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This is a scene from Sarah Ruhl’s Eurydice, staged by Tanghalang Pilipino. Here, Orpheus (Marco Viaña) returns to the underworld, clueless of the second death of his wife Eurydice (Lhorvie Nuevo). As memories are impermissible in the underworld, they can no longer recognise their bond in the living world nor feel the warmth of their short-lived love.
Photo by I.R. Arenas
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Aunty Jee… I have a problem
Dear Aunty Jee,
My wife and I have been married for three years and it all started so well, but now it is turning very sour. Whenever we talk, whatever we say, even about domestic matters, if the conversation does not go in my wife’s favour, she goes to her room and stays there sulking for days, sometimes for a week, only coming out during the night to visit the loo and to eat. Simple questions such as “Shall we clean out the cupboard under the stairs” – if our views do not coincide, off she goes. She simply will not communicate. I have tried to suggest that we go for counselling but this sets her off again. We are in our latter life and it makes me depressed to think that my final years will be spent like this. I feel an anger rising in me and fear that soon I might lose my temper and lash out.
I have tried to be nice when she is like this to no effect – now I am trying to be intolerant and angry, but neither seems to work. What can I do?
Anonymous male, 64, Wakefield
Dear Anonymous Male,
You have tried being nice and you have tried getting angry and neither of these have worked. That is because you are investing too much time and effort into your wife’s sulking. If she is treating you this way then you need to act as if it doesn’t bother you, otherwise she will continue.
Try acting indifferent. When she starts sulking, just ignore her. If you have done nothing wrong then you shouldn’t be made to feel as if you have done something wrong. It is unfair and cruel to you.
On the other hand you could try to explain to her how you are feeling in a calm and friendly way. Try to express your concerns to her in a way that she will understand that what she is doing is hurting you.
Your situation is a sensitive one, it isn’t easy at all I imagine. But just try what I have said and I hope that it works out for you.
Best of luck
Dear Aunty Jee,
I have recently found out that my dad cheated on my mum when I was younger. They don’t know that I know and they seem happy together. But only recently when I found out that my dad has cheated again… but this time with my sister in law! I really love my dad to bits but don’t really get on well with my mum. I just do not know whether to tell her about him cheating on her because I am scared of them splitting up. On the other hand, I do not want to upset my dad because he has never done anything wrong to me and I care about him a lot. I am also scared that my mum will flip and won’t believe me if I tell her. I need help as I am worrying about it and making myself ill. Do you have any suggestions?
Anonymous female, 22, Dewsbury
Beti, your situation breaks my heart.
It cannot be easy to know that your father has been having an affair.
I would suggest that you speak to your father and tell him that you know what has been happening. It is his place to speak to your mother about this.
This is difficult for me to give advice on as the situation is so very sensitive, but I would say that you start by speaking with your father.
Tell him that you know what he has been doing and how it is effecting you. Tread with caution as the situation is delicate, but this is one of the only things you can do at this moment.
Beyond this, it is up to your father and mother.
I wish you the best of luck beti.
Dear Aunty Jee,
This is such a tricky one. My friend Sara is a nice girl but despite being clean in every other way and always changing her clothes on a daily basis, her breath smells awful. I offer her mints and sweets as an indirect statement but she doesn’t understand. I mean, her breath is really, really bad and makes my eyes water when I smell it. It stinks in a big way. Sometimes she sits next to me and it wafts in my direction or she wants to give me a quick hug as she leaves me or a kiss on the cheek and I hate to tell her why I pull away. Nobody else has said anything to me about it but she needs to know, so please advise me on how can I tell her?
Anonymous female, 20, Leeds
Beti,
If the situation has got this bad it may be that she has a dental problem and needs to speak to a dentist.
Speak to her alone, without anyone else there so she doesn’t feel embarrassed or attacked. Try not to be judgemental when you speak to her. Be understanding and speak to her as a friend that is concerned.
Best of luck!
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TITLE: Smallest n for which G embeds in $S_n$?
QUESTION [33 upvotes]: Question: Given a finite group $G$, how do I find the smallest $n$ for which $G$ embeds in $S_n$?
Equivalently, what is the smallest set $X$ on which $G$ acts faithfully by permutations?
This looks like a basic question, but I seem not to be able to find answers or even this question in the literature. If this is known to be hard, is there at least a good strategy that would give a small (if not the smallest) $n$ for many groups?
Note: I do not care whether $G$ acts transitively on $X$, so for example for $G=C_6$ the answer is $n=5$ (mapping the generator to (123)(45)), not $n=6$ (regular action).
Edit: If this is not specific enough, is there a method that could find the smallest $n$ (or one close to the smallest one) for any group of size $\le 10^7$ in 5 seconds on some computer algebra system?
REPLY [5 votes]: Let $\mu(G) = \min\{n \mid G \text{ embeds in } S_n\}$. Here are some results on $\mu(G)$ from this paper by O. Becker:
$\mu(G)$ is known for abelian groups.
It is known eactly when $\mu(G) = |G|$. If $\mu(G) < |G|$, then $\mu(G) \le \frac{5}{6}|G|$.
The identity $\mu(G\times H) = \mu(G) + \mu(H)$ holds for a wide family of groups, for instance - for all $G,H$ with central socle.
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Let me know what I missed, and what should be higher or lower, but here is this year's version of my annual list of the year's top ten US legal ethics stories, along with seven honorable mentions.
Top Ten
- SCOTUS’s Hamdan decision rejected the controversial legal theories articulated in the OLC’s “torture memorandums”; in response, Congress passed the Military Commission Act, severely restricting detainees’ right to counsel.
- The plaintiff’s law firm Milberg Weiss was indicted.
- Corporate lawyers under focus: HP lawyers were caught up in the pretexting scandal; lawyers drew attention in the options backdating scandals.
- Mike Nifong’s prosecution of the Duke lacrosse players generated ethics controversies.
- DOJ rethought the Thompson/Holder approach to coercing corporations into waiving attorney client privilege; issued the McNulty memo; Senator Specter proposed a bill protecting privilege as well.
- McKesson successfully moved to disqualify Duane Morris even after signing a consent letter.
- SCOTUS’s decision in Gonzalez-Lopez reaffirmed the importance of a defendant’s right to counsel of the defendant’s choice.
- Law school news: pressure was placed on the ABA’s exercise of its role in accrediting law schools; Harvard and Stanford announced law school curriculum revisions; UC-Riverside announced its plans to open a new law school.
- California Supreme Court ruled in SF v. Cobra Systems that ethical screens don’t work for government lawyers where the “poisoned lateral” is the head of the office.
- State regulation of advertising: New York proposed stringent new rules on lawyer advertising; New Jersey outlawed participation in “Super Lawyer” campaign.
Honorable Mention
- Massachusetts required, and California considered requiring, mandatory disclosure of lawyers’ insurance coverage.
- California quietly continued its major overhaul of the one set of legal ethics rules that is still largely divorced from the ABA Model Rules and Model Code.
- Pop culture: an intoxicated lawyer was caught on Las Vegas court video tape and widely viewed on Youtube; Anna Nicole Smith gave birth to her lawyer’s child; an old Joe Jamail deposition excerpt was widely viewed on Youtube.
- Inadvertent disclosure and metadata issues generated controversy -- and conflicting ethics opinions.
- California Supreme Court ruled that a lawyer’s demand letter was criminal extortion (Flatley v. Mauro).
- Activist lawyer Lynne Stewart was sentenced to 28 months in prison.
- Alito’s Supreme Court confirmation hearings focused on judicial ethics.
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\begin{document}
\pagestyle{plain}
\title{An \'Etale Realization which does NOT exist}
\author{Jesse Leo Kass}
\address{Current: J.~L.~Kass, Dept.~of Mathematics, University of South Carolina, 1523 Greene Street, Columbia, SC 29208, United States of America}
\email{kassj@math.sc.edu}
\urladdr{http://people.math.sc.edu/kassj/}
\author{Kirsten Wickelgren}
\address{Current: K.~Wickelgren, School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160}
\email{kwickelgren3@math.gatech.edu}
\urladdr{http://people.math.gatech.edu/~kwickelgren3/}
\subjclass[2010]{Primary 14F42; Secondary 55P91,14F05.}
\date{\today}
\begin{abstract}
For a global field, local field, or finite field $k$ with infinite Galois group, we show that there can not exist a functor from the Morel--Voevodsky $\bbb{A}^1$-homotopy category of schemes over $k$ to a genuine Galois equivariant homotopy category satisfying a list of hypotheses one might expect from a genuine equivariant category and an \'etale realization functor. For example, these hypotheses are satisfied by genuine $\Z/2$-spaces and the $\bbb{R}$-realization functor constructed by Morel--Voevodsky. This result does not contradict the existence of \'etale realization functors to (pro-)spaces, (pro-)spectra or complexes of modules with actions of the absolute Galois group when the endomorphisms of the unit is not enriched in a certain sense. It does restrict enrichments to representation rings of Galois groups.
\end{abstract}
\maketitle
{\parskip=12pt
\section{Introduction}
Grothendieck envisioned that cohomological invariants of algebraic varieties are controlled by certain motives, with transcendental invariants, such as the Galois action on \'etale cohomology, being recovered by realization functors. Work of Morel--Voevodsky on $\bbb{A}^1$-homotopy theory provides candidate categories of motives, and realization functors have been constructed from the Morel--Voevodsky $\bbb{A}^1$-homotopy category and its stabilization \cite{morelvoevodsky1998} \cite{Isaksen-etale_realization} \cite{Quick-stable_realization} \cite{Ayoub-realization_etale}, \cite[Section~7.2]{CD-etale_motives}.
For example, Morel--Voevodsky construct an $\bbb{R}$-realization functor \cite[3.3]{morelvoevodsky1998}. On the level of schemes, this functor takes an $\bbb{R}$-scheme $X$ to the topological space (or simplicial set) of its complex points $X(\bbb{C})$ together with the action of the Galois group $\Gal(\bbb{C}/\bbb{R})$ on $X(\bbb{C})$. The association $X \mapsto X(\bbb{C})$ gives rise to functors from the $\bbb{A}^1$-homotopy categories of spaces and of $\bbb{P}^1$-spectra. The target category of this functor may be taken to be the genuine $\Gal(\bbb{C}/\bbb{R})$-homotopy category in the former case and the genuine $\Gal(\bbb{C}/\bbb{R})$-equivariant stable homotopy category in the latter. The adjective genuine (or {\em fine} in the terminology of \cite[3.3]{morelvoevodsky1998}) refers to the homotopy theory in which a weak equivalence of $\Gal(\bbb{C}/\bbb{R})$-spaces $X \to Y$ is not only an equivariant map which is a weak equivalence in the non-equivariant sense, but also satisfies the requirement that the map on fixed points $X^H \to Y^H$ is an equivalence for all subgroups of the group acting on the spaces, in this case $\Gal(\bbb{C}/\bbb{R})$ \cite{LewisMaySteinberger}. In the stable case, more notation would be required to define genuine $\Gal(\bbb{C}/\bbb{R})$-equivariant spectra, but one consequence of the more stringent notion of weak equivalence is that the group of endomorphisms of the sphere spectrum encodes interesting information about $\Gal(\bbb{C}/\bbb{R})$-sets, in contrast to $\bbb{Z}$, which is isomorphic to the endomorphisms of the sphere in ordinary stable homotopy. It follows that the Euler characteristic in genuine equivariant homotopy theory can be enriched from an element of $\bbb{Z}$ to an element of the Burnside ring $\A(\Gal(\bbb{C}/\bbb{R}))$ of formal differences of finite $\Gal(\bbb{C}/\bbb{R})$-sets.
One might hope to generalize this construction to fields $k$ with infinite Galois groups. Namely, let $G=\Gal_k=\Gal(\kbar/k)$ denote the absolute Galois group of $k$. One could hope to construct a category of genuine $G$-spectra or $G$-pro-spectra receiving an \'etale realization functor from $\bbb{P}^1$-spectra over $k$. Indeed, for profinite groups, homotopy theories of genuine $G$-(pro)-spectra are constructed in \cite{Fausk} and \cite{BarwickI}, and \cite{Quick-Profinite_G-spectra} contains a construction of a homotopy theory of profinite $G$-equivariant spectra. \'Etale realization functors have also been constructed over $k$ \cite{Isaksen-etale_realization} \cite{Quick-stable_realization} \cite{Ayoub-realization_etale}, but not so that the target is a genuine $G$-equivariant homotopy theory.
We show here that there is a reason for this lacuna. Namely, for $k$ a global field, local field, or finite field with infinite Galois group, there does not exist a symmetric monoidal \'etale realization functor from $\bbb{P}^1$-spectra over $k$ to a genuine $G$-equivariant homotopy theory under a set of hypothesis on what the terms {\em \'etale realization} and {\em genuine} are expected to imply. The hypotheses are best explained by the proof, and the proof is based on a result of Hoyois \cite[Theorem 1.5]{Hoyois_lef}. Namely, Hoyois shows that the $\bbb{A}^1$-Euler characteristic of a smooth proper variety with trivial cotangent sheaf is $0$, which implies, in particular, that the $\bbb{A}^1$-Euler characteristic of an elliptic curve $E$ is $0$. The outline of the proof is as follows. Suppose there is a symmetric monoidal \'etale realization functor $L \Et$. Then the Euler characteristic of $L \Et (E)$ is well-defined and equal to $0$. On the other hand, in genuine $G$-spectra, one could expect that the Euler characteristic is related to the alternating sum of the cohomology groups (with coefficients in some ring or field) considered with their $G$-actions. For example, this is the case for $G$ a finite group (Proposition \ref{assocchiX=repchi}). The theory of weights for Galois representations shows that the alternating sum of the \'etale cohomology groups for $E$ is non-zero in a representation ring. Since one could expect the cohomology groups of the \'etale realization to be the \'etale cohomology groups, at least for finite coefficients (as in \cite[Proposition 5.9]{Friedlander}), this is a contradiction.
A precise formulation is as follows. First, we introduce a definition. Let $k$ be a field and let $R$ be a ring. Let $\operatorname{Funct}(\Gal_k, R)$ denote the ring of functions $\Gal_k \to R$ with point-wise addition and multiplication.
\begin{df}\label{df:representation_ring}
By a {\em representation ring} of $\Gal_k$ with coefficients in $R$, we mean a ring $\Rep( \Gal_k, R)$ such that:
\begin{enumerate}
\item The isomorphism class of a finitely generated $R$-module $A$ with a continuous $\Gal_k$-action given by $A = \rH^i_{\et}(X_{\kbar}, R)$, where $X$ is a smooth, proper variety over $k$, determines an element $[A]$ of $\Rep( \Gal_k, R)$.
\item The trace map which takes $A$ to the function $\Gal_k \to R$ defined by taking $g$ in $G$ to the trace of $g$ acting on $A$, $$g \mapsto \Tr g \vert A$$ extends to a homomorphism $\Rep(\Gal_k, R) \to \operatorname{Funct}(\Gal_k, R)$.
\end{enumerate}
\end{df}
\begin{tm}\label{into:mainthm}
Let $k$ be a global field, local field with infinite Galois group, or finite field. Let $p$ be a prime different from the characteristic of $k$. It is impossible to simultaneously construct all of the following \begin{enumerate}
\item \label{mainthm_HgalK} A symmetric monoidal category $\calH_{\Gal_k}(\Spt)$, enriched over abelian groups. Let $\A(\Gal_k)= \End_{\calH_{\Gal_k}(\Spt)}(\Sphere)$ denote the endomorphisms of the symmetric monoidal unit $\Sphere$.
\item \label{mainthm_Rep} For all $n$, a representation ring $\Rep( \Gal_k, \Z/{p^n})$ with coefficients in $\Z/{p^n}$.
\item \label{mainthm_realization} A symmetric monoidal additive functor $L \Et: \calH_{\bbb{A}^1}(\Spt^{\bbb{P}^1}(k)) \to \calH_{\Gal_k}(\Spt)$, and ring homomorphisms $\assoc_n: \A(G) \to \Rep(\Gal_k, \Z/p^n)$ such that $$\assoc_n \chi (L\Et \Sigma^{\infty}_{\bbb{P}^1}X_+) = \sum_{i=-\infty}^{\infty} (-1)^i [H^i_{\et} (X_{\kbar}, \Z/p^n)]$$ for smooth proper schemes $X$ over $k$.
\end{enumerate}
\end{tm}
We are thinking of $\calH_{\Gal_k}(\Spt)$ as a genuine $\Gal_k$-equivariant stable homotopy category, $\A(\Gal_k)$ as a Burnside ring of $\Gal_k$, $\Rep( \Gal_k, \Z/{p^n})$ as a representation ring of $\Z/(p^n)$-modules with $\Gal_k$-action, and $L \Et$ as an \'etale realization functor. Since there are interesting contexts in which one may require a continuous Galois representation to satisfy additional conditions (e.g., be de Rham, crystalline, semi-stable, potentially semi-stable, etc.), we do not require that $\Rep( \Gal_k, \Z/{p^n})$ be the Grothendieck ring of continuous Galois representations, although this is certainly a possibility for \eqref{mainthm_Rep}. Theorem \ref{into:mainthm} is proven as Theorem \ref{tm:nonexistence} below and there is another variant given in Theorem \ref{tm:nonexistencev2}, where the representation rings involved only have coefficients in fields.
In Section \ref{GR-realization}, we show that the constructions of Theorem \ref{into:mainthm} do exist when $k=\bbb{R}$, the realization functor \eqref{mainthm_realization} being that of Morel--Voevodsky \cite[3.3]{morelvoevodsky1998}, and \eqref{mainthm_HgalK} and \eqref{mainthm_Rep} being the usual genuine $\Z/2$-equivariant stable homotopy category and $\Z/2$-representation rings, respectively. Section \ref{Gk-realization} contains the main result. As described above, the basic idea of the proof is that the $\bbb{A}^1$-Euler characteristic of an elliptic curve and the alternating sum of its \'etale cohomology groups are incompatible as Galois representations.
\section{Enriched Euler characteristics and genuine $\Gal_{\bbb{R}}$-realization}\label{GR-realization}
Let $G$ be a finite group. Let $\A(G)$ denote the Burnside ring, defined as the group completion (or Grothendieck group) of the semi-ring of isomorphism classes of finite $G$-sets with addition and multiplication given by disjoint union and product respectively.
The Burnside ring is the endomorphisms of the monoidal unit in the Burnside category $\Burn(G)$, whose definition we now recall. The objects of $\Burn(G)$ are finite $G$-sets. Given finite $G$-sets $T$ and $S$, consider the set of equivalence classes of diagrams of $G$-sets $$S \leftarrow U \rightarrow T$$ under the equivalence relation $[S \leftarrow U \rightarrow T]\sim[S \leftarrow U' \rightarrow T] $ when there is a $G$-equivariant bijection $\alpha: U \to U'$ such that the diagram $$\xymatrix{ & \ar[dl] U \ar[dd]_{\cong}^{\alpha} \ar[dr]& \\ S & &T \\ & \ar[ul] U' \ar[ur]&}$$ commutes. The morphisms $\Burn(G)(S,T)$ from $S$ to $T$ are the elements of the group completion of this set of equivalence classes under the operation $$[S \leftarrow U \rightarrow T] + [S \leftarrow U' \rightarrow T] = [S \leftarrow U \coprod U' \rightarrow T].$$ Composition of morphisms $S \leftarrow U \rightarrow T$ and $T \leftarrow U' \rightarrow Q$ is given by pullback \begin{equation}\label{BurnGcomp}\xymatrix{ && \ar[dl] V \ar[dr] && \\& \ar[dr]\ar[dl]U& &U' \ar[dr]\ar[dl] &\\ S&& T&& Q}\end{equation} and the composition $$\Burn(G)(S,T)\times \Burn(G)(T,Q) \to \Burn(G)(S,Q)$$ is defined by \eqref{BurnGcomp} and bilinearity. The symmetric monoidal structure takes $G$-sets $S$ and $T$ to their product $S \times T$, so $\A(G) = \Burn(G)(\ast, \ast)$, where $\ast$ denotes the one-point $G$-set.
An object $X$ of a symmetric monoidal category with unit $\Sphere$ and monoidal product $\wedge$ is said to be fully dualizable if there is a dual object $DX$ together with coevaluation and evaluation morphisms $$\eta: \Sphere \to X \wedge DX, ~~\epsilon: DX \wedge X \to \Sphere $$ such that the compositions $$X \stackrel{\eta \wedge 1_X}{\to} X \wedge DX \wedge X \stackrel{1_X \wedge \epsilon}{\to} X ,$$ $$DX \stackrel{1_{DX} \wedge \eta }{\to} DX \wedge X \wedge DX \stackrel{ \epsilon \wedge 1_{DX}}{\to} DX $$ are the identities. For example, every element of $\Burn(G)$ is fully dualizable with $DX = X$.
Given an endomorphism $f$ of a fully dualizable object $X$, the trace $\Tr (f)$ is the endomorphism of $\Sphere$ determined by the composition $$ \Sphere \stackrel{\eta}{\to} X \wedge DX \stackrel{f \wedge 1_{DX} }{\to} X \wedge DX\stackrel{\tau}{\to} DX \wedge X \stackrel{\epsilon}{\to} \Sphere,$$ where $\tau$ is the map which switches the order of the two factors. The Euler characteristic $\chi(X)$ is the trace of the identity $\chi(X) = \Tr (1_X)$. The following is a straightforward consequence of the definitions and is not original, but it is included for convenience and clarity.
\begin{lm}\label{lm:TrTUT} Let $T$ and $U$ be finite $G$-sets, and let $T \stackrel{i}{\leftarrow} U \stackrel{j}{\rightarrow} T$ be a diagram of finite $G$-sets. For elements $t$ and $t'$ in $T$, let $U_{t,t'}$ be the set $U_{t,t'} = i^{-1}(t) \cap j^{-1}(t')$. Then there are the following equalities in the Burnside ring $\A(G)$.
\begin{enumerate}
\item\label{TrUinBurnG} $\Tr[T \leftarrow U \rightarrow T] = \coprod_{t \in T} U_{t,t}$.
\item\label{chiT=T} $\chi (T) = T$.
\end{enumerate}
\end{lm}
\begin{proof}
We first show \eqref{TrUinBurnG}. Let $\Delta: T \to T \times X$ denote the diagonal. Then $\eta$ and $\epsilon$ are the morphisms $\ast \leftarrow T \stackrel{\Delta}{\to} T \times T $ and $ T \times T\stackrel{\Delta}{\leftarrow} T \to \ast$ respectively. The composition of $[T \leftarrow U \rightarrow T] \times 1_T $ and $\tau$ is $T \times T \stackrel{i \times 1_T}{\leftarrow} U \times T \stackrel{1_T \times j}\rightarrow T \times T$. The pullback diagram $$\xymatrix{ U\ar[d]^i\ar[r]_{1_U \times i} &U \times T \ar[d]^{i \times 1_T} \\T \ar[r]_{\Delta} & T \times T} $$ shows that the composition $\eta \circ ([T \leftarrow U \rightarrow T] \times 1_T) \circ \tau$ is $\ast \leftarrow U \stackrel{i \times j}{\to} T \times T$. The composition of this last map with $\eta$ shows $\Tr[T \leftarrow U \rightarrow T] = \coprod_{t \in T} U_{t,t}$. \eqref{chiT=T} follows from \eqref{TrUinBurnG}.
\end{proof}
Let $\calH_{G}(\Spt)$ denote the homotopy category of genuine $G$-spectra \cite[XII 5]{Alaska}, $\Sphere$ denote the sphere spectrum, and $\calH_{G}(\Spt)^d$ denote the full subcategory of fully dualizable objects.
It is a result of Segal \cite{SegalICM} and tom Dieck \cite{tomDieck_Trans} that there is a ring isomorphism \begin{equation}\label{AGcongEnd}\A(G) \cong \End_{\calH_{G}(\Spt)}(\Sphere),\end{equation} so for $X$ in $\calH_{G}(\Spt)^d$ we consider $\chi(X)$ to be an element of the Burnside ring $\A(G)$. The ring homomorphism \eqref{AGcongEnd} is induced from the fully faithful, symmetric, monoidal functor $\Sigma^{\infty}_+: \Burn(G) \to \calH_{G}(\Spt)$ taking a finite $G$-set $T$ to the suspension spectrum of $T_+$ \cite[Corollary 3.2 XIX]{Alaska}.
Let $R$ be a ring. Let $\Rep(G,R)$ denote the ring with generators given by isomorphism classes of finitely generated $R$-modules with $G$-action subject to the relation that $B = A \oplus C$ when there is a short exact sequence $$A \to B \to C.$$ Addition and multiplication in $\Rep(G,R)$ are induced from $\oplus$ and $\otimes$ respectively. For $X$ in $\calH_{G}(\Spt)$, the cohomology groups $H^i(X,R)$ determine elements of $\Rep(G,R)$.
\begin{df}\label{assoc_def_rmk}
Let $\assoc: \A(G) \to \Rep(G,R)$ be the map determined by taking a finite $G$-set $T$ to the permutation representation on $\oplus_T R$.
\end{df}
\begin{pr}\label{assocchiX=repchi}
\begin{enumerate}
\item \label{pr:assocchiX=repchi:case1} Let $R$ be any ring. For all finite $G$-CW complexes $X$ we have that $$\assoc \chi (X) = \sum_{i=-\infty}^{\infty} (-1)^i H^i(X,R).$$
\item \label{pr:assocchiX=repchi:case2} Let $R$ be a field of characteristic not dividing the order of $G$. Then for all $X$ in $\calH_{G}(\Spt)^d$ $$\assoc \chi (X) = \sum_{i=-\infty}^{\infty} (-1)^i H^i(X,R),$$ and in particular the right hand side is a well-defined element of $\Rep(G,R)$ in the sense that only finitely many of the summands are non-zero and the non-zero summands are finitely generated $R$-modules.
\end{enumerate}
\end{pr}
The proof of Proposition \ref{assocchiX=repchi} \eqref{pr:assocchiX=repchi:case1} is a straightforward induction on the cells of $X$. To prove \eqref{pr:assocchiX=repchi:case2}, we will have use of the following lemmas.
\begin{lm}\label{lm:int_Tr-to-perm}
Let $G$ be a finite cyclic group. Let $R$ be a field of characteristic not dividing the order of $G$. Let $T$ be a finite $G$-set, and $V \subseteq \assoc T$ be a $G$-fixed subspace such that the corresponding representation $\rho: G \to \GL V$ satisfies the property that $\Tr(\rho(g))$ is the image of an integer under the canonical map $\mathbb{Z} \to R$ for all $g$ in $G$. Then there are $G$-sets $E$ and $I$ such that $V \oplus \assoc E \cong \assoc I$.
\end{lm}
\begin{proof}
If $G$ has order $1$, then the lemma is true. Let $G$ have order $n>1$ and assume inductively that the lemma holds for cyclic groups of smaller orders. Because matrices which are similar over the algebraic closure of $R$ are similar over $R$, we may assume $R$ is algebraically closed. Thus $\assoc T$ and $V$ decompose into simultaneous $1$-dimensional eigenspaces whose eigenvalues are $n$th roots of unity. Let $d$ denote the largest order of an eigenvalue of $V$, and note that $d$ divides $n$. If $d=1$, then $V\cong \assoc I$ where $I$ is a set with trivial $G$-action of cardinality equal to the dimension of $V$, so the lemma is true.
We now induct on $d$. Let $g$ denote a generator of $G$. The condition that $\Tr(\rho(g))$ is an integer implies that the eigenspaces of $g$ in $V$ associated to the primitive $d$th roots of $1$ are all the same dimension. Call this dimension $a$. Let $d=\prod_{j=1}^{l} p_j^{e_j}$ be the prime factorization of $d$. Let $E_j$ denote the $G$-set consisting of a single orbit of cardinality $d/p_j$, and let $E' = \coprod_{i=1}^a \coprod_{j=1}^l E_j$. Then for each $d$th root of unity, we may choose an $a$ dimensional eigenspace of $V \oplus \assoc E'$. Let $V'$ denote the direct sum of these chosen eigenspaces and let $V'' \subset V \oplus \assoc E'$ denote a complementary $G$-representation to $V'$. By construction $V' \cong \assoc (\coprod_{j=1}^a I')$, where $I'$ is the $G$-set with a single orbit of cardinality $d$. In particular, this representation satisfies the property that the trace of every element of $G$ is an integer. It follows that the same property holds for $V''$. By induction on $d$, there are $G$-sets $E''$ and $I''$ such that $V'' \oplus \assoc E'' \cong \assoc I''$. Thus $V \oplus \assoc E' \oplus \assoc E'' \cong \assoc(\coprod_{j=1}^a I') \oplus \assoc I''$. We may therefore let $E = E' \coprod E''$ and $I = (\coprod_{j=1}^a I') \coprod I''$, and the lemma is true.
\end{proof}
\begin{lm}\label{assoc_det_set_G=Z/n}
Let $G$ be a finite cyclic group. Assume $R$ has a prime ideal of residue characteristic not dividing the order of $G$. If $T$ and $T'$ are two finite $G$-sets such that $\assoc T \cong \assoc T'$, then $T \cong T'$.
\end{lm}
\begin{rmk}
Lemma \ref{assoc_det_set_G=Z/n} becomes false when $G$ is $\Z/2\times \Z/2$ \cite[\S 7 Example 2]{Roberts-Equivariant_Milnor}.
\end{rmk}
\begin{proof}
By assumption, there is a ring map $R \to K$ where $K$ is an algebraically closed field of characteristic not dividing the order of $G$. We may replace $R$ by $K$, allowing us to decompose $\assoc T \cong \assoc T'$ into simultaneous $1$-dimensional eigenspaces whose eigenvalues are $n$th roots of unity. Let $d$ denote the largest order of such an eigenvalue. If $d=1$, then both $T$ and $T'$ have trivial $G$-actions and $\vert T \vert = \vert T' \vert = \dim \assoc T'$, so the lemma is true. Assume by induction that the lemma holds for all smaller values of $d$. Let $g$ be a generator of $G$ and $\zeta$ be a primitive $d$th root of unity. The number of $d$-cycles in $\assoc T$ is the dimension of the $\zeta$-eigenspace of $T$. Since the same holds for $T'$, it follows that $T$ and $T'$ contain the same number of $d$-cycles. Furthermore, removing the $d$-cycles from both $T$ and $T'$ results in finite $G$-sets with isomorphic associated permutation representations for which the lemma holds inductively. Thus $T \cong T'$.
\end{proof}
\begin{lm}\label{fBurnside_idempotentTr(f)}
Let $G$ be a finite group and let $f:T \to T$ be an idempotent in the Burnside category of $G$. Let $R$ be a field of characteristic not dividing the order of $G$. Then $\assoc \Tr(f) = \Image H^0(f)$ in $\Rep(G,R)$.
\end{lm}
We establish a few preliminaries before proving Lemma \ref{fBurnside_idempotentTr(f)}.
\begin{rmk}\label{calculate_trImageH0(f)andTrf}
We may directly calculate the trace functions $\chi_{\assoc \Tr(f)}$ and $\chi_{\Image H^0(f)}$ of $\assoc \Tr(f)$ and $\Image H^0(f)$. Since $f$ is idempotent, $$\chi_{\Image H^0(f)}(g) = \Tr(H^0(f) \vert \Image H^0(f) )= \Tr(H^0(g)H^0(f) \vert H^0(T)).$$ Let $f = [T \stackrel{i}{\leftarrow} U \stackrel{j}{\rightarrow} T] - [T \stackrel{i}{\leftarrow} V \stackrel{j}{\rightarrow} T]$. Let $\delta_t$ be the function $\delta_t: T \to R$ such that $\delta_t(t) = 1$ and $\delta_t(t') = 0$ for $t' \neq t$. In the below, $i_*$ denotes the push forward on $H^0$ associated to $i$, and $g^*, j^*$ are the corresponding pullbacks. We compute \begin{align*}
\chi_{\Image H^0(f)}(g) = \Tr(H^0(g)H^0(f) \vert H^0(T)) = \sum_{t \in T} (g^* i_* j^* \delta_t)(t) = \sum_{t \in T} (i_* g^* j^* \delta_t)(t) \\
= \sum_{t \in T} (\sum_{u \in U : i(u) = t} (g^* j^* \delta_t)(u) - \sum_{v \in V : i(v) = t} (g^* j^* \delta_t)(v)) \\
= \sum_{t \in T} (\sum_{u \in U : i(u) = t} \delta_t(gju) - \sum_{v \in V : i(v) = t} \delta_t(gjv)) \\
= \vert \{u \in U : i(u) = gj(u) \} \vert - \vert \{v \in V : i(v) = gj(v) \} \vert .
\end{align*}
It follows from Lemma \ref{lm:TrTUT} that \begin{align*} \chi_{\assoc \Tr(f)}(g) = \vert \{u \in U : gu=u, i(u) = j(u) \} \vert - \vert \{v \in V : gv =v, i(v) = j(v) \} \vert.\end{align*}
It is not clear from these calculations that $\chi_{\Image H^0(f)} = \chi_{\assoc \Tr(f)}$. We will show in Lemma \ref{fBurnside_idempotentTr(f)} that they are.
It is clear that $\chi_{\Image H^0(f)}$ takes values in the integers, and it follows from $$\chi_{\Image H^0(f)} + \chi_{\Ker H^0(f)} = \chi_{H^0(T)},$$ that $\chi_{\Ker H^0(f)}$ also takes values in the integers.
\end{rmk}
To prove Lemma \ref{fBurnside_idempotentTr(f)}, we will make use of the following variation of the Burnside category.
\begin{df}
Define the category $\Burn(G,R)$ to have objects $G$-sets, and for $G$-sets $S$ and $T$, the morphisms $\Burn(G,R)(S, T)$ are $R$-linear combinations of equivalence classes of diagrams of $G$-sets $S \leftarrow U \rightarrow T$. Composition is defined to be $R$-bilinear and induced from \eqref{BurnGcomp} as in the definition of the Burnside category.
\end{df}
$\Burn(G,R)(S, T)$ is a symmetric, monoidal category with the monoidal structure defined by the product of $G$-sets, and $\Tr[T \leftarrow U \rightarrow T] = \coprod_{t \in T} U_{t,t}$ by the proof of Lemma \ref{lm:TrTUT}.
\begin{lm}\label{TrAB=TrBA}
Let $A$ and $B$ be morphisms in $\Burn(G,R)(T, T)$. Then $\Tr(A \circ B)= \Tr (B\circ A)$.
\end{lm}
\begin{proof} Since composition of morphisms is $R$-bilinear, and $\Tr$ is $R$-linear, it suffices to prove the claim where $A$ and $B$ are determined by diagrams of $G$-sets $T \leftarrow A \rightarrow T$ and $T \leftarrow B \rightarrow T$, respectively. Then:
\begin{align*} \Tr(A \circ B)& \cong \coprod_{t \in T} (A \circ B)_{tt} \\
&\cong \coprod_{t \in T} \coprod_{t' \in T} A_{t, t'} \times B_{t',t} \\
& \cong \coprod_{t , t' \in T} A_{t, t'} \times B_{t',t} \\
& \cong \coprod_{t , t' \in T} B_{t',t} \times A_{t, t'} \\
&\cong \coprod_{t' \in T} \coprod_{t \in T} B_{t',t} \times A_{t, t'} \\
&\cong \coprod_{t' \in T} (A \circ B)_{t',t'} \cong \Tr(B \circ A) \end{align*}
\end{proof}
\begin{proof} (of Lemma \ref{fBurnside_idempotentTr(f)})
By Maschke's theorem \cite[Theorem 3.1, Theorem 3.5]{Etingofetal_Intro_Rep_thy}, it suffices to show that the trace functions $\chi_{\assoc \Tr(f)}$ and $\chi_{\Image H^0(f)}$ of $\assoc \Tr(f)$ and $\Image H^0(f)$ are the same. To do this, we may show for each $g$ in $G$ that $\chi_{\assoc \Tr(f)}(g) = \chi_{\Image H^0(f)}(g)$, thus reducing to the case where $G$ is a finite cyclic group.
By Remark \ref{fBurnside_idempotentTr(f)}, $\chi_{\Image H^0(f)}$ takes values in the integers, so we may apply Lemma \ref{lm:int_Tr-to-perm} to $\Image H^0(f) \subseteq \assoc T$. Thus there are $G$-sets $E$ and $I$ such that $\Image H^0(f) \oplus \assoc E \cong \assoc I$. Let $f': T \coprod E \to T \coprod E$ be the idempotent in the Burnside category of $G$ defined $f'=f \coprod 1_E$. Applying Remark \ref{fBurnside_idempotentTr(f)} and Lemma \ref{lm:int_Tr-to-perm} to $\Ker H^0(f') \subseteq \assoc (T \coprod)$, there are $G$-sets $E'$ and $I'$ such that $\Ker H^0(f') \oplus \assoc E' \cong \assoc I'$. Define $f'': T \coprod E \coprod E' \to T \coprod E \coprod E' $ be the idempotent in the Burnside category of $G$ defined $f''=f' \coprod 0_{E'}$.
By Lemma \ref{lm:TrTUT}, $\Tr f'' \cong E \coprod \Tr f$, and $\Image H^0(f'') \cong \Image H^0(f) \oplus \assoc E$. Therefore it suffices to prove the lemma with $f$ replaced by $f''$.
Since $f''$ is an idempotent, $$\assoc ( T \coprod E \coprod E' \to T) \cong \Ker H^0(f'') \oplus \Image H^0(f'').$$ By construction, $\Image H^0(f'') \cong \Image H^0(f') \cong \Image H^0(f) \oplus \assoc E \cong I$, and $\Ker H^0(f'') \cong \Ker H^0(f') \oplus \assoc E' \cong \assoc I'$. Replacing $T$ by $T \coprod E \coprod E'$ and $f$ by $f''$, we may therefore assume that there exist $G$-sets $I$ and $I'$ such that $\Image H^0(f) \cong \assoc I$ and $\Ker H^0(f) \cong \assoc I'$.
Since $f$ is an idempotent, there is an isomorphism $\assoc T \cong \Image H^0(f) \oplus \Ker H^0(f)$, whence an isomorphism $\eta: \assoc T \to \assoc (I \coprod I')$. Define $T'= I \coprod I'$. For $t \in T$ and $t' \in T'$, let $\eta_{t' ,t}$ denote the corresponding entry of the matrix of $\eta$ with respect to the bases $T$ and $T'$, and similarly define $(\eta^{-1})_{t,t'}$ to be the matrix entry of the inverse of $\eta$. Note that $\eta_{t' ,t} (t',t)$ is an element of $\assoc (T' \times T)$. The subset $\coprod_{t \in T,t' \in T'} \eta_{t' ,t} (t',t)$ of $\assoc (T' \times T)$ is invariant under the action of $G$ because $\eta$ is a $G$-isomorphism. It follows that we may view $\coprod_{t \in T,t' \in T'} \eta_{t' ,t} (t',t)$ as an $R$-linear combination of $G$-subsets of $T' \times T$. The analogous assertion holds for $\eta'$.
The $R$-linear combination of $G$-invariant subsets of $T' \times T$ associated to $\coprod_{t \in T,t' \in T'} \eta_{t' ,t} (t',t)$ determines a morphism in $\Burn(G,R)(T, T')$, which we will denote by $\Burn(\eta)$. Similarly define $\Burn(\eta^{-1})$ in $\Burn(G,R)(T', T)$. By construction, $\Burn(\eta) \circ f \circ \Burn(\eta^{-1}) = I$ where the two maps to $T'$ are both the canonical inclusion $I \to T'$.
We claim that Lemma \ref{TrAB=TrBA} implies that \begin{align*}\Tr (\Burn(\eta) \circ f \circ \Burn(\eta^{-1})) = \Tr (f) \end{align*} To see that we may apply Lemma \ref{TrAB=TrBA} in this case, note that by Lemma \ref{assoc_det_set_G=Z/n}, we may fix an isomorphism of $G$-sets $T \cong T'$, and consider $\Burn(\eta)$ and $\Burn(\eta^{-1})$ to both be elements of $\Burn(G,R)(T, T)$ allowing us to conclude $\Tr (\Burn(\eta) \circ f \circ \Burn(\eta^{-1}))= \Tr(\Burn(\eta^{-1})\circ \Burn(\eta) \circ f)$. Thus $\Tr (f) = I$. Since $\assoc I \cong \Image H^0(f)$, we have shown the lemma.
\end{proof}
The following corollary of Lemma \ref{fBurnside_idempotentTr(f)} is not needed for the rest of the article, but is included as a curiosity. To put this in context, we remark that there are many interesting idempotents of the Burnside ring of a finite group $G$. The idempotents of $\A(G) \otimes \Q$ were computed by \cite{Solomon}, \cite{Yoshida}, and \cite{Gluck-idempotent}. The integral idempotents, i.e., the idempotents of $\A(G)$, were computed by Dress \cite{Dress-solvable}, see \cite[Theorem 3.3.7, Corollary 3.3.9]{Bouc-Burnside_rings}.
\begin{co}\label{co:associdempAg}
Let $G$ be a finite group and let $R$ be a field of characteristic not dividing the order of $G$. Suppose $f$ is an idempotent in the Burnside ring $\A(G)$. Then $\assoc f$ is either $0$ or the trivial representation in $\Rep(G,R)$.
\end{co}
\begin{proof}
Since $\Image H^0(f)$ is a submodule of $H^0(\ast)$, we have that $\Image H^0(f)$ is either $0$ or $R$ with the trivial action. The corollary then follows from Lemma \ref{fBurnside_idempotentTr(f)}.
\end{proof}
We wish to thank Serge Bouc and Alexander Duncan for useful correspondence concerning Corollary \ref{co:associdempAg}. In particular, they explicitly computed an interesting (integral) idempotent of the Burnside ring of the alternating group $G=A_5$ such that the associated element of $\Rep(G, \Q)$ is $0$.
\begin{proof} (of Proposition \ref{assocchiX=repchi}) We first make an observation useful for proving both \eqref{pr:assocchiX=repchi:case1} and \eqref{pr:assocchiX=repchi:case2}: For a cofiber sequence $X \to Y \to Z$ in $\calH_{G}(\Spt)^d$, there is an equality $\chi(X) + \chi(Z) = \chi(Y)$ in $\A(G)$ (see for example \cite[XVII Theorem 1.6]{Alaska}), and therefore an equality $\assoc \chi(X) + \assoc \chi(Z) = \assoc \chi(Y)$.
Since a short exact sequence $A \to B \to C$ of $G$-modules induces the relation $A + C = B$ in $\Rep(G,R)$, it follows by induction that an exact sequence $$ 0 \to M_{n} \to M_{n+1} \to \ldots \to M_{m-1} \to M_m \to 0$$ of $G$-modules induces the relation $\sum_{i=-\infty}^{\infty} (-1)^i M_i = 0$. The cofiber sequence $X \to Y \to Z$ induces the long exact sequence $$ \ldots \to H^i(Z,R) \to H^i(Y,R) \to H^i(X,R) \to H^{i+1}(Z,R) \to \ldots $$ in cohomology, whence we have that $$ \sum_{i=-\infty}^{\infty} (-1)^i H^i(X,R) +\sum_{i=-\infty}^{\infty} (-1)^i H^i(Z,R)=\sum_{i=-\infty}^{\infty} (-1)^i H^i(Y,R)$$ under the hypothesis that for two (and thus all three) of $X$, $Y$, and $Z$, only finitely many terms in each sum are non-zero. Therefore if the claim holds for any two of $X$, $Y$, and $Z$ in a cofiber sequence $X \to Y \to Z,$ it holds for the third.
We now prove \eqref{pr:assocchiX=repchi:case1}: By induction on the number of cells of the finite $G$-CW complex $X$, it therefore suffices to show the claim for $X$ a finite $G$-set. This then follows from Lemma~\ref{lm:TrTUT}~\eqref{chiT=T}.
We now prove \eqref{pr:assocchiX=repchi:case2}: Let $A$ be a $G$-CW complex which is a retract in the homotopy category of a finite $G$-CW complex $Y$. Let $\iota$ be the endomorphism of $Y$ given by the composition of the retract and the inclusion $\iota:Y \to A \to Y$. It is formal that $\Tr(\iota) = \chi(A)$. We likewise have that the image $\Image H^i(\iota)$ of $H^i(\iota)$ is a direct summand of $H^i(Y,R)$ isomorphic to $H^i(A,R)$. We show the claim for $A$ by proving that for a $G$-equivariant map $f:Y \to Y$ which is idempotent in the homotopy category, we have \begin{equation}\label{Trf=sumimage}\assoc \Tr(f) = \sum_{i=-\infty}^{\infty} (-1)^i [\Image H^i(f)],\end{equation} and applying \eqref{Trf=sumimage} for $f=\iota$. Both sides of are unchanged under equivariant homotopy, and it follows that we may assume that $f$ is cellular \cite[I. Theorem 3.4]{Alaska}. This allows us to induct on the dimension of the top cells of $Y$, reducing to the case where $Y = T_+ \wedge S^n$ for a $G$-set $T$. Since replacing $f$ by its suspension multiplies both sides of \eqref{Trf=sumimage} by $-1$, we replace $f$ by $f \wedge S^{-n}$, reducing to the case where $Y= T_+$ and the map $f$ is now is the stable homotopy category. Then as remarked above $f$ corresponds to an idempotent $f: T \to T$ in $\Burn(G)$ \cite[Corollary 3.2 XIX]{Alaska}. Thus $ \sum_{i=-\infty}^{\infty} (-1)^i [\Image H^i(f)] = [\Image H^0(f)] $. The equality \eqref{Trf=sumimage} then follows by Lemma \ref{fBurnside_idempotentTr(f)}.
Let $V$ be an $n$-dimensional representation of $G$ and let $S^{-V}$ be the dual of the one-point compactification $S^V$ of $V$. If $X$ is a strongly dualizable $G$-spectra, then so is $DX$ and $\chi(X) = \chi(DX)$. Thus $\chi(S^{-V})=\chi(S^V)$. Since $S^V$ is a finite $G$-CW complex, we have that $$\assoc \chi(S^V) = \sum_{i=-\infty}^{\infty} (-1)^i H^i(S^V,R) = R + (-1)^n R,$$ where the action of $G$ on the first summand is trivial, and the action of $G$ on the second summand is via the sign on the determinant, i.e. $g \in G$ acts by $\pm 1$ depending on if $g$ preserves or reverses the orientation of $S^V$. Combining with the previous, we have $$\assoc \chi(S^{-V}) = R + (-1)^n R.$$ By definition, $H^i (DX, R) = \pi_{-i} F(DX, R) = \pi_{-i}(X \wedge R) = H_{-i} (X,R)$. Thus $$H^i(S^{-V}, R) = \begin{cases} R &\mbox{if } i= 0 \\
R & \mbox{if } i = -n \\
0&\mbox{otherwise }\end{cases},$$ where for $i=0$, the action is trivial, and for $i=-n$, $G$ acts on $R$ by the sign of the determinant. Thus the claim holds for $S^{-V}$.
By \cite[Proposition 2.1]{Fausk_Lewis_May}, $X$ in $\calH_{G}(\Spt)^d$ is equivalent to $\Sigma^{-V} \Sigma^{\infty} A$ where $A$ is a finitely dominated based $G$-CW complex and $V$ is a representation of $G$. Since $A$ is finitely dominated, by definition we have a finite $G$-CW complex $Y$ such that $A$ is a retract of $Y$ in the homotopy category of $G$-CW complexes. By the above, the claim holds for $A$ and $S^{-V}$. Since the smash product of two endomorphisms of $\Sphere$ induces the multiplication of $\A(G)$, we have $\chi(\Sigma^{-V} \Sigma^{\infty} A) = \chi(S^{-V}) \chi(A)$. Thus $\assoc \chi(\Sigma^{-V} \Sigma^{\infty} A) = \assoc \chi(S^{-V}) \assoc \chi(A)$. By the K\"unneth spectral sequence, we have the equality $$\sum_{i=-\infty}^{\infty} (-1)^i H^i(\Sigma^{-V} \Sigma^{\infty} A,R) = (\sum_{i=-\infty}^{\infty} (-1)^i H^i(S^{-V},R))(\sum_{i=-\infty}^{\infty} (-1)^i H^i(A,R))$$ in $\Rep(G,R)$. The claim for $X$ then follows from the claim for $S^{-V}$ and $A$.
\end{proof}
This concludes the facts we need about genuine $G$-spectra. We now turn to the relationship between $\bbb{A}^1$-homotopy theory over $\bbb{R}$ and genuine $\Gal_{\bbb{R}}$-spectra.
Let $\calH_{\bbb{A}^1}(\spaces{\bbb{R}})$ denote the $\bbb{A}^1$-homotopy category of simplicial presheaves on smooth schemes over $\bbb{R}$ in the sense of Morel-Voevodsky. Let $ \calH_{\Gal_{\bbb{R}}}(\sSet)$ denote the homotopy category of genuine $G$-spaces for $G = \Gal_{\bbb{R}}$. There is a symmetric monoidal functor $$LB:\calH_{\bbb{A}^1}(\spaces{\bbb{R}}) \to \calH_{\Gal_{\bbb{R}}}(\sSet),$$ called Betti realization, which takes a smooth scheme $X$ over $\bbb{R}$ to the complex points $X(\bbb{C})$ with the $\Gal_{\bbb{R}}$-action induced from the tautological action of $\Gal_{\bbb{R}}$ on $\bbb{C}$. See \cite[3, 3.3]{morelvoevodsky1998}.
$LB$ furthermore determines a functor after stabilization. Namely, let $\calH_{\bbb{A}^1}(\Spt^{\bbb{P}^1}(\bbb{R}))$ denote the $\bbb{A}^1$-homotopy category of $\bbb{P}^1$-spectra over $\bbb{R}$. As above, let $ \calH_{\Gal_{\bbb{R}}}(\Spt)$ denote the homotopy category of genuine $G$-spectra for $G = \Gal_{\bbb{R}}$. There is a commutative diagram of Betti-realization functors \cite[4.4]{HO_Galois} $$\xymatrix{\calH_{\bbb{A}^1}(\spaces{\bbb{R}}) \ar[rr]^{LB} \ar[d]^{(\Sigma^{\infty}_{\bbb{P}^1})_+}&& \ar[d]^{(\Sigma^{\infty}_{S^{\bbb{C}}})_+}\calH_{\Gal_{\bbb{R}}}(\sSet)\\ \calH_{\bbb{A}^1}(\Spt^{\bbb{P}^1}(\bbb{R})) \ar[rr]^{LB}&& \calH_{\Gal_{\bbb{R}}}(\Spt).}$$
For $k$ a field, Morel has shown that the endomorphisms of the sphere $ \End_{\calH_{\bbb{A}^1}(\Spt^{\bbb{P}^1}(k))}(\Sphere)$ are isomorphic to the Grothendieck-Witt group $\GW(k)$ of $k$ \cite[Corollary 1.24]{morel} (see \cite[footnote p~2]{Hoyois_lef} about the non-perfect case), defined to be the group completion of the semi-ring of symmetric bilinear forms. $\GW(k)$ is generated by $\langle a \rangle$ for $a \in k^*/(k^*)^2$ and has relations given by $\langle u \rangle + \langle -u \rangle = \langle 1 \rangle + \langle -1 \rangle$ and $\langle u \rangle + \langle v \rangle = \langle u + v\rangle + \langle (u+v)uv \rangle$ for $u,v\in k^*$ and $u + v \neq 0$. See \cite[Lemma 3.9]{morel} who cites \cite{milnor73}. The element $\langle a \rangle$ corresponds to the bilinear form $(x,y) \mapsto a xy$. We will compare $\GW(\bbb{R})$ and $\A(\Gal_{\bbb{R}})$ and for this, we need the following well-known constructions.
For a separable field extension $k \subseteq L$, there is a map $\Tr_{L/k}: \GW(L) \to \GW(k)$ defined by sending a symmetric bilinear form $b: V \times V \to L$ to the composition $\Tr_{L/k} b : V \times V \to k$ of the form $b$ with the field trace map $\Tr_{L/k}: L \to k$. In $\Tr_{L/k} b$, the vector space $V$ is now viewed as a vector space over $k$.
The map sending a bilinear form to the ordered pair of its rank and signature determines an isomorphism $$\xymatrix{\GW(\bbb{R}) \ar[rrr]_{\cong}^{\text{rank} \times \text{signature}}&&& \Z \times \Z},$$ as can be checked using the generators and relations described above and Sylvester's law of inertia.
Since $LB$ is a functor, it determines a map $$LB: \GW(\bbb{R}) \cong \End_{\calH_{\bbb{A}^1}(\Spt^{\bbb{P}^1}(\bbb{R}))}(\Sphere) \to \End_{\calH_{\Gal_{\bbb{R}}}(\Spt)}(\Sphere) \cong \A(\Gal_{\bbb{R}}).$$ Since $LB$ is symmetric monoidal, $LB(\chi Y) = \chi(LB Y)$ for every strongly dualizable object $Y$ of $\calH_{\bbb{A}^1}(\Spt^{\bbb{P}^1}(\bbb{R}))$. It is a result of Hoyois \cite[Theorem 1.9]{Hoyois_lef} that for a separable field extension $k \subseteq L$, the fully dualizable spectrum $\Sigma^{\infty}_{\bbb{P}^1} \Spec L_+$ has Euler characteristic $\chi \Spec L = \Tr_{L/k} \langle 1 \rangle$. Thus in $\End_{\calH_{\bbb{A}^1}(\Spt^{\bbb{P}^1}(\bbb{R}))}(\Sphere)$, we have that $\chi(\Sigma^{\infty}_{\bbb{P}^1} \Spec \C_+) = \langle 1 \rangle + \langle -1 \rangle$. Since $LB (\Sigma^{\infty}_{\bbb{P}^1} \Spec \C_+) $ is the element of $\calH_{\Gal_{\bbb{R}}}(\Spt)$ corresponding to the finite $ \Gal_{\bbb{R}}$-set given by $\Gal_{\bbb{R}}$ with left translation, it follows that the map $$ \GW(\bbb{R}) \to \A(\Gal_{\bbb{R}})$$ is determined by $$\xymatrix{ \GW(\bbb{R}) \ar[d]_{\text{rank} \times \text{signature}} \ar[r] & \A(\Gal_{\bbb{R}}) \ar[d]^{(T \mapsto \vert T \vert) \times (T \mapsto \vert T^{\Gal_{\bbb{R}} }\vert)} \\ \bbb{Z}^2 \ar[r]^1 & \bbb{Z}^2},$$ and in particular is an isomorphism. Here the notation $\vert T \vert$ for a finite set $T$ denotes the cardinality of $T$. We remark that there is much more to say about the relationship between the Burnside ring and the Grothendieck-Witt group. See, for example, the recent work of Kyle Ormsby and Jeremiah Heller \cite{HO_Galois}.
The following proposition about an elliptic curve is one consequence of the existence of a realization over $k=\mathbb{R}$. In the next section, we show that the analogous result often fails to hold when $k$ is a more complicated field and then use this fact to show that a suitable realization functor cannot exist.
\begin{pr}\label{chirER=0}
Let $E$ be an elliptic curve over $\R$. Then \begin{enumerate}
\item \label{chiRepEC=0} For any ring $R$, we have $\sum_{i=0}^{2}(-1)^i H^i (E(\bbb{C}),R) = 0$ in $\Rep(\Gal_{\bbb{R}},R).$
\item \label{chiRepetEC=0} For a finite ring $R$, we have $\sum_{i=0}^{2}(-1)^i H^i_{\et}(E_{\bbb{C}},R) = 0$ in $\Rep(\Gal_{\bbb{R}},R).$
\end{enumerate}
\end{pr}
\begin{proof}
$E$ is strongly dualizable in $\calH_{\bbb{A}^1}(\spaces{\bbb{R}})$, and by \cite[Theorem 1.5]{Hoyois_lef}, the Euler characteristic of $E$ is $0$. Since $LB$ is a symmetric monoidal functor, it follows that $LB E$ is strongly dualizable in $\calH_{\Gal_{\bbb{R}}}(\Spt)$ and that $\chi LB E = 0$.
\noindent \eqref{chiRepEC=0}: By \cite{Illman}, $LB E$ is a finite $\Gal_{\bbb{R}}$-CW complex, so we may apply Proposition~\ref{assocchiX=repchi} \eqref{pr:assocchiX=repchi:case1} to $LB E$. It follows that $ \sum_{i=-\infty}^{\infty} (-1)^i H^i(LB E,R) = 0$. Since $LB E = E(\C)$ and $H^i(E(\C), R) = 0$ for $i<0$ and $i>2$, we have $\sum_{i=0}^{2}(-1)^i H^i (E(\bbb{C}),R) = 0$ as claimed.
\noindent \eqref{chiRepetEC=0}: For $R$ finite, we have a natural isomorphism $H^i (E(\bbb{C}),R) \cong H^i_{\et}(E_{\bbb{C}},R) $ by \cite[III Theorem 3.12]{MilneECbook} \cite[XI]{sga4III}, so \eqref{chiRepetEC=0} follows from \eqref{chiRepEC=0}.
\end{proof}
\begin{rmk} We can also verify Proposition \eqref{chirER=0} directly. Indeed, when $R=\mathbb{C}$, the representations $H^{i}(E(\mathbb{C}), R)$ can be described explicitly as follows.
Let $\C$ denote the trivial representation and $\C(1)$ denote the sign representation in $\Rep(\Gal_{\bbb{R}},\C)$. Then in $\Rep(\Gal_{\bbb{R}},\C)$, we have equalities $H^0(E(\bbb{C}),\C) = \C$, $H^2(E(\bbb{C}),\C) = \C(1)$, $H^1(E(\bbb{C}),\C) = \C + \C(1)$. Indeed, the first equality follows because $\Gal_{\R}$ acts trivially on the single connected component of $E(\C)$; the second equality follows because the non-trivial element of $\Gal_{\R}$ reverses orientation; the third equality follows because the cup product gives an isomorphism of $\Gal_{\R}$-representations $\wedge^2 H^1(E(\bbb{C}),\C) \cong H^2(E(\bbb{C}),\C)$, and in the representation ring, $H^1(E(\bbb{C}),\C)$ is a direct sum of one-dimensional representations.
\end{rmk}
\section{Enriched Euler characteristics and restrictions on genuine $\Gal_{k}$-realization}\label{Gk-realization}
Now let $G$ be a profinite group, or more specifically a Galois group. One could hope to construct a homotopy category $\calH_{G}(\Spt)$ of genuine $G$-spectra or $G$-pro-spectra, and a Burnside ring $\A(G) = \End_{\calH_{G}(\Spt)}(\Sphere)$. For example, homotopy theories of $G$-spectra are constructed in \cite{Fausk} \cite{Quick-Profinite_G-spectra} and \cite{BarwickI}. We have in mind that there is some appropriate sort of $G$-set and corresponding Burnside category such that a suspension spectrum functor is fully faithful into $\calH_{G}(\Spt)$, and that objects in $\calH_{G}(\Spt)$ can be constructed from colimits of suspensions of these $G$-sets, as is the case when $G$ is finite. The point here being that the Euler characteristic of a $G$-set would be itself, and spaces would be built from $G$-sets, giving credibility to an analogue of Proposition \ref{assocchiX=repchi}. For any profinite group, a category of genuine $G$-equivariant spectra has been constructed from a Burnside category by Barwick in \cite{BarwickI}, and studied by Barwick, Glasman, and Shah in \cite{BGSII}.
Let $k$ be a field, and let $\kbar$ denote an algebraic closure of $k$ and $\Gal_k = \Gal(\kbar/k)$. One could furthermore hope to construct an {\em \'etale realization functor} $$L \Et: \calH_{\bbb{A}^1}(\Spt^{\bbb{P}^1}(k)) \to \calH_{\Gal_k}(\Spt)$$ from the stable $\bbb{A}^1$-homotopy category of $\bbb{P}^1$-spectra over $k$ to genuine $\Gal_k$-spectra or pro-spectra, appropriately completed away from the characteristic of $k$. For example, Quick has constructed an \'etale realization functor from the stable $\bbb{A}^1$-homotopy category to a stable homotopy category of profinite spaces in \cite{Quick-stable_realization}. \'Etale realization functors have been constructed and studied by Ayoub \cite{Ayoub-realization_etale} and \cite[Section~7.2]{CD-etale_motives} in a generalization of the following context. The category of Voevodsky motives is analogous to the stable $\bbb{A}^1$-category \cite[Section 2]{AyoubICM}. Let $\Lambda$ be the ring $\Lambda= \bbb{Z}/\ell$ for a prime $\ell$ different from the characteristic of $k$, and assume that $k$ is perfect. Let $D(\Sheaves(k_{\et}, \Lambda))$ denote the derived category of sheaves of $\Lambda$-modules on the small \'etale site of $\Spec k$. There is an \'etale realization functor from Voevodsky motives to $D(\Sheaves(k_{\et}, \Lambda))$. Since a sheaf of $\Lambda$-modules on the small \'etale site of $\Spec k$ is a $\Lambda$-module with an action of $\Gal_k$, the derived category $D(\Sheaves(k_{\et}, \Lambda))$ is similar to spectra (of $H \Lambda$-modules) equipped with an action of $\Gal_k$. We wish to draw a similarity between $D(\Sheaves(k_{\et}, \Lambda))$ and a homotopy theory of spectra with a $\Gal_k$-action and contrast $D(\Sheaves(k_{\et}, \Lambda))$ with a notion of genuine $\Gal_k$-spectra. For example, the endomorphisms of the symmetric monoidal unit of $D(\Sheaves(k_{\et}, \Lambda))$ is $\Lambda$, in contrast to the Burnside ring. Since \'etale realization functors exist in powerful contexts, one could hope for an \'etale realization functor to genuine Galois equivariant spectra or pro-spectra.
We have in mind that applying the \'etale realization functor $L \Et$ to the suspension spectrum of a smooth scheme gives an appropriate suspension spectrum of the \'etale topological type of Artin-Mazur \cite{Artin-Mazur} and Friedlander \cite{Friedlander}. In other words, we have in mind that applying $L \Et$ is compatible with an unstable \'etale realization functor $L \Et$ as constructed by Isaksen \cite{Isaksen-etale_realization} in the non-equivariant context. From this, we are lead to the following two expectations.
\begin{itemize}
\item Since the \'etale topological type $\Et ((X \times Y)_{\kbar})$ is equivalent to the product $\Et (X_{\kbar}) \times \Et (Y_{\kbar})$, cf. \cite[Corollarie~1.11]{sga4andhalf}, it is reasonable to hope that $L \Et$ is symmetric monoidal.
\item Since the \'etale cohomology of a smooth scheme $X$ with finite coefficients $R$ is isomorphic to the cohomology of the \'etale topological type $\Et X$ with coefficients in $R$, this would result in an isomorphism of $\Gal_k$-representations $$H^i_{\et}(X_{\kbar}, R) \cong H^i(L \Et X_{\kbar}, R).$$ This implies that if the Euler characteristic of a genuine $G$-spectrum or pro-spectrum is connected to the $\Gal_k$-representations given by its cohomology groups as in Proposition \ref{assocchiX=repchi}, then applying a homomorphism $\assoc$ from $\A(G)$ to a representation ring sends this Euler characteristic to $\sum_{i=-\infty}^{\infty} H^i_{\et}(X_{\kbar}, R).$
\end{itemize}
The purpose of this paper is to show that, for many fields $k$, it is impossible to simultaneously satisfy these hopes, i.e., such an \'etale realization functor does not exist.
\begin{tm}\label{tm:nonexistence}
Let $k$ be a global field, local field with infinite Galois group, or finite field. Let $p$ be a prime different from the characteristic of $k$. It is impossible to simultaneously construct all of the following \begin{enumerate}
\item A symmetric monoidal category $\calH_{\Gal_k}(\Spt)$, enriched over abelian groups. Let $\A(\Gal_k)= \End_{\calH_{\Gal_k}(\Spt)}(\Sphere)$ denote the endomorphisms of the symmetric monoidal unit $\Sphere$.
\item For all $n$, a representation ring $\Rep( \Gal_k, \Z/{p^n})$ with coefficients in $\Z/{p^n}$.
\item \label{thm:nonexistence:hyp:realization} A symmetric monoidal additive functor $L \Et: \calH_{\bbb{A}^1}(\Spt^{\bbb{P}^1}(k)) \to \calH_{\Gal_k}(\Spt)$, and ring homomorphisms $\assoc_n: \A(G) \to \Rep(\Gal_k, \Z/p^n)$ such that $$\assoc_n \chi (L\Et \Sigma^{\infty}_{\bbb{P}^1}X_+) = \sum_{i=-\infty}^{\infty} (-1)^i [H^i_{\et} (X_{\kbar}, \Z/p^n)]$$ for smooth proper schemes $X$ over $k$.
\end{enumerate}
\end{tm}
\begin{lm}\label{chirepEFellneq0pn}
Let $E$ be an elliptic curve over a finite field $\bbb{F}_{\ell}$, and let $p$ be any prime not dividing $\ell$. Suppose that for all $n$, we have a ring $\Rep( \Gal_k, \Z/{p^n})$ as in Theorem \ref{tm:nonexistence}. There exists a positive integer $n$, such that $\sum_{i=-\infty}^{\infty} (-1)^i [H^i_{\et} (E_{\kbar}, \Z/p^n)]$ is non-zero in $\Rep(\Gal_k,\bbb{Z}/p^n)$.
\end{lm}
\begin{proof}
Suppose the contrary. Then for all $g$ in $\Gal_k$ and all $n$, we have the equality $$\Tr g \vert H^1(E_{\kbar}, \Z/p^n) = \Tr g \vert H^0(E_{\kbar}, \Z/p^n) + \Tr g \vert H^2(E_{\kbar}, \Z/p^n)$$ in $\Z/p^n$. By definition, $H^i(E_{\kbar}, \Q_p) = \Q_p \otimes \varprojlim_n H^i(E_{\kbar}, \Z/p^n)$ \cite[V I]{MilneECbook}. Therefore \begin{equation}\label{TrgHiEQp}\Tr g \vert H^1(E_{\kbar}, \Q_p) = \Tr g \vert H^0(E_{\kbar}, \Q_p) + \Tr g \vert H^2(E_{\kbar}, \Q_p)\end{equation} in $\Q_p$. Let $F$ be the geometric Frobenius in $\Gal_k$, i.e., the inverse of the arithmetic Frobenius $a \mapsto a^{\ell}$ for all $a$ in $\kbar$. By the Weil Conjectures, $\Tr F \vert H^1(E_{\kbar}, \Q_p) = a_1 + a_2$, where $a_i$ are algebraic integers with absolute value $\ell^{1/2}$ \cite[Theorem~IV.1.2]{Freitag_Kiehl}. Since $\Gal_k$ acts trivially on $H^0(E_{\kbar}, \Q_p)$ and there is an isomorphism $H^2(E_{\kbar}, \Q_p) \cong \Q_p(1)$, Equation~\eqref{TrgHiEQp} for $g=F$ becomes $$ a_1 + a_2 = 1+\ell.$$ Taking the absolute value of both sides, we have $$ 2 \ell^{1/2}= \vert a_1 \vert + \vert a_2 \vert \geq \vert a_1 + a_2 \vert = 1 + \ell ,$$ which is impossible for any prime power $\ell$.
\end{proof}
\begin{proof}
(of Theorem \ref{tm:nonexistence}) Suppose $k$ is a finite field $k=\bbb{F}_{\ell}$. Choose an elliptic curve over $k=\bbb{F}_{\ell}$. To see that this is indeed possible, note that the Weierstra\ss ~equation $y^2 = x (x-1)(x+1)$ determines an elliptic curve when $\bbb{F}_{\ell}$ has odd characteristic. When $\ell=2^d$, the equation $y^2+y=x^3$ produces the desired elliptic curve. By Lemma \ref{chirepEFellneq0pn}, there exists a positive integer $n$ such that $\sum_{i=-\infty}^{\infty} (-1)^i [H^i_{\et} (E_{\kbar}, \Z/p^n)]$ is non-zero in $\Rep(\Gal_k,\bbb{Z}/p^n)$. By hypothesis \eqref{thm:nonexistence:hyp:realization} of the theorem, $\assoc_n \chi (L\Et \Sigma^{\infty}_{\bbb{P}^1}E_+) = \sum_{i=-\infty}^{\infty} (-1)^i [H^i_{\et} (E_{\kbar}, \Z/p^n)]$. It follows that $\chi (L\Et \Sigma^{\infty}_{\bbb{P}^1}E_+) $ is non-zero. Since $L \Et$ is a symmetric monoidal functor (by \eqref{thm:nonexistence:hyp:realization}), it follows that $L\Et \chi (\Sigma^{\infty}_{\bbb{P}^1}E_+) = \chi (L\Et \Sigma^{\infty}_{\bbb{P}^1}E_+)$, and therefore $\chi (\Sigma^{\infty}_{\bbb{P}^1}E_+)$ is non-zero. This contradicts \cite[Theorem 1.5]{Hoyois_lef}.
Suppose $k$ is a local field with infinite Galois group, i.e., $k$ is a finite extension of $\Q_{\ell}$ or $\bbb{F}_{\ell}((t))$, and let $\mathcal{O}_{k}$ denote the integral closure of $\Z_{\ell}$ or $\bbb{F}_{\ell}[[t]]$ in $k$. We may choose an elliptic curve $E$ over $k$ with good reduction by choosing an elliptic curve over the residue field (as discussed in the previous paragraph) and then applying \cite[III 7.3]{sga1}. Thus we have an elliptic curve $f: \mathcal{E} \to \Spec \mathcal{O}_k$. Let $E_0 = \mathcal{E} \times_{\Spec \mathcal{O}_k} \Spec \bbb{F}_{\ell}$ denote the fiber over the closed point. By smooth-proper base change, \cite[VI \S4 Corollary 4.2]{MilneECbook} $R^if_* \mathbb{Z}/p^n$ is a locally constant sheaf on $\Spec \mathcal{O}_k$ whose stalk at a geometric point $s: \Spec \Omega \to \Spec \mathcal{O}_k$ is $H^i(\mathcal{E}_s, \mathbb{Z}/p^n)$, where $\mathcal{E}_s = \mathcal{E} \times_{\Spec \mathcal{O}_k} \Spec \Omega$. It follows that the $\Gal_{k}$-action on $H^i(E_{\kbar}, \Z/p^n)$ factors through the quotient map $$\Gal_{k} \to \Gal(k^{\text{un}}/k) \cong \pi_1(\Spec \mathcal{O}_k) \cong \Gal_{\bbb{F}_{\ell^d}},$$ where $k^{\text{un}}$ is the maximal unramified extension of $k$ and $\bbb{F}_{\ell^d}$ is the residue field. It likewise follows that $H^i(E_{\kbar}, \Z/p^n)$ is isomorphic to $H^i((E_0)_{\overline{\bbb{F}_{\ell}}}, \Z/p^n)$ as $\Gal_{\bbb{F}_{\ell^d}}$-modules. This reduces the claim to the case where $k$ is a finite field.
Lastly, suppose that $k$ is a global field. Choose an elliptic curve $E$ over $k$, and let $\nu$ be a place not dividing the discriminant, so $E$ has good reduction at $\nu$. Let $k_{\nu}$ be the completion of $k$ at $\nu$. The decomposition group of $\nu$ in $\Gal_k$ gives an injection $\Gal_{k_{\nu}} \to \Gal_k$, and the $\Gal_{k_{\nu}}$-action on $H^i(E_{\kbar}, \Z/p^n)$ is isomorphic to the $\Gal_{k_{\nu}}$-action on $H^i(E_{\overline{k_{\nu}}}, \Z/p^n)$. By the previous paragraph, we may again reduce the claim to the case where $k$ is a finite field.
\end{proof}
We give a slight variation on Theorem \ref{tm:nonexistence}, where only the representation rings $\Rep( \Gal_k, \Z/p)$ with $p$ prime appear.
\begin{tm}\label{tm:nonexistencev2}
Let $k$ be a global field, a local field with infinite Galois group, or a finite field. It is impossible to simultaneously construct all of the following \begin{enumerate}
\item A symmetric monoidal category $\calH_{\Gal_k}(\Spt)$, enriched over abelian groups. Let $\A(\Gal_k)= \End_{\calH_{\Gal_k}(\Spt)}(\Sphere)$ denote the endomorphisms of the symmetric monoidal unit $\Sphere$.
\item \label{trassumption2} For infinitely many primes $p$, a representation ring $\Rep( \Gal_k, \Z/p)$ with coefficients in $\Z/p$.
\item \label{trassumption3} A symmetric monoidal additive functor $L \Et: \calH_{\bbb{A}^1}(\Spt^{\bbb{P}^1}(k)) \to \calH_{\Gal_k}(\Spt)$, and a homomorphism of rings $\assoc: \A(G) \to \Rep(\Gal_k, \Z/p)$ such that $\assoc \chi (L\Et \Sigma^{\infty}_{\bbb{P}^1}X_+) = \sum_{i=-\infty}^{\infty} (-1)^i H^i_{\et} (X_{\kbar}, \Z/p)$ for smooth proper schemes $X$ over $k$.
\end{enumerate}
\end{tm}
\begin{proof}
Replace Lemma \ref{chirepEFellneq0pn} with Lemma \ref{chirepEFellneq0p} below in the proof of Theorem \ref{tm:nonexistence}.
\end{proof}
\begin{lm}\label{chirepEFellneq0p}
Let $E$ be an elliptic curve over a finite field $\bbb{F}_{\ell}$. Let $p$ be any prime greater than the cardinality of $E(\bbb{F}_{\ell})$. Then $\sum_{i=-\infty}^{\infty} (-1)^i [H^i_{\et} (E_{\kbar}, \Z/p) ]$ is non-zero in $\Rep(\Gal_k,\bbb{Z}/p)$.
\end{lm}
\begin{proof}
Let $k= \bbb{F}_{\ell}$ and let $F$ be the geometric Frobenius in $\Gal_k$, i.e., the inverse of the arithmetic Frobenius $a \mapsto a^{\ell}$ for all $a$ in $\kbar$. By assumption \eqref{trassumption2}, it suffices to show that $\sum_{i=0}^{2} (-1)^i \Tr F\vert H^i_{\et} (E_{\kbar}, \Z/p) $ is non-zero in $\Z/p$. By \cite[pg.~292]{MilneECbook} the trace of $F$ equals the trace of the Frobenius endomorphism of $E$. By the Lefschetz Trace Formula, $$\sum_{i=0}^{2} (-1)^i \Tr F\vert H^i_{\et} (E_{\kbar}, \Z/p) = \vert E(\bbb{F}_{\ell}) \vert$$ in $\Z/p$ (see \cite[Proposition~7.1]{WIN3-2} for further explanation on the mod $p$ version; it is contained in \cite{sga4andhalf} and can also be deduced from \cite{Hoyois_lef}). Since $p$ is greater than $ \vert E(\bbb{F}_{\ell}) \vert$ by assumption and all elliptic curves contain at least one rational point, it follows that $\sum_{i=0}^{2} (-1)^i \Tr F\vert H^i_{\et} (E_{\kbar}, \Z/p)$ is non-zero as desired.
\end{proof}
\begin{rmk}
By Section \ref{GR-realization}, when $k=\R$, there does exist a symmetric monoidal category $\calH_{\Gal_k}(\Spt)$ and a symmetric monoidal functor $L \Et: \calH_{\bbb{A}^1}(\Spt^{\bbb{P}^1}(k)) \to \calH_{\Gal_k}(\Spt)$ satisfying the conditions of Theorems \ref{tm:nonexistence} and \ref{tm:nonexistencev2}.
\end{rmk}
\section{Acknowledgements}
We wish to thank Joseph Ayoub, Gunnar Carlsson, Akhil Mathew, and all the participants of the AIM workshop on Derived Equivariant Algebraic Geometry, June 13--17, 2016, organized by Andrew Blumberg, Teena Gerhardt, Michael Hill, and Kyle Ormsby, and additionally including the participation of Clark Barwick, Elden Elmanto, Saul Glasman, Jeremiah Heller, Denis Nardin, and Jay Shah. It was a pleasure discussing these ideas with you. It is also a pleasure to thank Serge Bouc for useful correspondence concerning Corollary \ref{co:associdempAg}, and Alexander Duncan for writing and running some GAP code to check an example of this corollary. The second named author also wishes to thank the Institut Mittag--Leffler for their hospitality during their special program ``Algebro-Geometric and Homotopical Methods," where she worked on this paper, and Tom Bachmann for interesting comments.
Jesse Leo Kass was partially sponsored by the Simons Foundation under Grant Number 429929, and the National Security Agency under Grant Number H98230-15-1-0264. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein. This manuscript is submitted for publication with the understanding that the United States Government is authorized to reproduce and distribute reprints.
Kirsten Wickelgren was partially supported by National Science Foundation Awards DMS-1406380 and DMS-1552730, and an AIM 5-year fellowship. She wishes to thank the Institut Mittag-Leffler for a very pleasant stay at the special program ÒAlgebro-geometric and homotopical methods,Ó while working on this project.
}
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TITLE: Does $x^n = 1$ for all $x$ in $G$ imply that $G$ is abelian?
QUESTION [0 upvotes]: Does $x^n = 1$ for all $x$ in $G$ imply that $G$ is abelian?
For $n = 2$ i know that this is correct. For odd prime numbers i know that $H_p$ (Heisenberg group modulo p) is a counter-example. But what about $n = mk$. Does this case have a counter-example too? If not, how can i go about proving this statement.
REPLY [3 votes]: Let $n\in\mathbb{N}$, and suppose $n>2$. Does there exist a non-abelian group $G$ in which $x^n=1$ for all $x\in G$? The answer is Yes.
For an odd prime $p$ you know that $H_p$ satisfies $x^p=1$ for all $x$.
For $n=4$ the Dihedral group $D_8$ of order $8$ (or the quaternion group) satisfies $x^4=1$ for all $x$.
Now let $n>4$. If $n$ is divisible by an odd prime $p$ so that $n=pk$, then the Heisenberg group $H_p$ satisfies $x^n=(x^{p})^k=1$ for all $x$.
If $n=2^k$ and $k\geqslant 2$ then $D_8$ satisfies $(x^4)^{2^{k-2}}=1$.
| 121,396
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These are my novels on Lulu, please check them out.
One of the great things about growing up in Wilkes-Barre, Pennsylvania was that every block had its own candy store. I had it even better; I had three candy stores, a movie theater, an ice cream counter, and a pizza parlor all in walking distance of my house. To top it off my house was next door to the city swimming pool; which also happen to be the gathering place for all the kids in the summer.
George’s store was a block away and he had all of the classic penny candy.
I must have run up to that store a thousand times with the coins that my mom could afford to give me. It was always a difficult choice. Would it be the milk chocolate squares called Grade A’s or the long lasting sweet balls called Jawbreakers, how about Mary Jane’s, love the peanut butter. Then there were things like the wax tubes with the different flavors of syrup in them and, of course, Bazooka bubble gum.
George’s also offered a nickel soda but you had to return the bottle as soon as you were finished with it. I spent quite some time standing outside that little store drinking a cherry or an orange soda.
The big candy bars were a nickel. I wouldn’t buy them unless I had lots of money, relatively speaking. Why would I buy one candy bar when I could get a variety of things? Those coins didn’t come easy and I had to get my bang for the buck, or should I say penny.
There was another little store on the same street. I don’t remember its name; it closed when I was still a kid. This store was the pot of gold at the end of the rainbow. In the summer, if I could get a hold of thirty cents, I would go and get the special ice cream cone. The man would take a rectangular pint of ice cream, that’s how pints were sold in those days, and he would use a big knife to cut it in thirds. I would get one third of a pint of ice cream in a cool rectangular topped code; that was a real treat.
Now you can get some of the same candy in big bags but it’s not the same. The adventure was standing in front of that glass staring at the choices and carefully pointing out the ones that you wanted. That is experience that will not be duplicated in a society of instant gratification.
Our neighborhood wasn’t very affluent, but we were all in the same boat. Every kid would get a few precious coins now and then and the candy is where we would go. Penny candy was a big deal, and the corner candy store was our dearest friend.
Priceless descriptions of ‘our neighborhoods’ Pete. Thanks. I can see … and smell … the penny candies as I read your words. And … I can feel the hair stand up on the back of my neck (my head is bald; no feeling there.) as I ‘hear’ the shrill chastisement of the George brothers annoyance coming down on me with a vengeance as I changed my selections repeatedly.
Thanks Dave. Yeah I remember being chased out of the store a few times myself.
| 281,056
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Tuesday, January 15, 2008
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| 53,705
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. Recent research has linked gum disease with heart disease, underscoring the importance of caring for your gums.
Gingivitis is often the first development in gum disease, and because gingivitis is inflammation, patients sometimes believe that treatment is unnecessary. But did you know that the bacteria responsible for gingivitis actually inhibit the immune system? Your Dallas dentist, Dr. Diep Truong, explains how gingivitis fools the immune system.
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The bacteria responsible for gingivitis are called porphyromonas gingivalis. These bacteria have an interesting effect on the body. Their presence triggers the production of an anti-inflammatory molecule called IL-10. This molecule, in turn, stops T-cells, blood cells which fight diseases and harmful substances, from working against the infection.
Fighting Gum Disease
Preventing gum disease involves keeping teeth clean of plaque and tartar, which contain harmful bacteria. To keep teeth clean, brush twice a day and floss once. When brushing, be sure to angle the bristles of the toothbrush toward the gum line. Changing your diet can also prove useful in the fight against gum disease. A diet consisting of too much sugar can feed bacteria and contribute to the formation of plaque. In addition to implementing good homecare and a healthy diet, visit your Dallas dentist, Dr. Truong, every six months for regular checkups and cleanings.
Schedule a Visit with Your Dallas Dentist
Have you had a dental visit in the past six months? Regular dental visits are an important part of your oral health regimen. If you’re due for a checkup,.
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| 310,962
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TITLE: Does this type of real functions have a name ? What are their characteristics?
QUESTION [6 upvotes]: Let $n > 0$ be an integer. Let $f:\mathbb R^+ \times \mathbb R^+ \rightarrow \mathbb R_0$ be a symmetric function such that the $(n+1) \times k$ matrix
\begin{equation}
\mathbb M=
\begin{pmatrix}
f(s_{k-n+1},s_1) & \ldots & f(s_{k-n+1},s_k)\\
\vdots & & \vdots \\
f(s_{k+1},s_1) & \ldots & f(s_{k+1},s_k)\\
\end{pmatrix}
\end{equation}
has at most $n$ linearly independent rows for any $k>n$ and real sequence $s_1<s_2<\ldots<s_{k+1}$. What does that say about $f$ ? Do such function have a name ? What are their characteristics/properties ? Is there an easier "definition" that would be equivalent, without matrices for example? Or a stronger "easier" property that would imply the above property ?
REPLY [1 votes]: This is a special case, but it is also a broad class. It isthe exact solution if $n=1$. If you take $f(x, y) = h(\min \{ x, y\} ) g(\max \{x,y\} ) $, then the first $(k-n+1) $ columns are multiple of $( g(s_{k+n-1}), \ldots , g(s_{k+1}) ) $ . The rank remain then unchanged if we discard the first $k-n$ columns; now we have $k-(k-n) = n $ columns, so that the rank is at most $n$ as desired. Note also that such functions are symmetric.
Conversely, suppose $n=1$ and set $k=2$. Taking determinant one obtains (the function is never zero)
$$ \frac{ f(s_2, s_2) }{ f(s_3, s_2) } = \frac{ f(s_2, s_1) }{f(s_3, s_1) } $$
Which means that the quotient on the right does not depend on $s_1 < s_2$. Taking $s_3 = L$ very big we get that for all $s_1 < s_2 < L$ :
$$ f(s_2, s_1) = \frac{ f(s_2, s_2) }{f(L, s_2) } f(L, s_1) = g(s_2) \cdot h(s_1) $$
By symmetry we will have that for $s_2 < s_1 < L$
$$ f(s_1, s_2) = f(s_2, s_1) = g(s_1) h(s_2) $$
So that $f(x, y) = h(\min \{x, y\}) g(\max \{x, y\}) $. We can repeat this argument for greater $L$, but since the new $h, g$ must agree with the old ones on $s_1 , s_2 < L$, we will get well defined functions $h, g$ on all of $\mathbb{R}^+$. Hope you enjoy!
| 160,077
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| 14,140
|
TITLE: possible determinants of permutations
QUESTION [4 upvotes]: this is taken from Gilbert Strang's Linear Algebra book:
What are all the possible $4\times4$ determinants of $I + P_{even}$? (P - permutation matrix)
I seem to be stuck on this question except for the one fact that the diagonal is always going contain $1$'s and that even permutations themselves have determinant $1.$
Thanks in advance
REPLY [2 votes]: Decompose the permutation into cycles and consider the cycle lengths. There are five possibilities:
1+1+1+1 (this notation means there are four 1-cycles). This is an even permutation and $P=I$. So, $\det(I+P)=2^4$.
1+1+2. This is an odd permutation.
1+3. This is an even permutation and $P$ is permutation similar to $\pmatrix{1\\ &0&1&0\\ &0&0&1\\ &1&0&0}$. Hence $\det(I+P)=2^2$.
2+2. This is an even permutation and $P$ is permutation similar to $\pmatrix{0&1\\ 1&0\\ &&0&1\\ &&1&0}$. Hence $\det(I+P)=0$.
4 . The permutation is a 4-cycle, hence odd.
In conclusion, $\det(I+P)$ can be $\color{red}{0,\,2^2}$ or $\color{red}{2^4}$.
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, but on those things which became their pet issues, they saw a binary choice — “you are either with us or against us.” The issues which they divided were held up to be gospel issues, issues in which there was no place for disagreement or a difference in perspective. Black and white, right or wrong. While some revisionist historians may argue that it was a disagreement understood at the time to be about cultural issues, even a cursory read of the letters of secession shows this to be wrong. While the things which caused the secession of 1857 were most certainly about nonessentials, the people involved saw them as things core to the gospel.
In 1882, there was another secessionist movement, this time over freemason lodge membership. There was nothing forcing boards of elders to allow members to be members of lodges, there was no statement by the General Synod allowing (or favoring) lodge membership. Indeed, the General Synod discouraged it. But because other churches somewhere else might allow their members to be members of lodges, a secession was required. Not because one is being forced to live and worship and practice their faith in a way that conflicted with their conscience, but because “somebody, somewhere might be doing or thinking something that I don’t like.” And so, this became a binary issue. Black or white, right or wrong. This became a gospel issue, and issue over which it was worth the risk of splitting the church apart again, leaving yet another wound in the body of Christ.
***
These are only two examples in my little corner of the Kingdom of God. Throughout history and across traditions, there have been topics, issues, that are held up as gospel issues that one must choose, you must choose this or that, black or white, right or wrong. No ability to wrestle, to struggle, to be in fellowship with disagreement. Whereas Joshua told his people to serve God or foreign gods (Josh 24:14-15), the narrative at times of tension and conflict are: choose this day your stance on this particular topic, because this topic determines whether or not you are a part of Christ.
This, however, is a false narrative, a false choice, a false dichotomy. To claim that we cannot be in relationship and fellowship and that we must break our covenantal promises because, while we all agree on the foundations of our faith and although we have all made the same promises, some see one topic differently.
This false narrative is rearing its ugly and sinful head in the Reformed Church yet again. One’s stance on human sexuality has become elevated to the single “gospel issue” which seems to matter by many in the fundamentalist/evangelical wing of the communion. The means of grace (the sacraments), the nature of covenant, salvation, or even the covenant promises that we had made to God and each other when we were ordained to ecclesiastical office, all these take second place. The narrative is that there must be a choice forced between two binary poles. This narrative, however, is artificial. This narrative is little more than a way to scorch the earth in order to try to force one into a sense of the worldly understanding of “victory.”
***
So often I hear, “We are tired of fighting!” To which I respond, “Then stop!” Stop fighting. Stop lobbing grenades over the walls, stop shooting artillery from your trenches. These are trenches that we have dug, they are walls that we have built, they are fights that we have initiated. Those who wish to cause the single issue of human sexuality to be the only thing that matters in covenantal fellowship wish to continue the fight until they either “win” or harm the church seeking a sense of victory. The goal is to continue the language of “us vs them” because it is known that if we are able to break free from this framework, that the fighting will stop, and no one except Christ and Christ’s church can claim victory.
And what about those who are not able or willing to make an artificial binary choice? What about those who think there is more to the church than sex, and who can have sex with whom? What about those who want to focus on living as disciples of Christ and living as a foretaste of the kingdom of God? What about those who want to love God and love others? What about those who are weary of the fighting, weary of the division, weary of the trenches and grenades and the war of attrition in which we are currently locked?
The choice is not binary. No two people can agree on everything, how much more for a church communion? The point is not to ignore differences, but to talk about them, even argue about them. For some, there are differences which are irreconcilable, but these are not the same for everyone. For one new and newly public faction, however, human sexuality seems to be a mark of the true church, but the means of grace are not. However, to insist that this must be the line in the sand for everyone is simply false.
***
So to those who wish to be the church, you are invited not into a faction, not into an alliance. You are invited, not by me, or by a leadership cadre. You are invited by Christ and by the saints who have gone before. You are invited into the church, you are invited into the Body of Christ, and into our corner of the Kingdom of Christ, the Reformed Church in America. Into this covenantal communion who have commitments to each other in the things that we see as essential (these can be found in the Government (and disciplinary and judicial procedures), the Liturgy, and the four Doctrinal Standards), while also allowing for difference with proper oversight (board of elders for members, consistory for church, classis for ministers and consistories), as well as ensuring that we live up to our covenantal promises, and fulfill the obligations which we have promised to fulfill (the synods, then, have a role in this).
There are those spinning this false narrative of an artificial binary choice which we must choose and choose in an instant, and if we allow this to control the conversation, we will never find peace, we will never find, unity, and we will never find purity. Indeed, there is no clear dividing line between the broad and problematic categories of “liberal,” “moderate,” and “conservative.” Indeed, there are conservatives who refuse to make this single issue the hill on which they are willing to die, and upon they are willing to, once again, carve up a part of Christ’s body.
We are not the world. We do not have parties, we do not have a binary opposition. We may disagree, but we are all working together for the same goal. Now we are to live into this. Understanding there are differences, and some of these differences are big. Understanding we can disagree about these differences and that we can even disagree strongly. But always understanding that Christ is far bigger than whether we sing hymns or Psalms or how we teach the Heidelberg Catechism, Christ is far bigger than the question of lodge membership, Christ is far bigger than human sexuality. Because if Christ is not enough to hold us together, then what is?
3 thoughts on “Rethinking the Artificial Binary”
Hi Matthew, I just had some clarifying questions for you, as I’m trying to better understand your post. First, if celebrating homosexuality was no different than membership in the masons, then yes, by all means let’s “stop fighting”. But isn’t this a much deeper, more foundational issue: an issue of sin? We have a portion of the denomination celebrating and affirming something that the vast majority of the universal church believes the Bible condemns and is sin. Would you be writing this same post if the issue was polyamory or pornography or adultery? These are also issues the vast majority of the universal church condems and considers to be sin. Would you herald polyamory, pornagraphy or adultery as non-binary choices for the sake of fellowship? Why or why not? Thanks in advance for any dialogue you can give regarding this. I really hope you are willing to answer these questions, as it will really help in better understanding the central argument that you made in this post.
A couple of things. First, you downplay, I think, the seriousness with which the seceders saw lodge membership. They also saw it as something which was completely incompatible with loyalty to Christ. Second, the question at the root of all this is this: where is the “church”? When the Belgic Confession speaks of the marks of the church, where are those marks to be seen? It’s certainly not in a synod, indeed, it is the local church. So we are a collection of churches which are in covenantal communion…what does that mean? The Reformed Church is not a hierarchy, there are no higher and lower assemblies in the usual sense of the term. At one level this is a discussion about human sexuality, at a deeper level, however, this is also about what it means to be the church, and this is the discussion that we are not having but need to have.
Thanks for the response. But to help me better understand you, could you answer the question of would you still write this article if the issue was adultery or polyamory? Would you herald these as non binary choices for the sake of a denomination staying together? Thanks in advance again for answering.
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\chapter{The extension theorem}\label{chap:extension}
\section{Morphisms of varieties with curves}
\subsection{Jets of \'etale morphisms}
Given a nonsingular variety $X$, let $\Delta_X : X \to X^\sharp$
denote the diagonal section. Since $X^\sharp$ is a bundle of formal discs,
the sheaf
$\underline\Aut_X(X^\sharp ; \Delta_X)$
is an affine group scheme over $X$, a Zariski-locally trivial twisted form
of the group of origin-preserving automorphisms of a formal disc
of dimension $\dim(X)$.
Given another
nonsingular variety $Y$ with $\dim(Y)=\dim(X)$, let
$$ X \xleftarrow{\pr_X} X \times Y \xrightarrow{\pr_Y} Y $$
be the two projections, and
consider the sheaves
\begin{eqnarray*}
\sF_{X,Y} &=& \underline\Hom_{X\times Y}(\pr_X^*X^\sharp,\pr_Y^*Y^\sharp ;
\pr_X^*\Delta_X,\pr_Y^*\Delta_Y) \\
\sE_{X,Y} &=& \underline\Isom_{X\times Y}(\pr_X^*X^\sharp,\pr_Y^*Y^\sharp ;
\pr_X^*\Delta_X,\pr_Y^*\Delta_Y)
\end{eqnarray*}
over $X\times Y$. Since $\pr_X^* X^\sharp$
and $\pr_Y^* Y^\sharp$ are bundles of formal discs
of equal dimension,
it follows that $\sF_{X,Y}$ is a Zariski-locally trivial $\AA^\infty$-bundle
over $X\times Y$, while
$\sE_{X,Y}$
is a Zariski-locally trivial right torsor
for $\underline\Aut_X(X^\sharp;\Delta_X)\times Y$, and
a Zariski-locally trivial left torsor for
$X\times\underline\Aut_Y(Y^\sharp;\Delta_Y)$.
\begin{lem}\label{lem:e-stratification}
$\sF_{X,Y} / X$ is equipped
with a natural stratification, restricting to $\sE_{X,Y}/X$.
\end{lem}
\begin{proof}
Let $\epsilon : \sF_{X,Y}\times_X X^\sharp \to Y^\sharp$ be the universal
map.
Consider the commutative diagram
$$\begin{CD}
\sF_{X,Y}\times_XX^\sharp\times_XX^\sharp @>{\id\times p_{13}}>>
\sF_{X,Y}\times_XX^\sharp @>{\epsilon}>> Y^\sharp \\
@A{\id\times\Delta_X}AA @. @AA{\Delta_Y}A \\
\sF_{X,Y} \times_XX^\sharp @>{\epsilon}>> Y^\sharp @>{p_2}>> Y
\end{CD}$$
where $p_2 :Y^\sharp \to Y$ is the right projection,
and $p_{13} : X^\sharp\times_XX^\sharp \to X^\sharp$ is the leftmost-rightmost projection.
By the universal property of $\sF_{X,Y}$, the top composite $\epsilon \circ (\id\times p_{13})$ defines a morphism
fitting into a diagram
$$\begin{CD}
\sF_{X,Y}\times_X X^\sharp @>>> \sF_{X,Y} \\
@VVV @VVV \\
X^\sharp @>>> X
\end{CD}
$$
where the bottom horizontal arrow is the \emph{right} structure map.
This gives the desired morphism $\sF_{X,Y}\times_XX^\sharp \to X^\sharp\times_X\sF_{X,Y}$
over $X^\sharp$. Since $\sE_{X,Y}$ is an open subscheme of $\sF_{X,Y}$, it
inherits a stratification.
\end{proof}
The sheaf $\sE_{X,Y}$ parametrises $\infty$-jets of formally-\'etale
maps $X \to Y$. Such maps induce isomorphisms of spaces of arcs:
the universal map
$$ \Phi_{X,Y} : \sE_{X,Y} \times_X X^\sharp \to \sE_{X,Y}\times_Y Y^\sharp $$
lifts to produce a commutative diagram
$$\begin{CD}
\sE_{X,Y} \times_X X^\sharp\times_X\Arc_X @>{\tilde\Phi_{X,Y}}>> \sE_{X,Y}\times_Y Y^\sharp\times_Y\Arc_Y \\
@VVV @VVV \\
\sE_{X,Y} \times_X X^\sharp @>{\Phi_{X,Y}}>> \sE_{X,Y}\times_Y Y^\sharp
\end{CD}$$
where both horizontal arrows are isomorphisms.
We now want to restrict to maps preserving
families of arcs induced by families of rational curves.
Let us for convenience introduce the following notion.
\begin{defn}
A \emph{good family} on a nonsingular projective variety $X$
is a irreducible component $\sM \subset \underline\Hom_\bir^n(\PP^1,X)$
such that the generic $\sM$-curve is minimal and unramified at $0$,
and the generic fibre of $\sM_1^\free \to X$ is geometrically irreducible and positive-dimensional.
\end{defn}
Given a nonsingular variety with
good family $(X,\sM)$, we will denote by $\hat\sM_1 \subset \Arc_X$
the closure of the image of the generic point of $\sM_1$ under the natural
monomorphism $\sM_1^\arc \to \Arc_X$ (in particular, $\sM_1$ and $\hat\sM_1$
are birational). Given another such pair $(Y,\sN)$,
with $\dim(Y)=\dim(X)$, consider the subsheaf
$\sC_{X,Y} \subset \sE_{X,Y}$ defined by
$$ \sC_{X,Y}(T) = \{ \varphi \in \sE_{X,Y}(T)\ |\
\varphi^*\tilde\Phi_{X,Y} : T \times_X X^\sharp \times_X \hat\sM_1
\xrightarrow{\simeq} T\times_Y Y^\sharp\times_Y \hat\sN_1
\}. $$
It is a closed subscheme of $\sE_{X,Y}$,
parametrising
$\infty$-jets of formally \'etale maps $X\to Y$
inducing isomorphisms of $\hat\sM_1$ onto $\hat\sN_1$.
We will also use the `generic locus'
$$ \sC_{X,Y}^\circ = x\times_X \sC_{X,Y}\times_Y y $$
where $x$, $y$ are the generic points of $X$, $Y$.
\begin{lem}\label{lem:c-stratification}
The stratification on $\sE_{X,Y} / X$ restricts to $\sC_{X,Y} / X$ and $\sC^\circ_{X,Y}/X$.
\end{lem}
\begin{proof}
It is enough to notice that
$\tilde\Phi_{X,Y}$ is horizontal with respect to the pullbacks of the canonical stratifications
on $X^\sharp\times_X\Arc_X$ and $Y^\sharp\times_Y\Arc_Y$.
\end{proof}
\begin{lem}\label{lem:c-generic}
$\sC_{X,Y}^\circ$ is dense in $\sC_{X,Y}$.
\end{lem}
\begin{proof}
Let $c_0 \in \sC_{X,Y}$ be a point with $\pr_X(c_0)=x_0$.
Let $I \subset \OO_{X,x_0}$ be the kernel of $\OO_{X,x_0} \to \pr_{X*}\OO_{\sC_{X,Y},c_0}$.
By Lemma \ref{lem:c-stratification}, $I \subset \cap_r \fm_{x_0}^r = 0$,
so that $x \times_X \Spec \OO_{\sC_{X,Y},c_0}$ is dense in $\Spec \OO_{\sC_{X,Y},c_0}$.
Now let $c \in x \times_X \Spec \OO_{\sC_{X,Y},c_0}$ be a point with
$\pr_Y(c)=y_0$.
Let $J \subset \OO_{Y,y_0}$ be the kernel of $\OO_{Y,y_0} \to \pr_{Y*} \OO_{\sC_{X,Y},c}$.
Since the definition of $\sC_{X,Y}$ is symmetric in $X$, $Y$, Lemma
\ref{lem:c-stratification} gives a stratification on $\sC_{X,Y}/Y$.
Hence $J \subset \cap_r \fm_{y_0}^r = 0$,
so that $\Spec\OO_{\sC_{X,Y},c}\times_Y y$ is dense in
$\Spec\OO_{\sC_{X,Y},c}$.
\end{proof}
\begin{lem}\label{lem:c-m1-n1}
There is a natural commutative diagram
$$\begin{CD}
\sC_{X,Y}^\circ \times_X X^\sharp \times_X m_1 @>>> \sC_{X,Y}^\circ\times_Y Y^\sharp\times_Y n_1 \\
@VVV @VVV \\
\sC_{X,Y}^\circ \times_X X^\sharp \times_X \hat\sM_1 @>{\tilde\Phi_{X,Y}}>> \sC_{X,Y}^\circ\times_Y Y^\sharp\times_Y\hat\sN_1
\end{CD}$$
where $m_1$, $n_1$ are the generic points of $\sM_1$, $\sN_1$.
\end{lem}
\begin{proof}
Recall that $m_1 \to \hat\sM_1$ and $n_1 \to \hat\sN_1$ are
inclusions of generic points, while the bottom horizontal arrow is an isomorphism.
The field extensions $\kappa(m_1)/\kappa(x)$ and $\kappa(n_1)/\kappa(y)$ are separable
by freeness of $m_1$, $n_1$.
It follows that
for every $c \in C^\circ_{X,Y}$ the induced isomorphism
$$\kappa(c)\otimes_{\kappa(x)}(x\times_X \hat\sM_1)
\xrightarrow{\tilde\Phi_{X,Y}} \kappa(c)\otimes_{\kappa(y)}(y\times_Y \hat\sN_1)$$
sends $c\times_xm_1$ to $c\times_yn_1$.
Since $\sC_{X,Y}^\circ\times_Y n_1$ is a localisation of
$\sC_{X,Y}^\circ\times_Y\hat\sN_1$, it follows that
$$ \sC^\circ_{X,Y}\times_X X^\sharp\times_X\hat\sM_1 \xrightarrow{\tilde\Phi_{X,Y}}
\sC^\circ_{X,Y}\times_Y Y^\sharp\times_Y\sN_1 $$
sends $\sC^\circ_{X,Y}\times_XX^\sharp\times_X m_1$
into $\sC^\circ_{X,Y}\times_YY^\sharp\times_Y n_1$.
\end{proof}
\subsection{Parallel transport along curves}
Note that given a morphism $f:T \to X$, the stratification on $\sC_{X,Y}/X$
induces one on the pullback $f^*\sC_{X,Y} / T$:
$$\begin{CD}
T^\sharp \times_T f^*\sC_{X,Y}@=
T^\sharp \times_X \sC_{X,Y} @=
T^\sharp \times_{X^\sharp} (X^\sharp\times_X\sC_{X,Y}) \\
@. @. @V{\simeq}VV \\
f^*\sC_{X,Y}\times_TT^\sharp
@= \sC_{X,Y}\times_XT^\sharp
@= (\sC_{X,Y}\times_XX^\sharp)\times_{X^\sharp}T^\sharp.
\end{CD}$$
The key point of this section is then the following `parallel transport' result,
an analogue of Hwang and Mok's analytic continuation along rational
curves~\cite{hwang-mok-extension}.
\begin{pro}\label{pro:extension-m1}
Let $(X,\sM)$ and $(Y,\sN)$ be a pair of nonsingular varieties
with good families, such that $\dim(X)=\dim(Y)$.
Let $m_2$ be the generic
point of $\sM_2$ together with evaluation maps $m_2 \rightrightarrows X$.
Then there is a natural isomorphism
$$ {\sC^\circ_{X,Y}}\times_Xm_2 \to m_2\times_X\sC^\circ_{X,Y} $$
horizontal over $m_2$ with respect to the induced stratifications.
\end{pro}
\begin{proof}
Let $\tilde m_1 \in \sM_2$
be the image of $m_1$ under the zero-section $\sM_1\to\sM_2$.
Set
$$ M = \Spec \OO_{\sM_2,\tilde m_1},\quad \hat M =\Spf\hat\OO_{\sM_2,\tilde m_1} $$
so that $M$ is the spectrum of a discrete valuation
ring with closed point $\tilde m_1$ and generic
point $m_2$, and $\hat M$ is its completion.
We will first construct a morphism
$$
\psi : \sC^\circ_{X,Y}\times_X X^\sharp \times_X M \to \sN_2.
$$
extending the canonical top horizontal arrow in the diagram
$$\begin{CD}
\sC^\circ_{X,Y}\times_X X^\sharp \times_X \hat M @>{\hat\psi}>> \hat\sN_2 \\
@VVV @VVV \\
\sC^\circ_{X,Y}\times_X X^\sharp \times_X \Arc_X \times_X X^\sharp @>>> \Arc_Y
\times_Y Y^\sharp
\end{CD}$$
where the bottom hotizontal arrow is induced by $\tilde \Phi$ and $\Phi$,
while the vertical arrows are induced by the natural
maps to the spaces of arcs and by evaluation at the second marked point.
With $(\sM_2/X)^2$ denoting the fibre product of $\sM_2$ with itself
with respect to the \emph{right} structure maps into $X$,
consider the natural diagram
$$ \sM_2 \overset{q_1}{\underset{\Delta}{\leftrightarrows}} (\sM_2/X)^\sharp \xrightarrow{q_2} X^\sharp\times_X \sM_2 $$
of morphisms over $X^\sharp$.
Its pullback by the rightmost structure
map $\sC^\circ_{X,Y}\times_XX^\sharp\to X$
gives
$$ \sC^\circ_{X,Y}\times_XX^\sharp\times_X M
\overset{\tilde q_1}{\underset{\tilde\Delta}{\leftrightarrows}}
\sC^\circ_{X,Y}\times_XX^\sharp\times_X (M/X)^\sharp
\xrightarrow{\tilde q_2}
\sC^\circ_{X,Y} \times_X X^\sharp \times_X X^\sharp \times_X M. $$
Let $\pi:\sM_2\to\sN_1$, $\varpi:\sN_2\to\sN_1$ and
$p_{13} : X^\sharp\times_XX^\sharp\to X^\sharp$ denote the natural
projections. By Lemma \ref{lem:c-m1-n1}, we have the composite
\begin{eqnarray*}
\nu : \sC^\circ_{X,Y}\times_X X^\sharp\times_X (M/X)^\sharp
&\xrightarrow{\tilde q_2}&
\sC^\circ_{X,Y} \times_X X^\sharp\times_X X^\sharp\times_X M \\
&\xrightarrow{p_{13*}}&
\sC^\circ_{X,Y}\times_XX^\sharp\times_X M \\
&\xrightarrow{\pi_*} &
\sC^\circ_{X,Y}\times_XX^\sharp\times_X m_1\\
&\xrightarrow{\tilde\Phi_{X,Y}}&
\sC^\circ_{X,Y}\times_YY^\sharp\times_Y n_1\\
&\xrightarrow{\pr_{n_1}}&
n_1.
\end{eqnarray*}
Consider now the pullback diagram
$$\begin{CD}
\tilde\Delta^*\nu^*\sN_2 @>>> \nu^*\sN_2 @>>> \sN_2 \\
@V{h}VV @VVV @V{\varpi}VV \\
\\
\sC^\circ_{X,Y}\times_XX^\sharp\times_X M
@>{\tilde\Delta}>>
\sC^\circ_{X,Y}\times_XX^\sharp\times_X (M/X)^\sharp
@>{\nu}>> \sN_1
\end{CD}$$
where the vertical arrows are $\PP^1$-bundles.
Viewing $\sN_2 \to \sN_1$ as the universal $\sN$-curve
in $Y$, let $N_{\sN_2/Y}$ be the universal normal sheaf on $\sN_2$.
Identifying $T_{M/X}$ with the pullback by $\Delta$ of the relative
tangent sheaf of $q_1$,
the pullback $\tilde\Delta^* d\nu$ defines by adjunction a map
$$
g : r_1^* T_{M/X} \to r_2^* N_{\sN_2/Y}
$$
of locally free sheaves on $\tilde\Delta^*\nu^*\sN_2$,
where.
$$ M \xleftarrow{r_1} \tilde\Delta^*\nu^*\sN_2 \xrightarrow{r_2} \sN_2, $$
are the natural projections.
\begin{lem}
The zero-locus of $g$
is the graph of a morphism
$$\psi: {\sC^\circ_{X,Y}}\times_XX^\sharp\times_XM \to \sN_2$$
lifting $\nu \circ \tilde\Delta$ and extending $\hat\psi$.
\end{lem}
\begin{proof}
Let ${\sC^\circ_{X,Y}}$ be the zero-locus of $g$.
Since $h$ is proper, so it $h|_{\sC^\circ_{X,Y}}$.
Since the generic $\sN$-curve is minimal,
the ideal sheaf $\sI_{\sC^\circ_{X,Y}}$ of ${\sC^\circ_{X,Y}}$ in $\tilde\Delta^*\nu^*\sN_2$
splits along the fibres of the $\PP^1$-bundle $h$
into invertible sheaves with degrees in $\{-1,0\}$, so that
$R^1h_*\sI_{\sC^\circ_{X,Y}}=0$ and the natural map
$$ \OO_{\sC^\circ_{X,Y}\times_XX^\sharp\times_XM} \to h_*\OO_{\sC^\circ_{X,Y}} $$
is surjective. It follows that every geometric fibre of $h|_{\sC^\circ_{X,Y}}$
is either empty, a single reduced point, or a whole $\PP^1$.
Consider the pullback
$$\begin{CD}
\iota^*\tilde\Delta^*\nu^*\hat\sN_2 @>{\tilde\iota}>> \tilde\Delta^*\nu^*\hat\sN_2
@>>> \hat\sN_2\\
@VVV @VVV @V{\varpi}VV\\
\sC^\circ_{X,Y}\times_X X^\sharp\times_X \hat M
@>{\iota}>> \sC^\circ_{X,Y}\times_X X^\sharp\times_X M
@>{\nu\circ \tilde\Delta}>> \sN_1
\end{CD}$$
where $\iota$ is the natural monomorphism.
By construction, the graph of
$$\hat\psi : \sC^\circ_{X,Y}\times_XX^\sharp\times_X\hat M \to \hat\sN_2$$
factors through ${\sC^\circ_{X,Y}}$.
Since the geometric generic fibre of $\sM_1^\free \to X$ is positive-dimensional,
the geometric fibres of $h|_{\sC^\circ_{X,Y}}$ over $\sC^\circ_{X,Y}\times_XX^\sharp\times_X \tilde m_1$
are single reduced points, so that
the restriction ${\sC^\circ_{X,Y}} \cap \iota^*\tilde\Delta^*\nu^*\sN_2$ actually coincides with
the graph of $\hat\psi$.
In particular, $\iota^*h|_{\sC^\circ_{X,Y}}$ is an isomorphism.
Since $\iota$ is an epimorphism of formal schemes,
and $\varpi$ is a $\PP^1$-bundle,
it follows that $\tilde\iota$ is an epimorphism of formal schemes.
Thus $h|_{\sC^\circ_{X,Y}}$ is a closed immersion, adic and admitting a section over $\iota$,
hence an isomorphism.
\end{proof}
We have thus constructed the map $\psi$,
which will allow us to produce
a morphism
$ \sC^\circ_{X,Y}\times_X M \to M\times_X \sC^\circ_{X,Y} $
whose restriction over $m_2$ gives the isomorphism announced in the Proposition.
By freeness of the generic
$\sM$-curve, we can choose an isomorphism
$$ \rho : \sC^\circ_{X,Y}\times_X M\times_XX^\sharp \to \sC^\circ_{X,Y}\times_X
X^\sharp\times_X M $$
over $\sC^\circ_{X,Y}\times X$ (leftmost-rightmost structure map).
Let $\phi$ be the composite
$$
\phi : \sC^\circ_{X,Y}\times_X M\times_XX^\sharp \xrightarrow{\rho}
\sC^\circ_{X,Y}\times_X X^\sharp\times_X M
\xrightarrow{\psi}
\sN_2 \to Y
$$
where the rightmost arrow is the right structure map.
Consider now the pair
$$\sC^\circ_{X,Y}\times_XM\times_XX^\sharp \overset{s^*\phi}{\underset{\phi}{\rightrightarrows}} Y $$
where
$s = \Delta_X \circ p_1:X^\sharp \to X^\sharp$
is the `retraction onto origin'.
These induce a morphism
$$ \theta = \langle \id , s^*\phi , \phi \rangle :
\sC^\circ_{X,Y}\times_XM\times_XX^\sharp
\to \sC^\circ_{X,Y}\times_X M \times_Y Y^\sharp $$
together with the map
$$ [\theta]:\sC^\circ_{X,Y} \times_X M \to M\times_X \sF_{X,Y} $$
defined by the universal property of $\sF_{X,Y}$.
Now, since $\psi$ is an extension of $\hat\psi$,
there is a commutative diagram
$$
\begin{CD}
\sC^\circ_{X,Y}\times_X \hat M \times_X X^\sharp
@>{\theta}>> \sC^\circ_{X,Y}\times_X \hat M \times Y^\sharp\\
@V{\id\times (p_{13} \circ e)}VV @VV{\id\times\pr_{Y^\sharp}}V \\
\sC^\circ_{X,Y}\times_X X^\sharp @>{\Phi_{X,Y}}>> \sC^\circ_{X,Y}\times_Y Y^\sharp
\end{CD}$$
where $e:\hat M \to \hat X$ is the restriction
of the structure morphism $\sM_2 \to X\times X$.
Hence the restriction of $[\theta]$
to $\sC^\circ_{X,Y}\times_X\hat M$
is a pullback of the stratifying isomorphism
$$ \sC^\circ_{X,Y}\times_X X^\sharp \to X^\sharp\times_X \sC^\circ_{X,Y},$$
and in particular it is horizontal and factors through $\hat M\times_X \sC^\circ_{X,Y}$.
Since $\sC^\circ_{X,Y}\times_X \hat M$ does not factor through
any proper subscheme of $\sC^\circ_{X,Y}\times_X M$,
it follows that $[\theta]$ factors through the open subscheme
$M\times_X \sE_{X,Y}\subset M\times_X \sF_{X,Y}$,
through the closed subscheme $M\times_X \sC_{X,Y} \subset M\times_X \sE_{X,Y}$,
and finally through the `generic locus' $M\times_X\sC^\circ_{X,Y} \subset
M\times_X \sC_{X,Y}$.
Letting $$\tau : \sC^\circ_{X,Y}\times_X M \to M \times_X {\sC^\circ_{X,Y}}$$ be the
map induced by the point-swapping involution on $M\subset\sM_2$, we have a pair of
morphisms
$$
\sC^\circ_{X,Y}\times_X M \overset{[\theta]}{\underset{\tau[\theta]\tau}{\rightleftarrows}}
M\times_X\sC^\circ_{X,Y}
$$
such that $\tau[\theta]$ and $[\theta]\tau$ restrict to identity
over $\sC^\circ_{X,Y}\times_X \hat M$ and $\hat M\times_X \sC^\circ_{X,Y}$.
Hence the above morphisms are mutual inverses, and
their restriction over
$m_2$ gives the isomorphism announced in the Proposition, thus concluding its proof.
\end{proof}
\subsection{Induction and descent}
We can now use Proposition \ref{pro:extension-m1} inductively to
trivialise $\sC^\circ_{X,Y}$ along generic chains of $\sM$-curves.
Under suitable conditions, the trivialisation descends generically
to the base.
\begin{pro}\label{pro:c-descent}
Let $(X,\sM)$ and $(Y,\sN)$ be a pair of nonsingular varieties with
good families such that $\dim(X)=\dim(Y)$.
Suppose that $X$ is simply-connected,
of Picard number $1$.
Let $\xi$ be the generic point of $X\times X$.
Then there is a natural isomorphism
$$\sC^\circ_{X,Y}\times_X \xi\to \xi\times_X{\sC^\circ_{X,Y}} $$
horizontal over $\xi$.
\end{pro}
We will need the following bit of commutative algebra.
\begin{lem}\label{lem:separable-closure}
Let $L/K$ be a finitely generated field extension, and $K^{s,L}$ the separable
algebraic closure of $K$ in $L$. Then the following is an equaliser diagram:
$$ K^{s,L} \to L \rightrightarrows \widehat{L\otimes_KL} $$
where we complete at the diagonal ideal.
\end{lem}
\begin{proof}
Let $K'\subset L$ be the equaliser of $L\rightrightarrows\widehat{L\otimes_KL}$.
We first observe that the claim is true in the following cases:
\begin{enumerate}
\item $L/K$ purely transcendental: then
$L\otimes_KL \to \widehat{L\otimes_KL}$ is injective,
so that $K'=K$.
\item $L/K$ purely inseparable: then $L\otimes_KL$ is Artinian, hence already complete,
and we argue as
above.
\item $L/K$ separable algebraic: then $L\otimes_KL$ is a product of finitely many
copies of $L$, so that the diagonal ideal is idempotent and $K'=L$.
\end{enumerate}
In the general case, by (2) we can assume that $L/K$ is separably generated, so that
there is an intermediate extension $K \subset L_0 \subset L$ such that $L/L_0$
is separable algebraic and $L_0/K$ purely transcendental. Suppose $x \in K'$.
Since $K'$ is invariant under the Galois group of $L/L_0$, the conjugates of $x$
are also contained in $K'$. It follows that the coefficients
of the minimal polynomial of $x$ in $L_0$ are contained in $K' \cap L_0$, and thus
in $K$ by (1). Hence $x$ is separable algebraic over $K$, and thus $K' \subset K^{s,L}$.
The converse follows by (3).
\end{proof}
\begin{proof}[Proof of Proposition \ref{pro:c-descent}.]
Let $m_2^i$ the generic point of $\sM_2^{i,\free}$. By Lemma \ref{lem:free-stuff},
$m_2^{i+1} \in m_2^i\times_X m_2$, so that Proposition \ref{pro:extension-m1}
and induction on $i$
gives a horizontal isomorphism
$$ \theta^i : \sC^\circ_{X,Y}\times_X m_2^i \to m_2^i\times_X\sC^\circ_{X,Y} $$
over $m_2^i$.
By Lemma \ref{lem:free-chains} we can choose $i$ such that $m_2^i$ maps to $\xi\in X\times X$.
By horizontality of $\theta^i$, the pullbacks
$$ \sC^\circ_{X,Y}\times_X (m_2^i / \xi)^\sharp \rightrightarrows
\sC^\circ_{X,Y} $$
of $\pr_{\sC^\circ_{X,Y}}\circ \theta^i$ by $(m_2^i / \xi)^\sharp \rightrightarrows m_2^i$
coincide.
Hence, by Lemma \ref{lem:separable-closure}, $\theta^i$ descends
to a morphism
$$
\bar\theta : \sC^\circ_{X,Y}\times_X \tilde\xi
\to \tilde\xi\times_X\sC^\circ_{X,Y}
$$
where $\tilde\xi$
is the spectrum of
the separable algebraic closure
of $\kappa(\xi)$ in $\kappa(m_2^i)$.
Being an algebraic subextension
of a finitely generated extension, $\kappa(\tilde\xi)/\kappa(\xi)$
is finite.
Horizontality and invertibility of $\bar\theta$
follows from that of $\theta^i$ by descent.
To show that $\bar\theta$ is in fact
defined over $\kappa(\xi)$,
we first
consider a geometric generic point $\bar\zeta$ of $X\times\sM_0$
and the corresponding rational curve
$$f : \PP^1_{\bar\zeta} \to \bar\zeta\times X \to X\times X.$$
\begin{lem}\label{lem:c-curves}
Let $W\subset f^*\tilde\xi$ be a connected component.
Then
$f^*\bar\theta:\sC^\circ_{X,Y} \times_X W \to W\times_X \sC^\circ_{X,Y}$
descends along
$W \to f^*\xi$.
\end{lem}
\begin{proof}
Let $\bar\vartheta = \pr_2\circ \bar\theta:\sC^\circ_{X,Y}\times_X\tilde\xi\to\sC^\circ_{X,Y}$.
Recall that in the Proof of Proposition \ref{pro:extension-m1} we have actually constructed
an isomorphism
$ \theta : \sC^\circ_{X,Y} \times_X M \to M\times_X \sC^\circ_{X,Y}$
horizontal over $M = \Spec \OO_{\sM_2,\tilde m_1}$,
where $\tilde m_1$ is the image of
$m_1$ under the zero-section $\sM_1 \to \sM_2$.
Consider the diagram
with Cartesian squares
$$\begin{CD}
\sC^\circ_{X,Y}\times_X (X \backslash \tilde\xi)^\sharp \times_{X\times X} \hat M @>>>
\sC^\circ_{X,Y}\times_X (X \backslash \tilde\xi)^2 \times_{X\times X} M @>{\Theta}>>
\sC^\circ_{X,Y}\times_XM\times_X\sC^\circ_{X,Y}\\
@VVV @VVV @VVV\\
\sC^\circ_{X,Y}\times_X (X\backslash \tilde\xi)^\sharp @>>>
\sC^\circ_{X,Y}\times_X (X\backslash \tilde\xi)^2
@>{\bar\vartheta\times\bar\vartheta}>> \sC^\circ_{X,Y}\times\sC^\circ_{X,Y}
\end{CD}$$
The left square is induced by the natural inclusion $(X\backslash\tilde\xi)^\sharp \to
(X\backslash\tilde\xi)^2$. By horizontality, the top horizontal composite
factors through the graph of $\theta$.
Identify $\PP^1_{\bar\zeta}$ and $\PP^1_{\bar\zeta}\times_{\bar\zeta}\PP^1_{\bar\zeta}$
with, respectively, $\bar\zeta \times_{\sM_0}\sM_1$ and
$\bar\zeta\times_{\sM_0}\sM_2$.
Consider $M_{\bar\zeta} = \bar\zeta\times_{\sM_0}M$
as a subscheme of $\PP^1_{\bar\zeta}\times_{\bar\zeta}\PP^1_{\bar\zeta}$.
The curve $f$ is identified with the natural morphism
$\bar\zeta\times_{\sM_0}\sM_1 \to X \times X$ induced
by $\bar\zeta\to X$ and $\sM_1 \to X$.
Let $W \subset f^*\tilde\xi$ be an irreducible component.
Pulling back the top row of the above diagram, we have that the composite
$$ \sC^\circ_{X,Y} \times_X (W/\bar\zeta)^\sharp \to
\sC^\circ_{X,Y} \times_{X} ( W / \bar\zeta )^2
\to \sC^\circ_{X,Y} \times \sC^\circ_{X,Y} $$
factors through the pullback of the diagram of $\theta$ by
$(W/\bar\zeta)^\sharp
\to M_{\bar\zeta}$. Hence so does the right arrow itself, and in particular
the restriction
$$ \sC^\circ_{X,Y} \times_X (W/\PP^1_{\bar\zeta})^2 \to \sC^\circ_{X,Y}\times\sC^\circ_{X,Y} $$
factors through the diagonal. Hence $f^*\bar\vartheta:\sC^\circ_{X,Y}\times W \to \sC^\circ_{X,Y}$
descends along $W \to f^*\xi$, and so does $f^*\bar\theta$.
\end{proof}
Continuing the proof of the Proposition,
fix a separable closure $\kappa(\bar\xi^s)$ of $\kappa(\xi)$.
The Galois group $\Gal(\bar\xi^s/\xi)$ acts on the set
$E$ of isomorphisms $\sC^\circ_{X,Y}\times_X \bar\xi^s\to\bar\xi^s\times_X\sC^\circ_{X,Y}$ horizontal over $\bar\xi^s$.
By the first part of the proof, there is an element
$\bar\theta \in E$ whose stabiliser
in $\Gal(\bar\xi^s/\xi)$ is of finite index.
Letting $\eta \simeq \bar\zeta\otimes k(t)$
be the generic point of $\PP^1_{\bar\zeta}$,
we have an extension $\kappa(\eta)/\kappa(\xi)$.
We can lift it to $\kappa(\bar\eta^s)/\kappa(\bar\xi^s)$
where $\bar\eta^s$ is a separable closure of $\eta$.
It then follows by Lemma \ref{lem:c-curves} that
the stabiliser of $\bar\theta$ in $\Gal(\bar\xi^s/\xi)$
contains the image of $\Gal(\bar\eta^s/\eta)$.
Let $\Gamma \to X\times X$ be a normal Galois cover corresponding to
the stabiliser of $\bar\theta$,
so that $\bar\theta$ is defined over the generic point of $\Gamma$, and
$f^*\Gamma$ is trivial by the previous paragraph.
We want to show that $\Gamma$ itself is trivial.
Since $X$ is simply-connected, it will be enough to show
that $\Gamma \to X\times X$ is \'etale.
Assuming the opposite, we have by the classical purity theorem
that it is ramified over a divisor $D \subset X\times X$.
Since the problem is symmetric under the transposition on $X\times X$,
we can assume that $D$ is not a pullback of a divisor
from the first factor.
Since
$X$ has Picard number $1$, the pullback $f^*D$ is positive,
and
there is a lift $\tilde f : \PP^1_{\bar\zeta} \to \Gamma$
of $f$ intersecting the ramification divisor.
It follows that $f$ is tangent to $D$ at the intersection points.
But by Lemma \ref{lem:free-intersection} a generic $\sM$-curve
intersects $D$ transversely, a contradiction.
Hence $\Gamma$ is trivial, $\bar\theta$ is invariant under $\Gal(\bar\xi^s/\xi)$,
and thus finally defined over $\xi$.
\end{proof}
\section{Extension}
\begin{thm}\label{thm:extension}
Let $(X,\sM)$ and $(Y,\sN)$ be a pair of simply-connected,
nonsingular projective Fano varieties of Picard number $1$ and
equal dimensions,
together with good families of rational curves.
Let $K$ be an algebraically closed field,
and $\bar c : \Spec K \to \sC^\circ_{X,Y}$ a geometric point.
Then there is an isomorphism
$$ \phi : X \otimes K \to Y \otimes K $$
extending the canonical isomorphism
$\bar c^*\Phi : \bar c^*X^\sharp\to\bar c^*Y^\sharp$.
\end{thm}
\begin{proof}
By Lemma \ref{lem:free-chains}, $\sM_2^{i,\free} \to X\times X$
is dominant for some $i>0$.
Let $X_K = X \otimes K$ and $Y_K = Y \otimes K$ with generic points
$x_K$, $y_K$.
Then, by Proposition \ref{pro:c-descent}, there is a horizontal section
$$ \sigma : \eta_{X_K} \to \sC^\circ_{X,Y} \otimes K. $$
Its composite with projection to $Y_K$ extends to a morphism
$$ \phi_0 : X_K \setminus W \to Y_K,\quad\quad \codim_{X_K}W \ge 2 $$
whose restriction to $x_K$ is formally \'etale and induces
an isomorphism $x_K^* X^\sharp \times_X \hat\sM_1\simeq
\phi_0|_{x_K}^* Y^\sharp\times_Y \hat\sN_1$.
It follows that
$\phi_0$ is dominant and generically \'etale. We will now show that
it is actally \'etale on entire $X_K\setminus W$.
Indeed, let $\bar\zeta$ be a geometric generic point
of $\sM_1 \otimes K$. Then the generic $\sM$-curve
$ f : \PP^1_\zeta \to X_K $
factors through $X_K \setminus W$ (by Lemma \ref{lem:free-intersection}),
and $\phi_0 \circ f : \PP^1_\zeta \to Y_K$ is a generic
$\sN$-curve (since its restriction to $\hat\PP^1_\zeta$
maps, as an unramified morphism from a formal disc,
to the generic point of $\hat\sN_1$). Now, if $\phi_0$ is not \'etale,
then, by the classical purity theorem, it is ramified over a divisor
$D \subset Y_K$. Since $Y$ has Picard number $1$, $\phi_0\circ f$
intersects $D$. But since $\phi_0\circ f$ is free, the intersection
is transverse (again by Lemma \ref{lem:free-intersection}).
It then follows that $f$ does not intersect the ramification divisor
in $X_K\setminus W$ -- a contradiction.
Hence $\phi_0$ is \'etale. Furthermore, since
$\phi_0 \circ f$ is free, it follows that the complement
of the image of $\phi_0$ has codimension at least $2$ in $Y_K$ (Lemma
\ref{lem:free-intersection}.
It then follows by simply-connectedness of $Y_K$ that $\phi_0$
is an isomorphism onto its image.
Now, since $X$ and $Y$ are Fano, we can find
an integer $d>0$ such that $-dK_X$ and $-dK_Y$ are both very ample.
Being an isomorphism of open subsets whose complements have codimension
at least $2$, $\phi_0$ induces an isomorphism of Picard
groups and
of spaces of global sections for any invertible sheaf.
Using the differential $d\phi_0$ to identify
$\phi_0^*K_Y$ with $K_X$,
we have a diagram
$$\begin{diagram}
\node{X_L}\arrow{s} \arrow{r,t,..}{\phi_0} \node{Y_L}\arrow{s} \\
\node{\PP H^0(X_K,\OO(-dK_X))^\vee} \arrow{=}
\node{\PP H^0(Y_K,\OO(-dK_Y))^\vee}
\end{diagram}$$
where the vertical arrows are the projective embeddings
indced by $-dK_X$ and $-dK_Y$. Hence
$\phi_0$ extends to an isomorphism $\phi:X_K\to Y_K$.
It remains to check that $\phi$ is an extension
of $\bar c^*\Phi$. Consider the lift of $\phi$ to a morphism
$\tilde\phi$, horizontal over $\sM_2^i$ and fitting into a
commutative diagram
$$\begin{CD}
\bar c \times_X \sM_2^i @>{\tilde\phi}>>\bar c\times_X \sM_2^i \times_X \sC_{X,Y} \\
@VVV @VVV \\
\bar c \times X @>{\phi}>> \bar c \times Y
\end{CD}$$
where the left vertical arrow is induced by the right structure map
$\sM_2^i\to X$.
Recall that in the proof of Proposition \ref{pro:extension-m1} we have
constructed an isomorphism
$$ \sC^\circ_{X,Y}\times_X M \to M \times_X \sC^\circ_{X,Y} $$
where $M = \Spec \OO_{\sM_2,\tilde m_1}$ and $\tilde m_1$
is the image of $m_1$ under the zero-section $\sM_1 \to \sM_2$.
Note that its restriction over $\tilde m_1$ is the
identity on $\sC^\circ_{X,Y} \times_X m_1$.
By induction, we have an isomorphism
$$ \tilde\theta^i: \sC^\circ_{X,Y} \times_X (X\backslash M/X)^i \to (X\backslash M/X)^i
\times_X \sC^\circ_{X,Y} $$
extending $\theta^i$ of the proof of Proposition \ref{pro:c-descent}.
It follows that we have a commutative diagram
$$\begin{CD}
\bar c \times_X m_2^i @>>> \bar c \times_X \sM_2^i \\
@VVV @V{\tilde\phi}VV \\
\bar c \times_X (X\backslash M/X)^i @>{\bar c^*\tilde\theta^i}>> \bar c\times_X\sM_2^i \times_X \sC_{X,Y}
\end{CD}$$
i.e. $\tilde\phi$ and $\bar c^*\tilde\theta^i$ agree
on $\bar c \times_X m_2^i$. Then, by irreducibility and reducedness
of $\bar c\times_X \sM_2^{i,\free}$ (cf. Lemma \ref{lem:free-stuff}),
they agree on $(X\backslash M/X)^i \subset \sM_2^i$. In particular,
the composite
$$ \bar c \times_X m_1 \xrightarrow{\bar c^*\langle \tilde m_1,\dots,\tilde m_1\rangle} \bar c \times_X \sM_2^i \xrightarrow{\tilde \phi}
\bar c \times_X \sM_2^i \times_X \sC_{X,Y} \to \bar c\times_X \sC_{X,Y} $$
factors through
the diagonal embedding $\bar c \to \bar c\times_X \sC_{X,Y}$.
Since $\tilde\phi$ is a horizontal lift of $\phi$, it follows that
we have a commutative diagram
$$\begin{CD}
(\bar c\times_X X^\sharp) \times_{X\times X}\sM_2^i @>{\tilde\phi}>>
\bar c \times_X
\sM_2^i \times_X \sC_{X,Y} \\
@VVV @VVV \\
\bar c\times_X X^\sharp @>{\bar c^*\Phi}>> \bar c \times Y
\end{CD}$$
Since the left vertical arrow is an epimorphism of formal schemes,
it follows that the composite
$$ \bar c \times_X X^\sharp \to \bar c\times X \xrightarrow{\phi} \bar c \times Y $$
coincides with $\bar c^*\Phi$.
\end{proof}
\begin{cor}\label{cor:extension}
Let $(X,\sM)$ and $(Y,\sN)$ be a pair of simply-connected,
nonsingular projective Fano varieties of Picard number $1$ and
equal dimensions,
together with good families of rational curves.
Let $K$ be an algebraically closed field, and
$\bar x_0:\Spec K \to X$, $\bar y_0:\Spec K \to Y$
a pair of geometric points such that there is an isomorphism
$\bar x_0^*X^\sharp \simeq \bar y_0^*Y^\sharp$
identifying $\bar x_0^*X^\sharp\times_X\hat\sM_1$ with
$\bar y_0^*Y^\sharp\times_Y\hat\sN_1$. Then
there is an isomorphism $X \simeq Y$ identifying
$\sM$ with $\sN$.
\end{cor}
\begin{proof}
By Lemma \ref{lem:c-generic}, $\sC^\circ_{X,Y}$ is nonempty,
so that there is an algebraically closed extension $L/K$
and a geometric point $\bar c : \Spec L \to \sC^\circ_{X,Y}$,
inducing by Theorem \ref{thm:extension} an isomorphism
$$ \phi_L : X\otimes L \to Y\otimes L $$
identifying $\hat\sM_1 \otimes L$ with $\hat\sN_1\otimes L$
and thus $\sM\otimes L$ with $\sN\otimes L$. Since $X$ and $Y$ are algebraic,
there is a subalgebra $A \subset L$, of finite type over $k$,
and such that $\phi_L$ is the base-change of an isomorphism
$$ \phi_A : X\otimes A \to Y\otimes A $$
identifying $\sM\otimes A$ with $\sN\otimes A$. Restricting
$\phi_A$ over a closed point of $\Spec A$ yields the desired
isomorphism $X\simeq Y$.
\end{proof}
\endinput
| 159,038
|
TITLE: If $T$ is a linear transformation from $V$ to $W$, find the transformation matrix with respect to a given basis of $V$ and a given basis of $W$
QUESTION [0 upvotes]: Suppose that $T(x, y) = (2x + 3y, 2x + 5y, 3x + 4y)$.
If $Bv$ is a basis of $V$ such that $Bv = \{(1, 2); (3, 1)\}$ and
if $Bw$ is a basis of $W$ such that $Bw = \{(1, 2, 3); (4, 1, 2); (3, 4, 1)\}$,
then what steps would I have to take to find the matrix of $T$ with respect to these bases? Please don't just give me the answer. I need to know how to get it. Thanks in advance!
REPLY [0 votes]: Assume $x \in V $. Then if we call $v_1, v_2$ the basis $B_V $ we know that $x = \alpha v_1 + \beta v_2$
Then calculating $T(x) = T(\alpha v_1 + \beta v_2) = T(\alpha v_1) + T(\beta v_2) = \alpha T(v_1) + \beta T(v_2) $
Thus to calculate the image of any vector $x $ we only need the values of $T(v_1)$ and $T(v_2) $.
But $T(v_1), T(v_2) $ are easily computable. Let us call them $v'_1$ and $v'_2$. Because each of the $v'$ is a vector from $W $, they can be written as a sum of the vectors from $B_W = \{w_1, w_2, w_3\} $.
This yields
$$\begin{cases}
T(v_1) = v'_1 = a_1 w_1 + a_2 w_2 + a_3 w_3\\
T(v_2) = v'_2 = b_1 w_1 + b_2 w_2 + b_3 w_3\end{cases}$$
Now we know that a linear transformation has, in its columns, the images of the different vectors from the basis, meaning the columns of the matrix you want are precisely the images of $v_1$ and $v_2$ written in the basis $B_W $, i.e. the matrix will take the form
$$\begin {bmatrix} a_1 & b_1\\
a_2 & b_2\\
a_3 & b_3\end {bmatrix} $$
All you have to do is compute those factors $a_i $ and $b_i $.
| 211,298
|
TITLE: Orderings on a space such that every initial segment has measure 0
QUESTION [0 upvotes]: Let $(\mu,X,\Sigma)$ be an atomless probability measure. Is it alway possible to find a well-ordering of $X$, $<$, such that for any $x\in X$, $Pr(\{y\mid y<x\})=0$?
(Edit: I'd also be interested in an answer to the weaker claim: it is always possible to find a well-ordering such that for every $x\in X$ such that $\{y\mid y<x\}$ is measureable, $Pr(\{y\mid y<x\})=0$.)
REPLY [1 votes]: Consistently no. For example, add $\omega_2$ random reals to a model of CH. In this model, there is a (Sierpinski) set $X$ of size $\omega_2 = 2^{\aleph_0}$ each one of whose uncountable subsets is non null so any well ordering of reals has an initial segment that contains exactly $\aleph_1$ points of $X$ and hence is proper and non null.
On the other hand, under CH, any measure space made by pasting together the product measures on $2^{\kappa}$'s cannot be a counterexample - Since we can partition them into $\aleph_1$ null sets and put them next to each other.
This doesn't rule out a ZFC counterexample. Unfortunately, I don't know any examples of atomless probability measure spaces which are essentially different from the above. By Maharam's theorem, we know that their measure algebras are essentially made of the measure algebras of $2^{\kappa}$'s but this doesn't seem to help.
If we only require that the measurable initial segments be null, then once again pasting together $2^{\kappa}$ doesn't work due to inner regularity. So I don't even know if this can be consistently false.
| 38,411
|
During an interview with the Better Government Association's Andy Shaw on Wednesday, Mayor Rahm Emanuel dismissed using surplus funds from tax increment financing districts to plug the budget hole for Chicago Public Schools.
Members of the parent group Raise Your Hand have pointed to $867 million in unallocated TIF funds and questioned why some of that money can't be declared a surplus. CPS would then be able to use a portion of those surplus funds rather than raise property taxes for the average homeowner by $84, they've argued. In the proposed budget to be voted on Aug. 24, school officials are hoping the higher taxes will bring in an extra $150 million to help close a $712 million budget shortfall.
| 252,104
|
Cousin Armadillo
Were you there when the lovely boy was born? Did you waddle toward the manger like a lump through the splendid host, to take your rightful turn? Did they laugh at you behind their lifted paw? Did they clear their throat behind their cloaking wing? Did they stare you down as though they dared refuse to let you spend your moment at the crib? Did you shut your naked ears to what they said: ''Just a turtle-shell around a baby pig!'' Did you lumber on and reach the shining hay where the mother watched the infant at her side and the infant watched the crowding company? Did you notice what pure light the child lay in? How he stirred in your direction - didn't he? And was seen to smile and welcome you as kin?
| 270,793
|
\begin{document}
\tcversion{}{\pagestyle{empty}}
\maketitle
\begin{abstract}
\input{abstract.tex}
\end{abstract}
\tcversion{}{\newpage \pagestyle{plain}\setcounter{page}{1}}
\section{Introduction}
\textit{Rounding} is a crucial step in the design of many approximation algorithms.
Given a fractional vector satisfying some constraints,
a rounding method produces an integer vector that satisfies those constraints, either exactly or
approximately.
\textit{Randomized rounding} \cite{RT} \cite[Chapter 5]{SW},
in which the coordinates of the fractional vector are rounded randomly and independently,
produces good integer vectors for many applications.
\textit{Dependent rounding} methods, in which the resulting integer vector does not have independent
coordinates, are important in many scenarios where naive randomized rounding does poorly.
Various techniques exist for designing dependent rounding methods
(see, e.g., the surveys \cite{SriSurvey,BansalSlides}).
It is common for a rounding scenario to involve two types of constraints:
hard constraints, which must be satisfied exactly by the integer solution,
and soft constraints, which must be approximately satisfied by the integer solution.
Low-congestion multi-path routing \cite{Sri01},
max cut with given sizes of parts \cite{AS},
thin spanning trees \cite{AGMOS}, and
submodular maximization under a matroid constraint \cite{CCPV,CVZFOCS} are
examples of problems whose solutions involve such a rounding scenario.
The hard constraint is often membership in an integer polytope
that is defined using combinatorial objects (e.g., matchings or matroids).
The soft constraints are usually simple linear inequalities.
With randomized rounding, the independent choices lead to
concentration of measure phenomena that are useful for handling soft constraints.
For example, Chernoff bounds are commonly used to show that linear inequalities
are approximately satisfied \cite{RT}.
The past decade has seen various uses of \emph{matrix} concentration bounds
(e.g., \cite{AW,RV07,Tropp11})
to show that linear matrix inequalities are approximately satisfied
by random sampling or rounding.
Such uses have occurred in many diverse areas:
graph sparsification \cite{SS08},
compressed sensing \cite{VershyninSurvey},
statistics \cite{TroppMasked},
machine learning \cite{Recht} and
numerical linear algebra \cite{Mahoney}.
With dependent rounding, concentration phenomena can also occur.
\emph{Pipage rounding}, \emph{swap rounding} and \emph{maximum entropy sampling}
are dependent rounding techniques
that have seen many important uses over the past decade
\cite{Sri01,AS,GandhiKPS06,CCPV,AGMOS,CVZFOCS}.
An important feature in some scenarios is that
any Chernoff bound that is valid under independent randomized rounding
remains valid under these dependent rounding techniques.
This fact is proven by showing that the rounded solution has a
\emph{negatively correlated distribution}, then appealing to the fact
that Chernoff bounds remain valid under such distributions~\cite{PS97}.
Unfortunately, commutativity plays a key role in proving that fact,
and these arguments do not seem to extend to matrix concentration bounds,
e.g., \cite{AW,Oliviera,RV07,Tropp11}.
Consequently, these matrix inequalities have so far not been combined with dependent rounding.
We prove the first result showing that matrix concentration bounds
are usable in a dependent rounding scenario.
Our technique is not based on negative correlation, but rather the
fortuitous interaction between pipage rounding and various pessimistic estimators.
In particular, we show that Tropp's matrix Chernoff bound \cite{Tropp11}
has a pessimistic estimator that decreases monotonically under pipage rounding.
As a consequence, we can extend the reach of pipage rounding
from soft constraints that are linear inequalities to soft constraints that are
linear matrix inequalities.
Our proof uses non-trivial techniques from matrix analysis and complex analysis;
in particular, we prove a new variant of Lieb's concavity theorem.
\subsection{Motivation and Results}
One key area where our techniques yield new results is for \emph{thin spanning trees}.
These are intriguing objects in graph theory that relate to
foundational topics, such as nowhere-zero flows \cite{Goddyn},
and the asymmetric traveling salesman problem \cite{AGMOS}.
Given a graph $G$ on $n$ nodes, a spanning tree $T$ of $G$ is \emph{$\alpha$-thin} if, for every
cut, the number of
edges of $T$ crossing the cut is at most $\alpha$ times the number of edges of $G$ crossing
the cut.
It has been conjectured that any graph with connectivity $\connec$ has an $f(\connec)$-thin
spanning tree where $f(\connec) = O(1/\connec)$.
This would imply a constant factor approximation algorithm for the asymmetric traveling salesman
problem~\cite{OS}.
Asadpour et al.~\cite{AGMOS} give a randomized algorithm to find a spanning tree that is $O(\frac{\log n}{\connec \log \log n})$-thin.
Later Chekuri et al.~\cite{CVZArxiv,CVZFOCS} gave a simpler algorithm using randomized pipage rounding
or swap rounding.
A \emph{spectrally-thin} spanning tree is a stronger notion that is naturally motivated by work on
spectral sparsification \cite{SS08,BSS09}.
A spanning tree $T$ is $\alpha$-spectrally-thin if $L_T \preceq \alpha L_G$, where $L_G$ refers to the Laplacian of $G$, and $\preceq$ to the L\"owner ordering of Hermitian matrices.
In \Section{thin}, we show a result on spectrally thin trees that strongly mirrors the result of
Asadpour et al.
\begin{theorem}\TheoremName{specthin}
There is a deterministic, polynomial-time algorithm that given any graph on $n$ nodes where every edge has effective conductance at least $\conduc$, constructs a
$O(\frac{\log n}{\conduc \log \log n})$-spectrally-thin spanning subtree.
\end{theorem}
This spectral notion of thinness seems to be an important one, as the recent breakthrough of Marcus et al.~\cite{MSS} implies that $O(1/\kappa)$-spectrally-thin trees exist. Details of this connection are given in \Appendix{kadison}.
It is unknown if similar techniques can show that $O(1/k)$-thin trees exist.
The best known algorithmic construction of spectrally-thin trees is still \Theorem{specthin}.
This result is a special case of a result in a more abstract geometric setting.
Suppose $V = \set{v_1,\ldots,v_m}$ are unit vectors in $\ell_2^n$ for which
$\sum_{i=1}^m v_i v_i \transpose$ is a multiple of the identity.
Does there exist a subset $V_B = \setst{ v_i }{ i \in B }$
that is a \emph{basis} of $\bR^n$
and for which the maximum eigenvalue of $\sum_{i \in B} v_i v_i \transpose$ is small?
The maximum eigenvalue is 1 if and only if $V_B$ is orthonormal,
but an arbitrary $V$ need not contain an orthonormal basis.
Again, the breakthrough of Marcus et al.~\cite{MSS} yields a non-constructive proof
of a basis with maximum eigenvalue $O(1)$; see \Appendix{kadison}.
In \Section{isotropic}, we show how to find in polynomial time a basis $V_B \subseteq V$
for which the maximum eigenvalue of $\sum_{i \in B} v_i v_i \transpose$ is $O(\log n / \log \log n)$.
Previous constructive techniques \cite{AW,Oliviera,RV07,Tropp11} only provide a bound of $O(\log n)$.
Our geometric result also relates to the \newterm{column subset selection} problem in numerical linear algebra \cite{BMD,Tropp09,BDM,DR}
which seeks to ``approximate'' a matrix $A$ by a small subset of its columns,
under various notions of approximation.
Define the \newterm{stable rank} of $A$ to be
the Frobenius norm divided by the spectral norm, all squared;
this roughly captures the rank of $A$, ignoring negligibly small singular values.
In numerical linear algebra \cite{BMD,BDM,DR}, the number of columns chosen
is typically much larger than the stable rank.
The operator theory community considers similar questions \cite{BT87,BT91,SSRI,Tropp09},
although the number of columns selected is typically much smaller than the stable rank.
In \Section{CSS},
we show that one can efficiently select a \emph{linearly independent} set of columns of size \emph{equal}
to the stable rank, while carefully controlling the maximum singular value.
\subsection{Techniques}
Our results are based on the pipage rounding technique \cite{AS,Sri01,GandhiKPS06,CCPV},
which has had several interesting uses in the recent literature.
Deterministic and randomized forms of pipage rounding exist;
our result applies to both of those, as well as to swap rounding.
Typical uses of pipage rounding involve some of the following ideas.
\begin{itemize}
\item There are processes that iteratively move a point in a matroid base polytope towards
an extreme point, while modifying only two coordinates at a time.
The exchange properties of matroid bases ensure that this is possible.
\item One can define a ``potential function'' on the matroid base polytope
(e.g., the ad hoc functions defined in \cite{AS},
or the multilinear extension of a submodular function \cite{CCPV})
such that the function is concave or convex in directions that increase one coordinate and decrease
another.
\item The randomized form of pipage rounding \cite{Sri01,GandhiKPS06,CVZFOCS}
outputs a matroid base whose elements are negatively correlated
(more precisely, \textit{negative cylinder dependent}).
This ensures that linear functions of that base satisfy the same
Chernoff-type concentration bounds that are satisfied under independent rounding.
\end{itemize}
Our aim is to show that, for various concentration bounds,
the final extreme point satisfies the same bounds that
would be achieved by independent randomized rounding.
For Chernoff bounds this follows from negative correlation,
but for other bounds such a result was not previously known.
\begin{itemize}
\item Let $f$ be a monotone submodular function defined on the ground set of the matroid.
When using randomized pipage rounding, does
the value of $f$ at the final extreme point satisfy the same lower tail bound
as when using independent rounding?
Chekuri et al.~\cite{CVZArxiv} conjectured this to be true,
and they proved such a result when using swap rounding.
\item Let $f$ be a linear function
mapping points in the matroid base polytope to symmetric matrices.
When using pipage rounding, can the value of $f$ at the final extreme point
be guaranteed to satisfy the same eigenvalue bounds
as when using independent rounding?
\end{itemize}
It does not seem easy to answer these questions using negative correlation properties.
We present a new approach that leads to a positive answer to both of these questions.
In both cases, we can define a pessimistic estimator \cite{R}
that bounds the probability that randomized rounding
fails to achieve the desired concentration.
We show that these pessimistic estimators are concave
when one element's sampling probability is increased
and another's is decreased by the same amount.
Due to that concavity property, the base output by randomized pipage rounding
satisfies the same concentration
bounds that would be satisfied under independent randomized rounding.
For the second question (matrix concentration), the pessimistic estimator
can be efficiently evaluated, so deterministic pipage rounding can also be used.
The concavity property of our pessimistic estimator
for matrix concentration is a non-trivial fact.
We establish that fact by proving a new variant of Lieb's concavity theorem \cite{Lieb},
which is a ``masterpiece of matrix analysis''~\cite{BS}
with deep applications in mathematical physics and quantum information theory
\cite{Carlen,Effros,NC}.
Although there is much interest in the mathematical physics community on
extensions and variants of Lieb's theorem,
our particular variant does not seem to appear in the literature.
\section{Preliminaries}
\SectionName{prelim}
Let $[m] = \set{1,\ldots,m}$.
For a set $S \subseteq [m]$, the vector $\chi(S) \in \bR^m$ is the characteristic vector of $S$.
For a vector $x \in \bR^m$ and a set $S \subseteq [m]$,
the notation $x(S)$ denotes $\sum_{i \in S} x_i$.
The vector $e_i$ denotes the $i\th$ standard basis vector of the
finite dimensional vector space that is apparent from context.
The vector $\vec{1}$ denotes a vector whose components are all ones
and whose dimension is apparent from context.
We will use $\bRnneg$ and $\bZnneg$ to denote the nonnegative and positive reals respectively.
Let $\Sym$ denote the space of symmetric, real matrices of size $n \times n$.
Let $\Psd, \Pd \subset \Sym$ respectively denote the cones of positive semidefinite
and positive definite matrices.
Let $\Diag \subseteq \Sym$ denote the space of $n \times n$ diagonal matrices.
Let $\preceq$ denote the L\"owner partial order on symmetric matrices,
i.e., $A \preceq B$ iff $B-A \in \Psd$.
Similarly, $A \prec B$ iff $B-A \in \Pd$.
For $A \in \Sym$, let $\lmax(A)$ and $\lmin(A)$ respectively denote the largest and smallest
eigenvalues of $A$.
For $B \in \Sym$, let $B^+$ denote its Moore-Penrose pseudoinverse.
For $B \in \Psd$, let $B^{+/2} \in \Psd$ denote the positive semidefinite square root of $B^+$.
The image of $B$ is $\image B$
and the orthogonal projection onto $\image B$ is $I_{\image B}$.
The notation $\norm{\cdot}$ denotes the $\ell_2$ norm
for vectors and the $\ell_2$ operator norm for matrices.
If $\cD$ is a distribution, $X \sim \cD$ means that
the random variable $X$ has distribution $\cD$.
\section{Concavity of Pessimistic \tcskip Estimators}
In this section we state the known results on pipage rounding
and our concavity of pessimistic estimators technique.
We then apply this technique in three scenarios, of increasing difficulty:
(1) Chernoff bounds, (2) submodular functions, and (3) matrix concentration.
The latter two results are new,
and in particular are not known to follow using negative correlation.
This pessimistic estimator for matrix concentration
underlies all applications in \Section{applications}.
\subsection{Pipage Rounding}
Pipage rounding is a dependent rounding process
originating in works of Ageev, Srinivasan and Sviridenko \cite{AS,Sri01}.
Calinescu et al.\ \cite{CCPV} generalized it to a matroid setting.
We now state the main results of randomized and deterministic pipage rounding;
a proof sketch is given in \Appendix{pip}.
Let $\mat$ be a matroid on $[m]$ and let $P \subset \bR^m$ be its base polytope.
For all algorithmic applications in this paper, $\mat$ can be presented to the
algorithm via an independence oracle.
A function $g : P \rightarrow \bR$ is said to be \textit{concave under swaps} if
\begin{equation}
\EquationName{fconcave}
\forall p \in P, ~\forall a,b \in [m],
\quad
z \mapsto g\big(p + z (e_a\kern-1pt - e_b)\kern-1pt\big)
~~\text{is concave}.
\end{equation}
\begin{theorem}[Randomized Pipage Rounding] \iftc{\mbox{}\\}
\TheoremName{randpip}
There is a randomized, polynomial-time algorithm that,
given $x \in P$, outputs an extreme point $\hat{x}$ of $P$
with $\expect{\hat{x}}=x$ and such that,
for any $g$ concave under swaps, $\expect{g(\hat{x})} \leq g(x)$.
\end{theorem}
\begin{theorem}[Deterministic Pipage Rounding] \iftc{\mbox{}\\}
\TheoremName{detpip}
There is a deterministic, polynomial-time algorithm that, given
$x \in P$ and a value oracle for a function $g$ that is concave under swaps,
outputs an extreme point $\hat{x}$ of $P$ with $g(\hat{x}) \leq g(x)$.
\end{theorem}
The swap rounding procedure of Chekuri et al.\ \cite{CVZArxiv,CVZFOCS}
also proves \Theorem{randpip} and \Theorem{detpip}.
For $x \in [0,1]^m$, let $\cD(x)$ be the product distribution on $\set{0,1}^m$ with marginals given by $x$,
i.e., $\probover{X \sim \cD(x)}{X_i=1} = x_i$.
Let $\cE \subseteq \set{0,1}^m$.
A \newterm{pessimistic estimator} \cite{R,SrinivasanNotes} for $\cE$
is a function $g : [0,1]^m \rightarrow \bR$
that
satisfies
\tcversion{\begin{gather}
\EquationName{PE}
\probover{X \sim \cD(x)}{X \in \cE} ~\leq~ g(x) \qquad\forall x \in [0,1]^m
\\\nonumber
\min \{ g(x \!-\! x_i e_i), g(x \!+\!(1\!-\!x_i)e_i) \} \leq g(x)
~~\forall x \!\in\! [0,1]^m , i \!\in\! [m].
\end{gather}}
{\begin{gather}
\EquationName{PE}
\probover{X \sim \cD(x)}{X \in \cE} ~\leq~ g(x) \quad\quad\qquad\forall x \in [0,1]^m
\\\nonumber
\min \big\{\: g(x - x_i e_i) ,\, g\big(x +(1 - x_i)e_i\big) \:\big\} ~\leq~ g(x)
~~~\forall x \in [0,1]^m,~ i \in [m].
\end{gather}
}
For uses of pessimistic estimators in derandomization,
the function $g$ is also required to be efficiently computable.
That is not required with their use in randomized pipage rounding
as $g$ is not even provided as input to the algorithm.
\begin{claim}[Concavity of Pessimistic Estimators]\iftc{\mbox{}\\}
\ClaimName{concPE}
Let $\cE \subseteq \set{0,1}^m$ and let $g$ be a function
that satisfies \eqref{eq:PE} and is concave under swaps.
Suppose randomized pipage rounding is started at an initial point
$x_0 \in P$, and let $\hat{x}$ be the (random) extreme point of $P$ that is output.
If $g(x_0) \leq \epsilon$ then $\prob{ \hat{x} \in \cE } \leq \epsilon$.
Suppose deterministic pipage rounding
is given oracle access to $g$ and an initial point $x_0 \in P$ with $g(x_0) < 1$.
Then the extreme point $\hat{x}$ of $P$ that is output satisfies $\hat{x} \not \in \cE$.
\end{claim}
We omit the proof of \Claim{concPE} as it is an easy consequence of
\Theorem{randpip} and \Theorem{detpip}.
\subsection{Chernoff bound}
Let us start with a simple result to illustrate the technique.
First we state the Chernoff bound in convenient notation.
We discuss only the right tail; an analogous result holds for the left tail.
Fix any vector $w \in [0,1]^m$.
For $t \in \bR$ and $\theta > 0$, define $g_{t,\theta} : [0,1]^m \rightarrow \bR$ by
$$
g_{t,\theta}(x) ~:=~
e^{-\theta t} \cdot \expectover{X \sim \cD(x)}{ e^{\theta w \transpose X}}.
$$
Let $\mu = w \transpose x$ and $\delta \geq 0$. Then
\tcversion{
\begin{align}\nonumber
\probover{X \sim \cD(x)}{ w\transpose X \geq t }
&~\leq~ \inf_{\theta>0} \, g_{t,\theta}(x)
\\
\EquationName{chernoffRHS}
\text{and}\tcversion{\quad}{\qquad\qquad}
g_{(1+\delta) \mu ,\, \ln(1+\delta)}(x)
&~\leq~ \Big( \frac{e^\delta}{(1+\delta)^{1+\delta}} \Big)^\mu.
\end{align}
}{
\begin{equation}
\EquationName{chernoffRHS}
\probover{X \sim \cD(x)}{ w\transpose X \geq t } ~\leq~ \inf_{\theta>0} \, g_{t,\theta}(x) \qquad
\text{and}\qquad g_{(1+\delta) \mu ,\, \ln(1+\delta)}(x) ~\leq~ \Big( \frac{e^\delta}{(1+\delta)^{1+\delta}} \Big)^\mu.
\end{equation}
}
The following claim is proven in \Appendix{concavity}.
\begin{claim}
\ClaimName{chernoffconcave}
$g_{t,\theta}$ is concave under swaps.
\end{claim}
Consequently, \Claim{concPE} implies the following result.
\begin{corollary}
If randomized pipage rounding starts at $x_0 \in P$
and outputs the extreme point $\hat{x}$ of $P$ then,
$\forall w \in [0,1]^m ,\, \delta \geq 0$,
\begin{equation}
\EquationName{pipageChernoffRHS}
\prob{ w \transpose \hat{x} \geq (1+\delta) \mu }
~\leq~ \Big( \frac{ e^\delta }{ (1+\delta)^{1+\delta} } \Big)^{\mu}
\end{equation}
where $\mu = w \transpose x_0$.
Furthermore, if this right-hand side is strictly less than $1$,
then deterministic pipage rounding outputs an extreme point $\hat{x}$ of $P$
with $w \transpose \hat{x} < (1+\delta) \mu$.
\end{corollary}
The key point is that the right-hand sides of \eqref{eq:chernoffRHS} and
\eqref{eq:pipageChernoffRHS} are the same.
Chekuri et al.~\cite{CVZArxiv} proved this fact using negative correlation of $\hat{x}$,
generalizing a result of Srinivasan~\cite{Sri01}.
\subsection{Submodular functions}
Chekuri et al.~\cite[Theorem 1.3]{CVZArxiv} prove an analog of the Chernoff bound
for concentration of submodular functions under independent rounding.
They show that the same bound remains true under swap rounding~\cite[Theorem 1.4]{CVZArxiv} and
ask whether it remains true under pipage rounding.
Formally, let $f : \set{0,1}^m \rightarrow \bR$
be a non-negative, monotone, submodular function with marginals in $[0,1]$.
The \textit{multilinear extension} of $f$ is $F : [0,1]^m \rightarrow \bR$ with
$ F(x) := \expectover{X \sim \cD(x)}{f(X)} $.
For $t \in \bR$ and $\theta<0$, define $g_{t,\theta} : [0,1]^m \rightarrow \bR$ by
\[
g_{t,\theta}(x) ~:=~ e^{-\theta t} \cdot \expectover{X \sim \cD(x)}{ e^{\theta f(X)} }.
\]
The left tail bound of Chekuri et al.\ is: with $\mu = F(x) ,\, \delta \in [0,1)$,
\tcversion{
\begin{align*}
\probover{X \sim \cD(x)}{ f(X) \leq t }
&~\leq~ \inf_{\theta<0} \, g_{t,\theta}(x) \\
\text{and}\tcversion{\quad}{\qquad\qquad}
g_{(1-\delta) \mu ,\, \ln(1-\delta)}(x)
&~\leq~ \exp( - \delta^2 \mu / 2 ).
\end{align*}
}{
\[
\probover{X \sim \cD(x)}{ f(X) \leq t }
~\leq~ \inf_{\theta<0} \, g_{t,\theta}(x)
\qquad\text{and}\qquad
g_{(1-\delta) \mu ,\, \ln(1-\delta)}(x)
~\leq~ \exp( - \delta^2 \mu / 2 ).
\]
}
The following claim is proven in \Appendix{concavity}.
\begin{claim}
\ClaimName{submodconcave}
$g_{t,\theta}$ is concave under swaps.
\end{claim}
\Claim{concPE} implies the following result,
answering an open question of Chekuri et al.\ \cite[p. 3]{CVZArxiv}.
\begin{corollary}
If randomized pipage rounding starts at $x_0 \in P$
and outputs the extreme point $\hat{x}$ of $P$ then,
letting $\mu = F(x_0)$,
we have
$ \prob{ f(\hat{x}) \leq (1-\delta) \mu } \leq \exp(-\delta^2 \mu / 2)$.
\end{corollary}
\noindent
Chekuri et al.~\cite[p.~583]{CVZFOCS} state that this fact
does not follow from negative correlation of $\hat{x}$.
\subsection{Matrix Concentration}
\onote{Modified}
Tropp~\cite{Tropp11}, improving on Ahlswede-Winter~\cite{AW} and Oliviera~\cite{Oliviera}, proves a beautiful analog of the Chernoff bound
for sums of independent random matrices.
We state a simplified form here.
\begin{theorem}
\TheoremName{tropp}
Let $M_1,\ldots,M_m \in \Psd$ satisfy $M_i \preceq R \cdot I$.
For $t \in \bR$ and $\theta>0$, define $g_{t,\theta} : [0,1]^m \rightarrow \bR$ by
$$
g_{t,\theta}(x) ~:=~ e^{-\theta t} \cdot
\trace \exp\Big(\sum_{i=1}^m \log \expectover{X \sim \cD(x)}{ e^{\theta X_i M_i} } \Big).
$$
Then, for $\mu \geq \norm{ \expectover{X \sim \cD(x)}{ \sum_i X_i M_i } }$ and $\delta \geq 0$,
\tcversion{
\begin{align*}
\probover{X \sim \cD(x)}{ \norm{\smallsum{i}{} X_i M_i} \geq t }
&~\leq~ \inf_{\theta>0} \, g_{t,\theta}(x) \\
\text{and}\tcversion{\quad}{\qquad\qquad\quad}
g_{(1+\delta) \mu ,\, \ln(1+\delta)}(x)
&~\leq~ n \cdot \Big( \frac{e^\delta}{(1+\delta)^{1+\delta}} \Big)^{\mu/R}.
\end{align*}
}{
$$
\probover{X \sim \cD(x)}{ \norm{\smallsum{i}{} X_i M_i} \geq t }
~\leq~ \inf_{\theta>0} \, g_{t,\theta}(x)
\qquad\text{and}\qquad
g_{(1+\delta) \mu ,\, \ln(1+\delta)}(x)
~\leq~ n \cdot \Big( \frac{e^\delta}{(1+\delta)^{1+\delta}} \Big)^{\mu/R}.
$$
}
\end{theorem}
The following is our main lemma on pessimistic estimators.
The proof is in \Appendix{concavity}.
\begin{lemma}
\LemmaName{pipage}
$g_{t,\theta}$ is concave under swaps.
\end{lemma}
Consequently, \Claim{concPE} implies the following result.
\begin{corollary}
\CorollaryName{matrixconcentration}
Let $P$ be a matroid base polytope and let $x_0 \in P$.
Let $M_1,\ldots,M_m \in \Psd$ satisfy $M_i \preceq R \cdot I$.
Let $\mu \geq \norm{\expectover{X \sim \cD(x_0)}{\sum_i X_i M_i}}$.
If randomized pipage rounding starts at $x_0$
and outputs the extreme point $\hat{x}=\chi(S)$ of $P$ then
we have
\begin{equation}
\EquationName{matrixPE}
\prob{ \norm{\smallsum{i \in S}{} M_i } \geq (1+\delta) \mu }
\:\leq\: n \cdot \Big( \frac{ e^\delta }{ (1+\delta)^{1+\delta} } \Big)^{\mu/R}.
\end{equation}
Furthermore, if this right-hand side is strictly less than $1$, then
deterministic pipage rounding outputs an extreme point $\hat{x} = \chi(S)$ of $P$ with
$\norm{\sum_{i \in S} M_i} < (1+\delta) \mu$.
\end{corollary}
The inequalities in \Theorem{tropp} involve non-trivial matrix analysis,
such as operator concavity of $\log$ and Lieb's celebrated concavity theorem \cite{Lieb}.
It seems that even those results do not suffice to prove \Lemma{pipage}.
To prove it, we derive a new variant of Lieb's theorem (\Theorem{liebvariant}).
Lieb \cite{Lieb} proved several related concavity theorems;
for us, the most relevant form is:
\begin{theorem}[Lieb \cite{Lieb}]
\TheoremName{lieb}
Let $L, K \in \Sym$ and $C \in \Pd$.
Then $z \mapsto \trace \exp\big( L + \log(C+zK) \big)$ is concave in a neighborhood of\/ $0$.
\end{theorem}
The main technical result of this paper is:
\begin{theorem}
\TheoremName{liebvariant}
\label{LIEBVARIANT}
Let $L \in \Sym$, $C_1, C_2 \in \Pd$ and $K_1, K_2 \in \Psd$.
Then the univariate function
\begin{equation}
\EquationName{exploglog}
z ~\mapsto~ \trace \exp\Big( L + \log( C_1 + z K_1 ) + \log( C_2 - z K_2 ) \Big)
\end{equation}
is concave in a neighborhood of\/ $0$.
\end{theorem}
There are several known approaches to proving Lieb's theorem.
The simplest is Tropp's approach \cite{Tropp12};
however, his proof is based on joint convexity of quantum entropy,
which is itself usually proven using Lieb's theorem.
We were unable to prove \Theorem{liebvariant} using Tropp's approach.
Lieb's original proof~\cite{Lieb},
which proves concavity by directly analyzing the second derivative,
involves numerous delicate steps of matrix analysis.
We were able to adapt this approach to prove a weaker form of \Theorem{liebvariant}
that requires some additional commutativity assumptions; details are in \Appendix{lieb}.
This weaker result suffices to prove \Lemma{pipage}.
Epstein~\cite{Epstein} gives an elegant approach to proving Lieb's theorem
using complex analysis, and in particular powerful results concerning \emph{Herglotz functions}.
Our proof of \Theorem{liebvariant}, which appears in \Appendix{epstein},
is an adaptation of Epstein's approach.
\medskip
\noindent \tcversion{\textsc{Remark.}}{\textbf{Remark.}}
Another well-known matrix concentration inequality is the Ahlswede-Winter \cite{AW} inequality,
for which pessimistic estimators were studied by Wigderson and Xiao \cite{WX}.
It is natural to wonder whether we could have used their pessimistic estimators instead.
Unfortunately they do not seem applicable for our scenario.
The issue is that the Ahlswede-Winter inequality is most effective
for analyzing sums of i.i.d.\ random matrices,
due to some inequalities that arise in their analysis.
In our scenario, due to the way that pipage rounding works,
we require non-i.i.d.\ product distributions,
so it is much more convenient to base our approach on \Theorem{tropp}.
\section{Applications}
\SectionName{applications}
\subsection{Rounding of semidefinite programs}
Let $\mat$ be a matroid and let $P \subset \bR^n$ be its base polytope.
Consider the spectrahedron
\begin{equation}
\EquationName{Qdef}
Q ~:=~ P \:\intersect\: \Bigl\{\,x \in \bR^m \;:\; \sum_{i=1}^m x_i A_i \preceq B \,\Bigr\},
\end{equation}
where each $A_1,\ldots,A_m, B \in \Psd$.
We think of $P$ as specifying ``hard'' constraints and the semidefinite constraint as being ``soft''.
\begin{theorem}
\TheoremName{main}
Suppose that $A_i \preceq B$ for all $i$.
If randomized pipage rounding starts at $x_0 \in Q$
and outputs the extreme point $\chi(S)$ of $P$, then
$ \prob{ \smallsum{i \in S}{} A_i \preceq \alpha B } \geq 1-1/n $,
for some $\alpha = O( \log n / \log \log n )$.
Furthermore, if deterministic pipage rounding starts at $x_0 \in Q$,
then it outputs an extreme point $\chi(S)$ of $P$ with
$\smallsum{i \in S}{} A_i \preceq \alpha B$.
\end{theorem}
This theorem is optimal with respect to $\alpha$, as discussed below.
The hypothesis that $A_i \preceq B$ is a ``width'' condition
that commonly arises in optimization and rounding.
\begin{proof}
Recall the notation defined in \Section{prelim}.
Let $M_i = B^{+/2} A_i B^{+/2}$.
By standard arguments,
\begin{align*}
\smallsum{i=1}{m} x_i A_i \preceq B
&\quad\iff\quad
\smallsum{i=1}{m} x_i M_i \preceq I_{\image B} \\
\text{and}\quad
\smallsum{i \in S}{} A_i \preceq \alpha B
&\quad\iff\quad
\smallsum{i \in S}{} M_i \preceq \alpha I_{\image B}.
\end{align*}
We assume that $A_i \preceq B$, so $\lmax(M_i) \leq 1$.
Apply \Corollary{matrixconcentration}
with $\delta = 4 \log n / \log \log n$, $\mu=1$ and $R=1$.
A standard calculation shows that the right-hand side of \eqref{eq:matrixPE} is less than $1/n$.
\end{proof}
Chekuri, Vondr\'ak and Zenklusen \cite{CVZArxiv,CVZFOCS} considered the problem of rounding a point
in a matroid polytope to an extreme point, subject to additional packing constraints.
Their result generalizes the low-congestion multi-path routing problem studied earlier by
Srinivasan et al.~\cite{Sri01,GandhiKPS06},
but it is itself a special case of \Theorem{main}
where the matrices $A_i$ and $B$ are diagonal.
The factor $\alpha = O(\log n / \log \log n)$ is optimal in \Theorem{main}
because it is optimal for rounding this low-congestion multi-path routing problem,
and even for the congestion minimization problem \cite{LRS}.
\subsection{Rounding an isotropic distribution to a \tcskip nearly orthonormal basis}
\SectionName{isotropic}
Let $w_1,\ldots,w_m \in \bR^n$ satisfy $\norm{w_i} = 1$ for all $i$.
Let $p_1,\ldots,p_m$ be a probability distribution on these vectors
such that the covariance matrix is $\sum_i p_i w_i w_i \transpose = I/n$.
A random vector drawn from that distribution is said to be in \newterm{isotropic position}.
\begin{theorem}
\TheoremName{isotropic}
There is a polynomial time algorithm (either randomized or deterministic)
to compute a subset $S \subseteq [m]$ such that $\setst{ w_i }{ i \in S }$ forms a
\emph{basis} of\/ $\bR^n$,
and for which $ \norm{ \sum_{i \in S} w_i w_i \transpose } \leq \alpha $,
where $\alpha = O(\log n / \log \log n)$.
\end{theorem}
As is discussed in \Appendix{kadison}, the recent breakthrough on the Kadison-Singer problem~\cite{MSS} implies the following existential result:
\begin{theorem}
\TheoremName{strongIsotropic}
There exists $S \subseteq [m]$ such that $\setst{ w_i }{ i \in S }$
forms a basis of\/ $\bR^n$, and for which
$\norm{ \sum_{i \in S} w_i w_i \transpose } = O(1)$.
\end{theorem}
We now prove \Theorem{isotropic} using \Theorem{main}.
Let $\mat$ be the linear matroid corresponding to the vectors $\set{w_1,\ldots,w_m}$.
Let $P$ be the base polytope of that linear matroid.
Let $r : 2^{[m]} \rightarrow \bZnneg$ be the rank function of that matroid,
i.e.,
$r(S) = \operatorname{dim} \big(\operatorname{span} \setst{ w_i }{ i \in S }\big)$.
Then
\[
P ~:=~ \setst{ x \in \bRnneg^n }{ x(J) \leq r(J) ~\forall J \subseteq [m] ~~\text{and}~~ x([m]) = r([m]) }.
\]
Define $A_i = w_i w_i \transpose$, $B = I$ and
\[
Q ~=~ P \:\intersect\: \Bigl\{\, x \in \bR^m \,:\, \sum_i x_i A_i \preceq B \,\Bigr\}.
\]
Let $x = n \cdot p$.
Then the following claim and the hypothesis that $\sum_i p_i w_i w_i \transpose = I/n$
show that $x \in Q$.
\begin{claim}
\ClaimName{pinP}
$x \in P$.
\end{claim}
Since $\norm{w_i}=1$, we have $A_i = w_i w_i \transpose \preceq I = B$.
\Theorem{main} gives an algorithm to construct an
extreme point $\chi(S)$ of $P$
for which $\sum_{i \in S} A_i \preceq \alpha \cdot B$,
with $\alpha = O(\log n / \log \log n)$.
Since $P$ is the base polytope of $\mat$,
$\setst{ w_i }{ i \in S }$ forms a basis of $\bR^n$.
Finally, $ \sum_{i \in S} w_i w_i \transpose \preceq \alpha \cdot I $.
This completes the proof of \Theorem{isotropic}, modulo the proof of \Claim{pinP}.
In \Appendix{Psubmodular}, we show that \Theorem{isotropic} can be generalized from a decomposition
of the identity into rank-one matrices $w_i w_i \transpose$ to a decomposition into matrices of
arbitrary rank.
The proof of \Claim{pinP} is analogous to the proof of \Claim{pinPprime}.
We remark that \Theorem{strongIsotropic} is not known to have a generalization
to matrices of arbitrary rank.
\subsection{Thin trees}
\SectionName{thin}
\label{THIN}
Let $G = (V,E)$ be a graph.
For convenience we assume that $V = [n]$.
The \newterm{cut} defined by $U \subseteq V$ is
\[
\delta_G(U) \:=\: \delta(U) \:=\: \setst{ uv \in E }{ \text{ exactly one of $u$ and $v$ is in $U$}}.
\]
For a subgraph $T$ of $G$,
let $\delta_T(U)$ denote all edges of $T$ with exactly one endpoint in $U$.
\begin{definition}
A subgraph $T$ of $G$ is called \newterm{$\epsilon$-thin} if
$\card{ \delta_T(U) } \leq \epsilon \cdot \card{ \delta_G(U) }$
for all $U \subseteq V$.
\end{definition}
\begin{conjecture}[Goddyn \cite{Goddyn}]
\ConjectureName{goddyn}
Every graph with connectivity at least $\connec$ has an $f(\connec)$-thin spanning subtree,
for some function $f$ that vanishes as $\connec$ tends to infinity.
\end{conjecture}
The crucial detail in this conjecture is that the function $f$ should not
depend on the size of the graph.
The best progress on this conjecture for general graphs is as follows.
\begin{theorem}[Asadpour et al.~\cite{AGMOS}]
\TheoremName{asadpour}
Let $G$ be a graph with $n$ vertices and connectivity $\connec$.
Then $G$ has a $O\big( \frac{\log n}{\connec \log \log n} \big)$-thin spanning subtree.
Moreover, there is a randomized, polynomial time algorithm to construct such a tree.
\end{theorem}
Now we define spectrally-thin trees and prove an analog of this theorem.
The Laplacian of $G$ is the symmetric matrix $L_G$ with rows and columns indexed by $V$
defined by
\[ L_G ~:=~ \sum_{uv \in E} (e_u - e_v) (e_u - e_v) \transpose. \]
\begin{definition}
Let $T$ be a spanning subtree of $G$ and let $L_T$ be the Laplacian of $T$.
The tree $T$ is \newterm{$\epsilon$-spectrally-thin} if
$L_T \preceq \epsilon L_G $.
\end{definition}
Any tree that is $\epsilon$-spectrally-thin is also $\epsilon$-thin,
because
\[
\card{\delta_T(U)}
~=~ \chi(U) \transpose \, L_T \, \chi(U)
~\leq~ \epsilon \cdot \chi(U) \transpose \, L_G \, \chi(U)
~=~ \epsilon \cdot \card{\delta_G(U)}.
\]
The converse is not true.
Moreover, the connectivity hypothesis in \Theorem{asadpour} does not suffice\footnote{
This result was independently observed by M.~de Carli Silva, N.~Harvey and C.~Sato,
and by M. Goemans~\cite{GoemansTalk}, using slightly different examples.
}
to obtain a good spectrally-thin tree.
The proof is in \Appendix{nospectrallythintree}.
\begin{theorem}
\TheoremName{nospectrallythintree}
For every $n, k \geq 1$, there exists a weighted graph with $n$ vertices and connectivity $\connec$
that does not have an $o( \sqrt{n}/\connec )$-spectrally-thin spanning subtree.
\end{theorem}
Nevertheless, if we strengthen the connectivity lower bound to a lower bound
on the effective conductances, then we have the following construction of spectrally-thin trees.
For an edge $e = uv \in E$, the \newterm{effective resistance} in $G$ between $u$ and $v$ is
$R_e := (e_u - e_v)\transpose L_G^+ (e_u - e_v)$.
The \newterm{effective conductance} in $G$ between $u$ and $v$ is $C_e := 1/R_e$.
\begin{theorem}
\TheoremName{conductanceweightedtree}
Let $G$ be a graph with $n$ vertices such that $\conduc \leq C_e$ for every edge $e$.
Then there is a polynomial time algorithm (either randomized or deterministic)
to construct a $O\big( \frac{\log n}{\conduc \log \log n} \big)$-spectrally-thin
spanning subtree of $G$.
\end{theorem}
\Theorem{conductanceweightedtree} follows directly from \Theorem{main},
letting $\mat$ be the graphic matroid corresponding to $G$.
It also follows from \Theorem{isotropic}, as we show in \Appendix{transitive}.
That viewpoint is advantageous, since \Theorem{strongIsotropic} then immediately implies
\begin{theorem}
\TheoremName{verythintree}
Let $G$ be a graph with $n$ vertices such that $\conduc \leq C_e$ for every edge $e$.
Then $G$ has a $O( 1/\conduc )$-spectrally-thin spanning subtree.
\end{theorem}
We are not aware of any formal connection between \Theorem{verythintree}
and \Conjecture{goddyn} or the traveling salesman problem.
Although \Theorem{asadpour} and \Theorem{conductanceweightedtree} are formally incomparable,
it is worth understanding their similarities and differences.
Both results have a seemingly suboptimal factor of $\log n / \log \log n$.
\Theorem{asadpour} requires only a connectivity lower bound,
which is important in applications \cite{Goddyn,AGMOS},
but the resulting tree is thin, not spectrally-thin; also, their algorithm is randomized.
\Theorem{conductanceweightedtree} requires a conductance lower bound
(which is stronger than a connectivity lower bound),
but the resulting tree is spectrally-thin (which is stronger than being thin);
also, our algorithm can be made deterministic.
The use of randomization seems quite inherent in the algorithms \cite{AGMOS,CVZFOCS}
for \Theorem{asadpour}, as the thinness condition involves controlling exponentially many cuts,
which seems difficult to accomplish by a deterministic, polynomial-time algorithm.
The quantities $\connec$ and $\conduc$ can be related in certain classes of graphs.
We say that a family of graphs has \newterm{nearly equal resistances} if
there is a constant $c$ (independent of the number of vertices)
such that $R_e \leq c R_f$ for all edges $e,f$.
For example, any Ramanujan graph has nearly equal resistances.
Edge-transitive graphs, such as hypercubes,
have nearly equal (in fact, exactly equal) resistances.
\begin{corollary}
\CorollaryName{transitive}
Let $G$ be a graph with $n$ vertices, nearly equal resistances,
and connectivity $\connec$.
Then there is a deterministic, polynomial time algorithm to construct a $O\big( \frac{\log n}{\connec \log \log n} \big)$-spectrally-thin tree of $G$.
\end{corollary}
The proof is in \Appendix{transitive}.
\subsection{Column-subset selection}
\SectionName{CSS}
\todo{Consequences of MSS here?}
Column-subset selection is an important topic in
numerical linear algebra \cite{BMD,Tropp09,DR,BDM}.
Similar questions are considered in operator theory \cite{BT87,BT91,SSRI,Tropp09,Youssef}.
In this section we prove a non-isotropic analog of \Theorem{isotropic},
which gives a new result on column-subset selection.
For a real matrix $A$, let $\norm{A}_F = \sqrt{ \trace A \transpose A }$ denote its Frobenius norm.
The \newterm{stable rank} of $A$ is $\stable(A) := \norm{A}_F^2/\norm{A}^2$.
\begin{theorem}
\TheoremName{kashin}
Let $A$ be a real matrix of size $n \times m$
whose columns are denoted $a_1,\ldots,a_m$.
Suppose that $\norm{a_i} = 1 ~\,\forall i$.
Then there is a deterministic, polynomial time algorithm to compute $S \subseteq [m]$
of size $\card{S} \geq \floor{ \stable(A) }$
such that $\setst{ a_i }{ i \in S }$ is linearly independent,
and $\norm{ \sum_{i \in S} a_i a_i \transpose } \leq O(\log n / \log \log n)$.
\end{theorem}
This is optimal with respect to $\card{S}$ as it can happen that $\stable(A)=\rank(A)$,
in which case $\setst{ a_i }{ i \in S }$ is linearly dependent whenever $\card{S} > \stable(A)$.
We now prove \Theorem{kashin} using \Theorem{main}.
Note that $\norm{A}_F^2=m$.
Let $p \in \bR^m$ be the vector with $p_i = \floor{m/\norm{A}^2}/m$ for all $i$.
Note that $\sum_i p_i = \floor{\stable(A)}$.
We claim that $p$ can be viewed as a ``fractional set'' of linearly independent vectors
of size $\floor{\stable(A)}$.
Formally, for any set $T \subseteq [m]$, let $A_T$ denote the submatrix of $A$
consisting of the columns in $T$.
Define the following family of sets
\[
\cB
~=~ \setst{ I \subseteq [m] }{ \rank A_I=\card{I}=\floor{\stable(A)} }.
\]
Then $\cB$ is the base family of the linear matroid corresponding to $A$,
truncated to rank $\floor{\stable(A)}$.
Let $\mat$ denote that matroid and let $P$ denote its base polytope.
\begin{claim}
\ClaimName{CSpinP}
$p \in P$.
\end{claim}
The proof is in \Appendix{CSS}.
Given this claim, all that remains is an easy application of \Theorem{main}.
Define $A_i = a_i a_i \transpose$, $B = I$ and
\[
Q ~=~ P \:\intersect\: \Bigl\{\: x \in \bR^m \;:\; \sum_i x_i A_i \preceq B \:\Bigr\}.
\]
We have $p \in Q$ by \Claim{CSpinP} and the fact that
\[
\sum_i p_i A_i
~\preceq~ \sum_i \frac{A_i}{\norm{A}^2}
~=~ \frac{A A \transpose}{\norm{A}^2}
~\preceq~ I ~=~ B.
\]
Note that $A_i = a_i a_i \transpose \preceq I = B$.
\Theorem{main} gives a deterministic algorithm
to construct an extreme point $\chi(S)$ of $P$
for which $\sum_{i \in S} A_i \preceq \alpha \cdot B$,
with $\alpha = O(\log n / \log \log n)$.
Since $S$ is a base of $\mat$,
the set $\setst{ a_i }{ i \in S }$ has rank equal to $\card{S} = \floor{\stable(A)}$.
This completes the proof of \Theorem{kashin}.
\paragraph*{Acknowledgements.}
N.~Harvey thanks Joel Friedman and Mohit Singh for numerous enlightening discussions.
We thank Isaac Fung for collaborating at a preliminary stage of this work.
We also thank Christos Boutsidis,
Joseph Cheriyan, Satoru Fujishige, Michel Goemans, Mary Beth Ruskai, Nikhil Srivastava,
Joel Tropp, Roman Vershynin and Jan Vondr\'ak for helpful discussions and suggestions.
| 159,583
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Tips for Conscience Building
Try these practices to help your child develop a healthy conscience. Remember to keep in mind your child's level of development and tailor your activities accordingly.
#1 Model Empathy
Model empathetic and caring behavior by treating your child with respect and kindness.
#2 Listen Actively
Listen closely to when your child expresses his or her feelings, particularly negative ones.
#3 Spend Time
Spend a lot of time with your child that is not oriented around discipline. Playtime is especially important for younger children. Chat time is important for teens.
#4 Set Limits
Set firm limits that focus on allowing children to experience the consequences of their mistakes. Promote responsibility.
#5 Do Not Tolerate Aggression
Take special care to notice any behaviors that infringe on the rights of others, even the family pet. Discuss and correct these behaviors through instruction. Keep in mind that aggression is never permissible, not for children and not for parents.
#6 Teach the Golden Rule
Take every opportunity to teach your child about empathy. The question to ask is "How would it feel to be in the other persons' place? What would that be like?"
#7 Allow Remorse
Allow and encourage children to feel remorse when they cause another person pain. Help them figure out how to make amends or repair the damage.
#8 Participate in Family Tasks
Promote your child's involvement in caring for the family. Let them assist with basic household tasks when you can. Establish chores when age appropriate. Help them to feel part of the "family team."
#9 Choose Media with Moral Messages
Make use of media that provides moral lessons. These can be books, television shows, movies, plays, etc. Discuss the stories to see that the lessons are understood.
| 383,846
|
Sku: CS-GS687
UPC: 049392270547
Collections: Animals and Insects, Puffer Toys, Toy Animals, Toys
Details: Puffer Ducks (1 Dozen) by CarnivalSource.com! Pull the soft tentacle hair on this puffer duck and they snap back into place.!
They were awesome! Better than I hoped. I purchased them for Duck Duck Jeep giveaways and they bring a smile to the Jeep owners who have received
The kids I sent these too absolutely loved them. They are squishy and fun to play with
| 121,177
|
TITLE: Torque due to magnetic field
QUESTION [5 upvotes]: My textbook states That the torque $\vec{\tau}$ experienced by a current carrying loop due to a magnetic field $\vec{B}$, is given by the equation $\vec{\tau} =\vec{M} \times \vec{B}$,where $\vec{M}$ denotes the magnetic moment of the current loop=$I\vec{A}$. However, it didnt specify about what point(s) is this equation valid. The most obvious choice seems to be the Center of mass of the loop.
For a uniform circular Loop, I was able to derive that this indeed holds true about the COM. However, we can extend this further for ANY random point:
Say the COM is $c$ and we wish to calculate the torque of magnetic field about a point say $p$. Take a current element $I\vec{dl}$. The force experienced by this Current element is $\vec{dF}$=$I\vec{dl} \times\vec{B}$. If the position vector Of point P wrt to the COM is $\vec{r_p}$, and that of the current element is $\vec{r_c}$, Then the torque, $\vec{dT}$ about point P is $(\vec{r_c}-\vec{r_p}) \times\vec{dF}$ = $\vec{r_c}\times\vec{dF}-\vec{r_p}\times\vec{dF}$. Integrating across the entire loop, The first integral becomes $\vec{M} \times \vec{B}$, while the second becomes zero.Which seems to suggest that the torque about ANY point is the same, and is $\vec{M} \times \vec{B}$.
1) How can we prove that for any , arbitrary loop,the Torue about COM is $\vec{M} \times \vec{B}$?
2)How do we show that the second integral cancels out? In the case I described above, the second integral had a term of $\int_{\theta=0} ^ {2\pi} \cos(\theta)\mathrm d\theta$ which became zero. Apart from "symmetry" how can we prove that the integral vanishes for a general case?
3) 1) and 2) together imply whatever be the loop, the Torque due to magnetic field about every point is the same. Can we generalize this further to the torque in an electric field given by $\vec{P} \times \vec{E}?$
REPLY [4 votes]: $\overrightarrow{\tau} = \overrightarrow{M} X \overrightarrow{B}$ is valid only in uniform magnetic field. Any point can be taken as origin and torque will be same.
To prove it, first of all prove it for a rectangular loop. Handling rectangular loop is easier than circular loop. You will also find that net force is zero. This means that this torque is a couple. Torque of a couple does not depend on origin. This is a direct result from rotational mechanics.
Now we can argue that any loop is made up of large number of rectangular loops! We get the required result immediately!
This result can be extended to $\overrightarrow{\tau} = \overrightarrow{p} X \overrightarrow{E}$ in uniform electric field.
| 25,934
|
TITLE: Where $a$ and $b$ are both constants, find the values of $a$ and $b$
QUESTION [0 upvotes]: Given
$$\frac{a}{6x-1}-\frac1{3x-1}\equiv\frac{b}{(6x-1)(3x+1)}$$
Where $a$ and $b$ are both constants, find the values of $a$ and $b$
REPLY [1 votes]: Assuming you mean that the denominator for the right side is to be $3x - 1$:
$$\frac{a}{6x-1}-\frac1{3x-1} = \frac{b}{(6x-1)(3x-1)}$$
$$\frac{a(3x-1)}{(6x-1)(3x-1)}-\frac{6x - 1}{(6x - 1)(3x-1)} = \frac{b}{(6x-1)(3x-1)}$$
$$ a(3x - 1) - 6x + 1 = b $$
Note that this is an identity, so it must hold for any value of $x$. Choose $x = 1/3$, which makes the $a$ term vanish:
$$ a(3(1/3) - 1) - 6(1/3) + 1 = b $$
$$ -2 + 1 = b $$
$$ -1 = b $$
Now returning to the equation:
$$ a(3x - 1) - 6x + 1 = b $$
$$ a(3x - 1) - 6x + 1 = -1 $$
Let $x = 1$. Then we have:
$$ a(3 \cdot 1 - 1) - 6 + 1 = -1 $$
$$ 2a - 5 = -1 $$
$$ 2a = 4 $$
$$ a = 2 $$
So:
$$ a = 2 $$
$$ b = -1 $$
Assuming you mean that the denominator for the left side is to be $3x + 1$:
$$\frac{a}{6x-1}-\frac{1}{3x+1} = \frac{b}{(6x-1)(3x+1)}$$
$$\frac{a(3x+1)}{(6x-1)(3x + 1)}-\frac{6x - 1}{(6x - 1)(3x+1)} = \frac{b}{(6x-1)(3x+1)}$$
$$a(3x + 1) - (6x - 1) = b$$
$$a(3x + 1) - 6x + 1 = b $$
Since this is an identity, it holds for any value of $x$. Choose $x = -1/3$, because it makes the $a$ term vanish. Then we have:
$$a(3 \cdot -1/3 + 1) - 6 \cdot -1/3 + 1 = b $$
$$2 + 1 = b$$
$$3 = b$$
Returning to:
$$a(3x + 1) - 6x + 1 = 3 $$
Let $x=1$. Now we have:
$$4a - 6 + 1 = 3$$
$$4a - 5 = 3$$
$$4a = 8$$
$$a = 2$$
So:
$$ a = 2 $$
$$ b = 3 $$
| 64,660
|
Mobile:
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| 396,906
|
Boys soccer Top 20 countdown: No. 14 Bergen Tech
Coach: Nelson Ramirez
Last year’s record: 15-6-3
2016 final ranking: 16
Best assets: The Knights might have the top goal-scorer in Bergen County by season’s end. Senior M Daniel Ryvkin’s 21 goals from last year is tied for the lead among all returning players from the county. “He is extremely smart with the ball,” Ramirez said. “He is one of those players that coaches say has his head on a swivel.” Ryvkin will be asked to go above and beyond again with Bergen Tech losing a pair of important scorers. Junior Carter Gwon is out 6-to-8 weeks with a back injury, while Sungjae Lee will play for Ramapo High School this fall. A season-ending ACL injury to senior defenseman Kaito Higashi only adds more adversity. The team hopes to get offensive production out of senior Alex Whang along with Eric Gulich, who may convert to striker. Senior D Corey Schultz is another strength in a big class that Ramirez is high on. “We’re bullish,” he said. “There’s no question about it. What these seniors have done is motivate us to work a little harder.”
Forecast: So much of the season will come down to how well Bergen Tech can overcome the obstacles and the changes on the field. If the Knights get offensive support for Ryvkin, they can contend for the Big North Liberty title and compete on the county level. Said Ramirez, “The fact that they are smart and willing to do the work makes us OK.”
50 H.S. Seniors to Watch: Nicole Gaito, Bergen Tech basketball
Non-Public Football Poll Countdown:: No. 5 Don Bosco
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| 334,135
|
TITLE: Sets whose $\epsilon$-neighborhoods have intersection of zero measure
QUESTION [1 upvotes]: Does there exist a characterization of
subsets of a segment such that the Lebesgue measure of its $\epsilon$-neighborhood tends to $0$ as $\epsilon\searrow 0$;
bounded functions that are continuous on the complement of a set of this type?
I mean either standard terminology or something which could be more convenient to work with.
REPLY [1 votes]: For any bounded set $A$, the limit of Lebesgue measure of its $\epsilon$-neighborhood as $\epsilon \to 0$ gives the Lebesgue measure of the closure $\overline{A}$. This is because the intersection of $\epsilon$-neighborhoods is the closure, and the measure is continuous on decreasing sequences of sets of finite measure. Therefore, the first condition amounts to $m(\overline{A})=0$.
The second condition can be rephrased as: bounded functions that are continuous on an open set of full measure. This property implies Riemann integrabiity, but is not equivalent to it (Thomae's function is Riemann integrable despite having a dense set of discontinuities).
| 136,771
|
SALT LAKE CITY – The days of underestimating Utah are over.
football
—
Super in Salt Lake City
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| 113,959
|
This first clip, from the old British Pathé newsreel, shows a canoe and kayak racing in Ausburg, Germany from 1957. British Pathe stopped producing in 1970 but until that time covered a wide range of clips in brief segments, giving the latest of news updates to everyone who might not enough reading the newspaper. Sometimes these clips would play before movies at the cinema, and later they made it to TV. The first event shown is apparently the Double-scull Canadian Canoe race.What exactly is the difference between canoe slalom and kayaking? I wondered the same thing, especially because the Olympic category for the sport includes both canoe and kayak sections.. The difference is small: in the canoe the paddler uses a single bladed paddle and kneels. In a kayak the paddler sits and uses and double ended paddle. Still this leads to two very different competitions, even if some may not be able to recognize the difference. Check out the second clip (it's in German, so be aware of that) to see a more thorough look at this 1957 competition! It's pretty cool to see just what this sport was like just a decade or so after the war ended, and before it developed into the Olympic sport it is today.
| 2,380
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TITLE: Can a term be its own coefficent in algebra?
QUESTION [3 upvotes]: I have a question in my math book; it asks me to find the coefficient of $b$ in the expression $3a+b+2c$. I thought, well, there is no coefficient of $b$, so I went on and then I wanted to go see if I was right at the back off the book and it says that $b$ counts as its own coefficient and I don't get it. Help explain it to me please; I`m so stuck. Thanks!
REPLY [3 votes]: It is an important basic fact that no matter what number $b$ is, $1b = b$.
So when you're asking about the "coefficient" of a certain term in a polynomial, and there isn't one written out explicitly, it makes the most sense to say that it is $1$.
If your book says otherwise, I recommend that you get a different book. Please let us know what book it is, so we can be sure not to use it.
| 163,966
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MISSOULA – The research coming out of the University of Montana is among the top in the nation and world, according to recent rankings by the National Taiwan University.
Of the more than 4,000 research institutions worldwide, NTU ranks the top 800 universities based on their production of scientific papers and the impact of those papers.
UM is highly ranked in the field of agriculture, which includes agricultural sciences, environment/ecology, and plant and animal science. UM also ranks highly in the subject areas of environment/ecology, geosciences, and plant and animal science.
“The NTU ranking is another indicator of the world class faculty at the University of Montana,” said Scott Whittenburg, UM vice president of research and creative scholarship. “Faculty publications, the citation of those articles by other researchers and the high impact of those journals are primary indicators of quality and demonstrate that our faculty and students are conducting research on par with leading institutions around the world.”
NTU ranking is considered a reliable source for universities devoted to scientific research. It is entirely based on scientific papers, reflecting scientific performance from three perspectives: research productivity – the number of faculty publishing research in journals; research impact – the number of citations those publications receive from other researchers; and research excellence.
The 2017 NTU rankings for UM follow:
- Field of agriculture, world ranking: 123, U.S. ranking: 45
- Subject of environment/ecology, world ranking: 73, U.S. ranking: 31
- Subject of geosciences, world ranking: 161, U.S. ranking: 56
- Subject of plant and animal science, world ranking: 165, U.S. ranking: 47
UM is the highest-ranked higher education institution in the state and improved in both world and national rankings in every field and subject category compared to the 2016 NTU rankings.
For more information visit.
###
| 252,607
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| 409,022
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Bob Black takes on another Leftist sacred cow: democracy. As he astutely points out, everyone interested in attaining/seizing and maintaining power is a democrat -- at least by their own reckoning. Researching the relevant literature of philosophy and political science, Black finds plenty of objections to institutionalized mob rule and the chaos which naturally follows, but he finds almost nothing written in its defense, merely its glorification. From the likes of Kim Jong Un to David Graeber, everybody claims to be in favor of democracy; since Everyone now accepts it as an inevitable feature of modern political life, there's no need to justify it. But democracy is not like gravity.
| 379,796
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FelBat still available.....
Share pet collecting news and advice.
Post Reply
10 posts •Page 1 of 1
- GrosBonda
- Posts:383
- Joined:October 1st, 2009
- Pet Score:10353
- BattleTag®:Grosbonda#1406
- Realm:Arygos-us
seems like they are still available...
- Kel
- Posts:457
- Joined:March 12th, 2014
- Pet Score:4788
- BattleTag®:TerribleTim#1747
- Realm:Ravencrest-us
Re: FelBat still available.....
Yeah, what do you mean by "still available"?
I mean, I have like 14 of them in the bank or something like that. So they will eventually go on the AH.
I mean, I have like 14 of them in the bank or something like that. So they will eventually go on the AH.
HayWire Motorsports
Tacoma, Washington
Tacoma, Washington
Re: FelBat still available.....
Just that the Nethershard vendor is still present in the capital city, so you can purchase them if you somehow still have shards.
- Thanks to Paladance for the sig!
Re: FelBat still available.....
That's fine, tried to adjust the numbers but still having 24 on one toon, can be spent on the bandages/felstones that being unique can't be purchased at once.
- Thesalesman
- Posts:52
- Joined:September 1st, 2016
- Pet Score:938
- BattleTag®:Nagex#1704
- Realm:Farstriders-us
Re: FelBat still available.....
Oh strange.... But so many people are hoarding them... Might take a while for price to go up.
Thesalesman
Re: FelBat still available.....
They are technically "still available", yes. But they aren't in that, someone can't today log on and decide they are going to farm up the shards to buy one. Though they'll be "available" via the AH for a long time to come.
Was nice of them to leave the vendors in for a bit so people can spend any of their excess shards.
Was nice of them to leave the vendors in for a bit so people can spend any of their excess shards.
Re: FelBat still available.....
This is actually good for me. I did several invasions on various characters, but only remembered to spend the shards from my main.
Re: FelBat still available.....
Same here. Wasn't interested in the invasions so never had any shards. Prices already went up from 300 to 800 so quickly bought one incase it rises even more. thus price increases) won't happen until the people who hoarded them have sold theirs off and the stockpiles are left with the few big AH players on each server, and that is likely to take a while. Hell, if anyone here missed getting one and still need it, just PM me and I'll give you one for free.
Post Reply
10 posts •Page 1 of 1
| 304,930
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\section{Introduction}
State estimation algorithms either compute a single state, a probability distribution of the state, or bound the set of possible states by sets. In stochastic approaches, measurement and process noises are modeled by provided statistical distributions (e.g., Gaussian \cite{conf:kfintro}). On the other hand, set-based approaches assume noises to be unknown but bounded by known bounds. Safety-critical applications require guarantees on the state estimation during operation -- such guarantees can be provided by the set-based approaches. Also, set-based approaches are traditionally used in fault detection by generating an adaptive threshold to check the consistency of the measurements with the estimated output set \cite{conf:intervalVSset-membership,conf:set_fault1,conf:faultcombastel1,conf:set_fault3,conf:actuator_fault,conf:fdi}. Among the family of set-based approaches, interval-based and set-membership observers have been introduced separately \cite{conf:intervalVSset-membership}. We focus in the subsequent literature survey on both approaches.
\footnotetext{\href{https://github.com/aalanwar/Distributed-Set-Based-Observers-Using-Diffusion-Strategy}{https://github.com/aalanwar/Distributed-Set-Based-Observers-Using-Diffusion-Strategy}}
\textbf{Interval-based observers:} These observers obtain possible sets of states by combining the model and the measurements through the observer gain \cite{conf:comparison} in order to bound state estimates by upper and lower bounds, which are obtained for instance from differential equations as in \cite{conf:interval1,conf:interval3}. Related work in \cite{conf:interval2asym} designs an exponentially stable interval observer for a two-dimensional time-invariant linear system. The aforementioned work is extended for arbitrary finite dimension in \cite{conf:interval4}. The previous works on linear systems have been extended to nonlinear systems in \cite{conf:interval7,conf:interval8}. Another observer was proposed based on Muller’s theorem for nonlinear uncertain systems in \cite{conf:interval6bioreactor}. Also, authors in \cite{conf:h_inf} introduces $H_\infty$ design into interval estimation. Interval observers for uncertain biological systems are proposed in \cite{conf:interval5bio}. By merging optimal and robust observer gain designs, a zonotopic Kalman Filter is proposed in \cite{conf:combastelzonotopes2}.
\textbf{Set-membership-based observers:} Unlike interval observers, which are based on observer gain derivation, set-membership-based observers intersect the set of states consistent with the model and the set consistent with the measurements to obtain the corrected state set \cite{conf:intervalVSset-membership}.
One early example of set-membership-based observers is a recursive algorithm bounding the state by ellipsoids \cite{conf:1968}. Another early example based on normalized least-mean-squares (NLMS) is presented in \cite{conf:setmem4LMS}. A set-membership state estimation algorithm based on
DC-programming is proposed in \cite{conf:setmem1dcprogramming}. Authors in \cite{conf:set_mem_discriptor} considers linear time-varying descriptor systems for set-membership estimation. Set-memberships observers for nonlinear models are investigated in \cite{conf:setmem5,conf:setmem6,conf:setmem7,conf:setmem8,conf:set-mem-event}. They are also used in applications such as underwater robotics \cite{conf:setmem2water}, a leader following consensus problem in networked multi-agent systems \cite{conf:setmem3eventleader} and localization \cite{conf:setloc}. Authors in \cite{conf:disevent} consider a class of discrete time-varying system with an event-based communication mechanism over sensor networks. Interconnected multi-rate system is considered in \cite{conf:intermulti}. Set-membership with affine-projection is considered in \cite{conf:affineset}. Finally, nonlinear kernel adaptive filtering is proposed in \cite{conf:nonlinearset}.
Different set representations have been used in set-based estimation, e.g, ellipsoids \cite{conf:ellipsoide,conf:set-ellipsoida,conf:dis_ellip_multirate},
orthotopes, and polytopes \cite{conf:orthotope,conf:polytope}. Zonotopes \cite{conf:zonotope_rep} are a special class of polytopes for which one can efficiently compute linear maps and Minkowski sums -- both are important operations for set-membership-based observers. A state bounding observer based on zonotopes is introduced in \cite{conf:zono1_Combastel}. Set-membership for discrete-time piecewise affine systems using zonotopes is studied in \cite{conf:zono2piecewise}. Another work considers discrete-time descriptor systems in \cite{conf:zono_18pages}. Set-based estimation of uncertain discrete-time systems using zonotopes is proposed also in \cite{conf:zono4}.
\textbf{Contributions:}
We propose distributed set-based estimators, where a set of nodes is required to collectively estimate the set of possible states of a linear dynamical system in a distributed fashion. In traditional distributed set-based estimation, every node in a sensor-network receives the estimates based on its measurements only; then, the node intersects its set with the estimated sets of its neighbors \cite{conf:dis-interconnected,conf:dis-kalmaninspired,conf:dis-partial}. However, we propose to let every node shares its measurements with its neighbor for faster convergence. We also supplement our newly proposed observers with a set-based diffusion step, which intersects the shared state sets. Unlike prior efforts, we propose a new zonotopes intersection technique in the diffusion step, which reduces the over-approximation of the intersection results. We use the term \textit{diffusion} since our intersection formula resembles the traditional diffusion step in stochastic Kalman filter. In set-based estimation, the center of the set is considered as a single point estimate. We show that our diffusion step enhances the single point estimate and decreases the volume of the estimated sets.
One main problem in distributed set-based estimation is the misalignment between the estimated sets by the distributed nodes, which would result in disagreements on fault detection results between nodes. This problem is usually solved by consensus methods \cite{conf:dis-consRR,conf:dis-cons2008,conf:dis-cons2013}. However, traditional consensus methods require the sensor network to perform several iterations before arriving at a consensus, which causes great overhead in set-based estimation. Our set-based diffusion step can be seen as lightweight approach to achieve partial consensus. One only obtains a partial consensus using our algorithms because every node has different neighbors with different measurements; thus, the resulting sets do not fully agree.
More specifically, we make the following contributions:
\begin{itemize}
\item We propose two distributed set-membership and interval-based algorithms combined with a new technique for intersection of zonotopes which is exploited in the proposed set-based diffusion step.
\item We provide closed-forms for our parameter-finding optimization problems to realize faster execution times.
\end{itemize}
The rest of the paper is organized as follows: System model and preliminaries are in Section \ref{sec:sysmodel}. In Section \ref{sec:setmem}, we present the distributed set-membership diffusion observer as our first algorithm. Our second solution is the distributed interval-based diffusion observer which is introduced in Section \ref{sec:berger}. Both algorithms are evaluated in Section \ref{sec:eval}. Finally, we conclude the paper in Section \ref{sec:conc}.
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\begin{document}
\title[Nonexistence of a crepant resolution of moduli spaces]
{Nonexistence of a crepant resolution of some moduli spaces of
sheaves on a K3 surface}
\date{}
\author{Jaeyoo Choy and Young-Hoon Kiem}
\address{Dept of Mathematics, Seoul National University,
Seoul 151-747, Korea} \email{donvosco@math.snu.ac.kr}
\email{kiem@math.snu.ac.kr}
\thanks{Young-Hoon Kiem was partially supported by KOSEF
R01-2003-000-11634-0; Jaeyoo Choy was partially supported by KRF
2003-070-C00001}
\keywords{Crepant resolution, irreducible symplectic variety,
moduli space, sheaf, K3 surface, desingularization, Hodge-Deligne
polynomial, Poincar\'{e} polynomial, stringy E-function}
\begin{abstract}
Let $M_c=M(2,0,c)$ be the moduli space of $\cO(1)$-semistable rank
2 torsion-free sheaves with Chern classes $c_1=0$ and $c_2=c$ on
a K3 surface $X$ where $\cO(1)$ is a generic ample line bundle on
$X$. When $c=2n\geq4$ is even, $M_c$ is a singular projective
variety equipped with a holomorphic symplectic structure on the
smooth locus. In particular, $M_c$ has trivial canonical divisor.
In \cite{ogrady}, O'Grady asks if
there is any symplectic desingularization of $M_{2n}$ for $n\ge
3$. In this paper, we show that there is no crepant resolution of
$M_{2n}$ for $n\geq 3$. This obviously implies that there is no
symplectic desingularization.
\end{abstract}
\maketitle
\section{Introduction}
Let $X$ be a complex projective K3 surface with polarization
$H=\cO_X(1)$ generic in the sense of \cite{ogrady} \S0. Let
$M(r,c_1,c_2)$ be the moduli space of rank $r$ $H$-semistable
torsion-free sheaves on $X$ with Chern classes $(c_1,c_2)$ in
$H^*(X,\zz)$. Let $M^s(r,c_1,c_2)$ be the open subscheme of
$H$-stable sheaves in $M(r,c_1,c_2)$. In \cite{Muk84}, Mukai shows
that $M^s(r,c_1,c_2)$ is smooth and has a holomorphic symplectic
structure. By \cite{Gi77}, if either $(c_1.H)$ or $c_2$ is an odd
number, then $M(2,c_1,c_2)$ is equal to $M^s(2,c_1,c_2) $ and thus
$M(2,c_1,c_2)$ is a smooth projective irreducible symplectic
variety. However if both $(c_1.H)$ and $c_2$ are even numbers then
generally $M(2,c_1,c_2)$ admits singularities. We restrict our
interest to the trivial determinant case i.e. $c_1=0$ and let
$M_c=M(2,0,c)$ where $c=2n$ ($n\geq2$). It is well-known that
$M_{2n}$ is an irreducible, normal (\cite{Yo01} Theorem 3.18) and
projective variety (\cite{HL97} Theorem 4.3.4) of dimension $8n-6$
(\cite{Muk84} Theorem 0.1) with only Gorenstein singularities
(\cite{HL97} Theorem 4.5.8, \cite{Ei95} Corollary 21.19). Since
$M_{2n}$ contains the smooth open subset $M^s_{2n}$, there arises
a natural question: does there exist a resolution of $M_{2n}$ such
that the Mukai form on $M^s_{2n}$ extends to the resolution
without degeneration? When $c=4$, O'Grady successfully extends the
Mukai form on $M^s_{2n}$ to some resolution without degeneration
(\cite{og97, ogrady}). At the same time, he conjectures
nonexistence of a symplectic desingularization of $M_{2n}$ for
$n\ge 3$ (\cite{ogrady}, (0.1)). Our main result in this paper is
the following.
\begin{theorem} \label{thm:main
result} If $n\geq3$, there is no crepant resolution of $M_{2n}$.
\end{theorem}
The highest exterior power of a symplectic form gives a
non-vanishing section of the canonical sheaf on $M_{2n}$. Likewise
any symplectic desingularization of $M_{2n}$ has trivial canonical
divisor and hence it must be a crepant resolution. Therefore,
O'Grady's conjecture is a consequence of Theorem \ref{thm:main
result}.
\begin{corollary} \label{cor:O'Grady's conjecture} If $n\geq3$,
there is no symplectic desingularization of $M_{2n}$.
\end{corollary}
The idea of the proof of Theorem \ref{thm:main result} is to use a
new invariant called the stringy E-function \cite{Bat98, DL99}.
Since $M_{2n}$ is normal irreducible variety with log terminal
singularities (\cite{ogrady}, 6.1), the stringy E-function of
$M_{2n}$ is a well-defined rational function. If there is a
crepant resolution $\tM_{2n}$ of $M_{2n}$, then the stringy
E-function of $M_{2n}$ is equal to the Hodge-Deligne polynomial
(E-polynomial) of $\tM_{2n}$ (Theorem \ref{thm:Batyrev's result}).
In particular, we deduce that the stringy E-function
$E_{st}(M_{2n};u,v)$ must be a polynomial. Therefore, Theorem
\ref{thm:main result} is a consequence of the following.
\begin{proposition}\label{prop:stringy E-function test} The stringy
E-function $E_{st}(M_{2n};u,v)$ is not a polynomial for $n\geq3$.
\end{proposition} To prove that
$E_{st}(M_{2n};u,v)$ is not a polynomial for $n\geq 3$, we show
that $E_{st}(M_{2n};z,z)$ is not a polynomial in $z$. Thanks to
the detailed analysis of Kirwan's desingularization in \cite{og97}
and \cite{ogrady} which is reviewed in section 4, we can find an
expression for $E_{st}(M_{2n};z,z)$ and then with some efforts on
the combinatorics of rational functions we show that
$E_{st}(M_{2n};z,z)$ is not a polynomial in section 3. In section
2, we recall basic facts on stringy E-function and in section 5 we
prove a lemma which computes the E-polynomial of a divisor.
In \cite{ogrady}, O'Grady gets a symplectic desingularization
$\tM_{2n}$ of $M_{2n}$ in the case when $n=2$. This turns out to
be a new irreducible symplectic variety, which means that it does
not come from a generalized Kummer variety nor from a Hilbert
scheme parameterizing 0-dimensional subschemes on a K3 surface
\cite{og98, Bea83}. Corollary \ref{cor:O'Grady's conjecture} shows
that unfortunately we cannot find any more irreducible symplectic
variety in this way.
After we finished the first draft of this paper, we learned that
Kaledin and Lehn \cite{KL04} proved Corollary \ref{cor:O'Grady's
conjecture} in a completely different way. We are grateful to D.
Kaledin for informing us of their approach. The second named
author thanks Professor Jun Li for useful discussions concerning
the article \cite{VW94}. Finally we would like to express our
gratitude to the referee for careful reading and challenging us
for many details which led us to improve the manuscript and
correct an error in Proposition 3.2.
\section{Preliminaries}
In this section we collect some facts that we shall use later.
For a topological space $V$, the Poincar\'e polynomial of $V$ is
defined as
\begin{equation} \label{eqn:Poincare polynomial}
P(V;z)=\sum_{i}(-1)^ib_i(V)z^i
\end{equation} where $b_i(V)$ is the $i$-th Betti number of $V$.
It is well-known from \cite{Go90} that the Betti numbers of the
Hilbert scheme of points $X^{[n]}$ in $X$ are given by the
following:
\begin{equation}\label{eqn:Betti for X[n]} \sum_{n\geq
0}P(X^{[n]};z)t^n=\prod_{k\geq1}
\prod_{i=0}^{4}(1-z^{2k-2+i}t^k)^{(-1)^{i+1}b_i(X)}.
\end{equation}
Next we recall the definition and basic facts about stringy
E-functions from \cite{Bat98,DL99}. Let $W$ be a normal
irreducible variety with at worst log-terminal singularities, i.e.
\begin{enumerate} \item W is $\qq$-Gorenstein;
\item for a resolution of singularities $\rho: V\to W$ such that
the exceptional locus of $\rho$ is a divisor $D$ whose irreducible
components $D_1,\cdots,D_r$ are smooth divisors with only normal
crossings, we have \[K_V=\rho^*K_W+\sum^r_{i=1} a_iD_i \] with
$a_i>-1$ for all $i$, where $D_i$ runs over all irreducible
components of $D$. The divisor $\sum^r_{i=1}a_iD_i$ is called the
\textit{discrepancy divisor}. \end{enumerate}
For each subset $J\subset I=\{1,2,\cdots,r\}$, define
$D_J=\cap_{j\in J}D_j$, $D_\emptyset=V$ and $D^0_J=D_J-\cup_{i\in
I-J}D_i$. Then the stringy E-function of $W$ is defined by
\begin{equation} \label{eqn:stringy E-function}
E_{st}(W;u,v)=\sum_{J\subset I}E(D^0_J;u,v)\prod_{j\in
J}\frac{uv-1}{(uv)^{a_j+1}-1} \end{equation} where \[ E(Z;u,v) =
\sum_{p,q}\sum_{k\geq 0} (-1)^kh^{p,q}(H^k_c(Z;\cc))u^pv^q \] is
the Hodge-Deligne polynomial for a variety $Z$. Note that the
Hodge-Deligne polynomials have \begin{enumerate} \item the
additive property: $E(Z;u,v)=E(U;u,v)+E(Z-U;u,v)$ if $U$ is a
smooth open subvariety of $Z$; \item the multiplicative property:
$E(Z;u,v)=E(B;u,v)E(F;u,v)$ if $Z$ is a Zariski locally trivial
$F$-bundle over $B$. \end{enumerate}
By \cite{Bat98} Theorem 6.27, the function $E_{st}$ is independent
of the choice of a resolution (Theorem 3.4 in \cite{Bat98}) and
the following holds.
\begin{theorem} \label{thm:Batyrev's result} (\cite{Bat98} Theorem
3.12) Suppose $W$ is a $\qq$-Gorenstein algebraic variety with at
worst log-terminal singularities. If $\rho:V\to W$ is a crepant
desingularization (i.e. $\rho^*K_W=K_V$) then
$E_{st}(W;u,v)=E(V;u,v)$. In particular, $E_{st}(W;u,v)$ is a
polynomial.
\end{theorem}
\section{Proof of Proposition \ref{prop:stringy E-function test}}
In this section we first find an expression for the stringy
E-function of the moduli space $M_{2n}$ for $n\geq 3$ by using the
detailed analysis of Kirwan's desingularization in \cite{og97,
ogrady}. Then we show that it cannot be a polynomial, which proves
Proposition \ref{prop:stringy E-function test}.
We fix a generic polarization of $X$ as in \cite{ogrady}. The
moduli space $M_{2n}$ has a stratification \[M_{2n}=M^s_{2n}\sqcup
(\Sigma-\Omega)\sqcup \Omega\] where $M^s_{2n}$ is the locus of
stable sheaves and $\Sigma\simeq(X^{[n]}\times X^{[n]})/{\rm
involution}$ is the locus of sheaves of the form $I_Z\oplus
I_{Z'}$ ($[Z],[Z']\in X^{[n]}$) while $\Omega\simeq X^{[n]}$ is
the locus of sheaves $I_Z\oplus I_{Z}$. For $n\geq 3$, Kirwan's
desingularization $\rho:\hM_{2n}\to M_{2n}$ is obtained by blowing
up $M_{2n}$ first along $\Omega$, next along the proper transform
of $\Sigma$ and finally along the proper transform of a subvariety
$\Delta$ in the exceptional divisor of the first blow-up. This is
indeed a desingularization by \cite{ogrady} Proposition 1.8.3.
Let $D_1=\hat{\Omega}$, $D_2=\hat{\Sigma}$ and $D_3=\hat{\Delta}$
be the (proper transforms of the) exceptional divisors of the
three blow-ups. Then they are smooth divisors with only normal
crossings as we will see in Proposition \ref{prop:analysis on exc}
and the discrepancy divisor of $\rho:\hM_{2n}\to M_{2n}$ is
(\cite{ogrady}, 6.1)
\[(6n-7)D_1+(2n-4)D_2+(4n-6)D_3.
\] Therefore the singularities are log-terminal for $n\geq 2$, and
from (\ref{eqn:stringy E-function}) the stringy E-function of
$M_{2n}$ is given by \begin{eqnarray}\label{eqn:stringy E-function
of M_c}
E(M^s_{2n};u,v)+E(D^0_1;u,v){\textstyle\frac{1-uv}{1-(uv)^{6n-6}}}
+E(D^0_2;u,v){\textstyle\frac{1-uv}{1-(uv)^{2n-3}}}\nonumber \\
+E(D^0_3;u,v) {\textstyle\frac{1-uv}{1-(uv)^{4n-5}}}
+E(D^0_{12};u,v){\textstyle\frac{1-uv}{1-(uv)^{6n-6}}\frac{1-uv}{1-(uv)^{2n-3}}} \\
+E(D^0_{23};u,v){\textstyle\frac{1-uv}{1-(uv)^{2n-3}}\frac{1-uv}{1-(uv)^{4n-5}}}
+E(D^0_{13};u,v){\textstyle\frac{1-uv}{1-(uv)^{4n-5}}\frac{1-uv}{1-(uv)^{6n-6}}}
\nonumber \\
+E(D^0_{123};u,v){\textstyle\frac{1-uv}{1-(uv)^{6n-6}}
\frac{1-uv}{1-(uv)^{2n-3}}\frac{1-uv}{1-(uv)^{4n-5}}} .\nonumber
\end{eqnarray}
We need to compute the Hodge-Deligne polynomials of $D^0_J$ for
$J\subset \{1,2,3\}$. Let $(\cc^{2n},\omega)$ be a symplectic
vector space. Let $\Gr^{\omega}(k,2n)$ be the Grassmannian of
$k$-dimensional subspaces of $\cc^{2n}$, isotropic with respect to
the symplectic form $\omega$ (i.e. the restriction of $\omega$ to
the subspace is zero).
\begin{lemma}\label{lem:Hodge poly of Gr} For $k\leq n$, the Hodge-Deligne polynomial of
$\Gr^\omega(k,2n)$ is
\[\prod_{1\leq i\leq k} \frac{1-(uv)^{2n-2k+2i}}{1-(uv)^i}. \]
\end{lemma}
\proof Consider the incidence variety \[ Z= \{(a,b)\in
\Gr^\omega(k-1,2n)\times \Gr^\omega(k,2n)|a\subset b\}. \] This is
a $\pp^{2n-2k+1}$-bundle over $\Gr^\omega(k-1,2n)$ and a
$\pp^{k-1}$-bundle over $\Gr^\omega(k,2n)$. We have the following
equalities between Hodge-Deligne polynomials: \begin{eqnarray*}
E(Z;u,v)&=&\frac{1-(uv)^{2n-2k+2}}{1-uv}
E(\Gr^\omega(k-1,2n);u,v)\\ &=& \frac{1-(uv)^{k}}{1-uv}
E(\Gr^\omega(k,2n);u,v). \end{eqnarray*} The desired formula
follows recursively from $\Gr^\omega(1,2n)=\pp^{2n-1}$. \qed\\
Let $\hat{\pp}^5$ be the blow-up of $\pp^5$ (projectivization of
the space of $3\times 3$ symmetric matrices) along $\pp^2$ (the
locus of rank 1 matrices). We have the following from \cite{og97}
and \cite{ogrady}. The proof will be presented in \S\ref{sec:
Proof of Lemma}.
\begin{proposition}\label{prop:analysis on exc}
Let $n\geq 3$.
(1) $D_1$ is a $\hat{\pp}^5$-bundle over a
$\Gr^\omega(3,2n)$-bundle over $X^{[n]}$.
(2) $D_2^0$ is a free $\zz_2$-quotient of a Zariski locally
trivial $I_{2n-3}$-bundle over $ X^{[n]}\times
X^{[n]}-\mathbf{\Delta} $ where $\mathbf{\Delta}$ is the diagonal
in $ X^{[n]}\times X^{[n]}$ and $I_{2n-3}$ is the incidence
variety given by
\[ I_{2n-3}=\{(p,H)\in \pp^{2n-3}\times \breve{\pp}^{2n-3}| p\in
H\}.
\]
(3) $D_3$ is a $\pp^{2n-4}$-bundle over a Zariski locally trivial
$ \pp^2$-bundle over a Zariski locally trivial
$\Gr^\omega(2,2n)$-bundle over $X^{[n]}$.
(4) $D_{12}$ is a $\pp^2$-bundle over a $\pp^2$-bundle over a
$\Gr^\omega(3,2n)$-bundle over $X^{[n]}$.
(5) $D_{23}$ is a $\pp^{2n-4}$-bundle over a $ \pp^1$-bundle over
a $\Gr^\omega(2,2n)$-bundle over $X^{[n]}$.
(6) $D_{13}$ is a $ \pp^2$-bundle over a $\pp^2$-bundle over a
$\Gr^\omega(3,2n)$-bundle over $X^{[n]}$.
(7) $D_{123}$ is a $\pp^1$-bundle over a $\pp^2$-bundle over
a $\Gr^\omega(3,2n)$-bundle over $X^{[n]}$. \\
All the above bundles except in (2) and (3) are Zariski locally
trivial. Moreover, $D_i$ ($i=1,2,3$) are smooth divisors such that
$D_1\cup D_2\cup D_3$ is normal crossing.
\end{proposition}
From Lemma \ref{lem:Hodge poly of Gr} and Proposition
\ref{prop:analysis on exc}, we have the following corollary by the
additive and multiplicative properties of the Hodge-Deligne
polynomial.
\begin{corollary}\label{eqn:computation of stringy E-function}
$$ E(D_1;u,v) = \Bigl({\textstyle
\frac{1-(uv)^6}{1-uv}-\!\frac{1-(uv)^3}{1-uv}+\!\bigl(\frac{1-(uv)^3}{1-uv}\bigr)^2}\Bigr)
\! \times\!\!\! \prod_{1\leq i\leq 3}\! \Bigl({\textstyle
\frac{1-(uv)^{2n-6+2i}}{1-(uv)^i}}\Bigr)\!\! \times\!
E(X^{[n]};u,v),$$
$$E(D_3;u,v) = {\textstyle
\frac{1-(uv)^{2n-3}}{1-uv}\cdot\frac{1-(uv)^3}{1-uv}} \times
\prod_{1\leq i\leq 2}\Bigl({\textstyle \frac{1-(uv)^{2n-4+2i}}
{1-(uv)^i}}\Bigr)\times E(X^{[n]};u,v), $$
$$ E(D_{12};u,v) = \Bigl({\textstyle
\frac{1-(uv)^3}{1-uv}}\Bigr)^2\times \prod_{1\leq i\leq
3}\Bigl({\textstyle \frac{1-(uv)^{2n-6+2i}}{1-(uv)^i}}\Bigr)
\times E(X^{[n]};u,v), $$
$$ E(D_{23};u,v) = {\textstyle
\frac{1-(uv)^{2n-3}}{1-uv}\cdot\frac{1-(uv)^2}{1-uv}}
\times\prod_{1\leq i\leq 2}\Bigl({\textstyle
\frac{1-(uv)^{2n-4+2i}}{1-(uv)^i}}\Bigr)\times E(X^{[n]};u,v), $$
$$E(D_{13};u,v) = {\textstyle \frac{
1-(uv)^3}{1-uv}\cdot\frac{1-(uv)^{2n-4}}{1-uv}} \times
\prod_{1\leq i\leq 2}\Bigl({\textstyle
\frac{1-(uv)^{2n-4+2i}}{1-(uv)^i}}\Bigr)\times E(X^{[n]};u,v), $$
$$ E(D^0_{123};u,v) ={\textstyle
\frac{1-(uv)^2}{1-uv}\cdot\frac{1-(uv)^{2n-4}}{1-uv}}\times
\prod_{1\leq i\leq 2}\Bigl({\textstyle
\frac{1-(uv)^{2n-4+2i}}{1-(uv)^i}}\Bigr)\times E(X^{[n]};u,v).$$
\end{corollary}
\proof Perhaps the only part that requires proof is the equation
for $E(D_3;u,v)$. From Proposition \ref{prop:analysis on exc} (3),
$D_3$ is a projective variety which is a $\pp^{2n-4}$-bundle over
a smooth projective variety, say $Y$, whose E-polynomial is
$$E(\pp^2;u,v)\times E(\mathrm{Gr}^\omega (2,2n);u,v)\times
E(X^{[n]};u,v).$$ By the Leray-Hirsch theorem (\cite{V02I} p.182),
we have
\begin{eqnarray*}H^*(D_3;\cc)\cong H^*(Y;\cc)\otimes
H^*(\pp^{2n-4};\cc)\cong H^*(Y;\cc)\otimes \cc
[\lambda]/(\lambda^{2n-3})\\ \cong H^*(Y;\cc)\oplus
H^*(Y;\cc)\lambda\oplus \cdots \oplus
H^*(Y;\cc)\lambda^{2n-4}\end{eqnarray*} where $\lambda$ is a class
of type $(1,1)$ which comes from the K\"ahler class. The above
determines the Hodge structure of $D_3$ because the Hodge
structure is compatible with the cup product. Therefore we deduce
that $$ E(D_3;u,v) = {\textstyle
\frac{1-(uv)^{2n-3}}{1-uv}\times E(Y;u,v)}. $$ \qed
For the E-polynomial of $D_2^0$ we have the following lemma whose
proof is presented in section \ref{sec: Computation of E-poly of
D_0^2}. Recall that $$I_{2n-3}=\{((x_i),(y_j))\in \pp^{2n-3}\times
\pp^{2n-3}\,|\, \sum_{i=0}^{2n-3} x_iy_i=0\}$$ and there is an
action of $\zz_2$ which interchanges $(x_i)$ and $(y_j)$. Let
$H^r(I_{2n-3})^+$ denote the $\zz_2$-invariant subspace of
$H^r(I_{2n-3})$ .
\begin{lemma}\label{lem: Hodge Deligne poly of D02}
\begin{eqnarray}\label{eqn: compute D02} \lefteqn{
E(D^0_2;z,z)=P(I_{2n-3};z) \Bigl(
\frac{P(X^{[n]};z)^2-P(X^{[n]};z^2)}2 \Bigr)} && \\ && +
P^+(I_{2n-3};z)\bigl(P(X^{[n]};z^2)-P(X^{[n]};z) \bigr)\nonumber
\end{eqnarray}
where $P^+(I_{2n-3};z)=\displaystyle \sum_{r\geq0}(-1)^rz^r\dim
H^r(I_{2n-3})^+$. Moreover
\begin{eqnarray}\label{eqn: E D02 is divisible by some Q}
E(D^0_2;z,z)=\frac{1-(z^2)^{2n-3}}{1-z^2} Q(z^2)\end{eqnarray} for some
polynomial $Q$.
\end{lemma}
\textit{Proof of Proposition \ref{prop:stringy E-function test}.}
Let us prove that \eqref{eqn:stringy E-function of M_c} cannot be
a polynomial. Let
$$S(z)=E_{st}(M_{2n};z,z)-E(M^s_{2n};z,z).$$ It suffices to show
that $S(z)$ is not a polynomial for all $n\geq3$ because
$E(M^s_{2n};z,z)$ is a polynomial.
Note that given any $n\geq 3$, we can explicitly compute
$E(X^{[n]};z,z)$ and $E(D^0_2;z,z)$ by (\ref{eqn:Betti for X[n]})
and Lemma \ref{lem: Hodge Deligne poly of D02}. If $n=3$, direct
calculation shows that $S(z)$ is as follows:
\begin{eqnarray*} S(z)& =&
1+46z^2+852z^4+12308z^6+111641z^8+886629z^{10}+4233151z^{12}\\
& & +4990239z^{14}+4999261z^{16}+4230852z^{18}+884441z^{20}+113877z^{22}\\
& & +12928z^{24}+3749z^{26}+3200z^{28}+2877z^{30}+299z^{32}+\cdots.
\end{eqnarray*} It is easy to see from (\ref{eqn:stringy E-function of M_c})
and Corollary \ref{eqn:computation of stringy E-function} that if
$S(z)$ were a polynomial, it should be of degree $\le 30$. Since
the series $S(z)$ has a nonzero coefficient of $z^{32}$, $S(z)$
cannot be a polynomial. So we assume from now on that $n\ge 4$.
Express the rational function $S(z)$ as
$$\frac{N(z)}{(1-(z^2)^{2n-3})(1-(z^2)^{4n-5})(1-(z^2)^{6n-6})}.$$
All we need to show is that the numerator $N(z)$ is not divisible
by the denominator
$(1-(z^2)^{2n-3})(1-(z^2)^{4n-5})(1-(z^2)^{6n-6})$.
As $E(X^{[n]};z,z)$ and $E(D^0_2;z,z)$ do not have nonzero terms
of odd degree by (\ref{eqn:Betti for X[n]}) and Lemma \ref{lem:
Hodge Deligne poly of D02}, all the nonzero terms in $S(z)$ are of
even degree by (\ref{eqn:stringy E-function of M_c}) and Corollary
\ref{eqn:computation of stringy E-function}. Hence, we can write
$S(z)=s(z^2)=s(t)$ for some rational function $s(t)$ in $t=z^2$.
The numerator $N(z)=n(z^2)=n(t)$ is not divisible by
$1-(z^2)^{2n-3}$ if and only if $n(t)$ is not divisible by
$1-t^{2n-3}$. By direct computation using (\ref{eqn:stringy
E-function of M_c}), Corollary \ref{eqn:computation of stringy
E-function} and Lemma \ref{lem: Hodge Deligne poly of D02}, $n(t)$
modulo $1-t^{2n-3}$ is congruent to
\begin{eqnarray}\label{eqn:denumerator modulo}
\shoveleft(1-t)^2(1-t^{4n-5})\times\Bigl({\textstyle
\frac{1-t^3}{1-t}}\Bigr)^2\times \prod_{1\leq i\leq
3}\Bigl({\textstyle \frac{1-t^{2n-6+2i}}{1-t^i}}\Bigr) \times
p(X^{[n]};t) \\ -(1-t)^2(1-t^{4n-5})\times {\textstyle
\frac{1-t^2}{1-t}\cdot\frac{1-t^{2n-4}}{1-t}}\times \prod_{1\leq
i\leq 2}\Bigl({\textstyle \frac{1-t^{2n-4+2i}}{1-t^i}}\Bigr)
\times p(X^{[n]};t) \nonumber \\ -(1-t)^2(1-t^{6n-6})\times
{\textstyle \frac{1-t^2}{1-t}\cdot\frac{1-t^{2n-4}}{1-t}}\times
\prod_{1\leq i\leq 2}\Bigl({\textstyle
\frac{1-t^{2n-4+2i}}{1-t^i}}\Bigr) \times p(X^{[n]};t) \nonumber
\\ + (1-t)^3\times{\textstyle \frac{1-t^2}{1-t}
\cdot\frac{1-t^{2n-4}}{1-t}}\times \prod_{1\leq i\leq
2}\Bigl({\textstyle \frac{1-t^{2n-4+2i}}{1-t^i}}\Bigr)\times
p(X^{[n]};t) \nonumber \end{eqnarray} where
$p(X^{[n]};t)=P(X^{[n]};z)$ with $t=z^2$. We write
(\ref{eqn:denumerator modulo}) as a product $\bar s(t)\cdot
p(X^{[n]};t)$ for some polynomial $\bar s(t)$. For the proof of
our claim for $n\geq 4$, it suffices to prove the following:
\begin{enumerate} \item if $n$ is not divisible by 3, then $1-t$
is the GCD of $1-t^{2n-3}$ and $\bar s(t)$, and
$\frac{1-t^{2n-3}}{1-t}$ does not divide $p(X^{[n]};t)$; \item if
$n$ is divisible by 3, then $1-t^3$ is the GCD of $1-t^{2n-3}$ and
$\bar s(t)$, and $\frac{1-t^{2n-3}}{1-t^3}$ does not divide
$p(X^{[n]};t)$.
\end{enumerate}
For (1), suppose $n$ is not divisible by 3. From
(\ref{eqn:denumerator modulo}), $\bar s(t)$ is divisible by $1-t$.
We claim that $\bar s(t)$ is not divisible by any irreducible
factor of $\frac{1-t^{2n-3}}{1-t}$, i.e. for any root $\alpha$ of
$1-t^{2n-3}$ which is not 1, $\bar s(\alpha)\neq 0$. Using the
relation $\alpha^{2n-3}=1$, we compute directly that
\begin{equation}\label{eqn:bar s} \bar s(\alpha)={\textstyle
-\frac{\alpha(1-\alpha^{-1}){(1-\alpha^3)}^2}{1+\alpha}},
\end{equation} which is not 0 because 3 does not divide $2n-3$.
Next we check that $\frac{1-t^{2n-3}}{1-t}$ does not divide
$p(X^{[n]};t)$.
We put
$${\displaystyle p(X^{[n]};t)=\sum_{0\leq i\leq 2n} c_it^i}$$ and
write $p(X^{[n]};t)$ as follows:
\begin{eqnarray} \label{eqn:p(t) when 3 NOT divides n} \lefteqn{
\sum_{0\leq i\leq 2n} c_it^i =
(c_0+c_{2n-3})+(c_1+c_{2n-2})t+(c_2+c_{2n-1})t^2 +(c_3+c_{2n})t^3
} \\ & & + \sum_{4\leq i\leq 2n-4} c_it^i + c_{2n-3}(t^{2n-3}-1)
+ c_{2n-2}t(t^{2n-3}-1)\nonumber \\ & & +
c_{2n-1}t^2(t^{2n-3}-1)+ c_{2n}t^3(t^{2n-3}-1).\nonumber
\end{eqnarray} Therefore, the divisibility
of $p(X^{[n]};t)$ by $\frac{1-t^{2n-3}}{1-t}$ is that of
$(c_0+c_{2n-3})+ (c_1+c_{2n-2})t+ (c_2+c_{2n-1})t^2
+(c_3+c_{2n})t^3 + {\displaystyle \sum_{4\leq i\leq 2n-4} c_it^i}$
by $\frac{1-t^{2n-3}}{1-t}$. Since
$\frac{1-t^{2n-3}}{1-t}={\displaystyle \sum_{0\leq i\leq 2n-4}
t^i}$, the polynomial $(c_0+c_{2n-3})+ (c_1+c_{2n-2})t+
(c_2+c_{2n-1})t^2 +(c_3+c_{2n})t^3 + {\displaystyle \sum_{4\leq
i\leq 2n-4} c_it^i}$ is divisible by $\frac{1-t^{2n-3}}{1-t}$ if
and only if it is a scalar multiple of ${\displaystyle \sum_{0\leq
i\leq 2n-4} t^i}$, i.e. $c_0+c_{2n-3}=c_1+c_{2n-2}=
c_2+c_{2n-1}=c_3+c_{2n}= c_4=\cdots =c_{2n-4}$ ($n\geq 4$).
Table \ref{table:list of ci} is the list of $c_i$ ($1\leq i\leq
4$) for $n\geq 3$, which comes from direct computation using the
generating functions (\ref{eqn:Betti for X[n]}) for the Betti
numbers of $X^{[n]}$. By Table \ref{table:list of ci}, we can
check that this is impossible. Indeed, for $n\geq 6$, $c_0=1$,
$c_1=23$, $c_2=300$ and $c_3=2876$, which implies $c_{2n-3}=2876$,
$c_{2n-2}=300$, $c_{2n-1}=23$ and $c_{2n-2}=1$ by Poincar\'{e}
duality. Thus $c_0+c_{2n-3}=2877$ while $c_1+c_{2n-2}=323$. For
$4\leq n\leq 5$, the proof is also direct computation using Table
\ref{table:list of ci}.
\begin{table}[t]
\begin{tabular}{c|cccccc}
&$n=3$& $n=4$ & $n=5$& $n=6$& $n=7$&$n\geq 8$ \\ \hline
$c_1$ & 23 & 23 & 23 & 23 & 23 & 23 \\
$c_2$ & 299 & 300 & 300 &300 &300 &300 \\
$c_3$ & 2554& 2852 & 2875 & 2876 &2876 &2876 \\
$c_4$ & 299 & 19298 &22127 &22426 &22449 &22450
\end{tabular}\caption{list of $c_i$ \label{table:list of ci}}
\end{table}
For (2), suppose 3 divides $n$ and $n\neq 3$. Then from
(\ref{eqn:bar s}), $(1-t^3)$ divides $\bar s(t)$. More precisely,
for a third root of unity $\alpha$, $\bar s(\alpha)=0$. On the
other hand, if $\alpha$ is a root of $1-t^{2n-3}$ but not a third
root of unity then we can observe that $\bar s(\alpha)\neq 0$ by
(\ref{eqn:bar s}). Therefore, since every root of $1-t^{2n-3}$ is
a simple root, any irreducible factor of
$\frac{1-t^{2n-3}}{1-t^3}$ does not divide $\bar s(t)$.
We next check that the polynomial $\frac{1-t^{2n-3}}{1-t^3}$ does
not divide $p(X^{[n]};t)$. Write $p(X^{[n]};t)=\displaystyle
\sum_{0\leq i\leq 2n} c_it^i$ as follows:
\begin{eqnarray}\label{eqn:p(t) when 3 divides n} \lefteqn{\sum_{0\leq i\leq 2n} c_it^i =
(c_0+c_{2n-3})+(c_1+c_{2n-2})t+(c_2+c_{2n-1})t^2 +(c_3+c_{2n})t^3}
\\ & & + \sum_{4\leq i\leq 2n-6} c_it^i - c_{2n-5}
\Bigl(\sum_{i=0}^{\frac{2n-9}3} t^{3i+1} \Bigr)-
c_{2n-4} \Bigl( \sum_{i=0}^{\frac{2n-9}3} t^{3i+2}\Bigr) \nonumber \\
& & + c_{2n-5}t\cdot{\textstyle \frac{1-t^{2n-3}}{1-t^3}} +
c_{2n-4}t^2\cdot{\textstyle \frac{1-t^{2n-3}}{1-t^3}} +
c_{2n-3}(t^{2n-3}-1) \nonumber \\ & & + c_{2n-2}t(t^{2n-3}-1) +
c_{2n-1}t^2(t^{2n-3}-1)+ c_{2n}t^3(t^{2n-3}-1) \nonumber
\end{eqnarray} where the equality comes from
\begin{eqnarray*} t^{2n-5} = -\sum_{i=0}^{\frac{2n-9}3} t^{3i+1}
+t\cdot{\textstyle \frac{1-t^{2n-3}}{1-t^3}}\ \ {\rm and}\ \
t^{2n-4} = -\sum_{i=0}^{\frac{2n-9}3} t^{3i+2} +
t^2\cdot{\textstyle \frac{1-t^{2n-3}}{1-t^3}} \end{eqnarray*}
since $\frac{1-t^{2n-3}}{1-t^3}=\displaystyle
\sum_{i=0}^{\frac{2n-6}3}t^{3i}$. Therefore, $p(X^{[n]};t)$ modulo
$\frac{1-t^{2n-3}}{1-t^3}$ is congruent to
\begin{eqnarray*} \lefteqn{R(t)=(c_0+c_{2n-3})+(c_1+c_{2n-2})t+(c_2+c_{2n-1})t^2
+(c_3+c_{2n})t^3} \\ & & + \displaystyle\sum_{4\leq i\leq 2n-6}
c_it^i - c_{2n-5} \Bigl(\displaystyle\sum_{i=0}^{\frac{2n-9}3}
t^{3i+1} \Bigr)- c_{2n-4} \Bigl(
\displaystyle\sum_{i=0}^{\frac{2n-9}3}
t^{3i+2}\Bigr).\end{eqnarray*} Now $R(t)$ is divisible by
$\frac{1-t^{2n-3}}{1-t^3}=\displaystyle
\sum_{i=0}^{\frac{2n-6}3}t^{3i}$ if and only if $R(t)$ is a scalar
multiple of $\displaystyle \sum_{i=0}^{\frac{2n-6}3}t^{3i}$
because $R(t)$ is of degree $\le 2n-6$. Thus the coefficient of
$R(t)$ with respect to $t^2$ should be 0 i.e.
$c_2+c_{2n-1}-c_{2n-4}=0$. However,
$c_2+c_{2n-1}-c_{2n-4}=c_2+c_1-c_4$ is not zero by Table
\ref{table:list of ci}. This proves Proposition \ref{prop:stringy
E-function test} for the case where 3 divides $n$ and $n\neq 3$.
So the proof of Proposition \ref{prop:stringy
E-function test} is completed for any $n\geq 3$. \qed\\
\begin{remark}
In case of smooth projective curves, we remark that the stringy
E-function of the moduli space of rank 2 bundles is explicitly
computed (\cite{kiem} and \cite{KL}). We were not able to compute
the stringy E-function of $M_{2n}$ precisely, because we do not
know how to compute the Hodge-Deligne polynomial $E(M^s_{2n};u,v)$
of the locus $M^s_{2n}$ of stable sheaves.
\end{remark}
\section{Analysis of Kirwan's desingularization}
\label{sec: Proof of Lemma}
This section is devoted to the proof of Proposition
\ref{prop:analysis on exc}. All can be extracted from \cite{og97}
but we spell out the details for reader's convenience.
To begin with, note that for each $Z\in X^{[n]}$, the tangent
space $T_{X^{[n]},Z}$ of the Hilbert scheme $X^{[n]}$ is
canonically isomorphic to $\Ext^1(I_Z,I_Z)$ where $I_Z$ is the
ideal sheaf of the 0-dimensional closed subscheme $Z$. By the
Yoneda pairing map and Serre duality, we have a skew-symmetric
pairing $\omega:\Ext^1(I_Z,I_Z)\otimes \Ext^1(I_Z,I_Z) \to
\Ext^2(I_Z,I_Z)\cong \cc$, which gives us a symplectic form
$\omega$ on the tangent bundle $T_{X^{[n]}}$ by \cite{Muk84}
Theorem 0.1.
Note that the Killing form on $sl(2)$ gives an isomorphism
$sl(2)^\vee\cong sl(2)$. Let $W=sl(2)^\vee\cong sl(2)\cong \cc^3$.
The adjoint action of $PGL(2)$ on $W$ gives us an identification
$SO(W)\cong PGL(2)$ (\cite{og97} \S1.5). For a symplectic vector
space $(V,\omega)$, let $\Hom^\omega(W,V)$ be the space of
homomorphisms from $W$ to $V$ whose image is isotropic. Let
$\Hom^\omega(W,T_{X^{[n]}}) $ be the bundle over $X^{[n]}$ whose
fiber over $Z\in X^{[n]}$ is $\Hom^\omega(W,T_{X^{[n]},Z})$.
Clearly $\Hom^\omega(W,T_{X^{[n]}})$ is Zariski locally trivial
over $X^{[n]}$. Let $\Hom_k^\omega(W,T_{X^{[n]}})$ be the
subbundle of $\Hom^\omega(W,T_{X^{[n]}})$ of rank $\leq k$
elements in $\Hom^\omega(W,T_{X^{[n]}})$. Also let
$\Gr^\omega(3,T_{X^{[n]}})$ be the relative Grassmannian of
isotropic 3-dimensional subspaces in $T_{X^{[n]}}$ and let $\cB$
denote the tautological rank 3 bundle on
$\Gr^\omega(3,T_{X^{[n]}})$. Obviously these bundles are all
Zariski locally trivial as well.
Let $\pp\Hom^\omega(W,T_{X^{[n]}})$ (resp.
$\pp\Hom_k^\omega(W,T_{X^{[n]}})$) be the projectivization of
$\Hom^\omega(W,T_{X^{[n]}})$ (resp.
$\Hom_k^\omega(W,T_{X^{[n]}})$). Likewise, let $\pp\Hom(W,\cB)$
and $\pp\Hom_k(W,\cB)$ denote the projectivizations of the bundles
$\Hom(W,\cB)$ and $\Hom_k(W,\cB)$. Note that there are obvious
forgetful maps
\begin{eqnarray*}f:\pp\Hom(W,\cB)\to\pp\Hom^\omega(W,T_{X^{[n]}})\ \mbox{\rm
and}\\
f_k:\pp\Hom_k(W,\cB)\to\pp\Hom_k^\omega(W,T_{X^{[n]}})\end{eqnarray*}
Since the pull-back of the defining ideal of
$\pp\Hom_1^\omega(W,T_{X^{[n]}})$ is the ideal of
$\pp\Hom_1(W,\cB)$ (both are actually given by the determinants of
$2\times 2$ minor matrices), $f$ gives rise to a map between
blow-ups
$$\overline{f}:Bl_{\pp\Hom_1(W,\cB)}\pp\Hom(W,\cB)\to
Bl_{\pp\Hom_1^\omega(W,T_{X^{[n]}})}\pp\Hom^\omega(W,T_{X^{[n]}}).$$
Let us denote $Bl_{\pp\Hom_1(W,\cB)}\pp\Hom(W,\cB)$ by $Bl^\cB$
and
$Bl_{\pp\Hom_1^\omega(W,T_{X^{[n]}})}\pp\Hom^\omega(W,T_{X^{[n]}})$
by $Bl^T$. We denote the proper transform of $\pp\Hom_2(W,\cB)$ in
$Bl^\cB$ by $Bl_2^\cB$ and the proper transform of
$\pp\Hom_2^\omega(W,T_{X^{[n]}})$ by $Bl_2^T$. Since $Bl_2^\cB$ is
a Cartier divisor which is mapped onto $Bl_2^T$ and the pull-back
of the defining ideal of $Bl_2^T$ is the ideal sheaf of
$Bl_2^\cB$, $\overline{f}$ lifts to
\begin{equation}\label{eq4.-2}
\hat{f}:Bl^\cB \to
Bl_{Bl_2^T}Bl^T.\end{equation} By \cite{og97} \S3.1 IV, $\hat f$
is an isomorphism on each fiber over $X^{[n]}$, so in particular
$\hat f$ is bijective.
Therefore, $\hat f$ is an isomorphism by Zariski's main theorem.
Note that $\pp\Hom(W,\cB)\git SO(W)$ (resp. $\pp\Hom_k(W,\cB)\git
SO(W)$) is isomorphic to the space of conics $\pp(S^2\cB)$ (resp.
rank $\leq k$ conics $\pp(S^2_k\cB)$) where the $SO(W)$-quotient
map is given by $[\alpha]\mapsto[\alpha\circ\alpha^t]$ where
$\alpha^t$ denotes the transpose of $\alpha\in \Hom(W,\cB)$
(\cite{og97} \S3.1). Let $\hat \pp(S^2\cB) =
Bl_{\pp(S^2_1\cB)}\pp(S^2\cB)$ denote the blow-up along the locus
of rank 1 conics. Then $Bl^\cB\git SO(W)$ is canonically
isomorphic to $\hat \pp(S^2\cB)$ by \cite{k2} Lemma 3.11. Since
$\cB$ is Zariski locally trivial, so is $\hat \pp(S^2\cB)$ over
$\Gr^\omega(3,T_{X^{[n]}})$.
Now consider Simpson's construction of the moduli space $M_{2n}$
(\cite{og97} \S1.1). Let $Q$ be the closure of the set of
semistable points $Q^{ss}$ in the Quot-scheme whose quotient by
the natural $PGL(N)$ action is $M_{2n}$ for some even integer $N$.
Then $Q^{ss}$ parameterizes semistable torsion-free sheaves $F$
together with surjective homomorphisms $h:\cO^{\oplus N}\to F(k)$
which induces an isomorphism $\cc^N\cong H^0(F(k))$ and
$H^1(F(k))=0$. Let $\Omega_Q$ denote the subset of $Q^{ss}$ which
parameterizes sheaves of the form $I_Z\oplus I_Z$ for some $Z\in
X^{[n]}$. This is precisely the locus of closed orbits with
maximal dimensional stabilizers, isomorphic to $PGL(2)$ and the
quotient of $\Omega_Q$ by $PGL(N)$ is $X^{[n]}$.
We can give a more precise description of $\Omega_Q$ as follows.
Let $\cL\to X^{[n]}\times X$ be the universal rank 1 sheaf such
that $\cL|_{Z\times X}$ is isomorphic to the ideal sheaf $I_Z$. By
\cite{HL97} Theorem 10.2.1, the tangent bundle $T_{X^{[n]}}$ is in
fact isomorphic to $\cE xt^1_{X^{[n]}}(\cL,\cL)$. Let
$p:X^{[n]}\times X\to X^{[n]}$ be the projection onto the first
component. For $k\gg 0$, $p_*\cL(k)$ is a vector bundle of rank
$N/2$. Let
\begin{equation}\label{eq4.-1}q:\pp \mathrm{Isom}(\cc^N, p_*\cL(k)\oplus
p_*\cL(k))\to X^{[n]}\end{equation} be the $PGL(N)$-bundle over
$X^{[n]}$ whose fiber over $Z$ is $\pp \mathrm{Isom}(\cc^N,
H^0(I_Z(k)\oplus I_Z(k)))$. Note that the standard action of
$GL(N)$ on $\cc^N$ and the obvious action of $GL(2)$ on
$p_*\cL(k)\oplus p_*\cL(k)$ induce a $PGL(N)\times
PGL(2)$-action on $\pp\mathrm{Isom}(\cc^N, p_*\cL(k)\oplus
p_*\cL(k))\to X^{[n]}$ .
\begin{lemma}\label{4.1} (1) $\Omega_Q\cong \pp
\mathrm{Isom}(\cc^N, p_*\cL(k)\oplus p_*\cL(k))\git SO(W).$\\
(2) Via the above isomorphism, the normal cone of $\Omega_Q$ in
$Q^{ss}$ is $$q^*\mathrm{Hom}^{\omega}(W,T_{X^{[n]}})\git SO(W)\to
\pp \mathrm{Isom}(\cc^N, p_*\cL(k)\oplus p_*\cL(k))\git SO(W)$$
whose fiber over a point lying over $Z\in X^{[n]}$ is $\mathrm{Hom}^{\omega}(W,T_{X^{[n]},Z})$.
\end{lemma}
\begin{proof}
(1) Let $\hat p:\pp \mathrm{Isom}(\cc^N, p_*\cL(k)\oplus
p_*\cL(k))\times X\to \pp \mathrm{Isom}(\cc^N, p_*\cL(k)\oplus
p_*\cL(k))$ be the obvious projection so that we have $q\circ \hat
p=p\circ (q\times 1_X)$. Let $H$ be the dual of the tautological
line bundle over $\pp \mathrm{Isom}(\cc^N, p_*\cL(k)\oplus
p_*\cL(k))$. There is a canonical isomorphism $\cO^{\oplus N}\cong
q^*(p_*\cL(k)\oplus p_*\cL(k))\otimes H$.
This induces a surjective homomorphism
\begin{eqnarray*}\cO^{\oplus N}\to \hat{p}^*q^*(p_*\cL(k)\oplus
p_*\cL(k))\otimes H =(q\times 1)^*(p^*p_*\cL(k)\oplus
p^*p_*\cL(k))\otimes H\\ \to (q\times 1)^*(\cL(k)\oplus
\cL(k))\otimes H\end{eqnarray*} over $\pp \mathrm{Isom}(\cc^N,
p_*\cL(k)\oplus p_*\cL(k))\times X$. By the universal property of
the Quot-scheme, we get a morphism $\pp \mathrm{Isom}(\cc^N,
p_*\cL(k)\oplus p_*\cL(k))\to Q^{ss}$ whose image is clearly
contained in $\Omega_Q$. This map is $PGL(2)$-invariant and hence
we get a morphism
\begin{equation}\label{eq4.00}\phi_\Omega:\pp \mathrm{Isom}(\cc^N,
p_*\cL(k)\oplus p_*\cL(k))\git SO(W)\to \Omega_Q.\end{equation} It
is easy to check that $\phi_\Omega$ is bijective. Since $\Omega_Q$
is smooth (\cite{og97} (1.5.1)), $\phi_\Omega$ is an isomorphism
by Zariski's main theorem.
(2) Let $\cO^{\oplus N} \to \cE(k)$ denote the universal quotient
sheaf on $Q^{ss}\times X$ restricted to $\Omega_Q$ and let $\cF$
be the kernel of the twisted homomorphism $\cO^{\oplus N}(-k) \to
\cE$ so that we have an exact sequence
$$0\to \cF \to \cO^{\oplus N}(-k) \to \cE\to 0$$ over
$\Omega_Q\times X$. The induced long exact sequence gives us
\begin{equation}\label{eqn: rel long exact sequence}
\HHom_{\Omega_Q}(\cO^{\oplus N}(-k),\cE)\to
\HHom_{\Omega_Q}(\cF,\cE)\to \EExt^1_{\Omega_Q}(\cE,\cE)\to
\EExt^1_{\Omega_Q}(\cO^{\oplus N}(-k),\cE) \end{equation} Let
$\pi:\Omega_Q\times X\to \Omega_Q$ be the obvious projection. Note
that $\EExt^1_{\Omega_Q}(\cO^{\oplus
N}(-k),\cE)=R^1\pi_*(\cE(k))^{\oplus N}=0$ and that
$\HHom_{\Omega_Q}(\cO^{\oplus N}(-k),\cE)\cong
\HHom_{\Omega_Q}(\cO^{\oplus N},\cE(k))$ is a vector bundle over
$\Omega_Q$ whose fiber is $gl(N)$ because $\cO_X^{\oplus N}\cong
H^0(E(k))$ for any $[\cO_X^{\oplus N}\to E(k)]\in Q^{ss}$. Let
$T^*_{Q^{ss}}, T^*_{\Omega_Q}$ be cotangent sheaves over $Q^{ss}$
and ${\Omega_Q}$ respectively.
By a famous result of Grothendieck
(\cite{Gr95} \S5) we know
$$(T^*_{Q^{ss}}|_{\Omega_Q})^\vee\cong
\HHom_{\Omega_Q}(\cF,\cE)$$ which contains the tangent bundle of
$\Omega_Q$ as a subbundle. So the first homomorphism in
\eqref{eqn: rel long exact sequence} is the tangent map of the
group action of $PGL(N)$\footnote{In fact the term prior to the
first term of \eqref{eqn: rel long exact sequence} is
$\HHom_{\Omega_Q}(\cE,\cE)$ which contains $\cO$ obviously and the
quotient of $\HHom_{\Omega_Q}(\cO^{\oplus N}(-k),\cE)$ by $\cO$ is
a vector bundle whose fiber is the Lie algebra of $PGL(N)$. } on
$\Omega_Q$ and the second homomorphism is the Kodaira-Spencer map.
Via the isomorphism $\phi_\Omega$
\eqref{eq4.00}, we have a map
$$\delta:\pp
\mathrm{Isom}(\cc^N, p_*\cL(k)\oplus p_*\cL(k))\to \pp
\mathrm{Isom}(\cc^N, p_*\cL(k)\oplus p_*\cL(k))\git SO(W)\cong
\Omega_Q.$$ From the proof of (1) above, the pull-back of $\cE$ by
$\delta\times 1$ is isomorphic to $(q\times 1)^*(\cL(k)\oplus
\cL(k))\otimes H$ and thus the vector bundle $\delta^*\cE
xt^1_{\Omega_Q}(\cE,\cE)$ is isomorphic to
$$q^*\cE xt^1_{X^{[n]}}(\cL, \cL)\otimes gl(2)\cong
q^*T_{X^{[n]}}\otimes gl(2).$$ The pull-back of the tangent sheaf
of $X^{[n]}$ sits in $q^*T_{X^{[n]}}\otimes gl(2)$ as
$q^*T_{X^{[n]}}\otimes \left(\begin{matrix}1&0\\
0&1\end{matrix}\right)$.
Hence the pull-back by $\delta$ of the
normal bundle of $\Omega_Q$ (in the sense of \cite{og97} \S1.3) is
isomorphic to
$$q^*T_{X^{[n]}}\otimes sl(2)\cong
q^*\mathrm{Hom}(W,T_{X^{[n]}}).$$ By \cite{og97} (1.5.10), the
normal cone is fiberwisely the same as the Hessian cone. (See
\cite{og97} \S1.3 for more details on the Hessian cone.) Since the
normal cone is contained in the Hessian cone, the normal cone is
equal to the Hessian cone which is the inverse image of zero by
the Yoneda square map $\Upsilon:\cE xt^1_{\Omega_Q}(\cE,\cE)\to
\cE xt^2_{\Omega_Q}(\cE,\cE)$. It is elementary to see that
$\delta^*\Upsilon^{-1}(0)$ is precisely
$q^*\mathrm{Hom}^\omega(W,T_{X^{[n]}}).$ Since $SO(W)$ acts freely
we obtain (2). See \cite{og97} (1.5.1) for a description of the
normal cone at each point.
\end{proof}
Let $\Sigma_Q$ denote the subset of $Q^{ss}$ whose sheaves are of
the form $I_Z\oplus I_W$ for some $Z,W\in X^{[n]}$. Then
$\Sigma_Q-\Omega_Q$ is precisely the locus of points in $Q^{ss}$
whose stabilizer is isomorphic to $\cc^*$. Let $\pi_R:R\to Q^{ss}$
be the blow-up of $Q^{ss}$ along $\Omega_Q$ and let $\Omega_R$
denote the exceptional divisor. By the above lemma,we have
\begin{equation}\label{eq4.0}\Omega_R\cong q^*\pp\mathrm{Hom}^{\omega}(W,T_{X^{[n]}})\git
SO(W).\end{equation} The following lemma is an easy exercise.
\begin{lemma}\label{4.2} (1) The locus of points
in $\pp\mathrm{Hom}^\omega(W,T_{X^{[n]},Z})^{ss}$ whose stabilizer
is 1-dimensional by the action of $SO(W)$ is precisely $\pp
\mathrm{Hom}^\omega _1(W,T_{X^{[n]},Z})^{ss}$.\\ (2) The locus of
nontrivial stabilizers is $\pp \mathrm{Hom}^\omega
_2(W,T_{X^{[n]},Z})^{ss}$.\end{lemma} Let
\begin{equation}\label{eq4.1}\Delta_R=q^*\pp\mathrm{Hom}^{\omega}_2(W,T_{X^{[n]}})\git
SO(W).\end{equation} Let $\Sigma_R$ be the proper transform of
$\Sigma_Q$. Then $\Sigma_R^{ss}$ is precisely the locus of points
in $R^{ss}$ with 1-dimensional stabilizers by \cite{k2}. Therefore
we have the following from Lemma \ref{4.2}.
\begin{corollary}\label{4.3}
$\Sigma_R^{ss}\cap
\Omega_R=q^*\pp\mathrm{Hom}^{\omega}_1(W,T_{X^{[n]}})^{ss}\git
SO(W).$\end{corollary}
We have an explicit description of
$\Sigma_R^{ss}$ from \cite{og97} \S1.7 III as follows. Let
$$\beta:\mathcal{X}^{[n]}\to X^{[n]}\times X^{[n]}$$
be the blow-up along the diagonal and let
$\mathcal{X}^{[n]}_0=X^{[n]}\times X^{[n]}-\mathbf{\Delta}$ where
$\mathbf{\Delta}$ is the diagonal. Let $\cL_1$ (resp. $\cL_2$) be
the pull-back to $\mathcal{X}^{[n]}\times X$ of the universal
sheaf $\cL\to X^{[n]}\times X$ by $p_{13}\circ (\beta\times 1)$
(resp. $p_{23}\circ (\beta\times 1)$) where $p_{ij}$ is the
projection onto the first (resp. second) and third components. Let
$p:\mathcal{X}^{[n]}\times X\to \mathcal{X}^{[n]}$ be the
projection onto the first component. Then for $k\gg0$,
$p_*\cL_1(k)\oplus p_*\cL_2(k)$ is a vector bundle of rank $N$.
Let
$$q:\pp\mathrm{Isom}(\cc^N,p_*\cL_1(k)\oplus p_*\cL_2(k))\to \mathcal{X}^{[n]}$$
be the $PGL(N)$-bundle. There is an action of $O(2)$ on
$\pp\mathrm{Isom}(\cc^N,p_*\cL_1(k)\oplus p_*\cL_2(k))$. We quote
\cite{og97} (1.7.10) and (1.7.1).
\begin{lemma}\label{4.4}
(1) $\Sigma_R^{ss}\cong\pp\mathrm{Isom}(\cc^N,p_*\cL_1(k)\oplus
p_*\cL_2(k))\git O(2)$\\
(2) The normal cone of $\Sigma_R^{ss}$ in $R^{ss}$ is a locally
trivial bundle over $\Sigma_R^{ss}$ with fiber the cone over a
smooth quadric in $\pp^{4n-5}$.
\end{lemma}
In fact we can give a more explicit description of the normal cone
when restricted to $\Sigma_R^0:=\Sigma_R^{ss}-\Omega_R$. Similarly
as in the proof of Lemma \ref{4.1}, the normal vector bundle to
$\Sigma_R^0$ is isomorphic to the vector bundle (of rank $4n-4$)
\begin{equation}\label{eq4.2} q^*[\cE
xt^1_{\mathcal{X}^{[n]}_0}(\cL_1,\cL_2)\oplus \cE
xt^1_{\mathcal{X}^{[n]}_0}(\cL_2,\cL_1)]\git O(2)\end{equation}
over $\pp\mathrm{Isom}(\cc^N,p_*\cL_1(k)\oplus p_*\cL_2(k))\git
O(2)$ where $O(2)$ acts as follows: if we realize $O(2)$ as the
subgroup of $PGL(2)$
generated by $$SO(2)=\{\theta_\alpha=\left(\begin{matrix}\alpha&0\\
0&\alpha^{-1}\end{matrix}\right)\}/\{\pm Id\},\qquad
\tau=\left(\begin{matrix} 0&1\\1&0\end{matrix}\right)$$
$\theta_\alpha$ multiplies $\alpha$ (resp. $\alpha^{-1}$) to
$\cL_1$ (resp. $\cL_2$) and $\tau$ interchanges $\cL_1$ and
$\cL_2$ by the induced action on $\mathcal{X}^{[n]}$ of
interchanging the first and second factors of $X^{[n]}\times
X^{[n]}$. The normal cone is the inverse image
$q^*\Upsilon^{-1}(0)$ of zero in terms of the Yoneda pairing
\begin{equation}\label{eq4.3}\Upsilon:\cE
xt^1_{\mathcal{X}^{[n]}_0}(\cL_1,\cL_2)\oplus \cE
xt^1_{\mathcal{X}^{[n]}_0}(\cL_2,\cL_1)\to \cE
xt^2_{\mathcal{X}^{[n]}_0}(\cL_1,\cL_1).\end{equation}
Let $\pi_S:S\to R^{ss}$ denote the blow-up of $R^{ss}$ along
$\Sigma_R^{ss}$ and let $\Sigma_S$ be the exceptional divisor of
$\pi_S$ while $\Omega_S$ (resp. $\Delta_S$) denotes the proper
transform of $\Omega_R$ (resp. $\Delta_R$). By \eqref{eq4.3}, we
have \begin{equation}\label{eq4.4} {\Sigma_S}
|_{\pi_S^{-1}(\Sigma_R^0)}\cong q^*\pp \Upsilon^{-1}(0)\git
O(2)\subset q^*\pp[\cE
xt^1_{\mathcal{X}^{[n]}_0}(\cL_1,\cL_2)\oplus \cE
xt^1_{\mathcal{X}^{[n]}_0}(\cL_2,\cL_1)]\git O(2).
\end{equation}
By
\cite{og97} (1.8.10), $S^s=S^{ss}$ and $S^s$ is smooth. The
quotient $S\git PGL(N)$ has only $\zz_2$-quotient singularities
along $\Delta_S\git PGL(N)$. Let $\pi_T:T\to S^{s}$ be the blow-up
of $S^{s}$ along $\Delta_S^{s}$. Then $T\git PGL(N)$ is
nonsingular and this is Kirwan's desingularization
$\rho:\hM_{2n}\to M_{2n}$.
Let $\Omega_T$ and $\Sigma_T$ denote the proper transforms of
$\Omega_S$ and $\Sigma_S$ respectively. Let $\Delta_T$ be the
exceptional divisor of $\pi_T$. Their quotients $\Omega_T\git
PGL(N)$, $\Sigma_T\git PGL(N)$ and $\Delta_T\git PGL(N)$ are
denoted by $D_1=\hat\Omega$, $D_2=\hat\Sigma$ and $D_3=\hat\Delta$
respectively.
With this preparation, we now embark on the proof of Proposition
\ref{prop:analysis on exc}.
\vspace{.5cm}\noindent \textbf{Proof of (1).} This is just
\cite{og97} (3.0.1). More precisely, by \eqref{eq4.0} and
Corollary \ref{4.3}, $\Omega_S$ is the blow-up of
$$q^*\pp\mathrm{Hom}^{\omega}(W,T_{X^{[n]}})\git SO(W)\text{ along
}q^*\pp\mathrm{Hom}^{\omega}_1(W,T_{X^{[n]}})\git SO(W).$$ By
\eqref{eq4.1}, $\Omega_T$ is the blow-up of $\Omega_S$ along the
proper transform of
$$q^*\pp\mathrm{Hom}^{\omega}_2(W,T_{X^{[n]}})\git SO(W)$$ and
$D_1=\hat\Omega$ is the quotient of $\Omega_T$ by the action of
$PGL(N)$. Since the action of $PGL(N)$ commutes with the action of
$SO(W)$, $D_1$ is in fact the quotient by $SO(W)\times PGL(N)$ of
the variety obtained from
$q^*\pp\mathrm{Hom}^{\omega}(W,T_{X^{[n]}})$ by two blow-ups. So
$D_1$ is also the consequence of taking the quotient by $PGL(N)$
first and then the quotient by $SO(W)$ second. Since $q$
\eqref{eq4.-1} is a principal $PGL(N)$ bundle, the result of the
first quotient is just $Bl_{Bl_2^T}Bl^T$ in \eqref{eq4.-2} which
is isomorphic to $Bl^\cB$. If we take further the quotient by
$SO(W)$, then as discussed above the result is $D_1=\hat\pp
(S^2\cB)$.
\vspace{.5cm}\noindent \textbf{Proof of (2).} We use Lemma
\ref{4.4}, \eqref{eq4.2}, and \eqref{eq4.4}. Note that
$\Sigma_R^0$ does not intersect with $\Omega_R$ and $\Delta_R$.
Hence $D_2^0$ is the quotient of $q^*\pp \Upsilon^{-1}(0)\git
O(2)$ which is a subset of $q^*\pp[\cE
xt^1_{\mathcal{X}^{[n]}_0}(\cL_1,\cL_2)\oplus \cE
xt^1_{\mathcal{X}^{[n]}_0}(\cL_2,\cL_1)]\git O(2)$, by the action
of $PGL(N)$. The above are bundles over the restriction of
$$\pp\mathrm{Isom}(\cc^N,p_*\cL_1(k)\oplus p_*\cL_2(k))\git O(2)$$
to the complement $\mathcal{X}^{[n]}_0$ of the diagonal
$\mathbf{\Delta}$ in $X^{[n]}\times X^{[n]}$. As in the proof of
(1), observe that $D_2^0$ is in fact the quotient of
$q^*\pp\Upsilon^{-1}(0)$ by the action of $PGL(N)\times O(2)$
since the actions commute. So we can first take the quotient by
the action of $PGL(N)$, then by the action of $SO(2)$, and finally
by the action of $\zz_2=O(2)/SO(2)$. Since
$\pp\mathrm{Isom}(\cc^N,p_*\cL_1(k)\oplus p_*\cL_2(k))$ is a
principal $PGL(N)$-bundle, the quotient by $PGL(N)$ gives us
$$\pp\Upsilon^{-1}(0)\subset \pp[\cE
xt^1_{\mathcal{X}^{[n]}_0}(\cL_1,\cL_2)\oplus \cE
xt^1_{\mathcal{X}^{[n]}_0}(\cL_2,\cL_1)]$$ over
$\mathcal{X}^{[n]}_0$. The algebraic vector bundles $\cE
xt^1_{\mathcal{X}^{[n]}_0}(\cL_1,\cL_2)$ and $\cE
xt^1_{\mathcal{X}^{[n]}_0}(\cL_2,\cL_1)$ are certainly Zariski
locally trivial and in fact these bundles are dual to each other
by the Yoneda pairing $\Upsilon$ which is non-degenerate (possibly
after tensoring with a line bundle). In particular,
$\Upsilon^{-1}(0)$ is Zariski locally trivial.
Next we take the quotient by the action of $SO(2)\cong \cc^*$.
This action is trivial on the base $\mathcal{X}^{[n]}_0$ and
$SO(2)$ acts on the fibers. Hence $\pp\Upsilon^{-1}(0)/SO(2)$ is a
Zariski locally trivial subbundle of
$$\pp[\cE xt^1_{\mathcal{X}^{[n]}_0}(\cL_1,\cL_2)\oplus \cE
xt^1_{\mathcal{X}^{[n]}_0}(\cL_2,\cL_1)]\git \cc^*\cong \pp\cE
xt^1_{\mathcal{X}^{[n]}_0}(\cL_1,\cL_2)\times_{\mathcal{X}^{[n]}_0}
\pp\cE xt^1_{\mathcal{X}^{[n]}_0}(\cL_2,\cL_1)$$ over
$\mathcal{X}^{[n]}_0$ given by the incidence relations in terms of
the identification $$\pp\cE
xt^1_{\mathcal{X}^{[n]}_0}(\cL_1,\cL_2)\cong \pp\cE
xt^1_{\mathcal{X}^{[n]}_0}(\cL_2,\cL_1)^\vee.$$ Finally, $D_2^0$
is the $\zz_2$-quotient of $\pp\Upsilon^{-1}(0)/SO(2)$.
\vspace{.5cm}\noindent \textbf{Proof of (3).} By \cite{og97}
(1.7.10), the intersection of $\Sigma_R^{ss}$ and $\Omega_R$ is
smooth. By Corollary \ref{4.3}, $\Delta_S$ is the blow-up of
$q^*\pp\mathrm{Hom}^{\omega}_2(W,T_{X^{[n]}})\git SO(W)$ along
$q^*\pp\mathrm{Hom}^{\omega}_1(W,T_{X^{[n]}})\git SO(W)$. Hence
$\Delta_S\git PGL(N)$ is the quotient of $$
Bl_{q^*\pp\mathrm{Hom}^{\omega}_1
(W,T_{X^{[n]}})}q^*\pp\mathrm{Hom}^{\omega}_2(W,T_{X^{[n]}})$$ by
the action of $SO(W)\times PGL(N)$. By taking the quotient by the
action of $PGL(N)$ we get
$$ Bl_{ \pp\mathrm{Hom}^{\omega}_1
(W,T_{X^{[n]}})} \pp\mathrm{Hom}^{\omega}_2(W,T_{X^{[n]}})$$since
$q$ is a principal $PGL(N)$-bundle. Next we take the quotient by
the action of $SO(W)$. Let $\mathrm{Gr}^\omega(2,T_{X^{[n]}})$ be
the relative Grassmannian of isotropic 2-dimensional subspaces in
$T_{X^{[n]}}$ and let $\mathcal A$ be the tautological rank 2
bundle on $\mathrm{Gr}^\omega(2,T_{X^{[n]}})$. We claim
\begin{equation}\label{}Bl_{\pp \Hom_1^\omega(W,T_{X^{[n]}})}
{\pp \Hom_2^\omega(W,T_{X^{[n]}})}\git SO(W)\simeq \pp(S^2\cA)
\end{equation} which is a $\pp^{2}$-bundle
over a $\Gr^\omega (2,2n)$-bundle over $X^{[n]}$. It is obvious
that the bundles are Zariski locally trivial.
There are forgetful maps \begin{eqnarray*}f:\pp\Hom(W,\cA)\to
\pp\Hom_2^\omega(W,T_{X^{[n]}}) \\ f_1:\pp\Hom_1(W,\cA)\to
\pp\Hom_1^\omega(W,T_{X^{[n]}}) \end{eqnarray*} where the
subscript 1 denotes the locus of rank $\leq1$ homomorphisms.
Because the ideal of $\pp\Hom^\omega_1(W,T_{[n]})$ pulls back to
the ideal of $\pp\Hom_1(W,\cA)$, $f$ lifts to $$ \hat
f:Bl_{\pp\Hom_1(W,\cA)} \pp\Hom(W,\cA)\to
Bl_{\pp\Hom_1^\omega(W,T_{X^{[n]}})}
{\pp\Hom_2^\omega(W,T_{X^{[n]}})}$$ This map is bijective
(\cite{og97} (3.5.1)) and hence $\hat f$ is an isomorphism by
Zariski's main theorem because the varieties are smooth. Now
observe that the quotient $\pp\Hom(W,\cA)\git SO(W)$ is
$\pp(S^2\cA)$ where the quotient map is given by $[\alpha]\mapsto
[\alpha\circ\alpha^t]$. Hence $\Delta_S\git PGL(N) $ is the
blow-up of $\pp\Hom(W,\cA)\git SO(W)\cong\pp(S^2\cA)$ along the
locus of rank 1 quadratic forms $\pp(S^2_1\cA)$ (\cite{k2} Lemma
3.11) which is a Cartier divisor. So we proved that
$$ \Delta_S\git PGL(N)\cong \pp(S^2\cA).$$
Finally $S\git PGL(N)$ is singular only along
$\Delta_S\git PGL(N)$ and the singularities are $\cc^{2n-3}/\{\pm
1\}$ by Luna's slice theorem \cite{og97} (1.2.1). Since $D_3$ is
the exceptional divisor of the blow-up of $S\git PGL(N)$ along
$\Delta_S\git PGL(N)$, we conclude that $D_3$ is a
$\pp^{2n-4}$-bundle over $\pp (S^2\mathcal A)$.
\vspace{.5cm}\noindent \textbf{Proof of (4).} By Corollary
\ref{4.3}, $\Sigma_S^s\cap\Omega_S$ is the exceptional divisor of
the blow-up
$Bl_{q^*\pp\mathrm{Hom}^{\omega}_1(W,T_{X^{[n]}})}q^*\pp\mathrm{Hom}^{\omega}(W,T_{X^{[n]}})\git
SO(W) $ and $\Sigma_T^s\cap \Omega_T$ is now the blow-up of the
exceptional divisor along the proper transform of
$q^*\pp\mathrm{Hom}^{\omega}_2(W,T_{X^{[n]}})\git SO(W)$. Using
the isomorphism \eqref{eq4.-2}, this is the exceptional divisor of
$$q^*Bl_{\pp (S^2_1\cB)}\pp (S^2\cB)\to q^*\pp (S^2\cB)$$ over
$\mathrm{Gr}^\omega(3,T_{X^{[n]}})$. Since $q$ is a principal
$PGL(N)$-bundle, $D_1\cap D_2=\Sigma_T^s\cap \Omega_T\git PGL(N)$
is the exceptional divisor of the blow-up $Bl_{\pp (S^2_1\cB)}\pp
(S^2\cB)$. Because the exceptional divisor is a Zariski locally
trivial $\pp^2 $-bundle over $\pp (S^2_1\cB)$ and $\pp
(S^2_1\cB)$ itself is a Zariski locally trivial $\pp^2$-bundle
over $\mathrm{Gr}^\omega(3,T_{X^{[n]}})$, we proved (4).
\vspace{.5cm}\noindent \textbf{Proof of (5).} From the above proof
of (3) it follows immediately that $\Sigma_S^s\cap \Delta_S\git
PGL(N)$ is $\pp (S^2_1\cA)$ and $D_2\cap D_3$ is a $\pp^{2n-4}$
bundle over $\pp (S^2_1\cA)$ which is Zariski locally trivial.
\vspace{.5cm}\noindent \textbf{Proof of (6).} As in the above
proof of (4), we start with \eqref{eq4.1} and use the isomorphism
\eqref{eq4.-2} to see that $D_1\cap D_3$ is the proper transform
of $\pp (S^2_2\cB)$ in the blow-up $Bl_{\pp (S^2_1\cB)}\pp
(S^2\cB)$. This is a Zariski locally trivial $\pp^2$-bundle over a
Zariski locally trivially $\pp^2$-bundle over
$\mathrm{Gr}^\omega(3,T_{X^{[n]}})$.
\vspace{.5cm}\noindent \textbf{Proof of (7).} This follows
immediately from the proof of (4) and (6).
\vspace{.5cm}
From the above descriptions, it is clear that $D_i$
($i=1,2,3$) are normal crossing smooth divisors. \qed
\section{Hodge-Deligne polynomial of $D_2^0 $}
\label{sec: Computation of E-poly of D_0^2}
In this section we prove Lemma \ref{lem: Hodge Deligne poly of
D02}. Recall $$I_{2n-3}=\{((x_i),(y_j))\in \pp^{2n-3}\times
\pp^{2n-3}\,|\, \sum_{i=0}^{2n-3} x_iy_i=0\}.$$ It is elementary
(\cite{GH78} p. 606) to see that
$$H^*(I_{2n-3};\qq)\cong \qq[a,b]/\langle a^{2n-2}, b^{2n-2},
a^{2n-3}+a^{2n-4}b+a^{2n-5}b^2+\cdots+b^{2n-3} \rangle$$ where $a$
(resp. $b$) is the pull-back of the first Chern class of the
tautological line bundle of the first (resp. second) $\pp^{2n-3}$.
The $\zz_2$-action interchanges $a$ and $b$ and the invariant
subspace of $H^*(I_{2n-3};\qq)$ is generated by classes of the
form $a^ib^j+a^jb^i$. As a vector space $H^*(I_{2n-3};\qq)$ is
\begin{equation}\label{eq5.0}\qq\text{-span}\{a^ib^j\,|\, 0\le i\le 2n-3, 0\le j\le
2n-4\}\end{equation} while the invariant subspace is
$$\qq\text{-span}\{a^ib^j+a^jb^i\,|\, 0\le i\le j\le 2n-4\}.$$
The index set $\{(i,j)\,|\, 0\le i\le j\le 2n-4\}$ is mapped to
its complement in $\{(i,j)\,|\, 0\le i\le 2n-3, 0\le j\le 2n-4\}$
by the map $(i,j)\mapsto (j+1,i)$. This immediately implies that
\begin{equation}\label{eq5.3}
P(I_{2n-3};z)=(1+z^2)P^+(I_{2n-3};z)\end{equation}
By \eqref{eq5.0} or the observation that $I_{2n-3}$ is the
Zariski locally trivial $\pp^{2n-4}$-bundle over $\pp^{2n-3}$, we
have
\begin{equation}\label{eq5.2}P(I_{2n-3};z)=\frac{1-(z^2)^{2n-2}}{1-z^2}\cdot
\frac{1-(z^2)^{2n-3}}{1-z^2}.\end{equation} Because $1+z^2$
divides $\frac{1-(z^2)^{2n-2}}{1-z^2}$,
$\frac{1-(z^2)^{2n-3}}{1-z^2}$ also divides $P^+(I_{2n-3};z)$.
Therefore, (\ref{eqn: E D02 is divisible by some Q}) is a direct
consequence of \eqref{eqn: compute D02} since $P(X^{[n]};z)$ has
no odd degree terms by \eqref{eqn:Betti for X[n]}.
Now let us prove \eqref{eqn: compute D02}. Let
$$\psi:\widetilde{D}_2^0:=\pp\Upsilon^{-1}(0)/SO(2)\to
\mathcal{X}^{[n]}_0=X^{[n]}\times X^{[n]}-\mathbf{\Delta}$$ be the
Zariski locally trivial $I_{2n-3}$-bundle in the proof of
Proposition \ref{prop:analysis on exc} (2) in \S4. Recall that
$D_2^0=\widetilde{D}_2^0/\zz_2$. We have seen in the proof of
Proposition \ref{prop:analysis on exc} (2) in \S4 that there is a
$\zz_2$-equivariant embedding
$$\imath:\widetilde{D}_2^0\hookrightarrow
\pp\cE
xt^1_{\mathcal{X}^{[n]}_0}(\cL_1,\cL_2)\times_{\mathcal{X}^{[n]}_0}
\pp\cE xt^1_{\mathcal{X}^{[n]}_0}(\cL_2,\cL_1)$$ where the
$\zz_2$-action interchanges $\cL_1$ and $\cL_2$.
Let $\lambda$ (resp. $\eta$) be the pull-back to
$\widetilde{D}_2^0$ of the first Chern class of the tautological
line bundle over $\pp\cE xt^1_{\mathcal{X}^{[n]}_0}(\cL_1,\cL_2)$
(resp. $\pp\cE xt^1_{\mathcal{X}^{[n]}_0}(\cL_2,\cL_1)$). By
definition, $\lambda$ and $\eta$ restrict to $a$ and $b$
respectively. The $\zz_2$-action interchanges $\lambda$ and
$\eta$. By the Leray-Hirsch theorem\footnote{The Leray-Hirsch
theorem in \cite{V02I} p.182 is stated for ordinary cohomology but
the statement holds also for compact support cohomology. See the
proof in \cite{V02I} p.195} we have an isomorphism
\begin{equation}\label{eq5.5} H^*_c(\widetilde{D}_2^0)\ \
\cong \ H^*_c(\mathcal{X}^{[n]}_0)\otimes
H^*(I_{2n-3}).\end{equation} As the pull-back and the cup product
preserve mixed Hodge structure, \eqref{eq5.5} determines the mixed
Hodge structure of $H^*_c(\widetilde{D}_2^0)$. The
$\zz_2$-invariant part is
\begin{equation}\label{eq5.6}
H^*_c(\widetilde{D}_2^0)^+\cong \left(
H^*_c(\mathcal{X}^{[n]}_0)^+\otimes H^*(I_{2n-3})^+\right) \oplus
\left( H^*_c(\mathcal{X}^{[n]}_0)^-\otimes H^*(I_{2n-3})^-\right)
\end{equation} where the superscript $\pm$ denotes the $\pm
1$-eigenspace of the $\zz_2$-action. Because $H^*_c(D_2^0)\cong
H^*_c(\widetilde{D}_2^0/\zz_2)\cong H^*_c(\widetilde{D}_2^0)^+$
(\cite{Gr57} Theorem 5.3.1 and Proposition 5.2.3), $E(D_2^0;u,v)$
is equal to
\begin{equation}\label{eq5.7}
E^+(\widetilde{D}_2^0;u,v)=E^+(\mathcal{X}^{[n]}_0;u,v)E^+(I_{2n-3};u,v)
+E^-(\mathcal{X}^{[n]}_0;u,v)E^-(I_{2n-3};u,v).
\end{equation}
where $E^\pm(Y;u,v)=\sum_{p,q}\sum_{k\geq0} (-1)^k
h^{p,q}(H^k_c(Y)^\pm) u^pv^q$.
It is easy to see \begin{eqnarray*}P^+(X^{[n]}\times
X^{[n]};z)=\frac{P(X^{[n]};z)^2+P(X^{[n]};z^2)}2,\\
{P^-(X^{[n]}\times
X^{[n]};z)=\frac{P(X^{[n]};z)^2-P(X^{[n]};z^2)}2}\end{eqnarray*}
(Macdonald's formula). Since $X^{[n]}\times X^{[n]}$ is smooth
projective, we have
\begin{eqnarray*} E^+(X^{[n]}\times
X^{[n]};z,z)=\frac{P(X^{[n]};z)^2+P(X^{[n]};z^2)}2\\
E^-(X^{[n]}\times
X^{[n]};z,z)=\frac{P(X^{[n]};z)^2-P(X^{[n]};z^2)}2
\end{eqnarray*}
Now as $\mathcal{X}^{[n]}_0=X^{[n]}\times X^{[n]}-\mathbf{\Delta}$
and $\mathbf{\Delta}\cong X^{[n]}$ is $\zz_2$-invariant, by the
additive property of the E-polynomial we have
\begin{eqnarray*}
E^+(\mathcal{X}^{[n]}_0;z,z)=E^+(X^{[n]}\times X^{[n]};z,z)
-E(X^{[n]};z,z)\\ =
\frac{P(X^{[n]};z)^2+P(X^{[n]};z^2)}2-P(X^{[n]};z),\end{eqnarray*}
\begin{eqnarray*}E^-(\mathcal{X}^{[n]}_0;z,z)=E^-(X^{[n]}\times X^{[n]};z,z)\\
=\frac{P(X^{[n]};z)^2-P(X^{[n]};z^2)}2.\end{eqnarray*} The
equation \eqref{eqn: compute D02} is an immediate consequence of
the above equations and \eqref{eq5.7}. \qed
\bibliographystyle{amsplain}
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Read & Watch: Del Potro Gets Personal In 'The Road To London'
Juan Martin del Potro is working hard to make his first appearance at The O2 in six years and let cameras in during his road to recovery.
In his new documentary “The Road to London: Juan Martin del Potro”, we went behind the scenes as the Argentine rehabbed from a fractured right kneecap he sustained last October at the Rolex Shanghai Masters, forcing him to miss the season-ending championships in London. As Del Potro spent grueling hours in the gym in his hometown of Tandil, returning to The O2 remained high on his mind.
"It’s one event that every player wants to be in,” said Del Potro. “It’s not easy to qualify for London, but it will be a good challenge to see if I can qualify once again.”
The documentary also shows a more relaxed side of Del Potro as he hosted a charity day event in his hometown, drank mate on a hill top and had a barbecue dinner with friends. Although he welcomed the time at home, the Argentine was eager to compete again and repeat the success of his 2018 season that saw him crack the Top 10 for the first time in four years.
“I never expected to be in the Top 10 again after all of my problems. It’s never easy to repeat a similar year, but you never know if I’m still in good shape and feeling healthy.”
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Kronos 360
Kronos 360 is a marketplace for buying and selling luxury watches: new, pre-owned, and collector models from more 80 major watch brands. It is a trusted third party that secures transactions and ensures the authentication and traceability of each luxury watch as well as logistics and delivery with a one-year warranty. Authentic luxury watches from Kronos 360 are put on sale by individuals and collectors.
For an inexperienced person, the purchase of a collector or vintage luxury watch from an individual can be considered a risk. Kronos 360 offers many advantages: a fair and reasonable price, quality after-sale service, and guarantee of authenticity and traceability.
Kronos 360 has the particularity of being at the meeting point of two distinct markets: the luxury one, strong and stable, and the second-hand market, which is in full expansion. Indeed, thanks to the rise of e-commerce, new consumption habits and growing watch enthusiast communities, the pre-owned luxury watch market has drastically transformed over recent years to become a booming industry.
Collector watches have been shown to be a safe investment that appreciates over time. Collector watches are among the many options for luxury investment, such as antique cars, works of art or fine wines, luxury watches have the advantage of being easy to store and sell. In recent years, there has been a strong increase in the sale of second-hand luxury watches, since a new luxury watch may initially lose value. The market is strong for collectible watches and the cost-to-value ratio is certifiable for each model.
OLMA invests in the fastest growing segments of the luxury market
Le Montrachet
Watchmaster
| 65,969
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During the Q3, the company launched its AED2.4 billion (USD653 million), 2,255 home master-plan development, Water's Edge, on the waterfront of Yas Island.
The first phase was launched at Cityscape Global in September and fully sold in a matter of days. Sales of homes at Water's Edge, West Yas, as well a land plot sale, took the total value of development sales during Q3 to AED 604 million (USD164.4 million).
Residential portfolio occupancy increased to 91 percent, while occupancy was stable in the office portfolio at 92 percent and at Yas Mall, which remained steady at 93 percent.
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Piedmont Culture
Piedmont is located in Northern Italy between Emilia Romagna and Switzerland, and it is a territory that boasts a rich cultural heritage. The history of the region is strongly related to the Savoy, a noble family that brought popularity and prestige to this part of Italy throughout time.
The Savoy Residences, declared as UNESCO world heritage sites, stand out as a testimony to the solid presence of the monarchy. Since the Middle Ages, the monarchy characterized the facets of the region and its importance in the entire country.
The domination of the Savoy dynasty, which reigned in Italy until the establishment of the Republic in 1946, has been a fundamental instrument in the cultural development of the region.
Piedmont was also a homeland and preferred destination of many illustrious characters through the centuries, and a few contemporary personalities were born in the region.
The capital of the region is Turin, a place that evolved yet maintained its original charm of a city of culture and art. Today, many stroll on the streets of modern Turin, or sip a cup of coffee or eat delicious dishes in historic cafés that once drew remarkable literates, poets, and scientists.
The literature in Turin, and in Piedmont, is an important part of the local culture. Apart from many literates and authors famous the world over, the city now hosts one of the most important international book conventions, the Salone del Libro.
Apart from literature, the region is linked to many other sides of the culture, such as the architecture and the arts, cinema, and even the music.
While Turin is a vibrant and lively city, the other cities of the region preserve different rhythms and stronger bonds with the past. The medieval charm and traditions are still preserved in cities like Casale Monferrato, Alessandria and Asti.
These areas are characterized by endless vineyards, bountiful agriculture and beautiful weather throughout the seasons. The local traditions still thrive in these areas and attract thousands to folkloristic events, such as the Palio of Asti or the Carnival of Ivrea.
As a strongly religious region, Piedmont is visited by many pilgrims and believers from all over the world. Catholicism is the dominant religion, yet the region also boasts an important Hebraic community and some communities of the Protestant confession.
Piedmont is also the homeland of two of the most iconic Italian products that turned the world’s head, the historic FIAT 500 and the delicious Nutella, a popular hazelnut cream spread.
Piedmont’s Art and Architecture
Piedmont’s art and architecture have ancient roots and early medieval-period testimonies. The first examples of artistic and architectural development in the region are the barbarian necropolis of Testona, near Moncalieri, and a Baptistery in Novara, together with a few remains of the churches that were built or rebuilt in the Romanesque period.
The unique Romanesque-Lombardy style that characterizes the Piedmont region still lives in the impressive architecture of the church of San Michele in Oleggio and in the Cathedral of Ivrea, an example of the first local Romanesque basilica in Piedmont.
But the crown of Romanesque art and architecture in the region is the majestic abbey Sacra di San Michele. The most impressive part of the abbey is, without a doubt, the Presbytery and its entrance that is flanked by capitals and pillars sculpted with images of the signs of the zodiac and constellations.
The cathedrals of Casale Monferrato and the Abbey of Santa Maria of Vezzolano are other significant examples of the Romanesque style.
Due to a weak urban development, only a few urban centers in Piedmont show traces of the medieval age. The traces are majorly constituted by abbeys and castles that were eventually destroyed or altered.
Between the 12th and 13th centuries, the cities that knew some development were Novara, Asti, and Vercelli. Alessandria, one of the most important cities of the region, together with Turin, developed as boroughs at first, then were developed from an economic and cultural point of view later on.
Regarding the art of the period, a few bas-reliefs in the basilica of Sant’Andrea in Vercelli and in the abbey of Sant’Antonio in Ranverso are some of the most representative examples of sculpture, while both edifices are splendid examples of Piedmont’s Gothic architecture.
The painting and sculpting remained modest in the following centuries, yet numerous buildings show the local interpretation of the Gothic style, in particular, some churches characterized by high cusps and overlapping portals, such as the cathedral of San Giovanni in Saluzzo and the cathedrals of Cirié and Chivasso, and many other settlements. The cathedrals of Asti and Novara are beautiful, although the most representative examples in terms of both architecture and sculpture are the cathedrals of Chieri and Susa.
In these cathedrals, part of the Gothic influence can be seen in the carved wood elements, such as the crucifixes, icons and the choir stalls.
The 15th and 16th centuries are dominated by feudalism and its persistence in the region is evidenced by a large number of castles located in various centers with a dominant position. Particularly well-preserved are the castles of Canavese and Monferrato. From the same era, the ancient center of Saluzzo has remained essentially unchanged.
In the second part of the 16th century, the political and social situation hindered the emergence of a more modern art manifested almost everywhere throughout Piedmont. The Renaissance style started to emerge in the Vercelli area, and was influenced by Lombardy’s art, which resulted in a remarkable school of painting being opened.
Giovanni Martino Spanzotti, Defendente Ferrari, and Gaudenzio Ferrari’s works can be admired in many churches throughout Vercelli, Novara, Ivrea, and Turin. From all the artworks, one of the most representative frescoes can be found in the chapel of Our Lady of Loreto, while other splendid examples are present in the church of Madonna delle Grazie in Varallo.
Turin started to meet a period of great artistic flowering in the second half of the 16th century, when the Savoy established the capital of the region in this city. The school of Turin was influenced by French art, and the local artists developed a unique interpretation of the Renaissance that is well visible in the Palace of Valentino.
Baroque emerged in Piedmont in the 17th century, and the greatest architect of the time is unanimously considered to be Guarino Guarini, even he was not from Piedmont. Among the edifices projected by the architect, the Chapel of the Holy Shroud, which is annexed to the Cathedral of Turin, is perhaps the most famous. As it is easy to imagine, the chapel houses the famous shroud Jesus presumably wore after death.
Besides the chapel, other important works of the architect include the church of San Lorenzo and the Carignano Palace in Turin. Inspired by the art of Borromini, the palace is a very bold interpretation of Baroque motifs.
In the 17th century painting and sculpture didn’t thrive, yet one of the most important artists of the era, whose works beautify various churches throughout the region, is Tanzio da Varallo.
The beginning of the 18th century represented a turning point in the architecture of Piedmont, thanks to the arrival of Filippo Juvarra. His works still impress locals and tourists alike, and among the most important edifices is the Palazzo Madama in Turin, Stupinigi Lodge in Nichelino and Basilica Superga in Turin. All these edifices are characterized by a perfect synthesis of space with Baroque or classic motifs.
Other important architectural works of the 18th century are the bell tower of the Basilica of San Gaudenzio in Novara, and the cathedral of Carignano.
While painting and sculpture were still shadowed by architecture, other arts and crafts started to arise in Piedmont, such as the production of furniture, tapestries, embroidery, and majolica, arts that transform the baroque of Piedmont in a splendid example of taste and sensitivity of the time.
While architecture dominated the region until the 19th century, the era of sculpture soon took over, represented by many monuments of Carlo Marochetti. The painting school of the time is also noteworthy and characterized mainly by the reproduction of romantic landscapes.
Nevertheless, great architecture emerges again in the second half of the 19th century. The greatest architect of the time is, without a doubt, Alessandro Antonelli, whose iconic Mole Antonelliana dominates the landscape of Turin and hosts the National Museum of Cinema. Another important work of the architect is the dome of the Basilica of San Gaudenzio in Novara, another splendid example of verticality and technical ability.
The beginning of the 20th century in Piedmont is characterized by economic recovery and Liberty style. The affirmation of this style in Turin sets the basis of the Great Modern Art Exhibition inaugurated in 1902. The architecture of the time is also characterized by the imaginative structures of Raimondo D’Aronco who built the main pavilion of the Exhibition.
Leonardo Bistolfi, a sculptor and exponent of Italian symbolism, is responsible for the suggestive transformations carried out in the art, while one of the most representative painters of the era, Giuseppe Pellizza da Volpedo, impresses with the Fourth State. The painting, now exhibited in the Gallery of Modern Art in Milan, is the result of a long elaboration in which the painter adapted the French post-impressionism to a new ethical content in the context of a social commitment characteristic of the Turin Socialist intellectuals.
The Liberty style had a relatively short life in Piedmont and knew its fall by the first decade of the 20th century. Piedmont is an industrial region, and the demands of industrial production were hardly compatible with the sumptuousness of the Liberty style.
From the second decade of the 20th century onwards, the architectural style of the region was dominated by industrial buildings and factories. Among the most representative examples is the FIAT building designed in 1919 by Giacomo Mattè-Trucco.
The Lingotto plant is particularly interesting, characterized by a structure in reinforced concrete with multiple planes connected by a helix ramp that leads to the roof. The edifice was reinvented in 1983 by architect Renzo Piano who added to Lingotto new spaces, such as offices, auditoriums, and more.
The most emblematic structure created by Renzo Piano in the reinvention of Lingotto is Bolla, a suspended meeting room that inspired the architect in some of his future works.
At the beginning of the 21st century, with the occasion of the 2006 Winter Olympics hosted by Turin, the regional architectural style was reinvented once more, and a number of facilities emerged, including the Olympic Village.
Science in Piedmont
Although modest from a science perspective, the Piedmont region boasts a few important names and was the birthplace of a number of people who actually made a difference in Italy and in the world.
One of these people is Joseph Louis Lagrange, unanimously considered one of the brightest mathematicians of the 18th century. Lagrange was born in Turin in 1736 and at the age of fourteen joined the University of Turin to undertake legal studies. His passion for geometry and experimental physics prevailed and the future genius of mathematics abandoned the legal studies to dedicate himself to the study of advanced mathematics.
Although he published his first scientific work exclusively in Italian, his vision brought him the attention of the International Scientific Community, and King Charles Emanuel III nominates him the Substitute of the Master of Mathematics despite his young age.
In 1766 Lagrange became the president of the Academy of Sciences in Berlin and continued his work that eventually led to the creation of the decimal metric system, which represents the basis of today’s International System.
To pay tribute to Lagrange’s great mind, the city of Turin commissioned a sculpture that was executed by Giovanni Albertoni. The sculpture dominates the center of Lagrange square.
Another important scientific figure of Piedmont is Rita Levi Montalcini, born in Turin in 1909. Montalcini was awarded the Nobel Prize for medicine in 1986, after years dedicated to the study of neurosciences.
She began her medical studies at the age of 20, and although she determined to pursue an academic career as an assistant in neurobiology and psychiatry, the racial laws that were emanating from the fascist regime in Italy blocked her.
Nevertheless, her passion for science pushed Montalcini to continue her research in an improvised home lab until an opportunity arose for her in Saint Louis, Missouri, USA.
The United States of America became the second home to Montalcini and she continues her research on the NFG molecule, a protein that plays an essential role in the differentiation of the various nerve cells.
Literature in Piedmont
Besides science, literature played an important role in Piedmont. One of the greatest names of the literature of the region is Umberto Eco, a critic, essayist, and writer famous all over the world. Eco was born in Alessandria in 1932 and graduated the University of Turin in 1954 with a thesis on the aesthetic thought of Tommaso Aquino.
His career started at a local TV agency where he collaborated with a few broadcasts while in parallel he began writing his first novels.
In 1980 Umberto Eco publishes The Name of the Rose, a novel that conquered the hearts of literary critics and readers all over the world. The novel, inspired by some of the most iconic landmarks of Piedmont, such as the Sacra of San Michele, became the inspiration for a splendid cinematic transposition featuring Sean Connery as one of the actors.
Besides Umberto Eco, Piedmont also boasts a few other important writers, such as Gianni Rodari, an author of novels and stories for children, Primo Levi, Cesare Pavese and Vittorio Alfieri.
A literary tour in the region leads to the discovery of all the places that inspired these great wordsmiths and is a beautiful way of discovering an alternative Piedmont, away from the masses.
Music in Piedmont
From a musical point of view, Piedmont is dominated by two distinctive composers and players, Paolo Conti and Ludovico Einaudi.
Paolo Conti was born in Asti in 1937 and as a teenager begins to cultivate his passion for the classic American jazz. He begins writing songs inspired by cinema, literature, and life, while in parallel he pursues a career as a solicitor.
In the 60s some of his creations were made famous by a series of important Italian musicians such as Adriano Celentano and Patty Bravo, while Conti launches his first jazz album in 1974. Since then, he influenced styles and impressed critics and enthusiasts with his timeless jazz songs.
Ludovico Einaudi, on the other hand, is an important composer and piano player who preserved in his musical expression an echo of the classical reality influenced by pop, jazz, and rock music. Because of his unique and distinctive style, Einaudi’s music is appreciated all over the world.
Movies and Cinema In Piedmont
Piedmont, and above all Turin, is also the birthplace of the Italian cinema. Since the beginning of the 20th century, the city played an important role in the film culture both in production and in the promotion of various Italian movies.
In fact, after the invention of the cinema in 1896, Turin was one of the first cities where the film industry started to develop, at first with silent movies, then with other styles of cinema.
One of the most iconic movies from the beginnings of cinema, Cabiria registered in 1914 by Giovanni Pastore, and it still represents a model of the evolution of cinema.
To discover the history of Turin and the evolution of cinema, visit the National Museum of Cinema, inaugurated in 1941. Today, the museum is located among the impressive scenery of Mole Antonelliana, and here you will be able to find out more about the history of cinema, its various genres, and the most famous movies and protagonists.
The museum is structured on four levels which are divided into the “archeology of cinema.” Exhibits include some pre-cinematic optic machines, magical lanterns, as well as ancient and modern cinematographic equipment. Among the most impressive objects in the collection are some decorations of the first Italian cinema hall seats.
The second level is dedicated to the machines and various phases of filmmaking, while the third level hosts a gallery.
The fourth level is probably the most impressive and it is unanimously considered the heart of the museum. The level is organized as a temple hall that offers access to ten different halls, each of them dedicated to a different cinematic genre.
The center of the edifice is equipped with comfortable chaise lounges where you can sit and watch two silent movies and a short brief of the history of Italian cinema. The museum also has a constantly expanding library.
A visit to the Mole Antonelliana also features a splendid panorama over the city and its surroundings.
The museum also hosts various events dedicated to cinema and film, including the Torino Film Festival, an event dedicated exclusively to Italian cinema.
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St. George Island Grocery Stores To Stock Up On Your Vacation
Situated along Florida’s famous ‘Gold Coast’, St. George Island is truly heaven on Earth. Our charming little beach community offers the perfect retreat for travelers that are seeking a quiet beach retreat.
The island itself is small but mighty, spanning just over 28 miles in total length. However, rest assured that St. George has everything you’ll need for a comfortable stay, including several grocery stores and convenience stores located within a short drive of your vacation home. And, many of St. George Island vacation rentals come with a full kitchen, which makes meal prep a breeze.
Whether you need to stock up on groceries for the week or you just need to grab a few essentials, check out these St. George Island grocery stores that are sure to meet all of your needs.
Grocery Stores In St. George Island
These grocery stores are located on St. George Island, just a quick drive from the area’s residential district and vacation rentals. All of the grocery stores on the island are centrally located along the main stretch in downtown St. George Island.
Piggy Wiggly Express
Photo Credit:
244 Franklin Boulevard St. George Island, Eastpoint, FL 32328 | (850) 927-2808
Hours: 7 a.m. – 10 p.m. Daily
Touted as St. George Island’s ‘hometown grocery store’, Piggy Wiggly does not disappoint. A popular regional chain, Piggy Wiggly carries everything you could need to fill your pantry during your island adventure, including an impressive selection of craft beer and wine, ready-made meals, and made-to-order gourmet pizzas.
Market Place
244 Franklin Blvd, Eastpoint, FL 32328
Hours: 7 a.m. – 10 p.m. Daily
If you’re seeking a local purveyor of great meats and seafood, Market Place is your destination. This local gem is conveniently situated downtown and offers a great value on chops and local seafood. They also offer all of the proper accompaniments, like sides, beverages, and pantry items for a fast and convenient meal.
Sparks & Sons’ Island Grocery
119 Franklin Blvd, St George Island, FL 32328 | (850) 927-2040
Hours: 8 a.m. – 10 p.m. Daily
Locally owned and operated, Spark & Sons’ Grocery is a new addition to the island and is quickly becoming a favorite among visitors and locals alike! This full-service grocer carries everything you could need to stock your kitchen during your getaway. From fresh produce to locally sourced seafood, you can find it all at Spark & Sons’!
SGI Fresh Market
Photo Credit:
119 Franklin Blvd, Eastpoint, FL 32328 | (850) 927-2258
Hours: 8 a.m. – 10 p.m. Daily
SGI Fresh Market is committed to providing quick, friendly service and offering the highest quality products available. The market offers everything from pantry items and produce to paper products, ready-made meals, and an impressive bakery. Not in the mood to shop? Don’t fret—you can order your groceries online and get delivery within the hour!
Grocery Stores Near St. George Island
Want to stock up on supplies before you hit the island? Check out one of these convenient grocery stores on the way to St. George Island.
Big Top Supermarket
357 US-319, Eastpoint, FL 32328 | (850) 670-8626
Hours: 7 a.m. – 10 p.m. Daily
Conveniently located in Eastpoint just before the St. George Island bridge, Big Top Supermarket is the perfect spot to stock up on vacation essentials. The expansive grocer offers everything you could need, from produce and fresh chops to pantry items, paper products, beverages, and more. Be sure to check out their deli section, with smoked ribs, boston butts, briskets, and more.
Piggy Wiggly
Photo Credit:
130 Avenue E, Apalachicola, FL 32320 | (850) 653-8768
Hours: 6 a.m. – 10 p.m. Daily
A tried and true Southern staple, Piggy Wiggly is a regional favorite and a great spot to grab groceries for the week. Piggy Wiggly is a full-service grocer, with pantry items, fresh produce, an impressive craft beer and wine selection, and an unrivaled deli and bakery.
Gulfside IGA
425 US-98, Apalachicola, FL 32320 | (850) 653-9695
Known for fast, friendly service, IGA is a great spot for savvy shoppers seeking a deal on groceries. IGA has all of the essentials for your stay, including produce and ready-made items, pantry essentials, paper products, and plenty of tasty snacks.
With all of these great local grocery options, it won’t be hard to stock up and settle into your new home away from home on St. George Island!
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TITLE: Riemann normal coordinates and inertial frames in general relativity
QUESTION [2 upvotes]: The weak equivalence principle states that for infinitesimal patches of space and time, the laws of nature are identical to that of special relativity. This is mathematically reflected in the existence of Riemann normal coordinates at every point on the manifold. Geodesics are straight lines in these coordinates as expected since particles are expected to move in straight lines in special relativity. However, I don't completely understand the relationship between these coordinates and inertial frames.
Consider the case of a particle moving along a geodesic; we can construct normal coordinates at every point on its trajectory. However, depending on how I define my tetrad basis, the coordinates of each of the points close to the particle will be different. It appears to me that there is a special orientation to the tetrad basis we can choose. We can define the "time" basis vector to be tangent to the trajectory of the particle and the other "space" vectors to be orthogonal to this vector. Then, as the particle moves along its path in an infinitesimal time, the magnitude of the "space" coordinates will not vary with only the "time" coordinate changing. This is an inertial frame in which geodesics are straight lines and the particle does not move in its own reference frame (for an infinitesimal time at least before we switch to a different set of normal coordinates adapted to a different point). Is this what we mean by an inertial frame? If so, what is the interpretation of the normal coordinates that can be constructed such that the "time" basis is not tangent to the trajectory of the particle, and where all components change as the particle moves?
Now on the other hand, say a particle is moving along a curve in the manifold with a non-zero four acceleration. At every point in time on its trajectory, it's still possible to construct normal coordinates at the position of the particle. If we follow the same procedure as before and define the "time" vector to be tangent to the trajectory of the particle at a point p, the geodesic along that tangent starting at p will appear to be parameterised by coordinates given by: $(t,0,0,0)$ (where t is the parameter). However, since the particle is not moving along a geodesic, the particles position cannot simply be parameterised by $(t,0,0,0)$ and there must be a change in position as well. This means that there are no normal coordinates in which a an accelerating particle can see geodesics as straight lines and simultaneously be at "rest" (in the normal coordinates). Is this a reflection of the fact that accelerating observers can't be in inertial reference frames? In that case, can the frame we constructed be considered a co-moving reference frame?
I'm not sure if this the right way to think about normal coordinates, so any help is appreciated.
REPLY [6 votes]: Is this what we mean by an inertial frame? If so, what is the interpretation of the normal coordinates that can be constructed such that the "time" basis is not tangent to the trajectory of the particle, and where all components change as the particle moves?
Yes, this is a (local) inertial frame. If the time direction is not aligned with the tangent vector to the geodesic, then the 4-velocity of the particle will not be $\mathbf u = \frac{\partial}{\partial t}$; in other words, you have constructed an inertial frame in which the particle in question is moving. This frame is related to the rest frame of the particle by a Lorentz boost.
Is this a reflection of the fact that accelerating observers can't be in inertial reference frames? In that case, can the frame we constructed be considered a co-moving reference frame?
Yes to both.
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TITLE: Is it possible for a solar system to have planets orbiting the star(s) in a spherical pattern?
QUESTION [1 upvotes]: By the question I mean that the planets spin in their ellipse but all ellipses describe the surface of a near-sphere shape around a star.
Same question but for a solar system with 2 stars of irrelevant sizes or nature
REPLY [7 votes]: I think what you mean is - is it possible for a planetary system to exist such that the planets do not orbit in a single plane, but the planets have a large scatter of inclination angles?
Our solar system has a relatively modest range, providing you ignore Pluto, of orbital inclination values (and eccentricities); zero to 7 degrees (Mercury). This is thought to be due to the way that the solar system was formed; from a rotating protoplanetary disk.
Other planetary systems are thought to form in the same way and indeed there is evidence from many of the multiple planetary systems discovered via the transiting technique, that many other planetary systems are also very "flat" and often flatter than our solar system.
(e.g. Fang & Margot 2012)
Nevertheless there are exceptions. One can use the Rossiter-Mclaughlin effect to estimate the projected orientation of a transiting planet's orbit to the equatorial plane of rotation of its parent star. There are many examples of planets which have orbits that go over the rotation poles of their parent stars or are even retrograde. For example: Anderson et al. 2010; Triaud's 2011 PhD thesis. About 1/3 of "hot Jupiters" are misaligned in this way.
The misalignment may be as a result of dynamical interaction with other planets or as a result of close fly-bys by other stars or interactions with a binary companion.
EDIT: The R-M effect is suggestive of non-coplanarity, but as only one planet is seen, it is not conclusive. There is at present I think only one solid example where the measurements suggest non-coplanarity of two planets and that is in the planetary system surrounding Upsilon And A.
Using radial velocities and astrometry from the fine guidance sensors on HST,
MacArthur et al. (2010) were able to establish that the c and d planets (i.e. the 2nd and 3rd planets in the system) were inclined at angles of $30\pm1$ degrees with respect to each other. This is much larger than the differences seen in our solar system (maybe 7 degrees).
The presence of two stars is something that is thought to be a key trigger of misalignment. Something called the Kozai mechanism can cause a periodic exchange between orbital inclination and eccentricity so that a planet flips backwards and forwards between two radically different orbits. It is quite possible for an outer planet to be affected by Kozai cycles whilst closer in, perhaps more massive planets contine on their more "usual" co-planar paths.
If you would like (much) more information then a good read is the review by Melvynn Davies et al. given at last year's "Protostars and Planets VI" conference. The talk can also be viewed here (he's a good speaker).
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DryHootch of America, A place for Veterans to Reconnect
Bob Curry, Vietnam combat veteran founded Dryhootch of America in 2008 as an accessible, collaborative network of people and nonprofit organizations to provide support and services for veterans and their families. The innovative organization operates out of coffee houses that serve as gathering places for veterans and families as well as the surrounding community. An acknowledged expert on war trauma and PTSD, Bob Curry is a renowned speaker who has spoken at conferences for military & VA, mental health, medical colleges, social work
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TITLE: Taking limit of $\left(1 - \frac{\lambda}{n}\right)^n$ where $\lambda$ and $n$ are related in context of Poisson distribution
QUESTION [0 upvotes]: Context of Poisson distribution: $n \rightarrow \infty$ and $\lambda := np$ where $p \in [0,1]$ can be thought of as probability of success, $n$ is the number of chances, and $\lambda$ as the average or expected value of success.
You can see the entire proof of turning a binomial distribution's probability mass function into Poisson's here in page 3, though I don't think you need to read it for this question:
https://mbernste.github.io/files/notes/Poisson.pdf
A key step within this proof is this relation:
$$e^{-\lambda} = \lim_{n\rightarrow \infty} \left(1-\frac{\lambda}{n}\right)^n$$
I understand this equality and its proof when $n$ and $\lambda$ are independent to each other, but have trouble accepting the equality when there is a relation, namely $\lambda = np$. Assuming $p$ is fixed and non-zero, $z:= -\frac{n}{\lambda}$, and using the limit definition of $e$, the proof for this equality goes like this:
$$\lim_{n\rightarrow \infty} \left(1 - \frac{\lambda}{n}\right)^n = \lim_{n\rightarrow \infty} \left(1 + \frac{1}{\left(-\frac{n}{\lambda}\right)}\right)^{\left(-\frac{n}{\lambda}\right)\left(-\lambda\right)}=\lim_{z\rightarrow \infty}\left(1+\frac{1}{z}\right)^{z\left(-\lambda\right)} = e^{-\lambda}$$
I feel this proof is incomplete or has some errors, specifically:
Since $z := -\frac{n}{\lambda}$, shouldn't the limit be $z \rightarrow -\infty$ instead? (The equality remains true)
Since $n \rightarrow \infty$ and $\lambda := np$, shouldn't $\lambda \rightarrow \infty$ as well such that $e^{-\lambda} = 0$?
REPLY [3 votes]: What you have is a special case of the Poisson limit theorem.
The parameter $\lambda$ here is a fixed number, and $np=\lambda$ should be understood as $np_n=\lambda$. Note in particular that $p$ is not fixed.
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Sports fantasy sports contests DraftKings has used in the past. Each contestant pays a $10,000 entry fee, then receives $5,000 in contest cash to make their wagers. Each player is required to make five wagers throughout the weekend, including one on Friday, two on Saturday, and two on Sunday.
In the case of this last weekend’s DraftKings contest, contestants were required to make wagers on the two games on Sunday: The Los Angeles Chargers visiting the New England Patriots and the Philadelphia Eagles visiting the New Orleans Saints.
At the end of the weekend, the sports bettors who still have bankrolls keep whatever cash they have left. Those at the top of the leaderboard receive a guaranteed total of $2.5 million, with the 1st place prize being $1 million.
Rufus Peabody Loses Shot at $1 Million
A contestant with the username rleejr86 made a final bet of $47,500 to boost his or her earnings to $101,474. That was enough for rleejr86 to take 1st place and win the $1,000,000 grand prize in the contest.
Rufus Peabody, a professional sports bettor with the username Opti5624, had a bankroll of almost $82,000 heading into Sunday’s NFC Championship Game between the New Orleans Saints and Philadelphia Eagles. A winning bet of $20,000 would have been enough to hand Mr. Peabody the million-dollar prize.
How Opti5624 Was in 1st Place
Peabody had achieved first place by winning a double-or-nothing bet on the New England Patriots in the first game. The Patriots were -3.5 favorites against the Los Angeles Chargers in the 1pm EST game on Sunday. The Patriots won the game 41-28, boosting Peabody’s total ahead of everyone else in the championship.
Unfortunately for Opti5624, the funds from the fourth leg in the contest were not released to his account before the Saints-Eagles game reached kickoff. The NFL staggers the schedule in the playoff rounds, so most games end 30 to 45 minutes before the next game on the schedule.
The Patriots-Chargers game ended close to the Saints-Eagles kickoff, but DraftKings had plenty of time to process wagers and allow contestants to bet their full bankroll on the final contest, which many would have done. Rufus Peabody has a total of $0 at kickoff, so he was unable to make a wager at all. (His nearly $82k was returned to his account after the game started.)
How Peabody Was Unable to Make a Bet
Rufus Peabody, the co-founder of Massey-Peabody (football analytics) and a onetime ESPN employee, said to ESPN’s Chalk, .”
DraftKings Statement on Sports Betting National Championship
James Chisolm, a DraftKings spokesman, released a statement on the controversy: “We recognize that in the rules the scheduled end of betting [kickoff of the NFC divisional-round game].”
DraftKings Sports Betting National Championship
Prior to the contest’s launch, DraftKings had touted the sports betting tournament as a good betting prospect for bettors. The guaranteed prize pool of $2.5 million needed approximately 500 entries in order to break even, given the fact many would win back some of their cash.
By Tuesday of last week, only 140 contestants had paid the $10,000 entry fee, producing a potential overlay. DraftKings mentioned this in public statements, hoping to drive high rollers to enter the contest. It still is not known how many entered, though one can assume DraftKings took a loss on the championship.
Now, it appears DraftKings took a loss and a public relations hit.
Add comment
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. We offer the Electro-Mechanical Technology course in either a 625 Hour complete program or the Electro-Mechanical Modular option, consisting of four modules and 221 hours of learning. These programs provide you with the option to enroll in the entire sequence of modules or take the modules that meet your specific career needs or interests.
Students.)
Watch our Advanced Manufacturing short-term career training information session to learn more.
Career Opportunities
Entry-level salaries for the following positions can be as high as $47,800 machineries, such as conveying systems, production machinery and packaging equipment. Millwrights install, dismantle, repair, reassemble and move machinery in factories, power plants and construction sites.
The Bureau of Labor Statistics projects employment of industrial machinery mechanics and machinery maintenance workers to grow 13 percent from 2019-2029. The increased adoption of sophisticated manufacturing machinery will require more mechanics to keep machines in good working order.
Topics covered in the Electro-Mechanical Technology 625 Hour Program:
- Power-Source & Process
Fundamentals Modules: 461 hours
- Technical Foundation: 100 hours
- Industry Overview: 16 hours
- Essential Skills: 48 hours the Electro-Mechanical Technology 625 Hour program, you will gain extensive knowledge of mechanical technology, including electrical and electronic circuits, and be prepared to operate, test and maintain unmanned, automated, robotic or electro-mechanical equipment.
The Electro-Mechanical Technology 625 Hour noncredit certificate program provides academic credits that can be used towards specific credit-bearing degree programs upon completion and enrollment in the credit degree program. Students are eligible for up to 27 college credits upon completion of this program and passing the corresponding certification exams. For more details, please view our Prior Learning Assessment.
Topics covered in the Electro-Mechanical Modular 221 Hour Program:
Electro-Mechanical Module: DC/AC Electricity
This DC-AC Electricity class covers the principles and application of alternating (AC) and direct (DC) current electricity. Topics include AC and DC circuit analysis and measurement in resistive, capacitive and inductive circuits.
You will be able to apply basic safety rules for working with electricity, describe the operation and function for power supplies, connect and test circuits composed of power supplies. This includes switches (NO and NC), resistors, capacitors, inductors, motors, transformers, buzzers, solenoids, lights, fuses, circuit breakers, and rheostats.
Electro-Mechanical Module: Electrical Wiring
This Electrical Wiring class covers the principles and application of electrical wiring as found in a typical manufacturing environment. Topics include electrical wiring practices, conduit and raceways and requirements for conductors, disconnects and raceways as specified by the National Electric Code (NEC).
Students will be able to apply basic safety rules for working with electrical wiring. Learn methods used for installing electrical outlets, switches and lighting appropriate to residential or light commercial construction, sizing and installing EMT and IMT conduit systems, design and install a wiring system in conduit in compliance with NEC requirements. Students will also be able to demonstrate methods of bundling, labeling, and terminating wires to construct an electrical control panel, wire electric motors, interconnect panels and motors, and wire a complete machine.
Electro-Mechanical Module: Mechanical Systems
This Mechanical Systems class covers the principles and applications of the most commonly found mechanical drive components in an industrial manufacturing environment. Topics include mechanical power transmission devices through an intermediate level along with related construction and troubleshooting techniques. All course material is supplemented with practical hands-on exposure to the items.
Students will be able to apply basic safety rules for working with mechanical equipment; install and align electric motors; describe the construction and operation of sprockets, master links, single roller, multiple strand and silent chain drives. You will be able to perform maintenance and troubleshooting operations on chain drive and v-belt drive systems.
Electro-Mechanical Module: Motor Controls
This Motor Controls class covers the principles and application of industrial sequential control and electrical controls construction as found in a typical manufacturing environment. Topics include AC fixed speed motor control, control transformers, relays, timers, and counters; mechanical, pneumatic and hydraulic input and output devices; sequencing and logic functions; introduction to component and systems troubleshooting.
Students will be able to apply basic safety rules for working with electrical equipment under 600 volts; describe the operation and function for 3 phase motors; manual, magnetic and reversing motor starters. Learn the operation and function for typical input and output devices that would be used for motor, electro-pneumatic, and electro-hydraulic control operations. You will be able to combine these control components with relays, timers and counters to create logic and sequential control circuits and develop the elementary ladder diagram. Students will also be exposed to demonstrating methods of troubleshooting and testing electrical components and systems.
When you complete this Electro-Mechanical Modular 221 Hour program, you will have knowledge of mechanical technology, including electrical and electronic circuits, and be prepared to operate, test and maintain unmanned, automated, and electro-mechanical equipment.
For comprehensive certification training, you should complete all four (4) modules. However, each module may be taken independently.
Program Qualifications
Successful applicants will have math skills, mechanical aptitude; and good hand-eye coordination. A math skills assessment is required for entry into the program.
Location
Training is conveniently located at the College’s Main Campus.
Financial Aid
Can I Receive Financial Aid for this program?
Only the Electro-Mechanical Technology Electro-Mechanical Technology.
Course Dates and Times
Course Hours:
221 Hours to complete the Electro-Mechanical Modular program. For comprehensive certification training, you should complete all four (4) modules. Each module may also be taken independently.
625 Hours to complete the Electro-Mechanical Technology (Industrial Maintenance) 625 Hour program.
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Project
Dainty Deer Christmas Card
- Designer: Marianne Fisher
- christmas, christmas card, deer, quick, speedy,
Marianne Fisher’s adorable deer card won’t take too long to make but it will certainly leave a lasting impression
What you need...
- Dies: Frantic Stamper, Tall Pine Trees; Simon Says Stamp, Kate Snowflake
- Ink-pad: Distress, Victorian Velvet
- Toppers: Daisy Mae Draws, Christmas Friends
- Patterned papers, First Edition, Wanderlust
- Card: white, kraft
- Paper, red
- Fun flock, white
- Sentiment
- Adhesives
instructions
Dainty Deer Card - 1 Fold a blank, 14.5cm square, from kraft card. Cut striped paper, 14cm square, ink the edges and attach to the card blank with the stripes running horizontally.
2 Trim red paper, 11cm square, then secure to the right-hand side of the card. Attach an offcut with vertical stripes across the bottom. Cut the right-hand edge of the circular topper, ink the edges and fix as shown.
3 Stick a sentiment over the edge of the topper. Attach a die-cut snowflake in the top-left and overlap trees in the bottom-left corner. Add PVA glue to the fur and bobble of the deer’s hat, then sprinkle with fun flock and gently press. Remove excess and press firmly.<<
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Anybody Want To Play Capture The Horse With Me?
Nickchat7 replied to Nickchat7's topic in Gift Shop - XBLASorry guys I was busy while you all commented. But I say we can meet today around 6:00 EST (3:00 PST). Is that good?
Heads In Love
Nickchat7 replied to TallishMass's topic in General Discussion - STEAM & XBLAThe Rose and Manbirth, because you know, mother's day father's day? No?
Anybody Want To Play Capture The Horse With Me?
Nickchat7 posted a topic in Gift Shop - XBLAI'm bored, I want to play BBT, I want to play with people on here. (My headset isn't with me) So, anyone want to play capture the horse?
-
- oh hey looky here im playing again, but why do you care vasdfgsfGFpfg 11
The Next Steamroller Victim
Nickchat7 replied to Nickchat7's topic in General Discussion - STEAM & XBLANo, this isn't about the next star head silly billy-er-knighty. It's just if they happen to release another victim, I'd like to know what shape everyone would like to see.
Guess Who's Back For National Dog Day?
Nickchat7 replied to Lindsay's topic in NewsAnd that head would be MEEEEEE naw jk im too stupid to ever be a star head
The Next Steamroller Victim
Nickchat7 posted a topic in General Discussion - STEAM & XBLAI mean, the star head was called Steamroller victim #1. There's gotta be another...?
Any Guesses For Today's Head?
Nickchat7 replied to Minecraft's topic in General Discussion - STEAM & XBLAI want another Steamroller victim... maybe the circle head.
Quiz For Gems!
Nickchat7 replied to BiPolarBear's topic in Gift Shop - XBLAWell then... It's tomorrow.
- Yay, no more nonsense! hey i felt smart there for a minu- NYAH NYAH TWO By the way lets not count 9001 as a record since we didnt really count all the way to 9001 k.
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Okay, you wont see me til Wednesday because I'm going on what's called a Schoolcation. Here's how it works: I am having my first day of school tomorrow, but when the day is over, I'm heading up to Maine. We won't be coming back 'til Wednesday, and we have no computers up in our house in Maine. Yes, I have a house in Maine. So don't expect any activity anytime soon.
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Dan's Special Message For Castle Crashers 5Th Year Anniversary
Nickchat7 replied to megan's topic in NewsMy new backup plan for everything: Castle Crashers
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TITLE: Is it true that the order of $ab$ is always equal to the order of $ba$?
QUESTION [18 upvotes]: How do I prove that if $a$, $b$ are elements of group, then $o(ab) = o(ba)$?
For some reason I end up doing the proof for abelian(ness?), i.e., I assume that the order of $ab$ is $2$ and do the steps that lead me to conclude that $ab=ba$, so the orders must be the same. Is that the right way to do it?
REPLY [2 votes]: (1)
$(ab)^n = e$
$\Rightarrow$
$(ba)^n = (ba)^nbb^{-1} = b(ab)^nb^{-1} = beb^{-1} = e$.
(2)
$(ba)^n = e$
$\Rightarrow$
$(ab)^n = (ab)^naa^{-1} = a(ba)^na^{-1} = aea^{-1} = e$.
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TITLE: Supremum over a compact set
QUESTION [0 upvotes]: Suppose $H$ is a separable Hilbert space and $F:H \to \mathbb{R}$ is a map (continuous if necessary). Let $K= \{a_n\}\cup \{a\}$ where $a_n \to a$ in $H$.
Isn't it always the case that
$$\sup_{h \in K} F(h) = \begin{cases}
F(a_{k_*}) &: \text{for some $k_* \in \mathbb{N}$}\\
F(a) &: \text{o/w}
\end{cases}$$
and hence, if the $F$ depends on a parameter $p$,
$$\lim_{p \to p_0} \sup_{h \in K} F(h) = \begin{cases}
\lim_{p \to p_0}F(a_{k_*}) &: \text{for some $k_* \in \mathbb{N}$}\\
\lim_{p \to p_0}F(a) &: \text{o/w}
\end{cases}$$
Is it correct?
REPLY [2 votes]: A convergent sequence together with its limit is always a compact space (at least in Hausdorff spaces). So the answer to your first question is "yes", because $F(K)$ is compact in $\mathbb{R}$. That's under assumption that $F$ is continous. Otherwise this is not true, $F(K)$ can be any countable subset of $\mathbb{R}$ (doesn't have to be closed, so $\sup$ doesn't have to belong to it).
The answer to the second question is "no", even when $F$ is continous and $H$ is well behaving space like $\mathbb{R}$. That's because the limit $\lim\sup F_p(K)$ doesn't even have to exist. For example take $a_n=\frac{1}{n}\in\mathbb{R}$. Then
$$K=\bigg\{0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots\bigg\}$$
Define
$$F_p:\mathbb{R}\to\mathbb{R}$$
$$F_p(x)=(-1)^p\cdot x$$
Then
$$\sup F_p(K)=\begin{cases}
0 &\mbox{ for odd } p \\
1 &\mbox{ for even } p
\end{cases}$$
So the limit doesn't exist.
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TITLE: Compact opens in sober $T_1$ are closed?
QUESTION [3 upvotes]: I am trying to establish some basic facts about spectral spaces. In relation to this I am looking for a proof of, or a counter example to, the statement that compact open subsets of a sober $T_1$ space are closed.
Recall that a topological space is sober if every closed irreducible subset has a generic point. Whence in a sober $T_1$ space the closed irreducible subsets are precisely the singletons.
Any help is appreciated.
REPLY [2 votes]: You can find counterexamples in some fairly typical sober $T_1$ spaces that are not Hausdorff. For example, consider $\mathbb{N}\cup\{x,y\}$ where every point of $\mathbb{N}$ is isolated and the neighbourhoods of $x$ and $y$ are the cofinite subsets containing $x$ and $y$ respectively. Then $\mathbb{N}\cup\{x\}$ is compact open but not closed.
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A historic handshake sealed the deal Thursday night that will bring a new professional sports team to Orlando.
Mayor Buddy Dyer and Orange County Mayor Teresa Jacobs have agreed to a proposal that will put a new $85 million stadium downtown.
Related: Follow WESH 2 sports director Larry Ridley on Twitter
The decision is the biggest step necessary to bring a Major League Soccer franchise to Orlando. Orange County has pledged $20 million in tourist taxes.
The deal also includes $25 million to finish the second phase of the Doctor Phillips Center for the Performing Arts.
The agreement comes just two days after talks about the stadium hit a snag, when some county leaders raised concerns about budgeting.
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\section{Examples}
Here we compute some examples.
Obviously,
If $G \not \in \cA$, $c_i(G) =0$ for all $i$.
\subsection{$\cA$ is the complement of triangle-free graphs}
$\cA$ consists of all graphs which have an induced triangle.
We compute the coefficients $c_i(G)$ of $\cP_{\cA}$ for various graphs $G$.
\begin{enumerate}[(i)]
\item
$G$ triangle-free. Then $c_i(G) =0$ for all $i$. This is trivially unimodal.
\item
$G = K_n$. Then $c_i(K_n) = {n \choose i}$ for $i \geq 3$ and $c_i(K_n) =0$ otherwise. This is clearly unimodal.
\item
$G=W_n=K_1\bowtie C_{n-1}$. Then $c_i(W_n)={{n-1} \choose {i-1}}-I_i(C_{n-1})$
where $I_i(C_{n-1})$ is the number of independent sets of size $i-1$ in $C_{n-1}$.
This can be computed explicitly via the independence polynomial of a cycle:
$$
I(C_n)=\frac{1}{2^{n-1}}[(1+2x+s)(1+s)^{n-2}+(1+2x-s)(1-s)^{n-2}]
$$
where $s=\sqrt{1+4x}$
see
\cite{levit2005independence}.
To prove that $c_i(W_n)$ is unimodal using Corollary \ref{co:3-4}
we need that
there are no subsets $S\subseteq V(W_n)$ such that $|S|=\lceil n/2 \rceil$ and
$W_n[S]\not\in \cA$.
However, every subset $S$ with $|S|=\lceil n/2 \rceil$ which contains the vertex of degree $n-1$
has an induced triangle.
Hence, Corollary \ref{co:3-4} cannot be applied, and we do not know whether $c_i(W_n)$ is unimodal.
\end{enumerate}
\begin{problem}
Let $\cA$ be the complement of triangle-free graphs.
Characterize the graphs $G$ for which $P_{\cA}(G;x)$ is unimodal.
\end{problem}
\ifskip\else
In particular, we have:
\begin{corollary}
\label{cor}
If $\cA$ is as above and $G$ is a graph of order $n$ such that
there are no subsets $S\subseteq V(G)$ such that $|S|=\lceil n/2 \rceil$ and
$G[S]\not\in \cA$, the sequence $\{c_i\}$ is unimodal with mode $\lceil n/2 \rceil$.
\end{corollary}
\begin{proof}
If there are no subsets $S\subseteq V(G)$ such that $|S|=\lceil n/2 \rceil$ and $G[S]\not\in \cA$,
then for $k=\lceil n/2 \rceil$, we have $\frac{c_k}{{n \choose k }}=1>\frac{\lfloor n/2 \rfloor}{\lceil n/2 \rceil+1}$.
\\
This can also be shown without using Lemma \ref{lemma2} by noting that in this case we have
$c_i(G) = {n \choose i}$,
for $i \geq \lceil n/2 \rceil$.
\end{proof}
\fi
\subsection{$\cA$ is the complement of $H$-free graphs}
Here $H$ is of order $h$.
\begin{enumerate}[(i)]
\item
$H =K_2$. Then $c_i(K_n) = {n \choose i}$ for $i \geq h$ and $c_i(K_n) =0$ otherwise. This is clearly unimodal.
\item
$H =K_2$. $c_i(K_{m,n})={n+m \choose i} - {m \choose i} - {n \choose i}$ for $i \geq 3$.
\item
$H =K_h$. Then $c_i(K_n) = {n \choose i}$ for $i \geq h$ and $c_i(K_n) =0$ otherwise. This is clearly unimodal.
\item
$H=K_{1,3}$. Then $\cA$ is the complement of claw-free graphs.
Then $c_i(K_{n,m})={{n+m} \choose i}-{m \choose i} - {n \choose i}$ for $i>4$,
$c_i(K_{n,m})={{n+m} \choose i}-{m \choose i} - {n \choose i}-{n \choose {i/2}}-{m \choose {i/2}}$
for $i=4$ and $c_i(K_{n,m})=0$ for $i<4$.
\end{enumerate}
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TITLE: Why does a scintillator need to be fast decaying?
QUESTION [0 upvotes]: I have two scintillators, say, one with a decay time of 1 ns vs. one with 100 ns. All other parameters like light yield, size of crystal, electronics used, source emission rate, are the same for both. How would their responses differ (say, for neutron irradiation)?
For purposes like active interrogation, would it really matter which one I use? Which is the limiting factor: timing of scintillator or timing of electronics?
REPLY [3 votes]: As I stated in my comment you need an estimate of the counting rate. If the time between detections is much larger than dead time, the dead time will not really be a factor. However, if the time between detections is comparable to the dead time, then a smaller deadtime will increase your detection efficiency. To determine the time between detections just take a sample count rate and divide it by count time. So take a sample spectrum for 10 secs say and determine how many detections were counted. Some electronics modules will have a count rate meter built in.
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The Global Impact of the Suez Canal blockage
The nearly weeklong Suez Canal blockage will have weeks and months-long implications for global trade. Rapid growth in global demand for container transportation means that carriers are overly reliant on a handful of routes for moving everything from agricultural commodities to consumer goods, e-Commerce packages and empties around the world.
During the crisis, there were two percentage figures that ruled the day. 12% of all global trade passes through the Suez over the course of a calendar year. The second figure – 30% – applies to both the total number of global container movements daily and to the amount of cargo destined for the US East Coast that traverses the Suez.
While the Ever Given is being examined and her owners have declared general average, the ships stuck on either side are now reaching their destinations and finding already occupied berths for which they must wait their turns.
Importers and exporters in the United States will find a series of slow-moving events that will unfold over the next few weeks because of this.
We at Bestway felt it was important that the blockage fallout be brought forward so that plans can be made and customers worldwide advised of potential delays in shipment departures and arrivals.
Congested European ports
The major Mediterranean and north European ports will have been waiting for the arrival of the ships trapped in the Red Sea. Terminals would be receiving export boxes, increasingly to the point of running out of space which meant exporters holding containers and chassis offsite and incurring additional costs.
The ships filled with imports from Asia also provide the necessary containers to load with European exports, creating an additional shortage in empty equipment at both quayside and inland depots.
Vessel integrity further damaged
Carriers operate fixed-day sailings on what are called “strings” and it takes multiple vessels to ensure that the port calls are made on the same schedule, week-in and week-out. The bottlenecks of multiple ships arriving at the same time at ports around the world mean that some vessels in the strings will be out of timing with their anticipated port calls.
To bring the ships back into alignment, we are anticipating a flurry of blanked sailings, or canceled port calls, to restore schedules. This will lead to rolled bookings, exacerbate out-of-location containers and drive up the spot rate market.
Air freight may be an option – but will be even more scarce
IATA’s figures show how much cargo is moving by air, but this must be viewed through the prism of passenger cancellations and so-called phreighter flights where passenger aircraft are used to fly cargo between points that would not normally be a scheduled commercial route.
The constrained capacity that has been tapped into by automotive and semiconductor companies because of a chip fire in Japan and consequential supply chain shortages means that companies who are looking to fly quantities of product to fill the gap before the arrival of a delayed ocean shipment are competing against more buyers for a limited set of options.
Communication remains the key
The more Bestway is a close partner in your supply chain from order placement to estimated fulfillment, the better positioned we are to provide the right kind of assistance. There are multiple alternatives available including utilizing different trade lanes, a combined sea / air service or leveraging available capacity into a city or region and then moving the cargo domestically to its final destination.
Speak to your Bestway representative for the solution that your particular situation demands.
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Mar. 10, 2021 | Corporate
MARION, N.Y. - Seneca Foods Corporation (NASDAQ: SENEA, SENEB) (“Seneca” or the “Company”), one of North America's leading providers of packaged fruits and vegetables with facilities located throughout the United States, today announced the final results of its modified Dutch auction tender offer, which expired at 6:00 p.m., New York City time, on Tuesday, March 9, 2021.
Based on the final count by Computershare Trust Company, the depositary for the tender offer (the “Depositary”), Seneca has accepted for purchase 531 shares of its Class A common stock, par value $0.25 per share (the “Class A Shares”), at a price of $46.00 per Class A Share, for an aggregate cost of $24,426, excluding fees and expenses relating to the tender offer. As Seneca accepted for purchase all the Class A Shares that were properly tendered and not properly withdrawn at a price at or below $46.00, there is no proration factor. The Company will promptly pay for the Class A Share repurchases with available cash.
“We believe our shares are undervalued, and we saw a tender offer as an opportunity to enhance value for our shareholders based upon the then current share price of our common stock. We believe the undersubscribed tender offer is clear evidence that shareholders share our view.”
-Paul Palmby, President and CEO
BofA Securities, Inc. acted as Dealer Manager for the tender offer, Georgeson LLC acted as Information Agent for the tender offer and Computershare Trust Company acted as the Depositary for the tender offer. All inquiries about the tender offer should be directed to the Dealer Manager or the Information Agent toll free at (888) 803-9655 or (866) 628-6079, respectively..
Contact: Timothy J. Benjamin, Chief Financial Officer 315-926-8100
view full pdf version here
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Debt Recovery during and after the Pandemic
6 May 2020 by Priya Patel
The Covid-19 pandemic and resulting lockdown has raised unprecedented challenges for businesses across the UK.
Some companies continue to operate with employees working from home, but others may have ceased to operate and we are seeing some major household retailers now ceasing to trade and making mass redundancies. Businesses’ cash flow has never been more important for their survival and especially so while the banking system and access to loans to bridge financial interruption is depressed.
Many businesses now operate on the basis that invoices are payable upon their presentation. It is therefore increasingly common for debts to rack up and in the current climate, default is rife.
Whilst this is perhaps understandable given the conditions, it is having a significant detrimental impact on cash-flow and the ultimate survival of the creditor businesses that are involved.
So what are some of the options for recovering these unpaid debts?
Every matter will, of course, turn on its own facts and, be advised upon according to the terms of the goods or services contract in play and the status of the company against whom the debt is being pursued will also be a relevant factor to consider.
In the majority of these cases though, there will be found to be a viable method of debt recovery. With the costs of the exercise in the round of considerations and the available methods of enforcement liable to take different periods to discharge and have potentially different prospects of succeeding depending on the debtor’s circumstances, it is important to select an appropriate method of debt recovery early on in the process.
Depending on the size of the debt, the threat of insolvency proceedings for a defaulting business (or individual) can carry with it a big deterrent for continuing to default and/or not enter into negotiations to settle.
What is a Statutory Demand?
A Statutory Demand, which is a written demand for payment of a debt served on either a company or an individual, can be used as a pre-cursor to winding-up or bankruptcy proceedings.
Payment of unsecured debts must be made within 21 days of issuing a Statutory Demand. An unpaid Statutory Demand is regarded as evidence of the debtor’s inability to pay its debts for the purposes of insolvency proceedings. After 21 days, if a qualifying debt remains unpaid, then an application to either bankrupt an individual or wind-up a company debtor can follow.
The process has the following potential advantages:
- It does not involve the courts from the outset;
- Preparing and serving a statutory demand is quick and inexpensive.
- It can either result in prompt payment of a debt, or flush out details of any dispute or cross-claim early on.
The consequences of not making the demanded payment can, therefore, be severe for a debtor.
The court will not generally make a winding-up order against a company where the petition debt is genuinely disputed by the debtor on substantial grounds though. It may, therefore, be that if such a challenge looks likely, other routes to recovery may be more advisable.
Are there any alternative routes?
An alternative route to recovery is to issue court proceedings using more traditional methods; by utilising a claim form (Part 7 proceedings).
The typical way to start a claim is to send a debtor a Letter Before Action (LBA), containing the concise details of the claim, pursuant to the Practice Direction on Pre Action Conduct contained in the Civil Procedure Rules.
That process seeks to elicit the response of the debtor, so that each party’s respective positions can be understood by each other in more detail, from the start.
Having information about the other sides’ case, early, can sometimes assist by bringing the parties in to view of what settlement options there could be available to them – to help avoid the issuing of the claim at court. Otherwise, it can be a way of engaging alternative dispute resolution in the case, early enough to settle matters before costs are run-up, or give the creditor the opportunity to use tactics to put the risk of having to pay a proportion of the creditor’s costs on the debtor.
Part 7 proceedings are more suitable for matters in relation to which there may be more likelihood of it being argued by the debtor that there is a dispute over there being a “debt” in the first place. There may also be arguments over a right to offset against what is owed, or a genuine cross or counter-claim which can also have sway.
If the debtor fails to act following the LBA, the next step would usually be to issue court proceedings.
Once a claim is issued, and upon allocation to track, if the value of a claim is £10,000 or less, it will most likely be allocated to the Small Claims Track and, generally, costs will not be recoverable by the winning party from the loser (unless there is a contractual right to recover costs).
If the value of a claim exceeds £10,000, a claim will be allocated to either the Fast Track or the Multi Track, under which tracks costs are recoverable by the winning party from the loser.
Enforcement
At this time The Ministry of Justice has advised that all bailiffs must follow the Government’s guidance during the pandemic. As a result, they have advised that bailiffs suspend all in-person visits.
The High Court Enforcement Officers Association states that all High Court Enforcement Officers (“HCEO”) members are also following this guidance.
Other methods of enforcement are still available, such as third party debt orders and charging orders.
Practical tips for Creditors
- Consider sending LBAs sooner in the credit-control process;
- Ensure contact details for customers are up-to-date, obtaining an email address where possible;
- If an agreement cannot be reached creditors should still consider taking legal action to avoid any delays caused by a post-lockdown bottleneck.
Fisher Jones Greenwood LLP’s Dispute Resolution team is able to assist clients with all their debt recovery queries and provide commercially focused, practical advice on the options available. For further advice on debt recovery please contact Fisher Jones Greenwood on 01206 700113 or email [email protected].
For more information on Debt & Tax legal issues that have arisen as a result of the COVID-19 pendemic, visit our Coronavirus Legal Advice hub.
Recent Posts
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American Reality star and entrepreneur, Kylie Jenner shows off her customized private jet as she goes on a vacation with her girls and daughter.
The young self-made billionaire took her girls and daughter on a trip in the personalized private jet.
Sophia Richie, Draya Michelle and other friends of Kylie’s were dressed in matching pink tracksuits.
ALSO READ: Media Personality Sets Instagram On Fire With Nearly Unclad Photo
she said in her caption
Let the adventures begin!! #kylieskinsummerTrip
photos below:
Watch the video below:
ALSO READ: BBNaija 2019: Tuoyo, Mercy Steal Show With Erotic Dance Moves
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This is not a political post even though that comes into it. I was born and raised in England, when I was young we had a Socialist Government. My Dad was into politics, he enjoyed the debate (much to my mums disgust) and wanted to do what he could in his work and living situations. So he was a member of the Labour Party. From what I can see, its not the same party now, much the same as here in the States, traditional values have changed. He was a union man. In his day the unions brought very necessary change to the lives of factory workers. Safety for machines that had often left people maimed due to lack of guards when they were needed. Accidents that could be prevented. No one though of suing back then, they fought to change things and make them better. Most of us started work at 14 or 15 years old but the old child labour laws had changed, so that children did not have to work in factories any longer. Going back to my genealogy children of 4 years old helped their parents to work from home. The women in my area were "plaiters" they plaited straw for the hat industry. A lot of women in my parents day did "Homework".......I know mum did several jobs from home. One I recall was putting birthday cards into the plastic sleeves or something like that, putting a card with an envelope and they got paid by the amount that they did. My Aunt Sally made hats. Putting feathers on them. Back then everyone wore hats.
Under the Labour Party many things were run by the government. That would never be popular here in the States. However, there were some advantages. The BBC ran the airwaves and one had a licence to own a TV or Radio. No big deal, a small fee. The electricity and Gas were controlled as were public transportation and so in my mind (I was a kid) everything ran smoothly. One put a coin in the meter and got a certain amount of electricity. I remember when someone forgot to put money in and there would be this clunking noise as the lights went out. You would have to search for a coin in the dark or go next door to ask for one. However, everyone had the necessary means to live. Trains and buses went everywhere one did not need a car. Coal and things like that used for fires. All went smoothly. People grew vegetables in their gardens or had an allotment.
When this changed and things were privatized so much happened, strikes for more money left us over the winters usually using candles and cooking over a coal fire in the living room. We learned to survive. God help us if the coal man went on strike.
The big thing in England was the National Health Care system. A wonderful piece of (?) legislation. Public schools that taught everything imaginable. (even though they had little success in teaching me math or spelling it was not their fault haha.)
Now..........to me, a country is not civilized if it does not give it's people health care from cradle to grave. Yes one pays higher taxes but few do not get back more than they pay in. You can opt out and pay insurance for private care always. People who live there complain. People always complain. You don't have to die because you cant go to the doctors though and throughout Europe you will see that most all countries have some form of health care far superior to the USA. I could go on. Doctors who made house calls. Midwives. Free dental and optical. If you could not pay, then free burial. The schools are free and superior to any here. Higher education as well.
So now.............here we are with a president who is taking away the most basic human rights for us living in the USA that is supposed to be the leader in the free world. What a joke that is. To increase his military budget he is taking away things like free school lunches, when sometimes that is the only good meal the child gets. Meals on wheels for the elderly. It will be a step back for us. The public schools are in a mess now and the teachers are beside themselves. Please read this
Michelle Charland
I can't help but have tears in my eyes and pain in my stomach. I've loved and taught too many children over the past 20 years to stand back and watch our education system get torn apart!!
Please call your Representative and tell them to vote No on House Bill 10. Our children deserve better than this!!!!!!
"To. Yes, there are all of these programs happening in our education system, in addition to just academics.
I was not going to get political but here we are.
The situation with health care could leave millions without insurance and those who have pre-existing conditions or in the middle of treatments. What are people supposed to do? Just die? Is that what Trump wants.
Sure Obama care may not be perfect. It has helped a lot, but others are paying way more than they can afford. So fix it.............don't do away with it. Face the problems and fix them. Or here's a thought. Universal Health and dental care. Who can afford a dentist. I sure can't.
It is a mess...............the greed is overwhelming. Where is the compassion for those less fortunate, yes including immigrants.
I know that in England (and Europe) the immigration problem is desperate. They flow in and are welcomed by the government who give them access to health care that they have not worked for or paid in to the system. True. What's the alternative? Let them die? No...........fix the laws that are flawed. Be sure some kind of work is found for them to earn their care. Educate them and move them forwards into society. In America, what happened to bring me your sick and dying, we have become a country of "I got mine, fuck you" (sorry)
What is the answer? Free enterprise is great. Becoming wealthy is great. You cant legislate compassion but people can shame them into paying a fair share in taxes with no offshore accounts, no loopholes and instilling into people the fact that the more you have the more is expected of you. Its all about power and possessions. How much money can a person use. Some great men like Bill Gates understands it, he remembers where he came from and there are many many like him, but not enough. This while people like Trump use the laws and loopholes to avoid paying taxes and yet earn billions ever year and are proud that they avoided taxes that go into the national budget to help those less fortunate. That is morally and ethically wrong to me.
When some have billions that they will never ever spend, nor will their kids or grandkids, and refuse to give up anything to help the homeless. Shame one them. It does not matter if anyone "deserves" to be helped, it's not something that one should judge. You don't know their story. That brings me to this, mental health. When the State Hospitals were closed nothing replaced them. Now people are out there (with guns) and no help because they cant afford the health care. Yet, the laws change so that even the mentally ill can purchase a gun or at least that is what it is coming to. WHAT THE HELL IS WRONG WITH THIS COUNTRY AND THE PEOPLE IN IT?
This president is also at war with the environment, destruction of the planet is fine by him. All that has been done to protect the public lands and animals, all in danger of being undone by one man who is the epitome of greed. Worse yet are the people who follow him.
Sorry, no nice pictures in this post. I could go on but will leave it at that.
><<
Thursday, March 16, 2017
Compassion...................
2 comments:.
Merle............
Very well said Janice!!! I just hope everything gets turned around for the better! I really feel for the United States people right now!!
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\section{Nonparametric consistency proofs}\label{section:consistency_proof}
In this appendix, I (i) present technical lemmas for regression, unconditional mean embeddings, and conditional mean embeddings; (ii) appeal to these lemmas to prove uniform consistency of dose response curves, incremental response curves, and counterfactual distributions.
\subsection{Lemmas}
\subsubsection{Regression}
Recall classic results for the kernel ridge regression estimator $\hat{\gamma}$ of $\gamma_0(w):=\mathbb{E}(Y|W=w)$. As in Section~\ref{sec:rkhs_main}, $W$ is the concatenation of regressors. Consider the following notation:
\begin{align*}
\gamma_0&=\argmin_{\gamma\in\mathcal{H}}\mathcal{E}(\gamma),\quad \mathcal{E}(\gamma)=\mathbb{E}[\{Y-\gamma(W)\}^2]; \\
\gamma_{\lambda}&=\argmin_{\gamma\in\mathcal{H}}\mathcal{E}_{\lambda}(\gamma),\quad \mathcal{E}_{\lambda}(\gamma)=\mathcal{E}(\gamma)+\lambda\|\gamma\|^2_{\mathcal{H}}; \\
\hat{\gamma}&=\argmin_{\gamma\in\mathcal{H}}\hat{\mathcal{E}}(\gamma),\quad \hat{\mathcal{E}}(\gamma)=\frac{1}{n}\sum_{i=1}^n\{Y_i-\gamma(W_i)\}^2+\lambda\|\gamma\|^2_{\mathcal{H}}.
\end{align*}
\begin{lemma}[Sampling error; Theorem 1 of \cite{smale2007learning}]\label{lemma:sampling}
Suppose Assumptions~\ref{assumption:RKHS} and~\ref{assumption:original} hold. Then with probability $1-\delta$,
$
\|\hat{\gamma}-\gamma_{\lambda}\|_{\mathcal{H}}\leq \frac{6 C \kappa_w \ln(2/\delta)}{\sqrt{n}\lambda}.
$
\end{lemma}
\begin{lemma}[Approximation error; Proposition 3 of \cite{caponnetto2007optimal}] \label{prop:approx_error}
Suppose Assumptions~\ref{assumption:original} and~\ref{assumption:smooth_gamma} hold. Then
$
\|\gamma_{\lambda}-\gamma_0\|_{\mathcal{H}}\leq \lambda^{\frac{c-1}{2}}\sqrt{\zeta}.
$
\end{lemma}
\begin{lemma}[Regression rate; Theorem D.1 of \cite{singh2020kernel}]\label{theorem:regression}
Suppose Assumptions~\ref{assumption:RKHS},~\ref{assumption:original}, and \ref{assumption:smooth_gamma} hold. Then with probability $1-\delta$,
$$
\|\hat{\gamma}-\gamma_0\|_{\mathcal{H}}\leq r_{\gamma}(n,\delta,c):=\dfrac{ \sqrt{\zeta}(c+1)}{4^{\frac{1}{c+1}}} \left\{\dfrac{6 C \kappa_w \ln(2/\delta)}{ \sqrt{n\zeta}(c-1)}\right\}^{\frac{c-1}{c+1}}.
$$
\end{lemma}
\begin{proof}
Immediate from triangle inequality using Lemmas~\ref{lemma:sampling} and~\ref{prop:approx_error}.
\end{proof}
\begin{remark}\label{remark:1}
In sample selection problems, I consider $\gamma_0(w)=\mathbb{E}[SY|W=w]$ where $W$ and hence $\kappa_w$ varies.
\begin{enumerate}
\item Static sample selection
\begin{enumerate}
\item $\theta_0^{ATE}$, $\theta_0^{DS}$, $\theta_0^{ATT}$: $\kappa_w=\kappa_s\kappa_d\kappa_x$;
\item $\theta_0^{CATE}$: $\kappa_w=\kappa_s\kappa_d\kappa_v\kappa_x$.
\end{enumerate}
\item Dynamic sample selection: $\kappa_w=\kappa_s\kappa_d\kappa_x\kappa_m$.
\end{enumerate}
\end{remark}
\subsubsection{Unconditional mean embedding}
Next, I recall classic results for the unconditional mean embedding estimator $\hat{\mu}_w$ for $\mu_w:=\mathbb{E}\{\phi(W)\}$. Let $W$ be a placeholder random variable as in Section~\ref{sec:rkhs_main}.
\begin{lemma}[Bennett inequality; Lemma 2 of \cite{smale2007learning}]\label{lemma:prob}
Let $(\xi_i)$ be i.i.d. random variables drawn from distribution $\mathbb{P}$ taking values in a real separable Hilbert space $\mathcal{K}$. Suppose there exists $ \tilde{M}$ such that
$
\|\xi_i\|_{\mathcal{K}} \leq \tilde{M}<\infty$ almost surely and $
\sigma^2(\xi_i):=\mathbb{E}(\|\xi_i\|_{\mathcal{K}}^2)$. Then $\forall n\in\mathbb{N}, \forall \eta\in(0,1)$,
$$
\mathbb{P}\left\{\left\|\dfrac{1}{n}\sum_{i=1}^n\xi_i-\mathbb{E}(\xi)\right\|_{\mathcal{K}}\leq\dfrac{2\tilde{M}\ln(2/\eta)}{n}+\sqrt{\dfrac{2\sigma^2(\xi)\ln(2/\eta)}{n}}\right\}\geq 1-\eta.
$$
\end{lemma}
\begin{lemma}[Mean embedding rate; Theorem D.2 of \cite{singh2020kernel}]\label{theorem:mean}
Suppose Assumptions~\ref{assumption:RKHS} and~\ref{assumption:original} hold. Then with probability $1-\delta$,
$$
\|\hat{\mu}_w-\mu_w\|_{\mathcal{H}_{\mathcal{W}}}\leq r_{\mu}(n,\delta):=\frac{4\kappa_w \ln(2/\delta)}{\sqrt{n}}.
$$
\end{lemma}
I quote a result that appeals to Lemma~\ref{lemma:prob}. \cite[Theorem 15]{altun2006unifying} originally prove this rate by McDiarmid inequality. See \cite[Theorem 2]{smola2007hilbert} for an argument via Rademacher complexity. See \cite[Proposition A.1]{tolstikhin2017minimax} for an improved constant and the proof that the rate is minimax optimal.
\begin{remark}\label{remark:2}
In various applications, $\kappa_w$ varies.
\begin{enumerate}
\item Static sample selection.
\begin{enumerate}
\item $\theta_0^{ATE}$: with probability $1-\delta$, $
\|\hat{\mu}_x-\mu_x\|_{\mathcal{H}_{\mathcal{X}}}\leq r_{\mu}(n,\delta):=\frac{4\kappa_x \ln(2/\delta)}{\sqrt{n}}.
$
\item $\theta_0^{DS}$: with probability $1-\delta$,
$
\|\hat{\nu}_x-\nu_x\|_{\mathcal{H}_{\mathcal{X}}}\leq r_{\nu}(\tilde{n},\delta):=\frac{4\kappa_x \ln(2/\delta)}{\sqrt{\tilde{n}}}.
$
\end{enumerate}
\item Dynamic sample selection.
\begin{enumerate}
\item $\theta_0^{ATE}$: with probability $1-\delta$, $\forall d\in\mathcal{D}$
$$
\left\|\frac{1}{n}\sum_{i=1}^n\{\phi(X_i) \otimes \mu_{m}(d,X_i)\}-\int \{\phi(x)\otimes \mu_{m}(d,x)\} \mathrm{d}\mathbb{P}(x)\right\|_{\mathcal{H}_{\mathcal{X}}\otimes \mathcal{H}_{\mathcal{M}}} \leq r^{ATE}_{\mu}(n,\delta):=\frac{4\kappa_x \kappa_m \ln(2/\delta)}{\sqrt{n}}.$$
\item $\theta_0^{DS}$: with probability $1-\delta$, $\forall d \in\mathcal{D}$
$$
\left\|\frac{1}{n}\sum_{i=1}^n\{\phi(\tilde{X}_{i}) \otimes \mu_{m}(d,\tilde{X}_{i})\}-\int \{\phi(x)\otimes \mu_{m}(d,x)\} \mathrm{d}\tilde{\mathbb{P}}(x)\right\|_{\mathcal{H}_{\mathcal{X}}\otimes \mathcal{H}_{\mathcal{M}}}
\leq r^{DS}_{\nu}(\tilde{n},\delta):=\frac{4\kappa_x\kappa_m \ln(2/\delta)}{\sqrt{\tilde{n}}}.$$
\end{enumerate}
\end{enumerate}
\end{remark}
\subsubsection{Conditional expectation operator and conditional mean embedding}
In Sections~\ref{section:static} and~\ref{section:dynamic} as well as Appendix~\ref{section:dist}, I consider the abstract conditonal expectation operator $E_j\in \mathcal{L}_2(\mathcal{H}_{\mathcal{A}_j},\mathcal{H}_{\mathcal{B}_j})$, where $\mathcal{A}_j$ and $\mathcal{B}_j$ are spaces that can be instantiated for different causal parameters.
\begin{lemma}[Conditional mean embedding rate; Theorem 2 of \cite{singh2019kernel}]\label{theorem:conditional}
Suppose Assumptions~\ref{assumption:RKHS},~\ref{assumption:original}, and~\ref{assumption:smooth_op} hold. Then with probability $1-\delta$,
$$
\|\hat{E}_j-E_j\|_{\mathcal{L}_2}\leq r_E(\delta,n,c_j):=\dfrac{ \sqrt{\zeta_j}(c_j+1)}{4^{\frac{1}{c_j+1}}} \left\{\dfrac{4\kappa_b(\kappa_a+\kappa_b \|E_j\|_{\mathcal{L}_2}) \ln(2/\delta)}{ \sqrt{n\zeta_j}(c_j-1)}\right\}^{\frac{c_j-1}{c_j+1}}.
$$
Moreover, $\forall b\in\mathcal{B}_j$
$$
\|\hat{\mu}_a(b)-\mu_a(b)\|_{\mathcal{H}_{\mathcal{A}_j}}\leq r_{\mu}(\delta,n,c_j):=\kappa_{b}\cdot
\dfrac{ \sqrt{\zeta_j}(c_j+1)}{4^{\frac{1}{c_j+1}}} \left\{\dfrac{4\kappa_b(\kappa_a+\kappa_b \|E_j\|_{\mathcal{L}_2}) \ln(2/\delta)}{ \sqrt{n\zeta_j}(c_j-1)}\right\}^{\frac{c_j-1}{c_j+1}}.
$$
\end{lemma}
\begin{remark}\label{remark:3}
Note that in various applications, $\kappa_a$ and $\kappa_b$ vary.
\begin{enumerate}
\item Static sample selection
\begin{enumerate}
\item $\theta_0^{ATT}$: $\kappa_a=\kappa_x$, $\kappa_b=\kappa_d$;
\item $\theta_0^{CATE}$: $\kappa_a=\kappa_x$, $\kappa_b=\kappa_v$;
\end{enumerate}
\item Dynamic sample selection:
\begin{enumerate}
\item $\theta_0^{ATE}$: $\kappa_a=\kappa_m$, $\kappa_b=\kappa_d\kappa_x$;
\item $\theta_0^{DS}$: $\kappa_a=\kappa_m$, $\kappa_b=\kappa_d\kappa_x$;
\end{enumerate}
\item Counterfactual distributions
\begin{enumerate}
\item Static $\theta_0^{D:ATE}$, $\theta_0^{D:DS}$, $\theta_0^{D:ATT}$: $\kappa_a=\kappa_y$, $\kappa_b=\kappa_s \kappa_d \kappa_x$.
\item Static $\theta_0^{D:CATE}$: $\kappa_a=\kappa_y$, $\kappa_b=\kappa_s \kappa_d \kappa_v \kappa_x$.
\item Dynamic $\theta_0^{D:ATE}$, $\theta_0^{D:DS}$: $\kappa_a=\kappa_y$, $\kappa_b=\kappa_s \kappa_d \kappa_x \kappa_m$.
\end{enumerate}
\end{enumerate}
\end{remark}
\subsection{Main results}
Appealing to Remarks~\ref{remark:notation},~\ref{remark:1},~\ref{remark:2}, and~\ref{remark:3}, I prove uniform consistency for (i) dose response curves, (ii) incremental response curves, and (iii) counterfactual distributions.
\subsubsection{Static sample selection}
\begin{proof}[Proof of Theorem~\ref{theorem:consistency_static}]
I generalize \cite[Theorem 6.5]{singh2020kernel} to the static sample selection problem.
Consider $\theta_0^{ATE}$.
\begin{align*}
&\hat{\theta}^{ATE}(d)-\theta_0^{ATE}(d)
=\langle \hat{\gamma} ,\phi(1)\otimes \phi(d)\otimes \hat{\mu}_x \rangle_{\mathcal{H}} - \langle \gamma_0 , \phi(1)\otimes \phi(d)\otimes \mu_x \rangle_{\mathcal{H}} \\
&=\langle \hat{\gamma} ,\phi(1)\otimes \phi(d)\otimes[\hat{\mu}_x-\mu_x] \rangle_{\mathcal{H}} + \langle [\hat{\gamma}-\gamma_0], \phi(1)\otimes \phi(d) \otimes \mu_x \rangle_{\mathcal{H}} \\
&=\langle [\hat{\gamma}-\gamma_0],\phi(1)\otimes \phi(d)\otimes[\hat{\mu}_x-\mu_x] \rangle_{\mathcal{H}} \\
&\quad + \langle \gamma_0,\phi(1)\otimes \phi(d)\otimes[\hat{\mu}_x-\mu_x] \rangle_{\mathcal{H}}\\
&\quad+\langle [\hat{\gamma}-\gamma_0],\phi(1)\otimes \phi(d) \otimes \mu_x \rangle_{\mathcal{H}}.
\end{align*}
Therefore by Lemmas~\ref{theorem:regression} and~\ref{theorem:mean}, with probability $1-2\delta$
\begin{align*}
&|\hat{\theta}^{ATE}(d)-\theta_0^{ATE}(d)|\leq
\|\hat{\gamma}-\gamma_0\|_{\mathcal{H}}\|\phi(1)\|_{\mathcal{H}_{\mathcal{S}}} \|\phi(d)\|_{\mathcal{H}_{\mathcal{D}}} \|\hat{\mu}_x-\mu_x\|_{\mathcal{H}_{\mathcal{X}}}\\
&\quad +
\|\gamma_0\|_{\mathcal{H}}\|\phi(1)\|_{\mathcal{H}_{\mathcal{S}}}\|\phi(d)\|_{\mathcal{H}_{\mathcal{D}}}\|\hat{\mu}_x-\mu_x\|_{\mathcal{H}_{\mathcal{X}}} \\
&\quad +
\|\hat{\gamma}-\gamma_0\|_{\mathcal{H}}\|\phi(1)\|_{\mathcal{H}_{\mathcal{S}}}\|\phi(d)\|_{\mathcal{H}_{\mathcal{D}}} \|\mu_x\|_{\mathcal{H}_{\mathcal{X}}}
\\
&\leq \kappa_s\kappa_d \cdot r_{\gamma}(n,\delta,c) \cdot r_{\mu}(n,\delta)+\kappa_s\kappa_d\cdot\|\gamma_0\|_{\mathcal{H}} \cdot r_{\mu}(n,\delta)+\kappa_s\kappa_d\kappa_x \cdot r_{\gamma}(n,\delta,c)\\
&=O\left(n^{-\frac{1}{2}\frac{c-1}{c+1}}\right).
\end{align*}
By the same argument, with probability $1-2\delta$
\begin{align*}
&|\hat{\theta}^{DS}(d,\tilde{\mathbb{P}})-\theta_0^{DS}(d,\tilde{\mathbb{P}})| \\
&\leq \kappa_s\kappa_d \cdot r_{\gamma}(n,\delta,c) \cdot r_{\nu}(\tilde{n},\delta)+\kappa_s\kappa_d\cdot\|\gamma_0\|_{\mathcal{H}} \cdot r_{\nu}(\tilde{n},\delta)+\kappa_s\kappa_d\kappa_x \cdot r_{\gamma}(n,\delta,c)\\
&=O\left( n^{-\frac{1}{2}\frac{c-1}{c+1}}+\tilde{n}^{-\frac{1}{2}}\right).
\end{align*}
Next, consider $\theta_0^{ATT}$.
\begin{align*}
&\hat{\theta}^{ATT}(d,d')-\theta_0^{ATT}(d,d') =\langle \hat{\gamma} ,\phi(1)\otimes \phi(d')\otimes \hat{\mu}_x(d) \rangle_{\mathcal{H}} - \langle \gamma_0 ,\phi(1)\otimes \phi(d')\otimes \mu_x(d) \rangle_{\mathcal{H}} \\
&=\langle \hat{\gamma} ,\phi(1)\otimes \phi(d')\otimes[\hat{\mu}_x(d)-\mu_x(d)] \rangle_{\mathcal{H}} + \langle [\hat{\gamma}-\gamma_0],\phi(1)\otimes \phi(d') \otimes \mu_x(d) \rangle_{\mathcal{H}} \\
&=\langle [\hat{\gamma}-\gamma_0], \phi(1)\otimes\phi(d')\otimes[\hat{\mu}_x(d)-\mu_x(d)] \rangle_{\mathcal{H}} \\
&\quad + \langle \gamma_0, \phi(1)\otimes\phi(d')\otimes[\hat{\mu}_x(d)-\mu_x(d)] \rangle_{\mathcal{H}}\\
&\quad +\langle [\hat{\gamma}-\gamma_0], \phi(1)\otimes\phi(d') \otimes \mu_x(d) \rangle_{\mathcal{H}}.
\end{align*}
Therefore by Lemmas~\ref{theorem:regression} and~\ref{theorem:conditional}, with probability $1-2\delta$
\begin{align*}
&|\hat{\theta}^{ATT}(d,d')-\theta_0^{ATT}(d,d')|
\leq
\|\hat{\gamma}-\gamma_0\|_{\mathcal{H}}\|\phi(1)\|_{\mathcal{H}_{\mathcal{S}}}\|\phi(d')\|_{\mathcal{H}_{\mathcal{D}}} \|\hat{\mu}_x(d)-\mu_x(d)\|_{\mathcal{H}_{\mathcal{X}}}\\
&\quad +\|\gamma_0\|_{\mathcal{H}}\|\phi(1)\|_{\mathcal{H}_{\mathcal{S}}}\|\phi(d')\|_{\mathcal{H}_{\mathcal{D}}}\|\hat{\mu}_x(d)-\mu_x(d)\|_{\mathcal{H}_{\mathcal{X}}} \\
&\quad +
\|\hat{\gamma}-\gamma_0\|_{\mathcal{H}}\|\phi(1)\|_{\mathcal{H}_{\mathcal{S}}}\|\phi(d')\|_{\mathcal{H}_{\mathcal{D}}} \|\mu_x(d)\|_{\mathcal{H}_{\mathcal{X}}}
\\
&\leq \kappa_s\kappa_d \cdot r_{\gamma}(n,\delta,c) \cdot r_{\mu}^{ATT}(n,\delta,c_1)+\kappa_s\kappa_d\cdot\|\gamma_0\|_{\mathcal{H}} \cdot r_{\mu}^{ATT}(n,\delta,c_1)
+\kappa_s\kappa_d\kappa_x \cdot r_{\gamma}(n,\delta,c)
\\
&=O\left(n^{-\frac{1}{2}\frac{c-1}{c+1}}+n^{-\frac{1}{2}\frac{c_1-1}{c_1+1}}\right).
\end{align*}
Finally, consider $\theta_0^{CATE}$.
\begin{align*}
&\hat{\theta}^{CATE}(d,v)-\theta_0^{CATE}(d,v)=\langle \hat{\gamma} , \phi(1)\otimes\phi(d)\otimes \phi(v)\otimes \hat{\mu}_{x}(v) \rangle_{\mathcal{H}} - \langle \gamma_0 ,\phi(1)\otimes \phi(d )\otimes \phi(v) \otimes \mu_{x}(v) \rangle_{\mathcal{H}} \\
&=\langle \hat{\gamma} ,\phi(1)\otimes \phi(d)\otimes \phi(v)\otimes[\hat{\mu}_{x}(v)-\mu_{x}(v)] \rangle_{\mathcal{H}} + \langle [\hat{\gamma}-\gamma_0],\phi(1)\otimes \phi(d)\otimes \phi(v) \otimes \mu_{x}(v) \rangle_{\mathcal{H}} \\
&=\langle [\hat{\gamma}-\gamma_0],\phi(1)\otimes \phi(d)\otimes \phi(v)\otimes[\hat{\mu}_{x}(v)-\mu_{x}(v)] \rangle_{\mathcal{H}} \\
&\quad + \langle \gamma_0,\phi(1)\otimes \phi(d)\otimes \phi(v)\otimes[\hat{\mu}_{x}(v)-\mu_{x}(v)] \rangle_{\mathcal{H}}\\
&\quad +\langle [\hat{\gamma}-\gamma_0],\phi(1)\otimes \phi(d)\otimes \phi(v) \otimes \mu_{x}(v) \rangle_{\mathcal{H}}.
\end{align*}
Therefore by Lemmas~\ref{theorem:regression} and~\ref{theorem:conditional}, with probability $1-2\delta$
\begin{align*}
&|\hat{\theta}^{CATE}(d,v)-\theta_0^{CATE}(d,v)|\leq
\|\hat{\gamma}-\gamma_0\|_{\mathcal{H}}\|\phi(1)\|_{\mathcal{H}_{\mathcal{S}}}\|\phi(d)\|_{\mathcal{H}_{\mathcal{D}}}\|\phi(v)\|_{\mathcal{H}_{\mathcal{V}}} \|\hat{\mu}_{x}(v)-\mu_{x}(v)\|_{\mathcal{H}_{\mathcal{X}}}\\
&\quad+
\|\gamma_0\|_{\mathcal{H}}\|\phi(1)\|_{\mathcal{H}_{\mathcal{S}}}\|\phi(d)\|_{\mathcal{H}_{\mathcal{D}}}\|\phi(v)\|_{\mathcal{H}_{\mathcal{V}}}\|\hat{\mu}_{x}(v)-\mu_{x}(v)\|_{\mathcal{H}_{\mathcal{X}}} \\
&\quad+
\|\hat{\gamma}-\gamma_0\|_{\mathcal{H}}\|\phi(1)\|_{\mathcal{H}_{\mathcal{S}}}\|\phi(d)\|_{\mathcal{H}_{\mathcal{D}}}\|\phi(v)\|_{\mathcal{H}_{\mathcal{V}}} \|\mu_{x}(v)\|_{\mathcal{H}_{\mathcal{X}}}
\\
&\leq \kappa_s\kappa_d\kappa_{v} \cdot r_{\gamma}(n,\delta,c) \cdot r_{\mu}^{CATE}(n,\delta,c_2)
+\kappa_s\kappa_d\kappa_{v}\cdot\|\gamma_0\|_{\mathcal{H}} \cdot r_{\mu}^{CATE}(n,\delta,c_2)
+\kappa_s\kappa_d\kappa_{v} \kappa_{x} \cdot r_{\gamma}(n,\delta,c)
\\
&=O\left(n^{-\frac{1}{2}\frac{c-1}{c+1}}+n^{-\frac{1}{2}\frac{c_2-1}{c_2+1}}\right).
\end{align*}
For incremental responses, replace $\phi(d)$ with $\nabla_d \phi(d)$ and hence replace $\|\phi(d)\|_{\mathcal{H}_{\mathcal{D}}}\leq \kappa_d$ with $\|\nabla_d \phi(d)\|_{\mathcal{H}_{\mathcal{D}}}\leq \kappa_d'$.
\end{proof}
\subsubsection{Dynamic sample selection}
To lighten notation, define
\begin{align*}
\Delta_p&:=\frac{1}{n}\sum_{i=1}^n \{\phi(X_i)\otimes \hat{\mu}_{m}(d,X_i) \}-\int \{\phi(x)\otimes \mu_{m}(d,x) \}\mathrm{d}\mathbb{P}(x), \\
\Delta_q&:=\frac{1}{\tilde{n}}\sum_{i=1}^{\tilde{n}} \{\phi(\tilde{X}_i)\otimes \hat{\nu}_{m}(d,\tilde{X}_i) \}-\int \{\phi(x)\otimes \nu_{m}(d,x) \}\mathrm{d}\tilde{\mathbb{P}}(x).
\end{align*}
\begin{proposition}\label{prop:delta_p}
Suppose Assumptions~\ref{assumption:RKHS} and~\ref{assumption:original} hold.
\begin{enumerate}
\item If in addition Assumption~\ref{assumption:smooth_op} holds with with $\mathcal{A}_4=\mathcal{X}$ and $\mathcal{B}_4=\mathcal{D}\times \mathcal{X}$ then with probability $1-2\delta$
\begin{align*}
&\left\|\Delta_p\right\|_{\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{M}}}\leq \kappa_x \cdot r^{ATE}_{\mu}(n,\delta,c_4)+r^{ATE}_{\mu}(n,\delta).
\end{align*}
\item If in addition Assumption~\ref{assumption:smooth_op} hold with $\mathcal{A}_5=\mathcal{X}$ and $\mathcal{B}_5=\mathcal{D}\times \mathcal{X}$ then with probability $1-2\delta$
\begin{align*}
&\left\|\Delta_q\right\|_{\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{M}}}\leq \kappa_x \cdot r^{DS}_{\nu}(\tilde{n},\delta,c_5)+r^{DS}_{\nu}(\tilde{n},\delta).
\end{align*}
\end{enumerate}
\end{proposition}
\begin{proof}
By triangle inequality,
\begin{align*}
\left\|\Delta_p\right\|_{\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{M}}}
&\leq \left\|\frac{1}{n}\sum_{i=1}^n \{\phi(X_i)\otimes \hat{\mu}_{m}(d,X_i) \}-\{\phi(X_i)\otimes \mu_{m}(d,X_i) \}\right\|_{\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{M}}} \\
&\quad + \left\|\frac{1}{n}\sum_{i=1}^n\{\phi(X_i) \otimes \mu_{m}(d,X_i)\}-\int \{\phi(x)\otimes \mu_{m}(d,x)\} \mathrm{d}\mathbb{P}(x)\right\|_{\mathcal{H}_{\mathcal{X}}\otimes \mathcal{H}_{\mathcal{M}}}.
\end{align*}
Focusing on the former term, by Lemma~\ref{theorem:conditional}
\begin{align*}
&\left\|\frac{1}{n}\sum_{i=1}^n \{\phi(X_i)\otimes \hat{\mu}_{m}(d,X_i) \}-\{\phi(X_i)\otimes \mu_{m}(d,X_i) \}\right\|_{\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{M}}} \\
&=\left\|\frac{1}{n}\sum_{i=1}^n \phi(X_i) \otimes \{\hat{\mu}_{m}(d,X_i)-\mu_{m}(d,X_i)\} \right\|_{\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{M}}} \\
&\leq \kappa_x \cdot \sup_{x\in\mathcal{X}}\left\| \hat{\mu}_{m}(d,x)-\mu_{m}(d,x)\right\|_{\mathcal{H}_{\mathcal{X}}} \\
&\leq \kappa_x \cdot r^{ATE}_{\mu}(n,\delta,c_4).
\end{align*}
Focusing on the latter term, by Lemma~\ref{theorem:mean}
\begin{align*}
\left\|\frac{1}{n}\sum_{i=1}^n\{\phi(X_i) \otimes \mu_{m}(d,X_i)\}-\int \{\phi(x)\otimes \mu_{m}(d,x)\} \mathrm{d}\mathbb{P}(x)\right\|_{\mathcal{H}_{\mathcal{X}}\otimes \mathcal{H}_{\mathcal{M}}}\leq r^{ATE}_{\mu}(n,\delta).
\end{align*}
The argument for $\theta_0^{DS}$ is identical.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{theorem:consistency_planning}]
I adapt the techniques of \cite[Theorem 7]{singh2021workshop} to the dynamic sample selection problem.
Consider $\theta_0^{ATE}$.
\begin{align*}
&\hat{\theta}^{ATE}(d)-\theta_0^{ATE}(d)\\
&=\langle \hat{\gamma}, \phi(1)\otimes \phi(d)\otimes \frac{1}{n}\sum_{i=1}^n\{\phi(X_i)\otimes \hat{\mu}_{m}(d,X_i)\} \rangle_{\mathcal{H}} - \langle \gamma_0,\phi(1) \otimes \phi(d)\otimes \int \phi(x)\otimes \mu_{m}(d,x) \mathrm{d}\mathbb{P}(x) \rangle_{\mathcal{H}} \\
&=\langle \hat{\gamma}, \phi(1)\otimes \phi(d)\otimes
\Delta_p \rangle_{\mathcal{H}} +\langle (\hat{\gamma}-\gamma_0), \phi(1)\otimes\phi(d) \otimes \int \phi(x)\otimes \mu_{m}(d,x) \mathrm{d}\mathbb{P}(x) \rangle_{\mathcal{H}} \\
&=\langle (\hat{\gamma}-\gamma_0), \phi(1)\otimes\phi(d)\otimes
\Delta_p \rangle_{\mathcal{H}}
\\
&\quad+\langle \gamma_0, \phi(1)\otimes\phi(d)\otimes
\Delta_p \rangle_{\mathcal{H}} \\
&\quad +\langle (\hat{\gamma}-\gamma_0), \phi(1)\otimes\phi(d) \otimes \int \phi(x)\otimes \mu_{m}(d,x) \mathrm{d}\mathbb{P}(x) \rangle_{\mathcal{H}}.
\end{align*}
Therefore by Lemmas~\ref{theorem:regression},~\ref{theorem:mean}, and~\ref{theorem:conditional} as well as~Proposition~\ref{prop:delta_p}, with probability $1-3\delta$
\begin{align*}
&|\hat{\theta}^{ATE}(d)-\theta^{ATE}_0(d)|\\
&\leq \|\hat{\gamma}-\gamma_0\|_{\mathcal{H}}\|\phi(1)\|_{\mathcal{H}_{\mathcal{S}}}\|\phi(d)\|_{\mathcal{H}_{\mathcal{D}}}
\left\|\Delta_p\right\|_{\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{M}}}
\\
&\quad +
\|\gamma_0\|_{\mathcal{H}}\|\phi(1)\|_{\mathcal{H}_{\mathcal{S}}}\|\phi(d)\|_{\mathcal{H}_{\mathcal{D}}}
\left\|\Delta_p\right\|_{\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{M}}}
\\
&\quad+
\|\hat{\gamma}-\gamma_0\|_{\mathcal{H}}\|\phi(1)\|_{\mathcal{H}_{\mathcal{S}}}\|\phi(d)\|_{\mathcal{H}_{\mathcal{D}}}\times \left\|\int \{\phi(x)\otimes \mu_{m}(d,x) \}\mathrm{d}\mathbb{P}(x)\right\|_{\mathcal{H}_{\mathcal{X}}\otimes\mathcal{H}_{\mathcal{M}}}
\\
&\leq \kappa_s\kappa_d \cdot r_{\gamma}(n,\delta,c) \cdot \{\kappa_x \cdot r^{ATE}_{\mu}(n,\delta,c_4)+r^{ATE}_{\mu}(n,\delta)\}\\
&\quad +\kappa_s\kappa_d\cdot\|\gamma_0\|_{\mathcal{H}} \cdot \{\kappa_x \cdot r^{ATE}_{\mu}(n,\delta,c_4)+r^{ATE}_{\mu}(n,\delta)\}\\
&\quad+\kappa_s\kappa_d\kappa_x\kappa_m \cdot r_{\gamma}(n,\delta,c)
\\
&=O\left(n^{-\frac{1}{2}\frac{c-1}{c+1}}+n^{-\frac{1}{2}\frac{c_4-1}{c_4+1}}\right).
\end{align*}
By the same argument
\begin{align*}
&|\hat{\theta}^{DS}(d)-\theta^{DS}_0(d)|\\
&\leq \kappa_s\kappa_d \cdot r_{\gamma}(n,\delta,c) \cdot \{\kappa_x \cdot r^{DS}_{\nu}(\tilde{n},\delta,c_5)+r^{DS}_{\nu}(\tilde{n},\delta)\}\\
&\quad +\kappa_s\kappa_d\cdot\|\gamma_0\|_{\mathcal{H}} \cdot \{\kappa_x \cdot r^{DS}_{\nu}(\tilde{n},\delta,c_5)+r^{DS}_{\nu}(\tilde{n},\delta)\}\\
&\quad +\kappa_s\kappa_d\kappa_x\kappa_m \cdot r_{\gamma}(n,\delta,c)
\\
&=O\left(n^{-\frac{1}{2}\frac{c-1}{c+1}}+\tilde{n}^{-\frac{1}{2}\frac{c_5-1}{c_5+1}}\right).
\end{align*}
For incremental responses, replace $\phi(d)$ with $\nabla_d \phi(d)$ and hence replace $\|\phi(d)\|_{\mathcal{H}_{\mathcal{D}}}\leq \kappa_d$ with $\|\nabla_d \phi(d)\|_{\mathcal{H}_{\mathcal{D}}}\leq \kappa_d'$.
\end{proof}
\subsubsection{Counterfactual distributions}
\begin{proof}[Proof of Theorem~\ref{theorem:consistency_dist}]
The argument is analogous to Theorems~\ref{theorem:consistency_static} and~\ref{theorem:consistency_planning}, replacing $\|\gamma_0\|_{\mathcal{H}}$ with $\|E_8\|_{\mathcal{L}_2}$ or $\|E_9\|_{\mathcal{L}_2}$ and replacing $r_{\gamma}(n,\delta,c)$ with $r_E(n,\delta,c_8)$ or $r_E(n,\delta,c_9)$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{theorem:conv_dist}]
I generalize \cite[Theorem A.7]{singh2020kernel} to sample selection problems.
Fix $d$. By Theorem~\ref{theorem:consistency_dist}
$$
\|\hat{\theta}^{ATE}(d)-\check{\theta}_0^{ATE}(d)\|_{\mathcal{H}_{\mathcal{Y}}}=O_p\left(n^{-\frac{1}{2}\frac{c_8-1}{c_8+1}}\right).
$$
Denote the samples constructed by Algorithm~\ref{algorithm:herding} by $(\tilde{Y}_j)^m_{j=1}$. Then by \cite[Section 4.2]{bach2012equivalence}
$$
\left\|\hat{\theta}^{ATE}(d)-\frac{1}{m}\sum_{j=1}^m \phi(\tilde{Y}_j)\right\|_{\mathcal{H}_{\mathcal{Y}}}=O(m^{-\frac{1}{2}}).
$$
Therefore by triangle inequality
$$
\left\|\frac{1}{m}\sum_{j=1}^m \phi(\tilde{Y}_j)-\check{\theta}_0^{ATE}(d)\right\|_{\mathcal{H}_{\mathcal{Y}}}=O_p\left(n^{-\frac{1}{2}\frac{c_8-1}{c_8+1}}+m^{-\frac{1}{2}}\right).
$$
The desired result follows from \cite{sriperumbudur2016optimal}, as quoted by \cite[Theorem 1.1]{simon2020metrizing}. The arguments for other counterfactual distributions are identical.
\end{proof}
| 30,125
|
camsdaddy
October 15, 2007, 08:58 AM
I have a 3913 TSW I have seen many positive things about this gun so I am thinking I am the weak link. It seams that I constantly shoot low I try to shoot with a standard 6 o clock hold only because I can see the target. I realize this will make my shots low. I have recently shot a friends Kimber same thing low left. I thought I was flinching but doesnt seem to be the prob (tried empty no jerk). Is there something I can change with my grip maybe that can change this I have filed the front sight down considerablly and still nothing its coming up but I think I am doing something basic that just isnt working out. Any ideas to try? If it were not multi gun issue I would say sight or ammo but it seems to be me.
| 112,143
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India-Inspired Designer Bangles
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Trading Texas Fashion for a front row seat alongside the runway at Perth Fashion Festival in Australia this fall for a sneak peek at Australia's up and coming local designers
Perth Designer Steph Audino hits the ground running after appearing on the Perth Fashion Festival runway now the spotlight is focused on a future fashion house success for Australia
| 29,675
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| 314,959
|
\begin{document}
\title{$\kappa$-Poincar\'e invariant quantum field theories\\
with KMS weight}
\author{Timoth\'e Poulain, Jean-Christophe Wallet}
\institute{
\textit{Laboratoire de Physique Th\'eorique, B\^at.\ 210\\
CNRS and Universit\'e Paris-Sud 11, 91405 Orsay Cedex, France}\\
e-mail: \href{mailto:timothe.poulain@th.u-psud.fr}{\texttt{timothe.poulain@th.u-psud.fr}}, \href{mailto:jean-christophe.wallet@th.u-psud.fr}{\texttt{jean-christophe.wallet@th.u-psud.fr}}\\[1ex]
}
\date{\today}
\maketitle
\begin{abstract}
A natural star product for 4-d $\kappa$-Minkowski space is used to investigate various classes of $\kappa$-Poincar\'e invariant scalar field theories with quartic interactions whose commutative limit coincides with the usual $\phi^4$ theory. $\kappa$-Poincar\'e invariance forces the integral involved in the actions to be a twisted trace, thus defining a KMS weight for the non-commutative (C*-)algebra modeling the $\kappa$-Minkowski space. In all the field theories, the twist generates different planar one-loop contributions to the 2-point function which are at most UV linearly diverging. Some of these theories are free of UV/IR mixing. In the others, UV/IR mixing shows up in non-planar contributions to the 2-point function at exceptional zero external momenta while staying finite at non-zero external momenta. These results are discussed together with the possibility for the KMS weight relative to the quantum space algebra to trigger the appearance of KMS state on the algebra of observables.
\end{abstract}
\vskip 1 true cm
\newpage
\section{Introduction.}
It is widely believed that the classical notion of space-time is no longer adequate at the Planck scale to reconcile gravity with quantum mechanics. One possible attempt to reach this goal comprises to trade the continuous smooth manifold describing the space-time by a non-commutative (quantum) space \cite{Doplich1}. In this spirit, the $\kappa$-Minkowski space-time appears in the physics literature to be one of the most studied non-commutative spaces with Lie algebra type non-commutativity and is sometimes regarded as a good candidate for a quantum space-time to be involved in a description of quantum gravity at least in some limit. Informally, it may be viewed as the enveloping algebra of the Lie algebra $[x_0,x_i]=i\kappa^{-1} x_i,\ [x_i,x_j]=0,\ i,j=1,\cdots, d$, where the deformation parameter $\kappa$ has dimension of a mass. The $\kappa$-Minkowski space-time has been characterized a long time ago in \cite{majid-ruegg} by exhibiting the Hopf algebra bicrossproduct structure of the $\kappa$-Poincar\'e quantum algebra \cite{luk1} which (co-)acts covariantly on it and may be viewed as describing its quantum symmetries. A considerable amount of literature has been devoted to the exploration of algebraic aspects related to $\kappa$-Minkowski space and $\kappa$-Poincar\'e algebra, in particular dealing with concepts inherited from quantum groups \cite{leningrad} as well as (twists) deformations. For a comprehensive recent review, on these algebraic developments, see e.g \cite{luk2} and the references therein. Besides, the possibility to have testable/observable consequences from related phenomenological models has raised a growing interest and resulted in many works dealing for instance with Doubly Special Relativity together with modified dispersion relations and relative locality \cite{ame-ca1,reloc}.\bigskip
Once the non-commutative nature of the space-time is assumed, Non-Commutative Field Theories (NCFT) arise naturally. For reviews on early studies, see e.g \cite{dnsw-rev} and references therein. Compared to the ordinary field theories, NCFT have their own salient features. In particular, many efforts have been focused on the exploration of their quantum behavior in order to obtain a good understanding of their renormalisation properties. The renormalisation of NCFT is known to be often a difficult task since most of these theories are non-local, thus precluding the use of the standard machinery controlling the ordinary local field theories. The technical hard points may even be complicated by the possible appearance of the UV/IR mixing, a typical phenomenon of NCFT which spoils renormalisability. For the popular Moyal spaces $\mathbb{R}^4_\theta$ and $\mathbb{R}^2_\theta$ as well as for $\mathbb{R}^3_\lambda$, a deformation of $\mathbb{R}^3$ \cite{lagraa}, it has been shown that this phenomenon and all the technical difficulties can be overcome from different ways within some NCFT as well as some non-commutative gauge models leading to renormalisable (or even finite in some instance) field theories on these quantum spaces \cite{Grosse:2003aj-pc,brol-1,vign-sym,thes-vt,wal-16,poulwal-1,vitwal2,vitwal1} and outlining the deep relationship between NCFT and matrix models \cite{Wallet:2007c,matrix1,matrix2}. Recall that the 4-d Moyal space can be viewed {\it{informally}} as $\mathbb{C}[x_\mu]/\mathcal{R}$, the quotient of the free algebra generated by 4 hermitean coordinates $(x_\mu)_{\mu=1,...,4}$ by the relation $\mathcal{R}$ defined by $[x_\mu,x_\nu]=i\theta_{\mu\nu}$ where $\theta_{\mu\nu}$ is a skew symmetric constant tensor. This deformation of $\mathbb{R}^4$ can be described as a (suitable) algebra of functions on $\mathbb{R}^4$ equipped with the popular Moyal product \cite{groen,Moyal} obtained from the Wigner-Weyl quantization scheme. For various presentations of the Moyal product, see e.g \cite{dnsw-rev}. The non-commutative $\mathbb{R}^3_\lambda$, another example of space with Lie algebra type non-commutativity, as the $\kappa$-Minkowski space-time also is, can be viewed {\it{informally}} as related to the universal enveloping algebra of $\mathfrak{su}(2)$, $U(\mathfrak{su}(2))\simeq\mathbb{C}[x_i]/\mathcal{R}'$, where the relation $\mathcal{R}'$ is defined by $[x_i,x_j]=i\varepsilon_{ijk}x_k$. For various derivations of star products related to $\mathbb{R}^3_\lambda$, see e.g \cite{lagraa,poulwal-1} and references therein. Note that these non-commutative spaces share a common underlying structure, each one being related to a group algebra. This latter corresponds, in the Moyal case, to the algebra for the Heisenberg group, which actually underlies the Weyl quantization, as it will be recalled below. For the space $\mathbb{R}^3_\lambda$, it is the convolution algebra of $SU(2)$, which has been shown to play an essential role in originating the special properties of $\mathbb{R}^3_\lambda$ \cite{vitwal1,vitwal2,wal-16}. In the case of $\kappa$-Minkowski space-time, the relevant group algebra is the convolution algebra of the affine group as it will be shown below.\bigskip
An important question to address is the fate of the symmetries of a non-commutative space-time. This has triggered a lot of works using various approaches which basically depend if one insists on preserving (almost all) the classical symmetries or if one considers deformed ones. For instance in \cite{Doplich1} the attention was focused on preserving the classical (undeformed) Lorentz or Poincar\'e symmetries for the Moyal space, as well as in \cite{dabrow} for $\kappa$-Minkowski space. In this latter work, the authors ensures classical covariance of $\kappa$-Minkowski space starting from a generalised version of it introduced in \cite{luk3}, i.e. $[x_\mu,x_\nu]=i\kappa^{-1}(v_\mu x_\nu - v_\nu x_\mu)$. They show that, under some assumptions, deformed (quantum) symmetries are not the only viable and consistent solution for treating such models. Note however that the original $\kappa$-Minkowski space \eqref{kappa-alg-d} (which we consider in this paper) does not fit in that description and breaks the classical relativity principle. This leads us to the other approach widely studied in the literature, namely the extension of the usual notion of Lie algebra symmetries to the one of (deformed) Hopf algebra symmetries aiming to encode the new (canonical) symmetries for the quantum space-times. This point of view is motivated by the fact that, in the commutative case, the Minkowski space-time can be regarded as the homogeneous space the Poincar\'e symmetry group acts on transitively. Hence, a deformation of the former should (in principle) implies a deformation of the latter and vice versa. This idea underlies the original derivation of $\kappa$-Minkowski as the homogeneous space associated to $\kappa$-Poincar\'e \cite{majid-ruegg}. Another interesting exemple (to put in perspective with \cite{Doplich1}) is given in \cite{chaichian}, where it is shown that the symmetries for the Moyal space can be obtained through formal (Drinfeld) twist deformation of the Lorentz sector of the Poincar\'e algebra while translation remains undeformed. General discussions on the fate of the Poincar\'e symmetries within the context of non-commutative (or quantum) space-times can be found in \cite{GAC2002:2} and references therein.\bigskip
NCFT on $\kappa$-Minkowski space have received a lot of interest from a long time, see for instance \cite{ital-1,habsb-imp,kappa-star1,hrvat-1}, but amazingly their quantum properties are not so widely explored, compared to the present status of the above mentioned NCFT. Nevertheless, the UV/IR mixing within some scalar field theories on $\kappa$-Minkowski has been examined a long time ago in \cite{gross-whl} and found to possibly occur. The corresponding analysis was based on a star product for the $\kappa$-deformation derived in \cite{star-spunz} from a general relationship between the Kontsevich formula and the Baker-Campbell-Hausdorff (BCH) formula that can be conveniently used when the non-commutativity is of Lie algebra type \cite{ypa}. NCFT considered in \cite{gross-whl} was $\kappa$-Poincar\'e invariant, which is a physically reasonable requirement, keeping in mind the important role played by the Poincar\'e invariance in ordinary field theories together with the fact that $\kappa$-Poincar\'e algebra can be viewed as describing the quantum symmetries of the $\kappa$-Minkowski space-time. \bigskip
It turns out that a very convenient star product for $\kappa$-Minkowski space can be obtained from a mere adaptation of the initial Wigner-Weyl quantization scheme which gives rise to the popular Moyal product. This can be illustrated schematically as follows. Recall that one important feature of this scheme is the notion of ``twisted convolution" of two functions{\footnote{with $f,g\in L^1(\mathbb{R}^2)$.}} $f$ and $g$ on the phase space $\mathbb{R}^2$, that we denote by $f\bullet g$, whose explicit expression was first given by von Neumann \cite{jvn}. This product is defined by $W(f\bullet g)=W(f)W(g)$ where $W(f)$ is the Weyl operator given by $W(f)=\int d\xi_1d\xi_2\ e^{i(\xi_1P+\xi_2Q)}f(\xi_1,\xi_2)$ in which the unitary operator in the integrand can be viewed as an element of the unimodular Heisenberg group{\footnote{To see that, use e.g the Glauber formula to reproduce the usual composition law for elements of the Heisenberg group.}}, obtained by exponentiating the Heisenberg algebra, says $[P,Q]=i\theta$ where $\theta$ is central. From this follows directly the Moyal product defining the deformation of $\mathbb{R}^2$. It is defined by $f\star g=\mathcal{F}^{-1}(\mathcal{F}f\bullet\mathcal{F}g)$ where the Weyl quantization map is $Q(f)=W(\mathcal{F}f)$ and $\mathcal{F}f$ is the Fourier transform of $f$.\bigskip
The natural extension of the above scheme to the construction of a star product for $\kappa$-Minkowski can then be achieved by simply replacing the Heisenberg group by the non-unimodular affine group as explained below, while $W(f)$ will be replaced by a representation of the convolution algebra of the affine group. Doing this, one can take advantage of the machinery of the harmonic analysis on Lie groups and, in particular, measures involved in action functionals are provided by Haar measures. Note that such a viewpoint has also been intensively used in \cite{vitwal2,wal-16} for $\mathbb{R}^3_\lambda$, the relevant group being $SU(2)$ reflecting the $\mathfrak{su}(2)$ non-commutativity of the quantum space and has provided the relationship between $\mathbb{R}^3_\lambda$ and the convolution algebra of $SU(2)$, the determination of the natural measure in the action functionals and by the way clarified the origin of the matrix basis used in \cite{vitwal1}. In the case of $\kappa$-Minkowski space, we note that such a natural construction has already been used in \cite{DS,matas} to derive a star product for a 2-dimensional $\kappa$-Minkowski space and to characterize a related multiplier algebra \cite{DS}. As far as we know, this product was amazingly not further exploited in the study of NCFT on $\kappa$-Minkowski space, despite its relatively simple expression and the associated tools of group harmonic analysis which make him well adapted to the study of quantum field theories. \bigskip
The construction of this natural star product defining the $\kappa$-deformation of the 4-d Minkowski space, considered with Euclidean signature in the present work, is presented in the section \ref{section2}. We then study in the section \ref{section3} different classes of $\kappa$-Poincar\'e invariant (complex) scalar field theories on the 4-d $\kappa$-Minkowski space whose commutative limit coincides with the usual $\phi^4$ theory. The kinetic operators are chosen to be square of Dirac operators. Requiring $\kappa$-Poincar\'e invariance forces the (Lebesgue) integral involved in the actions to be a twisted trace with respect to the star product. This therefore defines a Kubo-Martin-Schwinger (KMS) weight on the non-commutative (C*-)algebra modeling the $\kappa$-Minkowski space. The associated modular group and Tomita modular operator are characterized. This is presented in the subsection \ref{section30} where we also discuss the possibility for the above KMS weight together with the associated modular data related on the non-commutative algebra modeling the $\kappa$-Minkowski space to generate the appearance of KMS states on the algebra of observables related to a global (observer-independent) time. The mathematical material as well as technical computations are collected in the appendix \ref{apendixB}.\\
The one-loop contributions to the 2-point functions of each of these theories are computed and their UV and IR behaviors are analyzed. The corresponding material is given in the subsections \ref{section31} and \ref{subsection32}. We find that the twist automorphism related to the twisted trace splits the planar contributions to the 2-point function into different IR finite contributions whose UV behavior is controlled by the twist. These contributions are found to be at most UV linearly diverging, some being UV finite. A part of scalar theories considered in this work cannot give rise to non-planar contributions to the 2-point function so that these theories are expected to be free of UV/IR mixing. Conversely, UV/IR mixing shows up in another class of theories for which we find that the non-planar contributions to the 2-point function, while finite at non zero external momenta, becomes singular at exceptional zero external momenta with polynomial singularity. These results are finally discussed in the section \ref{section4}.
\section{$\kappa$-Minkowski space as a group algebra.}\label{section2}
\subsection{Convolution algebras and $\kappa$-Minkowski spaces.}\label{section2-1}
A convenient presentation of the $\kappa$-Minkowski space can be achieved by exploiting standard objects of the framework of group algebras and (C*-)dynamical systems \cite{dana}. This approach, which has been used in \cite{DS,matas} is the one we mainly follow in this paper. This framework has also been used in recent studies on $\mathbb{R}^3_\lambda$ spaces \cite{vitwal2,wal-16} related to the convolution algebra of the compact $SU(2)$ Lie group. Here, the relevant group is (related to) the affine group of the real line in the 2-dimensional case, i.e. a semi direct product of the two abelian groups $\mathbb{R}$, which extends in the $(d+1)$-dimensional case to $\mathbb{R}\ltimes_{\phi}\mathbb{R}^d$. We now collect the suitable material for the ensuing analysis.\bigskip
First, recall that the $\kappa$-deformation of the Minkowski space can be {\it{informally}} viewed as related to the universal enveloping algebra of the Lie algebra $\mathfrak{g}$ defined by:
\begin{equation}
[x_0,x_i]=\frac{i}{\kappa}x_i,\ \ [x_i,x_j]=0,\ \ i,j=1,\cdots, d.\label{kappa-alg-d}
\end{equation}
Here, $\kappa$ is a real number ($\kappa>0$) and the coordinates $x_0,\ x_i$ are assumed to be self-adjoint operators acting on some suitable Hilbert space. It turns out that $\mathfrak{g}$ is solvable. This can be easily deduced from the so-called derived Lie algebra $[\mathfrak{g},\mathfrak{g}]$ which is readily seen to be nilpotent. This is equivalent to have solvable $\mathfrak{g}$. Hence the associated Lie group, hereafter denoted by $\mathcal{G}_{d+1}$, is solvable (see e.g Theorem 5.9 of \cite{ypa-2}). We use this property below to characterize the relevant algebra modelling the non-commutative space. \\
Notice that any Lie group of the form $A\ltimes_{\phi}B$, $\phi:A\to \mathrm{Aut}(B)$, where $A$ and $B$ are Abelian connected Lie groups, is solvable and connected (and is simply connected whenever $A$ and $B$ are simply connected). This is the case for $\mathcal{G}_{d+1}=\mathbb{R}\ltimes_{\phi}\mathbb{R}^d$, relevant to describe the $(d+1)$-dimensional $\kappa$-Minkowski spaces. This group is not unimodular signaling the existence of distinct left and right-invariant Haar measures, denoted respectively by $d\mu$ and $d\nu$. They are related by the modular function of $\mathcal{G}_d$, a continuous group homomorphism $\Delta_{\mathcal{G}_{d+1}}:\mathcal{G}_{d+1}\to\mathbb{R}^+_{/0}$, by $d\nu(s)=\Delta_{\mathcal{G}_{d+1}}(s^{-1})d\mu(s)$ for any $s\in\mathcal{G}_{d+1}$.\bigskip
For the moment, we assume $d=1$, the extension to $d=3$ is straightforward and will be exploited below. $\mathcal{G}_2$ is known to be the orientation-preserving affine group of the real line, i.e. the ``$(ax+b)$-group", $a>0$, widely studied in the mathematical literature. For basic mathematical
details, see e.g \cite{dana,khalil} and references therein. For our present purpose, this (non-abelian simply connected) Lie group can be conveniently characterized by defining
\begin{equation}
W(p^0,p^1):=e^{ip^1x_1}e^{ip^0x_0}\label{group-elem},
\end{equation}
where $p^0,\ p^1\in\mathbb{R}$ can be interpreted as momenta. The group elements \eqref{group-elem} are related to the more traditional exponential form of the Lie algebra \eqref{kappa-alg-d} through a mere redefinition of $p^1$. Indeed, by using in \eqref{group-elem} the simplified BCH formula $e^Xe^Y=e^{\lambda(u)X+Y}$, valid whenever $[X,Y]=uX$, see \cite{vanbrunt}, where $\lambda(u)=\frac{ue^u}{e^u-1}$, one obtains $W(p^0,p^1)=e^{i(p^0x_0+\lambda(\frac{p^0}{\kappa})p^1x_1)}$. However, \eqref{group-elem} is easier to manipulate for the ensuing computations. Now, upon using $e^Xe^Y=e^{Y}e^{e^uX}$ which holds true when again $[X,Y]=uX$, one obtains from \eqref{group-elem} the group product on $\mathcal{G}_2$ given by
\begin{equation}
W(p^0,p^1)W(q^0,q^1)=W(p^0+q^0,p^1+e^{-p^0/\kappa}q^1).\label{grouplaw}
\end{equation}
The unit element and inverse are respectively given by
\begin{equation}
\bbone_{G}=W(0,0),\ W^{-1}(p^0,p^1)=W(-p^0,-e^{p^0/\kappa}p^1)\label{group-unit-invers}.
\end{equation}
At this point, some remarks are in order.
\begin{itemize}
\item {First, observe that the usual composition law for the $(ax+b)$-group can be obtained from \eqref{grouplaw} by representing the group elements \eqref{group-elem} as
\begin{equation}
W(p^0,b)=\begin{pmatrix}e^{-p^0/\kappa}&b\\0&1\end{pmatrix}
\end{equation}
and setting $a:=e^{-p^0/\kappa}$. That latter rewriting exhibits clearly the semi-direct product structure of $\mathcal{G}_2$ as
\begin{equation}
\mathcal{G}_2=\mathbb{R}^+_{/0}\ltimes_{\check{\phi}}\mathbb{R},\label{semi-dir}
\end{equation}
with $\check{\phi}:\mathbb{R}^+_{/0}\to\mathrm{Aut}(\mathbb{R})$ being given by the adjoint action of $\mathbb{R}^+_{/0}$ on $\mathbb{R}$. Indeed, the identifications $a\mapsto (a,0)$ and $b\mapsto (1,b)$ yield respectively the factors $\mathbb{R}^+_{/0}$ and $\mathbb{R}$ appearing in \eqref{semi-dir} while the action $\check{\phi}$, defined by $\check{\phi}(a)b=(a,0)(1,b)(a^{-1},0)$, is reflected at the level of \eqref{grouplaw} in
\begin{equation}
\phi:\mathbb{R}\to \mathrm{Aut}(\mathbb{R}),\qquad \phi(p^0)q=e^{-p^0/\kappa}q . \label{autom}
\end{equation}
}
\item {Next, note that the energy-momentum composition law is essentially given by the BCH formula for the Lie group underlying the non-commutative space-times whose algebras of coordinates are of Lie algebra type. This is the case for $\kappa$-Minkowski, see eqn. \eqref{kappa-alg-d}, as well as for the Moyal plane (resp. $\mathbb{R}^3_\lambda$) whose algebra of coordinate operators is given by the Heisenberg algebra $[x_\mu,x_\nu]=i\theta$ (resp. $\mathfrak{su}(2)$ algebra $[x_\mu,x_\nu]=i\lambda\varepsilon_{\mu\nu}^{\hspace{9pt}\rho} x_\rho$). Here, the composition law can be directly read from \eqref{grouplaw} and reflects the non trivial coproduct structure of the $\kappa$-Poincar\'e algebra, see \eqref{hopf1}.
}
\end{itemize}
Let $\pi_U:\mathcal{G}_2\to\mathcal{B}(\mathcal{H})$ denote a (strongly continuous) unitary representation of $\mathcal{G}_2$ where $\mathcal{H}$ is some suitable Hilbert space and $\mathcal{B}(\mathcal{H})$ is the (C*-)algebra of bounded operators on $\mathcal{H}$. A star product defining the 2-dimensional $\kappa$-Minkowski space can be obtained in a way similar to the usual Weyl quantization leading to the construction of Moyal product on the Moyal plane $\mathbb{R}^2_\theta$, see \cite{hennings}, the Heisenberg algebra and Heisenberg group being replaced now by \eqref{kappa-alg-d} and $\mathcal{G}_2$ \eqref{semi-dir} respectively. Accordingly, it is convenient to start from $L^1(\mathcal{G}_2)$, the convolution algebra of $\mathcal{G}_2$. Recall that it is a $^*$-algebra made of the set of integrable complex-valued functions on $\mathcal{G}_2$ with respect to some Haar measure equipped with the related convolution product{\footnote{Recall that $L^1(\mathcal{G}_2)$ is isomorphic to the completion w.r.t. the norm $||f||_1=\int_{\mathcal{G}_2}d\nu(s)f(s)$ of the algebra of compactly supported complex-valued functions on $\mathcal{G}_2$.}}. From now on, it will be assumed to be the right-invariant measure. Accordingly, the convolution product is defined by $(f\circ g)(t)=\int_{\mathcal{G}_2}d\nu(s)\ f(ts^{-1})g(s)$ for any $t\in\mathcal{G}_2$, $f, g\in{L^1(\mathcal{G}_2)}$. The involutive structure of the algebra can be ensured by any element of the one-parameter family of involutions defined $\forall t\in\mathcal{G}_2$ by $f^*(t):=\bar{f}(t^{-1})\Delta_{\mathcal{G}_2}^\alpha(t)$, $\alpha\in\mathbb{R}$. It turns out that the choice $\alpha=1$, assumed from now on, ensures that any representation of the convolution algebra defined for any $f\in{L^1(\mathcal{G}_2)}$ by
\begin{equation}
\pi:L^1(\mathcal{G}_2)\to \mathcal{B}(\mathcal{H}),\quad \pi(f)=\int_{\mathcal{G}_2}d\nu(s)f(s)\pi_U(s),\label{unireps}
\end{equation}
is a non-degenerate $^*$-representation. Indeed, a simple computation yields
\begin{equation}
\langle u,\pi(f)^\dag v\rangle=\langle \pi(f)u,v\rangle=\int_{\mathcal{G}_2}d\nu(s)\bar{f}(s)\langle u,\pi_U(s^{-1})v \rangle,\label{equai}
\end{equation}
where anti-linearity of the Hilbert product $\langle\cdot,\cdot\rangle$ and unitary property of $\pi_U$ have been used. Note that in \eqref{equai} the symbol $^\dag$ denotes the adjoint operation acting on operators, the nature of the various involutions should be obvious from the context. On the other hand, one computes
\begin{equation}
\langle u,\pi(f^*) v\rangle=\int_{\mathcal{G}_2}d\nu(s)\Delta_{\mathcal{G}_2}^\alpha(s)\bar{f}(s^{-1})\langle u,\pi_U(s)v\rangle,\label{equaii}
\end{equation}
which combined with the relation $d\nu(s^{-1})=\Delta_{\mathcal{G}_2}(s)d\nu(s)$ is equal to \eqref{equai} provided $\alpha=1$.\bigskip
To summarize:
\begin{equation}
\pi(f)^\dag=\pi(f^*),
\end{equation}
and one can easily check that
\begin{equation}
\pi(f\circ g)=\pi(f)\pi(g), \label{alg-morph-conv}
\end{equation}
for any $f, g\in{L^1(\mathcal{G}_2)}$.
\subsection{Quantization map and star product.}\label{2-2}
Let $\mathcal{F}f(p^0,p^1):=\int_{\mathbb{R}^2}dx_0dx_1e^{-i(p^0x_0+p^1x_1)}f(x_0,x_1)$ be the Fourier transform of $f\in L^1(\mathbb{R}^2)$. In the following, $\mathcal{S}_c$ denotes the space of Schwartz functions on $\mathbb{R}^2$ with compact support in the first variable.\bigskip
The quantization map is defined \cite{DS,matas} upon identifying functions on $\mathcal{G}_2$ with functions on $\mathbb{R}^2$ in view of \eqref{group-elem}-\eqref{group-unit-invers}. Namely, for any $f\in L^1(\mathbb{R}^2)\cap\mathcal{F}^{-1}(L^1(\mathbb{R}^2))$, we define
\begin{equation}
Q(f):=\pi(\mathcal{F}f),\label{quantizmap}
\end{equation}
where $\pi$ is the representation given by \eqref{unireps}. Notice that in view of \eqref{group-elem}, functions involved in the convolution product and involution map defined above are interpreted as Fourier transforms of functions of space-time coordinates. Hence, the occurrence of $\mathcal{F}f$ in the RHS of \eqref{quantizmap}. Then, since $Q$ must be a morphism of algebra, one writes
\begin{equation}
Q(f\star g)=Q(f)Q(g)=\pi(\mathcal{F}f)\pi(\mathcal{F}g)=\pi(\mathcal{F}f\circ\mathcal{F}g)\label{intermed1}
\end{equation}
where \eqref{alg-morph-conv} has been used to obtain the last equality in \eqref{intermed1}, which compared with $Q(f\star g)=\pi(\mathcal{F}(f\star g))$ stemming from \eqref{quantizmap} and using the non-degeneracy of \eqref{unireps}, yields
\begin{equation}
f\star g=\mathcal{F}^{-1}(\mathcal{F}f\circ\mathcal{F}g)\label{star prod-gene},
\end{equation}
where $\mathcal{F}^{-1}$ is the inverse Fourier transform on $\mathbb{R}^2$. In the same way, the requirement for $Q$ to be a $^*$-morphism yields
\begin{equation}
f^\dag=\mathcal{F}^{-1}(\mathcal{F}(f)^*)\label{involgen}.
\end{equation}
Note that both the star product and the involution are representation independent despite the fact that the quantization map $Q$ depends on $\pi$.
Finally, by using the fact that the right-invariant measure on $\mathcal{G}_2$ is $d\nu(p^0,p^1)=dp^0dp^1$, i.e. the Lebesgue measure, with the modular function given by
\begin{equation}
\Delta_{\mathcal{G}_2}(p^0,p^1)=e^{p^0/\kappa},\label{modul-2d}
\end{equation}
and combining the definition of the right-convolution product given above with eqns. \eqref{group-elem}-\eqref{group-unit-invers}, a simple calculation yields, for any $f,g\in\mathcal{F}(\mathcal{S}_c)$,
\begin{equation}
(f\star g)(x_0,x_1)=\int \frac{dp^0}{2\pi} dy_0\ e^{-iy_0p^0}f(x_0+y_0,x_1)g(x_0,e^{-p^0/\kappa}x_1) \label{starpro},
\end{equation}
with $f\star g\in\mathcal{F}(\mathcal{S}_c)$,
and
\begin{equation}
f^\dag(x_0,x_1)= \int \frac{dp^0}{2\pi} dy_0\ e^{-iy_0p^0}{\bar{f}}(x_0+y_0,e^{-p^0/\kappa}x_1),\ \ f^\dag\in\mathcal{F}(\mathcal{S}_c)\label{invol},
\end{equation}
which coincide with the star product and involution of \cite{DS,matas}. \bigskip
\noindent At this point, some comments are in order.
\begin{itemize}
\item First, it is instructive to get more insight on $C^*(\mathcal{G}_2)$, the C$^*$-algebra which models the $\kappa$-Minkowski space. Indeed, the completion of $L^1(\mathcal{G}_2)$ with respect to the norm related to the left regular representation on $L^2(\mathcal{G}_2)$ yields the reduced group C$^*$-algebra, $C^*_{red}(\mathcal{G}_2)$. Furthermore, since $\mathcal{G}_2$ is amenable as any solvable (locally compact) group, one has $C^*_{red}(\mathcal{G}_2)\simeq C^*(\mathcal{G}_2)$, involving as dense $^*$-subalgebra the set of Schwartz functions with compact support equipped with the above convolution product.
\item Eqns. \eqref{starpro} and \eqref{invol} can be extended \cite{DS} to (a subalgebra of) the multiplier algebra{\footnote{It involves the smooth functions on $\mathbb{R}^2$ satisfying standard polynomial bounds together with all the derivatives, with Fourier transform having compact support in the first variable.}} of $\mathcal{F}(\mathcal{S}_c)$ involving in particular $x_0$ and $x_1$ and the unit function. From \eqref{starpro} and \eqref{invol}, one easily obtains
\begin{equation}
x_0\star x_1=x_0x_1+\frac{i}{\kappa}x_1,\ x_1\star x_0=x_0x_1,\ x_\mu^\dag=x_\mu,\ \mu=1,2\label{def-relations},
\end{equation}
consistent with the defining
relation \eqref{kappa-alg-d} (for $d=1$).
\end{itemize}
The extension of the above construction to the 4-dimensional case is straightforward. Indeed, the group law becomes now $W(p^0,\vec{p})W(q^0,\vec{q})=W(p^0+q^0,\vec{p}+e^{-p^0/\kappa}\vec{q})$ with $W(p^0,\vec{p}):=e^{ip^ix_i}e^{ip^0x_0}$, $\vec{p}=(p^i,\ i=1,2,3)$ and $W^{-1}(p^0,\vec{p})=W(-p^0,-e^{p^0/\kappa}\vec{p})$. This entails the semi-direct product structure $\mathcal{G}_4=\mathbb{R}\ltimes_{\phi}\mathbb{R}^3$ where $\phi$ is still given by \eqref{autom}. Then, the construction leading to \eqref{starpro} and \eqref{invol} can be thoroughly reproduced, replacing $\mathbb{R}^2$ by $\mathbb{R}^4$ and \eqref{modul-2d} by
\begin{equation}
\Delta_{\mathcal{G}_4}(p^0,\vec{p})=e^{3p^0/\kappa}\label{modul-4d}.
\end{equation}
Setting for short $x:=(x_0,\vec{x})$, one obtains
\begin{align}
(f\star g)(x)&=\int \frac{dp^0}{2\pi} dy_0\ e^{-iy_0p^0}f(x_0+y_0,\vec{x})g(x_0,e^{-p^0/\kappa}\vec{x}) \label{starpro-4d},\\
f^\dag(x)&= \int \frac{dp^0}{2\pi} dy_0\ e^{-iy_0p^0}{\bar{f}}(x_0+y_0,e^{-p^0/\kappa}\vec{x})\label{invol-4d},
\end{align}
for any functions $f,g\in\mathcal{F}(\mathcal{S}_c)$ and one still has $f\star g\in\mathcal{F}(\mathcal{S}_c)$ and $f^\dag\in\mathcal{F}(\mathcal{S}_c)$. Here, $\mathcal{S}_c$ is now the set of Schwartz functions of $\mathbb{R}^4$ with compact support in the $p^0$ variable. Of course, comments similar to the one given above for $C^*(\mathcal{G}_2)$ and \eqref{def-relations} apply to the 4-dimensional case on which we focus in the rest of this paper. Notice that the functions in $\mathcal{F}(\mathcal{S}_c)$ are by construction analytic in the variable $x_0$, being Fourier transforms of functions with compact support in the variable $p^0$, thanks to the Paley-Wiener theorem.\bigskip
For the ensuing discussion, it will be sufficient to consider the algebra $\mathcal{F}(\mathcal{S}_c)$ unless otherwise stated, which will be denoted hereafter by $\mathcal{M}_\kappa$.
\subsection{$\kappa$-Poincar\'e invariant actions.}\label{section22}
In this section, we discuss general properties shared by $\kappa$-Poincar\'e invariant action functionals for complex-valued scalar fields, denoted generically by $S_\kappa(\phi)$.\\
Let $\mathcal{P}_\kappa$ denote the $\kappa$-Poincar\'e algebra. We will demand that the action functional $S_\kappa(\phi)$ obeys the following two conditions:
\begin{enumerate}
\item $S_\kappa(\phi)$ is $\mathcal{P}_\kappa$-invariant which is expressed as
\begin{equation}
h\triangleright S_\kappa(\phi)=\epsilon(h)S_\kappa(\phi),\label{invarquant}
\end{equation}
for any $h$ in the Hopf algebra $\mathcal{P}_\kappa$ where $\epsilon$ is the co-unit of $\mathcal{P}_\kappa$ (see appendix \ref{apendixA}),
\item $S_\kappa(\phi)$ reduces to the standard $\phi^4$ scalar field theory in the commutative limit $\kappa\to\infty$.
\end{enumerate}
Recall that $\mathcal{P}_\kappa$ has a natural action on $\mathcal{M}_\kappa$ which informally may be viewed as the action of a quantum symmetry on the corresponding quantum (non-commutative) space modelled by $\mathcal{M}_\kappa$, reflecting the fact that the algebra $\mathcal{M}_\kappa$ is a left-module over the Hopf algebra $\mathcal{P}_\kappa$. A convenient presentation of $\mathcal{P}_\kappa$ can be obtained from the 11 elements $(P_i, N_i,M_i, \mathcal{E},\mathcal{E}^{-1})$, $i=1,2,3$, which are respectively the momenta, the boost and the rotations together with the invertible element
\begin{equation}
\mathcal{E}:=e^{-P_0/\kappa},\label{Erond}
\end{equation}
to be discussed at length in a while. The relations between these elements which characterize the Hopf algebra structure together with the duality between the Hopf subalgebra describing the ``deformed translation algebra" and the $\kappa$-Minkowski space are presented in the appendix \ref{apendixA} for the sake of completeness.\bigskip
Going back to the condition a), it is known that the invariance of $S_\kappa(\phi)$ under $\mathcal{P}_\kappa$ is automatically achieved by considering action functionals of the form
\begin{equation}
S_\kappa(\phi)=\int d^4x\ \mathcal{L}(\phi),\label{weight}
\end{equation}
where $\phi\in\mathcal{F}(\mathcal{S}_c)$ so that $\mathcal{L}(\phi)\in\mathcal{F}(\mathcal{S}_c)$ in view of \eqref{starpro}. Indeed, by using \eqref{left-module0}-\eqref{left-modules1bis}, one has plainly
\begin{equation}
P_\mu\triangleright S_\kappa(\phi):=\int d^4x\ P_\mu \triangleright \mathcal{L}(\phi)=0,\ \ M_i\triangleright S_\kappa(\phi):=\int d^4x\ M_i \triangleright \mathcal{L}(\phi)=0,
\end{equation}
while
\begin{equation}
\mathcal{E}\triangleright S_\kappa(\phi):=\int d^4x\ \mathcal{E}\triangleright \mathcal{L}(\phi)=S_\kappa(\phi)
\end{equation}
where the last equality stems from the Cauchy theorem. Next, one obtains from \eqref{left-module2}
\begin{equation}
N\triangleright S_\kappa(\phi):=\int d^4x\ ([\frac{\kappa}{2}L_{x_i}(\mathcal{E}-\mathcal{E}^{-1})+L_{x_0}P_i\mathcal{E}+L_{x_i}\vec{P}^2\mathcal{E}])\triangleright \mathcal{L}(\phi).
\end{equation}
By using \eqref{left-module0}, \eqref{left-module1}, one easily checks that the last two terms in the right hand side vanish as integrals of total derivative of Schwartz functions while the 2 contributions of the first term balance each other thanks to the Cauchy theorem.\bigskip
For further use, we quote useful formulas
\begin{align}
\int d^4x\ (f\star g^\dag)(x)&=\int d^4x\ f(x){\bar{g}}(x),\label{algeb-1}\\
\int d^4x\ f^\dag(x)&=\int d^4x\ {\bar{f}}(x),\label{int-form1}
\end{align}
stemming from mere changes of variables and the use of the Cauchy theorem as it can be easily verified. We note that a mere consequence of \eqref{algeb-1} is
\begin{equation}
\int d^4x\ f\star f^\dag\ge0,\ \ \int d^4x\ f^\dag\star f\ge0,\label{positivity}
\end{equation}
thus defining two positive maps $\int d^4x:\mathcal{M}_{\kappa+}\to \mathbb{R}^+$ where $\mathcal{M}_{\kappa+}$ denotes the set of positive elements of $\mathcal{M}_\kappa$.\bigskip
It turns out that the Lebesgue integral does not define a trace. Indeed, a simple computation yields
\begin{equation}
\int d^4x\ f\star g=\int d^4x\ (\sigma\triangleright g)\star f\label{twistrace},
\end{equation}
where we define for further convenience
\begin{equation}
\sigma\triangleright f:=\mathcal{E}^3\triangleright f=e^{-\frac{3P_0}{\kappa}}\triangleright f\label{twistoperator},
\end{equation}
in which $\mathcal{E}$ is given by \eqref{Erond}. Hence, the Lebesgue integral cannot define a trace since cyclicity is lost in view of \eqref{twistrace}. Instead, it defines a twisted trace. \bigskip
Recall that a twisted trace (on an algebra) is defined in the mathematical literature as a linear positive map $\textrm{Tr}$ satisfying $\textrm{Tr}(a\star b)=\textrm{Tr}((\sigma\triangleright b)\star a)$, where $\sigma$ is an automorphism of the algebra called the twist. This is verified by the Lebesgue integral in view of \eqref{positivity}, \eqref{twistrace} where the corresponding twist is explicitly given by \eqref{twistoperator}, which will be discussed in the subsection \ref{section30}. This loss of cyclicity has often been considered as a troublesome feature of $\kappa$-Poincar\'e invariant field theories, this having probably discouraged the pursue of many studies of their properties at the quantum level.\bigskip
However, whenever there is a twisted trace, there is a related KMS condition (up to additional technical requirements that will not be essential for the ensuing discussion), a fact that is known in the mathematical literature. The relevant technical material needed for the discussion is presented in the appendix \ref{apendixB}. In the subsection \ref{section30}, we discuss some possible consequences of this KMS condition on field theories on $\kappa$-Minkowski space. In the subsections \ref{section31} and \ref{subsection32}, we construct a family of scalar field theories on 4-d $\kappa$-Minkowski space and study the UV and IR behaviour of the corresponding 2-point functions at one-loop order.
\section{Scalar field theories on 4-d $\kappa$-Minkowski space.}\label{section3}
\subsection{Trading cyclicity for KMS condition.}\label{section30}
To see that the Lebesgue integral actually defines a twisted trace, one key observation is to notice that \eqref{twistrace} and \eqref{twistoperator} can be interpreted as a KMS weight on $\mathcal{M}_\kappa$ for the group of $^*$-automorphisms of $\mathcal{M}_\kappa$
defined by
\begin{equation}
\sigma_t(f):=e^{it\frac{3P_0}{\kappa}}\triangleright f=e^{\frac{3t}{\kappa}\partial_0}\triangleright f, \label{sigmat-modul}
\end{equation}
for any $t\in\mathbb{R}$ and $f\in\mathcal{M}_\kappa$. This group of automorphisms is called the modular group for the KMS weight{\footnote{Roughtly speaking, a weight differs only from a state by an overall normalisation.}}. The corresponding mathematical details, technical computations and related references are collected in the appendix \ref{apendixB}. \\
The modular group, whose \eqref{sigmat-modul} is an example, is at the corner stone of the Modular Theory of Tomita-Takesaki, an essential tool in the area of von Neumann algebras. For details, see e.g \cite{takesaki} and references therein. It turns out that one of the initial motivations of Tomita to construct the Modular Theory was related to the harmonic analysis of (locally compact) non-unimodular group, as the one underlying the present study. In particular for these groups, the word ``modular'' refers to the modular function of the group, here \eqref{modul-4d}, while the Tomita modular operator is simply the multiplication by the modular function \eqref{modul-4d}. Recall that Modular Theory, KMS condition and twisted trace are rigidly linked. Hence, it is not surprising that these structures underlie the present framework since the requirement of $\kappa$-Poincar\'e invariance of the action functional fixes the trace involved in it to be twisted. \bigskip
To summarize the analysis of appendix \ref{apendixB}, the KMS weight $\varphi$ is simply given by the map
\begin{equation}
\varphi(f):=\int d^4x\ f(x),\label{varphi}
\end{equation}
for any $f\in\mathcal{M}_\kappa$ which verifies
\begin{equation}
\varphi(\sigma_t f)=\varphi(f),\ \ \varphi\big((e^{i\frac{3}{2\kappa}\partial_0}\triangleright f)\star(e^{-i\frac{3}{2\kappa}\partial_0}\triangleright f^\dag)\big)=\varphi(f^\dag\star f).\label{defining-properties}
\end{equation}
These 2 properties are actually defining properties for a KMS weight. Note that the $\kappa$-Poincar\'e invariance is crucial to insure that \eqref{defining-properties} holds true. It follows obviously that any action functional for a $\kappa$-Poincar\'e invariant theory is related to a KMS weight. Hence, the requirement of $\kappa$-Poincar\'e invariance trades the cyclicity of the trace for a KMS condition.\bigskip
Now, from a general theorem (Theorem [6.36] of 1st of ref. \cite{kuster}), one concludes that $\varphi$ must obey a KMS condition. Indeed, one defines{\footnote{ Note that $f_{a,b}(t)$ is continuous and bounded owing to the properties of the star product \eqref{starpro-4d}. As already mentioned at the end of Section \ref{2-2}, analyticity of $f_{a,b}$ stems from the Paley-Wiener theorem.}}
\begin{equation}
f_{a,b}(t):=\int d^4x\ \sigma_t(a)\star b,\label{eq0}
\end{equation}
for any $a,b\in\mathcal{M}_\kappa$. Then, by using the algebraic properties of the twist and $\sigma_t$, one computes
\begin{equation}
f_{a,b}(t)=\int d^4x\ \sigma_t(a)\star b =\int d^4x\ \sigma_i(b)\star \sigma_t(a)
=\int d^4x\ \sigma_i(b\star \sigma_{t-i}(a))=\int d^4x\ b\star \sigma_{t-i}(a).\label{eq1}
\end{equation}
in which we used \eqref{b10}. From this follows that
\begin{equation}
f_{a,b}(t+i)=\int d^4x\ b\star \sigma_{t}(a),\label{eq2}
\end{equation}
which verifies the above mentioned theorem (see appendix \ref{apendixB}). \bigskip
As pointed out in the appendix \ref{apendixB}, \eqref{eq0} and \eqref{eq2} represent an abstract version of the KMS condition introduced a long time ago as a tool to characterize equilibrium temperature states of quantum systems in field theory and statistical physics. To see that, set formally $f_{a,b}(t)=\langle \sigma_t(a)\star b\rangle$; then, \eqref{eq1} implies $\langle \sigma_t(a)\star b\rangle=\langle b\star\sigma_{t-i}(a)\rangle$ which bears some formal similarity with the usual form of the KMS condition for the quantum systems. Notice that $\sigma_t$ actually represents a ``time-translation operator" related to the Tomita operator $\Delta_T=e^{3P_0/\kappa}$ via $\sigma_t=(\Delta_T)^{it}$, as shown at the end of the appendix \ref{apendixB}. \bigskip
However, in the case of quantum systems or quantum field theory, $f_{A,B}(t)$ corresponds to a correlation function $\langle \Sigma_t(A)B\rangle_{\Omega}$ computed for some (thermal) vacuum $\Omega$ where $A$ and $B$ are now function(al)s (operators) of the fields and $\Sigma_t$ is the (Heisenberg) evolution operator, hence elements pertaining to the algebra of observables of the theory. But whenever a KMS condition holds true on the {\it{algebra of observables of a quantum system or a quantum field theory}}, the flow generated by the modular group, i.e. the Tomita flow, may be used to define a global (observer-independent) time which can be interpreted as the ``physical time". This reflects the deep correspondence between KMS condition and dynamics. This observation underlies the interesting proposal about the thermal origin of time introduced in \cite{ConRove}.\bigskip
While it would be tempting to interpret $\sigma_t$ \eqref{sigmat-modul} as defining (or generating) a ``physical time" for the present system, akin to the thermal time mentioned above, no conclusion can yet be drawn. In fact, eqn. \eqref{KMS-abst} linked to the modular group and its associated KMS condition \eqref{eq1}, \eqref{eq2} only holds at the level of $\mathcal{M}_\kappa$, the algebra modelling the $\kappa$-Minkowski space. To show that a natural global time can be defined requires to determine if \eqref{eq1}, \eqref{eq2} force a KMS condition to hold true at the level of the algebra of observables. This could be achieved by actually showing the existence of some KMS state(s) on this latter algebra built from the path integral machinery. In view of the possibility to associate to $\kappa$-Poincar\'e invariant non-commutative field theories a natural global time, a physically appealing property, the implications of the KMS condition \eqref{eq1}, \eqref{eq2} shared by all these theories obviously deserves further study. The full analysis is beyond the scope of the present paper. \bigskip
We now pass to the construction of reasonable $\kappa$-Poincar\'e invariant action functionals and the study of the UV and IR property of their one-loop 2-point functions, adopting the standard viewpoint of the non-commutative field theories, namely representing the non-commutative action functional as an action functional describing non-local commutative field theories. It turns out that the use of the star product introduced in the section \ref{section2} simplifies the computations of the correlation functions. This will be exemplified by explicit computations of 2-point functions in the subsection \ref{subsection32}. We first introduce the main elements of our framework and analyse carefully the corresponding properties.
\subsection{Construction of real action functionals.}\label{section31}
\subsubsection{Preliminary considerations.}\label{section311}
It is convenient to begin by introducing the following Hilbert product on $\mathcal{M}_\kappa$
\begin{equation}
\langle f,g\rangle:=\int d^4x\left(f^\dag\star g\right)(x)=\int d^4x\ {\bar{f}}(x)(\sigma\triangleright g)(x),\ \forall f,g\in\mathcal{M}_\kappa. \label{hilbert-product}
\end{equation}
To check that \eqref{hilbert-product} defines actually a Hilbert product, one observes that positivity is apparent from \eqref{positivity} while ${\overline{\langle f,g \rangle}}=\langle g,f \rangle $ stems from \eqref{int-form1} applied to $f^\dag\star g=(g^\dag\star f)^\dag$. Furthermore, the corresponding Hilbert space $\mathcal{H}$ can be shown to be (unitarily) isomorphic to $L^2(\mathbb{R}^4)$, i.e. $\mathcal{H}\simeq L^2(\mathbb{R}^4)$. The proof is given in the appendix \ref{apendixC}.\\
One can verify that $P_i$, $i=1,2,3$, and $\mathcal{E}$ are self-adjoint w.r.t. the Hilbert product \eqref {hilbert-product}. Indeed, one computes
\begin{align}
\langle f,P_i^\dag\triangleright g\rangle&=\langle P_i\triangleright f,g \rangle=-\int d^4x\ (\mathcal{E}^{-1}P_i\triangleright(f^\dag))\star g=-\int d^4x\ (P_i\triangleright(f^\dag))\star (\mathcal{E}\triangleright g)\nonumber\\
&=\int d^4x\ (\mathcal{E}\triangleright(f^\dag))\star (P_i\mathcal{E}\triangleright g)=\int d^4x\ f^\dag\star (P_i\triangleright g)\nonumber\\
&=\langle f,P_i\triangleright g\rangle,
\end{align}
where we have successively used \eqref{dag-hopfoperat}, the $\kappa$-Poincar\'e invariance \eqref{invarquant}, \eqref{deriv-twist1} and \eqref{relation-calE}. Hence $P_i$ is self-adjoint. Self-adjointness of $P_0$ and $\mathcal{E}$ can be shown similarly.\bigskip
In order to construct real action functionals, notice that \eqref{hilbert-product} is $\mathbb{R}$-valued for any $f, g\in\mathcal{F}(\mathcal{S}_c)$ satisfying $\langle f,g\rangle=\langle g,f\rangle $. Hence, reality condition for kinetic term of the form $\langle f,K_\kappa f\rangle$ is automatically verified providing that the kinetic operator $K_\kappa$ (assumed in the following to have dense domain in $\mathcal{H}$) is self-adjoint since this implies $\langle f,K_\kappa f\rangle=\langle K_\kappa f,f\rangle $.\\
We further assume the kinetic operator $K_\kappa$ to be a pseudo-differential operator, i.e.
\begin{equation}
(K_\kappa f)(x)=\int \frac{d^4p}{(2\pi)^4} d^4y\ \mathcal{K}_\kappa(p)f(y)e^{ip(x-y)},\label{pseudoper}
\end{equation}
for any $f$ in the domain of $K_\kappa$, where $\mathcal{K}_\kappa(p)$ is some rational fraction of $(p^0,\vec{p})$. Note that self-adjointness for $K_\kappa$ requires
\begin{equation}
{\overline{\mathcal{K}_\kappa}}(p^0,\vec{p})=\mathcal{K}_\kappa(p^0,\vec{p})\label{constr-K},
\end{equation}
a condition that will be fulfilled in the situations we will consider below. Indeed, a simple computation yields
\begin{equation}
\langle f,K_\kappa f\rangle=\int d^4xd^4y\frac{d^4p}{(2\pi)^4}\ \bar{f}(x)f(y)e^{ip(x-y)}e^{-3p^0/\kappa}\mathcal{K}_\kappa(p^0,\vec{p}),\label{algeb2}
\end{equation}
while
\begin{equation}
\langle K_\kappa f,f\rangle=\int d^4xd^4y\frac{d^4p}{(2\pi)^4}\ \bar{f}(x)f(y)e^{ip(x-y)}e^{-3p^0/\kappa}{\overline{\mathcal{K}_\kappa}}(p^0,\vec{p}),\label{algeb3}
\end{equation}
proving the above statement.\bigskip
We are now in position to construct $\kappa$-Poincar\'e invariant action functionals $S_\kappa(\phi^\dag,\phi)$ such that
\begin{equation}\label{action-limit}
\lim_{\kappa\to\infty}S_\kappa(\phi^\dag,\phi)=\int d^4x \left(\bar{\phi}(-\partial_\mu\partial^\mu+m^2)\phi+\lambda\bar{\phi}\phi\bar{\phi}\phi\right)(x),\ \lambda\in\mathbb{R},
\end{equation}
i.e. fulfilling the condition b) introduced in the section \ref{section22}. We assume the following usual generic form for the action functionals
\begin{equation}\label{star-act}
S_\kappa(\phi^\dag,\phi)=S_\kappa^\text{kin}(\phi^\dag,\phi)+S_\kappa^\text{int}(\phi^\dag,\phi),
\end{equation}
where $S_\kappa^\text{int}(\phi^\dag,\phi)$ is a quartic polynomial in the fields and $S_\kappa^\text{kin}(\phi^\dag,\phi)$ is the kinetic term. For the theories under consideration, the mass dimension for the fields and parameters are respectively $[\phi]=[\phi^\dag]=1$, $[\lambda]=0$ and $[m]=1$.
\subsubsection{Derivation of the kinetic term.}
Let us first discuss the kinetic term $S_\kappa^\text{kin}(\phi^\dag,\phi)$.\\According to the above discussion, admissible real kinetic terms are of the form
\begin{equation}
\langle \phi, K_\kappa\phi \rangle,\ \ \langle \phi^\dag,K_\kappa\phi^\dag \rangle,\label{realkineticterm}
\end{equation}
where $K_\kappa$ is self-adjoint. Its explicit expression will be given in a while. We also incorporate all possible ``mass terms" of similar forms, namely $m^2\langle \phi,\phi \rangle$ and $m^2\langle\phi^\dag,\phi^\dag \rangle$.\bigskip
A first natural choice for the kinetic operator is provided by the first Casimir of the $\kappa$-Poincar\'e algebra $\mathcal{P}_\kappa$. This latter is given in the Majid-Ruegg basis by
\begin{equation}
\mathcal{C}_\kappa(P_\mu)=4\kappa^2\sinh^2\left(\frac{P_0}{2\kappa}\right) + e^{P_0/\kappa} \vec{P}^2 \label{Casimir},
\end{equation}
or equivalently
\begin{equation}\label{Casimir2}
\mathcal{C}_\kappa(P_\mu)= e^{P_0/\kappa}\left(\kappa^2\left(1-e^{-P_0/\kappa}\right)^2+\vec{P}^2\right).
\end{equation}
The Casimir operator \eqref{Casimir} can be put into the form
\begin{equation}
\mathcal{C}_\kappa(P_\mu)=D_0^2+D_iD^i,\label{casimir-carre}
\end{equation}
with
\begin{equation}
D_0:=\kappa \mathcal{E}^{-1/2}(1-\mathcal{E}),\ \ D_i:= \mathcal{E}^{-1/2}P_i,\ i=1,2,3\label{dirac},
\end{equation}
where $D_0$ and $D_i$ define self-adjoint operators. To see that, first observe that one has $\int d^4x\ D_\mu f=0$ for any $f\in\mathcal{M}_\kappa$ and use \eqref{pairing-involution}, \eqref{invarquant} and \eqref{deriv-twist1} to compute for instance
\begin{align}
\langle D_if,g \rangle&=\int d^4x\ (D_if)^\dag\star g=-\int d^4x\ (\mathcal{E}^{-1/2}P_i\triangleright f^\dag)\star g\nonumber\\
&=-\int d^4x\ (P_i\triangleright f^\dag)\star
(\mathcal{E}^{1/2}\triangleright g)=\int d^4x\ f^\dag\star
(\mathcal{E}^{-1/2}P_i\triangleright g)\nonumber\\
&=\langle f,D_ig \rangle,
\end{align}
for any $f,g\in\mathcal{M}_\kappa$. The computation for $D_0$ is similar. Note by the way that $D_0$ and $D_i$ are not derivations of the algebra $\mathcal{M}_\kappa$. \bigskip
A second possible natural choice is given by the square of the equivariant Dirac operator involved in the construction of an equivariant spectral triple for the $\kappa$-Minkowski space \cite{frans-2}. It is given by
\begin{equation}
K^{eq}_\kappa(P_\mu):=\mathcal{C}_\kappa(P_\mu)+\frac{1}{4\kappa^2}\mathcal{C}_\kappa(P_\mu)^2.\label{kinetic}
\end{equation}
For latter convenience, we quote here a useful factorization of the kinetic operator \eqref{kinetic} supplemented by a mass term, assuming $0\leq m \leq \kappa$,
\begin{equation}
K^{eq}_\kappa(P_\mu)+m^2 = \frac{e^{2P_0/\kappa}}{4\kappa^2}\left(\vec{P}^{\hspace{2pt}2}+\kappa^2\mu^2_{+}\right)\left(\vec{P}^{\hspace{2pt}2}+\kappa^2\mu^2_{-}\right),\label{factor-propa}
\end{equation}
where the two positive functions $\mu^2_{+}$ and $\mu^2_{-}$ are given by
\begin{equation}
\mu^2_{\pm}(m,P_0):=\left[1 \pm 2e^{-P_0/\kappa} \sqrt{1-\left(\frac{m}{\kappa}\right)^2} + e^{-2P_0/\kappa}\right].
\end{equation}
Again, one can write
\begin{equation}
K^{eq}_\kappa(P_\mu)=D^{eq}_0D^{eq}_0+\sum_i D^{eq}_iD^{eq}_i,\label{carre-equiv}
\end{equation}
where
\begin{equation}
D^{eq}_0:=\frac{\mathcal{E}^{-1}}{2}\left(\kappa(1-\mathcal{E}^2)-\frac{1}{\kappa}\vec{P}^2\right) \ \ ,\ D^{eq}_i:=\mathcal{E}^{-1}P_i,\label{equivdirac}
\end{equation}
which can be easily verified to be self-adjoint using successively eqn. \eqref{pairing-involution}, the $\kappa$-Poincar\'e invariance \eqref{invarquant} and the twisted Leibnitz rules for the $P_\mu$ \eqref{deriv-twist1}, \eqref{deriv-twist2}.\bigskip
Now, recall that one has for any of the operators \eqref{dirac}, \eqref{equivdirac}, the following usefull formula
\begin{equation}
\langle \mathcal{D}_\mu f,g\rangle=\langle f,\mathcal{D}_\mu g \rangle,\label{byparts}
\end{equation}
for any $f,g\in\mathcal{M}_\kappa$, in which $\mathcal{D}_\mu$ denotes any of the operators \eqref{dirac}, \eqref{equivdirac}, stemming from the self-adjointness of these operators. From \eqref{realkineticterm}, \eqref{casimir-carre}, \eqref{carre-equiv} and \eqref{byparts}, a suitable form for the kinetic term is then given by
\begin{align}
S^\text{kin}_\kappa(\phi^\dag,\phi)&=\langle \phi, (K_\kappa+m^2)\phi \rangle+\langle \phi^\dag,(K_\kappa+m^2)\phi^\dag \rangle\nonumber\\
&=\int d^4x\ \phi^\dag\star(K_\kappa+m^2)\phi+\int d^4x\ \phi\star(K_\kappa+m^2)\phi^\dag\nonumber\\
&=\int d^4x\ \phi^\dag\star(1+\sigma^{-1})(K_\kappa+m^2)\phi,\label{kinetic-term}
\end{align}
where $K_\kappa=\mathcal{D}_\mu\mathcal{D}^\mu$ is any of the 2nd order operators \eqref{Casimir}, \eqref{kinetic}. Note by the way that, ignoring the mass terms, one has
\begin{equation}
\langle \phi, K_\kappa\phi \rangle=\langle \mathcal{D}_\mu\phi, \mathcal{D}_\mu\phi \rangle,
\end{equation}
and similarly for $\phi\to\phi^\dag$.\bigskip
It is important to realise that the analysis of the quantum behaviour of the NCFT under consideration can be conveniently carried out within the present framework by expressing the non-commutative action functional $S_\kappa(\phi^\dag,\phi)$ (involving star products) as a non-local ordinary quantum field theory $S_\kappa(\bar{\phi},\phi)$ (involving pointwise products). This can be achieved by making use of the integral forms for the star product \eqref{starpro-4d} and the involution \eqref{invol-4d} in the expression for the action functional \eqref{star-act}, \eqref{kinetic-term}, \eqref{interac-type1} and \eqref{interac-type2}.\\Applying this procedure to $S^\text{kin}_\kappa$ leads to a great simplification in the computation of the propagator, despite the fact that the star product \eqref{starpro-4d} is not closed w.r.t. the Lebesgue integral, i.e. $\int d^4x\ (f\star g)(x)\ne\int d^4x\ f(x)g(x)$. Indeed, further using \eqref{pseudoper}, we obtain
\begin{align}
S^\text{kin}_\kappa(\bar{\phi},\phi) &= \int d^4x_1d^4x_2 \ \bar{\phi}(x_1)\phi(x_2) \mathcal{K}_\kappa(x_1-x_2),\label{pratik-kinet-act}\\
\text{with}\ \ \mathcal{K}_\kappa(x_1-x_2) &= \int \frac{d^4p}{(2\pi)^4} \left(1+e^{-3p^0/\kappa}\right) \left(\mathcal{K}_\kappa(p)+m^2\right) e^{ip\cdot(x_1-x_2)}.\label{pratik-kinet}
\end{align}
The corresponding propagator can be derived by solving $\int d^4yd^4z\ \mathcal{K}_\kappa(x-y)P_\kappa(y-z)f(z)=
\int d^4z\ \delta(x-z)f(z)$ for any suitable test function $f(z)$, which amounts to invert $\mathcal{K}_\kappa(x-y)$.\\
Finally, assuming $K_\kappa=K^{eq}_\kappa$, eqn. \eqref{kinetic}, one obtains
\begin{equation}
P_\kappa(x_1-x_2) = \int \frac{d^4p}{(2\pi)^4} \ \frac{e^{ip\cdot(x_1-x_2)}}{\left(1+e^{-3p^0/\kappa}\right) \left(\mathcal{K}^{eq}_\kappa(p)+m^2\right)},\label{intermed2}
\end{equation}
which, combining \eqref{intermed2} with \eqref{factor-propa}, yields
\begin{equation} \label{propagator}
P_{\kappa}(x_1-x_2)=\int \frac{d^4p}{(2\pi)^4} \ \frac{e^{-2p^0/\kappa}}{1+e^{-3p^0/\kappa}} \ \frac{(2\kappa)^2e^{ip\cdot(x_1-x_2)}}{\left(\vec{p}^{\hspace{2pt}2}+\kappa^2\mu^2_{+}\right)\left(\vec{p}^{\hspace{2pt}2}+\kappa^2\mu^2_{-}\right)}.
\end{equation}
While assuming $K_\kappa=\mathcal{C}_\kappa$, eqn. \eqref{Casimir}, leads in a similar manner to
\begin{align}
P_{\kappa}(x_1-x_2)&=\int \frac{d^4p}{(2\pi)^4} \ \frac{e^{-p^0/\kappa}}{1+e^{-3p^0/\kappa}}\ \frac{e^{ip\cdot(x_1-x_2)}}{\vec{p}^{\hspace{2pt}2}+\kappa^2 \mu^2},\\
\mu^2(m,p^0)&=\left(m/\kappa\right)^2 e^{-p^0/\kappa}+\left(1-e^{-p^0/\kappa}\right)^2.
\end{align}
\subsubsection{Derivation of the interaction term.}
Let us now discuss the interaction part $S^\text{int}_\kappa(\phi^\dag,\phi)$.\\
In view of \eqref{hilbert-product}, the requirement for the action functional to be real forces to use the natural involution \eqref{invol-4d} in the construction of $S_\kappa$\footnote{Notice that $S_\kappa$ describes {\it{a priori}} the dynamics of a complex-valued field (obvious from \eqref{invol-4d}) unless one imposes the additional constraint $\bar{\phi}=\phi$, which therefore would give rise to a NCFT for a real-valued field.}. Recall that this involution is rigidly linked to the construction of the C*-algebra modeling the $\kappa$-Minkowski space (see section \ref{section2-1}). Hence, according to the discussion given section \ref{section311} and eqn. \eqref{positivity}, admissible (positive) real quartic (star) polynomial interactions are of the form $\langle f,f\rangle$, with $f\in\mathcal{M}_\kappa$ a polynomial in the fields $\phi$ and $\phi^\dag$. They are given by
\begin{align}
\langle\phi^\dag\star\phi,\phi^\dag\star\phi\rangle,\ \langle\phi^\dag\star\phi^\dag,\phi^\dag\star\phi^\dag\rangle,\ \langle\phi\star\phi^\dag,\phi\star\phi^\dag\rangle,\ \langle\phi\star\phi,\phi\star\phi\rangle,
\end{align}
leading respectively to the following real interactions terms, $\lambda\in\mathbb{R}$,
\begin{align}
S_{1;\kappa}^\text{int}&=\lambda\int d^4x\left(\phi^\dag\star\phi\star\phi^\dag\star\phi\right)(x),\ S_{2;\kappa}^\text{int}=\lambda\int d^4x\left(\phi\star\phi\star\phi^\dag\star\phi^\dag\right)(x),\label{interac-type1}\\
S_{3;\kappa}^\text{int}&=\lambda\int d^4x\left(\phi\star\phi^\dag\star\phi\star\phi^\dag\right)(x),\ S_{4;\kappa}^\text{int}=\lambda\int d^4x\left(\phi^\dag\star\phi^\dag\star\phi\star\phi\right)(x).\label{interac-type2}
\end{align}
The existence of these four different families of interactions reflects the non-commutativity of the star product, as well as the non-cyclicity of the integral, involved in $S_\kappa$, although they all admit the same commutative limit $\lambda\vert\phi\vert^4$, eqn. \eqref{action-limit}. In fact, the second set of interactions \eqref{interac-type2} differs from the first one \eqref{interac-type1} by some power of the twist factor $\sigma$, eqns. \eqref{twistrace}, \eqref{twistoperator}, as it can be easily realised upon using \eqref{twistrace} in \eqref{interac-type2}\footnote{Straightforward application of the twisted trace property of the integral \eqref{twistrace} in \eqref{interac-type2} yields
\begin{equation}\label{interac-type2bis}
S_{3;\kappa}^\text{int}=\lambda\int d^4x\left((\sigma\triangleright\phi^\dag)\star\phi\star\phi^\dag\star\phi\right)(x),\ \ S_{4;\kappa}^\text{int}=\lambda\int d^4x\left((\sigma\triangleright(\phi\star\phi))\star\phi^\dag\star\phi^\dag\right)(x).
\end{equation}
\label{foot-twist}}. The actual non-equivalence of the four interactions becomes more apparent after using the integral expressions for the star product \eqref{starpro-4d} and involution \eqref{invol-4d} in \eqref{interac-type1} and \eqref{interac-type2}, leading to the expressions for the corresponding vertex-functions, eqns. \eqref{vertex-p1}-\eqref{vertex-p4}. Anticipating the results of the section \ref{subsection32}, it will be shown that each of these theories leads to different quantum (one-loop) corrections to the 2-point functions.\\
Notice that $S_{1;\kappa}^\text{int}$ and $S_{3;\kappa}^\text{int}$ (resp. $S_{2;\kappa}^\text{int}$ and $S_{4;\kappa}^\text{int}$) may be viewed as so-called orientable (resp. non-orientable) interaction, according to the standard liturgy of NCFT, each type leading to its own quantum behavior for the corresponding NCFT. For more technical details on the diagrammatic, see e.g \cite{vign-sym,thes-vt} for NCFT on Moyal space and \cite{poulwal-1} for the $\mathbb{R}^3_\theta$ case and references therein.\\
Notice also that these four interactions obviously reduce to a single one whenever $\phi$ satisfies $\phi^\dag=\phi$. The resulting interaction actually coincides with the quartic interaction considered in \cite{GAC2002} only when the field $\phi$ satisfies the additional constraint $\bar{\phi}=\phi$, i.e $\phi$ is real-valued. This can be explicitly verified by standard computation from eqns. \eqref{vertex-p1}-\eqref{vertex-p4} given below. Recall that in \cite{GAC2002}, a nice use is made of path integral quantization methods to investigate some properties of real-valued scalar NCFT with quartic interaction on $\kappa$-Minkowski space, in particular the non-linear conservation law characterizing the interaction.\\
As we did for the kinetic term, it is convenient to express $S_{I;\kappa}^\text{int}$, $I=1,2,3,4$, as (commutative) non-local interaction terms involving $\phi$ and $\bar{\phi}$. This is achieved by successively using \eqref{algeb-1} and \eqref{starpro-4d}, \eqref{invol-4d} in \eqref{interac-type1} and \eqref{interac-type2}. Standard computations yield
\begin{equation}
S_{I;\kappa}^\text{int}=(2\pi)^4\lambda\int \left[\prod_{i=1}^4d^4x_i\right] \ \bar{\phi}(x_1) \phi(x_2) \bar{\phi}(x_3) \phi(x_4) \mathcal{V}_{I;\kappa}(x_1,x_2,x_3,x_4), \label{inter-real-gene}
\end{equation}
where the vertex function takes the form
\begin{equation}
\mathcal{V}_{I;\kappa}(x_1,x_2,x_3,x_4) = \int \left[\prod_{i=1}^4\frac{d^4p_i}{(2\pi)^4}\right] \ \widetilde{\mathcal{V}}_{I;\kappa}(p_1,p_2,p_3,p_4)e^{i(p_1\cdot x_1 - p_2\cdot x_2 + p_3\cdot x_3 - p_4\cdot x_4)}.\label{real-xvertex-gene}
\end{equation}
The explicit expressions for the vertex functions characterising the above interactions, \eqref{interac-type1} and \eqref{interac-type2}, are given by
\begin{align}
\widetilde{\mathcal{V}}_{1;\kappa}(\lbrace p_i\rbrace)& = \delta\left(p_2^0-p_1^0+p_4^0-p_3^0\right)\delta^{(3)}\left(\left(\vec{p}_2-\vec{p}_1\hspace{2pt}\right)e^{p_1^0/\kappa}+\left(\vec{p}_4-\vec{p}_3\hspace{2pt}\right)e^{p_4^0/\kappa}\right),\label{vertex-p1}\\
\widetilde{\mathcal{V}}_{2;\kappa}(\lbrace p_i\rbrace) &= \delta\left(p_2^0-p_1^0+p_4^0-p_3^0\right)\delta^{(3)}\left(\vec{p}_2-\vec{p}_1+\vec{p}_4e^{-p_2^0/\kappa}-\vec{p}_3e^{-p_1^0/\kappa}\right)\label{vertex-p2},\\
\widetilde{\mathcal{V}}_{3;\kappa}(\lbrace p_i\rbrace) &=\delta\left(p_2^0-p_1^0+p_4^0-p_3^0\right)\delta^{(3)}\left(\vec{p}_2-\vec{p}_3+\vec{p}_4e^{(p_4^0-p_3^0)/\kappa}-\vec{p}_1e^{(p_1^0-p_2^0)/\kappa}\right), \label{vertex-p3}\\
\widetilde{\mathcal{V}}_{4;\kappa}(\lbrace p_i\rbrace) &= \delta\left(p_2^0-p_1^0+p_4^0-p_3^0\right) \delta^{(3)}\left((\vec{p}_2+\vec{p}_4e^{-p_4^0/\kappa})e^{-p_2^0/\kappa}-(\vec{p}_1+\vec{p}_3e^{-p_3^0/\kappa})e^{-p_1^0/\kappa}\right)\label{vertex-p4}.
\end{align}
Equations \eqref{vertex-p1}-\eqref{vertex-p4} exhibit the energy-momentum conservation laws for each of those theories. As expected, the conservation law for the energy (time-like momenta) sector is the standard one while the 3-momentum conservation law becomes non linear. This stems from the semi-direct product structure underlying the non-commutative C*-algebra modelling $\kappa$-Minkowski and reflects the (Hopf algebraic) structure of the $\kappa$-Poincar\'e algebra underlying its (quantum) symmetries. Note this is sometimes geometrically interpreted (for instance in the context of relative locality, see e.g. \cite{reloc}) as reflecting the existence of a curvature of the energy-momentum space at very high (i.e. of order $\kappa$) energy.\bigskip
Finally, in view of the above discussions and the explicit expressions for the $\widetilde{\mathcal{V}}_{I;\kappa}$'s characterising the various models, one easily convinces ourself that it is not possible to reduce the four (tree level) vertex functions \eqref{vertex-p1}-\eqref{vertex-p4} into one unique vertex (involving a unique delta function). This is obvious when considering two theories of different nature (i.e. either orientable or non-orientable). On the other hand, $S_{1;\kappa}$ and $S_{3;\kappa}$ (resp. $S_{2;\kappa}$ and $S_{4;\kappa}$) differ from each other by some power of the twist factor (see e.g. footnote \ref{foot-twist}). Hence, there is a priori no reason for the different interactions to describe equivalent theories. Computations of first order corrections to the 2-point functions (reported section \ref{subsection32}) will show that the different models studied have indeed very different quantum behaviours, the twist playing an important role in their actual UV behaviour. In particular, UV/IR mixing shows up for non-orientable theories albeit absent in the orientable models.
\subsection{One-loop 2-point functions.}\label{subsection32}
In this section, we present the computation of the one-loop 2-point functions for each of the two field theories characterized respectively by the interaction terms $S_{1;\kappa}^\text{int}$ and $S_{2;\kappa}^\text{int}$, \eqref{interac-type1}, both with kinetic term corresponding to \eqref{kinetic}. We have verified that the field theories corresponding to interaction terms $S_{3;\kappa}^\text{int}$ or $S_{4;\kappa}^\text{int}$, \eqref{interac-type2}, exhibit a similar behaviour regarding the structure of the contributions received by the 2-point functions and their respective UV and IR behaviours. Their analysis can be obtained from straightforward adaptations of the material presented below. Anticipating the results, we find that the 2-point function for each of the theories \eqref{interac-type1} receives 4 types of contributions, hereafter denoted by Type-I, Type-II, Type-III and Type-IV. Type-I contributions can be interpreted as standard planar contributions while Type-II and Type-III contributions can be viewed as planar contributions stemming from the fact that the Lebesgue integral involved in the action is a twisted trace. The Type-IV contributions can be viewed as non-planar contributions which exhibit UV/IR mixing. Changing the kinetic term \eqref{kinetic} to \eqref{Casimir} does not modify noticeably the conclusions on the UV and IR behaviour of the field theories.
This will be discussed in the Section \ref{section4}.
\subsubsection{Preliminary considerations.}
To deal with the perturbative expansion, we follow the usual route used in (most of) the studies of NCFT, which we briefly recall now. Namely, first by making use of \eqref{starpro-4d} and \eqref{invol-4d}, the action functional $S_\kappa(\phi^\dag,\phi)$ involving star products is represented as an ordinary, albeit non local, action functional $S_\kappa(\bar{\phi},\phi)$ depending on $\phi$, $\bar{\phi}$ and the ordinary (commutative) product among functions, hence describing the dynamics of a complex scalar field. Accordingly, the perturbative expansion related to the NCFT is nothing but a usual perturbative expansion for an ordinary (complex scalar) field theory, stemming from the generating functional of the connected correlation functions
\begin{equation}
W_{I}[\bar{J},J]:=\ln\left(\mathcal{Z}_{I}[\bar{J},J]\right)
\end{equation}
with
\begin{equation}
\mathcal{Z}_{I}[\bar{J},J]:=\int d\bar{\phi}d\phi \ e^{-S^\text{kin}_\kappa(\bar{\phi},\phi)-S^\text{int}_{I;\kappa}(\bar{\phi},\phi)+\int d^4x \ \bar{J}(x)\phi(x) + \int d^4x \ J(x)\bar{\phi}(x)}, \ I=1,2,
\end{equation}
in which the functional measure is merely the ordinary functional measure for a scalar field theory $S_\kappa(\bar{\phi},\phi)$ implementing formally the integration over the field configurations $\phi$ and $\bar{\phi}$. Accordingly, correlation functions built from $\phi$ and $\bar{\phi}$ are then generated by the repeated action of standard functional derivatives with respect to $J$ and $\bar{J}$ satisfying the usual functional rule
\begin{equation}
\frac{\delta J(p)}{\delta J(q)}=\delta^{(4)}(p-q).
\end{equation}
Note that there is no need to introduce a notion of non-commutative (star) functional derivative in the present approach. \bigskip
Let us recall, for the sake of completeness, the main steps of the derivation of the contributions to the one-loop 2-point functions. This can be achieved by first rewriting the interaction term $S^\text{int}_{I;\kappa}$ replacing the fields $\phi$ and $\bar{\phi}$ by the functional derivatives w.r.t. their corresponding sources $\bar{J}$ and $J$ respectively, then computing the Gaussian integral for the free field theory. This leads to
\begin{align}
&W_{0}[\bar{J},J]:=\int \frac{d^4p}{(2\pi)^4} \ \overline{\mathcal{F}J}(p)P_\kappa(p)\mathcal{F}J(p),\\
W_{I}[\bar{J},J]=\ln N&+W_{0}[\bar{J},J]+\ln\left(1+e^{-W_{0}}\left(e^{-S_{I;\kappa}^\text{int}[\frac{\delta}{\delta \mathcal{F}J},\frac{\delta}{\delta\overline{\mathcal{F}J}}]}-1\right)e^{W_{0}}\right),\label{expansion1}
\end{align}
with $N$ some normalisation constant, $P_\kappa(p)$ the Fourier transform of \eqref{propagator} and where we have switched from position to momentum representation for computational convenience.\\Now expanding the last logarithm in \eqref{expansion1} up to the first order in the coupling constant $\lambda$ and defining the effective action $\Gamma$ as the Legendre transform of $W_{I}$,
\begin{equation}
\Gamma[\bar{\phi},\phi]:=\int \frac{d^4p}{(2\pi)^4} \left(\overline{\mathcal{F}J}(p)\mathcal{F}\phi(p)+\mathcal{F}J(p)\overline{\mathcal{F}\phi}(p)\right)-W_{I}[\bar{J},J],
\end{equation}
one finds, after standard computation, the following expression for the one-loop quadratic part of $\Gamma$
\begin{equation}
\Gamma^{(2)}_1[\bar{\phi},\phi]:=\int\frac{d^4p_3}{(2\pi)^4}\frac{d^4p_4}{(2\pi)^4}\ \overline{\mathcal{F}\phi}(p_3)\mathcal{F}\phi(p_4) \Gamma^{(2)}_1(p_3,p_4),
\end{equation}
with
\begin{align}
\Gamma^{(2)}_1(p_3,p_4):=\lambda\int &\frac{d^4p_1}{(2\pi)^4}\frac{d^4p_2}{(2\pi)^4}\ P_\kappa(p_1) \delta^{(4)}(p_2-p_1)\times\nonumber\\
&\times\Big[\widetilde{\mathcal{V}}_{I;\kappa}(p_1,p_2,p_3,p_4)+\widetilde{\mathcal{V}}_{I;\kappa}(p_3,p_4,p_1,p_2)+\nonumber\\
&\hspace{1cm}+\widetilde{\mathcal{V}}_{I;\kappa}(p_3,p_2,p_1,p_4)+\widetilde{\mathcal{V}}_{I;\kappa}(p_1,p_4,p_3,p_2)\Big],\label{gamma2}
\end{align}
The various contributions mentioned at the beginning of this section are then obtained by replacing $\widetilde{\mathcal{V}}_{I;\kappa}$ by the different expressions for the vertex function \eqref{vertex-p1}-\eqref{vertex-p4} in \eqref{gamma2}.
\subsubsection{Scalar theory with $\phi^\dag\star\phi\star\phi^\dag\star\phi$ interaction.}\label{321}
The relevant classical action functional is $S^\text{kin}_\kappa+S_{1;\kappa}^\text{int}$, see \eqref{pratik-kinet-act}, \eqref{pratik-kinet}, \eqref{interac-type1}. By a simple inspection of \eqref{gamma2}, one easily realizes that the one-loop 2-point function receives two types of contribution, hereafter called Type-I and Type-II contributions.\\
The contributions of Type-I are nothing but the usual planar contributions, according to the usual denomination prevailing in the non-commutative field theories. The Type-II contributions, while similar to the planar contributions in that they do not depend on the external momenta, are a new type of contributions generated by the twist which arises in the vertex functions, thus altering some diagrams with ``planar topology". No non-planar contributions (namely, depending on the external momenta) can be obtained within the present model so that no IR singularity related to the UV/IR mixing can occur in the 2-point function. Let us now study the UV behaviour of these contributions.\bigskip
Typical Type-I contribution to the one-loop effective action can be written as
\begin{equation}
\Gamma^{(2)}_{1;(I)}(p_3,p_4)= e^{-3p_3^0/\kappa}\delta^{(4)}(p_4-p_3)\Sigma_{(I)},\label{type1-struct}
\end{equation}
in which
\begin{equation} \label{Sigma1}
\Sigma_{(I)} := \lambda \int \frac{d^4p}{(2\pi)^4} \ \frac{e^{-2p^0/\kappa}}{1+e^{-3p^0/\kappa}} \ \frac{4\kappa^2}{\left(\vec{p}^{\hspace{2pt}2}+\kappa^2\mu^2_{+}\right)\left(\vec{p}^{\hspace{2pt}2}+\kappa^2\mu^2_{-}\right)}.
\end{equation}
Because of the strong decay of the propagator at large momentum $\vec{p}$ (, $\sim 1/\vec{p}^{\hspace{2pt}4}$), the spatial integral is finite and the integration over the 3-momentum $d^3\vec{p}$ can be performed by making use of the two following relations
\begin{align}
\frac{1}{A^aB^b}=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} & \int_0^1 du \ \frac{u^{a-1}(1-u)^{b-1}}{\left(uA+(1-u)B\right)^{a+b}}, \ a,b>0, \\
\int \frac{d^np}{(2\pi)^n} \frac{1}{(p^2+M^2)^{m}}&=M^{n-2m}\frac{\Gamma(m-n/2)}{(4\pi)^{n/2} \Gamma(m)}, \ m>n/2> 0,
\end{align}
where $\Gamma(z)$ is the Euler gamma function. This leads to
\begin{equation}
\Sigma_{(I)} = \frac{2\kappa^2\lambda}{(2\pi)^2} \int_\mathbb{R} dp^0\ \frac{e^{-2p^0/\kappa}}{\left(1+e^{-3p^0/\kappa}\right)} \ \frac{\sqrt{\mu^2_{+}}-\sqrt{\mu^2_{-}}}{\mu^2_{+}-\mu^2_{-}},
\end{equation}
with $\mu^2_{+}-\mu^2_{-}=4\sqrt{1-\left(\frac{m}{\kappa}\right)^2}\ e^{-p^0/\kappa}$. By finally performing the change of variables,
\begin{equation}
y=e^{-p^0/\kappa}, \label{change-var}
\end{equation}
$\Sigma_{(I)}$ reduces to
\begin{align}
&\Sigma_{(I)} = C \int_0^\infty dy \left[\frac{\sqrt{1+2\sqrt{1-\left(\frac{m}{\kappa}\right)^2}y+y^2}}{1+y^3}-\frac{\sqrt{1-2\sqrt{1-\left(\frac{m}{\kappa}\right)^2}y+y^2}}{1+y^3}\right], \label{typeI} \\
&\text{with}\ \ C:=\frac{\lambda}{(2\pi)^2}\frac{\kappa^3}{2\sqrt{\kappa^2-m^2}}, \label{constanteC}
\end{align}
whose UV behaviour can easily be inferred by use of the d'Alembert criterion as shown below. Before proceeding to that analysis, some comments are in order:
\begin{itemize}
\item First, notice that, due to the change of variables \eqref{change-var}, both the lower $(0)$ and upper $(\infty)$ bounds of integration in \eqref{typeI} correspond to the UV (large $|p^0|$) regime.
\item Next, some of the integrals w.r.t. the $y$ variable appearing in the computation of the one-loop order corrections to the 2-point function, have to be understood as regularized integrals. One way of regularizing them amounts to introduce a cut-off for $y$. Motivated by the Hopf algebraic structure of the $\kappa$-Poincar\'e algebra (in particular the deformed translation algebra), which is generated by $P_i$ and $\mathcal{E}=e^{-P_0/\kappa}$ (for more details see appendix \ref{apendixA}), it is natural to interpret $y=e^{-p^0/\kappa}$ as related to the ``physical" quantity replacing $p^0$ in the NCFT. More precisely, having in mind the expression for the 1st Casimir operator of the $\kappa$-Poincar\'e algebra, \eqref{Casimir2}, one can interpret the quantity
\begin{equation}
\mathcal{P}^0(\kappa):=\kappa(1-y)
\end{equation}
as the relevant quantity for the $\kappa$-Poincar\'e covariant quantum field theories, which reduces to $p^0$ when taking the formal commutative limit ($\kappa\to\infty$). Assuming $\vert\mathcal{P}^0\vert\leq\Lambda_0$, it follows that one can derive an appropriate cut-off for $y$. This is achieved by noticing that the introduction of $\Lambda_0$ induces a cut-off for $p^0$, say $M_\kappa(\Lambda_0)$, which is easily shown to be related to $\Lambda_0$ by
\begin{equation}
M_\kappa(\Lambda_0) = \kappa \ln\left(1+\frac{\Lambda_0}{\kappa}\right),
\end{equation}
with the limit $M_\kappa(\Lambda_0)\to\Lambda_0$ when $\kappa\to\infty$. Thus,
\begin{equation}
\frac{\kappa}{\kappa+\Lambda_0} \leq y \leq \frac{\kappa+\Lambda_0}{\kappa}.
\end{equation}
\end{itemize}
Having in mind these two comments, we can now study the UV behaviour of the scalar field theories under consideration.\\
When $y\to \infty$, one can check that
\begin{equation}
\sqrt{\mu^2_{+}}-\sqrt{\mu^2_{-}}=2\sqrt{1-\left(\frac{m}{\kappa}\right)^2}+\mathcal{O}(\frac{1}{y^2}),\label{mumu}
\end{equation}
so that the integrand in eqn. \eqref{typeI} behaves like $\sim y^{-3}$. Meanwhile, when $y\to 0$, one verifies that the integrand behaves like $\sim y$. Hence, the integral is convergent, showing that typical Type-I contribution given by $\Sigma^{(I)}$ is (UV) finite. \bigskip
By performing a similar computation, one finds that typical contribution of Type-II have the same structure than those of Type-I \eqref{type1-struct}, still independent of external momenta, but receiving an extra contribution proportional to some power of $e^{-3p^0/\kappa}$ stemming from the twist $\sigma$, as indicated above. Indeed, the one-loop effective action can be cast into the form
\begin{equation}
\Gamma^{(2)}_{1;(I{\hspace{-2pt}I})}(p_3,p_4)=\delta^{(4)}(p_4-p_3)\Sigma_{(I{\hspace{-2pt}I})},
\end{equation}
where
\begin{equation}
\Sigma_{(I{\hspace{-2pt}I})} = \lambda \int \frac{d^4p}{(2\pi)^4} \ \frac{e^{-5p^0/\kappa}}{1+e^{-3p^0/\kappa}} \ \frac{4\kappa^2}{\left(\vec{p}^{\hspace{2pt}2}+\kappa^2\mu^2_{+}\right)\left(\vec{p}^{\hspace{2pt}2}+\kappa^2\mu^2_{-}\right)}. \label{Sigma2}
\end{equation}
Observe from \eqref{Sigma2} and \eqref{Sigma1} that one has formally
\begin{equation}
\Sigma_{(I)}=\int \frac{d^4p}{(2\pi)^4}\ \mathcal{I}(p),\label{type1-sig}
\end{equation}
where the integrand $\mathcal{I}$ can be read off from \eqref{Sigma1}, while
\begin{equation}
\Sigma_{(I{\hspace{-2pt}I})}=\int \frac{d^4p}{(2\pi)^4}\ e^{-3p^0/\kappa}\mathcal{I}(p),\label{type2-sig}
\end{equation}
in which the extra factor $e^{-3p^0/\kappa}$ is generated by a twist factor.\bigskip
Now, performing the change of variable \eqref{change-var} in \eqref{Sigma2} and characterizing the UV (large $|p^0|$) regime as done above for the Type-I contributions, one easily finds that the integral in \eqref{Sigma2} reduces to
\begin{equation}
\Sigma_{(I{\hspace{-2pt}I})} = C \int_0^\infty dy \ y^3 \left[\frac{\sqrt{1+2\sqrt{1-\left(\frac{m}{\kappa}\right)^2}y+y^2}}{1+y^3}-\frac{\sqrt{1-2\sqrt{1-\left(\frac{m}{\kappa}\right)^2}y+y^2}}{1+y^3}\right],
\end{equation}
where the constant $C$ is given by \eqref{constanteC}. The integral is still convergent for $y\to 0$ since the twist contributes by a factor $y^3$ at the numerator while the integrand behaves now like \eqref{mumu} when $y\to\infty$, instead of the convergent behaviour of the Type-I contribution. Hence,
\begin{equation}
\Sigma_{(I{\hspace{-2pt}I})} \sim \frac{\lambda\kappa}{(2\pi)^2} \ \Lambda_0 + \lbrace \text{finite terms}\rbrace,
\end{equation}
which exhibits a linear UV divergence essentially produced by the twist in view of \eqref{type1-sig} and \eqref{type2-sig}.\bigskip
To summarize the results, we have found that within the field theory described by the action functional $S^\text{kin}_\kappa+S_{1;\kappa}^\text{int}$, the twist splits the planar contributions to the 2-point function into two different planar-like contributions which are IR finite and whose UV behaviour is affected by the twist. Note that all the contributions to the 2-point functions are independent of the external momenta so that no IR singularities at exceptional (zero) momenta, related to UV/IR mixing, can occur (there is no non-planar contributions).
\subsubsection{Scalar theory with $\phi\star\phi\star\phi^\dag\star\phi^\dag$ interaction.}\label{322}
The relevant classical action functional is now $S^\text{kin}_\kappa+S_{2;\kappa}^\text{int}$, see \eqref{pratik-kinet-act}, \eqref{pratik-kinet}, \eqref{interac-type1}. From the perturbative expansion of the corresponding partition function, one finds that the one-loop 2-point function receives three types of contribution, hereafter called Type-I, Type-III and Type-IV contributions.\\
The Type-I and Type-III are planar type, i.e. independent of the external momenta but differing from each other by its own contribution coming from the twist $\sigma$. This results in different powers of the factor $e^{-3p^0/\kappa}$ in the integrands of the various contributions, hence the denomination ``Type-III" since this factor is different from the one for Type-II contributions exhibited in the subsection \ref{321}. As for the field theory examined in subsection \ref{321}, Type-I contributions are found to be UV finite. The Type-IV contributions can be actually interpreted as non-planar contributions. This signals that the corresponding field theory has UV/IR mixing since Type-IV contributions evaluated at exceptional zero external momentum are divergent.\bigskip
Let us start by considering planar contributions. Typical Type-III contribution to the one-loop effective action can be written as
\begin{equation}
\Gamma^{(2)}_{1;(I{\hspace{-2pt}I}{\hspace{-2pt}I})}(p_3,p_4)= \delta^{(4)}(p_4-p_3)\Sigma_{(I{\hspace{-2pt}I}{\hspace{-2pt}I})},
\end{equation}
in which $\Sigma_{(I{\hspace{-2pt}I}{\hspace{-2pt}I})}=\int d^4p\ e^{3p^0/\kappa}\mathcal{I}(p)$, with $\mathcal{I}(p)$ defined in \eqref{type1-sig}. After performing the integration over $d^3\vec{p}$ and the change of variable \eqref{change-var}, one obtains
\begin{equation}
\Sigma_{(I{\hspace{-2pt}I}{\hspace{-2pt}I})} =C \int_0^\infty \frac{dy}{y^3} \left[\frac{\sqrt{1+2\sqrt{1-\left(\frac{m}{\kappa}\right)^2}y+y^2}}{1+y^3}-\frac{\sqrt{1-2\sqrt{1-\left(\frac{m}{\kappa}\right)^2}y+y^2}}{1+y^3}\right].\label{type3-div}
\end{equation}
Using \eqref{mumu}, one easily finds that the integrand in \eqref{type3-div} behaves like $\sim y^{-6}$ when $y\to\infty$ while it behaves like $\sim y^{-2}$ for $y\to0$, such that
\begin{equation}
\Sigma_{(I{\hspace{-2pt}I}{\hspace{-2pt}I})}\sim \frac{\lambda\kappa}{(2\pi)^2} \ \Lambda_0 + \lbrace \text{finite terms}\rbrace ,
\end{equation}
indicating that \eqref{type3-div} has a UV linear divergence (as for Type-II contribution of the field theory considered in the previous subsection).\bigskip
Finally, let us consider the non-planar Type-IV contributions. That latter can be written as
\begin{equation}
\Gamma^{(2)}_{1;(I{\hspace{-2pt}V})}(p_3,p_4)=\delta(p_4^0-p_3^0)\Sigma_{(I{\hspace{-2pt}V})}(p_3,p_4),\label{decadix1}
\end{equation}
where
\begin{equation}
\Sigma_{(I{\hspace{-2pt}V})}(p_3,p_4) = (2\kappa)^2\lambda \int \frac{d^4p}{(2\pi)^4} \ \frac{e^{-2p^0/\kappa}}{1+e^{-3p^0/\kappa}} \frac{\delta^{(3)}\left(\left(1-e^{-p_3^0/\kappa}\right)\vec{p}+\vec{p}_4e^{-p^0/\kappa}-\vec{p}_3\right)}{\left(\vec{p}^{\hspace{2pt}2}+\kappa^2\mu^2_{+}\right)\left(\vec{p}^{\hspace{2pt}2}+\kappa^2\mu^2_{-}\right)}. \label{nonplanar}
\end{equation}
Note that $\Sigma_{(I{\hspace{-2pt}V})}(p_3,p_4)$ depends on two (external) momenta which however are not independent, due to the (non-linear) momentum conservation ensured by the delta functions. This dependence by the way signals that the effective action functional \eqref{decadix1} is non-local.
Let's first examine the infrared sector. Setting $(p^0_3,\vec{p}_3)\to(0,\vec{0})$ in \eqref{nonplanar} leads to
\begin{equation}
\Sigma_{(I{\hspace{-2pt}V})}(0,p_4) = (2\kappa)^2\lambda \int \frac{d^4p}{(2\pi)^4} \ \frac{e^{-2p^0/\kappa}}{1+e^{-3p^0/\kappa}}\frac{e^{3p^0/\kappa}}{\left(\vec{p}^{\hspace{2pt}2}+\kappa^2\mu^2_{+}\right)\left(\vec{p}^{\hspace{2pt}2}+\kappa^2\mu^2_{-}\right)} \ \delta^{(3)}(\vec{p}_4),
\end{equation}
such that
\begin{equation} \label{mixingIII}
\Sigma_{(I{\hspace{-2pt}V})}(0,p_4)=\delta^{(3)}(\vec{p}_4)\Sigma_{(I{\hspace{-2pt}I}{\hspace{-2pt}I})},
\end{equation}
indicating that the conservation law is preserved, namely $p_4\to 0$ when $p_3\to 0$, and that the non-planar contributions tends toward (Type-III) planar contributions in the limit of vanishing external momenta.
To study the UV behaviour of \eqref{nonplanar}, we perform the integration over $d^3\vec{p}$ together with the change of variables \eqref{change-var}. Standard computation yield
\begin{equation}\label{typeIV}
\Sigma_{(I{\hspace{-2pt}V})}(p_3,p_4) = \frac{\kappa^2\lambda}{4\pi^4} \left|1-e^{-p_3^0/\kappa}\right| \int_0^\infty dy \ \frac{y}{\left(1+y^3\right)\Omega_{+}(y)\Omega_{-}(y)},
\end{equation}
with
\begin{equation}
\Omega_{\pm}(y) = \left(y\vec{p}_4-\vec{p}_3\right)^2+\kappa^2\left(1-e^{-p_3^0/\kappa}\right)^2\mu_{\pm}^2(y).
\end{equation}
Now, one can easily check that the integrand in \eqref{typeIV} behaves like $\sim y$ when $y\to0$, while it behaves like $\sim y^{-6}$ when $y\to\infty$.
Therefore, one concludes that Type-IV contributions are finite for any (non zero) external 4-momenta while $\lim_{p_3\to 0}\Sigma_{(I{\hspace{-2pt}V})}(p_3,p_4)\sim-\lambda\kappa\Lambda_0$, namely diverges (UV) linearly. This last phenomenon reflects the existence of perturbative UV/IR mixing when considering interactions of the form of $S^\text{int}_{2;\kappa}$. The same result occurs for interactions given by $S^\text{int}_{4;\kappa}$.
\section{Discussion and conclusion.}\label{section4}
The Weyl quantization scheme provides a natural framework to describe $\kappa$-deformations of the Minkowski space-time. A well controlled star product for $\kappa$-Minkowski space is easily obtained from the representations of the convolution algebra of the affine group which here replaces the Heisenberg group underlying the popular quantization of a phase space. Owing to the fact that the $\kappa$-Minkowski space supports a natural action of a deformation of the Poincar\'e Lie algebra, the $\kappa$-Poincar\'e algebra playing the role of the algebra of symmetry of the quantum space, it is physically relevant to require $\kappa$-Poincar\'e invariance of any physically reasonable action functional. Doing this necessarily implies that the trace building the action functional is twisted, stemming simply from the peculiar behavior of the star product w.r.t. the Lebesgue integral involved in the action.\bigskip
We have examined various classes of (complex) scalar field theories on 4-d $\kappa$-Minkowski space, considering all possible types of quartic interaction allowed by reality condition of the action functional, and whose commutative limit coincides with the standard (commutative) complex $\phi^4$ theory. The kinetic operators were chosen to be square of different Dirac operators. The use of algebraic properties of the twisted trace leads to an easy computation of the corresponding propagators, despite the fact that the star product is not closed w.r.t. the integral.\bigskip
Focusing first on a kinetic operator \eqref{kinetic} related to the Dirac operator of an equivariant spectral triple considered in \cite{frans-2}, we have analyzed the one-loop UV and IR behavior of the 2-point functions for each of these theories, presenting in details the technical analysis for representative classes of theories \eqref{interac-type1} in the subsection \ref{subsection32}. We find that the twist splits the planar contributions to the 2-point function into different {\it{IR finite}} contributions whose UV behavior depends on the power of the twist factor arising, technically speaking, from the respective positions of the contracted fields in the interaction combined with the non-cyclicity of the trace. The UV behavior of these contributions ranges from UV finitude to at most UV linear divergence, which is slightly milder than in the commutative scalar theory. The interaction term of the scalar theory considered in the subsection \ref{321} cannot produce non-planar contributions, since the interaction is orientable (in the terminology of non-commutative field theories). Hence, no UV/IR mixing is expected to occur in this field theory which therefore should be perturbatively renormalizable to all orders.\\
It turns out that the computation of the 1-loop contributions to the 4-point function for this NCFT shows that this latter is UV finite. The full derivation is cumbersome and will be reported elsewhere \cite{PW-1} together with the analysis of 2- and 4-point functions for the other NCFT considered in this paper. The UV finiteness is partly due to the large spatial momentum behavior of the propagator which decays as $1/\vec{p}^{\ 4}$. This yields finite spatial integrals for all the contributions while each of the remaining integrals over $y$ is found to be finite by a mere use of d'Alembert criterion. This additional observation together with the strong decay of the propagator at large (spatial) momenta makes very likely the perturbative renormalisability of this NCFT to all orders.\bigskip
UV/IR mixing is expected to occur in the scalar theory of subsection \ref{322} (the interaction is no longer orientable). Indeed, we find that the so-called Type-IV contribution, which depends on the external momenta, is finite for non zero external moment while it becomes singular at exceptional zero external momenta, see for instance \eqref{mixingIII}. It would be interesting to examine if this UV/IR mixing could be removed by using procedures similar to the one used to deal with the mixing within non-commutative field theories
on Moyal spaces \cite{Grosse:2003aj-pc}. The above conclusions apply to the 2-point functions of the field theories \eqref{interac-type2}, whose analysis can be obtained from straightforward adaptations of subsection \ref{subsection32}. In the same way, changing the kinetic term \eqref{kinetic} to \eqref{Casimir} does not modify significantly the conclusions on the UV and IR behavior of these field theories. For instance, for the theory considered section \ref{321}, the Type-I contribution remains finite whereas the Type-II contribution diverges quadratically. Note that our conclusions qualitatively agree with those obtained a long time ago in \cite{gross-whl} where a scalar field theory built from another (albeit presumably equivalent) star product and a different kinetic operator has been considered. Again, linear UV divergences for planar-type contributions together with UV/IR mixing in non-planar contributions was shown to occur in that model. The precise comparison between both work is however drastically complicated by the technical approach used in \cite{gross-whl} leading to very involved formulas.\bigskip
An immediate natural extension of this analysis is the computation of the one-loop corrections to the vertex functions and beta functions in the above field theories. The corresponding work will be reported elsewhere \cite{PW-1}. The extension of the present work to the case of gauge theories defined on $\kappa$-Minkowski spaces is an interesting issue \cite{PSW-1}. In view of the natural action of the $\kappa$-Poincar\'e algebra, the framework of bicovariant differential calculus \cite{Maj-sit} seems to be better suited here than the standard derivation-based differential calculus with which most of the non-commutative gauge models on $\mathbb{R}^4_\theta$ or $\mathbb{R}^3_\lambda$ have been built \cite{mdv-jcw}. A suitable framework should presumably take into account algebras of twisted derivations as well as twisted gauge transformations.\bigskip
To conclude, we mention that the star product considered in this paper could be used in the construction of other (even non $\kappa$-Poincar\'e invariant) NCFT or gauge versions of them and should prove convenient to compute related quantum corrections. We note that the NCFT with orientable interaction \eqref{vertex-p1} provides an explicit example (as far as we know the first one) of a UV/IR mixing free NCFT on the 4-d $\kappa$-Minkowski space which is very likely renormalisable to all orders. It would be very interesting to show if some KMS condition stemming from the twisted trace rules the correlation functions of this NCFT which would signal the appearance of an observer-independent time within this theory and would then give to the NCFT on $\kappa$-Minkowski space a new impulse toward potential applications to fundamental physics.
\vskip 0,5 true cm
{\bf{Acknowledgments:}} J.-C. Wallet thanks N. Franco for discussions related to the present work. T. Poulain is grateful to A. Sitarz for discussions on material of ref. \cite{DS}.
\appendix
\section{Basics on $\kappa$-Poincar\'e algebra and deformed translations.}\label{apendixA}
Let $\mathcal{P}_\kappa$ denote the $\kappa$-Poincar\'e algebra. Let $\Delta:\mathcal{P}_\kappa\otimes\mathcal{P}_\kappa\to\mathcal{P}_\kappa$, $\epsilon:\mathcal{P}_\kappa\to\mathbb{C}$ and $S:\mathcal{P}_\kappa\to\mathcal{P}_\kappa$ be respectively the coproduct, counit and antipode, thus endowing $\mathcal{P}_\kappa$ with a Hopf algebra structure. A convenient presentation of $\mathcal{P}_\kappa$ is obtained from the 11 elements $(P_i, N_i,M_i, \mathcal{E},\mathcal{E}^{-1})$, $i=1,2,3$, respectively the momenta, the boost, the rotations and $\mathcal{E}:=e^{-P_0/\kappa}$ satisfying the Lie algebra relations{\footnote{In the following, Greek (resp. Latin) indices label as usual space-time (resp. purely spatial) coordinates.}}
\begin{equation}
[M_i,M_j]= i\epsilon_{ij}^{\hspace{5pt}k}M_k,\ [M_i,N_j]=i\epsilon_{ij}^{\hspace{5pt}k}N_k,\ [N_i,N_j]=-i\epsilon_{ij}^{\hspace{5pt}k}M_k\label{poinc1},
\end{equation}
\begin{equation}
[M_i,P_j]= i\epsilon_{ij}^{\hspace{5pt}k}P_k,\ [P_i,\mathcal{E}]=[M_i,\mathcal{E}]=0,\ [N_i,\mathcal{E}]=-\frac{i}{\kappa}P_i\mathcal{E}\label{poinc2},
\end{equation}
\begin{equation}
[N_i,P_j]=-\frac{i}{2}\delta_{ij}\left(\kappa(1-\mathcal{E}^{2})+\frac{1}{\kappa}\vec{P}^2\right)+\frac{i}{\kappa}P_iP_j\label{poinc3},
\end{equation}
with the Hopf algebra structure defined by
\begin{align}
\Delta P_0&=P_0\otimes\bbone+\bbone\otimes P_0,\ \Delta P_i=P_i\otimes\bbone+\mathcal{E}\otimes P_i,\label{hopf1}\\
\Delta \mathcal{E}&=\mathcal{E}\otimes\mathcal{E},\ \Delta M_i=M_i\otimes\bbone+\bbone\otimes M_i,\label{hopf1bis}\\
\Delta N_i&=N_i\otimes \bbone+\mathcal{E}\otimes N_i-\frac{1}{\kappa}\epsilon_{i}^{\hspace{2pt}jk}P_j\otimes M_k,\label{hopf2}
\end{align}
and
\begin{align}
\epsilon(P_0)&=\epsilon(P_i)=\epsilon(M_i)=\epsilon(N_i)=0,\ \epsilon(\mathcal{E})=1\label{hopf3},\\
S(P_0)&=-P_0,\ S(\mathcal{E})=\mathcal{E}^{-1},\ S(P_i)=-\mathcal{E}^{-1}P_i,\label{hopf4}\\
S(M_i)&=-M_i,\ S(N_i)=-\mathcal{E}^{-1}(N_i-\frac{1}{\kappa}\epsilon_{i}^{\hspace{2pt}jk}P_jM_k)\label{hopf4bis}.
\end{align}
Recall that the $\kappa$-Minkowski space can be viewed as the dual of the Hopf subalgebra generated by $P_\mu$, $\mathcal{E}$, sometimes called the ``deformed translation algebra". This latter becomes a $^*$-Hopf algebra through: $P_\mu^\dag=P_\mu$, $\mathcal{E}^\dag=\mathcal{E}$. Then, by promoting the above duality to a duality between $^*$-algebras insuring compatibility among the involutions, one obtains
\begin{equation}
(t\triangleright f)^\dag=S(t)^\dag\triangleright f,\label{pairing-involution}
\end{equation}
which holds true for any $t$ in the deformed translation algebra and for any $f\in\mathcal{M}_\kappa$. This, combined with \eqref{hopf4} implies
\begin{equation}
(P_0\triangleright f)^\dag=-P_0\triangleright(f^\dag),\ (P_i\triangleright f)^\dag=-\mathcal{E}^{-1}P_i\triangleright(f^\dag),\ (\mathcal{E}\triangleright f)^\dag=\mathcal{E}^{-1}\triangleright(f^\dag)\label{dag-hopfoperat}.
\end{equation}
It must be stressed that the $P_i$'s act as twisted derivations on $\mathcal{M}_\kappa$ while $P_0$ remains untwisted as it can be readily seen from \eqref{hopf1}. One has for any $f,g\in\mathcal{M}_\kappa$
\begin{align}
P_i\triangleright(f\star g)&=(P_i\triangleright f)\star g+(\mathcal{E}\triangleright f)\star (P_i\triangleright g)\label{deriv-twist1},\\
P_0\triangleright(f\star g)&=(P_0\triangleright f)\star g+f\star(P_0\triangleright g )\label{deriv-twist2}.
\end{align}
Note that $\mathcal{E}$ is not a derivation of $\mathcal{M}_\kappa$ since one has
\begin{equation}
\mathcal{E}\triangleright(f\star g)=(\mathcal{E}\triangleright f)\star(\mathcal{E}\triangleright g).\label{relation-calE}
\end{equation}
The structure of $\mathcal{M}_\kappa$ as left-module over the Hopf algebra $\mathcal{P}_\kappa$ can be expressed, for any $f\in\mathcal{F}(\mathcal{S}_c)$, in terms of the bicrossproduct basis $(M_i,N_i,P_\mu)$, \cite{majid-ruegg}, by
\begin{align}
(\mathcal{E}\triangleright f)(x)&=f(x_0+\frac{i}{\kappa},\vec{x})\label{left-module0},\\
(P_\mu\triangleright f)(x)&=-i(\partial_\mu f)(x),\label{left-module1}\\
(M_i\triangleright f)(x)&=\left(\epsilon_{ijk}L_{x_j}P_k\triangleright f\right)(x),\label{left-modules1bis}\\
(N_i\triangleright f)(x)&=\bigg(\big(\frac{1}{2}L_{x_i}(\kappa(1-\mathcal{E}^2)+\frac{1}{\kappa}\vec{P}^{2})+L_{x_0}P_i-\frac{i}{\kappa}L_{x_k}P_kP_i \big)\triangleright f\bigg)(x),\label{left-module2}
\end{align}
where $L_a$ denotes the left (standard) multiplication operator, i.e. $L_af:=af$.
\section{KMS weight and twisted trace.}\label{apendixB}
A KMS weight on a (C*-)algebra $\mathbb{A}$ for a modular group of $^*$-automorphisms $\{\sigma_t\}_{t\in\mathbb{R}}$ is defined \cite{kuster} as a (densely defined) linear
map $\varphi:\mathbb{A}_+\to\mathbb{R}^+$ ($\mathbb{A}_+$ is the set of positive elements of $\mathbb{A}$) such that $\{\sigma_t\}_{t\in\mathbb{R}}$ admits an analytic extension, still a one-parameter group, $\{\sigma_z\}_{z\in\mathbb{C}}$ acting on $\mathbb{A}$ satisfying the following two conditions{\footnote{Some alternative equivalent definitions exist, which however are less convenient for the present discussion. The above definition \cite{kuster} also require that $\varphi$ is lower semi-continuous and that $\{\sigma_z\}$ is norm-continuous, two conditions which are fortunately fulfilled in this paper.}}:
\begin{equation}
{\textrm{i)}}\ \ \varphi\circ\sigma_z=\varphi,\ \ {\textrm{ii)}}\ \ \varphi(a^\dag \star a)=\varphi(\sigma_{\frac{i}{2}}(a)\star(\sigma_{\frac{i}{2}}(a))^\dag),\label{prop-kmsweight}
\end{equation}
for any $a$ in the domain of $\sigma_{\frac{i}{2}}$. The notion of weight on a C*-algebra extends the usual notion of state, since a state can be viewed (up to technical subtleties) as a weight with unit norm. In the present situation, the characterization of the relevant C*-algebra has been discussed in the section \ref{section2}. For our purpose, it will be sufficient to keep in mind that it involves $\mathcal{M}_\kappa$ as a dense $^*$-subalgebra. For more mathematical details on KMS weights, see e.g \cite{kuster}. Note that the notion of KMS weight related to the present twisted trace has been already used in \cite{matas} to construct a modular spectral triple for $\kappa$-Minkowski space.\bigskip
To verify that the twisted trace \eqref{twistrace}, \eqref{twistoperator} is actually a KMS weight, we first characterize the properties of $\sigma_t$ \eqref{sigmat-modul}. From \eqref{sigmat-modul}, \eqref{starpro-4d} and \eqref{invol-4d}, one obtains
\begin{equation}
\sigma_{t_1}\sigma_{t_2}=\sigma_{t_1+t_2},\ \sigma^{-1}_t=\sigma_{-t},\ \ \forall t,t_1,t_2\in\mathbb{R}, \label{modulargroup}
\end{equation}
and
\begin{equation}
\sigma_t(f\star g)=\sigma_t(f)\star\sigma_t(g),\ \ \sigma_t(f^\dag)=(\sigma_t(f))^\dag,\ \forall t\in\mathbb{R}\label{modular-sigma},
\end{equation}
for any $f,g\in\mathcal{M}_\kappa$. Hence $\sigma_t$ \eqref{sigmat-modul} defines a group of $^*$-automorphisms of $\mathcal{M}_\kappa$.
Next, set $\varphi(f):=\int d^4x\ f(x)$. Then, $\varphi$ verifies the property i) of \eqref{prop-kmsweight} as a mere consequence of \eqref{invarquant}, i.e. the $\kappa$-Poincar\'e invariance of the action functional. Namely
\begin{equation}
\varphi(\sigma_t f)=\sigma_t\triangleright\int d^4x\ f(x)=(\mathcal{E})^{-i3t}\triangleright\int d^4x\ f(x)=\epsilon(\mathcal{E})^{-i3t}\int d^4x\ f(x)=\varphi(f),
\end{equation}
for any $f\in\mathcal{M}_\kappa$ where the action of $\mathcal{E}$ has been extended to the one of $\sigma_t$ by using the functional calculus. \\
Before we verify the property ii) of \eqref{prop-kmsweight}, one remark is in order. Extend $\sigma_t$ \eqref{modular-sigma} to
\begin{equation}
\sigma_z(f):=e^{iz\frac{3P_0}{\kappa}}\triangleright f=e^{\frac{3z}{\kappa}\partial_0}\triangleright f,\ \forall z\in\mathbb{C}\label{sigmat-modul-z},
\end{equation}
for any $f\in\mathcal{M}_\kappa$. Then, one can easily verify that \eqref{modulargroup} and \eqref{modular-sigma} extend respectively to
\begin{equation}
\sigma_{z_1}\sigma_{z_2}=\sigma_{z_1+z_2},\ \sigma^{-1}_z=\sigma_{-z},\ \ \forall z,z_1,z_2\in\mathbb{C}, \label{prop-modulargroup-z}
\end{equation}
and
\begin{equation}
\sigma_z(f\star g)=\sigma_z(f)\star\sigma_z(g)\label{morphalg-modul-z},
\end{equation}
while $\sigma_z$ is no longer an automorphism of $^*$-algebra. Namely, one has
\begin{equation}
\sigma_z(f^\dag)=(\sigma_{\bar{z}}(f))^\dag,\ \forall z\in\mathbb{C}\label{modulargroup-z}.
\end{equation}
In particular, the twist $\sigma$ \eqref{twistoperator} is recovered for $z=i$, i.e.
\begin{equation}
\sigma=\sigma_{z=i}\label{b10}
\end{equation}
and one has $\sigma(f^\dag)=(\sigma^{-1}(f))^\dag$. This type of automorphim is known as a regular automorphim in the mathematical literature and occurs in the framework of twisted spectral triples. It has been introduced in \cite{como-1} in conjunction with the assumption of the existence of a distinguished group of ($^*$-)automorphisms of the algebra indexed by one real parameter, says $t$, i.e. the modular group, such that the analytic extension $\sigma_{t=i}$ coincides precisely with the regular automorphism. Here, the modular group linked with the twisted trace is defined by $(\sigma_t)_{t\in\mathbb{R}}$ while the twist $\sigma=\sigma_{t=i}$ defines the related regular automorphism.\bigskip
To verify the 2nd property of \eqref{prop-kmsweight}, we use \eqref{morphalg-modul-z}, \eqref{modulargroup-z} and \eqref{twistrace} to compute the RHS of ii) \eqref{prop-kmsweight}. One has
\begin{align}
\varphi(\sigma_{\frac{i}{2}}(f)\star(\sigma_{\frac{i}{2}}(f))^\dag)&=\int d^4x\ \sigma_{\frac{i}{2}}(f)\star\sigma_{-\frac{i}{2}}(f^\dag)=\int d^4x\ \sigma_{\frac{i}{2}}(f\star\sigma_{-i}(f^\dag))\nonumber\\
&=\int d^4x\ f\star\sigma_{-i}(f^\dag)=\int d^4x\ \sigma(\sigma_{-i}(f^\dag))\star f
=\varphi(f^\dag\star f)\label{proof-kmsweight}.
\end{align}
Hence $\varphi(f)=\int d^4x\ f(x)$ for any $f\in\mathcal{M}_\kappa$ defines a KMS weight. \bigskip
Now, the Theorem [6.36] of the 1st of ref. \cite{kuster} guarantees, for each pair $(a,b)\in\mathbb{A}$, the existence of a bounded continuous function $f:\Sigma\to\mathbb{C}$, where $\Sigma$ is the strip defined by $\{z\in\mathbb{C},\ 0\le\textrm{Im}(z)\le 1\}$, such that one has
\begin{equation}
f(t)=\varphi(\sigma_t(a)\star b),\ \ f(t+i)=\varphi(b\star\sigma_t(a)),\label{KMS-abst}
\end{equation}
where it is easy to realize that eqn. \eqref{KMS-abst} is an abstract version of the KMS condition.\bigskip
Note that $\sigma_t$ \eqref{sigmat-modul} defines ``time translations" since one has $\sigma_t(\phi)(x_0,\vec{x})=\phi(x_0+\frac{3t}{\kappa},\vec{x})$. Now we introduce the GNS representation of $\mathcal{M}_\kappa$, $\pi_{GNS}:\mathcal{F}(\mathcal{S}_c)\to\mathcal{B}(\mathcal{H})$, defined as usual by $\pi_{GNS}(\phi)\cdot v=\phi\star v$ for any $v\in\mathcal{H}$ where $\mathcal{H}$ is the Hilbert space unitary equivalent to $L^2(\mathbb{R}^4)$ discussed in the subsection \ref{section31}. Then, we compute
\begin{eqnarray}
\pi_{GNS}(\sigma_t\phi)\cdot\omega&=&(\sigma_t\phi)\star\omega=\sigma_t(\phi\star(\sigma_t^{-1}\omega))
=(\sigma_t\odot\pi_{GNS}(\phi)\odot\sigma_t^{-1})\cdot(\omega)\nonumber\\
&=&((\Delta_T)^{it}\odot\pi_{GNS}(\phi)\odot(\Delta_T)^{-it})\cdot\omega\label{compo2},
\end{eqnarray}
for any $\omega\in\mathcal{H}$ and any $\phi\in\mathcal{M}_\kappa$, where $\odot$ stands for the composition law of maps (not to be confused with the convolution product) and $\Delta_T$ is the Tomita operator given by
\begin{equation}
\Delta_T=e^{\frac{3P_0}{\kappa}},
\end{equation}
which coincides with \eqref{modul-4d} and such that $\sigma_t=(\Delta_T)^{it}$. Eqn. \eqref{compo2} indicates that the modular group defined by $\{\sigma_t\}_{t\in\mathbb{R}}$ generates a ``temporal" evolution for the operators stemming from the Weyl quantization map $Q$. \bigskip
\section{Characterization of the Hilbert space.}\label{apendixC}
The Hilbert space $\mathcal{H}$ related to the Hilbert product \eqref{hilbert-product} can be obtained canonically from the GNS construction by completing the linear space $\mathcal{F}(\mathcal{S}_c)$ with respect to the natural norm
\begin{equation}
||f||^2=\langle f,f\rangle=\int d^4x\ \left(f^\dag\star f\right)(x)=\int d^4x\ |f^\dag(x)|^2.
\end{equation}
Unitary equivalence between $\mathcal{H}$ and $L^2(\mathbb{R}^4)$ can be easily shown by considering the (invertible) intertwiner map $A_\kappa:\mathcal{F}(\mathcal{S}_c)\to L^2(\mathbb{R}^4)$ which is defined for any $f\in\mathcal{F}(\mathcal{S}_c)$ by
\begin{equation}
(A_\kappa f)(x)=\int \frac{dp^0}{2\pi} dy_0\ e^{iy_0p^0}{f}(x_0+y_0,e^{-p^0/\kappa}\vec{x}),\label{intertwin}
\end{equation}
with
\begin{equation}
{\overline{(A_\kappa f)}}(x)=f^\dag(x)\label{conjug-unitar}.
\end{equation}
It follows immediately that $||A_\kappa f||^2_2=\int d^4x\ {\overline{(A_\kappa f)}}(x)(A_\kappa f)(x)=\int d^4x\ |f^\dag(x)|^2=||f||^2$. Therefore $A_\kappa $ defines an isometry which, owing to the density of $\mathcal{F}(\mathcal{S}_c)$ in $\mathcal{H}$, extends to $\mathcal{H}\to L^2(\mathbb{R}^4)$ while surjectivity of $A_\kappa$ stems directly from the existence of $A_\kappa^{-1}$ together with density of $\mathcal{F}(\mathcal{S}_c)$ in $L^2(\mathbb{R}^4)$. This proves that $A_\kappa$ is unitary together with the unitary equivalence mentioned above. Note that one verifies that $A_\kappa^{-1}$ is simply given by
\begin{equation}
(A_\kappa^{-1}f)(x)=\int \frac{dp^0}{2\pi} dy_0e^{-ip^0y_0}f(x_0+y_0,e^{-p^0/\kappa}\vec{x}),\label{intertwin-invers}
\end{equation}
for any $f\in\mathcal{F}(\mathcal{S}_c)$ so that \eqref{conjug-unitar} takes the convenient form $f^\dag(x)=(A_\kappa^{-1}\bar{f})(x)$.
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\section{An intuitive view of SVRG as approximate stochastic Newton descent}
\label{append:sec:svrg-intuition}
Here we present an intuitive view of SVRG as approximate stochastic Newton descent,
which is the inspiration behind our work.
Gradient descent solves the optimization problem $\htheta = \arg \min_\theta f(\theta)$,
where the function is a sum of $n$ functions $f(\theta) = \frac{1}{n} \sum_{i=1}^n f_i(\theta)$, using
\begin{align*}
\theta_{t+1} = \theta_t - \eta \nabla f(\theta_t),
\end{align*}
and stochastic gradient descent uniformly samples a random index at each step
\begin{align*}
\theta_{t+1} = \theta_t - \eta_t \nabla f_i(\theta_t).
\end{align*}
\begin{center}
\scalebox{1.0}{\framebox{
\begin{minipage}{.5\textwidth}
\begin{itemize}
\item \textbf{Outer loop:}
\item $g \gets \nabla f(\theta_t) = \sum_{i=1}^n \nabla f_i(\theta_t)$
\item Let $d$ be the descent direction
\item \begin{itemize}
\item \textbf{Inner loop:}
\item Choose a random index $k$
\item $d \gets d - \eta ( \nabla f_k(\theta_t + d) - \nabla f_k(\theta_t) + g)$
\end{itemize}
\item $\theta_{t+1} = \theta_t + d$
\end{itemize}
\end{minipage}
}}
\end{center}
SVRG \cite{johnson2013accelerating} improves gradient descent and SGD by having an outer loop and an inner loop.
Here, we give an intuitive explanation of SVRG as stochastic proximal Newton descent,
by arguing that
\begin{itemize}
\item each outer loop approximately computes the Newton direction $-(\nabla^2 f)^{-1} \nabla f$
\item the inner loops can be viewed as SGD steps solving a proximal Newton step $
\min_d \langle \nabla f, d \rangle + \frac{1}{2} d^\top (\nabla^2 f) d$
\end{itemize}
First, it is well known \cite{bubeck2015convex} that the Newton direction is exactly the solution of
\begin{align}
\min_d \langle \nabla f(\theta), d \rangle + \frac{1}{2} d^\top [\nabla^2 f(\theta)] d .
\label{eq:newton-dir}
\end{align}
Next, let's consider solving \eqref{eq:newton-dir} using gradient descent on a function of $d$,
and notice that its gradient with respect to $d$ is
\begin{align*}
\nabla f(\theta) + [\nabla^2 f(\theta)] d ,
\end{align*}
which can be approximated through $f$'s Taylor expansion ($[\nabla^2 f(\theta)] d \approx \nabla f(\theta + d) - \nabla f(\theta)$) as
\begin{align*}
\nabla f(\theta) + [\nabla f(\theta + d) - \nabla f(\theta)] .
\end{align*}
Thus, SVRG's inner loops can be viewed as using SGD to solve proximal Newton steps in outer loops.
And it can be viewed as the power series identity for matrix inverse $H^{-1} = \sum_{i=0}^\infty (I - \eta H)$, which corresponds to unrolling the gradient descent recursion for the optimization problem $H^{-1} = \arg \min_\Omega \operatorname{Tr}\left( \frac{1}{2} \Omega^\top H \Omega - \Omega \right) $.
| 206,606
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Crystal Peak Minerals Inc. Announces Transfer of Listing to NEX
TORONTO, Nov. 11, 2020 (GLOBE NEWSWIRE) — Crystal Peak Minerals Inc. (Crystal Peak or the Company) (TSXV: CPM, OTCQB: CPMMF) today announced that, effective at the opening of trading on Friday, November 13, 2020, the listing of its common shares will be transferred from the TSX Venture Exchange (TSXV) to the NEX board of the TSXV. This is a result of the restructuring transaction described in the Company’s press release dated October 20, 2020. The trading symbol of the Corporation will also change from CPM to CPM.H. There is no change in the Corporation’s name or CUSIP number.
Crystal Peak is in the process of working to identify and secure a new project with the focus being on the gold and base metals sectors.
CBJ Newsmakers
| 178,023
|
\begin{document}
\bibliographystyle{amsplain}
\title{Character degree sums and real representations of finite classical groups of odd characteristic}
\author {C. Ryan Vinroot}
\date{}
\maketitle
\begin{abstract}
Let $\FF_q$ be a finite field with $q$ elements, where $q$ is the power of an odd prime, and let $\GSp(2n, \FF_q)$ and $\GO^{\pm}(2n, \FF_q)$ denote the symplectic and orthogonal groups of similitudes over $\FF_q$, respectively. We prove that every real-valued irreducible character of $\GSp(2n, \FF_q)$ or $\GO^{\pm}(2n, \FF_q)$ is the character of a real representation, and we find the sum of the dimensions of the real representations of each of these groups. We also show that if $\bG$ is a classical connected group defined over $\FF_q$ with connected center, with dimension $d$ and rank $r$, then the sum of the degrees of the irreducible characters of $\bG(\FF_q)$ is bounded above by $(q+1)^{(d+r)/2}$. Finally, we show that if $\bG$ is any connected reductive group defined over $\FF_q$, for any $q$, the sum of the degrees of the irreducible characters of $\bG(\FF_q)$ is bounded below by $q^{(d-r)/2}(q-1)^r$. We conjecture that this sum can always be bounded above by $q^{(d-r)/2}(q+1)^r$.
\\
\\
2000 {\it Mathematics Subject Classification:} 20C33, 20G40
\end{abstract}
\section{Introduction}
Given a finite group $G$, and an irreducible complex representation $(\pi, V)$ of $G$ which is self-dual (that is, has a real-valued character), one may ask whether $(\pi, V)$ is a real representation. Frobenius and Schur gave a method of answering this question by introducing an invariant, which we denote $\ep(\pi)$, or $\ep(\chi)$ when $\chi$ is the character of $\pi$ (called the Frobenius-Schur indicator), which gives the value $0$ when $\chi$ is not real-valued, $1$ when $\pi$ is a real representation, and $-1$ when $\chi$ is real-valued but $\pi$ is not a real representation (see \cite[Chapter 4]{Is76}). For example, if $S_n$ is the symmetric group on $n$ elements, it is known that all of the representations of $S_n$ are real (and, in fact, rational). It follows from results of Frobenius and Schur that the sum of all of the dimensions of the irreducible representations of $S_n$ is equal to the number of elements in $S_n$ which square to the identity element.
Now consider the group $\GL(n, \FF_q)$ of invertible linear transformations on an $n$-dimensional vector space over the field $\FF_q$ with $q$ elements, where $q$ is the power of a prime $p$. Not every irreducible character of $\GL(n, \FF_q)$ is real-valued, however, every real-valued irreducible character of $\GL(n, \FF_q)$ is the character of a real representation. That is, $\ep(\pi) = 0$ or $1$ for every irreducible representation $\pi$ of $\GL(n, \FF_q)$. This follows from Ohmori's result \cite{Oh77} that every irreducible character of $\GL(n, \FF_q)$ has rational Schur index 1 (also proved by Zelevinsky \cite{Ze81}), and a direct proof is given by Prasad \cite{Pr98}. This result also follows from a theorem of Gow \cite{Go83}, which states that when $q$ is the power of an odd prime, the sum of the dimensions of the irreducible representations of $\GL(n, \FF_q)$ is equal to the number of symmetric matrices in the group (also obtained by Klyachko \cite{Kl83} and Macdonald \cite{Mac95} for all $q$). Gow's proof actually implies that the {\em twisted} Frobenius-Schur indicator (see Section \ref{realreps}) of an irreducible representation of $\GL(n, \FF_q)$, with respect to the transpose-inverse automorphism, is always $1$. Using the result that every real-valued character of $\GL(n, \FF_q)$ is the character of a real representation, in Section \ref{realsums}, we compute the sum of the dimensions of the real representations of the group. This gives one way to compare the size of the set of real representations to the size of the set of all representations of $\GL(n, \FF_q)$.
In Section \ref{realreps} of this paper, we consider the groups of symplectic and orthogonal similitudes over a finite field, denoted $\GSp(2n, \FF_q)$ and $\GO^{\pm}(2n, \FF_q)$ respectively, where $q$ is the power of an odd prime. The main results of Section \ref{realreps}, which are Theorems \ref{GSpReal} and \ref{GOReal}, state that, as in the case of the group $\GL(n, \FF_q)$, every real-valued irreducible character of $\GSp(2n, \FF_q)$ or $\GO^{\pm}(2n, \FF_q)$ is the character of a real representation. There is a very direct proof in the case of the groups $\GO^{\pm}(2n, \FF_q)$, which comes from the fact that every irreducible representation of the orthogonal groups $\gO^{\pm}(2n, \FF_q)$, where $q$ is the power of an odd prime, is a real representation, another result of Gow \cite{Go85}. Also in \cite{Go85}, Gow proves that a real-valued character of the symplectic group $\Sp(2n, \FF_q)$, where $q$ is the power of an odd prime, is the character of real representation if and only if the central element $-I$ acts trivially, and in particular, has irreducible real-valued characters which are not the characters of real representations. It may seem to be a surprising result, then, that there are no such characters for the group of symplectic similitudes $\GSp(2n, \FF_q)$ by Theorem \ref{GSpReal}. However, the situation is similar for the finite special linear group, $\SL(n, \FF_q)$, where in the case that $n$ is congruent to $2$ mod $4$ and $q$ is congruent to $1$ mod $4$ the group has real-valued characters which are not characters of real representations (see \cite{Go81}), while all irreducible real-valued characters of $\GL(n, \FF_q)$ are characters of real representations. So, in this instance, the relationship between the characters of $\GSp(2n, \FF_q)$ and $\Sp(2n, \FF_q)$ is similar to the relationship between the characters of $\GL(n, \FF_q)$ and $\SL(n, \FF_q)$.
In Section \ref{realsums}, we apply the results of Section \ref{realreps} to compute expressions for the sum of the dimensions of the real representations of the groups $\GSp(2n, \FF_q)$ and $\GO^{\pm}(2n, \FF_q)$, in Theorems \ref{GSpRealSum} and \ref{GORealSum}. By results of Frobenius and Schur and Theorems \ref{GSpReal} and \ref{GOReal}, we need only count the number of elements in each group which square to the identity element. In the resulting expressions, we obtain a term which is the sum of all of the dimensions of the representations of $\Sp(2n, \FF_q)$ and $\gO^{\pm}(2n, \FF_q)$. This sum has been computed for $\Sp(2n, \FF_q)$ by Gow in the case that $q$ is congruent to $1$ mod $4$ in \cite{Go85}, and by the author in the case that $q$ is congruent to $3$ mod $4$ in \cite{Vi05}. This sum has not been computed in the case of the orthogonal groups $\gO^{\pm}(n, \FF_q)$, however, and so we do this in Section \ref{OrthogSums}, with the result given in Theorem \ref{OrthSum}. We also give several corollaries for the special orthogonal groups in Section \ref{OrthogSums}.
In the last three sections, we focus on the following result of Kowalski \cite[Proposition 5.5]{Ko08}, which he obtains in the context of sieving applications.
\begin{theorem} [Kowalski] \label{KoThm} Let $\bG$ be a split connected reductive group with connected center over $\bar{\FF}_q$, defined over $\FF_q$. Let $d$ be the dimension of $\bG$, let $r$ be the rank of $\bG$, let $W$ be the Weyl group of $\bG$, and let $G = \bG(\FF_q)$. Then, the sum of the degrees of the irreducible complex characters of $G$ is bounded above as follows:
$$ \sum_{\chi \in {\rm Irr}(G)} \chi(1) \leq (q+1)^{(d+r)/2} \left(1 + \frac{2r|W|}{q-1} \right).$$
\end{theorem}
Kowalski also notes that in the cases of the groups $\GL(n, \FF_q)$ and $\GSp(2n, \FF_q)$ ($q$ the power of an odd prime), by results of Gow \cite{Go83} and the author \cite{Vi05}, the factor $1 + \frac{2r|W|}{q-1}$ may be removed from the bound in Theorem \ref{KoThm}. In Section \ref{IneqLemmas}, we prove several inequalities which we apply to improve Theorem \ref{KoThm} in the cases of orthogonal groups. In particular, in Theorem \ref{MainBound}, which is the main result of Section \ref{Bound}, we prove that the factor $1 + \frac{2r|W|}{q-1}$ may be removed from the bound in Theorem \ref{KoThm} for any connected classical group (removing the restriction of being split) with connected center which is defined over $\FF_q$, with $q$ the power of an odd prime. In the case of the unitary group $\U(n, \FF_{q^2})$, this follows from a specific formula for the sum of the degrees of the irreducible characters due to Thiem and the author \cite{ThVi07}. The other groups to check are the special orthogonal groups $\SO(2n+1, \FF_q)$ and the connected orthogonal similitude groups $\GO^{\pm, \circ}(2n, \FF_q)$, which is where our results from Sections \ref{OrthogSums} and \ref{IneqLemmas} are applied.
Finally, in Section \ref{Lower}, we turn to the general case of a finite reductive group with connected center. We use the Gelfand-Graev character to prove, in Proposition \ref{LowerProp}, that if $\bG$ is a connected reductive group with connected center defined over $\FF_q$ (for any $q$) with dimension $d$ and rank $r$, then the sum of the degrees of the irreducible complex characters of $\bG(\FF_q)$ is bounded below by $q^{(d-r)/2}(q-1)^r$. We notice that in the cases of the finite general linear, unitary, and symplectic similitude groups (for $q$ odd), the sum of the degrees of the irreducible characters can actually be bounded above by $q^{(d-r)/2}(q+1)^r$, and we conjecture that this is the case for all finite reductive groups with connected center.
\\
\\
{\bf Acknowledgments. } The author thanks Julio Brau for checking some of the calculations made in Section \ref{OrthogSums}.
\section{Real representations of finite classical groups} \label{realreps}
Let $G$ be a finite group and $\sigma$ an automorphism of $G$ such that $\sigma^2$ is the identity. Let $(\pi, W)$ be an irreducible complex representation of $G$, and suppose that ${^\sigma \pi} \cong \hat{\pi}$, where $\hat{\pi}$ is the contragredient representation of $\pi$ and ${^\sigma \pi}$ is defined by ${^\sigma \pi}(g) = \pi(\sigma(g))$. If $\chi$ is the character of $\pi$, note that ${^\sigma \pi} \cong \hat{\pi}$ if and only if ${^\sigma \chi} = \bar{\chi}$. From the isomorphism ${^\sigma \pi} \cong \hat{\pi}$, there must exist a nondegenerate bilinear form $B_{\sigma}: W \times W \rightarrow \CC$, unique up to scalar by Schur's Lemma, such that
$$ B_{\sigma}({^\sigma \pi}(g) u, \pi(g) v) = B_{\sigma}(u, v), \;\; \text{ for all } g \in G, \, u, v \in W.$$
Since switching the variables of $B_{\sigma}$ gives a bilinear form with the same property, and the bilinear form is unique up to scalar, then $B_{\sigma}$ must be either symmetric or skew-symmetric. That is, we have
$$B_{\sigma}(u, v) = \varepsilon_{\sigma}(\pi) B_{\sigma}(v, u) \;\; \text{ for all } u, v \in W,$$
where $\ep_{\sigma}(\pi) = \pm 1$. If ${^\sigma \pi} \not\cong \hat{\pi}$, then define $\ep_{\sigma}(\pi) = 0$. We will also write $\ep_{\sigma}(\pi) = \ep_{\sigma}(\chi)$. In the case that $\sigma$ is the identity automorphism, then $\ep_{\sigma}(\pi) = \ep(\pi)$ is just the classical Frobenius-Schur indicator of $\pi$. In this case, if $\chi$ is the character of $\pi$, then $\ep(\chi) = 0$ exactly when $\chi$ is not real-valued, and when $\chi = \bar{\chi}$, then $\ep(\chi) = 1$ if $\chi$ is the character of a real representation, and $\ep(\chi) = -1$ otherwise. In the case that $\sigma$ is an order $2$ automorphism, the invariant $\ep_{\sigma}(\pi)$ is called the {\em twisted} Frobenius-Schur indicator of $\pi$, which was first considered by Mackey \cite{Ma58}, and later studied in more detail by Kawanaka and Matsuyama \cite{KaMa90}. For any finite group $G$, let $\Irr(G)$ denote the set of complex irreducible characters of $G$. The twisted Frobenius-Schur indicators satisfy the following identity (see \cite{KaMa90}), which reduces to the classical Frobenius-Schur involution formula in the case that $\sigma$ is trivial:
$$ \sum_{\chi \in \Irr(G)} \ep_{\sigma}(\chi) \chi(1) = \big| \{ g \in G \, \mid \, \sigma(g) = g^{-1} \} \big|.$$
If $G$ is a finite group with $(\pi, W)$ an irreducible complex representation with character $\chi$, and $z$ is a central element of $G$, then by Schur's Lemma $\pi(z)$ acts by a scalar on $W$. We let $\omega_{\pi}(z)$, or $\omega_{\chi}(z)$, denote this scalar. If $\sigma$ is an order $2$ automorphism of $G$ and $z$ is a central element of $G$, we define $G(\sigma, z)$ to be the following central extension of $G$:
$$ G(\sigma, z) = \langle G, \tau \; \mid \; \tau^2 = z, \tau^{-1} g \tau = \sigma(g) \text{ for all } g \in G \rangle.$$
Note that $G$ is an index $2$ subgroup of $G(\sigma, z)$. In the case that $z = 1$, we just have that $G(\sigma, 1)$ is the split extension of $G$ by $\sigma$. We have the following properties of the representations of $G$ and $G(\sigma, z)$, and their Frobenius-Schur indicators, which were proven in \cite[Section 2]{Vi05}.
\begin{proposition} \label{induce} Let $\chi$ be an irreducible character of $G$, and $\chi^{+}$ the induced character ${\rm Ind}_G^{G(\sigma, z)}(\chi)$. Then:
\begin{enumerate}
\item $\chi^{+}$ is irreducible if and only if ${^\sigma \chi} \neq \chi$. In this case, we have $\ep(\chi^{+}) = \ep(\chi) + \omega_{\chi}(z) \ep_{\sigma}(\chi)$.
\item If ${^\sigma \chi} = \chi$, then $\chi^+ = \psi_1 + \psi_2$, where $\psi_1$ and $\psi_2$ are irreducible characters of $G(\sigma, z)$. In this case, we have $\ep(\psi_1) + \ep(\psi_2) = \ep(\chi) + \omega_{\chi}(z) \ep_{\sigma}(\chi)$. Also, each $\psi_i$ is an extension of the character $\chi$, and $\ep(\psi_1) = \ep(\psi_2)$.
\end{enumerate}
\end{proposition}
In statement 2 in the proposition above, although the last claim is not specifically proven in \cite[Section 2]{Vi05}, it follows from the definition of the induced character that each $\psi_i$ is an extension of $\chi$, and it follows from the same calculation as in \cite[Lemma 2.3]{Vi05} that we have $\ep(\psi_i) = \frac{1}{2}(\ep(\chi) + \omega_{\chi}(z)\ep_{\sigma}(\chi))$ for $i=1,2$, so that $\ep(\psi_1) = \ep(\psi_2)$. The next lemma is used in the proof of the first main result of this section.
\begin{lemma} \label{MainLemma}
Let $G$ be a finite group, and $\sigma$ an order 2 automorphism of $G$. Suppose that every character of $G(\sigma, z)$ is real-valued, and that every irreducible character $\chi$ of $G$ satisfies $\ep_{\sigma}(\chi) = \omega_{\chi}(z)$. Then, every real-valued character $\chi$ of $G$ satisfies $\ep(\chi) = 1$, and every irreducible character $\psi$ of $G(\sigma, z)$ satisfies $\ep(\psi) = 1$.
\end{lemma}
\begin{proof} Let $\chi$ be an irreducible real-valued character of $G$, so that $\ep(\chi) = \pm 1$. We are assuming that $\ep_{\sigma}(\chi) = \omega_{\chi}(z) = \pm 1$, so that in particular we have ${^\sigma \chi} = \bar{\chi}$. Since $\chi$ is real-valued, then ${^\sigma \chi} = \chi$, and by Proposition \ref{induce}, $\chi^+ = \psi_1 + \psi_2$, where
$$ \ep(\psi_1) + \ep(\psi_2) = \ep(\chi) + \omega_{\chi}(z) \ep_{\sigma}(\chi).$$
Now, we have $\omega_{\chi}(z) \ep_{\sigma}(\chi) = 1$ by assumption, and $\ep(\chi) = \pm 1$, $\ep(\psi_1) = \ep(\psi_2) = \pm 1$, by Proposition \ref{induce} and by assumption. The only possibility is $\ep(\chi) = 1$ and $\ep(\psi_i) = 1$.
For the second statement, every irreducible character of $G(\sigma, z)$ is either extended or induced from an irreducible character of $G$, since $G$ is an index $2$ subgroup. We have just checked that whenever $\psi$ is an irreducible of $G(\sigma, z)$ which is extended from $G$, then $\ep(\psi) = 1$. Now assume that $\psi = {\rm Ind}_G^{G(\sigma, z)}(\chi)$ for some irreducible $\chi$ of $G$. By Proposition \ref{induce}, we must have ${^\sigma \chi} \neq \chi$, and so $\chi \neq \bar{\chi}$ since we are assuming $\ep_{\sigma}(\chi) = \pm 1$. From Proposition \ref{induce}(1), we have $\ep(\psi) = \ep(\chi) + \omega_{\chi}(z) \ep_{\sigma}(\chi)$. Now, $\ep(\chi) = 0$ since $\chi \neq \bar{\chi}$, and $\omega_{\chi}(z) \ep_{\sigma}(\chi) = 1$ by assumption. Thus $\ep(\psi) = 1$.
\end{proof}
Let $\FF_q$ be a finite field with $q$ elements, where $q$ is the power of an odd prime. Let $V = \FF_q^{2n}$, and let $\langle \cdot, \cdot \rangle$ be a nondegenerate skew-symmetric form on $V$ (of which there is only one equivalence class, by \cite[Theorem 2.10]{Gr02}). The group of all elements $g$ of $\GL(V)$ which leave $\langle \cdot, \cdot \rangle$ invariant up to a scalar multiple is called the {\em symplectic group of similitudes on $\FF_q^{2n}$} (or the {\em conformal symplectic group on $\FF_q^{2n}$}) which we will denote as $\GSp(2n, \FF_q)$ (this group is also denoted as $\CSp(2n, \FF_q)$). That is, $\GSp(2n, \FF_q)$ is the group of elements $g$ of $\GL(2n, \FF_q)$ such that $\langle gv, gw \rangle = \mu(g) \langle v, w \rangle$ for all $v, w \in V$, where $\mu(g) \in \FF_q^{\times}$ is a scalar depending only on $g$. Then $\mu: \GSp(2n, \FF_q) \rightarrow \FF_q^{\times}$ is a character called the {\em similitude character}, and the symplectic group over $\FF_q$, denoted $\Sp(2n, \FF_q)$, is the kernel of $\mu$.
Now consider the case that $G = \GSp(2n, \FF_q)$, where $q$ is the power of an odd prime, and we let $\sigma$ be the order $2$ automorphism of $G$ defined by $\sigma(g) = \mu(g)^{-1} g$, where $\mu$ is the similitude character. We have the following results for the group $\GSp(2n, \FF_q)$ proven previously by the author, the first statement in \cite[Theorem 6.2]{Vi05}, and the second in \cite{Vi04}.
\begin{proposition} \label{factor}
Let $G = \GSp(2n, \FF_q)$, where $q$ is the power of an odd prime. Then:
\begin{enumerate}
\item Every irreducible character $\chi$ of $G$ satisfies $\ep_{\sigma}(\chi) = \omega_{\chi}(-I)$, where $\sigma(g) = \mu(g)^{-1} g$.
\item For any $g \in G$, we may write $g = h_1 h_2$ such that $h_1^2 = I$, $\mu(h_1) = -1$, and $h_2^2 = \mu(g) I, \mu(h_2) = - \mu(g)$.
\end{enumerate}
\end{proposition}
We are now ready to prove the following result.
\begin{theorem} \label{GSpReal} Let $G = \GSp(2n, \FF_q)$, and $q$ the power of an odd prime. Then:
\begin{enumerate}
\item[i.] If $\chi$ is an irreducible real-valued character of $G$, then $\ep(\chi) = 1$.
\item[ii.] If $\sigma$ is the order $2$ automorphism of $G$ defined by $\sigma(g) = \mu(g)^{-1} g$, then every irreducible character $\psi$ of $G(\sigma, -I)$ satisfies $\ep(\psi) = 1$.
\end{enumerate}
\end{theorem}
\begin{proof} Since $\ep_{\sigma}(\chi) = \omega_{\chi}(-I)$ for every irreducible character $\chi$ of $G$, then by Lemma \ref{MainLemma}, it is enough to show that every irreducible character of $G(\sigma, -I)$ is real-valued. Equivalently, we must show that every element of $G(\sigma, -I)$ is conjugate to its inverse. We prove this by applying the factorization given by Proposition \ref{factor}(2).
First, let $g \in G \subset G(\sigma, -I)$. Write $g = h_1 h_2$ as in Proposition \ref{factor}(2). Then,
$$g^{-1} = h_2^{-1} h_1^{-1} = \mu(g)^{-1} h_2 h_1.$$
If we conjugate $g$ by $h_1 \tau$, we obtain
$$ (h_1 \tau) g (h_1 \tau)^{-1} = h_1 (\tau g \tau^{-1}) h_1 = h_1 (\mu(g)^{-1} h_1 h_2) h_1 = \mu(g)^{-1} h_2 h_1 = g^{-1},$$
and so $g$ is conjugate to $g^{-1}$. Now let $g \tau \in G(\sigma, -I) \setminus G$, and again write $g = h_1 h_2$ as in Proposition \ref{factor}(2). In this case,
$$(g\tau)^{-1} = \tau^{-1}\mu(g)^{-1} h_2 h_1 = -\mu(g) \tau h_2 h_1 = -\mu(g) \mu(h_2)^{-1} h_2 h_1 \tau = h_2 h_1 \tau,$$
since $\mu(h_2) = - \mu(g)$. Now conjugate $g\tau$ by $h_1$, and we have
$$ h_1(g\tau)h_1 = h_2 \tau h_1 = h_2 h_1 \tau = (g \tau)^{-1}.$$
So, $g\tau$ is conjugate to its inverse, and all characters of $G(\sigma, -I)$ are real-valued.
\end{proof}
Gow \cite{Go83} proved that the split extension of $\GL(n, \FF_q)$ by the transpose-inverse automorphism has the property that all of its characters are characters of real representations. This is analogous to our result for the central extension of $\GSp(2n, \FF_q)$ in Theorem \ref{GSpReal}(ii).
Now consider a nondegenerate symmetric form $\langle \cdot, \cdot \rangle$ on a vector space $V = \FF_q^n$ over a finite field $\FF_q$, where $q$ is odd. Similar to the symplectic case, the {\em orthogonal group of similitudes on $\langle \cdot, \cdot \rangle$} ({\em or conformal orthogonal group}) is the group of elements $g \in GL(V)$ which leave the form $\langle \cdot, \cdot \rangle$ invariant up to a scalar multiple. When the form does not need to be emphasized, we will denote this group by $\GO(n, \FF_q)$ (the group is sometimes denoted ${\rm CO}(n, \FF_q)$). As in the symplectic case, we let $\mu$ denote the similitude character, where if $g \in \GO(n, \FF_q)$, then $\langle gu, gv \rangle = \mu(g) \langle u, v \rangle$ for all $u, v \in V$. In particular, the orthogonal group $\gO(n, \FF_q)$ for the form $\langle \cdot, \cdot \rangle$ is the kernel of the homomorphism $\mu: \GO(n, \FF_q) \rightarrow \FF_q^{\times}$.
When $n = 2m+1$ is odd, any symmetric form on $V$ gives an isomorphic group (see \cite[Chapter 9]{Gr02}), and in fact the group $\GO(2m+1, \FF_q)$ is just the direct product of its center with the special orthogonal group $\SO(2m+1, \FF_q)$ \cite[Lemma 1.3]{Sh80}. So, we will not consider the odd case. When $n = 2m$ is even, however, there are two different equivalence classes of symmetric forms, split and nonsplit (see \cite[Chapter 9]{Gr02} or \cite[Section 15.3]{DiMi91}), giving two non-isomorphic groups denoted $\GO^+(2m, \FF_q)$ and $\GO^-(2m, \FF_q)$, respectively. When considering both cases, and the type of form makes a difference in the result, we will write $\GO^{\pm}(2m, \FF_q)$. Similarly, we let ${\rm O}^{+}(n, \FF_q)$ denote a split orthogonal group, ${\rm O}^-(n, \FF_q)$ the non-split orthogonal group, and in the case that $n$ is odd, we just write $\gO(n, \FF_q)$ for the unique orthogonal group. When considering the split and non-split orthogonal groups at the same time, we use the notation $\gO^{\pm}(n, \FF_q)$. We also use $\gO(n, \FF_q)$ for any of these orthogonal groups when the general case is considered.
We now study the real-valued characters of the groups $\GO^{\pm}(2n, \FF_q)$. We could proceed in a fashion similar to the case of the group $\GSp(2n, \FF_q)$, by defining an automorphism $\sigma(g) = \mu(g)^{-1} g$ on $\GO^{\pm}(2n, \FF_q)$. The exact same argument will work, since in \cite{Vi06} the author proved the appropriate factorization result, and $\ep_{\sigma}(\chi) = 1$ for every irreducible character $\chi$ of any orthogonal similitude group. However, a more direct argument is possible here, coming from the result of Gow \cite[Theorem 1]{Go85} that for odd $q$, every irreducible character of any orthogonal group $\gO(n, \FF_q)$ has Frobenius-Schur indicator $1$. We give this direct argument in the following, and the proof is very similar to that of \cite[Theorem 2]{Vi06}.
\begin{theorem} \label{GOReal} Let $G = \GO^{\pm}(2n, \FF_q)$, where $q$ is the power of an odd prime. Then every irreducible real-valued character $\chi$ of $G$ satisfies $\ep(\chi) = 1$.
\end{theorem}
\begin{proof} Let $\chi$ be an irreducible real-valued character of $G$, which is the character of the representation $(\pi, W)$. Then there is a nondegenerate bilinear form $B$ on $W$, unique up to scalar, such that
$$B(\pi(g) u, \pi(g) v) = B(u, v) \quad \text{for all } g \in G, \, u, v \in W,$$
which must be either symmetric or skew-symmetric, and $\ep(\chi) = 1$ is equivalent to $B$ being a symmetric form.
Now let $Z$ be the center of $G$, which consists of all nonzero scalar matrices, where we have $\mu(bI) = b^2$ for any $b \in \FF_q^{\times}$. Let $H = Z \cdot \gO^{\pm}(2n, \FF_q)$, which is an index $2$ subgroup of $G$, since $G/H \cong \FF_q^{\times}/ (\FF_q^{\times})^2$. Note that any irreducible representation of $H$ is an extension of an irreducible of $\gO^{\pm}(2n, \FF_q)$, since $Z$ is central.
Since $H$ is an index $2$ subgroup of $G$, then $\pi$ is either extended or induced from an irreducible representation of $H$. First suppose that $\pi$ is extended from an irreducible $\rho$ of $H$, with character $\psi$. Then $\psi$ is the extension of a real-valued irreducible character $\theta$ of $\gO^{\pm}(2n, \FF_q)$, and say $\theta$ is the character of the representation $(\phi, W)$. By the result of Gow \cite{Go85}, we have $\ep(\theta) = 1$, which means there is a nondegenerate bilinear form on $W$, unique up to scalar, which is invariant under the action of $G$ under $\phi$, which must be symmetric. But the form $B$ is such a form, since $(\pi, W)$ is an extension of $(\rho, W)$. Therefore, $B$ must be symmetric and $\ep(\chi) = 1$ in this case.
Now suppose that $\pi$ is induced from an irreducible representation of $H$, which means that $\pi$ restricted to $H$ is isomorphic to the direct sum of two irreducible representations, $(\rho_1, W_1)$ and $(\rho_2, W_2)$ of $H$, which are extended from irreducible representations $(\phi_1, W_1)$ and $(\phi_2, W_2)$ of $\gO^{\pm}(2n, \FF_q)$, respectively. If $\theta_1$ is the character of $\phi_1$, then again by Gow's result, $\ep(\theta_1) = 1$, and there is a nondegenerate bilinear form on $W_1$, unique up to scalar, which is invariant under the action of $\phi_1$ and is symmetric. We can make the same conclusion as in the previous case if we can show that the form $B$ is nondegenerate on $W_1$, since then $B$ would be symmetric on a nonzero subspace, and thus symmetric everywhere. If $B$ is nondegenerate on $W_1 \times W_2$, then for $u \in W_1$, $v \in W_2$, and $g \in G$, we would have $B(\pi(g) u, \pi(g) v) = B(\phi_1(g) u_1, \phi_2(g) u_2)$, which would imply $\phi_1 \cong \hat{\phi_2} \cong \phi_2$. This, in turn, would imply that $\rho_1 \cong \rho_2$, but it is impossible for an irreducible representation to restrict to the direct sum of $2$ isomorphic representations of an index $2$ subgroup \cite[Corollary 6.19]{Is76}. Then $B$ must be degenerate on $W_1 \times W_2$, and so must be nondegenerate on $W_1 \times W_1$, since it is nondegenerate on $W$. Thus, $B$ must be symmetric, and $\ep(\chi) = 1$.
\end{proof}
In summary, when $q$ is odd, all real-valued irreducible characters of the groups $\GL(n, \FF_q)$ (see the Introduction), $\GSp(2n, \FF_q)$, and $\GO^{\pm}(2n, \FF_q)$ have Frobenius-Schur indicator $1$, and the same is also true for the special orthogonal groups $\SO(2n+1, \FF_q)$ \cite[Theorem 2]{Go85}. These are all examples of groups of $\FF_q$-points of classical groups with connected center (although the algebraic groups $\GO^{\pm}(2n)$ are disconnected). Another example one might consider is the finite unitary group $\U(n, \FF_{q^2})$, but this is known to have irreducible characters with Frobenius-Schur indicator equal to $-1$ (see \cite{Oh96}, for example).
\section{Degree sums of real-valued characters} \label{realsums}
As mentioned in the Introduction, every real-valued irreducible character of the group $\GL(n, \FF_q)$ is the character of a real representation. It follows from the classical Frobenius-Schur involution formula (see \cite[Chapter 4]{Is76}) that when $G = \GL(n, \FF_q)$, we have
$$ \sum_{\chi \in {\rm Irr}(G) \atop{ \chi \; \RR \text{-valued}}} \chi(1) = | \{ g \in G \, \mid \; g^2 = I \}|.$$
We may count the number of elements in $\GL(n, \FF_q)$ which square to the identity by summing the indices of the centralizer of an element in each order 2 conjugacy class. We will make such counts for several groups, and the following notation will be helpful for the expressions obtained. For any $q > 1$, and any integers $m, k \geq 0$ such that $m \geq k$, the {\em $q$-binomial} or {\em Gaussian binomial coefficients} are defined as
$$ \binom{m}{k}_q = \frac{(q^m - 1) (q^{m-1} - 1) \cdots (q^{m-k+1} - 1)}{(q^k -1)(q^{k-1} -1) \cdots (q-1)}.$$
It follows from induction and the identity $\binom{m}{k}_q = \binom{m-1}{k}_q + q^{m-k} \binom{m-1}{k-1}_q$, $m \geq 1$, that the Gaussian binomial coefficients are polynomials in $q$. The Gaussian binomial coefficients have analogous properties to the standard binomial coefficients, for example, we have $\binom{m}{k}_q = \binom{m}{m-k}_q$.
We now count the number of elements in $\GL(n, \FF_q)$ which square to the identity, when $q$ is the power of an odd prime. Each such element in $\GL(n, \FF_q)$ must have elementary divisors only of the form $x \pm 1$, and so is conjugate to a diagonal element with only $1$'s and $-1$'s on the diagonal. An element in the conjugacy class of elements with exactly $k$ eigenvalues which are equal to $1$, and $n-k$ eigenvalues equal to $-1$, has centralizer isomorphic to $\GL(k, \FF_q) \times \GL(n-k, \FF_q)$. Since $|\GL(m, \FF_q)| = q^{m(m-1)/2} \prod_{i=1}^m (q^i -1)$, then the size of the conjugacy class containing the elements whose square is $I$ and which have exactly $k$ eigenvalues equal to $1$ is
$$ \frac{|\GL(n, \FF_q)|}{|\GL(k, \FF_q) \times \GL(n-k, \FF_q)|} = q^{k(n-k)} \frac{\prod_{i=1}^n (q^i -1)}{\prod_{i=1}^k (q^i -1) \prod_{i=1}^{n-k} (q^i -1)} = q^{k(n-k)} \binom{n}{k}_q.$$
Summing the size of each conjugacy class, for $0 \leq k \leq n$, gives the result
\begin{equation} \label{GLReal}
\sum_{\chi \in {\rm Irr}(G) \atop{ \chi \; \RR \text{-valued}}} \chi(1) = \sum_{k=0}^n q^{k(n-k)} \binom{n}{k}_q.
\end{equation}
Using the fact that as a polynomial in $q$, the degree of $\binom{n}{k}_q$ is $k(n-k)$, we calculate that the sum in (\ref{GLReal}), as a polynomial in $q$, has degree $n^2/2$ when $n$ is even, and degree $(n^2 - 1)/2$ when $n$ is odd. On the other hand, from a result of Gow \cite[Theorem 4]{Go83}, the sum of the degrees of all of the irreducible characters of $\GL(n,\FF_q)$ as a polynomial in $q$ has degree $(n^2 + n)/2$. The ratio of the sum of the degrees of real-valued characters over the sum of all degrees in this case is roughly $q^{n/2}$ when $n$ is even and $q^{(n+1)/2}$ when $n$ is odd. This gives a quick comparison in measure of the set of real-valued irreducible characters of $\GL(n,\FF_q)$ to the set of all irreducible characters, relative to degree size.
By Theorem \ref{GSpReal}, when $q$ is odd, every real-valued character of $\GSp(2n, \FF_q)$ is the character of a real representation, and so the sum of the degrees of the real-valued characters of $\GSp(2n, \FF_q)$ is equal to the number of elements in $\GSp(2n, \FF_q)$ which square to $I$. Similar to the case of $\GL(n, \FF_q)$, we may count these elements to obtain an expression in $q$ for this character degree sum.
\begin{theorem} \label{GSpRealSum}
Let $q$ be the power of an odd prime, let $G = \GSp(2n, \FF_q)$, and let $H = \Sp(2n, \FF_q)$. Then, the sum of the degrees of the real-valued characters of $G$ is given by
\begin{align*}
\sum_{\chi \in {\rm Irr}(G) \atop{ \chi \; \RR \text{-valued}}} \chi(1) = \sum_{k=0}^n q^{2k(n-k)} \binom{n}{k}_{q^2} + \sum_{ \chi \in {\rm Irr}(H)} \chi(1) & = \sum_{k=0}^n q^{2k(n-k)} \binom{n}{k}_{q^2} + \frac{|\Sp(2n, \FF_q)|}{|\GL(n, \FF_q)|} \\
& = \sum_{k=0}^n q^{2k(n-k)} \binom{n}{k}_{q^2} + q^{n(n+1)/2} \prod_{i=1}^n (q^i + 1).
\end{align*}
\end{theorem}
\begin{proof} We must count the number of elements of $\GSp(2n, \FF_q)$ such that $g^2 = I$. Such an element must satisfy $\mu(g) = \pm 1$. If $\mu(g) = 1$, then we have $g \in \Sp(2n, \FF_q)$, and we may count such elements using results on the conjugacy classes and their centralizers in $\Sp(2n, \FF_q)$ due to Wall \cite{Wa62}. An element in $\Sp(2n, \FF_q)$ which squares to the identity must have elementary divisors only of the form $x \pm 1$, and thus must be conjugate in $\GL(2n, \FF_q)$ to a diagonal element with only $1$'s and $-1$'s on the diagonal. By \cite[p. 36, Case B]{Wa62}, the only such elements which are in $\Sp(2n, \FF_q)$ must have an even number $2k$ of eigenvalues equal to $1$, and so $2(n-k)$ eigenvalues equal to $-1$, and there is a unique such conjugacy class. Also by \cite[p. 36, Case B]{Wa62}, an element in this conjugacy class has centralizer in $\Sp(2n, \FF_q)$ with size equal to $|\Sp(2k, \FF_q) \times \Sp(2(n-k), \FF_q)|$. Using the fact that $|\Sp(2m, \FF_q)| = q^{m^2} \prod_{i=1}^m (q^{2i} - 1)$, we have that the size of the conjugacy class of $\Sp(2n, \FF_q)$ consisting of elements with $2k$ eigenvalues equal to $1$ and $2(n-k)$ eigenvalues equal to $-1$ and which square to the identity is
$$ \frac{|\Sp(2n, \FF_q)|}{|\Sp(2k, \FF_q) \times \Sp(2(n-k), \FF_q)|} = q^{2k(n-k)} \frac{\prod_{i=1}^n (q^{2i} - 1)}{\prod_{i=1}^k (q^{2i}-1) \prod_{i=1}^{n-k} (q^{2i}-1)} = q^{2k(n-k)} \binom{n}{k}_{q^2}.$$
Thus, the total number of elements in $\GSp(2n, \FF_q)$ such that $g^2 = I$ and $\mu(g) = 1$ is
\begin{equation} \label{FirstTerm}
\sum_{k=0}^n q^{2k(n-k)} \binom{n}{k}_{q^2}.
\end{equation}
Now suppose $g \in \GSp(2n, \FF_q)$ such that $g^2 = I$ and $\mu(g) = -1$. In other words, $g^2 = -\mu(g) I$. By \cite[Proposition 4]{Vi04}, there is a unique such conjugacy class in $\GSp(2n, \FF_q)$, and by \cite[Proposition 3.2]{Vi05}, the centralizer in $\GSp(2n, \FF_q)$ of any element in this conjugacy class has size $(q-1)|\GL(n, \FF_q)|$. Therefore, the total number of elements in $\GSp(2n, \FF_q)$ such that $g^2 = I$ and $\mu(g)=-1$ is
\begin{equation} \label{SecondTerm}
\frac{|\GSp(2n, \FF_q)|}{(q-1)|\GL(n, \FF_q)|} = \frac{|\Sp(2n, \FF_q)|}{|\GL(n, \FF_q)|} = q^{n(n+1)/2} \prod_{i=1}^n (q^i + 1).
\end{equation}
From \cite[Theorem 3]{Go85} and \cite[Corollary 6.1]{Vi05}, the expression (\ref{SecondTerm}) is also the sum of the degrees of all irreducible characters of the symplectic group $\Sp(2n, \FF_q)$ when $q$ is odd. Finally, the total number of elements in $\GSp(2n, \FF_q)$ which square to $I$ is the sum of (\ref{FirstTerm}) and (\ref{SecondTerm}), as desired.
\end{proof}
From Theorem \ref{GSpRealSum}, the sum of the degrees of the real-valued characters of $\GSp(2n, \FF_q)$ is a polynomial in $q$ of degree $n^2 + n$, while from \cite[Corollary 6.1]{Vi05}, the sum of the degrees of all of the irreducible characters of $\GSp(2n, \FF_q)$ is a polynomial in $q$ of degree $n^2 + n + 1$. In this case, the ratio of the sum of all degrees to the sum of the degrees of real-valued characters is roughly $q$, which is quite different from the $\GL(n, \FF_q)$ case.
If we let $H = \Sp(2n, \FF_q)$, $q$ odd, then it follows from the Frobenius-Schur involution formula that the expression (\ref{FirstTerm}) is exactly
\begin{equation} \label{SpReal}
\sum_{k=0}^n q^{2k(n-k)} \binom{n}{k}_{q^2} = \sum_{\chi \in \Irr(H)} \ep(\chi) \chi(1) = \sum_{\chi \in \Irr(H) \atop{ \ep(\chi) = 1 }} \chi(1) - \sum_{\chi \in \Irr(H) \atop{ \ep(\chi) = -1}} \chi(1).
\end{equation}
In the case that $q \equiv 1($mod $4)$, it follows from a result of Wonenburger \cite[Theorem 2]{Wo66} and the fact that $-1$ is a square in $\FF_q$, that $\ep(\chi) = \pm 1$ for every irreducible character of $\Sp(2n, \FF_q)$. When $q \equiv 3($mod $4)$, however, there are characters of $\Sp(2n, \FF_q)$ which are not real-valued, by \cite[Lemma 5.3]{FeZu82}. Combining these facts with (\ref{SpReal}) and Theorem \ref{GSpRealSum}, we obtain the following.
\begin{corollary} \label{SpSum}
Let $q$ be the power of an odd prime, let $H = \Sp(2n, \FF_q)$, and let $G = \GSp(2n, \FF_q)$. Then,
$$ 2 \sum_{\chi \in \Irr(H) \atop{ \ep(\chi) = 1 }} \chi(1) + \sum_{\chi \in \Irr(H) \atop{ \ep(\chi) = 0 }} \chi(1) = \sum_{\chi \in {\rm Irr}(G) \atop{ \chi \; \RR \text{-valued}}} \chi(1).$$
If $q \equiv 1($mod $4)$, then for $\delta = 1$ or $-1$, we have
$$ \sum_{\chi \in \Irr(H) \atop{ \ep(\chi) = \delta }} \chi(1) = \frac{1}{2} \left(q^{n(n+1)/2} \prod_{i=1}^n (q^i + 1) + \delta \sum_{k=0}^n q^{2k(n-k)} \binom{n}{k}_{q^2} \right).$$
\end{corollary}
We now turn to the orthogonal similitude groups, $\GO^{\pm}(2n, \FF_q)$. By Theorem \ref{GOReal}, every real-valued character of $\GO^{\pm}(2n, \FF_q)$ has Frobenius-Schur indicator $1$, and so again we find the sum of the degrees of the real-valued characters of this group by counting the number of elements which square to the identity. Just as in the case of $\GSp(2n, \FF_q)$, such an element must either be orthogonal, or skew-orthogonal (that is, has similitude $-1$). We prove that in the group $\GO^+(2n, \FF_q)$, the latter elements form a single conjugacy class, whereas there are no such elements in $\GO^-(2n, \FF_q)$. These facts follow from the classification of conjugacy classes in $\GO^{\pm}(2n, \FF_q)$ due to Shinoda \cite{Sh80}, although we are also able to give an elementary proof, which is essentially the same as the proof of a result of Gow \cite[Lemma 1]{Go88} for symplectic groups. We also find the centralizer of a skew-orthogonal order $2$ element in the group $\GO^+(2n, \FF_q)$. The following result could certainly be obtained for a more general type of conjugacy class in the group of orthogonal similitudes over an arbitrary field (with characteristic not $2$), similar to the results for groups of symplectic similitudes in \cite[Proposition 4]{Vi04} and \cite[Proposition 3.2]{Vi05}, but we do not need such a result here.
\begin{lemma} \label{GOConjClass} Let $G = \GO^{\pm}(2n, \FF_q)$, where $q$ is the power of an odd prime. Consider the set $K = \{ g \in G \, \mid \, g^2 = I, \mu(g) = -1 \}$. Then we have the following:
\begin{enumerate}
\item If $G = \GO^+(2n, \FF_q)$, then the set $K$ forms a single conjugacy class in $G$. The centralizer in $G$ of any element of $K$ is isomorphic to $\FF_q^{\times} \times \GL(n, \FF_q)$.
\item If $G = \GO^-(2n, \FF_q)$, then the set $K$ is empty.
\end{enumerate}
\end{lemma}
\begin{proof} Let $V = \FF_q^{2n}$, let $\langle \cdot, \cdot \rangle$ denote a nondegenerate symmetric form on $V$, and let $G$ be the orthogonal group of similitudes on $\langle \cdot, \cdot \rangle$. If $g \in G$ such that $g^2 = I$ and $\mu(g) = -1$, then the only eigenvalues of $g$ are $1$ and $-1$. Let $V_1$ and $V_{-1}$ be the eigenspaces of $g$ for $1$ and $-1$, respectively, and note that the assumption $\mu(g) = -1$ forces $V_1$ and $V_{-1}$ to be totally isotropic spaces with respect to $\langle \cdot, \cdot \rangle$. This implies both $V_1$ and $V_{-1}$ have dimension $n$, and through the inner product $\langle \cdot, \cdot \rangle$, $V_{-1}$ is isomorphic to the dual space of $V_1$. If $v_1, \ldots v_n$ is any basis of $V_1$, choose $w_1, \ldots, w_n$ to be a dual basis of $V_{-1}$, so that $\langle v_i, w_j \rangle = \delta_{ij}$. With respect to this basis, the form $\langle \cdot, \cdot \rangle$ can be represented by the matrix $\left( \begin{array} {cc} 0 & I \\ I & 0 \end{array} \right)$, which means that it must be a split form on $V$. That is, assuming that an element $g \in G$ with the above properties exists, we cannot have that $G$ is a group corresponding to a non-split form, which proves statement $2$. We may now assume $G = \GO^+(2n, \FF_q)$. With respect to the basis we have chosen, $g$ must be the element $\left( \begin{array} {cc} I & 0 \\ 0 & -I \end{array} \right)$, thus determining the conjugacy class of $g$ uniquely.
Now, without loss of generality, we may assume the form $\langle \cdot, \cdot \rangle$ is given by the matrix $\left( \begin{array} {cc} 0 & I \\ I & 0 \end{array} \right)$, and the element $g = \left( \begin{array} {cc} I & 0 \\ 0 & -I \end{array} \right)$. It is a direct computation that the centralizer of $g$ in $G$ consists of the following set of elements:
$$ \left\{ \left( \begin{array} {cc} A & 0 \\ 0 & \lambda ({^T A}^{-1}) \end{array} \right) \, \mid \, A \in \GL(n, \FF_q), \lambda \in \FF_q^{\times} \right\}.$$
By mapping the element $\left( \begin{array} {cc} A & 0 \\ 0 & \lambda ({^T A}^{-1}) \end{array} \right)$ to $(\lambda, A)$, we see that this group is isomorphic to $\FF_q^{\times} \times \GL(n, \FF_q)$, completing the proof of statement 1.
\end{proof}
We now use the above result to give expressions for the sums of the degrees of real-valued characters of the groups $\GO^{\pm}(2n, \FF_q)$. The key result due to Gow \cite[Theorem 1]{Go85} we use here, which we also used in the previous section, is that any irreducible character $\chi$ of any finite orthogonal group $\gO(n, \FF_q)$ (where $q$ is odd) satisfies $\ep(\chi) = 1$.
\begin{theorem} \label{GORealSum} The sum of the degrees of the irreducible real-valued characters of the groups $\GO^{\pm}(2n, \FF_q)$, where $q$ is the power of an odd prime, are given as follows:
\begin{enumerate}
\item If $G = \GO^+(2n, \FF_q)$ and $H = \gO^+(2n, \FF_q)$, then
$$\sum_{\chi \in {\rm Irr}(G) \atop{ \chi \; \RR \text{-valued}}} \chi(1) = \sum_{ \chi \in {\rm Irr}(H)} \chi(1) + \frac{|\gO^+(2n, \FF_q)|}{|\GL(n, \FF_q)|} = \sum_{ \chi \in {\rm Irr}(H)} \chi(1) + 2q^{n(n-1)/2} \prod_{i=1}^{n-1} (q^i + 1).$$
\item If $G = \GO^-(2n, \FF_q)$, and $H = \gO^-(2n, \FF_q)$, then
$$ \sum_{\chi \in {\rm Irr}(G) \atop{ \chi \; \RR \text{-valued}}} \chi(1) = \sum_{ \chi \in {\rm Irr}(H)} \chi(1).$$
\end{enumerate}
\end{theorem}
\begin{proof} From Theorem \ref{GOReal}, the sum of the degrees of the real-valued characters of $\GO^{\pm}(2n, \FF_q)$ is equal to the number of elements $g$ in the group such that $g^2 = I$. In the case $G = \GO^+(2n, \FF_q)$, we add the number of such elements in the group $\gO^+(2n, \FF_q)$ to the number of these elements such that $\mu(g) = -1$. The number of elements in $\gO^+(2n, \FF_q)$ which square to the identity is equal to the sum of the degrees of all of the irreducible characters of $\gO^{+}(2n, \FF_q)$, by the Frobenius-Schur formula and \cite[Theorem 1]{Go85}. By Lemma \ref{GOConjClass}(1), the number of order $2$ elements in $\GO^+(2n, \FF_q)$ such that $\mu(g) = -1$ is equal to the index of the centralizer of the unique conjugacy class of such elements in the group, which is given by
$$ \frac{|\GO^+(2n, \FF_q)|}{|\FF_q^{\times} \times \GL(n, \FF_q)|} = \frac{|\gO^+(2n, \FF_q)|}{|\GL(n, \FF_q)|} = \frac{ 2q^{n(n-1)} (q^n - 1) \prod_{i=1}^{n-1} (q^{2i} - 1)}{ q^{n(n-1)/2} \prod_{i=1}^n(q^i - 1)} = 2q^{n(n-1)/2} \prod_{i=1}^{n-1} (q^i + 1).$$
Adding this quantity to the sum of the degrees of the irreducible characters of $\gO^+(2n, \FF_q)$ gives the result for this case.
In the case $G = \GO^-(2n, \FF_q)$, by Lemma \ref{GOConjClass}(2), the only elements in $G$ which square to the identity are in the group $H = \gO^-(2n, \FF_q)$. So, the sum of the degrees of the real-valued irreducible characters of $G$ is equal to the number of elements in $H$ which square to the identity, which, as in the previous case, is equal to the sum of the degrees of the irreducible characters of $H$.
\end{proof}
In Theorem \ref{OrthSum} below, we find expressions in $q$ for the sums of the degrees of the orthogonal groups $\gO(n, \FF_q)$, which, combined with Theorem \ref{GORealSum}, give us expressions for the sums of the degrees of the real-valued irreducible characters of $\GO^{\pm}(2n, \FF_q)$ as polynomials in $q$.
There are several similarities between Theorems \ref{GSpRealSum} and \ref{GORealSum}(1). Both sums have a part corresponding to the sum of the degrees of the irreducible characters of the symplectic or orthogonal subgroups, respectively. Also, both sums have a part which is the index of a general linear group as a subgroup of the symplectic or orthogonal subgroups, which corresponds to the size of a unique conjugacy class. This part of the sum is conveniently factorizable as a polynomial in $q$. The main difference is that this part of the sum corresponds to the sum of the degrees of the irreducible characters of the symplectic group in Theorem \ref{GSpRealSum}, but it does not correspond to the sum of the degrees for the orthogonal group in Theorem \ref{GORealSum}(1). This transposed difference of the role of this part of the sum is essentially due to the difference between the defining bilinear forms, one being skew-symmetric, and the other symmetric, which switches the roles of the two different types of order $2$ elements which must be counted in each case. As a result, we will see in the next section that the sum of the degrees of the irreducible characters of a finite orthogonal group is not, in general, a conveniently factorizable polynomial in $q$, but rather another sum involving Gaussian binomial coefficients.
\section{Character degree sums for orthogonal groups} \label{OrthogSums}
In this section we compute an expression for the sum of the degrees of the irreducible characters of the orthogonal groups over finite fields of odd characteristic. By the main theorem of \cite{Go85}, every irreducible character of such a group is the character of a real representation, and so by the classical Frobenius-Schur involution formula, the sum of the degrees of the irreducible characters is equal to the number of elements which square to the identity in the group.
So, in order to find the character degree sum for these finite orthogonal groups, we must count the number of involutions, which we do using the results on conjugacy classes in these groups due to Wall \cite{Wa62}.
If $g \in {\rm O}(n, \FF_q)$ and $g^2 = I$, then any elementary divisor of $g$ must be either $x+1$ or $x-1$, so over $\GL(n,\FF_q)$ is conjugate to a diagonal matrix with only $1$'s and $-1$'s on the diagonal. By the results of Wall \cite[p. 36, Case B]{Wa62} (see also \cite[Section 2.2]{Fu00}), if there are $j$ eigenvalues of $g$ equal to $1$, and $n-j$ eigenvalues equal to $-1$, where $0 < j < n$, then there are exactly $2$ conjugacy classes in ${\rm O}(n, \FF_q)$ of such elements which square to the identity. Let us label these conjugacy classes by the notation $(j, n-j)^{\pm}$, where $j$ is the number of eigenvalues equal to $1$, $n-j$ the number of eigenvalues equal to $-1$, and the sign $\pm$ distinguishes the two corresponding conjugacy classes. The following summarizes the results we use from \cite{Wa62} for these conjugacy classes and the order of the centralizers of elements in these conjugacy classes.
\begin{proposition} [Wall] \label{OrthCent}
Let $G$ be an orthogonal group over $\FF_q$ with $q$ the power of an odd prime.
\begin{enumerate}
\item If $G = {\rm O}(n, \FF_q)$ with $n$ odd, and $j$ is even, then an element in the conjugacy class $(j, n-j)^{\pm}$ has centralizer size $|\gO^{\pm}(k, \FF_q) \times \gO(n-j, \FF_q)|$. If $j$ is odd, an element in the conjugacy class $(j, n-j)^{\pm}$ has centralizer size $|\gO(j, \FF_q) \times \gO^{\pm}(n-j, \FF_q)|$.
\item If $G = \gO^+(n, \FF_q)$ with $n$ even, and $j$ is even, then an element in the conjugacy class $(j, n-j)^{\pm}$ has centralizer size $|\gO^{\pm}(j, \FF_q) \times \gO^{\pm}(n-j, \FF_q)|$. If $j$ is odd, an element in either of the conjugacy classes $(j, n-j)^{\pm}$ has centralizer size $|\gO(j,\FF_q) \times \gO(n-j, \FF_q)|$.
\item If $G = \gO^-(n, \FF_q)$ with $n$ even, and $j$ is even, then an element in the conjugacy class $(j, n-j)^{\pm}$ has centralizer size $|\gO^{\pm}(j, \FF_q) \times \gO^{\mp}(n-j, \FF_q)|$. If $j$ is odd, an element in either of the conjugacy classes $(j, n-j)^{\pm}$ has centralizer size $|\gO(j,\FF_q) \times \gO(n-j, \FF_q)|$.
\end{enumerate}
\end{proposition}
Using Proposition \ref{OrthCent}, and that the orders of the orthogonal groups are given by
$$ |\gO(2m+1, \FF_q)| = 2q^{m^2} \prod_{i=1}^m (q^{2i}-1), \quad \text{ and } \quad |\gO^{\pm}(2m, \FF_q)| = 2q^{m(m-1)} (q^m \mp 1) \prod_{i=1}^{m-1} (q^{2i} -1),$$
when $q$ is the power of an odd prime, we may obtain the following.
\begin{theorem} \label{OrthSum}
The sums of the character degrees of the orthogonal groups over $\FF_q$, where $q$ is the power of an odd prime, are given as follows:
\begin{enumerate}
\item If $G = {\rm O}(n, \FF_q)$, where $n = 2m+1$ is odd, then
$$ \sum_{\chi \in \Irr(G)} \chi(1) = 2 \sum_{k=0}^m q^{2k(m-k+1)} \binom{m}{k}_{q^2}.$$
\item If $G = {\rm O}^{\pm}(n, \FF_q)$, where $n=2m$ is even, then
$$ \sum_{\chi \in \Irr(G)} \chi(1) = \sum_{k=0}^m q^{2k(m-k)} \binom{m}{k}_{q^2} + q^{m-1} (q^m \mp 1) \sum_{k=0}^{m-1} q^{2k(m-k-1)} \binom{m-1}{k}_{q^2}.$$
\end{enumerate}
\end{theorem}
\begin{proof} In each case, we must sum the indices of the centralizers of the conjugacy classes of elements which square to the identity, using Proposition \ref{OrthCent}. In case 1, when $G = \gO(2m+1, \FF_q)$, first consider the two conjugacy classes of type $(j, 2m+1-j)^{\pm}$, when $j=2k$ is even, $0 < k \leq m$. Then, the sizes of the two centralizers $C^{\pm}$ of these classes are
$$|C^{\pm}| = |\gO^{\pm}(2k, \FF_q) \times \gO(2(m-k)+1, \FF_q)| = 4q^{k(k-1) + (m-k)^2} (q^k \mp 1) \prod_{i=1}^{k-1} (q^{2i}-1) \prod_{i=1}^{m-k} (q^{2i} - 1).$$
Computing the sum of the indices of these centralizers, we obtain
$$ \frac{|G|}{|C^{+}|} + \frac{|G|}{|C^{-}|} = q^{2k(m-k+1)} \frac{\prod_{i=1}^m (q^{2i} - 1)}{\prod_{i=1}^k (q^{2i} - 1) \prod_{i=1}^{m-k} (q^{2i}-1)} = q^{2k(m-k+1)} \binom{m}{k}_{q^2}.$$
Consider now the two conjugacy classes of type $(j, 2m+1-j)^{\pm}$, when $j = 2k+1$ is odd, so $2m+1 - j = 2(m-k)$, and $0 \leq k < m$. The orders of the two centralizers $C^{\pm}$ in this case are
$$ |C^{\pm}| = |\gO(2k+1, \FF_q) \times \gO^{\pm}(2(m-k), \FF_q)|,$$
which are the same as in the previous case, except that $k$ is replaced by $m-k$. Thus, the sum of the indices of these centralizers is
$$ \frac{|G|}{|C^+|} + \frac{|G|}{|C^-|} = q^{2(m-k)(k+1)} \binom{m}{m-k}_{q^2}.$$
Taking the sum of these terms, and replacing $k$ by $m-k$ in the sum, we obtain
$$ \sum_{k=0}^{m-1} q^{2(m-k)(k+1)} \binom{m}{m-k}_{q^2} = \sum_{k=1}^m q^{2k(m-k+1)} \binom{m}{k}_{q^2}$$
elements from these conjugacy classes which square to the identity. Adding in the two central elements $\pm I$, we obtain that the total number of elements in $\gO(2m+1, \FF_q)$ which square to the identity, and so the sum of the degrees of the irreducible characters is
$$ 2 \sum_{k=1}^m q^{2k(m-k+1)} \binom{m}{k}_{q^2} + 2 = 2\sum_{k=0}^m q^{2k(m-k+1)} \binom{m}{k}_{q^2}.$$
Now consider case 2, when $G = \gO^{\pm}(2m, \FF_q)$. If $G = \gO^+(2m, \FF_q)$, then the two conjugacy classes of type $(j, 2m-j)^{\pm}$, where $j = 2k$ is even, $0 < k < m$, have centralizers $C^{\pm}$ with orders $|\gO^{\pm}(2k, \FF_q) \times \gO^{\pm}(2(m-k), \FF_q)|$, so
$$|C^{\pm}| = 4 q^{k(k-1) + (m-k)(m-k-1)} (q^k \mp 1)(q^{m-k} \mp 1) \prod_{i=1}^{k-1}(q^{2i}-1) \prod_{i=1}^{m-k-1}(q^{2i}-1).$$
If $G = \gO^-(2m, \FF_q)$, the two corresponding conjugacy classes have centralizers $C^{\pm}$ with orders $|\gO^{\pm}(2k, \FF_q) \times \gO^{\mp}(2(m-k), \FF_q)|$, which is the same as above, except that $q^{m-k} \mp 1$ is replaced by $q^{m-k} \pm 1$. In both cases, the sum of the indices of these centralizers is computed to be
$$ \frac{|G|}{|C^+|} + \frac{|G|}{|C^-|} = q^{2k(m-k)} \binom{m}{k}_{q^2}.$$
Adding in the two central elements, these conjugacy classes contribute exactly
\begin{equation} \label{EvenCon}
\sum_{k=1}^{m-1} q^{2k(m-k)} \binom{m}{k}_{q^2} + 2 = \sum_{k=0}^m q^{2k(m-k)} \binom{m}{k}_{q^2}.
\end{equation}
When $G = \gO^{\pm}(2m, \FF_q)$, the two conjugacy classes of type $(2k+1, 2m-(2k+1))^{\pm}$, $0 \leq k \leq m-1$, have centralizers of the same order $|\gO(2k+1, \FF_q) \times \gO(2(m-k-1) + 1, \FF_q)|$. The union of these two conjugacy classes thus has cardinality
$$ \frac{2|\gO^{\pm}(2m, \FF_q)|}{|\gO(2k+1, \FF_q) \times \gO(2(m-k-1) + 1, \FF_q)|} = q^{2k(m-k-1) + m -1} (q^m \mp 1) \binom{m-1}{k}_{q^2}.$$
Taking the sum of these contributions from $k=0$ to $k=m-1$, and adding to (\ref{EvenCon}), the result is obtained.
\end{proof}
In the case that $n=2m+1$ is odd, the orthogonal group $\gO(n, \FF_q)$ is just the direct product $\{\pm I\} \times \SO(2m+1, \FF_q)$ of the center with the special orthogonal group. The sum of the degrees of the characters of $\SO(2m+1, \FF_q)$ is thus exactly half the sum for $\gO(2m+1, \FF_q)$ obtained in case 1 of Theorem \ref{OrthSum}. That is, we have the following.
\begin{corollary} \label{SOcor}
Let $q$ be the power of an odd prime, and let $G = \SO(2m+1, \FF_q)$. Then
$$ \sum_{\chi \in \Irr(G)} \chi(1) = \sum_{k = 0}^m q^{2k(m-k+1)} \binom{m}{k}_{q^2}.$$
\end{corollary}
When $n = 2m$ is even, Gow \cite[Theorem 2]{Go85} proved that every real-valued character of $\SO^{\pm}(2m, \FF_q)$ is the character of a real representation, and in the case $n = 4l$ is divisible by $4$, every character of $\SO^{\pm}(4l, \FF_q)$ is real-valued. To find the sum of the degrees of the real-valued characters of $\SO^{\pm}(2m, \FF_q)$ in these cases, then, we have to count the number of elements of the group which square to the identity. This is exactly the number of elements in $\gO^{\pm}(2m, \FF_q)$ which square to the identity and which have determinant $1$, and so have an even number of eigenvalues equal to $-1$ (and an even number equal to $1$). This is just the first part of the sum obtained in case 2 of Theorem \ref{OrthSum}. Curiously, this is exactly what is obtained as the sum of the degrees of the real-valued characters of $\GL(m, \FF_{q^2})$ in (\ref{GLReal}). We summarize these observations below.
\begin{corollary} \label{SOcor2}
Let $q$ be the power of an odd prime, let $H = \SO^{\pm}(2m, \FF_q)$, and let $G = \GL(m, \FF_{q^2})$. Then,
$$\sum_{\chi \in {\rm Irr}(H) \atop{ \chi \; \RR \text{-valued}}} \chi(1) = \sum_{k=0}^m q^{2k(m-k)} \binom{m}{k}_{q^2} = \sum_{\chi \in {\rm Irr}(G) \atop{ \chi \; \RR \text{-valued}}} \chi(1).$$
In the case that $m = 2l$ is also even, this is the sum of the degrees of all of the characters of the group $\SO^{\pm}(4l, \FF_q)$.
\end{corollary}
\section{Inequality Lemmas} \label{IneqLemmas}
In this section, we prove several inequalities in preparation for the results in Section \ref{Bound}. We begin with an elementary bound for the Gaussian binomial coefficients.
\begin{lemma} \label{binomineq} For any integers $m \geq 1$, $1 \leq k \leq m$, we have
$$ \binom{m}{k}_q \leq q^{k(m-k) - m+1} (q+1)^{m-1}.$$
\end{lemma}
\begin{proof} The inequality reduces to $1 \leq 1$ when $m = 1$. If $k = 1$, then for any $m \geq 1$, we have $\binom{m}{1}_q = q^{m-1} + \cdots + q + 1 \leq (q+1)^{m-1}$. So, we may assume $k \geq 2$. Assume the inequality holds for $m=n-1$, for any $k \geq 1$. We use the identity $\binom{n}{k}_q = \binom{n-1}{k}_q + q^{n-k}\binom{n-1}{k-1}_q$, $k \geq 1$, which was mentioned in Section \ref{realsums}. By this identity and the induction hypothesis, and with $k \geq 2$, we have
\begin{align*}
\binom{n}{k}_q & \leq (q^{k(n-1-k)-n+2} + q^{k(n-k) -n+2})(q+1)^{n-2} = q^{k(n-1-k) - n+2} (q^k + 1)(q+1)^{n-2} \\
& \leq q^{k(n-1-k)-n+2}q^{k-1}(q+1)(q+1)^{n-2} = q^{k(n-k) - n+1} (q+1)^{n-1},
\end{align*}
as desired.
\end{proof}
The next two Lemmas will be used to bound the expressions obtained in Section \ref{OrthogSums}.
\begin{lemma} \label{EvenDimIneq}
For any integer $m \geq 1$, and any $q > 1$,
$$ \sum_{k=0}^m q^{2k(m-k)} \binom{m}{k}_{q^2} \leq \left\{ \begin{array}{ll} 2(q+1)^{m^2-1} & \text{if $m$ is odd} \\ (q+1)^{m^2} & \text{if $m$ is even.} \end{array} \right.$$
\end{lemma}
\begin{proof}
From the symmetry in $k$ and $m-k$ in the sum, we have
\begin{equation} \label{SymmSum}
\sum_{k=0}^m q^{2k(m-k)} \binom{m}{k}_{q^2} = \left\{ \begin{array}{ll} 2\sum_{k=0}^{(m-1)/2} q^{2k(m-k)} \binom{m}{k}_{q^2} & \text{if $m$ is odd} \\ q^{m^2/2} \binom{m}{m/2}_{q^2} + 2\sum_{k=0}^{(m/2) - 1} q^{2k(m-k)} \binom{m}{k}_{q^2} & \text{if $m$ is even.} \end{array} \right.
\end{equation}
From Lemma \ref{binomineq}, we have, for any $k \geq 1$,
$$ \binom{m}{k}_{q^2} \leq q^{2k(m-k)-2m+2} (q^2 + 1)^{m-1} \leq q^{2k(m-k)-m+1} (q+1)^{m-1},$$
since $q^2 + 1 \leq q(q+1)$. Now, for $s = (m-1)/2$ if $m$ is odd, or $s = (m/2) - 1$ if $m$ is even, we have
\begin{align} \label{Intermed}
\sum_{k=0}^s q^{2k(m-k)} \binom{m}{k}_{q^2} & \leq (q+1)^{m-1} \left( 1 + \sum_{k=1}^s q^{4k(m-k) - m+1} \right) \notag\\
& \leq (q+1)^{m-1} (q+1)^{4s(m-s) - m+1} = (q+1)^{4s(m-s)},
\end{align}
since the exponent of $q$ in the sum is maximum when $k=s$. When $m$ is odd, then substituting $s=(m-1)/2$ and applying (\ref{SymmSum}) gives the desired result. When $m$ is even, then applying (\ref{Intermed}) with $s = (m/2) - 1$, and Lemma \ref{binomineq}, we have
\begin{align*}
q^{m^2/4} \binom{m}{m/2}_{q^2} + 2\sum_{k=0}^{(m/2) - 1} q^{2k(m-k)} \binom{m}{k}_{q^2} & \leq q^{m^2 - m + 1}(q+1)^{m-1} + 2(q+1)^{m^2 - 4} \\
& \leq (q+1)^{m^2 - 4}(q^4 + 2) \leq (q+1)^{m^2},
\end{align*}
as claimed for $m$ even.
\end{proof}
\begin{lemma} \label{OddDimIneq}
For any integer $m \geq 0$, and any $q > 1$,
$$ \sum_{k=0}^m q^{2k(m-k+1)} \binom{m}{k}_{q^2} \leq (q+1)^{m^2 + m}.$$
\end{lemma}
\begin{proof} We may assume $m \geq 1$. Note that by switching the roles of $k$ and $m-k$, we obtain
$$ \sum_{k=0}^m q^{2k(m-k+1)} \binom{m}{k}_{q^2} = \sum_{k=0}^m q^{2k} q^{2k(m-k)} \binom{m}{k}_{q^2} = \sum_{k=0}^m q^{2(m-k)} q^{2k(m-k)} \binom{m}{k}_{q^2}.$$
It follows that we have
$$2\sum_{k=0}^m q^{2k(m-k+1)} \binom{m}{k}_{q^2} = \sum_{k=0}^m (q^{2k}+q^{2(m-k)})q^{2k(m-k)} \binom{m}{k}_{q^2}.$$
From the symmetry in $k$ and $m-k$ in the right-hand side of the above equation, we have
\begin{equation*}
\sum_{k=0}^m q^{2k(m-k+1)} \binom{m}{k}_{q^2} =
\end{equation*}
\begin{equation} \label{SymmSum2}
=\left\{ \begin{array}{ll} \sum_{k=0}^{(m-1)/2}(q^{2k} + q^{2(m-k)})q^{2k(m-k)}\binom{m}{k}_{q^2} & \text{if $m$ is odd} \\ q^{m^2/2 + m}\binom{m}{m/2}_{q^2} + \sum_{k=0}^{(m/2)-1} (q^{2k} + q^{2(m-k)}) q^{2k(m-k)} \binom{m}{k}_{q^2} & \text{if $m$ is even}. \end{array} \right.
\end{equation}
Let $s = (m-1)/2$ if $m$ is odd, and $s = (m/2)-1$ if $m$ is even. In the sums (\ref{SymmSum2}), the term $q^{2k} + q^{2(m-k)}$ takes its maximum value when $k=s$. Applying this, and the inequality (\ref{Intermed}), we have
\begin{align} \label{Intermed2}
\sum_{k=0}^s (q^{2k} + q^{2(m-k)}) q^{2k(m-k)} \binom{m}{k}_{q^2} & \leq (q^{2s} + q^{2(m-s)}) \sum_{k=0}^s q^{2k(m-k)} \binom{m}{k}_{q^2} \notag \\
& \leq (q^{2s} + q^{2(m-s)})(q+1)^{4s(m-s)}.
\end{align}
When $m$ is odd and $s = (m-1)/2$, then by (\ref{Intermed2}) and (\ref{SymmSum2}),
$$\sum_{k=0}^m q^{2k(m-k+1)} \binom{m}{k}_{q^2} \leq (q^{m-1} + q^{m+1})(q+1)^{(m^2-1)/2} = q^{m-1}(q+1)^{m^2 + 1} \leq (q+1)^{m^2 + m},$$
as claimed. When $m$ is even, with $s = (m/2) -1$, apply (\ref{Intermed2}), together with (\ref{SymmSum2}) and Lemma \ref{binomineq} to obtain
\begin{align*}
\sum_{k=0}^m q^{2k(m-k+1)} \binom{m}{k}_{q^2} & \leq q^{m^2 + 1} (q+1)^{m-1} + (q^{m-2} + q^{m+2})(q+1)^{m^2 - 4} \\
& \leq q^{m^2 + 1} (q+1)^{m-1} + q^{m-2} (q^4 + 1)(q + 1)^{m^2 - 4} \\
& \leq q^2 (q+1)^{m^2 + m - 2} + (q+1)^{m^2 +m -2} \leq (q+1)^{m^2+m},
\end{align*}
as desired.
\end{proof}
\section{An upper bound for character degree sums} \label{Bound}
Let $\bG$ be a connected algebraic group with connected center over $\bar{\FF}_q$, defined over $\FF_q$ by some Frobenius map. By the {\em dimension} of $\bG$, we mean the dimension of $\bG$ as an algebraic variety over $\bar{\FF}_q$. The {\em rank} of $\bG$ is the dimension of a maximal torus of $\bG$. That is, if the rank of $\bG$ is $r$, then a maximal torus of $\bG$ is isomorphic to $(\bar{\FF}_q^{\times})^r$.
In this section, we will consider the case when $\bG$ is a connected {\em classical} group with connected center, when $q$ is the power of an odd prime. For us, these are the groups $\GL(n, \bar{\FF}_q)$, $\SO(2n+1, \bar{\FF}_q)$, $\GSp(2n, \bar{\FF}_q)$, and a certain index $2$ subgroup of $\GO(2n, \bar{\FF}_q)$, denoted $\GO^{\circ}(2n, \bar{\FF}_q)$, which we describe now.
Let $V$ be a $2n$-dimensional vector space over $\bar{\FF}_q$, with $q$ odd, and consider a nondegenerate symmetric form on $V$. Since $\bar{\FF}_q$ is algebraically closed, there is only one equivalence class of forms (by \cite[Theorem 4.4]{Gr02}, for example), which we may assume corresponds to the matrix $J = \left( \begin{array} {cc} 0 & I \\ I & 0 \end{array} \right)$. Consider the orthogonal group of similitudes $\GO(2n, \bar{\FF}_q)$ of this form. For any $g \in \GO(2n, \bar{\FF}_q)$, then, we have ${^T g} = J \mu(g) g^{-1} J$. Since $g$ is conjugate to its transpose in $\GL(V)$, it follows that $g$ is conjugate to $\mu(g) g^{-1}$ in $\GL(V)$, and so they have the same determinant. In particular, we have ${\rm det}(g)^2 = \mu(g)^{2n}$. The subgroup consisting of elements with the property that ${\rm det}(g) = \mu(g)^n$ is a connected algebraic group (by an argument similar to that given in \cite[Section 15.2]{DiMi91}). So, we define
\begin{equation} \label{GOconnDefn}
\GO^{\circ} (2n, \bar{\FF}_q) = \{ g \in \GO(2n, \bar{\FF}_q) \, \mid \, {\rm det}(g) = \mu(g)^n \},
\end{equation}
the connected component of the identity in the orthogonal group of similitudes. Note that $\SO(2n, \bar{\FF}_q)$ is contained in $\GO^{\circ}(2n, \bar{\FF}_q)$, and its center consists of all scalar matrices.
If we define the $\FF_q$-structure of $\GO^{\circ}(2n, \bar{\FF}_q)$ by the standard Frobenius map $F$, which raises entries to the power of $q$, we get that the group of $\FF_q$-points is the index $2$ subgroup of the split orthogonal group of similitudes over $\FF_q$ satisfying the condition in (\ref{GOconnDefn}), which we denote $\GO^{+, \circ}(2n, \FF_q)$. If we compose the standard Frobenius map with conjugation by an orthogonal reflection defined over $\FF_q$ (see \cite[Section 15.3]{DiMi91}), we get that the group of $\FF_q$-points of $\GO^{\circ}(2n, \bar{\FF}_q)$ is the index $2$ subgroup of the non-split orthogonal group of similitudes over $\FF_q$ which satisfies (\ref{GOconnDefn}), which we denote $\GO^{-, \circ}(2n, \FF_q)$. We may relate the character degree sums for the groups $\GO^{\pm, \circ}(2n, \FF_q)$ to those for the orthogonal groups $\gO^{\pm}(2n, \FF_q)$ in the following way.
\begin{lemma} \label{GOSumLemma} Let $q$ be the power of an odd prime, let $G = \GO^{\pm, \circ}(2m, \FF_q)$, and let $H = \gO^{\pm}(2m, \FF_q)$. Then
$$ \sum_{ \chi \in {\rm Irr}(G)} \chi(1) \leq (q-1) \sum_{\chi \in {\rm Irr}(H)} \chi(1).$$
\end{lemma}
\begin{proof} Let $S = \SO^{\pm}(2m, \FF_q)$ be the special orthogonal group. Then $S$ is a normal subgroup of $G$ of index $q-1$, with $G/S \cong \FF_q^{\times}$ cyclic. From Clifford theory (see \cite[Chapter 6 and Theorem 11.7]{Is76}), the restriction of any irreducible character of $G$ to $S$ has a multiplicity-free decomposition, and each irreducible character of $S$ appears in the restriction to $S$ of at most $q-1$ characters of $G$. It follows that we have
\begin{equation} \label{IneqOne}
\sum_{\chi \in {\rm Irr}(G)} \chi(1) \leq (q-1) \sum_{\chi \in {\rm Irr}(S)} \chi(1).
\end{equation}
Since $S$ is an index $2$ subgroup of $H$, every irreducible character of $H$ is extended or induced from an irreducible character of $S$. A character of $H$ which is extended from a character of $S$ has the same degree as that character of $H$, while that character of $S$ may be extended to give a second distinct irreducible of $H$. A character of $H$ which is induced from a character of $S$ has degree twice that of the character of $S$, but can also be induced by exactly one other distinct character of $S$. It follows that we have
\begin{equation} \label{IneqTwo}
\sum_{\chi \in {\rm Irr}(S)} \chi(1) \leq \sum_{\chi \in {\rm Irr}(H)} \chi(1).
\end{equation}
The result follows from (\ref{IneqOne}) and (\ref{IneqTwo}).
\end{proof}
We may now prove the main result of this section, in which we improve Kowalski's Theorem \ref{KoThm} in the case of any connected classical group with connected center, as follows.
\begin{theorem} \label{MainBound} Let $q$ be the power of an odd prime, and let $\bG$ be a connected classical group with connected center defined over $\FF_q$, where the rank of $\bG$ is $r$, and the dimension of $\bG$ is $d$. Then the sum of the degrees of the irreducible characters of the finite group $\bG(\FF_q)$ may be bounded as follows:
$$ \sum_{\chi \in {\rm Irr}(\bG(\FF_q))} \chi(1) \leq (q+1)^{(d+r)/2}.$$
\end{theorem}
\begin{proof} The dimensions and ranks of finite classical groups may be computed directly from their definitions (see \cite[Chapter 15] {DiMi91}, for example). If $\bG = \GL(n, \bar{\FF}_q)$, then $d = n^2$ and $r = n$. In this case, $\bG$ may have $\FF_q$-structure given by the standard Frobenius map, in which case $\bG(\FF_q) = \GL(n, \FF_q)$, or the standard Frobenius map composed with the transpose-inverse automorphism, in which case $\bG(\FF_q) = {\rm U}(n, \FF_{q^2})$, the finite unitary group defined over $\FF_q$. The case $\bG(\FF_q) = \GL(n, \FF_q)$ was considered by Kowalski \cite[p. 80]{Ko08}, and he showed that the sum of the degrees of the characters is indeed bounded above by $(q+1)^{(d+r)/2}$. When $\bG(\FF_q) = {\rm U}(n, \FF_{q^2})$, the sum of the degrees of the irreducible characters of $\bG(\FF_q)$ was computed by Thiem and the author \cite[Theorem 5.2]{ThVi07} to be
\begin{align*}
\sum_{\chi \in {\rm Irr}(\bG(\FF_q))} \chi(1) & = (q+1)q^2(q^3+1)q^4 \cdots (q^n + (1-(-1)^n)/2)\\
& \leq \prod_{i=1}^n (q+1)^i = (q+1)^{(n^2 + n)/2} = (q+1)^{(d+r)/2}.
\end{align*}
If $\bG = \GSp(2n, \bar{\FF}_q)$, then $d = 2n^2 + n +1$ and $r = n+1$, and we may assume $\bG$ has $\FF_q$-structure given by the standard Frobenius map, so that $\bG(\FF_q) = \GSp(2n, \FF_q)$. This case was also considered by Kowalski, and he noticed that by the formula given in \cite[Corollary 6.1]{Vi05}, the sum of the degrees of $\GSp(2n, \FF_q)$ is bounded above by $(q+1)^{(d+r)/2}$. We note that the results in \cite{Vi05} are proven only when $q$ is odd, and so we can only obtain this bound in the case $q$ is odd.
If $\bG = \SO(2n+1, \bar{\FF}_q)$, then $d = 2n^2 + n$ and $r = n$, and again we may assume $\bG$ has $\FF_q$-structure given by the standard Frobenius map, and so $\bG(\FF_q) = \SO(2n+1, \FF_q)$. By Corollary \ref{SOcor} and Lemma \ref{OddDimIneq}, we have
$$ \sum_{\chi \in {\rm Irr}(\bG(\FF_q))} \chi(1) = \sum_{k=0}^n q^{2k(n-k+1)} \binom{n}{k}_{q^2} \leq (q+1)^{n^2 + n} = (q+1)^{(d+r)/2}. $$
If $\bG = \GO^{\circ}(2n, \bar{\FF}_q)$, then $d = 2n^2 - n + 1$ and $r = n+1$. In this case, as explained above, $\bG$ can have $\FF_q$-structure given by the standard Frobenius map, so that $\bG(\FF_q) = \GO^{+, \circ}(2n, \FF_q)$, or by the standard Frobenius map composed with conjugation by an orthogonal reflection defined over $\FF_q$, in which case $\bG(\FF_q) = \GO^{-, \circ}(2n, \FF_q)$. We consider both cases at once, so let $\bG(\FF_q) = \GO^{\pm, \circ}(2n, \FF_q)$. By Theorem \ref{OrthSum}(2) and Lemma \ref{GOSumLemma}, we have
\begin{equation} \label{GOIntermed}
\sum_{\chi \in {\rm Irr}(\bG(\FF_q))} \chi(1) \leq (q-1) \left(\sum_{k=0}^n q^{2k(n-k)} \binom{n}{k}_{q^2} + q^{n-1} (q^n + 1) \sum_{k=0}^{n-1} q^{2k(n-k-1)} \binom{n-1}{k}_{q^2} \right).
\end{equation}
If $n$ is even, then by (\ref{GOIntermed}) and Lemma \ref{EvenDimIneq}, we have
\begin{align*}
\sum_{\chi \in {\rm Irr}(\bG(\FF_q))} \chi(1) & \leq (q-1)\left( (q+1)^{n^2} + q^{n-1}(q^n + 1) 2(q+1)^{(n-1)^2-1} \right) \\
& \leq (q-1) \left( (q+1)^{n^2} + 2(q+1)^{n^2 - 1} \right) = (q+1)^{n^2 - 1}(q-1)(q+3) \\
& = (q+1)^{n^2 -1} (q^2 + 2q -3) \leq (q+1)^{n^2 + 1} = (q+1)^{(d+r)/2},
\end{align*}
as required. Similarly, when $n$ is odd, then by (\ref{GOIntermed}) and Lemma \ref{EvenDimIneq}, we have
\begin{align*}
\sum_{\chi \in {\rm Irr}(\bG(\FF_q))} \chi(1) & \leq (q-1)\left( 2(q+1)^{n^2-1} + q^{n-1}(q^n + 1)(q+1)^{(n-1)^2} \right) \\
& \leq (q-1) \left( 2(q+1)^{n^2-1} + (q+1)^{n^2} \right) = (q+1)^{n^2 - 1}(q-1)(q+3) \\
& = (q+1)^{n^2 -1} (q^2 + 2q -3) \leq (q+1)^{n^2 + 1} = (q+1)^{(d+r)/2},
\end{align*}
which was the last case to consider.
\end{proof}
\section{A lower bound and a conjecture} \label{Lower}
We now consider the case that $\bG$ is any connected reductive group over $\bar{\FF}_q$ with connected center, defined over $\FF_q$, where $q$ is the power of some prime $p$ (we allow $p=2$ here). If $G = \bG(\FF_q)$, and $N$ is a maximal unipotent subgroup of $G$, then $N$ is a Sylow $p$-subgroup of $G$ \cite[Proposition 3.19(i)]{DiMi91}. The {\em Gelfand-Graev character} of $G$ is constructed by inducing a {\em nondegenerate} linear character from $N$ to $G$. For a detailed discussion on Gelfand-Graev characters and nondegenerate characters, see \cite[Section 8.1]{Ca85} or \cite[Chapter 14]{DiMi91}. The main result on the Gelfand-Graev character which we need is that its decomposition into a sum of irreducible characters of $G$ is multiplicity-free, a result which is due to Steinberg \cite{St67} in the general case. By applying this fact, we obtain the following lower bound for the sum of the degrees of the irreducible characters of $G = \bG(\FF_q)$.
\begin{proposition} \label{LowerProp} Let $\bG$ be a connected reductive group over $\bar{\FF}_q$ with connected center, defined over $\FF_q$, of dimension $d$ and rank $r$. Then the sum of the dimensions of the irreducible representations of the group $G = \bG(\FF_q)$ is bounded below as follows:
$$ \sum_{\chi \in {\rm Irr}(G)} \chi(1) \geq q^{(d-r)/2} (q-1)^r.$$
\end{proposition}
\begin{proof} Since the Gelfand-Graev character of $G = \bG(\FF_q)$ is multiplicity-free, then the degree of the Gelfand-Graev character gives a lower bound for the sum of the degrees of all of the irreducible characters of $G$. We give a lower bound for the degree of the Gelfand-Graev character to prove our claim.
The {\em semisimple rank} of $\bG$, which we denote by $l$, is defined as the rank of $\bG/R(\bG)$, where $R(\bG)$ is the radical of $\bG$, which is the maximal closed, connected, normal, solvable subgroup of $\bG$ (see \cite[6.4.14]{Sp98}). The degree of the Gelfand-Graev character is the $p'$-part of the order of $G$, since it is induced from a linear character of a Sylow $p$-subgroup of $G$. By \cite[Section 2.9]{Ca85}, the $p'$-part of the order of $G$ is given by
$$ |\bZ(\FF_q)| \prod_{i=1}^l (q^{d_i} - \omega_i),$$
where $\bZ$ is the center of $\bG$, the $d_i$ are the degrees of the generators of the ring of polynomial invariants of the Weyl group $W$ of $\bG$, and the $\omega_i$ are roots of unity which are eigenvalues of a linear map on the algebra of polynomial invariants of $W$. Now, we have $|q^{d_i} - \omega_i| \geq q^{d_i} - 1 \geq q^{d_i - 1}(q-1)$. From \cite[Proposition 2.4.1(iv)]{Ca85}, we have $\sum_{i=1}^l (d_i - 1)$ is equal to the number of positive roots of the root system corresponding to the Weyl group $W$, and by \cite[Corollary 8.1.3(ii)]{Sp98}, this is equal to $(d-r)/2$. These observations give
\begin{equation} \label{firstpart}
\prod_{i=1}^l (q^{d_i} - \omega_i) \geq q^{\sum_{i=1}^l (d_i - 1)} (q-1)^l = q^{(d-r)/2} (q-1)^l.
\end{equation}
By \cite[Proposition 7.3.1(i)]{Sp98}, since $\bG$ is assumed to be a connected reductive group with connected center $\bZ$, we have $R(\bG) = \bZ$, where $\bZ$ is itself a torus. In particular, we have $r = l + {\rm dim}(\bZ)$, by the definition of semisimple rank. By \cite[Proposition 3.3.7]{Ca85} and the formula for the order of the $\FF_q$-points of a maximal torus \cite[Proposition 3.3.5]{Ca85}, like in \cite[p. 75]{Ko08}, this implies we have $|\bZ(\FF_q)| \geq (q-1)^{r - l}$. Combining this with (\ref{firstpart}) gives us the desired bound.
\end{proof}
Let us return to the question of a general upper bound for the sum of the degrees of the irreducible characters of $\bG(\FF_q)$. When $\bG(\FF_q) = \GL(n, \FF_q)$, then the sum of the character degrees of $\bG(\FF_q)$ is given by (see \cite{Go83, Kl83, Mac95})
$$(q-1)q^2(q^3-1) \cdots (q^n - (1 - (-1)^n)/2),$$
which is, of course, bounded above by $q^{(n^2+n)/2} < q^{(n^2 - n)/2}(q+1)^n$. When $\bG(\FF_q) = {\rm U}(n, \FF_{q^2})$, then by the sum of the character degrees given in the proof of Theorem \ref{MainBound}, this can be seen to be bounded above by $q^{(n^2 - n)/2}(q+1)^n$ as well. Note that in these two cases, we have $(n^2 - n)/2 = (d-r)/2$ and $n = r$.
When $\bG(\FF_q) = {\rm GSp}(2n, \FF_q)$, with $q$ odd, the sum of the degrees of the irreducible characters of $\bG(\FF_q)$, by \cite[Corollary 6.1]{Vi05}, is
$$ \frac{1}{2} q^{(n^2 + n)/2}(q-1) \left( \prod_{i=1}^n (q^i + 1) + \prod_{i=1}^n (q^i + (-1)^i) \right) \leq q^{n^2} (q+1)^{n+1},$$
where $n+1=r$ and $n^2 = (d-r)/2$.
If $\bG(\FF_q)$ is $\SO(2n+1, \FF_q)$ or $\GO^{\pm, \circ}(2n, \FF_q)$, with $q$ odd, then we may check directly for small values of $n$ that the sum of the character degrees of these groups are bounded by $q^{(d-r)/2}(q+1)^r$ as well. We could perhaps prove this for all $n$ for these groups, although we would have to tighten the bounds found in Section \ref{IneqLemmas}. It would be much more satisfying to have a general proof using Deligne-Lusztig theory, along the same lines as Kowalski's proof of Theorem \ref{KoThm} in \cite{Ko08}.
Based on these examples, we conclude by making the following conjecture for an upper bound for the sum of the degrees of the irreducible characters of a finite reductive group. We include the lower bound obtained in Proposition \ref{LowerProp} in the statement to stress the symmetry in these bounds.
\begin{conjecture} Let $\bG$ be a connected reductive group over $\bar{\FF}_q$ with connected center, defined over $\FF_q$, of dimension $d$ and rank $r$. Then the sum of the degrees of the irreducible complex characters of $G = \bG(\FF_q)$ may be bounded as follows:
$$ q^{(d-r)/2}(q-1)^r \leq \sum_{\chi \in {\rm Irr}(G)} \chi(1) \leq q^{(d-r)/2}(q+1)^r.$$
\end{conjecture}
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Brief description of the package:
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Roemi Ice Cream9:25:00 PM
March 3rd 2012
After lunch together, my high school friend had another business so we splited, and so did Aji Adi and Bilal. Its only me, Hesna and Agung with no where to go. When Agung recomended an ice cream place..we're directly went there. yuhuuuu~
So its Roemi Ice cream. Another line of business by the owner of Mirota. After torn between sat inside or outside, and Thank God its gave me time to took some picture, we're finally sat outside and not on the sofa. hahaha
Oh how i love Jogjakarta. I felt like so damn rich ! hahaha. Only its not my money that gets a lot, its the price thats oh so cheap, possible me to buy a lot. Its a big ice cream cafe, but the price is so friendly. Check the menu. It will be hard to find something this cheap in Jakarta.
We ordered Blueberry Riffle, Bitter Chocolate and Strawberry gelato. And...its so yum yum <3
God, i ate a lot in this trip. No wonder i gain a lot :(
This Ice cream cafe also sale cakes, unique chocolates, snack and milk powder from Mirota. And let me say it clearly..everything is cheap. Those beautiful cakes cost only Rp 50.000 ! haa, Jogjaa..jogjaaaa..
I bough one chocolate. It tasted good, but cadburry blackforest is my own personal chocolate heaven. Nothing compare to it, except feroro rocher :p
For this delicious ice cream and chocolate, each of us only spend under Rp20.000. Extra wow ! hahaha
Later i'll compare to another Jogjakarta's ice cream cafe. But for now..let me said GOOD NIGHT :D
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The tuba is the largest and lowest-pitched musical instrument in the brass family. As with all brass instruments, the sound a tuba makes is produced by lip vibration into a large mouthpiece. The tuba first appeared in the mid-19th century, making it one of the newer instruments in the modern orchestra and concert band.
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\begin{document}
\title{The allelic partition for coalescent point processes}
\author{\textsc{By Amaury Lambert
}
}
\date{\today}
\maketitle
\noindent\textsc{Laboratoire de Probabilités et Modèles Aléatoires\\
UMR 7599 CNRS and UPMC Univ Paris 06\\
Case courrier 188\\
4, Place Jussieu\\
F-75252 Paris Cedex 05, France}\\
\textsc{E-mail: }amaury.lambert@upmc.fr\\
\textsc{URL: }http://ecologie.snv.jussieu.fr/amaury/
\begin{abstract}
\noindent
Assume that individuals alive at time $t$ in some population can be ranked in such a way that the coalescence times between consecutive individuals are i.i.d. The ranked sequence of these branches is called a coalescent point process. We have shown in a previous work \cite{L} that splitting trees are important instances of such populations.
Here, individuals are given DNA sequences, and for a sample of $n$ DNA sequences belonging to distinct individuals, we consider the number $S_n$ of polymorphic sites (sites at which at least two sequences differ), and the number $A_n$ of distinct haplotypes (sequences differing at one site at least).
It is standard to assume that mutations arrive at constant rate (on germ lines), and never hit the same site on the DNA sequence.
We study the mutation pattern associated with coalescent point processes under this assumption. Here, $S_n$ and $A_n$ grow linearly as $n$ grows, with explicit rate. However, when the branch lengths have infinite expectation, $S_n$ grows more rapidly, e.g. as $n \ln(n)$ for critical birth--death processes.
Then, we study the frequency spectrum of the sample, that is, the numbers of polymorphic sites/haplotypes carried by $k$ individuals in the sample. These numbers are shown to grow also linearly with sample size, and we provide simple explicit formulae for mutation frequencies and haplotype frequencies. For critical birth--death processes, mutation frequencies are given by the harmonic series and haplotype frequencies by Fisher's logarithmic series.
\end{abstract}
\medskip
\textit{Running head.} The allelic partition for coalescent point processes.\\
\textit{MSC Subject Classification (2000).} Primary 92D10; secondary
60-06, 60G10, 60G51, 60G55, 60G70, 60J10, 60J80, 60J85.\\
\textit{Key words and phrases.} coalescent point process -- splitting tree -- Crump--Mode--Jagers process -- linear birth--death process -- Yule process -- allelic partition -- infinite site model -- infinite allele model -- Poisson point process -- Lévy process -- scale function -- law of large numbers -- Kingman coalescent -- Fisher logarithmic series.
\section{Introduction}
\subsection{The coalescent point process}
Splitting trees are those random trees where individuals give birth at constant rate $b$ during a lifetime with general distribution $\Lambda(\cdot)/b$, to i.i.d. copies of themselves (see \cite{GK}), where $\Lambda$ is a positive measure on $(0,\infty]$ with total mass $b$ called the \emph{lifespan measure}. In \cite{L}, we have shown that if the splitting tree is started from one individual with known birth time, say $0$, and known death time, then individuals alive at time $t$ can be ranked in such a way that the coalescence times between consecutive individuals are i.i.d.
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\caption{ A coalescent point process for $n=16$ individuals.}
\label{fig : coalpointproc}
\end{figure}
Specifically, let $N_t$ be the number of individuals alive at time $t$. The process $(N_t;t\ge 0)$ is a (homogeneous, binary) Crump--Mode--Jagers process, and is not Markovian unless $\Lambda$ has an exponential density or is a point mass at $\infty$. To these $N_t$ individuals, give labels $0,1,\ldots, N_t-1$ according to the (unique) order complying with the following rule : `any individual comes before her own descendants, but after her younger siblings and their descendants'. For any integers $i,k$ such that $0\le i<i+k< N_t$, we let $C_{i,i+k}$ be the \emph{coalescence time} (or \emph{divergence time}) between individual $i$ and individual $i+k$, that is, the time elapsed since the lineages of individual $i$ and $i+k$ have diverged. Also define $H_{i+1}:=C_{i,i+1}$. Then recall from \cite{L} that for a splitting tree,
\begin{equation}
\label{eqn : def coal}
C_{i,i+k}=\max\{H_{i+1},\ldots,H_{i+k}\}
\end{equation}
and conditional on $\{N_t\not=0\}$, the sequence $(H_i;1\le i \le N_t-1)$ has the same law as a sequence of i.i.d. r.v. killed at its first value $\ge t$. As a by-product, we get that the law of $N_t$ conditional on $\{N_t\not=0\}$ is geometric.
The aforementioned property comes from the fact that the jumping contour process of the splitting tree is a Lévy process $X=(X_s;s\ge0)$ with Lévy measure $\Lambda$ and drift coefficient $-1$. Then the excursions of the contour process between consecutive visits of points at height $t$ are i.i.d. excursions of $X$. As a consequence, the $(H_i)$ are also i.i.d., and their common distribution is that of $H':=t-\inf_s X_s$, where $X$ is started at $t$ and killed upon hitting $\{0\}\cup(t,+\infty)$. Note that all branch lengths but the last one are distributed as some r.v. $H$ which is $H'$ conditioned to be smaller than $t$.
The distribution of $H'$ can be expressed in terms of a nonnegative, nondecreasing, differentiable function $W$, called the \emph{scale function} of $X$, such that $W(0)=1$
\begin{equation}
\label{eqn : scale}
\PP(H'> x)=\frac{1}{W(x)}\qquad x\ge 0.
\end{equation}
The scale function $W$ is characterised by its Laplace transform (see e.g. \cite{B})
\begin{equation}
\label{eqn : LT scale}
\intgen dx\, e^{-\lbd x} \, W(x) = \left(\lbd -\intgen \Lambda(dx) (1-e^{-\lbd x}) \right)^{-1}.
\end{equation}
From now on, with no need to refer to the framework of splitting trees, we will consider the genealogy of what we call a \emph{coalescent point process} (originating from \cite{P} where $\Lambda (dx)=b^2\exp(-bx) dx$) :
\begin{enumerate}
\item let $H_1, H_2,\ldots$ be a sequence of independent random variables called \emph{branch lengths} all distributed as some positive r.v. $H$, and set $H_0$ to equal $+\infty$.
\item
the genealogy of the population $\{0,1,2,\ldots\}$ is given by \eqref{eqn : def coal}.
\end{enumerate}
We will stick to the notation
$$
W(x):=\frac{1}{\PP(H> x)}\qquad x\ge 0.
$$
It will always be implicit that a \emph{sample} of $n$ individuals refers to the \emph{first} $n$ individuals $\{0,1,\ldots,n-1\}$.
\begin{rem}
\label{rem : H' et H}
In the case of splitting trees, conditional on $\{N_t\not=0\}$, $N_t$ is geometric with success probability $\PP(H'>t)$, and conditional on $\{N_t=n\}$, the branch lengths $(H_i;1\le i \le n-1)$ are i.i.d. with distribution $\PP(H'\in\cdot\mid H'<t)$. In what follows, we will repeatedly refer to the genealogy of a splitting tree with $n$ leaves by setting the r.v. $H$ to equal $H'$, without the conditioning (i.e. $t\to\infty$). In the subcritical case, this amounts to considering quasi-stationary populations, which are those populations conditioned to be still alive at time $t$, as $t\tendinfty$ (see e.g. \cite{L9}). Another possibility would be, as in \cite{AP}, to give a prior distribution to the time $t$ of origin, and condition the whole tree on $\{N_t=n\}$. Then as $n\to\infty$, the posterior distribution of $t$ goes to $\infty$, and we would be left with a (possibly different) distribution of $H$ charging the whole half-line.
\end{rem}
\begin{rem}
No distribution of edge lengths can make the coalescent point process coincide with the Kingman coalescent \cite{Ki}. Indeed, here, the smallest branch length in a sample of $n$ individuals is the minimum of $n-1$ i.i.d. random variables, whereas in the Kingman coalescent, it is the minimum of $n(n-1)/2$ i.i.d. random variables (with exponential distribution).
\end{rem}
Our goal is to characterise the mutation pattern for samples of $n$ individuals, mainly as $n$ gets large. We specify the mutation scheme in the next subsection.
Works studying mutation patterns arising from random genealogies are numerous. Mutation patterns related to populations with fixed size (Wright--Fisher model, Kingman coalescent) are well-known and culminate in \emph{Ewens' sampling formula} (see \cite{D1} for a comprehensive account on that subject). More recent works concern mutation patterns related to more general coalescents \cite{BG,M}, to branching populations \cite{AD, Bclusters}, or to both \cite{BBS}.
\subsection{Mutation scheme}
We adopt two classical assumptions on mutation schemes from population genetics (see e.g. \cite{Ew})
\begin{enumerate}
\item
mutations occur at \emph{constant rate} $\theta$ on germ lines,
\item
mutations are \emph{neutral}, that is, they have no effect on birth rates and lifetimes.
\end{enumerate}
As is usual, we assume that mutations are point substitutions occurring at a single site on the DNA sequence, and that each site can be hit at most once by a mutation. This last assumption is known as the \emph{infinitely-many sites model} (ISM). Instances of DNA sequence are called \emph{alleles} or \emph{haplotypes}, so that under the ISM, each mutation yields a new allele. Without reference to DNA sequences, this last assumption by itself is known as the \emph{infinitely-many alleles model} (IAM).\\
\\
Specifically, we let $({\cal P}_i;i=0,1,2\ldots)$ be independent Poisson measures on $(0,\infty)$ with intensity $\theta$ (cf. assumption 1). For each $i$ we denote the atoms of ${\cal P}_i$ by $\ell_{i1}<\ell_{i2}<\cdots$ and call them \emph{mutations}. Now let $H_1, H_2, \ldots$ be an independent coalescent point process (cf. assumption 2). In agreement with the genealogical structure of a coalescent point process explained in the beginning of this section, we will say that individual $i+k$ \emph{carries} (or \emph{bears}) mutation $\ell_{ij}$ if $k\ge 0$ and
$$
\max\{H_{i+1},\ldots H_{i+k} \}< \ell_{ij} <H_i,
$$
where we agree that $\max\emptyset=0$ and $H_0=+\infty$. The second inequality is trivially due to the fact that we throw away all atoms $\ell_{ij}$ such that $H_i\le\ell_{ij}$. The set of mutations that an individual bears is her \emph{allele} or her \emph{haplotype}, or merely her type.\\
\\
For a sample of $n$ individuals, we call $S_n$ the number of \emph{polymorphic sites}, that is, the number of mutations $(\ell_{ij}; 0\le i\le n-1, j\ge 1)$ that are carried by at least one individual and at most $n-1$. Formally, this yields
$$
S_n=\mbox{Card}\{\ell_{ij}< H_i, 1\le i\le n-1, j\ge 1\}+\mbox{Card}\{\ell_{0j}<\max\{H_1,\ldots, H_{n-1}\}, j\ge 1\}.
$$
Further, we define $S_n(k)$ as the number of mutations carried by $k$ individuals in the sample. In particular,
$$
S_n=\sum_{k=1}^{n-1}S_n(k).
$$
The sequence $(S_n(1),\ldots, S_n(n-1))$ is called the \emph{site frequency spectrum} of the sample.
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\caption{ A coalescent point process with mutations for a sample of $n=9$ individuals. Site $a$ is \emph{not} polymorphic because \emph{no} individual in the sample carries a mutation at that site; site $g$ is \emph{not} polymorphic because \emph{all} individuals in the sample carry the mutation at that site. The number of polymorphic sites is $S_n=6$. The number of distinct haplotypes is $A_n=5$.}
\label{fig : mutations}
\end{figure}
Similarly, we call $A_n$ the number of \emph{distinct haplotypes} in a sample of $n$ individuals, that is, the number of alleles that are carried by at least one individual, and $A_n(k)$ as the number of alleles carried by $k$ individuals. In particular, we have
$$
A_n=\sum_{k=1}^{n}A_n(k) \mbox{ and }\sum_{k=1}^n kA_n(k) = n.
$$
The sequence $(A_n(1),\ldots, A_n(n))$ is called the \emph{allele frequency spectrum} of the sample.
\begin{rem}
One always has the inequality $S_n\ge A_n -1$. Indeed, apart from the ancestral haplotype, each new haplotype independent of at least one new mutation.
\end{rem}
\subsection{Examples of coalescent point processes}
Before going into the main part of this work, we provide a few simple examples of coalescent point processes derived from splitting trees, in part for application purposes.
\paragraph{Yule tree.} When $\Lambda$ is a point mass at $\infty$, the splitting tree is a Yule tree, and $(N_t;t\ge0)$ is a pure-birth binary process with birth rate, say $a$. Then $W(x)=e^{ax}$, and $H$ has an exponential distribution with parameter $a$ (see \cite{P}).
\paragraph{Birth--death process.}
When $\Lambda$ has an exponential density, $(N_t;t\ge 0)$ is a Markovian birth--death process with (birth rate $b$ and) death rate, say $d$. Then it is known (see \cite{L} for example) that if $b\not=d$, then
$$
W(x) = \frac{d-be^{(b-d)x}}{d-b}\qquad x\ge 0,
$$
whereas if $b=d=:a$,
$$
W(x)=1+ax\qquad x\ge 0.
$$
Notice that in the subcritical case ($b<d$), $H$ can take the value $\infty$ with probability $1-(b/d)$, which is due to the constrained size of quasi-stationary populations (see Remark \ref{rem : H' et H}). Elementary calculations show that $H$ conditioned to be finite has the same law as the branch length of a \emph{supercritical} birth--death process with birth rate $d$ and death rate $b$.
\paragraph{Consistency and sampling.}
The genealogy associated with a coalescent point process is \emph{consistent} in the sense that the genealogy of a sample of $n$ individuals has the same law as that of a sample of $n+1$ individuals from which the \emph{last} individual has been withdrawn (in the splitting tree framework, the last individual is the individual who has no descendants in the sample, and whose ancestors have no elder sibling with descendants in the sample). This property would not hold any longer if the withdrawn individual was chosen at random.
On the other hand, if all individuals in the population are censused \emph{independently} with probability $c$, then the genealogy of the census is still that of a coalescent point process. Indeed, the typical branch length is $H''$, where
$$
H''\stackrel{\cal L}{=}\max\{H_1,\ldots, H_K\},
$$
and $K$ is an independent (modified) geometric r.v., that is, $\PP(K=j)=c(1-c)^{j-1}$. As a consequence,
$$
\frac{1}{W_c(x)}:=\PP(H''> x)=1-\sum_{j\ge 1}c(1-c)^{j-1}\PP(H\le x)^{j}\qquad x\ge 0.
$$
This last equation also reads
$$
W_c = 1-c +cW.
$$
Applying this Bernoulli sampling procedure with intensity $c$ to the previous examples yields the following elementary results.
\begin{description}
\item[--] the census of a Yule population has the genealogy of a birth--death process population, with birth rate $ac$ and death rate $a(1-c)$
\item[--] the census of a birth--death process population has the genealogy of another birth--death process population with birth rate $bc$ and death rate $d-b(1-c)$. In particular, censusing a critical birth--death process population with rate $b=d=:a$ amounts to replacing $a$ with $ac$.
\end{description}
\paragraph{Infinite lifespan measure.}
Actually, everything that was stated about splitting trees still holds if the lifespan measure is infinite, provided the lifespans of children remain summable, that is $\intgen(1\wedge r)\Lambda(dr)<\infty$. In particular, one still has $W(0)=1$, and the number of individuals alive at a fixed time remains a.s. finite.
On the contrary, it is a completely different task to define the real tree whose jumping contour process is a Lévy process with no negative jumps but \emph{infinite variation} (see \cite{B}). However, in our setting, this only requires replacing the coalescent point process $H_1, H_2,\ldots$ with a true Poisson point process with intensity measure $ds\,\nu(dx)$, where $\nu$ is a $\sigma$-finite positive measure defined as the push forward of the excursion measure of $X$ away from $\{t\}$ by the function which maps an excursion $\eps$ into $t-\inf_s\eps_s$. Similarly as in the finite variation case,
$$
W(x):=\frac{1}{\nu((x,\infty))}\qquad x\ge0.
$$
In the Brownian case, for example $\nu(dx)= x^{-2}dx$ (again, see \cite{P}), that is, $W(x)=x$.
Here, the analogue of Bernoulli sampling with intensity $c$ consists in taking the maximum $H''$ of the point process on an interval with exponential length of parameter $c$ (instead of a geometric length). Now $c$ can take any positive value. Standard calculations then yield
$$
\frac{1}{W_c(x)}:=\PP(H''> x)=1-\frac{c}{c+\nu((x,\infty))}\qquad x\ge 0,
$$
so that
$$
W_c=1+cW.
$$
As far as splitting trees with infinite variation are concerned, we will only focus on the stable case, where $W(x)=x^{\alpha-1}$ for some $\alpha\in(1,2]$, the Brownian case corresponding to $\alpha=2$. In particular, we see that the Brownian coalescent point process censused with intensity $c$ has the same law as the coalescent point process associated with a critical birth--death process with rate $c$.
\subsection{Statements, outline, examples}
Our results regarding polymorphic sites are stated in Section \ref{sec : S}.
In the first two subsections of Section \ref{sec : S}, we assume that $\EE(H)$ is finite. Theorem \ref{thm : LLN+CLT} provides a law of large numbers and a central limit theorem (if $H$ has a second moment) on the number $S_n$ of polymorphic sites. In particular,
\begin{equation}
\lim_{n\tendinfty}\frac{S_n}{n}=\theta\,\EE(H)\qquad\mbox{ a.s. }
\end{equation}
We also give exact explicit formulae for the \emph{expectation} of the number $S_n(k)$ of mutations carried by $k$ individuals in a sample of $n$.
In the third subsection, we make the less stringent assumption that $\EE(\min (H_1, H_2))$ is finite. Theorem \ref{thm : SFS large} then gives the asymptotic behaviour of the site frequency spectrum of large samples via the following a.s. convergence
\begin{equation}
\label{eqn : SFS large}
\lim_{n\tendinfty}\frac{S_n(k)}{n} = \theta\,\intgen \frac{dx}{W(x)^2}\left(1-\frac{1}{W(x)}\right)^{k-1}.
\end{equation}
In the fourth subsection, we treat the case of stable laws with parameter $\alpha$, that is, $W$ is given by $W(x)=1+c x^{\alpha -1}$, where $\alpha\in(1,2]$ and $c$ is some positive parameter that can be interpreted as a sampling intensity. Since here $\EE(H)=\infty$, the only result holding in the stable case is \eqref{eqn : SFS large}, and only for $\alpha>3/2$. Theorems \ref{thm : brown} and \ref{thm : stable} give the asymptotic behaviour of $S_n$. When $\alpha=2$, $S_n/n\ln (n)$ converges in probability (to $\theta/c$), and when $\alpha\not=2$, $S_n/n^{\beta}$ converges in distribution, with $\beta=1/(\alpha-1)$.\\
\\
Section \ref{sec : A} displays our results regarding distinct haplotypes. The trick is to characterise the law of the branch length $H^\theta$ of the next individual bearing no mutation other than those carried by, say, individual 0. Proposition \ref{prop : next branch with no} does this as follows
$$
\frac{1}{\PP(H_\theta>x)}=:W_\theta(x)=1+\int_0^x W'(u)e^{-\theta u}\, du\qquad x\ge 0.
$$
Theorem \ref{thm : AFS large} states a.s. convergences without moment existence assumptions. Specifically,
\begin{equation}
\lim_{n\tendinfty}\frac{A_n}{n}=\EE\left(1-e^{-\theta H^\theta}\right)\qquad\mbox{ a.s., }
\end{equation}
and the allele frequency spectrum for large samples is given by the following a.s. convergence
\begin{equation}
\lim_{n\tendinfty}\frac{A_n(k)}{n} = \intgen dx \,\theta\,e^{-\theta x} \frac{1}{W_\theta(x)^2}\left(1-\frac{1}{W_\theta(x)}\right)^{k-1}.
\end{equation}
Before ending this last subsection, we want to point out that in some cases, more explicit formulae can be computed. First, for the \emph{Yule process} with birth rate 1, (or with parameter $a$, but after replacing $\theta$ with $a\theta$), that is, when $W(x) = e^x$, one gets easily
$$
\lim_{n\tendinfty}\frac{S_n}{n}=\theta\quad\mbox{ and }\quad \lim_{n\tendinfty}\frac{S_n(k)}{n}=\frac{\theta}{k(k+1)}\ .
$$
Computations are not as straightforward for the number of haplotypes.
Second, for the \emph{critical birth--death process} with birth rate 1 (or with parameter $a$, but after replacing $\theta$ with $a\theta$), that is, when $W(x) = 1+x$, one gets
$$
\lim_{n\tendinfty}\frac{S_n}{n \ln(n)}=\theta\quad\mbox{ and }\quad \lim_{n\tendinfty}\frac{S_n(k)}{n}=\frac{\theta}{k}\ .
$$
In addition,
$$
\lim_{n\tendinfty}\frac{A_n}{n}=\theta \ln\left(1+\theta^{-1}\right)\quad\mbox{ and }\quad \lim_{n\tendinfty}\frac{A_n(k)}{n}=\frac{\theta}{k} \left(1+\theta\right)^{-k}\ .
$$
\begin{rem}
It is amusing to notice that the rescaled number $A_n(k)$ of haplotypes with $k$ representatives is also the probability that a species has $k$ representatives in Fisher's \emph{log-series of species abundance} \cite{FCW}. In Fisher's model, a given species has an unknown density which is assumed to be drawn from a Gamma distribution with parameter $a$. As a result of Bernoulli sampling in a large population, it is then assumed that given the value $d$ of this density, the number $X$ of individuals spotted from this species is Poisson with parameter $\rho d$, where $\rho$ is the sampling intensity.
It can then be shown that as $a\downarrow 0$, conditional on $\{X\ge 1\}$ (since at least one individual must be spotted for the species to be recorded), $\PP(X=k)$ goes to $C(1+1/\rho)^{-k}/k$, for some normalising constant $C$.
\end{rem}
\begin{rem}
In a coalescent point process, divergence times are on average deeper than in the Kingman coalescent (our trees are more `star-like'). This forbids convergence of our statistics without rescaling (by the sample size $n$ or by $n\ln(n)$). In particular, notice that the asymptotic proportion of individuals in a cluster of size greater than $K$, i.e. $\lim_n n^{-1}\sum_{k\ge K}A_n(k)$, vanishes as $K$ grows to $\infty$. This shows that the largest cluster in a sample of $n$ has neglectable size w.r.t. $n$, which contrasts with the Kingman coalescent, where the allele frequency spectrum is given by Ewens' sampling formula (see \cite{D1, Ew}). As $n\to\infty$, the numbers of haplotypes $A_n(k)$ carried by $k$ individuals \cite{ABT} converge to independent Poisson r.v. with parameter $\theta/k$, and the $i$-th eldest haplotype \cite{DT} is carried by approximately $P_in$ individuals, where $(P_i;i\ge 1)$ is a Poisson--Dirichlet r.v.
\end{rem}
\section{Number of polymorphic sites}
\label{sec : S}
Results for polymorphic sites depend on integrability assumptions on $H$. Of course these are always fulfilled if the time $t$ when the population was founded is known, since then $H\le t$ a.s.
We will see that the critical assumptions are either $\EE(\min(H_1,H_2))<\infty$, or the more stringent $\EE (H)<\infty$.
Notice that the first assumption is equivalent to the integrability of $1/W^2$, and the second one to the integrability of $1/W$.
\subsection{Law of large numbers and central limit theorem}
Recall that $S_n$ is the number of polymorphic sites in the sample of $n$ individuals.
\begin{thm}
\label{thm : LLN+CLT}
If $\EE(H)<\infty$, then
$$
\lim_{n\tendinfty} n^{-1}S_n = \theta\, \EE(H)\qquad\mbox{a.s. and in }L^1.
$$
If in addition $\EE(H^{2})<\infty$, then
$$
\sqrt{n}\left(n^{-1}S_n-\theta\,\EE(H)\right)
$$
converges in distribution to a centered normal variable with variance $\theta\,\EE(H)+\theta^2\mbox{Var}(H)$.
\end{thm}
\paragraph{Proof.}
Set $Y_n:=\max\{H_1, \ldots, H_{n-1}\}$.
Recall from the Introduction that
$$
S_n=\sum_{i=1}^{n-1} Q_i +R_n,
$$
where $Q_i$ is the number of points of the Poisson point process ${\cal P}_i$ in $(0,H_i)$, and $R_n$ is the number of points of the Poisson point process ${\cal P}_0$ in $(0,Y_n)$.
By the strong law of large numbers, we know that
$$
\lim_{n\tendinfty} n^{-1}\sum_{i=1}^{n-1} Q_i = \theta\, \EE(H)\qquad\mbox{a.s. and in }L^1,
$$
so we need to prove that
$$
\lim_{n\tendinfty} n^{-1} R_n = 0\qquad\mbox{a.s. and in }L^1.
$$
Now because $R_n/Y_n$ converges to $\theta$ a.s. and in $L^1$, it is sufficient to prove that
$$
\lim_{n\tendinfty} n^{-1} Y_n = 0\qquad\mbox{a.s. and in }L^1.
$$
Because $Y_n<\sum_{i=1}^{n-1}H_i$,
$$
\limsup_{n\tendinfty} n^{-1} Y_n =:Y<\infty\mbox{ a.s.}
$$
By the 0-1 law, $Y$ is not random. To prove that $Y=0$, we let $Y_n^{(1)}$ (resp. $Y_n^{(2)}$) be the maximum of the $H_i$'s indexed by odd (resp. even) numbers. Then it is clear that $Y_n=\max(Y_n^{(1)}, Y_n^{(2)})$, and that $n^{-1}Y_n^{(1)}$ as well as $n^{-1}Y_n^{(2)}$ both converge to $Y/2$. This shows that $Y=Y/2$, so that $Y=0$.
For convergence in $L^1$, pick any $x>0$, and notice that
\debeq
n^{-1}\EE(Y_n) &=& n^{-1}\EE(Y_n,Y_n\le x)+n^{-1}\EE(Y_n, Y_n>x)\\
&\le& n^{-1}x+n^{-1}\EE\left(\sum_{i=1}^{n-1}H_i\indic{H_i>x}\right)\\
&\le& n^{-1}x+\EE(H,H>x).
\fineq
Since $\EE(H)<\infty$, this last inequality shows that $n^{-1}\EE(Y_n)$ vanishes as $n\tendinfty$.
Now we prove the central limit theorem for $S_n$. It is elementary to compute $\mbox{Var}(Q_1)$ as $\theta\,\EE(H)+\theta^2\mbox{Var}(H)$, so by the classical central limit theorem applied to the sum of $Q_i$'s, we only have to prove that $R_n/\sqrt{n}$ converges to 0 in probability. For any $\lbd>0$,
$$
\EE\left(\exp\left(-\lbd R_n/\sqrt{n}\right)\right)=\EE\left(\exp\left(-\theta Y_n\left(1-e^{-\lbd/\sqrt{n}}\right)\right)\right),
$$
which shows it is sufficient to prove that $Y_n/\sqrt{n}$ converges to 0 in probability. As previously, we write
\debeq
n^{-1}\EE\left(Y_n^2\right) &=& n^{-1}\EE\left(Y_n^2,Y_n\le x\right)+n^{-1}\EE\left(Y_n^2, Y_n>x\right)\\
&\le& n^{-1}x^2+n^{-1}\EE\left(\sum_{i=1}^{n-1}H_i^2\indic{H_i>x}\right)\\
&\le& n^{-1}x^2+\EE\left(H^2,H>x\right).
\fineq
Thus, convergence of $Y_n/\sqrt{n}$ to 0 holds in $L^2$, and subsequently, it holds in probability.
\hfill$\Box$
\subsection{Explicit formulae for the expected frequency spectrum}
Recall that $S_n(k)$ denotes the number of mutant sites that are carried by exactly $k$ individuals in the sample of $n$ individuals (and since we only count polymorphic sites, $S_n(n)=0$).
\begin{thm}
\label{thm : exact exp}
For all $1\le k\le n-1$,
\begin{equation*}
\EE(S_n(k)) = \theta\,\intgen dx \left(1-\frac{1}{W(x)}\right)^{k-1}
\left( \frac{n-k-1}{W(x)^2}+\frac{2}{W(x)} \right),
\end{equation*}
which is finite if and only if $\EE(H)<\infty$.
Then in particular,
$$
\lim_{n\tendinfty}n^{-1}\EE(S_n(k)) = \theta\,\intgen \frac{dx}{W(x)^2}\left(1-\frac{1}{W(x)}\right)^{k-1}.
$$
\end{thm}
\begin{rem}
Taking the sum over $k$ in the r.h.s. of the last equality of the theorem, one gets $\theta\,\EE(H)$, so that, thanks to the $L^1$ convergence in Theorem \ref{thm : LLN+CLT},
$$
\lim_{n\tendinfty}n^{-1}\sum_{k=1}^{n-1}\EE(S_n(k))=\theta\,\EE(H)=\sum_{k\ge 1}\lim_{n\tendinfty} n^{-1}\EE (S_n(k)).
$$
\end{rem}
Before giving a proof of the previous theorem, we want to make a point that will also be useful in the next subsection. For any tree with point mutations, a mutation is carried by $k$ individuals if and only of it is in the part of the tree subtending $k$ leaves. Then in any given tree with edge lengths and Poisson point process of mutations (with rate $\theta$) independent of the genealogy (as in our situation), the expectation of the number of mutations carried by $k$ individuals is $\theta L_k$, where $L_k$ is the Lebesgue measure of the part of the tree subtending $k$ leaves (i.e., tips).
In our setting, we will call $L_k(n)$, for $k\le n-1$, the Lebesgue measure of the part of the tree subtending $k$ tips among individuals $\{0,1,\ldots, n-1\}$, so that
$$
\EE(S_n(k)) = \theta\, \EE(L_k(n))\qquad 1\le k\le n-1.
$$
\begin{rem}
The last equality along with more specific considerations given in the next subsection provide a less analytic and more transparent proof than the proof we give hereafter. However, we stick to it for the interest of the method itself.
\end{rem}
\paragraph{Proof of Theorem \ref{thm : exact exp}}
We set $N(x)$ to be the smallest $i\ge 1$ such that $H_i> x$. The proof relies on the fact that
$$
\EE(S_n(k)) = \lim_{x\tendinfty}\theta\,\EE(L_k(N(x)) \mid N(x)=n)\qquad 1\le k\le n-1.
$$
On the event $\{N(x)=n\}$, we will need to extend the definition of $L_k(N(x))$ to $k=n$, as being the Lebesgue measure of the part of the tree \emph{up to time $-x$} subtending all tips $\{0,1,\ldots, n-1\}$, that is, $L_n(N(x))= x-\max_{i=1,\ldots, n-1}H_i$.
For editing reasons, we will prefer to write $F(x)=\PP(H>x)$, instead of $1/W(x)$. Since $F$ is a.e. differentiable and our goal is to let $x\tendinfty$, we can set $f(x) := -F'(x)$ without loss of generality.
We let $\tilde{H}$ denote the branch length $H_{N(x)}$, and we set
$$
\tilde{N}:=\min\{k\ge 1:H_{N(x)+k}>x+dx\},
$$
as well as $\tilde{L}_k$ the Lebesgue measure of the part of the tree subtending $k$ leaves among individuals $\{N(x),N(x)+1,\ldots, N(x+dx)-1\}$.
Note that $(\tilde{N}, \tilde{L}_k, \tilde{H})$ are independent of $(N(x), L_k(N(x)))$; that $\tilde{H}$ is distributed as $H$ conditional on $\{H>x\}$; and that ($\tilde{N},\tilde{L}_k)$ is independent of $\tilde{H}$ and distributed as $(N(x+dx), L_k(N(x+dx))$. Next observe that if $\tilde{H}>x+dx$, then $N(x+dx)=N(x)$ and $L_k(N(x+dx))=L_k(N(x))$, except if $k=n$, where by definition $L_n(N(x+dx))=L_n(N(x))+dx$. On the other hand, if $\tilde{H}\in dx$, $L_k(N(x+dx))$ is the sum of measures of edges subtending $k$ tips in $\{0,1,\ldots,N(x)-1\}$ with measures of edges subtending $k$ tips in $\{N(x),\ldots,N(x)+\tilde{N}-1\}$. This reads
\begin{multline*}
L_k(N(x+dx))\indic{N(x+dx)=n} = \indic{\tilde{H}>x+dx}\indic{N(x)=n}\left(L_k(N(x))+dx\indicbis{k=n}\right)\\
+\indic{\tilde{H}\le x+dx}\sum_{j=1}^{n-1}\indic{N(x)=j}\indic{\tilde{N}=n-j}\left(L_k(N(x))+\tilde{L}_k(\tilde{N})\right),
\end{multline*}
where we have used the extension of the definition of $L_k$ specified earlier (cases when $k=j$ or $k=n-j$ in the sum). Now set
$$
U_{k,n}(x):= \EE(L_k(N(x)), N(x)=n).
$$
By the independences stated previously, taking expectations, we get
$$
U_{k,n}'(x+)= -U_{k,n}(x)\frac{f}{F}(x)+\indicbis{k=n}\PP(N(x)=k)
+2\sum_{j=1}^{n-1}U_{k,j}(x)\PP(N(x)=n-j)\frac{f}{F}(x).
$$
Setting
$$
V_k(x;s):=\sum_{n\ge k}U_{k,n}(x)s^n\qquad s\in[0,1),
$$
and observing that $\vert U_{k,n}(x)\vert\le nx$, and (so) that $\vert U_{k,n}'(x)\vert \le c(x)n^2$ for some positive $c(x)$ independent of $k$ and $n$, we get
$$
\frac{\partial V_k}{\partial x}(x;s)= -\frac{f}{F}(x)V_k(x;s)+\PP(N(x)=k) s^k + 2 \frac{f}{F}(x)\sum_{n\ge k}s^n\sum_{j=1}^{n-1}U_{k,j}(x)\PP(N(x)=n-j).
$$
Since $U_{k,j}(x)=0$ when $j\le k-1$, the last term equals
\debeq
2 \frac{f}{F}(x)\sum_{n\ge k+1}s^n\sum_{j=k}^{n-1}U_{k,j}(x)\PP(N(x)=n-j)&=& 2 \frac{f}{F}(x)\sum_{j\ge k}U_{k,j}(x)s^j\sum_{n\ge j+1}s^{n-j}\PP(N(x)=n-j)\\
&=& 2 \frac{f}{F}(x)V_k(x;s)\sum_{n\ge 1}s^{n}\PP(N(x)=n).
\fineq
As a consequence, we get the following differential equation
$$
\frac{\partial V_k}{\partial x}(x;s)= G_k(x;s)V_k(x;s)+\PP(N(x)=k)s^k,
$$
where we have put
$$
G_k(x;s):=\left(2\EE\left(s^{N(x)}\right)-1\right)\frac{f}{F}(x).
$$
Now since $\PP(N(x)=k)=F(x)(1-F(x))^{k-1}$, we easily get
$$
\int_0^x G_k(y;s) \, dy = \ln\left[\frac{F(x)}{\left( 1-s+sF(x)\right)^2}\right].
$$
This allows us to integrate the differential equation in $V_k(.;s)$ to finally arrive at
$$
V_k(x;s)=\frac{s^k F(x)}{(1-s+sF(x))^2}
\int_0^x(1-F(y))^{k-1}(1-s+sF(y))^2\, dy.
$$
With the shortcuts $u:=1-F(x)$ and $v:=1-F(y)$, and using the series expansion of $(1-us)^{-2}$, we get
$$
V_k(x;s)=s^k (1-u)\int_0^x v^{k-1} (1-vs)^2 \sum_{j\ge 1} ju^{j-1}s^{j-1}\, dy.
$$
It is elementary algebra to compute the following equality
$$
(1-vs)^2 \sum_{j\ge 1} ju^{j-1}s^{j-1} = 1+\sum_{j\ge 1}s^j u^{j-2}\left(j(u-v)^2+u^2-v^2\right),
$$
which yields
$$
V_k(x;s)= s^k(1-u)\int_0^x v^{k-1}\, dy+(1-u)\sum_{j\ge 1}\int_0^x v^{k-1}s^{k+j} u^{j-2}\left(j(u-v)^2+u^2-v^2\right)\, dy.
$$
Identifying this entire series with the definition of $V_k$, we get for all $1\le k\le n-1$,
\begin{multline*}
U_{k,n}(x)= F(x)(1-F(x))^{n-k-2}\int_0^x (1-F(y))^{k-1}\times\\
\times\left((n-k)(F(y)-F(x))^2+(1-F(x))^2- (1-F(y))^2 \right)\, dy.
\end{multline*}
As a consequence,
\begin{multline*}
\EE(L_k(N(x)) \mid N(x)=n)=(1-F(x))^{-k-1}\int_0^x (1-F(y))^{k-1}\times\\
\times\left((n-k)(F(y)-F(x))^2+(1-F(x))^2- (1-F(y))^2 \right)\, dy.
\end{multline*}
which, by Beppo Levi's theorem, converges, as $x\tendinfty$, to
$$
\theta^{-1} \EE(S_n(k))=\intgen (1-F(y))^{k-1}
\left((n-k)F(y)^2+1- (1-F(y))^2 \right)\, dy,
$$
and this finishes the proof.\hfill$\Box$
\subsection{Site frequency spectrum of large samples}
Here, we assume that $\EE(\min(H_1,H_2))<\infty$, that is, $1/W^2$ is integrable.
\begin{thm}
\label{thm : SFS large}
For all $1\le k\le n-1$, the following convergence holds a.s. (and in $L^1$ as well if $\EE(H)<\infty$)
\begin{eqnarray*}
\lim_{n\tendinfty}n^{-1}S_n(k) &=& \theta\, \EE\left(\left( \min\{H_1,H_{k+1}\}-\max\{H_2,\ldots,H_k\}\right)^+\right)\\
&=& \theta\,\intgen \frac{dx}{W(x)^2}\left(1-\frac{1}{W(x)}\right)^{k-1}.
\end{eqnarray*}
\end{thm}
\paragraph{Proof.}
Reasoning similarly as in the previous subsection, we see that a point mutation occurring on branch $i$ is carried by $k$ individuals if and only if it is carried by individuals $i,i+1,\ldots, i+k-1$, and by no one else. This happens if and only if this mutation, corresponding to the atom $\ell_{ij}$, say, of ${\cal P}_i$, has
$$
\max\{H_{i+1},\ldots, H_{i+k-1}\}<\ell_{ij}<H_i,
$$
for the mutation to be carried by individuals $i,i+1,\ldots, i+k-1$, along with
$$
\ell_{ij}<H_{i+k},
$$
for the mutation not to be carried by others.
More formally, we set ${\cal F}$ the space of point processes on $(0,\infty)$, and ${F}_k$ the set of $(k+1)$-dimensional arrays with values in ${\cal F}\times (0,\infty)$. Next, for any $\Xi\in F_k$, written as $\Xi = ((p_0,x_0),\ldots, (p_k, x_k))$ we define
$$
G(\Xi):= \mbox{Card}\left(p_0\cap \left(\max\{x_{1},\ldots, x_{k-1}\}, \min\{x_0, x_{k}\}\right)\right),
$$
where it is understood that the interval $(a,b)$ is empty if $a\ge b$. Then the number of mutations carried by $k$ individuals among the first $n$ can be written as
$$
S_n(k)=\sum_{i=0}^{n-k}G(\Xi_i),
$$
where
$$
\Xi_i:=(({\cal P}_i, H_i),\ldots,({\cal P}_{i+k}, H_{i+k}))
$$
and, for the last term of the sum to be correctly written, $H_{n}$ is set to $+\infty$ (as $H_0$). Next, observe that
$$
\EE(G(\Xi_1)) =\theta\, \EE\left(\left( \min\{H_1,H_{k+1}\}-\max\{H_2,\ldots,H_k\}\right)^+\right),
$$
so that $G(\Xi_1)$ is integrable (assumption stated before the theorem). Now for any $0\le r\le k$, the random values $G(\Xi_i)$, for $i$ such that $i=r\; [k+1]$ (standing for mod $(k+1)$), are i.i.d. and integrable, so by the strong law of large numbers, we have the following a.s. convergence
$$
\lim_{n\tendinfty} n^{-1}\sum_{0\le i= r [k+1]\le n-k}G(\Xi_i)=\frac{1}{k+1}\;\EE(G(\Xi_1)).
$$
Actually, the convergence would also hold in $L^1$ if we had discarded mutations carried by individual $0$ and individual $n-k$, which involve terms that are not integrable if $\EE(H)=\infty$. If $\EE(H)<\infty$, then convergence holds in $L^1$.
Summing over $r$ these $k+1$ equalities, we get the convergence of $n^{-1}S_n(k)$ to $\EE(G(\Xi_1))$, and
\debeq
\EE(G(\Xi_1)) &=&\theta\, \EE\left(\left( \min\{H_1,H_{k+1}\}-\max\{H_2,\ldots,H_k\}\right)^+\right)\\
&=&\theta\,\EE\intgen dx\,\indicbis{x<\min\{H_1,H_{k+1}\}}\,\indicbis{x>\max\{H_2,\ldots,H_k\}}\\
&=&\theta\,\intgen dx\,\PP(H>x)^2\,\PP(H<x)^{k-1},
\fineq
which ends the proof.\hfill$\Box$
\subsection{Stable laws}
Here, we tackle the case when $H$ is in the domain of attraction of a stable law, which happens in particular for a splitting tree whose contour process is a stable Lévy process with no negative jumps with index $\alpha\in(1,2]$. If such a population is censused with intensity $c>0$ then the corresponding function $W$ (see Introduction) is
$$
W(x)=1+cx^{\alpha-1}\qquad x\ge 0.
$$
From now on, we will assume that $W$ has the form given in the foregoing display.
Recall that $1/W(x)$ is the probability that a branch has length greater than $x$.
Observe that here $H$ is not integrable, so that Theorems \ref{thm : LLN+CLT} and \ref{thm : exact exp} do not apply. However, asymptotic results for the site frequency spectrum of large samples given in Theorem \ref{thm : SFS large} apply for $\alpha>3/2$.
\subsubsection{Brownian case}
Here, we assume that $\alpha=2$, which corresponds both to a (censused) Brownian population and to the (censused or not) population of a critical birth--death process.
\begin{thm}
\label{thm : brown}
When $W(x)=1+cx$, we have the following convergence in probability
$$
\lim_{n\tendinfty} \frac{S_n}{n\ln (n)} = \theta/c.
$$
\end{thm}
\paragraph{Proof.}
Recall that $S_n$ is to be written as
$$
S_n=\sum_{i=1}^{n-1} Q_i +R_n,
$$
where $Q_i$ is the number of points of the Poisson point process ${\cal P}_i$ in $(0,H_i)$, and $R_n$ is the number of points of the Poisson point process ${\cal P}_0$ in $(0,Y_n)$, where $Y_n=\max\{H_1, \ldots, H_{n-1}\}$.
Now observe that
$$
\PP(Y_n>\vareps n\ln (n)) = 1-\left(1-\frac{1}{1+c\vareps n\ln (n)} \right)^{n-1},
$$
which vanishes as $n\tendinfty$, so that $Y_n/n\ln (n)$ converges to 0 in probability. This implies in turn that $R_n/n\ln (n)$ also converges to 0 in probability. As a consequence, we can focus on the sum of $Q_i$'s. Pick any $\lbd>0$ and check that
$$
\EE\left(\exp-\frac{\lbd}{n\ln(n)} \sum_{i=1}^{n-1} Q_i\right)=\left(\EE\left(\exp-\theta H\left(1-e^{-\lbd/n\ln(n)}\right)\right)\right)^{n-1},
$$
We are bound to study the behaviour of $\EE(\exp-yH)$ as $y\to 0$.
\debeq
\EE(\exp-yH) &=& 1-y\int_0^\infty\frac{e^{-yx}}{W(x)}\, dx\\
&=& 1-y\int_0^\infty\frac{e^{-u}}{y+cu}\, du\\
&=& 1-y\int_1^\infty\frac{e^{-u}}{y+cu}\,du +y \int_0^1\frac{1-e^{-u}}{y+cu}\,du- yc^{-1}\ln((y+c)/y)\\
&=& 1+c^{-1}y\ln(y) + O(y),
\fineq
where $O(y)/y$ is bounded near 0. Setting $u_n:=\theta\left(1-e^{-\lbd/n\ln(n)}\right)$, there is a vanishing sequence $v_n$ such that
\debeq
\EE\left(\exp-\lbd\frac{S_n}{n\ln(n)} \right) &=&\left(1+c^{-1}u_n\ln(u_n)+O(u_n)\right)^n(1+v_n)\\
&=&\exp\left(c^{-1}nu_n\ln(u_n)+O(nu_n)\right)(1+v_n),
\fineq
which converges to $\exp(-\lbd\theta/c)$.\hfill$\Box$
\subsubsection{Stable case $\alpha\not= 2$}
Here, we assume that $W(x)=1+cx^{\alpha-1}$, for some $\alpha\in(1,2)$.
\begin{thm}
\label{thm : stable}
When $W(x)=1+cx^{\alpha-1}$, we have the following convergence in distribution
$$
\lim_{n\tendinfty} \frac{S_n}{n^{1/(\alpha-1)}} = Z_{\varphi(\mathbf{e})},
$$
where $(Z_t;t\ge 0)$ is the stable subordinator with Laplace exponent $\lbd\mapsto c^{-1}\theta^{\alpha-1}\lbd^{\alpha - 1}$, $\mathbf{e}$ is an independent exponential r.v. with parameter 1, and $\varphi$ is defined by
$$
\varphi(x)=x^{1-\alpha}\,e^{-x}+\int_0^x ds\, s^{1-\alpha}\,e^{-s}\qquad x>0.
$$
\end{thm}
\begin{rem}
Observe that $\varphi$ decreases on $(0,\infty)$ from $+\infty$ to a positive limit, equal to $\Gamma(2-\alpha)$. Also, recall that $S_n=\sum_{i=1}^{n-1} Q_i +R_n$, where $R_n$ is the extra contribution from the maximum branch length. Then it is possible to see by the same kind of proof as that of the theorem, that $\sum_{i=1}^{n-1} Q_i$ converges in distribution to $Z_{\Gamma(2-\alpha)}$. This indicates that, opposite to the Brownian case, the (double) contribution of the maximum branch length is not negligible here.
\end{rem}
\paragraph{Proof.}
Let us compute the limiting distribution of $n^{-1/(\alpha -1)}(Y_n+\sum_{i=1}^{n-1}H_i)$, where $Y_n=\max\{H_1, \ldots, H_{n-1}\}$. Set $\beta:=1/(\alpha-1)$, as well as
$$
I_n(\lbd):=\EE\left(\exp-\lbd\, n^{-\beta}\left(Y_n+\sum_{i=1}^{n-1}H_i\right)\right).
$$
Then
$$
I_n(\lbd)=\intgen\PP(Y_n \in dz) e^{-2\lbd n^{-\beta} z} \left( \EE\left(e^{-\lbd n^{-\beta}H_z'}\right)\right)^{n-2},
$$
where $H_z'$ has the law of $H$ conditioned on being smaller than $z$. Next, we have
$$
\PP(Y_n\in dz) = \left(\frac{cz^{\alpha-1}}{1+cz^{\alpha-1}}\right)^{n-2}\frac{c(n-1)(\alpha-1)z^{\alpha-2}}{(1+cz^{\alpha-1})^2}\,dz\qquad z>0
$$
and
$$
\PP(H_z'\in dx) = \frac{c(\alpha-1)x^{\alpha-2}}{(1+cx^{\alpha-1})^2}\,\frac{1+cz^{\alpha-1}}{cz^{\alpha-1}}\,dx\qquad 0< x < z,
$$
so we get
$$
I_n(\lbd)=\intgen dz \,\frac{c(n-1)(\alpha-1)z^{\alpha-2}}{(1+cz^{\alpha-1})^2}e^{-2\lbd n^{-\beta} z} \left( \int_0^z dx\, \frac{c(\alpha-1)x^{\alpha-2}}{(1+cx^{\alpha-1})^2}\,e^{-\lbd n^{-\beta}x}\right)^{n-2}
$$
Changing variables, this also reads
$$
I_n(\lbd)=c^{-1} (1-n^{-1})(\alpha-1)\lbd^{\alpha-1}\intgen dv \,\frac{v^{-\alpha} \,e^{-2v}}{\left(1+n^{-1}c^{-1}\lbd^{\alpha-1}v^{1-\alpha}\right)^2} J_n(v;\lbd)^{n-2}
$$
where
\debeq
J_n(v;\lbd) &=& (\alpha-1)cn\lbd^{1-\alpha}\int_0^v du\, \frac{u^{\alpha-2}\, e^{-u}}{(1+cn\lbd^{1-\alpha}u^{\alpha-1})^2}\\
&=& \left[\frac{- e^{-u}}{1+cn\lbd^{1-\alpha}u^{\alpha-1}}\right]_0^v- \int_0^v du\,\frac{ e^{-u}}{1+cn\lbd^{1-\alpha}u^{\alpha-1}}\\
&=&1-\frac{ e^{-v}}{1+cn\lbd^{1-\alpha}v^{\alpha-1}}- \int_0^v du\,\frac{ e^{-u}}{1+cn\lbd^{1-\alpha}u^{\alpha-1}}\\
&=& 1-n^{-1}K_n(v;\lbd),
\fineq
where $K_n(v;\lbd)$ is positive and converges to $c^{-1}\lbd^{\alpha-1}\varphi(v)$ as $n\tendinfty$.
By the Lebesgue convergence theorem, we get the convergence of $I_n(\lbd)$ to
$$
c^{-1}(\alpha-1)\lbd^{\alpha-1}\intgen dv \,v^{-\alpha} \,e^{-2v} \exp (-c^{-1}\lbd^{\alpha-1}\varphi(v)).
$$
Integrating by parts with $\varphi'(v)=(1-\alpha)v^{-\alpha}e^{-v}$, we finally get
$$
\lim_{n\tendinfty} I_n(\lbd) = \intgen dv \,e^{-v} \exp (-c^{-1}\lbd^{\alpha-1}\varphi(v)).
$$
The last step is the same as in the foregoing proof, that is
\debeq
\lim_{n\tendinfty} \EE\left(\exp-\lbd\, n^{-1/(\alpha-1)}S_n\right)
&=&\lim_{n\tendinfty} \EE\left(\exp-\theta\,\left(1-e^{-\lbd n^{-1/(\alpha-1)}}\right)\left(Y_n+\sum_{i=1}^{n-1}H_i\right)\right)\\
&=&\lim_{n\tendinfty} I_n(\theta\lbd )\\
&=&\intgen dv \,e^{-v} \exp (-c^{-1}\theta^{\alpha-1}\lbd^{\alpha-1}\varphi(v)),
\fineq
which is the desired result. \hfill $\Box$
\section{Number of distinct haplotypes}
\label{sec : A}
\subsection{The next branch with no extra mutation}
We let ${\cal E}^\theta$ denote the set of individuals who \emph{carry no more mutations than} individual $0$ (some of and at most exactly the mutations carried by $0$, but no other mutation). Set $K^\theta_0:=0$ and for $i\ge 1$, define $K^\theta_i$ as the $i$-th individual in ${\cal E}^\theta$, and $H^\theta_i:=H_{K^\theta_i}$ the associated branch length.
We write $H^\theta$ in lieu of $H^\theta_1$ and we define the function $W_\theta$ by
$$
\PP(H^\theta>x)=\frac{1}{W_\theta(x)}\qquad x\ge 0.
$$
\begin{prop}
\label{prop : next branch with no}
The bivariate sequence $((K^\theta_i-K^\theta_{i-1},H^\theta_i);i\ge 1)$ is a sequence of i.i.d. random pairs.
The function $W_\theta$ is given by
$$
W_\theta(x)=1+\int_0^x W'(u)e^{-\theta u}\, du\qquad x\ge 0.
$$
\end{prop}
\begin{rem}
In the case when the coalescent process is derived from a splitting tree with lifespan measure $\Lambda$, the calculation of $W_\theta$ is straightforward. Indeed, it can be seen in that case that the point process $(H^\theta_i;i\ge 1)$ is the coalescent point process of the splitting tree obtained from the initial splitting tree with mutations after throwing away all points above a mutation. But this new tree is again a splitting tree, since lifespans are i.i.d. and terminate either at death time or at the first point mutation, so the lifespan measure is now $\Lambda_\theta (dx)=e^{-\theta x}\,\Lambda (dx)+ \theta e^{-\theta x} \Lambda((x,\infty))\, dx$. As a consequence, $W_\theta$ is here the scale function characterised as in \eqref{eqn : LT scale} by its Laplace transform
\debeq
\intgen dx\, e^{-\lbd x} \, W_\theta(x) &=& \left(\lbd -\intgen \Lambda_\theta(dx) (1-e^{-\lbd x}) \right)^{-1}\\
&=&\frac{\lbd+\theta}{\lbd}\left(\lbd+\theta -\intgen \Lambda(dx) (1-e^{-(\lbd +\theta) x}) \right)^{-1}\\
&=&\frac{\lbd+\theta}{\lbd}
\intgen dx\, e^{-(\lbd+\theta) x} \, W(x),
\fineq
which yields the equality given in the statement.
\end{rem}
\paragraph{Proof.}
First observe that the pair $(K^\theta_1, H^\theta_1)$ does not depend on the haplotype of individual $0$, and that the $i$-th individual with no mutation other than those carried by individual $0$ is also the next individual after $K^\theta_{i-1}$ with no mutation other than those carried by individual $K^\theta_{i-1}$. This ensures that $(K^\theta_i-K^\theta_{i-1},H^\theta_i)$ has the same law as $(K^\theta_1,H^\theta_1)$, and the independence between $(K^\theta_i-K^\theta_{i-1},H^\theta_i)$ and previous pairs is due to the independence of branch lengths and the fact that new mutations can only occur on branches with labels strictly greater than $K^\theta_{i-1}$.
Now the event $\{H^\theta\in dx\}$ can be decomposed according to: the value of $H_1$; conditional on $H_1=z$, the value of the age $V_z$ of the oldest mutation on $H_1$; conditional on $V_z=y$, the value $H_y'$ of the branch length associated with the first individual in ${\cal E}^\theta_1$ with branch length greater than $y$. Indeed, $H^\theta\in dx$ if: $H_1\in dx$ and there is no mutation in $H_1$ (then $K^\theta_0=1$); or $H_1\in dx$, the age of the oldest mutation on $H_1=x$ is $V_x=y<x$ and the next individual with no mutation other than those carried by individual $1$ and branch length $H_y'>y$ has $H_y'<x$; or $H_1=z<x$, the age of the oldest mutation on $H_1=z$ is $V_z=y<z$ and the next individual with no mutation other than those carried by individual $1$ and branch length $H_y'>y$ has $H_y'\in dx$.
\begin{multline*}
\PP(H^\theta\in dx)= \PP(H_1\in dx)e^{-\theta x}+\PP(H_1\in dx)\int_0^x\PP(V_x\in dy)\PP(H_y'<x)\\
+\int_0^x\PP(H_1\in dz)\int_0^z\PP(V_z\in dy)\PP(H_y'\in dx).
\end{multline*}
Thanks to the first statement of the proposition, $H_y'$ has the same law as $H^\theta$ conditioned on being greater than $y$. Then since $\PP(V_z\in dy)=\theta \,e^{-\theta(z-y)}\,dy$, we get
$$
\PP(H^\theta\in dx)
=\PP(H_1\in dx)(1-\PP(H^\theta>x) f(x))+\PP(H^\theta \in dx)\int_0^x \PP(H_1\in dz) f(z),
$$
where we have set
$$
f(x):=\int_0^x\, dy\, \theta \,e^{-\theta (x-y)}\,W_\theta (y)\qquad x\ge 0.
$$
We can drop the index 1 of $H_1$, since only its law now matters.
We can rewrite the last result as
$$
\PP(H\in dx)=\PP(H^\theta\in dx)(1-\int_0^x \PP(H\in dz) f(z))+\PP(H\in dx)\PP(H^\theta>x) f(x),
$$
which can be integrated as
$$
\PP(H > x)=\PP(H^\theta >x)(1-\int_0^x \PP(H\in dz) f(z)).
$$
Defining now the function $G$ as
$$
G(x):=\PP(H>x)(W_\theta(x)-f(x)),
$$
we get, thanks to the last integration,
$$
G(x)=1-\int_0^x \PP(H\in dz) f(z)-\PP(H>x)f(x).
$$
Integrating by parts yields
$$
G(x)=1-\int_0^x dz\,\PP(H>z) f'(z)= 1-\int_0^x dz\,\PP(H>z)(-\theta f(z) +\theta W_\theta (z))=1-\theta \int_0^x dz\, G(z),
$$
which shows that $G(x)=e^{-\theta x}$. This reads
$$
W(x)=e^{\theta x} W_\theta(x)-\theta \int_0^x\, dy \,e^{\theta y}\,W_\theta (y).
$$
One differentiation and one integration provide the result.\hfill $\Box$
\subsection{Main result}
\subsubsection{Statement}
Recall that $A_n(k)$ denotes the number of haplotypes carried by $k$ individuals in a sample of $n$.
\begin{thm}
\label{thm : AFS large}
For all $k\ge 1$, the following convergence holds a.s.
$$
\lim_{n\tendinfty} n^{-1}A_n(k) = \intgen dx \,\theta\,e^{-\theta x} \frac{1}{W_\theta(x)^2}\left(1-\frac{1}{W_\theta(x)}\right)^{k-1}.
$$
In addition,
$$
\lim_{n\tendinfty} n^{-1}A_n = \intgen dx \,\theta\,e^{-\theta x} \frac{1}{W_\theta(x)}=\EE\left(1-e^{-\theta H^\theta}\right).
$$
\end{thm}
Before proving this statement, we insert a (sub)subsection in which we state and prove a preliminary key result.
\subsubsection{The key lemma}
Recall that $\ell_{1i}$ denotes the (time elapsed since the) $i$-th (most recent) mutation on the first branch length. In particular, the mutations carried by individual 1 and not by individual 0 are exactly those $\ell_{1i}$ such that $\ell_{1i}<H_1$ (the other points of the process are thrown away).
Let $N_i$ denote the number of individuals whose \emph{most recent mutation} is $\ell_{1i}$.
\begin{lem}
\label{lem : key}
In an infinite sample, for any integer $k\ge 1$,
$$
\sum_{i\ge 1} \PP(N_i=k)= \intgen \theta \,e^{-\theta z} \, dz \,\frac{1}{W_\theta(z)^2}\left(1-\frac{1}{W_\theta(z)}\right)^{k-1}
$$
\end{lem}
\paragraph{Proof.}
In the first place, not to care for the fact that only mutations with $\ell_{1i}<H_1$ contribute, we consider the number $N_i'$ of individuals whose most recent mutation is $\ell_{0i}$, and we condition on $\ell_{0j}=v_j$, $j\ge 1$.
We will use later the fact that the law of $N_i$ conditional on $\ell_{1j}=v_j$, $j\ge 1$, is that of $N_i'\indicbis{v_i<H}$, where $H$ is independent of $N_i'$ and the point process $(\ell_{0i};i\ge 1)$.
Recall from the previous subsection that ${\cal E}^\theta$ is the set of individuals who carry no more mutations than individual 0, that $K_i^\theta$ is the $i$-th individual in ${\cal E}^\theta$, and $H_i^\theta:=H_{K_i^\theta}$. Then set $D_0:=0$ and
$$
D_i:= \inf\{j\ge 1 : H_j^\theta > v_{i-1}\}\qquad i\ge 1.
$$
Now observe that $N_i'=D_{i}-D_{i-1}$ for all $i\ge 1$ (for $N_1'$, the count includes individual 0). As an application of Proposition \ref{prop : next branch with no}, we get that conditional on $\ell_{0j}=v_j$, $j\ge 1$,
$$
\PP(N_1'=k)= \PP(H^\theta <v_1 )^{k-1}\PP(H^\theta >v_1),
$$
whereas for any $i\ge 2$,
$$
\PP(N_i'\not=0)= \PP(H^\theta <v_i \mid H^\theta > v_{i-1})
\quad \mbox{ and }\quad \PP(N_i'=k \mid N_i' \not=0)= \PP(H^\theta <v_i )^{k-1}\PP(H^\theta >v_i).
$$
Recalling the relation between the laws of $N_i$ and $N_i'$ mentioned in the beginning of the proof, we get that conditional on $\ell_{1j}=v_j$, $j\ge 1$,
$$
\PP(N_1=k)= \PP(H^\theta <v_1 )^{k-1}\PP(H^\theta >v_1)\PP(H >v_1).
$$
whereas for any $i\ge 2$,
$$
\PP(N_i\not=0)= \PP(H^\theta <v_i \mid H^\theta > v_{i-1})\PP(H>v_i).
$$
Now $\PP(N_i'=k \mid N_i' \not=0)=\PP(N_i=k \mid N_i \not=0)$, so we finally get (for $i\ge 2$)
\debeq
\PP(N_i=k) &=&\PP(H^\theta <v_i )^{k-1}\PP(H^\theta <v_i \mid H^\theta > v_{i-1})\PP(H^\theta >v_i)\PP(H>v_i)\\ &=&\left(1-\frac{1}{W_\theta(v_i)}\right)^{k-1}\left(1-\frac{W_\theta(v_{i-1})}{W_\theta(v_i)}\right)\frac{1}{W(v_i)W_\theta(v_i)} .
\fineq
It is well-known that for the Poisson point process of mutations,
$$
\PP(\ell_{1,i-1}\in dx, \ell_{1i} \in dz)= \frac{\theta^i x^{i-2}}{(i-2)!}\,e^{-\theta z}\, dx\, dz\qquad 0<x<z, i\ge 2,
$$
so that
\debeq
\sum_{i\ge 2}\PP(N_i=k) &=& \sum_{i\ge 2} \intgen dz \int_0^z dx\, \frac{\theta^i x^{i-2}}{(i-2)!} \, e^{-\theta z}\, \left(1-\frac{1}{W_\theta(z)}\right)^{k-1}\left(1-\frac{W_\theta(x)}{W_\theta(z)}\right)\frac{1}{W(z)W_\theta(z)}\\
&=&\intgen dz\, \theta \, e^{-\theta z} \left(1-\frac{1}{W_\theta(z)}\right)^{k-1}\frac{1}{W(z)W_\theta(z)} \int_0^z dx\, \theta\,e^{\theta x} \left(1-\frac{W_\theta(x)}{W_\theta(z)}\right).
\fineq
Now thanks to Proposition \ref{prop : next branch with no}, we can perform the following integration by parts on the last integral in the last display
\debeq
\int_0^z dx\, \theta\,e^{\theta x} \left(1-\frac{W_\theta(x)}{W_\theta(z)}\right)
&=& \left[ e^{\theta x} \left(1-\frac{W_\theta(x)}{W_\theta(z)}\right)\right]_0^z
+\frac{1}{W_\theta(z)}\int_0^z dx\, e^{\theta x} W_\theta'(x)\\
&=& -1+\frac{1}{W_\theta(z)}+ \frac{1}{W_\theta(z)}\int_0^z dx\, W'(x)\\
&=& \frac{W(z)}{W_\theta(z)}-1.
\fineq
This entails
$$
\sum_{i\ge 2}\PP(N_i=k) = \intgen dz\, \theta \, e^{-\theta z} \left(1-\frac{1}{W_\theta(z)}\right)^{k-1}\frac{1}{W(z)W_\theta(z)}\left(\frac{W(z)}{W_\theta(z)}-1\right).
$$
But since
$$
\PP(N_1=k)=\intgen dz\, \theta \, e^{-\theta z} \left(1-\frac{1}{W_\theta(z)}\right)^{k-1}\frac{1}{W(z)W_\theta(z)},
$$
the result follows.\hfill$\Box$
\subsubsection{Proof of Theorem \ref{thm : AFS large}}
For each individual $i\ge 0$, we denote by ${\cal A}_{ij}$ the set of individuals bearing the unique haplotype whose most recent mutation is $\ell_{ij}$. In particular, it is understood that ${\cal A}_{ij}=\emptyset$ whenever $\ell_{ij}>H_i$ (because no such haplotype exists).
Now fix $M\ge 1$. Similarly as in the proof of Theorem \ref{thm : SFS large}, we can define
$$
G_M(\Xi_i):= \mbox{ Card } \{j\ge 1: \mbox{ Card } {\cal A}_{ij} \cap\{i,\ldots,i+M\} \ge k\},
$$
where
$$
\Xi_i:=(({\cal P}_i, H_i),\ldots,({\cal P}_{i+M}, H_{i+M})).
$$
Observe that $G_M$ is bounded from above, so that $G_M(\Xi_i)$ is integrable for all $i\ge 0$.
Now for any $0\le r\le M$, the random variables $G_M(\Xi_i)$, for $i$ such that $i=r\; [M+1]$ (standing for mod $(M+1)$), are i.i.d. and integrable, so by the strong law of large numbers, we have the following convergence a.s. (and in $L^1$)
$$
\lim_{n\tendinfty} n^{-1}\sum_{0\le i= r [M+1]\le n-M}G_M(\Xi_i)=\frac{1}{M+1}\;\EE(G_M(\Xi_1)).
$$
Summing over $r$ these $M+1$ equalities, we get the following convergence a.s. (and in $L^1$)
$$
\lim_{n\tendinfty} n^{-1}\sum_{i=0}^{n-M}G_M(\Xi_i)=\EE(G_M(\Xi_1)).
$$
Our goal is now to let $M\tendinfty$. First define
$$
A_n'(k):=\sum_{i=0}^n\mbox{ Card } \{j\ge 1: \mbox{ Card } {\cal A}_{ij} \cap\{i,\ldots,n\} \ge k\}.
$$
Notice that
$$
A_n'(k) = \sum_{h\ge k} A_n(h).
$$
Then for any $i=0,\ldots, n-M$, for any $j\ge 1$, if $\mbox{ Card } {\cal A}_{ij} \cap\{i,\ldots,i+M\} \ge k$, then $\mbox{ Card } {\cal A}_{ij} \cap\{i,\ldots,n\} \ge k$, so that $A_n'(k)\ge \sum_{i=1}^{n-M}G_M(\Xi_i)$, and
$$
\liminf_{n\tendinfty} n^{-1} A_n'(k)\ge \liminf_{n\tendinfty} n^{-1}\sum_{i=1}^{n-M}G_M(\Xi_i)= \EE(G_M(\Xi_1)).
$$
Letting $M\tendinfty$, Beppo Levi's theorem yields
$$
\liminf_{n\tendinfty} n^{-1} A_n'(k) \ge \EE\;\big[ \mbox{Card } \{j\ge 1: \mbox{ Card } {\cal A}_{1j} \ge k\}\big]= \sum_{j\ge 1} \PP(\mbox{Card } {\cal A}_{1j} \ge k)=:y_k.
$$
In the notation of the previous subsection $\mbox{Card } {\cal A}_{1j}=N_j$, so by Fubini--Tonelli's theorem,
$$
y_k=\sum_{j\ge 1}\PP(N_j\ge k)=\sum_{j\ge 1}\sum_{h\ge k}\PP(N_j=h)=\sum_{h\ge k} x_h,
$$
where $x_k:= \sum_{j\ge 1}\PP(N_j=k)$. Thanks to Lemma \ref{lem : key} we have the following explicit expression for $x_k$
$$
x_k=\intgen \theta \,e^{-\theta z} \, dz \,\frac{1}{W_\theta(z)^2}\left(1-\frac{1}{W_\theta(z)}\right)^{k-1}.
$$
Now recall that $\sum_{k\ge 1}A_n'(k)=\sum_{h\ge 1}hA_n(h)=n$. Since it is easily seen that $\sum_{k\ge 1}y_k=\sum_{h\ge 1}hx_h=1$, by Fatou's lemma
$$
1=\sum_{k\ge 1}y_k \le \sum_{k\ge 1}\liminf_n n^{-1}A_n'(k)\le \liminf_n n^{-1}\sum_{k\ge 1} A_n'(k)=1.
$$
Then we would get a contradiction if there was $k_0$ such that $\liminf_n n^{-1}A_n'(k_0)>y_{k_0}$, so that for all $k\ge 1$ a.s.,
$$
\lim_{n\tendinfty} n^{-1}A_n'(k)=y_{k}.
$$
The first equation of the theorem stems from the fact that $A_n(k) = A_n'(k) -A_n'(k+1)$ and the second one by taking $k=1$ in the last display.
It takes an elementary integration by parts to check that
$$
y_1 = \intgen dx \,\theta\,e^{-\theta x} \frac{1}{W_\theta(x)}=
\EE\left(1-e^{-\theta H^\theta}\right).
$$
\paragraph{Acknowledgments.} This work was partially funded by the project MAEV `Modèles Aléatoires de l'\'Evolution du Vivant' of ANR (French national research agency).
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TITLE: Resultants - if $a,b$ have degrees $m,n$, why are there polynomials $u,v$ of degrees $m-1,n-1$ so that $Res(a,b)=av-bu$?
QUESTION [2 upvotes]: Let $R$ be a commutative ring. Suppose I have polynomials $a\left(x\right),b\left(x\right) \in R\left[x\right]$ of degrees $m,n$, respectively. My reading states that
there then exist polynomials $u\left(x\right),v\left(x\right) \in R\left[x\right]$ of degrees $m-1,n-1$, respectively, such that $a(x)v(x)-b(x)u(x) = \operatorname{Res}_{m,n}(a(x), b(x))$.
My question is why must this be true and how can it be proven? I have tried small examples by hand and it seems to be true, but I have no idea how to approach a proof. I think there might be a way via the division algorithm, but that isn't obvious to me.
REPLY [2 votes]: You cannot demand $u\left( x\right) $ and $v\left( x\right) $ to have
degrees exactly $m-1$ and $n-1$. Indeed, if $R=\mathbb{Q}$ and $m=2$ and $n=2$
and $a\left( x\right) =x^{2}$ and $b\left( x\right) =x^{2}-1$, then the
only pair $\left( u\left( x\right) ,\ v\left( x\right) \right) $ of two
polynomials $u\left( x\right) $ and $v\left( x\right) $ of degree $\leq1$
satisfying $a\left( x\right) v\left( x\right) -b\left( x\right) u\left(
x\right) =\operatorname{Res}_{2,2}\left( a\left( x\right)
,\ b\left( x\right) \right) =1$ is $\left( 1,\ 1\right) $, so the degrees
are $0$ here.
So the appropriate thing to ask for is to have two polynomials $u\left(
x\right) $ and $v\left( x\right) $ of degrees $\leq m-1$ and $\leq n-1$,
respectively, that satisfy
\begin{align}
a\left( x\right) v\left( x\right) -b\left( x\right) u\left( x\right)
=\operatorname{Res}_{m,n}\left( a\left( x\right) ,\ b\left(
x\right) \right) .
\end{align}
To prove that two such polynomials exist (in full generality, assuming that $m+n > 0$), we recall the
linear-algebraic meaning of the resultant first.
For each nonnegative integer $k$, we let $R\left[ x\right] _{\deg<k}$ denote
the $R$-submodule of $R\left[ x\right] $ spanned by $x^{0},x^{1}
,\ldots,x^{k-1}$. This is a free $R$-module of rank $k$; it consists of those
polynomials that have degree $<k$. Now, consider the map
\begin{align*}
\Phi:R\left[ x\right] _{\deg<n}\times R\left[ x\right] _{\deg<m} &
\rightarrow R\left[ x\right] _{\deg<n+m},\\
\left( r\left( x\right) ,\ s\left( x\right) \right) & \mapsto a\left(
x\right) r\left( x\right) +b\left( x\right) s\left( x\right) .
\end{align*}
This is an $R$-linear map between two free $R$-modules of the same rank.
Moreover, with respect to the "monomial" bases of both $R$-modules $R\left[
x\right] _{\deg<n}\times R\left[ x\right] _{\deg<m}$ and $R\left[
x\right] _{\deg<n+m}$ (that is, the bases formed by the monomials), this
linear map $\Phi$ is represented precisely by the Sylvester matrix
\begin{align}
S:=\left(
\begin{array}
[c]{c}
\begin{array}
[c]{ccccccccc}
a_{0} & 0 & 0 & \cdots & 0 & b_{0} & 0 & \cdots & 0\\
a_{1} & a_{0} & 0 & \cdots & 0 & b_{1} & b_{0} & \cdots & 0\\
\vdots & a_{1} & a_{0} & \cdots & 0 & \vdots & b_{1} & \ddots & \vdots\\
\vdots & \vdots & a_{1} & \ddots & \vdots & \vdots & \vdots & \ddots & b_{0}\\
a_{m} & \vdots & \vdots & \ddots & a_{0} & \vdots & \vdots & \ddots & b_{1}\\
0 & a_{m} & \vdots & \ddots & a_{1} & b_{n} & \vdots & \ddots & \vdots\\
\vdots & \vdots & \ddots & \ddots & \vdots & 0 & b_{n} & \ddots & \vdots\\
0 & 0 & 0 & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 & \cdots & a_{m} & 0 & 0 & \cdots & b_{n}
\end{array}
\\
\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }
_{n\text{ columns}}
\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }
_{m\text{ columns}}
\end{array}
\right)
\end{align}
(where $a_{0},a_{1},\ldots,a_{m}$ are the coefficients of $a\left( x\right)
$, and where $b_{0},b_{1},\ldots,b_{n}$ are the coefficients of $b\left(
x\right) $). The determinant $\det S$ of this matrix $S$ is precisely the
resultant $\operatorname{Res}_{m,n}\left( a\left( x\right)
,\ b\left( x\right) \right) $.
Now, we need the following fact:
Lemma 1. Let $M$ and $N$ be two free $R$-modules of the same rank, which
is finite. Let $f:M\rightarrow N$ be an $R$-linear map. Let $A$ be a matrix
that represents this map $f$ with respect to some pair of bases of $M$ and of
$N$. Then, $\left( \det A\right) \cdot v\in f\left( M\right) $ for each
$v\in N$.
Proof of Lemma 1. Let $k$ be the rank of the free modules $M$ and $N$. We
WLOG assume that both $M$ and $N$ are the $R$-module $R^{k}$ of column vectors
of size $k$, and that the $R$-linear map $f$ is actually the left
multiplication by the matrix $A$ -- that is, we have $f\left( w\right) =Aw$
for each vector $w\in R^{k}$. (This is an assumption we can make, because we
can always replace $M$ and $N$ by $R^{k}$ via the two bases we have chosen,
and then the linear map $f$ becomes the left multiplication by the matrix that
represents it with respect to these bases; but this matrix is precisely $A$.)
Let $\operatorname*{adj}A$ be the adjugate of the matrix $A$. It is well-known
that $A\cdot\operatorname*{adj}A=\left( \det A\right) \cdot I_{k}$ (where
$I_{k}$ denotes the $k\times k$ identity matrix). Now, let $v\in N$. Then,
$v\in N=R^{k}$ and $\left( \operatorname*{adj}A\right) \cdot v\in R^{k}=M$.
Furthermore,
\begin{align*}
f\left( \left( \operatorname*{adj}A\right) \cdot v\right) &
=\underbrace{A\cdot\left( \operatorname*{adj}A\right) }_{=\left( \det
A\right) \cdot I_{k}}\cdot v\ \ \ \ \ \ \ \ \ \ \left( \text{since }f\left(
w\right) =Aw\text{ for each }w\in R^{k}\right) \\
& =\left( \det A\right) \cdot I_{k}v=\left( \det A\right) \cdot v,
\end{align*}
so that
\begin{align}
\left( \det A\right) \cdot v=f\left( \left( \operatorname*{adj}A\right)
\cdot v\right) \in f\left( M\right) .
\end{align}
This proves Lemma 1. $\blacksquare$
We can now easily finish our proof: Applying Lemma 1 to $M=R\left[ x\right]
_{\deg<n}\times R\left[ x\right] _{\deg<m}$ and $N=R\left[ x\right]
_{\deg<n+m}$ and $f=\Phi$ and $A=S$ and $v=1$, we conclude that
\begin{align}
\left( \det S\right) \cdot1\in\Phi\left( R\left[ x\right] _{\deg<n}\times
R\left[ x\right] _{\deg<m}\right)
\end{align}
(since $R\left[ x\right] _{\deg<n}\times R\left[ x\right] _{\deg<m}$ and
$R\left[ x\right] _{\deg<n+m}$ are two free $R$-modules of the same rank,
which is finite). Since $\left( \det S\right) \cdot1=\det
S=\operatorname{Res}_{m,n}\left( a\left( x\right) ,\ b\left(
x\right) \right) $, we can rewrite this as
\begin{align}
\operatorname{Res}_{m,n}\left( a\left( x\right) ,\ b\left(
x\right) \right) =\Phi\left( R\left[ x\right] _{\deg<n}\times R\left[
x\right] _{\deg<m}\right) .
\end{align}
In other words, there exists a pair $\left( r\left( x\right) ,\ s\left(
x\right) \right) \in R\left[ x\right] _{\deg<n}\times R\left[ x\right]
_{\deg<m}$ such that $\operatorname{Res}_{m,n}\left( a\left(
x\right) ,\ b\left( x\right) \right) =\Phi\left( r\left( x\right)
,\ s\left( x\right) \right) $. Consider this pair. Then,
\begin{align*}
\operatorname{Res}_{m,n}\left( a\left( x\right) ,\ b\left(
x\right) \right) & =\Phi\left( r\left( x\right) ,\ s\left( x\right)
\right) \\
& =a\left( x\right) r\left( x\right) +b\left( x\right) s\left(
x\right) \ \ \ \ \ \ \ \ \ \ \left( \text{by the definition of }\Phi\right)
\\
& =a\left( x\right) r\left( x\right) -b\left( x\right) \left( -s\left(
x\right) \right) .
\end{align*}
Moreover, from $\left( r\left( x\right) ,\ s\left( x\right) \right) \in
R\left[ x\right] _{\deg<n}\times R\left[ x\right] _{\deg<m}$, we obtain
$r\left( x\right) \in R\left[ x\right] _{\deg<n}$, so that $\deg\left(
r\left( x\right) \right) <n$ and thus $\deg\left( r\left( x\right)
\right) \leq n-1$. Similarly, $\deg\left( s\left( x\right) \right) \leq
m-1$, so that $\deg\left( -s\left( x\right) \right) \leq m-1$ as well.
Thus, we have found two polynomials $u\left( x\right) $ and $v\left(
x\right) $ of degrees $\leq m-1$ and $\leq n-1$, respectively, that satisfy
\begin{align}
a\left( x\right) v\left( x\right) -b\left( x\right) u\left( x\right)
=\operatorname{Res}_{m,n}\left( a\left( x\right) ,\ b\left(
x\right) \right)
\end{align}
(namely, $u\left( x\right) =-s\left( x\right) $ and $v\left( x\right)
=r\left( x\right) $).
| 137,654
|
\begin{document}
\title[Classification problems in $2$-representation theory]
{Classification problems in\\ $2$-representation theory}
\author{Volodymyr Mazorchuk}
\begin{abstract}
This article surveys recent advances and future challenges
in the $2$-representation theory of finitary $2$-categories with
a particular emphasis on problems related to classification of
various classes of $2$-representations.
\end{abstract}
\maketitle
\section{Introduction}\label{s1}
Higher representation theory emerged as an offspring of categorification.
The latter term traditionally describes the approach, originated in
\cite{CF,Cr}, of upgrading set-theoretical notions to category theoretical with
a hope to create more structure. The major breakthrough of categorification
was invention of Khovanov homology in \cite{Kh}. After that several other
spectacular applications followed, for example, to Brou{\'e}'s abelian defect
group conjecture in \cite{CR}, to study of modules over Lie superalgebras
in \cite{BS,BLW} and to various other problems. The common feature of all these
and many other applications is that construction and comparison of functorial
actions on different categories was the key part of the argument.
Functorial actions form the red thread of \cite{BFK} leading to an alternative
reformulation of Khovanov homology in \cite{St} using BGG category $\mathcal{O}$.
In \cite{CR} and, later on, in \cite{Ro}, functorial actions were abstractly
reformulated in terms of representation theory of certain $2$-categories.
This direction of study was subsequently called {\em higher representation
theory} or, alternatively, just {\em $2$-representation theory} to emphasize that, so far,
it directs only at this second level of general higher categories. However, one has to
note that these and many further papers like \cite{KL,El} and other mainly study
special examples of $2$-categories which originate in topologically motivated
diagrammatic calculus.
The series \cite{MM1,MM2,MM3,MM4,MM5,MM6} of papers took higher representation theory
to a more abstract level. These papers started a systematic study of abstract
$2$-analogues of finite dimensional algebras, called {\em finitary $2$-categories},
and their representation theory. Despite of the fact that the $2$-category of
$2$-representations of an abstract finitary $2$-category is much more complicated
than the category of modules over a finite dimensional algebra (in particular,
it is not abelian), it turned out
that, in many cases, it is possible to {\em construct}, {\em compare} and even
{\em classify} various classes of $2$-representations. These kinds of problems
were recently studied in a number of papers, see \cite{MM5,MM6,Zh2,MZ1,MaMa,KMMZ,MT,MMMT,MMZ,MZ2}.
The aim of the present article is to give an overview of these results with a particular
emphasis on open problems and future challenges.
We start in Section~\ref{s2} with a brief description of main objects of study.
Section~\ref{s3} lists a number of classical examples. Sections~\ref{s4},
\ref{s5} and \ref{s6} then concentrate on special classes of $2$-representations.
In particular, Sections~\ref{s4} addresses {\em cell $2$-representations}
and Sections~\ref{s5} is devoted to {\em simple transitive $2$-representations}.
We do not provide any proofs but rather give explicit references to original
sources.
The present paper might serve as a complement reading to the series of lectures on
higher representation theory which the author gave during
{\em Brazilian Algebra Meeting} in Diamantina, Brazil, in August 2016.
\vspace{5mm}
{\bf Acknowledgments.} The author is partially supported by the
Swedish Research Council and G{\"o}ran Gustafsson Stiftelse.
The authors thanks the organizers of the Brazilian Algebra Meeting
for invitation to give the series of lecture on higher representation theory.
\section{Finitary $2$-categories and their $2$-representations}\label{s2}
\subsection{Finitary $2$-categories}\label{s2.1}
A {\em $2$-category} is a category which is enriched over the monoidal category
$\mathbf{Cat}$ of small categories (the monoidal structure of the latter category is given by
Cartesian product). This means that a $2$-category $\cC$ consists of
\begin{itemize}
\item objects, denoted $\mathtt{i}$, $\mathtt{j}$, $\mathtt{k}$,\dots;
\item small morphism categories $\cC(\mathtt{i},\mathtt{j})$, for all $\mathtt{i},\mathtt{j}\in\cC$;
\item bifunctorial compositions $\cC(\mathtt{j},\mathtt{k})\times \cC(\mathtt{i},\mathtt{j})
\to \cC(\mathtt{i},\mathtt{k})$, for all $\mathtt{i},\mathtt{j},\mathtt{k}\in\cC$;
\item and identity objects $\mathbbm{1}_{\mathtt{i}}\in \cC(\mathtt{i},\mathtt{i})$,
for all $\mathtt{i}\in\cC$;
\end{itemize}
which satisfy the obvious collection of strict axioms. The following terminology is standard:
\begin{itemize}
\item objects in $\cC(\mathtt{i},\mathtt{j})$ are called {\em $1$-morphisms} of $\cC$
and will be denoted by $\mathrm{F}$, $\mathrm{G}$, etc.;
\item morphisms in $\cC(\mathtt{i},\mathtt{j})$ are called {\em $2$-morphisms} of $\cC$
and will be denoted by $\alpha$, $\beta$, etc.;
\item composition of $2$-morphisms inside $\cC(\mathtt{i},\mathtt{j})$ is called {\em vertical}
and will be denoted by $\circ_v$;
\item composition of $2$-morphisms coming from
$\cC(\mathtt{j},\mathtt{k})\times \cC(\mathtt{i},\mathtt{j})
\to \cC(\mathtt{i},\mathtt{k})$ is called {\em horizontal}
and will be denoted by $\circ_h$.
\end{itemize}
As usual, for a $1$-morphism $\mathrm{F}$, the identity $2$-morphism for $\mathrm{F}$
is denoted $\mathrm{id}_{\mathrm{F}}$, moreover, for a $2$-morphism $\alpha$,
the compositions $\mathrm{id}_{\mathrm{F}}\circ_h\alpha$ and $\mathrm{id}_{\mathrm{F}}\circ_h\alpha$
are denoted by $\mathrm{F}(\alpha)$ and $\alpha_{\mathrm{F}}$, respectively.
We refer the reader to \cite{Le,Mc} for more details on $2$-categories and for various generalizations,
in particular, for the corresponding {\em non-strict} notion of a {\em bicategory}.
Let $\Bbbk$ be an algebraically closed field. Recall that a category $\mathcal{C}$ is called
{\em finitary $\Bbbk$-linear} provided that it is equivalent to the category of projective modules
over a finite dimensional (associative) $\Bbbk$-algebra. Each such category is $\Bbbk$-linear,
that is enriched over the category of $\Bbbk$-vector spaces, moreover, it is idempotent split and
Krull-Schmidt and has finitely many isomorphism classes of indecomposable
objects and finite dimensional (over $\Bbbk$) spaces of morphisms.
Following \cite[Subsection~2.2]{MM1}, we will say that a {\em $2$-category} $\cC$ is
{\em finitary over $\Bbbk$} provided that
\begin{itemize}
\item each $\cC(\mathtt{i},\mathtt{j})$ is finitary $\Bbbk$-linear;
\item all compositions are biadditive and $\Bbbk$-bilinear, whenever appropriate;
\item all $\mathbbm{1}_{\mathtt{i}}$ are indecomposable.
\end{itemize}
The last condition is a technical condition which makes the life easier at many occasions. From the representation
theoretic prospective, this condition is not restrictive as, starting from a $2$-category satisfying all
other conditions, one can use idempotent splitting to produce a finitary $2$-category with essentially
the same representation theory.
In what follows, we will simply say that $\cC$ is {\em finitary} as our field $\Bbbk$ will be
fixed throughout the paper (with the exception of examples related to Soergel bimodules where
$\Bbbk=\mathbb{C}$).
\subsection{$2$-representations}\label{s2.2}
For two $2$-categories $\cA$ and $\cC$, a {\em $2$-functor} $\Phi:\cA\to \cC$ is a functor which
respects all $2$-categorical structure. This means that $\Phi$
\begin{itemize}
\item maps $1$-morphisms to $1$-morphisms;
\item maps $2$-morphisms to $2$-morphisms;
\item is compatible with composition of $1$-morphisms;
\item is compatible with both horizontal and vertical composition of $2$-morphisms;
\item sends identity $1$-morphisms to identity $1$-morphisms;
\item sends identity $2$-morphisms to identity $2$-morphisms.
\end{itemize}
A {\em $2$-representation} of a $2$-category $\cC$ is a $2$-functor to some fixed $2$-category.
Classical examples of such target $2$-categories are:
\begin{itemize}
\item the $2$-category $\mathbf{Cat}$ of small categories, here $1$-morphisms are functors and $2$-morphisms
are natural transformations;
\item the $2$-category $\mathfrak{A}^{f}_{\Bbbk}$ of {\em finitary $\Bbbk$-linear} categories, here
$1$-morphisms are additive $\Bbbk$-linear functors and $2$-morphisms are natural transformations;
\item the $2$-category $\mathfrak{R}_{\Bbbk}$ of {\em finitary abelian $\Bbbk$-linear} categories, here
objects are categories equivalent to module categories of finite dimensional (associative) $\Bbbk$-algebras,
$1$-morphisms are right exact additive $\Bbbk$-linear functors and $2$-morphisms are natural transformations.
\end{itemize}
All $2$-representations of a $2$-category $\cC$ (in a fixed target $2$-category) form a $2$-category.
In this $2$-category we have:
\begin{itemize}
\item $1$-morphisms are (strong) $2$-natural transformations;
\item $2$-morphisms are modifications.
\end{itemize}
One has to make a choice for the level of strictness for $1$-morphisms in the
$2$-category of $2$-representations of $\cC$. In the language of \cite[Subsection~1.2]{Le}, this corresponds
to choosing between the so-called strong or strict transformations. Strict transformations were considered
in the paper \cite{MM1} and the setup was changed to strong transformations in \cite{MM3}. The latter
allows for more flexibility and more reasonable results (for example, the relation of equivalence of
two $2$-representations becomes symmetric).
For a finitary $2$-category $\cC$, its $2$-representations in $\mathfrak{A}^{f}_{\Bbbk}$ are called
{\em finitary additive} $2$-representations and the corresponding $2$-category of $2$-representations
is denoted by $\cC\text{-}\mathrm{afmod}$. Further, $2$-representations of $\cC$ in $\mathfrak{R}_{\Bbbk}$
are called {\em abelian} $2$-representations and the corresponding $2$-category of $2$-representations
is denoted by $\cC\text{-}\mathrm{mod}$. Note that neither $\cC\text{-}\mathrm{afmod}$ nor
$\cC\text{-}\mathrm{mod}$ are abelian categories.
We will usually denote $2$-representations of $\cC$ by $\mathbf{M}$, $\mathbf{N}$ etc. For the
sake of readability, it is often convenient to use the actions notation $\mathrm{F}\, X$ instead
of the representation notation $\mathbf{M}(\mathrm{F})\big( X\big)$.
Here is an example of a $2$-representation: for $\mathtt{i}\in\cC$, the {\em principal}
$2$-representation $\mathbf{P}_{\mathtt{i}}$ is defined to be the Yoneda $2$-representation
$\cC(\mathtt{i},{}_-)$. If $\cC$ is finitary, we have $\mathbf{P}_{\mathtt{i}}\in \cC\text{-}\mathrm{afmod}$.
Two $2$-representations $\mathbf{M}$ and $\mathbf{N}$ of $\cC$ are called {\em equivalent} provided
that there is a $2$-natural transformation $\Phi:\mathbf{M}\to\mathbf{N}$ such that
$\Phi_{\mathtt{i}}$ is an equivalence of categories, for each $\mathtt{i}\in\cC$.
\subsection{Abelianization}\label{s2.3}
Given a finitary $\Bbbk$-linear category $\mathcal{C}$, the {\em diagrammatic abe\-li\-a\-ni\-za\-tion} of
$\mathcal{C}$ is the category $\overline{\mathcal{C}}$ of diagrams of the form
$X\overset{\alpha}{\longrightarrow} Y$ over $\mathcal{C}$ with morphisms being the obvious commutative
squares modulo the projective homotopy relations. The category $\overline{\mathcal{C}}$ is abelian
and is equivalent to the category of modules over a finite dimensional $\Bbbk$-algebra.
The original category $\mathcal{C}$ canonically embeds into $\overline{\mathcal{C}}$ via diagrams
of the form $0\to Y$ and this embedding provides an equivalence between $\mathcal{C}$ and the
category of projective objects in $\overline{\mathcal{C}}$. We refer the reader to \cite{Fr} for details.
For a finitary $2$-category $\cC$, using diagrammatic abelianization and component-wise action on
diagrams defines a $2$-functor
\begin{displaymath}
\overline{\hspace{1mm}\cdot\hspace{1mm}}: \cC\text{-}\mathrm{afmod}\to \cC\text{-}\mathrm{mod},
\end{displaymath}
called {\em abelianization}, see \cite[Subsection~3.1]{MM1}. In \cite[Section~3]{MMMT} one finds
a more advanced refinement of this construction which is way more technical but also
has some extra nice properties.
\subsection{Fiat $2$-categories}\label{s2.4}
As we will see, many examples of finitary $2$-categories have additional structure which plays
very important role and significantly simplifies many arguments. This additional structure, on a finitary
$2$-category $\cC$, consists of
\begin{itemize}
\item a {\em weak involution} $\star$ which inverts the direction of both $1$- and $2$-morphisms,
\item {\em adjunction morphisms}
$\varepsilon^{(\mathrm{F})}:\mathrm{F}\circ \mathrm{F}^{\star}\to\mathbbm{1}_{\mathtt{j}}$
and $\eta^{(\mathrm{F})}:\mathbbm{1}_{\mathtt{i}}\to \mathrm{F}^{\star}\circ \mathrm{F}$,
for each $\mathrm{F}\in\cC(\mathtt{i},\mathtt{j})$, which make $(\mathrm{F},\mathrm{F}^{\star})$
a pair of adjoint $1$-morphisms in the sense that
\begin{displaymath}
\mathrm{id}_{\mathrm{F}}=\varepsilon^{(\mathrm{F})}_{\mathrm{F}}\circ_v\mathrm{F}(\eta^{(\mathrm{F})})
\quad\text{ and }\quad
\mathrm{id}_{\mathrm{F}}^{\star}=\mathrm{F}^{\star}(\varepsilon^{(\mathrm{F})})\circ_v
\eta^{(\mathrm{F})}_{\mathrm{F}^{\star}}.
\end{displaymath}
\end{itemize}
A $2$-category $\cC$ having such an additional structure is called {\em fiat}, where
``f'' stands for {\em finitary}, ``i'' stands for {\em involution}, ``a'' stands for
{\em adjunction} and ``t'' stands for {\em $2$-category}, see \cite[Subsection~2.4]{MM1}.
If a similar structure exists for a not necessarily involutive anti-autoequivalence $\star$,
the $2$-category $\cC$ is called {\em weakly fiat}, see \cite[Subsection~7.3]{MM2} and \cite[Appendix]{MM6}.
In many situations, involutions in $2$-categories change the direction of $1$-morphisms but preserve the
direction of $2$-morphisms, see e.g. \cite[Page~3]{Le}. The above definition, in which both the
directions of $1$- and $2$-morphisms get reversed, is motivated by the
$2$-category of endofunctors of $A$-mod, for a finite dimensional $\Bbbk$-algebra $A$. For each
pair $(\mathrm{F},\mathrm{G})$ of adjoint endofunctors of $A$-mod, there is an $A$-$A$-bimodule $Q$ such that
$\mathrm{F}$ is isomorphic to $Q\otimes_A{}_-$ and $\mathrm{G}$ is isomorphic to
$\mathrm{Hom}_{A-}(Q,{}_-)$, see \cite[Chapter~I]{Ba}. Natural transformations between functors
correspond to homomorphisms between bimodules. When taking the adjoint functor, the bimodule $Q$
ends up on the contravariant place of the bifunctor $\mathrm{Hom}$ and hence the direction of natural
transformations gets reversed.
In the literature one could find similar structures under the name of {\em rigid}
tensor categories categories, see e.g. \cite{EGNO}.
\subsection{Grothendieck decategorification}\label{s2.5}
For an finitary $\Bbbk$-linear category $\mathcal{C}$, let $[\mathcal{C}]_{\oplus}$ denote the
{\em split Grothendieck group} of $\mathcal{C}$. Then $[\mathcal{C}]_{\oplus}$ is a free abelian group
which has a canonical generating set given by isomorphism classes of indecomposable objects in $\mathcal{C}$.
For a category $\mathcal{C}$ equivalent to $A$-mod, for some finite dimensional $\Bbbk$-algebra $A$,
let $[\mathcal{C}]$ denote the {\em Grothendieck group} of $\mathcal{C}$. Then $[\mathcal{C}]$ is a free
abelian group which has a canonical generating set given by isomorphism classes of simple objects in $\mathcal{C}$.
Let $\cC$ be a finitary $2$-category and $\mathbf{M}\in \cC$-afmod. Let $[\cC]_{\oplus}$ denote the
ordinary category which has the same objects as $\cC$ and in which morphisms are given by
$[\cC]_{\oplus}(\mathtt{i},\mathtt{j}):=[\cC(\mathtt{i},\mathtt{j})]_{\oplus}$ with induced composition.
The category $[\cC]_{\oplus}$ is called the {\em Grothendieck decategorification} of $\cC$. The category
$[\cC]_{\oplus}$ acts on abelian groups $[\mathbf{M}(\mathtt{i})]_{\oplus}$, where $\mathtt{i}\in \cC$,
which in this way defines the {\em Grothendieck decategorification} $[\mathbf{M}]_{\oplus}$ of $\mathbf{M}$.
If $\cC$ is fiat and $\mathbf{M}\in \cC$-mod, then $[\cC]_{\oplus}$ acts on abelian groups
$[\mathbf{M}(\mathtt{i})]$, where $\mathtt{i}\in \cC$,
which in this way defines the {\em Grothendieck decategorification} $[\mathbf{M}]$ of $\mathbf{M}$.
We note that there are alternative decategorifications, notably, the {\em trace decategorification}
introduced in \cite{BGHL}.
\section{Examples of finitary $2$-categories}\label{s3}
\subsection{Set-theoretic issues}\label{s3.1}
There are some set-theoretic complications due to the fact that, by definition,
each $\cC(\mathtt{i},\mathtt{j})$ of a $2$-category $\cC$ has to be small. This
prevents us to consider, for example, the category of all $A$-$A$--bimodules, for
a finite dimensional $\Bbbk$-algebra $A$, as a $2$-category (with one formal object
that can be identified with $A$-mod). The reason for that is the observation
that the category of all bimodules is not small.
In what follows we will give many examples by considering all endofunctors (of some type)
of some category $\mathcal{C}$. To avoid the above problem, we will always assume that
$\mathcal{C}$ is small. This, however, creates a choice. For example, one has to choose
a small category $\mathcal{C}$ equivalent to $A$-mod. Different choices of $\mathcal{C}$
lead to different, however, usually {\em biequivalent}, $2$-categories.
\subsection{Projective endofunctors of $A$-mod}\label{s3.2}
Let $A$ be a finite dimensional, basic, connected $\Bbbk$-algebra. Fix a small category
$\mathcal{C}$ equivalent to $A$-mod. Recall that a {\em projective} $A$-$A$-bimodule
is an $A$-$A$-bimodule from the additive closure $\mathrm{add}(A\otimes_{\Bbbk}A)$
of ${}_AA\otimes_{\Bbbk}A_A$. We also have the {\em regular} or {\em identity} $A$-$A$-bimodule
${}_AA_A$.
Denote by $\cC_A=\cC_{A,\mathcal{C}}$ the $2$-category which has
\begin{itemize}
\item one object $\mathtt{i}$ (which should be though of as $\mathcal{C}$);
\item as $1$-morphisms, endofunctors of $\mathcal{C}$ from the additive closure of
endofunctors given by tensoring with projective or regular $A$-$A$-bimodules;
\item as $2$-morphisms, all natural transformations of functors.
\end{itemize}
This $2$-category appears in \cite[Subsection~7.3]{MM1}.
If $A$ is simple, then $A\cong\Bbbk$ and $\cC_A$ has a unique (up to isomorphism)
indecomposable $1$-morphism, namely $\mathbb{1}_{\mathtt{i}}$.
If $A$ is not simple, let $e_1+e_2+\dots+e_n=1$ be a primitive decomposition of the
identity $1\in A$. Then, apart from $\mathbb{1}_{\mathtt{i}}$, the $2$-category $\cC_A$ has
$n^2$ additional indecomposable $1$-morphisms $\mathrm{F}_{ij}$, where,
for $i,j=1,2,\dots,n$, the morphism $\mathrm{F}_{ij}$ corresponds to tensoring
with $Ae_i\otimes_{\Bbbk}e_jA$. We note that
\begin{displaymath}
\mathrm{F}_{ij}\circ\mathrm{F}_{st}\cong \mathrm{F}_{it}^{\dim(e_jAe_s)}\quad\text{ and }\quad
\mathrm{F}\circ \mathrm{F}\cong\mathrm{F}^{\dim(A)},\quad\text{ for }\quad
\mathrm{F}:=\bigoplus_{i,j=1}^n\mathrm{F}_{ij}.
\end{displaymath}
The $2$-category $\cC_A$ is always finitary. It is weakly fiat if and only if $A$ is self-injective.
It is fiat if and only if $A$ is weakly symmetric. In the latter case,
$(\mathrm{F}_{ij},\mathrm{F}_{ji})$ forms an adjoint pair of $1$-morphisms, for all $i$ and $j$.
\subsection{Finitary $2$-categories of all bimodules}\label{s3.3}
For $n=1,2,\dots$, let $A_n$ denote the $\Bbbk$-algebra given as the quotient of the
path algebra of the quiver
\begin{equation}\label{eq1}
\xymatrix{1\ar[rr]&&2\ar[rr]&&3\ar[rr]&&\dots\ar[rr]&&n}
\end{equation}
by the relations that the product of any two arrows is zero. Let $\mathcal{C}$ be a small
category equivalent to $A$-mod.
Denote by $\cF_{A_n}=\cF_{A_n,\mathcal{C}}$ the $2$-category which has
\begin{itemize}
\item one object $\mathtt{i}$ (which should be though of as $\mathcal{C}$);
\item as $1$-morphisms, all right exact endofunctors of $\mathcal{C}$;
\item as $2$-morphisms, all natural transformations of functors.
\end{itemize}
The $2$-category $\cF_{A_n}$ is finitary, see \cite[Section~2]{MZ2}. The reason for that is the fact that
the enveloping algebra $A_n\otimes_{\Bbbk}A_n^{\mathrm{op}}$ of $A_n$ is a special biserial algebra
in the sense of \cite{BR,WW}, so one can use the classification of indecomposable modules for this
algebra to check that it has, in fact, only finitely many of them, up to isomorphism. Unless $n=1$,
the $2$-category $\cF_{A_n}$ is neither fiat nor weakly fiat.
Note that a similar $2$-category $\cF_{A}$ can be defined for any connected and basic $\Bbbk$-algebra $A$.
However, if $\cF_{A}$ is fiat, then $A\cong A_n$, for some $n$, see \cite[Theorem~1]{MZ2}.
\subsection{Subbimodules of the identity bimodule}\label{s3.4}
Let $\Gamma$ be a finite oriented tree and $A$ be the path algebra of $\Gamma$.
Let $\mathcal{C}$ be a small category equivalent to $A$-mod.
Denote by $\cG_{A}=\cG_{A,\mathcal{C}}$ the $2$-category which has
\begin{itemize}
\item one object $\mathtt{i}$ (which should be though of as $\mathcal{C}$);
\item as $1$-morphisms, all endofunctors of $\mathcal{C}$ from the additive closure of
subfunctors of the identity functor;
\item as $2$-morphisms, all natural transformations of functors.
\end{itemize}
The $2$-category $\cG_{A}$ is finitary due to the fact that the regular $A$-$A$-bimodule is
multiplicity free and hence has only finitely many subbimodules. Unless $\Gamma$ has one vertex,
the $2$-category $\cF_{A_n}$ is neither fiat nor weakly fiat.
The $2$-category $\cG_{A}$ first appeared, slightly disguised, in \cite{GM1}
(inspired by \cite{Gr}), for $\Gamma$ being the quiver in \eqref{eq1}.
It was further studied, for various special types of trees, in \cite{GM2} and \cite{Zh1,Zh2}.
\subsection{Soergel bimodules for finite Coxeter systems}\label{s3.5}
Let $(W,S)$ be a finite Coxeter system and $\mathfrak{h}$ be a reflection faithful complexified $W$-module.
Let, further, $\mathtt{C}$ be the corresponding coinvariant algebra, that is the quotient of
$\mathbb{C}[\mathfrak{h}]$ by the ideal generated by homogeneous $W$-invariant polynomials of positive degree.
Then $\mathtt{C}$ is a finite dimensional algebra which carries the natural structure of a
regular $W$-module, in particular, $\dim(\mathtt{C})=|W|$.
For $s\in S$, denote by $\mathtt{C}^s$ the subalgebra of $s$-invariants in $\mathtt{C}$. Then
$\mathtt{C}$ is a free $\mathtt{C}^s$-module of rank two. Both algebras $\mathtt{C}$ and
$\mathtt{C}^s$ are symmetric (recall that $A$ is symmetric provided that ${}_AA_A\cong
\mathrm{Hom}_{\Bbbk}({}_AA_A,\Bbbk)$). We refer to \cite{Hi} for details.
For $w\in W$ with reduced decomposition $w=s_1s_2\cdots s_k$, define the {\em Bott-Samelson}
$\mathtt{C}$-$\mathtt{C}$-bimodule
\begin{displaymath}
\hat{B}_w:=\mathtt{C}\otimes_{\mathtt{C}^{s_1}}\mathtt{C}\otimes_{\mathtt{C}^{s_2}}\mathtt{C}
\otimes_{\mathtt{C}^{s_3}}\dots \otimes_{\mathtt{C}^{s_k}}\mathtt{C}.
\end{displaymath}
Let $\mathcal{C}$ be a small category equivalent to $\mathtt{C}$-mod.
Denote by $\cS_{W}=\cS_{W,S,V,\mathcal{C}}$ the $2$-category which has
\begin{itemize}
\item one object $\mathtt{i}$ (which should be though of as $\mathcal{C}$);
\item as $1$-morphisms, all endofunctors of $\mathcal{C}$ coming from tensoring with bimodules
from the additive closure of all Bott-Samelson bimodules;
\item as $2$-morphisms, all natural transformations of functors.
\end{itemize}
The $2$-category $\cS_{W}$ is called the $2$-category of {\em Soergel bimodules} over $\mathtt{C}$.
The non-trivial point here is the fact that it is closed under composition of functors. This is
shown in \cite{So2}. Moreover, it is also shown in \cite{So2} that, for each $w\in W$, the
bimodule $\hat{B}_w$ contains a unique indecomposable summand $B_w$ which does not belong to the
additive closure of all $\hat{B}_x$, where the length of $x$ is strictly smaller than that of $w$.
Bimodules $B_w$ are usually called {\em Soergel bimodules} (although the name is also used
for any direct sum of such bimodules). The $2$-category $\cS_{W}$ is both
finitary and fiat, where $(B_w,B_{w^{-1}})$ forms an adjoint pair of $1$-morphisms, for all $w$.
The theory is inspired by \cite{So1} where, for finite {\em Weyl} groups, Soergel bimodules
appear as ``combinatorial description'' of indecomposable projective functors on the principal
block of the BGG category $\mathcal{O}$ associated with a triangular decomposition of the
simple finite dimensional Lie algebra corresponding to $W$, see \cite{BGG,BG,Hu} for details on the latter.
An explicit connection between the $2$-category of Soergel bimodules and the
Kazhdan-Lusztig basis of the Hecke algebra of $(W,S)$ was established in \cite{EW}.
\subsection{Singular Soergel bimodules}\label{s3.6}
Let $(W,S)$ be a finite Coxeter system, $\mathfrak{h}$ a reflection faithful complexified $W$-module and
$\mathtt{C}$ the corresponding coinvariant algebra. For each $T\subset S$, let
$W^T$ be the subgroup of $W$ generated by all $t\in T$ and $\mathtt{C}^T$ the subalgebra of
$W^T$-invariants in $\mathtt{C}$. Note that $\mathtt{C}^{\varnothing}=\mathtt{C}$ while
$\mathtt{C}^S=\mathbb{C}$.
For each $T\subset S$, let $\mathcal{C}^T$ be a small category equivalent to $\mathtt{C}^T$-mod.
Denote by $\cS\cS_{W}=\cS\cS_{W,S,V,\mathcal{C}}$ the $2$-category which has
\begin{itemize}
\item objects $\mathtt{i}^T$, where $T\subset S$ (each $\mathtt{i}^T$ should be
though of as $\mathcal{C}^T$);
\item as $1$-morphisms, all endofunctors from $\mathcal{C}^T$ to $\mathcal{C}^R$
given by
\begin{displaymath}
\mathrm{Res}^{\mathtt{C}}_{\mathtt{C}^R} \circ\mathrm{F}\circ \mathrm{Ind}^{\mathtt{C}}_{\mathtt{C}^T},
\end{displaymath}
where $\mathrm{F}$ is given by a usual Soergel bimodule;
\item as $2$-morphisms, all natural transformations of functors.
\end{itemize}
The $2$-category $\cS\cS_{W}$ is called the $2$-category of {\em singular Soergel bimodules} over $\mathtt{C}$.
The $2$-category $\cS\cS_{W}$ is both finitary and fiat.
The $2$-category $\cS\cS_{W}$ also admits an alternative description using projective
functors between {\em singular} blocks of the BGG category $\mathcal{O}$, see \cite{So1,BGG,BG}.
\subsection{Finitary quotients of $2$-Kac-Moody algebras}\label{s3.7}
Let $\mathfrak{g}$ be a simple complex finite dimensional Lie algebra and $\dot{U}_{\mathfrak{g}}$ the
idempotented version of the universal enveloping algebra of $\mathfrak{g}$, see \cite{Lu}.
The papers \cite{KL,Ro} introduce certain (not finitary) $2$-categories whose Grothendieck decategorification
is isomorphic to the integral form of $\dot{U}_{\mathfrak{g}}$. In \cite{Br} it is further shown that the
two (slightly different) constructions in \cite{KL,Ro} give, in fact, biequivalent $2$-categories.
These are the so-called {\em $2$-Kac-Moody algebras} of finite type which we will denote $\cU_{\mathfrak{g}}$.
Each simple finite dimensional $\mathfrak{g}$-module $V(\lambda)$, where $\lambda$ is the highest weight,
admits a {\em categorification} in the sense that there exists a $2$-representation $\mathbf{M}_{\lambda}$
of $\cU_{\mathfrak{g}}$ (even a unique one, up to equivalence, under the additional assumption that the
object $\lambda$ is represented by a non-zero semi-simple category) whose Grothendieck decategorification
is isomorphic to the integral form of $V(\lambda)$. Let $\cI_{\lambda}$ be the kernel of $\mathbf{M}_{\lambda}$.
Then $\cI_{\lambda}$ is a two-sided $2$-ideal of $\cU_{\mathfrak{g}}$ and the quotient $2$-category
$\cU_{\mathfrak{g}}/\cI_{\lambda}$ is both, finitary and fiat.
\section{Cells and cell $2$-representations}\label{s4}
\subsection{Cells}\label{s4.1}
For a finitary $2$-category $\cC$, denote by $\mathcal{S}[\cC]$ the set of isomorphism classes of indecomposable
$1$-objects in $\cC$. The set $\mathcal{S}[\cC]$ is finite and has the natural multivalued operation
$\bullet$ given, for $[\mathrm{F}],[\mathrm{G}]\in \mathcal{S}[\cC]$, by
\begin{displaymath}
[\mathrm{F}]\bullet[\mathrm{G}]=
\{[\mathrm{H}]\,:\,\mathrm{H}\text{ is isomorphic to a summand of } \mathrm{F}\circ \mathrm{G}\}.
\end{displaymath}
The operation $\bullet$ is associative (as a multivalued operation) and hence defines on $\mathcal{S}[\cC]$
the structure of a {\em multisemigroup}, see \cite[Section~3]{MM2} (see also \cite{KuMa} for more details on multisemigroups).
The {\em left} partial pre-order $\leq_L$ on $\mathcal{S}[\cC]$ is defined by setting
$[\mathrm{F}]\leq_L[\mathrm{G}]$, for $[\mathrm{F}],[\mathrm{G}]\in \mathcal{S}[\cC]$, provided
that $\mathrm{G}$ is isomorphic to a summand of $\mathrm{H}\circ \mathrm{F}$, for some $1$-morphism
$\mathrm{H}$. Equivalence classes with respect to $\leq_L$ are called {\em left cells} and the corresponding
equivalence relation is denoted $\sim_L$.
The {\em right} partial pre-order $\leq_R$, the {\em right cells} and the corresponding
equivalence relation $\sim_R$ are defined similarly using multiplication with $\mathrm{H}$ on the right
of $\mathrm{F}$. The {\em two-sided} partial pre-order $\leq_J$, the {\em two-sided cells} and the corresponding
equivalence relation $\sim_J$ are defined similarly using multiplication with $\mathrm{H}_1$ and
$\mathrm{H}_2$ on both sides of $\mathrm{F}$.
These notions are similar and spirit to and generalize the notions of {\em Green's} relations and partial
orders for semigroups, see \cite{Gre}, and also the notions of Kazhdan-Lusztig cells and order in
\cite{KaLu}, see also \cite{KM2}. We refer the reader to \cite{KuMa} for more details and to
\cite{KM2} for a generalization to positively based algebras.
For simplicity, we will say ``left cells in $\cC$'' instead of ``left cells in $\mathcal{S}[\cC]$''
and similarly for right and $2$-sided cells.
A two-sided cell $\mathcal{J}$ is said to be {\em regular} provided that
\begin{itemize}
\item any pair of left cells inside $\mathcal{J}$ is incomparable with respect to the left order;
\item any pair of right cells inside $\mathcal{J}$ is incomparable with respect to the right order.
\end{itemize}
A two-sided cell $\mathcal{J}$ is said to be {\em strongly regular} provided that
it is regular and $|\mathcal{L}\cap\mathcal{R}|=1$, for any left cell $\mathcal{L}$ in $\mathcal{J}$
and any right cell $\mathcal{R}$ in $\mathcal{J}$. We refer to \cite[Subsection~4.8]{MM1} for details.
A two-sided cell $\mathcal{J}$ is said to be {\em idempotent} provided that it contains three
elements $\mathrm{F}$, $\mathrm{G}$ and $\mathrm{H}$ (not necessarily distinct) such that
$\mathrm{F}$ is isomorphic to a direct summand of $\mathrm{G}\circ \mathrm{H}$,
see \cite[Subsection~2.3]{CM}. The following is \cite[Corollary~19]{KM2}.
\begin{proposition}\label{prop5}
Each idempotent two-sided cell of finitary $2$-category is regular, in particular,
each two-sided cell of (weakly) fiat $2$-category is regular.
\end{proposition}
\subsection{Cell $2$-representations}\label{s4.2}
Let $\cC$ be a finitary $2$-category and $\mathcal{L}$ a left cell in $\cC$. Then there is
an object $\mathtt{i}=\mathtt{i}_{\mathcal{L}}\in\cC$ such that all $1$-morphisms in $\mathcal{L}$
start from $\mathtt{i}$. Consider the principal $2$-representation $\mathbf{P}_{\mathtt{i}}$.
For each $\mathtt{j}\in\cC$, denote by $\mathbf{M}(\mathtt{j})$ the additive closure in
$\mathbf{P}_{\mathtt{i}}(\mathtt{j})$ of all indecomposable $1$-morphisms
$\mathrm{F}\in \mathbf{P}_{\mathtt{i}}(\mathtt{j})=\cC(\mathtt{i},\mathtt{j})$ satisfying $\mathcal{L}\leq_L \mathrm{F}$
(note that the latter notation makes sense as $\mathcal{L}$ is a left cell). Then $\mathbf{M}$
has the natural structure of a $2$-representation of $\cC$ which is inherited from
$\mathbf{P}_{\mathtt{i}}$ by restriction. The following lemma can be found e.g. in \cite[Lemma~3]{MM5}.
\begin{lemma}\label{lem1}
The $2$-representation $\mathbf{M}$ has a unique maximal
$\cC$-invariant ideal $\mathbf{I}$.
\end{lemma}
The quotient $2$-representation $\mathbf{M}/\mathbf{I}$ is called the {\em cell $2$-representation}
corresponding to $\mathcal{L}$ and denoted $\mathbf{C}_{\mathcal{L}}$. The construction presented here
first appears in \cite[Subsection~6.5]{MM2}.
It follows directly from the construction that isomorphism classes of indecomposable objects in
\begin{displaymath}
\coprod_{\mathtt{j}\in\ccC} \mathbf{C}_{\mathcal{L}}(\mathtt{j})
\end{displaymath}
correspond bijectively to elements in $\mathcal{L}$. Note that two cell $2$-representations
$\mathbf{C}_{\mathcal{L}}$ and $\mathbf{C}_{\mathcal{L}'}$ might be equivalent even
in the case $\mathcal{L}\neq \mathcal{L}'$. The following is \cite[Theorem~3.1]{MM6}.
\begin{theorem}\label{thm7}
Let $\cC$ be a weakly fiat $2$-category in which all two-sided cells are strongly regular.
Then, for any left cells $\mathcal{L}$ and $\mathcal{L}'$ in $\cC$, we have
$\mathbf{C}_{\mathcal{L}}\cong \mathbf{C}_{\mathcal{L}'}$ if and only if
$\mathcal{L}$ and $\mathcal{L}'$ belong to the same two-sided cell in $\cC$.
\end{theorem}
Cell $2$-representations can be viewed as natural $2$-analogues of semigroups representations
associated to left cells, see \cite[Subsection~11.2]{GM}.
\subsection{Basic properties of cell $2$-representations}\label{s4.3}
Let $\cC$ be a finitary $2$-category and $\mathbf{M}\in \cC$-afmod. We will say that $\mathbf{M}$
is {\em transitive} provided that, for any $\mathtt{i},\mathtt{j}\in \cC$ and for any
indecomposable $X\in \mathbf{M}(\mathtt{i})$ and $Y\in \mathbf{M}(\mathtt{j})$, there is a
$1$-morphism $\mathrm{F}\in\cC(\mathtt{i},\mathtt{j})$ such that $Y$ is isomorphic to a
summand of $\mathrm{F}\, X$.
Directly from the definition of left cells and the construction of cell $2$-representations
it follows that each cell $2$-representation is transitive.
We will say that $\mathbf{M}$ is {\em simple} provided that $\mathbf{M}$ has no proper
$\cC$-stable ideals. Note that every simple $2$-representation is automatically transitive.
Directly from construction of cell $2$-representations and Lemma~\ref{lem1}
it follows that each cell $2$-representation is simple. So, we have the following claim
(see \cite[Section~3]{MM5}):
\begin{proposition}\label{prop2}
Every cell $2$-representation of a finitary $2$-category is both simple and transitive.
\end{proposition}
\subsection{Alternative construction of cell $2$-representations for fiat $2$-categories}\label{s4.4}
Let $\cC$ be a fiat $2$-category, $\mathcal{L}$ a left cell in $\cC$ and
$\mathtt{i}=\mathtt{i}_{\mathcal{L}}$. Consider the principal $2$-representation $\mathbf{P}_{\mathtt{i}}$
and its abelianization $\overline{\mathbf{P}}_{\mathtt{i}}$. For $\mathtt{j}\in\cC$, projective objects
in $\overline{\mathbf{P}}_{\mathtt{i}}(\mathtt{j})$ correspond (up to isomorphism) to $1$-morphisms
$\mathrm{F}\in\cC(\mathtt{i},\mathtt{j})$ and are denoted $P_{\mathrm{F}}$. The simple top of
$P_{\mathrm{F}}$ is denoted $L_{\mathrm{F}}$.
In the fiat case, \cite[Section~4]{MM1} provides an alternative construction of cell
$2$-representation which is based on the notion of {\em Duflo involution}.
The following is \cite[Proposition~17]{MM1}:
\begin{proposition}\label{prop3}
The left cell $\mathcal{L}$ contains a unique element
$\mathrm{G}=\mathrm{G}_{\mathcal{L}}$, called the {\em Duflo involution} in $\mathcal{L}$,
such that there is a sub-object $K$ of the projective object $P_{\mathbbm{1}_{\mathtt{i}}}$
satisfying the following conditions:
\begin{enumerate}[$($a$)$]
\item\label{prop3.1} $\mathrm{F}\, (P_{\mathbbm{1}_{\mathtt{i}}}/K)=0$, for all $\mathrm{F}\in \mathcal{L}$;
\item\label{prop3.2} $\mathrm{F}\, \mathrm{top}(K)\neq 0$, for all $\mathrm{F}\in \mathcal{L}$;
\item\label{prop3.3} $\mathrm{top}(K)\cong L_{\mathrm{G}}$.
\end{enumerate}
\end{proposition}
For $\mathtt{j}\in\cC$, denote by $\mathbf{N}(\mathtt{j})$ the additive closure, in
$\overline{\mathbf{P}}_{\mathtt{i}}(\mathtt{j})$, of all elements of the form
$\mathrm{F}\, L_{\mathrm{G}}$, where $\mathrm{F}\in\mathcal{L}\cap\cC(\mathtt{i},\mathtt{j})$.
The following is \cite[Proposition~22]{MM2}:
\begin{proposition}\label{prop4}
{\hspace{2mm}}
\begin{enumerate}[$($i$)$]
\item\label{prop4.1} The assignment $\mathbf{N}$ inherits, by restriction from $\overline{\mathbf{P}}_{\mathtt{i}}$ ,
the natural structure of a $2$-representation of $\cC$.
\item\label{prop4.2} The $2$-representations $\mathbf{N}$ and $\mathbf{C}_{\mathcal{L}}$
are equivalent.
\end{enumerate}
\end{proposition}
The notion of Duflo involution was generalized to some non-fiat $2$-categories in \cite{Zh1}.
In \cite[Example~8]{Xa} (see also \cite[Subsection~9.3]{KMMZ}) one finds an example of a
fiat $2$-category with a left cell $\mathcal{L}$ such that the corresponding Duflo
involution $\mathrm{G}$ satisfies $\mathrm{G}\not\cong\mathrm{G}^{\star}$.
\section{Simple transitive $2$-representations}\label{s5}
\subsection{Weak Jordan-H{\"o}lder theory}\label{s5.1}
In this subsection we overview the weak Jordan-H{\"o}lder theory for additive $2$-representations
of finitary $2$-categories developed in \cite[Section~4]{MM5}. Here
simple transitive $2$-representations play a crucial role. We start with the following observation
which is just a variation of Lemma~\ref{lem1}, see \cite[Lemma~4]{MM5}.
\begin{lemma}\label{lem11}
Every transitive $2$-representation of a finitary $2$-category has a unique simple transitive quotient.
\end{lemma}
Let $\cC$ be a finitary $2$-category and $\mathbf{M}\in\cC$-afmod. Let $\mathrm{Ind}(\mathbf{M})$ be the
(finite!) set of isomorphism classes of indecomposable objects in
\begin{displaymath}
\coprod_{\mathtt{i}\in\ccC} \mathbf{M}(\mathtt{i}).
\end{displaymath}
The {\em action pre-order} $\to_{\ccC}$ on $\mathrm{Ind}(\mathbf{M})$ is defined as follows:
$X\to_{\ccC}Y$ provided that $Y$ is isomorphic to a direct summand of $\mathrm{F}\, X$, for some
$1$-morphism $\mathrm{F}$ in $\cC$. Consider a filtration
\begin{equation}\label{eq2}
\varnothing=\mathcal{Q}_0\subsetneq \mathcal{Q}_1\subsetneq \dots \subsetneq \mathcal{Q}_m=\mathrm{Ind}(\mathbf{M})
\end{equation}
such that, for each $i=1,2,\dots,m$,
\begin{itemize}
\item the set $\mathcal{Q}_i\setminus \mathcal{Q}_{i-1}$ is an equivalence class with respect to $\to_{\ccC}$,
\item the set $\mathcal{Q}_i$ has the property that, for all $X\in \mathcal{Q}_i$ and $Y\in \mathrm{Ind}(\mathbf{M})$
such that $X\to_{\ccC}Y$, we have $Y\in \mathcal{Q}_i$.
\end{itemize}
For $i=1,2,\dots,m$ and $\mathtt{j}\in\cC$, let $\mathbf{M}_i(\mathtt{j})$ denote the additive closure in
$\mathbf{M}(\mathtt{j})$ of all objects from $\mathcal{Q}_i\cap \mathbf{M}(\mathtt{j})$. Furthermore, we
denote by $\mathbf{I}_i(\mathtt{j})$ the ideal of $\mathbf{M}_i(\mathtt{j})$ generated by all
objects from $\mathcal{Q}_{i-1}\cap \mathbf{M}(\mathtt{j})$.
The assignment $\mathbf{M}_i$ inherits, by restriction from $\mathbf{M}$, the structure of a $2$-representation
of $\cC$, moreover, $\mathbf{I}_i$ is a $\cC$-stable ideal in $\mathbf{M}_i$. Hence we have a filtration
\begin{displaymath}
0\subset \mathbf{M}_1\subset \mathbf{M}_2\subset \dots\subset \mathbf{M}_m=\mathbf{M}
\end{displaymath}
of $\mathbf{M}$ by $2$-representations. This is a {\em weak Jordan-H{\"o}lder } series of $\mathbf{M}$.
By construction, the $2$-representation $\mathbf{M}_i/\mathbf{I}_i$ is transitive and we can denote by
$\mathbf{L}_i$ its unique simple transitive quotient, given by Lemma~\ref{lem11}. For a $2$-representation
$\mathbf{N}$, denote by $[\mathbf{N}]$ the equivalence class of $\mathbf{N}$. The multiset
$\{[\mathbf{L}_i]\,:\,i=1,2,\dots,m\}$ is then the multiset of {\em weak Jordan-H{\"o}lder subquotients}
of $\mathbf{M}$. The following is \cite[Theorem~8]{MM5}.
\begin{theorem}\label{thm12}
The multiset $\{[\mathbf{L}_i]\,:\,i=1,2,\dots,m\}$ does not depend on the choice of the
filtration \eqref{eq2}.
\end{theorem}
\subsection{How many simple transitive $2$-representations do we have?}\label{s5.2}
Theorem~\ref{thm12} motivates the study of simple transitive $2$-representations for finitary
$2$-categories. Note that simple $2$-representations are automatically transitive and hence
the name is slightly redundant. However, the name {\em simple transitive} has an advantage that
it addresses both layers of the representation structure:
\begin{itemize}
\item transitivity refers to the discrete layer of objects;
\item simplicity refers to the $\Bbbk$-linear layer of morphisms.
\end{itemize}
For an arbitrary finitary $2$-category $\cC$, we thus have a general problem:
\begin{problem}\label{prob14}
Classify all simple transitive $2$-representations of $\cC$, up to equivalence.
\end{problem}
Later on we will survey the cases in which this problem is solved, however, for general $\cC$,
even for general fiat $\cC$, it is wide open.
Of course, one could try to draw parallels with finite dimensional algebras.
One very easy fact from the classical representation theory is that every finite dimensional
$\Bbbk$-algebra has only a finite number of isomorphism classes of simple modules.
The corresponding statement in $2$-representation theory is still open. At the
moment this seems to be one of the major challenges in this theory.
\begin{question}\label{quest15}
Is it true that, for any finitary (fiat) $2$-category $\cC$, the number of equivalence classes of
simple transitive $2$-representations of $\cC$ is finite?
\end{question}
In all cases in which the answer to Problem~\ref{prob14} is known (see below), the number of
of equivalence classes of simple transitive $2$-representations is indeed finite.
\subsection{Finitary $2$-categories of type I}\label{s5.3}
A finitary $2$-category $\cC$ is said to be of {\em type} I provided that every
simple transitive $2$-representation of $\cC$ is equivalent to a cell $2$-representations.
Thus, for a finitary $2$-category of type I, Problem~\ref{prob14} has a fairly easy answer
(modulo comparison of cell $2$-representations with each other). Moreover, for a finitary
$2$-category of type I, Question~\ref{quest15} has positive answer. The first example
of finitary $2$-categories of type I is provided by \cite[Theorem~18]{MM5} and \cite[Theorem~33]{MM6}.
\begin{theorem}\label{thm16}
Every weakly fiat $2$-category with strongly regular two-sided cells is of type I.
\end{theorem}
Here we also note that many results in \cite{MM1}--\cite{MM5} assume that a certain {\em numerical condition}
is satisfied. This assumption was rendered superfluous by \cite[Proposition~1]{MM6}.
As a special case of Theorem~\ref{thm16}, we have that, the $2$-category $\cC_A$,
for a self-injective finite dimensional $\Bbbk$-algebra $A$, see Subsection~\ref{s3.2},
is of type I. Another special case of Theorem~\ref{thm16} is that the $2$-category $\cS_{W}$
of Soergel bimodules in Weyl type $A$ (that is when $W$ is isomorphic to the symmetric group),
see Subsection~\ref{s3.5}, is of type I. Furthermore, all finitary quotients of
$2$-Kac-moody algebras from Subsection~\ref{s3.7} are of type I, see \cite[Subsection~7.2]{MM5}
for details.
Simple transitive $2$-representations of the $2$-category $\cC_A$ were also studied
for some $A$ which are not self-injective (in this case $\cC_A$ is not weakly fiat).
The first result in this direction was the following statement, which is the main result of \cite{MZ1}.
\begin{theorem}\label{thm18}
For $A=A_2$ or $A_3$ as in Subsection~\ref{s3.3}, the corresponding $2$-category $\cC_A$ is of type I.
\end{theorem}
Despite of the fact that $A$ as in Theorem~\ref{thm18} is not self-injective, it has a non-zero
projective-injective module. The latter plays a crucial role in the arguments. Recently, based on
some progress made in \cite{KM2} and \cite{KMMZ}, Theorem~\ref{thm18} was generalized in \cite{MZ2}
as follows:
\begin{theorem}\label{thm19}
Let $A$ be a basic connected $\Bbbk$-algebra which has a non-zero projective-injective module
and which is directed in the sense that the Gabriel quiver of the algebra $A$ has neither loops nor
oriented cycles. Then the corresponding $2$-category $\cC_A$ is of type I.
\end{theorem}
When $A$ does not have any non-zero projective-injective module, the approach of \cite{MM5,MZ1,MZ2}
fails. Only one special case (the smallest one) was recently completed in \cite[Theorem~6]{MMZ} (partially based on
\cite[Subsection~7.1]{MM5}).
\begin{theorem}\label{thm20}
For $A=\Bbbk[x,y]/(x^2,y^2,xy)$, the corresponding $2$-category $\cC_A$ is of type I.
\end{theorem}
After all the cases listed above, the following question is rather natural:
\begin{question}\label{quest17}
Is it true that the $2$-category $\cC_A$ is of type I, for any $A$?
\end{question}
Apart from the cases listed above, there is a number of other type I examples.
The following result in \cite[Theorem~6.1]{Zi}.
\begin{theorem}\label{thm21}
The $2$-category of Soergel bimodules in Weyl type $B_2$ is of type I.
\end{theorem}
One interesting difference of the latter case compared to all other cases listed above
is the fact that, for the $2$-category of Soergel bimodules in Weyl type $B_2$, there are
non-equivalent cell $2$-representations which correspond to left cells inside the same two-sided cell.
Let $(W,S)$ be a finite Coxeter system. The corresponding $2$-category $\cS_W$ of Soergel bimodules,
see Subsection~\ref{s3.5}, has a unique minimum two-sided cell consisting of the identity $1$-morphism.
If we take this minimum two-sided cell away, in what remains there is again a unique minimum two-sided
cell $\mathcal{J}$. This two-sided cell contains, in particular, all Soergel bimodules of the form
$\mathtt{C}\otimes_{\mathtt{C}^s}\mathtt{C}$, where $s\in S$. There is a unique $2$-ideal $\cI$ in
$\cS_W$ which is maximal, with respect to inclusions, in the set of all $2$-ideals in
$\cS_W$ that do not contain identity $2$-morphisms for $1$-morphisms in $\mathcal{J}$. The quotient
$\underline{\cS_W}:=\cS_W/\cJ$ is called the {\em small quotient} of $\cS_W$, see \cite[Subsection~3.2]{KMMZ}.
The $2$-category $\underline{\cS_W}$ inherits from $\cS_W$ the structure of a fiat $2$-category.
The following result can be found in \cite[Sections~6, 7 and 8]{KMMZ}.
\begin{theorem}\label{thm22}
{\hspace{2mm}}
\begin{enumerate}[$($i$)$]
\item\label{thm22.1} If $|S|>2$, then $\underline{\cS_W}$ is of type I.
\item\label{thm22.2} If $(W,S)$ is of Coxeter type $I_2(n)$, with $n>4$, then
$\underline{\cS_W}$ is of type I if and only if $n$ is odd.
\end{enumerate}
\end{theorem}
\subsection{Finitary $2$-categories that are not of type I}\label{s5.4}
First, rather degenerate examples of finitary $2$-categories which are not of type I were constructed
already in \cite[Subsection~3.2]{MM5}. They are inspired by transitive group actions. Each finite group
structure can be extended to a fiat $2$-category in a fairly obvious way (by adding formal direct sums
of elements and linearizing spaces of identity $2$-morphisms). The resulting $2$-category has just one left cell
and the corresponding cell $2$-representation is, morally, the left regular representation of the group.
However, simple transitive $2$-representations correspond to transitive actions of the original group
on sets. The latter are given by action on (left) cosets modulo subgroups. In particular, we get a lot of
simple transitive $2$-representations which are not cell $2$-representations. The example, however, feels
rather artificial.
The first more ``natural'' example was constructed in \cite{MaMa}. Let $(W,S)$ be a Coxeter system
of type $I_2(n)$, with $n>3$. Consider the small quotient $\underline{\cS_W}$ of the
$2$-category of Soergel bimodules. This is a fiat $2$-category with two two-sided cells. The minimum
one consists just of the identity $1$-morphisms. The maximum one is not strongly regular. Let $S=\{s,t\}$.
Denote by $\cQ_n$ the $2$-full sub-$2$-category of $\underline{\cS_W}$ given by all $1$-morphisms in the additive
closure of the identity $1$-morphisms and of all $1$-morphisms $\mathrm{F}$ which lie in the same right
cell and in the same left cell as the $1$-morphisms given by $\mathtt{C}\otimes_{\mathtt{C}^s}\mathtt{C}$.
The main result of \cite{MaMa} is the following:
\begin{theorem}\label{thm23}
{\hspace{2mm}}
\begin{enumerate}[$($i$)$]
\item\label{thm23.1} The $2$-category $\cQ_5$ is of type I.
\item\label{thm23.2} The $2$-category $\cQ_4$ is not of type I. In fact, $\cQ_4$ has a unique
(up to equivalence) simple transitive $2$-representation which is not equivalent to any of two
cell $2$-representations.
\end{enumerate}
\end{theorem}
The major part of \cite{MaMa} is devoted to an explicit construction of this additional
simple transitive $2$-representation of $\cQ_4$. The construction is very technical and is based
on the following idea: One of the two cell $2$-representations of $\cQ_4$ has an invertible
automorphism which swaps the isomorphism classes of the two non-isomorphic
indecomposable objects in the underlying
category of this $2$-representation. The additional
simple transitive $2$-representation is constructed using the orbit category with respect to this
non-trivial automorphism. The main issue is that this automorphism is not strict (as homomorphisms
between $2$-rep\-re\-sen\-ta\-tions are not strict) and so it requires a lot of technical effort to
go around this complication. Based on this construction, the following statement was proved
in \cite[Sections~7]{KMMZ}.
\begin{theorem}\label{thm24}
Let $(W,S)$ be of Coxeter type $I_2(n)$, with $n>4$ even. Then $\underline{\cS_W}$ is not of type I.
Moreover, the following holds:
\begin{enumerate}[$($i$)$]
\item\label{thm24.1} Two of the three cell $2$-representations of $\underline{\cS_W}$ have an invertible
automorphism which is not isomorphic to the identity. The orbit construction as in \cite{MaMa}
with respect to this automorphism produces a new simple transitive $2$-representation of $\underline{\cS_W}$.
\item\label{thm24.2} If $n\neq 12,18,30$, then every
simple transitive $2$-representation of $\underline{\cS_W}$ is equivalent to either a cell $2$-representation
or one of the $2$-representations constructed in \eqref{thm24.1}.
\end{enumerate}
\end{theorem}
\subsection{Schur's lemma}\label{s5.5}
As we saw in the previous subsection, endomorphisms of cell $2$-representations play an important role
in this study. This naturally raises the following problem:
\begin{problem}\label{quest25}
Describe the (properties of the) bicategory of endomorphisms of a simple transitive
$2$-representation of a finitary $2$-category.
\end{problem}
The only known result in this direction is the following statement which is \cite[Theorem~16]{MM3}.
\begin{theorem}\label{thm26}
Let $\cC$ be a fiat $2$-category, $\mathcal{J}$ a strongly regular two-sided cell in $\cC$
and $\mathcal{L}$ a left cell in $\mathcal{J}$. Then any endomorphism of the cell
$2$-representation $\mathbf{C}_{\mathcal{L}}$ is isomorphic to the direct sum of a number
of copies of the identity endomorphism $\mathrm{ID}_{\mathbf{C}_{\mathcal{L}}}$.
Moreover, the endomorphism space (given by all modifications)
of $\mathrm{ID}_{\mathbf{C}_{\mathcal{L}}}$ consist just of scalar multiplies of the identity
modification.
\end{theorem}
\subsection{Apex}\label{s5.6}
The following is \cite[Lemma~1]{CM}.
\begin{lemma}\label{lem31}
Let $\cC$ be a finitary $2$-category and $\mathbf{M}$ a transitive $2$-representation of $\cC$.
There is a unique two sided cell $\mathcal{J}=\mathcal{J}_{\mathbf{M}}$ which is maximal, with respect
to the two-sided order, in the set of all two-sided cells that contain $1$-morphisms which are not
annihilated by $\mathbf{M}$. The two-sided cell $\mathcal{J}$ is idempotent.
\end{lemma}
The two-sided cell $\mathcal{J}$ is called the {\em apex} of $\mathbf{M}$. The general problem of
classification of all simple transitive $2$-representations of $\cC$ thus splits naturally into
subproblems to classify simple transitive $2$-representations of $\cC$ with a fixed apex
$\mathcal{J}$ (which should be an idempotent two-sided cell). Quite often, this simplifies the problem,
due to the following result which is proved analogously to \cite[Theorem~18]{MM5}.
\begin{theorem}\label{thm31}
Let $\cC$ be a weakly fiat $2$-category, $\mathcal{J}$ a strongly regular two-sided cell in $\cC$ and
$\mathbf{M}$ a simple transitive $2$-representation of $\cC$ with apex $\mathcal{J}$.
Then $\mathbf{M}$ is equivalent to a cell $2$-representation.
\end{theorem}
\subsection{Connection to integral matrices}\label{s5.7}
Let $\cC$ be a finitary $2$-category and $\mathbf{M}\in\cC$-afmod. Let $\mathrm{Ind}(\mathbf{M})$ be
as in Subsection~\ref{s5.1}. Then, to any $1$-morphism $\mathrm{F}$ in $\cC$, one can associate a matrix
$[\mathrm{F}]_{\mathbf{M}}$ whose rows and columns are indexed by elements in $\mathrm{Ind}(\mathbf{M})$
and the intersection of the $X$-row and $Y$-column gives the multiplicity of $X$ as a summand of
$\mathrm{F}\, Y$. The following observation is \cite[Lemma~11(ii)]{MM5}.
\begin{lemma}\label{lem32}
Assume that $\mathbf{M}$ is transitive and that $\mathrm{F}$ contains, as summands, representatives
from all isomorphism classes of indecomposable $1$-morphisms in $\cC$. Then
all coefficients in $[\mathrm{F}]_{\mathbf{M}}$ are positive.
\end{lemma}
This observation allows one to use the classical Perron-Frobenius Theorem (see \cite{Pe,Fro1,Fro2}) in
the study of simple transitive $2$-representations. This is an important ingredient in the arguments
in \cite{MM5,MZ1,Zi,MMZ}.
The above observation also provides some evidence for the general positive answer to
Question~\ref{quest15}.
Indeed, the Grothendieck decategorification of a finitary $2$-category $\cC$
gives a finite dimensional $\Bbbk$-algebra, call it $A$. For each $1$-morphism $\mathrm{F}$ in $\cC$,
we thus have the minimal polynomial $g_{\mathrm{F}}(\lambda)$ for the class $[\mathrm{F}]_{\oplus}$ in $A$.
Now, if $\mathbf{M}\in\cC$-afmod, then $[\mathbf{M}]_{\oplus}$ gives rise to an $A$-module and hence
\begin{displaymath}
g_{\mathrm{F}}([\mathrm{F}]_{\mathbf{M}})=0.
\end{displaymath}
Therefore, if $\mathbf{M}$ is transitive and $\mathrm{F}$ contains, as summands, representatives
from all isomorphism classes of indecomposable $1$-morphisms in $\cC$, then, because of
Lemma~\ref{lem32} and \cite[Theorem~3.2]{Zi2}, there are only finitely many possibilities
for the matrix $[\mathrm{F}]_{\mathbf{M}}$. Consequently, there are only finitely many possibilities
for matrices $[\mathrm{G}]_{\mathbf{M}}$, where $\mathrm{G}$ is an indecomposable $1$-morphism in $\cC$.
In many papers, for instance, in \cite{MM5,MM6,MZ1,MZ2,Zi}, the classification problem was approached
in two steps. The first step addressed classification of all possibilities for matrices
$[\mathrm{F}]_{\mathbf{M}}$. The second step studied actual $2$-representations for each solution
provided by the first step. In case of the $2$-category $\cC_A$, the first step studied matrices
$M$ with positive integer coefficients satisfying $M^2=\dim(A) M$. This is an interesting combinatorial
problem which was investigated in detail in \cite{Zi2}.
\subsection{Approach using (co)algebra objects}\label{s5.8}
The story with the cases $n=12$, $18$, $30$ in Theorem~\ref{thm24}\eqref{thm24.2} was quite interesting.
The detailed study of integral matrices, as outlined in Subsection~\ref{s5.7}, suggested in these cases
possibility of existence of simple transitive $2$-representations of $\cS_W$ which are neither
cell $2$-representations nor the ones constructed in Theorem~\ref{thm24}\eqref{thm24.1}.
These $2$-representations were constructed in \cite{MT}, based on \cite{El}, using diagrammatic calculus.
Under the additional assumption of gradeability, it was shown that, together with
cell $2$-representations and the $2$-representations constructed in Theorem~\ref{thm24}\eqref{thm24.1},
these exhaust all simple transitive $2$-representations of $\cS_W$.
Unfortunately, the diagrammatic calculus is not really compatible with our definitions of $2$-categories.
This raised a natural problem to reformulate the results of \cite{MT} in some language compatible
with our definitions. This was achieved in \cite{MMMT} using ideas of \cite{EGNO} related to the study of
algebra and coalgebra objects in $2$-categories. More precisely, the following is \cite[Theorem~9]{MMMT}.
\begin{theorem}\label{thm33}
Let $\cC$ be a fiat $2$-category and $\mathbf{M}$ a transitive $2$-representations of $\cC$. Then there
is a coalgebra object $A$ in the injective abelianization $\underline{\cC}$ of $\cC$ such that
$\mathbf{M}$ is equivalent to the $2$-representation of $\cC$ given by the action of $\cC$ on the
category of injective right $A$-comodule objects in $\underline{\cC}$.
\end{theorem}
For cell $2$-representations, the corresponding coalgebra objects turn out to be related to
Duflo involutions, see \cite[Subsection~6.3]{MMMT}.
Theorem~\ref{thm33} motivates the following general problem:
\begin{problem}\label{prob34}
Classify all coalgebra objects in $\underline{\cC}$, up to isomorphism.
\end{problem}
In this general formulation, Problem~\ref{prob34} is certainly more difficult than Problem~\ref{prob14}.
However, the very useful side of Theorem~\ref{thm33} is that one can construct simple transitive
$2$-representations by guessing the corresponding coalgebra objects (as it was done for the ``exotic''
simple transitive $2$-representations of $\cS_W$ in types $I_2(12)$, $I_2(18)$ and $I_2(30)$ in
\cite{MMMT}).
\section{Other classes of $2$-representations and related questions}\label{s6}
\subsection{Isotypic $2$-representations}\label{s6.1}
Let $\cC$ be a finitary $2$-category and $\mathbf{M}$ a $2$-rep\-re\-sen\-ta\-tion of $\cC$.
We will say that $\mathbf{M}$ is {\em isotypic} provided that all weak Jordan-H{\"o}lder
subquotients of $\mathbf{M}$ are equivalent, see \cite[Subsection~4.3.4]{Ro2} and \cite[Subsection~3.6]{MM6}.
For any $2$-representation $\mathbf{M}$ of $\cC$ and any finitary $\Bbbk$-linear
$2$-category $\mathcal{A}$ one defines the {\em inflation} $\mathbf{M}^{\boxtimes\mathcal{A}}$
of $\mathbf{M}$ by $\mathcal{A}$ as the $2$-representation of $\cC$ which sends each
$\mathtt{i}\in \cC$ to the tensor product $\mathbf{M}(\mathtt{i})\boxtimes \mathcal{A}$
and defines the action of $\cC$ on the objects and morphisms in these tensor products
by acting on the first component, see \cite[Subsection~3.6]{MM6} for details.
The following result is \cite[Theorem~4]{MM6}.
\begin{theorem}\label{thm37}
Let $\cC$ be a weakly fiat $2$-category with a unique maximal two-sided cell
$\mathcal{J}$. Let $\mathcal{L}$ be a left cell in $\mathcal{J}$. Assume that
$\mathcal{J}$ is strongly regular and that any non-zero $2$-ideal of $\cC$ contains the identity
$2$-morphism, for some $1$-morphism in $\mathcal{J}$.
Then any isotypic faithful $2$-representation of $\cC$ is equivalent to an inflation of
the cell $2$-representation $\mathbf{C}_{\mathcal{L}}$.
\end{theorem}
For finitary quotients of $2$-Kac-Moody algebras, the statement of Theorem~\ref{thm37}
can be deduced from \cite[Subsection~4.3.4]{Ro2}. Compared to the general case, the case of
$2$-Kac-Moody algebras is substantially simplified by existence of idempotent $1$-morphisms
in each two-sided cell. A challenging problem related to isotypic $2$-representation is the following:
\begin{problem}\label{thm37}
Classify faithful isotypic $2$-representation for an arbitrary
weakly fiat $2$-category $\cC$ with unique maximal two-sided cell $\mathcal{J}$
under the assumption that that $\mathcal{J}$ is strongly regular.
\end{problem}
In the easiest case, this problem will appear in the next subsection.
\subsection{All $2$-representations}\label{s6.2}
The question of classification of all $2$-representations, for a given finitary $2$-category
$\cC$, is open in all non-trivial case. The only {\em trivial} case is the case when the only indecomposable
$1$-morphisms in $\cC$ are the identities, up to isomorphism. It is certainly enough to consider
the case when $\cC$ has one object, say $\mathtt{i}$. Up to biequivalence, we may also assume
that $\mathbb{1}_{\mathtt{i}}$ is the only indecomposable $1$-morphism in $\cC$
(on the nose and not just up to isomorphism). Then, directly from the definition,
we have that a $2$-representation of such $\cC$ is given by a pair $(Q,\varphi)$,
where $Q$ is a finite dimensional $\Bbbk$-algebra and $\varphi$ is an algebra homomorphism from
$\mathrm{End}(\mathbb{1}_{\mathtt{i}})$ to the center of $Q$.
Then $\cC$ acts on a small category equivalent to
$Q$-proj in the obvious way. Note that all $2$-representations
of $\cC$ are isotypic. Furthermore, $\cC$ satisfies the assumptions of Theorem~\ref{thm37}
if and only if $\mathrm{End}(\mathbb{1}_{\mathtt{i}})\cong\Bbbk$.
The first non-trivial case to consider would be the following:
\begin{problem}\label{prop45}
Classify, up to equivalence, all finitary additive $2$-representations of $\cC_D$,
where $D=\mathbb{C}[x]/(x^2)$.
\end{problem}
\subsection{Discrete extensions between $2$-representations}\label{s6.3}
A major challenge in $2$-representation theory is the following:
\begin{problem}\label{prob46}
Develop a sensible homological theory for the study of $2$-rep\-re\-sen\-ta\-tions.
\end{problem}
A fairly naive attempt to define some analogue of $\mathrm{Ext}^1$ for $2$-representations
was made in \cite{CM}.
Let $\cC$ be a finitary $2$-category and $\mathbf{M}\in\cC$-afmod. For each $\mathtt{i}\in\cC$,
choose a full, additive, idempotents split and isomorphism closed subcategory $\mathbf{K}(\mathtt{i})$
of $\mathbf{M}(\mathtt{i})$ such that $\mathbf{K}$ becomes a sub-$2$-representation of $\cC$
by restriction. Let $\mathbf{I}$ be the ideal of $\mathbf{M}$ generated by $\mathbf{K}$
and $\mathbf{N}:=\mathbf{M}/\mathbf{I}$. Then the sequence
\begin{equation}\label{eq3}
0\to \mathbf{K}\overset{\Phi}{\longrightarrow}\mathbf{M}\overset{\Psi}{\longrightarrow}\mathbf{N}\to 0,
\end{equation}
where $\Phi$ is the natural inclusion and $\Psi$ is the natural projection, will be called a
{\em short exact sequence} of $2$-representations of $\cC$. The {\em discrete extension}
$\Theta$ {\em realized} by \eqref{eq3} is the subset of $\mathcal{S}[\cC]$ that consists of all
classes $[\mathrm{F}]$ for which there exist an indecomposable object $X$ in some
$\mathbf{M}(\mathtt{i})\setminus \mathbf{K}(\mathtt{i})$ and an indecomposable object $Y$ in some
$\mathbf{K}(\mathtt{j})$ such that $Y$ is isomorphic to a summand of $\mathrm{F}\, X$.
For $\mathbf{K}',\mathbf{N}'\in\cC$-afmod, the set $\mathrm{Dext}(\mathbf{N}',\mathbf{K}')$ of
{\em discrete extensions} from $\mathbf{N}'$ to $\mathbf{K}'$
consists of all possible $\Theta$ which are realized by some short
exact sequence \eqref{eq3} with $\mathbf{K}$ equivalent to $\mathbf{K}'$ and
$\mathbf{N}$ equivalent to $\mathbf{N}'$.
In many case, a very useful piece of information is to know whether
$\mathrm{Dext}(\mathbf{N},\mathbf{K})$ is empty (i.e. the first discrete extension {\em vanishes})
or not. In a number of cases, one could also either explicitly describe all elements in
$\mathrm{Dext}(\mathbf{N},\mathbf{K})$ or at least give a reasonable estimate of how they look like.
Vanishing of discrete extensions between certain simple transitive $2$-representations appears
in a disguised form and is an essential part of the arguments in \cite{MM5,MM6}.
The following is \cite[Theorem~25]{CM}.
\begin{theorem}\label{thm47}
Let $\cC$ be a weakly fiat $2$-category,
$\mathbf{K}$ a transitive $2$-representation of $\cC$ with apex $\mathcal{J}_{\mathbf{K}}$,
and $\mathbf{N}$ a transitive $2$-representation of $\cC$ with apex $\mathcal{J}_{\mathbf{N}}$.
Assume that, for any left cell $\mathcal{L}$ in $\mathcal{J}_{\mathbf{N}}$,
there exists a left cell $\mathcal{L}'$ in $\mathcal{J}_{\mathbf{K}}$ such that
$\mathcal{L}\geq_L\mathcal{L}'$. Then $\mathrm{Dext}(\mathbf{N},\mathbf{K})=\varnothing$.
\end{theorem}
As a consequence of Theorem~\ref{thm47}, all discrete self-extensions for transitive
$2$-rep\-re\-sen\-ta\-tions of weakly fiat $2$-categories vanish.
The results in \cite[Subsection~7.2]{CM} suggest that the answer to Question~\ref{quest48}
might be interesting and is not obvious.
\begin{question}\label{quest48}
What is $\mathrm{Dext}(\mathbf{N},\mathbf{K})$, for any pair
$(\mathbf{N},\mathbf{K})$ of simple transitive $2$-representations of the
$2$-category $\cS_W$ of Soergel bimodules in Weyl type $A$?
\end{question}
\subsection{Applications}\label{s6.4}
The first, rather spectacular, application of classification of certain classes of $2$-representations
appears in \cite{CR}. More precisely, \cite[Proposition~5.26]{CR} classifies, up to equivalence,
all $2$-representations of the $2$-Kac-Moody version of $\mathfrak{sl}_2$ which satisfy a number
of natural assumptions. This is an essential ingredient in the proof of derived equivalence for certain
blocks of the symmetric group, see \cite[Theorem~7.6]{CR}. Similar ideology was used, in particular,
to describe blocks of Lie superalgebras, see, for example, \cite{BS,BLW} and references therein.
In \cite{KM1}, classification of simple transitive $2$-representation for the $2$-category of
Soergel bimodules in type $A$ (cf. Theorem~\ref{thm16}) was used to classify indecomposable projective functors on the
principal block of BGG category $\mathcal{O}$ for $\mathfrak{sl}_n$.
| 6,181
|
TITLE: How can I prove the countability of a set which is an union of the set itself and it cartesian product with itself?
QUESTION [0 upvotes]: Assume we need to prove the countability of $(\Bbb{N}\times\Bbb{N}) \cup \Bbb{N}$
As far as I know, $\Bbb{N}\times\Bbb{N}$ will result in pairs such as $\langle\,1,1\,\rangle$ for all possible $\Bbb{N}$.
The resulting set of the union, would then contains all possible $\Bbb{N}$ values plus all the pairs.
To prove the countability of the sets, I usually draw in extension mode a bijective function, and then I can assume that it is countable as the cardinality are the same.
The problem here is that I have no idea how to map to a set which has pairs and singleton - is that even possible ? Any hint ?
REPLY [2 votes]: Guide:
Prove that $\mathbb{N} \times \mathbb{N}$ is countable.
Prove that union of countable set is countable.
Remark: $\mathbb{N} \times \mathbb{N}$ is known as the cartesian product rather than the power set.
| 39,935
|
\section{Proof of cut elmination and the Church-Rosser property}
\label{sec-proof-ce}
In this appendix we show that the cut elimination procedure terminates and
that the rewrite system induced by the cut elimination rewrites and the
communting conversion has the Church-Rosser property. We begin by proving
that cut elimination is a terminating procedure.
\begin{remark}[Generalized rewrites] \label{remark-genre}
The number of rewritings in our system motivates the use of a ``generalized''
system of rewritings which help to reduce this number. Let $\alpha(f)$
denote any of the morphims
\[\alpha\{a_i \mapsto f_i\}_i, \qquad \alpha[a]f, \qquad
\alpha\<(\alpha_i) \mapsto f\>, \qquad
\alpha\<\alpha_i \mid \Lambda_i \mapsto f_i\>_i
\]
Then $\alpha(f);_\gamma g \Lra \alpha(f ;_\gamma g)$ will be used
respectively to denote:
\[\begin{array}{rcl}
\alpha\{a_i \mapsto f_i\}_i ;_\gamma g &\xymatrix{\ar@{=>}[r]^{(3)}&}&
\alpha\{a_i \mapsto f_i ;_\gamma g\}_i \medskip\\
\alpha[a]f ;_\gamma g &\xymatrix{\ar@{=>}[r]^{(5)}&}&
\alpha[a](f ;_\gamma g) \medskip\\
\alpha\<(\alpha_i) \mapsto f\> ;_\gamma g &\xymatrix{\ar@{=>}[r]^{(7)}&}&
\alpha\<(\alpha_i) \mapsto f;_\gamma g\> \smallskip\\
\alpha\<\alpha_i \mid \Lambda_i \mapsto f_i\>_i ;_\gamma g
&\xymatrix{\ar@{=>}[r]^{(9)}&}&
\alpha\left\<\begin{array}{l}
\alpha_i \mid \Lambda_i \mapsto f_i \\
\alpha_k \mid \Lambda_k \cup \Lambda(g) \setminus \{\gamma\}
\mapsto f_k ;_\gamma g
\end{array}\right\>_{i \neq k}
\end{array}
\]
Dually $\xymatrix@1{g ;_\gamma \alpha(f) \ar@{=>}[r] & \alpha(g;_\gamma f)}$
will denote any of the rewrites (4), (6), (7), or (10). The rewrites (11)
and (13) (and their duals) may be represented as $\xymatrix@1{\gamma(f)
;_\gamma \gamma(g) \ar@{=>}[r] & f ;_\gamma g}$ and the permuting
conversions (15) through (24) may be represented as
$\xymatrix@1{\alpha(\beta(f)) \ar@{|=|}[r] & \beta(\alpha(f))}$.
\end{remark}
\subsection{The cut measure on terms}
The purpose of this section is to show that the cut elimination procedure
terminates. To this end we define a bag of cut heights and show that the
bag is strictly reduced on each of the cut elmination rewrites.
We begin by defining the multiset ordering of Dershowitz and
Manna~\cite{dershowitz79:proving}. Let $(S,\succ)$ be a partially-ordered
set, and let $\script{M}(S)$ denote the multisets (or bags) over $S$. For
$M,N \in \script{M}(S)$, $M > N$ (``$>$'' is called the \textbf{multiset}
(or \textbf{bag}) \textbf{ordering}), if there are multisets $X,Y \in
\script{M}(S)$, where $\emptyset \neq X \subseteq M$, such that
\[N = (M \bs X) \cup Y \quad \text{and} \quad (\forall y \in Y)
(\exists x \in X)\ x \succ y
\]
where $\cup$ here is the multiset union.
For example,
\[[3] > [2,2,2,1], \quad [7,3] > [7], \quad [5,2] > [5,1]
\]
Recall from~\cite{dershowitz79:proving} that if $(S,\succ)$ is a total order
(linear order) then $\script{M}(S)$ is a total order. To see this consider
$M,N \in \script{M}(S)$. To determine whether $M > N$ sort the elements of
both $M$ and $N$ and then compare the two sorted sequences lexicograpically.
We now define the \textbf{height} of a term as:
\begin{itemize}
\item $\hgt[a] = 1$ when $a$ is an atomic map (or an identity)
\item $\hgt[\alpha\{a_i \mapsto f_i\}_{i \in I}] = 1+ \max\{\hgt[f_i] \mid
i \in I\}$
\item $\hgt[\alpha[a_k] \cdot f] = 1+\hgt[f]$
\item $\hgt[\alpha\<(\alpha_i)_{i \in I} \mapsto f\>] = 1 + \hgt[f]$
\item $\hgt[\alpha\<\alpha_i \mid \Omega_i \mapsto f_i\>_{i \in I}] =
1 + \sum_{i \in I} \hgt[f_i]$
\item $\hgt[f ; g] = \hgt[f] + \hgt[g]$
\end{itemize}
The \textbf{height of a cut} is defined simply as its height, e.g.,
$\cuthgt[f;g] = \hgt[f;g]$. Define a function $\Lambda:T \ra \bag(\mathbb{N})$
which takes a term to its bag of cut heights.
\begin{proposition} \quad
\begin{enumerate}[{\upshape (i)}]
\item If $\xymatrix{t_1 \ar@{=>}[r]& t_2}$ then $\Lambda(t_1) > \Lambda(t_2)$.
\item If $\xymatrix{t_1 \ar@{|=|}[r]^{(a)}& t_2}$ and $(a)$ is an
interchange which does not involve the nullary cotuple or tuple then
$\Lambda(t_1) = \Lambda(t_2)$.
\end{enumerate}
\end{proposition}
\begin{proof}
We begin with the proof of part (i). There are three properties that must be
shown: $\hgt[t_1] \geq \hgt[t_2]$, the height of each non-principal cut does
not increase, and the height of any cut produced from the principal cut is
strictly less than the height of the principal cut.
A simple examination of the rewrites will confirm
that if $t_1 \Lra t_2$ then $\hgt[t_1] \geq \hgt[t_2]$:
\begin{equation*} \tag*{(1), dually (2)}
\hgt[f;1] = \hgt[f] + \hgt[1] > \hgt[f]
\end{equation*}
\begin{align*} \tag*{(3), dually (4)}
\hgt[\alpha\{a_i \mapsto f_i\}_{i \in I} ;_\gamma g]
&= \hgt[\alpha\{a_i \mapsto f_i\}_{i \in I}] + \hgt[g] \\
&= 1+ \max\{\hgt[f_i] \mid i \in I\} + \hgt[g] \\
&= 1+ \max\{\hgt[f_i] + \hgt[g] \mid i \in I\} \\
&= \hgt[\alpha\{a_i \mapsto f_i ;_\gamma g\}_{i \in I}]
\end{align*}
If $I = \emptyset$ then
\[\hgt[\alpha\{\,\} ;_\gamma g] =
\hgt[\alpha\{\,\}] + \hgt[g] > \hgt[\alpha\{\,\}]
\]
\begin{align*} \tag*{(5), dually (6)}
\hgt[\alpha[a_k]f ;_\gamma g] &= \hgt[\alpha[a_k]f]+\hgt[g] \\
&= 1+ \hgt[f] + \hgt[g] \\
&= \hgt[\alpha[a_k](f ;_\gamma g)]
\end{align*}
\begin{align*} \tag*{(7), dually (8)}
\hgt[\alpha\<(\alpha_i)_{i \in I} \mapsto f\> ;_\gamma g]
&= \hgt[\alpha\<(\alpha_i)_{i \in I} \mapsto f\> +\hgt[g] \\
&= 1 + \hgt[f] + \hgt[g] \\
&= \hgt[\alpha\<(\alpha_i)_{i \in I} \mapsto f ;_\gamma g\>
\end{align*}
\begin{align*} \tag*{(9), dually (10)}
\hgt[\alpha\<\alpha_i \mid \Omega_i \mapsto f_i\>_{i \in I} ;_\gamma g]
&= \hgt[\alpha\<\alpha_i \mid \Omega_i \mapsto f_i\>_{i \in I}]+\hgt[g]\\
&= 1 + \sum_{k \neq i \in I} \hgt[f_i] + \hgt[f_k] + \hgt[g] \\
&= \hgt\left[\alpha\left\<\begin{array}{l}
\alpha_i \mid \Omega_i \mapsto f_i \\
\alpha_k \mid \Omega_k \mapsto f_k ;_\gamma g
\end{array} \right\>_{k \neq i \in I}\right]
\end{align*}
\begin{align*} \tag*{(11), dually (12)}
\hgt[\gamma[a_k]f ;_\gamma \gamma\{a_i \mapsto g_i\}_{i \in I}]
&= \hgt[\gamma[a_k]f] + \hgt[\gamma\{a_i \mapsto g_i\}_{i \in I}] \\
&= 1+ \hgt[f] + 1 + \max\{\hgt[g_i] \mid i \in I\} \\
&> \hgt[f] + \max\{\hgt[g_i] \mid i \in I\} \\
&\geq \hgt[f] + \hgt[g_k] \\
&= \hgt[f ;_\gamma g_k]
\end{align*}
\begin{align*} \tag*{(13), dually (14)}
\hgt[\gamma\<\alpha_i \mid \Omega_i \mapsto f_i\>_i ;_\gamma
\gamma\<(\alpha_i)_i \mapsto g\>]
&= \hgt[\gamma\<\alpha_i \mid \Omega_i \mapsto f_i\>_i] +
\hgt[\alpha\<(\alpha_i)_i \mapsto g\>] \\
&= 1 + \sum_i \hgt[f_i] + 1 + \hgt[g] \\
&> \sum_i \hgt[f_i] + \hgt[g] \\
&= \hgt\left[f_n ;_{\alpha_n} ( \cdots (f_2 ;_{\alpha_2} (f_1 ;_{\alpha_1} g))
\cdots)\right]
\end{align*}
If $I = \emptyset$ then
\begin{align*}
\hgt[\gamma\<\,\>_i ;_\gamma \gamma\<(\,) \mapsto g\>]
&= \hgt[\gamma\<\,\>_i] + \hgt[\gamma\<(\,) \mapsto g\>] \\
&= 1+1+\hgt[g] \\
&> \hgt[g]
\end{align*}
Moreover, this implies that cuts below and cuts above the redex will not
increase their cut height on a rewriting.
Finally, consider the principal cut of the reduction. Rewrite (1) (dually (2))
removes a cut and so strictly reduces the bag of cut heights. It is an easy
observation that (5), (7), (9), and (11) (and their duals) each replace a
cut with one of lesser height, and that (3) and (13) (and their duals)
replace a cut with zero or more cuts of lesser height. Thus applying any of
the rewrites strictly reduces the bag.
We now prove part (ii). For the equations (15), (16), (17), and (18) we
assume that the index sets are non-empty. This then implies that the
commuting conversions are all of the form $\alpha(\beta(f))$ and thus
\[\hgt[\alpha(\beta(f))] = 1 + \hgt[\beta(f)] = 1 + 1+ \hgt[f] =
\hgt[\beta(\alpha(f))]
\]
which proves that the height does not change across these (non-empty
(co)tuple) interchanges.
\end{proof}
To see that the height is not invariant across the emtpy cotuple (dually the
tuple) rule recall one of the nullary versions of the rewrite (15):
\[\alpha\{\,\} \pc \beta\{b_j \mapsto \alpha\{\,\}\}_j
\]
The height on the left-hand side is one, while on the right-hand side the
height is two.
\subsection{Proof of the Church-Rosser property} \label{sec-church-rosser}
In this section we present a proof of the Church-Rosser property for
morphisms. We wish to show that given any two morphisms related by a series
of reductions and permuting conversions
\[\xymatrix{t_1 \ar@{<=}[r] & t_2 \ar@{|=|}[r] & t_3 \ar@{=>}[r] &
\quad \cdots \quad &
\ar@{=>}[l] t_{n-2} \ar@{|=|}[r] & t_{n-1} \ar@{=>}[r] & t_n}
\]
there is an alternative way of arranging the reductions and permuting
conversions so that $t_1$ and $t_n$ can be reduced to terms which are
related by the permuting conversions alone. That is, we wish to show that
there is a convergence of the following form:
\[\xymatrix{t_1 \ar@{=>}[dr]_{*} & && & t_n \ar@{=>}[dl]^{*} \\
& t_1' \ar@{|=|}[rr]_{*} && t_n'}
\]
When the rewriting system terminates (in the appropriate sense) this allows
the decision procedure for the equality of $\Sigma\Pi$-terms to be reduced
to the decision procedure for the permuting conversions (see
Section~\ref{sec-dp}). In order to test the equality of two terms, one can
rewrite both terms into a reduced form (one from which there are no further
reductions), and these will be equal if and only if the two reduced forms
are equivalent through the permuting conversions alone. In the current
situation the reduction process is, of course, the cut-elimination procedure.
Following~\cite{cockett01:finite} we say a rewrite system is \textbf{locally
confluent modulo equations} if any (one step) divergence of the following
form
\[\vcenter{\xymatrix{& t_0 \ar@{=>}[dl] \ar@{=>}[dr] \\ t_1 && t_2}}
\qquad \text{or} \qquad
\vcenter{\xymatrix{& t_0 \ar@{=>}[dl] \ar@{|=|}[dr] \\ t_1 && t_2}}
\]
(where ``$\xymatrix@1{\ar@{=>}[r]&}$'' denotes a reduction and
``$\xymatrix@1{t_1 \ar@{|=|}[r]& t_2}$'' an equation) has a convergence,
respectively, of the form
\[
\vcenter{\xymatrix{t_1 \ar@{|=>}[dr]_{*} && t_2 \ar@{|=>}[dl]^{*} \\ & t'}}
\qquad \text{and} \qquad
\vcenter{\xymatrix{t_1 \ar@{|=>}[ddr]_{*} && t_2 \ar@{=>}[d] \\
&& t_2' \ar@{|=>}[dl]^{*} \\ & t'}}
\]
where the new arrow ``$\xymatrix@1{\ar@{|=>}[r]&}$'' indicates either
an equality or a reduction in the indicated direction.
This gives:
\begin{proposition} \label{prop-conf}
Suppose $(N,\mathcal{R},\mathcal{E})$ is a rewriting system with the
equations equipped with a well-ordered measure on the rewrite arrows such
that the measure of the divergences is strictly greater than the measure of
the convergences then the system is confluent modulo equations if and only if
it is locally confluent modulo equations.
\end{proposition}
\begin{proof}
If the system is confluent modulo equations it is certainly locally
confluent modulo equations. Conversely suppose we have a chain of reductions,
equations, and expansions. We may associate with it the bag of measures of
the arrows of the sequence.
The idea will be to show that replacing any local divergence in this chain
by a local confluence will result in a new chain whose bag measure is
strictly smaller. However, this can be seen by inspection as we are removing
the arrows associated with the divergence and replacing them with the arrows
associated with the convergence. The measure on the arrows associated with
the divergence is strictly greater then that of the measure on the arrows
associated with the convergence.
Thus, each rewriting reduces the measure and, therefore, any sequence
of rewriting on such a chain must terminate. However, it can only terminate
when there are no local divergences to resolve. This then implies that the
end result must be a confluence modulo equations.
\end{proof}
\subsubsection{Resolving critical pairs locally}
The proof of Church-Rosser involves examining all the possible critical
pairs involving reductions or reductions and conversions, and showing
that they are all of the form shown above and that they may be resolved in
the way shown above. It then must be shown that there is some measure on the
arrows which decreases when replacing a divergence with a convergences.
This will then suffice to show that our system is locally confluent modulo
equations, so that by Proposition~\ref{prop-conf}, it is confluent modulo
equations. The rewrites (1)-(12) are the ``reductions'' and the commuting
conversions (13)-(24) are the ``equations''. The resolutions of the critical
pairs will be presented using the ``generalized'' rewrites (see
Remark~\ref{remark-genre}). For the additive rewrites see~\cite{pastro:msc}
where they have been written out in detail.
The reductions are as follows. Note that these reductions assume that all
index sets are non-empty. The reductions when the index set is empty are
handled separately below.
\begin{description}
\item[Reduction diagram 0:] $\xymatrix{1;1 \ar@{=>}[r]_{(2)}^{(1)} & 1}$
\item[Reduction diagram 1:] Substitute (3), (5), (7), or (9) for $(a)$ to
get the reduction diagrams for (1)-(3), (1)-(5), (1)-(7), and (1)-(9).
\[\xymatrix@R=5ex@C=7ex{
& \alpha(f) ;_\gamma 1 \ar@{=>}[dl]_-{(1)} \ar@{=>}[dr]^-{(a)} \\
\alpha(f) && \alpha(f ;_\gamma 1) \ar@{=>}[ll]^-{\alpha\{(1)\}} }
\]
Substituting the dual rewrites in the mirror image of the diagram above give
the dual reduction diagrams.
\item[Reduction diagram 2:] each row in the table corresponds to the
resolution of the critical pair ($a$)-($b$).
\[\begin{array}{c|c|c}
a & b & c \\ \hline
3 & 4 & 15 \\
3 & 6 & 16 \\
3 & 8 & 17 \\
3 & 10 & 18 \\
5 & 6 & 19 \\
5 & 8 & 20 \\
5 & 10 & 21 \\
7 & 8 & 22 \\
7 & 10 & 23 \\
9 & 10 & 24
\end{array}
\qqqquad
\vcenter{\xymatrix@M=1ex@C=20ex@R=10ex@!0{
& \alpha(f) ;_\gamma \beta(g) \ar@{=>}[dl]_{(a)} \ar@{=>}[dr]^{(b)} \\
\alpha(f ;_\gamma \beta(g)) \ar@{=>}[d]_{\alpha((b))}
&& \beta(\alpha(f) ;_\gamma g) \ar@{=>}[d]^{\beta((a))} \\
\alpha(\beta(f ;_\gamma g)) \ar@{|=|}[rr]_{(c)}
&& \beta(\alpha(f ;_\gamma g))}}
\]
\item[Reduction diagram 3:] each row in the table corresponds to the
resolution of the critical pair ($a$)-($b$).
\[\begin{array}{c|c|c}
a & b & c \\ \hline
3 & 15 & 3 \\
3 & 16 & 5 \\
3 & 17 & 7 \\
3 & 18 & 9 \\
5 & 16 & 3 \\
5 & 19 & 5 \\
5 & 20 & 7 \\
5 & 21 & 9 \\
\end{array}
\quad
\begin{array}{c|c|c}
a & b & c \\ \hline
7 & 17 & 3 \\
7 & 20 & 5 \\
7 & 22 & 7 \\
7 & 23 & 9 \\
9 & 18 & 3 \\
9 & 21 & 5 \\
9 & 23 & 7 \\
9 & 24 & 9
\end{array}
\qqqquad
\vcenter{\xymatrix@M=1ex@C=20ex@R=10ex@!0{
& \alpha(\beta(f)) ;_\gamma g \ar@{=>}[dl]_{(a)} \ar@{|=|}[dr]^{(b);1} \\
\alpha(\beta(f) ;_\gamma g) \ar@{=>}[dd]_{\alpha((c))}
&& \beta(\alpha(f)) ;_\gamma g \ar@{=>}[d]^{(c)} \\
&& \beta(\alpha(f) ;_\gamma g) \ar@{=>}[d]^{\beta((a))} \\
\alpha(\beta(f ;_\gamma g)) \ar@{|=|}[rr]_{(d)}
&& \beta(\alpha(f ;_\gamma g))}}
\]
\item[Reduction diagram 4:] each row in the table corresponds to the
resolution of the critical pair ($a$)-($b$).
\[\begin{array}{c|c|c|c}
a & b & c & d\\ \hline
3 & w & 4 & 15 \\
3 & w & 6 & 16 \\
3 & w & 8 & 17 \\
3 & w & 10 & 18 \\
5 & x & 4 & 16 \\
5 & x & 6 & 19 \\
5 & x & 8 & 20\\
5 & x & 10 & 21
\end{array}
\quad
\begin{array}{c|c|c|c}
a & b & c & d \\ \hline
7 & y & 4 & 17 \\
7 & y & 6 & 20 \\
7 & y & 8 & 22 \\
7 & y & 10 & 23 \\
9 & z & 4 & 18 \\
9 & z & 6 & 21 \\
9 & z & 8 & 23 \\
9 & z & 10 & 24
\end{array}
\qqqquad
\vcenter{\xymatrix@M=1ex@C=20ex@R=10ex@!0{
& \alpha(\gamma(f)) ;_\gamma \beta(g)
\ar@{=>}[dl]_{(a)} \ar@{|=|}[dr]^{(b);1} \\
\alpha(\gamma(f) ;_\gamma \beta(g)) \ar@{=>}[ddd]_{\alpha((c))}
&& \gamma(\alpha(f)) ;_\gamma \beta(g) \ar@{=>}[d]^{(c)} \\
&& \beta(\gamma(\alpha(f)) ;_\gamma g) \ar@{|=|}[d]^{\beta((b);1)} \\
&& \beta(\alpha(\gamma(f)) ;_\gamma g) \ar@{=>}[d]^{\beta((a))} \\
\alpha(\beta(\gamma(f) ;_\gamma g)) \ar@{|=|}[rr]_{(d)}
&& \beta(\alpha(\gamma(f) ;_\gamma g))}}
\]
where $w \in \{15, 16, 17, 18\}$, $x \in \{16, 19, 20, 21\}$, $y \in
\{17,20,22,23\}$, and $z \in \{18,21,23,24\}$. This means that there are 64
reductions that fit this general case!
\item[Reduction diagram 5:] each row in the table corresponds to the
resolution of the critical pair ($a$)-($b$).
\[\begin{array}{c|c|c|c}
a & b & c & c'\\ \hline
11 & 16 & 3 & 3 \\
11 & 19 & 5 & 5 \\
11 & 20 & 7 & 7 \\
11 & 21 & 9 & 9 \\
13 & 18 & 3 & 3^+,4^* \\
13 & 21 & 5 & 5^+,6^* \\
13 & 23 & 7 & 7^+,8^* \\
13 & 24 & 9 & 9^+,10^*
\end{array}
\qqqquad
\vcenter{\xymatrix@M=1ex@C=20ex@R=10ex@!0{
& \gamma(\alpha(f)) ;_\gamma \gamma(g)
\ar@{=>}[dl]_{(a)} \ar@{|=|}[dr]^{(b);1} \\
\alpha(f) ;_\gamma g \ar@{=>}[d]_{(c')}
&& \alpha(\gamma(f)) ;_\gamma \gamma(g) \ar@{=>}[d]^{(c)} \\
\alpha(\gamma(f) ;_\gamma g) && \alpha(\gamma(f) ;_\gamma \gamma(g))
\ar@{=>}[ll]^-{\alpha((a))}}}
\]
where $x^+,y^*$ means zero or one application of the rewrite ($x$) and zero
or more applications of the rewrite ($y$).
This may be a good time for a concrete example. Suppose $(a,b,c,c') =
(13,21,5,(5^+,6^*))$, $i \in \{1,\ldots,n\}$, and $\alpha \in \Omega_k$.
The reduction diagram for this case is:
\[\xymatrix@M=1ex@C=27ex@R=12ex@!0{
& {\gamma\left\<\begin{array}{l}
\gamma_i \mid \Omega_i \mapsto f_i \\
\gamma_k \mid \Omega_k \mapsto \alpha[a]f_k
\end{array}\right\>_{i \neq k} ;_\gamma \gamma\<(\gamma_i)_i \mapsto g\>}
\ar@{=>}[dl]_-{(13)} \ar@{|=|}[dr]^-{(21);1} \\
f_n ;_{\gamma_n} (\cdots (\alpha[a]f_k ;_{\gamma_k} (\cdots (f_1
;_\gamma g)\cdots))\cdots) \ar@{=>}[d]_{(5)}
&& \alpha[a]\gamma\<\gamma_i \mid \Omega_i \mapsto f_i\>_i ;_\gamma
\gamma\<(\gamma_i)_i \mapsto g\> \ar@{=>}[dd]^{(5)} \\
f_n ;_{\gamma_n} (\cdots (\alpha[a](f_k ;_{\gamma_k} (\cdots (f_1
;_\gamma g)\cdots)))\cdots) \ar@{=>}[d]_{(6)^*} \\
\alpha[a](f_n ;_{\gamma_n} (\cdots (f_k ;_{\gamma_k} (\cdots (f_1
;_\gamma g)\cdots)))\cdots)
&& \alpha[a](\gamma\<\gamma_i \mid \Omega_i \mapsto f_i\>_i ;_\gamma
\gamma\<(\gamma_i)_i \mapsto g\>)
\ar@{=>}[ll]^-{\alpha((13))}}
\]
\item[Reduction diagram 6:] each row in the table corresponds to the
resolution of the critical pair ($a$)-($b$).
\[\begin{array}{c|c|c|c}
a & b & c & c' \\
\hline
11 & 15 & 4 & 4 \\
11 & 16 & 6 & 4 \\
11 & 17 & 8 & 8 \\
11 & 18 & 10 & 10 \\
13 & 17 & 4 & 4^+,3^* \\
13 & 20 & 6 & 6^+,5^* \\
13 & 22 & 8 & 8^+,7^* \\
13 & 23 & 10 & 10^+,9^*
\end{array}
\qqqquad
\vcenter{\xymatrix@M=1ex@C=20ex@R=10ex@!0{
& \gamma(f) ;_\gamma \gamma(\beta(g))
\ar@{=>}[dl]_{(a)} \ar@{|=|}[dr]^{1;(b)} \\
f ;_\gamma \beta(g) \ar@{=>}[d]_{(c')}
&& \gamma(f) ;_\gamma \beta(\gamma(g)) \ar@{=>}[d]^{(c)} \\
\beta(f ;_\gamma g) && \beta(\gamma(f) ;_\gamma \gamma(g))
\ar@{=>}[ll]^-{\beta((a))}}}
\]
where $x^+,y^*$ means zero or one application of the rewrite ($x$) and zero
or more applications of the rewrite ($y$).
\end{description}
We now explore the cases when the index sets may be empty. For the empty
ltensor (dually rpar) rule the rewrites fit the cases above. The cases for
the empty cotuple (dually tuple) and rtensor (dually lpar) however do not.
We start by first examing what happens to the reduction diagrams for the
case of the empty cotuple (dually tuple).
\begin{description}
\item[Reduction diagram 1:]
$\xymatrix{\alpha\{\,\};1 \ar@{=>}[r]_{(3)}^{(1)} & \alpha\{\,\}}$
\item[Reduction diagram 2:] there are two (non-dual) cases corresponding
to only $I = \emptyset$ and both $I = J = \emptyset$. Each row in the table
corresponds to the resolution of the critical pair (3)-($a$).
\[\begin{array}{c|c}
a & b \\ \hline
4 & 15 \\
6 & 16 \\
8 & 17 \\
10 & 18
\end{array}
\qqqquad
\vcenter{\xymatrix@M=1ex@C=12ex@R=9ex@!0{
& \alpha\{\,\} ;_\gamma \beta(g) \ar@{=>}[dl]_{(3)} \ar@{=>}[dr]^{(a)} \\
\alpha\{\,\} \ar@{|=|}[dr]_-{(b)}
&& \beta(\alpha\{\,\} ;_\gamma g) \ar@{=>}[dl]^{\beta((3))} \\
& \beta(\alpha\{\,\})}}
\]
Dual to the above diagram is the nullary redutions for $\beta$. If both
$\alpha$ and $\beta$ have empty index sets:
\[\xymatrix@R=4ex@C=5ex{
& \alpha\{\,\} ;_\gamma \beta\{\,\} \ar@{=>}[dl]_{(3)} \ar@{=>}[dr]^{(4)} \\
\alpha\{\,\} \ar@{|=|}[rr]_-{(15)} && \beta\{\,\}}
\]
\item[Reduction diagram 3:] there are three cases. The first we describe is
when both $I = J = \emptyset$. The resolution is as follows:
\[\xymatrix@M=1ex@C=12ex@R=9ex@!0{
& \alpha\{\,\} ;_\gamma g \ar@{=>}[dl]_{(3)} \ar@{|=|}[dr]^{(15)} \\
\alpha\{\,\} \ar@{|=|}[dr]_-{(15)}
&& \beta\{\,\} ;_\gamma g \ar@{=>}[dl]^{(3)} \\
& \beta\{\,\}}
\]
The two remaining cases corresponding to whether the apex (of the reduction
diagram) starts with $\alpha\{\,\}$ or with $\beta(\alpha\{\,\})$. Each row
in the table corresponds to the resolution of the critical pair (3)-($b$) in
the reduction diagram on the left and ($a$)-($b$) in the reduction diagram on
the right.
\[\vcenter{\xymatrix@M=1ex@C=12ex@R=9ex@!0{
& \alpha\{\,\} ;_\gamma g \ar@{=>}[dl]_{(3)} \ar@{|=|}[dr]^{(b);1} \\
\alpha\{\,\} \ar@{|=|}[d]_{(a)}
&& \beta(\alpha\{\,\}) ;_\gamma g \ar@{=>}[d]^{(a)} \\
\beta(\alpha\{\,\})
&& \beta(\alpha\{\,\} ;_\gamma g) \ar@{=>}[ll]^{\beta((3))}}}
\qqqquad
\begin{array}{c|c}
a & b \\ \hline
3 & 15 \\
5 & 16 \\
7 & 17 \\
9 & 18
\end{array}
\qqqquad
\vcenter{\xymatrix@M=1ex@C=12ex@R=9ex@!0{
& \beta(\alpha\{\,\}) ;_\gamma g \ar@{=>}[dl]_{(a)}
\ar@{|=|}[dr]^{(b);1} \\
\beta(\alpha\{\,\} ;_\gamma g) \ar@{=>}[d]_{\beta((3))}
&& \alpha\{\,\} ;_\gamma g \ar@{=>}[d]^{(3)} \\
\beta(\alpha\{\,\}) && \alpha\{\,\} \ar@{|=|}[ll]^{(b)}}}
\]
\item[Reduction diagram 4:] The channel $\alpha$ is non-empty and if channel
$\gamma$ is empty the reduction diagram is identical.
\item[Reduction diagram 5:] each row in the table corresponds to the
resolution of the critical pair ($a$)-($b$).
\[\begin{array}{c|c|c}
a & b & c \\ \hline
11 & 16 & 3 \\
13 & 18 & 3^+,4^*
\end{array}
\qqqquad
\vcenter{\xymatrix@M=1ex@C=11ex@R=8ex@!0{
& \gamma(\alpha\{\,\}) ;_\gamma \gamma(g)
\ar@{=>}[dl]_-{(a)} \ar@{|=|}[dr]^-{(b);1} \\
\alpha\{\,\} ;_\gamma g \ar@{=>}[dr]_-{(c)}
&& \alpha\{\,\} ;_\gamma \gamma(g) \ar@{=>}[dl]^-{(3)} \\
& \alpha\{\,\}}}
\]
where $3^+,4^*$ means zero or one application of the rewrite (3) and zero
or more applications of the rewrite (4).
\item[Reduction diagram 6:] each row in the table corresponds to the
resolution of the critical pair ($a$)-($b$).
\[\begin{array}{c|c|c}
a & b & c \\ \hline
11 & 15 & 4 \\
13 & 17 & 4^*
\end{array}
\qqqquad
\vcenter{\xymatrix@M=1ex@C=11ex@R=8ex@!0{
& \gamma(f) ;_\gamma \gamma(\beta\{\,\}) \ar@{=>}[dl]_-{(a)}
\ar@{|=|}[dr]^-{1;(b)} \\
f ;_\gamma \beta\{\,\} \ar@{=>}[dr]_-{(c)}
&& \gamma(f) ;_\gamma \beta\{\,\} \ar@{=>}[dl]^-{(3)} \\
& \beta\{\,\}}}
\]
where $4^*$ means zero or more applications of the rewrite (4).
\end{description}
We now examing what happens to the reduction diagrams for the case of the
empty rtensor (dually lpar). Due to the typing constraints the only
reduction diagrams that need be considered are 1 and 6.
\begin{description}
\item[Reduction diagram 1:]
$\xymatrix{\alpha\<\,\>;1 \ar@{=>}[r]_{(13)}^{(1)} & \alpha\<\,\>}$
\item[Reduction diagram 6:] each row in the table corresponds to the
resolution of the critical pair (13)-($a$).
\[\begin{array}{c|c}
a & b \\ \hline
17 & 4 \\
20 & 6 \\
22 & 8 \\
23 & 10
\end{array}
\qqqquad
\vcenter{\xymatrix@M=1ex@C=11ex@R=8ex@!0{
& \gamma\<\,\> ;_\gamma \gamma\<(\,) \mapsto \beta(g)\>
\ar@{=>}[dl]_-{(13)} \ar@{|=|}[dr]^-{1;(a)} \\
\beta(g)
&& \gamma\<\,\> ;_\gamma \beta(\gamma\<(\,) \mapsto g\>) \ar@{=>}[dl]^-{(b)} \\
& \beta(\gamma\<\,\> ;_\gamma \gamma\<(\,) \mapsto g\>)
\ar@{=>}[ul]^-{\delta(13)}}}
\]
\end{description}
\subsubsection{The measure on the rewriting arrows}
We define a measure $\lambda:A \ra \bag(\mathbb{N})$ on the rewriting arrows
as follows:
\begin{itemize}
\item if $\xymatrix{t_1 \ar@{=>}[r]^x & t_2}$ then $\lambda(x) =
\min\{\Lambda(t_1),\ \Lambda(t_2)\}$
\item if $\xymatrix{t_1 \ar@{|=|}[r]^x & t_2}$ then $\lambda(x) =
\max\{\Lambda(t_1),\ \Lambda(t_2)\}$
\end{itemize}
where $\Lambda(t)$ is the bag of cut heights of $t$.
A quick examination of the reduction diagrams now confirms that this measure
will decrease when we replace a divergence with a convergence.
This completes the proof of the proposition:
\begin{proposition}
$\channel(\cat{A})$ under the rewrites (1)-(14) is confluent modulo the
equations (15)-(24).
\end{proposition}
| 11,093
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\begin{document}
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\put(-0.7,2.0){\chudam The 21$^{st}$ Annual Meeting in Mathematics (AMM 2016)} \put(-0.7,1.5){\chudam Annual Pure and Applied Mathematics Conference 2016 (APAM 2016)}
\put(-0.7,1.0){\chudams Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University}
\put(-0.7,0.5){\chudams Speaker: M.~Tuntapthai}
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\centerline {\bf \chuto Bounds of the Normal Approximation for }
\medskip
\centerline {\bf \chuto Linear Recursions with Two Effects }
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\vskip.8cm
\centerline {Mongkhon Tuntapthai{\footnote{\textit{Corresponding author}}}{\footnote{\textit{The author is supported by the young researcher development project of Khon Kaen University}}}
}
\renewcommand{\thefootnote}{\arabic{footnote}}
\vskip.5cm
\centerline{Department of Mathematics, Faculty of Science, Khon Kaen University
}
\centerline{\texttt{mongkhon@kku.ac.th}
}
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\begin{abstract}
Let $X_0$ be a non-constant random variable with finite variance.
Given an integer $k\ge2$, define a sequence $\set{X_n}_{n=1}^\infty$ of approximately linear recursions
with small perturbations $\set{\Delta_n}_{n=0}^\infty$ by
\[
X_{n+1} = \sum_{i=1}^k a_{n,i} X_{n,i} + \Delta_n
\quad \text{for all } n\ge0
\]
where $X_{n,1},\dots,X_{n,k}$ are independent copies of the $X_n$ and $a_{n,1},\dots,a_{n,k}$ are real numbers.
In 2004, Goldstein obtained bounds on the Wasserstein distance between the standard normal distribution and the law of $X_n$ which is in the form $C \gamma^n$ for some constants $C>0$ and $0 < \gamma < 1$.
In this article, we extend the results to the case of two effects by studying a linear model $Z_n=X_n+Y_n$ for all $n\ge0$,
where $\set{Y_n}_{n=1}^\infty$ is a sequence of approximately linear recursions with an initial random variable $Y_0$ and perturbations $\set{\Lambda_n}_{n=0}^\infty$,
i.e., for some $\ell \ge2$,
\[
Y_{n+1} = \sum_{j=1}^\ell b_{n,j} Y_{n,j} + \Lambda_n
\quad \text{for all } n\ge0
\]
where $Y_n$ and $Y_{n,1},\dots,Y_{n,\ell}$ are independent and identically distributed random variables
and $b_{n,1},\dots,b_{n,\ell}$ are real numbers.
Applying the zero bias transformation in the Stein\rq s equation, we also obtain the bound for $Z_n$.
Adding further conditions that the two models $(X_n,\Delta_n)$ and $(Y_n,\Lambda_n)$ are independent and that the difference between variance of $X_n$ and $Y_n$ is smaller than the sum of variances of their perturbation parts, our result is the same as previous work.
\end{abstract}
\medskip
\noindent
{\bf Mathematics Subject Classification:}
60F05, 60G18
\smallskip
\noindent
{\bf Keywords:}
Hierarchical sequence, Stein\rq s method, Zero bias
\section{Introduction and Main Theorem}
Let $Z$ be a standard normally distributed random variable and
$X_0$ a non-constant random variable with finite variance. For a positive integer $k\ge2$, we consider a sequence $\{X_n\}_{n=1}^\infty$ of approximately linear recursions with perturbations $\{\Delta_n\}_{n=0}^\infty$,
\[
X_{n+1} = \sum_{i=1}^k a_{n,i} X_{n,i} + \Delta_n
\quad \text{ for all } n \ge 0
\]
where the $X_n$ and $X_{n,1},\dots,X_{n,k}$ are independent and identically distributed random variables
and $a_{n,1},\dots,a_{n,k}$ are real numbers.
For all integers $n\ge0$, we introduce some notation for the model $\paren{X_n,a_n,\Delta_n}$,
\[
\lambda_{a,n}^2
= \sum_{i=1}^k a_{n,i}^2,
\;\;
\varphi_{a,n}
= \sum_{i=1}^k \frac{|a_{n,i}|^3}{\lambda_{a,n}^3},
\;\;
\var{X_n}
= \sigma_{X,n}^2
\]
and
\[
\widetilde{X}_n
= \frac{X_n-\E X_n}{\sigma_{X,n}}.
\]
Arising originally from statistical physics, the approximately linear recursions are special type of hierarchical strucutres and often applied to the conductivity of random mediums.
A natural way in the classical probability theory is to study limit theorems for the distributions of these models.
A strong law of large numbers for the hierarchical structure was obtained by \cite{Wehr1997, LiRogers1999, Jordan2002}.
The central limit theorem for recursions was first introduced by \cite{WehrWoo2001}
and the bounds to normal approximation based on the Wasserstein distance were obtained by \cite{Goldstein2004}.
The following two conditions were used in the last work.
\begin{condition}
\label{condition1}
For each $i=1,\dots,k$, the sequence $\{a_{n,i}\}_{n=0}^\infty$ converges to some real number $a_i$ satisfying that at least two of the $a_i$\rq s are nonzero.
Set $\lambda_{a}^2 = \sum_{i=1}^k a_i^2$.
There exist $0 < \delta_{X,2} < \delta_{\Delta,2} < 1$ and positive constants $C_{X,2}$, $C_{\Delta,2}$ such that for all $n\ge 0$,
\[
\var{X_n}
\ge C_{X,2}^2 \lambda_{a}^{2n} \paren{1-\delta_{X,2}}^{2n},
\]
\[
\var{\Delta_n}
\le C_{\Delta,2}^2 \lambda_{a}^{2n} \paren{1-\delta_{\Delta,2}}^{2n}.
\]
\end{condition}
\begin{condition}
\label{condition2}
With $\delta_{X,2}$, $\delta_{\Delta,2}$ and $\lambda_{a}$ as in the Condition \ref{condition1}, there exists $\delta_{X,4} \ge 0$ and $\delta_{\Delta,4} \ge 0$ such that
\[
\phi_{X,\Delta,2}
= \frac{\paren{1-\delta_{\Delta,2}} \paren{1+\delta_{X,4}}^3} {\paren{1-\delta_{X,2}}^4}
< 1
\;\; \text{ and } \;\;
\phi_{X,\Delta,4}
= \paren{\frac{1-\delta_{\Delta,4}}{1-\delta_{X,2}} }^2
< 1
\]
and positive constants $C_{X,4}$, $C_{\Delta,4}$ such that
\[
\E \paren{X_n - \E X_n}^4
\le C_{X,4}^4 \lambda_{a}^{4n}
\paren{1+\delta_{X,4}}^{4n},
\]
\[
\E \paren{\Delta_n - \E \Delta_n}^4
\le C_{\Delta,4}^4 \lambda_{a}^{4n}
\paren{1-\delta_{\Delta,4}}^{4n} .
\]
\end{condition}
Recall that the Wasserstien distance or $L^1$-distance between two laws $F$ and $G$ is given by
\[
\norm{F-G}_1
= \int_{-\infty}^\infty \abs{F(t) - G(t)} \,dt.
\]
For any random variable $X$, the law or cumulative distribution function of $X$ is denoted by $\mathcal L(X)$.
\begin{thm} \textnormal{\cite{Goldstein2004}}
\label{intro:linearrecursion:Goldstein}
Under Conditions \ref{condition1} and \ref{condition2},
there exist constants $C>0$ and $\gamma \in (0,1)$ such that
\[
\norm{\mathcal L (\widetilde{X}_n ) - \mathcal L (Z)}_1
\le C \gamma^n.
\]
\end{thm}
In this article, we extend the bounds to the case of two effects.
Let $\{Z_n\}_{n=0}^\infty$ be a sequence of linear model with two effects given by
\[
Z_n = X_n + Y_n
\quad \text{ for all } n \ge 0
\]
where $Y_0$ is a non-degenerated random and for some integer $\ell \ge2$,
\[
Y_{n+1} = \sum_{j=1}^\ell b_{n,j} Y_{n,j} + \Lambda_n
\quad \text{for all } n\ge0
\]
where $b_{n,1},\dots,b_{n,\ell}$ are real numbers,
$Y_{n,1},\dots,Y_{n,\ell}$ are independent copy of the $Y_n$ and $\Lambda_n$ is a small perturbation.
Note that the perturbations $\Delta_n$ and $\Lambda_n$ always depend on $X_n$ and $Y_n$, respectively.
From now on, we assume that random variables from two models of recursions $(X_n,\Delta_n)$ and $(Y_n,\Lambda_n)$ are independent for all $n\ge0$, and denote
\[
\lambda_n^2
= \sum_{i=1}^k a_{n,i}^2 + \sum_{j=1}^\ell b_{n,j}^2,
\quad
\var{Z_n}
= \sigma_{X,n}^2 + \sigma_{Y,n}^2
= \sigma_{n}^2
\]
and
\[
\widetilde Z_n
= \frac{Z_n -\E Z_n}{\sigma_n}.
\]
The bound for linear recursions with two effects is derived by adding further assumption that the difference between variances of two models $(X_n,\Delta_n)$, $(Y_n,\Lambda_n)$, is smaller than variances of perturbations, the following is our main theorem.
\begin{thm}
\label{intro:twoeffect:mainthm}
With constants $\delta_{X,2}$, $\delta_{X,4}$, $\delta_{\Delta,2}$
and $\delta_{Y,2}$, $\delta_{Y,4}$, $\delta_{\Lambda,2}$
as in Condition \ref{condition1} and \ref{condition2}
for the models $\paren{X_n,\Delta_n}$
and $\paren{Y_n,\Lambda_n}$, suppose that
\[
\psi_{X,Y,\Lambda}
= \frac{\paren{1-\delta_{\Lambda,2}} \paren{1+\delta_{X,4}}^3 }{\paren{1-\delta_{Y,2}}\paren{1-\delta_{X,2}}^3}
< 1
\;\; \text{ and } \;\;
\psi_{Y,X,\Delta}
= \frac{\paren{1-\delta_{\Delta,2}} \paren{1+\delta_{Y,4}}^3 }{\paren{1-\delta_{X,2}}\paren{1-\delta_{Y,2}}^3}
< 1
\]
and that
\[
\abs{\var{X_n} - \var{Y_n}}
\le \frac{\var{\Delta_n} + \var{\Lambda_n}}{\max\{\lambda_{a,n}^2,\lambda_{b,n}^2 \} } ,
\]
then there exist constants $C>0$ and $\gamma\in(0,1)$ such that
\[
\norm{\mathcal L (\widetilde Z_n) - \mathcal L (Z)}_1
\le C \gamma^n.
\]
\end{thm}
\section{Auxiliary Results}
\label{auxresult}
Before proving the main theorem, we present some results for the models $(X_n,\Delta_n)$ and $(Y_n,\Lambda_n)$.
For all $n\ge 0$, let
\[
r_{X,n }
= \frac{\lambda_n \sigma_{X,n}}{\sigma_{n+1}},
\quad
r_{Y,n }
= \frac{\lambda_n \sigma_{Y,n}}{\sigma_{n+1}} .
\]
We begin with the bounds of $r_{X,n}$ and $r_{Y,n}$.
\begin{lem}
\label{auxresult:lem:boundrn}
With constants $\delta_{X,2}$, $\delta_{\Delta,2}$ and $\delta_{Y,2}$, $\delta_{\Lambda,2}$
as in Condition \ref{condition1} for the models
$(X_n,\Delta_n)$ and $(Y_n,\Lambda_n)$, and suppose that
\[
\abs{\var{X_n} - \var{Y_n}}
\le \frac{\var{\Delta_n} + \var{\Lambda_n}}{\max\{\lambda_{a,n}^2,\lambda_{b,n}^2 \} } ,
\]
then for an integer $p\ge1$, there exists a positive constant $C_{r,p}$ such that
\[
\abs{r_{X,n}^p -1}
\le C_{r,p} \set{\paren{\frac{1-\delta_{\Delta,2}}{1-\delta_{X,2}}}^n
+ \paren{\frac{1-\delta_{\Lambda,2}}{1-\delta_{Y,2}}}^n }
\]
and
\[
\abs{r_{Y,n}^p -1}
\le C_{r,p} \set{\paren{\frac{1-\delta_{\Delta,2}}{1-\delta_{X,2}}}^n
+ \paren{\frac{1-\delta_{\Lambda,2}}{1-\delta_{Y,2}}}^n } .
\]
\end{lem}
\proof
Following the argument of \cite[Lemma 6]{WehrWoo2001}, we consider the variances of linear model of recursions
\begin{eqnarray*}
\sigma_{n+1}^2
&= &
\var{Z_{n+1}}
\\
&= &
\lambda_{a,n}^2 \var{X_n}
+ \lambda_{b,n}^2 \var{Y_n}
+ \var{\Delta_n}
+ \var{\Lambda_n}
\\
&= &
\lambda_n^2 \sigma_{X,n}^2
+ \lambda_{b,n}^2 \set{ \var{Y_n} - \var{X_n} }
+ \var{\Delta_n}
+ \var{\Lambda_n} ,
\end{eqnarray*}
The triangle inequality yields
\begin{eqnarray*}
\sigma_{n+1}
&\le &
\lambda_n \sigma_{X,n}
+ \sqrt{\lambda_{b,n}^2 \abs{ \var{Y_n} - \var{X_n} }}
+ \sqrt{\var{\Delta_n} + \var{\Lambda_n} }
\\
&\le &
\lambda_n \sigma_{X,n}
+ 2 \sqrt{\var{\Delta_n} + \var{\Lambda_n} } .
\end{eqnarray*}
Also, we note that
\begin{eqnarray*}
\lambda_{a,n}^2 \sigma_{X,n}^2
&= &
\sigma_{n+1}^2
- \lambda_{b,n}^2 \set{ \var{Y_n} - \var{X_n} }
- \var{\Delta_n}
- \var{\Lambda_n}
\\
&\le &
\sigma_{n+1}^2
+ \lambda_{b,n}^2 \abs{ \var{Y_n} - \var{X_n} }
+ \var{\Delta_n}
+ \var{\Lambda_n} ,
\end{eqnarray*}
which implies that
\begin{eqnarray*}
\lambda_n \sigma_{X,n}
&\le &
\sigma_{n+1}
+ \sqrt{\lambda_{b,n}^2 \abs{ \var{Y_n} - \var{X_n} }}
+ \sqrt{\var{\Delta_n} + \var{\Lambda_n} }
\\
&\le &
\lambda_n \sigma_{X,n}
+ 2 \sqrt{\var{\Delta_n} + \var{\Lambda_n} } .
\end{eqnarray*}
Then there exists a constant $C_{r,1}$ such that
\begin{eqnarray*}
\abs{r_{X,n}-1}
&= &
\frac{\abs{ \lambda_n \sigma_{X,n} - \sigma_{n+1} } }{\sigma_{n+1}}
\\
&\le &
\frac{2 \sqrt{\var{\Delta_n} + \var{\Lambda_n} } }{\sigma_{n+1}}
\\
&\le &
2 \sqrt{\frac{\var{\Delta_n}}{\var{X_{n+1}}} }
+ 2 \sqrt{ \frac{\var{\Lambda_n}}{\var{Y_{n+1}}} }
\\
&\le &
\frac{2C_{\Delta,2} \paren{1-\delta_{\Delta,2}}^n }
{C_{X,2} \lambda_a \paren{1-\delta_{X,2} }^{n+1} }
+ \frac{2C_{\Lambda,2} \paren{1-\delta_{\Lambda,2}}^n }
{C_{Y,2} \lambda_b \paren{1-\delta_{Y,2} }^{n+1} }
\\
&\le &
C_{r,1} \set{\paren{\frac{1-\delta_{\Delta,2}}{1-\delta_{X,2}}}^n
+ \paren{\frac{1-\delta_{\Lambda,2}}{1-\delta_{Y,2}}}^n } .
\end{eqnarray*}
Now, since
\[
\abs{r^p-1}
= \abs{\paren{r-1+1}^p-1} \le \sum_{j=1}^p \binom{p}{j} \abs{r-1}^j
\]
and the assumption that
$0<\delta_{X,2}<\delta_{\Delta,2}<1$
and $0<\delta_{Y,2}<\delta_{\Lambda,2}<1$,
there are constants $C_{r,p}$ such that
\[
\abs{r_{X,n}^p -1}
\le C_{r,p} \set{\paren{\frac{1-\delta_{\Delta,2}}{1-\delta_{X,2}}}^n
+ \paren{\frac{1-\delta_{\Lambda,2}}{1-\delta_{Y,2}}}^n }
\]
and similarly, we can see that
\[
\abs{r_{Y,n}^p -1}
\le C_{r,p} \set{\paren{\frac{1-\delta_{\Delta,2}}{1-\delta_{X,2}}}^n
+ \paren{\frac{1-\delta_{\Lambda,2}}{1-\delta_{Y,2}}}^n }
\]
for all $p=1,2,3,\dots$.
\endproof
For all $n\ge 0$, let
\[
U_n = U_{X,n} + U_{Y,n}
\]
where
\[
U_{X,n+1}
= \sum_{i=1}^k \frac{a_{n,i}}{\lambda_n}
\paren{ \frac{X_{n,i} - \E X_{n,i}}{\sigma_{X,n}} }
\;\; \text{ and } \;\;
U_{Y,n+1}
= \sum_{j=1}^\ell \frac{b_{n,j}}{\lambda_n}
\paren{ \frac{Y_{n,j} - \E Y_{n,j}}{\sigma_{Y,n}} } .
\]
Next, we follow the proof of \cite[Lemma 4.1]{ChenEt2010} to prepare an inequality for the Wasserstein distance between laws of $U_n$ and its zero bias transformation.
\begin{lem}
\label{auxresult:lem:boundLU}
For all integers $n\ge1$ and the zero bias transformation
$U_n^\ast$, $\widetilde X_n^\ast$, $\widetilde Y_n^\ast$
of the $U_n$, $\widetilde X_n$, $\widetilde Y_n$, respectively,
we have
\[
\norm{\mathcal L (U_n) - \mathcal L (U_n^\ast) }_1
\le \norm{\mathcal L (\widetilde X_n) - \mathcal L (\widetilde X_n^\ast)}_1
+ \norm{\mathcal L (\widetilde Y_n) - \mathcal L (\widetilde Y_n^\ast)}_1 .
\]
\end{lem}
\proof
Set $m=k+\ell$. Let
\[
\xi_i =
\begin{cases}
\paren{ X_{n,i} - \E X_{n,i} } / \sigma_{X,n}
& \text{ for } i = 1,\dots, k
\\
\paren{ Y_{n,i-k} - \E Y_{n,i-k} } / \sigma_{Y,n}
& \text{ for } i = k+1,\dots,m
\end{cases}
\]
and
\[
\alpha_{n,i} =
\begin{cases}
a_{n,i}
& \text{ for } i = 1,\dots, k
\\
b_{n,i-k}
& \text{ for } i = k+1,\dots,m.
\end{cases}
\]
Note that $U_{n+1}$ is a sum of independent random variables and can be written as
\[
U_{n+1}
= \sum_{i=1}^m \frac{\alpha_{n,i}}{\lambda_n} \xi_i.
\]
Let $I$ be a random index independent of all other variables and satisfying that
\[
\PP{I = i}
= \frac{\alpha_{n,i}^2}{\lambda_n^2}
\quad \text{for } i = 1,\dots,m.
\]
By the result of \cite[Lemma 2.8]{ChenEt2010},
the random variable
\[
U_{n+1}^\ast
= U_{n+1} - \frac{\alpha_{n,I}}{\lambda_n} \paren{\xi_{I}^\ast - \xi_{I} }
\]
has the $U_{n+1}$-zero biased distribution.
By taking the dual form of the $L^1$-distance discussed in \cite{Rachev1984}, we can see that
\[
\norm{\mathcal L (U_{n+1}) - \mathcal L (U_{n+1}^\ast) }_1
\; = \; \inf \E \abs{X-Y}
\; \le \; \E \abs{U_{n+1} - U_{n+1}^\ast}
\]
where the infimum is taken over all coupling of $X$ and $Y$ having the joint distribution with $\mathcal L (U_{n+1})$ and its zero bias distribution.
Let $V_1,\dots, V_m$ be independent uniformly distributed random variables on $[0, 1]$.
For $i=1,\dots,m$, let $\xi_i^\ast$ be the zero bias transformation of $\xi_i$.
Let $F_\xi$ and $F_{\xi^\ast}$ be the distribution functions of $\xi$ and $\xi^\ast$, respectively.
Set
\[
(\xi_i,\xi_i^\ast)
= \paren{F_{\xi_i}^{-1} (V_i) , F_{\xi_i^\ast}^{-1} (V_i) }
\quad \text{ for all }
i=1,\dots,m.
\]
By the results of \cite{Rachev1984}, we obtain that
\[
\E \abs{\xi_i - \xi_i^\ast} =
\begin{cases}
\norm{\mathcal L (\widetilde X_n) - \mathcal L (\widetilde X_n^\ast)}_1
& \text{ for } i=1,\dots,k
\\
\norm{\mathcal L (\widetilde Y_n) - \mathcal L (\widetilde Y_n^\ast)}_1
& \text{ for } i=k+1,\dots,m.
\end{cases}
\]
Now, we obtain
\begin{eqnarray*}
&&\hskip-.75cm
\norm{\mathcal L (U_{n+1}) - \mathcal L (U_{n+1}^\ast) }_1
\\
&\le &
\E \abs{U_{n+1} - U_{n+1}^\ast}
\\
&= &
\E \sum_{i=1}^m \frac{|\alpha_{n,i}|}{\lambda_n}
\abs{\xi_i - \xi_i^\ast} \boldsymbol{1} \paren{I = i}
\\
&= &
\sum_{i=1}^m \frac{|\alpha_{n,i}|^3}{\lambda_n^3}
\E \abs{\xi_i - \xi_i^\ast}
\\
&= &
\sum_{i=1}^k \frac{|a_{n,i}|^3}{\lambda_n^3}
\norm{\mathcal L (\widetilde X_n) - \mathcal L (\widetilde X_n^\ast)}_1
+ \sum_{j=1}^\ell \frac{|b_{n,j}|^3}{\lambda_n^3}
\norm{\mathcal L (\widetilde Y_n) - \mathcal L (\widetilde Y_n^\ast)}_1
\\
&= &
\frac{\lambda_{a,n}^3 \, \varphi_{a,n}}{\lambda_n^3}
\norm{\mathcal L (\widetilde X_n) - \mathcal L (\widetilde X_n^\ast)}_1
+ \frac{\lambda_{b,n}^3 \, \varphi_{b,n}}{\lambda_n^3}
\norm{\mathcal L (\widetilde Y_n) - \mathcal L (\widetilde Y_n^\ast)}_1
\\
&\le &
\norm{\mathcal L (\widetilde X_n) - \mathcal L (\widetilde X_n^\ast)}_1
+ \norm{\mathcal L (\widetilde Y_n) - \mathcal L (\widetilde Y_n^\ast)}_1.
\end{eqnarray*}
\endproof
\section{Proof of Main Theorem}
\proof[Proof of Theorem \ref{intro:twoeffect:mainthm}]
By the results of \cite[Theorem 4.1]{ChenEt2010},
we can calculate the bound on $L^1$-distance by using the zero bias transformation as follows
\begin{eqnarray}
\norm{\mathcal L (\widetilde Z_n) - \mathcal L (Z)}_1
& \le &
2 \norm{\mathcal L(\widetilde Z_n) - \mathcal L(\widetilde Z_n^\ast) }_1 .
\label{proof:eq:LZn-LZ}
\end{eqnarray}
Moreover, we can use equivalent forms of the $L^1$-distance found in \cite{Rachev1984} and given by
\[
\norm{\mathcal L(\widetilde Z_n) - \mathcal L(\widetilde Z_n^\ast) }_1
\; =\;
\sup_{h\in \mathfrak{Lip}} \abs{\E h(\widetilde Z_n) - \E h(\widetilde Z_n^\ast)}
\; =\;
\sup_{f\in \mathfrak{F_{ac}}} \abs{\E f'(\widetilde Z_n) - \E f'(\widetilde Z_n^\ast)}
\]
where
\(
\mathfrak{Lip}
= \set{h\colon\R\to\R : \abs{h(x)-h(y)} \le |x-y| \; \text{ for all } x,y\in \R}
\)
\noindent
and
\(
\mathfrak{F_{ac}}
= \set{f\colon\R\to\R : f \text{ is absolutely continuous, } f(0)=f'(0)=0, \, f'\in \mathfrak{Lip}} .
\)
Now, we present some facts about the Stein\rq s method for normal approximation.
For each $f \in\mathcal F$, define $h\colon\R\to\R$ by
\[
h(w) = f'(w) - wf(w).
\]
By the characterization of normal distribution, $\E h(Z)=0$.
Also, we observe that
\[
\abs{h'(w)}
\; = \;
\abs{f''(w) - wf'(w) - f(w)}
\; \le \;
1 + w^2 + \frac{w^2}{2}
\]
and hence
\[
\abs{h(w) - h(u) }
= \abs{\int_{u}^{w} h'(t)\,dt}
\le \abs{w-u} +\frac{1}{2}\abs{w^3-u^3}.
\]
From the definition of zero bias transformation and that $\var{\widetilde Z_{n+1}} = 1$, we have
\begin{eqnarray}
&&\hskip-.75cm
\abs{\E f'(\widetilde Z_{n+1}) - \E f'(\widetilde Z_{n+1}^\ast)}
\notag\\
&=&
\abs{\E f'(\widetilde Z_{n+1}) - \E \widetilde Z_{n+1} f(\widetilde Z_{n+1})}
\notag\\
&=&
\abs{\E h(\widetilde Z_{n+1})}
\notag\\
&\le &
\abs{\E h(\widetilde Z_{n+1}) - \E h(U_{n+1}) }
+ \abs{\E h(U_{n+1}) }
\notag\\
&\le &
\E \abs{\widetilde Z_{n+1} - U_{n+1}}
+ \frac{1}{2} \E \abs{\widetilde Z_{n+1}^3 - U_{n+1}^3}
+ \abs{\E h(U_{n+1}) }
\notag\\
&= &
\beta_n
+ \abs{\E f'(U_{n+1}) - \E f'(U_{n+1}^\ast)}
\notag\\
&\le &
\beta_n
+ \norm{\mathcal L (U_{n+1}) - \mathcal L (U_{n+1}^\ast) }_1
\notag\\
&\le &
\beta_n
+ \norm{\mathcal L (\widetilde X_{n}) - \mathcal L (\widetilde X_{n}^\ast) }_1
+ \norm{\mathcal L (\widetilde Y_{n}) - \mathcal L (\widetilde Y_{n}^\ast) }_1
\label{proof:eq:EfZ-EfZstar}
\end{eqnarray}
where we apply Lemma \ref{auxresult:lem:boundLU} in the last inequality and denote for all $n\ge0$,
\begin{eqnarray}
\beta_n
= \E \abs{\widetilde Z_{n+1} - U_{n+1}}
+ \frac{1}{2} \E \abs{\widetilde Z_{n+1}^3 - U_{n+1}^3} .
\label{proof:eq:betan}
\end{eqnarray}
By (\ref{proof:eq:LZn-LZ}) and taking the supremum of (\ref{proof:eq:EfZ-EfZstar}) over $f \in \mathfrak{F_{ac}}$, we obtain
\begin{eqnarray*}
\norm{\mathcal L (\widetilde Z_{n+1}) - \mathcal L (Z) }_1
&\le &
2 \norm{\mathcal L (\widetilde Z_{n+1}) - \mathcal L (\widetilde Z_{n+1}^\ast)}_1
\\
&\le &
2 \beta_n
+ 2 \norm{\mathcal L (\widetilde X_{n}) - \mathcal L (\widetilde X_{n}^\ast) }_1
+ 2 \norm{\mathcal L (\widetilde Y_{n}) - \mathcal L (\widetilde Y_{n}^\ast) }_1 .
\end{eqnarray*}
Applying the Condition \ref{condition1} and \ref{condition2}
for the models $(X_n,\Delta_n)$ and $(Y_n,\Lambda_n)$ in Theorem \ref{intro:linearrecursion:Goldstein},
there exist positive constants $C_{X,a,\Delta}$, $C_{Y,b\Lambda}$
and $\gamma_{X,a,\Delta}\in(0,1)$, $\gamma_{Y,b,\Lambda}\in(0,1)$ such that for all $n\ge 0$,
\[
\norm{\mathcal L (\widetilde X_{n}) - \mathcal L (\widetilde X_{n}^\ast) }_1
\le C_{X,a,\Delta} \paren{\gamma_{X,a,\Delta} }^n
\]
and
\[
\norm{\mathcal L (\widetilde Y_{n}) - \mathcal L (\widetilde Y_{n}^\ast) }_1
\le C_{Y,b,\Lambda} \paren{\gamma_{Y,b,\Lambda} }^n .
\]
We remain to show that $\beta_n \le C_\beta \gamma_\beta^n$ for some $C_\beta>0$ and $\gamma_\beta\in(0,1)$ and the proof is completed by choosing $C=C_{X,a,\Delta} + C_{Y,b,\Lambda} + C_\beta$ and $\gamma=\max\set{\gamma_{X,a,\Delta},\gamma_{Y,b,\Lambda},\gamma_\beta}$.
\medskip
Recalling the definition of $r_{X,n}$, $r_{Y,n}$ and $U_{X,n}$, $U_{Y,n}$ in Lemma \ref{auxresult:lem:boundrn} and \ref{auxresult:lem:boundLU}, respectively, the linear model of recursions can be written as
\begin{eqnarray*}
\widetilde Z_{n+1}
&=&
\frac{Z_{n+1} - \E Z_{n+1}}{\sigma_{n+1}}
\notag\\
&=&
\frac{X_{n+1} - \E X_{n+1}}{\sigma_{n+1}}
+ \frac{Y_{n+1} - \E Y_{n+1}}{\sigma_{n+1}}
\notag\\
&=&
\frac{\sigma_{X,n}}{\sigma_{n+1}} \set{ \sum_{i=1}^k a_{n,i} \paren{\frac{X_{n,i} - \E X_{n,i} }{\sigma_{X,n} } }
+ \frac{\Delta_n - \E \Delta_n}{\sigma_{X,n}} }
\notag\\
&&
+ \frac{\sigma_{Y,n}}{\sigma_{n+1}} \set{ \sum_{j=1}^\ell b_{n,j} \paren{\frac{Y_{n,j} - \E Y_{n,j} }{\sigma_{Y,n}} }
+ \frac{\Lambda_n - \E \Lambda_n}{\sigma_{Y,n}} }
\notag\\
&=&
r_{X,n} {U}_{X,n+1}
+ r_{Y,n} {U}_{Y,n+1}
+ \Gamma_n
\end{eqnarray*}
where
$\Gamma_n = \Gamma_{X,\Delta,n} + \Gamma_{Y,\Lambda,n}$,
\[
\Gamma_{X,\Delta,n}
= \frac{\sigma_{X,n}}{\sigma_{n+1}} \paren{\frac{\Delta_n - \E \Delta_n}{\sigma_{X,n}} }
\;\; \text{and } \;\;
\Gamma_{Y,\Lambda,n}
= \frac{\sigma_{Y,n}}{\sigma_{n+1}} \paren{\frac{\Delta_n - \E \Delta_n}{\sigma_{Y,n}} } .
\]
Using Conditions \ref{condition1} and \ref{condition2} for the models $(X_n,\Delta_n)$ and $(Y_n,\Lambda_n)$, the result of \cite[Lemma 6]{WehrWoo2001} gives that the limits
\[
\lim_{n\to \infty} \frac{\sigma_{X,n}}{\lambda_{a,0} \dots \lambda_{a,n-1}}
\;\; \text{and} \;\;
\lim_{n\to \infty} \frac{\sigma_{Y,n}}{\lambda_{b,0} \dots \lambda_{b,n-1}}
\]
exist in $(0,1)$,
so we have
\[
\lim_{n\to\infty} \frac{\sigma_{X,n+1}}{\sigma_{X,n}} = \lambda_a
\;\; \text{ and } \;\;
\lim_{n\to\infty} \frac{\sigma_{Y,n+1}}{\sigma_{Y,n}} = \lambda_b .
\]
Therefore, there exist positive constants $C_{\Gamma,X,\Delta,2}$ and $C_{\Gamma,Y,\Lambda,2}$ such that
\[
\E \Gamma_{X,\Delta,n}^2
\le \paren{\frac{\sigma_{X,n}}{\sigma_{X,n+1}}}^2 \frac{\var{\Delta_n}}{\var{X_n}}
\le C_{\Gamma,X,\Delta,2}^2 \paren{\frac{1-\delta_{\Delta,2}}{1-\delta_{X,2}}}^{2n}
\]
\[
\E \Gamma_{Y,\Lambda,n}^2
\le \paren{\frac{\sigma_{Y,n}}{\sigma_{Y,n+1}}}^2 \frac{\var{\Lambda_n}}{\var{Y_n}}
\le C_{\Gamma,Y,\Lambda,2}^2 \paren{\frac{1-\delta_{\Lambda,2}}{1-\delta_{Y,2}}}^{2n}.
\]
Moreover, there exist positive constants $C_{\Gamma,X,\Delta,4}$ and $C_{\Gamma,Y,\Lambda,4}$ such that
\[
\E \Gamma_{X,\Delta,n}^4
\le \paren{\frac{\sigma_{X,n}}{\sigma_{X,n+1}}}^4
\E \paren{\frac{\Delta_n - \E \Delta_n}{\sigma_{X,n}} }^4
\le C_{\Gamma,X,\Delta,4}^4 \paren{\frac{1-\delta_{\Delta,4}}{1-\delta_{X,2}}}^{4n}
\]
\[
\E \Gamma_{Y,\Lambda,n}^4
\le \paren{\frac{\sigma_{Y,n}}{\sigma_{Y,n+1}}}^4
\E \paren{\frac{\Lambda_n - \E \Lambda_n}{\sigma_{Y,n}} }^4
\le C_{\Gamma,Y,\Lambda,4}^4 \paren{\frac{1-\delta_{\Lambda,4}}{1-\delta_{Y,2}}}^{4n} .
\]
By independence for $X_{n,i}$\rq s and $Y_{n,j}$\rq s, there exist positive constants $C_{U,X}$ and $C_{U,Y}$ such that
\begin{eqnarray*}
\E U_{X,n+1}^2
&= &
\frac{\lambda_{a,n}^2}{\lambda_n^2}
\E \paren{\frac{X_n - \E X_n}{\sigma_{X,n}} }^2
\le 1
\\
\E U_{Y,n+1}^2
&= &
\frac{\lambda_{b,n}^2}{\lambda_n^2}
\E \paren{\frac{Y_n - \E Y_n}{\sigma_{Y,n}} }^2
\le 1
\\
\E U_{X,n+1}^4
&\le &
8 \sum_{i=1}^k \frac{a_{n,i}^4}{\lambda_{n}^4}
\E \paren{\frac{X_n - \E X_n}{\sigma_{X,n}} }^4
\le C_{U,X}^4 \paren{\frac{1+\delta_{X,4}}{1-\delta_{X,2}}}^{4n}
\\
\E U_{Y,n+1}^4
&\le &
8 \sum_{j=1}^\ell \frac{b_{n,j}^4}{\lambda_{n}^4}
\E \paren{\frac{Y_n - \E Y_n}{\sigma_{Y,n}} }^4
\le C_{U,Y}^4 \paren{\frac{1+\delta_{Y,4}}{1-\delta_{Y,2}}}^{4n} .
\end{eqnarray*}
From Lemma \ref{auxresult:lem:boundrn} and Condition \ref{condition1} and \ref{condition2},
the following results will be often used for all $n\ge0$ and $p=1,2,3$,
\begin{eqnarray}
\abs{r_{X,n}^p -1}
\le C_{r,p} \paren{\phi_{X,\Delta,2}^n + \phi_{Y,\Lambda,2}^n}
\label{proof:eq:rXnp-1}
\\
\abs{r_{Y,n}^p -1}
\le C_{r,p} \paren{\phi_{X,\Delta,2}^n + \phi_{Y,\Lambda,2}^n} .
\label{proof:eq:rYnp-1}
\end{eqnarray}
Now, considering the first term of $\beta_n$ in (\ref{proof:eq:betan}),
\begin{eqnarray*}
&&\hskip-.75cm
\E \abs{\widetilde Z_{n+1} - U_{n+1}}
\\
&=&
\E \abs{\paren{r_{X,n}-1} U_{X,n+1}
+ \paren{r_{Y,n} - 1} U_{Y,n+1}
+ \Gamma_{X,\Delta,n}
+ \Gamma_{Y,\Lambda,n} }
\\
&\le &
\abs{r_{X,n} - 1} \sqrt{ \E U_{X,n+1}^2 }
+ \abs{r_{Y,n} - 1} \sqrt{ \E U_{Y,n+1}^2 }
+ \sqrt{\E \Gamma_{X,\Delta,n}^2}
+ \sqrt{\E \Gamma_{Y,\Lambda,n}^2}
\\
&\le &
2 C_{r,1} \paren{\phi_{X,\Delta,2}^n + \phi_{Y,\Lambda,2}^n}
+ C_{\Gamma,X,\Delta,2} \paren{\frac{1-\delta_{\Delta,2}}{1-\delta_{X,2}} }^n
+ C_{\Gamma,Y,\Lambda,2} \paren{\frac{1-\delta_{\Lambda,2}}{1-\delta_{Y,2}} }^n
\\
&\le &
C_0 \paren{\phi_{X,\Delta,2}^n + \phi_{Y,\Lambda,2}^n} .
\end{eqnarray*}
\noindent
For the second term of $\beta_n$,
\begin{eqnarray*}
&& \hskip-.75cm
\E \abs{\widetilde Z_{n+1}^3 - U_{n+1}^3}
\\
&= &
\E \abs{ \paren{ r_{X,n} U_{X,n+1}
+ r_{Y,n} U_{Y,n+1}
+ \Gamma_n }^3
- U_{n+1}^3 }
\\
&= &
\E \left| \paren{ r_{X,n} U_{X,n+1}
+ r_{Y,n} U_{Y,n+1} }^3
+ 3 \paren{ r_{X,n} U_{X,n+1} + r_{Y,n} U_{Y,n+1} }^2 \Gamma_n \right.
\\
&& \quad
\left. \phantom{\bigg|}
+ 3 \paren{ r_{X,n} U_{X,n+1}
+ r_{Y,n} U_{Y,n+1} } \Gamma_n^2
+ \Gamma_n^3
- U_{n+1}^3 \right|
\\
&\le &
\E \abs{ \paren{ r_{X,n} U_{X,n+1}
+ r_{Y,n} U_{Y,n+1} }^3 - U_{n+1}^3 }
+ 3 \E \abs{\paren{ r_{X,n} U_{X,n+1}
+ r_{Y,n} U_{Y,n+1} }^2 \Gamma_n }
\\
&&
+ 3 \E \abs{\paren{ r_{X,n} U_{X,n+1}
+ r_{Y,n} U_{Y,n+1} } \Gamma_n^2 }
+ \E \abs{\Gamma_n}^3
\\
& := &
A_1 + A_2 + A_3 +A_4 .
\end{eqnarray*}
Notice that
\begin{eqnarray*}
A_1
&= &
\E \abs{ \paren{ r_{X,n} U_{X,n+1}
+ r_{Y,n} U_{Y,n+1} }^3 - \paren{U_{X,n+1}+U_{Y,n+1}}^3 }
\\
&= &
\E \left| \paren{r_{X,n}^3 - 1} U_{X,n+1}^3
+ 3\paren{r_{X,n}^2r_{Y,n}-1} U_{X,n+1}^2 U_{Y,n+1}
\right.
\\
&& \quad \;\;
\left.
+ 3\paren{r_{X,n}r_{Y,n}^2-1} U_{X,n+1} U_{Y,n+1}^2
+ \paren{r_{Y,n}^3 - 1} U_{Y,n+1}^3 \right|
\\
&\le &
\E \abs{\paren{r_{X,n}^3 - 1} U_{X,n+1}^3 }
+ 3 \E \abs{\paren{r_{X,n}^2r_{Y,n} - r_{Y,n} + r_{Y,n} -1} U_{X,n+1}^2 U_{Y,n+1} }
\\
&&
+ 3 \E \abs{\paren{r_{X,n} r_{Y,n}^2 - r_{X,n} + r_{X,n} -1} U_{X,n+1} U_{Y,n+1}^2 }
+ \E \abs{\paren{r_{Y,n}^3 - 1} U_{Y,n+1}^3}
\\
&\le &
\E \abs{\paren{r_{X,n}^3 - 1} U_{X,n+1}^3 }
+ \E \abs{\paren{r_{Y,n}^3 - 1} U_{Y,n+1}^3}
\\
&&
+ 3 \E \abs{\paren{r_{X,n}^2-1} r_{Y,n} U_{X,n+1}^2 U_{Y,n+1} }
+ 3 \E \abs{\paren{r_{Y,n}-1} U_{X,n+1}^2 U_{Y,n+1} }
\\
&&
+ 3 \E \abs{\paren{r_{Y,n}^2-1} r_{X,n} U_{X,n+1} U_{Y,n+1}^2 }
+ 3 \E \abs{\paren{r_{X,n}-1} U_{X,n+1} U_{Y,n+1}^2 }
\\
&\le &
\abs{r_{X,n}^3-1} \paren{\E U_{X,n}^4}^{3/4}
+ \abs{r_{Y,n}^3-1} \paren{\E U_{Y,n}^4}^{3/4}
\\
&&
+ 3 \abs{r_{X,n}^2-1} r_{Y,n} \E U_{X,n}^2 \sqrt{\E U_{Y,n}^2}
+ 3 r_{X,n} \abs{r_{Y,n}^2-1} \sqrt{\E U_{X,n}^2} \E U_{Y,n}^2
\\
&&
+ 3 \abs{r_{Y,n}-1} \E U_{X,n}^2 \sqrt{\E U_{Y,n}^2}
+ 3 \abs{r_{X,n}-1} \sqrt{\E U_{X,n}^2} \E U_{Y,n}^2
\\
&\le &
8^{3/4} C_{r,3} C_{X,4}^3 \set{ \paren{\frac{\paren{1-\delta_{\Delta,2}}\paren{1+\delta_{X,4}}^3}{\paren{1-\delta_{X,2}}^4} }^n
+ \paren{\frac{\paren{1-\delta_{\Lambda,2}}\paren{1+\delta_{X,4}}^3}{\paren{1-\delta_{Y,2}}\paren{1-\delta_{X,2}}^3} }^n }
\\
&&
+ 8^{3/4} C_{r,3} C_{Y,4}^3 \set{ \paren{\frac{\paren{1-\delta_{\Delta,2}}\paren{1+\delta_{Y,4}}^3}{\paren{1-\delta_{X,2}}\paren{1-\delta_{Y,2}}^3} }^n
+ \paren{\frac{\paren{1-\delta_{\Lambda,2}}\paren{1+\delta_{Y,4}}^3}{\paren{1-\delta_{Y,2}}^4} }^n }
\\
&&
+ 6 C_{r,2} \paren{1+2C_{r,1}}
\paren{\phi_{X,\Delta,2}^n + \phi_{Y,\Lambda,2}^n}
+ 6 C_{r,1} \paren{\phi_{X,\Delta,2}^n + \phi_{Y,\Lambda,2}^n}
\\
&\le &
C_1 \paren{\phi_{X,\Delta,2}^n + \phi_{Y,\Lambda,2}^n
+ \psi_{X,Y,\Lambda}^n + \psi_{Y,X,\Delta}^n }.
\end{eqnarray*}
As a special case of (\ref{proof:eq:rXnp-1}) and (\ref{proof:eq:rYnp-1}) when $p=1$, we can see that for all $n\ge0$
\[
r_{X,n}
\le 1 + C_{r,1} \paren{\phi_{X,\Delta,2}^n + \phi_{Y,\Lambda,2}^n}
\le 1 + 2 C_{r,1}
\]
\[
r_{Y,n}
\le 1 + C_{r,1} \paren{\phi_{X,\Delta,2}^n + \phi_{Y,\Lambda,2}^n}
\le 1 + 2 C_{r,1}.
\]
So, we have that
\begin{eqnarray*}
A_2
&= &
3 \E \abs{\paren{ r_{X,n} U_{X,n+1}
+ r_{Y,n} U_{Y,n+1} }^2 \paren{\Gamma_{X,\Delta,n}+\Gamma_{Y,\Lambda,n} } }
\\
&\le &
6 \E \abs{\paren{ r_{X,n}^2 U_{X,n+1}^2
+ r_{Y,n}^2 U_{Y,n+1}^2 } \paren{\Gamma_{X,\Delta,n}+\Gamma_{Y,\Lambda,n} } }
\\
&\le &
6 r_{X,n}^2 \sqrt{\E U_{X,n+1}^4 \E \Gamma_{X,\Delta,n}^2 }
+ 6 r_{X,n}^2 \E U_{X,n+1}^2 \sqrt{\E \Gamma_{Y,\Lambda,n}^2}
\\
&&
+ 6 r_{Y,n}^2 \E U_{Y,n+1}^2 \sqrt{\E \Gamma_{X,\Delta,n}^2}
+ 6 r_{Y,n}^2 \sqrt{\E U_{Y,n+1}^4 \E \Gamma_{Y,\Lambda,n}^2 }
\\
&\le &
12\sqrt 2 \paren{1+C_{r,1}}^2 C_{U,X}^2 C_{\Gamma,X,\Delta,2} \paren{\frac{\paren{1-\delta_{\Delta,2}}\paren{1+\delta_{X,4}}^3}{\paren{1-\delta_{X,2}}^4} }^n
\\
&&
+ 6 \paren{1+C_{r,1}}^2 C_{\Gamma,X,\Delta,2} \paren{\frac{1-\delta_{\Delta,2}}{1-\delta_{X,2}} }^n
+ 6 \paren{1+C_{r,1}}^2 C_{\Gamma,Y,\Lambda,2} \paren{\frac{1-\delta_{\Lambda,2}}{1-\delta_{Y,2}} }^n
\\
&&
+ 12\sqrt 2 \paren{1+C_{r,1}}^2 C_{U,Y}^2 C_{\Gamma,Y,\Lambda,2} \paren{\frac{\paren{1-\delta_{\Lambda,2}}\paren{1+\delta_{Y,4}}^3}{\paren{1-\delta_{Y,2}}^4} }^n
\\
&\le &
C_2 \paren{\phi_{X,\Delta,2}^n + \phi_{Y,\Lambda,2}^n}
\end{eqnarray*}
and that
\begin{eqnarray*}
A_3
&= &
3 \E \abs{\paren{ r_{X,n} U_{X,n+1}
+ r_{Y,n} U_{Y,n+1} } \paren{\Gamma_{X,\Delta,n}+\Gamma_{Y,\Lambda,n}}^2 }
\\
&\le &
6 \E \abs{\paren{ r_{X,n} U_{X,n+1} + r_{Y,n} U_{Y,n+1} }
\paren{\Gamma_{X,\Delta,n}^2+\Gamma_{Y,\Lambda,n}^2 } }
\\
&\le &
6 r_{X,n} \sqrt{ \E U_{X,n+1}^2 \E \Gamma_{X,\Delta,n}^4 }
+ 6 r_{X,n} \sqrt{\E U_{X,n+1}^2} \E \Gamma_{Y,\Lambda,n}^2
\\
&&
+ 6 r_{Y,n} \sqrt{\E U_{Y,n+1}^2} \E \Gamma_{X,\Delta,n}^2
+ 6 r_{Y,n} \sqrt{ \E U_{Y,n+1}^2 \E \Gamma_{Y,\Lambda,n}^4}
\\
&\le &
6 \paren{1+C_{r,1}} \paren{ C_{\Gamma,X,\Delta,4}^2 + C_{\Gamma,X,\Delta,2}^2 }
\paren{\frac{1-\delta_{\Delta,2}}{1-\delta_{X,2}} }^{2n}
\\
&&
+ 6 \paren{1+C_{r,1}} \paren{ C_{\Gamma,Y,\Lambda,2}^2 + C_{\Gamma,Y,\Lambda,4}^2 }
\paren{\frac{1-\delta_{\Lambda,2}}{1-\delta_{Y,2}} }^{2n}
\\
&\le &
C_3 \paren{\phi_{X,\Delta,2}^{2n} + \phi_{Y,\Lambda,2}^{2n}} .
\end{eqnarray*}
Lastly,
\begin{eqnarray*}
A_4
&\le &
\paren{\E \Gamma_n^4}^{3/4}
\\
&\le &
8^{3/4} \paren{\E \Gamma_{X,\Delta,n}^4
+ \E \Gamma_{Y,\Lambda,n}^4 }^{3/4}
\\
&\le &
8^{3/4} \set{ C_{\Gamma,X,\Delta,4}^4
\paren{ \frac{1-\delta_{\Delta,4}}{1-\delta_{X,2}} }^{4n}
+ C_{\Gamma,Y,\Lambda,4}^4
\paren{ \frac{1-\delta_{\Lambda,4}}{1-\delta_{Y,2}} }^{4n} }^{3/4}
\\
&\le &
C_4 \paren{ \phi_{X,\Delta,4}^{3n/2}
+ \phi_{Y,\Lambda,4}^{3n/2} }.
\end{eqnarray*}
Setting $\gamma_\beta = \max\set{\phi_{X,\Delta,2},\phi_{Y,\Lambda,2},\phi_{X,\Delta,4}^{3/2},\phi_{Y,\Lambda,4}^{3/2},\psi_{X,Y,\Lambda},\phi_{Y,X,\Delta} } \in(0,1)$
\noindent
and $C_\beta = 2C_0+4C_1+2C_2+2C_3+2C_4$,
we obtain the claim for $\beta_n$.
\endproof
\newpage
| 48,191
|
I am closing out Top Shelf month with an e-book I bought from Comixology for my Kindle Fire. Yes, they sell digital, too!
The facile way to describe this book would be to call it Stephen King's "The Body" meets District 9 set in Australia (I almost wrote Alien Nation, but that just means I am OLD). But Blue.
This deceptively complex book disarms the reader with a very cartoonish style, and writer/artist Pat Grant demonstrations some great storytelling skills. It is an outstanding debut work, one that smacks of technical and artistic expertise. That said, this Melbourne-based comics creator is a relative newcomer and has worked on a variety of small projects listed here. He also is interested in surf culture and has an excellent essay about the history of Australian surfing and underground comix as his influences in the back pages of Blue.
Reviews I have read online have been very complimentary about this freshman effort. Andy Khouri called it "an uncommonly sophisticated look at prejudice and localism." Bill Sherman wrote, "Sharply unsentimental and often darkly funny, it makes a powerful debut for artist, writer and, zinemaker Grant—a must read for anyone invested in following literary comics." Slate's Dan Kois called it "thoughful and complicated" and added that it "really stuck with me after I read it."
A preview and much more is available here from the publisher.
The whole book is also available online here, though I think it is still well worth its purchase prices. Support an author who is starting out!
Also, if I have not a great job selling you on the book - or if you are already onboard - you should check out Pat Grant's SHAMELESS PUBLICITY DOCUMENT!
| 20,737
|
How do we conceptualise our psychological, existential and political condition?
Anxiety and depression are on the rise in Australia and across the globe; digital media has created a pandemic of loneliness and disconnection; ideological extremism is widening our divisions and threatening our democracies – and all the while, the wellness industry is spinning everything from mindfulness to minimalism into big business.
Where does this leave us?
Griffith Review 72: States of Mind will explore the parameters of our cognitive landscapes and how far they might take us.
RRP: 27.99 / Publication Date: Apr 2021 / ISBN: 978-1-922212-59-7 / Extent: 264pp / Formats: Paperback (234 x 153mm), eBook
| 346,732
|
These are the first two pieces in my smaller set of landscape series. They are 10X10 fiber art mounted on a 12X12 canvas. The first one is a picture from my own imagination. It reminds me of my own private retreat; somewhere I would like to go to just enjoy nature's beauty. It's almost like a hideaway. I have been stuggling for a name for this piece and as of yet have not found one. Any suggestions would be welcome. The second piece was composed from a picture that a friend took while on vacation. It is reminiscent of a peaceful oasis garden and hence I have named it "The Oasis." Funny how some artwork seem to be born with names while others are harder to convey with titles and more easily expressed with thoughts and images. I would like to put forth a few more additions to this series and see where it takes me. At the moment, I am thinking I like the images born of my own imagination better than those set before me. What do they say? The images concocted from your mind are always better than the images you see before you? I wonder if that is really true.
| 289,872
|
TITLE: Eigenvectors and $x_{n}$
QUESTION [0 upvotes]: I've got the following problem in linear algebra under chapter eigenvector and eigenvalue.
Determine $x_{n}$ if
$$\left\{\begin{matrix}
x_{1}=3\\
x_{2}=0\\
x_{n}=x_{n-1}+2x_{n-2},\ n=3,4...
\end{matrix}\right.$$
The answer is:
$$x_{n}=2(-1)^{n-1}+2^{n-1}$$
I do not understand the problem. How should I handle this?
REPLY [1 votes]: The equation is linear, and this is why it is relevant in the context of linear algebra: if you find several independent solutions, their linear combination will also be a solution.
Now, to get some inspiration about the possible shape of a solution, let us consider the simpler
$$z_0=1,z_n=2z_{n-1}.$$
The first terms are $z_1=2,z_2=4,z_3=8,\cdots$ and the obvious pattern $z_n=2^n$ appears. Hence, for the original problem we will try
$$x_n=r^n.$$
Plugging in the recurrence,
$$r^n=r^{n-1}+2r^{n-2},$$ or after simplification
$$r^2=r+2.$$
Solving this quadratic equation, we have established that $2^n$ and $(-1)^n$ are two solutions, and any linear combination $$a\,2^n+b\,(-1)^n$$ is also a solution. You can determine $a,b$ by using the extra information available.
Theory shows that a linear recurrence of order $k$ (largest difference between the term indexes, here $k=2$) has exactly $k$ independent solutions, which form a basis for the general solution.
| 29,587
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Fifteen Reactions to the Gay Cat Marriage
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13. If they can find love, can Lonesome George? The incredibly sad and old Galapagos tortoise?
14. Lonesome George found love!
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\begin{document}
\title{\textbf{The homotopy theory of
dg-categories and derived Morita theory}}
\bigskip
\bigskip
\author{\bigskip\\
Bertrand To\"en \\
\small{Laboratoire Emile Picard UMR CNRS 5580} \\
\small{Universit\'{e} Paul Sabatier, Bat 1R2} \\
\small{Toulouse Cedex 9}
\small{France}}
\date{September 2006}
\maketitle
\begin{abstract}
The main purpose of this work is to study the homotopy theory
of dg-categories up to quasi-equivalences.
Our main result is a description
of the mapping spaces between two dg-categories $C$ and $D$ in terms
of the nerve of a certain category of $(C,D)$-bimodules. We also prove that the
homotopy category $Ho(dg-Cat)$ possesses
internal Hom's relative to the (derived) tensor product
of dg-categories. We use these
two results in order to prove a derived version of Morita theory,
describing the morphisms between
dg-categories of modules over two dg-categories $C$ and $D$ as
the dg-category of $(C,D)$-bi-modules. Finally,
we give three applications of our results. The first one expresses Hochschild
cohomology as endomorphisms of the identity functor, as well as
higher homotopy groups of the
\emph{classifying space of dg-categories} (i.e. the nerve of
the category of dg-categories and quasi-equivalences between them).
The second application is the existence of a good theory of localization for
dg-categories, defined in terms of a natural universal property. Our last application
states that the dg-category of (continuous) morphisms between the dg-categories of
quasi-coherent (resp. perfect) complexes on two schemes (resp. smooth and proper schemes) is
quasi-equivalent to the
dg-category of quasi-coherent (resp. perfect) complexes on their product.
\end{abstract}
\medskip
\tableofcontents
\bigskip
\section{Introduction}
Let $A$ and $B$ be two associative algebras (over some field $k$), and
$A-Mod$ and $B-Mod$ be their categories of right modules. It is well known that
any functor $A-Mod \longrightarrow B-Mod$ which commutes with colimits is of the form
$$\begin{array}{ccc}
A-Mod & \longrightarrow & B-Mod \\
M & \mapsto & M \otimes_{A} P,
\end{array}$$
for some $A^{op}\otimes B$-module $P$. More generally, there exists a natural equivalence
of categories between $(A^{op}\otimes B)-Mod$ and the category of
all colimit preserving functors $A-Mod \longrightarrow B-Mod$. This is known
as Morita theory for rings.
Now, let $A$ and $B$ be two
associative dg-algebras (say over some field $k$), together with their triangulated
derived category of right (unbounded) dg-modules $D(A)$ and $D(B)$.
A natural way of constructing triangulated functors
from $D(A)$ to $D(B)$ is by choosing $P$ a left
$A^{op}\otimes B$-dg-module, and considering the derived functor
$$\begin{array}{ccc}
D(A) & \longrightarrow & D(B) \\
M & \mapsto & M\otimes_{A}^{\mathbb{L}}P.
\end{array}$$
However, it is well known that there exist
triangulated functors $D(A) \longrightarrow D(B)$ that do not
arise from a $A^{op}\otimes B$-dg-module (see e.g. \cite[2.5, 6.8]{ds}). The situation is even
worse, as the functor
$$D(A^{op}\otimes B) \longrightarrow Hom_{tr}(D(A),D(B))$$
is not expected to be reasonable in any sense
as the right hand side simply does not possess a natural triangulated structure.
Therefore,
triangulated categories do not
appear as the right object to consider if one is looking for
an extension of Morita theory to dg-algebras.
The main purpose of this work is to provide
a solution to this problem by replacing
the notion of triangulated categories by the notion of
dg-categories. \\
\begin{center} \textit{dg-Categories} \end{center}
A dg-category is a category which is enriched over the monoidal category of
complexes over some base ring $k$. It consists of a set of objects
together with complexes $C(x,y)$ for two any objects $x$ and $y$, and
composition morphisms $C(x,y)\otimes C(y,z) \longrightarrow C(x,z)$ (assumed to be
associative and unital). As linear categories can be
understood as \emph{rings with several objects}, dg-categories can be thought as
\emph{dg-algebras with several objects}, the precise statement being that
dg-algebras are exactly dg-categories having a unique object.
From a dg-category
$C$ one can form a genuine category $[C]$ by keeping the same set of objects
and defining the set of morphisms between $x$ and $y$ in $[C]$ to be
$H^{0}(C(x,y))$. In turns out that a lot of triangulated categories
appearing in geometric contexts are of the form $[C]$ for some
natural dg-category $C$ (this is for example the case for
the derived category of a reasonable abelian category, as well as for the derived
category of dg-modules over some dg-algebra).
The new feature of
dg-categories is the notion of \emph{quasi-equivalences}, a mixture between
quasi-isomorphisms and categorical equivalences and which turns out to be the
right notion of equivalences between
dg-categories. Precisely, a morphism $f : C \longrightarrow D$ between two
dg-categories is a quasi-equivalence if it satisfies the following two conditions
\begin{itemize}
\item For any objects $x$ and $y$ in $C$ the induced morphism
$C(x,y) \longrightarrow D(f(x),f(y))$ is a quasi-isomorphism.
\item The induced functor $[C] \longrightarrow [D]$
is an equivalence of categories.
\end{itemize}
In practice we are only interested in dg-categories up to quasi-equivalences, and the main
object of study is thus the localized category $Ho(dg-Cat)$ of dg-categories
with respect to quasi-equivalences, or
better its refined simplicial version $L(dg-Cat)$
of Dwyer and Kan (see \cite{dk2}). The main purpose of this paper
is to study the simplicial category $L(dg-Cat)$,
and to show that a derived version
of Morita theory can be extracted from it. The key tool for us will be the existence
of a model structure on the category of dg-categories (see \cite{tab}), which will
allow us to use standard constructions of homotopical algebra (mapping spaces,
homotopy limits and colimits \dots) in order
to describe $L(dg-Cat)$. \\
\begin{center} \textit{Statement of the results} \end{center}
Let $C$ and $D$ be two dg-categories, considered as objects in $L(dg-Cat)$.
A first invariant is the homotopy type of the simplicial set of morphism
$L(dg-Cat)(C,D)$, which is well known to be weakly equivalent to the mapping space
$Map(C,D)$ computed in the model category of dg-categories (see \cite{dk,dk2}).
From $C$ and $D$ one can
form the tensor product $C\otimes D^{op}$ (suitably derived if necessary),
as well as the category $(C\otimes D^{op})-Mod$ of $C\otimes D^{op}$-modules
(these are enriched functors from $C\otimes D^{op}$ to the category of complexes).
There exists an obvious notion of quasi-isomorphism between $C\otimes D^{op}$-modules,
and thus a homotopy category $Ho((C\otimes D^{op})-Mod)$. Finally, inside
$Ho((C\otimes D^{op})-Mod)$ is a certain full sub-category of
\emph{right quasi-representable objects}, consisting of modules $F$ such that
for any $x\in C$ the induced $D^{op}$-module $F(x,-)$ is quasi-isomorphic to a
$D^{op}$-module of the form $D(-,y)$ for some $y\in D$ (see \S 3 for details).
One can then consider the category $\mathcal{F}(C,D)$ consisting of all
right quasi-representable $C\otimes D^{op}$-modules and quasi-isomorphisms between them.
The main result of this work is the following.
\begin{thm}\label{ti1}{(See Thm. \ref{t1})}
There exists a natural weak equivalence of simplicial sets
$$Map(C,D)\simeq N(\mathcal{F}(C,D))$$
where $N(\mathcal{F}(C,D))$ is the nerve of the
category $\mathcal{F}(C,D)$.
\end{thm}
We would like to mention that this theorem does not simply follow from the
existence of the model structure on dg-categories. Indeed, this model structure
is not simplicially enriched (even in some weak sense, as
the model category of complexes is for example), and there is no obvious
manner to compute the mapping spaces $Map(C,D)$. \\
As an important corollary one gets the following result.
\begin{cor}\label{cti1}
\begin{enumerate}
\item There is a natural bijection between $[C,D]$, the set of morphisms
between $C$ and $D$ in $Ho(dg-Cat)$, and
the isomorphism classes of right quasi-representable objects in
$Ho((C\otimes D^{op})-Mod)$.
\item For two morphism $f,g : C\longrightarrow D$ there is a natural weak equivalence
$$\Omega_{f,g}Map(C,D) \simeq Map(\phi(f),\phi(g))$$
where $Map(\phi(f),\phi(g))$ is the mapping space between the $C\otimes D^{op}$-modules corresponding to $f$ and $g$.
\end{enumerate}
\end{cor}
The tensor product of dg-categories, suitably derived, induces a
symmetric monoidal structure on $Ho(dg-Cat)$. Our second main result states
that this monoidal structure is closed.
\begin{thm}\label{ti2}{(See Thm. \ref{t2})}
The symmetric monoidal category $Ho(dg-Cat)$ is closed. More precisely,
for any three dg-categories $A$, $B$ and $C$, there exists
a dg-category $\mathbb{R}\underline{Hom}(B,C)$ and
functorial isomorphisms in $Ho(SSet)$
$$Map(A,\mathbb{R}\underline{Hom}(B,C))\simeq Map(A\otimes^{\mathbb{L}} B,C).$$
Furthermore, $\mathbb{R}\underline{Hom}(B,C)$ is naturally
isomorphic in $Ho(dg-Cat)$ to the dg-category of
cofibrant right quasi-representable $B\otimes C^{op}$-modules.
\end{thm}
Finally, Morita theory can be expressed in the following terms. Let
us use the notation
$\widehat{C}:=\mathbb{R}\underline{Hom}(C^{op},Int(C(k)))$, where
$Int(C(k))$ is the dg-category of cofibrant complexes. Note that by our theorem
\ref{ti2} $\widehat{C}$ is also quasi-equivalent to the dg-category of
cofibrant $C^{op}$-modules.
\begin{thm}\label{ti3}{(See Thm. \ref{t3} and Cor. \ref{ct3})}
There exists a natural isomorphism in $Ho(dg-Cat)$
$$\mathbb{R}\underline{Hom}_{c}(\widehat{C},\widehat{D})\simeq
\widehat{C^{op}\otimes^{\mathbb{L}} D},$$
where $\mathbb{R}\underline{Hom}_{c}(\widehat{C},\widehat{D})$ is the full sub-dg-category
of $\mathbb{R}\underline{Hom}(\widehat{C},\widehat{D})$ consisting of morphisms commuting with infinite
direct sums.
\end{thm}
As a corollary we obtain the following result.
\begin{cor}\label{cti3}
There is natural bijection between
$[\widehat{C},\widehat{D}]_{c}$, the sub-set of $[\widehat{C},\widehat{D}]$ consisting
of direct sums preserving morphisms, and
the isomorphism classes in $Ho((C\otimes^{\mathbb{L}} D^{op})-Mod)$.
\end{cor}
\bigskip
\begin{center} \textit{Three applications} \end{center}
\smallskip
We will give three applications of our general results. The first
one is a description of the homotopy groups
of the classifying space of dg-categories $|dg-Cat|$, defined
as the nerve of the category of quasi-equivalences between dg-categories.
For this, recall that the Hochschild cohomology of a dg-category
$C$ is defined by
$$\mathbb{HH}^{i}:=[C,C[i]]_{C\otimes^{\mathbb{L}}C^{op}-Mod},$$
where $C$ is the $C\otimes^{\mathbb{L}}C^{op}$-module
sending $(x,y)\in C\otimes C^{op}$ to $C(y,x)$.
\begin{cor}{(See Cor. \ref{chh'}, \ref{crpic})}
\begin{enumerate}
For any dg-category $C$ one has
\item
$$\mathbb{HH}^{*}(C)\simeq
H^{*}(\mathbb{R}\underline{Hom}(C,C)(Id,Id)).$$
\item
$$\pi_{i}(|dg-Cat|,C)\simeq \mathbb{HH}^{2-i}(C) \qquad\forall i>2.$$
\item $$\pi_{2}((|dg-Cat|,C)\simeq Aut_{Ho(C\otimes C^{op}-Mod)}(C)\simeq
\mathbb{HH}^{0}(C)^{*}$$
\item $$\pi_{1}(|dg-Cat|,\widehat{BA})\simeq RPic(A),$$
where $A$ is a dg-algebra, $BA$ the dg-category
with a unique object and $A$ as its endomorphism, and
where $RPic(A)$ is the derived Picard group
of $A$ as defined for example in \cite{rz,ke2,yek}.
\end{enumerate}
\end{cor}
Our second application is the existence of localization for
dg-categories. For this, let $C$ be any
dg-category and $S$ be a set of morphisms in $[C]$.
For any dg-category $D$ we define
$Map_{S}(C,D)$ as the sub-simplicial set of
$Map(C,D)$ consisting of morphisms sending
$S$ to isomorphisms in $[D]$.
\begin{cor}{(See Cor. \ref{cloc})}
The $Ho(SSet_{\mathbb{U}})$-enriched functor
$$Map_{S}(C,-) : Ho(dg-Cat_{\mathbb{U}}) \longrightarrow
Ho(SSet_{\mathbb{U}})$$
is co-represented by an object $L_{S}(C) \in Ho(dg-Cat_{\mathbb{U}})$.
\end{cor}
Our final application will provide a proof of the
following fact, which can be considered as a possible answer to a folklore question
to know whether or not all triangulated functors between derived categories
of varieties are induced by some object in the derived category of their product
(see e.g. \cite{o} where this is proved for
triangulated equivalences between derived categories of
smooth projective varieties).
\begin{cor}\label{cti3'}{(See Thm. \ref{tfour})}
Let $X$ and $Y$ be two quasi-compact and separated $k$-schemes, one of them
being flat over $Spec\, k$, and let $L_{qcoh}(X)$ and
$L_{qcoh}(Y)$ their dg-categories of
(fibrant) quasi-coherent complexes. Then, one has a natural
isomorphism in $Ho(dg-Cat)$
$$L_{qcoh}(X\times_{k} Y)\simeq \mathbb{R}\underline{Hom}_{c}(L_{qcoh}(X),L_{qcoh}(Y)).$$
In particular, there is a natural bijection between
$[L_{qcoh}(X),L_{qcoh}(Y)]_{c}$ and set of isomorphism classes of
objects in the category $D_{qcoh}(X\times Y)$.
If furthermore $X$ and $Y$ are smooth and proper over $Spec\, k$,
then one has
a natural
isomorphism in $Ho(dg-Cat)$
$$L_{parf}(X\times_{k} Y)\simeq
\mathbb{R}\underline{Hom}(L_{parf}(X),L_{parf}(Y)),$$
where $L_{parf}(X)$ (resp. $L_{parf}(Y)$) is the full sub-dg-category
of $L_{qcoh}(X)$ (resp. of $L_{qcoh}(Y)$) consisting of
perfect complexes.
\end{cor}
\bigskip
\begin{center} \textit{Related works} \end{center}
The fact that dg-categories provide natural and
interesting enhancement of derived categories
has been recognized for some times,
and in particular in \cite{bk}. They have been
used more recently in \cite{bll} in which a very special
case of our theorem \ref{tfour} is proved
for smooth projective varieties.
The present work
follows the same philosophy that dg-categories are
the \emph{true derived categories} (though I do not like
very much this expression).
Derived equivalences between (non-dg) algebras have been
heavily studied by J. Rickard (see e.g. \cite{ri1,ri2}),
and the results obtained have been commonly called
\emph{Morita theory for derived categories}.
The present work can be considered as
a continuation of this fundamental work, though our techniques
and our purposes are rather different. Indeed, in our mind
the word \emph{derived} appearing in our
title does not refer to generalizing Morita theory from module categories to
derived categories, but to generalizing
Morita theory from algebras to dg-algebras.
Morita theory for dg-algebras and ring spectra
has been approached recently using
model category techniques in \cite{ss}. The results
obtained this way state in particular that
two ring spectra have Quillen equivalent
model categories of modules if and only if
a certain bi-module exists. This approach, however, does
not say anything about \emph{higher homotopies}, in the
sense that it seems hard (or even impossible) to compare
the whole model category of bi-modules
with the category of Quillen equivalences, already simply
because a model category of Quillen functors does
not seem to exist in any reasonable sense. This is another
incarnation of the principle that
model category theory does not work very well
as soon as categories of functors are involved, and
that some sort of higher categorical structures
are then often needed
(see e.g. \cite[\S 1]{to2}).
A relation between the derived Picard group
and Hochschild cohomology is given in \cite{ke2}, and
is somehow close to our Corollary \ref{chh'}. An interpretation
of Hochschild cohomology as first order deformations of
dg-categories is also given in \cite{hagII}.
There has been many works on dg-categories (as well as its weakened, but
after all equivalent, notion
of $A_{\infty}$-categories) in which several universal constructions, such as
reasonable dg-categories of dg-functors or quotient and localization
of dg-categories, have been studied (see for example \cite{dr,ke,ly,ly2}). Of course,
when compared in a correct way, our constructions give back the same objects as the ones
considered in these papers, but I would like to point out that the two approaches
are different
and that our results can not be deduced from these previous works.
Indeed, the universal properties of the constructions of
\cite{dr,ke,ly,ly2} are expressed in a somehow
un-satisfactory manner (at least for my personal taste)
as they are stated in terms of certain dg-categories of dg-functors
that are not themselves defined by some universal properties (except an obvious one
with respect to themselves !)\footnote{The situation is very comparable to the situation where one tries
to explain why categories of functors give the \emph{right notion}: expressing universal
properties using itself categories of functors is not helpful.}.
In some sense, the results proved in these papers are more
properties satisfied by certain constructions rather than existence theorems.
On the contrary our results truly are existence theorems and our dg-categories of dg-functors,
or our localized dg-categories,
are constructed as solution to a universal problem inside the category
$Ho(dg-Cat)$ (or rather inside the simplicial category $L(dg-Cat)$).
As far as I know, these
universal properties were not known to be satisfied by the constructions of
\cite{dr,ke,ly,ly2}.
The results of the present work can also be generalized in an obvious way to other
contexts, as for example simplicially enriched categories, or even
spectral categories. Indeed, the key tool that makes the proofs working is
the existence of a nice model category structure on enriched categories.
For simplicial categories this model structure is known to exist by a recent
work of J. Bergner, and our theorems \ref{t1} and \ref{t2} can be easily shown to
be true in this setting (essentially the same proofs work). Theorem
\ref{t3} also stays correct for simplicial categories except that one
needs to replace the notion of continuous morphisms by the more elaborated notion
of colimit preserving morphisms. More recently,
J. Tapia has done some progress
for proving the existence of a model category structure on $M$-enriched categories
for very general monoidal model categories $M$, including for example
spectral categories (i.e. categories enriched in symmetric spectra). I am convinced
that theorems \ref{t1} and \ref{t2}, as well
as the correct modification of theorem \ref{t3}, stay correct in this general setting.
As a consequence one would get a Morita theory for symmetric ring spectra.
Finally, I did not investigate at all the question of the behavior of
the equivalence of theorem \ref{t1} with respect to composition of morphisms. Of course,
on the level of bi-modules composition is given by the tensor product, but
the combinatorics of these compositions are not an easy question. This is related to the
question: \emph{What do dg-categories form ?} It is commonly expected that the
answer is \emph{an $E_{2}$-category}, whatever this means. The point of view of this work
is to avoid this difficulty by stating that
another possible answer is \emph{a simplicially enriched category} (precisely
the Dwyer-Kan localization $L(dg-Cat)$), which is a perfectly well understood structure.
Our theorem \ref{t2}, as well as its corollary \ref{cp5'} state that
the simplicial category $L(dg-Cat)$ is enriched over itself
in a rather strong sense. In fact, one can show that
$L(dg-Cat)$ is a \emph{symmetric monoidal simplicial category} in the
sense of Segal monoids explained in \cite{kt}, and I believe that
another equivalent way to talk about
$E_{2}$-categories is by considering $L(dg-Cat)$-enriched simplicial categories, again
in some Segal style of definitions (see for example \cite{to}). In other words,
I think the $E_{2}$-category of dg-categories should be completely determined by the
symmetric monoidal simplicial category $L(dg-Cat)$.
\bigskip
\textbf{Acknowledgments:} I am very grateful to M. Anel, C. Barwick, L. Katzarkov, T. Pantev,
M. Spitzweck, J. Tapia, M. Vaqui\'e and G. Vezzosi for their participation in
the small workshop on non-abelian Hodge theory which took place in Toulouse during
the spring 2004.
I also would like to thank warmly C. Simpson for his
participation to this workshop via
some tricky but enjoyable video-conference meeting. It has been during one of the
informal conversation of this workshop that the general ideas for a proof
of theorem \ref{tfour} have been found, and I think this
particular theorem should be attributed to the all of us.
I would like to thank B. Keller for several comments
on earlier versions of this work, and for pointing out to
me related references. I also thank the referee for
his careful reading.
\bigskip
\bigskip
\textbf{Conventions:}
All along this work universes will be denoted
by $\mathbb{U}\in \mathbb{V}\in \mathbb{W} \dots$.
We will always assume that they satisfy the
infinite axiom.
We use the notion of model categories
in the sense of \cite{ho}. The expression
\emph{equivalence} always refer to
weak equivalence in a model category.
For a model category $M$, we will denote by
$Map_{M}$ (or $Map$ if $M$ is clear) its mapping spaces
as defined in \cite{ho}. We will always consider
$Map_{M}(x,y)$ as an object in the homotopy category
$Ho(SSet)$. In the same way, the set of morphisms
in the homotopy category $Ho(M)$ will be denoted
by $[-,-]_{M}$, or by $[-,-]$ if $M$ is clear.
The natural $Ho(SSet)$-tensor structure
on $Ho(M)$ will be denoted
by $K\otimes^{\mathbb{L}}X$, for $K$ a simplicial set
and $X$ an object in $M$. In the same way, the
$Ho(SSet)$-cotensor structure will be denoted
by $X^{\mathbb{R}K}$. The homotopy fiber products will
be denoted by $x\times^{h}_{z}y$, and dually the
homotopy push-outs will be denoted by
$x\coprod^{\mathbb{L}}_{z}y$.
For all along this work, we fix an
associative, unital and commutative ring $k$. We denote
by $C(k)_{\mathbb{U}}$ the category of $\mathbb{U}$-small
(un-bounded) complexes of $k$-modules, for some
universe $\mathbb{U}$ with $k\in \mathbb{U}$.
The category $C(k)_{\mathbb{U}}$ is a
symmetric monoidal model category, where one uses
the projective model structures for which fibrations
are epimorphisms and equivalences are quasi-isomorphisms
(see e.g. \cite{ho}).
When the universe $\mathbb{U}$
is irrelevant we will simply write $C(k)$ for
$C(k)_{\mathbb{U}}$. The monoidal structure on $C(k)$ is the
usual tensor product of complexes over $k$, and will be denoted by
$\otimes$. Its derived version will be denoted by $\otimes^{\mathbb{L}}$.
\section{The model structure}
Recall that a $\mathbb{U}$-small $dg$-category $C$ consists
of the following data.
\begin{itemize}
\item A $\mathbb{U}$-small set of objects
$Ob(C)$, also sometimes denoted by $C$ itself.
\item For any pair of objects
$(x,y)\in Ob(C)^{2}$ a complex
$C(x,y)\in C(k)$.
\item For any triple $(x,y,z)\in Ob(C)^{3}$
a composition morphism
$C(x,y)\otimes C(y,z) \longrightarrow C(x,z)$,
satisfying the usual associativity
condition.
\item For any object $x\in Ob(C)$, a
morphism $k \longrightarrow C(x,x)$, satisfying the
usual unit condition with respect to the
above composition.
\end{itemize}
For two dg-categories $C$ and $D$, a
morphism of dg-categories (or simply a dg-functor)
$f : C \longrightarrow D$ consists
of the following data.
\begin{itemize}
\item A map of sets $f : Ob(C) \longrightarrow Ob(D)$.
\item For any pair of objects $(x,y)\in Ob(C)^{2}$,
a morphism in $C(k)$
$$f_{x,y} : C(x,y) \longrightarrow D(f(x),f(y))$$
satisfying the usual unit and associativity conditions.
\end{itemize}
The $\mathbb{U}$-small dg-categories and
dg-functors do form a category $dg-Cat_{\mathbb{U}}$.
When the universe $\mathbb{U}$ is irrelevant, we will simply write
$dg-Cat$ for $dg-Cat_{\mathbb{U}}$.
\\
We define a functor
$$[-] : dg-Cat_{\mathbb{U}} \longrightarrow Cat_{\mathbb{U}},$$
from $dg-Cat_{\mathbb{U}}$ to the category of $\mathbb{U}$-small
categories by the following construction.
For $C\in dg-Cat_{\mathbb{U}}$, the set of object of $[C]$
is simply the set of object of $C$. For two
object $x$ and $y$ in $[C]$, the set of morphisms
from $x$ to $y$ in $[C]$
is defined by
$$[C](x,y):=H^{0}(C(x,y)).$$
Composition of morphisms in $[C]$ is given by the natural
morphism
$$[C](x,y)\times [C](y,z) =
H^{0}(C(x,y)) \times H^{0}(C(x,y)) \longrightarrow
H^{0}(C(x,y)\otimes^{\mathbb{L}}C(y,z)) \longrightarrow
H^{0}(C(x,z))=[C](x,z).$$
The unit of an object $x$ in $[C]$ is simply given by
the point in $[k,C(x,x)]=H^{0}(C(x,x))$ image of the unit morphism
$k \longrightarrow C(x,x)$ in $M$.
This construction, provides a functor
$C \mapsto [C]$ from $dg-Cat_{\mathbb{U}}$ to the category
of $\mathbb{U}$-small categories. For a morphism $f : C \longrightarrow D$
in $dg-Cat$, we will denote by $[f] : [C] \longrightarrow [D]$
the corresponding morphism in $Cat$.
\begin{df}\label{d1}
Let $f : C \longrightarrow D$ be a morphism in
$dg-Cat$.
\begin{enumerate}
\item The morphism $f$ is \emph{quasi-fully faithful}
if for any two objects $x$ and $y$ in $C$
the morphism $f_{x,y} : C(x,y) \longrightarrow D(f(x),f(y))$
is a quasi-isomorphism.
\item The morphism $f$ is \emph{quasi-essentially surjective}
if the induced functor $[f] : [C] \longrightarrow [D]$
is essentially surjective.
\item The morphism $f$ is a \emph{quasi-equivalence}
if it is quasi-fully faithful and quasi-essentially surjective.
\item The morphism $f$ is a \emph{fibration} if
it satisfies the following two conditions.
\begin{enumerate}
\item For any $x$ and $y$ in $C$ the morphism
$f_{x,y} : C(x,y) \longrightarrow D(f(x),f(y))$
is a fibration in $C(k)$ (i.e. is an epimorphism).
\item For any $x\in C$, and any isomorphism
$v : [f](x) \rightarrow y'$ in $[D]$, there
exists an isomorphism $u : x \rightarrow y$ in $[C]$
such that $[f](u)=v$.
\end{enumerate}
\end{enumerate}
\end{df}
In \cite{tab} it is proved that
the above notions of fibrations and quasi-equivalences in
$dg-Cat$ form a model category structure. The
model category $dg-Cat_{\mathbb{U}}$ is furthermore
$\mathbb{U}$-cofibrantly generated in the sense
of \cite[Appendix]{hagI}. Moreover,
for $\mathbb{U}\in \mathbb{V}$, the set of
generators for the cofibrations and trivial cofibrations
can be chosen to be the same for $dg-Cat_{\mathbb{U}}$ and for
$dg-Cat_{\mathbb{V}}$. As a consequence we get that
the natural inclusion functor
$$Ho(dg-Cat_{\mathbb{U}}) \longrightarrow Ho(dg-Cat_{\mathbb{V}})$$
is fully faithful. This inclusion functor also induces
natural equivalences on mapping spaces
$$Map_{dg-Cat_{\mathbb{U}}}(C,D)\simeq
Map_{dg-Cat_{\mathbb{V}}}(C,D),$$
for two $\mathbb{U}$-small dg-categories $C$ and $D$.
As a consequence we see that we can change our universe
without any serious harm.
Note also that the functor
$$[-] : dg-Cat \longrightarrow Cat$$
induces a functor
$$Ho(dg-Cat) \longrightarrow Ho(Cat),$$
where $Ho(Cat)$ is the category of small categories and
isomorphism classes of functors between them. In other words,
any morphism $C \rightarrow D$ in $Ho(dg-Cat)$ induces
a functor $[C] \rightarrow [D]$ well defined up to a
non-unique isomorphism. This lack of uniqueness will not
be so much of a trouble as we will essentially be interested
in properties of functors which are invariant
by isomorphisms (e.g. being fully faithful, being an equivalence \dots).
\begin{df}\label{dqess}
Let $f : C \longrightarrow D$ be a morphism of dg-categories.
The \emph{quasi-essential image of $f$} is the full sub-dg-category
of $D$ consisting of all objects $x\in D$ whose image in
$[D]$ lies in the essential image of the functor
$[f] : [C] \rightarrow [D]$.
\end{df}
The model category $dg-Cat$
also satisfies the following additional properties.
\begin{prop}\label{p0}
\begin{enumerate}
\item Any object $C\in dg-Cat$ is fibrant.
\item There exists a cofibrant replacement functor
$Q$ on $dg-Cat$, such that for any $C\in dg-Cat$ the natural
morphism $Q(C) \longrightarrow C$ induces the identity of the
sets of objects.
\item If $C$ is a cofibrant object in $dg-Cat$ and
$x$ and $y$ are two objects in $C$, then
$C(x,y)$ is a cofibrant object in $C(k)$.
\end{enumerate}
\end{prop}
\textit{Sketch of proof:} $(1)$ is clear by definition. $(2)$ simply follows from the fact that
one can choose the generating cofibrations $A\rightarrow B$ to induce the identity on the set
of objects (see \cite{tab} for details). Finally, for $(3)$, one uses that any cofibrant object
can be written as a transfinite composition of push-outs along the generating
cofibrations. As the functor $C \mapsto C(x,y)$ commutes with filtered colimits, and that
a filtered colimit of cofibrations stays a cofibration, one sees that it is enough to
prove that the property $(3)$ is preserved by push-outs along a generating cofibration.
But this can be easily checked by an explicit description of such a push-out
(see \cite{tab} proof of Lem. 2.2. for more details). \hfill $\Box$ \\
To finish this paragraph, recall that a morphism
$x \rightarrow y$ in a model category $M$ is called a \emph{homotopy monomorphism}
if for any $z\in M$ the induced morphism
$$Map_{M}(z,x) \longrightarrow Map_{M}(z,y)$$
induces an injection on $\pi_{0}$ and isomorphisms on all
$\pi_{i}$ for $i>0$ (for all base points). This is also equivalent to say that
the natural morphism
$$x \longrightarrow x\times^{h}_{y}x$$
is an isomorphism in $Ho(M)$.
The following lemma will be used
implicitly in the sequel.
\begin{lem}\label{lmono}
A morphism $f : C \longrightarrow D$ in $dg-Cat$ is a homotopy monomorphism if and
only if it is quasi-fully faithful.
\end{lem}
\textit{Proof:} We can of course suppose that the morphism
$f$ is a fibration in $dg-Cat$. Then, $f$ is a homotopy monomorphism
if and only if the induced morphism
$$\Delta : C \longrightarrow C\times_{D}C$$
is a quasi-equivalence.
Let us first assume that $f$ is quasi-fully faithful.
For any $x$ and $y$ in $C$ the induced morphism by $\Delta$ is
the diagonal of $C(x,y)$
$$\Delta(x,y) : C(x,y) \longrightarrow C(x,y)\times_{D(f(x),f(y))}C(x,y).$$
As $f$ is a fibration, the morphism $C(x,y) \longrightarrow D(f(x),f(y))$
is a trivial fibration, and thus the morphism $\Delta(x,y)$
is a quasi-isomorphism.
This shows that $\Delta$ is quasi-fully faithful.
Now, let $t$ be an object in $C\times_{D}C$, corresponding to
two points $x$ and $y$ in $C$ such that $f(x)=f(y)$. We consider the
identity morphism $f(x) \rightarrow f(y)$ in $[D]$. As $[C] \rightarrow [D]$ is
fully faithful, the identity can be lifted to an isomorphism in $[C]$
$u : x \rightarrow y$. Furthermore, as $C(x,y) \longrightarrow D(f(x),f(y))$
is a fibration, the morphism $u$ can be represented by
a zero cycle $u\in Z^{0}(C(x,y))$ whose image by $f$ is the identity.
This implies that the point $t$ is isomorphic in $[C\times_{D}C]$ to
the image of the point $x\in C$ by $\Delta$, and thus
that $\Delta$ is quasi-essentially surjective. We have shown
that $\Delta$ is a quasi-equivalence and therefore that $f$
is a homotopy monomorphism.
Conversely, let us assume that $f$ is a homotopy monomorphism. Then,
for any $x$ and $y$ in $C$ the natural morphism
$$C(x,y) \longrightarrow C(x,y)\times_{D(f(x),f(y))}C(x,y)$$
is a quasi-isomorphism, and thus the morphism $C(x,y) \longrightarrow D(f(x),f(y))$
is a homotopy monomorphism in $C(k)$. As $C(k)$ is a stable model category
(see \cite[\S 7]{ho})this clearly implies that $C(x,y) \longrightarrow D(f(x),f(y))$
is in fact a quasi-isomorphism. \hfill $\Box$ \\
\begin{cor}\label{clmono}
Let $C \longrightarrow D$ be a quasi-fully faithful morphism in $dg-Cat$
and $B$ be any dg-category. Then, the induced morphism
$$Map(B,C) \longrightarrow Map(B,D)$$
induces an injection on $\pi_{0}$ and an isomorphism
on $\pi_{i}$ for $i>0$. Furthermore, the image of
$$\pi_{0}(Map(B,C))=[B,C] \longrightarrow
[B,D]=\pi_{0}(Map(B,D))$$ consists of all morphism such that the
induced functor $[B] \rightarrow [D]$ factors through
the essential image of $[C] \rightarrow [D]$.
\end{cor}
\textit{Proof:} Only the last statement requires a proof.
For this we can of course assume that $B$ is cofibrant. Furthermore, one can
replace $C$ by its quasi-essential image in $D$. The statement is then clear
by the description of $[B,C]$ and $[B,D]$ as
homotopy classes of morphisms between $B$ and $C$ or $D$. \hfill $\Box$ \\
\section{Modules over dg-categories}
Let $C \in dg-Cat_{\mathbb{U}}$
be a fixed $\mathbb{U}$-small $dg$-category. Recall that
a $\mathbb{U}$-small $C$-dg-module $F$ (or simply
a $C$-module) consists of the following data.
\begin{itemize}
\item For any object $x\in C$ a complex $F(x)\in C(k)_{\mathbb{U}}$.
\item For any two objects
$x$ and $y$ in $C$, a morphism of complexes
$$ C(x,y)\otimes F(x) \longrightarrow F(y),$$
satisfying the usual associativity and unit conditions.
\end{itemize}
Note that a $C$-module is nothing else than
a morphism of dg-categories
$F : C \longrightarrow C(k)$, where
$C(k)$ is a dg-category in the obvious way, or equivalently
as a $C(k)$-enriched functor from $C$ to $C(k)$.
For two $C$-dg-modules $F$ and $G$, a morphism
from $F$ to $G$ is simply the data of
morphisms $f_{x} : F(x) \longrightarrow G(x)$
commuting with the structure morphisms. This
is nothing else than a $C(k)$-enriched natural transformation
between the corresponding
$C(k)$-enriched functors. The $\mathbb{U}$-small $C$-modules
and morphisms between them form
a category, denoted by $C-Mod_{\mathbb{U}}$.
Once again, when the universe $\mathbb{U}$ is irrelevant
we will simply write $C-Mod$ for $C-Mod_{\mathbb{U}}$. \\
Let $z\in C$ be an object in $C$. One defines
a $C$-module
$\underline{h}^{z}\in C-Mod$, by the formula
$\underline{h}^{z}(x):=C(z,x)$, and with
structure morphisms
$$C(z,x)\otimes C(x,y) \longrightarrow C(z,y)$$
being the composition in $C$.
\begin{df}\label{d2}
Let $C\in dg-Cat$ and $f : F \longrightarrow G$ be a morphism
of $C$-modules. The morphism $f$ is
an \emph{equivalence} (resp. a \emph{fibration})
if for any $x\in C$ the morphism
$$f_{x} : F(x) \longrightarrow G(x)$$
is an equivalence (resp. a fibration) in $C(k)$.
\end{df}
We recall that as $C(k)$ is cofibrantly generated,
the above definition endows $C-Mod$ with a structure of a
cofibrantly generated model category (see for example
\cite[\S 11]{hi}). The natural $C(k)$-enrichment of
$C-Mod$ endows furthermore $C-Mod$ with a structure
of a $C(k)$-model category in the sense of \cite[4.2.18]{ho}.
The $C(k)$-enriched $Hom$'s of the category
$C-Mod$ will be denoted by $\underline{Hom}$, and its
derived version by
$$\mathbb{R}\underline{Hom} : Ho(C-Mod)^{op}\times Ho(C-Mod) \longrightarrow
Ho(C(k)).$$
The notion of modules over dg-categories
has the following natural generalization.
Let $M$ be a $C(k)_{\mathbb{U}}$-model category in the sense
of \cite[4.2.18]{ho}, and let us suppose that it
is $\mathbb{U}$-cofibrantly generated in the
sense of \cite[Appendix A]{hagI}. Then, for a
$\mathbb{U}$-small dg-category
$C$ one has a category of $C(k)$-enriched
functors $M^{C}$ from $C$ to $M$.
Furthermore, it can be endowed with a structure
of a $\mathbb{U}$-cofibrantly generated model
category for which equivalences and fibrations are
defined levelwise in $M$ (see e.g. \cite[11.6]{hi}).
The category
$M^{C}$ has itself a natural $C(k)$-enrichment induced
from the one on $M$,
making it into a $C(k)$-model category.
When $M=C(k)_{\mathbb{U}}$ itself, the model category
$M^{C}$ can be identified with $C-Mod_{\mathbb{U}}$.
Let $f : C \longrightarrow D$ be a morphism in $dg-Cat$.
Composing with $f$ gives a restriction functor
$$f^{*} : M^{D} \longrightarrow M^{C}.$$
This functor has a left adjoint
$$f_{!} : M^{C} \longrightarrow M^{D}.$$
The adjunction $(f_{!},f^{*})$ is clearly a Quillen adjunction, compatible
with the $C(k)$-enrichment.
\begin{prop}\label{p1}
Let $f : C \longrightarrow D$ be a quasi-equivalence
between $\mathbb{U}$-small dg-categories. Let
$M$ be a $\mathbb{U}$-cofibrantly generated
$C(k)$-model category, such that the
domain and codomain of a set of generating
cofibrations are cofibrant objects in $M$. We assume that
one of the following conditions is satisfied.
\begin{enumerate}
\item For
any cofibrant object $A\in M$, and any
quasi-isomorphism $X \longrightarrow Y$ in $C(k)$, the
induced morphism
$$X\otimes A \longrightarrow Y\otimes A$$
is an equivalence in $M$.
\item All the complexes of morphisms of $C$ and
$D$ are cofibrant objects in $C(k)$.
\end{enumerate}
Then
the Quillen adjunction $(f_{!},f^{*})$ is
a Quillen equivalence.
\end{prop}
\textit{Proof:} The functor $f^{*}$ clearly preserves
equivalences. Furthermore, as $f$ is quasi-essentially
surjective,
the functor $f^{*} : Ho(M^{D}) \longrightarrow Ho(M^{C})$
is easily seen to be conservative. Therefore, one is reduced to check that
the adjunction morphism $Id\Rightarrow f^{*}\mathbb{L}f_{!} $
is an isomorphism.
For $x\in C$, and $A\in M$, one writes
$\underline{h}^{x}\otimes A \in M^{C}$ for the object
defined by
$$\begin{array}{cccc}
\underline{h}^{x}\otimes A & C & \longrightarrow & M \\
& y & \mapsto & C(x,y)\otimes A.
\end{array}$$
The model category $M^{C}$ is itself
cofibrantly generated, and a set of
generating cofibration can be chosen
to consist of morphisms of the form
$$\underline{h}^{x}\otimes A \longrightarrow \underline{h}^{x}\otimes B$$
for some generating cofibration $A\longrightarrow B$ in $M$.
By assumption on $M$, any object $F\in Ho(M^{C})$ can thus
be written as a
homotopy colimit of objects of the form
$\underline{h}^{x}\otimes A$, for
certain cofibrant $A\in M$, and certain $x\in C$. As
the two functors $f^{*}$ and $\mathbb{L}f_{!}$ commute
with homotopy colimits it is then enough to show that
the natural morphism
$$\underline{h}^{x}\otimes A
\longrightarrow f^{*}\mathbb{L}f_{!}
(\underline{h}^{x}\otimes A)
$$
is an isomorphism in $Ho(M^{C})$. By adjunction, one clearly has
$\mathbb{L}f_{!}(\underline{h}^{x}\otimes A)\simeq
\underline{h}^{f(x)}\otimes A$.
Therefore, the adjunction morphism
$$\underline{h}^{x}\otimes A
\longrightarrow f^{*}\mathbb{L}f_{!}
(\underline{h}^{x}\otimes A)\simeq f^{*}(\underline{h}^{f(x)}\otimes A)
$$
evaluated at $y\in C$ is the morphism
$$f_{x,y}\otimes Id_{A} : C(x,y) \otimes A \longrightarrow
D(f(x),f(y))\otimes A.$$
The fact that this is an isomorphism in $Ho(M)$
follows from the fact that
$f$ is quasi-fully faithful, one of our hypothesis $(1)$ and $(2)$, and the fact that
$M$ is a $C(k)$-model category. \hfill $\Box$ \\
Another important property of the model category
$M^{C}$ is the following.
\begin{prop}\label{p2}
Let $C$ be a $\mathbb{U}$-small dg-category
with cofibrant complexes of morphisms
(i.e. $C(x,y)$ is cofibrant in $C(k)$ for all
$x$ and $y$),
and $M$ be a $\mathbb{U}$-cofibrantly generated
$C(k)$-model category. Then,
for any $x\in C$ the evaluation functor
$$\begin{array}{cccc}
x^{*} : & M^{C} & \longrightarrow & M \\
& F & \mapsto & F(x)
\end{array}$$
preserves fibrations, cofibrations and equivalences.
\end{prop}
\textit{Proof:} For fibrations and equivalences this
is clear by definition. The
functor $x^{*}$ commutes with colimits, and thus
by a small object argument one is
reduced to show that $x^{*}$ sends
generating cofibrations to cofibrations.
One
knows that the generating set of cofibrations
in $M^{C}$ can be chosen to consist of morphisms
of the form $\underline{h}^{z}\otimes A \longrightarrow
\underline{h}^{z}\otimes B$ for some cofibration
$A \longrightarrow B$ in $M$. The image by $x^{*}$
of such a morphism is
$$C(z,x)\otimes A \longrightarrow
C(z,x)\otimes B.$$
As by assumption $C(z,x)$ is a cofibrant object in $C(k)$, one
sees that this morphism is a cofibration in $M$. \hfill $\Box$ \\
Two important cases of application of proposition \ref{p2} is when
$C$ itself is a cofibrant dg-category (see Prop. \ref{p0}), or
when $k$ is a field. \\
\begin{cor}\label{cp2}
The conclusion of Prop. \ref{p1} is satisfied
when $M$ is of the form $D-Mod_{\mathbb{U}}$, for
a $\mathbb{U}$-small dg-category $D$ with
cofibrant complexes of morphisms (in particular for
$M=C(k)$).
\end{cor}
\textit{Proof:} This follows easily from Prop. \ref{p2} and the
fact that $C(k)$ itself satisfies the hypothesis $(1)$
of Prop. \ref{p1}. \hfill $\Box$ \\
Let $\mathbb{U} \in \mathbb{V}$ be two universes. Let $M$ be
a $C(k)_{\mathbb{U}}$-model category which is supposed to be
furthermore $\mathbb{V}$-small.
We define a $\mathbb{V}$-small dg-category
$Int(M)$ in the following way\footnote{The notation \emph{Int} is taken from
\cite{hs}. As far as I understand it stands for \emph{internal}.}. The set of objects
of $Int(M)$ is the set of fibrant and cofibrant objects
in $M$. For two such objects $F$ and $E$
one sets
$$Int(M)(E,F):=\underline{Hom}(E,F)\in C(k)_{\mathbb{U}},$$
where $\underline{Hom}(E,F)$ is the $C(k)$-valued
Hom of the category $M$.
The dg-category $Int(M)$ is of course
only $\mathbb{V}$-small as its sets of objects is only $\mathbb{V}$-small.
However, for any $E$ and $F$ in $Int(M)$
the complex $Int(M)(E,F)$ is in fact
$\mathbb{U}$-small. \\
The following is a general fact about
$C(k)$-enriched model categories.
\begin{prop}\label{p3}
There exists a natural equivalence of categories
$$[Int(M)]\simeq Ho(M).$$
\end{prop}
\textit{Proof:} This follows from the formula
$$H^{0}(\mathbb{R}\underline{Hom}(X,Y))\simeq [k,\mathbb{R}\underline{Hom}(X,Y)]_{C(k)}\simeq
[X,Y]_{M},$$
for two objects $X$ and $Y$ in $M$. \hfill $\Box$ \\
For $x\in C$, the object $\underline{h}^{x} \in C-Mod_{\mathbb{U}}$ is
cofibrant and fibrant, and therefore
the construction $x \mapsto
\underline{h}^{x}$, provides a morphism of dg-categories
$$\underline{h}^{-} : C^{op} \longrightarrow Int(C-Mod_{\mathbb{U}}),$$
where $C^{op}$ is the opposite dg-category of $C$ ($C^{op}$ has the same set of objects than $C$ and
$C^{op}(x,y):=C(y,x)$).
The morphism $\underline{h}^{-}$ can also be written dually as
$$\underline{h}_{-} : C \longrightarrow
Int(C^{op}-Mod_{\mathbb{U}}).$$
The dg-functor $\underline{h}^{-}$ will be considered as
a morphism in $dg-Cat_{\mathbb{V}}$, and is clearly
quasi-fully faithful by an application of the $C(k)$-enriched
Yoneda lemma.
\begin{df}\label{d3}
\begin{enumerate}
\item
Let $C\in dg-Cat_{\mathbb{U}}$, and
$F\in C^{op}-Mod_{\mathbb{U}}$ be a $C^{op}$-module.
The object $F$ is called \emph{representable} (resp.
\emph{quasi-representable}) if it is isomorphic
in $C^{op}-Mod_{\mathbb{U}}$ (resp.
in $Ho(C^{op}-Mod_{\mathbb{U}})$) to $\underline{h}_{x}$
for some object $x\in C$.
\item Dually, let $C\in dg-Cat_{\mathbb{U}}$, and
$F\in C-Mod_{\mathbb{U}}$ be a $C$-module.
The object $F$ is called \emph{corepresentable} (resp.
\emph{quasi-corepresentable}) if it is isomorphic
in $C-Mod_{\mathbb{U}}$ (resp.
in $Ho(C-Mod_{\mathbb{U}})$) to $\underline{h}^{x}$
for some object $x\in C$.
\end{enumerate}
\end{df}
As the morphism $\underline{h}_{-}$ is quasi-fully faithful,
it induces a quasi-equivalence between $C$ and the
full dg-category of $Int(C^{op}-Mod_{\mathbb{U}})$
consisting of quasi-representable objects. This quasi-equivalence is
a morphism in $dg-Cat_{\mathbb{V}}$. \\
\section{Mapping spaces and bi-modules}
Let $C$ and $D$ be two objects in $dg-Cat$. One has
a tensor product $C\otimes D \in dg-Cat$ defined in the following way.
The set of objects of $C\otimes D$ is $Ob(C)\times Ob(D)$, and
for $(x,y)$ and $(x',y')$ two objects in $Ob(C\otimes D)$ one sets
$$(C\otimes D)((x,y),(x',y')):=C(x,y)\otimes D(x',y').$$
Composition in $C\otimes D$ is given by the obvious
formula.
This defines a symmetric monoidal structure on
$dg-Cat$, which is easily seen to be
closed. The unit of this structure will be
denoted by $\mathbf{1}$, and is the dg-category with
a unique object and $k$ as its endomorphism ring.
The model category $dg-Cat$ together with
the symmetric monoidal structure $-\otimes -$ is \emph{not}
a symmetric monoidal model category, as
the tensor product of two cofibrant objects in
$dg-Cat$ is not cofibrant in general. A direct consequence
of this fact is that the internal Hom object
between cofibrant-fibrant objects in $dg-Cat$
can not be invariant by quasi-equivalences, and thus does not
provide internal Hom's for the homotopy categories
$Ho(dg-Cat)$. This fact is the main difficulty in
computing the mapping spaces in $dg-Cat$, as the naive
approach simply does not work.
However, it is true that the monoidal structure
$\otimes$ on $dg-Cat$ is closed, and that
$dg-Cat$ has corresponding internal Hom objects
$C^{D}$ satisfying the usual adjunction rule
$$Hom_{dg-Cat}(A\otimes B,C)\simeq Hom(A,C^{B}).$$
This gives a natural equivalence of categories
$$M^{C\otimes D}\simeq (M^{C})^{D}$$
for any $C(k)$-enriched category $M$. Furthermore, when
$M$ is a $\mathbb{U}$-cofibrantly generated model category,
this last equivalence is compatible with the
model structures on both sides. \\
The functor $-\otimes -$ can be derived into a functor
$$-\otimes^{\mathbb{L}} - : dg-Cat\times dg-Cat \longrightarrow dg-Cat$$
defined by the formula
$$C\otimes^{\mathbb{L}}D:=
Q(C)\otimes D$$
where $Q$ is a cofibrant replacement in $dg-Cat$ which acts by
the identity on the sets of objects. Clearly,
the functor $-\otimes^{\mathbb{L}} -$ preserves quasi-equivalences and
passes through the homotopy categories
$$-\otimes^{\mathbb{L}} - : Ho(dg-Cat)\times Ho(dg-Cat) \longrightarrow
Ho(dg-Cat).$$
Note that when $C$ is cofibrant, one has
a natural quasi-equivalence $C\otimes^{\mathbb{L}}D \longrightarrow C\otimes D$.
We now consider $(C\otimes D^{op})-Mod$, the category of
$(C\otimes D^{op})$-modules. For any
object $x\in C$, there exists a natural
morphism of dg-categories $D^{op} \longrightarrow (C\otimes D^{op})$ sending
$y\in D$ to the object $(x,y)$, and
$$D^{op}(y,z) \longrightarrow (C\otimes D^{op})((x,y),(x,z))=
C(x,x)\otimes D^{op}(y,z)$$
being the tensor product of the unit $k \longrightarrow C(x,x)$ and
the identity on $D^{op}(y,z)$. As $C$ and $Q(C)$ has the same
set of objects, one sees that for any $x\in C$ one also gets
a natural morphism of dg-categories
$$i_{x} : D^{op} \longrightarrow Q(C)\otimes D^{op}=C\otimes^{\mathbb{L}}D^{op}.$$
\begin{df}\label{d4}
Let $C$ and $D$ be two dg-categories.
An object $F \in (C\otimes^{\mathbb{L}}D^{op})-Mod$ is called
\emph{right quasi-representable}, if for any $x\in C$, the $D^{op}$-module
$i_{x}^{*}(F)\in D^{op}-Mod$ is quasi-representable in the sense
of Def. \ref{d3}.
\end{df}
We now let $\mathbb{U}\in \mathbb{V}$ be two universes, and
let $C$ and $D$ be two $\mathbb{U}$-small dg-categories.
Let
$\Gamma^{*}$ be a co-simplicial resolution functor in $dg-Cat_{\mathbb{U}}$
in the sense of \cite[\S 16.1]{hi}. Recall that
$\Gamma^{*}$ is a functor from $dg-Cat_{\mathbb{U}}$ to
$dg-Cat^{\Delta}_{\mathbb{U}}$, equipped with a natural
augmentation $\Gamma^{0} \longrightarrow Id$, and such the
following two conditions are satisfied.
\begin{itemize}
\item For any $n$, and any $C\in dg-Cat_{\mathbb{U}}$ the morphism
$\Gamma^{n}(C) \rightarrow C$
is a quasi-equivalence.
\item For any $C\in dg-Cat_{\mathbb{U}}$, the object
$\Gamma^{*}(C) \in dg-Cat_{\mathbb{U}}^{\Delta}$ is cofibrant
for the Reedy model structure.
\item The morphism $\Gamma^{0}(C) \longrightarrow C$
is equal to $Q(C) \longrightarrow C$.
\end{itemize}
The left mapping space between $C$ and $D$ is by definition the
$\mathbb{U}$-small simplicial set
$$\begin{array}{cccc}
Map^{l}(C,D):=Hom(\Gamma^{*}(C),D) : & \Delta^{op} & \longrightarrow & Set_{\mathbb{U}} \\
& [n] & \mapsto & Hom(\Gamma^{n}(C),D).
\end{array}$$
Note that the mapping space $Map^{l}(C,D)$ defined above has
the correct homotopy type as all objects are fibrant
in $dg-Cat_{\mathbb{U}}$.
For any $[n] \in \Delta$, one considers the
(non-full) sub-category $\mathcal{M}(\Gamma^{n}(C),D)$
of $(\Gamma^{n}(C)\otimes D^{op})-Mod_{\mathbb{U}}$
defined in the following way. The objects
of $\mathcal{M}(\Gamma^{n}(C),D)$ are the
$(\Gamma^{n}(C)\otimes D^{op})$-modules $F$ such that
$F$ is right quasi-representable, and for any
$x\in \Gamma^{n}(C)$ the $D^{op}$-module $F(x,-)$
is cofibrant in $D^{op}-Mod_{\mathbb{U}}$. The morphisms
in $\mathcal{M}(\Gamma^{n}(C),D)$ are simply the
equivalences in $(\Gamma^{n}(C)\otimes D^{op})-Mod_{\mathbb{U}}$.
The nerve of the category $\mathcal{M}(\Gamma^{n}(C),D)$ gives
a $\mathbb{V}$-small simplicial set
$N(\mathcal{M}(\Gamma^{n}(C),D))$. For $[n] \rightarrow [m]$
a morphism in $\Delta$, one has a natural morphism
of dg-categories $\Gamma^{n}(C)\otimes D^{op} \longrightarrow
\Gamma^{m}(C)\otimes D^{op}$, and thus a well defined
morphism of simplicial sets
$$N(\mathcal{M}(\Gamma^{m}(C),D)) \longrightarrow N(\mathcal{M}(\Gamma^{n}(C),D))$$
obtained by pulling back the modules from
$\Gamma^{m}(C)\otimes D^{op}$ to $\Gamma^{n}(C)\otimes D^{op}$. This
defines a functor
$$\begin{array}{cccc}
N(\mathcal{M}(\Gamma^{*}(C),D)) : & \Delta^{op} & \longrightarrow &
SSet_{\mathbb{V}} \\
& [n] & \mapsto & N(\mathcal{M}(\Gamma^{n}(C),D)).
\end{array}$$
The set of zero simplices in $N(\mathcal{M}(\Gamma^{n}(C),D))$
is the set of all objects in the category
$\mathcal{M}(\Gamma^{n}(C),D)$. Therefore,
one defines a natural morphism of sets
$$Hom(\Gamma^{n}(C),D) \longrightarrow
N(\mathcal{M}(\Gamma^{n}(C),D))_{0}$$
by sending a morphism of dg-categories $f : \Gamma^{n}(C) \longrightarrow D$,
to the $(\Gamma^{n}(C)\otimes D^{op})$-module $\phi(f)$
defined by $\phi(f)(x,y):=D(y,f(x))$ and the natural
transition morphisms. Note that $\phi(f)$ belongs to
the sub-category $\mathcal{M}(\Gamma^{n}(C),D)$ as
for any $x\in \Gamma^{n}(C)$ the $D^{op}$-module
$\phi(f)(x,-)=\underline{h}_{f(x)}$
is representable and thus
quasi-representable and cofibrant.
By adjunction, this morphism of sets can also
be considered as a morphism of simplicial sets
$$\phi : Hom(\Gamma^{n}(C),D) \longrightarrow
N(\mathcal{M}(\Gamma^{n}(C),D)),$$
where the set $Hom(\Gamma^{n}(C),D)$ is considered as a constant
simplicial set.
This construction is clearly functorial
in $n$, and gives a well defined morphism of
bi-simplicial sets
$$\phi : Hom(\Gamma^{*}(C),D) \longrightarrow N(\mathcal{M}(\Gamma^{*}(C),D)).$$
Passing to the diagonal one gets a morphism in
$SSet_{\mathbb{V}}$
$$\phi : Map^{l}(C,D) \longrightarrow
d(N(\mathcal{M}(\Gamma^{*}(C),D))).$$
Finally, the diagonal $d(N(\mathcal{M}(\Gamma^{*}(C),D)))$ receives a natural
morphism
$$\psi : N(\mathcal{M}(\Gamma^{0}(C),D))=N(\mathcal{M}(Q(C),D)) \longrightarrow
d(N(\mathcal{M}(\Gamma^{*}(C),D))).$$
Clearly, the diagram of simplicial sets
$$\xymatrix{
Map^{l}(C,D) \ar[r]^-{\phi} & d(N(\mathcal{M}(\Gamma^{*}(C),D))) &
\ar[l]_-{\psi} N(\mathcal{M}(Q(C),D))}$$
is functorial in $C$. \\
The main theorem of this work is the following.
\begin{thm}\label{t1}
The two morphisms in $SSet_{\mathbb{V}}$
$$\xymatrix{
Map^{l}(C,D) \ar[r]^-{\phi} & d(N(\mathcal{M}(\Gamma^{*}(C),D))) &
\ar[l]_-{\psi} N(\mathcal{M}(Q(C),D))}$$
are weak equivalences.
\end{thm}
\textit{Proof:} For any $n$, the morphism
$\Gamma^{n}(C)\otimes D^{op} \longrightarrow Q(C)\otimes D^{op}$
is a quasi-equivalence of dg-categories. Therefore,
Prop. \ref{cp2} implies that the pull-back functor
$$(Q(C)\otimes D^{op})-Mod \longrightarrow
(\Gamma^{n}(C)\otimes D^{op})-Mod$$
is the right adjoint of a Quillen equivalence. As these functors
obviously preserve the notion of being right quasi-representable,
one finds that the induced morphism
$$N(\mathcal{M}(Q(C),D)) \longrightarrow
N(\mathcal{M}(\Gamma^{n}(C),D))$$
is a weak equivalence. This clearly implies that
the morphism $\psi$ is a weak equivalence.
It remains to show that the morphism $\phi$ is also
a weak equivalence. For this, we start by proving that
it induces an isomorphism on connected components.
\begin{lem}\label{lt1}
The induced morphism
$$\pi_{0}(\phi) : [C,D]\simeq \pi_{0}(Map^{l}(C,D))\longrightarrow
\pi_{0}(d(N(\mathcal{M}(\Gamma^{*}(C),D))))$$
is an isomorphism.
\end{lem}
\textit{Proof:} First of all, replacing $C$ by
$Q(C)$ one can suppose that $Q(C)=C$
(one can do this because of Prop. \ref{cp2}).
One then has $\pi_{0}(Map^{l}(C,D))\simeq [C,D]$, and
$\pi_{0}(d(N(\mathcal{M}(\Gamma^{*}(C),D))))\simeq
\pi_{0}(N(\mathcal{M}(C,D)))$ is the set of
isomorphism classes in $Ho((C\otimes D^{op})-Mod)^{rqr}$,
the full sub-category of $Ho((C\otimes D^{op})-Mod)$
consisting of all right quasi-representable objects. The morphism
$\phi$ naturally gives a morphism
$$\phi : [C,D] \longrightarrow Iso(Ho((C\otimes D^{op})-Mod)^{rqr})$$
which can be described as follows. For
any $f\in [C,D]$, represented by $f : C \longrightarrow D$
in $Ho(dg-Cat)$, $\phi(f)$ is the $C\otimes D^{op}$-module
defined by $\phi(f)(x,y):=D(y,f(x))$.
\begin{sublem}\label{sl2}
With the same notations as above,
let
$M$ be a $\mathbb{U}$-cofibrantly generated
$C(k)_{\mathbb{U}}$-model category, which is
furthermore $\mathbb{V}$-small. Let
$Iso(Ho(M^{C}))$ be the set of isomorphism classes
of objects in $Ho(M^{C})$.
Then, the natural morphism
$$Hom(C,Int(M)) \longrightarrow Iso(Ho(M^{C}))$$
is surjective.
\end{sublem}
\textit{Proof of sub-lemma \ref{sl2}:} Of course, the morphism
$$Hom(C,Int(M)) \longrightarrow Iso(Ho(M^{C}))$$
sends a morphism of dg-categories $C \longrightarrow Int(M)$
to the corresponding object in $M^{C}$.
Let $F\in Ho(M^{C})$ be
a any cofibrant and fibrant object.
This object is given by
a $C(k)$-enriched functor $F : C \longrightarrow M$.
Furthermore, as $F$ is fibrant and cofibrant,
Prop. \ref{p2} tells us that $F(x)$ is fibrant and cofibrant in $M$
for any $x\in C$.
The object $F$ can therefore be naturally considered as a morphism
of $\mathbb{V}$-small dg-categories
$$F : C \longrightarrow Int(M),$$
which gives an element in $Hom(C,Int(M))$ sent to $F$ by the map of the lemma.
\hfill $\Box$ \\
Let us now prove that the morphism $\phi$ is surjective
on connected component.
For this, let $F\in Ho((C\otimes D^{op})-Mod_{\mathbb{U}})$ be
a right quasi-representable object. One needs to show that
$F$ is isomorphic to some $\phi(f)$ for some
morphism of dg-categories $f : C \longrightarrow D$.
Sub-lemma \ref{sl2} applied to $M=D^{op}-Mod_{\mathbb{U}}$ implies
that $F$ corresponds to a morphism
of $\mathbb{V}$-small dg-categories
$$F : C \longrightarrow Int(D^{op}-Mod_{\mathbb{U}})^{qr},$$
where $Int(D^{op}-Mod)^{qr}$ is the full sub-dg-category
of $Int(D^{op}-Mod_{\mathbb{U}})$ consisting of all
quasi-representable objects.
One has a diagram
in $dg-Cat_{\mathbb{V}}$
$$\xymatrix{
C \ar[r]^-{F} & Int(D^{op}-Mod_{\mathbb{U}})^{qr} \\
& D \ar[u]_-{\underline{h}}.}$$
As the morphism $\underline{h}$ is a quasi-equivalence, and
as $C$ is cofibrant, one finds a morphism of dg-categories
$f : C \longrightarrow D$, such that the two morphisms
$$F : C \longrightarrow Int(D^{op}-Mod_{\mathbb{U}})^{qr} \qquad
\underline{h}_{f(-)}=\phi(f) : C \longrightarrow Int(D^{op}-Mod_{\mathbb{U}})^{qr}$$
are homotopic in $dg-Cat_{\mathbb{V}}$. Let
$$\xymatrix{
C \ar[d]_-{i_{0}}\ar[rd]^-{F} & \\
C' \ar[r]^-{H} & Int(D^{op}-Mod_{\mathbb{U}})^{qr} \\
C\ar[u]^-{i_{1}} \ar[ru]_-{\phi(f)} & }$$
be a homotopy in $dg-Cat_{\mathbb{V}}$. Note that
$C'$ is a cylinder object for $C$, and thus can be
chosen to be cofibrant and $\mathbb{U}$-small. We let
$p : C' \longrightarrow C$ the natural projection, such that
$p\circ i_{0}=p\circ i_{1}=Id$. This diagram gives rise to an
equivalence of categories (by Prop. \ref{cp2})
$$i_{0}^{*}\simeq i_{1}^{*}\simeq (p^{*})^{-1} : Ho((C'\otimes D^{op})-Mod_{\mathbb{U}}) \longrightarrow
Ho((C\otimes D^{op})-Mod_{\mathbb{U}}).$$
Furthermore, one has
$$F\simeq i_{0}^{*}(H)\simeq i_{1}^{*}(H)\simeq \phi(f).$$
This shows that the two $C\otimes D^{op}$-modules
$F$ and $\phi(f)$ are
isomorphic in $Ho((C\otimes D^{op})-Mod_{\mathbb{U}})$, or in other
words that $\phi(f)=F$ in $Iso(Ho((C\otimes D^{op})-Mod_{\mathbb{U}}))$.
This finishes the proof of the surjectivity part of the
lemma \ref{lt1}. \\
Let us now prove that $\phi$ is injective. For this, let
$f,g : C \longrightarrow D$ be two morphisms of dg-categories, such that
the two $(C\otimes D^{op})$-modules $\phi(f)$ and $\phi(g)$ are
isomorphic in $Ho((C\otimes D^{op})-Mod_{\mathbb{U}}))$. Composing
$f$ and $g$ with
$$\underline{h} : D \longrightarrow Int(D^{op}-Mod_{\mathbb{U}})$$
one gets two new morphisms of dg-categories
$$f',g' : C \longrightarrow Int(D^{op}-Mod_{\mathbb{U}}).$$
Using that $\underline{h}$ is quasi-fully faithful
Cor. \ref{clmono}
implies that
if $f'$ and $g'$ are homotopic morphisms in $dg-Cat_{\mathbb{V}}$, then
$f$ and $g$ are equal as morphisms in $Ho(dg-Cat_{\mathbb{V}})$.
As the inclusion $Ho(dg-Cat_{\mathbb{U}}) \longrightarrow
Ho(dg-Cat_{\mathbb{V}})$ is fully faithful
(see remark after Def. \ref{d1}), we see it is enough to show that
$f'$ and $g'$ are homotopic in $dg-Cat_{\mathbb{V}}$.
\begin{sublem}\label{sl1}
Let $M$ be a $C(k)_{\mathbb{U}}$-model category which is
$\mathbb{U}$-cofibrantly generated and $\mathbb{V}$-small.
Let $u$ and $v$ be two morphisms in $dg-Cat_{\mathbb{V}}$
$$u,v : C \longrightarrow Int(M)$$
such that the corresponding objects
in $Ho(M^{C})$ are isomorphic. Then
$u$ and $v$ are homotopic as morphisms in $dg-Cat_{\mathbb{V}}$.
\end{sublem}
\textit{Proof of sub-lemma \ref{sl1}:} First of all, any isomorphism in
$Ho(M^{C})$ between levelwise cofibrant and fibrant objects can be represented as a string of
trivial cofibrations and trivial fibrations between levelwise cofibrant and fibrant objects.
Therefore, sub-lemma \ref{sl2} shows that one is
reduced
to the case where there exists
an equivalence $\alpha : u \longrightarrow v$ in
$M^{C}$ which is either a fibration or a cofibration.
Let us start with the case where $\alpha$ is a cofibration
in $M^{C}$. The morphism $\alpha$ can also be considered
as an object in $(M^{I})^{C}$, where
$I$ is the category with two
objects $0$ and $1$ and a unique morphism
$0\rightarrow 1$. The category
$M^{I}$, which is the category
of morphisms in $M$, is endowed with its projective model structure, for
which fibrations and equivalences are defined on the underlying
objects in $M$. As the morphism $\alpha$ is a cofibration in
$M^{C}$, we see that for $x\in C$ the
corresponding morphism $\alpha_{x} : u(x) \rightarrow v(x)$
is a cofibration in $M$, and thus is a cofibrant (and fibrant)
object in $M^{I}$ because of proposition \ref{p2}. This implies that $\alpha$ gives rise to
a morphism of dg-categories
$$\alpha : C \longrightarrow Int(M^{I}).$$
Now, let $Int(M) \longrightarrow Int(M^{I})$ be the natural
inclusion morphism, sending a cofibrant and fibrant object in $M$ to the
identity morphism. This a morphism in $dg-Cat_{\mathbb{V}}$ which
is easily seen to be quasi-fully faithful. We let
$C'\subset Int(M^{I})$ be the quasi-essential image of
$Int(M)$ in $Int(M^{I})$. It is easy to check that
$C'$ is the full sub-dg-category of $Int(M^{I})$ consisting of
all objects in $M^{I}$ corresponding to equivalences in $M$.
The morphism $\alpha : C \longrightarrow Int(M^{I})$
thus factors through the sub-dg-category $C'\subset Int(M^{I})$.
The two objects $0$ and $1$ of $I$ give two projections
$$C' \subset Int(M^{I}) \rightrightarrows Int(M),$$
both of them having the natural inclusion $Int(M)
\longrightarrow Int(M^{I})$ as a section.
We have thus constructed a commutative diagram in $dg-Cat_{\mathbb{V}}$
$$\xymatrix{
& Int(M) \\
C \ar[r]^-{\alpha} \ar[ru]^-{u} \ar[rd]_-{v} & C' \ar[u] \ar[d] \\
& Int(M)}$$
which provides a homotopy between $u$ and $v$ in $dg-Cat_{\mathbb{V}}$.
For the case where $\alpha$ is a fibration in $M^{C}$, one
uses the same argument, but endowing $M^{I}$ with its
injective model structure, for which equivalences and cofibrations
are defined levelwise. We leave the details to the reader.
\hfill $\Box$ \\
We have finished the proof of sub-lemma \ref{sl1}, which applied
to $M=D^{op}-Mod_{\mathbb{U}}$ finishes the proof of the
injectivity on connected components, and thus of lemma \ref{lt1}.
\hfill $\Box$ \\
In order to finish the proof of the theorem, one uses
the functoriality of the morphisms $\phi$ and $\psi$
with respect to $D$. First of all, the
simplicial set $Map^{l}(C,D)=Hom(\Gamma^{*}(C),D)$ is
obviously functorial in $D$. One thus has a functor
$$\begin{array}{cccc}
Map^{l}(C,-) : & dg-Cat_{\mathbb{U}} & \longrightarrow &
SSet_{\mathbb{V}} \\
& D & \mapsto & Map^{l}(C,D).
\end{array}$$
The functoriality of $N(\mathcal{M}(C,D))$ in $D$ is slightly
more complicated. Let $u : D \longrightarrow E$ a morphism
in $dg-Cat_{\mathbb{U}}$. One has a functor
$$(Id\otimes u_{!}) : (C\otimes D^{op})-Mod_{\mathbb{U}} \longrightarrow
(C\otimes E^{op})-Mod_{\mathbb{U}}.$$
This functor can also be described as
$$(u_{!})^{C} : (D^{op}-Mod_{\mathbb{U}})^{C} \longrightarrow
(E^{op}-Mod_{\mathbb{U}})^{C},$$
the natural extension of the functor $u_{!} : D^{op}-Mod_{\mathbb{U}}
\longrightarrow E^{op}-Mod_{\mathbb{U}}$.
Clearly, the functor $(u_{!})^{C}$ sends the sub-category
$\mathcal{M}(C,D)$ to the sub-category
$\mathcal{M}(C,E)$ (here one uses that
$(u_{!})^{C}$ preserves equivalences
because the object $F\in \mathcal{M}(C,D)$
are such that $F(x,-)$ is cofibrant in $D^{op}-Mod_{\mathbb{U}}$). Unfortunately,
this does not define a
presheaf of categories $\mathcal{M}(C,-)$ on
$dg-Cat_{\mathbb{U}}$, as for two
morphisms
$$\xymatrix{
D \ar[r]^-{u} & E \ar[r]^-{v} & F}$$
of dg-categories one only has a natural isomorphism
$(v\circ u)_{!}\simeq (v_{!})\circ u_{!}$ which in general
is not an identity. However, these natural isomorphisms
makes $D \mapsto \mathcal{M}(C,D)$ into a
lax functor from $dg-Cat_{\mathbb{U}}$ to $Cat_{\mathbb{V}}$.
Using the standard rectification procedure, one can
replace up to a natural equivalence the lax functor
$\mathcal{M}(C,-)$ by a true presheaf of categories
$\mathcal{M}'(C,-)$.
Furthermore, the natural morphism
$$Hom(C,D) \longrightarrow \mathcal{M}(C,D)$$
from the set of morphisms $Hom(C,D)$, considered
as a discrete category, to the
category $\mathcal{M}(C,D)$ clearly gives a
morphism of lax functors
$$Hom(C,-) \longrightarrow \mathcal{M}(C,-).$$
By rectification this also induces a natural morphism
of presheaves of categories
$$Hom(C,-) \longrightarrow \mathcal{M}'(C,-).$$
Passing to the nerve one gets a morphism of functors
from $dg-Cat_{\mathbb{U}}$ to $SSet_{\mathbb{V}}$
$$Hom(C,-) \longrightarrow N(\mathcal{M}'(C,-)).$$
This morphism being functorial in $C$ give a
diagram in $(SSet_{\mathbb{V}})^{dg-Cat_{\mathbb{U}}}$
$$\xymatrix{
Map^{l}(C,-)=Hom(\Gamma^{*}(C),-) \ar[r]^-{\phi'} &
d(N(\mathcal{M}'(\Gamma^{*}(C),-))) & \ar[l]_-{\psi'}
N(\mathcal{M}'(Q(C),-)).}$$
These morphisms, evaluated at an object $D\in dg-Cat_{\mathbb{U}}$
gives a diagram of simplicial sets
$$\xymatrix{
Map^{l}(C,D) \ar[r] &
d(N(\mathcal{M}'(\Gamma^{*}(C),D)) & \ar[l]
N(\mathcal{M}'(Q(C),D)),}$$
weakly equivalent to the diagram
$$\xymatrix{
Map^{l}(C,D) \ar[r] &
d(N(\mathcal{M}(\Gamma^{*}(C),D)) & \ar[l]
N(\mathcal{M}(Q(C),D)).}$$
In order to finish the proof of the theorem it is therefore
enough to show that the two
morphism $\phi'$ and $\psi'$ are weak equivalences
of diagrams of simplicial sets. We already know that
$\psi'$ is a weak equivalence, and thus we obtain a
morphism well defined in $Ho((SSet_{\mathbb{V}})^{dg-Cat_{\mathbb{U}}})$
$$k : (\psi')^{-1}\circ \phi :
Map^{l}(C,-) \longrightarrow N(\mathcal{M}'(Q(C),-)).$$
Using our corollary \ref{cp2} it is easy to see that
the functor $N(\mathcal{M}'(Q(C),-))$ sends quasi-equivalences
to weak equivalences. Furthermore, the standard
properties of mapping spaces imply that so does the
functor $Map^{l}(C,-)$.
\begin{sublem}\label{sl3}
Let $k : F \longrightarrow G$ be a morphism in
$(SSet_{\mathbb{V}})^{dg-Cat_{\mathbb{U}}}$. Assume the following
conditions are satisfied.
\begin{enumerate}
\item Both functors $F$ and $G$ send quasi-equivalences
to weak equivalences.
\item
For any diagram in $dg-Cat_{\mathbb{U}}$
$$\xymatrix{
& C\ar[d]^-{p} \\
D \ar[r] &E}$$
with $p$ a fibration, the commutative diagrams
$$\xymatrix{
F(C\times_{E}D) \ar[r] \ar[d] & F(C) \ar[d] & & & G(C\times_{E}D) \ar[r] \ar[d] & G(C) \ar[d] \\
F(D) \ar[r] & F(E) & & & G(D) \ar[r] & G(E)}$$
are homotopy cartesian.
\item $F(*)\simeq G(*)\simeq *$, where $*$ is the final
object in $dg-Cat$.
\item For any $C\in dg-Cat_{\mathbb{U}}$ the morphism
$k_{C} : \pi_{0}(F(C)) \longrightarrow \pi_{0}(G(C))$
is an isomorphism.
\end{enumerate}
Then, for any $C\in dg-Cat_{\mathbb{U}}$ the natural
morphism
$$k_{C} : F(C) \longrightarrow G(C)$$
is a weak equivalence.
\end{sublem}
\textit{Proof of sub-lemma \ref{sl3}:} Condition
$(1)$ implies that the induced functors
$$Ho(F),Ho(G) : Ho(dg-Cat_{\mathbb{U}}) \longrightarrow
Ho(SSet_{\mathbb{V}})$$
have natural structures of $Ho(SSet_{\mathbb{U}})$-enriched
functors (see for example \cite[Thm. 2.3.5]{hagI}). In particular, for any
$K\in Ho(SSet_{\mathbb{U}})$, and any $C\in Ho(dg-Cat_{\mathbb{U}})$
one has natural morphisms in $Ho(SSet_{\mathbb{U}})$
$$F(C^{\mathbb{R}K})\longrightarrow
Map(K,F(C)) \qquad G(C^{\mathbb{R}K})\longrightarrow
Map(K,G(C)).$$
Our hypothesis $(2)$ and $(3)$ tells us that
when $K$ is a finite simplicial set,
these morphisms are in fact isomorphisms, as
the object $C^{\mathbb{R}K}$ can be functorially constructed using
successive homotopy products and homotopy fiber products. Therefore, conditions
$(4)$ implies that for any finite $K\in Ho(SSet_{\mathbb{U}})$
and any $C\in dg-Cat_{\mathbb{U}}$, the morphism $k_{C}$
induces an isomorphism
$$k_{C^{\mathbb{R}K}} : \pi_{0}(F(C^{\mathbb{R}K}))\simeq
[K,F(C)] \longrightarrow [K,G(C)]\simeq \pi_{0}(G(C^{\mathbb{R}K})).$$
This of course implies that $F(C) \longrightarrow G(C)$
is a weak equivalence. \hfill $\Box$ \\
In order to finish the proof of theorem \ref{t1} it remains to
show that the two functors $Map^{l}(C,-)$ and
$N(\mathcal{M}'(Q(C),-))$ satisfy the conditions
of sub-lemma \ref{sl3}. The case of
$Map^{l}(C,-)$ is clear by the standard properties
of mapping spaces (see \cite[\S 5.4]{ho} or \cite[\S 17]{hi}).
It only remains to show property
$(2)$ of sub-lemma \ref{sl3} for the functor $N(\mathcal{M}'(Q(C),-))$.
\begin{sublem}\label{sl5}
Let $C$ be a cofibrant $\mathbb{U}$-small dg-category,
and let
$$\xymatrix{
D \ar[r]^-{u} \ar[d]_-{v} & D_{1} \ar[d]^-{p} \\
D_{2} \ar[r]_-{q} & D_{3}}$$
be a cartesian diagram in $dg-Cat_{\mathbb{U}}$
with $p$ a fibration. Then, the
square
$$\xymatrix{
N(\mathcal{M}'(C,D)) \ar[r] \ar[d] & N(\mathcal{M}'(C,D_{1})) \ar[d] \\
N(\mathcal{M}'(C,D_{2})) \ar[r] & N(\mathcal{M}'(C,D_{3}))}$$
is homotopy cartesian.
\end{sublem}
\textit{Proof:} We start by showing that the morphism
$$N(\mathcal{M}'(C,D)) \longrightarrow
N(\mathcal{M}'(C,D_{1}))\times^{h}_{N(\mathcal{M}'(C,D_{3}))}
N(\mathcal{M}'(C,D_{2}))$$
induces an injection on $\pi_{0}$ and an isomorphism
on all $\pi_{i}$ for $i>0$. For this, we consider the
induced diagram
of dg-categories
$$\xymatrix{
C\otimes D^{op} \ar[r]^-{u} \ar[d]_-{v} & C\otimes D^{op}_{1} \ar[d]^-{p} \\
C\otimes D^{op}_{2} \ar[r]_-{q} & C\otimes D^{op}_{3},}$$
where we keep the same names for the induced morphisms
after tensoring with $C$. It is then enough to show that
for $F$ and $G$ in $\mathcal{M}(C,D)$ the square
of path spaces
$$\xymatrix{
\Omega_{F,G}N(\mathcal{M}'(C,D)) \ar[r] \ar[d] &
\Omega_{u_{!}F,u_{!}G}N(\mathcal{M}'(C,D_{1})) \ar[d] \\
\Omega_{v_{!}F,v_{!}G}N(\mathcal{M}'(C,D_{2})) \ar[r] &
\Omega_{w_{!}F,w_{!}G}N(\mathcal{M}'(C,D_{3})),}$$
is homotopy cartesian (where $w=p\circ u$). Using
the natural equivalence between path spaces
in nerves of sub-categories of equivalences
in model categories
and mapping spaces of equivalences
(see \cite{dk}, and also \cite[Appendix A]{hagII}), one finds that
the previous diagram is in fact equivalent to the
following one
$$\xymatrix{
Map^{eq}(F,G) \ar[r] \ar[d] &
Map^{eq}(u_{!}F,u_{!}G) \ar[d] \\
Map^{eq}(v_{!}F,v_{!}G) \ar[r] &
Map^{eq}(w_{!}F,w_{!}G),}$$
where $Map^{eq}$ denotes the sub-simplicial set
of the mapping spaces consisting of all connected
components corresponding to equivalences.
By adjunction, this last diagram is equivalent to
$$\xymatrix{
Map^{eq}(F,G) \ar[r] \ar[d] &
Map^{eq}(F,u^{*}u_{!}G) \ar[d] \\
Map^{eq}(F,v^{*}v_{!}G) \ar[r] &
Map^{eq}(F,w^{*}w_{!}G).}$$
Therefore, to show that this last square is homotopy
cartesian, it is enough to prove that for any $G\in \mathcal{M}(C,D)$ the
natural morphism
$$G\longrightarrow u^{*}u_{!}G \times^{h}_{w^{*}w_{!}G}v^{*}v_{!}G$$
is an equivalence in $C \otimes D^{op}-Mod_{\mathbb{U}}$.
As this can be tested by fixing some object $x\in C$ and considering
the corresponding morphism
$$G(x,-)\longrightarrow (u^{*}u_{!}G \times^{h}_{w^{*}w_{!}G}v^{*}v_{!}G)(x,-)$$
in $D^{op}-Mod_{\mathbb{U}}$, we see that one can
assume that $C=\mathbf{1}$.
One can then write
$G=\underline{h}_{x}$ for some point $x\in D$. For $z\in D$, one has natural
isomorphisms
$$u^{*}u_{!}G(z)=D_{1}(u(z),u(x))
\qquad v^{*}v_{!}G(z)=D_{2}(v(z),v(x)) \qquad
w^{*}w_{!}G(z)=D_{3}(w(z),w(x)).$$
We therefore find that for any $z\in D$ the morphism
$$G(z)\longrightarrow (u^{*}u_{!}G \times^{h}_{w^{*}w_{!}G}v^{*}v_{!}G)(z)$$
can be written as
$$D(z,x)\longrightarrow D_{1}(u(z),u(x)) \times_{D_{3}(w(z),w(x))}^{h}
D_{2}(v(z),v(x)),$$
which by assumption on the morphism $p$ is a quasi-isomorphism
of complexes. This implies that
the morphism
$$G\longrightarrow u^{*}u_{!}G \times^{h}_{w^{*}w_{!}G}v^{*}v_{!}G$$
is an equivalence, and thus that
$$N(\mathcal{M}'(C,D)) \longrightarrow
N(\mathcal{M}'(C,D_{1}))\times^{h}_{N(\mathcal{M}'(C,D_{3}))}
N(\mathcal{M}'(C,D_{2}))$$
induces an injection on $\pi_{0}$ and an isomorphisms
on all $\pi_{i}$ for $i>0$. It only remains to show that the
above morphism is also surjective on connected components.
The set $\pi_{0}(N(\mathcal{M}'(C,D_{1}))\times^{h}_{N(\mathcal{M}'(C,D_{3}))}
N(\mathcal{M}'(C,D_{2})))$ can be described in the
following way. We consider
a category $\mathcal{N}$ whose objects are 5-tuples
$(F_{1},F_{2},F_{3};a,b)$, with
$F_{i}\in \mathcal{M}(C,D_{i})$ and where $a$ and $b$ are two morphisms
in $\mathcal{M}(C,D_{3})$
$$a : p_{!}(F_{1}) \longrightarrow F_{3} \longleftarrow
q_{!}(F_{2}) : b.$$
Morphisms in $\mathcal{N}$ are defined in the obvious way,
as morphisms $F_{i} \rightarrow G_{i}$ in
$\mathcal{M}(C,D_{i})$, commuting with the morphisms
$a$ and $b$. It is not hard
to check that $\pi_{0}(N(\mathcal{N}))$ is naturally
isomorphic to
$\pi_{0}(N(\mathcal{M}'(C,D_{1}))\times^{h}_{N(\mathcal{M}'(C,D_{3}))}
N(\mathcal{M}'(C,D_{2})))$. Furthermore, the natural
map
$$\pi_{0}(N(\mathcal{M}(C,D)))
\longrightarrow
\pi_{0}(N(\mathcal{N}))$$
is induced by the functor
$\mathcal{M}(C,D)
\longrightarrow \mathcal{N}$ that
sends an object
$F\in \mathcal{M}(C,D)$ to
$(u_{!}F,v_{!}F,w_{!}F;a,b)$ where
$a$ and $b$ are the two natural
isomorphisms
$$p_{!}u_{!}(F) \simeq w_{!}(F)\simeq q_{!}v_{!}(F).$$
Now, let $(F_{1},F_{2},F_{3};a,b) \in \mathcal{N}$,
and let us define an object $F\in Ho((C\otimes D^{op})-Mod_{\mathbb{U}})$
by the following formula
$$F:=u^{*}(F_{1})\times^{h}_{w^{*}(F_{3})}v^{*}(F_{1}).$$
Clearly, one has natural morphisms
in $Ho(C\otimes D_{i}^{op}-Mod_{\mathbb{U}})$
$$\mathbb{L}u_{!}(F)\rightarrow F_{1} \qquad \mathbb{L}v_{!}(F)\rightarrow F_{2}
\qquad \mathbb{L}w_{!}(F)\rightarrow F_{3}.$$
We claim that $F$ is right quasi-representable and that
these morphisms are in fact
isomorphisms. This will clearly finish the proof of
the surjectivity on connected components.
For this one can clearly assume that $C=\mathbf{1}$. One can then
write
$F_{i}=\underline{h}_{x_{i}}$, for some
$x_{i}\in D_{i}$. As $p$ is a fibration, the equivalence
$$a : p_{!}(\underline{h}_{x_{1}})=\underline{h}_{p(x_{1})} \longrightarrow
\underline{h}_{x_{3}}$$
can be lifted to an equivalence
$\underline{h}_{x_{1}} \longrightarrow \underline{h}_{x_{1}'}$
in $D^{op}_{1}-Mod$.
Replacing $x_{1}$ by $x_{1}'$ one can suppose that
$p(x_{1})=x_{3}$ and $a=id$. In the same way, the equivalence
$$b : q_{!}(\underline{h}_{x_{2}}) \longrightarrow\underline{h}_{p(x_{1})}$$
can be lifted to an equivalence
$\underline{h}_{x_{1}''} \longrightarrow \underline{h}_{x_{1}}$
in $D^{op}_{1}-Mod$.
Thus, replacing $x_{1}$ by $x_{1}''$ one can suppose that
$q(x_{2})=p(x_{1})=x_{3}$ and that $a$ and $b$ are the
identity morphisms. Then, clearly
$F\simeq \underline{h}_{x}$, where $x\in D$
is the point given by $(x_{1},x_{2},x_{3})$.
This shows that $F$ is right quasi-representable, and also that
the natural morphisms
$$u_{!}(F)\rightarrow F_{1} \qquad
v_{!}(F)\rightarrow F_{2}
\qquad w_{!}(F)\rightarrow F_{3}$$
are equivalences. \hfill $\Box$ \\
We have now finished the proof of sub-lemma \ref{sl5}
and thus of theorem \ref{t1}. \hfill $\Box$ \\
Recall that $\mathcal{M}(Q(C),D)$ has been defined as the category
of equivalences between right quasi-representable
$Q(C)\otimes D^{op}$-modules $F$ such that $F(x,-)$ is cofibrant
in $D^{op}-Mod$ for any $x\in C$. This last condition is only
technical and useful for functorial reasons and does not affect the
nerve. Indeed, let $\mathcal{F}(Q(C),D)$ be the category of
all equivalences between right quasi-representable $(Q(C)\otimes D^{op})$-modules.
The natural inclusion functor
$$\mathcal{M}(Q(C),D) \longrightarrow \mathcal{F}(Q(C),D)$$
induces a weak equivalence on the corresponding nerves
as there exists a functor in the other direction just by taking
a cofibrant replacement (note that a cofibrant
$(Q(C)\otimes D^{op})$-module $F$ is such that
$F(x,-)$ is cofibrant for any $x\in Q(C)$, because of
Prop. \ref{p2}). In particular, theorem \ref{t1} implies the
existence of a string of weak equivalences
$$\xymatrix{
Map^{l}(C,D) \ar[r] & d(N(\mathcal{M}(\Gamma^{*}(C),D))) & \ar[l] N(\mathcal{M}(Q(C),D)) \ar[r] &
N(\mathcal{F}(Q(C),D)).}$$
The following corollary is a direct consequence of theorem
\ref{t1} and the above remark.
\begin{cor}\label{ct1}
Let $C$ and $D$ be two $\mathbb{U}$-small dg-categories.
Then, there exists a functorial
bijection between the set of maps $[C,D]$ in $Ho(dg-Cat_{\mathbb{U}})$, and
the set of isomorphism classes of right quasi-representable
objects in $Ho((C\otimes^{\mathbb{L}} D^{op})-Mod_{\mathbb{U}})$.
\end{cor}
Another important corollary of Theorem \ref{t1} is the
following.
\begin{cor}\label{ct1'}
Let $C$ be a $\mathbb{U}$-small dg-categories. Then, there exists
a functorial isomorphism between the set
$[\mathbf{1},C]$ and
the set of isomorphism classes of the category
$[C]$.
\end{cor}
\textit{Proof:} The Yoneda embedding
$\underline{h} : C \longrightarrow Int(C^{op}-Mod_{\mathbb{U}})$
induces a fully faithful functor
$$[C] \longrightarrow [Int(C^{op}-Mod_{\mathbb{U}})].$$
The essential image of this functor clearly is the
sub-category of quasi-representable $C^{op}$-modules. Therefore,
$[\underline{h}]$ induces a natural bijection between the
isomorphism classes of $[C]$ and the isomorphism classes
of quasi-representable objects in $[Int(C^{op}-Mod_{\mathbb{U}})]$.
As one has a natural equivalence
$[Int(C^{op}-Mod_{\mathbb{U}})]\simeq Ho(C^{op}-Mod_{\mathbb{U}})$
corollary \ref{ct1} implies the result. \hfill $\Box$ \\
More generally, one can describe the higher homotopy groups
of the mapping spaces by the following formula.
\begin{cor}\label{cint}
Let $C$ be a $\mathbb{U}$-small dg-category, and $x\in C$ be an object.
Then, one has natural isomorphisms of groups
$$\pi_{1}(Map(\mathbf{1},C),x)\simeq Aut_{[C]}(x) \qquad
\pi_{i}(Map(\mathbf{1},C),x)\simeq H^{1-i}(C(x,x)) \; \forall \; i>1.$$
\end{cor}
\textit{Proof:} We use the general formula
$$\pi_{1}(N(W),x)\simeq Aut_{Ho(M)}(x) \qquad
\pi_{i}(N(W),x)\simeq \pi_{i-1}(Map_{M}(x,x),Id) \; \forall \; i>1,$$
for a model category $M$, its sub-category of equivalences $W$ and
a point $x\in M$ (see e.g. \cite[Cor. A.0.4]{hagII}). Applied to
$M=C^{op}-Mod_{\mathbb{U}}$ and using theorem \ref{t1} one finds
$$\pi_{1}(Map(\mathbf{1},C),x)\simeq Aut_{Ho(C^{op}-Mod)}(\underline{h}_{x}) \qquad
\pi_{i}(Map(\mathbf{1},C),x)\simeq \pi_{i-1}(Map_{C^{op}-Mod}(\underline{h}_{x},\underline{h}_{x}),Id) \; \forall \; i>1.$$
Using that the morphism $\underline{h}$ is quasi-fully faithful one finds
$$Aut_{Ho(C^{op}-Mod)}(\underline{h}_{x}) \simeq Aut_{[C]}(x) \qquad
\pi_{i-1}(Map_{C^{op}-Mod}(\underline{h}_{x},\underline{h}_{x}),Id)\simeq
H^{1-i}(C(x,x)).$$
\hfill $\Box$ \\
\begin{cor}\label{ct1''}
Let $C$ and $D$ be two $\mathbb{U}$-small dg-categories.
Let $Int((C\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}}^{rqr})$
be the full sub-dg-category of
$Int((C\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}})$ consisting
of all right quasi-representable objects. Then,
$Int((C\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}}^{rqr})$
is isomorphic in $Ho(dg-Cat_{\mathbb{V}})$ to a $\mathbb{U}$-small
dg-category.
\end{cor}
\textit{Proof:} Indeed, we know by corollary \ref{ct1} that
the set of isomorphism classes of
$[Int((C\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}}^{rqr})]$
is essentially $\mathbb{U}$-small, as it is
in bijection with $[C,D]$. Let us choose an essentially $\mathbb{U}$-small
full sub-dg-category $E$ in $Int((C\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}}^{rqr})$ which
contains a set of representatives of isomorphism
classes of objects. As we already know that
the complexes of morphisms in $Int((C\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}}^{rqr})$
are $\mathbb{U}$-small, the dg-category $E$ is
essentially $\mathbb{U}$-small, and thus isomorphic to a
$\mathbb{U}$-small dg-category. As $E$ is quasi-equivalent to
$Int((C\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}}^{rqr})$
this implies the result.
\hfill $\Box$ \\
We finish by the following last corollary.
\begin{cor}\label{cint2}
Let $C$ and $D$ be two $\mathbb{U}$-small dg-categories, and
let $f,g : C \longrightarrow D$ be two morphisms
with corresponding $(C\otimes^{\mathbb{L}} D^{op})$-modules
$\phi(f)$ and $\phi(g)$. Then, there exists a natural weak equivalence
of simplicial sets
$$\Omega_{f,g}Map_{dg-Cat}(C,D)\simeq Map_{(C\otimes^{\mathbb{L}} D^{op})-Mod}^{eq}(\phi(f),\phi(g)),$$
where $Map^{eq}(\phi(f),\phi(g))$ is the sub-simplicial set of
$Map(\phi(f),\phi(g))$ consisting of equivalences.
\end{cor}
\textit{Proof:} This follows immediately from
theorem \ref{t1} and the standard relations between
path spaces of nerves of equivalences in a model category and
its mapping spaces (see e.g. \cite[Appendix A]{hagII}). \hfill $\Box$ \\
\section{The simplicial structure}
Let $K\in SSet_{\mathbb{U}}$ be a $\mathbb{U}$-small
simplicial set and $C\in dg-Cat_{\mathbb{U}}$. One can form
the derived tensor product $K\otimes^{\mathbb{L}}C \in Ho(dg-Cat_{\mathbb{U}})$,
as well as the derived exponential $C^{\mathbb{R}K}$. One has the usual
adjunction isomorphism
$$[K\otimes^{\mathbb{L}}C,D]\simeq [C,D^{\mathbb{R}K}]\simeq [K,Map(C,D)].$$
Let $\Delta(K)$ be the
simplex category of $K$. An object of $\Delta(K)$
is therefore a pair $(n,a)$ with $n\in \Delta$ and $x\in K_{n}$.
A morphism $(n,x) \rightarrow (m,y)$ is the data
of a morphism $u : [n] \rightarrow [m]$ in $\Delta$
such that $u^{*}(y)=x$. The simplicial set
$K$ is then naturally weakly equivalent to the
homotopy colimit of the constant diagram
$$\Delta(K) \longrightarrow * \in SSet.$$
In other words, one has a natural weak equivalence
$$N(\Delta(K))\simeq K.$$
We now consider
$\Delta(K)_{k}$ the $k$-linear category
freely generated by the category $\Delta(K)$, and
consider $\Delta(K)_{k}$ as an object
in $dg-Cat_{\mathbb{U}}$.
\begin{thm}\label{p5}
Let $C$ and $D$ be two $\mathbb{U}$-small
dg-categories, and $K\in SSet_{\mathbb{U}}$.
Then, there exists a functorial injective map
$$[K\otimes^{\mathbb{L}}C,D] \longrightarrow [\Delta(K)_{k}\otimes^{\mathbb{L}}C,D].$$
Moreover, the image of this map consists exactly of all
morphism $\Delta(K)_{k}\otimes^{\mathbb{L}}C\longrightarrow D$
in $Ho(dg-Cat_{\mathbb{U}})$ such that for any
$c\in C$ the induced functor
$$\Delta(K)_{k} \longrightarrow [D]$$
sends all morphisms in $\Delta(K)_{k}$ to
isomorphisms in $[D]$.
\end{thm}
\textit{Proof:} Using our theorem \ref{t1} one finds natural
equivalences
$$[K\otimes^{\mathbb{L}}C,D]\simeq [K,Map(C,D)]\simeq
[K,N(\mathcal{M}(Q(C),D))].$$
We then use the next technical lemma.
\begin{lem}\label{lp5}
Let $M$ be a $\mathbb{V}$-small $\mathbb{U}$-combinatorial model category
and $K\in SSet_{\mathbb{U}}$. Let $W\subset M$ be the sub-category
of equivalences in $M$. Then, there exists a natural bijection
between $[K,N(W)]_{SSet_{\mathbb{V}}}$ and the set of isomorphism
classes of objects
$F \in Ho(M^{\Delta(K)})$ corresponding to functors $F : \Delta(K)
\longrightarrow M$
sending all morphisms of $\Delta(K)$ to
equivalences in $M$.
\end{lem}
\textit{Proof:} First of all, the lemma is invariant by
changing $M$ up to a Quillen equivalence, and thus
by \cite{du} one can suppose that
$M$ is a simplicial model category.
The proof of the lemma will use some techniques of
simplicial localizations \`a la Dwyer-Kan,
as well as some result about $S$-categories. We start by
a short digression on the subject.
We recall the existence of a model category of
$S$-categories, as shown in \cite{be}, and which is similar
to the one we use on dg-categories. This model category
will be denoted by $S-Cat$ (or $S-Cat_{\mathbb{V}}$ if one needs
to specify the universe). For any
$\mathbb{V}$-small category $C$ with a sub-category $S\subset C$, one can
form a $\mathbb{V}$-small $S$-category $L(C,S)$ by formally inverting the
morphisms in $S$ in a homotopy meaningful way (see e.g. \cite{dk2}).
Using the language of model categories, this means that for any
$\mathbb{V}$-small $S$-category $T$, there exists functorial isomorphisms
between $[L(C,S),T]_{S-Cat}$ and the subset of $[C,T]_{S-Cat}$ consisting
of all morphisms sending $S$ to
isomorphisms in $[T]$ (the category $[T]$ is defined by taking connected
component of simplicial sets of morphisms in $T$). Finally, one can define
a functor $N : Ho(S-Cat_{\mathbb{V}}) \longrightarrow Ho(SSet_{\mathbb{V}})$
by sending an $S$-category to its nerve. It is well known that
the functor $N$ becomes an equivalence when restricted to
$S$-categories $T$ such that $[T]$ is a groupoid (this is
just another way to state delooping theory). Finally, for any
category $C$ with a sub-category $S\subset C$, one has a natural
weak equivalence $N(L(C,S))\simeq N(C)$.
Now, as explained in \cite[Prop. A.0.6]{hagII},
$N(W)$ can be also interpreted as the nerve
of the $S$-category $\mathcal{G}(M)$, of cofibrant and
fibrant objects in $M$ together with their simplicial sets of equivalences.
One therefore has natural isomorphism
$$[K,N(W)]\simeq [N(\Delta(K)),N(\mathcal{G}(M))]\simeq [L(\Delta(K),\Delta(K)),\mathcal{G}(M)].$$
Furthermore, as all morphisms in $[\mathcal{G}(M)]$ are isomorphisms one finds
a bijection between $[K,N(W)]$ and $[\Delta(K),\mathcal{G}(M)]$.
Let $Int(M)$ be the $S$-category of fibrant and cofibrant objects
in $M$ together with their simplicial sets of morphisms.
Then, as $\mathcal{G}(M)$ is precisely the sub-$S$-category of
$Int(M)$ consisting of equivalences, the set $[\Delta(K),\mathcal{G}(M)]$ is also
the subset of $[\Delta(K),Int(M)]$ consisting of all morphisms such that the induced
functor $\Delta(K) \longrightarrow [Int(M)]\simeq Ho(M)$
sends all morphisms to isomorphisms.
Finally,
it turns out that the same results as
our lemmas \ref{sl2} and \ref{sl1} are valid in the context of $S$-categories (their proofs
are exactly the same). Therefore, we see that $[\Delta(K),Int(M)]$ is in a natural bijection with
isomorphism classes of objects in $Ho(M^{\Delta(K)})$. Putting all of this together
gives the lemma. \hfill $\Box$ \\
We apply the previous lemma to the case where
$M:=(C\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}}$, and we find a
natural injection $[K,N(W)] \hookrightarrow
Iso(Ho(M^{\Delta(K)}))$, whose image consists of all functors
$\Delta(K) \rightarrow M$ sending all morphisms of $\Delta(K)$ to
equivalences in $M$. Composing with the natural inclusion
$\mathcal{M}(Q(C),D)\subset M$ provides a natural injection of
$$[K,N(\mathcal{M}(Q(C),D))] \subset [K,N(W)] \subset
Iso(Ho(M^{\Delta(K)})).$$
By the construction of the bijection of lemma \ref{lp5} one easily sees
that the image of this inclusion consists of all
functors $F : \Delta(K) \longrightarrow W$ such that
for any $k\in K$ one has $F(k)\in \mathcal{M}(Q(C),D)$.
Finally, one clearly has a natural equivalence of categories, compatible
with the model structures
$$M^{\Delta(K)}\simeq (C\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}}^{\Delta(K)_{k}}\simeq
(\Delta(K)_{k}\otimes C\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}},$$
inducing a bijection between
$Iso(Ho(M^{\Delta(K)}))$ and the isomorphism classes of
objects in $Ho((\Delta(K)_{k}\otimes C\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}})$.
Another application of theorem \ref{t1} easily implies the result.
\hfill $\Box$ \\
\section{Internal Hom's}
Let us recall that $Ho(dg-Cat_{\mathbb{U}})$ is endowed with
the symmetric monoidal structure $\otimes^{\mathbb{L}}$.
Recall that the monoidal structure $\otimes^{\mathbb{L}}$
is said to be closed if for any two objects
$C$ and $D$ in $Ho(dg-Cat_{\mathbb{U}})$ the functor
$A \mapsto [A\otimes^{\mathbb{L}}C,D]$ is representable
by an object $\mathbb{R}\underline{Hom}(C,D)\in Ho(dg-Cat_{\mathbb{U}})$.
Recall also from
corollary \ref{ct1''} that the $\mathbb{V}$-small dg-category
$Int((C\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}}^{rqr})$
is essentially $\mathbb{U}$-small and therefore can be
considered as an object in $Ho(dg-Cat_{\mathbb{U}})$.
\begin{thm}\label{t2}
The monoidal category $(Ho(dg-Cat_{\mathbb{U}}),\otimes^{\mathbb{L}})$
is closed. Furthermore, for any two $\mathbb{U}$-small dg-categories
$C$ and $D$ one has a natural isomorphism in $Ho(dg-Cat_{\mathbb{U}})$
$$\mathbb{R}\underline{Hom}(C,D)\simeq
Int((C\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}}^{rqr}).$$
\end{thm}
\textit{Proof:} The proof is essentially the same
as for theorem \ref{t1} and is also based on the same
lemmas \ref{sl2} and \ref{sl1}. Indeed, from these
two lemmas one extracts the following result.
\begin{lem}\label{l3}
Let $M$ be $C(k)_{\mathbb{U}}$-enriched
$\mathbb{U}$-cofibrantly generated model category
which is $\mathbb{V}$-small. We assume that
the domain and codomain of a set of generating
cofibrations are cofibrant in $M$.
Let $M_{0}$ be
a full sub-category of $M$ which is closed by equivalences, and
$Int(M_{0})$ be the full sub-dg-category of
$Int(M)$ consisting of all objects belonging to $M_{0}$.
Let $A$ be a cofibrant and
$\mathbb{U}$-small dg-category, and let $Ho(M_{0}^{A})$ be the
full sub-category of $Ho(M^{A})$ consisting of objects
$F\in Ho(M^{A})$ such that $F(a)\in M_{0}$ for any $a\in A$.
Then, one has a natural
isomorphism
$$\phi : [A,Int(M_{0})]\simeq Iso(Ho(M_{0}^{A})).$$
\end{lem}
\textit{Proof:} The morphism
$$\phi : [A,Int(M_{0})]\longrightarrow Iso(Ho(M_{0}^{A}))$$
simply sends a morphism $A \longrightarrow
Int(M_{0})$ to the corresponding object in $M_{0}^{A}$.
Using our proposition \ref{p1} it is easy to see that
this maps sends homotopic morphisms to isomorphic
objects in $Ho(M_{0}^{A})$, and is therefore well defined.
As for the proof of lemma \ref{sl2}, the morphism
$\phi$ is clearly surjective. Let $u,v : A \longrightarrow
Int(M_{0})$ be two morphisms of dg-categories such that
the corresponding objects in $Ho(M_{0}^{A})$ are isomorphic.
Then, these objects are isomorphic in $Ho(M^{A})$, which
implies by lemma \ref{sl1} that the two compositions
$$u',v' : A \longrightarrow Int(M_{0}) \longrightarrow Int(M)$$
are homotopic in $dg-Cat_{\mathbb{V}}$. Let
$$\xymatrix{
A \ar[rd]^-{u'} \ar[d] & \\
A' \ar[r]^-{H} & Int(M) \\
A \ar[ru]_-{v'} \ar[u]}$$
be a homotopy between $u'$ and $v'$. As $M_{0}$ is closed by equivalences
in $M$ one clearly sees that the morphism $H$ factors through the
sub-dg-category $Int(M_{0})$, showing that
$u$ and $v$ are homotopic. \hfill $\Box$ \\
We come back to the proof of theorem \ref{t2}.
Using our theorem \ref{t1} one has a natural
isomorphism
$$[A\otimes^{\mathbb{L}}C,D]\simeq
Iso(Ho(((A\otimes^{\mathbb{L}} C)\otimes^{\mathbb{L}} D^{op})-Mod_{\mathbb{U}}^{rqr}))\simeq
Iso(Ho(((C\otimes^{\mathbb{L}} D^{op})-Mod_{\mathbb{U}}^{rqr})^{A})).$$
An application of lemma \ref{l3} (with
$M=(C\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}}$ and
$M_{0}$ the full sub-category of right quasi-representable
objects)
shows that
one has a natural isomorphism
$$[A,Int((C\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}}^{rqr})]\simeq
Iso(Ho(((C\otimes^{\mathbb{L}} D^{op})-Mod_{\mathbb{U}}^{rqr})^{A})).$$
Putting this together one finds a natural isomorphism
$$[A\otimes^{\mathbb{L}}C,D]\simeq [A,Int((C\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}}^{rqr})]$$
showing the theorem. \hfill $\Box$ \\
\begin{cor}\label{cp5}
For any $C$ and $D$ two $\mathbb{U}$-small
dg-categories, and any $K\in SSet_{\mathbb{U}}$, one has
a functorial isomorphism in $Ho(dg-Cat_{\mathbb{U}})$
$$K\otimes^{\mathbb{L}}(C\otimes^{\mathbb{L}}D)\simeq
(K\otimes^{\mathbb{L}}C)\otimes^{\mathbb{L}}D.$$
\end{cor}
\textit{Proof:} This follows easily from Thm. \ref{p5},
Thm. \ref{t2} and the Yoneda lemma applied to $Ho(dg-Cat_{\mathbb{U}})$.
\hfill $\Box$ \\
\begin{cor}\label{cp5'}
For any $C$, $D$ and $E$ three $\mathbb{U}$-small
dg-categories one has
a functorial isomorphism in $Ho(SSet_{\mathbb{U}})$
$$Map(C\otimes^{\mathbb{L}}D,E)\simeq
Map(C,\mathbb{R}\underline{Hom}(D,E)).$$
\end{cor}
\textit{Proof:} By Cor. \ref{cp5}, for any $K\in SSet_{\mathbb{U}}$, one has
functorial isomorphisms
$$[K,Map(C\otimes^{\mathbb{L}}D,E)]\simeq
[K\otimes^{\mathbb{L}}(C\otimes^{\mathbb{L}}D),E]\simeq
[(K\otimes^{\mathbb{L}}C)\otimes^{\mathbb{L}}D,E]\simeq$$
$$[K\otimes^{\mathbb{L}}C,\mathbb{R}\underline{Hom}(D,E)]\simeq
[K,Map(C,\mathbb{R}\underline{Hom}(D,E))].$$
\hfill $\Box$ \\
\begin{cor}\label{cp5''}
Let $C\in dg-Cat_{\mathbb{U}}$ be
a dg-category. Then the functor
$$-\otimes^{\mathbb{L}}C : dg-Cat_{\mathbb{U}} \longrightarrow dg-Cat_{\mathbb{U}}$$
commutes with homotopy colimits.
\end{cor}
\textit{Proof:} This follows formally from Cor. \ref{cp5'}. \hfill $\Box$ \\
\begin{cor}\label{cp5'''}
Let $C \longrightarrow D$ be a quasi-fully faithful morphism in $dg-Cat_{\mathbb{U}}$. Then, for
any $B\in dg-Cat_{\mathbb{U}}$ the induced morphism
$$\mathbb{R}\underline{Hom}(B,C) \longrightarrow \mathbb{R}\underline{Hom}(B,D)$$
is quasi-fully faithful.
\end{cor}
\textit{Proof:} Using Lem. \ref{lmono} it is enough to show that
$\mathbb{R}\underline{Hom}(B,-)$ preserves homotopy monomorphisms. But this
follows formally from Cor. \ref{cp5'}. \hfill $\Box$ \\
\section{Morita morphisms and bi-modules}
In this paragraph we will use the following notations. For any
$C\in dg-Cat_{\mathbb{U}}$ one sets
$$\widehat{C}:=Int(C^{op}-Mod_{\mathbb{U}})\in dg-Cat_{\mathbb{V}}.$$
By theorem \ref{t2} and lemma \ref{l3}, one has an isomorphism in
$Ho(dg-Cat_{\mathbb{V}})$
$$\widehat{C}\simeq \mathbb{R}\underline{Hom}(C^{op},Int(C(k)_{\mathbb{U}})) \in Ho(dg-Cat_{\mathbb{V}}).$$
Indeed, lemma \ref{l3} implies that for any $A\in dg-Cat_{\mathbb{U}}$ one has
$$[A,\widehat{C}]\simeq Iso(Ho((A\otimes^{\mathbb{L}}C^{op})-Mod_{\mathbb{U}}))\simeq
[A\otimes^{\mathbb{L}}C^{op},\widehat{\mathbf{1}}].$$
Note also that
$$Int(C(k)_{\mathbb{U}})\simeq \widehat{\mathbf{1}}.$$
We will also consider $\widehat{C}_{pe}$ the full sub-dg-category of
$\widehat{C}$ consisting of $C^{op}$-modules which are
homotopically finitely presented. In other words,
a $C^{op}$-module $F$ is in $\widehat{C}_{pe}$ if for any
filtered diagram of objects $G_{i}$ in $C^{op}-Mod_{\mathbb{U}}$, the natural morphism
$$Colim_{i}Map(F,G_{i}) \longrightarrow Map(F,Colim_{i}G_{i})$$
is a weak equivalence. It is easy to check that the
objects in $\widehat{C}_{pe}$ are precisely the objects equivalent to retracts of finite
cell $C^{op}$-modules. To be more precise, an object $F\in Ho(\widehat{C})$ is
in $Ho(\widehat{C}_{pe})$ if and only if it is a retract in $Ho(\widehat{C})$ of
an object $G$ for which there exists a finite sequence of morphisms of $C^{op}$-modules
$$\xymatrix{
0 \ar[r] & G_{1} \ar[r] & G_{2} \ar[r] & \dots \ar[r] & G_{n}=G,}$$
in such a way that for any $i$ there exists a push-out square
$$\xymatrix{G_{i} \ar[r] & G_{i+1} \\
A\otimes \underline{h}_{x} \ar[u] \ar[r] & \ar[u] B\otimes \underline{h}_{x}}$$
for some $x\in C$, and some cofibration $A\rightarrow B$ in $C(k)$ with
$A$ and $B$ bounded complexes of projective modules of finite type.
Objects in $\widehat{C}_{pe}$ will also be called
\emph{compact} or \emph{perfect} (note that they are precisely
the compact objects in the triangulated category $[\widehat{C}]$, in the
usual sense). More generally, for any dg-category $T$, we will write
$T_{pe}$ for the full sub-dg-category of $T$ consisting of compact objects
(i.e. the objects $x$ such that $[T](x,-)$ commutes with
(infinite) direct sums). \\
Let us consider $C$ and $D$ two $\mathbb{U}$-small dg-categories,
and $u : \widehat{C} \longrightarrow \widehat{D}$ a morphism in $Ho(dg-Cat_{\mathbb{V}})$.
Then, $u$ induces a functor, well defined up to an (non-unique) isomorphism
$$[u] : [\widehat{C}]\longrightarrow [\widehat{D}].$$
We will say that the morphism $u$ is \emph{continuous} if the functor
$[u]$ commutes with $\mathbb{U}$-small direct sums. Note that
$[\widehat{C}]$ and $[\widehat{D}]$ are the homotopy categories of the model
categories of $C^{op}$-modules and $D^{op}$-modules, and thus these two
categories always have direct sums. More generally, we will denote by
$\mathbb{R}\underline{Hom}_{c}(\widehat{C},\widehat{D})$ the full sub-dg-category
of $\mathbb{R}\underline{Hom}(\widehat{C},\widehat{D})$ consisting of
continuous morphisms.
\begin{df}\label{d5}
Let $C$ and $D$ be two $\mathbb{U}$-small dg-categories.
\begin{enumerate}
\item
The dg-category of \emph{Morita morphisms} from $C$ to $D$ is
$\mathbb{R}\underline{Hom}_{c}(\widehat{C},\widehat{D})$.
\item The dg-category of \emph{perfect Morita morphisms} from $C$ to $D$ is
$\mathbb{R}\underline{Hom}(\widehat{C}_{pe},\widehat{D}_{pe})$.
\end{enumerate}
\end{df}
We warn the reader that there are in general no relations between
the dg-category $\mathbb{R}\underline{Hom}(\widehat{C}_{pe},\widehat{D}_{pe})$ and
$\mathbb{R}\underline{Hom}_{c}(\widehat{C},\widehat{D})_{pe}$. An example where these
two objects agree will be given in Thm. \ref{tfour2}. \\
\begin{thm}\label{t3}
Let $C\in dg-Cat_{\mathbb{U}}$, and let us consider the Yoneda
embedding $\underline{h} : C \longrightarrow \widehat{C}$. Let
$D$ be any $\mathbb{U}$-small dg-category.
\begin{enumerate}
\item
The pull-back functor
$$\underline{h}^{*} : \mathbb{R}\underline{Hom}_{c}(\widehat{C},\widehat{D}) \longrightarrow
\mathbb{R}\underline{Hom}(C,\widehat{D})$$
is an isomorphism in $Ho(dg-Cat_{\mathbb{V}})$.
\item The pull-back functor
$$\underline{h}^{*} : \mathbb{R}\underline{Hom}(\widehat{C}_{pe},\widehat{D}_{pe}) \longrightarrow
\mathbb{R}\underline{Hom}(C,\widehat{D}_{pe})$$
is an isomorphism in $Ho(dg-Cat_{\mathbb{V}})$.
\end{enumerate}
\end{thm}
\textit{Proof:} We start by proving $(1)$. \\
Using the universal properties of internal Hom's one reduces the problem to show that for
any $A\in dg-Cat_{\mathbb{U}}$, the morphism\footnote{We prefer to change
notation from $\underline{h}$ to $l$ during the proof, just
in order to avoid future confusions.}
$$l:=\underline{h} : C \longrightarrow \widehat{C}$$
induces a bijective morphism
$$l^{*} : [\widehat{C}\otimes^{\mathbb{L}} A,\widehat{D}]_{c} \longrightarrow
[C\otimes^{\mathbb{L}} A,\widehat{D}],$$
where by definition $[\widehat{C}\otimes^{\mathbb{L}} A,\widehat{D}]_{c}$
is the subset of $[\widehat{C}\otimes^{\mathbb{L}} A,\widehat{D}]$ consisting
of morphisms $f : \widehat{C}\otimes^{\mathbb{L}} A \longrightarrow \widehat{D}$
such that for any object $a\in A$ the induced morphism $f(-,a) : \widehat{C} \longrightarrow
\widehat{D}$ is continuous. Now, as $\widehat{D}=\mathbb{R}\underline{Hom}(D^{op},\widehat{\mathbf{1}})$,
one has natural bijections
$$[C\otimes^{\mathbb{L}} A,\widehat{D}]\simeq [C,\widehat{A^{op}\otimes^{\mathbb{L}}D}] \qquad
[\widehat{C}\otimes^{\mathbb{L}} A,\widehat{D}]_{c} \simeq [\widehat{C},\widehat{A^{op}\otimes^{\mathbb{L}}D}]_{c}.$$
Therefore, we have to prove that for any $A$ the induced morphism
$$l^{*} : [\widehat{C},\widehat{A^{op}\otimes^{\mathbb{L}}D}]_{c} \longrightarrow
[C,\widehat{A^{op}\otimes^{\mathbb{L}}D}],$$
is bijective. For this, we consider the
quasi-fully faithful morphism in $dg-Cat_{\mathbb{W}}$
for some universe $\mathbb{V}\in \mathbb{W}$
$$\widehat{A^{op}\otimes^{\mathbb{L}}D}\simeq Int((A\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}}) \longrightarrow
\widehat{A^{op}\otimes^{\mathbb{L}}D}_{\mathbb{V}}:=
Int((A\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{V}}).$$
One has a commutative square
$$\xymatrix{
[\widehat{C},\widehat{A^{op}\otimes^{\mathbb{L}}D}]_{c} \ar[r] \ar[d] &
[\widehat{C},\widehat{A^{op}\otimes^{\mathbb{L}}D}_{\mathbb{V}}]_{c} \ar[d] \\
[C,\widehat{A^{op}\otimes^{\mathbb{L}}D}] \ar[r] &
[C,\widehat{A^{op}\otimes^{\mathbb{L}}D}_{\mathbb{V}}].}$$
We claim that the right vertical morphism is bijective. For this,
we use lemma \ref{l3} which implies that it is enough to show
the following lemma.
\begin{lem}\label{lt3}
Let $C$ be a $\mathbb{U}$-small dg-category and
$M$ a $\mathbb{V}$-combinatorial $C(k)_{\mathbb{V}}$-model category
which is $\mathbb{W}$-small for some $\mathbb{V}\in \mathbb{W}$.
We assume that the domain and codomain of a set of
generating cofibrations are cofibrant in $M$.
We also assume that for any cofibrant object $X\in M$, and any
quasi-isomorphism $Z\longrightarrow Z'$ in $C(k)$, the
induced morphism
$$Z\otimes X \longrightarrow Z'\otimes X$$
is an equivalence in $M$.
Then, the Quillen adjunction
$$l_{!} : M^{C} \longrightarrow M^{\widehat{C}} \qquad
M^{C} \longleftarrow M^{\widehat{C}} : l^{*}$$
induces a fully faithful functor
$$\mathbb{L}l_{!} : Ho(M^{C}) \longrightarrow
Ho(M^{\widehat{C}})$$
whose essential image consists of all $\widehat{C}$-modules
corresponding to continuous morphisms in $Ho(dg-Cat_{\mathbb{W}})$.
\end{lem}
\textit{Proof:} First of all, the modules $F \in Ho(M^{\widehat{C}})$
corresponding to continuous morphisms are precisely the ones
for which for any $\mathbb{U}$-small family of objects $x_{i}\in \widehat{C}$, the
natural morphism
$$\bigoplus^{\mathbb{L}} F(x_{i}) \longrightarrow F(\oplus_{i}x_{i})$$
is an isomorphism in $Ho(M)$.
We start by showing that $\mathbb{L}l_{!}$ is fully faithful.
As both functors $\mathbb{L}l_{!}$ and $l^{*}$ commute
with homotopy colimits, it is enough to show that
for any $x\in C$ and any $X\in M$, the adjunction morphism
$$X\otimes^{\mathbb{L}} \underline{h}^{x} \longrightarrow l^{*}\mathbb{L}l_{!}(
X\otimes^{\mathbb{L}} \underline{h}^{x})$$
is an isomorphism in $Ho(M^{C})$. But this follows immediately from the fact that
the morphism of dg-categories $l$ is fully faithful
and our hypothesis on $M$.
It remains to show that for any $F\in Ho(M^{\widehat{C}})$, corresponding to
a continuous morphism, the adjunction morphism
$$\mathbb{L}l_{!}l^{*}(F) \longrightarrow F$$
is an isomorphism in $Ho(M^{\widehat{C}})$. As we already
know that $\mathbb{L}l_{!}$ is fully faithful it is enough to
show that the functor $l^{*}$ is conservative when restricted
to the sub-category of modules corresponding to continuous functors.
Let $u : F \longrightarrow G$ be morphism between such modules, and let us
assume that $l^{*}(F) \longrightarrow l^{*}(G)$
is an isomorphism in $Ho(M^{C})$. We need to show that
$u$ itself is an isomorphism in $Ho(M^{\widehat{C}})$.
\begin{sublem}\label{sllt3}
Let $F : \widehat{C} \longrightarrow M$ be a morphism of
dg-categories corresponding to a continuous morphism.
\begin{enumerate}
\item
Let $X : I \longrightarrow C^{op}-Mod_{\mathbb{U}}$
be a $\mathbb{U}$-small diagram of cofibrant objects in
$C^{op}-Mod_{\mathbb{U}}$. Then, the natural morphism
$$Hocolim_{i}F(X_{i}) \longrightarrow
F(Hocolim_{i}X_{i})$$
is an isomorphism in $Ho(M)$.
\item Let $Z\in C(k)_{\mathbb{U}}$ and $X\in M$. Then, the natural
morphism
$$Z\otimes^{\mathbb{L}}F(X) \longrightarrow F(Z\otimes^{\mathbb{L}}X)$$
is an isomorphism in $Ho(M)$.
\end{enumerate}
\end{sublem}
\textit{Proof of sub-lemma \ref{sllt3}:}
$(1)$ As any homotopy colimit is a composition of
homotopy push-outs and infinite (homotopy) sums, it is enough to check the
sub-lemma for one of these colimits. For the direct sum case this is
our hypothesis on $F$. It remains to show that $F$ commutes with
homotopy push-outs. For this we assume that $F$ is fibrant and cofibrant, and thus
is given by a morphism of dg-categories $\widehat{C} \longrightarrow Int(M)$.
We consider the commutative diagram of dg-categories
$$\xymatrix{
(\widehat{C})^{op} \ar[r]^-{F} \ar[d] & Int(M)^{op} \ar[d] \\
Int(\widehat{C}-Mod_{\mathbb{V}}) \ar[r]_-{F_{!}} &
Int(Int(M)-Mod_{\mathbb{V}}),}$$
where the vertical morphisms are the dual Yoneda embeddings $\underline{h}^{(-)}$.
The functor $F_{!}$ being left Quillen clearly commutes, up to equivalences,
with homotopy push-outs. Furthermore, as the model categories
$\widehat{C}-Mod_{\mathbb{V}}$ and $Int(M)-Mod_{\mathbb{V}}$ are stable model categories, this
implies that $F_{!}$ also commutes, up to equivalence, with homotopy pull-backs.
Furthermore, the morphism $\underline{h}^{(-)}$ sends
homotopy push-out squares to homotopy pull-back squares, and moreover
a square in $Int(M)$ is a homotopy push-out square if and only if
its image by $\underline{h}$ is a homotopy pull-back square
in $Int(M)-Mod_{\mathbb{V}}$. We deduce from these remarks that
$F$ preserves homotopy push-out squares. \\
$(2)$ Any complex $Z$ can be constructed from the trivial
complex $k$ using homotopy colimits and loop objects.
As we already know that $F$ commutes with homotopy colimits, it
is enough to see that it also commutes with loop objects.
But the loop functor is inverse, up to equivalence, to
the suspension functor. The suspension being a homotopy push-out,
$F$ commutes with it, and therefore $F$ commutes with the loop functor.
\hfill $\Box$ \\
Now, let us come back to our morphism
$u : F \longrightarrow G$ such that $l^{*}(u)$ is
an equivalence. Let $X$ be an object in $\widehat{C}$. We know that
$X$ can be written as the homotopy colimit of
objects of the form $Z\otimes^{\mathbb{L}}\underline{h}_{x}$
with $x\in C$ and $Z\in C(k)$.
Therefore, one has a commutative
diagram in $Ho(M)$
$$\xymatrix{
Hocolim_{i} F(Z_{i}\otimes^{\mathbb{L}}\underline{h}_{x_{i}}) \ar[d] \ar[r]^-{u}
& Hocolim_{i} G(Z_{i}\otimes^{\mathbb{L}}\underline{h}_{x_{i}}) \ar[d] \\
F(X) \ar[r]^-{u} & G(X).}$$
By the sub-lemma $(1)$ the vertical morphisms are isomorphisms in $Ho(M)$, and
the top horizontal morphism is also by hypothesis
and the sub-lemma $(2)$.
Thus, the bottom horizontal morphism is an isomorphism in $Ho(M)$, and this
for any $X\in \widehat{C}$. This shows that $l^{*}$ is conservative when
restricted to continuous morphisms, and thus finishes the proof of the
lemma \ref{lt3}. \hfill $\Box$ \\
We come back to our commutative diagram
$$\xymatrix{
[\widehat{C},\widehat{A^{op}\otimes^{\mathbb{L}}D}]_{c} \ar[r] \ar[d] &
[\widehat{C},\widehat{A^{op}\otimes^{\mathbb{L}}D}_{\mathbb{V}}]_{c} \ar[d] \\
[C,\widehat{A^{op}\otimes^{\mathbb{L}}D}] \ar[r] &
[C,\widehat{A^{op}\otimes^{\mathbb{L}}D}_{\mathbb{V}}].}$$
Lemma \ref{lt3} shows that the right vertical
morphism is bijective, and corollary \ref{clmono} implies that
the horizontal morphisms are injective. It remains to show that
a morphism $u\in [\widehat{C},\widehat{A^{op}\otimes^{\mathbb{L}}D}_{\mathbb{V}}]_{c}$,
whose restriction $C \longrightarrow \widehat{A^{op}\otimes^{\mathbb{L}}D}_{\mathbb{V}}$
factors thought
$\widehat{A^{op}\otimes^{\mathbb{L}}D}$, itself factors through
$\widehat{A^{op}\otimes^{\mathbb{L}}D}$. But this is true
as by sub-lemma \ref{sllt3} the image by $u$ of any
$C^{op}$-module can be written as a $\mathbb{U}$-small homotopy colimit
of objects of the form $Z\otimes^{\mathbb{L}}u(l(x)) $
for $Z\in C(k)_{\mathbb{U}}$ and $x\in C$. Therefore,
if the restriction of $u$ to $C$ has $\mathbb{U}$-small images, then so does
$u$ itself. This finishes the proof of theorem \ref{t3} $(1)$. \\
$(2)$ We consider the quasi-fully faithful morphism
$\widehat{D}_{pe} \longrightarrow \widehat{D}$. We therefore have
a homotopy commutative diagram
$$\xymatrix{
\mathbb{R}\underline{Hom}(\widehat{C}_{pe},\widehat{D}_{pe}) \ar[r]
\ar[d] & \ar[d] \mathbb{R}\underline{Hom}(\widehat{C}_{pe},\widehat{D}) \\
\mathbb{R}\underline{Hom}(C,\widehat{D}_{pe}) \ar[r] & \mathbb{R}\underline{Hom}(C,\widehat{D}),}$$
where the horizontal morphisms are quasi-fully faithful by Cor. \ref{cp5'''}. We claim that the
right vertical morphism is a quasi-equivalence. For this, using the universal
properties of internal Hom's, it is enough to show that the induced morphism
$$[\widehat{C}_{pe},\widehat{D}] \longrightarrow [C,\widehat{D}]$$
is bijective for any $D$. Using our lemma \ref{l3} one sees that it is enough to prove the following
lemma.
\begin{lem}\label{lt3'}
Let $C$ be a cofibrant and $\mathbb{U}$-small dg-category and
$M$ a $\mathbb{V}$-combinatorial $C(k)_{\mathbb{V}}$-model category
satisfying the same assumption as in lemma \ref{lt3}.
\begin{enumerate}
\item
Then, the Quillen adjunction
$$l_{!} : M^{C} \longrightarrow M^{\widehat{C}_{pe}} \qquad
M^{C} \longleftarrow M^{\widehat{C}_{pe}} : l^{*}$$
is a Quillen equivalence.
\item For any $F\in M^{\widehat{C}_{pe}}$, and
any a $\mathbb{U}$-small diagram of perfect and cofibrant objects in
$C^{op}-Mod_{\mathbb{U}}$,
$X : I \longrightarrow C^{op}-Mod_{\mathbb{U}}$,
the natural morphism
$$Hocolim_{i}F(X_{i}) \longrightarrow
F(Hocolim_{i}X_{i})$$
is an isomorphism in $Ho(M)$.
\item For any $F\in M^{\widehat{C}_{pe}}$, and any
perfect complex $Z\in C(k)_{\mathbb{U}}$ and any $X\in M$, the natural
morphism
$$Z\otimes^{\mathbb{L}}F(X) \longrightarrow F(Z\otimes^{\mathbb{L}}X)$$
is an isomorphism in $Ho(M)$.
\end{enumerate}
\end{lem}
\textit{Proof:} This is the same as for lemma \ref{lt3}
and sub-lemma \ref{sllt3}. \hfill $\Box$ \\
Coming back to our square of dg-categories one sees that
the horizontal morphisms are quasi-fully faithful and that the right vertical morphism
is a quasi-equivalence. This formally implies that the left vertical morphism
is quasi-fully faithful. We now consider the square of sets
$$\xymatrix{
[\widehat{C}_{pe},\widehat{D}_{pe}] \ar[r] \ar[d] &
[\widehat{C}_{pe},\widehat{D}] \ar[d] \\
[C,\widehat{D}_{pe}] \ar[r] & [C,\widehat{D}],}$$
obtained from the square of dg-categories by passing to equivalence classes
of objects. Again, the right vertical morphism is a bijection and the horizontal
morphisms are injective.
For $u\in [C,\widehat{D}_{pe}]$, its image in $[C,\widehat{D}]$
comes from an element $v\in [\widehat{C}_{pe},\widehat{D}]$.
For any $x\in C$,
$v(l(x)) \in \widehat{D}$ is a perfect $D^{op}$-module, and thus
so is $v(Z\otimes^{\mathbb{L}}l(x))\simeq Z\otimes^{\mathbb{L}}v(l(x))$
for any perfect complex $Z$ of $k$-modules.
As any perfect $C^{op}$-module is constructed as a retract
of a finite homotopy colimit of objects of the form
$Z\otimes^{\mathbb{L}}l(x)$, we deduce that
$v(X)$ is a perfect $D^{op}$-module for any
$X\in \widehat{C }_{pe}$. Therefore, Cor. \ref{clmono} implies that
$v$ comes in fact from an element in $[\widehat{C}_{pe},\widehat{D}_{pe}]$.
This shows that
$[\widehat{C}_{pe},\widehat{D}_{pe}] \longrightarrow [C,\widehat{D}_{pe}]$ is
surjective, and thus
that
$$\mathbb{R}\underline{Hom}(\widehat{C}_{pe},\widehat{D}_{pe}) \longrightarrow
\mathbb{R}\underline{Hom}(C,\widehat{D}_{pe})$$
is quasi-essentially surjective. This finishes the proof of the theorem. \hfill $\Box$ \\
The following corollary is the promised derived version of Morita theory.
\begin{cor}\label{ct3}
Let $C$ and $D$ be two $\mathbb{U}$-small dg-categories, then there exists a natural
isomorphism in $Ho(dg-Cat_{\mathbb{V}})$
$$\mathbb{R}\underline{Hom}_{c}(\widehat{C},\widehat{D})\simeq
\widehat{C^{op}\otimes^{\mathbb{L}}D}\simeq
Int((C\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}}).$$
In particular, there exists a natural weak equivalence
$$Map_{c}(\widehat{C},\widehat{D})\simeq
|(C\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}}|,$$
where $Map_{c}(\widehat{C},\widehat{D})$ is the sub-simplicial set
of continuous morphisms in $Map(\widehat{C},\widehat{D})$ and where
$|(C\otimes^{\mathbb{L}}D^{op})-Mod_{\mathbb{U}}|$ is the nerve
of the sub-category of equivalences between $C\otimes^{\mathbb{L}}D^{op}$-modules.
\end{cor}
\textit{Proof:} The first part follows from the universal properties
of internal Hom's, as by theorem \ref{t3}
$$\mathbb{R}\underline{Hom}_{c}(\widehat{C},\widehat{D})\simeq
\mathbb{R}\underline{Hom}(C,\mathbb{R}\underline{Hom}(D^{op},\widehat{\mathbf{1}}))\simeq
\mathbb{R}\underline{Hom}(C\otimes^{\mathbb{L}}D^{op},\widehat{\mathbf{1}})\simeq
\widehat{C^{op}\otimes^{\mathbb{L}}D}.$$
The second part follows from the relation between mapping spaces and
internal Hom's, as well as Prop. \ref{cp5'}. Indeed, one has
$$Map_{c}(\widehat{C},\widehat{D}) \simeq Map(\mathbf{1},\mathbb{R}\underline{Hom}_{c}(\widehat{C},\widehat{D}))\simeq
Map(\mathbf{1},\mathbb{R}\underline{Hom}(C\otimes^{\mathbb{L}}D^{op},\widehat{\mathbf{1}}))
\simeq
Map(\mathbf{1},\widehat{C^{op}\otimes^{\mathbb{L}}D}).$$
By theorem \ref{t1} this last simplicial set is weakly equivalent to
the nerve of the category of equivalences between
quasi-representable $\mathbb{V}$-small $\widehat{C^{op}\otimes^{\mathbb{L}}D}$-modules. The
enriched Yoneda
lemma for the model category $C\otimes^{\mathbb{L}}D^{op}-Mod$ easily implies that
this nerve is weakly equivalent to the nerve of equivalences
between $\mathbb{U}$-small $C\otimes^{\mathbb{L}}D^{op}$-modules. \hfill $\Box$ \\
\section{Applications}
In this last section we present three kinds of applications
of our main results. A first application explains the relation
between Hochschild cohomology and internal Hom's of dg-categories.
In the same spirit, we present a relation between the negative
part of Hochschild cohomology and the higher homotopy groups of
the \emph{classifying space of dg-categories}, as well as an
interpretation of the fundamental group of this space
as the so-called \emph{derived Picard group}.
As a second application, we present a proof of the existence of
a good localization functor for dg-categories. This implies for
example the existence of a quotient of a dg-category by
a full sub-dg-category, satisfying the required universal
property. Finally, our last application states that the
(derived) dg-category of morphisms between the dg-categories of quasi-coherent
complexes over some (reasonable) schemes is naturally equivalent to
the dg-category of quasi-coherent complexes over their product.
Under smoothness and properness conditions
the same statement stays correct
when one replaces \emph{quasi-coherent} by \emph{perfect}.
This last result can be considered as a solution to
a question of D. Orlov, concerning the existence of
representative objects for triangulated functors between
derived categories of smooth projective varieties.
\subsection{Hochschild cohomology, classifying space of dg-categories,
and derived Picard groups}
As a first application we give a formula relating higher homotopy groups
of mapping spaces between dg-categories and Hochschild cohomology. For this,
let us recall that for any $\mathbb{U}$-small dg-category $C$, one defines its
Hochschild cohomology groups as
$$\mathbb{HH}^{i}(C):=H^{i}(\mathbb{R}\underline{Hom}_{C\otimes^{\mathbb{L}}C^{op}}(C,C)),$$
where $C$ is the $C\otimes^{\mathbb{L}}C^{op}$-module defined by the trivial formula
$C(x,y):=C(x,y)$, and where $\mathbb{R}\underline{Hom}_{C\otimes^{\mathbb{L}}C^{op}}$ are the $Ho(C(k))$-enriched
Hom's of the category $Ho(C\otimes^{\mathbb{L}}C^{op}-Mod_{\mathbb{U}})$.
More generally, the Hochschild complex of $C$ is defined by
$$\mathbb{HH}(C):=\mathbb{R}\underline{Hom}_{C\otimes^{\mathbb{L}}C^{op}}(C,C)),$$
which is a well defined object in the derived category $Ho(C(k))$ of complexes
of $k$-modules.
\begin{cor}\label{chh-}
With the notation above, there exists an isomorphism in $Ho(C(k))$
$$\mathbb{HH}(C)\simeq \mathbb{R}\underline{Hom}(C,C)(Id,Id),$$
where $Id$ is the identity of $C$, considered as an
object of the dg-category $\mathbb{R}\underline{Hom}(C,C)$.
In particular, one has
$$\mathbb{HH}^{i}(C)\simeq H^{i}(\mathbb{R}\underline{Hom}(C,C)(Id,Id)).$$
\end{cor}
\textit{Proof:} Using Thm. \ref{t2}, one has
$$\mathbb{R}\underline{Hom}(C,C)(Id,Id)\simeq
Int(C\otimes^{\mathbb{L}}C^{op}-Mod_{\mathbb{U}}^{rqr}).$$
Furthermore, through this identification the identity morphism
of $C$ goes to the bi-module $C$ itself. This implies the result by
the definition of Hochschild cohomology.
\hfill $\Box$ \\
An important consequence of Cor. \ref{chh-} is the following Morita invariance
of Hochschild cohomology.
\begin{cor}\label{chh--}
With the notation above, there exists an isomorphism in $Ho(C(k))$
$$\mathbb{HH}(C)\simeq \mathbb{HH}(\widehat{C}).$$
\end{cor}
\textit{Proof:} Indeed, the identity of $\widehat{C}$ is clearly
continuous, and thus by Thm. \ref{t3} (1) one has
$$\mathbb{HH}(\widehat{C})\simeq
\mathbb{R}\underline{Hom}(\widehat{C},\widehat{C})(Id,Id)
\simeq \mathbb{R}\underline{Hom}(C,\widehat{C})(\underline{h},\underline{h}),$$
where $\underline{h} : C \longrightarrow \widehat{C}$ is the Yoneda
embedding. As the morphism $\underline{h}$ is quasi-fully faithful,
Cor. \ref{cp5'''} implies that the
morphism
$$\underline{h}^{*} : \mathbb{R}\underline{Hom}(C,\widehat{C})(\underline{h},\underline{h})
\longrightarrow \mathbb{R}\underline{Hom}(C,C)(Id,Id)$$
is a quasi-isomorphism. Cor. \ref{chh-} implies the result. \hfill $\Box$ \\
\begin{cor}\label{chh}
With the notation above one has isomorphisms of groups
$$\pi_{i}(Map(C,C),Id)\simeq \mathbb{HH}^{1-i}(C)$$
for any $i>1$. For $i=1$, one has an isomorphism of groups
$$\pi_{1}(Map(C,C),Id)\simeq \mathbb{HH}^{0}(C)^{*}=Aut_{Ho(C\otimes^{\mathbb{L}}C^{op}-Mod_{\mathbb{U}})}(C).$$
\end{cor}
\textit{Proof:} This follows immediately from Thm. \ref{t1},
the well-known relations between mapping spaces and classifying
spaces of model categories (see e.g. \cite[Cor. A.0.4]{hagII}) and the formula
$$H^{-i}(\mathbb{R}\underline{Hom}_{C\otimes^{\mathbb{L}}C^{op}}(C,C))\simeq
\pi_{i}(Map_{C\otimes^{\mathbb{L}}C^{op}-Mod_{\mathbb{U}}}(C,C)).$$
\hfill $\Box$ \\
Let $|dg-Cat|$ be the nerve of the category of quasi-equivalences in
$dg-Cat_{\mathbb{U}}$. Using the usual relations between
mapping spaces in model category and nerve of categories of equivalences
(see e.g. \cite[Appendix A]{hagII}) one finds the following consequence.
\begin{cor}\label{chh'}
For a $\mathbb{U}$-small dg-category $C$, one has
$$\pi_{i}(|dg-Cat|,C)\simeq \mathbb{HH}^{2-i}(C) \qquad \forall \; i >2.$$
Moreover, one has
$$\pi_{2}(|dg-Cat|,C)\simeq \mathbb{HH}^{0}(C)^{*}.$$
\end{cor}
\begin{rmk}
\emph{The above corollary only gives an interpretation
of negative Hochschild cohomology groups. The positive
part of the Hochschild cohomology can also be
interpreted in terms of deformation theory of
dg-categories as done for example in \cite[\S 8.5]{hagII}.}
\end{rmk}
For a ($\mathbb{U}$-small) dg-algebra $A$, one can define
the derived Picard group $RPic(A)$ of $A$, as done
for example in \cite{rz,ke2,yek}.
Using our notations and definitions, the group
$RPic(A)$ can be defined in the following way.
To simplify notations let us assume that
the underlying complex of $A$ is cofibrant, and we will
consider $A$ as a dg-category with a unique object
which we denote by $BA$.
Note that the category
$(A\otimes A^{op})-Mod_{\mathbb{U}}$,
of $A\otimes A^{op}$-dg-modules, is also
the category $(BA\otimes BA^{op})-Mod_{\mathbb{U}}$.
This category can be
endowed with the following
monoidal structure. For $X$ and $Y$ two
$(A\otimes A^{op})$-dg-modules,
we can form the internal tensor product
$X\otimes_{A}Y \in (A\otimes A^{op})-Mod_{\mathbb{U}}$
as the coequalizer of the two natural morphisms
$$(X\otimes A \otimes Y) \rightrightarrows X\otimes Y.$$
This endows the model category
$(A\otimes A^{op})-Mod_{\mathbb{U}}$ with
a structure of monoidal model category (see for
example \cite{kt} where the simplicial analog
is considered). Deriving this monoidal
structure provides a monoidal category
$(Ho((A\otimes A^{op})-Mod_{\mathbb{U}}),\otimes_{A}^{\mathbb{L}})$.
By definition,
the group $RPic(A)$ is the group of isomorphism
classes of objects in $Ho((A\otimes A^{op})-Mod_{\mathbb{U}})$
which are invertible for the monoidal structure $\otimes_{A}^{\mathbb{L}}$.
\begin{cor}\label{crpic}
There is
a group isomorphism
$$RPic(A)\simeq \pi_{1}(|dg-Cat_{\mathbb{V}}|,\widehat{BA}).$$
\end{cor}
\textit{Proof:} This easily follows from the formula
$$\pi_{1}(|dg-Cat_{\mathbb{V}}|,\widehat{C}) \simeq
Aut_{Ho(dg-Cat)}(\widehat{C})$$
and Cor. \ref{ct3}. \hfill $\Box$ \\
\subsection{Localization and quotient of dg-categories}
Let $C$ be a $\mathbb{U}$-small dg category, and $S$ be a set of
morphisms in $[C]$. For any $\mathbb{U}$-small dg-category $D$, we consider
$Map_{S}(C,D)$ the sub-simplicial set of
$Map(C,D)$ being the union of all connected components corresponding to
morphisms $f : C \longrightarrow D$ in $Ho(dg-Cat)$ such that
$[f] : [C] \longrightarrow [D]$ sends $S$ to isomorphisms in $[D]$.
\begin{cor}\label{cloc}
The $Ho(SSet_{\mathbb{U}})$-enriched functor
$$Map_{S}(C,-) : Ho(dg-Cat_{\mathbb{U}}) \longrightarrow
Ho(SSet_{\mathbb{U}})$$
is co-represented by an object $L_{S}(C) \in Ho(dg-Cat_{\mathbb{U}})$.
\end{cor}
\textit{Proof:} Let $I_{k}$ be the dg-category with two objects $0$ and $1$, and
freely generated by a unique morphism $0 \rightarrow 1$. Using
theorem \ref{t1} one easily sees that $Map(I_{k},C)$ can be identified
with the nerve of the category $(C^{op}-Mod_{\mathbb{U}})_{rqr}^{I}$, of morphisms
between quasi-representable $C^{op}$-modules. Using the dg-Yoneda
lemma one sees that $[I_{k},C]$ is in a natural bijection with
isomorphism classes of morphisms in $[C]$. In particular, the set
$S$ can be classified by a morphism in $Ho(dg-Cat_{\mathbb{U}})$
$$S : \coprod_{f\in S}I_{k} \longrightarrow C.$$
We consider the natural morphism $I_{k} \longrightarrow \mathbf{1}$,
and we define $L_{S}C$ to be the homotopy push-out
$$\xymatrix{
\coprod_{f\in S}I_{k} \ar[r] \ar[d] & C \ar[d] \\
\coprod_{f\in S}\mathbf{1} \ar[r] & L_{S}C.}$$
For any $D$ one has a homotopy pull-back diagram
$$\xymatrix{
Map(L_{S}C,D) \ar[r] \ar[d] & \prod_{f\in S}Map(\mathbf{1},D) \ar[d] \\
Map(C,D) \ar[r] & \prod_{f\in S}Map(I_{k},D).}$$
Therefore, in order to see that $L_{S}C$ has the correct universal property, it is
enough to check that $Map(\mathbf{1},D) \longrightarrow Map(I_{k},D)$ induces an injection
on $\pi_{0}$, a bijection on $\pi_{i}$ for $i>0$, and that its image in
$[I_{k},D]$ consists of all morphisms in $[D]$ which are isomorphisms.
Using theorem \ref{t1} once again we see that this follows from the following
very general fact: if $M$ is a model category, then
the Quillen adjunction $Mor(M) \leftrightharpoons M$ (where $Mor(M)$ is the model
category of morphisms in $M$), sending a morphism in $M$ to its
target, induces a fully faithful functor $Ho(M) \longrightarrow Ho(Mor(M))$, whose essential image
consists of all equivalences in $M$. \hfill $\Box$.
\begin{cor}\label{cloc2}
Let $C\in dg-Cat_{\mathbb{U}}$ be a
dg-category and $S$ a set of morphisms in $[C]$.
Then, the natural morphism $C \longrightarrow L_{S}C$
induces for any $D\in dg-Cat_{\mathbb{U}}$
a quasi-fully faithful morphism
$$\mathbb{R}\underline{Hom}(L_{S}C,D) \longrightarrow
\mathbb{R}\underline{Hom}(C,D),$$
whose quasi-essential image consists
of all morphisms $C \rightarrow D$ in $Ho(dg-Cat)$
sending $S$ to isomorphisms in $[D]$.
\end{cor}
\textit{Proof:} This follows formally from Cor. \ref{cloc},
Thm. \ref{t2} and Lem. \ref{lmono}. \hfill $\Box$ \\
One important example of application of the localization construction
is the existence of a good theory of quotients of dg-categories.
For this, let $C$ be a $\mathbb{U}$-small dg-category, and
$\{X_{i}\}_{i\in I}$ be a sub-set of objects in $C$.
We assume that $[C]$ has a zero object $0$.
One consider $S$ the set of morphisms in $[C]$ consisting
of all $X_{i} \rightarrow 0$. The dg-category
$L_{S}C$ is then denoted by $C/<X_{i}>$, and is called
the quotient of $C$ by the sub-set of objects $\{X_{i}\}_{i\in I}$.
This terminology is justified by the fact that for any
dg-category $D$ with a zero object, the morphism
$$l^{*} : \mathbb{R}\underline{Hom}(C/<X_{i}>,D) \longrightarrow
\mathbb{R}\underline{Hom}(C,D)$$
is quasi-fully faithful, and its image consists of all
morphisms $f : C \longrightarrow D$ such that for all $i\in I$
$[f(X_{i})]\simeq 0$ in $[D]$.
\subsection{Maps between dg-categories of quasi-coherent complexes}
We now pass to our last application describing maps between
dg-categories of quasi-coherent complexes on $k$-schemes.
For this, let $X$
be a quasi-compact and separated scheme over $Spec\, k$. We consider
$QCoh(X)$ the category of $\mathbb{U}$-small quasi-coherent
sheaves on $X$. As this is a Grothendieck category we know that
there exists a $\mathbb{U}$-cofibrantly generated model category
$C(QCoh(X))$ of (unbounded) complexes of quasi-coherent sheaves
on $X$ (the cofibrations being the monomorphisms and the
equivalences being the quasi-isomorphisms, see e.g. \cite{ho2}).
It is easy to check that the natural
$C(k)_{\mathbb{U}}$-enrichment of $C(QCoh(X))$ makes it into a
$C(k)_{\mathbb{U}}$-model category, and thus as explained in
\S 3 we can construct a $\mathbb{V}$-small dg-category
$Int(C(QCoh(X))$. This dg-category will be denoted by
$L_{qcoh}(X)$. Note that $[L_{qcoh}(X)]$ is naturally equivalent to
the (unbounded)
derived category of quasi-coherent sheaves $D_{qcoh}(X)$, and will be
identified with it.
We need to recall that an object $E$ in $L_{qcoh}(X)$
is homotopically finitely presented, or perfect in the sense of \S 7,
if and only if it is a compact object of $D_{qcoh}(X)$, and thus
if and only if it is a perfect complex on $X$ (see for example
\cite{bv}). We will use this fact implicitly in the sequel.
\\
\begin{thm}\label{tfour}
Let $X$ and $Y$ be two quasi-compact and separated schemes over $k$,
and assume that one of them is flat over $Spec\, k$.
Then, there
exists an isomorphism in $Ho(dg-Cat_{\mathbb{V}})$
$$\mathbb{R}\underline{Hom}_{c}(L_{qcoh}(X),L_{qcoh}(Y))\simeq
L_{qcoh}(X\times_{k} Y).$$
\end{thm}
\textit{Proof:} We start noticing that the model
categories $C(QCoh(X))$ and $C(QCoh(Y))$ are stable,
proper, cofibrantly generated,
and admit a compact generator (see \cite{bv}). Therefore, they satisfy the conditions of the
main theorem of \cite{ss}, and thus one can find two objects
$E_{X}$ and $E_{Y}$ in $L_{qcoh}(X)$ and $L_{qcoh}(Y)$, and two Quillen
equivalences
$$C(QCoh(X)) \leftrightharpoons A_{X}^{op}-Mod_{\mathbb{U}} \qquad
C(QCoh(Y)) \leftrightharpoons A_{Y}^{op}-Mod_{\mathbb{U}}$$
where
$A_{X}$ (resp. $A_{Y}$) is the full sub-dg-category of
$L_{qcoh}(X)$ (resp. of $L_{qcoh}(Y)$) consisting of $E_{X}$
(resp. $E_{Y}$) only (in other words, $A_{X}$ is the dg-category with a unique
object and $\mathbb{R}\underline{End}(E_{X})$ as endomorphism dg-algebra).
In the following we will write $A_{X}$ for both, the dg-category
and the corresponding dg-algebra $\mathbb{R}\underline{End}(E_{X})$
(and the same with $A_{Y}$).
These Quillen equivalences
are $C(k)$-enriched Quillen equivalences, and with
a bit of care one can check that they
provide natural isomorphisms in $Ho(dg-Cat_{\mathbb{V}})$
$$L_{qcoh}(X)\simeq \widehat{A_{X}} \qquad L_{qcoh}(Y)\simeq \widehat{A_{Y}}.$$
\begin{lem}\label{ldual}
There exists an isomorphism in $Ho(dg-Cat_{\mathbb{V}})$
$$\widehat{A_{Y}}\simeq \widehat{A_{Y}^{op}}.$$
\end{lem}
\textit{Proof:} By the general theory of \cite{ss} it is enough to show that
the triangulated category $D_{qcoh}(Y)\simeq [L_{qcoh}(Y)]$ possesses
a compact generator $F_{Y}$ such that
the dg-algebra $\mathbb{R}\underline{End}(F_{Y})$ is
naturally equivalent to $\mathbb{R}\underline{End}(E_{Y})^{op}$.
For this we take $F_{Y}=E_{Y}^{\vee}$ to be the dual perfect complex
of $E_{Y}$.
Let $<F_{Y}>$ be the smallest thick triangulated sub-category
of $D_{parf}(Y)$ containing $F_{Y}$. We let
$\phi : D_{parf}(Y) \longrightarrow D_{parf}(Y)^{op}$ be the
involution sending a perfect complex $E$ to its dual $E^{\vee}$.
Then, clearly $\phi(<F_{Y}>)=<E_{Y}>=D_{parf}(Y)$.
This shows that $F_{Y}$ classically generates
$D_{parf}(Y)$, and thus by \cite[Thm. 2.1.2]{bv} that
$F_{Y}$ is a compact generator of $D_{qcoh}(Y)$. \hfill $\Box$ \\
\begin{lem}\label{ldiag}
There exists an isomorphism in $Ho(dg-Cat_{\mathbb{V}})$
$$\widehat{A_{X}\otimes_{k}^{\mathbb{L}} A_{Y}}\simeq L_{qcoh}(X\times_{k} Y).$$
\end{lem}
\textit{Proof:} This follows from the fact that the external product
$E_{X}\boxtimes E_{Y}$ is a compact generator of
$D_{qcoh}(X\times_{k} Y)$, as explained in \cite[Lem. 3.4.1]{bv}. Indeed,
flat base change induces a natural quasi-isomorphism of dg-algebras
(one uses here that either $X$ or $Y$ is flat over $k$)
$$\mathbb{R}\underline{End}(E_{X}\boxtimes E_{Y})\simeq
\mathbb{R}\underline{End}(E_{X})\otimes_{k}^{\mathbb{L}} \mathbb{R}\underline{End}(E_{Y})\simeq
A_{X}\otimes_{k}^{\mathbb{L}} A_{Y}.$$
\hfill $\Box$ \\
We are now ready to prove theorem \ref{tfour}. Indeed, using theorem \ref{t3} one finds
$$\mathbb{R}\underline{Hom}_{c}(L_{qcoh}(X),L_{qcoh}(Y))\simeq
\mathbb{R}\underline{Hom}_{c}(\widehat{A_{X}},\widehat{A_{Y}})\simeq
\mathbb{R}\underline{Hom}(A_{X},\widehat{A_{Y}}).$$
Lemma \ref{ldual} and the universal properties of internal Hom's give an isomorphism
$$\mathbb{R}\underline{Hom}(A_{X},\widehat{A_{Y}})\simeq
\mathbb{R}\underline{Hom}(A_{X},\widehat{A_{Y}^{op}})\simeq
\widehat{A_{X}\otimes_{k}^{\mathbb{L}} A_{Y}}.$$
Finally lemma \ref{ldiag} implies the theorem. \hfill $\Box$ \\
\begin{cor}\label{cfour-1}
Under the same conditions as in Thm. \ref{tfour},
there
exists a bijection
between $[L_{qcoh}(X),L_{qcoh}(Y)]_{c}$, the sub-set of
$[L_{qcoh}(X),L_{qcoh}(Y)]$ consisting of continuous morphisms,
and the isomorphism classes of objects in the
derived category $D_{qcoh}(X\times_{k} Y)$.
\end{cor}
\textit{Proof:} Readily follows from theorem \ref{tfour} and the fact that
$[L_{qcoh}(X\times_{k} Y)]\simeq D_{qcoh}(X\times_{k} Y)$. \hfill $\Box$ \\
Tracking back the construction of the equivalence in theorem
\ref{tfour} one sees that the bijection of corollary
\ref{cfour-1} can be described as follows. Let
$E \in D_{qcoh}(X\times_{k} Y)$ be an object, and let
us consider the two projections
$$p_{X} : X\times_{k}Y \longrightarrow X \qquad
p_{Y} : X\times_{k}Y \longrightarrow Y.$$
We consider the functor
$$\phi_{E} : D_{qcoh}(X) \longrightarrow D_{qcoh}(Y)$$
defined by
$$\phi_{E}(F):=\mathbb{R}(p_{Y})_{*}(\mathbb{L}p_{X}^{*}(F)\otimes^{\mathbb{L}}
E),$$
for any $F\in D_{qcoh}(X)$. Then, the functor $\phi_{E}$ is the natural
functor induced by the morphism $L_{qcoh}(X) \longrightarrow L_{qcoh}(Y)$
in $Ho(dg-Cat)$, corresponding to $E$ via the bijection
of Cor. \ref{cfour-1}.
\begin{cor}\label{cfour}
Let $X$ be a quasi-compact and separated scheme, flat over $Spec\, k$.
Then, one has
$$\pi_{1}(Map(L_{qcoh}(X),L_{qcoh}(X),Id))\simeq \mathcal{O}_{X}(X)^{*}$$
$$\pi_{i}(Map(L_{qcoh}(X),L_{qcoh}(X),Id))\simeq \mathbb{HH}^{1-i}(A_{X}) \simeq 0 \; \forall \; i>1.$$
\end{cor}
\textit{Proof:} Indeed theorem \ref{tfour}, theorem \ref{t1}, corollary
\ref{cint} and corollary \ref{cp5'} give
$$Map(L_{qcoh}(X),L_{qcoh}(X))\simeq Map(*,L_{qcoh}(X\times_{k} X)).$$
Furthermore, the identity on the right is clearly sent to the
diagonal $\Delta_{X}$ in $L_{qcoh}(X\times_{k} X)$.
Therefore, one finds for any $i>1$
$$\pi_{i}(Map(L_{qcoh}(X),L_{qcoh}(X)),Id))\simeq
\pi_{i}(Map(*,L_{qcoh}(X\times_{k} X)),\Delta_{X})\simeq$$
$$H^{1-i}(L_{qcoh}(X\times_{k} X)(\Delta_{X},\Delta_{X}))\simeq
Ext^{1-i}_{X\times_{k} X}(\Delta(X),\Delta(X))\simeq
0.$$
For $i=1$, one has
$$\pi_{1}(Map(L_{qcoh}(X),L_{qcoh}(X)),Id))\simeq
\pi_{1}(Map(*,L_{qcoh}(X\times_{k} X)),\Delta_{X})\simeq
Aut_{D_{qcoh}(X\times_{k} X)}(\Delta_{X})\simeq \mathcal{O}_{X}(X)^{*}.$$
\hfill $\Box$ \\
Corollary \ref{cfour}
combined with the usual relations between mapping spaces and nerves
of categories of equivalences also has the following important consequence.
\begin{cor}\label{cfour'}
Let $X$ be a quasi-compact and separated scheme, flat over $k$. Then, one has
$$\pi_{i}(|dg-Cat|,L_{qcoh}(X))\simeq 0 \qquad \forall \; i>2.$$
In particular, the sub-simplicial set of $|dg-Cat|$ corresponding to dg-categories
of the form $L_{qcoh}(X)$, for $X$ a quasi-compact and separated scheme
flat over $k$, is a $2$-truncated simplicial set.
\end{cor}
We finish by a refined version of theorem \ref{tfour} involving
only perfect complexes instead of all quasi-coherent complexes.
For this, we will denote by $L_{parf}(X)$ the full sub-dg-category of
$L_{qcoh}(X)$ consisting of all perfect complexes.
\begin{thm}\label{tfour2}
Let $X$ and $Y$ be two smooth and proper schemes over $Spec\, k$.
Then, there
exists an isomorphism in $Ho(dg-Cat_{\mathbb{V}})$
$$\mathbb{R}\underline{Hom}(L_{parf}(X),L_{parf}(Y))\simeq
L_{parf}(X\times_{k} Y).$$
\end{thm}
\textit{Proof:} The triangulated category
$D_{qcoh}(X)$ being generated by its compact objects, one sees
that the Yoneda embedding
$$L_{qcoh}(X) \longrightarrow \widehat{L_{parf}(X)}$$
is an isomorphism in $Ho(dg-Cat_{\mathbb{V}})$. Using
our Thm. \ref{t3} we see that
$\mathbb{R}\underline{Hom}(L_{parf}(X),L_{parf}(Y))$ can be
identified, up to quasi-equivalence, with the full sub-dg-category
of $\mathbb{R}\underline{Hom}_{c}(L_{qcoh}(X),L_{qcoh}(Y))$
consisting of all morphisms $L_{qcoh}(X) \longrightarrow
L_{qcoh}(Y)$ which preserve perfect complexes. Using Thm. \ref{tfour},
we see that $\mathbb{R}\underline{Hom}(L_{parf}(X),L_{parf}(Y))$
is quasi-equivalent to the full sub-dg-category
of $L_{qcoh}(X\times_{k}Y)$ consisting of objects
$E$ such that for any perfect complex $F$ on $X$, the
complex $\mathbb{R}(p_{Y})_{*}(p_{X}^{*}(F)\otimes^{\mathbb{L}}E)$
is perfect on $Y$. To finish the proof we thus need to
show that an object $E\in D_{qcoh}(X\times_{k}Y)$ is perfect
if and only if the functor
$$\Phi_{E}:=\mathbb{R}(p_{Y})_{*}(p_{X}^{*}(-)\otimes^{\mathbb{L}}E) :
D_{qcoh}(X) \longrightarrow D_{qcoh}(Y)$$
preserves perfect objects. Clearly, as $X$ is flat and proper
over $Spec\, k$, $\Phi_{E}$ preserves perfect complexes if
$E$ is itself perfect.
Conversely, let $E$ be an object in $D_{qcoh}(X\times_{k}Y)$
such that $\Phi_{E}$ preserves perfect complexes.
\begin{lem}\label{ltfour2}
Let $Z$ be a smooth and proper scheme over $Spec\, k$, and
$E\in D_{qcoh}(Z)$. If for any perfect complex $F$ on $Z$, the
complex of $k$-modules $\mathbb{R}\underline{Hom}(F,E)$ is
perfect, then $E$ is perfect on $Z$.
\end{lem}
\textit{Proof of the lemma:} We let $A_{Z}$ be a dg-algebra
over $k$ such that $L_{qcoh}(Z)$ is quasi-equivalent to
$\widehat{A_{Z}}$ (with the same abuse of notations that
$A_{Z}$ also means the dg-category with a unique object
and $A_{Z}$ as endomorphism dg-algebra). As $Z$ is flat and
proper over $Spec\, k$, the underlying complex of $k$-modules
of $A_{Z}$ is perfect. Furthermore, as $Z$ is smooth, the diagonal
$\Delta : Z \hookrightarrow Z\times_{k}Z$ is a local complete intersection
morphism, and thus $\Delta_{*}(\mathcal{O}_{Z})$ is a perfect complex
on $Z$. Equivalently, the $A_{Z}\otimes^{\mathbb{L}}_{k}A_{Z}^{op}$-dg-module
$A_{Z}$ is perfect, or equivalently lies in the smallest
sub-dg-category of $\widehat{A_{Z}^{op}\otimes^{\mathbb{L}}_{k}A_{Z}}$
containing $A_{Z}\otimes^{\mathbb{L}}A_{Z}^{op}$ and which is
stable by retracts, homotopy push-outs and the loop functor
(or the shift functor).
We now apply our theorem \ref{t3} in order to
translate this last fact in terms of dg-categories of morphisms.
Let $F : \widehat{A_{Z}} \longrightarrow \widehat{A_{Z}}$
be the morphism of dg-categories sending
an $A_{Z}^{op}$-dg-module $M$ to the free $A_{Z}^{op}$-dg-module
$$F(M):=\underline{M}\otimes^{\mathbb{L}}A_{Z}^{op},$$
where $\underline{M}$ is the underlying complex of
$k$-modules of $M$. By what we have seen, the identity morphism lies in the
smallest sub-dg-category of $\mathbb{R}\underline{Hom}_{c}(\widehat{A_{Z}},\widehat{A_{Z}})$ containing
the object $F$ and which is stable by retracts, homotopy push-outs
and the loop functor.
Evaluating the identity of the dg-category
$\widehat{A_{Z}}$ at an object $M$, we get that the
object $M\in \widehat{A_{Z}}$ lies in the smallest sub-dg-category
of $\widehat{A_{Z}}$ containing $\underline{M}\otimes^{\mathbb{L}} A_{Z}^{op}$
and stable by retracts, homotopy push-outs and by the loop functor.
Now, by our hypothesis
the object $E$ corresponds to $M\in \widehat{A_{Z}}$ such that
$\underline{M}$ is a perfect complex
of $k$-modules. Therefore, $M$ itself belongs to the smallest
sub-dg-category of $\widehat{A_{Z}}$ containing $A_{Z}^{op}$ and
which stable by retracts, homotopy push-outs and the loop functor.
By definition
of being perfect, this implies that $M\in \widehat{A_{Z}}_{pe}$, and thus
that $E$ is a perfect complex on $Z$. \hfill $\Box$ \\
Let now $E_{X}$ and $E_{Y}$ be compact generators of $D_{qcoh}(X)$
and $D_{qcoh}(Y)$. Then, by the projection formula one has
$$\mathbb{R}(p_{Y})_{*}((E_{X}^{\vee}\boxtimes E_{Y}^{\vee})\otimes^{\mathbb{L}}E)
\simeq \mathbb{R}(p_{Y})_{*}(p_{X}^{*}(E_{X}^{\vee})\otimes^{\mathbb{L}}E)
\otimes^{\mathbb{L}}E_{Y}^{\vee}$$
which is perfect on $Y$. This implies in particular that
$$\mathbb{R}\underline{Hom}(E_{X}\boxtimes E_{Y},E)\simeq
\mathbb{R}\Gamma(Y,\mathbb{R}(p_{Y})_{*}((E_{X}^{\vee}\boxtimes E_{Y}^{\vee})\otimes^{\mathbb{L}}E)$$
is a perfect complex of $k$-modules.
As the perfect complex $E_{X}\boxtimes E_{Y}$
is a generator on $D_{qcoh}(X\times_{k}Y)$,
one sees that
for any perfect complex $F$ on $X\times_{k}Y$, the complex
of $k$-modules $\mathbb{R}\underline{Hom}(F,E)$ is perfect.
The lemma \ref{ltfour2} implies that $E$ is perfect on
$X\times_{k}Y$, which
finishes the proof of the theorem. \hfill $\Box$ \\
| 42,804
|
:: Gauss Lemma and Law of Quadratic Reciprocity
:: by Li Yan , Xiquan Liang and Junjie Zhao
environ
vocabularies NUMBERS, SUBSET_1, INT_1, NAT_1, FINSEQ_1, NEWTON, ARYTM_3,
ARYTM_1, RELAT_1, CARD_1, FUNCT_1, CARD_3, FUNCOP_1, ORDINAL4, XBOOLE_0,
TARSKI, XXREAL_0, FINSEQ_5, SQUARE_1, INT_2, COMPLEX1, ABIAN, FINSEQ_2,
FUNCT_7, PARTFUN1, FINSET_1, RFINSEQ, CALCUL_2, REAL_1, ZFMISC_1, INT_5,
ORDINAL1;
notations XREAL_0, GROUP_4, INT_1, RVSUM_1, FINSET_1, ORDINAL1, CARD_1, NAT_D,
INT_2, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, NAT_3, ABIAN, GR_CY_1,
FINSEQ_5, FINSEQ_2, EULER_2, NEWTON, FINSEQ_7, REAL_1, PEPIN, TARSKI,
XBOOLE_0, SUBSET_1, XCMPLX_0, NUMBERS, XXREAL_0, NAT_1, DOMAIN_1,
FINSEQ_1, RFINSEQ, FUNCOP_1, CALCUL_2, ZFMISC_1, CARD_3, WSIERP_1,
BINOP_1, RECDEF_1;
constructors ABIAN, WSIERP_1, PEPIN, NAT_3, NAT_D, REALSET1, GR_CY_1,
FINSEQ_5, EULER_2, RFINSEQ, GROUP_4, FINSEQ_7, REAL_1, WELLORD2,
CALCUL_2, PROB_3, RECDEF_1, BINOP_1, BINOP_2, CLASSES1, NUMBERS;
registrations XXREAL_0, MEMBERED, RELAT_1, FINSEQ_1, ORDINAL1, WSIERP_1,
NUMBERS, XBOOLE_0, XREAL_0, NAT_1, INT_1, FINSET_1, NAT_3, RVSUM_1,
FUNCT_1, CARD_1, NEWTON, SUBSET_1, VALUED_0, VALUED_1, FINSEQ_2,
PRE_POLY, RELSET_1;
requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM;
definitions TARSKI;
equalities INT_1, XCMPLX_0, SQUARE_1, FINSEQ_2, CALCUL_2, CARD_3;
expansions INT_1, FUNCT_1, TARSKI;
theorems FINSEQ_1, FINSEQ_7, RELAT_1, FINSEQ_3, FINSEQ_2, ABSVALUE, FINSEQ_5,
EULER_2, SEQ_2, FINSEQ_4, FUNCT_2, XBOOLE_0, NAT_3, NEWTON, INT_2, NAT_2,
WSIERP_1, CARD_1, PEPIN, NAT_1, XCMPLX_1, XREAL_1, HEYTING3, ORDINAL1,
EULER_1, XBOOLE_1, TARSKI, RVSUM_1, NUMBERS, INT_1, FUNCT_1, FUNCOP_1,
XXREAL_0, NAT_D, HILBERT3, SERIES_2, GRAPH_5, HILBERT2, RFINSEQ,
ZFMISC_1, CARD_2, XREAL_0, FINSEQ_6, CALCUL_2, NAT_4, PROB_3, TOPREAL7,
CARD_3, REAL_3, PRE_FF, INT_4, VALUED_1, PRE_POLY, XTUPLE_0;
schemes NAT_1, FUNCT_2, RECDEF_1, FINSEQ_2, FUNCT_7, FINSEQ_1;
begin
reserve i,i1,i2,i3,i4,i5,j,r,a,b,x,y for Integer,
d,e,k,n for Nat,
fp,fk for FinSequence of INT,
f,f1,f2 for FinSequence of REAL,
p for Prime;
theorem Th1:
i1 divides i2 & i1 divides i3 implies i1 divides i2-i3
proof
assume that
A1: i1 divides i2 and
A2: i1 divides i3;
consider i4 such that
A3: i2=i1*i4 by A1;
consider i5 such that
A4: i3=i1*i5 by A2;
i2-i3=i1*(i4-i5) by A3,A4;
hence thesis;
end;
theorem Th2:
i divides a & i divides (a-b) implies i divides b
proof
assume that
A1: i divides a and
A2: i divides (a-b);
A3: b=-(a-b)+a;
i divides -(a-b) by A2,INT_2:10;
hence thesis by A1,A3,WSIERP_1:4;
end;
Lm1: (x divides y implies y mod x = 0) &
(x<>0 & y mod x = 0 implies x divides y)
proof
thus x divides y implies y mod x = 0
proof
assume x divides y;
then consider i such that
A1: y = x * i;
y mod x = (x * i + 0) mod x by A1
.= 0 mod x by EULER_1:12
.= 0 by INT_1:73;
hence thesis;
end;
assume that
A2: x<>0 and
A3: y mod x = 0;
y = (y div x) * x + (y mod x) by A2,INT_1:59
.= (y div x) * x by A3;
hence thesis;
end;
definition
let fp;
func Poly-INT fp -> Function of INT,INT means
:Def1:
for x being Element of
INT holds ex fr being FinSequence of INT st len fr = len fp & (for d st d in
dom fr holds fr.d = (fp.d) * x|^(d-'1)) & it.x = Sum fr;
existence
proof
defpred F[Element of INT,set] means ex fr being FinSequence of INT st len
fr = len fp & (for d st d in dom fr holds fr.d = (fp.d) * $1|^(d-'1)) & $2 =
Sum fr;
A1: for x being Element of INT ex y being Element of INT st F[x,y]
proof
let x be Element of INT;
deffunc G(Nat) = (fp.$1) * x|^($1-'1);
consider fr be FinSequence such that
A2: len fr = len fp & for d being Nat st d in dom fr holds fr.d = G(
d) from FINSEQ_1:sch 2;
for d being Nat st d in dom fr holds fr.d in INT
proof
let d be Nat;
assume d in dom fr;
then fr.d = (fp.d) * x|^(d-'1) by A2;
hence thesis by INT_1:def 2;
end;
then reconsider fr as FinSequence of INT by FINSEQ_2:12;
take Sum fr;
take fr;
thus thesis by A2;
end;
consider f being Function of INT,INT such that
A3: for x being Element of INT holds F[x,f.x] from FUNCT_2:sch 3(A1);
take f;
thus thesis by A3;
end;
uniqueness
proof
let f1 be Function of INT,INT;
let f2 be Function of INT,INT;
assume that
A4: for x being Element of INT holds ex fr1 being FinSequence of INT
st len fr1 = len fp & (for d st d in dom fr1 holds fr1.d = (fp.d) * x|^(d-'1))
& f1.x = Sum fr1 and
A5: for x being Element of INT holds ex fr2 being FinSequence of INT
st len fr2 = len fp & (for d st d in dom fr2 holds fr2.d = (fp.d) * x|^(d-'1))
& f2.x = Sum fr2;
for x being Element of INT holds f1.x = f2.x
proof
let x be Element of INT;
consider fr1 being FinSequence of INT such that
A6: len fr1 = len fp and
A7: for d st d in dom fr1 holds fr1.d = (fp.d) * x|^(d-'1) and
A8: f1.x = Sum fr1 by A4;
consider fr2 being FinSequence of INT such that
A9: len fr2 = len fp and
A10: for d st d in dom fr2 holds fr2.d = (fp.d) * x|^(d-'1) and
A11: f2.x = Sum fr2 by A5;
A12: dom fr1 = dom fr2 by A6,A9,FINSEQ_3:29;
for d being Nat st d in dom fr1 holds fr1.d = fr2.d
proof
let d be Nat;
assume
A13: d in dom fr1;
hence fr2.d = (fp.d) * x|^(d-'1) by A10,A12
.= fr1.d by A7,A13;
end;
hence thesis by A8,A11,A12,FINSEQ_1:13;
end;
hence thesis by FUNCT_2:63;
end;
end;
theorem Th3:
len fp = 1 implies (Poly-INT fp) = INT --> fp.1
proof
assume
A1: len fp = 1;
for x being object st x in dom Poly-INT fp holds (Poly-INT fp).x = fp.1
proof
let x be object;
assume x in dom Poly-INT fp;
then reconsider x as Element of INT;
consider fr being FinSequence of INT such that
A2: len fr = len fp and
A3: for d st d in dom fr holds fr.d = (fp.d) * x|^(d-'1) and
A4: (Poly-INT fp).x = Sum fr by Def1;
1 in dom fr by A1,A2,FINSEQ_3:25;
then
A5: fr.1 = (fp.1) * x|^(1-'1) by A3
.= fp.1 * x|^0 by XREAL_1:232
.= fp.1 * 1 by NEWTON:4;
fr = <*fr.1*> by A1,A2,FINSEQ_1:40;
hence thesis by A4,A5,RVSUM_1:73;
end;
then Poly-INT fp = dom Poly-INT fp --> fp.1 by FUNCOP_1:11;
hence thesis by FUNCT_2:def 1;
end;
theorem
len fp = 1 implies for x being Element of INT holds (Poly-INT fp).x = fp.1
proof
assume
A1: len fp = 1;
let x be Element of INT;
consider fr being FinSequence of INT such that
A2: len fr = len fp and
A3: for d st d in dom fr holds fr.d = (fp.d) * x|^(d-'1) and
A4: (Poly-INT fp).x = Sum fr by Def1;
1 in dom fr by A1,A2,FINSEQ_3:25;
then
A5: fr.1 = (fp.1) * x|^(1-'1) by A3
.= fp.1 * x|^0 by XREAL_1:232
.= fp.1 * 1 by NEWTON:4;
fr = <*fr.1*> by A1,A2,FINSEQ_1:40;
hence thesis by A4,A5,RVSUM_1:73;
end;
reserve fr for FinSequence of REAL;
theorem Th5:
for f,f1,f2 st len f = n+1 & len f1 = len f & len f2 = len f & (
for d st d in dom f holds f.d = f1.d - f2.d) holds ex fr st len fr = len f - 1
& (for d st d in dom fr holds fr.d = f1.d - f2.(d + 1)) & Sum f = Sum fr + f1.(
n+1) - f2.1
proof
defpred P[Nat] means for f,f1,f2 st len f = $1 + 1 & len f1 = len f & len f2
= len f & (for d st d in dom f holds f.d = f1.d - f2.d) holds ex fr st len fr =
len f - 1 & (for d st d in dom fr holds fr.d = f1.d - f2.(d + 1)) & Sum f = Sum
fr + f1.($1+1) - f2.1;
A1: for n st P[n] holds P[n+1]
proof
let n;
assume
A2: P[n];
let f,f1,f2;
assume that
A3: len f = (n+1)+1 and
A4: len f1 = len f and
A5: len f2 = len f and
A6: for d st d in dom f holds f.d = f1.d - f2.d;
set ff1 = f1|Seg(n+1);
reconsider ff1 as FinSequence of REAL by FINSEQ_1:18;
A7: len ff1 = n+1 by A3,A4,FINSEQ_3:53;
set ff2 = f2|Seg(n+1);
reconsider ff2 as FinSequence of REAL by FINSEQ_1:18;
A8: f2 = ff2^<*f2.((n+1)+1)*> by A3,A5,FINSEQ_3:55;
A9: len ff2 = n+1 by A3,A5,FINSEQ_3:53;
then ff2 <> {};
then 1 in dom ff2 by FINSEQ_5:6;
then
A10: ff2.1 = f2.1 by A8,FINSEQ_1:def 7;
A11: f1 = ff1^<*f1.((n+1)+1)*> by A3,A4,FINSEQ_3:55;
(n+1)+1 in Seg ((n+1)+1) by FINSEQ_1:4;
then ((n+1)+1) in dom f by A3,FINSEQ_1:def 3;
then
A12: f.((n+1)+1) = f1.((n+1)+1) - f2.((n+1)+1) by A6;
set f3 = f|Seg(n+1);
reconsider f3 as FinSequence of REAL by FINSEQ_1:18;
A13: dom f3 = Seg(n+1) by A3,FINSEQ_3:54;
then
A14: len f3 = n+1 by FINSEQ_1:def 3;
A15: f = f3^<*f.(n+1+1)*> by A3,FINSEQ_3:55;
A16: for d st d in dom f3 holds f3.d = ff1.d - ff2.d
proof
let d;
A17: dom f3 c= dom f by A15,FINSEQ_1:26;
assume
A18: d in dom f3;
then
A19: d in dom ff2 by A13,A9,FINSEQ_1:def 3;
d in dom ff1 by A13,A7,A18,FINSEQ_1:def 3;
then
A20: f1.d = ff1.d by A11,FINSEQ_1:def 7;
f3.d = f.d by A15,A18,FINSEQ_1:def 7
.= f1.d - f2.d by A6,A18,A17;
hence thesis by A8,A19,A20,FINSEQ_1:def 7;
end;
ff1 <> {} by A7;
then (n+1) in dom ff1 by A7,FINSEQ_5:6;
then ff1.(n+1) = f1.(n+1) by A11,FINSEQ_1:def 7;
then consider f4 being FinSequence of REAL such that
A21: len f4 = len f3 - 1 and
A22: for d st d in dom f4 holds f4.d=ff1.d - ff2.(d + 1) and
A23: Sum f3 = Sum f4 + f1.(n+1) - f2.1 by A2,A14,A7,A9,A16,A10;
take f5 = f4^<*f1.(n+1) - f2.(n+2)*>;
f1.(n+1) - f2.(n+2) is Element of REAL by XREAL_0:def 1;
then <*f1.(n+1) - f2.(n+2)*> is FinSequence of REAL by FINSEQ_1:74;
then reconsider f5 as FinSequence of REAL by FINSEQ_1:75;
A24: Sum f = Sum f4 + f1.(n+1) - f2.1 + f.(n+1+1) by A15,A23,RVSUM_1:74
.= Sum f4 + (f1.(n+1) - f2.(n+2)) + f1.((n+1)+1) - f2.1 by A12
.= Sum f5 + f1.((n+1)+1) - f2.1 by RVSUM_1:74;
A25: len f4 + 1 = n + 1 by A13,A21,FINSEQ_1:def 3;
A26: for d st d in dom f5 holds f5.d = f1.d - f2.(d + 1)
proof
let d;
assume d in dom f5;
then d in Seg len f5 by FINSEQ_1:def 3;
then d in Seg (len f4 + 1) by FINSEQ_2:16;
then d in Seg len f4 \/{len f4 + 1} by FINSEQ_1:9;
then
A27: d in Seg len f4 or d in {len f4 + 1} by XBOOLE_0:def 3;
per cases by A27,TARSKI:def 1;
suppose
A28: d in Seg len f4;
then d+1 in Seg(len f4 + 1) by FINSEQ_1:60;
then d+1 in Seg len ff2 by A3,A5,A14,A21,FINSEQ_3:53;
then
A29: d+1 in dom ff2 by FINSEQ_1:def 3;
A30: d in dom f4 by A28,FINSEQ_1:def 3;
len f4 <= len ff1 by A14,A7,A21,XREAL_1:147;
then dom f4 c= dom ff1 by FINSEQ_3:30;
then
A31: f1.d = ff1.d by A11,A30,FINSEQ_1:def 7;
f5.d = f4.d by A30,FINSEQ_1:def 7
.= ff1.d - ff2.(d+1) by A22,A30;
hence thesis by A8,A31,A29,FINSEQ_1:def 7;
end;
suppose
A32: d = len f4 + 1;
1 in Seg 1 by FINSEQ_1:2,TARSKI:def 1;
then 1 in dom <*f1.(n+1) - f2.(n+2)*> by FINSEQ_1:38;
then f5.d = <*f1.(n+1) - f2.(n+2)*>.1 by A32,FINSEQ_1:def 7
.= f1.d - f2.(d+1) by A25,A32,FINSEQ_1:40;
hence thesis;
end;
end;
len f5 = len f4 + 1 by FINSEQ_2:16
.= len f - 1 by A3,A13,A21,FINSEQ_1:def 3;
hence thesis by A26,A24;
end;
A33: P[0]
proof
let f,f1,f2;
assume that
A34: len f = 0+1 and
len f1 = len f and
len f2 = len f and
A35: for d st d in dom f holds f.d = f1.d - f2.d;
take <*>REAL;
0+1 in Seg (0+1) by FINSEQ_1:4;
then 1 in dom f by A34,FINSEQ_1:def 3;
then f.1 = f1.1 - f2.1 by A35;
then f = <*f1.1 - f2.1*> by A34,FINSEQ_1:40;
hence thesis by A34,RVSUM_1:72,73;
end;
for n holds P[n] from NAT_1:sch 2(A33,A1);
hence thesis;
end;
theorem Th6:
len fp = n+2 implies for a being Integer holds ex fr being
FinSequence of INT, r being Integer st len fr = n+1 & (for x being Element of
INT holds (Poly-INT fp).x = (x-a)*(Poly-INT fr).x + r) & fp.(n+2) = fr.(n+1)
proof
assume
A1: len fp = n+2;
(n+1)+1 in Seg ((n+1)+1) by FINSEQ_1:4;
then n+2 in dom fp by A1,FINSEQ_1:def 3;
then reconsider A = fp.(n+2) as Element of INT by FINSEQ_2:11;
reconsider n1=n+1 as Element of NAT;
let a be Integer;
defpred P[Nat,Integer,set] means $3 = fp.(n+2-$1) +a*$2;
A2: for d being Nat st 1 <= d & d < n1 holds for x being Element
of INT ex y being Element of INT st P[d,x,y]
proof
let d be Nat;
assume that
1 <= d and
d < n1;
let x be Element of INT;
set y = fp.(n+2-d) +a*x;
reconsider y as Element of INT by INT_1:def 2;
take y;
thus thesis;
end;
consider p being FinSequence of INT such that
A3: len p = n1 & (p.1 = A or n1 = 0) & for d being Nat st 1
<= d & d < n1 holds P[d,p.d,p.(d+1)] from RECDEF_1:sch 4(A2);
take fr = Rev p;
take r = fp.1 + a*fr.1;
A4: len fr = n+1 by A3,FINSEQ_5:def 3;
for x being Element of INT holds (Poly-INT fp).x = (x-a)*(Poly-INT fr).x + r
proof
let x be Element of INT;
deffunc F(Nat) = (fr.$1)*x|^$1;
deffunc FF(Nat) = a*(fr.$1) * x|^($1-'1);
consider f1 being FinSequence of INT such that
A5: len f1 = len fp and
A6: for d st d in dom f1 holds f1.d = (fp.d) * x|^(d-'1) and
A7: (Poly-INT fp).x = Sum f1 by Def1;
A8: f1 <> {} by A1,A5;
then n+2 in dom f1 by A1,A5,FINSEQ_5:6;
then f1.(n+2)=(fp.(n+2))*x|^((n+1)+1-'1) by A6;
then
A9: f1.(n+2)=(fp.(n+2))*x|^(n+1) by NAT_D:34;
f1.1 = (fp.1)*x|^(1-'1) by A6,A8,FINSEQ_5:6;
then f1.1=(fp.1)*x|^0 by XREAL_1:232;
then
A10: f1.1 = (fp.1)*1 by NEWTON:4;
reconsider n as Element of NAT by ORDINAL1:def 12;
consider f4 being FinSequence such that
A11: len f4 = n+1 & for d being Nat st d in dom f4 holds f4.d=F(d)
from FINSEQ_1:sch 2;
A12: for d being Nat st d in dom f4 holds f4.d in INT
proof
let d be Nat;
reconsider d1 = d as Element of NAT by ORDINAL1:def 12;
assume d in dom f4;
then f4.d1 = (fr.d1)*x|^d1 by A11;
hence thesis by INT_1:def 2;
end;
f4 <> {} by A11;
then n+1 in dom f4 by A11,FINSEQ_5:6;
then f4.(n+1)=(fr.(n+1))*x|^(n+1) by A11;
then
A13: f4.(n+1) = (fp.(n+2))*x|^(n+1) by A3,FINSEQ_5:62;
reconsider f4 as FinSequence of INT by A12,FINSEQ_2:12;
consider f5 being FinSequence such that
A14: len f5 = n+1 & for d being Nat st d in dom f5 holds f5.d=FF(d)
from FINSEQ_1:sch 2;
A15: for d being Nat st d in dom f5 holds f5.d in INT
proof
let d be Nat;
assume d in dom f5;
then f5.d = a*(fr.d) * x|^(d-'1) by A14;
hence thesis by INT_1:def 2;
end;
f5 <> {} by A14;
then 1 in dom f5 by FINSEQ_5:6;
then f5.1 = a*(fr.1) * x|^(1-'1) by A14;
then f5.1 = a*(fr.1)*x|^0 by XREAL_1:232;
then
A16: f5.1 = a*(fr.1)*1 by NEWTON:4;
reconsider f5 as FinSequence of INT by A15,FINSEQ_2:12;
A17: f4 is FinSequence of REAL by FINSEQ_3:117;
consider f2 being FinSequence of INT such that
A18: len f2 = len fr and
A19: for d st d in dom f2 holds f2.d = (fr.d) * x|^(d-'1) and
A20: (Poly-INT fr).x = Sum f2 by Def1;
set f3 = (x-a)*f2;
A21: dom f3 = dom f2 by VALUED_1:def 5;
then
A22: len f3 = len f2 by FINSEQ_3:29;
A23: dom f3 = dom f4 by A4,A18,A11,A21,FINSEQ_3:29;
A24: for k being Element of NAT st k in dom f3 holds f3.k = (fr.k)*x|^k -
a*(fr.k) * x|^(k-'1)
proof
let k be Element of NAT;
assume
A25: k in dom f3;
then
A26: k >= 1 by FINSEQ_3:25;
A27: k in dom f2 by A25,VALUED_1:def 5;
thus f3.k = (x-a)*(f2.k) by A25,VALUED_1:def 5
.= (x-a)*((fr.k) * x|^(k-'1)) by A19,A27
.= (fr.k) * (x|^(k-'1)*x) - a*(fr.k) * x|^(k-'1)
.= (fr.k)*x|^(k-'1+1) - a*(fr.k) * x|^(k-'1) by NEWTON:6
.= (fr.k)*x|^k - a*(fr.k) * x|^(k-'1) by A26,XREAL_1:235;
end;
A28: dom f3 = dom f5 by A4,A18,A14,A21,FINSEQ_3:29;
A29: for d st d in dom f3 holds f3.d = f4.d - f5.d
proof
let d;
assume
A30: d in dom f3;
then
A31: f5.d = a*(fr.d) * x|^(d-'1) by A14,A28;
f4.d = (fr.d)*x|^d by A11,A23,A30;
hence thesis by A24,A30,A31;
end;
f5 is FinSequence of REAL by FINSEQ_3:117;
then consider f6 being FinSequence of REAL such that
A32: len f6 = len f3 - 1 and
A33: for d st d in dom f6 holds f6.d = f4.d - f5.(d + 1) and
A34: Sum f3 = Sum f6 + f4.(n+1) - f5.1 by A4,A18,A11,A14,A22,A29,A17,Th5;
A35: len f6 <= len f3 by A4,A18,A22,A32,XREAL_1:145;
then
A36: dom f6 c= dom f3 by FINSEQ_3:30;
A37: for d being Element of NAT st d in dom f6 holds f6.d = f1.(d+1)
proof
let d be Element of NAT;
A38: dom f6 c= dom p by A3,A4,A18,A22,A35,FINSEQ_3:30;
assume
A39: d in dom f6;
then
A40: d in Seg n by A4,A18,A22,A32,FINSEQ_1:def 3;
then
A41: d <= n by FINSEQ_1:1;
then
A42: n-d >= 0 by XREAL_1:48;
then reconsider d9=n-d+1 as Element of NAT by INT_1:3;
d >= 1 by A40,FINSEQ_1:1;
then n-d <= n-1 by XREAL_1:10;
then d9 <= (n-1)+1 by XREAL_1:6;
then
A43: d9 < n+1 by XREAL_1:145;
d9 >= 0+1 by A42,XREAL_1:6;
then
A44: p.(d9+1) = fp.(n+2-d9) +a*(p.d9) by A3,A43;
d < n+1 by A41,XREAL_1:145;
then
A45: d+1 in Seg (n+1) by FINSEQ_3:11;
then
A46: d+1 in dom f5 by A14,FINSEQ_1:def 3;
d+0 < n+2 by A41,XREAL_1:8;
then d+1 in Seg (n+2) by FINSEQ_3:11;
then
A47: d+1 in dom f1 by A1,A5,FINSEQ_1:def 3;
A48: d+1 in dom p by A3,A45,FINSEQ_1:def 3;
thus f6.d = f4.d - f5.(d + 1) by A33,A39
.= (fr.d)*x|^d - f5.(d+1) by A11,A23,A36,A39
.= (fr.d)*x|^d - a*fr.(d+1) * x|^(d+1-'1) by A14,A46
.= (fr.d)*x|^d - a*(fr.(d+1)) * x|^d by NAT_D:34
.= ((fr.d) - a*fr.(d+1))* x|^d
.= (p.((n+1)-d+1) - a*fr.(d+1))* x|^d by A3,A39,A38,FINSEQ_5:58
.= (p.((n-d+1)+1)-a*p.((n+1)-(d+1)+1))* x|^d by A3,A48,FINSEQ_5:58
.= fp.(d+1)* x|^(d+1-'1) by A44,NAT_D:34
.= f1.(d+1) by A6,A47;
end;
f1 = <*f1.1*>^f6^<*f1.(n+2)*>
proof
set K = <*f1.1*>^f6^<*f1.(n+2)*>;
A49: for d being Nat st d in dom f1 holds f1.d = K.d
proof
let d be Nat;
assume
A50: d in dom f1;
then
A51: d>=1 by FINSEQ_3:25;
A52: d<=n+2 by A1,A5,A50,FINSEQ_3:25;
per cases by A51,A52,XXREAL_0:1;
suppose
A53: d=1;
hence K.d = (<*f1.1*>^(f6^<*f1.(n+2)*>)).1 by FINSEQ_1:32
.= f1.d by A53,FINSEQ_1:41;
end;
suppose
A54: d>1 & d<n+2;
then reconsider w=d-1 as Element of NAT by INT_1:3;
d-1<n+2-1 by A54,XREAL_1:9;
then
A55: d-1<=n+1-1 by INT_1:7;
d-1>=0+1 by A54,INT_1:7,XREAL_1:50;
then w in Seg n by A55,FINSEQ_1:1;
then
A56: w in dom f6 by A4,A18,A22,A32,FINSEQ_1:def 3;
then
A57: w in dom (f6^<*f1.(n+2)*>) by FINSEQ_2:15;
thus K.d = (<*f1.1*>^(f6^<*f1.(n+2)*>)).(w+1) by FINSEQ_1:32
.= (f6^<*f1.(n+2)*>).w by A57,FINSEQ_3:103
.= f6.w by A56,FINSEQ_1:def 7
.= f1.(w+1) by A37,A56
.= f1.d;
end;
suppose
A58: d=n+2;
set K1 = <*f1.1*>^f6;
thus K.d = K.((n+1)+1) by A58
.= K.(len K1 +1) by A4,A18,A22,A32,FINSEQ_5:8
.= f1.d by A58,FINSEQ_1:42;
end;
end;
len K = len (<*f1.1*>^(f6^<*f1.(n+2)*>)) by FINSEQ_1:32
.= 1+len (f6^<*f1.(n+2)*>) by FINSEQ_5:8
.= 1+len f6 +1 by FINSEQ_2:16
.= len f1 by A1,A4,A5,A18,A22,A32;
hence thesis by A49,FINSEQ_3:29;
end;
then Sum f1 = Sum (<*f1.1*>^(f6^<*f1.(n+2)*>)) by FINSEQ_1:32
.= f1.1 + Sum (f6^<*f1.(n+2)*>) by RVSUM_1:76
.= f1.1 + (Sum f6 + f1.(n+2)) by RVSUM_1:74
.= Sum ((x-a)*f2) + r by A10,A9,A13,A16,A34
.= (x-a) * (Poly-INT fr).x + r by A20,RVSUM_1:87;
hence thesis by A7;
end;
hence thesis by A3,FINSEQ_5:62,def 3;
end;
theorem Th7:
p divides i*j implies p divides i or p divides j
proof
assume
A1: p divides i*j;
per cases;
suppose
i>=0 & j>=0;
then reconsider i,j as Element of NAT by INT_1:3;
p divides i*j by A1;
hence thesis by NEWTON:80;
end;
suppose
i>=0 & j<0;
then reconsider i,j9=-j as Element of NAT by INT_1:3;
p divides -(i*j) by A1,INT_2:10;
then p divides i*j9;
then p divides i or p divides j9 by NEWTON:80;
hence thesis by INT_2:10;
end;
suppose
i<0 & j>=0;
then reconsider i9=-i,j as Element of NAT by INT_1:3;
p divides -(i*j) by A1,INT_2:10;
then p divides i9*j;
then p divides i9 or p divides j by NEWTON:80;
hence thesis by INT_2:10;
end;
suppose
i<0 & j<0;
then reconsider i9=-i,j9=-j as Element of NAT by INT_1:3;
p divides i9*j9 by A1;
then p divides i9 or p divides j9 by NEWTON:80;
hence thesis by INT_2:10;
end;
end;
reserve fr,f for FinSequence of INT;
theorem Th8:
for fp st len fp = n+1 & p>2 & not p divides fp.(n+1) holds for
fr st (for d st d in dom fr holds (Poly-INT fp).(fr.d) mod p=0) & (for d,e st d
in dom fr & e in dom fr & d<>e holds not fr.d,fr.e are_congruent_mod p) holds
len fr <= n
proof
defpred P[Nat] means for fp st len fp = $1+1 & p>2 & not p divides fp.($1+1)
holds for fr st (for d st d in dom fr holds (Poly-INT fp).(fr.d) mod p = 0) & (
for d,e st d in dom fr & e in dom fr & d<>e holds not fr.d,fr.e
are_congruent_mod p) holds len fr <= $1;
A1: for n st P[n] holds P[n+1]
proof
let n;
assume
A2: P[n];
let fp;
assume that
A3: len fp = (n+1)+1 and
A4: p>2 and
A5: not p divides fp.(n+1+1);
per cases;
suppose
A6: for x holds (Poly-INT fp).x mod p <> 0;
assume ex fr st (for d st d in dom fr holds (Poly-INT fp).(fr.d) mod p
=0) & (for d,e st d in dom fr & e in dom fr & d<>e holds not fr.d,fr.e
are_congruent_mod p) & len fr > n+1;
then consider fr such that
A7: for d st d in dom fr holds (Poly-INT fp).(fr.d) mod p=0 and
for d,e st d in dom fr & e in dom fr & d<>e holds not fr.d,fr.e
are_congruent_mod p and
A8: len fr > n+1;
fr <> {} by A8;
then (Poly-INT fp).(fr.1) mod p=0 by A7,FINSEQ_5:6;
hence contradiction by A6;
end;
suppose
ex a being Integer st (Poly-INT fp).a mod p = 0;
then consider a being Integer such that
A9: (Poly-INT fp).a mod p=0;
assume ex f st (for d st d in dom f holds (Poly-INT fp).(f.d) mod p =
0) & (for d,e st d in dom f & e in dom f & d<>e holds not f.d,f.e
are_congruent_mod p) & len f > n+1;
then consider f such that
A10: for d st d in dom f holds (Poly-INT fp).(f.d) mod p = 0 and
A11: for d,e st d in dom f & e in dom f & d<>e holds not f.d,f.e
are_congruent_mod p and
A12: len f > n+1;
consider fk,r such that
A13: len fk = n+1 and
A14: for x being Element of INT holds (Poly-INT fp).x =(x-a)*(
Poly-INT fk).x +r and
A15: fp.(n+2)=fk.(n+1) by A3,Th6;
a is Element of INT by INT_1:def 2;
then
A16: (Poly-INT fp).a mod p =((a-a)*(Poly-INT fk).a +r) mod p by A14
.= r mod p;
A17: for d being Element of NAT st d in dom f holds p divides (f.d -a)*(
Poly-INT fk).(f.d)
proof
let d be Element of NAT;
f.d is Element of INT by INT_1:def 2;
then
A18: (Poly-INT fp).(f.d) mod p = (((f.d-a)*(Poly-INT fk).(f.d)) +r)
mod p by A14
.=(((f.d-a)*(Poly-INT fk).(f.d)) mod p + (r mod p)) mod p by NAT_D:66
.= ((f.d-a)*(Poly-INT fk).(f.d)) mod p by A9,A16,NAT_D:65;
assume d in dom f;
then ((f.d-a)*(Poly-INT fk).(f.d)) mod p = 0 by A10,A18;
hence thesis by INT_1:62;
end;
per cases;
suppose
A19: for d st d in dom f holds not p divides (f.d - a);
for d st d in dom f holds (Poly-INT fk).(f.d) mod p = 0
proof
let d;
assume
A20: d in dom f;
then p divides (f.d -a)*(Poly-INT fk).(f.d) by A17;
then p divides (f.d -a) or p divides (Poly-INT fk).(f.d) by Th7;
hence thesis by A19,A20,INT_1:62;
end;
then len f <= n by A2,A4,A5,A13,A15,A11;
hence contradiction by A12,XREAL_1:145;
end;
suppose
ex d st d in dom f & p divides (f.d - a);
then consider d9 being Element of NAT such that
A21: d9 in dom f and
A22: p divides (f.d9 - a);
set f9 = f - {f.d9};
A23: for d st d in dom f9 holds not p divides (f9.d - a)
proof
given k being Nat such that
A24: k in dom f9 and
A25: p divides (f9.k -a);
f9.k in rng f9 by A24,FUNCT_1:3;
then
A26: f9.k in rng f \ {f.d9} by FINSEQ_3:65;
then f9.k in rng f by XBOOLE_0:def 5;
then consider w being object such that
A27: w in dom f and
A28: f.w = f9.k by FUNCT_1:def 3;
reconsider w as Element of NAT by A27;
p divides ((f.w - a)-(f.d9 - a)) by A22,A25,A28,Th1; then
A29: f.w,f.d9 are_congruent_mod p;
not f9.k in {f.d9} by A26,XBOOLE_0:def 5;
then w <> d9 by A28,TARSKI:def 1;
hence contradiction by A11,A21,A27,A29;
end;
A30: for d st d in dom f9 holds (Poly-INT fk).(f9.d) mod p = 0
proof
let d;
assume
A31: d in dom f9;
then f9.d in rng f9 by FUNCT_1:3;
then f9.d in rng f \ {f.d9} by FINSEQ_3:65;
then f9.d in rng f by XBOOLE_0:def 5;
then ex w being object st w in dom f & f.w = f9.d by FUNCT_1:def 3;
then p divides (f9.d -a)*(Poly-INT fk).(f9.d) by A17;
then p divides (f9.d -a) or p divides (Poly-INT fk).(f9.d) by Th7;
hence thesis by A23,A31,INT_1:62;
end;
A32: f is one-to-one
by A11,INT_1:11;
then
A33: f9 is one-to-one by FINSEQ_3:87;
A34: for d,e st d in dom f9 & e in dom f9 & d<>e holds not f9.d,f9.e
are_congruent_mod p
proof
let d,e;
assume that
A35: d in dom f9 and
A36: e in dom f9 and
A37: d<>e;
f9.e in rng f9 by A36,FUNCT_1:3;
then f9.e in rng f \ {f.d9} by FINSEQ_3:65;
then f9.e in rng f by XBOOLE_0:def 5;
then consider w2 being object such that
A38: w2 in dom f and
A39: f9.e = f.w2 by FUNCT_1:def 3;
f9.d in rng f9 by A35,FUNCT_1:3;
then f9.d in rng f \ {f.d9} by FINSEQ_3:65;
then f9.d in rng f by XBOOLE_0:def 5;
then consider w1 being object such that
A40: w1 in dom f and
A41: f9.d = f.w1 by FUNCT_1:def 3;
reconsider w1,w2 as Element of NAT by A40,A38;
w1 <> w2 by A33,A35,A36,A37,A41,A39;
hence thesis by A11,A40,A41,A38,A39;
end;
f.d9 in rng f by A21,FUNCT_1:3;
then len f9 = len f - 1 by A32,FINSEQ_3:90;
then len f9 > (n+1)-1 by A12,XREAL_1:9;
hence contradiction by A2,A4,A5,A13,A15,A30,A34;
end;
end;
end;
A42: P[0]
proof
let fp;
assume that
A43: len fp = 0+1 and
p>2 and
A44: not p divides fp.(0+1);
assume ex fr st (for d st d in dom fr holds (Poly-INT fp).(fr.d) mod p=0)
& (for d,e st d in dom fr & e in dom fr & d<>e holds not fr.d,fr.e
are_congruent_mod p) & len fr > 0;
then consider fr such that
A45: for d st d in dom fr holds (Poly-INT fp).(fr.d) mod p=0 and
for d,e st d in dom fr & e in dom fr & d<>e holds not fr.d,fr.e
are_congruent_mod p and
A46: len fr > 0;
fr <> {} by A46;
then
A47: (Poly-INT fp).(fr.1) mod p=0 by A45,FINSEQ_5:6;
A48: fr.1 in INT by INT_1:def 2;
(Poly-INT fp).(fr.1) = (INT --> fp.1).(fr.1) by A43,Th3
.= fp.1 by A48,FUNCOP_1:7;
hence contradiction by A44,A47,Lm1;
end;
for n holds P[n] from NAT_1:sch 2(A42,A1);
hence thesis;
end;
definition
let a be Integer, m be natural Number;
pred a is_quadratic_residue_mod m means
ex x being Integer st (x^2 - a) mod m = 0;
end;
reserve b,m for Nat;
theorem Th9:
a^2 is_quadratic_residue_mod m
proof
(a^2 - a^2) mod m = 0 by INT_1:73;
hence thesis;
end;
theorem
1 is_quadratic_residue_mod 2
proof
1^2 is_quadratic_residue_mod 2 by Th9;
hence thesis;
end;
theorem Th11:
i is_quadratic_residue_mod m & i,j are_congruent_mod m implies
j is_quadratic_residue_mod m
proof
assume that
A1: i is_quadratic_residue_mod m and
A2: i,j are_congruent_mod m;
consider x being Integer such that
A3: (x^2 - i) mod m = 0 by A1;
A4: (i - j) mod m = 0 by Lm1,A2;
(x^2 - j) mod m = ((x^2 - i) + (i - j)) mod m
.= (((x^2 - i) mod m) + ((i - j) mod m)) mod m by NAT_D:66
.= 0 by A3,A4,NAT_D:65;
hence thesis;
end;
Lm2: i,p are_coprime or p divides i
proof
per cases;
suppose
i>=0;
then reconsider i as Element of NAT by INT_1:3;
i,p are_coprime or (i gcd p) = p by PEPIN:2;
hence thesis by NAT_D:def 5;
end;
suppose
A1: i<0;
then reconsider m = -i as Element of NAT by INT_1:3;
A2: m,p are_coprime or (m gcd p) = p by PEPIN:2;
per cases by A2,NAT_D:def 5;
suppose
A3: m,p are_coprime;
m = |.i.| by A1,ABSVALUE:def 1;
then i gcd p = m gcd |.p.| by INT_2:34
.= m gcd p by ABSVALUE:def 1
.= 1 by A3,INT_2:def 3;
hence thesis by INT_2:def 3;
end;
suppose
p divides m;
then consider t being Nat such that
A4: m = p * t by NAT_D:def 3;
i = p * (-t) by A4;
hence thesis;
end;
end;
end;
theorem Th12:
i divides j implies i gcd j = |.i.|
proof
assume i divides j;
then |.i.| gcd |.j.| = |.i.| by NEWTON:49,INT_2:16;
hence thesis by INT_2:34;
end;
theorem Th13:
for i,j,m being Integer st i mod m = j mod m holds i|^n mod m = j|^n mod m
proof
let i,j,m be Integer;
defpred P[Nat] means i|^$1 mod m = j|^$1 mod m;
assume
A1: i mod m = j mod m;
A2: for n being Nat st P[n] holds P[n+1]
proof
let n be Nat;
assume
A3: P[n];
thus i|^(n+1) mod m = ((i|^n)*i) mod m by NEWTON:6
.= (((j|^n) mod m)*(j mod m)) mod m by A1,A3,NAT_D:67
.= ((j|^n)*j) mod m by NAT_D:67
.= j|^(n+1) mod m by NEWTON:6;
end;
i|^0 = 1 by NEWTON:4;
then
A4: P[0] by NEWTON:4;
for n being Nat holds P[n] from NAT_1:sch 2(A4,A2);
hence thesis;
end;
theorem Th14:
a gcd p = 1 & (x^2 - a) mod p = 0 implies x,p are_coprime
proof
assume that
A1: a gcd p = 1 and
A2: (x^2 - a) mod p = 0;
assume not x,p are_coprime;
then
A3: p divides x^2 by Lm2,INT_2:2;
p divides (x^2 - a) by A2,Lm1;
then p divides (x^2 - (x^2 - a)) by A3,Th1;
then p gcd a = |.p.| by Th12
.= p by ABSVALUE:def 1;
hence contradiction by A1,INT_2:def 4;
end;
theorem
p > 2 & a gcd p = 1 & a is_quadratic_residue_mod p implies ex x,y
being Integer st (x^2 - a) mod p = 0 & (y^2 - a) mod p = 0 & not x,y
are_congruent_mod p
proof
assume that
A1: p > 2 and
A2: a gcd p = 1 and
A3: a is_quadratic_residue_mod p;
consider x such that
A4: (x^2 - a) mod p = 0 by A3;
take x;
take -x;
not x,(-x) are_congruent_mod p
proof
assume x,(-x) are_congruent_mod p;
then
A5: p divides 2*x;
2,p are_coprime by A1,INT_2:28,30;
then 2 gcd p = 1 by INT_2:def 3;
then p divides x by A5,WSIERP_1:29;
then consider i being Integer such that
A6: x = p * i;
x gcd p = p*i gcd p*1 by A6
.= p*(i gcd 1) by EULER_1:15
.= p*1 by WSIERP_1:8;
then x gcd p <> 1 by INT_2:def 4;
then not x,p are_coprime by INT_2:def 3;
hence contradiction by A2,A4,Th14;
end;
hence thesis by A4;
end;
theorem Th16:
p>2 implies ex fp being FinSequence of NAT st len fp = (p-'1)
div 2 & (for d st d in dom fp holds fp.d gcd p = 1) & (for d st d in dom fp
holds fp.d is_quadratic_residue_mod p) & for d,e st d in dom fp & e in dom fp &
d<>e holds not fp.d,fp.e are_congruent_mod p
proof
deffunc F(Nat) = $1^2;
consider fp being FinSequence such that
A1: len fp = (p-'1) div 2 & for d being Nat st d in dom fp holds fp.d =
F(d) from FINSEQ_1:sch 2;
for d being Nat st d in dom fp holds fp.d in NAT
proof
let d be Nat;
assume d in dom fp;
then fp.d = d^2 by A1;
hence thesis;
end;
then reconsider fp as FinSequence of NAT by FINSEQ_2:12;
A2: p>1 by INT_2:def 4;
then
A3: p-'1 = p-1 by XREAL_1:233;
assume p > 2;
then p is odd by PEPIN:17;
then p-1 is even by HILBERT3:2;
then 2 divides (p-'1) by A3,PEPIN:22;
then (p-'1) mod 2 = 0 by PEPIN:6;
then
A4: (p-'1) div 2 = (p-'1)/2 by PEPIN:63;
A5: for d,e st d in dom fp & e in dom fp & d<>e holds not fp.d,fp.e
are_congruent_mod p
proof
p-1>0 by A2,XREAL_1:50;
then (p-1)/2 < (p-1)/1 by XREAL_1:76;
then (p-'1) div 2 < p by A3,A4,XREAL_1:147;
then
A6: ((p-'1) div 2) - 1 < p by XREAL_1:147;
let d,e;
assume that
A7: d in dom fp and
A8: e in dom fp and
A9: d<>e;
A10: e in Seg ((p-'1) div 2) by A1,A8,FINSEQ_1:def 3;
then
A11: e <= (p-'1) div 2 by FINSEQ_1:1;
A12: d in Seg ((p-'1) div 2) by A1,A7,FINSEQ_1:def 3;
then
A13: d >= 1 by FINSEQ_1:1;
then 1-((p-'1) div 2) <= d-e by A11,XREAL_1:13;
then
A14: -(((p-'1) div 2) - 1) <= d-e;
A15: d <= (p-'1) div 2 by A12,FINSEQ_1:1;
then d+e <= (p-'1) div 2 + ((p-'1) div 2) by A11,XREAL_1:7;
then d+e < p by A3,A4,XREAL_1:147;
then (d+e),p are_coprime by A13,EULER_1:2;
then
A16: (d+e) gcd p = 1 by INT_2:def 3;
assume fp.d,fp.e are_congruent_mod p;
then p divides (d^2 - fp.e) by A1,A7;
then p divides (d^2 - e^2) by A1,A8;
then
A17: p divides (d - e)*(d+e);
d-e <> 0 by A9;
then |.p.| <= |.d-e.| by A16,A17,INT_4:6,WSIERP_1:29;
then
A18: p <= |.d-e.| by ABSVALUE:def 1;
e >= 1 by A10,FINSEQ_1:1;
then d-e <= ((p-'1) div 2) - 1 by A15,XREAL_1:13;
then |.d-e.| <= ((p-'1) div 2) - 1 by A14,ABSVALUE:5;
hence contradiction by A18,A6,XXREAL_0:2;
end;
A19: for d st d in dom fp holds d gcd p = 1
proof
let d;
A20: 1*d <= 2*d by XREAL_1:64;
assume d in dom fp;
then
A21: d in Seg ((p-'1) div 2) by A1,FINSEQ_1:def 3;
then d <= (p-'1) div 2 by FINSEQ_1:1;
then 2*d <= (p-'1)/2*2 by A4,XREAL_1:64;
then d <= p-'1 by A20,XXREAL_0:2;
then
A22: d < p by A3,XREAL_1:147;
d >= 1 by A21,FINSEQ_1:1;
then d,p are_coprime by A22,EULER_1:2;
hence thesis by INT_2:def 3;
end;
A23: for d st d in dom fp holds fp.d gcd p = 1
proof
let d;
assume
A24: d in dom fp;
then d gcd p = 1 by A19;
then d^2 gcd p = 1 by WSIERP_1:7;
hence thesis by A1,A24;
end;
take fp;
for d st d in dom fp holds fp.d is_quadratic_residue_mod p
proof
let d;
assume
A25: d in dom fp;
d^2 is_quadratic_residue_mod p by Th9;
hence thesis by A1,A25;
end;
hence thesis by A1,A23,A5;
end;
::Euler Criterion
theorem Th17:
p > 2 & a gcd p = 1 & a is_quadratic_residue_mod p implies
a|^((p-'1) div 2) mod p = 1
proof
assume that
A1: p > 2 and
A2: a gcd p = 1 and
A3: a is_quadratic_residue_mod p;
consider s being Integer such that
A4: (s^2 - a) mod p = 0 by A3;
A5: p>1 by INT_2:def 4;
p is odd by A1,PEPIN:17;
then p-1 is even by HILBERT3:2;
then p-'1 is even by A5,XREAL_1:233;
then 2 divides (p-'1) by PEPIN:22;
then
A6: p-'1 = 2*((p-'1) div 2) by NAT_D:3;
s^2,a are_congruent_mod p by A4,INT_1:62;
then a mod p = s^2 mod p by NAT_D:64;
then
A7: a|^((p-'1) div 2) mod p = (s^2)|^((p-'1) div 2) mod p by Th13
.= (s|^2)|^((p-'1) div 2) mod p by NEWTON:81
.= s|^(p-'1) mod p by A6,NEWTON:9;
A8: s,p are_coprime by A2,A4,Th14;
per cases;
suppose
s>=0;
then reconsider s as Element of NAT by INT_1:3;
s,p are_coprime by A2,A4,Th14;
hence thesis by A7,PEPIN:37;
end;
suppose
A9: s<0;
then reconsider s9 = -s as Element of NAT by INT_1:3;
A10: |.p.| = p by ABSVALUE:def 1;
s9 gcd p = |.s.| gcd p by A9,ABSVALUE:def 1
.= s gcd p by A10,INT_2:34
.= 1 by A8,INT_2:def 3;
then s9,p are_coprime by INT_2:def 3;
then
A11: s9|^(p-'1) mod p = 1 by PEPIN:37;
s|^(p-'1) mod p = (s|^2)|^((p-'1) div 2) mod p by A6,NEWTON:9
.= ((-s)|^2)|^((p-'1) div 2) mod p by WSIERP_1:1
.= 1 by A6,A11,NEWTON:9;
hence thesis by A7;
end;
end;
theorem Th18:
p>2 & b gcd p = 1 & not b is_quadratic_residue_mod p implies
b|^((p-'1) div 2) mod p = p - 1
proof
assume that
A1: p>2 and
A2: b gcd p = 1 and
A3: not b is_quadratic_residue_mod p;
reconsider b as Element of NAT by ORDINAL1:def 12;
A4: p>1 by INT_2:def 4;
then
A5: 1 mod p = 1 by NAT_D:14;
p is odd by A1,PEPIN:17;
then p-1 is even by HILBERT3:2;
then p-'1 is even by A4,XREAL_1:233;
then 2 divides (p-'1) by PEPIN:22;
then p-'1 = 2*((p-'1) div 2) by NAT_D:3;
then
A6: b|^(p-'1) - 1 = (b|^((p-'1) div 2))|^2 - 1 by NEWTON:9
.= (b|^((p-'1) div 2))^2 - 1 by NEWTON:81
.= (b|^((p-'1) div 2) +1)*(b|^((p-'1) div 2)-1);
b,p are_coprime by A2,INT_2:def 3;
then (b|^(p-'1)) mod p = 1 by PEPIN:37;
then (b|^(p-'1) - 1) mod p = 0 by A5,INT_4:22;
then
A7: p divides (b|^((p-'1) div 2) +1)*(b|^((p-'1) div 2)-1) by A6,Lm1;
p-1 > 2-1 by A1,XREAL_1:9;
then p-1 >= 1 + 1 by INT_1:7;
then p-'1 >= 2 by A4,XREAL_1:233;
then (p-'1) div 2 >= 2 div 2 by NAT_2:24;then
A8: (p-'1) div 2 >= 1 by PEPIN:44;
per cases by A8,XXREAL_0:1;
suppose
A9: (p-'1) div 2 = 1;
A10: now
assume p divides (b - 1);
then p divides -(b - 1) by INT_2:10;
then (1^2 - b) mod p = 0 by Lm1;
hence contradiction by A3;
end;
p divides (b +1)*(b|^1 -1) by A7,A9;
then p divides (b - (-1)) by A10,Th7;
then b,(-1) are_congruent_mod p;
then
A11: b mod p = (-1) mod p by NAT_D:64;
-p < -2 by A1,XREAL_1:24;
then -p < -2+1 by XREAL_1:39;
then b mod p = p - 1 by A11,NAT_D:63;
hence thesis by A9;
end;
suppose
A12: (p-'1) div 2 > 1;
set l = (p-'1) div 2;
0 is Element of INT by INT_1:def 2;then
A13: (l-'1) |-> 0 is FinSequence of INT by FINSEQ_2:63;
set K1 = <*-1*>^((l-'1) |-> 0);
A14: len ((l-'1) |-> 0) = l-'1 by CARD_1:def 7;
A15: len K1 = 1 + (l-'1) by CARD_1:def 7
.= l by A12,XREAL_1:235;
set fs = <*-1*>^((l-'1) |-> 0)^<*1*>;
1 is Element of INT by INT_1:def 2;then
A16: <*1*> is FinSequence of INT by FINSEQ_1:74;
-1 is Element of INT by INT_1:def 2;
then <*-1*> is FinSequence of INT by FINSEQ_1:74;
then <*-1*>^((l-'1) |-> 0) is FinSequence of INT by A13,FINSEQ_1:75;
then reconsider fs as FinSequence of INT by A16,FINSEQ_1:75;
A17: len fs = 1 + ((l-'1) +1) by CARD_1:def 7
.= 1 + l by A12,XREAL_1:235;
A18: fs.1 = (<*-1*>^(((l-'1) |-> 0)^<*1*>)).1 by FINSEQ_1:32
.= -1 by FINSEQ_1:41;
A19: for x being Element of INT holds (Poly-INT fs).x = x|^l -1
proof
let x be Element of INT;
consider fr such that
A20: len fr = len fs and
A21: for d st d in dom fr holds fr.d = (fs.d) * x|^(d-'1) and
A22: (Poly-INT fs).x = Sum fr by Def1;
fr = <*-1*>^((l-'1) |-> 0)^<*x|^l*>
proof
set K = <*-1*>^((l-'1) |-> 0)^<*x|^l*>;
A23: for d being Nat st d in dom fr holds fr.d = K.d
proof
let d be Nat;
assume
A24: d in dom fr; then
A25: d in Seg(l + 1) by A17,A20,FINSEQ_1:def 3; then
A26: d>=1 by FINSEQ_1:1;
A27: d<=l+1 by A25,FINSEQ_1:1;
per cases by A26,A27,XXREAL_0:1;
suppose
A28: d=1; then
A29: fr.1 = (fs.1) * x|^(1-'1) by A21,A24
.= (fs.1) * x|^0 by XREAL_1:232
.= (fs.1) * 1 by NEWTON:4
.= -1 by A18;
K.1 = (<*-1*>^(((l-'1) |-> 0)^<*x|^l*>)).1 by FINSEQ_1:32
.= fr.1 by A29,FINSEQ_1:41;
hence thesis by A28;
end;
suppose
A30: d>1 & d<l+1;
then reconsider w = d-1 as Element of NAT by INT_1:3;
d-1<l+1-1 by A30,XREAL_1:9; then
A31: w <= l-'1 by NAT_D:49;
A32: w >= 0+1 by A30,INT_1:7,XREAL_1:50;
A33: ((l-'1) |-> 0).w = 0;
w in Seg(l-'1) by A31,A32,FINSEQ_1:1; then
A34: w in dom ((l-'1) |-> 0) by A14,FINSEQ_1:def 3; then
A35: w in dom (((l-'1) |-> 0)^<*1*>) by FINSEQ_2:15;
A36: w in dom (((l-'1) |-> 0)^<*x|^l*>) by A34,FINSEQ_2:15;
A37: fs.d = (<*-1*>^(((l-'1) |-> 0)^<*1*>)).(w+1) by FINSEQ_1:32
.= (((l-'1) |-> 0)^<*1*>).w by A35,FINSEQ_3:103
.= 0 by A33,A34,FINSEQ_1:def 7;
thus K.d = (<*-1*>^(((l-'1) |-> 0)^<*x|^l*>)).(w+1) by FINSEQ_1:32
.= (((l-'1) |-> 0)^<*x|^l*>).w by A36,FINSEQ_3:103
.= (fs.d)*x|^(d-'1) by A33,A34,A37,FINSEQ_1:def 7
.= fr.d by A21,A24;
end;
suppose
A38: d=l+1;
then d in dom fs by A17,FINSEQ_5:6;
then
A39: d in dom fr by A20,FINSEQ_3:29;
fs.d = 1 by A15,A38,FINSEQ_1:42;
hence fr.d = 1 * x|^(l+1-'1) by A21,A38,A39
.= x|^l by NAT_D:34
.= K.d by A15,A38,FINSEQ_1:42;
end;
end;
len K = 1 + ((l-'1) +1) by CARD_1:def 7
.= len fr by A12,A17,A20,XREAL_1:235;
hence thesis by A23,FINSEQ_3:29;
end;
then Sum fr = Sum (<*-1*>^(((l-'1) |-> 0)^<*x|^l*>)) by FINSEQ_1:32
.= -1 + Sum (((l-'1) |-> 0)^<*x|^l*>) by RVSUM_1:76
.= -1 + (Sum ((l-'1) |-> 0) + x|^l) by RVSUM_1:74
.= -1 + ((l-'1)*0 + x|^l) by RVSUM_1:80;
hence thesis by A22;
end;
consider fp being FinSequence of NAT such that
A40: len fp = l and
A41: for d st d in dom fp holds fp.d gcd p = 1 and
A42: for d st d in dom fp holds fp.d is_quadratic_residue_mod p and
A43: for d,e st d in dom fp & e in dom fp & d<>e holds not fp.d,fp.e
are_congruent_mod p by A1,Th16;
A44: fs.(l+1) = 1 by A15,FINSEQ_1:42;
now
assume p divides (b|^l - 1);
then
A45: (b|^l - 1) mod p = 0 by Lm1;
reconsider b as Element of INT by INT_1:def 2;
set f = fp^<*b*>;
<*b*> is FinSequence of NAT by FINSEQ_1:74;
then reconsider f as FinSequence of NAT by FINSEQ_1:75;
A46: len f = l+1 by A40,FINSEQ_2:16;
A47: for d,e st d in dom f & e in dom f & d<>e holds not f.d,f.e
are_congruent_mod p
proof
let d,e;
assume that
A48: d in dom f and
A49: e in dom f and
A50: d<>e;
A51: e>=1 by A49,FINSEQ_3:25;
A52: d<= l+1 by A46,A48,FINSEQ_3:25;
A53: e<=l+1 by A46,A49,FINSEQ_3:25;
per cases by A48,A52,FINSEQ_3:25,XXREAL_0:1;
suppose
A54: d>=1 & d<l+1;
then d<=l by NAT_1:13;
then
A55: d in dom fp by A40,A54,FINSEQ_3:25;
then
A56: f.d = fp.d by FINSEQ_1:def 7;
per cases by A49,A53,FINSEQ_3:25,XXREAL_0:1;
suppose
A57: e>=1 & e<l+1;
then e<=l by NAT_1:13;
then
A58: e in dom fp by A40,A57,FINSEQ_3:25;
then not fp.d,fp.e are_congruent_mod p by A43,A50,A55;
hence thesis by A56,A58,FINSEQ_1:def 7;
end;
suppose
A59: e=l+1;
not f.d,b are_congruent_mod p
proof
f.d is_quadratic_residue_mod p by A42,A55,A56;
then consider j being Integer such that
A60: (j^2 - f.d) mod p = 0;
assume a61: f.d,b are_congruent_mod p;
p divides (j^2 - f.d) by A60,INT_1:62;
then p divides ((j^2 - f.d)+(f.d - b)) by a61,WSIERP_1:4;
then (j^2 - b) mod p = 0 by INT_1:62;
hence contradiction by A3;
end;
hence thesis by A40,A59,FINSEQ_1:42;
end;
end;
suppose
A62: d=l+1;
then e<=l by A50,A53,NAT_1:8;
then
A63: e in dom fp by A40,A51,FINSEQ_3:25;
then f.e = fp.e by FINSEQ_1:def 7;
then f.e is_quadratic_residue_mod p by A42,A63;
then consider j being Integer such that
A64: (j^2 - f.e) mod p = 0;
A65: p divides (j^2 - f.e) by A64,INT_1:62;
not b,f.e are_congruent_mod p
proof
assume b,f.e are_congruent_mod p;
then p divides ((j^2 - f.e) - (b - f.e)) by A65,Th1;
then (j^2 - b) mod p = 0 by INT_1:62;
hence contradiction by A3;
end;
hence thesis by A40,A62,FINSEQ_1:42;
end;
end;
A66: (Poly-INT fs).b mod p = 0 by A19,A45;
A67: for d st d in dom f holds (Poly-INT fs).(f.d) mod p = 0
proof
let d;
assume d in dom f;
then
A68: d in Seg (l+1) by A46,FINSEQ_1:def 3;
then
A69: d<=l+1 by FINSEQ_1:1;
per cases by A68,A69,FINSEQ_1:1,XXREAL_0:1;
suppose
A70: d>=1 & d<l+1;
reconsider k = fp.d as Element of INT by INT_1:def 2;
d<=l by A70,NAT_1:13; then
A71: d in dom fp by A40,A70,FINSEQ_3:25;
then fp.d gcd p = 1 by A41;
then (fp.d)|^l mod p = 1 mod p by A1,A5,A42,A71,Th17;
then (k|^l - 1) mod p = 0 by INT_4:22;
then (Poly-INT fs).k mod p = 0 by A19;
hence thesis by A71,FINSEQ_1:def 7;
end;
suppose
d=l+1;
hence thesis by A40,A66,FINSEQ_1:42;
end;
end;
reconsider f as FinSequence of INT by FINSEQ_2:24,NUMBERS:17;
not p divides fs.(l+1) by A4,A44,NAT_D:7;
then len f <= l by A1,A17,A67,A47,Th8;
hence contradiction by A46,XREAL_1:29;
end;
then p divides (b|^l + 1) by A7,Th7;
then consider k being Nat such that
A72: (b|^l + 1) = p*k by NAT_D:def 3;
-p < -1 by A4,XREAL_1:24;
then
A73: (-1) mod p = (-1) + p by NAT_D:63;
b|^l mod p = (p*k + (- 1)) mod p by A72
.= p - 1 by A73,NAT_D:61;
hence thesis;
end;
end;
theorem Th19:
p>2 & a gcd p = 1 & not a is_quadratic_residue_mod p implies
a|^((p-'1) div 2) mod p = p - 1
proof
assume that
A1: p>2 and
A2: a gcd p = 1 and
A3: not a is_quadratic_residue_mod p;
set l = a mod p;
reconsider l as Element of NAT by INT_1:3,57;
A4: l mod p = a mod p by NAT_D:65;
then
A5: l,a are_congruent_mod p by NAT_D:64;
then l gcd p = 1 by A2,WSIERP_1:43;
then l|^((p-'1) div 2) mod p = p - 1 by A1,A3,A5,Th11,Th18;
hence thesis by A4,Th13;
end;
theorem Th20:
p > 2 & a gcd p = 1 & a is_quadratic_residue_mod p implies (a|^(
(p-'1) div 2) - 1) mod p = 0
proof
assume that
A1: p > 2 and
A2: a gcd p = 1 and
A3: a is_quadratic_residue_mod p;
A4: p>1 by INT_2:def 4;
a|^((p-'1) div 2) mod p = 1 by A1,A2,A3,Th17;
then a|^((p-'1) div 2) mod p = 1 mod p by A4,NAT_D:14;
then a|^((p-'1) div 2),1 are_congruent_mod p by NAT_D:64;
hence thesis by INT_1:62;
end;
theorem Th21:
p > 2 & a gcd p = 1 & not a is_quadratic_residue_mod p implies
(a|^((p-'1) div 2) + 1) mod p = 0
proof
assume that
A1: p > 2 and
A2: a gcd p = 1 and
A3: not a is_quadratic_residue_mod p;
A4: p-1<p by XREAL_1:146;
a|^((p-'1) div 2) mod p = p-1 by A1,A2,A3,Th19;
then a|^((p-'1) div 2) mod p = (p-1) mod p by A4,NAT_D:63;
then a|^((p-'1) div 2),(p-1) are_congruent_mod p by NAT_D:64;
then p divides -((a|^((p-'1) div 2)+1) - p) by INT_2:10;
then p divides (p-(a|^((p-'1) div 2)+1));
then p divides (a|^((p-'1) div 2)+1) by Th2;
hence thesis by INT_1:62;
end;
reserve b for Integer;
theorem
a is_quadratic_residue_mod p & b is_quadratic_residue_mod p implies
a*b is_quadratic_residue_mod p
proof
assume that
A1: a is_quadratic_residue_mod p and
A2: b is_quadratic_residue_mod p;
consider i being Integer such that
A3: (i^2 - a) mod p =0 by A1;
consider j being Integer such that
A4: (j^2 - b) mod p =0 by A2;
A5: j^2,b are_congruent_mod p by A4,INT_1:62;
i^2,a are_congruent_mod p by A3,INT_1:62;
then (i^2)*(j^2),a*b are_congruent_mod p by A5,INT_1:18;
then ((i*j)^2 - a*b) mod p = 0 by INT_1:62;
hence thesis;
end;
theorem
p>2 & a gcd p = 1 & b gcd p = 1 & a is_quadratic_residue_mod p & not b
is_quadratic_residue_mod p implies not a*b is_quadratic_residue_mod p
proof
assume that
A1: p>2 and
A2: a gcd p = 1 and
A3: b gcd p = 1 and
A4: a is_quadratic_residue_mod p and
A5: not b is_quadratic_residue_mod p;
A6: a*b gcd p = 1 by A2,A3,WSIERP_1:6;
set l = (p-'1) div 2;
(b|^l + 1) mod p = 0 by A1,A3,A5,Th21; then
A7: p divides (b|^l + 1) by INT_1:62;
A8: (a|^l -1)*(b|^l +1) = a|^l * b|^l +a|^l*1 -1*b|^l -1*1
.= (a*b)|^l +a|^l*1 -1*b|^l -1*1 by NEWTON:7
.= ((a*b)|^l -1) +(a|^l - 1) -(b|^l - 1);
(a|^l -1) mod p = 0 by A1,A2,A4,Th20; then
A9: p divides (a|^l -1) by INT_1:62; then
A10: p divides (a|^l -1)*(b|^l +1) by INT_2:2;
assume a*b is_quadratic_residue_mod p;
then ((a*b)|^l -1) mod p = 0 by A1,A6,Th20;
then p divides ((a*b)|^l -1) by INT_1:62;
then p divides ((a*b)|^l -1) +(a|^l - 1) by A9,WSIERP_1:4;
then p divides (b|^l - 1) by A10,A8,Th2;
then p divides ((b|^l +1) - (b|^l -1)) by A7,Th1;
hence contradiction by A1,NAT_D:7;
end;
theorem
p>2 & a gcd p = 1 & b gcd p = 1 & not a is_quadratic_residue_mod p &
not b is_quadratic_residue_mod p implies a*b is_quadratic_residue_mod p
proof
assume that
A1: p>2 and
A2: a gcd p = 1 and
A3: b gcd p = 1 and
A4: not a is_quadratic_residue_mod p and
A5: not b is_quadratic_residue_mod p;
A6: a*b gcd p = 1 by A2,A3,WSIERP_1:6;
set l = (p-'1) div 2;
(b|^l + 1) mod p = 0 by A1,A3,A5,Th21;
then
A7: p divides (b|^l + 1) by INT_1:62;
A8: (a|^l +1)*(b|^l +1) = a|^l * b|^l +a|^l*1 +1*b|^l +1*1
.= (a*b)|^l +a|^l + b|^l +1 by NEWTON:7
.= ((a*b)|^l +1) +(a|^l + 1) - (1- b|^l);
(a|^l +1) mod p = 0 by A1,A2,A4,Th21; then
A9: p divides (a|^l +1) by INT_1:62; then
A10: p divides (a|^l +1)*(b|^l +1) by INT_2:2;
now
assume not a*b is_quadratic_residue_mod p;
then ((a*b)|^l +1) mod p = 0 by A1,A6,Th21;
then p divides ((a*b)|^l +1) by INT_1:62;
then p divides ((a*b)|^l +1) +(a|^l + 1) by A9,WSIERP_1:4;
then p divides (1- b|^l) by A10,A8,Th2;
then p divides ((b|^l +1) + (1- b|^l)) by A7,WSIERP_1:4;
hence contradiction by A1,NAT_D:7;
end;
hence thesis;
end;
definition
::$N Legendre symbol
let a be Integer, p be Prime;
func Lege (a,p) -> Integer equals :Def3:
1 if (a is_quadratic_residue_mod p & a mod p <> 0),
0 if (a is_quadratic_residue_mod p & a mod p = 0)
otherwise -1;
coherence;
consistency;
end;
theorem Th25:
Lege (a,p) = 1 or Lege (a,p) = 0 or Lege (a,p) = -1
proof
per cases;
suppose a is_quadratic_residue_mod p & a mod p <> 0;
hence thesis by Def3;
end;
suppose a is_quadratic_residue_mod p & a mod p = 0;
hence thesis by Def3;
end;
suppose not a is_quadratic_residue_mod p;
hence thesis by Def3;
end;
end;
theorem Th26:
a mod p <> 0 implies Lege (a^2,p) = 1
proof
assume a mod p <> 0; then
not p divides a by INT_1:62; then
not p divides a^2 by Th7; then
a^2 mod p <> 0 by INT_1:62;
hence thesis by Def3,Th9;
end;
theorem
Lege (1,p) = 1
proof
1 < p by INT_2:def 4; then
1 mod p <> 0 by NAT_D:14; then
Lege (1^2,p) = 1 by Th26;
hence thesis;
end;
Lm3: a gcd p = 1 implies not p divides a
proof
assume
A1: a gcd p = 1;
assume p divides a; then
p divides (p gcd a) by INT_2:def 2; then
p = 1 by A1,WSIERP_1:15;
hence thesis by INT_2:def 4;
end;
theorem Th28:
p>2 & a gcd p = 1 implies Lege (a,p),a|^((p-'1) div 2) are_congruent_mod p
proof
assume that
A1: p>2 and
A2: a gcd p = 1;
not p divides a by Lm3,A2; then
A3: a mod p <> 0 by INT_1:62;
A4: p>1 by INT_2:def 4;
then -p < -1 by XREAL_1:24; then
A5: (-1) mod p = p+(-1) by NAT_D:63;
per cases;
suppose
A6: a is_quadratic_residue_mod p;
then a|^((p-'1) div 2) mod p = 1 by A1,A2,Th17;
then a|^((p-'1) div 2) mod p = 1 mod p by A4,NAT_D:14;
then a|^((p-'1) div 2) mod p = Lege (a,p) mod p by A6,Def3,A3;
hence thesis by NAT_D:64;
end;
suppose
A7: not a is_quadratic_residue_mod p;
then a|^((p-'1) div 2) mod p = p-1 by A1,A2,Th19
.= Lege (a,p) mod p by A5,A7,Def3;
hence thesis by NAT_D:64;
end;
end;
theorem
p>2 & a gcd p =1 & a,b are_congruent_mod p implies Lege (a,p) = Lege (b,p)
proof
assume that
A1: p>2 and
A2: a gcd p = 1 and
A3: a,b are_congruent_mod p;
Lege (a,p),a|^((p-'1) div 2) are_congruent_mod p by A1,A2,Th28; then
A4: Lege (a,p) mod p = a|^((p-'1) div 2) mod p by NAT_D:64;
b gcd p = 1 by A2,A3,WSIERP_1:43;
then Lege (b,p),b|^((p-'1) div 2) are_congruent_mod p by A1,Th28;
then
A5: Lege (b,p) mod p = b|^((p-'1) div 2) mod p by NAT_D:64;
a mod p = b mod p by A3,NAT_D:64;
then Lege (a,p) mod p = Lege (b,p) mod p by A4,A5,Th13;
then Lege (a,p),Lege (b,p) are_congruent_mod p by NAT_D:64;
then
A6: p divides (Lege (a,p) - Lege (b,p));
per cases by Th25;
suppose
A7: Lege (a,p) = 1;
A8: now assume Lege (b,p) = 0; then
p = 1 by A6,A7,WSIERP_1:15;
hence contradiction by A1;
end;
Lege (b,p) <> -1 by A7,A1,A6,NAT_D:7;
hence thesis by A7,A8,Th25;
end;
suppose
A9: Lege (a,p) = 0;
A10: now assume Lege (b,p) = 1; then
p = 1 by WSIERP_1:15,A6,A9,INT_2:10;
hence contradiction by A1;
end;
now assume Lege (b,p) = -1; then
p = 1 by A6,A9,WSIERP_1:15;
hence contradiction by A1;
end;
hence thesis by A9,Th25,A10;
end;
suppose
A11: Lege (a,p) = -1;
A12: now
assume Lege (b,p) = 1;
then p divides -2 by A6,A11;
then p divides 2 by INT_2:10;
hence contradiction by A1,NAT_D:7;
end;
now assume Lege (b,p) = 0; then
p = 1 by WSIERP_1:15,A6,A11,INT_2:10;
hence contradiction by A1;
end;
hence thesis by A11,Th25,A12;
end;
end;
theorem
p>2 & a gcd p=1 & b gcd p=1 implies Lege(a*b,p)=Lege(a,p)*Lege(b,p)
proof
assume that
A1: p>2 and
A2: a gcd p=1 and
A3: b gcd p=1;
A4: Lege(b,p),b|^((p-'1) div 2) are_congruent_mod p by A1,A3,Th28;
Lege(a,p),a|^((p-'1) div 2) are_congruent_mod p by A1,A2,Th28;
then Lege(a,p)*Lege(b,p),(a|^((p-'1) div 2))*(b|^((p-'1) div 2))
are_congruent_mod p by A4,INT_1:18;
then Lege(a,p)*Lege(b,p),(a*b)|^((p-'1) div 2) are_congruent_mod p by
NEWTON:7;
then
A5: (a*b)|^((p-'1) div 2),Lege(a,p)*Lege(b,p) are_congruent_mod p by INT_1:14;
a*b gcd p = 1 by A2,A3,WSIERP_1:6;
then Lege(a*b,p),(a*b)|^((p-'1) div 2) are_congruent_mod p by A1,Th28;
then Lege(a*b,p),Lege(a,p)*Lege(b,p) are_congruent_mod p by A5,INT_1:15;
then
A6: p divides (Lege(a*b,p)-Lege(a,p)*Lege(b,p));
A7: Lege(b,p) = 1 or Lege(b,p) = -1 or Lege(b,p) = 0 by Th25;
A8: Lege(a,p) = 1 or Lege(a,p) = -1 or Lege(a,p) = 0 by Th25;
per cases by Th25;
suppose
A9: Lege(a*b,p) = 1;
now
assume Lege(a,p) = 0 or Lege(b,p) = 0; then
p = 1 by A6,A9,WSIERP_1:15;
hence contradiction by A1;
end;
hence thesis by A8,A7,A1,A6,A9,NAT_D:7;
end;
suppose
A10: Lege(a*b,p) = 0;
A11: now
assume Lege(a,p) * Lege(b,p) = -1; then
p <= 1 by A6,A10,NAT_D:7; then
p < 1+1 by NAT_1:13;
hence contradiction by A1;
end;
now
assume Lege(a,p) * Lege(b,p) = 1; then
p divides 1 by A6,A10,INT_2:10; then
p <= 1 by NAT_D:7; then
p < 1+1 by NAT_1:13;
hence contradiction by A1;
end;
hence thesis by A8,A7,A11,A10;
end;
suppose
A12: Lege(a*b,p) = -1;
A13: now
assume Lege(a,p) = 0 or Lege(b,p) = 0; then
p = 1 or p = -1 by A6,A12,INT_2:13;
hence contradiction by INT_2:def 4;
end;
now
assume Lege(a,p) * Lege(b,p) = 1; then
p divides (-2) by A6,A12; then
p divides 2 by INT_2:10;
hence contradiction by A1,NAT_D:7;
end;
hence thesis by A12,A13,A7,A8;
end;
end;
theorem Th31:
(for d st d in dom fr holds fr.d = 1 or fr.d = 0 or fr.d = -1) implies
Product fr = 1 or Product fr = 0 or Product fr = -1
proof
defpred P[FinSequence of INT] means (for d st d in dom $1 holds $1.d = 1 or
$1.d = 0 or $1.d = -1) implies Product $1 = 1 or Product $1 = 0 or
Product $1 = -1;
A1: for p being FinSequence of INT, n being Element of INT st P[p] holds P[p
^<*n*>]
proof
let p be FinSequence of INT,i be Element of INT;
set p1 = p^<*i*>;
assume
A2: P[p];
P[p1]
proof
assume
A3: for d st d in dom p1 holds p1.d = 1 or p1.d = 0 or p1.d = -1;
A4: for d st d in dom p holds p.d = 1 or p.d = 0 or p.d = -1
proof
let d;
assume
A5: d in dom p;
then p1.d = 1 or p1.d = 0 or p1.d = -1 by A3,FINSEQ_2:15;
hence thesis by A5,FINSEQ_1:def 7;
end;
A6: len p1 in dom p1 by FINSEQ_5:6;
A7: Product p1 = (Product p)*i by RVSUM_1:96;
len p1 =len p +1 by FINSEQ_2:16;
then
A8: p1.(len p + 1) = 1 or p1.(len p + 1) = 0 or
p1.(len p + 1) = -1 by A3,A6;
per cases by A2,A4,A8,FINSEQ_1:42;
suppose
Product p = 1 & i =1;
hence thesis by A7;
end;
suppose
Product p = 1 & i =0;
hence thesis by A7;
end;
suppose
Product p = 1 & i = -1;
hence thesis by A7;
end;
suppose
Product p = -1 & i = 1;
hence thesis by A7;
end;
suppose
Product p = -1 & i = 0;
hence thesis by A7;
end;
suppose
Product p = -1 & i = -1;
hence thesis by A7;
end;
suppose
Product p = 0 & i = 1;
hence thesis by A7;
end;
suppose
Product p = 0 & i = 0;
hence thesis by A7;
end;
suppose
Product p = 0 & i = -1;
hence thesis by A7;
end;
end;
hence thesis;
end;
A9: P[<*>INT] by RVSUM_1:94;
for p being FinSequence of INT holds P[p] from FINSEQ_2:sch 2(A9,A1 );
hence thesis;
end;
reserve m for Integer;
theorem Th32:
for f,fr st len f = len fr & (for d st d in dom f holds f.d,fr.d
are_congruent_mod m) holds Product f,Product fr are_congruent_mod m
proof
defpred P[Nat] means for f,fr st len f =$1 & len f=len fr & (for d st d in
dom f holds f.d,fr.d are_congruent_mod m) holds Product f,Product fr
are_congruent_mod m;
A1: for n be Nat st P[n] holds P[n+1]
proof
let n be Nat;
assume
A2: P[n];
P[n+1]
proof
let f,fr;
assume that
A3: len f = n+1 and
A4: len f = len fr and
A5: for d st d in dom f holds f.d,fr.d are_congruent_mod m;
consider fr1 being FinSequence of INT,b being Element of INT such that
A6: fr = fr1^<*b*> by A3,A4,FINSEQ_2:19;
f <> {} by A3; then
A7: (n+1) in dom f by A3,FINSEQ_5:6;
consider f1 being FinSequence of INT,a being Element of INT such that
A8: f = f1^<*a*> by A3,FINSEQ_2:19;
A9: n+1 = len fr1 +1 by A3,A4,A6,FINSEQ_2:16; then
A10: fr.(n+1) = b by A6,FINSEQ_1:42;
A11: n+1 = len f1 +1 by A3,A8,FINSEQ_2:16; then
A12: dom f1 = dom fr1 by A9,FINSEQ_3:29;
for d st d in dom f1 holds f1.d,fr1.d are_congruent_mod m
proof
let d;
assume
A13: d in dom f1; then
A14: f.d = f1.d by A8,FINSEQ_1:def 7;
fr.d = fr1.d by A6,A12,A13,FINSEQ_1:def 7;
hence thesis by A5,A8,A13,A14,FINSEQ_2:15;
end;
then
A15: Product f1,Product fr1 are_congruent_mod m by A2,A11,A9;
f.(n+1) = a by A8,A11,FINSEQ_1:42;
then a,b are_congruent_mod m by A5,A7,A10;
then (Product f1)*a,(Product fr1)*b are_congruent_mod m by A15,INT_1:18;
then Product f,(Product fr1)*b are_congruent_mod m by A8,RVSUM_1:96;
hence thesis by A6,RVSUM_1:96;
end;
hence thesis;
end;
A16: P[0]
proof
let f,fr;
assume that
A17: len f = 0 and
A18: len f = len fr;
A19: f = <*>INT by A17;
fr = <*>INT by A17,A18;
hence thesis by A19,INT_1:11;
end;
for n be Nat holds P[n] from NAT_1:sch 2(A16,A1);
hence thesis;
end;
theorem Th33:
for f,fr st len f = len fr & (for d st d in dom f holds f.d,-fr.
d are_congruent_mod m) holds Product f,((-1)|^(len f))*Product fr
are_congruent_mod m
proof
defpred P[Nat] means for f,fr st len f =$1 & len f=len fr & (for d st d in
dom f holds f.d,-fr.d are_congruent_mod m) holds Product f,((-1)|^(len f))*
Product fr are_congruent_mod m;
A1: for n be Nat st P[n] holds P[n+1]
proof
let n be Nat;
assume
A2: P[n];
P[n+1]
proof
let f,fr;
assume that
A3: len f = n+1 and
A4: len f = len fr and
A5: for d st d in dom f holds f.d,-fr.d are_congruent_mod m;
consider fr1 be FinSequence of INT,b be Element of INT such that
A6: fr = fr1^<*b*> by A3,A4,FINSEQ_2:19;
f <> {} by A3; then
A7: (n+1) in dom f by A3,FINSEQ_5:6;
consider f1 be FinSequence of INT,a be Element of INT such that
A8: f = f1^<*a*> by A3,FINSEQ_2:19;
A9: n+1 = len fr1 +1 by A3,A4,A6,FINSEQ_2:16; then
A10: fr.(n+1) = b by A6,FINSEQ_1:42;
A11: n+1 = len f1 +1 by A3,A8,FINSEQ_2:16;
then
A12: dom f1 = dom fr1 by A9,FINSEQ_3:29;
for d st d in dom f1 holds f1.d,-fr1.d are_congruent_mod m
proof
let d;
assume
A13: d in dom f1;
then
A14: f.d = f1.d by A8,FINSEQ_1:def 7;
fr.d = fr1.d by A6,A12,A13,FINSEQ_1:def 7;
hence thesis by A5,A8,A13,A14,FINSEQ_2:15;
end;
then
A15: Product f1,((-1)|^(len f1))*Product fr1 are_congruent_mod m by A2,A11,A9;
f.(n+1) = a by A8,A11,FINSEQ_1:42;
then a,-b are_congruent_mod m by A5,A7,A10;
then (Product f1)*a,((-1)|^(len f1))*(Product fr1)*(-b)
are_congruent_mod m by A15,INT_1:18; then
Product f,((-1)|^(len f1))*(-1)*((Product fr1)*b) are_congruent_mod
m by A8,RVSUM_1:96;
then Product f,((-1)|^(len f1 + 1))*((Product fr1)*b) are_congruent_mod
m by NEWTON:6;
hence thesis by A3,A6,A11,RVSUM_1:96;
end;
hence thesis;
end;
A16: P[0]
proof
let f,fr;
assume that
A17: len f = 0 and
A18: len f = len fr;
A19: f = <*>INT by A17;
A20: (-1)|^(len f) = 1 by A17,NEWTON:4;
fr = <*>INT by A17,A18;
hence thesis by A19,A20,INT_1:11;
end;
for n be Nat holds P[n] from NAT_1:sch 2(A16,A1);
hence thesis;
end;
reserve fp for FinSequence of NAT;
theorem Th34:
p>2 & (for d st d in dom fp holds fp.d gcd p = 1) implies ex fr
being FinSequence of INT st len fr = len fp & (for d st d in dom fr holds fr.d
= Lege (fp.d,p)) & Lege (Product fp,p) = Product fr
proof
assume
A1: p>2;
deffunc F(Nat) = Lege (fp.$1,p);
set k = (p-'1) div 2;
assume
A2: for d st d in dom fp holds fp.d gcd p = 1;
set f = fp|^k;
reconsider f as FinSequence of INT by FINSEQ_2:24,NUMBERS:17;
consider fr being FinSequence such that
A3: len fr = len fp & for d being Nat st d in dom fr holds fr.d = F(d)
from FINSEQ_1:sch 2;
for d being Nat st d in dom fr holds fr.d in INT
proof
let d be Nat;
assume d in dom fr;
then fr.d = Lege (fp.d,p) by A3;
hence thesis by INT_1:def 2;
end;
then reconsider fr as FinSequence of INT by FINSEQ_2:12;
A4: fp is FinSequence of REAL by FINSEQ_2:24,NUMBERS:19;
A5: len f = len fp by NAT_3:def 1;
for d st d in dom fr holds fr.d,f.d are_congruent_mod p
proof
let d;
assume
A6: d in dom fr;
then d in dom fp by A3,FINSEQ_3:29;
then fp.d gcd p = 1 by A2;
then Lege (fp.d,p),(fp.d)|^k are_congruent_mod p by A1,Th28; then
A7: fr.d,(fp.d)|^k are_congruent_mod p by A3,A6;
d in dom f by A3,A5,A6,FINSEQ_3:29;
hence thesis by A7,NAT_3:def 1;
end; then
A8: Product f,Product fr are_congruent_mod p by A3,A5,Th32,INT_1:14;
Product(fp) gcd p = 1 by A2,WSIERP_1:36;
then
Lege (Product fp,p),(Product fp)|^((p-'1) div 2) are_congruent_mod p by A1
,Th28;
then Lege (Product fp,p),Product f are_congruent_mod p by A4,NAT_3:15;
then Lege (Product fp,p),Product fr are_congruent_mod p by A8,INT_1:15;
then
A9: p divides (Lege (Product fp,p) - Product fr);
take fr;
A10: for d st d in dom fr holds fr.d = 1 or fr.d = 0 or fr.d = -1
proof
let d;
assume d in dom fr;
then fr.d = Lege (fp.d,p) by A3;
hence thesis by Th25;
end;
per cases by Th25;
suppose
A11: Lege (Product fp,p) = 1;
then A12: Product fr <> -1 by A1,A9,NAT_D:7;
now
assume Product fr = 0;
then p = 1 by A9,A11,WSIERP_1:15;
hence contradiction by A1;
end;
hence thesis by A3,A10,A11,Th31,A12;
end;
suppose
A13: Lege (Product fp,p) = 0;
A14: now
assume Product fr = -1;
then p = 1 by A9,A13,WSIERP_1:15;
hence contradiction by A1;
end;
now
assume Product fr = 1;
then p divides 1 by A9,A13,INT_2:10;
then p = 1 by WSIERP_1:15;
hence contradiction by A1;
end;
hence thesis by A3,A10,A13,Th31,A14;
end;
suppose
A15: Lege (Product fp,p) = -1;
A16: now
assume Product fr = 1;
then p divides -2 by A9,A15;
then p divides 2 by INT_2:10;
hence contradiction by A1,NAT_D:7;
end;
now
assume Product fr = 0;
then p divides 1 by A9,A15,INT_2:10;
then p = 1 by WSIERP_1:15;
hence contradiction by A1;
end;
hence thesis by A3,A10,A15,Th31,A16;
end;
end;
theorem
p > 2 & d gcd p = 1 & e gcd p = 1 implies Lege((d^2)*e,p) = Lege(e,p)
proof
assume that
A1: p > 2 and
A2: d gcd p = 1 and
A3: e gcd p = 1;
reconsider d2=d^2, e as Element of NAT by ORDINAL1:def 12;
set fp = <*d2,e*>;
reconsider fp as FinSequence of NAT by FINSEQ_2:13;
not p divides d by A2,Lm3; then
d mod p <> 0 by INT_1:62; then
A4: Lege(d^2,p) = 1 by Th26;
reconsider p as prime Element of NAT by ORDINAL1:def 12;
for k st k in dom fp holds fp.k gcd p = 1
proof
let k;
assume k in dom fp;
then k in Seg(len fp) by FINSEQ_1:def 3;
then
A5: k in Seg 2 by FINSEQ_1:44;
per cases by A5,FINSEQ_1:2,TARSKI:def 2;
suppose
k = 1;
then fp.k = d^2 by FINSEQ_1:44;
hence thesis by A2,WSIERP_1:7;
end;
suppose
k = 2;
hence thesis by A3,FINSEQ_1:44;
end;
end;
then consider fr be FinSequence of INT such that
A6: len fr = len fp and
A7: for k be Nat st k in dom fr holds fr.k = Lege (fp.k,p) and
A8: Lege (Product fp,p) = Product fr by A1,Th34;
A9: len fr = 2 by A6,FINSEQ_1:44;
then 2 in dom fr by FINSEQ_3:25;
then fr.2 = Lege(fp.2,p) by A7;
then
A10: fr.2 = Lege(e,p) by FINSEQ_1:44;
fr.1 = Lege(fp.1,p) by A7,A9,FINSEQ_3:25;
then fr.1 = Lege(d^2,p) by FINSEQ_1:44;
then fr = <*1,Lege(e,p)*> by A4,A9,A10,FINSEQ_1:44;
then Product fr = 1 * Lege(e,p) by RVSUM_1:99;
hence thesis by A8,RVSUM_1:99;
end;
theorem Th36:
p>2 implies Lege (-1,p) = (-1)|^((p-'1) div 2)
proof
assume
A1: p>2;
|.(-1)|^((p-'1) div 2).| = 1 by SERIES_2:1;
then
A2: (-1)|^((p-'1) div 2) =1 or -(-1)|^((p-'1) div 2) =1 by ABSVALUE:1;
(-1) gcd p = |.(-1)|^1.| gcd |.p.| by INT_2:34
.= 1 gcd |.p.| by SERIES_2:1
.= 1 by NEWTON:51;
then
A3: Lege (-1,p),(-1)|^((p-'1) div 2) are_congruent_mod p by A1,Th28;
per cases by A2;
suppose
A4: (-1)|^((p-'1) div 2) = 1;
then
A5: p divides (Lege (-1,p) - 1) by A3;
A6: now
assume Lege(-1,p) = -1;
then p divides -2 by A5;
then p divides 2 by INT_2:10;
hence contradiction by A1,NAT_D:7;
end;
now
assume Lege(-1,p) = 0;
then p divides 1 by A5,INT_2:10; then
p <= 1 by NAT_D:7; then
p < 1+1 by NAT_1:13;
hence contradiction by A1;
end;
hence thesis by A4,Th25,A6;
end;
suppose
A7: (-1)|^((p-'1) div 2) = -1;
then A8: p divides (Lege (-1,p) - (-1)) by A3;
then A9: Lege(-1,p) <> 1 by A1,NAT_D:7;
now
assume Lege(-1,p) = 0; then
p <= 1 by A8,NAT_D:7; then
p < 1+1 by NAT_1:13;
hence contradiction by A1;
end;
hence thesis by A7,Th25,A9;
end;
end;
theorem
p>2 & p mod 4 = 1 implies (-1) is_quadratic_residue_mod p
proof
assume that
A1: p>2 and
A2: p mod 4 = 1;
p>1 by INT_2:def 4;
then
A3: p-'1 = p-1 by XREAL_1:233;
p = (p div 4)*4 + 1 by A2,NAT_D:2;
then p-'1 = 2*(2*(p div 4)) by A3;
then (-1)|^((p-'1) div 2) = (-1)|^(2*(p div 4)) by NAT_D:18
.= ((-1)|^2)|^(p div 4) by NEWTON:9
.= (1|^2)|^(p div 4) by WSIERP_1:1
.= 1;
then Lege(-1,p) = 1 by A1,Th36;
hence thesis by Def3;
end;
theorem
p>2 & p mod 4 = 3 implies not (-1) is_quadratic_residue_mod p
proof
assume that
A1: p>2 and
A2: p mod 4 = 3;
p>1 by INT_2:def 4; then
A3: p-'1 = p-1 by XREAL_1:233;
p = (p div 4)*4 + 3 by A2,NAT_D:2;
then p-'1 = 2*(2*(p div 4) + 1) by A3;
then (-1)|^((p-'1) div 2) = (-1)|^(2*(p div 4) + 1) by NAT_D:18
.= (-1)|^(2*(p div 4)) * (-1) by NEWTON:6
.= ((-1)|^2)|^(p div 4) *(-1) by NEWTON:9
.= (1|^2)|^(p div 4) *(-1) by WSIERP_1:1
.= 1 *(-1);
then Lege(-1,p) = -1 by A1,Th36; then
not (-1 is_quadratic_residue_mod p & -1 mod p <> 0) &
not (-1 is_quadratic_residue_mod p & -1 mod p = 0) by Def3;
hence thesis;
end;
begin :: Gauss Lemma and Law of Quadratic Reciprocity
theorem Th39:
for D being non empty set,f being FinSequence of D, i,j being
Nat holds f is one-to-one iff Swap(f,i,j) is one-to-one
proof
let D be non empty set,f be FinSequence of D,i,j be Nat;
thus f is one-to-one implies Swap(f,i,j) is one-to-one
proof
set ff = Swap(f,i,j);
A1: rng ff = rng f by FINSEQ_7:22;
assume f is one-to-one;
then
A2: card(rng f) = len f by FINSEQ_4:62;
len ff = len f by FINSEQ_7:18;
hence thesis by A2,A1,FINSEQ_4:62;
end;
assume Swap(f,i,j) is one-to-one;
then card(rng Swap(f,i,j)) = len Swap(f,i,j) by FINSEQ_4:62;
then card(rng f) = len Swap(f,i,j) by FINSEQ_7:22;
then card(rng f) = len f by FINSEQ_7:18;
hence thesis by FINSEQ_4:62;
end;
theorem Th40:
for f being FinSequence of NAT st len f = n & (for d st d in dom
f holds f.d>0 & f.d<=n) & f is one-to-one holds rng f = Seg n
proof
defpred P[Nat] means for f being FinSequence of NAT st len f = $1 & (for d
st d in dom f holds f.d>0 & f.d<=$1) & f is one-to-one holds rng f = Seg $1;
A1: for n be Nat st P[n] holds P[n+1]
proof
let n be Nat;
assume
A2: P[n];
P[n+1]
proof
let f be FinSequence of NAT;
assume that
A3: len f = n+1 and
A4: for d st d in dom f holds f.d>0 & f.d<=n+1 and
A5: f is one-to-one;
A6: f <> {} by A3;
then
A7: n+1 in dom f by A3,FINSEQ_5:6;
then
A8: f.(n+1) > 0 by A4;
consider f1 being FinSequence of NAT,a being Element of NAT such that
A9: f = f1^<*a*> by A6,HILBERT2:4;
A10: f1 is one-to-one by A5,A9,FINSEQ_3:91;
A11: len f = len f1 + 1 by A9,FINSEQ_2:16;
f.(n+1) <= n+1 by A4,A7;
then
A12: a<=n+1 by A3,A9,A11,FINSEQ_1:42;
per cases by A3,A9,A11,A8,A12,FINSEQ_1:42,XXREAL_0:1;
suppose
A13: a = n+1;
for d st d in dom f1 holds f1.d>0 & f1.d<=n
proof
let d;
assume
A14: d in dom f1;
then
A15: d in dom f by A9,FINSEQ_2:15;
A16: now
d <= n by A3,A11,A14,FINSEQ_3:25;
then d < n+1 by XREAL_1:145;
then f.d <> f.(n+1) by A5,A7,A15;
then
A17: f1.d <> f.(n+1) by A9,A14,FINSEQ_1:def 7;
assume f1.d = n+1;
hence contradiction by A3,A9,A11,A13,A17,FINSEQ_1:42;
end;
f.d<=n+1 by A4,A15;
then f1.d<=n+1 by A9,A14,FINSEQ_1:def 7;
then
A18: f1.d < n+1 by A16,XXREAL_0:1;
f.d>0 by A4,A15;
hence thesis by A9,A14,A18,FINSEQ_1:def 7,NAT_1:13;
end;
then rng f1 = Seg n by A2,A3,A11,A10;
then rng f1 \/ {a} = Seg (n+1) by A13,FINSEQ_1:9;
then rng f1 \/ rng <*a*> = Seg (n+1) by FINSEQ_1:38;
hence thesis by A9,FINSEQ_1:31;
end;
suppose
A19: a > 0 & a < n+1;
ex d st d in dom f1 & f1.d = n+1
proof
assume
A20: for d st d in dom f1 holds f1.d <> n+1;
for d being Nat st d in dom f holds f.d in Seg n
proof
let d be Nat;
assume
A21: d in dom f;
then
A22: d in Seg(n+1) by A3,FINSEQ_1:def 3;
then
A23: d<=n+1 by FINSEQ_1:1;
per cases by A22,A23,FINSEQ_1:1,XXREAL_0:1;
suppose
d=n+1;
then
A24: f.d = a by A3,A9,A11,FINSEQ_1:42;
then
A25: f.d<=n by A19,NAT_1:13;
f.d>=0+1 by A19,A24,NAT_1:13;
hence thesis by A25,FINSEQ_1:1;
end;
suppose
A26: d>=1 & d<n+1;
then d<=n by NAT_1:13;
then d in Seg n by A26,FINSEQ_1:1;
then
A27: d in dom f1 by A3,A11,FINSEQ_1:def 3;
then f1.d <> n+1 by A20;
then
A28: f.d <> n+1 by A9,A27,FINSEQ_1:def 7;
f.d <=n+1 by A4,A21;
then f.d <n+1 by A28,XXREAL_0:1;
then
A29: f.d<=n by NAT_1:13;
f.d>0 by A4,A21;
then f.d>=0+1 by NAT_1:13;
hence thesis by A29,FINSEQ_1:1;
end;
end;
then f is FinSequence of Seg n by FINSEQ_2:12;
then rng f c= Seg n by FINSEQ_1:def 4;
then card rng f <= card Seg n by NAT_1:43;
then n+1 <= card Seg n by A3,A5,FINSEQ_4:62;
then n+1 <= n+0 by FINSEQ_1:57;
hence contradiction by XREAL_1:6;
end;
then consider d1 being Element of NAT such that
A30: d1 in dom f1 and
A31: f1.d1 = n+1;
d1<=n by A3,A11,A30,FINSEQ_3:25;
then
A32: d1 <= len f by A3,NAT_1:13;
A33: 0+1 <= n+1 by XREAL_1:6;
set f2 = Swap(f,d1,n+1);
A34: len f2 = n+1 by A3,FINSEQ_7:18;
then
A35: f2 <> {};
then consider
f3 being FinSequence of NAT,b being Element of NAT such that
A36: f2 = f3^<*b*> by HILBERT2:4;
A37: n+1 = len f3 +1 by A34,A36,FINSEQ_2:16;
A38: 1 <= d1 by A30,FINSEQ_3:25;
then f2/.(n+1) = f/.d1 by A3,A32,A33,FINSEQ_7:31;
then f2/.(n+1) = f.d1 by A38,A32,FINSEQ_4:15;
then f2.(n+1) = f.d1 by A34,A33,FINSEQ_4:15;
then
A39: f2.(n+1) = n+1 by A9,A30,A31,FINSEQ_1:def 7;
then
A40: b = n+1 by A36,A37,FINSEQ_1:42;
A41: f2 is one-to-one by A5,Th39;
A42: for d st d in dom f3 holds f3.d>0 & f3.d<=n
proof
let d;
assume
A43: d in dom f3;
then
A44: d in dom f2 by A36,FINSEQ_2:15;
A45: now
d<=n by A37,A43,FINSEQ_3:25;
then
A46: d < n+1 by XREAL_1:145;
assume f3.d=n+1;
then
A47: f2.d = n+1 by A36,A43,FINSEQ_1:def 7;
n+1 in dom f2 by A34,A35,FINSEQ_5:6;
hence contradiction by A39,A41,A44,A47,A46;
end;
f2.d in rng f2 by A44,FUNCT_1:3;
then f2.d in rng f by FINSEQ_7:22;
then
A48: ex e being Nat st e in dom f & f2.d = f.e by FINSEQ_2:10;
then f2.d<=n+1 by A4;
then f3.d<=n+1 by A36,A43,FINSEQ_1:def 7;
then
A49: f3.d<n+1 by A45,XXREAL_0:1;
f2.d >0 by A4,A48;
hence thesis by A36,A43,A49,FINSEQ_1:def 7,NAT_1:13;
end;
f3 is one-to-one by A36,A41,FINSEQ_3:91;
then
A50: rng f3 = Seg n by A2,A37,A42;
rng f2 = rng f3 \/ rng <*b*> by A36,FINSEQ_1:31
.= Seg n \/ {n+1} by A40,A50,FINSEQ_1:38
.= Seg (n+1) by FINSEQ_1:9;
hence thesis by FINSEQ_7:22;
end;
end;
hence thesis;
end;
A51: P[0]
proof
let f be FinSequence of NAT;
assume len f = 0;
then f = {};
hence thesis;
end;
for n be Nat holds P[n] from NAT_1:sch 2(A51,A1);
hence thesis;
end;
reserve a,m for Nat;
theorem Th41:
for f be FinSequence of NAT st p>2 & a gcd p = 1 & f = a*(idseq
((p-'1) div 2)) & m=card{k where k is Element of NAT:k in rng (f mod p) & k > p
/2} holds Lege(a,p) = (-1)|^m
proof
let f be FinSequence of NAT;
assume that
A1: p>2 and
A2: a gcd p = 1 and
A3: f = a*(idseq ((p-'1) div 2)) and
A4: m=card{k where k is Element of NAT:k in rng (f mod p) & k > p/2};
set f1 = f mod p;
A5: len f1 = len f by EULER_2:def 1;
set X = {k where k is Element of NAT:k in rng f1 & k > p/2};
for x being object st x in X holds x in rng f1
proof
let x be object;
assume x in X;
then ex k being Element of NAT st x = k & k in rng f1 & k > p/2;
hence thesis;
end;
then
A6: X c= rng f1;
then reconsider X as finite set;
reconsider X as finite Subset of NAT by A6,XBOOLE_1:1;
card X is Element of NAT;
then reconsider m as Element of NAT by A4;
A7: rng f1 \ X c= rng f1 by XBOOLE_1:36;
reconsider Y = rng f1 \ X as finite Subset of NAT;
A8: a|^((p-'1) div 2),Lege(a,p) are_congruent_mod p by A1,A2,Th28,INT_1:14;
set f2 = Sgm rng f1;
(Product f1) mod p = (Product f) mod p by EULER_2:11;
then
A9: Product f1,Product f are_congruent_mod p by NAT_D:64;
A10: p>1 by INT_2:def 4;
then
A11: p-'1 = p-1 by XREAL_1:233;
then
A12: p-'1>0 by A10,XREAL_1:50;
set p9 = (p-'1) div 2;
rng idseq p9 = Seg p9;
then reconsider I = idseq p9 as FinSequence of NAT by FINSEQ_1:def 4;
dom f = dom I by A3,VALUED_1:def 5;
then
A13: len f = len I by FINSEQ_3:29
.= p9 by CARD_1:def 7;
p >= 2+1 by A1,NAT_1:13;
then p-1 >= 3-1 by XREAL_1:9;
then f1 <> {} by A13,A11,A5,NAT_2:13;
then rng f1 is non empty Subset of NAT;
then consider n1 being Element of NAT such that
A14: rng f1 c= Seg n1 \/ {0} by HEYTING3:1;
I is Element of p9-tuples_on REAL by FINSEQ_2:109,NUMBERS:19;
then
A15: Product f = (Product (p9|->a))*(Product I) by A3,RVSUM_1:108
.= (a|^p9) * (Product I) by NEWTON:def 1;
p is odd by A1,PEPIN:17;
then
A16: p-'1 is even by A11,HILBERT3:2;
then
A17: p9 = ((p-'1)+1) div 2 by NAT_2:26
.= p div 2 by A10,XREAL_1:235;
2 divides (p-'1) by A16,PEPIN:22;
then
A18: p-'1 = 2*p9 by NAT_D:3;
then p9 divides (p-'1);
then p9 <= (p-'1) by A12,NAT_D:7;
then
A19: p9 < p by A11,XREAL_1:146,XXREAL_0:2;
for d being Nat st d in dom I holds I.d gcd p = 1
proof
let d be Nat;
assume d in dom I;
then
A20: d in Seg len I by FINSEQ_1:def 3;
then
A21: d in Seg p9 by CARD_1:def 7;
then
A22: I.d = d by FINSEQ_2:49;
d <= p9 by A21,FINSEQ_1:1;
then
A23: d < p by A19,XXREAL_0:2;
d >= 1 by A20,FINSEQ_1:1;
then d,p are_coprime by A23,EULER_1:2;
hence thesis by A22,INT_2:def 3;
end;
then
A24: (Product I) gcd p = 1 by WSIERP_1:36;
A25: for d st d in dom f holds f.d = a*d
proof
let d;
assume
A26: d in dom f;
then d in dom I by A3,VALUED_1:def 5;
then d in Seg len I by FINSEQ_1:def 3;
then
A27: d is Element of Seg p9 by CARD_1:def 7;
thus f.d = a * I.d by A3,A26,VALUED_1:def 5
.= a*d by A27;
end;
A28: for d,e being Nat st 1<=d & d<e & e<=len f1 holds f1.d <> f1 .e
proof
let d,e be Nat;
assume that
A29: 1<=d and
A30: d<e and
A31: e<=len f1;
A32: e<=len f by A31,EULER_2:def 1;
1<=e by A29,A30,XXREAL_0:2;
then
A33: e in dom f by A32,FINSEQ_3:25;
then
A34: f1.e = (f.e) mod p by EULER_2:def 1;
d<len f by A30,A32,XXREAL_0:2;
then
A35: d in dom f by A29,FINSEQ_3:25;
then
A36: f1.d = (f.d) mod p by EULER_2:def 1;
now
assume f1.d = f1.e;
then f.e,f.d are_congruent_mod p by A36,A34,NAT_D:64;
then p divides (a*e - f.d) by A25,A33;
then p divides (a*e - a*d) by A25,A35;
then
A37: p divides a*(e-d);
A38: p9-1<p by A19,XREAL_1:147;
reconsider dd = e-d as Element of NAT by A30,NAT_1:21;
A39: |.p.| = p by ABSVALUE:def 1;
A40: |.dd.| = dd by ABSVALUE:def 1;
A41: dd <= p9 - 1 by A13,A5,A29,A31,XREAL_1:13;
dd <> 0 by A30;
then p <= dd by A2,A37,A39,A40,INT_4:6,WSIERP_1:29;
hence contradiction by A41,A38,XXREAL_0:2;
end;
hence thesis;
end;
then
A42: len f1 = card rng f1 by GRAPH_5:7;
then
A43: f1 is one-to-one by FINSEQ_4:62;
A44: dom f1 = dom f by A5,FINSEQ_3:29;
not 0 in rng f1
proof
reconsider a as Element of NAT by ORDINAL1:def 12;
assume 0 in rng f1;
then consider n be Nat such that
A45: n in dom f1 and
A46: f1.n = 0 by FINSEQ_2:10;
0 = (f.n) mod p by A44,A45,A46,EULER_2:def 1
.= (a*n) mod p by A25,A44,A45;
then
A47: p divides (a*n) by PEPIN:6;
n >= 1 by A45,FINSEQ_3:25;
then
A48: p <= n by A2,A47,NAT_D:7,WSIERP_1:30;
n <= p9 by A13,A5,A45,FINSEQ_3:25;
hence contradiction by A19,A48,XXREAL_0:2;
end;
then
A49: {0} misses rng f1 by ZFMISC_1:50;
then
A50: f2 is one-to-one by A14,FINSEQ_3:92,XBOOLE_1:73;
A51: rng f1 c= Seg n1 by A14,A49,XBOOLE_1:73;
then
A52: X c= Seg n1 by A6;
len f = card rng f1 by A5,A28,GRAPH_5:7;
then reconsider n = p9 - m as Element of NAT by A4,A13,A6,NAT_1:21,43;
A53: Y c= Seg n1 by A51,A7;
A54: rng f1 = rng f2 by A51,FINSEQ_1:def 13;
then
A55: Product f1 = Product f2 by A43,A50,EULER_2:10,RFINSEQ:26;
set f3 = (len (f2/^n) |-> p) - (f2/^n);
set f4 = (f2|n) ^ f3;
A56: f2/^n is FinSequence of INT by FINSEQ_2:24,NUMBERS:17;
A57: dom f3 = dom(len (f2/^n) |-> p)/\dom(f2/^n) by VALUED_1:12
.= (Seg len (len(f2/^n) |-> p))/\dom(f2/^n) by FINSEQ_1:def 3
.= dom(f2/^n) /\ dom(f2/^n) by FINSEQ_1:def 3,CARD_1:def 7
.= dom(f2/^n);
then
A58: len f3 = len(f2/^n) by FINSEQ_3:29;
for k,l being Nat st k in Y & l in X holds k < l
proof
let k,l be Nat;
assume that
A59: k in Y and
A60: l in X;
A61: not k in X by A59,XBOOLE_0:def 5;
A62: ex l1 being Element of NAT st l1 = l & l1 in rng f1 & l1>p/2 by A60;
k in rng f1 by A59,XBOOLE_0:def 5;
then k <= p/2 by A61;
hence thesis by A62,XXREAL_0:2;
end;
then Sgm (Y\/X) = (Sgm Y)^(Sgm X) by A52,A53,FINSEQ_3:42;
then Sgm (rng f1 \/ X) = (Sgm Y)^(Sgm X) by XBOOLE_1:39;
then
A63: f2 = (Sgm Y)^(Sgm X) by A6,XBOOLE_1:12;
A64: for d st d in dom f3 holds f3.d = p - (f2/^n).d
proof
let d;
assume
A65: d in dom f3;
then d in Seg len(f2/^n) by A57,FINSEQ_1:def 3;
then (len (f2/^n) |-> p).d = p by FINSEQ_2:57;
hence thesis by A65,VALUED_1:13;
end;
A66: len Sgm Y = card Y by A51,A7,FINSEQ_3:39,XBOOLE_1:1
.= p9 - m by A4,A13,A5,A6,A42,CARD_2:44;
then
A67: f2/^n = Sgm X by A63,FINSEQ_5:37;
A68: for d st d in dom f3 holds f3.d > 0 & f3.d <= p9
proof
let d;
reconsider w = f3.d as Element of INT by INT_1:def 2;
assume
A69: d in dom f3;
then (Sgm X).d in rng Sgm X by A67,A57,FUNCT_1:3;
then (Sgm X).d in X by A52,FINSEQ_1:def 13;
then
A70: ex ll be Element of NAT st ll = (Sgm X).d & ll in rng f1 & ll > p/2;
then consider e being Nat such that
A71: e in dom f1 and
A72: f1.e = (f2/^n).d by A67,FINSEQ_2:10;
(f2/^n).d = f.e mod p by A44,A71,A72,EULER_2:def 1;
then
A73: (f2/^n).d < p by NAT_D:1;
A74: f3.d = p - (f2/^n).d by A64,A69;
then w < p - p/2 by A67,A70,XREAL_1:10;
hence thesis by A17,A74,A73,INT_1:54,XREAL_1:50;
end;
A75: rng f3 c= INT by RELAT_1:def 19;
for d being Nat st d in dom f3 holds f3.d in NAT
proof
let d be Nat;
assume
A76: d in dom f3;
f3.d > 0 by A68,A76;
hence thesis by A75,INT_1:3;
end;
then reconsider f3 as FinSequence of NAT by FINSEQ_2:12;
|.(-1)|^m.| = 1 by SERIES_2:1;
then
A77: (-1)|^m = 1 or -((-1)|^m) = 1 by ABSVALUE:1;
f3 is FinSequence of NAT;
then reconsider f4 as FinSequence of NAT by FINSEQ_1:75;
A78: f2|n = Sgm Y by A63,A66,FINSEQ_3:113,FINSEQ_6:10;
A79: for d st d in dom f4 holds f4.d>0 & f4.d <= p9
proof
let d;
assume
A80: d in dom f4;
per cases by A80,FINSEQ_1:25;
suppose
A81: d in dom(f2|n);
reconsider d as Element of NAT by ORDINAL1:def 12;
(f2|n).d in rng Sgm Y by A78,A81,FUNCT_1:3;
then
A82: (f2|n).d in Y by A53,FINSEQ_1:def 13;
then
A83: (f2|n).d in rng f1 by XBOOLE_0:def 5;
not (f2|n).d in X by A82,XBOOLE_0:def 5;
then (f2|n).d <= p/2 by A83;
then
A84: (f2|n).d <= p9 by A17,INT_1:54;
not (f2|n).d in {0} by A49,A83,XBOOLE_0:3;
then (f2|n).d <> 0 by TARSKI:def 1;
hence thesis by A81,A84,FINSEQ_1:def 7;
end;
suppose
ex l being Nat st l in dom f3 & d=len(f2|n)+ l;
then consider l be Element of NAT such that
A85: l in dom f3 and
A86: d = len(f2|n)+ l;
f4.d = f3.l by A85,A86,FINSEQ_1:def 7;
hence thesis by A68,A85;
end;
end;
A87: f2 = (f2|n)^(f2/^n) by RFINSEQ:8;
then
A88: f2/^n is one-to-one by A50,FINSEQ_3:91;
for d,e being Nat st 1<=d & d<e & e<=len f3 holds f3.d <> f3.e
proof
let d,e be Nat;
assume that
A89: 1<=d and
A90: d<e and
A91: e<=len f3;
1<=e by A89,A90,XXREAL_0:2;
then
A92: e in dom f3 by A91,FINSEQ_3:25;
then
A93: f3.e = p - (f2/^n).e by A64;
d<len f3 by A90,A91,XXREAL_0:2;
then
A94: d in dom f3 by A89,FINSEQ_3:25;
then f3.d = p - (f2/^n).d by A64;
hence thesis by A88,A57,A90,A94,A92,A93;
end;
then len f3 = card rng f3 by GRAPH_5:7;
then
A95: f3 is one-to-one by FINSEQ_4:62;
A96: len f2 = p9 by A13,A5,A14,A49,A42,FINSEQ_3:39,XBOOLE_1:73;
then
A97: n <= len f2 by XREAL_1:43;
A98: rng(f2|n) misses rng f3
proof
assume rng(f2|n) meets rng f3;
then consider x be object such that
A99: x in rng(f2|n) and
A100: x in rng f3 by XBOOLE_0:3;
consider e being Nat such that
A101: e in dom f3 and
A102: f3.e = x by A100,FINSEQ_2:10;
x = p - (f2/^n).e by A64,A101,A102;
then
A103: x = p - f2.(e+n) by A97,A57,A101,RFINSEQ:def 1;
e+n in dom f2 by A57,A101,FINSEQ_5:26;
then consider e1 be Nat such that
A104: e1 in dom f1 and
A105: f1.e1 = f2.(e+n) by A54,FINSEQ_2:10,FUNCT_1:3;
A106: e1 in dom f by A5,A104,FINSEQ_3:29;
A107: e1 <= p9 by A13,A5,A104,FINSEQ_3:25;
rng(f2|n) c= rng f2 by FINSEQ_5:19;
then consider d1 be Nat such that
A108: d1 in dom f1 and
A109: f1.d1 = x by A54,A99,FINSEQ_2:10;
d1 <= p9 by A13,A5,A108,FINSEQ_3:25;
then d1+e1 <= p9+p9 by A107,XREAL_1:7;
then
A110: d1+e1 < p by A11,A18,XREAL_1:146,XXREAL_0:2;
x = f.d1 mod p by A44,A108,A109,EULER_2:def 1;
then (f.d1 mod p) + f2.(e+n) = p by A103;
then (f.d1 mod p) + (f.e1 mod p) = p by A105,A106,EULER_2:def 1;
then ((f.d1 mod p)+(f.e1 mod p)) mod p = 0 by NAT_D:25;
then (f.d1 + f.e1) mod p = 0 by EULER_2:6;
then p divides (f.d1 + f.e1) by PEPIN:6;
then p divides (d1*a + f.e1) by A25,A44,A108;
then p divides (d1*a + e1*a) by A25,A106;
then
A111: p divides (d1+e1)*a;
d1 >= 1 by A108,FINSEQ_3:25;
hence contradiction by A2,A111,A110,NAT_D:7,WSIERP_1:30;
end;
f2|n is one-to-one by A50,A87,FINSEQ_3:91;
then
A112: f4 is one-to-one by A95,A98,FINSEQ_3:91;
A113: for d st d in dom f3 holds f3.d,-(f2/^n).d are_congruent_mod p
proof
let d;
assume d in dom f3;
then f3.d mod p = (p - (f2/^n).d) mod p by A64
.= (1*p + (-(f2/^n).d)) mod p
.= (-(f2/^n).d) mod p by EULER_1:12;
hence thesis by NAT_D:64;
end;
A114: len(f2/^n) = (len f2 -' n) by RFINSEQ:29
.= len f2 - n by A96,XREAL_1:43,233
.= m by A96;
len(f2|n) = n by A96,FINSEQ_1:59,XREAL_1:43;
then len f4 = n + m by A58,A114,FINSEQ_1:22
.= len f by A13;
then rng f4 = rng I by A13,A112,A79,Th40;
then Product f4 = Product I by A112,EULER_2:10,RFINSEQ:26;
then
A115: (Product(f2|n)) * (Product f3) = Product I by RVSUM_1:97;
f3 is FinSequence of INT by FINSEQ_2:24,NUMBERS:17;
then (Product f3)*(Product(f2|n)),(-1)|^m * Product(f2/^n)*(Product(f2|n)
) are_congruent_mod p by A58,A114,A56,A113,Th33,INT_4:11;
then (Product f3)*(Product(f2|n)),(-1)|^m * ((Product(f2|n))*Product(f2/^
n ) ) are_congruent_mod p;
then Product I,(-1)|^m * Product((f2|n)^(f2/^n)) are_congruent_mod p by
A115,RVSUM_1:97;
then
A116: Product I,(-1)|^m * Product f1 are_congruent_mod p by A55,RFINSEQ:8;
(-1)|^m * Product f1,(-1)|^m * Product f are_congruent_mod p by A9,
INT_4:11;
then Product I,(-1)|^m * (a|^p9) * (Product I) are_congruent_mod p by A15
,A116,INT_1:15;
then p divides (1-((-1)|^m * (a|^p9)))* Product I;
then p divides (1-((-1)|^m * (a|^p9))) by A24,WSIERP_1:29;
then p divides (-1)|^m * (1-((-1)|^m * (a|^p9))) by INT_2:2;
then
A117: p divides ((-1)|^m - (-1)|^m * (-1)|^m * (a|^p9));
(-1)|^m * (-1)|^m = (-1)|^(m+m) by NEWTON:8
.= (-1)|^(2*m)
.= ((-1)|^2)|^m by NEWTON:9
.= (1|^2)|^m by WSIERP_1:1
.= 1;
then (-1)|^m,a|^p9 are_congruent_mod p by A117;
then
A118: (-1)|^m,Lege(a,p) are_congruent_mod p by A8,INT_1:15;
per cases by A77;
suppose
A119: (-1)|^m = 1;
then
A120: Lege(a,p) <> -1 by A118,A1,NAT_D:7;
now
assume Lege(a,p) = 0;
then p divides (1-0) by A118,A119;
then p = 1 by WSIERP_1:15;
hence contradiction by A1;
end;
hence thesis by A119,Th25,A120;
end;
suppose
A121: (-1)|^m = -1;
A122: now
assume Lege(a,p) = 1;
then p divides (-2) by A118,A121;
then p divides 2 by INT_2:10;
hence contradiction by A1,NAT_D:7;
end;
now
assume Lege(a,p) = 0;
then p divides (-1-0) by A118,A121;
then p = 1 by WSIERP_1:15,INT_2:10;
hence contradiction by A1;
end;
hence thesis by A121,Th25,A122;
end;
end;
theorem Th42:
p>2 implies Lege(2,p) = (-1)|^((p^2 -'1) div 8)
proof
set p9 = (p-'1) div 2;
set I = idseq p9;
set fp = 2 * I;
set nn = p div 8;
A1: p>1 by INT_2:def 4;
then
A2: p-1 = p-'1 by XREAL_1:233;
A3: for d st d in dom fp holds fp.d = 2*d
proof
let d;
assume
A4: d in dom fp;
then d in dom I by VALUED_1:def 5;
then d in Seg len I by FINSEQ_1:def 3;
then
A5: d is Element of Seg p9 by CARD_1:def 7;
thus fp.d = 2 * I.d by A4,VALUED_1:def 5
.= 2 * d by A5;
end;
for d being Nat st d in dom fp holds fp.d in NAT;
then reconsider fp as FinSequence of NAT by FINSEQ_2:12;
set f = fp mod p;
set X = {k where k is Element of NAT:k in rng f & k > p/2};
set m = card X;
dom fp = dom I by VALUED_1:def 5;
then
A6: len fp = len I by FINSEQ_3:29
.= p9 by CARD_1:def 7;
set Y = {d where d is Element of NAT:d in dom f & f.d > p/2};
for x be object st x in Y holds x in dom f
proof
let x be object;
assume x in Y;
then ex k be Element of NAT st x = k & k in dom f & f.k > p/ 2;
hence thesis;
end;
then Y c= dom f;
then reconsider Y as finite Subset of NAT by XBOOLE_1:1;
set Z = seq((p div 4),p9-'(p div 4));
A7: p mod 8 <= 8-1 by INT_1:52,NAT_D:1;
8 = 2*4;
then
A8: 2 divides 8;
A9: now
assume p mod 8 = 0;
then 8 divides p by PEPIN:6;
then p = 8 by INT_2:def 4;
hence contradiction by A8,NAT_4:12;
end;
for x being object st x in X holds x in rng f
proof
let x be object;
assume x in X;
then ex k being Element of NAT st x = k & k in rng f & k > p /2;
hence thesis;
end;
then
A10: X c= rng f;
then reconsider X as finite set;
card X is Element of NAT;
then reconsider m as Element of NAT;
A11: len f = len fp by EULER_2:def 1;
then
A12: dom f = dom fp by FINSEQ_3:29;
assume
A13: p>2;
then 2,p are_coprime by EULER_1:2;
then
A14: 2 gcd p = 1 by INT_2:def 3;
then
A15: Lege(2,p) = (-1)|^m by A13,Th41;
p is odd by A13,PEPIN:17;
then
A16: p-1 is even by HILBERT3:2;
then
A17: p9 = ((p-'1)+1) div 2 by A2,NAT_2:26
.= p div 2 by A1,XREAL_1:235;
then
A18: f <> {} by A13,A6,A11,NAT_2:13;
then reconsider U=dom f as non empty finite Subset of NAT;
2 divides (p-'1) by A16,A2,PEPIN:22;
then
A19: p-'1 = 2*p9 by NAT_D:3;
A20: for d st d in dom f holds f.d = 2*d
proof
let d;
assume
A21: d in dom f;
then d<= p9 by A6,A11,FINSEQ_3:25;
then 2*d<=(p-'1) by A19,XREAL_1:64;
then 2*d<p by NAT_2:9,XXREAL_0:2;
hence 2*d = 2*d mod p by NAT_D:24
.= fp.d mod p by A3,A12,A21
.= f.d by A12,A21,EULER_2:def 1;
end;
A22: for d1,d2,k1,k2 be Nat st 1 <= d1 & d1 < d2 & d2 <= len f & k1=f.d1 &
k2=f.d2 holds k1 < k2
proof
let d1,d2,k1,k2 be Nat;
assume that
A23: 1 <= d1 and
A24: d1 < d2 and
A25: d2 <= len f and
A26: k1=f.d1 and
A27: k2=f.d2;
1 <= d2 by A23,A24,XXREAL_0:2;
then d2 in dom f by A25,FINSEQ_3:25;
then
A28: k2 = 2*d2 by A20,A27;
d1 <= len f by A24,A25,XXREAL_0:2;
then d1 in dom f by A23,FINSEQ_3:25;
then k1 = 2*d1 by A20,A26;
hence thesis by A24,A28,XREAL_1:68;
end;
rng f is non empty Subset of NAT by A18;
then consider n1 be Element of NAT such that
A29: rng f c= Seg n1 \/ {0} by HEYTING3:1;
reconsider X as finite Subset of NAT by A10,XBOOLE_1:1;
Z, ((p-'1) div 2)-'(p div 4) are_equipotent by CALCUL_2:6; then
A30: card Z = ((p-'1) div 2)-'(p div 4) by CARD_1:def 2;
not 0 in rng f
proof
assume 0 in rng f;
then consider n be Nat such that
A31: n in dom f and
A32: f.n = 0 by FINSEQ_2:10;
2*n =0 by A20,A31,A32;
hence contradiction by A31,FINSEQ_3:25;
end;
then
A33: {0} misses rng f by ZFMISC_1:50;
then rng f c= Seg n1 by A29,XBOOLE_1:73;
then
A34: Sgm rng f = f by A22,FINSEQ_1:def 13;
A35: X,Y are_equipotent
proof
deffunc F(Element of U) = f.$1;
set YY = {d where d is Element of U:F(d) in X};
A36: now
let x be set;
assume x in X;
then consider y be Element of NAT such that
A37: y=x and
A38: y in rng f and
y > p/2;
consider d being Nat such that
A39: d in U and
A40: f.d = y by A38,FINSEQ_2:10;
reconsider d as Element of U by A39;
take d;
thus x=F(d) by A37,A40;
end;
A41: Y c= YY
proof
let x be object;
assume x in Y;
then
A42: ex d be Element of NAT st d = x & d in dom f & f.d > p/ 2;
then reconsider x as Element of U;
reconsider f as FinSequence of NAT qua set;
f.x in rng f by FUNCT_1:3;
then F(x) in X by A42;
hence thesis;
end;
now
let x be object;
assume x in YY;
then consider d be Element of U such that
A43: d = x and
A44: f.d in X;
ex k be Element of NAT st k = f.d & k in rng f & k > p/ 2 by A44;
hence x in Y by A43;
end;
then
A45: YY c= Y;
A46: for d1,d2 being Element of U st F(d1) = F(d2) holds d1 = d2
proof
let d1,d2 be Element of U;
assume
A47: F(d1)=F(d2);
f is one-to-one by A29,A33,A34,FINSEQ_3:92,XBOOLE_1:73;
hence thesis by A47;
end;
X,YY are_equipotent from FUNCT_7:sch 3(A36,A46);
hence thesis by A41,A45,XBOOLE_0:def 10;
end;
p div 2 < p by INT_1:56;
then (p div 2) div 2 <= p div 2 by NAT_2:24;
then
A48: p div (2*2) <= p div 2 by NAT_2:27;
A49: Z c= Y
proof
let x be object;
assume
A50: x in Z;
then reconsider x as Element of NAT;
A51: x>=(p div 4)+1 by A50,CALCUL_2:1;
then (p div 4)+x >= (p div 4)+1 by NAT_1:12;
then
A52: x >= 1 by XREAL_1:6;
x<=((p-'1) div 2)-'(p div 4)+(p div 4) by A50,CALCUL_2:1;
then x<=((p-'1) div 2) by A17,A48,XREAL_1:235;
then
A53: x in dom f by A6,A11,A52,FINSEQ_3:25;
x > p/4 by A51,INT_1:29,XXREAL_0:2;
then 2*x > 2*(p/4) by XREAL_1:68;
then f.x > p/2 by A20,A53;
hence thesis by A53;
end;
now
let x be object;
A54: p/4 >= [\p/4/] by INT_1:def 6;
assume x in Y;
then consider x1 be Element of NAT such that
A55: x1=x and
A56: x1 in dom f and
A57: f.x1 >p/2;
2*x1>p/2 by A20,A56,A57;
then x1 > (p/2)/2 by XREAL_1:83;
then x1 > [\p/4/] by A54,XXREAL_0:2;
then
A58: x1 >= (p div 4) + 1 by NAT_1:13;
x1 <= p9 by A6,A11,A56,FINSEQ_3:25;
then x1 <= (p9-'(p div 4) + (p div 4)) by A17,A48,XREAL_1:235;
hence x in Z by A55,A58;
end;
then Y c= Z;
then Y = Z by A49,XBOOLE_0:def 10;
then
A59: m = ((p-'1) div 2)-'(p div 4) by A30,A35,CARD_1:5;
A60: now
assume p mod 8 = 2;
then 8 divides (p - 2) by PEPIN:8;
then 2 divides (p - 2) by A8,INT_2:9;
then 2 divides -(p-2) by INT_2:10;
then 2 divides (2 - p);
then 2 divides p by Th2;
hence contradiction by A13,NAT_4:12;
end;
A61: now
assume p mod 8 = 4;
then 8 divides (p - 4) by PEPIN:8;
then 2 divides (p - 4) by A8,INT_2:9;
then 2 divides -(p - 4) by INT_2:10;
then
A62: 2 divides (4 - p);
4 = 2*2;
then 2 divides 4;
then 2 divides p by A62,Th2;
hence contradiction by A13,NAT_4:12;
end;
A63: now
assume p mod 8 = 6;
then 8 divides (p - 6) by PEPIN:8;
then 2 divides (p - 6) by A8,INT_2:9;
then 2 divides -(p - 6) by INT_2:10;
then
A64: 2 divides (6-p);
6=2*3;
then 2 divides 6;
then 2 divides p by A64,Th2;
hence contradiction by A13,NAT_4:12;
end;
p mod 8 = 0 or ... or p mod 8 = 7 by A7;
then per cases by A9,A60,A61,A63;
suppose
p mod 8 = 1;
then
A65: p=8*nn+1 by NAT_D:2;
then p-'1 = 2*(4*nn) by A2;
then
A66: (p-'1) div 2 = 4*nn by NAT_D:18;
p div 4 = (4*(2*nn)+1) div 4 by A65
.= 2*nn+(1 div 4) by NAT_D:61
.= 2*nn+0 by NAT_D:27;
then m = 4*nn - 2*nn by A59,A66,XREAL_1:64,233
.= 2*nn;
then
A67: Lege(2,p) =((-1)|^2)|^nn by A15,NEWTON:9
.= (1|^2)|^nn by WSIERP_1:1
.= 1;
(p^2 -'1) div 8 = (((8*nn)^2 + 2*(8*nn)) + 1-'1) div 8 by A65
.= 8*(8*nn^2 + 2*nn) div 8 by NAT_D:34
.= 8*nn^2 + 2*nn by NAT_D:18;
hence (-1)|^((p^2 -'1) div 8) = (-1)|^(2*(4*nn^2 + nn))
.= ((-1)|^2)|^(4*nn^2 + nn) by NEWTON:9
.= (1|^2)|^(4*nn^2 + nn) by WSIERP_1:1
.= Lege(2,p) by A67;
end;
suppose
p mod 8 = 3;
then
A68: p=8*nn+3 by NAT_D:2;
then p-'1 = 2*(4*nn+1) by A2;
then
A69: (p-'1) div 2 = 4*nn+1 by NAT_D:18;
A70: 4*nn>=2*nn by XREAL_1:64;
p div 4 = (4*(2*nn)+3) div 4 by A68
.= 2*nn+(3 div 4) by NAT_D:61
.= 2*nn+0 by NAT_D:27;
then m = 4*nn+1-2*nn by A59,A69,A70,NAT_1:12,XREAL_1:233
.= 2*nn+1;
then
A71: Lege(2,p) = (-1)|^(2*nn)*(-1) by A15,NEWTON:6
.= ((-1)|^2)|^nn*(-1) by NEWTON:9
.= (1|^2)|^nn*(-1) by WSIERP_1:1
.= -1;
(p^2 -'1) div 8 = (8*(8*nn^2)+8*(6*nn)+3*3-1) div 8 by A68,NAT_1:12
,XREAL_1:233
.= 8*(8*nn^2+6*nn+1) div 8
.= 8*nn^2+6*nn+1 by NAT_D:18;
hence (-1)|^((p^2 -'1) div 8) = (-1)|^(2*(4*nn^2+3*nn))*(-1) by NEWTON:6
.= ((-1)|^2)|^(4*nn^2+3*nn)*(-1) by NEWTON:9
.= (1|^2)|^(4*nn^2+3*nn)*(-1) by WSIERP_1:1
.= Lege(2,p) by A71;
end;
suppose
p mod 8 = 5;
then
A72: p=8*nn+5 by NAT_D:2;
then p-'1 = 2*(4*nn+2) by A2;
then
A73: (p-'1) div 2 = 4*nn+2 by NAT_D:18;
A74: 4*nn>=2*nn by XREAL_1:64;
p div 4 = (4*(2*nn+1)+1) div 4 by A72
.= 2*nn+1+(1 div 4) by NAT_D:61
.= 2*nn+1+0 by NAT_D:27;
then m = 4*nn+2-(2*nn+1) by A59,A73,A74,XREAL_1:7,233
.= 2*nn+1;
then
A75: Lege(2,p) = (-1)|^(2*nn)*(-1) by A15,NEWTON:6
.= ((-1)|^2)|^nn*(-1) by NEWTON:9
.= (1|^2)|^nn*(-1) by WSIERP_1:1
.= -1;
(p^2 -'1) div 8=(8*(8*nn^2)+8*(10*nn)+25-1) div 8 by A72,NAT_1:12
,XREAL_1:233
.= 8*(8*nn^2+10*nn+3) div 8
.= 8*nn^2+10*nn+3 by NAT_D:18;
hence (-1)|^((p^2 -'1) div 8) = (-1)|^(2*(4*nn^2)+2*(5*nn)+2*1+1)
.= (-1)|^(2*(4*nn^2+5*nn+1))*(-1) by NEWTON:6
.= ((-1)|^2)|^(4*nn^2+5*nn+1)*(-1) by NEWTON:9
.= (1|^2)|^(4*nn^2+5*nn+1)*(-1) by WSIERP_1:1
.= Lege(2,p) by A75;
end;
suppose
p mod 8 = 7;
then
A76: p=8*nn+7 by NAT_D:2;
then p-'1 = 2*(4*nn+3) by A2;
then
A77: (p-'1) div 2 = 4*nn+3 by NAT_D:18;
A78: 4*nn>=2*nn by XREAL_1:64;
p div 4 = (4*(2*nn+1)+3) div 4 by A76
.= 2*nn+1+(3 div 4) by NAT_D:61
.= 2*nn+1+0 by NAT_D:27;
then m = 4*nn+3-(2*nn+1) by A59,A77,A78,XREAL_1:7,233
.= 2*nn+2;
then
A79: Lege(2,p) = (-1)|^(2*(nn+1)) by A13,A14,Th41
.= ((-1)|^2)|^(nn+1) by NEWTON:9
.= (1|^2)|^(nn+1) by WSIERP_1:1
.= 1;
(p^2 -'1) div 8=(8*(8*nn^2)+8*(14*nn)+49-1) div 8 by A76,NAT_1:12
,XREAL_1:233
.= 8*(8*nn^2+14*nn+6) div 8
.= 8*nn^2+14*nn+6 by NAT_D:18;
hence (-1)|^((p^2 -'1) div 8) = (-1)|^(2*(4*nn^2+7*nn+3))
.= ((-1)|^2)|^(4*nn^2+7*nn+3) by NEWTON:9
.= (1|^2)|^(4*nn^2+7*nn+3) by WSIERP_1:1
.= Lege(2,p) by A79;
end;
end;
theorem
p>2 & (p mod 8 = 1 or p mod 8 = 7) implies 2 is_quadratic_residue_mod p
proof
assume that
A1: p>2 and
A2: p mod 8 = 1 or p mod 8 = 7;
set nn = p div 8;
per cases by A2;
suppose
p mod 8 = 1;
then p = 8*nn+1 by NAT_D:2;
then (p^2 -'1) div 8 = (((8*nn)^2 + 2*(8*nn)) + 1-'1) div 8
.= 8*(8*nn^2 + 2*nn) div 8 by NAT_D:34
.= 2*(4*nn^2 + nn) by NAT_D:18;
then Lege(2,p) = (-1)|^(2*(4*nn^2 + nn)) by A1,Th42
.= ((-1)|^2)|^(4*nn^2 + nn) by NEWTON:9
.= (1|^2)|^(4*nn^2 + nn) by WSIERP_1:1
.= 1;
hence thesis by Def3;
end;
suppose
p mod 8 = 7;
then p = 8*nn+7 by NAT_D:2;
then (p^2 -'1) div 8=(8*(8*nn^2)+8*(14*nn)+49-1) div 8 by NAT_1:12
,XREAL_1:233
.= 8*(8*nn^2+14*nn+6) div 8
.= 2*(4*nn^2+7*nn+3) by NAT_D:18;
then Lege(2,p) = (-1)|^(2*(4*nn^2+7*nn+3)) by A1,Th42
.= ((-1)|^2)|^(4*nn^2+7*nn+3) by NEWTON:9
.= (1|^2)|^(4*nn^2+7*nn+3) by WSIERP_1:1
.= 1;
hence thesis by Def3;
end;
end;
theorem
p>2 & (p mod 8 = 3 or p mod 8 = 5) implies not 2 is_quadratic_residue_mod p
proof
assume that
A1: p>2 and
A2: p mod 8 = 3 or p mod 8 = 5;
set nn = p div 8;
per cases by A2;
suppose
p mod 8 = 3;
then p = 8*nn+3 by NAT_D:2;
then (p^2 -'1) div 8 = (8*(8*nn^2)+8*(6*nn)+3*3-1) div 8 by NAT_1:12
,XREAL_1:233
.= 8*(8*nn^2+6*nn+1) div 8
.= 8*nn^2+6*nn+1 by NAT_D:18;
then Lege(2,p) = (-1)|^(8*nn^2+6*nn+1) by A1,Th42
.= (-1)|^(2*(4*nn^2+3*nn))*(-1) by NEWTON:6
.= ((-1)|^2)|^(4*nn^2+3*nn)*(-1) by NEWTON:9
.= (1|^2)|^(4*nn^2+3*nn)*(-1) by WSIERP_1:1
.= -1; then
not (2 is_quadratic_residue_mod p & 2 mod p <> 0) &
not (2 is_quadratic_residue_mod p & 2 mod p = 0) by Def3;
hence thesis;
end;
suppose
p mod 8 = 5;
then p = 8*nn+5 by NAT_D:2;
then (p^2 -'1) div 8=(8*(8*nn^2)+8*(10*nn)+25-1) div 8 by NAT_1:12
,XREAL_1:233
.= 8*(8*nn^2+10*nn+3) div 8
.= 8*nn^2+10*nn+3 by NAT_D:18;
then Lege(2,p) = (-1)|^(2*(4*nn^2)+2*(5*nn)+2*1+1) by A1,Th42
.= (-1)|^(2*(4*nn^2+5*nn+1))*(-1) by NEWTON:6
.= ((-1)|^2)|^(4*nn^2+5*nn+1)*(-1) by NEWTON:9
.= (1|^2)|^(4*nn^2+5*nn+1)*(-1) by WSIERP_1:1
.= -1; then
not (2 is_quadratic_residue_mod p & 2 mod p <> 0) &
not (2 is_quadratic_residue_mod p & 2 mod p = 0) by Def3;
hence thesis;
end;
end;
theorem Th45:
for a,b be Nat st a mod 2 = b mod 2 holds (-1)|^a = (-1)|^b
proof
let a,b be Nat;
assume a mod 2 = b mod 2;
then a,b are_congruent_mod 2 by NAT_D:64;
then
A1: 2 divides (a-b);
per cases;
suppose
a>=b;
then reconsider l=a-b as Element of NAT by NAT_1:21;
consider n be Nat such that
A2: l=2*n by A1,NAT_D:def 3;
(-1)|^a = (-1)|^(b + (2*n)) by A2
.= ((-1)|^b) * ((-1)|^(2*n)) by NEWTON:8
.= (-1)|^b * ((-1)|^2)|^n by NEWTON:9
.= (-1)|^b * (1|^2)|^n by WSIERP_1:1
.= (-1)|^b * 1;
hence thesis;
end;
suppose a<b;
then reconsider l=b-a as Element of NAT by NAT_1:21;
2 divides -(a-b) by A1,INT_2:10;
then consider n be Nat such that
A3: l=2*n by NAT_D:def 3;
(-1)|^b = (-1)|^(a+2*n) by A3
.= (-1)|^a * (-1)|^(2*n) by NEWTON:8
.= (-1)|^a * ((-1)|^2)|^n by NEWTON:9
.= (-1)|^a * (1|^2)|^n by WSIERP_1:1
.= (-1)|^a * 1;
hence thesis;
end;
end;
reserve f,g,h,k for FinSequence of REAL;
theorem Th46:
len f = len h & len g = len k implies f^g-h^k = (f-h)^(g-k)
proof
assume that
A1: len f = len h and
A2: len g = len k;
A3: len(f-h) = len f by A1,TOPREAL7:7;
len(f^g) = len h + len k by A1,A2,FINSEQ_1:22;
then len(f^g) = len(h^k) by FINSEQ_1:22;
then
A4: len(f^g-h^k) = len(f^g) by TOPREAL7:7;
A5: len(g-k) = len g by A2,TOPREAL7:7;
then len((f-h)^(g-k)) = len f + len g by A3,FINSEQ_1:22;
then len(f^g-h^k) = len((f-h)^(g-k)) by A4,FINSEQ_1:22;
then
A6: dom(f^g-h^k) = dom((f-h)^(g-k)) by FINSEQ_3:29;
for d being Nat st d in dom((f-h)^(g-k)) holds ((f-h)^(g-k)).d = (f^g-h ^k).d
proof
let d be Nat;
assume
A7: d in dom((f-h)^(g-k));
per cases by A7,FINSEQ_1:25;
suppose
A8: d in dom(f-h);
then
A9: ((f-h)^(g-k)).d = (f-h).d by FINSEQ_1:def 7
.= f.d - h.d by A8,VALUED_1:13;
A10: dom f = dom(f-h) by A1,TOPREAL7:7;
A11: dom h = dom(f-h) by A1,A3,FINSEQ_3:29;
(f^g-h^k).d = (f^g).d - (h^k).d by A6,A8,FINSEQ_2:15,VALUED_1:13
.= f.d - (h^k).d by A8,A10,FINSEQ_1:def 7
.= f.d - h.d by A8,A11,FINSEQ_1:def 7;
hence thesis by A9;
end;
suppose
ex e being Nat st e in dom(g-k) & d = len(f-h) + e;
then consider e such that
A12: e in dom(g-k) and
A13: d = len(f-h) + e;
e in dom g by A2,A12,TOPREAL7:7;
then
A14: (f^g).d = g.e by A3,A13,FINSEQ_1:def 7;
e in dom k by A2,A5,A12,FINSEQ_3:29;
then
A15: (h^k).d = k.e by A1,A3,A13,FINSEQ_1:def 7;
((f-h)^(g-k)).d = (g-k).e by A12,A13,FINSEQ_1:def 7
.=g.e - k.e by A12,VALUED_1:13;
hence thesis by A6,A12,A13,A14,A15,FINSEQ_1:28,VALUED_1:13;
end;
end;
hence thesis by A6;
end;
theorem Th47:
for f be FinSequence of REAL,m be Real holds Sum(((len f) |-> m)
- f) = (len f)*m - Sum f
proof
defpred P[Nat] means for f be FinSequence of REAL,m be Real st len f = $1
holds Sum(($1|-> m) - f) = $1 * m - Sum f;
A1: for n be Nat st P[n] holds P[n+1]
proof
let n be Nat;
assume
A2: P[n];
P[n+1]
proof
let f be FinSequence of REAL,m be Real;
A3: len<*m*> = 1 by FINSEQ_1:39;
assume
A4: len f = n+1;
then f <> {};
then consider f1 be FinSequence of REAL,x be Element of REAL such that
A5: f = f1^<*x*> by HILBERT2:4;
reconsider mm=m as Element of REAL by XREAL_0:def 1;
A6: n + 1 = len f1 + 1 by A4,A5,FINSEQ_2:16;
then
A7: len(n|-> m)=len f1 by CARD_1:def 7;
A8: len<*x*> = 1 by FINSEQ_1:39;
((n+1)|-> m)-f = (n|-> m)^<*m*> - f1^<*x*> by A5,FINSEQ_2:60
.= ((n|-> mm)-f1) ^ (<*mm*>-<*x*>) by A7,A8,A3,Th46
.= ((n|-> m)-f1) ^ <*m-x*> by RVSUM_1:29;
hence Sum(((n+1)|-> m)-f) = Sum((n|-> m)-f1) + (m-x) by RVSUM_1:74
.= n*m - Sum f1 + (m - x) by A2,A6
.= (n+1)*m - (Sum f1 + x)
.= (n+1)*m - Sum f by A5,RVSUM_1:74;
end;
hence thesis;
end;
A9: P[0]
proof
let f be FinSequence of REAL,m be Real;
assume len f = 0;
then Sum f = 0 by PROB_3:62;
hence thesis by RVSUM_1:28,72;
end;
for n be Nat holds P[n] from NAT_1:sch 2(A9,A1);
hence thesis;
end;
reserve X for finite set,
F for FinSequence of bool X;
definition
let X, F;
redefine func Card F -> Cardinal-yielding FinSequence of NAT;
coherence
proof
rng Card F c= NAT
proof
let y be object;
assume y in rng Card F;
then consider x being object such that
A1: x in dom Card F and
A2: y = (Card F).x by FUNCT_1:def 3;
A3: x in dom F by A1,CARD_3:def 2;
then F.x in rng F by FUNCT_1:3;
then reconsider Fx = F.x as finite set;
y = card Fx by A2,A3,CARD_3:def 2;
hence thesis;
end;
hence thesis by FINSEQ_1:def 4;
end;
end;
theorem Th48:
for f be FinSequence of bool X st (for d,e st d in
dom f & e in dom f & d<>e holds f.d misses f.e) holds card union rng f = Sum
Card f
proof
defpred P[Nat] means for f be FinSequence of bool X st len f = $1 & (for d,e
st d in dom f & e in dom f & d<>e holds f.d misses f.e) holds card union rng f
= Sum Card f;
A1: for n be Nat st P[n] holds P[n+1]
proof
let n be Nat;
assume
A2: P[n];
P[n+1]
proof
let f be FinSequence of bool X;
assume that
A3: len f = n+1 and
A4: for d,e st d in dom f & e in dom f & d<>e holds f.d misses f.e;
A5: f <> {} by A3;
then consider
f1 be FinSequence of bool X,Y be Element of bool X such that
A6: f = f1^<*Y*> by HILBERT2:4;
reconsider F1 = union(rng f1) as finite set;
A7: union(rng f) = union((rng f1) \/ (rng <*Y*>)) by A6,FINSEQ_1:31
.= union((rng f1) \/ {Y}) by FINSEQ_1:38
.= F1 \/ union {Y} by ZFMISC_1:78
.= F1 \/ Y by ZFMISC_1:25;
A8: n+1 = len f1 +1 by A3,A6,FINSEQ_2:16;
F1 misses Y
proof
A9: n+1 in dom f by A3,A5,FINSEQ_5:6;
assume F1 meets Y;
then consider x be object such that
A10: x in F1 /\ Y by XBOOLE_0:4;
x in F1 by A10,XBOOLE_0:def 4;
then consider Z be set such that
A11: x in Z and
A12: Z in rng f1 by TARSKI:def 4;
consider k be Nat such that
A13: k in dom f1 and
A14: f1.k = Z by A12,FINSEQ_2:10;
k <= n by A8,A13,FINSEQ_3:25;
then
A15: k < n + 1 by NAT_1:13;
k in dom f by A6,A13,FINSEQ_2:15;
then f.(n+1) misses f.k by A4,A15,A9;
then Y misses f.k by A6,A8,FINSEQ_1:42;
then
A16: Y misses Z by A6,A13,A14,FINSEQ_1:def 7;
x in Y \/ Z by A11,XBOOLE_0:def 3;
then not x in Y by A11,A16,XBOOLE_0:5;
hence contradiction by A10,XBOOLE_0:def 4;
end;
then
A17: card F1 + card Y = card(F1\/Y) by CARD_2:40;
reconsider gg = <*card Y*> as FinSequence of NAT;
A18: Card f = Card f1 ^ Card<*Y*> by A6,PRE_POLY:25
.= Card f1 ^ gg by PRE_POLY:24;
for d,e st d in dom f1 & e in dom f1 & d<>e holds f1.d misses f1.e
proof
let d,e;
assume that
A19: d in dom f1 and
A20: e in dom f1 and
A21: d<>e;
A22: f.e = f1.e by A6,A20,FINSEQ_1:def 7;
A23: e in dom f by A6,A20,FINSEQ_2:15;
A24: d in dom f by A6,A19,FINSEQ_2:15;
f.d = f1.d by A6,A19,FINSEQ_1:def 7;
hence thesis by A4,A21,A22,A24,A23;
end;
then card union rng f1 = Sum Card f1 by A2,A8;
hence thesis by A17,A18,A7,RVSUM_1:74;
end;
hence thesis;
end;
A25: P[0]
proof
let f be FinSequence of bool X;
assume that
A26: len f = 0 and
for d,e st d in dom f & e in dom f & d<>e holds f.d misses f.e;
A27: Card {} = {};
f = {} by A26;
hence thesis by A27,CARD_1:27,RVSUM_1:72,ZFMISC_1:2;
end;
let f be FinSequence of bool X;
for n be Nat holds P[n] from NAT_1:sch 2(A25,A1); then
P[len f];
hence thesis;
end;
Lm4: Sum(fp) is Element of NAT;
reserve q for Prime;
::$N The law of quadratic reciprocity
theorem Th49:
p>2 & q>2 & p<>q implies
Lege(p,q)*Lege(q,p)=(-1)|^(((p-'1) div 2)*((q-'1) div 2))
proof
assume that
A1: p>2 and
A2: q>2 and
A3: p<>q;
A4: q,p are_coprime by A3,INT_2:30;
then
A5: q gcd p = 1 by INT_2:def 3;
reconsider p,q as prime Element of NAT by ORDINAL1:def 12;
set p9 = (p-'1) div 2;
A6: p>1 by INT_2:def 4;
then
A7: p-'1 = p-1 by XREAL_1:233;
then
A8: p-'1>0 by A6,XREAL_1:50;
p is odd by A1,PEPIN:17;
then
A9: p-'1 is even by A7,HILBERT3:2;
then
A10: 2 divides (p-'1) by PEPIN:22;
then
A11: p-'1 = 2*p9 by NAT_D:3;
then p9 divides (p-'1);
then p9 <= (p-'1) by A8,NAT_D:7;
then
A12: p9 < p by A7,XREAL_1:146,XXREAL_0:2;
set f1 = q*idseq p9;
A13: for d st d in dom f1 holds f1.d = q*d
proof
let d;
assume
A14: d in dom f1;
then d in dom idseq p9 by VALUED_1:def 5;
then d in Seg len idseq p9 by FINSEQ_1:def 3;
then
A15: d is Element of Seg p9 by CARD_1:def 7;
f1.d = q * (idseq p9).d by A14,VALUED_1:def 5;
hence thesis by A15;
end;
A16: for d being Nat st d in dom f1 holds f1.d in NAT;
dom f1 = dom idseq p9 by VALUED_1:def 5;
then
A17: len f1 = len idseq p9 by FINSEQ_3:29;
then
A18: len f1 = p9 by CARD_1:def 7;
set q9 = (q-'1) div 2;
set g1 = p*idseq q9;
A19: for d st d in dom g1 holds g1.d = p*d
proof
let d;
assume
A20: d in dom g1;
then d in dom idseq q9 by VALUED_1:def 5;
then d in Seg len idseq q9 by FINSEQ_1:def 3;
then
A21: d is Element of Seg q9 by CARD_1:def 7;
g1.d = p * (idseq q9).d by A20,VALUED_1:def 5;
hence thesis by A21;
end;
A22: for d being Nat st d in dom g1 holds g1.d in NAT;
dom g1 = dom idseq q9 by VALUED_1:def 5;
then len g1 = len idseq q9 by FINSEQ_3:29;
then
A23: len g1 = q9 by CARD_1:def 7;
reconsider g1 as FinSequence of NAT by A22,FINSEQ_2:12;
set g3 = g1 mod q;
set g4 = Sgm rng g3;
A24: len g3 = len g1 by EULER_2:def 1;
then
A25: dom g1 = dom g3 by FINSEQ_3:29;
set XX = {k where k is Element of NAT:k in rng g4 & k>q/2};
for x being object st x in XX holds x in rng g4
proof
let x be object;
assume x in XX;
then ex k being Element of NAT st x = k & k in rng g4 & k > q/2;
hence thesis;
end;
then
A26: XX c= rng g4;
reconsider f1 as FinSequence of NAT by A16,FINSEQ_2:12;
deffunc F(Nat) = f1.$1 div p;
consider f2 be FinSequence such that
A27: len f2 = p9 & for d being Nat st d in dom f2 holds f2.d = F(d) from
FINSEQ_1:sch 2;
A28: q>1 by INT_2:def 4;
then
A29: q-'1 = q-1 by XREAL_1:233;
then
A30: q-'1 >0 by A28,XREAL_1:50;
q >= 2+1 by A2,NAT_1:13;
then q-1 >= 3-1 by XREAL_1:9;
then
A31: q9 >= 1 by A29,NAT_2:13;
then len g3 >=1 by A23,EULER_2:def 1;
then g3 <> {};
then rng g3 is finite non empty Subset of NAT;
then consider n2 be Element of NAT such that
A32: rng g3 c= Seg n2 \/ {0} by HEYTING3:1;
deffunc F(Nat) = g1.$1 div q;
consider g2 be FinSequence such that
A33: len g2 = q9 & for d being Nat st d in dom g2 holds g2.d = F(d)
from FINSEQ_1:sch 2;
for d being Nat st d in dom g2 holds g2.d in NAT
proof
let d be Nat;
assume d in dom g2;
then g2.d = g1.d div q by A33;
hence thesis;
end;
then reconsider g2 as FinSequence of NAT by FINSEQ_2:12;
A34: dom g1 = dom g2 by A23,A33,FINSEQ_3:29;
A35: for d st d in dom g1 holds g1.d = g2.d * q + g3.d
proof
let d;
assume
A36: d in dom g1;
then
A37: g3.d = g1.d mod q by EULER_2:def 1;
g2.d = g1.d div q by A33,A34,A36;
hence thesis by A37,NAT_D:2;
end;
q is odd by A2,PEPIN:17;
then
A38: q-'1 is even by A29,HILBERT3:2;
then
A39: 2 divides (q-'1) by PEPIN:22;
then
A40: q-'1 = 2*q9 by NAT_D:3;
then q9 divides (q-'1);
then q9 <= (q-'1) by A30,NAT_D:7;
then
A41: q9 < q by A29,XREAL_1:146,XXREAL_0:2;
not 0 in rng g3
proof
assume 0 in rng g3;
then consider a be Nat such that
A42: a in dom g3 and
A43: g3.a = 0 by FINSEQ_2:10;
a in dom g1 by A24,A42,FINSEQ_3:29;
then
A44: g1.a = g2.a * q + 0 by A35,A43;
a in dom g1 by A24,A42,FINSEQ_3:29;
then p*a = g2.a * q by A19,A44;
then
A45: q divides p*a;
a >= 1 by A42,FINSEQ_3:25;
then
A46: q <= a by A4,A45,NAT_D:7,PEPIN:3;
a <= q9 by A23,A24,A42,FINSEQ_3:25;
hence contradiction by A41,A46,XXREAL_0:2;
end;
then
A47: {0} misses rng g3 by ZFMISC_1:50;
then
A48: g4 is one-to-one by A32,FINSEQ_3:92,XBOOLE_1:73;
A49: for d,e st d in dom g1 & e in dom g1 & q divides (g1.d-g1.e) holds d=e
proof
A50: q,(p qua Integer) are_coprime by A3,INT_2:30;
let d,e;
assume that
A51: d in dom g1 and
A52: e in dom g1 and
A53: q divides (g1.d-g1.e);
A54: g1.e = p*e by A19,A52;
g1.d = p*d by A19,A51;
then
A55: q divides (d-e)*p by A53,A54;
now
assume d <> e;
then d-e <> 0;
then |.q.| <= |.d-e.| by A55,A50,INT_2:25,INT_4:6;
then
A56: q <= |.d-e.| by ABSVALUE:def 1;
A57: e>=1 by A52,FINSEQ_3:25;
A58: d>=1 by A51,FINSEQ_3:25;
e<=q9 by A23,A52,FINSEQ_3:25;
then
A59: d-e>=1-q9 by A58,XREAL_1:13;
A60: q9-1<q by A41,XREAL_1:147;
d<=q9 by A23,A51,FINSEQ_3:25;
then d-e<=q9-1 by A57,XREAL_1:13;
then
A61: d-e < q by A60,XXREAL_0:2;
-(q9-1) > -q by A60,XREAL_1:24;
then d-e > -q by A59,XXREAL_0:2;
hence contradiction by A56,A61,SEQ_2:1;
end;
hence thesis;
end;
for x,y be object st x in dom g3 & y in dom g3 & g3.x=g3.y holds x=y
proof
let x,y be object;
assume that
A62: x in dom g3 and
A63: y in dom g3 and
A64: g3.x=g3.y;
reconsider x,y as Element of NAT by A62,A63;
A65: g1.y = g2.y * q + g3.y by A25,A35,A63;
g1.x = g2.x * q + g3.x by A25,A35,A62;
then g1.x - g1.y = (g2.x - g2.y) * q by A64,A65;
then q divides (g1.x - g1.y);
hence thesis by A49,A25,A62,A63;
end;
then
A66: g3 is one-to-one;
then len g3 = card rng g3 by FINSEQ_4:62;
then
A67: len g4 = q9 by A23,A24,A32,A47,FINSEQ_3:39,XBOOLE_1:73;
reconsider XX as finite Subset of NAT by A26,XBOOLE_1:1;
set mm = card XX;
reconsider YY = rng g4 \ XX as finite Subset of NAT;
A68: g3 is Element of NAT* by FINSEQ_1:def 11;
len g3 = q9 by A23,EULER_2:def 1;
then g3 in q9-tuples_on NAT by A68;
then
A69: g3 is Element of q9-tuples_on REAL by FINSEQ_2:109,NUMBERS:19;
for d being Nat st d in dom idseq q9 holds (idseq q9).d in NAT;
then idseq q9 is FinSequence of NAT by FINSEQ_2:12;
then reconsider N = Sum idseq q9 as Element of NAT by Lm4;
A70: 2,q are_coprime by A2,EULER_1:2;
dom(q*g2) = dom g2 by VALUED_1:def 5;
then
A71: len(q*g2) = q9 by A33,FINSEQ_3:29;
q*g2 is Element of NAT* by FINSEQ_1:def 11;
then q*g2 in q9-tuples_on NAT by A71;
then
A72: q*g2 is Element of q9-tuples_on REAL by FINSEQ_2:109,NUMBERS:19;
A73: dom (q*g2+g3) = dom (q*g2) /\ dom g3 by VALUED_1:def 1
.= dom g2 /\ dom g3 by VALUED_1:def 5
.= dom g1 by A25,A34;
for d being Nat st d in dom g1 holds g1.d = (q*g2+g3).d
proof
let d be Nat;
assume
A74: d in dom g1;
then
A75: d in dom (q*g2) by A34,VALUED_1:def 5;
(q*g2+g3).d = (q*g2).d + g3.d by A73,A74,VALUED_1:def 1;
hence (q*g2+g3).d = q * g2.d + g3.d by A75,VALUED_1:def 5
.= g1.d by A35,A74;
end;
then g1 = q*g2 + g3 by A73;
then
A76: Sum g1 = Sum(q*g2) + Sum g3 by A72,A69,RVSUM_1:89
.= q*(Sum g2) + Sum g3 by RVSUM_1:87;
A77: rng g3 c= Seg n2 by A32,A47,XBOOLE_1:73;
then
A78: rng g4 = rng g3 by FINSEQ_1:def 13;
then
A79: XX c= Seg n2 by A77,A26;
A80: len g3 = card rng g4 by A66,A78,FINSEQ_4:62;
mm <= card rng g4 by A26,NAT_1:43;
then mm <= q9 by A23,A80,EULER_2:def 1;
then reconsider nn = q9 - mm as Element of NAT by NAT_1:21;
A81: g4 = (g4|nn)^(g4/^nn) by RFINSEQ:8;
then
A82: g4/^nn is one-to-one by A48,FINSEQ_3:91;
A83: q9 = ((q-'1)+1) div 2 by A38,NAT_2:26
.= q div 2 by A28,XREAL_1:235;
A84: g3 is FinSequence of REAL by FINSEQ_2:24,NUMBERS:19;
g4 is FinSequence of REAL by FINSEQ_2:24,NUMBERS:19;
then
A85: Sum g4 = Sum g3 by A66,A78,A48,RFINSEQ:9,26,A84;
A86: rng g4 \ XX c= rng g4 by XBOOLE_1:36;
then
A87: YY c= Seg n2 by A77,A78;
for k,l being Nat st k in YY & l in XX holds k < l
proof
let k,l be Nat;
assume that
A88: k in YY and
A89: l in XX;
A90: not k in XX by A88,XBOOLE_0:def 5;
A91: ex l1 being Element of NAT st l1 = l & l1 in rng g4 & l1>q/2 by A89;
k in rng g4 by A88,XBOOLE_0:def 5;
then k <= q/2 by A90;
hence thesis by A91,XXREAL_0:2;
end;
then Sgm (YY\/XX) = (Sgm YY)^(Sgm XX) by A87,A79,FINSEQ_3:42;
then Sgm (rng g4 \/ XX) = (Sgm YY)^(Sgm XX) by XBOOLE_1:39;
then
A92: g4 = (Sgm YY)^(Sgm XX) by A78,A26,XBOOLE_1:12;
then Sum g4 = Sum(Sgm YY) + Sum(Sgm XX) by RVSUM_1:75;
then
A93: p*(Sum idseq q9)=q*(Sum g2)+Sum(Sgm YY)+Sum(Sgm XX) by A76,A85,RVSUM_1:87;
A94: len Sgm YY = card YY by A77,A78,A86,FINSEQ_3:39,XBOOLE_1:1
.= q9 - mm by A23,A24,A26,A80,CARD_2:44;
then
A95: g4/^nn = Sgm XX by A92,FINSEQ_5:37;
for d being Nat st d in dom f2 holds f2.d in NAT
proof
let d be Nat;
assume d in dom f2;
then f2.d = f1.d div p by A27;
hence thesis;
end;
then reconsider f2 as FinSequence of NAT by FINSEQ_2:12;
set f3 = f1 mod p;
A96: len f3 = len f1 by EULER_2:def 1;
then
A97: dom f1 = dom f3 by FINSEQ_3:29;
set f4 = Sgm rng f3;
p >= 2+1 by A1,NAT_1:13;
then
A98: p-1 >= 3-1 by XREAL_1:9;
then f3 <> {} by A18,A7,A96,NAT_2:13;
then rng f3 is finite non empty Subset of NAT;
then consider n1 be Element of NAT such that
A99: rng f3 c= Seg n1 \/ {0} by HEYTING3:1;
A100: dom f1 = dom f2 by A18,A27,FINSEQ_3:29;
A101: for d st d in dom f1 holds f1.d = f2.d * p + f3.d
proof
let d;
assume
A102: d in dom f1;
then
A103: f3.d = f1.d mod p by EULER_2:def 1;
f2.d = f1.d div p by A27,A100,A102;
hence thesis by A103,NAT_D:2;
end;
not 0 in rng f3
proof
assume 0 in rng f3;
then consider a be Nat such that
A104: a in dom f3 and
A105: f3.a = 0 by FINSEQ_2:10;
f1.a = f2.a * p + 0 by A97,A101,A104,A105;
then q*a = f2.a *p by A13,A97,A104;
then
A106: p divides q*a;
a >= 1 by A104,FINSEQ_3:25;
then
A107: p <= a by A4,A106,NAT_D:7,PEPIN:3;
a <= p9 by A18,A96,A104,FINSEQ_3:25;
hence contradiction by A12,A107,XXREAL_0:2;
end;
then
A108: {0} misses rng f3 by ZFMISC_1:50;
then
A109: f4 is one-to-one by A99,FINSEQ_3:92,XBOOLE_1:73;
A110: for d,e st d in dom f1 & e in dom f1 & p divides (f1.d-f1.e) holds d=e
proof
A111: q,(p qua Integer) are_coprime by A3,INT_2:30;
let d,e;
assume that
A112: d in dom f1 and
A113: e in dom f1 and
A114: p divides (f1.d-f1.e);
A115: f1.e = q*e by A13,A113;
f1.d = q*d by A13,A112;
then
A116: p divides (d-e)*q by A114,A115;
now
assume d <> e;
then d-e <> 0;
then |.p.| <= |.d-e.| by A116,A111,INT_2:25,INT_4:6;
then
A117: p <= |.d-e.| by ABSVALUE:def 1;
A118: e>=1 by A113,FINSEQ_3:25;
A119: d>=1 by A112,FINSEQ_3:25;
e<=p9 by A18,A113,FINSEQ_3:25;
then
A120: d-e>=1-p9 by A119,XREAL_1:13;
A121: p9-1<p by A12,XREAL_1:147;
d<=p9 by A18,A112,FINSEQ_3:25;
then d-e<=p9-1 by A118,XREAL_1:13;
then
A122: d-e < p by A121,XXREAL_0:2;
-(p9-1) > -p by A121,XREAL_1:24;
then d-e > -p by A120,XXREAL_0:2;
hence contradiction by A117,A122,SEQ_2:1;
end;
hence thesis;
end;
for x,y be object st x in dom f3 & y in dom f3 & f3.x=f3.y holds x=y
proof
let x,y be object;
assume that
A123: x in dom f3 and
A124: y in dom f3 and
A125: f3.x=f3.y;
reconsider x,y as Element of NAT by A123,A124;
A126: f1.y = f2.y * p + f3.y by A97,A101,A124;
f1.x = f2.x * p + f3.x by A97,A101,A123;
then f1.x - f1.y = (f2.x - f2.y) * p by A125,A126;
then p divides (f1.x - f1.y);
hence thesis by A110,A97,A123,A124;
end;
then
A127: f3 is one-to-one;
then len f3 = card rng f3 by FINSEQ_4:62;
then
A128: len f4 = p9 by A18,A96,A99,A108,FINSEQ_3:39,XBOOLE_1:73;
A129: g4|nn = Sgm YY by A92,A94,FINSEQ_3:113,FINSEQ_6:10;
A130: g4|nn is one-to-one by A48,A81,FINSEQ_3:91;
A131: Lege(p,q) = (-1)|^(Sum g2)
proof
set g5 = (mm|->q)-(g4/^nn);
set g6 = (g4|nn)^g5;
A132: g4/^nn is FinSequence of REAL by FINSEQ_2:24,NUMBERS:19;
A133: len(g4|nn) = nn by A67,FINSEQ_1:59,XREAL_1:43;
A134: len(g4/^nn) = (len g4 -' nn) by RFINSEQ:29
.= len g4 - nn by A67,XREAL_1:43,233
.= mm by A67;
A135: dom g5 = dom(mm |-> q) /\ dom(g4/^nn) by VALUED_1:12
.= (Seg len (mm |-> q)) /\ dom(g4/^nn) by FINSEQ_1:def 3
.= dom(g4/^nn) /\ dom(g4/^nn) by FINSEQ_1:def 3,A134,CARD_1:def 7
.= dom(g4/^nn);
then
A136: len g5 = len(g4/^nn) by FINSEQ_3:29;
A137: for d st d in dom g5 holds g5.d = q - (g4/^nn).d
proof
let d;
assume
A138: d in dom g5;
then d in Seg mm by A134,A135,FINSEQ_1:def 3;
then (mm |-> q).d = q by FINSEQ_2:57;
hence thesis by A138,VALUED_1:13;
end;
A139: for d st d in dom g5 holds g5.d > 0 & g5.d <= q9
proof
let d;
reconsider w = g5.d as Element of INT by INT_1:def 2;
assume
A140: d in dom g5;
then (Sgm XX).d in rng Sgm XX by A95,A135,FUNCT_1:3;
then (Sgm XX).d in XX by A79,FINSEQ_1:def 13;
then
A141: ex ll be Element of NAT st ll = (Sgm XX).d & ll in rng g3 &
ll > q/2 by A78;
then consider e being Nat such that
A142: e in dom g3 and
A143: g3.e=(g4/^nn).d by A95,FINSEQ_2:10;
(g4/^nn).d = g1.e mod q by A25,A142,A143,EULER_2:def 1;
then
A144: (g4/^nn).d < q by NAT_D:1;
A145: g5.d = q - (g4/^nn).d by A137,A140;
then w < q - q/2 by A95,A141,XREAL_1:10;
hence thesis by A83,A145,A144,INT_1:54,XREAL_1:50;
end;
A146: rng g5 c= INT by RELAT_1:def 19;
for d being Nat st d in dom g5 holds g5.d in NAT
proof
let d be Nat;
assume
A147: d in dom g5;
g5.d > 0 by A139,A147;
hence thesis by A146,INT_1:3;
end;
then reconsider g5 as FinSequence of NAT by FINSEQ_2:12;
g5 is FinSequence of NAT;
then reconsider g6 as FinSequence of NAT by FINSEQ_1:75;
A148: g6 is FinSequence of REAL by FINSEQ_2:24,NUMBERS:19;
A149: nn <= len g4 by A67,XREAL_1:43;
A150: rng(g4|nn) misses rng g5
proof
assume not rng(g4|nn) misses rng g5;
then consider x be object such that
A151: x in rng(g4|nn) and
A152: x in rng g5 by XBOOLE_0:3;
consider e being Nat such that
A153: e in dom g5 and
A154: g5.e = x by A152,FINSEQ_2:10;
x = q - (g4/^nn).e by A137,A153,A154;
then
A155: x = q - g4.(e+nn) by A149,A135,A153,RFINSEQ:def 1;
e+nn in dom g4 by A135,A153,FINSEQ_5:26;
then consider e1 be Nat such that
A156: e1 in dom g3 and
A157: g3.e1 = g4.(e+nn) by A78,FINSEQ_2:10,FUNCT_1:3;
A158: e1 <= q9 by A23,A24,A156,FINSEQ_3:25;
rng(g4|nn) c= rng g4 by FINSEQ_5:19;
then consider d1 be Nat such that
A159: d1 in dom g3 and
A160: g3.d1 = x by A78,A151,FINSEQ_2:10;
d1 <= q9 by A23,A24,A159,FINSEQ_3:25;
then d1+e1 <= q9+q9 by A158,XREAL_1:7;
then
A161: d1+e1 < q by A29,A40,XREAL_1:146,XXREAL_0:2;
A162: e1 in dom g1 by A24,A156,FINSEQ_3:29;
then
A163: g4.(e+nn) = g1.e1 mod q by A157,EULER_2:def 1;
A164: d1 in dom g1 by A24,A159,FINSEQ_3:29;
then x = g1.d1 mod q by A160,EULER_2:def 1;
then ((g1.d1 mod q)+(g1.e1 mod q)) mod q = 0 by A155,A163,NAT_D:25;
then (g1.d1 + g1.e1) mod q = 0 by EULER_2:6;
then q divides (g1.d1 + g1.e1) by PEPIN:6;
then q divides (d1*p + g1.e1) by A19,A164;
then q divides (d1*p + e1*p) by A19,A162;
then
A165: q divides (d1+e1)*p;
d1 >= 1 by A159,FINSEQ_3:25;
hence contradiction by A4,A165,A161,NAT_D:7,PEPIN:3;
end;
for d,e being Nat st 1<=d & d<e & e<=len g5 holds g5.d
<> g5. e
proof
let d,e be Nat;
assume that
A166: 1<=d and
A167: d<e and
A168: e<=len g5;
1<=e by A166,A167,XXREAL_0:2;
then
A169: e in dom g5 by A168,FINSEQ_3:25;
then
A170: g5.e = q - (g4/^nn).e by A137;
d<len g5 by A167,A168,XXREAL_0:2;
then
A171: d in dom g5 by A166,FINSEQ_3:25;
then g5.d = q - (g4/^nn).d by A137;
hence thesis by A82,A135,A167,A171,A169,A170;
end;
then len g5 = card rng g5 by GRAPH_5:7;
then g5 is one-to-one by FINSEQ_4:62;
then
A172: g6 is one-to-one by A130,A150,FINSEQ_3:91;
A173: for d st d in dom g6 holds g6.d>0 & g6.d <= q9
proof
let d;
assume
A174: d in dom g6;
per cases by A174,FINSEQ_1:25;
suppose
A175: d in dom(g4|nn);
then (g4|nn).d in rng Sgm YY by A129,FUNCT_1:3;
then
A176: (g4|nn).d in YY by A87,FINSEQ_1:def 13;
then
A177: (g4|nn).d in rng g4 by XBOOLE_0:def 5;
not (g4|nn).d in XX by A176,XBOOLE_0:def 5;
then (g4|nn).d <= q/2 by A177;
then
A178: (g4|nn).d <= q9 by A83,INT_1:54;
not (g4|nn).d in {0} by A47,A78,A177,XBOOLE_0:3;
then (g4|nn).d <> 0 by TARSKI:def 1;
hence thesis by A175,A178,FINSEQ_1:def 7;
end;
suppose
ex l being Nat st l in dom g5 & d=len(g4|nn)+ l;
then consider l be Element of NAT such that
A179: l in dom g5 and
A180: d = len(g4|nn)+ l;
g6.d = g5.l by A179,A180,FINSEQ_1:def 7;
hence thesis by A139,A179;
end;
end;
A181: idseq q9 is FinSequence of REAL by RVSUM_1:145;
len g6 = len(g4|nn) + len g5 by FINSEQ_1:22
.= q9 by A133,A134,A136;
then rng g6 = rng idseq q9 by A172,A173,Th40;
then N = Sum g6 by A172,A148,A181,RFINSEQ:9,26
.= Sum(g4|nn) + Sum g5 by RVSUM_1:75
.= Sum(g4|nn) + (mm*q - Sum(g4/^nn)) by A134,A132,Th47
.= Sum(g4|nn) + mm*q - Sum(g4/^nn);
then (p-1)*N = q*(Sum g2) + 2*Sum(Sgm XX) - mm*q by A93,A95,A129;
then
A182: (p-'1)*N mod 2 = ((q*(Sum g2)-mm*q) + 2*Sum(Sgm XX)) mod 2 by A6,
XREAL_1:233
.= (q*(Sum g2)-mm*q) mod 2 by EULER_1:12;
2 divides (p-'1)*N by A10,NAT_D:9;
then (q*((Sum g2)-mm)) mod 2 = 0 by A182,PEPIN:6;
then 2 divides (q*((Sum g2)-mm)) by Lm1;
then 2 divides ((Sum g2) - mm) by A70,INT_2:25;
then (Sum g2),mm are_congruent_mod 2;
then (Sum g2) mod 2 = mm mod 2 by NAT_D:64;
then (-1)|^(Sum g2) = (-1)|^mm by Th45;
hence thesis by A2,A5,A78,Th41;
end;
for d being Nat st d in dom idseq p9 holds (idseq p9).d in NAT;
then idseq p9 is FinSequence of NAT by FINSEQ_2:12;
then reconsider M = Sum idseq p9 as Element of NAT by Lm4;
A183: 2,p are_coprime by A1,EULER_1:2;
set X = {k where k is Element of NAT:k in rng f4 & k>p/2};
for x being object st x in X holds x in rng f4
proof
let x be object;
assume x in X;
then ex k being Element of NAT st x = k & k in rng f4 & k > p/2;
hence thesis;
end;
then
A184: X c= rng f4;
A185: p9 >= 1 by A7,A98,NAT_2:13;
A186: (Sum f2) + (Sum g2) = p9 * q9
proof
reconsider A = Seg p9,B = Seg q9 as non empty finite Subset of NAT by A185
,A31;
deffunc F(Element of A,Element of B) = $1/p - $2/q;
A187: for x be Element of A, y be Element of B holds F(x,y) in REAL by
XREAL_0:def 1;
consider z being Function of [:A,B:], REAL such that
A188: for x be Element of A, y be Element of B holds z.(x,y) = F(x,y)
from FUNCT_7:sch 1(A187);
defpred G[set,set] means ex x be Element of A st $1=x & $2 = {[x,y] where
y is Element of B:z.(x,y)>0};
A189: for d being Nat st d in Seg p9 ex x1 be Element of bool dom z st G[d ,x1]
proof
let d be Nat;
assume d in Seg p9;
then reconsider d as Element of A;
take x1 = {[d,y] where y is Element of B:z.(d,y)>0};
x1 c= dom z
proof
let l be object;
assume l in x1;
then ex yy be Element of B st [d,yy] = l & z.(d,yy) > 0;
then l in [:A,B:];
hence thesis by FUNCT_2:def 1;
end;
hence thesis;
end;
consider Pr be FinSequence of bool dom z such that
A190: dom Pr = Seg p9 & for d being Nat st d in Seg p9 holds G[d,Pr.d]
from FINSEQ_1:sch 5(A189);
A191: dom Card Pr = dom Pr by CARD_3:def 2
.= dom f2 by A27,A190,FINSEQ_1:def 3;
for d being Nat st d in dom Card Pr holds (Card Pr).d = f2.d
proof
let d be Nat;
assume
A192: d in dom Card Pr;
then d in Seg p9 by A27,A191,FINSEQ_1:def 3;
then consider m be Element of A such that
A193: m = d and
A194: Pr.d = {[m,y] where y is Element of B:z.(m,y)>0} by A190;
Pr.d = [:{m},Seg(f2.m):]
proof
set L = [:{m},Seg(f2.m):];
A195: L c= Pr.d
proof
now
assume q mod p = 0;
then
A196: p divides q by PEPIN:6;
then p <= q by NAT_D:7;
then p < q by A3,XXREAL_0:1;
hence contradiction by A6,A196,NAT_4:12;
end;
then
A197: -(q div p) = (-q) div p + 1 by WSIERP_1:41;
2 divides (p-'1)*q by A10,NAT_D:9;
then (p-'1)*q mod 2 = 0 by PEPIN:6;
then ((p-'1)*q) div 2 = (p-'1)*q/2 by REAL_3:4;
then
A198: (p9*q) div p = ((p-1)*q) div (2*p) by A7,A11,NAT_2:27
.= ((p*q - q) div p) div 2 by PRE_FF:5
.= (q+(-(q div p)-1)) div 2 by A197,NAT_D:61
.= (2*q9+(-(q div p))) div 2 by A29,A40
.= q9+((-(q div p)) div 2) by NAT_D:61;
A199: (p9*q) div p <= q9
proof
per cases;
suppose
(q div p) mod 2 = 0;
then (-(q div p)) div 2 = -((q div p) div 2) by WSIERP_1:42
.= -(q div (2*p)) by NAT_2:27;
then (p9*q) div p = q9-(q div (2*p)) by A198;
hence thesis by XREAL_1:43;
end;
suppose
(q div p) mod 2 <> 0;
then -((q div p) div 2) = (-(q div p)) div 2 +1 by WSIERP_1:41;
then (-(q div p)) div 2 = -((q div p) div 2)-1
.= -(q div (2*p)) - 1 by NAT_2:27;
then (p9*q) div p = q9 -((q div (2*p)) + 1) by A198;
hence thesis by XREAL_1:43;
end;
end;
m <= p9 by FINSEQ_1:1;
then m*q <= p9*q by XREAL_1:64;
then (m*q) div p <= (p9*q) div p by NAT_2:24;
then
A200: (m*q) div p <= q9 by A199,XXREAL_0:2;
m in Seg p9;
then
A201: m in dom f1 by A18,FINSEQ_1:def 3;
then
A202: f2.m = f1.m div p by A27,A100
.= (m*q) div p by A13,A201;
now
assume m*q/p is integer;
then
A203: p divides m*q by WSIERP_1:17;
A204: m <= p9 by FINSEQ_1:1;
0+1 <= m by FINSEQ_1:1;
then p <= m by A5,A203,NAT_D:7,WSIERP_1:30;
hence contradiction by A12,A204,XXREAL_0:2;
end;
then
A205: [\m*q/p/] < m*q/p by INT_1:26;
let l be object;
assume l in L;
then consider x,y be object such that
A206: x in {m} and
A207: y in Seg(f2.m) and
A208: l = [x,y] by ZFMISC_1:def 2;
reconsider y as Element of NAT by A207;
A209: 1 <= y by A207,FINSEQ_1:1;
y <= f2.m by A207,FINSEQ_1:1;
then y <= q9 by A200,A202,XXREAL_0:2;
then reconsider y as Element of B by A209,FINSEQ_1:1;
y <= [\m*q/p/] by A207,A202,FINSEQ_1:1;
then y < m*q/p by A205,XXREAL_0:2;
then y*p < (m*q)/p*p by XREAL_1:68;
then y*p < m*q by XCMPLX_1:87;
then y/q < m/p by XREAL_1:106;
then m/p - y/q > 0 by XREAL_1:50;
then z.(m,y) > 0 by A188;
then [m,y] in Pr.d by A194;
hence thesis by A206,A208,TARSKI:def 1;
end;
Pr.d c= L
proof
let l be object;
A210: m in {m} by TARSKI:def 1;
m in Seg p9;
then
A211: m in dom f1 by A18,FINSEQ_1:def 3;
assume l in Pr.d;
then consider y1 be Element of B such that
A212: l = [m,y1] and
A213: z.(m,y1) > 0 by A194;
m/p - y1/q > 0 by A188,A213;
then m/p - y1/q + y1/q > 0 + y1/q by XREAL_1:6;
then m/p*q > y1/q*q by XREAL_1:68;
then (m*q)/p > y1 by XCMPLX_1:87;
then ((m*q) div p) >= y1 by INT_1:54;
then ((f1.m) div p) >= y1 by A13,A211;
then
A214: y1 <= f2.m by A27,A100,A211;
y1 >= 1 by FINSEQ_1:1;
then y1 in Seg(f2.m) by A214,FINSEQ_1:1;
hence thesis by A212,A210,ZFMISC_1:def 2;
end;
hence thesis by A195,XBOOLE_0:def 10;
end;
then card(Pr.d) = card [:Seg(f2.m),{m}:] by CARD_2:4
.= card Seg(f2.m) by CARD_1:69;
then
A215: card(Pr.d) = card(f2.d) by A193,FINSEQ_1:55
.= f2.d;
d in dom Pr by A192,CARD_3:def 2;
hence thesis by A215,CARD_3:def 2;
end;
then
A216: Card Pr = f2 by A191;
defpred K[set,set] means ex y be Element of B st $1=y & $2 = {[x,y] where
x is Element of A:z.(x,y)<0};
A217: for d being Nat st d in Seg q9 ex x1 be Element of bool dom z st K[d ,x1]
proof
let d be Nat;
assume d in Seg q9;
then reconsider d as Element of B;
take x1 = {[x,d] where x is Element of A:z.(x,d)<0};
x1 c= dom z
proof
let l be object;
assume l in x1;
then ex xx be Element of A st [xx,d] = l & z.(xx,d) < 0;
then l in [:A,B:];
hence thesis by FUNCT_2:def 1;
end;
hence thesis;
end;
consider Pk be FinSequence of bool dom z such that
A218: dom Pk = Seg q9 & for d being Nat st d in Seg q9 holds K[d,Pk.d]
from FINSEQ_1:sch 5(A217);
A219: dom Card Pk = Seg(len g2) by A33,A218,CARD_3:def 2
.= dom g2 by FINSEQ_1:def 3;
A220: for d being Nat st d in dom Card Pk holds (Card Pk).d = g2.d
proof
let d be Nat;
assume
A221: d in dom Card Pk;
then d in Seg q9 by A33,A219,FINSEQ_1:def 3;
then consider n be Element of B such that
A222: n = d and
A223: Pk.d = {[x,n] where x is Element of A:z.(x,n)<0} by A218;
Pk.d = [:Seg(g2.n),{n}:]
proof
set L = [:Seg(g2.n),{n}:];
A224: L c= Pk.d
proof
now
assume p mod q = 0;
then
A225: q divides p by PEPIN:6;
then q <= p by NAT_D:7;
then q < p by A3,XXREAL_0:1;
hence contradiction by A28,A225,NAT_4:12;
end;
then
A226: -(p div q) = (-p) div q + 1 by WSIERP_1:41;
2 divides (q-'1)*p by A39,NAT_D:9;
then (q-'1)*p mod 2 = 0 by PEPIN:6;
then ((q-'1)*p) div 2 = (q-'1)*p/2 by REAL_3:4;
then
A227: (q9*p) div q = ((q-1)*p) div (2*q) by A29,A40,NAT_2:27
.= ((q*p - p) div q) div 2 by PRE_FF:5
.= (p+(-(p div q)-1)) div 2 by A226,NAT_D:61
.= (2*p9-(p div q)) div 2 by A7,A11
.= p9+((-(p div q)) div 2) by NAT_D:61;
A228: (q9*p) div q <= p9
proof
per cases;
suppose
(p div q) mod 2 = 0;
then (-(p div q)) div 2 = -((p div q) div 2) by WSIERP_1:42
.= -(p div (2*q)) by NAT_2:27;
then (q9*p) div q = p9-(p div (2*q)) by A227;
hence thesis by XREAL_1:43;
end;
suppose
(p div q) mod 2 <> 0;
then -((p div q) div 2) = (-(p div q)) div 2 +1 by WSIERP_1:41;
then (-(p div q)) div 2 = -((p div q) div 2)-1
.= -(p div (2*q)) - 1 by NAT_2:27;
then (q9*p) div q = p9 -((p div (2*q)) + 1) by A227;
hence thesis by XREAL_1:43;
end;
end;
n in Seg q9;
then
A229: n in dom g1 by A23,FINSEQ_1:def 3;
then
A230: g2.n = g1.n div q by A33,A34
.= (n*p) div q by A19,A229;
let l be object;
assume l in L;
then consider x,y be object such that
A231: x in Seg(g2.n) and
A232: y in {n} and
A233: l = [x,y] by ZFMISC_1:def 2;
reconsider x as Element of NAT by A231;
A234: x <= g2.n by A231,FINSEQ_1:1;
n <= q9 by FINSEQ_1:1;
then n*p <= q9*p by XREAL_1:64;
then (n*p) div q <= (q9*p) div q by NAT_2:24;
then (n*p) div q <= p9 by A228,XXREAL_0:2;
then
A235: x <= p9 by A230,A234,XXREAL_0:2;
1 <= x by A231,FINSEQ_1:1;
then reconsider x as Element of A by A235,FINSEQ_1:1;
now
assume n*p/q is integer;
then
A236: q divides n*p by WSIERP_1:17;
A237: n <= q9 by FINSEQ_1:1;
0+1 <= n by FINSEQ_1:1;
then q <= n by A5,A236,NAT_D:7,WSIERP_1:30;
hence contradiction by A41,A237,XXREAL_0:2;
end;
then [\n*p/q/] < n*p/q by INT_1:26;
then x < n*p/q by A230,A234,XXREAL_0:2;
then x*q < (n*p)/q*q by XREAL_1:68;
then x*q < n*p by XCMPLX_1:87;
then x/p - n/q < 0 by XREAL_1:49,106;
then z.(x,n) < 0 by A188;
then [x,n] in Pk.d by A223;
hence thesis by A232,A233,TARSKI:def 1;
end;
Pk.d c= L
proof
let l be object;
A238: n in {n} by TARSKI:def 1;
n in Seg q9;
then
A239: n in dom g1 by A23,FINSEQ_1:def 3;
assume l in Pk.d;
then consider x be Element of A such that
A240: l = [x,n] and
A241: z.(x,n) < 0 by A223;
x/p - n/q < 0 by A188,A241;
then x/p - n/q + n/q < 0 + n/q by XREAL_1:6;
then x/p*p < n/q*p by XREAL_1:68;
then x < (n*p)/q by XCMPLX_1:87;
then x <= (n*p) div q by INT_1:54;
then ((g1.n) div q) >= x by A19,A239;
then
A242: x <= g2.n by A33,A34,A239;
x >= 1 by FINSEQ_1:1;
then x in Seg(g2.n) by A242,FINSEQ_1:1;
hence thesis by A240,A238,ZFMISC_1:def 2;
end;
hence thesis by A224,XBOOLE_0:def 10;
end;
then card(Pk.d) = card Seg(g2.n) by CARD_1:69;
then
A243: card(Pk.d) = card(g2.d) by A222,FINSEQ_1:55
.= g2.d;
d in dom Pk by A221,CARD_3:def 2;
hence thesis by A243,CARD_3:def 2;
end;
reconsider U1 = union rng Pr, U2 = union rng Pk as finite Subset of dom z
by PROB_3:48;
dom z c= U1 \/ U2
proof
let l be object;
assume l in dom z;
then consider x,y be object such that
A244: x in A and
A245: y in B and
A246: l = [x,y] by ZFMISC_1:def 2;
reconsider y as Element of B by A245;
reconsider x as Element of A by A244;
A247: z.(x,y) <> 0
proof
assume z.(x,y) = 0;
then x/p - y/q = 0 by A188;
then x*q = y*p by XCMPLX_1:95;
then
A248: p divides x*q;
A249: x <= p9 by FINSEQ_1:1;
x >= 0 + 1 by FINSEQ_1:1;
then p <= x by A5,A248,NAT_D:7,WSIERP_1:30;
hence contradiction by A12,A249,XXREAL_0:2;
end;
per cases by A247;
suppose
A250: z.(x,y) > 0;
G[x,Pr.x] by A190;
then l in Pr.x by A246,A250;
then l in Union Pr by A190,PROB_3:49;
hence thesis by XBOOLE_0:def 3;
end;
suppose
A251: z.(x,y) < 0;
K[y,Pk.y] by A218;
then l in Pk.y by A246,A251;
then l in Union Pk by A218,PROB_3:49;
hence thesis by XBOOLE_0:def 3;
end;
end;
then
A252: U1 \/ U2 = dom z by XBOOLE_0:def 10;
A253: U1 misses U2
proof
assume U1 meets U2;
then consider l be object such that
A254: l in U1 and
A255: l in U2 by XBOOLE_0:3;
l in Union Pk by A255;
then consider k2 be Nat such that
A256: k2 in dom Pk and
A257: l in Pk.k2 by PROB_3:49;
l in Union Pr by A254;
then consider k1 be Nat such that
A258: k1 in dom Pr and
A259: l in Pr.k1 by PROB_3:49;
reconsider k1,k2 as Element of NAT by ORDINAL1:def 12;
consider n1 be Element of B such that
n1 = k2 and
A260: Pk.k2 = {[x,n1] where x is Element of A:z.(x,n1)<0} by A218,A256;
consider n2 be Element of A such that
A261: l = [n2,n1] and
A262: z.(n2,n1) < 0 by A257,A260;
consider m1 be Element of A such that
m1 = k1 and
A263: Pr.k1 = {[m1,y] where y is Element of B:z.(m1,y)>0} by A190,A258;
A264: ex m2 be Element of B st l = [m1,m2] & z.(m1,m2) > 0 by A259,A263;
then m1 = n2 by A261,XTUPLE_0:1;
hence contradiction by A264,A261,A262,XTUPLE_0:1;
end;
A265: for d,e st d in dom Pk & e in dom Pk & d<>e holds Pk.d misses Pk.e
proof
let d,e;
assume that
A266: d in dom Pk and
A267: e in dom Pk and
A268: d<>e;
consider y2 be Element of B such that
A269: y2=e and
A270: Pk.e = {[x,y2] where x is Element of A:z.(x,y2)<0} by A218,A267;
consider y1 be Element of B such that
A271: y1=d and
A272: Pk.d = {[x,y1] where x is Element of A:z.(x,y1)<0} by A218,A266;
now
assume not Pk.d misses Pk.e;
then consider l be object such that
A273: l in Pk.d and
A274: l in Pk.e by XBOOLE_0:3;
A275: ex x2 be Element of A st l = [x2,y2] & z.(x2,y2) < 0 by A270,A274;
ex x1 be Element of A st l = [x1,y1] & z.(x1,y1) < 0 by A272,A273;
hence contradiction by A268,A271,A269,A275,XTUPLE_0:1;
end;
hence thesis;
end;
A276: card union rng Pk = Sum Card Pk by A265,Th48;
A277: for d,e st d in dom Pr & e in dom Pr & d<>e holds Pr.d misses Pr.e
proof
let d,e;
assume that
A278: d in dom Pr and
A279: e in dom Pr and
A280: d<>e;
consider x2 be Element of A such that
A281: x2=e and
A282: Pr.e = {[x2,y] where y is Element of B:z.(x2,y)>0} by A190,A279;
consider x1 be Element of A such that
A283: x1=d and
A284: Pr.d = {[x1,y] where y is Element of B:z.(x1,y)>0} by A190,A278;
now
assume not Pr.d misses Pr.e;
then consider l be object such that
A285: l in Pr.d and
A286: l in Pr.e by XBOOLE_0:3;
A287: ex y2 be Element of B st l = [x2,y2] & z.(x2,y2) > 0 by A282,A286;
ex y1 be Element of B st l = [x1,y1] & z.(x1,y1) > 0 by A284,A285;
hence contradiction by A280,A283,A281,A287,XTUPLE_0:1;
end;
hence thesis;
end;
card union rng Pr = Sum Card Pr by A277,Th48;
then card(U1 \/ U2) = (Sum Card Pr) + (Sum Card Pk) by A276,A253,CARD_2:40;
then (Sum Card Pr) + (Sum Card Pk) = card [:A,B:] by A252,FUNCT_2:def 1
.= card A * card B by CARD_2:46
.= p9* card B by FINSEQ_1:57
.= p9 * q9 by FINSEQ_1:57;
hence thesis by A216,A219,A220,FINSEQ_1:13;
end;
dom(p*f2) = dom f2 by VALUED_1:def 5;
then
A288: len(p*f2) = p9 by A27,FINSEQ_3:29;
p*f2 is Element of NAT* by FINSEQ_1:def 11;
then p*f2 in p9-tuples_on NAT by A288;
then
A289: p*f2 is Element of p9-tuples_on REAL by FINSEQ_2:109,NUMBERS:19;
A290: p9 = ((p-'1)+1) div 2 by A9,NAT_2:26
.= p div 2 by A6,XREAL_1:235;
reconsider X as finite Subset of NAT by A184,XBOOLE_1:1;
set m = card X;
reconsider Y = rng f4 \ X as finite Subset of NAT;
A291: f3 is Element of NAT* by FINSEQ_1:def 11;
len f3 = p9 by A17,A96,CARD_1:def 7;
then f3 in p9-tuples_on NAT by A291;
then
A292: f3 is Element of p9-tuples_on REAL by FINSEQ_2:109,NUMBERS:19;
A293: rng f3 c= Seg n1 by A99,A108,XBOOLE_1:73;
then
A294: rng f4 = rng f3 by FINSEQ_1:def 13;
then
A295: X c= Seg n1 by A293,A184;
A296: dom (p*f2+f3) = dom (p*f2) /\ dom f3 by VALUED_1:def 1
.= dom f2 /\ dom f3 by VALUED_1:def 5
.= dom f1 by A97,A100;
for d being Nat st d in dom f1 holds f1.d = (p*f2+f3).d
proof
let d be Nat;
assume
A297: d in dom f1;
then
A298: d in dom (p*f2) by A100,VALUED_1:def 5;
(p*f2+f3).d = (p*f2).d + f3.d by A296,A297,VALUED_1:def 1;
hence (p*f2+f3).d = p * f2.d + f3.d by A298,VALUED_1:def 5
.= f1.d by A101,A297;
end;
then f1 = p*f2 + f3 by A296;
then
A299: Sum f1 = Sum(p*f2) + Sum f3 by A289,A292,RVSUM_1:89
.= p*(Sum f2) + Sum f3 by RVSUM_1:87;
A300: rng f4 \ X c= rng f4 by XBOOLE_1:36;
then
A301: Y c= Seg n1 by A293,A294;
A302: len f3 = card rng f4 by A127,A294,FINSEQ_4:62;
then reconsider n = p9 - m as Element of NAT by A18,A96,A184,NAT_1:21,43;
A303: f4 = (f4|n)^(f4/^n) by RFINSEQ:8;
then
A304: f4/^n is one-to-one by A109,FINSEQ_3:91;
A305: f3 is FinSequence of REAL by FINSEQ_2:24,NUMBERS:19;
f4 is FinSequence of REAL by FINSEQ_2:24,NUMBERS:19;
then
A306: Sum f4 = Sum f3 by A127,A294,A109,RFINSEQ:9,26,A305;
for k,l being Nat st k in Y & l in X holds k < l
proof
let k,l be Nat;
assume that
A307: k in Y and
A308: l in X;
A309: not k in X by A307,XBOOLE_0:def 5;
A310: ex l1 being Element of NAT st l1 = l & l1 in rng f4 & l1>p/2 by A308;
k in rng f4 by A307,XBOOLE_0:def 5;
then k <= p/2 by A309;
hence thesis by A310,XXREAL_0:2;
end;
then Sgm (Y\/X) = (Sgm Y)^(Sgm X) by A295,A301,FINSEQ_3:42;
then Sgm (rng f4 \/ X) = (Sgm Y)^(Sgm X) by XBOOLE_1:39;
then
A311: f4 = (Sgm Y)^(Sgm X) by A294,A184,XBOOLE_1:12;
then Sum f4 = Sum(Sgm Y) + Sum(Sgm X) by RVSUM_1:75;
then
A312: q*(Sum idseq p9)=p*(Sum f2)+Sum(Sgm Y) + Sum(Sgm X) by A299,A306,
RVSUM_1:87;
A313: len Sgm Y = card Y by A293,A294,A300,FINSEQ_3:39,XBOOLE_1:1
.= p9 - m by A18,A96,A184,A302,CARD_2:44;
then
A314: f4/^n = Sgm X by A311,FINSEQ_5:37;
A315: f4|n = Sgm Y by A311,A313,FINSEQ_3:113,FINSEQ_6:10;
A316: f4|n is one-to-one by A109,A303,FINSEQ_3:91;
Lege(q,p) = (-1)|^(Sum f2)
proof
set f5 = (m|->p)-(f4/^n);
set f6 = (f4|n)^f5;
A317: f4/^n is FinSequence of REAL by FINSEQ_2:24,NUMBERS:19;
A318: len(f4|n) = n by A128,FINSEQ_1:59,XREAL_1:43;
A319: len(f4/^n) = (len f4 -' n) by RFINSEQ:29
.= len f4 - n by A128,XREAL_1:43,233
.= m by A128;
A320: dom f5 = dom(m |-> p) /\ dom(f4/^n) by VALUED_1:12
.= (Seg len (m |-> p)) /\ dom(f4/^n) by FINSEQ_1:def 3
.= dom(f4/^n) /\ dom(f4/^n) by FINSEQ_1:def 3,A319,CARD_1:def 7
.= dom(f4/^n);
then
A321: len f5 = len(f4/^n) by FINSEQ_3:29;
A322: for d st d in dom f5 holds f5.d = p - (f4/^n).d
proof
let d;
assume
A323: d in dom f5;
then d in Seg m by A319,A320,FINSEQ_1:def 3;
then (m |-> p).d = p by FINSEQ_2:57;
hence thesis by A323,VALUED_1:13;
end;
A324: for d st d in dom f5 holds f5.d > 0 & f5.d <= p9
proof
let d;
reconsider w = f5.d as Element of INT by INT_1:def 2;
assume
A325: d in dom f5;
then (Sgm X).d in rng Sgm X by A314,A320,FUNCT_1:3;
then (Sgm X).d in X by A295,FINSEQ_1:def 13;
then
A326: ex ll be Element of NAT st ll = (Sgm X).d & ll in rng f3 &
ll > p/2 by A294;
then consider e being Nat such that
A327: e in dom f3 and
A328: f3.e = (f4/^n).d by A314,FINSEQ_2:10;
(f4/^n).d = f1.e mod p by A97,A327,A328,EULER_2:def 1;
then
A329: (f4/^n).d < p by NAT_D:1;
A330: f5.d = p - (f4/^n).d by A322,A325;
then w < p - p/2 by A314,A326,XREAL_1:10;
hence thesis by A290,A330,A329,INT_1:54,XREAL_1:50;
end;
A331: rng f5 c= INT by RELAT_1:def 19;
for d being Nat st d in dom f5 holds f5.d in NAT
proof
let d be Nat;
assume
A332: d in dom f5;
f5.d > 0 by A332,A324;
hence thesis by A331,INT_1:3;
end;
then reconsider f5 as FinSequence of NAT by FINSEQ_2:12;
f5 is FinSequence of NAT;
then reconsider f6 as FinSequence of NAT by FINSEQ_1:75;
A333: f6 is FinSequence of REAL by FINSEQ_2:24,NUMBERS:19;
A334: n <= len f4 by A128,XREAL_1:43;
A335: rng(f4|n) misses rng f5
proof
assume not rng(f4|n) misses rng f5;
then consider x be object such that
A336: x in rng(f4|n) and
A337: x in rng f5 by XBOOLE_0:3;
consider e being Nat such that
A338: e in dom f5 and
A339: f5.e = x by A337,FINSEQ_2:10;
x = p - (f4/^n).e by A322,A338,A339;
then
A340: x = p - f4.(e+n) by A334,A320,A338,RFINSEQ:def 1;
e+n in dom f4 by A320,A338,FINSEQ_5:26;
then consider e1 be Nat such that
A341: e1 in dom f3 and
A342: f3.e1 = f4.(e+n) by A294,FINSEQ_2:10,FUNCT_1:3;
A343: e1 <= p9 by A18,A96,A341,FINSEQ_3:25;
rng(f4|n) c= rng f4 by FINSEQ_5:19;
then consider d1 be Nat such that
A344: d1 in dom f3 and
A345: f3.d1 = x by A294,A336,FINSEQ_2:10;
d1 <= p9 by A18,A96,A344,FINSEQ_3:25;
then d1+e1 <= p9+p9 by A343,XREAL_1:7;
then
A346: d1+e1 < p by A7,A11,XREAL_1:146,XXREAL_0:2;
x = f1.d1 mod p by A97,A344,A345,EULER_2:def 1;
then (f1.d1 mod p) + f4.(e+n) = p by A340;
then (f1.d1 mod p) + (f1.e1 mod p) = p by A97,A341,A342,EULER_2:def 1;
then ((f1.d1 mod p)+(f1.e1 mod p)) mod p = 0 by NAT_D:25;
then (f1.d1 + f1.e1) mod p = 0 by EULER_2:6;
then p divides (f1.d1 + f1.e1) by PEPIN:6;
then p divides (d1*q + f1.e1) by A13,A97,A344;
then p divides (d1*q + e1*q) by A13,A97,A341;
then
A347: p divides (d1+e1)*q;
d1 >= 1 by A344,FINSEQ_3:25;
hence contradiction by A4,A347,A346,NAT_D:7,PEPIN:3;
end;
for d,e being Nat st 1<=d & d<e & e<=len f5 holds f5.d
<> f5. e
proof
let d,e be Nat;
assume that
A348: 1<=d and
A349: d<e and
A350: e<=len f5;
1<=e by A348,A349,XXREAL_0:2;
then
A351: e in dom f5 by A350,FINSEQ_3:25;
then
A352: f5.e = p - (f4/^n).e by A322;
d<len f5 by A349,A350,XXREAL_0:2;
then
A353: d in dom f5 by A348,FINSEQ_3:25;
then f5.d = p - (f4/^n).d by A322;
hence thesis by A304,A320,A349,A353,A351,A352;
end;
then len f5 = card rng f5 by GRAPH_5:7;
then f5 is one-to-one by FINSEQ_4:62;
then
A354: f6 is one-to-one by A316,A335,FINSEQ_3:91;
A355: for d st d in dom f6 holds f6.d>0 & f6.d <= p9
proof
let d;
assume
A356: d in dom f6;
per cases by A356,FINSEQ_1:25;
suppose
A357: d in dom(f4|n);
then (f4|n).d in rng Sgm Y by A315,FUNCT_1:3;
then
A358: (f4|n).d in Y by A301,FINSEQ_1:def 13;
then
A359: (f4|n).d in rng f4 by XBOOLE_0:def 5;
not (f4|n).d in X by A358,XBOOLE_0:def 5;
then (f4|n).d <= p/2 by A359;
then
A360: (f4|n).d <= p9 by A290,INT_1:54;
not (f4|n).d in {0} by A108,A294,A359,XBOOLE_0:3;
then (f4|n).d <> 0 by TARSKI:def 1;
hence thesis by A357,A360,FINSEQ_1:def 7;
end;
suppose
ex l being Nat st l in dom f5 & d=len(f4|n)+ l;
then consider l be Element of NAT such that
A361: l in dom f5 and
A362: d = len(f4|n)+ l;
f6.d = f5.l by A361,A362,FINSEQ_1:def 7;
hence thesis by A324,A361;
end;
end;
A363: idseq p9 is FinSequence of REAL by RVSUM_1:145;
len f6 = len(f4|n) + len f5 by FINSEQ_1:22
.= p9 by A318,A319,A321;
then rng f6 = rng idseq p9 by A354,A355,Th40;
then M = Sum f6 by A363,A354,A333,RFINSEQ:9,26
.= Sum(f4|n) + Sum f5 by RVSUM_1:75
.= Sum(f4|n) + (m*p - Sum(f4/^n)) by A319,A317,Th47
.= Sum(f4|n) + m*p - Sum(f4/^n);
then (q-1)*M = p*(Sum f2) + 2*Sum(Sgm X) - m*p by A312,A314,A315;
then
A364: (q-'1)*M mod 2 = ((p*(Sum f2)-m*p) + 2*Sum(Sgm X)) mod 2 by A28,
XREAL_1:233
.= (p*(Sum f2)-m*p) mod 2 by EULER_1:12;
2 divides (q-'1)*M by A39,NAT_D:9;
then (q-'1)*M mod 2 = 0 by PEPIN:6;
then 2 divides (p*((Sum f2)-m)) by A364,Lm1;
then (Sum f2),m are_congruent_mod 2 by A183,INT_2:25;
then (Sum f2) mod 2 = m mod 2 by NAT_D:64;
then (-1)|^(Sum f2) = (-1)|^m by Th45;
hence thesis by A1,A5,A294,Th41;
end;
hence thesis by A131,A186,NEWTON:8;
end;
theorem
p>2 & q>2 & p<>q & p mod 4 = 3 & q mod 4 = 3 implies Lege(p,q) = -Lege (q,p)
proof
assume that
A1: p>2 and
A2: q>2 and
A3: p<>q and
A4: p mod 4 = 3 and
A5: q mod 4 = 3;
q>1 by INT_2:def 4;
then
A6: q-'1 = q-1 by XREAL_1:233;
q = 4*(q div 4)+3 by A5,NAT_D:2;
then q-'1 = 2*(2*(q div 4)+1) by A6;
then
A7: (q-'1) div 2=2*(q div 4)+1 by NAT_D:18;
p>1 by INT_2:def 4;
then
A8: p-'1 = p-1 by XREAL_1:233;
p=4*(p div 4)+3 by A4,NAT_D:2;
then p-'1 = 2*(2*(p div 4)+1) by A8;
then (p-'1) div 2=2*(p div 4)+1 by NAT_D:18;
then
A9: Lege(p,q)*Lege(q,p) =(-1)|^((2*(p div 4)+1)*(2*(q div 4)+1)) by A1,A2,A3,A7
,Th49
.=((-1)|^(2*(p div 4)+1))|^(2*(q div 4)+1) by NEWTON:9
.= ((-1)|^(2*(p div 4))*(-1))|^(2*(q div 4)+1) by NEWTON:6
.= ((((-1)|^2)|^(p div 4))*(-1))|^(2*(q div 4)+1) by NEWTON:9
.= ((1|^2)|^(p div 4)*(-1))|^(2*(q div 4)+1) by WSIERP_1:1
.= (-1)|^(2*(q div 4))*(-1) by NEWTON:6
.= ((-1)|^2)|^(q div 4) * (-1) by NEWTON:9
.= (1|^2)|^(q div 4) *(-1) by WSIERP_1:1
.= 1*(-1);
per cases by Th25;
suppose
Lege(p,q) = 1;
hence thesis by A9;
end;
suppose
Lege(p,q) = 0;
hence thesis by A9;
end;
suppose
Lege(p,q) = -1;
hence thesis by A9;
end;
end;
theorem
p>2 & q>2 & p<>q & (p mod 4 = 1 or q mod 4 = 1) implies Lege(p,q) = Lege(q,p)
proof
assume that
A1: p>2 and
A2: q>2 and
A3: p<>q and
A4: p mod 4 = 1 or q mod 4 = 1;
p>1 by INT_2:def 4;
then
A5: p-'1 = p-1 by XREAL_1:233;
q>1 by INT_2:def 4;
then
A6: q-'1 = q-1 by XREAL_1:233;
per cases by A4;
suppose
p mod 4 = 1;
then p=4*(p div 4)+1 by NAT_D:2;
then p-'1 = 2*(2*(p div 4)) by A5;
then (p-'1) div 2 = 2*(p div 4) by NAT_D:18;
then
A7: Lege(p,q)*Lege(q,p) = (-1)|^((2*(p div 4))*((q-'1) div 2)) by A1,A2,A3,Th49
.= ((-1)|^(2*(p div 4)))|^((q-'1) div 2) by NEWTON:9
.= (((-1)|^2)|^(p div 4))|^((q-'1) div 2) by NEWTON:9
.= ((1|^2)|^(p div 4))|^((q-'1) div 2) by WSIERP_1:1
.= 1;
per cases by Th25;
suppose
Lege(p,q)=1;
hence thesis by A7;
end;
suppose
Lege(p,q)=0;
hence thesis by A7;
end;
suppose
Lege(p,q)=-1;
hence thesis by A7;
end;
end;
suppose
q mod 4 = 1;
then q=4*(q div 4)+1 by NAT_D:2;
then q-'1 = 2*(2*(q div 4)) by A6;
then (q-'1) div 2 = 2*(q div 4) by NAT_D:18;
then
A8: Lege(p,q)*Lege(q,p) = (-1)|^((2*(q div 4))*((p-'1) div 2)) by A1,A2,A3,Th49
.= ((-1)|^(2*(q div 4)))|^((p-'1) div 2) by NEWTON:9
.= (((-1)|^2)|^(q div 4))|^((p-'1) div 2) by NEWTON:9
.= ((1|^2)|^(q div 4))|^((p-'1) div 2) by WSIERP_1:1
.= 1;
per cases by Th25;
suppose
Lege(p,q)=1;
hence thesis by A8;
end;
suppose
Lege(p,q)=0;
hence thesis by A8;
end;
suppose
Lege(p,q)=-1;
hence thesis by A8;
end;
end;
end;
| 46,127
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The recipe I used called for pecans in the topping but I didn't have any on hand. But I'm sure they would be a great addition to the sweet and crunchy topping.
These are a great way to start the day, especially if you have a breakfast sweet tooth like I do. Do you prefer sweet or savory breakfast?
Please head over to the Brown Eyed Baker's blog for the recipe: Pumpkin and Cream Cheese Muffins.
Thanks for stopping by, have a great Saturday!
8 comments:
ooh i must make these!
mmmmmm i ♥ streusel stopping!!!
Wow!! They look very yummy!!
I need to try these one day too!!
TFS!!
hi Mara! Your muffins look delicious!! I like a cookie (or a little something sweet) with my coffee in the morning too. :)
I just have to tell you that my daughter made these yesterday, and they were divine! I think they were gone within a half an hour. Next time we'll have to double the recipe (or triple, or quadruple...). They were so good. Thank you for sharing.
man...i'm so jealous! all that yummy baking! those look SO yummy!
What a winning recipe! These look deeeelicious, Mara!
Mara you are awesome and these muffins are beyond delicious! Recipe keeper for sure!
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(I hope you’ll forgive me for talking mostly about books the last two weeks. 🙂 I happened to finish several recently and I’m trying to finish off my Spring Reading Thing before it ends.)
I’ve mentioned before the importance of reading missionary biographies, for our own growth and inspiration and to keep before us those names in church history that need to be remembered just like Washington, Lincoln, and others need to be remembered in our secular history.
Hudson Taylor is one of those names for several reasons. suffered much hardship uncomplainingly and purposefully lived as simple a life as possible. resp0nsible in its habits led to Hudson beginning the China Inland Mission. There were a few missionaries in the bigger cities, but China wanted to go inland where the gospel had not been preached. Probably the most notable aspects of Hudson, however, were his simple childlike (but not childish) faith and his unswerving obedience to what he perceived God wanted him to do.
For these reasons I was very glad to see It Is Not Death to Die: A New Biography of Hudson Taylor by Jim Cromarty. There are two older well-known biographies of Hudson Taylor. One and they can sometimes be hard to find (Amazon only had used copies but I found them on sale just now here.) These are excellent and easily readable though they were written over a hundred years ago. The other well-known biography of Taylor is Hudson Taylor’s Spiritual Secret, also written by his daughter and son-in-law, but much more compact at 272 pages and still printed regularly today.
I had high hopes that this new biography by Cromarty would bridge the gap between these two and bring Hudson’s life before a modern audience that might not seek out the older books. And while it is a faithful representation with much research evidently behind it and I can recommend it, I wish it were more dynamically written. It’s a good reference book for people who want to know more about Taylor, but I don’t know if it would draw in those who are unfamiliar with him or those who do not like to read biographies.
Biographers do have it a little rough: they can write in a story form, which is more interesting but tends to be less accurate as the biographer has to invent conversations and situations to bring out the points he needs to; or they can right a factual version which can tend to be more encyclopedic and accurate, but which doesn’t appeal to the average modern reader. This one is in the style of the latter. I think it could have been much more condensed: there are many descriptions of various CIM missionaries’ travels which could have been left out or at least summarized. The book is 481 pages, not including indexes and end notes, and I have to admit I got bogged down in places.
But I do recommend the book. If you persevere, you will find great nuggets about Taylor’s character. He was not unflawed: he was very human and he would never have wanted people to think he was some super-Christian. But he loved and followed the Lord in an exemplary and humble way.
I marked way too many places to share, especially in a review that is long already:
But here are a few places that stood out to me:
His health, as he described it, could “not be called robust” (p. 49), but I hadn’t realized he struggled so much with his health through the years, including regular bouts of dysentery. as).
I had thought that the title of this book came from the hymn, “It is Not Death to Die,” originally written in 1832 and recently updated. But in writing of Hudson’s death, Cromarty cites the Banner of Truth 1977 publication of Pilgrim’s Progress, at the section where Mr. Valiant-For-Truth dies, and the line “It Is Not Death to Die” is in the passage he quotes but I have not found it in the online versions of Pilgrim’s Progress. Nevertheless, the sentiment is true. Dying to self and living for Christ, which Hudson Taylor exemplified, is true life, just as dying to this body makes way for heaven for those who have trusted Christ as Savior.
(For a more positive review that brings out some different things about Cromarty’s book and Taylor’s life, see my friend Debbie’s review here.)
I love it when a book is marked up like that! It means that you got a lot out of it! Thanks for sharing this great book!
Thanks for the review and recommendation. I have learned quite about Hudson Taylor from friends, but I have not yet read any biographies on his life. I will add it to my list. 🙂
I just finished a book about China (by a Christian journalist who used to work for NPR – how is that for unlikely?) and it whetted my appetite for all things Hudson Taylor. I was delighted to read your review. I think I’ll start with the earlier biographies, but this one is on my TBR list. Thank you!
Now there’s a rich review of a book! I appreciate your excerpts — they help me get a feel for the writing style and level of detail.
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| 304,810
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TITLE: Can path integrals be used to understand entanglement?
QUESTION [15 upvotes]: I like path integrals. I prefer to try to understand quantum phenomena in terms of path integrals rather than Hamiltonian mechanics. However, most of the standard texts on quantum mechanics start from Hamiltonian mechanics and mention the path integrals as a side note. This makes it difficult for me to gain insight into common phenomena (with a few exceptions like certain areas of optics) from the perspective of path integrals formulations. I am aware of some exceptions to this rule (books on path integrals in particular), but I have only seen entanglement explained from the perspective of eigenstates of a Hamiltonian.
I was wondering how entanglement would be interpreted if you started from a path integral formulation. I would be more than happy to hear from a quantum field theoretic perspective as well, but the usual quantum mechanical path integral would certainly satisfy my curiosity.
REPLY [3 votes]: Path integrals have been used to discuss entanglement of quantum fields. The path integral is used to trace out part of the system, so that the entanglement is given in terms of Von Neumann entropy, in terms of the trace of the partial density matrix.
You can find an example here for instance:
http://arxiv.org/abs/hep-th/9401125
| 207,338
|
TITLE: Is SO(2n+1)/U(n) a symmetric space?
QUESTION [15 upvotes]: I am a physics student with only a rudimentary knowledge of differential geometry, so please feel free to point out if I miss something elementary / trivial.
According to https://arxiv.org/abs/1408.2760, $ SO(2n+1)/U(n) $ is not a symmetric space because it does not have the right Cartan decomposition of the Lie algebra. That is, suppose that $ \mathfrak{g} $ is the Lie algebra of $ SO(2n+1)$ and $ \mathfrak{h} $ is the Lie algebra of $ U(n) $. There is a decomposition $$ \mathfrak{g} = \mathfrak{h} + \mathfrak{p} $$ for some $ \mathfrak{p} $ such that $ [\mathfrak{h},\mathfrak{h}] \subset \mathfrak{h}$ and $[\mathfrak{h},\mathfrak{p}] \subset \mathfrak{p}$. But $ [\mathfrak{p},\mathfrak{p}] $ is not in $ \mathfrak{h} $, so we do not have a Cartan involution on this space.
I'm wondering if it is that simple. I'll be grateful if someone can clear up my confusion below.
I think that $SO(2n+1)/U(n)$ and $SO(2n+2)/U(n+1)$ are diffeomorphic. For instance, this book shows that the two are the same homogeneous spaces by showing that
Any element $SO(2n+1)$ can be written as an ordered product of two elements, one in $SO(2n+1)$ and another in $U(n+1)$. They write this as $SO(2n+2)=SO(2n+1)\cdot U(n+1)$
The quotient $X = SO(2n+2)/U(n+1) = SO(2n+1)\cdot U(n+1) /
U(n+1)$ can be thought of as $ SO(2n+1)/U(n)$ because the
$SO(2n+1)$-action on $X$ is transitive and the stabilizer of the identity
$eU(n+1)$ of $X$ are the elements of $SO(2n+1)$ that are also in
$U(n+1)$: $SO(2n+1) \cap U(n+1) = U(n)$.
An example in low dimensions is $ SO(5)/U(2) = SO(6)/U(3) = \mathbb{C}P^3$, a complex projective space.
But $SO(2n+2)/U(n+1)$ is a symmetric space. So why is $SO(2n+1)/U(n)$ not?
REPLY [3 votes]: The homogeneous space $M=G/K=SO(2n+1)/U(n)$ is not a symmetric space but a generalized flag manifold of the compact simple Lie group $SO(2n+1)$.
It arises from the more general family
$$
M_{n, p}:=SO(2n+1)/(U(p) \times SO(2(n-p)+1) )
$$
for $p=n$.
Notice that for any $2\leq p\leq n$ the isotropy representation of $M_{n, p}$ decomposes intro two isotropy summands:
$${\frak m}\cong T_{o}M_{n, p}=\frak{m}_{1}\oplus\frak{m}_{2}.$$
This is true, since in this case you paint black in the Dynkin diagram of $G=SO(2n+1)$ a simple root of Dynkin mark equal to 2.
It is easy to see that $[\frak{m}, \frak{m}]\neq \frak{k}$, in particular one computes
$$
[\frak{m}_{1}, \frak{m}_{1}]\subset\frak{k}\oplus\frak{m}_{2}, \quad [\frak{m}_{1}, \frak{m}_{2}]\subset\frak{m}_{1},\quad [\frak{m}_{2}, \frak{m}_{2}]\subset\frak{k}
$$
For $p=1$ one gets the Hermitian symmetric space $SO(2n+1)/SO(2)\times SO(2n-1)$, which of course is isotropy irreducible (in this case, in the Dynkin diagram of $G=SO(2n+1)$ we paint black the first simple root, which has Dynkin mark equal to 1, recall that the highest root of $SO(2n+1)$ is given by $\tilde{\alpha}=\alpha_1+2\alpha_2+\cdots+2\alpha_{n}$ where $\{\alpha_1, \ldots, \alpha_{n}\}$ is a basis of simple roots. The coefficients $\{1, 2, \ldots, 2\}$ are the so called Dynkin marks).
In fact, the only generalized flag manifolds which are the same time symmetric spaces are the (compact) Hermitian symmetric spaces and these are the unique flag manifolds which are isotropy irreducible.
(see Which Kahler Manifolds are also Einstein Manifolds? for more details)
Finally notice that a smooth manifold can has more than one expressions as a homogeneous space, $M=G/K=G'/K'$.
For example
$$S^{6}=SO(7)/SO(6)=G_2/SU(3), \quad S^{7}=SO(8)/SO(7)=Sp(2)/Sp(1)=Spin(7)/G_2.$$
However, only the pairs $(SO(7), SO(6))$ and $(SO(8), SO(7))$ are symmetric pairs (and so passing to the double coverings you get simply-connected symmetric spaces). And similar in your example: the first complex projective space $$SO(5)/U(2)=ℂP^{3}_{{\frak m}_{1}\oplus{\frak m}_{2}}$$ is not a symmetric space but a flag manifold with two isotropy summands, but $SU(4)/U(3)=ℂP^{3}$ is a Hermitian symmetric space (irreducible).
| 151,947
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| 183,469
|
A Survey on Packaging Materials and Technologies for Commercial Food Products
1University of Parma
1University of Parma
1University of Parma
1University of Naples Federico II
Citation Information: International Journal of Food Engineering. Volume 7, Issue 1, Pages –, ISSN (Online) 1556-3758, DOI: 10.2202/1556-3758.1687, January 2011
- Published Online:
- 2011-01-11
This paper presents the outcomes of a comprehensive and detailed analysis of the Italian food market, targeting the Fast Moving Consumer Goods products and aimed at defining the existing relationships between product characteristics and packaging technologies. A sample of 175 products has been examined, and the corresponding characteristics and packaging technologies were derived from an in-field investigation performed in large and small Italian retailers. Subsequent statistical analyses were first focused on providing descriptive statistics concerning the packaging materials used for food products. Then, cluster analysis was exploited to group products on the basis of the material used for the primary packaging. Discriminant analysis was also adopted to identify product characteristics that significantly contribute to the choice of packaging material. Finally, based on the clusters obtained, contingency tables were used to explore possible relationships among packaging material, packaging techniques and relevant product characteristics. Results of this study could provide useful guidelines to food manufacturers to identify the most suitable packaging technology for new food products.
Keywords: packaging materials; packaging technologies; survey; Fast Moving Consumer Goods; food products
| 303,863
|
\begin{document}
\maketitle
\noindent{\bf Abstract:} In applied probability, the normal approximation is often used for the distribution of data with assumed additive structure. This tradition is based on the central limit theorem for sums of (independent) random variables. However, it is practically impossible to check the conditions providing the validity of the central limit theorem when the observed sample size is limited. Therefore it is very important to know what the real accuracy of the normal approximation is in the cases where it is used despite it is theoretically inapplicable. Moreover, in some situations related with computer simulation, if the distributions of separate summands in the sum belong to the domain of attraction of a stable law with characteristic exponent less than two, then the observed distance between the distribution of the normalized sum and the normal law first decreases as the number of summands grows and begins to increase only when the number of summands becomes large enough. In the present paper an attempt is undertaken to give some theoretical explanation to this effect.
\smallskip
\noindent{\bf Key words:} central limit theorem; accuracy of the normal approximation; heavy tails; uniform distance; stable distribution
\smallskip
\noindent{\bf AMS 2000 subject classification:} 60F05, 60G50, 60G55, 62E20, 62G30
\section{Introduction}
In applied studies, the normal approximation is often used for the distribution of data with (at least assumed) additive structure. This tradition is based on the central limit theorem of probability theory which states that the distributions of sums of (independent) random variables satisfying certain conditions (say, the Lindeberg condition) converge to the normal law as the number of summands infinitely increases. However, it is practically impossible to check the conditions providing the validity of the central limit theorem when the observed sample size is limited. In particular, with moderate sample size, the histogram constructed from the sample from the Cauchy distribution whose tails are so heavy that even the mathematical expectation does not exist, is practically visually indistinguishable from the normal (Gaussian) density. Therefore it is very important to know what the real accuracy of the normal approximation is in the cases where it is used despite it is theoretically inapplicable. Moreover, in some situations related with computer simulation, if the distributions of separate summands in the sum belong to the domain of attraction of a stable law with characteristic exponent less than two, then the observed distance between the distribution of the normalized sum and the normal law first decreases as the number of summands grows and begins to increase only when the number of summands becomes large enough. In the present paper an attempt is undertaken to give some theoretical explanation to this effect. In Section 2 we introduce the notation, give necessary definitions and formulate some auxiliary results. In Section 3 the theorem is proved presenting the upper bound for the accuracy of the invalid normal approximation. In Section 4 the problem of evaluation of the threshold number of summands providing best possible accuracy of the invalid normal approximation is considered.
\section{Notation, definitions and auxiliary results}
Throughout the paper we assume that all the random variables are defined on the same probability space $(\Omega,\mathfrak{F},{\sf P})$. The mathematical expectation and variance with respect to the probability measure ${\sf P}$ will be denoted ${\sf E}$ and ${\sf D}$, respectively. The symbol $\eqd$ means the coincidence of distributions.
For $n\in\mathbb{N}$, let $X_1,\ldots,X_n$ be a homogeneous sample, that is, a set of independent identically distributed random variables with common distribution function $F(x)={\sf P}(X_1<x)$, $x\in\R$. For simplicity, without serious loss of generality we will assume that $F(x)$ is continuous.
We will follow the lines of approach described in \cite{Korolev2020}. Denote $S_n=X_1+\ldots+X_n$. The indicator of a set (event) $A\in\mathfrak{F}$ will be denoted $\I_A=\I_A(\omega)$, $\omega\in\Omega$:
$$
\I_A(\omega)=\begin{cases}1, & \omega\in A,\vspace{1mm}\cr 0, &\omega\notin A.\end{cases}
$$
Consider $u>0$ such that $0<F(u)<1$. It is obvious that $X_j=X_j\I_{\{|X_j|\le u\}}+X_j\I_{\{|X_j|> u\}}$. Then
$$
S_n=\sum\nolimits_{j=1}^n X_j\I_{\{|X_j|\le u\}}+\sum\nolimits_{j=1}^n X_j\I_{\{|X_j|> u\}}\equiv
S_n^{(\le u)}+S_n^{(> u)}.
$$
The number $N_n(u)$ of non-zero summands in the sum $S_n^{(\le u)}$ is a random variable that has the binomial distribution with parameters $n$ (``number of trials'') and $p=p(u)={\sf P}(|X_1|\le u)=F(u)-F(-u)$ (probability of ``success''). Note that, as $u$ infinitely grows, the parameter $p$ tends to 1. So, for $x\in\R$ we can write
$$
{\sf P}(S_n^{(\le u)}<x)\eqd {\sf P}\Big(\sum\nolimits_{j=0}^{N_n(u)}X_j^{(\le
u)}<x\Big),\eqno(1)
$$
where the random variables $X_1^{(\le u)},X_2^{(\le u)},\ldots$ are independent and have one and the same distribution function
$$
F^{(\le u)}(x)\equiv {\sf P}(X_1^{(\le u)}<x)={\sf
P}\big(X_1\I_{\{|X_1|\le u\}}<x\big|\,|X_1|\le u\big)=
$$
$$
=\frac{{\sf P}(X_1<x;\,|X_1|\le u)}{{\sf P}(|X_1|\le u)}=\begin{cases}1, &
x> u;\vspace{1mm}\\{\displaystyle\frac{F(x)-F(-u)}{F(u)-F(-u)}}, & |x|\le u; \vspace{1mm}\\0, & x<-u.\end{cases}\eqno(2)
$$
Moreover, the random variable $N_n(u)$ can be assumed to be independent of the sequence $X_1^{(\le u)},X_2^{(\le u)},\ldots$ For definiteness, if $N_n(u)=0$, then the sum $S_n^{(\le u)}$ is set equal to zero.
Similarly, for $x\in\R$ we have
$$
{\sf P}(S_n^{(>u)}<x)\eqd {\sf P}\Big(\sum\nolimits_{j=0}^{n-N_n(u)}X_j^{(>u)}<x\Big),\eqno(3)
$$
where $N_n(u)$ is {\it the same as in} (1) and is independent of the independent random variables $X_1^{(>u)},X_2^{(>u)},\ldots$ that have one and the same distribution function
$$
F^{(>u)}(x)\equiv {\sf P}(X_1^{(>u)}<x)={\sf P}\big(X_1\I_{\{|X_1|>u\}}<x\big|\,|X_1|>u\big)=
$$
$$
=\frac{{\sf P}(X_1<x;\,|X_1|> u)}{{\sf P}(|X_1|> u)}=\begin{cases}{\displaystyle\frac{F(x)}{F(-u)+1-F(u)}}, &
x<-u;\vspace{2mm}\\{\displaystyle\frac{F(-u)}{F(-u)+1-F(u)}}, & |x|\le u;\vspace{2mm}\\{\displaystyle\frac{F(-u)+F(x)-F(u)}{F(-u)+1-F(u)}}, & x>u.
\end{cases}\eqno(4)
$$
For definiteness, if $N_n(u)=n$, then the sum $S_n^{(>u)}$ is set equal to zero. Moreover, in (1) and (3) the random variables $X_1^{(\le u)},X_2^{(\le u)},\ldots,X_1^{(>u)},X_2^{(>u)},\ldots$ can be assumed to be jointly independent while the random variables $S_n^{(\le u)}$ and $S_n^{(> u)}$ are {\it not} independent and are related by the random variable $N_n(u)$.
\smallskip
{\sc Lemma 1.} {\it Let $A\in\mathfrak{F}$, $B\in\mathfrak{F}$. Then ${\sf P}(AB)\ge{\sf P}(A)-{\sf P}(\overline{B})$.}
\smallskip
The {\sc proof} is elementary.
\smallskip
The uniform (Kolmogorov) distance between the distribution functions $F_{\xi}$ and $F_{\eta}$ of random variables $\xi$ and $\eta$ will be denoted $\rho(F_{\xi},\,F_{\eta})$, $\rho(F_{\xi},\,F_{\eta})=\sup_x|F_{\xi}(x)-F_{\eta}(x)|$. The normal distribution function with expectation $a\in\R$ and variance $\sigma^2>0$ will be denoted $\Phi_{a,\sigma}(x)$,
$$
\Phi_{a,\sigma}(x)=\frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{x}\exp\Big\{-\frac{(z-a)^2}{2\sigma^2}\Big\}dz=\Phi_{0,1}\Big(\frac{x-a}{\sigma}\Big)=
\Phi_{0,\sigma}(x-a),\ \ \ x\in\R.
$$
\smallskip
{\sc Lemma 2.} {\it For any $a\in\R$, $\sigma>0$, $b\in\R$}
$$
\rho(\Phi_{a+b,\,\sigma},\,\Phi_{a,\,\sigma})= 2\Phi_{0,\sigma}\big({\textstyle\frac{|b|}{2}}\big)-1.
$$
\smallskip
{\sc Proof.} First, note that if $H(x)$ and $G(x)$ are two differentiable distribution functions, then $\rho(H,G)$ is realized (the supremum in $\sup_x|H(x)-G(x)|$ is attained) at one of the points $x$ where $F'(x)=G'(x)$. Indeed, we have
$$
\rho(H,G)=\sup_x|H(x)-G(x)|=\max\big\{\max_x\big[H(x)-G(x)\big], \max_x\big[G(x)-H(x)\big]\big\},
$$
and the extremum of each of the expressions in braces on the right-hand side is attained at the point where the derivative of the corresponding expression is equal to zero, which is equivalent to the equality of the derivatives of the distribution functions $H$ and $G$, that is, to the equality of the corresponding densities. In the case under consideration the latter condition is equivalent to that
$$
\frac{1}{\sigma\sqrt{2\pi}}\exp\Big\{-\frac12\Big(\frac{x-a-b}{\sigma}\Big)^2\Big\}=
\frac{1}{\sigma\sqrt{2\pi}}\exp\Big\{-\frac12\Big(\frac{x-a}{\sigma}\Big)^2\Big\},
$$
or $\big(x-(a+b)\big)^2=(x-a)^2$. Solving this equation we obtain $x-a=\frac{b}{2}$ yielding the desired result with the account of the relation $\Phi_{0,\sigma}(-|b|)=1-\Phi_{0,\sigma}(|b|)$.
\smallskip
Using the Lagrange formula, it is easy to deduce from Lemma 2 that
$$
\rho(\Phi_{a+b,\,\sigma},\,\Phi_{a,\,\sigma})\le \frac{|b|}{\sigma\sqrt{2\pi}}
$$
(see, e. g., inequality (3.4) in \cite{Petrov1972}).
\smallskip
{\sc Lemma 3}. {\it For $n\in\mathbb{N}$ let $\xi_1,\ldots,\xi_n$ be random variables, $a_1,\ldots,a_n$ be positive numbers such that
$a_1+\ldots+a_n=1$. Then for any $x>0$
$$
{\sf P}\Big(\Big|\sum\nolimits_{j=1}^n\xi_j\Big|\ge x\Big)\le\sum\nolimits_{j=1}^n{\sf P}(|\xi_j|\ge a_jx).
$$
If, in addition, the random variables $\xi_1,\ldots,\xi_n$ are identically distributed, then
$$
{\sf P}\Big(\Big|\sum\nolimits_{j=1}^n\xi_j\Big|\ge x\Big)\le n{\sf P}\big(|\xi_1|\ge {\textstyle\frac{x}{n}}\big).
$$
}
\smallskip
{\sc Proof}. First, note that
$$
{\sf P}\Big(\Big|\sum\nolimits_{j=1}^n\xi_j\Big|\ge x\Big)\le{\sf P}\Big(\sum\nolimits_{j=1}^n|\xi_j|\ge x\Big).
$$
Next, from geometrical considerations it follows that
$$
\Big\{\omega:\, \sum\nolimits_{j=1}^n|\xi_j(\omega)|\ge x\Big\}\subseteq\big\{\omega:\,|\xi_1(\omega)|\ge a_1x\big\}\bigcup
\Big\{\omega:\, \sum\nolimits_{j=2}^n|\xi_j(\omega)|\ge (1-a_1)x\Big\}\subseteq
$$
$$
\subseteq\big\{\omega:\,|\xi_1(\omega)|\ge a_1x\big\}\bigcup\big\{\omega:\,|\xi_2(\omega)|\ge a_2x\big\}\bigcup
\Big\{\omega:\, \sum\nolimits_{j=3}^n|\xi_j(\omega)|\ge (1-a_1-a_2)x\Big\}\subseteq\ldots
$$
$$
\ldots\subseteq\bigcup_{j=1}\nolimits^n\big\{\omega:\,|\xi_j(\omega)|\ge a_jx\big\}.
$$
Therefore,
$$
{\sf P}\Big(\Big|\sum\nolimits_{j=1}^n\xi_j\Big|\ge x\Big)\le{\sf P}\Big(\sum\nolimits_{j=1}^n|\xi_j|\ge x\Big)\le
$$
$$
\le{\sf P}\Big(\bigcup\nolimits_{j=1}^n\big\{\omega:\,|\xi_j(\omega)|\ge a_jx\big\}\Big)\le\sum\nolimits_{j=1}^n{\sf P}(|\xi_j|\ge a_jx).
$$
The lemma is proved.
\smallskip
{\sc Lemma 4}. {\it For $n\in\mathbb{N}$ let $\xi_1,\ldots,\xi_n$ be random variables such that ${\sf E}|\xi_j|^{\delta}<\infty$ for some $\delta>0$, $j=1,\ldots,n$. Denote $\theta_n=\xi_1+\ldots+\xi_n$.
\noindent {\rm(i)} If $0<\delta\le1$, then
$$
{\sf E}|\theta_n|^{\delta}\le \sum\nolimits_{j=1}^n{\sf E}|\xi_j|^{\delta}.
$$
\noindent {\rm (ii)} If $1\le\delta\le2$, the random variables $\xi_1,\ldots,\xi_n$ are independent and ${\sf E}\xi_j=0$, $j=1,\ldots,n$, then
$$
{\sf E}|\theta_n|^{\delta}\le \Big(2-\frac1n\Big)\sum\nolimits_{j=1}^n{\sf E}|\xi_j|^{\delta}.
$$
}
\smallskip
{\sc Proof}. Statement (i) is elementary, statement (ii) was proved in \cite{BahrEsseen1965}.
\section{Main results}
Consider the upper bound for the uniform distance between the distribution of the normalized sum
$$
S_n^*=\frac{1}{\sqrt{n}}\sum\nolimits_{j=1}^nX_j
$$
and the normal law with some expectation $a\in\R$ and variance $\sigma^2>0$. The choice of concrete values of $a$ and $\sigma^2$ will be discussed later.
From what has been said it follows that
$$
S_n^*\eqd\frac{S_n^{(\le u)}}{\sqrt{n}}+\frac{S_n^{(> u)}}{\sqrt{n}}.
$$
For brevity and convenience, we will use the notation
$$
\zeta_n=\frac{S_n^{(\le u)}}{\sqrt{n}},\ \ \ \eta_n=\frac{S_n^{(> u)}}{\sqrt{n}}.
$$
\smallskip
{\sc Theorem 1.} {\it Let $u>0$ be arbitrary. Then for any $a\in\R$ and $\sigma>0$ we have}
$$
\rho(F_{\zeta_n+\eta_n},\,\Phi_{a,\,\sigma})\le\rho(F_{\zeta_n},\,\Phi_{a,\,\sigma})+n\big(F(-u)+1-F(u)\big).\eqno(5)
$$
\smallskip
{\sc Proof}. Let $\epsilon>0$ be arbitrary. According to Lemma 1 we have
$$
{\sf P}(\zeta_n+\eta_n<x)={\sf P}(\zeta_n+\eta_n<x;\,|\eta_n|\le\epsilon)+{\sf P}(\zeta_n+\eta_n<x;\,|\eta_n|>\epsilon)\ge
$$
$$
\ge{\sf P}(\zeta_n<x-\eta_n;\,|\eta_n|\le\epsilon)\ge{\sf P}(\zeta_n<x-\epsilon;\,|\eta_n|\le\epsilon)\ge{\sf P}(\zeta_n<x-\epsilon)-{\sf P}(|\eta_n|\ge\epsilon).\eqno(6)
$$
On the other hand, obviously,
$$
{\sf P}(\zeta_n+\eta_n<x)={\sf P}(\zeta_n<x-\eta_n;\,|\eta_n|\le\epsilon)+{\sf P}(\zeta_n+\eta_n<x;\,|\eta_n|>\epsilon)\le
$$
$$
\le{\sf P}(\zeta_n<x+\epsilon;\,|\eta_n|\le\epsilon)+{\sf P}(\zeta_n+\eta_n<x;\,|\eta_n|>\epsilon)\le{\sf P}(\zeta_n<x+\epsilon)+{\sf P}(|\eta_n|>\epsilon).\eqno(7)
$$
It is easy to see that
$$
|{\sf P}(\zeta_n+\eta_n<x)-\Phi_{a,\,\sigma}(x)|=\max\big\{{\sf P}(\zeta_n+\eta_n<x)-\Phi_{a,\,\sigma}(x),\,\Phi_{a,\,\sigma}(x)-{\sf P}(\zeta_n+\eta_n<x)\big\}.\eqno(8)
$$
Using (7) and Lemma 2 we obtain
$$
{\sf P}(\zeta_n+\eta_n<x)-\Phi_{a,\,\sigma}(x)\le {\sf P}(|\eta_n|>\epsilon)+\big[{\sf P}(\zeta_n<x+\epsilon)-\Phi_{a,\,\sigma}(x+\epsilon)\big]+
$$
$$
+\big[\Phi_{a,\,\sigma}(x+\epsilon)-\Phi_{a,\,\sigma}(x)\big]\le{\sf P}(|\eta_n|>\epsilon)+\rho(F_{\zeta_n},\,\Phi_{a,\,\sigma})+
\big[2\Phi_{0,\sigma}\big({\textstyle\frac{\epsilon}{2}}\big)-1\big].\eqno(9)
$$
Using (6) and Lemma 2 we obtain
$$
\Phi_{a,\,\sigma}(x)-{\sf P}(\zeta_n+\eta_n<x)\le \Phi_{a,\,\sigma}(x)-{\sf P}(\zeta_n<x-\epsilon)+{\sf P}(|\eta_n|>\epsilon)=
$$
$$
=\big[\Phi_{a,\,\sigma}(x)-\Phi_{a,\,\sigma}(x-\epsilon)\big]-\big[{\sf P}(\zeta_n<x-\epsilon)-\Phi_{a,\,\sigma}(x-\epsilon)\big]+ {\sf P}(|\eta_n|>\epsilon) \le
$$
$$
\le\rho(F_{\zeta_n},\,\Phi_{a,\,\sigma})+{\sf P}(|\eta_n|>\epsilon)+\big[2\Phi_{0,\sigma}\big({\textstyle\frac{\epsilon}{2}}\big)-1\big].\eqno(10)
$$
Substituting (9) and (10) in (8) we obtain
$$
\rho(F_{\zeta_n+\eta_n},\,\Phi_{a,\,\sigma})\le\rho(F_{\zeta_n},\,\Phi_{a,\,\sigma})+\big[2\Phi_{0,\sigma}\big({\textstyle\frac{\epsilon}{2}}\big)-1\big]+
{\sf P}(|\eta_n|>\epsilon).\eqno(11)
$$
To estimate the last term on the right-hand side of (11) use representation (3) for $S_n^{(>u)}$. With the account of the convention $\sum_{j=0}^0=0$, by the formula of total probability we have
$$
{\sf P}(|\eta_n|>\epsilon)={\sf P}\Big(\Big|\sum\nolimits_{j=1}^{n-N_n(u)}X_j^{(>u)}\Big|>\epsilon\sqrt{n}\Big)=
$$
$$
=\sum_{k=1}^nC_n^k\big(F(-u)+1-F(u)\big)^k
\big(F(u)-F(-u)\big)^{n-k}{\sf P}\Big(\Big|\sum\nolimits_{j=1}^kX_j^{(>u)}\Big|>\epsilon\sqrt{n}\Big).\eqno(13)
$$
Estimating the probability in the last expression by Lemma 3, continuing (13) we obtain
$$
{\sf P}(|\eta_n|>\epsilon)\le\sum_{k=1}^nC_n^k\big(F(-u)+1-F(u)\big)^k
\big(F(u)-F(-u)\big)^{n-k}k{\sf P}\big(|X_j^{(>u)}|>{\textstyle\frac{\epsilon\sqrt{n}}{k}}\big)\le
$$
$$
\le{\sf P}\big(|X_j^{(>u)}|>{\textstyle\frac{\epsilon}{\sqrt{n}}}\big)\cdot\sum_{k=1}^nC_n^k\big(F(-u)+1-F(u)\big)^k
\big(F(u)-F(-u)\big)^{n-k}k=
$$
$$
=n\big(F(-u)+1-F(u)\big){\sf P}\big(|X_j^{(>u)}|>{\textstyle\frac{\epsilon}{\sqrt{n}}}\big).
$$
Substitution of this bound in (11) yields
$$
\rho(F_{\zeta_n+\eta_n},\,\Phi_{a,\,\sigma})\le\rho(F_{\zeta_n},\,\Phi_{a,\,\sigma})+\big[2\Phi_{0,\sigma}\big({\textstyle\frac{\epsilon}{2}}\big)-1\big]+
n\big(F(-u)+1-F(u)\big){\sf P}\big(|X_j^{(>u)}|>{\textstyle\frac{\epsilon}{\sqrt{n}}}\big).\eqno(14)
$$
Now let $\epsilon\to 0$ in (14) and obtain the desired result. The theorem is proved.
\smallskip
In practice, the values of the parameters $a$ and $\sigma$ can be chosen by the following reasoning. It is easy to verify (say, by the consideration of characteristic functions) that
$$
S_n^{(\le u)}\eqd\sum\nolimits_{j=1}^n\widetilde{X}_j^{(\le u)},
$$
where $\widetilde{X}_1^{(\le u)},\ldots,\widetilde{X}_n^{(\le u)}$ are independent identically distributed random variables,
$$
\widetilde{X}_j^{(\le u)}=\begin{cases}X_j^{(\le u)} & \text{ with probability } F(u)-F(-u);\vspace{2mm}\\ 0 & \text{ with probability } F(-u)+1-F(u).\end{cases}
$$
Then in accordance with (3), the parameter $a$ can be defined as
$$
a=a(u)={\sf E}\widetilde{X}_1^{(\le u)}=[F(u)-F(-u)]{\sf E}X_1^{(\le u)},
$$
and the parameter $\sigma^2$ can be defined as
$$
\sigma^2=\sigma^2(u)={\sf D}\widetilde{X}_1^{(\le u)}=[F(u)-F(-u)]{\sf D}X_1^{(\le u)}+[F(-u)+1-F(u)]\big({\sf E}X_1^{(\le u)}\big)^2.
$$
With these values of $a$ and $\sigma$ the first term on the right-hand side of (5) will tend to zero by the central limit theorem as $n\to\infty$, and can be estimated by the standard techniques, say, by the Berry--Esseen inequality for binomial random sums, see \cite{KorolevDorofeeva2017, KorolevShevtsova2012}.
As regards the second term on the right-hand side of (5), with large $u$, $p=F(u)-F(-u)$ close to one and moderate (but large enough) $n$ the term $\eta_n$ may be small due to that the sum $S_n^{(>u)}$ contains very few summands. Moreover, in the case of light tails, putting $u=u_n$ so that $n[F(-u)+1-F(u_n)]\to 0$ as $n\to\infty$, it is possible to make sure that the right-hand side of (5) can be made arbitrarily small by the choice of arbitrarily large $n$ so that the limit distribution for the normalized sum $S_n^*$ will be normal.
Under some additional conditions, at the expense of introducing additional parameter, the dependence of the second term of the bound given in Theorem 1 on $n$ can be made better.
\smallskip
For $c\in(0,2]$ let $h(c)=\I_{(1,2]}(c)$.
\smallskip
{\sc Theorem 2}. {\it Assume that the distribution function $F(x)$ belongs to the domain of attraction of a stable law with characteristic exponent $\alpha\in(0,2)$. If, moreover, $\alpha\ge1$, then additionally assume that $F$ is symmetric $($that is, $F(-x)=1-F(x)$ for $x>0$$)$. Then for any $u>0$ $\epsilon>0$ and $\delta\in(0,\alpha)$ we have}
$$
\rho(F_{\zeta_n+\eta_n},\,\Phi_{a,\,\sigma})\le\rho(F_{\zeta_n},\,\Phi_{a,\,\sigma})+\big[2\Phi_{0,\sigma}\big({\textstyle\frac{\epsilon}{2}}\big)-1\big]+
2^{h(\delta)}\epsilon^{-\delta}n^{1-\delta/2}\big(F(-u)+1-F(u)\big){\sf E}\big|X_1^{(>u)}\big|^{\delta}.\eqno(15)
$$
\smallskip
{\sc Proof}. The starting point of the proof is inequality (11). In accordance with the result of \cite{Tucker1975}, in the case under consideration ${\sf E}|X_1|^{\delta}<\infty$ for any $\delta\in(0,\alpha)$, and hence, ${\sf E}|X_1^{(>u)}|^{\delta}<\infty$. Moreover, if $\alpha>1$, then the mathematical expectation of $X_1$ exists and, due to the assumption that in that case the distribution of $X_1$ is symmetric, ${\sf E}X_1=0$. Therefore by representation (3), the Markov inequality, and Lemma 4, continuing (13) we obtain
$$
{\sf P}(|\eta_n|>\epsilon)\le\frac{1}{\epsilon^{\delta}n^{\delta/2}}\sum\nolimits_{k=0}^{n}C_n^k\big(F(-u)+1-F(u)\big)^k\big(F(u)-F(-u)\big)^{n-k}
{\sf E}\Big|\sum\nolimits_{j=0}^kX_j^{(>u)}\Big|^{\delta}\le
$$
$$
\le\frac{2^{h(\delta)}{\sf E}\big|X_1^{(>u)}\big|^{\delta}}{\epsilon^{\delta}n^{\delta/2}}\sum\nolimits_{k=0}^nC_n^kk\big(F(-u)+1-F(u)\big)^k\big(F(u)-F(-u)\big)^{n-k}=
$$
$$
=2^{h(\delta)}\epsilon^{-\delta}n^{1-\delta/2}\big(F(-u)+1-F(u)\big){\sf E}\big|X_1^{(>u)}\big|^{\delta}.\eqno(16)
$$
The theorem is proved.
\smallskip
We see that in (15) the exponent of $n$ is less than that in (5). However, in (15) an additional parameter $\epsilon$ appeared. The second term on the right-hand side of (15) can be made arbitrarily small by the appropriate choice of $\epsilon$. With $n$ and $\epsilon$ fixed, the third term on the right-hand side of (15) can be made arbitrarily small by the choice of large enough $u$.
Actually Theorems 1 and 2 are simple variants of a so-called pre-limit theorem, see \cite{Klebanov1999}.
As an illustration of how Theorem 2 acts, consider the following example.
\smallskip
{\sc Example}. Assume that the random variables $X_1,X_2,\ldots$ have common probability density
$$
f(x)=\frac{3}{4(1+|x|)^{5/2}},\ \ \ x\in\R.
$$
The corresponding distribution function has the form
$$
F(x)=\begin{cases}{\displaystyle\frac{1}{2(|x|+1)^{3/2}}}, & x<0;\vspace{2mm}\\
{\displaystyle 1-\frac{1}{2(x+1)^{3/2}}}, & x\ge0.\end{cases}
$$
This distribution function belongs to the domain of attraction of a stable law with characteristic exponent $\frac32$. It is easy to make sure that in this case, according to (4), we have
$$
{\sf E}|X_1^{(>u)}|=2\int_{0}^{\infty}xdF^{(>u)}(x)={\textstyle\frac32}(u+1)^{3/2}\int_u^{\infty}\frac{xdx}{(1+x)^{5/2}}=3(u+1)-1.
$$
Hence, choosing $\delta=1$ we see that the right-hand side of (16) is
$$
\frac{\sqrt{n}(1-p){\sf E}|X_1^{(>u)}|}{\epsilon}=\frac{\sqrt{n}(3u-2)}{2\epsilon(u+1)^{3/2}}=\frac{\sqrt{n}}{2\epsilon}\Big[\frac{3}{\sqrt{u+1}}+
o\Big(\frac{1}{\sqrt{u}}\Big)\Big]
$$
as $u\to\infty$. Therefore, with fixed $\epsilon>0$ and $n$ by choosing $u$ large enough the third summand on the right-hand side of (15) can be made arbitrarily small.
\section{On the threshold value of the number of summands}
Consider the problem of determination of $n_0$ such that for $n$ growing from 1 to $n_0$ the distance $\rho(F_{\zeta_n+\eta_n},\,\Phi_{a,\sigma})$ decreases and for $n>n_0$ this distance increases. Assume that the first summand on the right-hand side of (5) is estimated by the Berry--Esseen inequality
$$
\rho(F_{\zeta_n},\,\Phi_{a,\sigma})\le \frac{C_0\widetilde{L}_3^{(\le u)}}{\sqrt{n}},
$$
where $\widetilde{L}_3^{(\le u)}$ is the Lyapunov fraction,
$$
\widetilde{L}_3^{(\le u)}=\frac{{\sf E}\big|\widetilde{X}_1^{(\le u)}-{\sf E}\widetilde{X}_1^{(\le u)}\big|^3}{\big({\sf D}\widetilde{X}_1^{(\le u)}\big)^{3/2}},
$$
$C_0>0$ is the absolute constant, $C_0\le 0.4690$ \cite{Shevtsova2014}. It is easy to verify that if $c>0$, $d>0$, then
$$
\mathrm{arg}\min_{z>0}\Big(\frac{c}{\sqrt{z}}+dz\Big)=\Big(\frac{c}{2d}\Big)^{2/3}.
$$
Putting $z=n$, $c=C_0\widetilde{L}_3^{(\le u)}$, $d=F(-u)+1-F(u)$, we see that the minimum of the upper bound for $\rho(F_{\zeta_n+\eta_n},\,\Phi_{a,\sigma})$ is attained at $n_0$ which is either the integer part of
$$
z_0=\bigg(\frac{C_0\widetilde{L}_3^{(\le u)}}{2\big(F(-u)+1-F(u)\big)}\bigg)^{2/3},
$$
or at $n_0+1$.
Now assume that conditions of Theorem 2 hold. Then we have
$$
\mathrm{arg}\min_{z>0}\Big(\frac{c}{\sqrt{z}}+dz^{1-\delta/2}\Big)=\bigg(\frac{c}{2d(1-\frac{\delta}{2})}\bigg)^{\frac{2}{3-\delta}}
$$
so that
$$
z_0=\bigg(\frac{C_0\widetilde{L}_3^{(\le u)}}{2(1-\frac{\delta}{2})\big(F(-u)+1-F(u)\big)}\bigg)^{\frac{2}{3-\delta}},
$$
\renewcommand{\refname}{References}
| 9,254
|
TITLE: Complex analysis computation of $Im(f(z)-f(z_0))$
QUESTION [2 upvotes]: I have a computation that I wanted to carry out and know the answer. My issue is that when I did the computation I did not arrive at the answer that I know to be true. I would really appreciate a fresh set of eyes to comment on my work.
For $f(z)= kz- \sqrt{z}$ we want to compute Im$[f(z)-f\left(\frac{1}{4k^2}\right)]=0$, where $k$ is a real number. What we should obtain is a parabola $x= \frac{1}{4k^2}-k^2y^2$.
My computations
Consider that Im$[f(z)-f\left(\frac{1}{4k^2}\right)]=0$ is the same as Im$[f(z)]=$Im$[f\left(\frac{1}{4k^2}\right)]$.
Then I compute
$f(\frac{1}{4k^2})=k \frac{1}{4k^2} - \sqrt{\frac{1}{4k^2}} =\frac{1}{4k}- \frac{1}{2k}= -\frac{1}{4k}$
Where we know that this is real thus we know the Im$[f\left(\frac{1}{4k^2}\right)]=0$
To compute Im$[f(z)]=0$ we let $z= x+iy$ then we observe
Im $[ k(x+iy) -\sqrt{x+iy}]=0$
Which is equivalent to Im$[k(x+iy)]=$Im$[\sqrt{x+iy}]$. Then to deal with the sqrt I use the exponential definition of complex numbers and see that $r= \sqrt{x^2+y^2}$ and $\theta$ be the corresponding angle. Thus Im$[k(x+iy)]=$Im$[\sqrt{x+iy}]$ is the same as $ky=$Im$[r^{1/2} e^{\theta/2}]$.
Then using Eulers identity to make $e^{\theta/2} =\cos \theta/2+i \sin \theta/2$.
Then we get that
$$ky= r^{1/2} \cos\theta/2.$$
Squaring both sides we obtain
$$k^2y^2= r \cos^2 \theta/2.$$
Using trig we get
$$k^2y^2=\frac{1}{2} r(1+ \cos \theta).$$
Now using that $\cos \theta = \frac{x}{r}$ we get
$$k^2y^2 = \frac{1}{2}(r+ x).$$
We can turn $r$ into $x$ and $y$,
$$k^2y^2= \frac{1}{2}(\sqrt{y^2+x^2} +x).$$
However that is not the answer that is given to be true, i.e. $x= \frac{1}{4k^2}-k^2y^2$. I would really appreciate comments and solutions.
REPLY [2 votes]: A mistake I see is that the imaginary part of $r^{1/2}e^{\theta/2}$ is $r^{1/2}\sin\theta/2$ (and not cosine). The change is minor, though: your last equality changes to
$$
k^2y^2= \frac{1}{2}(\sqrt{y^2+x^2} -x).
$$
So
$$
\sqrt{x^2+y^2}=x+2k^2y^2.
$$
Squaring,
$$
x^2+y^2=x^2+4xk^2y^2+4k^4y^4,
$$
so, after subtracting $x^2$ and dividing by $y^2$,
$$
1=4xk^2+4k^4y^2.
$$
Solving for $x$,
$$
x=\frac1{4k^2}-k^2y^2.
$$
| 120,080
|
Well, it’s been a few weeks since my last entry here, which normally indicates a lack of activity on my part. In this case though, nothing could be further from the truth. Firstly there was a couple of trips to the wall that were, well, uneventful. And then there was a four day trip to the Montserrat Massif with the chaps at climbcatalunya last weekend, which turned out to be one of the greatest things I’ve ever done, culminating in a multi-pitch to the top of one of the pinnacles that make up the startlingly dramatic landscape of the summit of the massif. But I’m getting ahead of myself.
The trip started pretty badly actually, with a long wait on the runway at Stansted due to fog in Barcelona. There were meant to be four of us climbing for the weekend - however, at that moment Sol & I were sitting at Stansted, while Eddie and Champ were sitting at Luton for exactly the same reason. Eventually Barcelona air traffic control declared that they could see well enough, and we got going, arriving there a couple of hours later to be met by our guides for the weekend, Gee and Carole. We all piled into Gee’s Landy, and headed straight from the airport up to the massif. The landscape is like nothing I’ve ever seen before. And rather than trying (and failing,no doubt) to describe it, let’s see if I can find a photo. Ah - yup, here we go.
(Image copyright someone else. No idea who. Hope they don’t mind me linking to it, but as I’m the only person ever likely to read this, I don’t suppose they’ll notice the extra bandwidth). Well, as you can guess, we were all pretty excited to see something like this, and couldn’t wait to get started. Important things first though, we stopped at the cafe and had coffee and a sandwich before heading up the funicular railway, and a short walk in to the base of six or seven climbs, all about 25 metres. Now, given that the only place I’ve ever clipped a bolt before now is at the wall at Hatfield, I had two immediate concerns. Firstly, where were the big plastic juggy holds? Secondly, why were the bolts four or five metres apart? I had about twenty minutes or so to ponder on this, as Sol decided to lead the first climb. This sort of technical fidgety climbing is right up his street, and he fairly whistled up the climb before tying off at the top and abbing back down with an enormous smile on his face. So… My turn. My first outdoor sports lead, on an unfamiliar rock type best describe as ‘entertainingly bolted’. I needn’t have worried. It took a few metres to get used to the technique for moving up on this rock (just look for any kind of feature, no matter how small, jam your toe on/into it and stand up. Don’t bother looking for handholds - there aren’t any. If you find a handhold, it’s usually cause for quite a celebration) but move up I did. The runout between the bolts really didn’t bother me actually, as I was enjoying the climb so much and concentrating on the movements. I made it to the top, clipped into the lower-off with a handy quickdraw, and turned around to look at the view before rethreading. Wow. Really wow*10^6 - now I’d cleared the trees at the base of the climb I could see clearly, and the view was stunning. Anyhow. Acutely aware that I was hanging on a single piece of protection, I decided that rethreading was a good idea, and then it hit me that my actions in the next minute or so would have a distinct effect on the length and excitement of the rest of my life… I checked the knot four times before shouting to Sol to take up the slack in the rope while I unclipped the quickdraw. I was back on the ground twenty seconds later, in complete control, with the biggest smile on my face. The rest of the afternoon was spent at the same face, trying different routes, and chilling out enjoying ourselves. I probably made about three climbs, before attempting one last one but dropping off half way through ‘cos I was hungry, and tired having been up since 4am in order to catch the plane. For tea, Gee and Carole took us to a local restaurant where we had cold beer, tapas, squid and chips. Perfect…
Next day dawned bright and beautiful, with some stunning views from our refuge down to the mist in the valley below. I’ve got some photos somewhere which I’ll upload when I get an hour spare. Anyhow, a long day on the south face of the massif beckoned, so we breakfasted, drank coffee, and all bundled into the car. The walk-in was a bit more strenuous this time, with a bit of ‘bushwhacking’ as Gee described it to get to the base of the first face. Gee scuttled off up a 6a+ climb to put a rope up for us, Sol led up a very technical 5+, while I settled for the easy option and led up a nicely situated 4+ just around the corner. In fact, I liked it so much I pulled the rope down, and led up the route immediately next to it as well. The sun was up, and I was feeling good. I then had a go at top-roping Gee’s 6a+ route, and suddenly realised that despite feeling good, I wasn’t Johnny Dawes. I got about 12 metres up, and just fell off. So I tried again, made it to about 13 metres, and this time just gave up. I’m not sure why, but this knocked me back a bit, and I completely failed to complete any further climbs that day, including a beautiful technical crack line that Gee picked out when he saw I was struggling with the technical climbing. I couldn’t help feeling that for some reason I was letting everyone else down by not giving it 100% effort. Sol & Eddie on the other hand were both giving it at least 100% effort, and proved it by both climbing a beautiful 30 metre neighbouring route involving a chimney, some bridging, laybacking from a hanging flake, traversing an undercut before clipping in to the lower-off a full 30 metres up. A beautiful route, but quite beyond me. I sat at the bottom, sulked a little, and realised that just because a line is there, I don’t *have* to climb it. I can appreciate it just as much by looking at it and working how I’d approach the climb if I was technically capable. Food. Beer. Zzzzzzz…
Next day, and another sector on the South face with another fearsome walk-in. First route up was a brutal looking chimney, straight up, a full 30 metres. I belayed while Eddie led it, and it was obviously a struggle, as it took him 30 minutes or so. My turn… I like this kind of climbing more than the technical nadgery stuff we’d been doing up to that point, so was expecting to enjoy this climb and do well. But, yesterday’s demons were still haunting me as I got to a tricky narrowing of the chimney about 15 metres up, and rather than press on, just bailed out when I looked down and got scared. I was furious with myself. Bloody furious. It was such a beautiful piece of rock, and well within my ability. So I cleaned the crap off my shoes, told the demons to piss off and got back on there. 10 minutes later I was at the lower-off, feeling bloody marvellous. From there, it was straight onto a very technical “six something plus I guess” that was next to me. And here, I had another epiphany. I got to a very reachy technical move which normally I’d just shrug my shoulders at and drop off saying “can’t do that”. This time, however, I had a big sweary moment. I really shouted. Lots. And, with a final scream of “f*cksocks” I pulled up on a handhold that was no wider than a pencil using a pebble hole no bigger than a large grape for my toe before driving straight through another less than enormous hold up to a “resting point” which was actually a small ledge smaller than my mobile phone. I closed my eyes, breathed very very deeply, and relaxed. At this moment Sol took a picture, so rather than capturing the most dynamically explosive move of the weekend, it actually looks like I’m having a snooze. I’m sure I climbed something else later that day, but I’m already having difficulties remembering exactly what I did and when I did it. What I do remember is eating sausage and chips for tea in a local bar and drinking a couple of cold beers before suggesting to Gee that we maybe have a pop at a small, easy, multi-pitch route the next day.
And, next day, I was standing at the bottom of one of the massif pinnacles after a 7c+ walk-in wondering what on earth I’d done. It was about a 100 metre climb in all, with a traverse, two 30 metre pitches and a final 20 metre scramble to the summit. Champ had flown home ill the day before, so the plan was for Gee to lead me, and then Sol and Eddie to follow us up exchanging the lead. There was a lot of ropework to learn, and Gee was brilliant at explaining everything at ground level, demonstrating as he went along. So, all geared up, he started off along the traverse. Eventually I heard the call of “safe” so took him off belay, shouted “that’s me” as the rope came taught, and started off. I’ve never done a traverse before, and bloody hell, this was a baptism of fire. It started off easy enough, but there were only three bolts over the entire length of the pitch. And the second one was immediately *before* the hardest set of moves, so if I’d slipped there I would have had a 10 metre swing out over the trees. It’s safe to say at this point that my mind was concentrated wonderfully. I made the belay ’stance’ (actually, it was technically a hanging belay, as there was no ledge at all, just a couple of bolts and a chain) clipped in, and breathed a huge sigh of relief. Gee then went through exactly what was going to happen next, and exactly what my responsibilities were as I belayed him up. Off he went. The first bolt was a good four metres above, so a fall while reaching to clip it would have been entertaining for both of us. Of course, he didn’t fall. He vanished round the corner, and 10 minutes later again I heard the call of “safe” and the process repeated itself. The climbing wasn’t too tricky - a couple of balancey moves, but really, a lot easier than we’d been doing the day before. The exposure was completely different though, as I was acutely aware that I was starting from a point 20 metres up, and climbing another 30 metres from there. It was fantastic. The next belay point was a cave, where we stopped, had a quick bite to eat and a drink, and again, a lesson in constructing safe anchors and belaying off them. I felt enormously privileged to be there - on a global scale, a mere handful of people had ever sat in this cave and seen this view. Next pitch was more of the same. 30 metres of quite simple climbing up to the next belay. At which point, rather than making myself safe, Gee just told me to scramble the final metres up to the summit being bloody careful not to fall off as the only protection was from his belay point. I crawled onto the top of the pinnacle, clove-hitched my rope to a rusty old piece of kit bolted on up there, and sat down. I’d made it. And just sitting here typing it up is enough to give me a huge surge of adrenaline. The feeling was completely and utterly indescribable. I don’t care that it wasn’t the highest pinnacle on the massif. It was the proudest moment of my life in terms of achievement, and I had about 30 minutes of complete solitude up there while Gee waited at the belay for Sol and Eddie to make their way up. I’m not a spiritual person, but sitting up there, completely alone, on the summit was the closest I’ve ever been to a spiritual revelation. Enough of this though. As anyone will tell you, a climb is not over until you’re safe on the ground again, and we had a two pitch abseil ahead of us, using full length ropes. I made a bit of a meal of this to be honest, as my belay plate doesn’t work too well for abbing on two full ropes, so I bounced my way down, and managed to jam my thumb in the bloody thing just to add injury to insult. (As I write this, I am debating what to replace it with). The last abseil pitch was the most risky part of the whole day, for two reasons. Firstly, the back of the chimney we were descending was very loose so a lot of rock came down. Secondly, there was a bees nest at the bottom. But, we all made it down, thanked Gee profusely for the help and guidance, and prepared for the flight home. (Actually, there was a lot of chilling out and chatting, but my typing finger is hurting now).
I got home after midnight. Tired, but immensely, immensely proud. Enormous thanks to Gee and Carole for their hospitality, assistance, good humour, coffee and encouragement. I’ll be back again. No doubt.
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| 338,586
|
barefoot sandals running Secrets
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(*
File: Pi_Transcendendtal.thy
Author: Manuel Eberl, TU München
A proof of the transcendence of pi
*)
section \<open>The Transcendence of $\pi$\<close>
theory Pi_Transcendental
imports
"E_Transcendental.E_Transcendental"
"Symmetric_Polynomials.Symmetric_Polynomials"
"HOL-Real_Asymp.Real_Asymp"
Pi_Transcendental_Polynomial_Library
begin
lemma ring_homomorphism_to_poly [intro]: "ring_homomorphism (\<lambda>i. [:i:])"
by standard auto
lemma (in ring_closed) coeff_power_closed:
"(\<And>m. coeff p m \<in> A) \<Longrightarrow> coeff (p ^ n) m \<in> A"
by (induction n arbitrary: m)
(auto simp: mpoly_coeff_1 coeff_mpoly_times intro!: prod_fun_closed)
lemma (in ring_closed) coeff_prod_closed:
"(\<And>x m. x \<in> X \<Longrightarrow> coeff (f x) m \<in> A) \<Longrightarrow> coeff (prod f X) m \<in> A"
by (induction X arbitrary: m rule: infinite_finite_induct)
(auto simp: mpoly_coeff_1 coeff_mpoly_times intro!: prod_fun_closed)
lemma map_of_rat_of_int_poly [simp]: "map_poly of_rat (of_int_poly p) = of_int_poly p"
by (intro poly_eqI) (auto simp: coeff_map_poly)
text \<open>
Given a polynomial with rational coefficients, we can obtain an integer polynomial that
differs from it only by a nonzero constant by clearing the denominators.
\<close>
lemma ratpoly_to_intpoly:
assumes "\<forall>i. poly.coeff p i \<in> \<rat>"
obtains q c where "c \<noteq> 0" "p = Polynomial.smult (inverse (of_nat c)) (of_int_poly q)"
proof (cases "p = 0")
case True
with that[of 1 0] show ?thesis by auto
next
case False
from assms obtain p' where p': "p = map_poly of_rat p'"
using ratpolyE by auto
define c where "c = Lcm ((nat \<circ> snd \<circ> quotient_of \<circ> poly.coeff p') ` {..Polynomial.degree p'})"
have "\<not>snd (quotient_of x) \<le> 0" for x
using quotient_of_denom_pos[of x, OF surjective_pairing] by auto
hence "c \<noteq> 0" by (auto simp: c_def)
define q where "q = Polynomial.smult (of_nat c) p'"
have "poly.coeff q i \<in> \<int>" for i
proof (cases "i > Polynomial.degree p'")
case False
define m n
where "m = fst (quotient_of (poly.coeff p' i))"
and "n = nat (snd (quotient_of (poly.coeff p' i)))"
have mn: "n > 0" "poly.coeff p' i = of_int m / of_nat n"
using quotient_of_denom_pos[of "poly.coeff p' i", OF surjective_pairing]
quotient_of_div[of "poly.coeff p' i", OF surjective_pairing]
by (auto simp: m_def n_def)
from False have "n dvd c" unfolding c_def
by (intro dvd_Lcm) (auto simp: c_def n_def o_def not_less)
hence "of_nat c * (of_int m / of_nat n) = (of_nat (c div n) * of_int m :: rat)"
by (auto simp: of_nat_div)
also have "\<dots> \<in> \<int>" by auto
finally show ?thesis using mn by (auto simp: q_def)
qed (auto simp: q_def coeff_eq_0)
with int_polyE obtain q' where q': "q = of_int_poly q'" by auto
moreover have "p = Polynomial.smult (inverse (of_nat c)) (map_poly of_rat (of_int_poly q'))"
unfolding smult_conv_map_poly q'[symmetric] p' using \<open>c \<noteq> 0\<close>
by (intro poly_eqI) (auto simp: coeff_map_poly q_def of_rat_mult)
ultimately show ?thesis
using q' p' \<open>c \<noteq> 0\<close> by (auto intro!: that[of c q'])
qed
lemma symmetric_mpoly_symmetric_sum:
assumes "\<And>\<pi>. \<pi> permutes A \<Longrightarrow> g \<pi> permutes X"
assumes "\<And>x \<pi>. x \<in> X \<Longrightarrow> \<pi> permutes A \<Longrightarrow> mpoly_map_vars \<pi> (f x) = f (g \<pi> x)"
shows "symmetric_mpoly A (\<Sum>x\<in>X. f x)"
unfolding symmetric_mpoly_def
proof safe
fix \<pi> assume \<pi>: "\<pi> permutes A"
have "mpoly_map_vars \<pi> (sum f X) = (\<Sum>x\<in>X. mpoly_map_vars \<pi> (f x))"
by simp
also have "\<dots> = (\<Sum>x\<in>X. f (g \<pi> x))"
by (intro sum.cong assms \<pi> refl)
also have "\<dots> = (\<Sum>x\<in>g \<pi>`X. f x)"
using assms(1)[OF \<pi>] by (subst sum.reindex) (auto simp: permutes_inj_on)
also have "g \<pi> ` X = X"
using assms(1)[OF \<pi>] by (simp add: permutes_image)
finally show "mpoly_map_vars \<pi> (sum f X) = sum f X" .
qed
(* TODO: The version of this theorem in the AFP is to weak and should be replaced by this one. *)
lemma symmetric_mpoly_symmetric_prod:
assumes "g permutes X"
assumes "\<And>x \<pi>. x \<in> X \<Longrightarrow> \<pi> permutes A \<Longrightarrow> mpoly_map_vars \<pi> (f x) = f (g x)"
shows "symmetric_mpoly A (\<Prod>x\<in>X. f x)"
unfolding symmetric_mpoly_def
proof safe
fix \<pi> assume \<pi>: "\<pi> permutes A"
have "mpoly_map_vars \<pi> (prod f X) = (\<Prod>x\<in>X. mpoly_map_vars \<pi> (f x))"
by simp
also have "\<dots> = (\<Prod>x\<in>X. f (g x))"
by (intro prod.cong assms \<pi> refl)
also have "\<dots> = (\<Prod>x\<in>g`X. f x)"
using assms by (subst prod.reindex) (auto simp: permutes_inj_on)
also have "g ` X = X"
using assms by (simp add: permutes_image)
finally show "mpoly_map_vars \<pi> (prod f X) = prod f X" .
qed
text \<open>
We now prove the transcendence of $i\pi$, from which the transcendence of $\pi$ will follow
as a trivial corollary. The first proof of this was given by von Lindemann~\cite{lindemann_pi82}.
The central ingredient is the fundamental theorem of symmetric functions.
The proof can, by now, be considered folklore and one can easily find many similar variants of
it, but we mostly follows the nice exposition given by Niven~\cite{niven_pi39}.
An independent previous formalisation in Coq that uses the same basic techniques was given by
Bernard et al.~\cite{bernard_pi16}. They later also formalised the much stronger
Lindemann--Weierstra{\ss} theorem~\cite{bernard_lw17}.
\<close>
lemma transcendental_i_pi: "\<not>algebraic (\<i> * pi)"
proof
\<comment> \<open>Suppose $i\pi$ were algebraic.\<close>
assume "algebraic (\<i> * pi)"
\<comment> \<open>We obtain some nonzero integer polynomial that has $i\pi$ as a root. We can assume
w.\,l.\,o.\,g.\ that the constant coefficient of this polynomial is nonzero.\<close>
then obtain p
where p: "poly (of_int_poly p) (\<i> * pi) = 0" "p \<noteq> 0" "poly.coeff p 0 \<noteq> 0"
by (elim algebraicE'_nonzero) auto
define n where "n = Polynomial.degree p"
\<comment> \<open>We define the sequence of the roots of this polynomial:\<close>
obtain root where "Polynomial.smult (Polynomial.lead_coeff (of_int_poly p))
(\<Prod>i<n. [:-root i :: complex, 1:]) = of_int_poly p"
using complex_poly_decompose'[of "of_int_poly p"] unfolding n_def by auto
note root = this [symmetric]
\<comment> \<open>We note that $i\pi$ is, of course, among these roots.\<close>
from p and root obtain idx where idx: "idx < n" "root idx = \<i> * pi"
by (auto simp: poly_prod)
\<comment> \<open>We now define a new polynomial \<open>P'\<close>, whose roots are all numbers that arise as a
sum of any subset of roots of \<open>p\<close>. We also count all those subsets that sum up to 0
and call their number \<open>A\<close>.\<close>
define root' where "root' = (\<lambda>X. (\<Sum>j\<in>X. root j))"
define P where "P = (\<lambda>i. \<Prod>X | X \<subseteq> {..<n} \<and> card X = i. [:-root' X, 1:])"
define P' where "P' = (\<Prod>i\<in>{0<..n}. P i)"
define A where "A = card {X\<in>Pow {..<n}. root' X = 0}"
have [simp]: "P' \<noteq> 0" by (auto simp: P'_def P_def)
\<comment> \<open>We give the name \<open>Roots'\<close> to those subsets that do not sum to zero and note that there
is at least one, namely $\{i\pi\}$.\<close>
define Roots' where "Roots' = {X. X \<subseteq> {..<n} \<and> root' X \<noteq> 0}"
have [intro]: "finite Roots'" by (auto simp: Roots'_def)
have "{idx} \<in> Roots'" using idx by (auto simp: Roots'_def root'_def)
hence "Roots' \<noteq> {}" by auto
hence card_Roots': "card Roots' > 0" by (auto simp: card_eq_0_iff)
have P'_altdef: "P' = (\<Prod>X\<in>Pow {..<n} - {{}}. [:-root' X, 1:])"
proof -
have "P' = (\<Prod>(i, X)\<in>(SIGMA x:{0<..n}. {X. X \<subseteq> {..<n} \<and> card X = x}). [:- root' X, 1:])"
unfolding P'_def P_def by (subst prod.Sigma) auto
also have "\<dots> = (\<Prod>X\<in>Pow{..<n} - {{}}. [:- root' X, 1:])"
using card_mono[of "{..<n}"]
by (intro prod.reindex_bij_witness[of _ "\<lambda>X. (card X, X)" "\<lambda>(_, X). X"])
(auto simp: case_prod_unfold card_gt_0_iff intro: finite_subset[of _ "{..<n}"])
finally show ?thesis .
qed
\<comment> \<open>Clearly, @{term A} is nonzero, since the empty set sums to 0.\<close>
have "A > 0"
proof -
have "{} \<in> {X\<in>Pow {..<n}. root' X = 0}"
by (auto simp: root'_def)
thus ?thesis by (auto simp: A_def card_gt_0_iff)
qed
\<comment> \<open>Since $e^{i\pi} + 1 = 0$, we know the following:\<close>
have "0 = (\<Prod>i<n. exp (root i) + 1)"
using idx by force
\<comment> \<open>We rearrange this product of sums into a sum of products and collect all summands that are
1 into a separate sum, which we call @{term A}:\<close>
also have "\<dots> = (\<Sum>X\<in>Pow {..<n}. \<Prod>i\<in>X. exp (root i))"
by (subst prod_add) auto
also have "\<dots> = (\<Sum>X\<in>Pow {..<n}. exp (root' X))"
by (intro sum.cong refl, subst exp_sum [symmetric])
(auto simp: root'_def intro: finite_subset[of _ "{..<n}"])
also have "Pow {..<n} = {X\<in>Pow {..<n}. root' X \<noteq> 0} \<union> {X\<in>Pow {..<n}. root' X = 0}"
by auto
also have "(\<Sum>X\<in>\<dots>. exp (root' X)) = (\<Sum>X | X \<subseteq> {..<n} \<and> root' X \<noteq> 0. exp (root' X)) +
(\<Sum>X | X \<subseteq> {..<n} \<and> root' X = 0. exp (root' X))"
by (subst sum.union_disjoint) auto
also have "(\<Sum>X | X \<subseteq> {..<n} \<and> root' X = 0. exp (root' X)) = of_nat A"
by (simp add: A_def)
\<comment> \<open>Finally, we obtain the fact that the sum of $\exp(u)$ with $u$ ranging over all the non-zero
roots of @{term P'} is a negative integer.\<close>
finally have eq: "(\<Sum>X | X \<subseteq> {..<n} \<and> root' X \<noteq> 0. exp (root' X)) = -of_nat A"
by (simp add: add_eq_0_iff2)
\<comment> \<open>Next, we show that \<open>P'\<close> is a rational polynomial since it can be written as a symmetric
polynomial expression (with rational coefficients) in the roots of \<open>p\<close>.\<close>
define ratpolys where "ratpolys = {p::complex poly. \<forall>i. poly.coeff p i \<in> \<rat>}"
have ratpolysI: "p \<in> ratpolys" if "\<And>i. poly.coeff p i \<in> \<rat>" for p
using that by (auto simp: ratpolys_def)
have "P' \<in> ratpolys"
proof -
define Pmv :: "nat \<Rightarrow> complex poly mpoly"
where "Pmv = (\<lambda>i. \<Prod>X | X \<subseteq> {..<n} \<and> card X = i. Const ([:0,1:]) -
(\<Sum>i\<in>X. monom (Poly_Mapping.single i 1) 1))"
define P'mv where "P'mv = (\<Prod>i\<in>{0<..n}. Pmv i)"
have "insertion (\<lambda>i. [:root i:]) P'mv \<in> ratpolys"
proof (rule symmetric_poly_of_roots_in_subring[where l = "\<lambda>x. [:x:]"])
show "ring_closed ratpolys"
by standard (auto simp: ratpolys_def coeff_mult)
then interpret ring_closed ratpolys .
show "\<forall>m. coeff P'mv m \<in> ratpolys"
by (auto simp: P'mv_def Pmv_def coeff_monom when_def mpoly_coeff_Const coeff_pCons' ratpolysI
intro!: coeff_prod_closed minus_closed sum_closed uminus_closed)
show "\<forall>i. [:poly.coeff (of_int_poly p) i:] \<in> ratpolys"
by (intro ratpolysI allI) (auto simp: coeff_pCons')
show "[:inverse (of_int (Polynomial.lead_coeff p)):] *
[:of_int (Polynomial.lead_coeff p) :: complex:] = 1"
using \<open>p \<noteq> 0\<close> by (auto intro!: poly_eqI simp: field_simps)
next
have "symmetric_mpoly {..<n} (Pmv k)" for k
unfolding symmetric_mpoly_def
proof safe
fix \<pi> :: "nat \<Rightarrow> nat" assume \<pi>: "\<pi> permutes {..<n}"
hence "mpoly_map_vars \<pi> (Pmv k) =
(\<Prod>X | X \<subseteq> {..<n} \<and> card X = k. Const [:0, 1:] -
(\<Sum>x\<in>X. MPoly_Type.monom (Poly_Mapping.single (\<pi> x) (Suc 0)) 1))"
by (simp add: Pmv_def permutes_bij)
also have "\<dots> = (\<Prod>X | X \<subseteq> {..<n} \<and> card X = k. Const [:0, 1:] -
(\<Sum>x\<in>\<pi>`X. MPoly_Type.monom (Poly_Mapping.single x (Suc 0)) 1))"
using \<pi> by (subst sum.reindex) (auto simp: permutes_inj_on)
also have "\<dots> = (\<Prod>X \<in> (\<lambda>X. \<pi>`X)`{X. X \<subseteq> {..<n} \<and> card X = k}. Const [:0, 1:] -
(\<Sum>x\<in>X. MPoly_Type.monom (Poly_Mapping.single x (Suc 0)) 1))"
by (subst prod.reindex) (auto intro!: inj_on_image permutes_inj_on[OF \<pi>])
also have "(\<lambda>X. \<pi>`X)`{X. X \<subseteq> {..<n} \<and> card X = k} = {X. X \<subseteq> \<pi> ` {..<n} \<and> card X = k}"
using \<pi> by (subst image_image_fixed_card_subset) (auto simp: permutes_inj_on)
also have "\<pi> ` {..<n} = {..<n}"
by (intro permutes_image \<pi>)
finally show "mpoly_map_vars \<pi> (Pmv k) = Pmv k" by (simp add: Pmv_def)
qed
thus "symmetric_mpoly {..<n} P'mv"
unfolding P'mv_def by (intro symmetric_mpoly_prod) auto
next
show vars_P'mv: "vars P'mv \<subseteq> {..<n}"
unfolding P'mv_def Pmv_def
by (intro order.trans[OF vars_prod] UN_least order.trans[OF vars_diff]
Un_least order.trans[OF vars_sum] order.trans[OF vars_monom_subset]) auto
qed (insert root, auto intro!: ratpolysI simp: coeff_pCons')
also have "insertion (\<lambda>i. [:root i:]) (Pmv k) = P k" for k
by (simp add: Pmv_def insertion_prod insertion_diff insertion_sum root'_def P_def
sum_to_poly del: insertion_monom)
(* TODO: insertion_monom should not be a simp rule *)
hence "insertion (\<lambda>i. [:root i:]) P'mv = P'"
by (simp add: P'mv_def insertion_prod P'_def)
finally show "P' \<in> ratpolys" .
qed
\<comment> \<open>We clear the denominators and remove all powers of $X$ from @{term P'} to obtain a new
integer polynomial \<open>Q\<close>.\<close>
define Q' where "Q' = (\<Prod>X\<in>Roots'. [:- root' X, 1:])"
have "P' = (\<Prod>X\<in>Pow {..<n}-{{}}. [:-root' X, 1:])"
by (simp add: P'_altdef)
also have "Pow {..<n}-{{}} = Roots' \<union>
{X. X \<in> Pow {..<n} - {{}} \<and> root' X = 0}" by (auto simp: root'_def Roots'_def)
also have "(\<Prod>X\<in>\<dots>. [:-root' X, 1:]) =
Q' * [:0, 1:] ^ card {X. X \<subseteq> {..<n} \<and> X \<noteq> {} \<and> root' X = 0}"
by (subst prod.union_disjoint) (auto simp: Q'_def Roots'_def)
also have "{X. X \<subseteq> {..<n} \<and> X \<noteq> {} \<and> root' X = 0} = {X. X \<subseteq> {..<n} \<and> root' X = 0} - {{}}"
by auto
also have "card \<dots> = A - 1" unfolding A_def
by (subst card_Diff_singleton) (auto simp: root'_def)
finally have Q': "P' = Polynomial.monom 1 (A - 1) * Q'"
by (simp add: Polynomial.monom_altdef)
have degree_Q': "Polynomial.degree P' = Polynomial.degree Q' + (A - 1)"
by (subst Q')
(auto simp: Q'_def Roots'_def degree_mult_eq Polynomial.degree_monom_eq degree_prod_eq)
have "\<forall>i. poly.coeff Q' i \<in> \<rat>"
proof
fix i :: nat
have "poly.coeff Q' i = Polynomial.coeff P' (i + (A - 1))"
by (simp add: Q' Polynomial.coeff_monom_mult)
also have "\<dots> \<in> \<rat>" using \<open>P' \<in> ratpolys\<close> by (auto simp: ratpolys_def)
finally show "poly.coeff Q' i \<in> \<rat>" .
qed
from ratpoly_to_intpoly[OF this] obtain c Q
where [simp]: "c \<noteq> 0" and Q: "Q' = Polynomial.smult (inverse (of_nat c)) (of_int_poly Q)"
by metis
have [simp]: "Q \<noteq> 0" using Q Q' by auto
have Q': "of_int_poly Q = Polynomial.smult (of_nat c) Q'"
using Q by simp
have degree_Q: "Polynomial.degree Q = Polynomial.degree Q'"
by (subst Q) auto
have "Polynomial.lead_coeff (of_int_poly Q :: complex poly) = c"
by (subst Q') (simp_all add: degree_Q Q'_def lead_coeff_prod)
hence lead_coeff_Q: "Polynomial.lead_coeff Q = int c"
using of_int_eq_iff[of "Polynomial.lead_coeff Q" "of_nat c"] by (auto simp del: of_int_eq_iff)
have Q_decompose: "of_int_poly Q =
Polynomial.smult (of_nat c) (\<Prod>X\<in>Roots'. [:- root' X, 1:])"
by (subst Q') (auto simp: Q'_def lead_coeff_Q)
have "poly (of_int_poly Q) (\<i> * pi) = 0"
using \<open>{idx} \<in> Roots'\<close> \<open>finite Roots'\<close> idx
by (force simp: root'_def Q_decompose poly_prod)
have degree_Q: "Polynomial.degree (of_int_poly Q :: complex poly) = card Roots'"
by (subst Q') (auto simp: Q'_def degree_prod_eq)
have "poly (of_int_poly Q) (0 :: complex) \<noteq> 0"
by (subst Q') (auto simp: Q'_def Roots'_def poly_prod)
hence [simp]: "poly Q 0 \<noteq> 0" by simp
have [simp]: "poly (of_int_poly Q) (root' Y) = 0" if "Y \<in> Roots'" for Y
using that \<open>finite Roots'\<close> by (auto simp: Q' Q'_def poly_prod)
\<comment> \<open>We find some closed ball that contains all the roots of @{term Q}.\<close>
define r where "r = Polynomial.degree Q"
have "r > 0" using degree_Q card_Roots' by (auto simp: r_def)
define Radius where "Radius = Max ((\<lambda>Y. norm (root' Y)) ` Roots')"
have Radius: "norm (root' Y) \<le> Radius" if "Y \<in> Roots'" for Y
using \<open>finite Roots'\<close> that by (auto simp: Radius_def)
from Radius[of "{idx}"] have "Radius \<ge> pi"
using idx by (auto simp: Roots'_def norm_mult root'_def)
hence Radius_nonneg: "Radius \<ge> 0" and "Radius > 0" using pi_gt3 by linarith+
\<comment> \<open>Since this ball is compact, @{term Q} is bounded on it. We obtain such a bound.\<close>
have "compact (poly (of_int_poly Q :: complex poly) ` cball 0 Radius)"
by (intro compact_continuous_image continuous_intros) auto
then obtain Q_ub
where Q_ub: "Q_ub > 0"
"\<And>u :: complex. u \<in> cball 0 Radius \<Longrightarrow> norm (poly (of_int_poly Q) u) \<le> Q_ub"
by (auto dest!: compact_imp_bounded simp: bounded_pos cball_def)
\<comment> \<open>Using this, define another upper bound that we will need later.\<close>
define fp_ub
where "fp_ub = (\<lambda>p. \<bar>c\<bar> ^ (r * p - 1) / fact (p - 1) * (Radius ^ (p - 1) * Q_ub ^ p))"
have fp_ub_nonneg: "fp_ub p \<ge> 0" for p
unfolding fp_ub_def using \<open>Radius \<ge> 0\<close> Q_ub
by (intro mult_nonneg_nonneg divide_nonneg_pos zero_le_power) auto
define C where "C = card Roots' * Radius * exp Radius"
\<comment> \<open>We will now show that any sufficiently large prime number leads to
\<open>C * fp_ub p \<ge> 1\<close>, from which we will then derive a contradiction.\<close>
define primes_at_top where "primes_at_top = inf_class.inf sequentially (principal {p. prime p})"
have "eventually (\<lambda>p. \<forall>x\<in>{nat \<bar>poly Q 0\<bar>, c, A}. p > x) sequentially"
by (intro eventually_ball_finite ballI eventually_gt_at_top) auto
hence "eventually (\<lambda>p. \<forall>x\<in>{nat \<bar>poly Q 0\<bar>, c, A}. p > x) primes_at_top"
unfolding primes_at_top_def eventually_inf_principal by eventually_elim auto
moreover have "eventually (\<lambda>p. prime p) primes_at_top"
by (auto simp: primes_at_top_def eventually_inf_principal)
ultimately have "eventually (\<lambda>p. C * fp_ub p \<ge> 1) primes_at_top"
proof eventually_elim
case (elim p)
hence p: "prime p" "p > nat \<bar>poly Q 0\<bar>" "p > c" "p > A" by auto
hence "p > 1" by (auto dest: prime_gt_1_nat)
\<comment> \<open>We define the polynomial $f(X) = \frac{c^s}{(p-1)!} X^{p-1} Q(X)^p$, where $c$ is
the leading coefficient of $Q$. We also define $F(X)$ to be the sum of all its
derivatives.\<close>
define s where "s = r * p - 1"
define fp :: "complex poly"
where "fp = Polynomial.smult (of_nat c ^ s / fact (p - 1))
(Polynomial.monom 1 (p - 1) * of_int_poly Q ^ p)"
define Fp where "Fp = (\<Sum>i\<le>s+p. (pderiv ^^ i) fp)"
define f F where "f = poly fp" and "F = poly Fp"
have degree_fp: "Polynomial.degree fp = s + p" using degree_Q card_Roots' \<open>p > 1\<close>
by (simp add: fp_def s_def degree_mult_eq degree_monom_eq
degree_power_eq r_def algebra_simps)
\<comment> \<open>Using the same argument as in the case of the transcendence of $e$, we now
consider the function
\[I(u) := e^u F(0) - F(u) = u \int\limits_0^1 e^{(1-t)x} f(tx)\,\textrm{d}t\]
whose absolute value can be bounded with a standard ``maximum times length'' estimate
using our upper bound on $f$. All of this can be reused from the proof for $e$, so
there is not much to do here. In particular, we will look at $\sum I(x_i)$ with the
$x_i$ ranging over the roots of $Q$ and bound this sum in two different ways.\<close>
interpret lindemann_weierstrass_aux fp .
have I_altdef: "I = (\<lambda>u. exp u * F 0 - F u)"
by (intro ext) (simp add: I_def degree_fp F_def Fp_def poly_sum)
\<comment> \<open>We show that @{term fp_ub} is indeed an upper bound for $f$.\<close>
have fp_ub: "norm (poly fp u) \<le> fp_ub p" if "u \<in> cball 0 Radius" for u
proof -
have "norm (poly fp u) = \<bar>c\<bar> ^ (r * p - 1) / fact (p - 1) * (norm u ^ (p - 1) *
norm (poly (of_int_poly Q) u) ^ p)"
by (simp add: fp_def f_def s_def norm_mult poly_monom norm_divide norm_power)
also have "\<dots> \<le> fp_ub p"
unfolding fp_ub_def using that Q_ub \<open>Radius \<ge> 0\<close>
by (intro mult_left_mono[OF mult_mono] power_mono zero_le_power) auto
finally show ?thesis .
qed
\<comment> \<open>We now show that the following sum is an integer multiple of $p$. This argument again
uses the fundamental theorem of symmetric functions, exploiting that the inner sums are
symmetric over the roots of $Q$.\<close>
have "(\<Sum>i=p..s+p. \<Sum>Y\<in>Roots'. poly ((pderiv ^^ i) fp) (root' Y)) / p \<in> \<int>"
proof (subst sum_divide_distrib, intro Ints_sum[of "{a..b}" for a b])
fix i assume i: "i \<in> {p..s+p}"
then obtain roots' where roots': "distinct roots'" "set roots' = Roots'"
using finite_distinct_list \<open>finite Roots'\<close> by metis
define l where "l = length roots'"
define fp' where "fp' = (pderiv ^^ i) fp"
define d where "d = Polynomial.degree fp'"
\<comment> \<open>We define a multivariate polynomial for the inner sum $\sum f(x_i)/p$ in order
to show that it is indeed a symmetric function over the $x_i$.\<close>
define R where "R = (smult (1 / of_nat p) (\<Sum>k\<le>d. \<Sum>i<l. smult (poly.coeff fp' k)
(monom (Poly_Mapping.single i k) (1 / of_int (c ^ k)))) :: complex mpoly)"
\<comment> \<open>The $j$-th coefficient of the $i$-th derivative of $f$ are integer multiples
of $c^j p$ since $i \geq p$.\<close>
have integer: "poly.coeff fp' j / (of_nat c ^ j * of_nat p) \<in> \<int>" if "j \<le> d" for j
proof -
define fp'' where "fp'' = Polynomial.monom 1 (p - 1) * Q ^ p"
define x
where "x = c ^ s * poly.coeff ((pderiv ^^ i) (Polynomial.monom 1 (p - 1) * Q ^ p)) j"
have "[:fact p:] dvd ([:fact i:] :: int poly)" using i
by (auto intro: fact_dvd)
also have "[:fact i:] dvd ((pderiv ^^ i) (Polynomial.monom 1 (p - 1) * Q ^ p))"
by (rule fact_dvd_higher_pderiv)
finally have "c ^ j * fact p dvd x" unfolding x_def of_nat_mult using that i
by (intro mult_dvd_mono)
(auto intro!: le_imp_power_dvd simp: s_def d_def fp'_def degree_higher_pderiv degree_fp)
hence "of_int x / (of_int (c ^ j * fact p) :: complex) \<in> \<int>"
by (intro of_int_divide_in_Ints) auto
also have "of_int x / (of_int (c ^ j * fact p) :: complex) =
poly.coeff fp' j / (of_nat c ^ j * of_nat p)" using \<open>p > 1\<close>
by (auto simp: fact_reduce[of p] fp'_def fp_def higher_pderiv_smult x_def field_simps
simp flip: coeff_of_int_poly higher_pderiv_of_int_poly)
finally show ?thesis .
qed
\<comment> \<open>Evaluating $R$ yields is an integer since it is symmetric.\<close>
have "insertion (\<lambda>i. c * root' (roots' ! i)) R \<in> \<int>"
proof (intro symmetric_poly_of_roots_in_subring_monic allI)
define Q' where "Q' = of_int_poly Q \<circ>\<^sub>p [:0, 1 / of_nat c :: complex:]"
show "symmetric_mpoly {..<l} R" unfolding R_def
by (intro symmetric_mpoly_smult symmetric_mpoly_sum[of "{..d}"] symmetric_mpoly_symmetric_sum)
(simp_all add: mpoly_map_vars_monom permutes_bij permutep_single bij_imp_bij_inv permutes_inv_inv)
show "MPoly_Type.coeff R m \<in> \<int>" for m unfolding R_def coeff_sum coeff_smult sum_distrib_left
using integer by (auto simp: R_def coeff_monom when_def intro!: Ints_sum)
show "vars R \<subseteq> {..<l}" unfolding R_def
by (intro order.trans[OF vars_smult] order.trans[OF vars_sum] UN_least
order.trans[OF vars_monom_subset]) auto
show "ring_closed \<int>" by standard auto
have "(\<Prod>i<l. [:- (of_nat c * root' (roots' ! i)), 1:]) =
(\<Prod>Y\<leftarrow>roots'. [:- (of_nat c * root' Y), 1:])"
by (subst prod_list_prod_nth) (auto simp: atLeast0LessThan l_def)
also have "\<dots> = (\<Prod>Y\<in>Roots'. [:- (of_nat c * root' Y), 1:])"
using roots' by (subst prod.distinct_set_conv_list [symmetric]) auto
also have "\<dots> = (\<Prod>Y\<in>Roots'. Polynomial.smult (of_nat c) ([:-root' Y, 1:])) \<circ>\<^sub>p [:0, 1 / c:]"
by (simp add: pcompose_prod pcompose_pCons)
also have "(\<Prod>Y\<in>Roots'. Polynomial.smult (of_nat c) ([:-root' Y, 1:])) =
Polynomial.smult (of_nat c ^ card Roots') (\<Prod>Y\<in>Roots'. [:-root' Y, 1:])"
by (subst prod_smult) auto
also have "\<dots> = Polynomial.smult (of_nat c ^ (card Roots' - 1))
(Polynomial.smult c (\<Prod>Y\<in>Roots'. [:-root' Y, 1:]))"
using \<open>finite Roots'\<close> and \<open>Roots' \<noteq> {}\<close>
by (subst power_diff) (auto simp: Suc_le_eq card_gt_0_iff)
also have "Polynomial.smult c (\<Prod>Y\<in>Roots'. [:-root' Y, 1:]) = of_int_poly Q"
using Q_decompose by simp
finally show "Polynomial.smult (of_nat c ^ (card Roots' - 1)) Q' =
(\<Prod>i<l. [:- (of_nat c * root' (roots' ! i)), 1:])"
by (simp add: pcompose_smult Q'_def)
fix i :: nat
show "poly.coeff (Polynomial.smult (of_nat c ^ (card Roots' - 1)) Q') i \<in> \<int>"
proof (cases i "Polynomial.degree Q" rule: linorder_cases)
case greater
thus ?thesis by (auto simp: Q'_def coeff_pcompose_linear coeff_eq_0)
next
case equal
thus ?thesis using \<open>Roots' \<noteq> {}\<close> degree_Q card_Roots' lead_coeff_Q
by (auto simp: Q'_def coeff_pcompose_linear lead_coeff_Q power_divide power_diff)
next
case less
have "poly.coeff (Polynomial.smult (of_nat c ^ (card Roots' - 1)) Q') i =
of_int (poly.coeff Q i) * (of_int (c ^ (card Roots' - 1)) / of_int (c ^ i))"
by (auto simp: Q'_def coeff_pcompose_linear power_divide)
also have "\<dots> \<in> \<int>" using less degree_Q
by (intro Ints_mult of_int_divide_in_Ints) (auto intro!: le_imp_power_dvd)
finally show ?thesis .
qed
qed auto
\<comment> \<open>Moreover, by definition, evaluating @{term R} gives us $\sum f(x_i)/p$.\<close>
also have "insertion (\<lambda>i. c * root' (roots' ! i)) R =
(\<Sum>Y\<leftarrow>roots'. poly fp' (root' Y)) / of_nat p"
by (simp add: insertion_sum R_def poly_altdef d_def sum_list_sum_nth atLeast0LessThan
l_def power_mult_distrib algebra_simps
sum.swap[of _ "{..Polynomial.degree fp'}"] del: insertion_monom)
also have "\<dots> = (\<Sum>Y\<in>Roots'. poly ((pderiv ^^ i) fp) (root' Y)) / of_nat p"
using roots' by (subst sum_list_distinct_conv_sum_set) (auto simp: fp'_def poly_pcompose)
finally show "\<dots> \<in> \<int>" .
qed
then obtain K where K: "(\<Sum>i=p..s+p. \<Sum>Y\<in>Roots'.
poly ((pderiv ^^ i) fp) (root' Y)) = of_int K * p"
using \<open>p > 1\<close> by (auto elim!: Ints_cases simp: field_simps)
\<comment> \<open>Next, we show that $F(0)$ is an integer and coprime to $p$.\<close>
obtain F0 :: int where F0: "F 0 = of_int F0" "coprime (int p) F0"
proof -
have "(\<Sum>i=p..s + p. poly ((pderiv ^^ i) fp) 0) / of_nat p \<in> \<int>"
unfolding sum_divide_distrib
proof (intro Ints_sum)
fix i assume i: "i \<in> {p..s+p}"
hence "fact p dvd poly ((pderiv ^^ i) ([:0, 1:] ^ (p - 1) * Q ^ p)) 0"
by (intro fact_dvd_poly_higher_pderiv_aux') auto
then obtain k where k: "poly ((pderiv ^^ i) ([:0, 1:] ^ (p - 1) * Q ^ p)) 0 = k * fact p"
by auto
have "(pderiv ^^ i) fp = Polynomial.smult (of_nat c ^ s / fact (p - 1))
(of_int_poly ((pderiv ^^ i) ([:0, 1:] ^ (p - 1) * Q ^ p)))"
by (simp add: fp_def higher_pderiv_smult Polynomial.monom_altdef
flip: higher_pderiv_of_int_poly)
also have "poly \<dots> 0 / of_nat p = of_int (c ^ s * k)"
using k \<open>p > 1\<close> by (simp add: fact_reduce[of p])
also have "\<dots> \<in> \<int>" by simp
finally show "poly ((pderiv ^^ i) fp) 0 / of_nat p \<in> \<int>" .
qed
then obtain S where S: "(\<Sum>i=p..s + p. poly ((pderiv ^^ i) fp) 0) = of_int S * p"
using \<open>p > 1\<close> by (auto elim!: Ints_cases simp: field_simps)
have "F 0 = (\<Sum>i\<le>s + p. poly ((pderiv ^^ i) fp) 0)"
by (auto simp: F_def Fp_def poly_sum)
also have "\<dots> = (\<Sum>i\<in>insert (p - 1) {p..s + p}. poly ((pderiv ^^ i) fp) 0)"
proof (intro sum.mono_neutral_right ballI)
fix i assume i: "i \<in> {..s + p} - insert (p - 1) {p..s + p}"
hence "i < p - 1" by auto
have "Polynomial.monom 1 (p - 1) dvd fp"
by (auto simp: fp_def intro: dvd_smult)
with i show "poly ((pderiv ^^ i) fp) 0 = 0"
by (intro poly_higher_pderiv_aux1'[of _ "p - 1"]) (auto simp: Polynomial.monom_altdef)
qed auto
also have "\<dots> = poly ((pderiv ^^ (p - 1)) fp) 0 + of_int S * of_nat p"
using \<open>p > 1\<close> S by (subst sum.insert) auto
also have "poly ((pderiv ^^ (p - 1)) fp) 0 = of_int (c ^ s * poly Q 0 ^ p)"
using poly_higher_pderiv_aux2[of "p - 1" 0 "of_int_poly Q ^ p :: complex poly"]
by (simp add: fp_def higher_pderiv_smult Polynomial.monom_altdef)
finally have "F 0 = of_int (S * int p + c ^ s * poly Q 0 ^ p)"
by simp
moreover have "coprime p c" "coprime (int p) (poly Q 0)"
using p by (auto intro!: prime_imp_coprime dest: dvd_imp_le_int[rotated])
hence "coprime (int p) (c ^ s * poly Q 0 ^ p)"
by auto
hence "coprime (int p) (S * int p + c ^ s * poly Q 0 ^ p)"
unfolding coprime_iff_gcd_eq_1 gcd_add_mult by auto
ultimately show ?thesis using that[of "S * int p + c ^ s * poly Q 0 ^ p"] by blast
qed
\<comment> \<open>Putting everything together, we have shown that $\sum I(x_i)$ is an integer coprime
to $p$, and therefore a nonzero integer, and therefore has an absolute value of at least 1.\<close>
have "(\<Sum>Y\<in>Roots'. I (root' Y)) = F 0 * (\<Sum>Y\<in>Roots'. exp (root' Y)) - (\<Sum>Y\<in>Roots'. F (root' Y))"
by (simp add: I_altdef sum_subtractf sum_distrib_left sum_distrib_right algebra_simps)
also have "\<dots> = -(of_int (F0 * int A) +
(\<Sum>i\<le>s+p. \<Sum>Y\<in>Roots'. poly ((pderiv ^^ i) fp) (root' Y)))"
using F0 by (simp add: Roots'_def eq F_def Fp_def poly_sum sum.swap[of _ "{..s+p}"])
also have "(\<Sum>i\<le>s+p. \<Sum>Y\<in>Roots'. poly ((pderiv ^^ i) fp) (root' Y)) =
(\<Sum>i=p..s+p. \<Sum>Y\<in>Roots'. poly ((pderiv ^^ i) fp) (root' Y))"
proof (intro sum.mono_neutral_right ballI sum.neutral)
fix i Y assume i: "i \<in> {..s+p} - {p..s+p}" and Y: "Y \<in> Roots'"
have "[:-root' Y, 1:] ^ p dvd of_int_poly Q ^ p"
by (intro dvd_power_same) (auto simp: dvd_iff_poly_eq_0 Y)
hence "[:-root' Y, 1:] ^ p dvd fp"
by (auto simp: fp_def intro!: dvd_smult)
thus "poly ((pderiv ^^ i) fp) (root' Y) = 0"
using i by (intro poly_higher_pderiv_aux1') auto
qed auto
also have "\<dots> = of_int (K * int p)" using K by simp
finally have "(\<Sum>Y\<in>Roots'. I (root' Y)) = -of_int (K * int p + F0 * int A)"
by simp
moreover have "coprime p A"
using p \<open>A > 0\<close> by (intro prime_imp_coprime) (auto dest!: dvd_imp_le)
hence "coprime (int p) (F0 * int A)"
using F0 by auto
hence "coprime (int p) (K * int p + F0 * int A)"
using F0 unfolding coprime_iff_gcd_eq_1 gcd_add_mult by auto
hence "K * int p + F0 * int A \<noteq> 0"
using p by (intro notI) auto
hence "norm (-of_int (K * int p + F0 * int A) :: complex) \<ge> 1"
unfolding norm_minus_cancel norm_of_int by linarith
ultimately have "1 \<le> norm (\<Sum>Y\<in>Roots'. I (root' Y))" by metis
\<comment> \<open>The M--L bound on the integral gives us an upper bound:\<close>
also have "norm (\<Sum>Y\<in>Roots'. I (root' Y)) \<le>
(\<Sum>Y\<in>Roots'. norm (root' Y) * exp (norm (root' Y)) * fp_ub p)"
proof (intro sum_norm_le lindemann_weierstrass_integral_bound fp_ub fp_ub_nonneg)
fix Y u assume *: "Y \<in> Roots'" "u \<in> closed_segment 0 (root' Y)"
hence "closed_segment 0 (root' Y) \<subseteq> cball 0 Radius"
using \<open>Radius \<ge> 0\<close> Radius[of Y] by (intro closed_segment_subset) auto
with * show "u \<in> cball 0 Radius" by auto
qed
also have "\<dots> \<le> (\<Sum>Y\<in>Roots'. Radius * exp (Radius) * fp_ub p)"
using Radius by (intro sum_mono mult_right_mono mult_mono fp_ub_nonneg \<open>Radius \<ge> 0\<close>) auto
also have "\<dots> = C * fp_ub p" by (simp add: C_def)
finally show "1 \<le> C * fp_ub p" .
qed
\<comment> \<open>It now only remains to show that this inequality is inconsistent for large $p$.
This is obvious, since the upper bound is an exponential divided by a factorial and
therefore clearly tends to zero.\<close>
have "(\<lambda>p. C * fp_ub p) \<in> \<Theta>(\<lambda>p. (C / (Radius * \<bar>c\<bar>)) * (p / 2 ^ p) *
((2 * \<bar>c\<bar> ^ r * Radius * Q_ub) ^ p / fact p))"
(is "_ \<in> \<Theta>(?f)") using degree_Q card_Roots' \<open>Radius > 0\<close>
by (intro bigthetaI_cong eventually_mono[OF eventually_gt_at_top[of 0]])
(auto simp: fact_reduce power_mult [symmetric] r_def
fp_ub_def power_diff power_mult_distrib)
also have "?f \<in> o(\<lambda>p. 1 * 1 * 1)"
proof (intro landau_o.big_small_mult landau_o.big_mult)
have "(\<lambda>x. (real_of_int (2 * \<bar>c\<bar> ^ r) * Radius * Q_ub) ^ x / fact x) \<longlonglongrightarrow> 0"
by (intro power_over_fact_tendsto_0)
thus "(\<lambda>x. (real_of_int (2 * \<bar>c\<bar> ^ r) * Radius * Q_ub) ^ x / fact x) \<in> o(\<lambda>x. 1)"
by (intro smalloI_tendsto) auto
qed real_asymp+
finally have "(\<lambda>p. C * fp_ub p) \<in> o(\<lambda>_. 1)" by simp
from smalloD_tendsto[OF this] have "(\<lambda>p. C * fp_ub p) \<longlonglongrightarrow> 0" by simp
hence "eventually (\<lambda>p. C * fp_ub p < 1) at_top"
by (intro order_tendstoD) auto
hence "eventually (\<lambda>p. C * fp_ub p < 1) primes_at_top"
unfolding primes_at_top_def eventually_inf_principal by eventually_elim auto
moreover note \<open>eventually (\<lambda>p. C * fp_ub p \<ge> 1) primes_at_top\<close>
\<comment> \<open>We therefore have a contradiction for any sufficiently large prime.\<close>
ultimately have "eventually (\<lambda>p. False) primes_at_top"
by eventually_elim auto
\<comment> \<open>Since sufficiently large primes always exist, this concludes the theorem.\<close>
moreover have "frequently (\<lambda>p. prime p) sequentially"
using primes_infinite by (simp add: cofinite_eq_sequentially[symmetric] Inf_many_def)
ultimately show False
by (auto simp: frequently_def eventually_inf_principal primes_at_top_def)
qed
theorem transcendental_pi: "\<not>algebraic pi"
using transcendental_i_pi by (simp add: algebraic_times_i_iff)
end
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