+1}{2}\right]$ for simplicity below, where $\left[x\right]$ is the largest integer smaller than $x$. Then we obtain
\[\mathcal R_\xi(F_D)\geqslant\sum_{m=1}^{B(D,d,\delta)}\left(r_1(D)-{d+m-1\choose m-1}\right)\]
by definition directly. And we have
\begin{eqnarray*}
\sum_{m=1}^{B(D,d,\delta)}\left(r_1(D)-{d+m-1\choose m-1}\right)&=&B(D,d,\delta)r_1(D)-{d+B(D,d,\delta)\choose d+1}-1\\
&\geqslant&B(D,d,\delta)r_1(D)-\frac{\left(B(D,d,\delta)+\frac{d}{2}\right)^{d+1}}{(d+1)!}-1.
\end{eqnarray*}
By some asymptotic estimates, we obtain
\begin{eqnarray*}
& &B(D,d,\delta)r_1(D)-\frac{\left(B(D,d,\delta)+\frac{d}{2}\right)^{d+1}}{(d+1)!}-1\\
&\geqslant&\left(\sqrt[d]{\delta}(D-\delta+2)+\frac{d+1}{2}\right)\frac{\delta(D-\delta+2)^d}{d!}-\frac{\left(\sqrt[d]{\delta}(D-\delta+2)+d+\frac{3}{2}\right)^{d+1}}{(d+1)!}-1\\
&\geqslant&\frac{d\delta^{1+\frac{1}{d}}}{(d+1)!}D^{d+1}+B_2(d,\delta)D^d,
\end{eqnarray*}
where $B_2(d,\delta)$ is a constant depending on $d$ and $\delta$ which can be given explicitly by the above argument, and we terminate the proof.
\end{proof}
\begin{rema}
We keep all the notations in Proposition \ref{explicit lower bound of R(E)}. In \cite[(3.11)]{Heath-Brown} and \cite[Proposition 3.5]{Chen2}, we have the same dominant term as that obtained in Proposition \ref{explicit lower bound of R(E)}. For some particular cases, we can obtain better estimates than that in Proposition \ref{explicit lower bound of R(E)}. For example, in \cite[Lemma 16.11]{Salberger2015} and the remark below it, if $X\hookrightarrow\mathbb P^n$ is a hypersurface of degree $\delta$ satisfying $2\leqslant \delta\leqslant2^{n-1}$, we have
\[\mathcal R_{\xi_j}(F_D)\geqslant\left(\frac{2\delta}{n!}+\frac{\delta\left(\frac{\delta}{2}\right)^{\frac{1}{n-2}}}{(n-1)!}\left(1-\frac{2}{n}\right)\right)D^n+O_{\delta,n}(D^{n-1}),\]
which has a better dominant term than that given in Proposition \ref{explicit lower bound of R(E)}. In fact, in \cite{Salberger2015}, Salberger applied the numerical inequality in Proposition \ref{R(E)-kernel} (provided in \cite[Lemma 16.10]{Salberger2015}) for $1\leqslant m\leqslant\left(\frac{\delta}{2}\right)^{\frac{1}{n-2}}$, and a very ingenious control in \cite[(16.5)]{Salberger2015} for those $\left(\frac{\delta}{2}\right)^{\frac{1}{n-2}}<m\leqslant 2n$. As an application, we consider the number of rational points with bounded height in the complement of the union of all lines on integral cubic surfaces in $\mathbb P^3_{\Q}$. The above calculation is the key ingredient of the proof of \cite[Theorem 16.1]{Salberger2015}, which refines the former results in \cite{Heath-Brown,Salberger2008,Salberger_preprint2013}.
\end{rema}
Compared with Proposition \ref{explicit lower bound of R(E)}, it is an important subject to give a better or even the optimal dominant term in the estimate of $\mathcal R_{\xi_j}(F_D)$ for more general cases. In fact, we will see that the lower bound of its dominant term given in Proposition \ref{explicit lower bound of R(E)} can also be given by a lower bound of a particular invariant about the positivity of certain line bundles, see \eqref{lower bound of I_X} and Theorem \ref{R(E) by GIT} below.
\subsubsection{Connection with Seshadri constant}
In this part, we will give a lower bound of the positive integer $m$ such that
\[\dim_K\left(F_{D}\cap m_{\xi_j}^m\right)_K=\dim_K\ker\left(F_{D,K}\rightarrow H^0\left(X,\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\otimes\O_{X}/\sn_{\eta_j}^m\right)\right)\]
are both zero, where all the above notations are same as those in Proposition \ref{R(E)-kernel}. For this target, we will introduce some notions on the geometric positivity of line bundles. We refer to \cite[\S5.1]{LazarsfeldI} for a systemic introduction to it.
Let $X$ be an closed integral projective scheme over a field, $L$ be a line bundle on $X$, and $\xi\in X$ be a regular point with the maximal ideal $\sn_\xi\subset \O_X$. We consider the natural map
\begin{equation}\label{separates s-jets}
H^0\left(X,L\right)\rightarrow H^0\left(X,L\otimes\O_X/\sn_\xi^{s+1}\right)
\end{equation}
taking the global sections of $L$ to their $s$-jets at $\xi$. By definition, the kernel of the map \eqref{separates s-jets} is $H^0\left(X, L\otimes \sn_\xi^{s+1}\right)$.
In addition, let $L$ be a nef line bundle on $X$. We fix a closed point $\xi\in X$, and let $\pi:\;\widetilde X\rightarrow X$ be the blow-up at $\xi$, and $E=\pi^{-1}(\xi)$ be the exceptional divisor. We define the \textit{Seshadri constant} of $L$ at $\xi$ as
\begin{equation}
\epsilon(X,L;\xi)=\epsilon(L,\xi)=\sup\{\epsilon>0|\;\pi^*L-\epsilon E\hbox{ is nef }\}.
\end{equation}
By \cite[Proposition 5.1.5]{LazarsfeldI}, we have
\begin{equation}\label{seshadri constant by intersection}
\epsilon(L;\xi)=\inf_{\xi\in C\subseteq X}\left\{\frac{(L\cdot C)}{\mu_\xi(C)}\right\},
\end{equation}
where $C$ takes over all integral curves $C\subseteq X$ passing through $\xi$, and $\mu_\xi(C)$ is the multiplicity of $\xi$ in $C$, see \S \ref{local algebra} for its precise definition.
Some properties of the Seshadri constant will be useful in the proof of the proposition below.
\begin{prop}\label{bound of finite sum}
With all the notations and conditions in Proposition \ref{R(E)-kernel}, when $m\geqslant \left[\sqrt[d]{\delta}D\right]+1$, we have
\[\ker\left(F_{D,K}\rightarrow H^0\left(X,\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\otimes\O_{X}/\sn_{\eta_j}^m\right)\right)=0,\]
where $\left[\ndot\right]$ denotes the largest integer smaller than $s$.
\end{prop}
\begin{proof}
By the definition of $F_{D,K}$ induced in \eqref{evaluation map}, the $K$-vector space $F_{D,K}$ is a sub-$K$-vector space of $H^0\left(X,\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\right)$, so it is enough to prove the bound for the $K$-linear map
\begin{equation*}
H^0\left(X,\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\right)\rightarrow H^0\left(X,\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\otimes\O_{X}/\sn_{\eta_j}^m\right).
\end{equation*}
In other words, we need a bound of $m\in\mathbb N$ such that $H^0\left(X, \O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\otimes \sn_{\eta_j}^{m}\right)$ is zero.
By definition, the space $H^0\left(X, \O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\otimes \sn_{\eta_j}^{m}\right)$ is zero when $m$ is strictly larger than the possible maximal multiplicity of the point $\eta_j$ in the divisors which are linearly equivalent to $\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}$. We denote by $\mu_{\eta_j}\left(\left|\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\right|\right)$ the above maximal multiplicity. By \cite[Corollary 12.4]{Fulton} and \eqref{seshadri constant by intersection}, we have
\begin{equation}\label{seshadri constant>multiplicity}
\mu_{\eta_j}\left(\left|\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\right|\right)\leqslant\epsilon\left(\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D},\eta_j\right),
\end{equation}
where we consider the intersection in the regular locus of $X$, and the multiplicity of a point in pure-dimensional schemes is considered at \cite[Corollary 12.4]{Fulton}. In addition, the multiplicity satisfies the additivity of cycles by \cite[Chap. VIII, \S 7, $\mathrm{n}^\circ$ 1, Prop. 3]{Bourbaki83}.
By \cite[Example 5.1.4]{LazarsfeldI}, we have
\begin{equation}\label{homogenesous of seshadri constant}
\epsilon\left(\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D};\eta_j\right)=D\epsilon\left(\O_{\mathbb P(\sE_K)}(1)|_X;\eta_j\right).
\end{equation} By \cite[Proposition 5.1.9]{LazarsfeldI}, we have
\begin{equation}\label{bound of seshadri constant}
\epsilon\left(\O_{\mathbb P(\sE_K)}(1)|_X;\eta_j\right)\leqslant\sqrt[d]{\frac{\O_{\mathbb P(\sE_K)}(1)|_X^d}{\mu_{\eta_j}\left(X\right)}}=\sqrt[d]{\delta},
\end{equation}
for $\eta_j$ is regular in $X$ and $\deg(X)=\delta$ with respect to $\O(1)$.
By \eqref{seshadri constant>multiplicity}, \eqref{homogenesous of seshadri constant} and \eqref{bound of seshadri constant}, when $m\geqslant \left[\sqrt[d]{\delta}D\right]+1$, we have $m>\mu_{\eta_j}\left(|\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}|\right)$, and we will have a trivial kernel in this case.
\end{proof}
\subsection{Invariants induced by blow-up}
Let $\overline \sE$ be a Hermitian vector bundle of rank $n+1$ on $\spec\O_K$, $X$ be a closed integral sub-scheme of $\mathbb P(\sE_K)$ of dimension $d$ and degree $\delta$, and $\mathscr X$ be the Zariski closure of $X$ in $\mathbb P(\sE)$. If the positive integer $D$ is large enough, then we have $F_D=H^0\left(\mathscr X,\O_{\mathbb P(\sE)}(1)|_{\mathscr X}^{\otimes D}\right)$ and $F_{D,K}=H^0\left(X,\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\right)$, where $F_D$ and $F_{D,K}$ are defined at Definition \ref{arithmetic hilbert function}. By this fact, we will give an alternative description of the term $\mathcal R_{\xi_j}(F_D)$ in Theorem \ref{semi-global determinant method}.
Let $\eta\in X(K)$ be non-singular, $\sn_{\eta}$ be the maximal ideal of $\O_X$ at the point $\eta$, and
\begin{equation}\label{blow-up at one point}
\pi:\widetilde{X}\rightarrow X
\end{equation}
be the blow-up of $X$ at $\eta$. Let $E=\pi^{-1}(\eta)$ be the exceptional divisor of the above blow-up morphism $\pi$, and $I_E\subset\O_{\widetilde{X}}$ be the ideal sheaf of $E\subset \widetilde{X}$. By the projection formula (cf. \cite[Chap. III, Exercise 8.3]{GTM52}) applied at \eqref{blow-up at one point}, we have $R^i\pi_*\left(\pi^*\left(\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\right)\right)=0$ for all $i\geqslant1$, and then we have $\pi_*\left(\pi^*\left(\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\right)\right)=\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}$. So we obtain
\[H^0\left(X,\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\right)\cong H^0\left(\widetilde{X},\pi^*\left(\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\right)\right).\]
From the above isomorphism, we have the commutative diagram
\[\xymatrix{H^0\left(X,\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\right)\ar[d]\ar[r]&H^0\left(X,\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\otimes\O_{X}/\sn_\eta^m\right)\ar[d]\\H^0\left(\widetilde{X},\pi^*\left(\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\right)\right)\ar[r]&H^0\left(\widetilde{X},\pi^*\left(\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\right)\otimes\O_{\widetilde{X}}/I_E^m\right),}\]
where the kernel of the bottom map is isomorphic to $H^0\left(\widetilde{X}, \pi^*\left(\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\right)\otimes I_E^m\right)$ for $m\geqslant1$. By the above argument, we have the following result.
\begin{prop}\label{kernel-blowup}
With all the above notations, we have
\begin{eqnarray*}
& &\dim_K\left(H^0\left(\widetilde{X}, \pi^*\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\otimes I_E^m\right)\right)\\
&=&\dim_K\ker\left(H^0\left(X,\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\right)\rightarrow H^0\left(X,\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\otimes\O_{X}/\sn_{\eta}^m\right)\right)
\end{eqnarray*}
for all $m\geqslant1$.
\end{prop}
\subsection{The volume of certain line bundles}
In this part, we will give a connection between the above invariant $\mathcal R_{\xi_j}(F_D)$ in Theorem \ref{semi-global determinant method} and the volume of certain line bundles.
\subsubsection{Definition of volume function}
In the first step, we will recall the definition of the volume of line bundles on projective varieties at \cite[Definition 2.2.31]{LazarsfeldI}. For more details about this notion, see \cite[\S 2.2.C]{LazarsfeldI}.
Let $X$ be a projective integral scheme of dimension $d$ over a field, and $L$ be a line bundle on $X$. Let $D\in \mathbb N^+$, and we denote by $h^0\left(X, L^{\otimes D}\right)=\dim H^0\left(X,L^{\otimes D}\right)$ for simplicity. Then the \textit{volume} of the line bundle $L$ is defined to be the non-negative number
\begin{equation}\label{definition of volume}
\vol\left(L\right)=\vol_X\left(L\right)=\limsup_{D\rightarrow\infty}\frac{h^0\left(X,L^{\otimes D}\right)}{D^d/d!}.
\end{equation}
Meanwhile, if $E$ is a Cartier divisor on $X$, we denote the volume by $\vol(E)$ or $\vol_X(E)$ for simplicity, or by passing $\O_X(E)$.
Let $NS(X)$ be the N\'eron-Severi group of $X$ (see \cite[Definition 1.1.15]{LazarsfeldI} for its definition). By \cite[Proposition 2.2.41]{LazarsfeldI}, the volume of a line bundle only depends on its class in N\'eron-Severi group. Let $NS(X)_{\mathbb R}=NS(X)\otimes_{\mathbb Z}\mathbb R$. By \cite[Corollary 2.2.45]{LazarsfeldI}, the above volume function can be extended uniquely to a continuous function
\begin{equation}\label{volume over R}
\vol:\; NS(X)_{\mathbb R}\rightarrow \mathbb R,
\end{equation}
where Cartier $\mathbb R$-divisors (see \cite[\S1.3.B]{LazarsfeldI} for its definition) are considered above.
\subsubsection{Dependence on the reduction}\label{depending on reduction}
We keep all the notations as above. Let $H$ be a Cartier divisor on $X$ given by a hyperplane section in $\mathbb P(\sE_K)$. Let $\eta_1,\eta_2\in X(K)$ be both non-singular, and $\pi_1:\; \widetilde{X}_1\rightarrow X$ and $\pi_2:\; \widetilde{X}_2\rightarrow X$ be the blow-up of $X$ at $\eta_1$ and $\eta_2$ respectively, with respect to the exceptional divisors $E_1\subset\widetilde{X}_1$ and $E_2\subset\widetilde{X}_2$. By Proposition \ref{R(E)-kernel} and Proposition \ref{kernel-blowup}, if two rational points of $\eta_1,\eta_2\in X(K)$ have the same non-singular specialization modulo a maximal ideal of $\O_K$ in the sense of Theorem \ref{semi-global determinant method}, then we have
\[h^0\left(\widetilde{X}_1, D\pi^*_1(H)-mE_1\right)=h^0\left(\widetilde{X}_2, D\pi^*_2(H)-mE_2\right)\]
for every $D,m\in\mathbb N$, which means it only depends on its specialization by the operation of Theorem \ref{semi-global determinant method}.
\subsubsection{Pseudo-effective thresholds}
By the fact stated in \S \ref{depending on reduction} above, we will introduce the following invariant.
\begin{defi}\label{I(X)}
Let $X$ be a closed integral sub-scheme of $\mathbb P(\sE_K)$ over the number field $K$, $\eta\in X(K)$ whose specialization modulo the maximal ideal $\p$ of $\O_K$ is the non-singular point $\xi$ in the sense of Theorem \ref{semi-global determinant method}, $\pi:\;\widetilde X\rightarrow X$ be the blow-up at $\eta$, and $E\subset\widetilde X$ be its exceptional divisor. Let $H$ be a Cartier divisor on $X$ given by a hyperplane section in $\mathbb P(\sE_K)$. We define
\[I_X(H,\xi)=\int_0^\infty\vol\left(\pi^*H-\lambda E\right)d\lambda,\]
where the above volume function $\vol(\ndot)$ follows the extended definition introduced at \eqref{volume over R} over $\widetilde X$.
\end{defi}
\subsection{The dominant term of $\mathcal R_{\xi_j}(F_D)$}
We keep all the above notations and conditions. We will give an asymptotic estimate of $\mathcal R_{\xi_j}(F_D)$ defined at \eqref{R(E)} by the invariant $I_X(H,\xi_j)$, where $j\in J$ is given in Theorem \ref{semi-global determinant method}.
\begin{theo}\label{R(E) by GIT}
Let $X$ be a closed integral sub-scheme of $\mathbb P(\sE_K)$ of dimension $d$ and degree $\delta$ over a number field $K$. Let $F_D$ be the same as that in Theorem \ref{semi-global determinant method}, $\mathcal R_{\xi_j}(F_D)$ be defined at \eqref{R(E)}, where $j\in J$ and $\xi_j\in\mathscr X(\f_{\p_j})$ are the same as those in Theorem \ref{semi-global determinant method}, and $H$ be a Cartier divisor on $X$ given by a hyperplane section in $\mathbb P(\sE_K)$. Then we have
\[\mathcal R_{\xi_j}(F_D)=\frac{D^{d+1}}{d!}I_X(H,\xi_j)+O_{d,\delta}(D^d),\]
where $I_X(H,\xi_j)$ is defined in Definition \ref{I(X)}.
\end{theo}
\begin{proof}
Let $\eta\in X(K)$ be a rational point whose reduction modulo $\p$ is $\xi$ in the sense of Theorem \ref{semi-global determinant method}, $\pi:\widetilde X\rightarrow X$ be the blow-up of $X$ at $\eta$, $E=\pi^{-1}(\eta)$ be the exceptional divisor of $\pi$. If denote $B=H^0(X,\O_X)$ and let $\dim B$ be the Krull dimension of the ring $B$, then by \cite[Lemma 2.1]{Poonen2004}, when $D\geqslant\dim B-1$, we have $F_D=H^0\left(\mathscr X,\O_{\mathbb P(\sE)}(1)|_{\mathscr X}^{\otimes D}\right)$ and $F_{D,K}=H^0\left(X,\O_{\mathbb P(\sE_K)}(1)|_X^{\otimes D}\right)$. Then by Proposition \ref{R(E)-kernel} and Proposition \ref{kernel-blowup}, we have
\[\mathcal R_{\xi_j}(F_D)\sim\sum_{m=1}^\infty h^0\left(\widetilde X, D\pi^*H-mE\right)\]
when $D$ tends into infinite.
It is evident that we have
\[h^0\left(X,DH\right)=h^0\left(\widetilde X, D\pi^*H\right)=\frac{D^d}{d!}\delta+O_{d,\delta}(D^{d-1}),\]
since $\vol(\pi^*H)=\vol(H)=\delta$. Meanwhile, if $m\geqslant1$, we have $0\leqslant h^0\left(\widetilde X, D\pi^*H-mE\right)\leqslant h^0\left(\widetilde X, D\pi^*H\right)$ and $\vol\left(\pi^*H-\frac{m}{D}E\right)\leqslant\vol\left(\pi^*H\right)=\delta$ for every $m\in\mathbb N$. Then by the definition of volume at \eqref{definition of volume}, when $m=1,\ldots,\left[\sqrt[d]{\delta}D\right]+1$, we have
\begin{equation}\label{volume of DH-mE}
h^0\left(\widetilde X, D\pi^*H-mE\right)=\frac{D^d}{d!}\vol\left(\pi^*H-\frac{m}{D}E\right)+O_{d,\delta}(D^{d-1}),
\end{equation}
where $[s]$ denotes the largest integer smaller than $s\in \mathbb R$.
By Proposition \ref{bound of finite sum}, we have
\[\sum_{m=1}^\infty h^0\left(\widetilde X, D\pi^*H-mE\right)=\sum_{m=1}^{\left[\sqrt[d]{\delta}D\right]+1} h^0\left(\widetilde X, D\pi^*H-mE\right).\]
By the estimate of remainder term in \eqref{volume of DH-mE} and Definition \ref{I(X)}, we have
\begin{eqnarray*}
\sum_{m=1}^{\left[\sqrt[d]{\delta}D\right]+1} h^0\left(\widetilde X, D\pi^*H-mE\right)&=&\frac{D^d}{d!}\sum_{m=1}^\infty\vol\left(\pi^*H-\frac{m}{D}E\right)+O_{d,\delta}(D^d)\\
&=&\frac{D^{d+1}}{d!}I_X(H,\xi_j)+O_{d,\delta}(D^d),
\end{eqnarray*}
and we obtain the result.
\end{proof}
\begin{rema}
By \cite[Corollary 4.2]{McKinnonRoth_2015}, when $X\hookrightarrow \mathbb P(\sE_K)$ is of degree $\delta$ with respect to $\O(1)$, we have the following lower bound of $I_X(H,\xi)$ introduced in Definition \ref{I(X)}, which is
\[ I_X(H,\xi)\geqslant\frac{d\vol(H)}{d+1}\sqrt[d]{\frac{\vol(H)}{\mu_\eta(X)}}\geqslant\frac{d}{d+1}\epsilon_\eta(H)\vol(H),\]
where the reduction of $\eta\in X(K)$ modulo the maximal ideal $\p$ of $\O_K$ is $\xi$ in the sense of Theorem \ref{semi-global determinant method}, $\mu_\eta(X)$ is the multiplicity of $\eta$ in $X$, and $\epsilon_\eta(H)$ is the Seshadri constant of $H$ at $\eta$. For the application in this article, we have
\begin{equation}\label{lower bound of I_X}
I_X(H,\xi)\geqslant\frac{d\vol(H)}{d+1}\sqrt[d]{\frac{\vol(H)}{\mu_\eta(X)}}=\frac{d\delta^{1+\frac{1}{d}}}{(d+1)},
\end{equation}
since the point $\eta$ is regular in $X$, and $\vol(H)=H^d=\delta$ by definition. Then by Theorem \ref{R(E) by GIT}, we have
\[\mathcal R_{\xi_j}(F_D)\geqslant\frac{d\delta^{1+\frac{1}{d}}}{(d+1)!}D^{d+1}+O_{d,\delta}(D^d),\]
which is the same as that obtained in Proposition \ref{explicit lower bound of R(E)} and some other former results, for example, in \cite[Main Lemma 2.5]{Salberger07}.
\end{rema}
\section{The number of auxiliary hypersurfaces}
In this section, for a closed integral sub-scheme $X$ of $\mathbb P(\sE_K)$, we will give an upper bound of the number of hypersurfaces which cover $S(X;B)=\left\{\xi\in X(K)|\;H_K(\xi)\leqslant B\right\}$ but do not contain the generic point of $X$. The height function $H_K(\ndot)=\exp\left([K:\Q]h(\ndot)\right)$, and $h(\ndot)$ follows the definition \eqref{arakelov height} by Arakelov theory with respect to the Hermitian vector bundle $\overline{\sE}$ on $\spec\O_K$.
\subsection{Application of the asymptotic estimate of $\mathcal R_{\xi_j}(F_D)$}
Let $\overline{\sE}$ be a Hermitian vector bundle of rank $n+1$ on $\spec\O_K$, $X$ be a closed integral sub-scheme of $\mathbb P(\sE_K)$, and $\mathscr X$ be the Zariski closure of $X$ in $\mathbb P(\sE)$. Let $\p$ be a maximal ideal of $\O_K$, and $\xi\in\mathscr X(\f_\p)$. We denote by $S(X;B,\xi)$ the sub-set of $S(X;B)$ whose reduction modulo $\p$ is $\xi$ in the sense of Theorem \ref{semi-global determinant method}.
\begin{lemm}\label{I_X(H,xi) same}
We keep all the notations and conditions in Theorem \ref{semi-global determinant method}. If $\bigcap\limits_{j\in J}S(X;B,\xi_j)$ is not empty, then for every $j\in J$, all the invariants $\left\{I_X(H,\xi_j)\right\}_{j\in J}$ are equal.
\end{lemm}
\begin{proof}
By Proposition \ref{R(E)-kernel}, the invariant $I_X(H,\xi_j)$ only depends on its specialization. Then we obtain the assertion from Proposition \ref{kernel-blowup} directly.
\end{proof}
We keep all the notations and conditions in Lemma \ref{I_X(H,xi) same}, and we define
\begin{equation}\label{I_X(H,xi_J)}
I_X(H,\xi_J)=I_X(H,\xi_j)
\end{equation}
for all $j\in J$. Then by the asymptotic estimate of $\mathcal R_\xi(F_D)$, we have the result below deduced from Theorem \ref{semi-global determinant method}.
\begin{theo}\label{semi-global determinant method mid}
We keep all the notations in Theorem \ref{semi-global determinant method}. Let $\{\p_j\}_{j\in J}$ be a family of maximal ideals of $\O_K$ and $B,\epsilon>0$. For every $j\in J$, let $\xi_j\in \mathscr X(\f_{\p_j})$ be a regular rational point. Let $I_X(H,\xi_J)$ be defined at \eqref{I_X(H,xi_J)} (by Lemma \ref{I_X(H,xi) same} it is well defined). If the inequality
\begin{equation}\label{hypothesis of step 1}
\sum_{j\in J}\log N(\p_j)\geqslant (1+\epsilon)\left(\log B +[K:\Q]\frac{\log\left((n+1)(d+1)\right)}{2}\right)\frac{\delta}{I_X(H,\xi_J)}
\end{equation}
is verified, then there exists a hypersurface of degree $O_{d,\delta,\epsilon}(1)$ in $\mathbb P(\sE_K)$, which contains the set $\bigcap\limits_{j\in J}S(X;B,\xi_j)$ but do not contain the generic point of $X$.
\end{theo}
\begin{proof}
We only need to prove the assertion for the case when $\bigcap\limits_{j\in J}S(X;B,\xi_j)\neq\emptyset$. Let $D\in\mathbb N^+$. Firstly, we suppose that such there does not exist such a hypersurface of degree $D$. By Theorem \ref{semi-global determinant method}, we have
\begin{equation}\label{first step of determinant method}
\frac{\log B}{[K:\Q]}\geqslant\frac{\wmu(\F_D)}{D}-\frac{\log r_1(D)}{2D}+\sum_{j\in J}\frac{\mathcal R_{\xi_j}(F_D)}{Dr_1(D)}\frac{\log N(\p_j)}{[K:\Q]}.
\end{equation}
By the fact that $\xi_j$ is regular for every $j\in J$, the fact
\[r_1(D)=\frac{\delta}{d!}D^d+O_{d,\delta}(D^{d-1}),\]
we apply Theorem \ref{R(E) by GIT} by combining the above to assertions, and then there exists a constant $C(d,\delta)$ depending on $d$ and $\delta$, such that
\[\frac{\mathcal R_{\xi_j}(F_D)}{Dr_1(D)}\geqslant \frac{I_X(H,\xi_J)}{\delta}+\frac{C(d,\delta)}{D}\]
is verified for each $D\geqslant1$ and $j\in J$. By \cite[\S1.2]{Chardin89}, we have
\[r_1(D)\leqslant\delta{D+d\choose D}\leqslant\delta(d+1)^D.\]
We combine the above arguments and the trivial lower bound of $\wmu(\F_D)$ introduced at \eqref{trivial lower bound of F_D}. From the inequality \eqref{first step of determinant method}, we deduce
\[\frac{\log B}{[K:\Q]}\geqslant-\frac{1}{2}\log(n+1)-\frac{\log\delta}{2D}-\frac{1}{2}\log(d+1)+\left(\frac{I_X(H,\xi_J)}{\delta}+\frac{C(d,\delta)}{D}\right)\sum_{j\in J}\frac{\log N(\p_j)}{[K:\Q]},\]
and we obtain
\begin{eqnarray*}
& &\left(\frac{I_X(H,\xi_J)}{\delta}\sum_{j\in J}\frac{\log N(\p_j)}{[K:\Q]}-\frac{\log B}{[K:\Q]}-\frac{1}{2}\log(n+1)-\frac{1}{2}\log(d+1)\right)D\\
&\leqslant&\left(-\frac{\log\delta}{2}+C(d,\delta)\right)\sum_{j\in J}\frac{\log N(\p_j)}{[K:\Q]}.
\end{eqnarray*}
By the hypothesis \eqref{hypothesis of step 1}, the left side of the above inequality is larger than or equal to
\[\frac{\epsilon}{1+\epsilon}\cdot\frac{I_X(H,\xi_J)}{\delta}\sum_{j\in J}\frac{\log N(\p_j)}{[K:\Q]}D,\]
which implies that
\[D\leqslant(\epsilon^{-1}+1)\frac{\delta}{I_X(H,\xi_J)}\left(-\frac{\log\delta}{2}+C(d,\delta)\right).\]
By \cite[Corollary 4.2]{McKinnonRoth_2015} (referred at \eqref{lower bound of I_X}), there exists a lower bound of $I_X(H,\xi_J)$ which only depends on the $d$ and $\delta$, since all $\xi_j$ is regular in $\mathscr X_{\f_{\p_j}}$ for each $j\in J$. Then we obtain a contradiction which terminates the proof.
\end{proof}
The following result can be considered as a generalization of \cite[Main Lemma 16.3.1]{Salberger2015}.
\begin{coro}\label{semi-global determinant method 2}
We keep all the notations and conditions in Theorem \ref{semi-global determinant method mid}. Let
\begin{equation}\label{inf of I_X}
I_X(H)=\inf_{\begin{subarray}{c}\eta\in S(X^\mathrm{reg};B)\end{subarray}}\{I_X(H,\eta)\}.
\end{equation}
Then if the inequality
\begin{equation*}
\sum_{j\in J}\log N(\p_j)\geqslant (1+\epsilon)\left(\log B +\frac{1}{2}[K:\Q]\log\left((n+1)(d+1)\right)\right)\frac{\delta}{I_X(H)}
\end{equation*}
is verified, then there exists a hypersurface of degree $O_{n,\delta,\epsilon}(1)$ in $\mathbb P(\sE_K)$, which contains $\bigcap\limits_{j\in J}S(X;B,\xi_j)$ but does not contain the generic point of $X$.
\end{coro}
\begin{proof}
By definition \eqref{inf of I_X}, we have
\[\frac{\delta}{I_X(H)}\geqslant\frac{\delta}{I_X(H,\xi_J)},\]
where $I_X(H,\xi_J)$ is defined in the assertion of Theorem \ref{semi-global determinant method mid}. Then we obtain the assertion from \eqref{hypothesis of step 1} in Theorem \ref{semi-global determinant method mid} directly.
\end{proof}
\subsection{Bertrand's postulate of number fields}
In order to apply Theorem \ref{semi-global determinant method mid} and Corollary \ref{semi-global determinant method 2}, we need some estimate about the distribution of prime ideals of rings of algebraic integers. In fact, we need an analogue of Bertrand's postulate for the case of number fields.
\begin{lemm}\label{Bertrand's postulate}
Let $K$ be a number field, and $\O_K$ be the ring of integers of $K$. There exists a constant $\alpha(K)\geqslant2$ depending on $K$, such that for all number $N_0\geqslant1$, there exists at least one maximal ideal $\p$ of $\O_K$, such that $N_0<N(\p)\leqslant\alpha(K)N_0$.
\end{lemm}
We refer to \cite{Lagarias-Odlyzko} or \cite[Th\'eor\`eme 2]{Serre-postulat} for a proof by admitting the generalized Riemann hypothesis, and to \cite[Th\'eor\`eme 1.7]{Winckler_these} without admitting it.
\subsection{Complexity of the singular locus}
Let $\overline{\sE}$ be a Hermitian vector bundle of rank $n+1$ on $\spec\O_K$, $X$ be a closed integral sub-scheme of $\mathbb P(\sE_K)$ of degree $\delta$ and dimension $d$. In order to give an upper bound of the number of auxiliary hypersurfaces which cover $S(X;B)$ but do not contain the generic point of $X$, we divide $S(X;B)$ into two part: the part of regular points and the part of singular points. In this part, we will deal with the singular part $S(X^{\mathrm{sing}};B)$.
By \cite[Theorem 3.10]{Chen1} (see also \cite[\S 2.6]{Chen2}), we have the following control to the complexity of the singular locus.
\begin{prop}\label{covering singular locus}
Let $\overline{\sE}$ be a Hermitian vector bundle of rank $n+1$ on $\spec\O_K$, and $X$ be a closed integral sub-scheme of $\mathbb P(\sE_K)$, which is of degree $\delta$ and of dimension $d$. Then there exists a hypersurface of degree $(\delta-1)(n-d)$ in $\mathbb P(\sE_K)$ which covers $S(X^{\mathrm{sing}};B)$ but do not contain the generic point of $X$.
\end{prop}
\subsection{Control of regular reductions}
Let $\p$ be a maximal ideal of $\O_K$, $S(X^{\mathrm{reg}};B)$ be the sub-set of $S(X;B)$ consisting of regular points, and $S(X;B,\xi)$ be the sub-set of $S(X;B)$ whose reduction modulo $\p$ is $\xi$, where the operation modulo $\p$ follows the sense of Theorem \ref{semi-global determinant method}. We denote
\[S(X^{\mathrm{reg}};B,\p)=\bigcup_{\begin{subarray}{c}\xi\in\mathscr X(\f_\p)\\ \mu_\xi\left(\mathscr X\right)=1\end{subarray}}S(X;B,\xi).\]
In other words, $S(X^{\mathrm{reg}};B,\p)$ is the sub-set of $S(X^{\mathrm{reg}};B)$ with regular reduction modulo $\p$.
In order to give a numerical description of the regular reductions, we introduce the following constants original from \cite[Notation 19]{Chen2}.
\begin{eqnarray*}
C_1&=&(d+2)\wmu_{\max}\left(\sym^\delta\left(\overline{\sE}^\vee\right)\right)+\frac{1}{2}(d+2)\log\rg\left(\sym^\delta \sE\right)\\
& &\;+\frac{\delta}{2}\log\left((d+2)(n-d)\right)+\frac{\delta}{2}(d+1)\log(n+1),
\end{eqnarray*}
\begin{equation*}
C_2=\frac{r}{2}\log \rg\left(\sym^\delta\sE\right)+\frac{1}{2}\log \rg\left(\wedge^{n-d}\sE\right)+\log\sqrt{(n-d)!}+(n-d)\log \delta,
\end{equation*}
and
\begin{equation}\label{constant C_3}
C_3=(n-d)C_1+C_2.
\end{equation}
The above constant $C_1$ is original from \cite[(21)]{Chen1}, and $C_2$ is from \cite[Remark 3.9]{Chen1}. The constant $C_3$ firstly appeared at \cite[Theorem 3.10]{Chen1}, and we have
\begin{equation}\label{estimate of C_3}
C_3\ll_{n,d}\delta.
\end{equation}
By the above notations, we state the following result.
\begin{lemm}[\cite{Chen2}, Lemma 4.1]\label{upper bound of regular reduction}
Let $N_0>0$ be a real number and $r$ be the integral part of
\begin{equation}\label{number of places for regular reductions}
\frac{(n-d)(\delta-1)\log B+\left((n-d)h_{\overline{\O_{\mathbb P(\sE)}(1)}}(X)+C_3\right)[K:\Q]}{\log N_0}+1,
\end{equation}
where the constant $C_3$ is defined at \eqref{constant C_3}, and the height $h_{\overline{\O_{\mathbb P(\sE)}(1)}}(X)$ is defined at \eqref{definition of arakelov height}. If $\p_1,\ldots, \p_r$ are distinct maximal ideals of $\O_K$ such that $N(\p_i)>N_0$ is verified for every $i=1,\ldots,r$, then
\[S(X^{\mathrm{reg}};B)=\bigcup_{i=1}^rS(X^{\mathrm{reg}};B;\p_i).\]
\end{lemm}
\subsection{An upper bound of the number of auxiliary hypersurfaces}
In this part, we will estimate the number of auxiliary hypersurfaces which cover $S(X;B)$ but do not contain the generic point of $X$. In fact, by Proposition \ref{covering singular locus}, we only need to consider the regular part $S(X^{\mathrm{reg}};B)$.
By \cite[Theorem 4.8]{Chen1} and \cite[Proposition 2.12]{Chen1}, the rational points with small height in $X$ can be covered by one hypersurface of degree $O_{n}(\delta)$ not containing the generic point of $X$, where the "small" height means that the bound $B$ is small compared with the height of $X$. We will use the above argument to deal with the points with small height and the method of Theorem \ref{semi-global determinant method} and Proposition \ref{semi-global determinant method 2} to deal with the regular points with large height, and combine it with Lemma \ref{upper bound of regular reduction}.
\begin{theo}\label{number of hypersurfaces}
Let $K$ be a number field and $\O_K$ be its ring of integers. Let $\overline{\sE}$ be a Hermitian vector bundle of rank $n+1$ on $\spec\O_K$, $X$ be a closed integral sub-scheme of $\mathbb P(\sE_K)$ of dimension $d$ and degree $\delta$, and $\epsilon>0$ be an arbitrary real number. Then there exists an explicit constant $C_4(\epsilon, \delta, n,d, K)$, such that for every $B\geqslant e^\epsilon$, the set $S(X;B)$ can be covered by no more than $C_4(\epsilon, \delta, n,d, K)B^{\frac{(1+\epsilon)d\delta}{I_X(H)}}$ hypersurfaces with degree of $O_{n,\delta,\epsilon}(1)$ which do not contain the generic point of $X$, where $I_X(H)$ is defined at \eqref{inf of I_X}.
\end{theo}
\begin{proof}
We divide this proof into two parts, for the case of large heights and the case of small heights.
\textbf{Part 1. Case of large height varieties. - }Suppose that the inequality
\[h_{\overline{\O_{\mathbb P(\sE)}(1)}}(X)>\frac{(2d+2)^{d+1}}{d!}\delta\left(\frac{\log B}{[K:\Q]}+\frac{3}{2}\log(n+1)+2^d\right)\]
is verified, where $h_{\overline{\O_{\mathbb P(\sE)}(1)}}(X)$ is defined at \eqref{definition of arakelov height}. Then by \cite[Theorem 4.8]{Chen1} and \cite[Proposition 2.12]{Chen1} (see also \S 2.1 and \S 2.3 of \cite{Chen2}), there exists a hypersurface in $\mathbb P(\sE_K)$ of degree $2(n-d)(\delta-1)+d+2$ which covers $S(X;B)$ but does not contain the generic point of $X$.
\textbf{Part 2. Case of small height varieties. - } Now we suppose that the inequality
\[h_{\overline{\O_{\mathbb P(\sE)}(1)}}(X)\leqslant\frac{(2d+2)^{d+1}}{d!}\delta\left(\frac{\log B}{[K:\Q]}+\frac{3}{2}\log(n+1)+2^d\right)\]
is verified. Let
\[\log N_0=(1+\epsilon)\left(\log B +\frac{1}{2}[K:\Q]\left(\log(n+1)+\log(d+1)\right)\right)\frac{\delta}{I_X(H)},\]
and $r$ be the positive integer defined at \eqref{number of places for regular reductions} in Lemma \ref{upper bound of regular reduction}. In this case, we have
\[r\leqslant \frac{A_1\log B+A_2}{\log N_0}+1,\]
where we denote
\[A_1=(n-d)(\delta-1)+\frac{(2d+2)^{d+1}}{d!}(n-d)\delta,\]
and
\[A_2=[K:\Q]\left(C_3+\frac{(2d+2)^{d+1}}{d!}\delta\left(\log(n+1)+\frac{1}{2}\log(d+1)+2^d\right)\right),\]
with the constant $C_3$ is defined at \eqref{constant C_3}. By the assumption that $\log B\geqslant\epsilon$, we obtain $r\leqslant A_3$, where
\[A_3=\frac{I_X(H)}{\delta}\left(A_1+\epsilon^{-1}A_2\right)+1.\]
By Bertrand's postulate (cf. Lemma \ref{Bertrand's postulate}), there exists a family of maximal ideals $\p_1,\ldots,\p_r$ of $\O_K$, such that
\[\alpha(K)^{i-1}N_0\leqslant N(\p_i)\leqslant\alpha(K)^iN_0\]
for every $i=1,\ldots,r$, where the constant $\alpha(K)\geqslant2$ depends only on the number field $K$.
For each $\p_i$, we have
\[\#\mathscr X\left(\f_{\p_i}\right)\leqslant\delta\left(N(\p_i)^d+\cdots+1\right)\leqslant\delta(d+1)N(\p_i)^d\leqslant\delta(d+1)\alpha(K)^{di}N_0^d,\]
and then we obtain the following upper bound of the number of auxiliary hypersurfaces which cover $S_1(X;B)$ but do not cover the generic point of $X$. The upper bound mentioned above is
\begin{eqnarray*}
\sum_{i=1}^r\#\mathscr X(\f_{\p_i})&\leqslant&\delta(d+1)N_0^d\sum_{i=1}^r\alpha(K)^{di}\\
&=&\delta(d+1)N_0^d\frac{\alpha(K)^d(\alpha(K)^{rd}-1)}{\alpha(K)^d-1}\\
&\leqslant& C'_4B^{\frac{(1+\epsilon)d\delta}{I_X(H)}},
\end{eqnarray*}
where the constant
\begin{eqnarray}\label{constant C''_4}
C'_4&=&\delta(d+1)\frac{\alpha(K)^d(\alpha(K)^{A_3d}-1)}{\alpha(K)^d-1}\left((d+1)(n+1)\right)^{\frac{(1+\epsilon)[K:\Q]d\delta}{2I_X(H)}}\nonumber\\
&\leqslant&\delta(d+1)\frac{\alpha(K)^d(\alpha(K)^{A_3d}-1)}{\alpha(K)^d-1}\left((d+1)(n+1)\right)^{\frac{1}{2}(1+\epsilon)[K:\Q](d+1)\delta^{-\frac{1}{d}}}\nonumber\\
&=:&C''_4(\epsilon, \delta, n,d, K).
\end{eqnarray}
In the above inequality, the second line is from the lower bound of $I_X(H)$ provided at \cite[Corollary 4.2]{McKinnonRoth_2015} (see \eqref{lower bound of I_X} for this lower bound in our application) and the definition of $I_X(H)$ at \eqref{inf of I_X}. Then we obtain the assertion by Corollary \ref{semi-global determinant method 2}.
\textbf{Conclusion. - }By the above argument, we obtain the assertion after combining it with Proposition \ref{covering singular locus}, where we choose the constant $C_4(\epsilon, \delta, n,d, K)=C''_4(\epsilon, \delta, n,d, K)+1$ introduced at \eqref{constant C''_4}.
\end{proof}
\begin{rema}
In the proof of Theorem \ref{number of hypersurfaces}, by the fact that $A_1\ll_{n,d}\delta$ and $A_2\ll_{n,d}\delta$, we have $A_3\ll_{n,d,\epsilon}\delta^{1+\frac{1}{d}}$, we obtain
\[\log C_4(\epsilon, \delta, n,d, K)\ll_{n,K,\epsilon}\delta^{1+\frac{1}{d}},\]
since we have $1\leqslant d\leqslant n-1$.
\end{rema}
\begin{rema}
In Theorem \ref{number of hypersurfaces}, we do not give an explicit upper bound of the degree of auxiliary hypersurfaces. The main obstruction is that in Theorem \ref{R(E) by GIT}, when we estimate $\mathcal R_{\xi_j}(F_D)$, until the author's knowledge, we cannot find an explicit lower bound of $\dim H^0\left(X,\mathscr L^{\otimes m}\right)$ for arbitrary line bundle $\mathscr L$. If $\mathscr L$ is ample, see \cite[Page 92]{Kollar07} for such an explicit lower bound. So by the strategy of this article, we are not able to control the dependence of $S(X;B)$ on the degree of $X$ due to the limit of the author's ability.
\end{rema}
\backmatter
\bibliography{liu}
\bibliographystyle{smfplain}
\end{document} | 59,833 |
Powertis, developer of large-scale PV projects in Europe and Latin America, will develop 2 GW of solar PV over the next three years between Brazil and Spain, 1 GW per country.
Headquartered in Madrid and less than a year old, Powertis has secured power purchase agreements (PPAs) for 1 GW in Brazil. In addition, Powertis is entering into the Spanish market where offers turnkey services ranging from feasibility study to project financing.
“In less than 15 seconds, enough energy from the sun reaches the Earth to keep the world running using clean energy. For this reason, we are convinced that the future will be solar. At Powertis, we focus on large projects that use the latest technology, minimizes the cost of solar, and let us negotiate sophisticated bilateral contracts, with the ultimate goal of offering a guaranteed and sustainable return to our investors,” says Pablo Otín, CEO of Powertis.
The construction of the first projects in Brazil will start in the first quarter of 2020 and the entire portfolio, which exceeds 1 GW, will be connected to the grid before December 2021. “Our main goal is to establish Powertis as the leading company in the bilateral market in Brazil”, states Otín. He also adds that this country, together with Spain, are two of the most active regions in the global PV market in bilateral PPAs. “We are taking advantage of these opportunities to showcase our team’s unique knowledge when it comes to contract and finance this type of projects.” | 68,838 |
Melanie did an apprenticeship at the TU Wien. She graduated in 2011 and continued working at the TU Wien as a secretary for a SFB FWF project at the Institute of Materials Chemistry for 7 years. She joined our team on 17. September 2018.
Student support
Teaching and course administration
Assistance Professor Reutterer
Back office
IT-Support | 261,986 |
*Hi there. I’ve taken a little pivot from my usual (but who can really call a blog I used to regularly write in two years ago ‘usual’) writing. This is really just for me now. I am trying to find my voice. To write regularly, and rawly (is that a word?) and to put it in a place where my mom (hi mom) and my sister (hi Amy) and maybe a few friends can see it. To hopefully be able to look back years from now and see how much I’ve grown as a person and a writer. I will not be tip-toeing around issues anymore or choose my words carefully because I think it has restricted me in the past. If you signed up for notifications years ago for this blog and were just alerted then, buckle up, cuz, well, sometimes it’s just real boring and sometimes I talk about Trump and how our city school system is real f-ed up. Welcome. *
It’s August 25, 2021. I was just looking up today on delta.com the amount of credit we have from a trip to NY we were supposed to take 18 months ago. 18 months ago. 18 months ago we had to cancel a trip to NY because coronavirus was killing people in New York City and they were running out of room in the morgues and putting people in refrigerated semi trucks. They were putting dead people in refrigerated trucks!
And last year, last Spring, we stayed home and we didn’t see our families and we didn’t go to work or church or the barely even the grocery store, unless we heard there might be a new shipment of toilet paper.
Now here we are. Here I am. So many long months later and it’s worse it’s so much worse. It’s what we thought was happening last year. But now we have a vaccine and people won’t take it because of the internet and fake news and because of “freedom.” And probably a little bit because of Donald Trump. And people are dying. Our hospitals are full again. This weekend, the city of Tallahassee ran out of oxygen completely. Orlando apparently uses oxygen to sanitize their water and they had to stop doing that.
Because of this virus. Because people won’t get the vaccine. Over and over I see hospital graphics about the number of people hospitalized with COVID that are vaccinated vs unvaccinated.
122 in patients being treated for covid, 10 vaccinated.
28 in patients being treated for COVID, 4 vaccinated.
10 deaths from COVID. All unvaccinated.
And on and on and on and on.
I saw something the other day that said that we weren’t meant to carry all this weight, we are not God. Well what are we supposed to do? People are dying, the physicians are exhausted, begging people to get the shot, Afghanistan has fallen to the Taliban and did you see that a 14 month old died of COVID just two weeks ago?
We weren’t meant to carry it all yet we are. Somewhere, it’s in there somewhere, right?
People are scared. My throat is sore, should I go get a COVID test? Mabry was exposed, and woke up with a runny nose. I got her a rapid test and it was negative, but should I have done a PCR? I sent her to school. Am I being responsible enough?
———————————————————————————————————-
There is so much to say, but how will I ever quiet my brain down enough to say it? There is so much I have to say.
There are stories in me about death and accidents and grief so deep you cannot see the bottom. About friendship, about the rarity that is the depth of decades-long friendship. They are my stories and I don’t know how to get them out of me. I want someone to tell me what to do. Please someone lay out a plan for me.
My friend Katie says, “you should be putting at least one body of work out into the world each week,” and I say, “yes, I know,” but how do I do it?
Where do I start? I am sitting in the kitchen and the dishwasher’s green light is screaming at me. The dishwasher is screaming at me to empty it. Who knew dishwashers could be so f-ing loud?
I am so busy. I am so busy perfecting my bicep curl and my parenting techniques. I am so busy cooking cow tongue and vacuuming the carpet. I cannot sit down and read even though I know it will make me want to write.
The laundry is screaming at me. Screaming! Why is the laundry so loud?
I have to read other people’s stories.
I have to take out the laundry’s vocal chords.
No one is going to tell me what to do.
I just have to do it.
4 thoughts on “August 25, 2021”
Oh my friend this is wonderful. I feel these thoughts. I can relate. Thank you for sharing.
Cheers to you cutting the bullshit. I love it, sis!
Keep writing!
So proud that you are my sister! This is the bravery we all need right now. ❤️ | 122,021 |
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Please be in prayer for Skylar who tested positive for COVID-19. Fortunately sbe does not have symptoms. Also add to those prayers some for the entire Dickason family because Skylar had just visited home this weekend for an early Thanksgiving which now places the entire family in quarantine for 14 days. | 392,390 |
TITLE: Application of Lindeberg-Feller Theorem
QUESTION [4 upvotes]: I want to prove that for independent variables $X_1,X_2,\dots$ $$\frac{\sum\limits_{k=1}^n X_k}{\sqrt{\mathbb V\left(\sum\limits_{k=1}^n X_k\right)}}\xrightarrow{n\to\infty}\mathcal N(0,1),$$
where $\mathbb P(X_n = n) = \mathbb P(X_n = -n) = \frac{1}{2}$. I cant use the normal CLT because the $X_n$ are not identically distributed. I have to use the Lindeberg-Feller Theorem. I know that $\mathbb E(X_n) = 0$ and that $\mathbb V(X_n) = n^2$ and that therefore $$\mathbb V\left(\sum\limits_{k=1}^n X_k\right) = \sum\limits_{k=1}^n\mathbb V(X_k) = \sum\limits_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}.$$
I know that I need to find $X_{n,k}$ with
$\sum\limits_{k=1}^{k_n}\mathbb E(X_{n,k}^2)\xrightarrow{n\to\infty}1$ and $\sum\limits_{k=1}^{k_n}\mathbb E(X_{n,k}^2 \mathbf 1_{\{|X_{n,k}|>\varepsilon\}})\xrightarrow{n\to\infty}0$
for all $\varepsilon >0$. How can I find such $X_{n,k}$. Any help is appreciated.
REPLY [3 votes]: Let $\sigma_n^2:={\mathbb V\left(\sum\limits_{k=1}^n X_k\right)}=n^3/(3+o(1))$
and define $Y_{n,k}:=X_k/\sigma_n$ for $k \le n$.
Then
$\sum\limits_{k=1}^{n}\mathbb E(Y_{n,k}^2)=1$.
Moreover, for all $\varepsilon >0$,
if $n$ is large enough, then $|Y_{n,k}|\le n/\sigma_n <\varepsilon$ for all $k \le n$, so $\sum\limits_{k=1}^{n}\mathbb E(Y_{n,k}^2 \mathbf 1_{\{|Y_{n,k}|>\varepsilon\}})=0$.
Thus the conditions of the Lindeberg-Feller Theorem are met. | 8,351 |
TITLE: Integral tends to infty
QUESTION [1 upvotes]: How can we prove that $\lim\limits_{\varepsilon\searrow 0} \dfrac{1}{\varepsilon} \displaystyle\int_0^1 \left [\dfrac{1}{4}-\dfrac{1}{\pi^2}\arctan^2\left (\dfrac{x}{\varepsilon}\right )\right ]^2 dx=+\infty$?
I came across this integral in studying the Heaviside aproximation function. I used the following sharp estimate:
$\arctan(x)<\dfrac{\pi^2 x}{4+\sqrt{32+(2\pi x)^2}}, \forall\ x>0$, but the limit goes to $0$ (I computed it exactly with wolfram). This estimation and many others can be found here:
https://journalofinequalitiesandapplications.springeropen.com/articles/10.1186/s13660-018-1734-7
The integral in the statement is simulated here:
https://www.wolframalpha.com/input/?i=int_0%5E1+1000000000+%281%2F4+-+%281%2Fpi%5E2%29+%28arctan%2810000000000+x%29%29%29%5E2+dx
But I wonder if it can be proved that it really tends to infinity. Maybe there are some singularities and the program gives some big errors.
REPLY [0 votes]: OHH, please sorry. I made a mistake putting an extra $2$. It should be:
$$\lim\limits_{\varepsilon\searrow 0} \dfrac{1}{\varepsilon} \displaystyle\int_0^1 \left [\dfrac{1}{4}-\dfrac{1}{\pi^2}\arctan^2\left (\dfrac{x}{\varepsilon}\right )\right ]^2 dx=+\infty$$ | 194,888 |
\begin{document}
\title[Curvature of $R^{n-p}f_*\Omega^p_{\cX/S}(\cK_{\cX/S}^{\otimes m})$
and applications]{Curvature of $R^{n-p}f_*\Omega^p_{\cX/S}(\cK_{\cX/S}^{\otimes m})$\\ and applications}
\author{Georg Schumacher}
\address{Fachbereich Mathematik und Informatik,
Philipps-Universit\"at Marburg, Lahnberge, Hans-Meerwein-Straße, D-35032
Marburg,Germany}
\email{[email protected]}
\date{}
\maketitle
\begin{abstract}
Given an effectively parameterized family $f:\cX\to S$ of canonically
polarized manifolds, the \ke metrics on the fibers induce a hermitian
metric on the relative canonical bundle $\cK_{\cX/S}$. We use a global
elliptic equation to show that this metric is strictly positive everywhere
and give estimates.
The direct images $R^{n-p}f_*\Omega^p_{\cX/S}(\cK_{\cX/S}^{\otimes m})$,
$m > 0$, carry induced natural hermitian metrics. We prove an explicit
formula for the curvature tensor of these direct images. The formula for
$p=n$, implies that $f_*(\cK_{\cX/S}^{\otimes (m+1)})$ is Nakano-positive
with estimates (for effectively parameterized families). We apply it to
the morphisms $S^p\cT_S \to R^pf_*\Lambda^p\cT_{\cX/S}$ induced by the \ks
map and obtain a differential geometric proof for hyperbolicity properties
of $\cM_{\text{can}}$. Similar results hold for families of polarized
Ricci-flat manifolds. These will appear elsewhere.\footnote{}
\end{abstract}
\footnotetext{Dissertation in progress.}
\section{Introduction}
For any holomorphic family $f: \cX \to S$ of canonically polarized,
complex manifolds, the unique \ke metrics on the fibers define an
intrinsic metric on the relative canonical bundle $\cK_{\cX/S}$. The
construction is functorial in the sense of compatibility with base
changes. By definition, its curvature form has at least as many positive
eigenvalues as the dimension of the fibers indicates. It turned
\cite{sch-preprint} out that it is {\em strictly positive}, provided the
induced deformation is not infinitesimally trivial at the corresponding
point of the base.
Actually the first variation of the metric tensor in a family of compact
\ke manifolds contains the information about the induced deformation, more
precisely, it contains the harmonic representatives $A_s=
A^\alpha_{s\ol\beta}\pt_\alpha dz^\ol\beta $ of the \ks classes
$\rho(\pt/\pt s)$. The positivity of the hermitian metric will be measured
in terms of a certain global function. Essential is an elliptic equation
on the fibers, which relates this function to the pointwise norm of the
harmonic \ks forms. The ''strict'' positivity of the corresponding
(fiberwise) operator $(\Box + id)^{-1}$, where $\Box$ is the complex
Laplacian, can be seen in a direct way (cf. \cite{sch-preprint}). For
families of compact Riemann surfaces the elliptic equation was previously
derived in terms of automorphic forms by Wolpert \cite{wo}. Later in
higher dimensions a similar equation arose in the work of Siu
\cite{siu:canlift} for families of canonical polarized manifolds.
In this article, we will reduce estimates for the positivity of the
curvature of $\cK_{\cX/S}$ on $\cX$ to estimates of the resolvent kernel
of the above integral operator, whose its positivity was already shown by
Yosida in \cite{yos}. Finally estimates for the resolvent kernel follow
from the estimates for the heat kernel, which were achieved by Cheeger and
Yau in \cite{c-y}.
The positivity of the relative canonical bundle is important, when
estimating the curvature of the direct image sheaves
$R^{n-p}f_*\Omega^p_{\cX/S}(\cK_{\cX/S}^{\otimes m})$. These are equipped
with a natural hermitian metric that is induced by the $L^2$-inner product
of harmonic tensors on the fibers of $f$. We will give an explicit formula
for the curvature tensor.
A main motivation of our approach is the study of the curvature of the
classical \wp metric, in particular the computation of the curvature
tensor by Tromba \cite{tr} and Wolpert \cite{wo}. It immediately implies
the hyperbolicity of the classical \tei space. The curvature tensor of
the generalized \wp metric for families of metrics with constant negative
Ricci curvature was explicitly computed by Siu in \cite{siu:canlift}; in
\cite{sch:curv} a formula in terms of elliptic operators and harmonic \ks
tensors was derived. However, the curvature of the generalized \wp metric
seems not to satisfy any negativity condition. This difficulty was
overcome in the work of Viehweg and Zuo in \cite{v-z}. Their approach to
hyperbolicity makes use of the period map in the sense of Griffiths.
On the other hand our results are motivated by Berndtsson's result on the
Nakano-positivity for certain direct images. In \cite{sch-preprint} we
showed that the positivity of $f_*\cK_{\cX/S}^{\otimes 2}$ together with
the curvature of the generalized \wp metric is sufficient to imply a
hyperbolicity result for moduli of canonically polarized complex manifolds
in dimension two.
For ample $K_X$ the cohomology groups $H^{n-p}(X,\Omega^p_X(K^{\otimes
m}_X))$ are critical with respect to the Kodaira-Nakano vanishing theorem.
The understanding of this situation is the other main motivation. We will
consider the relative case. Let $A_{i \ol\beta}^\alpha(z,s) \pt_\alpha
dz^{\ol\beta}$ be a harmonic \ks form. Then for $s \in S$ the cup product
together with the contraction defines
\begin{eqnarray*}
A_{i \ol\beta}^\alpha \pt_\alpha dz^{\ol\beta}\cup \textvisiblespace :
\cA^{0,n-p}(\cX_s,\Omega^p_{\cX_s}(\cK_{\cX_s}^{\otimes m})) &\to&
\cA^{0,n-p+1}(\cX_s,\Omega^{p-1}_{\cX_s}(\cK_{\cX_s}^{\otimes m}))\\ A_{\ol \jmath
\alpha}^\ol\beta \pt_\ol\beta dz^{\alpha}\cup \textvisiblespace :
\cA^{0,n-p}(\cX_s,\Omega^p_{\cX_s}(\cK_{\cX_s}^{\otimes m})) &\to&
\cA^{0,n-p-1}(\cX_s,\Omega^{p+1}_{\cX_s}(\cK_{\cX_s}^{\otimes m})).
\end{eqnarray*}
\enlargethispage{.5cm} We will apply the above product to harmonic
$(0,n-p)$-forms. In general the result is not harmonic. We denote the
pointwise $L^2$ inner product by a dot.
\begin{maintheorem}
The curvature tensor for $R^{n-p}f_*\Omega^p_{\cX/S}(\cK_{\cX/S}^{\otimes
m})$ is given by
\begin{eqnarray*}
R_{i\ol\jmath}^{\phantom{{i\ol\jmath}}\ol\ell k}(s)&=& m \int_{\cX_s}
\left( \Box + 1 \right)^{-1}(A_i\cdot A_\ol\jmath) \cdot(\psi^k \cdot
\psi^\ol\ell) g\/ dV\nonumber\\
&& \quad + m \int_{\cX_s} \left( \Box + m \right)^{-1} (A_i\cup\psi^k)
\cdot (A_\ol\jmath \cup \psi^\ol\ell) g\/ dV \\
&& \quad + m \int_{\cX_s} \left( \Box - m \right)^{-1}
(A_i\cup\psi^\ol\ell)\cdot (A_\ol\jmath \cup \psi^k) g\/ dV.
\nonumber
\end{eqnarray*}
\end{maintheorem}
The potentially negative third term is not present for $p=n$, i.e.\ for
$f_*\cK_{\cX/S}^{\otimes(m+1)}$. From the main theorem we get immediately
a fact which is contained in Berndtsson's theorem \cite{berndtsson}:
{\em The locally free sheaf $f_*\cK_{\cX/S}^{\otimes(m+1)}$ is
Nakano-positive.}
(Very recently Berndtsson considered the curvature of $f_* (\cK_{\cX/S}
\otimes \cL)$, \cite{berndtsson-pre}.)
We prove the estimate
$$
R(A,\ol A, \psi,\ol\psi) \geq P_n(d(\cX_s)) \cdot \|A\|^2\cdot
\|\psi\|^2,
$$
where the coefficient $P_n(d(\cX_s))>0$ is an explicit function depending
on the dimension and the diameter of the fibers.
By Serre duality the theorem yields the following version, which contains
for $p=1$ the curvature formula for the generalized \wp metric. Again a
tangent vector of the base is identified with a harmonic \ks form $A_i$,
and $\nu_k$ stands for a section of the relevant sheaf:
\begin{maintheorem}
The curvature for $R^pf_*\Lambda^p\cT_{\cX/S}$ equals
\begin{eqnarray*}
R_{i\ol\jmath k \ol\ell }(s)&=&- \int_{\cX_s}
\left( \Box + 1 \right)^{-1}(A_i\cdot A_\ol\jmath)
\cdot(\nu_k \cdot \nu_\ol\ell) g\/ dV\nonumber\\
&& \quad - \int_{\cX_s} \left( \Box + 1 \right)^{-1} (A_i\wedge\nu_\ol\ell)
\cdot (A_\ol\jmath \wedge \nu_k) g\/ dV \\
&& \quad - \int_{\cX_s} \left( \Box - 1 \right)^{-1}
(A_i\wedge \nu_k)\cdot (A_\ol\jmath \wedge \nu_\ol\ell) g\/ dV.
\nonumber
\end{eqnarray*}
\end{maintheorem}
In order to prove a result about hyperbolicity of moduli spaces we use
Demailly's approach. An upper semi-continuous Finsler metric of negative
holomorphic curvature on a relatively compact subspace of the moduli stack
of canonically polarized varieties can be constructed so that any such
space is hyperbolic with respect to the orbifold structure.
We get immediately the following fact related to Shafarevich's
hyperbolicity conjecture for higher dimensions solved by Migliorini
\cite{m}, Kovács \cite{kv1,kv2,kv3}, Bedulev-Viehweg \cite{b-v}, and
Viehweg-Zuo \cite{v-z,v-z2}.
{\bf Application.} {\it Let $\cX \to C$ be a non-isotrivial holomorphic
family of canonically polarized manifolds over a curve. Then $g(C)>1$.}
{\it Acknowledgements.} This work was begun during a visit to Harvard
University. The author would like to thank Professor Yum-Tong Siu for his
cordial hospitality and many discussions. His thanks also go to Bo
Berndtsson, Jeff Cheeger, Jean-Pierre Demailly, Gordon Heier, Stefan
Kebekus, Janos Kollár, Sándor Kovács, and Thomas Peternell for discussions
and remarks.
\bigskip
\bigskip
\section{Fiber integration and Lie derivatives}
\subsection{Definition of fiber integrals and basic properties}
Let denote $\{\cX_s\}_{s\in S}$ a holomorphic family of compact complex
manifolds $\cX_s$ of dimension $n>0$ parameterized by a reduced complex
space $S$. By definition, it is given by a proper holomorphic submersion
$f:\cX \to S$, such that the $\cX_s = f^{-1}(s)$ for $s\in S$. In case of
a smooth space $S$, if $\eta$ is a differential form of class $\cinf$ of
degree $2n+r$ the fiber integral
$$
\int_{\cX/S} \eta
$$
is a differential form of degree $r$ on $S$. It can be defined as follows:
Fix a point $s_0\in S$ and denote by $X=\cX_{s_0}$ the fiber. Let $U
\subset S$ be an open neighborhood of $s_0$ such that there exists a
$\cinf$ trivialization of the family:
$$
\xymatrix{f^{-1}U \ar[d]_{f|f^{-1}U } & X \ar[l]_{\Phi}^\sim\ar[dl]^{pr}\times U\\ U }
$$
Let $z=(z^1,\ldots,z^n)$ and $s=(s^1,\ldots,s^k)$ denote local
(holomorphic) coordinates on $X$ and $S$ resp.
The pull-back $\Phi^*\eta$ possesses a summand $\eta'$, which is of the
form $\sum \eta_k(z,s) dV_z \wedge d\sigma^{k_1}\wedge\ldots\wedge
d\sigma^{k_r}$, where the $\sigma^\kappa$ run through the real and complex
parts of $s^j$, and where $dV_z$ denotes the relative Euclidean volume
element. Now
$$
\int_{\cX/S} \eta := \int_{X\times S/S} \Phi^*\eta := \sum_{k=(k_1,\ldots,k_r)} \left(\int_{\cX_s}
\eta_k(z,s) dV_z \right) d\sigma^{k_1}\wedge\ldots\wedge d\sigma^{k_r}.
$$
The definition is independent of the choice of coordinates and
differentiable trivializations. The fiber integral coincides with the
push-forward of the corresponding current. Hence, if $\eta$ is a
differentiable form of type $(n+r,n+s)$, then the fiber integral is of
type $(r,s)$.
Singular base spaces are treated as follows: Using deformation theory, we
can assume that $S\subset W$ is a closed subspace of some open set $W
\subset \mathbb C^N$, and that an almost complex structure is defined on
$X\times S$ so that $\cX$ is the integrable locus. Then by definition, a
differential form of class $\cinf$ on $\cX$ will be given on the whole
ambient space $X \times W$ (with a type decomposition defined on $\cX$).
Fiber integration commutes with taking exterior derivatives:
\begin{equation}\label{eq:fibintallg}
d \int_{\cX/S} \eta = \int_{\cX/S} d\eta,
\end{equation}
and since it preserves the type (or to be seen explicitly in local
holomorphic coordinates), the same equation holds true for $\pt$ and
$\ol\pt$ instead of $d$.
A \ka form $\omega_\cX$ on a singular space, by definition is a form that
possesses locally a $\pt\ol\pt$-potential, which is the restriction of a
$\cinf$ function on a smooth ambient space. It follows from the above
facts that given a \ka form $\omega_\cX$ on the total space, the fiber
integral
\begin{equation}
\int_{\cX/S} \omega_\cX^{n+1}
\end{equation}
is a \ka form on the base space $S$, which possesses locally a smooth
$\pt\ol\pt$-potential, even if the base space of the smooth family is
singular.
For the actual computation of exterior derivatives of fiber integrals
\eqref{eq:fibintallg}, in particular of functions, given by integrals of
$(n,n)$-forms on the fibers, the above definition seems to be less
suitable. Instead the problem is reduced to derivatives of the form
\begin{equation}\label{eq:derfibint}
\frac{\pt}{\pt s^i} \int_{\cX_s} \eta,
\end{equation}
where only the vertical components of $\eta$ contribute to the integral.
Here and later we will always use the summation convention.
\begin{lemma}\label{le:intLie}
Let
$$
w_i=\left.\left(\frac{\pt}{\pt s^i} + b^\alpha_i(z,s)\frac{\pt}{\pt z^\alpha}
+ c^\ol\beta(z,s)\frac{\pt}{\pt z^\ol\beta} \right)\right|_s
$$
be differentiable vector fields, whose projection to $S$ equal
$\frac{\pt}{\pt s^i}$. Then
$$
\frac{\pt}{\pt s^i} \int_{\cX_s} \eta =
\int_{\cX_s} L_{w_i}\left(\eta\right),
$$
where $L_{w_i}$ denotes the Lie derivative.
\end{lemma}
Concerning singular base spaces, observe that it is sufficient that the
above equation is given on the first infinitesimal neighborhood of $s$ in
$S$.
\begin{proof}
Because of linearity, one may consider the real and imaginary parts of
$\pt/\pt s^i$ and $w_i$ resp.\ separately.
Let $\pt/\pt t$ stand for $Re(\pt/\pt s^i)$, and let $\Phi_t: X \to \cX_t$
be the one parameter family of diffeomorphisms generated by $Re(w_i)$.
Then
$$
\frac{d}{d t} \int_{\cX_s} \eta = \int_X \frac{d}{dt} \Phi^*_t \eta = \int_X L_{Re(w^i)}(\eta).
$$
It is known that the vector fields $Re(w_i)$ and $Im(w_i)$ need not
commute.
\end{proof}
In our applications, the form $\eta$ will typically consist of inner
products of differential forms with values in hermitian vector bundles,
whose factors need to be treated separately. This will be achieved by the
Lie derivatives. In this context, we will have to use covariant
derivatives with respect to the given hermitian vector bundle on the total
space and to the \ka metrics on the fibers.
\subsection{Direct images and differential forms}\label{ss:dolb}
Let $(\cE, h)$ be a hermitian, holomorphic vector bundle on $\cX$, whose
direct image $R^q f_*\cE$ is {\em locally free}. Furthermore we assume
that for all $s\in S$ the cohomology $H^{q+1}(\cX_s, \cE \otimes
\cO_{\cX_s})$ {\em vanishes}. Then the statement of the
Grothendieck-Grauert comparison theorem holds for $R^q f_*\cE$, in
particular $R^q f_*\cE\otimes_{\cO_S} \C(s)$ can be identified with
$H^{q}(\cX_s, \cE \otimes_{\cO_\cX}\cO_{\cX_s})$.
For simplicity, we assume that he base space $S$ is smooth. Locally, after
replacing $S$ by a neighborhood of any given point, we can represent
sections of the $q$-th direct image sheaf in terms of Dolbeault cohomology
by $\ol\pt$-closed $(0,q)$-forms. On the other hand, fiberwise, we have
harmonic representatives of cohomology classes with respect to the \ka
form and hermitian metric on the fibers. The following fact will be
essential.
\begin{lemma}\label{le:dolb}
Let $\wt\psi \in R^q f_*\cE(S)$ be a section. Let $\psi_s \in
\cA^{0,q}(\cX_s,\cE_s)$ be the harmonic representatives of the cohomology
classes $\wt\psi|\cX_s$.
Then locally with respect to $S$ there exists a $\ol\pt$-closed form
$\psi\in \cA^{0,q}(\cX,\cE)$, which represents $\wt \psi$, and whose
restrictions to the fibers $\cX_s$ equal $\psi_s$.
\end{lemma}
We omit the simple proof.
In this way, the relative Serre duality can be treated in terms of such
differential forms. Let $\cE^\vee = \textit{Hom}_\cX(\cE,\cO_\cX)$. Then
$$
R^pf_*\cE \otimes_{\cO_S} R^{n-p}f_*(\cE^\vee \otimes_{\cO_\cX} \cK_{\cX/S}) \to \cO_S
$$
is given by the fiber integral of the wedge product of $\ol\pt$-closed
differential forms in the sense of Lemma~\ref{le:dolb}. By
\eqref{eq:fibintallg} (for the operator $\ol\pt$), the result is a
$\ol\pt$-closed $0$-form i.e.\ holomorphic function.
\section{Estimates for resolvent and heat kernel}
Let $(X,\omega_X)$ be a compact \ka manifold.
The Laplace operator for differentiable functions is given by $\Box =
\ol\pt\ol\pt^*+\ol\pt^*\ol\pt$, where the adjoint $\ol\pt^*$ is the formal
adjoint operator. The Laplacian is self-adjoint with non-negative
eigenvalues.
The corresponding resolvent operator $(id + \Box)^{-1}$ is defined on the
space of continuous functions and bounded.
First, we observe that the resolvent operator is positive: If $\chi \geq
0$ everywhere on $X$, then the function given by $(\Box + id)^{-1}\chi$ is
non-negative. This fact follows immediately from the minimum principle
applied to the elliptic equation
$$
\Box \phi + \phi = \chi.
$$
So the integral kernel $P(z,w)$ must be non-negative for all $z$ and $w$.
For any function $\chi(z)$
$$
(\Box + id)^{-1}(\chi)(z)= \int_X P(z,w)\chi(w) g(w)dV_w
$$
holds. In a similar way we denote by $P(t,z,w)$ the integral kernel for
the heat operator
$$
\frac{d}{dt} + \Box
$$
so that the solution of the heat equation with initial function $\chi(z)$
for $t=0$ is given by
$$
\int_X P(t,z,w)\chi(w) g(w)dV_w.
$$
The explicit representation of the above operators in terms of eigen
functions of the Laplacian yields the following relation.
\begin{lemma}
\label{le:heat} Let $P(z,w)$ be the integral kernel of the resolvent
operator and denote by $P(t,z,w)$ the heat kernel. Then
\begin{equation}\label{eq:resheatest}
P(z,w)= \int_0^\infty e^{-t}P(t,z,w)dt.
\end{equation}
\end{lemma}
\begin{proof}
Let $\{\chi_\nu\}$ be a set of eigenfunctions of the Laplacian with
eigenvalues $\lambda_\nu$ so that
$$
P(z,w)= \sum_\nu \frac{1}{1+\lambda_\nu} \chi_\nu(z)\chi_\nu(w)
$$
and
$$
P(t,z,w)= \sum_\nu e^{-t\lambda_\nu} \chi_\nu(z)\chi_\nu(w).
$$
Then, since the eigenvalues are non-negative,
$$
\int_0^\infty e^{-t(\lambda + 1)} dt = \frac{1}{1+\lambda}
$$
implies \eqref{eq:resheatest}, (cf.\ also \cite[(3.13)]{c-y}).
\end{proof}
We now apply the lower estimates for the heat kernel by Cheeger and Yau
\cite{c-y} to the resolvent kernel. Assuming constant negative Ricci
curvature $-1$, we use the estimates from \cite[(4.3) Corollary]{st}.
\begin{equation}\label{eq:hker}
P(t,z,w)\geq Q_{n}(t,r(z,w)):=\frac{1}{(2\pi t)^n} e^{- \frac{r^2(z,w)}{t}} e^{-\frac{2n-1}{4}t},
\end{equation}
Where $r=r(z,w)$ denotes the geodesic distance (and $n=\dim X$).
Let
\begin{equation}\label{eq:hker1}
P_{n}(r)= \int_0^\infty e^{-t} Q_n(t,r) dt>0.
\end{equation}
Using Lemma~\ref{le:heat} and \eqref{eq:hker} we get
\begin{equation}\label{eq:hker2}
P(z,w) \geq P_n(r(z,w)) \geq P_n(d(X)),
\end{equation}
where $d(X)$ denotes the diameter of $X$. However, $\lim_{r\to
\infty}P_n(r)=0$.
\begin{proposition}\label{pr:resol}
Let $(X,\omega_X)$ be a \ke manifold of constant Ricci curvature $-1$ with
volume element $g\/ dV$ and diameter $d(X)$. Let $\chi$ be a non-negative
continuous function. Let
\begin{equation}\label{eq:ellip}
(1+\Box)\phi = \chi.
\end{equation}
Then
\begin{equation}
\phi(z)\geq P_n(d(X))\cdot \int_X \chi g\/ dV
\end{equation}
for all $z\in X$.
\end{proposition}
Conversely let for all solutions of \eqref{eq:ellip} an estimate
$\phi(z)\geq P \cdot \int_X \chi g\/ dV$ hold. for some number $P$. Then
$P\leq\inf P(z,w)$ follows immediately.
\medskip
\begin{tiny}
We mention that symbolic integration of \eqref{eq:hker} with
\eqref{eq:hker1} yields an explicit estimate.
$$
P_{n}(r) \geq \frac{1}{(2\pi)^n} \frac{(2n+3)^{\frac{n-1}{2}}}{2^{n-2}} \frac{1}{r^{n-1}} \text{BesselK}\left( n-1, \sqrt{2n+3} r \right)
$$
\end{tiny}
\section{Positivity of $K_{\cX/S}$}\label{se:posi}
Let $X$ be a canonically polarized manifold of dimension $n$, equipped
with a \ke metric $\omega_X$. In terms of local holomorphic coordinates
$(z^1,\ldots, z^n)$ we write
$$
\omega_X=\ii g_{\alpha\ol\beta}(z)\; dz^\alpha\wedge dz^\ol\beta
$$
so that the \ke equation reads
\begin{equation}\label{eq:ke}
\omega_X=-{\rm Ric}(\omega_X), \text{ i.e. } \omega_X= \ddb \log g(z),
\end{equation}
where $g:=\det g_{\alpha\ol\beta}$. We consider $g$ as a hermitian metric
on the anti-canonical bundle $K_X^{-1}$.
For any holomorphic family of compact, canonically polarized manifolds $f:
\cX \to S$ of dimension $n$ with fibers $\cX_s$ for $s\in S$ the \ke forms
$\omega_{\cX_s}$ depend differentiably on the parameter $s$. The resulting
relative \ka form will be denoted by
$$
\omega_{\cX/S} = \ii g_{\alpha,\ol\beta}(z,s)\;dz^\alpha\wedge dz^\ol\beta.
$$
The corresponding hermitian metric on the relative anti-canonical bundle
is given by $g=\det \gab(z,s)$. We consider the real $(1,1)$-form
$$
\omega_\cX= \ddb \log g(z,s)
$$
on the total space $\cX$. We will discuss the question, whether
$\omega_\cX$ is a \ka form on the total space.
The \ke equation \eqref{eq:ke} implies that
$$
\omega_\cX|\cX_s = \omega_{\cX_s}
$$
for all $s\in S$. In particular $\omega_\cX$, restricted to any fiber, is
positive definite. Our result is the following statement (cf.\ Main
Theorem).
\begin{theorem}\label{th:main}
Let $\cX \to S$ be a holomorphic family of canonically polarized, compact,
complex manifolds. Then the hermitian metric on $\cK_{\cX/S}$ induced by
the \ke metrics on the fibers is semi-positive and strictly positive on
all fibers. It is strictly positive in horizontal directions, for which
the family is not infinitesimally trivial.
\end{theorem}
Both the statement of the Theorem and the methods are valid for smooth,
proper families of singular (even non-reduced) complex spaces (for the
necessary theory cf.\ \cite{f-s:extremal}).
It is sufficient to prove the theorem for base spaces of dimension one
assuming $S\subset \C$. (In order to treat singular base spaces, the claim
can be reduced to the case where the base is a double point $(0,\mathbb
C[s]/(s^2))$. The arguments below will still be meaningful and can be
applied literally.)
We denote the \ks map for the family $f:\cX \to S$ at a given point
$s_0\in S$ by
$$
\rho_{s_0} :T_{s_0} \to H^1(X, \cT_X)
$$
where $X=\cX_{s_0}$. The family is called {\it effectively parameterized}
at $s_0$, if $\rho_{s_0}$ is injective. The \ks map is induced as edge
homomorphism by the short exact sequence
$$
0 \to \cT_{\cX/S} \to \cT_\cX \to f^*\cT_S \to 0.
$$
If $v \in T_{s_0}S$ is a tangent vector, say $v=\frac{\pt}{\pt s}|_{s_0}$
and $\frac{\pt}{\pt s} + b^\alpha \frac{\pt}{\pt z^\alpha}$ is any lift of
class $\cinf$ to $\cX$ along $X$, then
$$
\ol\pt\left(\frac{\pt}{\pt s} + b^\alpha(z) \frac{\pt}{\pt z^\alpha}\right)=
\frac{\partial b^\alpha(z)}{\partial z^\ol\beta}
\frac{\pt}{\pt z^\alpha} dz^\ol\beta
$$
is a $\ol\pt$-closed form on $X$, which represents $\rho_{s_0}(\pt / \pt
s)$. Observe that $b^\alpha$ cannot be a tensor on $X$, unless the family
is infinitesimally trivial.
We will use the semi-colon notation as well as raising and lowering of
indices for covariant derivatives with respect to the {\it \ke metrics on
the fibers}. The $s$-direction will be indicated by the index $s$. In this
sense the coefficients of $\omega_\cX$ will be denoted by $g_{s\ol s}$,
$g_{\alpha\ol s}$, $\gab$ etc.
Next, we define {\it canonical lifts} of tangent vectors of $S$ as
differentiable vector fields on $\cX$ along the fibers of $f$ in the sense
of Siu \cite{siu:canlift}. By definition these satisfy the property that
the induced representative of the \ks class is {\it harmonic}.
Since the form $\omega_\cX$ is positive, when restricted to fibers, {\em
horizontal lifts} of tangent vectors with respect to the pointwise
sesquilinear form $\langle-,-\rangle_{\omega_\cX}$ are well-defined (cf.\
also \cite{sch:curv}).
\begin{lemma}\label{le:canlift}
The horizontal lift of $\pt/\pt s$ equals
$$
v = \pt_s + a_s^\alpha \pt_\alpha,
$$
where
$$
a_s^\alpha = - g^{\ol\beta \alpha} g_{s \ol \beta}.
$$
\end{lemma}
\begin{proposition}\label{pr:harmrep}
The horizontal lift induces the harmonic representative of
$\rho_{s_0}(\pt/\pt s)$.
\end{proposition}
\begin{proof}
The \ks form of the tangent vector $\pt/\pt_{s_0}$is given by $\ol\pt
v|\cX_s = a^\alpha_{s;\ol\beta}\pt_\alpha dz^\ol\beta$. We consider the
tensor
$$
A^\alpha_{s\ol\beta}:= a^\alpha_{s;\ol\beta}|{\cX_{s_0}}
$$
on $X$. Then
\begin{gather*}
g^{\ol\beta\gamma} A^\alpha_{s\ol\beta;\gamma}= - g^{\ol\beta\gamma}
g^{\ol\delta \alpha} g_{s\ol\delta;\ol\beta\gamma} = - g^{\ol\beta\gamma}
g^{\ol\delta \alpha} g_{s\ol\beta;\ol\delta\gamma} =
-g^{\ol\beta\gamma}g^{\ol\delta \alpha} \left(
g_{s\ol\beta;\gamma\ol\delta} -
g_{s\ol\tau}R^\ol\tau_{\; \ol\beta\ol\delta\gamma} \right)\\
=-g^{\ol\delta\alpha}\left(\left({\pt\log g}/{\pt s} \right)_{;\ol\delta}
+ g_{s\ol \tau}R^\ol\tau_{\; \ol\delta}\right) = 0.
\end{gather*}
\end{proof}
It follows immediately from the proposition that the harmonic \ks forms
induce symmetric tensors. This fact reflects the close relationship
between the \ks tensors and the \ke metrics.
\begin{corollary}\label{co:symm}
Let $A_{s \ol\beta\,\ol\delta}= g_{\alpha\ol\beta}A^\alpha_{s\ol\delta}$.
Then
\begin{equation}
A_{s \ol\beta\,\ol\delta}=A_{s \ol\delta\,\ol\beta}.
\end{equation}
\end{corollary}
Next, we introduce a {\it global} function $\varphi(z,s)$, which is by
definition the {\em pointwise inner product of the canonical lift $v$ of
$\pt/\pt s$ at $s\in S$} with itself with respect to $\omega_\cX$.
\begin{definition}\label{de:varphi}
\begin{equation}\label{eq:varphidef}
\varphi(z,s) := \langle \pt_s + a_s^\alpha \pt_\alpha, \pt_s + a_s^\beta
\pt_\beta \rangle_{\omega_\cX}
\end{equation}
\end{definition}
Since $\omega_\cX$ is not known to be positive definite in all directions,
$\varphi\geq 0$ is not known at this point.
\begin{lemma}\label{le:varphi_0}
\begin{equation}\label{eq:varphi}
\varphi = g_{s\ol s} - g_{\alpha\ol s}
g_{s\ol\beta} g{^{\ol\beta\alpha}}
\end{equation}
\end{lemma}
\begin{proof}
The proof follows from Lemma~\ref{le:canlift} and
$$
\varphi = g_{s\ol s} + g_{s\ol\beta}a^\ol\beta_{\ol s} + a_s^\alpha g_{\alpha\ol s}
+ a_s^\alpha a_{\ol s}^\ol\beta \gab.
$$
\end{proof}
Denote by $\omega^{n+1}_\cX$ the $(n+1)$-fold exterior product, divided by
$(n+1)!$ and by $dV$ the Euclidean volume element in fiber direction. Then
the global real function $\varphi$ satisfies the following property:
\begin{lemma}\label{le:varphi}
$$
\omega^{n+1}_\cX= \varphi \cdot g \cdot dV\ii ds\wedge \ol{ds}.
$$
\end{lemma}
\begin{proof}
Compute the following $(n+1)\times(n+1)$-determinant
$$ \det
\left(
\begin{array}{cc}
g_{s\ol s} & g_{s\ol\beta}\\ g_{\alpha\ol s}& \gab
\end{array}
\right),
$$
where $\alpha,\beta=1,\ldots,n$.
\end{proof}
So far we are looking at {\it local} computations, which essentially only
involve derivatives of certain tensors. The only {\it global ingredient}\/
is the fact that we are given global solutions of the \ke equation.
The key quantity is the differentiable function $\varphi$ on $\cX$.
Restricted to any fiber it ties together the yet to be proven positivity
of the hermitian metric on the relative canonical bundle and the canonical
lift of tangent vectors, which is related to the harmonic \ks forms.
We use the Laplacian operators $\Box_{g,s}$ with non-negative eigenvalues
on the fibers $\cX_s$ so that for a real valued function $\chi$ the
Laplacian equals $\Box_{g,s}\chi
= - g^{\ol\beta\alpha}\chi_{;\alpha\ol\beta}$.
\begin{proposition}\label{pr:elleq}
The following elliptic equation holds fiberwise:
\begin{equation}\label{eq:phiA}
(\Box_{g,s} + {\rm id})\varphi(z,s) = \|A_s(z,s)\|^2,
\end{equation}
where
$$
A_s=A^\alpha_{s\ol\beta} \frac{\pt}{\pt z^\alpha}dz^\ol\beta
$$
is the harmonic representative of the \ks class $\rho_s(\frac{\pt}{\pt
s})$ as above.
\end{proposition}
\begin{proof}
The essence to prove an elliptic equation for $\varphi$ involving tensors
on the fibers is to eliminate the second order derivatives with respect to
the base parameter. This is achieved by the left hand side of
\eqref{eq:phiA}. First,
\begin{eqnarray*}
g^{\ol\delta\gamma}g_{s\ol s;\gamma\ol\delta}
&=&g^{\ol\delta\gamma}\partial_s\partial_\ol s g_{\gamma\ol\delta}\\
&=&\partial_s(g^{\ol\delta\gamma}\partial_\ol s g_{\gamma\ol\delta})
-a_s^{\gamma;\ol\delta}\partial_\ol s g_{\gamma\ol\delta}\\
&=&\partial_s\partial_\ol s \log g
+a_s^{\gamma;\ol\delta} a_{\ol s\gamma;\ol\delta}\\
&=& g_{s \ol s}
+a_s^\sigma{}_{;\gamma} a_{\ol s\sigma;\ol\delta} g^{\ol\delta\gamma}.
\end{eqnarray*}
Next
\begin{eqnarray*}
(a_s^\sigma a_{\ol s\sigma})_{;\gamma\ol\delta}g^{\ol\delta\gamma}
&=\left(a_s^\sigma{}_{;\gamma\ol\delta} a_{\ol s\sigma}
+A_{s\ol\delta}^\sigma A_{\ol s\sigma\gamma}
+a_{s;\gamma}^\sigma a_{\ol s\sigma;\ol\delta}
+a_s^\sigma A_{\ol s\sigma\gamma;\ol\delta}
\right)g^{\ol\delta\gamma} .
\end{eqnarray*}
The last term vanishes because of the harmonicity of $A_s$, and
\begin{eqnarray*}
a_{s;\gamma\ol\delta}^\sigma g^{\ol\delta\gamma}
&=&A_{s\ol\delta;\gamma}^\sigma g^{\ol\delta\gamma}
+a_s^\lambda R^\sigma{}_{\lambda\gamma\ol\delta}g^{\ol\delta\gamma}\\
&=&0-a_s^\lambda R^\sigma{}_\lambda\\
&=& a_s^\sigma .
\end{eqnarray*}
\end{proof}
\begin{definition}\label{de:wpherm}
The \wp hermitian product on $T_sS$ is given by the $L^2$-inner product of
harmonic \ks forms:
\begin{equation}\label{eq:wpherm}
\Big\|\frac{\pt}{\pt s}\Big\|^2_{WP}:= \int_{\cX_s} A^\alpha_{s\ol\beta} A^\ol\delta_{\ol s\gamma }
g_{\alpha\ol\delta}g^{\ol\beta\gamma} g \, dV =
\int_{\cX_s} A^\alpha_{s\ol\beta} A^\ol\beta_{\ol s\alpha } g \, dV
\end{equation}
If $\frac{\pt}{\pt s^i}\in T_sS$ are part of a basis, we denote by
$G^{WP}_{i\ol\jmath}(s)$ the inner product, and set
$$
\omega^{WP}:= \ii G^{WP}_{i\ol\jmath} ds^i\we ds^{\ol\jmath}
$$
\end{definition}
Observe that the generalized \wp form is equal to a fiber integral:
\begin{proposition}[cf.\ \cite{f-s:extremal}]
\begin{equation}\label{eq:wpfib}
\omega^{WP} = \int_{\cX/S} \omega^{n+1}_\cX.
\end{equation}
\end{proposition}
The proposition implies the \ka property of $\omega^{WP}$ immediately. The
{\it proof} follows from Lemma~\ref{le:varphi} and
Proposition~\ref{pr:elleq}.
\section{Curvature of $R^{n-p}f_*\Omega^p_{\cX/S}(\cK_{\cX/S}^{\otimes m})$
-- Statement of the theorem and Applications}
\subsection{Statement of the theorem}\label{ss:curv} We consider an effectively
parameterized family $\cX \to S$ of canonically polarized manifolds,
equipped with \ke metrics of constant Ricci curvature $-1$. For any $m>0$
the direct image sheaves
$f_*\cK_{\cX/S}^{\otimes(m+1)}=f_*\Omega^n_{\cX/S}(\cK_{\cX/S}^{\otimes
m})$ are locally free. For values of $p$ other than $n$ we assume local
freeness of
$$
R^{n-p}f_*\Omega^p_{\cX/S}(\cK_{\cX/S}^{\otimes m}).
$$
The assumptions of Section~\ref{ss:dolb} are satisfied by Kodaira-Nakano
vanishing so that we can apply Lemma~\ref{le:dolb}. If necessary, we
replace $S$ by a (Stein) open subset, such that the direct image is
actually free, and denote by $\{\psi^1,\ldots,\psi^r\}\subset
R^{n-p}f_*\Omega^p_{\cX/S}(\cK_{\cX/S}^{\otimes m})(S)$ a basis of the
corresponding free $\cO_S$-module, and at a given point $s\in S$ we denote
by $\{(\pt/\pt s_i)|_s; i=1,\ldots,M\}$ a basis of the complex tangent
space $T_sS$ of $S$ over $\C$, where the $s_i$ are holomorphic coordinate
functions of a minimal smooth ambient space $U\subset \C^M$.
Let $A_{i \ol\beta}^\alpha(z,s) \pt_\alpha dz^{\ol\beta}$ be a harmonic
\ks form. Then for $s \in S$ the cup product together with the
contraction defines
\begin{footnotesize}
\begin{eqnarray}
A_{i \ol\beta}^\alpha \pt_\alpha dz^{\ol\beta}\cup \textvisiblespace :
\cA^{0,n-p}(\cX_s,\Omega^p_{\cX_s}(\cK_{\cX_s}^{\otimes m})) &\to&
\cA^{0,n-p+1}(\cX_s,\Omega^{p-1}_{\cX_s}(\cK_{\cX_s}^{\otimes m}))\label{eq:cup1}\\ A_{\ol \jmath
\alpha}^\ol\beta \pt_\ol\beta dz^{\alpha}\cup \textvisiblespace :
\cA^{0,n-p}(\cX_s,\Omega^p_{\cX_s}(\cK_{\cX_s}^{\otimes m})) &\to&
\cA^{0,n-p-1}(\cX_s,\Omega^{p+1}_{\cX_s}(\cK_{\cX_s}^{\otimes m})).\label{eq:cup2}
\end{eqnarray}
\end{footnotesize}
\enlargethispage{.5cm} We will apply the above product to harmonic
$(0,n-p)$-forms. In general the result is not harmonic. We use the
notation $\psi^{\ol\ell}:= \ol{\psi^\ell}$ for sections $\psi_k$ (and a
notation of similar type for tensors on the fibers):
\begin{theorem}\label{th:curvgen}
The curvature tensor for $R^{n-p}f_*\Omega^p_{\cX/S}(\cK_{\cX/S}^{\otimes
m})$ is given by
\begin{eqnarray}\label{eq:curvgen}
R_{i\ol\jmath}^{\phantom{{i\ol\jmath}}\ol\ell k}(s)&=& m \int_{\cX_s}
\left( \Box + 1 \right)^{-1}(A_i\cdot A_\ol\jmath) \cdot(\psi^k \cdot
\psi^\ol\ell) g\/ dV\nonumber\\
&& \quad + m \int_{\cX_s} \left( \Box + m \right)^{-1} (A_i\cup\psi^k)
\cdot (A_\ol\jmath \cup \psi^\ol\ell) g\/ dV \\
&& \quad + m \int_{\cX_s} \left( \Box - m \right)^{-1}
(A_i\cup\psi^\ol\ell)\cdot (A_\ol\jmath \cup \psi^k) g\/ dV.
\nonumber
\end{eqnarray}
The only contribution in \eqref{eq:curvgen}, which may be negative,
originates from the harmonic parts in the third term. It equals
$$
- \int_{\cX_s} H(A_i\cup\psi^\ol\ell) \ol{H(A_j\cup\psi^\ol k)} g dV.
$$
\end{theorem}
Concerning the third term, the theorem contains the fact that the positive
eigenvalues of the Laplacian are larger than $m$.
Now the pointwise estimate \eqref{eq:hker2} of the resolvent kernel (cf.\
also Proposition~\ref{pr:resol}) translates into an estimate.
\begin{proposition}\label{pr:est1}
Let $f:\cX \to S$ be a family of canonically polarized manifolds, and $s
\in S$. Let a tangent vector of $S$ at $s$ be given by a harmonic \ks form
$A$ and let $\psi$ be a harmonic $(p,n-p)$-form on $\cX_s$ with values in
the $m$-canonical bundle. Then
\begin{equation}\label{eq:est0}
R(A,\ol A, \psi,\ol\psi) \geq P_n(d(\cX_s)) \cdot \|A\|^2\cdot
\|\psi\|^2 - \|H(A\cup \ol\psi)\|^2.
\end{equation}
\end{proposition}
For $p=n$ we obtain the following result.
\begin{corollary}\label{co:curvmcan}
For $f_*\cK_{\cX/S}^{\otimes(m+1)}$ the curvature equals
\begin{eqnarray}\label{eq:curvmcan}
R_{i\ol\jmath}^{\phantom{i\ol\jmath}\ol\ell k}(s)&=& m \int_{\cX_s}
\left( \Box + m \right)^{-1} (A_i\cup\psi^k)
\cdot (A_\ol\jmath \cup \psi^\ol\ell) g\/ dV\nonumber\\ && \quad +
m \int_{\cX_s} \left( \Box + 1 \right)^{-1}(A_i\cdot A_\ol\jmath)
\cdot(\psi^k \cdot \psi^\ol\ell)
g\/ dV.
\end{eqnarray}
\end{corollary}
The first term in \eqref{eq:curvmcan} yields immediately Nakano
semipositivity, since the operator $(\Box +m)^{-1}$ is positive on the
respective tensors. In fact more can be shown for the curvature of the
direct image of relative $m$-canonical bundles.
Let
\begin{equation}\label{eq:inpro}
H^{\ol\ell k} = \int_{\cX_s} \psi^k \cdot \psi^\ol\ell g\, dV.
\end{equation}
\begin{corollary}\label{co:curv1}
Let $s\in S$ be any point. Let $\xi^i_k\in \C$. Then
\begin{equation}
R_{i\ol\jmath}^{\phantom{i\ol\jmath}\ol\ell k}(s) \xi^i_k\ol{\xi^j_\ell}
\geq m \cdot P_n(d(\cX_s) )\cdot G_{i\ol\jmath}^{WP}\cdot H^{\ol\ell k}
\cdot \xi^i_k\ol{\xi^j_\ell}.
\end{equation}
In particular the curvature is strictly Nakano-positive with the above
estimate.
\end{corollary}
Next, we set $m=1$ and take a dual basis $\{\nu_i\}\subset R^p
f_*\Lambda^p\cT_{\cX/S}(S)$ of the $\{\psi^k\}$ and normal coordinates at
a given point $s_0\in S$. Observing that the role of conjugate and
non-conjugate tensors is being switched, we compute the curvature as
follows.
\begin{theorem}\label{th:curvgendual}
The curvature of $R^pf_*\Lambda^p\cT_{\cX/S}$ equals
\begin{eqnarray}\label{eq:curvgendual}
R_{i\ol\jmath k \ol\ell }(s)&=&- \int_{\cX_s}
\left( \Box + 1 \right)^{-1}(A_i\cdot A_\ol\jmath)
\cdot(\nu_k \cdot \nu_\ol\ell) g\/ dV\nonumber\\
&& \quad - \int_{\cX_s} \left( \Box + 1 \right)^{-1} (A_i\wedge\nu_\ol\ell)
\cdot (A_\ol\jmath \wedge \nu_k) g\/ dV \\
&& \quad - \int_{\cX_s} \left( \Box - 1 \right)^{-1}
(A_i\wedge \nu_k)\cdot (A_\ol\jmath \wedge \nu_\ol\ell) g\/ dV.
\nonumber
\end{eqnarray}
The only possible positive contribution arises from
$$
\int_{\cX_s}H(A_i\wedge \nu_k) H(A_\ol\jmath \wedge \nu_\ol\ell) g\/ dV.
$$
\end{theorem}
We observe that for $n=1$ the third term in \eqref{eq:curvgendual} is not
present and we have the formula for the classical \wp metric on \tei
space: It is known from the results of Wolpert that the classical \wp
metric for families of Riemann surfaces of genus larger than one has
negative curvature: According to \cite{wo} the sectional curvature is
negative, and the holomorphic sectional curvature is bounded from above by
a negative constant. A stronger curvature property, which is related to
strong rigidity, was shown in \cite{sch:teich}. The strongest result on
curvature by Liu, Sun, and Yau now follows immediately from
Corollary~\ref{co:curvmcan}:
\begin{corollary}[\cite{yau}]
The \wp metric on the \tei space of Riemann surfaces of genus $p>1$ is
dual Nakano negative.
\end{corollary}
\begin{proof}
Observe that for a universal family $f:\cX\to S$ the classical \wp metric
on $R^1f_* \cT_{\cX/S}$ corresponds to the $L^2$ metric on its dual bundle
$f_*(\cK_{\cX/S}^{\otimes2})$, which is Nakano positive according to
Corollary~\ref{co:curv1}.
\end{proof}
For $p=1$ we obtain the curvature for the generalized \wp metric from
\cite{sch:curv}, (cf.\ \cite{siu:canlift}). Again we can estimate the
curvature like in Proposition~\ref{pr:est1}.
The following case is of particular interest.
\begin{proposition}\label{pr:est2}
Let $f:\cX \to S$ be a family of canonically polarized manifolds and $s
\in S$. Let a tangent vectors of $S$ at $s$ be given by harmonic \ks forms
$A, A_1,\ldots, A_p$ on $\cX_s$. Let $R$ denote the curvature tensor for
$R^pf_*\Lambda^p\cT_{\cX/S}$. Then we have in terms of the \wp norms:
\begin{gather}\label{eq:est1}
R(A,\ol A, H(A_1\wedge \ldots \wedge A_p) ,\ol{H(A_1\wedge \ldots \wedge
A_p)}) \leq \nonumber \hspace{10cm} \\ \qquad - P_n(d(\cX_s)) \cdot
\|A\|^2\cdot \| H(A_1\wedge \ldots \wedge A_p) \|^2 + \|H(A\wedge
A_1\wedge \ldots \wedge A_p)\|^2.
\end{gather}
\end{proposition}
\begin{proof}
Since the $A$ and $A_i$ are $\ol\pt$-closed forms, we have $H(A\wedge
H(A_1\wedge \ldots\wedge A_p))=H(A\wedge A_1 \wedge \ldots \wedge A_p)$.
\end{proof}
Next, we define higher \ks maps defined on the symmetric powers of the
tangent bundle of the base. For $p>0$ we let the morphism
\begin{equation}\label{eq:rhop}
\rho^p: S^p \cT_{S} \to R^pf_*\Lambda^p \cT_{\cX/S}
\end{equation}
send a symmetric power
$$
\frac{\pt}{\pt s^{i_1}}\otimes \ldots \otimes \frac{\pt}{\pt s^{i_p}}
$$
to the class of
$$
A_{i_1}\wedge \ldots \wedge A_{i_p}:=
A_{i_1\ol\beta_1}^{\alpha_1}\pt_{\alpha_1}dz^{\ol\beta_1}
\wedge \ldots \wedge A_{i_p\ol\beta_p}^{\alpha_p}\pt_{\alpha_p}dz^{\ol\beta_p}.
$$
\begin{definition}
Let the tangent vector $\pt/\pt s$ correspond to the harmonic \ks tensor
$A_s$. The generalized \wp function of degree $p$ on the tangent space is
\begin{gather*}
\| \pt/\pt s \|^{WP}_p := \|A_s\|_p := \| \underbrace{A_s \wedge \ldots
\wedge A_s}_p \|^{1/p}\hspace{7cm} \\
:= \left( \int_{\cX_s} H(A_s \wedge \ldots \wedge A_s)\cdot \ol{H(A_s
\wedge \ldots \wedge A_s)} g \, dV \right) ^{1/2p}
\end{gather*}
\end{definition}
For the computation of the curvature, we assumed that the sheaves
$R^p\Lambda^p\cT_{\cX/S}$ are locally free.
Observe that the coefficient $P_n(d(\cX_s))$ in \eqref{eq:est1} remains
bounded as long as the fibers are smooth independent of the local freeness
of the direct image sheaves.
Given a family over a curve, the $p$-th \wp function of tangent vectors
defines a hermitian (pseudo)metric on the curve, which we denote by $G_p$.
\begin{lemma}\label{le:curvgp}
The curvature $K_{G_p}$ of $G_p$, at points with $G_p(s)\neq 0$ satisfies
\begin{eqnarray}\label{eq:curvgp}
&&\\
K_{G_p} &\leq& \frac{1}{p}\left( - \frac{1}{c_{p,n}} P_n(d(\cX_s)) + \max_{A\neq
0}\{({\|A\|_{p+1}}/{\|A\|_{p}})^{2p+2}\}\right)
\text{\quad for \quad} p< n \nonumber \\
&&\\
K_{G_p} &\leq& -\frac{1}{p \cdot c_{p,n}} P_n(d(\cX_s)) \text{\quad for \quad} p= n \text{, or if }\/ G_{p+1}\equiv 0
\nonumber
\end{eqnarray}
for some $c_{p,n}>0$. These estimates are uniform on any relatively
compact subset of the moduli space.
\end{lemma}
\begin{proof}
Let $A^p$ be the {\em harmonic projection} of the $p$-fold exterior
product of $A_s$. Then the curvature tensor for $R^pf_*
\Lambda^p\cT_{\cX/C}$ satisfies
\begin{gather}
R(\pt_s,\pt_\ol s, A^p, \ol{A^p}) \geq -\frac{\pt^2}{\pt s\ol{\pt s}}
\log(G_p^p)\cdot (G_p^p)\cdot A^p\, \ol{A^p} =\\ \nonumber- p
\frac{\pt^2}{\pt s\ol{\pt s}} \log(G_p) \cdot \|A\|^{2p}_p = p G_p K_{G_p}
\|A\|^{2p}_p .
\end{gather}
Here $R(\pt_s,\pt_\ol s,\textvisiblespace,\textvisiblespace)$ is the
curvature form applied to the tangent vectors $\pt/\pt s$ and $\pt/\ol{\pt
s}$ resp. With respect to $G_p$, we identify $G_p=\|\pt/\pt
s\|^2_p=\|A_s\|^2_p$ so that
$$
R(A_s,\ol{A_s},A^p, \ol{A^p}) \geq p K_{G_p} \|A\|^{2p+2}_p.
$$
Now the estimate of Proposition~\ref{pr:est2} implies
\begin{gather*}
K_{G_p} \leq \frac{1}{p}\left( -P_n(d(\cX_s)) \|A\|_1^2 \|A\|_p^{2p} +
\|A\|_{p+1}^{2(p+1)} \right)\Big/\|A\|_p^{2(p+1)}.
\end{gather*}
The second term is not present for $p=n$.
Now the proof follows from Lemma~\ref{le:estwedge} below.
\end{proof}
\begin{lemma}\label{le:estwedge}
$$
\|A\|_p \leq c_{p,n}\|A\|_1.
$$
for some $c_{p,n}>0$.
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{le:estwedge}]
Since $A_s$ is harmonic and $\| A_s\we\ldots\we A_s\|^2 \geq \|
H(A_s\we\ldots\we A_s)\|^2$, it is sufficient to show the pointwise
estimate (up to a constant that only depends upon the dimension and
degree)
$$
\|A_s \cdot A_{\ol s}\|^p(z) \gtrsim \|(A_s \we \ldots \we A_s) \cdot
\ol{(A_s \we \ldots \we A_s)}\|(z).
$$
We set
$$
B^\alpha_\gamma = A^\alpha_{s\ol\beta}(z) A^\ol\beta_{\ol s \gamma}(z)
$$
and use the symmetry $A_{a\ol\beta\ol\delta}$. It follows that
$$
\|(A_s \we
\ldots \we A_s) \cdot \ol{(A_s \we \ldots \we A_s)}\|(z)
$$
equals the determinant type expression
$$
\underline B(z)=\sum_{\sigma\in \mathfrak{S}_p} \epsilon(\sigma) B^{\alpha_1}_{\alpha_{\sigma(1)}}
\cdot\ldots\cdot B^{\alpha_p}_{\alpha_{\sigma(p)}}
$$
where the inner summations take place with respect to the indices
$\alpha_1,\ldots, \alpha_p$. At the given point $z$ we may assume that
$B^\alpha_\gamma$ is diagonal with non-negative entries
$\lambda_1,\ldots,\lambda_n$. Now it is easy to see that $\underline B(z)$
equals
$$
\sum_{1\leq i_1< \ldots i_p \leq n} \lambda_{i_1} \cdot \ldots \cdot \lambda_{i_p}
$$
up to a numerical constant depending on $p$ and $n$. Again, up to a
numerical constant this can be estimated from below by
$$
\big(\sum^n_{i=1} \lambda_i\big)^p = \|A_s \cdot A_\ol s\|^p(z) .
$$
\end{proof}
\begin{lemma}\label{le:estquot}
For any family over a base space $S$, which is mapped to a relatively
compact subset of the moduli space,
$$
\max_{A\neq 0}\{{\|A\|_{p+1}}/{\|A\|_{p}}\}
$$
is finite.
\end{lemma}
\begin{proof}
First we take a modification $\wt S \to S$ such that the pull backs of the
direct image sheaves $R^pf_*\Lambda^p\cT_{\cX/S}$ modulo torsion are
locally free. Then, by continuity the above quotients are bounded from
above. Next, we restrict the original family to the image of the locus of
torsion, i.e.\ to the support of the annihilator ideal, and repeat the
process.
\end{proof}
\subsection{Hyperbolicity conjecture of Shafarevich}
In \cite[3.2]{demasantacruz} Demailly gives a proof of the Ahlfors lemma
for hermitian metrics of negative curvature in the context of currents
using an approximation argument. Our argument depends upon the following
special case:
\begin{proposition}[Demailly]\label{pr:ahlschw}
Let $\gamma=\gamma(s) \ii ds\wedge \ol{ds}$, $\gamma(s)\geq 0$ be given on
an open disk $\Delta_R=\{|s|<R\}$, where $\log \gamma(s)$ is a subharmonic
function such that $(\ddb(\log \gamma) \geq A\, \gamma$ in the sense of
currents for some $A>0$. Let $\rho$ denote the Poincaré metric on
$\Delta_R$. Then $\gamma\leq \rho/A$ holds.
\end{proposition}
Before we construct a Finsler metric of negative holomorphic curvature, we
will treat families over curves directly.
Let $C$ be a compact smooth complex curve and $f: \cX\to C$ a holomorphic
family of smooth canonically polarized varieties. Let $0 \neq \pt/\pt s$
be a local coordinate vector field on $C$ and $A_s$ the corresponding
family of harmonic \ks forms. Then we set
$$
p_0 := \max \{p; H(\underbrace{A_s\wedge \ldots A_s}_p) \not\equiv 0 \text{ on some open }U \subset C \}.
$$
Observe that the zero set of $H(A_s \wedge \ldots \wedge A_s)$ is
analytic.
We do not have to assume that the direct images
$R^pf_*\Lambda^p\cT_{\cX/C}$ are locally free. This is the case over the
complement $C'=C\backslash \{c_1,\ldots,c_k\}$ of a finite set of points.
Near points $c_j$ we can compute \ks forms $B_s$, $s\in C$ in terms o a
differential trivialization so that the $L^2$-norms of the $B_s$ are
bounded near the critical points. Hence the norms of the harmonic
representatives $A_s$ are bounded near $c_j$. The same holds for the wedge
product of these. Because of the boundedness $\log G_{p_0}$ is subharmonic
on all of $C$.
\begin{proposition}\label{pr:nonisotr}
Let $f: \cX\to C$ a non-isotrivial holomorphic family of smooth
canonically polarized varieties over a curve, and let $0<p_0\leq n$ be
chosen as above. Then $\log G_{p_0}$ is subharmonic, and for the curvature
\begin{equation}\label{eq:curvgp1}
K_{G_{p_0}} \leq - \frac{1}{p \cdot c_{p,n}} P_n(d(\cX_s))
\end{equation}
holds so that the curvature current is negative.
\end{proposition}
As long as the fibers are smooth, the diameters $d(\cX_s)$ are bounded
from above. The \!\! {\em proof}\/ of the estimate follows from
Lemma~\ref{le:curvgp} at points, where the direct image of order $p_0$ is
locally free.
At points with {\em singular fibers}, $\log G_{p_0}$ need no longer be
subharmonic, unless current of integration is added. An elementary
(counter$\text{-}$)ex\-am\-ple in fiber dimension zero is the map from a
hyperelliptic to a rational curve.
By Proposition~\ref{pr:ahlschw}) the genus of the base must be larger than
one. This gives a short proof of the following version of Shafarevich's
hyperbolicity conjecture for canonically polarized varieties \cite{b-v,
keko,kk, kv1, kv2, m, v-z, v-z2}.
\smallskip
{\bf Application.} {\it If a compact curve $C$ parameterizes a
non-isotrivial family of canonically polarized manifolds, its genus must
be greater than one.}
\subsection{Finsler metric on the moduli stack} Different notions are
common. We do not assume the triangle inequality/convexity. Such metrics
are also called {\em pseudo-metrics} (cf.\ \cite{kobook}).
\begin{definition}\label{de:fins}
Let $Z$ be a reduced complex space and $TZ$ it Zariski tangent fiber
bundle. An upper semi-continuous function
$$
F:TZ \to [0,\infty)
$$
is called Finsler pseudo-metric (or pseudo-length function), if
$$
F(av)=|a|F(v) \text{ for all } a\in \C, v\in TZ.
$$
\end{definition}
The triangle inequality on the fibers not required for the definition of
the ''holomorphic'' (or ''sectional'') curvature.
All metrics $G_p$ from Section~\ref{ss:curv} are (upper semi-continuous)
Finsler pseudo-metrics.
A pseudometric $\gamma$ for a curve $C$ like in
Proposition~\ref{pr:ahlschw} and\ref{pr:nonisotr} may have isolated
zeroes.
We will use the fact that the holomorphic curvature of a Finsler metric at
a certain point $p$ in the direction of a tangent vector $v$ is the
supremum of the curvatures of the pull-back of the given Finsler metric to
a holomorphic disk through $p$ and tangent to $v$ (cf.\
\cite{abate-patrizio}). (For a hermitian metric, the holomorphic curvature
is known to be equal to the holomorphic sectional curvature.) In view of
Demailly's theorem, a Finsler metric may be defined in the above sense (as
long as the fibers are smooth, which is always the case. Furthermore, any
convex sum $G=\sum_j a_j G_j$, $a_j>0$ is upper semi-continuous and has
the property that $\log G$ restricted to a curve is subharmonic.
\begin{lemma}[cf.\ {\cite[Lemma~3]{sch:framas}}]\label{le:convsum}
Let $C$ be a complex curve and $G_j$ a collection of pseudo-metrics of
bounded curvature, whose sum has no common zero. Then the curvatures $K$
satisfy the following equation.
\begin{equation}\label{eq:curvest}
K_{\sum_{j=1}^k G_j} \leq \sum_{j=1}^k \frac{G_j^2}{(\sum_{i=1}^k G_i)^2} K_{G_j} .
\end{equation}
\end{lemma}
Now with Lemma~\ref{le:convsum} and Lemma~\ref{le:curvgp} we can construct
convex sums of the metrics $G_p$ with negative holomorphic curvature. In
this way we arrive at a (upper semi-continuous) Finsler metric rather than
a pseudo-metric. The convex sum accounts both cases where some $G_p$
vanishes or not. The metric is primarily given on local universal
families, but intrinsically given. It descends to the coarse moduli space
in the orbifold sense.
\begin{theorem}\label{pr:exfins}
On any relatively compact subset of the moduli space of canonically
polarized manifolds there exists a Finsler orbifold metric, i.e.\ a
Finsler metric on the moduli stack, whose holomorphic curvature is bounded
from above by a negative constant.
\end{theorem}
\section{Computation of the curvature}
We know from Lemma~\ref{le:dolb} that the metric tensor for
$R^{n-p}f_*\Omega^p_{\cX/S}(\cK_{\cX/S})$ on the base space $S$ is given
in terms of an integral which involves harmonic representatives of certain
cohomology classes and that these are the restrictions of certain
$\ol\pt$-closed differential forms on the total space. We already saw that
these give rise to fiber integrals. When we actually compute derivatives
with respect to the base, we will apply Lie derivatives with respect to
horizontal lifts of tangent vectors of the base. At this point we need to
take into account that the exterior derivatives $\pt$ has to be taken with
respect to the hermitian metric on the relative canonical line bundle.
Here covariant derivatives with respect to the total space occur (at least
in an implicit way). Since we are dealing with alternating forms we may
use covariant derivatives with respect to the \ka structure on the fibers,
which is necessary to somewhat simplify the computations. Again, we will
use the semi-colon notation for covariant derivatives and use a $|$-symbol
for ordinary derivatives, if necessary. Greek indices are being used for
fiber coordinates, Latin indices indicate the base direction. Dealing with
alternating forms, for instance of degree $(p,q)$, extra coefficients of
the form $1/p!q!$ are sometimes customary; these play a role, when the
coefficients of an alternating form are turned into skew-symmetric tensors
by taking the average. However, for the sake of a halfway simple notation,
we follow the better part of the literature and leave these to the reader.
\subsection{Setup}
As above, we denote by $f:\cX \to S$ a smooth family of canonically
polarized manifolds and we pick up the notation from
Section~\ref{se:posi}. The fiber coordinates were denoted by $z^\alpha$
and the coordinates of the base by $s^i$. We set $\pt_i=\pt/\pt s^i$,
$\pt_\alpha=\pt/\pt z^\alpha$.
Again we have {\em horizontal lifts of tangent vectors and coordinate
vector fields on the base}
$$
v_i= \pt_i + a_i^\alpha \pt_\alpha.
$$
As above we have the corresponding harmonic representatives
$$
A_i=A^\alpha_{i\ol\beta}\pt_\alpha dz^{\ol \beta}
$$
of the \ks classes $\rho(\pt_i|_{s_0})$.
For the computation of the curvature it is sufficient to treat the case
where $\dim S =1$. We set $s=s_1$ and $v_s=v_1$ etc. In this case we write
$s$ and $\ol s$ for the indices $1$ and $\ol 1$ so that
$$
v_s= \pt_s + a_s^\alpha \pt_\alpha
$$
etc.
Sections of \RP will be denoted by letters like $\psi$.
\begin{gather*}
\psi|_{\cX_s} =
\psi_{\alpha_1,\ldots,\alpha_p,\ol\beta_{p+1},\ldots,\ol\beta_n}
dz^{\alpha_1}\wedge \ldots \wedge dz^{\alpha_p}\wedge
dz^{\ol\beta_{p+1}}\we
\ldots \we dz^{\ol\beta_n} \\
= \psi_{A_p\ol B_{n-p}} dz^{A_p}\we dz^{\ol B_{n-p}} \hspace{5cm}
\end{gather*}
where $A_p=(\alpha_1,\ldots,\alpha_p)$ and $\ol B_{n-p}=(\ol\beta_{p+1},
\ldots,\ol\beta_n)$. The further component of $\psi$ is
$$
\psi_{\alpha_1,\ldots,\alpha_p,\ol\beta_{p+1},\ldots,\ol\beta_{n-1},\ol s}
dz^{\alpha_1}\wedge \ldots \wedge dz^{\alpha_p} \we dz^{\ol\beta_{p+1}}\we
\ldots \we dz^{\ol\beta_{n-1}}\we \ol{ds}.
$$
Now Lemma~\ref{le:dolb} implies
\begin{equation}
\psi_{\alpha_1,\ldots,\alpha_p,\ol \beta_{p+1}, \ldots, \ol\beta_n |\ol s}
= \sum_{j=p+1}^n (-1)^{n-j}
\psi_{\alpha_1,\ldots,\alpha_p,\ol \beta_{p+1}, \ldots, \wh{\ol\beta}_j, \ldots, \ol\beta_n ,\ol s|\ol\beta_j }.
\end{equation}
Since these are the coefficients of alternating forms, on the right-hand
side, we may also take the covariant derivatives with respect to the given
structure on the fibers
$$
\psi_{\alpha_1,\ldots,\alpha_p,\ol \beta_{p+1},
\ldots, \wh{\ol\beta}_j, \ldots, \ol\beta_n ,\ol s;\ol\beta_j}.
$$
\subsection{Cup-Product}
We define the cup-product of a differential form with values in the
relative holomorphic tangent bundle and an (line bundle valued)
differential form now in terns of local coordinates.
\begin{definition}\label{de:cup}
Let
$$
\mu= \mu^\sigma_{\alpha_1,\ldots,\alpha_p,\ol\beta_1,\ldots, \ol\beta_q}\pt_\sigma \,
dz^{\alpha_1}\we\ldots\we dz^{\alpha_p}\we dz^{\ol\beta_1}\we\ldots\we dz^{\ol\beta_q},
$$
and
$$
\nu= \nu_{\gamma_1,\ldots,\gamma_a,\ol\delta_1,\ldots,\ol\delta_b}
dz^{\gamma_1}\we\ldots\we dz^{\gamma_a}\we dz^{\ol\delta_1}
\we\ldots\we dz^\ol{\delta_b}
$$
Then
\begin{gather}\label{eq:cup}
\mu\cup\nu := \mu^\sigma_{\alpha_1,\ldots,\alpha_p,\ol\beta_1,\ldots,
\ol\beta_q}
\nu_{\sigma\gamma_2,\ldots,\gamma_a,\ol\delta_1,\ldots,\ol\delta_b}
dz^{\alpha_1}\we\ldots\we dz^{\alpha_p} \\
\nonumber \hspace{3cm}\we dz^{\ol\beta_1}\we\ldots\we dz^{\ol\beta_q}\we
dz^{\gamma_2}\we\ldots\we dz^{\gamma_q}\we dz^{\ol\delta_1} \we\ldots\we
dz^\ol{\delta_b}
\end{gather}
\end{definition}
\subsection{Lie derivatives}
Let again the base be smooth, $\dim S=1$ with local coordinate $s$. Then
the induced metric on \RP is given by \eqref{eq:inpro}, where the
pointwise inner product equals
$$
\psi^k \cdot \psi^\ol\ell g\, dV = (\ii)^n (-1)^{n(n-p)} \frac{1}{g^m} \psi^k \we \psi^\ol\ell,
$$
and where $1/g^m$ stands for the hermitian metric on the $m$-th canonical
bundle on the fibers.
\begin{lemma}\label{le:Lieder}
$$
\frac{\pt}{\pt s} H^{\ol\ell k} = \int_{\cX_s}
L_v(\psi^k \cdot \psi^\ol\ell) g\, dV = \langle L_v \psi^k, \psi^\ell \rangle + \langle \psi^k, L_\ol v \psi^\ell \rangle ,
$$
where $L_v$ denotes the Lie derivative with respect to the canonical lift
$v$ of the coordinate vector field $\pt/\pt s$.
\end{lemma}
\begin{proof}
Taking the Lie derivative is not type-preserving. We need the
$(1,1)$-component: $L_v(g_{\alpha\ol\beta})= \big[ \pt_s + a^\alpha_s
\pt_\alpha , g_{\alpha,\ol\beta} \big]_{\alpha\ol\beta} =
g_{\alpha\ol\beta|s} + a^\gamma_{s}g_{\alpha\ol\beta;\gamma}+
a^\gamma_{s;\alpha}g_{\gamma\ol\beta} =-
a_{s\ol\beta;\alpha}+a^\gamma_{s;\alpha}g_{\gamma\ol\beta}=0 $. So
$L_v(det(g_{\alpha\ol\beta}))=0$.
\end{proof}
\begin{equation}
L_v\psi = L_v\psi' + L_v\psi'',
\end{equation}
where $L_v\psi'$ is of type $(p,n-p)$ and $L_v\psi''$ is of type
$(p-1,n-p+1)$. We have
\begin{eqnarray}
L_v\psi' &=& \big[\pt_s + a^\alpha_s \pt_\alpha, \psi_{A_p\ol B_{n-p}} dz^{A_p } dz^{\ol B_{n-q}}\big]_{(p,n-p)}
\nonumber \\
&=& (\psi_{;s} + a^\alpha_s \psi_{;\alpha} + \sum_{j=1}^p a^\alpha_{s;\alpha_j}
\psi_{
{\tiny\vtop{
\hbox{$\alpha_1,\ldots,\alpha,\ldots,\alpha_p\ol B_{n-p}\;$}\vskip-.8mm
\hbox{$\phantom{\alpha_1,\ldots,}{|\atop j} $}}}}) dz^{A_p}\we dz^{\ol B_{n-p}} \label{eq:lvprime} \\
L_v\psi'' &=&\big[\pt_s + a^\alpha_s \pt_\alpha, \psi_{A_p\ol B_{n-p}} dz^{A_p } dz^{\ol B_{n-q}}\big]_{(p-1,n-p+1)}
\nonumber \\
&=& \sum^p_{j=1} A^\alpha_{s\ol\beta_p}
\psi^k_{ {\tiny\vtop{ \hbox{$\alpha_1,\ldots,\alpha,\ldots,\alpha_p\ol
B_{n-p}$\;}\vskip-.8mm \hbox{$\phantom{\alpha_1,\ldots,}{|\atop j} $}}}} \nonumber \\
&& \quad
\vtop{\hbox{$dz^{\alpha_1}\we\ldots\we dz^{\ol\beta_p}\we\ldots\we
dz^{\alpha_p} \we dz^{\ol\beta_{p+1}}\we\ldots\we
dz^{\ol\beta_n}$}\hbox{$\phantom{dz^{\alpha_1}\we\ldots\we \; }{|\atop j} $}} \label{eq:lvsecond}
\end{eqnarray}
We also note the values for the derivatives with respect to $\ol v$.
\begin{eqnarray}
L_\ol v\psi' &=& \big[\pt_\ol s + a^\ol\beta_\ol s \pt_\ol\beta, \psi_{A_p\ol B_{n-p}}
dz^{A_p } dz^{\ol B_{n-q}}\big]_{(p,n-p)} \nonumber \\
&=& (\psi_{;\ol s} + a^\ol\beta_\ol s \psi_{;\ol\beta} + \sum_{j=1}^p a^\ol\beta_{\ol s;\ol\beta_j}
\psi_{
{\tiny\vtop{
\hbox{$A_p \ol\beta_{p+1},\ldots,\ol\beta,\ldots,{\ol\beta}_n\;$}\vskip-.8mm
\hbox{$\phantom{A_p \ol\beta_{p+1},\ldots,}{|\atop j}$}}}})
dz^{A_p}\we dz^{\ol B_{n-p}} \label{eq:lvbprime} \\
L_\ol v\psi'' &=& \big[\pt_\ol s + a^\ol\beta_\ol s \pt_\ol\beta, \psi_{A_p\ol B_{n-p}} dz^{A_p }
dz^{\ol B_{n-q}}\big]_{(p+1,n-p-1)} \nonumber \\
&=& \sum^n_{j=p+1} A^\ol\beta_{\ol s \alpha_{p+1}}
\psi^k_{\tiny\vtop{ \hbox{$\alpha_1,\ldots,\alpha_p,\ol\beta_{p+1},\ldots,\ol \beta,\ldots,\ol\beta_n\;$}\vskip-.8mm
\hbox{$\phantom{\alpha_1,\ldots,\alpha_p,\ol\beta_{p+1},\ldots,}{|\atop j}$}}} \nonumber \\ && \quad
\vtop{\hbox{$dz^{\alpha_1}\we\ldots\we dz^{\alpha_p}\we dz^{\ol\beta_1}\we\ldots\we dz^{\alpha_{p+1}}\we\ldots
\we dz^{\ol\beta_n} $}
\hbox{$\phantom{dz^{\alpha_1}\we\ldots\we dz^{\alpha_p}\we dz^{\ol\beta_1}\we\ldots\we dz}{|\atop j}$}} \label{eq:lvbsecond}
\end{eqnarray}
\begin{lemma}
\begin{eqnarray}
(L_v\psi^k)'' &=& A_s \cup \psi^k\label{eq:2} \\
(L_\ol v\psi^k)'' &=& (-1)^p A_\ol s \cup \psi^k \label{eq:3}
\end{eqnarray}
\end{lemma}
\begin{proof}[Proof of \eqref{eq:2}.]
By \eqref{eq:lvsecond} we have
\begin{gather*}
L_v\psi'' = \hspace{10cm}\\ =\sum^p_{j=1}
A^\alpha_{s\ol\beta_p}\psi^k_{\alpha_1,\ldots,
\wh\alpha_j,\ldots,\alpha_p,\alpha, \ol B_{n-p}} dz^{\alpha_1}\we
\ldots\we \wh{dz^{\alpha_j}}\we\ldots \we dz^{\alpha_p} \we
dz^{\ol\beta_p}\we
\ldots\we dz^{\ol\beta_n}\\
= (-1)^{p-1}\sum^p_{j=1}
A^\alpha_{s\ol\beta_p}\psi^k_{\alpha,\alpha_1,\ldots,\alpha_{p-1},\ol\beta_{p},\ldots,\ol\beta_{n}}
dz^{\alpha_1}\we\ldots\we\ldots \we dz^{\alpha_{p-1}} \we
dz^{\ol\beta_p}\we \ldots\we dz^{\ol\beta_n}.
\end{gather*}
\end{proof}
\begin{proof}[Proof of \eqref{eq:3}]
The claim follows in a similar way from \eqref{eq:lvbsecond}.
\end{proof}
The situation is not quite symmetric because of Lemma~\ref{le:dolb}, which
implies that the contraction of the global $(0,n-p)$-form $\psi$ with
values in $\Omega^p_{\cX/S}(\cK_{\cX/S})$ is well-defined. Like in
Definition~\ref{de:cup} we have a cup-product on the total space
(restricted to the fibers).
\begin{eqnarray*}
\ol v \cup \psi &=& (\pt_\ol s + a^\ol\beta_\ol s \pt_\ol\beta) \cup \psi \\
&=&\psi_{A_p,\ol s, \ol \beta_{p+1},\ldots,\ol\beta_{n-1}} +
a^\ol\beta_\ol s \psi_{A_p,\ol\beta,\ol\beta_{p+1},\ldots,\ol \beta_{n-1} }
\end{eqnarray*}
\begin{lemma}\label{le:lvpsi1}
\begin{equation}\label{eq:lvpsi1}
L_\ol v\psi'= (-1)^p\ol\pt (\ol v \cup \psi).
\end{equation}
\end{lemma}
\begin{proof}
The proof follows from the fact that, according to Lemma~\ref{le:dolb},
$\psi$ is given by a $\ol\pt$-closed $(0,n-p)$-form on the total space
$\cX$ with values in a certain holomorphic vector bundle.
\end{proof}
We will need that the forms $\psi$ on the fibers are {\em also harmonic
with respect to $\pt$} (which was defined as the connection of the line
bundle $\cK^{\otimes m}_{\cX/S}$). First, we note the following fact,
which immediately follows from the fact that the curvature of
$(\cK_{\cX/S},g^{-1})$ equals $-\omega_\cX$. We will need this fact for
both the total space and the restriction to fibers.
\begin{lemma}\label{le:ddb}
\begin{equation}\label{eq:ddb}
\ii [\ol\pt, \pt] = - m L_\cX,
\end{equation}
where $L_\cX$ denotes the multiplication with $\omega_{\cX}$.
\end{lemma}
Now:
\begin{lemma}\label{le:boxdboxdb}
The following equation holds on $\cA^{(p,q)}(\cK^{\otimes m}_{\cX_s})$.
\begin{equation}
\Box_\pt = \Box_\ol\pt + m\cdot (n-p-q) \cdot id.
\end{equation}
In particular, the harmonic forms $\psi \in \cA^{(p,n-q)}(\cK_{\cX_s})$
are also harmonic with respect to $\pt$.
\end{lemma}
\begin{proof}
We use the formulas
$$
\ii\, \ol\pt^* = [\Lambda,\pt] \text{\quad and \quad} -\ii \pt^* = [\Lambda,\ol \pt],
$$
where $\Lambda$ denotes the adjoint operator to $L$. Then
$$
\Box_\pt-\Box_\ol\pt = [\Lambda,\ii (\pt \ol\pt + \ol\pt \pt)]=[\Lambda,m\cdot \omega_\cX] =m\cdot(n-p-q)\cdot id.
$$
\end{proof}
Now we compute the curvature in the following way. Because of
\eqref{eq:lvpsi1}
$$
\langle \psi^k, L_\ol v (\psi^\ell)' \rangle = 0
$$
holds for all $s\in S$ so that by Lemma~\ref{le:Lieder}
\begin{gather*}
\frac{\pt}{\pt s} H^{\ol\ell k} = \langle L_v \psi^k, \psi^\ell \rangle +
\langle \psi^k, L_\ol v \psi^\ell \rangle
=\langle (L_v \psi^k)', \psi^\ell \rangle + \langle \psi^k, (L_\ol v
\psi^\ell)' \rangle \\
=\langle (L_v \psi^k)', \psi^\ell \rangle.
\end{gather*}
Later in the computation we will use normal coordinates (of the second
kind) at a given point $s_0\in S$. The condition $(\pt/\pt s)H^{\ol\ell
k}|_{s_0}=0$ for all $k,\ell$ means that for $s=s_0$ the harmonic
projection
\begin{equation}\label{eq:HLv}
H((L_v \psi^k)') =0
\end{equation}
vanishes for all $k$.
In order to compute the second order derivative of $H^{\ol\ell k}$ we
begin with
\begin{equation}\label{eq:dsH}
\frac{\pt}{\pt s} H^{\ol\ell k} = \langle L_v \psi^k, \psi^\ell \rangle.
\end{equation}
which contains both $(L_v \psi^k)'$ and $(L_v \psi^k)''$. Now
\begin{gather*}
\pt_\ol s\pt_s \langle \psi^k,\psi^\ell \rangle = \langle L_\ol v L_v
\psi^k , \psi^\ell \rangle +\langle L_v \psi^k,L_v\psi^\ell \rangle \\ =
\langle L_{[\ol v,v]}\psi^k , \psi^\ell\rangle + \langle L_vL_\ol
v\psi^k,\psi^\ell \rangle + \langle L_v \psi^k,L_v\psi^\ell \rangle \\ =
\langle L_{[\ol v,v]}\psi^k , \psi^\ell\rangle + \pt_s\langle L_\ol v
\psi^k, \psi^\ell\rangle -\langle L_\ol v\psi^k,L_\ol v\psi^\ell\rangle +
\langle L_v\psi^k,L_v\psi^\ell\rangle
\end{gather*}
We just saw that $\langle L_\ol v\psi^k , \psi^\ell \rangle \equiv 0$.
Hence for all $s\in S$
\begin{equation}\label{eq:Lolvv}
\pt_\ol s\pt_s \langle \psi^k,\psi^\ell \rangle = \langle L_{[\ol
v,v]}\psi^k , \psi^\ell\rangle -\langle L_\ol v\psi^k,L_\ol
v\psi^\ell\rangle + \langle L_v\psi^k,L_v\psi^\ell\rangle
\end{equation}
The fact that we are computing Lie-derivatives of $n$-forms (with values
in some line bundle) implies that
$$
\langle L_v\psi^k,L_v\psi^\ell\rangle = \langle (L_v\psi^k)',(L_v\psi^\ell)'\rangle
- \langle (L_v\psi^k)'',(L_v\psi^\ell)''\rangle,
$$
and
$$
\langle L_\ol v\psi^k,L_\ol v\psi^\ell\rangle = \langle (L_\ol v\psi^k)',(L_\ol v\psi^\ell)'\rangle
- \langle (L_\ol v\psi^k)'',(L_\ol v\psi^\ell)''\rangle.
$$
\begin{lemma}\label{le:vvb}
Restricted to the fibers $\cX_s$ the following equation holds for $L_{[\ol
v, v]}$ applied to $\cK^{\otimes m}_{\cX/S}$-valued functions and
differential forms resp.
\begin{equation}\label{eq:vvb}
L_{[\ol v, v]} =
\big[-\varphi^{;\alpha}\pt_\alpha + \varphi^{;\ol\beta}\pt_{\ol\beta},\; \textvisiblespace \;\big]
- m\cdot \varphi \cdot id
\end{equation}
\end{lemma}
\begin{proof}
We first compute the vector field $[\ol v, v]$ on the fibers:
\begin{gather*}
[\ol v, v]= [\pt_\ol s + a^\ol\beta_\ol s \pt_\ol\beta, \pt_{s} +
a^\alpha_{s} \pt_\alpha ]\hspace{5cm} \\= \left(\pt_\ol s (a^\alpha_ s) +
a^\ol\beta_\ol s a^\alpha_{s|\ol\beta}\right)\pt_\alpha - \left( \pt_s
(a^\ol\beta_\ol s) + a^\alpha_{s}a^\ol\beta_{\ol
s|\alpha}\right)\pt_\ol\beta.
\end{gather*}
Now
\begin{gather*}
\pt_\ol s (a^\alpha_ s) = -\pt_\ol s (g^{\ol\beta\alpha}g_{s\ol\beta}) =
g^{\ol\beta\sigma} g_{\sigma\ol s| \ol \tau}g^{\ol\tau
\alpha}g_{s\ol\beta} - g^{\ol\beta\alpha}g_{s\ol \beta|\ol s} \\
\hspace{5cm} = g^{\ol\beta\sigma}a_{\ol s
\sigma;\ol\tau}g^{\ol\tau\alpha}a_{s\ol\beta} - g^{\ol\beta\alpha} g_{s\ol
s; \ol\beta}
\end{gather*}
Now \eqref{eq:varphi} implies that the coefficient of $\pt_\alpha$ is
$-\varphi^{;\alpha}$. In the same way the coefficient of $\pt_\ol \beta$
is computed.
Next, we compute the contribution of the connection on $\cK^{\otimes
m}_{\cX/S}$ which we denote by $[\ol v, v]_{\cK^{\otimes m}_{\cX/S}}$. We
use \eqref{eq:ddb}:
\begin{gather*}
[\pt_\ol s + a^\ol\beta_\ol s \pt_\ol\beta, \pt_{s} + a^\alpha_{s}
\pt_\alpha]_{\cK^{\otimes m}_{\cX/S}}\hspace{5cm} \\ =-m\left(g_{s\ol s}
+ a^\ol\beta_{\ol s} g_{s\ol\beta} + a^\alpha_s g_{\alpha\ol s} +
a^\ol\beta_\ol s a^\alpha_s g_{\alpha\ol\beta} \right) = -m \varphi.
\end{gather*}
\end{proof}
\begin{lemma}\label{le:lvvpsi}
\begin{equation}\label{eq:lvvpsi}
\langle L_{[\ol v, v]} \psi^k , \psi^\ell \rangle = -m \langle \varphi \psi^k, \psi^\ell\rangle
= - m \int_{\cX_s} (\Box + 1)^{-1}(A_s \cdot A_\ol s) \psi^k \psi^\ol\ell \, g \, dV
\end{equation}
\end{lemma}
\begin{proof}
The $\pt$-closedness of the $\psi^k$ can be read as
$$
\psi^k_{;\alpha} = \sum_{j=1}^p
\psi_{
{\tiny\vtop{
\hbox{$\alpha_1,\ldots,\alpha,\ldots,\alpha_p\ol B_{n-p};\alpha_j\;$}\vskip-.8mm
\hbox{$\phantom{\alpha_1,\ldots,}{|\atop j} $}}}}.
$$
Hence
\begin{eqnarray*}
[\varphi^{;\alpha}\pt_\alpha, \psi^k_{A_p\ol B_{n-p}}]'& = &
\varphi^{;\alpha}\psi_{;\alpha} + \sum_{j=1}^p \varphi^{;\alpha}_{\;
;\alpha_j} \psi^k_{ {\tiny\vtop{
\hbox{$\alpha_1,\ldots,\alpha,\ldots,\alpha_p\ol B_{n-p}\;$}\vskip-.8mm
\hbox{$\phantom{\alpha_1,\ldots,}{|\atop j} $}}}}\\
& = & \sum_{j=1}^p \big( \varphi^{;\alpha} \psi^k_{ {\tiny\vtop{
\hbox{$\alpha_1,\ldots,\alpha,\ldots,\alpha_p\ol B_{n-p}\;$}\vskip-.8mm
\hbox{$\phantom{\alpha_1,\ldots,}{|\atop j} $}}}}\big)_{;\alpha_j}\\
&=& \pt\big( \varphi^{;\alpha}\pt _\alpha\cup \psi^k\big).
\end{eqnarray*}
Now
\begin{gather*}
\langle [\varphi^{;\alpha}\pt_\alpha, \psi^k_{A_p\ol
B_{n-p}}],\psi^\ell\rangle = \langle [\varphi^{;\alpha}\pt_\alpha,
\psi^k_{A_p\ol B_{n-p}}]',\psi^\ell\rangle \qquad \\ \qquad=\langle
\pt\big( \varphi^{;\alpha}\pt _\alpha\cup \psi^k\big),\psi^\ell \rangle =
\langle \varphi^{;\alpha}\pt _\alpha\cup \psi^k , \pt^* \psi^\ell \rangle
=0.
\end{gather*}
In the same way we get
$$
\langle [\varphi^{;\ol\beta}\pt_\ol\beta, \psi^k_{A_p\ol
B_{n-p}}],\psi^\ell\rangle =0,
$$
and, according to Lemma~\ref{le:vvb}, we are left with the desired term.
\end{proof}
\begin{proposition}\label{pr:baseq}
In view of \eqref{eq:cup1} and \eqref{eq:cup2} we have
\begin{eqnarray}
\ol\pt(L_v\psi^k)'&=& \pt(A_s\cup \psi^k)\label{eq:0} \\
\ol\pt^*(L_v\psi^k)'&=& 0 \label{eq:4} \\
\pt^*(A_s\cup\psi^k) &=&0 \label{eq:5} \\
\ol\pt^* (L_\ol v\psi^k)'&=& \pt^* (A_\ol s \cup \psi^k) \label{eq:6}\\
\ol\pt(L_\ol v\psi^k)'&=& 0 \label{eq:1} \\
\ol\pt^*(A_\ol s \cup \psi^k) &=&0 \label{eq:7}
\end{eqnarray}
\end{proposition}
The proof of the above proposition is the technical part of this article
and will be given at the end of the manuscript.
{\it Proof of Theorem~\ref{th:curvgen}.} Again, we may set $i=j=s$ and use
normal coordinates at a given point $s_0\in S$.
We continue with \eqref{eq:Lolvv} and apply \eqref{eq:lvvpsi}. Let $G_\pt$
and $G_\ol\pt$ denote the Green's operators on the spaces of
differentiable $\cK_{\cX_s}$-valued $(p,q)$-forms on the fibers with
respect to $\Box_\pt$ and $\Box_\ol\pt$ resp. We know from
Lemma~\ref{le:boxdboxdb} that for $p+q=n$ the Green's operators $G_\pt$
and $G_\ol\pt$ coincide.
We compute $\langle (L_v\psi^k)', L_v\psi^\ell)'\rangle$: Since the
harmonic projection\\ $H ((L_v\psi^k)')=0$ vanishes for $s=s_0$, we have
\begin{gather*}
(L_v\psi^k)'= G_\ol\pt \Box_\ol\pt (L_v\psi^k)'= G_\ol\pt \ol\pt^*\ol\pt
(L_v\psi^k)' = \ol\pt^*G_\ol\pt \pt(A_s\cup\psi^k)
\end{gather*}
by \eqref{eq:4} and \eqref{eq:0}. The form $\ol\pt(L_v\psi^k)'=
\pt(A_s\cup \psi^k)$ is of type $(p,n-p+1)$ so that by
Lemma~\ref{le:boxdboxdb} on this space of such forms $G_\ol\pt=(\Box_\pt
+m)^{-1} $ holds.
Now
\begin{gather*}
\langle (L_v\psi^k)', (L_v\psi^\ell)'\rangle =\langle \ol\pt^*G_\ol\pt
\pt(A_s\cup\psi^k), (L_v\psi^\ell)' \rangle\\ = \langle G_\ol\pt
\pt(A_s\cup\psi^k), \pt (A_s \cup \psi^\ell) \rangle = \langle (\Box_\pt
+m)^{-1} \pt (A_s\cup \psi^k), \pt (A_s\cup \psi^\ell)\rangle\\
= \langle \pt^* (\Box_\pt +m)^{-1} \pt (A_s\cup \psi^k), A_s\cup
\psi^\ell\rangle.
\end{gather*}
Because of \eqref{eq:5}
\begin{gather*}
\langle (L_v\psi^k)', (L_v\psi^\ell)'\rangle = \langle (\Box_\pt
+m)^{-1}\Box_\pt (A_s \cup \psi^k) , A_s\cup \psi^\ell\rangle\\
= \langle A_s\cup \psi^k, A_s \cup\psi^\ell\rangle -m \langle (\Box+m
)^{-1}(A_s \cup \psi_k), A_s \cup \psi^\ell\rangle.
\end{gather*}
(For $(p-1, n-p+1)$-forms, we write $\Box=\Box_\pt=\Box_\ol\pt$.)
Altogether we have
\begin{equation}\label{eq:part2}
\langle L_v \psi^k , L_v \psi^\ell \rangle|_{s_0} = - m
\int_{\cX_s} (\Box +m)^{-1}(A_s\cup\psi^k)\cdot (A_\ol s\cup \psi^\ol\ell)\, g\, dV.
\end{equation}
Finally we need to compute $\langle L_\ol v\psi^k, L_\ol v \psi^\ell
\rangle$.
By equation \eqref{eq:3} we have that $(\langle L_\ol v\psi^k)'', (L_\ol
v \psi^\ell)'' \rangle = \langle A_\ol s \cup \psi^k , A_\ol s \cup
\psi^\ell \rangle$. Now Lemma~\ref{le:lvpsi1} implies that the harmonic
projections of the $(L_\ol v \psi^k)'$ vanish for all parameters $s$. So
\begin{gather*}
\langle (L_\ol v \psi^k)', (L_\ol v \psi^\ell)'\rangle = \langle G_\ol\pt
\Box_\ol\pt (L_\ol v \psi^k)', (L_\ol v \psi^\ell)'\rangle \\
\vtop{\hbox{$=$}\vskip-4mm\hbox{\tiny$\!\! \eqref{eq:1}$} } \langle
(G_\ol\pt \ol\pt^*\ol\pt L_\ol v \psi^k)', (L_\ol v \psi^\ell)'\rangle =
\langle
(G_\ol\pt \ol\pt L_\ol v \psi^k)', \ol\pt (L_\ol v \psi^\ell)'\rangle\\
\vtop{\hbox{$=$}\vskip-4mm\hbox{\tiny$\!\! \eqref{eq:6}$} } \langle
G_\ol\pt \pt^*(A_\ol s \cup \psi^k), \pt^*(A_\ol\ s\cup \psi^\ell)
\rangle. \hspace{3cm}
\end{gather*}
Now the $(p+1, n-p)$-form $\ol\pt^*(L_\ol v\psi^k)'= \pt^* (A_\ol s \cup
\psi^k)$ is orthogonal to both the spaces of $\ol\pt$- and $\pt$-harmonic
forms. On these, we have by Lemma~\ref{le:boxdboxdb}
$$
\Box_\ol\pt=\Box_\pt - m\cdot id.
$$
We see that all eigenvalues of $\Box_\pt$ are larger or equal to $m$ for
$(p,n-p-1)$-forms.
{\bf Claim.} {\it Let $\sum_\nu \lambda_\nu \rho_\nu$ be the eigenfunction
decomposition of $A_\ol s\cup \psi^k$. Then all $\lambda_\nu > m$ or
$\lambda_0=0$. In particular $(\Box - m)^{-1}(A_\ol s\cup \psi^k)$
exists.}
In order to verify the claim, we consider $\pt^*(A_\ol s\cup \psi^k)=
\sum_\nu \pt^*(\rho_\nu)$ with
$$
\Box_\pt \pt^*(\rho_\nu) = \lambda_\nu \pt^*(\rho_\nu) = \Box_\ol\pt
\pt^*(\rho_\nu) + m \cdot\pt^*(\rho_\nu).
$$
This fact implies that $\sum_\nu \pt^*(\rho_\nu)$ is also the
eigenfunction expansion with respect to $\Box_\ol\pt$ and eigenvalues
$\lambda_\nu -m\geq 0$ of $\pt^*(A_\ol s\cup \psi^k)=\ol\pt^*(L_\ol v
\psi^k)$. The latter is orthogonal to the space of $\ol\pt$-harmonic
functions so that $\lambda_\nu-m=0$ does not occur. (The harmonic part of
$A_\ol v\cup \psi^k$ may be present though.) This shows the claim.
Now
$$
G_\ol\pt \pt^*(A_\ol s \cup \psi^k) = (\Box_\pt -m)^{-1} \pt^*(A_\ol s \cup \psi^k)
$$
so that \eqref{eq:7} implies
\begin{gather*}
\langle (L_\ol v \psi^k)', (L_\ol v \psi^\ell)'\rangle = \langle
(\Box_\pt -m)^{-1} \Box_\pt (A_\ol s \cup \psi^k) ,A_\ol s \cup \psi^\ell
\rangle \\
= \langle A_\ol s \cup \psi^k ,A_\ol s \cup \psi^\ell \rangle + m\cdot
\langle (\Box_\pt -m)^{-1} (A_\ol s \cup \psi^k) ,A_\ol s \cup \psi^\ell
\rangle.
\end{gather*}
Now \eqref{eq:3} yields the final equation (again with $\Box_\ol\pt =
\Box_\pt = \Box$ for $(p+1,n-p-1)$-forms)
\begin{equation}\label{eq:part3}
\langle L_\ol v \psi^k, L_\ol v \psi^\ell\rangle = m \int_{\cX_s} (\Box - m)^{-1}( A_\ol s \cup \psi^k)
\cdot (A_s \cup \psi^\ol\ell) \, g\, dV.
\end{equation}
The main theorem follows from \eqref{eq:lvvpsi}, \eqref{eq:part2},
\eqref{eq:Lolvv}, and \eqref{eq:part3}. \qed
\begin{proof}[Proof of Proposition~\ref{pr:baseq}]
We verify \eqref{eq:0}: We will need various identities. For simplicity,
we drop the superscript $k$. The tensors below are meant to be
coefficients of alternating forms on the fibers, i.e.\ skew-symmetrized.
\begin{equation}\label{eq:aux1}
\psi_{;s\ol\beta_{n+1}}= \psi_{;\ol\beta_{n+1}s} - m \cdot g_{s\ol\beta_{n+1}} \psi=
m\cdot a_{s\ol\beta_{n+1}}
\end{equation}
\begin{gather} \label{eq:aux2}
\psi_{;\alpha\ol\beta_{n+1}}= \psi_{;\ol\beta_{n+1}\alpha}- m\cdot
g_{\alpha\ol\beta_{n+1}}\psi \hspace{5cm} \\ \nonumber - \sum^p_{j=1}
\psi_{\tiny \vtop{ \hbox{$\alpha_1,\ldots,\sigma,\ldots,\alpha_p, \ol
B_{n-p}$}\vskip-1.5mm\hbox{$ \phantom{\alpha_1,\ldots,}{|\atop j}$}}
}R^\sigma_{\; \alpha_j\alpha\ol\beta_{n+1}}-\sum^n_{j=p+1}
\psi_{\tiny\vtop{\hbox{$A_p \ol\beta_{p+1},
\ldots,\ol\tau,\ldots,\ol\beta_{n} $}\vskip-1.5mm\hbox{$\phantom{A_p
\ol\beta_{p+1},
\ldots,}{|\atop j} $}}} R^\ol\tau_{\:\ol\beta_j\alpha\ol\beta_n }\\
\nonumber = -m \cdot a_{s\ol\beta_{n+1}}\psi - \sum^p_{j=1}
a^\alpha_s\psi_{\alpha_1,\ldots,\sigma,\ldots,\alpha_p\ol
B_{n-p}}R^\sigma_{\; \alpha_j\alpha\ol\beta_{n+1}}
\end{gather}
\begin{gather}\label{eq:aux3}
a^\alpha_{s;\alpha_j\ol \beta_{n+1}} = A^\alpha_{s\ol\beta_{n+1};\alpha_j}
+ a^\sigma_s R^\alpha_{\; \sigma\alpha_j\ol\beta_{n+1}}
\end{gather}
Now, starting from \eqref{eq:lvprime} we get, using \eqref{eq:aux1},
\eqref{eq:aux2}, and \eqref{eq:aux3},
\begin{gather*}
\ol\pt L_v\psi' = \Big( \psi_{;s\ol\beta_{n+1}} +
A^\alpha_{s\ol\beta_{n+1}}\psi_{;\alpha} + a^\alpha_s
\psi_{;\alpha\ol\beta_{n+1}} +
\sum^p_{j=1}a^\alpha_{s;\alpha_j\ol\beta_{n+1}}\psi_{\alpha_1,\ldots,\alpha,\ldots,\alpha_p,\ol
B_{n-p}}\\ \nonumber + \sum^p_{j=1} a^\alpha_{s;\alpha_j}
\psi_{\alpha_1,\ldots,\alpha,\ldots,\alpha_p,\ol B_{n-p};\ol\beta_{n+1}}
\Big) dz^{\ol\beta_{n+1}}\we dz^{A_p}\we dz^{\ol B_{n-p}}
\\ \nonumber
= \Big(A^\alpha_{s\ol\beta_{n+1}}\psi_{;\alpha} + \sum^p_{j=1}
A^\alpha_{s\ol\beta_{n+1};\alpha_j}
\psi_{\alpha_1,\ldots,\alpha,\ldots,\alpha_p,\ol
B_{n-p}}\Big)dz^{\ol\beta_{n+1}}\we dz^{A_p}\we dz^{\ol B_{n-p}}
\end{gather*}
Because of the fiberwise $\pt$-closedness of $\psi$ this equals
\begin{gather*}
\sum^p_{j=1} \big(A^\alpha_{s\ol\beta_{n+1}} \psi_{\tiny \vtop{
\hbox{$\alpha_1,\ldots,\alpha,\ldots,\alpha_p, \ol
B_{n-p}$}\vskip-1.5mm\hbox{$ \phantom{\alpha_1,\ldots,}{|\atop j}$}} }
\big)_{;\alpha_j}
dz^{\ol\beta_{n+1}}\we dz^{A_p}\we dz^{\ol B_{n-p}} \\
= (-1)^n \sum^p_{j=1} \big(A^\alpha_{s\ol\beta_{n+1}}
\psi_{\alpha,\alpha_2,\ldots,\alpha_p, \ol B_{n-p}} \big)_{;\alpha_1}
dz^{\alpha_1}\we dz^{A_{p-1}}\we dz^{\ol\beta_1}\we\ldots\we
dz^{\ol\beta_{n+1}}\\ \nonumber
=\pt \Big((-1)^n A^\alpha_{s\ol\beta_{n+1}}
\psi_{\alpha,\alpha_2,\ldots,\alpha_p,\ol\beta_{p+1},\ldots,\ol\beta_n}
dz^{A_{p-1}}\we dz^{\ol B_{n+1}}\Big) = \pt\big( A_s \cup \psi\big).
\end{gather*}
This shows \eqref{eq:0}.
Next, we prove \eqref{eq:4}. We begin with \eqref{eq:lvbprime}. We first
note
\begin{equation*}\label{eq:dsGamma}
\pt_s(\Gamma^\sigma_{\alpha\gamma})= -a^\sigma_{s; \alpha\gamma}
\end{equation*}
which follows in a straightforward way. Now this equation implies
\begin{gather*}
\pt_s(\psi_{;\gamma}) = \psi_{;s\gamma} - \sum^p_{j=1}
a^\sigma_{s;\alpha_j\gamma}\psi_{\tiny\vtop{\hbox{$\alpha_1,
\ldots,\sigma,\ldots,\alpha_p \ol B_{n-p}
$}\vskip-1.5mm\hbox{$\phantom{\alpha_1, \ldots,}{|\atop j} $} }}
\end{gather*}
so that (with $g^{\ol\beta_n\gamma}\psi_{;\gamma}=0$ and $\pt_s
g^{\ol\beta_n\gamma}= g^{\ol\beta_n\sigma} a^\gamma_{s;\sigma} $)
\begin{gather}\label{eq:aux01}
g^{\ol\beta_n\gamma}\psi_{;s\gamma} = -\psi_{;\gamma}
g^{\ol\beta_n\sigma} a^\gamma_{s;\sigma} +
\sum^p_{j=1}g^{\ol\beta_n\gamma}
a^\sigma_{s;\alpha_j\gamma}\psi_{\tiny\vtop{\hbox{$\alpha_1,
\ldots,\sigma,\ldots,\alpha_p \ol B_{n-p}
$}\vskip-1.5mm\hbox{$\phantom{\alpha_1, \ldots,}{|\atop j} $} }}
\end{gather}
follows. Next, since fiberwise $\psi$ is $\ol\pt^*$-closed,
\begin{gather}\label{eq:aux02}
g^{\ol\beta_n\gamma} (a^\alpha_s\psi_{;\alpha})_{;\gamma}=
g^{\ol\beta_n\gamma} a^\alpha_{s;\gamma}\psi_{;\alpha},
\end{gather}
and with the same argument
\begin{gather}\label{eq:aux03}
g^{\ol\beta_n\gamma} \big(\sum^p_{j=1}
a^\sigma_{s;\alpha_j}\psi_{\tiny\vtop{\hbox{$\alpha_1,
\ldots,\sigma,\ldots,\alpha_p \ol B_{n-p}
$}\vskip-1.5mm\hbox{$\phantom{\alpha_1, \ldots,}{|\atop j} $} }}
\big)_{;\gamma} = g^{\ol\beta_n\gamma} \sum^p_{j=1}
a^\sigma_{s;\alpha_j\gamma}\psi_{\tiny\vtop{\hbox{$\alpha_1,
\ldots,\sigma,\ldots,\alpha_p \ol B_{n-p}
$}\vskip-1.5mm\hbox{$\phantom{\alpha_1, \ldots,}{|\atop j} $} }}.
\end{gather}
Now $\ol\pt^*(L_v\psi')=0$ follows from \eqref{eq:aux01},
\eqref{eq:aux02}, and \eqref{eq:aux03}.
We come to the $\pt^*$-closedness \eqref{eq:5} of $A_s\cup \psi$. We need
to show that
$$
\big(A^\alpha_{s\ol\beta_{n+1}}\psi_{\alpha,\alpha_2,\ldots,\alpha_p,\ol\beta_{p+1},\ldots,\beta_n}
\big)_{;\ol\delta}g^{\ol\delta\alpha_p}
$$
vanishes. Since $\pt^*\psi=0$ the above quantity equals
$$
A^\alpha_{s\ol\beta_{n+1};\ol\delta} \psi_{\alpha,\alpha_2,\ldots,\alpha_p,\ol B_{n-p}}g^{\ol\delta\alpha_p}.
$$
Because of the $\ol\pt$-closedness of $A_s$ this equals
$$
(A^{\alpha\alpha_p}_{s})_{;\ol \beta_{n+1}}\psi_{\alpha,\alpha_2,\ldots,\alpha_p,\ol B_{n-p}}g^{\ol\delta\alpha_p}.
$$
However,
$$
A^{\alpha\alpha_p}_s=A^{\alpha_p\alpha}_s
$$
whereas $\psi$ is skew-symmetric so that also this contribution vanishes.
The proof of \eqref{eq:6}, \eqref{eq:1}, and \eqref{eq:7} is similar, we
remark that \eqref{eq:1} follows from Lemma~\ref{le:lvpsi1}.
\end{proof} | 126,292 |
Boken är slut online
From music festivals and golf under the midnight sun to cities, castles, waterways and mountains, it's all found in this new edition Sweden Berlitz Pocket Guide. Clear colour-coded sections enable you to locate the information you need easily, with sections on history, where to go, what to do, eating out, handy travel tips, hotels & restaurants and other insightful features, allowing you to make the most of your leisure time. Written by a local expert, this guide offers comprehensive coverage of Sweden's attractions, identifying the top 10 sights the country has to offer, as well as highlighting the country's best shopping, sports, children's activities, entertainment, festivals, national parks and much more. Shopaholics are well catered for with a dedicated section on shopping, with advice on the best places to find crafts, clothes, glass, porcelain, textiles and much more. An 'Eating Out' section provides information on where to go and what the local specialities are. Insightful cultural advice on Vikings, Sami and sour herring, gives you a deeper understanding of Sweden's heritage. Accommodation for all budgets is also recommended, along with an A-Z of practical travel tips covering currency, etiquette, language tips, translations and how to get around the country, supported with detailed mapping to aid orientation. A city plan of Stockholm is also included in this guide.
- Format: Häftad
- ISBN: 9789812685131
- Språk: Engelska
- Utgivningsdatum: 2008-10-28
- Del i serien: Berlitz reseguider
- Förlag: Reseförlaget | 369,752 |
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GSAs reduce suicide risks study finds
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My generation working to end discrimination
SMYAL’s GSA camp taught me that equality is within reach
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Education Dept. affirms students’ right to form GSAs
Sec'y Duncan issues guidance to schools throughout the country
June 14, 2011 |
© Copyright Brown, Naff, Pitts Omnimedia, Inc. 2014. All rights reserved. | 393,969 |
\begin{document}
\title{Inverse system characterizations of the (hereditarily) just infinite property in profinite groups}
\author{Colin D. Reid}
\maketitle
\begin{abstract}We give criteria on an inverse system of finite groups that ensure the limit is just infinite or hereditarily just infinite. More significantly, these criteria are `universal' in that all (hereditarily) just infinite profinite groups arise as limits of the specified form.
This is a corrected and revised version of \cite{ReidInvLim}.\end{abstract}
\section{Introduction}
\begin{notn}In this paper, all groups will be profinite groups, all homomorphisms are required to be continuous, and all subgroups are required to be closed; in particular, all references to commutator subgroups are understood to mean the closures of the corresponding abstractly defined subgroups. For an inverse system
\[
\Lambda = \{(G_n)_{n > 0},\rho_n: G_{n+1} \twoheadrightarrow G_n\}
\]
of finite groups, we require all the homomorphisms $\rho_n$ to be surjective. A subscript $o$ will be used to indicate open inclusion, for instance $A \leq_o B$ means that $A$ is an open subgroup of $B$. We use `pronilpotent' and `prosoluble' to mean a group that is the inverse limit of finite nilpotent groups or finite soluble groups respectively, and `$G$-invariant subgroup of $H$' to mean a subgroup of $H$ normalized by $G$.
\end{notn}
A profinite group $G$ is \defbold{just infinite} if it is infinite, and every nontrivial normal subgroup of $G$ is of finite index; it is \defbold{hereditarily just infinite} if in addition every open subgroup of $G$ is just infinite.
At first sight the just infinite property is a qualitative one, like that of simplicity: either a group has nontrivial normal subgroups of infinite index, or it does not. However, it has been shown by Barnea, Gavioli, Jaikin-Zapirain, Monti and Scoppola (see \cite{BGJMS}, especially Theorem 36) and the present author (\cite{Rei}) that the just infinite property in profinite groups can be characterized by properties of the lattice of \emph{open} normal subgroups, and as such may be regarded as a kind of boundedness property on the finite images (see \cite[Theorems B1 and B2]{Rei}); moreover, it suffices to consider any collection of finite images that form an inverse system for the group. Similar considerations apply to the hereditarily just infinite property.
The weakness of the characterization given in \cite{Rei} is that the conditions imposed on the finite images are asymptotic: no correspondence is established between the (hereditarily) just infinite property \textit{per se} and the structure of any given finite image. In a sense this is unavoidable, as any finite group appears as the image of a hereditarily just infinite profinite group (see Example \ref{primex}). By contrast in \cite{BGJMS}, a strong `periodic' structure of some just infinite pro-$p$ groups is described, but this structure can only exist for a certain special class of just infinite virtually pro-$p$ groups, those which have a property known as finite obliquity (essentially in the sense of \cite{KLP}). In the present paper we therefore take a new approach, which is to show the existence of an inverse system with certain specified properties for any (hereditarily) just infinite profinite group, and in turn to show that the specified properties imply the (hereditarily) just infinite property in the limit. The inverse system characterizations in this paper are essentially variations on the following:
\begin{thm}\label{introthm}Let $\Lambda = \{(G_n)_{n > 0},\rho_n: G_{n+1} \rightarrow G_n\}$ be an inverse system of finite groups. Let $A_n$ be a nontrivial normal subgroup of $G_n$ and let $P_n = \rho_n(A_{n+1})$ for $n > 0$. Suppose furthermore, for each $n > 0$:
\begin{enumerate}[(i)]
\item $A_{n+1} > \ker\rho_n \ge P_{n+1}$;
\item $A_n$ has a unique maximal $G_n$-invariant subgroup;
\item Each normal subgroup of $G_n$ contains $P_n$ or is contained in $A_n$ (or both).
\end{enumerate}
Then $G = \varprojlim G_n$ is just infinite. Conversely, every just infinite profinite group is the limit of such an inverse system.\end{thm}
A similar characterization applies to hereditarily just infinite groups: see Theorem~\ref{mainjithm:hji}. As shown by Wilson in \cite{Wil}, such groups are not necessarily virtually pro-$p$; we derive one of Wilson's constructions as a special case.
We will also discuss some examples of hereditarily just infinite profinite groups that are not virtually prosoluble, illustrating some features of this class. In particular, the following serves as a source of examples of such groups:
\begin{defn}\label{def:subprim}Let $X$ be a set and let $G$ be a group acting on $X$, with kernel $K$. We say $G$ acts \defbold{subprimitively} on $X$ if for every normal subgroup $H$ of $G$, then $H/(H \cap K)$ acts faithfully on every $H$-orbit.\end{defn}
\begin{prop}\label{intro:subprim}Let $G$ be a just infinite profinite group. Suppose there are infinitely many nonabelian chief factors $R/S$ of $G$ for which $G$ permutes the simple factors of $R/S$ subprimitively. Then $G$ is hereditarily just infinite.\end{prop}
\begin{rem}The following classes of permutation groups are all special cases of transitive subprimitive permutation groups: regular permutation groups, primitive permutation groups, quasiprimitive groups in the sense of \cite{Praeger} and semiprimitive groups in the sense of \cite{BM}. An intransitive faithful action is subprimitive if and only if the group acts faithfully and subprimitively on every orbit.\end{rem}
\paragraph{Acknowledgements}
My thanks go to Davide Veronelli, who pointed out some errors in the original article and proposed some corrections; the need to fix the errors in the original article was the immediate motivation for this revised article. I also thank John Wilson for showing me a preliminary version of \cite{Wil}; Charles Leedham-Green for his helpful comments; and Laurent Bartholdi for his suggestion to look at \cite{Lucchini} and determine whether the group in question is hereditarily just infinite.
\section{Preliminaries}
\begin{defn}Let $G$ be a profinite group. A \defbold{chief factor} of $G$ is a quotient $K/L$ of some normal subgroup $K$ of $G$, such that $L \unlhd G$ and there are no normal subgroups $M$ of $G$ satisfying $K < M < L$.
We say two chief factors $K_1/L_1$ and $K_2/L_2$ of $G$ are \defbold{associated} to each other if
\[
K_1L_2 = K_2L_1; \; K_i \cap L_1L_2 = L_i \; (i = 1,2).
\]
This also implies that $K_1L_2=K_1K_2$.
Given a normal subgroup $N$ of $G$, say $N$ \defbold{covers} the chief factor $K/L$ if $NL \ge K$.\end{defn}
The association relation is not transitive in general. For instance, if $G$ is the Klein $4$-group with subgroups $\{H_1,H_2,H_3\}$ of order $2$, then the pairs of associated chief factors are those of the form $\{G/H_i,H_j/\triv\}$ for $i \neq j$. Thus $H_1/\triv$ is associated to $G/H_2$ and $G/H_2$ is associated to $H_3/\triv$, but $H_1/\triv$ and $H_3/\triv$ are not associated to each other.
\emph{Nonabelian} chief factors however are much better behaved under association. A theory of association classes of nonabelian chief factors is developed in a much more general context in \cite{RW_Polish}.
Parts (i) and (ii) of the following lemma are standard facts about profinite groups and will be used without further comment.
\begin{lem}\label{chieflem}Let $G$ be a profinite group.
\begin{enumerate}[(i)]
\item Let $K$ and $L$ be normal subgroups of $G$ such that $K > L$. Then there exists $L \le M < K$ such that $K/M$ is a chief factor of $G$.
\item Every chief factor of $G$ is finite.
\item Let $K_1/L_1$ and $K_2/L_2$ be associated chief factors of $G$. Then $K_1/L_1 \cong K_2/L_2$ and $\CC_G(K_1/L_1) = \CC_G(K_2/L_2)$.
\end{enumerate}
\end{lem}
\begin{proof}Given normal subgroups $K$ and $L$ of $G$ such that $K > L$, then $L$ is an intersection of open normal subgroups of $G$, so there is an open normal subgroup $O$ of $G$ containing $L$ but not $K$. Then $K \cap O \unlhd G$ and $K \cap O <_o K$. In particular, $K/(K \cap O)$ is a finite normal factor of $G$. This implies (i) and (ii).
For (iii), let $H = K_1K_2=K_1L_2=K_2L_1$. Since $K_1 \cap L_1L_2 = L_1$, the kernel of the natural homomorphism from $K_1$ to $H/L_1L_2$ is exactly $L_1$; since $H = K_1L_2$, this homomorphism is also surjective. Thus $K_1/L_1 \cong H/L_1L_2$. Let $T$ be a subgroup of $G$. If $[H,T] \le L_1L_2$ then $[K_1,T] \le L_1L_2 \cap K_1 = L_1$, and conversely if $[K_1,T] \le L_1$ then $[H,T]$ is contained in the normal closure of $L_1 \cup [L_2,T]$ (since $H = K_1L_2$) and thus in $L_1L_2$. Hence $\CC_G(K_1/L_1) = \CC_G(H/L_1L_2)$. Similarly $K_2/L_2 \cong H/L_1L_2$ and $\CC_G(K_2/L_2) = \CC_G(H/L_1L_2)$.
\end{proof}
We will also need some definitions and results from \cite{Rei}.
\begin{defn}The \emph{cosocle} or \emph{Mel'nikov subgroup} $\M(G)$ of $G$ is the intersection of all maximal open normal subgroups of $G$. \end{defn}
\begin{lem}[{see \cite[Lemma~2.2]{Rei} and its corollary; see also \cite{Zal}}]\label{melfin} Let $G$ be a just infinite profinite group, and let $H$ be an open subgroup of $G$. Then $|H:\M(H)|$ is finite.\end{lem}
\begin{defn}Given a profinite group $G$ and subgroup $H$, we define $\Ob_G(H)$ and $\Ob^*_G(H)$ as follows:
\[ \Ob_G(H) := H \cap \bigcap \{K \unlhd_o G \mid K \not\le H\}\]
\[ \Ob^*_G(H) := H \cap \bigcap \{K \leq_o G \mid H \le \N_G(K), \; K \not\le H\}. \]
Note that $\Ob_G(H)$ and $\Ob^*_G(H)$ have finite index in $H$ if and only if the relevant intersections are finite.\end{defn}
\begin{thm}[{\cite[Theorem~36]{BGJMS}, \cite[Theorem~A and Corollary~2.6]{Rei}}]\label{genob} Let $G$ be a just infinite profinite group, and let $H$ be an open subgroup of $G$. Then $|G:\Ob_G(H)|$ is finite. If $G$ is hereditarily just infinite, then $|G:\Ob^*_G(H)|$ is finite.\end{thm}
\begin{cor}\label{chiefin}Let $G$ be a just infinite profinite group and let $H$ be an open subgroup of $G$. Then $K \le H$ for all but finitely many chief factors $K/L$ of $G$.\end{cor}
\begin{proof}By Theorem \ref{genob}, there are only finitely many normal subgroups of $G$ not contained in $H$. In turn, if $K/L$ is a chief factor of $G$ then $L \ge \M(K)$, so by Lemma \ref{melfin}, the quotients of a given open normal subgroup of $G$ can only produce finitely many chief factors of $G$.\end{proof}
\section{Narrow subgroups}
The key idea in this paper is that of a `narrow' subgroup associated to a chief factor. These are a general feature of profinite groups, but they have further properties that will be useful in establishing the just infinite property. Throughout this section, $G$ will be a profinite group.
\begin{defn}Let $1 < A \unlhd G$, and define $\M_G(A)$ to be the intersection of all maximal open $G$-invariant subgroups of $A$. Note that $\M(A) \le \M_G(A) < A$. Say $A$ is \emph{narrow} in $G$ and write $A \nar G$ if there is a unique maximal $G$-invariant subgroup of $A$, in other words $\M_G(A)$ is the maximal $G$-invariant subgroup of $A$.
Note that if $A \nar G$, then $A/N \nar G/N$ and $\M_G(A/N) = \M_G(A)/N$ for any $N \unlhd G$ such that $N < A$.
Given $A \nar G$ and a chief factor $K/L$ of $G$, we will say $A$ is associated to $K/L$ to mean $A/\M_G(A)$ is associated to $K/L$.\end{defn}
\begin{lem}\label{critlem}Let $A$ and $K$ be normal subgroups of the profinite group $G$.
\begin{enumerate}[(i)]
\item We have $K\M_G(A) \ge A$ if and only if $K \ge A$.
\item Suppose $A$ is narrow in $G$ and that $KN \ge A$ for some proper $G$-invariant subgroup of $A$. Then $K \ge A$.
\end{enumerate}
\end{lem}
\begin{proof}
(i)
Suppose $K \ngeq A$. Then $K \cap A$ is contained in a maximal $G$-invariant subgroup $R$ of $A$. By the modular law, $K\M_G(A) \cap A = \M_G(A)(K \cap A) \le R < A$, so $K\M_G(A) \ngeq A$. The converse is clear.
(ii)
Since $A$ is narrow in $G$, the group $\M_G(A)$ contains every proper $G$-invariant subgroup of $A$. In particular, $\M_G(A) \ge N$, so $K\M_G(A) \ge KN \ge A$ and hence $K \ge A$ by part (i).\end{proof}
The existence of narrow subgroups in profinite groups is shown by a compactness argument.
\begin{lem}\label{narrowassoc}Given any chief factor $K/L$ of $G$, there is a narrow subgroup $A$ of $G$ associated to $K/L$. Those $A \nar G$ associated to $K/L$ are precisely those narrow subgroups of $G$ contained in $K$ but not $L$, and it follows in this case that $A \cap L = \M_G(A)$. In particular, every nontrivial normal subgroup of $G$ contains a narrow subgroup of $G$.\end{lem}
\begin{proof}Suppose $A \nar G$ and $A$ is associated to $K/L$. Then $K\M_G(A) \ge A$, so $K \ge A$ by Lemma \ref{critlem}. Also $AL \ge K$, so $A \not\le L$. Conversely, let $A \nar G$ such that $A \le K$ and $A \not\le L$. Then $AL = K$ since $K/L$ is a chief factor of $G$, and clearly then $AL = K(A \cap L)$ and $(A \cap L)L < AK$. To show $A \cap L = \M_G(A)$ and that $A/\M_G(A)$ is associated to $K/L$, it remains to show that $A/A \cap L$ is a chief factor of $G$: this is the case as any $G/A \cap L$-invariant subgroup of $A/A \cap L$ would correspond via the isomorphism theorems to a $G/L$-invariant subgroup of $AL/L = K/L$.
It remains to show that narrow subgroups with the specified properties exist. Let $\mc{K}(G,K)$ be the set of normal subgroups of $G$ contained in $K$ and let $\mc{D} = \mc{K}(G,K) \setminus \mc{K}(G,L)$. Given that $K \setminus L$ is compact, one sees that the intersection of any descending chain in $\mc{D}$ is not contained in $L$, and is thus an element of $\mc{D}$. Hence $\mc{D}$ has a minimal element $A$ by Zorn's lemma. Now any normal subgroup of $G$ properly contained in $A$ must be contained in $L$ by the minimality of $A$ in $\mc{D}$. Thus $A \cap L$ is the unique maximal $G$-invariant subgroup of $A$ and so $A \nar G$.\end{proof}
We now define a relation on chief factors which gives the intuition underlying the rest of this paper.
\begin{defn}Given chief factors $a = K_1/L_1$ and $b = K_2/L_2$, say $a \gar b$ if $L_1 \ge K_2$ and $\M_{G/L_2}(K_1/L_2) = L_1/L_2$ (in particular, $K_1/L_2 \nar G/L_2$).\end{defn}
\begin{prop}\label{transprop}
\begin{enumerate}[(i)]
\item Let $K_1/L_1$ and $K_2/L_2$ be chief factors of $G$. Let $N$ be a normal subgroup of $G$ that covers $K_1/L_1$ and suppose $K_1/L_1 \gar K_2/L_2$. Then $N$ covers $K_2/L_2$.
\item The relation $\gar$ is a strict partial order.
\item Suppose $K_1/L_1 \gar K_2/L_2 \gar \dots$ is a descending sequence of open chief factors of $G$ such that $\bigcap_i L_i = \triv$. Then $K_1 \nar G$.\end{enumerate}
\end{prop}
\begin{proof}(i) We have $\M_{G/L_2}(K_1/L_2) = L_1/L_2$, so $NL_2/L_2 \ge K_1/L_2$ by Lemma \ref{critlem}, that is, $NL_2 \ge K_1$. As $K_2 \le K_1$, this implies that $N$ covers $K_2/L_2$.
(ii) It is clear that $\gar$ is antisymmetric and antireflexive. Suppose $K_1/L_1 \gar K_2/L_2$ and $K_2/L_2 \gar K_3/L_3$. It remains to show that $L_1$ is the unique maximal proper $G$-invariant subgroup of $K_1$ containing $L_3$. Suppose there is another such subgroup $R$. Since $M_{G/L_2}(K_1/L_2) = L_1/L_2$ and $R$ is not contained in $L_1$, we must have $RL_2 = K_1$. In particular, $RL_2 \ge K_2$. Since $L_2/L_3 = \M_{G/L_3}(K_2/L_3)$, we have $R/L_3 \ge K_2/L_3$ by Lemma \ref{critlem}, that is $R \ge K_2$. But then $L_2 \le R$, so in fact $R = K_1$, a contradiction.
(iii) It follows from (ii) that $K_1/L_i \nar G/L_i$ for all $i$. Suppose there is a maximal proper $G$-invariant subgroup $N$ of $K_1$ other than $L_1$. Then $N \not\le L_1$, so $NL_i \not\le L_1$ for all $i$. Since $K_1/L_i \nar G/L_i$ with $\M_{G/L_i}(K_1/L_i) = L_1/L_i$, this forces $NL_i = K_1$ for all $i$. It follows by a standard compactness argument that $N = K_1$, a contradiction.\end{proof}
The relation $\gar$ can be used to obtain some restrictions on the just infinite images of a profinite group.
\begin{thm}\label{narchthm}Let $G$ be a profinite group.
\begin{enumerate}[(i)]
\item Suppose $K_1/L_1 \gar K_2/L_2 \gar \dots$ is a descending sequence of open chief factors of $G$ and let $L = \bigcap_i L_i$. Then there is a normal subgroup $K\ge L$ of $G$ such that $G/K$ is just infinite and such that for all $i$, $K$ does not cover $K_i/L_i$. Indeed, it suffices for $K \ge L$ to be maximal subject to not covering the factors $K_i/L_i$.
\item Every just infinite image $G/K$ of $G$ arises in the manner described in (i) with $K=L$.\end{enumerate}
\end{thm}
\begin{proof}(i) Let $\mc{N}$ be the set of all normal subgroups of $G$ which contain $L$ and which do not cover $K_i/L_i$ for any $i$, let $\mc{C}$ be a chain in $\mc{N}$, let $R = \overline{\bigcup \mc{C}}$ and let $i \in \bN$. Then $RL_i = (\bigcup \mc{C})L_i = CL_i$ for some $C \in \mc{C}$, since $L_i$ is already open and in particular of finite index in $G$; so $RL_i$ does not contain $K_i$, which ensures $R \in \mc{N}$. Hence $\mc{N}$ has a maximal element $K$ by Zorn's lemma. Since $K$ does not cover any of the factors $K_i/L_i$, we have $KL_1 > KL_2 > KL_3 > \dots$ and so $|G:K|$ is infinite. Let $P$ be a normal subgroup of $G$ properly containing $K$. Then $P$ covers $K_i/L_i$ for some $i$ by the maximality of $K$. Moreover $M_{G/K}(K_i/L) = L_iL/L$ by Proposition \ref{transprop} and $P \ge L$, so $P \ge K_i$ by Lemma \ref{critlem}. In particular $|G:P| \le |G:K_i| < \infty$. Hence $G/K$ is just infinite.
(ii) By Lemma \ref{narrowassoc}, there are narrow subgroups of $G/K$ associated to every chief factor of $G/K$. Let $R/S$ be a chief factor of $G$ such that $S \ge K$ and let $K_1/K$ be a narrow subgroup of $G/K$ associated to $R/S$, with $\M_{G/K}(K_1/K) = L_1/K$. Thereafter we choose $K_{i+1}/K$ to be a narrow subgroup associated to a chief factor $R/S$ such that $S \ge K$ and $R \le L_i$. It is clear that the chief factors $K_i/L_i$ will have the required properties.\end{proof}
\section{Characterizations of the just infinite property and control over chief factors}
Here is a universal inverse limit construction for just infinite profinite groups along the lines of Theorem \ref{introthm}, also incorporating some information about chief factors.
\begin{thm}\label{mainjithm}Let $G$ be a just infinite profinite group. Let $\mc{C}_1,\mc{C}_2,\dots$ be a sequence of classes of finite groups, such that $G$ has infinitely many chief factors in $\mc{C}_n$ for all $n$. Then $G$ is the limit of an inverse system $\Lambda = \{(G_n)_{n > 0},\rho_n: G_{n+1} \twoheadrightarrow G_n\}$ as follows:
Each $G_n$ has a specified nontrivial normal subgroup $A_n$ such that, setting $P_n = \rho_{n}(A_{n+1})$:
\begin{enumerate}[(i)]
\item $\M_{G_{n+1}}(A_{n+1}) \ge \ker\rho_{n} \ge P_{n+1}$;
\item Each normal subgroup of $G_n$ contains $P_n$ or is contained in $A_n$ (or both);
\item $P_n$ is a minimal normal subgroup of $G_n$;
\item $P_n \in \mc{C}_n$ for all $n$.
\end{enumerate}
Conversely, any inverse system satisfying conditions (i) and (ii) (for some choice of nontrivial normal subgroups $A_n$) for all but finitely many $n$ has a limit that is just infinite.\end{thm}
\begin{proof}Suppose that $G$ is just infinite with the specified chief factors. We will obtain an infinite descending chain $(K_n)_{n \ge 0}$ of narrow subgroups of $G$, and then use these to construct the required inverse system.
Let $K_0 = G$. Suppose $K_n$ has been chosen, and let $L= \Ob_G(\M_G(K_n))$. Then by Theorem \ref{genob}, $L$ is open in $G$, and hence by Corollary \ref{chiefin}, all but finitely many chief factors $R/S$ of $G$ satisfy $R \le L$; note also that $L$ is a proper subgroup of $K_n$. Let $R/S$ be a chief factor such that $R \le L$ and $R/S \in \mc{C}_n$, and let $K_{n+1}$ be a narrow subgroup of $G$ associated to $R/S$. Then $K_{n+1} \le L$ and $K_{n+1}/\M_G(K_{n+1}) \cong R/S$. Since $K_{n+1} \le L$, every normal subgroup of $G$ contains $K_{n+1}$ or is contained in $\M_G(K_n)$ (or both).
Set $G_n = G/\M_G(K_{n+1})$, set $P_n = K_{n+1}/\M_G(K_{n+1})$, set $A_n = K_{n}/\M_G(K_{n+1})$ and let the maps $\rho_n$ be the natural quotient maps. The inverse limit of $(G_n,\rho_n)$ is then an infinite quotient of $G$, which is then equal to $G$ since $G$ is just infinite. Given a normal subgroup $N/\M_G(K_{n+1})$ of $G_n$ such that $N/\M_G(K_{n+1}) \nleq A_{n}$, then $N \nleq K_{n}$, so $N \ge K_{n+1}$ and hence $N/\M_G(K_{n+1}) \ge P_n$, so condition (ii) is satisfied. The other conditions are clear.
Now suppose we are given an inverse system satisfying (i) and (ii) for $n \ge n_0>1$, with inverse limit $G$. Let $\pi_n: G \rightarrow G_n$ be the surjections associated to the inverse limit. Note that condition (i) ensures that the groups $\pi\inv_n(A_n)$ form a descending chain of open normal subgroups of $G$; since $P_{n+1} \le \ker\rho_n$, we see that $\pi\inv_{n+2}(A_{n+2}) \le \ker\pi_n$, and consequently the subgroups $\pi\inv_n(A_n)$ have trivial intersection. The fact that $A_{n}$ is nontrivial ensures that $\pi\inv_n(A_n)$ is nontrivial; since the descending chain $(\pi\inv_n(A_n))$ has trivial intersection, $G$ must be infinite. Let $N$ be a nontrivial normal subgroup of $G$. Then there is some $n_1 \ge n_0$ such that for all $n \ge n_1$, $N$ is not contained in $\pi\inv(A_n)$. By (ii) it follows that $\pi_n(N)$ contains $P_n$; hence $\rho_n(\pi_{n+1}(N) \cap A_{n+1})=P_n$ and thus
\[
A_{n+1} = (\pi_{n+1}(N) \cap A_{n+1})\ker\rho_n = (\pi_{n+1}(N) \cap A_{n+1})\M_{G_{n+1}}(A_{n+1}),
\]
using (i). We conclude by Lemma~\ref{critlem} that $\pi_{n+1}(N) \ge A_{n+1}$. Hence $\pi_{n+1}(N) \ge \ker\rho_{n}$ for all $n \ge n_1$, so in fact $N \ge \ker\pi_{n_1}$. In particular, $N$ is open in $G$, showing that $G$ is just infinite.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{introthm}]
Let $G$ be a just infinite profinite group and let $A_n$ and $P_n$ be as obtained in Theorem~\ref{mainjithm}. Conditions (i) and (iii) of Theorem~\ref{mainjithm} together imply that $A_{n+1} > \M_{G_{n+1}}(A_{n+1}) = \ker\rho_{n}$ and that $\ker\rho_n$ is a maximal $G_{n+1}$-invariant subgroup of $A_{n+1}$, so $A_{n+1}$ is a narrow normal subgroup of $G_{n+1}$, as required for condition (ii) of the present theorem. The other two conditions are now clear. Thus $G$ is the limit of an inverse system satisfying the given conditions.
Conversely, let $\Lambda = \{(G_n)_{n > 0},\rho_n: G_{n+1} \twoheadrightarrow G_n\}$ be an inverse system of finite groups with subgroups $A_n$ and $P_n$ of $G_n$ satisfying the given conditions, and let $G = \varprojlim G_n$. Conditions (i) and (ii) together of the present theorem imply condition (i) of Theorem~\ref{mainjithm}, and condition (iii) of the present theorem is condition (ii) of Theorem~\ref{mainjithm}. Thus $G$ is just infinite by Theorem~\ref{mainjithm}.
\end{proof}
\begin{rem}
A similar statement was given in the original published version of this article (\cite[Theorem~4.1]{ReidInvLim}). However, the statement and proof given in \cite{ReidInvLim} fail to take sufficient account of the kernels of the maps $\rho_n$. We rectify this issue here and in the results derived from Theorem~\ref{mainjithm} by explicitly imposing a condition on $\ker\rho_n$.
\end{rem}
If $G$ is pronilpotent, then all chief factors of $G_n$ are central. But if $G$ is not virtually pronilpotent, it could be helpful to replace condition (ii) of Theorem \ref{mainjithm} with the stronger but easier-to-verify condition that $A_n$ contains the centralizer of the chief factor $P_n/\M_G(P_n)$. Indeed, this can always be arranged, with some inevitable adjustments to the classes $\mc{C}_n$.
\begin{defn}Given a profinite group $G$ and a prime $p$, write $\OO^p(G)$ for the intersection of all normal subgroups of $G$ of $p$-power index.\end{defn}
\begin{lem}\label{opsch}Let $G$ be a finite group and let $p$ be a prime. Suppose that all chief factors of $G$ of exponent $p$ are central, and that $p$ does not divide the order of the Schur multiplier of any nonabelian composition factor of $G$. Then $\OO^p(G)$ has no composition factors of order $p$.\end{lem}
\begin{proof}It suffices to show $\OO^p(G) < G$ whenever $G$ has a composition factor of order $p$, as then one can repeat the argument for $\OO^p(G)$ and obtain the conclusion from the fact that $\OO^p(\OO^p(G))=\OO^p(G)$. Let $N$ be a normal subgroup of largest order such that $\OO^p(N) < N$. Suppose $N < G$, and let $K/N$ be a minimal normal subgroup of $G/N$; note that every $G$-chief factor of $N/\OO^p(N)$ is central in $G$, so $K/\OO^p(N)$ is an iterated central extension of $K/N$. If $K/N$ is abelian, then $[K,K] \leq N$, so $K/\OO^p(N)$ is nilpotent. On the other hand, if $K/N$ is nonabelian, then it is a direct power of a nonabelian finite simple group $S$, such that the Schur multiplier of $S$ has order coprime to $p$; it follows that $K/[K,K]\OO^p(N)$ is a nontrivial $p$-group. In either case $\OO^p(K) < K$, contradicting the choice of $N$.\end{proof}
We now specify a condition $\condb$ on classes of finite groups $\mc{C}$:
$\condb$ The class $\mc{C}$ consists of finite characteristically simple groups. For each prime $p$, if $\mc{C}$ contains some elementary abelian $p$-group, then within the class of finite groups, $\mc{C}$ contains all elementary abelian $p$-groups and all direct powers of all finite simple groups $S$ such that $p$ divides the Schur multiplier of $S$.
\begin{lem}\label{seclem}Let $G$ be a just infinite profinite group that is not virtually pronilpotent and let $H$ be an open subgroup of $G$. Let $\mc{C}$ be a class of groups satisfying $\condb$. Let $\mc{D}$ be the set of chief factors of $G$ belonging to $\mc{C}$, and suppose $\mc{D}$ is infinite. Then $K\CC_G(K/L) \le H$ for infinitely many $K/L \in \mc{D}$.\end{lem}
\begin{proof}By Corollary \ref{chiefin}, $K \le H$ for all but finitely many chief factors $K/L$ of $G$. It thus suffices to assume that, for all but finitely many $K/L \in \mc{D}$, $\CC_G(K/L)$ is not contained in $H$, and derive a contradiction. As $\CC_G(K/L)$ is a normal subgroup of $G$, by Theorem \ref{genob} there are only finitely many possibilities for the subgroups $\CC_G(K/L)$ that are not contained in $H$. Thus $R = \bigcap_{K/L \in \mc{D}} \CC_G(K/L)$ has finite index in $G$. Moreover, given $K/L \in \mc{D}$ such that $K \le R$, then $K/L$ is central in $R$ and in particular abelian. Let $N$ be the smallest $G$-invariant subgroup of $R$ such that $R/N$ has a $G$-chief series whose factors are all in $\mc{D}$ (equivalently, the intersection of all such subgroups). Then $R/N$ is pronilpotent, which means $G/N$ is virtually pronilpotent.
As $G$ itself is not virtually pronilpotent, it follows that $N$ has finite index in $G$, so there is some $K/L \in \mc{D}$ with $K \le N$. As $N$ centralizes all such chief factors, $K/L$ is abelian, say of exponent $p$, while all nonabelian simple composition factors of $N$ have Schur multipliers of order coprime to $p$. Thus, $\OO^p(N) < N$ by Lemma~\ref{opsch}. As $\OO^p(N)$ is characteristic in $N$, it is normal in $G$. But then $R/\OO^p(N)$ is an image of $R$ with a $G$-chief series whose factors are all in $\mc{D}$, contradicting the definition of $N$.\end{proof}
\begin{cor}\label{mainjithm:centralizers}Let $G$ be a just infinite profinite group that is not virtually pronilpotent, and let $\mc{C}_1,\mc{C}_2,\dots$ be a sequence of classes of finite groups, all accounting for infinitely many chief factors of $G$. Suppose each class $\mc{C}_n$ also satisfies $\condb$. Then $G$ is the limit of an inverse system of the form specified in Theorem~\ref{mainjithm}, with the additional condition that $\CC_{G_n}(P_n) < A_n$ for all $n$.\end{cor}
\begin{proof}
In the construction of the inverse system in Theorem~\ref{mainjithm}, we were free to choose any chief factor $R/S$ such that $R \le L$ and $R/S \in \mc{C}_{n+1}$. It follows immediately from Corollary~\ref{chiefin} and Lemma~\ref{seclem} that there will be a choice for which $\CC_G(R/S) \le L$ and hence $\CC_G(K/\M_G(K)) \le L$, where $K$ is the associated narrow subgroup, which implies $\CC_{G_n}(P_n) < A_n$ in the inverse limit construction.
\end{proof}
The need for a condition like $\condb$ becomes clear when one considers an iterated transitive wreath product $G$ of copies of some finite perfect group $P$ with nontrivial centre. Although $G$ is just infinite, has infinitely many abelian chief factors and is not virtually pronilpotent, all abelian chief factors of $G$ are central factors.
\section{Characterizations of the hereditarily just infinite property}
We can adapt Theorem~\ref{mainjithm} (using an idea from \cite{Wil}) to obtain a universal construction for hereditarily just infinite groups.
\begin{defn}A finite group $H$ is a \emph{central product} of subgroups $\{H_i \mid i \in I\}$ if these subgroups generate $H$, and whenever $i \not=j$ then $[H_i,H_j]=1$. If $H$ is a normal subgroup of a group $G$, we say $H$ is a \defbold{basal central product} in $G$ if in addition the groups $\{H_i \mid i \in I\}$ form a conjugacy class of subgroups in $G$.
Say $H$ is \emph{centrally indecomposable} if it cannot be expressed as a central product of proper subgroups, and \emph{basally centrally indecomposable in $G$} if it cannot be expressed as a basal central product in $G$.\end{defn}
\begin{thm}\label{mainjithm:hji}Let $G$ be a hereditarily just infinite profinite group. Let $\mc{C}_1,\mc{C}_2,\dots$ be a sequence of classes of finite groups, such that $G$ has infinitely many chief factors in $\mc{C}_n$ for all $n$. Then $G$ is the limit of an inverse system $\Lambda = \{(G_n)_{n > 0},\rho_n: G_{n+1} \twoheadrightarrow G_n\}$ as follows:
Each $G_n$ has a specified nontrivial normal subgroup $A_n$ such that, setting $P_n = \rho_{n}(A_{n+1})$:
\begin{enumerate}[(i)]
\item $\M_{G_{n+1}}(A_{n+1}) \ge \ker\rho_{n} \ge P_{n+1}$;
\item Each normal subgroup of $G_n$ contains $P_n$ or is contained in $A_n$ (or both);
\item $P_n$ is a minimal normal subgroup of $G_n$;
\item $P_n \in \mc{C}_n$ for all $n$;
\item In $G_n$, every normal subgroup containing $A_n$ is basally centrally indecomposable.
\end{enumerate}
Conversely, any inverse system satisfying conditions (i) and (ii) for all but finitely many $n$ and (v) for infinitely many $n$ has a limit that is hereditarily just infinite.\end{thm}
\begin{proof}Suppose that $G$ is hereditarily just infinite with the specified chief factors. We will obtain an infinite descending chain $(K_n)_{n \ge 0}$ of narrow subgroups of $G$ as in the proof of Theorem~\ref{mainjithm}, additionally ensuring that (v) will be satisfied.
Suppose that $G$ is not virtually abelian; note that in this case, $|H:[H,H]|$ is finite for every nontrivial normal subgroup $H$ of $G$. Let $K_0 = G$. Suppose $K_n$ has been chosen for some $n \ge 0$; let $M = \Ob^*_G(\M(K_n))$, and let $L$ be a normal subgroup of $G$ properly contained in $[M,M]$. As in the proof of Theorem~\ref{mainjithm}, there is a chief factor $R/S$ of $G$ such that $R \le L$ and $R/S \in \mc{C}_n$, and $K_{n+1} \nar G$ associated to $R/S$. Then $K_{n+1} \le L$ and $K_{n+1}/\M_G(K_{n+1}) \cong R/S$. Set $G_n = G/\M_G(K_{n+1})$, set $P_n = K_{n+1}/\M_G(K_{n+1})$, set $A_n = K_{n}/\M_G(K_{n+1})$ and let the maps $\rho_n$ be the natural quotient maps.
If instead $G$ is virtually abelian, then it is easy to see that in fact $G$ has a unique largest abelian open normal subgroup $N$, such that $\CC_G(N) = N \cong \bZ_p$ for some prime $p$. Note that $\mc{C}_n$ must contain the cyclic group of order $p$. Let $K_0 = N$ and suppose $K_n$ has been chosen for some $n \ge 0$. Then there are open subgroups $L_n$ and $K_{n+1}$ of $K_n$ such that for all $g \in G \setminus N$ we have
\[
[\Ob_G(\M(K_n)),\langle g \rangle] \ge L_n \text{ and } [L_n,\langle g \rangle] \ge K_{n+1}.
\]
(If $G = N$, we simply take $K_{n+1} = \M(K_n)$.) We then define $G_n$, $P_n$, $A_n$ and $\rho_n$ as before.
In both cases, the inverse limit of $(G_n,\rho_n)$ is an infinite quotient of $G$, which is in fact equal to $G$ since $G$ is just infinite. Conditions (i)--(iv) are satisfied as in the proof of Theorem~\ref{mainjithm}. Let $\{ \pi_n: G \rightarrow G_n \mid n > 0\}$ be the surjections associated to the inverse limit. We now prove by contradiction that (v) is satisfied. Let $U$ be a normal subgroup of $G_n$ containing $A_n$, and suppose $U$ is a proper basal central product in $G_n$: say $U$ is the normal closure of $V$, where $V < U$ and the distinct $G_n$-conjugates of $V$ centralize each other. Note that $V$ is normal in $U$ and hence normalized by $A_n$; thus $\pi\inv_n(V)$ is normalized by $K_n$.
If $G$ is not virtually abelian, then $V$ is not contained in $\M(K_n)$; it follows that $\pi\inv_n(V) \ge \Ob^*_G(\M(K_n))$. By the same argument, $\pi\inv_n(gVg\inv) \ge \Ob^*_G(\M(K_n))$ for every $g \in G$. Thus all $G_n$-conjugates of $V$ contain the group $\pi_n(\Ob^*_G(\M(K_n)))$, which is nonabelian as a consequence of the choice of $L$. If $G$ is virtually abelian, the existence of a nontrivial decomposition ensures that $U$ is not cyclic of prime power order, so $\pi\inv_n(U) \not\le N$. Thus $\pi\inv_n(U)$ contains $K_n$ and $\pi\inv_n(V)$ contains some $g \in G \setminus N$. We then have $\pi\inv_n(V) \ge [K_n,\langle g \rangle] \ge L_n$; indeed $\pi\inv_n(hVh\inv) \ge L_n$ for all $h \in G_n$. The choice of $L_n$ ensures that $L_n/\M_G(K_{n+1})$ is a noncentral subgroup of $hVh\inv$ for all $h \in G_n$. In either case we obtain a subgroup of $\bigcap_{h \in G_n}hVh\inv$ that is not centralized by $V$, contradicting the hypothesis that the conjugates of $V$ form a basal central decomposition of $U$. Thus (v) holds for all $n > 0$.
\
Conversely, suppose we are given an inverse system satisfying (i) and (ii) for all $n \ge n_0>1$, and also (v) for infinitely many $n$. Let $G$ be the inverse limit and let $\pi_n: G \rightarrow G_n$ be the surjections associated to the inverse limit. Then $G$ is just infinite by Theorem~\ref{mainjithm}. As in the proof of Theorem~\ref{mainjithm}, the groups $\pi\inv(A_n)$ for $n \ge n_0$ form a descending chain of open normal subgroups of $G$ with trivial intersection. If $G$ is not hereditarily just infinite, then it would have a nontrivial subgroup $V$ of infinite index such that the distinct conjugates of $V$ centralize each other: this is clear if $G$ is virtually abelian, and given by \cite[2.1]{Wil} if $G$ is not virtually abelian. Let $U$ be the normal closure of $V$ in $G$. Then for all but finitely many $n$, we see that $\pi\inv_n(A_n) \le U$ and $\pi_n(V) < \pi_n(U)$, so there exists $n$ such that (v) holds, $A_n \le \pi_n(U)$ and $\pi_n(V) < \pi_n(U)$. But then $\pi_n(U)$ is the normal closure of $\pi_n(V)$ in $G_n$, and the distinct conjugates of $\pi_n(V)$ centralize each other, contradicting (v). From this contradiction, we conclude that $G$ is hereditarily just infinite.
\end{proof}
We derive a construction from \cite{Wil} as a special case. (Actually the conclusion is slightly stronger than stated in \cite{Wil}: the inverse limit is hereditarily just infinite, even if it is virtually abelian.)
\begin{thm}[{See Wilson \cite[2.2]{Wil}}]\label{wilthm} Let $G$ be the inverse limit of a sequence $(H_n)_{n \geq 0}$ of finite groups and surjective homomorphisms $\phi_n: H_n \twoheadrightarrow H_{n-1}$. For each $n \geq 1$ write $K_n = \ker\phi_n$, and suppose that for all $L \unlhd H_n$ such that $L \nleq K_n$ the following assertions hold:
\begin{enumerate}[(i)]
\item $K_n < L$;
\item $L$ has no proper subgroup whose distinct $H_n$-conjugates centralize each other and generate $L$.
\end{enumerate}
Then $G$ is a hereditarily just infinite profinite group.\end{thm}
\begin{proof}
After relabelling, we may assume that none of the homomorphisms $H_n \twoheadrightarrow H_{n-1}$ are injective. Set
\[
G_n = H_{2n+2}; \; P_n = \ker\phi_{2n+2}; \; Q_n = \ker(\phi_{2n+1}\phi_{2n+2}); \; A_n = \ker(\phi_{2n}\phi_{2n+1}\phi_{2n+2}).
\]
Notice that $\rho_n := \phi_{2n+3}\phi_{2n+4}$ is a surjective homomorphism from $G_{n+1}$ to $G_n$ with kernel $Q_{n+1}$. We see that every normal subgroup $U$ of $G_n$ containing $A_n$ properly contains $P_n$, and hence by (ii), $U$ is basally centrally indecomposable in $G_n$. Thus condition (v) of Theorem~\ref{mainjithm:hji} is satisfied. Let $M$ be a maximal $G_{n+1}$-invariant subgroup of $A_{n+1}$. Then $M$ is not properly contained in $Q_{n+1}$; by considering the quotient $H_{2n+3}$ of $G_{n+1}$ and applying (i), we see that $M \ge Q_{n+1}$. Thus $M_{G_{n+1}}(A_{n+1}) \ge \ker\rho_n$; clearly also $\ker\rho_n \ge P_{n+1}$. Thus condition (i) of Theorem~\ref{mainjithm:hji} is satisfied. Condition (ii) of Theorem~\ref{mainjithm:hji} also follows from condition (i) of the present theorem. Hence by Theorem~\ref{mainjithm:hji}, $G$ is hereditarily just infinite.
\end{proof}
For virtually pronilpotent groups, there is an alternative way to specialize Theorem~\ref{mainjithm} to a characterization of the hereditarily just infinite property, without having to check that a collection of normal subgroups are basally centrally indecomposable. We appeal to some results from \cite{ReiFJ}.
\begin{lem}\label{fijilem}
Let $G$ be a profinite group with no nontrivial finite normal subgroups.
\begin{enumerate}[(i)]
\item (\cite[Lemma~4]{ReiFJ}) If $G$ has a (hereditarily) just infinite open subgroup, then $G$ is (hereditarily) just infinite.
\item (\cite[Theorem~2]{ReiFJ}) If $G$ is pro-$p$ and $\Phi(G)$ is just infinite, then $G$ is hereditarily just infinite.\end{enumerate}
\end{lem}
This leads naturally to our modified construction.
\begin{thm}\label{hjivp}
Let $\Lambda = \{(G_n)_{n > 0},\rho_n: G_{n+1} \twoheadrightarrow G_n\}$ be an inverse system of finite groups. Let $F_n = \Phi(\OO_p(G_n))$, let $\triv < A_n \unlhd F_n$ and let $P_{n} = \rho_{n}(A_{n+1})$ for $n > 0$. Suppose that the following conditions hold for all $n > 0$:
\begin{enumerate}[(i)]
\item $\M_{F_{n+1}}(A_{n+1}) \ge \ker\rho_n \ge P_{n+1}$;
\item Each normal subgroup of $F_n$ contains $P_n$ or is contained in $A_n$ (or both);
\item $F_{n+1} = \rho\inv_n(F_n)$;
\item Every minimal normal subgroup of $G_1$ is contained in $F_1$.
\end{enumerate}
Then $G$ is hereditarily just infinite and virtually pro-$p$. Conversely, every hereditarily just infinite virtually pro-$p$ group is the limit of such an inverse system.
\end{thm}
\begin{proof}
Suppose $\Lambda$ is an inverse system as specified, with inverse limit $G$. Let $F = \Phi(\OO_p(G))$. By condition (iv), $F = \varprojlim F_n$ and $|G:F| = |G_1:F_1| < \infty$, so $F$ is an open pro-$p$ subgroup of $G$. Moreover, $F$ is just infinite by Theorem~\ref{mainjithm}. Let $K$ be a nontrivial normal subgroup of $G$. If $F \cap K = \triv$, then $\pi_1(FK) = F_1 \times \pi_1(K)$ where $\pi_1: G \rightarrow G_1$ is the natural projection. Since $\ker\pi_1 \le F$, the group $\pi_1(K)$ is nontrivial and we have a contradiction to (iv). Thus every nontrivial normal subgroup of $G$ intersects $F$ nontrivially, and is hence infinite; by Lemma~\ref{fijilem} it follows that $G$ is hereditarily just infinite.
Now let $G$ be a hereditarily just infinite virtually pro-$p$ group. Let $F = \Phi(\OO_p(G))$. Then $F$ is open in $G$ and in particular just infinite. The construction of a suitable inverse system is analogous to that in the proof of Theorem~\ref{mainjithm}.
Choose $R_1 \nar F$ such that $R_1 \le \M(\Ob^*_G(F))$ and $S_1$ to be the core of $\M_F(R_1)$ in $G$. Thereafter, choose $R_{n+1} \nar F$ such that $R_{n+1} \le \Ob_F(\M_F(S_n))$ and set $S_{n+1}$ to be the core of $\M_F(R_{n+1})$ in $G$.
Set $G_n = G/S_{n+1}$, $A_n = R_n/S_{n+1}$ and $P_n = R_{n+1}/S_{n+1}$. Then conditions (i) and (ii) are satisfied for the same reasons as in Theorem~\ref{mainjithm}. The choice of $S_1$ ensures conditions (iii) and (iv).
\end{proof}
\section{Further remarks on nonabelian chief factors}
Given a narrow normal subgroup $A$ of $G$, there are some alternative descriptions of $\M_G(A)$ in the case that $A/\M_G(A)$ is either central in $G$ or topologically perfect.
\begin{prop}\label{nonabmel}Let $G$ be a profinite group and let $1 < A \unlhd G$.
\begin{enumerate}[(i)]
\item We have $[A,\M_G(A)] \le \M(A)$. If $A/\M_G(A)$ is topologically perfect, then $\M_G(A) = \M(A)$.
\item Suppose $A > \M(A)[A,G]$. Then $A \nar G$ if and only if $|A:\M(A)[A,G]|$ is prime and every chief factor of $G$ occurring as a quotient of $A$ is central. If $A \nar G$ then $\M_G(A) = \M(A)[A,G]$.
\item If $G$ is pronilpotent, then $\M_G(A) = \M(A)[A,G]$.
\end{enumerate}
\end{prop}
\begin{proof}
(i)
Let $K$ be a normal subgroup of $A$ such that $A/K$ is a nonabelian simple group and let $L$ be the core of $K$ in $G$. Then $A/L$ is isomorphic to a subdirect product of copies of $A/K$; since $A/K$ is a nonabelian simple group, in fact $A/L$ is isomorphic to a direct product of copies of $A/K$, say $A/L = F_1 \times F_2 \times \dots \times F_n$ where $F_i \cong A/K$. We see that $K/L$ contains all but one of the simple direct factors of $A/L$, say $K/L = F_2 \times \dots \times F_n$. Suppose $\M_G(A) \nleq L$; then there exists $M < A$ such that $M$ is $G$-invariant and $L < M$. Then either $F_1 \nleq M/L$, in which case $M \le K$, or else $F_i \nleq M/L$ for $i > 1$, in which case $\CC_{G/L}(M/L)$ is a nontrivial $G$-invariant subgroup of $K/L$. In either case we have a contradiction to the fact that $L$ is the core of $K$ in $G$. Thus in fact $\M_G(A) \le L$ and in particular $\M_G(A) \le K$, showing that $A/\M_G(A)$ accounts for all nonabelian simple quotients of $A$. The group $A/\M(A)$ is by construction a subdirect product of finite simple groups. From there, it is easily seen that in fact $A/\M(A) = \Z(A/\M(A)) \times P/\M(A)$ where $P/\M(A)$ is perfect; indeed $P/\M(A)$ is a direct product of finite simple groups. Since $A/\M_G(A)$ accounts for all nonabelian finite simple quotients, we must have $\M_G(A) \cap P \le \M(A)$, so $\M_G(A)/\M(A)$ is central in $A/\M(A)$ as required. If $A/\M_G(A)$ is topologically perfect, then there are no $G$-invariant subgroups $N$ of $A$ such that $A/N$ is an abelian chief factor of $G$. Since $P$ is characteristic in $A$ and $A/P$ is abelian, it follows that $P = A$ and hence $\M_G(A) = \M(A)$.
(ii)
Let $B = \M(A)[A,G]$. Then $A/B$ is central in $G/B$ and $\M(A/B)=1$, so $A/B$ is residually an abelian simple group.
If $|A:B|$ is not prime, then $A/B$ is of the form $K_1/B \times K_2/B$ where $K_1/B$ is of prime order and $K_2/B$ is nontrivial. Thus $K_1$ and $K_2$ are proper subgroups of $A$ normalized by $G$ and $K_1K_2 = A$. If instead there is a chief factor $A/C$ of $G$ that is not central, then $[A,G] \not< C$, so $BC = A$, but both $B$ and $C$ are proper subgroups of $A$. In either case $A$ is not narrow in $G$.
Now suppose $|A:B|=p$ and that any chief factor of $G$ of the form $A/C$ is central. Let $K < A$ such that $A/K$ is a chief factor of $G$. Then $A/K$ is a direct product of simple groups, so $K \ge \M(A)$, and $A/K \le \Z(G/K)$, so $K \ge [A,G]$ and hence $K \ge B$. Since $|A:B|$ is prime, in fact we must have $K=B$. Thus every proper $G$-invariant subgroup of $A$ is contained in $B$, in other words $A \nar G$ with $\M_G(A)=B$.
(iii)
Suppose $G$ is pronilpotent. Then every chief factor of $G$ is central, so $\M_G(A) \ge [A,G]$; clearly also $\M_G(A) \ge \M(A)$, so $\M_G(A) \ge \M(A)[A,G]$. On the other hand, the quotient $V = A/\M(A)[A,G]$ is a central factor of $G$, such that every Sylow subgroup of $V$ is elementary abelian; it then follows easily that for every nontrivial element $v$ of $V$, there is some maximal subgroup $W$ of $V$ that does not contain $v$. Thus the $G$-invariant quotients of $A$ of prime order separate elements of $V$, so $\M_G(A) = \M(A)[A,G]$.
\end{proof}
We now give an inverse system characterization of a class of hereditarily just infinite groups that are not virtually prosoluble. The class obtained is less general than that given by Theorem~\ref{mainjithm:hji}, but the conditions here are easier to check and still provide some interesting examples. Note that the conditions in brackets are automatic, by the positive answer to the Schreier conjecture; Theorem~\ref{primhji} has been stated in such a way to avoid using any deep results about finite simple groups. Proposition~\ref{intro:subprim} will follow immediately given the positive answer to the Schreier conjecture.
\begin{thm}\label{primhji}Let $\Lambda = \{(G_n)_{n > 0},\rho_n: G_{n+1} \twoheadrightarrow G_n\}$ be an inverse system of finite groups. Let $A_n$ be a normal subgroup of $G_n$ and set $P_{n} = \rho_n(A_{n+1})$ for $n > 0$. Suppose the following conditions hold for sufficiently large $n$:
\begin{enumerate}[(i)]
\item $\M(A_{n+1}) = \ker\rho_{n} \ge P_{n+1}$;
\item $A_n > \CC_{G_n}(P_n)$;
\item $P_n$ is a direct product of nonabelian simple groups, such that $G_n$ permutes the simple factors of $P_n$ subprimitively by conjugation (and $\N_{G_n}(F)/F\CC_{G_n}(F)$ is soluble for each simple factor $F$ of $P_n$).\end{enumerate}
Then $G = \varprojlim G_n$ is hereditarily just infinite and not virtually prosoluble. Moreover, this construction accounts for all profinite groups $G$ for which the following is true:
$\condc$ $G$ is just infinite, and there are infinitely many nonabelian chief factors $R/S$ of $G$ such that the simple factors of $R/S$ are permuted subprimitively by conjugation in $G/S$ (and such that $\N_{G/S}(F)/F\CC_{G/S}(F)$ is soluble for each simple factor $F$ of $R/S$).\end{thm}
For the proof, we note a characterization of subprimitive actions.
\begin{lem}\label{lem:subprim}
Let $G$ be a group acting on a set $X$. Then the following are equivalent:
\begin{enumerate}[(i)]
\item $G$ acts subprimitively on $X$;
\item For every $L \unlhd K \unlhd G$, either $L$ acts trivially on $X$, or $L$ fixes no point in $X$.
\end{enumerate}
\end{lem}
\begin{proof}
Let $N$ be the kernel of the action of $G$ on $X$.
Suppose $G$ acts subprimitively on $X$; let $L \unlhd K \unlhd G$, and let $Y$ be the set of fixed points of $L$. Then $Y$ is a $K$-invariant set; thus if $Y$ is nonempty, then $K/(K \cap N)$ acts faithfully on $Y$. Since $L$ acts trivially on $Y$, we conclude that $L \le K \cap N$, so $L$ acts trivially on $X$. Thus (i) implies (ii).
Conversely, suppose that $G$ does not act subprimitively on $X$. Then there is a normal subgroup $K$ and a $K$-orbit $Y$, such that $K/(K \cap N)$ does not act faithfully on $Y$. The fixator $L$ of $Y$ in $K$ is then a subgroup of $G$ that acts nontrivially on $X$ (since $K \ge L > K \cap N$), has a fixed point in $X$ and satisfies $L \unlhd K \unlhd G$. Thus (ii) implies (i).
\end{proof}
\begin{proof}[Proof of Theorem~\ref{primhji}]Suppose $\Lambda$ is an inverse system as specified, with inverse limit $G$; let $\pi_n: G \rightarrow G_n$ be the associated projection map. By renumbering, we may assume that the given conditions hold for all $n$. Note that conditions (i) and (iii) imply that $A_n$ is perfect and $\M(A_n) = \M_{G_n}(A_n)$ for all $n > 1$. Condition (ii) ensures that $P_n$ is nontrivial; conditions (i) and (iii) then ensure that $G$ has a composition series with infinitely many nonabelian factors, that is, $G$ is infinite and not virtually prosoluble.
We claim now that $G$ is hereditarily just infinite. To prove this claim, it suffices to show (given Lemma~\ref{fijilem}) that every open normal subgroup of $G$ is just infinite, that is, given any nontrivial subgroup $H$ of $G$ such that $H \unlhd K \unlhd_o G$ for some $K$, then $H$ is open in $G$. Letting $H_n = \pi_n(H)$, there exists $n_0$ such that $H_n$ is not contained in $A_n$, for all $n \ge n_0$, so by condition (i), $H_n$ does not centralize $P_n$. This implies $H_n \cap P_n$ is a nontrivial subnormal subgroup of $G_n$, which is therefore insoluble, so $H$ cannot be prosoluble. Thus, the intersection $I$ of the terms in the derived series of $H$ is nontrivial. Clearly $I$ is also normal in $K$, so without loss of generality we may assume $H=I$, that is, $H$ is perfect. We can now draw a stronger conclusion about $H_n$ for sufficiently large $n$: since $H_n$ is perfect and not contained in $P_n\CC_{G_n}(P_n)$, the quotient $H_n/(H_n \cap P_n\CC_{G_n}(P_n))$ is insoluble. This ensures that $H_n$ acts nontrivially on the set $\Omega_n$ of simple direct factors of $P_n$. Since $H_n \unlhd \pi_n(K) \unlhd G_n$, it follows by Lemma~\ref{lem:subprim} that $H_n$ acts with no fixed points on $\Omega_n$ and hence $[H_n,P_n] = P_n$. In the case when $H$ is normal in $G$, we conclude that $P_n \le H_n$ for $n$ sufficiently large. In particular, we must have $P_n \le \pi_n(K)$ for $n$ sufficiently large. But then the fact that $H_n$ is normal in $\pi_n(K)$ ensures that $P_n$ normalizes $H_n$, so $P_n \le H_n$. It then follows that $A_{n+1} \le H_{n+1}\ker\rho_n = H_{n+1}\M_{G_{n+1}}(A_{n+1})$, so $A_{n+1} \le H_{n+1}$ by Lemma~\ref{critlem}; in particular, $\ker\rho_n \le H_{n+1}$ for $n$ sufficiently large, ensuring that $H$ is open in $G$ as desired.
Conversely, let $G$ be a profinite group satisfying $\condc$. As in previous proofs, we construct a descending chain $(K_n)$ of open normal subgroups of $G$.
Let $K_0 = G$. Suppose $K_n$ has been chosen. Then $\M(K_n) \leq_o G$ by Lemma \ref{melfin} so $\Ob_G(\M(K_n)) \le_o G$ by Theorem \ref{genob}. Given Lemma~\ref{narrowassoc}, we can find $K_{n+1} \nar G$ such that $K_{n+1} \le \Ob_G(\M(K_n))$, $K_{n+1}/\M_G(K_{n+1})$ is perfect, the simple factors of $K_{n+1}/\M_G(K_{n+1})$ are permuted subprimitively by conjugation in $G/\M_G(K_{n+1})$, and the group of outer automorphisms induced by $G$ on each simple factor is soluble; additionally $\M_G(K_{n+1})=\M(K_{n+1})$ by Proposition \ref{nonabmel}. Moreover, we have infinitely many choices for $K_{n+1}$; suppose that for all but finitely many of them, $\CC_G(K_{n+1}/\M(K_{n+1})) \not\le \M(K_n)$. Then by Theorem \ref{genob}, $D = \bigcap_K \CC_G(K/\M(K))$ would have finite index, where $K$ ranges over all possible choices for $K_{n+1}$. By Theorem~\ref{genob} again, this would imply in turn that $K\le D$ for all but finitely many choices of $K$, which is absurd given that $K/\M(K)$ is perfect and $D$ centralizes $K/\M(K)$. We can therefore ensure $\CC_G(K_{n+1}/\M(K_{n+1})) \le \M(K_n)$. Now set $G_n = G/\M(K_{n+1})$, set $A_n = K_n/\M(K_{n+1})$, set $P_n = K_{n+1}/\M(K_{n+1})$ and let the maps $\rho_n$ be the natural quotient maps $G_{n+1} \rightarrow G_n$. This produces an inverse system for $G$ with all the required properties.
\end{proof}
\begin{rem}A just infinite branch group $G$ can certainly have infinitely many nonabelian chief factors. However, given condition $\condc$, the permutation action of $G$ on the simple factors of a chief factor can only be subprimitive for finitely many of these chief factors.\end{rem}
The following shows the flexibility of the conditions in Theorem~\ref{primhji}.
\begin{ex}\label{primex}Let $X_0$ be a finite set with at least two elements, let $G_{-1}$ be a nontrivial subprimitive subgroup of $\mathrm{Sym}(X_0)$, let $S_0,S_1,\dots$ be a sequence of nonabelian finite simple groups and let $F_0,F_1,\dots$ be a sequence of nontrivial finite perfect groups. Set $G_0 = S_0 \wr_{X_0} G_{-1}$, that is, the wreath product of $S_0$ with $G_{-1}$ where the wreathing action is given by the natural action of $G_{-1}$ on $X_0$. Thereafter $H_n$ is constructed from $G_n$ as $H_n = F_n \wr_{Y_n} G_n$, where the action of $G_n$ on $Y_n$ is subprimitive and faithful, and $G_{n+1}$ is constructed from $H_{n}$ as $G_{n+1} = S_{n+1} \wr_{X_{n+1}} H_n$ where the action of $H_n$ on $X_{n+1}$ is subprimitive and faithful. Set $\rho_n:G_{n+1} \rightarrow G_n$ to be the natural quotient map from $S_{n+1} \wr_{X_{n+1}} (F_n \wr_{Y_n} G_n)$ to $G_n$ for $n \ge 0$, and let $G$ be the inverse limit arising from these homomorphisms. For $n \ge 0$ set $A_{n+1}$ to be the kernel of the natural projection of $G_{n+1}$ onto $H_{n-1}$; thus $A_{n+1}$ is a direct product of $|X_{n}|$ copies of $S_{n+1} \wr_{X_{n+1}} (F_n \wr_{Y_{n}} S_n)$. We observe that $A_n$ is perfect and $\M(A_n) = \ker\rho_{n-1}$. Moreover, $P_n$ is a normal subgroup of $G$ isomorphic to a power of $S_n$, with $G_n$ permuting the copies of $S_n$ subprimitively and $\CC_{G_n}(P_n) = \triv$. Thus $G$ is hereditarily just infinite by Theorem \ref{primhji}.
Note the following:
\begin{enumerate}[(i)]
\item Every nontrivial finite group can occur as the initial permutation group $G_{-1}$. There is also a great deal of freedom in the choice of the groups $F_n$, as there are general methods to embed a finite group as a `large' subnormal factor of a finite perfect group. For instance if $B$ is a finite group, then $B \wr A_5$ has a perfect normal subgroup of the form $K \rtimes A_5$ where $K = \{f: \{1,2,\dots,5\} \rightarrow B \mid \prod^5_{i=1}f(i) \in [B,B]\}$; clearly $B$ appears as a quotient of $K$. The groups $F_n$ could thus be chosen so that in the resulting inverse limit $G$, every finite group appears as a subnormal factor $K/L$, such that $L \unlhd K \unlhd H \unlhd G$.
\item Provided the $S_n$ or $F_n$ are chosen to be `universal' (that is, so that for every finite group $F$ there are infinitely many $n$ such that $F$ embeds into $S_n$ or $F_n$), then every countably based profinite group embeds into every open subgroup of $G$, because $G$ contains a closed subgroup $\prod_n S_n \times \prod_n F_n$ (take the `diagonal' subgroup at each level of the wreath product). This gives an alternative proof of Theorem A of \cite{Wil}.
\item The group constructed by Lucchini in \cite{Lucchini} is a special case of the construction under discussion, and indeed an example of the previous observation. This is especially interesting as Lucchini's paper predates \cite{Wil}, and appears to be the earliest example in the literature of a hereditarily just infinite profinite group that is not virtually pro-$p$. (It is proven in \cite{Lucchini} that the group is just infinite, but the hereditarily just infinite property is not considered.)
\item There are exactly $2^{\aleph_0}$ commensurability classes of groups in this family of examples. Consider for instance the set $\mc{S}$ of nonabelian finite simple groups occurring infinitely many times in a composition series for $G$. Then $\mc{S}$ only depends on $G$ up to commensurability, but can be chosen to be an arbitrary nonempty subset of the set of all isomorphism classes of nonabelian finite simple groups. On the other hand there cannot be more than $2^{\aleph_0}$ examples, since every just infinite profinite group is countably based and there are only $2^{\aleph_0}$ countably based profinite groups up to isomorphism.
\item There are interesting infinite ascending chains inside this family of examples. For instance, let $T_k$ be the group formed in the inverse limit if one replaces $G_k$ with $S_k \wr_{X_k} \mathrm{Sym}(X_{k})$ in the construction and extends from there as before. Then $G$ embeds naturally into $T_k$ as an open subgroup, and if $k' > k$ then $T_k$ embeds naturally into $T_{k'}$ as an open subgroup. Taking the direct limit of the groups $T_k$, one obtains a totally disconnected, locally compact group $T$. This group has a simple locally finite abstract subgroup $A$, formed as the union of the groups $\mathrm{Alt}(X_{k})$, where $\mathrm{Alt}(X_{k}) \le \mathrm{Sym}(X_{k}) \le T_k$. The smallest closed normal subgroup $N$ of $T$ containing $A$ will then be a topologically simple open subgroup of $T$. We thus obtain a totally disconnected, locally compact, topologically simple group $N$ such that every closed subgroup with open normalizer is open, but also (for a suitable choice of $S_n$ or $F_n$ as in (ii)) such that every identity neighbourhood in $N$ contains a copy of every countably based profinite group as a closed subgroup.
\end{enumerate}
\end{ex} | 48,578 |
To find conservations laws directly from the equations of motion, we begin by rewriting \eqref{eqn:nlnlA} and \eqref{eqn:nlnlB} as
\begin{eqnarray}
&&\dt\sint{\partial_z^{-1}\varphi} = \oint_{\partial\mathscr{D}}\phi\left(\sigma_3\grad\varphi\right)\dotn\,ds + 2\odint{\phi\varphi_{z}\,dx}\label{eqn:nlnlAA}\\
\nonumber\\
&&\dt\oint_{\partial\mathscr{D}}{\left(\phi\left(\sigma_3\grad\varphi\right)\dotn\right)\,ds}=-\oint_{\partial\mathscr{D}}{\left[(p/\rho + gz)\left(\sigma_3\grad\varphi\right)\dotn +2\phi\varphi_{zz}(\grad\phi\dotn)\right]\,ds},\label{eqn:nlnlBB}
\end{eqnarray}
where \(\partial_z^{-1}\varphi\) is the anti-derivative of \(\varphi\) with respect to \(z\) such that \(\partial_z^{-1}\varphi(x,z)\bigg\vert_{z = 0} = 0\) for all \(x\) and \(\sigma_3\) is the third Pauli matrix so that
\[
\sigma_3\grad\varphi = \begin{bmatrix}1 & 0\\0&-1\end{bmatrix}\begin{bmatrix}\varphi_{x}\\\varphi_{z} \end{bmatrix}= \begin{bmatrix}\varphi_{x}\\-\varphi_{z}\end{bmatrix}.
\]
In \eqref{eqn:nlnlAA} and \eqref{eqn:nlnlBB}, we have assumed that \(\grad\phi\dotn = 0\) on \(\Gamma\) as described in \Cref{sec:background}.
With this simplification, we now proceed to compute the eight conservation laws as computed by \cite{olver} by choosing a sequence of harmonic polynomials in the form \[\varphi_n = \frac{1}{n!}(x + iz)^n.\] As noted in \cite{olver}, there is a recursive nature where \emph{higher-order} conserved densities are found in terms of lower-order densities. With this in mind, we define the quantities \(A\) and \(B\) as
\begin{equation}
A = \sint{\partial_z^{-1}\varphi}, \qquad B = \odint{\phi(\sigma_3\grad\varphi)\dotn}\,ds.\label{eqn:ABEqns}
\end{equation}
With these definitions, Equations \eqref{eqn:nlnlAA} and \eqref{eqn:nlnlBB} can be represented in the following compact form that is conducive to back-substituting lower-order conserved densities.
\begin{eqnarray}
\frac{dB}{dt} &=& -\odint{(p/\rho + gz)(\sigma_3\grad\varphi)\dotn}\,ds - 2\sint{\phi\,\eta_t\varphi_{zz}},\label{eqn:nlnlBBB}\\
\frac{dA}{dt} &=& B + 2\odint{\phi\varphi_{z}}\,dx. \label{eqn:nlnlAAA}
\end{eqnarray}
\Cref{tab:summary} shows a summary of the conserved densities at the free surface along with the corresponding test functions. The boundary fluxes are given in the details presented below.
\subsection{\(\varphi_0 = 1\)}
Equations \eqref{eqn:nlnlBBB} and \eqref{eqn:nlnlAAA} serve as our foundation for deriving the conservation laws. As stated earlier, we assume that \(\phi\) and \(\eta\) decay rapidly enough as \(\vert x \vert \to \infty\) such that the corresponding integrals make sense. Substituting \(\varphi_0 = 1\) into the expressions for \(A\) and \(B\) yields
\[A_0 = \sint{\eta} \qquad B_0 = 0.\]
Using \eqref{eqn:nlnlBBB} yields \(0 = 0\), while substitution into \eqref{eqn:nlnlAAA} yields
\begin{equation}
\dt A_0 = \dt\sint{\eta} = 0 \qquad \Rightarrow\qquad \dt\wint{\eta} = 0. \qquad \label{eqn:T3}\tag{\(T_3\)}
\end{equation}
where we have used the same numbering convention for the conserved densities as in \cite{olver}.
\subsection{\(\varphi_1 = x + iz\)}
Substituting \(\varphi = x + iz\) into the expressions for \(A\) and \(B\) given by \eqref{eqn:ABEqns} we find
\[A_1 = \sint{\left(x\eta + \frac{i}{2}\eta^2\right)}\,dx \qquad B_1 = \odint{\phi\begin{bmatrix}1\\-i\end{bmatrix}\dotn}\,ds.\]
Using \eqref{eqn:nlnlBBB} we find
\begin{eqnarray*}
\dt B_1
&=& igA_0-i\bint{p/\rho} + g\odint{z}\,dz\\
&=& i\left(gA_0 - \bint{p/\rho}\right)\\
&=& i\dt\left(tgA_0\right)- i\bint{p/\rho} - igt\dt\left(A_0\right)
\end{eqnarray*}
As previously stated, we assume that \(\eta\to 0\) rapidly so that \(\eta^2\) vanishes in the limit as \(|x|\to\infty\). Using the fact that \(\dt A_0 = 0\) from \eqref{eqn:T3}, the real and imaginary parts yield the following conservation laws with the appropriate boundary-flux terms denoted:
\begin{equation}
\dt\odint{-\phi}\,dz = 0,\label{eqn:T1}\tag{\(T_1\)}
\end{equation}
and
\begin{equation}
\dt\odint{-\phi - tgz}\,dx = -\bint{p/\rho}.\label{eqn:T4}\tag{\(T_4\)}
\end{equation}
It is worth noting that in deriving \eqref{eqn:T1} and \eqref{eqn:T4}, the previously determined conservation law \eqref{eqn:T3} was needed. This will be a recurring theme as we move up the hierarchy; conservation laws determined from lower-order harmonic polynomials will be needed to move to higher-order harmonic polynomials.
Repeating the same process with \(A_1\), and using the integral identities generated in \eqref{eqn:nlnl:idB}, upon separating real and imaginary parts, we find
\begin{equation}
\displaystyle \dt\odint{ xz\,dx + t\phi\,dz} = 0,\label{eqn:T5}\tag{\(T_5\)}
\end{equation}
and
\begin{equation}
\dt\sint{-\frac{1}{2}\eta^2} + \dt\odint{ t \phi\,dx + \frac{1}{2}gt^2 z\,dx} = 0.\label{eqn:T6}\tag{\(T_6\)}
\end{equation}
\subsection{\(\varphi_2 = \frac{1}{2}\left(x + iz\right)^2\)}
To continue in the hierarchy, we continue the process by beginning with \(\varphi_2 = \frac{1}{2}(x + iz)^2\) substituted into \eqref{eqn:nlnlBBB}. Separating real and imaginary parts, we find
\begin{equation}
\dt\odint{\phi(z\,dx - x\,dz) -\left(4t\mathscr{H} -\frac{7}{2}gz^2 - \frac{7}{6}g^2t^3 z\right)\,dx} = \bint{hp/\rho}, \label{eqn:T7}\tag{\(T_7\)}
\end{equation}
\begin{equation}
\dt\odint{\phi(x\,dx + z\,dz) + gtxz\,dx + \frac{1}{2}gt^2\phi\,dz} = -\bint{x p/\rho },\label{eqn:T8}\tag{\(T_8\)}
\end{equation}
where we have used the fact that \[\dt \mathcal{H} = \dt\sint{\mathscr{H}} = \dt\sint{\underbrace{\frac{1}{2}\left(\phi\grad\phi\dotn + gz^2\right)}_{\mathscr{H}}} = 0.\]
\begin{remark}
Knowing the Hamiltonian at the outset does simplify the calculations. However, a priori knowledge of \(\mathscr{H}\) is not necessary. Instead, the Hamiltonian arises naturally when one considers the integral identities presented in \Cref{app:simplification1d}.
\end{remark}
\subsection{Summary of Conservation Laws}
At this point, we have computed the eight conservation laws presented in \cite{olver} for the case with no surface tension. The corresponding densities at the free surface are given in \Cref{tab:summary}. The corresponding boundary flux terms can be deduced from Equations \eqref{eqn:T1} - \eqref{eqn:T8}.
\begin{table}[H]
\hrule ~\\
\caption{Table of Conserved Densities from \cite{olver} (Thm. 6.2) along with a summary of how they were found via the weak formulation.}\label{tab:summary}
\vspace*{.05in}
\renewcommand{\arraystretch}{1.8}
\hrule
\small{
\begin{tabular}{l|l}
\(\displaystyle T_{1,s} = -\eta_x q\) & Imaginary part of \eqref{eqn:nlnlBB} with \(\varphi = \frac{1}{2}(x + iz)^2\)\\
\(\displaystyle T_{2,s} = \frac{1}{2}q\eta_t + \frac{1}{2}g\eta^2\) & Embedded in \eqref{eqn:nlnlBB} with \(\varphi = \frac{1}{6}(x+iz)^3\)\\
\(\displaystyle T_{3,s} = \eta\) & Imaginary part of \eqref{eqn:nlnlAA} with \(\varphi = x +iz\)\\
\(\displaystyle T_{4,s} = q + gt\eta\) &Real part of \eqref{eqn:nlnlBB} with \(\varphi =\frac{1}{2} (x + iz)^2\)\\
\(\displaystyle T_{5,s} = x\eta + t\eta_x q\) &Imaginary part of \eqref{eqn:nlnlAA} with \(\varphi = \frac{1}{2}(x + iz)^2\)\\
\(\displaystyle T_{6,s} = \frac{1}{2}\eta^2 - tq - \frac{1}{2}gt^2\eta\) &Real part of \eqref{eqn:nlnlAA} with \(\varphi = \frac{1}{2}(x+iz)^2\)\\
\(\displaystyle T_{7,s} = q(\eta - x\eta_x) - 4tT_2 + \frac{7}{2}gt\eta^2 - \frac{7}{2}gt^2q - \frac{7}{6}g^2t^3\eta\) &Imaginary part of \eqref{eqn:nlnlBB} with \(\varphi = \frac{1}{6}(x + iz)^3\)\\
\(\displaystyle T_{8,s} = (x + \eta\eta_x)q + gtx\eta +\frac{1}{2}t^2g \eta_x q\) &Real part of \eqref{eqn:nlnlBB} with \(\varphi = \frac{1}{6}(x + iz)^3\)\\
\end{tabular}}
\hrule
\end{table}
As in \cite{olver,olver1983conservation}, deriving \eqref{eqn:T7} requires additional manipulations. While we do not need to take specific linear combinations of separate one-forms, we do instead need to find and extract the Hamiltonian. This is aided by the observation of the term \(2\eta_t\,\phi\,\varphi_{zzz}(x,\eta)\) as a \emph{Legendre-type transform} lurking within Equation \eqref{eqn:nlnlBBB}. At this point, we have derived the 8 conservation laws for a one-dimensional free-surface on the whole-line (see \Cref{tab:summary} for details) without surface-tension. The computed values for \(T_1\) - \(T_8\) are in agreement with those presented in \cite{olver,olver1983conservation}. There is nothing that necessarily stops us from continuing the process. For example, we have yet to use the cubic test function in \eqref{eqn:nlnlAA}. However, without introducing nonlocal operators of some variety, it appears to be impossible to continue the aforementioned process as expected via the results of \cite{olver1983conservation}.
It is a rather remarkable fact that by considering the harmonic polynomials of the form \[\varphi_n = \frac{1}{n!}(x + iz)^n,\] we are able to immediately recover the various contour integrals used to prove Theorem 6.1 in \S6 of \cite{olver}. Specifically, \(I^1, I^3, I^4, \ldots, I^8\) are found via the left-hand sides of Equations \eqref{eqn:nlnlAA} - \eqref{eqn:nlnlBB}, while the closed differentials \(G^1, G^3, G^4, \ldots, G^8\) are found directly via \eqref{eqn:nlnl:idA} and \eqref{eqn:nlnl:idB} as described in \Cref{app:simplification1d}.
\subsection{Including Surface Tension}
When surface tension is included, the dynamic boundary condition \eqref{eqn:dynamic1d} is replaced by
\begin{equation}
\phi_t + \frac{1}{2}\vert\grad\phi\vert^2 + g\eta =\frac{\sigma\eta_{xx}}{\left(1 + \eta_x^2\right)^{3/2}}\label{eqn:dynamic1DST},
\end{equation}
where \(\sigma\) represents the coefficient of surface-tension. The resulting formulations still hold with the slight modification as presented below:
\begin{eqnarray}
&&\dt\odint{\partial_z^{-1}\varphi}\,dx = \oint_{\partial\mathscr{D}}\phi\grad\phi\dotn\,ds\label{eqn:nlnlAAST}\\ \nonumber\\
&&\dt\oint_{\partial\mathscr{D}}{\left(\phi\left(\sigma_3\grad\varphi\right)\dotn\right)\,ds}=-\oint_{\partial\mathscr{D}}{\left[(p/\rho + gz + S)\left(\sigma_3\grad\varphi\right)\dotn +2\phi\varphi_{zz}(\grad\phi\dotn)\right]\,ds},\label{eqn:nlnlBBST}
\end{eqnarray}
where \[S = -\dx\frac{\sigma\eta_x}{\sqrt{1+\eta_x^2}}\] represent the surface tension force acting only at the free surface \(\mathscr{S}\).
We find the same results as described in \Cref{tab:summary} with the exception of two changes. First, the resulting Hamiltonian or \(T_2\) is modified to include the additional term
\[T_2 = \frac{1}{2}q\eta_t + \frac{g}{2}\eta^2 + \sigma\left(\sqrt{1 + \eta_x^2} - 1\right).\] Second, when attempting to recover \eqref{eqn:T7}, one encounters an additional term of the form \[\dt\sint{\sigma\sqrt{1+\eta_x^2}}\neq 0.\] This results in the loss of \(T_7\) as a conserved density in the presence of surface tension and is to be expected (see \cite{olver, olver1983conservation}). Physically, it makes sense as to why \eqref{eqn:T7} is no longer conserved in the presence of surface tension as it does not make sense for the arc-length of the interface to remain constant for all time. | 181,242 |
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\begin{document}
\selectlanguage{english}
\title[Moment Polytopes of rank one]{Momentum polytopes of rank one for multiplicity free quasi-Hamiltonian manifolds}
\author{Kay Paulus}
\address{Department of Mathematics, Friedrich-Alexander-University Erlangen-Nuremberg, Cauerstr. 11, 91058 Erlangen}
\email{[email protected]}
\date{\today}
\begin{abstract}
We classify all momentum polytopes of rank one for multiplicity free quasi-Hamiltonian $K$-manifolds for simple and simply connected Lie groups $K$ by using the methods developed in \cite{Kno14}. This leads to lots of new concrete examples of multiplicity free quasi-Hamiltonian manifolds or equivalently, Hamiltonian loop group actions.\\
\end{abstract}
\maketitle
\selectlanguage{english}
\part{Introduction}
Let $K$ be a simply connected compact Lie group with Lie algebra $\mathfrak{k}$ and complexification $G$, $T_\R$ a maximal torus for $K$ and $T$ its complexification, let $\mathfrak{t}$ be the Cartan of $\mathfrak{k}$.
A quasi-Hamiltonian manifold $M$ is a concept introduced by Alekseev, Malkin and Meinrenken in \cite{AMM98} as a setting equivalent to Hamiltonian Loop group actions.\\
We choose a scalar product on $\mathfrak{k}$, let $\tau$ be an automorphism of $K$ leaving this scalar product on $\mathfrak{k}$ invariant. There are two canonical 1-forms $\theta$ and $\bar{\theta}$ on $\mathfrak{k}$ defined by $\theta(k\xi)=\xi=\bar \theta(\xi k)$ with $k\in K$ and $\xi \in \mathfrak{k}$. Then we take $\Theta_\tau=\frac 12 (\bar \theta + ^{\tau^{-1}}\theta)$ with $\Theta_\tau(k\xi)=\frac 12 (\operatorname{Ad}(k) \xi+^{\tau^{-1}}\xi)$ where $^\tau k$ denotes the operation of $\tau$ on $k\in K$. We use the scalar product on $\mathfrak{k}$ to define the canonical bi-invariant closed 3-form
\[
\chi:=\frac{1}{12}\la \theta,[\theta,\theta]\ra_{\mathfrak{k}}=\frac{1}{12}\la \bar\theta,[\bar\theta,\bar\theta]\ra_{\mathfrak{k}}
\]
We write $K\tau$ for the twisted $K$-action, i.e. $x\mapsto kx\tau(k)^{-1}$ to avoid confusions with the adjoint action of $K$.
\begin{defn}[\cite{Kno14}, Definition 2.3.]
A quasi-Hamiltonian $K\tau$-manifold is a smooth manifold $M$ equipped with a $K$-action, a 2-form $w$, and a smooth map $m:M\to K\tau$, called the (group valued) moment map which have the following properties:
\begin{enumerate}[label=\alph*)]
\item $m$ is $K$-equivariant
\item the form $w$ is $K$-invariant and satisfies $dw=-m^* \chi$
\item $w(\xi x,\eta)=\la \xi, m^*\Theta_\tau (\eta)\ra_{\mathfrak{k}}$ for all $\xi \in \mathfrak{k}, \eta\in T_x M$
\item $\operatorname{ker}w_x=\{\xi x \in T_x M \mid \xi \in \mathfrak{k}$ with $^{m(x)\tau}\xi+\xi=0\}$
\end{enumerate}
\end{defn}
We call a quasi-Hamiltonian manifold (from now on: q-Hamiltonian) {\bf multiplicity free} if its symplectic reductions are discrete, i.e. points. Being multiplicity free implies that the image of the moment map $\P_M$ is convex and locally polytopal.\\
For this paper, a ``quasi-Hamiltonian $K\tau$-manifold'' is always multiplicity free and compact. This is the setting of \cite{Kno14}. Knop showed that quasi-Hamiltonian manifolds are uniquely determined by their moment polytope $\P$ and principal isotropy group $L_S$ which can be encoded in some lattice $\L_S$. Knop also answers which pairs $(\P,\L_S)$ can arise:
Let $Z$ be a smooth affine spherical variety. We denote by $\aaa_\P\subseteq \mathfrak{t}$ the affine space spanned by $\P$. Any $a\in \P$ gives rise to an isotropy group $K(a)$ with respect to the twisted action. We denote by $C_Z$ the convex cone generated by the weight monoid of $Z$ and by $C_a\P$ the tangent cone of $\P$ in $a\in \A$.
\begin{defn}\label{sphpair}(cf. \cite{Kno14}, Definition 6.6)
Let $K$ be a simply connected compact Lie group with automorphism $\tau$ and fundamental alcove $\A$. Let $\P\subseteq \A$ be a locally convex subset and $\Lambda_S \subseteq \bar\aaa_\P$ a lattice. Then $(\P,\Lambda_S)$ is called {\bf spherical in $a\in \P$} if
\begin{enumerate}
\item $\P$ is polyhedral in $a$, i.e.
\[
P\cap U = (a+C_a\P)\cap U
\]
for a neighborhood $U$ of $a$ in $\A$, and
\item there is a smooth affine spherical $K(a)_\C$-variety $Z$ with weight monoid $\G_Z$ such that
\[
C_a\P\cap \Lambda_S = \G_Z
\]
\end{enumerate}
The pair $(\P,\Lambda_S)$ is called a {\bf spherical pair} if it is spherical for all $a\in\P$.\\
The variety $Z$ is called a {\bf local model} in $a$ if it fullfills the conditions above in $a$. By abuse of notation, we shall also call the corresponding weight monoid (cf. \cref{sphericaldata}) and the set of its generators a local model, as a smooth affine spherical variety is uniquely determined by its weight monoid (cf. \cite{Los06}).
\end{defn}
\begin{thm}(\cite{Kno14}, thm. 6.7)
Let $K$ be a a simply connected compact Lie group with twist $\tau$. Then the map $M\to (\P_M, \Lambda_M)$ furnishes a bijection between
\begin{itemize}
\item isomorphism classes of convex multiplicity free quasi-Hamiltonian $K\tau$-manifolds and
\item spherical pairs $(\P_M, \Lambda_M)$.
\end{itemize}
Under this correspondence, $M$ is compact if and only if $\P_M$ is closed in $\A$.
\end{thm}
We also recall that every Hamiltonian manifold carries a quasi-Hamiltonian structure.
\begin{defn}
We call a q-Hamiltonian {\bf genuine} if the moment polytope touches every wall of the fundamental alcove at least once. That means that genuine q-Hamiltonian manifolds are exactly those that do not carry a Hamiltonian structure.
\end{defn}
The aim of this paper is to classify all quasi-Hamiltonian manifolds of rank one for $K$ simple, genuine or not.
This paper discusses a part of my nearly-finished doctoral project conducted under the supervision of Friedrich Knop at FAU Erlangen-N\"urnberg. Therefore it clearly has substantial, partially word-by-word, overlap with my doctoral thesis \cite{Pau}.
{\bf Acknowledgment:} \\
I would like to thank Friedrich Knop, Guido Pezzini, Bart Van Steirteghem and Wolfgang Ruppert for countless explanations, discussions and remarks.
\part{Generalities}
\section{Affine root systems}
First we shall recall some well known facts about affine root systems that are needed for the rest of the paper. We follow the introduction in \cite{Kno14} that is mainly based on \cite{Mac72} and \cite{Mac03}.\\
Let $\overline{\aaa}$ be a Euclidean vector space with an associated affine space $\aaa$. We denote by $A(\aaa)$ the set of affine linear functions on $\aaa$, the gradient of $\a\in A(\aaa)$ is denoted by $\aa \in \overline{\aaa}$, and it has the property
\begin{equation}\label{afflin}
\a(x+t)=\a(x)+\la \aa, t\ra, ~ x\in \aaa, t\in \overline{\aaa}
\end{equation}
Let $O(\bar\aaa)$ be the orthogonal group of the vector space $\overline\aaa$ and $M(\aaa)$ the isometries of $\aaa$, and let $M(\aaa)\to O(\overline{\aaa}), w\mapsto \overline{w}$ be the natural projection between these two sets.
A motion $s\in M(\aaa)$ is called a reflection if its fixed point set is an affine hyperplane $H$. We can write this reflection $s_\a(x)=x-\a(x)\aa^\vee$ with the usual convention $\aa^\vee=\frac{2\aa}{||\aa||^2}$. Its action on an affine linear function $\beta\in A(\aaa)$ is:
\[
s_\a(\beta)=\beta-\la\overline{\beta}, \aa^\vee\ra \a
\]
\begin{defn}
An affine root system on an affine space $\aaa$ is a subset $\Phi\subset A(\aaa)$ such that:
\begin{enumerate}
\item $\R1\cap \Phi=\emptyset$
\item $s_\a(\Phi)=\Phi$ for all $\a\in\Phi$
\item $\la \bar \beta, \bar \alpha ^\vee \ra \in \Z$ for all $\a,\beta \in \Phi$
\item the Weyl Group $W_\Phi:=\la s_\a,\a\in\Phi\ra\subset M(\aaa)$ is an Euclidean reflection group, which means in particular that $W$ acts properly discontinuously.
\item The affine root system $\Phi$ is called reduced if $\R\a\cap\Phi=\{+\a, -\a\}$ for all $\a\in\Phi$
\end{enumerate}
\end{defn}
Note that we do not ask $\Phi$ to generate the affine space. Also, all finite (classical) root systems are affine in this definition.
\begin{defn}
The set of non-decomposable roots of $\Phi$ (with respect to one fixed alcove) is called the set of {\bf simple roots}. We denote the set of all simple roots by $S$.
A root is called positive if it is a linear combination of simple roots with non-negative coefficients.
\end{defn}
We shall name the affine root systems and order the simple roots just as in Tables Aff 1 to Aff 3 in \cite{Kac90} resp. \cite{Bou81}.
\begin{defn}[\cite{Kno14}, 3.3]
Let $\Phi\subset A(\aaa)$ be an affine root system.
\begin{enumerate}
\item A {\bf weight lattice} for $\Phi$ is a lattice $\Lambda \subseteq A(\aaa)$ with $\bar\Phi\subset \Lambda$ and $\bar\Phi^\vee \subseteq \Lambda^\vee$ where $\Lambda^\vee = \{t \in \bar\aaa \mid \la t, \Lambda \ra \subseteq \Z\}$ is the dual lattice for $\Phi$.
\item An {\bf integral root system} is a pair $(\Phi, \Lambda)$ where $\Phi\subset A(\aaa)$ is an affine root system and $\Lambda \subseteq \bar\aaa$ is a weight lattice for $\Phi$.
\end{enumerate}
\end{defn}
The connected components of the complement of the union of all reflection hyperplanes in $\aaa$ are called the {\bf chambers} of $\Phi$ (or $W$). The closure $\bar{\mathcal{A}}$ of a single chamber is called an {\bf alcove}.\\
There is a well known theory of Euclidean reflection groups stating that $W$ acts simply transitively on the set of alcoves, that each alcove is a fundamental domain for the action of $W$ and that the group $W$ is generated by the finitely many reflections about the faces of codimension one of any alcove.
\section{Basic combinatorial invariants of spherical varieties}\label{sphericaldata}
In this section, we recall some basic facts, notations and invariants from the (combinatorial) theory of spherical varieties. We particularly focus on smooth affine spherical varieties of rank one and try to avoid digging into the depth of the general theory as much as possible to keep the paper accessible to a broad range of readers. For a more detailed introduction, we refer to \cite{Lun01}. The author also recommends \cite{Pez10} as a general introduction to spherical varieties and \cite{BL10} as an introduction to the combinatorial theory of spherical varieties containing numerous instructive examples.
Let $B$ be a Borel subgroup of $G$. Remember that an irreducible $G$-variety is called {\bf spherical} if it is normal and has an open $B$-orbit. A closed subgroup $H$ is called spherical if $G/H$ is spherical.
The {\bf weight monoid} $\G(Z)$ of a complex affine algebraic $G$-variety $Z$ is the set of isomorphism classes of irreducible representations of $G$ in its coordinate ring.
\begin{defn}\label{combin}
Let $Z$ be a spherical $G$-variety with open orbit $G/H$. The basic invariants the theory of spherical varieties uses about $Z$ are:
\begin{enumerate}
\item The {\bf (weight) lattice} of $Z$, called $\Lambda(Z)$, is the subgroup of the character group of the Borel $B$ (that can be identified with the character group of the maximal torus $T$) consisting of the $B$-weights of $B$-eigenvectors in the field of rational functions $\C(Z)$.
\item The rank of $\Lambda(Z)$ is called the {\bf rank} of $Z$.
\item Let $\G$ be a set of dominant weights of $G$. The set of {\bf simple roots orthogonal to $\G$} is
\[
S^p(\G):=\{\a\in S: \la \lambda, \alpha^\vee \ra =0 \forall \lambda \in \G\}
\]
\item By \cite{Bri90}, the set of $G$-invariant $\Q$-valued discrete valuations of $\C(Z)$, called $\V(Z)$, is a co-simplicial cone. The set of {\bf spherical roots} $\S(Z)$ of $Z$ is the minimal set of primitive elements of $\Lambda(Z)$ such that
\[
\V(Z)=\{\eta\in\Hom_\Z(\Lambda(Z),\Q)\mid\la \eta, \sigma \ra \le 0 \forall \sigma \in \S(Z)\}
\]
\end{enumerate}
\end{defn}
Note that these invariants only depend on the open $G$-orbit $G/H$ of $Z$.\\
Let us recall the list of spherical roots of smooth affine spherical varieties that is a subset of Akhiezer's classification of smooth spherical varieties of rank one in \cite{Akh83}. They are known to be exactly the weights of homogeneous smooth affine spherical varieties of rank one.\\
We also give the so-called Luna Diagrams of the spherical roots, see e.g. \cite{BL10} for a closer explanation of this diagram notation. Note that the factors in [] apply to the whole sums and are optional, meaning that the spherical roots with and without this factor exist.
\begin{lis}\label{lunad}{Diagrams of spherical roots}
\begin{center}
\begin{longtable}{p{6cm}p{6cm}}
Spherical root & Diagram\\\endhead
\hline
$\a_1$ & $\begin{picture}(-1800,2400)(0,0) \put(0,0){\usebox{\aone}}\end{picture}$\\
$2\a_1$& $\begin{picture}(-1800,2400)(0,0) \put(0,0){\usebox{\aprime}}\end{picture}$\\
$[\frac 12]\a+\a'$ & $\begin{picture}(-1800,2400)(0,-900) \put(0,0){\usebox{\vertex}}
\put (2700,0){\usebox \vertex}
\multiput(0,0)(2700,0){2}{\usebox\wcircle}
\multiput(0,-300)(2700,0){2}{\line(0,-1){600}}
\put(0,-900){\line(1,0){2700}}
\put(900,-1800){\tiny $[\nicefrac12]$}
\end{picture}$\\
$\a_1+\dots+\a_r$ & $\begin{picture}(-1800,2400)(0,0) \put(0,0){\usebox{\mediumam}}\end{picture}$\\
$[\frac12]\a_1+2\a_2+\a_3$ & $\begin{picture}(-1800,2400)(0,0) \put(0,0){\usebox{\dthree}}
\put (1200,900){\tiny$[\nicefrac12]$}\end{picture}$\\
$\a_1+\dots+\a_r$ & $\begin{picture}(-1800,2400)(0,0) \put(0,0){\usebox{\shortbm}}\end{picture}$\\
$2\a_1+\dots+2\a_r$& $\begin{picture}(-1800,2400)(0,0) \put(0,0){\usebox{\shortbprimem}}\end{picture}$\\
$[\frac 12]\a_1+2\a_2+3\a_3$ & $\begin{picture}(-1800,2400)(0,0) \put(0,0){\usebox{\bthirdthree}}\put (2700,900){\tiny$[\nicefrac12]$}\end{picture}$\\
$\a_1+2\a_2+\dots+2\a_{n-1}+\a_r$ & $\begin{picture}(-1800,2400)(0,0) \put(0,0){\usebox{\shortcm}}\end{picture}$\\
$[\frac12]2\a_1+\dots+2\a_{r-2}+\a_{r-1}+\a_r$ & $\begin{picture}(-1800,2400)(0,0) \put(0,0){\usebox{\shortdm}}\put (-600,900){\tiny$[\nicefrac12]$}\end{picture}$\\
$\a_1+2\a_2+3\a_3+2\a_4$ & $\begin{picture}(-1800,2400)(0,0)\put(0,0){\usebox{\ffour}}\end{picture}$\\
$[2]2\a_1+\a_2$ & $\begin{picture}(-1800,2400)(0,0) \put(0,0){\usebox{\gtwo}}\put(-300,600){\tiny [2]}\end{picture}$\\
\end{longtable}
\end{center}
\end{lis}
\begin{defn}
We will use the abbreviation $\aa_{i,j}$ for the sum $\aa_i+\aa_{i+1}+\dots+\aa_j$.
\end{defn}
\section{Local models and twisted conjugacy classes}
Every twisted q-Hamiltonian manifold locally looks like a Hamiltonian one (cf. \cite{Kno14}, part 5) and Sjamaar \cite{Sja98} showed that smooth affine spherical varieties are local models for (multiplicity-free) Hamiltonian manifolds. Their general structure is:
\begin{thm}[cf. \cite{KVS05}, cor. 2.2]
Let $Z$ be a smooth affine spherical $G$-variety. Then $Z\cong G\times ^H V$ where $H$ is a reductive subgroup of $G$ such that $G/H$ is spherical and $V$ is a spherical $H$-module.
\end{thm}
A smooth affine spherical variety corresponds to a triple $(G,H,V)$. Hence a complete classification of smooth affine spherical varieties would require giving all these triples. We refer to \cite{KVS05}, Examples 2.3 - 2.6 as an illustration of which problems appear. The actual objects of this classification are:
\begin{defn}[\cite{KVS05}, Def. 2.7.]\leavevmode\\
\begin{enumerate}
\item Let $\hh\subseteq \gg$ be semisimple Lie algebras and let $V$ be a representation of $\hh$. For $\mathfrak{s}$, a Cartan subalgebra of the centralizer $\mathfrak{c_g(h)}$ of $\hh$, put $\bar\hh:=\hh \oplus \mathfrak{s}$, a maximal central extension of $\hh$ in $\gg$. Let $\mathfrak{z}$ be a Cartan subalgebra of $\gll(V)^\hh$ (the centralizer of $\hh$ in $\gll(V)$). We call $(\gg,hh,V)$ a {\bf spherical triple} if there exists a Borel subalgebra $\mathfrak{b}$ of $\gg$ and a vector $v\in V$ such that:
\begin{enumerate}
\item $\mathfrak{b}+\bar \hh =\gg$ and
\item $[(\mathfrak{b}\cap \bar \hh + \mathfrak{z}]v=V$ where $\mathfrak{s}$ acts as via any homomorphism $\mathfrak{s}\to \mathfrak{z}$ on $V$.
\end{enumerate}
\item Two triples $(\gg_i, \hh_i, \V_i), i=1,2$ are {\em isomorphic} if there exist linear bijections $\a:\gg_1\to \gg_2$ and $\beta:V_1\to V_2$ such that:
\begin{enumerate}
\item $\a$ is a Lie algebra homomorphism
\item $\a(\hh_1)=\hh_2$
\item $\b(\xi v)=\alpha(\xi)\beta(v)$ for all $\xi \in \hh_1$ and $b\in V_1$.
\end{enumerate}
\item Triples of the form $(\gg_1\oplus \gg_2, \hh_1\oplus \hh_2, V_1\oplus V_2)$ with $(\gg_i, \hh_i, V_i)\ne(0,0,0)$ are called {\em decomposable}.
\item Triples of the form $(\mathfrak{k},\mathfrak{k}, 0)$ and $(0,0,V)$ are said to be {\em trivial}.
\item A pair $(\gg,\hh)$ of semisimple Lie algebras is called {\em spherical} if $(\gg,\hh, 0)$ is a spherical triple.
\item A spherical triple (or pair) is {\em primitive} if it is non-trivial an indecomposable.
\end{enumerate}
\end{defn}
Let us particularly recall that a summand of the form $(\gg, \hh, 0)$ corresponds to a homogeneous variety and a factor $(\hh,\hh,V)$ corresponds to a spherical module.
\begin{thm}[\cite{KVS05}, thm. 2.9.]
If $G\times^H V$ is a smooth affine spherical variety then $(\gg', \hh', V)$ is a spherical triple. Moreover, it follows from the classification in \cite{KVS05} that every spherical triple arises this way.
\end{thm}
We shall add the spherical triples of the local models in \cref{ListHom} and \cref{ham1}.
The local models are closely related to the local root systems from \cite{Kno14}. We shall give a brief overview how they are constructed.
Let us fix some notation: We have a simply connected compact Lie group $K$ that operates on a smooth manifold $M$ by twisted conjugation. Let $\mathfrak{t}\subset \mathfrak{k}$ be a Cartan subalgebra and $\Phi_K$ the corresponding root system. Let us remember some {\bf facts about twisted conjugacy classes} from \cite{Kno14}:
\begin{thm}\label{levi}[cf. \cite{Kno14},thm. 4.2.]
Let $K$ be a simply connected compact Lie group and $\tau$ an automorphism of $K$. Then there is a $\tau$-stable maximal torus $T \subseteq K$ and an integral affine root system $(\Phi_\tau, \Lambda_\tau)$ on $\aaa=\mathfrak{t}^\tau$, the $\tau$-fixed part of $\operatorname{Lie} T$, such that the following holds:
\begin{enumerate}[label=\alph*)]
\item Let $\operatorname{pr}^\tau:\mathfrak{t}\to \aaa$ be the orthogonal projection. Then $\bar\Phi_\tau=\operatorname{pr}^\tau\Phi(\mathfrak{k,t})$ and $\Lambda_\tau=\operatorname{pr}^\tau\Xi(T)$. Moreover, $\Lambda_\tau$ is also the weight lattice (dual of the coroot lattice) of $\bar\Phi_\tau$.
\item For any alcove $\A\subseteq \aaa$ of $\Phi_\tau$ the composition
\[
c:\A\hookrightarrow\aaa \overset{exp}{\to} K \to K\tau/K
\]
is a homeomorphism.
\item For $a\in\A$ let $u:=\exp a\in K$. Then the twisted centralizer
\[
K(a):=K_u=\{ k \in K: ku^\tau k^{-1}=u\}=K^{\bar\tau} \qquad \text{ where } \bar\tau:=\operatorname{Ad}(u)\circ\tau
\]
is a connected subgroup of $K$ with maximal torus $S:=\exp \aaa=(T^\tau)^0$. Its root datum is $(\bar\Phi_\tau(a), \Lambda_\tau)$ where $\Phi_\tau(a):=\{\a\in\Phi:\alpha(a)=0\}$.
\end{enumerate}
\end{thm}
\begin{defn}
We denote $S(a)=\{\a\in S: \a(a)=0\}$ for $a\in \bar\A$
\end{defn}
The alcove and the momentum image of a q-Hamiltonian manifold $M$ have the following relation: As $\bar\A$ is a fundamental domain for the Weyl group $W^\tau$, the map $\bar\A \to \mathfrak{t}/W^\tau$ is a homeomorphism. Having a moment map $m:M \to K$, we can introduce the invariant moment map $m_+:M\to\bar\A$ with the property that the it makes the following diagram commute:
\[
\begin{xy}
\xymatrix{
M \ar[r]^m \ar[d]_{m_+} & K \ar@{->>}[d] \\
\bar\A \ar[r]_{\text{bij}} & K/_\tau K
}
\end{xy}
\]
The bijection between the alcove and the set of twisted conjugacy classes was already known before, cf. \cite{MW04}. The image of $m_+$, called $\P_M=m_+(M)\subseteq \bar\A$ determines the actual image of $m$ via $m(M)=K\cdot \exp(\P_M)$. It follows that $\P_M$ is a convex polytope lying in $ \bar\A$ and all fibers of $m_+$ (and $m$) are connected (\cite{Kno14}, 4.4).
Another important definition is the one of a local root system: Here, we use $\Phi_X:=\{\a\in\Phi:\a(X)=0\}$ for $X\in\P$.
\begin{defn}[\cite{Kno14}, definition 7.2.]
A local system of roots (or local root system) $\Phi(*)$ on $\P$ is a family $(\Phi(X))_{X\in\P}$ of root systems on $\aaa$ such that for each $X\in\P$:
\begin{enumerate}
\item $\Phi_Y=\Phi(X)_Y$ for all $Y$ in a sufficiently small neighborhood of $X$ in $\P$.
\item Every root $\a\in\Phi(X)$ is either non-negative or non-positive on $\P$.
\end{enumerate}
An {\bf integral local root system on $\P$} is a pair $(\Phi(*), \Lambda)$ such that $(\Phi(x),\L)$ is an integral root system for every $x\in\P$.
\end{defn}
\begin{rem}\leavevmode\\
We observe that setting $X=Y$ in the definition leads to $\Phi(X)=\{\a\in \Phi: \a(X)=0\}$.
\end{rem}
\part{Genuine Quasi-Hamiltonian Manifolds of Rank one}
This part is devoted to the classification of all compact genuine quasi-Hamiltonian $K$-manifolds for a simple and simply connected Lie group $K$ such that the corresponding spherical pair $(\P,\L_S)$ (ref. \cref{sphpair}) has the following properties:
\begin{itemize}
\item The moment polytope $\P$ is a line segment $[X_1X_2], X_i\in \bar\A \setminus \A$ that touches every wall of the fundamental alcove $\A$.
\item The lattice $\L_S$ is a rank-one-lattice and the generator $\w$ spans $P$ as a line.
\end{itemize}
For the rest of this part, we assume our polytopes and local models to have this property.
\begin{prop}
Let $\N\w$ be the weight monoid of a local model for a quasi-Hamiltonian manifold $M$ of rank one in $X$. Then the moment polytope can be written as $\{X+\R\w\}\cap \A$.
\end{prop}
\begin{proof}
The cone spanned by $\w$ is the cone of the corresponding spherical variety via \cref{sphpair}. It is the line segment defined by $X+\R\w\cap \A$.
\end{proof}
\section{Local models of rank one}\label{rank1model}
For rank reasons, it follows that for smooth affine spherical varieties $Z$ of rank one, only the two extreme cases for $Z:=G\times^H V$ can show up:
\begin{lem}\label{rank1var}
A smooth affine spherical $G$-variety $Z$ of rank one is of exactly one of the following types:
\begin{enumerate}
\item $Z=G/H, V=0$. We will call this the ``homogeneous case'', as it means that $Z$ is a homogeneous spherical variety.
\item $Z=V=\C^n, H=G=\SL(n,\C) \text{ or } V=\C^{2n}, H=G=\SP(2n, \C)$. We call this the ``inhomogeneous case''.
\end{enumerate}
\end{lem}
\begin{proof}
As the homogeneous space $G/H$ is the image of $Z=G\times^H V$ under the projection $Z\to G/H$, the rank of the homogeneous space $G/H$ is at most the rank of $Z$, so either $0$ or $1$.
If it is $0$, then $G/H$ is projective by \cite{Tim11}, Proposition 10.1, but, being also affine, it is a single point, i.e. $G=H$. We deduce $Z=V$, i.e. $Z$ is a spherical module of rank $1$. By the classification of spherical modules (cf. \cite{Kno98}), the module $V$ and the group $G$ must be of the form in statement (2) of the lemma.
Supose now $G/H$ has rank $1$, that means $Z$ and $G/H$ have the same rank. Let $K(a)\subseteq G$ be the stabilizer of a point $a$ in the open $G$-orbit of $Z$ such that $K\subseteq H$. By \cite{Gan10}, Lemma 2.4, the quotient $K/H$ is finite. This implies that the projection $Z\to G/H$ has finite fibers, i.e. $V$ contains only finitely many points. Hence $V=0$ and $Z$ is of the form in statement (1) of the lemma.
\end{proof}
Inhomogeneous smooth affine spherical varieties are are only possible for type $A$ or $C$.
\begin{lem}\label{inh1}
The generators of the weight monoids of smooth affine spherical homogeneous varieties are the $\w$ with the following values paired with the corresponding coroots of the drawn simple roots. We use the following diagrams for them:
\begin{center}
\begin{tabular}{cc}
Value on simple roots & diagram\\
$\begin{picture}(12000,1000)
\put(0,0){\usebox\dynkinathree}
\put(3600,0){\usebox\susp}
\put(7200,0){\usebox\dynkinathree}
\put(0,300){\tiny $1$}
\multiput(1800,300)(1800,0){2}{\tiny $0$}
\multiput(7200,300)(1800,0){3}{\tiny $0$}
\end{picture}$&$\begin{picture}(12000,1000)
\put(0,0){\usebox\dynkinathree}
\put(3600,0){\usebox\susp}
\put(7200,0){\usebox\dynkinathree}
\put(-350,-300){$\btr$}
\end{picture}$\\
$\begin{picture}(12000,1000)
\put(0,0){\usebox\dynkinathree}
\put(3600,0){\usebox\susp}
\put(7200,0){\usebox\dynkincthree}
\put(0,300){\tiny $1$}
\multiput(1800,300)(1800,0){2}{\tiny $0$}
\multiput(7200,300)(1800,0){3}{\tiny $0$}
\end{picture}$&
$\begin{picture}(12000,1000)
\put(0,0){\usebox\dynkinathree}
\put(3600,0){\usebox\susp}
\put(7200,0){\usebox\dynkincthree}
\put(-350,-300){$\btr$}
\end{picture}$
\end{tabular}
\end{center}
\end{lem}
\begin{proof}
We already saw that inhomogeneous smooth affine spherical varieties of rank one are spherical modules. The classification of spherical modules in \cite{Kno98} gives the desired result.
\end{proof}
Note that there is a certain analogy to Luna's diagrams. Again, the simple roots in the support of $\w$ whose coroots pair zero with $\w$ are the ones in the diagram that are not marked. In that sense, the triangles in this diagrams have a similar meaning to the circles in Luna diagrams.
\begin{defn}
Let $X\in\bar\A$ and suppose $\Phi(X):=\{S\setminus\{\a_{n_1},\a_{n_2},\dots,\a_{n_m}\}\}:=S_{n_1, \dots, n_m}$ We denote the {\bf set of all homogeneous local models in $X$} by
\[
H(X):=H(n_1, \dots, n_m)
\]
For example, by stating $\w\in H(0)$ we say that $\w$ is (the weight of) a homogeneous local model for $\Phi(X)=S \setminus \{\a_0\}$.\\
We also can define the {\bf set of inhomogeneous local models $w$ fulfilling $\la w, \aa_k^\vee \ra =1 $ in $X$} as
\[
I_k(X):=I_k(n_1, \dots, n_m)
\]
\end{defn}
So, for example, $\w\in I_1(0)$ means that $\w$ is an inhomogeneous local model on $\Phi(X):= S \setminus \{\a_0\}$ and that $\w(\a_1):=\la \w, \aa_1^\vee\ra=1$. (cf. \cref{inh1})
Now let us look at some examples of moment polytopes of rank one.
Let us assume that $\w$ is the weight of a {\bf homogeneous smooth affine spherical variety} of rank one in $X_1$ where $\P=[X_1X_2]$ is the moment polytope of a q-Hamiltonian manifold. Then $\w$ and the local model $\w^\sharp$ in $X_2$ must generate the same lattice, hence $\w^\sharp=\w-t\delta$ for some $t\in \R$ and $\delta$ the root given by the Dynkin labels. We call that the case of a ``bi-homogeneous'' manifold (or momentum polytope).
\begin{example}
We fix the affine root system $A_n^{(1)}$. We take $X_1=(0,0,0,\dots,0)$, the local root system is $S(X_1)=S\setminus\{\alpha_0\}$. We think about the local model given by $\w=\aa_1+\aa_2+\dots+\aa_n=(1,0,\dots, 0, -1)$ which corresponds to the variety $\nicefrac{\SL(n+1)}{\GL(n)}$. We want our polytope to touch every wall of the alcove, hence we demand $\a_0 \in S(X_2)$. Then the other point where the moment polytope hits the walls of the alcove is $X_2=(1,0,\dots,0,-1)$ with local root system $\Phi\setminus\{\a_1, \a_n\}$. Because $\w^\sharp:=\w-\delta=-\aa_1-\aa_2-\dots-\aa_n=\aa_0$ is a spherical root of rank one in this local root system corresponding to $\nicefrac{\SL(2)}{T}$, we have found a bi-homogeneous variety. (Technically, the acting group is $\SL(2)\times \SL(n)$ with a trivial operation of the second factor. We will always leave out components that act trivially.)
\end{example}
\begin{example}
We look at $A_n^{(1)}$ and again, $X_1=(0,0,0,\dots,0)$ with local root system $\Phi(X_1)=S\setminus\{\alpha_0\}$. The first fundamental weight $\w_1$ of $A_n$ is an inhomogeneous local model here, corresponding to $\SL(n+1)\times^{\SL(n+1)} \C^{n+1}$. The local root system of $X_2$ is $S\setminus\{\a_1\}$ (we shall have a general theorem for that later: \cref{inthm}) which also permits an inhomogeneous local model. We verify by elementary calculations that $-\w$ is an inhomogeneous local model there.
\end{example}
\begin{example}
We consider $\operatorname{G}_2^{(1)}$. Then $\w=\aa_2+2\aa_1$ is a homogeneous model for $X_1=(0,0,0)$, the corresponding variety is $\nicefrac{\operatorname{G}_2}{\SL(3)}$. Hence $ S(X_2)=\{\a_0,\a_2\}$, and $-\w$ is an inhomogeneous model corresponding to $\SL(3)\times^{\SL(3)} \C^3$ there.
\end{example}
These three examples lead to the following definition:
\begin{defn}
Let the line segment $[X_1X_2]=(X_1+\R\w)\cap\bar\A$ be the moment polytope of a quasi-Hamiltonian manifold.
\begin{enumerate}
\item We call this moment polytope {\bf bi-homogeneous} if $\w$ is the generator of the weight monoid of a homogeneous smooth affine spherical variety in $X_1$ and $\w^\sharp :=t\d-\w$ for $t \in \R$ is the generator of the weight monoid of a homogeneous smooth affine spherical variety in $X_2$.
\item We call this moment polytope {\bf bi-inhomogeneous} if $\w$ is the generator of the weight monoid of an inhomogeneous smooth affine spherical variety in $X_1$ and $-\w$ is the generator of the weight monoid of an inhomogeneous smooth affine spherical variety in $X_2$.
\item We call this moment polytope {\bf mixed} if $\w$ is the generator of the weight monoid of a homogeneous smooth affine spherical variety in $X_1$ and $-\w$ is the generator of the weight monoid of an inhomogeneous smooth affine spherical variety in $X_2$.
\end{enumerate}
\end{defn}
Of course, part 3 of this definition is symmetric in switching $X_1$ and $X_2$.
\section{Structure of bi-homogeneous polytopes}
We first think about the structure of bi-homogeneous moment polytopes. Recall that $S(X_1):=\{\a\in S: \a(X_1)=0\}$. We want to prove:
\begin{thm}\label{homthm}
For all bi-homogeneous moment polytopes of rank one, $S(X_1)$ and $ S(X_2)$ contain at least $n-1$ roots.
\end{thm}
The rest of this section is to prove this statement.
\begin{defn}\leavevmode\\
\begin{enumerate}
\item Two simple roots are called neighbors if they are connected in the affine Dynkin diagram.
\item We call a simple root {\em at the end} of a diagram if it is only connected to one other root.
\item If we say ``left'' or ``right'', we always mean left or right in the corresponding affine Dynkin diagram in \cite{Kac90}.
\item We call the root given by the Dynkin labels of an affine root system $\delta$, e.g. for $A_n^{(1)}$, we have $\d=\sum_{i=0}^n \a_i$.
\item We say a simple root $\a_i$ is supported in $\w$ if its coefficient in $\w=\sum k_i \a_i$ is nonzero.
\end{enumerate}
\end{defn}
We remember that we asked our moment polytopes to ``touch every wall of the alcove'', which means every simple root must be included in $S(X_1)$ or $ S(X_2)$.
\begin{lem}\label{trick}
Let $\alpha$ be a simple root, $\w$ a dominant weight of a smooth affine spherical variety of rank one that is a local model in $X_1$. Suppose $X_2=X_1+c\cdot \w$ for some $c \in \R$. Then:
\begin{enumerate}
\item Let $\a \in \Phi(X_1)$ and $\a\in S^p(\w)$. Then $\a\in\Phi(X_2)$.
\item Let $\a \in \Phi(X_1)$ and $\alpha \notin S^p(\w)$. Then $\a \notin \Phi(X_2)$.
\item Let $\a \notin \Phi(X_1)$ and $\alpha \in S^p(\w)$. Then $\a \notin \Phi(X_2)$.
\item Let $\a \notin \Phi(X_1)$ and $\alpha \notin S^p(\w)$. Then $\a\in\Phi(X_2)$ if and only if $c=-\frac{\a(X_1)}{\la \w, \a^\vee\ra}$
\end{enumerate}
\end{lem}
\begin{proof}\leavevmode \\
We use the arithmetic of affine combinations, namely equation \eqref{afflin}.
\begin{enumerate}
\item Remember that $\a\in S^p(\w)$ means that $\la \w, \alpha ^\vee\ra=0$. Having $\a \in \Phi(X_1)$ means $\alpha(X_1)=0.$ Hence
$
\alpha(X_2)=\underbrace{\alpha(X_1)}_{=0}+c \cdot \underbrace {\la \w, \alpha^\vee\ra}_{=0}=0,
$
so $\alpha$ is in $\Phi(X_2)$.
\item
In this case, we have
$
\a(X_2)=\underbrace{\alpha(X_1)}_{=0}+c\cdot\underbrace{\la \w, \alpha^\vee\ra}_{\ne 0}\ne 0
$,
so $\a$ is not in $\Phi(X_2)$.
\item In this case, we have
$
\a(X_2)=\underbrace{\alpha(X_1)}_{\ne 0}+c\cdot\underbrace{\la \w, \alpha^\vee\ra}_{=0}\ne 0
$,
so $\a$ is not in $\Phi(X_2)$.
\item
In this case, we have
\[
\a(X_2)=\underbrace{\alpha(X_1)}_{\ne 0}+c\cdot\underbrace{\la \w, \alpha^\vee\ra}_{\ne 0}
\]
We have $\a \in \Phi(X_2)$ if and only if this equation is zero. Hence $c=-\frac{\a(X_1)}{\la \w, \a^\vee\ra}$.
\end{enumerate}
\end{proof}
\begin{rem}
This means in particular: If $S(X_1)=S\setminus \{\a\}$, meaning there is exactly one simple root not in $S(X_1)$, we can always realize $\a\in S(X_2)$.
\end{rem}
Let us now discuss what $t\in\R$ can appear in $\w^\sharp=t\delta-\w$.
As spherical roots are linear combinations of simple roots and the coefficients in these linear combinations are in $\N[\frac12]$ and $\le 3$, $t$ could only be chosen such that all coefficients of simple roots in $\delta$ are in $\N[\frac12]$ and no coefficient of $t\delta$ is bigger then twice the biggest coefficient of a spherical root possible in any subsystem of the affine root system (e.g. 6 in root systems with subsystems of type $B_3$ or $F$). \\
\begin{lem}\label{homlem1}
Let $\A^0$ be the fundamental alcove of an affine root system, and $\w\in H(X_1)$. Then the roots not supported in $\w$ must all be contained in one component of connection of $ S(X_2)$, or they must form $A_1\times A_1$ in $ S(X_2)$.
\end{lem}
\begin{proof}
This is because the weights of homogeneous smooth affine spherical varieties are exactly the spherical roots of rank one. Inspecting \cref{lunad} gives the desired result.
\end{proof}
We shall use this to prove the following:
\begin{thm}
Let $\Phi$ be an affine root system not of type $A_n^{(1)}$, $E_i^{(1)}, i\in \{6,7,8\}$. If $|S(X_1)|<n-1$, no bi-homogeneous polytopes of rank one are possible.
\end{thm}
\begin{proof}
We first state that for $D_4^{(3)}$ and $G_2^{(1)}$, the theorem is trivially true. So let us assume the local root system contains more than three simple roots.
\begin{rem}
The arguments in the following proof do not see the length of the several simple roots involved. So we only use the corresponding simply laced diagrams for instructive drawings. We mark the simple roots not in $(X_1)$ with a cross above.
\end{rem}
We assume that three roots are missing in $S(X_1)$. The following things can happen:
\begin{itemize}
\item The three missing roots are neighbors. In root systems that do not have an ending of type $D$, the middle root of the missing roots is always in $S^p$ for any spherical root we could choose, so it is never in $ S(X_2)$.\\
If the local root system in $X_1$ had an ending of type $D$ at the (without loss of generality) left end and we had chosen $\{\a_{2}, \a_{1}, \a_0\}\notin S(X_1)$, then $\a_{1}$ and $\a_0 \in S^p\forall w$ on $S(X_1)$ and hence not in $ S(X_2)$ by \cref{trick}.
If we had chosen $X_1$ such that $\{\a_{3},\a_{2},\a_{0}\}\notin S(X_1)$, we had $\a_0\in S^p$ for every $\w$ we could choose and hence $\a_0\notin S(X_2)$.
\item Two of the missing roots are neighbors at the, without loss of generality, right end of the root system, meaning $\a_{n-1}, \a_n$ for ending $B$ and $C$ and w.l.o.g $\a_{n-2}, \a_n$ for type $D$. For every $\w$ we could choose, $\a_n \notin S(X_2)$ by \cref{trick}.
\item Two are neighbors,but not at the end of $S(X_1)$:
First assume the missing roots $\a_{n-2}, \a_{n-1}$ are surrounded by $A_1\times A_1$:
\[
\begin{picture}(2400,1800)(-300,-900)
\put(-1800,0){\dots}
\put(0,0){\usebox\dynkinafour}
\put(5400,0){\usebox\dynkinatwo}
\put(-600,600){$\times$}
\put(3000,600){$\times$}
\put(2500,-1200){\tiny $n-2$}
\put(4800, -1200){\tiny $n-1$}
\put(4800,600){$\times$}
\end{picture}
\]
Then $\aa_{n-4},\aa_{n-2},\aa_{n-1}$ needed to be supported in $\d-t\w$ and are neither connected nor $A_1\times A_1$, as $\aa_{n-3}\notin S(X_2)$, contradiction.\\
Assume now the two missing roots $\aa_{d},\aa_{d+1}$ are neighbored by something that is not $A_1\times A_1$. For a spherical root $\w$ supported on the component left of $\a_d$ we get $\a_{d+1}\in S^p$, so $\a_{d+1}$ will never be in $ S(X_2)$ by \cref{trick}, same the component right of $\a_{d+1}$ and $\a_d$.
\[
\begin{picture}(2400,1800)(-300,-900)
\put(0,0){\usebox\dynkinafour}
\put(5400,0){\usebox\dynkinatwo}
\put(-1800,0){$\dots$}
\put(8100,0){$\dots$}
\put(3000,600){$\times$}
\put(3000,-1200){\tiny $d$}
\put(4800, -1200){\tiny $d+1$}
\put(4800,600){$\times$}
\end{picture}
\]
\item None of them are neighbors.\\
If no ending of $\Phi$ is of type $D$, the three roots $\notin S(X_1)$ could cut out an $A_1\times A_1$:
\[
\begin{picture}(2400,1800)(-300,-900)
\put(0,0){\usebox\dynkinafour}
\put(5400,0){\usebox\dynkinatwo}
\put(-600,600){$\times$}
\put(3000,600){$\times$}
\put(6600,600){$\times$}
\put(1500,-900){$\tiny \a$}
\put(5100, -900){$\tiny \a'$}
\end{picture}
\]
If there are exactly five roots in $\Phi$, like in the instructive drawing above, the roots not supported in $\w=[\frac12]\a+\a'$ form an $A_1\times A_1\times A_1$, contradicting \cref{homlem1}.
Suppose now $\Phi$ has more then five simple roots and no ending of $S(X_1)$ is of type $D$:
\[
\begin{picture}(2400,1800)(-300,-900)
\put(-1800,0){$\dots$}
\put(9000,0){$\dots$}
\multiput(0,0)(3600,0){3}{\usebox\dynkinatwo}
\multiput(1800,0)(3600,0){2}{\usebox\shortsusp}
\multiput(1200,900)(3600,0){3}{$\times$}
\put(1200,-1200){$d$}
\put(4800,-1200){$e$}
\put(8400,-1200){$f$}
\put(000,600){\textrm{I}}
\put(3600,600){\textrm{II}}
\put(7000,600){\textrm{III}}
\put(10800,600){\textrm{IV}}
\end{picture}
\]
We number the connected components from left to right. A $\w$ for component {\textrm I} or {\textrm I+II} or \textrm{II} has $\aa_f\in S^p$ and hence $\aa_f\notin S(X_2)$. Analogous for an $\w$ for component {\textrm III, III+IV} or {\textrm IV} and $\aa_d$. If component {\textrm I} and {\textrm IV} form $A_1\times A_1$, $\aa_e\notin S(X_2)$ for $\w$ of type $D_2$. If component {\textrm I} and {\textrm II} form $A_1 \times A_1$, $\aa_f\notin S(X_2)$ for such an $\w$. If component {\textrm I} and {\textrm III} form $A_1\times A_1$, the roots not supported in $\w$ of type $D_2$ supported on these two components are distributed over two connected components in $ S(X_2)$ that are not of type $A_1$, contradicting \cref{lunad}.\\
If the local root system had one ending of type $D$ and exactly 4 simple roots (which is the case for $B_3^{(1)}$ or $A_{3}^{(2)}$), we could have $\{\aa_0,\aa_1,\aa_3\}\notin S(X_1)$. But then these roots have to be in $ S(X_2)$ and form three connected components. Impossible.\\
If there is an ending of type $D$ on the w.l.o.g. left, it could happen that $S(X_1)$ only has two components. Let us assume $\a_0, \a_1, \a_d$ with $2<d<n-1$ are $\notin S(X_1)$.
\[
\begin{picture}(0,2400)(0,0)
\thicklines\put(1800,1200){\usebox\susp}
\put(5400,1200){\usebox{\shortsusp}}
\put(1800,1200){\line(-1,1){1200}}
\put(1800,1200){\line(-1,-1){1200}}
\put(-300,00){$\times$}
\put(-300,2400){$\times$}
\put(5400,900){$\times$}
\put(5400,-300){$\a_d$}
\end{picture}
\]
One possible weight on the component right of $\a_d$ leads to $\a_0, \a_1\notin S(X_2)$ by \cref{trick}.
The other possibility is something supporting $\{\a_2,\dots,\a_{d-1}\}$. But then $\delta-\w$ consists of three connected components of type either $A_1\times A_1 \times $something$ $ each containing one missing root if we had $\w=\aa_2+\dots+\aa_{d-1}$ or $A_3\times $something each containing a missing root if we had $\w=[\frac12]\aa_2+2\aa_3+\aa_4$ or $\w=[2]\aa_2$. Both oppose \cref{lunad}.\\
If $\aa_0, \aa_1, \aa_n$ were $\notin S(X_1)$, the roots not supported in $\w$ were contained in two different components of $ S(X_2)$ that are not $A_1\times A_1$, contradicting\cref{homlem1}.\\
Let us now assume that we are in $D_n^{(1)}$ and three roots at the ends are $\notin S(X_1)$, without loss of generality $\a_0, \a_1, \a_n \notin S(X_1)$:
\[
\begin{picture}(0,3600)(0,0)
\thicklines\put(1800,1200){\usebox\susp}
\put(5400,1200){\usebox{\bifurc}}
\put(1800,1200){\line(-1,1){1200}}
\put(1800,1200){\line(-1,-1){1200}}
\put(-300,0){$\times$}
\put(-300,2400){$\times$}
\put(6300,2400){$\times$}
\end{picture}
\]
If we chose $\w=\aa_2+\dots+\aa_{n-1}$, we had $\aa_2, \aa_{n-1} \notin S(X_2)$ via \cref{trick}, and hence the roots not supported in $\w$ are in three components of connection of $ S(X_2)$, contradicting \cref{homlem1}.
If we chose $[\frac12]\aa_2+2\aa_3+\aa_4$ in $D_5^{(1)}$, we had $ S(X_2)$ of type $A_3\times A_1 \times A_1$, where the missing roots are distributed over one $A_1$ and $A_3$, contradicting \cref{homlem1}.
\end{itemize}
This finishes the proof.
\end{proof}
Let us now settle the remaining cases. The following lemma is more general than needed to prove \cref{homthm} above.
\begin{lem}\label{homlem3}
Let $\Phi$ be an affine root system of type $E_i^{(1)}, i \in \{6,7,8\}$. Then there are no bi-homogeneous manifolds of rank one.
\end{lem}
\begin{proof}
We remember that for a bi-homogeneous polytope generated by a weight $\w$, we have $\w+(\w^\sharp)=t\cdot \delta$ for some coefficient $t\in \R^\times$. As coefficients of simple roots in spherical roots are $\frac12,1,2,3$, $t$ must be chosen such that all coefficients of $t\delta$ are in $\N[\frac12]$ and $\le 4$.
\begin{defn}
Let $k_(\aa_i)$ be the coefficient of $\aa_i$ in $\d$. For $\w$ the weight of a homogeneous variety of rank one, let $k_\w(\aa_i)$ be the coefficient of $\aa_i$ in $\w$, that is $\w=\sum k_\w(\aa_i) \cdot \aa_i$.
\end{defn}
We show the proof for $E_6^{(1)}$, the proof for the other cases is similar and can be found in \cite{Pau}.
For this case, the coefficients of simple roots in $\d$ are $1,2,3$. That means $t\in\{\frac12, 1\}$.
\begin{itemize}
\item $t=\frac12$. Then $k(\aa_4)=\frac32$. As $E_6^{(1)}$ does not contain a subsystem of type $B_3$, it follows that w.l.o.g $k_\w(\aa_4)=1$ and $k_{\w^\sharp}(\aa_4)=\frac12$. That means that $\aa_4$ must be contained in $S(X_1)$ and $ S(X_2)$, meaning that $\aa_4$ is in $S^p$ by \cref{trick}. The remaining possibilities for $\w$ are the following (up to diagram automorphisms):
\begin{itemize}
\item $\w=\aa_1+2\aa_3+\aa_4$.
\item $\w=\aa_3+\aa_4+\aa_5$.
\item $\w=\frac12(\aa_3+\aa_5+2\aa_4+2\aa_2)$
\end{itemize}
All are in contradiction to \cref{homlem1}
\item $t=1$. Then $\w$ and $\w^\sharp$ must support $\aa_4$, w.l.o.g. $k_\w(\aa_4)=2$ and $k_{\w^\sharp}(\aa_4)=1$. This again forces $\aa_4\in S^p$. The only possibility for $\w$ is $\w=\aa_3+\aa_5+2\aa_4+2\aa_2$ is in contradiction to \cref{homlem1}.
\end{itemize}
So there are no bi-homogeneous polytopes for $E_6^{(1)}$.
\end{proof}
Finally, it remains to prove the theorem for $A_n^{(1)}$:
\begin{lem}\label{homlem5}
We consider an affine root system of type $A_n^{(1)}$. If three or more roots are $\notin S(X_1)$, then there is no bi-homogeneous moment polytope.
\end{lem}
\begin{proof} We distinguish the following cases:
\begin{enumerate}
\item Assume there are three missing roots that are neighbors, without loss of generality we assume $\aa_k, \aa_{k+1}, \aa_{k+2} \notin S(X_1)$. But then $\aa_{k+1}\in S^p(\w), \aa_{k+1}(X_1)>0$. Hence $\aa_{k+1}\notin S(X_2)$ by \cref{trick}.
\item Assume that three roots are missing, two of the missing roots $\aa_k, \aa_{k+1}$ are neighbors, and a third one, $\aa_l$ is not $\aa_{k-1}$ or $\aa_{k+2}$. Then we have to consider:
\begin{itemize}
\item $\w$ has $\aa_k+2, \dots, \aa_{l-1}$ in its support. But then $\aa_{k}\notin S(X_2)$ because $\aa_k(X_1)\ne 0, \aa_k\in S^p$ and \cref{trick}.
\item $\w$ has $\aa_{l+1}, \dots, \aa_{k-1}$ in its support. But then $\aa_{k+1}\notin S(X_2)$ because $\aa_{k+1}(X_1) \ne 0, \aa_{k+1}\in S^p$ and \cref{trick}.
\end{itemize}
\item Assume none of the three missing roots are neighbors (and hence $n\ge 5$), we call them $\aa_k, \aa_l, \aa_m$ clockwise. We assume $\w$ has $\aa_{k+1},\dots, \aa_{l-1}$ in its support. But then $\aa_m\notin S(X_2)$ because $\aa_l(X_1)\ne 0, \aa_l\in S^p$ and \cref{trick}.
If the three roots $\notin S(X_1)$ are w.l.o.g. $\aa_0, \aa_{2}, \aa_{4}$ and $n\ge 5$, then the roots not in the support of $\w$ are in two components of support $A_1\times A_{\ge 3}$ in $ S(X_2)$, and there is no spherical root with this support via \cref{lunad}.
\end{enumerate}
\end{proof}
This finishes the proof of \cref{homthm}.
\section{Structure of bi-inhomogeneous and mixed polytopes}
Fortunately, we can find an even stronger necessary condition for the cases of bi-inhomogeneous and mixed polytopes.
Remember that we ask $\w$ to touch every wall of the alcove at least once.
We want to prove the following theorem which is an inhomogeneous analogon to \cref{trick}:
\begin{thm}[Structure of inhomogeneous polytopes]\label{inthm}
Let $\w$ be an inhomogeneous local model for a genuine quasi-Hamiltonian manifold of rank one for some $X_1 \in \bar \A_0$, and $\w\in I_k(S(X_1))$, meaning $\la \w, \aa_k^\vee \ra=1$. Then:\\
The local root system $ S(X_2):=\Phi(X_1+c\w)$ is
\[
\Phi(X_2)=S \setminus \{\a_k\}
\]
\end{thm}
\begin{proof}
Choose some subset $S(X_1)\subset S$ such that there is an inhomogeneous local model for $S(X_1)$. After renumbering, we can assume that $S(X_1)=\{\a_0, \dots, \a_p\}$ for some $p<n$. We recognize that there are $n-p-1$ free coordinates in $\w$. We also know from \cref{inh1} that exactly one $\a_k(\w)>0$, we can assume $k=p$. So, $\a_i(\w)=0$ for $i=0, \dots, p-1$ and $\a_p(\w)>0$. We consider now the point $X_2=X_1+c\w$ for some positive real number $c$, the second point where $\P$ intersects the walls of the alcove. We have by \cref{afflin}:
\[
\a_i(X_2)=\a_i(X_1)+c\cdot \aa_i(\w)
\]
By construction, $\a_i(X_1)=0$ for $i\in\{0,...,p\}$. But $\aa_i(\w)$ is also zero for $i=0, \dots, p-1$. Furthermore, $\aa_p(\w)>0$. It follows:
\begin{equation}
\a_0, \dots, \a_{p-1} \in S(X_2), \a_p \notin S(X_2)
\end{equation}
We want that $\w$ touches every wall of the alcove at least once. It remains to prove that we can always find an $\w$ that leads to an $X_2$ such that all simple roots not in $S(X_1)$ are in $ S(X_2)$, that is that our $\P$ is genuine:
Recall that, by using the relations of coordinates given by $\a_0, \dots, \a_{p}$, we defined $p+1$ coordinates of $X_1$, $n-p-1$ coordinates of $X_1$ are still free. So we have $n-p-1+1=n-p$ free coordinates in $X_2$, because by going from $X_1$ to $X_2=X_1+c\w$, we get one additional parameter $c$. We have $n+1-p+1=n-p$ equations to solve, coming from the roots $\a_{p+1}, \dots, \a_n$. So, in general, we face an (in general inhomogeneous) system of $n-p$ equations in $n-p$ unknowns. Because the simple roots are linearly independent, this system has full rank and it is well known that it is always solvable with exactly one solution. So, it is always possible to have all roots $\a_{p+1}, \dots, \a_n$ in $ S(X_2)$.
Concluding, we have seen that $ S(X_2)$ is $S\setminus\{\a_k\}$: All simple roots supported in $\w$ w but $\a_k$ are still in $S(X_2)$, and all roots not in $S(X_1)$ are in $S(X_2)$.
\end{proof}
\begin{cor}[bi-inhomogeneous polytopes]\label{incor1}
Bi-inhomogeneous polytopes are only possible if $S(X_1)=S\setminus\{\a_j\}$ and $ S(X_2):=S \setminus\{\a_k\}$ each contain $n$ roots. If $\w\in I_k(j)$, meaning $\la\w, \a_k^\vee\ra=1$ for $\a_k\in S(X_1)$, then $ S(X_2)=
S\setminus \{\a_k\}$ and there we must have $-\w\in I_j(k)$, and vice versa.
\end{cor}
\begin{proof}
This is an immediate consequence of \cref{inthm}. Starting in without loss of generality $X_1$ with an inhomogeneous $\w$, $X_2$ is n-elemental and by the proof of the \cref{inthm}, $\a_k$ is not in $\Phi(X_2)$. As $\w$ takes the value zero on all roots of $ S(X_2)$ but $\a_j$, a bi-inhomogeneous polytope has only a chance to take value 1 there.
\end{proof}
\begin{rem}
This means: if $\P=[X_1X_2]$ is a bi-inhomogeneous momentum polytope, then $X_1$ and $X_2$ are vertices of $\bar \A$.
\end{rem}
It also follows: in case of a bi-inhomogeneous polytope, $\w$ is completely determined by the root $\a_k$ with $\la \w, \aa_k^\vee\ra=1$ and the one simple root not contained in the local root system $S(X_1)$. There are no free parameters! Hence instead of writing $\w\in I_k(j)$, we can and will write $\w=I_k(j)$.
The fact $|S(X_1)|=| S(X_2)|=n$ leads to a strong tool for finding bi-inhomogeneous polytopes: It is necessary that the root system has two n-elemental subsets that allow the existence of inhomogeneous local models, meaning that there are components supporting $n$ roots and have one connected component of type $A_l$ or $C_l$. We also know from \cref{incor1} which roots have to be or cannot be in the one or the other. That significantly reduces the number of cases remaining to check.
We also find a necessary condition for mixed polytopes:
\begin{cor}[mixed momentum polytopes]\label{mixcor2}
Mixed momentum polytopes (homogeneous in $X_1$, inhomogeneous in $X_2$) can only exist if $|S(X_1)|=n$.
\end{cor}
\begin{proof}
Let $\w$ be an inhomogeneous local model in $X_2$. Then $|S(X_1)|=n$ by \cref{inthm}.
\end{proof}
\section{Explicit determination of moment polytopes of rank one}
We first sum up the last three sections:
\begin{thm}\label{qhamnot}
For every bi-homogeneous moment polytope $[X_1X_2]$ of any compact genuine quasi-Hamiltonian $K$-manifold of rank one, $S(X_1)$ and $ S(X_2)$ contain at least $n-1$ roots ($n$ denotes the rank of $K$). For every mixed momentum polytope with homogeneous local model in $X_1$, the local root system $S(X_1)$ contains $n$ roots. For every bi-homogeneous moment polytope, we have $|S(X_1)|=|S(X_2)|=n$.
\end{thm}
\begin{proof}
This is the combination of \cref{homthm}, \cref{incor1} and \cref{mixcor2}.
\end{proof}
In general, for a fixed local root system $\Phi_X$, there are only finitely many local models possible, namely the homogeneous spherical varieties found in \cite{Akh83} or the spherical modules of rank one for $A$ and $C$. So which local models are candidates in one vertex of the alcove is determined by the local root systems and the known classification of smooth affine spherical varieties, we discussed this in detail in \cref{rank1model}.
We shall establish a short notation for going through our algorithm. We will use the following tables:
\begin{center}
\begin{longtable}{|c|c|c|c|c|}
$S(X_1)$ & choice of $\w$ & $ S(X_2)$ & $\w^\sharp$ hom.? & $-w$ inhom.?\\\endhead
\end{longtable}
\end{center}
In the first column, we choose the subsystem we shall start with as a local root system in one ``vertex'' of our polytope by denoting its simple roots. That corresponds to some wall of the alcove. Then we denote all possible choices we have for the local model there in column two where we write $I_k$ for an inhomogeneous model being $1$ on the root $\a_k$ and zero elsewhere. The next column gives the local root system at the other end of our polytope, in $X_2$ which is easy to see via \cref{trick}, \cref{incor1} or elementary calculations.
We will not explicitly mention cases that only differ by a diagram automorphism.
As the case-by-case-studies are pretty similar for all affine root systems, we shall only discuss two instructive cases in detail. We refer to \cite{Pau} for a detailed study of all affine root systems.
\subsection{$A_n^{(1)}$}
Note that in $A_n^{(1)}$, all subsystems of $n$ roots only differ by a diagram automorphism. We will choose $S\setminus\{a_0\}$ as a representative for our examination.
As this root system has the structure of a circle, all sums below have to be read ``modulo $n+1$''.
\begin{center}
\tiny{
\begin{longtable}{|p{2.5cm}|p{3cm}|p{2cm}|p{3cm}|p{3cm}|}
\hline
$S(X_1)$ & choice of $\w$ & $ S(X_2)$ & $\w^\sharp$ hom.? & $-w$ inhom.?\\\endhead
\hline
\multicolumn{5}{|c|}{$n=1$} \\
\hline
$\a_1$ & $I_1$ & $\a_0$ & & $-\w=I_0$\\
\hline
& $\aa_1$ & $\a_0$ & $\w^\sharp=\a_0$ & \\
\hline
& $2\aa_1$ & $\a_0$ & $\w^\sharp=2\aa_0$ & \\
\hline
\multicolumn{5}{|c|}{$n\ge 2$} \\
\hline
$S\setminus \{\a_0\}$ & $\a_{1,n}$ & $S\setminus\{\a_1, \a_n\}$ &$\w^\sharp=\aa_0$ & \\
\hline
& $I_1$ & $S\setminus\{\a_1\}$ & &$-\w=I_0$ \\
\hline
$S\setminus \{\a_0\}, n=3$ & $[\frac12]\aa_1+2\aa_2+\aa_3$ & $S\setminus\{\a_2\}$ &$\w^\sharp=[\frac 12]\aa_3+2\aa_0+\aa_1$ & \\
\hline
$S\setminus\{\aa_1,\aa_n\}$ & $2\aa_0$ & $S\setminus\{\a_0\}$ & no: $\w^\sharp=2\a_{1,n}$, \cref{lunad} & no: $\la -\w, \aa_1^\vee\ra = \la-\w, \aa_n^\vee\ra>0$, \cref{inh1}\\
\hline
$S\setminus \{\a_d, \a_e\};\linebreak d\le e$ & $\aa_{d+1,e-1}$ & $S\setminus \{\a_{d+1}, \a_{e-1}\}$& $\w^\sharp=\a_{e,d}$ & \\
\hline
$S\setminus\{\a_0, \a_4\}$, $n\ge 4$ & $[\frac 12] \aa_1+2\aa_2+\aa_3$ & $S \setminus \{\a_2\}$ & no: $\w^\sharp=[\frac 12] 2\aa_0+\aa_1+\aa_3+2\aa_4+\dots+2\aa_n$, \cref{lunad} & no:$\la-\w,\a_3^\vee\ra=0,\linebreak \la -\w, \aa_4^\vee\ra=[\frac 12]1$, \cref{inh1}\\
\hline
$S\setminus\{\a_1,\a_3\}, n=3$ & $[\frac 12] \aa_1+\aa_3$ & $\a_2, \a_0$ & $\w^\sharp=[\frac 12] \aa_2+\aa_0$ & \\
\hline
$S\setminus\{\a_1,\a_3\}, n>3$ & $[\frac 12] \aa_1+\aa_3$ & $\ssm{1}{3}$ & no: $\w^\sharp=[\frac 12] \aa_0+\aa_{2,n}$, \cref{lunad} & no: $\Sp{2}>0, \Sp{3}>0$, \cref{inh1} \\
\hline
\end{longtable}
}
\end{center}
\subsection{$D_n^{(1)}, n\ge 4$}
We first argue that there are no bi-inhomogeneous polytopes.
\begin{itemize}
\item $S(X_1)=S\setminus\{\a_0\}$ or another root at the end. Then $S(X_1)$ is of type $D$, where no inhomogeneous model is possible via \cref{inh1}.
\item $S(X_1)=S\setminus\{\a_2\}$. Then $\w=I_0$ (or $w=I_1$,or if $n=4$: $\w=I_3$ or $\w=I_4$) is possible. But then, $ S(X_2)=S\setminus\{\a_0\}$ is of type $D$.
\item $S(X_1)=S\setminus\{\a_d\}, 2<d<n-1$. But then $S(X_1)$ is of type $D_l\times D_k$ for some $l,k$.
\end{itemize}
Now let us continue for bi-homogeneous and mixed polytopes.
\begin{center}
\tiny{
\begin{longtable}{|p{2.5cm}|p{3cm}|p{2cm}|p{3cm}|p{3cm}|}
\hline
$S(X_1)$ & choice of $\w$ & $ S(X_2)$ & $\w^\sharp$ hom.? & $-w$ inhom.?\\\endhead
\hline
\multicolumn{5}{|c|}{$n=4$} \\
\hline
$\Ssm\{\a_0\}$ & $[\frac12]2\aa_3+2\aa_2+\aa_4+\aa_1$ & $\Ssm\{\a_3\}$ & $\w^\sharp=[\frac12]2\aa_0+2\aa_2+\aa_1+\aa_3$ & \\
\hline
$\Ssm\{\a_0,\a_1\}$ & $ [\frac12]\aa_4+2\aa_2+\aa_3$ & $\Ssm\{\a_2\}$ & $\w^\sharp=[\frac12]\aa_0+\aa_1$ & \\
\hline
& $\aa_4+\aa_2+\aa_4$ & $\Ssm\{\a_3,\a_4\}$ & $\w^\sharp=\aa_0+\aa_2+\aa_1$ & \\
\hline
$\Ssm\{\a_0,\a_2\}$ &$[2]\aa_1$ & elementary: never contains both missing roots & & \\
\hline
$\Ssm\{\a_0,\a_2\}$ &$[\frac12]\aa_1+\aa_4$ & elementary: never contains both missing roots & & \\
\hline
$\Ssm\{\a_2\}$ & $[\frac12]\aa_0+\aa_1$ & $\Ssm\{\a_0,\a_2\}$ & already seen: $\w^\sharp=[\frac12]\aa_4+2\aa_2+\aa_3$ & \\
\hline
& $[2]\aa_1$ & $\Ssm\{\a_1\}$ & no: $\w^\sharp=[2]\aa_0+2\aa_2+\aa_3+\aa_4$, \cref{lunad} & no: System of type $D$, \cref{inh1}\\
\hline
\multicolumn{5}{|c|}{$n\ge 5$} \\
\hline
$\Ssm\{\a_0\}$ & $[\frac12]2\aa_{1,n-2}+\aa_{n-1}+\aa_n$ & $\Ssm\{\a_1\}$ & $\w^\sharp=[\frac12]2\aa_0+2\aa_{2,n-2}+\aa_{n-1}+\aa_n$ & \\
\hline
$\Ssm\{\a_0,\a_1\}$ & $[\frac12]2\aa_{2, n-2}+\aa_{n-1}+\aa_n$ & $\Ssm\{\a_2\}$ & $\w^\sharp=[\frac12]\aa_0+\aa_2$ & \\
\hline
$\Ssm\{\a_0,\a_2\}$ & $[2]\aa_1$ & $\a_0\notin S(X_2)$, \cref{trick} & & \\
\hline
& $\w \in C(n)$ & $\a_0\notin S(X_2)$, \cref{trick} & & \\
\hline
$\Ssm\{\a_0, \a_d\}, 2<d<n-1$ & $\aa_{1,d-1}$ & $\Ssm\{\a_1,\a_{d-1}\}$ & no: $\w^\sharp=\aa_0+2\aa_{2,n-2}+\aa_{n-1}+\aa_n$, \cref{lunad} &no: $\Sp{0}>0, \Sp{d}>0$, \cref{mixcor2}\\
\hline
$d=4$ & $[\frac12]\aa_1+2\aa_2+\aa_3$ & $\Ssm\{\a_2\}$ & no: $\S(X_2)$ neither connected nor $A_1\times A_1$, \cref{homlem1} & no: $\Sp{0}>0,\Sp{4}>0$, \cref{mixcor2}\\
\hline
& $\w \in C(n)$ & $\a_0\notin S(X_2)$ \cref{trick} & & \\
\hline
$\Ssm\{\a_0,\a_n\}$ & $\aa_{1,n-1}$ & $\Ssm\{\a_1,\a_{n-1}\}$ & $\w^\sharp=\aa_0+\aa_{2,n-2}+\aa_{n}$ & \\
\hline
$S\setminus\{\a_d\}$ with w.l.o.g $2\le d \le n/2$ & $[\frac12]\aa_0+\aa_1+2\aa_{2,d-1}$ & $\Ssm\{\a_{d-1}\}$ & $\w^\sharp=[\frac12]2\aa_{d,n-2}+\aa_{n-1}+\aa_n$ & \\
\hline$\d-\w$ neither connected nor $A_1\times A_1$
& $[\frac12]2\aa_{d+1,n-2}+\aa_{n-1}+\aa_n$ & $\Ssm\{\a_{d+1}$ & $\w^\sharp=[\frac12]\aa_0+\aa_1+2\aa_{2,d}$ & \\
\hline
$d=2$ & $[2]\aa_1$ & $\Ssm\{\a_1\}$ & no: $\w^\sharp=[2]\aa_0+2\aa_{2,n-2}+\aa_{n-1}+\aa_n$, \cref{lunad} & no: $\la-\w,\aa_3^\vee\ra=1$, \cref{inh1}\\
\hline
$\ssm{d}{e}, 1<d,d+1< e<n-1$ & $\w\in C(0)$ & $\a_e\notin S(X_2)$& &\\
\hline
& $\w\in C(n)$ & $\a_d\notin S(X_2)$ \cref{trick}& &\\
\hline
&$\w\in C(d+1)$ & & no: \cref{homlem1} & no: \cref{mixcor2} \\
\hline
$\ssm{d}{d+1}, 1<d<n-2$ & $\w\in C(0)$ & $\a_{d+1}\notin S(X_2)$, \cref{trick}& &\\
\hline
& $\w\in C(n)$ & $\a_d\notin S(X_2)$, \cref{trick} & &\\
\hline
$\ssm{2}{n}$ & $\w\in C(0)$ or $\w\in C(0)+C(1)$ & $\a_n\notin S(X_2)$ & & \\
\hline
& $\w\in C(3)$ & & no: $\w^\sharp$ not $\in$ \cref{lunad} & no: $\Sp{2}>0, \Sp{n}<0$, \cref{mixcor2} \\
\hline
\end{longtable}
}
\end{center}
\section{List of genuine Moment Polytopes of Rank one}
\begin{thm}[Momentum Polytopes of Rank one]
A compact multiplicity free genuine quasi-Hamiltonian $K$-manifold of rank one for $K$ simple corresponds to a spherical pair $(\P,\L_S)$ with the following properties:
\begin{itemize}
\item $\P$ is a line segment $[X_1X_2]$ that is spanned as a line by $X_1+\R\w$ for some $\w$ found in \cref{ListHom} and $X_1$ such that $S(X_1)$ is the corresponding local root system.
\item We have $\a(X_1)=0$ or $\a(X_2)=0$ for every simple root $\a_0, \a_1, \dots, \a_n$
\item The lattice $\L_S$ is generated by this $\w$
\end{itemize}
\end{thm}
All parameters are assumed to be non-negative integers here. We mark inhomogeneous local models by printing them in {\bf bold font}. A $[2]$ in front of a momentum polytope says that not only $\w$ and $\w^\sharp$, but also $2\w$ and $2 \w^\sharp$ is possible. In the "local model"-columns we list the primitive spherical triple, containing the information about the semisimple type of the model. Note that we will only note the components of the local model with non-trivial operation.\\
We also investigated the operation of the connected center that could appear in the cases where the diagram of the primitive triples shows an asterisk. If these $\tt^1$ acted via a scalar $N$, we put an $\C^k_N$ in the $V$-entry of the triple. We omit this if $N=1$ which stands for the "standard" operation.
To find this scalar $N$, we determine a linearly independent set of primitive $\zeta\in\Z\S$ corresponding to the center of the Levi of the local model. It is defined by the fact that its elements are perpendicular to all simple roots in the local root system. Then it is well known that $N=\la \w, \zeta^\vee\ra$.\\
Remarkably, it turned out that $N=1$ for all genuine quasi-Hamiltonian Manifolds of rank one. We shall see that this is not true for those quasi-Hamiltonians of rank one that are not genuine in the next part.
\pagebreak
\begin{lis}\label{ListHom}
\begin{center}
List of quasi-Hamiltonian manifolds of rank one
\tiny{
\begin{longtable}{|p{0.7cm}|p{2cm}|p{3.3cm}|p{2cm}|p{3.3cm}||p{2cm}|}
\hline
&$\w$ &local model & $\w^\sharp$ &local model & remarks\\\endhead
\hline
\multicolumn{6}{|c|}{$A_1^{(1)}$} \\
\hline
A1)&$2\aa_0 \in H(1)$ & $(\sll(2),0,0)$ & $2\aa_1\in H(0)$ &$(\sll(2),0,0)$ & \\
\hline
\multicolumn{6}{|c|}{$A_3^{(1)}$} \\
\hline
A2)&$[\frac12](\aa_1+\aa_3) \in H(0,2)$&$(\sll(2)\times\sll(2),\Delta\sll(2),0)$ & $[\frac 12](\aa_0+\aa_2) \in H(1,3)$&$(\sll(2)\times\sll(2),\Delta\sll(2),0)$ & \\
\hline
A3)&$[2]\frac12(\aa_i+2\aa_{i+1}+\aa_{i+2}) \in H(0)$ & $(\sll(4),\sp(4),0)$ & $[2]\frac12(\aa_{i+2}+2\aa_{i+3}+\aa_{i})\in H(2)$&$(\sll(4),\sp(4),0)$& $i\in\{0,1,2,3\}$, everything $\mod 4$ \\
\hline
\multicolumn{6}{|c|}{$A_n^{(1)}, n\ge 1$} \\
\hline
A4)&$\aa_d+\dots+\aa_e \in H(d-1,e+1)$ & $(\sll(e-d+1),\tt^1+\sll(e-d),0) $ & $\aa_{e+1}+ \dots+\aa_{d-1} \in H(d,e)$ & $(\sll(n-(e-d)),\tt^1+\sll(n-(e-d+1)),0)$ & $0\le d \le e \le n$, everything $\mod n+1$. \\
\hline
A5)&$\boldsymbol{\w = I_k(k+1)}$ & $(\sll(n+1),\sll(n+1),\C^{n+1})$ & $\boldsymbol{-\w = I_{k+1}(k)}$ &$(\sll(n+1),\sll(n+1),\C^{n+1})$ & $0\le k \le n$, everything $\mod n+1$.\\
\hline
\multicolumn{6}{|c|}{$B_3^{(1)}$} \\
\hline
B1)&$\frac12 (\aa_0+2\aa_2+3\aa_3) \in H(1)$ &$(\soo(7),G_2,0)$& $\boldsymbol{-\w \in I_1(3)}$ &$(\sll(4),\sll(4),\C^4) $& or 0 and 1 switched \\
\hline
B2)&$\frac 12 (\aa_1+\aa_3)\in H(2)$ &$(\sll(2)\times \sll(2),\Delta\sll(2),0)$& $\boldsymbol{-\w \in I_2(1,3)}$ &$(\sll(3),\sll(3),\C^3)$ & or 0 and 1 switched \\
\hline
\multicolumn{6}{|c|}{$B_n^{(1)}$} \\
\hline
B3)&$[2](\aa_1+\dots+\aa_n) \in H(0)$ &$(\soo(2n+1),\soo(2n),0)$ &$[2](\aa_0+\aa_2+ \dots+\aa_n) \in H(1)$ &$(\soo(2n+1),\soo(2n),0)$ & \\
\hline
B4)&$[2](\frac 12(\aa_0+\aa_1)+\aa_2+\dots+\aa_d)\in \break H(d+1)$ &$(\soo(2k+2),\soo(2k+1),0)$ & $[2](\aa_{d+1}+\dots+\aa_n)\in H(d)$&$(\soo(2n-2k+1),\soo(2n-2k),0)$& $1\le d < n$\\
\hline
\addlinespace[0.3cm]
\hline
\multicolumn{6}{|c|}{$C_2^{(1)}$} \\
\hline
C1)&$[2]\frac12(\aa_0+\aa_2) \in H(1)$ &$(\sll(2)\times \sll(2),\Delta\sll(2),0)$& $[2]\aa_{1} \in H(0,2)$ &$(\sll(2),0,0)$& \\
\hline
C2)&$[2](\aa_0+\aa_1) \in H(2)$ &$(\soo(5),\soo(4),0)$& $[2](\aa_{1}+\aa_2) \in H(0)$ &$(\soo(5),\soo(4),0$& \\
\hline
\multicolumn{6}{|c|}{$C_n^{(1)}$} \\
\hline
C3)&$\frac 12 (\aa_0+\aa_n) \in H(1, n-1)$ &$(\sll(2)\times \sll(2),\Delta\sll(2),0)$ & $\aa_1+\dots+\aa_{n-1} \in H(0,n)$ & $(\sll(n-1),\tt^1+\sll(n-2),0)$&see also $n=2$ \\
\hline
C4)&$\aa_0+2\aa_1+\dots+2\aa_{d-1}+\aa_d\in H(d+1)$ &$(\sp(2d+2),(\sll(2)+\sp(2d)),0)$& $\aa_{d}+2\aa_{d+1}+\dots+2\aa_{n-1}+\aa_n \in H(d-1)$ &$(\sp(2n-2d+2),(\sll(2)+\sp(2n-2d)),0)$&$1 \le d <n$ \\
\hline
C5)&$\boldsymbol{\w= I_k-1(k)}$ &$(\sp(2k),\sp(2k),\C^{2k})$& $\boldsymbol{\w = I_{k}(k-1)}$&$(\sp(2n-2k+2),\sp(2n-2k+2),\C^{2n-2k+2})$ &$1 \le k \le n$ \\
\hline
\pagebreak
\hline
\multicolumn{6}{|c|}{$D_4^{(1)}$} \\
\hline
D1)&$[2] \aa_i+\aa_2+\frac 12 \aa_j +\frac 12 \aa_k \in H(l)$ &$(\soo(8),\soo(7),0)$& [2] $\aa_l+\aa_2+\frac 12\aa_j+\frac 12 \aa_k\in H(i)$ &$(\soo(8),\soo(7),0)$& $i \ne j\ne k \ne l$;\break $i,j,k,l\in 0,1,3,4$ \\
\hline
D2)&$[1/2]\aa_i+2\aa_2+\aa_j \in H(k,l)$&$(\sll(4),\sp(4),0)$ &$[1/2] \aa_k+\aa_l\in H(2)$&$(\sll(4),\sp(4),0)$& $i,j,k,l \in 0,1,3,4$\break $i\ne j\ne k \ne l$ \\
\hline
D3)&$\aa_i+\aa_2+\aa_j \in H(k,l)$ &$(\sll(4),\tt^1+\sll(3),0)$ &$\aa_k +\aa_2+\aa_l\in H(i,j)$ &$(\sll(4),\tt^1+\sll(3),0)$ & $i,j,k,l \in 0,1,3,4$\break $i\ne j\ne k \ne l$ \\
\hline
\multicolumn{6}{|c|}{$D_n^{(1)}$} \\
\hline
&$\w$ &local model & $\w^\sharp$ &local model & remarks\\\hline
D4)&$[2](\aa_1+\aa_2+\dots+\aa_{n-2}+\frac12 \aa_{n-1}+\frac12{\aa_n}) \in H(0)$ &$(\soo(2n),\soo(2n-1),0)$ & $[2](\aa_0+\aa_2+\dots+\aa_{n-2}+\frac 12\aa_{n-1}+\frac 12\aa_n) \in H(1)$ &$(\soo(2n),\soo(2n-1),0)$& \\
\hline
D5)&$[2](\aa_{n-1}+\aa_{n-2}+\dots+\aa_2+\frac12 \aa_0+\frac 12 \aa_1) \in H(n)$ &$(\soo(2n),\soo(2n-1),0)$& $[2](\aa_n+\aa_{n-2}+\dots+\aa_2+\frac 12 \aa_1 +\frac 12 \aa_0)\in H(n-1)$ &$(\soo(2n),\soo(2n-1),0)$& \\
\hline
D6)&$[2](\frac 12\aa_0+\frac 12 \aa_1+\aa_2+\dots+\aa_{d}) \in H(d+1)$ &$(\soo(2d+2),\soo(2d+1),0)$& $[2](\aa_{d+1}+\dots+\aa_{n-2}+\frac 12\aa_{n-1}+\frac 12 \aa_n)\break \in H(d)$ &$(\soo(2n-2d),\soo(2n-2d-1),0)$& $1\le d \le n-2$ \\
\hline
D7)&$\aa_1+ \dots+\aa_{n-1} \in H(0,n)$ &$(\sll(n),\tt^1+\sll(n-1),0)$ & $\aa_{0}+\aa_2+\dots+\aa_{n-2}+ \aa_n \break \in H(1,n-1)$ &$(\sll(n),\tt^1+\sll(n-1),0)$& or $\a_n$ and $\a_{n-1}$ switched \\
\hline
\addlinespace[0.3cm]
\hline
\multicolumn{6}{|c|}{$F_4^{(1)}$} \\
\hline
F1)&$\aa_1+2\aa_2+3\aa_3+2\aa_4\in H(0)$ &$(F_4,\soo(9),0)$& $\aa_0+\aa_1+\aa_2+\aa_3 \in H(4)$&$(\soo(9),\soo(8),0)$& \\
\hline
F2)&$\aa_2+2\aa_3+\aa_4 \in H(1)$ &$(\sll(4),\sp(4),0)$& $\frac 12 \aa_o+\aa_1+\frac 12 \aa_2 \in H(3)$ &$(\sll(4),\sp(4),0)$& \\
\hline
F3)&{$\aa_3+\aa_4\in H(2)$} &$(\sll(3),\gll(2),0)$&$\boldsymbol{-\w \in I_2(3,4)}$&$(\gll(4),\gll(4),\C^4)$&\\
\hline
\addlinespace[0.3cm]
\hline
\multicolumn{6}{|c|}{$G_2^{(1)}$} \\
\hline
&$\w$ &local model & $\w^\sharp$ &local model & remarks\\\hline
G1)&{$\aa_2+2\aa_1 \in H(0)$} &$(G_2,\sll(3),0)$& $\boldsymbol{-\w = I_0(1)}$&$(\sll(3),\sll(3),\C^3)$ & \\
\hline
G2)&{$\aa_1\in H(2)$}&$(\sll(2),0,0)$ & $\boldsymbol{-\w= I_2(1)}$ &$(\sll(3),\sll(3),\C^3)$& \\
\hline
G3)&{$\frac 12(\aa_0+\aa_1) \in H(2)$}&$(\sll(2)\times \sll(2),\Delta\sll(2),0)$& $\boldsymbol{-\w\in I_2(0,1)}$ &$(\gll(2),\gll(2),\C^2)$& \\
\hline
\addlinespace[0.3cm]
\hline
\multicolumn{6}{|c|}{$A_2^{(2)}$} \\
\hline
A6)&$2\aa_0 \in H(1)$ &$(\sll(2),0,0)$& $\aa_1\in H(0)$ &$(\sll(2),0,0)$& \\
\hline
A7)&{$\a_0 \in H(1)$} &$(\sll(2),0,0)$& $\boldsymbol{-\w= I_1(0)}$ &$(\sll(2),\sll(2),\C^2)$&\\
\hline
\pagebreak
\hline
\multicolumn{6}{|c|}{$A_{2n}^{(2)}$} \\
\hline
A8)&$2\aa_0+2\aa_1+\dots+2\aa_{n-1} \in H(n)$ &$(\soo(2n+1),\soo(2n),0)$& $\aa_n\in H(n-1)$&$(\sll(2),0,0)$& \\
\hline
A9)&{$\a_0+\aa_1+\dots+\aa_d \in H(d+1)$} &$(\soo(2(d+1)+1),\soo(2(d+1)),0)$&$\boldsymbol{-\w= I_{d+1}(d)}$&$(\sp(2n-2d),\sp(2n-2d),\C^{2n-2d})$ &$0\le d < n$\\
\hline
\addlinespace[0.3cm]
\hline
\multicolumn{6}{|c|}{$A_{2n-1}^{(2)}$} \\
\hline
{\tiny A10)}&$\aa_1+2\aa_2+\dots+2\aa_{n-1}+\aa_n \in H(0)$ &$(\sp(2n),(\sll(2)+\sp(2n-2)),0)$& $\aa_0\in H(2)$&$(\sll(2),0,0)$ &or $\aa_0$ and $\aa_1$ switched \\
\hline
{\tiny A11)}&$\aa_0+\aa_1+2\aa_2+\dots+2\aa_{n-1}\in H(n)$ &$(\soo(2n),\soo(2n-1),0)$& $\aa_n\in H(n-1)$ &$(\sll(2),0,0)$& \\
\hline
{\tiny A12)}&$\aa_0+\aa_1+\aa_2 \in H(3)$ &$(\sll(4),\gll(3),0)$& $\aa_2+2\aa_3+\dots +2\aa_{n-1}+\aa_n \in H(0,1)$ &$(\sp(2(n-1),(\sll(2)+\sp(2(n-2))),0)$& \\
\hline
{\tiny A13)}&$\boldsymbol{\w = I_1(0)}$ &$(\sp(2n),\sp(2n),\C^{2n})$&$\boldsymbol{-w= I_0(1)}$ &$(\sp(2n),\sp(2n),\C^{2n})$& \\
\hline
{\tiny A14)}&{$\frac 12 \aa_0+\frac 12 \aa_1+\aa_2+\dots+\aa_d\in H(d+1)$}&$(\soo(2(d+1)),\soo(2(d+1)-1),0)$&$\boldsymbol{-\w = I_{d+1}(d)}$ &$(\sp(2n-2d),\sp(2n-2d),\C^{2n-2d})$&$1\le d \le n$\\
\hline
\multicolumn{6}{|c|}{$D_{3}^{(2)}$} \\
\hline
D8)&$\frac{1}{2}(\aa_0+\aa_2)\in H(1)$ &$(\sll(2)\times \sll(2),\Delta\sll(2),0)$& $\boldsymbol {-\w \in I_1(0,2)}$ &$(\gll(2),\gll(2),\C^2)$ & \\
\hline
D9)&$\boldsymbol{\w\in I_0(2)}$ &$(\sll(3),\sll(3),C^3)$& $\boldsymbol{-\w\in I_2(0)}$ &$(\sll(3),\sll(3),\C^3)$& \\
\hline
\multicolumn{6}{|c|}{$D_4^{(2)}$} \\
\hline
{\tiny D10)}&$[2]\frac12(\aa_0+\aa_2)\in H(1,3)$ &$(\sll(2)\times \sll(2),\Delta\sll(2),0)$& $[2]\frac12(\aa_1+\aa_3)\in H(0,2)$ &$(\sll(2)\times \sll(2),\Delta\sll(2),0)$& \\
\hline
{\tiny D11)}&$[2]\frac12(3\aa_0+2\aa_1+\aa_2) \in H(3)$ &$(\soo(7),G_2,0)$& $[2]\frac12(\aa_1+2\aa_2+3\aa_3) \in H(0)$ &$(\soo(7),G_2,0)$& \\
\hline
\multicolumn{6}{|c|}{$D_{n+1}^{(2)}$} \\
\hline
{\tiny D12)}&$[2](\aa_0+\dots+\aa_d)\in H(d+1)$ &$(\soo(2d+3),\soo(2d+2),0)$& $[2](\aa_{d+1}+\dots+\aa_n)\in H(d)$ &$(\soo(2n-2d),\soo(2n-2d-1),0)$& $0\le d < n$ \\
\hline
{\tiny D13)}&$\aa_1+\dots+\aa_{n-1}\in H(0,n)$ &$(\sll(n-1),\gll(n-2),0)$& $\aa_0+\aa_n\in H(1,n-1)$&$(\sll(2)\times \sll(2),\Delta\sll(2),0)$& \\
\hline
\addlinespace[0.3cm]
\hline
\multicolumn{6}{|c|}{$E_{6}^{(2)}$} \\
\hline
E1)&$2\aa_1+3\aa_2+2\aa_3+\aa_4\in H(0)$ &$(F_4,\soo(9),0)$& $\aa_0\in H(1)$ &$(\sll(2),0,0)$& \\
\hline
E2)&$\aa_2+\aa_3+\aa_4\in H(1)$ &$(\soo(7),\soo(6),0)$& $\aa_0+2\aa_1+2\aa_2+\aa_3\in H(4)$ &$(\sp(8),(\sll(2)+\sp(6)),0)$& \\
\hline
E3)&{$\aa_0+\aa_1+\aa_2\in H(3)$} &$(\sll(4),\gll(3),0)$&$\boldsymbol{-\w\in I_3(0,2)}$ &$(\gll(2),\gll(2),C^2)$& \\
\hline
\pagebreak
\hline
\multicolumn{6}{|c|}{$D_{4}^{(3)}$} \\
\hline
{\tiny D14)}&$[2](2\aa_1+\aa_2) \in H(0)$ &$(G_2,\sll(3),0)$& $[2]\aa_0\in H(1)$ &$(\sll(2),0,0)$& \\
\hline
{\tiny D15)}&$[2]\frac 12(\aa_0+\aa_2) \in H(1)$ &$(\sll(2)\times \sll(2),\Delta\sll(2),0)$& $[2]\aa_1\in H(0,2)$ &$(\sll(2),0,0)$& \\
\hline
{\tiny D16)}&{$\aa_0+\aa_1\in H(2)$} &$(\sll(3),\gll(2),0)$& $\boldsymbol{-\w\in I_2(0,1)}$ &$(\sll(2),\sll(2),\C^2)$& \\
\hline
\end{longtable}
}
\end{center}
\end{lis}
We continue with a list of the corresponding diagrams:
\begin{longtable}{c|c|c|c|c}
$\begin{picture}(2400,1800)(-300,-900)
\put(0,0){\usebox{\leftrightbiedge}}
\put(0,0){\usebox{\aprime}}
\put(1800,0){\usebox{\aprime}}
\end{picture}$&$\begin{picture}(4500,3600)(-2350,-900)
\put(-2500,600){\tiny{[1/2]}}
\multiput(0,0)(0,1800){2}{\usebox{\edge}}
\multiput(0,1800)(1800,0){2}{\usebox{\vedge}}
\put(0,0){\usebox{\wcircle}}
\put(1800,0){\usebox{\wcircle}}
\put(0,1800){\usebox{\wcircle}}
\put(1800,1800){\usebox{\wcircle}}
\put(200,200){\line(1,1){1400}}
\put(200,1600){\line(1,-1){1400}}
\end{picture}$
&$\begin{picture}(4500,1800)(-2350,-900)
\put(-2500,600){\tiny{[1/2]}}
\multiput(0,0)(0,1800){2}{\usebox{\edge}}
\multiput(0,1800)(1800,0){2}{\usebox{\vedge}}
\put(0,0){\circle*{600}}
\put(1800,1800){\circle*{600}}
\end{picture}$&$\begin{picture}(10800,3600)(-600,-900)
\thicklines
\multiput(0,0)(5400,0){2}{\usebox{\shortam}}
\put(3600,0){\usebox{\edge}}
\multiput(-600,0)(9600,0){2}{\line(1,0){600}}
\multiput(-600,0)(10200,0){2}{\line(0,-1){1200}}
\put (-650,-1200){\line(1,0){10250}}
\end{picture}$ &
$
\begin{picture}(10800,1800)(-600,-900)
\thicklines
\multiput(0,0)(5400,0){2}{\usebox{\dynkinathree}}
\put(3600,0){\usebox{\shortsusp}}
\multiput(-600,0)(9600,0){2}{\line(1,0){600}}
\multiput(-600,0)(10200,0){2}{\line(0,-1){1200}}
\put (-650,-1200){\line(1,0){10250}}
\put(-350,-400){$\btr$}
\put(8250, -400){$\btl$}
\end{picture}
$\\
A1)&A2)&A3)&A4)&A5\\
\hline
$\begin{picture}(4500,3600)(0,-1800)
\thicklines
\put(0,0){\usebox\dynkinbthree}
\put(1800,0){\usebox\vedge}
\put(3600,0){\circle*{600}}
\put(-350,-350){$\btr$}
\put(3300,600){\tiny $\nicefrac12$}
\end{picture}$
&
$\begin{picture}(4500,3600)(-900,-900)
\thicklines
\put(0,0){\usebox\dynkinbthree}
\put(1800,0){\usebox\vedge}
\put(0,0){\circle{600}}
\put(3700,0){\circle{600}}
\multiput(0,300)(3700,0){2}{\line(0,1){300}}
\put(0,600){\line(1,0){3700}}
\put(1800,1200){\tiny $\nicefrac12$}
\put(1300,-250){$\btd$}
\end{picture}$&&$\begin{picture}(2400,2700)(1500,-1800)
\put(-2100,0){\tiny{[2]}}
\multiput(0,0)(1800,0){2}{\usebox{\vertex}}
\put(1800,-1800){\usebox{\vertex}}
\put(0,0){\line(1,0){1800}}
\put(1800,0){\line(0,-1){1800}}
\put(1800,0){\usebox{\susp}}
\put(5400,0){\usebox{\rightbiedge}}
\put(0,0){\circle*{600}}
\put(1800,-1800){\circle*{600}}
\end{picture}$&
$\begin{picture}(2400,2700)(3300,-900)
\put(0,1200){[\nicefrac12]}
\multiput(0,0)(1800,0){2}{\usebox{\vertex}}
\multiput(1800,-1800)(1800,1800){2}{\usebox{\vertex}}
\multiput(0,0)(1800,0){2}{\line(1,0){1800}}
\put(1800,0){\line(0,-1){1800}}
\put(3600,0){\usebox{\shortsusp}}
\put(5400,0){\usebox{\edge}}
\put(7200,0){\usebox{\shortsusp}}
\put(9000,0){\usebox{\rightbiedge}}
\put(5400,0){\circle*{600}}
\put(7200,0){\circle*{600}}
\put(7200,600){\tiny 2}
\end{picture}$\\
B1) & B2) && B3) & B4)\\
\hline
$
\begin{picture}(3600,2400)(-300,-900)
\put(200,500){\tiny{[\nicefrac12]}}
\put(0,0){\usebox{\rightbiedge}}
\put(1800,0){\usebox{\leftbiedge}}
\multiput(0,0)(3600,0){2}{\usebox{\wcircle}}
\put(1800,-600){\usebox{\wcircle}}
\multiput(0,-250)(3600,0){2}{\line(0,-1){950}}
\put(0,-1200){\line(1,0){3600}}
\end{picture}
$&
$
\begin{picture}(5400,1800)(-1500,-900)
\put(-1800,0){\tiny{[2]}}
\put(0,0){\usebox{\rightbiedge}}
\put(1800,0){\usebox{\leftbiedge}}
\multiput(0,0)(3600,0){2}{\circle*{600}}
\end{picture}$&
$
\begin{picture}(7500,2400)(-300,-900)
\put(3000, -900){\tiny 1/2}
\put(0,0){\usebox{\rightbiedge}}
\put(5400,0){\usebox{\leftbiedge}}
\put(1800,0){\usebox{\shortam}}
\multiput(0,0)(7200,0){2}{\usebox{\wcircle}}
\multiput(0,-250)(7200,0){2}{\line(0,-1){950}}
\put(0,-1200){\line(1,0){7200}}
\end{picture}
$&
$
\begin{picture}(9000,1800)(900,-900)
\put(0,0){\usebox{\rightbiedge}}
\put(9000,0){\usebox{\leftbiedge}}
\put(1800,0){\usebox{\shortsusp}}
\put(3600,0){\usebox{\dynkinathree}}
\put(7200,0){\usebox{\shortsusp}}
\multiput(3600,0)(3600,0){2}{\circle*{600}}
\end{picture}
$&$
\begin{picture}(7200,2400)(-300,-900)
\put(0,0){\usebox{\rightbiedge}}
\put(5400,0){\usebox{\leftbiedge}}
\put(1800,0){\usebox{\susp}}
\put(1500,-350){$\btr$}
\put(5199,-350){$\btl$}
\end{picture}
$
\\
C1) & C2) & C3) & C4) & C5\\
\hline
$\begin{picture}(2400,5000)(1500,-1500)
\put(1800,2100){\tiny[1/2]}
\multiput(900,0)(1800,0){2}{\usebox{\edge}}
\put(2700,0){\usebox{\vedge}}
\put(2700,1800){\usebox{\vedge}}
\put(2700,1800){\usebox{\vertex}}
\put(900,0){\circle*{600}}
\put(2700,-1800){\circle*{600}}
\end{picture}$
&$\begin{picture}(5400,5000)(-1500,-1500)
\put(-1800,600){\tiny[1/2]}
\multiput(0,0)(1800,0){2}{\usebox{\edge}}
\put(1800,0){\usebox{\vedge}}
\put(1800,1800){\usebox{\vedge}}
\put(1800,1800){\usebox{\vertex}}
\multiput(1800,1800)(0,-3600){2}{\circle{600}}
\put(1800,0){\circle*{600}}
\multiput(2050,1800)(0,-3600){2}{\line(1,0){2150}}
\put(4200, 1800){\line(0,-1){3600}}
\end{picture}$&$ \begin{picture}(2400,5000)(-0000,-300)
\put(-2700,-200){\tiny[1/2]}
\put(0,0){\usebox{\vertex}}
\multiput(-1200,1200)(0,-2400){2}{\usebox{\vertex}}
\put(-1200,-1200){\line(1,1){1200}}
\put(-1200,1200){\line(1,-1){1200}}
\put(000,0){\usebox{\susp}}
\put(3600,0){\usebox{\bifurc}}
\multiput(-1200,1200)(0,-2400){2}{\circle*{600}}
\end{picture}$ & $\begin{picture}(9000,1800)(00,-900)
\put(-900,2400){\tiny[1/2]}
\put(0,0){\usebox{\vertex}}
\multiput(-1200,1200)(0,-2400){2}{\usebox{\vertex}}
\put(-1200,-1200){\line(1,1){1200}}
\put(-1200,1200){\line(1,-1){1200}}
\put(0,0){\usebox{\shortsusp}}
\put(1800,0){\usebox{\dynkinafour}}
\put(7200,0){\usebox{\shortsusp}}
\put(9000,0){\usebox{\bifurc}}
\multiput(3600,0)(1800,0){2}{\circle*{600}}
\end{picture}$&$\begin{picture}(9000,5000)(-2100,-900)
\put(0,0){\usebox{\vertex}}
\multiput(-1200,1200)(0,-2400){2}{\usebox{\vertex}}
\put(-1200,-1200){\line(1,1){1200}}
\put(-1200,1200){\line(1,-1){1200}}
\put(000,0){\usebox{\susp}}
\put(3600,0){\usebox{\bifurc}}
\multiput(-1200,1200)(0,-2400){2}{\circle{600}}
\multiput(4800,1200)(0,-2400){2}{\circle{600}}
\multiput(-1200,-1200)(25,0){20}{\circle*{70}}
\multiput(-700,-1200)(0,25){12}{\circle*{70}}
\multiput(-700,-900)(25,0){12}{\circle*{70}}
\multiput(-400,-900)(0,25){12}{\circle*{70}}
\multiput(-400,-600)(25,0){12}{\circle*{70}}
\multiput(-100,-600)(0,25){12}{\circle*{70}}
\multiput(-100,-300)(25,0){12}{\circle*{70}}
\multiput(200,-300)(25,25){8}{\circle*{70}}
\multiput(400,-100)(25,-25){8}{\circle*{70}}
\multiput(600,-300)(25,25){8}{\circle*{70}}
\multiput(800,-100)(25,-25){8}{\circle*{70}}
\multiput(1000,-300)(25,25){8}{\circle*{70}}
\multiput(1200,-100)(25,-25){8}{\circle*{70}}
\multiput(1400,-300)(25,25){8}{\circle*{70}}
\multiput(1600,-100)(25,-25){8}{\circle*{70}}
\multiput(1800,-300)(25,25){8}{\circle*{70}}
\multiput(2000,-100)(25,-25){8}{\circle*{70}}
\multiput(2200,-300)(25,25){8}{\circle*{70}}
\multiput(2400,-100)(25,-25){8}{\circle*{70}}
\multiput(2600,-300)(25,25){8}{\circle*{70}}
\multiput(2800,-100)(25,-25){8}{\circle*{70}}
\multiput(3000,-300)(25,25){8}{\circle*{70}}
\multiput(3200,-100)(25,-25){8}{\circle*{70}}
\multiput(3400,-300)(25,0){12}{\circle*{70}}
\multiput(3700,-300)(0,-25){12}{\circle*{70}}
\multiput(3700,-600)(25,0){12}{\circle*{70}}
\multiput(4000,-600)(0,-25){12}{\circle*{70}}
\multiput(4000,-900)(25,0){12}{\circle*{70}}
\multiput(4300,-900)(0,-25){12}{\circle*{70}}
\multiput(4300,-1200)(25,0){20}{\circle*{70}}
\multiput(-1200,+1200)(25,0){20}{\circle*{70}}
\multiput(-700,+1200)(0,-25){12}{\circle*{70}}
\multiput(-700,+900)(25,0){12}{\circle*{70}}
\multiput(-400,900)(0,-25){12}{\circle*{70}}
\multiput(-400,600)(25,0){12}{\circle*{70}}
\multiput(-100,600)(0,-25){12}{\circle*{70}}
\multiput(-100,300)(25,0){12}{\circle*{70}}
\multiput(200,300)(25,-25){8}{\circle*{70}}
\multiput(400,100)(25,25){8}{\circle*{70}}
\multiput(600,300)(25,-25){8}{\circle*{70}}
\multiput(800,100)(25,25){8}{\circle*{70}}
\multiput(1000,300)(25,-25){8}{\circle*{70}}
\multiput(1200,100)(25,25){8}{\circle*{70}}
\multiput(1400,300)(25,-25){8}{\circle*{70}}
\multiput(1600,100)(25,25){8}{\circle*{70}}
\multiput(1800,300)(25,-25){8}{\circle*{70}}
\multiput(2000,100)(25,25){8}{\circle*{70}}
\multiput(2200,300)(25,-25){8}{\circle*{70}}
\multiput(2400,100)(25,25){8}{\circle*{70}}
\multiput(2600,300)(25,-25){8}{\circle*{70}}
\multiput(2800,100)(25,25){8}{\circle*{70}}
\multiput(3000,300)(25,-25){8}{\circle*{70}}
\multiput(3200,100)(25,25){8}{\circle*{70}}
\multiput(3400,300)(25,0){12}{\circle*{70}}
\multiput(3700,300)(0,25){12}{\circle*{70}}
\multiput(3700,600)(25,0){12}{\circle*{70}}
\multiput(4000,600)(0,25){12}{\circle*{70}}
\multiput(4000,900)(25,0){12}{\circle*{70}}
\multiput(4300,900)(0,25){12}{\circle*{70}}
\multiput(4300,1200)(25,0){20}{\circle*{70}}
\end{picture}$
\\
D1) & D2) & D4)& D5) & D3), D6)\\
\hline
&&$\begin{picture}(2400,1800)(2400,-900)
\put(0,0){\usebox{\dynkinatwo}}
\put(1800,0){\usebox{\dynkinffour}}
\put(0,0){\circle*{600}}
\put(7200,0){\circle*{600}}
\end{picture}
$
&
$
\begin{picture}(2400,1800)(2400,-200)
\put(1200,600){\tiny[1/2]}
\put(0,0){\usebox{\dynkinatwo}}
\put(1800,0){\usebox{\dynkinffour}}
\put(1800,0){\circle*{600}}
\put(5400,0){\circle*{600}}
\end{picture}
$ &$\begin{picture}(2400,1800)(2400,-900)
\put(0,0){\usebox{\dynkinatwo}}
\put(1800,0){\usebox{\dynkinffour}}
\put(5400,0){\usebox\atwo}
\put(3300,-350){$\btl$}
\end{picture}
$\\
&&F1) & F2)&F3)\\
\hline
$ \begin{picture}(3600,1800)(0,-900)
\put(0,0){\usebox\dynkinathree}
\multiput(1800,200)(0,-400){2}{\line(1,0){1800}}
\put(3300,0){\line(-1,1){500}}
\put(3300,0){\line(-1,-1){500}}
\put(3600,0){\circle*{600}}
\put(-300,-350){$\btr$}
\end{picture}$
&
$ \begin{picture}(3600,1800)(0,-500)
\put(0,0){\usebox\dynkinathree}
\multiput(1800,200)(0,-400){2}{\line(1,0){1800}}
\put(3300,0){\line(-1,1){500}}
\put(3300,0){\line(-1,-1){500}}
\put(3600,0){\usebox\aone}
\put(1300,-350){$\btl$}
\end{picture}$
&
$\begin{picture}(3600,1800)(0,-500)
\put(0,0){\usebox\dynkinathree}
\multiput(1800,200)(0,-400){2}{\line(1,0){1800}}
\put(3300,0){\line(-1,1){500}}
\put(3300,0){\line(-1,-1){500}}
\multiput(0,0)(3600,0){2}{\circle{600}}
\multiput(0,300)(3600,0){2}{\line(0,1){600}}
\put(0,900){\line(1,0){3600}}
\put(1400,-200){$\bt$}
\end{picture}$&&\\
G1) & G2) & G3)&&\\
\hline
$\begin{picture}(2400,2400)(-300,-900)
\multiput(0,0)(1800,0){2}{\circle*{300}}\thicklines
\multiput(0,-60)(0,120){2}{\line(1,0){1800}}
\multiput(0,-180)(0,360){2}{\line(1,0){1800}}
\multiput(150,0)(25,25){20}{\circle*{50}}
\multiput(150,0)(25,-25){20}{\circle*{50}}
\put(0,0){\usebox{\aprime}}
\put(1800,0){\usebox{\aone}}
\end{picture}$ &
$\begin{picture}(2400,2400)(-300,-900)
\multiput(0,0)(1800,0){2}{\circle*{300}}\thicklines
\multiput(0,-60)(0,120){2}{\line(1,0){1800}}
\multiput(0,-180)(0,360){2}{\line(1,0){1800}}
\multiput(150,0)(25,25){20}{\circle*{50}}
\multiput(150,0)(25,-25){20}{\circle*{50}}
\put(1500,-350){$\btl$}
\put(0,0){\usebox{\aone}}
\end{picture}$&$\begin{picture}(7200,2400)(-300,-900)
\multiput(0,0)(5400,0){2}{\usebox{\leftbiedge}}
\put(1800,0){\usebox{\susp}}
\put(0,0){\usebox{\vertex}}
\put(5400,0){\circle*{600}}
\put(5400,600){\tiny 2}
\put(7200,0){\usebox{\aone}}
\end{picture}$
&
$\begin{picture}(7200,1800)(1500,-900)
\multiput(0,0)(9000,0){2}{\usebox{\leftbiedge}}
\put(1800,0){\usebox{\susp}}
\put(3600,0){\usebox\dynkinatwo}
\put(5400,0){\usebox\susp}
\put(0,0){\usebox{\vertex}}
\put(3600,0){\circle*{600}}
\put(5300,-350){$\btr$}
\end{picture}$&$\begin{picture}(2400,2400)(-300,-900)
\put(-1800,0){\usebox{\edge}}
\put(0,0){\usebox{\vedge}}
\put(0,0){\usebox{\shortsusp}}
\put(1800,0){\usebox{\leftbiedge}}
\put(-1800,0){\usebox{\aone}}
\put(0,0){\circle*{600}}
\end{picture}
$\\
A6) & A7)& A8) & A9)&A10)\\
\hline
&$
\begin{picture}(3600,3600)(-900,-900)
\put(-1800,0){\usebox{\edge}}
\put(0,0){\usebox{\vedge}}
\put(0,0){\usebox{\shortsusp}}
\put(1800,0){\usebox{\leftbiedge}}
\put(1800,0){\circle*{600}}
\put(3600,0){\usebox{\aone}}
\end{picture}$&
$\begin{picture}(3600,3600)(-300,-900)
\put(-1800,0){\usebox{\edge}}
\put(0,0){\usebox{\vedge}}
\put(0,0){\usebox{\edge}}
\put(1800,0){\usebox{\shortsusp}}
\put(3600,0){\usebox{\leftbiedge}}
\put(-1800,0){\circle{600}}
\put(0,-1800){\circle{600}}
\put(1800,0){\circle*{600}}
\multiput(-1800,0)(25,-25){13}{\circle*{70}}
\multiput(-1475,-325)(25,25){7}{\circle*{70}}
\multiput(-1300,-150)(25,-25){7}{\circle*{70}}
\multiput(-1125,-325)(25,25){7}{\circle*{70}}
\multiput(-950,-150)(25,-25){7}{\circle*{70}}
\multiput(-775,-325)(25,25){7}{\circle*{70}}
\multiput(-600,-150)(25,-25){7}{\circle*{70}}
\multiput(-425,-325)(25,25){7}{\circle*{70}}
\multiput(-250,-150)(25,-25){7}{\circle*{70}}
\multiput(-75,-325)(-25,-25){7}{\circle*{70}}
\multiput(-250,-500)(25,-25){7}{\circle*{70}}
\multiput(-75,-675)(-25,-25){7}{\circle*{70}}
\multiput(-250,-850)(25,-25){7}{\circle*{70}}
\multiput(-75,-1025)(-25,-25){7}{\circle*{70}}
\multiput(-250,-1200)(25,-25){7}{\circle*{70}}
\multiput(-75,-1375)(-25,-25){7}{\circle*{70}}
\multiput(-250,-1550)(25,-25){12}{\circle*{70}}
\end{picture}$& $\begin{picture}(7200,3000)(-300,-1900)
\put(-1800,0){\usebox{\edge}}
\put(0,0){\usebox{\vedge}}
\put(0,0){\usebox{\edge}}
\put(1800,0){\usebox{\susp}}
\put(5400,0){\usebox{\leftbiedge}}
\put(-2100,-350){$\btr$}
\put(-500,-2100){$\bt$}
\end{picture}$
&
$\begin{picture}(7200,3000)(-300,-1900)
\put(-1800,0){\usebox{\edge}}
\put(0,0){\usebox{\vedge}}
\put(0,0){\usebox{\edge}}
\put(1800,0){\usebox{\shortsusp}}
\put(3600,0){\usebox{\dynkinatwo}}
\put(5400,0){\usebox{\shortsusp}}
\put(7200,0){\usebox{\leftbiedge}}
\put(3600,0){\circle*{600}}
\put(5100,-350){$\btr$}
\end{picture}$\\
& A11) & A12)&A13)&A14)\\
\hline
$ \begin{picture}(3600,2400)(-300,00)
\put(0,0){\usebox{\vertex}}
\put(0,0){\usebox{\leftbiedge}}
\put(1800,0){\usebox{\rightbiedge}}
\multiput(0,0)(3600,0){2}{\circle{600}}
\multiput(0,300)(3600,0){2}{\line(0,1){300}}
\put(0,600){\line(1,0){3600}}
\put(1500,1200){\tiny $\nicefrac12$}
\put(1500,-350){$\btl$}
\end{picture}$&
$ \begin{picture}(3600,2400)(0,-900)
\put(0,0){\usebox{\vertex}}
\put(0,0){\usebox{\leftbiedge}}
\put(1800,0){\usebox{\rightbiedge}}
\put(-200,-350){$\btr$}
\put(3400,-350){$\btl$}
\end{picture}$&$ \begin{picture}(3600,2400)(-150,-900)
\put(-2400,0){\tiny[1/2]}
\put(0,0){\usebox{\vertex}}
\put(0,0){\usebox{\leftbiedge}}
\put(3600,0){\usebox{\rightbiedge}}
\put(1800,0){\usebox{\dynkinatwo}}
\multiput(0,0)(3600,0){2}{\usebox{\wcircle}}
\multiput(1800,0)(3600,0){2}{\usebox{\wcircle}}
\multiput(0,-250)(3600,0){2}{\line(0,-1){950}}
\multiput(1800,250)(3600,0){2}{\line(0,1){950}}
\put(0,-1200){\line(1,0){3600}}
\put(1800,1200){\line(1,0){3600}}
\end{picture}$
&
$\begin{picture}(2400,1800)(1500,-900)
\put(-2400,0){\tiny[1/2]}
\put(0,0){\usebox{\vertex}}
\put(0,0){\usebox{\leftbiedge}}
\put(3600,0){\usebox{\rightbiedge}}
\put(1800,0){\usebox{\dynkinatwo}}
\multiput(0,0)(5400,0){2}{\circle*{600}}
\end{picture}$\\
D8) & D9) &D10)& D11& \\
\hline
&&$
\begin{picture}(9000,1800)(-300,-00)
\put(0,900){\tiny [2]}
\put(0,0){\usebox{\vertex}}
\put(0,0){\usebox{\leftbiedge}}
\put(7200,0){\usebox{\rightbiedge}}
\multiput(1800,0)(3600,0){2}{\usebox{\shortsusp}}
\put(3600,0){\usebox{\dynkinatwo}}
\put(3600,0){\circle*{600}}
\put(5400,0){\circle*{600}}
\end{picture}
$
&
$
\begin{picture}(9000,1800)(-300,00)
\put(0,0){\usebox{\vertex}}
\put(0,0){\usebox{\leftbiedge}}
\put(5400,0){\usebox{\rightbiedge}}
\put(1800,0){\usebox{\shortam}}
\multiput(0,0)(7200,0){2}{\usebox{\wcircle}}
\multiput(0,250)(7200,0){2}{\line(0,1){950}}
\put(0,1200){\line(1,0){7200}}
\end{picture}
$&\\
&&D12) & D13) & \\
\hline
&&$
\begin{picture}(9000,2400)(-300,-900)
\put(0,0){\usebox{\dynkinathree}}
\put(3600,0){\usebox{\leftbiedge}}
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&&E1) & E2) & E3)\\
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&&D14) & D15) & D16)\\
\hline
\end{longtable}
\begin{rem}\leavevmode\\
\begin{enumerate}
\item Knop showed in \cite{Kno14} that the case A2 (without the factor 1/2) corresponds to the disymmetric manifold $\nicefrac{\SU(4)}{SO(4)}\times \nicefrac{\SU(4)}{\SP(4)}$.
\item The three possible cases $\w(A5),2\w(A4),4\w(A1)$ for $A_1^{(1)}$ correspond, in that order, to the manifolds $S^4$ (the so-called "spinning 4-sphere", \cite{AMW02}), $S^2\times S^2$ and $\PP^2(\C)$ by \cite{Kno14}, pp. 36.
\item Knop \cite{Kno14} determined the Manifold that corresponds to case C5). More precisely, this spherical pair comes from the $\SP(2n)$-manifold structure on the quaternionic Grassmanian: $M=\operatorname{Gr}_k(\mathbb{H}^{n+1})$. This is a generalization of a result from \cite{Esh09}.
\end{enumerate}
\end{rem}
\part{Hamiltonian Manifolds of Rank one}
Let us recall that every q-Hamiltonian manifold localy looks like a Hamiltonian manifold, cf. \cite{Kno14}. Conversely, every Hamiltonian manifolds carries a quasi-Hamiltonian structure. We shall use this to apply our theory to classify momentum polytopes of compact Hamiltonian manifolds of rank one.\\
Hence, speaking in the quasi-Hamiltonian setting, we now want to look at momentum polytopes of rank one that do not hit the wall of the fundamental alcove corresponding to $\a_0$, but every other wall. That means we can forget about the existence of this one wall and find ourselves in the Weyl chamber of a classical root system that is in bijection with the orbit space $\mathfrak{k^*}/K$ of a classical Hamiltonian manifold, more details in \cite{Kno11}. We again have the classification theorem that this manifolds are characterized by spherical pairs.\\
We classify all compact convex Hamiltonian manifolds such that the corresponding spherical pair $(\P,\L_S)$ has the following properties:
\begin{itemize}
\item The moment polytope $\P=[X_1X_2]$ is a line segment touching every wall of the dominant chamber, meaning $\a(X_1)=0$ or $\a(X_2)=0$ for everey simple root $\a\in\{\a_1, \dots, \a_n\}$, that does not touch $\mathbf 0$.
\item The lattice $\L_S$ has rank one and its generator $\w$ generates $\P$ as a line segment.
\end{itemize}
Most of our theory now works similar. We have the following structure theorems:
\begin{thm}
There are no bi-homogeneous polytopes of rank one for compact Hamiltonian $K$-manifolds.
\end{thm}
\begin{proof}
Suppose there was a bi-homogeneous polytope. Hence the weight $\w_1$ in $X_1$ and the weight $\w_2$ in $X_2$ are $\Z$-linear combinations of simple roots. As we suppose our polytope to hit every wall of the alcove, $\w_2=-\w_1$ could only be possible if there was a linear dependence between the simple roots (Which is given via the root $\d$ in the affine case!). But as the simple roots of a classical root system are well known to be linearly independent, this is not possible.
\end{proof}
We can also find an analogon for \cref{inthm}:
\begin{thm}
Let $G$ be simple of rank $n$ and $\w$ the weight of an inhomogeneous smooth affine spherical variety of rank one that is a local model in $X_1$ with $\la \w, \a_k^\vee \ra =1$. Then, for $X_2=X_1+c\w$, we have
\[
S(X_2)= S \setminus \{\a_k\}
\]
\end{thm}
\begin{proof}
This is just the proof of \cref{inthm} with some slight adjustments: We again assume $S(X_1) =\{\a_1, \dots, \a_k\}$ after renumbering. This time, our $\w$ has $n-k$ free coordinates, and $X_1$ also has $n-k$ free parameters. We argue as above that $\a_1, \dots, \a_{k-1}\in S(X_2), \a_k \notin S(X_2)$. It is also possible for the polytope to touch every wall of the alcove because for that, it is necessary to solve a homogeneous system of $n-k$ equations with $2(n-k)+1$ free parameters. As the equations are determined by simple roots (which are linearly independent), the system is always solvable.
\end{proof}
We deduce the same way as for genuine q-Hamiltonians:
\begin{cor} \leavevmode\\
\begin{enumerate}
\item Bi-inhomogeneous polytopes are only possible if $|S(X_1)|=| S(X_2)|=n-1$ and, if $S(X_1)=S \setminus \a_k , \la \w, \a_j^\vee \ra =1$ we must have $ S(X_2)=S \setminus \a_j$ and $\la -\w, \a_k^\vee \ra =1$.
\item Let $\w$ be a homogeneous polytope for $X_1$. Mixed polytopes are only possible if $|S(X_1)|=n-1$.
\end{enumerate}
\end{cor}
\begin{proof}\leavevmode\\
\begin{enumerate}
\item Reason the same way as in the proof of \cref{incor1}
\item Same reasons as for \cref{mixcor2}
\end{enumerate}
\end{proof}
It follows:
\begin{cor}
Let us consider a compact Hamiltonian $K$-manifold of rank one for $K$ simple such that the corresponding moment polytope touches every wall of the chamber, but not in the origin. Then we can find its spherical pair by examining all $X_1$ with $|S(X_1)|=n-1$.
\end{cor}
We go on and explicitly determine these polytopes, using the same short notation we used for the q-Hamiltonians. If something does not work, the reason is always \cref{inh1}. We shall again give the case-by-case-study for $A_n$ and refer to \cite{Pau} for details on the other cases (that work very similar).
For $A_n$, we have:
\begin{center}
\tiny{
\begin{longtable}{|p{3cm}|p{3cm}|p{3cm}|p{4cm}|}
\hline
$S(X_1)$ & choice of $\w$ & $ S(X_2)$ & $-w$ inhom.?\\\endhead
\hline
$\sm{1}$ & $\aa_{1,n}$ & $\ssm{1}{n}$& $-\w\in I_1$ \\
\hline
$n=2$ & 2$\a_2$ &$\sm{2}$ & no: $\Sp{1}=2$ \\
\hline
$n=4$ & $\aa_2+2\aa_3+\aa_4$ & $\sm{3}$ & $-w\in I_4$ \\
\hline
$n=4$ & $\frac12(\aa_2+2\aa_3+\aa_4)$ & $\sm{3}$ & no: $\Sp{4}=\frac12$, \\
\hline
$\sm{d}, 0<d<n$ & $\a_{1,d}$ & $\sm{1,d}$ & $-\w\in I_{d-1}$ \\
\hline
$2<d<n-1$& $I_1$ & $\sm{1}$ & no: $\Sp{d}>0$ \\
\hline
& $I_{d-1}$ & $\sm{d-1}$ & $-\w\in I_d$ \\
\hline
$d=2$ & $I_1$ & $\sm{1}$ & $-\w\in I_2$ \\
\hline
$d=4$ & $[\frac12]\aa_1+2\aa_2+\aa_3$ & $\sm{2}$ &no: $\Sp{4}>0$\\
\hline
$d=2$ & $2\a_1$ & $\sm{1}$ & no: $\Sp{2}=2$ \\
\hline
$n=3, d=2$ & $\aa_1+\aa_3$ & $\a_2$ & no: $\Sp{2}=2$ \\
\hline
& $\frac12(\aa_1+\aa_3)$ & $\a-2$ & $-\w\in I_2$\\
\hline
\end{longtable}
}
\end{center}
\section{List of Hamiltonian manifolds of rank one}
\begin{thm}[Momentum Polytopes of Rank one]
A compact multiplicity free Hamiltonian $K$-manifold of rank one for $K$ simple with moment polytope $\P=[X_1X_2]$ such that $\a(X_1)=0$ or $\a(X_2)=0$ for every simple root $\a_1, \dots, \a_n$ corresponds to a spherical pair $(\P,\L_S)$ with the following properties:
\begin{itemize}
\item $\P$ is a line segment $[X_1X_2]:=(X_1+\R\w)\cap \bar\A$ for some $\w$ found in the list below and $X_1$ such that $S(X_1)$ is the local root system indicated.
\item The lattice $\L_S$ is generated by this $\w$
\end{itemize}
\end{thm}
\begin{lis}\label{ham1} List of Hamiltonian manifolds of rank one
\tiny{
\begin{longtable}{|p{0.7cm}|p{2cm}|p{3.2cm}|p{2cm}|p{3.2cm}||p{1.5cm}|}
\hline
&$\w$ &local model & $-\w$ &local model & remarks\\\endhead
\hline
\multicolumn{6}{|c|}{$A_{3}$} \\
\hline
A1)&$\frac 12 (\a_1+\a_3)\in H(2)$ &$(\sll(2)\times \sll(2),\Delta\sll(2),0)$& $\boldsymbol{-w\in I_2(1,3)}$ &$(\tt^1+\sll(2)+\tt^1,\tt^1+\sll(2)+\tt^1,\C^2_{\nicefrac12,-\nicefrac12})$ & \\
\hline
\multicolumn{6}{|c|}{$A_{4}$} \\
\hline
A2)&$\a_1+2\a_2+\a_3\in H(4)$ &$(\sll(4),\sp(4),0)$& $\boldsymbol{-w\in I_4(2)}$ &$(\tt^1+\sll(3),\tt^1+\sll(3),\C^3_2)$ & \\
\hline
\multicolumn{6}{|c|}{$A_{n}$} \\
\hline
A3)&$\a_1+\dots+\a_k\in H(k+1)$ &$(\sll(k),\gll(k-1),0)$ & $\boldsymbol{-w\in}\linebreak \boldsymbol{I_{k+1}(1,k)}$ &$(\tt^1+\sll(n-k)+\tt^1,\tt^1+\sll(n-k)+\tt^1,\C^{n-k}_{1,1})$ & \\
\hline
A4)&$ \boldsymbol{\w\in I_k(k+1)}$ &$(\tt^1+\sll(k),\tt^1+\sll(k),\C^k_{-\nicefrac{k}{n+1}})$ & $\boldsymbol{-\w\in I_{k+1}(k)}$ & $(\tt^1+\sll(n-k),\tt^1+\sll(n-k),\C^{n-k}_{\nicefrac{n-k}{n+1}})$& \\
\hline
\addlinespace[0.3cm]
\hline
\multicolumn{6}{|c|}{$B_{3}$} \\
\hline
B1)&$\frac 12 (\a_1+\a_3)\in H(2)$ & $(\sll(2)\times \sll(2),\Delta\sll(2),0)$ & $\boldsymbol{-w\in I_2(1,3)}$ &$(\tt^1+\sll(2)+\tt^1,\tt^1+\sll(2)+\tt^1,\C^2_{\nicefrac12, -\nicefrac12})$ & \\
\hline
B2)&$\boldsymbol{\w \in I_3(1)}$ &$(\tt^1+\sp(4),\tt^1+\sp(4),\C^4_{\nicefrac12})$ & $\boldsymbol{-\w \in I_1(3)}$ & $(\tt^1+\sp(4),\tt^1+\sp(4),\C^4_{-\nicefrac12})$ &\\
\hline
\multicolumn{6}{|c|}{$B_{4}$} \\
\hline
B3)&$\a_2+2\a_3+3\a_4 \in H(1)$ &$(\soo(7),G_2,0)$ & $\boldsymbol{-w\in I_1(4)}$ & $(\tt^1+\sll(4),\tt^1+\sll(4),\C^4_{-3})$ & \\
\hline
B4)&$\frac 12 \a_1+\a_2+\frac 12 \a_3 \in H(4)$ & $(\sll(4),\sp(4),0)$ & $\boldsymbol{-\w \in I_4(2)}$ & $(\tt^1+\sp(4),\tt^1+\sp(4),\C^4_{-1})$ & \\
\hline
\multicolumn{6}{|c|}{$B_{n}$} \\
\hline
B5)&$\a_{k+1}+\dots+\a_n\in H(k)$ &$(\soo(2(n-k)+1),\soo(2(n-k)),0)$ & $\boldsymbol{-w\in}\linebreak \boldsymbol{I_k(k+1)}$ &$(\tt^1+\sll(k),\tt^1+\sll(k),\C^k_{-1}) $ & \\
\hline
B6)&$\boldsymbol{\w\in I_{n-1}(n)}$ & $(\tt^1+\sll(n),\tt^1\sll(n),\C^n)$ & $\boldsymbol{-\w\in}\linebreak \boldsymbol{ I_n(n-1)}$ & $(\tt^1+\sll(2),\tt^1+\sll(2),\C^2_{-\nicefrac32})$ & \\
\hline
\addlinespace[0.3cm]
\hline
\multicolumn{6}{|c|}{$C_{n}$} \\
\hline
C1)&$\a_1+\dots+\a_{k}\in H(k+1)$ & $(\sll(k),\gll(k-1),0)$ & $\boldsymbol{-w\in}\linebreak\boldsymbol{ I_{k+1}(1,k)}$ & $(\tt^1+\sp(2n-2k),\tt^1+\sp(2n-2k),\C^{2n-2k}_{-1})$ & $1\le k <n$ \\
\hline
C2) &$\a_2+2\a_3+\dots+2\a_{n-1}+\a_n \in H(1)$& $(\sp(2(n-1)),\sll(2)+\sp(2n-3),0)$ & $\boldsymbol{-\w\in I_1(3)}$ &$(\tt^1+\sll(3),\tt^1+\sll(3),\C^3_{-1})$ & \\
\hline
C3)&$\boldsymbol{\w \in I_k(k+1)}$ &$(\tt^1+\sll(k+1),\tt^1+\sll(k+1),\C^{k+1}_{-1})$ & $\boldsymbol{-\w\in I_{k+1}(k)}$ &$(\tt^1+\sp(2(n-k)),\tt^1+\sp(2(n-k)),\C^{2n-2k})$ & \\
\hline
\addlinespace[0.3cm]
\hline
\multicolumn{6}{|c|}{$D_{4}$} \\
\hline
D1)&$\frac 12 (\a_i+2\a_2+\a_j) \in H(4)$ &$(\sll(4),\sp(k),0)$ & $\boldsymbol{-w\in I_k(2)}$ &$(\tt^1+\sll(2),\tt^1+\sll(2),\C^2_{-1})$ & $i,j,k\in 1,3,4; i \ne j \ne k$ \\
\hline
D2)&$\frac 12 (\a_i+\a_j) \in H(2)$ & $(\sll(2)\times \sll(2),\Delta\sll(2),0)$& $ \boldsymbol{-\w \in I_2(i,j)}$ & $(\tt^1+\sll(3)+\tt^1,\tt^1+\sll(3)+\tt^1,\C^3_{-\nicefrac12,-\nicefrac12})$ & $i,j\in 1,3,4; i \ne j$\\
\hline
D3)&$\a_i+\a_2+\a_j \in H(k)$ &$(\sll(4),\gll(3),0)$& $\boldsymbol{-\w \in I_k(i,j)}$&$(\tt^1+\sll(3)+\tt^1,\tt^1+\sll(3)+\tt^1,\C^3_{1,-1})$ & $i,j,k\in 1,3,4; i \ne j \ne k$\\
\hline
\multicolumn{6}{|c|}{$D_{n}$} \\
\hline
D4)&$\a_{k+1}+\dots+\a_{n-2}+\frac 12 \a_{n-1}+\frac 12 \a_n \in H(k)$ &$(\soo(2(n-k)),\soo(2(n-k)-1),0)$\ & $\boldsymbol{-\w \in}\linebreak \boldsymbol{ I_k(k+1)}$& $(\tt^1+\sll(k),\tt^1+\sll(k),\C^{k}_{-1})$ & \\
\hline
D5)&$\a_1+\dots + \a_{n-1} \in H(n)$ & $(\sll(n),\gll(n-1),0)$ & $\boldsymbol{-\w \in}\linebreak\boldsymbol{ I_n(1,n-1)}$ &$(\tt^1+\sll(n-1)+\tt^1,\tt^1+\sll(n-1)+\tt^1,\C^{n-1}_{1,-1})$ & or n and n-1 switched \\
\hline
D6)&$\boldsymbol{\w\in I_{n-1}(n)}$ & $(\sll(n),\sll(n),\C^n_{-\nicefrac12})$ & $\boldsymbol{-w\in}\linebreak\boldsymbol{ I_n(n-1)}$ &$(\sll(n),\sll(n),\C^n_{-\nicefrac12})$ & or n and n-1 switched \\
\hline
\pagebreak
\hline
\multicolumn{6}{|c|}{$F_{4}$} \\
\hline
F1)&$\a_2+2\a_3+\a_4 \in H(1)$ &$(\sp(6),\sll(2)+\sp(4),0)$ & $\boldsymbol{-w\in I_1(3)}$ &$(\tt^1+\sll(3),\tt^1+\sll(3),\C^3_{-2})$ & \\
\hline
F2)&$\a_3+\a_4\in H(2)$ & $(\sll(3),\gll(2),0)$ & $ \boldsymbol{-\w \in I_2(3,4)}$ &$(\tt^1+\sll(3)+\tt^1,\tt^1+\sll(3)+\tt^1,\C^3_{-1,-1})$& \\
\hline
F3)&$\a_1+\a_2+\a_3 \in H(4)$ &$(\soo(7),\soo(6),0)$ & $\boldsymbol{-\w \in I_4(1)}$ & $(\tt^1+\sp(6),\tt^1+\sp(6),\C^6_{-1})$ & \\
\hline
F4)&$\boldsymbol{\w \in I_3(2)}$ &$(\sll(3),\sll(3),\C^3_{2})$ & $\boldsymbol{-\w \in I_2(3)}$ &$(\tt^1+\sll(3),\tt^1+\sll(3),\C^3_{-2})$ & \\
\hline
\addlinespace[0.3cm]
\hline
\multicolumn{6}{|c|}{$G_{2}$} \\
\hline
G1)&$\a_1\in H(2)$ &$(\sll(2),0,0)$ & $\boldsymbol{-w\in I_2(1)}$ & $(\tt^1+\sll(2),\tt^1+\sll(2),\C^2_{-1})$ & \\
\hline
\end{longtable}
}
\end{lis}
\begin{longtable}{c|c|c|c|c}
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A1)&A2)&A3)&A4)&B1)\\
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B2) &B3) & B4) & B5) & B6)\\
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&& $
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$
&
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\put(0,0){\usebox\dynkinffour}
\put(1500,-350){$\btl$}
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\end{picture}
$&$\begin{picture}(5000,3000)
\put(0,0){\usebox\dynkingtwo}
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F1&F2&F3) & F4) & G1) \\
\end{longtable} | 195,127 |
Etsy Finds: 10 Handmade Tabletop Accessories
Last week we shared 10 of our favorite kitchen finds, and now we move into the dining room with these ceramic, metal, and fabric tabletop accessories. Handmade by sellers on Etsy, these would certainly add a little warmth and elegance to the table!
TOP ROW
1 Butter Crock, $28 from Abby T Pottery
2 Hibiscus Stay on Coasters, $24 (set of 6) from DimmalimmHome
3 Cream Pansy Ring/Flower Holder, $28 from Janson Pottery
4 Stoneware Olive Dish, $27 from Glazed Over
5 Sphere Creamer, $18 from Rou Designs
BOTTOM ROW
6 Salt and Pepper Cellars, $32 from paulova
7 Organic Cloth Napkins, $39 (set of 12) from Charlotte Handmade
8 Custom Napkin Rings, $24 (set of 4) from Sierra Metal Design
9 Striped Wool Felt Table Runner, $52 from Inbound Thread
10 Coffee Bean Placemat, $28 (set of 4) from Kainkain
Related: Etsy Finds: 10 Handmade Kitchen Accessories
(Images: As linked) | 274,652 |
Ideal Network is a cause-marketing platform. As stated on its website, it uses collective purchasing power to benefit local businesses, schools and non-profits. Like other daily deal sites, such as Groupon, Ideal Network allows you to get great discounts from your favorite local merchants and stores. However, instead of your money going directly to a company, 20%-25% of your purchase automatically is allocated to a local cause of your choice. You are able to support non-profits and schools while still saving money. Everybody wins-- the seller and the buyer, as well as the third-party charity!
It works like this:
1. First, you sign up for ideal network (idealnetwork.com). You immediately start receiving and e-mail daily with a great discount.
2. If you decide to purchase the "ideal" deal, 20-25% of your purchase price goes straight to the campaign of your choice.
3. The cause is able to create meaningful changes within your community!
The causes include breast cancer, schools, wildlife conservation, housing and poverty reduction.
Ideal Network is able to give merchants the opportunity to earn customer's goodwill over time by appealing to a sense of personal ethics. As a result, Ideal Network appeals to merchants who otherwise may not have offered their product at a discount.
For instance, Theo chocolate was the first company to offer a deal through Ideal Network. $20 worth of chocolate were offered for $10. $5 was given to Theo Chocolates while the non-profit and Ideal Network split the remaining $5. Although, offering chocolate at a 50% discount does eat at Theo Chocolate's incoming profit, Joe Whinney the founder stated that, "if it helps to remind customer of what our business values are, it is worth spending money on."
Because the goal, "to raise modest amounts of money for a cause with measurable needs," is attainable and those benefiting are local organizations, Ideal Network is beginning to expand into cities all over the country.
To check it out go to:
-Gabrielle Gurian
SOURCE: IDEAL NETWORK and CROSS CUT | 48,896 |
Visual SQL Query Builder and Parser for .NET, Java and Delphi – Active Query Builder
Active Query Builder – Visual SQL query building component with user-friendly graphical interface and powerul API to build, parse and analyze complex SQL queries
Active Query Builder is a query builder component suite for Borland Delphi 5-7, 2005 and C++ Builder 5,, MySQL, PostgreSQL, InterBase, Informix, SQLite, and ANSI-92 SQL dialects. The sophisticated Active SQL parser can translate any valid SQL statement into it’s visual representation. You can use full range of database connectivity components as well.
Tags: visual query designer, Delphi query builder, sql query designer, graphical query builder, SQL text editor, ASP.NET query builder, sql parser, sql query builder, VCL query builder, visual query builder, Active Query Builder, .NET query builder
| 287,286 |
TITLE: How to prove that the given set is not uncountable?
QUESTION [2 upvotes]: I was trying to solve the question given in my assignment on metric spaces.
Let $S$ be a subset of $R$. Let $C$ be the set of points $x$ in $R$ with the property that $S\cap (x-\delta,x+\delta )$ is uncountable for every $\delta > 0.$ Prove that $S - C$ is finite or countable.
I started like this:
Let $x$ $\in$ $S-C$ $=>$ $S\cap (x-\delta,x+\delta )$ is countable for some $\delta>0$.
Also, $(S-C)\cap(x-\delta,x+\delta )\subset S\cap (x-\delta,x+\delta )$.
So, $(S-C)\cap(x-\delta,x+\delta )$ is countable.
Hence, for every $x$ $ \in$ $S-C$ , $\exists$ a $\delta>0$ such that $(S-C)\cap(x-\delta,x+\delta )$ is countable.
But after that I couldn't advance. Any help would be appreciated.
Thanks.
REPLY [1 votes]: For any $x\in S\setminus C$, let $a_x,b_x$ be defined so that:
$$a_x=\inf\{a\mid S\cap (a,x) \text{ is countable or finite}\}\\
b_x=\sup\{b\mid S\cap (x,b)\text{ is countable or finite}\}$$
There is always such $a_x,b_x$ because $x\notin C$ so $(x-\delta,x+\delta)$ is finite or countable for some $\delta$, and we can show that $(a_x,b_x)\cap S$ is countable or finite, since it is the countable union of $S\cap\left(a_x+\frac{1}{n},b_x-\frac{1}{n}\right)$ each of which must be countable.
The values $a_x,b_x$ can be $-\infty$ or $+\infty$, respectively.
Now, if $x,y\in S\setminus C$ and $y\in (a_x,b_x)$ then $(a_y,b_y)=(a_x,b_x)$.
So we have an equivalence relation on $S\setminus C$ defined as $x\sim y$ if and only if $(a_x,b_x)=(a_y,b_y)$. Modulo this equivalence relation, there is only countably many classes, because $x\sim y$, and the interals $(a_x,b_x)$ and $(a_y,b_y)$ are disjoint of the are not equal.
But then:
$$S\setminus C \subseteq S\cap \bigcup_{i=1}^{\infty}(a_{x_i},b_{x_i})=\bigcup_{i=1}^{\infty} S\cap (a_{x_i},b_{x_i})$$ is countable or finite.
A more direct way to write this same proof is to use that $\mathbb R$ has a countable basis. This also lets you generalize the theorem.
For each $x\in S\setminus C$, let $\delta_x>0$ be chosen so that $I_x=(x-\delta_x,x+\delta_x)$ has the property that $I_x\cap S$ is finite or countable.
Then let $U=\bigcup_{x\in S\setminus C} I_x$.
We see immediately that $S\setminus C\subseteq U$, and that $U$ is open, since it is a union of open sets.
Let $U_1,U_2,U_3,\dots$ be a countable basis for $\mathbb R$. Then pick the countable subset $V_1,V_2,\dots$ such that $V_i\subseteq I_x$ for some $x\in S\setminus C$. Show $U=\bigcup V_i$.
Now, since $V_i\subseteq I_x$ for some $x$, $V_i\cap S\subseteq I_x\cap S$ which is finite or countable.
So $S\cap U=S\cap\bigcup V_i= \bigcup S\cap V_i$ is finite or countable.
But $S\setminus C\subseteq S\cap U$. So $S\setminus C$ is also countable or finite.
Generalization: Let $X$ be a topological space with a countable basis. Let $S\subseteq X$ and let $C$ be the set of elements of $x\in X$ such that $U\cap S$ is uncountable for every open set $U$ of $X$ containing $x$. Then $S\setminus C$ is countable or finite. | 164,291 |
TITLE: how to prove that $f_n\to f$ in $L^1$
QUESTION [1 upvotes]: Let Q be the unit square in $R^2$. Consider functions $f_n\in L^1(Q)$ such that $f_n\to f$ almost everywhere in Q and $\int_{Q}|f_n|\to \int_{Q} |f|<\infty $
(a)prove that $\int_A|f_n|\to \int_A |f| $ for every measurable subset A of Q
(b)prove that $f_n \to f$ in $L^1$
I am just thinking about whether I can use Fatou's theorem to prove part(a), but still have no idea how to prove it.
REPLY [1 votes]: Hint: Apply Fatou's Lemma to $|f_n| + |f| -|f_n-f|.$ (A classic application, but not so obvious on your first encounter.) | 22,652 |
A desktop celebrity: Leo Gomes
If you’re one of our Desktop blog regulars, then you’ve probably already virtually talked to Leo Gomes.
Leo recently hit the 1000-mark in the comments section of this blog. 1000-plus comments is a lot! And, with Leo, it isn’t just a matter of quantity, but also quality – we’re pleased with the engagement he’s shown by objectively and constructively discussing ideas with developers and users alike.
Say “Hi!” to Leo if you pass by him in the comments section. He’s the one with this avatar:
If you’re curious, his avatar is a character from a music video by the Japanese band Scandal. It took a while before we spotted it. Can you? | 139,868 |
The term market correction refers to the downward movement of a financial market or individual security. A market correction is classified as a secondary trend, since they are usually short in duration.
Financial markets, such as commodities, bonds and stocks, typically demonstrate an upward or downward trend over time. Secular trends can last as long as 25 years, while primary trends will last for twelve months or more. Market corrections are secondary trends, which last from as few as a couple of weeks to several months, and may represent the reversal of a bull market.
While there is no strict definition, a commonly used description of a correction would be a decline of ten percent or more in a financial market or individual security. A correction may be a precursor to the start of a bear market, or merely represent an opportunity investors took to lock in capital gains in what might be perceived as an over-valued or over-bought market.
dead cat bounce, bull market, bear market, market trend, market sentiment | 247,264 |
\begin{document}
\title[Classical limit of the quantum Zeno effect]{Classical limit of the quantum Zeno effect}
\author{Paolo Facchi$^{1,2}$, Sandro Graffi$^{3}$, Marilena Ligab\`o$^{1}$,}
\address{$^1$Dipartimento di Matematica, Universit\`a di Bari, I-70125 Bari, Italy}
\address{$^2$Istituto Nazionale di Fisica Nucleare, Sezione di Bari, I-70126 Bari, Italy}
\address{$^3$Dipartimento di Matematica, Universit\`{a} di Bologna, I-40127 Bologna, Italy}
\begin{abstract}
The evolution of a quantum system subjected to infinitely many measurements in a finite time interval is confined in a proper subspace of the Hilbert space. This phenomenon is called ``quantum Zeno effect": a particle under intensive observation does not evolve. This effect is at variance with the classical evolution, which obviously is not affected by any observations. By a semiclassical analysis we will show that the quantum Zeno effect vanishes at all orders, when the Planck constant tends to zero, and thus it is a purely quantum phenomenon without classical analog, at the same level of tunneling.
\end{abstract}
\pacs{03.65.Sq, 03.65.Xp}
\date{\today}
\maketitle
\maketitle
\section{Introduction}
In this paper we study the classical limit of the quantum Zeno
effect in its simplest formulation, namely a free particle subjected to position measurements. The presence of any smooth and bounded potential does not affect our results.
Therefore, let us consider a free quantum particle in $\RM^n$. Its states are
described by vectors in the Hilbert space $\mathcal{H}=L^2(\RM^n)$ and the Schr\"odinger operator is $H=-\hbar^2 \Delta/2m$ with domain the Sobolev space $D(H)=H^2(\RM^n)$.
Let $P=\chi_{\Omega}$ be the orthogonal projection onto a compact set $\Omega \subset \RM^n$ with regular boundary. Here $\chi_{\Omega}$ denotes the characteristic function of the set $\Omega$ ($\chi_{\Omega}(x)$ equals $1$ for $x\in\Omega$ and $0$ otherwise). $P$ is the observable associated to a measurement that ascertains whether or not the particle is in the spatial region $\Omega$.
If one performs $N$ measurements on the particle at regular time intervals of length $t/N$, at the end of this procedure the state of the system is, up to a normalization,
\begin{equation}
\psi_N(t)=(P\, U(t/N)\, P)^N\psi,
\end{equation}
where $\psi$ is the initial state of the particle and $U(t)=e^{-itH/\hbar}$ is the evolution group generated by $H$.
Let
\begin{equation}
V_{N}(t)=(Pe^{-i t H /\hbar N}P)^N.
\label{eq:productformula}
\end{equation}
We are interested in the limit $N \to +\infty$ of the product formula $V_{N}(t)$. In Ref.~\cite{exner} it has been proved that
\begin{theorem}
There exists a set $M \subset \RM$ of Lebesgue measure zero and a strictly increasing sequence $\{N\}$ of positive integers along which we have
\label{lim. Zeno}
\begin{equation*}
\lim_{N \to +\infty}V_{N}(t) \psi =e^{-itH_{\Omega}/\hbar} P \psi,
\end{equation*}
for all $\psi \in \mathcal{H}$ and for all $t \in \RM \setminus M$, where $H_{\Omega}=-\hbar^2 \Delta_{\Omega}/2m$, and $\Delta_{\Omega}$ is the Laplace operator with Dirichlet boundary condition on $\partial \Omega$, that is $D(H_{\Omega}) = H^2(\Omega)\cap H^1_0 (\Omega)$.
\end{theorem}
The limit in Theorem~\ref{lim. Zeno} implies that, if it is possible to perform infinitely many position measurements in the finite time interval $[0,t]$ the probability of finding the particle in the region $\Omega$ in \emph{each} of these measurements reads
\begin{equation}
p_N(t) = \langle \psi_N(t), \psi_N(t)\rangle = \| V_N(t) \psi \|^2 \to \| P \psi \|^2 =1,
\label{eq:QZE}
\end{equation}
for $N\to+\infty$.
This peculiar quantum behavior was named \emph{quantum Zeno effect} by Misra and Sudarshan \cite{misra}.
Since then, the quantum Zeno effect has received constant attention by physicists and mathematicians. For an up-to-date review of the main mathematical and physical aspects, see \cite{ZenoMP} and references therein.
The effect has been observed experimentally in a variety of systems, on
experiments involving photons \cite{kwiat}, nuclear spins \cite{Chapovsky}, ions \cite{balzer2002}, optical pumping \cite{molhave2000}, photons in a cavity \cite{haroche}, ultracold atoms \cite{raizenlatest} and Bose-Einstein condensates \cite{ketterle}. Moreover, these ideas might lead to remarkable applications, e.g.\ in quantum computation and in the control of decoherence.
Of course, the behavior in (\ref{eq:QZE}) is at complete variance with that of a classical particle. Indeed, a free particle with a nonzero initial momentum will eventually escape from the region
$\Omega$ and obviously its motion is not modified by any observations.
More precisely, a particle with initial momentum $\xi\neq 0$ after a time
\begin{equation}
T_\xi = m \delta(\Omega)/ |\xi|
\end{equation}
will be surely found outside $\Omega$, independently of its initial position $x\in\Omega$, where
\begin{equation}
\delta(\Omega)=\sup_{x,y\in\Omega} |x-y|
\end{equation}
is the diameter of $\Omega$.
In this paper we will prove that the quantum Zeno effect is a purely quantum phenomenon, at the same level of tunneling; namely, it cannot be observed at any finite order in $\hbar$, in the limit $\hbar\to 0$.
Notice that, in order to compare classical and quantum dynamics one has to describe them in the same space. In fact, by the Wigner--Moyal formalism, one can give a description of quantum mechanics in classical phase space, which is completely equivalent to the usual description in Hilbert space. Functions $\tau(x,\xi)$ on the phase space $\mathbb{R}^{2n}$ (classical observables) are mapped into operators $T=\textrm{Op}^{W}(\tau)$ on the Hilbert space $L^2(\mathbb{R}^{n})$ (quantum observables) via the Weyl quantization map.
In particular, the noncommutative product of two quantum operators $T_1 T_2$ corresponds to the twisted convolution product (definition recalled below) $\tau_1\sharp\tau_2$ of the classical observables $\tau_1$ and $\tau_2$, while the commutator $[T_1,T_2]$ corresponds to the Moyal bracket $\{\tau_1,\tau_2\}_M$. The main point here is that both $\tau_1\sharp\tau_2$ and $\{\tau_1,\tau_2\}_M$ depend on the Planck constant $\hbar$. When $\hbar\to 0$ they reduce to commutative multiplication and Poisson bracket, respectively, thus restoring classical mechanics. Semiclassical analysis deals with all quantum corrections to classical mechanics at each order in $\hbar$, which are encoded in asymptotic power series in $\hbar$ of $\tau_1\sharp\tau_2$ and $\{\tau_1,\tau_2\}_M$.
In the following we will analyze (a suitable regularization of) the product formula in (\ref{eq:productformula}) with the above-mentioned tools.
Let
\begin{equation}
\tilde{V}_N(t)= P_N(t)\, P_N\left( \frac{N-1}{N}t\right) \dots P_N\left( \frac{2}{N}t\right) P_N\left( \frac{1}{N}t\right) P_N(0),
\label{eq:regproductformula}
\end{equation}
where $P_{N}$ is the multiplication operator by a suitable $C^{\infty}$ mollification of the characteristic function $\chi_{\Omega}$ (see section \ref{sec:reg. proj.}) and $P_{N}(s)$, $s \in \RM$, is the evolution of $P_{N}$ in the Heisenberg picture. Our goal is to prove the following
\begin{theorem}\label{main th.}
Let $M \subset \RM$, $\{N\}$ and $H_{\Omega}$ be as in the Theorem \ref {lim. Zeno}.
\begin{enumerate}
\item
\begin{equation*}
\lim_{N \to +\infty}\tilde{V}_{N}(t) \psi= e^{itH/\hbar} e^{-itH_{\Omega}/\hbar} P \psi,
\end{equation*}
for all $\psi \in \mathcal{H}$ and for all $t \in \RM \setminus M$.
\item $\tilde{V}_{N}(t)$ has a semiclassical symbol $\Theta_N $ and
\begin{eqnarray*}
\Theta_N (x,\xi;t) \sim \sum_{j=0}^{+\infty} \hbar^j \,\Theta_{j,N} (x,\xi;t) ,
\end{eqnarray*}
for $\hbar\to0$ and for all $x , \xi \in \RM^n$.
\item For every $\xi \in \RM^{n}$, $\xi\neq0$, if $t > T_{\xi}:= m \delta(\Omega)/ |\xi|$, one has
\begin{equation*}
\lim_{N \to +\infty}\Theta_{j,N}(x,\xi;t)= 0,
\end{equation*}
uniformly in $x \in \RM$ and $j \in \NM$.
\end{enumerate}
\end{theorem}
Statement (i) of Theorem~\ref{main th.} allows one to replace the product formula (\ref{eq:productformula}) with its regularized version (\ref{eq:regproductformula}), which is more suitable to a semiclassical analysis. Notice, indeed, that, for any $\psi\in\mathcal{H}$, also $\tilde{\psi}_N(t) =\tilde{V}_N(t)\psi$ satisfies Eq.~(\ref{eq:QZE}) and thus is related to the probability of finding the particle in $\Omega$ in all $N$ measurements.
Statement (ii) says that the quantum product formula $\tilde{V}_N(t)$ has a classical counterpart $\Theta_N (x,\xi;t)$ that admits an asymptotic expansion in $\hbar$, and, finally, statement (iii) asserts that each term $\Theta_{j,N}$ of the expansion identically vanishes for $t>T_\xi$ in the limit $N\to\infty$.
This last statement is the main result of this paper. Its physical meaning is the following: we consider the asymptotic expansion of the product formula for $\hbar\to 0$.
About the zero-th order, classical, term we have already discussed: given an initial momentum $\xi\neq 0$, at times $t>T_\xi$ we get $\lim_{N\to \infty} \Theta_{0,N} (x,\xi;t) = 0$, uniformly in $x\in\Omega$, that is the classical particle, initially in $\Omega$, has eventually escaped from that region. Statement (iii) asserts that the \emph{same} feature is shared by \emph{all} quantum corrections, independently of the order in $\hslash$.
\section{Weyl's quantization and Egorov's theorem}
In this section, mainly intended as a set up of the notation, we will briefly recall the tools needed in the following. For all details and proofs we refer to \cite{Sjostrand, Folland, Robert1, Robert2}.
Let us start with Weyl's quantization.
Let $\tau$ be a function in the Schwartz space $\mathcal{S}(\RM^{2n})$. We can define the following operator on $L^{2}(\RM^n)$
\begin{equation}
\textrm{Op}^{W}(\tau)=\int_{\RM^{2n}} e^{i(\xi \cdot X+y \cdot p)}\hat{\tau}(\xi,y)\; \frac{d\xi dy} {(2\pi )^n},
\end{equation}
where $X$ is the position operator $X\psi(x)=x\psi(x)$, $p=\hbar D_x/i$ the momentum operator, with $D_x$ the $n$-dimensional gradient, and the Fourier transform is defined by
\begin{equation}
\hat{\tau}(\xi,y)=\int_{\RM^{2n}} \tau(x, \eta) e^{-i(\xi \cdot x+\eta \cdot y)}\; \frac{dx d\eta}{(2\pi)^n}.
\end{equation}
It is easy to check that if $\tau$ is real then $\textrm{Op}^{W}(\tau)$ is a bounded self-adjoint operator.
The operator $\textrm{Op}^{W}(\tau)$ is called the (Weyl) quantization of the symbol $\tau$. Physically, it is interpreted as the quantum observable corresponding to the classical observable $\tau$.
One can prove that, for any $\psi\in\mathcal{S}(\RM^{n})$
\begin{equation}
(\textrm{Op}^{W}(\tau)\psi)(x)=\int_{\RM^{2n}}
\tau\left(\frac{x+y}{2},\xi\right)e^{-i\xi\cdot(y-x)/\hbar}\psi(y) \;\frac{d\xi dy}{(2\pi \hbar)^n}.
\label{eq:intkern}
\end{equation}
Equation (\ref{eq:intkern}) allows one to extend the quantization map to tempered distributions $\tau$.
We also recall the definition of the twisted convolution product between two symbols $\tau_{1}$ and $\tau_{2}$\begin{equation}
\fl \qquad \tau_{1} \sharp \tau_{2} (x,\xi)=\int_{\RM^{4n}}
\tau_{1}(x_1,\xi_1)\tau_{2}(x_2,\xi_2) e^{\frac{2i}{\hbar}[(x-x_1)\cdot(\xi-\xi_2)-(x-x_2)\cdot(\xi-\xi_1)]}\; \frac{dx_1 d\xi_1 dx_2 d\xi_2}{(\pi \hbar)^{2n}}.
\label{eq:twistedconv}
\end{equation}
The twisted product is the image on the space of symbols of the noncommutative operator product, namely
\begin{equation}
\textrm{Op}^{W}(\tau_{1}\sharp \tau_{2})=\textrm{Op}^{W}(\tau_{1})\, \textrm{Op}^{W}(\tau_{2}).
\end{equation}
The last ingredient we need in our analysis is Egorov's theorem that tell us how the time evolution and the Weyl quantization are related.
We will focus our attention to the case we are interested in, i.e.\ the free Hamiltonian.
In this case the Schr\"odinger operator $H=\textrm{Op}^{W}(\mathcal{H})$ is the Weyl quantization of the Hamiltonian $\mathcal{H}(x,\xi)=\xi^2/2m$.
The time evolution of a classical bounded observable is
$\tau_{t}:=\tau \circ \phi^{\mathcal{H}}_{t}$, where
\begin{equation}
\phi^{\mathcal{H}}_{t}:\RM^{2n}\to \RM^{2n} \quad \phi^{\mathcal{H}}_{t}(x,\xi)=\left(x+\frac{\xi t}{m}, \xi\right)
\end{equation}
is the Hamiltonian flow.
On the other hand, the quantum time evolution of a bounded observable $T=\textrm{Op}^{W}(\tau)$ is
\begin{equation}
T(t)=e^{itH/\hbar}Te^{-itH/\hbar},
\end{equation}
which is a solution of the equation
\begin{equation}\label{eq.quant.}
\dot{T}(t)=\frac{i}{\hbar}[H,T(t)] .
\end{equation}
Let $\tau_{t}^{\hbar}(x,\xi)$ be the symbol of $T(t)$, namely $T(t)=\textrm{Op}^{W}(\tau_{t}^{\hbar})$.
Equation (\ref{eq.quant.}) is mirrored into the following equation for the symbol $\tau_{t}^{\hbar}$ on the phase space
\begin{equation}\label{eq.cl.}
\dot{\tau}_{t}^{\hbar}=\{\mathcal{H}, \tau_{t}^{\hbar}\}_{M}
\end{equation}
with the initial condition $\tau_{0}^{\hbar}= \tau $, where
\begin{equation}
\{f,g\}_{M}=f \sharp g- g\sharp f ,
\end{equation}
is the Moyal bracket.
Solving equation (\ref{eq.cl.}) one finds that, since $\mathcal{H}$ is quadratic in $(x,\xi)$
\begin{equation}\label{eg.th.}
\tau_{t}^{\hbar}= \tau_{t}= \tau\circ \phi^{\mathcal{H}}_{t},
\end{equation}
namely
\begin{equation}
T(t)=e^{itH/\hbar}Te^{-itH/\hbar}=\textrm{Op}^{W}(\tau_{t}^{\hbar})=\textrm{Op}^{W}(\tau \circ \phi^{\mathcal{H}}_{t}).
\end{equation}
Thus, in this case time evolution and quantization commute. For general non quadratic Hamiltonian, the semiclassical Egorov theorem (see \cite{Robert1}) states that (\ref{eg.th.}) holds only at order $0$ in $\hbar$.
\section{A modified product formula}\label{sec:reg. proj.}
The projection $P$ can be considered as a pseudodifferential operator whose symbol is the characteristic function $\varsigma(x,\xi)=\chi_{\Omega}(x)$ of the set $\Omega\times\RM^n$ in the phase space.
However, in order to have a sufficiently smooth symbol, instead of the projection $P$, we consider an operator $0\leq P_{N}\leq 1$ as the Weyl quantization of a symbol which is a $C^{\infty}$ mollification of the characteristic function $\chi_{\Omega}$. Namely, given an $\varepsilon$-neighbourhood of the domain $\Omega$
\begin{equation}
\Omega_{\varepsilon}=\{x\in\RM^n \,|\, d(x,\Omega)<\varepsilon\},
\end{equation}
with $d(x,\Omega)=\inf_{y\in\Omega} |x-y|$ (see Fig.~\ref{fig:mollified}),
we take
\begin{equation}
P_{N} = \textrm{Op}^{W}(\vartheta^{(N)}), \qquad \vartheta^{(N)} (x,\xi)= \chi^{(N)}_\Omega(x) ,
\label{eq:tildeP}
\end{equation}
where
\begin{equation}
\chi_\Omega \leq \chi^{(N)}_\Omega \leq \chi_{\Omega_{1/N^3}}, \qquad
\chi^{(N)}_\Omega \in C^{\infty}(\RM^{n})
\label{eq:chiN}
\end{equation}
is a smoothed approximation of the characteristic function $\chi_\Omega$ supported in $\Omega_{\varepsilon_N}$, with $\varepsilon_N=1/N^3$. See Fig.~\ref{fig:mollified}.
\begin{figure}
\begin{center}
\includegraphics[width=0.7\textwidth]{mollification}
\caption{$\varepsilon$-neighborhood of the compact set $\Omega$ and mollified characteristic function $\chi_{\Omega}^{(N)}$, with $\varepsilon_N=1/N^3$.}
\label{fig:mollified}
\end{center}
\end{figure}
Observe now that, since $N(P_{N}-P) u \to 0$ when $N\to\infty$, one has that
\begin{equation}
N(P_N e^{-it H/\hbar N}P_N - P e^{-it H/\hbar N}P) u \to 0,
\end{equation}
for any $u$ such that $u, Pu \in H^2(\RM^n)$, which is a dense subset of $L^2(\RM^n)$. Therefore, the limit generators of the two discrete semigroups coincide.
By Theorem~\ref{lim. Zeno} it follows that
\begin{lemma}
Let $M$, $\{N\}$ and $H_{\Omega}$ be as in the Theorem \ref {lim. Zeno}. One has
\label{lemma1}
\begin{equation*}
\lim_{N\to +\infty} \left(P_{N}e^{-itH/\hbar N}P_{N}\right)^{N} \psi
=e^{-itH_{\Omega}/\hbar} P\psi,
\end{equation*}
for all $\psi \in \mathcal{H}$ and for all $t \in \RM \setminus M$.
\end{lemma}
\noindent Therefore, in our analysis of the quantum Zeno effect we can use the sequence $\{P_{N}\}$ in place of the projection $P$.
Note that, while the projection $P$ is associated with a yes/no spatial measurement which ascertains whether or not the particle is in the region $\Omega$, its smoothed version $P_{N}$ corresponds to a fuzzy spatial measurement which is not sharp at the boundary of the region. Thus, the physical meaning of the above statement is that the interference effects arising from a small smoothing of the projection do not affect the overall phenomenon.
First let us rewrite $V_{N}(t)$ in a more convenient way.
By using the evolution of $P$ in the Heisenberg picture,
\begin{equation}
P(s)=e^{isH/\hbar}Pe^{-isH/\hbar},
\end{equation}
we obtain
\begin{eqnarray}
V_{N}(t) & = & (Pe^{-itH/N\hbar}P)^N \nonumber\\
& = & e^{-itH/\hbar}P(t)\,P\left( \frac{N-1}{N}t\right) \dots P\left( \frac{2}{N}t\right) P\left( \frac{1}{N}t\right) P(0).
\label{eq:VNnew}
\end{eqnarray}
Now let us substitute in the above equation the projection $P$ with the positive operator $P_{N}$ given by Eq.~(\ref{eq:tildeP}) and neglect the final (trivial) unitary evolution in (\ref{eq:VNnew}). We end up with the following product formula
\begin{equation}\label{op}
\tilde{V}_N(t)= P_N(t)\, P_N\left( \frac{N-1}{N}t\right) \dots P_N\left( \frac{2}{N}t\right) P_N\left( \frac{1}{N}t\right) P_N(0).
\end{equation}
This is the main object of our investigation.
Notice now that from Theorem~\ref{lim. Zeno} and Lemma~\ref{lemma1} we immediately get the following
\begin{corollary}\label{mod.prod.form.}
One gets
\begin{equation*}
\lim_{N \to +\infty}\tilde{V}_{N}(t) \psi= e^{itH/\hbar} e^{-itH_{\Omega}/\hbar} P \psi,
\end{equation*}
for all $\psi \in \mathcal{H}$ and for all $t \in \RM \setminus M$.
\end{corollary}
\noindent This is a reformulation of the quantum Zeno effect: the strong limit of the product formula (\ref{op}) exists and yields a nontrivial evolution. In particular, notice that, for any $\psi\in\mathcal{H}$, also $\tilde{\psi}_N(t) =\tilde{V}_N(t)\psi$ satisfies Eq.~(\ref{eq:QZE}). Observe that Corollary~\ref{mod.prod.form.} is statement $(i)$ of Theorem~\ref{main th.}.
\section{Semiclassical analysis of the quantum Zeno effect}
Now we have all the ingredients to prove the last statements of Theorem~\ref{main th.}. Let us focus on the classical limit of the product formula (\ref{op}).
First we can construct $\vartheta^{(N)}(x,\xi;t)=(\vartheta^{(N)} \circ \phi^{\mathcal{H}}_{t})(x,\xi)$, which, since the Hamiltonian is quadratic, coincides with the symbol of the Heisenberg evolution of $P_N$.
Define for all $k=0, \dots, N$,
\begin{equation}
\vartheta_{k}(x,\xi):=\vartheta^{(N)}\left(x,\xi; {\frac{k t}{N}}\right),
\label{eq:thetak}
\end{equation}
so that the symbol of the operator (\ref{op}), $\tilde{V}_N(t)= \textrm{Op}^{W}(\Theta_N)$, is given by
\begin{equation}\label{twisted N convolution}
\Theta_N = \vartheta_{N}\sharp \vartheta_{N-1}\sharp \dots \sharp \vartheta_{1}\sharp \vartheta_{0}.
\end{equation}
From Eq.~(\ref{eq:twistedconv}), it is not difficult to show that \cite{Robert1}
\begin{eqnarray}
\phi_{1} \sharp \phi_{2} & \sim & \sum_{j=0}^{+\infty}\left(\frac{i \hbar}{2}\right)^j \frac{1}{j!}
\left(D_{x,\phi_{1}}\cdot D_{\xi,\phi_{2}}-D_{x,\phi_{2}}\cdot D_{\xi,\phi_{1}}\right)^{j}\phi_{1} \phi_{2}\nonumber\\
& = & \sum_{j=0}^{+\infty}\left(\frac{i \hbar}{2}\right)^j \frac{1}{j!} \phi_{1} \sharp_{j} \phi_{2} ,
\label{eq:asymconv}
\end{eqnarray}
where
\begin{equation}
\phi_{1} \sharp_{j} \phi_{2}:=\left(D_{x,\phi_{1}}\cdot D_{\xi,\phi_{2}}-D_{x,\phi_{2}}\cdot D_{\xi,\phi_{1}}\right)^{j}\phi_{1} \phi_{2}.
\end{equation}
Here, the subscripts $\phi_{1}$ and $\phi_2$ indicate that the differentiation is to be applied only to $\phi_{1}$ or $\phi_2$.
By plugging (\ref{eq:asymconv}) into (\ref{twisted N convolution}) we finally obtain the desired asymptotic power series in $\hbar$ of the symbol $\Theta_N(t)$ of the product formula $\tilde{V}_N(t)$ in (\ref{op}):
\begin{eqnarray}
\Theta_N (x,\xi;t) \sim \sum_{j=0}^{+\infty}
\hbar^j \,
\Theta_{j,N} (x,\xi;t) ,
\end{eqnarray}
where
\begin{equation}
\Theta_{j,N}:=\frac{i^j}{2^j j!} \sum_{ j_{1},\dots, j_{N}} \delta_{j_{1}+\dots +j_{N},j} \;
\vartheta_{N}\sharp_{j_N}(\vartheta_{N-1}\sharp_{j_{N-1}}( \dots ( \vartheta_{1}\sharp_{j_1} \vartheta_{0}) \dots )) ,
\label{eq:ThetajN}
\end{equation}
with $\delta_{k,l}$ the Kronecker delta.
This is statement $(ii)$ of Theorem~\ref{main th.}.
Observe that $\Theta_{j,N}$ is a function of $x$ and $\xi$ and $t$, namely is a function of the initial position and
momentum of the particle and of the total time of the experiment.
We want to prove that at each order $j$, whatever the initial nonzero momentum, after a certain time the particle is no longer confined in the region of observation. Precisely, we will prove the last statement of Theorem~\ref{main th.}:
\begin{proposition}
For every $\xi \in \RM^{n}$, $\xi\neq0$,
one gets that, for all $t > T_{\xi}:=m \delta(\Omega)/ |\xi|$,
\begin{equation*}
\lim_{N \to +\infty} \Theta_{j,N}(x,\xi;t)=0,
\end{equation*}
uniformly in $x \in \RM$ and $j \in \NM$.
\end{proposition}
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{phase_space}
\caption{Phase space representation of $\Theta_{0,N}(x,\xi)$. The dashed line denotes the initial momentum $\xi\neq0$ of the particle.}
\label{fig:phase_space}
\end{center}
\end{figure}
\begin{proof}
Let us fix the initial momentum of the particle $\xi \neq 0$.
Consider first the case $j=0$,
\begin{equation}
\Theta_{0,N}=\vartheta_{N}\dots\vartheta_{1}\vartheta_{0}.
\end{equation}
By making use of (\ref{eq:thetak}) and (\ref{eq:tildeP}) we get
\begin{equation}
\vartheta_{k}(x,\xi) = \chi^{(N)}_\Omega \left(x+ \frac{k t \xi}{Nm} \right),
\end{equation}
so that
\begin{equation}
\Theta_{0,N}(x,\xi;t) = \chi^{(N)}_\Omega (x+\xi t/m)\dots \chi^{(N)}_\Omega (x+\xi t/(mN)) \chi^{(N)}_\Omega (x).
\end{equation}
Since by Eq.~(\ref{eq:chiN}) $\chi^{(N)}_\Omega$ is a mollification of the characteristic function $\chi_\Omega$, we get that the supports satisfy the equation
\begin{equation}
\fl \qquad \mathrm{supp} [\Theta_{0,N}(\cdot,\xi;t)]\subset
\mathrm{supp} [\vartheta_N(\cdot,\xi)\, \vartheta(\cdot,\xi) ]=
\mathrm{supp} [\chi^{(N)}_\Omega (\cdot+\xi t/m)\,\chi^{(N)}_\Omega].
\end{equation}
Observe that
\begin{equation}
\textrm{supp}[\chi^{(N)}_\Omega (\cdot+\xi t/m)]=\textrm{supp}[\chi^{(N)}_\Omega]-\xi t/m \subset \Omega_{\varepsilon_N} -\xi t/m
\end{equation}
by Eq.~(\ref{eq:chiN}).
Therefore, the support
\begin{equation}
\mathrm{supp} [\Theta_{0,N}(\cdot,\xi;t)]\subset \left(\Omega_{\varepsilon_N}-\xi t/m \right)
\end{equation}
is empty if $t> T_{\xi}^N:=m \delta(\Omega_{\varepsilon_N})/ |\xi|$. See Fig.~\ref{fig:phase_space}.
Therefore, since $T_{\xi}^N \to T_{\xi}:=m \delta(\Omega)/ |\xi|$, we have proved that for any $t > T_{\xi}$
\begin{equation}
\Theta_{0,N}(\cdot,\xi;t)\equiv 0,
\end{equation}
for sufficiently large $N$.
Let us consider now $j > 0$.
Observe that
\begin{equation}
\mathrm{supp} [\Theta_{j,N}(\cdot,\xi;t)]\subset \mathrm{supp} [\Theta_{0,N}(\cdot,\xi;t)]
\end{equation}
therefore, also in this case we have that for $t > T_{\xi}$
\begin{equation}
\Theta_{j,N}(\cdot,\xi;t)\equiv 0,
\end{equation}
for sufficiently large $N$.
\end{proof}
Notice that this result holds for all $N \in \NM$ and $t \in \RM$, thus in particular it holds if we restrict $N$ and $t$ as in the hypothesis of Theorem~\ref{main th.}.
\section{Concluding remarks}\label{sec:conclusion}
We have shown that the quantum Zeno effect vanishes at all orders in $\hbar$, when $\hbar\to0$,
and thus it is a purely quantum phenomenon without classical analog. Remark that, typically, quantum observables have instead non-zero asymptotic expansions in $\hbar$: elementary examples are (see e.g.\cite{LL1965}, \S\S 50,51) the transition probabilities and the Bohr frequency condition. In the first case the asymptotic expansion yields the quantum corrections to the classical observable evolved along the classical motion, and in the second case all quantum corrections to the classical frequencies.
The quantum Zeno effect is at variance with the above examples. As such, it represents the counterpart of quantum tunneling through a confining barrier: in the quantum realm the first yields perfect localization, while the latter yields leakage and also the tunnelling amplitude vanishes to all orders in $\hbar$. And conversely in the classical realm. However, the analogy we have drawn is not yet totally symmetric. Indeed, quantum tunneling is known to be of order $e^{-1/\hbar}$. In this respect it would be very interesting to know whether the quantum Zeno effect is also exponentially vanishing.
\ack This work is partly supported by the European Community
through the Integrated Project EuroSQIP.
\section*{References} | 106,819 |
Jillian and Jonathan had their destination Modern Jewish Wedding at the Lonesome Valley Resort. Their chuppah over looked an incredible view of the mountains of North Carolina. We were struck by the natural decor the couple used in their reception, especially the old wooden window used as their Jewish Wedding Program. Wood played a prominent role in all of the decor. There was a wood table used as a unique guest book and even a wooden wagon made an appearance during the processional carrying the two cutest wedding party attendants.
Jillian shares the couples’ story.
Jonathan and I went to high school together in Tampa, Florida. Although we were friends throughout high school, we never dated. After graduation, we lost touch as we both went to different colleges. Five years later, I ran into Jonathan while I was in South Florida for law school. As it turned out, he was now living and working in South Florida for his father’s company. Jonathan and I picked up our friendship from where it left off. Eventually, our friendship developed into a committed relationship and he proposed while on a trip in San Juan, Puerto Rico.
Although Jonathan and I are both originally from Florida, we couldn’t find any wedding venues that we loved. We didn’t want to get married on the beach and in Florida that left us with very few options. I spent my summers growing up in Cashiers, North Carolina and it has always been a very special place for me. Jonathan had never been to Cashiers and since we both had vacation time available, we decided to take a trip to Western Carolina to check out some wedding venues.
Thankfully, Jonathan instantly fell in love with Cashiers and the neighboring town of Highlands. We happened upon Canyon Kitchen at Lonesome Valley while we were looking to make dinner reservations. Both of us are “foodies” and couldn’t help but notice their James Beard Nominated Chef, Chef Fleer. After checking out their website, I also realized that they did weddings. We made an appointment with their onsite coordinator, Sarah Jennings, and after walking around the property and taking in the
absolutely amazing view, we knew we had found our venue. Everyone at Canyon Kitchen was easy to work and their food is spectacular.
After securing our venue, we were lucky enough to find our wedding planner, Becca Knuth of Asheville Event Co. Becca helped reign in our vision and provided us with some of the most fabulous vendors and priceless advice. Our wedding didn’t necessarily have a theme to begin with, but we knew we wanted it to be rustic and elegant. We also wanted it to reflect our love of wine and food. The space at Canyon Kitchen was already warm and inviting, so we added elements of birch wood, mercury glass, and wine bottle lanterns as decor. Flora provided a gorgeous display of green, orange, and white flowers as wells as the chuppah for our outdoor Jewish ceremony. “We Got the Beat” Band rocked the whole evening and our wonderful photographer, Jeremy Russell, as well as our videographer, Timm Young, captured all of the greatest moments of the night. Although many of our guests thought we were crazy for having our wedding in a small mountain town, we continue to receive compliments months later thanks largely in part to Becca and our team of vendors. It was truly magical.”
Vendor Resources: Wedding Venue: Canyon Kitchen at Lonesome Valley//Wedding Planner: Becca Knuth, Asheville Event Co//Rehearsal Dinner + Guest Accommodations: Old Edwards Inn//
Floral Design: Flora//Rentals: Classic Event Rental//Photographer: Jeremy Russell//Video: Timm Young Films//Ceremony & Reception Music: We Got the Beat, East Coast Entertainment//Cake Designer: Artista Cakes//Hair: Lalena Poff//Transportation: Young//Ketubah: Simply Two Rings – Lapis Blue Ketubah Artist: Michelle Rummel exclusively sold on Ketubah.com.
{Wedding Short} //Featuring Jillian & Jonathan// from Timm Young Films on Vimeo. | 384,800 |
I caught up with VH1 reality star LaLa Vasquez Anthony recently and I asked her about her bestie Kim Kardashian’s incident of being “flour-bombed” on the red carpet: Before we discuss the Kim K incident, let's pause to chat about Lala's cute purse. She told me that she saw it in Vogue and she" had to have it". And peep how the colors on the purse are the same as the Knicks colors (the team her hubby plays basketball for) So cute!
Back to Kim K, for those of you that don’t know, Kim was walking the red carpet last week to promote her new fragrance line and someone poured floor on her while she was talking to the press. Scandal!
Lala told me this about the incident also known flour-gate:
“It is what it is. You know it’s unfortunate that someone took those measures to make a point but I appreciated the way she handled it. Dusted herself and came back out and kept moving forward”
What do you think of the flour-bomb incident? Should Kim K file charges or let bygones be bygones?
Note: Only a member of this blog may post a comment. | 258,330 |
I've been lamenting for the longest time that our English standards are declining. The clearest sign of this is when official channels start using bad English. I'm not talking about your neighbourhood kopitiam or your bubble tea shop. I'm referring to organisations like SBS Transit and town councils. Check out this sign, also from Mr Brown:
Sometimes, I suspect people have gotten so used to reading superficially, amidst the barrage of information, that words have lost their meaning. In the MRT the other day, I saw this advertisement:
And then, there's the local media. Mediacorp is one of the worst culprits of bad English - I've lost count of the number of times I've spotted mistakes in their news commentary and ticker tape. We used to be able to rely on the Straits Times as being the pillar of good English but not anymore. I don't read the Straits Times regularly but I picked up an issue to see if I could catch them out. It took me all of 10 minutes to spot this error:
If you want to see more grammatical errors in the Straits Times, check out this blog. Some of the errors are really unforgivable.
When I speak to corporate employers, they all sing the same refrain: the new generation of workers is unable to speak and write English competently. Forget about being able to compose fancy marketing speak, it would appear that even stringing a grammatical sentence together is a challenge for some. Some marketing agencies even tell me they've switched from hiring Singaporeans to Hong Kongers because the latter have a better grasp of English.
For shame. What's the point of boasting about our high level of distinctions in the national exams when it doesn't translate into a better grasp of English? It's our first language afterall.
Maybe the answer lies in re-examining the way English is taught in schools as well as better training for English teachers. In the meantime, perhaps posting errors on sites like Mr Brown is the way to go. I'm sure such publicity will motivate establishments to check their signs and publications more conscientiously.
5 comments:
About that Aberdeen MRT poster - I could never figure out what was that thing they showed in the ad - some metallic conical shape stuff. At least for their "spices" ad, it was a pot of curry; that I could relate.
K
THANK YOU! That 'short and sweet' advertisement annoys me every single time I use the MRT.
it's a good time for schools to start asking kids to take pictures of signboards and clips from ST to begin peer-editing. sure to improve the kids' english!
I don't think The Straits Times understands the rule of proximity.
Quince: Yeah, we need a 12-year-old to explain it to ST :D | 48,677 |
'Boring Formless Nonsense' intervenes in an aesthetics of failure that has largely been delimited by the visual arts and its avant-garde legacies. It focuses on contemporary experimental composition in which failure rubs shoulders with the categories of chance, noise, and obscurity
Available: Newton Park
Priest is at his best in [the] musicological sections, lucidly explaining the processes and structures of complex works . . . His judgments lack the obtuseness that dogs writing on lowercase music and his prose style ... has a flair and rhythm that compels attention
The Wire | 23,071 |
TITLE: What are topoi?
QUESTION [17 upvotes]: I have been hearing a lot about the concept of "topos". I asked a friend of mine in the know and he said that topoi are a generalization of sheaves on a topological space. In particular, topoi were usefull when an actual topology was not available. Can anyone elaborate on this or make this idea more clear?
REPLY [13 votes]: Topoi can be looked at from many points of view.
Topoi can be seen as categories of sheaves on (generalized) spaces. Indeed, the premier example of a (Grothendieck) topos is the category $\mathrm{Sh}(X)$ of set-valued sheaves on a topological space $X$. Instead of spaces, also sites work.
Topoi can be seen as generalized spaces. For instance we have a functor from the category of topological spaces to the category of topoi, namely the functor $X \mapsto \mathrm{Sh}(X)$. This functor is fully faithful if we restrict to sober topological spaces. (Soberness is a very weak separation axiom. Every Hausdorff space is sober and so is every scheme from algebraic geometry.)
Many geometrical concepts generalize to topoi, for instance there are: point of a topos, open and closed subtopos, connected topos, continuous map between topoi, coverings of topoi, ...
Topoi can be seen as alternate mathematical universes. The special topos $\mathrm{Set}$, the category of sets and maps, is the usual universe. Any topos admits an "internal language" which can be used for working inside of a topos as if it consisted of plain sets. Any theorem which admits an intuitionistic proof (a proof not using the law of excluded middle or axiom of choice) is valid in any topos.
For instance the statement "For any short exact sequence $0 \to M' \to M \to M'' \to 0$ of modules, the module $M$ is finitely generated if $M'$ and $M''$ are" is such a theorem and therefore also holds in the topos of sheaves on a ringed space. In this way it automatically yields the statement "For any short exact sequence $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F''} \to 0$ of sheaves of $\mathcal{O}_X$-modules, the sheaf $\mathcal{F}$ is of finite type if $\mathcal{F}'$ and $\mathcal{F}''$ are".
In the internal language of some topoi, exotic statements such as "any map $\mathbb{R} \to \mathbb{R}$ is smooth" or "there exists a real number $\varepsilon$ such that $\varepsilon^2 = 0$ but $\varepsilon \neq 0$" hold. This is useful for synthetic differential geometry.
Topoi can be seen as embodiments of logical theories: For any (so-called "geometric") theory $\mathbb{T}$ there is a classifying topos $\mathrm{Set}[\mathbb{T}]$ whose points are precisely the models of $\mathbb{T}$ in the category of sets, and conversely any (Grothendieck) topos is the classifying topos of some theory. The classifying topoi of two theories are equivalent if and only if the theories are Morita-equivalent.
I learned this from the nLab entry on topoi. The main examples for topoi are:
The category $\mathrm{Set}$ of sets and maps.
The category $\mathrm{Sh}(X)$ of set-valued sheaves on any site. Grothendieck conceived topoi because of this example – he needed it for etalé cohomology. The "etalé topology" on a scheme is not an honest topology, but a Grothendieck site.
The effective topos associated to any model of computation. In the internal language of such a topos, the statement "for any natural number $n$, there is a prime number $p > n$" holds if and only of there is a program in the given model of computation which computes, given any number $n$, a prime number $p > n$. The statement "any map $\mathbb{N} \to \mathbb{N}$ whatsoever is given by a Turing machine" is true in many of those topoi.
Topoi can for instance be used
in algebraic geometry to work with generalized topologies like the etalé topology,
in logic to construct interesting models of theories,
in computer science to compare models of computation,
as tools to build bridges between different subjects of mathematics.
Very fine resources for learning about topoi include:
Tom Leinster's informal introduction to topos theory. Start here!
The textbook Sheaves in Geometry and Logic by Saunders Mac Lane and Ieke Moerdijk.
The reference Sketches of an Elephant: A Topos Theory Compendium by Peter Johnstone.
If you are in a hurry, then enjoy Luc Illusie's two-page note in the AMS series "What is …?". | 128,612 |
Poppies on the Wheat
By Helen Hunt Jackson.
--------
I've been promising Daniel to take him out for bagels since he came home for the holidays, but things kept getting in the way -- first family stuff, then travel, then weather -- so on Monday we were finally going to go get our bagels. Except Daniel is spending the next couple of days with Paul's parents in Pennsylvania, so he had to cancel a planned lunch with my father on Tuesday, and we all ended up at Ambrosia because we decided it made more sense to get bagels when Daniel is back and can have them for lunch over the weekend. I dragged Dad and Daniel into Target to get laundry stuff, and after I got home, I picked up Adam from school and took him to the post office to submit photos to a Scholastic contest since the entry had to be postmarked today, then we went to get gas since we've been advised to do that in addition to turning on our taps and various other things in anticipation of the bitter cold that has arrived in the past couple of hours. We also watched The Brothers Grimm, just because we were in the mood, and the new Almost Human. Here are some photos from when we went to the Mormon Temple holiday lights:
I am of SUCH a double mind about the newest Sherlock. Without any spoilers, let me say that I have quibbles but for the most part I howled through the whole thing (in a good way this time). And I am kind of boggled that the writers have now spent 2/3 of the season largely in service to what they think women viewers want to see. It's nearly the opposite of what Paramount has always done to Star Trek, namely sucking up to the young male demographic with vocalized contempt for any other audience (particularly due to the oft-cited belief that women viewers remain loyal even if a show is bending over double to try to please the boys). Gatiss and Moffat could have opted to try to impress the male demographic, yet instead they seem to be writing primarily for an audience of women who already adore them and would stick with them through a much higher concentration of blokey bullshit. As on Doctor Who, this isn't translating into awesome female characters but it is producing a show that's more about relationships than trying to be impressive within its genre as the source material once did. And I find I can't complain about that. | 379,206 |
Clarins Friends and Family Event 20% Off Coupon Code
I’m thinking you should be hauling yourself a little Clarins Palazzo d’Oro…..wanna know why?
Jump for it!
Because you can save 20% Off your entire Clarins purchase using promo code FBFF1109. This ends tomorrow so start stocking up right quickly dear friends.
If you’re not up for online hauling you can step into your local Macy’s (print the coupon below) and get two free skin care samples personalized to your skin type PLUS a free 20 minute skin time facial treatment. Sweet right?
Happy Hauling!
us.clarins.com
Hey Muse.
I kind of miss your old writing style from around three months ago – before you switched to this plain white layout, I think. Your content seems so overly commercialized now, which is sad, because the reason why I subscribed to your blog in the first place was because of your pro-consumer (and not pro-company) posts.
I still think your blog is fab, but I don’t enjoy it as much anymore.
hi d!
Sorry to hear you aren’t loving the blog anymore. That sure is sad! I think the blog has def evolved and gotten larger over time and I can see changes in my writing alot as before I was a bit more creative but sometimes I’m so busy with not only blogging but life and running my office that posts aren’t as “lively” as they once were. Not sure about the pro company posts…if you mean the 20% off from Clarins I was thinking everyone would like to take advantage of that sale for sure. As for pro company in general I don’t think I’m doing anything different than I always did which is relate info about products and such back to you about what I think.
I’m so sorry you aren’t enjoying it as much as you did. I’d love to hear any suggestion about what you’d like to see that would change that!
PS as for the plan white layout it’s temporary
a new layout is in the works. I moved to wordpress so the design process is taking longer than it would from within blogger.
Sad that I took this coupon straight to Macy’s last night and the Clarins lady had left at 6 pm for the night, and the Clinique lady had NO idea what I was talking about or what to give me. I got 2 random products she scrounged up from behind the counter and I didn’t get a facial. Yeah, I’ll be going back some time soon (and hopefully the Clarins lady will be there).
aw shoot andrea! I’d def try again! Says until supplies run out so it’s not like it’s an expiration date or anything. My macy’s has the same prob, Clarins girl is never in, it’s very weird, Macy’s that is, they sometimes don’t have anyone manning the counters
| 332,201 |
SECRECY NEWS
from the FAS Project on Government Secrecy
Volume 2011, Issue No. 79
August 22, 2011
Secrecy News Blog:
- SOME NEW WRINKLES IN NUCLEAR WEAPONS SECRECY
- STERLING DEFENSE ARGUES AGAINST SECRET EVIDENCE
- EUROPEAN UNION SECURITY POLICY, AND MORE FROM CRS
SOME NEW WRINKLES IN NUCLEAR WEAPONS SECRECY
A newly released intelligence guide to document classification markings explains the meaning and proper use of control markings to designate classified information. See "Authorized Classification and Control Markings Register," CAPCO, Volume 4, Edition 2, May 31, 2011:).
STERLING DEFENSE ARGUES AGAINST SECRET EVIDENCE
Prosecutors 'gray."
EUROPEAN UNION SECURITY POLICY, AND MORE FROM CRS
Recent reports from the Congressional Research Service that have not been made readily available to the public include the following.
"The European Union: Foreign and Security Policy," August 15, 2011:
"Standard & Poor's Downgrade of U.S. Government Long-Term Debt," August 9, 2011:
"The Obama Administration's Cybersecurity Proposal: Criminal Provisions," July 29,: | 229,555 |
\begin{document}
\maketitle
\begin{abstract}
We introduce a new approach to an enumerative problem closely linked with the geometry of branched coverings;
that is, we study the number $\Pkhur{\a}{i_2,i_3,\ldots}$ of ways a given permutation (with cycles described
by the partition $\a$) can be decomposed into a product of exactly $i_2$ 2-cycles, $i_3$ 3-cycles,
\emph{etc.}, with certain minimality and transitivity conditions imposed on the factors. The method is to
encode such factorizations as planar maps with certain \emph{descent structure} and apply a new combinatorial
decomposition to make their enumeration more manageable. We apply our technique to determine
$\Pkhur{\a}{i_2,i_3,\ldots}$ when $\a$ has one or two parts, extending earlier work of Goulden and Jackson.
We also show how these methods are readily modified to count \emph{inequivalent} factorizations, where
equivalence is defined by permitting commutations of adjacent disjoint factors. Our technique permits a
substantial generalization of recent work of Goulden, Jackson, and Latour, while allowing for a considerable
simplification of their analysis.
\end{abstract}
\section{Introduction}
We begin with a brief review of standard notation. We write $\a \ptn n$ (respectively, $\a \cmp n$) to
indicate that $\a$ is a partition (composition) of $n$, and denote by $\len{\a}$ the number of parts of $\a$.
The partition with $m_i$ parts equal to $i$ is denoted by $[1^{m_1} 2^{m_2} \cdots]$.
The \emph{cycle type} of a permutation $\p$ in the symmetric group $\Sym{n}$ is the partition of $n$
determined by the lengths of the disjoint cycles comprising $\p$. The conjugacy class of $\Sym{n}$ consisting
of all permutations of cycle type $\l \ptn n$ is denoted by $\Class{\l}$. This notation is extended to allow
$\l$ to be a composition, in which case the ordering of the parts of $\l$ is simply ignored. Members of
$\Class{[1^{n-k}\,k]}$ are called \emph{cycles of length $k$}, or \emph{$k$-cycles}, while elements of
$\Class{[n]}$ are \emph{full cycles} in $\Sym{n}$.
For vectors $\mathbf{j}=(j_1,\ldots,j_m)$ and
$\mathbf{x}=(x_1,\ldots,x_m)$ we use the abbreviations
$\mathbf{x}^\mathbf{j} = x_1^{j_1} \cdots x_m^{j_m}$ and
$\mathbf{j}! = j_1! \cdots j_m!$. Finally, if $f \in
\Q[[\mathbf{x}]]$ is a formal power series, then we write
$[\mathbf{x}^{\mathbf{j}}]\,f(\mathbf{x})$ for the coefficient of
the monomial $\mathbf{x}^{\mathbf{j}}$ in $f(\mathbf{x})$.
\subsection{Factorizations of Permutations into Cycles}
\label{ssec:factorizations}
A \bold{transitive factorization} of a permutation $\p \in \Sym{n}$
is a tuple $F=(\s_r,\ldots,\s_1)$ of permutations $\s_i \in \Sym{n}$
such that (1) $\p = \s_r \cdots \s_1$, and (2) the group
$\ang{\s_1,\ldots,\s_r}$ generated by the factors acts transitively
on $\Sym{n}$. If $\s_i \in \Class{\b_i}$ for $1 \leq i \leq r$ and
$\p \in \Class{\a}$, then one can show~\cite{gj-transitive} that
\begin{equation}
\label{eq:genusdefn}
nr - \sum_{i=1}^r \len{\b_i} \geq n + \len{\a} - 2.
\end{equation}
In the case of equality above, $F$ is said to be \bold{minimal transitive}. For example, since
\begin{align}
\label{eq:cycexmp}
(1\,2\,3\,4\,5)(6\,7\,8)(9)
&=
(3\,4\,7\,9) \cdot
(6\,9\,7\,8) \cdot
(1\,2) \cdot
(2\,4\,5) \cdot (2\,6),
\end{align}
the tuple $((3479),(6978),(12),(245),(26))$ is a factorization of
$(12345)(678)(9)$. (Fixed points have been suppressed in the
factors.) This factorization is easily verified to be minimal
transitive.
Minimal transitive factorizations have been very well studied, with much of this attention stemming from the
fact that they serve geometers as combinatorial models for branched coverings of the sphere by the sphere. In
this context, transitivity guarantees connectedness of the associated covering, while minimality implies the
covering surface is the sphere. For further information on these connections see~\cite{melou-schaeffer} and
the references therein.
The focus of this paper is the class of \bold{cycle factorizations},
by which we mean minimal transitive factorizations, such
as~\eqref{eq:cycexmp} above, whose factors are all cycles of length
at least two.\footnote{This condition avoids the triviality of
having factors equal to the identity.} In particular, given a
composition $\a$ and a sequence $\vec{i}=(i_2,i_3,\ldots)$ of
nonnegative integers (called the \bold{cycle index}), we wish to
determine the number $\Phur{\a}{i}$ of cycle factorizations of any
fixed permutation $\p \in \Class{\a}$ into exactly $i_2$ 2-cycles,
$i_3$ 3-cycles, \emph{etc}. Our results will be formulated in terms
of the generating series
\begin{equation}
\label{eq:hurwitzgs}
\PshurGS{m}(\vec{x},\vec{q}, u) :=
\sum_{n \geq 1} \, \sum_{\vec{i} \,\geq\, \vec{0}}
\sum_{\substack{\a \cmp n \\ \len{\a} = m}}
\Phur{\a}{\vec{i}} \, \vec{q}^{\vec{i}}\,
\frac{x_1^{\a_1}}{\a_1} \cdots \frac{x_m^{\a_m}}{\a_m}
\frac{u^{r(\vec{i})}}{r(\vec{i})!}, \quad\qquad m \geq 1.
\end{equation}
Here, and throughout, $r(\vec{i}) := i_2+i_3 + \cdots$ denotes the
total number of factors in any factorization counted by
$\Phur{\a}{\vec{i}}$, and $\vec{q} = (q_2, q_3, \ldots)$ is a vector
of indeterminates.
The structure of generic cycle factorizations is not well understood and, aside from explicit evaluations of
$\Phur{[n]}{\vec{i}}$ (see~\cite{gj-cactus, springer} and Theorem~\ref{thm:springer} of this paper), little
work has been done on their enumeration. However, a significant effort has been directed toward a natural
specialization of this problem, which is to count what we call \bold{$k$-cycle factorizations}. These are
cycle factorizations whose factors are all $k$-cycles for some fixed $k$.
The case $k=2$ (transposition factors) is particularly important geometrically. Counting \mbox{2-cycle}
factorizations of permutations is known as the \emph{Hurwitz problem}, and dates back to Hurwitz's original
investigations into the classification of almost simple ramified coverings of the sphere by the
sphere~\cite{hurwitz}. The following formula, suggested but not completely proved by Hurwitz himself, gives
the number of \mbox{2-cycle} factorizations of any permutation of cycle type $(\a_1,\ldots,\a_m)$:
\begin{equation}
\label{eq:hurwitz}
n^{m-3} (n+m-2)! \prod_{i=1}^m \frac{\a_i^{\a_i+1}}{\a_i!}.
\end{equation}
Although it has been extensively studied from various points of view, ranging from analytic to combinatorial
(see \cite{melou-schaeffer, goryunov-lando,gj-transitive}, for example), no purely bijective proof of this
striking enumerative formula is known. Such a proof would be of tremendous interest as it could provide
further insight into the underlying geometry. This is particularly true in light of a recent celebrated
result of Ekedahl, Lando, Shapiro, and Vainshtein~\cite{elsv} that identifies the enumeration of
\mbox{2-cycle} factorizations (\emph{i.e.} almost simple coverings) with the evaluation of certain Hodge
integrals, objects of great interest in the intersection theory of the moduli space of curves.
See~\cite{gj-gromov} for further details on these fascinating connections. Indeed, the ultimate goal of our
approach to factorization problems is a full combinatorialization of Hodge integrals.
For arbitrary $k > 2$, counting \mbox{$k$-cycle} factorizations
has not been as thoroughly examined and appears quite difficult.
Substantial progress was made in~\cite{gj-aspects}, where generating
series formulations for the number of such factorizations of
permutations with up to three cycles are given. In particular,
letting $\PkshurGS{m}{k}$ denote the series obtained by specializing
$\PshurGS{m}$ at $u=q_k=1$ and $q_i = 0$ for $i \neq k$, it is shown
there that $x\frac{d}{dx} \PkshurGS{1}{k}(x) = s(x)$ and
\begin{equation}
\label{eq:gjresult}
\PkshurGS{2}{k}(x_1,x_2) =
\log\pr{\frac{s(x_1)-s(x_2)}{x_1-x_2}}
- \frac{s(x_1)^{k} - s(x_2)^k}{s(x_1)-s(x_2)},
\end{equation}
where $s \in \Q[[x]]$ (which depends on $k$) is the unique series
satisfying
\begin{equation}
\label{eq:seqn}
s = x e^{s^{k-1}}.
\end{equation}
A more complicated expression for $\PkshurGS{m}{3}$ is also given in terms of $s$, but the calculations
necessary to evaluate $\PkshurGS{m}{k}$ for $k \geq 4$ become intractable.
\subsection{Equivalence up to Commutation of Disjoint Factors}
\label{ssec:equivalence}
There is a natural equivalence relation on cycle factorizations induced by permitting commutations of
disjoint factors. That is, we say two cycle factorizations are \bold{equivalent} if one can be obtained from
the other by repeatedly exchanging adjacent factors that are disjoint in the sense that no symbol is moved by
both. For example, the following factorizations are equivalent:
\begin{align*}
(3\,4\,7\,9) \cdot
(6\,9\,7\,8) \cdot
(1\,2) \cdot
(2\,4\,5) \cdot (2\,6)
\sim
(1\,2) \cdot
(3\,4\,7\,9) \cdot
(2\,4\,5) \cdot
(6\,9\,7\,8) \cdot
(2\,6).
\end{align*}
Let $\IPhur{\a}{i}$ denote the number of inequivalent cycle
factorizations of $\p \in \Class{\a}$ with cycle index $\vec{i}$. We
shall study these numbers through the series
\begin{equation}
\label{eq:inequivgs}
\IPshurGS{m}(\vec{x},\vec{q}, u) :=
\sum_{n \geq 1} \, \sum_{\vec{i} \,\geq\, \vec{0}}
\sum_{\substack{\a \cmp n \\ \len{\a} = m}}
\IPhur{\a}{\vec{i}} \, \vec{q}^{\vec{i}}\,
\frac{x_1^{\a_1}}{\a_1} \cdots \frac{x_m^{\a_m}}{\a_m}
u^{r(\vec{i})}, \quad\qquad m \geq
1.
\end{equation}
As before, let $\IPkshurGS{m}{k}(\vec{x})$ be the restricted series counting inequivalent \mbox{$k$-cycle}
factorizations obtained from $\IPshurGS{m}$ by setting $u=q_k=1$ and $q_i = 0$ for $i \neq k$ in
$\IPshurGS{m}$.
The problem of counting factorizations up to commutation can
apparently be traced back to Stanley, who originally posed it in the
context of \mbox{2-cycle} factorizations. The first result along
these lines came from Eidswick~\cite{eidswick} and
Longyear~\cite{longyear}, who proved (independently) that the number
of inequivalent \mbox{2-cycle} factorizations of the full cycle
$(1\,2\,\cdots\,n)$ is the generalized Catalan number
\begin{equation}
\label{eq:catalan}
\frac{1}{2n-1}\binom{3n-3}{n-1}.
\end{equation}
Longyear's approach involved commutation of factorizations into a
canonical form. This led to the following cubic functional equation
for the generating series $h(x) = \frac{d}{dx} \PkshurGS{1}{2}(x)$,
from which~\eqref{eq:catalan} is easily deduced:
\begin{equation}
\label{eq:hdefn}
h(x) = 1 + xh(x)^3.
\end{equation}
Springer~\cite{springer} later generalized Longyear's argument to obtain an explicit formula for
$\IPhur{[n]}{i}$. His result is recovered here as Theorem~\ref{thm:ispringer}. Also see~\cite{gj-macdonald}
for an alternative derivation of the number of inequivalent \mbox{$k$-cycle} factorizations of a full cycle.
More recently, Goulden, Jackson, and Latour~\cite{glj-inequivalent}
counted inequivalent \mbox{2-cycle} factorizations of any
permutation $\p \in \Class{[n,m]}$, proving that
\begin{equation}
\label{eq:igjresult}
\PkshurGS{2}{2}(x_1,x_2)
=
\log\pr{1+x_1 x_2 h(x_1) h(x_2) \frac{h(x_1)-h(x_2)}{x_1-x_2}},
\end{equation}
where $h$ is defined by~\eqref{eq:hdefn}. Their method again relies on commutation to canonical form, with
the additional aid of a clever combinatorial construction and an intricate inclusion-exclusion argument.
\subsection{Outline of the Paper and Statement of Results}
\label{ssec:outline}
The primary goal of this paper is to introduce a new technique in the enumeration of both cycle
factorizations and their equivalence classes under commutation. We believe our approach to be of interest
for two principal reasons: First, it conveniently allows one to ignore the fine detail of factorizations
(\emph{i.e.} element-wise analysis) and focus on the grander structure, and second, it makes clearer the
structural parallels between the enumeration of factorizations and their equivalence classes. What follows is
a brief overview of the paper highlighting our main results.
In Section~\ref{sec:preliminaries} the reader is introduced to various constructs and conventions that are
used extensively throughout the paper. Our analysis of cycle factorizations then begins in
Section~\ref{sec:cyclefacts}, where we describe a graphical representation that allows~\eqref{eq:hurwitzgs}
to be viewed as a generating series for a special class of labelled planar maps. Cycle factorizations of full
cycles are then seen to correspond with particularly simple maps, namely \emph{cacti}, which are a natural
generalization of trees. This leads to the recovery of a known explicit formula for the number
$\Phur{[n]}{i}$ of cycle factorizations of a full cycle (see Theorem~\ref{thm:springer}). It also marks our
first encounter with the series $\rt = \rt(x,\vec{q},u) \in \Q[\vec{q},u][[x]]$ defined as the unique
solution of the functional equation
\begin{equation}
\label{eq:rcdefn}
\rt = x e^{u Q(\rt)},
\end{equation}
where $Q(z) \in \Q[\vec{q}][[z]]$ is given by
\begin{equation}
\label{eq:qdefn}
Q(z) := \sum_{k \geq 2} q_k z^{k-1}.
\end{equation}
Clearly $\rt$ is a generalization of the series $s$ given by~\eqref{eq:seqn}, making it no surprise that it
plays a central role in our analysis. In particular, we shall present a graphical decomposition of maps
(called \emph{pruning}) that identifies an algebraic dependence of $\PshurGS{m}$ on $\rt$. This is the
content of Theorem~\ref{thm:cactuspruning}, and is the centerpiece of our method. By exploiting the pruning
decomposition we deduce following extension of~\eqref{eq:gjresult} to factorizations of arbitrary cycle
index.
\begin{thm}
\label{thm:mainthm1} Let $\rt$ and $Q$ be defined as above and, for
$i=1,2$, set $\rt_i = \rt(x_i, \vec{q},u)$. Then
\begin{align*}
\PshurGS{2}(x_1,x_2,\vec{q},u) &= \log\pr{\frac{\rt_1-\rt_2}{x_1-x_2}}
- u\frac{\rt_1 Q(\rt_1) - \rt_2 Q(\rt_2)}{\rt_1-\rt_2}.
\end{align*}
\qed
\end{thm}
In the end, our proof of Theorem~\ref{thm:mainthm1} still rests on an \emph{ad hoc} enumeration that we have not
been able to generalize. Thus a formulation of $\PshurGS{m}$ for arbitrary $m$ remains out of reach. We believe
that it will be more tedious than difficult to extend our methods to arrive at an expression for $\PshurGS{3}$,
but this has only yet been done in a special case. We comment further on these developments in
\S\ref{ssec:furtherresults}.
In Section~\ref{sec:inequiv} we turn to the enumeration of cycle factorizations up to the equivalence defined
in~\S\ref{ssec:equivalence}. By modifying our graphical representation of cycle factorizations to allow for
commutations of adjacent factors, we are led to Springer's formula~\cite{springer} for the number of inequivalent
cycle factorizations of a full cycle (Theorem~\ref{thm:ispringer}). We discover that the unique solution $\irt =
\irt(x,\vec{q},u) \in \Q[\vec{q},u][[x]]$ of the functional equation
\begin{equation}
\label{eq:ircdefn}
\irt = 1 + u \irt Q(x\irt^2)
\end{equation}
plays a role directly analogous to that of $\rt$ in the enumeration of ordered cycle factorizations. Again we
develop a pruning decomposition (Theorem~\ref{thm:inequivpruning}) from which we deduce the following
generalization of~\eqref{eq:igjresult} to factorizations of arbitrary cycle index.
\begin{thm}
\label{thm:mainthm2} Let $\irt$ and $Q$ be defined as above
and, for $i=1,2$, set $\irt_i = \irt(x_i, \vec{q},u)$. Then
\begin{equation*}
\IPshurGS{2}(x_1,x_2,\vec{q},u)
=
\log \pr{\frac{(x_1 \irt_1 - x_2 \irt_2)^2}{(x_1-x_2)(x_1\irt_1^2-x_2\irt_2^2)}}.
\end{equation*}
\qed
\end{thm}
Upon setting $u=q_2=1$ and $q_k = 0$ for $k \neq 2$, the defining equation~\eqref{eq:ircdefn} of $\irt$ transforms
to $\irt = 1 + x \irt^3$, thus identifying $\irt$ it with Longyear's series $h(x)$ (see~\eqref{eq:hdefn}). Under
these same specializations, it is easy to check that Theorem~\ref{thm:mainthm2} reduces to~\eqref{eq:igjresult}.
Again, we have been unable to extend Theorem~\ref{thm:mainthm2} to give a general expression for $\IPshurGS{m}$.
However, we have used our method to deduce a raw form of $\IPkshurGS{3}{2}$, thus extending the
Goulden-Jackson-Latour result in a different direction. See \S\ref{ssec:ifurtherresults} for further comments on
this and related matters.
\section{Preliminaries}
\label{sec:preliminaries}
The definitions and notational conventions described in this section
are used extensively throughout the remainder. We warn the reader
that some of these conventions are nonstandard.
\subsection{Cyclic Lists}
\label{ssec:cycliclists}
A \bold{cyclic list} is an equivalence class under the relation identifying finite sequences that are cyclic
shifts of one another. We write $\olist{a_1,\ldots,a_n}$ for the cyclic list with representative sequence
$(a_1,\ldots,a_n)$. Thus $\olist{a_1,a_2,a_3,a_4} = \olist{a_3,a_4,a_1,a_2}$. Some liberties will be taken
with this notation; for instance, when considering the list $\olist{a_1,a_2,a_3,a_4}$, we adopt the
convention that $a_5$ is to be interpreted as $a_1$, while $a_0=a_4$, \emph{etc.} A cyclic list
$L=\olist{a_1,\ldots,a_n}$ of real numbers is \bold{increasing} (respectively, \bold{nondecreasing}) if one
of its representative sequences is strictly increasing (nondecreasing). If $a_{i-1} \geq a_i$ then the pair
$(a_{i-1},a_i)$ is called a \bold{descent} of $L$.
\subsection{Maps and Polymaps}
\label{ssec:maps}
Recall that a \bold{planar map} (subsequently, a \bold{map}) is a 2-dimensional cellular complex whose polyhedron
is homeomorphic to the sphere. The 0-cells, 1-cells, and 2-cells of a map are its \bold{vertices}, \bold{edges},
and \bold{faces}, respectively. An \bold{isomorphism} of two maps is an orientation-preserving homeomorphism
between their polyhedra which sends $i$-cells to $i$-cells and preserves incidence. We always consider isomorphic
maps to be indistinguishable. If a map is labelled, then this definition of isomorphism is amplified to preserve
all labels.
The \bold{boundary walk} of a face $f$ of a map is the cyclic sequence $\olist{(v_0,e_0),\ldots,(v_{k},e_{k})}$ of
alternating vertices and edges listed in order as they are encountered along a counterclockwise traversal of the
boundary of $f$. (Counterclockwise here means that $f$ is always kept to the left of the line of traversal.) A
subsequence $(e_{i-1},v_i,e_i)$ consisting of two consecutive edges and their common incident vertex is called a
\bold{corner} of $f$.
A map is \bold{2-coloured} if its faces have been painted black and white so every edge is incident with both
a black face and a white face (so no two similarly coloured faces are adjacent). We are interested in the
special class 2-coloured maps for which the boundary walk of every black face is a cycle (\emph{i.e.}
contains no repeated vertices or edges). We call these \bold{polymaps}, and make the following supporting
definitions:
\begin{itemize}
\setlength{\itemsep}{0pt}
\item The black faces of a polymap are called \bold{polygons}. An \bold{$m$-gon} is a black face of
degree $m$.
\item The white faces of a polymap are referred to simply as \bold{faces}.
\item A \bold{corner} of a polymap always refers to a corner of a (white) face.
\item The \bold{rotator} of a vertex $v$ in a polymap is the unique cyclic list of polygons encountered along a
clockwise tour of small radius about $v$.
\end{itemize}
For example, a polymap with 9 polygons and 3 faces is illustrated in
Figure~\ref{fig:polymap}A. The rotator of vertex $v$ is
$\olist{p_1,p_2,p_3}$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.65\textwidth]{CactiPruning-Polymaps.eps}
\caption{(A) A polymap with 8 polygons and 3 faces. (B) The descent structure of a polymap}
\label{fig:polymap}
\end{center}
\end{figure}
\subsection{Descent Structure}
\label{ssec:descents}
Let $\amap$ be a polymap whose polygons are labelled with integers. We shall always regard the edges of
$\amap$ as being labelled, with each edge inheriting its label from the unique polygon that it borders. Let
$f$ be a face of the $\amap$ with boundary walk $\olist{(v_0,e_0),\ldots,(v_k,e_k)}$. If $e_{i-1} \geq e_i$,
then the pair $(e_{i-1},e_i)$ a \bold{descent} of $f$. We say $v_i$ is \bold{at} this descent, and the
corner $(e_{i-1},v_i,e_i)$ is called a \bold{descent corner}. The \bold{descent set} of $f$ is the set
$\{v_i \,:\, e_{i-1} \geq e_i\}$ of all vertices at descents of $f$. The cyclic list obtained by listing the
vertices of this set in the order in which they appear along the boundary walk of $f$ is called the
\bold{descent cycle} of $f$.
\begin{exmp} Consider the outer face $f$ of the polymap in Figure~\ref{fig:polymap}B. The cyclic list of edge labels encountered along
its boundary walk is $\olist{1,2,3,3,2,3,1,3,4}$, so $f$ has 4
descents, namely $(3,3), (3,2), (3,1)$, $(4,1)$. The hollow vertices
$a,b,c$, and $d$ are at these descents, so the descent set of $f$ is
$\{a,b,c,d\}$ and its descent cycle is $\olist{a,b,c,d}$. The
descent corners of the other faces are marked with crosses. \qed
\end{exmp}
\subsection{Constellations}
\label{ssec:constellations}
An \bold{$r$-constellation} on $n$ vertices is a polymap whose vertices are distinctly labelled
$1,2,\ldots,n$, and whose polygons are labelled $1,\ldots,r$ such that the rotator of every vertex is
$\olist{1,2,\ldots,r}$. Figure~\ref{fig:constellation} illustrates a $3$-constellation on $5$ vertices.
\begin{figure}[t]
\begin{center}
\includegraphics[width=.35\textwidth]{CactiPruning-Constellation.eps}
\caption{A 3-constellation on 5 vertices.} \label{fig:constellation}
\end{center}
\end{figure}
Our interest in this special class of polymaps stems from a well
known connection between maps and factorizations. A complete
description of this correspondence can be found
in~\cite{melou-schaeffer} (albeit in dual form\footnote{We have
borrowed the term ``constellation'' from~\cite{melou-schaeffer},
though we caution that the term is used there for an object that is
dual to those considered here.}), so here it will suffice to outline
a version particularly suited to our needs.
\begin{prop}
\label{prop:mainbijection} Minimal transitive factorizations in
$\Sym{n}$ with $r$ factors are in bijection with
\mbox{$r$-constellations} on $n$ vertices. In particular,
factorizations of $\p$ correspond with constellations whose decent
cycles are exactly the cycles of $\p$.
\end{prop}
\begin{proof}[Sketch proof:]
Let $\amap[C]$ be an $r$-constellation on $n$ vertices. From $\amap[C]$ define permutations
$\s_1,\ldots,\s_r$ as follows: To form $\s_i$, first obtain a collection of disjoint cycles by listing the
vertices of each polygon labelled $i$ as they appear in clockwise order around its perimeter, and then let
$\s_i$ be the product of these cycles. (See Example~\ref{exmp:constellation}, below.)
Now set $\p=\s_r \cdots \s_1$. Regarding $\p$ as the sequential product of $\s_1,\ldots,\s_r$, observe that
it acts on a vertex $v$ to move it clockwise around the polygons of $\amap[C]$ following edges labelled
$1,2,\ldots,r,$ in turn. That is, $v$ starts at the descent corner $(r,v,1)$ of some face and is moved by
$\pi$ along the boundary walk until it lands at the next descent corner $(r,v',1)$ of that face. Since
$\p(v)=v'$, the cycles of $\p$ are the descent cycles of $\amap[C]$.
Set $F=(\s_r,\ldots,\s_1)$. We claim $\amap[C] \leftrightarrow F$ is the desired correspondence. Clearly $F$
is a factorization of $\p$, and it is transitive because $\amap[C]$ is connected. Note that $\amap[C]$ has
$n$ vertices and $nr$ edges. If $\p \in \Class{\a}$ and $\s_i \in \Class{\b_i}$ for all $i$ then $\amap[C]$
has $\len{\a} + \sum_i \len{\b_i}$ total faces (white faces $+$ polygons). Since $\amap[C]$ is planar,
Euler's polyhedral formula ($\#\text{vertices}-\#\text{edges}+\#\text{faces}=2$) gives equality
in~\eqref{eq:genusdefn}, so $F$ is minimal transitive.
\end{proof}
\begin{exmp} \label{exmp:constellation} The $3$-constellation of Figure~\ref{fig:constellation} corresponds to the factorization
$F = (\s_3, \s_2, \s_1)$ of $\p=(1\,2\,4)(3)(5)$, where $\s_1 = (1\,5)(2\,4\,3)$, $\s_2=(1)(2\,5)(3)(4)$, and
$\s_3=(1\,5\,3)(2)(4)$. Note that vertices $1,2$ and $4$ are at descents of the outer face, and the cycle
$(1\,2\,4)$ of $\p$ coincides with the descent cycle of that face.\qed
\end{exmp}
\section{Cycle Factorizations}
\label{sec:cyclefacts}
In this section we shall introduce a convenient graphical representation of cycle factorizations as decorated
polymaps, and proceed to give a decomposition of these polymaps that can simplify their enumeration. The
section culminates in a proof of Theorem~\ref{thm:mainthm1}.
\subsection{Proper and $\a$-Proper Polymaps}
\label{ssec:properpolymaps}
Let $F=(\s_r,\ldots,\s_1)$ be a cycle factorization and let $\amap[C]$ be the constellation corresponding to $F$
through the bijection of Proposition~\ref{prop:mainbijection}. Then exactly $r$ of the polygons of $\amap[C]$ are
of degree two or more, while the remainder are 1-gons bounded by loops. Let $\amap_F$ be the polymap obtained by
removing all 1-gons. For example, Figure~\ref{fig:properpolymap}A
\begin{figure}[t]
\begin{center}
\includegraphics[width=.85\textwidth]{CactiPruning-ProperPolymaps.eps}
\caption{(A) The polymap of a cycle factorization. (B) A $(3,1,5)$-proper polymap. } \label{fig:properpolymap}
\end{center}
\end{figure}
shows $\amap_F$ when $F$ is the factorization
\begin{equation}
\label{eq:cycexmp2}
(1\,5\,7\,8\,4)(2)(3\,9\,6)
=
(2\,7\,8\,6) \cdot (2\,6\,3\,9)\cdot(1\,5)\cdot(4\,5\,8)\cdot(5\,9).
\end{equation}
Notice that all rotators of $\amap_F$ are increasing, since every rotator of $\amap[C]$ is
$\olist{1,\ldots,r}$. In fact, the removal of loops does not affect descent structure, in the sense that
$\amap_F$ has precisely the same descent cycles as $\amap[C]$. Moreover, $\amap[C]$ is easily recovered from
$\amap_F$ by attaching loops in the unique manner which makes each rotator $\olist{1,\ldots,r}$.
We say a loopless polymap is \bold{proper} if its polygons are labelled with distinct integers so that all
rotators are increasing. (The polygon labels are assumed to be $1,\ldots,r$, for some $r$, unless otherwise
specified.) Let the \bold{poly-index} of a loopless polymap be the vector $\vec{i}=(i_2,i_3,\ldots)$, where
$i_k$ is the number of $k$-gons it contains. With these definitions, we have:
\begin{prop}
\label{prop:mapping} The mapping $F \mapsto \amap_F$ is a bijection between cycle factorizations in $\Sym{n}$ and
proper polymaps with vertices labelled $1,\ldots,n$. In particular, if $F$ is a factorization of $\p$ of cycle
index $\vec{i}$, then $\amap_F$ is a polymap of poly-index $\vec{i}$ whose descent cycles are precisely the cycles
of $\p$. \qed
\end{prop}
\newcommand{\Pai}{\mathcal{P}_{\!\a}(\vec{i})}
Let $\a=(\a_1,\ldots,\a_m)$ be a composition, and let $\Pai$ be the set of
vertex-labelled proper polymaps of poly-index $\vec{i}$ whose
descent sets are $\Dset{\a}{1},\ldots,\Dset{\a}{m}$, where
\begin{align}
\label{eq:canonicalsets}
\Dset{\a}{j} := \{k \in \N \,:\, \a_1+\cdots+\a_{j-1} < k \leq \a_1+\cdots+\a_j\}.
\end{align}
Under the mapping $F \mapsto \amap_F$, members of $\Pai$ correspond with factorizations of permutations whose
orbits are $\Dset{\a}{1},\ldots,\Dset{\a}{m}$. Since there are $\prod_j (\a_j-1)!$ such permutations, each
admitting the same number of factorizations of given cycle index, we conclude that $|\Pai| = \Phur{\a}{i}
\cdot \prod_j (\a_j-1)!$.
Let us say a proper polymap is \bold{$\a$-proper} if its faces are labelled $1,\ldots,m$ so that face $j$ has
$\a_j$ descents, for $1 \leq j \leq m$. (See Figure~\ref{fig:properpolymap}B, for example.) Notice that any
$\a$-proper polymap of poly-index $\vec{i}$ can be transformed into a member of $\Pai$ by first labelling its
vertices in any of the $\smash{\prod_j \a_j!}$ ways that make $\Dset{\a}{j}$ the descent set of face $j$ for
$j=1,\ldots,m$, and then stripping face labels. In fact, no members of $\Pai$ are duplicated in this process
when $\len{\a} \geq 2$, since the face and polygon labels of $\a$-proper polymaps preclude nontrivial
automorphisms.\footnote{This is not so when $\len{\a}=1$, since a polymap composed of a single polygon has
rotational symmetry.} So for $\len{\a} \geq 2$ we have $|\Pai| = \Pshur{\a}{i} \cdot \prod_j \a_j!$, where
$\Pshur{\a}{i}$ is the number of $\a$-proper polymaps of poly-index $\vec{i}$.
Comparing the expressions above for $|\Pai|$ gives $\Phur{\a}{i} =
\a_1\cdots\a_m \Pshur{\a}{i}$ when $\len{\a} \geq 2$.
Thus~\eqref{eq:hurwitzgs} becomes
\begin{equation}
\label{eq:propergs}
\PshurGS{m}(\vec{x},\vec{q}, u) =
\sum_{n \geq 1} \, \sum_{\vec{i} \,\geq\, \vec{0}}
\sum_{\substack{\a \cmp n \\ \len{\a} = m}}
\Pshur{\a}{i} \, \vec{q}^{\vec{i}}\,
\vec{x}^{\pmb{\a}}
\frac{u^{r(\vec{i})}}{r(\vec{i})!},\qquad\text{for $m \geq
2$}.
\end{equation}
That is, $\PshurGS{m}(\vec{x},\vec{q},u)$ is the generating series
for $\a$-proper polymaps, where $u$ is an exponential marker for labelled polygons, $x_j$
marks descents of face $j$ (for $1 \leq j \leq m$), and $q_k$
records the number of $k$-gons (for $k \geq 2$).
\subsection{Proper Cacti and Factorizations of Full Cycles}
\label{ssec:fullcycles}
\newcommand{\hrt}{\bar{\rt}}
A \bold{cactus} is a polymap with only one face. Hence Proposition~\ref{prop:mapping} implies $\Phur{[n]}{i}$
is the number of proper cacti of poly-index $\vec{i}$ whose sole faces have descent cycle
$(1\,2\,\cdots\,n)$. However, observe that the fixed descent cycle of such a cactus forces all vertex labels
once the location of one is known. Thus $\Phur{[n]}{i}$ is the number of \emph{vertex-rooted} proper cacti of
poly-index $\vec{i}$. (That is, the root marks the location of a canonical label, say 1.)
Let $\rt=\rt(x,\vec{q},u)$ be the generating series for
vertex-rooted proper cacti with respect to total vertices (marked by
$x$), labelled polygons (marked by $u$), and poly-index (marked by
$\vec{q}$). That is,
\begin{equation} \label{eq:rcexpansion}
\rt
:=\sum_{n \geq 1} \sum_{\vec{i} \geq \vec{0}} \Phur{[n]}{\vec{i}} \,\vec{q}^{\vec{i}}
{x^n} \frac{u^{r(\vec{i})}}{r(\vec{i})!}
= x\frac{d}{dx} \PshurGS{1}(x,\vec{q},u).
\end{equation}
Though this may appear to be in conflict with our earlier definition~\eqref{eq:rcdefn} of $\rt$, we now give
a combinatorial decomposition of cacti to show that the two definitions coincide.
Suppose that the root $v$ of a rooted proper cactus $C$ is incident
with $m$ polygons. Detaching these polygons from $v$ results in a
collection $\{C_1,\ldots,C_m\}$ of rooted proper cacti, where the
root of each $C_i$ is incident with only one polygon. (See
Figure~\ref{fig:polycactusdecomp1}.)
\begin{figure}[t]
\begin{center}
\includegraphics[width=\textwidth]{CactiPruning-CactusDecomp.eps}
\caption{Decomposition of a rooted cactus.} \label{fig:polycactusdecomp1}
\end{center}
\end{figure}
The ordering of $C_1,\ldots,C_m$ around $v$ need not be recorded, as it can be deduced from the increasing rotator
condition. Thus
\begin{equation}
\label{eq:hrt}
\rt = x \sum_{m \geq 0} \frac{\hrt^m}{m!} = x e^{\hrt},
\end{equation}
where the series $\hrt=\hrt(x,\vec{q},u)$ counts rooted proper cacti
such as $C_i$ in which $x$ marks only \emph{nonroot} vertices. But
if the root of $C_i$ is incident with a $k$-gon, then removal of
this polygon leaves a $(k-1)$-tuple $C^{1}_{i},\ldots,C^{k-1}_i$ of
rooted proper cacti. (See Figure~\ref{fig:polycactusdecomp1},
right-hand side.) This accounts for a contribution $u q_k \rt^{k-1}$
to $\hrt$, and summing over $k$ yields $\hrt=uQ(\rt)$, where $Q$ is
defined as in~\eqref{eq:qdefn}. Thus~\eqref{eq:hrt} gives $\rt =
xe^{uQ(\rt)}$, in agreement with~\eqref{eq:rcdefn}.
The following result was originally proved bijectively by Springer~\cite{springer}. In fact, the
correspondence between factorizations and cacti employed here was also developed in~\cite{springer}, but from
a less general point of view.
\begin{thm}[Springer~\cite{springer}]
\label{thm:springer} Let $\vec{i}=(i_2,i_3,\ldots)$ be a sequence of nonnegative integers and set
$r=r(\vec{i})=i_1+i_2+\cdots$. Then the number of cycle factorizations of $(1\,2\,\cdots\,n)$ with cycle index
$\vec{i}$ is
$$
\Phur{[n]}{\vec{i}} = \frac{n^{r-1}\,r!}{\prod_{k \geq 2} i_k!}
$$
in the case that $n + r - 1 = \sum_{k \geq 2} ki_k$, and zero otherwise. \qed
\end{thm}
\begin{proof}
From~\eqref{eq:rcexpansion} we have $\Phur{[n]}{\vec{i}} = r!\,[x^n
\vec{q}^{\vec{i}} u^r]\,\rt(x,\vec{q},u)$. This coefficient can be
obtained from $~\eqref{eq:rcdefn}$ through a straightforward
application of Lagrange inversion~\cite{goulden-jackson}. An
alternative proof that relies on a Pr\"ufer-like encoding of cacti
can be found in~\cite{springer}.
\end{proof}
\subsection{Smooth Polymaps, Cores, and Branches}
\label{ssec:smoothpolymaps}
Given~\eqref{eq:propergs}, we now wish to count $\a$-proper polymaps for general $\a$ with $\len{\a} \geq 2$.
To do so we invoke a technique we call \emph{pruning}, whereby polymaps are simplified through the removal of
cacti. In this section we lay the groundwork for this approach, which is then executed in the next.
A \bold{leaf} of a polymap is a polygon that shares exactly one vertex with another polygon, and a polymap
is \bold{smooth} if it does not have any leaves. If $\amap$ is a polymap with at least two faces, then
iteratively removing its leaves results in a unique smooth polymap called the \bold{core} of $\amap$ and
denoted by $\acore$. Labels of $\acore$ are inherited from $\amap$ in the obvious way. See
Figure~\ref{fig:polymapcore}, for example. If $p$ is a polygon of $\amap$ sharing only one vertex $v$ with
$\acore$, then separating $p$ from $v$ results in two components, one of which is a rooted cactus $B$ whose
root vertex is incident only with $p$. We call $B$ a \bold{branch} of $\amap$, and refer to $p$ as its
\bold{root polygon}. The \bold{base corner} of $B$ is the corner of $\acore$ in which $p$ was attached. (See
Figure~\ref{fig:polymapcore}.)
\begin{figure}[t]
\begin{center}
\psfrag{frag:m}{$\amap$} \psfrag{frag:b}{$B$} \psfrag{frag:mc}{$\acore$}
\includegraphics[width=.85\textwidth]{CactiPruning-PolymapCore.eps}
\caption{A proper polymap, its core, and one of its branches.} \label{fig:polymapcore}
\end{center}
\end{figure}
Consider now the case when $\amap$ is proper, with at least two
faces. Let $B$ and $p$ be as described above, and let $f$ be the
face of $\acore$ in which $B$ was attached. Let
$\olist{(v_0,e_0),\ldots,(v_k,e_k)}$ be the boundary walk of $f$,
where the indexing has been chosen so that the sequence
$(e_0,\ldots,e_k)$ is as small as possible in lexicographic order.
It is not difficult to see that this condition specifies
$e_0,\ldots,e_k$ uniquely, and hence also determines a unique value
of $b$, with $0 \leq b \leq k$, such that $(e_{b-1},v_b,e_b)$ is the
base corner of $B$.
The key observation here is that the increasing rotator condition on $\amap$ implies $\olist{e_{b-1},p,e_b}$
is nondecreasing, and that this puts strong restrictions on the possible values of $b$, as is demonstrated by
the following lemma.
\begin{lem}
\label{lem:cyclic} Let $L = \olist{a_0,\ldots,a_k}$ be a cyclic list of real numbers with $d$ descents. If $a \in
\R$ is not in $L$, then there are exactly $d$ values of $i$ with $0 \leq i \leq k$ such that
$\olist{a_{i-1},a,a_{i}}$ is nondecreasing.
\end{lem}
\begin{proof}
Let $\mathcal{P}$ be the polygonal path in the plane connecting the
points $(0,a_0),\ldots,(k,a_k),(k+1,a_0)$, in that order. Let $s_i$
be the $i$-th step of $\mathcal{P}$. Call $s_i$ an \emph{up step}
if $a_i > a_{i-1}$ and a \emph{down step} otherwise. Note that
$\olist{a_{i-1},a,a_i}$ is nondecreasing if and only if either
$a_{i-1} < a< a_i$, or $a > a_{i-1} \geq a_i$, or $a_{i-1} \geq a_i
> a$. Plainly, one of these conditions holds if and only if either
(1) $s_i$ is an up step which the line $y = a$ crosses, or (2) $s_i$
is a down step which this line misses. Since $\mathcal{P}$ begins
and ends at the same $y$-coordinate, the numbers of up steps and
down steps crossed by $y=a$ must be equal. Thus the number of
indices $i$ for which (1) or (2) is satisfied is equal to number of
down steps of $P$. Since down steps reflect descents of $L$, this
completes the proof. See Figure~\ref{fig:cyclelemma} for an
illustration; the dashed line is $a=4$, and steps for which
$\olist{a_{i-1},a,a_i}$ is nondecreasing have been thickened.
\end{proof}
\begin{figure}[t]
\begin{center}
\includegraphics[width=.35\textwidth]{CactiPruning-CycleLemma.eps}
\caption{The proof of Lemma~\ref{lem:cyclic}, with $L=\olist{2,7,1,3,6,5,7,3}$ and $a=4$.} \label{fig:cyclelemma}
\end{center}
\end{figure}
In particular, the lemma implies $\olist{e_{s-1},p,e_s}$ is
increasing for exactly $d$ values of $s$ in the range $0 \leq s \leq
k$, where $d$ is the number of descents of face $f$. Let these
possible $s$-values be $s_1 < \cdots < s_d$. Then since
$\olist{e_{b-1},p,e_b}$ is nondecreasing, we have $b=s_i$ for a
unique $i \in \{1,\ldots,d\}$. We call $i$ the \bold{index} of the
branch $B$.
\begin{exmp} Consider branch $B$ of $\amap$ shown in
Figure~\ref{fig:polymapcore}, and let $f$ be the face of $\acore$ in
which $B$ was attached. The boundary walk of $f$ is
$\olist{(v_0,e_0),\ldots,(v_5,e_5)}$, where we choose
$(e_0,\ldots,e_5)=(4,4,14,8,12,12)$ to be lexicographically minimal.
The base corner of $B$ is $(e_2,v_3,e_3)=(14,v_3,8)$ and the polygon
attached to its root has label 2, so in the notation used above we
have $b=3$ and $p=2$. Note that $f$ has $d=4$ descents, namely
$(14,8)$, $(12,12)$, $(12,4)$, and $(4,4)$. In accordance with the
lemma, $\olist{e_{s-1}, p, e_s}$ is nondecreasing for exactly $d=4$
values of $s \in \{0,\ldots,5\}$, namely $s_1=1$, $s_2=3$, $s_3=4$,
and $s_4=5$. Since $b=s_2$, the index of $B$ is $i=2$. \qed
\end{exmp}
To reiterate, Lemma~\ref{lem:cyclic} guarantees that a given branch can be attached to a face with $d$
descents in \emph{exactly} $d$ positions so as to maintain increasing rotators, and the index of a branch
records which of these positions it occupies.
\subsection{Pruning Cacti}
\label{ssec:pruningcacti}
Let $\Psmhur{\th}{i}$ denote the number of smooth $\th$-proper
polymaps with poly-index $\vec{i}$. For $m \geq 2$ we define the
following smooth polymap analogue of the series $\PshurGS{m}$:
\begin{equation}
\label{eq:smoothgs}
\PsmhurGS{m}(\vec{x},\vec{q},u) :=
\sum_{n \geq 1} \, \sum_{\vec{i} \,\geq\, \vec{0}}
\sum_{\substack{\th \cmp n \\ \len{\th} = m}}
\Psmhur{\th}{i} \, \vec{q}^{\vec{i}} \,
\vec{x}^{\pmb{\th}}
\frac{u^{r(\vec{i})}}{r(\vec{i})!}.
\end{equation}
The following theorem is the centrepiece of our approach to counting cycle factorizations. At first glance
it appears to be the rather transparent statement that arbitrary polymaps can be viewed as the composition of
smooth polymaps with cacti. However, it is worth emphasizing that we are concerned with the decomposition of
\emph{proper} polymaps, and therefore must maintain control of descent structure through the pruning
process. It is Lemma~\ref{lem:cyclic}, and the resulting ``index of a branch'', that enables us to deal with
this subtlety.
\begin{thm}
\label{thm:cactuspruning} Fix $m \geq 2$ and set $\rt_i =
\rt(x_i,\vec{q},u)$ for $i=1,\ldots,m$, where $\rt$ is given
by~\eqref{eq:rcdefn}. Then
\begin{equation*}
\label{eq:cactipruning}
\PshurGS{m}(\vec{x}, \vec{q}, u) =
\PsmhurGS{m}(\vec{\rt}, \vec{q}, u),
\end{equation*}
where $\vec{x} = (x_1,\ldots,x_m)$ and $\vec{\rt} = (\rt_1,\ldots,\rt_m)$.
\end{thm}
\begin{proof}
Let $\a$ be an $m$-part composition, let $\amap$ be an $\a$-proper
polymap. Suppose $\acore$ has $\th_j$ descents in face $j$, for $1
\leq j \leq m$. Then for all $1 \leq j \leq m$ and $1 \leq i \leq
\th_j$, let $\aset{B}^j_i$ be the set of all branches of index $i$
in face $f$ of $\amap$. Assemble all branches of $\aset{B}^j_i$ into
a single rooted proper cactus $C^j_i$ by identifying their root
vertices, and let $\aset{O}_j$ be the ordered forest
$(C^j_1,\ldots,C^j_{\th_j})$. See Figure~\ref{fig:pruningcacti} for
an example of these constructions.
\begin{figure}[t]
\begin{center}
\psfrag{frag:m}{$\amap$} \psfrag{frag:mc}{$\acore$} \psfrag{frag:f1}{$\aset{O}_1$} \psfrag{frag:f2}{$\aset{O}_2$}
\psfrag{frag:b11}{$\aset{B}^1_1$} \psfrag{frag:b12}{$\aset{B}^1_2$} \psfrag{frag:b13}{$\aset{B}^1_3$}
\psfrag{frag:b14}{$\aset{B}^1_4$} \psfrag{frag:b21}{$\aset{B}^2_1$} \psfrag{frag:b22}{$\aset{B}^2_3$}
\psfrag{frag:b23}{$\aset{B}^2_3$}
\includegraphics[width=\textwidth]{CactiPruning-PruningExample.eps}
\caption{Pruning cacti from an $\a$-proper polymap.} \label{fig:pruningcacti}
\end{center}
\end{figure}
We claim that the mapping $\Phi \,:\, \amap
\mapsto (\aset{O}_1,\ldots,\aset{O}_m, \acore)$ is a polygon-preserving
bijection between $\a$-proper polymaps and tuples
$(\aset{F}_1,\ldots,\aset{F}_m,
\amap[S] )$ satisfying the following properties:
\begin{itemize}
\setlength{\itemsep}{0pt}
\item[(a)] $\aset{F}_j$ is an ordered forest containing $\th_j$ rooted proper cacti with a total of $\a_j$ vertices.
\item[(b)] $\amap[S]$ is a smooth $\th$-proper polymap, where $\th=(\th_1,\ldots,\th_m)$.
\item[(c)] The polygon labels of $\amap[S]$ and $\aset{F}_1,\ldots,\aset{F}_m$ together
partition $\{1,\ldots,r\}$.
\end{itemize}
The fact that $\Phi(\amap)$ is indeed a tuple of this type follows from two simple observations: (1) a vertex is
at a descent of face $f$ of $\amap$ if and only if it is at a descent of face $f$ of $\acore$, and (2) every
non-root vertex of a branch of $\amap$ is at a descent of the face in which it lies.
The injectivity of $\Phi$ is immediate because two $\a$-proper polymaps are isomorphic if and only if their
cores are isomorphic and all branches in corresponding faces agree. To prove $\Phi$ is also surjective, let
$(\aset{F}_1,\ldots,\aset{F}_m,\amap[S])$ satisfy (a) through (c), where $\smash{\aset{F}_j =
(C^j_1,\ldots,C^j_{\th_j})}$ for $1 \leq j \leq m$. For fixed $j$ and $i$ with $1 \leq i \leq \th_j$, the
rooted cactus $C^j_i$ can unambiguously be viewed as a collection of branches whose roots have been
identified. Take any such branch, $B$, and let $p$ be the label of its root polygon. Now let
$\olist{(v_0,e_0),\ldots,(v_k,e_k)}$ be the boundary walk of face $j$ of $\amap[S]$, with $(e_0,\ldots,e_k)$
lexicographically minimal. Then $\olist{e_0,\ldots,e_k}$ has $\th_j$ descents according to (b). Since (c)
ensures that $p$ is distinct from $e_0,\ldots,e_k$, Lemma~\ref{lem:cyclic} implies $\olist{e_{s-1},p,e_s}$ is
nondecreasing for exactly $\th_j$ values of $s$, say $0 \leq s_1 < \cdots < s_{\th_j} \leq k$. Attach $B$ to
face $j$ of $\amap[S]$ at corner $(e_{s_i-1},v_{s_i},e_{s_i})$, doing so in the unique manner that leaves the
rotator of $v_{s_i}$ increasing. Repeat this process for all $i$, $j$ and $B$ to iteratively build a polymap
$\amap$. Note that the order in which branches are attached is immaterial, so $\amap$ is well defined. The
fact that $\amap$ is $\a$-proper follows from conditions (a)--(c) and observations (1) and (2) made above.
Since $\Phi(\amap)=(\aset{F}_1,\ldots,\aset{F}_m, \amap[S])$, by construction, we conclude that $\Phi$ is
surjective.
This bijection shows $\Pshur{\a}{i}$ to be the number of tuples $(\aset{F}_1,\ldots,\aset{F}_m,\amap[S])$
satisfying (a)--(c) and with poly-index $\vec{i}$. The result now follows by comparing~\eqref{eq:propergs}
and~\eqref{eq:smoothgs}, and recalling from~\S\ref{ssec:fullcycles} that $\rt$ is the generating series for
rooted proper cacti.
\end{proof}
\subsection{Factorizations of Permutations with Two Cycles}
\label{ssec:twofacepolymaps}
We now conclude this section with a proof of Theorem~\ref{thm:mainthm1}. The proof relies on
Theorem~\ref{thm:cactuspruning} and the following well known result, which can be found
in~\cite{goulden-jackson}. Recall that a \emph{circular permutation} of $\{1,\ldots,n\}$ is a cyclic list
$\olist{a_1,\ldots,a_{n}}$ such that $\{a_1,\ldots,a_{n}\} = \{1,\ldots,n\}$.
\begin{lem}
\label{lem:circperm} There are
$$
n!\,[x^{n-d} y^d]\,\log\pr{\frac{x-y}{xe^y-ye^x}}
$$
circular permutations of $\{1,\ldots,n\}$ having exactly $d$
descents. \qed
\end{lem}
\vspace*{1.5mm}
\noindent\textbf{Proof of Theorem~\ref{thm:mainthm1}:} We first
determine the series $\PsmhurGS{2}(z_1,z_2,\vec{q},u)$ counting
smooth $\a$-proper polymaps with $\len{\a}=2$. Let $\amap[S]$ be
such a polymap, noting that this means $\amap[S]$ is simply a closed
chain of polygons, each incident with exactly two others. (See
Figure~\ref{fig:2facepolymap}. Vertex labels have been suppressed
for clarity.)
\begin{figure}[t]
\begin{center}
\includegraphics[width=.25\textwidth]{CactiPruning-2FacePolymap.eps}
\caption{A smooth proper polymap.} \label{fig:2facepolymap}
\end{center}
\end{figure}
Let $\olist{l_1,\ldots,l_r}$ be the cyclic list of distinct polygon labels encountered along the boundary walk of
face 1. Set $\g_i=(j_1,j_2)$ if the polygon with label $l_i$ is a $(j_1+j_2)$-gon that has $j_1-1$ vertices
incident only with face $1$ and $j_2-1$ incident only with face $2$. Then $\amap[S]$ is fully specified by
$\olist{l_1,\g_1,\ldots,l_r,\g_r}$. For example, the polymap of Figure~\ref{fig:2facepolymap} corresponds with the
cyclic list
$$
\olist{1,(1,1),3,(2,1),5,(3,3),2,(1,2),4,(3,2)}.
$$
Let us say a vertex incident with only one face is \emph{internal} to that face; the remaining vertices are
\emph{extremal}. Extremal vertices are hollow in Figure~\ref{fig:2facepolymap}.
Clearly all vertices internal to a given face are at descents of
that face. Therefore, temporarily ignoring extremal vertices, a
polygon with $j_f-1$ vertices internal to face $f$ (for $f=1,2$)
contributes $u q_{j_1+j_2} z_1^{j_1-1} z_2^{j_2-1}$ to
$\PsmhurGS{2}(z_1,z_2,\vec{q},u)$. Sum over $j_1, j_2 \geq 1$ to
define
\begin{align*}
\delta := \sum_{j_1, j_2 \geq 1} u q_{j_1+j_2} z_1^{j_1-1} z_2^{j_2-1}
= u\frac{Q(z_1)-Q(z_2)}{z_1-z_2}.
\end{align*}
Now the sole extremal vertex incident with polygons $l_{i-1}$ and
$l_i$ is at a descent of face 1 if $(l_{i-1},l_i)$ is a descent of
$\olist{l_1,\ldots,l_r}$, and at a descent of face 2 otherwise. Thus
Lemma~\ref{lem:circperm} gives
\begin{align*}
\PsmhurGS{2}(z_1,z_2, \vec{q}, u)
&= \log\pr{\frac{x-y}{xe^y-ye^x}}\Bigg|_{x = z_1\delta,\; y = z_2\delta} \\
&= \log\pr{\frac{z_1-z_2}{z_1e^{-uQ(z_1)}-z_2e^{-uQ(z_2)}}}
- u\pr{\frac{z_1 Q(z_1)-z_2 Q(z_2)}{z_1-z_2}}.
\end{align*}
The result follows immediately from Theorem~\ref{thm:cactuspruning}
and identity~\eqref{eq:rcdefn}. \qed
\subsection{Further Results}
\label{ssec:furtherresults}
As mentioned in the introduction, we have been unable to extend the \emph{ad hoc} enumeration applied in our proof
of Theorem~\ref{thm:mainthm1} to give a general formulation of $\PshurGS{m}$ for $m > 3$. The problem, of course,
is that structure of smooth polymaps on $m$ faces gets far more complicated as $m$ increases; in particular,
polygons can be incident with three or more faces. The case $m=3$ appears within reach (though tedious), and the
more refined enumeration offered by $\PshurGS{3}$ (as compared with Goulden and Jackson's series $\PshurGS{3}{k}$)
may provide some insight into the more general structure.
We have had somewhat more success in applying our techniques to analyze the Hurwitz problem. Observe that the
polymaps associated with \mbox{2-cycle} factorizations (through Proposition~\ref{prop:mapping}) can be simplified
by collapsing \mbox{2-gons} to single edges, thereby eliminating all polygons and leaving \emph{maps} in the
ordinary sense. In particular, cacti become trees, so \S\ref{ssec:fullcycles} gives a bijection between
\mbox{2-cycle} factorizations of $(1\,2\,\cdots\,n)$ and rooted trees with edges labelled $1,\ldots,n-1$. These
trees, in turn, correspond with trees on $n$ labelled vertices, as can be seen by ``pushing'' edge labels away
from the root onto vertices and assigning label $n$ to the root. Since there are $n^{n-2}$ trees on $n$ labelled
vertices, we have an elegant bijective proof of Hurwitz's formula~\eqref{eq:hurwitz} in the case $m=1$. This
bijection is equivalent to that given by Moszkowski~\cite{moszkowski}.
In fact, the argument used in~\S\ref{ssec:fullcycles} to associate factorizations of full cycles with rooted
cacti applies more generally to give a bijection between cycle factorizations of any $\p \in \Class{\a}$ and
\mbox{$\a$-proper} polymaps with one distinguished descent in each face. We have used this correspondence
(restricted to \mbox{2-cycle} factorizations) together with the pruning bijection established in
Theorem~\ref{thm:cactuspruning} to give the first bijective proofs of~\eqref{eq:hurwitz} when $m=2$ and
$m=3$. (Details can be found in~\cite{irvingphd}.) Thus the simplicity of Hurwitz's formula in these cases is
explained by a corresponding simplicity in the structure of certain smooth maps with two and three faces. We
have as yet been unable to extend our methods to $m \geq 4$, but we hope that the shift in viewpoint afforded
by pruning can be further exploited in this direction.
We should mention that the techniques employed here apply equally well to transitive factorizations
$F=(\s_r,\ldots,\s_1)$ not constrained by the minimality condition. If $\s_i \in \Class{\b_i}$ and
$\s_r\cdots\s_1 \in \Class{\a}$, then a parity argument applied to~\eqref{eq:genusdefn} shows $nr - \sum_{i=1}^r
\len{\b_i} = n + \len{\a} - 2 + 2g$ for a unique integer $g \geq 0$ called the \emph{genus} of $F$. Thus minimal
transitive factorizations are of genus 0. Unsurprisingly, genus $g$ factorizations correspond with genus $g$
polymaps, and many of the results described here (in particular, Theorem~\ref{thm:cactuspruning} and
Theorem~\ref{thm:inequivpruning} of the next section) have obvious higher genus analogues. Our focus on minimal
transitive factorizations reflects our belief that understanding the combinatorics in genus 0 is key to an
understanding in all genera.
The graphical approach can also be used to study what we call \emph{$\b$-factorizations}, which are minimal
transitive factorizations $(\rho,\t_r,\ldots,\t_1)$ such that $\rho \in \Class{\b}$ and the factors
$\t_1,\ldots,\t_r$ are transpositions. Thus $[1^n]$-factorizations are synonymous with \mbox{2-cycle}
factorizations. Determining the number of $\b$-factorizations of a given $\p \in \Class{\a}$ is known as the
\emph{double Hurwitz problem}. It has been the object of recent attention because of known and conjectural links
with geometry, in particular intersection theory; further information can be found in~\cite{gj-vakil}. Though we
have recovered a number of the results of~\cite{gj-vakil} through combinatorial methods, we have not made further
headway.
\section{Inequivalent Cycle Factorizations}
\label{sec:inequiv}
We now turn to the enumeration of equivalence classes of cycle factorizations under the commutation relation
introduced in \S\ref{ssec:equivalence}. The graphical methods developed in the previous section will be
reconsidered in this new context, and an appropriate pruning mechanism will be developed. The section ends
with a proof of Theorem~\ref{thm:mainthm2}.
\subsection{Marked and $\a$-Marked Polymaps}
\label{ssec:markedpolymaps}
\newcommand{\imap}{\widetilde{\mathscr{M}}}
Our starting point is the observation that two cycle factorizations are equivalent if and only if (1) they
have precisely the same factors, and (2) the factors moving any given symbol appear in the same order in both
factorizations. Interpreting this through the lens of our graphical correspondence $F \mapsto \amap_F$
(Proposition~\ref{prop:mapping}), we see that permissible commutations of a cycle factorization $F$
correspond with relabellings of the polygons of $\amap_F$ that preserve the relative order of the polygons
incident with any given vertex.
In particular, \emph{two factorizations are equivalent if and only if their corresponding polymaps have the same
descent structure.} Thus the equivalence class $[F]$ containing $F$ is naturally represented by the polymap
$\imap_F$ that results from stripping the polygon labels of $\amap_F$ and instead recording only the location of
descents.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\textwidth]{CactiPruning-InequivMaps.eps}
\caption{Polymaps corresponding to equivalent cycle
factorizations..} \label{fig:inequivalentmaps}
\end{center}
\end{figure}
\begin{exmp} Consider the following equivalent cycle factorizations of $(1\,2\,3)(4\,5)(6\,7\,8)$:
\label{exmp:inequiv}
\begin{align*}
F &= ((2\,8\,6), (3\,5\,7), (4\,6), (5\,6), (1\,4\,3), (2\,7)) \\
G &= ((3\,5\,7), (2\,8\,6), (2\,7), (4\,6), (1\,4\,3), (5\,6)).
\end{align*}
Factors $(5\,6)$, $(4\,6)$, and $(2\,8\,6)$ move symbol 6, and they
appear in this same right-to-left order in both factorizations. The
corresponding polymaps $\amap_F$ and $\amap_G$ are shown in
Figures~\ref{fig:inequivalentmaps}A and~\ref{fig:inequivalentmaps}B,
respectively, where vertex labels have been placed in descent
corners. Note that the descent structure of these polymaps is
identical. The class $[F]$ (equivalently, $[G]$) is represented by
the polymap $\imap_F$ (equivalently, $\imap_G$) shown in
Figure~\ref{fig:inequivalentmaps}C, where again the location of
descents is recorded by the placement of vertex labels. \qed
\end{exmp}
Define a \bold{marked} polymap to be a loopless polymap $\amap$ in
which certain corners have been distinguished subject to the
requirement that every vertex is at exactly one distinguished corner. A marked polymap
$\amap$ is \bold{properly marked} if its distinguished corners coincide with the descent corners under some
polygon labelling of $\amap$. That is, properly marked
polymaps are exactly those objects that result from marking descents and
deleting polygon labels from proper polymaps. (Note that not every marked polymap is properly marked.)
We refer to the distinguished corners of a properly marked polymap
as \bold{descent corners}, and interpret terms such as \bold{descent
cycle} in the obvious way.
See Figure~\ref{fig:markedpolymaps}A for an example of a properly marked
polymap. (Two polygon labellings which validate the indicated descent corners
are shown in Figure~\ref{fig:inequivalentmaps}.)
\begin{figure}[t]
\begin{center}
\includegraphics[width=\textwidth]{CactiPruning-MarkedPolymaps.eps}
\caption{(A,B) A properly marked polymap. (B) The rotator of $v$ is $(p_1,p_2,p_3)$. (C) A \mbox{$(3,2,3)$-marked} polymap.}
\label{fig:markedpolymaps}
\end{center}
\end{figure}
It will be convenient to redefine the \bold{rotator} of a vertex $v$
in a properly marked polymap to be the tuple $(P_1,\ldots,P_m)$ of polygons
incident with $v$, listed in order as they are encountered along a
clockwise tour about $v$ beginning in the unique descent corner
containing $v$. In Figure~\ref{fig:markedpolymaps}B we have
indicated polygons $p_1,p_2$, and $p_3$ such that the rotator of
vertex $v$ is $(p_1,p_2,p_3)$.
Properly marked polymaps will play the same role in the enumeration of
inequivalent factorizations to follow as proper polymaps did in our
approach to counting ordered factorizations. Our observations thus
far are summarized by the following analogue of
Proposition~\ref{prop:mapping}:
\begin{prop}
\label{prop:imapping} The mapping $[F] \mapsto \imap_F$ is a bijection between equivalence classes of cycle
factorizations and vertex-labelled properly marked polymaps. If $F$ is a factorization of $\p$ of cycle index $\vec{i}$,
then $\imap_F$ is of poly-index $\vec{i}$ with descent cycles equal to the cycles of $\p$. \qed
\end{prop}
Let $\a=(\a_1,\ldots,\a_m) \cmp n$. An \bold{$\a$-marked} polymap is a vertex-labelled properly marked
polymap whose faces are labelled $1,\ldots,m$ so that face $j$ has descent set $\Dset{\a}{j}$, where
$\Dset{\a}{j}$ is defined as in~\eqref{eq:canonicalsets}. For example, under the convention that the
locations of vertex labels indicate descent corners, the polymap of Figure~\ref{fig:markedpolymaps}C is seen
to be $(3,2,3)$-marked. Of course, the face labels of $\a$-marked polymaps are superfluous, since they are
determined by the descent sets. They are included in the definition only as a matter of
convenience.\footnote{Note the contrast between this definition of $\a$-marked polymaps and that of
$\a$-proper polymaps in~\S\ref{ssec:properpolymaps}. In the latter case, \emph{only} face labels were used.
Here we do not have that luxury, because face-labelled marked polymaps can admit nontrivial automorphisms,
even when two or more faces are present.}
Let $\IPshur{\a}{i}$ denote the number of $\a$-marked polymaps with poly-index $\vec{i}$. Under
Proposition~\ref{prop:imapping}, $\a$-marked polymaps correspond with factorizations of permutations whose
orbits are $\Dset{\a}{1},\ldots,\Dset{\a}{m}$. There are $\prod_j (\a_j-1)!$ such permutations, so
$\IPshur{\a}{i} = \Phur{\a}{i} \cdot \prod_j (\a_j-1)!$. Thus~\eqref{eq:inequivgs} becomes
\begin{equation}
\label{eq:amarkedpolymaps}
\IPshurGS{m}(\vec{x},\vec{q}, u) =
\sum_{n \geq 1} \, \sum_{\vec{i} \,\geq\, \vec{0}}
\sum_{\substack{\a \cmp n \\ \len{\a} = m}}
\IPhur{\a}{\vec{i}} \, \vec{q}^{\vec{i}}\,
\frac{\vec{x}^{\pmb{\a}}}{\pmb{\a}!} u^{r(\vec{i})}.
\end{equation}
That is, $\IPshurGS{m}(\vec{x},\vec{q},u)$ is the generating series for $\a$-marked polymaps, where $u$ marks
polygons, $\vec{q}$ records poly-index, and $x_j$ is an exponential marker for labelled vertices at descents
of face $j$.
\subsection{Marked Cacti and Inequivalent Factorizations of Full Cycles}
\label{ssec:icacti}
Observe that every marked cactus is properly marked. The argument used in~\S\ref{ssec:fullcycles} to show
$\Phur{[n]}{i}$ is the number of vertex-rooted proper cacti of poly-index $\vec{i}$ therefore applies
\emph{mutatis mutandis} to identify $\IPhur{[n]}{i}$ as the number of vertex-rooted marked cacti of
poly-index $\vec{i}$. Thus the series $\irt=\irt(x,\vec{p},u)$ defined by
\begin{equation}
\label{eq:ircdefn2}
\irt := \sum_{n \geq 1} \sum_{\vec{i \geq 0}}
\IPhur{[n]}{i} x^{n-1} \vec{q}^{\vec{i}}
u^{r(\vec{i})} = \frac{d}{dx} \IPshurGS{1}(x,\vec{q},u)
\end{equation}
counts such cacti with respect to the number of \emph{non-root} vertices (marked by $x$), polygons (marked by
$u$), and poly-index (marked by $\vec{q}$). As in~\S\ref{ssec:fullcycles}, we now consolidate this
definition of $\irt$ with the one given in the introduction by describing a decomposition of marked cacti
that proves their generating series satisfies~\eqref{eq:ircdefn}.
Let $C$ be a rooted marked cactus, and suppose its root vertex has
rotator $(p_1,\ldots,p_m)$. Detach $p_1$ from $C$ to obtain a
rooted marked cactus $C'$ whose root has rotator $(p_2,\ldots,p_m)$,
as shown in Figure~\ref{fig:inequivalentcactus}. (If $m=1$ then $C'$
consists of a single vertex.)
\begin{figure}[t]
\begin{center}
\includegraphics[width=\textwidth]{CactiPruning-InequivCactus.eps}
\caption{Decomposition of a descent-marked cactus.} \label{fig:inequivalentcactus}
\end{center}
\end{figure}
Now focus on polygon $p_1$. Suppose $p_1$ is a $k$-gon, and let $v$
be one of its $k-1$ non-root vertices. Let
$(p_{v}^1,\ldots,p_{v}^r,p_1,p_{v}^{r+1},\ldots,p_{v}^{s})$ be the
rotator of $v$, where the degenerate cases $r=0$ and $s=r$ are
possible. Detach the polygons $p_v^i$ from $v$ to form two
cacti, $C^1_v$ and $C^2_v$, whose roots have rotators
$(p_{v}^1,\ldots,p_{v}^r)$ and $(p_{v}^{r+1},\ldots,p_{v}^{s})$,
respectively. See Figure~\ref{fig:inequivalentcactus}, far right.
Thus $C$ decomposes into $C'$ together with $k$-gon $p_1$
and an ordered list of triples $(v,C^1_v,C^2_v)$, one for each of
the $k-1$ non-root vertices of $p_1$. Allowing for all possible $k$
gives
$$
\irt = 1 + \sum_{k \geq 2} \irt \cdot u q_k \cdot (x\irt^2)^{k-1}
= 1+ u\irt Q(x \irt^2),
$$
where the summand 1 accounts for the case in which $C$ consists of a
single vertex, and $Q$ is defined as in~\eqref{eq:qdefn}. This is in
agreement with~\eqref{eq:ircdefn}, as claimed.
The following result first appeared in~\cite{springer}. The bijective proof given there relied on the
commutation of cycle factorizations into canonical forms, together with a correspondence between these
canonical forms and a class of trees for which enumerative formulae are known. In hindsight, our use of
marked cacti can be viewed as a high level graphical interpretation of this bijection that bypasses the
element-wise analysis necessary to establish the canonical forms.
\begin{thm}[Springer~\cite{springer}]
\label{thm:ispringer} Let $\vec{i}=(i_2,i_3,\ldots)$ be a sequence
of nonnegative integers, not all zero, and set $r=r(\vec{i})=i_2 +
i_3 + \cdots$. Then the number of inequivalent cycle factorizations
of $(1\,2\,\cdots\,n)$ of cycle index $\vec{i}$ is
$$
\IPhur{[n]}{i} = \frac{(2n+r-2)!}{(2n-1)! \, \prod_{k \geq 2} i_k!}
$$
in the case that $n + r - 1 = \sum_{k \geq 2} ki_k$, and zero otherwise.
\end{thm}
\begin{proof}Set $v =
\irt-1$. Then~\eqref{eq:ircdefn2} and~\eqref{eq:ircdefn} give
$\IPhur{[n]}{i}=[x^{n-1} u^r \vec{q}^\vec{i}]\, v$ with $v = u (1+v)
Q(x(1+v)^2).$ By Lagrange inversion~\cite{goulden-jackson} we have
\begin{align*}
[x^{n-1} u^r \vec{q}^\vec{i}]\,v
&= [x^{n-1} \vec{q}^\vec{i}]\,\,\frac{1}{r}\, [\l^{r-1}]\,
(1+\l)^r Q(x(1+\l)^2)^r \\
&= \frac{1}{r}\, [\l^{r-1}]\,(1+\l)^{r+2n-2}
\binom{r}{i_2,\, i_3,\, i_4,\,\ldots},
\end{align*}
and the result follows upon simplification.
\end{proof}
Specializing the previous theorem to count inequivalent
$k$-cycle factorizations entails setting $i_k=r$ and $i_j=0$ for $j
\neq k$. Doing so, we find that the number of inequivalent
$k$-cycle factorizations of $(1\,2\,\cdots\,n)$ is
$$
\frac{1}{2n-1}\binom{2n+r-2}{r}
$$
when $n=1+r(k-1)$ for some integer $r$. This formula first appeared
in~\cite{gj-macdonald}. Setting $k=2$ yields the
result~\eqref{eq:catalan} of Longyear mentioned in the introduction.
\subsection{Pruning Cacti}
\label{ssec:Ipruningcacti}
Let $\a=(\a_1,\ldots,\a_m)$ be a composition. Define the \bold{face
degree sequence} of an $\a$-marked polymap to be $(d_1,\ldots,d_m)$,
where $d_j$ is the degree of face $j$, for $1 \leq j \leq m$.
Alternatively, $d_j$ is the number of corners in face $j$. Let
$\IPsmhur{\a}{i}{d}$ denote the number of smooth $\a$-marked
polymaps with poly-index $\vec{i}$ and face degree sequence
$\vec{d}$. Then, for $m \geq 1$, define
\begin{equation*}
\IPsmhurGS{m}(\vec{x},\vec{t},\vec{q}, u) :=
\sum_{n \geq 1} \, \sum_{\vec{i},\vec{d} \,\geq\, \vec{0}}
\sum_{\substack{\a \cmp n \\ \len{\a} = m}}
\IPsmhur{\a}{i}{d} \,
\frac{\vec{x}^{\pmb{\a}}}{\pmb{\a}!} \,
\vec{q}^{\vec{i}}\, \vec{t}^{\vec{d}}
u^{r(\vec{i})}.
\end{equation*}
With these definitions we have the following marked polymap analogue of
Theorem~\ref{thm:cactuspruning}.
\begin{thm}
\label{thm:inequivpruning} Let $m \geq 1$ and set $\irt_i =
\irt(x_i,\vec{q},u)$ for $1 \leq i \leq m$, where $\irt$ is given
by~\eqref{eq:ircdefn}. Then
\begin{equation*}
\IPshurGS{m}(\vec{x}, \vec{q}, u) =
\IPsmhurGS{m}(\vec{x}\circ \vec{\irt},\; \vec{\irt},\; \vec{q},\; u),
\end{equation*}
where $\vec{x}=(x_1,\ldots,x_m)$, $\vec{\irt} = (\irt_1,\ldots,\irt_m)$ and
$\vec{x}\circ\vec{\irt} = (x_1\irt_1,\ldots,x_m\irt_m)$.
\end{thm}
\begin{proof}
Let $\amap$ be a properly marked polymap with at least two faces and let $v$ be a vertex of $\acore$ incident with face $f$. Let $(p_1,\ldots,p_k)$ be the
rotator of $v$. We wish to prune from $\amap$ all branches in $f$ attached at $v$.
Suppose first that $v$ is at a descent of $f$. Then there exist $i,j$ with $1 \leq i \leq j \leq k$ such
that $p_1,\ldots,p_{i-1}, p_{j+1},\ldots,p_k$ are the root polygons of the branches attached to $v$ in face
$f$. (See Figure~\ref{fig:inequivpruning}A. For clarity, most descent corners have not been indicated.)
Detach these branch from $\amap$ to form a new polymap $\amap'$ and two rooted marked cacti, $C_1$ and
$C_2$, whose roots have rotators $(p_1,\ldots,p_{i-1})$ and $(p_{j+1},\ldots,p_k)$, respectively. Note that
cases $i=1$ and $j=k$ yield trivial cacti $C_1$ and $C_2$, respectively. The descent corner of $f$ at $v$ is
inherited by $\amap'$, and it is easy to check that this makes $\amap'$ properly marked.
Figure~\ref{fig:inequivpruning}A illustrates this pruning mechanism.
\begin{figure}[t]
\begin{center}
\psfrag{frag:m}{$\amap$}
\psfrag{frag:mp}{$\amap'$}
\includegraphics[width=.85\textwidth]{CactiPruning-InequivPruning.eps}
\caption{Pruning cacti from a marked polymap.} \label{fig:inequivpruning}
\end{center}
\end{figure}
Now suppose $v$ is not at a descent of $f$. Then there exist $i,j$ with $1 \leq i < j \leq k$ such that
$p_{i+1},\ldots,p_{j-1}$ are the root polygons of the branches attached to $v$ in $f$. (See
Figure~\ref{fig:inequivpruning}B.) Detach these branches from $\amap$ to form a properly marked polymap
$\amap'$ and a rooted marked cactus $C$ whose root has rotator $(p_{i+1},\ldots,p_{j-1})$. When $i=j$ there
are no branches attached to $v$ in $f$, and $C$ is trivial.
Repeating this pruning process for each incident face-vertex pair $(v,f)$ of $\amap$ (that is, for each
corner of $\amap$) results in a smooth properly marked polymap $\amap[S]$ and a collection of rooted marked
cacti, each of which is naturally associated with a particular face of $\amap[S]$. In fact, if a face of
$\amap[S]$ is of degree $D$ and has $d$ descents, then it has $D-d$ non-descent corners and thus exactly
$2d+(D-d)=D+d$ associated pruned cacti. Each non-root vertex of these cacti contributes one descent to the
corresponding face of $\amap$. The pruning of all cacti from a properly marked polymap is illustrated in
Figure~\ref{fig:inequivpruning}C.
To complete the proof, observe that if $\amap$ is $\a$-marked, where $m=\len{\a} \geq 2$, then its corners
are distinguishable and thus the pruning process described above is reversible. Recall that
$\irt(x,\vec{q},u)$ counts rooted marked cacti, with $x$ recording non-root vertices. Since rooting
eliminates nontrivial automorphisms, these vertices can be freely labelled. From~\eqref{eq:amarkedpolymaps}
it follows that $\IPshurGS{m}(\vec{x},\vec{q},u)$ is obtained from $\IPsmhurGS{m}(\vec{z},\vec{t},\vec{q},u)$
through the substitutions $z_j \mapsto x_j\irt_j$ and $t_j \mapsto \irt_j$, for $1 \leq j \leq m$.
\end{proof}
\subsection{Inequivalent Factorizations of Permutations with Two Cycles}
\label{ssec:Itwofacepolymaps}
We now use the pruning of marked polymaps to prove Theorem~\ref{thm:mainthm2}. As mentioned in the
introduction, this theorem generalizes the main result of~\cite{glj-inequivalent}, and it is worth noting
that the intricate inclusion-exclusion argument used there is avoided entirely by our method. In particular,
the following lemma (which is a routine exercise in exponential generating series) is seen to be a more
natural source of the logarithm appearing in~\eqref{eq:igjresult}.
\begin{lem}
\label{lem:beadlemma} For $m,n \geq 1$, let $c_{n,m}$ be the number
of distinct necklaces made of $n$ labelled white beads and $m$
(independently) labelled black beads, counted up to rotational
symmetry. Then
\begin{equation}
\label{eq:beadeqn}
\sum_{n,m \geq 1} c_{n,m} \frac{x^n}{n!}\frac{y^m}{m!} z^{n+m}
= \log\bigg(1+\frac{xy z^2}{1-z(x+y)}\bigg).
\end{equation}
\qed
\end{lem}
\vspace*{1.5mm}
\noindent\textbf{Proof of Theorem~\ref{thm:mainthm2}:} We begin by determining
$\IPsmhurGS{2}(z_1,z_2,t_1,t_2,\vec{q},u)$.
Let $\amap[S]$ be a smooth \mbox{$\a$-marked} polymap, where $\len{\a}=2$.
Then $\amap[S]$ is a closed chain of polygons, each incident with exactly
two others. As in the proof of Theorem~\ref{thm:mainthm1}, we say a vertex incident with two polygons
is \emph{extremal}. Since $\amap[S]$ is properly marked, at least one extremal
vertex is at a descent of each face.
Let $L=\olist{v_1,\ldots,v_r}$ be the cyclic list of extremal
vertices encountered along the boundary walk of face 1 of
$\amap[S]$. Regard those $v_i$ that are at descents of face 1 as
white beads and those at descents of face 2 as black beads, so $L$
corresponds with a necklace of the type counted by
Lemma~\ref{lem:beadlemma}. (See Figure~\ref{fig:maptonecklace} for
an illustration, where extremal vertices are indicated in grey, but
vertex and bead labels are not shown.)
\begin{figure}[t]
\begin{center}
\includegraphics[width=.55\textwidth]{CactiPruning-TwoFaceMarkedPolymap.eps}
\caption{A smooth marked polymap and its associated necklace.}
\label{fig:maptonecklace}
\end{center}
\end{figure}
Let $M$ be the monomial of
$\IPsmhurGS{2}(z_1,z_2,t_1,t_2,\vec{q},u)$ corresponding to
$\amap[S]$. An extremal vertex of $\amap[S]$ contributes the factor
$z_1 t_1 t_2$ to $M$ if it is at a descent of face 1, and
contributes $z_2 t_1 t_2$ otherwise. A polygon of $\amap[S]$ with
$j_f-1$ vertices incident only with face $f$, for $f=1,2$, further
contributes the factor $u q_{j_1+j_2} (t_1 z_1)^{j_1-1}
(t_2z_2)^{j_2-1}$. Thus $\IPsmhurGS{2}(z_1,z_2,t_1,t_2,\vec{q},u)$
is obtained from~\eqref{eq:beadeqn} by performing the substitutions
$x \mapsto z_1 t_1 t_2$, $y \mapsto z_2 t_1 t_2$, and
\begin{align*}
z \mapsto \sum_{j_1, j_2 \geq 1} u q_{j_1+j_2} (t_1z_1)^{j_1-1} (t_2z_2)^{j_2-1}
= u\frac{Q(t_1z_1)-Q(t_2z_2)}{t_1z_1-t_2z_2}.
\end{align*}
But Theorem~\ref{thm:inequivpruning} implies
$\IPshurGS{2}(x_1,x_2,\vec{q},u)$ is got from $\IPsmhurGS{2}(z_1,
z_2, t_1, t_2, \vec{q}, u)$ through the substitutions $z_i \mapsto
x_i \irt_i$ and $t_i \mapsto \irt_i$. Performing this chain of
substitutions leaves
\begin{align*}
\IPshurGS{2}(x_1,x_2,\vec{q},u)
= \log\pr{1 + \frac{\delta^2 x_1 x_2 \irt_1 \irt_2}{1-\delta(x_1 \irt_1 + x_2
\irt_2)}},
\end{align*}
where
$$
\delta
:=
u\irt_1\irt_2\frac{Q(x_1\irt_1^2)-Q(x_2\irt_2^2)}{x_1\irt_1^2-x_2\irt_2^2}.
$$
Theorem~\ref{thm:mainthm2} now follows upon simplification,
using~\eqref{eq:ircdefn} to write $uQ(x_i\irt_i^2)=1-\irt_i^{-1}$.
\qed
\subsection{Further Results} \label{ssec:ifurtherresults}
\newcommand{\note}[1]{\vspace*{2mm}\noindent\textbf{#1}}
We have been unable to generalize the proof of Theorem~\ref{thm:mainthm2} to obtain expressions for $\IPshurGS{m}$
for any $m > 2$. However, Theorem~\ref{thm:inequivpruning} has been used~\cite{irvingphd} to give a generating
series formulation for the number of inequivalent \mbox{2-cycle} factorizations of permutations composed of three
cycles. The expression for $\IPkshurGS{3}{2}$ thus obtained is far more complicated than the very
succinct~\eqref{eq:igjresult} and we have been unable to simplify it in a meaningful way. Finding a natural
expression for $\IPkshurGS{3}{2}$ remains an intriguing open problem, as it would grant far greater insight into
the structure of inequivalent factorizations in general. We note that an approach from the point of view
of~\cite{glj-inequivalent} involves significant technical hurdles that do not surface in the simpler known cases.
Inspired by the definition of $\b$-factorizations (see \S\ref{ssec:furtherresults}) we can extend
Theorem~\ref{thm:ispringer} in the following way. Let $\b = [1^{j_1} 2^{j_2} \cdots] \ptn n$ and consider the set
of minimal transitive factorizations $(\rho,\s_r,\ldots,\s_1)$ of the full cycle $(1\,2\,\cdots\,n)$ such that
$\rho \in \Class{\b}$ and the $\s_i$ are all cycles of length at least two. Say two such factorizations are
equivalent if one can be obtained from the other by commutation of adjacent disjoint \emph{cycle factors}
$\s_1,\ldots,\s_m$. That is, the position of $\rho$ is fixed. Then the number of inequivalent factorizations of
this type with exactly $i_k$ cycle factors being \mbox{$k$-cycles} for $k \geq 2$ is
\begin{equation*}
\frac{n(r-1)!\,(\len{\b}-1)!}{
\prod_{k \geq 2} i_k! \cdot \prod_{k \geq 1} j_k!}
\binom{r+\len{\b}+n-2}{r-1},
\end{equation*}
where $r=i_2+i_3+\cdots$, subject to the necessary condition $\sum
ki_k = r+\len{\b}-1$. This result is readily proved through a
refinement of the decomposition of marked cacti given in
\S\ref{ssec:icacti}. Note that Theorem~\ref{thm:ispringer} is
recovered by setting $\b=[1^n]$.
We conclude by noting that our definition of equivalence of
factorizations may not seem the most natural. For instance, one
could allow all pairs of adjacent commuting factors to be
interchanged, whether or not they are disjoint. A general statement
of the corresponding enumerative problem might then be the
following: For a multiset $B$ of partitions of $n$, find the number
of inequivalent minimal transitive factorizations of a given $\p \in
\Sym{n}$ whose factors have cycle types specified by $B$. We have
not yet investigated this problem in any detail.
\section*{Acknowledgements}
The bulk of the research for this article was completed during the author's doctoral studies under the
supervision of David Jackson, whose support and guidance has been greatly appreciated.
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2} | 204,084 |
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(Moscow Times – themoscowtimes.com – September 12, 2013) Russia’s Prosecutor General’s Office and the U.S. Justice Department have agreed to sign a memorandum on cooperation in order to work together on criminal matters, in a move that could put the thorny issue of extradition between the two countries up for discussion.
The agreement was reached at a meeting that took place between both sides on the sidelines of the International Conference of Prosecutors, currently being held in Moscow, Interfax reported Wednesday.
“The two sides continued discussions about international cooperation, primarily in connection with the fight against crime and the defense of human rights,” Russian prosecutors said in a statement.
The participants in the meeting agreed on the need to collaborate more effectively on criminal matters, including the institution of extradition and mutual legal assistance.
Deputy Assistant Attorney General Bruce Swartz, who headed the U.S. delegation, said the American side would soon put forward their proposal outlining how they think cooperation could be implemented.
The Russian side in the discussion was led by Deputy Prosecutor General Alexander Zvyagintsev.
Comment with Wordpress | 291,095 |
:: Double Sequences and Limits
:: by Noboru Endou , Hiroyuki Okazaki and Yasunari Shidama
environ
vocabularies NUMBERS, SUBSET_1, FUNCT_1, RELAT_1, XBOOLE_0, ARYTM_3, ARYTM_1,
XXREAL_0, CARD_1, TARSKI, NAT_1, ZFMISC_1, VALUED_0, ORDINAL2, SEQ_2,
XREAL_0, COMPLEX1, XCMPLX_0, XXREAL_2, FINSET_1, MEMBERED, MESFUNC9,
BHSP_3, SEQ_1, VALUED_1, DBLSEQ_1, FUNCOP_1, BINOP_1, BINOP_2, REAL_1;
notations ZFMISC_1, SUBSET_1, ORDINAL1, XXREAL_2, RELAT_1, PARTFUN1, RELSET_1,
FUNCT_1, FUNCT_2, NUMBERS, BINOP_1, XCMPLX_0, TARSKI, XBOOLE_0, FUNCOP_1,
FUNCT_3, FINSEQOP, MEMBERED, COMPLEX1, XXREAL_0, XREAL_0, SEQ_1, NAT_1,
MESFUNC9, VALUED_1, SEQ_2, FINSET_1, BINOP_2;
constructors TOPMETR, NAT_D, MESFUNC9, SEQ_2, COMSEQ_2, SEQ_4, REAL_1,
FINSEQOP, FUNCT_3, BINOP_2;
registrations ORDINAL1, XXREAL_0, XREAL_0, XBOOLE_0, NUMBERS, XCMPLX_0,
FINSET_1, SUBSET_1, MEMBERED, VALUED_0, XXREAL_2, RELSET_1, MEASURE6,
SEQ_4, FUNCT_1, FUNCT_2, VALUED_1, BINOP_2, NAT_1, SEQ_2, SEQ_1;
requirements SUBSET, REAL, BOOLE, NUMERALS, ARITHM;
equalities BINOP_1;
expansions MEASURE6;
theorems TARSKI, XREAL_0, NAT_1, FUNCT_1, XXREAL_0, ZFMISC_1, XREAL_1,
FUNCT_2, RELAT_1, MESFUNC9, ORDINAL1, HEINE, NUMBERS, XCMPLX_1, VALUED_1,
ABSVALUE, COMPLEX1, MEMBERED, XXREAL_2, RINFSUP1, SEQM_3, SEQ_2, SEQ_4,
FUNCOP_1, FINSEQOP, XBOOLE_1, BINOP_2;
schemes FUNCT_2, FUNCT_3, FRAENKEL, ASYMPT_0;
begin
reserve Rseq, Rseq1, Rseq2 for Function of [:NAT,NAT:],REAL;
reserve rseq1,rseq2 for convergent Real_Sequence;
reserve n,m,N,M for Nat;
reserve e,r for Real;
:: Convergence in the Pringsheim sense
definition let Rseq;
attr Rseq is P-convergent means
ex p be Real st
for e be Real st 0<e
ex N be Nat st
for n,m be Nat st n>=N & m>=N holds
|.Rseq.(n,m) - p.| < e;
end;
definition let Rseq;
assume A1: Rseq is P-convergent;
func P-lim Rseq -> Real means :def6:
for e be Real st 0<e
ex N be Nat st
for n,m be Nat st n>=N & m>=N holds
|. Rseq.(n,m) - it .| < e;
existence by A1;
uniqueness
proof
given g1,g2 be Real such that
A2: for e st 0<e
ex N st for n,m st n>=N & m>=N holds |. Rseq.(n,m)-g1 .| < e and
A3: for e st 0<e
ex N st for n,m st n>=N & m>=N holds |. Rseq.(n,m)-g2 .| < e and
A4: g1<>g2;
A5: now assume |.g1-g2.|=0; then
g1-g2 = 0 by ABSVALUE:2;
hence contradiction by A4;
end;
A6: 0<=|.g1-g2.| by COMPLEX1:46; then
consider N1 be Nat such that
A7: for n,m st N1<=n & N1<=m holds |. Rseq.(n,m)-g1 .| < |.g1-g2.|/4 by A2,A5;
consider N2 be Nat such that
A8: for n,m st N2<=n & N2<=m holds |.Rseq.(n,m)-g2.| < |.g1-g2.|/4 by A3,A5,A6;
N1+N2 >= N1 & N1+N2 >= N2 by NAT_1:11; then
|.Rseq.(N1+N2,N1+N2)-g1.| < |.g1-g2.|/4 &
|.Rseq.(N1+N2,N1+N2)-g2.| < |.g1-g2.|/4 by A7,A8; then
A10: |.Rseq.(N1+N2,N1+N2)-g2.| + |.Rseq.(N1+N2,N1+N2)-g1.|
< |.g1-g2.|/4 + |.g1-g2.|/4 by XREAL_1:8;
|.g1-g2.| = |. (-(Rseq.(N1+N2,N1+N2)-g1)) + (Rseq.(N1+N2,N1+N2)-g2).|; then
|.g1-g2.| <= |.-(Rseq.(N1+N2,N1+N2)-g1).| + |.Rseq.(N1+N2,N1+N2)-g2.|
by COMPLEX1:56; then
A11: |.g1-g2.| <= |.Rseq.(N1+N2,N1+N2)-g1.| + |.Rseq.(N1+N2,N1+N2)-g2.|
by COMPLEX1:52;
|.g1-g2.|/2 < |.g1-g2.| by A5,A6,XREAL_1:216;
hence contradiction by A10,A11,XXREAL_0:2;
end;
end;
definition let Rseq;
attr Rseq is convergent_in_cod1 means
for m be Element of NAT holds ProjMap2(Rseq,m) is convergent;
attr Rseq is convergent_in_cod2 means
for n be Element of NAT holds ProjMap1(Rseq,n) is convergent;
end;
definition let Rseq;
func lim_in_cod1 Rseq -> Function of NAT,REAL means :def32:
for m be Element of NAT holds it.m = lim ProjMap2(Rseq,m);
existence
proof
defpred P[Element of NAT,set] means $2 = lim ProjMap2(Rseq,$1);
a1:now let m be Element of NAT;
reconsider n = lim ProjMap2(Rseq,m) as Element of REAL by XREAL_0:def 1;
take n;
thus P[m,n];
end;
consider IT be Function of NAT,REAL such that
a2: for m be Element of NAT holds P[m,IT.m] from FUNCT_2:sch 3(a1);
take IT;
thus thesis by a2;
end;
uniqueness
proof
let f1,f2 be Function of NAT,REAL;
assume that
a3: for m be Element of NAT holds f1.m = lim ProjMap2(Rseq,m) and
a4: for m be Element of NAT holds f2.m = lim ProjMap2(Rseq,m);
now let m be Element of NAT;
thus f1.m = lim ProjMap2(Rseq,m) by a3 .= f2.m by a4;
end;
hence f1 = f2 by FUNCT_2:63;
end;
end;
definition let Rseq;
func lim_in_cod2 Rseq -> Function of NAT,REAL means :def33:
for n be Element of NAT holds it.n = lim ProjMap1(Rseq,n);
existence
proof
defpred P[Element of NAT,set] means $2 = lim ProjMap1(Rseq,$1);
a1:now let m be Element of NAT;
reconsider n = lim ProjMap1(Rseq,m) as Element of REAL by XREAL_0:def 1;
take n;
thus P[m,n];
end;
consider IT be Function of NAT,REAL such that
a2: for m be Element of NAT holds P[m,IT.m] from FUNCT_2:sch 3(a1);
take IT;
thus thesis by a2;
end;
uniqueness
proof
let f1,f2 be Function of NAT,REAL;
assume that
a3: for n be Element of NAT holds f1.n = lim ProjMap1(Rseq,n) and
a4: for n be Element of NAT holds f2.n = lim ProjMap1(Rseq,n);
now let n be Element of NAT;
thus f1.n = lim ProjMap1(Rseq,n) by a3 .= f2.n by a4;
end;
hence f1 = f2 by FUNCT_2:63;
end;
end;
definition let Rseq;
assume a2: lim_in_cod1 Rseq is convergent;
func cod1_major_iterated_lim Rseq -> Real means :def34:
for e be Real st 0<e
ex M be Nat st
for m be Nat st m>=M holds
|.(lim_in_cod1 Rseq).m - it.| < e;
existence by a2,SEQ_2:def 6;
uniqueness
proof
let z1,z2 be Real;
assume that
a4: for e st 0<e
ex M st for m st m>=M holds |.(lim_in_cod1 Rseq).m - z1.| < e
and
a5: for e st 0<e
ex M st for m st m>=M holds |.(lim_in_cod1 Rseq).m - z2.| < e;
assume a6: z1 <> z2;
set p = |.z1 - z2.|/2;
a7: |.z1-z2.| > 0 by a6,COMPLEX1:62; then
consider M1 be Nat such that
a8: for m st M1<=m holds |.(lim_in_cod1 Rseq).m-z1.| < p by a4;
consider M2 be Nat such that
a9: for m st M2<=m holds |.(lim_in_cod1 Rseq).m-z2.| < p by a5,a7;
set M = max(M1,M2);
a0: M is Nat by TARSKI:1;
for m st M <= m holds 2*p<2*p
proof
let m;
set s = lim_in_cod1 Rseq;
assume a10: M <= m;
M2<=M by XXREAL_0:25; then
M + M2 <= M + m by a10,XREAL_1:7; then
M2<=m by XREAL_1:6; then
a11: |. s.m - z2.|< p by a9;
|.z1 - z2.| = |.z1 - s.m - (z2 - s.m).|; then
|.z1 - z2.| <= |.z1-s.m.| + |.z2-s.m.| by COMPLEX1:57; then
a12: |.z1 - z2.| <= |.s.m - z1.| + |.z2-s.m.| by COMPLEX1:60;
M1<=M by XXREAL_0:25; then
M + M1 <= M + m by a10,XREAL_1:7; then
M1<=m by XREAL_1:6; then
|.s.m - z1.|< p by a8; then
|.s.m - z1.| + |.s.m - z2.| <p + p by a11,XREAL_1:8;
hence thesis by a12,COMPLEX1:60;
end;
hence contradiction by a0;
end;
end;
definition let Rseq;
assume a2: lim_in_cod2 Rseq is convergent;
func cod2_major_iterated_lim Rseq -> Real means :def35:
for e be Real st 0<e
ex N be Nat st
for n be Nat st n>=N holds
|.(lim_in_cod2 Rseq).n - it.| < e;
existence by a2,SEQ_2:def 6;
uniqueness
proof
let z1,z2 be Real;
assume that
a4: for e st 0<e
ex M st for m st m>=M holds |.(lim_in_cod2 Rseq).m - z1.| < e
and
a5: for e st 0<e
ex M st for m st m>=M holds |.(lim_in_cod2 Rseq).m - z2.| < e;
assume a6: z1 <> z2;
set p = |.z1 - z2.|/2;
a7: |.z1-z2.| > 0 by a6,COMPLEX1:62; then
consider M1 be Nat such that
a8: for m st M1<=m holds |.(lim_in_cod2 Rseq).m-z1.| < p by a4;
consider M2 be Nat such that
a9: for m st M2<=m holds |.(lim_in_cod2 Rseq).m-z2.| < p by a5,a7;
set M = max(M1,M2);
a0: M is Nat by TARSKI:1;
for m st M <= m holds 2*p<2*p
proof
let m;
set s = lim_in_cod2 Rseq;
assume a10: M <= m;
M2<=M by XXREAL_0:25; then
M + M2 <= M + m by a10,XREAL_1:7; then
M2<=m by XREAL_1:6; then
a11: |. s.m - z2.|< p by a9;
|.z1 - z2.| = |.z1 - s.m - (z2 - s.m).|; then
|.z1 - z2.| <= |.z1-s.m.| + |.z2-s.m.| by COMPLEX1:57; then
a12: |.z1 - z2.| <= |.s.m - z1.| + |.z2-s.m.| by COMPLEX1:60;
M1<=M by XXREAL_0:25; then
M + M1 <= M + m by a10,XREAL_1:7; then
M1<=m by XREAL_1:6; then
|.s.m - z1.|< p by a8; then
|.s.m - z1.| + |.s.m - z2.| <p + p by a11,XREAL_1:8;
hence thesis by a12,COMPLEX1:60;
end;
hence contradiction by a0;
end;
end;
definition
let Rseq be Function of [:NAT,NAT:],REAL;
attr Rseq is uniformly_convergent_in_cod1 means
Rseq is convergent_in_cod1 &
for e be Real st e>0
ex M be Nat st
for m be Nat st m>=M holds
for n be Nat holds
|. Rseq.(n,m) - (lim_in_cod1 Rseq).n .| < e;
end;
definition
let Rseq be Function of [:NAT,NAT:],REAL;
attr Rseq is uniformly_convergent_in_cod2 means
Rseq is convergent_in_cod2 &
for e be Real st e>0
ex N be Nat st
for n be Nat st n>=N holds
for m be Nat holds
|. Rseq.(n,m) - (lim_in_cod2 Rseq).m .| < e;
end;
definition let Rseq;
attr Rseq is non-decreasing means
for n1,m1,n2,m2 be Nat st
n1>=n2 & m1>=m2 holds Rseq.(n1,m1) >= Rseq.(n2,m2);
attr Rseq is non-increasing means
for n1,m1,n2,m2 be Nat st
n1>=n2 & m1>=m2 holds Rseq.(n1,m1) <= Rseq.(n2,m2);
end;
theorem th28:
for a,b,c be Real st a <= b & b <= c holds
|.b.| <= |.a.| or |.b.| <= |.c.|
proof
let a,b,c be Real;
assume a1: a<=b & b<=c;
per cases;
suppose b >= 0; then
|.b.| = b & |.c.| = c by a1,ABSVALUE:def 1;
hence thesis by a1;
end;
suppose b < 0; then
|.a.| = -a & |.b.| = -b by a1,ABSVALUE:def 1;
hence thesis by a1,XREAL_1:24;
end;
end;
registration
cluster non-decreasing P-convergent -> bounded_below bounded_above
for Function of [:NAT,NAT:],REAL;
coherence
proof let Rseq be Function of [:NAT,NAT:],REAL;
assume a1: Rseq is non-decreasing P-convergent; then
consider p be Real such that
a3: for e st 0<e
ex N st for n,m st n>=N & m>=N holds |.Rseq.(n,m)-p.| < e;
consider N such that
a4: for n,m st n>=N & m>=N holds |.Rseq.(n,m)-p.| < 1 by a3;
a5:for n,m st n>=N & m>=N holds |.Rseq.(n,m).| < 1 + |.p.|
proof
let n,m;
assume n>=N & m>=N; then
a6: |.Rseq.(n,m)-p.| < 1 by a4;
|.Rseq.(n,m)-p.| >= |.Rseq.(n,m).| - |.p.| by COMPLEX1:59; then
|.Rseq.(n,m).| - |.p.| < 1 by a6,XXREAL_0:2;
hence |.Rseq.(n,m).| < 1 + |.p.| by XREAL_1:19;
end;
deffunc F(Element of NAT) = $1;
deffunc F1(Nat) = |.Rseq.(1,$1).|;
reconsider E2 = {F(m) where m is Element of NAT: m <= N} as
finite non empty Subset of NAT from ASYMPT_0:sch 2;
reconsider EE = [:E2,E2:] as finite set;
c1:E2 is finite;
deffunc F(Element of NAT,Element of NAT) = |.Rseq.($1,$2).|;
a9:{F(m,n) where m is Element of NAT, n is Element of NAT :
m in E2 & n in E2} is finite from FRAENKEL:sch 22(c1,c1);
set F = {F(m,n) where m is Element of NAT, n is Element of NAT :
m in E2 & n in E2};
N is Element of NAT by ORDINAL1:def 12; then
N in E2; then
b1: |.Rseq.(N,N).| in F;
now let x be object;
assume x in F; then
consider p be Element of NAT, q be Element of NAT such that
a10: x = F(p,q) & p in E2 & q in E2;
thus x is real by a10;
end; then
reconsider F as non empty real-membered set by b1,MEMBERED:def 3;
reconsider M1 = sup F as Element of REAL by a9,XXREAL_2:16;
set M = max(M1,1 + |.p.|);
a14:M >= 1+ |.p.| & 1+ |.p.| >= 1 & M >= M1 & M >= M1
by XXREAL_0:25,XREAL_1:31,COMPLEX1:46;
c1:for n,m holds |.Rseq.(n,m).| <= M
proof
let n,m;
c0: n is Element of NAT & m is Element of NAT & N is Element of NAT
by ORDINAL1:def 12;
per cases;
suppose n >= N & m >= N; then
|.Rseq.(n,m).| < 1 + |.p.| by a5;
hence |.Rseq.(n,m).| <= M by a14,XXREAL_0:2;
end;
suppose n < N & m < N; then
n in E2 & m in E2 by c0; then
|.Rseq.(n,m).| in F; then
a15: |.Rseq.(n,m).| <= M1 by XXREAL_2:4;
M >= M1 by XXREAL_0:25;
hence |.Rseq.(n,m).| <= M by a15,XXREAL_0:2;
end;
suppose a18: n < N & m >= N; then
a19: Rseq.(n,N) <= Rseq.(n,m) & Rseq.(n,m) <= Rseq.(N,m) by a1;
n in E2 & N in E2 by a18,c0; then
|.Rseq.(n,N).| in F; then
a20: |.Rseq.(n,N).| <= M1 by XXREAL_2:4;
a21: |.Rseq.(N,m).| < 1+ |.p.| by a5,a18;
per cases by a19,th28;
suppose |.Rseq.(n,m).| <= |.Rseq.(n,N).|; then
|.Rseq.(n,m).| <= M1 by a20,XXREAL_0:2;
hence |.Rseq.(n,m).| <= M by a14,XXREAL_0:2;
end;
suppose |.Rseq.(n,m).| <= |.Rseq.(N,m).|; then
|.Rseq.(n,m).| <= 1+ |.p.| by a21,XXREAL_0:2;
hence |.Rseq.(n,m).| <= M by a14,XXREAL_0:2;
end;
end;
suppose a22: n >= N & m < N; then
a23: Rseq.(N,m) <= Rseq.(n,m) & Rseq.(n,m) <= Rseq.(n,N) by a1;
N in E2 & m in E2 by a22,c0; then
|.Rseq.(N,m).| in F; then
a24: |.Rseq.(N,m).| <= M1 by XXREAL_2:4;
a25: |.Rseq.(n,N).| < 1+ |.p.| by a5,a22;
per cases by a23,th28;
suppose |.Rseq.(n,m).| <= |.Rseq.(N,m).|; then
|.Rseq.(n,m).| <= M1 by a24,XXREAL_0:2;
hence |.Rseq.(n,m).| <= M by a14,XXREAL_0:2;
end;
suppose |.Rseq.(n,m).| <= |.Rseq.(n,N).|; then
|.Rseq.(n,m).| <= 1+ |.p.| by a25,XXREAL_0:2;
hence |.Rseq.(n,m).| <= M by a14,XXREAL_0:2;
end;
end;
end;
now let a be ExtReal;
assume a in Rseq.: [:NAT,NAT:]; then
consider x be Element of [:NAT,NAT qua non empty set:] such that
b2: x in [:NAT,NAT:] & a = Rseq.x by FUNCT_2:65;
consider n,m be object such that
b3: n in NAT & m in NAT & x = [n,m] by ZFMISC_1:def 2;
reconsider n,m as Element of NAT by b3;
|.Rseq.(n,m).| <= M by c1;
hence -M <= a & a <= M by b2,b3,ABSVALUE:5;
end; then
(for a be ExtReal st a in Rseq.: [:NAT,NAT:] holds -M <= a) &
(for a be ExtReal st a in Rseq.: [:NAT,NAT:] holds a <= M); then
-M is LowerBound of Rseq.: [:NAT,NAT:] &
M is UpperBound of Rseq.: [:NAT,NAT:] by XXREAL_2:def 1,def 2;
hence thesis by XXREAL_2:def 9,def 10;
end;
end;
registration
cluster non-increasing P-convergent -> bounded_below bounded_above
for Function of [:NAT,NAT:],REAL;
coherence
proof
let Rseq be Function of [:NAT,NAT:],REAL;
assume a1: Rseq is non-increasing P-convergent; then
consider p be Real such that
a3: for e st 0<e
ex N st for n,m st n>=N & m>=N holds |.Rseq.(n,m)-p.| < e;
consider N such that
a4: for n,m st n>=N & m>=N holds |.Rseq.(n,m)-p.| < 1 by a3;
a5:for n,m st n>=N & m>=N holds |.Rseq.(n,m).| < 1 + |.p.|
proof
let n,m;
assume n>=N & m>=N; then
a6: |.Rseq.(n,m)-p.| < 1 by a4;
|.Rseq.(n,m)-p.| >= |.Rseq.(n,m).| - |.p.| by COMPLEX1:59; then
|.Rseq.(n,m).| - |.p.| < 1 by a6,XXREAL_0:2;
hence |.Rseq.(n,m).| < 1 + |.p.| by XREAL_1:19;
end;
deffunc F1(Element of NAT) = |.Rseq.(1,$1).|;
deffunc F(Element of NAT) = $1;
reconsider E2 = {F(m) where m is Element of NAT: m <= N} as
finite non empty Subset of NAT from ASYMPT_0:sch 2;
reconsider EE = [:E2,E2:] as finite set;
c1:E2 is finite;
deffunc F(Element of NAT,Element of NAT) = |.Rseq.($1,$2).|;
a9:{F(m,n) where m is Element of NAT, n is Element of NAT :
m in E2 & n in E2} is finite from FRAENKEL:sch 22(c1,c1);
set F = {F(m,n) where m is Element of NAT, n is Element of NAT :
m in E2 & n in E2};
N is Element of NAT by ORDINAL1:def 12; then
N in E2; then
b1: |.Rseq.(N,N).| in F;
now let x be object;
assume x in F; then
consider p be Element of NAT, q be Element of NAT such that
a10: x = F(p,q) & p in E2 & q in E2;
thus x is real by a10;
end; then
reconsider F as non empty real-membered set by b1,MEMBERED:def 3;
reconsider M1 = sup F as Element of REAL by a9,XXREAL_2:16;
set M = max(M1,1+ |.p.|);
a14:M >= 1+ |.p.| & 1+ |.p.| >= 1 & M >= M1 & M >= M1
by XXREAL_0:25,XREAL_1:31,COMPLEX1:46;
c1:for n,m holds |.Rseq.(n,m).| <= M
proof
let n,m;
c0: n is Element of NAT & m is Element of NAT & N is Element of NAT
by ORDINAL1:def 12;
per cases;
suppose n >= N & m >= N; then
|.Rseq.(n,m).| < 1 + |.p.| by a5;
hence |.Rseq.(n,m).| <= M by a14,XXREAL_0:2;
end;
suppose n < N & m < N; then
n in E2 & m in E2 by c0; then
|.Rseq.(n,m).| in F; then
a15: |.Rseq.(n,m).| <= M1 by XXREAL_2:4;
M >= M1 by XXREAL_0:25;
hence |.Rseq.(n,m).| <= M by a15,XXREAL_0:2;
end;
suppose a18: n < N & m >= N; then
a19: Rseq.(n,N) >= Rseq.(n,m) & Rseq.(n,m) >= Rseq.(N,m) by a1;
n in E2 & N in E2 by a18,c0; then
|.Rseq.(n,N).| in F; then
a20: |.Rseq.(n,N).| <= M1 by XXREAL_2:4;
a21: |.Rseq.(N,m).| < 1+ |.p.| by a5,a18;
per cases by a19,th28;
suppose |.Rseq.(n,m).| <= |.Rseq.(n,N).|; then
|.Rseq.(n,m).| <= M1 by a20,XXREAL_0:2;
hence |.Rseq.(n,m).| <= M by a14,XXREAL_0:2;
end;
suppose |.Rseq.(n,m).| <= |.Rseq.(N,m).|; then
|.Rseq.(n,m).| <= 1+ |.p.| by a21,XXREAL_0:2;
hence |.Rseq.(n,m).| <= M by a14,XXREAL_0:2;
end;
end;
suppose a22: n >= N & m < N; then
a23: Rseq.(N,m) >= Rseq.(n,m) & Rseq.(n,m) >= Rseq.(n,N) by a1;
N in E2 & m in E2 by a22,c0; then
|.Rseq.(N,m).| in F; then
a24: |.Rseq.(N,m).| <= M1 by XXREAL_2:4;
a25: |.Rseq.(n,N).| < 1+ |.p.| by a5,a22;
per cases by a23,th28;
suppose |.Rseq.(n,m).| <= |.Rseq.(N,m).|; then
|.Rseq.(n,m).| <= M1 by a24,XXREAL_0:2;
hence |.Rseq.(n,m).| <= M by a14,XXREAL_0:2;
end;
suppose |.Rseq.(n,m).| <= |.Rseq.(n,N).|; then
|.Rseq.(n,m).| <= 1+ |.p.| by a25,XXREAL_0:2;
hence |.Rseq.(n,m).| <= M by a14,XXREAL_0:2;
end;
end;
end;
now let a be ExtReal;
assume a in Rseq.: [:NAT,NAT:]; then
consider x be Element of [:NAT,NAT:] such that
b2: x in [:NAT,NAT:] & a = Rseq.x by FUNCT_2:65;
consider n,m be object such that
b3: n in NAT & m in NAT & x = [n,m] by ZFMISC_1:def 2;
reconsider n,m as Element of NAT by b3;
|.Rseq.(n,m).| <= M by c1;
hence -M <= a & a <= M by b3,b2,ABSVALUE:5;
end; then
(for a be ExtReal st a in Rseq.: [:NAT,NAT:] holds -M <= a) &
(for a be ExtReal st a in Rseq.: [:NAT,NAT:] holds a <= M); then
-M is LowerBound of Rseq.: [:NAT,NAT:] &
M is UpperBound of Rseq.: [:NAT,NAT:] by XXREAL_2:def 1,def 2;
hence thesis by XXREAL_2:def 9,def 10;
end;
end;
LM111:
for r be Element of REAL holds
[:NAT,NAT:]-->r is P-convergent & [:NAT,NAT:]-->r is convergent_in_cod1
& [:NAT,NAT:]-->r is convergent_in_cod2
proof
let r be Element of REAL;
set Rseq = [:NAT,NAT:] --> r;
a1:for n,m be Nat holds Rseq.(n,m) = r
proof
let n,m be Nat;
n is Element of NAT & m is Element of NAT by ORDINAL1:def 12;
hence Rseq.(n,m) = r by FUNCOP_1:70;
end;
a3:now let e be Real;
assume a2: 0<e;
a4: now let n,m such that n>=0 & m>=0;
Rseq.(n,m) = r by a1;
hence |. Rseq.(n,m) - r.| < e by a2,COMPLEX1:44;
end;
reconsider N = 0 as Nat;
take N;
thus for n,m st n>=N & m>=N holds |. Rseq.(n,m) - r.| < e by a4;
end;
b1:now let m be Element of NAT;
now let e be Real;
assume b3: 0<e;
now let n be Nat;
assume 0<=n;
b4: n is Element of NAT by ORDINAL1:def 12; then
ProjMap2(Rseq,m).n = Rseq.(n,m) by MESFUNC9:def 7; then
ProjMap2(Rseq,m).n = r by b4,FUNCOP_1:70;
hence |.ProjMap2(Rseq,m).n - r.| < e by b3,ABSVALUE:2;
end;
hence ex N be Nat st for n be Nat st N<=n
holds |.ProjMap2(Rseq,m).n-r.| < e;
end;
hence ProjMap2(Rseq,m) is convergent by SEQ_2:def 6;
end;
now let n be Element of NAT;
now let e be Real;
assume c3: 0<e;
now let m be Nat;
assume 0<=m;
c4: m is Element of NAT by ORDINAL1:def 12; then
ProjMap1(Rseq,n).m = Rseq.(n,m) by MESFUNC9:def 6; then
ProjMap1(Rseq,n).m = r by c4,FUNCOP_1:70;
hence |.ProjMap1(Rseq,n).m - r.| < e by c3,ABSVALUE:2;
end;
hence ex M be Nat st for m be Nat st M<=m
holds |.ProjMap1(Rseq,n).m-r.| < e;
end;
hence ProjMap1(Rseq,n) is convergent by SEQ_2:def 6;
end;
hence thesis by a3,b1;
end;
registration
let r be Element of REAL;
cluster [:NAT,NAT:] --> r -> P-convergent convergent_in_cod1
convergent_in_cod2 for Function of [:NAT,NAT:],REAL;
coherence by LM111;
end;
theorem
for r be Element of REAL holds P-lim ([:NAT,NAT:] --> r) = r
proof
let r be Element of REAL;
set Rseq = [:NAT,NAT:] --> r;
a1:for n,m be Nat holds Rseq.(n,m) = r
proof
let n,m be Nat;
n is Element of NAT & m is Element of NAT by ORDINAL1:def 12;
hence Rseq.(n,m) = r by FUNCOP_1:70;
end;
now let e be Real;
assume a2: 0<e;
a4: now let n,m such that n>=0 & m>=0;
Rseq.(n,m) = r by a1;
hence |. Rseq.(n,m) - r.| < e by a2,COMPLEX1:44;
end;
reconsider N = 0 as Nat;
take N;
thus for n,m st n>=N & m>=N holds |. Rseq.(n,m) - r.| < e by a4;
end;
hence P-lim ([:NAT,NAT:] --> r) = r by def6;
end;
registration
cluster P-convergent convergent_in_cod1 convergent_in_cod2
for Function of [:NAT,NAT:],REAL;
existence
proof
a1: 1 is Element of REAL by XREAL_0:def 1; then
reconsider f = [:NAT,NAT:] --> 1 as Function of [:NAT,NAT:],REAL
by FUNCOP_1:46;
f is P-convergent convergent_in_cod1 convergent_in_cod2
by a1;
hence thesis;
end;
end;
reserve Pseq for P-convergent Function of [:NAT,NAT:],REAL;
registration
let Pseq2 be P-convergent convergent_in_cod2 Function of [:NAT,NAT:],REAL;
cluster lim_in_cod2 Pseq2 -> convergent;
coherence
proof
Pseq2 is P-convergent; then
consider z be Real such that
a3: for e st 0<e
ex N1 be Nat st
for n,m st n>=N1 & m>=N1 holds |.Pseq2.(n,m)-z.| < e;
for e st 0<e
ex N st for n st n>=N holds |.(lim_in_cod2 Pseq2).n - z.| < e
proof
let e;
assume a8: 0 < e; then
consider N1 be Nat such that
a15: for n,m st n>=N1 & m>=N1 holds |.Pseq2.(n,m)-z.| < e/2 by a3;
a12:for n be Element of NAT st n >= N1 holds
ex N2 be Nat st
for m st m>=N2 holds
|.ProjMap1(Pseq2,n).m - (lim_in_cod2 Pseq2).n.| < e/2
proof
let n be Element of NAT;
assume n >= N1;
Pseq2 is convergent_in_cod2; then
ProjMap1(Pseq2,n) is convergent; then
consider N2 be Nat such that
a9: for m be Nat st m>=N2 holds
|.ProjMap1(Pseq2,n).m - lim ProjMap1(Pseq2,n).| < e/2
by a8,SEQ_2:def 7;
take N2;
thus for m st m >= N2 holds
|.ProjMap1(Pseq2,n).m - (lim_in_cod2 Pseq2).n .| < e/2
proof
let m;
assume m >=N2; then
|.ProjMap1(Pseq2,n).m - lim ProjMap1(Pseq2,n).| < e/2 by a9;
hence thesis by def33;
end;
end;
take N1;
thus for n st n>=N1 holds |.(lim_in_cod2 Pseq2).n - z.| < e
proof
let n;
reconsider n1=n as Element of NAT by ORDINAL1:def 12;
assume a14: n>=N1; then
consider N2 be Nat such that
a13: for m st m>=N2 holds
|.ProjMap1(Pseq2,n1).m - (lim_in_cod2 Pseq2).n1 .| < e/2
by a12;
reconsider M=max(N2,N1) as Element of NAT by ORDINAL1:def 12;
a17: ProjMap1(Pseq2,n1).M = Pseq2.(n,M) by MESFUNC9:def 6;
a16: M>=N2 & M>=N1 by XXREAL_0:25;
a18: |.(lim_in_cod2 Pseq2).n - z.|
<= |.(lim_in_cod2 Pseq2).n - ProjMap1(Pseq2,n1).M .|
+ |.Pseq2.(n,M) - z.| by a17,COMPLEX1:63;
|.(lim_in_cod2 Pseq2).n - ProjMap1(Pseq2,n1).M .|
= |.ProjMap1(Pseq2,n1).M - (lim_in_cod2 Pseq2).n .| by COMPLEX1:60; then
a20: |.(lim_in_cod2 Pseq2).n - ProjMap1(Pseq2,n1).M .| < e/2
by a13,XXREAL_0:25;
|.Pseq2.(n1,M) - z.| < e/2 by a15,a16,a14; then
|.(lim_in_cod2 Pseq2).n - ProjMap1(Pseq2,n1).M .|
+ |.Pseq2.(n,M) - z.| < e/2 + e/2 by a20,XREAL_1:8;
hence |.(lim_in_cod2 Pseq2).n - z.| < e by a18,XXREAL_0:2;
end;
end;
hence lim_in_cod2 Pseq2 is convergent by SEQ_2:def 6;
end;
end;
theorem
Rseq is P-convergent convergent_in_cod2 implies
P-lim Rseq = cod2_major_iterated_lim Rseq
proof
assume a1: Rseq is P-convergent convergent_in_cod2; then
consider z be Real such that
a3: for e st 0<e
ex N1 be Nat st
for n,m st n>=N1 & m>=N1 holds |.Rseq.(n,m)-z.| < e;
for e st 0<e
ex N st for n st n>=N holds |.(lim_in_cod2 Rseq).n - z.| < e
proof
let e;
assume a8: 0 < e; then
consider N1 be Nat such that
a15: for n,m st n>=N1 & m>=N1 holds |.Rseq.(n,m)-z.| < e/2 by a3;
a12:for n be Element of NAT st n >= N1 holds
ex N2 be Nat st
for m st m>=N2 holds |.ProjMap1(Rseq,n).m - (lim_in_cod2 Rseq).n.| < e/2
proof
let n be Element of NAT;
assume n >= N1;
ProjMap1(Rseq,n) is convergent by a1; then
consider N2 be Nat such that
a9: for m be Nat st m>=N2 holds
|.ProjMap1(Rseq,n).m - lim ProjMap1(Rseq,n).| < e/2
by a8,SEQ_2:def 7;
take N2;
thus for m st m >= N2 holds
|.ProjMap1(Rseq,n).m - (lim_in_cod2 Rseq).n .| < e/2
proof
let m;
assume m >=N2; then
|.ProjMap1(Rseq,n).m - lim ProjMap1(Rseq,n).| < e/2 by a9;
hence thesis by def33;
end;
end;
take N1;
thus for n st n>=N1 holds |.(lim_in_cod2 Rseq).n - z.| < e
proof
let n;
reconsider n1=n as Element of NAT by ORDINAL1:def 12;
assume a14: n>=N1; then
consider N2 be Nat such that
a13: for m st m>=N2 holds
|.ProjMap1(Rseq,n1).m - (lim_in_cod2 Rseq).n1 .| < e/2
by a12;
reconsider M=max(N2,N1) as Element of NAT by ORDINAL1:def 12;
a17: ProjMap1(Rseq,n1).M = Rseq.(n,M) by MESFUNC9:def 6;
a16: M>=N2 & M>=N1 by XXREAL_0:25;
a18: |.(lim_in_cod2 Rseq).n - z.|
<= |.(lim_in_cod2 Rseq).n - ProjMap1(Rseq,n1).M .|
+ |.Rseq.(n,M) - z.| by a17,COMPLEX1:63;
|.(lim_in_cod2 Rseq).n - ProjMap1(Rseq,n1).M .|
= |.ProjMap1(Rseq,n1).M - (lim_in_cod2 Rseq).n .| by COMPLEX1:60; then
a20: |.(lim_in_cod2 Rseq).n - ProjMap1(Rseq,n1).M .| < e/2 by a13,XXREAL_0:25;
|.Rseq.(n1,M) - z.| < e/2 by a15,a16,a14; then
|.(lim_in_cod2 Rseq).n - ProjMap1(Rseq,n1).M .|
+ |.Rseq.(n,M) - z.| < e/2 + e/2 by a20,XREAL_1:8;
hence |.(lim_in_cod2 Rseq).n - z.| < e by a18,XXREAL_0:2;
end;
end; then
a21:lim_in_cod2 Rseq is convergent by SEQ_2:def 6;
for e st 0<e
ex N st for n st n>=N holds |.(lim_in_cod2 Rseq).n - P-lim Rseq.| < e
proof
let e;
assume a22: 0<e; then
consider N1 be Nat such that
a23: for n,m st n>=N1 & m>=N1 holds |.Rseq.(n,m)-P-lim Rseq.| < e/2 by a1,def6;
take N = N1;
hereby let n;
assume n>=N; then
a27: for m st m >= N1 holds |.Rseq.(n,m) - P-lim Rseq.| < e/2 by a23;
reconsider n1=n as Element of NAT by ORDINAL1:def 12;
ProjMap1(Rseq,n1) is convergent by a1; then
consider N3 be Nat such that
a29: for m st m>=N3 holds
|.ProjMap1(Rseq,n1).m - lim ProjMap1(Rseq,n1).| < e/2
by a22,SEQ_2:def 7;
reconsider M = max(N1,N3) as Element of NAT by ORDINAL1:def 12;
a31: |.Rseq.(n,M) - P-lim Rseq.| < e/2 by a27,XXREAL_0:25;
ProjMap1(Rseq,n1).M = Rseq.(n,M) by MESFUNC9:def 6; then
|.Rseq.(n,M) - lim ProjMap1(Rseq,n1).| < e/2 by a29,XXREAL_0:25; then
|.Rseq.(n,M) - (lim_in_cod2 Rseq).n .| < e/2 by def33; then
|.(lim_in_cod2 Rseq).n - Rseq.(n,M).| < e/2 by COMPLEX1:60; then
a32: |.(lim_in_cod2 Rseq).n - Rseq.(n,M).| + |.Rseq.(n,M) - P-lim Rseq.|
< e/2 + e/2 by a31,XREAL_1:8;
|.(lim_in_cod2 Rseq).n - P-lim Rseq.|
<= |.(lim_in_cod2 Rseq).n - Rseq.(n,M).|
+ |. Rseq.(n,M) - P-lim Rseq.| by COMPLEX1:63;
hence |.(lim_in_cod2 Rseq).n - P-lim Rseq.| < e by a32,XXREAL_0:2;
end;
end;
hence thesis by a21,def35;
end;
registration
let Pseq1 be P-convergent convergent_in_cod1 Function of [:NAT,NAT:],REAL;
cluster lim_in_cod1 Pseq1 -> convergent;
coherence
proof
Pseq1 is P-convergent; then
consider z be Real such that
a3: for e st 0<e
ex N1 be Nat st
for n,m st n>=N1 & m>=N1 holds |.Pseq1.(n,m)-z.| < e;
for e st 0<e
ex N st for n st n>=N holds |.(lim_in_cod1 Pseq1).n - z.| < e
proof
let e;
assume a8: 0 < e; then
consider N1 be Nat such that
a15: for n,m st n>=N1 & m>=N1 holds |.Pseq1.(n,m)-z.| < e/2 by a3;
a12:for n be Element of NAT st n >= N1 holds
ex N2 be Nat st
for m st m>=N2 holds
|.ProjMap2(Pseq1,n).m - (lim_in_cod1 Pseq1).n .| < e/2
proof
let n be Element of NAT;
assume n >= N1;
Pseq1 is convergent_in_cod1; then
ProjMap2(Pseq1,n) is convergent; then
consider N2 be Nat such that
a9: for m st m>=N2 holds
|.ProjMap2(Pseq1,n).m - lim ProjMap2(Pseq1,n).| < e/2
by a8,SEQ_2:def 7;
take N2;
thus for m st m >= N2 holds
|.ProjMap2(Pseq1,n).m - (lim_in_cod1 Pseq1).n .| < e/2
proof
let m;
assume m >=N2; then
|.ProjMap2(Pseq1,n).m - lim ProjMap2(Pseq1,n).| < e/2 by a9;
hence thesis by def32;
end;
end;
take N1;
thus for n st n>=N1 holds |.(lim_in_cod1 Pseq1).n - z.| < e
proof
let n be Nat;
reconsider n1=n as Element of NAT by ORDINAL1:def 12;
assume a14: n>=N1; then
consider N2 be Nat such that
a13: for m st m>=N2 holds
|.ProjMap2(Pseq1,n1).m - (lim_in_cod1 Pseq1).n .| < e/2 by a12;
reconsider M=max(N2,N1) as Element of NAT by ORDINAL1:def 12;
a17: ProjMap2(Pseq1,n1).M = Pseq1.(M,n) by MESFUNC9:def 7;
a16: M>=N2 & M>=N1 by XXREAL_0:25;
a18: |.(lim_in_cod1 Pseq1).n - z.|
<= |.(lim_in_cod1 Pseq1).n - ProjMap2(Pseq1,n1).M .|
+ |.Pseq1.(M,n) - z.| by a17,COMPLEX1:63;
|.(lim_in_cod1 Pseq1).n - ProjMap2(Pseq1,n1).M .|
= |.ProjMap2(Pseq1,n1).M - (lim_in_cod1 Pseq1).n .| by COMPLEX1:60; then
a20: |.(lim_in_cod1 Pseq1).n - ProjMap2(Pseq1,n1).M .| < e/2
by a13,XXREAL_0:25;
|.Pseq1.(M,n1) - z.| < e/2 by a15,a16,a14; then
|.(lim_in_cod1 Pseq1).n - ProjMap2(Pseq1,n1).M .|
+ |.Pseq1.(M,n) - z.| < e/2 + e/2 by a20,XREAL_1:8;
hence |.(lim_in_cod1 Pseq1).n - z.| < e by a18,XXREAL_0:2;
end;
end;
hence lim_in_cod1 Pseq1 is convergent by SEQ_2:def 6;
end;
end;
theorem
Rseq is P-convergent convergent_in_cod1 implies
P-lim Rseq = cod1_major_iterated_lim Rseq
proof
assume a1: Rseq is P-convergent convergent_in_cod1; then
consider z be Real such that
a3: for e st 0<e
ex N1 be Nat st
for n,m st n>=N1 & m>=N1 holds |.Rseq.(n,m)-z.| < e;
for e st 0<e
ex N st for n st n>=N holds |.(lim_in_cod1 Rseq).n - z.| < e
proof
let e;
assume a8: 0 < e; then
consider N1 be Nat such that
a15: for n,m st n>=N1 & m>=N1 holds |.Rseq.(n,m)-z.| < e/2 by a3;
a12:for n be Element of NAT st n >= N1 holds
ex N2 be Nat st
for m st m>=N2 holds |.ProjMap2(Rseq,n).m - (lim_in_cod1 Rseq).n .| < e/2
proof
let n be Element of NAT;
assume n >= N1;
ProjMap2(Rseq,n) is convergent by a1; then
consider N2 be Nat such that
a9: for m st m>=N2 holds |.ProjMap2(Rseq,n).m - lim ProjMap2(Rseq,n).| < e/2
by a8,SEQ_2:def 7;
take N2;
thus for m st m >= N2 holds
|.ProjMap2(Rseq,n).m - (lim_in_cod1 Rseq).n .| < e/2
proof
let m;
assume m >=N2; then
|.ProjMap2(Rseq,n).m - lim ProjMap2(Rseq,n).| < e/2 by a9;
hence thesis by def32;
end;
end;
take N1;
thus for n st n>=N1 holds |.(lim_in_cod1 Rseq).n - z.| < e
proof
let n be Nat;
reconsider n1=n as Element of NAT by ORDINAL1:def 12;
assume a14: n>=N1; then
consider N2 be Nat such that
a13: for m st m>=N2 holds
|.ProjMap2(Rseq,n1).m - (lim_in_cod1 Rseq).n .| < e/2 by a12;
reconsider M=max(N2,N1) as Element of NAT by ORDINAL1:def 12;
a17: ProjMap2(Rseq,n1).M = Rseq.(M,n) by MESFUNC9:def 7;
a16: M>=N2 & M>=N1 by XXREAL_0:25;
a18: |.(lim_in_cod1 Rseq).n - z.|
<= |.(lim_in_cod1 Rseq).n - ProjMap2(Rseq,n1).M .|
+ |.Rseq.(M,n) - z.| by a17,COMPLEX1:63;
|.(lim_in_cod1 Rseq).n - ProjMap2(Rseq,n1).M .|
= |.ProjMap2(Rseq,n1).M - (lim_in_cod1 Rseq).n .| by COMPLEX1:60; then
a20: |.(lim_in_cod1 Rseq).n - ProjMap2(Rseq,n1).M .| < e/2 by a13,XXREAL_0:25;
|.Rseq.(M,n1) - z.| < e/2 by a15,a16,a14; then
|.(lim_in_cod1 Rseq).n - ProjMap2(Rseq,n1).M .|
+ |.Rseq.(M,n) - z.| < e/2 + e/2 by a20,XREAL_1:8;
hence |.(lim_in_cod1 Rseq).n - z.| < e by a18,XXREAL_0:2;
end;
end; then
a21:lim_in_cod1 Rseq is convergent by SEQ_2:def 6;
for e st 0<e
ex N st for n st n>=N holds |.(lim_in_cod1 Rseq).n - P-lim Rseq.| < e
proof
let e;
assume a22: 0<e; then
consider N1 be Nat such that
a23: for n,m st n>=N1 & m>=N1 holds
|.Rseq.(n,m)-P-lim Rseq.| < e/2 by a1,def6;
take N = N1;
hereby let n;
assume n>=N; then
a27: for m st m >= N1 holds |.Rseq.(m,n) - P-lim Rseq.| < e/2 by a23;
reconsider n1=n as Element of NAT by ORDINAL1:def 12;
ProjMap2(Rseq,n1) is convergent by a1; then
consider N3 be Nat such that
a29: for m st m>=N3 holds
|.ProjMap2(Rseq,n1).m - lim ProjMap2(Rseq,n1).| < e/2
by a22,SEQ_2:def 7;
reconsider M = max(N1,N3) as Element of NAT by ORDINAL1:def 12;
a31: |.Rseq.(M,n) - P-lim Rseq.| < e/2 by a27,XXREAL_0:25;
ProjMap2(Rseq,n1).M = Rseq.(M,n) by MESFUNC9:def 7; then
|.Rseq.(M,n) - lim ProjMap2(Rseq,n1).| < e/2 by a29,XXREAL_0:25; then
|.Rseq.(M,n) - (lim_in_cod1 Rseq).n .| < e/2 by def32; then
|.(lim_in_cod1 Rseq).n - Rseq.(M,n).| < e/2 by COMPLEX1:60; then
a32: |.(lim_in_cod1 Rseq).n - Rseq.(M,n).| + |.Rseq.(M,n) - P-lim Rseq.|
< e/2 + e/2 by a31,XREAL_1:8;
|.(lim_in_cod1 Rseq).n - P-lim Rseq.|
<= |.(lim_in_cod1 Rseq).n - Rseq.(M,n).|
+ |.Rseq.(M,n) - P-lim Rseq.| by COMPLEX1:63;
hence |.(lim_in_cod1 Rseq).n - P-lim Rseq.| < e by a32,XXREAL_0:2;
end;
end;
hence thesis by a21,def34;
end;
LM112:
Rseq is non-decreasing bounded_above implies
Rseq is P-convergent convergent_in_cod1 convergent_in_cod2
proof
assume that
a1: Rseq is non-decreasing and
a2: Rseq is bounded_above;
reconsider M = sup(Rseq.: [:NAT,NAT:]) as Element of REAL by a2,XXREAL_2:16;
b2:[:NAT,NAT:] = dom Rseq by FUNCT_2:def 1; then
b3:rng Rseq = Rseq.: [:NAT,NAT:] by RELAT_1:113;
a3:for e st 0<e ex N st Rseq.(N,N) > M-e
proof
let e;
assume a4: 0<e;
assume a7: for n holds Rseq.(n,n) <= M-e;
now let a be ExtReal;
assume a in Rseq.: [:NAT,NAT:]; then
consider x be object such that
a5: x in dom Rseq & a = Rseq.x by b3,FUNCT_1:def 3;
consider i,j be object such that
a6: i in NAT & j in NAT & x = [i,j] by a5,ZFMISC_1:def 2;
reconsider i,j as Nat by a6;
a0: max(i,j) is Nat by TARSKI:1;
max(i,j) >= i & max(i,j) >= j by XXREAL_0:25; then
a8: Rseq.(max(i,j),max(i,j)) >= Rseq.(i,j) by a1,a0;
Rseq.(max(i,j),max(i,j)) <= M-e by a7,a0;
hence a <= M-e by a5,a6,a8,XXREAL_0:2;
end; then
M-e is UpperBound of Rseq.: [:NAT,NAT:] by XXREAL_2:def 1;
hence contradiction by a4,XREAL_1:44,XXREAL_2:def 3;
end;
for e st 0<e
ex N st for n,m st n>=N & m>=N holds |.Rseq.(n,m) - M.| < e
proof
let e;
assume a5: 0<e; then
consider N such that
a10: Rseq.(N,N) > M - e by a3;
take N;
hereby let n,m;
assume n>=N & m>=N; then
a11: Rseq.(N,N) <= Rseq.(n,m) by a1;
n in NAT & m in NAT by ORDINAL1:def 12; then
[n,m] in [:NAT,NAT:] by ZFMISC_1:def 2; then
a12: Rseq.(n,m) <= M by b2,b3,FUNCT_1:3,XXREAL_2:4;
M < M+e by a5,XREAL_1:29; then
M - e < Rseq.(n,m) & Rseq.(n,m) < M + e by a10,a11,a12,XXREAL_0:2;
hence |.Rseq.(n,m) - M.| < e by RINFSUP1:1;
end;
end;
hence Rseq is P-convergent;
for m be Element of NAT holds ProjMap2(Rseq,m) is convergent
proof
let m be Element of NAT;
NAT = dom ProjMap2(Rseq,m) by FUNCT_2:def 1; then
c3: rng ProjMap2(Rseq,m) = ProjMap2(Rseq,m).:NAT by RELAT_1:113;
now let a be object;
assume a in ProjMap2(Rseq,m).:NAT; then
consider i be object such that
c4: i in dom ProjMap2(Rseq,m) & a = ProjMap2(Rseq,m).i by c3,FUNCT_1:def 3;
reconsider i as Element of NAT by c4;
[i,m] in [:NAT,NAT:] by ZFMISC_1:def 2; then
Rseq.(i,m) in Rseq.: [:NAT,NAT:] by b2,b3,FUNCT_1:3;
hence a in Rseq.: [:NAT,NAT:] by c4,MESFUNC9:def 7;
end;
then
c5: ProjMap2(Rseq,m) is bounded_above by a2,XXREAL_2:43,TARSKI:def 3;
now let n;
n is Element of NAT by ORDINAL1:def 12; then
c6: ProjMap2(Rseq,m).n = Rseq.(n,m) & ProjMap2(Rseq,m).(n+1) = Rseq.(n+1,m)
by MESFUNC9:def 7;
n<=n+1 by NAT_1:11;
hence ProjMap2(Rseq,m).n <= ProjMap2(Rseq,m).(n+1) by a1,c6;
end; then
ProjMap2(Rseq,m) is non-decreasing by SEQM_3:def 8;
hence ProjMap2(Rseq,m) is convergent by c5;
end;
hence Rseq is convergent_in_cod1;
for m be Element of NAT holds ProjMap1(Rseq,m) is convergent
proof
let m be Element of NAT;
NAT = dom ProjMap1(Rseq,m) by FUNCT_2:def 1; then
c3: rng ProjMap1(Rseq,m) = ProjMap1(Rseq,m).:NAT by RELAT_1:113;
now let a be object;
assume a in ProjMap1(Rseq,m).:NAT; then
consider i be object such that
c4: i in dom ProjMap1(Rseq,m) & a = ProjMap1(Rseq,m).i by c3,FUNCT_1:def 3;
reconsider i as Element of NAT by c4;
[m,i] in [:NAT,NAT:] by ZFMISC_1:def 2; then
Rseq.(m,i) in Rseq.: [:NAT,NAT:] by b2,b3,FUNCT_1:3;
hence a in Rseq.: [:NAT,NAT:] by c4,MESFUNC9:def 6;
end; then
c5: ProjMap1(Rseq,m) is bounded_above by a2,XXREAL_2:43,TARSKI:def 3;
now let n;
n is Element of NAT by ORDINAL1:def 12; then
c6: ProjMap1(Rseq,m).n = Rseq.(m,n) & ProjMap1(Rseq,m).(n+1) = Rseq.(m,n+1)
by MESFUNC9:def 6;
n<=n+1 by NAT_1:11;
hence ProjMap1(Rseq,m).n <= ProjMap1(Rseq,m).(n+1) by a1,c6;
end; then
ProjMap1(Rseq,m) is non-decreasing by SEQM_3:def 8;
hence ProjMap1(Rseq,m) is convergent by c5;
end;
hence Rseq is convergent_in_cod2;
end;
registration
cluster non-decreasing bounded_above ->
P-convergent convergent_in_cod1 convergent_in_cod2
for Function of [:NAT,NAT:],REAL;
coherence by LM112;
end;
LM113:
Rseq is non-increasing bounded_below implies
Rseq is P-convergent convergent_in_cod1 convergent_in_cod2
proof
assume that
a1: Rseq is non-increasing and
a2: Rseq is bounded_below;
reconsider M = inf(Rseq.: [:NAT,NAT:]) as Element of REAL by a2,XXREAL_2:15;
b2:[:NAT,NAT:] = dom Rseq by FUNCT_2:def 1; then
b3:rng Rseq = Rseq.: [:NAT,NAT:] by RELAT_1:113;
a3:for e st 0<e ex N st Rseq.(N,N) < M+e
proof
let e;
assume a4: 0<e;
assume a7: for n holds Rseq.(n,n) >= M+e;
now let a be ExtReal;
assume a in Rseq.: [:NAT,NAT:]; then
consider x be object such that
a5: x in dom Rseq & a = Rseq.x by b3,FUNCT_1:def 3;
consider i,j be object such that
a6: i in NAT & j in NAT & x = [i,j] by a5,ZFMISC_1:def 2;
reconsider i,j as Nat by a6;
a0: max(i,j) is Nat by TARSKI:1;
max(i,j) >= i & max(i,j) >= j by XXREAL_0:25; then
a8: Rseq.(max(i,j),max(i,j)) <= Rseq.(i,j) by a1,a0;
Rseq.(max(i,j),max(i,j)) >= M+e by a7,a0;
hence a >= M+e by a5,a6,a8,XXREAL_0:2;
end; then
M+e is LowerBound of Rseq.: [:NAT,NAT:] by XXREAL_2:def 2;
hence contradiction by a4,XREAL_1:29,XXREAL_2:def 4;
end;
for e st 0<e
ex N st for n,m st n>=N & m>=N holds |.Rseq.(n,m) - M.| < e
proof
let e;
assume a5: 0<e; then
consider N such that
a10: Rseq.(N,N) < M+e by a3;
take N;
hereby let n,m;
assume n>=N & m>=N; then
a11: Rseq.(N,N) >= Rseq.(n,m) by a1;
n in NAT & m in NAT by ORDINAL1:def 12; then
[n,m] in [:NAT,NAT:] by ZFMISC_1:def 2; then
a12: Rseq.(n,m) >= M by b2,b3,FUNCT_1:3,XXREAL_2:3;
M > M-e by a5,XREAL_1:44; then
M - e < Rseq.(n,m) & Rseq.(n,m) < M + e by a10,a11,a12,XXREAL_0:2;
hence |.Rseq.(n,m) - M.| < e by RINFSUP1:1;
end;
end;
hence Rseq is P-convergent;
for m be Element of NAT holds ProjMap2(Rseq,m) is convergent
proof
let m be Element of NAT;
NAT = dom ProjMap2(Rseq,m) by FUNCT_2:def 1; then
c3: rng ProjMap2(Rseq,m) = ProjMap2(Rseq,m).:NAT by RELAT_1:113;
now let a be object;
assume a in ProjMap2(Rseq,m).:NAT; then
consider i be object such that
c4: i in dom ProjMap2(Rseq,m) & a = ProjMap2(Rseq,m).i by c3,FUNCT_1:def 3;
reconsider i as Element of NAT by c4;
[i,m] in [:NAT,NAT:] by ZFMISC_1:def 2; then
Rseq.(i,m) in Rseq.: [:NAT,NAT:] by b2,b3,FUNCT_1:3;
hence a in Rseq.: [:NAT,NAT:] by c4,MESFUNC9:def 7;
end; then
c5: ProjMap2(Rseq,m) is bounded_below by a2,XXREAL_2:44,TARSKI:def 3;
now let n;
n is Element of NAT by ORDINAL1:def 12; then
c6: ProjMap2(Rseq,m).n = Rseq.(n,m) & ProjMap2(Rseq,m).(n+1) = Rseq.(n+1,m)
by MESFUNC9:def 7;
n<=n+1 by NAT_1:11;
hence ProjMap2(Rseq,m).n >= ProjMap2(Rseq,m).(n+1) by a1,c6;
end; then
ProjMap2(Rseq,m) is non-increasing by SEQM_3:def 9;
hence ProjMap2(Rseq,m) is convergent by c5;
end;
hence Rseq is convergent_in_cod1;
for m be Element of NAT holds ProjMap1(Rseq,m) is convergent
proof
let m be Element of NAT;
NAT = dom ProjMap1(Rseq,m) by FUNCT_2:def 1; then
c3: rng ProjMap1(Rseq,m) = ProjMap1(Rseq,m).:NAT by RELAT_1:113;
now let a be object;
assume a in ProjMap1(Rseq,m).:NAT; then
consider i be object such that
c4: i in dom ProjMap1(Rseq,m) & a = ProjMap1(Rseq,m).i by c3,FUNCT_1:def 3;
reconsider i as Element of NAT by c4;
[m,i] in [:NAT,NAT:] by ZFMISC_1:def 2; then
Rseq.(m,i) in Rseq.: [:NAT,NAT:] by b2,b3,FUNCT_1:3;
hence a in Rseq.: [:NAT,NAT:] by c4,MESFUNC9:def 6;
end; then
c5: ProjMap1(Rseq,m) is bounded_below by a2,XXREAL_2:44,TARSKI:def 3;
now let n;
n is Element of NAT by ORDINAL1:def 12; then
c6: ProjMap1(Rseq,m).n = Rseq.(m,n) & ProjMap1(Rseq,m).(n+1) = Rseq.(m,n+1)
by MESFUNC9:def 6;
n<=n+1 by NAT_1:11;
hence ProjMap1(Rseq,m).n >= ProjMap1(Rseq,m).(n+1) by a1,c6;
end; then
ProjMap1(Rseq,m) is non-increasing by SEQM_3:def 9;
hence ProjMap1(Rseq,m) is convergent by c5;
end;
hence Rseq is convergent_in_cod2;
end;
registration
cluster non-increasing bounded_below ->
P-convergent convergent_in_cod1 convergent_in_cod2
for Function of [:NAT,NAT:],REAL;
coherence by LM113;
end;
theorem
Rseq is uniformly_convergent_in_cod1 & lim_in_cod1 Rseq is convergent implies
Rseq is P-convergent & P-lim Rseq = cod1_major_iterated_lim Rseq
proof
assume that
a3: Rseq is uniformly_convergent_in_cod1 and
a2: lim_in_cod1 Rseq is convergent;
a4:now let e be Real;
assume a5: 0<e; then
consider N1 be Nat such that
a6: for m st m >= N1 holds
for n holds |.Rseq.(n,m) - (lim_in_cod1 Rseq).n .| < e/2 by a3;
consider z be Real such that
a7: for e st e>0
ex N2 be Nat st
for m st m>=N2 holds |.(lim_in_cod1 Rseq).m - z.| < e by a2,SEQ_2:def 6;
a8: z = cod1_major_iterated_lim Rseq by a2,a7,def34;
consider N2 be Nat such that
a9: for n st n>=N2 holds |.(lim_in_cod1 Rseq).n - z.| < e/2 by a5,a7;
set N = max(N1,N2);
a0: N is Nat by TARSKI:1;
for n,m st n>=N & m>=N holds
|.Rseq.(n,m) - cod1_major_iterated_lim Rseq.| < e
proof
let n,m;
assume a10: n>=N & m>=N;
N >= N1 & N >= N2 by XXREAL_0:25; then
n >= N2 & m >= N1 by a10,XXREAL_0:2; then
|.Rseq.(n,m) - (lim_in_cod1 Rseq).n .| <e/2
& |.(lim_in_cod1 Rseq).n - z.| < e/2 by a6,a9; then
a12: |.Rseq.(n,m) - (lim_in_cod1 Rseq).n .|
+ |.(lim_in_cod1 Rseq).n - z.| < e/2 + e/2 by XREAL_1:8;
|.Rseq.(n,m) - cod1_major_iterated_lim Rseq.|
<= |.Rseq.(n,m) - (lim_in_cod1 Rseq).n .|
+ |.(lim_in_cod1 Rseq).n - cod1_major_iterated_lim Rseq.|
by COMPLEX1:63;
hence |.Rseq.(n,m) - cod1_major_iterated_lim Rseq.| < e
by a8,a12,XXREAL_0:2;
end;
hence ex N st for n,m st n>=N & m>=N holds
|.Rseq.(n,m) - cod1_major_iterated_lim Rseq.| < e by a0;
end;
hence Rseq is P-convergent;
hence thesis by a4,def6;
end;
theorem
Rseq is uniformly_convergent_in_cod2 & lim_in_cod2 Rseq is convergent implies
Rseq is P-convergent & P-lim Rseq = cod2_major_iterated_lim Rseq
proof
assume that
a3: Rseq is uniformly_convergent_in_cod2 and
a2: lim_in_cod2 Rseq is convergent;
a4:now let e;
assume a5: 0<e; then
consider N1 be Nat such that
a6: for n st n >= N1 holds
for m holds |.Rseq.(n,m) - (lim_in_cod2 Rseq).m .| < e/2 by a3;
consider z be Real such that
a7: for e st e>0
ex N2 be Nat st
for n st n>=N2 holds
|.(lim_in_cod2 Rseq).n - z.| < e by a2,SEQ_2:def 6;
a8: z = cod2_major_iterated_lim Rseq by a2,a7,def35;
consider N2 be Nat such that
a9: for n st n>=N2 holds |.(lim_in_cod2 Rseq).n - z.| < e/2 by a5,a7;
set N = max(N1,N2);
a0: N is Nat by TARSKI:1;
for n,m st n>=N & m>=N holds
|.Rseq.(n,m) - cod2_major_iterated_lim Rseq.| < e
proof
let n,m;
assume a10: n>=N & m>=N;
N >= N1 & N >= N2 by XXREAL_0:25; then
n >= N1 & m >= N2 by a10,XXREAL_0:2; then
|.Rseq.(n,m) - (lim_in_cod2 Rseq).m .| <e/2
& |.(lim_in_cod2 Rseq).m - z.| < e/2 by a6,a9; then
a12: |.Rseq.(n,m) - (lim_in_cod2 Rseq).m .|
+ |.(lim_in_cod2 Rseq).m - z.| < e/2 + e/2 by XREAL_1:8;
|.Rseq.(n,m) - cod2_major_iterated_lim Rseq.|
<= |.Rseq.(n,m) - (lim_in_cod2 Rseq).m .|
+ |.(lim_in_cod2 Rseq).m - cod2_major_iterated_lim Rseq.|
by COMPLEX1:63;
hence |.Rseq.(n,m) - cod2_major_iterated_lim Rseq.| < e
by a8,a12,XXREAL_0:2;
end;
hence ex N st
for n,m st n>=N & m>=N holds
|.Rseq.(n,m) - cod2_major_iterated_lim Rseq.| < e by a0;
end;
hence Rseq is P-convergent;
hence thesis by a4,def6;
end;
definition
let Rseq;
attr Rseq is Cauchy means
for e be Real st e>0
ex N be Nat st
for n1,n2,m1,m2 be Nat st N<=n1 & n1<=n2 & N<=m1 & m1<=m2
holds |.Rseq.(n2,m2) - Rseq.(n1,m1).| < e;
end;
theorem
Rseq is P-convergent iff Rseq is Cauchy
proof
hereby assume a1: Rseq is P-convergent;
now let e;
assume a2: e > 0;
consider z be Real such that
a3: for e st 0<e
ex N st
for n,m st n>=N & m>=N holds |.Rseq.(n,m) - z.| < e by a1;
consider N such that
a4: for n,m st n>=N & m>=N holds |.Rseq.(n,m) - z.| < e/2 by a2,a3;
now let n1,n2,m1,m2 be Nat;
assume b1: N<=n1 & n1<=n2 & N<=m1 & m1<=m2; then
N<=n2 & N<=m2 by XXREAL_0:2; then
|.Rseq.(n1,m1) - z.| < e/2 & |.Rseq.(n2,m2) - z.| < e/2 by a4,b1; then
|.Rseq.(n2,m2) - z.| + |.Rseq.(n1,m1) - z.| < e/2 + e/2
by XREAL_1:8; then
a5: |.Rseq.(n2,m2) - z.| + |.z - Rseq.(n1,m1).| < e by COMPLEX1:60;
|.Rseq.(n2,m2) - Rseq.(n1,m1).|
<= |.Rseq.(n2,m2) - z.| + |.z - Rseq.(n1,m1).| by COMPLEX1:63;
hence |.Rseq.(n2,m2) - Rseq.(n1,m1).| < e by a5,XXREAL_0:2;
end;
hence ex N st
for n1,n2,m1,m2 be Nat st N<=n1 & n1<=n2 & N<=m1 & m1<=m2
holds |.Rseq.(n2,m2) - Rseq.(n1,m1).| < e;
end;
hence Rseq is Cauchy;
end;
assume a6: Rseq is Cauchy;
deffunc F(Element of NAT) = Rseq.($1,$1);
consider seq be Function of NAT,REAL such that
a7: for n be Element of NAT holds seq.n = F(n) from FUNCT_2:sch 4;
reconsider seq as Real_Sequence;
now let e;
assume e > 0; then
consider N such that
a8: for n1,n2,m1,m2 be Nat st N<=n1 & n1<=n2 & N<=m1 & m1<=m2
holds |. Rseq.(n2,m2) - Rseq.(n1,m1).| < e by a6;
take N;
hereby let n;
c1: n is Element of NAT & N is Element of NAT by ORDINAL1:def 12;
assume n>=N; then
|. Rseq.(n,n) - Rseq.(N,N).| < e by a8; then
|. seq.n - Rseq.(N,N).| < e by a7,c1;
hence |. seq.n - seq.N .| < e by a7,c1;
end;
end; then
a11: seq is convergent by SEQ_4:41;
reconsider z = lim seq as Complex;
for e st 0<e
ex N st for n,m st n>=N & m>=N holds |.Rseq.(n,m) - z.| < e
proof
let e;
assume a12: 0<e; then
consider N1 be Nat such that
a13: for n st n>=N1 holds |. seq.n - lim seq.| < e/2 by a11,SEQ_2:def 7;
consider N2 be Nat such that
a14: for n1,n2,m1,m2 be Nat st N2<=n1 & n1<=n2 & N2<=m1 & m1<=m2
holds |. Rseq.(n2,m2) - Rseq.(n1,m1).| < e/2 by a6,a12;
reconsider N = max(N1,N2) as Nat by TARSKI:1;
take N;
hereby let n,m;
c2: N is Element of NAT by ORDINAL1:def 12;
assume a15: n>=N & m>=N;
a18: Rseq.(N,N) = seq.N by a7,c2;
N>=N1 & N>=N2 by XXREAL_0:25; then
|. Rseq.(N,N) - z.| < e/2 & |. Rseq.(n,m) - Rseq.(N,N).| < e/2
by a13,a14,a18,a15; then
b1: |. Rseq.(n,m) - Rseq.(N,N).| + |. Rseq.(N,N) - z.| < e/2+e/2
by XREAL_1:8;
|. Rseq.(n,m) - z.| <= |. Rseq.(n,m) - Rseq.(N,N).| + |. Rseq.(N,N) - z.|
by COMPLEX1:63;
hence |. Rseq.(n,m) - z.| < e by b1,XXREAL_0:2;
end;
end;
hence Rseq is P-convergent;
end;
theorem
for Rseq be Function of [:NAT,NAT:],REAL st
Rseq is non-decreasing or Rseq is non-increasing holds
Rseq is P-convergent iff Rseq is bounded_below bounded_above;
definition
let X,Y be non empty set;
let H be BinOp of Y;
let f,g be Function of X,Y;
redefine func H*(f,g) -> Function of [:X,X:],Y;
coherence
proof
a1:dom H = [:Y,Y:] & dom f = X & dom g = X by FUNCT_2:def 1;
a2:rng [:f,g:] c= [:Y,Y:];
dom(H*(f,g)) = dom(H*[:f,g:]) by FINSEQOP:def 4; then
dom(H*(f,g)) = dom [:f,g:] by a1,a2,RELAT_1:27; then
a3:dom(H*(f,g)) = [:X,X:] by FUNCT_2:def 1;
rng(H*(f,g)) = rng(H*[:f,g:]) by FINSEQOP:def 4; then
rng (H*(f,g)) c= rng H & rng H c= Y by RELAT_1:26,def 19;
hence H*(f,g) is Function of [:X,X:],Y by a3,FUNCT_2:2,XBOOLE_1:1;
end;
end;
theorem
multreal*(rseq1,rseq2) is convergent_in_cod1 convergent_in_cod2
& lim_in_cod1 (multreal*(rseq1,rseq2)) is convergent
& cod1_major_iterated_lim (multreal*(rseq1,rseq2)) = lim rseq1 * lim rseq2
& lim_in_cod2 (multreal*(rseq1,rseq2)) is convergent
& cod2_major_iterated_lim (multreal*(rseq1,rseq2)) = lim rseq1 * lim rseq2
& multreal*(rseq1,rseq2) is P-convergent
& P-lim (multreal*(rseq1,rseq2)) = lim rseq1 * lim rseq2
proof
set Rseq = multreal*(rseq1,rseq2);
a2:for n,m holds Rseq.(n,m) = rseq1.n * rseq2.m
proof
let n,m;
a1: n in NAT & m in NAT by ORDINAL1:def 12;
dom Rseq = [:NAT,NAT:] by FUNCT_2:def 1; then
[n,m] in dom Rseq by a1,ZFMISC_1:def 2; then
Rseq.(n,m) = multreal.(rseq1.n,rseq2.m) by FINSEQOP:77;
hence Rseq.(n,m) = rseq1.n * rseq2.m by BINOP_2:def 11;
end;
x1:for m be Element of NAT, e be Real st 0<e
ex N st
for n st n>=N holds |. ProjMap2(Rseq,m).n - (lim rseq1)*rseq2.m .| < e
proof
let m be Element of NAT, e be Real;
assume a3: 0<e;
per cases;
suppose rseq2.m <> 0; then
a4: |.rseq2.m .| > 0 by COMPLEX1:47; then
consider N such that
a5: for n st n>=N holds |. rseq1.n - lim rseq1.| < e/ |. rseq2.m .|
by a3,SEQ_2:def 7;
take N;
hereby let n;
assume n>=N; then
a6: |. rseq1.n - lim rseq1.| < e/ |. rseq2.m .| by a5;
n is Element of NAT by ORDINAL1:def 12; then
|.ProjMap2(Rseq,m).n - (lim rseq1)*rseq2.m .|
= |. Rseq.(n,m) - (lim rseq1)*rseq2.m .| by MESFUNC9:def 7
.= |. rseq1.n * rseq2.m - (lim rseq1)*rseq2.m .| by a2
.= |.(rseq1.n - lim rseq1)*rseq2.m .|
.= |. rseq1.n - lim rseq1.| * |. rseq2.m .| by COMPLEX1:65; then
|.ProjMap2(Rseq,m).n - (lim rseq1)*rseq2.m .|
< e/ |. rseq2.m .| * |. rseq2.m .| by a4,a6,XREAL_1:68;
hence |.ProjMap2(Rseq,m).n - (lim rseq1)*rseq2.m .| < e
by a4,XCMPLX_1:87;
end;
end;
suppose a7: rseq2.m = 0;
take 0;
hereby let n;
assume n >= 0;
n is Element of NAT by ORDINAL1:def 12; then
|.ProjMap2(Rseq,m).n - (lim rseq1)*rseq2.m .|
= |. Rseq.(n,m) - (lim rseq1)*rseq2.m .| by MESFUNC9:def 7
.= |. rseq1.n * rseq2.m - (lim rseq1)*rseq2.m .| by a2
.= 0 by a7,COMPLEX1:44;
hence |.ProjMap2(Rseq,m).n - (lim rseq1)*rseq2.m .| < e by a3;
end;
end;
end;
p1:for m be Element of NAT holds ProjMap2(Rseq,m) is convergent
proof
let m be Element of NAT;
for e st 0<e
ex N st for n st n>=N holds
|.ProjMap2(Rseq,m).n - (lim rseq1)*rseq2.m .| < e by x1;
hence ProjMap2(Rseq,m) is convergent by SEQ_2:def 6;
end;
hence Rseq is convergent_in_cod1;
x2:for m be Element of NAT, e be Real st 0<e
ex N st for n st n>=N holds
|. ProjMap1(Rseq,m).n - (rseq1.m)*(lim rseq2).| < e
proof
let m be Element of NAT, e be Real;
assume a3: 0<e;
per cases;
suppose rseq1.m <> 0; then
a4: |. rseq1.m .| > 0 by COMPLEX1:47; then
consider N be Nat such that
a5: for n be Nat st n>=N holds
|. rseq2.n - lim rseq2.| < e/ |. rseq1.m .| by a3,SEQ_2:def 7;
take N;
hereby let n;
assume n>=N; then
a6: |. rseq2.n - lim rseq2.| < e/ |. rseq1.m .| by a5;
n is Element of NAT by ORDINAL1:def 12; then
|. ProjMap1(Rseq,m).n - (lim rseq2)*rseq1.m .|
= |. Rseq.(m,n) - (lim rseq2)*rseq1.m .| by MESFUNC9:def 6
.= |. rseq2.n * rseq1.m - (lim rseq2)*rseq1.m .| by a2
.= |. (rseq2.n - lim rseq2)*rseq1.m .|
.= |. rseq2.n - lim rseq2.| * |. rseq1.m .| by COMPLEX1:65; then
|. ProjMap1(Rseq,m).n - (lim rseq2)*rseq1.m .|
< e/ |. rseq1.m .| * |. rseq1.m .| by a4,a6,XREAL_1:68;
hence |. ProjMap1(Rseq,m).n - (rseq1.m)*(lim rseq2).| < e
by a4,XCMPLX_1:87;
end;
end;
suppose a7: rseq1.m = 0;
take 0;
hereby let n;
assume n >= 0;
n is Element of NAT by ORDINAL1:def 12; then
|. ProjMap1(Rseq,m).n - (lim rseq2)*rseq1.m .|
= |. Rseq.(m,n) - (lim rseq2)*rseq1.m .| by MESFUNC9:def 6
.= |. rseq2.n * rseq1.m - (lim rseq2)*rseq1.m .| by a2
.= 0 by a7,COMPLEX1:44;
hence |. ProjMap1(Rseq,m).n - (rseq1.m)*(lim rseq2).| < e by a3;
end;
end;
end;
p2:for m be Element of NAT holds ProjMap1(Rseq,m) is convergent
proof
let m be Element of NAT;
for e st 0<e ex N st for n st n>=N holds
|.ProjMap1(Rseq,m).n - (rseq1.m)*(lim rseq2).| < e by x2;
hence ProjMap1(Rseq,m) is convergent by SEQ_2:def 6;
end;
hence Rseq is convergent_in_cod2;
x3:for e st 0<e ex N st for n st n>=N holds
|.(lim_in_cod1 Rseq).n - (lim rseq1)*(lim rseq2).| < e
proof
let e;
assume a3: 0<e;
a6: for n holds (lim_in_cod1 Rseq).n = (lim rseq1)*(rseq2.n)
proof
let n;
reconsider n1=n as Element of NAT by ORDINAL1:def 12;
a4: ProjMap2(Rseq,n1) is convergent by p1;
a5: (lim_in_cod1 Rseq).n = lim ProjMap2(Rseq,n1) by def32;
for e st 0<e ex N st for m st m>=N holds
|.ProjMap2(Rseq,n1).m - (lim rseq1)*rseq2.n .| < e by x1;
hence (lim_in_cod1 Rseq).n = (lim rseq1)*(rseq2.n) by a5,a4,SEQ_2:def 7;
end;
per cases by COMPLEX1:46;
suppose b1: |.lim rseq1.| > 0; then
consider N such that
a7: for n st n>=N holds |. rseq2.n - lim rseq2.| < e / |.lim rseq1.|
by a3,SEQ_2:def 7;
take N;
hereby let n;
assume n>=N; then
a8: |. rseq2.n - lim rseq2.| < e / |.lim rseq1.| by a7;
|.(lim_in_cod1 Rseq).n - (lim rseq1)*(lim rseq2).|
= |.(lim rseq1)*(rseq2.n) - (lim rseq1)*(lim rseq2).| by a6
.= |.(lim rseq1)*(rseq2.n - lim rseq2).|
.= |.lim rseq1.| * |. rseq2.n - lim rseq2.| by COMPLEX1:65;
hence |.(lim_in_cod1 Rseq).n - (lim rseq1)*(lim rseq2).| < e
by a8,b1,XREAL_1:79;
end;
end;
suppose a9: |.lim rseq1.| = 0;
take 0;
hereby let n;
assume n >= 0;
|.(lim_in_cod1 Rseq).n - (lim rseq1)*(lim rseq2).|
= |.(lim rseq1)*(rseq2.n) - (lim rseq1)*(lim rseq2).| by a6
.= |.(lim rseq1)*(rseq2.n - lim rseq2).|
.= |.lim rseq1.| * |. rseq2.n - lim rseq2.| by COMPLEX1:65
.= 0 by a9;
hence |.(lim_in_cod1 Rseq).n - (lim rseq1)*(lim rseq2).| < e by a3;
end;
end;
end;
hence lim_in_cod1 Rseq is convergent by SEQ_2:def 6;
hence cod1_major_iterated_lim Rseq = lim rseq1 * lim rseq2 by x3,def34;
x4:for e st 0<e ex N st for n st n>=N holds
|.(lim_in_cod2 Rseq).n - (lim rseq1)*(lim rseq2).| < e
proof
let e;
assume a3: 0<e;
a6: for n holds (lim_in_cod2 Rseq).n = (rseq1.n)*(lim rseq2)
proof
let n;
reconsider n1=n as Element of NAT by ORDINAL1:def 12;
a4: ProjMap1(Rseq,n1) is convergent by p2;
a5: (lim_in_cod2 Rseq).n = lim ProjMap1(Rseq,n1) by def33;
for e st 0<e ex N st for m st m>=N holds
|.ProjMap1(Rseq,n1).m - (rseq1.n)*(lim rseq2).| < e by x2;
hence (lim_in_cod2 Rseq).n = (rseq1.n)*(lim rseq2) by a5,a4,SEQ_2:def 7;
end;
per cases by COMPLEX1:46;
suppose b1: |.lim rseq2.| > 0; then
consider N such that
a7: for n st n>=N holds |. rseq1.n - lim rseq1.| < e / |.lim rseq2.|
by a3,SEQ_2:def 7;
take N;
hereby let n;
assume n>=N; then
a8: |. rseq1.n - lim rseq1.| < e / |.lim rseq2.| by a7;
|.(lim_in_cod2 Rseq).n - (lim rseq1)*(lim rseq2).|
= |.(rseq1.n)*(lim rseq2) - (lim rseq1)*(lim rseq2).| by a6
.= |.(rseq1.n - lim rseq1)*(lim rseq2).|
.= |.lim rseq2.| * |. rseq1.n - lim rseq1.| by COMPLEX1:65;
hence |.(lim_in_cod2 Rseq).n - (lim rseq1)*(lim rseq2).| < e
by a8,b1,XREAL_1:79;
end;
end;
suppose a9: |.lim rseq2.| = 0;
take 0;
hereby let n;
assume n >= 0;
|.(lim_in_cod2 Rseq).n - (lim rseq1)*(lim rseq2).|
= |.(rseq1.n)*(lim rseq2) - (lim rseq1)*(lim rseq2).| by a6
.= |.(lim rseq2)*(rseq1.n - lim rseq1).|
.= |.lim rseq2.| * |. rseq1.n - lim rseq1.| by COMPLEX1:65
.= 0 by a9;
hence |.(lim_in_cod2 Rseq).n - (lim rseq1)*(lim rseq2).| < e by a3;
end;
end;
end;
hence lim_in_cod2 Rseq is convergent by SEQ_2:def 6;
hence cod2_major_iterated_lim Rseq = lim rseq1 * lim rseq2
by x4,def35;
x5:for e st 0<e ex N st for n,m st n>=N & m>=N holds
|. Rseq.(n,m) - (lim rseq1)*(lim rseq2).| < e
proof
let e;
assume c1: 0<e;
consider K be Real such that
c2: 0 < K & for n holds |. rseq1.n .| < K by SEQ_2:3;
set b = max(K,|.lim rseq2.|);
c10:b >= K & b >= |.lim rseq2.| by XXREAL_0:25; then
consider N1 be Nat such that
c4: for n st n>=N1 holds |. rseq1.n - lim rseq1.| < e/(2*b)
by c1,c2,SEQ_2:def 7;
consider N2 be Nat such that
c5: for n st n>=N2 holds |. rseq2.n - lim rseq2.| < e/(2*b)
by c1,c2,c10,SEQ_2:def 7;
reconsider N = max(N1,N2) as Nat by TARSKI:1;
take N;
thus for n,m st n>=N & m>=N holds
|. Rseq.(n,m) - (lim rseq1)*(lim rseq2).| < e
proof
let n,m;
assume c13: n >= N & m >= N;
max(N1,N2) >= N1 & max(N1,N2) >= N2 by XXREAL_0:25; then
c6: n >= N1 & m >= N2 by c13,XXREAL_0:2;
c7: |. Rseq.(n,m) - (lim rseq1)*(lim rseq2).|
<= |. Rseq.(n,m) - (rseq1.n)*(lim rseq2).|
+ |. (rseq1.n)*(lim rseq2) - (lim rseq1)*(lim rseq2).| by COMPLEX1:63;
c11: |. rseq1.n - lim rseq1.| >= 0 & |. rseq2.m - lim rseq2.| >= 0
by COMPLEX1:46;
c12: |. rseq1.n .| < b & |.lim rseq2.| <= b by c2,c10,XXREAL_0:2;
|. Rseq.(n,m) - (rseq1.n)*(lim rseq2).|
= |.(rseq1.n)*(rseq2.m) - (rseq1.n)*(lim rseq2).| by a2
.= |.(rseq1.n)*(rseq2.m - lim rseq2).|
.= |. rseq1.n .| * |. rseq2.m - lim rseq2.| by COMPLEX1:65; then
c8: |. Rseq.(n,m) - (rseq1.n)*(lim rseq2).|
<= b * |. rseq2.m - lim rseq2.| by c11,c12,XREAL_1:64;
|. rseq2.m - lim rseq2.| < e/(2*b) by c5,c6; then
b * |. rseq2.m - lim rseq2.| < b * (e/(2*b)) by c2,c10,XREAL_1:68; then
|. Rseq.(n,m) - (rseq1.n)*(lim rseq2).| < b * (e/(2*b))
by c8,XXREAL_0:2; then
|. Rseq.(n,m) - (rseq1.n)*(lim rseq2).| < e/(2*b/b) by XCMPLX_1:81; then
c15: |. Rseq.(n,m) - (rseq1.n)*(lim rseq2).| < e/2 by c2,c10,XCMPLX_1:89;
|. (rseq1.n)*(lim rseq2) - (lim rseq1)*(lim rseq2).|
= |.(lim rseq2)*(rseq1.n - lim rseq1).|
.= |.lim rseq2.| * |. rseq1.n - lim rseq1.| by COMPLEX1:65; then
c9: |.(rseq1.n)*(lim rseq2) - (lim rseq1)*(lim rseq2).|
<= b * |. rseq1.n - lim rseq1.| by c11,XXREAL_0:25,XREAL_1:64;
|. rseq1.n - lim rseq1.| < e/(2*b) by c4,c6; then
b * |. rseq1.n - lim rseq1.| < b * (e/(2*b)) by c2,c10,XREAL_1:68; then
|.(rseq1.n)*(lim rseq2) - (lim rseq1)*(lim rseq2).| < b * (e/(2*b))
by c9,XXREAL_0:2; then
|.(rseq1.n)*(lim rseq2) - (lim rseq1)*(lim rseq2).| < e/(2*b/b)
by XCMPLX_1:81; then
|.(rseq1.n)*(lim rseq2) - (lim rseq1)*(lim rseq2).| < e/2
by c2,c10,XCMPLX_1:89; then
|. Rseq.(n,m) - (rseq1.n)*(lim rseq2).|
+ |.(rseq1.n)*(lim rseq2) - (lim rseq1)*(lim rseq2).| < e/2 + e/2
by c15,XREAL_1:8;
hence |. Rseq.(n,m) - (lim rseq1)*(lim rseq2).| < e by c7,XXREAL_0:2;
end;
end;
hence Rseq is P-convergent;
hence P-lim Rseq = lim rseq1 * lim rseq2 by x5,def6;
end;
theorem
addreal*(rseq1,rseq2) is convergent_in_cod1 convergent_in_cod2
& lim_in_cod1 (addreal*(rseq1,rseq2)) is convergent
& cod1_major_iterated_lim (addreal*(rseq1,rseq2)) = lim rseq1 + lim rseq2
& lim_in_cod2 (addreal*(rseq1,rseq2)) is convergent
& cod2_major_iterated_lim (addreal*(rseq1,rseq2)) = lim rseq1 + lim rseq2
& addreal*(rseq1,rseq2) is P-convergent
& P-lim (addreal*(rseq1,rseq2)) = lim rseq1 + lim rseq2
proof
set Rseq = addreal*(rseq1,rseq2);
a2:for n,m holds Rseq.(n,m) = rseq1.n + rseq2.m
proof
let n,m;
a1: n in NAT & m in NAT by ORDINAL1:def 12;
dom Rseq = [:NAT,NAT:] by FUNCT_2:def 1; then
[n,m] in dom Rseq by a1,ZFMISC_1:def 2; then
Rseq.(n,m) = addreal.(rseq1.n,rseq2.m) by FINSEQOP:77;
hence Rseq.(n,m) = rseq1.n + rseq2.m by BINOP_2:def 9;
end;
x1:for m be Element of NAT, e be Real st 0<e
ex N be Nat st
for n be Nat st n>=N holds
|.ProjMap2(Rseq,m).n - (lim rseq1 + rseq2.m).| < e
proof
let m be Element of NAT, e be Real;
assume 0<e; then
consider N be Nat such that
a3: for n be Nat st n>=N holds
|.rseq1.n - lim rseq1.| < e by SEQ_2:def 7;
take N;
hereby let n be Nat;
assume a4: n>=N;
n is Element of NAT by ORDINAL1:def 12; then
|.ProjMap2(Rseq,m).n - (lim rseq1 + rseq2.m).|
= |. Rseq.(n,m) - (lim rseq1 + rseq2.m).| by MESFUNC9:def 7
.= |.(rseq1.n + rseq2.m) - (lim rseq1 + rseq2.m).| by a2
.= |. rseq1.n - lim rseq1.|;
hence |. ProjMap2(Rseq,m).n - (lim rseq1 + rseq2.m).|< e
by a3,a4;
end;
end;
p1:for m be Element of NAT holds ProjMap2(Rseq,m) is convergent
proof
let m be Element of NAT;
for e st 0<e ex N st for n st n>=N holds
|. ProjMap2(Rseq,m).n - (lim rseq1 + rseq2.m).| < e by x1;
hence ProjMap2(Rseq,m) is convergent by SEQ_2:def 6;
end;
hence Rseq is convergent_in_cod1;
x2:for m be Element of NAT, e be Real st 0<e
ex N st for n st n>=N holds
|. ProjMap1(Rseq,m).n - (rseq1.m + lim rseq2).| < e
proof
let m be Element of NAT, e be Real;
assume 0<e; then
consider N such that
a3: for n st n>=N holds |. rseq2.n - lim rseq2.| < e by SEQ_2:def 7;
take N;
hereby let n;
assume a4: n>=N;
n is Element of NAT by ORDINAL1:def 12; then
|. ProjMap1(Rseq,m).n - (lim rseq2 + rseq1.m).|
= |. Rseq.(m,n) - (lim rseq2 + rseq1.m).| by MESFUNC9:def 6
.= |. rseq2.n + rseq1.m - (lim rseq2 + rseq1.m).| by a2
.= |. rseq2.n - lim rseq2.|;
hence |. ProjMap1(Rseq,m).n - (rseq1.m + lim rseq2).| < e
by a3,a4;
end;
end;
p2:for m be Element of NAT holds ProjMap1(Rseq,m) is convergent
proof
let m be Element of NAT;
for e st 0<e ex N st for n st n>=N holds
|.ProjMap1(Rseq,m).n - (rseq1.m + lim rseq2).| < e by x2;
hence ProjMap1(Rseq,m) is convergent by SEQ_2:def 6;
end;
hence Rseq is convergent_in_cod2;
x3:for e st 0<e ex N st for n st n>=N holds
|.(lim_in_cod1 Rseq).n - (lim rseq1 + lim rseq2).| < e
proof
let e;
assume 0<e; then
consider N such that
a3: for n st n>=N holds |. rseq2.n - lim rseq2.| < e by SEQ_2:def 7;
take N;
hereby let n;
assume n>=N; then
a4: |. rseq2.n - lim rseq2.| < e by a3;
reconsider n1 = n as Element of NAT by ORDINAL1:def 12;
a5: ProjMap2(Rseq,n1) is convergent by p1;
a6: (lim_in_cod1 Rseq).n = lim ProjMap2(Rseq,n1) by def32;
for e st 0<e ex N st for m st m>=N holds
|. ProjMap2(Rseq,n1).m - (lim rseq1 + rseq2.n).| < e by x1; then
|.(lim_in_cod1 Rseq).n - (lim rseq1 + lim rseq2).|
= |.(lim rseq1 + rseq2.n) - (lim rseq1 + lim rseq2).|
by a5,a6,SEQ_2:def 7;
hence |. (lim_in_cod1 Rseq).n - (lim rseq1 + lim rseq2).| < e by a4;
end;
end;
hence lim_in_cod1 Rseq is convergent by SEQ_2:def 6;
hence cod1_major_iterated_lim Rseq = lim rseq1 + lim rseq2 by x3,def34;
x4:for e st 0<e ex N st for n st n>=N holds
|. (lim_in_cod2 Rseq).n - (lim rseq1 + lim rseq2).| < e
proof
let e;
assume 0<e; then
consider N such that
a3: for n st n>=N holds |. rseq1.n - lim rseq1.| < e by SEQ_2:def 7;
take N;
hereby let n;
assume n>=N; then
a4: |. rseq1.n - lim rseq1.| < e by a3;
reconsider n1 = n as Element of NAT by ORDINAL1:def 12;
a5: ProjMap1(Rseq,n1) is convergent by p2;
a6: (lim_in_cod2 Rseq).n = lim ProjMap1(Rseq,n1) by def33;
for e st 0<e ex N st for m st m>=N holds
|.ProjMap1(Rseq,n1).m - (rseq1.n + lim rseq2).| < e by x2; then
|.(lim_in_cod2 Rseq).n - (lim rseq1 + lim rseq2).|
= |.(rseq1.n + lim rseq2) - (lim rseq1 + lim rseq2).|
by a5,a6,SEQ_2:def 7;
hence |.(lim_in_cod2 Rseq).n - (lim rseq1 + lim rseq2).| < e by a4;
end;
end;
hence lim_in_cod2 Rseq is convergent by SEQ_2:def 6;
hence cod2_major_iterated_lim Rseq = lim rseq1 + lim rseq2
by x4,def35;
x5:for e st 0<e ex N st for n,m st n>=N & m>=N holds
|. Rseq.(n,m) - (lim rseq1 + lim rseq2).| < e
proof
let e;
assume c1: 0<e; then
consider N1 be Nat such that
c4: for n st n>=N1 holds |. rseq1.n - lim rseq1.| < e/2 by SEQ_2:def 7;
consider N2 be Nat such that
c5: for n st n>=N2 holds |. rseq2.n - lim rseq2.| < e/2 by c1,SEQ_2:def 7;
reconsider N = max(N1,N2) as Nat by TARSKI:1;
take N;
thus for n,m st n>=N & m>=N holds
|. Rseq.(n,m) - (lim rseq1 + lim rseq2).| < e
proof
let n,m;
assume c13: n>=N & m>=N;
max(N1,N2) >= N1 & max(N1,N2) >= N2 by XXREAL_0:25; then
c6: n >= N1 & m >= N2 by c13,XXREAL_0:2;
|. Rseq.(n,m) - (lim rseq1 + lim rseq2).|
= |. rseq1.n + rseq2.m - (lim rseq1 + lim rseq2).| by a2
.= |.(rseq1.n - lim rseq1) + (rseq2.m - lim rseq2).|; then
a8: |. Rseq.(n,m) - (lim rseq1 + lim rseq2).|
<= |. rseq1.n - lim rseq1.| + |. rseq2.m - lim rseq2.| by COMPLEX1:56;
|. rseq1.n - lim rseq1.| < e/2 & |. rseq2.m - lim rseq2.| < e/2
by c4,c5,c6; then
|. rseq1.n - lim rseq1.| + |. rseq2.m - lim rseq2.| < e/2 + e/2
by XREAL_1:8;
hence |. Rseq.(n,m) - (lim rseq1 + lim rseq2).| < e by a8,XXREAL_0:2;
end;
end;
hence Rseq is P-convergent;
hence P-lim Rseq = lim rseq1 + lim rseq2 by x5,def6;
end;
lmADD:
for Rseq1,Rseq2 be Function of [:NAT,NAT:],REAL holds
dom(Rseq1+Rseq2) = [:NAT,NAT:] & dom(Rseq1-Rseq2) = [:NAT,NAT:] &
(for n,m be Nat holds
(Rseq1+Rseq2).(n,m) = Rseq1.(n,m) + Rseq2.(n,m)) &
(for n,m be Nat holds
(Rseq1-Rseq2).(n,m) = Rseq1.(n,m) - Rseq2.(n,m))
proof
let Rseq1,Rseq2 be Function of [:NAT,NAT:],REAL;
thus
a1: dom(Rseq1+Rseq2) = [:NAT,NAT:] & dom(Rseq1-Rseq2) = [:NAT,NAT:]
by FUNCT_2:def 1;
hereby let n,m;
n in NAT & m in NAT by ORDINAL1:def 12;
hence (Rseq1+Rseq2).(n,m) = Rseq1.(n,m) + Rseq2.(n,m)
by VALUED_1:def 1,a1,ZFMISC_1:87;
end;
hereby let n,m;
n in NAT & m in NAT by ORDINAL1:def 12;
hence (Rseq1-Rseq2).(n,m) = Rseq1.(n,m) - Rseq2.(n,m)
by VALUED_1:13,a1,ZFMISC_1:87;
end;
end;
theorem
Rseq1 is P-convergent & Rseq2 is P-convergent implies
Rseq1+Rseq2 is P-convergent & P-lim(Rseq1+Rseq2) = P-lim Rseq1 + P-lim Rseq2
proof
assume a1: Rseq1 is P-convergent & Rseq2 is P-convergent;
a2:now let e;
assume a4: 0 < e; then
consider N1 be Nat such that
a5: for n,m st n>=N1 & m>=N1 holds
|. Rseq1.(n,m) - P-lim Rseq1.| < e/2 by a1,def6;
consider N2 be Nat such that
a6: for n,m st n>=N2 & m>=N2 holds
|. Rseq2.(n,m) - P-lim Rseq2.| < e/2 by a1,a4,def6;
reconsider N=max(N1,N2) as Nat by TARSKI:1;
take N;
now let n,m;
assume a8: n>=N & m>=N;
N>=N1 & N>=N2 by XXREAL_0:25; then
n>=N1 & n>=N2 & m>=N1 & m>=N2 by a8,XXREAL_0:2; then
|. Rseq1.(n,m) - P-lim Rseq1.| < e/2 &
|. Rseq2.(n,m) - P-lim Rseq2.| < e/2 by a5,a6; then
a9: |. Rseq1.(n,m) - P-lim Rseq1.| + |. Rseq2.(n,m) - P-lim Rseq2.| < e/2+e/2
by XREAL_1:8;
(Rseq1+Rseq2).(n,m) = Rseq1.(n,m) + Rseq2.(n,m) by lmADD; then
(Rseq1+Rseq2).(n,m) - (P-lim Rseq1 + P-lim Rseq2)
= (Rseq1.(n,m) - P-lim Rseq1) + (Rseq2.(n,m) - P-lim Rseq2); then
|.(Rseq1+Rseq2).(n,m) - (P-lim Rseq1 + P-lim Rseq2).|
<= |. Rseq1.(n,m) - P-lim Rseq1.|
+ |. Rseq2.(n,m) - P-lim Rseq2.| by COMPLEX1:56;
hence |.(Rseq1+Rseq2).(n,m) - (P-lim Rseq1 + P-lim Rseq2).| < e
by a9,XXREAL_0:2;
end;
hence for n,m be Nat st n>=N & m>=N holds
|.(Rseq1+Rseq2).(n,m) - (P-lim Rseq1 + P-lim Rseq2).| < e;
end;
hence Rseq1+Rseq2 is P-convergent;
hence thesis by a2,def6;
end;
theorem th54b:
Rseq1 is P-convergent & Rseq2 is P-convergent implies
Rseq1-Rseq2 is P-convergent & P-lim(Rseq1-Rseq2) = P-lim Rseq1 - P-lim Rseq2
proof
assume a1: Rseq1 is P-convergent & Rseq2 is P-convergent;
a2:now let e;
assume a4: 0 < e; then
consider N1 be Nat such that
a5: for n,m st n>=N1 & m>=N1 holds
|. Rseq1.(n,m) - P-lim Rseq1.| < e/2 by a1,def6;
consider N2 be Nat such that
a6: for n,m st n>=N2 & m>=N2 holds
|. Rseq2.(n,m) - P-lim Rseq2.| < e/2 by a1,a4,def6;
reconsider N=max(N1,N2) as Nat by TARSKI:1;
take N;
now let n,m;
assume a8: n>=N & m>=N;
N>=N1 & N>=N2 by XXREAL_0:25; then
n>=N1 & n>=N2 & m>=N1 & m>=N2 by a8,XXREAL_0:2; then
a10: |. Rseq1.(n,m) - P-lim Rseq1.| < e/2 &
|. Rseq2.(n,m) - P-lim Rseq2.| < e/2 by a5,a6; then
|. P-lim Rseq2 - Rseq2.(n,m).| < e/2 by COMPLEX1:60; then
a9: |. Rseq1.(n,m) - P-lim Rseq1.| + |.P-lim Rseq2 - Rseq2.(n,m) .| < e/2+e/2
by a10,XREAL_1:8;
(Rseq1-Rseq2).(n,m) = Rseq1.(n,m) - Rseq2.(n,m) by lmADD; then
(Rseq1-Rseq2).(n,m) - (P-lim Rseq1 - P-lim Rseq2)
= (Rseq1.(n,m) - P-lim Rseq1) + (P-lim Rseq2 - Rseq2.(n,m)); then
|.(Rseq1-Rseq2).(n,m) - (P-lim Rseq1 - P-lim Rseq2).|
<= |. Rseq1.(n,m) - P-lim Rseq1.|
+ |.P-lim Rseq2 - Rseq2.(n,m).| by COMPLEX1:56;
hence |. (Rseq1-Rseq2).(n,m) - (P-lim Rseq1 - P-lim Rseq2).| < e
by a9,XXREAL_0:2;
end;
hence for n,m be Nat st n>=N & m>=N holds
|.(Rseq1-Rseq2).(n,m) - (P-lim Rseq1 - P-lim Rseq2).| < e;
end;
hence Rseq1-Rseq2 is P-convergent;
hence thesis by a2,def6;
end;
lm55a:
for a be Real st
(for n,m be Nat holds Rseq.(n,m) = a) holds
Rseq is P-convergent & P-lim Rseq = a
proof
let a be Real;
assume a1: for n,m be Nat holds Rseq.(n,m) = a;
a3:now let e be Real;
assume a2: 0<e;
a4: now let n,m such that n>=0 & m>=0;
Rseq.(n,m) = a by a1;
hence |. Rseq.(n,m) - a.| < e by a2,COMPLEX1:44;
end;
reconsider N = 0 as Nat;
take N;
thus for n,m st n>=N & m>=N holds |. Rseq.(n,m) - a.| < e by a4;
end;
hence Rseq is P-convergent;
hence P-lim Rseq = a by a3,def6;
end;
theorem
for Rseq be Function of [:NAT,NAT:],REAL, r be Real st
Rseq is P-convergent holds
r(#)Rseq is P-convergent & P-lim (r(#)Rseq) = r * P-lim Rseq
proof
let Rseq be Function of [:NAT,NAT:],REAL;
let r be Real;
assume a1: Rseq is P-convergent;
a4:now assume a2: r=0;
a3: now let n,m;
(r(#)Rseq).(n,m) = r * Rseq.(n,m) by VALUED_1:6;
hence (r(#)Rseq).(n,m) = 0 by a2;
end;
hence r(#)Rseq is P-convergent by lm55a;
thus P-lim (r(#)Rseq) = 0 by a3,lm55a;
end;
now assume r <> 0; then
a5: |.r.| > 0 by COMPLEX1:47;
a7: now let e be Real;
assume 0 < e; then
consider N such that
a6: for n,m st n>=N & m>=N holds |. Rseq.(n,m) - P-lim Rseq.| < e/ |.r.|
by a1,a5,def6;
take N;
hereby let n,m;
assume n>=N & m>=N; then
|. Rseq.(n,m) - P-lim Rseq.| < e/ |.r.| by a6; then
|.r.| * |.Rseq.(n,m) - P-lim Rseq.| < e/ |.r.| * |.r.|
by a5,XREAL_1:68; then
|.r.| * |. Rseq.(n,m) - P-lim Rseq.| < |.r.| / (|.r.| / e)
by XCMPLX_1:79; then
|.r.| * |. Rseq.(n,m) - P-lim Rseq.| < e by a5,XCMPLX_1:52; then
|.r*(Rseq.(n,m) - P-lim Rseq).| < e by COMPLEX1:65; then
|.r*Rseq.(n,m) - r * P-lim Rseq.| < e;
hence |.(r(#)Rseq).(n,m) - r * P-lim Rseq.| < e by VALUED_1:6;
end;
end;
hence r(#)Rseq is P-convergent;
hence P-lim (r(#)Rseq) = r * P-lim Rseq by a7,def6;
end;
hence thesis by a4;
end;
theorem th55b:
Rseq is P-convergent &(for n,m be Nat holds Rseq.(n,m) >= r)
implies P-lim Rseq >= r
proof
assume a1: Rseq is P-convergent;
assume a2: for n,m be Nat holds Rseq.(n,m) >= r;
assume not P-lim Rseq >= r; then
r - P-lim Rseq > 0 by XREAL_1:50; then
consider N such that
a3: for n,m st n>=N & m>=N holds
|. Rseq.(n,m) - P-lim Rseq.| < r - P-lim Rseq by a1,def6;
|. Rseq.(N,N) - P-lim Rseq.| < r - P-lim Rseq by a3; then
P-lim Rseq + (r - P-lim Rseq) > Rseq.(N,N) by RINFSUP1:1;
hence contradiction by a2;
end;
theorem
Rseq1 is P-convergent & Rseq2 is P-convergent &
(for n,m be Nat holds Rseq1.(n,m) <= Rseq2.(n,m)) implies
P-lim Rseq1 <= P-lim Rseq2
proof
assume a1: Rseq1 is P-convergent & Rseq2 is P-convergent;
assume a2: for n,m be Nat holds Rseq1.(n,m) <= Rseq2.(n,m);
a3:Rseq2 - Rseq1 is P-convergent &
P-lim(Rseq2-Rseq1) = P-lim Rseq2 - P-lim Rseq1 by a1,th54b;
now let n,m;
(Rseq2-Rseq1).(n,m) = Rseq2.(n,m) - Rseq1.(n,m) by lmADD;
hence (Rseq2-Rseq1).(n,m) >= 0 by a2,XREAL_1:48;
end;
hence P-lim Rseq1 <= P-lim Rseq2 by a3,th55b,XREAL_1:49;
end;
theorem
Rseq1 is P-convergent & Rseq2 is P-convergent & P-lim Rseq1 = P-lim Rseq2 &
(for n,m be Nat holds
Rseq1.(n,m) <= Rseq.(n,m) & Rseq.(n,m) <= Rseq2.(n,m) )
implies Rseq is P-convergent & P-lim Rseq = P-lim Rseq1
proof
assume that
a1: Rseq1 is P-convergent & Rseq2 is P-convergent and
a2: P-lim Rseq1 = P-lim Rseq2 and
a3: for n,m be Nat holds
Rseq1.(n,m) <= Rseq.(n,m) & Rseq.(n,m) <= Rseq2.(n,m);
a4:for e st 0<e
ex N st for n,m st n>=N & m>=N holds |. Rseq.(n,m) - P-lim Rseq1.| < e
proof
let e;
assume a5: 0<e; then
consider N1 be Nat such that
a6: for n,m st n>=N1 & m>=N1 holds |. Rseq1.(n,m) - P-lim Rseq1.| < e
by a1,def6;
consider N2 be Nat such that
a7: for n,m st n>=N2 & m>=N2 holds |. Rseq2.(n,m) - P-lim Rseq1.| < e
by a1,a2,a5,def6;
reconsider N = max(N1,N2) as Nat by TARSKI:1;
take N;
a8: max(N1,N2) >= N1 & max(N1,N2) >= N2 by XXREAL_0:25;
hereby let n,m;
assume n>=N & m>=N; then
n>=N1 & m>=N1 & n>=N2 & m>=N2 by a8,XXREAL_0:2; then
a9: |. Rseq1.(n,m) - P-lim Rseq1.| < e & |. Rseq2.(n,m) - P-lim Rseq1.| < e
by a6,a7; then
a10: -e < - |. Rseq1.(n,m) - P-lim Rseq1.| by XREAL_1:24;
- |. Rseq1.(n,m) - P-lim Rseq1.| <= Rseq1.(n,m) - P-lim Rseq1
by ABSVALUE:4; then
a11: -e < Rseq1.(n,m) - P-lim Rseq1 by a10,XXREAL_0:2;
Rseq2.(n,m) - P-lim Rseq1 <= |. Rseq2.(n,m) - P-lim Rseq1.|
by ABSVALUE:4; then
a12: Rseq2.(n,m) - P-lim Rseq1 < e by a9,XXREAL_0:2;
a13: Rseq1.(n,m) - P-lim Rseq1 <= Rseq.(n,m) - P-lim Rseq1 &
Rseq.(n,m) - P-lim Rseq1 <= Rseq2.(n,m) - P-lim Rseq1
by a3,XREAL_1:9; then
-e < Rseq.(n,m) - P-lim Rseq1 & Rseq.(n,m) - P-lim Rseq1 < e
by a11,a12,XXREAL_0:2; then
a14: |. Rseq.(n,m) - P-lim Rseq1.| <= e by ABSVALUE:5;
-(Rseq.(n,m) - P-lim Rseq1) <> e by a3,a11,XREAL_1:9; then
|. Rseq.(n,m) - P-lim Rseq1.| <> e by a13,a12,ABSVALUE:1;
hence |. Rseq.(n,m) - P-lim Rseq1.| < e by a14,XXREAL_0:1;
end;
end;
hence Rseq is P-convergent;
hence P-lim Rseq = P-lim Rseq1 by a4,def6;
end;
definition
let X be non empty set, seq be Function of [:NAT,NAT:],X;
mode subsequence of seq -> Function of [:NAT,NAT:],X means :def9:
ex N,M be increasing sequence of NAT st
for n,m be Nat holds it.(n,m) = seq.(N.n,M.m);
existence
proof
deffunc F(object,object) = seq.((id NAT).$1,(id NAT).$2);
consider s be Function such that
a1: dom s = [:NAT,NAT:] & for n,m be object st n in NAT & m in NAT holds
s.(n,m) = F(n,m) from FUNCT_3:sch 2;
for z be object st z in [:NAT,NAT:] holds s.z in X
proof
let z be object;
assume z in [:NAT,NAT:]; then
consider n,m be object such that
a2: n in NAT & m in NAT & z = [n,m] by ZFMISC_1:def 2;
reconsider n,m as Nat by a2;
s.z = s.(n,m) by a2; then
s.z = seq.((id NAT).n,(id NAT).m) by a1,a2;
hence s.z in X;
end; then
reconsider s as Function of [:NAT,NAT:],X by a1,FUNCT_2:3;
take s,(id NAT),(id NAT);
hereby let n,m be Nat;
n is Element of NAT & m is Element of NAT by ORDINAL1:def 12;
hence s.(n,m) = seq.((id NAT).n,(id NAT).m) by a1;
end;
end;
end;
lem01:
for seq be increasing sequence of NAT, n be Nat holds
n <= seq.n
proof
let seq be increasing sequence of NAT;
let n be Nat;
a1:seq is Real_Sequence by FUNCT_2:7,NUMBERS:19;
rng seq c= NAT by RELAT_1:def 19;
hence n <= seq.n by a1,HEINE:2;
end;
LM114:
for Rseq1 be subsequence of Rseq st Rseq is P-convergent holds
Rseq1 is P-convergent & P-lim Rseq1 = P-lim Rseq
proof
let Rseq1 be subsequence of Rseq;
assume a1: Rseq is P-convergent;
a2:for e st 0<e
ex N st for n,m st n>=N & m>=N holds |. Rseq1.(n,m) - P-lim Rseq.| < e
proof
let e;
assume 0<e; then
consider N such that
a3: for n,m st n>=N & m>=N holds |. Rseq.(n,m) - P-lim Rseq.| < e by a1,def6;
take N;
hereby let n,m;
assume a4: n>=N & m>=N;
consider I1,I2 be increasing sequence of NAT such that
a5: for n,m holds Rseq1.(n,m) = Rseq.(I1.n,I2.m) by def9;
I1.n >= n & I2.m >= m by lem01; then
I1.n >= N & I2.m >= N by a4,XXREAL_0:2; then
|. Rseq.(I1.n,I2.m) - P-lim Rseq.| < e by a3;
hence |. Rseq1.(n,m) - P-lim Rseq.| < e by a5;
end;
end;
hence Rseq1 is P-convergent;
hence P-lim Rseq1 = P-lim Rseq by a2,def6;
end;
th63d:
Rseq is convergent_in_cod1 implies
for Rseq1 be subsequence of Rseq holds Rseq1 is convergent_in_cod1
proof
assume a1: Rseq is convergent_in_cod1;
hereby let Rseq1 be subsequence of Rseq;
consider I1,I2 be increasing sequence of NAT such that
a7: for n,m holds Rseq1.(n,m) = Rseq.(I1.n,I2.m) by def9;
for m be Element of NAT holds ProjMap2(Rseq1,m) is convergent
proof
let m be Element of NAT;
reconsider m2=I2.m as Element of NAT;
a4: ProjMap2(Rseq,m2) is convergent by a1;
now let e be Real;
assume 0<e; then
consider N such that
a5: for n st n>=N holds
|.ProjMap2(Rseq,m2).n - lim ProjMap2(Rseq,m2).| < e
by a4,SEQ_2:def 7;
take N;
hereby let n;
assume a6: n >= N;
x1: I1.n is Element of NAT & n is Element of NAT by ORDINAL1:def 12;
I1.n >= n by lem01; then
I1.n >= N by a6,XXREAL_0:2; then
|.ProjMap2(Rseq,m2).(I1.n) - lim ProjMap2(Rseq,m2).| < e by a5; then
|. Rseq.(I1.n,I2.m)-lim ProjMap2(Rseq,m2).| < e
by MESFUNC9:def 7; then
|. Rseq1.(n,m) - lim ProjMap2(Rseq,m2).| < e by a7;
hence |. ProjMap2(Rseq1,m).n - lim ProjMap2(Rseq,m2).| < e
by x1,MESFUNC9:def 7;
end;
end;
hence ProjMap2(Rseq1,m) is convergent by SEQ_2:def 6;
end;
hence Rseq1 is convergent_in_cod1;
end;
end;
registration
let Pseq;
cluster -> P-convergent for subsequence of Pseq;
coherence by LM114;
end;
theorem
for Psubseq be subsequence of Pseq holds
P-lim Psubseq = P-lim Pseq by LM114;
registration
let Rseq be convergent_in_cod1 Function of [:NAT,NAT:],REAL;
cluster -> convergent_in_cod1 for subsequence of Rseq;
coherence by th63d;
end;
theorem
for Rseq1 be subsequence of Rseq st
Rseq is convergent_in_cod1 & lim_in_cod1 Rseq is convergent holds
lim_in_cod1 Rseq1 is convergent &
cod1_major_iterated_lim Rseq1 = cod1_major_iterated_lim Rseq
proof
let Rseq1 be subsequence of Rseq;
assume that
a1: Rseq is convergent_in_cod1 & lim_in_cod1 Rseq is convergent;
consider I1,I2 be increasing sequence of NAT such that
a7: for n,m be Nat holds Rseq1.(n,m) = Rseq.(I1.n,I2.m) by def9;
a8:Rseq1 is convergent_in_cod1 by a1;
a10:for e st 0<e
ex N st for m st m >= N holds
|.(lim_in_cod1 Rseq1).m - cod1_major_iterated_lim Rseq.| < e
proof
let e;
assume 0<e; then
consider N such that
a11: for m st m >= N holds
|.(lim_in_cod1 Rseq).m - cod1_major_iterated_lim Rseq.| < e by a1,def34;
take N;
hereby let m;
assume a12: m >= N;
reconsider m2 = I2.m as Element of NAT;
reconsider m1 = m as Element of NAT by ORDINAL1:def 12;
x2: ProjMap2(Rseq1,m1) is convergent by a8;
for p be Real st 0<p
ex K be Nat st
for n be Nat st n>=K holds
|.ProjMap2(Rseq1,m1).n - lim ProjMap2(Rseq,m2).| < p
proof
let p be Real;
assume b1: 0<p;
ProjMap2(Rseq,m2) is convergent by a1; then
consider K be Nat such that
b2: for n st n>=K holds |.ProjMap2(Rseq,m2).n - lim ProjMap2(Rseq,m2).| < p
by b1,SEQ_2:def 7;
take K;
hereby let n;
assume b3: n >= K;
x2: n is Element of NAT &
I1.n is Element of NAT & I2.m is Element of NAT by ORDINAL1:def 12;
I1.n >= n by lem01; then
I1.n >= K by b3,XXREAL_0:2; then
|.ProjMap2(Rseq,m2).(I1.n)-lim ProjMap2(Rseq,m2).| < p by b2; then
|. Rseq.(I1.n,I2.m)-lim ProjMap2(Rseq,m2).| <p
by MESFUNC9:def 7; then
|. Rseq1.(n,m) - lim ProjMap2(Rseq,m2).| < p by a7;
hence |. ProjMap2(Rseq1,m1).n - lim ProjMap2(Rseq,m2).| <p
by x2,MESFUNC9:def 7;
end;
end; then
c1: lim ProjMap2(Rseq1,m1) = lim ProjMap2(Rseq,m2) by x2,SEQ_2:def 7;
I2.m >= m by lem01; then
I2.m >= N by a12,XXREAL_0:2; then
a13: |.(lim_in_cod1 Rseq).(I2.m) - cod1_major_iterated_lim Rseq.| < e by a11;
(lim_in_cod1 Rseq).(I2.m) = lim ProjMap2(Rseq,m2) by def32;
hence |.(lim_in_cod1 Rseq1).m - cod1_major_iterated_lim Rseq.| < e
by def32,a13,c1;
end;
end;
hence lim_in_cod1 Rseq1 is convergent by SEQ_2:def 6;
hence thesis by a10,def34;
end;
th63c:
Rseq is convergent_in_cod2 implies
for Rseq1 be subsequence of Rseq holds Rseq1 is convergent_in_cod2
proof
assume a1: Rseq is convergent_in_cod2;
hereby let Rseq1 be subsequence of Rseq;
consider I1,I2 be increasing sequence of NAT such that
a7: for n,m holds Rseq1.(n,m) = Rseq.(I1.n,I2.m) by def9;
for m be Element of NAT holds ProjMap1(Rseq1,m) is convergent
proof
let m be Element of NAT;
reconsider m1 = I1.m as Element of NAT;
a4: ProjMap1(Rseq,m1) is convergent by a1;
now let e;
assume 0<e; then
consider N such that
a5: for n st n>=N holds
|. ProjMap1(Rseq,m1).n - lim ProjMap1(Rseq,m1).| < e
by a4,SEQ_2:def 7;
take N;
hereby let n;
assume a6: n >= N;
x2: n is Element of NAT &
I1.m is Element of NAT & I2.n is Element of NAT by ORDINAL1:def 12;
I2.n >= n by lem01; then
I2.n >= N by a6,XXREAL_0:2; then
|. ProjMap1(Rseq,m1).(I2.n) - lim ProjMap1(Rseq,m1).| < e by a5; then
|. Rseq.(I1.m,I2.n)-lim ProjMap1(Rseq,m1).| < e
by MESFUNC9:def 6; then
|. Rseq1.(m,n) - lim ProjMap1(Rseq,m1).| < e by a7;
hence |.ProjMap1(Rseq1,m).n - lim ProjMap1(Rseq,m1).| < e
by x2,MESFUNC9:def 6;
end;
end;
hence ProjMap1(Rseq1,m) is convergent by SEQ_2:def 6;
end;
hence Rseq1 is convergent_in_cod2;
end;
end;
registration
let Rseq be convergent_in_cod2 Function of [:NAT,NAT:],REAL;
cluster -> convergent_in_cod2 for subsequence of Rseq;
coherence by th63c;
end;
theorem
for Rseq1 be subsequence of Rseq st
Rseq is convergent_in_cod2 & lim_in_cod2 Rseq is convergent holds
lim_in_cod2 Rseq1 is convergent &
cod2_major_iterated_lim Rseq1 = cod2_major_iterated_lim Rseq
proof
let Rseq1 be subsequence of Rseq;
assume that
a1: Rseq is convergent_in_cod2 & lim_in_cod2 Rseq is convergent;
consider I1,I2 be increasing sequence of NAT such that
a7: for n,m holds Rseq1.(n,m) = Rseq.(I1.n,I2.m) by def9;
a8:Rseq1 is convergent_in_cod2 by a1;
a10:for e st 0<e ex N st for m st m >= N holds
|.(lim_in_cod2 Rseq1).m - cod2_major_iterated_lim Rseq.| < e
proof
let e;
assume 0<e; then
consider N such that
a11: for m st m >= N holds
|.(lim_in_cod2 Rseq).m - cod2_major_iterated_lim Rseq.| < e
by a1,def35;
take N;
hereby let m;
reconsider mm = m as Element of NAT by ORDINAL1:def 12;
assume a12: m >= N;
reconsider m1 = I1.m as Element of NAT;
x2: ProjMap1(Rseq1,mm) is convergent by a8;
for p be Real st 0<p
ex K be Nat st
for n be Nat st n>=K holds
|.ProjMap1(Rseq1,mm).n - lim ProjMap1(Rseq,m1).| < p
proof
let p be Real;
assume b1: 0<p;
ProjMap1(Rseq,m1) is convergent by a1; then
consider K be Nat such that
b2: for n st n>=K holds
|. ProjMap1(Rseq,m1).n - lim ProjMap1(Rseq,m1).| < p
by b1,SEQ_2:def 7;
take K;
hereby let n;
assume b3: n >= K;
x3: n is Element of NAT &
I1.m is Element of NAT & I2.n is Element of NAT by ORDINAL1:def 12;
I2.n >= n by lem01; then
I2.n >= K by b3,XXREAL_0:2; then
|. ProjMap1(Rseq,m1).(I2.n)-lim ProjMap1(Rseq,m1).| < p by b2; then
|. Rseq.(I1.m,I2.n)-lim ProjMap1(Rseq,m1).| <p
by MESFUNC9:def 6; then
|. Rseq1.(m,n) - lim ProjMap1(Rseq,m1).| < p by a7;
hence |.ProjMap1(Rseq1,mm).n - lim ProjMap1(Rseq,m1).| <p
by x3,MESFUNC9:def 6;
end;
end; then
c1: lim ProjMap1(Rseq1,mm) = lim ProjMap1(Rseq,m1) by x2,SEQ_2:def 7;
I1.m >= m by lem01; then
I1.m >= N by a12,XXREAL_0:2; then
a13: |.(lim_in_cod2 Rseq).(I1.m) - cod2_major_iterated_lim Rseq.| < e by a11;
(lim_in_cod2 Rseq).(I1.m) = lim ProjMap1(Rseq,m1) by def33;
hence |.(lim_in_cod2 Rseq1).m - cod2_major_iterated_lim Rseq.| < e
by def33,a13,c1;
end;
end;
hence lim_in_cod2 Rseq1 is convergent by SEQ_2:def 6;
hence thesis by a10,def35;
end;
| 77,634 |
TITLE: An Integral Inequality
QUESTION [4 upvotes]: Let $f$ and $g$ be real functions such that $\int_0^\infty(f(x))^2dx<\infty$ and $\int_0^\infty(g(x)^2dx<\infty$. Prove that:
$$\left(\int_0^\infty\int_0^\infty\frac{f(x)g(y)}{x+y}dxdy \right)^2\leq C\int_0^\infty(f(x))^2dx\int_0^\infty(g(y))^2dy$$
where $C$ is an universal constant (independent of $f$ and $g$)
My attempt: using Cauchy-Schwarz inequality I evaluate:
$$LHS\leq(\int_0^\infty (g(y))^2dy)\times (\int_0^\infty (\int_0^\infty \frac{f(x)}{x+y}dx)^2dy)\leq$$ $$\leq(\int_0^\infty(g(y))^2dy)\times(\int_0^\infty((\int_0^\infty(f(x))^2dx)\times(\int_0^\infty\frac{dx}{(x+y)^2}))dy)\leq$$ $$\leq\left(\int_0^\infty(f(x))^2dx\int_0^\infty(g(y))^2dy\right)\times\int_0^\infty\int_0^\infty\frac{dxdy}{(x+y)^2}$$
which is useless since $\int_0^\infty\int_0^\infty\frac{dxdy}{(x+y)^2}$ is not finite.
REPLY [5 votes]: Assume without loss of generality that $f,g$ are nonnegative (in order to use safely Fubini's Theorem), and denote by $I$ the double integral in the left-hand side. We have
\begin{eqnarray} I&=&\int_0^\infty g(y)\left(\int_y^\infty f(u-y)\,\frac{du}u \right)dy\\
&=&\int_0^\infty g(y)\left(\int_1^\infty f(y(v-1))\,\frac{dv}v \right)dy\\
&=&\int_0^\infty g(y)\left(\int_0^\infty f(yu)\frac{du}{u+1} \right)dy\\
&=&\int_0^\infty\left(\int_0^\infty g(y)f(uy)\, dy\right)\frac{du}{u+1}\cdot
\end{eqnarray}
Now, apply Cauchy-Scharz to the inner integral: this gives
\begin{eqnarray} \int_0^\infty g(y)f(uy)\, dy&\leq& \Vert g\Vert_2\times \left(\int_0^\infty f(uy)^2 dy\right)^{1/2}\\
&=&\Vert g\Vert_2\times \left(\int_0^\infty f(t)^2 \frac{dt}{u}\right)^{1/2}\\
&=&\Vert g\Vert_2\times \Vert f\Vert_2\times \frac{1}{\sqrt u}\cdot
\end{eqnarray}
Altogether, we obtain
$$I\leq \Vert g\Vert_2\times \Vert f\Vert_2\times\int_0^\infty\frac{du}{(u+1)\sqrt u}\, , $$
which gives the result since the integral in the right-hand side is finite. | 91,610 |
Okay first off…WHAT THE HELL IS THIS GAME ABOUT?!? It looks amazing, stars The Walking Dead‘s Norman Reedus, but looks very weird. Tell us what this game is about Kojima!!!
Mads Mikkelsen, Léa Seydoux, Lindsay Wagner, Emily O’Brien, and Troy Baker meaning. Another element talked about is the existence of a type of rain, called “Timefall”, with the ability to age or deteriorate whatever it hits. Another gameplay element talked about is the player’s ability to interact with the environment and wander outside of the character’s body, as well as recovering items when they die.
| 98,185 |
FOR RELEASE: April 19, 2007
CONTACT: Pamela Williams
Office of Communications
405/271-5601
Rabies Season is Here Remember to Vaccinate Your Pets
Vaccinating pets against rabies is crucial for their protection as well as the protection of you and your family, according to public health officials with the Oklahoma State Department of Health (OSDH). In the first four months of 2007, there have been 31 confirmed cases of animal rabies reported in Oklahoma. So far, 27 skunks, 3 cattle and 1 dog have tested positive for rabies.
The OSDH also reports a noticeable increase of rabies cases in the Garfield County and Vance Air Force Base areas, with five cases reported. Other counties with increased rabies activity include Blaine and Grady counties, with three cases each.
In 2006, a total of 69 animal rabies cases were reported statewide for the entire year. “Our animal disease surveillance is indicating nearly double the number of animal rabies cases in 2007 compared to the number we observed in 2006 at this same time of year,” said Deputy State Epidemiologist and State Public Health Veterinarian Dr. Kristy Bradley. “We want to remind animal owners of the importance of keeping their pets up-to-date on rabies vaccinations – not only to protect their pets, but to protect their families from potential exposure to rabies and costly, post-exposure rabies shots.”
In Oklahoma and the central plains states, skunks are the primary wild animal source of rabies virus. Bats can also be infected with and transmit rabies. Rabid wildlife spread the disease to other wild animals or unvaccinated pets and livestock when they bite, spreading the virus through saliva into the fresh bite wound.
Although most rabies cases in Oklahoma occur in skunks, most human exposures to rabies result from contact with rabid pets or livestock that developed rabies because they were not vaccinated and had an encounter with a rabid wild animal.
Rabies is a viral disease that affects the central nervous system and is almost always fatal once symptoms of the disease have started. Rabies virus is found in the brain, spinal cord and saliva of infected animals and is transmitted through a bite or opening in the skin or mucous membrane (eyes, nose, or mouth).
Oklahoma rules and regulations require that owners have their dogs, cats, and ferrets vaccinated against rabies by a veterinarian by the time the animal is four months of age. The interval between rabies vaccinations and boosters will depend upon the age of the animal, type of vaccine administered, and city licensing codes. Rabies vaccines labeled for use in horses, sheep, and cattle are also available and recommended for show animals and all valuable breeding stock.
Other ways that pet owners can reduce the risk of pet exposure to rabies is to keep their dogs and cats close to home to reduce contact with other animals. Outdoor dogs should be kenneled, or kept within a fenced-in yard. Cats should be kept indoors as much as possible and not allowed to roam freely at night. Do not keep pet food outdoors for extended periods of time and keep trashcans tightly sealed to avoid attracting hungry wildlife.
In addition, parents should teach their children to never handle wild animals, including bats found on the ground, or approach unfamiliar dogs or cats. If you suspect your animals have been exposed to rabies, immediately contact the county health department or call the local animal control officer.
For more information about rabies, contact your veterinarian or the county health department. Rabies information is also available on the OSDH Web site at.
### | 391,789 |
Sep 102013
Do you remember the Indie title Amnesia the Dark Descent ?
A great Survival-horror game released in 2010 that has had a lot of success, and if you don’t know it, check it on the net, it’s probable that with the release of the new game you can find it at a discounted price.
But I’m here to talk about Amnesia: A Machine for Pigs, from the creators of Amnesia: The Dark Descent and Dear Esther comes this new first-person horrorgame that will drag you to the depths of greed, power and madness, sound interesting enough ? | 199,027 |
Mod developer BlackCaesar has announced the release of beta version 2.0 of Men of War mod Global War, available for download now. This update adds the UK faction and two new maps.
N4G is a community of gamers posting and discussing the latest game news. It’s part of NewsBoiler, a network of social news sites covering today’s pop culture. | 372,623 |
McIntosh County Sheriff's Office, Oklahoma
End of Watch Saturday, May 8, 1920
Jack Hunter
Deputy Sheriff Jack Hunter was shot and killed by an inmate during an escape attempt from the county jail.
The suspect, who was being held for officials in Seattle, Washington, on charges of forgery, bigamy, and draft dodging, had a pistol slipped to him by a friend. Deputy Hunter was picking up dirty dishes from the inmates when the man produced the hidden gun and shot him. The suspect escaped and fled to Alma, Arkansas, where he was shot and killed by the McIntosh County sheriff and his deputy three weeks later.
Deputy Hunter had served with the McIntosh County Sheriff's Office for 10 years. He was survived by his wife and three children.
Bio
- Age 43
- Tour 10 years
- Badge Not available
Incident Details
- Cause Gunfire
- Weapon Handgun; Pistol
- Offender Shot and killed
Most Recent ReflectionView all 3 Reflections
Your heroism and service is honored today, the 52
May | 141,116 |
Ted McCain
First and foremost, Ted McCain is an educator. He has taught high school students at Maple Ridge Secondary School for twenty-five years
First and foremost, Ted McCain is an educator. He has taught high school students at Maple Ridge Secondary School for twenty-five years. Although he has had several opportunities to take other jobs both inside education and in the private sector, he has felt his primary calling is to help prepare teenagers for success as they move into adult life.
Ted McCain has been an innovator and pioneer in technology education. In 1997, Ted received the Prime Minister's Award For Teaching Excellence. Ted was awarded this prestigious Canadian national award for his work in developing a real-world technology curriculum for grade 11 and 12 students that prepares them for employment in the areas of website design and computer networking directly out of high school. Ted was recognized for his work in creating his innovative Instructional Technology for Maple Ridge Secondary School in Vancouver, B.C. Ted McCain has also taught at the junior college level.
Ted is also an author. He has written or co-written six books on the future, effective teaching, McCain has also consulted with school districts and businesses on effective teaching for the digital generation and the implementation of instructional technology. His clients have included Apple Computer, Microsoft, Aldus, and Toyota, as well as many school districts and educational associations in both the United States and Canada.
Ted has now joined the Thornburg Center For Professional Development in Chicago, Illinois as an associate director. In this role, Ted has expanded his work as an educational futurist. Ted focuses on the impact on students and learning from the astounding changes taking place in the world today as a consequence of technological development. Ted McCain is passionate in his belief that schools must change so they can effectively prepare students for the rest of their lives.
Videos
Related Speakers View all
More like Ted | 188,355 |
TITLE: Is this set the Power Set of Natural Numbers?
QUESTION [0 upvotes]: I have a school problem and one of the affirmations I have to prove about it seems -to me- contradictory to what my intuition says.
Let $Q_n:\{m \in \mathbb{N}: m>n\}$ and $P_n$ the family of all subsets of $\{1,...,n\}$. Let us define
$L_n=\{L:L \in P_n \enspace or \enspace L=P \cup Q_n, P \in P_n\}.$
My intuition tells me that $\cup_{n=1}^{\infty}L_n=P[\mathbb{N}]$. Where $P[\mathbb{N}]$ is the Power Set of Natural Numbers.
Am I right about this?
REPLY [1 votes]: For every $n$, every set in $L_n$ will either be finite or will include every natural number greater than $n$.
Every set then in $\bigcup\limits_{n=1}^\infty L_n$ will either be finite or will have some value of $n$ for which every natural number greater than $n$ appears. Remember... $A\in \bigcup\limits_{n=1}^\infty L_n$ is true if and only if there exists some $n$ for which $A\in L_n$.
As such, the set of even natural numbers is not in your set as it is neither finite nor has a value $n$ for which every number greater than $n$ appears in it. | 64,908 |
TITLE: $U(1)$ Faddeev-Popov formalism
QUESTION [3 upvotes]: What is the correct series expansion for the $U(1)$ Faddeev-Popov ghosts?
I know that the $U(1)$ ghosts are only a phase such that they can be neglected in most cases but it turns out that this is not true in curved spaces even for $U(1)$ theories so please don't answer this...
In this thread Faddeev-Popov ghost propagator in canonical quantization I found that $c$ is hermitian and $\bar{c}$ anti-hermitian which makes sense since $\bar{c} = c^\dagger \gamma_0$.
But in the $U(1)$ case the ghost are Grassmann variables such that $\bar{c} = c^\dagger \gamma_0$ doesn't make sense does it?
For those willing to help me even more. I think that the source of my problem is a poor understanding of the Faddeev-Popov mechanism. More precisely, what happens when $\det(\square)$ is written as a path integral? What exactly do the $c$ and $\bar{c}$ fields mean? Why is it said that one is a ghost and the other an anti ghost?
When quantizing them I obtain $\{ c_k , \bar{c_{k'}}\} = -\delta(k-k')$ how does this tell us anything regarding the norm of these ghosts?
I read Peskin and Schroeder but they do not answer this question (or I missed it).
Finally, my sincere aplogies for this "all over the place" type question. I fail to pinpoint the exact sources of my confusion that's why my question is rather broad. I hope that someone more experiences can pinpoint it with the above information.
REPLY [4 votes]: As discussed in Kugo and Ojima 1979, "ghost is Hermitian, anti-ghost is anti-Hermitian" is just a convention, another being that both fields are Hermitian, which results in a factor of $i$ in the FP-ghost term so that the Lagrangian is still Hermitian. In their notation $c,\,\overline{c}$ are both Hermitian while $C:=c,\,\overline{C}:=i\overline{c}$ provide a half-Hermitian convention. Then $$\mathcal{L}_{FP}=-i\partial_\mu\overline{c}D^\mu c=-\partial_\mu\overline{C}D^\mu C.$$ | 133,574 |
Believe it or not but I was reading the Koran the other night for something to do when I came across this verse: Koran . Muslims are permitted to lie under certain circumstances, known as taqiyya to advance the cause of Islam – by gaining the trust of non-believers in order to draw out their vulnerability and defeat them. I then asked myself, who does that remind me of? Voila, Waleed Aly. Capiche
Advertisements | 216,336 |
\begin{document}
\renewcommand{\thefootnote}{}
\footnotetext{{\it Date:} April 3, 2016
\qquad {\it Last update:}
\today\ (\now\ JST)\vspace{-1\smallskipamount}
}
\footnotetext{{\it 2010 Mathematical Subject Classification:}
03E75, 46C05\vspace{-1\smallskipamount}}
\footnotetext{{\it Keywords:}
pre-Hilbert spaces, orthonormal basis, elementary submodels,
Singular Compactness Theorem,
Fodor-type Reflection Principle\vspace{-2\smallskipamount}}
\footnotetext{}
\maketitle
\renewcommand{\thefootnote}{$\ast$}
\footnotetext{Graduate School of System Informatics, Kobe University,
Kobe, Japan, \\
\mbox{}\hspace{4.2ex}E-Mail: {\tt [email protected]}\\
The research was partially supported by Grant-in-Aid for Exploratory
Research No.\ 26610040 of the Ministry of Education,
Culture, Sports, Science and Technology Japan (MEXT).
The author would like to thank Hiroshi Fujita of Ehime University who brought
the author's attention to the maximal orthonormal system in a pre-Hilbert
space in \Exof{example-1} which is not an orthonormal basis. Fujita also
suggested the
possibility of a construction of pre-Hilbert spaces without orthonormal bases
based on this example. Thanks are also due to Hiroshi Ando of Chiba
University and Hiroshi Sakai of Kobe University as well as Ilijas Farah for
valuable comments.
}
\begin{abstract}
We give an algebraic characterization of pre-Hilbert spaces with an orthonormal
basis. This characterization is used to show that there are pre-Hilbert spaces
$X$ of dimension and density $\lambda$ for any uncountable $\lambda$ without
any orthonormal basis.
Let us call a pre-Hilbert space without any orthonormal bases pathological.
The pair of the cardinals $\kappa\leq\lambda$ such that there is a
pre-Hilbert space of dimension $\kappa$ and density $\lambda$ are known to be
characterized
by the inequality $\lambda\leq\kappa^{\aleph_0}$. Our result implies that there
are pathological pre-Hilbert spaces with dimension $\kappa$ and
density $\lambda$ for all combinations of such $\kappa$ and $\lambda$ including
the case $\kappa=\lambda$.
A Singular Compactness Theorem on pathology of pre-Hilbert spaces is obtained.
A reflection theorem asserting that for any pathological pre-Hilbert space $X$
there are stationarily many pathological sub-inner-product-spaces $Y$ of $X$ of
smaller density is shown to be equivalent with Fodor-type Reflection
Principle (FRP).
\end{abstract}
\begin{quotation}
\noindent
\mbox{}\hfill {\bf Contents}\hfill \mbox{}\medskip\\
\small
\noindent
\ref{intro}\ \ Introduction\ \ \dotfill\ \ \pageref{intro}\\
\ref{section-2}\ \ Pathological pre-Hilbert spaces constructed from a pre-ladder
system\ \ \dotfill\ \ \pageref{section-2}\\
\ref{char}\ \ A characterization of the non-pathology\ \ \dotfill\ \ \pageref{char}\\
\ref{dim-den}\ \ Dimension and density of pre-Hilbert spaces\ \ \dotfill\ \
\pageref{dim-den}\\
\ref{direct-sum}\ \ Orthogonal direct sum\ \ \dotfill\ \ \pageref{direct-sum}\\
\ref{reflection}\ \ Reflection and non-reflection of pathology\ \ \dotfill\ \
\pageref{reflection}\\
\ref{singular}\ \ A Singular Compactness Theorem\ \ \dotfill\ \
\pageref{singular}\\
\ref{FRP}\ \ Reflection of pathology and Fodor-type Reflection
Principle\ \ \dotfill\ \
\pageref{FRP}\\
\phantom{\ref{intro}}\ \ References\ \ \dotfill\ \ \pageref{literature}\\
\end{quotation}
\section{Introduction}
\label{intro}
An inner product space whose topology is not necessarily
complete is often called a {\it pre-Hilbert space}.
In a pre-Hilbert space $X$, a maximal orthonormal system $S$ of $X$ does
not necessarily span a dense subspace of $X$, that is, such $S$ does not need
to be {\it an orthonormal basis}\/ (see \Exof{example-1} below). It is known that
it is even possible that there is no orthonormal basis at all in some pre-Hilbert
space (see \Lemmaof{halmos}).
Let us call a pre-Hilbert space {\it pathological\/} if it
does not have any orthonormal bases. If $X$ is not pathological, i.e.\ if it does
have an
orthonormal basis, then we say that such $X$ is {\it non-pathological}.
By Bessel's inequality, it is easy to see that all maximal
orthonormal system $S$ of a pre-Hilbert space $X$ has the same cardinality
independently of whether $S$ is a basis of $X$ or not.
This cardinality is called the {\it dimension} of the pre-Hilbert space $X$ and
denoted by $\dim(X)$.
In the following, we fix the scalar field $K$ of the pre-Hilbert spaces we
consider in this paper to be $\reals$ or
$\complexnrs$ throughout.
For an infinite set $S$, let
\begin{xitemize}
\xitem[] $\ell_2(S)=\setof{\bbu \in\fnsp{S}{K}}{\sum_{x\in S}(\bbu (x))^2
<\infty}$,
\end{xitemize}
where $\sum_{x\in S}(\bbu (x))^2$ is defined as
$\sup\setof{\sum_{x\in A}(\bbu (x))^2}{A\in[S]^{<\aleph_0}}$. $\ell_2(S)$ is
endowed with a natural structure of inner product space with coordinatewise
addition and scalar multiplication, as
well as the inner product defined by
\begin{xitemize}
\xitem[]
$(\bbu,\bbv)=\sum_{x\in S}\bbu(x)\overline{\bbv(x)}$\ \ for $\bbu$, $\bbv\in\ell_2(S)$.
\end{xitemize}
It is easy to see that $\ell_2(S)$ is a/the Hilbert space of density $\cardof{S}$.
Note that any pre-Hilbert space $X$ of density $\lambda$ can be embedded densely into
$\ell_2(\lambda)$ as a sub-inner-product-space. Here we call a subspace $Y$ of a
(pre-)Hilbert space $X$ a sub-inner-product-space of $X$ if $Y$ is a linear
subspace of $X$ with the inner product which is the restriction of the inner
product of $X$ to $Y$.
For a pre-Hilbert space $X$ and $S\subseteq X$, we denote by $[S]_X$ the
sub-inner-product-space of $X$ whose underlying set is the linear subspace of $X$
spanned by $S$.
If $U$ is a subset of $\ell_2(S)$, we denote with
$\cls_{\ell_2(S)}(U)$ the topological closure of $[U]_{\ell_2(S)}$ in $\ell_2(S)$. We write
simply $\cls(U)$ if it is clear in which $\ell_2(S)$ we are working.
For $x\in S$, let $\bbe^S_x\in\ell_2(S)$ be the standard unit vector at $x$ defined by
\begin{xitemize}
\xitem[]
$\bbe^S_x(y)=\delta_{x,y}$\ \ for $y\in S$.
\end{xitemize}
For $\bba\in\ell_2(S)$, the support of $\bba$ is defined by
\begin{xitemize}
\xitem[]
$\supp(\bba)=\setof{x\in S}{\bba(x)\not=0}$\ \
($=\setof{x\in S}{(\bba,\bbe^S_x)\not=0}$).
\end{xitemize}
By the definition of $\ell^2(S)$, $\supp(\bba)$ is a countable subset of $S$ for
all $\bba\in\ell^2(S)$.
For a subset $U$ of $\ell_2(S)$ the support of $U$ is the set
$\supp(U)=\bigcup\setof{\supp(\bba)}{\bba\in U}$.
For $X\subseteq\ell_2(S)$ and $S'\subseteq S$, let
$X\downarrow S'=\setof{\bbu\in X}{\supp(\bbu)\subseteq S'}$.
For $\bbu\in\ell_2(S)$, let $\bbu\downarrow S'\in\ell_2(S)$ be defined by
\begin{xitemize}
\xitem[]
$\left(\bbu\downarrow S'\right)(x)=\left\{\,
\begin{array}{@{}ll}
\bbu(x) &\mbox{if }x\in S'\\[\jot]
0 &\mbox{otherwise}
\end{array}
\right.$
\end{xitemize}
for $x\in S$.
Note that $X\downarrow S'$ is not necessarily equal to
$\setof{\bbu\downarrow S'}{\bbu\in X}$\ \ (e.g., we have
$X\downarrow\omega\not=\setof{\bbu\downarrow\omega}{\bbu\in X}$ where $X$ is the
pre-Hilbert space defined in \Exof{example-1} below).
\begin{Ex}
\label{example-1}
Let $X$ be the sub-inner-product-space of $\ell_2(\omega+1)$ spanned by
$\setof{\bbe^{\omega+1}_n}{n\in\omega}\cup\ssetof{\bbb}$ where
$\bbb\in\ell_2(\omega+1)$
is defined by
\begin{xitemize}
\xitem[] $\bbb(\omega)=1$;
\xitem[] $\bbb(n)=\frac{1}{n+2}$\ \ for $n\in\omega$.
\end{xitemize}
Then $\setof{\bbe^{\omega+1}_n}{n\in\omega}$ is a maximal orthonormal system in $X$
but it is not a basis of $X$.
\end{Ex}
\prf If $\setof{\bbe^{\omega+1}_n}{n\in\omega}$ were not maximal, then there
would be an element $\bbc$ of X represented as a
linear combination of $\bbb$ and some of $\bbe^{\omega+1}_n$'s ($n\in\omega$)
\st\ $\bbc$ is orthogonal to all $\bbe^{\omega+1}_n$,
$n\in\omega$. However, any of such linear combinations has an infinite support
and hence is not orthogonal to $\bbe^{\omega+1}_n$ for any $n$ in the support.
$\setof{\bbe^{\omega+1}_n}{n\in\omega}$ is not an orthonormal basis of $X$
since
$\cls_{\ell_2(\omega+1)}(\setof{\bbe^{\omega+1}_n}{n\in\omega})
=\ell_2(\omega+1)\downarrow\omega\not=\ell_2(\omega+1)$.
\qedofEx
\qedskip
For all separable pre-Hilbert spaces (including the $X$ in
\Exof{example-1}), we can always find an orthonormal basis: suppose that $X$ is
separable and let $\setof{\bba_n}{n\in\omega}$ be dense in $X$. Then, by
Gram-Schmidt orthonormalization process, we can find an orthonormal system
$\setof{\bbb_n}{n\in\omega}$ which spans the same dense sub-inner-product-space as that
spanned by $\setof{\bba_n}{n\in\omega}$. Thus there are no separable
pathological pre-Hilbert spaces.
The situation is different if we consider non-separable pre-Hilbert spaces.
\begin{Lemma}{\rm (P.\,Halmos, see Gudder \cite{gudder})}
\label{halmos}
There are pre-Hilbert spaces $X$ of dimension $\aleph_0$ and
density $\lambda$ for any $\aleph_0<\lambda\leq 2^{\aleph_0}$.
\end{Lemma}
Note that a pre-Hilbert space $X$ with $\dim(X)<d(X)$ cannot have any orthonormal
basis, that is, such a pre-Hilbert space is pathological.
For any two pre-Hilbert spaces $X$, $Y$, the {\it orthogonal direct sum of\/ $X$
and\/ $Y$} is
the direct sum $X\oplus Y=\setof{\pairof{\bbx,\bby}}{\bbx\in X, \bby\in Y}$ of $X$ and $Y$ as
linear spaces together with the inner product defined by
$(\pairof{\bbx_0,\bby_0},\pairof{\bbx_1,\bby_1})=(\bbx_0,\bbx_1)+(\bby_0,\bby_1)$
for $\bbx_0$, $\bbx_1\in X$
and $\bby_0$, $\bby_1\in Y$. A sub-inner-product-space $X_0$ of a pre-Hilbert
space $X$ is an {\it orthogonal direct summand of\/ $X$} if there is a
sub-inner-product-space $X_1$ of $X$ \st\ the mapping
$\mapping{\varphi}{X_0\oplus X_1}{X}$; $\pairof{\bbx_0,\bbx_1}\mapsto \bbx_0+\bbx_1$ is an
isomorphism of pre-Hilbert spaces.
If this holds, we usually identify $X_0\oplus X_1$ with $X$ by $\varphi$ as above.
\qedskip
\noindent
{\bf Proof of \bfLemmaof{halmos}.}
Let $B$ be a linear basis (Hamel basis) of the linear space $\ell_2(\omega)$ extending
$\setof{\bbe^\omega_n}{n\in\omega}$. Note that $\cardof{B}=2^{\aleph_0}$ (Let
$\calA$ be an almost disjoint family of infinite subsets of $\omega$ of
cardinality $2^{\aleph_0}$. For each $a\in\calA$ let $\bbb_a\in\ell_2(\omega)$ be
\st\ $\supp(\bbb_a)=a$. Then $\setof{\bbb_a}{a\in\calA}$ is a linearly
independent subset of $\ell_2(\omega)$ of cardinality $2^{\aleph_0}$\,).
Let $\mapping{f}{B}{
\setof{\bbe^\lambda_\alpha}{\alpha<\lambda}\cup\ssetof{\bbzero_{\ell_2(\lambda)}}}$ be a
surjection \st\ $f(\bbe^\omega_n)=\bbzero_{\ell_2(\lambda)}$ for all $n\in\omega$.
Note that $f$ generates a linear mapping from the linear space $\ell_2(\omega)$
to a dense subspace of $\ell_2(\lambda)$.
Let $U=\setof{\pairof{\bbb,f(\bbb)}}{\bbb\in B}$ and
$X=[U]_{\ell_2(\omega)\oplus\ell_2(\lambda)}$. Then this $X$ is as desired since
$\setof{\pairof{\bbe^\omega_n,\bbzero}}{n\in\omega}$ is a maximal orthonormal system in
$X$ while we have
$\cls_{\ell_2(\omega)\oplus\ell_2(\lambda)}(X)=\ell_2(\omega)\oplus\ell_2(\lambda)$
and hence $d(X)=\lambda$.
\qedof{\Lemmaof{halmos}}
\qedskip
For sub-inner-product-spaces $X_0$, $X_1$ of a pre-Hilbert space $X$, we have
$[X_0\cup X_1]_X\cong X_0\oplus X_1$ with the isomorphism extending
\begin{xitemize}
\xitem[]
$i_{X_0\cup X_1}=\setof{\pairof{\bbx_0,\pairof{\bbx_0,\bbzero}}}{\bbx\in X_0}
\cup\setof{\pairof{\bbx_1,\pairof{\bbzero,\bbx_1}}}{\bbx_1\in X_1}$,
\end{xitemize}
if we have
\begin{xitemize}
\xitem[p-0]
$(\bbx_0,\bbx_1)=0$
for any
$\bbx_0\in X_0$ and $\bbx_1\in X_1$.
\end{xitemize}
Sub-inner-product-spaces $X_0$ and $X_1$ of a pre-Hilbert space $X$ with
\xitemof{p-0} are said to be {\it orthogonal} to each other and this is denoted by
$X_0\perp X_1$.
If $X_0$ and $X_1$ are sub-inner-product-spaces of $X$ and $X_0\perp X_1$, we
identify $[X_0\cup X_1]_X$ with $X_0\oplus X_1$ by the isomorphism extending the
$i_{X_0\cup X_1}$ as above and write $[X_0\cup X_1]_X=X_0\oplus X_1$.
Similarly, if $X_i$, $i\in I$ are sub-inner-product-spaces of $X$ we denote
$\oplus_{i\in I}X_i=[\bigcup_{i\in I}X_i]_X$ if $X_i$, $i\in I$ are pairwise
orthogonal, that is, if
we have $X_i\perp X_j$ for all distinct $i$, $j\in I$.
For pairwise orthogonal sub-inner-product-paces $X_i$, $i\in I$ of $X$, we denote
with $\oplusbar^X_{i\in I}X_i$ the maximal linear subspace $X'$ of $X$ \st\ $X'$
contains $\oplus_{i\in I}X_i$ as a dense subset of $X'$. Thus, we
have $X=\oplusbar^X_{i\in I}X_i$ if $\oplus_{i\in I}X_i$ is dense in $X$. If it is
clear in which $X$ we are working we drop the superscript $X$ and simply write
$\oplusbar_{i\in I}X_i$.
An easy but very important fact for us is that
\begin{xitemize}
\xitem[orthonormal-basis]
if $X_i$, $i\in I$ are all
non-pathological with orthonormal bases $B_i$ for $X_i$, $i\in I$ and
$X=\oplusbar_{i\in I}X_i$, then $X$ is also non-pathological with
the orthonormal basis $\bigcup_{i\in I}B_i$.
\end{xitemize}
In the following we show that there are also pathological pre-Hilbert spaces $X$ with
$\dim(X)=d(X)=\lambda$ for an uncountable $\lambda$. For regular $\lambda$ this
is shown in \Thmof{main-thm} and the general case in \Corof{oplus-2}.
In \sectionof{char} we prove an algebraic characterization of pre-Hilbert spaces
with orthonormal bases.
In \sectionof{dim-den}, we give a proof of the theorem by Buhagiara, Chetcutib
and Weber asserting that $\kappa\leq\lambda$ are dimension and density
of a pre-Hilbert space if and only if $\lambda\leq\kappa^{\aleph_0}$ holds
(see \Thmof{dim-d-0}). \Corof{oplus-2} implies that there are pathological
pre-Hilbert spaces with $\dim(X)=\kappa$ and $d(X)=\lambda$ for all such $\kappa$
and $\lambda$.
In sections \ref{reflection}, \ref{singular}, \ref{FRP} we study the
set-theoretic reflection
of the pathology of pre-Hilbert spaces.
Our set-theoretic notation is quite standard. For the basic notions and notation
in set-theory we do not explain here, the reader may consult Jech \cite{millenium-book} or
Kunen \cite{kunen}.
\section{Pathological pre-Hilbert spaces constructed from a pre-ladder system}
\label{section-2}
For a cardinals $\lambda$, $\kappa$, let
\begin{xitemize}
\xitem[]
$E^\kappa_\lambda=\setof{\alpha<\lambda}{\cf(\alpha)=\kappa}$.
\end{xitemize}
For $E\subseteq E^\omega_\lambda$,
$\calA=\seqof{A_\alpha}{\alpha\in E}$ is said to be a {\it ladder system}
on $E$ if
\begin{xitemize}
\xitem[p-1] $A_\alpha\subseteq\alpha$ for all $\alpha\in E$;
\xitem[p-1-0]
$A_\alpha$ is cofinal in $\alpha$ for all $\alpha\in E$; and
\xitem[p-2] $\otp(A_\alpha)=\omega$ for all $\alpha\in E$.
\end{xitemize}
Note that, for any ladder system $\seqof{A_\alpha}{\alpha\in E}$, the sequence
$\seqof{A_\alpha}{\alpha\in E}$ is pairwise almost disjoint. We shall call a
sequence $\seqof{A_\alpha}{\alpha\in E}$ of countable subsets
of $\lambda$ a {\it pre-ladder system} if \xitemof{p-1} holds and \st\ it is
pairwise almost disjoint.
\begin{Thm}
\label{main-thm}
Suppose that $\kappa$ is a regular cardinal $>\omega_1$,
$E\subseteq E^\omega_\kappa$ is stationary and
$\seqof{A_\xi}{\xi\in E}$ is a pre-ladder system \st\
\begin{xitemize}
\xitem[m-4]
$A_\xi\subseteq\xi$ consists of successor ordinals for all $\xi\in E$.
\end{xitemize}
If $\seqof{\bbu_\xi}{\xi<\kappa}$ is a sequence of elements
of $\ell_2(\kappa)$ \st\
\begin{xitemize}
\xitem[m-5] $\bbu_\xi=\bbe^\kappa_\xi$ for all $\xi\in\kappa\setminus E$,
\xitem[m-6] $\supp(\bbu_\xi)=A_\xi\cup\ssetof{\xi}$ for all $\xi\in E$.
\end{xitemize}
Then, letting $U=\setof{\bbu_\xi}{\xi<\kappa}$, $X=[U]_{\ell_2(\kappa)}$ is a
pathological pre-Hilbert space of dimension and density $\kappa$.
\end{Thm}
\prf
We have $d(X)=\kappa$ since $\cls(X)=\ell_2(\kappa)$.
$\dim(X)\leq\dim(\ell_2(\kappa))=\kappa$ since $X$ is a sub-inner-product-space of
$\ell_2(\kappa)$ and $\dim(X)\geq\kappa$ since
$\setof{\bbu_\alpha}{\alpha\in\kappa\setminus E}$ is an orthonormal system
$\subseteq X$ of cardinality $\kappa$.
To show that $X$ is pathological, suppose toward a contradiction that
$\seqof{\bbb_\xi}{\xi<\kappa}$ is an orthonormal basis of $X$.
Let $\chi$ be a sufficiently large regular cardinal and let
$\seqof{M_\alpha}{\alpha<\kappa}$ be a continuously increasing sequence of
elementary submodels of $\calH(\chi)$ \st\
\begin{xitemize}
\xitem[m-7] $\cardof{M_\alpha}<\kappa$ for all $\alpha<\kappa$,
\xitem[m-8] $\seqof{A_\xi}{\xi\in E}$, $\seqof{\bbu_\xi}{\xi<\kappa}$,
$\seqof{\bbb_\xi}{\xi<\kappa}\in M_0$,
\xitem[m-9] $\kappa_\alpha=\kappa\cap M_\alpha\in\kappa$ for all
$\alpha<\kappa$ and $\seqof{\kappa_\alpha}{\alpha<\kappa}$ is a strictly
increasing sequence of ordinals cofinal in $\kappa$.
\end{xitemize}
For $\alpha<\kappa$, let
$H_\alpha=\ell_2(\kappa)\downarrow\kappa_\alpha$. Note that
$H_\alpha$ is a closed sub-inner-product-space of $\ell_2(\kappa)$ isomorphic to
$\ell_2(\kappa_\alpha)$.
Let $B_\alpha=\setof{\bbb_\xi}{\xi<\kappa_\alpha}$ for $\alpha<\kappa$.
\begin{Claim}
\label{claim-1}
$\supp(B_\alpha)\subseteq\kappa_\alpha$ and $B_\alpha$ is an
orthonormal basis of $H_\alpha$.
\end{Claim}
\prfofClaim
For $\xi<\kappa_\alpha$ $\bbb_\xi\in M_\alpha$ by \xitemof{m-8}. Hence
$\supp(\bbb_\xi)\in M_\alpha$. Since $\supp(\bbb_\xi)$ is countable it follows
that $\supp(\bbb_\xi)\subseteq\kappa\cap M_\alpha=\kappa_\alpha$. Thus we have
$\supp(B_\alpha)\subseteq\kappa_\alpha$.
For $\eta<\kappa_\alpha$, we have
\begin{xitemize}
\xitem[m-10]
$\calH(\chi)\models{}$``\,there are $A\in[\kappa]^{\aleph_0}$ and $c\in\fnsp{A}{K}$ \st\
$\sum_{\xi\in A}c(\xi)\bbb_\xi=\bbu_\eta$\,''
\end{xitemize}
since $\seqof{\bbb_\xi}{\xi<\kappa}$ is an orthonormal basis.
By \xitemof{m-8} and elementarity, it follows that
\begin{xitemize}
\xitem[m-11]
$M_\alpha\models{}$``\,there are $A\in[\kappa]^{\aleph_0}$ and $c\in\fnsp{A}{K}$ \st\
$\sum_{\xi\in A}c(\xi)\bbb_\xi=\bbu_\eta$\,''.
\end{xitemize}
Let $A\in[\kappa]^{\aleph_0}\cap M_\alpha$ and
$c\in\fnsp{A}{\kappa}\cap M_\alpha$ be witnesses of \xitemof{m-11}. Since
$A$ is countable we have $A\subseteq M_\alpha$.
Thus $\bbu_\eta$ is a limit of linear combinations of elements of $B_\alpha$.
It follows that $\cls\left([B_\alpha]_{H_\alpha}\right)
\supseteq\cls(\setof{\bbu_\xi}{\xi<\kappa_\alpha})=H_\alpha$.
\qedofClaim\qedskip
Since $E$ is stationary, there is an
$\alpha^*<\kappa$ \st\ $\kappa_{\alpha^*}\in E$. Let $\kappa^*=\kappa_{\alpha^*}$.
\begin{Claim}
\label{claim-2}
For any nonzero $\bba\in X$ represented as a linear combination of finitely
many elements of $U$ including (a non-zero multiple of)
$\bbu_{\kappa*}$, there is $\xi<\kappa^*$ \st\
$(\bba, \bbb_\xi)\not=0$.
\end{Claim}
\prfofClaim
Suppose that
\begin{xitemize}
\xitem[m-12]
$\bba=c\bbu_{\kappa*}+\sum_{\xi\in s}a_\xi\bbu_\xi+\sum_{\eta\in t}b_\eta\bbu_\eta$
\end{xitemize}
where $s\in[\kappa^*]^{<\aleph_0}$,
$t\in[\kappa\setminus(\kappa^*+1)]^{<\aleph_0}$ and
$c$, $a_\xi$, $b_\eta\in K\setminus\ssetof{0}$ for $\xi\in s$ and $\eta\in t$.
Since $\supp(\bbu_\xi)$, $\xi\in s$ are bounded subsets of $\kappa^*$ and
$\supp(\bbu_\eta)\cap\kappa^*$, $\eta\in t$ are finite,
$\supp(\bba)\cap\kappa^*$ contains an end-segment of $A_{\kappa^*}$ and in
particular it is non-empty.
Thus
$\bba\downarrow\kappa^*$ is a non-zero element of $H_{\alpha^*}$. By \Claimof{claim-1}, it
follows that there is $\xi<\kappa^*$ \st\
$(\bba,\bbb_\xi)=(\bba\downarrow\kappa^*,\bbb_\xi)\not=0$.
\qedofClaim\qedskip
By \Claimabove, there are no $\bba\in X$ as in the assertion of \Claimabove\
among $\bbb_\xi$, $\xi<\kappa$. It follows that
$\kappa^*\not\in\bigcup\setof{\supp(\bbb_\xi)}{\xi<\kappa}$. This is a
contradiction to the assumption that $\setof{\bbb_\xi}{\xi<\kappa}$ is an
orthonormal basis of $X$ and hence of $\ell_2(\kappa)$.
\qedofThm\qedskip
The construction of $X$ in \Thmabove\ can be further modified to obtain the
following additional property of $X$: there is
$\calS\subseteq[U]^{<\kappa}$
\st\
\begin{xitemize}
\xitem[m-2] $\calS$ is a stationary subset of $[U]^{<\kappa}$,
\xitem[m-3] for all $A$, $B\in\calS$ with
$A\subseteq B$, $[A]_X$ is an orthogonal direct summand of $[B]_X$.
\end{xitemize}
For \xitemof{m-2} and \xitemof{m-2}, we can just
start from a stationary and co-stationary $E$ and let
\begin{xitemize}
\xitem[p-14]
$\calS=\setof{U_\gamma}{\gamma\in\kappa\setminus E}$
\end{xitemize}
where $U_\gamma=\setof{\bbu_\xi}{\xi<\gamma}$.
Then $U$ and this $\calS$ are as desired:
$\calS$ is a stationary subset of $[U]^{<\kappa}$ by the choice of $E$.
For $U_{\gamma_0}$, $U_{\gamma_1}\in\calS$
with $\gamma_0<\gamma_1$,
we have
$\bbu_\xi\downarrow(\kappa\setminus\gamma_0)\in[U_{\gamma_1}]_X$ for all
$\xi\in\gamma_1\setminus\gamma_0$.
Hence
\begin{xitemize}
\xitem[]
$[U_{\gamma_1}]_X
=[U_{\gamma_0}]_X
\oplus[\setof{\bbu_\xi\downarrow(\kappa\setminus\gamma_0)}{
\xi\in\gamma_1\setminus\gamma_0}]_X$.
\end{xitemize}
\Thmof{main-thm} applied to $\kappa=\omega_1$ gives pathological pre-Hilbert
spaces with interesting properties. Note that for a stationary subset $E$ of
$\omega_1$ there is a \po\ which ``shoots'' a club subset inside $E$ while
preserving all cardinals (e.g. the shooting a club forcing with finite
conditions).
If $X$ is a pre-Hilbert space constructed as in \Thmof{main-thm} for stationary
and co-stationary $E\subseteq E^\omega_{\omega_1}$ and a pre-ladder system
on $E$, letting $U\subseteq\ell_2(\omega_1)$ be the generator of $X$ as in
\Thmof{main-thm}, we have that
$X\downarrow \alpha$ is non-pathological for all $\alpha<\omega_1$ since
$X\downarrow\alpha$ is separable.
If we shoot a club subset of $\omega_1\setminus E$,
we obtain a continuously increasing sequence of non-pathological
sub-inner-product-spaces $\seqof{X_\alpha}{\alpha<\omega_1}$ of $X$ \st\
$\bigcup_{\alpha<\omega_1} X_\alpha=X$ and that $X_\alpha$ is an orthogonal
direct summand of $X_{\alpha+1}$ for all $\alpha<\omega_1$. It follows that $X$
is non pathological in such a generic extension. Thus we obtain:
\begin{Cor}
\label{cor-1}
\assertof{1} There is a pathological pre-Hilbert space $X$ of dimension and
density $\aleph_1$ \st\ there is a \po\ $\poP$ preserving all cardinals \st\
$\forces{\poP}{X\xmbox{ has an ortho\-normal basis}}$.\smallskip
\assertof{2} There is a pathological pre-Hilbert space $X$ of dimension and
density $\aleph_1$ \st, for any \po\ $\poP$ preserving $\omega_1$, we
have $\forces{\poP}{X\mbox{ is pathological\/}}$.
\end{Cor}
\prf A proof of \assertof{1} is already explained above.
For \assertof{2}, we can use the club set $E=E^\omega_{\omega_1}$ in the
construction of the
proof of \Thmof{main-thm}. The pre-Hilbert space $X$ constructed in the proof of
\Thmof{main-thm} with this $E$ is as desired: since $E^*$ remains stationary in
any generic extension preserving $\omega_1$, $X$ remains pathological there.
\qedofCor
\qedskip
\section{A Characterization of the non-pathology}
\label{char}
Using some of the ideas in the proof of \Thmof{main-thm}, we obtain an
``algebraic'' characterization of pre-Hilbert spaces with
orthonormal bases (see \Thmof{T-char}).
This characterization is used in later sections.
\begin{Lemma}
\label{L-char-1}
Suppose that $X$ is a pre-Hilbert space and $X$ is a dense sub-inner-product-space
of $\ell_2(S)$. If $\calB\subseteq X$ is an orthonormal basis then, for any
$S_0\subseteq S$, there is an $A\subseteq S$ \st\
$S_0\subseteq A$, $\cardof{A}=\cardof{S_0}+\aleph_0$,
$X\downarrow A$ is a dense sub-inner-product-space of $\ell_2(S)\downarrow A$,
$\calB_A=\setof{\bbb\in\calB}{\supp(\bbb)\subseteq A}$ is an orthonormal basis
of $X\downarrow A$ and $\calB^-_A=\calB\setminus\calB_A$ is an orthonormal
basis of $X\downarrow (S\setminus A)$. In particular, we have
$X=(X\downarrow A)\oplus(X\downarrow (S\setminus A))$.
\end{Lemma}
\prf Let $\chi$ be a sufficiently large regular cardinal and let
$M\prec\calH(\chi)$ be \st\
\begin{xitemize}
\xitem[chr-1]
$K$, $X$, $S$, $\calB\in M$, $S_0\subseteq M$ and $\cardof{M}=\cardof{S_0}+\aleph_0$.
\end{xitemize}
We show that $A=S\cap M$ is as desired. Since $S_0\subseteq M$, we have
$S_0\subseteq S\cap M=A$.
Since $\calB$ is also an orthonormal basis of $\ell_2(S)$,
we have
\begin{xitemize}
\xitem[chr-1-0]
$\calH(\chi)\models$``there is a $B\in[\calB]^{\aleph_0}$ and $c\in\fnsp{B}{K}$
\st\ $\sum_{\bbu\in B}c(\bbu)\bbu=\bbe^S_s$\,''
\end{xitemize}
for all $s\in A$.
By elementarity, it follows that
\begin{xitemize}
\xitem[chr-2]
$M\models$``there is a $B\in[\calB]^{\aleph_0}$ and $c\in\fnsp{B}{K}$
\st\ $\sum_{\bbu\in B}c(\bbu)\bbu=\bbe^S_s$\,''.
\end{xitemize}
Let $B\in[\calB]^{\aleph_0}\cap M$ and $c\in\fnsp{B}{K}\cap M$ be witnesses
of \xitemof{chr-2}. By $B\in M$ and since $B$ is countable, we have $B\subseteq M$.
For each $\bbb\in B$, since $\bbb\in M$ and $\supp(\bbb)$ is countable, we have
$\supp(\bbb)\subseteq M$.
It follows that $B\subseteq\calB_A$ and
$\bbe^S_s\in\cls_{\ell_2(S)\downarrow A}(\calB_A)$ for all $s\in A$.
Thus
$\setof{\bbe^S_s}{s\in A}\subseteq
\cls_{\ell_2(S)\downarrow A}(\calB_A)$
and hence
\begin{xitemize}
\xitem[chr-3]
$\cls_{\ell_2(S)\downarrow A}(\calB_A)=\ell_2(S)\downarrow A$.
\end{xitemize}
Since $\calB_A\subseteq X\downarrow A$, \xitemof{chr-3} implies that
$X\downarrow A$ is dense in $\ell_2(S)\downarrow A$.
For any $\bbb\in\calB^-_A$, we have $\supp(\bbb)\subseteq S\setminus A$:
otherwise, $\bbb\downarrow A\not=\bbzero_{\ell_2(S)}$. By \xitemof{chr-3} it
follows that there is a $\bbc\in\calB_A$ \st\
$(\bbb,\bbc)=(\bbb\downarrow A,\bbc)\not=0$. This is a contradiction to the
orthonormality of $\calB$.
Thus $\calB^-_A\subseteq X\downarrow(S\setminus A)$.
$\calB^-_A$ is an
orthonormal basis of $X\downarrow(S\setminus A)$: similarly to the argument
above, it is enough to show
that, for each $s\in S\setminus A$, $\bbe^S_s$ can be obtained as a (possibly
infinite) sum of elements in $\calB^-_A$ in $\ell_2(S)\downarrow(S\setminus A)$.
Since $\calB$ is an orthonormal basis
of $\ell_2(S)$,
we have $\bbe^S_s=\sum_{\bbb\in B}(\bbe^S_s,\bbb)\bbb$ where
$B=\setof{\bbb\in\calB}{(\bbe^S_s,\bbb)\not=0}$. Since
$\supp(\bbb)\not\subseteq A$ for all $\bbb\in B$, we have $B\subseteq\calB^-_A$.
\qedofLemma
\begin{Lemma}
\label{L-char-2}
Suppose that $X$ is a non-pathological pre-Hilbert space and $X$ is a dense
sub-inner-product space of $\ell_2(S)$ for some infinite set $S$. Then there is a
partition $\calP$ of
$S$ into countable subsets \st\ $X=\oplusbar_{A\in\calP}X\downarrow A$.
\end{Lemma}
\prf Let $\cardof{S}=\kappa$ and $\calB=\setof{\bbb_\alpha}{\alpha<\kappa}$ be an
orthonormal basis
of $X$. Let $S=\setof{s_\alpha}{\alpha<\kappa}$.
We define by induction on $\alpha\in\kappa$ the sequences
$\seqof{S_\alpha}{\alpha<\kappa}$ and $\seqof{A_\alpha}{\alpha<\kappa}$ of
subsets of $S$ \st:
\begin{xitemize}
\xitem[chr-4] $S_0=S$;
\xitem[chr-5] $A_\alpha\in[S_\alpha]^{\aleph_0}$ for all $\alpha\in\kappa$;
\xitem[chr-6] $S_{\alpha+1}=S_\alpha\setminus A_\alpha$ for all $\alpha\in\kappa$;
\xitem[chr-7] $S_\gamma=\bigcap_{\alpha<\gamma}S_\alpha$ for all limit
$\gamma\in\kappa$;
\xitem[chr-8] $s_\alpha\in\bigcup_{\beta\leq\alpha}A_\alpha$ for
all $\alpha\in\kappa$;
\xitem[chr-9] $\calB\cap(X\downarrow S_\alpha)$ is an orthonormal basis of
$X\downarrow S_\alpha$ for
all $\alpha\in\kappa$; and
\xitem[chr-10] $\calB\cap(X\downarrow A_\alpha)$ is an orthonormal basis of
$X\downarrow A_\alpha$ for
all $\alpha\in\kappa$.
\end{xitemize}
The construction of $A_\alpha$ and $S_{\alpha+1}$ is possible by \Lemmaof{L-char-1}.
We just have to check that the construction of $S_\gamma$ at limit steps
$\gamma<\kappa$ works.
For a limit $\gamma<\kappa$ we have $S_\gamma=\bigcap_{\alpha<\gamma}S_\alpha$ by
\xitemof{chr-7}. For each $s\in S_\gamma$ and $\alpha<\gamma$ there are a countable
$B_\alpha\subseteq\calB\cap(X\downarrow S_\alpha)$ and a sequence
$\seqof{a^\alpha_\bbb}{\bbb\in B_\alpha}$ in $K$ \st\
\begin{xitemize}
\xitem[chr-11]
$\bbe^S_s=\sum_{\bbb\in B_\alpha}a^\alpha_\bbb\bbb$.
\end{xitemize}
By the uniqueness of the representation of elements of $\ell_2(S)$ as an infinite
linear combination of elements of $\calB$. It follows that there is a countable
$B^*\subseteq\calB\cap (X\downarrow S_\alpha)$ and a sequence
$\seqof{a_\bbb}{\bbb\in B^*}$ \st\
$B_\alpha=B^*$ for all $\alpha<\gamma$ and $a^\alpha_\bbb=a_\bbb$ for all
$\alpha<\gamma$ and $\bbb\in B^*$. It follows that
$B^*\subseteq\calB\cap(X\downarrow B_\gamma)$.
Thus, we have
$\bbe^S_s\in\cls_{\ell_2(S)}[\calB\cap(X\downarrow S_\gamma)]$, for all
$s\in S_\gamma$. It follows that
$\calB\cap(X\downarrow S_\gamma)$ is an orthonormal basis of
$X\downarrow S_\gamma$, i.e.\ $S_\gamma$ satisfies \xitemof{chr-9}.
$\calP=\setof{A_\alpha}{\alpha<\kappa}$ is then a partition of $S$ as desired. \qedofLemma
\begin{Thm}
\label{T-char}
Suppose that $X$ is a pre-Hilbert space. Then $X$ is non-pathological if and only
if there are separable sub-inner-product-spaces $X_\alpha$, $\alpha<\delta$ of
$X$ \st\ $X=\oplusbar_{\alpha<\delta}X_\alpha$.
\end{Thm}
\prf If $X$ is separable then the claim is trivial with $\delta=1$.
Suppose that $X$ is non-separable.
If $X$ is non-pathological then there are separable
sub-inner-product-spaces $X_\alpha$, $\alpha<\kappa$ for $\kappa=d(X)$ with
$X=\oplusbar_{\alpha<\kappa}X_\alpha$ by \Lemmaof{L-char-2}.
Conversely, if there are $X_\alpha$, $\alpha<\delta$ as above, then
each $X_\alpha$ for $\alpha\in\delta$ has an orthonormal basis $B_\alpha$.
$\calB=\bigcup_{\alpha<\delta}B_\alpha$ is then an orthonormal basis of $X$. \qedofThm
\begin{Lemma}
\label{filtration}
Suppose that $X$ is a non-pathological pre-Hilbert space and $X$ is a dense
sub-inner-product space of $\ell_2(S)$ for some uncountable set $S$. Then there
is a filtration $\seqof{S_\alpha}{\alpha<\kappa}$ of $S$ for
$\kappa=\cf(\cardof{S})$ \st\
$X\downarrow S_\alpha$ is an orthogonal direct summand of $X$ for all
$\alpha<\kappa$.
\end{Lemma}
\prf By \Lemmaof{L-char-2} there is a partition $\calP$ of $S$ into countable
subsets \st\ $X=\oplusbar_{P\in\calP}X\downarrow P$. Let
$\seqof{\calP_\alpha}{\alpha<\kappa}$ be a filtration of $\calP$ and let
$S_\alpha=\bigcup\calP_\alpha$ for $\alpha<\kappa$. Then
$\seqof{S_\alpha}{\alpha<\kappa}$ is as desired. \qedofLemma
\qedskip
The following Lemmas are used in \sectionof{FRP}. We put them together here since they
stand in a similar context as that of previous results in this section.
\begin{Lemma}
\label{summand}
Suppose that $X$ is a pre-Hilbert-space which is a dense
sub-inner-product-space of $\ell_2(S)$. For $S'\subseteq S$ \st\
\begin{xitemize}
\xitem[chr-15-0]
$X\downarrow S'$ is dense in $\ell_2\downarrow S'$,
\end{xitemize}
$X\downarrow S'$ is not an
orthogonal direct summand of $X$ if and only if there is $\bba\in X$ \st\
\begin{xitemize}
\xitem[chr-16]
$\bba\downarrow S'\not\in X$.
\end{xitemize}
\end{Lemma}
\prf If there is no $\bba\in X$ with \xitemof{chr-16} then we clearly have
$X=(X\downarrow S')\otimes(X\downarrow S\setminus S')$.
Suppose that $\bba\in X$ satisfies \xitemof{chr-16}. Note that then we have
$\supp(\bba)\nsubseteq S'$ and $\supp(\bba)\cap S'\not=\emptyset$.
Suppose toward a contradiction that there is a sub-inner-product space $X''$ of
$X$ \st\
\begin{xitemize}
\xitem[chr-17]
$X=(X\downarrow S')\oplus X''$.
\end{xitemize}
Then there are $\bba'\in X\downarrow S'$
and $\bba''\in X''$ \st\ $\bba=\bba'+\bba''$. So $\bba''=\bba-\bba'$. It follows
that $\bba''\downarrow S'\not=\bbzero$ by \xitemof{chr-16}. By \xitemof{chr-15-0},
there is some $\bbb\in X\downarrow S'$ \st\
$(\bba'',\bbb)=(\bba''\downarrow S',\bbb)\not=0$. This is a contradiction to
\xitemof{chr-17}.\qedofLemma
\begin{Lemma}
\label{L-dense-0}
Suppose that $X$ is a pre-Hilbert-space which is a dense
sub-inner-product-space of $\ell_2(S)$. For a sufficiently large regular $\chi$
and $M\prec\calH(\chi)$ with $K$, $X$, $S\in M$ , $X\downarrow (S\cap M)$ is
dense in $\ell_2(S)\downarrow(S\cap M)$.
\end{Lemma}
\prf For $s\in S\cap M$, we have
\begin{xitemize}
\xitem[chr-18] $\calH(\chi)\models$ there are $A\in[X]^{\aleph_0}$ and $c\in\fnsp{A}{K}$
\st\ $\bbe^S_s=\sum_{\bbb\in A}c(\bbb)\bbb$.
\end{xitemize}
By elementarity it follows that
\begin{xitemize}
\xitem[chr-19] $M\models$ there are $A\in[X]^{\aleph_0}$ and $c\in\fnsp{A}{K}$
\st\ $\bbe^S_s=\sum_{\bbb\in A}c(\bbb)\bbb$.
\end{xitemize}
Let $A\in[X]^{\aleph_0}\cap M$ and $c\in\fnsp{A}{K}\cap M$ be witnesses of
\xitemof{chr-19}. By the countability of $A$ we have $A\subseteq M$ and, for each
$\bbb\in A$, $\supp(\bbb)\subseteq M$ since $\supp(\bbb)$ is countable.
This shows that $\bbe^S_s\in\cls(X\downarrow (S\cap M))$.
\qedofLemma
\begin{Lemma}
\label{L-dense-1}
Suppose that $X$ is a pre-Hilbert-space which is a dense
sub-inner-product-space of $\ell_2(S)$ for an uncountable S. Then there is a
filtration $\seqof{S_\alpha}{\alpha<\kappa}$ of $S$ \st\
$X\downarrow S_\alpha$ dense in $\ell_2(S)\downarrow S_\alpha$ for all $\alpha<\kappa$
\end{Lemma}
\prf Let $\chi$ be a sufficiently large regular cardinal. Let
$\kappa=\cf(\cardof{S})$ and let $\seqof{M_\alpha}{\alpha<\kappa}$ be a
continuously increasing sequence of elementary submodels of $\calH(\chi)$ \st\
\begin{xitemize}
\xitem[chr-20] $K$, $X$, $S\in M_0$,
\xitem[chr-21] $\cardof{M_\alpha}<\cardof{S}$ for all $\alpha<\kappa$,
\xitem[chr-22] $S\subseteq \bigcup_{\alpha<\kappa}M_\alpha$.
\end{xitemize}
Letting $S_\alpha=S\cap M_\alpha$ for $\alpha<\kappa$, the sequence
$\seqof{S_\alpha}{\alpha<\kappa}$ is as desired by \Lemmaof{L-dense-0}.
\qedofLemma
\section{Dimension and density of pre-Hilbert spaces}
\label{dim-den}
The proof of \Lemmaof{halmos} actually yields pre-Hilbert spaces
of the following combinations of dimension and density:
\begin{Lemma}[A generalization of \bfLemmaof{halmos}]
\label{halmos-0}
For any cardinal $\kappa$ and $\lambda$ with
$\kappa<\lambda\leq\kappa^{\aleph_0}$, there are (pathological) pre-Hilbert spaces of
dimension $\kappa$ and density $\lambda$.\qed
\end{Lemma}
On the other hand if $\kappa^{\aleph_0}<\lambda$ there are no pre-Hilbert space
$X$ with dimension $\kappa$ and density $\lambda$.
\begin{Prop}{\rm (David Buhagiara, Emmanuel Chetcutib and Hans Weber
\cite{buhagiara-etal}, see also \cite{farah})}
\label{dim-d}
For any pre-Hilbert space $X$, we have
$d(X)\leq\cardof{X}\leq (\dim(X))^{\aleph_0}$.
\end{Prop}
\prf Let $X$ be a pre-Hilbert space. We may assume \wolog\ that $X$ is a dense
sub-inner-product-space of the Hilbert space $\ell_2(\kappa)$ for
$\kappa=d(X)>\dim(X)\geq\aleph_0$.
Let $\calB=\seqof{\bbb_\xi}{\xi<\kappa}$ be a maximal orthonormal system in $X$
and
$D=\bigcup\setof{\supp(\bbb_\xi)}{\xi<\kappa}$. By the assumption we have
$\cardof{D}=\kappa$.
\begin{Claim}
For any distinct $\bba_0$, $\bba_1\in X$ we have
$\bba_0\restr D\not=\bba_1\restr D$.
\end{Claim}
\prf Suppose that there were $\bba_0$, $\bba_1\in X$ \st\ $\bba_0\not=\bba_1$ but
$\bba_0\restr D=\bba_1\restr D$. Then $\bba_2=\bba_1-\bba_0$ would be a non-zero element
of $X$ orthogonal to all $\bbb_\xi$, $\xi<\kappa$. This is a contradiction to the
maximality of $\calB$.\qedofClaim\qedskip
Let $\mapping{\varphi}{\ell_2(D)}{X}$ be defined by
\begin{xitemize}
\xitem[m-13] $\varphi(\bbc)=\left\{\,
\begin{array}{@{}ll}
\mbox{the unique }\bba\in X\mbox{ \st\ }\bbc=\bba\restr D;
&\mbox{if there is such }\bba\in X,\\[\jot]
\bbzero; &\mbox{otherwise}
\end{array}
\right.$
\end{xitemize}
for $\bbc\in\ell_2(D)$. $\varphi$ is well-defined by \Claimabove\ and it is
surjective. Thus we have
\begin{xitemize}
\xitem[]
$d(X)\leq\cardof{X}\leq \cardof{\ell_2(D)}=(\dim(X))^{\aleph_0}$.\\\qedofProp
\end{xitemize}
The following theorem will be yet extended in \Corof{cor-3}.
\begin{Thm}
\label{dim-d-0}
For any cardinal $\kappa\leq\lambda$ there is a pre-Hilbert space of dimension
$\kappa$ and density $\lambda$ if and only if $\lambda\leq \kappa^{\aleph_0}$ holds.
\end{Thm}
\prf For $\kappa=\lambda$, $\ell_2(\kappa)$ is an example of pre-Hilbert space
of dimension and density $\kappa$ and $\lambda$. If
$\kappa<\lambda<\kappa^{\aleph_0}$, \Lemmaof{halmos-0} provides an example.
The converse also holds by \Propof{dim-d}.\qedofThm
\section{Orthogonal direct sum}
\label{direct-sum}
In a variety $\calV$ of algebraic structures it can happen that there is a non
free algebra $A\in\calV$ \st\ the product $A\otimes F$ is free for some free algebra
$F\in\calV$. For example, it is known that there are non-free projective algebra $B$
in the variety $\calB$ of Boolean algebras but free product $B\oplus F$ of any
projective algebra $B$ with a sufficiently large free Boolean algebra $F$ is
free.
In contrast, the pathology of pre-Hilbert space remains by orthogonal direct sum.
\begin{Thm}
\label{T-direct-sum}
For any pre-Hilbert spaces $X_0$ and $X_1$, the orthogonal direct sum
$X_0\oplus X_1$ is pathological if and only if at least one of $X_0$ and $X_1$
is pathological.
\end{Thm}
\prf If $X_0$ and $X_1$ are both non-pathological and $\calB_0$ and $\calB_1$ are
orthonormal bases of $X_0$ and $X_1$ respectively, then
$\calB_0\times\ssetof{\bbzero_{X_2}}\cup\ssetof{\bbzero_{X_1}}\times\calB_1$ is an
orthonormal basis of $X_0\oplus X_1$.
Conversely, suppose that $X_0\oplus X_1$ is non-pathological and $\calB$ is an
orthonormal basis of $X=X_0\oplus X_1$. \Wolog, we may assume that there are $S$,
$S^0$, $S^1$ \st\ $S=S^0\cup S^1$, $S^0\cap S^1=\emptyset$, $X_i$ is a dense
sub-inner-product-space of $\ell_2(S)\downarrow S^i$ for $i\in 2$ and
$X_0\oplus X_1=[X_0\cup X_1]_{\ell_2(S)}$.
By \Lemmaof{L-char-2}, there is a partition $\seqof{A_\alpha}{\alpha<\delta}$ of
$S$ into countable sets \st\ $X=\oplusbar_{\alpha\in\kappa}X\downarrow A_\alpha$. We
may assume that the elements of partition $A_\alpha$ in the proof of
\Lemmaof{L-char-2} is obtained in the construction as the intersection of
$S_\alpha$ (in the proof of \Lemmaof{L-char-2}) and countable
$M_\alpha\prec\calH(\chi)$ \st\ $\calB$, $X_0$, $X_1$, $S^0$,
$S^1\ctenten\in M_\alpha$.
Then as in the proof of \Lemmaof{L-char-1}, we have
$X\downarrow A_\alpha= (X_0\downarrow(A_{0,\alpha}))\oplus (X_1\downarrow(A_{1,\alpha}))$
where $A_{i,\alpha}=A_\alpha\cap S^i$ for $i\in 2$.
Let
$\calP_i=\setof{A_{i,\alpha}}{\alpha<\kappa,\,A_{i,\alpha}\not=\emptyset}$
for $i\in 2$. Then $X_i=\oplusbar_{P\in\calP_i}X_i\downarrow P$ for
$i\in 2$. Thus $X_i$, $i\in 2$ are non-pathological.
\qedofThm
\iffalse
\begin{Thm}
\label{oplus-1}
Let $X$ be a pre-Hilbert space constructed in the proof of \Lemmaof{halmos}.
Then, for any cardinal pre-Hilbert space $Y$, the orthogonal direct sum
$X\oplus Y$ is a pathological pre-Hilbert space.
\end{Thm}
\prf Suppose that $X$ is the pre-Hilbert space with dimension $\aleph_0$ and
density $\lambda$ with
$\aleph_1\leq\lambda\leq 2^{\aleph_0}$ constructed in the proof of
\Lemmaof{halmos} and $Y$ a pre-Hilbert space with $d(Y)=\mu$.
Let $\kappa=\sup\ssetof{\lambda,\mu}$.
$X\oplus Y$ may be seen as a sub-inner-product space
of $\ell_2(S)$ for $S=\omega\cup S_0\cup S_1$ where
\begin{xitemize}
\xitem[x-0] $\omega$, $S_0$, $S_1$ are pairwise disjoint sets with
$\cardof{S_0}=\lambda$ and $\cardof{S_1}=\mu$;
\xitem[x-1] $X\oplus Y=[U]_{\ell_2(S)}$ where
$U=\setof{\bbe^S_n}{n\in\omega}\cup\setof{\bbf_\bbx}{\bbx\in B}
\cup\setof{\bbg_\xi}{\xi\in\delta}$
\xitem[x-2]
$B\subseteq (\ell_2(U)\downarrow\omega)\setminus\setof{\bbe^S_n}{n\in\omega}$
and $B\cup\setof{\bbe^S_n}{n\in\omega}$ is a linear basis of
$\ell_2(S)\downarrow\omega$;
\xitem[x-3] There is a surjection $\mapping{f}{B}{S_0}$ \st, for each $\bbx\in B$,
$\bbf_\bbx\in\ell_2(S)$ is \st\ $\supp(\bbf_\bbx)=\supp(\bbx)\cup\ssetof{f(\bbx)}$,
$\bbf_\bbx\restr\omega= \bbx\restr\omega$ and $\bbf_\bbx(f(\bbx))=1$;
\xitem[x-4] $\setof{\bbg_\xi}{\xi<\delta}$ is a linear basis of
$X\oplus Y\downarrow S_1\cong Y$.
\end{xitemize}
Clearly we have $\cls(X)=\ell_2(S)$ and hence $d(X)=\kappa$.
Suppose, toward a contradiction, that $\seqof{\bbb_\alpha}{\alpha<\kappa}$ is an
orthonormal basis of $X$.
Let $\chi$ be a sufficiently large regular cardinal and let $M\prec\calH(\chi)$
be countable \st\
\begin{xitemize}
\xitem[] $\seqof{\bbb_\alpha}{\alpha<\kappa}$, $S_0$, $S_1$, $B$, $f$,
$\seqof{\bbf_\bbx}{\bbx\in B}$, $\seqof{\bbg_\xi}{\xi\in\delta}\in M$.
\end{xitemize}
Note that, similarly to \Claimof{claim-1}, we can prove that
\begin{xitemize}
\xitem[x-5]
$\setof{\bbb_\alpha}{\alpha\in\kappa\cap M}$ is an orthonormal basis of
$X\downarrow (S\cap M)$.
\end{xitemize}
Similarly to the proof of \Thmof{main-thm}, the following claim leads to a
contradiction:
\begin{Claim}
For any $\bbx^*\in B\setminus M$ and for any $\bbu\in X$ represented by a
linear combination of elements of $U$ containing (a non-zero multiple of)
$\bbf_{\bbx^*}$, there is $\alpha\in\kappa\cap M$
\st\ $(\bbu,\bbb_\alpha)\not=0$.
\end{Claim}
\prfofClaim Suppose that
\begin{xitemize}
\xitem[x-6]
$\bbu=a\bbf_{\bbx^*}+\sum_{\bbx\in u}a_\bbx\bbf_\bbx+\sum_{n\in v}b_n\bbe^S_n
+\sum_{\xi\in w}c_\xi\bbg_\xi$
\end{xitemize}
where $u\in[B\setminus\ssetof{\bbx^*}]^{<\aleph_0}$, $v\in[\omega]^{<\aleph_0}$,
$w\in[\delta]^{<\aleph_0}$, $a$, $a_\bbx$, $b_n$,
$c_\xi\in K\setminus\ssetof{0}$ for $\bbx\in u$, $n\in v$ and $\xi\in w$.
We have
\begin{xitemize}
\xitem[x-7]
$\bbu\downarrow (S\cap M)=\left(a \bbx^*+\sum_{\bbx\in u\cap M}a_\bbx\bbf_\bbx
+\sum_{n\in v}b_n\bbe^S_n\right)
+\sum_{\xi\in w\cap M}c_\xi\bbg_\xi
$.
\end{xitemize}
By \xitemof{x-0} $\sim$ \xitemof{x-4} and by the choice of $\bbx^*$, it follows that
$\bbu\downarrow(S\cap M)\not=\bbzero$. By \xitemof{x-5}, there is
$\alpha\in\kappa\cap M$ \st\
\begin{xitemize}
\xitem[]
$(\bbu,\bbb_\alpha)=(\bbu\downarrow (S\cap M), \bbb_\alpha)\not=0$.
\end{xitemize}
\qedofClaim\\
\qedofThm
\fi
\begin{Cor}
\label{oplus-2}
For any uncountable cardinal $\lambda$, there is a pathological pre-Hilbert space $Z$
of dimension and density $\lambda$.
\end{Cor}
\prf Let $X$ be any pathological pre-Hilbert space with density $\aleph_1$.
Then $Z=X\oplus\ell_2(\lambda)$ has
dimension and density $\lambda$. $Z$ is pathological by \Thmof{T-direct-sum}. \qedofCor
\begin{Cor}
\label{cor-2}
For any infinite cardinals $\kappa$ and $\lambda$ with
$\kappa\leq \lambda\leq \kappa^{\aleph_0}$ there is a pathological pre-Hilbert space of
dimension $\kappa$ and density $\lambda$.
\end{Cor}
\prf By \Lemmaof{halmos-0} and \Corof{oplus-2}.\qedofCor
\begin{Cor}
\label{cor-3}
\assertof{1}
For any infinite cardinals $\kappa$ and $\lambda$ with
$\kappa\leq \lambda\leq \kappa^{\aleph_0}$ there is a pathological pre-Hilbert
space of
dimension $\kappa$ and density $\lambda$ \st\ there is a \po\ $\poP$ preserving
all cardinals \st\ $\forces{\poP}{X\mbox{ is non-pathological\/}}$. \smallskip
\assertof{2} For any infinite cardinals $\kappa$ and $\lambda$ with
$\kappa\leq \lambda\leq \kappa^{\aleph_0}$ there is a pathological pre-Hilbert
space of
dimension $\kappa$ and density $\lambda$ which remains pathological in any
generic extension preserving $\omega_1$.
\end{Cor}
\prf The pre-Hilbert space of the form $X\oplus Y$ will do where $X$ is as in
\Corof{cor-1},\assertof{1} or \assertof{2} and $Y$ is as in
\Corof{cor-2}.\qedofCor
\section{Reflection and non-reflection of pathology}
\label{reflection}
For any pre-Hilbert space $X$ all sub-inner-product-spaces of $X$ of density
$\aleph_0$ are non-pathological. If $S\subseteq E^\omega_{\omega_2}$ is
non-reflecting stationary set, then the sub-inner-product-space of
$\ell_2(\omega_2)$ constructed from a ladder system on $S$,
there are club many $\beta<\omega_2$ \st\
$X\downarrow \beta$ is non-pathological.
A similar non-reflection theorem holds at an arbitrary regular uncountable
cardinal $\kappa>\aleph_1$ under a weak form of the square principle at $\kappa$.
For a regular cardinal $\kappa$, $\ADS^-(\kappa)$ is the assertion
that there is a stationary set $S\subseteq E^\omega_\kappa$ and a sequence
$\seqof{A_\alpha}{\alpha\in S}$ \st
\begin{xitemize}
\xitem[ads-0] $A_\alpha\subseteq\alpha$ and $\otp(A_\alpha)=\omega$ for all
$\alpha\in S$;
\xitem[ads-1] for any $\beta<\kappa$, there is a mapping
$\mapping{f}{S\cap\beta}{\beta}$ \st\ $f(\alpha)<\sup(A_\alpha)$ for all
$\alpha\in S\cap\beta$ and $A_\alpha\setminus f(\alpha)$,
$\alpha\in S\cap\beta$ are pairwise disjoint
\end{xitemize}
(for more about $\ADS^-(\kappa)$, see Fuchino, Juha\'asz, Soukup,
Szentmikl\'ossy, Usuba \cite{fuchino-juhasz-etal} and Fuchino, Sakai, Soukup \cite{more} ).
We shall call $\seqof{A_\alpha}{\alpha\in S}$ as above
an $\ADS^-(\kappa)$-sequence. Note that it follows from \xitemof{ads-0} and
\xitemof{ads-1} that $A_\alpha$, $\alpha\in S$ are pairwise almost disjoint.
Under $\ADS^-(\kappa)$, we may further assume that the $\ADS^-(\kappa)$-sequence
$\seqof{A_\alpha}{\alpha\in S}$ satisfies that
$A_\alpha\subseteq\alpha\setminus\Lim$ for all $\alpha\in S$.
Since an $\ADS^-(\kappa)$-sequence is a pre-ladder system, we can apply the
construction of pre-Hilbert spaces in the proof of \Thmof{main-thm} to the
sequence and obtain the following:
\begin{Thm}
\label{T-refl-1}
Assume that $\ADS^-(\kappa)$ holds for a regular cardinal $\kappa>\omega_1$.
Then there is a pathological dense sub-inner-product-space $X$ of $\ell_2(\kappa)$
\st\ $X\downarrow \beta$ is non-pathological for all $\beta<\kappa$. Furthermore
for any regular $\lambda<\kappa$,
$\setof{S\in[\kappa]^\lambda}{X\downarrow S\mbox{ is non-pathological\/}}$
contains a club subset of $[\kappa]^\lambda$.
\end{Thm}
\prf
Let $\seqof{A_\alpha}{\alpha\in E}$ be an $\ADS^-(\kappa)$-sequence on a
stationary $E\subseteq E^\omega_\kappa$. Let
$\seqof{\bbu_\xi}{\xi<\kappa}$ be a sequence of elements of $\ell_2(\kappa)$ with
\xitemof{m-5} and \xitemof{m-6}, $U=\setof{\bbu_\xi}{\xi<\kappa}$ and
$X=[U]_{\ell_2(\kappa)}$. Then $X$ is pathological by \Thmof{main-thm}.
For $\beta<\kappa$ let $U_\beta=\setof{\bbu_\xi}{\xi<\beta}$. We show that
$X_\beta=[U_\beta]_{\ell_2(\kappa)}$ is non-pathological.
Let $\mapping{f}{E\cap\beta}{\beta}$ be as in \xitemof{ads-1}.
For each $\alpha\in E\cap\beta$, let
$B_\alpha=(A_\alpha\setminus f(\alpha))\cup\ssetof{\alpha}$. Then $B_\alpha$,
$\alpha\in E\cap\beta$ are pairwise disjoint. Let
$C=\beta\setminus(\bigcup_{\alpha\in E\cap\beta}B_\alpha)$.
Note that
\begin{xitemize}
\xitem[ads-2] $\bbu'_\alpha=\bbu_\alpha
-\sum_{\xi\in A_\alpha\cap f(\alpha)}\bbu_\alpha(\xi)\bbe^\kappa_\xi$
\end{xitemize}
is an element of $X$ and $\supp(\bbu'_\alpha)=B_\alpha$. It follows that $X_\beta$ is
the orthogonal sum of the sub-inner-product-spaces $X\downarrow C$, $X\downarrow B_\alpha$,
$\alpha\in E\cap\beta$. In particular, we have
$X_\beta=\oplusbar_{\alpha\in E\cap\beta}X\downarrow B_\alpha\ \oplus\ X\downarrow C$.
Note that from this it follows that $X_\beta=X\downarrow\beta$.
Now $X\downarrow B_\alpha$, $\alpha\in E\cap\beta$
are non-pathological since they are separable. Let $U_\alpha$ be an orthonormal
basis of $X\downarrow B_\alpha$ for $\alpha\in E\cap\beta$. Also $X\downarrow C$ is
non-pathological with the orthonormal basis
$\setof{\bbe^\kappa_\alpha}{\alpha\in C}$. Thus
$\bigcup_{\alpha\in E\cap\beta}U_\alpha\cup\setof{\bbe^\kappa_\alpha}{\alpha\in C}$
is an orthonormal basis of $X_\beta$.
The same argument shows that $X\downarrow S$ is non-pathological for any
bounded subset $S$ of $\kappa$ closed \wrt\
the sequence $\seqof{A_\alpha}{\alpha\in E}$ (that is,
$A_\alpha\subseteq S$ for all $\alpha\in E\cap S$). Note that, for all regular
$\lambda<\kappa$ there are club many such $S$ of cardinality $\lambda$.
\qedofThm\qedskip
Under the consistency strength of certain very large cardinals we obtain reflection
theorems for pathology of pre-Hilbert spaces.
\begin{Thm}
\label{T-refl-2}
Suppose that $\kappa$ is a supercompact cardinal. Then for any pathological
pre-Hilbert space $X$, there are stationarily many pathological
sub-inner-product-spaces $Y$ of $X$ of size $<\kappa$.
\end{Thm}
\prf Suppose that $X$ is a pathological pre-Hilbert space of size $\lambda$.
We may assume that the underlying set of $X$ is $\lambda$.
If $\lambda<\kappa$ then the statement of the theorem is trivial. So we assume that
$\lambda\geq\kappa$. Let $\calC\subseteq[\lambda]^{<\kappa}$ be a club set.
Let $\elembed{j}{V}{M}$ be an elementary embedding with $\crit(j)=\kappa$,
$j(\kappa)>\lambda$ and $\fnsp{\lambda}{M}\subseteq M$. Then we have
$j\imageof X\in M$ and $j\imageof X\in j(\calC)$: the latter is because
$M\models\mbox{``}j(\calC)$ is a club subset of $[j(\lambda)]^{<j(\kappa)}$'' and
$\calD=\setof{j(Y)}{Y\in\calC}\subseteq j(\calC)$ is of
cardinality $<j(\kappa)$ with $\bigcup\calD=j\imageof X$.
We have $V\models j\imageof X\cong X$ and hence
$V\models\mbox{``}j\imageof X\mbox{ is pathological''}$. It
follows that $M\models\mbox{``}j\imageof X\mbox{ is pathological''}$.
Putting these facts together, we obtain
\begin{xitemize}
\xitem[refl-0]
$M\models$``\
\parbox[t]{\textwidth}{$j\imageof X$ is a sub-inner-product space of $j(X)$,
$j\imageof X\in j(\calC)$ and\\
$j\imageof X$ is pathological''.}
\end{xitemize}
Thus,
\begin{xitemize}
\xitem[refl-1]
$M\models$``there is a pathological sub-inner-product-space $Y$ of $j(X)$ with
$Y\in j(\calC)$.
\end{xitemize}
By elementarity if follows
\begin{xitemize}
\xitem[refl-2]
$V\models$``there is a pathological sub-inner-product-space $Y$ of $X$ with $Y\in \calC$.
\qedofThm\qedskip
\end{xitemize}
\begin{Thm}
\label{T-refl-3}
Suppose that $X$ is a pathological pre-Hilbert space and $X$ is a dense
sub-inner-product-space of $\ell_2(S)$ for some infinite set $S$. Then for any
ccc \po\ $\poP$ we have
$\forces{\poP}{[X]_{\ell_2(S)}\mbox{ is pathological\/}}$.
\end{Thm}
\prf Suppose that $X$ is a pre-Hilbert space and there is a ccc \po\ $\poP$ \st\
\begin{xitemize}
\xitem[refl-3]
$\forces{\poP}{[X]_{\ell_2(S)}\mbox{ is non-pathological\/}}$.
\end{xitemize}
We show that $X$ is then non-pathological.
By \Thmof{L-char-2} and the Maximal Principle there is
a $\poP$-name $\utilde{\calP}$ of partition of $S$ into countable sets \st\
\begin{xitemize}
\xitem[refl-4]
$\forces{\poP}{[X]_{\ell_2(S)}
=\oplusbar_{P\in\utilde{\calP}}[X]_{\ell_2(S)}\downarrow P}$.
\end{xitemize}
\begin{Claim}
\label{T-refl-3-C-1}
There is a partition $\calP'$ of $\kappa$ into countable sets \st, for each
$P\in\calP'$, we have
$\forces{\poP}{P\mbox{ is a countable union of elements of }\utilde{\calP}}$.
\end{Claim}
\prfofClaim
Let $\sim$ be the transitive closure of the relation
\begin{xitemize}
\xitem[refl-5]
$\sim_0=\setof{\pairof{s,t}\in S}{
\begin{array}[t]{@{}l}
\mbox{there is }p\in\poP\mbox{ \st}\\[\jot]
p\forces{\poP}{s\mbox{ and }t\mbox{ belong to the same
set }\in\utilde{\poP}}.}
\end{array}
$
\end{xitemize}
By the ccc of $\poP$, $Q_s=\setof{t\in S}{s\sim_0 t}$ is countable for all
$s\in S$. Hence all equivalence classes of $\sim$ are also countable.
Let $\calP'$ be the partition of $S$ into equivalence classes of $\sim$.
Let $P\in\calP'$. We show that
$\forces{\calP}{P\xmbox{ is a union of elements of }\utilde{\calP}}$. Let $G$ be
an arbitrary $(V,\poP)$-generic set In $V[G]$ suppose that $s\in P$, $s\in Q$ for
some $Q\in\utilde{\calP}^G$ and $t\in Q$. Then there is some $p\in G$ \st\
$p\forces{\poP}{s\mbox{ and }t\mbox{ are in the same element of }\utilde{\calP}}$.
It follows that $s\sim_0 t$ and $t\in P$.
\qedofClaim
\qedskip
\begin{Claim}
\label{T-refl-3-C-2}
For $P\in\calP'$ we have $X=(X\downarrow P)\oplus(X\downarrow (S\setminus P))$.
\end{Claim}
\prfofClaim Let $G$ be a $(V,\poP)$-generic set. In $V[G]$, we have
$[X]_{\ell_2(S)}
=([X]_{\ell_2(S)}\downarrow P) \oplus ([X]_{\ell_2(S)}\downarrow (S\setminus P))$ by
\Claimof{T-refl-3-C-1}. Hence, in $V$, we have
$X=(X\downarrow P) \oplus (X\downarrow (S\setminus P))$
\qedofClaim
\qedskip
It follows from \Claimabove\ that
$X=\oplusbar_{P\in\calP'}X\downarrow P$. Thus, by \Thmof{T-char}, $X$ is has an
orthonormal basis. \qedofThm
\qedskip
The Cohen forcing $\Fn(\kappa,2)$ in the following theorem can be replaced by may
other c.c.c.\ forcing notions which can be seen as iterations with certain
coherence (see Dow, Tall Weiss \cite{dow-etal}).
\begin{Thm}
Assume that $\kappa$ is a supercompact cardinal and let $\poP=\Fn(\kappa,2)$.
Then we have
\begin{xitemize}
\xitem[refl-6]
$\forces{\poP}{
\begin{array}[t]{@{}l}
\mbox{for every pathological pre-Hilbert space }X\mbox{ which is a dense sub-inner-}\\
\mbox{product space of }\ell_2(\lambda)\mbox{ for some infinite }\lambda,
\mbox{ there are stationarily many}\\
S\in[\lambda]^{<2^{\aleph_0}}\mbox{ \st\ }
X\downarrow S\mbox{ is pathological\/}}.
\end{array}$
\end{xitemize}
\end{Thm}
\prf Let $G$ be a $(V,\poP)$-generic filter. Working in $V[G]$, let $X$ be a
pre-Hilbert space which is a dense sub-inner-product-space of $\ell_2(\lambda)$.
If $\lambda<\kappa=\left(2^{\aleph_0}\right)^{V[G]}$ then the assertion is
trivial. Thus we assume $\lambda\geq\kappa$. Let $\calC\subseteq[\lambda]^{<\kappa}$ be a
club set. It is enough to show that there is some $S\in\calC$ \st\ $X\downarrow S$
is pathological.
Back in $V$, let $\elembed{j}{V}{M}$ be a $\lambda$-supercompact embedding. That
is, the elementary embedding $j$ is \st\ $M\subseteq V$ is a transitive class
$\crit(j)=\kappa$, $j(\kappa)>\lambda$ and $\fnsp{\lambda}{M}\subseteq M$.
Let $\poP^*=\Fn(j(\kappa),2)=j(\poP)$ and let $G^*$ be a $(V,\poP^*)$-generic
filter with $G^*\supseteq G$.
Let $\elembed{j^*}{V[G]}{M[G^*]}$ be the extension of $j$ defined by
\begin{xitemize}
\xitem[refl-7]
$j^*([\utilde{a}]^G)=[j(\utilde{a})]^{G^*}$
\end{xitemize}
for each $\poP$-name $\utilde{a}$. It is easy to check that $j^*$ is
well-defined and
\begin{xitemize}
\xitem[refl-8]
$\left(\fnsp{\lambda}{M[G^*]}\right)^{V[G^*]}\subseteq M[G^*]$.
\end{xitemize}
It follows that $j^*\imageof X\in M[G^*]$ and
$\supp(j^*\imageof X)=j\imageof\lambda\in j^*(\calC)$.
Since $V[G^*]$ is a c.c.c.\ extension of $V[G]$,
by
\Lemmaof{T-refl-3}, we have
\begin{xitemize}
\xitem[]
$V[G^*]\models\mbox{``} [j\imageof X]_{\ell_2(j(\lambda))}\mbox{ is pathological''}$.
\end{xitemize}
It follows that
\begin{xitemize}
\xitem[refl-9]
$M[G^*]\models\mbox{``} [j\imageof X]_{\ell_2(j(\lambda))}\mbox{ is pathological''}$
\end{xitemize}
by the same argument as right after \xitemof{refl-0}.
Thus we have
\begin{xitemize}
\xitem[refl-10]
$M[G^*]\models$``there is $S\in j^*(\calC)$ \st\ $j(X)\downarrow S$ is pathological''.
\end{xitemize}
By elementarity it follows that
\begin{xitemize}
\xitem[refl-11]
$V[G]\models$``there is $S\in\calC$ \st\ $X\downarrow S$ is pathological''.
\end{xitemize}
\qedofThm
\section{A Singular Compactness Theorem}
\label{singular}
The proof of the following theorem follows closely the proof of
Shelah's Singular Compactness Theorem given in Hodges \cite{hodges}. A similar
Singular Compactness Theorem in the context of (non-)freeness of modules is given
in Eklof \cite{eklof}.
\begin{Thm}
\label{Th-SC}
Suppose that $\lambda$ is a singular cardinal and $X$ is a pre-Hilbert space
which is a dense sub-inner-product-space of $\ell_2(\lambda)$. If $X$ is
pathological then there is a cardinal $\lambda'<\lambda$ \st\
\begin{xitemize}
\xitem[sc-1]
$\setof{u\in[\lambda]^{\kappa^+}}{X\downarrow u\mbox{ is a pathological
pre-Hilbert space}}$
\end{xitemize}
is stationary in $[\lambda]^{\kappa^+}$ for all $\lambda'\leq\kappa<\lambda$.
\end{Thm}
In the following we shall prove the contraposition of
the statement of the theorem:
\begin{list}{}{\setlength{\leftmargin}{0pt}}
\item {\bf \bfThmof{Th-SC}${}^{\mbox{\bf*}}$}
\it
For any singular $\lambda$ and any pre-Hilbert space $X$ which is a dense
sub-inner-product-space of $\ell_2(\lambda)$, if
\item
\begin{xitemize}
\xitem[sc-2]
$\calN^X_{\kappa}=\setof{u\in[\lambda]^{\kappa^+}}{X\downarrow u
\mbox{ is a non-pathological pre-Hilbert space}}$ contains a club in
$[\lambda]^{\kappa^+}$ for cofinally many $\kappa<\lambda$,
\end{xitemize}
then $X$ is non-pathological.
\end{list}
For a dense sub-inner-product space $X$ of $\ell_2(\lambda)$ and $v$,
$v'\subseteq \lambda$, we write $u'\osmmdin{X}u$ if
$u\subseteq u'$, $X\downarrow u$ and $X\downarrow u'$ are dense in
$\ell_2(\lambda)\downarrow u$ and $\ell_2(\lambda)\downarrow u'$ respectively;
and
$X\downarrow u$ is an orthogonal direct summand of $X\downarrow u'$, i.e.\ if
$X\downarrow u'=(X\downarrow u)\oplus (X\downarrow (u'\setminus u))$, see
\Lemmaof{summand}.
For a cardinal $\kappa$, the $\kappa$-Shelah game over
$X\subseteq\ell_2(\lambda)$ (notation $\calG_\kappa(X)$) is the game whose
matches $\calM$ are $\omega$-sequences of
moves by Players I and II
\begin{xitemize}
\item[] $\calM:\ \ \
\begin{array}{lllll}
{\rm I}\quad &u_0 &u_1 &u_2 &\cdots\\
{\rm II}\quad &v_0 &v_1 &v_2 &\cdots\\
\end{array}
$
\end{xitemize}
where $u_i$, $v_i\in[\lambda]^{\kappa}$ for $i\in\omega$ and
$u_0\subseteq v_0\subseteq u_1\subseteq v_1\subseteq u_2\subseteq v_2\subseteq\cdots$.
Player II wins if $X\downarrow v_i$ is non-pathological and
$v_{i+1}\osmmdin{X}v_i$ for all $i\in\omega$.
Note that, if Player II wins in a match $\calM$ with the moves
$u_0\subseteq v_0\subseteq u_1\subseteq v_1\subseteq u_2\subseteq v_2\subseteq\cdots$,
then $X\downarrow w$ for $w=\bigcup_{i\in\omega}u_i=\bigcup_{i\in\omega}v_i$ is
non-pathological.
\begin{Lemma}
\label{L-sc-0}
$\kappa$-Shelah game over $X\subseteq\ell_2(\lambda)$ is determined for
regular $\kappa$.
\end{Lemma}
\prf Since the game is open for Player I, the proof of Gale-Stewart Theorem
applies (see e.g.\ Kanamori \cite{kanamori} or Hodges \cite{hodges}). \qedofLemma
\qedskip
\begin{Lemma}
\label{L-sc-1}
Suppose that $X$ is
a dense sub-inner-product-space of $\ell_2(\lambda)$ for a cardinal
$\lambda$. For a cardinal $\kappa<\lambda$, if $\calN^X_{\kappa}$ contains a
club subset of $[\lambda]^{\kappa^+}$,
then Player II has a winning strategy in $\calG_\kappa(X)$.
\end{Lemma}
\prf By \Lemmaof{L-sc-0}, it is enough to show that the Player I does not have a
winning strategy.
Suppose that $\sigma$ is a strategy for Player I. We show that it is not
winning.
Let $\calC\subseteq\calM^X_\kappa$ be club in $[\lambda]^{\kappa^+}$.
Let $\chi$ be a sufficiently large regular cardinal and let
$\seqof{M_\alpha}{\alpha<\kappa^+}$ be a continuously increasing chain of
elementary submodels of $\calH(\theta)$ \st\
\begin{xitemize}
\xitem[sc-2-0] $\sigma$, $X$, $\lambda$, $\kappa$, $\calC\ctenten$\ $\in M_0$;
\xitem[sc-3] $\cardof{M_\alpha}=\kappa$ and $M_\alpha\in M_\alpha+1$ for all $\alpha<\kappa^+$;
\xitem[sc-3-0] $\alpha\subseteq M_\alpha$ for all $\alpha<\kappa^+$
\xitem[sc-4] For any finite subsequence $\calG_0$ of
$\seqof{M_\beta}{\beta\leq\alpha}$, if $\calG_0$ is the moves of Player II in an
initial segment $\calM_0$ of a match in $\calG_\kappa(X)$ where the Player I
has played according to $\sigma$ and the last member of $\calG_0$ is the last
move in $\calM_0$, then $\sigma(\calM_0)\in M_{\alpha+1}$ and
$\sigma(\calM_0)\subseteq M_{\alpha+1}$.
\end{xitemize}
Let $M=\bigcup_{\alpha<\kappa^+}M_\alpha$. By \xitemof{sc-2-0}, \xitemof{sc-3}
and \xitemof{sc-3-0}, we have
\begin{xitemize}
\xitem[sc-5] $\lambda\cap M\in\calC$.
\end{xitemize}
By \Thmof{T-char}, there is a partition $\calP$ of $\lambda\cap M$ into countable
sets \st\ $X\downarrow(\lambda\cap M)=\oplusbar_{A\in\calP}X\downarrow A$.
Let
$C=\setof{\alpha<\kappa^+}{\lambda\cap M_\alpha\xmbox{ is a union of some elements of }
\calP}$. Then $C$ is a club set $\subseteq\kappa^+$,
\begin{xitemize}
\xitem[sc-5-0]
$M\downarrow(\lambda\cap M_\alpha)$ is non-pathological for all $\alpha\in C$ and
\xitem[sc-6]
$(\lambda\cap M_\alpha)\osmmdin{X}(\lambda\cap M)$ for every $\alpha\in C$.
\end{xitemize}
Let $\alpha_i$, $i\in\omega$ be the first $\omega$ elements of $C$ and
$v_i=\lambda\cap M_{\alpha_i}$ for $i\in\omega$.
By \xitemof{sc-4}, there is a match $\calM$ in $\calG_\kappa(X)$ in which Player
I has chosen his moves according to $\sigma$ and $\seqof{v_i}{i\in\omega}$ is
the moves of Player II. Player II wins in this match $\calM$ by \xitemof{sc-6}.
This shows that $\sigma$ is not a winning strategy of Player I.\qedofLemma
\qedskip
\noindent
{\bf Proof of \bfThmof{Th-SC}${}^{\mbox{\bf*}}$:}\ \ Suppose that $X$ and $\lambda$ are as in
\Thmof{Th-SC}${}^{\mbox{*}}$. Let $\delta=\cf(\lambda)$ and let
$\seqof{\lambda_\xi}{\xi<\delta}$ be a continuously increasing sequence of
cardinals below $\lambda$ \st\
\begin{xitemize}
\xitem[sc-7] $\delta<\lambda_0$;
\xitem[sc-8] $\calN^X_{\lambda_\xi}$ (defined in \xitemof{sc-2}) contains a club subset
$\subseteq[\lambda]^{(\lambda_\xi)^+}$ for all successor $\xi<\delta$.
\end{xitemize}
The condition \xitemof{sc-8} is possible by our assumption \xitemof{sc-7}.
In the following, we construct $u^i_\xi$, $\tilde{u}^i_\xi$, $v^i_\xi$ for
$\xi<\delta$ and $i\in\omega$ \st\
\begin{xitemize}
\xitem[sc-9]
$\lambda_\xi=u^0_\xi\subseteq\tilde{u}^0_\xi\subseteq v^0_\xi\subseteq
u^1_\xi\subseteq\tilde{u}^1_\xi\subseteq v^1_\xi\subseteq
u^2_\xi\subseteq\tilde{u}^2_\xi\subseteq v^2_\xi\subseteq\cdots$
\end{xitemize}
and,
letting $w_\xi=\bigcup_{i\in\omega}u^i_\xi=\bigcup_{i\in\omega}\tilde{u}^i_\xi
=\bigcup_{i\in\omega}v^i_\xi$, we have
\begin{xitemize}
\xitem[sc-10] $\seqof{w_\xi}{\xi\in\delta}$ is a filtration of $\lambda$;
\xitem[sc-11] $X\downarrow w_\xi$ is non-pathological for all $\xi\in\delta$;
\xitem[sc-12] $w_\eta\osmmdin{X}w_\xi$ for all $\xi<\eta<\delta$.
\end{xitemize}
From \xitemof{sc-10}, \xitemof{sc-11} and \xitemof{sc-12}, it follows immediately
that $X$ is non-pathological.
For the construction of $u^i_\xi$, $\tilde{u}^i_\xi$, $v^i_\xi$ for
$\xi<\delta$ and $i\in\omega$, we fix winning strategies $\sigma_\xi$ for Player
II in $\calG_{\lambda_\xi}(X)$ for all successor $\xi<\delta$. We have such strategies by
\xitemof{sc-8} and \Lemmaof{L-sc-1}.
The following describes the inductive construction:
\begin{xitemize}
\xitem[sc-13]
$\cardof{u^i_\xi}=\cardof{\tilde{u}^i_\xi}=\cardof{v^i_\xi}=\lambda_\xi$ for
all $\xi<\delta$;
\xitem[sc-14] The sequence $\tilde{u}^0_\xi$, $v^0_\xi$, $\tilde{u}^1_\xi$,
$v^1_\xi$, $\tilde{u}^2_\xi$, $v^2_\xi$\ctenten\ is a match in
$\calG_{\lambda_\xi}(X)$ in which Player II has played according
to $\sigma_\xi$ for all successor $\xi<\delta$ (\xitemof{sc-11} for all
successor $\xi<\delta$ follows from this);
\xitem[sc-15] When
$\seqof{u^k_\xi}{k\leq i,\,\xi<\delta}$,
$\seqof{\tilde{u}^j_\xi}{j<i,\,\xi<\delta}$ and
$\seqof{v^j_\xi}{j<i,\,\xi<\delta}$ have been chosen (according to all the
conditions described here) for
an $i\in\omega$ then $\tilde{u}^i_\xi$ for each $\xi<\delta$ is \st\
$\tilde{u}^i_\xi\supseteq\bigcup_{\eta\leq\xi}u^i_\eta$ holds
(note that $\cardof{\bigcup_{\eta\leq\xi}u^i_\eta}=\lambda_\xi$ by
\xitemof{sc-13}. This condition guarantees that the sequence
$\seqof{w_\xi}{\xi<\delta}$ is going to be increasing);
\end{xitemize}
For each successor $\xi<\delta$ and $i\in\omega$, if $v^i_\xi$ has been chosen according to
the conditions described here, $X\downarrow v^i_\xi$ is non-pathological by
\xitemof{sc-14}. Thus we can find a partition $\calP^i_\xi$ of $v^i_\xi$ into
countable sets \st\ $X\downarrow v^i_\xi=\oplusbar_{A\in\calP^i_\xi}X\downarrow A$
by \Thmof{T-char}. If $i>0$ then we may choose $\calP^i_\xi$ \st\
$\calP^{i-1}_\xi\subseteq \calP^i_\xi$ (this is possible since
$v^i_\xi\osmmdin{X}v^{i-1}_\xi$ by \xitemof{sc-14}).
\begin{xitemize}
\xitem[sc-16] (a continuation of \xitemof{sc-15}) When
$\seqof{u^k_\xi}{k\leq i,\,\xi<\delta}$,
$\seqof{\tilde{u}^j_\xi}{j<i,\,\xi<\delta}$ and
$\seqof{v^j_\xi}{j<i,\,\xi<\delta}$ have been chosen (according to all the
conditions described here) for
an $i\in\omega$ then we choose $\tilde{u}^i_\xi$ also \st\
$\tilde{u}^i_\xi\cap v^k_{\xi+1}$ is a union of some elements of $\calP^k_\xi$ for
all $k<i$ for all (not necessarily successor) $\xi<\delta$
(this makes $w_{\xi+1}\osmmdin{X}w_\xi$ for all $\xi<\delta$);
\end{xitemize}
For each $\xi<\delta$ and $i\in\omega$, when $v^i_\xi$ has been chosen, we enumerate it as
$v^i_\xi=\setof{\beta_{i,\xi,\eta}}{\eta<\lambda_\xi}$.
\begin{xitemize}
\xitem[sc-17] When $\seqof{u^j_\xi}{j< i,\,\xi<\delta}$,
$\seqof{\tilde{u}^j_\xi}{j<i,\,\xi<\delta}$ and
$\seqof{v^j_\xi}{j<i,\,\xi<\delta}$ have been chosen (according to all the
conditions described here) for
an $i\in\omega$ then we let
$u^{i+1}_\xi=\setof{\beta_{i,\xi,\eta}}{\xi<\delta,\,\eta<\lambda_\xi}\cup v^i_\xi$ \\
(this is possible since the set on the right side of the inequality has size
$\leq\lambda_\xi$. This condition makes the
sequence $\seqof{w_\xi}{\xi<\delta}$ continuous).
\end{xitemize}
To see that \xitemof{sc-17} makes the sequence $\seqof{w_\xi}{\xi<\delta}$
continuous, suppose that $\nu\in w_\gamma$ for a limit $\gamma<\delta$.
Then there is $i^*\in\omega$ \st\ $\nu\in v^{i^*}_\gamma$. Hence there is
$\eta^*<\lambda_\gamma$ \st\ $\nu=\beta_{i^*,\gamma,\eta^*}$. Let $\xi<\gamma$
be \st\ $\eta^*<\lambda_\xi$. Then by \xitemof{sc-17} we have
$\nu=\beta_{i^*,\gamma,\eta^*}\in u^{i^*+1}_\xi\subseteq w_\xi$.
As noted above, the choice of
$u^i_\xi$, $\tilde{u}^i_\xi$, $v^i_\xi$ for
$\xi<\delta$ and $i\in\omega$ with
\xitemof{sc-9}, \xitemof{sc-13} $\sim$ \xitemof{sc-17} makes
$\seqof{w_\xi}{\xi<\delta}$ satisfy the conditions
\xitemof{sc-10}, \xitemof{sc-11} for all successor $\xi<\delta$ and
\xitemof{sc-12} for all $\xi<\delta$ and $\eta=\xi+1$.
By the continuity of
$\seqof{w_\xi}{\xi<\delta}$ we can then prove inductively that \xitemof{sc-11} and
\xitemof{sc-12} hold for all $\xi<\eta<\delta$. \qedof{\Thmof{Th-SC}${}^{\mbox{*}}$}
\section{Reflection of pathology and Fodor-type Reflection
Principle}
\label{FRP}
In this section we prove the following theorem which gives characterizations of
\FRP\ in terms of pathology of pre-Hilbert spaces.
\begin{Thm}
\label{main-thm-1} Each of the following assertions is equivalent to \FRP:
\begin{xitemize}
\xitem[chr-23] For any regular $\kappa>\omega_1$ and any dense sub-inner-product-space
$X$ of $\ell_2(\kappa)$, if $X$ is pathological then
\begin{xitemize}
\item[] $S_X=\setof{\alpha<\kappa}{X\downarrow \alpha\mbox{ is pathological\/}}$
\end{xitemize}
is stationary in $\kappa$.
\xitem[chr-24] For any regular $\kappa>\omega_1$ and any dense sub-inner-product-space
$X$ of $\ell_2(\kappa)$, if $X$ is pathological then
\begin{xitemize}
\item[] $S^{\aleph_1}_X=\setof{U\in[\kappa]^{\aleph_1}}{
X\downarrow U\mbox{ is pathological\/}}$
\end{xitemize}
is stationary in $[\kappa]^{\aleph_1}$.
\end{xitemize}
\end{Thm}
First let us review some facts around the reflection principle \FRP\ needed
for the proof of \Thmof{main-thm-1}.
One of the combinatorial statements equivalent to \FRP\ we are going to use below is as
follows:
\begin{xitemize}
\item[(\FRP)] For any regular $\kappa>\omega_1$, any stationary
$E\subseteq E^\omega_\kappa$ and any mapping $\mapping{g}{E}{[\kappa]^{\aleph_0}}$,
there is $\alpha^*\in E^{\omega_1}_\kappa$ \st\
\begin{xitemize}
\xitem[frp-0]
$\alpha^*$ is closed \wrt\ $g$
(that is, $g(\alpha)\subseteq\alpha^*$ for all $\alpha\in E\cap\alpha^*$) and, for
any $I\in[\alpha^*]^{\aleph_1}$ closed \wrt\ $g$, closed in $\alpha^*$ \wrt\
the order topology and with $\sup(I)=\alpha^*$, if
$\seqof{I_\alpha}{\alpha<\omega_1}$ is a filtration of $I$ then
$\sup(I_\alpha)\in E$ and
$g(\sup(I_\alpha))\cap\sup(I_\alpha)\subseteq I_\alpha$ hold for stationarily many
$\alpha<\omega_1$
\end{xitemize}
\end{xitemize}
(see Fuchino, Sakai, Soukup \cite{more}).
\FRP\ was invented by Lajos Soukup and the author in 2008 and then published in
Fuchino Juha\'asz, Soukup,
Szentmikl\'ossy, Usuba \cite{fuchino-juhasz-etal} by a formulation slightly
different from the one given above.
In Fuchino, Sakai, Soukup \cite{more} it is proved that \FRP\ is equivalent to
the statement that
$\ADS^-(\kappa)$ fails for all regular $\kappa>\omega_1$. This characterization
of \FRP\ is used to show the equivalence of \FRP\ to many mathematical reflection
statements
in Fuchino \cite{balogh}, Fuchino, Sakai, Soukup \cite{more}, Fuchino, Rinot \cite{rinot}.
One of the typical mathematical assertion equivalent with \FRP\ is:
\begin{xitemize}
\item[] For every non-metrizable countably compact topological space $X$ there is
a non-metrizable subspace of $X$ of cardinality $\leq\aleph_1$ (see \cite{more}).
\end{xitemize}
Our present result adds another couple of mathematical
reflection statements to the long list of the statements equivalent to \FRP.
For the proof of \Thmof{main-thm-1} we need the following easy observations:
\begin{Lemma}{\rm (cf. Lemma 6.1 in \cite{fuchino-juhasz-etal})}
\label{frp-2}
Suppose that $\kappa$ is a regular cardinal $>\aleph_1$, $C\subseteq\kappa$
club, $E\subseteq C$
stationary and $a_\eta\in[\kappa]^{\aleph_0}$ for $\eta\in E$. Then there is a
stationary $E'\subseteq E^\omega_\kappa\cap C$ and a mapping
$\mapping{\overline{\eta}}{E'}{E}$; $\xi\mapsto\eta_\xi$
\st, for all $\xi\in E'$, we
have $\xi\leq\eta_\xi$ and $a_{\eta_\xi}\cap\xi=a_{\eta_\xi}\cap\eta_\xi$.
\end{Lemma}
\prf
We prove the Lemma in the following two cases:
\noindent
{\bf Case I.} $E\cap E^\omega_\kappa$ is stationary.
Then
$E'=E\cap E^\omega_\kappa$ with $\bar{\eta}=\id_{E'}$ is as
desired. \smallskip
\noindent
{\bf Case II.} $E\cap E^\omega_\kappa$ is non-stationary.
Then $E''=E\setminus E^\omega_\kappa$ is stationary. For each $\eta\in E''$ we have
$\sup(a_\eta\cap\eta)<\eta$. By Fodor's Lemma there are $\eta_0<\kappa$ and stationary
$E'''\subseteq E''$ \st\ $\sup(a_\eta\cap\eta)\leq\eta_0$ for all $\eta\in E'''$.
Let $E'=(E^\omega_\kappa\cap C)\setminus \eta_0$ and, for each $\xi\in E'$, let
$\eta_\xi=\min(E'''\setminus \xi)$. Then this $E'$ with
$\mapping{\bar{\eta}}{E'}{E}$; $\xi\mapsto\eta_\xi$ is as desired.
\qedofLemma
\begin{Lemma}
\label{frp-2-0}
Suppose that $\kappa$ and $\lambda$ are regular cardinals with
$\aleph_0<\kappa<\lambda$ and $A$ a set of size $\geq\lambda$.
If $S\subseteq[A]^{<\lambda}$ is stationary in $[A]^{<\lambda}$ and
$U_s\subseteq[s]^{<\kappa}$ is stationary for all $s\in S$, then
$\bigcup_{s\in S}U_s$ is stationary in $[A]^{<\kappa}$.
\end{Lemma}
\prf
Suppose that $C\subseteq[A]^{<\kappa}$ is club. Then there is
$\mapping{f}{[A]^{<\aleph_0}}{[A]^{<\kappa}}$ \st\
\begin{xitemize}
\xitem[frp-2-1]
$C_f=\setof{x\in[A]^{<\kappa}}{x\mbox{ is closed \wrt\ }f}\subseteq C$
\end{xitemize}
(see e.g. Lemma 8.26 in Jech \cite{millenium-book}).
Note that then
\begin{xitemize}
\xitem[frp-2-2]
$C^{<\lambda}_f=\setof{y\in[A]^{<\lambda}}{y\mbox{ is closed \wrt\ }f}$
\end{xitemize}
is a club $\subseteq[A]^{<\lambda}$.
Since $S$ is a stationary subset of $[A]^{<\lambda}$, there is
$s^*\in S\cap C^{<\lambda}_f$. Now
$C_f\cap[s^*]^{<\kappa}$ is a club in $[s^*]^{<\kappa}$ and $U_{s^*}$ is
stationary in $[s^*]^{<\kappa}$.
Thus there is
$u^*\in U_{s^*}\cap (C_f\cap[s^*]^{<\kappa})\subseteq (\bigcup_{s\in S}U_s)\cap C_f
\subseteq (\bigcup_{s\in S}U_s)\cap C$. \\
\qedofLemma
\qedskip
\noindent
{\bf Proof of \bfThmof{main-thm-1}:} First we show that \FRP\ implies \xitemof{chr-23}.
Assume that \FRP\ holds.
Suppose that $X$ is a dense
sub-inner-product-space of $\ell_2(\kappa)$ for a regular cardinal
$\kappa>\aleph_1$. We assume that $S_X$ (in \xitemof{chr-23}) is non-stationary
and drive a contradiction.
By the assumption there is a club set $C\subseteq\kappa$ \st\ $X\downarrow\alpha$
is non-pathological for all $\alpha\in C$. By \Lemmaof{L-dense-1} we may assume
that $X\downarrow\alpha$ is dense in $\ell_2(\kappa)\downarrow\alpha$ for all
$\alpha\in C$.
Since $X$ is pathological,
\begin{xitemize}
\xitem[frp-3]
$E=\setof{\alpha\in C}{X\downarrow\alpha
\mbox{ is not an orthogonal direct summand of }X}$
\end{xitemize}
is stationary. By \Lemmaof{summand}, there is $\bba_\alpha\in X$ \st\
$\bba_\alpha\downarrow\alpha\not\in X\downarrow\alpha$ for all $\alpha\in E$.
Let $A_\alpha=\supp(\bba_\alpha)$ for $\alpha\in E$. By
\Lemmaof{frp-2}, we may assume \wolog\ that $E\subseteq C\cap E^\omega_\kappa$.
By \FRP, there is $\alpha^*\in E^{\omega_1}_\kappa$ \st\ \xitemof{frp-0} holds
for $\mapping{g}{E}{[\kappa]^{\aleph_0}}$; $\alpha\mapsto A_\alpha$.
Now since $E\cap\alpha^*$ is unbounded in $\alpha^*$, we have
$\alpha^*\in C$. Thus $X\downarrow\alpha^*$ is non-pathological. Hence by
\Thmof{T-char} there are club many $I\in[\alpha^*]^{\aleph_1}$ \st\
$X\downarrow I$ is non-pathological and $X\downarrow I$ is dense in
$\ell_2(\kappa)\downarrow I$. It follows that there is $I^*\in[\alpha^*]^{\aleph_1}$
\st\
\begin{xitemize}
\xitem[frp-4] $I^*$ is closed \wrt\ $g$ and closed in $\alpha^*$ \wrt\ the order topology;
\xitem[frp-4-0] $X\downarrow I^*$ is non-pathological;
\xitem[frp-4-1] $X\downarrow I^*$ is dense in $\ell_2(\kappa)\downarrow I^*$ and
\xitem[frp-5] $\sup(I^*)=\alpha^*$.
\end{xitemize}
Let $\seqof{I_\alpha}{\alpha<\omega_1}$ be a filtration of $I^*$ \st\
$X\downarrow I_\alpha$ is dense in $\ell_2(\kappa)\downarrow I_\alpha$.
By \xitemof{frp-0},
\begin{xitemize}
\xitem[frp-6]
$E_0=\setof{\alpha\in\omega_1}{\sup(I_\alpha)\in E,\,
A_{\sup(I_\alpha)}\cap\sup(I_\alpha)\subseteq I_\alpha}$
\end{xitemize}
is stationary. By \Lemmaof{filtration} and \Lemmaof{summand} this is a
contradiction to \xitemof{frp-4-0}.
This proves that \FRP\ implies \xitemof{chr-23}.
Since \FRP\ is equivalent to the
global negation of $\ADS^-(\kappa)$. \Thmof{T-refl-1} implies the converse.
For the equivalence of \FRP\ and \xitemof{chr-24}, it is enough by virtue of the
second part of \Thmof{T-refl-1} to show that
\xitemof{chr-23} implies \xitemof{chr-24}.
Assume that \xitemof{chr-23} holds. We prove that \xitemof{chr-24} holds for all
uncountable $\kappa$ by induction on $\kappa$: if $\kappa$ is $\aleph_1$ there is
nothing to prove.
Suppose that $\kappa>\aleph_1$ and \xitemof{chr-24} has been
established for all infinite cardinals $<\kappa$.
If $\kappa$ is a regular cardinal then
\xitemof{chr-24} for $\kappa$ follows from \xitemof{chr-23}, the induction hypothesis and
\Lemmaof{frp-2-0}. If $\kappa$ is a singular cardinal then \xitemof{chr-24} for
$\kappa$ follows from \Thmof{Th-SC}, the induction hypothesis and \Lemmaof{frp-2-0}.
\qedof{\Thmof{main-thm-1}}
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